LMFDB - Embedded newform 4056.2.c.p.337.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(337,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4056.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.c (of order $2$, degree $1$, not 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:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.649638144.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} + 75x^{4} - 170x^{2} + 169 $ x^8 - 14x^6 + 75x^4 - 170*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.5
Root			$2.34138 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	4056.337
Dual form			4056.2.c.p.337.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.475353i q^{5} +4.55539i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.475353i q^{5} +4.55539i q^{7} +1.00000 q^{9} +1.65202i q^{11} +0.475353i q^{15} -7.93945 q^{17} -2.60272i q^{19} +4.55539i q^{21} -3.81208 q^{23} +4.77404 q^{25} +1.00000 q^{27} +0.823335 q^{29} -1.79260i q^{31} +1.65202i q^{33} -2.16541 q^{35} +1.87263i q^{37} +10.7022i q^{41} +9.89016 q^{43} +0.475353i q^{45} +7.29869i q^{47} -13.7515 q^{49} -7.93945 q^{51} -10.0063 q^{53} -0.785291 q^{55} -2.60272i q^{57} +1.48660i q^{59} -11.7403 q^{61} +4.55539i q^{63} -3.32352i q^{67} -3.81208 q^{69} +1.90676i q^{71} -6.14686i q^{73} +4.77404 q^{75} -7.52558 q^{77} -10.7515 q^{79} +1.00000 q^{81} -14.7628i q^{83} -3.77404i q^{85} +0.823335 q^{87} +7.30404i q^{89} -1.79260i q^{93} +1.23721 q^{95} -4.18490i q^{97} +1.65202i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + 8 * q^9	$ 8 q + 8 q^{3} + 8 q^{9} - 24 q^{17} + 4 q^{23} - 4 q^{25} + 8 q^{27} - 12 q^{29} - 20 q^{35} + 16 q^{43} - 36 q^{49} - 24 q^{51} + 4 q^{53} + 20 q^{55} - 4 q^{61} + 4 q^{69} - 4 q^{75} + 56 q^{77} - 12 q^{79} + 8 q^{81} - 12 q^{87} + 68 q^{95}+O(q^{100}) $ 8 * q + 8 * q^3 + 8 * q^9 - 24 * q^17 + 4 * q^23 - 4 * q^25 + 8 * q^27 - 12 * q^29 - 20 * q^35 + 16 * q^43 - 36 * q^49 - 24 * q^51 + 4 * q^53 + 20 * q^55 - 4 * q^61 + 4 * q^69 - 4 * q^75 + 56 * q^77 - 12 * q^79 + 8 * q^81 - 12 * q^87 + 68 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$.

$n$	$1015$	$2029$	$2705$	$3889$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0.475353i	0.212584i	0.994335	+	0.106292i	$0.0338979\pi$
$5$	0.475353i	0.212584i	−0.994335	+	0.106292i	$0.966102\pi$
$6$	0	0
$7$	4.55539i	1.72177i	0.508796	+	0.860887i	$0.330091\pi$
$7$	4.55539i	1.72177i	−0.508796	+	0.860887i	$0.669909\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.65202i	0.498102i	0.968490	+	0.249051i	$0.0801187\pi$
$11$	1.65202i	0.498102i	−0.968490	+	0.249051i	$0.919881\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.475353i	0.122736i
$16$	0	0
$17$	−7.93945	−1.92560	−0.962800	−	0.270214i	$-0.912905\pi$
$17$	−7.93945	−1.92560	−0.962800	+	0.270214i	$0.912905\pi$
$18$	0	0
$19$	− 2.60272i	− 0.597106i	−0.954393	−	0.298553i	$-0.903496\pi$
$19$	− 2.60272i	− 0.597106i	0.954393	−	0.298553i	$-0.0965039\pi$
$20$	0	0
$21$	4.55539i	0.994067i
$22$	0	0
$23$	−3.81208	−0.794874	−0.397437	−	0.917629i	$-0.630100\pi$
$23$	−3.81208	−0.794874	−0.397437	+	0.917629i	$0.630100\pi$
$24$	0	0
$25$	4.77404	0.954808
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.823335	0.152889	0.0764447	−	0.997074i	$-0.475643\pi$
$29$	0.823335	0.152889	0.0764447	+	0.997074i	$0.475643\pi$
$30$	0	0
$31$	− 1.79260i	− 0.321960i	−0.986958	−	0.160980i	$-0.948535\pi$
$31$	− 1.79260i	− 0.321960i	0.986958	−	0.160980i	$-0.0514654\pi$
$32$	0	0
$33$	1.65202i	0.287579i
$34$	0	0
$35$	−2.16541	−0.366022
$36$	0	0
$37$	1.87263i	0.307858i	0.988082	+	0.153929i	$0.0491928\pi$
$37$	1.87263i	0.307858i	−0.988082	+	0.153929i	$0.950807\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.7022i	1.67141i	0.549179	+	0.835705i	$0.314941\pi$
$41$	10.7022i	1.67141i	−0.549179	+	0.835705i	$0.685059\pi$
$42$	0	0
$43$	9.89016	1.50824	0.754118	−	0.656739i	$-0.228065\pi$
$43$	9.89016	1.50824	0.754118	+	0.656739i	$0.228065\pi$
$44$	0	0
$45$	0.475353i	0.0708614i
$46$	0	0
$47$	7.29869i	1.06462i	0.846549	+	0.532311i	$0.178676\pi$
$47$	7.29869i	1.06462i	−0.846549	+	0.532311i	$0.821324\pi$
$48$	0	0
$49$	−13.7515	−1.96451
$50$	0	0
$51$	−7.93945	−1.11175
$52$	0	0
$53$	−10.0063	−1.37447	−0.687234	−	0.726436i	$-0.741175\pi$
$53$	−10.0063	−1.37447	−0.687234	+	0.726436i	$0.741175\pi$
$54$	0	0
$55$	−0.785291	−0.105889
$56$	0	0
$57$	− 2.60272i	− 0.344739i
$58$	0	0
$59$	1.48660i	0.193539i	0.995307	+	0.0967696i	$0.0308510\pi$
$59$	1.48660i	0.193539i	−0.995307	+	0.0967696i	$0.969149\pi$
$60$	0	0
$61$	−11.7403	−1.50319	−0.751595	−	0.659625i	$-0.770715\pi$
$61$	−11.7403	−1.50319	−0.751595	+	0.659625i	$0.770715\pi$
$62$	0	0
$63$	4.55539i	0.573925i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 3.32352i	− 0.406033i	−0.979175	−	0.203016i	$-0.934926\pi$
$67$	− 3.32352i	− 0.406033i	0.979175	−	0.203016i	$-0.0650745\pi$
$68$	0	0
$69$	−3.81208	−0.458921
$70$	0	0
$71$	1.90676i	0.226291i	0.993578	+	0.113145i	$0.0360926\pi$
$71$	1.90676i	0.226291i	−0.993578	+	0.113145i	$0.963907\pi$
$72$	0	0
$73$	− 6.14686i	− 0.719435i	−0.933061	−	0.359718i	$-0.882873\pi$
$73$	− 6.14686i	− 0.719435i	0.933061	−	0.359718i	$-0.117127\pi$
$74$	0	0
$75$	4.77404	0.551259
$76$	0	0
$77$	−7.52558	−0.857619
$78$	0	0
$79$	−10.7515	−1.20964	−0.604821	−	0.796361i	$-0.706755\pi$
$79$	−10.7515	−1.20964	−0.604821	+	0.796361i	$0.706755\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 14.7628i	− 1.62043i	−0.586135	−	0.810213i	$-0.699351\pi$
$83$	− 14.7628i	− 1.62043i	0.586135	−	0.810213i	$-0.300649\pi$
$84$	0	0
$85$	− 3.77404i	− 0.409352i
$86$	0	0
$87$	0.823335	0.0882708
$88$	0	0
$89$	7.30404i	0.774226i	0.922032	+	0.387113i	$0.126528\pi$
$89$	7.30404i	0.774226i	−0.922032	+	0.387113i	$0.873472\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 1.79260i	− 0.185884i
$94$	0	0
$95$	1.23721	0.126935
$96$	0	0
$97$	− 4.18490i	− 0.424912i	−0.977171	−	0.212456i	$-0.931854\pi$
$97$	− 4.18490i	− 0.424912i	0.977171	−	0.212456i	$-0.0681463\pi$
$98$	0	0
$99$	1.65202i	0.166034i
$100$	0	0
$101$	5.51930	0.549191	0.274595	−	0.961560i	$-0.411456\pi$
$101$	5.51930	0.549191	0.274595	+	0.961560i	$0.411456\pi$
$102$	0	0
$103$	−11.8902	−1.17157	−0.585786	−	0.810466i	$-0.699214\pi$
$103$	−11.8902	−1.17157	−0.585786	+	0.810466i	$0.699214\pi$
$104$	0	0
$105$	−2.16541	−0.211323
$106$	0	0
$107$	5.96823	0.576971	0.288486	−	0.957484i	$-0.406848\pi$
$107$	5.96823	0.576971	0.288486	+	0.957484i	$0.406848\pi$
$108$	0	0
$109$	− 9.79260i	− 0.937961i	−0.883209	−	0.468980i	$-0.844622\pi$
$109$	− 9.79260i	− 0.937961i	0.883209	−	0.468980i	$-0.155378\pi$
$110$	0	0
$111$	1.87263i	0.177742i
$112$	0	0
$113$	−1.93945	−0.182449	−0.0912243	−	0.995830i	$-0.529078\pi$
$113$	−1.93945	−0.182449	−0.0912243	+	0.995830i	$0.529078\pi$
$114$	0	0
$115$	− 1.81208i	− 0.168978i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 36.1673i	− 3.31545i
$120$	0	0
$121$	8.27084	0.751894
$122$	0	0
$123$	10.7022i	0.964989i
$124$	0	0
$125$	4.64611i	0.415561i
$126$	0	0
$127$	8.87263	0.787319	0.393659	−	0.919256i	$-0.371209\pi$
$127$	8.87263	0.787319	0.393659	+	0.919256i	$0.371209\pi$
$128$	0	0
$129$	9.89016	0.870780
$130$	0	0
$131$	−3.97750	−0.347516	−0.173758	−	0.984788i	$-0.555591\pi$
$131$	−3.97750	−0.347516	−0.173758	+	0.984788i	$0.555591\pi$
$132$	0	0
$133$	11.8564	1.02808
$134$	0	0
$135$	0.475353i	0.0409118i
$136$	0	0
$137$	17.4968i	1.49485i	0.664345	+	0.747426i	$0.268711\pi$
$137$	17.4968i	1.49485i	−0.664345	+	0.747426i	$0.731289\pi$
$138$	0	0
$139$	−20.2771	−1.71988	−0.859941	−	0.510393i	$-0.829500\pi$
$139$	−20.2771	−1.71988	−0.859941	+	0.510393i	$0.829500\pi$
$140$	0	0
$141$	7.29869i	0.614660i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0.391374i	0.0325019i
$146$	0	0
$147$	−13.7515	−1.13421
$148$	0	0
$149$	7.68471i	0.629556i	0.949165	+	0.314778i	$0.101930\pi$
$149$	7.68471i	0.629556i	−0.949165	+	0.314778i	$0.898070\pi$
$150$	0	0
$151$	− 1.65202i	− 0.134439i	−0.997738	−	0.0672196i	$-0.978587\pi$
$151$	− 1.65202i	− 0.134439i	0.997738	−	0.0672196i	$-0.0214128\pi$
$152$	0	0
$153$	−7.93945	−0.641867
$154$	0	0
$155$	0.852115	0.0684436
$156$	0	0
$157$	−13.4855	−1.07626	−0.538132	−	0.842861i	$-0.680870\pi$
$157$	−13.4855	−1.07626	−0.538132	+	0.842861i	$0.680870\pi$
$158$	0	0
$159$	−10.0063	−0.793550
$160$	0	0
$161$	− 17.3655i	− 1.36859i
$162$	0	0
$163$	− 18.9423i	− 1.48368i	−0.670578	−	0.741839i	$-0.733954\pi$
$163$	− 18.9423i	− 1.48368i	0.670578	−	0.741839i	$-0.266046\pi$
$164$	0	0
$165$	−0.785291	−0.0611348
$166$	0	0
$167$	2.79455i	0.216249i	0.994137	+	0.108125i	$0.0344845\pi$
$167$	2.79455i	0.216249i	−0.994137	+	0.108125i	$0.965515\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 2.60272i	− 0.199035i
$172$	0	0
$173$	−6.95071	−0.528452	−0.264226	−	0.964461i	$-0.585117\pi$
$173$	−6.95071	−0.528452	−0.264226	+	0.964461i	$0.585117\pi$
$174$	0	0
$175$	21.7476i	1.64396i
$176$	0	0
$177$	1.48660i	0.111740i
$178$	0	0
$179$	−9.94573	−0.743379	−0.371689	−	0.928357i	$-0.621221\pi$
$179$	−9.94573	−0.743379	−0.371689	+	0.928357i	$0.621221\pi$
$180$	0	0
$181$	12.9887	0.965446	0.482723	−	0.875773i	$-0.339648\pi$
$181$	12.9887	0.965446	0.482723	+	0.875773i	$0.339648\pi$
$182$	0	0
$183$	−11.7403	−0.867867
$184$	0	0
$185$	−0.890159	−0.0654458
$186$	0	0
$187$	− 13.1161i	− 0.959146i
$188$	0	0
$189$	4.55539i	0.331356i
$190$	0	0
$191$	−11.6467	−0.842723	−0.421362	−	0.906893i	$-0.638448\pi$
$191$	−11.6467	−0.842723	−0.421362	+	0.906893i	$0.638448\pi$
$192$	0	0
$193$	− 0.621280i	− 0.0447207i	−0.999750	−	0.0223603i	$-0.992882\pi$
$193$	− 0.621280i	− 0.0447207i	0.999750	−	0.0223603i	$-0.00711811\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.85641i	0.132263i	0.997811	+	0.0661317i	$0.0210658\pi$
$197$	1.85641i	0.132263i	−0.997811	+	0.0661317i	$0.978934\pi$
$198$	0	0
$199$	−13.7916	−0.977658	−0.488829	−	0.872379i	$-0.662576\pi$
$199$	−13.7916	−0.977658	−0.488829	+	0.872379i	$0.662576\pi$
$200$	0	0
$201$	− 3.32352i	− 0.234423i
$202$	0	0
$203$	3.75061i	0.263241i
$204$	0	0
$205$	−5.08734	−0.355315
$206$	0	0
$207$	−3.81208	−0.264958
$208$	0	0
$209$	4.29974	0.297420
$210$	0	0
$211$	22.8627	1.57393	0.786966	−	0.616996i	$-0.211651\pi$
$211$	22.8627	1.57393	0.786966	+	0.616996i	$0.211651\pi$
$212$	0	0
$213$	1.90676i	0.130649i
$214$	0	0
$215$	4.70131i	0.320627i
$216$	0	0
$217$	8.16597	0.554342
$218$	0	0
$219$	− 6.14686i	− 0.415366i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	7.38801i	0.494738i	0.968921	+	0.247369i	$0.0795660\pi$
$223$	7.38801i	0.494738i	−0.968921	+	0.247369i	$0.920434\pi$
$224$	0	0
$225$	4.77404	0.318269
$226$	0	0
$227$	9.55734i	0.634343i	0.948368	+	0.317172i	$0.102733\pi$
$227$	9.55734i	0.634343i	−0.948368	+	0.317172i	$0.897267\pi$
$228$	0	0
$229$	17.6017i	1.16315i	0.813492	+	0.581575i	$0.197563\pi$
$229$	17.6017i	1.16315i	−0.813492	+	0.581575i	$0.802437\pi$
$230$	0	0
$231$	−7.52558	−0.495147
$232$	0	0
$233$	−19.3391	−1.26695	−0.633473	−	0.773765i	$-0.718371\pi$
$233$	−19.3391	−1.26695	−0.633473	+	0.773765i	$0.718371\pi$
$234$	0	0
$235$	−3.46945	−0.226322
$236$	0	0
$237$	−10.7515	−0.698387
$238$	0	0
$239$	18.6417i	1.20583i	0.797805	+	0.602916i	$0.205994\pi$
$239$	18.6417i	1.20583i	−0.797805	+	0.602916i	$0.794006\pi$
$240$	0	0
$241$	− 0.0605458i	− 0.00390010i	−0.999998	−	0.00195005i	$-0.999379\pi$
$241$	− 0.0605458i	− 0.00390010i	0.999998	−	0.00195005i	$-0.000620720\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	− 6.53683i	− 0.417623i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 14.7628i	− 0.935554i
$250$	0	0
$251$	18.5974	1.17386	0.586928	−	0.809639i	$-0.300337\pi$
$251$	18.5974	1.17386	0.586928	+	0.809639i	$0.300337\pi$
$252$	0	0
$253$	− 6.29763i	− 0.395929i
$254$	0	0
$255$	− 3.77404i	− 0.236340i
$256$	0	0
$257$	−29.9063	−1.86550	−0.932750	−	0.360523i	$-0.882598\pi$
$257$	−29.9063	−1.86550	−0.932750	+	0.360523i	$0.882598\pi$
$258$	0	0
$259$	−8.53055	−0.530063
$260$	0	0
$261$	0.823335	0.0509631
$262$	0	0
$263$	−10.0668	−0.620747	−0.310373	−	0.950615i	$-0.600454\pi$
$263$	−10.0668	−0.620747	−0.310373	+	0.950615i	$0.600454\pi$
$264$	0	0
$265$	− 4.75651i	− 0.292190i
$266$	0	0
$267$	7.30404i	0.447000i
$268$	0	0
$269$	28.1336	1.71534	0.857669	−	0.514201i	$-0.171912\pi$
$269$	28.1336	1.71534	0.857669	+	0.514201i	$0.171912\pi$
$270$	0	0
$271$	26.0492i	1.58238i	0.611573	+	0.791188i	$0.290537\pi$
$271$	26.0492i	1.58238i	−0.611573	+	0.791188i	$0.709463\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.88680i	0.475592i
$276$	0	0
$277$	7.85156	0.471755	0.235877	−	0.971783i	$-0.424204\pi$
$277$	7.85156	0.471755	0.235877	+	0.971783i	$0.424204\pi$
$278$	0	0
$279$	− 1.79260i	− 0.107320i
$280$	0	0
$281$	1.26064i	0.0752037i	0.999293	+	0.0376018i	$0.0119719\pi$
$281$	1.26064i	0.0752037i	−0.999293	+	0.0376018i	$0.988028\pi$
$282$	0	0
$283$	2.61932	0.155703	0.0778513	−	0.996965i	$-0.475194\pi$
$283$	2.61932	0.155703	0.0778513	+	0.996965i	$0.475194\pi$
$284$	0	0
$285$	1.23721	0.0732861
$286$	0	0
$287$	−48.7528	−2.87779
$288$	0	0
$289$	46.0349	2.70794
$290$	0	0
$291$	− 4.18490i	− 0.245323i
$292$	0	0
$293$	25.4386i	1.48614i	0.669214	+	0.743070i	$0.266631\pi$
$293$	25.4386i	1.48614i	−0.669214	+	0.743070i	$0.733369\pi$
$294$	0	0
$295$	−0.706661	−0.0411434
$296$	0	0
$297$	1.65202i	0.0958598i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	45.0535i	2.59684i
$302$	0	0
$303$	5.51930	0.317075
$304$	0	0
$305$	− 5.58078i	− 0.319554i
$306$	0	0
$307$	− 15.5100i	− 0.885203i	−0.896718	−	0.442601i	$-0.854056\pi$
$307$	− 15.5100i	− 0.885203i	0.896718	−	0.442601i	$-0.145944\pi$
$308$	0	0
$309$	−11.8902	−0.676408
$310$	0	0
$311$	19.0829	1.08209	0.541047	−	0.840993i	$-0.318028\pi$
$311$	19.0829	1.08209	0.541047	+	0.840993i	$0.318028\pi$
$312$	0	0
$313$	12.3406	0.697535	0.348767	−	0.937209i	$-0.386600\pi$
$313$	12.3406	0.697535	0.348767	+	0.937209i	$0.386600\pi$
$314$	0	0
$315$	−2.16541	−0.122007
$316$	0	0
$317$	19.5636i	1.09880i	0.835559	+	0.549401i	$0.185144\pi$
$317$	19.5636i	1.09880i	−0.835559	+	0.549401i	$0.814856\pi$
$318$	0	0
$319$	1.36016i	0.0761545i
$320$	0	0
$321$	5.96823	0.333114
$322$	0	0
$323$	20.6642i	1.14979i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 9.79260i	− 0.541532i
$328$	0	0
$329$	−33.2483	−1.83304
$330$	0	0
$331$	25.7291i	1.41420i	0.707115	+	0.707099i	$0.249996\pi$
$331$	25.7291i	1.41420i	−0.707115	+	0.707099i	$0.750004\pi$
$332$	0	0
$333$	1.87263i	0.102619i
$334$	0	0
$335$	1.57985	0.0863162
$336$	0	0
$337$	4.90141	0.266997	0.133498	−	0.991049i	$-0.457379\pi$
$337$	4.90141	0.266997	0.133498	+	0.991049i	$0.457379\pi$
$338$	0	0
$339$	−1.93945	−0.105337
$340$	0	0
$341$	2.96140	0.160369
$342$	0	0
$343$	− 30.7559i	− 1.66066i
$344$	0	0
$345$	− 1.81208i	− 0.0975593i
$346$	0	0
$347$	18.9618	1.01792	0.508962	−	0.860789i	$-0.330029\pi$
$347$	18.9618	1.01792	0.508962	+	0.860789i	$0.330029\pi$
$348$	0	0
$349$	− 12.0141i	− 0.643102i	−0.946892	−	0.321551i	$-0.895796\pi$
$349$	− 12.0141i	− 0.643102i	0.946892	−	0.321551i	$-0.104204\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 14.2113i	− 0.756394i	−0.925725	−	0.378197i	$-0.876544\pi$
$353$	− 14.2113i	− 0.756394i	0.925725	−	0.378197i	$-0.123456\pi$
$354$	0	0
$355$	−0.906383	−0.0481058
$356$	0	0
$357$	− 36.1673i	− 1.91418i
$358$	0	0
$359$	− 14.2766i	− 0.753488i	−0.926317	−	0.376744i	$-0.877044\pi$
$359$	− 14.2766i	− 0.753488i	0.926317	−	0.376744i	$-0.122956\pi$
$360$	0	0
$361$	12.2258	0.643465
$362$	0	0
$363$	8.27084	0.434106
$364$	0	0
$365$	2.92192	0.152941
$366$	0	0
$367$	17.2371	0.899768	0.449884	−	0.893087i	$-0.351465\pi$
$367$	17.2371	0.899768	0.449884	+	0.893087i	$0.351465\pi$
$368$	0	0
$369$	10.7022i	0.557137i
$370$	0	0
$371$	− 45.5825i	− 2.36652i
$372$	0	0
$373$	9.53055	0.493473	0.246737	−	0.969083i	$-0.420642\pi$
$373$	9.53055	0.493473	0.246737	+	0.969083i	$0.420642\pi$
$374$	0	0
$375$	4.64611i	0.239924i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	2.25241i	0.115698i	0.998325	+	0.0578492i	$0.0184243\pi$
$379$	2.25241i	0.115698i	−0.998325	+	0.0578492i	$0.981576\pi$
$380$	0	0
$381$	8.87263	0.454559
$382$	0	0
$383$	− 37.9901i	− 1.94120i	−0.240696	−	0.970601i	$-0.577376\pi$
$383$	− 37.9901i	− 1.94120i	0.240696	−	0.970601i	$-0.422624\pi$
$384$	0	0
$385$	− 3.57730i	− 0.182316i
$386$	0	0
$387$	9.89016	0.502745
$388$	0	0
$389$	32.0413	1.62456	0.812280	−	0.583267i	$-0.198226\pi$
$389$	32.0413	1.62456	0.812280	+	0.583267i	$0.198226\pi$
$390$	0	0
$391$	30.2659	1.53061
$392$	0	0
$393$	−3.97750	−0.200638
$394$	0	0
$395$	− 5.11077i	− 0.257151i
$396$	0	0
$397$	34.7002i	1.74155i	0.491681	+	0.870776i	$0.336383\pi$
$397$	34.7002i	1.74155i	−0.491681	+	0.870776i	$0.663617\pi$
$398$	0	0
$399$	11.8564	0.593563
$400$	0	0
$401$	28.8027i	1.43834i	0.694835	+	0.719169i	$0.255477\pi$
$401$	28.8027i	1.43834i	−0.694835	+	0.719169i	$0.744523\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0.475353i	0.0236205i
$406$	0	0
$407$	−3.09362	−0.153345
$408$	0	0
$409$	29.3455i	1.45104i	0.688200	+	0.725521i	$0.258401\pi$
$409$	29.3455i	1.45104i	−0.688200	+	0.725521i	$0.741599\pi$
$410$	0	0
$411$	17.4968i	0.863053i
$412$	0	0
$413$	−6.77205	−0.333231
$414$	0	0
$415$	7.01753	0.344477
$416$	0	0
$417$	−20.2771	−0.992975
$418$	0	0
$419$	−22.7417	−1.11101	−0.555503	−	0.831515i	$-0.687474\pi$
$419$	−22.7417	−1.11101	−0.555503	+	0.831515i	$0.687474\pi$
$420$	0	0
$421$	− 0.831698i	− 0.0405345i	−0.999795	−	0.0202672i	$-0.993548\pi$
$421$	− 0.831698i	− 0.0405345i	0.999795	−	0.0202672i	$-0.00645170\pi$
$422$	0	0
$423$	7.29869i	0.354874i
$424$	0	0
$425$	−37.9033	−1.83858
$426$	0	0
$427$	− 53.4815i	− 2.58815i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 26.2766i	− 1.26570i	−0.774275	−	0.632849i	$-0.781886\pi$
$431$	− 26.2766i	− 1.26570i	0.774275	−	0.632849i	$-0.218114\pi$
$432$	0	0
$433$	5.46448	0.262606	0.131303	−	0.991342i	$-0.458084\pi$
$433$	5.46448	0.262606	0.131303	+	0.991342i	$0.458084\pi$
$434$	0	0
$435$	0.391374i	0.0187650i
$436$	0	0
$437$	9.92180i	0.474624i
$438$	0	0
$439$	13.3046	0.634993	0.317497	−	0.948259i	$-0.397158\pi$
$439$	13.3046	0.634993	0.317497	+	0.948259i	$0.397158\pi$
$440$	0	0
$441$	−13.7515	−0.654835
$442$	0	0
$443$	−14.0879	−0.669336	−0.334668	−	0.942336i	$-0.608624\pi$
$443$	−14.0879	−0.669336	−0.334668	+	0.942336i	$0.608624\pi$
$444$	0	0
$445$	−3.47199	−0.164588
$446$	0	0
$447$	7.68471i	0.363474i
$448$	0	0
$449$	− 9.11077i	− 0.429964i	−0.976618	−	0.214982i	$-0.931031\pi$
$449$	− 9.11077i	− 0.429964i	0.976618	−	0.214982i	$-0.0689693\pi$
$450$	0	0
$451$	−17.6803	−0.832533
$452$	0	0
$453$	− 1.65202i	− 0.0776186i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.57161i	0.400963i	0.979697	+	0.200481i	$0.0642506\pi$
$457$	8.57161i	0.400963i	−0.979697	+	0.200481i	$0.935749\pi$
$458$	0	0
$459$	−7.93945	−0.370582
$460$	0	0
$461$	39.5655i	1.84275i	0.388677	+	0.921374i	$0.372932\pi$
$461$	39.5655i	1.84275i	−0.388677	+	0.921374i	$0.627068\pi$
$462$	0	0
$463$	11.4836i	0.533688i	0.963740	+	0.266844i	$0.0859808\pi$
$463$	11.4836i	0.533688i	−0.963740	+	0.266844i	$0.914019\pi$
$464$	0	0
$465$	0.852115	0.0395159
$466$	0	0
$467$	−25.9125	−1.19909	−0.599545	−	0.800341i	$-0.704652\pi$
$467$	−25.9125	−1.19909	−0.599545	+	0.800341i	$0.704652\pi$
$468$	0	0
$469$	15.1399	0.699097
$470$	0	0
$471$	−13.4855	−0.621381
$472$	0	0
$473$	16.3387i	0.751255i
$474$	0	0
$475$	− 12.4255i	− 0.570121i
$476$	0	0
$477$	−10.0063	−0.458156
$478$	0	0
$479$	− 34.3102i	− 1.56767i	−0.620968	−	0.783836i	$-0.713260\pi$
$479$	− 34.3102i	− 1.56767i	0.620968	−	0.783836i	$-0.286740\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 17.3655i	− 0.790158i
$484$	0	0
$485$	1.98930	0.0903296
$486$	0	0
$487$	40.6582i	1.84240i	0.389092	+	0.921199i	$0.372789\pi$
$487$	40.6582i	1.84240i	−0.389092	+	0.921199i	$0.627211\pi$
$488$	0	0
$489$	− 18.9423i	− 0.856602i
$490$	0	0
$491$	16.8964	0.762526	0.381263	−	0.924467i	$-0.375489\pi$
$491$	16.8964	0.762526	0.381263	+	0.924467i	$0.375489\pi$
$492$	0	0
$493$	−6.53683	−0.294404
$494$	0	0
$495$	−0.785291	−0.0352962
$496$	0	0
$497$	−8.68602	−0.389621
$498$	0	0
$499$	− 24.6899i	− 1.10527i	−0.833422	−	0.552637i	$-0.813622\pi$
$499$	− 24.6899i	− 1.10527i	0.833422	−	0.552637i	$-0.186378\pi$
$500$	0	0
$501$	2.79455i	0.124851i
$502$	0	0
$503$	−31.4363	−1.40167	−0.700837	−	0.713322i	$-0.747190\pi$
$503$	−31.4363	−1.40167	−0.700837	+	0.713322i	$0.747190\pi$
$504$	0	0
$505$	2.62361i	0.116749i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	26.5415i	1.17643i	0.808704	+	0.588216i	$0.200169\pi$
$509$	26.5415i	1.17643i	−0.808704	+	0.588216i	$0.799831\pi$
$510$	0	0
$511$	28.0013	1.23870
$512$	0	0
$513$	− 2.60272i	− 0.114913i
$514$	0	0
$515$	− 5.65202i	− 0.249058i
$516$	0	0
$517$	−12.0576	−0.530291
$518$	0	0
$519$	−6.95071	−0.305102
$520$	0	0
$521$	16.0605	0.703625	0.351813	−	0.936070i	$-0.385565\pi$
$521$	16.0605	0.703625	0.351813	+	0.936070i	$0.385565\pi$
$522$	0	0
$523$	0.995028	0.0435095	0.0217548	−	0.999763i	$-0.493075\pi$
$523$	0.995028	0.0435095	0.0217548	+	0.999763i	$0.493075\pi$
$524$	0	0
$525$	21.7476i	0.949143i
$526$	0	0
$527$	14.2322i	0.619966i
$528$	0	0
$529$	−8.46802	−0.368175
$530$	0	0
$531$	1.48660i	0.0645131i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	2.83702i	0.122655i
$536$	0	0
$537$	−9.94573	−0.429190
$538$	0	0
$539$	− 22.7178i	− 0.978524i
$540$	0	0
$541$	− 10.8825i	− 0.467874i	−0.972252	−	0.233937i	$-0.924839\pi$
$541$	− 10.8825i	− 0.467874i	0.972252	−	0.233937i	$-0.0751610\pi$
$542$	0	0
$543$	12.9887	0.557401
$544$	0	0
$545$	4.65494	0.199396
$546$	0	0
$547$	18.1985	0.778111	0.389056	−	0.921214i	$-0.372801\pi$
$547$	18.1985	0.778111	0.389056	+	0.921214i	$0.372801\pi$
$548$	0	0
$549$	−11.7403	−0.501063
$550$	0	0
$551$	− 2.14291i	− 0.0912911i
$552$	0	0
$553$	− 48.9774i	− 2.08273i
$554$	0	0
$555$	−0.890159	−0.0377852
$556$	0	0
$557$	− 5.59682i	− 0.237145i	−0.992945	−	0.118572i	$-0.962168\pi$
$557$	− 5.59682i	− 0.237145i	0.992945	−	0.118572i	$-0.0378318\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 13.1161i	− 0.553763i
$562$	0	0
$563$	24.4763	1.03155	0.515776	−	0.856723i	$-0.327504\pi$
$563$	24.4763	1.03155	0.515776	+	0.856723i	$0.327504\pi$
$564$	0	0
$565$	− 0.921925i	− 0.0387857i
$566$	0	0
$567$	4.55539i	0.191308i
$568$	0	0
$569$	17.8682	0.749074	0.374537	−	0.927212i	$-0.377802\pi$
$569$	17.8682	0.749074	0.374537	+	0.927212i	$0.377802\pi$
$570$	0	0
$571$	−1.74669	−0.0730968	−0.0365484	−	0.999332i	$-0.511636\pi$
$571$	−1.74669	−0.0730968	−0.0365484	+	0.999332i	$0.511636\pi$
$572$	0	0
$573$	−11.6467	−0.486547
$574$	0	0
$575$	−18.1990	−0.758952
$576$	0	0
$577$	10.6236i	0.442267i	0.975244	+	0.221133i	$0.0709756\pi$
$577$	10.6236i	0.442267i	−0.975244	+	0.221133i	$0.929024\pi$
$578$	0	0
$579$	− 0.621280i	− 0.0258195i
$580$	0	0
$581$	67.2502	2.79001
$582$	0	0
$583$	− 16.5306i	− 0.684625i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 44.8051i	− 1.84930i	−0.380815	−	0.924651i	$-0.624356\pi$
$587$	− 44.8051i	− 1.84930i	0.380815	−	0.924651i	$-0.375644\pi$
$588$	0	0
$589$	−4.66563	−0.192244
$590$	0	0
$591$	1.85641i	0.0763624i
$592$	0	0
$593$	34.9834i	1.43660i	0.695736	+	0.718298i	$0.255079\pi$
$593$	34.9834i	1.43660i	−0.695736	+	0.718298i	$0.744921\pi$
$594$	0	0
$595$	17.1922	0.704812
$596$	0	0
$597$	−13.7916	−0.564451
$598$	0	0
$599$	21.9569	0.897133	0.448566	−	0.893749i	$-0.351935\pi$
$599$	21.9569	0.897133	0.448566	+	0.893749i	$0.351935\pi$
$600$	0	0
$601$	−5.77404	−0.235528	−0.117764	−	0.993042i	$-0.537573\pi$
$601$	−5.77404	−0.235528	−0.117764	+	0.993042i	$0.537573\pi$
$602$	0	0
$603$	− 3.32352i	− 0.135344i
$604$	0	0
$605$	3.93156i	0.159841i
$606$	0	0
$607$	−19.4455	−0.789269	−0.394635	−	0.918838i	$-0.629129\pi$
$607$	−19.4455	−0.789269	−0.394635	+	0.918838i	$0.629129\pi$
$608$	0	0
$609$	3.75061i	0.151982i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	9.48266i	0.383001i	0.981493	+	0.191500i	$0.0613354\pi$
$613$	9.48266i	0.383001i	−0.981493	+	0.191500i	$0.938665\pi$
$614$	0	0
$615$	−5.08734	−0.205141
$616$	0	0
$617$	− 43.4822i	− 1.75053i	−0.483646	−	0.875263i	$-0.660688\pi$
$617$	− 43.4822i	− 1.75053i	0.483646	−	0.875263i	$-0.339312\pi$
$618$	0	0
$619$	5.40607i	0.217288i	0.994081	+	0.108644i	$0.0346509\pi$
$619$	5.40607i	0.217288i	−0.994081	+	0.108644i	$0.965349\pi$
$620$	0	0
$621$	−3.81208	−0.152974
$622$	0	0
$623$	−33.2727	−1.33304
$624$	0	0
$625$	21.6617	0.866466
$626$	0	0
$627$	4.29974	0.171715
$628$	0	0
$629$	− 14.8677i	− 0.592812i
$630$	0	0
$631$	− 16.7101i	− 0.665219i	−0.943065	−	0.332609i	$-0.892071\pi$
$631$	− 16.7101i	− 0.665219i	0.943065	−	0.332609i	$-0.107929\pi$
$632$	0	0
$633$	22.8627	0.908710
$634$	0	0
$635$	4.21763i	0.167371i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.90676i	0.0754302i
$640$	0	0
$641$	−0.0937441	−0.00370267	−0.00185133	−	0.999998i	$-0.500589\pi$
$641$	−0.0937441	−0.00370267	−0.00185133	+	0.999998i	$0.500589\pi$
$642$	0	0
$643$	− 17.1699i	− 0.677115i	−0.940946	−	0.338558i	$-0.890061\pi$
$643$	− 17.1699i	− 0.677115i	0.940946	−	0.338558i	$-0.109939\pi$
$644$	0	0
$645$	4.70131i	0.185114i
$646$	0	0
$647$	32.2448	1.26767	0.633837	−	0.773467i	$-0.281479\pi$
$647$	32.2448	1.26767	0.633837	+	0.773467i	$0.281479\pi$
$648$	0	0
$649$	−2.45590	−0.0964023
$650$	0	0
$651$	8.16597	0.320050
$652$	0	0
$653$	6.64238	0.259936	0.129968	−	0.991518i	$-0.458512\pi$
$653$	6.64238	0.259936	0.129968	+	0.991518i	$0.458512\pi$
$654$	0	0
$655$	− 1.89071i	− 0.0738763i
$656$	0	0
$657$	− 6.14686i	− 0.239812i
$658$	0	0
$659$	−9.57058	−0.372817	−0.186408	−	0.982472i	$-0.559685\pi$
$659$	−9.57058	−0.372817	−0.186408	+	0.982472i	$0.559685\pi$
$660$	0	0
$661$	29.6632i	1.15376i	0.816828	+	0.576882i	$0.195731\pi$
$661$	29.6632i	1.15376i	−0.816828	+	0.576882i	$0.804269\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.63597i	0.218554i
$666$	0	0
$667$	−3.13862	−0.121528
$668$	0	0
$669$	7.38801i	0.285637i
$670$	0	0
$671$	− 19.3952i	− 0.748742i
$672$	0	0
$673$	−11.1436	−0.429554	−0.214777	−	0.976663i	$-0.568902\pi$
$673$	−11.1436	−0.429554	−0.214777	+	0.976663i	$0.568902\pi$
$674$	0	0
$675$	4.77404	0.183753
$676$	0	0
$677$	−5.44763	−0.209369	−0.104685	−	0.994505i	$-0.533383\pi$
$677$	−5.44763	−0.209369	−0.104685	+	0.994505i	$0.533383\pi$
$678$	0	0
$679$	19.0638	0.731603
$680$	0	0
$681$	9.55734i	0.366238i
$682$	0	0
$683$	29.2894i	1.12073i	0.828246	+	0.560364i	$0.189339\pi$
$683$	29.2894i	1.12073i	−0.828246	+	0.560364i	$0.810661\pi$
$684$	0	0
$685$	−8.31715	−0.317782
$686$	0	0
$687$	17.6017i	0.671545i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	23.7383i	0.903049i	0.892259	+	0.451524i	$0.149120\pi$
$691$	23.7383i	0.903049i	−0.892259	+	0.451524i	$0.850880\pi$
$692$	0	0
$693$	−7.52558	−0.285873
$694$	0	0
$695$	− 9.63878i	− 0.365620i
$696$	0	0
$697$	− 84.9700i	− 3.21847i
$698$	0	0
$699$	−19.3391	−0.731472
$700$	0	0
$701$	7.62417	0.287961	0.143980	−	0.989581i	$-0.454010\pi$
$701$	7.62417	0.287961	0.143980	+	0.989581i	$0.454010\pi$
$702$	0	0
$703$	4.87394	0.183824
$704$	0	0
$705$	−3.46945	−0.130667
$706$	0	0
$707$	25.1425i	0.945582i
$708$	0	0
$709$	− 30.4988i	− 1.14540i	−0.819763	−	0.572702i	$-0.805895\pi$
$709$	− 30.4988i	− 1.14540i	0.819763	−	0.572702i	$-0.194105\pi$
$710$	0	0
$711$	−10.7515	−0.403214
$712$	0	0
$713$	6.83353i	0.255918i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	18.6417i	0.696187i
$718$	0	0
$719$	6.63057	0.247279	0.123639	−	0.992327i	$-0.460543\pi$
$719$	6.63057	0.247279	0.123639	+	0.992327i	$0.460543\pi$
$720$	0	0
$721$	− 54.1643i	− 2.01718i
$722$	0	0
$723$	− 0.0605458i	− 0.00225172i
$724$	0	0
$725$	3.93063	0.145980
$726$	0	0
$727$	17.4607	0.647583	0.323792	−	0.946128i	$-0.395042\pi$
$727$	17.4607	0.647583	0.323792	+	0.946128i	$0.395042\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−78.5225	−2.90426
$732$	0	0
$733$	23.3284i	0.861653i	0.902435	+	0.430826i	$0.141778\pi$
$733$	23.3284i	0.861653i	−0.902435	+	0.430826i	$0.858222\pi$
$734$	0	0
$735$	− 6.53683i	− 0.241115i
$736$	0	0
$737$	5.49052	0.202246
$738$	0	0
$739$	2.17187i	0.0798936i	0.999202	+	0.0399468i	$0.0127188\pi$
$739$	2.17187i	0.0798936i	−0.999202	+	0.0399468i	$0.987281\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	46.5728i	1.70859i	0.519788	+	0.854296i	$0.326011\pi$
$743$	46.5728i	1.70859i	−0.519788	+	0.854296i	$0.673989\pi$
$744$	0	0
$745$	−3.65295	−0.133834
$746$	0	0
$747$	− 14.7628i	− 0.540142i
$748$	0	0
$749$	27.1876i	0.993414i
$750$	0	0
$751$	42.3995	1.54718	0.773590	−	0.633686i	$-0.218459\pi$
$751$	42.3995	1.54718	0.773590	+	0.633686i	$0.218459\pi$
$752$	0	0
$753$	18.5974	0.677726
$754$	0	0
$755$	0.785291	0.0285797
$756$	0	0
$757$	−0.805251	−0.0292673	−0.0146337	−	0.999893i	$-0.504658\pi$
$757$	−0.805251	−0.0292673	−0.0146337	+	0.999893i	$0.504658\pi$
$758$	0	0
$759$	− 6.29763i	− 0.228589i
$760$	0	0
$761$	1.20545i	0.0436974i	0.999761	+	0.0218487i	$0.00695521\pi$
$761$	1.20545i	0.0436974i	−0.999761	+	0.0218487i	$0.993045\pi$
$762$	0	0
$763$	44.6091	1.61496
$764$	0	0
$765$	− 3.77404i	− 0.136451i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	15.2279i	0.549134i	0.961568	+	0.274567i	$0.0885345\pi$
$769$	15.2279i	0.549134i	−0.961568	+	0.274567i	$0.911466\pi$
$770$	0	0
$771$	−29.9063	−1.07705
$772$	0	0
$773$	18.1394i	0.652430i	0.945296	+	0.326215i	$0.105773\pi$
$773$	18.1394i	0.652430i	−0.945296	+	0.326215i	$0.894227\pi$
$774$	0	0
$775$	− 8.55793i	− 0.307410i
$776$	0	0
$777$	−8.53055	−0.306032
$778$	0	0
$779$	27.8550	0.998008
$780$	0	0
$781$	−3.15000	−0.112716
$782$	0	0
$783$	0.823335	0.0294236
$784$	0	0
$785$	− 6.41039i	− 0.228797i
$786$	0	0
$787$	17.5504i	0.625605i	0.949818	+	0.312802i	$0.101268\pi$
$787$	17.5504i	0.625605i	−0.949818	+	0.312802i	$0.898732\pi$
$788$	0	0
$789$	−10.0668	−0.358388
$790$	0	0
$791$	− 8.83496i	− 0.314135i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 4.75651i	− 0.168696i
$796$	0	0
$797$	−19.3587	−0.685721	−0.342861	−	0.939386i	$-0.611396\pi$
$797$	−19.3587	−0.685721	−0.342861	+	0.939386i	$0.611396\pi$
$798$	0	0
$799$	− 57.9476i	− 2.05004i
$800$	0	0
$801$	7.30404i	0.258075i
$802$	0	0
$803$	10.1547	0.358352
$804$	0	0
$805$	8.25474	0.290941
$806$	0	0
$807$	28.1336	0.990351
$808$	0	0
$809$	10.4686	0.368055	0.184028	−	0.982921i	$-0.441086\pi$
$809$	10.4686	0.368055	0.184028	+	0.982921i	$0.441086\pi$
$810$	0	0
$811$	38.1332i	1.33904i	0.742795	+	0.669518i	$0.233500\pi$
$811$	38.1332i	1.33904i	−0.742795	+	0.669518i	$0.766500\pi$
$812$	0	0
$813$	26.0492i	0.913585i
$814$	0	0
$815$	9.00429	0.315407
$816$	0	0
$817$	− 25.7413i	− 0.900576i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 10.9439i	− 0.381946i	−0.981595	−	0.190973i	$-0.938836\pi$
$821$	− 10.9439i	− 0.381946i	0.981595	−	0.190973i	$-0.0611642\pi$
$822$	0	0
$823$	39.5156	1.37743	0.688714	−	0.725033i	$-0.258175\pi$
$823$	39.5156	1.37743	0.688714	+	0.725033i	$0.258175\pi$
$824$	0	0
$825$	7.88680i	0.274583i
$826$	0	0
$827$	− 38.3552i	− 1.33374i	−0.745174	−	0.666870i	$-0.767633\pi$
$827$	− 38.3552i	− 1.33374i	0.745174	−	0.666870i	$-0.232367\pi$
$828$	0	0
$829$	−1.09716	−0.0381058	−0.0190529	−	0.999818i	$-0.506065\pi$
$829$	−1.09716	−0.0381058	−0.0190529	+	0.999818i	$0.506065\pi$
$830$	0	0
$831$	7.85156	0.272368
$832$	0	0
$833$	109.180	3.78285
$834$	0	0
$835$	−1.32840	−0.0459711
$836$	0	0
$837$	− 1.79260i	− 0.0619612i
$838$	0	0
$839$	11.4660i	0.395849i	0.980217	+	0.197924i	$0.0634201\pi$
$839$	11.4660i	0.395849i	−0.980217	+	0.197924i	$0.936580\pi$
$840$	0	0
$841$	−28.3221	−0.976625
$842$	0	0
$843$	1.26064i	0.0434189i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	37.6769i	1.29459i
$848$	0	0
$849$	2.61932	0.0898949
$850$	0	0
$851$	− 7.13862i	− 0.244709i
$852$	0	0
$853$	18.6403i	0.638232i	0.947716	+	0.319116i	$0.103386\pi$
$853$	18.6403i	0.638232i	−0.947716	+	0.319116i	$0.896614\pi$
$854$	0	0
$855$	1.23721	0.0423117
$856$	0	0
$857$	−41.9413	−1.43269	−0.716344	−	0.697747i	$-0.754186\pi$
$857$	−41.9413	−1.43269	−0.716344	+	0.697747i	$0.754186\pi$
$858$	0	0
$859$	8.80656	0.300476	0.150238	−	0.988650i	$-0.451996\pi$
$859$	8.80656	0.300476	0.150238	+	0.988650i	$0.451996\pi$
$860$	0	0
$861$	−48.7528	−1.66149
$862$	0	0
$863$	20.0833i	0.683643i	0.939765	+	0.341822i	$0.111044\pi$
$863$	20.0833i	0.683643i	−0.939765	+	0.341822i	$0.888956\pi$
$864$	0	0
$865$	− 3.30404i	− 0.112341i
$866$	0	0
$867$	46.0349	1.56343
$868$	0	0
$869$	− 17.7617i	− 0.602525i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 4.18490i	− 0.141637i
$874$	0	0
$875$	−21.1648	−0.715502
$876$	0	0
$877$	− 33.5669i	− 1.13347i	−0.823899	−	0.566737i	$-0.808206\pi$
$877$	− 33.5669i	− 1.13347i	0.823899	−	0.566737i	$-0.191794\pi$
$878$	0	0
$879$	25.4386i	0.858023i
$880$	0	0
$881$	−35.2560	−1.18781	−0.593903	−	0.804537i	$-0.702414\pi$
$881$	−35.2560	−1.18781	−0.593903	+	0.804537i	$0.702414\pi$
$882$	0	0
$883$	7.59185	0.255486	0.127743	−	0.991807i	$-0.459227\pi$
$883$	7.59185	0.255486	0.127743	+	0.991807i	$0.459227\pi$
$884$	0	0
$885$	−0.706661	−0.0237541
$886$	0	0
$887$	31.8146	1.06823	0.534115	−	0.845412i	$-0.320645\pi$
$887$	31.8146	1.06823	0.534115	+	0.845412i	$0.320645\pi$
$888$	0	0
$889$	40.4182i	1.35558i
$890$	0	0
$891$	1.65202i	0.0553447i
$892$	0	0
$893$	18.9965	0.635692
$894$	0	0
$895$	− 4.72773i	− 0.158031i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1.47591i	− 0.0492243i
$900$	0	0
$901$	79.4444	2.64668
$902$	0	0
$903$	45.0535i	1.49929i
$904$	0	0
$905$	6.17424i	0.205239i
$906$	0	0
$907$	−24.8639	−0.825593	−0.412796	−	0.910823i	$-0.635448\pi$
$907$	−24.8639	−0.825593	−0.412796	+	0.910823i	$0.635448\pi$
$908$	0	0
$909$	5.51930	0.183064
$910$	0	0
$911$	28.9183	0.958105	0.479052	−	0.877786i	$-0.340980\pi$
$911$	28.9183	0.958105	0.479052	+	0.877786i	$0.340980\pi$
$912$	0	0
$913$	24.3884	0.807138
$914$	0	0
$915$	− 5.58078i	− 0.184495i
$916$	0	0
$917$	− 18.1190i	− 0.598343i
$918$	0	0
$919$	−19.2512	−0.635039	−0.317519	−	0.948252i	$-0.602850\pi$
$919$	−19.2512	−0.635039	−0.317519	+	0.948252i	$0.602850\pi$
$920$	0	0
$921$	− 15.5100i	− 0.511072i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	8.94001i	0.293946i
$926$	0	0
$927$	−11.8902	−0.390524
$928$	0	0
$929$	15.1667i	0.497604i	0.968554	+	0.248802i	$0.0800368\pi$
$929$	15.1667i	0.497604i	−0.968554	+	0.248802i	$0.919963\pi$
$930$	0	0
$931$	35.7914i	1.17302i
$932$	0	0
$933$	19.0829	0.624747
$934$	0	0
$935$	6.23478	0.203899
$936$	0	0
$937$	49.2136	1.60774	0.803869	−	0.594807i	$-0.202771\pi$
$937$	49.2136	1.60774	0.803869	+	0.594807i	$0.202771\pi$
$938$	0	0
$939$	12.3406	0.402722
$940$	0	0
$941$	46.2058i	1.50627i	0.657868	+	0.753133i	$0.271459\pi$
$941$	46.2058i	1.50627i	−0.657868	+	0.753133i	$0.728541\pi$
$942$	0	0
$943$	− 40.7978i	− 1.32856i
$944$	0	0
$945$	−2.16541	−0.0704409
$946$	0	0
$947$	− 20.3651i	− 0.661778i	−0.943670	−	0.330889i	$-0.892651\pi$
$947$	− 20.3651i	− 0.661778i	0.943670	−	0.330889i	$-0.107349\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	19.5636i	0.634394i
$952$	0	0
$953$	−31.6467	−1.02514	−0.512568	−	0.858647i	$-0.671306\pi$
$953$	−31.6467	−1.02514	−0.512568	+	0.858647i	$0.671306\pi$
$954$	0	0
$955$	− 5.53627i	− 0.179150i
$956$	0	0
$957$	1.36016i	0.0439678i
$958$	0	0
$959$	−79.7047	−2.57380
$960$	0	0
$961$	27.7866	0.896342
$962$	0	0
$963$	5.96823	0.192324
$964$	0	0
$965$	0.295327	0.00950691
$966$	0	0
$967$	16.7482i	0.538585i	0.963058	+	0.269293i	$0.0867899\pi$
$967$	16.7482i	0.538585i	−0.963058	+	0.269293i	$0.913210\pi$
$968$	0	0
$969$	20.6642i	0.663830i
$970$	0	0
$971$	−5.40263	−0.173378	−0.0866892	−	0.996235i	$-0.527629\pi$
$971$	−5.40263	−0.173378	−0.0866892	+	0.996235i	$0.527629\pi$
$972$	0	0
$973$	− 92.3701i	− 2.96125i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	38.5158i	1.23223i	0.787657	+	0.616114i	$0.211294\pi$
$977$	38.5158i	1.23223i	−0.787657	+	0.616114i	$0.788706\pi$
$978$	0	0
$979$	−12.0664	−0.385644
$980$	0	0
$981$	− 9.79260i	− 0.312654i
$982$	0	0
$983$	− 32.5378i	− 1.03779i	−0.854837	−	0.518897i	$-0.826343\pi$
$983$	− 32.5378i	− 1.03779i	0.854837	−	0.518897i	$-0.173657\pi$
$984$	0	0
$985$	−0.882448	−0.0281171
$986$	0	0
$987$	−33.2483	−1.05831
$988$	0	0
$989$	−37.7021	−1.19886
$990$	0	0
$991$	−0.982470	−0.0312092	−0.0156046	−	0.999878i	$-0.504967\pi$
$991$	−0.982470	−0.0312092	−0.0156046	+	0.999878i	$0.504967\pi$
$992$	0	0
$993$	25.7291i	0.816487i
$994$	0	0
$995$	− 6.55586i	− 0.207835i
$996$	0	0
$997$	−50.1073	−1.58691	−0.793457	−	0.608627i	$-0.791721\pi$
$997$	−50.1073	−1.58691	−0.793457	+	0.608627i	$0.791721\pi$
$998$	0	0
$999$	1.87263i	0.0592474i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.c.p.337.5		8
13.3	even	3		312.2.bf.b.121.3	yes	8
13.4	even	6		312.2.bf.b.49.2	✓	8
13.5	odd	4		4056.2.a.bd.1.3		4
13.8	odd	4		4056.2.a.be.1.2		4
13.12	even	2	inner	4056.2.c.p.337.4		8
39.17	odd	6		936.2.bi.c.361.3		8
39.29	odd	6		936.2.bi.c.433.2		8
52.3	odd	6		624.2.bv.g.433.3		8
52.31	even	4		8112.2.a.cq.1.3		4
52.43	odd	6		624.2.bv.g.49.2		8
52.47	even	4		8112.2.a.cs.1.2		4
156.95	even	6		1872.2.by.m.1297.3		8
156.107	even	6		1872.2.by.m.433.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.bf.b.49.2	✓	8	13.4	even	6
312.2.bf.b.121.3	yes	8	13.3	even	3
624.2.bv.g.49.2		8	52.43	odd	6
624.2.bv.g.433.3		8	52.3	odd	6
936.2.bi.c.361.3		8	39.17	odd	6
936.2.bi.c.433.2		8	39.29	odd	6
1872.2.by.m.433.2		8	156.107	even	6
1872.2.by.m.1297.3		8	156.95	even	6
4056.2.a.bd.1.3		4	13.5	odd	4
4056.2.a.be.1.2		4	13.8	odd	4
4056.2.c.p.337.4		8	13.12	even	2	inner
4056.2.c.p.337.5		8	1.1	even	1	trivial
8112.2.a.cq.1.3		4	52.31	even	4
8112.2.a.cs.1.2		4	52.47	even	4

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:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.475353i q^{5} +4.55539i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.475353i q^{5} +4.55539i q^{7} +1.00000 q^{9} +1.65202i q^{11} +0.475353i q^{15} -7.93945 q^{17} -2.60272i q^{19} +4.55539i q^{21} -3.81208 q^{23} +4.77404 q^{25} +1.00000 q^{27} +0.823335 q^{29} -1.79260i q^{31} +1.65202i q^{33} -2.16541 q^{35} +1.87263i q^{37} +10.7022i q^{41} +9.89016 q^{43} +0.475353i q^{45} +7.29869i q^{47} -13.7515 q^{49} -7.93945 q^{51} -10.0063 q^{53} -0.785291 q^{55} -2.60272i q^{57} +1.48660i q^{59} -11.7403 q^{61} +4.55539i q^{63} -3.32352i q^{67} -3.81208 q^{69} +1.90676i q^{71} -6.14686i q^{73} +4.77404 q^{75} -7.52558 q^{77} -10.7515 q^{79} +1.00000 q^{81} -14.7628i q^{83} -3.77404i q^{85} +0.823335 q^{87} +7.30404i q^{89} -1.79260i q^{93} +1.23721 q^{95} -4.18490i q^{97} +1.65202i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + 8 * q^9	\( 8 q + 8 q^{3} + 8 q^{9} - 24 q^{17} + 4 q^{23} - 4 q^{25} + 8 q^{27} - 12 q^{29} - 20 q^{35} + 16 q^{43} - 36 q^{49} - 24 q^{51} + 4 q^{53} + 20 q^{55} - 4 q^{61} + 4 q^{69} - 4 q^{75} + 56 q^{77} - 12 q^{79} + 8 q^{81} - 12 q^{87} + 68 q^{95}+O(q^{100}) \) 8 * q + 8 * q^3 + 8 * q^9 - 24 * q^17 + 4 * q^23 - 4 * q^25 + 8 * q^27 - 12 * q^29 - 20 * q^35 + 16 * q^43 - 36 * q^49 - 24 * q^51 + 4 * q^53 + 20 * q^55 - 4 * q^61 + 4 * q^69 - 4 * q^75 + 56 * q^77 - 12 * q^79 + 8 * q^81 - 12 * q^87 + 68 * q^95

Introduction

L-functions

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