LMFDB - Embedded newform 144.7.o.a.31.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,7,Mod(31,144)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 2]))

N = Newforms(chi, 7, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("144.31");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	144.o (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$33.1277880413$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.11
Character	$\chi$	$=$	144.31
Dual form			144.7.o.a.79.11

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(18.8245 + 19.3555i) q^{3} +(119.170 - 206.408i) q^{5} +(297.697 - 171.875i) q^{7} +(-20.2728 + 728.718i) q^{9} +O(q^{10})$	\(q+(18.8245 + 19.3555i) q^{3} +(119.170 - 206.408i) q^{5} +(297.697 - 171.875i) q^{7} +(-20.2728 + 728.718i) q^{9} +(1236.99 - 714.175i) q^{11} +(473.058 - 819.360i) q^{13} +(6238.46 - 1578.95i) q^{15} -6404.62 q^{17} -2546.17i q^{19} +(8930.75 + 2526.60i) q^{21} +(-17689.0 - 10212.7i) q^{23} +(-20590.4 - 35663.7i) q^{25} +(-14486.3 + 13325.4i) q^{27} +(1735.29 + 3005.61i) q^{29} +(21927.8 + 12660.0i) q^{31} +(37108.9 + 10498.5i) q^{33} -81929.5i q^{35} +66492.6 q^{37} +(24764.2 - 6267.80i) q^{39} +(-58777.3 + 101805. i) q^{41} +(50903.9 - 29389.4i) q^{43} +(147998. + 91025.8i) q^{45} +(43680.1 - 25218.7i) q^{47} +(257.833 - 446.580i) q^{49} +(-120564. - 123965. i) q^{51} +110518. q^{53} -340433. i q^{55} +(49282.5 - 47930.5i) q^{57} +(-27925.3 - 16122.7i) q^{59} +(77336.7 + 133951. i) q^{61} +(119214. + 220422. i) q^{63} +(-112748. - 195286. i) q^{65} +(-138114. - 79740.3i) q^{67} +(-135314. - 534630. i) q^{69} +52596.8i q^{71} +494799. q^{73} +(302684. - 1.06989e6i) q^{75} +(245498. - 425215. i) q^{77} +(-532196. + 307264. i) q^{79} +(-530619. - 29546.3i) q^{81} +(929041. - 536382. i) q^{83} +(-763238. + 1.32197e6i) q^{85} +(-25509.1 + 90166.7i) q^{87} +626436. q^{89} -325228. i q^{91} +(167740. + 662745. i) q^{93} +(-525551. - 303427. i) q^{95} +(286108. + 495554. i) q^{97} +(495355. + 915893. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 48 q^{3} - 72 q^{5} + 360 q^{7} - 1452 q^{9}+O(q^{10}) $ 24 * q - 48 * q^3 - 72 * q^5 + 360 * q^7 - 1452 * q^9	$ 24 q - 48 q^{3} - 72 q^{5} + 360 q^{7} - 1452 q^{9} + 864 q^{11} - 840 q^{13} + 11544 q^{15} - 12888 q^{17} + 21792 q^{21} - 60264 q^{23} - 42828 q^{25} - 39312 q^{27} - 5760 q^{29} + 18360 q^{31} + 54252 q^{33} + 49728 q^{37} - 31704 q^{39} - 52164 q^{41} + 283968 q^{45} - 104760 q^{47} + 236004 q^{49} - 305664 q^{51} + 134352 q^{53} - 325524 q^{57} - 280368 q^{59} + 76440 q^{61} - 266760 q^{63} - 22752 q^{65} - 1158048 q^{67} - 446904 q^{69} + 43800 q^{73} - 979632 q^{75} + 652104 q^{77} + 225576 q^{79} - 1391652 q^{81} + 306288 q^{83} - 414000 q^{85} + 1793928 q^{87} + 2486304 q^{89} - 1107576 q^{93} - 1538784 q^{95} - 365916 q^{97} + 1331280 q^{99}+O(q^{100}) $ 24 * q - 48 * q^3 - 72 * q^5 + 360 * q^7 - 1452 * q^9 + 864 * q^11 - 840 * q^13 + 11544 * q^15 - 12888 * q^17 + 21792 * q^21 - 60264 * q^23 - 42828 * q^25 - 39312 * q^27 - 5760 * q^29 + 18360 * q^31 + 54252 * q^33 + 49728 * q^37 - 31704 * q^39 - 52164 * q^41 + 283968 * q^45 - 104760 * q^47 + 236004 * q^49 - 305664 * q^51 + 134352 * q^53 - 325524 * q^57 - 280368 * q^59 + 76440 * q^61 - 266760 * q^63 - 22752 * q^65 - 1158048 * q^67 - 446904 * q^69 + 43800 * q^73 - 979632 * q^75 + 652104 * q^77 + 225576 * q^79 - 1391652 * q^81 + 306288 * q^83 - 414000 * q^85 + 1793928 * q^87 + 2486304 * q^89 - 1107576 * q^93 - 1538784 * q^95 - 365916 * q^97 + 1331280 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	18.8245	+	19.3555i	0.697205	+	0.716871i
$3$	18.8245	+	19.3555i	0.697205	+	0.716871i
$4$		0			0
$5$	119.170	−	206.408i	0.953359	−	1.65127i	0.215281	−	0.976552i	$-0.430933\pi$
$5$	119.170	−	206.408i	0.953359	−	1.65127i	0.738079	−	0.674715i	$-0.235733\pi$
$6$		0			0
$7$	297.697	−	171.875i	0.867921	−	0.501095i	0.00126437	−	0.999999i	$-0.499598\pi$
$7$	297.697	−	171.875i	0.867921	−	0.501095i	0.866657	+	0.498905i	$0.166264\pi$
$8$		0			0
$9$	−20.2728	+	728.718i	−0.0278091	+	0.999613i
$10$		0			0
$11$	1236.99	−	714.175i	0.929367	−	0.536570i	0.0427554	−	0.999086i	$-0.486386\pi$
$11$	1236.99	−	714.175i	0.929367	−	0.536570i	0.886611	+	0.462516i	$0.153053\pi$
$12$		0			0
$13$	473.058	−	819.360i	0.215320	−	0.372945i	−0.738052	−	0.674744i	$-0.764254\pi$
$13$	473.058	−	819.360i	0.215320	−	0.372945i	0.953371	+	0.301799i	$0.0975873\pi$
$14$		0			0
$15$	6238.46	−	1578.95i	1.84843	−	0.467836i
$16$		0			0
$17$	−6404.62			−1.30361			−0.651804	−	0.758388i	$-0.725987\pi$
$17$	−6404.62			−1.30361			−0.651804	+	0.758388i	$0.725987\pi$
$18$		0			0
$19$		−	2546.17i		−	0.371216i	−0.982624	−	0.185608i	$-0.940575\pi$
$19$		−	2546.17i		−	0.371216i	0.982624	−	0.185608i	$-0.0594255\pi$
$20$		0			0
$21$	8930.75	+	2526.60i	0.964340	+	0.272822i
$22$		0			0
$23$	−17689.0	−	10212.7i	−1.45385	−	0.839380i	−0.455152	−	0.890414i	$-0.650415\pi$
$23$	−17689.0	−	10212.7i	−1.45385	−	0.839380i	−0.998697	+	0.0510336i	$0.983748\pi$
$24$		0			0
$25$	−20590.4	−	35663.7i	−1.31779	−	2.28248i
$26$		0			0
$27$	−14486.3	+	13325.4i	−0.735983	+	0.677000i
$28$		0			0
$29$	1735.29	+	3005.61i	0.0711505	+	0.123236i	0.899406	−	0.437115i	$-0.144000\pi$
$29$	1735.29	+	3005.61i	0.0711505	+	0.123236i	−0.828255	+	0.560351i	$0.810666\pi$
$30$		0			0
$31$	21927.8	+	12660.0i	0.736056	+	0.424962i	0.820634	−	0.571455i	$-0.193621\pi$
$31$	21927.8	+	12660.0i	0.736056	+	0.424962i	−0.0845776	+	0.996417i	$0.526954\pi$
$32$		0			0
$33$	37108.9	+	10498.5i	1.03261	+	0.292137i
$34$		0			0
$35$		−	81929.5i		−	1.91089i
$36$		0			0
$37$	66492.6			1.31271			0.656355	−	0.754453i	$-0.272098\pi$
$37$	66492.6			1.31271			0.656355	+	0.754453i	$0.272098\pi$
$38$		0			0
$39$	24764.2	−	6267.80i	0.417476	−	0.105663i
$40$		0			0
$41$	−58777.3	+	101805.i	−0.852822	+	1.47713i	0.0258291	+	0.999666i	$0.491777\pi$
$41$	−58777.3	+	101805.i	−0.852822	+	1.47713i	−0.878651	+	0.477465i	$0.841556\pi$
$42$		0			0
$43$	50903.9	−	29389.4i	0.640244	−	0.369645i	−0.144464	−	0.989510i	$-0.546146\pi$
$43$	50903.9	−	29389.4i	0.640244	−	0.369645i	0.784709	+	0.619865i	$0.212813\pi$
$44$		0			0
$45$	147998.	+	91025.8i	1.62412	+	0.998911i
$46$		0			0
$47$	43680.1	−	25218.7i	0.420717	−	0.242901i	−0.274667	−	0.961539i	$-0.588568\pi$
$47$	43680.1	−	25218.7i	0.420717	−	0.242901i	0.695384	+	0.718638i	$0.255234\pi$
$48$		0			0
$49$	257.833	−	446.580i	0.00219155	−	0.00379587i
$50$		0			0
$51$	−120564.	−	123965.i	−0.908882	−	0.934519i
$52$		0			0
$53$	110518.			0.742347			0.371173	−	0.928564i	$-0.378956\pi$
$53$	110518.			0.742347			0.371173	+	0.928564i	$0.378956\pi$
$54$		0			0
$55$		−	340433.i		−	2.04618i
$56$		0			0
$57$	49282.5	−	47930.5i	0.266114	−	0.258814i
$58$		0			0
$59$	−27925.3	−	16122.7i	−0.135969	−	0.0785020i	0.430472	−	0.902604i	$-0.358347\pi$
$59$	−27925.3	−	16122.7i	−0.135969	−	0.0785020i	−0.566442	+	0.824102i	$0.691680\pi$
$60$		0			0
$61$	77336.7	+	133951.i	0.340719	+	0.590142i	0.984566	−	0.175012i	$-0.0559963\pi$
$61$	77336.7	+	133951.i	0.340719	+	0.590142i	−0.643848	+	0.765154i	$0.722663\pi$
$62$		0			0
$63$	119214.	+	220422.i	0.476765	+	0.881521i
$64$		0			0
$65$	−112748.	−	195286.i	−0.410554	−	0.711101i
$66$		0			0
$67$	−138114.	−	79740.3i	−0.459213	−	0.265127i	0.252500	−	0.967597i	$-0.418747\pi$
$67$	−138114.	−	79740.3i	−0.459213	−	0.265127i	−0.711713	+	0.702470i	$0.752080\pi$
$68$		0			0
$69$	−135314.	−	534630.i	−0.411904	−	1.62744i
$70$		0			0
$71$			52596.8i			0.146955i	0.997297	+	0.0734775i	$0.0234097\pi$
$71$			52596.8i			0.146955i	−0.997297	+	0.0734775i	$0.976590\pi$
$72$		0			0
$73$	494799.			1.27192			0.635961	−	0.771721i	$-0.280604\pi$
$73$	494799.			1.27192			0.635961	+	0.771721i	$0.280604\pi$
$74$		0			0
$75$	302684.	−	1.06989e6i	0.717473	−	2.53604i
$76$		0			0
$77$	245498.	−	425215.i	0.537745	−	0.931401i
$78$		0			0
$79$	−532196.	+	307264.i	−1.07942	+	0.623203i	−0.930740	−	0.365682i	$-0.880836\pi$
$79$	−532196.	+	307264.i	−1.07942	+	0.623203i	−0.148680	+	0.988885i	$0.547502\pi$
$80$		0			0
$81$	−530619.	−	29546.3i	−0.998453	−	0.0555967i
$82$		0			0
$83$	929041.	−	536382.i	1.62480	−	0.938080i	0.639192	−	0.769047i	$-0.279269\pi$
$83$	929041.	−	536382.i	1.62480	−	0.938080i	0.985610	−	0.169033i	$-0.0540645\pi$
$84$		0			0
$85$	−763238.	+	1.32197e6i	−1.24281	+	2.15260i
$86$		0			0
$87$	−25509.1	+	90166.7i	−0.0387380	+	0.136927i
$88$		0			0
$89$	626436.			0.888601			0.444301	−	0.895878i	$-0.353452\pi$
$89$	626436.			0.888601			0.444301	+	0.895878i	$0.353452\pi$
$90$		0			0
$91$		−	325228.i		−	0.431582i
$92$		0			0
$93$	167740.	+	662745.i	0.208539	+	0.823943i
$94$		0			0
$95$	−525551.	−	303427.i	−0.612977	−	0.353902i
$96$		0			0
$97$	286108.	+	495554.i	0.313484	+	0.542970i	0.979114	−	0.203312i	$-0.0651705\pi$
$97$	286108.	+	495554.i	0.313484	+	0.542970i	−0.665630	+	0.746282i	$0.731837\pi$
$98$		0			0
$99$	495355.	+	915893.i	0.510518	+	0.943929i
$100$		0			0
$101$	−440747.	−	763396.i	−0.427784	−	0.740944i	0.568891	−	0.822413i	$-0.307372\pi$
$101$	−440747.	−	763396.i	−0.427784	−	0.740944i	−0.996676	+	0.0814682i	$0.974039\pi$
$102$		0			0
$103$	1.21906e6	+	703824.i	1.11561	+	0.644098i	0.940277	−	0.340411i	$-0.110566\pi$
$103$	1.21906e6	+	703824.i	1.11561	+	0.644098i	0.175334	+	0.984509i	$0.443899\pi$
$104$		0			0
$105$	1.58579e6	−	1.54229e6i	1.36986	−	1.33228i
$106$		0			0
$107$		−	182957.i		−	0.149347i	−0.997208	−	0.0746735i	$-0.976209\pi$
$107$		−	182957.i		−	0.149347i	0.997208	−	0.0746735i	$-0.0237915\pi$
$108$		0			0
$109$	−244357.			−0.188688			−0.0943441	−	0.995540i	$-0.530075\pi$
$109$	−244357.			−0.188688			−0.0943441	+	0.995540i	$0.530075\pi$
$110$		0			0
$111$	1.25169e6	+	1.28700e6i	0.915228	+	0.941043i
$112$		0			0
$113$	−733410.	+	1.27030e6i	−0.508290	+	0.880384i	0.491664	+	0.870785i	$0.336389\pi$
$113$	−733410.	+	1.27030e6i	−0.508290	+	0.880384i	−0.999954	+	0.00959925i	$0.996944\pi$
$114$		0			0
$115$	−4.21599e6	+	2.43410e6i	−2.77208	+	1.60046i
$116$		0			0
$117$	587492.	+	361336.i	0.366813	+	0.225608i
$118$		0			0
$119$	−1.90664e6	+	1.10080e6i	−1.13143	+	0.653231i
$120$		0			0
$121$	134311.	−	232633.i	0.0758148	−	0.131315i
$122$		0			0
$123$	−3.07695e6	+	778773.i	−1.65350	+	0.418500i
$124$		0			0
$125$	−6.09098e6			−3.11858
$126$		0			0
$127$		−	2.90681e6i		−	1.41908i	−0.704667	−	0.709538i	$-0.748904\pi$
$127$		−	2.90681e6i		−	1.41908i	0.704667	−	0.709538i	$-0.251096\pi$
$128$		0			0
$129$	1.52709e6	+	432030.i	0.711370	+	0.201254i
$130$		0			0
$131$	1.30910e6	+	755809.i	0.582316	+	0.336200i	0.762053	−	0.647514i	$-0.224191\pi$
$131$	1.30910e6	+	755809.i	0.582316	+	0.336200i	−0.179737	+	0.983715i	$0.557525\pi$
$132$		0			0
$133$	−437624.	−	757987.i	−0.186014	−	0.322186i
$134$		0			0
$135$	1.02414e6	+	4.57809e6i	0.416252	+	1.86073i
$136$		0			0
$137$	−1.52718e6	−	2.64515e6i	−0.593921	−	1.02870i	−0.993698	−	0.112089i	$-0.964246\pi$
$137$	−1.52718e6	−	2.64515e6i	−0.593921	−	1.02870i	0.399777	−	0.916612i	$-0.369088\pi$
$138$		0			0
$139$	3.16098e6	+	1.82499e6i	1.17700	+	0.679542i	0.955319	−	0.295577i	$-0.0955120\pi$
$139$	3.16098e6	+	1.82499e6i	1.17700	+	0.679542i	0.221682	+	0.975119i	$0.428845\pi$
$140$		0			0
$141$	1.31038e6	+	370720.i	0.467455	+	0.132248i
$142$		0			0
$143$		−	1.35138e6i		−	0.462137i
$144$		0			0
$145$	827177.			0.271328
$146$		0			0
$147$	13497.4	−	3416.17i	0.00424911	−	0.00107544i
$148$		0			0
$149$	−1.53836e6	+	2.66452e6i	−0.465049	+	0.805489i	−0.999204	−	0.0398977i	$-0.987297\pi$
$149$	−1.53836e6	+	2.66452e6i	−0.465049	+	0.805489i	0.534154	+	0.845387i	$0.320630\pi$
$150$		0			0
$151$	2.37059e6	−	1.36866e6i	0.688533	−	0.397525i	−0.114529	−	0.993420i	$-0.536536\pi$
$151$	2.37059e6	−	1.36866e6i	0.688533	−	0.397525i	0.803062	+	0.595895i	$0.203203\pi$
$152$		0			0
$153$	129840.	−	4.66716e6i	0.0362521	−	1.30310i
$154$		0			0
$155$	5.22628e6	−	3.01739e6i	1.40345	−	0.810283i
$156$		0			0
$157$	−3.32390e6	+	5.75716e6i	−0.858913	+	1.48768i	0.0140546	+	0.999901i	$0.495526\pi$
$157$	−3.32390e6	+	5.75716e6i	−0.858913	+	1.48768i	−0.872967	+	0.487779i	$0.837807\pi$
$158$		0			0
$159$	2.08046e6	+	2.13914e6i	0.517568	+	0.532167i
$160$		0			0
$161$	−7.02128e6			−1.68244
$162$		0			0
$163$			6.64679e6i			1.53479i	0.641174	+	0.767396i	$0.278448\pi$
$163$			6.64679e6i			1.53479i	−0.641174	+	0.767396i	$0.721552\pi$
$164$		0			0
$165$	6.58925e6	−	6.40849e6i	1.46685	−	1.42661i
$166$		0			0
$167$	−2.16723e6	−	1.25125e6i	−0.465325	−	0.268656i	0.248956	−	0.968515i	$-0.419913\pi$
$167$	−2.16723e6	−	1.25125e6i	−0.465325	−	0.268656i	−0.714281	+	0.699859i	$0.753246\pi$
$168$		0			0
$169$	1.96584e6	+	3.40493e6i	0.407275	+	0.705421i
$170$		0			0
$171$	1.85544e6	+	51618.1i	0.371072	+	0.0103232i
$172$		0			0
$173$	−713118.	−	1.23516e6i	−0.137728	−	0.238552i	0.788908	−	0.614511i	$-0.210647\pi$
$173$	−713118.	−	1.23516e6i	−0.137728	−	0.238552i	−0.926636	+	0.375959i	$0.877313\pi$
$174$		0			0
$175$	−1.22594e7	−	7.07798e6i	−2.28747	−	1.32067i
$176$		0			0
$177$	−213618.	−	844010.i	−0.0385228	−	0.152205i
$178$		0			0
$179$		−	1.21323e6i		−	0.211536i	−0.994391	−	0.105768i	$-0.966270\pi$
$179$		−	1.21323e6i		−	0.211536i	0.994391	−	0.105768i	$-0.0337301\pi$
$180$		0			0
$181$	296213.			0.0499539			0.0249769	−	0.999688i	$-0.492049\pi$
$181$	296213.			0.0499539			0.0249769	+	0.999688i	$0.492049\pi$
$182$		0			0
$183$	−1.13687e6	+	4.01846e6i	−0.185505	+	0.655702i
$184$		0			0
$185$	7.92392e6	−	1.37246e7i	1.25148	−	2.16763i
$186$		0			0
$187$	−7.92244e6	+	4.57402e6i	−1.21153	+	0.699477i
$188$		0			0
$189$	−2.02223e6	+	6.45678e6i	−0.299534	+	0.956380i
$190$		0			0
$191$	−548352.	+	316591.i	−0.0786972	+	0.0454358i	−0.538832	−	0.842413i	$-0.681134\pi$
$191$	−548352.	+	316591.i	−0.0786972	+	0.0454358i	0.460135	+	0.887849i	$0.347801\pi$
$192$		0			0
$193$	−5.66007e6	+	9.80353e6i	−0.787317	+	1.36367i	0.140288	+	0.990111i	$0.455197\pi$
$193$	−5.66007e6	+	9.80353e6i	−0.787317	+	1.36367i	−0.927605	+	0.373563i	$0.878136\pi$
$194$		0			0
$195$	1.65743e6	−	5.85848e6i	0.223527	−	0.790098i
$196$		0			0
$197$	1.97684e6			0.258566			0.129283	−	0.991608i	$-0.458732\pi$
$197$	1.97684e6			0.258566			0.129283	+	0.991608i	$0.458732\pi$
$198$		0			0
$199$			3.80885e6i			0.483320i	0.970361	+	0.241660i	$0.0776919\pi$
$199$			3.80885e6i			0.483320i	−0.970361	+	0.241660i	$0.922308\pi$
$200$		0			0
$201$	−1.05652e6	−	4.17435e6i	−0.130104	−	0.514044i
$202$		0			0
$203$	1.03318e6	+	596507.i	0.123506	+	0.0713062i
$204$		0			0
$205$	1.40090e7	+	2.42643e7i	1.62609	+	2.81647i
$206$		0			0
$207$	7.80081e6	−	1.26832e7i	0.879486	−	1.42994i
$208$		0			0
$209$	−1.81841e6	−	3.14958e6i	−0.199183	−	0.344996i
$210$		0			0
$211$	−1.04522e7	−	6.03456e6i	−1.11265	−	0.642389i	−0.173136	−	0.984898i	$-0.555390\pi$
$211$	−1.04522e7	−	6.03456e6i	−1.11265	−	0.642389i	−0.939515	+	0.342509i	$0.888723\pi$
$212$		0			0
$213$	−1.01804e6	+	990111.i	−0.105348	+	0.102458i
$214$		0			0
$215$		−	1.40093e7i		−	1.40962i
$216$		0			0
$217$	8.70380e6			0.851785
$218$		0			0
$219$	9.31437e6	+	9.57710e6i	0.886791	+	0.911804i
$220$		0			0
$221$	−3.02976e6	+	5.24769e6i	−0.280692	+	0.486174i
$222$		0			0
$223$	9.91395e6	−	5.72382e6i	0.893989	−	0.516145i	0.0187436	−	0.999824i	$-0.494033\pi$
$223$	9.91395e6	−	5.72382e6i	0.893989	−	0.516145i	0.875245	+	0.483680i	$0.160700\pi$
$224$		0			0
$225$	2.64062e7	−	1.42816e7i	2.31824	−	1.25380i
$226$		0			0
$227$	1.05785e7	−	6.10752e6i	0.904374	−	0.522141i	0.0257573	−	0.999668i	$-0.491800\pi$
$227$	1.05785e7	−	6.10752e6i	0.904374	−	0.522141i	0.878617	+	0.477528i	$0.158467\pi$
$228$		0			0
$229$	2.55787e6	−	4.43037e6i	0.212996	−	0.368921i	−0.739654	−	0.672987i	$-0.765011\pi$
$229$	2.55787e6	−	4.43037e6i	0.212996	−	0.368921i	0.952651	+	0.304066i	$0.0983444\pi$
$230$		0			0
$231$	1.28517e7	−	3.25274e6i	1.04261	−	0.263884i
$232$		0			0
$233$	−4.13676e6			−0.327034			−0.163517	−	0.986541i	$-0.552284\pi$
$233$	−4.13676e6			−0.327034			−0.163517	+	0.986541i	$0.552284\pi$
$234$		0			0
$235$		−	1.20213e7i		−	0.926288i
$236$		0			0
$237$	−1.59656e7	−	4.51684e6i	−1.19933	−	0.339304i
$238$		0			0
$239$	207614.	+	119866.i	0.0152077	+	0.00878016i	0.507585	−	0.861602i	$-0.330538\pi$
$239$	207614.	+	119866.i	0.0152077	+	0.00878016i	−0.492377	+	0.870382i	$0.663872\pi$
$240$		0			0
$241$	−2.05216e6	−	3.55444e6i	−0.146609	−	0.253934i	0.783363	−	0.621564i	$-0.213502\pi$
$241$	−2.05216e6	−	3.55444e6i	−0.146609	−	0.253934i	−0.929972	+	0.367630i	$0.880169\pi$
$242$		0			0
$243$	−9.41678e6	−	1.08266e7i	−0.656271	−	0.754525i
$244$		0			0
$245$	−61451.9	−	106438.i	−0.00417866	−	0.00723765i
$246$		0			0
$247$	−2.08623e6	−	1.20449e6i	−0.138443	−	0.0799301i
$248$		0			0
$249$	2.78707e7	+	7.88493e6i	1.80530	+	0.510740i
$250$		0			0
$251$			2.62075e6i			0.165731i	0.996561	+	0.0828655i	$0.0264072\pi$
$251$			2.62075e6i			0.165731i	−0.996561	+	0.0828655i	$0.973593\pi$
$252$		0			0
$253$	−2.91747e7			−1.80154
$254$		0			0
$255$	−3.99550e7	+	1.01126e7i	−2.40963	+	0.609875i
$256$		0			0
$257$	−2.38542e6	+	4.13166e6i	−0.140529	+	0.243403i	−0.927696	−	0.373337i	$-0.878214\pi$
$257$	−2.38542e6	+	4.13166e6i	−0.140529	+	0.243403i	0.787167	+	0.616740i	$0.211547\pi$
$258$		0			0
$259$	1.97947e7	−	1.14285e7i	1.13933	−	0.657791i
$260$		0			0
$261$	−2.22542e6	+	1.20360e6i	−0.125167	+	0.0676959i
$262$		0			0
$263$	2.85537e7	−	1.64855e7i	1.56962	−	0.906222i	0.573409	−	0.819269i	$-0.305620\pi$
$263$	2.85537e7	−	1.64855e7i	1.56962	−	0.906222i	0.996212	−	0.0869525i	$-0.0277129\pi$
$264$		0			0
$265$	1.31705e7	−	2.28119e7i	0.707723	−	1.22581i
$266$		0			0
$267$	1.17924e7	+	1.21250e7i	0.619538	+	0.637013i
$268$		0			0
$269$	−3.67419e7			−1.88758			−0.943789	−	0.330548i	$-0.892766\pi$
$269$	−3.67419e7			−1.88758			−0.943789	+	0.330548i	$0.892766\pi$
$270$		0			0
$271$			1.68695e7i			0.847605i	0.905755	+	0.423803i	$0.139305\pi$
$271$			1.68695e7i			0.847605i	−0.905755	+	0.423803i	$0.860695\pi$
$272$		0			0
$273$	6.29496e6	−	6.12227e6i	0.309389	−	0.300902i
$274$		0			0
$275$	−5.09402e7	−	2.94103e7i	−2.44942	−	1.41417i
$276$		0			0
$277$	2.56226e6	+	4.43797e6i	0.120555	+	0.208807i	0.919987	−	0.391950i	$-0.128199\pi$
$277$	2.56226e6	+	4.43797e6i	0.120555	+	0.208807i	−0.799432	+	0.600757i	$0.794866\pi$
$278$		0			0
$279$	−9.67014e6	+	1.57226e7i	−0.445267	+	0.723954i
$280$		0			0
$281$	−5.49551e6	−	9.51850e6i	−0.247679	−	0.428993i	0.715202	−	0.698917i	$-0.246334\pi$
$281$	−5.49551e6	−	9.51850e6i	−0.247679	−	0.428993i	−0.962881	+	0.269925i	$0.913001\pi$
$282$		0			0
$283$	2.21732e6	+	1.28017e6i	0.0978294	+	0.0564818i	0.548117	−	0.836402i	$-0.315345\pi$
$283$	2.21732e6	+	1.28017e6i	0.0978294	+	0.0564818i	−0.450287	+	0.892884i	$0.648678\pi$
$284$		0			0
$285$	−4.02027e6	−	1.58842e7i	−0.173668	−	0.686168i
$286$		0			0
$287$			4.04095e7i			1.70938i
$288$		0			0
$289$	1.68816e7			0.699392
$290$		0			0
$291$	−4.20585e6	+	1.48664e7i	−0.170677	+	0.603289i
$292$		0			0
$293$	−1.33423e7	+	2.31095e7i	−0.530428	+	0.918728i	0.468942	+	0.883229i	$0.344635\pi$
$293$	−1.33423e7	+	2.31095e7i	−0.530428	+	0.918728i	−0.999370	+	0.0354991i	$0.988698\pi$
$294$		0			0
$295$	−6.65570e6	+	3.84267e6i	−0.259255	+	0.149681i
$296$		0			0
$297$	−8.40276e6	+	2.68291e7i	−0.320740	+	1.02409i
$298$		0			0
$299$	−1.67358e7	+	9.66243e6i	−0.626085	+	0.361470i
$300$		0			0
$301$	1.01026e7	−	1.74983e7i	0.370454	−	0.641646i
$302$		0			0
$303$	6.47907e6	−	2.29015e7i	0.232908	−	0.823257i
$304$		0			0
$305$	3.68648e7			1.29931
$306$		0			0
$307$			1.80526e7i			0.623912i	0.950096	+	0.311956i	$0.100984\pi$
$307$			1.80526e7i			0.623912i	−0.950096	+	0.311956i	$0.899016\pi$
$308$		0			0
$309$	9.32534e6	+	3.68447e7i	0.316074	+	1.24882i
$310$		0			0
$311$	3.62362e7	+	2.09210e7i	1.20465	+	0.695506i	0.961586	−	0.274503i	$-0.0885135\pi$
$311$	3.62362e7	+	2.09210e7i	1.20465	+	0.695506i	0.243066	+	0.970010i	$0.421847\pi$
$312$		0			0
$313$	−1.93993e7	−	3.36005e7i	−0.632633	−	1.09575i	−0.987011	−	0.160651i	$-0.948641\pi$
$313$	−1.93993e7	−	3.36005e7i	−0.632633	−	1.09575i	0.354378	−	0.935102i	$-0.384693\pi$
$314$		0			0
$315$	5.97035e7	+	1.66094e6i	1.91015	+	0.0531402i
$316$		0			0
$317$	1.69786e7	+	2.94079e7i	0.532997	+	0.923178i	0.999257	+	0.0385306i	$0.0122677\pi$
$317$	1.69786e7	+	2.94079e7i	0.532997	+	0.923178i	−0.466260	+	0.884648i	$0.654399\pi$
$318$		0			0
$319$	4.29306e6	+	2.47860e6i	0.132250	+	0.0763544i
$320$		0			0
$321$	3.54122e6	−	3.44407e6i	0.107063	−	0.104126i
$322$		0			0
$323$			1.63073e7i			0.483920i
$324$		0			0
$325$	−3.89618e7			−1.13498
$326$		0			0
$327$	−4.59991e6	−	4.72965e6i	−0.131554	−	0.135265i
$328$		0			0
$329$	8.66896e6	−	1.50151e7i	0.243433	−	0.421638i
$330$		0			0
$331$	−3.05366e6	+	1.76303e6i	−0.0842047	+	0.0486156i	−0.541511	−	0.840694i	$-0.682148\pi$
$331$	−3.05366e6	+	1.76303e6i	−0.0842047	+	0.0486156i	0.457306	+	0.889309i	$0.348814\pi$
$332$		0			0
$333$	−1.34799e6	+	4.84544e7i	−0.0365052	+	1.31220i
$334$		0			0
$335$	−3.29181e7	+	1.90053e7i	−0.875590	+	0.505522i
$336$		0			0
$337$	−2.81383e7	+	4.87370e7i	−0.735205	+	1.27341i	0.219429	+	0.975629i	$0.429581\pi$
$337$	−2.81383e7	+	4.87370e7i	−0.735205	+	1.27341i	−0.954633	+	0.297784i	$0.903753\pi$
$338$		0			0
$339$	−3.83935e7	+	9.71735e6i	−0.985505	+	0.249430i
$340$		0			0
$341$	3.61659e7			0.912088
$342$		0			0
$343$			4.02647e7i			0.997796i
$344$		0			0
$345$	−1.26477e8	−	3.57818e7i	−3.08004	−	0.871375i
$346$		0			0
$347$	−2.15843e7	−	1.24617e7i	−0.516594	−	0.298255i	0.218946	−	0.975737i	$-0.429738\pi$
$347$	−2.15843e7	−	1.24617e7i	−0.516594	−	0.298255i	−0.735540	+	0.677481i	$0.763071\pi$
$348$		0			0
$349$	4.85248e6	+	8.40475e6i	0.114153	+	0.197719i	0.917441	−	0.397872i	$-0.130251\pi$
$349$	4.85248e6	+	8.40475e6i	0.114153	+	0.197719i	−0.803288	+	0.595591i	$0.796918\pi$
$350$		0			0
$351$	4.06542e6	+	1.81732e7i	0.0940121	+	0.420252i
$352$		0			0
$353$	3.39254e6	+	5.87605e6i	0.0771259	+	0.133586i	0.902009	−	0.431718i	$-0.142092\pi$
$353$	3.39254e6	+	5.87605e6i	0.0771259	+	0.133586i	−0.824883	+	0.565304i	$0.808759\pi$
$354$		0			0
$355$	1.08564e7	+	6.26796e6i	0.242662	+	0.140101i
$356$		0			0
$357$	−5.71981e7	−	1.61820e7i	−1.25712	−	0.355653i
$358$		0			0
$359$			5.77924e7i			1.24907i	0.780996	+	0.624536i	$0.214712\pi$
$359$			5.77924e7i			1.24907i	−0.780996	+	0.624536i	$0.785288\pi$
$360$		0			0
$361$	4.05629e7			0.862199
$362$		0			0
$363$	7.03106e6	−	1.77955e6i	0.146995	−	0.0372041i
$364$		0			0
$365$	5.89652e7	−	1.02131e8i	1.21260	−	2.10028i
$366$		0			0
$367$	−5.91261e7	+	3.41365e7i	−1.19614	+	0.690591i	−0.959692	−	0.281054i	$-0.909316\pi$
$367$	−5.91261e7	+	3.41365e7i	−1.19614	+	0.690591i	−0.236446	+	0.971645i	$0.575983\pi$
$368$		0			0
$369$	−7.29958e7	−	4.48960e7i	−1.45284	−	0.893570i
$370$		0			0
$371$	3.29010e7	−	1.89954e7i	0.644299	−	0.371986i
$372$		0			0
$373$	1.29579e7	−	2.24437e7i	0.249693	−	0.432481i	−0.713748	−	0.700403i	$-0.753004\pi$
$373$	1.29579e7	−	2.24437e7i	0.249693	−	0.432481i	0.963441	+	0.267922i	$0.0863369\pi$
$374$		0			0
$375$	−1.14660e8	−	1.17894e8i	−2.17429	−	2.23562i
$376$		0			0
$377$	3.28357e6			0.0612804
$378$		0			0
$379$		−	5.83783e7i		−	1.07234i	−0.844109	−	0.536171i	$-0.819870\pi$
$379$		−	5.83783e7i		−	1.07234i	0.844109	−	0.536171i	$-0.180130\pi$
$380$		0			0
$381$	5.62629e7	−	5.47194e7i	1.01729	−	0.989387i
$382$		0			0
$383$	2.77446e7	+	1.60184e7i	0.493836	+	0.285116i	0.726165	−	0.687521i	$-0.241301\pi$
$383$	2.77446e7	+	1.60184e7i	0.493836	+	0.285116i	−0.232328	+	0.972637i	$0.574634\pi$
$384$		0			0
$385$	−5.85120e7	−	1.01346e8i	−1.02533	−	1.77592i
$386$		0			0
$387$	2.03846e7	+	3.76904e7i	0.351698	+	0.650276i
$388$		0			0
$389$	3.10417e7	+	5.37658e7i	0.527347	+	0.913393i	0.999492	+	0.0318714i	$0.0101467\pi$
$389$	3.10417e7	+	5.37658e7i	0.527347	+	0.913393i	−0.472145	+	0.881521i	$0.656520\pi$
$390$		0			0
$391$	1.13291e8	+	6.54087e7i	1.89525	+	1.09422i
$392$		0			0
$393$	1.00141e7	+	3.95661e7i	0.164981	+	0.651846i
$394$		0			0
$395$			1.46466e8i			2.37655i
$396$		0			0
$397$	1.04878e7			0.167615			0.0838075	−	0.996482i	$-0.473292\pi$
$397$	1.04878e7			0.167615			0.0838075	+	0.996482i	$0.473292\pi$
$398$		0			0
$399$	6.43317e6	−	2.27392e7i	0.101276	−	0.357978i
$400$		0			0
$401$	4.96212e6	−	8.59464e6i	0.0769545	−	0.133289i	−0.824980	−	0.565162i	$-0.808814\pi$
$401$	4.96212e6	−	8.59464e6i	0.0769545	−	0.133289i	0.901935	+	0.431873i	$0.142147\pi$
$402$		0			0
$403$	2.07463e7	−	1.19779e7i	0.316975	−	0.183005i
$404$		0			0
$405$	−6.93324e7	+	1.06003e8i	−1.04369	+	1.59571i
$406$		0			0
$407$	8.22505e7	−	4.74874e7i	1.21999	−	0.704360i
$408$		0			0
$409$	−5.87888e6	+	1.01825e7i	−0.0859260	+	0.148828i	−0.905785	−	0.423737i	$-0.860718\pi$
$409$	−5.87888e6	+	1.01825e7i	−0.0859260	+	0.148828i	0.819859	+	0.572565i	$0.194052\pi$
$410$		0			0
$411$	2.24499e7	−	7.93532e7i	0.323362	−	1.14298i
$412$		0			0
$413$	−1.10844e7			−0.157348
$414$		0			0
$415$		−	2.55682e8i		−	3.57731i
$416$		0			0
$417$	2.41803e7	+	9.55370e7i	0.333467	+	1.31754i
$418$		0			0
$419$	−1.26782e8	−	7.31976e7i	−1.72352	−	0.995073i	−0.911331	−	0.411674i	$-0.864944\pi$
$419$	−1.26782e8	−	7.31976e7i	−1.72352	−	0.995073i	−0.812186	−	0.583399i	$-0.801722\pi$
$420$		0			0
$421$	−3.11860e7	−	5.40158e7i	−0.417940	−	0.723894i	0.577792	−	0.816184i	$-0.303914\pi$
$421$	−3.11860e7	−	5.40158e7i	−0.417940	−	0.723894i	−0.995732	+	0.0922905i	$0.970581\pi$
$422$		0			0
$423$	1.74918e7	+	3.23417e7i	0.231107	+	0.427309i
$424$		0			0
$425$	1.31874e8	+	2.28412e8i	1.71788	+	2.97545i
$426$		0			0
$427$	4.60458e7	+	2.65845e7i	0.591434	+	0.341465i
$428$		0			0
$429$	2.61567e7	−	2.54392e7i	0.331292	−	0.322204i
$430$		0			0
$431$		−	2.02542e7i		−	0.252978i	−0.991968	−	0.126489i	$-0.959629\pi$
$431$		−	2.02542e7i		−	0.252978i	0.991968	−	0.126489i	$-0.0403709\pi$
$432$		0			0
$433$	1.42026e8			1.74946			0.874732	−	0.484607i	$-0.161037\pi$
$433$	1.42026e8			1.74946			0.874732	+	0.484607i	$0.161037\pi$
$434$		0			0
$435$	1.55712e7	+	1.60104e7i	0.189171	+	0.194507i
$436$		0			0
$437$	−2.60034e7	+	4.50392e7i	−0.311591	+	0.539692i
$438$		0			0
$439$	1.21442e8	−	7.01145e7i	1.43541	−	0.828732i	0.437881	−	0.899033i	$-0.355729\pi$
$439$	1.21442e8	−	7.01145e7i	1.43541	−	0.828732i	0.997526	+	0.0703005i	$0.0223958\pi$
$440$		0			0
$441$	320204.	+	196941.i	0.00373346	+	0.00229626i
$442$		0			0
$443$	−3.69677e7	+	2.13433e7i	−0.425217	+	0.245499i	−0.697307	−	0.716773i	$-0.745619\pi$
$443$	−3.69677e7	+	2.13433e7i	−0.425217	+	0.245499i	0.272090	+	0.962272i	$0.412285\pi$
$444$		0			0
$445$	7.46524e7	−	1.29302e8i	0.847157	−	1.46732i
$446$		0			0
$447$	−8.05321e7	+	2.03826e7i	−0.901667	+	0.228211i
$448$		0			0
$449$	−7.63226e7			−0.843168			−0.421584	−	0.906789i	$-0.638526\pi$
$449$	−7.63226e7			−0.843168			−0.421584	+	0.906789i	$0.638526\pi$
$450$		0			0
$451$			1.67909e8i			1.83039i
$452$		0			0
$453$	7.11163e7	+	2.01196e7i	0.765023	+	0.216433i
$454$		0			0
$455$	−6.71297e7	−	3.87574e7i	−0.712657	−	0.411453i
$456$		0			0
$457$	−5.36449e7	−	9.29157e7i	−0.562056	−	0.973510i	−0.997317	−	0.0732054i	$-0.976677\pi$
$457$	−5.36449e7	−	9.29157e7i	−0.562056	−	0.973510i	0.435261	−	0.900304i	$-0.356656\pi$
$458$		0			0
$459$	9.27796e7	−	8.53441e7i	0.959433	−	0.882543i
$460$		0			0
$461$	−1.75328e7	−	3.03678e7i	−0.178957	−	0.309963i	0.762566	−	0.646910i	$-0.223939\pi$
$461$	−1.75328e7	−	3.03678e7i	−0.178957	−	0.309963i	−0.941524	+	0.336947i	$0.890606\pi$
$462$		0			0
$463$	−4.99139e7	−	2.88178e7i	−0.502897	−	0.290348i	0.227012	−	0.973892i	$-0.427104\pi$
$463$	−4.99139e7	−	2.88178e7i	−0.502897	−	0.290348i	−0.729909	+	0.683544i	$0.760438\pi$
$464$		0			0
$465$	1.56786e8	+	4.43563e7i	1.55936	+	0.441160i
$466$		0			0
$467$		−	1.29340e8i		−	1.26994i	−0.772536	−	0.634970i	$-0.781012\pi$
$467$		−	1.29340e8i		−	1.26994i	0.772536	−	0.634970i	$-0.218988\pi$
$468$		0			0
$469$	−5.48216e7			−0.531414
$470$		0			0
$471$	−1.74004e8	+	4.40402e7i	−1.66531	+	0.421489i
$472$		0			0
$473$	4.19783e7	−	7.27085e7i	0.396681	−	0.687072i
$474$		0			0
$475$	−9.08058e7	+	5.24268e7i	−0.847292	+	0.489184i
$476$		0			0
$477$	−2.24052e6	+	8.05367e7i	−0.0206440	+	0.742060i
$478$		0			0
$479$	−3.36830e7	+	1.94469e7i	−0.306482	+	0.176947i	−0.645351	−	0.763886i	$-0.723289\pi$
$479$	−3.36830e7	+	1.94469e7i	−0.306482	+	0.176947i	0.338869	+	0.940833i	$0.389956\pi$
$480$		0			0
$481$	3.14548e7	−	5.44814e7i	0.282652	−	0.489568i
$482$		0			0
$483$	−1.32172e8	−	1.35900e8i	−1.17300	−	1.20609i
$484$		0			0
$485$	1.36382e8			1.19545
$486$		0			0
$487$		−	8.04552e7i		−	0.696574i	−0.937388	−	0.348287i	$-0.886763\pi$
$487$		−	8.04552e7i		−	0.696574i	0.937388	−	0.348287i	$-0.113237\pi$
$488$		0			0
$489$	−1.28652e8	+	1.25123e8i	−1.10025	+	1.07007i
$490$		0			0
$491$	−3.14375e7	−	1.81504e7i	−0.265585	−	0.153336i	0.361295	−	0.932452i	$-0.382335\pi$
$491$	−3.14375e7	−	1.81504e7i	−0.265585	−	0.153336i	−0.626880	+	0.779116i	$0.715668\pi$
$492$		0			0
$493$	−1.11139e7	−	1.92498e7i	−0.0927523	−	0.160652i
$494$		0			0
$495$	2.48079e8	+	6.90153e6i	2.04538	+	0.0569023i
$496$		0			0
$497$	9.04010e6	+	1.56579e7i	0.0736384	+	0.127545i
$498$		0			0
$499$	6.95533e7	+	4.01566e7i	0.559779	+	0.323188i	0.753057	−	0.657956i	$-0.228579\pi$
$499$	6.95533e7	+	4.01566e7i	0.559779	+	0.323188i	−0.193278	+	0.981144i	$0.561912\pi$
$500$		0			0
$501$	−1.65785e7	−	6.55022e7i	−0.131836	−	0.520886i
$502$		0			0
$503$			1.38884e8i			1.09131i	0.838009	+	0.545656i	$0.183719\pi$
$503$			1.38884e8i			1.09131i	−0.838009	+	0.545656i	$0.816281\pi$
$504$		0			0
$505$	−2.10095e8			−1.63133
$506$		0			0
$507$	−2.88982e7	+	1.02146e8i	−0.221742	+	0.783787i
$508$		0			0
$509$	4.47554e7	−	7.75187e7i	0.339385	−	0.587832i	−0.644932	−	0.764240i	$-0.723114\pi$
$509$	4.47554e7	−	7.75187e7i	0.339385	−	0.587832i	0.984317	+	0.176408i	$0.0564478\pi$
$510$		0			0
$511$	1.47300e8	−	8.50438e7i	1.10393	−	0.637353i
$512$		0			0
$513$	3.39287e7	+	3.68847e7i	0.251313	+	0.273209i
$514$		0			0
$515$	2.90550e8	−	1.67749e8i	2.12716	−	1.22811i
$516$		0			0
$517$	3.60211e7	−	6.23905e7i	0.260667	−	0.451488i
$518$		0			0
$519$	1.04830e7	−	3.70541e7i	0.0749865	−	0.265054i
$520$		0			0
$521$	−8.93181e7			−0.631577			−0.315789	−	0.948830i	$-0.602269\pi$
$521$	−8.93181e7			−0.631577			−0.315789	+	0.948830i	$0.602269\pi$
$522$		0			0
$523$			9.96143e7i			0.696333i	0.937433	+	0.348166i	$0.113196\pi$
$523$			9.96143e7i			0.696333i	−0.937433	+	0.348166i	$0.886804\pi$
$524$		0			0
$525$	−9.37800e7	−	3.70527e8i	−0.648086	−	2.56060i
$526$		0			0
$527$	−1.40440e8	−	8.10828e7i	−0.959528	−	0.553984i
$528$		0			0
$529$	1.34582e8	+	2.33103e8i	0.909118	+	1.57464i
$530$		0			0
$531$	1.23150e7	−	2.00228e7i	0.0822528	−	0.133734i
$532$		0			0
$533$	5.56101e7	+	9.63196e7i	0.367259	+	0.636111i
$534$		0			0
$535$	−3.77638e7	−	2.18029e7i	−0.246612	−	0.142381i
$536$		0			0
$537$	2.34827e7	−	2.28385e7i	0.151644	−	0.147484i
$538$		0			0
$539$		−	736552.i		−	0.00470367i
$540$		0			0
$541$	−7.95037e6			−0.0502106			−0.0251053	−	0.999685i	$-0.507992\pi$
$541$	−7.95037e6			−0.0502106			−0.0251053	+	0.999685i	$0.507992\pi$
$542$		0			0
$543$	5.57608e6	+	5.73337e6i	0.0348281	+	0.0358105i
$544$		0			0
$545$	−2.91200e7	+	5.04373e7i	−0.179888	+	0.311575i
$546$		0			0
$547$	−1.55761e8	+	8.99286e7i	−0.951692	+	0.549460i	−0.893606	−	0.448852i	$-0.851833\pi$
$547$	−1.55761e8	+	8.99286e7i	−0.951692	+	0.549460i	−0.0580857	+	0.998312i	$0.518500\pi$
$548$		0			0
$549$	−9.91804e7	+	5.36411e7i	−0.599389	+	0.324176i
$550$		0			0
$551$	7.65279e6	−	4.41834e6i	0.0457473	−	0.0264122i
$552$		0			0
$553$	−1.05622e8	+	1.82943e8i	−0.624568	+	1.08178i
$554$		0			0
$555$	4.14812e8	−	1.04988e8i	2.42645	−	0.614133i
$556$		0			0
$557$	−8.76720e7			−0.507335			−0.253668	−	0.967291i	$-0.581637\pi$
$557$	−8.76720e7			−0.507335			−0.253668	+	0.967291i	$0.581637\pi$
$558$		0			0
$559$		−	5.56115e7i		−	0.318368i
$560$		0			0
$561$	−2.37669e8	−	6.72390e7i	−1.34612	−	0.380832i
$562$		0			0
$563$	9.14612e7	+	5.28051e7i	0.512521	+	0.295904i	0.733869	−	0.679291i	$-0.237712\pi$
$563$	9.14612e7	+	5.28051e7i	0.512521	+	0.295904i	−0.221348	+	0.975195i	$0.571046\pi$
$564$		0			0
$565$	1.74801e8	+	3.02764e8i	0.969166	+	1.67865i
$566$		0			0
$567$	−1.63042e8	+	8.24045e7i	−0.894438	+	0.452066i
$568$		0			0
$569$	−1.44922e8	−	2.51013e8i	−0.786681	−	1.36257i	−0.927990	−	0.372606i	$-0.878464\pi$
$569$	−1.44922e8	−	2.51013e8i	−0.786681	−	1.36257i	0.141308	−	0.989966i	$-0.454869\pi$
$570$		0			0
$571$	−2.91357e8	−	1.68215e8i	−1.56501	−	0.903558i	−0.996737	−	0.0807168i	$-0.974279\pi$
$571$	−2.91357e8	−	1.68215e8i	−1.56501	−	0.903558i	−0.568271	−	0.822841i	$-0.692388\pi$
$572$		0			0
$573$	−1.64503e7	−	4.65396e6i	−0.0874398	−	0.0247376i
$574$		0			0
$575$			8.41139e8i			4.42450i
$576$		0			0
$577$	−2.90800e7			−0.151379			−0.0756897	−	0.997131i	$-0.524116\pi$
$577$	−2.90800e7			−0.151379			−0.0756897	+	0.997131i	$0.524116\pi$
$578$		0			0
$579$	−2.96301e8	+	7.49933e7i	−1.52650	+	0.386355i
$580$		0			0
$581$	1.84382e8	−	3.19359e8i	0.940134	−	1.62836i
$582$		0			0
$583$	1.36710e8	−	7.89294e7i	0.689912	−	0.398321i
$584$		0			0
$585$	1.44594e8	−	7.82028e7i	0.722243	−	0.390620i
$586$		0			0
$587$	9.96478e7	−	5.75317e7i	0.492667	−	0.284441i	−0.233013	−	0.972474i	$-0.574859\pi$
$587$	9.96478e7	−	5.75317e7i	0.492667	−	0.284441i	0.725680	+	0.688032i	$0.241525\pi$
$588$		0			0
$589$	3.22346e7	−	5.58320e7i	0.157753	−	0.273236i
$590$		0			0
$591$	3.72131e7	+	3.82627e7i	0.180274	+	0.185359i
$592$		0			0
$593$	−7.96643e7			−0.382032			−0.191016	−	0.981587i	$-0.561178\pi$
$593$	−7.96643e7			−0.382032			−0.191016	+	0.981587i	$0.561178\pi$
$594$		0			0
$595$			5.24728e8i			2.49105i
$596$		0			0
$597$	−7.37223e7	+	7.16999e7i	−0.346478	+	0.336973i
$598$		0			0
$599$	3.07735e7	+	1.77671e7i	0.143185	+	0.0826677i	0.569881	−	0.821727i	$-0.306989\pi$
$599$	3.07735e7	+	1.77671e7i	0.143185	+	0.0826677i	−0.426696	+	0.904395i	$0.640323\pi$
$600$		0			0
$601$	−5.91559e7	−	1.02461e8i	−0.272505	−	0.471993i	0.696997	−	0.717074i	$-0.254519\pi$
$601$	−5.91559e7	−	1.02461e8i	−0.272505	−	0.471993i	−0.969503	+	0.245081i	$0.921186\pi$
$602$		0			0
$603$	6.09081e7	−	9.90298e7i	0.277794	−	0.451662i
$604$		0			0
$605$	−3.20116e7	−	5.54456e7i	−0.144557	−	0.250381i
$606$		0			0
$607$	−3.33321e8	−	1.92443e8i	−1.49038	−	0.860471i	−0.490441	−	0.871475i	$-0.663164\pi$
$607$	−3.33321e8	−	1.92443e8i	−1.49038	−	0.860471i	−0.999939	+	0.0110032i	$0.996497\pi$
$608$		0			0
$609$	7.90345e6	+	3.12267e7i	0.0349917	+	0.138253i
$610$		0			0
$611$		−	4.77196e7i		−	0.209206i
$612$		0			0
$613$	2.68018e8			1.16354			0.581772	−	0.813352i	$-0.302359\pi$
$613$	2.68018e8			1.16354			0.581772	+	0.813352i	$0.302359\pi$
$614$		0			0
$615$	−2.05935e8	+	7.27915e8i	−0.885329	+	3.12936i
$616$		0			0
$617$	−1.46408e8	+	2.53587e8i	−0.623319	+	1.07962i	0.365545	+	0.930794i	$0.380883\pi$
$617$	−1.46408e8	+	2.53587e8i	−0.623319	+	1.07962i	−0.988863	+	0.148826i	$0.952451\pi$
$618$		0			0
$619$	−2.72469e8	+	1.57310e8i	−1.14880	+	0.663260i	−0.948595	−	0.316493i	$-0.897494\pi$
$619$	−2.72469e8	+	1.57310e8i	−1.14880	+	0.663260i	−0.200206	+	0.979754i	$0.564161\pi$
$620$		0			0
$621$	3.92338e8	−	8.77674e7i	1.63827	−	0.366487i
$622$		0			0
$623$	1.86488e8	−	1.07669e8i	0.771236	−	0.445273i
$624$		0			0
$625$	−4.04136e8	+	6.99985e8i	−1.65534	+	2.86714i
$626$		0			0
$627$	2.67310e7	−	9.44857e7i	0.108446	−	0.383322i
$628$		0			0
$629$	−4.25860e8			−1.71126
$630$		0			0
$631$			2.28584e8i			0.909823i	0.890537	+	0.454912i	$0.150329\pi$
$631$			2.28584e8i			0.909823i	−0.890537	+	0.454912i	$0.849671\pi$
$632$		0			0
$633$	−7.99552e7	−	3.15905e8i	−0.315236	−	1.24550i
$634$		0			0
$635$	−5.99990e8	−	3.46404e8i	−2.34327	−	1.35289i
$636$		0			0
$637$	−243940.	−	422516.i	−0.000943766	−	0.00163465i
$638$		0			0
$639$	−3.83283e7	−	1.06629e6i	−0.146898	−	0.00408668i
$640$		0			0
$641$	−3.91187e7	−	6.77556e7i	−0.148529	−	0.257259i	0.782155	−	0.623084i	$-0.214120\pi$
$641$	−3.91187e7	−	6.77556e7i	−0.148529	−	0.257259i	−0.930684	+	0.365824i	$0.880787\pi$
$642$		0			0
$643$	1.81207e8	+	1.04620e8i	0.681620	+	0.393533i	0.800465	−	0.599379i	$-0.204586\pi$
$643$	1.81207e8	+	1.04620e8i	0.681620	+	0.393533i	−0.118845	+	0.992913i	$0.537919\pi$
$644$		0			0
$645$	2.71158e8	−	2.63719e8i	1.01052	−	0.982794i
$646$		0			0
$647$		−	4.71445e8i		−	1.74068i	−0.492454	−	0.870338i	$-0.663900\pi$
$647$		−	4.71445e8i		−	1.74068i	0.492454	−	0.870338i	$-0.336100\pi$
$648$		0			0
$649$	−4.60576e7			−0.168487
$650$		0			0
$651$	1.63845e8	+	1.68467e8i	0.593869	+	0.610620i
$652$		0			0
$653$	−1.27587e8	+	2.20987e8i	−0.458211	+	0.793645i	−0.998867	−	0.0475991i	$-0.984843\pi$
$653$	−1.27587e8	+	2.20987e8i	−0.458211	+	0.793645i	0.540655	+	0.841244i	$0.318176\pi$
$654$		0			0
$655$	3.12010e8	−	1.80139e8i	1.11031	−	0.641039i
$656$		0			0
$657$	−1.00310e7	+	3.60569e8i	−0.0353710	+	1.27143i
$658$		0			0
$659$	−3.80068e7	+	2.19433e7i	−0.132802	+	0.0766734i	−0.564929	−	0.825139i	$-0.691097\pi$
$659$	−3.80068e7	+	2.19433e7i	−0.132802	+	0.0766734i	0.432127	+	0.901813i	$0.357763\pi$
$660$		0			0
$661$	1.51278e8	−	2.62022e8i	0.523808	−	0.907263i	−0.475808	−	0.879549i	$-0.657844\pi$
$661$	1.51278e8	−	2.62022e8i	0.523808	−	0.907263i	0.999616	−	0.0277132i	$-0.00882251\pi$
$662$		0			0
$663$	−1.58606e8	+	4.01429e7i	−0.544224	+	0.137742i
$664$		0			0
$665$	−2.08607e8			−0.709354
$666$		0			0
$667$		−	7.08882e7i		−	0.238889i
$668$		0			0
$669$	2.97413e8	+	8.41413e7i	0.993303	+	0.281016i
$670$		0			0
$671$	1.91329e8	+	1.10464e8i	0.633305	+	0.365639i
$672$		0			0
$673$	3.62925e6	+	6.28605e6i	0.0119062	+	0.0206221i	0.871917	−	0.489654i	$-0.162877\pi$
$673$	3.62925e6	+	6.28605e6i	0.0119062	+	0.0206221i	−0.860011	+	0.510276i	$0.829543\pi$
$674$		0			0
$675$	7.73513e8	+	2.42261e8i	2.51511	+	0.787720i
$676$		0			0
$677$	6.92116e7	+	1.19878e8i	0.223056	+	0.386344i	0.955734	−	0.294231i	$-0.0950635\pi$
$677$	6.92116e7	+	1.19878e8i	0.223056	+	0.386344i	−0.732679	+	0.680575i	$0.761730\pi$
$678$		0			0
$679$	1.70347e8	+	9.83499e7i	0.544158	+	0.314170i
$680$		0			0
$681$	3.17350e8	+	8.97818e7i	1.00484	+	0.284281i
$682$		0			0
$683$		−	3.48842e8i		−	1.09488i	−0.836844	−	0.547441i	$-0.815602\pi$
$683$		−	3.48842e8i		−	1.09488i	0.836844	−	0.547441i	$-0.184398\pi$
$684$		0			0
$685$	−7.27976e8			−2.26488
$686$		0			0
$687$	1.33903e8	−	3.38906e7i	0.412971	−	0.104522i
$688$		0			0
$689$	5.22815e7	−	9.05543e7i	0.159842	−	0.276854i
$690$		0			0
$691$	3.12929e7	−	1.80670e7i	0.0948443	−	0.0547584i	−0.451828	−	0.892105i	$-0.649228\pi$
$691$	3.12929e7	−	1.80670e7i	0.0948443	−	0.0547584i	0.546672	+	0.837347i	$0.315894\pi$
$692$		0			0
$693$	3.04885e8	+	1.87519e8i	0.916087	+	0.563438i
$694$		0			0
$695$	7.53386e8	−	4.34968e8i	2.24421	−	1.29569i
$696$		0			0
$697$	3.76447e8	−	6.52025e8i	1.11175	−	1.92560i
$698$		0			0
$699$	−7.78726e7	−	8.00691e7i	−0.228010	−	0.234441i
$700$		0			0
$701$	−4.82722e8			−1.40134			−0.700670	−	0.713486i	$-0.747115\pi$
$701$	−4.82722e8			−1.40134			−0.700670	+	0.713486i	$0.747115\pi$
$702$		0			0
$703$		−	1.69302e8i		−	0.487299i
$704$		0			0
$705$	2.32678e8	−	2.26295e8i	0.664029	−	0.645813i
$706$		0			0
$707$	−2.62418e8	−	1.51507e8i	−0.742566	−	0.428721i
$708$		0			0
$709$	7.94025e7	+	1.37529e8i	0.222790	+	0.385883i	0.955654	−	0.294492i	$-0.0951503\pi$
$709$	7.94025e7	+	1.37529e8i	0.222790	+	0.385883i	−0.732864	+	0.680375i	$0.761817\pi$
$710$		0			0
$711$	−2.13119e8	−	3.94050e8i	−0.592945	−	1.09633i
$712$		0			0
$713$	−2.58587e8	−	4.47887e8i	−0.713410	−	1.23566i
$714$		0			0
$715$	−2.78937e8	−	1.61044e8i	−0.763111	−	0.440582i
$716$		0			0
$717$	1.58817e6	+	6.27491e6i	0.00430864	+	0.0170235i
$718$		0			0
$719$			3.75480e8i			1.01018i	0.863066	+	0.505091i	$0.168541\pi$
$719$			3.75480e8i			1.01018i	−0.863066	+	0.505091i	$0.831459\pi$
$720$		0			0
$721$	4.83880e8			1.29102
$722$		0			0
$723$	3.01671e7	−	1.06631e8i	0.0798214	−	0.282143i
$724$		0			0
$725$	7.14607e7	−	1.23774e8i	0.187523	−	0.324799i
$726$		0			0
$727$	−3.63797e8	+	2.10038e8i	−0.946795	+	0.546632i	−0.892084	−	0.451870i	$-0.850757\pi$
$727$	−3.63797e8	+	2.10038e8i	−0.946795	+	0.546632i	−0.0547113	+	0.998502i	$0.517424\pi$
$728$		0			0
$729$	3.22881e7	−	3.86073e8i	0.0833412	−	0.996521i
$730$		0			0
$731$	−3.26020e8	+	1.88228e8i	−0.834627	+	0.481872i
$732$		0			0
$733$	−2.20496e8	+	3.81910e8i	−0.559872	+	0.969726i	0.437635	+	0.899153i	$0.355816\pi$
$733$	−2.20496e8	+	3.81910e8i	−0.559872	+	0.969726i	−0.997507	+	0.0705732i	$0.977517\pi$
$734$		0			0
$735$	903356.	−	3.19308e6i	0.00227508	−	0.00804169i
$736$		0			0
$737$	−2.27794e8			−0.569036
$738$		0			0
$739$		−	2.18237e8i		−	0.540749i	−0.962755	−	0.270374i	$-0.912853\pi$
$739$		−	2.18237e8i		−	0.540749i	0.962755	−	0.270374i	$-0.0871475\pi$
$740$		0			0
$741$	−1.59589e7	−	6.30540e7i	−0.0392236	−	0.154974i
$742$		0			0
$743$	−3.89192e8	−	2.24700e8i	−0.948850	−	0.547819i	−0.0561268	−	0.998424i	$-0.517875\pi$
$743$	−3.89192e8	−	2.24700e8i	−0.948850	−	0.547819i	−0.892724	+	0.450605i	$0.851208\pi$
$744$		0			0
$745$	3.66652e8	+	6.35061e8i	0.886718	+	1.53584i
$746$		0			0
$747$	3.72037e8	+	6.87883e8i	0.892533	+	1.65026i
$748$		0			0
$749$	−3.14457e7	−	5.44656e7i	−0.0748370	−	0.129621i
$750$		0			0
$751$	−2.43189e8	−	1.40405e8i	−0.574148	−	0.331484i	0.184656	−	0.982803i	$-0.440883\pi$
$751$	−2.43189e8	−	1.40405e8i	−0.574148	−	0.331484i	−0.758804	+	0.651319i	$0.774216\pi$
$752$		0			0
$753$	−5.07259e7	+	4.93344e7i	−0.118808	+	0.115549i
$754$		0			0
$755$		−	6.52412e8i		−	1.51594i
$756$		0			0
$757$	3.32455e8			0.766382			0.383191	−	0.923669i	$-0.374825\pi$
$757$	3.32455e8			0.766382			0.383191	+	0.923669i	$0.374825\pi$
$758$		0			0
$759$	−5.49201e8	−	5.64692e8i	−1.25605	−	1.29148i
$760$		0			0
$761$	2.14007e8	−	3.70670e8i	0.485594	−	0.841073i	−0.514269	−	0.857629i	$-0.671937\pi$
$761$	2.14007e8	−	3.70670e8i	0.485594	−	0.841073i	0.999863	+	0.0165555i	$0.00527003\pi$
$762$		0			0
$763$	−7.27443e7	+	4.19989e7i	−0.163767	+	0.0945507i
$764$		0			0
$765$	−9.47869e8	−	5.82986e8i	−2.11721	−	1.30219i
$766$		0			0
$767$	−2.64205e7	+	1.52539e7i	−0.0585538	+	0.0338060i
$768$		0			0
$769$	−7.54555e7	+	1.30693e8i	−0.165925	+	0.287391i	−0.936983	−	0.349374i	$-0.886394\pi$
$769$	−7.54555e7	+	1.30693e8i	−0.165925	+	0.287391i	0.771058	+	0.636764i	$0.219728\pi$
$770$		0			0
$771$	−1.24875e8	+	3.16057e7i	−0.272466	+	0.0689608i
$772$		0			0
$773$	−2.61066e8			−0.565212			−0.282606	−	0.959236i	$-0.591199\pi$
$773$	−2.61066e8			−0.565212			−0.282606	+	0.959236i	$0.591199\pi$
$774$		0			0
$775$		−	1.04270e9i		−	2.24004i
$776$		0			0
$777$	5.93829e8	+	1.68001e8i	1.26590	+	0.358136i
$778$		0			0
$779$	2.59214e8	+	1.49657e8i	0.548335	+	0.316581i
$780$		0			0
$781$	3.75633e7	+	6.50616e7i	0.0788517	+	0.136575i
$782$		0			0
$783$	−6.51889e7	−	2.04169e7i	−0.135797	−	0.0425309i
$784$		0			0
$785$	7.92218e8	+	1.37216e9i	1.63770	+	2.83659i
$786$		0			0
$787$	2.56186e8	+	1.47909e8i	0.525572	+	0.303439i	0.739211	−	0.673474i	$-0.235199\pi$
$787$	2.56186e8	+	1.47909e8i	0.525572	+	0.303439i	−0.213640	+	0.976913i	$0.568532\pi$
$788$		0			0
$789$	8.56596e8	+	2.42340e8i	1.74399	+	0.493394i
$790$		0			0
$791$			5.04221e8i			1.01881i
$792$		0			0
$793$	1.46339e8			0.293454
$794$		0			0
$795$	6.89465e8	−	1.74503e8i	1.37218	−	0.347297i
$796$		0			0
$797$	3.45926e8	−	5.99161e8i	0.683294	−	1.18350i	−0.290675	−	0.956822i	$-0.593880\pi$
$797$	3.45926e8	−	5.99161e8i	0.683294	−	1.18350i	0.973970	−	0.226679i	$-0.0727867\pi$
$798$		0			0
$799$	−2.79755e8	+	1.61516e8i	−0.548450	+	0.316648i
$800$		0			0
$801$	−1.26996e7	+	4.56496e8i	−0.0247112	+	0.888258i
$802$		0			0
$803$	6.12060e8	−	3.53373e8i	1.18208	−	0.682475i
$804$		0			0
$805$	−8.36725e8	+	1.44925e9i	−1.60397	+	2.77815i
$806$		0			0
$807$	−6.91650e8	−	7.11159e8i	−1.31603	−	1.35315i
$808$		0			0
$809$	8.30305e8			1.56817			0.784083	−	0.620656i	$-0.213133\pi$
$809$	8.30305e8			1.56817			0.784083	+	0.620656i	$0.213133\pi$
$810$		0			0
$811$		−	2.46518e8i		−	0.462154i	−0.972935	−	0.231077i	$-0.925775\pi$
$811$		−	2.46518e8i		−	0.462154i	0.972935	−	0.231077i	$-0.0742249\pi$
$812$		0			0
$813$	−3.26518e8	+	3.17560e8i	−0.607624	+	0.590955i
$814$		0			0
$815$	1.37195e9	+	7.92098e8i	2.53435	+	1.46321i
$816$		0			0
$817$	−7.48304e7	−	1.29610e8i	−0.137218	−	0.237669i
$818$		0			0
$819$	2.36999e8	+	6.59329e6i	0.431415	+	0.0120019i
$820$		0			0
$821$	−1.60349e8	−	2.77733e8i	−0.289759	−	0.501878i	0.683993	−	0.729489i	$-0.260242\pi$
$821$	−1.60349e8	−	2.77733e8i	−0.289759	−	0.501878i	−0.973752	+	0.227611i	$0.926909\pi$
$822$		0			0
$823$	4.94858e8	+	2.85706e8i	0.887731	+	0.512532i	0.873200	−	0.487363i	$-0.162041\pi$
$823$	4.94858e8	+	2.85706e8i	0.887731	+	0.512532i	0.0145312	+	0.999894i	$0.495374\pi$
$824$		0			0
$825$	−3.89674e8	−	1.53961e9i	−0.693968	−	2.74188i
$826$		0			0
$827$			3.13199e8i			0.553737i	0.960908	+	0.276868i	$0.0892966\pi$
$827$			3.13199e8i			0.553737i	−0.960908	+	0.276868i	$0.910703\pi$
$828$		0			0
$829$	9.66387e8			1.69624			0.848120	−	0.529804i	$-0.177735\pi$
$829$	9.66387e8			1.69624			0.848120	+	0.529804i	$0.177735\pi$
$830$		0			0
$831$	−3.76658e7	+	1.33137e8i	−0.0656363	+	0.232004i
$832$		0			0
$833$	−1.65132e6	+	2.86018e6i	−0.00285692	+	0.00494832i
$834$		0			0
$835$	−5.16538e8	+	2.98223e8i	−0.887244	+	0.512251i
$836$		0			0
$837$	−4.86355e8	+	1.08799e8i	−0.829424	+	0.185545i
$838$		0			0
$839$	8.93011e7	−	5.15580e7i	0.151207	−	0.0872992i	−0.422488	−	0.906369i	$-0.638843\pi$
$839$	8.93011e7	−	5.15580e7i	0.151207	−	0.0872992i	0.573694	+	0.819069i	$0.305510\pi$
$840$		0			0
$841$	2.91389e8	−	5.04701e8i	0.489875	−	0.848489i
$842$		0			0
$843$	8.07852e7	−	2.85550e8i	0.134849	−	0.476650i
$844$		0			0
$845$	9.37075e8			1.55312
$846$		0			0
$847$		−	9.23387e7i		−	0.151962i
$848$		0			0
$849$	1.69617e7	+	6.70161e7i	0.0277170	+	0.109511i
$850$		0			0
$851$	−1.17619e9	−	6.79072e8i	−1.90848	−	1.10186i
$852$		0			0
$853$	−2.78379e7	−	4.82167e7i	−0.0448528	−	0.0776874i	0.842727	−	0.538341i	$-0.180949\pi$
$853$	−2.78379e7	−	4.82167e7i	−0.0448528	−	0.0776874i	−0.887580	+	0.460653i	$0.847615\pi$
$854$		0			0
$855$	2.31767e8	−	3.76827e8i	0.370812	−	0.602898i
$856$		0			0
$857$	−4.54438e8	−	7.87110e8i	−0.721992	−	1.25053i	−0.960200	−	0.279313i	$-0.909893\pi$
$857$	−4.54438e8	−	7.87110e8i	−0.721992	−	1.25053i	0.238208	−	0.971214i	$-0.423440\pi$
$858$		0			0
$859$	8.03338e8	+	4.63807e8i	1.26741	+	0.731742i	0.974498	−	0.224396i	$-0.0720408\pi$
$859$	8.03338e8	+	4.63807e8i	1.26741	+	0.731742i	0.292917	+	0.956138i	$0.405374\pi$
$860$		0			0
$861$	−7.82148e8	+	7.60691e8i	−1.22540	+	1.19179i
$862$		0			0
$863$		−	8.55042e8i		−	1.33032i	−0.746703	−	0.665158i	$-0.768364\pi$
$863$		−	8.55042e8i		−	1.33032i	0.746703	−	0.665158i	$-0.231636\pi$
$864$		0			0
$865$	−3.39929e8			−0.525218
$866$		0			0
$867$	3.17789e8	+	3.26753e8i	0.487620	+	0.501374i
$868$		0			0
$869$	−4.38880e8	+	7.60162e8i	−0.668784	+	1.15837i
$870$		0			0
$871$	−1.30672e8	+	7.54435e7i	−0.197755	+	0.114174i
$872$		0			0
$873$	−3.66919e8	+	1.98446e8i	−0.551477	+	0.298263i
$874$		0			0
$875$	−1.81327e9	+	1.04689e9i	−2.70668	+	1.56271i
$876$		0			0
$877$	1.61167e7	−	2.79149e7i	0.0238933	−	0.0413844i	−0.853832	−	0.520549i	$-0.825727\pi$
$877$	1.61167e7	−	2.79149e7i	0.0238933	−	0.0413844i	0.877725	+	0.479165i	$0.159060\pi$
$878$		0			0
$879$	−6.98458e8	+	1.76779e8i	−1.02843	+	0.260294i
$880$		0			0
$881$	−5.65989e8			−0.827714			−0.413857	−	0.910342i	$-0.635819\pi$
$881$	−5.65989e8			−0.827714			−0.413857	+	0.910342i	$0.635819\pi$
$882$		0			0
$883$			4.04660e8i			0.587772i	0.955841	+	0.293886i	$0.0949485\pi$
$883$			4.04660e8i			0.587772i	−0.955841	+	0.293886i	$0.905051\pi$
$884$		0			0
$885$	−1.99667e8	−	5.64881e7i	−0.288056	−	0.0814942i
$886$		0			0
$887$	4.11050e8	+	2.37320e8i	0.589012	+	0.340066i	0.764707	−	0.644379i	$-0.222884\pi$
$887$	4.11050e8	+	2.37320e8i	0.589012	+	0.340066i	−0.175695	+	0.984445i	$0.556217\pi$
$888$		0			0
$889$	−4.99609e8	−	8.65349e8i	−0.711091	−	1.23165i
$890$		0			0
$891$	−6.77470e8	+	3.42406e8i	−0.957761	+	0.484070i
$892$		0			0
$893$	−6.42112e7	−	1.11217e8i	−0.0901688	−	0.156177i
$894$		0			0
$895$	−2.50421e8	−	1.44580e8i	−0.349302	−	0.201670i
$896$		0			0
$897$	−5.02065e8	−	1.42040e8i	−0.695637	−	0.196803i
$898$		0			0
$899$			8.78754e7i			0.120945i
$900$		0			0
$901$	−7.07829e8			−0.967729
$902$		0			0
$903$	5.28865e8	−	1.33855e8i	0.718260	−	0.181791i
$904$		0			0
$905$	3.52997e7	−	6.11409e7i	0.0476240	−	0.0824872i
$906$		0			0
$907$	−6.47026e8	+	3.73561e8i	−0.867162	+	0.500656i	−0.866404	−	0.499344i	$-0.833574\pi$
$907$	−6.47026e8	+	3.73561e8i	−0.867162	+	0.500656i	−0.000757622		1.00000i	$0.500241\pi$
$908$		0			0
$909$	5.65235e8	−	3.05704e8i	0.752554	−	0.407014i
$910$		0			0
$911$	2.93280e8	−	1.69325e8i	0.387907	−	0.223958i	−0.293346	−	0.956006i	$-0.594769\pi$
$911$	2.93280e8	−	1.69325e8i	0.387907	−	0.223958i	0.681253	+	0.732048i	$0.261435\pi$
$912$		0			0
$913$	7.66141e8	−	1.32700e9i	1.00669	−	1.74364i
$914$		0			0
$915$	6.93963e8	+	7.13538e8i	0.905886	+	0.931438i
$916$		0			0
$917$	5.19620e8			0.673872
$918$		0			0
$919$			5.27845e8i			0.680080i	0.940411	+	0.340040i	$0.110441\pi$
$919$			5.27845e8i			0.680080i	−0.940411	+	0.340040i	$0.889559\pi$
$920$		0			0
$921$	−3.49417e8	+	3.39831e8i	−0.447265	+	0.434995i
$922$		0			0
$923$	4.30957e7	+	2.48813e7i	0.0548061	+	0.0316423i
$924$		0			0
$925$	−1.36911e9	−	2.37137e9i	−1.72987	−	2.99623i
$926$		0			0
$927$	−5.37603e8	+	8.74081e8i	−0.674873	+	1.09727i
$928$		0			0
$929$	−2.14026e8	−	3.70703e8i	−0.266943	−	0.462359i	0.701128	−	0.713036i	$-0.252680\pi$
$929$	−2.14026e8	−	3.70703e8i	−0.266943	−	0.462359i	−0.968071	+	0.250677i	$0.919347\pi$
$930$		0			0
$931$	−1.13707e6	−	656487.i	−0.00140909	−	0.000813537i
$932$		0			0
$933$	2.77194e8	+	1.09520e9i	0.341302	+	1.34849i
$934$		0			0
$935$			2.18034e9i			2.66741i
$936$		0			0
$937$	−1.26186e9			−1.53389			−0.766945	−	0.641713i	$-0.778224\pi$
$937$	−1.26186e9			−1.53389			−0.766945	+	0.641713i	$0.778224\pi$
$938$		0			0
$939$	2.85173e8	−	1.00800e9i	0.344439	−	1.21748i
$940$		0			0
$941$	−4.21493e8	+	7.30047e8i	−0.505850	+	0.876158i	0.494127	+	0.869390i	$0.335488\pi$
$941$	−4.21493e8	+	7.30047e8i	−0.505850	+	0.876158i	−0.999977	+	0.00676797i	$0.997846\pi$
$942$		0			0
$943$	2.07942e9	−	1.20056e9i	2.47975	−	1.43168i
$944$		0			0
$945$	1.09174e9	+	1.18686e9i	1.29367	+	1.40638i
$946$		0			0
$947$	9.87771e7	−	5.70290e7i	0.116307	−	0.0671500i	−0.440718	−	0.897646i	$-0.645276\pi$
$947$	9.87771e7	−	5.70290e7i	0.116307	−	0.0671500i	0.557025	+	0.830496i	$0.311943\pi$
$948$		0			0
$949$	2.34068e8	−	4.05419e8i	0.273870	−	0.474357i
$950$		0			0
$951$	−2.49589e8	+	8.82220e8i	−0.290191	+	1.02574i
$952$		0			0
$953$	−2.92357e8			−0.337781			−0.168890	−	0.985635i	$-0.554018\pi$
$953$	−2.92357e8			−0.337781			−0.168890	+	0.985635i	$0.554018\pi$
$954$		0			0
$955$			1.50913e8i			0.173267i
$956$		0			0
$957$	3.28403e7	+	1.29753e8i	0.0374689	+	0.148041i
$958$		0			0
$959$	−9.09274e8	−	5.24970e8i	−1.03095	−	0.595221i
$960$		0			0
$961$	−1.23198e8	−	2.13386e8i	−0.138814	−	0.240434i
$962$		0			0
$963$	1.33324e8	+	3.70905e6i	0.149289	+	0.00415320i
$964$		0			0
$965$	1.34902e9	+	2.33657e9i	1.50119	+	2.60014i
$966$		0			0
$967$	6.26838e8	+	3.61905e8i	0.693227	+	0.400235i	0.804820	−	0.593519i	$-0.202262\pi$
$967$	6.26838e8	+	3.61905e8i	0.693227	+	0.400235i	−0.111593	+	0.993754i	$0.535595\pi$
$968$		0			0
$969$	−3.15636e8	+	3.06977e8i	−0.346908	+	0.337392i
$970$		0			0
$971$		−	6.27074e8i		−	0.684954i	−0.939526	−	0.342477i	$-0.888734\pi$
$971$		−	6.27074e8i		−	0.684954i	0.939526	−	0.342477i	$-0.111266\pi$
$972$		0			0
$973$	1.25468e9			1.36206
$974$		0			0
$975$	−7.33439e8	−	7.54127e8i	−0.791317	−	0.813637i
$976$		0			0
$977$	−6.56152e7	+	1.13649e8i	−0.0703592	+	0.121866i	−0.899059	−	0.437828i	$-0.855748\pi$
$977$	−6.56152e7	+	1.13649e8i	−0.0703592	+	0.121866i	0.828700	+	0.559694i	$0.189081\pi$
$978$		0			0
$979$	7.74894e8	−	4.47385e8i	0.825836	−	0.476797i
$980$		0			0
$981$	4.95380e6	−	1.78067e8i	0.00524725	−	0.188615i
$982$		0			0
$983$	−1.40095e9	+	8.08836e8i	−1.47489	+	0.851530i	−0.999600	−	0.0282964i	$-0.990992\pi$
$983$	−1.40095e9	+	8.08836e8i	−1.47489	+	0.851530i	−0.475294	+	0.879827i	$0.657658\pi$
$984$		0			0
$985$	2.35580e8	−	4.08036e8i	0.246507	−	0.426962i
$986$		0			0
$987$	4.53814e8	−	1.14860e8i	0.471983	−	0.119458i
$988$		0			0
$989$	−1.20058e9			−1.24109
$990$		0			0
$991$			1.35082e9i			1.38796i	0.719993	+	0.693981i	$0.244145\pi$
$991$			1.35082e9i			1.38796i	−0.719993	+	0.693981i	$0.755855\pi$
$992$		0			0
$993$	−9.16081e7	−	2.59169e7i	−0.0935592	−	0.0264689i
$994$		0			0
$995$	7.86179e8	+	4.53901e8i	0.798090	+	0.460778i
$996$		0			0
$997$	4.05565e8	+	7.02459e8i	0.409237	+	0.708819i	0.994804	−	0.101805i	$-0.0324617\pi$
$997$	4.05565e8	+	7.02459e8i	0.409237	+	0.708819i	−0.585568	+	0.810624i	$0.699128\pi$
$998$		0			0
$999$	−9.63236e8	+	8.86041e8i	−0.966131	+	0.888704i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.7.o.a.31.11	✓	24
3.2	odd	2		432.7.o.c.415.1		24
4.3	odd	2		144.7.o.c.31.2	yes	24
9.2	odd	6		432.7.o.b.127.1		24
9.7	even	3		144.7.o.c.79.2	yes	24
12.11	even	2		432.7.o.b.415.1		24
36.7	odd	6	inner	144.7.o.a.79.11	yes	24
36.11	even	6		432.7.o.c.127.1		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.7.o.a.31.11	✓	24	1.1	even	1	trivial
144.7.o.a.79.11	yes	24	36.7	odd	6	inner
144.7.o.c.31.2	yes	24	4.3	odd	2
144.7.o.c.79.2	yes	24	9.7	even	3
432.7.o.b.127.1		24	9.2	odd	6
432.7.o.b.415.1		24	12.11	even	2
432.7.o.c.127.1		24	36.11	even	6
432.7.o.c.415.1		24	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	144.o (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(18.8245 + 19.3555i) q^{3} +(119.170 - 206.408i) q^{5} +(297.697 - 171.875i) q^{7} +(-20.2728 + 728.718i) q^{9} +O(q^{10})\)	\(q+(18.8245 + 19.3555i) q^{3} +(119.170 - 206.408i) q^{5} +(297.697 - 171.875i) q^{7} +(-20.2728 + 728.718i) q^{9} +(1236.99 - 714.175i) q^{11} +(473.058 - 819.360i) q^{13} +(6238.46 - 1578.95i) q^{15} -6404.62 q^{17} -2546.17i q^{19} +(8930.75 + 2526.60i) q^{21} +(-17689.0 - 10212.7i) q^{23} +(-20590.4 - 35663.7i) q^{25} +(-14486.3 + 13325.4i) q^{27} +(1735.29 + 3005.61i) q^{29} +(21927.8 + 12660.0i) q^{31} +(37108.9 + 10498.5i) q^{33} -81929.5i q^{35} +66492.6 q^{37} +(24764.2 - 6267.80i) q^{39} +(-58777.3 + 101805. i) q^{41} +(50903.9 - 29389.4i) q^{43} +(147998. + 91025.8i) q^{45} +(43680.1 - 25218.7i) q^{47} +(257.833 - 446.580i) q^{49} +(-120564. - 123965. i) q^{51} +110518. q^{53} -340433. i q^{55} +(49282.5 - 47930.5i) q^{57} +(-27925.3 - 16122.7i) q^{59} +(77336.7 + 133951. i) q^{61} +(119214. + 220422. i) q^{63} +(-112748. - 195286. i) q^{65} +(-138114. - 79740.3i) q^{67} +(-135314. - 534630. i) q^{69} +52596.8i q^{71} +494799. q^{73} +(302684. - 1.06989e6i) q^{75} +(245498. - 425215. i) q^{77} +(-532196. + 307264. i) q^{79} +(-530619. - 29546.3i) q^{81} +(929041. - 536382. i) q^{83} +(-763238. + 1.32197e6i) q^{85} +(-25509.1 + 90166.7i) q^{87} +626436. q^{89} -325228. i q^{91} +(167740. + 662745. i) q^{93} +(-525551. - 303427. i) q^{95} +(286108. + 495554. i) q^{97} +(495355. + 915893. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 48 q^{3} - 72 q^{5} + 360 q^{7} - 1452 q^{9}+O(q^{10}) \) 24 * q - 48 * q^3 - 72 * q^5 + 360 * q^7 - 1452 * q^9	\( 24 q - 48 q^{3} - 72 q^{5} + 360 q^{7} - 1452 q^{9} + 864 q^{11} - 840 q^{13} + 11544 q^{15} - 12888 q^{17} + 21792 q^{21} - 60264 q^{23} - 42828 q^{25} - 39312 q^{27} - 5760 q^{29} + 18360 q^{31} + 54252 q^{33} + 49728 q^{37} - 31704 q^{39} - 52164 q^{41} + 283968 q^{45} - 104760 q^{47} + 236004 q^{49} - 305664 q^{51} + 134352 q^{53} - 325524 q^{57} - 280368 q^{59} + 76440 q^{61} - 266760 q^{63} - 22752 q^{65} - 1158048 q^{67} - 446904 q^{69} + 43800 q^{73} - 979632 q^{75} + 652104 q^{77} + 225576 q^{79} - 1391652 q^{81} + 306288 q^{83} - 414000 q^{85} + 1793928 q^{87} + 2486304 q^{89} - 1107576 q^{93} - 1538784 q^{95} - 365916 q^{97} + 1331280 q^{99}+O(q^{100}) \) 24 * q - 48 * q^3 - 72 * q^5 + 360 * q^7 - 1452 * q^9 + 864 * q^11 - 840 * q^13 + 11544 * q^15 - 12888 * q^17 + 21792 * q^21 - 60264 * q^23 - 42828 * q^25 - 39312 * q^27 - 5760 * q^29 + 18360 * q^31 + 54252 * q^33 + 49728 * q^37 - 31704 * q^39 - 52164 * q^41 + 283968 * q^45 - 104760 * q^47 + 236004 * q^49 - 305664 * q^51 + 134352 * q^53 - 325524 * q^57 - 280368 * q^59 + 76440 * q^61 - 266760 * q^63 - 22752 * q^65 - 1158048 * q^67 - 446904 * q^69 + 43800 * q^73 - 979632 * q^75 + 652104 * q^77 + 225576 * q^79 - 1391652 * q^81 + 306288 * q^83 - 414000 * q^85 + 1793928 * q^87 + 2486304 * q^89 - 1107576 * q^93 - 1538784 * q^95 - 365916 * q^97 + 1331280 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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