LMFDB - Embedded newform 1089.3.b.h.485.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(485,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1089.485");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.b (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:	$29.6731007888$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.3317760000.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 12x^{6} + 23x^{4} + 12x^{2} + 1 $ x^8 + 12x^6 + 23x^4 + 12*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.6
Root			$-0.837556i$ of defining polynomial
Character	$\chi$	$=$	1089.485
Dual form			1089.3.b.h.485.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.356394i q^{2} +3.87298 q^{4} +3.44572i q^{5} -3.96812 q^{7} +2.80588i q^{8} +O(q^{10})$	$q+0.356394i q^{2} +3.87298 q^{4} +3.44572i q^{5} -3.96812 q^{7} +2.80588i q^{8} -1.22803 q^{10} +4.75615 q^{13} -1.41421i q^{14} +14.4919 q^{16} -29.1280i q^{17} +14.8644 q^{19} +13.3452i q^{20} +22.9639i q^{23} +13.1270 q^{25} +1.69506i q^{26} -15.3685 q^{28} -4.63312i q^{29} +27.4919 q^{31} +16.3884i q^{32} +10.3810 q^{34} -13.6730i q^{35} +36.8569 q^{37} +5.29760i q^{38} -9.66829 q^{40} +68.3199i q^{41} +26.8968 q^{43} -8.18418 q^{46} +42.9425i q^{47} -33.2540 q^{49} +4.67839i q^{50} +18.4205 q^{52} +78.2194i q^{53} -11.1341i q^{56} +1.65122 q^{58} +92.7992i q^{59} -40.5612 q^{61} +9.79796i q^{62} +52.1270 q^{64} +16.3884i q^{65} -54.3327 q^{67} -112.812i q^{68} +4.87298 q^{70} -91.0029i q^{71} +109.347 q^{73} +13.1356i q^{74} +57.5697 q^{76} -143.292 q^{79} +49.9351i q^{80} -24.3488 q^{82} -44.0003i q^{83} +100.367 q^{85} +9.58587i q^{86} -64.7730i q^{89} -18.8730 q^{91} +88.9386i q^{92} -15.3044 q^{94} +51.2187i q^{95} +16.8891 q^{97} -11.8515i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{16} + 136 q^{25} + 96 q^{31} + 176 q^{34} + 16 q^{37} - 328 q^{49} + 416 q^{58} + 448 q^{64} + 216 q^{67} + 8 q^{70} + 208 q^{82} - 120 q^{91} + 352 q^{97}+O(q^{100}) $ 8 * q - 8 * q^16 + 136 * q^25 + 96 * q^31 + 176 * q^34 + 16 * q^37 - 328 * q^49 + 416 * q^58 + 448 * q^64 + 216 * q^67 + 8 * q^70 + 208 * q^82 - 120 * q^91 + 352 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$1$	$-1$

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 $

$2$	0.356394i	0.178197i	0.996023	+	0.0890985i	$0.0283986\pi$
$2$	0.356394i	0.178197i	−0.996023	+	0.0890985i	$0.971601\pi$
$3$	0	0
$3$	0	0
$4$	3.87298	0.968246
$5$	3.44572i	0.689144i	0.938760	+	0.344572i	$0.111976\pi$
$5$	3.44572i	0.689144i	−0.938760	+	0.344572i	$0.888024\pi$
$6$	0	0
$7$	−3.96812	−0.566874	−0.283437	−	0.958991i	$-0.591475\pi$
$7$	−3.96812	−0.566874	−0.283437	+	0.958991i	$0.591475\pi$
$8$	2.80588i	0.350735i
$9$	0	0
$10$	−1.22803	−0.122803
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.75615	0.365858	0.182929	−	0.983126i	$-0.441442\pi$
$13$	4.75615	0.365858	0.182929	+	0.983126i	$0.441442\pi$
$14$	− 1.41421i	− 0.101015i
$15$	0	0
$16$	14.4919	0.905746
$17$	− 29.1280i	− 1.71341i	−0.515804	−	0.856706i	$-0.672507\pi$
$17$	− 29.1280i	− 1.71341i	0.515804	−	0.856706i	$-0.327493\pi$
$18$	0	0
$19$	14.8644	0.782339	0.391169	−	0.920319i	$-0.372071\pi$
$19$	14.8644	0.782339	0.391169	+	0.920319i	$0.372071\pi$
$20$	13.3452i	0.667261i
$21$	0	0
$22$	0	0
$23$	22.9639i	0.998429i	0.866479	+	0.499214i	$0.166378\pi$
$23$	22.9639i	0.998429i	−0.866479	+	0.499214i	$0.833622\pi$
$24$	0	0
$25$	13.1270	0.525081
$26$	1.69506i	0.0651948i
$27$	0	0
$28$	−15.3685	−0.548873
$29$	− 4.63312i	− 0.159763i	−0.996804	−	0.0798814i	$-0.974546\pi$
$29$	− 4.63312i	− 0.159763i	0.996804	−	0.0798814i	$-0.0254542\pi$
$30$	0	0
$31$	27.4919	0.886837	0.443418	−	0.896315i	$-0.353766\pi$
$31$	27.4919	0.886837	0.443418	+	0.896315i	$0.353766\pi$
$32$	16.3884i	0.512137i
$33$	0	0
$34$	10.3810	0.305325
$35$	− 13.6730i	− 0.390658i
$36$	0	0
$37$	36.8569	0.996131	0.498066	−	0.867139i	$-0.334044\pi$
$37$	36.8569	0.996131	0.498066	+	0.867139i	$0.334044\pi$
$38$	5.29760i	0.139410i
$39$	0	0
$40$	−9.66829	−0.241707
$41$	68.3199i	1.66634i	0.553019	+	0.833169i	$0.313476\pi$
$41$	68.3199i	1.66634i	−0.553019	+	0.833169i	$0.686524\pi$
$42$	0	0
$43$	26.8968	0.625508	0.312754	−	0.949834i	$-0.398748\pi$
$43$	26.8968	0.625508	0.312754	+	0.949834i	$0.398748\pi$
$44$	0	0
$45$	0	0
$46$	−8.18418	−0.177917
$47$	42.9425i	0.913670i	0.889552	+	0.456835i	$0.151017\pi$
$47$	42.9425i	0.913670i	−0.889552	+	0.456835i	$0.848983\pi$
$48$	0	0
$49$	−33.2540	−0.678654
$50$	4.67839i	0.0935678i
$51$	0	0
$52$	18.4205	0.354240
$53$	78.2194i	1.47584i	0.674889	+	0.737919i	$0.264191\pi$
$53$	78.2194i	1.47584i	−0.674889	+	0.737919i	$0.735809\pi$
$54$	0	0
$55$	0	0
$56$	− 11.1341i	− 0.198823i
$57$	0	0
$58$	1.65122	0.0284692
$59$	92.7992i	1.57287i	0.617674	+	0.786434i	$0.288075\pi$
$59$	92.7992i	1.57287i	−0.617674	+	0.786434i	$0.711925\pi$
$60$	0	0
$61$	−40.5612	−0.664937	−0.332469	−	0.943114i	$-0.607882\pi$
$61$	−40.5612	−0.664937	−0.332469	+	0.943114i	$0.607882\pi$
$62$	9.79796i	0.158032i
$63$	0	0
$64$	52.1270	0.814485
$65$	16.3884i	0.252129i
$66$	0	0
$67$	−54.3327	−0.810935	−0.405468	−	0.914109i	$-0.632891\pi$
$67$	−54.3327	−0.810935	−0.405468	+	0.914109i	$0.632891\pi$
$68$	− 112.812i	− 1.65900i
$69$	0	0
$70$	4.87298	0.0696140
$71$	− 91.0029i	− 1.28173i	−0.767653	−	0.640866i	$-0.778576\pi$
$71$	− 91.0029i	− 1.28173i	0.767653	−	0.640866i	$-0.221424\pi$
$72$	0	0
$73$	109.347	1.49791	0.748954	−	0.662622i	$-0.230556\pi$
$73$	109.347	1.49791	0.748954	+	0.662622i	$0.230556\pi$
$74$	13.1356i	0.177508i
$75$	0	0
$76$	57.5697	0.757496
$77$	0	0
$78$	0	0
$79$	−143.292	−1.81383	−0.906913	−	0.421318i	$-0.861568\pi$
$79$	−143.292	−1.81383	−0.906913	+	0.421318i	$0.861568\pi$
$80$	49.9351i	0.624189i
$81$	0	0
$82$	−24.3488	−0.296936
$83$	− 44.0003i	− 0.530124i	−0.964231	−	0.265062i	$-0.914608\pi$
$83$	− 44.0003i	− 0.530124i	0.964231	−	0.265062i	$-0.0853924\pi$
$84$	0	0
$85$	100.367	1.18079
$86$	9.58587i	0.111464i
$87$	0	0
$88$	0	0
$89$	− 64.7730i	− 0.727786i	−0.931441	−	0.363893i	$-0.881447\pi$
$89$	− 64.7730i	− 0.727786i	0.931441	−	0.363893i	$-0.118553\pi$
$90$	0	0
$91$	−18.8730	−0.207395
$92$	88.9386i	0.966724i
$93$	0	0
$94$	−15.3044	−0.162813
$95$	51.2187i	0.539144i
$96$	0	0
$97$	16.8891	0.174115	0.0870573	−	0.996203i	$-0.472254\pi$
$97$	16.8891	0.174115	0.0870573	+	0.996203i	$0.472254\pi$
$98$	− 11.8515i	− 0.120934i
$99$	0	0
$100$	50.8407	0.508407
$101$	− 164.568i	− 1.62939i	−0.579889	−	0.814696i	$-0.696904\pi$
$101$	− 164.568i	− 1.62939i	0.579889	−	0.814696i	$-0.303096\pi$
$102$	0	0
$103$	145.381	1.41147	0.705733	−	0.708478i	$-0.250618\pi$
$103$	145.381	1.41147	0.705733	+	0.708478i	$0.250618\pi$
$104$	13.3452i	0.128319i
$105$	0	0
$106$	−27.8769	−0.262990
$107$	189.414i	1.77022i	0.465377	+	0.885112i	$0.345919\pi$
$107$	189.414i	1.77022i	−0.465377	+	0.885112i	$0.654081\pi$
$108$	0	0
$109$	−57.5418	−0.527906	−0.263953	−	0.964536i	$-0.585026\pi$
$109$	−57.5418	−0.527906	−0.263953	+	0.964536i	$0.585026\pi$
$110$	0	0
$111$	0	0
$112$	−57.5057	−0.513444
$113$	29.7769i	0.263512i	0.991282	+	0.131756i	$0.0420616\pi$
$113$	29.7769i	0.263512i	−0.991282	+	0.131756i	$0.957938\pi$
$114$	0	0
$115$	−79.1270	−0.688061
$116$	− 17.9440i	− 0.154690i
$117$	0	0
$118$	−33.0731	−0.280280
$119$	115.583i	0.971289i
$120$	0	0
$121$	0	0
$122$	− 14.4558i	− 0.118490i
$123$	0	0
$124$	106.476	0.858676
$125$	131.375i	1.05100i
$126$	0	0
$127$	−12.5923	−0.0991520	−0.0495760	−	0.998770i	$-0.515787\pi$
$127$	−12.5923	−0.0991520	−0.0495760	+	0.998770i	$0.515787\pi$
$128$	84.1312i	0.657275i
$129$	0	0
$130$	−5.84072	−0.0449286
$131$	20.3597i	0.155418i	0.996976	+	0.0777089i	$0.0247605\pi$
$131$	20.3597i	0.155418i	−0.996976	+	0.0777089i	$0.975240\pi$
$132$	0	0
$133$	−58.9839	−0.443488
$134$	− 19.3638i	− 0.144506i
$135$	0	0
$136$	81.7298	0.600955
$137$	− 35.9170i	− 0.262168i	−0.991371	−	0.131084i	$-0.958154\pi$
$137$	− 35.9170i	− 0.262168i	0.991371	−	0.131084i	$-0.0418458\pi$
$138$	0	0
$139$	−101.091	−0.727273	−0.363637	−	0.931541i	$-0.618465\pi$
$139$	−101.091	−0.727273	−0.363637	+	0.931541i	$0.618465\pi$
$140$	− 52.9554i	− 0.378253i
$141$	0	0
$142$	32.4329	0.228401
$143$	0	0
$144$	0	0
$145$	15.9644	0.110100
$146$	38.9707i	0.266923i
$147$	0	0
$148$	142.746	0.964500
$149$	− 210.311i	− 1.41148i	−0.708469	−	0.705742i	$-0.750614\pi$
$149$	− 210.311i	− 1.41148i	0.708469	−	0.705742i	$-0.249386\pi$
$150$	0	0
$151$	212.078	1.40449	0.702246	−	0.711934i	$-0.252180\pi$
$151$	212.078	1.40449	0.702246	+	0.711934i	$0.252180\pi$
$152$	41.7079i	0.274394i
$153$	0	0
$154$	0	0
$155$	94.7295i	0.611158i
$156$	0	0
$157$	119.173	0.759066	0.379533	−	0.925178i	$-0.376085\pi$
$157$	119.173	0.759066	0.379533	+	0.925178i	$0.376085\pi$
$158$	− 51.0685i	− 0.323218i
$159$	0	0
$160$	−56.4697	−0.352936
$161$	− 91.1233i	− 0.565983i
$162$	0	0
$163$	266.268	1.63355	0.816773	−	0.576959i	$-0.195761\pi$
$163$	266.268	1.63355	0.816773	+	0.576959i	$0.195761\pi$
$164$	264.602i	1.61342i
$165$	0	0
$166$	15.6814	0.0944665
$167$	113.090i	0.677184i	0.940933	+	0.338592i	$0.109951\pi$
$167$	113.090i	0.677184i	−0.940933	+	0.338592i	$0.890049\pi$
$168$	0	0
$169$	−146.379	−0.866148
$170$	35.7702i	0.210413i
$171$	0	0
$172$	104.171	0.605645
$173$	117.615i	0.679856i	0.940452	+	0.339928i	$0.110403\pi$
$173$	117.615i	0.679856i	−0.940452	+	0.339928i	$0.889597\pi$
$174$	0	0
$175$	−52.0896	−0.297655
$176$	0	0
$177$	0	0
$178$	23.0847	0.129689
$179$	− 231.526i	− 1.29344i	−0.762727	−	0.646721i	$-0.776140\pi$
$179$	− 231.526i	− 1.29344i	0.762727	−	0.646721i	$-0.223860\pi$
$180$	0	0
$181$	−10.8105	−0.0597265	−0.0298633	−	0.999554i	$-0.509507\pi$
$181$	−10.8105	−0.0597265	−0.0298633	+	0.999554i	$0.509507\pi$
$182$	− 6.72622i	− 0.0369572i
$183$	0	0
$184$	−64.4339	−0.350184
$185$	126.998i	0.686478i
$186$	0	0
$187$	0	0
$188$	166.315i	0.884657i
$189$	0	0
$190$	−18.2540	−0.0960739
$191$	− 49.5203i	− 0.259269i	−0.991562	−	0.129634i	$-0.958620\pi$
$191$	− 49.5203i	− 0.259269i	0.991562	−	0.129634i	$-0.0413803\pi$
$192$	0	0
$193$	8.40418	0.0435450	0.0217725	−	0.999763i	$-0.493069\pi$
$193$	8.40418	0.0435450	0.0217725	+	0.999763i	$0.493069\pi$
$194$	6.01918i	0.0310267i
$195$	0	0
$196$	−128.792	−0.657104
$197$	− 125.190i	− 0.635481i	−0.948178	−	0.317741i	$-0.897076\pi$
$197$	− 125.190i	− 0.635481i	0.948178	−	0.317741i	$-0.102924\pi$
$198$	0	0
$199$	255.728	1.28506	0.642532	−	0.766259i	$-0.277884\pi$
$199$	255.728	1.28506	0.642532	+	0.766259i	$0.277884\pi$
$200$	36.8329i	0.184164i
$201$	0	0
$202$	58.6512	0.290353
$203$	18.3848i	0.0905654i
$204$	0	0
$205$	−235.411	−1.14835
$206$	51.8129i	0.251519i
$207$	0	0
$208$	68.9259	0.331374
$209$	0	0
$210$	0	0
$211$	149.332	0.707736	0.353868	−	0.935295i	$-0.384866\pi$
$211$	149.332	0.707736	0.353868	+	0.935295i	$0.384866\pi$
$212$	302.942i	1.42897i
$213$	0	0
$214$	−67.5060	−0.315449
$215$	92.6789i	0.431065i
$216$	0	0
$217$	−109.091	−0.502725
$218$	− 20.5075i	− 0.0940713i
$219$	0	0
$220$	0	0
$221$	− 138.537i	− 0.626866i
$222$	0	0
$223$	−211.270	−0.947400	−0.473700	−	0.880686i	$-0.657082\pi$
$223$	−211.270	−0.947400	−0.473700	+	0.880686i	$0.657082\pi$
$224$	− 65.0310i	− 0.290317i
$225$	0	0
$226$	−10.6123	−0.0469571
$227$	− 175.091i	− 0.771325i	−0.922640	−	0.385662i	$-0.873973\pi$
$227$	− 175.091i	− 0.771325i	0.922640	−	0.385662i	$-0.126027\pi$
$228$	0	0
$229$	−142.460	−0.622095	−0.311047	−	0.950394i	$-0.600680\pi$
$229$	−142.460	−0.622095	−0.311047	+	0.950394i	$0.600680\pi$
$230$	− 28.2004i	− 0.122610i
$231$	0	0
$232$	13.0000	0.0560345
$233$	264.240i	1.13408i	0.823692	+	0.567038i	$0.191911\pi$
$233$	264.240i	1.13408i	−0.823692	+	0.567038i	$0.808089\pi$
$234$	0	0
$235$	−147.968	−0.629650
$236$	359.410i	1.52292i
$237$	0	0
$238$	−41.1932	−0.173081
$239$	− 176.997i	− 0.740573i	−0.928918	−	0.370287i	$-0.879259\pi$
$239$	− 176.997i	− 0.740573i	0.928918	−	0.370287i	$-0.120741\pi$
$240$	0	0
$241$	−14.6084	−0.0606156	−0.0303078	−	0.999541i	$-0.509649\pi$
$241$	−14.6084	−0.0606156	−0.0303078	+	0.999541i	$0.509649\pi$
$242$	0	0
$243$	0	0
$244$	−157.093	−0.643823
$245$	− 114.584i	− 0.467690i
$246$	0	0
$247$	70.6976	0.286225
$248$	77.1392i	0.311045i
$249$	0	0
$250$	−46.8213	−0.187285
$251$	− 217.159i	− 0.865174i	−0.901592	−	0.432587i	$-0.857601\pi$
$251$	− 217.159i	− 0.865174i	0.901592	−	0.432587i	$-0.142399\pi$
$252$	0	0
$253$	0	0
$254$	− 4.48782i	− 0.0176686i
$255$	0	0
$256$	178.524	0.697360
$257$	− 33.4151i	− 0.130020i	−0.997885	−	0.0650099i	$-0.979292\pi$
$257$	− 33.4151i	− 0.130020i	0.997885	−	0.0650099i	$-0.0207079\pi$
$258$	0	0
$259$	−146.252	−0.564681
$260$	63.4719i	0.244123i
$261$	0	0
$262$	−7.25608	−0.0276950
$263$	− 335.360i	− 1.27513i	−0.770396	−	0.637566i	$-0.779941\pi$
$263$	− 335.360i	− 1.27513i	0.770396	−	0.637566i	$-0.220059\pi$
$264$	0	0
$265$	−269.522	−1.01706
$266$	− 21.0215i	− 0.0790282i
$267$	0	0
$268$	−210.429	−0.785185
$269$	− 337.759i	− 1.25561i	−0.778371	−	0.627805i	$-0.783954\pi$
$269$	− 337.759i	− 1.25561i	0.778371	−	0.627805i	$-0.216046\pi$
$270$	0	0
$271$	436.501	1.61071	0.805353	−	0.592796i	$-0.201976\pi$
$271$	436.501	1.61071	0.805353	+	0.592796i	$0.201976\pi$
$272$	− 422.121i	− 1.55192i
$273$	0	0
$274$	12.8006	0.0467176
$275$	0	0
$276$	0	0
$277$	199.130	0.718881	0.359440	−	0.933168i	$-0.382968\pi$
$277$	199.130	0.718881	0.359440	+	0.933168i	$0.382968\pi$
$278$	− 36.0282i	− 0.129598i
$279$	0	0
$280$	38.3649	0.137018
$281$	250.442i	0.891253i	0.895219	+	0.445627i	$0.147019\pi$
$281$	250.442i	0.891253i	−0.895219	+	0.445627i	$0.852981\pi$
$282$	0	0
$283$	43.0813	0.152231	0.0761153	−	0.997099i	$-0.475748\pi$
$283$	43.0813	0.152231	0.0761153	+	0.997099i	$0.475748\pi$
$284$	− 352.453i	− 1.24103i
$285$	0	0
$286$	0	0
$287$	− 271.101i	− 0.944604i
$288$	0	0
$289$	−559.441	−1.93578
$290$	5.68963i	0.0196194i
$291$	0	0
$292$	423.500	1.45034
$293$	− 7.51229i	− 0.0256392i	−0.999918	−	0.0128196i	$-0.995919\pi$
$293$	− 7.51229i	− 0.0256392i	0.999918	−	0.0128196i	$-0.00408072\pi$
$294$	0	0
$295$	−319.760	−1.08393
$296$	103.416i	0.349378i
$297$	0	0
$298$	74.9536	0.251522
$299$	109.220i	0.365283i
$300$	0	0
$301$	−106.730	−0.354584
$302$	75.5835i	0.250276i
$303$	0	0
$304$	215.414	0.708600
$305$	− 139.762i	− 0.458238i
$306$	0	0
$307$	−360.275	−1.17353	−0.586767	−	0.809756i	$-0.699599\pi$
$307$	−360.275	−1.17353	−0.586767	+	0.809756i	$0.699599\pi$
$308$	0	0
$309$	0	0
$310$	−33.7610	−0.108907
$311$	− 415.622i	− 1.33641i	−0.743979	−	0.668203i	$-0.767064\pi$
$311$	− 415.622i	− 1.33641i	0.743979	−	0.668203i	$-0.232936\pi$
$312$	0	0
$313$	−63.2863	−0.202193	−0.101096	−	0.994877i	$-0.532235\pi$
$313$	−63.2863	−0.202193	−0.101096	+	0.994877i	$0.532235\pi$
$314$	42.4727i	0.135263i
$315$	0	0
$316$	−554.969	−1.75623
$317$	− 422.491i	− 1.33278i	−0.745604	−	0.666389i	$-0.767839\pi$
$317$	− 422.491i	− 1.33278i	0.745604	−	0.666389i	$-0.232161\pi$
$318$	0	0
$319$	0	0
$320$	179.615i	0.561297i
$321$	0	0
$322$	32.4758	0.100857
$323$	− 432.972i	− 1.34047i
$324$	0	0
$325$	62.4341	0.192105
$326$	94.8963i	0.291093i
$327$	0	0
$328$	−191.698	−0.584444
$329$	− 170.401i	− 0.517936i
$330$	0	0
$331$	−233.077	−0.704159	−0.352079	−	0.935970i	$-0.614525\pi$
$331$	−233.077	−0.704159	−0.352079	+	0.935970i	$0.614525\pi$
$332$	− 170.412i	− 0.513290i
$333$	0	0
$334$	−40.3045	−0.120672
$335$	− 187.215i	− 0.558851i
$336$	0	0
$337$	−622.254	−1.84645	−0.923226	−	0.384257i	$-0.874458\pi$
$337$	−622.254	−1.84645	−0.923226	+	0.384257i	$0.874458\pi$
$338$	− 52.1686i	− 0.154345i
$339$	0	0
$340$	388.720	1.14329
$341$	0	0
$342$	0	0
$343$	326.394	0.951585
$344$	75.4694i	0.219388i
$345$	0	0
$346$	−41.9173	−0.121148
$347$	648.204i	1.86802i	0.357241	+	0.934012i	$0.383717\pi$
$347$	648.204i	1.86802i	−0.357241	+	0.934012i	$0.616283\pi$
$348$	0	0
$349$	−304.125	−0.871419	−0.435709	−	0.900087i	$-0.643502\pi$
$349$	−304.125	−0.871419	−0.435709	+	0.900087i	$0.643502\pi$
$350$	− 18.5644i	− 0.0530412i
$351$	0	0
$352$	0	0
$353$	− 250.753i	− 0.710350i	−0.934800	−	0.355175i	$-0.884421\pi$
$353$	− 250.753i	− 0.710350i	0.934800	−	0.355175i	$-0.115579\pi$
$354$	0	0
$355$	313.571	0.883297
$356$	− 250.865i	− 0.704676i
$357$	0	0
$358$	82.5145	0.230487
$359$	114.425i	0.318732i	0.987220	+	0.159366i	$0.0509450\pi$
$359$	114.425i	0.318732i	−0.987220	+	0.159366i	$0.949055\pi$
$360$	0	0
$361$	−140.048	−0.387946
$362$	− 3.85280i	− 0.0106431i
$363$	0	0
$364$	−73.0948	−0.200810
$365$	376.780i	1.03227i
$366$	0	0
$367$	39.9818	0.108942	0.0544711	−	0.998515i	$-0.482653\pi$
$367$	39.9818	0.108942	0.0544711	+	0.998515i	$0.482653\pi$
$368$	332.791i	0.904323i
$369$	0	0
$370$	−45.2615	−0.122328
$371$	− 310.384i	− 0.836614i
$372$	0	0
$373$	−442.965	−1.18757	−0.593787	−	0.804622i	$-0.702368\pi$
$373$	−442.965	−1.18757	−0.593787	+	0.804622i	$0.702368\pi$
$374$	0	0
$375$	0	0
$376$	−120.492	−0.320456
$377$	− 22.0358i	− 0.0584505i
$378$	0	0
$379$	−225.413	−0.594758	−0.297379	−	0.954760i	$-0.596113\pi$
$379$	−225.413	−0.594758	−0.297379	+	0.954760i	$0.596113\pi$
$380$	198.369i	0.522024i
$381$	0	0
$382$	17.6487	0.0462009
$383$	− 182.924i	− 0.477608i	−0.971068	−	0.238804i	$-0.923245\pi$
$383$	− 182.924i	− 0.477608i	0.971068	−	0.238804i	$-0.0767554\pi$
$384$	0	0
$385$	0	0
$386$	2.99520i	0.00775958i
$387$	0	0
$388$	65.4113	0.168586
$389$	− 619.589i	− 1.59277i	−0.604787	−	0.796387i	$-0.706742\pi$
$389$	− 619.589i	− 1.59277i	0.604787	−	0.796387i	$-0.293258\pi$
$390$	0	0
$391$	668.892	1.71072
$392$	− 93.3070i	− 0.238028i
$393$	0	0
$394$	44.6169	0.113241
$395$	− 493.745i	− 1.24999i
$396$	0	0
$397$	−45.3649	−0.114269	−0.0571347	−	0.998366i	$-0.518196\pi$
$397$	−45.3649	−0.114269	−0.0571347	+	0.998366i	$0.518196\pi$
$398$	91.1398i	0.228995i
$399$	0	0
$400$	190.236	0.475590
$401$	− 361.330i	− 0.901073i	−0.892758	−	0.450536i	$-0.851233\pi$
$401$	− 361.330i	− 0.901073i	0.892758	−	0.450536i	$-0.148767\pi$
$402$	0	0
$403$	130.756	0.324456
$404$	− 637.371i	− 1.57765i
$405$	0	0
$406$	−6.55222	−0.0161385
$407$	0	0
$408$	0	0
$409$	683.744	1.67175	0.835873	−	0.548922i	$-0.184962\pi$
$409$	683.744	1.67175	0.835873	+	0.548922i	$0.184962\pi$
$410$	− 83.8991i	− 0.204632i
$411$	0	0
$412$	563.058	1.36665
$413$	− 368.238i	− 0.891618i
$414$	0	0
$415$	151.613	0.365332
$416$	77.9456i	0.187369i
$417$	0	0
$418$	0	0
$419$	413.509i	0.986895i	0.869775	+	0.493448i	$0.164264\pi$
$419$	413.509i	0.986895i	−0.869775	+	0.493448i	$0.835736\pi$
$420$	0	0
$421$	−328.460	−0.780189	−0.390095	−	0.920775i	$-0.627558\pi$
$421$	−328.460	−0.780189	−0.390095	+	0.920775i	$0.627558\pi$
$422$	53.2211i	0.126116i
$423$	0	0
$424$	−219.475	−0.517629
$425$	− 382.364i	− 0.899680i
$426$	0	0
$427$	160.952	0.376936
$428$	733.597i	1.71401i
$429$	0	0
$430$	−33.0302	−0.0768145
$431$	542.164i	1.25792i	0.777438	+	0.628960i	$0.216519\pi$
$431$	542.164i	1.25792i	−0.777438	+	0.628960i	$0.783481\pi$
$432$	0	0
$433$	−716.903	−1.65567	−0.827833	−	0.560975i	$-0.810426\pi$
$433$	−716.903	−1.65567	−0.827833	+	0.560975i	$0.810426\pi$
$434$	− 38.8795i	− 0.0895840i
$435$	0	0
$436$	−222.858	−0.511143
$437$	341.345i	0.781110i
$438$	0	0
$439$	196.790	0.448269	0.224135	−	0.974558i	$-0.428044\pi$
$439$	196.790	0.448269	0.224135	+	0.974558i	$0.428044\pi$
$440$	0	0
$441$	0	0
$442$	49.3739	0.111706
$443$	− 524.354i	− 1.18364i	−0.806069	−	0.591821i	$-0.798409\pi$
$443$	− 524.354i	− 1.18364i	0.806069	−	0.591821i	$-0.201591\pi$
$444$	0	0
$445$	223.190	0.501549
$446$	− 75.2954i	− 0.168824i
$447$	0	0
$448$	−206.846	−0.461710
$449$	− 313.024i	− 0.697159i	−0.937279	−	0.348580i	$-0.886664\pi$
$449$	− 313.024i	− 0.697159i	0.937279	−	0.348580i	$-0.113336\pi$
$450$	0	0
$451$	0	0
$452$	115.325i	0.255145i
$453$	0	0
$454$	62.4013	0.137448
$455$	− 65.0310i	− 0.142925i
$456$	0	0
$457$	4.06821	0.00890200	0.00445100	−	0.999990i	$-0.498583\pi$
$457$	4.06821	0.00890200	0.00445100	+	0.999990i	$0.498583\pi$
$458$	− 50.7718i	− 0.110855i
$459$	0	0
$460$	−306.458	−0.666212
$461$	− 99.7105i	− 0.216292i	−0.994135	−	0.108146i	$-0.965509\pi$
$461$	− 99.7105i	− 0.216292i	0.994135	−	0.108146i	$-0.0344914\pi$
$462$	0	0
$463$	326.079	0.704273	0.352137	−	0.935949i	$-0.385455\pi$
$463$	326.079	0.704273	0.352137	+	0.935949i	$0.385455\pi$
$464$	− 67.1429i	− 0.144704i
$465$	0	0
$466$	−94.1734	−0.202089
$467$	422.171i	0.904007i	0.892016	+	0.452003i	$0.149291\pi$
$467$	422.171i	0.904007i	−0.892016	+	0.452003i	$0.850709\pi$
$468$	0	0
$469$	215.598	0.459698
$470$	− 52.7348i	− 0.112202i
$471$	0	0
$472$	−260.384	−0.551661
$473$	0	0
$474$	0	0
$475$	195.126	0.410791
$476$	447.653i	0.940447i
$477$	0	0
$478$	63.0807	0.131968
$479$	− 467.797i	− 0.976611i	−0.872673	−	0.488305i	$-0.837615\pi$
$479$	− 467.797i	− 0.976611i	0.872673	−	0.488305i	$-0.162385\pi$
$480$	0	0
$481$	175.297	0.364443
$482$	− 5.20633i	− 0.0108015i
$483$	0	0
$484$	0	0
$485$	58.1952i	0.119990i
$486$	0	0
$487$	−414.175	−0.850463	−0.425231	−	0.905085i	$-0.639807\pi$
$487$	−414.175	−0.850463	−0.425231	+	0.905085i	$0.639807\pi$
$488$	− 113.810i	− 0.233217i
$489$	0	0
$490$	40.8371	0.0833410
$491$	176.284i	0.359031i	0.983755	+	0.179516i	$0.0574530\pi$
$491$	176.284i	0.359031i	−0.983755	+	0.179516i	$0.942547\pi$
$492$	0	0
$493$	−134.954	−0.273740
$494$	25.1962i	0.0510044i
$495$	0	0
$496$	398.411	0.803249
$497$	361.110i	0.726580i
$498$	0	0
$499$	605.317	1.21306	0.606530	−	0.795061i	$-0.292561\pi$
$499$	605.317	1.21306	0.606530	+	0.795061i	$0.292561\pi$
$500$	508.813i	1.01763i
$501$	0	0
$502$	77.3941	0.154171
$503$	180.872i	0.359587i	0.983704	+	0.179793i	$0.0575429\pi$
$503$	180.872i	0.359587i	−0.983704	+	0.179793i	$0.942457\pi$
$504$	0	0
$505$	567.057	1.12288
$506$	0	0
$507$	0	0
$508$	−48.7698	−0.0960035
$509$	− 23.8193i	− 0.0467962i	−0.999726	−	0.0233981i	$-0.992551\pi$
$509$	− 23.8193i	− 0.0467962i	0.999726	−	0.0233981i	$-0.00744853\pi$
$510$	0	0
$511$	−433.903	−0.849126
$512$	400.150i	0.781543i
$513$	0	0
$514$	11.9089	0.0231691
$515$	500.942i	0.972704i
$516$	0	0
$517$	0	0
$518$	− 52.1235i	− 0.100624i
$519$	0	0
$520$	−45.9839	−0.0884305
$521$	198.640i	0.381267i	0.981661	+	0.190633i	$0.0610542\pi$
$521$	198.640i	0.381267i	−0.981661	+	0.190633i	$0.938946\pi$
$522$	0	0
$523$	−1000.04	−1.91212	−0.956059	−	0.293173i	$-0.905289\pi$
$523$	−1000.04	−1.91212	−0.956059	+	0.293173i	$0.905289\pi$
$524$	78.8529i	0.150483i
$525$	0	0
$526$	119.520	0.227225
$527$	− 800.786i	− 1.51952i
$528$	0	0
$529$	1.66120	0.00314027
$530$	− 96.0561i	− 0.181238i
$531$	0	0
$532$	−228.444	−0.429405
$533$	324.940i	0.609643i
$534$	0	0
$535$	−652.668	−1.21994
$536$	− 152.451i	− 0.284424i
$537$	0	0
$538$	120.375	0.223746
$539$	0	0
$540$	0	0
$541$	−491.351	−0.908227	−0.454113	−	0.890944i	$-0.650044\pi$
$541$	−491.351	−0.908227	−0.454113	+	0.890944i	$0.650044\pi$
$542$	155.566i	0.287023i
$543$	0	0
$544$	477.361	0.877502
$545$	− 198.273i	− 0.363803i
$546$	0	0
$547$	−195.142	−0.356750	−0.178375	−	0.983963i	$-0.557084\pi$
$547$	−195.142	−0.356750	−0.178375	+	0.983963i	$0.557084\pi$
$548$	− 139.106i	− 0.253843i
$549$	0	0
$550$	0	0
$551$	− 68.8688i	− 0.124989i
$552$	0	0
$553$	568.601	1.02821
$554$	70.9687i	0.128102i
$555$	0	0
$556$	−391.524	−0.704179
$557$	− 615.014i	− 1.10416i	−0.833793	−	0.552078i	$-0.813835\pi$
$557$	− 615.014i	− 1.10416i	0.833793	−	0.552078i	$-0.186165\pi$
$558$	0	0
$559$	127.925	0.228847
$560$	− 198.149i	− 0.353837i
$561$	0	0
$562$	−89.2561	−0.158819
$563$	328.204i	0.582955i	0.956578	+	0.291477i	$0.0941468\pi$
$563$	328.204i	0.582955i	−0.956578	+	0.291477i	$0.905853\pi$
$564$	0	0
$565$	−102.603	−0.181598
$566$	15.3539i	0.0271270i
$567$	0	0
$568$	255.344	0.449549
$569$	830.739i	1.46000i	0.683448	+	0.729999i	$0.260480\pi$
$569$	830.739i	1.46000i	−0.683448	+	0.729999i	$0.739520\pi$
$570$	0	0
$571$	−1048.66	−1.83652	−0.918262	−	0.395973i	$-0.870407\pi$
$571$	−1048.66	−1.83652	−0.918262	+	0.395973i	$0.870407\pi$
$572$	0	0
$573$	0	0
$574$	96.6189	0.168326
$575$	301.447i	0.524256i
$576$	0	0
$577$	−518.202	−0.898096	−0.449048	−	0.893508i	$-0.648237\pi$
$577$	−518.202	−0.898096	−0.449048	+	0.893508i	$0.648237\pi$
$578$	− 199.382i	− 0.344951i
$579$	0	0
$580$	61.8300	0.106603
$581$	174.598i	0.300513i
$582$	0	0
$583$	0	0
$584$	306.816i	0.525370i
$585$	0	0
$586$	2.67734	0.00456883
$587$	1016.52i	1.73173i	0.500282	+	0.865863i	$0.333230\pi$
$587$	1016.52i	1.73173i	−0.500282	+	0.865863i	$0.666770\pi$
$588$	0	0
$589$	408.652	0.693807
$590$	− 113.961i	− 0.193153i
$591$	0	0
$592$	534.127	0.902242
$593$	1073.37i	1.81007i	0.425341	+	0.905033i	$0.360154\pi$
$593$	1073.37i	1.81007i	−0.425341	+	0.905033i	$0.639846\pi$
$594$	0	0
$595$	−398.268	−0.669358
$596$	− 814.532i	− 1.36666i
$597$	0	0
$598$	−38.9252	−0.0650923
$599$	− 767.105i	− 1.28064i	−0.768107	−	0.640322i	$-0.778801\pi$
$599$	− 767.105i	− 1.28064i	0.768107	−	0.640322i	$-0.221199\pi$
$600$	0	0
$601$	752.890	1.25273	0.626365	−	0.779530i	$-0.284542\pi$
$601$	752.890	1.25273	0.626365	+	0.779530i	$0.284542\pi$
$602$	− 38.0379i	− 0.0631858i
$603$	0	0
$604$	821.376	1.35989
$605$	0	0
$606$	0	0
$607$	−503.535	−0.829548	−0.414774	−	0.909925i	$-0.636139\pi$
$607$	−503.535	−0.829548	−0.414774	+	0.909925i	$0.636139\pi$
$608$	243.604i	0.400664i
$609$	0	0
$610$	49.8105	0.0816566
$611$	204.241i	0.334273i
$612$	0	0
$613$	748.338	1.22078	0.610390	−	0.792101i	$-0.291013\pi$
$613$	748.338	1.22078	0.610390	+	0.792101i	$0.291013\pi$
$614$	− 128.400i	− 0.209120i
$615$	0	0
$616$	0	0
$617$	− 634.767i	− 1.02880i	−0.857552	−	0.514398i	$-0.828016\pi$
$617$	− 634.767i	− 1.02880i	0.857552	−	0.514398i	$-0.171984\pi$
$618$	0	0
$619$	−684.329	−1.10554	−0.552769	−	0.833334i	$-0.686429\pi$
$619$	−684.329	−1.10554	−0.552769	+	0.833334i	$0.686429\pi$
$620$	366.886i	0.591751i
$621$	0	0
$622$	148.125	0.238143
$623$	257.027i	0.412563i
$624$	0	0
$625$	−124.506	−0.199210
$626$	− 22.5549i	− 0.0360301i
$627$	0	0
$628$	461.556	0.734963
$629$	− 1073.57i	− 1.70678i
$630$	0	0
$631$	−266.046	−0.421627	−0.210813	−	0.977526i	$-0.567611\pi$
$631$	−266.046	−0.421627	−0.210813	+	0.977526i	$0.567611\pi$
$632$	− 402.061i	− 0.636173i
$633$	0	0
$634$	150.573	0.237497
$635$	− 43.3895i	− 0.0683300i
$636$	0	0
$637$	−158.161	−0.248291
$638$	0	0
$639$	0	0
$640$	−289.893	−0.452957
$641$	− 450.083i	− 0.702158i	−0.936346	−	0.351079i	$-0.885815\pi$
$641$	− 450.083i	− 0.702158i	0.936346	−	0.351079i	$-0.114185\pi$
$642$	0	0
$643$	142.988	0.222376	0.111188	−	0.993799i	$-0.464534\pi$
$643$	142.988	0.222376	0.111188	+	0.993799i	$0.464534\pi$
$644$	− 352.919i	− 0.548011i
$645$	0	0
$646$	154.309	0.238868
$647$	796.539i	1.23113i	0.788088	+	0.615563i	$0.211071\pi$
$647$	796.539i	1.23113i	−0.788088	+	0.615563i	$0.788929\pi$
$648$	0	0
$649$	0	0
$650$	22.2511i	0.0342325i
$651$	0	0
$652$	1031.25	1.58167
$653$	54.6582i	0.0837032i	0.999124	+	0.0418516i	$0.0133257\pi$
$653$	54.6582i	0.0837032i	−0.999124	+	0.0418516i	$0.986674\pi$
$654$	0	0
$655$	−70.1539	−0.107105
$656$	990.087i	1.50928i
$657$	0	0
$658$	60.7298	0.0922946
$659$	333.527i	0.506111i	0.967452	+	0.253056i	$0.0814356\pi$
$659$	333.527i	0.506111i	−0.967452	+	0.253056i	$0.918564\pi$
$660$	0	0
$661$	−489.397	−0.740389	−0.370195	−	0.928954i	$-0.620709\pi$
$661$	−489.397	−0.740389	−0.370195	+	0.928954i	$0.620709\pi$
$662$	− 83.0671i	− 0.125479i
$663$	0	0
$664$	123.460	0.185933
$665$	− 203.242i	− 0.305627i
$666$	0	0
$667$	106.394	0.159512
$668$	437.994i	0.655680i
$669$	0	0
$670$	66.7223	0.0995856
$671$	0	0
$672$	0	0
$673$	−699.341	−1.03914	−0.519570	−	0.854428i	$-0.673908\pi$
$673$	−699.341	−1.03914	−0.519570	+	0.854428i	$0.673908\pi$
$674$	− 221.768i	− 0.329032i
$675$	0	0
$676$	−566.923	−0.838644
$677$	389.391i	0.575171i	0.957755	+	0.287585i	$0.0928525\pi$
$677$	389.391i	0.575171i	−0.957755	+	0.287585i	$0.907148\pi$
$678$	0	0
$679$	−67.0180	−0.0987011
$680$	281.618i	0.414144i
$681$	0	0
$682$	0	0
$683$	332.412i	0.486693i	0.969939	+	0.243347i	$0.0782453\pi$
$683$	332.412i	0.486693i	−0.969939	+	0.243347i	$0.921755\pi$
$684$	0	0
$685$	123.760	0.180672
$686$	116.325i	0.169570i
$687$	0	0
$688$	389.787	0.566551
$689$	372.024i	0.539947i
$690$	0	0
$691$	873.853	1.26462	0.632310	−	0.774715i	$-0.282107\pi$
$691$	873.853	1.26462	0.632310	+	0.774715i	$0.282107\pi$
$692$	455.521i	0.658267i
$693$	0	0
$694$	−231.016	−0.332876
$695$	− 348.331i	− 0.501196i
$696$	0	0
$697$	1990.02	2.85512
$698$	− 108.388i	− 0.155284i
$699$	0	0
$700$	−201.742	−0.288203
$701$	630.096i	0.898853i	0.893317	+	0.449426i	$0.148372\pi$
$701$	630.096i	0.898853i	−0.893317	+	0.449426i	$0.851628\pi$
$702$	0	0
$703$	547.856	0.779312
$704$	0	0
$705$	0	0
$706$	89.3670	0.126582
$707$	653.027i	0.923660i
$708$	0	0
$709$	−1007.61	−1.42117	−0.710587	−	0.703609i	$-0.751571\pi$
$709$	−1007.61	−1.42117	−0.710587	+	0.703609i	$0.751571\pi$
$710$	111.755i	0.157401i
$711$	0	0
$712$	181.745	0.255260
$713$	631.321i	0.885443i
$714$	0	0
$715$	0	0
$716$	− 896.697i	− 1.25237i
$717$	0	0
$718$	−40.7803	−0.0567971
$719$	170.989i	0.237815i	0.992905	+	0.118907i	$0.0379391\pi$
$719$	170.989i	0.237815i	−0.992905	+	0.118907i	$0.962061\pi$
$720$	0	0
$721$	−576.889	−0.800124
$722$	− 49.9124i	− 0.0691308i
$723$	0	0
$724$	−41.8689	−0.0578300
$725$	− 60.8191i	− 0.0838884i
$726$	0	0
$727$	408.204	0.561490	0.280745	−	0.959782i	$-0.409418\pi$
$727$	408.204	0.561490	0.280745	+	0.959782i	$0.409418\pi$
$728$	− 52.9554i	− 0.0727409i
$729$	0	0
$730$	−134.282	−0.183948
$731$	− 783.451i	− 1.07175i
$732$	0	0
$733$	−1178.02	−1.60713	−0.803563	−	0.595220i	$-0.797065\pi$
$733$	−1178.02	−1.60713	−0.803563	+	0.595220i	$0.797065\pi$
$734$	14.2493i	0.0194132i
$735$	0	0
$736$	−376.340	−0.511332
$737$	0	0
$738$	0	0
$739$	−938.068	−1.26938	−0.634688	−	0.772769i	$-0.718871\pi$
$739$	−938.068	−1.26938	−0.634688	+	0.772769i	$0.718871\pi$
$740$	491.863i	0.664679i
$741$	0	0
$742$	110.619	0.149082
$743$	− 1050.56i	− 1.41394i	−0.707242	−	0.706971i	$-0.750061\pi$
$743$	− 1050.56i	− 1.41394i	0.707242	−	0.706971i	$-0.249939\pi$
$744$	0	0
$745$	724.673	0.972716
$746$	− 157.870i	− 0.211622i
$747$	0	0
$748$	0	0
$749$	− 751.617i	− 1.00349i
$750$	0	0
$751$	1074.61	1.43091	0.715456	−	0.698658i	$-0.246219\pi$
$751$	1074.61	1.43091	0.715456	+	0.698658i	$0.246219\pi$
$752$	622.320i	0.827553i
$753$	0	0
$754$	7.85344	0.0104157
$755$	730.763i	0.967898i
$756$	0	0
$757$	−518.488	−0.684924	−0.342462	−	0.939532i	$-0.611261\pi$
$757$	−518.488	−0.684924	−0.342462	+	0.939532i	$0.611261\pi$
$758$	− 80.3359i	− 0.105984i
$759$	0	0
$760$	−143.714	−0.189097
$761$	− 1304.01i	− 1.71355i	−0.515690	−	0.856775i	$-0.672464\pi$
$761$	− 1304.01i	− 1.71355i	0.515690	−	0.856775i	$-0.327536\pi$
$762$	0	0
$763$	228.333	0.299256
$764$	− 191.791i	− 0.251036i
$765$	0	0
$766$	65.1930	0.0851083
$767$	441.367i	0.575446i
$768$	0	0
$769$	−141.048	−0.183418	−0.0917088	−	0.995786i	$-0.529233\pi$
$769$	−141.048	−0.183418	−0.0917088	+	0.995786i	$0.529233\pi$
$770$	0	0
$771$	0	0
$772$	32.5492	0.0421622
$773$	234.469i	0.303323i	0.988432	+	0.151661i	$0.0484624\pi$
$773$	234.469i	0.303323i	−0.988432	+	0.151661i	$0.951538\pi$
$774$	0	0
$775$	360.887	0.465661
$776$	47.3889i	0.0610682i
$777$	0	0
$778$	220.818	0.283828
$779$	1015.54i	1.30364i
$780$	0	0
$781$	0	0
$782$	238.389i	0.304845i
$783$	0	0
$784$	−481.915	−0.614688
$785$	410.638i	0.523106i
$786$	0	0
$787$	1454.04	1.84757	0.923784	−	0.382915i	$-0.125080\pi$
$787$	1454.04	1.84757	0.923784	+	0.382915i	$0.125080\pi$
$788$	− 484.858i	− 0.615302i
$789$	0	0
$790$	175.968	0.222744
$791$	− 118.158i	− 0.149378i
$792$	0	0
$793$	−192.915	−0.243273
$794$	− 16.1678i	− 0.0203624i
$795$	0	0
$796$	990.429	1.24426
$797$	325.332i	0.408195i	0.978951	+	0.204098i	$0.0654260\pi$
$797$	325.332i	0.408195i	−0.978951	+	0.204098i	$0.934574\pi$
$798$	0	0
$799$	1250.83	1.56549
$800$	215.130i	0.268913i
$801$	0	0
$802$	128.776	0.160568
$803$	0	0
$804$	0	0
$805$	313.985	0.390044
$806$	46.6006i	0.0578171i
$807$	0	0
$808$	461.760	0.571485
$809$	− 894.483i	− 1.10567i	−0.833292	−	0.552833i	$-0.813547\pi$
$809$	− 894.483i	− 1.10567i	0.833292	−	0.552833i	$-0.186453\pi$
$810$	0	0
$811$	−92.4668	−0.114016	−0.0570079	−	0.998374i	$-0.518156\pi$
$811$	−92.4668	−0.114016	−0.0570079	+	0.998374i	$0.518156\pi$
$812$	71.2039i	0.0876896i
$813$	0	0
$814$	0	0
$815$	917.485i	1.12575i
$816$	0	0
$817$	399.806	0.489359
$818$	243.682i	0.297900i
$819$	0	0
$820$	−911.743	−1.11188
$821$	− 616.078i	− 0.750399i	−0.926944	−	0.375200i	$-0.877574\pi$
$821$	− 616.078i	− 0.750399i	0.926944	−	0.375200i	$-0.122426\pi$
$822$	0	0
$823$	1584.99	1.92587	0.962937	−	0.269727i	$-0.0869336\pi$
$823$	1584.99	1.92587	0.962937	+	0.269727i	$0.0869336\pi$
$824$	407.922i	0.495051i
$825$	0	0
$826$	131.238	0.158884
$827$	− 776.862i	− 0.939374i	−0.882833	−	0.469687i	$-0.844367\pi$
$827$	− 776.862i	− 0.939374i	0.882833	−	0.469687i	$-0.155633\pi$
$828$	0	0
$829$	−1047.54	−1.26361	−0.631807	−	0.775126i	$-0.717687\pi$
$829$	−1047.54	−1.26361	−0.631807	+	0.775126i	$0.717687\pi$
$830$	54.0338i	0.0651010i
$831$	0	0
$832$	247.924	0.297986
$833$	968.624i	1.16281i
$834$	0	0
$835$	−389.675	−0.466677
$836$	0	0
$837$	0	0
$838$	−147.372	−0.175862
$839$	− 197.206i	− 0.235049i	−0.993070	−	0.117524i	$-0.962504\pi$
$839$	− 197.206i	− 0.235049i	0.993070	−	0.117524i	$-0.0374958\pi$
$840$	0	0
$841$	819.534	0.974476
$842$	− 117.061i	− 0.139027i
$843$	0	0
$844$	578.362	0.685263
$845$	− 504.381i	− 0.596901i
$846$	0	0
$847$	0	0
$848$	1133.55i	1.33673i
$849$	0	0
$850$	136.272	0.160320
$851$	846.375i	0.994566i
$852$	0	0
$853$	−1389.41	−1.62885	−0.814423	−	0.580271i	$-0.802947\pi$
$853$	−1389.41	−1.62885	−0.814423	+	0.580271i	$0.802947\pi$
$854$	57.3622i	0.0671688i
$855$	0	0
$856$	−531.474	−0.620881
$857$	218.943i	0.255476i	0.991808	+	0.127738i	$0.0407717\pi$
$857$	218.943i	0.255476i	−0.991808	+	0.127738i	$0.959228\pi$
$858$	0	0
$859$	679.659	0.791221	0.395611	−	0.918418i	$-0.370533\pi$
$859$	679.659	0.791221	0.395611	+	0.918418i	$0.370533\pi$
$860$	358.944i	0.417377i
$861$	0	0
$862$	−193.224	−0.224158
$863$	719.743i	0.834002i	0.908906	+	0.417001i	$0.136919\pi$
$863$	719.743i	0.834002i	−0.908906	+	0.417001i	$0.863081\pi$
$864$	0	0
$865$	−405.268	−0.468518
$866$	− 255.500i	− 0.295035i
$867$	0	0
$868$	−422.509	−0.486761
$869$	0	0
$870$	0	0
$871$	−258.414	−0.296687
$872$	− 161.456i	− 0.185155i
$873$	0	0
$874$	−121.653	−0.139191
$875$	− 521.312i	− 0.595785i
$876$	0	0
$877$	920.439	1.04953	0.524766	−	0.851247i	$-0.324153\pi$
$877$	920.439	1.04953	0.524766	+	0.851247i	$0.324153\pi$
$878$	70.1349i	0.0798802i
$879$	0	0
$880$	0	0
$881$	− 309.389i	− 0.351180i	−0.984463	−	0.175590i	$-0.943817\pi$
$881$	− 309.389i	− 0.351180i	0.984463	−	0.175590i	$-0.0561832\pi$
$882$	0	0
$883$	−856.028	−0.969454	−0.484727	−	0.874665i	$-0.661081\pi$
$883$	−856.028	−0.969454	−0.484727	+	0.874665i	$0.661081\pi$
$884$	− 536.553i	− 0.606960i
$885$	0	0
$886$	186.877	0.210922
$887$	508.096i	0.572826i	0.958106	+	0.286413i	$0.0924629\pi$
$887$	508.096i	0.572826i	−0.958106	+	0.286413i	$0.907537\pi$
$888$	0	0
$889$	49.9677	0.0562067
$890$	79.5434i	0.0893746i
$891$	0	0
$892$	−818.246	−0.917316
$893$	638.316i	0.714799i
$894$	0	0
$895$	797.774	0.891368
$896$	− 333.843i	− 0.372592i
$897$	0	0
$898$	111.560	0.124232
$899$	− 127.373i	− 0.141684i
$900$	0	0
$901$	2278.38	2.52872
$902$	0	0
$903$	0	0
$904$	−83.5505	−0.0924231
$905$	− 37.2500i	− 0.0411602i
$906$	0	0
$907$	−509.667	−0.561927	−0.280963	−	0.959719i	$-0.590654\pi$
$907$	−509.667	−0.561927	−0.280963	+	0.959719i	$0.590654\pi$
$908$	− 678.124i	− 0.746832i
$909$	0	0
$910$	23.1767	0.0254689
$911$	451.807i	0.495946i	0.968767	+	0.247973i	$0.0797645\pi$
$911$	451.807i	0.495946i	−0.968767	+	0.247973i	$0.920236\pi$
$912$	0	0
$913$	0	0
$914$	1.44989i	0.00158631i
$915$	0	0
$916$	−551.744	−0.602341
$917$	− 80.7898i	− 0.0881023i
$918$	0	0
$919$	−1645.29	−1.79030	−0.895152	−	0.445762i	$-0.852933\pi$
$919$	−1645.29	−1.79030	−0.895152	+	0.445762i	$0.852933\pi$
$920$	− 222.021i	− 0.241327i
$921$	0	0
$922$	35.5362	0.0385426
$923$	− 432.824i	− 0.468932i
$924$	0	0
$925$	483.820	0.523049
$926$	116.212i	0.125499i
$927$	0	0
$928$	75.9293	0.0818204
$929$	1005.87i	1.08274i	0.840783	+	0.541372i	$0.182095\pi$
$929$	1005.87i	1.08274i	−0.840783	+	0.541372i	$0.817905\pi$
$930$	0	0
$931$	−494.303	−0.530937
$932$	1023.40i	1.09806i
$933$	0	0
$934$	−150.459	−0.161091
$935$	0	0
$936$	0	0
$937$	209.683	0.223781	0.111891	−	0.993721i	$-0.464309\pi$
$937$	209.683	0.223781	0.111891	+	0.993721i	$0.464309\pi$
$938$	76.8380i	0.0819168i
$939$	0	0
$940$	−573.077	−0.609656
$941$	601.375i	0.639081i	0.947573	+	0.319541i	$0.103529\pi$
$941$	601.375i	0.639081i	−0.947573	+	0.319541i	$0.896471\pi$
$942$	0	0
$943$	−1568.89	−1.66372
$944$	1344.84i	1.42462i
$945$	0	0
$946$	0	0
$947$	− 18.4760i	− 0.0195101i	−0.999952	−	0.00975504i	$-0.996895\pi$
$947$	− 18.4760i	− 0.0195101i	0.999952	−	0.00975504i	$-0.00310517\pi$
$948$	0	0
$949$	520.073	0.548022
$950$	69.5416i	0.0732017i
$951$	0	0
$952$	−324.314	−0.340666
$953$	− 1285.82i	− 1.34923i	−0.738169	−	0.674616i	$-0.764309\pi$
$953$	− 1285.82i	− 1.34923i	0.738169	−	0.674616i	$-0.235691\pi$
$954$	0	0
$955$	170.633	0.178673
$956$	− 685.506i	− 0.717057i
$957$	0	0
$958$	166.720	0.174029
$959$	142.523i	0.148616i
$960$	0	0
$961$	−205.194	−0.213521
$962$	62.4747i	0.0649426i
$963$	0	0
$964$	−56.5780	−0.0586908
$965$	28.9584i	0.0300088i
$966$	0	0
$967$	−401.260	−0.414953	−0.207477	−	0.978240i	$-0.566525\pi$
$967$	−401.260	−0.414953	−0.207477	+	0.978240i	$0.566525\pi$
$968$	0	0
$969$	0	0
$970$	−20.7404	−0.0213819
$971$	1310.59i	1.34973i	0.737942	+	0.674864i	$0.235798\pi$
$971$	1310.59i	1.34973i	−0.737942	+	0.674864i	$0.764202\pi$
$972$	0	0
$973$	401.141	0.412272
$974$	− 147.610i	− 0.151550i
$975$	0	0
$976$	−587.810	−0.602264
$977$	− 1220.43i	− 1.24916i	−0.780960	−	0.624581i	$-0.785270\pi$
$977$	− 1220.43i	− 1.24916i	0.780960	−	0.624581i	$-0.214730\pi$
$978$	0	0
$979$	0	0
$980$	− 443.782i	− 0.452839i
$981$	0	0
$982$	−62.8266	−0.0639782
$983$	− 89.1639i	− 0.0907059i	−0.998971	−	0.0453530i	$-0.985559\pi$
$983$	− 89.1639i	− 0.0907059i	0.998971	−	0.0453530i	$-0.0144412\pi$
$984$	0	0
$985$	431.369	0.437938
$986$	− 48.0967i	− 0.0487796i
$987$	0	0
$988$	273.810	0.277136
$989$	617.655i	0.624525i
$990$	0	0
$991$	48.9921	0.0494370	0.0247185	−	0.999694i	$-0.492131\pi$
$991$	48.9921	0.0494370	0.0247185	+	0.999694i	$0.492131\pi$
$992$	450.548i	0.454181i
$993$	0	0
$994$	−128.698	−0.129474
$995$	881.166i	0.885594i
$996$	0	0
$997$	209.770	0.210401	0.105201	−	0.994451i	$-0.466451\pi$
$997$	209.770	0.210401	0.105201	+	0.994451i	$0.466451\pi$
$998$	215.731i	0.216163i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.b.h.485.6	yes	8
3.2	odd	2	inner	1089.3.b.h.485.3	✓	8
11.10	odd	2	inner	1089.3.b.h.485.4	yes	8
33.32	even	2	inner	1089.3.b.h.485.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1089.3.b.h.485.3	✓	8	3.2	odd	2	inner
1089.3.b.h.485.4	yes	8	11.10	odd	2	inner
1089.3.b.h.485.5	yes	8	33.32	even	2	inner
1089.3.b.h.485.6	yes	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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.356394i q^{2} +3.87298 q^{4} +3.44572i q^{5} -3.96812 q^{7} +2.80588i q^{8} +O(q^{10})\)	\(q+0.356394i q^{2} +3.87298 q^{4} +3.44572i q^{5} -3.96812 q^{7} +2.80588i q^{8} -1.22803 q^{10} +4.75615 q^{13} -1.41421i q^{14} +14.4919 q^{16} -29.1280i q^{17} +14.8644 q^{19} +13.3452i q^{20} +22.9639i q^{23} +13.1270 q^{25} +1.69506i q^{26} -15.3685 q^{28} -4.63312i q^{29} +27.4919 q^{31} +16.3884i q^{32} +10.3810 q^{34} -13.6730i q^{35} +36.8569 q^{37} +5.29760i q^{38} -9.66829 q^{40} +68.3199i q^{41} +26.8968 q^{43} -8.18418 q^{46} +42.9425i q^{47} -33.2540 q^{49} +4.67839i q^{50} +18.4205 q^{52} +78.2194i q^{53} -11.1341i q^{56} +1.65122 q^{58} +92.7992i q^{59} -40.5612 q^{61} +9.79796i q^{62} +52.1270 q^{64} +16.3884i q^{65} -54.3327 q^{67} -112.812i q^{68} +4.87298 q^{70} -91.0029i q^{71} +109.347 q^{73} +13.1356i q^{74} +57.5697 q^{76} -143.292 q^{79} +49.9351i q^{80} -24.3488 q^{82} -44.0003i q^{83} +100.367 q^{85} +9.58587i q^{86} -64.7730i q^{89} -18.8730 q^{91} +88.9386i q^{92} -15.3044 q^{94} +51.2187i q^{95} +16.8891 q^{97} -11.8515i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{16} + 136 q^{25} + 96 q^{31} + 176 q^{34} + 16 q^{37} - 328 q^{49} + 416 q^{58} + 448 q^{64} + 216 q^{67} + 8 q^{70} + 208 q^{82} - 120 q^{91} + 352 q^{97}+O(q^{100}) \) 8 * q - 8 * q^16 + 136 * q^25 + 96 * q^31 + 176 * q^34 + 16 * q^37 - 328 * q^49 + 416 * q^58 + 448 * q^64 + 216 * q^67 + 8 * q^70 + 208 * q^82 - 120 * q^91 + 352 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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