LMFDB - Embedded newform 1089.3.b.h.485.8

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			485.8
Root			$-0.319918i$ of defining polynomial
Character	$\chi$	$=$	1089.485
Dual form			1089.3.b.h.485.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.80588i q^{2} -3.87298 q^{4} +2.03151i q^{5} +0.504017 q^{7} +0.356394i q^{8} +O(q^{10})$	$q+2.80588i q^{2} -3.87298 q^{4} +2.03151i q^{5} +0.504017 q^{7} +0.356394i q^{8} -5.70017 q^{10} -22.0767 q^{13} +1.41421i q^{14} -16.4919 q^{16} -11.9816i q^{17} +5.92017 q^{19} -7.86799i q^{20} +42.7628i q^{23} +20.8730 q^{25} -61.9445i q^{26} -1.95205 q^{28} -36.4765i q^{29} -3.49193 q^{31} -44.8489i q^{32} +33.6190 q^{34} +1.02391i q^{35} -32.8569 q^{37} +16.6113i q^{38} -0.724016 q^{40} -27.2102i q^{41} -58.0737 q^{43} -119.988 q^{46} +11.8298i q^{47} -48.7460 q^{49} +58.5672i q^{50} +85.5025 q^{52} -61.7877i q^{53} +0.179629i q^{56} +102.349 q^{58} -59.9359i q^{59} -49.5055 q^{61} -9.79796i q^{62} +59.8730 q^{64} -44.8489i q^{65} +108.333 q^{67} +46.4045i q^{68} -2.87298 q^{70} -51.4049i q^{71} -123.204 q^{73} -92.1925i q^{74} -22.9287 q^{76} -9.12820 q^{79} -33.5035i q^{80} +76.3488 q^{82} +88.2722i q^{83} +24.3407 q^{85} -162.948i q^{86} +4.52349i q^{89} -11.1270 q^{91} -165.620i q^{92} -33.1930 q^{94} +12.0269i q^{95} +71.1109 q^{97} -136.776i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.80588i	1.40294i	0.712698	+	0.701471i	$0.247473\pi$
$2$	2.80588i	1.40294i	−0.712698	+	0.701471i	$0.752527\pi$
$3$	0	0
$3$	0	0
$4$	−3.87298	−0.968246
$5$	2.03151i	0.406301i	0.979148	+	0.203151i	$0.0651181\pi$
$5$	2.03151i	0.406301i	−0.979148	+	0.203151i	$0.934882\pi$
$6$	0	0
$7$	0.504017	0.0720025	0.0360012	−	0.999352i	$-0.488538\pi$
$7$	0.504017	0.0720025	0.0360012	+	0.999352i	$0.488538\pi$
$8$	0.356394i	0.0445492i
$9$	0	0
$10$	−5.70017	−0.570017
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−22.0767	−1.69820	−0.849102	−	0.528228i	$-0.822857\pi$
$13$	−22.0767	−1.69820	−0.849102	+	0.528228i	$0.822857\pi$
$14$	1.41421i	0.101015i
$15$	0	0
$16$	−16.4919	−1.03075
$17$	− 11.9816i	− 0.704799i	−0.935850	−	0.352400i	$-0.885366\pi$
$17$	− 11.9816i	− 0.704799i	0.935850	−	0.352400i	$-0.114634\pi$
$18$	0	0
$19$	5.92017	0.311588	0.155794	−	0.987790i	$-0.450206\pi$
$19$	5.92017	0.311588	0.155794	+	0.987790i	$0.450206\pi$
$20$	− 7.86799i	− 0.393399i
$21$	0	0
$22$	0	0
$23$	42.7628i	1.85925i	0.368502	+	0.929627i	$0.379871\pi$
$23$	42.7628i	1.85925i	−0.368502	+	0.929627i	$0.620129\pi$
$24$	0	0
$25$	20.8730	0.834919
$26$	− 61.9445i	− 2.38248i
$27$	0	0
$28$	−1.95205	−0.0697161
$29$	− 36.4765i	− 1.25781i	−0.777482	−	0.628905i	$-0.783503\pi$
$29$	− 36.4765i	− 1.25781i	0.777482	−	0.628905i	$-0.216497\pi$
$30$	0	0
$31$	−3.49193	−0.112643	−0.0563215	−	0.998413i	$-0.517937\pi$
$31$	−3.49193	−0.112643	−0.0563215	+	0.998413i	$0.517937\pi$
$32$	− 44.8489i	− 1.40153i
$33$	0	0
$34$	33.6190	0.988793
$35$	1.02391i	0.0292547i
$36$	0	0
$37$	−32.8569	−0.888023	−0.444011	−	0.896021i	$-0.646445\pi$
$37$	−32.8569	−0.888023	−0.444011	+	0.896021i	$0.646445\pi$
$38$	16.6113i	0.437140i
$39$	0	0
$40$	−0.724016	−0.0181004
$41$	− 27.2102i	− 0.663665i	−0.943338	−	0.331832i	$-0.892333\pi$
$41$	− 27.2102i	− 0.663665i	0.943338	−	0.331832i	$-0.107667\pi$
$42$	0	0
$43$	−58.0737	−1.35055	−0.675276	−	0.737565i	$-0.735976\pi$
$43$	−58.0737	−1.35055	−0.675276	+	0.737565i	$0.735976\pi$
$44$	0	0
$45$	0	0
$46$	−119.988	−2.60843
$47$	11.8298i	0.251697i	0.992049	+	0.125849i	$0.0401654\pi$
$47$	11.8298i	0.251697i	−0.992049	+	0.125849i	$0.959835\pi$
$48$	0	0
$49$	−48.7460	−0.994816
$50$	58.5672i	1.17134i
$51$	0	0
$52$	85.5025	1.64428
$53$	− 61.7877i	− 1.16581i	−0.812542	−	0.582903i	$-0.801917\pi$
$53$	− 61.7877i	− 1.16581i	0.812542	−	0.582903i	$-0.198083\pi$
$54$	0	0
$55$	0	0
$56$	0.179629i	0.00320765i
$57$	0	0
$58$	102.349	1.76463
$59$	− 59.9359i	− 1.01586i	−0.861398	−	0.507931i	$-0.830410\pi$
$59$	− 59.9359i	− 1.01586i	0.861398	−	0.507931i	$-0.169590\pi$
$60$	0	0
$61$	−49.5055	−0.811565	−0.405782	−	0.913970i	$-0.633001\pi$
$61$	−49.5055	−0.811565	−0.405782	+	0.913970i	$0.633001\pi$
$62$	− 9.79796i	− 0.158032i
$63$	0	0
$64$	59.8730	0.935515
$65$	− 44.8489i	− 0.689983i
$66$	0	0
$67$	108.333	1.61691	0.808453	−	0.588561i	$-0.200305\pi$
$67$	108.333	1.61691	0.808453	+	0.588561i	$0.200305\pi$
$68$	46.4045i	0.682419i
$69$	0	0
$70$	−2.87298	−0.0410426
$71$	− 51.4049i	− 0.724013i	−0.932176	−	0.362007i	$-0.882092\pi$
$71$	− 51.4049i	− 0.724013i	0.932176	−	0.362007i	$-0.117908\pi$
$72$	0	0
$73$	−123.204	−1.68772	−0.843861	−	0.536561i	$-0.819723\pi$
$73$	−123.204	−1.68772	−0.843861	+	0.536561i	$0.819723\pi$
$74$	− 92.1925i	− 1.24584i
$75$	0	0
$76$	−22.9287	−0.301694
$77$	0	0
$78$	0	0
$79$	−9.12820	−0.115547	−0.0577734	−	0.998330i	$-0.518400\pi$
$79$	−9.12820	−0.115547	−0.0577734	+	0.998330i	$0.518400\pi$
$80$	− 33.5035i	− 0.418793i
$81$	0	0
$82$	76.3488	0.931083
$83$	88.2722i	1.06352i	0.846895	+	0.531760i	$0.178469\pi$
$83$	88.2722i	1.06352i	−0.846895	+	0.531760i	$0.821531\pi$
$84$	0	0
$85$	24.3407	0.286361
$86$	− 162.948i	− 1.89475i
$87$	0	0
$88$	0	0
$89$	4.52349i	0.0508257i	0.999677	+	0.0254129i	$0.00809004\pi$
$89$	4.52349i	0.0508257i	−0.999677	+	0.0254129i	$0.991910\pi$
$90$	0	0
$91$	−11.1270	−0.122275
$92$	− 165.620i	− 1.80022i
$93$	0	0
$94$	−33.1930	−0.353117
$95$	12.0269i	0.126599i
$96$	0	0
$97$	71.1109	0.733102	0.366551	−	0.930398i	$-0.380539\pi$
$97$	71.1109	0.733102	0.366551	+	0.930398i	$0.380539\pi$
$98$	− 136.776i	− 1.39567i
$99$	0	0
$100$	−80.8407	−0.808407
$101$	− 56.7909i	− 0.562287i	−0.959666	−	0.281143i	$-0.909286\pi$
$101$	− 56.7909i	− 0.562287i	0.959666	−	0.281143i	$-0.0907136\pi$
$102$	0	0
$103$	168.619	1.63708	0.818539	−	0.574452i	$-0.194785\pi$
$103$	168.619	1.63708	0.818539	+	0.574452i	$0.194785\pi$
$104$	− 7.86799i	− 0.0756537i
$105$	0	0
$106$	173.369	1.63556
$107$	− 138.818i	− 1.29736i	−0.761061	−	0.648680i	$-0.775321\pi$
$107$	− 138.818i	− 1.29736i	0.761061	−	0.648680i	$-0.224679\pi$
$108$	0	0
$109$	130.288	1.19530	0.597651	−	0.801756i	$-0.296101\pi$
$109$	130.288	1.19530	0.597651	+	0.801756i	$0.296101\pi$
$110$	0	0
$111$	0	0
$112$	−8.31222	−0.0742162
$113$	8.56369i	0.0757848i	0.999282	+	0.0378924i	$0.0120644\pi$
$113$	8.56369i	0.0757848i	−0.999282	+	0.0378924i	$0.987936\pi$
$114$	0	0
$115$	−86.8730	−0.755417
$116$	141.273i	1.21787i
$117$	0	0
$118$	168.173	1.42520
$119$	− 6.03893i	− 0.0507473i
$120$	0	0
$121$	0	0
$122$	− 138.907i	− 1.13858i
$123$	0	0
$124$	13.5242	0.109066
$125$	93.1912i	0.745530i
$126$	0	0
$127$	−146.756	−1.15556	−0.577781	−	0.816192i	$-0.696081\pi$
$127$	−146.756	−1.15556	−0.577781	+	0.816192i	$0.696081\pi$
$128$	− 11.3989i	− 0.0890536i
$129$	0	0
$130$	125.841	0.968006
$131$	182.026i	1.38951i	0.719246	+	0.694756i	$0.244488\pi$
$131$	182.026i	1.38951i	−0.719246	+	0.694756i	$0.755512\pi$
$132$	0	0
$133$	2.98387	0.0224351
$134$	303.969i	2.26842i
$135$	0	0
$136$	4.27017	0.0313983
$137$	156.416i	1.14172i	0.821046	+	0.570861i	$0.193391\pi$
$137$	156.416i	1.14172i	−0.821046	+	0.570861i	$0.806609\pi$
$138$	0	0
$139$	−34.0090	−0.244669	−0.122334	−	0.992489i	$-0.539038\pi$
$139$	−34.0090	−0.244669	−0.122334	+	0.992489i	$0.539038\pi$
$140$	− 3.96560i	− 0.0283257i
$141$	0	0
$142$	144.236	1.01575
$143$	0	0
$144$	0	0
$145$	74.1022	0.511050
$146$	− 345.695i	− 2.36778i
$147$	0	0
$148$	127.254	0.859825
$149$	− 134.377i	− 0.901859i	−0.892559	−	0.450930i	$-0.851093\pi$
$149$	− 134.377i	− 0.901859i	0.892559	−	0.450930i	$-0.148907\pi$
$150$	0	0
$151$	−163.581	−1.08332	−0.541659	−	0.840598i	$-0.682204\pi$
$151$	−163.581	−1.08332	−0.541659	+	0.840598i	$0.682204\pi$
$152$	2.10991i	0.0138810i
$153$	0	0
$154$	0	0
$155$	− 7.09388i	− 0.0457670i
$156$	0	0
$157$	−175.173	−1.11575	−0.557877	−	0.829924i	$-0.688384\pi$
$157$	−175.173	−1.11575	−0.557877	+	0.829924i	$0.688384\pi$
$158$	− 25.6127i	− 0.162105i
$159$	0	0
$160$	91.1108	0.569442
$161$	21.5532i	0.133871i
$162$	0	0
$163$	−144.268	−0.885080	−0.442540	−	0.896749i	$-0.645923\pi$
$163$	−144.268	−0.885080	−0.442540	+	0.896749i	$0.645923\pi$
$164$	105.385i	0.642590i
$165$	0	0
$166$	−247.681	−1.49206
$167$	260.059i	1.55724i	0.627496	+	0.778620i	$0.284080\pi$
$167$	260.059i	1.55724i	−0.627496	+	0.778620i	$0.715920\pi$
$168$	0	0
$169$	318.379	1.88390
$170$	68.2971i	0.401748i
$171$	0	0
$172$	224.919	1.30767
$173$	− 269.404i	− 1.55725i	−0.627489	−	0.778625i	$-0.715917\pi$
$173$	− 269.404i	− 1.55725i	0.627489	−	0.778625i	$-0.284083\pi$
$174$	0	0
$175$	10.5203	0.0601162
$176$	0	0
$177$	0	0
$178$	−12.6924	−0.0713056
$179$	34.3460i	0.191877i	0.995387	+	0.0959386i	$0.0305852\pi$
$179$	34.3460i	0.191877i	−0.995387	+	0.0959386i	$0.969415\pi$
$180$	0	0
$181$	−243.190	−1.34359	−0.671794	−	0.740738i	$-0.734476\pi$
$181$	−243.190	−1.34359	−0.671794	+	0.740738i	$0.734476\pi$
$182$	− 31.2211i	− 0.171545i
$183$	0	0
$184$	−15.2404	−0.0828284
$185$	− 66.7489i	− 0.360805i
$186$	0	0
$187$	0	0
$188$	− 45.8165i	− 0.243705i
$189$	0	0
$190$	−33.7460	−0.177610
$191$	137.156i	0.718094i	0.933320	+	0.359047i	$0.116898\pi$
$191$	137.156i	0.718094i	−0.933320	+	0.359047i	$0.883102\pi$
$192$	0	0
$193$	133.624	0.692352	0.346176	−	0.938170i	$-0.387480\pi$
$193$	133.624	0.692352	0.346176	+	0.938170i	$0.387480\pi$
$194$	199.529i	1.02850i
$195$	0	0
$196$	188.792	0.963226
$197$	166.299i	0.844160i	0.906559	+	0.422080i	$0.138700\pi$
$197$	166.299i	0.844160i	−0.906559	+	0.422080i	$0.861300\pi$
$198$	0	0
$199$	−309.728	−1.55642	−0.778211	−	0.628004i	$-0.783872\pi$
$199$	−309.728	−1.55642	−0.778211	+	0.628004i	$0.783872\pi$
$200$	7.43901i	0.0371950i
$201$	0	0
$202$	159.349	0.788855
$203$	− 18.3848i	− 0.0905654i
$204$	0	0
$205$	55.2778	0.269648
$206$	473.125i	2.29672i
$207$	0	0
$208$	364.087	1.75042
$209$	0	0
$210$	0	0
$211$	207.470	0.983271	0.491635	−	0.870801i	$-0.336399\pi$
$211$	207.470	0.983271	0.491635	+	0.870801i	$0.336399\pi$
$212$	239.303i	1.12879i
$213$	0	0
$214$	389.506	1.82012
$215$	− 117.977i	− 0.548731i
$216$	0	0
$217$	−1.75999	−0.00811057
$218$	365.573i	1.67694i
$219$	0	0
$220$	0	0
$221$	264.514i	1.19689i
$222$	0	0
$223$	−288.730	−1.29475	−0.647376	−	0.762171i	$-0.724134\pi$
$223$	−288.730	−1.29475	−0.647376	+	0.762171i	$0.724134\pi$
$224$	− 22.6046i	− 0.100913i
$225$	0	0
$226$	−24.0287	−0.106322
$227$	− 400.444i	− 1.76407i	−0.471184	−	0.882035i	$-0.656173\pi$
$227$	− 400.444i	− 1.76407i	0.471184	−	0.882035i	$-0.343827\pi$
$228$	0	0
$229$	12.4597	0.0544090	0.0272045	−	0.999630i	$-0.491339\pi$
$229$	12.4597	0.0544090	0.0272045	+	0.999630i	$0.491339\pi$
$230$	− 243.755i	− 1.05981i
$231$	0	0
$232$	13.0000	0.0560345
$233$	− 71.3406i	− 0.306183i	−0.988212	−	0.153091i	$-0.951077\pi$
$233$	− 71.3406i	− 0.306183i	0.988212	−	0.153091i	$-0.0489229\pi$
$234$	0	0
$235$	−24.0323	−0.102265
$236$	232.131i	0.983604i
$237$	0	0
$238$	16.9445	0.0711955
$239$	− 132.906i	− 0.556093i	−0.960568	−	0.278046i	$-0.910313\pi$
$239$	− 132.906i	− 0.556093i	0.960568	−	0.278046i	$-0.0896869\pi$
$240$	0	0
$241$	−130.884	−0.543087	−0.271543	−	0.962426i	$-0.587534\pi$
$241$	−130.884	−0.543087	−0.271543	+	0.962426i	$0.587534\pi$
$242$	0	0
$243$	0	0
$244$	191.734	0.785794
$245$	− 99.0277i	− 0.404195i
$246$	0	0
$247$	−130.698	−0.529140
$248$	− 1.24450i	− 0.00501816i
$249$	0	0
$250$	−261.484	−1.04594
$251$	− 166.247i	− 0.662339i	−0.943571	−	0.331169i	$-0.892557\pi$
$251$	− 166.247i	− 0.662339i	0.943571	−	0.331169i	$-0.107443\pi$
$252$	0	0
$253$	0	0
$254$	− 411.781i	− 1.62119i
$255$	0	0
$256$	271.476	1.06045
$257$	389.435i	1.51531i	0.652655	+	0.757655i	$0.273655\pi$
$257$	389.435i	1.51531i	−0.652655	+	0.757655i	$0.726345\pi$
$258$	0	0
$259$	−16.5604	−0.0639398
$260$	173.699i	0.668073i
$261$	0	0
$262$	−510.744	−1.94940
$263$	272.114i	1.03465i	0.855788	+	0.517327i	$0.173073\pi$
$263$	272.114i	1.03465i	−0.855788	+	0.517327i	$0.826927\pi$
$264$	0	0
$265$	125.522	0.473668
$266$	8.37238i	0.0314751i
$267$	0	0
$268$	−419.571	−1.56556
$269$	− 237.350i	− 0.882341i	−0.897423	−	0.441171i	$-0.854563\pi$
$269$	− 237.350i	− 0.882341i	0.897423	−	0.441171i	$-0.145437\pi$
$270$	0	0
$271$	−301.401	−1.11218	−0.556091	−	0.831122i	$-0.687699\pi$
$271$	−301.401	−1.11218	−0.556091	+	0.831122i	$0.687699\pi$
$272$	197.600i	0.726469i
$273$	0	0
$274$	−438.885	−1.60177
$275$	0	0
$276$	0	0
$277$	−15.5326	−0.0560743	−0.0280371	−	0.999607i	$-0.508926\pi$
$277$	−15.5326	−0.0560743	−0.0280371	+	0.999607i	$0.508926\pi$
$278$	− 95.4252i	− 0.343256i
$279$	0	0
$280$	−0.364917	−0.00130327
$281$	211.250i	0.751781i	0.926664	+	0.375890i	$0.122663\pi$
$281$	211.250i	0.751781i	−0.926664	+	0.375890i	$0.877337\pi$
$282$	0	0
$283$	29.6649	0.104823	0.0524114	−	0.998626i	$-0.483309\pi$
$283$	29.6649	0.104823	0.0524114	+	0.998626i	$0.483309\pi$
$284$	199.090i	0.701023i
$285$	0	0
$286$	0	0
$287$	− 13.7144i	− 0.0477855i
$288$	0	0
$289$	145.441	0.503258
$290$	207.922i	0.716973i
$291$	0	0
$292$	477.166	1.63413
$293$	440.744i	1.50425i	0.659022	+	0.752123i	$0.270970\pi$
$293$	440.744i	1.50425i	−0.659022	+	0.752123i	$0.729030\pi$
$294$	0	0
$295$	121.760	0.412746
$296$	− 11.7100i	− 0.0395608i
$297$	0	0
$298$	377.046	1.26526
$299$	− 944.061i	− 3.15739i
$300$	0	0
$301$	−29.2702	−0.0972431
$302$	− 458.989i	− 1.51983i
$303$	0	0
$304$	−97.6350	−0.321168
$305$	− 100.571i	− 0.329740i
$306$	0	0
$307$	−114.307	−0.372336	−0.186168	−	0.982518i	$-0.559607\pi$
$307$	−114.307	−0.372336	−0.186168	+	0.982518i	$0.559607\pi$
$308$	0	0
$309$	0	0
$310$	19.9046	0.0642084
$311$	492.303i	1.58297i	0.611190	+	0.791484i	$0.290691\pi$
$311$	492.303i	1.58297i	−0.611190	+	0.791484i	$0.709309\pi$
$312$	0	0
$313$	−202.714	−0.647648	−0.323824	−	0.946117i	$-0.604969\pi$
$313$	−202.714	−0.647648	−0.323824	+	0.946117i	$0.604969\pi$
$314$	− 491.516i	− 1.56534i
$315$	0	0
$316$	35.3534	0.111878
$317$	400.582i	1.26366i	0.775105	+	0.631832i	$0.217697\pi$
$317$	400.582i	1.26366i	−0.775105	+	0.631832i	$0.782303\pi$
$318$	0	0
$319$	0	0
$320$	121.632i	0.380101i
$321$	0	0
$322$	−60.4758	−0.187813
$323$	− 70.9330i	− 0.219607i
$324$	0	0
$325$	−460.806	−1.41786
$326$	− 404.800i	− 1.24172i
$327$	0	0
$328$	9.69757	0.0295658
$329$	5.96241i	0.0181228i
$330$	0	0
$331$	433.077	1.30839	0.654194	−	0.756327i	$-0.273008\pi$
$331$	433.077	1.30839	0.654194	+	0.756327i	$0.273008\pi$
$332$	− 341.877i	− 1.02975i
$333$	0	0
$334$	−729.696	−2.18472
$335$	220.078i	0.656951i
$336$	0	0
$337$	−403.120	−1.19620	−0.598100	−	0.801421i	$-0.704078\pi$
$337$	−403.120	−1.19620	−0.598100	+	0.801421i	$0.704078\pi$
$338$	893.334i	2.64300i
$339$	0	0
$340$	−94.2710	−0.277268
$341$	0	0
$342$	0	0
$343$	−49.2656	−0.143632
$344$	− 20.6971i	− 0.0601661i
$345$	0	0
$346$	755.917	2.18473
$347$	104.418i	0.300916i	0.988616	+	0.150458i	$0.0480747\pi$
$347$	104.418i	0.300916i	−0.988616	+	0.150458i	$0.951925\pi$
$348$	0	0
$349$	−49.2133	−0.141012	−0.0705062	−	0.997511i	$-0.522461\pi$
$349$	−49.2133	−0.141012	−0.0705062	+	0.997511i	$0.522461\pi$
$350$	29.5189i	0.0843396i
$351$	0	0
$352$	0	0
$353$	234.322i	0.663801i	0.943314	+	0.331901i	$0.107690\pi$
$353$	234.322i	0.663801i	−0.943314	+	0.331901i	$0.892310\pi$
$354$	0	0
$355$	104.429	0.294167
$356$	− 17.5194i	− 0.0492118i
$357$	0	0
$358$	−96.3709	−0.269193
$359$	227.101i	0.632594i	0.948660	+	0.316297i	$0.102440\pi$
$359$	227.101i	0.632594i	−0.948660	+	0.316297i	$0.897560\pi$
$360$	0	0
$361$	−325.952	−0.902913
$362$	− 682.361i	− 1.88498i
$363$	0	0
$364$	43.0948	0.118392
$365$	− 250.289i	− 0.685724i
$366$	0	0
$367$	−509.982	−1.38960	−0.694798	−	0.719205i	$-0.744506\pi$
$367$	−509.982	−1.38960	−0.694798	+	0.719205i	$0.744506\pi$
$368$	− 705.242i	− 1.91642i
$369$	0	0
$370$	187.290	0.506188
$371$	− 31.1421i	− 0.0839409i
$372$	0	0
$373$	−416.132	−1.11564	−0.557818	−	0.829963i	$-0.688361\pi$
$373$	−416.132	−1.11564	−0.557818	+	0.829963i	$0.688361\pi$
$374$	0	0
$375$	0	0
$376$	−4.21606	−0.0112129
$377$	805.279i	2.13602i
$378$	0	0
$379$	−372.587	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$379$	−372.587	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$380$	− 46.5798i	− 0.122578i
$381$	0	0
$382$	−384.844	−1.00744
$383$	− 649.614i	− 1.69612i	−0.529899	−	0.848061i	$-0.677770\pi$
$383$	− 649.614i	− 1.69612i	0.529899	−	0.848061i	$-0.322230\pi$
$384$	0	0
$385$	0	0
$386$	374.933i	0.971330i
$387$	0	0
$388$	−275.411	−0.709823
$389$	− 448.470i	− 1.15288i	−0.817140	−	0.576439i	$-0.804442\pi$
$389$	− 448.470i	− 1.15288i	0.817140	−	0.576439i	$-0.195558\pi$
$390$	0	0
$391$	512.367	1.31040
$392$	− 17.3728i	− 0.0443183i
$393$	0	0
$394$	−466.617	−1.18431
$395$	− 18.5440i	− 0.0469468i
$396$	0	0
$397$	−6.63508	−0.0167131	−0.00835653	−	0.999965i	$-0.502660\pi$
$397$	−6.63508	−0.0167131	−0.00835653	+	0.999965i	$0.502660\pi$
$398$	− 869.060i	− 2.18357i
$399$	0	0
$400$	−344.236	−0.860590
$401$	16.2649i	0.0405609i	0.999794	+	0.0202804i	$0.00645590\pi$
$401$	16.2649i	0.0405609i	−0.999794	+	0.0202804i	$0.993544\pi$
$402$	0	0
$403$	77.0902	0.191291
$404$	219.950i	0.544432i
$405$	0	0
$406$	51.5855	0.127058
$407$	0	0
$408$	0	0
$409$	−80.9908	−0.198021	−0.0990107	−	0.995086i	$-0.531568\pi$
$409$	−80.9908	−0.198021	−0.0990107	+	0.995086i	$0.531568\pi$
$410$	155.103i	0.378300i
$411$	0	0
$412$	−653.058	−1.58509
$413$	− 30.2087i	− 0.0731446i
$414$	0	0
$415$	−179.325	−0.432109
$416$	990.113i	2.38008i
$417$	0	0
$418$	0	0
$419$	156.122i	0.372607i	0.982492	+	0.186303i	$0.0596508\pi$
$419$	156.122i	0.372607i	−0.982492	+	0.186303i	$0.940349\pi$
$420$	0	0
$421$	−173.540	−0.412210	−0.206105	−	0.978530i	$-0.566079\pi$
$421$	−173.540	−0.412210	−0.206105	+	0.978530i	$0.566079\pi$
$422$	582.137i	1.37947i
$423$	0	0
$424$	22.0208	0.0519358
$425$	− 250.092i	− 0.588451i
$426$	0	0
$427$	−24.9516	−0.0584347
$428$	537.638i	1.25616i
$429$	0	0
$430$	331.030	0.769838
$431$	204.134i	0.473629i	0.971555	+	0.236814i	$0.0761033\pi$
$431$	204.134i	0.473629i	−0.971555	+	0.236814i	$0.923897\pi$
$432$	0	0
$433$	−345.097	−0.796990	−0.398495	−	0.917170i	$-0.630467\pi$
$433$	−345.097	−0.796990	−0.398495	+	0.917170i	$0.630467\pi$
$434$	− 4.93834i	− 0.0113787i
$435$	0	0
$436$	−504.603	−1.15735
$437$	253.163i	0.579321i
$438$	0	0
$439$	−688.693	−1.56878	−0.784388	−	0.620270i	$-0.787023\pi$
$439$	−688.693	−1.56878	−0.784388	+	0.620270i	$0.787023\pi$
$440$	0	0
$441$	0	0
$442$	−742.194	−1.67917
$443$	− 702.545i	− 1.58588i	−0.609300	−	0.792940i	$-0.708549\pi$
$443$	− 702.545i	− 1.58588i	0.609300	−	0.792940i	$-0.291451\pi$
$444$	0	0
$445$	−9.18950	−0.0206506
$446$	− 810.142i	− 1.81646i
$447$	0	0
$448$	30.1770	0.0673594
$449$	471.864i	1.05092i	0.850818	+	0.525461i	$0.176107\pi$
$449$	471.864i	1.05092i	−0.850818	+	0.525461i	$0.823893\pi$
$450$	0	0
$451$	0	0
$452$	− 33.1670i	− 0.0733784i
$453$	0	0
$454$	1123.60	2.47489
$455$	− 22.6046i	− 0.0496804i
$456$	0	0
$457$	−170.345	−0.372746	−0.186373	−	0.982479i	$-0.559673\pi$
$457$	−170.345	−0.372746	−0.186373	+	0.982479i	$0.559673\pi$
$458$	34.9604i	0.0763327i
$459$	0	0
$460$	336.458	0.731430
$461$	279.960i	0.607289i	0.952785	+	0.303645i	$0.0982036\pi$
$461$	279.960i	0.607289i	−0.952785	+	0.303645i	$0.901796\pi$
$462$	0	0
$463$	147.921	0.319485	0.159742	−	0.987159i	$-0.448934\pi$
$463$	147.921	0.319485	0.159742	+	0.987159i	$0.448934\pi$
$464$	601.568i	1.29648i
$465$	0	0
$466$	200.173	0.429557
$467$	826.636i	1.77010i	0.465497	+	0.885050i	$0.345876\pi$
$467$	826.636i	1.77010i	−0.465497	+	0.885050i	$0.654124\pi$
$468$	0	0
$469$	54.6015	0.116421
$470$	− 67.4317i	− 0.143472i
$471$	0	0
$472$	21.3608	0.0452559
$473$	0	0
$474$	0	0
$475$	123.572	0.260151
$476$	23.3887i	0.0491359i
$477$	0	0
$478$	372.919	0.780166
$479$	− 531.483i	− 1.10957i	−0.831994	−	0.554784i	$-0.812801\pi$
$479$	− 531.483i	− 1.10957i	0.831994	−	0.554784i	$-0.187199\pi$
$480$	0	0
$481$	725.370	1.50804
$482$	− 367.245i	− 0.761919i
$483$	0	0
$484$	0	0
$485$	144.462i	0.297860i
$486$	0	0
$487$	−607.825	−1.24810	−0.624050	−	0.781385i	$-0.714514\pi$
$487$	−607.825	−1.24810	−0.624050	+	0.781385i	$0.714514\pi$
$488$	− 17.6434i	− 0.0361546i
$489$	0	0
$490$	277.860	0.567062
$491$	127.294i	0.259255i	0.991563	+	0.129628i	$0.0413782\pi$
$491$	127.294i	0.259255i	−0.991563	+	0.129628i	$0.958622\pi$
$492$	0	0
$493$	−437.046	−0.886504
$494$	− 366.722i	− 0.742353i
$495$	0	0
$496$	57.5887	0.116106
$497$	− 25.9090i	− 0.0521307i
$498$	0	0
$499$	380.683	0.762893	0.381446	−	0.924391i	$-0.375426\pi$
$499$	380.683	0.762893	0.381446	+	0.924391i	$0.375426\pi$
$500$	− 360.928i	− 0.721856i
$501$	0	0
$502$	466.470	0.929223
$503$	141.680i	0.281670i	0.990033	+	0.140835i	$0.0449788\pi$
$503$	141.680i	0.281670i	−0.990033	+	0.140835i	$0.955021\pi$
$504$	0	0
$505$	115.371	0.228458
$506$	0	0
$507$	0	0
$508$	568.385	1.11887
$509$	527.724i	1.03679i	0.855143	+	0.518393i	$0.173470\pi$
$509$	527.724i	1.03679i	−0.855143	+	0.518393i	$0.826530\pi$
$510$	0	0
$511$	−62.0968	−0.121520
$512$	716.134i	1.39870i
$513$	0	0
$514$	−1092.71	−2.12589
$515$	342.550i	0.665146i
$516$	0	0
$517$	0	0
$518$	− 46.4666i	− 0.0897039i
$519$	0	0
$520$	15.9839	0.0307382
$521$	20.4491i	0.0392496i	0.999807	+	0.0196248i	$0.00624717\pi$
$521$	20.4491i	0.0392496i	−0.999807	+	0.0196248i	$0.993753\pi$
$522$	0	0
$523$	404.213	0.772873	0.386437	−	0.922316i	$-0.373706\pi$
$523$	404.213	0.772873	0.386437	+	0.922316i	$0.373706\pi$
$524$	− 704.984i	− 1.34539i
$525$	0	0
$526$	−763.520	−1.45156
$527$	41.8389i	0.0793907i
$528$	0	0
$529$	−1299.66	−2.45683
$530$	352.201i	0.664529i
$531$	0	0
$532$	−11.5565	−0.0217227
$533$	600.711i	1.12704i
$534$	0	0
$535$	282.009	0.527119
$536$	38.6091i	0.0720319i
$537$	0	0
$538$	665.976	1.23787
$539$	0	0
$540$	0	0
$541$	−35.1928	−0.0650514	−0.0325257	−	0.999471i	$-0.510355\pi$
$541$	−35.1928	−0.0650514	−0.0325257	+	0.999471i	$0.510355\pi$
$542$	− 845.697i	− 1.56033i
$543$	0	0
$544$	−537.361	−0.987796
$545$	264.681i	0.485653i
$546$	0	0
$547$	368.347	0.673395	0.336698	−	0.941613i	$-0.390690\pi$
$547$	368.347	0.673395	0.336698	+	0.941613i	$0.390690\pi$
$548$	− 605.797i	− 1.10547i
$549$	0	0
$550$	0	0
$551$	− 215.947i	− 0.391918i
$552$	0	0
$553$	−4.60077	−0.00831965
$554$	− 43.5826i	− 0.0786689i
$555$	0	0
$556$	131.716	0.236900
$557$	178.620i	0.320683i	0.987062	+	0.160341i	$0.0512595\pi$
$557$	178.620i	0.320683i	−0.987062	+	0.160341i	$0.948741\pi$
$558$	0	0
$559$	1282.07	2.29351
$560$	− 16.8863i	− 0.0301541i
$561$	0	0
$562$	−592.744	−1.05470
$563$	171.436i	0.304505i	0.988342	+	0.152252i	$0.0486527\pi$
$563$	171.436i	0.304505i	−0.988342	+	0.152252i	$0.951347\pi$
$564$	0	0
$565$	−17.3972	−0.0307915
$566$	83.2362i	0.147060i
$567$	0	0
$568$	18.3204	0.0322542
$569$	− 653.652i	− 1.14877i	−0.818584	−	0.574386i	$-0.805241\pi$
$569$	− 653.652i	− 1.14877i	0.818584	−	0.574386i	$-0.194759\pi$
$570$	0	0
$571$	172.238	0.301642	0.150821	−	0.988561i	$-0.451808\pi$
$571$	172.238	0.301642	0.150821	+	0.988561i	$0.451808\pi$
$572$	0	0
$573$	0	0
$574$	38.4811	0.0670402
$575$	892.588i	1.55233i
$576$	0	0
$577$	628.202	1.08874	0.544369	−	0.838846i	$-0.316769\pi$
$577$	628.202	1.08874	0.544369	+	0.838846i	$0.316769\pi$
$578$	408.092i	0.706041i
$579$	0	0
$580$	−286.997	−0.494822
$581$	44.4907i	0.0765761i
$582$	0	0
$583$	0	0
$584$	− 43.9091i	− 0.0751868i
$585$	0	0
$586$	−1236.68	−2.11037
$587$	122.740i	0.209097i	0.994520	+	0.104549i	$0.0333398\pi$
$587$	122.740i	0.209097i	−0.994520	+	0.104549i	$0.966660\pi$
$588$	0	0
$589$	−20.6728	−0.0350982
$590$	341.645i	0.579059i
$591$	0	0
$592$	541.873	0.915326
$593$	365.467i	0.616302i	0.951338	+	0.308151i	$0.0997101\pi$
$593$	365.467i	0.616302i	−0.951338	+	0.308151i	$0.900290\pi$
$594$	0	0
$595$	12.2681	0.0206187
$596$	520.440i	0.873222i
$597$	0	0
$598$	2648.93	4.42964
$599$	− 317.385i	− 0.529859i	−0.964268	−	0.264929i	$-0.914651\pi$
$599$	− 317.385i	− 0.529859i	0.964268	−	0.264929i	$-0.0853486\pi$
$600$	0	0
$601$	296.732	0.493731	0.246866	−	0.969050i	$-0.420599\pi$
$601$	296.732	0.493731	0.246866	+	0.969050i	$0.420599\pi$
$602$	− 82.1287i	− 0.136426i
$603$	0	0
$604$	633.546	1.04892
$605$	0	0
$606$	0	0
$607$	829.161	1.36600	0.682999	−	0.730419i	$-0.260675\pi$
$607$	829.161	1.36600	0.682999	+	0.730419i	$0.260675\pi$
$608$	− 265.513i	− 0.436699i
$609$	0	0
$610$	282.190	0.462606
$611$	− 261.162i	− 0.427434i
$612$	0	0
$613$	824.364	1.34480	0.672401	−	0.740187i	$-0.265263\pi$
$613$	824.364	1.34480	0.672401	+	0.740187i	$0.265263\pi$
$614$	− 320.733i	− 0.522366i
$615$	0	0
$616$	0	0
$617$	147.294i	0.238725i	0.992851	+	0.119363i	$0.0380851\pi$
$617$	147.294i	0.238725i	−0.992851	+	0.119363i	$0.961915\pi$
$618$	0	0
$619$	454.329	0.733972	0.366986	−	0.930227i	$-0.380390\pi$
$619$	454.329	0.733972	0.366986	+	0.930227i	$0.380390\pi$
$620$	27.4745i	0.0443137i
$621$	0	0
$622$	−1381.35	−2.22081
$623$	2.27992i	0.00365958i
$624$	0	0
$625$	332.506	0.532010
$626$	− 568.791i	− 0.908612i
$627$	0	0
$628$	678.444	1.08032
$629$	393.677i	0.625878i
$630$	0	0
$631$	36.0464	0.0571258	0.0285629	−	0.999592i	$-0.490907\pi$
$631$	36.0464	0.0571258	0.0285629	+	0.999592i	$0.490907\pi$
$632$	− 3.25323i	− 0.00514752i
$633$	0	0
$634$	−1123.99	−1.77285
$635$	− 298.136i	− 0.469506i
$636$	0	0
$637$	1076.15	1.68940
$638$	0	0
$639$	0	0
$640$	23.1568	0.0361826
$641$	− 1209.52i	− 1.88692i	−0.331486	−	0.943460i	$-0.607550\pi$
$641$	− 1209.52i	− 1.88692i	0.331486	−	0.943460i	$-0.392450\pi$
$642$	0	0
$643$	1057.01	1.64388	0.821938	−	0.569577i	$-0.192893\pi$
$643$	1057.01	1.64388	0.821938	+	0.569577i	$0.192893\pi$
$644$	− 83.4752i	− 0.129620i
$645$	0	0
$646$	199.030	0.308096
$647$	− 730.812i	− 1.12954i	−0.825249	−	0.564770i	$-0.808965\pi$
$647$	− 730.812i	− 1.12954i	0.825249	−	0.564770i	$-0.191035\pi$
$648$	0	0
$649$	0	0
$650$	− 1292.97i	− 1.98918i
$651$	0	0
$652$	558.748	0.856975
$653$	493.064i	0.755076i	0.925994	+	0.377538i	$0.123229\pi$
$653$	493.064i	0.755076i	−0.925994	+	0.377538i	$0.876771\pi$
$654$	0	0
$655$	−369.787	−0.564560
$656$	448.750i	0.684069i
$657$	0	0
$658$	−16.7298	−0.0254253
$659$	887.112i	1.34615i	0.739575	+	0.673074i	$0.235027\pi$
$659$	887.112i	1.34615i	−0.739575	+	0.673074i	$0.764973\pi$
$660$	0	0
$661$	−574.603	−0.869293	−0.434647	−	0.900601i	$-0.643127\pi$
$661$	−574.603	−0.869293	−0.434647	+	0.900601i	$0.643127\pi$
$662$	1215.16i	1.83559i
$663$	0	0
$664$	−31.4597	−0.0473790
$665$	6.06174i	0.00911540i
$666$	0	0
$667$	1559.84	2.33859
$668$	− 1007.20i	− 1.50779i
$669$	0	0
$670$	−617.514	−0.921663
$671$	0	0
$672$	0	0
$673$	311.362	0.462647	0.231324	−	0.972877i	$-0.425694\pi$
$673$	311.362	0.462647	0.231324	+	0.972877i	$0.425694\pi$
$674$	− 1131.11i	− 1.67820i
$675$	0	0
$676$	−1233.08	−1.82408
$677$	1283.45i	1.89580i	0.318573	+	0.947898i	$0.396796\pi$
$677$	1283.45i	1.89580i	−0.318573	+	0.947898i	$0.603204\pi$
$678$	0	0
$679$	35.8411	0.0527851
$680$	8.67487i	0.0127572i
$681$	0	0
$682$	0	0
$683$	− 1285.45i	− 1.88206i	−0.338319	−	0.941031i	$-0.609858\pi$
$683$	− 1285.45i	− 1.88206i	0.338319	−	0.941031i	$-0.390142\pi$
$684$	0	0
$685$	−317.760	−0.463883
$686$	− 138.234i	− 0.201507i
$687$	0	0
$688$	957.748	1.39208
$689$	1364.07i	1.97978i
$690$	0	0
$691$	−171.853	−0.248702	−0.124351	−	0.992238i	$-0.539685\pi$
$691$	−171.853	−0.248702	−0.124351	+	0.992238i	$0.539685\pi$
$692$	1043.40i	1.50780i
$693$	0	0
$694$	−292.984	−0.422167
$695$	− 69.0894i	− 0.0994092i
$696$	0	0
$697$	−326.022	−0.467750
$698$	− 138.087i	− 0.197832i
$699$	0	0
$700$	−40.7451	−0.0582073
$701$	− 690.179i	− 0.984564i	−0.870436	−	0.492282i	$-0.836163\pi$
$701$	− 690.179i	− 0.984564i	0.870436	−	0.492282i	$-0.163837\pi$
$702$	0	0
$703$	−194.518	−0.276697
$704$	0	0
$705$	0	0
$706$	−657.480	−0.931274
$707$	− 28.6236i	− 0.0404860i
$708$	0	0
$709$	479.613	0.676464	0.338232	−	0.941063i	$-0.390171\pi$
$709$	479.613	0.676464	0.338232	+	0.941063i	$0.390171\pi$
$710$	293.017i	0.412700i
$711$	0	0
$712$	−1.61215	−0.00226425
$713$	− 149.325i	− 0.209432i
$714$	0	0
$715$	0	0
$716$	− 133.022i	− 0.185784i
$717$	0	0
$718$	−637.220	−0.887493
$719$	− 1025.44i	− 1.42620i	−0.701064	−	0.713099i	$-0.747291\pi$
$719$	− 1025.44i	− 1.42620i	0.701064	−	0.713099i	$-0.252709\pi$
$720$	0	0
$721$	84.9868	0.117874
$722$	− 914.582i	− 1.26673i
$723$	0	0
$724$	941.869	1.30092
$725$	− 761.373i	− 1.05017i
$726$	0	0
$727$	−250.204	−0.344159	−0.172079	−	0.985083i	$-0.555049\pi$
$727$	−250.204	−0.344159	−0.172079	+	0.985083i	$0.555049\pi$
$728$	− 3.96560i	− 0.00544725i
$729$	0	0
$730$	702.282	0.962030
$731$	695.816i	0.951869i
$732$	0	0
$733$	−945.472	−1.28987	−0.644933	−	0.764239i	$-0.723115\pi$
$733$	−945.472	−1.28987	−0.644933	+	0.764239i	$0.723115\pi$
$734$	− 1430.95i	− 1.94952i
$735$	0	0
$736$	1917.87	2.60580
$737$	0	0
$738$	0	0
$739$	654.012	0.884996	0.442498	−	0.896769i	$-0.354092\pi$
$739$	654.012	0.884996	0.442498	+	0.896769i	$0.354092\pi$
$740$	258.517i	0.349348i
$741$	0	0
$742$	87.3810	0.117764
$743$	1183.38i	1.59270i	0.604837	+	0.796349i	$0.293238\pi$
$743$	1183.38i	1.59270i	−0.604837	+	0.796349i	$0.706762\pi$
$744$	0	0
$745$	272.988	0.366427
$746$	− 1167.62i	− 1.56517i
$747$	0	0
$748$	0	0
$749$	− 69.9664i	− 0.0934132i
$750$	0	0
$751$	75.3851	0.100380	0.0501898	−	0.998740i	$-0.484017\pi$
$751$	75.3851	0.100380	0.0501898	+	0.998740i	$0.484017\pi$
$752$	− 195.096i	− 0.259436i
$753$	0	0
$754$	−2259.52	−2.99671
$755$	− 332.316i	− 0.440153i
$756$	0	0
$757$	488.488	0.645294	0.322647	−	0.946519i	$-0.395427\pi$
$757$	488.488	0.645294	0.322647	+	0.946519i	$0.395427\pi$
$758$	− 1045.43i	− 1.37920i
$759$	0	0
$760$	−4.28630	−0.00563987
$761$	187.728i	0.246685i	0.992364	+	0.123343i	$0.0393614\pi$
$761$	187.728i	0.246685i	−0.992364	+	0.123343i	$0.960639\pi$
$762$	0	0
$763$	65.6673	0.0860647
$764$	− 531.203i	− 0.695291i
$765$	0	0
$766$	1822.74	2.37956
$767$	1323.18i	1.72514i
$768$	0	0
$769$	−257.324	−0.334621	−0.167311	−	0.985904i	$-0.553508\pi$
$769$	−257.324	−0.334621	−0.167311	+	0.985904i	$0.553508\pi$
$770$	0	0
$771$	0	0
$772$	−517.523	−0.670367
$773$	− 475.467i	− 0.615093i	−0.951533	−	0.307546i	$-0.900492\pi$
$773$	− 475.467i	− 0.615093i	0.951533	−	0.307546i	$-0.0995079\pi$
$774$	0	0
$775$	−72.8871	−0.0940478
$776$	25.3435i	0.0326591i
$777$	0	0
$778$	1258.35	1.61742
$779$	− 161.089i	− 0.206790i
$780$	0	0
$781$	0	0
$782$	1437.64i	1.83842i
$783$	0	0
$784$	803.915	1.02540
$785$	− 355.866i	− 0.453332i
$786$	0	0
$787$	−290.097	−0.368612	−0.184306	−	0.982869i	$-0.559004\pi$
$787$	−290.097	−0.368612	−0.184306	+	0.982869i	$0.559004\pi$
$788$	− 644.075i	− 0.817354i
$789$	0	0
$790$	52.0323	0.0658636
$791$	4.31625i	0.00545669i
$792$	0	0
$793$	1092.92	1.37820
$794$	− 18.6173i	− 0.0234474i
$795$	0	0
$796$	1199.57	1.50700
$797$	792.022i	0.993754i	0.867821	+	0.496877i	$0.165520\pi$
$797$	792.022i	0.993754i	−0.867821	+	0.496877i	$0.834480\pi$
$798$	0	0
$799$	141.740	0.177396
$800$	− 936.130i	− 1.17016i
$801$	0	0
$802$	−45.6374	−0.0569045
$803$	0	0
$804$	0	0
$805$	−43.7855	−0.0543919
$806$	216.306i	0.268370i
$807$	0	0
$808$	20.2399	0.0250494
$809$	− 1021.86i	− 1.26311i	−0.775331	−	0.631556i	$-0.782417\pi$
$809$	− 1021.86i	− 1.26311i	0.775331	−	0.631556i	$-0.217583\pi$
$810$	0	0
$811$	113.251	0.139644	0.0698221	−	0.997559i	$-0.477757\pi$
$811$	113.251	0.139644	0.0698221	+	0.997559i	$0.477757\pi$
$812$	71.2039i	0.0876896i
$813$	0	0
$814$	0	0
$815$	− 293.082i	− 0.359609i
$816$	0	0
$817$	−343.806	−0.420816
$818$	− 227.251i	− 0.277813i
$819$	0	0
$820$	−214.090	−0.261085
$821$	344.122i	0.419150i	0.977793	+	0.209575i	$0.0672080\pi$
$821$	344.122i	0.419150i	−0.977793	+	0.209575i	$0.932792\pi$
$822$	0	0
$823$	121.006	0.147031	0.0735153	−	0.997294i	$-0.476578\pi$
$823$	121.006	0.147031	0.0735153	+	0.997294i	$0.476578\pi$
$824$	60.0948i	0.0729306i
$825$	0	0
$826$	84.7621	0.102618
$827$	251.924i	0.304624i	0.988332	+	0.152312i	$0.0486718\pi$
$827$	251.924i	0.304624i	−0.988332	+	0.152312i	$0.951328\pi$
$828$	0	0
$829$	−226.464	−0.273177	−0.136589	−	0.990628i	$-0.543614\pi$
$829$	−226.464	−0.273177	−0.136589	+	0.990628i	$0.543614\pi$
$830$	− 503.166i	− 0.606224i
$831$	0	0
$832$	−1321.80	−1.58870
$833$	584.054i	0.701146i
$834$	0	0
$835$	−528.312	−0.632708
$836$	0	0
$837$	0	0
$838$	−438.061	−0.522746
$839$	580.612i	0.692028i	0.938229	+	0.346014i	$0.112465\pi$
$839$	580.612i	0.692028i	−0.938229	+	0.346014i	$0.887535\pi$
$840$	0	0
$841$	−489.534	−0.582086
$842$	− 486.934i	− 0.578306i
$843$	0	0
$844$	−803.528	−0.952048
$845$	646.789i	0.765431i
$846$	0	0
$847$	0	0
$848$	1019.00i	1.20165i
$849$	0	0
$850$	701.728	0.825562
$851$	− 1405.05i	− 1.65106i
$852$	0	0
$853$	1056.85	1.23898	0.619491	−	0.785003i	$-0.287339\pi$
$853$	1056.85	1.23898	0.619491	+	0.785003i	$0.287339\pi$
$854$	− 70.0113i	− 0.0819804i
$855$	0	0
$856$	49.4738	0.0577964
$857$	− 1471.20i	− 1.71669i	−0.513071	−	0.858346i	$-0.671492\pi$
$857$	− 1471.20i	− 1.71669i	0.513071	−	0.858346i	$-0.328508\pi$
$858$	0	0
$859$	−1109.66	−1.29180	−0.645902	−	0.763421i	$-0.723518\pi$
$859$	−1109.66	−1.29180	−0.645902	+	0.763421i	$0.723518\pi$
$860$	456.924i	0.531307i
$861$	0	0
$862$	−572.776	−0.664474
$863$	− 369.201i	− 0.427811i	−0.976854	−	0.213906i	$-0.931382\pi$
$863$	− 369.201i	− 0.427811i	0.976854	−	0.213906i	$-0.0686185\pi$
$864$	0	0
$865$	547.297	0.632713
$866$	− 968.301i	− 1.11813i
$867$	0	0
$868$	6.81643	0.00785303
$869$	0	0
$870$	0	0
$871$	−2391.62	−2.74584
$872$	46.4338i	0.0532498i
$873$	0	0
$874$	−710.347	−0.812754
$875$	46.9700i	0.0536800i
$876$	0	0
$877$	84.1500	0.0959521	0.0479761	−	0.998848i	$-0.484723\pi$
$877$	84.1500	0.0959521	0.0479761	+	0.998848i	$0.484723\pi$
$878$	− 1932.39i	− 2.20090i
$879$	0	0
$880$	0	0
$881$	763.999i	0.867195i	0.901107	+	0.433598i	$0.142756\pi$
$881$	763.999i	0.867195i	−0.901107	+	0.433598i	$0.857244\pi$
$882$	0	0
$883$	−3.97183	−0.00449811	−0.00224906	−	0.999997i	$-0.500716\pi$
$883$	−3.97183	−0.00449811	−0.00224906	+	0.999997i	$0.500716\pi$
$884$	− 1024.46i	− 1.15889i
$885$	0	0
$886$	1971.26	2.22490
$887$	− 1020.39i	− 1.15038i	−0.818021	−	0.575189i	$-0.804929\pi$
$887$	− 1020.39i	− 1.15038i	0.818021	−	0.575189i	$-0.195071\pi$
$888$	0	0
$889$	−73.9677	−0.0832033
$890$	− 25.7847i	− 0.0289715i
$891$	0	0
$892$	1118.25	1.25364
$893$	70.0343i	0.0784259i
$894$	0	0
$895$	−69.7741	−0.0779599
$896$	− 5.74522i	− 0.00641208i
$897$	0	0
$898$	−1324.00	−1.47438
$899$	127.373i	0.141684i
$900$	0	0
$901$	−740.315	−0.821660
$902$	0	0
$903$	0	0
$904$	−3.05205	−0.00337616
$905$	− 494.041i	− 0.545902i
$906$	0	0
$907$	−672.333	−0.741271	−0.370635	−	0.928778i	$-0.620860\pi$
$907$	−672.333	−0.741271	−0.370635	+	0.928778i	$0.620860\pi$
$908$	1550.91i	1.70805i
$909$	0	0
$910$	63.4259	0.0696988
$911$	− 320.354i	− 0.351650i	−0.984421	−	0.175825i	$-0.943741\pi$
$911$	− 320.354i	− 0.351650i	0.984421	−	0.175825i	$-0.0562594\pi$
$912$	0	0
$913$	0	0
$914$	− 477.969i	− 0.522941i
$915$	0	0
$916$	−48.2561	−0.0526813
$917$	91.7443i	0.100048i
$918$	0	0
$919$	−755.334	−0.821908	−0.410954	−	0.911656i	$-0.634804\pi$
$919$	−755.334	−0.821908	−0.410954	+	0.911656i	$0.634804\pi$
$920$	− 30.9610i	− 0.0336533i
$921$	0	0
$922$	−785.536	−0.851992
$923$	1134.85i	1.22952i
$924$	0	0
$925$	−685.820	−0.741428
$926$	415.050i	0.448218i
$927$	0	0
$928$	−1635.93	−1.76285
$929$	653.730i	0.703692i	0.936058	+	0.351846i	$0.114446\pi$
$929$	653.730i	0.703692i	−0.936058	+	0.351846i	$0.885554\pi$
$930$	0	0
$931$	−288.584	−0.309972
$932$	276.301i	0.296460i
$933$	0	0
$934$	−2319.45	−2.48335
$935$	0	0
$936$	0	0
$937$	−1051.46	−1.12216	−0.561078	−	0.827763i	$-0.689613\pi$
$937$	−1051.46	−1.12216	−0.561078	+	0.827763i	$0.689613\pi$
$938$	153.206i	0.163332i
$939$	0	0
$940$	93.0766	0.0990176
$941$	− 503.345i	− 0.534904i	−0.963571	−	0.267452i	$-0.913818\pi$
$941$	− 503.345i	− 0.534904i	0.963571	−	0.267452i	$-0.0861817\pi$
$942$	0	0
$943$	1163.59	1.23392
$944$	988.458i	1.04710i
$945$	0	0
$946$	0	0
$947$	369.018i	0.389671i	0.980836	+	0.194836i	$0.0624173\pi$
$947$	369.018i	0.389671i	−0.980836	+	0.194836i	$0.937583\pi$
$948$	0	0
$949$	2719.93	2.86610
$950$	346.728i	0.364976i
$951$	0	0
$952$	2.15224	0.00226075
$953$	852.586i	0.894634i	0.894375	+	0.447317i	$0.147620\pi$
$953$	852.586i	0.894634i	−0.894375	+	0.447317i	$0.852380\pi$
$954$	0	0
$955$	−278.633	−0.291762
$956$	514.743i	0.538435i
$957$	0	0
$958$	1491.28	1.55666
$959$	78.8364i	0.0822068i
$960$	0	0
$961$	−948.806	−0.987312
$962$	2035.30i	2.11570i
$963$	0	0
$964$	506.911	0.525841
$965$	271.458i	0.281304i
$966$	0	0
$967$	−683.004	−0.706312	−0.353156	−	0.935564i	$-0.614892\pi$
$967$	−683.004	−0.706312	−0.353156	+	0.935564i	$0.614892\pi$
$968$	0	0
$969$	0	0
$970$	−405.344	−0.417881
$971$	869.351i	0.895315i	0.894205	+	0.447657i	$0.147742\pi$
$971$	869.351i	0.895315i	−0.894205	+	0.447657i	$0.852258\pi$
$972$	0	0
$973$	−17.1411	−0.0176168
$974$	− 1705.49i	− 1.75101i
$975$	0	0
$976$	816.441	0.836517
$977$	743.912i	0.761425i	0.924694	+	0.380712i	$0.124321\pi$
$977$	743.912i	0.761425i	−0.924694	+	0.380712i	$0.875679\pi$
$978$	0	0
$979$	0	0
$980$	383.533i	0.391360i
$981$	0	0
$982$	−357.173	−0.363720
$983$	352.071i	0.358159i	0.983835	+	0.179080i	$0.0573120\pi$
$983$	352.071i	0.358159i	−0.983835	+	0.179080i	$0.942688\pi$
$984$	0	0
$985$	−337.838	−0.342983
$986$	− 1226.30i	− 1.24371i
$987$	0	0
$988$	506.190	0.512338
$989$	− 2483.40i	− 2.51102i
$990$	0	0
$991$	1939.01	1.95662	0.978309	−	0.207152i	$-0.0664195\pi$
$991$	1939.01	1.95662	0.978309	+	0.207152i	$0.0664195\pi$
$992$	156.609i	0.157872i
$993$	0	0
$994$	72.6976	0.0731364
$995$	− 629.214i	− 0.632376i
$996$	0	0
$997$	115.855	0.116204	0.0581020	−	0.998311i	$-0.481495\pi$
$997$	115.855	0.116204	0.0581020	+	0.998311i	$0.481495\pi$
$998$	1068.15i	1.07029i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.b.h.485.8	yes	8
3.2	odd	2	inner	1089.3.b.h.485.1	✓	8
11.10	odd	2	inner	1089.3.b.h.485.2	yes	8
33.32	even	2	inner	1089.3.b.h.485.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1089.3.b.h.485.1	✓	8	3.2	odd	2	inner
1089.3.b.h.485.2	yes	8	11.10	odd	2	inner
1089.3.b.h.485.7	yes	8	33.32	even	2	inner
1089.3.b.h.485.8	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)

\(f(q)\)	\(=\)	\(q+2.80588i q^{2} -3.87298 q^{4} +2.03151i q^{5} +0.504017 q^{7} +0.356394i q^{8} +O(q^{10})\)	\(q+2.80588i q^{2} -3.87298 q^{4} +2.03151i q^{5} +0.504017 q^{7} +0.356394i q^{8} -5.70017 q^{10} -22.0767 q^{13} +1.41421i q^{14} -16.4919 q^{16} -11.9816i q^{17} +5.92017 q^{19} -7.86799i q^{20} +42.7628i q^{23} +20.8730 q^{25} -61.9445i q^{26} -1.95205 q^{28} -36.4765i q^{29} -3.49193 q^{31} -44.8489i q^{32} +33.6190 q^{34} +1.02391i q^{35} -32.8569 q^{37} +16.6113i q^{38} -0.724016 q^{40} -27.2102i q^{41} -58.0737 q^{43} -119.988 q^{46} +11.8298i q^{47} -48.7460 q^{49} +58.5672i q^{50} +85.5025 q^{52} -61.7877i q^{53} +0.179629i q^{56} +102.349 q^{58} -59.9359i q^{59} -49.5055 q^{61} -9.79796i q^{62} +59.8730 q^{64} -44.8489i q^{65} +108.333 q^{67} +46.4045i q^{68} -2.87298 q^{70} -51.4049i q^{71} -123.204 q^{73} -92.1925i q^{74} -22.9287 q^{76} -9.12820 q^{79} -33.5035i q^{80} +76.3488 q^{82} +88.2722i q^{83} +24.3407 q^{85} -162.948i q^{86} +4.52349i q^{89} -11.1270 q^{91} -165.620i q^{92} -33.1930 q^{94} +12.0269i q^{95} +71.1109 q^{97} -136.776i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 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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