LMFDB - Embedded newform 675.4.b.i.649.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(649,675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("675.649"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-40,0,0,0,0,0,0,0,0,0,0,0,-176,0,0,-116] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	675.649
Dual form			675.4.b.i.649.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.24264i q^{2} -10.0000 q^{4} +11.0000i q^{7} +8.48528i q^{8} -16.9706 q^{11} -29.0000i q^{13} +46.6690 q^{14} -44.0000 q^{16} +50.9117i q^{17} -29.0000 q^{19} +72.0000i q^{22} +84.8528i q^{23} -123.037 q^{26} -110.000i q^{28} +271.529 q^{29} -268.000 q^{31} +254.558i q^{32} +216.000 q^{34} +83.0000i q^{37} +123.037i q^{38} +271.529 q^{41} +232.000i q^{43} +169.706 q^{44} +360.000 q^{46} +390.323i q^{47} +222.000 q^{49} +290.000i q^{52} +305.470i q^{53} -93.3381 q^{56} -1152.00i q^{58} +288.500 q^{59} +767.000 q^{61} +1137.03i q^{62} +728.000 q^{64} -511.000i q^{67} -509.117i q^{68} -712.764 q^{71} -137.000i q^{73} +352.139 q^{74} +290.000 q^{76} -186.676i q^{77} +475.000 q^{79} -1152.00i q^{82} +576.999i q^{83} +984.293 q^{86} -144.000i q^{88} -254.558 q^{89} +319.000 q^{91} -848.528i q^{92} +1656.00 q^{94} +821.000i q^{97} -941.866i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 40 q^{4} - 176 q^{16} - 116 q^{19} - 1072 q^{31} + 864 q^{34} + 1440 q^{46} + 888 q^{49} + 3068 q^{61} + 2912 q^{64} + 1160 q^{76} + 1900 q^{79} + 1276 q^{91} + 6624 q^{94}+O(q^{100}) $ 4 * q - 40 * q^4 - 176 * q^16 - 116 * q^19 - 1072 * q^31 + 864 * q^34 + 1440 * q^46 + 888 * q^49 + 3068 * q^61 + 2912 * q^64 + 1160 * q^76 + 1900 * q^79 + 1276 * q^91 + 6624 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 4.24264i	− 1.50000i	−0.661438	−	0.750000i	$-0.730053\pi$
$2$	− 4.24264i	− 1.50000i	0.661438	−	0.750000i	$-0.269947\pi$
$3$	0	0
$3$	0	0
$4$	−10.0000	−1.25000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	11.0000i	0.593944i	0.954886	+	0.296972i	$0.0959768\pi$
$7$	11.0000i	0.593944i	−0.954886	+	0.296972i	$0.904023\pi$
$8$	8.48528i	0.375000i
$9$	0	0
$10$	0	0
$11$	−16.9706	−0.465165	−0.232583	−	0.972577i	$-0.574718\pi$
$11$	−16.9706	−0.465165	−0.232583	+	0.972577i	$0.574718\pi$
$12$	0	0
$13$	− 29.0000i	− 0.618704i	−0.950948	−	0.309352i	$-0.899888\pi$
$13$	− 29.0000i	− 0.618704i	0.950948	−	0.309352i	$-0.100112\pi$
$14$	46.6690	0.890916
$15$	0	0
$16$	−44.0000	−0.687500
$17$	50.9117i	0.726347i	0.931722	+	0.363173i	$0.118307\pi$
$17$	50.9117i	0.726347i	−0.931722	+	0.363173i	$0.881693\pi$
$18$	0	0
$19$	−29.0000	−0.350161	−0.175080	−	0.984554i	$-0.556019\pi$
$19$	−29.0000	−0.350161	−0.175080	+	0.984554i	$0.556019\pi$
$20$	0	0
$21$	0	0
$22$	72.0000i	0.697748i
$23$	84.8528i	0.769262i	0.923070	+	0.384631i	$0.125671\pi$
$23$	84.8528i	0.769262i	−0.923070	+	0.384631i	$0.874329\pi$
$24$	0	0
$25$	0	0
$26$	−123.037	−0.928056
$27$	0	0
$28$	− 110.000i	− 0.742430i
$29$	271.529	1.73868	0.869339	−	0.494216i	$-0.164545\pi$
$29$	271.529	1.73868	0.869339	+	0.494216i	$0.164545\pi$
$30$	0	0
$31$	−268.000	−1.55272	−0.776358	−	0.630292i	$-0.782935\pi$
$31$	−268.000	−1.55272	−0.776358	+	0.630292i	$0.782935\pi$
$32$	254.558i	1.40625i
$33$	0	0
$34$	216.000	1.08952
$35$	0	0
$36$	0	0
$37$	83.0000i	0.368787i	0.982853	+	0.184393i	$0.0590321\pi$
$37$	83.0000i	0.368787i	−0.982853	+	0.184393i	$0.940968\pi$
$38$	123.037i	0.525241i
$39$	0	0
$40$	0	0
$41$	271.529	1.03429	0.517143	−	0.855899i	$-0.326996\pi$
$41$	271.529	1.03429	0.517143	+	0.855899i	$0.326996\pi$
$42$	0	0
$43$	232.000i	0.822783i	0.911459	+	0.411391i	$0.134957\pi$
$43$	232.000i	0.822783i	−0.911459	+	0.411391i	$0.865043\pi$
$44$	169.706	0.581456
$45$	0	0
$46$	360.000	1.15389
$47$	390.323i	1.21137i	0.795704	+	0.605686i	$0.207101\pi$
$47$	390.323i	1.21137i	−0.795704	+	0.605686i	$0.792899\pi$
$48$	0	0
$49$	222.000	0.647230
$50$	0	0
$51$	0	0
$52$	290.000i	0.773380i
$53$	305.470i	0.791690i	0.918317	+	0.395845i	$0.129548\pi$
$53$	305.470i	0.791690i	−0.918317	+	0.395845i	$0.870452\pi$
$54$	0	0
$55$	0	0
$56$	−93.3381	−0.222729
$57$	0	0
$58$	− 1152.00i	− 2.60802i
$59$	288.500	0.636601	0.318300	−	0.947990i	$-0.396888\pi$
$59$	288.500	0.636601	0.318300	+	0.947990i	$0.396888\pi$
$60$	0	0
$61$	767.000	1.60991	0.804953	−	0.593338i	$-0.202190\pi$
$61$	767.000	1.60991	0.804953	+	0.593338i	$0.202190\pi$
$62$	1137.03i	2.32908i
$63$	0	0
$64$	728.000	1.42188
$65$	0	0
$66$	0	0
$67$	− 511.000i	− 0.931770i	−0.884845	−	0.465885i	$-0.845736\pi$
$67$	− 511.000i	− 0.931770i	0.884845	−	0.465885i	$-0.154264\pi$
$68$	− 509.117i	− 0.907934i
$69$	0	0
$70$	0	0
$71$	−712.764	−1.19140	−0.595701	−	0.803207i	$-0.703126\pi$
$71$	−712.764	−1.19140	−0.595701	+	0.803207i	$0.703126\pi$
$72$	0	0
$73$	− 137.000i	− 0.219653i	−0.993951	−	0.109826i	$-0.964971\pi$
$73$	− 137.000i	− 0.219653i	0.993951	−	0.109826i	$-0.0350295\pi$
$74$	352.139	0.553180
$75$	0	0
$76$	290.000	0.437701
$77$	− 186.676i	− 0.276282i
$78$	0	0
$79$	475.000	0.676477	0.338238	−	0.941060i	$-0.390169\pi$
$79$	475.000	0.676477	0.338238	+	0.941060i	$0.390169\pi$
$80$	0	0
$81$	0	0
$82$	− 1152.00i	− 1.55143i
$83$	576.999i	0.763059i	0.924357	+	0.381529i	$0.124603\pi$
$83$	576.999i	0.763059i	−0.924357	+	0.381529i	$0.875397\pi$
$84$	0	0
$85$	0	0
$86$	984.293	1.23417
$87$	0	0
$88$	− 144.000i	− 0.174437i
$89$	−254.558	−0.303181	−0.151591	−	0.988443i	$-0.548440\pi$
$89$	−254.558	−0.303181	−0.151591	+	0.988443i	$0.548440\pi$
$90$	0	0
$91$	319.000	0.367476
$92$	− 848.528i	− 0.961578i
$93$	0	0
$94$	1656.00	1.81706
$95$	0	0
$96$	0	0
$97$	821.000i	0.859381i	0.902976	+	0.429690i	$0.141377\pi$
$97$	821.000i	0.859381i	−0.902976	+	0.429690i	$0.858623\pi$
$98$	− 941.866i	− 0.970845i
$99$	0	0
$100$	0	0
$101$	543.058	0.535013	0.267506	−	0.963556i	$-0.413800\pi$
$101$	543.058	0.535013	0.267506	+	0.963556i	$0.413800\pi$
$102$	0	0
$103$	− 839.000i	− 0.802613i	−0.915944	−	0.401306i	$-0.868556\pi$
$103$	− 839.000i	− 0.802613i	0.915944	−	0.401306i	$-0.131444\pi$
$104$	246.073	0.232014
$105$	0	0
$106$	1296.00	1.18753
$107$	763.675i	0.689975i	0.938607	+	0.344987i	$0.112117\pi$
$107$	763.675i	0.689975i	−0.938607	+	0.344987i	$0.887883\pi$
$108$	0	0
$109$	−218.000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−218.000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0	0
$112$	− 484.000i	− 0.408337i
$113$	1510.38i	1.25739i	0.777654	+	0.628693i	$0.216410\pi$
$113$	1510.38i	1.25739i	−0.777654	+	0.628693i	$0.783590\pi$
$114$	0	0
$115$	0	0
$116$	−2715.29	−2.17335
$117$	0	0
$118$	− 1224.00i	− 0.954901i
$119$	−560.029	−0.431410
$120$	0	0
$121$	−1043.00	−0.783621
$122$	− 3254.11i	− 2.41486i
$123$	0	0
$124$	2680.00	1.94090
$125$	0	0
$126$	0	0
$127$	1244.00i	0.869190i	0.900626	+	0.434595i	$0.143109\pi$
$127$	1244.00i	0.869190i	−0.900626	+	0.434595i	$0.856891\pi$
$128$	− 1052.17i	− 0.726562i
$129$	0	0
$130$	0	0
$131$	2511.64	1.67514	0.837570	−	0.546330i	$-0.183976\pi$
$131$	2511.64	1.67514	0.837570	+	0.546330i	$0.183976\pi$
$132$	0	0
$133$	− 319.000i	− 0.207976i
$134$	−2167.99	−1.39765
$135$	0	0
$136$	−432.000	−0.272380
$137$	− 1340.67i	− 0.836070i	−0.908431	−	0.418035i	$-0.862719\pi$
$137$	− 1340.67i	− 0.836070i	0.908431	−	0.418035i	$-0.137281\pi$
$138$	0	0
$139$	−947.000	−0.577867	−0.288933	−	0.957349i	$-0.593301\pi$
$139$	−947.000	−0.577867	−0.288933	+	0.957349i	$0.593301\pi$
$140$	0	0
$141$	0	0
$142$	3024.00i	1.78710i
$143$	492.146i	0.287800i
$144$	0	0
$145$	0	0
$146$	−581.242	−0.329479
$147$	0	0
$148$	− 830.000i	− 0.460984i
$149$	−576.999	−0.317246	−0.158623	−	0.987339i	$-0.550705\pi$
$149$	−576.999	−0.317246	−0.158623	+	0.987339i	$0.550705\pi$
$150$	0	0
$151$	−2311.00	−1.24547	−0.622737	−	0.782431i	$-0.713979\pi$
$151$	−2311.00	−1.24547	−0.622737	+	0.782431i	$0.713979\pi$
$152$	− 246.073i	− 0.131310i
$153$	0	0
$154$	−792.000	−0.414423
$155$	0	0
$156$	0	0
$157$	1622.00i	0.824520i	0.911066	+	0.412260i	$0.135261\pi$
$157$	1622.00i	0.824520i	−0.911066	+	0.412260i	$0.864739\pi$
$158$	− 2015.25i	− 1.01472i
$159$	0	0
$160$	0	0
$161$	−933.381	−0.456899
$162$	0	0
$163$	− 2243.00i	− 1.07782i	−0.842362	−	0.538912i	$-0.818836\pi$
$163$	− 2243.00i	− 1.07782i	0.842362	−	0.538912i	$-0.181164\pi$
$164$	−2715.29	−1.29286
$165$	0	0
$166$	2448.00	1.14459
$167$	− 2732.26i	− 1.26604i	−0.774135	−	0.633020i	$-0.781815\pi$
$167$	− 2732.26i	− 1.26604i	0.774135	−	0.633020i	$-0.218185\pi$
$168$	0	0
$169$	1356.00	0.617205
$170$	0	0
$171$	0	0
$172$	− 2320.00i	− 1.02848i
$173$	− 1357.65i	− 0.596646i	−0.954465	−	0.298323i	$-0.903573\pi$
$173$	− 1357.65i	− 0.596646i	0.954465	−	0.298323i	$-0.0964273\pi$
$174$	0	0
$175$	0	0
$176$	746.705	0.319801
$177$	0	0
$178$	1080.00i	0.454772i
$179$	−2341.94	−0.977903	−0.488952	−	0.872311i	$-0.662621\pi$
$179$	−2341.94	−0.977903	−0.488952	+	0.872311i	$0.662621\pi$
$180$	0	0
$181$	−1591.00	−0.653360	−0.326680	−	0.945135i	$-0.605930\pi$
$181$	−1591.00	−0.653360	−0.326680	+	0.945135i	$0.605930\pi$
$182$	− 1353.40i	− 0.551214i
$183$	0	0
$184$	−720.000	−0.288473
$185$	0	0
$186$	0	0
$187$	− 864.000i	− 0.337871i
$188$	− 3903.23i	− 1.51421i
$189$	0	0
$190$	0	0
$191$	4768.73	1.80656	0.903280	−	0.429051i	$-0.141152\pi$
$191$	4768.73	1.80656	0.903280	+	0.429051i	$0.141152\pi$
$192$	0	0
$193$	3481.00i	1.29828i	0.760669	+	0.649140i	$0.224871\pi$
$193$	3481.00i	1.29828i	−0.760669	+	0.649140i	$0.775129\pi$
$194$	3483.21	1.28907
$195$	0	0
$196$	−2220.00	−0.809038
$197$	2392.85i	0.865398i	0.901538	+	0.432699i	$0.142439\pi$
$197$	2392.85i	0.865398i	−0.901538	+	0.432699i	$0.857561\pi$
$198$	0	0
$199$	−2351.00	−0.837477	−0.418739	−	0.908107i	$-0.637528\pi$
$199$	−2351.00	−0.837477	−0.418739	+	0.908107i	$0.637528\pi$
$200$	0	0
$201$	0	0
$202$	− 2304.00i	− 0.802519i
$203$	2986.82i	1.03268i
$204$	0	0
$205$	0	0
$206$	−3559.58	−1.20392
$207$	0	0
$208$	1276.00i	0.425359i
$209$	492.146	0.162883
$210$	0	0
$211$	1703.00	0.555637	0.277818	−	0.960634i	$-0.410389\pi$
$211$	1703.00	0.555637	0.277818	+	0.960634i	$0.410389\pi$
$212$	− 3054.70i	− 0.989612i
$213$	0	0
$214$	3240.00	1.03496
$215$	0	0
$216$	0	0
$217$	− 2948.00i	− 0.922227i
$218$	924.896i	0.287348i
$219$	0	0
$220$	0	0
$221$	1476.44	0.449394
$222$	0	0
$223$	− 1388.00i	− 0.416804i	−0.978043	−	0.208402i	$-0.933174\pi$
$223$	− 1388.00i	− 0.416804i	0.978043	−	0.208402i	$-0.0668263\pi$
$224$	−2800.14	−0.835234
$225$	0	0
$226$	6408.00	1.88608
$227$	4717.82i	1.37944i	0.724077	+	0.689719i	$0.242266\pi$
$227$	4717.82i	1.37944i	−0.724077	+	0.689719i	$0.757734\pi$
$228$	0	0
$229$	−434.000	−0.125238	−0.0626191	−	0.998038i	$-0.519945\pi$
$229$	−434.000	−0.125238	−0.0626191	+	0.998038i	$0.519945\pi$
$230$	0	0
$231$	0	0
$232$	2304.00i	0.652004i
$233$	3461.99i	0.973403i	0.873568	+	0.486701i	$0.161800\pi$
$233$	3461.99i	0.973403i	−0.873568	+	0.486701i	$0.838200\pi$
$234$	0	0
$235$	0	0
$236$	−2885.00	−0.795751
$237$	0	0
$238$	2376.00i	0.647114i
$239$	−2817.11	−0.762443	−0.381222	−	0.924484i	$-0.624497\pi$
$239$	−2817.11	−0.762443	−0.381222	+	0.924484i	$0.624497\pi$
$240$	0	0
$241$	−2095.00	−0.559962	−0.279981	−	0.960006i	$-0.590328\pi$
$241$	−2095.00	−0.559962	−0.279981	+	0.960006i	$0.590328\pi$
$242$	4425.07i	1.17543i
$243$	0	0
$244$	−7670.00	−2.01238
$245$	0	0
$246$	0	0
$247$	841.000i	0.216646i
$248$	− 2274.06i	− 0.582269i
$249$	0	0
$250$	0	0
$251$	−203.647	−0.0512114	−0.0256057	−	0.999672i	$-0.508151\pi$
$251$	−203.647	−0.0512114	−0.0256057	+	0.999672i	$0.508151\pi$
$252$	0	0
$253$	− 1440.00i	− 0.357834i
$254$	5277.85	1.30379
$255$	0	0
$256$	1360.00	0.332031
$257$	1798.88i	0.436619i	0.975880	+	0.218309i	$0.0700542\pi$
$257$	1798.88i	0.436619i	−0.975880	+	0.218309i	$0.929946\pi$
$258$	0	0
$259$	−913.000	−0.219039
$260$	0	0
$261$	0	0
$262$	− 10656.0i	− 2.51271i
$263$	373.352i	0.0875357i	0.999042	+	0.0437679i	$0.0139362\pi$
$263$	373.352i	0.0875357i	−0.999042	+	0.0437679i	$0.986064\pi$
$264$	0	0
$265$	0	0
$266$	−1353.40	−0.311964
$267$	0	0
$268$	5110.00i	1.16471i
$269$	7484.02	1.69631	0.848157	−	0.529744i	$-0.177712\pi$
$269$	7484.02	1.69631	0.848157	+	0.529744i	$0.177712\pi$
$270$	0	0
$271$	−3319.00	−0.743966	−0.371983	−	0.928239i	$-0.621322\pi$
$271$	−3319.00	−0.743966	−0.371983	+	0.928239i	$0.621322\pi$
$272$	− 2240.11i	− 0.499364i
$273$	0	0
$274$	−5688.00	−1.25410
$275$	0	0
$276$	0	0
$277$	8354.00i	1.81207i	0.423203	+	0.906035i	$0.360906\pi$
$277$	8354.00i	1.81207i	−0.423203	+	0.906035i	$0.639094\pi$
$278$	4017.78i	0.866800i
$279$	0	0
$280$	0	0
$281$	2579.53	0.547621	0.273811	−	0.961784i	$-0.411716\pi$
$281$	2579.53	0.547621	0.273811	+	0.961784i	$0.411716\pi$
$282$	0	0
$283$	6208.00i	1.30398i	0.758226	+	0.651992i	$0.226066\pi$
$283$	6208.00i	1.30398i	−0.758226	+	0.651992i	$0.773934\pi$
$284$	7127.64	1.48925
$285$	0	0
$286$	2088.00	0.431699
$287$	2986.82i	0.614308i
$288$	0	0
$289$	2321.00	0.472420
$290$	0	0
$291$	0	0
$292$	1370.00i	0.274566i
$293$	6194.26i	1.23506i	0.786548	+	0.617529i	$0.211866\pi$
$293$	6194.26i	1.23506i	−0.786548	+	0.617529i	$0.788134\pi$
$294$	0	0
$295$	0	0
$296$	−704.278	−0.138295
$297$	0	0
$298$	2448.00i	0.475869i
$299$	2460.73	0.475946
$300$	0	0
$301$	−2552.00	−0.488687
$302$	9804.74i	1.86821i
$303$	0	0
$304$	1276.00	0.240736
$305$	0	0
$306$	0	0
$307$	− 2320.00i	− 0.431301i	−0.976471	−	0.215650i	$-0.930813\pi$
$307$	− 2320.00i	− 0.431301i	0.976471	−	0.215650i	$-0.0691872\pi$
$308$	1866.76i	0.345353i
$309$	0	0
$310$	0	0
$311$	−797.616	−0.145430	−0.0727149	−	0.997353i	$-0.523166\pi$
$311$	−797.616	−0.145430	−0.0727149	+	0.997353i	$0.523166\pi$
$312$	0	0
$313$	− 1307.00i	− 0.236026i	−0.993012	−	0.118013i	$-0.962348\pi$
$313$	− 1307.00i	− 0.236026i	0.993012	−	0.118013i	$-0.0376524\pi$
$314$	6881.56	1.23678
$315$	0	0
$316$	−4750.00	−0.845596
$317$	2070.41i	0.366832i	0.983035	+	0.183416i	$0.0587155\pi$
$317$	2070.41i	0.366832i	−0.983035	+	0.183416i	$0.941284\pi$
$318$	0	0
$319$	−4608.00	−0.808773
$320$	0	0
$321$	0	0
$322$	3960.00i	0.685348i
$323$	− 1476.44i	− 0.254338i
$324$	0	0
$325$	0	0
$326$	−9516.24	−1.61674
$327$	0	0
$328$	2304.00i	0.387857i
$329$	−4293.55	−0.719487
$330$	0	0
$331$	−5173.00	−0.859014	−0.429507	−	0.903063i	$-0.641313\pi$
$331$	−5173.00	−0.859014	−0.429507	+	0.903063i	$0.641313\pi$
$332$	− 5769.99i	− 0.953824i
$333$	0	0
$334$	−11592.0	−1.89906
$335$	0	0
$336$	0	0
$337$	2621.00i	0.423665i	0.977306	+	0.211832i	$0.0679431\pi$
$337$	2621.00i	0.423665i	−0.977306	+	0.211832i	$0.932057\pi$
$338$	− 5753.02i	− 0.925808i
$339$	0	0
$340$	0	0
$341$	4548.11	0.722270
$342$	0	0
$343$	6215.00i	0.978363i
$344$	−1968.59	−0.308544
$345$	0	0
$346$	−5760.00	−0.894970
$347$	7704.64i	1.19195i	0.803003	+	0.595975i	$0.203234\pi$
$347$	7704.64i	1.19195i	−0.803003	+	0.595975i	$0.796766\pi$
$348$	0	0
$349$	−1955.00	−0.299853	−0.149927	−	0.988697i	$-0.547904\pi$
$349$	−1955.00	−0.299853	−0.149927	+	0.988697i	$0.547904\pi$
$350$	0	0
$351$	0	0
$352$	− 4320.00i	− 0.654139i
$353$	1187.94i	0.179115i	0.995982	+	0.0895576i	$0.0285453\pi$
$353$	1187.94i	0.179115i	−0.995982	+	0.0895576i	$0.971455\pi$
$354$	0	0
$355$	0	0
$356$	2545.58	0.378977
$357$	0	0
$358$	9936.00i	1.46685i
$359$	−560.029	−0.0823320	−0.0411660	−	0.999152i	$-0.513107\pi$
$359$	−560.029	−0.0823320	−0.0411660	+	0.999152i	$0.513107\pi$
$360$	0	0
$361$	−6018.00	−0.877387
$362$	6750.04i	0.980039i
$363$	0	0
$364$	−3190.00	−0.459345
$365$	0	0
$366$	0	0
$367$	7895.00i	1.12293i	0.827500	+	0.561465i	$0.189762\pi$
$367$	7895.00i	1.12293i	−0.827500	+	0.561465i	$0.810238\pi$
$368$	− 3733.52i	− 0.528868i
$369$	0	0
$370$	0	0
$371$	−3360.17	−0.470219
$372$	0	0
$373$	− 9803.00i	− 1.36080i	−0.732839	−	0.680402i	$-0.761805\pi$
$373$	− 9803.00i	− 1.36080i	0.732839	−	0.680402i	$-0.238195\pi$
$374$	−3665.64	−0.506807
$375$	0	0
$376$	−3312.00	−0.454264
$377$	− 7874.34i	− 1.07573i
$378$	0	0
$379$	−10505.0	−1.42376	−0.711881	−	0.702300i	$-0.752156\pi$
$379$	−10505.0	−1.42376	−0.711881	+	0.702300i	$0.752156\pi$
$380$	0	0
$381$	0	0
$382$	− 20232.0i	− 2.70984i
$383$	− 1086.12i	− 0.144903i	−0.997372	−	0.0724516i	$-0.976918\pi$
$383$	− 1086.12i	− 0.144903i	0.997372	−	0.0724516i	$-0.0230823\pi$
$384$	0	0
$385$	0	0
$386$	14768.6	1.94742
$387$	0	0
$388$	− 8210.00i	− 1.07423i
$389$	2155.26	0.280915	0.140458	−	0.990087i	$-0.455143\pi$
$389$	2155.26	0.280915	0.140458	+	0.990087i	$0.455143\pi$
$390$	0	0
$391$	−4320.00	−0.558751
$392$	1883.73i	0.242711i
$393$	0	0
$394$	10152.0	1.29810
$395$	0	0
$396$	0	0
$397$	12422.0i	1.57038i	0.619253	+	0.785192i	$0.287436\pi$
$397$	12422.0i	1.57038i	−0.619253	+	0.785192i	$0.712564\pi$
$398$	9974.45i	1.25622i
$399$	0	0
$400$	0	0
$401$	−15511.1	−1.93164	−0.965819	−	0.259216i	$-0.916536\pi$
$401$	−15511.1	−1.93164	−0.965819	+	0.259216i	$0.916536\pi$
$402$	0	0
$403$	7772.00i	0.960672i
$404$	−5430.58	−0.668766
$405$	0	0
$406$	12672.0	1.54902
$407$	− 1408.56i	− 0.171547i
$408$	0	0
$409$	−7265.00	−0.878316	−0.439158	−	0.898410i	$-0.644723\pi$
$409$	−7265.00	−0.878316	−0.439158	+	0.898410i	$0.644723\pi$
$410$	0	0
$411$	0	0
$412$	8390.00i	1.00327i
$413$	3173.50i	0.378105i
$414$	0	0
$415$	0	0
$416$	7382.19	0.870053
$417$	0	0
$418$	− 2088.00i	− 0.244324i
$419$	−3173.50	−0.370013	−0.185006	−	0.982737i	$-0.559231\pi$
$419$	−3173.50	−0.370013	−0.185006	+	0.982737i	$0.559231\pi$
$420$	0	0
$421$	3413.00	0.395106	0.197553	−	0.980292i	$-0.436701\pi$
$421$	3413.00	0.395106	0.197553	+	0.980292i	$0.436701\pi$
$422$	− 7225.22i	− 0.833455i
$423$	0	0
$424$	−2592.00	−0.296884
$425$	0	0
$426$	0	0
$427$	8437.00i	0.956194i
$428$	− 7636.75i	− 0.862468i
$429$	0	0
$430$	0	0
$431$	12677.0	1.41678	0.708388	−	0.705824i	$-0.249423\pi$
$431$	12677.0	1.41678	0.708388	+	0.705824i	$0.249423\pi$
$432$	0	0
$433$	− 8642.00i	− 0.959141i	−0.877503	−	0.479570i	$-0.840792\pi$
$433$	− 8642.00i	− 0.959141i	0.877503	−	0.479570i	$-0.159208\pi$
$434$	−12507.3	−1.38334
$435$	0	0
$436$	2180.00	0.239457
$437$	− 2460.73i	− 0.269366i
$438$	0	0
$439$	−524.000	−0.0569685	−0.0284842	−	0.999594i	$-0.509068\pi$
$439$	−524.000	−0.0569685	−0.0284842	+	0.999594i	$0.509068\pi$
$440$	0	0
$441$	0	0
$442$	− 6264.00i	− 0.674091i
$443$	− 18158.5i	− 1.94749i	−0.227649	−	0.973743i	$-0.573104\pi$
$443$	− 18158.5i	− 1.94749i	0.227649	−	0.973743i	$-0.426896\pi$
$444$	0	0
$445$	0	0
$446$	−5888.79	−0.625206
$447$	0	0
$448$	8008.00i	0.844514i
$449$	3309.26	0.347825	0.173913	−	0.984761i	$-0.444359\pi$
$449$	3309.26	0.347825	0.173913	+	0.984761i	$0.444359\pi$
$450$	0	0
$451$	−4608.00	−0.481114
$452$	− 15103.8i	− 1.57173i
$453$	0	0
$454$	20016.0	2.06916
$455$	0	0
$456$	0	0
$457$	− 9466.00i	− 0.968930i	−0.874811	−	0.484465i	$-0.839014\pi$
$457$	− 9466.00i	− 0.968930i	0.874811	−	0.484465i	$-0.160986\pi$
$458$	1841.31i	0.187857i
$459$	0	0
$460$	0	0
$461$	3241.38	0.327475	0.163738	−	0.986504i	$-0.447645\pi$
$461$	3241.38	0.327475	0.163738	+	0.986504i	$0.447645\pi$
$462$	0	0
$463$	− 11315.0i	− 1.13575i	−0.823115	−	0.567875i	$-0.807766\pi$
$463$	− 11315.0i	− 1.13575i	0.823115	−	0.567875i	$-0.192234\pi$
$464$	−11947.3	−1.19534
$465$	0	0
$466$	14688.0	1.46010
$467$	− 17462.7i	− 1.73036i	−0.501462	−	0.865180i	$-0.667204\pi$
$467$	− 17462.7i	− 1.73036i	0.501462	−	0.865180i	$-0.332796\pi$
$468$	0	0
$469$	5621.00	0.553419
$470$	0	0
$471$	0	0
$472$	2448.00i	0.238725i
$473$	− 3937.17i	− 0.382730i
$474$	0	0
$475$	0	0
$476$	5600.29	0.539262
$477$	0	0
$478$	11952.0i	1.14366i
$479$	8926.52	0.851488	0.425744	−	0.904844i	$-0.360012\pi$
$479$	8926.52	0.851488	0.425744	+	0.904844i	$0.360012\pi$
$480$	0	0
$481$	2407.00	0.228170
$482$	8888.33i	0.839943i
$483$	0	0
$484$	10430.0	0.979527
$485$	0	0
$486$	0	0
$487$	− 18493.0i	− 1.72073i	−0.509674	−	0.860367i	$-0.670234\pi$
$487$	− 18493.0i	− 1.72073i	0.509674	−	0.860367i	$-0.329766\pi$
$488$	6508.21i	0.603715i
$489$	0	0
$490$	0	0
$491$	−12812.8	−1.17766	−0.588831	−	0.808256i	$-0.700412\pi$
$491$	−12812.8	−1.17766	−0.588831	+	0.808256i	$0.700412\pi$
$492$	0	0
$493$	13824.0i	1.26288i
$494$	3568.06	0.324969
$495$	0	0
$496$	11792.0	1.06749
$497$	− 7840.40i	− 0.707626i
$498$	0	0
$499$	12256.0	1.09951	0.549753	−	0.835327i	$-0.314722\pi$
$499$	12256.0	1.09951	0.549753	+	0.835327i	$0.314722\pi$
$500$	0	0
$501$	0	0
$502$	864.000i	0.0768171i
$503$	− 7382.19i	− 0.654385i	−0.944958	−	0.327193i	$-0.893897\pi$
$503$	− 7382.19i	− 0.654385i	0.944958	−	0.327193i	$-0.106103\pi$
$504$	0	0
$505$	0	0
$506$	−6109.40	−0.536751
$507$	0	0
$508$	− 12440.0i	− 1.08649i
$509$	−20992.6	−1.82806	−0.914028	−	0.405652i	$-0.867044\pi$
$509$	−20992.6	−1.82806	−0.914028	+	0.405652i	$0.867044\pi$
$510$	0	0
$511$	1507.00	0.130461
$512$	− 14187.4i	− 1.22461i
$513$	0	0
$514$	7632.00	0.654928
$515$	0	0
$516$	0	0
$517$	− 6624.00i	− 0.563488i
$518$	3873.53i	0.328558i
$519$	0	0
$520$	0	0
$521$	−12269.7	−1.03176	−0.515879	−	0.856661i	$-0.672535\pi$
$521$	−12269.7	−1.03176	−0.515879	+	0.856661i	$0.672535\pi$
$522$	0	0
$523$	− 6833.00i	− 0.571293i	−0.958335	−	0.285646i	$-0.907792\pi$
$523$	− 6833.00i	− 0.571293i	0.958335	−	0.285646i	$-0.0922083\pi$
$524$	−25116.4	−2.09392
$525$	0	0
$526$	1584.00	0.131304
$527$	− 13644.3i	− 1.12781i
$528$	0	0
$529$	4967.00	0.408235
$530$	0	0
$531$	0	0
$532$	3190.00i	0.259970i
$533$	− 7874.34i	− 0.639917i
$534$	0	0
$535$	0	0
$536$	4335.98	0.349414
$537$	0	0
$538$	− 31752.0i	− 2.54447i
$539$	−3767.46	−0.301069
$540$	0	0
$541$	−10555.0	−0.838808	−0.419404	−	0.907800i	$-0.637761\pi$
$541$	−10555.0	−0.838808	−0.419404	+	0.907800i	$0.637761\pi$
$542$	14081.3i	1.11595i
$543$	0	0
$544$	−12960.0	−1.02143
$545$	0	0
$546$	0	0
$547$	17291.0i	1.35157i	0.737098	+	0.675786i	$0.236196\pi$
$547$	17291.0i	1.35157i	−0.737098	+	0.675786i	$0.763804\pi$
$548$	13406.7i	1.04509i
$549$	0	0
$550$	0	0
$551$	−7874.34	−0.608817
$552$	0	0
$553$	5225.00i	0.401790i
$554$	35443.0	2.71810
$555$	0	0
$556$	9470.00	0.722334
$557$	− 10335.1i	− 0.786196i	−0.919497	−	0.393098i	$-0.871403\pi$
$557$	− 10335.1i	− 0.786196i	0.919497	−	0.393098i	$-0.128597\pi$
$558$	0	0
$559$	6728.00	0.509059
$560$	0	0
$561$	0	0
$562$	− 10944.0i	− 0.821432i
$563$	16427.5i	1.22973i	0.788633	+	0.614864i	$0.210789\pi$
$563$	16427.5i	1.22973i	−0.788633	+	0.614864i	$0.789211\pi$
$564$	0	0
$565$	0	0
$566$	26338.3	1.95598
$567$	0	0
$568$	− 6048.00i	− 0.446775i
$569$	15697.8	1.15656	0.578282	−	0.815837i	$-0.303723\pi$
$569$	15697.8	1.15656	0.578282	+	0.815837i	$0.303723\pi$
$570$	0	0
$571$	−4075.00	−0.298658	−0.149329	−	0.988788i	$-0.547711\pi$
$571$	−4075.00	−0.298658	−0.149329	+	0.988788i	$0.547711\pi$
$572$	− 4921.46i	− 0.359749i
$573$	0	0
$574$	12672.0	0.921462
$575$	0	0
$576$	0	0
$577$	6995.00i	0.504689i	0.967637	+	0.252345i	$0.0812016\pi$
$577$	6995.00i	0.504689i	−0.967637	+	0.252345i	$0.918798\pi$
$578$	− 9847.17i	− 0.708630i
$579$	0	0
$580$	0	0
$581$	−6346.99	−0.453214
$582$	0	0
$583$	− 5184.00i	− 0.368266i
$584$	1162.48	0.0823697
$585$	0	0
$586$	26280.0	1.85259
$587$	5583.32i	0.392586i	0.980545	+	0.196293i	$0.0628904\pi$
$587$	5583.32i	0.392586i	−0.980545	+	0.196293i	$0.937110\pi$
$588$	0	0
$589$	7772.00	0.543701
$590$	0	0
$591$	0	0
$592$	− 3652.00i	− 0.253541i
$593$	− 14968.0i	− 1.03653i	−0.855219	−	0.518266i	$-0.826578\pi$
$593$	− 14968.0i	− 1.03653i	0.855219	−	0.518266i	$-0.173422\pi$
$594$	0	0
$595$	0	0
$596$	5769.99	0.396557
$597$	0	0
$598$	− 10440.0i	− 0.713919i
$599$	−18192.4	−1.24094	−0.620470	−	0.784230i	$-0.713058\pi$
$599$	−18192.4	−1.24094	−0.620470	+	0.784230i	$0.713058\pi$
$600$	0	0
$601$	−6550.00	−0.444559	−0.222280	−	0.974983i	$-0.571350\pi$
$601$	−6550.00	−0.444559	−0.222280	+	0.974983i	$0.571350\pi$
$602$	10827.2i	0.733031i
$603$	0	0
$604$	23110.0	1.55684
$605$	0	0
$606$	0	0
$607$	12827.0i	0.857713i	0.903373	+	0.428857i	$0.141083\pi$
$607$	12827.0i	0.857713i	−0.903373	+	0.428857i	$0.858917\pi$
$608$	− 7382.19i	− 0.492414i
$609$	0	0
$610$	0	0
$611$	11319.4	0.749480
$612$	0	0
$613$	− 18767.0i	− 1.23653i	−0.785970	−	0.618264i	$-0.787836\pi$
$613$	− 18767.0i	− 1.23653i	0.785970	−	0.618264i	$-0.212164\pi$
$614$	−9842.93	−0.646951
$615$	0	0
$616$	1584.00	0.103606
$617$	7551.90i	0.492752i	0.969174	+	0.246376i	$0.0792398\pi$
$617$	7551.90i	0.492752i	−0.969174	+	0.246376i	$0.920760\pi$
$618$	0	0
$619$	−24581.0	−1.59611	−0.798056	−	0.602583i	$-0.794138\pi$
$619$	−24581.0	−1.59611	−0.798056	+	0.602583i	$0.794138\pi$
$620$	0	0
$621$	0	0
$622$	3384.00i	0.218145i
$623$	− 2800.14i	− 0.180073i
$624$	0	0
$625$	0	0
$626$	−5545.13	−0.354038
$627$	0	0
$628$	− 16220.0i	− 1.03065i
$629$	−4225.67	−0.267867
$630$	0	0
$631$	−18223.0	−1.14968	−0.574838	−	0.818267i	$-0.694935\pi$
$631$	−18223.0	−1.14968	−0.574838	+	0.818267i	$0.694935\pi$
$632$	4030.51i	0.253679i
$633$	0	0
$634$	8784.00	0.550248
$635$	0	0
$636$	0	0
$637$	− 6438.00i	− 0.400444i
$638$	19550.1i	1.21316i
$639$	0	0
$640$	0	0
$641$	−1595.23	−0.0982963	−0.0491481	−	0.998792i	$-0.515651\pi$
$641$	−1595.23	−0.0982963	−0.0491481	+	0.998792i	$0.515651\pi$
$642$	0	0
$643$	26296.0i	1.61277i	0.591389	+	0.806386i	$0.298580\pi$
$643$	26296.0i	1.61277i	−0.591389	+	0.806386i	$0.701420\pi$
$644$	9333.81	0.571124
$645$	0	0
$646$	−6264.00	−0.381507
$647$	25659.5i	1.55916i	0.626301	+	0.779582i	$0.284568\pi$
$647$	25659.5i	1.55916i	−0.626301	+	0.779582i	$0.715432\pi$
$648$	0	0
$649$	−4896.00	−0.296125
$650$	0	0
$651$	0	0
$652$	22430.0i	1.34728i
$653$	441.235i	0.0264423i	0.999913	+	0.0132212i	$0.00420855\pi$
$653$	441.235i	0.0264423i	−0.999913	+	0.0132212i	$0.995791\pi$
$654$	0	0
$655$	0	0
$656$	−11947.3	−0.711071
$657$	0	0
$658$	18216.0i	1.07923i
$659$	27051.1	1.59903	0.799515	−	0.600647i	$-0.205090\pi$
$659$	27051.1	1.59903	0.799515	+	0.600647i	$0.205090\pi$
$660$	0	0
$661$	623.000	0.0366594	0.0183297	−	0.999832i	$-0.494165\pi$
$661$	623.000	0.0366594	0.0183297	+	0.999832i	$0.494165\pi$
$662$	21947.2i	1.28852i
$663$	0	0
$664$	−4896.00	−0.286147
$665$	0	0
$666$	0	0
$667$	23040.0i	1.33750i
$668$	27322.6i	1.58255i
$669$	0	0
$670$	0	0
$671$	−13016.4	−0.748872
$672$	0	0
$673$	− 27875.0i	− 1.59659i	−0.602269	−	0.798293i	$-0.705737\pi$
$673$	− 27875.0i	− 1.59659i	0.602269	−	0.798293i	$-0.294263\pi$
$674$	11120.0	0.635497
$675$	0	0
$676$	−13560.0	−0.771507
$677$	6601.55i	0.374768i	0.982287	+	0.187384i	$0.0600009\pi$
$677$	6601.55i	0.374768i	−0.982287	+	0.187384i	$0.939999\pi$
$678$	0	0
$679$	−9031.00	−0.510424
$680$	0	0
$681$	0	0
$682$	− 19296.0i	− 1.08340i
$683$	− 15680.8i	− 0.878491i	−0.898367	−	0.439245i	$-0.855246\pi$
$683$	− 15680.8i	− 0.878491i	0.898367	−	0.439245i	$-0.144754\pi$
$684$	0	0
$685$	0	0
$686$	26368.0	1.46754
$687$	0	0
$688$	− 10208.0i	− 0.565663i
$689$	8858.63	0.489822
$690$	0	0
$691$	−10600.0	−0.583564	−0.291782	−	0.956485i	$-0.594248\pi$
$691$	−10600.0	−0.583564	−0.291782	+	0.956485i	$0.594248\pi$
$692$	13576.5i	0.745808i
$693$	0	0
$694$	32688.0	1.78792
$695$	0	0
$696$	0	0
$697$	13824.0i	0.751250i
$698$	8294.36i	0.449780i
$699$	0	0
$700$	0	0
$701$	−13593.4	−0.732406	−0.366203	−	0.930535i	$-0.619342\pi$
$701$	−13593.4	−0.732406	−0.366203	+	0.930535i	$0.619342\pi$
$702$	0	0
$703$	− 2407.00i	− 0.129135i
$704$	−12354.6	−0.661407
$705$	0	0
$706$	5040.00	0.268673
$707$	5973.64i	0.317768i
$708$	0	0
$709$	33523.0	1.77572	0.887858	−	0.460117i	$-0.152193\pi$
$709$	33523.0	1.77572	0.887858	+	0.460117i	$0.152193\pi$
$710$	0	0
$711$	0	0
$712$	− 2160.00i	− 0.113693i
$713$	− 22740.6i	− 1.19445i
$714$	0	0
$715$	0	0
$716$	23419.4	1.22238
$717$	0	0
$718$	2376.00i	0.123498i
$719$	31870.7	1.65310	0.826549	−	0.562865i	$-0.190301\pi$
$719$	31870.7	1.65310	0.826549	+	0.562865i	$0.190301\pi$
$720$	0	0
$721$	9229.00	0.476707
$722$	25532.2i	1.31608i
$723$	0	0
$724$	15910.0	0.816700
$725$	0	0
$726$	0	0
$727$	− 13084.0i	− 0.667481i	−0.942665	−	0.333741i	$-0.891689\pi$
$727$	− 13084.0i	− 0.667481i	0.942665	−	0.333741i	$-0.108311\pi$
$728$	2706.80i	0.137803i
$729$	0	0
$730$	0	0
$731$	−11811.5	−0.597626
$732$	0	0
$733$	19222.0i	0.968596i	0.874903	+	0.484298i	$0.160925\pi$
$733$	19222.0i	0.968596i	−0.874903	+	0.484298i	$0.839075\pi$
$734$	33495.6	1.68440
$735$	0	0
$736$	−21600.0	−1.08178
$737$	8671.96i	0.433427i
$738$	0	0
$739$	−6320.00	−0.314594	−0.157297	−	0.987551i	$-0.550278\pi$
$739$	−6320.00	−0.314594	−0.157297	+	0.987551i	$0.550278\pi$
$740$	0	0
$741$	0	0
$742$	14256.0i	0.705329i
$743$	4157.79i	0.205295i	0.994718	+	0.102648i	$0.0327314\pi$
$743$	4157.79i	0.205295i	−0.994718	+	0.102648i	$0.967269\pi$
$744$	0	0
$745$	0	0
$746$	−41590.6	−2.04121
$747$	0	0
$748$	8640.00i	0.422339i
$749$	−8400.43	−0.409806
$750$	0	0
$751$	20333.0	0.987965	0.493982	−	0.869472i	$-0.335541\pi$
$751$	20333.0	0.987965	0.493982	+	0.869472i	$0.335541\pi$
$752$	− 17174.2i	− 0.832818i
$753$	0	0
$754$	−33408.0	−1.61359
$755$	0	0
$756$	0	0
$757$	− 14011.0i	− 0.672706i	−0.941736	−	0.336353i	$-0.890806\pi$
$757$	− 14011.0i	− 0.672706i	0.941736	−	0.336353i	$-0.109194\pi$
$758$	44568.9i	2.13564i
$759$	0	0
$760$	0	0
$761$	25981.9	1.23764	0.618820	−	0.785533i	$-0.287611\pi$
$761$	25981.9	1.23764	0.618820	+	0.785533i	$0.287611\pi$
$762$	0	0
$763$	− 2398.00i	− 0.113779i
$764$	−47687.3	−2.25820
$765$	0	0
$766$	−4608.00	−0.217355
$767$	− 8366.49i	− 0.393867i
$768$	0	0
$769$	6289.00	0.294912	0.147456	−	0.989069i	$-0.452892\pi$
$769$	6289.00	0.294912	0.147456	+	0.989069i	$0.452892\pi$
$770$	0	0
$771$	0	0
$772$	− 34810.0i	− 1.62285i
$773$	− 7229.46i	− 0.336385i	−0.985754	−	0.168192i	$-0.946207\pi$
$773$	− 7229.46i	− 0.336385i	0.985754	−	0.168192i	$-0.0537930\pi$
$774$	0	0
$775$	0	0
$776$	−6966.42	−0.322268
$777$	0	0
$778$	− 9144.00i	− 0.421373i
$779$	−7874.34	−0.362166
$780$	0	0
$781$	12096.0	0.554198
$782$	18328.2i	0.838127i
$783$	0	0
$784$	−9768.00	−0.444971
$785$	0	0
$786$	0	0
$787$	− 25675.0i	− 1.16292i	−0.813576	−	0.581458i	$-0.802482\pi$
$787$	− 25675.0i	− 1.16292i	0.813576	−	0.581458i	$-0.197518\pi$
$788$	− 23928.5i	− 1.08175i
$789$	0	0
$790$	0	0
$791$	−16614.2	−0.746817
$792$	0	0
$793$	− 22243.0i	− 0.996056i
$794$	52702.1	2.35558
$795$	0	0
$796$	23510.0	1.04685
$797$	3326.23i	0.147831i	0.997265	+	0.0739154i	$0.0235495\pi$
$797$	3326.23i	0.147831i	−0.997265	+	0.0739154i	$0.976451\pi$
$798$	0	0
$799$	−19872.0	−0.879876
$800$	0	0
$801$	0	0
$802$	65808.0i	2.89746i
$803$	2324.97i	0.102175i
$804$	0	0
$805$	0	0
$806$	32973.8	1.44101
$807$	0	0
$808$	4608.00i	0.200630i
$809$	10080.5	0.438087	0.219043	−	0.975715i	$-0.429706\pi$
$809$	10080.5	0.438087	0.219043	+	0.975715i	$0.429706\pi$
$810$	0	0
$811$	14312.0	0.619682	0.309841	−	0.950788i	$-0.399724\pi$
$811$	14312.0	0.619682	0.309841	+	0.950788i	$0.399724\pi$
$812$	− 29868.2i	− 1.29085i
$813$	0	0
$814$	−5976.00	−0.257320
$815$	0	0
$816$	0	0
$817$	− 6728.00i	− 0.288106i
$818$	30822.8i	1.31747i
$819$	0	0
$820$	0	0
$821$	−2766.20	−0.117590	−0.0587948	−	0.998270i	$-0.518726\pi$
$821$	−2766.20	−0.117590	−0.0587948	+	0.998270i	$0.518726\pi$
$822$	0	0
$823$	33343.0i	1.41223i	0.708098	+	0.706114i	$0.249553\pi$
$823$	33343.0i	1.41223i	−0.708098	+	0.706114i	$0.750447\pi$
$824$	7119.15	0.300980
$825$	0	0
$826$	13464.0	0.567158
$827$	18379.1i	0.772799i	0.922332	+	0.386399i	$0.126281\pi$
$827$	18379.1i	0.772799i	−0.922332	+	0.386399i	$0.873719\pi$
$828$	0	0
$829$	−3593.00	−0.150531	−0.0752654	−	0.997164i	$-0.523980\pi$
$829$	−3593.00	−0.150531	−0.0752654	+	0.997164i	$0.523980\pi$
$830$	0	0
$831$	0	0
$832$	− 21112.0i	− 0.879720i
$833$	11302.4i	0.470114i
$834$	0	0
$835$	0	0
$836$	−4921.46	−0.203603
$837$	0	0
$838$	13464.0i	0.555019i
$839$	−17140.3	−0.705301	−0.352651	−	0.935755i	$-0.614720\pi$
$839$	−17140.3	−0.705301	−0.352651	+	0.935755i	$0.614720\pi$
$840$	0	0
$841$	49339.0	2.02300
$842$	− 14480.1i	− 0.592658i
$843$	0	0
$844$	−17030.0	−0.694546
$845$	0	0
$846$	0	0
$847$	− 11473.0i	− 0.465427i
$848$	− 13440.7i	− 0.544287i
$849$	0	0
$850$	0	0
$851$	−7042.78	−0.283694
$852$	0	0
$853$	4741.00i	0.190303i	0.995463	+	0.0951517i	$0.0303336\pi$
$853$	4741.00i	0.190303i	−0.995463	+	0.0951517i	$0.969666\pi$
$854$	35795.2	1.43429
$855$	0	0
$856$	−6480.00	−0.258740
$857$	− 11981.2i	− 0.477562i	−0.971073	−	0.238781i	$-0.923252\pi$
$857$	− 11981.2i	− 0.477562i	0.971073	−	0.238781i	$-0.0767478\pi$
$858$	0	0
$859$	−6887.00	−0.273552	−0.136776	−	0.990602i	$-0.543674\pi$
$859$	−6887.00	−0.273552	−0.136776	+	0.990602i	$0.543674\pi$
$860$	0	0
$861$	0	0
$862$	− 53784.0i	− 2.12516i
$863$	− 8400.43i	− 0.331349i	−0.986181	−	0.165674i	$-0.947020\pi$
$863$	− 8400.43i	− 0.331349i	0.986181	−	0.165674i	$-0.0529801\pi$
$864$	0	0
$865$	0	0
$866$	−36664.9	−1.43871
$867$	0	0
$868$	29480.0i	1.15278i
$869$	−8061.02	−0.314674
$870$	0	0
$871$	−14819.0	−0.576490
$872$	− 1849.79i	− 0.0718370i
$873$	0	0
$874$	−10440.0	−0.404048
$875$	0	0
$876$	0	0
$877$	13475.0i	0.518835i	0.965765	+	0.259418i	$0.0835306\pi$
$877$	13475.0i	0.518835i	−0.965765	+	0.259418i	$0.916469\pi$
$878$	2223.14i	0.0854527i
$879$	0	0
$880$	0	0
$881$	5243.90	0.200535	0.100268	−	0.994961i	$-0.468030\pi$
$881$	5243.90	0.200535	0.100268	+	0.994961i	$0.468030\pi$
$882$	0	0
$883$	7909.00i	0.301426i	0.988578	+	0.150713i	$0.0481569\pi$
$883$	7909.00i	0.301426i	−0.988578	+	0.150713i	$0.951843\pi$
$884$	−14764.4	−0.561742
$885$	0	0
$886$	−77040.0	−2.92123
$887$	− 35672.1i	− 1.35034i	−0.737662	−	0.675171i	$-0.764070\pi$
$887$	− 35672.1i	− 1.35034i	0.737662	−	0.675171i	$-0.235930\pi$
$888$	0	0
$889$	−13684.0	−0.516250
$890$	0	0
$891$	0	0
$892$	13880.0i	0.521005i
$893$	− 11319.4i	− 0.424175i
$894$	0	0
$895$	0	0
$896$	11573.9	0.431538
$897$	0	0
$898$	− 14040.0i	− 0.521738i
$899$	−72769.8	−2.69967
$900$	0	0
$901$	−15552.0	−0.575041
$902$	19550.1i	0.721670i
$903$	0	0
$904$	−12816.0	−0.471520
$905$	0	0
$906$	0	0
$907$	− 16999.0i	− 0.622318i	−0.950358	−	0.311159i	$-0.899283\pi$
$907$	− 16999.0i	− 0.622318i	0.950358	−	0.311159i	$-0.100717\pi$
$908$	− 47178.2i	− 1.72430i
$909$	0	0
$910$	0	0
$911$	39032.3	1.41954	0.709768	−	0.704435i	$-0.248800\pi$
$911$	39032.3	1.41954	0.709768	+	0.704435i	$0.248800\pi$
$912$	0	0
$913$	− 9792.00i	− 0.354948i
$914$	−40160.8	−1.45339
$915$	0	0
$916$	4340.00	0.156548
$917$	27628.1i	0.994939i
$918$	0	0
$919$	28348.0	1.01753	0.508767	−	0.860904i	$-0.330101\pi$
$919$	28348.0	1.01753	0.508767	+	0.860904i	$0.330101\pi$
$920$	0	0
$921$	0	0
$922$	− 13752.0i	− 0.491213i
$923$	20670.1i	0.737125i
$924$	0	0
$925$	0	0
$926$	−48005.5	−1.70363
$927$	0	0
$928$	69120.0i	2.44502i
$929$	33160.5	1.17111	0.585554	−	0.810633i	$-0.300877\pi$
$929$	33160.5	1.17111	0.585554	+	0.810633i	$0.300877\pi$
$930$	0	0
$931$	−6438.00	−0.226635
$932$	− 34619.9i	− 1.21675i
$933$	0	0
$934$	−74088.0	−2.59554
$935$	0	0
$936$	0	0
$937$	− 133.000i	− 0.00463706i	−0.999997	−	0.00231853i	$-0.999262\pi$
$937$	− 133.000i	− 0.00463706i	0.999997	−	0.00231853i	$-0.000738011\pi$
$938$	− 23847.9i	− 0.830129i
$939$	0	0
$940$	0	0
$941$	48790.4	1.69024	0.845122	−	0.534573i	$-0.179527\pi$
$941$	48790.4	1.69024	0.845122	+	0.534573i	$0.179527\pi$
$942$	0	0
$943$	23040.0i	0.795637i
$944$	−12694.0	−0.437663
$945$	0	0
$946$	−16704.0	−0.574095
$947$	− 45328.4i	− 1.55541i	−0.628629	−	0.777705i	$-0.716383\pi$
$947$	− 45328.4i	− 1.55541i	0.628629	−	0.777705i	$-0.283617\pi$
$948$	0	0
$949$	−3973.00	−0.135900
$950$	0	0
$951$	0	0
$952$	− 4752.00i	− 0.161779i
$953$	43122.2i	1.46576i	0.680360	+	0.732878i	$0.261823\pi$
$953$	43122.2i	1.46576i	−0.680360	+	0.732878i	$0.738177\pi$
$954$	0	0
$955$	0	0
$956$	28171.1	0.953054
$957$	0	0
$958$	− 37872.0i	− 1.27723i
$959$	14747.4	0.496579
$960$	0	0
$961$	42033.0	1.41093
$962$	− 10212.0i	− 0.342255i
$963$	0	0
$964$	20950.0	0.699952
$965$	0	0
$966$	0	0
$967$	22061.0i	0.733644i	0.930291	+	0.366822i	$0.119554\pi$
$967$	22061.0i	0.733644i	−0.930291	+	0.366822i	$0.880446\pi$
$968$	− 8850.15i	− 0.293858i
$969$	0	0
$970$	0	0
$971$	−33449.0	−1.10549	−0.552744	−	0.833351i	$-0.686419\pi$
$971$	−33449.0	−1.10549	−0.552744	+	0.833351i	$0.686419\pi$
$972$	0	0
$973$	− 10417.0i	− 0.343221i
$974$	−78459.2	−2.58110
$975$	0	0
$976$	−33748.0	−1.10681
$977$	− 20602.3i	− 0.674642i	−0.941390	−	0.337321i	$-0.890479\pi$
$977$	− 20602.3i	− 0.674642i	0.941390	−	0.337321i	$-0.109521\pi$
$978$	0	0
$979$	4320.00	0.141029
$980$	0	0
$981$	0	0
$982$	54360.0i	1.76649i
$983$	11930.3i	0.387098i	0.981091	+	0.193549i	$0.0619999\pi$
$983$	11930.3i	0.387098i	−0.981091	+	0.193549i	$0.938000\pi$
$984$	0	0
$985$	0	0
$986$	58650.3	1.89433
$987$	0	0
$988$	− 8410.00i	− 0.270807i
$989$	−19685.9	−0.632936
$990$	0	0
$991$	−35017.0	−1.12245	−0.561227	−	0.827662i	$-0.689670\pi$
$991$	−35017.0	−1.12245	−0.561227	+	0.827662i	$0.689670\pi$
$992$	− 68221.7i	− 2.18351i
$993$	0	0
$994$	−33264.0	−1.06144
$995$	0	0
$996$	0	0
$997$	13646.0i	0.433474i	0.976230	+	0.216737i	$0.0695413\pi$
$997$	13646.0i	0.433474i	−0.976230	+	0.216737i	$0.930459\pi$
$998$	− 51997.8i	− 1.64926i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.b.i.649.2		4
3.2	odd	2	inner	675.4.b.i.649.4		4
5.2	odd	4		675.4.a.n.1.2		2
5.3	odd	4		27.4.a.c.1.1	✓	2
5.4	even	2	inner	675.4.b.i.649.3		4
15.2	even	4		675.4.a.n.1.1		2
15.8	even	4		27.4.a.c.1.2	yes	2
15.14	odd	2	inner	675.4.b.i.649.1		4
20.3	even	4		432.4.a.q.1.2		2
35.13	even	4		1323.4.a.t.1.1		2
40.3	even	4		1728.4.a.bk.1.1		2
40.13	odd	4		1728.4.a.bp.1.1		2
45.13	odd	12		81.4.c.e.55.2		4
45.23	even	12		81.4.c.e.55.1		4
45.38	even	12		81.4.c.e.28.1		4
45.43	odd	12		81.4.c.e.28.2		4
60.23	odd	4		432.4.a.q.1.1		2
105.83	odd	4		1323.4.a.t.1.2		2
120.53	even	4		1728.4.a.bp.1.2		2
120.83	odd	4		1728.4.a.bk.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.a.c.1.1	✓	2	5.3	odd	4
27.4.a.c.1.2	yes	2	15.8	even	4
81.4.c.e.28.1		4	45.38	even	12
81.4.c.e.28.2		4	45.43	odd	12
81.4.c.e.55.1		4	45.23	even	12
81.4.c.e.55.2		4	45.13	odd	12
432.4.a.q.1.1		2	60.23	odd	4
432.4.a.q.1.2		2	20.3	even	4
675.4.a.n.1.1		2	15.2	even	4
675.4.a.n.1.2		2	5.2	odd	4
675.4.b.i.649.1		4	15.14	odd	2	inner
675.4.b.i.649.2		4	1.1	even	1	trivial
675.4.b.i.649.3		4	5.4	even	2	inner
675.4.b.i.649.4		4	3.2	odd	2	inner
1323.4.a.t.1.1		2	35.13	even	4
1323.4.a.t.1.2		2	105.83	odd	4
1728.4.a.bk.1.1		2	40.3	even	4
1728.4.a.bk.1.2		2	120.83	odd	4
1728.4.a.bp.1.1		2	40.13	odd	4
1728.4.a.bp.1.2		2	120.53	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.24264i q^{2} -10.0000 q^{4} +11.0000i q^{7} +8.48528i q^{8} -16.9706 q^{11} -29.0000i q^{13} +46.6690 q^{14} -44.0000 q^{16} +50.9117i q^{17} -29.0000 q^{19} +72.0000i q^{22} +84.8528i q^{23} -123.037 q^{26} -110.000i q^{28} +271.529 q^{29} -268.000 q^{31} +254.558i q^{32} +216.000 q^{34} +83.0000i q^{37} +123.037i q^{38} +271.529 q^{41} +232.000i q^{43} +169.706 q^{44} +360.000 q^{46} +390.323i q^{47} +222.000 q^{49} +290.000i q^{52} +305.470i q^{53} -93.3381 q^{56} -1152.00i q^{58} +288.500 q^{59} +767.000 q^{61} +1137.03i q^{62} +728.000 q^{64} -511.000i q^{67} -509.117i q^{68} -712.764 q^{71} -137.000i q^{73} +352.139 q^{74} +290.000 q^{76} -186.676i q^{77} +475.000 q^{79} -1152.00i q^{82} +576.999i q^{83} +984.293 q^{86} -144.000i q^{88} -254.558 q^{89} +319.000 q^{91} -848.528i q^{92} +1656.00 q^{94} +821.000i q^{97} -941.866i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 40 q^{4} - 176 q^{16} - 116 q^{19} - 1072 q^{31} + 864 q^{34} + 1440 q^{46} + 888 q^{49} + 3068 q^{61} + 2912 q^{64} + 1160 q^{76} + 1900 q^{79} + 1276 q^{91} + 6624 q^{94}+O(q^{100}) \) 4 * q - 40 * q^4 - 176 * q^16 - 116 * q^19 - 1072 * q^31 + 864 * q^34 + 1440 * q^46 + 888 * q^49 + 3068 * q^61 + 2912 * q^64 + 1160 * q^76 + 1900 * q^79 + 1276 * q^91 + 6624 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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