LMFDB - Embedded newform 432.5.g.h.271.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,5,Mod(271,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("432.271");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	432.g (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:	$44.6558240522$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.280120707.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 10x^{4} + 5x^{3} + 83x^{2} - 18x + 4 $ x^6 - x^5 + 10x^4 + 5x^3 + 83x^2 - 18x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{43}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.1
Root			$1.72219 + 2.98292i$ of defining polynomial
Character	$\chi$	$=$	432.271
Dual form			432.5.g.h.271.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-26.1823 q^{5} -22.7768i q^{7} +O(q^{10})$	$q-26.1823 q^{5} -22.7768i q^{7} +108.483i q^{11} -283.091 q^{13} +373.628 q^{17} +535.882i q^{19} -611.572i q^{23} +60.5132 q^{25} -211.061 q^{29} -564.547i q^{31} +596.349i q^{35} -330.118 q^{37} +779.429 q^{41} -223.961i q^{43} -2562.48i q^{47} +1882.22 q^{49} +4868.26 q^{53} -2840.34i q^{55} -4507.24i q^{59} +3966.12 q^{61} +7411.98 q^{65} +2586.70i q^{67} +3793.73i q^{71} -1477.03 q^{73} +2470.90 q^{77} +2801.96i q^{79} -3756.18i q^{83} -9782.44 q^{85} +7193.58 q^{89} +6447.91i q^{91} -14030.6i q^{95} -9980.33 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 42 q^{5}+O(q^{10}) $ 6 * q + 42 * q^5	$ 6 q + 42 q^{5} + 168 q^{13} + 624 q^{17} + 1632 q^{25} + 1596 q^{29} - 2652 q^{37} + 9828 q^{41} - 4908 q^{49} + 13158 q^{53} + 3936 q^{61} + 38472 q^{65} + 3606 q^{73} + 47826 q^{77} - 5760 q^{85} + 52620 q^{89} + 13290 q^{97}+O(q^{100}) $ 6 * q + 42 * q^5 + 168 * q^13 + 624 * q^17 + 1632 * q^25 + 1596 * q^29 - 2652 * q^37 + 9828 * q^41 - 4908 * q^49 + 13158 * q^53 + 3936 * q^61 + 38472 * q^65 + 3606 * q^73 + 47826 * q^77 - 5760 * q^85 + 52620 * q^89 + 13290 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−26.1823	−1.04729	−0.523646	−	0.851936i	$-0.675429\pi$
$5$	−26.1823	−1.04729	−0.523646	+	0.851936i	$0.675429\pi$
$6$	0	0
$7$	− 22.7768i	− 0.464833i	−0.972616	−	0.232416i	$-0.925337\pi$
$7$	− 22.7768i	− 0.464833i	0.972616	−	0.232416i	$-0.0746632\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	108.483i	0.896556i	0.893894	+	0.448278i	$0.147963\pi$
$11$	108.483i	0.896556i	−0.893894	+	0.448278i	$0.852037\pi$
$12$	0	0
$13$	−283.091	−1.67510	−0.837548	−	0.546363i	$-0.816012\pi$
$13$	−283.091	−1.67510	−0.837548	+	0.546363i	$0.816012\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	373.628	1.29283	0.646415	−	0.762986i	$-0.276268\pi$
$17$	373.628	1.29283	0.646415	+	0.762986i	$0.276268\pi$
$18$	0	0
$19$	535.882i	1.48444i	0.670157	+	0.742219i	$0.266227\pi$
$19$	535.882i	1.48444i	−0.670157	+	0.742219i	$0.733773\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 611.572i	− 1.15609i	−0.816005	−	0.578045i	$-0.803816\pi$
$23$	− 611.572i	− 1.15609i	0.816005	−	0.578045i	$-0.196184\pi$
$24$	0	0
$25$	60.5132	0.0968212
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−211.061	−0.250964	−0.125482	−	0.992096i	$-0.540048\pi$
$29$	−211.061	−0.250964	−0.125482	+	0.992096i	$0.540048\pi$
$30$	0	0
$31$	− 564.547i	− 0.587457i	−0.955889	−	0.293729i	$-0.905104\pi$
$31$	− 564.547i	− 0.587457i	0.955889	−	0.293729i	$-0.0948962\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	596.349i	0.486816i
$36$	0	0
$37$	−330.118	−0.241138	−0.120569	−	0.992705i	$-0.538472\pi$
$37$	−330.118	−0.241138	−0.120569	+	0.992705i	$0.538472\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	779.429	0.463670	0.231835	−	0.972755i	$-0.425527\pi$
$41$	779.429	0.463670	0.231835	+	0.972755i	$0.425527\pi$
$42$	0	0
$43$	− 223.961i	− 0.121126i	−0.998164	−	0.0605629i	$-0.980710\pi$
$43$	− 223.961i	− 0.121126i	0.998164	−	0.0605629i	$-0.0192896\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2562.48i	− 1.16002i	−0.814610	−	0.580009i	$-0.803049\pi$
$47$	− 2562.48i	− 1.16002i	0.814610	−	0.580009i	$-0.196951\pi$
$48$	0	0
$49$	1882.22	0.783931
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4868.26	1.73309	0.866547	−	0.499096i	$-0.166335\pi$
$53$	4868.26	1.73309	0.866547	+	0.499096i	$0.166335\pi$
$54$	0	0
$55$	− 2840.34i	− 0.938957i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 4507.24i	− 1.29481i	−0.762145	−	0.647406i	$-0.775854\pi$
$59$	− 4507.24i	− 1.29481i	0.762145	−	0.647406i	$-0.224146\pi$
$60$	0	0
$61$	3966.12	1.06588	0.532938	−	0.846154i	$-0.321088\pi$
$61$	3966.12	1.06588	0.532938	+	0.846154i	$0.321088\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7411.98	1.75432
$66$	0	0
$67$	2586.70i	0.576232i	0.957596	+	0.288116i	$0.0930288\pi$
$67$	2586.70i	0.576232i	−0.957596	+	0.288116i	$0.906971\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3793.73i	0.752576i	0.926503	+	0.376288i	$0.122800\pi$
$71$	3793.73i	0.752576i	−0.926503	+	0.376288i	$0.877200\pi$
$72$	0	0
$73$	−1477.03	−0.277169	−0.138585	−	0.990351i	$-0.544255\pi$
$73$	−1477.03	−0.277169	−0.138585	+	0.990351i	$0.544255\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2470.90	0.416749
$78$	0	0
$79$	2801.96i	0.448961i	0.974479	+	0.224480i	$0.0720685\pi$
$79$	2801.96i	0.448961i	−0.974479	+	0.224480i	$0.927932\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 3756.18i	− 0.545243i	−0.962121	−	0.272622i	$-0.912109\pi$
$83$	− 3756.18i	− 0.545243i	0.962121	−	0.272622i	$-0.0878908\pi$
$84$	0	0
$85$	−9782.44	−1.35397
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7193.58	0.908166	0.454083	−	0.890959i	$-0.349967\pi$
$89$	7193.58	0.908166	0.454083	+	0.890959i	$0.349967\pi$
$90$	0	0
$91$	6447.91i	0.778640i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 14030.6i	− 1.55464i
$96$	0	0
$97$	−9980.33	−1.06072	−0.530361	−	0.847772i	$-0.677944\pi$
$97$	−9980.33	−1.06072	−0.530361	+	0.847772i	$0.677944\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11656.0	1.14264	0.571318	−	0.820729i	$-0.306432\pi$
$101$	11656.0	1.14264	0.571318	+	0.820729i	$0.306432\pi$
$102$	0	0
$103$	− 19438.7i	− 1.83229i	−0.400851	−	0.916143i	$-0.631285\pi$
$103$	− 19438.7i	− 1.83229i	0.400851	−	0.916143i	$-0.368715\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 15787.9i	− 1.37898i	−0.724297	−	0.689488i	$-0.757835\pi$
$107$	− 15787.9i	− 1.37898i	0.724297	−	0.689488i	$-0.242165\pi$
$108$	0	0
$109$	9645.68	0.811857	0.405929	−	0.913905i	$-0.366948\pi$
$109$	9645.68	0.811857	0.405929	+	0.913905i	$0.366948\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	23324.3	1.82664	0.913318	−	0.407246i	$-0.133511\pi$
$113$	23324.3	1.82664	0.913318	+	0.407246i	$0.133511\pi$
$114$	0	0
$115$	16012.4i	1.21077i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 8510.05i	− 0.600950i
$120$	0	0
$121$	2872.37	0.196187
$122$	0	0
$123$	0	0
$124$	0	0
$125$	14779.6	0.945892
$126$	0	0
$127$	11065.9i	0.686089i	0.939319	+	0.343044i	$0.111458\pi$
$127$	11065.9i	0.686089i	−0.939319	+	0.343044i	$0.888542\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	30441.5i	1.77388i	0.461887	+	0.886939i	$0.347172\pi$
$131$	30441.5i	1.77388i	−0.461887	+	0.886939i	$0.652828\pi$
$132$	0	0
$133$	12205.7	0.690015
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15683.4	−0.835601	−0.417800	−	0.908539i	$-0.637199\pi$
$137$	−15683.4	−0.835601	−0.417800	+	0.908539i	$0.637199\pi$
$138$	0	0
$139$	− 18140.3i	− 0.938888i	−0.882962	−	0.469444i	$-0.844454\pi$
$139$	− 18140.3i	− 0.938888i	0.882962	−	0.469444i	$-0.155546\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 30710.7i	− 1.50182i
$144$	0	0
$145$	5526.05	0.262832
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2611.65	0.117636	0.0588182	−	0.998269i	$-0.481267\pi$
$149$	2611.65	0.117636	0.0588182	+	0.998269i	$0.481267\pi$
$150$	0	0
$151$	5467.16i	0.239777i	0.992787	+	0.119889i	$0.0382538\pi$
$151$	5467.16i	0.239777i	−0.992787	+	0.119889i	$0.961746\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	14781.1i	0.615240i
$156$	0	0
$157$	−11636.6	−0.472091	−0.236045	−	0.971742i	$-0.575851\pi$
$157$	−11636.6	−0.472091	−0.236045	+	0.971742i	$0.575851\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−13929.7	−0.537389
$162$	0	0
$163$	− 46215.6i	− 1.73945i	−0.493532	−	0.869727i	$-0.664295\pi$
$163$	− 46215.6i	− 1.73945i	0.493532	−	0.869727i	$-0.335705\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8416.27i	− 0.301777i	−0.988551	−	0.150889i	$-0.951786\pi$
$167$	− 8416.27i	− 0.301777i	0.988551	−	0.150889i	$-0.0482135\pi$
$168$	0	0
$169$	51579.7	1.80595
$170$	0	0
$171$	0	0
$172$	0	0
$173$	44502.7	1.48694	0.743471	−	0.668768i	$-0.233178\pi$
$173$	44502.7	1.48694	0.743471	+	0.668768i	$0.233178\pi$
$174$	0	0
$175$	− 1378.30i	− 0.0450056i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18823.1i	0.587468i	0.955887	+	0.293734i	$0.0948980\pi$
$179$	18823.1i	0.587468i	−0.955887	+	0.293734i	$0.905102\pi$
$180$	0	0
$181$	−22837.0	−0.697079	−0.348539	−	0.937294i	$-0.613322\pi$
$181$	−22837.0	−0.697079	−0.348539	+	0.937294i	$0.613322\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8643.25	0.252542
$186$	0	0
$187$	40532.4i	1.15910i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 2313.01i	− 0.0634033i	−0.999497	−	0.0317016i	$-0.989907\pi$
$191$	− 2313.01i	− 0.0634033i	0.999497	−	0.0317016i	$-0.0100926\pi$
$192$	0	0
$193$	−55820.3	−1.49857	−0.749286	−	0.662247i	$-0.769603\pi$
$193$	−55820.3	−1.49857	−0.749286	+	0.662247i	$0.769603\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−68607.5	−1.76783	−0.883913	−	0.467652i	$-0.845100\pi$
$197$	−68607.5	−1.76783	−0.883913	+	0.467652i	$0.845100\pi$
$198$	0	0
$199$	61160.6i	1.54442i	0.635367	+	0.772210i	$0.280849\pi$
$199$	61160.6i	1.54442i	−0.635367	+	0.772210i	$0.719151\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4807.28i	0.116656i
$204$	0	0
$205$	−20407.3	−0.485598
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−58134.3	−1.33088
$210$	0	0
$211$	− 81638.3i	− 1.83370i	−0.399229	−	0.916851i	$-0.630722\pi$
$211$	− 81638.3i	− 1.83370i	0.399229	−	0.916851i	$-0.369278\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5863.83i	0.126854i
$216$	0	0
$217$	−12858.6	−0.273069
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−105771.	−2.16562
$222$	0	0
$223$	82248.7i	1.65394i	0.562248	+	0.826969i	$0.309937\pi$
$223$	82248.7i	1.65394i	−0.562248	+	0.826969i	$0.690063\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	33569.7i	0.651472i	0.945461	+	0.325736i	$0.105612\pi$
$227$	33569.7i	0.651472i	−0.945461	+	0.325736i	$0.894388\pi$
$228$	0	0
$229$	−787.962	−0.0150257	−0.00751284	−	0.999972i	$-0.502391\pi$
$229$	−787.962	−0.0150257	−0.00751284	+	0.999972i	$0.502391\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1022.71	−0.0188383	−0.00941914	−	0.999956i	$-0.502998\pi$
$233$	−1022.71	−0.0188383	−0.00941914	+	0.999956i	$0.502998\pi$
$234$	0	0
$235$	67091.6i	1.21488i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	50109.2i	0.877246i	0.898671	+	0.438623i	$0.144534\pi$
$239$	50109.2i	0.877246i	−0.898671	+	0.438623i	$0.855466\pi$
$240$	0	0
$241$	43582.2	0.750369	0.375185	−	0.926950i	$-0.377579\pi$
$241$	43582.2	0.750369	0.375185	+	0.926950i	$0.377579\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−49280.8	−0.821005
$246$	0	0
$247$	− 151704.i	− 2.48658i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	117597.i	1.86659i	0.359107	+	0.933296i	$0.383081\pi$
$251$	117597.i	1.86659i	−0.359107	+	0.933296i	$0.616919\pi$
$252$	0	0
$253$	66345.4	1.03650
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6069.05	0.0918871	0.0459436	−	0.998944i	$-0.485371\pi$
$257$	6069.05	0.0918871	0.0459436	+	0.998944i	$0.485371\pi$
$258$	0	0
$259$	7519.03i	0.112089i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 49410.9i	− 0.714350i	−0.934037	−	0.357175i	$-0.883740\pi$
$263$	− 49410.9i	− 0.714350i	0.934037	−	0.357175i	$-0.116260\pi$
$264$	0	0
$265$	−127462.	−1.81506
$266$	0	0
$267$	0	0
$268$	0	0
$269$	48002.4	0.663373	0.331687	−	0.943390i	$-0.392382\pi$
$269$	48002.4	0.663373	0.331687	+	0.943390i	$0.392382\pi$
$270$	0	0
$271$	− 36694.7i	− 0.499648i	−0.968291	−	0.249824i	$-0.919627\pi$
$271$	− 36694.7i	− 0.499648i	0.968291	−	0.249824i	$-0.0803728\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6564.68i	0.0868056i
$276$	0	0
$277$	70296.5	0.916166	0.458083	−	0.888909i	$-0.348536\pi$
$277$	70296.5	0.916166	0.458083	+	0.888909i	$0.348536\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−100362.	−1.27103	−0.635516	−	0.772088i	$-0.719212\pi$
$281$	−100362.	−1.27103	−0.635516	+	0.772088i	$0.719212\pi$
$282$	0	0
$283$	− 61603.8i	− 0.769192i	−0.923085	−	0.384596i	$-0.874341\pi$
$283$	− 61603.8i	− 0.769192i	0.923085	−	0.384596i	$-0.125659\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 17752.9i	− 0.215529i
$288$	0	0
$289$	56076.8	0.671409
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−92661.6	−1.07936	−0.539678	−	0.841872i	$-0.681454\pi$
$293$	−92661.6	−1.07936	−0.539678	+	0.841872i	$0.681454\pi$
$294$	0	0
$295$	118010.i	1.35605i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	173131.i	1.93656i
$300$	0	0
$301$	−5101.12	−0.0563032
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−103842.	−1.11628
$306$	0	0
$307$	− 27023.0i	− 0.286719i	−0.989671	−	0.143360i	$-0.954209\pi$
$307$	− 27023.0i	− 0.286719i	0.989671	−	0.143360i	$-0.0457906\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 179192.i	− 1.85267i	−0.376697	−	0.926336i	$-0.622940\pi$
$311$	− 179192.i	− 1.85267i	0.376697	−	0.926336i	$-0.377060\pi$
$312$	0	0
$313$	127234.	1.29872	0.649359	−	0.760482i	$-0.275037\pi$
$313$	127234.	1.29872	0.649359	+	0.760482i	$0.275037\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−58572.2	−0.582872	−0.291436	−	0.956590i	$-0.594133\pi$
$317$	−58572.2	−0.582872	−0.291436	+	0.956590i	$0.594133\pi$
$318$	0	0
$319$	− 22896.5i	− 0.225003i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	200221.i	1.91913i
$324$	0	0
$325$	−17130.8	−0.162185
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−58365.1	−0.539214
$330$	0	0
$331$	138606.i	1.26511i	0.774517	+	0.632553i	$0.217993\pi$
$331$	138606.i	1.26511i	−0.774517	+	0.632553i	$0.782007\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 67725.9i	− 0.603483i
$336$	0	0
$337$	41344.7	0.364049	0.182024	−	0.983294i	$-0.441735\pi$
$337$	41344.7	0.364049	0.182024	+	0.983294i	$0.441735\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	61243.9	0.526689
$342$	0	0
$343$	− 97558.0i	− 0.829229i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	105131.i	0.873116i	0.899676	+	0.436558i	$0.143803\pi$
$347$	105131.i	0.873116i	−0.899676	+	0.436558i	$0.856197\pi$
$348$	0	0
$349$	−137366.	−1.12779	−0.563896	−	0.825846i	$-0.690698\pi$
$349$	−137366.	−1.12779	−0.563896	+	0.825846i	$0.690698\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	160592.	1.28877	0.644384	−	0.764702i	$-0.277114\pi$
$353$	160592.	1.28877	0.644384	+	0.764702i	$0.277114\pi$
$354$	0	0
$355$	− 99328.7i	− 0.788167i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 227947.i	− 1.76867i	−0.466858	−	0.884333i	$-0.654614\pi$
$359$	− 227947.i	− 1.76867i	0.466858	−	0.884333i	$-0.345386\pi$
$360$	0	0
$361$	−156849.	−1.20356
$362$	0	0
$363$	0	0
$364$	0	0
$365$	38672.2	0.290277
$366$	0	0
$367$	102329.i	0.759743i	0.925039	+	0.379872i	$0.124032\pi$
$367$	102329.i	0.759743i	−0.925039	+	0.379872i	$0.875968\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 110883.i	− 0.805598i
$372$	0	0
$373$	−203355.	−1.46163	−0.730813	−	0.682577i	$-0.760859\pi$
$373$	−203355.	−1.46163	−0.730813	+	0.682577i	$0.760859\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	59749.4	0.420389
$378$	0	0
$379$	191182.i	1.33097i	0.746410	+	0.665487i	$0.231776\pi$
$379$	191182.i	1.33097i	−0.746410	+	0.665487i	$0.768224\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 22578.2i	− 0.153919i	−0.997034	−	0.0769593i	$-0.975479\pi$
$383$	− 22578.2i	− 0.153919i	0.997034	−	0.0769593i	$-0.0245212\pi$
$384$	0	0
$385$	−64693.9	−0.436458
$386$	0	0
$387$	0	0
$388$	0	0
$389$	102591.	0.677969	0.338984	−	0.940792i	$-0.389917\pi$
$389$	102591.	0.677969	0.338984	+	0.940792i	$0.389917\pi$
$390$	0	0
$391$	− 228500.i	− 1.49463i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 73361.9i	− 0.470193i
$396$	0	0
$397$	123707.	0.784898	0.392449	−	0.919774i	$-0.371628\pi$
$397$	123707.	0.784898	0.392449	+	0.919774i	$0.371628\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	197064.	1.22551	0.612757	−	0.790271i	$-0.290060\pi$
$401$	197064.	1.22551	0.612757	+	0.790271i	$0.290060\pi$
$402$	0	0
$403$	159818.i	0.984048i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 35812.3i	− 0.216194i
$408$	0	0
$409$	−12465.9	−0.0745209	−0.0372604	−	0.999306i	$-0.511863\pi$
$409$	−12465.9	−0.0745209	−0.0372604	+	0.999306i	$0.511863\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−102660.	−0.601871
$414$	0	0
$415$	98345.5i	0.571029i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	302372.i	1.72232i	0.508334	+	0.861160i	$0.330262\pi$
$419$	302372.i	1.72232i	−0.508334	+	0.861160i	$0.669738\pi$
$420$	0	0
$421$	296949.	1.67540	0.837700	−	0.546131i	$-0.183900\pi$
$421$	296949.	1.67540	0.837700	+	0.546131i	$0.183900\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	22609.4	0.125173
$426$	0	0
$427$	− 90335.6i	− 0.495454i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 56130.2i	− 0.302163i	−0.988521	−	0.151082i	$-0.951724\pi$
$431$	− 56130.2i	− 0.302163i	0.988521	−	0.151082i	$-0.0482757\pi$
$432$	0	0
$433$	118092.	0.629861	0.314930	−	0.949115i	$-0.398019\pi$
$433$	118092.	0.629861	0.314930	+	0.949115i	$0.398019\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	327731.	1.71615
$438$	0	0
$439$	− 285863.i	− 1.48330i	−0.670786	−	0.741651i	$-0.734043\pi$
$439$	− 285863.i	− 1.48330i	0.670786	−	0.741651i	$-0.265957\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24605.8i	0.125381i	0.998033	+	0.0626903i	$0.0199680\pi$
$443$	24605.8i	0.125381i	−0.998033	+	0.0626903i	$0.980032\pi$
$444$	0	0
$445$	−188345.	−0.951115
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10169.4	0.0504430	0.0252215	−	0.999682i	$-0.491971\pi$
$449$	10169.4	0.0504430	0.0252215	+	0.999682i	$0.491971\pi$
$450$	0	0
$451$	84555.1i	0.415706i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 168821.i	− 0.815463i
$456$	0	0
$457$	208738.	0.999468	0.499734	−	0.866179i	$-0.333431\pi$
$457$	208738.	0.999468	0.499734	+	0.866179i	$0.333431\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	273836.	1.28851	0.644255	−	0.764810i	$-0.277167\pi$
$461$	273836.	1.28851	0.644255	+	0.764810i	$0.277167\pi$
$462$	0	0
$463$	68980.0i	0.321782i	0.986972	+	0.160891i	$0.0514367\pi$
$463$	68980.0i	0.321782i	−0.986972	+	0.160891i	$0.948563\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 369919.i	− 1.69618i	−0.529849	−	0.848092i	$-0.677751\pi$
$467$	− 369919.i	− 1.69618i	0.529849	−	0.848092i	$-0.322249\pi$
$468$	0	0
$469$	58916.8	0.267851
$470$	0	0
$471$	0	0
$472$	0	0
$473$	24296.1	0.108596
$474$	0	0
$475$	32428.0i	0.143725i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 333869.i	− 1.45514i	−0.686033	−	0.727570i	$-0.740649\pi$
$479$	− 333869.i	− 1.45514i	0.686033	−	0.727570i	$-0.259351\pi$
$480$	0	0
$481$	93453.5	0.403929
$482$	0	0
$483$	0	0
$484$	0	0
$485$	261308.	1.11089
$486$	0	0
$487$	− 13483.0i	− 0.0568500i	−0.999596	−	0.0284250i	$-0.990951\pi$
$487$	− 13483.0i	− 0.0568500i	0.999596	−	0.0284250i	$-0.00904917\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 249584.i	− 1.03527i	−0.855601	−	0.517636i	$-0.826812\pi$
$491$	− 249584.i	− 1.03527i	0.855601	−	0.517636i	$-0.173188\pi$
$492$	0	0
$493$	−78858.1	−0.324453
$494$	0	0
$495$	0	0
$496$	0	0
$497$	86409.1	0.349822
$498$	0	0
$499$	− 141894.i	− 0.569855i	−0.958549	−	0.284928i	$-0.908030\pi$
$499$	− 141894.i	− 0.569855i	0.958549	−	0.284928i	$-0.0919695\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 182514.i	− 0.721373i	−0.932687	−	0.360687i	$-0.882542\pi$
$503$	− 182514.i	− 0.721373i	0.932687	−	0.360687i	$-0.117458\pi$
$504$	0	0
$505$	−305182.	−1.19667
$506$	0	0
$507$	0	0
$508$	0	0
$509$	210276.	0.811623	0.405812	−	0.913957i	$-0.366989\pi$
$509$	210276.	0.811623	0.405812	+	0.913957i	$0.366989\pi$
$510$	0	0
$511$	33642.1i	0.128837i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	508951.i	1.91894i
$516$	0	0
$517$	277986.	1.04002
$518$	0	0
$519$	0	0
$520$	0	0
$521$	459002.	1.69098	0.845491	−	0.533990i	$-0.179308\pi$
$521$	459002.	1.69098	0.845491	+	0.533990i	$0.179308\pi$
$522$	0	0
$523$	− 509485.i	− 1.86263i	−0.364208	−	0.931317i	$-0.618660\pi$
$523$	− 509485.i	− 1.86263i	0.364208	−	0.931317i	$-0.381340\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 210930.i	− 0.759483i
$528$	0	0
$529$	−94179.5	−0.336546
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−220650.	−0.776692
$534$	0	0
$535$	413364.i	1.44419i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	204189.i	0.702838i
$540$	0	0
$541$	130861.	0.447110	0.223555	−	0.974691i	$-0.428234\pi$
$541$	130861.	0.447110	0.223555	+	0.974691i	$0.428234\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−252546.	−0.850252
$546$	0	0
$547$	− 168658.i	− 0.563681i	−0.959461	−	0.281840i	$-0.909055\pi$
$547$	− 168658.i	− 0.563681i	0.959461	−	0.281840i	$-0.0909448\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 113104.i	− 0.372540i
$552$	0	0
$553$	63819.8	0.208692
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−191979.	−0.618789	−0.309394	−	0.950934i	$-0.600126\pi$
$557$	−191979.	−0.618789	−0.309394	+	0.950934i	$0.600126\pi$
$558$	0	0
$559$	63401.5i	0.202897i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	395138.i	1.24661i	0.781978	+	0.623306i	$0.214211\pi$
$563$	395138.i	1.24661i	−0.781978	+	0.623306i	$0.785789\pi$
$564$	0	0
$565$	−610685.	−1.91302
$566$	0	0
$567$	0	0
$568$	0	0
$569$	149013.	0.460255	0.230128	−	0.973160i	$-0.426086\pi$
$569$	149013.	0.460255	0.230128	+	0.973160i	$0.426086\pi$
$570$	0	0
$571$	438428.i	1.34470i	0.740233	+	0.672351i	$0.234715\pi$
$571$	438428.i	1.34470i	−0.740233	+	0.672351i	$0.765285\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 37008.2i	− 0.111934i
$576$	0	0
$577$	−404675.	−1.21550	−0.607750	−	0.794128i	$-0.707928\pi$
$577$	−404675.	−1.21550	−0.607750	+	0.794128i	$0.707928\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−85553.8	−0.253447
$582$	0	0
$583$	528125.i	1.55382i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 359707.i	− 1.04393i	−0.852966	−	0.521966i	$-0.825199\pi$
$587$	− 359707.i	− 1.04393i	0.852966	−	0.521966i	$-0.174801\pi$
$588$	0	0
$589$	302530.	0.872044
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−26999.2	−0.0767789	−0.0383895	−	0.999263i	$-0.512223\pi$
$593$	−26999.2	−0.0767789	−0.0383895	+	0.999263i	$0.512223\pi$
$594$	0	0
$595$	222813.i	0.629370i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 240755.i	− 0.670998i	−0.942041	−	0.335499i	$-0.891095\pi$
$599$	− 240755.i	− 0.670998i	0.942041	−	0.335499i	$-0.108905\pi$
$600$	0	0
$601$	149042.	0.412628	0.206314	−	0.978486i	$-0.433853\pi$
$601$	149042.	0.412628	0.206314	+	0.978486i	$0.433853\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−75205.2	−0.205465
$606$	0	0
$607$	155634.i	0.422404i	0.977442	+	0.211202i	$0.0677378\pi$
$607$	155634.i	0.422404i	−0.977442	+	0.211202i	$0.932262\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	725416.i	1.94314i
$612$	0	0
$613$	151355.	0.402787	0.201393	−	0.979510i	$-0.435453\pi$
$613$	151355.	0.402787	0.201393	+	0.979510i	$0.435453\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	206149.	0.541515	0.270757	−	0.962648i	$-0.412726\pi$
$617$	206149.	0.541515	0.270757	+	0.962648i	$0.412726\pi$
$618$	0	0
$619$	− 719388.i	− 1.87751i	−0.344587	−	0.938754i	$-0.611981\pi$
$619$	− 719388.i	− 1.87751i	0.344587	−	0.938754i	$-0.388019\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 163847.i	− 0.422145i
$624$	0	0
$625$	−424784.	−1.08745
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−123341.	−0.311750
$630$	0	0
$631$	− 110761.i	− 0.278181i	−0.990280	−	0.139091i	$-0.955582\pi$
$631$	− 110761.i	− 0.278181i	0.990280	−	0.139091i	$-0.0444179\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 289731.i	− 0.718535i
$636$	0	0
$637$	−532839.	−1.31316
$638$	0	0
$639$	0	0
$640$	0	0
$641$	232198.	0.565123	0.282561	−	0.959249i	$-0.408816\pi$
$641$	232198.	0.565123	0.282561	+	0.959249i	$0.408816\pi$
$642$	0	0
$643$	− 97718.2i	− 0.236349i	−0.992993	−	0.118174i	$-0.962296\pi$
$643$	− 97718.2i	− 0.236349i	0.992993	−	0.118174i	$-0.0377042\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	468010.i	1.11801i	0.829164	+	0.559006i	$0.188817\pi$
$647$	468010.i	1.11801i	−0.829164	+	0.559006i	$0.811183\pi$
$648$	0	0
$649$	488960.	1.16087
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−157237.	−0.368747	−0.184373	−	0.982856i	$-0.559026\pi$
$653$	−157237.	−0.368747	−0.184373	+	0.982856i	$0.559026\pi$
$654$	0	0
$655$	− 797029.i	− 1.85777i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	131927.i	0.303783i	0.988397	+	0.151892i	$0.0485365\pi$
$659$	131927.i	0.303783i	−0.988397	+	0.151892i	$0.951464\pi$
$660$	0	0
$661$	231429.	0.529681	0.264840	−	0.964292i	$-0.414681\pi$
$661$	231429.	0.529681	0.264840	+	0.964292i	$0.414681\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−319573.	−0.722648
$666$	0	0
$667$	129079.i	0.290137i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	430258.i	0.955617i
$672$	0	0
$673$	841920.	1.85883	0.929417	−	0.369031i	$-0.120310\pi$
$673$	841920.	1.85883	0.929417	+	0.369031i	$0.120310\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−719682.	−1.57023	−0.785115	−	0.619350i	$-0.787396\pi$
$677$	−719682.	−1.57023	−0.785115	+	0.619350i	$0.787396\pi$
$678$	0	0
$679$	227320.i	0.493058i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 244051.i	− 0.523166i	−0.965181	−	0.261583i	$-0.915755\pi$
$683$	− 244051.i	− 0.523166i	0.965181	−	0.261583i	$-0.0842446\pi$
$684$	0	0
$685$	410627.	0.875118
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.37816e6	−2.90310
$690$	0	0
$691$	− 260932.i	− 0.546477i	−0.961946	−	0.273238i	$-0.911905\pi$
$691$	− 260932.i	− 0.546477i	0.961946	−	0.273238i	$-0.0880948\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	474954.i	0.983291i
$696$	0	0
$697$	291217.	0.599447
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−301721.	−0.614001	−0.307001	−	0.951709i	$-0.599325\pi$
$701$	−301721.	−0.614001	−0.307001	+	0.951709i	$0.599325\pi$
$702$	0	0
$703$	− 176904.i	− 0.357954i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 265487.i	− 0.531135i
$708$	0	0
$709$	−907338.	−1.80500	−0.902499	−	0.430693i	$-0.858269\pi$
$709$	−907338.	−1.80500	−0.902499	+	0.430693i	$0.858269\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−345261.	−0.679154
$714$	0	0
$715$	804077.i	1.57284i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	527835.i	1.02103i	0.859867	+	0.510517i	$0.170546\pi$
$719$	527835.i	1.02103i	−0.859867	+	0.510517i	$0.829454\pi$
$720$	0	0
$721$	−442752.	−0.851707
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−12772.0	−0.0242986
$726$	0	0
$727$	− 74068.0i	− 0.140140i	−0.997542	−	0.0700700i	$-0.977678\pi$
$727$	− 74068.0i	− 0.140140i	0.997542	−	0.0700700i	$-0.0223223\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 83678.2i	− 0.156595i
$732$	0	0
$733$	846522.	1.57554	0.787772	−	0.615967i	$-0.211235\pi$
$733$	846522.	1.57554	0.787772	+	0.615967i	$0.211235\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−280614.	−0.516624
$738$	0	0
$739$	− 61089.6i	− 0.111861i	−0.998435	−	0.0559304i	$-0.982187\pi$
$739$	− 61089.6i	− 0.111861i	0.998435	−	0.0559304i	$-0.0178125\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 41231.0i	− 0.0746873i	−0.999302	−	0.0373436i	$-0.988110\pi$
$743$	− 41231.0i	− 0.0746873i	0.999302	−	0.0373436i	$-0.0118896\pi$
$744$	0	0
$745$	−68378.9	−0.123200
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−359598.	−0.640993
$750$	0	0
$751$	208283.i	0.369296i	0.982805	+	0.184648i	$0.0591145\pi$
$751$	208283.i	0.369296i	−0.982805	+	0.184648i	$0.940886\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 143143.i	− 0.251117i
$756$	0	0
$757$	−239582.	−0.418083	−0.209042	−	0.977907i	$-0.567034\pi$
$757$	−239582.	−0.418083	−0.209042	+	0.977907i	$0.567034\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−277152.	−0.478574	−0.239287	−	0.970949i	$-0.576914\pi$
$761$	−277152.	−0.478574	−0.239287	+	0.970949i	$0.576914\pi$
$762$	0	0
$763$	− 219698.i	− 0.377378i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.27596e6i	2.16893i
$768$	0	0
$769$	576316.	0.974559	0.487279	−	0.873246i	$-0.337989\pi$
$769$	576316.	0.974559	0.487279	+	0.873246i	$0.337989\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	725252.	1.21375	0.606876	−	0.794797i	$-0.292423\pi$
$773$	725252.	1.21375	0.606876	+	0.794797i	$0.292423\pi$
$774$	0	0
$775$	− 34162.5i	− 0.0568783i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	417682.i	0.688290i
$780$	0	0
$781$	−411557.	−0.674727
$782$	0	0
$783$	0	0
$784$	0	0
$785$	304672.	0.494417
$786$	0	0
$787$	944022.i	1.52417i	0.647478	+	0.762084i	$0.275824\pi$
$787$	944022.i	1.52417i	−0.647478	+	0.762084i	$0.724176\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 531253.i	− 0.849080i
$792$	0	0
$793$	−1.12277e6	−1.78544
$794$	0	0
$795$	0	0
$796$	0	0
$797$	469517.	0.739153	0.369577	−	0.929200i	$-0.379503\pi$
$797$	469517.	0.739153	0.369577	+	0.929200i	$0.379503\pi$
$798$	0	0
$799$	− 957414.i	− 1.49971i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 160234.i	− 0.248498i
$804$	0	0
$805$	364711.	0.562803
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−735826.	−1.12429	−0.562145	−	0.827039i	$-0.690024\pi$
$809$	−735826.	−1.12429	−0.562145	+	0.827039i	$0.690024\pi$
$810$	0	0
$811$	− 892746.i	− 1.35733i	−0.734447	−	0.678666i	$-0.762558\pi$
$811$	− 892746.i	− 1.35733i	0.734447	−	0.678666i	$-0.237442\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.21003e6i	1.82172i
$816$	0	0
$817$	120017.	0.179804
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−850226.	−1.26139	−0.630693	−	0.776032i	$-0.717229\pi$
$821$	−850226.	−1.26139	−0.630693	+	0.776032i	$0.717229\pi$
$822$	0	0
$823$	1.26407e6i	1.86625i	0.359552	+	0.933125i	$0.382930\pi$
$823$	1.26407e6i	1.86625i	−0.359552	+	0.933125i	$0.617070\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 311680.i	− 0.455720i	−0.973694	−	0.227860i	$-0.926827\pi$
$827$	− 311680.i	− 0.455720i	0.973694	−	0.227860i	$-0.0731729\pi$
$828$	0	0
$829$	198506.	0.288844	0.144422	−	0.989516i	$-0.453868\pi$
$829$	198506.	0.288844	0.144422	+	0.989516i	$0.453868\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	703249.	1.01349
$834$	0	0
$835$	220357.i	0.316049i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 697934.i	− 0.991495i	−0.868467	−	0.495747i	$-0.834894\pi$
$839$	− 697934.i	− 0.991495i	0.868467	−	0.495747i	$-0.165106\pi$
$840$	0	0
$841$	−662734.	−0.937017
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.35048e6	−1.89136
$846$	0	0
$847$	− 65423.3i	− 0.0911939i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	201891.i	0.278777i
$852$	0	0
$853$	−617443.	−0.848591	−0.424296	−	0.905524i	$-0.639478\pi$
$853$	−617443.	−0.848591	−0.424296	+	0.905524i	$0.639478\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−789988.	−1.07562	−0.537810	−	0.843066i	$-0.680748\pi$
$857$	−789988.	−1.07562	−0.537810	+	0.843066i	$0.680748\pi$
$858$	0	0
$859$	554120.i	0.750961i	0.926830	+	0.375480i	$0.122522\pi$
$859$	554120.i	0.750961i	−0.926830	+	0.375480i	$0.877478\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 124143.i	− 0.166687i	−0.996521	−	0.0833436i	$-0.973440\pi$
$863$	− 124143.i	− 0.166687i	0.996521	−	0.0833436i	$-0.0265599\pi$
$864$	0	0
$865$	−1.16518e6	−1.55726
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−303966.	−0.402519
$870$	0	0
$871$	− 732273.i	− 0.965244i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 336631.i	− 0.439682i
$876$	0	0
$877$	−305131.	−0.396722	−0.198361	−	0.980129i	$-0.563562\pi$
$877$	−305131.	−0.396722	−0.198361	+	0.980129i	$0.563562\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−147217.	−0.189674	−0.0948368	−	0.995493i	$-0.530233\pi$
$881$	−147217.	−0.189674	−0.0948368	+	0.995493i	$0.530233\pi$
$882$	0	0
$883$	512247.i	0.656988i	0.944506	+	0.328494i	$0.106541\pi$
$883$	512247.i	0.656988i	−0.944506	+	0.328494i	$0.893459\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	452530.i	0.575176i	0.957754	+	0.287588i	$0.0928533\pi$
$887$	452530.i	0.575176i	−0.957754	+	0.287588i	$0.907147\pi$
$888$	0	0
$889$	252046.	0.318916
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.37319e6	1.72198
$894$	0	0
$895$	− 492831.i	− 0.615251i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	119154.i	0.147431i
$900$	0	0
$901$	1.81892e6	2.24059
$902$	0	0
$903$	0	0
$904$	0	0
$905$	597925.	0.730045
$906$	0	0
$907$	− 633048.i	− 0.769524i	−0.923016	−	0.384762i	$-0.874284\pi$
$907$	− 633048.i	− 0.769524i	0.923016	−	0.384762i	$-0.125716\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	553885.i	0.667395i	0.942680	+	0.333698i	$0.108296\pi$
$911$	553885.i	0.667395i	−0.942680	+	0.333698i	$0.891704\pi$
$912$	0	0
$913$	407483.	0.488842
$914$	0	0
$915$	0	0
$916$	0	0
$917$	693360.	0.824556
$918$	0	0
$919$	492823.i	0.583526i	0.956491	+	0.291763i	$0.0942419\pi$
$919$	492823.i	0.583526i	−0.956491	+	0.291763i	$0.905758\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1.07397e6i	− 1.26064i
$924$	0	0
$925$	−19976.5	−0.0233473
$926$	0	0
$927$	0	0
$928$	0	0
$929$	759240.	0.879726	0.439863	−	0.898065i	$-0.355027\pi$
$929$	759240.	0.879726	0.439863	+	0.898065i	$0.355027\pi$
$930$	0	0
$931$	1.00865e6i	1.16370i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1.06123e6i	− 1.21391i
$936$	0	0
$937$	1.27179e6	1.44856	0.724280	−	0.689506i	$-0.242172\pi$
$937$	1.27179e6	1.44856	0.724280	+	0.689506i	$0.242172\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−639225.	−0.721896	−0.360948	−	0.932586i	$-0.617547\pi$
$941$	−639225.	−0.721896	−0.360948	+	0.932586i	$0.617547\pi$
$942$	0	0
$943$	− 476677.i	− 0.536045i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 98219.3i	− 0.109521i	−0.998500	−	0.0547604i	$-0.982560\pi$
$947$	− 98219.3i	− 0.109521i	0.998500	−	0.0547604i	$-0.0174395\pi$
$948$	0	0
$949$	418136.	0.464285
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.36949e6	1.50790	0.753951	−	0.656931i	$-0.228146\pi$
$953$	1.36949e6	1.50790	0.753951	+	0.656931i	$0.228146\pi$
$954$	0	0
$955$	60560.1i	0.0664018i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	357217.i	0.388414i
$960$	0	0
$961$	604808.	0.654894
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.46150e6	1.56944
$966$	0	0
$967$	− 1.52061e6i	− 1.62616i	−0.582149	−	0.813082i	$-0.697788\pi$
$967$	− 1.52061e6i	− 1.62616i	0.582149	−	0.813082i	$-0.302212\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1.44116e6i	− 1.52852i	−0.644906	−	0.764262i	$-0.723103\pi$
$971$	− 1.44116e6i	− 1.52852i	0.644906	−	0.764262i	$-0.276897\pi$
$972$	0	0
$973$	−413177.	−0.436426
$974$	0	0
$975$	0	0
$976$	0	0
$977$	595108.	0.623458	0.311729	−	0.950171i	$-0.399092\pi$
$977$	595108.	0.623458	0.311729	+	0.950171i	$0.399092\pi$
$978$	0	0
$979$	780384.i	0.814222i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 947648.i	− 0.980709i	−0.871523	−	0.490354i	$-0.836867\pi$
$983$	− 947648.i	− 0.980709i	0.871523	−	0.490354i	$-0.163133\pi$
$984$	0	0
$985$	1.79630e6	1.85143
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−136969.	−0.140032
$990$	0	0
$991$	− 17609.3i	− 0.0179306i	−0.999960	−	0.00896531i	$-0.997146\pi$
$991$	− 17609.3i	− 0.0179306i	0.999960	−	0.00896531i	$-0.00285378\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1.60132e6i	− 1.61746i
$996$	0	0
$997$	−1.47639e6	−1.48529	−0.742643	−	0.669688i	$-0.766428\pi$
$997$	−1.47639e6	−1.48529	−0.742643	+	0.669688i	$0.766428\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.5.g.h.271.1	yes	6
3.2	odd	2		432.5.g.g.271.5	✓	6
4.3	odd	2	inner	432.5.g.h.271.2	yes	6
12.11	even	2		432.5.g.g.271.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.5.g.g.271.5	✓	6	3.2	odd	2
432.5.g.g.271.6	yes	6	12.11	even	2
432.5.g.h.271.1	yes	6	1.1	even	1	trivial
432.5.g.h.271.2	yes	6	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	432.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-26.1823 q^{5} -22.7768i q^{7} +O(q^{10})\)	\(q-26.1823 q^{5} -22.7768i q^{7} +108.483i q^{11} -283.091 q^{13} +373.628 q^{17} +535.882i q^{19} -611.572i q^{23} +60.5132 q^{25} -211.061 q^{29} -564.547i q^{31} +596.349i q^{35} -330.118 q^{37} +779.429 q^{41} -223.961i q^{43} -2562.48i q^{47} +1882.22 q^{49} +4868.26 q^{53} -2840.34i q^{55} -4507.24i q^{59} +3966.12 q^{61} +7411.98 q^{65} +2586.70i q^{67} +3793.73i q^{71} -1477.03 q^{73} +2470.90 q^{77} +2801.96i q^{79} -3756.18i q^{83} -9782.44 q^{85} +7193.58 q^{89} +6447.91i q^{91} -14030.6i q^{95} -9980.33 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 42 q^{5}+O(q^{10}) \) 6 * q + 42 * q^5	\( 6 q + 42 q^{5} + 168 q^{13} + 624 q^{17} + 1632 q^{25} + 1596 q^{29} - 2652 q^{37} + 9828 q^{41} - 4908 q^{49} + 13158 q^{53} + 3936 q^{61} + 38472 q^{65} + 3606 q^{73} + 47826 q^{77} - 5760 q^{85} + 52620 q^{89} + 13290 q^{97}+O(q^{100}) \) 6 * q + 42 * q^5 + 168 * q^13 + 624 * q^17 + 1632 * q^25 + 1596 * q^29 - 2652 * q^37 + 9828 * q^41 - 4908 * q^49 + 13158 * q^53 + 3936 * q^61 + 38472 * q^65 + 3606 * q^73 + 47826 * q^77 - 5760 * q^85 + 52620 * q^89 + 13290 * q^97

Introduction

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