LMFDB - Embedded newform 1120.3.o.a.911.36

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,3,Mod(911,1120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1120.911");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1120.o (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:	$30.5177896084$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			911.36
Character	$\chi$	$=$	1120.911
Dual form			1120.3.o.a.911.35

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.90764 q^{3} -2.23607i q^{5} -2.64575i q^{7} -0.545650 q^{9} +O(q^{10})$	$q+2.90764 q^{3} -2.23607i q^{5} -2.64575i q^{7} -0.545650 q^{9} -3.72992 q^{11} +11.4554i q^{13} -6.50167i q^{15} -21.0578 q^{17} -34.1011 q^{19} -7.69288i q^{21} +13.2839i q^{23} -5.00000 q^{25} -27.7553 q^{27} -16.0429i q^{29} +34.3776i q^{31} -10.8452 q^{33} -5.91608 q^{35} -55.8425i q^{37} +33.3080i q^{39} +60.4834 q^{41} -43.8665 q^{43} +1.22011i q^{45} +54.2795i q^{47} -7.00000 q^{49} -61.2286 q^{51} +2.77880i q^{53} +8.34035i q^{55} -99.1537 q^{57} -47.3973 q^{59} +46.1788i q^{61} +1.44365i q^{63} +25.6149 q^{65} +58.3870 q^{67} +38.6248i q^{69} +61.6792i q^{71} +50.9950 q^{73} -14.5382 q^{75} +9.86843i q^{77} -44.8179i q^{79} -75.7914 q^{81} -146.534 q^{83} +47.0868i q^{85} -46.6468i q^{87} -64.3103 q^{89} +30.3080 q^{91} +99.9574i q^{93} +76.2524i q^{95} +33.0183 q^{97} +2.03523 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 144 q^{9}+O(q^{10}) $ 48 * q + 144 * q^9	$ 48 q + 144 q^{9} - 32 q^{11} + 64 q^{19} - 240 q^{25} - 192 q^{27} - 32 q^{33} - 96 q^{41} + 96 q^{43} - 336 q^{49} - 192 q^{51} + 160 q^{57} + 576 q^{59} + 160 q^{67} + 656 q^{81} - 224 q^{99}+O(q^{100}) $ 48 * q + 144 * q^9 - 32 * q^11 + 64 * q^19 - 240 * q^25 - 192 * q^27 - 32 * q^33 - 96 * q^41 + 96 * q^43 - 336 * q^49 - 192 * q^51 + 160 * q^57 + 576 * q^59 + 160 * q^67 + 656 * q^81 - 224 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	2.90764	0.969212	0.484606	−	0.874733i	$-0.338963\pi$
$3$	2.90764	0.969212	0.484606	+	0.874733i	$0.338963\pi$
$4$	0	0
$5$	− 2.23607i	− 0.447214i
$5$	− 2.23607i	− 0.447214i
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	−0.545650	−0.0606278
$10$	0	0
$11$	−3.72992	−0.339083	−0.169542	−	0.985523i	$-0.554229\pi$
$11$	−3.72992	−0.339083	−0.169542	+	0.985523i	$0.554229\pi$
$12$	0	0
$13$	11.4554i	0.881181i	0.897708	+	0.440590i	$0.145231\pi$
$13$	11.4554i	0.881181i	−0.897708	+	0.440590i	$0.854769\pi$
$14$	0	0
$15$	− 6.50167i	− 0.433445i
$16$	0	0
$17$	−21.0578	−1.23870	−0.619348	−	0.785116i	$-0.712603\pi$
$17$	−21.0578	−1.23870	−0.619348	+	0.785116i	$0.712603\pi$
$18$	0	0
$19$	−34.1011	−1.79480	−0.897398	−	0.441222i	$-0.854545\pi$
$19$	−34.1011	−1.79480	−0.897398	+	0.441222i	$0.854545\pi$
$20$	0	0
$21$	− 7.69288i	− 0.366328i
$22$	0	0
$23$	13.2839i	0.577561i	0.957395	+	0.288781i	$0.0932499\pi$
$23$	13.2839i	0.577561i	−0.957395	+	0.288781i	$0.906750\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	−27.7553	−1.02797
$28$	0	0
$29$	− 16.0429i	− 0.553202i	−0.960985	−	0.276601i	$-0.910792\pi$
$29$	− 16.0429i	− 0.553202i	0.960985	−	0.276601i	$-0.0892081\pi$
$30$	0	0
$31$	34.3776i	1.10895i	0.832199	+	0.554477i	$0.187082\pi$
$31$	34.3776i	1.10895i	−0.832199	+	0.554477i	$0.812918\pi$
$32$	0	0
$33$	−10.8452	−0.328644
$34$	0	0
$35$	−5.91608	−0.169031
$36$	0	0
$37$	− 55.8425i	− 1.50926i	−0.656152	−	0.754628i	$-0.727817\pi$
$37$	− 55.8425i	− 1.50926i	0.656152	−	0.754628i	$-0.272183\pi$
$38$	0	0
$39$	33.3080i	0.854051i
$40$	0	0
$41$	60.4834	1.47520	0.737602	−	0.675235i	$-0.235958\pi$
$41$	60.4834	1.47520	0.737602	+	0.675235i	$0.235958\pi$
$42$	0	0
$43$	−43.8665	−1.02015	−0.510075	−	0.860130i	$-0.670382\pi$
$43$	−43.8665	−1.02015	−0.510075	+	0.860130i	$0.670382\pi$
$44$	0	0
$45$	1.22011i	0.0271136i
$46$	0	0
$47$	54.2795i	1.15488i	0.816432	+	0.577442i	$0.195949\pi$
$47$	54.2795i	1.15488i	−0.816432	+	0.577442i	$0.804051\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	−61.2286	−1.20056
$52$	0	0
$53$	2.77880i	0.0524302i	0.999656	+	0.0262151i	$0.00834549\pi$
$53$	2.77880i	0.0524302i	−0.999656	+	0.0262151i	$0.991655\pi$
$54$	0	0
$55$	8.34035i	0.151643i
$56$	0	0
$57$	−99.1537	−1.73954
$58$	0	0
$59$	−47.3973	−0.803344	−0.401672	−	0.915784i	$-0.631571\pi$
$59$	−47.3973	−0.803344	−0.401672	+	0.915784i	$0.631571\pi$
$60$	0	0
$61$	46.1788i	0.757030i	0.925595	+	0.378515i	$0.123565\pi$
$61$	46.1788i	0.757030i	−0.925595	+	0.378515i	$0.876435\pi$
$62$	0	0
$63$	1.44365i	0.0229152i
$64$	0	0
$65$	25.6149	0.394076
$66$	0	0
$67$	58.3870	0.871448	0.435724	−	0.900080i	$-0.356492\pi$
$67$	58.3870	0.871448	0.435724	+	0.900080i	$0.356492\pi$
$68$	0	0
$69$	38.6248i	0.559780i
$70$	0	0
$71$	61.6792i	0.868721i	0.900739	+	0.434361i	$0.143026\pi$
$71$	61.6792i	0.868721i	−0.900739	+	0.434361i	$0.856974\pi$
$72$	0	0
$73$	50.9950	0.698562	0.349281	−	0.937018i	$-0.386426\pi$
$73$	50.9950	0.698562	0.349281	+	0.937018i	$0.386426\pi$
$74$	0	0
$75$	−14.5382	−0.193842
$76$	0	0
$77$	9.86843i	0.128161i
$78$	0	0
$79$	− 44.8179i	− 0.567315i	−0.958926	−	0.283658i	$-0.908452\pi$
$79$	− 44.8179i	− 0.567315i	0.958926	−	0.283658i	$-0.0915479\pi$
$80$	0	0
$81$	−75.7914	−0.935696
$82$	0	0
$83$	−146.534	−1.76547	−0.882734	−	0.469874i	$-0.844300\pi$
$83$	−146.534	−1.76547	−0.882734	+	0.469874i	$0.844300\pi$
$84$	0	0
$85$	47.0868i	0.553962i
$86$	0	0
$87$	− 46.6468i	− 0.536170i
$88$	0	0
$89$	−64.3103	−0.722587	−0.361294	−	0.932452i	$-0.617665\pi$
$89$	−64.3103	−0.722587	−0.361294	+	0.932452i	$0.617665\pi$
$90$	0	0
$91$	30.3080	0.333055
$92$	0	0
$93$	99.9574i	1.07481i
$94$	0	0
$95$	76.2524i	0.802657i
$96$	0	0
$97$	33.0183	0.340395	0.170197	−	0.985410i	$-0.445559\pi$
$97$	33.0183	0.340395	0.170197	+	0.985410i	$0.445559\pi$
$98$	0	0
$99$	2.03523	0.0205579
$100$	0	0
$101$	− 63.9852i	− 0.633516i	−0.948506	−	0.316758i	$-0.897406\pi$
$101$	− 63.9852i	− 0.633516i	0.948506	−	0.316758i	$-0.102594\pi$
$102$	0	0
$103$	− 172.641i	− 1.67613i	−0.545571	−	0.838065i	$-0.683687\pi$
$103$	− 172.641i	− 1.67613i	0.545571	−	0.838065i	$-0.316313\pi$
$104$	0	0
$105$	−17.2018	−0.163827
$106$	0	0
$107$	27.6358	0.258278	0.129139	−	0.991626i	$-0.458779\pi$
$107$	27.6358	0.258278	0.129139	+	0.991626i	$0.458779\pi$
$108$	0	0
$109$	− 40.5524i	− 0.372040i	−0.982546	−	0.186020i	$-0.940441\pi$
$109$	− 40.5524i	− 0.372040i	0.982546	−	0.186020i	$-0.0595590\pi$
$110$	0	0
$111$	− 162.370i	− 1.46279i
$112$	0	0
$113$	111.982	0.990993	0.495496	−	0.868610i	$-0.334986\pi$
$113$	111.982	0.990993	0.495496	+	0.868610i	$0.334986\pi$
$114$	0	0
$115$	29.7037	0.258293
$116$	0	0
$117$	− 6.25062i	− 0.0534241i
$118$	0	0
$119$	55.7138i	0.468183i
$120$	0	0
$121$	−107.088	−0.885023
$122$	0	0
$123$	175.864	1.42979
$124$	0	0
$125$	11.1803i	0.0894427i
$126$	0	0
$127$	− 6.61733i	− 0.0521049i	−0.999661	−	0.0260525i	$-0.991706\pi$
$127$	− 6.61733i	− 0.0521049i	0.999661	−	0.0260525i	$-0.00829369\pi$
$128$	0	0
$129$	−127.548	−0.988742
$130$	0	0
$131$	−43.4236	−0.331478	−0.165739	−	0.986170i	$-0.553001\pi$
$131$	−43.4236	−0.331478	−0.165739	+	0.986170i	$0.553001\pi$
$132$	0	0
$133$	90.2231i	0.678369i
$134$	0	0
$135$	62.0627i	0.459724i
$136$	0	0
$137$	−190.903	−1.39346	−0.696728	−	0.717336i	$-0.745361\pi$
$137$	−190.903	−1.39346	−0.696728	+	0.717336i	$0.745361\pi$
$138$	0	0
$139$	49.7124	0.357643	0.178821	−	0.983882i	$-0.442772\pi$
$139$	49.7124	0.357643	0.178821	+	0.983882i	$0.442772\pi$
$140$	0	0
$141$	157.825i	1.11933i
$142$	0	0
$143$	− 42.7275i	− 0.298794i
$144$	0	0
$145$	−35.8729	−0.247400
$146$	0	0
$147$	−20.3535	−0.138459
$148$	0	0
$149$	− 210.354i	− 1.41177i	−0.708325	−	0.705886i	$-0.750549\pi$
$149$	− 210.354i	− 1.41177i	0.708325	−	0.705886i	$-0.249451\pi$
$150$	0	0
$151$	249.557i	1.65270i	0.563160	+	0.826348i	$0.309585\pi$
$151$	249.557i	1.65270i	−0.563160	+	0.826348i	$0.690415\pi$
$152$	0	0
$153$	11.4902	0.0750995
$154$	0	0
$155$	76.8705	0.495939
$156$	0	0
$157$	− 286.796i	− 1.82673i	−0.407145	−	0.913363i	$-0.633476\pi$
$157$	− 286.796i	− 1.82673i	0.407145	−	0.913363i	$-0.366524\pi$
$158$	0	0
$159$	8.07975i	0.0508160i
$160$	0	0
$161$	35.1459	0.218298
$162$	0	0
$163$	−305.698	−1.87545	−0.937724	−	0.347382i	$-0.887071\pi$
$163$	−305.698	−1.87545	−0.937724	+	0.347382i	$0.887071\pi$
$164$	0	0
$165$	24.2507i	0.146974i
$166$	0	0
$167$	− 132.454i	− 0.793136i	−0.918005	−	0.396568i	$-0.870201\pi$
$167$	− 132.454i	− 0.793136i	0.918005	−	0.396568i	$-0.129799\pi$
$168$	0	0
$169$	37.7749	0.223520
$170$	0	0
$171$	18.6073	0.108815
$172$	0	0
$173$	314.979i	1.82069i	0.413854	+	0.910343i	$0.364182\pi$
$173$	314.979i	1.82069i	−0.413854	+	0.910343i	$0.635818\pi$
$174$	0	0
$175$	13.2288i	0.0755929i
$176$	0	0
$177$	−137.814	−0.778611
$178$	0	0
$179$	45.7818	0.255764	0.127882	−	0.991789i	$-0.459182\pi$
$179$	45.7818	0.255764	0.127882	+	0.991789i	$0.459182\pi$
$180$	0	0
$181$	193.246i	1.06766i	0.845592	+	0.533829i	$0.179247\pi$
$181$	193.246i	1.06766i	−0.845592	+	0.533829i	$0.820753\pi$
$182$	0	0
$183$	134.271i	0.733722i
$184$	0	0
$185$	−124.868	−0.674960
$186$	0	0
$187$	78.5440	0.420021
$188$	0	0
$189$	73.4336i	0.388537i
$190$	0	0
$191$	− 236.860i	− 1.24010i	−0.784561	−	0.620052i	$-0.787112\pi$
$191$	− 236.860i	− 1.24010i	0.784561	−	0.620052i	$-0.212888\pi$
$192$	0	0
$193$	169.608	0.878800	0.439400	−	0.898292i	$-0.355191\pi$
$193$	169.608	0.878800	0.439400	+	0.898292i	$0.355191\pi$
$194$	0	0
$195$	74.4789	0.381943
$196$	0	0
$197$	− 101.250i	− 0.513957i	−0.966417	−	0.256978i	$-0.917273\pi$
$197$	− 101.250i	− 0.513957i	0.966417	−	0.256978i	$-0.0827269\pi$
$198$	0	0
$199$	− 133.520i	− 0.670953i	−0.942049	−	0.335476i	$-0.891103\pi$
$199$	− 133.520i	− 0.670953i	0.942049	−	0.335476i	$-0.108897\pi$
$200$	0	0
$201$	169.768	0.844618
$202$	0	0
$203$	−42.4454	−0.209091
$204$	0	0
$205$	− 135.245i	− 0.659732i
$206$	0	0
$207$	− 7.24837i	− 0.0350163i
$208$	0	0
$209$	127.194	0.608585
$210$	0	0
$211$	−101.033	−0.478830	−0.239415	−	0.970917i	$-0.576956\pi$
$211$	−101.033	−0.478830	−0.239415	+	0.970917i	$0.576956\pi$
$212$	0	0
$213$	179.341i	0.841975i
$214$	0	0
$215$	98.0884i	0.456225i
$216$	0	0
$217$	90.9545	0.419145
$218$	0	0
$219$	148.275	0.677055
$220$	0	0
$221$	− 241.225i	− 1.09152i
$222$	0	0
$223$	202.640i	0.908700i	0.890823	+	0.454350i	$0.150129\pi$
$223$	202.640i	0.908700i	−0.890823	+	0.454350i	$0.849871\pi$
$224$	0	0
$225$	2.72825	0.0121256
$226$	0	0
$227$	−267.140	−1.17683	−0.588414	−	0.808560i	$-0.700248\pi$
$227$	−267.140	−1.17683	−0.588414	+	0.808560i	$0.700248\pi$
$228$	0	0
$229$	− 206.318i	− 0.900952i	−0.892788	−	0.450476i	$-0.851254\pi$
$229$	− 206.318i	− 0.900952i	0.892788	−	0.450476i	$-0.148746\pi$
$230$	0	0
$231$	28.6938i	0.124216i
$232$	0	0
$233$	−211.775	−0.908904	−0.454452	−	0.890771i	$-0.650165\pi$
$233$	−211.775	−0.908904	−0.454452	+	0.890771i	$0.650165\pi$
$234$	0	0
$235$	121.373	0.516480
$236$	0	0
$237$	− 130.314i	− 0.549849i
$238$	0	0
$239$	289.274i	1.21035i	0.796092	+	0.605175i	$0.206897\pi$
$239$	289.274i	1.21035i	−0.796092	+	0.605175i	$0.793103\pi$
$240$	0	0
$241$	4.15604	0.0172450	0.00862249	−	0.999963i	$-0.497255\pi$
$241$	4.15604	0.0172450	0.00862249	+	0.999963i	$0.497255\pi$
$242$	0	0
$243$	29.4237	0.121085
$244$	0	0
$245$	15.6525i	0.0638877i
$246$	0	0
$247$	− 390.640i	− 1.58154i
$248$	0	0
$249$	−426.067	−1.71111
$250$	0	0
$251$	44.7337	0.178222	0.0891110	−	0.996022i	$-0.471597\pi$
$251$	44.7337	0.178222	0.0891110	+	0.996022i	$0.471597\pi$
$252$	0	0
$253$	− 49.5479i	− 0.195841i
$254$	0	0
$255$	136.911i	0.536907i
$256$	0	0
$257$	237.117	0.922635	0.461318	−	0.887235i	$-0.347377\pi$
$257$	237.117	0.922635	0.461318	+	0.887235i	$0.347377\pi$
$258$	0	0
$259$	−147.745	−0.570445
$260$	0	0
$261$	8.75379i	0.0335394i
$262$	0	0
$263$	250.284i	0.951649i	0.879540	+	0.475824i	$0.157850\pi$
$263$	250.284i	0.951649i	−0.879540	+	0.475824i	$0.842150\pi$
$264$	0	0
$265$	6.21359	0.0234475
$266$	0	0
$267$	−186.991	−0.700341
$268$	0	0
$269$	168.319i	0.625722i	0.949799	+	0.312861i	$0.101287\pi$
$269$	168.319i	0.625722i	−0.949799	+	0.312861i	$0.898713\pi$
$270$	0	0
$271$	− 238.366i	− 0.879580i	−0.898101	−	0.439790i	$-0.855053\pi$
$271$	− 238.366i	− 0.879580i	0.898101	−	0.439790i	$-0.144947\pi$
$272$	0	0
$273$	88.1247	0.322801
$274$	0	0
$275$	18.6496	0.0678167
$276$	0	0
$277$	481.489i	1.73823i	0.494612	+	0.869114i	$0.335310\pi$
$277$	481.489i	1.73823i	−0.494612	+	0.869114i	$0.664690\pi$
$278$	0	0
$279$	− 18.7581i	− 0.0672334i
$280$	0	0
$281$	−378.549	−1.34715	−0.673575	−	0.739119i	$-0.735242\pi$
$281$	−378.549	−1.34715	−0.673575	+	0.739119i	$0.735242\pi$
$282$	0	0
$283$	304.185	1.07486	0.537429	−	0.843309i	$-0.319396\pi$
$283$	304.185	1.07486	0.537429	+	0.843309i	$0.319396\pi$
$284$	0	0
$285$	221.714i	0.777945i
$286$	0	0
$287$	− 160.024i	− 0.557575i
$288$	0	0
$289$	154.433	0.534370
$290$	0	0
$291$	96.0052	0.329915
$292$	0	0
$293$	262.660i	0.896449i	0.893921	+	0.448224i	$0.147943\pi$
$293$	262.660i	0.896449i	−0.893921	+	0.448224i	$0.852057\pi$
$294$	0	0
$295$	105.984i	0.359266i
$296$	0	0
$297$	103.525	0.348569
$298$	0	0
$299$	−152.172	−0.508936
$300$	0	0
$301$	116.060i	0.385581i
$302$	0	0
$303$	− 186.046i	− 0.614012i
$304$	0	0
$305$	103.259	0.338554
$306$	0	0
$307$	−80.8679	−0.263413	−0.131707	−	0.991289i	$-0.542046\pi$
$307$	−80.8679	−0.263413	−0.131707	+	0.991289i	$0.542046\pi$
$308$	0	0
$309$	− 501.978i	− 1.62453i
$310$	0	0
$311$	− 112.701i	− 0.362384i	−0.983448	−	0.181192i	$-0.942004\pi$
$311$	− 112.701i	− 0.362384i	0.983448	−	0.181192i	$-0.0579956\pi$
$312$	0	0
$313$	−9.04215	−0.0288887	−0.0144443	−	0.999896i	$-0.504598\pi$
$313$	−9.04215	−0.0288887	−0.0144443	+	0.999896i	$0.504598\pi$
$314$	0	0
$315$	3.22811	0.0102480
$316$	0	0
$317$	− 94.9343i	− 0.299477i	−0.988726	−	0.149739i	$-0.952157\pi$
$317$	− 94.9343i	− 0.299477i	0.988726	−	0.149739i	$-0.0478432\pi$
$318$	0	0
$319$	59.8385i	0.187582i
$320$	0	0
$321$	80.3548	0.250326
$322$	0	0
$323$	718.096	2.22321
$324$	0	0
$325$	− 57.2768i	− 0.176236i
$326$	0	0
$327$	− 117.912i	− 0.360586i
$328$	0	0
$329$	143.610	0.436505
$330$	0	0
$331$	41.9978	0.126882	0.0634408	−	0.997986i	$-0.479793\pi$
$331$	41.9978	0.126882	0.0634408	+	0.997986i	$0.479793\pi$
$332$	0	0
$333$	30.4705i	0.0915029i
$334$	0	0
$335$	− 130.557i	− 0.389723i
$336$	0	0
$337$	−86.3522	−0.256238	−0.128119	−	0.991759i	$-0.540894\pi$
$337$	−86.3522	−0.256238	−0.128119	+	0.991759i	$0.540894\pi$
$338$	0	0
$339$	325.603	0.960482
$340$	0	0
$341$	− 128.225i	− 0.376028i
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	86.3677	0.250341
$346$	0	0
$347$	96.7644	0.278860	0.139430	−	0.990232i	$-0.455473\pi$
$347$	96.7644	0.278860	0.139430	+	0.990232i	$0.455473\pi$
$348$	0	0
$349$	− 201.576i	− 0.577581i	−0.957392	−	0.288791i	$-0.906747\pi$
$349$	− 201.576i	− 0.577581i	0.957392	−	0.288791i	$-0.0932532\pi$
$350$	0	0
$351$	− 317.946i	− 0.905830i
$352$	0	0
$353$	346.313	0.981058	0.490529	−	0.871425i	$-0.336804\pi$
$353$	346.313	0.981058	0.490529	+	0.871425i	$0.336804\pi$
$354$	0	0
$355$	137.919	0.388504
$356$	0	0
$357$	161.996i	0.453769i
$358$	0	0
$359$	− 468.773i	− 1.30577i	−0.757455	−	0.652887i	$-0.773557\pi$
$359$	− 468.773i	− 1.30577i	0.757455	−	0.652887i	$-0.226443\pi$
$360$	0	0
$361$	801.886	2.22129
$362$	0	0
$363$	−311.372	−0.857775
$364$	0	0
$365$	− 114.028i	− 0.312406i
$366$	0	0
$367$	443.307i	1.20792i	0.797015	+	0.603960i	$0.206411\pi$
$367$	443.307i	1.20792i	−0.797015	+	0.603960i	$0.793589\pi$
$368$	0	0
$369$	−33.0028	−0.0894384
$370$	0	0
$371$	7.35202	0.0198168
$372$	0	0
$373$	− 522.058i	− 1.39962i	−0.714329	−	0.699810i	$-0.753268\pi$
$373$	− 522.058i	− 1.39962i	0.714329	−	0.699810i	$-0.246732\pi$
$374$	0	0
$375$	32.5084i	0.0866890i
$376$	0	0
$377$	183.777	0.487471
$378$	0	0
$379$	677.341	1.78718	0.893590	−	0.448885i	$-0.148179\pi$
$379$	677.341	1.78718	0.893590	+	0.448885i	$0.148179\pi$
$380$	0	0
$381$	− 19.2408i	− 0.0505007i
$382$	0	0
$383$	395.089i	1.03156i	0.856720	+	0.515782i	$0.172499\pi$
$383$	395.089i	1.03156i	−0.856720	+	0.515782i	$0.827501\pi$
$384$	0	0
$385$	22.0665	0.0573155
$386$	0	0
$387$	23.9357	0.0618495
$388$	0	0
$389$	− 47.9128i	− 0.123169i	−0.998102	−	0.0615845i	$-0.980385\pi$
$389$	− 47.9128i	− 0.123169i	0.998102	−	0.0615845i	$-0.0196154\pi$
$390$	0	0
$391$	− 279.731i	− 0.715424i
$392$	0	0
$393$	−126.260	−0.321273
$394$	0	0
$395$	−100.216	−0.253711
$396$	0	0
$397$	− 212.853i	− 0.536155i	−0.963397	−	0.268077i	$-0.913612\pi$
$397$	− 212.853i	− 0.536155i	0.963397	−	0.268077i	$-0.0863883\pi$
$398$	0	0
$399$	262.336i	0.657484i
$400$	0	0
$401$	−154.562	−0.385441	−0.192720	−	0.981254i	$-0.561731\pi$
$401$	−154.562	−0.385441	−0.192720	+	0.981254i	$0.561731\pi$
$402$	0	0
$403$	−393.807	−0.977188
$404$	0	0
$405$	169.475i	0.418456i
$406$	0	0
$407$	208.288i	0.511764i
$408$	0	0
$409$	293.278	0.717062	0.358531	−	0.933518i	$-0.383278\pi$
$409$	293.278	0.717062	0.358531	+	0.933518i	$0.383278\pi$
$410$	0	0
$411$	−555.078	−1.35055
$412$	0	0
$413$	125.401i	0.303636i
$414$	0	0
$415$	327.660i	0.789541i
$416$	0	0
$417$	144.545	0.346632
$418$	0	0
$419$	−143.577	−0.342665	−0.171333	−	0.985213i	$-0.554807\pi$
$419$	−143.577	−0.342665	−0.171333	+	0.985213i	$0.554807\pi$
$420$	0	0
$421$	− 670.441i	− 1.59250i	−0.604969	−	0.796249i	$-0.706815\pi$
$421$	− 670.441i	− 1.59250i	0.604969	−	0.796249i	$-0.293185\pi$
$422$	0	0
$423$	− 29.6176i	− 0.0700180i
$424$	0	0
$425$	105.289	0.247739
$426$	0	0
$427$	122.178	0.286130
$428$	0	0
$429$	− 124.236i	− 0.289594i
$430$	0	0
$431$	− 166.175i	− 0.385557i	−0.981242	−	0.192779i	$-0.938250\pi$
$431$	− 166.175i	− 0.385557i	0.981242	−	0.192779i	$-0.0617499\pi$
$432$	0	0
$433$	526.096	1.21500	0.607502	−	0.794318i	$-0.292172\pi$
$433$	526.096	1.21500	0.607502	+	0.794318i	$0.292172\pi$
$434$	0	0
$435$	−104.305	−0.239783
$436$	0	0
$437$	− 452.996i	− 1.03660i
$438$	0	0
$439$	364.132i	0.829458i	0.909945	+	0.414729i	$0.136124\pi$
$439$	364.132i	0.829458i	−0.909945	+	0.414729i	$0.863876\pi$
$440$	0	0
$441$	3.81955	0.00866112
$442$	0	0
$443$	165.964	0.374637	0.187319	−	0.982299i	$-0.440020\pi$
$443$	165.964	0.374637	0.187319	+	0.982299i	$0.440020\pi$
$444$	0	0
$445$	143.802i	0.323151i
$446$	0	0
$447$	− 611.633i	− 1.36831i
$448$	0	0
$449$	125.174	0.278785	0.139392	−	0.990237i	$-0.455485\pi$
$449$	125.174	0.278785	0.139392	+	0.990237i	$0.455485\pi$
$450$	0	0
$451$	−225.598	−0.500217
$452$	0	0
$453$	725.621i	1.60181i
$454$	0	0
$455$	− 67.7708i	− 0.148947i
$456$	0	0
$457$	507.504	1.11051	0.555256	−	0.831680i	$-0.312620\pi$
$457$	507.504	1.11051	0.555256	+	0.831680i	$0.312620\pi$
$458$	0	0
$459$	584.466	1.27335
$460$	0	0
$461$	− 631.319i	− 1.36946i	−0.728799	−	0.684728i	$-0.759921\pi$
$461$	− 631.319i	− 1.36946i	0.728799	−	0.684728i	$-0.240079\pi$
$462$	0	0
$463$	314.239i	0.678702i	0.940660	+	0.339351i	$0.110207\pi$
$463$	314.239i	0.678702i	−0.940660	+	0.339351i	$0.889793\pi$
$464$	0	0
$465$	223.512	0.480670
$466$	0	0
$467$	246.565	0.527977	0.263988	−	0.964526i	$-0.414962\pi$
$467$	246.565	0.527977	0.263988	+	0.964526i	$0.414962\pi$
$468$	0	0
$469$	− 154.477i	− 0.329376i
$470$	0	0
$471$	− 833.899i	− 1.77049i
$472$	0	0
$473$	163.618	0.345916
$474$	0	0
$475$	170.506	0.358959
$476$	0	0
$477$	− 1.51625i	− 0.00317873i
$478$	0	0
$479$	353.677i	0.738365i	0.929357	+	0.369182i	$0.120362\pi$
$479$	353.677i	0.738365i	−0.929357	+	0.369182i	$0.879638\pi$
$480$	0	0
$481$	639.695	1.32993
$482$	0	0
$483$	102.192	0.211577
$484$	0	0
$485$	− 73.8311i	− 0.152229i
$486$	0	0
$487$	− 80.5255i	− 0.165350i	−0.996577	−	0.0826751i	$-0.973654\pi$
$487$	− 80.5255i	− 0.165350i	0.996577	−	0.0826751i	$-0.0263464\pi$
$488$	0	0
$489$	−888.858	−1.81771
$490$	0	0
$491$	527.421	1.07418	0.537089	−	0.843526i	$-0.319524\pi$
$491$	527.421	1.07418	0.537089	+	0.843526i	$0.319524\pi$
$492$	0	0
$493$	337.828i	0.685250i
$494$	0	0
$495$	− 4.55091i	− 0.00919376i
$496$	0	0
$497$	163.188	0.328346
$498$	0	0
$499$	−421.653	−0.844996	−0.422498	−	0.906364i	$-0.638847\pi$
$499$	−421.653	−0.844996	−0.422498	+	0.906364i	$0.638847\pi$
$500$	0	0
$501$	− 385.127i	− 0.768717i
$502$	0	0
$503$	370.294i	0.736171i	0.929792	+	0.368086i	$0.119987\pi$
$503$	370.294i	0.736171i	−0.929792	+	0.368086i	$0.880013\pi$
$504$	0	0
$505$	−143.075	−0.283317
$506$	0	0
$507$	109.836	0.216639
$508$	0	0
$509$	612.684i	1.20370i	0.798609	+	0.601850i	$0.205570\pi$
$509$	612.684i	1.20370i	−0.798609	+	0.601850i	$0.794430\pi$
$510$	0	0
$511$	− 134.920i	− 0.264032i
$512$	0	0
$513$	946.486	1.84500
$514$	0	0
$515$	−386.038	−0.749588
$516$	0	0
$517$	− 202.458i	− 0.391602i
$518$	0	0
$519$	915.844i	1.76463i
$520$	0	0
$521$	−1026.96	−1.97114	−0.985568	−	0.169280i	$-0.945856\pi$
$521$	−1026.96	−1.97114	−0.985568	+	0.169280i	$0.945856\pi$
$522$	0	0
$523$	622.716	1.19066	0.595331	−	0.803481i	$-0.297021\pi$
$523$	622.716	1.19066	0.595331	+	0.803481i	$0.297021\pi$
$524$	0	0
$525$	38.4644i	0.0732656i
$526$	0	0
$527$	− 723.917i	− 1.37366i
$528$	0	0
$529$	352.538	0.666423
$530$	0	0
$531$	25.8623	0.0487050
$532$	0	0
$533$	692.858i	1.29992i
$534$	0	0
$535$	− 61.7955i	− 0.115506i
$536$	0	0
$537$	133.117	0.247890
$538$	0	0
$539$	26.1094	0.0484405
$540$	0	0
$541$	− 554.563i	− 1.02507i	−0.858666	−	0.512535i	$-0.828707\pi$
$541$	− 554.563i	− 1.02507i	0.858666	−	0.512535i	$-0.171293\pi$
$542$	0	0
$543$	561.890i	1.03479i
$544$	0	0
$545$	−90.6779	−0.166381
$546$	0	0
$547$	−316.468	−0.578552	−0.289276	−	0.957246i	$-0.593414\pi$
$547$	−316.468	−0.578552	−0.289276	+	0.957246i	$0.593414\pi$
$548$	0	0
$549$	− 25.1975i	− 0.0458971i
$550$	0	0
$551$	547.080i	0.992885i
$552$	0	0
$553$	−118.577	−0.214425
$554$	0	0
$555$	−363.070	−0.654180
$556$	0	0
$557$	764.633i	1.37277i	0.727238	+	0.686385i	$0.240804\pi$
$557$	764.633i	1.37277i	−0.727238	+	0.686385i	$0.759196\pi$
$558$	0	0
$559$	− 502.506i	− 0.898937i
$560$	0	0
$561$	228.377	0.407090
$562$	0	0
$563$	16.0309	0.0284741	0.0142370	−	0.999899i	$-0.495468\pi$
$563$	16.0309	0.0284741	0.0142370	+	0.999899i	$0.495468\pi$
$564$	0	0
$565$	− 250.400i	− 0.443185i
$566$	0	0
$567$	200.525i	0.353660i
$568$	0	0
$569$	−1025.19	−1.80175	−0.900874	−	0.434080i	$-0.857073\pi$
$569$	−1025.19	−1.80175	−0.900874	+	0.434080i	$0.857073\pi$
$570$	0	0
$571$	−816.632	−1.43018	−0.715090	−	0.699033i	$-0.753614\pi$
$571$	−816.632	−1.43018	−0.715090	+	0.699033i	$0.753614\pi$
$572$	0	0
$573$	− 688.702i	− 1.20192i
$574$	0	0
$575$	− 66.4196i	− 0.115512i
$576$	0	0
$577$	291.896	0.505885	0.252943	−	0.967481i	$-0.418602\pi$
$577$	291.896	0.505885	0.252943	+	0.967481i	$0.418602\pi$
$578$	0	0
$579$	493.159	0.851743
$580$	0	0
$581$	387.692i	0.667284i
$582$	0	0
$583$	− 10.3647i	− 0.0177782i
$584$	0	0
$585$	−13.9768	−0.0238920
$586$	0	0
$587$	−690.362	−1.17609	−0.588043	−	0.808830i	$-0.700101\pi$
$587$	−690.362	−1.17609	−0.588043	+	0.808830i	$0.700101\pi$
$588$	0	0
$589$	− 1172.31i	− 1.99034i
$590$	0	0
$591$	− 294.397i	− 0.498133i
$592$	0	0
$593$	−697.130	−1.17560	−0.587799	−	0.809007i	$-0.700005\pi$
$593$	−697.130	−1.17560	−0.587799	+	0.809007i	$0.700005\pi$
$594$	0	0
$595$	124.580	0.209378
$596$	0	0
$597$	− 388.226i	− 0.650296i
$598$	0	0
$599$	− 596.289i	− 0.995475i	−0.867328	−	0.497737i	$-0.834164\pi$
$599$	− 596.289i	− 0.995475i	0.867328	−	0.497737i	$-0.165836\pi$
$600$	0	0
$601$	−704.967	−1.17299	−0.586495	−	0.809953i	$-0.699493\pi$
$601$	−704.967	−1.17299	−0.586495	+	0.809953i	$0.699493\pi$
$602$	0	0
$603$	−31.8589	−0.0528340
$604$	0	0
$605$	239.455i	0.395794i
$606$	0	0
$607$	234.801i	0.386823i	0.981118	+	0.193411i	$0.0619552\pi$
$607$	234.801i	0.386823i	−0.981118	+	0.193411i	$0.938045\pi$
$608$	0	0
$609$	−123.416	−0.202653
$610$	0	0
$611$	−621.791	−1.01766
$612$	0	0
$613$	791.727i	1.29156i	0.763523	+	0.645781i	$0.223468\pi$
$613$	791.727i	1.29156i	−0.763523	+	0.645781i	$0.776532\pi$
$614$	0	0
$615$	− 393.243i	− 0.639420i
$616$	0	0
$617$	−1079.35	−1.74935	−0.874675	−	0.484710i	$-0.838925\pi$
$617$	−1079.35	−1.74935	−0.874675	+	0.484710i	$0.838925\pi$
$618$	0	0
$619$	744.432	1.20264	0.601318	−	0.799010i	$-0.294642\pi$
$619$	744.432	1.20264	0.601318	+	0.799010i	$0.294642\pi$
$620$	0	0
$621$	− 368.699i	− 0.593718i
$622$	0	0
$623$	170.149i	0.273112i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	369.835	0.589848
$628$	0	0
$629$	1175.92i	1.86951i
$630$	0	0
$631$	− 560.358i	− 0.888047i	−0.896015	−	0.444023i	$-0.853551\pi$
$631$	− 560.358i	− 0.888047i	0.896015	−	0.444023i	$-0.146449\pi$
$632$	0	0
$633$	−293.768	−0.464088
$634$	0	0
$635$	−14.7968	−0.0233020
$636$	0	0
$637$	− 80.1875i	− 0.125883i
$638$	0	0
$639$	− 33.6553i	− 0.0526687i
$640$	0	0
$641$	−376.327	−0.587094	−0.293547	−	0.955945i	$-0.594836\pi$
$641$	−376.327	−0.587094	−0.293547	+	0.955945i	$0.594836\pi$
$642$	0	0
$643$	619.457	0.963385	0.481693	−	0.876340i	$-0.340022\pi$
$643$	619.457	0.963385	0.481693	+	0.876340i	$0.340022\pi$
$644$	0	0
$645$	285.205i	0.442179i
$646$	0	0
$647$	1213.64i	1.87579i	0.346919	+	0.937895i	$0.387228\pi$
$647$	1213.64i	1.87579i	−0.346919	+	0.937895i	$0.612772\pi$
$648$	0	0
$649$	176.788	0.272401
$650$	0	0
$651$	264.462	0.406240
$652$	0	0
$653$	902.189i	1.38161i	0.723043	+	0.690803i	$0.242743\pi$
$653$	902.189i	1.38161i	−0.723043	+	0.690803i	$0.757257\pi$
$654$	0	0
$655$	97.0982i	0.148242i
$656$	0	0
$657$	−27.8255	−0.0423523
$658$	0	0
$659$	−971.542	−1.47427	−0.737134	−	0.675747i	$-0.763821\pi$
$659$	−971.542	−1.47427	−0.737134	+	0.675747i	$0.763821\pi$
$660$	0	0
$661$	483.212i	0.731032i	0.930805	+	0.365516i	$0.119108\pi$
$661$	483.212i	0.731032i	−0.930805	+	0.365516i	$0.880892\pi$
$662$	0	0
$663$	− 701.395i	− 1.05791i
$664$	0	0
$665$	201.745	0.303376
$666$	0	0
$667$	213.112	0.319508
$668$	0	0
$669$	589.204i	0.880723i
$670$	0	0
$671$	− 172.243i	− 0.256696i
$672$	0	0
$673$	−289.578	−0.430279	−0.215140	−	0.976583i	$-0.569021\pi$
$673$	−289.578	−0.430279	−0.215140	+	0.976583i	$0.569021\pi$
$674$	0	0
$675$	138.776	0.205595
$676$	0	0
$677$	559.941i	0.827091i	0.910483	+	0.413546i	$0.135710\pi$
$677$	559.941i	0.827091i	−0.910483	+	0.413546i	$0.864290\pi$
$678$	0	0
$679$	− 87.3582i	− 0.128657i
$680$	0	0
$681$	−776.746	−1.14060
$682$	0	0
$683$	−249.736	−0.365646	−0.182823	−	0.983146i	$-0.558524\pi$
$683$	−249.736	−0.365646	−0.182823	+	0.983146i	$0.558524\pi$
$684$	0	0
$685$	426.873i	0.623172i
$686$	0	0
$687$	− 599.898i	− 0.873214i
$688$	0	0
$689$	−31.8322	−0.0462005
$690$	0	0
$691$	−542.895	−0.785666	−0.392833	−	0.919610i	$-0.628505\pi$
$691$	−542.895	−0.785666	−0.392833	+	0.919610i	$0.628505\pi$
$692$	0	0
$693$	− 5.38471i	− 0.00777015i
$694$	0	0
$695$	− 111.160i	− 0.159943i
$696$	0	0
$697$	−1273.65	−1.82733
$698$	0	0
$699$	−615.764	−0.880921
$700$	0	0
$701$	− 756.696i	− 1.07945i	−0.841841	−	0.539726i	$-0.818528\pi$
$701$	− 756.696i	− 1.07945i	0.841841	−	0.539726i	$-0.181472\pi$
$702$	0	0
$703$	1904.29i	2.70881i
$704$	0	0
$705$	352.908	0.500578
$706$	0	0
$707$	−169.289	−0.239447
$708$	0	0
$709$	− 180.770i	− 0.254964i	−0.991841	−	0.127482i	$-0.959310\pi$
$709$	− 180.770i	− 0.254964i	0.991841	−	0.127482i	$-0.0406896\pi$
$710$	0	0
$711$	24.4549i	0.0343951i
$712$	0	0
$713$	−456.668	−0.640489
$714$	0	0
$715$	−95.5416	−0.133625
$716$	0	0
$717$	841.103i	1.17309i
$718$	0	0
$719$	− 517.175i	− 0.719297i	−0.933088	−	0.359649i	$-0.882897\pi$
$719$	− 517.175i	− 0.719297i	0.933088	−	0.359649i	$-0.117103\pi$
$720$	0	0
$721$	−456.766	−0.633517
$722$	0	0
$723$	12.0842	0.0167140
$724$	0	0
$725$	80.2143i	0.110640i
$726$	0	0
$727$	817.989i	1.12516i	0.826744	+	0.562579i	$0.190191\pi$
$727$	817.989i	1.12516i	−0.826744	+	0.562579i	$0.809809\pi$
$728$	0	0
$729$	767.676	1.05305
$730$	0	0
$731$	923.733	1.26366
$732$	0	0
$733$	1076.44i	1.46854i	0.678859	+	0.734269i	$0.262475\pi$
$733$	1076.44i	1.46854i	−0.678859	+	0.734269i	$0.737525\pi$
$734$	0	0
$735$	45.5117i	0.0619207i
$736$	0	0
$737$	−217.779	−0.295493
$738$	0	0
$739$	−402.196	−0.544243	−0.272122	−	0.962263i	$-0.587725\pi$
$739$	−402.196	−0.544243	−0.272122	+	0.962263i	$0.587725\pi$
$740$	0	0
$741$	− 1135.84i	− 1.53285i
$742$	0	0
$743$	− 1118.95i	− 1.50598i	−0.658030	−	0.752992i	$-0.728610\pi$
$743$	− 1118.95i	− 1.50598i	0.658030	−	0.752992i	$-0.271390\pi$
$744$	0	0
$745$	−470.366	−0.631364
$746$	0	0
$747$	79.9562	0.107036
$748$	0	0
$749$	− 73.1174i	− 0.0976200i
$750$	0	0
$751$	− 1150.33i	− 1.53173i	−0.643001	−	0.765865i	$-0.722311\pi$
$751$	− 1150.33i	− 1.53173i	0.643001	−	0.765865i	$-0.277689\pi$
$752$	0	0
$753$	130.069	0.172735
$754$	0	0
$755$	558.026	0.739108
$756$	0	0
$757$	652.519i	0.861980i	0.902357	+	0.430990i	$0.141835\pi$
$757$	652.519i	0.861980i	−0.902357	+	0.430990i	$0.858165\pi$
$758$	0	0
$759$	− 144.067i	− 0.189812i
$760$	0	0
$761$	−1017.69	−1.33731	−0.668653	−	0.743574i	$-0.733129\pi$
$761$	−1017.69	−1.33731	−0.668653	+	0.743574i	$0.733129\pi$
$762$	0	0
$763$	−107.292	−0.140618
$764$	0	0
$765$	− 25.6929i	− 0.0335855i
$766$	0	0
$767$	− 542.953i	− 0.707891i
$768$	0	0
$769$	−631.277	−0.820906	−0.410453	−	0.911882i	$-0.634629\pi$
$769$	−631.277	−0.820906	−0.410453	+	0.911882i	$0.634629\pi$
$770$	0	0
$771$	689.451	0.894229
$772$	0	0
$773$	408.165i	0.528028i	0.964519	+	0.264014i	$0.0850465\pi$
$773$	408.165i	0.528028i	−0.964519	+	0.264014i	$0.914954\pi$
$774$	0	0
$775$	− 171.888i	− 0.221791i
$776$	0	0
$777$	−429.590	−0.552883
$778$	0	0
$779$	−2062.55	−2.64769
$780$	0	0
$781$	− 230.058i	− 0.294569i
$782$	0	0
$783$	445.274i	0.568677i
$784$	0	0
$785$	−641.296	−0.816937
$786$	0	0
$787$	632.596	0.803806	0.401903	−	0.915682i	$-0.368349\pi$
$787$	632.596	0.803806	0.401903	+	0.915682i	$0.368349\pi$
$788$	0	0
$789$	727.734i	0.922350i
$790$	0	0
$791$	− 296.277i	− 0.374560i
$792$	0	0
$793$	−528.995	−0.667080
$794$	0	0
$795$	18.0669	0.0227256
$796$	0	0
$797$	− 242.744i	− 0.304572i	−0.988336	−	0.152286i	$-0.951337\pi$
$797$	− 242.744i	− 0.304572i	0.988336	−	0.152286i	$-0.0486635\pi$
$798$	0	0
$799$	− 1143.01i	− 1.43055i
$800$	0	0
$801$	35.0909	0.0438089
$802$	0	0
$803$	−190.207	−0.236871
$804$	0	0
$805$	− 78.5887i	− 0.0976257i
$806$	0	0
$807$	489.411i	0.606457i
$808$	0	0
$809$	287.478	0.355349	0.177675	−	0.984089i	$-0.443143\pi$
$809$	287.478	0.355349	0.177675	+	0.984089i	$0.443143\pi$
$810$	0	0
$811$	−395.709	−0.487928	−0.243964	−	0.969784i	$-0.578448\pi$
$811$	−395.709	−0.487928	−0.243964	+	0.969784i	$0.578448\pi$
$812$	0	0
$813$	− 693.083i	− 0.852500i
$814$	0	0
$815$	683.561i	0.838725i
$816$	0	0
$817$	1495.90	1.83096
$818$	0	0
$819$	−16.5376	−0.0201924
$820$	0	0
$821$	− 613.731i	− 0.747540i	−0.927521	−	0.373770i	$-0.878065\pi$
$821$	− 613.731i	− 0.747540i	0.927521	−	0.373770i	$-0.121935\pi$
$822$	0	0
$823$	868.140i	1.05485i	0.849602	+	0.527424i	$0.176842\pi$
$823$	868.140i	1.05485i	−0.849602	+	0.527424i	$0.823158\pi$
$824$	0	0
$825$	54.2262	0.0657287
$826$	0	0
$827$	−1327.86	−1.60564	−0.802819	−	0.596223i	$-0.796668\pi$
$827$	−1327.86	−1.60564	−0.802819	+	0.596223i	$0.796668\pi$
$828$	0	0
$829$	1459.23i	1.76023i	0.474758	+	0.880116i	$0.342535\pi$
$829$	1459.23i	1.76023i	−0.474758	+	0.880116i	$0.657465\pi$
$830$	0	0
$831$	1400.00i	1.68471i
$832$	0	0
$833$	147.405	0.176957
$834$	0	0
$835$	−296.175	−0.354701
$836$	0	0
$837$	− 954.159i	− 1.13997i
$838$	0	0
$839$	− 718.578i	− 0.856469i	−0.903668	−	0.428235i	$-0.859136\pi$
$839$	− 718.578i	− 0.856469i	0.903668	−	0.428235i	$-0.140864\pi$
$840$	0	0
$841$	583.627	0.693967
$842$	0	0
$843$	−1100.68	−1.30567
$844$	0	0
$845$	− 84.4673i	− 0.0999613i
$846$	0	0
$847$	283.327i	0.334507i
$848$	0	0
$849$	884.458	1.04176
$850$	0	0
$851$	741.807	0.871689
$852$	0	0
$853$	− 1327.41i	− 1.55616i	−0.628165	−	0.778080i	$-0.716194\pi$
$853$	− 1327.41i	− 1.55616i	0.628165	−	0.778080i	$-0.283806\pi$
$854$	0	0
$855$	− 41.6072i	− 0.0486633i
$856$	0	0
$857$	−743.575	−0.867649	−0.433824	−	0.900997i	$-0.642836\pi$
$857$	−743.575	−0.867649	−0.433824	+	0.900997i	$0.642836\pi$
$858$	0	0
$859$	−1212.71	−1.41177	−0.705886	−	0.708325i	$-0.749451\pi$
$859$	−1212.71	−1.41177	−0.705886	+	0.708325i	$0.749451\pi$
$860$	0	0
$861$	− 465.292i	− 0.540408i
$862$	0	0
$863$	− 866.196i	− 1.00370i	−0.864954	−	0.501852i	$-0.832652\pi$
$863$	− 866.196i	− 1.00370i	0.864954	−	0.501852i	$-0.167348\pi$
$864$	0	0
$865$	704.314	0.814236
$866$	0	0
$867$	449.035	0.517918
$868$	0	0
$869$	167.167i	0.192367i
$870$	0	0
$871$	668.843i	0.767903i
$872$	0	0
$873$	−18.0164	−0.0206374
$874$	0	0
$875$	29.5804	0.0338062
$876$	0	0
$877$	− 1068.53i	− 1.21839i	−0.793019	−	0.609197i	$-0.791492\pi$
$877$	− 1068.53i	− 1.21839i	0.793019	−	0.609197i	$-0.208508\pi$
$878$	0	0
$879$	763.718i	0.868849i
$880$	0	0
$881$	233.716	0.265285	0.132643	−	0.991164i	$-0.457654\pi$
$881$	233.716	0.265285	0.132643	+	0.991164i	$0.457654\pi$
$882$	0	0
$883$	415.264	0.470287	0.235144	−	0.971961i	$-0.424444\pi$
$883$	415.264	0.470287	0.235144	+	0.971961i	$0.424444\pi$
$884$	0	0
$885$	308.162i	0.348205i
$886$	0	0
$887$	− 408.232i	− 0.460239i	−0.973162	−	0.230119i	$-0.926088\pi$
$887$	− 408.232i	− 0.460239i	0.973162	−	0.230119i	$-0.0739117\pi$
$888$	0	0
$889$	−17.5078	−0.0196938
$890$	0	0
$891$	282.696	0.317279
$892$	0	0
$893$	− 1850.99i	− 2.07278i
$894$	0	0
$895$	− 102.371i	− 0.114381i
$896$	0	0
$897$	−442.461	−0.493267
$898$	0	0
$899$	551.514	0.613475
$900$	0	0
$901$	− 58.5156i	− 0.0649452i
$902$	0	0
$903$	337.460i	0.373709i
$904$	0	0
$905$	432.112	0.477471
$906$	0	0
$907$	−914.218	−1.00796	−0.503979	−	0.863716i	$-0.668131\pi$
$907$	−914.218	−1.00796	−0.503979	+	0.863716i	$0.668131\pi$
$908$	0	0
$909$	34.9135i	0.0384087i
$910$	0	0
$911$	364.468i	0.400075i	0.979788	+	0.200037i	$0.0641064\pi$
$911$	364.468i	0.400075i	−0.979788	+	0.200037i	$0.935894\pi$
$912$	0	0
$913$	546.559	0.598641
$914$	0	0
$915$	300.240	0.328131
$916$	0	0
$917$	114.888i	0.125287i
$918$	0	0
$919$	654.446i	0.712128i	0.934462	+	0.356064i	$0.115882\pi$
$919$	654.446i	0.712128i	−0.934462	+	0.356064i	$0.884118\pi$
$920$	0	0
$921$	−235.134	−0.255303
$922$	0	0
$923$	−706.557	−0.765501
$924$	0	0
$925$	279.213i	0.301851i
$926$	0	0
$927$	94.2018i	0.101620i
$928$	0	0
$929$	1009.59	1.08675	0.543373	−	0.839491i	$-0.317147\pi$
$929$	1009.59	1.08675	0.543373	+	0.839491i	$0.317147\pi$
$930$	0	0
$931$	238.708	0.256399
$932$	0	0
$933$	− 327.695i	− 0.351227i
$934$	0	0
$935$	− 175.630i	− 0.187839i
$936$	0	0
$937$	791.301	0.844505	0.422252	−	0.906478i	$-0.361240\pi$
$937$	791.301	0.844505	0.422252	+	0.906478i	$0.361240\pi$
$938$	0	0
$939$	−26.2913	−0.0279992
$940$	0	0
$941$	1423.78i	1.51305i	0.653968	+	0.756523i	$0.273103\pi$
$941$	1423.78i	1.51305i	−0.653968	+	0.756523i	$0.726897\pi$
$942$	0	0
$943$	803.456i	0.852021i
$944$	0	0
$945$	164.202	0.173759
$946$	0	0
$947$	812.534	0.858009	0.429004	−	0.903302i	$-0.358864\pi$
$947$	812.534	0.858009	0.429004	+	0.903302i	$0.358864\pi$
$948$	0	0
$949$	584.166i	0.615559i
$950$	0	0
$951$	− 276.034i	− 0.290257i
$952$	0	0
$953$	−968.257	−1.01601	−0.508005	−	0.861354i	$-0.669617\pi$
$953$	−968.257	−1.01601	−0.508005	+	0.861354i	$0.669617\pi$
$954$	0	0
$955$	−529.634	−0.554591
$956$	0	0
$957$	173.989i	0.181806i
$958$	0	0
$959$	505.083i	0.526677i
$960$	0	0
$961$	−220.816	−0.229777
$962$	0	0
$963$	−15.0795	−0.0156588
$964$	0	0
$965$	− 379.256i	− 0.393011i
$966$	0	0
$967$	586.922i	0.606952i	0.952839	+	0.303476i	$0.0981472\pi$
$967$	586.922i	0.606952i	−0.952839	+	0.303476i	$0.901853\pi$
$968$	0	0
$969$	2087.96	2.15476
$970$	0	0
$971$	−888.710	−0.915252	−0.457626	−	0.889145i	$-0.651300\pi$
$971$	−888.710	−0.915252	−0.457626	+	0.889145i	$0.651300\pi$
$972$	0	0
$973$	− 131.527i	− 0.135176i
$974$	0	0
$975$	− 166.540i	− 0.170810i
$976$	0	0
$977$	−856.004	−0.876155	−0.438078	−	0.898937i	$-0.644340\pi$
$977$	−856.004	−0.876155	−0.438078	+	0.898937i	$0.644340\pi$
$978$	0	0
$979$	239.872	0.245017
$980$	0	0
$981$	22.1274i	0.0225560i
$982$	0	0
$983$	1640.26i	1.66863i	0.551288	+	0.834315i	$0.314137\pi$
$983$	1640.26i	1.66863i	−0.551288	+	0.834315i	$0.685863\pi$
$984$	0	0
$985$	−226.401	−0.229849
$986$	0	0
$987$	417.566	0.423066
$988$	0	0
$989$	− 582.718i	− 0.589199i
$990$	0	0
$991$	− 1293.79i	− 1.30554i	−0.757555	−	0.652771i	$-0.773606\pi$
$991$	− 1293.79i	− 1.30554i	0.757555	−	0.652771i	$-0.226394\pi$
$992$	0	0
$993$	122.114	0.122975
$994$	0	0
$995$	−298.559	−0.300059
$996$	0	0
$997$	− 299.484i	− 0.300385i	−0.988657	−	0.150193i	$-0.952011\pi$
$997$	− 299.484i	− 0.300385i	0.988657	−	0.150193i	$-0.0479894\pi$
$998$	0	0
$999$	1549.92i	1.55148i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.3.o.a.911.36		48
4.3	odd	2		280.3.o.a.211.7	✓	48
8.3	odd	2	inner	1120.3.o.a.911.35		48
8.5	even	2		280.3.o.a.211.8	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.3.o.a.211.7	✓	48	4.3	odd	2
280.3.o.a.211.8	yes	48	8.5	even	2
1120.3.o.a.911.35		48	8.3	odd	2	inner
1120.3.o.a.911.36		48	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:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1120.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.90764 q^{3} -2.23607i q^{5} -2.64575i q^{7} -0.545650 q^{9} +O(q^{10})\)	\(q+2.90764 q^{3} -2.23607i q^{5} -2.64575i q^{7} -0.545650 q^{9} -3.72992 q^{11} +11.4554i q^{13} -6.50167i q^{15} -21.0578 q^{17} -34.1011 q^{19} -7.69288i q^{21} +13.2839i q^{23} -5.00000 q^{25} -27.7553 q^{27} -16.0429i q^{29} +34.3776i q^{31} -10.8452 q^{33} -5.91608 q^{35} -55.8425i q^{37} +33.3080i q^{39} +60.4834 q^{41} -43.8665 q^{43} +1.22011i q^{45} +54.2795i q^{47} -7.00000 q^{49} -61.2286 q^{51} +2.77880i q^{53} +8.34035i q^{55} -99.1537 q^{57} -47.3973 q^{59} +46.1788i q^{61} +1.44365i q^{63} +25.6149 q^{65} +58.3870 q^{67} +38.6248i q^{69} +61.6792i q^{71} +50.9950 q^{73} -14.5382 q^{75} +9.86843i q^{77} -44.8179i q^{79} -75.7914 q^{81} -146.534 q^{83} +47.0868i q^{85} -46.6468i q^{87} -64.3103 q^{89} +30.3080 q^{91} +99.9574i q^{93} +76.2524i q^{95} +33.0183 q^{97} +2.03523 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 144 q^{9}+O(q^{10}) \) 48 * q + 144 * q^9	\( 48 q + 144 q^{9} - 32 q^{11} + 64 q^{19} - 240 q^{25} - 192 q^{27} - 32 q^{33} - 96 q^{41} + 96 q^{43} - 336 q^{49} - 192 q^{51} + 160 q^{57} + 576 q^{59} + 160 q^{67} + 656 q^{81} - 224 q^{99}+O(q^{100}) \) 48 * q + 144 * q^9 - 32 * q^11 + 64 * q^19 - 240 * q^25 - 192 * q^27 - 32 * q^33 - 96 * q^41 + 96 * q^43 - 336 * q^49 - 192 * q^51 + 160 * q^57 + 576 * q^59 + 160 * q^67 + 656 * q^81 - 224 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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