LMFDB - Embedded newform 684.7.h.c.37.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,7,Mod(37,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("684.37");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	684.h (of order $2$, degree $1$, 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:	$157.356993196$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5090x^{6} + 8905881x^{4} + 5831691048x^{2} + 827887219200 $ x^8 + 5090x^6 + 8905881x^4 + 5831691048*x^2 + 827887219200
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{3}\cdot 5 $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.8
Root			$-40.3415i$ of defining polynomial
Character	$\chi$	$=$	684.37
Dual form			684.7.h.c.37.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+186.796 q^{5} +202.293 q^{7} +O(q^{10})$	$q+186.796 q^{5} +202.293 q^{7} +1591.27 q^{11} +143.277i q^{13} -4771.21 q^{17} +(-1010.48 - 6784.16i) q^{19} -7737.11 q^{23} +19267.6 q^{25} -8820.72i q^{29} -51550.3i q^{31} +37787.4 q^{35} -94640.6i q^{37} -78747.2i q^{41} -129047. q^{43} +97220.5 q^{47} -76726.6 q^{49} +46143.8i q^{53} +297243. q^{55} -61729.4i q^{59} -85634.0 q^{61} +26763.5i q^{65} -49044.4i q^{67} -408517. i q^{71} -130029. q^{73} +321903. q^{77} -106942. i q^{79} +12258.3 q^{83} -891242. q^{85} +561920. i q^{89} +28983.8i q^{91} +(-188753. - 1.26725e6i) q^{95} -851962. i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{5} + 362 q^{7}+O(q^{10}) $ 8 * q - 2 * q^5 + 362 * q^7	$ 8 q - 2 q^{5} + 362 q^{7} - 902 q^{11} - 1550 q^{17} + 6232 q^{19} + 18820 q^{23} - 12158 q^{25} + 101762 q^{35} - 335042 q^{43} + 570394 q^{47} + 448182 q^{49} + 1089198 q^{55} - 632014 q^{61} - 852938 q^{73} - 1850530 q^{77} - 441200 q^{83} - 1828374 q^{85} - 627950 q^{95}+O(q^{100}) $ 8 * q - 2 * q^5 + 362 * q^7 - 902 * q^11 - 1550 * q^17 + 6232 * q^19 + 18820 * q^23 - 12158 * q^25 + 101762 * q^35 - 335042 * q^43 + 570394 * q^47 + 448182 * q^49 + 1089198 * q^55 - 632014 * q^61 - 852938 * q^73 - 1850530 * q^77 - 441200 * q^83 - 1828374 * q^85 - 627950 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	186.796			1.49437			0.747183	−	0.664618i	$-0.231406\pi$
$5$	186.796			1.49437			0.747183	+	0.664618i	$0.231406\pi$
$6$		0			0
$7$	202.293			0.589775			0.294888	−	0.955532i	$-0.404718\pi$
$7$	202.293			0.589775			0.294888	+	0.955532i	$0.404718\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1591.27			1.19555			0.597773	−	0.801666i	$-0.296053\pi$
$11$	1591.27			1.19555			0.597773	+	0.801666i	$0.296053\pi$
$12$		0			0
$13$			143.277i			0.0652146i	0.999468	+	0.0326073i	$0.0103811\pi$
$13$			143.277i			0.0652146i	−0.999468	+	0.0326073i	$0.989619\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4771.21			−0.971141			−0.485570	−	0.874198i	$-0.661388\pi$
$17$	−4771.21			−0.971141			−0.485570	+	0.874198i	$0.661388\pi$
$18$		0			0
$19$	−1010.48	−	6784.16i	−0.147321	−	0.989089i
$19$	−1010.48	−	6784.16i	−0.147321	−	0.989089i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−7737.11			−0.635910			−0.317955	−	0.948106i	$-0.602996\pi$
$23$	−7737.11			−0.635910			−0.317955	+	0.948106i	$0.602996\pi$
$24$		0			0
$25$	19267.6			1.23313
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	8820.72i		−	0.361668i	−0.983514	−	0.180834i	$-0.942120\pi$
$29$		−	8820.72i		−	0.361668i	0.983514	−	0.180834i	$-0.0578796\pi$
$30$		0			0
$31$		−	51550.3i		−	1.73040i	−0.501428	−	0.865199i	$-0.667192\pi$
$31$		−	51550.3i		−	1.73040i	0.501428	−	0.865199i	$-0.332808\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	37787.4			0.881340
$36$		0			0
$37$		−	94640.6i		−	1.86841i	−0.356735	−	0.934206i	$-0.616110\pi$
$37$		−	94640.6i		−	1.86841i	0.356735	−	0.934206i	$-0.383890\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	78747.2i		−	1.14257i	−0.820751	−	0.571286i	$-0.806445\pi$
$41$		−	78747.2i		−	1.14257i	0.820751	−	0.571286i	$-0.193555\pi$
$42$		0			0
$43$	−129047.			−1.62308			−0.811542	−	0.584294i	$-0.801371\pi$
$43$	−129047.			−1.62308			−0.811542	+	0.584294i	$0.801371\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	97220.5			0.936406			0.468203	−	0.883621i	$-0.344902\pi$
$47$	97220.5			0.936406			0.468203	+	0.883621i	$0.344902\pi$
$48$		0			0
$49$	−76726.6			−0.652165
$50$		0			0
$51$		0			0
$52$		0			0
$53$			46143.8i			0.309946i	0.987919	+	0.154973i	$0.0495290\pi$
$53$			46143.8i			0.309946i	−0.987919	+	0.154973i	$0.950471\pi$
$54$		0			0
$55$	297243.			1.78658
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	61729.4i		−	0.300563i	−0.988643	−	0.150282i	$-0.951982\pi$
$59$		−	61729.4i		−	0.300563i	0.988643	−	0.150282i	$-0.0480181\pi$
$60$		0			0
$61$	−85634.0			−0.377274			−0.188637	−	0.982047i	$-0.560407\pi$
$61$	−85634.0			−0.377274			−0.188637	+	0.982047i	$0.560407\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			26763.5i			0.0974545i
$66$		0			0
$67$		−	49044.4i		−	0.163067i	−0.996671	−	0.0815334i	$-0.974018\pi$
$67$		−	49044.4i		−	0.163067i	0.996671	−	0.0815334i	$-0.0259817\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	408517.i		−	1.14139i	−0.821161	−	0.570696i	$-0.806673\pi$
$71$		−	408517.i		−	1.14139i	0.821161	−	0.570696i	$-0.193327\pi$
$72$		0			0
$73$	−130029.			−0.334250			−0.167125	−	0.985936i	$-0.553448\pi$
$73$	−130029.			−0.334250			−0.167125	+	0.985936i	$0.553448\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	321903.			0.705103
$78$		0			0
$79$		−	106942.i		−	0.216904i	−0.994102	−	0.108452i	$-0.965411\pi$
$79$		−	106942.i		−	0.216904i	0.994102	−	0.108452i	$-0.0345893\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	12258.3			0.0214386			0.0107193	−	0.999943i	$-0.496588\pi$
$83$	12258.3			0.0214386			0.0107193	+	0.999943i	$0.496588\pi$
$84$		0			0
$85$	−891242.			−1.45124
$86$		0			0
$87$		0			0
$88$		0			0
$89$			561920.i			0.797085i	0.917150	+	0.398542i	$0.130484\pi$
$89$			561920.i			0.797085i	−0.917150	+	0.398542i	$0.869516\pi$
$90$		0			0
$91$			28983.8i			0.0384620i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−188753.	−	1.26725e6i	−0.220152	−	1.47806i
$96$		0			0
$97$		−	851962.i		−	0.933480i	−0.884395	−	0.466740i	$-0.845428\pi$
$97$		−	851962.i		−	0.933480i	0.884395	−	0.466740i	$-0.154572\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.73735e6			1.68625			0.843127	−	0.537714i	$-0.180712\pi$
$101$	1.73735e6			1.68625			0.843127	+	0.537714i	$0.180712\pi$
$102$		0			0
$103$		−	1.34950e6i		−	1.23498i	−0.786577	−	0.617492i	$-0.788149\pi$
$103$		−	1.34950e6i		−	1.23498i	0.786577	−	0.617492i	$-0.211851\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1.32161e6i			1.07883i	0.842042	+	0.539413i	$0.181354\pi$
$107$			1.32161e6i			1.07883i	−0.842042	+	0.539413i	$0.818646\pi$
$108$		0			0
$109$			1.99638e6i			1.54157i	0.637093	+	0.770787i	$0.280137\pi$
$109$			1.99638e6i			1.54157i	−0.637093	+	0.770787i	$0.719863\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1.42114e6i			0.984920i	0.870335	+	0.492460i	$0.163902\pi$
$113$			1.42114e6i			0.984920i	−0.870335	+	0.492460i	$0.836098\pi$
$114$		0			0
$115$	−1.44526e6			−0.950282
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−965182.			−0.572754
$120$		0			0
$121$	760581.			0.429328
$122$		0			0
$123$		0			0
$124$		0			0
$125$	680431.			0.348381
$126$		0			0
$127$			3.62081e6i			1.76764i	0.467823	+	0.883822i	$0.345039\pi$
$127$			3.62081e6i			1.76764i	−0.467823	+	0.883822i	$0.654961\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.12259e6			1.83382			0.916909	−	0.399096i	$-0.130676\pi$
$131$	4.12259e6			1.83382			0.916909	+	0.399096i	$0.130676\pi$
$132$		0			0
$133$	−204412.	−	1.37239e6i	−0.0868864	−	0.583340i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.08355e6			−1.19920			−0.599598	−	0.800302i	$-0.704673\pi$
$137$	−3.08355e6			−1.19920			−0.599598	+	0.800302i	$0.704673\pi$
$138$		0			0
$139$	−1.59964e6			−0.595631			−0.297816	−	0.954623i	$-0.596258\pi$
$139$	−1.59964e6			−0.595631			−0.297816	+	0.954623i	$0.596258\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			227992.i			0.0779671i
$144$		0			0
$145$		−	1.64767e6i		−	0.540464i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.38080e6			−1.02202			−0.511011	−	0.859574i	$-0.670729\pi$
$149$	−3.38080e6			−1.02202			−0.511011	+	0.859574i	$0.670729\pi$
$150$		0			0
$151$			3.08302e6i			0.895458i	0.894169	+	0.447729i	$0.147767\pi$
$151$			3.08302e6i			0.895458i	−0.894169	+	0.447729i	$0.852233\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	9.62938e6i		−	2.58585i
$156$		0			0
$157$	3.24712e6			0.839072			0.419536	−	0.907739i	$-0.362193\pi$
$157$	3.24712e6			0.839072			0.419536	+	0.907739i	$0.362193\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.56516e6			−0.375044
$162$		0			0
$163$	−2.32461e6			−0.536769			−0.268385	−	0.963312i	$-0.586490\pi$
$163$	−2.32461e6			−0.536769			−0.268385	+	0.963312i	$0.586490\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			274073.i			0.0588460i	0.999567	+	0.0294230i	$0.00936698\pi$
$167$			274073.i			0.0588460i	−0.999567	+	0.0294230i	$0.990633\pi$
$168$		0			0
$169$	4.80628e6			0.995747
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	2.32755e6i		−	0.449532i	−0.974413	−	0.224766i	$-0.927838\pi$
$173$		−	2.32755e6i		−	0.449532i	0.974413	−	0.224766i	$-0.0721617\pi$
$174$		0			0
$175$	3.89771e6			0.727269
$176$		0			0
$177$		0			0
$178$		0			0
$179$			6.83222e6i			1.19125i	0.803263	+	0.595625i	$0.203095\pi$
$179$			6.83222e6i			1.19125i	−0.803263	+	0.595625i	$0.796905\pi$
$180$		0			0
$181$		−	9.40294e6i		−	1.58573i	−0.609400	−	0.792863i	$-0.708590\pi$
$181$		−	9.40294e6i		−	1.58573i	0.609400	−	0.792863i	$-0.291410\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	1.76785e7i		−	2.79209i
$186$		0			0
$187$	−7.59229e6			−1.16104
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.12644e7			1.61662			0.808309	−	0.588759i	$-0.200383\pi$
$191$	1.12644e7			1.61662			0.808309	+	0.588759i	$0.200383\pi$
$192$		0			0
$193$			1.26456e6i			0.175901i	0.996125	+	0.0879504i	$0.0280317\pi$
$193$			1.26456e6i			0.175901i	−0.996125	+	0.0879504i	$0.971968\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.92683e6			0.775218			0.387609	−	0.921824i	$-0.373301\pi$
$197$	5.92683e6			0.775218			0.387609	+	0.921824i	$0.373301\pi$
$198$		0			0
$199$	5.49979e6			0.697889			0.348945	−	0.937143i	$-0.386540\pi$
$199$	5.49979e6			0.697889			0.348945	+	0.937143i	$0.386540\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	1.78437e6i		−	0.213303i
$204$		0			0
$205$		−	1.47096e7i		−	1.70742i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.60794e6	−	1.07954e7i	−0.176129	−	1.18250i
$210$		0			0
$211$		−	1.20580e7i		−	1.28359i	−0.766876	−	0.641795i	$-0.778190\pi$
$211$		−	1.20580e7i		−	1.28359i	0.766876	−	0.641795i	$-0.221810\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.41053e7			−2.42548
$216$		0			0
$217$		−	1.04283e7i		−	1.02055i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	683603.i		−	0.0633326i
$222$		0			0
$223$			1.50611e7i			1.35813i	0.734077	+	0.679066i	$0.237615\pi$
$223$			1.50611e7i			1.35813i	−0.734077	+	0.679066i	$0.762385\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	1.46485e7i		−	1.25232i	−0.779694	−	0.626161i	$-0.784625\pi$
$227$		−	1.46485e7i		−	1.25232i	0.779694	−	0.626161i	$-0.215375\pi$
$228$		0			0
$229$	1.69472e7			1.41121			0.705603	−	0.708607i	$-0.250676\pi$
$229$	1.69472e7			1.41121			0.705603	+	0.708607i	$0.250676\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.81937e6			0.301942			0.150971	−	0.988538i	$-0.451760\pi$
$233$	3.81937e6			0.301942			0.150971	+	0.988538i	$0.451760\pi$
$234$		0			0
$235$	1.81604e7			1.39933
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.51010e7			−1.83864			−0.919322	−	0.393507i	$-0.871262\pi$
$239$	−2.51010e7			−1.83864			−0.919322	+	0.393507i	$0.871262\pi$
$240$		0			0
$241$		−	7.55161e6i		−	0.539496i	−0.962931	−	0.269748i	$-0.913060\pi$
$241$		−	7.55161e6i		−	0.539496i	0.962931	−	0.269748i	$-0.0869404\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.43322e7			−0.974574
$246$		0			0
$247$	972011.	−	144778.i	0.0645031	−	0.00960750i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.68658e7			1.69894			0.849471	−	0.527635i	$-0.176921\pi$
$251$	2.68658e7			1.69894			0.849471	+	0.527635i	$0.176921\pi$
$252$		0			0
$253$	−1.23118e7			−0.760259
$254$		0			0
$255$		0			0
$256$		0			0
$257$			1.72065e7i			1.01366i	0.862046	+	0.506831i	$0.169183\pi$
$257$			1.72065e7i			1.01366i	−0.862046	+	0.506831i	$0.830817\pi$
$258$		0			0
$259$		−	1.91451e7i		−	1.10194i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	7.35861e6			0.404510			0.202255	−	0.979333i	$-0.435173\pi$
$263$	7.35861e6			0.404510			0.202255	+	0.979333i	$0.435173\pi$
$264$		0			0
$265$			8.61947e6i			0.463173i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	2.68576e7i		−	1.37978i	−0.723913	−	0.689891i	$-0.757658\pi$
$269$		−	2.68576e7i		−	1.37978i	0.723913	−	0.689891i	$-0.242342\pi$
$270$		0			0
$271$	−1.84648e6			−0.0927762			−0.0463881	−	0.998923i	$-0.514771\pi$
$271$	−1.84648e6			−0.0927762			−0.0463881	+	0.998923i	$0.514771\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.06600e7			1.47426
$276$		0			0
$277$	1.00493e7			0.472819			0.236410	−	0.971653i	$-0.424029\pi$
$277$	1.00493e7			0.472819			0.236410	+	0.971653i	$0.424029\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	1.79351e7i		−	0.808321i	−0.914688	−	0.404160i	$-0.867564\pi$
$281$		−	1.79351e7i		−	0.808321i	0.914688	−	0.404160i	$-0.132436\pi$
$282$		0			0
$283$	−3.11035e7			−1.37230			−0.686151	−	0.727460i	$-0.740701\pi$
$283$	−3.11035e7			−1.37230			−0.686151	+	0.727460i	$0.740701\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	1.59300e7i		−	0.673860i
$288$		0			0
$289$	−1.37309e6			−0.0568860
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	2.91396e7i		−	1.15846i	−0.815165	−	0.579229i	$-0.803354\pi$
$293$		−	2.91396e7i		−	1.15846i	0.815165	−	0.579229i	$-0.196646\pi$
$294$		0			0
$295$		−	1.15308e7i		−	0.449152i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	1.10855e6i		−	0.0414706i
$300$		0			0
$301$	−2.61052e7			−0.957254
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.59961e7			−0.563785
$306$		0			0
$307$			4.02183e7i			1.38998i	0.719020	+	0.694989i	$0.244591\pi$
$307$			4.02183e7i			1.38998i	−0.719020	+	0.694989i	$0.755409\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.73795e7			0.577771			0.288885	−	0.957364i	$-0.406715\pi$
$311$	1.73795e7			0.577771			0.288885	+	0.957364i	$0.406715\pi$
$312$		0			0
$313$	3.25313e7			1.06089			0.530443	−	0.847721i	$-0.322026\pi$
$313$	3.25313e7			1.06089			0.530443	+	0.847721i	$0.322026\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	2.86504e7i		−	0.899400i	−0.893180	−	0.449700i	$-0.851531\pi$
$317$		−	2.86504e7i		−	0.899400i	0.893180	−	0.449700i	$-0.148469\pi$
$318$		0			0
$319$		−	1.40361e7i		−	0.432390i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.82120e6	+	3.23687e7i	0.143070	+	0.960544i
$324$		0			0
$325$			2.76060e6i			0.0804181i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.96670e7			0.552269
$330$		0			0
$331$			4.74259e7i			1.30777i	0.756594	+	0.653885i	$0.226862\pi$
$331$			4.74259e7i			1.30777i	−0.756594	+	0.653885i	$0.773138\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	9.16129e6i		−	0.243681i
$336$		0			0
$337$		−	2.20894e7i		−	0.577157i	−0.957456	−	0.288579i	$-0.906817\pi$
$337$		−	2.20894e7i		−	0.577157i	0.957456	−	0.288579i	$-0.0931827\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	8.20305e7i		−	2.06877i
$342$		0			0
$343$	−3.93208e7			−0.974406
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.73262e7			−0.654019			−0.327009	−	0.945021i	$-0.606041\pi$
$347$	−2.73262e7			−0.654019			−0.327009	+	0.945021i	$0.606041\pi$
$348$		0			0
$349$	4.12946e7			0.971442			0.485721	−	0.874114i	$-0.338557\pi$
$349$	4.12946e7			0.971442			0.485721	+	0.874114i	$0.338557\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	5.44872e7			1.23871			0.619356	−	0.785110i	$-0.287394\pi$
$353$	5.44872e7			1.23871			0.619356	+	0.785110i	$0.287394\pi$
$354$		0			0
$355$		−	7.63092e7i		−	1.70566i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.80050e7			0.389144			0.194572	−	0.980888i	$-0.437668\pi$
$359$	1.80050e7			0.389144			0.194572	+	0.980888i	$0.437668\pi$
$360$		0			0
$361$	−4.50038e7	+	1.37105e7i	−0.956593	+	0.291428i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.42888e7			−0.499491
$366$		0			0
$367$	−8.33600e7			−1.68640			−0.843198	−	0.537603i	$-0.819330\pi$
$367$	−8.33600e7			−1.68640			−0.843198	+	0.537603i	$0.819330\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			9.33456e6i			0.182798i
$372$		0			0
$373$		−	2.46066e7i		−	0.474159i	−0.971490	−	0.237080i	$-0.923810\pi$
$373$		−	2.46066e7i		−	0.474159i	0.971490	−	0.237080i	$-0.0761902\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.26380e6			0.0235860
$378$		0			0
$379$		−	1.80987e7i		−	0.332452i	−0.986088	−	0.166226i	$-0.946842\pi$
$379$		−	1.80987e7i		−	0.332452i	0.986088	−	0.166226i	$-0.0531581\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			4.68217e7i			0.833395i	0.909045	+	0.416697i	$0.136812\pi$
$383$			4.68217e7i			0.833395i	−0.909045	+	0.416697i	$0.863188\pi$
$384$		0			0
$385$	6.01300e7			1.05368
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.49752e7			0.424288			0.212144	−	0.977238i	$-0.431955\pi$
$389$	2.49752e7			0.424288			0.212144	+	0.977238i	$0.431955\pi$
$390$		0			0
$391$	3.69154e7			0.617558
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	1.99763e7i		−	0.324134i
$396$		0			0
$397$	1.03759e6			0.0165827			0.00829133	−	0.999966i	$-0.497361\pi$
$397$	1.03759e6			0.0165827			0.00829133	+	0.999966i	$0.497361\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			1.02883e8i			1.59555i	0.602953	+	0.797777i	$0.293991\pi$
$401$			1.02883e8i			1.59555i	−0.602953	+	0.797777i	$0.706009\pi$
$402$		0			0
$403$	7.38595e6			0.112847
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	1.50599e8i		−	2.23377i
$408$		0			0
$409$		−	7.78751e7i		−	1.13823i	−0.822259	−	0.569113i	$-0.807287\pi$
$409$		−	7.78751e7i		−	1.13823i	0.822259	−	0.569113i	$-0.192713\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	1.24874e7i		−	0.177265i
$414$		0			0
$415$	2.28980e6			0.0320371
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.20628e8			1.63986			0.819932	−	0.572461i	$-0.194011\pi$
$419$	1.20628e8			1.63986			0.819932	+	0.572461i	$0.194011\pi$
$420$		0			0
$421$		−	9.94059e7i		−	1.33219i	−0.745867	−	0.666095i	$-0.767965\pi$
$421$		−	9.94059e7i		−	1.33219i	0.745867	−	0.666095i	$-0.232035\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−9.19301e7			−1.19754
$426$		0			0
$427$	−1.73231e7			−0.222507
$428$		0			0
$429$		0			0
$430$		0			0
$431$			3.81811e7i			0.476888i	0.971156	+	0.238444i	$0.0766373\pi$
$431$			3.81811e7i			0.476888i	−0.971156	+	0.238444i	$0.923363\pi$
$432$		0			0
$433$			1.37273e8i			1.69091i	0.534043	+	0.845457i	$0.320672\pi$
$433$			1.37273e8i			1.69091i	−0.534043	+	0.845457i	$0.679328\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7.81817e6	+	5.24898e7i	0.0936830	+	0.628971i
$438$		0			0
$439$			9.17437e6i			0.108438i	0.998529	+	0.0542192i	$0.0172670\pi$
$439$			9.17437e6i			0.108438i	−0.998529	+	0.0542192i	$0.982733\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.79271e6			−0.0206204			−0.0103102	−	0.999947i	$-0.503282\pi$
$443$	−1.79271e6			−0.0206204			−0.0103102	+	0.999947i	$0.503282\pi$
$444$		0			0
$445$			1.04964e8i			1.19114i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	6.43497e6i		−	0.0710898i	−0.999368	−	0.0355449i	$-0.988683\pi$
$449$		−	6.43497e6i		−	0.0710898i	0.999368	−	0.0355449i	$-0.0113167\pi$
$450$		0			0
$451$		−	1.25308e8i		−	1.36600i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			5.41406e6i			0.0574763i
$456$		0			0
$457$	−1.28044e8			−1.34156			−0.670779	−	0.741658i	$-0.734040\pi$
$457$	−1.28044e8			−1.34156			−0.670779	+	0.741658i	$0.734040\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.10758e7			−0.317190			−0.158595	−	0.987344i	$-0.550696\pi$
$461$	−3.10758e7			−0.317190			−0.158595	+	0.987344i	$0.550696\pi$
$462$		0			0
$463$	8.91490e7			0.898201			0.449100	−	0.893481i	$-0.351745\pi$
$463$	8.91490e7			0.898201			0.449100	+	0.893481i	$0.351745\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.26320e7			0.222214			0.111107	−	0.993808i	$-0.464560\pi$
$467$	2.26320e7			0.222214			0.111107	+	0.993808i	$0.464560\pi$
$468$		0			0
$469$		−	9.92134e6i		−	0.0961727i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.05348e8			−1.94047
$474$		0			0
$475$	−1.94695e7	−	1.30715e8i	−0.181666	−	1.21967i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.34746e8			1.22605			0.613026	−	0.790062i	$-0.289952\pi$
$479$	1.34746e8			1.22605			0.613026	+	0.790062i	$0.289952\pi$
$480$		0			0
$481$	1.35598e7			0.121848
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	1.59143e8i		−	1.39496i
$486$		0			0
$487$		−	4.94767e7i		−	0.428365i	−0.976794	−	0.214182i	$-0.931291\pi$
$487$		−	4.94767e7i		−	0.428365i	0.976794	−	0.214182i	$-0.0687087\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.86984e8			−1.57965			−0.789825	−	0.613333i	$-0.789828\pi$
$491$	−1.86984e8			−1.57965			−0.789825	+	0.613333i	$0.789828\pi$
$492$		0			0
$493$			4.20855e7i			0.351230i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	8.26400e7i		−	0.673165i
$498$		0			0
$499$	−6.68560e6			−0.0538070			−0.0269035	−	0.999638i	$-0.508565\pi$
$499$	−6.68560e6			−0.0538070			−0.0269035	+	0.999638i	$0.508565\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.07046e8			0.841139			0.420570	−	0.907260i	$-0.361830\pi$
$503$	1.07046e8			0.841139			0.420570	+	0.907260i	$0.361830\pi$
$504$		0			0
$505$	3.24529e8			2.51988
$506$		0			0
$507$		0			0
$508$		0			0
$509$			8.78261e7i			0.665994i	0.942928	+	0.332997i	$0.108060\pi$
$509$			8.78261e7i			0.665994i	−0.942928	+	0.332997i	$0.891940\pi$
$510$		0			0
$511$	−2.63039e7			−0.197132
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	2.52081e8i		−	1.84552i
$516$		0			0
$517$	1.54704e8			1.11952
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	5.29466e7i		−	0.374391i	−0.982323	−	0.187195i	$-0.940060\pi$
$521$		−	5.29466e7i		−	0.374391i	0.982323	−	0.187195i	$-0.0599397\pi$
$522$		0			0
$523$			2.04408e8i			1.42887i	0.699701	+	0.714436i	$0.253317\pi$
$523$			2.04408e8i			1.42887i	−0.699701	+	0.714436i	$0.746683\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			2.45958e8i			1.68046i
$528$		0			0
$529$	−8.81730e7			−0.595619
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.12826e7			0.0745124
$534$		0			0
$535$			2.46871e8i			1.61216i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.22093e8			−0.779693
$540$		0			0
$541$	1.07904e7			0.0681469			0.0340735	−	0.999419i	$-0.489152\pi$
$541$	1.07904e7			0.0681469			0.0340735	+	0.999419i	$0.489152\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			3.72916e8i			2.30368i
$546$		0			0
$547$			5.89052e7i			0.359908i	0.983675	+	0.179954i	$0.0575949\pi$
$547$			5.89052e7i			0.359908i	−0.983675	+	0.179954i	$0.942405\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−5.98411e7	+	8.91313e6i	−0.357722	+	0.0532814i
$552$		0			0
$553$		−	2.16336e7i		−	0.127924i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.50709e7			−0.260814			−0.130407	−	0.991461i	$-0.541628\pi$
$557$	−4.50709e7			−0.260814			−0.130407	+	0.991461i	$0.541628\pi$
$558$		0			0
$559$		−	1.84893e7i		−	0.105849i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			2.83179e8i			1.58685i	0.608670	+	0.793424i	$0.291703\pi$
$563$			2.83179e8i			1.58685i	−0.608670	+	0.793424i	$0.708297\pi$
$564$		0			0
$565$			2.65463e8i			1.47183i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			2.97743e8i			1.61624i	0.589021	+	0.808118i	$0.299514\pi$
$569$			2.97743e8i			1.61624i	−0.589021	+	0.808118i	$0.700486\pi$
$570$		0			0
$571$	4.51806e7			0.242685			0.121343	−	0.992611i	$-0.461280\pi$
$571$	4.51806e7			0.242685			0.121343	+	0.992611i	$0.461280\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.49076e8			−0.784159
$576$		0			0
$577$	−1.15048e8			−0.598896			−0.299448	−	0.954113i	$-0.596802\pi$
$577$	−1.15048e8			−0.598896			−0.299448	+	0.954113i	$0.596802\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.47977e6			0.0126440
$582$		0			0
$583$			7.34273e7i			0.370554i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.59622e8			−0.789183			−0.394592	−	0.918857i	$-0.629114\pi$
$587$	−1.59622e8			−0.789183			−0.394592	+	0.918857i	$0.629114\pi$
$588$		0			0
$589$	−3.49726e8	+	5.20904e7i	−1.71152	+	0.254925i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6.60997e7			0.316983			0.158491	−	0.987360i	$-0.449337\pi$
$593$	6.60997e7			0.316983			0.158491	+	0.987360i	$0.449337\pi$
$594$		0			0
$595$	−1.80292e8			−0.855905
$596$		0			0
$597$		0			0
$598$		0			0
$599$			1.30989e8i			0.609475i	0.952436	+	0.304738i	$0.0985688\pi$
$599$			1.30989e8i			0.609475i	−0.952436	+	0.304738i	$0.901431\pi$
$600$		0			0
$601$			2.96588e8i			1.36625i	0.730303	+	0.683124i	$0.239379\pi$
$601$			2.96588e8i			1.36625i	−0.730303	+	0.683124i	$0.760621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.42073e8			0.641573
$606$		0			0
$607$			7.29083e7i			0.325995i	0.986626	+	0.162997i	$0.0521162\pi$
$607$			7.29083e7i			0.325995i	−0.986626	+	0.162997i	$0.947884\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			1.39294e7i			0.0610674i
$612$		0			0
$613$	3.92971e8			1.70600			0.853000	−	0.521910i	$-0.174780\pi$
$613$	3.92971e8			1.70600			0.853000	+	0.521910i	$0.174780\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.27576e8			0.968884			0.484442	−	0.874823i	$-0.339023\pi$
$617$	2.27576e8			0.968884			0.484442	+	0.874823i	$0.339023\pi$
$618$		0			0
$619$	−1.93252e8			−0.814804			−0.407402	−	0.913249i	$-0.633565\pi$
$619$	−1.93252e8			−0.814804			−0.407402	+	0.913249i	$0.633565\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			1.13672e8i			0.470101i
$624$		0			0
$625$	−1.73955e8			−0.712521
$626$		0			0
$627$		0			0
$628$		0			0
$629$			4.51551e8i			1.81449i
$630$		0			0
$631$	1.93900e7			0.0771773			0.0385887	−	0.999255i	$-0.487714\pi$
$631$	1.93900e7			0.0771773			0.0385887	+	0.999255i	$0.487714\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			6.76353e8i			2.64151i
$636$		0			0
$637$		−	1.09931e7i		−	0.0425307i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	3.28211e8i		−	1.24617i	−0.782152	−	0.623087i	$-0.785878\pi$
$641$		−	3.28211e8i		−	1.24617i	0.782152	−	0.623087i	$-0.214122\pi$
$642$		0			0
$643$	2.37495e7			0.0893349			0.0446675	−	0.999002i	$-0.485777\pi$
$643$	2.37495e7			0.0893349			0.0446675	+	0.999002i	$0.485777\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.17583e7			−0.0434143			−0.0217072	−	0.999764i	$-0.506910\pi$
$647$	−1.17583e7			−0.0434143			−0.0217072	+	0.999764i	$0.506910\pi$
$648$		0			0
$649$		−	9.82282e7i		−	0.359337i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.27271e8			−0.457079			−0.228539	−	0.973535i	$-0.573395\pi$
$653$	−1.27271e8			−0.457079			−0.228539	+	0.973535i	$0.573395\pi$
$654$		0			0
$655$	7.70082e8			2.74040
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	2.32452e8i		−	0.812227i	−0.913823	−	0.406114i	$-0.866884\pi$
$659$		−	2.32452e8i		−	0.812227i	0.913823	−	0.406114i	$-0.133116\pi$
$660$		0			0
$661$		−	3.14463e8i		−	1.08884i	−0.838812	−	0.544421i	$-0.816750\pi$
$661$		−	3.14463e8i		−	1.08884i	0.838812	−	0.544421i	$-0.183250\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.81833e7	−	2.56356e8i	−0.129840	−	0.871723i
$666$		0			0
$667$			6.82469e7i			0.229988i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.36267e8			−0.451048
$672$		0			0
$673$		−	2.76451e8i		−	0.906929i	−0.891274	−	0.453465i	$-0.850188\pi$
$673$		−	2.76451e8i		−	0.906929i	0.891274	−	0.453465i	$-0.149812\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	4.08200e8i		−	1.31555i	−0.753215	−	0.657775i	$-0.771498\pi$
$677$		−	4.08200e8i		−	1.31555i	0.753215	−	0.657775i	$-0.228502\pi$
$678$		0			0
$679$		−	1.72346e8i		−	0.550543i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			3.61070e8i			1.13326i	0.823973	+	0.566630i	$0.191753\pi$
$683$			3.61070e8i			1.13326i	−0.823973	+	0.566630i	$0.808247\pi$
$684$		0			0
$685$	−5.75995e8			−1.79204
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.61133e6			−0.0202130
$690$		0			0
$691$	3.47750e8			1.05398			0.526991	−	0.849871i	$-0.323320\pi$
$691$	3.47750e8			1.05398			0.526991	+	0.849871i	$0.323320\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.98806e8			−0.890091
$696$		0			0
$697$			3.75720e8i			1.10960i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	8.95408e7			0.259936			0.129968	−	0.991518i	$-0.458513\pi$
$701$	8.95408e7			0.259936			0.129968	+	0.991518i	$0.458513\pi$
$702$		0			0
$703$	−6.42057e8	+	9.56321e7i	−1.84802	+	0.275257i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.51453e8			0.994511
$708$		0			0
$709$	1.93298e8			0.542362			0.271181	−	0.962528i	$-0.412586\pi$
$709$	1.93298e8			0.542362			0.271181	+	0.962528i	$0.412586\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			3.98851e8i			1.10038i
$714$		0			0
$715$			4.25879e7i			0.116511i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.52698e8			0.948891			0.474445	−	0.880285i	$-0.342649\pi$
$719$	3.52698e8			0.948891			0.474445	+	0.880285i	$0.342649\pi$
$720$		0			0
$721$		−	2.72994e8i		−	0.728363i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	1.69954e8i		−	0.445983i
$726$		0			0
$727$	−2.02827e8			−0.527865			−0.263932	−	0.964541i	$-0.585020\pi$
$727$	−2.02827e8			−0.527865			−0.263932	+	0.964541i	$0.585020\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.15709e8			1.57624
$732$		0			0
$733$	−4.60785e8			−1.17000			−0.585001	−	0.811033i	$-0.698906\pi$
$733$	−4.60785e8			−1.17000			−0.585001	+	0.811033i	$0.698906\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	7.80430e7i		−	0.194954i
$738$		0			0
$739$	−2.61415e8			−0.647734			−0.323867	−	0.946103i	$-0.604983\pi$
$739$	−2.61415e8			−0.647734			−0.323867	+	0.946103i	$0.604983\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	3.25053e8i		−	0.792480i	−0.918147	−	0.396240i	$-0.870315\pi$
$743$		−	3.25053e8i		−	0.792480i	0.918147	−	0.396240i	$-0.129685\pi$
$744$		0			0
$745$	−6.31519e8			−1.52728
$746$		0			0
$747$		0			0
$748$		0			0
$749$			2.67352e8i			0.636264i
$750$		0			0
$751$			2.90899e8i			0.686787i	0.939192	+	0.343393i	$0.111576\pi$
$751$			2.90899e8i			0.686787i	−0.939192	+	0.343393i	$0.888424\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			5.75895e8i			1.33814i
$756$		0			0
$757$	7.60667e8			1.75350			0.876752	−	0.480943i	$-0.159706\pi$
$757$	7.60667e8			1.75350			0.876752	+	0.480943i	$0.159706\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	3.20532e8			0.727306			0.363653	−	0.931534i	$-0.381529\pi$
$761$	3.20532e8			0.727306			0.363653	+	0.931534i	$0.381529\pi$
$762$		0			0
$763$			4.03854e8i			0.909182i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.84438e6			0.0196011
$768$		0			0
$769$	4.20733e8			0.925183			0.462592	−	0.886571i	$-0.346920\pi$
$769$	4.20733e8			0.925183			0.462592	+	0.886571i	$0.346920\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	6.21759e8i		−	1.34612i	−0.739588	−	0.673060i	$-0.764980\pi$
$773$		−	6.21759e8i		−	1.34612i	0.739588	−	0.673060i	$-0.235020\pi$
$774$		0			0
$775$		−	9.93253e8i		−	2.13381i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.34233e8	+	7.95722e7i	−1.13010	+	0.168325i
$780$		0			0
$781$		−	6.50061e8i		−	1.36459i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6.06548e8			1.25388
$786$		0			0
$787$			1.97860e7i			0.0405913i	0.999794	+	0.0202956i	$0.00646075\pi$
$787$			1.97860e7i			0.0405913i	−0.999794	+	0.0202956i	$0.993539\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			2.87486e8i			0.580881i
$792$		0			0
$793$		−	1.22693e7i		−	0.0246038i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	5.31780e8i		−	1.05041i	−0.850977	−	0.525203i	$-0.823990\pi$
$797$		−	5.31780e8i		−	1.05041i	0.850977	−	0.525203i	$-0.176010\pi$
$798$		0			0
$799$	−4.63860e8			−0.909382
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.06911e8			−0.399610
$804$		0			0
$805$	−2.92366e8			−0.560452
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−8.55853e8			−1.61642			−0.808209	−	0.588896i	$-0.799563\pi$
$809$	−8.55853e8			−1.61642			−0.808209	+	0.588896i	$0.799563\pi$
$810$		0			0
$811$			6.99910e8i			1.31214i	0.754701	+	0.656069i	$0.227782\pi$
$811$			6.99910e8i			1.31214i	−0.754701	+	0.656069i	$0.772218\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.34228e8			−0.802130
$816$		0			0
$817$	1.30398e8	+	8.75472e8i	0.239115	+	1.60537i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.84681e8			−1.05655			−0.528274	−	0.849074i	$-0.677161\pi$
$821$	−5.84681e8			−1.05655			−0.528274	+	0.849074i	$0.677161\pi$
$822$		0			0
$823$	4.37701e8			0.785196			0.392598	−	0.919710i	$-0.371576\pi$
$823$	4.37701e8			0.785196			0.392598	+	0.919710i	$0.371576\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1.36193e8i			0.240790i	0.992726	+	0.120395i	$0.0384162\pi$
$827$			1.36193e8i			0.240790i	−0.992726	+	0.120395i	$0.961584\pi$
$828$		0			0
$829$			7.38167e7i			0.129566i	0.997899	+	0.0647830i	$0.0206355\pi$
$829$			7.38167e7i			0.129566i	−0.997899	+	0.0647830i	$0.979364\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.66079e8			0.633344
$834$		0			0
$835$			5.11957e7i			0.0879375i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			4.83661e8i			0.818946i	0.912322	+	0.409473i	$0.134287\pi$
$839$			4.83661e8i			0.818946i	−0.912322	+	0.409473i	$0.865713\pi$
$840$		0			0
$841$	5.17018e8			0.869196
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8.97793e8			1.48801
$846$		0			0
$847$	1.53860e8			0.253207
$848$		0			0
$849$		0			0
$850$		0			0
$851$			7.32245e8i			1.18814i
$852$		0			0
$853$	−1.52710e8			−0.246048			−0.123024	−	0.992404i	$-0.539259\pi$
$853$	−1.52710e8			−0.246048			−0.123024	+	0.992404i	$0.539259\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1.05617e9i			1.67799i	0.544138	+	0.838996i	$0.316857\pi$
$857$			1.05617e9i			1.67799i	−0.544138	+	0.838996i	$0.683143\pi$
$858$		0			0
$859$	3.53777e8			0.558150			0.279075	−	0.960269i	$-0.409972\pi$
$859$	3.53777e8			0.558150			0.279075	+	0.960269i	$0.409972\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1.01186e8i			0.157431i	0.996897	+	0.0787153i	$0.0250818\pi$
$863$			1.01186e8i			0.157431i	−0.996897	+	0.0787153i	$0.974918\pi$
$864$		0			0
$865$		−	4.34776e8i		−	0.671765i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1.70174e8i		−	0.259318i
$870$		0			0
$871$	7.02692e6			0.0106343
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.37646e8			0.205466
$876$		0			0
$877$		−	1.10266e9i		−	1.63472i	−0.576127	−	0.817360i	$-0.695437\pi$
$877$		−	1.10266e9i		−	1.63472i	0.576127	−	0.817360i	$-0.304563\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−9.09645e7			−0.133028			−0.0665142	−	0.997785i	$-0.521188\pi$
$881$	−9.09645e7			−0.133028			−0.0665142	+	0.997785i	$0.521188\pi$
$882$		0			0
$883$	−3.92658e8			−0.570338			−0.285169	−	0.958477i	$-0.592050\pi$
$883$	−3.92658e8			−0.570338			−0.285169	+	0.958477i	$0.592050\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	3.46247e8i		−	0.496152i	−0.968741	−	0.248076i	$-0.920202\pi$
$887$		−	3.46247e8i		−	0.496152i	0.968741	−	0.248076i	$-0.0797983\pi$
$888$		0			0
$889$			7.32465e8i			1.04251i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−9.82390e7	−	6.59559e8i	−0.137952	−	0.926189i
$894$		0			0
$895$			1.27623e9i			1.78016i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4.54711e8			−0.625830
$900$		0			0
$901$		−	2.20162e8i		−	0.301001i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	1.75643e9i		−	2.36965i
$906$		0			0
$907$			4.69987e8i			0.629888i	0.949110	+	0.314944i	$0.101986\pi$
$907$			4.69987e8i			0.629888i	−0.949110	+	0.314944i	$0.898014\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			2.84138e8i			0.375816i	0.982187	+	0.187908i	$0.0601706\pi$
$911$			2.84138e8i			0.375816i	−0.982187	+	0.187908i	$0.939829\pi$
$912$		0			0
$913$	1.95063e7			0.0256308
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8.33971e8			1.08154
$918$		0			0
$919$	1.06131e9			1.36740			0.683700	−	0.729763i	$-0.260370\pi$
$919$	1.06131e9			1.36740			0.683700	+	0.729763i	$0.260370\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5.85309e7			0.0744355
$924$		0			0
$925$		−	1.82350e9i		−	2.30399i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−5.01662e8			−0.625697			−0.312848	−	0.949803i	$-0.601283\pi$
$929$	−5.01662e8			−0.625697			−0.312848	+	0.949803i	$0.601283\pi$
$930$		0			0
$931$	7.75304e7	+	5.20526e8i	0.0960778	+	0.645049i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.41821e9			−1.73502
$936$		0			0
$937$	−6.16919e8			−0.749910			−0.374955	−	0.927043i	$-0.622342\pi$
$937$	−6.16919e8			−0.749910			−0.374955	+	0.927043i	$0.622342\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1.00994e9i		−	1.21206i	−0.795441	−	0.606031i	$-0.792761\pi$
$941$		−	1.00994e9i		−	1.21206i	0.795441	−	0.606031i	$-0.207239\pi$
$942$		0			0
$943$			6.09276e8i			0.726572i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−6.91940e8			−0.814739			−0.407369	−	0.913263i	$-0.633554\pi$
$947$	−6.91940e8			−0.814739			−0.407369	+	0.913263i	$0.633554\pi$
$948$		0			0
$949$		−	1.86301e7i		−	0.0217980i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			4.95481e8i			0.572464i	0.958160	+	0.286232i	$0.0924028\pi$
$953$			4.95481e8i			0.572464i	−0.958160	+	0.286232i	$0.907597\pi$
$954$		0			0
$955$	2.10414e9			2.41582
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.23781e8			−0.707256
$960$		0			0
$961$	−1.76993e9			−1.99428
$962$		0			0
$963$		0			0
$964$		0			0
$965$			2.36215e8i			0.262860i
$966$		0			0
$967$	−1.38648e9			−1.53332			−0.766661	−	0.642052i	$-0.778083\pi$
$967$	−1.38648e9			−1.53332			−0.766661	+	0.642052i	$0.778083\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	3.47483e8i		−	0.379556i	−0.981827	−	0.189778i	$-0.939223\pi$
$971$		−	3.47483e8i		−	0.379556i	0.981827	−	0.189778i	$-0.0607769\pi$
$972$		0			0
$973$	−3.23595e8			−0.351288
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.77225e9i		−	1.90038i	−0.311664	−	0.950192i	$-0.600886\pi$
$977$		−	1.77225e9i		−	1.90038i	0.311664	−	0.950192i	$-0.399114\pi$
$978$		0			0
$979$			8.94167e8i			0.952951i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			2.01843e8i			0.212497i	0.994340	+	0.106248i	$0.0338839\pi$
$983$			2.01843e8i			0.212497i	−0.994340	+	0.106248i	$0.966116\pi$
$984$		0			0
$985$	1.10711e9			1.15846
$986$		0			0
$987$		0			0
$988$		0			0
$989$	9.98447e8			1.03213
$990$		0			0
$991$			7.84564e8i			0.806135i	0.915170	+	0.403067i	$0.132056\pi$
$991$			7.84564e8i			0.806135i	−0.915170	+	0.403067i	$0.867944\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.02734e9			1.04290
$996$		0			0
$997$	1.23743e9			1.24863			0.624316	−	0.781172i	$-0.285378\pi$
$997$	1.23743e9			1.24863			0.624316	+	0.781172i	$0.285378\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.7.h.c.37.8		8
3.2	odd	2		76.7.c.b.37.7	yes	8
12.11	even	2		304.7.e.c.113.2		8
19.18	odd	2	inner	684.7.h.c.37.7		8
57.56	even	2		76.7.c.b.37.2	✓	8
228.227	odd	2		304.7.e.c.113.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.7.c.b.37.2	✓	8	57.56	even	2
76.7.c.b.37.7	yes	8	3.2	odd	2
304.7.e.c.113.2		8	12.11	even	2
304.7.e.c.113.7		8	228.227	odd	2
684.7.h.c.37.7		8	19.18	odd	2	inner
684.7.h.c.37.8		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	684.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+186.796 q^{5} +202.293 q^{7} +O(q^{10})\)	\(q+186.796 q^{5} +202.293 q^{7} +1591.27 q^{11} +143.277i q^{13} -4771.21 q^{17} +(-1010.48 - 6784.16i) q^{19} -7737.11 q^{23} +19267.6 q^{25} -8820.72i q^{29} -51550.3i q^{31} +37787.4 q^{35} -94640.6i q^{37} -78747.2i q^{41} -129047. q^{43} +97220.5 q^{47} -76726.6 q^{49} +46143.8i q^{53} +297243. q^{55} -61729.4i q^{59} -85634.0 q^{61} +26763.5i q^{65} -49044.4i q^{67} -408517. i q^{71} -130029. q^{73} +321903. q^{77} -106942. i q^{79} +12258.3 q^{83} -891242. q^{85} +561920. i q^{89} +28983.8i q^{91} +(-188753. - 1.26725e6i) q^{95} -851962. i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{5} + 362 q^{7}+O(q^{10}) \) 8 * q - 2 * q^5 + 362 * q^7	\( 8 q - 2 q^{5} + 362 q^{7} - 902 q^{11} - 1550 q^{17} + 6232 q^{19} + 18820 q^{23} - 12158 q^{25} + 101762 q^{35} - 335042 q^{43} + 570394 q^{47} + 448182 q^{49} + 1089198 q^{55} - 632014 q^{61} - 852938 q^{73} - 1850530 q^{77} - 441200 q^{83} - 1828374 q^{85} - 627950 q^{95}+O(q^{100}) \) 8 * q - 2 * q^5 + 362 * q^7 - 902 * q^11 - 1550 * q^17 + 6232 * q^19 + 18820 * q^23 - 12158 * q^25 + 101762 * q^35 - 335042 * q^43 + 570394 * q^47 + 448182 * q^49 + 1089198 * q^55 - 632014 * q^61 - 852938 * q^73 - 1850530 * q^77 - 441200 * q^83 - 1828374 * q^85 - 627950 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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