LMFDB - Embedded newform 1089.3.b.c.485.1

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:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 16x^{2} + 49 $ x^4 + 16*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.1
Root			$-3.44572i$ of defining polynomial
Character	$\chi$	$=$	1089.485
Dual form			1089.3.b.c.485.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.44572i q^{2} -7.87298 q^{4} +6.27415i q^{5} -8.87298 q^{7} +13.3452i q^{8} +O(q^{10})$	$q-3.44572i q^{2} -7.87298 q^{4} +6.27415i q^{5} -8.87298 q^{7} +13.3452i q^{8} +21.6190 q^{10} -24.7460 q^{13} +30.5738i q^{14} +14.4919 q^{16} -4.68030i q^{17} +29.2379 q^{19} -49.3963i q^{20} +27.7454i q^{23} -14.3649 q^{25} +85.2677i q^{26} +69.8569 q^{28} -34.5584i q^{29} +24.7298 q^{31} +3.44572i q^{32} -16.1270 q^{34} -55.6704i q^{35} -1.36492 q^{37} -100.746i q^{38} -83.7298 q^{40} -50.9673i q^{41} +62.8407 q^{43} +95.6028 q^{46} +23.5027i q^{47} +29.7298 q^{49} +49.4975i q^{50} +194.825 q^{52} -4.32105i q^{53} -118.412i q^{56} -119.079 q^{58} -69.2965i q^{59} -36.9839 q^{61} -85.2121i q^{62} +69.8407 q^{64} -155.260i q^{65} -9.12702 q^{67} +36.8480i q^{68} -191.825 q^{70} -33.2454i q^{71} -8.76210 q^{73} +4.70312i q^{74} -230.190 q^{76} -3.01613 q^{79} +90.9245i q^{80} -175.619 q^{82} -43.9746i q^{83} +29.3649 q^{85} -216.531i q^{86} -105.449i q^{89} +219.571 q^{91} -218.439i q^{92} +80.9839 q^{94} +183.443i q^{95} +102.095 q^{97} -102.441i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4} - 20 q^{7}+O(q^{10}) $ 4 * q - 16 * q^4 - 20 * q^7	$ 4 q - 16 q^{4} - 20 q^{7} + 40 q^{10} - 68 q^{13} - 4 q^{16} + 24 q^{19} + 20 q^{25} + 140 q^{28} - 56 q^{31} - 80 q^{34} + 72 q^{37} - 180 q^{40} - 12 q^{43} + 212 q^{46} - 36 q^{49} + 392 q^{52} - 120 q^{58} - 24 q^{61} + 16 q^{64} - 52 q^{67} - 380 q^{70} - 128 q^{73} - 456 q^{76} - 136 q^{79} - 656 q^{82} + 40 q^{85} + 460 q^{91} + 200 q^{94} + 176 q^{97}+O(q^{100}) $ 4 * q - 16 * q^4 - 20 * q^7 + 40 * q^10 - 68 * q^13 - 4 * q^16 + 24 * q^19 + 20 * q^25 + 140 * q^28 - 56 * q^31 - 80 * q^34 + 72 * q^37 - 180 * q^40 - 12 * q^43 + 212 * q^46 - 36 * q^49 + 392 * q^52 - 120 * q^58 - 24 * q^61 + 16 * q^64 - 52 * q^67 - 380 * q^70 - 128 * q^73 - 456 * q^76 - 136 * q^79 - 656 * q^82 + 40 * q^85 + 460 * q^91 + 200 * q^94 + 176 * 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 $

$2$	− 3.44572i	− 1.72286i	−0.507877	−	0.861430i	$-0.669569\pi$
$2$	− 3.44572i	− 1.72286i	0.507877	−	0.861430i	$-0.330431\pi$
$3$	0	0
$3$	0	0
$4$	−7.87298	−1.96825
$5$	6.27415i	1.25483i	0.778685	+	0.627415i	$0.215887\pi$
$5$	6.27415i	1.25483i	−0.778685	+	0.627415i	$0.784113\pi$
$6$	0	0
$7$	−8.87298	−1.26757	−0.633785	−	0.773510i	$-0.718499\pi$
$7$	−8.87298	−1.26757	−0.633785	+	0.773510i	$0.718499\pi$
$8$	13.3452i	1.66815i
$9$	0	0
$10$	21.6190	2.16190
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−24.7460	−1.90354	−0.951768	−	0.306819i	$-0.900735\pi$
$13$	−24.7460	−1.90354	−0.951768	+	0.306819i	$0.900735\pi$
$14$	30.5738i	2.18384i
$15$	0	0
$16$	14.4919	0.905746
$17$	− 4.68030i	− 0.275312i	−0.990480	−	0.137656i	$-0.956043\pi$
$17$	− 4.68030i	− 0.275312i	0.990480	−	0.137656i	$-0.0439568\pi$
$18$	0	0
$19$	29.2379	1.53884	0.769418	−	0.638745i	$-0.220546\pi$
$19$	29.2379	1.53884	0.769418	+	0.638745i	$0.220546\pi$
$20$	− 49.3963i	− 2.46981i
$21$	0	0
$22$	0	0
$23$	27.7454i	1.20632i	0.797620	+	0.603161i	$0.206092\pi$
$23$	27.7454i	1.20632i	−0.797620	+	0.603161i	$0.793908\pi$
$24$	0	0
$25$	−14.3649	−0.574597
$26$	85.2677i	3.27953i
$27$	0	0
$28$	69.8569	2.49489
$29$	− 34.5584i	− 1.19167i	−0.803107	−	0.595835i	$-0.796821\pi$
$29$	− 34.5584i	− 1.19167i	0.803107	−	0.595835i	$-0.203179\pi$
$30$	0	0
$31$	24.7298	0.797737	0.398868	−	0.917008i	$-0.369403\pi$
$31$	24.7298	0.797737	0.398868	+	0.917008i	$0.369403\pi$
$32$	3.44572i	0.107679i
$33$	0	0
$34$	−16.1270	−0.474324
$35$	− 55.6704i	− 1.59058i
$36$	0	0
$37$	−1.36492	−0.0368896	−0.0184448	−	0.999830i	$-0.505872\pi$
$37$	−1.36492	−0.0368896	−0.0184448	+	0.999830i	$0.505872\pi$
$38$	− 100.746i	− 2.65120i
$39$	0	0
$40$	−83.7298	−2.09325
$41$	− 50.9673i	− 1.24310i	−0.783373	−	0.621552i	$-0.786502\pi$
$41$	− 50.9673i	− 1.24310i	0.783373	−	0.621552i	$-0.213498\pi$
$42$	0	0
$43$	62.8407	1.46141	0.730706	−	0.682692i	$-0.239191\pi$
$43$	62.8407	1.46141	0.730706	+	0.682692i	$0.239191\pi$
$44$	0	0
$45$	0	0
$46$	95.6028	2.07832
$47$	23.5027i	0.500058i	0.968238	+	0.250029i	$0.0804402\pi$
$47$	23.5027i	0.500058i	−0.968238	+	0.250029i	$0.919560\pi$
$48$	0	0
$49$	29.7298	0.606731
$50$	49.4975i	0.989949i
$51$	0	0
$52$	194.825	3.74663
$53$	− 4.32105i	− 0.0815292i	−0.999169	−	0.0407646i	$-0.987021\pi$
$53$	− 4.32105i	− 0.0815292i	0.999169	−	0.0407646i	$-0.0129794\pi$
$54$	0	0
$55$	0	0
$56$	− 118.412i	− 2.11450i
$57$	0	0
$58$	−119.079	−2.05308
$59$	− 69.2965i	− 1.17452i	−0.809400	−	0.587258i	$-0.800207\pi$
$59$	− 69.2965i	− 1.17452i	0.809400	−	0.587258i	$-0.199793\pi$
$60$	0	0
$61$	−36.9839	−0.606293	−0.303146	−	0.952944i	$-0.598037\pi$
$61$	−36.9839	−0.606293	−0.303146	+	0.952944i	$0.598037\pi$
$62$	− 85.2121i	− 1.37439i
$63$	0	0
$64$	69.8407	1.09126
$65$	− 155.260i	− 2.38861i
$66$	0	0
$67$	−9.12702	−0.136224	−0.0681121	−	0.997678i	$-0.521698\pi$
$67$	−9.12702	−0.136224	−0.0681121	+	0.997678i	$0.521698\pi$
$68$	36.8480i	0.541882i
$69$	0	0
$70$	−191.825	−2.74035
$71$	− 33.2454i	− 0.468245i	−0.972207	−	0.234123i	$-0.924778\pi$
$71$	− 33.2454i	− 0.468245i	0.972207	−	0.234123i	$-0.0752218\pi$
$72$	0	0
$73$	−8.76210	−0.120029	−0.0600144	−	0.998198i	$-0.519115\pi$
$73$	−8.76210	−0.120029	−0.0600144	+	0.998198i	$0.519115\pi$
$74$	4.70312i	0.0635557i
$75$	0	0
$76$	−230.190	−3.02881
$77$	0	0
$78$	0	0
$79$	−3.01613	−0.0381789	−0.0190895	−	0.999818i	$-0.506077\pi$
$79$	−3.01613	−0.0381789	−0.0190895	+	0.999818i	$0.506077\pi$
$80$	90.9245i	1.13656i
$81$	0	0
$82$	−175.619	−2.14169
$83$	− 43.9746i	− 0.529815i	−0.964274	−	0.264907i	$-0.914659\pi$
$83$	− 43.9746i	− 0.529815i	0.964274	−	0.264907i	$-0.0853414\pi$
$84$	0	0
$85$	29.3649	0.345470
$86$	− 216.531i	− 2.51781i
$87$	0	0
$88$	0	0
$89$	− 105.449i	− 1.18482i	−0.805638	−	0.592409i	$-0.798177\pi$
$89$	− 105.449i	− 1.18482i	0.805638	−	0.592409i	$-0.201823\pi$
$90$	0	0
$91$	219.571	2.41286
$92$	− 218.439i	− 2.37434i
$93$	0	0
$94$	80.9839	0.861530
$95$	183.443i	1.93098i
$96$	0	0
$97$	102.095	1.05252	0.526262	−	0.850323i	$-0.323593\pi$
$97$	102.095	1.05252	0.526262	+	0.850323i	$0.323593\pi$
$98$	− 102.441i	− 1.04531i
$99$	0	0
$100$	113.095	1.13095
$101$	90.8461i	0.899466i	0.893163	+	0.449733i	$0.148481\pi$
$101$	90.8461i	0.899466i	−0.893163	+	0.449733i	$0.851519\pi$
$102$	0	0
$103$	−17.6351	−0.171214	−0.0856072	−	0.996329i	$-0.527283\pi$
$103$	−17.6351	−0.171214	−0.0856072	+	0.996329i	$0.527283\pi$
$104$	− 330.240i	− 3.17539i
$105$	0	0
$106$	−14.8891	−0.140463
$107$	4.08583i	0.0381853i	0.999818	+	0.0190927i	$0.00607775\pi$
$107$	4.08583i	0.0381853i	−0.999818	+	0.0190927i	$0.993922\pi$
$108$	0	0
$109$	95.6976	0.877959	0.438980	−	0.898497i	$-0.355340\pi$
$109$	95.6976	0.877959	0.438980	+	0.898497i	$0.355340\pi$
$110$	0	0
$111$	0	0
$112$	−128.587	−1.14810
$113$	− 118.804i	− 1.05136i	−0.850682	−	0.525681i	$-0.823811\pi$
$113$	− 118.804i	− 1.05136i	0.850682	−	0.525681i	$-0.176189\pi$
$114$	0	0
$115$	−174.079	−1.51373
$116$	272.078i	2.34550i
$117$	0	0
$118$	−238.776	−2.02353
$119$	41.5283i	0.348977i
$120$	0	0
$121$	0	0
$122$	127.436i	1.04456i
$123$	0	0
$124$	−194.698	−1.57014
$125$	66.7261i	0.533809i
$126$	0	0
$127$	112.508	0.885890	0.442945	−	0.896549i	$-0.353934\pi$
$127$	112.508	0.885890	0.442945	+	0.896549i	$0.353934\pi$
$128$	− 226.869i	− 1.77241i
$129$	0	0
$130$	−534.982	−4.11524
$131$	− 15.0403i	− 0.114811i	−0.998351	−	0.0574056i	$-0.981717\pi$
$131$	− 15.0403i	− 0.114811i	0.998351	−	0.0574056i	$-0.0182828\pi$
$132$	0	0
$133$	−259.427	−1.95058
$134$	31.4491i	0.234695i
$135$	0	0
$136$	62.4597	0.459262
$137$	199.470i	1.45598i	0.685586	+	0.727991i	$0.259546\pi$
$137$	199.470i	1.45598i	−0.685586	+	0.727991i	$0.740454\pi$
$138$	0	0
$139$	152.808	1.09934	0.549671	−	0.835381i	$-0.314753\pi$
$139$	152.808	1.09934	0.549671	+	0.835381i	$0.314753\pi$
$140$	438.292i	3.13066i
$141$	0	0
$142$	−114.554	−0.806721
$143$	0	0
$144$	0	0
$145$	216.825	1.49534
$146$	30.1917i	0.206793i
$147$	0	0
$148$	10.7460	0.0726079
$149$	270.664i	1.81653i	0.418391	+	0.908267i	$0.362594\pi$
$149$	270.664i	1.81653i	−0.418391	+	0.908267i	$0.637406\pi$
$150$	0	0
$151$	94.2863	0.624413	0.312206	−	0.950014i	$-0.398932\pi$
$151$	94.2863	0.624413	0.312206	+	0.950014i	$0.398932\pi$
$152$	390.186i	2.56701i
$153$	0	0
$154$	0	0
$155$	155.159i	1.00102i
$156$	0	0
$157$	186.157	1.18571	0.592857	−	0.805307i	$-0.298000\pi$
$157$	186.157	1.18571	0.592857	+	0.805307i	$0.298000\pi$
$158$	10.3927i	0.0657769i
$159$	0	0
$160$	−21.6190	−0.135118
$161$	− 246.184i	− 1.52910i
$162$	0	0
$163$	−44.3327	−0.271979	−0.135990	−	0.990710i	$-0.543421\pi$
$163$	−44.3327	−0.271979	−0.135990	+	0.990710i	$0.543421\pi$
$164$	401.265i	2.44673i
$165$	0	0
$166$	−151.524	−0.912796
$167$	− 152.039i	− 0.910415i	−0.890385	−	0.455208i	$-0.849565\pi$
$167$	− 152.039i	− 0.910415i	0.890385	−	0.455208i	$-0.150435\pi$
$168$	0	0
$169$	443.363	2.62345
$170$	− 101.183i	− 0.595196i
$171$	0	0
$172$	−494.744	−2.87642
$173$	256.038i	1.47999i	0.672613	+	0.739995i	$0.265172\pi$
$173$	256.038i	1.47999i	−0.672613	+	0.739995i	$0.734828\pi$
$174$	0	0
$175$	127.460	0.728341
$176$	0	0
$177$	0	0
$178$	−363.347	−2.04127
$179$	− 113.405i	− 0.633548i	−0.948501	−	0.316774i	$-0.897400\pi$
$179$	− 113.405i	− 0.633548i	0.948501	−	0.316774i	$-0.102600\pi$
$180$	0	0
$181$	160.270	0.885471	0.442735	−	0.896652i	$-0.354008\pi$
$181$	160.270	0.885471	0.442735	+	0.896652i	$0.354008\pi$
$182$	− 756.579i	− 4.15702i
$183$	0	0
$184$	−370.268	−2.01233
$185$	− 8.56369i	− 0.0462902i
$186$	0	0
$187$	0	0
$188$	− 185.037i	− 0.984238i
$189$	0	0
$190$	632.093	3.32680
$191$	75.0218i	0.392784i	0.980525	+	0.196392i	$0.0629225\pi$
$191$	75.0218i	0.392784i	−0.980525	+	0.196392i	$0.937077\pi$
$192$	0	0
$193$	−118.903	−0.616079	−0.308039	−	0.951374i	$-0.599673\pi$
$193$	−118.903	−0.616079	−0.308039	+	0.951374i	$0.599673\pi$
$194$	− 351.790i	− 1.81335i
$195$	0	0
$196$	−234.062	−1.19420
$197$	− 187.271i	− 0.950613i	−0.879820	−	0.475306i	$-0.842337\pi$
$197$	− 187.271i	− 0.950613i	0.879820	−	0.475306i	$-0.157663\pi$
$198$	0	0
$199$	−29.5706	−0.148596	−0.0742979	−	0.997236i	$-0.523672\pi$
$199$	−29.5706	−0.148596	−0.0742979	+	0.997236i	$0.523672\pi$
$200$	− 191.703i	− 0.958514i
$201$	0	0
$202$	313.030	1.54965
$203$	306.636i	1.51052i
$204$	0	0
$205$	319.776	1.55988
$206$	60.7656i	0.294978i
$207$	0	0
$208$	−358.617	−1.72412
$209$	0	0
$210$	0	0
$211$	−393.825	−1.86647	−0.933234	−	0.359270i	$-0.883026\pi$
$211$	−393.825	−1.86647	−0.933234	+	0.359270i	$0.883026\pi$
$212$	34.0195i	0.160469i
$213$	0	0
$214$	14.0786	0.0657879
$215$	394.272i	1.83382i
$216$	0	0
$217$	−219.427	−1.01119
$218$	− 329.747i	− 1.51260i
$219$	0	0
$220$	0	0
$221$	115.819i	0.524066i
$222$	0	0
$223$	−121.016	−0.542673	−0.271337	−	0.962485i	$-0.587466\pi$
$223$	−121.016	−0.542673	−0.271337	+	0.962485i	$0.587466\pi$
$224$	− 30.5738i	− 0.136490i
$225$	0	0
$226$	−409.365	−1.81135
$227$	− 160.388i	− 0.706554i	−0.935519	−	0.353277i	$-0.885067\pi$
$227$	− 160.388i	− 0.706554i	0.935519	−	0.353277i	$-0.114933\pi$
$228$	0	0
$229$	201.000	0.877729	0.438865	−	0.898553i	$-0.355381\pi$
$229$	201.000	0.877729	0.438865	+	0.898553i	$0.355381\pi$
$230$	599.826i	2.60794i
$231$	0	0
$232$	461.190	1.98789
$233$	− 66.7261i	− 0.286378i	−0.989695	−	0.143189i	$-0.954264\pi$
$233$	− 66.7261i	− 0.286378i	0.989695	−	0.143189i	$-0.0457357\pi$
$234$	0	0
$235$	−147.460	−0.627488
$236$	545.570i	2.31174i
$237$	0	0
$238$	143.095	0.601238
$239$	− 337.726i	− 1.41308i	−0.707673	−	0.706540i	$-0.750255\pi$
$239$	− 337.726i	− 1.41308i	0.707673	−	0.706540i	$-0.249745\pi$
$240$	0	0
$241$	18.9516	0.0786373	0.0393187	−	0.999227i	$-0.487481\pi$
$241$	18.9516	0.0786373	0.0393187	+	0.999227i	$0.487481\pi$
$242$	0	0
$243$	0	0
$244$	291.173	1.19333
$245$	186.529i	0.761344i
$246$	0	0
$247$	−723.520	−2.92923
$248$	330.025i	1.33075i
$249$	0	0
$250$	229.919	0.919677
$251$	247.487i	0.986005i	0.870028	+	0.493003i	$0.164101\pi$
$251$	247.487i	0.986005i	−0.870028	+	0.493003i	$0.835899\pi$
$252$	0	0
$253$	0	0
$254$	− 387.671i	− 1.52626i
$255$	0	0
$256$	−502.363	−1.96235
$257$	330.733i	1.28690i	0.765488	+	0.643450i	$0.222498\pi$
$257$	330.733i	1.28690i	−0.765488	+	0.643450i	$0.777502\pi$
$258$	0	0
$259$	12.1109	0.0467602
$260$	1222.36i	4.70138i
$261$	0	0
$262$	−51.8246	−0.197804
$263$	− 275.658i	− 1.04813i	−0.851679	−	0.524064i	$-0.824415\pi$
$263$	− 275.658i	− 1.04813i	0.851679	−	0.524064i	$-0.175585\pi$
$264$	0	0
$265$	27.1109	0.102305
$266$	893.914i	3.36058i
$267$	0	0
$268$	71.8569	0.268123
$269$	− 141.004i	− 0.524177i	−0.965044	−	0.262088i	$-0.915589\pi$
$269$	− 141.004i	− 0.524177i	0.965044	−	0.262088i	$-0.0844112\pi$
$270$	0	0
$271$	−47.3165	−0.174600	−0.0872998	−	0.996182i	$-0.527824\pi$
$271$	−47.3165	−0.174600	−0.0872998	+	0.996182i	$0.527824\pi$
$272$	− 67.8267i	− 0.249363i
$273$	0	0
$274$	687.317	2.50845
$275$	0	0
$276$	0	0
$277$	257.919	0.931117	0.465558	−	0.885017i	$-0.345854\pi$
$277$	257.919	0.931117	0.465558	+	0.885017i	$0.345854\pi$
$278$	− 526.535i	− 1.89401i
$279$	0	0
$280$	742.933	2.65333
$281$	111.455i	0.396637i	0.980138	+	0.198318i	$0.0635480\pi$
$281$	111.455i	0.396637i	−0.980138	+	0.198318i	$0.936452\pi$
$282$	0	0
$283$	116.490	0.411625	0.205813	−	0.978591i	$-0.434016\pi$
$283$	116.490	0.411625	0.205813	+	0.978591i	$0.434016\pi$
$284$	261.741i	0.921622i
$285$	0	0
$286$	0	0
$287$	452.232i	1.57572i
$288$	0	0
$289$	267.095	0.924203
$290$	− 747.117i	− 2.57626i
$291$	0	0
$292$	68.9839	0.236246
$293$	274.233i	0.935950i	0.883742	+	0.467975i	$0.155016\pi$
$293$	274.233i	0.935950i	−0.883742	+	0.467975i	$0.844984\pi$
$294$	0	0
$295$	434.776	1.47382
$296$	− 18.2151i	− 0.0615375i
$297$	0	0
$298$	932.631	3.12963
$299$	− 686.586i	− 2.29628i
$300$	0	0
$301$	−557.585	−1.85244
$302$	− 324.884i	− 1.07578i
$303$	0	0
$304$	423.714	1.39380
$305$	− 232.042i	− 0.760794i
$306$	0	0
$307$	−506.587	−1.65012	−0.825060	−	0.565045i	$-0.808859\pi$
$307$	−506.587	−1.65012	−0.825060	+	0.565045i	$0.808859\pi$
$308$	0	0
$309$	0	0
$310$	534.633	1.72462
$311$	− 522.247i	− 1.67925i	−0.543166	−	0.839625i	$-0.682775\pi$
$311$	− 522.247i	− 1.67925i	0.543166	−	0.839625i	$-0.317225\pi$
$312$	0	0
$313$	11.3186	0.0361616	0.0180808	−	0.999837i	$-0.494244\pi$
$313$	11.3186	0.0361616	0.0180808	+	0.999837i	$0.494244\pi$
$314$	− 641.446i	− 2.04282i
$315$	0	0
$316$	23.7460	0.0751455
$317$	− 5.85640i	− 0.0184745i	−0.999957	−	0.00923723i	$-0.997060\pi$
$317$	− 5.85640i	− 0.0184745i	0.999957	−	0.00923723i	$-0.00294034\pi$
$318$	0	0
$319$	0	0
$320$	438.191i	1.36935i
$321$	0	0
$322$	−848.282	−2.63442
$323$	− 136.842i	− 0.423660i
$324$	0	0
$325$	355.474	1.09377
$326$	152.758i	0.468582i
$327$	0	0
$328$	680.169	2.07369
$329$	− 208.539i	− 0.633859i
$330$	0	0
$331$	−599.617	−1.81153	−0.905766	−	0.423779i	$-0.860703\pi$
$331$	−599.617	−1.81153	−0.905766	+	0.423779i	$0.860703\pi$
$332$	346.211i	1.04281i
$333$	0	0
$334$	−523.885	−1.56852
$335$	− 57.2642i	− 0.170938i
$336$	0	0
$337$	544.571	1.61594	0.807968	−	0.589226i	$-0.200567\pi$
$337$	544.571	1.61594	0.807968	+	0.589226i	$0.200567\pi$
$338$	− 1527.70i	− 4.51983i
$339$	0	0
$340$	−231.190	−0.679969
$341$	0	0
$342$	0	0
$343$	170.984	0.498495
$344$	838.623i	2.43786i
$345$	0	0
$346$	882.236	2.54981
$347$	− 626.271i	− 1.80482i	−0.430882	−	0.902408i	$-0.641797\pi$
$347$	− 626.271i	− 1.80482i	0.430882	−	0.902408i	$-0.358203\pi$
$348$	0	0
$349$	2.25198	0.00645268	0.00322634	−	0.999995i	$-0.498973\pi$
$349$	2.25198	0.00645268	0.00322634	+	0.999995i	$0.498973\pi$
$350$	− 439.190i	− 1.25483i
$351$	0	0
$352$	0	0
$353$	− 565.154i	− 1.60100i	−0.599332	−	0.800501i	$-0.704567\pi$
$353$	− 565.154i	− 1.60100i	0.599332	−	0.800501i	$-0.295433\pi$
$354$	0	0
$355$	208.587	0.587568
$356$	830.196i	2.33201i
$357$	0	0
$358$	−390.762	−1.09151
$359$	151.210i	0.421197i	0.977573	+	0.210598i	$0.0675412\pi$
$359$	151.210i	0.421197i	−0.977573	+	0.210598i	$0.932459\pi$
$360$	0	0
$361$	493.855	1.36802
$362$	− 552.246i	− 1.52554i
$363$	0	0
$364$	−1728.68	−4.74911
$365$	− 54.9747i	− 0.150616i
$366$	0	0
$367$	−8.45762	−0.0230453	−0.0115226	−	0.999934i	$-0.503668\pi$
$367$	−8.45762	−0.0230453	−0.0115226	+	0.999934i	$0.503668\pi$
$368$	402.084i	1.09262i
$369$	0	0
$370$	−29.5081	−0.0797515
$371$	38.3406i	0.103344i
$372$	0	0
$373$	483.363	1.29588	0.647940	−	0.761692i	$-0.275631\pi$
$373$	483.363	1.29588	0.647940	+	0.761692i	$0.275631\pi$
$374$	0	0
$375$	0	0
$376$	−313.649	−0.834173
$377$	855.181i	2.26839i
$378$	0	0
$379$	412.458	1.08828	0.544139	−	0.838995i	$-0.316856\pi$
$379$	412.458	1.08828	0.544139	+	0.838995i	$0.316856\pi$
$380$	− 1444.24i	− 3.80064i
$381$	0	0
$382$	258.504	0.676712
$383$	529.141i	1.38157i	0.723060	+	0.690785i	$0.242735\pi$
$383$	529.141i	1.38157i	−0.723060	+	0.690785i	$0.757265\pi$
$384$	0	0
$385$	0	0
$386$	409.707i	1.06142i
$387$	0	0
$388$	−803.790	−2.07162
$389$	− 606.884i	− 1.56011i	−0.625709	−	0.780057i	$-0.715190\pi$
$389$	− 606.884i	− 1.56011i	0.625709	−	0.780057i	$-0.284810\pi$
$390$	0	0
$391$	129.857	0.332115
$392$	396.751i	1.01212i
$393$	0	0
$394$	−645.282	−1.63777
$395$	− 18.9237i	− 0.0479080i
$396$	0	0
$397$	474.571	1.19539	0.597696	−	0.801723i	$-0.296083\pi$
$397$	474.571	1.19539	0.597696	+	0.801723i	$0.296083\pi$
$398$	101.892i	0.256010i
$399$	0	0
$400$	−208.175	−0.520439
$401$	− 115.682i	− 0.288483i	−0.989543	−	0.144242i	$-0.953926\pi$
$401$	− 115.682i	− 0.288483i	0.989543	−	0.144242i	$-0.0460742\pi$
$402$	0	0
$403$	−611.964	−1.51852
$404$	− 715.230i	− 1.77037i
$405$	0	0
$406$	1056.58	2.60242
$407$	0	0
$408$	0	0
$409$	−56.9859	−0.139330	−0.0696649	−	0.997570i	$-0.522193\pi$
$409$	−56.9859	−0.139330	−0.0696649	+	0.997570i	$0.522193\pi$
$410$	− 1101.86i	− 2.68746i
$411$	0	0
$412$	138.841	0.336992
$413$	614.866i	1.48878i
$414$	0	0
$415$	275.903	0.664827
$416$	− 85.2677i	− 0.204970i
$417$	0	0
$418$	0	0
$419$	− 752.136i	− 1.79507i	−0.440938	−	0.897537i	$-0.645354\pi$
$419$	− 752.136i	− 1.79507i	0.440938	−	0.897537i	$-0.354646\pi$
$420$	0	0
$421$	137.129	0.325722	0.162861	−	0.986649i	$-0.447928\pi$
$421$	137.129	0.325722	0.162861	+	0.986649i	$0.447928\pi$
$422$	1357.01i	3.21566i
$423$	0	0
$424$	57.6653	0.136003
$425$	67.2322i	0.158193i
$426$	0	0
$427$	328.157	0.768518
$428$	− 32.1677i	− 0.0751581i
$429$	0	0
$430$	1358.55	3.15942
$431$	− 183.355i	− 0.425416i	−0.977116	−	0.212708i	$-0.931772\pi$
$431$	− 183.355i	− 0.425416i	0.977116	−	0.212708i	$-0.0682284\pi$
$432$	0	0
$433$	−36.4274	−0.0841279	−0.0420640	−	0.999115i	$-0.513393\pi$
$433$	−36.4274	−0.0841279	−0.0420640	+	0.999115i	$0.513393\pi$
$434$	756.085i	1.74213i
$435$	0	0
$436$	−753.425	−1.72804
$437$	811.217i	1.85633i
$438$	0	0
$439$	120.730	0.275011	0.137506	−	0.990501i	$-0.456092\pi$
$439$	120.730	0.275011	0.137506	+	0.990501i	$0.456092\pi$
$440$	0	0
$441$	0	0
$442$	399.079	0.902893
$443$	543.277i	1.22636i	0.789943	+	0.613180i	$0.210110\pi$
$443$	543.277i	1.22636i	−0.789943	+	0.613180i	$0.789890\pi$
$444$	0	0
$445$	661.601	1.48674
$446$	416.988i	0.934950i
$447$	0	0
$448$	−619.696	−1.38325
$449$	− 85.2677i	− 0.189906i	−0.995482	−	0.0949529i	$-0.969730\pi$
$449$	− 85.2677i	− 0.189906i	0.995482	−	0.0949529i	$-0.0302700\pi$
$450$	0	0
$451$	0	0
$452$	935.341i	2.06934i
$453$	0	0
$454$	−552.651	−1.21729
$455$	1377.62i	3.02773i
$456$	0	0
$457$	−291.554	−0.637975	−0.318987	−	0.947759i	$-0.603343\pi$
$457$	−291.554	−0.637975	−0.318987	+	0.947759i	$0.603343\pi$
$458$	− 692.590i	− 1.51220i
$459$	0	0
$460$	1370.52	2.97939
$461$	61.4741i	0.133349i	0.997775	+	0.0666747i	$0.0212390\pi$
$461$	61.4741i	0.133349i	−0.997775	+	0.0666747i	$0.978761\pi$
$462$	0	0
$463$	71.6633	0.154780	0.0773901	−	0.997001i	$-0.475341\pi$
$463$	71.6633	0.154780	0.0773901	+	0.997001i	$0.475341\pi$
$464$	− 500.818i	− 1.07935i
$465$	0	0
$466$	−229.919	−0.493389
$467$	212.044i	0.454055i	0.973888	+	0.227027i	$0.0729007\pi$
$467$	212.044i	0.454055i	−0.973888	+	0.227027i	$0.927099\pi$
$468$	0	0
$469$	80.9839	0.172673
$470$	508.105i	1.08107i
$471$	0	0
$472$	924.776	1.95927
$473$	0	0
$474$	0	0
$475$	−420.000	−0.884211
$476$	− 326.951i	− 0.686873i
$477$	0	0
$478$	−1163.71	−2.43454
$479$	− 8.84164i	− 0.0184585i	−0.999957	−	0.00922927i	$-0.997062\pi$
$479$	− 8.84164i	− 0.0184585i	0.999957	−	0.00922927i	$-0.00293781\pi$
$480$	0	0
$481$	33.7762	0.0702208
$482$	− 65.3019i	− 0.135481i
$483$	0	0
$484$	0	0
$485$	640.557i	1.32074i
$486$	0	0
$487$	−483.252	−0.992304	−0.496152	−	0.868236i	$-0.665254\pi$
$487$	−483.252	−0.992304	−0.496152	+	0.868236i	$0.665254\pi$
$488$	− 493.558i	− 1.01139i
$489$	0	0
$490$	642.728	1.31169
$491$	317.006i	0.645634i	0.946461	+	0.322817i	$0.104630\pi$
$491$	317.006i	0.645634i	−0.946461	+	0.322817i	$0.895370\pi$
$492$	0	0
$493$	−161.744	−0.328081
$494$	2493.05i	5.04665i
$495$	0	0
$496$	358.383	0.722547
$497$	294.986i	0.593533i
$498$	0	0
$499$	503.631	1.00928	0.504640	−	0.863330i	$-0.331625\pi$
$499$	503.631	1.00928	0.504640	+	0.863330i	$0.331625\pi$
$500$	− 525.333i	− 1.05067i
$501$	0	0
$502$	852.772	1.69875
$503$	116.661i	0.231931i	0.993253	+	0.115965i	$0.0369962\pi$
$503$	116.661i	0.231931i	−0.993253	+	0.115965i	$0.963004\pi$
$504$	0	0
$505$	−569.982	−1.12868
$506$	0	0
$507$	0	0
$508$	−885.774	−1.74365
$509$	293.193i	0.576017i	0.957628	+	0.288009i	$0.0929932\pi$
$509$	293.193i	0.576017i	−0.957628	+	0.288009i	$0.907007\pi$
$510$	0	0
$511$	77.7460	0.152145
$512$	823.527i	1.60845i
$513$	0	0
$514$	1139.61	2.21715
$515$	− 110.645i	− 0.214845i
$516$	0	0
$517$	0	0
$518$	− 41.7307i	− 0.0805612i
$519$	0	0
$520$	2071.98	3.98457
$521$	− 96.5030i	− 0.185226i	−0.995702	−	0.0926132i	$-0.970478\pi$
$521$	− 96.5030i	− 0.185226i	0.995702	−	0.0926132i	$-0.0295220\pi$
$522$	0	0
$523$	201.427	0.385138	0.192569	−	0.981283i	$-0.438318\pi$
$523$	201.427	0.385138	0.192569	+	0.981283i	$0.438318\pi$
$524$	118.412i	0.225977i
$525$	0	0
$526$	−949.839	−1.80578
$527$	− 115.743i	− 0.219626i
$528$	0	0
$529$	−240.806	−0.455211
$530$	− 93.4165i	− 0.176258i
$531$	0	0
$532$	2042.47	3.83922
$533$	1261.23i	2.36629i
$534$	0	0
$535$	−25.6351	−0.0479160
$536$	− 121.802i	− 0.227243i
$537$	0	0
$538$	−485.859	−0.903083
$539$	0	0
$540$	0	0
$541$	−866.375	−1.60143	−0.800716	−	0.599044i	$-0.795547\pi$
$541$	−866.375	−1.60143	−0.800716	+	0.599044i	$0.795547\pi$
$542$	163.039i	0.300811i
$543$	0	0
$544$	16.1270	0.0296453
$545$	600.421i	1.10169i
$546$	0	0
$547$	461.145	0.843044	0.421522	−	0.906818i	$-0.361496\pi$
$547$	461.145	0.843044	0.421522	+	0.906818i	$0.361496\pi$
$548$	− 1570.42i	− 2.86573i
$549$	0	0
$550$	0	0
$551$	− 1010.42i	− 1.83379i
$552$	0	0
$553$	26.7621	0.0483944
$554$	− 888.718i	− 1.60418i
$555$	0	0
$556$	−1203.06	−2.16377
$557$	− 285.694i	− 0.512916i	−0.966555	−	0.256458i	$-0.917445\pi$
$557$	− 285.694i	− 0.512916i	0.966555	−	0.256458i	$-0.0825555\pi$
$558$	0	0
$559$	−1555.05	−2.78185
$560$	− 806.772i	− 1.44066i
$561$	0	0
$562$	384.042	0.683349
$563$	321.739i	0.571473i	0.958308	+	0.285736i	$0.0922382\pi$
$563$	321.739i	0.571473i	−0.958308	+	0.285736i	$0.907762\pi$
$564$	0	0
$565$	745.393	1.31928
$566$	− 401.391i	− 0.709172i
$567$	0	0
$568$	443.667	0.781104
$569$	− 963.864i	− 1.69396i	−0.531624	−	0.846980i	$-0.678418\pi$
$569$	− 963.864i	− 1.69396i	0.531624	−	0.846980i	$-0.321582\pi$
$570$	0	0
$571$	−1014.45	−1.77663	−0.888313	−	0.459238i	$-0.848122\pi$
$571$	−1014.45	−1.77663	−0.888313	+	0.459238i	$0.848122\pi$
$572$	0	0
$573$	0	0
$574$	1558.26	2.71475
$575$	− 398.560i	− 0.693148i
$576$	0	0
$577$	362.903	0.628948	0.314474	−	0.949266i	$-0.398172\pi$
$577$	362.903	0.628948	0.314474	+	0.949266i	$0.398172\pi$
$578$	− 920.334i	− 1.59227i
$579$	0	0
$580$	−1707.06	−2.94320
$581$	390.186i	0.671577i
$582$	0	0
$583$	0	0
$584$	− 116.932i	− 0.200226i
$585$	0	0
$586$	944.931	1.61251
$587$	− 567.835i	− 0.967351i	−0.875247	−	0.483676i	$-0.839301\pi$
$587$	− 567.835i	− 0.967351i	0.875247	−	0.483676i	$-0.160699\pi$
$588$	0	0
$589$	723.048	1.22759
$590$	− 1498.12i	− 2.53918i
$591$	0	0
$592$	−19.7803	−0.0334126
$593$	− 203.728i	− 0.343555i	−0.985136	−	0.171777i	$-0.945049\pi$
$593$	− 203.728i	− 0.343555i	0.985136	−	0.171777i	$-0.0549510\pi$
$594$	0	0
$595$	−260.554	−0.437907
$596$	− 2130.93i	− 3.57539i
$597$	0	0
$598$	−2365.78	−3.95616
$599$	− 591.566i	− 0.987589i	−0.869579	−	0.493795i	$-0.835609\pi$
$599$	− 591.566i	− 0.987589i	0.869579	−	0.493795i	$-0.164391\pi$
$600$	0	0
$601$	−719.157	−1.19660	−0.598301	−	0.801272i	$-0.704157\pi$
$601$	−719.157	−1.19660	−0.598301	+	0.801272i	$0.704157\pi$
$602$	1921.28i	3.19150i
$603$	0	0
$604$	−742.314	−1.22900
$605$	0	0
$606$	0	0
$607$	−16.0000	−0.0263591	−0.0131796	−	0.999913i	$-0.504195\pi$
$607$	−16.0000	−0.0263591	−0.0131796	+	0.999913i	$0.504195\pi$
$608$	100.746i	0.165700i
$609$	0	0
$610$	−799.552	−1.31074
$611$	− 581.598i	− 0.951879i
$612$	0	0
$613$	−317.740	−0.518336	−0.259168	−	0.965832i	$-0.583448\pi$
$613$	−317.740	−0.518336	−0.259168	+	0.965832i	$0.583448\pi$
$614$	1745.56i	2.84292i
$615$	0	0
$616$	0	0
$617$	629.469i	1.02021i	0.860112	+	0.510105i	$0.170393\pi$
$617$	629.469i	1.02021i	−0.860112	+	0.510105i	$0.829607\pi$
$618$	0	0
$619$	243.982	0.394155	0.197077	−	0.980388i	$-0.436855\pi$
$619$	243.982	0.394155	0.197077	+	0.980388i	$0.436855\pi$
$620$	− 1221.56i	− 1.97026i
$621$	0	0
$622$	−1799.52	−2.89311
$623$	935.645i	1.50184i
$624$	0	0
$625$	−777.772	−1.24444
$626$	− 39.0006i	− 0.0623013i
$627$	0	0
$628$	−1465.61	−2.33378
$629$	6.38823i	0.0101562i
$630$	0	0
$631$	658.236	1.04316	0.521581	−	0.853201i	$-0.325342\pi$
$631$	658.236	1.04316	0.521581	+	0.853201i	$0.325342\pi$
$632$	− 40.2509i	− 0.0636882i
$633$	0	0
$634$	−20.1795	−0.0318289
$635$	705.892i	1.11164i
$636$	0	0
$637$	−735.693	−1.15493
$638$	0	0
$639$	0	0
$640$	1423.41	2.22407
$641$	464.074i	0.723985i	0.932181	+	0.361993i	$0.117903\pi$
$641$	464.074i	0.723985i	−0.932181	+	0.361993i	$0.882097\pi$
$642$	0	0
$643$	218.000	0.339036	0.169518	−	0.985527i	$-0.445779\pi$
$643$	218.000	0.339036	0.169518	+	0.985527i	$0.445779\pi$
$644$	1938.21i	3.00964i
$645$	0	0
$646$	−471.520	−0.729907
$647$	1135.45i	1.75494i	0.479629	+	0.877471i	$0.340771\pi$
$647$	1135.45i	1.75494i	−0.479629	+	0.877471i	$0.659229\pi$
$648$	0	0
$649$	0	0
$650$	− 1224.86i	− 1.88440i
$651$	0	0
$652$	349.030	0.535322
$653$	− 793.419i	− 1.21504i	−0.794305	−	0.607519i	$-0.792165\pi$
$653$	− 793.419i	− 1.21504i	0.794305	−	0.607519i	$-0.207835\pi$
$654$	0	0
$655$	94.3649	0.144069
$656$	− 738.614i	− 1.12594i
$657$	0	0
$658$	−718.569	−1.09205
$659$	− 87.6613i	− 0.133022i	−0.997786	−	0.0665109i	$-0.978813\pi$
$659$	− 87.6613i	− 0.133022i	0.997786	−	0.0665109i	$-0.0211867\pi$
$660$	0	0
$661$	1071.04	1.62033	0.810165	−	0.586202i	$-0.199377\pi$
$661$	1071.04	1.62033	0.810165	+	0.586202i	$0.199377\pi$
$662$	2066.11i	3.12101i
$663$	0	0
$664$	586.851	0.883811
$665$	− 1627.69i	− 2.44765i
$666$	0	0
$667$	958.837	1.43754
$668$	1197.00i	1.79192i
$669$	0	0
$670$	−197.317	−0.294502
$671$	0	0
$672$	0	0
$673$	−402.883	−0.598637	−0.299319	−	0.954153i	$-0.596759\pi$
$673$	−402.883	−0.598637	−0.299319	+	0.954153i	$0.596759\pi$
$674$	− 1876.44i	− 2.78403i
$675$	0	0
$676$	−3490.59	−5.16359
$677$	123.859i	0.182953i	0.995807	+	0.0914765i	$0.0291586\pi$
$677$	123.859i	0.182953i	−0.995807	+	0.0914765i	$0.970841\pi$
$678$	0	0
$679$	−905.885	−1.33415
$680$	391.881i	0.576296i
$681$	0	0
$682$	0	0
$683$	− 621.918i	− 0.910567i	−0.890346	−	0.455284i	$-0.849538\pi$
$683$	− 621.918i	− 0.910567i	0.890346	−	0.455284i	$-0.150462\pi$
$684$	0	0
$685$	−1251.50	−1.82701
$686$	− 589.162i	− 0.858837i
$687$	0	0
$688$	910.683	1.32367
$689$	106.928i	0.155194i
$690$	0	0
$691$	1058.90	1.53242	0.766209	−	0.642591i	$-0.222141\pi$
$691$	1058.90	1.53242	0.766209	+	0.642591i	$0.222141\pi$
$692$	− 2015.78i	− 2.91298i
$693$	0	0
$694$	−2157.96	−3.10945
$695$	958.743i	1.37949i
$696$	0	0
$697$	−238.542	−0.342242
$698$	− 7.75971i	− 0.0111171i
$699$	0	0
$700$	−1003.49	−1.43355
$701$	− 875.810i	− 1.24937i	−0.780876	−	0.624686i	$-0.785227\pi$
$701$	− 875.810i	− 1.24937i	0.780876	−	0.624686i	$-0.214773\pi$
$702$	0	0
$703$	−39.9073	−0.0567671
$704$	0	0
$705$	0	0
$706$	−1947.36	−2.75830
$707$	− 806.076i	− 1.14014i
$708$	0	0
$709$	286.661	0.404318	0.202159	−	0.979353i	$-0.435204\pi$
$709$	286.661	0.404318	0.202159	+	0.979353i	$0.435204\pi$
$710$	− 718.731i	− 1.01230i
$711$	0	0
$712$	1407.24	1.97645
$713$	686.139i	0.962326i
$714$	0	0
$715$	0	0
$716$	892.836i	1.24698i
$717$	0	0
$718$	521.026	0.725663
$719$	− 273.950i	− 0.381015i	−0.981686	−	0.190507i	$-0.938987\pi$
$719$	− 273.950i	− 0.381015i	0.981686	−	0.190507i	$-0.0610133\pi$
$720$	0	0
$721$	156.476	0.217026
$722$	− 1701.69i	− 2.35690i
$723$	0	0
$724$	−1261.80	−1.74282
$725$	496.429i	0.684729i
$726$	0	0
$727$	1276.84	1.75631	0.878154	−	0.478377i	$-0.158775\pi$
$727$	1276.84	1.75631	0.878154	+	0.478377i	$0.158775\pi$
$728$	2930.22i	4.02502i
$729$	0	0
$730$	−189.427	−0.259490
$731$	− 294.114i	− 0.402344i
$732$	0	0
$733$	−1064.27	−1.45193	−0.725966	−	0.687731i	$-0.758607\pi$
$733$	−1064.27	−1.45193	−0.725966	+	0.687731i	$0.758607\pi$
$734$	29.1426i	0.0397038i
$735$	0	0
$736$	−95.6028	−0.129895
$737$	0	0
$738$	0	0
$739$	−830.794	−1.12421	−0.562107	−	0.827064i	$-0.690009\pi$
$739$	−830.794	−1.12421	−0.562107	+	0.827064i	$0.690009\pi$
$740$	67.4218i	0.0911105i
$741$	0	0
$742$	132.111	0.178047
$743$	1335.35i	1.79724i	0.438729	+	0.898619i	$0.355429\pi$
$743$	1335.35i	1.79724i	−0.438729	+	0.898619i	$0.644571\pi$
$744$	0	0
$745$	−1698.18	−2.27944
$746$	− 1665.53i	− 2.23262i
$747$	0	0
$748$	0	0
$749$	− 36.2535i	− 0.0484025i
$750$	0	0
$751$	−877.917	−1.16900	−0.584499	−	0.811395i	$-0.698709\pi$
$751$	−877.917	−1.16900	−0.584499	+	0.811395i	$0.698709\pi$
$752$	340.600i	0.452926i
$753$	0	0
$754$	2946.72	3.90811
$755$	591.566i	0.783531i
$756$	0	0
$757$	1075.54	1.42079	0.710394	−	0.703804i	$-0.248517\pi$
$757$	1075.54	1.42079	0.710394	+	0.703804i	$0.248517\pi$
$758$	− 1421.21i	− 1.87495i
$759$	0	0
$760$	−2448.08	−3.22116
$761$	− 351.679i	− 0.462127i	−0.972939	−	0.231064i	$-0.925779\pi$
$761$	− 351.679i	− 0.462127i	0.972939	−	0.231064i	$-0.0742205\pi$
$762$	0	0
$763$	−849.123	−1.11287
$764$	− 590.645i	− 0.773096i
$765$	0	0
$766$	1823.27	2.38025
$767$	1714.81i	2.23573i
$768$	0	0
$769$	−466.423	−0.606532	−0.303266	−	0.952906i	$-0.598077\pi$
$769$	−466.423	−0.606532	−0.303266	+	0.952906i	$0.598077\pi$
$770$	0	0
$771$	0	0
$772$	936.123	1.21259
$773$	− 258.978i	− 0.335030i	−0.985870	−	0.167515i	$-0.946426\pi$
$773$	− 258.978i	− 0.335030i	0.985870	−	0.167515i	$-0.0535742\pi$
$774$	0	0
$775$	−355.242	−0.458377
$776$	1362.48i	1.75577i
$777$	0	0
$778$	−2091.15	−2.68786
$779$	− 1490.18i	− 1.91293i
$780$	0	0
$781$	0	0
$782$	− 447.450i	− 0.572187i
$783$	0	0
$784$	430.843	0.549544
$785$	1167.98i	1.48787i
$786$	0	0
$787$	169.270	0.215083	0.107541	−	0.994201i	$-0.465702\pi$
$787$	169.270	0.215083	0.107541	+	0.994201i	$0.465702\pi$
$788$	1474.38i	1.87104i
$789$	0	0
$790$	−65.2056	−0.0825388
$791$	1054.15i	1.33267i
$792$	0	0
$793$	915.202	1.15410
$794$	− 1635.24i	− 2.05949i
$795$	0	0
$796$	232.808	0.292473
$797$	− 888.999i	− 1.11543i	−0.830032	−	0.557716i	$-0.811678\pi$
$797$	− 888.999i	− 1.11543i	0.830032	−	0.557716i	$-0.188322\pi$
$798$	0	0
$799$	110.000	0.137672
$800$	− 49.4975i	− 0.0618718i
$801$	0	0
$802$	−398.607	−0.497016
$803$	0	0
$804$	0	0
$805$	1544.60	1.91875
$806$	2108.66i	2.61620i
$807$	0	0
$808$	−1212.36	−1.50045
$809$	− 998.765i	− 1.23457i	−0.786740	−	0.617284i	$-0.788233\pi$
$809$	− 998.765i	− 1.23457i	0.786740	−	0.617284i	$-0.211767\pi$
$810$	0	0
$811$	514.097	0.633905	0.316952	−	0.948441i	$-0.397340\pi$
$811$	514.097	0.633905	0.316952	+	0.948441i	$0.397340\pi$
$812$	− 2414.14i	− 2.97308i
$813$	0	0
$814$	0	0
$815$	− 278.150i	− 0.341288i
$816$	0	0
$817$	1837.33	2.24887
$818$	196.357i	0.240046i
$819$	0	0
$820$	−2517.59	−3.07023
$821$	− 984.093i	− 1.19865i	−0.800505	−	0.599326i	$-0.795435\pi$
$821$	− 984.093i	− 1.19865i	0.800505	−	0.599326i	$-0.204565\pi$
$822$	0	0
$823$	−636.083	−0.772883	−0.386442	−	0.922314i	$-0.626296\pi$
$823$	−636.083	−0.772883	−0.386442	+	0.922314i	$0.626296\pi$
$824$	− 235.344i	− 0.285612i
$825$	0	0
$826$	2118.66	2.56496
$827$	− 994.996i	− 1.20314i	−0.798820	−	0.601570i	$-0.794542\pi$
$827$	− 994.996i	− 1.20314i	0.798820	−	0.601570i	$-0.205458\pi$
$828$	0	0
$829$	−768.996	−0.927619	−0.463809	−	0.885935i	$-0.653518\pi$
$829$	−768.996	−0.927619	−0.463809	+	0.885935i	$0.653518\pi$
$830$	− 950.685i	− 1.14540i
$831$	0	0
$832$	−1728.28	−2.07725
$833$	− 139.145i	− 0.167040i
$834$	0	0
$835$	953.917	1.14242
$836$	0	0
$837$	0	0
$838$	−2591.65	−3.09266
$839$	927.065i	1.10496i	0.833525	+	0.552482i	$0.186319\pi$
$839$	927.065i	1.10496i	−0.833525	+	0.552482i	$0.813681\pi$
$840$	0	0
$841$	−353.284	−0.420076
$842$	− 472.508i	− 0.561174i
$843$	0	0
$844$	3100.57	3.67367
$845$	2781.72i	3.29198i
$846$	0	0
$847$	0	0
$848$	− 62.6203i	− 0.0738447i
$849$	0	0
$850$	231.663	0.272545
$851$	− 37.8701i	− 0.0445008i
$852$	0	0
$853$	160.889	0.188616	0.0943078	−	0.995543i	$-0.469936\pi$
$853$	160.889	0.188616	0.0943078	+	0.995543i	$0.469936\pi$
$854$	− 1130.74i	− 1.32405i
$855$	0	0
$856$	−54.5262	−0.0636989
$857$	808.391i	0.943280i	0.881791	+	0.471640i	$0.156338\pi$
$857$	808.391i	0.943280i	−0.881791	+	0.471640i	$0.843662\pi$
$858$	0	0
$859$	−1311.06	−1.52627	−0.763133	−	0.646241i	$-0.776340\pi$
$859$	−1311.06	−1.52627	−0.763133	+	0.646241i	$0.776340\pi$
$860$	− 3104.10i	− 3.60941i
$861$	0	0
$862$	−631.788	−0.732933
$863$	− 183.805i	− 0.212984i	−0.994314	−	0.106492i	$-0.966038\pi$
$863$	− 183.805i	− 0.212984i	0.994314	−	0.106492i	$-0.0339618\pi$
$864$	0	0
$865$	−1606.42	−1.85713
$866$	125.519i	0.144941i
$867$	0	0
$868$	1727.55	1.99026
$869$	0	0
$870$	0	0
$871$	225.857	0.259308
$872$	1277.10i	1.46457i
$873$	0	0
$874$	2795.23	3.19820
$875$	− 592.059i	− 0.676639i
$876$	0	0
$877$	1049.04	1.19617	0.598083	−	0.801434i	$-0.295929\pi$
$877$	1049.04	1.19617	0.598083	+	0.801434i	$0.295929\pi$
$878$	− 416.001i	− 0.473805i
$879$	0	0
$880$	0	0
$881$	624.821i	0.709218i	0.935015	+	0.354609i	$0.115386\pi$
$881$	624.821i	0.709218i	−0.935015	+	0.354609i	$0.884614\pi$
$882$	0	0
$883$	−1336.79	−1.51391	−0.756957	−	0.653465i	$-0.773315\pi$
$883$	−1336.79	−1.51391	−0.756957	+	0.653465i	$0.773315\pi$
$884$	− 911.838i	− 1.03149i
$885$	0	0
$886$	1871.98	2.11285
$887$	1268.64i	1.43025i	0.698995	+	0.715127i	$0.253631\pi$
$887$	1268.64i	1.43025i	−0.698995	+	0.715127i	$0.746369\pi$
$888$	0	0
$889$	−998.282	−1.12293
$890$	− 2279.69i	− 2.56145i
$891$	0	0
$892$	952.758	1.06811
$893$	687.171i	0.769508i
$894$	0	0
$895$	711.520	0.794995
$896$	2013.00i	2.24665i
$897$	0	0
$898$	−293.808	−0.327181
$899$	− 854.624i	− 0.950638i
$900$	0	0
$901$	−20.2238	−0.0224460
$902$	0	0
$903$	0	0
$904$	1585.46	1.75383
$905$	1005.56i	1.11111i
$906$	0	0
$907$	1470.26	1.62102	0.810509	−	0.585726i	$-0.199190\pi$
$907$	1470.26	1.62102	0.810509	+	0.585726i	$0.199190\pi$
$908$	1262.73i	1.39067i
$909$	0	0
$910$	4746.88	5.21636
$911$	− 1344.21i	− 1.47554i	−0.675055	−	0.737768i	$-0.735880\pi$
$911$	− 1344.21i	− 1.47554i	0.675055	−	0.737768i	$-0.264120\pi$
$912$	0	0
$913$	0	0
$914$	1004.61i	1.09914i
$915$	0	0
$916$	−1582.47	−1.72759
$917$	133.452i	0.145531i
$918$	0	0
$919$	−1013.26	−1.10256	−0.551282	−	0.834319i	$-0.685861\pi$
$919$	−1013.26	−1.10256	−0.551282	+	0.834319i	$0.685861\pi$
$920$	− 2323.12i	− 2.52513i
$921$	0	0
$922$	211.823	0.229742
$923$	822.690i	0.891322i
$924$	0	0
$925$	19.6069	0.0211967
$926$	− 246.931i	− 0.266665i
$927$	0	0
$928$	119.079	0.128317
$929$	578.763i	0.622995i	0.950247	+	0.311498i	$0.100831\pi$
$929$	578.763i	0.622995i	−0.950247	+	0.311498i	$0.899169\pi$
$930$	0	0
$931$	869.238	0.933660
$932$	525.333i	0.563662i
$933$	0	0
$934$	730.643	0.782273
$935$	0	0
$936$	0	0
$937$	−679.472	−0.725157	−0.362578	−	0.931953i	$-0.618103\pi$
$937$	−679.472	−0.725157	−0.362578	+	0.931953i	$0.618103\pi$
$938$	− 279.048i	− 0.297492i
$939$	0	0
$940$	1160.95	1.23505
$941$	− 1046.59i	− 1.11221i	−0.831112	−	0.556106i	$-0.812295\pi$
$941$	− 1046.59i	− 1.11221i	0.831112	−	0.556106i	$-0.187705\pi$
$942$	0	0
$943$	1414.11	1.49958
$944$	− 1004.24i	− 1.06381i
$945$	0	0
$946$	0	0
$947$	377.752i	0.398893i	0.979909	+	0.199447i	$0.0639144\pi$
$947$	377.752i	0.398893i	−0.979909	+	0.199447i	$0.936086\pi$
$948$	0	0
$949$	216.827	0.228479
$950$	1447.20i	1.52337i
$951$	0	0
$952$	−554.204	−0.582147
$953$	− 939.364i	− 0.985692i	−0.870117	−	0.492846i	$-0.835957\pi$
$953$	− 939.364i	− 0.985692i	0.870117	−	0.492846i	$-0.164043\pi$
$954$	0	0
$955$	−470.698	−0.492877
$956$	2658.91i	2.78129i
$957$	0	0
$958$	−30.4658	−0.0318015
$959$	− 1769.89i	− 1.84556i
$960$	0	0
$961$	−349.435	−0.363616
$962$	− 116.383i	− 0.120981i
$963$	0	0
$964$	−149.206	−0.154778
$965$	− 746.016i	− 0.773074i
$966$	0	0
$967$	−519.663	−0.537397	−0.268699	−	0.963224i	$-0.586594\pi$
$967$	−519.663	−0.537397	−0.268699	+	0.963224i	$0.586594\pi$
$968$	0	0
$969$	0	0
$970$	2207.18	2.27544
$971$	461.769i	0.475560i	0.971319	+	0.237780i	$0.0764198\pi$
$971$	461.769i	0.475560i	−0.971319	+	0.237780i	$0.923580\pi$
$972$	0	0
$973$	−1355.87	−1.39349
$974$	1665.15i	1.70960i
$975$	0	0
$976$	−535.968	−0.549147
$977$	− 1613.28i	− 1.65126i	−0.564211	−	0.825631i	$-0.690819\pi$
$977$	− 1613.28i	− 1.65126i	0.564211	−	0.825631i	$-0.309181\pi$
$978$	0	0
$979$	0	0
$980$	− 1468.54i	− 1.49851i
$981$	0	0
$982$	1092.31	1.11234
$983$	1662.92i	1.69168i	0.533437	+	0.845840i	$0.320900\pi$
$983$	1662.92i	1.69168i	−0.533437	+	0.845840i	$0.679100\pi$
$984$	0	0
$985$	1174.96	1.19286
$986$	557.324i	0.565238i
$987$	0	0
$988$	5696.26	5.76545
$989$	1743.54i	1.76293i
$990$	0	0
$991$	−68.7903	−0.0694150	−0.0347075	−	0.999398i	$-0.511050\pi$
$991$	−68.7903	−0.0694150	−0.0347075	+	0.999398i	$0.511050\pi$
$992$	85.2121i	0.0858993i
$993$	0	0
$994$	1016.44	1.02257
$995$	− 185.530i	− 0.186462i
$996$	0	0
$997$	−758.510	−0.760792	−0.380396	−	0.924824i	$-0.624212\pi$
$997$	−758.510	−0.760792	−0.380396	+	0.924824i	$0.624212\pi$
$998$	− 1735.37i	− 1.73885i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.b.c.485.1	✓	4
3.2	odd	2	inner	1089.3.b.c.485.4	yes	4
11.10	odd	2		1089.3.b.d.485.4	yes	4
33.32	even	2		1089.3.b.d.485.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1089.3.b.c.485.1	✓	4	1.1	even	1	trivial
1089.3.b.c.485.4	yes	4	3.2	odd	2	inner
1089.3.b.d.485.1	yes	4	33.32	even	2
1089.3.b.d.485.4	yes	4	11.10	odd	2

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)

\(f(q)\)	\(=\)	\(q-3.44572i q^{2} -7.87298 q^{4} +6.27415i q^{5} -8.87298 q^{7} +13.3452i q^{8} +O(q^{10})\)	\(q-3.44572i q^{2} -7.87298 q^{4} +6.27415i q^{5} -8.87298 q^{7} +13.3452i q^{8} +21.6190 q^{10} -24.7460 q^{13} +30.5738i q^{14} +14.4919 q^{16} -4.68030i q^{17} +29.2379 q^{19} -49.3963i q^{20} +27.7454i q^{23} -14.3649 q^{25} +85.2677i q^{26} +69.8569 q^{28} -34.5584i q^{29} +24.7298 q^{31} +3.44572i q^{32} -16.1270 q^{34} -55.6704i q^{35} -1.36492 q^{37} -100.746i q^{38} -83.7298 q^{40} -50.9673i q^{41} +62.8407 q^{43} +95.6028 q^{46} +23.5027i q^{47} +29.7298 q^{49} +49.4975i q^{50} +194.825 q^{52} -4.32105i q^{53} -118.412i q^{56} -119.079 q^{58} -69.2965i q^{59} -36.9839 q^{61} -85.2121i q^{62} +69.8407 q^{64} -155.260i q^{65} -9.12702 q^{67} +36.8480i q^{68} -191.825 q^{70} -33.2454i q^{71} -8.76210 q^{73} +4.70312i q^{74} -230.190 q^{76} -3.01613 q^{79} +90.9245i q^{80} -175.619 q^{82} -43.9746i q^{83} +29.3649 q^{85} -216.531i q^{86} -105.449i q^{89} +219.571 q^{91} -218.439i q^{92} +80.9839 q^{94} +183.443i q^{95} +102.095 q^{97} -102.441i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4} - 20 q^{7}+O(q^{10}) \) 4 * q - 16 * q^4 - 20 * q^7	\( 4 q - 16 q^{4} - 20 q^{7} + 40 q^{10} - 68 q^{13} - 4 q^{16} + 24 q^{19} + 20 q^{25} + 140 q^{28} - 56 q^{31} - 80 q^{34} + 72 q^{37} - 180 q^{40} - 12 q^{43} + 212 q^{46} - 36 q^{49} + 392 q^{52} - 120 q^{58} - 24 q^{61} + 16 q^{64} - 52 q^{67} - 380 q^{70} - 128 q^{73} - 456 q^{76} - 136 q^{79} - 656 q^{82} + 40 q^{85} + 460 q^{91} + 200 q^{94} + 176 q^{97}+O(q^{100}) \) 4 * q - 16 * q^4 - 20 * q^7 + 40 * q^10 - 68 * q^13 - 4 * q^16 + 24 * q^19 + 20 * q^25 + 140 * q^28 - 56 * q^31 - 80 * q^34 + 72 * q^37 - 180 * q^40 - 12 * q^43 + 212 * q^46 - 36 * q^49 + 392 * q^52 - 120 * q^58 - 24 * q^61 + 16 * q^64 - 52 * q^67 - 380 * q^70 - 128 * q^73 - 456 * q^76 - 136 * q^79 - 656 * q^82 + 40 * q^85 + 460 * q^91 + 200 * q^94 + 176 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more