LMFDB - Embedded newform 675.3.c.p.26.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,3,Mod(26,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, 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("675.26");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	675.c (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:	$18.3924178443$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	675.26
Dual form			675.3.c.p.26.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61803i q^{2} -2.85410 q^{4} +9.70820 q^{7} -3.00000i q^{8} +14.1803i q^{11} +5.70820 q^{13} -25.4164i q^{14} -19.2705 q^{16} +5.94427i q^{17} +32.4164 q^{19} +37.1246 q^{22} -1.58359i q^{23} -14.9443i q^{26} -27.7082 q^{28} +2.06888i q^{29} +49.2492 q^{31} +38.4508i q^{32} +15.5623 q^{34} +26.5836 q^{37} -84.8673i q^{38} +65.2361i q^{41} -49.3738 q^{43} -40.4721i q^{44} -4.14590 q^{46} -70.1378i q^{47} +45.2492 q^{49} -16.2918 q^{52} -28.6393i q^{53} -29.1246i q^{56} +5.41641 q^{58} -95.2361i q^{59} +19.0000 q^{61} -128.936i q^{62} +23.5836 q^{64} -107.666 q^{67} -16.9656i q^{68} -75.2624i q^{71} +96.7902 q^{73} -69.5967i q^{74} -92.5197 q^{76} +137.666i q^{77} -101.833 q^{79} +170.790 q^{82} +44.6656i q^{83} +129.262i q^{86} +42.5410 q^{88} -56.2918i q^{89} +55.4164 q^{91} +4.51973i q^{92} -183.623 q^{94} -132.584 q^{97} -118.464i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 12 q^{7} - 4 q^{13} - 10 q^{16} + 76 q^{19} + 68 q^{22} - 84 q^{28} + 36 q^{31} + 22 q^{34} + 160 q^{37} + 44 q^{43} - 30 q^{46} + 20 q^{49} - 92 q^{52} - 32 q^{58} + 76 q^{61} + 148 q^{64}+ \cdots - 584 q^{97}+O(q^{100}) $ 4 * q + 2 * q^4 + 12 * q^7 - 4 * q^13 - 10 * q^16 + 76 * q^19 + 68 * q^22 - 84 * q^28 + 36 * q^31 + 22 * q^34 + 160 * q^37 + 44 * q^43 - 30 * q^46 + 20 * q^49 - 92 * q^52 - 32 * q^58 + 76 * q^61 + 148 * q^64 - 216 * q^67 + 92 * q^73 - 142 * q^76 - 300 * q^79 + 388 * q^82 + 36 * q^88 + 168 * q^91 - 332 * q^94 - 584 * q^97

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 $

$2$	− 2.61803i	− 1.30902i	−0.756055	−	0.654508i	$-0.772876\pi$
$2$	− 2.61803i	− 1.30902i	0.756055	−	0.654508i	$-0.227124\pi$
$3$	0	0
$3$	0	0
$4$	−2.85410	−0.713525
$5$	0	0
$5$	0	0
$6$	0	0
$7$	9.70820	1.38689	0.693443	−	0.720511i	$-0.256093\pi$
$7$	9.70820	1.38689	0.693443	+	0.720511i	$0.256093\pi$
$8$	− 3.00000i	− 0.375000i
$9$	0	0
$10$	0	0
$11$	14.1803i	1.28912i	0.764553	+	0.644561i	$0.222960\pi$
$11$	14.1803i	1.28912i	−0.764553	+	0.644561i	$0.777040\pi$
$12$	0	0
$13$	5.70820	0.439093	0.219546	−	0.975602i	$-0.429542\pi$
$13$	5.70820	0.439093	0.219546	+	0.975602i	$0.429542\pi$
$14$	− 25.4164i	− 1.81546i
$15$	0	0
$16$	−19.2705	−1.20441
$17$	5.94427i	0.349663i	0.984598	+	0.174832i	$0.0559381\pi$
$17$	5.94427i	0.349663i	−0.984598	+	0.174832i	$0.944062\pi$
$18$	0	0
$19$	32.4164	1.70613	0.853063	−	0.521807i	$-0.174742\pi$
$19$	32.4164	1.70613	0.853063	+	0.521807i	$0.174742\pi$
$20$	0	0
$21$	0	0
$22$	37.1246	1.68748
$23$	− 1.58359i	− 0.0688518i	−0.999407	−	0.0344259i	$-0.989040\pi$
$23$	− 1.58359i	− 0.0688518i	0.999407	−	0.0344259i	$-0.0109603\pi$
$24$	0	0
$25$	0	0
$26$	− 14.9443i	− 0.574780i
$27$	0	0
$28$	−27.7082	−0.989579
$29$	2.06888i	0.0713408i	0.999364	+	0.0356704i	$0.0113567\pi$
$29$	2.06888i	0.0713408i	−0.999364	+	0.0356704i	$0.988643\pi$
$30$	0	0
$31$	49.2492	1.58868	0.794342	−	0.607470i	$-0.207816\pi$
$31$	49.2492	1.58868	0.794342	+	0.607470i	$0.207816\pi$
$32$	38.4508i	1.20159i
$33$	0	0
$34$	15.5623	0.457715
$35$	0	0
$36$	0	0
$37$	26.5836	0.718475	0.359238	−	0.933246i	$-0.383037\pi$
$37$	26.5836	0.718475	0.359238	+	0.933246i	$0.383037\pi$
$38$	− 84.8673i	− 2.23335i
$39$	0	0
$40$	0	0
$41$	65.2361i	1.59112i	0.605872	+	0.795562i	$0.292824\pi$
$41$	65.2361i	1.59112i	−0.605872	+	0.795562i	$0.707176\pi$
$42$	0	0
$43$	−49.3738	−1.14823	−0.574114	−	0.818775i	$-0.694654\pi$
$43$	−49.3738	−1.14823	−0.574114	+	0.818775i	$0.694654\pi$
$44$	− 40.4721i	− 0.919821i
$45$	0	0
$46$	−4.14590	−0.0901282
$47$	− 70.1378i	− 1.49229i	−0.665782	−	0.746146i	$-0.731902\pi$
$47$	− 70.1378i	− 1.49229i	0.665782	−	0.746146i	$-0.268098\pi$
$48$	0	0
$49$	45.2492	0.923454
$50$	0	0
$51$	0	0
$52$	−16.2918	−0.313304
$53$	− 28.6393i	− 0.540365i	−0.962809	−	0.270182i	$-0.912916\pi$
$53$	− 28.6393i	− 0.540365i	0.962809	−	0.270182i	$-0.0870840\pi$
$54$	0	0
$55$	0	0
$56$	− 29.1246i	− 0.520082i
$57$	0	0
$58$	5.41641	0.0933863
$59$	− 95.2361i	− 1.61417i	−0.590435	−	0.807085i	$-0.701044\pi$
$59$	− 95.2361i	− 1.61417i	0.590435	−	0.807085i	$-0.298956\pi$
$60$	0	0
$61$	19.0000	0.311475	0.155738	−	0.987798i	$-0.450225\pi$
$61$	19.0000	0.311475	0.155738	+	0.987798i	$0.450225\pi$
$62$	− 128.936i	− 2.07962i
$63$	0	0
$64$	23.5836	0.368494
$65$	0	0
$66$	0	0
$67$	−107.666	−1.60695	−0.803475	−	0.595339i	$-0.797018\pi$
$67$	−107.666	−1.60695	−0.803475	+	0.595339i	$0.797018\pi$
$68$	− 16.9656i	− 0.249494i
$69$	0	0
$70$	0	0
$71$	− 75.2624i	− 1.06003i	−0.847987	−	0.530017i	$-0.822186\pi$
$71$	− 75.2624i	− 1.06003i	0.847987	−	0.530017i	$-0.177814\pi$
$72$	0	0
$73$	96.7902	1.32589	0.662947	−	0.748666i	$-0.269306\pi$
$73$	96.7902	1.32589	0.662947	+	0.748666i	$0.269306\pi$
$74$	− 69.5967i	− 0.940497i
$75$	0	0
$76$	−92.5197	−1.21736
$77$	137.666i	1.78787i
$78$	0	0
$79$	−101.833	−1.28902	−0.644511	−	0.764595i	$-0.722939\pi$
$79$	−101.833	−1.28902	−0.644511	+	0.764595i	$0.722939\pi$
$80$	0	0
$81$	0	0
$82$	170.790	2.08281
$83$	44.6656i	0.538140i	0.963121	+	0.269070i	$0.0867163\pi$
$83$	44.6656i	0.538140i	−0.963121	+	0.269070i	$0.913284\pi$
$84$	0	0
$85$	0	0
$86$	129.262i	1.50305i
$87$	0	0
$88$	42.5410	0.483421
$89$	− 56.2918i	− 0.632492i	−0.948677	−	0.316246i	$-0.897577\pi$
$89$	− 56.2918i	− 0.632492i	0.948677	−	0.316246i	$-0.102423\pi$
$90$	0	0
$91$	55.4164	0.608972
$92$	4.51973i	0.0491275i
$93$	0	0
$94$	−183.623	−1.95344
$95$	0	0
$96$	0	0
$97$	−132.584	−1.36684	−0.683421	−	0.730025i	$-0.739508\pi$
$97$	−132.584	−1.36684	−0.683421	+	0.730025i	$0.739508\pi$
$98$	− 118.464i	− 1.20882i
$99$	0	0
$100$	0	0
$101$	24.0000i	0.237624i	0.992917	+	0.118812i	$0.0379085\pi$
$101$	24.0000i	0.237624i	−0.992917	+	0.118812i	$0.962091\pi$
$102$	0	0
$103$	78.8328	0.765367	0.382684	−	0.923879i	$-0.375000\pi$
$103$	78.8328	0.765367	0.382684	+	0.923879i	$0.375000\pi$
$104$	− 17.1246i	− 0.164660i
$105$	0	0
$106$	−74.9787	−0.707346
$107$	59.6656i	0.557623i	0.960346	+	0.278811i	$0.0899404\pi$
$107$	59.6656i	0.557623i	−0.960346	+	0.278811i	$0.910060\pi$
$108$	0	0
$109$	−22.9149	−0.210228	−0.105114	−	0.994460i	$-0.533521\pi$
$109$	−22.9149	−0.210228	−0.105114	+	0.994460i	$0.533521\pi$
$110$	0	0
$111$	0	0
$112$	−187.082	−1.67038
$113$	− 128.387i	− 1.13617i	−0.822971	−	0.568084i	$-0.807685\pi$
$113$	− 128.387i	− 1.13617i	0.822971	−	0.568084i	$-0.192315\pi$
$114$	0	0
$115$	0	0
$116$	− 5.90481i	− 0.0509035i
$117$	0	0
$118$	−249.331	−2.11298
$119$	57.7082i	0.484943i
$120$	0	0
$121$	−80.0820	−0.661835
$122$	− 49.7426i	− 0.407727i
$123$	0	0
$124$	−140.562	−1.13357
$125$	0	0
$126$	0	0
$127$	−40.7477	−0.320848	−0.160424	−	0.987048i	$-0.551286\pi$
$127$	−40.7477	−0.320848	−0.160424	+	0.987048i	$0.551286\pi$
$128$	92.0608i	0.719225i
$129$	0	0
$130$	0	0
$131$	260.164i	1.98599i	0.118176	+	0.992993i	$0.462295\pi$
$131$	260.164i	1.98599i	−0.118176	+	0.992993i	$0.537705\pi$
$132$	0	0
$133$	314.705	2.36620
$134$	281.872i	2.10352i
$135$	0	0
$136$	17.8328	0.131124
$137$	164.331i	1.19950i	0.800188	+	0.599749i	$0.204733\pi$
$137$	164.331i	1.19950i	−0.800188	+	0.599749i	$0.795267\pi$
$138$	0	0
$139$	34.4984	0.248190	0.124095	−	0.992270i	$-0.460397\pi$
$139$	34.4984	0.248190	0.124095	+	0.992270i	$0.460397\pi$
$140$	0	0
$141$	0	0
$142$	−197.039	−1.38760
$143$	80.9443i	0.566044i
$144$	0	0
$145$	0	0
$146$	− 253.400i	− 1.73562i
$147$	0	0
$148$	−75.8723	−0.512651
$149$	− 38.1803i	− 0.256244i	−0.991758	−	0.128122i	$-0.959105\pi$
$149$	− 38.1803i	− 0.256244i	0.991758	−	0.128122i	$-0.0408949\pi$
$150$	0	0
$151$	182.997	1.21190	0.605950	−	0.795503i	$-0.292793\pi$
$151$	182.997	1.21190	0.605950	+	0.795503i	$0.292793\pi$
$152$	− 97.2492i	− 0.639798i
$153$	0	0
$154$	360.413	2.34035
$155$	0	0
$156$	0	0
$157$	144.705	0.921689	0.460844	−	0.887481i	$-0.347547\pi$
$157$	144.705	0.921689	0.460844	+	0.887481i	$0.347547\pi$
$158$	266.602i	1.68735i
$159$	0	0
$160$	0	0
$161$	− 15.3738i	− 0.0954897i
$162$	0	0
$163$	−16.8328	−0.103269	−0.0516344	−	0.998666i	$-0.516443\pi$
$163$	−16.8328	−0.103269	−0.0516344	+	0.998666i	$0.516443\pi$
$164$	− 186.190i	− 1.13531i
$165$	0	0
$166$	116.936	0.704435
$167$	− 36.5805i	− 0.219045i	−0.993984	−	0.109522i	$-0.965068\pi$
$167$	− 36.5805i	− 0.219045i	0.993984	−	0.109522i	$-0.0349321\pi$
$168$	0	0
$169$	−136.416	−0.807198
$170$	0	0
$171$	0	0
$172$	140.918	0.819290
$173$	− 32.3313i	− 0.186886i	−0.995625	−	0.0934430i	$-0.970213\pi$
$173$	− 32.3313i	− 0.186886i	0.995625	−	0.0934430i	$-0.0297873\pi$
$174$	0	0
$175$	0	0
$176$	− 273.262i	− 1.55263i
$177$	0	0
$178$	−147.374	−0.827943
$179$	− 136.774i	− 0.764100i	−0.924142	−	0.382050i	$-0.875218\pi$
$179$	− 136.774i	− 0.764100i	0.924142	−	0.382050i	$-0.124782\pi$
$180$	0	0
$181$	45.9180	0.253690	0.126845	−	0.991923i	$-0.459515\pi$
$181$	45.9180	0.253690	0.126845	+	0.991923i	$0.459515\pi$
$182$	− 145.082i	− 0.797154i
$183$	0	0
$184$	−4.75078	−0.0258194
$185$	0	0
$186$	0	0
$187$	−84.2918	−0.450758
$188$	200.180i	1.06479i
$189$	0	0
$190$	0	0
$191$	− 68.3870i	− 0.358047i	−0.983845	−	0.179024i	$-0.942706\pi$
$191$	− 68.3870i	− 0.358047i	0.983845	−	0.179024i	$-0.0572938\pi$
$192$	0	0
$193$	−77.7082	−0.402633	−0.201317	−	0.979526i	$-0.564522\pi$
$193$	−77.7082	−0.402633	−0.201317	+	0.979526i	$0.564522\pi$
$194$	347.108i	1.78922i
$195$	0	0
$196$	−129.146	−0.658908
$197$	− 103.997i	− 0.527903i	−0.964536	−	0.263952i	$-0.914974\pi$
$197$	− 103.997i	− 0.527903i	0.964536	−	0.263952i	$-0.0850259\pi$
$198$	0	0
$199$	239.751	1.20478	0.602389	−	0.798203i	$-0.294216\pi$
$199$	239.751	1.20478	0.602389	+	0.798203i	$0.294216\pi$
$200$	0	0
$201$	0	0
$202$	62.8328	0.311054
$203$	20.0851i	0.0989416i
$204$	0	0
$205$	0	0
$206$	− 206.387i	− 1.00188i
$207$	0	0
$208$	−110.000	−0.528846
$209$	459.676i	2.19941i
$210$	0	0
$211$	−10.1672	−0.0481857	−0.0240929	−	0.999710i	$-0.507670\pi$
$211$	−10.1672	−0.0481857	−0.0240929	+	0.999710i	$0.507670\pi$
$212$	81.7395i	0.385564i
$213$	0	0
$214$	156.207	0.729938
$215$	0	0
$216$	0	0
$217$	478.122	2.20332
$218$	59.9919i	0.275192i
$219$	0	0
$220$	0	0
$221$	33.9311i	0.153534i
$222$	0	0
$223$	−72.2492	−0.323988	−0.161994	−	0.986792i	$-0.551792\pi$
$223$	−72.2492	−0.323988	−0.161994	+	0.986792i	$0.551792\pi$
$224$	373.289i	1.66647i
$225$	0	0
$226$	−336.122	−1.48726
$227$	26.4752i	0.116631i	0.998298	+	0.0583155i	$0.0185729\pi$
$227$	26.4752i	0.116631i	−0.998298	+	0.0583155i	$0.981427\pi$
$228$	0	0
$229$	−280.495	−1.22487	−0.612435	−	0.790521i	$-0.709810\pi$
$229$	−280.495	−1.22487	−0.612435	+	0.790521i	$0.709810\pi$
$230$	0	0
$231$	0	0
$232$	6.20665	0.0267528
$233$	115.082i	0.493914i	0.969026	+	0.246957i	$0.0794307\pi$
$233$	115.082i	0.493914i	−0.969026	+	0.246957i	$0.920569\pi$
$234$	0	0
$235$	0	0
$236$	271.813i	1.15175i
$237$	0	0
$238$	151.082	0.634798
$239$	147.390i	0.616695i	0.951274	+	0.308347i	$0.0997759\pi$
$239$	147.390i	0.616695i	−0.951274	+	0.308347i	$0.900224\pi$
$240$	0	0
$241$	−62.5805	−0.259670	−0.129835	−	0.991536i	$-0.541445\pi$
$241$	−62.5805	−0.259670	−0.129835	+	0.991536i	$0.541445\pi$
$242$	209.658i	0.866353i
$243$	0	0
$244$	−54.2279	−0.222246
$245$	0	0
$246$	0	0
$247$	185.039	0.749148
$248$	− 147.748i	− 0.595757i
$249$	0	0
$250$	0	0
$251$	309.580i	1.23339i	0.787203	+	0.616694i	$0.211528\pi$
$251$	309.580i	1.23339i	−0.787203	+	0.616694i	$0.788472\pi$
$252$	0	0
$253$	22.4559	0.0887584
$254$	106.679i	0.419995i
$255$	0	0
$256$	335.353	1.30997
$257$	75.7477i	0.294738i	0.989082	+	0.147369i	$0.0470805\pi$
$257$	75.7477i	0.294738i	−0.989082	+	0.147369i	$0.952920\pi$
$258$	0	0
$259$	258.079	0.996444
$260$	0	0
$261$	0	0
$262$	681.118	2.59969
$263$	367.974i	1.39914i	0.714564	+	0.699570i	$0.246625\pi$
$263$	367.974i	1.39914i	−0.714564	+	0.699570i	$0.753375\pi$
$264$	0	0
$265$	0	0
$266$	− 823.909i	− 3.09740i
$267$	0	0
$268$	307.289	1.14660
$269$	− 279.358i	− 1.03850i	−0.854621	−	0.519252i	$-0.826211\pi$
$269$	− 279.358i	− 1.03850i	0.854621	−	0.519252i	$-0.173789\pi$
$270$	0	0
$271$	62.4164	0.230319	0.115159	−	0.993347i	$-0.463262\pi$
$271$	62.4164	0.230319	0.115159	+	0.993347i	$0.463262\pi$
$272$	− 114.549i	− 0.421137i
$273$	0	0
$274$	430.225	1.57016
$275$	0	0
$276$	0	0
$277$	−26.6262	−0.0961233	−0.0480617	−	0.998844i	$-0.515304\pi$
$277$	−26.6262	−0.0961233	−0.0480617	+	0.998844i	$0.515304\pi$
$278$	− 90.3181i	− 0.324885i
$279$	0	0
$280$	0	0
$281$	412.869i	1.46929i	0.678454	+	0.734643i	$0.262650\pi$
$281$	412.869i	1.46929i	−0.678454	+	0.734643i	$0.737350\pi$
$282$	0	0
$283$	−104.292	−0.368522	−0.184261	−	0.982877i	$-0.558989\pi$
$283$	−104.292	−0.368522	−0.184261	+	0.982877i	$0.558989\pi$
$284$	214.807i	0.756361i
$285$	0	0
$286$	211.915	0.740961
$287$	633.325i	2.20671i
$288$	0	0
$289$	253.666	0.877736
$290$	0	0
$291$	0	0
$292$	−276.249	−0.946059
$293$	311.243i	1.06226i	0.847289	+	0.531131i	$0.178233\pi$
$293$	311.243i	1.06226i	−0.847289	+	0.531131i	$0.821767\pi$
$294$	0	0
$295$	0	0
$296$	− 79.7508i	− 0.269428i
$297$	0	0
$298$	−99.9574	−0.335428
$299$	− 9.03947i	− 0.0302323i
$300$	0	0
$301$	−479.331	−1.59246
$302$	− 479.092i	− 1.58640i
$303$	0	0
$304$	−624.681	−2.05487
$305$	0	0
$306$	0	0
$307$	−454.158	−1.47934	−0.739671	−	0.672969i	$-0.765019\pi$
$307$	−454.158	−1.47934	−0.739671	+	0.672969i	$0.765019\pi$
$308$	− 392.912i	− 1.27569i
$309$	0	0
$310$	0	0
$311$	− 211.957i	− 0.681535i	−0.940148	−	0.340768i	$-0.889313\pi$
$311$	− 211.957i	− 0.681535i	0.940148	−	0.340768i	$-0.110687\pi$
$312$	0	0
$313$	−284.827	−0.909989	−0.454995	−	0.890494i	$-0.650359\pi$
$313$	−284.827	−0.909989	−0.454995	+	0.890494i	$0.650359\pi$
$314$	− 378.843i	− 1.20651i
$315$	0	0
$316$	290.641	0.919751
$317$	252.056i	0.795128i	0.917574	+	0.397564i	$0.130144\pi$
$317$	252.056i	0.795128i	−0.917574	+	0.397564i	$0.869856\pi$
$318$	0	0
$319$	−29.3375	−0.0919670
$320$	0	0
$321$	0	0
$322$	−40.2492	−0.124998
$323$	192.692i	0.596569i
$324$	0	0
$325$	0	0
$326$	44.0689i	0.135181i
$327$	0	0
$328$	195.708	0.596671
$329$	− 680.912i	− 2.06964i
$330$	0	0
$331$	−177.915	−0.537507	−0.268754	−	0.963209i	$-0.586612\pi$
$331$	−177.915	−0.537507	−0.268754	+	0.963209i	$0.586612\pi$
$332$	− 127.480i	− 0.383977i
$333$	0	0
$334$	−95.7690	−0.286733
$335$	0	0
$336$	0	0
$337$	−462.450	−1.37225	−0.686127	−	0.727482i	$-0.740691\pi$
$337$	−462.450	−1.37225	−0.686127	+	0.727482i	$0.740691\pi$
$338$	357.143i	1.05664i
$339$	0	0
$340$	0	0
$341$	698.371i	2.04801i
$342$	0	0
$343$	−36.4133	−0.106161
$344$	148.122i	0.430586i
$345$	0	0
$346$	−84.6443	−0.244637
$347$	− 493.384i	− 1.42186i	−0.703265	−	0.710928i	$-0.748275\pi$
$347$	− 493.384i	− 1.42186i	0.703265	−	0.710928i	$-0.251725\pi$
$348$	0	0
$349$	62.0820	0.177885	0.0889427	−	0.996037i	$-0.471651\pi$
$349$	62.0820	0.177885	0.0889427	+	0.996037i	$0.471651\pi$
$350$	0	0
$351$	0	0
$352$	−545.246	−1.54899
$353$	− 123.836i	− 0.350810i	−0.984496	−	0.175405i	$-0.943877\pi$
$353$	− 123.836i	− 0.350810i	0.984496	−	0.175405i	$-0.0561235\pi$
$354$	0	0
$355$	0	0
$356$	160.663i	0.451299i
$357$	0	0
$358$	−358.079	−1.00022
$359$	243.994i	0.679648i	0.940489	+	0.339824i	$0.110368\pi$
$359$	243.994i	0.679648i	−0.940489	+	0.339824i	$0.889632\pi$
$360$	0	0
$361$	689.823	1.91087
$362$	− 120.215i	− 0.332085i
$363$	0	0
$364$	−158.164	−0.434517
$365$	0	0
$366$	0	0
$367$	236.371	0.644062	0.322031	−	0.946729i	$-0.395634\pi$
$367$	236.371	0.644062	0.322031	+	0.946729i	$0.395634\pi$
$368$	30.5166i	0.0829256i
$369$	0	0
$370$	0	0
$371$	− 278.036i	− 0.749424i
$372$	0	0
$373$	−301.167	−0.807419	−0.403709	−	0.914887i	$-0.632279\pi$
$373$	−301.167	−0.807419	−0.403709	+	0.914887i	$0.632279\pi$
$374$	220.679i	0.590050i
$375$	0	0
$376$	−210.413	−0.559610
$377$	11.8096i	0.0313252i
$378$	0	0
$379$	−102.003	−0.269137	−0.134569	−	0.990904i	$-0.542965\pi$
$379$	−102.003	−0.269137	−0.134569	+	0.990904i	$0.542965\pi$
$380$	0	0
$381$	0	0
$382$	−179.039	−0.468690
$383$	− 705.158i	− 1.84114i	−0.390574	−	0.920572i	$-0.627723\pi$
$383$	− 705.158i	− 1.84114i	0.390574	−	0.920572i	$-0.372277\pi$
$384$	0	0
$385$	0	0
$386$	203.443i	0.527054i
$387$	0	0
$388$	378.407	0.975276
$389$	− 346.043i	− 0.889570i	−0.895637	−	0.444785i	$-0.853280\pi$
$389$	− 346.043i	− 0.889570i	0.895637	−	0.444785i	$-0.146720\pi$
$390$	0	0
$391$	9.41330	0.0240749
$392$	− 135.748i	− 0.346295i
$393$	0	0
$394$	−272.267	−0.691034
$395$	0	0
$396$	0	0
$397$	−172.128	−0.433571	−0.216786	−	0.976219i	$-0.569557\pi$
$397$	−172.128	−0.433571	−0.216786	+	0.976219i	$0.569557\pi$
$398$	− 627.676i	− 1.57707i
$399$	0	0
$400$	0	0
$401$	− 436.407i	− 1.08830i	−0.838989	−	0.544148i	$-0.816853\pi$
$401$	− 436.407i	− 1.08830i	0.838989	−	0.544148i	$-0.183147\pi$
$402$	0	0
$403$	281.125	0.697580
$404$	− 68.4984i	− 0.169551i
$405$	0	0
$406$	52.5836	0.129516
$407$	376.964i	0.926202i
$408$	0	0
$409$	−353.663	−0.864701	−0.432350	−	0.901706i	$-0.642316\pi$
$409$	−353.663	−0.864701	−0.432350	+	0.901706i	$0.642316\pi$
$410$	0	0
$411$	0	0
$412$	−224.997	−0.546109
$413$	− 924.571i	− 2.23867i
$414$	0	0
$415$	0	0
$416$	219.485i	0.527609i
$417$	0	0
$418$	1203.45	2.87906
$419$	503.060i	1.20062i	0.799768	+	0.600310i	$0.204956\pi$
$419$	503.060i	1.20062i	−0.799768	+	0.600310i	$0.795044\pi$
$420$	0	0
$421$	−448.745	−1.06590	−0.532951	−	0.846146i	$-0.678917\pi$
$421$	−448.745	−1.06590	−0.532951	+	0.846146i	$0.678917\pi$
$422$	26.6180i	0.0630759i
$423$	0	0
$424$	−85.9180	−0.202637
$425$	0	0
$426$	0	0
$427$	184.456	0.431981
$428$	− 170.292i	− 0.397878i
$429$	0	0
$430$	0	0
$431$	− 263.204i	− 0.610681i	−0.952243	−	0.305341i	$-0.901230\pi$
$431$	− 263.204i	− 0.610681i	0.952243	−	0.305341i	$-0.0987703\pi$
$432$	0	0
$433$	−155.702	−0.359589	−0.179794	−	0.983704i	$-0.557543\pi$
$433$	−155.702	−0.359589	−0.179794	+	0.983704i	$0.557543\pi$
$434$	− 1251.74i	− 2.88419i
$435$	0	0
$436$	65.4013	0.150003
$437$	− 51.3344i	− 0.117470i
$438$	0	0
$439$	44.0031	0.100235	0.0501174	−	0.998743i	$-0.484040\pi$
$439$	44.0031	0.100235	0.0501174	+	0.998743i	$0.484040\pi$
$440$	0	0
$441$	0	0
$442$	88.8328	0.200979
$443$	842.902i	1.90271i	0.308090	+	0.951357i	$0.400310\pi$
$443$	842.902i	1.90271i	−0.308090	+	0.951357i	$0.599690\pi$
$444$	0	0
$445$	0	0
$446$	189.151i	0.424105i
$447$	0	0
$448$	228.954	0.511059
$449$	− 505.656i	− 1.12618i	−0.826395	−	0.563091i	$-0.809612\pi$
$449$	− 505.656i	− 1.12618i	0.826395	−	0.563091i	$-0.190388\pi$
$450$	0	0
$451$	−925.070	−2.05115
$452$	366.430i	0.810685i
$453$	0	0
$454$	69.3131	0.152672
$455$	0	0
$456$	0	0
$457$	−414.577	−0.907172	−0.453586	−	0.891213i	$-0.649855\pi$
$457$	−414.577	−0.907172	−0.453586	+	0.891213i	$0.649855\pi$
$458$	734.346i	1.60338i
$459$	0	0
$460$	0	0
$461$	− 103.909i	− 0.225398i	−0.993629	−	0.112699i	$-0.964050\pi$
$461$	− 103.909i	− 0.225398i	0.993629	−	0.112699i	$-0.0359496\pi$
$462$	0	0
$463$	−814.997	−1.76025	−0.880126	−	0.474740i	$-0.842542\pi$
$463$	−814.997	−1.76025	−0.880126	+	0.474740i	$0.842542\pi$
$464$	− 39.8684i	− 0.0859234i
$465$	0	0
$466$	301.289	0.646542
$467$	32.8096i	0.0702561i	0.999383	+	0.0351281i	$0.0111839\pi$
$467$	32.8096i	0.0702561i	−0.999383	+	0.0351281i	$0.988816\pi$
$468$	0	0
$469$	−1045.24	−2.22866
$470$	0	0
$471$	0	0
$472$	−285.708	−0.605314
$473$	− 700.138i	− 1.48021i
$474$	0	0
$475$	0	0
$476$	− 164.705i	− 0.346019i
$477$	0	0
$478$	385.872	0.807264
$479$	574.456i	1.19928i	0.800269	+	0.599641i	$0.204690\pi$
$479$	574.456i	1.19928i	−0.800269	+	0.599641i	$0.795310\pi$
$480$	0	0
$481$	151.745	0.315477
$482$	163.838i	0.339913i
$483$	0	0
$484$	228.562	0.472236
$485$	0	0
$486$	0	0
$487$	−189.131	−0.388359	−0.194179	−	0.980966i	$-0.562204\pi$
$487$	−189.131	−0.388359	−0.194179	+	0.980966i	$0.562204\pi$
$488$	− 57.0000i	− 0.116803i
$489$	0	0
$490$	0	0
$491$	453.437i	0.923496i	0.887011	+	0.461748i	$0.152778\pi$
$491$	453.437i	0.923496i	−0.887011	+	0.461748i	$0.847222\pi$
$492$	0	0
$493$	−12.2980	−0.0249452
$494$	− 484.440i	− 0.980647i
$495$	0	0
$496$	−949.058	−1.91342
$497$	− 730.663i	− 1.47015i
$498$	0	0
$499$	100.088	0.200578	0.100289	−	0.994958i	$-0.468023\pi$
$499$	100.088	0.200578	0.100289	+	0.994958i	$0.468023\pi$
$500$	0	0
$501$	0	0
$502$	810.492	1.61453
$503$	− 564.692i	− 1.12265i	−0.827596	−	0.561324i	$-0.810292\pi$
$503$	− 564.692i	− 1.12265i	0.827596	−	0.561324i	$-0.189708\pi$
$504$	0	0
$505$	0	0
$506$	− 58.7902i	− 0.116186i
$507$	0	0
$508$	116.298	0.228933
$509$	− 839.522i	− 1.64935i	−0.565603	−	0.824677i	$-0.691357\pi$
$509$	− 839.522i	− 1.64935i	0.565603	−	0.824677i	$-0.308643\pi$
$510$	0	0
$511$	939.659	1.83886
$512$	− 509.721i	− 0.995550i
$513$	0	0
$514$	198.310	0.385817
$515$	0	0
$516$	0	0
$517$	994.577	1.92375
$518$	− 675.659i	− 1.30436i
$519$	0	0
$520$	0	0
$521$	380.227i	0.729802i	0.931046	+	0.364901i	$0.118897\pi$
$521$	380.227i	0.729802i	−0.931046	+	0.364901i	$0.881103\pi$
$522$	0	0
$523$	194.207	0.371332	0.185666	−	0.982613i	$-0.440556\pi$
$523$	194.207	0.371332	0.185666	+	0.982613i	$0.440556\pi$
$524$	− 742.535i	− 1.41705i
$525$	0	0
$526$	963.368	1.83150
$527$	292.751i	0.555504i
$528$	0	0
$529$	526.492	0.995259
$530$	0	0
$531$	0	0
$532$	−898.200	−1.68835
$533$	372.381i	0.698651i
$534$	0	0
$535$	0	0
$536$	322.997i	0.602606i
$537$	0	0
$538$	−731.368	−1.35942
$539$	641.649i	1.19044i
$540$	0	0
$541$	159.836	0.295445	0.147723	−	0.989029i	$-0.452806\pi$
$541$	159.836	0.295445	0.147723	+	0.989029i	$0.452806\pi$
$542$	− 163.408i	− 0.301491i
$543$	0	0
$544$	−228.562	−0.420151
$545$	0	0
$546$	0	0
$547$	647.295	1.18335	0.591677	−	0.806175i	$-0.298466\pi$
$547$	647.295	1.18335	0.591677	+	0.806175i	$0.298466\pi$
$548$	− 469.018i	− 0.855873i
$549$	0	0
$550$	0	0
$551$	67.0658i	0.121716i
$552$	0	0
$553$	−988.614	−1.78773
$554$	69.7082i	0.125827i
$555$	0	0
$556$	−98.4621	−0.177090
$557$	− 901.161i	− 1.61788i	−0.587889	−	0.808942i	$-0.700041\pi$
$557$	− 901.161i	− 1.61788i	0.587889	−	0.808942i	$-0.299959\pi$
$558$	0	0
$559$	−281.836	−0.504179
$560$	0	0
$561$	0	0
$562$	1080.91	1.92332
$563$	529.003i	0.939615i	0.882769	+	0.469807i	$0.155677\pi$
$563$	529.003i	0.939615i	−0.882769	+	0.469807i	$0.844323\pi$
$564$	0	0
$565$	0	0
$566$	273.039i	0.482402i
$567$	0	0
$568$	−225.787	−0.397513
$569$	− 493.623i	− 0.867527i	−0.901027	−	0.433764i	$-0.857185\pi$
$569$	− 493.623i	− 0.867527i	0.901027	−	0.433764i	$-0.142815\pi$
$570$	0	0
$571$	−760.246	−1.33143	−0.665715	−	0.746206i	$-0.731873\pi$
$571$	−760.246	−1.33143	−0.665715	+	0.746206i	$0.731873\pi$
$572$	− 231.023i	− 0.403887i
$573$	0	0
$574$	1658.07	2.88862
$575$	0	0
$576$	0	0
$577$	38.4559	0.0666480	0.0333240	−	0.999445i	$-0.489391\pi$
$577$	38.4559	0.0666480	0.0333240	+	0.999445i	$0.489391\pi$
$578$	− 664.105i	− 1.14897i
$579$	0	0
$580$	0	0
$581$	433.623i	0.746339i
$582$	0	0
$583$	406.115	0.696596
$584$	− 290.371i	− 0.497210i
$585$	0	0
$586$	814.845	1.39052
$587$	548.489i	0.934394i	0.884153	+	0.467197i	$0.154736\pi$
$587$	548.489i	0.934394i	−0.884153	+	0.467197i	$0.845264\pi$
$588$	0	0
$589$	1596.48	2.71050
$590$	0	0
$591$	0	0
$592$	−512.279	−0.865337
$593$	9.97058i	0.0168138i	0.999965	+	0.00840690i	$0.00267603\pi$
$593$	9.97058i	0.0168138i	−0.999965	+	0.00840690i	$0.997324\pi$
$594$	0	0
$595$	0	0
$596$	108.971i	0.182837i
$597$	0	0
$598$	−23.6656	−0.0395746
$599$	− 507.755i	− 0.847670i	−0.905739	−	0.423835i	$-0.860684\pi$
$599$	− 507.755i	− 0.847670i	0.905739	−	0.423835i	$-0.139316\pi$
$600$	0	0
$601$	−680.416	−1.13214	−0.566070	−	0.824357i	$-0.691537\pi$
$601$	−680.416	−1.13214	−0.566070	+	0.824357i	$0.691537\pi$
$602$	1254.91i	2.08456i
$603$	0	0
$604$	−522.292	−0.864722
$605$	0	0
$606$	0	0
$607$	305.374	0.503087	0.251544	−	0.967846i	$-0.419062\pi$
$607$	305.374	0.503087	0.251544	+	0.967846i	$0.419062\pi$
$608$	1246.44i	2.05006i
$609$	0	0
$610$	0	0
$611$	− 400.361i	− 0.655255i
$612$	0	0
$613$	876.584	1.42999	0.714995	−	0.699130i	$-0.246429\pi$
$613$	876.584	1.42999	0.714995	+	0.699130i	$0.246429\pi$
$614$	1189.00i	1.93648i
$615$	0	0
$616$	412.997	0.670450
$617$	740.489i	1.20014i	0.799946	+	0.600072i	$0.204862\pi$
$617$	740.489i	1.20014i	−0.799946	+	0.600072i	$0.795138\pi$
$618$	0	0
$619$	−309.902	−0.500650	−0.250325	−	0.968162i	$-0.580537\pi$
$619$	−309.902	−0.500650	−0.250325	+	0.968162i	$0.580537\pi$
$620$	0	0
$621$	0	0
$622$	−554.912	−0.892141
$623$	− 546.492i	− 0.877195i
$624$	0	0
$625$	0	0
$626$	745.686i	1.19119i
$627$	0	0
$628$	−413.003	−0.657648
$629$	158.020i	0.251224i
$630$	0	0
$631$	−324.830	−0.514786	−0.257393	−	0.966307i	$-0.582863\pi$
$631$	−324.830	−0.514786	−0.257393	+	0.966307i	$0.582863\pi$
$632$	305.498i	0.483384i
$633$	0	0
$634$	659.890	1.04084
$635$	0	0
$636$	0	0
$637$	258.292	0.405482
$638$	76.8065i	0.120386i
$639$	0	0
$640$	0	0
$641$	− 951.246i	− 1.48400i	−0.670398	−	0.742002i	$-0.733877\pi$
$641$	− 951.246i	− 1.48400i	0.670398	−	0.742002i	$-0.266123\pi$
$642$	0	0
$643$	−835.289	−1.29905	−0.649525	−	0.760341i	$-0.725032\pi$
$643$	−835.289	−1.29905	−0.649525	+	0.760341i	$0.725032\pi$
$644$	43.8785i	0.0681343i
$645$	0	0
$646$	504.474	0.780920
$647$	− 247.393i	− 0.382370i	−0.981554	−	0.191185i	$-0.938767\pi$
$647$	− 247.393i	− 0.382370i	0.981554	−	0.191185i	$-0.0612330\pi$
$648$	0	0
$649$	1350.48	2.08086
$650$	0	0
$651$	0	0
$652$	48.0426	0.0736849
$653$	228.293i	0.349606i	0.984603	+	0.174803i	$0.0559288\pi$
$653$	228.293i	0.349606i	−0.984603	+	0.174803i	$0.944071\pi$
$654$	0	0
$655$	0	0
$656$	− 1257.13i	− 1.91636i
$657$	0	0
$658$	−1782.65	−2.70919
$659$	907.512i	1.37710i	0.725187	+	0.688552i	$0.241753\pi$
$659$	907.512i	1.37710i	−0.725187	+	0.688552i	$0.758247\pi$
$660$	0	0
$661$	−468.152	−0.708248	−0.354124	−	0.935199i	$-0.615221\pi$
$661$	−468.152	−0.708248	−0.354124	+	0.935199i	$0.615221\pi$
$662$	465.787i	0.703606i
$663$	0	0
$664$	133.997	0.201803
$665$	0	0
$666$	0	0
$667$	3.27627	0.00491195
$668$	104.404i	0.156294i
$669$	0	0
$670$	0	0
$671$	269.426i	0.401530i
$672$	0	0
$673$	195.629	0.290682	0.145341	−	0.989382i	$-0.453572\pi$
$673$	195.629	0.290682	0.145341	+	0.989382i	$0.453572\pi$
$674$	1210.71i	1.79630i
$675$	0	0
$676$	389.346	0.575956
$677$	− 114.158i	− 0.168623i	−0.996439	−	0.0843116i	$-0.973131\pi$
$677$	− 114.158i	− 0.168623i	0.996439	−	0.0843116i	$-0.0268691\pi$
$678$	0	0
$679$	−1287.15	−1.89565
$680$	0	0
$681$	0	0
$682$	1828.36	2.68088
$683$	1240.50i	1.81625i	0.418705	+	0.908123i	$0.362484\pi$
$683$	1240.50i	1.81625i	−0.418705	+	0.908123i	$0.637516\pi$
$684$	0	0
$685$	0	0
$686$	95.3313i	0.138967i
$687$	0	0
$688$	951.459	1.38293
$689$	− 163.479i	− 0.237270i
$690$	0	0
$691$	487.669	0.705743	0.352872	−	0.935672i	$-0.385205\pi$
$691$	487.669	0.705743	0.352872	+	0.935672i	$0.385205\pi$
$692$	92.2767i	0.133348i
$693$	0	0
$694$	−1291.70	−1.86123
$695$	0	0
$696$	0	0
$697$	−387.781	−0.556357
$698$	− 162.533i	− 0.232855i
$699$	0	0
$700$	0	0
$701$	− 925.384i	− 1.32009i	−0.751226	−	0.660046i	$-0.770537\pi$
$701$	− 925.384i	− 1.32009i	0.751226	−	0.660046i	$-0.229463\pi$
$702$	0	0
$703$	861.745	1.22581
$704$	334.423i	0.475033i
$705$	0	0
$706$	−324.207	−0.459216
$707$	232.997i	0.329557i
$708$	0	0
$709$	−616.322	−0.869283	−0.434642	−	0.900603i	$-0.643125\pi$
$709$	−616.322	−0.869283	−0.434642	+	0.900603i	$0.643125\pi$
$710$	0	0
$711$	0	0
$712$	−168.875	−0.237185
$713$	− 77.9907i	− 0.109384i
$714$	0	0
$715$	0	0
$716$	390.367i	0.545205i
$717$	0	0
$718$	638.784	0.889671
$719$	− 404.067i	− 0.561984i	−0.959710	−	0.280992i	$-0.909337\pi$
$719$	− 404.067i	− 0.561984i	0.959710	−	0.280992i	$-0.0906634\pi$
$720$	0	0
$721$	765.325	1.06148
$722$	− 1805.98i	− 2.50136i
$723$	0	0
$724$	−131.055	−0.181015
$725$	0	0
$726$	0	0
$727$	834.991	1.14854	0.574271	−	0.818665i	$-0.305285\pi$
$727$	834.991	1.14854	0.574271	+	0.818665i	$0.305285\pi$
$728$	− 166.249i	− 0.228364i
$729$	0	0
$730$	0	0
$731$	− 293.492i	− 0.401493i
$732$	0	0
$733$	678.480	0.925620	0.462810	−	0.886457i	$-0.346841\pi$
$733$	678.480	0.925620	0.462810	+	0.886457i	$0.346841\pi$
$734$	− 618.827i	− 0.843088i
$735$	0	0
$736$	60.8905	0.0827316
$737$	− 1526.74i	− 2.07155i
$738$	0	0
$739$	−284.580	−0.385089	−0.192544	−	0.981288i	$-0.561674\pi$
$739$	−284.580	−0.385089	−0.192544	+	0.981288i	$0.561674\pi$
$740$	0	0
$741$	0	0
$742$	−727.909	−0.981009
$743$	489.947i	0.659418i	0.944083	+	0.329709i	$0.106951\pi$
$743$	489.947i	0.659418i	−0.944083	+	0.329709i	$0.893049\pi$
$744$	0	0
$745$	0	0
$746$	788.466i	1.05692i
$747$	0	0
$748$	240.577	0.321628
$749$	579.246i	0.773359i
$750$	0	0
$751$	−1180.49	−1.57189	−0.785945	−	0.618297i	$-0.787823\pi$
$751$	−1180.49	−1.57189	−0.785945	+	0.618297i	$0.787823\pi$
$752$	1351.59i	1.79733i
$753$	0	0
$754$	30.9180	0.0410053
$755$	0	0
$756$	0	0
$757$	−358.255	−0.473257	−0.236628	−	0.971600i	$-0.576042\pi$
$757$	−358.255	−0.473257	−0.236628	+	0.971600i	$0.576042\pi$
$758$	267.048i	0.352306i
$759$	0	0
$760$	0	0
$761$	− 428.115i	− 0.562569i	−0.959624	−	0.281285i	$-0.909239\pi$
$761$	− 428.115i	− 0.562569i	0.959624	−	0.281285i	$-0.0907605\pi$
$762$	0	0
$763$	−222.462	−0.291562
$764$	195.183i	0.255476i
$765$	0	0
$766$	−1846.13	−2.41009
$767$	− 543.627i	− 0.708770i
$768$	0	0
$769$	421.656	0.548318	0.274159	−	0.961684i	$-0.411601\pi$
$769$	421.656	0.548318	0.274159	+	0.961684i	$0.411601\pi$
$770$	0	0
$771$	0	0
$772$	221.787	0.287289
$773$	− 792.246i	− 1.02490i	−0.858718	−	0.512449i	$-0.828738\pi$
$773$	− 792.246i	− 1.02490i	0.858718	−	0.512449i	$-0.171262\pi$
$774$	0	0
$775$	0	0
$776$	397.751i	0.512565i
$777$	0	0
$778$	−905.951	−1.16446
$779$	2114.72i	2.71466i
$780$	0	0
$781$	1067.25	1.36651
$782$	− 24.6443i	− 0.0315145i
$783$	0	0
$784$	−871.976	−1.11221
$785$	0	0
$786$	0	0
$787$	−218.456	−0.277581	−0.138790	−	0.990322i	$-0.544321\pi$
$787$	−218.456	−0.277581	−0.138790	+	0.990322i	$0.544321\pi$
$788$	296.818i	0.376672i
$789$	0	0
$790$	0	0
$791$	− 1246.41i	− 1.57574i
$792$	0	0
$793$	108.456	0.136767
$794$	450.636i	0.567552i
$795$	0	0
$796$	−684.273	−0.859640
$797$	− 78.5341i	− 0.0985371i	−0.998786	−	0.0492686i	$-0.984311\pi$
$797$	− 78.5341i	− 0.0985371i	0.998786	−	0.0492686i	$-0.0156890\pi$
$798$	0	0
$799$	416.918	0.521800
$800$	0	0
$801$	0	0
$802$	−1142.53	−1.42460
$803$	1372.52i	1.70924i
$804$	0	0
$805$	0	0
$806$	− 735.994i	− 0.913144i
$807$	0	0
$808$	72.0000	0.0891089
$809$	922.233i	1.13997i	0.821656	+	0.569983i	$0.193050\pi$
$809$	922.233i	1.13997i	−0.821656	+	0.569983i	$0.806950\pi$
$810$	0	0
$811$	1199.82	1.47944	0.739719	−	0.672916i	$-0.234959\pi$
$811$	1199.82	1.47944	0.739719	+	0.672916i	$0.234959\pi$
$812$	− 57.3251i	− 0.0705974i
$813$	0	0
$814$	986.906	1.21241
$815$	0	0
$816$	0	0
$817$	−1600.52	−1.95902
$818$	925.901i	1.13191i
$819$	0	0
$820$	0	0
$821$	460.663i	0.561099i	0.959840	+	0.280550i	$0.0905167\pi$
$821$	460.663i	0.561099i	−0.959840	+	0.280550i	$0.909483\pi$
$822$	0	0
$823$	1375.20	1.67096	0.835478	−	0.549523i	$-0.185191\pi$
$823$	1375.20	1.67096	0.835478	+	0.549523i	$0.185191\pi$
$824$	− 236.498i	− 0.287013i
$825$	0	0
$826$	−2420.56	−2.93046
$827$	− 497.800i	− 0.601935i	−0.953634	−	0.300968i	$-0.902690\pi$
$827$	− 497.800i	− 0.601935i	0.953634	−	0.300968i	$-0.0973096\pi$
$828$	0	0
$829$	−1339.24	−1.61549	−0.807744	−	0.589533i	$-0.799312\pi$
$829$	−1339.24	−1.61549	−0.807744	+	0.589533i	$0.799312\pi$
$830$	0	0
$831$	0	0
$832$	134.620	0.161803
$833$	268.974i	0.322898i
$834$	0	0
$835$	0	0
$836$	− 1311.96i	− 1.56933i
$837$	0	0
$838$	1317.03	1.57163
$839$	297.613i	0.354723i	0.984146	+	0.177362i	$0.0567563\pi$
$839$	297.613i	0.354723i	−0.984146	+	0.177362i	$0.943244\pi$
$840$	0	0
$841$	836.720	0.994910
$842$	1174.83i	1.39528i
$843$	0	0
$844$	29.0182	0.0343817
$845$	0	0
$846$	0	0
$847$	−777.453	−0.917890
$848$	551.894i	0.650819i
$849$	0	0
$850$	0	0
$851$	− 42.0976i	− 0.0494684i
$852$	0	0
$853$	−887.647	−1.04062	−0.520309	−	0.853978i	$-0.674183\pi$
$853$	−887.647	−1.04062	−0.520309	+	0.853978i	$0.674183\pi$
$854$	− 482.912i	− 0.565470i
$855$	0	0
$856$	178.997	0.209109
$857$	− 623.977i	− 0.728094i	−0.931381	−	0.364047i	$-0.881395\pi$
$857$	− 623.977i	− 0.728094i	0.931381	−	0.364047i	$-0.118605\pi$
$858$	0	0
$859$	−1015.24	−1.18189	−0.590945	−	0.806712i	$-0.701245\pi$
$859$	−1015.24	−1.18189	−0.590945	+	0.806712i	$0.701245\pi$
$860$	0	0
$861$	0	0
$862$	−689.076	−0.799392
$863$	1480.72i	1.71578i	0.513833	+	0.857890i	$0.328225\pi$
$863$	1480.72i	1.71578i	−0.513833	+	0.857890i	$0.671775\pi$
$864$	0	0
$865$	0	0
$866$	407.633i	0.470708i
$867$	0	0
$868$	−1364.61	−1.57213
$869$	− 1444.02i	− 1.66171i
$870$	0	0
$871$	−614.577	−0.705600
$872$	68.7446i	0.0788355i
$873$	0	0
$874$	−134.395	−0.153770
$875$	0	0
$876$	0	0
$877$	253.957	0.289575	0.144788	−	0.989463i	$-0.453750\pi$
$877$	253.957	0.289575	0.144788	+	0.989463i	$0.453750\pi$
$878$	− 115.202i	− 0.131209i
$879$	0	0
$880$	0	0
$881$	490.502i	0.556756i	0.960472	+	0.278378i	$0.0897968\pi$
$881$	490.502i	0.556756i	−0.960472	+	0.278378i	$0.910203\pi$
$882$	0	0
$883$	881.878	0.998730	0.499365	−	0.866392i	$-0.333567\pi$
$883$	881.878	0.998730	0.499365	+	0.866392i	$0.333567\pi$
$884$	− 96.8429i	− 0.109551i
$885$	0	0
$886$	2206.75	2.49069
$887$	− 267.827i	− 0.301947i	−0.988538	−	0.150973i	$-0.951759\pi$
$887$	− 267.827i	− 0.301947i	0.988538	−	0.150973i	$-0.0482407\pi$
$888$	0	0
$889$	−395.587	−0.444979
$890$	0	0
$891$	0	0
$892$	206.207	0.231173
$893$	− 2273.61i	− 2.54604i
$894$	0	0
$895$	0	0
$896$	893.745i	0.997483i
$897$	0	0
$898$	−1323.82	−1.47419
$899$	101.891i	0.113338i
$900$	0	0
$901$	170.240	0.188946
$902$	2421.86i	2.68499i
$903$	0	0
$904$	−385.161	−0.426063
$905$	0	0
$906$	0	0
$907$	−493.240	−0.543815	−0.271907	−	0.962323i	$-0.587654\pi$
$907$	−493.240	−0.543815	−0.271907	+	0.962323i	$0.587654\pi$
$908$	− 75.5630i	− 0.0832192i
$909$	0	0
$910$	0	0
$911$	− 1798.73i	− 1.97445i	−0.159328	−	0.987226i	$-0.550933\pi$
$911$	− 1798.73i	− 1.97445i	0.159328	−	0.987226i	$-0.449067\pi$
$912$	0	0
$913$	−633.374	−0.693728
$914$	1085.38i	1.18750i
$915$	0	0
$916$	800.562	0.873976
$917$	2525.73i	2.75434i
$918$	0	0
$919$	−338.000	−0.367791	−0.183896	−	0.982946i	$-0.558871\pi$
$919$	−338.000	−0.367791	−0.183896	+	0.982946i	$0.558871\pi$
$920$	0	0
$921$	0	0
$922$	−272.036	−0.295050
$923$	− 429.613i	− 0.465453i
$924$	0	0
$925$	0	0
$926$	2133.69i	2.30420i
$927$	0	0
$928$	−79.5503	−0.0857223
$929$	− 1205.17i	− 1.29728i	−0.761096	−	0.648639i	$-0.775339\pi$
$929$	− 1205.17i	− 1.29728i	0.761096	−	0.648639i	$-0.224661\pi$
$930$	0	0
$931$	1466.82	1.57553
$932$	− 328.456i	− 0.352420i
$933$	0	0
$934$	85.8967	0.0919665
$935$	0	0
$936$	0	0
$937$	−1187.65	−1.26750	−0.633750	−	0.773538i	$-0.718485\pi$
$937$	−1187.65	−1.26750	−0.633750	+	0.773538i	$0.718485\pi$
$938$	2736.47i	2.91735i
$939$	0	0
$940$	0	0
$941$	1037.42i	1.10247i	0.834350	+	0.551235i	$0.185843\pi$
$941$	1037.42i	1.10247i	−0.834350	+	0.551235i	$0.814157\pi$
$942$	0	0
$943$	103.307	0.109552
$944$	1835.25i	1.94412i
$945$	0	0
$946$	−1832.98	−1.93762
$947$	− 304.180i	− 0.321203i	−0.987019	−	0.160602i	$-0.948657\pi$
$947$	− 304.180i	− 0.321203i	0.987019	−	0.160602i	$-0.0513435\pi$
$948$	0	0
$949$	552.498	0.582190
$950$	0	0
$951$	0	0
$952$	173.125	0.181854
$953$	− 1308.97i	− 1.37353i	−0.726881	−	0.686763i	$-0.759031\pi$
$953$	− 1308.97i	− 1.37353i	0.726881	−	0.686763i	$-0.240969\pi$
$954$	0	0
$955$	0	0
$956$	− 420.666i	− 0.440028i
$957$	0	0
$958$	1503.95	1.56988
$959$	1595.36i	1.66357i
$960$	0	0
$961$	1464.49	1.52392
$962$	− 397.272i	− 0.412965i
$963$	0	0
$964$	178.611	0.185281
$965$	0	0
$966$	0	0
$967$	−769.489	−0.795749	−0.397874	−	0.917440i	$-0.630252\pi$
$967$	−769.489	−0.795749	−0.397874	+	0.917440i	$0.630252\pi$
$968$	240.246i	0.248188i
$969$	0	0
$970$	0	0
$971$	− 73.4164i	− 0.0756091i	−0.999285	−	0.0378045i	$-0.987964\pi$
$971$	− 73.4164i	− 0.0756091i	0.999285	−	0.0378045i	$-0.0120364\pi$
$972$	0	0
$973$	334.918	0.344212
$974$	495.151i	0.508369i
$975$	0	0
$976$	−366.140	−0.375143
$977$	− 865.384i	− 0.885756i	−0.896582	−	0.442878i	$-0.853957\pi$
$977$	− 865.384i	− 0.885756i	0.896582	−	0.442878i	$-0.146043\pi$
$978$	0	0
$979$	798.237	0.815359
$980$	0	0
$981$	0	0
$982$	1187.11	1.20887
$983$	− 1728.98i	− 1.75888i	−0.476013	−	0.879438i	$-0.657919\pi$
$983$	− 1728.98i	− 1.75888i	0.476013	−	0.879438i	$-0.342081\pi$
$984$	0	0
$985$	0	0
$986$	32.1966i	0.0326538i
$987$	0	0
$988$	−528.122	−0.534536
$989$	78.1880i	0.0790577i
$990$	0	0
$991$	445.675	0.449722	0.224861	−	0.974391i	$-0.427807\pi$
$991$	445.675	0.449722	0.224861	+	0.974391i	$0.427807\pi$
$992$	1893.67i	1.90895i
$993$	0	0
$994$	−1912.90	−1.92445
$995$	0	0
$996$	0	0
$997$	−78.2253	−0.0784607	−0.0392303	−	0.999230i	$-0.512491\pi$
$997$	−78.2253	−0.0784607	−0.0392303	+	0.999230i	$0.512491\pi$
$998$	− 262.034i	− 0.262560i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.3.c.p.26.1		4
3.2	odd	2	inner	675.3.c.p.26.4		4
5.2	odd	4		675.3.d.i.674.4		4
5.3	odd	4		675.3.d.e.674.1		4
5.4	even	2		135.3.c.c.26.4	yes	4
15.2	even	4		675.3.d.e.674.2		4
15.8	even	4		675.3.d.i.674.3		4
15.14	odd	2		135.3.c.c.26.1	✓	4
20.19	odd	2		2160.3.l.g.161.4		4
45.4	even	6		405.3.i.c.26.1		8
45.14	odd	6		405.3.i.c.26.4		8
45.29	odd	6		405.3.i.c.296.1		8
45.34	even	6		405.3.i.c.296.4		8
60.59	even	2		2160.3.l.g.161.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.3.c.c.26.1	✓	4	15.14	odd	2
135.3.c.c.26.4	yes	4	5.4	even	2
405.3.i.c.26.1		8	45.4	even	6
405.3.i.c.26.4		8	45.14	odd	6
405.3.i.c.296.1		8	45.29	odd	6
405.3.i.c.296.4		8	45.34	even	6
675.3.c.p.26.1		4	1.1	even	1	trivial
675.3.c.p.26.4		4	3.2	odd	2	inner
675.3.d.e.674.1		4	5.3	odd	4
675.3.d.e.674.2		4	15.2	even	4
675.3.d.i.674.3		4	15.8	even	4
675.3.d.i.674.4		4	5.2	odd	4
2160.3.l.g.161.2		4	60.59	even	2
2160.3.l.g.161.4		4	20.19	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	675.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.61803i q^{2} -2.85410 q^{4} +9.70820 q^{7} -3.00000i q^{8} +14.1803i q^{11} +5.70820 q^{13} -25.4164i q^{14} -19.2705 q^{16} +5.94427i q^{17} +32.4164 q^{19} +37.1246 q^{22} -1.58359i q^{23} -14.9443i q^{26} -27.7082 q^{28} +2.06888i q^{29} +49.2492 q^{31} +38.4508i q^{32} +15.5623 q^{34} +26.5836 q^{37} -84.8673i q^{38} +65.2361i q^{41} -49.3738 q^{43} -40.4721i q^{44} -4.14590 q^{46} -70.1378i q^{47} +45.2492 q^{49} -16.2918 q^{52} -28.6393i q^{53} -29.1246i q^{56} +5.41641 q^{58} -95.2361i q^{59} +19.0000 q^{61} -128.936i q^{62} +23.5836 q^{64} -107.666 q^{67} -16.9656i q^{68} -75.2624i q^{71} +96.7902 q^{73} -69.5967i q^{74} -92.5197 q^{76} +137.666i q^{77} -101.833 q^{79} +170.790 q^{82} +44.6656i q^{83} +129.262i q^{86} +42.5410 q^{88} -56.2918i q^{89} +55.4164 q^{91} +4.51973i q^{92} -183.623 q^{94} -132.584 q^{97} -118.464i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 12 q^{7} - 4 q^{13} - 10 q^{16} + 76 q^{19} + 68 q^{22} - 84 q^{28} + 36 q^{31} + 22 q^{34} + 160 q^{37} + 44 q^{43} - 30 q^{46} + 20 q^{49} - 92 q^{52} - 32 q^{58} + 76 q^{61} + 148 q^{64}+ \cdots - 584 q^{97}+O(q^{100}) \) 4 * q + 2 * q^4 + 12 * q^7 - 4 * q^13 - 10 * q^16 + 76 * q^19 + 68 * q^22 - 84 * q^28 + 36 * q^31 + 22 * q^34 + 160 * q^37 + 44 * q^43 - 30 * q^46 + 20 * q^49 - 92 * q^52 - 32 * q^58 + 76 * q^61 + 148 * q^64 - 216 * q^67 + 92 * q^73 - 142 * q^76 - 300 * q^79 + 388 * q^82 + 36 * q^88 + 168 * q^91 - 332 * q^94 - 584 * q^97

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