LMFDB - Embedded newform 392.6.i.b.177.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,6,Mod(177,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("392.177");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	392.i (of order $3$, degree $2$, 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:	$62.8704573667$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			177.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	392.177
Dual form			392.6.i.b.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-10.0000 - 17.3205i) q^{3} +(37.0000 - 64.0859i) q^{5} +(-78.5000 + 135.966i) q^{9} +O(q^{10})$	\(q+(-10.0000 - 17.3205i) q^{3} +(37.0000 - 64.0859i) q^{5} +(-78.5000 + 135.966i) q^{9} +(-62.0000 - 107.387i) q^{11} +478.000 q^{13} -1480.00 q^{15} +(599.000 + 1037.50i) q^{17} +(-1522.00 + 2636.18i) q^{19} +(-92.0000 + 159.349i) q^{23} +(-1175.50 - 2036.03i) q^{25} -1720.00 q^{27} -3282.00 q^{29} +(2864.00 + 4960.59i) q^{31} +(-1240.00 + 2147.74i) q^{33} +(-5163.00 + 8942.58i) q^{37} +(-4780.00 - 8279.20i) q^{39} -8886.00 q^{41} -9188.00 q^{43} +(5809.00 + 10061.5i) q^{45} +(-11832.0 + 20493.6i) q^{47} +(11980.0 - 20750.0i) q^{51} +(-5843.00 - 10120.4i) q^{53} -9176.00 q^{55} +60880.0 q^{57} +(-8438.00 - 14615.0i) q^{59} +(9241.00 - 16005.9i) q^{61} +(17686.0 - 30633.1i) q^{65} +(7766.00 + 13451.1i) q^{67} +3680.00 q^{69} -31960.0 q^{71} +(2443.00 + 4231.40i) q^{73} +(-23510.0 + 40720.5i) q^{75} +(-22280.0 + 38590.1i) q^{79} +(36275.5 + 62831.0i) q^{81} +67364.0 q^{83} +88652.0 q^{85} +(32820.0 + 56845.9i) q^{87} +(-35997.0 + 62348.6i) q^{89} +(57280.0 - 99211.9i) q^{93} +(112628. + 195077. i) q^{95} +48866.0 q^{97} +19468.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 20 q^{3} + 74 q^{5} - 157 q^{9}+O(q^{10}) $ 2 * q - 20 * q^3 + 74 * q^5 - 157 * q^9	$ 2 q - 20 q^{3} + 74 q^{5} - 157 q^{9} - 124 q^{11} + 956 q^{13} - 2960 q^{15} + 1198 q^{17} - 3044 q^{19} - 184 q^{23} - 2351 q^{25} - 3440 q^{27} - 6564 q^{29} + 5728 q^{31} - 2480 q^{33} - 10326 q^{37} - 9560 q^{39} - 17772 q^{41} - 18376 q^{43} + 11618 q^{45} - 23664 q^{47} + 23960 q^{51} - 11686 q^{53} - 18352 q^{55} + 121760 q^{57} - 16876 q^{59} + 18482 q^{61} + 35372 q^{65} + 15532 q^{67} + 7360 q^{69} - 63920 q^{71} + 4886 q^{73} - 47020 q^{75} - 44560 q^{79} + 72551 q^{81} + 134728 q^{83} + 177304 q^{85} + 65640 q^{87} - 71994 q^{89} + 114560 q^{93} + 225256 q^{95} + 97732 q^{97} + 38936 q^{99}+O(q^{100}) $ 2 * q - 20 * q^3 + 74 * q^5 - 157 * q^9 - 124 * q^11 + 956 * q^13 - 2960 * q^15 + 1198 * q^17 - 3044 * q^19 - 184 * q^23 - 2351 * q^25 - 3440 * q^27 - 6564 * q^29 + 5728 * q^31 - 2480 * q^33 - 10326 * q^37 - 9560 * q^39 - 17772 * q^41 - 18376 * q^43 + 11618 * q^45 - 23664 * q^47 + 23960 * q^51 - 11686 * q^53 - 18352 * q^55 + 121760 * q^57 - 16876 * q^59 + 18482 * q^61 + 35372 * q^65 + 15532 * q^67 + 7360 * q^69 - 63920 * q^71 + 4886 * q^73 - 47020 * q^75 - 44560 * q^79 + 72551 * q^81 + 134728 * q^83 + 177304 * q^85 + 65640 * q^87 - 71994 * q^89 + 114560 * q^93 + 225256 * q^95 + 97732 * q^97 + 38936 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$297$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$	−10.0000	−	17.3205i	−0.641500	−	1.11111i	−0.985098	−	0.171994i	$-0.944979\pi$
$3$	−10.0000	−	17.3205i	−0.641500	−	1.11111i	0.343598	−	0.939117i	$-0.388354\pi$
$4$		0			0
$5$	37.0000	−	64.0859i	0.661876	−	1.14640i	−0.318246	−	0.948008i	$-0.603094\pi$
$5$	37.0000	−	64.0859i	0.661876	−	1.14640i	0.980122	−	0.198395i	$-0.0635729\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−78.5000	+	135.966i	−0.323045	+	0.559531i
$10$		0			0
$11$	−62.0000	−	107.387i	−0.154493	−	0.267590i	0.778381	−	0.627792i	$-0.216041\pi$
$11$	−62.0000	−	107.387i	−0.154493	−	0.267590i	−0.932874	+	0.360202i	$0.882708\pi$
$12$		0			0
$13$	478.000			0.784458			0.392229	−	0.919868i	$-0.371704\pi$
$13$	478.000			0.784458			0.392229	+	0.919868i	$0.371704\pi$
$14$		0			0
$15$	−1480.00			−1.69837
$16$		0			0
$17$	599.000	+	1037.50i	0.502695	+	0.870693i	0.999995	+	0.00311466i	$0.000991427\pi$
$17$	599.000	+	1037.50i	0.502695	+	0.870693i	−0.497300	+	0.867579i	$0.665675\pi$
$18$		0			0
$19$	−1522.00	+	2636.18i	−0.967232	+	1.67529i	−0.263737	+	0.964595i	$0.584955\pi$
$19$	−1522.00	+	2636.18i	−0.967232	+	1.67529i	−0.703495	+	0.710700i	$0.748378\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−92.0000	+	159.349i	−0.0362634	+	0.0628100i	−0.883587	−	0.468266i	$-0.844879\pi$
$23$	−92.0000	+	159.349i	−0.0362634	+	0.0628100i	0.847324	+	0.531076i	$0.178212\pi$
$24$		0			0
$25$	−1175.50	−	2036.03i	−0.376160	−	0.651528i
$26$		0			0
$27$	−1720.00			−0.454066
$28$		0			0
$29$	−3282.00			−0.724676			−0.362338	−	0.932047i	$-0.618021\pi$
$29$	−3282.00			−0.724676			−0.362338	+	0.932047i	$0.618021\pi$
$30$		0			0
$31$	2864.00	+	4960.59i	0.535265	+	0.927106i	0.999150	+	0.0412109i	$0.0131216\pi$
$31$	2864.00	+	4960.59i	0.535265	+	0.927106i	−0.463886	+	0.885895i	$0.653545\pi$
$32$		0			0
$33$	−1240.00	+	2147.74i	−0.198215	+	0.343319i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5163.00	+	8942.58i	−0.620009	+	1.07389i	0.369475	+	0.929241i	$0.379538\pi$
$37$	−5163.00	+	8942.58i	−0.620009	+	1.07389i	−0.989483	+	0.144646i	$0.953796\pi$
$38$		0			0
$39$	−4780.00	−	8279.20i	−0.503230	−	0.871620i
$40$		0			0
$41$	−8886.00			−0.825556			−0.412778	−	0.910832i	$-0.635442\pi$
$41$	−8886.00			−0.825556			−0.412778	+	0.910832i	$0.635442\pi$
$42$		0			0
$43$	−9188.00			−0.757792			−0.378896	−	0.925439i	$-0.623696\pi$
$43$	−9188.00			−0.757792			−0.378896	+	0.925439i	$0.623696\pi$
$44$		0			0
$45$	5809.00	+	10061.5i	0.427632	+	0.740680i
$46$		0			0
$47$	−11832.0	+	20493.6i	−0.781292	+	1.35324i	0.149897	+	0.988702i	$0.452106\pi$
$47$	−11832.0	+	20493.6i	−0.781292	+	1.35324i	−0.931189	+	0.364536i	$0.881228\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	11980.0	−	20750.0i	0.644958	−	1.11710i
$52$		0			0
$53$	−5843.00	−	10120.4i	−0.285724	−	0.494888i	0.687061	−	0.726600i	$-0.258901\pi$
$53$	−5843.00	−	10120.4i	−0.285724	−	0.494888i	−0.972784	+	0.231712i	$0.925567\pi$
$54$		0			0
$55$	−9176.00			−0.409022
$56$		0			0
$57$	60880.0			2.48192
$58$		0			0
$59$	−8438.00	−	14615.0i	−0.315580	−	0.546601i	0.663981	−	0.747750i	$-0.268866\pi$
$59$	−8438.00	−	14615.0i	−0.315580	−	0.546601i	−0.979561	+	0.201149i	$0.935532\pi$
$60$		0			0
$61$	9241.00	−	16005.9i	0.317976	−	0.550751i	−0.662090	−	0.749425i	$-0.730330\pi$
$61$	9241.00	−	16005.9i	0.317976	−	0.550751i	0.980066	+	0.198674i	$0.0636635\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	17686.0	−	30633.1i	0.519214	−	0.899305i
$66$		0			0
$67$	7766.00	+	13451.1i	0.211354	+	0.366076i	0.952139	−	0.305667i	$-0.0988794\pi$
$67$	7766.00	+	13451.1i	0.211354	+	0.366076i	−0.740785	+	0.671743i	$0.765546\pi$
$68$		0			0
$69$	3680.00			0.0930519
$70$		0			0
$71$	−31960.0			−0.752421			−0.376210	−	0.926534i	$-0.622773\pi$
$71$	−31960.0			−0.752421			−0.376210	+	0.926534i	$0.622773\pi$
$72$		0			0
$73$	2443.00	+	4231.40i	0.0536558	+	0.0929345i	0.891606	−	0.452812i	$-0.149579\pi$
$73$	2443.00	+	4231.40i	0.0536558	+	0.0929345i	−0.837950	+	0.545747i	$0.816246\pi$
$74$		0			0
$75$	−23510.0	+	40720.5i	−0.482614	+	0.835911i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−22280.0	+	38590.1i	−0.401650	+	0.695678i	−0.993925	−	0.110058i	$-0.964896\pi$
$79$	−22280.0	+	38590.1i	−0.401650	+	0.695678i	0.592275	+	0.805736i	$0.298230\pi$
$80$		0			0
$81$	36275.5	+	62831.0i	0.614329	+	1.06405i
$82$		0			0
$83$	67364.0			1.07333			0.536664	−	0.843796i	$-0.319684\pi$
$83$	67364.0			1.07333			0.536664	+	0.843796i	$0.319684\pi$
$84$		0			0
$85$	88652.0			1.33089
$86$		0			0
$87$	32820.0	+	56845.9i	0.464880	+	0.805195i
$88$		0			0
$89$	−35997.0	+	62348.6i	−0.481716	+	0.834357i	−0.999780	−	0.0209851i	$-0.993320\pi$
$89$	−35997.0	+	62348.6i	−0.481716	+	0.834357i	0.518064	+	0.855342i	$0.326653\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	57280.0	−	99211.9i	0.686745	−	1.18948i
$94$		0			0
$95$	112628.	+	195077.i	1.28038	+	2.21768i
$96$		0			0
$97$	48866.0			0.527324			0.263662	−	0.964615i	$-0.415070\pi$
$97$	48866.0			0.527324			0.263662	+	0.964615i	$0.415070\pi$
$98$		0			0
$99$	19468.0			0.199633
$100$		0			0
$101$	−25803.0	−	44692.1i	−0.251690	−	0.435941i	0.712301	−	0.701874i	$-0.247653\pi$
$101$	−25803.0	−	44692.1i	−0.251690	−	0.435941i	−0.963991	+	0.265934i	$0.914320\pi$
$102$		0			0
$103$	−90212.0	+	156252.i	−0.837860	+	1.45122i	0.0538212	+	0.998551i	$0.482860\pi$
$103$	−90212.0	+	156252.i	−0.837860	+	1.45122i	−0.891681	+	0.452665i	$0.850473\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	32850.0	−	56897.9i	0.277381	−	0.480437i	−0.693352	−	0.720599i	$-0.743867\pi$
$107$	32850.0	−	56897.9i	0.277381	−	0.480437i	0.970733	+	0.240162i	$0.0772004\pi$
$108$		0			0
$109$	56353.0	+	97606.3i	0.454308	+	0.786885i	0.998648	−	0.0519796i	$-0.0165531\pi$
$109$	56353.0	+	97606.3i	0.454308	+	0.786885i	−0.544340	+	0.838865i	$0.683220\pi$
$110$		0			0
$111$	206520.			1.59094
$112$		0			0
$113$	−23502.0			−0.173145			−0.0865723	−	0.996246i	$-0.527591\pi$
$113$	−23502.0			−0.173145			−0.0865723	+	0.996246i	$0.527591\pi$
$114$		0			0
$115$	6808.00	+	11791.8i	0.0480037	+	0.0831449i
$116$		0			0
$117$	−37523.0	+	64991.7i	−0.253415	+	0.438928i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	72837.5	−	126158.i	0.452264	−	0.783343i
$122$		0			0
$123$	88860.0	+	153910.i	0.529595	+	0.917285i
$124$		0			0
$125$	57276.0			0.327867
$126$		0			0
$127$	−94592.0			−0.520409			−0.260205	−	0.965553i	$-0.583790\pi$
$127$	−94592.0			−0.520409			−0.260205	+	0.965553i	$0.583790\pi$
$128$		0			0
$129$	91880.0	+	159141.i	0.486124	+	0.841991i
$130$		0			0
$131$	−35146.0	+	60874.7i	−0.178936	+	0.309926i	−0.941516	−	0.336967i	$-0.890599\pi$
$131$	−35146.0	+	60874.7i	−0.178936	+	0.309926i	0.762580	+	0.646893i	$0.223932\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−63640.0	+	110228.i	−0.300535	+	0.520543i
$136$		0			0
$137$	−138645.	−	240140.i	−0.631107	−	1.09311i	−0.987326	−	0.158706i	$-0.949268\pi$
$137$	−138645.	−	240140.i	−0.631107	−	1.09311i	0.356219	−	0.934402i	$-0.384066\pi$
$138$		0			0
$139$	−130308.			−0.572050			−0.286025	−	0.958222i	$-0.592334\pi$
$139$	−130308.			−0.572050			−0.286025	+	0.958222i	$0.592334\pi$
$140$		0			0
$141$	473280.			2.00480
$142$		0			0
$143$	−29636.0	−	51331.1i	−0.121194	−	0.209913i
$144$		0			0
$145$	−121434.	+	210330.i	−0.479645	+	0.830770i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	200765.	−	347735.i	0.740836	−	1.28317i	−0.211278	−	0.977426i	$-0.567763\pi$
$149$	200765.	−	347735.i	0.740836	−	1.28317i	0.952115	−	0.305740i	$-0.0989040\pi$
$150$		0			0
$151$	37988.0	+	65797.1i	0.135583	+	0.234836i	0.925820	−	0.377965i	$-0.123376\pi$
$151$	37988.0	+	65797.1i	0.135583	+	0.234836i	−0.790237	+	0.612801i	$0.790043\pi$
$152$		0			0
$153$	−188086.			−0.649573
$154$		0			0
$155$	423872.			1.41712
$156$		0			0
$157$	197161.	+	341493.i	0.638369	+	1.10569i	0.985791	+	0.167979i	$0.0537240\pi$
$157$	197161.	+	341493.i	0.638369	+	1.10569i	−0.347421	+	0.937709i	$0.612943\pi$
$158$		0			0
$159$	−116860.	+	202407.i	−0.366584	+	0.634941i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	5862.00	−	10153.3i	0.0172813	−	0.0299321i	−0.857255	−	0.514891i	$-0.827832\pi$
$163$	5862.00	−	10153.3i	0.0172813	−	0.0299321i	0.874537	+	0.484959i	$0.161166\pi$
$164$		0			0
$165$	91760.0	+	158933.i	0.262388	+	0.454469i
$166$		0			0
$167$	−551928.			−1.53141			−0.765705	−	0.643192i	$-0.777610\pi$
$167$	−551928.			−1.53141			−0.765705	+	0.643192i	$0.777610\pi$
$168$		0			0
$169$	−142809.			−0.384626
$170$		0			0
$171$	−238954.	−	413880.i	−0.624919	−	1.08239i
$172$		0			0
$173$	−216447.	+	374897.i	−0.549840	+	0.952351i	0.448445	+	0.893810i	$0.351978\pi$
$173$	−216447.	+	374897.i	−0.549840	+	0.952351i	−0.998285	+	0.0585404i	$0.981355\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−168760.	+	292301.i	−0.404889	+	0.701289i
$178$		0			0
$179$	−279810.	−	484645.i	−0.652726	−	1.13055i	−0.982459	−	0.186480i	$-0.940292\pi$
$179$	−279810.	−	484645.i	−0.652726	−	1.13055i	0.329733	−	0.944074i	$-0.393041\pi$
$180$		0			0
$181$	604710.			1.37199			0.685995	−	0.727607i	$-0.259367\pi$
$181$	604710.			1.37199			0.685995	+	0.727607i	$0.259367\pi$
$182$		0			0
$183$	−369640.			−0.815927
$184$		0			0
$185$	382062.	+	661751.i	0.820738	+	1.42156i
$186$		0			0
$187$	74276.0	−	128650.i	0.155326	−	0.269033i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	204576.	−	354336.i	0.405762	−	0.702800i	−0.588648	−	0.808390i	$-0.700340\pi$
$191$	204576.	−	354336.i	0.405762	−	0.702800i	0.994410	+	0.105589i	$0.0336729\pi$
$192$		0			0
$193$	−270433.	−	468404.i	−0.522596	−	0.905164i	−0.999654	−	0.0262917i	$-0.991630\pi$
$193$	−270433.	−	468404.i	−0.522596	−	0.905164i	0.477058	−	0.878872i	$-0.341703\pi$
$194$		0			0
$195$	−707440.			−1.33230
$196$		0			0
$197$	−629898.			−1.15639			−0.578195	−	0.815898i	$-0.696243\pi$
$197$	−629898.			−1.15639			−0.578195	+	0.815898i	$0.696243\pi$
$198$		0			0
$199$	−141524.	−	245127.i	−0.253336	−	0.438791i	0.711106	−	0.703085i	$-0.248195\pi$
$199$	−141524.	−	245127.i	−0.253336	−	0.438791i	−0.964442	+	0.264293i	$0.914861\pi$
$200$		0			0
$201$	155320.	−	269022.i	0.271167	−	0.469675i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−328782.	+	569467.i	−0.546416	+	0.946420i
$206$		0			0
$207$	−14444.0	−	25017.7i	−0.0234294	−	0.0405810i
$208$		0			0
$209$	377456.			0.597724
$210$		0			0
$211$	142756.			0.220744			0.110372	−	0.993890i	$-0.464796\pi$
$211$	142756.			0.220744			0.110372	+	0.993890i	$0.464796\pi$
$212$		0			0
$213$	319600.	+	553563.i	0.482678	+	0.836023i
$214$		0			0
$215$	−339956.	+	588821.i	−0.501564	+	0.868735i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	48860.0	−	84628.0i	0.0688404	−	0.119235i
$220$		0			0
$221$	286322.	+	495924.i	0.394343	+	0.683022i
$222$		0			0
$223$	889696.			1.19806			0.599031	−	0.800726i	$-0.295553\pi$
$223$	889696.			1.19806			0.599031	+	0.800726i	$0.295553\pi$
$224$		0			0
$225$	369107.			0.486067
$226$		0			0
$227$	−571578.	−	990002.i	−0.736226	−	1.27518i	−0.954184	−	0.299222i	$-0.903273\pi$
$227$	−571578.	−	990002.i	−0.736226	−	1.27518i	0.217958	−	0.975958i	$-0.430060\pi$
$228$		0			0
$229$	347893.	−	602568.i	0.438386	−	0.759307i	−0.559179	−	0.829047i	$-0.688884\pi$
$229$	347893.	−	602568.i	0.438386	−	0.759307i	0.997565	+	0.0697396i	$0.0222169\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	173563.	−	300620.i	0.209444	−	0.362767i	−0.742096	−	0.670294i	$-0.766168\pi$
$233$	173563.	−	300620.i	0.209444	−	0.362767i	0.951539	+	0.307527i	$0.0995014\pi$
$234$		0			0
$235$	875568.	+	1.51653e6i	1.03424	+	1.79135i
$236$		0			0
$237$	891200.			1.03063
$238$		0			0
$239$	−1.64296e6			−1.86051			−0.930255	−	0.366912i	$-0.880415\pi$
$239$	−1.64296e6			−1.86051			−0.930255	+	0.366912i	$0.880415\pi$
$240$		0			0
$241$	583719.	+	1.01103e6i	0.647383	+	1.12130i	0.983746	+	0.179568i	$0.0574699\pi$
$241$	583719.	+	1.01103e6i	0.647383	+	1.12130i	−0.336363	+	0.941733i	$0.609197\pi$
$242$		0			0
$243$	516530.	−	894656.i	0.561151	−	0.971942i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−727516.	+	1.26009e6i	−0.758753	+	1.31420i
$248$		0			0
$249$	−673640.	−	1.16678e6i	−0.688541	−	1.19259i
$250$		0			0
$251$	−790612.			−0.792098			−0.396049	−	0.918229i	$-0.629619\pi$
$251$	−790612.			−0.792098			−0.396049	+	0.918229i	$0.629619\pi$
$252$		0			0
$253$	22816.0			0.0224098
$254$		0			0
$255$	−886520.	−	1.53550e6i	−0.853764	−	1.47876i
$256$		0			0
$257$	64895.0	−	112401.i	0.0612884	−	0.106155i	−0.833753	−	0.552137i	$-0.813812\pi$
$257$	64895.0	−	112401.i	0.0612884	−	0.106155i	0.895042	+	0.445983i	$0.147146\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	257637.	−	446240.i	0.234103	−	0.405478i
$262$		0			0
$263$	−35444.0	−	61390.8i	−0.0315975	−	0.0547286i	0.849794	−	0.527115i	$-0.176726\pi$
$263$	−35444.0	−	61390.8i	−0.0315975	−	0.0547286i	−0.881392	+	0.472386i	$0.843393\pi$
$264$		0			0
$265$	−864764.			−0.756455
$266$		0			0
$267$	1.43988e6			1.23608
$268$		0			0
$269$	−895087.	−	1.55034e6i	−0.754197	−	1.30631i	−0.945773	−	0.324829i	$-0.894693\pi$
$269$	−895087.	−	1.55034e6i	−0.754197	−	1.30631i	0.191576	−	0.981478i	$-0.438640\pi$
$270$		0			0
$271$	886808.	−	1.53600e6i	0.733511	−	1.27048i	−0.221863	−	0.975078i	$-0.571214\pi$
$271$	886808.	−	1.53600e6i	0.733511	−	1.27048i	0.955374	−	0.295400i	$-0.0954528\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−145762.	+	252467.i	−0.116228	+	0.201314i
$276$		0			0
$277$	137725.	+	238547.i	0.107848	+	0.186799i	0.914898	−	0.403684i	$-0.132271\pi$
$277$	137725.	+	238547.i	0.107848	+	0.186799i	−0.807050	+	0.590483i	$0.798937\pi$
$278$		0			0
$279$	−899296.			−0.691659
$280$		0			0
$281$	594170.			0.448895			0.224448	−	0.974486i	$-0.427942\pi$
$281$	594170.			0.448895			0.224448	+	0.974486i	$0.427942\pi$
$282$		0			0
$283$	−546214.	−	946070.i	−0.405412	−	0.702194i	0.588957	−	0.808164i	$-0.299539\pi$
$283$	−546214.	−	946070.i	−0.405412	−	0.702194i	−0.994369	+	0.105970i	$0.966205\pi$
$284$		0			0
$285$	2.25256e6	−	3.90155e6i	1.64272	−	2.84528i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−7673.50	+	13290.9i	−0.00540442	+	0.00936073i
$290$		0			0
$291$	−488660.	−	846384.i	−0.338278	−	0.585915i
$292$		0			0
$293$	333654.			0.227053			0.113527	−	0.993535i	$-0.463785\pi$
$293$	333654.			0.227053			0.113527	+	0.993535i	$0.463785\pi$
$294$		0			0
$295$	−1.24882e6			−0.835500
$296$		0			0
$297$	106640.	+	184706.i	0.0701502	+	0.121504i
$298$		0			0
$299$	−43976.0	+	76168.7i	−0.0284471	+	0.0492718i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−516060.	+	893842.i	−0.322919	+	0.559312i
$304$		0			0
$305$	−683834.	−	1.18444e6i	−0.420921	−	0.729057i
$306$		0			0
$307$	1.05997e6			0.641872			0.320936	−	0.947101i	$-0.396003\pi$
$307$	1.05997e6			0.641872			0.320936	+	0.947101i	$0.396003\pi$
$308$		0			0
$309$	3.60848e6			2.14995
$310$		0			0
$311$	668244.	+	1.15743e6i	0.391773	+	0.678570i	0.992683	−	0.120746i	$-0.0385287\pi$
$311$	668244.	+	1.15743e6i	0.391773	+	0.678570i	−0.600911	+	0.799316i	$0.705195\pi$
$312$		0			0
$313$	−822093.	+	1.42391e6i	−0.474308	+	0.821525i	−0.999567	−	0.0294171i	$-0.990635\pi$
$313$	−822093.	+	1.42391e6i	−0.474308	+	0.821525i	0.525260	+	0.850942i	$0.323968\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	861849.	−	1.49277e6i	0.481707	−	0.834341i	−0.518072	−	0.855337i	$-0.673350\pi$
$317$	861849.	−	1.49277e6i	0.481707	−	0.834341i	0.999780	+	0.0209956i	$0.00668359\pi$
$318$		0			0
$319$	203484.	+	352445.i	0.111958	+	0.193916i
$320$		0			0
$321$	−1.31400e6			−0.711759
$322$		0			0
$323$	−3.64671e6			−1.94489
$324$		0			0
$325$	−561889.	−	973220.i	−0.295082	−	0.511096i
$326$		0			0
$327$	1.12706e6	−	1.95213e6i	0.582878	−	1.00957i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.37481e6	+	2.38125e6i	−0.689722	+	1.19463i	0.282206	+	0.959354i	$0.408934\pi$
$331$	−1.37481e6	+	2.38125e6i	−0.689722	+	1.19463i	−0.971928	+	0.235279i	$0.924399\pi$
$332$		0			0
$333$	−810591.	−	1.40398e6i	−0.400582	−	0.693828i
$334$		0			0
$335$	1.14937e6			0.559561
$336$		0			0
$337$	−3.41489e6			−1.63796			−0.818978	−	0.573824i	$-0.805459\pi$
$337$	−3.41489e6			−1.63796			−0.818978	+	0.573824i	$0.805459\pi$
$338$		0			0
$339$	235020.	+	407067.i	0.111072	+	0.192383i
$340$		0			0
$341$	355136.	−	615114.i	0.165390	−	0.286464i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	136160.	−	235836.i	0.0615888	−	0.106675i
$346$		0			0
$347$	−365382.	−	632860.i	−0.162901	−	0.282153i	0.773007	−	0.634398i	$-0.218752\pi$
$347$	−365382.	−	632860.i	−0.162901	−	0.282153i	−0.935908	+	0.352245i	$0.885418\pi$
$348$		0			0
$349$	−2.29749e6			−1.00969			−0.504847	−	0.863209i	$-0.668451\pi$
$349$	−2.29749e6			−1.00969			−0.504847	+	0.863209i	$0.668451\pi$
$350$		0			0
$351$	−822160.			−0.356196
$352$		0			0
$353$	585359.	+	1.01387e6i	0.250026	+	0.433058i	0.963533	−	0.267590i	$-0.0862274\pi$
$353$	585359.	+	1.01387e6i	0.250026	+	0.433058i	−0.713507	+	0.700649i	$0.752894\pi$
$354$		0			0
$355$	−1.18252e6	+	2.04818e6i	−0.498009	+	0.862578i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.94327e6	+	3.36584e6i	−0.795787	+	1.37834i	0.126552	+	0.991960i	$0.459609\pi$
$359$	−1.94327e6	+	3.36584e6i	−0.795787	+	1.37834i	−0.922339	+	0.386383i	$0.873724\pi$
$360$		0			0
$361$	−3.39492e6	−	5.88017e6i	−1.37108	−	2.37477i
$362$		0			0
$363$	−2.91350e6			−1.16051
$364$		0			0
$365$	361564.			0.142054
$366$		0			0
$367$	−466520.	−	808036.i	−0.180803	−	0.313160i	0.761351	−	0.648339i	$-0.224536\pi$
$367$	−466520.	−	808036.i	−0.180803	−	0.313160i	−0.942154	+	0.335180i	$0.891203\pi$
$368$		0			0
$369$	697551.	−	1.20819e6i	0.266692	−	0.461924i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	196109.	−	339671.i	0.0729836	−	0.126411i	−0.827224	−	0.561872i	$-0.810081\pi$
$373$	196109.	−	339671.i	0.0729836	−	0.126411i	0.900208	+	0.435461i	$0.143415\pi$
$374$		0			0
$375$	−572760.	−	992049.i	−0.210327	−	0.364297i
$376$		0			0
$377$	−1.56880e6			−0.568477
$378$		0			0
$379$	−4.72930e6			−1.69122			−0.845608	−	0.533805i	$-0.820762\pi$
$379$	−4.72930e6			−1.69122			−0.845608	+	0.533805i	$0.820762\pi$
$380$		0			0
$381$	945920.	+	1.63838e6i	0.333843	+	0.578233i
$382$		0			0
$383$	−948672.	+	1.64315e6i	−0.330460	+	0.572374i	−0.982602	−	0.185723i	$-0.940537\pi$
$383$	−948672.	+	1.64315e6i	−0.330460	+	0.572374i	0.652142	+	0.758097i	$0.273871\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	721258.	−	1.24926e6i	0.244801	−	0.424008i
$388$		0			0
$389$	1.86148e6	+	3.22417e6i	0.623711	+	1.08030i	0.988789	+	0.149323i	$0.0477093\pi$
$389$	1.86148e6	+	3.22417e6i	0.623711	+	1.08030i	−0.365077	+	0.930977i	$0.618957\pi$
$390$		0			0
$391$	−220432.			−0.0729177
$392$		0			0
$393$	1.40584e6			0.459150
$394$		0			0
$395$	1.64872e6	+	2.85567e6i	0.531685	+	0.920905i
$396$		0			0
$397$	−1.66904e6	+	2.89086e6i	−0.531484	+	0.920557i	0.467841	+	0.883813i	$0.345032\pi$
$397$	−1.66904e6	+	2.89086e6i	−0.531484	+	0.920557i	−0.999325	+	0.0367445i	$0.988301\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.13745e6	+	3.70217e6i	−0.663796	+	1.14973i	0.315814	+	0.948821i	$0.397722\pi$
$401$	−2.13745e6	+	3.70217e6i	−0.663796	+	1.14973i	−0.979610	+	0.200908i	$0.935611\pi$
$402$		0			0
$403$	1.36899e6	+	2.37116e6i	0.419893	+	0.727275i
$404$		0			0
$405$	5.36877e6			1.62644
$406$		0			0
$407$	1.28042e6			0.383149
$408$		0			0
$409$	1.28660e6	+	2.22845e6i	0.380306	+	0.658710i	0.991106	−	0.133075i	$-0.0424852\pi$
$409$	1.28660e6	+	2.22845e6i	0.380306	+	0.658710i	−0.610800	+	0.791785i	$0.709152\pi$
$410$		0			0
$411$	−2.77290e6	+	4.80280e6i	−0.809710	+	1.40246i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	2.49247e6	−	4.31708e6i	0.710410	−	1.23047i
$416$		0			0
$417$	1.30308e6	+	2.25700e6i	0.366970	+	0.635611i
$418$		0			0
$419$	5.26828e6			1.46600			0.732999	−	0.680230i	$-0.238120\pi$
$419$	5.26828e6			1.46600			0.732999	+	0.680230i	$0.238120\pi$
$420$		0			0
$421$	−973354.			−0.267649			−0.133824	−	0.991005i	$-0.542726\pi$
$421$	−973354.			−0.267649			−0.133824	+	0.991005i	$0.542726\pi$
$422$		0			0
$423$	−1.85762e6	−	3.21750e6i	−0.504786	−	0.874314i
$424$		0			0
$425$	1.40825e6	−	2.43916e6i	0.378187	−	0.655040i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−592720.	+	1.02662e6i	−0.155491	+	0.269319i
$430$		0			0
$431$	−1.77868e6	−	3.08076e6i	−0.461216	−	0.798850i	0.537806	−	0.843069i	$-0.319253\pi$
$431$	−1.77868e6	−	3.08076e6i	−0.461216	−	0.798850i	−0.999022	+	0.0442188i	$0.985920\pi$
$432$		0			0
$433$	−1.95496e6			−0.501092			−0.250546	−	0.968105i	$-0.580610\pi$
$433$	−1.95496e6			−0.501092			−0.250546	+	0.968105i	$0.580610\pi$
$434$		0			0
$435$	4.85736e6			1.23077
$436$		0			0
$437$	−280048.	−	485057.i	−0.0701502	−	0.121504i
$438$		0			0
$439$	1.64840e6	−	2.85512e6i	0.408228	−	0.707071i	−0.586464	−	0.809976i	$-0.699480\pi$
$439$	1.64840e6	−	2.85512e6i	0.408228	−	0.707071i	0.994691	+	0.102905i	$0.0328136\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.52910e6	−	4.38053e6i	0.612289	−	1.06052i	−0.378565	−	0.925575i	$-0.623582\pi$
$443$	2.52910e6	−	4.38053e6i	0.612289	−	1.06052i	0.990854	−	0.134941i	$-0.0430844\pi$
$444$		0			0
$445$	2.66378e6	+	4.61380e6i	0.637673	+	1.10448i
$446$		0			0
$447$	−8.03060e6			−1.90099
$448$		0			0
$449$	2.12730e6			0.497981			0.248990	−	0.968506i	$-0.419901\pi$
$449$	2.12730e6			0.497981			0.248990	+	0.968506i	$0.419901\pi$
$450$		0			0
$451$	550932.	+	954242.i	0.127543	+	0.220911i
$452$		0			0
$453$	759760.	−	1.31594e6i	0.173953	−	0.301295i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−144565.	+	250394.i	−0.0323797	+	0.0560833i	−0.881761	−	0.471696i	$-0.843642\pi$
$457$	−144565.	+	250394.i	−0.0323797	+	0.0560833i	0.849381	+	0.527779i	$0.176975\pi$
$458$		0			0
$459$	−1.03028e6	−	1.78450e6i	−0.228257	−	0.395352i
$460$		0			0
$461$	2.66870e6			0.584854			0.292427	−	0.956288i	$-0.405537\pi$
$461$	2.66870e6			0.584854			0.292427	+	0.956288i	$0.405537\pi$
$462$		0			0
$463$	7.58619e6			1.64464			0.822321	−	0.569024i	$-0.192679\pi$
$463$	7.58619e6			1.64464			0.822321	+	0.569024i	$0.192679\pi$
$464$		0			0
$465$	−4.23872e6	−	7.34168e6i	−0.909081	−	1.57457i
$466$		0			0
$467$	709806.	−	1.22942e6i	0.150608	−	0.260860i	−0.780843	−	0.624727i	$-0.785210\pi$
$467$	709806.	−	1.22942e6i	0.150608	−	0.260860i	0.931451	+	0.363867i	$0.118544\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	3.94322e6	−	6.82986e6i	0.819028	−	1.41860i
$472$		0			0
$473$	569656.	+	986673.i	0.117074	+	0.202778i
$474$		0			0
$475$	7.15644e6			1.45534
$476$		0			0
$477$	1.83470e6			0.369207
$478$		0			0
$479$	942032.	+	1.63165e6i	0.187597	+	0.324928i	0.944449	−	0.328659i	$-0.106597\pi$
$479$	942032.	+	1.63165e6i	0.187597	+	0.324928i	−0.756851	+	0.653587i	$0.773263\pi$
$480$		0			0
$481$	−2.46791e6	+	4.27455e6i	−0.486371	+	0.842419i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.80804e6	−	3.13162e6i	0.349023	−	0.604526i
$486$		0			0
$487$	3.00694e6	+	5.20817e6i	0.574516	+	0.995091i	0.996094	+	0.0882991i	$0.0281431\pi$
$487$	3.00694e6	+	5.20817e6i	0.574516	+	0.995091i	−0.421578	+	0.906792i	$0.638524\pi$
$488$		0			0
$489$	−234480.			−0.0443439
$490$		0			0
$491$	4.29232e6			0.803504			0.401752	−	0.915749i	$-0.368401\pi$
$491$	4.29232e6			0.803504			0.401752	+	0.915749i	$0.368401\pi$
$492$		0			0
$493$	−1.96592e6	−	3.40507e6i	−0.364291	−	0.630970i
$494$		0			0
$495$	720316.	−	1.24762e6i	0.132133	−	0.228860i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−672546.	+	1.16488e6i	−0.120912	+	0.209426i	−0.920128	−	0.391618i	$-0.871915\pi$
$499$	−672546.	+	1.16488e6i	−0.120912	+	0.209426i	0.799215	+	0.601045i	$0.205249\pi$
$500$		0			0
$501$	5.51928e6	+	9.55967e6i	0.982399	+	1.70157i
$502$		0			0
$503$	202008.			0.0355999			0.0177999	−	0.999842i	$-0.494334\pi$
$503$	202008.			0.0355999			0.0177999	+	0.999842i	$0.494334\pi$
$504$		0			0
$505$	−3.81884e6			−0.666352
$506$		0			0
$507$	1.42809e6	+	2.47352e6i	0.246738	+	0.427362i
$508$		0			0
$509$	−4.89172e6	+	8.47271e6i	−0.836887	+	1.44953i	0.0555970	+	0.998453i	$0.482294\pi$
$509$	−4.89172e6	+	8.47271e6i	−0.836887	+	1.44953i	−0.892484	+	0.451078i	$0.851040\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	2.61784e6	−	4.53423e6i	0.439187	−	0.760695i
$514$		0			0
$515$	6.67569e6	+	1.15626e7i	1.10912	+	1.92105i
$516$		0			0
$517$	2.93434e6			0.482818
$518$		0			0
$519$	8.65788e6			1.41089
$520$		0			0
$521$	5.24152e6	+	9.07857e6i	0.845985	+	1.46529i	0.884763	+	0.466041i	$0.154320\pi$
$521$	5.24152e6	+	9.07857e6i	0.845985	+	1.46529i	−0.0387785	+	0.999248i	$0.512347\pi$
$522$		0			0
$523$	−3.10509e6	+	5.37817e6i	−0.496386	+	0.859766i	−0.999991	−	0.00416822i	$-0.998673\pi$
$523$	−3.10509e6	+	5.37817e6i	−0.496386	+	0.859766i	0.503605	+	0.863934i	$0.332007\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.43107e6	+	5.94279e6i	−0.538150	+	0.932103i
$528$		0			0
$529$	3.20124e6	+	5.54472e6i	0.497370	+	0.861470i
$530$		0			0
$531$	2.64953e6			0.407787
$532$		0			0
$533$	−4.24751e6			−0.647614
$534$		0			0
$535$	−2.43090e6	−	4.21044e6i	−0.367183	−	0.635980i
$536$		0			0
$537$	−5.59620e6	+	9.69290e6i	−0.837447	+	1.45050i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−2.54044e6	+	4.40017e6i	−0.373178	+	0.646363i	−0.990052	−	0.140699i	$-0.955065\pi$
$541$	−2.54044e6	+	4.40017e6i	−0.373178	+	0.646363i	0.616875	+	0.787061i	$0.288398\pi$
$542$		0			0
$543$	−6.04710e6	−	1.04739e7i	−0.880132	−	1.52443i
$544$		0			0
$545$	8.34024e6			1.20278
$546$		0			0
$547$	3.34687e6			0.478267			0.239133	−	0.970987i	$-0.423137\pi$
$547$	3.34687e6			0.478267			0.239133	+	0.970987i	$0.423137\pi$
$548$		0			0
$549$	1.45084e6	+	2.51292e6i	0.205441	+	0.355835i
$550$		0			0
$551$	4.99520e6	−	8.65195e6i	0.700929	−	1.21405i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	7.64124e6	−	1.32350e7i	1.05301	−	1.82386i
$556$		0			0
$557$	−3.50189e6	−	6.06545e6i	−0.478260	−	0.828371i	0.521429	−	0.853295i	$-0.325399\pi$
$557$	−3.50189e6	−	6.06545e6i	−0.478260	−	0.828371i	−0.999689	+	0.0249237i	$0.992066\pi$
$558$		0			0
$559$	−4.39186e6			−0.594456
$560$		0			0
$561$	−2.97104e6			−0.398567
$562$		0			0
$563$	6.49094e6	+	1.12426e7i	0.863052	+	1.49485i	0.868969	+	0.494867i	$0.164783\pi$
$563$	6.49094e6	+	1.12426e7i	0.863052	+	1.49485i	−0.00591737	+	0.999982i	$0.501884\pi$
$564$		0			0
$565$	−869574.	+	1.50615e6i	−0.114600	+	0.198493i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−949709.	+	1.64494e6i	−0.122973	+	0.212996i	−0.920939	−	0.389707i	$-0.872576\pi$
$569$	−949709.	+	1.64494e6i	−0.122973	+	0.212996i	0.797966	+	0.602703i	$0.205910\pi$
$570$		0			0
$571$	831498.	+	1.44020e6i	0.106726	+	0.184855i	0.914442	−	0.404717i	$-0.132630\pi$
$571$	831498.	+	1.44020e6i	0.106726	+	0.184855i	−0.807716	+	0.589572i	$0.799297\pi$
$572$		0			0
$573$	−8.18304e6			−1.04119
$574$		0			0
$575$	432584.			0.0545633
$576$		0			0
$577$	−4.38672e6	−	7.59802e6i	−0.548530	−	0.950082i	−0.998376	−	0.0569756i	$-0.981854\pi$
$577$	−4.38672e6	−	7.59802e6i	−0.548530	−	0.950082i	0.449845	−	0.893106i	$-0.351479\pi$
$578$		0			0
$579$	−5.40866e6	+	9.36807e6i	−0.670491	+	1.16133i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−724532.	+	1.25493e6i	−0.0882849	+	0.152914i
$584$		0			0
$585$	2.77670e6	+	4.80939e6i	0.335459	+	0.581032i
$586$		0			0
$587$	5.18393e6			0.620961			0.310480	−	0.950580i	$-0.399510\pi$
$587$	5.18393e6			0.620961			0.310480	+	0.950580i	$0.399510\pi$
$588$		0			0
$589$	−1.74360e7			−2.07090
$590$		0			0
$591$	6.29898e6	+	1.09102e7i	0.741825	+	1.28488i
$592$		0			0
$593$	−4.24929e6	+	7.35998e6i	−0.496226	+	0.859489i	−0.999991	−	0.00435235i	$-0.998615\pi$
$593$	−4.24929e6	+	7.35998e6i	−0.496226	+	0.859489i	0.503765	+	0.863841i	$0.331948\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−2.83048e6	+	4.90254e6i	−0.325031	+	0.562970i
$598$		0			0
$599$	−5.62355e6	−	9.74027e6i	−0.640388	−	1.10918i	−0.985346	−	0.170567i	$-0.945440\pi$
$599$	−5.62355e6	−	9.74027e6i	−0.640388	−	1.10918i	0.344958	−	0.938618i	$-0.387893\pi$
$600$		0			0
$601$	−3.46439e6			−0.391238			−0.195619	−	0.980680i	$-0.562672\pi$
$601$	−3.46439e6			−0.391238			−0.195619	+	0.980680i	$0.562672\pi$
$602$		0			0
$603$	−2.43852e6			−0.273108
$604$		0			0
$605$	−5.38997e6	−	9.33571e6i	−0.598685	−	1.03695i
$606$		0			0
$607$	499856.	−	865776.i	0.0550647	−	0.0953748i	−0.837179	−	0.546929i	$-0.815797\pi$
$607$	499856.	−	865776.i	0.0550647	−	0.0953748i	0.892244	+	0.451554i	$0.149130\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.65570e6	+	9.79595e6i	−0.612891	+	1.06156i
$612$		0			0
$613$	−4.90670e6	−	8.49865e6i	−0.527398	−	0.913480i	−0.999490	−	0.0319305i	$-0.989834\pi$
$613$	−4.90670e6	−	8.49865e6i	−0.527398	−	0.913480i	0.472092	−	0.881549i	$-0.343499\pi$
$614$		0			0
$615$	1.31513e7			1.40210
$616$		0			0
$617$	−5.34745e6			−0.565501			−0.282751	−	0.959193i	$-0.591247\pi$
$617$	−5.34745e6			−0.565501			−0.282751	+	0.959193i	$0.591247\pi$
$618$		0			0
$619$	3.41384e6	+	5.91295e6i	0.358110	+	0.620265i	0.987645	−	0.156707i	$-0.0500878\pi$
$619$	3.41384e6	+	5.91295e6i	0.358110	+	0.620265i	−0.629535	+	0.776972i	$0.716755\pi$
$620$		0			0
$621$	158240.	−	274080.i	0.0164660	−	0.0285199i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	5.79265e6	−	1.00332e7i	0.593167	−	1.02740i
$626$		0			0
$627$	−3.77456e6	−	6.53773e6i	−0.383440	−	0.664138i
$628$		0			0
$629$	−1.23705e7			−1.24670
$630$		0			0
$631$	−3.60970e6			−0.360909			−0.180455	−	0.983583i	$-0.557757\pi$
$631$	−3.60970e6			−0.360909			−0.180455	+	0.983583i	$0.557757\pi$
$632$		0			0
$633$	−1.42756e6	−	2.47261e6i	−0.141607	−	0.245271i
$634$		0			0
$635$	−3.49990e6	+	6.06201e6i	−0.344447	+	0.596599i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	2.50886e6	−	4.34547e6i	0.243066	−	0.421003i
$640$		0			0
$641$	6.69267e6	+	1.15920e7i	0.643361	+	1.11433i	0.984678	+	0.174385i	$0.0557937\pi$
$641$	6.69267e6	+	1.15920e7i	0.643361	+	1.11433i	−0.341317	+	0.939948i	$0.610873\pi$
$642$		0			0
$643$	−9.91115e6			−0.945358			−0.472679	−	0.881235i	$-0.656713\pi$
$643$	−9.91115e6			−0.945358			−0.472679	+	0.881235i	$0.656713\pi$
$644$		0			0
$645$	1.35982e7			1.28701
$646$		0			0
$647$	8.91796e6	+	1.54464e7i	0.837539	+	1.45066i	0.891946	+	0.452141i	$0.149340\pi$
$647$	8.91796e6	+	1.54464e7i	0.837539	+	1.45066i	−0.0544074	+	0.998519i	$0.517327\pi$
$648$		0			0
$649$	−1.04631e6	+	1.81227e6i	−0.0975101	+	0.168892i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.16162e6	−	3.74403e6i	0.198379	−	0.343603i	−0.749624	−	0.661864i	$-0.769766\pi$
$653$	2.16162e6	−	3.74403e6i	0.198379	−	0.343603i	0.948003	+	0.318261i	$0.103099\pi$
$654$		0			0
$655$	2.60080e6	+	4.50472e6i	0.236867	+	0.410266i
$656$		0			0
$657$	−767102.			−0.0693330
$658$		0			0
$659$	1.97858e7			1.77476			0.887382	−	0.461035i	$-0.152522\pi$
$659$	1.97858e7			1.77476			0.887382	+	0.461035i	$0.152522\pi$
$660$		0			0
$661$	−7.88858e6	−	1.36634e7i	−0.702255	−	1.21634i	−0.967673	−	0.252209i	$-0.918843\pi$
$661$	−7.88858e6	−	1.36634e7i	−0.702255	−	1.21634i	0.265417	−	0.964134i	$-0.414490\pi$
$662$		0			0
$663$	5.72644e6	−	9.91849e6i	0.505942	−	0.876318i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	301944.	−	522982.i	0.0262792	−	0.0455169i
$668$		0			0
$669$	−8.89696e6	−	1.54100e7i	−0.768557	−	1.33118i
$670$		0			0
$671$	−2.29177e6			−0.196501
$672$		0			0
$673$	6.78762e6			0.577670			0.288835	−	0.957379i	$-0.406732\pi$
$673$	6.78762e6			0.577670			0.288835	+	0.957379i	$0.406732\pi$
$674$		0			0
$675$	2.02186e6	+	3.50196e6i	0.170801	+	0.295837i
$676$		0			0
$677$	7.49711e6	−	1.29854e7i	0.628669	−	1.08889i	−0.359150	−	0.933280i	$-0.616933\pi$
$677$	7.49711e6	−	1.29854e7i	0.628669	−	1.08889i	0.987819	−	0.155607i	$-0.0497334\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−1.14316e7	+	1.98000e7i	−0.944578	+	1.63606i
$682$		0			0
$683$	−5.77902e6	−	1.00096e7i	−0.474026	−	0.821038i	0.525531	−	0.850774i	$-0.323867\pi$
$683$	−5.77902e6	−	1.00096e7i	−0.474026	−	0.821038i	−0.999558	+	0.0297363i	$0.990533\pi$
$684$		0			0
$685$	−2.05195e7			−1.67086
$686$		0			0
$687$	−1.39157e7			−1.12490
$688$		0			0
$689$	−2.79295e6	−	4.83754e6i	−0.224138	−	0.388219i
$690$		0			0
$691$	110078.	−	190661.i	0.00877012	−	0.0151903i	−0.861607	−	0.507576i	$-0.830542\pi$
$691$	110078.	−	190661.i	0.00877012	−	0.0151903i	0.870377	+	0.492386i	$0.163875\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.82140e6	+	8.35090e6i	−0.378626	+	0.655800i
$696$		0			0
$697$	−5.32271e6	−	9.21921e6i	−0.415003	−	0.718806i
$698$		0			0
$699$	−6.94252e6			−0.537433
$700$		0			0
$701$	4.78933e6			0.368111			0.184056	−	0.982916i	$-0.441077\pi$
$701$	4.78933e6			0.368111			0.184056	+	0.982916i	$0.441077\pi$
$702$		0			0
$703$	−1.57162e7	−	2.72212e7i	−1.19938	−	2.07740i
$704$		0			0
$705$	1.75114e7	−	3.03306e7i	1.32693	−	2.29831i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−2.13446e6	+	3.69699e6i	−0.159468	+	0.276206i	−0.934677	−	0.355499i	$-0.884311\pi$
$709$	−2.13446e6	+	3.69699e6i	−0.159468	+	0.276206i	0.775209	+	0.631705i	$0.217644\pi$
$710$		0			0
$711$	−3.49796e6	−	6.05864e6i	−0.259502	−	0.449471i
$712$		0			0
$713$	−1.05395e6			−0.0776421
$714$		0			0
$715$	−4.38613e6			−0.320860
$716$		0			0
$717$	1.64296e7	+	2.84569e7i	1.19352	+	2.06723i
$718$		0			0
$719$	8.09798e6	−	1.40261e7i	0.584190	−	1.01185i	−0.410786	−	0.911732i	$-0.634746\pi$
$719$	8.09798e6	−	1.40261e7i	0.584190	−	1.01185i	0.994976	−	0.100115i	$-0.0319211\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	1.16744e7	−	2.02206e7i	0.830593	−	1.43863i
$724$		0			0
$725$	3.85799e6	+	6.68224e6i	0.272594	+	0.472147i
$726$		0			0
$727$	6.53426e6			0.458522			0.229261	−	0.973365i	$-0.426369\pi$
$727$	6.53426e6			0.458522			0.229261	+	0.973365i	$0.426369\pi$
$728$		0			0
$729$	−3.03131e6			−0.211257
$730$		0			0
$731$	−5.50361e6	−	9.53254e6i	−0.380938	−	0.659804i
$732$		0			0
$733$	−6.58086e6	+	1.13984e7i	−0.452400	+	0.783579i	−0.998535	−	0.0541179i	$-0.982765\pi$
$733$	−6.58086e6	+	1.13984e7i	−0.452400	+	0.783579i	0.546135	+	0.837697i	$0.316099\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	962984.	−	1.66794e6i	0.0653056	−	0.113113i
$738$		0			0
$739$	7.11738e6	+	1.23277e7i	0.479412	+	0.830366i	0.999721	−	0.0236117i	$-0.00751652\pi$
$739$	7.11738e6	+	1.23277e7i	0.479412	+	0.830366i	−0.520309	+	0.853978i	$0.674183\pi$
$740$		0			0
$741$	2.91006e7			1.94696
$742$		0			0
$743$	−2.15835e7			−1.43434			−0.717168	−	0.696901i	$-0.754562\pi$
$743$	−2.15835e7			−1.43434			−0.717168	+	0.696901i	$0.754562\pi$
$744$		0			0
$745$	−1.48566e7	−	2.57324e7i	−0.980684	−	1.69859i
$746$		0			0
$747$	−5.28807e6	+	9.15921e6i	−0.346734	+	0.600560i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−9.32969e6	+	1.61595e7i	−0.603625	+	1.04551i	0.388642	+	0.921389i	$0.372944\pi$
$751$	−9.32969e6	+	1.61595e7i	−0.603625	+	1.04551i	−0.992267	+	0.124121i	$0.960389\pi$
$752$		0			0
$753$	7.90612e6	+	1.36938e7i	0.508131	+	0.880109i
$754$		0			0
$755$	5.62222e6			0.358956
$756$		0			0
$757$	−2.56681e6			−0.162800			−0.0813999	−	0.996682i	$-0.525939\pi$
$757$	−2.56681e6			−0.162800			−0.0813999	+	0.996682i	$0.525939\pi$
$758$		0			0
$759$	−228160.	−	395185.i	−0.0143759	−	0.0248998i
$760$		0			0
$761$	1.29793e7	−	2.24808e7i	0.812436	−	1.40718i	−0.0987188	−	0.995115i	$-0.531474\pi$
$761$	1.29793e7	−	2.24808e7i	0.812436	−	1.40718i	0.911155	−	0.412065i	$-0.135192\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−6.95918e6	+	1.20537e7i	−0.429937	+	0.744672i
$766$		0			0
$767$	−4.03336e6	−	6.98599e6i	−0.247559	−	0.428785i
$768$		0			0
$769$	5.53267e6			0.337380			0.168690	−	0.985669i	$-0.446046\pi$
$769$	5.53267e6			0.337380			0.168690	+	0.985669i	$0.446046\pi$
$770$		0			0
$771$	−2.59580e6			−0.157266
$772$		0			0
$773$	−4.16470e6	−	7.21347e6i	−0.250689	−	0.434206i	0.713027	−	0.701137i	$-0.247324\pi$
$773$	−4.16470e6	−	7.21347e6i	−0.250689	−	0.434206i	−0.963716	+	0.266931i	$0.913990\pi$
$774$		0			0
$775$	6.73326e6	−	1.16624e7i	0.402691	−	0.697480i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.35245e7	−	2.34251e7i	0.798504	−	1.38305i
$780$		0			0
$781$	1.98152e6	+	3.43209e6i	0.116244	+	0.201341i
$782$		0			0
$783$	5.64504e6			0.329051
$784$		0			0
$785$	2.91798e7			1.69009

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:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	392.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-10.0000 - 17.3205i) q^{3} +(37.0000 - 64.0859i) q^{5} +(-78.5000 + 135.966i) q^{9} +O(q^{10})\)	\(q+(-10.0000 - 17.3205i) q^{3} +(37.0000 - 64.0859i) q^{5} +(-78.5000 + 135.966i) q^{9} +(-62.0000 - 107.387i) q^{11} +478.000 q^{13} -1480.00 q^{15} +(599.000 + 1037.50i) q^{17} +(-1522.00 + 2636.18i) q^{19} +(-92.0000 + 159.349i) q^{23} +(-1175.50 - 2036.03i) q^{25} -1720.00 q^{27} -3282.00 q^{29} +(2864.00 + 4960.59i) q^{31} +(-1240.00 + 2147.74i) q^{33} +(-5163.00 + 8942.58i) q^{37} +(-4780.00 - 8279.20i) q^{39} -8886.00 q^{41} -9188.00 q^{43} +(5809.00 + 10061.5i) q^{45} +(-11832.0 + 20493.6i) q^{47} +(11980.0 - 20750.0i) q^{51} +(-5843.00 - 10120.4i) q^{53} -9176.00 q^{55} +60880.0 q^{57} +(-8438.00 - 14615.0i) q^{59} +(9241.00 - 16005.9i) q^{61} +(17686.0 - 30633.1i) q^{65} +(7766.00 + 13451.1i) q^{67} +3680.00 q^{69} -31960.0 q^{71} +(2443.00 + 4231.40i) q^{73} +(-23510.0 + 40720.5i) q^{75} +(-22280.0 + 38590.1i) q^{79} +(36275.5 + 62831.0i) q^{81} +67364.0 q^{83} +88652.0 q^{85} +(32820.0 + 56845.9i) q^{87} +(-35997.0 + 62348.6i) q^{89} +(57280.0 - 99211.9i) q^{93} +(112628. + 195077. i) q^{95} +48866.0 q^{97} +19468.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 20 q^{3} + 74 q^{5} - 157 q^{9}+O(q^{10}) \) 2 * q - 20 * q^3 + 74 * q^5 - 157 * q^9	\( 2 q - 20 q^{3} + 74 q^{5} - 157 q^{9} - 124 q^{11} + 956 q^{13} - 2960 q^{15} + 1198 q^{17} - 3044 q^{19} - 184 q^{23} - 2351 q^{25} - 3440 q^{27} - 6564 q^{29} + 5728 q^{31} - 2480 q^{33} - 10326 q^{37} - 9560 q^{39} - 17772 q^{41} - 18376 q^{43} + 11618 q^{45} - 23664 q^{47} + 23960 q^{51} - 11686 q^{53} - 18352 q^{55} + 121760 q^{57} - 16876 q^{59} + 18482 q^{61} + 35372 q^{65} + 15532 q^{67} + 7360 q^{69} - 63920 q^{71} + 4886 q^{73} - 47020 q^{75} - 44560 q^{79} + 72551 q^{81} + 134728 q^{83} + 177304 q^{85} + 65640 q^{87} - 71994 q^{89} + 114560 q^{93} + 225256 q^{95} + 97732 q^{97} + 38936 q^{99}+O(q^{100}) \) 2 * q - 20 * q^3 + 74 * q^5 - 157 * q^9 - 124 * q^11 + 956 * q^13 - 2960 * q^15 + 1198 * q^17 - 3044 * q^19 - 184 * q^23 - 2351 * q^25 - 3440 * q^27 - 6564 * q^29 + 5728 * q^31 - 2480 * q^33 - 10326 * q^37 - 9560 * q^39 - 17772 * q^41 - 18376 * q^43 + 11618 * q^45 - 23664 * q^47 + 23960 * q^51 - 11686 * q^53 - 18352 * q^55 + 121760 * q^57 - 16876 * q^59 + 18482 * q^61 + 35372 * q^65 + 15532 * q^67 + 7360 * q^69 - 63920 * q^71 + 4886 * q^73 - 47020 * q^75 - 44560 * q^79 + 72551 * q^81 + 134728 * q^83 + 177304 * q^85 + 65640 * q^87 - 71994 * q^89 + 114560 * q^93 + 225256 * q^95 + 97732 * q^97 + 38936 * q^99

Introduction

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