LMFDB - Embedded newform 224.2.b.b.113.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,2,Mod(113,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("224.113");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	224.b (of order $2$, degree $1$, not 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:	$1.78864900528$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2312.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x + 4 $ x^4 - x^3 - 2*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.2
Root			$1.28078 + 0.599676i$ of defining polynomial
Character	$\chi$	$=$	224.113
Dual form			224.2.b.b.113.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.936426i q^{3} +3.33513i q^{5} +1.00000 q^{7} +2.12311 q^{9} +O(q^{10})$	$q-0.936426i q^{3} +3.33513i q^{5} +1.00000 q^{7} +2.12311 q^{9} +4.27156i q^{11} -3.33513i q^{13} +3.12311 q^{15} +2.00000 q^{17} -0.936426i q^{19} -0.936426i q^{21} +3.12311 q^{23} -6.12311 q^{25} -4.79741i q^{27} -1.87285i q^{29} -6.24621 q^{31} +4.00000 q^{33} +3.33513i q^{35} +1.87285i q^{37} -3.12311 q^{39} -12.2462 q^{41} +4.27156i q^{43} +7.08084i q^{45} +1.00000 q^{49} -1.87285i q^{51} -8.54312i q^{53} -14.2462 q^{55} -0.876894 q^{57} -7.60669i q^{59} -3.33513i q^{61} +2.12311 q^{63} +11.1231 q^{65} -15.7392i q^{67} -2.92456i q^{69} +8.00000 q^{71} -6.00000 q^{73} +5.73384i q^{75} +4.27156i q^{77} +1.87689 q^{81} +9.47954i q^{83} +6.67026i q^{85} -1.75379 q^{87} +0.246211 q^{89} -3.33513i q^{91} +5.84912i q^{93} +3.12311 q^{95} -4.24621 q^{97} +9.06897i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7} - 8 q^{9}+O(q^{10}) $ 4 * q + 4 * q^7 - 8 * q^9	$ 4 q + 4 q^{7} - 8 q^{9} - 4 q^{15} + 8 q^{17} - 4 q^{23} - 8 q^{25} + 8 q^{31} + 16 q^{33} + 4 q^{39} - 16 q^{41} + 4 q^{49} - 24 q^{55} - 20 q^{57} - 8 q^{63} + 28 q^{65} + 32 q^{71} - 24 q^{73} + 24 q^{81} - 40 q^{87} - 32 q^{89} - 4 q^{95} + 16 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 - 8 * q^9 - 4 * q^15 + 8 * q^17 - 4 * q^23 - 8 * q^25 + 8 * q^31 + 16 * q^33 + 4 * q^39 - 16 * q^41 + 4 * q^49 - 24 * q^55 - 20 * q^57 - 8 * q^63 + 28 * q^65 + 32 * q^71 - 24 * q^73 + 24 * q^81 - 40 * q^87 - 32 * q^89 - 4 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 0.936426i	− 0.540646i	−0.962770	−	0.270323i	$-0.912870\pi$
$3$	− 0.936426i	− 0.540646i	0.962770	−	0.270323i	$-0.0871305\pi$
$4$	0	0
$5$	3.33513i	1.49152i	0.666217	+	0.745758i	$0.267913\pi$
$5$	3.33513i	1.49152i	−0.666217	+	0.745758i	$0.732087\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.12311	0.707702
$10$	0	0
$11$	4.27156i	1.28792i	0.765058	+	0.643962i	$0.222710\pi$
$11$	4.27156i	1.28792i	−0.765058	+	0.643962i	$0.777290\pi$
$12$	0	0
$13$	− 3.33513i	− 0.924999i	−0.886619	−	0.462500i	$-0.846953\pi$
$13$	− 3.33513i	− 0.924999i	0.886619	−	0.462500i	$-0.153047\pi$
$14$	0	0
$15$	3.12311	0.806382
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	− 0.936426i	− 0.214831i	−0.994214	−	0.107415i	$-0.965742\pi$
$19$	− 0.936426i	− 0.214831i	0.994214	−	0.107415i	$-0.0342575\pi$
$20$	0	0
$21$	− 0.936426i	− 0.204345i
$22$	0	0
$23$	3.12311	0.651213	0.325606	−	0.945505i	$-0.394432\pi$
$23$	3.12311	0.651213	0.325606	+	0.945505i	$0.394432\pi$
$24$	0	0
$25$	−6.12311	−1.22462
$26$	0	0
$27$	− 4.79741i	− 0.923262i
$28$	0	0
$29$	− 1.87285i	− 0.347780i	−0.984765	−	0.173890i	$-0.944366\pi$
$29$	− 1.87285i	− 0.347780i	0.984765	−	0.173890i	$-0.0556337\pi$
$30$	0	0
$31$	−6.24621	−1.12185	−0.560926	−	0.827866i	$-0.689555\pi$
$31$	−6.24621	−1.12185	−0.560926	+	0.827866i	$0.689555\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	3.33513i	0.563740i
$36$	0	0
$37$	1.87285i	0.307895i	0.988079	+	0.153948i	$0.0491987\pi$
$37$	1.87285i	0.307895i	−0.988079	+	0.153948i	$0.950801\pi$
$38$	0	0
$39$	−3.12311	−0.500097
$40$	0	0
$41$	−12.2462	−1.91254	−0.956268	−	0.292490i	$-0.905516\pi$
$41$	−12.2462	−1.91254	−0.956268	+	0.292490i	$0.905516\pi$
$42$	0	0
$43$	4.27156i	0.651407i	0.945472	+	0.325703i	$0.105601\pi$
$43$	4.27156i	0.651407i	−0.945472	+	0.325703i	$0.894399\pi$
$44$	0	0
$45$	7.08084i	1.05555i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 1.87285i	− 0.262252i
$52$	0	0
$53$	− 8.54312i	− 1.17349i	−0.809773	−	0.586744i	$-0.800410\pi$
$53$	− 8.54312i	− 1.17349i	0.809773	−	0.586744i	$-0.199590\pi$
$54$	0	0
$55$	−14.2462	−1.92096
$56$	0	0
$57$	−0.876894	−0.116147
$58$	0	0
$59$	− 7.60669i	− 0.990307i	−0.868806	−	0.495153i	$-0.835112\pi$
$59$	− 7.60669i	− 0.990307i	0.868806	−	0.495153i	$-0.164888\pi$
$60$	0	0
$61$	− 3.33513i	− 0.427020i	−0.976941	−	0.213510i	$-0.931510\pi$
$61$	− 3.33513i	− 0.427020i	0.976941	−	0.213510i	$-0.0684896\pi$
$62$	0	0
$63$	2.12311	0.267486
$64$	0	0
$65$	11.1231	1.37965
$66$	0	0
$67$	− 15.7392i	− 1.92285i	−0.275061	−	0.961427i	$-0.588698\pi$
$67$	− 15.7392i	− 1.92285i	0.275061	−	0.961427i	$-0.411302\pi$
$68$	0	0
$69$	− 2.92456i	− 0.352075i
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	5.73384i	0.662087i
$76$	0	0
$77$	4.27156i	0.486789i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.87689	0.208544
$82$	0	0
$83$	9.47954i	1.04052i	0.854009	+	0.520258i	$0.174164\pi$
$83$	9.47954i	1.04052i	−0.854009	+	0.520258i	$0.825836\pi$
$84$	0	0
$85$	6.67026i	0.723492i
$86$	0	0
$87$	−1.75379	−0.188026
$88$	0	0
$89$	0.246211	0.0260983	0.0130492	−	0.999915i	$-0.495846\pi$
$89$	0.246211	0.0260983	0.0130492	+	0.999915i	$0.495846\pi$
$90$	0	0
$91$	− 3.33513i	− 0.349617i
$92$	0	0
$93$	5.84912i	0.606525i
$94$	0	0
$95$	3.12311	0.320424
$96$	0	0
$97$	−4.24621	−0.431137	−0.215569	−	0.976489i	$-0.569161\pi$
$97$	−4.24621	−0.431137	−0.215569	+	0.976489i	$0.569161\pi$
$98$	0	0
$99$	9.06897i	0.911466i
$100$	0	0
$101$	− 7.08084i	− 0.704570i	−0.935893	−	0.352285i	$-0.885405\pi$
$101$	− 7.08084i	− 0.704570i	0.935893	−	0.352285i	$-0.114595\pi$
$102$	0	0
$103$	14.2462	1.40372	0.701860	−	0.712314i	$-0.252353\pi$
$103$	14.2462	1.40372	0.701860	+	0.712314i	$0.252353\pi$
$104$	0	0
$105$	3.12311	0.304784
$106$	0	0
$107$	7.19612i	0.695675i	0.937555	+	0.347837i	$0.113084\pi$
$107$	7.19612i	0.695675i	−0.937555	+	0.347837i	$0.886916\pi$
$108$	0	0
$109$	5.61856i	0.538160i	0.963118	+	0.269080i	$0.0867197\pi$
$109$	5.61856i	0.538160i	−0.963118	+	0.269080i	$0.913280\pi$
$110$	0	0
$111$	1.75379	0.166462
$112$	0	0
$113$	13.1231	1.23452	0.617259	−	0.786760i	$-0.288243\pi$
$113$	13.1231	1.23452	0.617259	+	0.786760i	$0.288243\pi$
$114$	0	0
$115$	10.4160i	0.971294i
$116$	0	0
$117$	− 7.08084i	− 0.654624i
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	−7.24621	−0.658746
$122$	0	0
$123$	11.4677i	1.03401i
$124$	0	0
$125$	− 3.74571i	− 0.335026i
$126$	0	0
$127$	−4.87689	−0.432754	−0.216377	−	0.976310i	$-0.569424\pi$
$127$	−4.87689	−0.432754	−0.216377	+	0.976310i	$0.569424\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	− 3.86098i	− 0.337336i	−0.985673	−	0.168668i	$-0.946053\pi$
$131$	− 3.86098i	− 0.337336i	0.985673	−	0.168668i	$-0.0539465\pi$
$132$	0	0
$133$	− 0.936426i	− 0.0811985i
$134$	0	0
$135$	16.0000	1.37706
$136$	0	0
$137$	0.246211	0.0210352	0.0105176	−	0.999945i	$-0.496652\pi$
$137$	0.246211	0.0210352	0.0105176	+	0.999945i	$0.496652\pi$
$138$	0	0
$139$	− 18.0227i	− 1.52866i	−0.644824	−	0.764331i	$-0.723069\pi$
$139$	− 18.0227i	− 1.52866i	0.644824	−	0.764331i	$-0.276931\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	14.2462	1.19133
$144$	0	0
$145$	6.24621	0.518720
$146$	0	0
$147$	− 0.936426i	− 0.0772351i
$148$	0	0
$149$	12.2888i	1.00674i	0.864071	+	0.503370i	$0.167907\pi$
$149$	12.2888i	1.00674i	−0.864071	+	0.503370i	$0.832093\pi$
$150$	0	0
$151$	−19.1231	−1.55622	−0.778108	−	0.628130i	$-0.783820\pi$
$151$	−19.1231	−1.55622	−0.778108	+	0.628130i	$0.783820\pi$
$152$	0	0
$153$	4.24621	0.343286
$154$	0	0
$155$	− 20.8319i	− 1.67326i
$156$	0	0
$157$	− 16.6757i	− 1.33086i	−0.746459	−	0.665431i	$-0.768248\pi$
$157$	− 16.6757i	− 1.33086i	0.746459	−	0.665431i	$-0.231752\pi$
$158$	0	0
$159$	−8.00000	−0.634441
$160$	0	0
$161$	3.12311	0.246135
$162$	0	0
$163$	8.01726i	0.627961i	0.949430	+	0.313980i	$0.101663\pi$
$163$	8.01726i	0.627961i	−0.949430	+	0.313980i	$0.898337\pi$
$164$	0	0
$165$	13.3405i	1.03856i
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	1.87689	0.144376
$170$	0	0
$171$	− 1.98813i	− 0.152036i
$172$	0	0
$173$	17.4968i	1.33026i	0.746729	+	0.665129i	$0.231623\pi$
$173$	17.4968i	1.33026i	−0.746729	+	0.665129i	$0.768377\pi$
$174$	0	0
$175$	−6.12311	−0.462863
$176$	0	0
$177$	−7.12311	−0.535405
$178$	0	0
$179$	− 2.39871i	− 0.179288i	−0.995974	−	0.0896438i	$-0.971427\pi$
$179$	− 2.39871i	− 0.179288i	0.995974	−	0.0896438i	$-0.0285729\pi$
$180$	0	0
$181$	13.7511i	1.02211i	0.859548	+	0.511056i	$0.170745\pi$
$181$	13.7511i	1.02211i	−0.859548	+	0.511056i	$0.829255\pi$
$182$	0	0
$183$	−3.12311	−0.230867
$184$	0	0
$185$	−6.24621	−0.459231
$186$	0	0
$187$	8.54312i	0.624735i
$188$	0	0
$189$	− 4.79741i	− 0.348960i
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	−2.87689	−0.207083	−0.103542	−	0.994625i	$-0.533018\pi$
$193$	−2.87689	−0.207083	−0.103542	+	0.994625i	$0.533018\pi$
$194$	0	0
$195$	− 10.4160i	− 0.745903i
$196$	0	0
$197$	− 1.05171i	− 0.0749309i	−0.999298	−	0.0374655i	$-0.988072\pi$
$197$	− 1.05171i	− 0.0749309i	0.999298	−	0.0374655i	$-0.0119284\pi$
$198$	0	0
$199$	1.75379	0.124323	0.0621614	−	0.998066i	$-0.480201\pi$
$199$	1.75379	0.124323	0.0621614	+	0.998066i	$0.480201\pi$
$200$	0	0
$201$	−14.7386	−1.03958
$202$	0	0
$203$	− 1.87285i	− 0.131448i
$204$	0	0
$205$	− 40.8427i	− 2.85258i
$206$	0	0
$207$	6.63068	0.460864
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	− 2.39871i	− 0.165134i	−0.996586	−	0.0825669i	$-0.973688\pi$
$211$	− 2.39871i	− 0.165134i	0.996586	−	0.0825669i	$-0.0263118\pi$
$212$	0	0
$213$	− 7.49141i	− 0.513303i
$214$	0	0
$215$	−14.2462	−0.971584
$216$	0	0
$217$	−6.24621	−0.424020
$218$	0	0
$219$	5.61856i	0.379667i
$220$	0	0
$221$	− 6.67026i	− 0.448691i
$222$	0	0
$223$	22.2462	1.48972	0.744858	−	0.667223i	$-0.232517\pi$
$223$	22.2462	1.48972	0.744858	+	0.667223i	$0.232517\pi$
$224$	0	0
$225$	−13.0000	−0.866667
$226$	0	0
$227$	12.4041i	0.823289i	0.911344	+	0.411645i	$0.135045\pi$
$227$	12.4041i	0.823289i	−0.911344	+	0.411645i	$0.864955\pi$
$228$	0	0
$229$	− 4.15628i	− 0.274655i	−0.990526	−	0.137327i	$-0.956149\pi$
$229$	− 4.15628i	− 0.274655i	0.990526	−	0.137327i	$-0.0438512\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	0.246211	0.0161298	0.00806492	−	0.999967i	$-0.497433\pi$
$233$	0.246211	0.0161298	0.00806492	+	0.999967i	$0.497433\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−23.6155	−1.52756	−0.763781	−	0.645476i	$-0.776659\pi$
$239$	−23.6155	−1.52756	−0.763781	+	0.645476i	$0.776659\pi$
$240$	0	0
$241$	−20.2462	−1.30417	−0.652087	−	0.758145i	$-0.726106\pi$
$241$	−20.2462	−1.30417	−0.652087	+	0.758145i	$0.726106\pi$
$242$	0	0
$243$	− 16.1498i	− 1.03601i
$244$	0	0
$245$	3.33513i	0.213074i
$246$	0	0
$247$	−3.12311	−0.198718
$248$	0	0
$249$	8.87689	0.562550
$250$	0	0
$251$	26.5658i	1.67682i	0.545042	+	0.838408i	$0.316514\pi$
$251$	26.5658i	1.67682i	−0.545042	+	0.838408i	$0.683486\pi$
$252$	0	0
$253$	13.3405i	0.838712i
$254$	0	0
$255$	6.24621	0.391153
$256$	0	0
$257$	−10.4924	−0.654499	−0.327250	−	0.944938i	$-0.606122\pi$
$257$	−10.4924	−0.654499	−0.327250	+	0.944938i	$0.606122\pi$
$258$	0	0
$259$	1.87285i	0.116373i
$260$	0	0
$261$	− 3.97626i	− 0.246125i
$262$	0	0
$263$	20.4924	1.26362	0.631808	−	0.775125i	$-0.282313\pi$
$263$	20.4924	1.26362	0.631808	+	0.775125i	$0.282313\pi$
$264$	0	0
$265$	28.4924	1.75028
$266$	0	0
$267$	− 0.230559i	− 0.0141100i
$268$	0	0
$269$	23.3459i	1.42343i	0.702470	+	0.711713i	$0.252080\pi$
$269$	23.3459i	1.42343i	−0.702470	+	0.711713i	$0.747920\pi$
$270$	0	0
$271$	6.24621	0.379430	0.189715	−	0.981839i	$-0.439244\pi$
$271$	6.24621	0.379430	0.189715	+	0.981839i	$0.439244\pi$
$272$	0	0
$273$	−3.12311	−0.189019
$274$	0	0
$275$	− 26.1552i	− 1.57722i
$276$	0	0
$277$	− 8.54312i	− 0.513306i	−0.966504	−	0.256653i	$-0.917380\pi$
$277$	− 8.54312i	− 0.513306i	0.966504	−	0.256653i	$-0.0826198\pi$
$278$	0	0
$279$	−13.2614	−0.793937
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	8.65840i	0.514688i	0.966320	+	0.257344i	$0.0828474\pi$
$283$	8.65840i	0.514688i	−0.966320	+	0.257344i	$0.917153\pi$
$284$	0	0
$285$	− 2.92456i	− 0.173236i
$286$	0	0
$287$	−12.2462	−0.722871
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	3.97626i	0.233093i
$292$	0	0
$293$	− 7.08084i	− 0.413667i	−0.978376	−	0.206833i	$-0.933684\pi$
$293$	− 7.08084i	− 0.413667i	0.978376	−	0.206833i	$-0.0663158\pi$
$294$	0	0
$295$	25.3693	1.47706
$296$	0	0
$297$	20.4924	1.18909
$298$	0	0
$299$	− 10.4160i	− 0.602371i
$300$	0	0
$301$	4.27156i	0.246209i
$302$	0	0
$303$	−6.63068	−0.380923
$304$	0	0
$305$	11.1231	0.636907
$306$	0	0
$307$	15.3287i	0.874853i	0.899254	+	0.437426i	$0.144110\pi$
$307$	15.3287i	0.874853i	−0.899254	+	0.437426i	$0.855890\pi$
$308$	0	0
$309$	− 13.3405i	− 0.758916i
$310$	0	0
$311$	−20.4924	−1.16202	−0.581009	−	0.813897i	$-0.697342\pi$
$311$	−20.4924	−1.16202	−0.581009	+	0.813897i	$0.697342\pi$
$312$	0	0
$313$	28.7386	1.62440	0.812202	−	0.583377i	$-0.198269\pi$
$313$	28.7386	1.62440	0.812202	+	0.583377i	$0.198269\pi$
$314$	0	0
$315$	7.08084i	0.398960i
$316$	0	0
$317$	21.8836i	1.22911i	0.788875	+	0.614554i	$0.210664\pi$
$317$	21.8836i	1.22911i	−0.788875	+	0.614554i	$0.789336\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	6.73863	0.376114
$322$	0	0
$323$	− 1.87285i	− 0.104208i
$324$	0	0
$325$	20.4214i	1.13277i
$326$	0	0
$327$	5.26137	0.290954
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 22.4095i	− 1.23174i	−0.787849	−	0.615869i	$-0.788805\pi$
$331$	− 22.4095i	− 1.23174i	0.787849	−	0.615869i	$-0.211195\pi$
$332$	0	0
$333$	3.97626i	0.217898i
$334$	0	0
$335$	52.4924	2.86797
$336$	0	0
$337$	−9.12311	−0.496967	−0.248484	−	0.968636i	$-0.579932\pi$
$337$	−9.12311	−0.496967	−0.248484	+	0.968636i	$0.579932\pi$
$338$	0	0
$339$	− 12.2888i	− 0.667437i
$340$	0	0
$341$	− 26.6811i	− 1.44486i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	9.75379	0.525126
$346$	0	0
$347$	− 9.06897i	− 0.486848i	−0.969920	−	0.243424i	$-0.921729\pi$
$347$	− 9.06897i	− 0.486848i	0.969920	−	0.243424i	$-0.0782706\pi$
$348$	0	0
$349$	26.2705i	1.40623i	0.711078	+	0.703113i	$0.248207\pi$
$349$	26.2705i	1.40623i	−0.711078	+	0.703113i	$0.751793\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	0	0
$353$	24.2462	1.29050	0.645248	−	0.763973i	$-0.276754\pi$
$353$	24.2462	1.29050	0.645248	+	0.763973i	$0.276754\pi$
$354$	0	0
$355$	26.6811i	1.41608i
$356$	0	0
$357$	− 1.87285i	− 0.0991219i
$358$	0	0
$359$	6.63068	0.349954	0.174977	−	0.984573i	$-0.444015\pi$
$359$	6.63068	0.349954	0.174977	+	0.984573i	$0.444015\pi$
$360$	0	0
$361$	18.1231	0.953848
$362$	0	0
$363$	6.78554i	0.356149i
$364$	0	0
$365$	− 20.0108i	− 1.04741i
$366$	0	0
$367$	6.24621	0.326050	0.163025	−	0.986622i	$-0.447875\pi$
$367$	6.24621	0.326050	0.163025	+	0.986622i	$0.447875\pi$
$368$	0	0
$369$	−26.0000	−1.35351
$370$	0	0
$371$	− 8.54312i	− 0.443537i
$372$	0	0
$373$	12.2888i	0.636291i	0.948042	+	0.318146i	$0.103060\pi$
$373$	12.2888i	0.636291i	−0.948042	+	0.318146i	$0.896940\pi$
$374$	0	0
$375$	−3.50758	−0.181131
$376$	0	0
$377$	−6.24621	−0.321696
$378$	0	0
$379$	23.4612i	1.20512i	0.798073	+	0.602561i	$0.205853\pi$
$379$	23.4612i	1.20512i	−0.798073	+	0.602561i	$0.794147\pi$
$380$	0	0
$381$	4.56685i	0.233967i
$382$	0	0
$383$	−28.4924	−1.45589	−0.727947	−	0.685633i	$-0.759526\pi$
$383$	−28.4924	−1.45589	−0.727947	+	0.685633i	$0.759526\pi$
$384$	0	0
$385$	−14.2462	−0.726054
$386$	0	0
$387$	9.06897i	0.461002i
$388$	0	0
$389$	7.72197i	0.391519i	0.980652	+	0.195760i	$0.0627173\pi$
$389$	7.72197i	0.391519i	−0.980652	+	0.195760i	$0.937283\pi$
$390$	0	0
$391$	6.24621	0.315884
$392$	0	0
$393$	−3.61553	−0.182379
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12.9300i	0.648936i	0.945897	+	0.324468i	$0.105185\pi$
$397$	12.9300i	0.648936i	−0.945897	+	0.324468i	$0.894815\pi$
$398$	0	0
$399$	−0.876894	−0.0438996
$400$	0	0
$401$	−9.12311	−0.455586	−0.227793	−	0.973710i	$-0.573151\pi$
$401$	−9.12311	−0.455586	−0.227793	+	0.973710i	$0.573151\pi$
$402$	0	0
$403$	20.8319i	1.03771i
$404$	0	0
$405$	6.25969i	0.311047i
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	− 0.230559i	− 0.0113726i
$412$	0	0
$413$	− 7.60669i	− 0.374301i
$414$	0	0
$415$	−31.6155	−1.55195
$416$	0	0
$417$	−16.8769	−0.826465
$418$	0	0
$419$	− 6.78554i	− 0.331495i	−0.986168	−	0.165748i	$-0.946996\pi$
$419$	− 6.78554i	− 0.331495i	0.986168	−	0.165748i	$-0.0530038\pi$
$420$	0	0
$421$	− 16.0345i	− 0.781475i	−0.920502	−	0.390738i	$-0.872220\pi$
$421$	− 16.0345i	− 0.781475i	0.920502	−	0.390738i	$-0.127780\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−12.2462	−0.594029
$426$	0	0
$427$	− 3.33513i	− 0.161398i
$428$	0	0
$429$	− 13.3405i	− 0.644087i
$430$	0	0
$431$	23.6155	1.13752	0.568760	−	0.822504i	$-0.307423\pi$
$431$	23.6155	1.13752	0.568760	+	0.822504i	$0.307423\pi$
$432$	0	0
$433$	14.4924	0.696461	0.348231	−	0.937409i	$-0.386783\pi$
$433$	14.4924	0.696461	0.348231	+	0.937409i	$0.386783\pi$
$434$	0	0
$435$	− 5.84912i	− 0.280444i
$436$	0	0
$437$	− 2.92456i	− 0.139901i
$438$	0	0
$439$	26.7386	1.27617	0.638083	−	0.769968i	$-0.279728\pi$
$439$	26.7386	1.27617	0.638083	+	0.769968i	$0.279728\pi$
$440$	0	0
$441$	2.12311	0.101100
$442$	0	0
$443$	− 6.14441i	− 0.291930i	−0.989290	−	0.145965i	$-0.953371\pi$
$443$	− 6.14441i	− 0.291930i	0.989290	−	0.145965i	$-0.0466287\pi$
$444$	0	0
$445$	0.821147i	0.0389261i
$446$	0	0
$447$	11.5076	0.544290
$448$	0	0
$449$	−32.7386	−1.54503	−0.772516	−	0.634996i	$-0.781002\pi$
$449$	−32.7386	−1.54503	−0.772516	+	0.634996i	$0.781002\pi$
$450$	0	0
$451$	− 52.3104i	− 2.46320i
$452$	0	0
$453$	17.9074i	0.841362i
$454$	0	0
$455$	11.1231	0.521459
$456$	0	0
$457$	−7.36932	−0.344722	−0.172361	−	0.985034i	$-0.555140\pi$
$457$	−7.36932	−0.344722	−0.172361	+	0.985034i	$0.555140\pi$
$458$	0	0
$459$	− 9.59482i	− 0.447848i
$460$	0	0
$461$	− 34.5830i	− 1.61069i	−0.592805	−	0.805346i	$-0.701979\pi$
$461$	− 34.5830i	− 1.61069i	0.592805	−	0.805346i	$-0.298021\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	−19.5076	−0.904642
$466$	0	0
$467$	− 0.936426i	− 0.0433326i	−0.999765	−	0.0216663i	$-0.993103\pi$
$467$	− 0.936426i	− 0.0433326i	0.999765	−	0.0216663i	$-0.00689714\pi$
$468$	0	0
$469$	− 15.7392i	− 0.726770i
$470$	0	0
$471$	−15.6155	−0.719526
$472$	0	0
$473$	−18.2462	−0.838962
$474$	0	0
$475$	5.73384i	0.263087i
$476$	0	0
$477$	− 18.1379i	− 0.830479i
$478$	0	0
$479$	6.24621	0.285397	0.142698	−	0.989766i	$-0.454422\pi$
$479$	6.24621	0.285397	0.142698	+	0.989766i	$0.454422\pi$
$480$	0	0
$481$	6.24621	0.284803
$482$	0	0
$483$	− 2.92456i	− 0.133072i
$484$	0	0
$485$	− 14.1617i	− 0.643049i
$486$	0	0
$487$	−25.3693	−1.14959	−0.574797	−	0.818296i	$-0.694919\pi$
$487$	−25.3693	−1.14959	−0.574797	+	0.818296i	$0.694919\pi$
$488$	0	0
$489$	7.50758	0.339504
$490$	0	0
$491$	22.1789i	1.00092i	0.865759	+	0.500461i	$0.166836\pi$
$491$	22.1789i	1.00092i	−0.865759	+	0.500461i	$0.833164\pi$
$492$	0	0
$493$	− 3.74571i	− 0.168698i
$494$	0	0
$495$	−30.2462	−1.35947
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	24.2824i	1.08703i	0.839400	+	0.543514i	$0.182906\pi$
$499$	24.2824i	1.08703i	−0.839400	+	0.543514i	$0.817094\pi$
$500$	0	0
$501$	7.49141i	0.334692i
$502$	0	0
$503$	−30.2462	−1.34861	−0.674306	−	0.738452i	$-0.735557\pi$
$503$	−30.2462	−1.34861	−0.674306	+	0.738452i	$0.735557\pi$
$504$	0	0
$505$	23.6155	1.05088
$506$	0	0
$507$	− 1.75757i	− 0.0780566i
$508$	0	0
$509$	− 32.9407i	− 1.46007i	−0.683408	−	0.730036i	$-0.739503\pi$
$509$	− 32.9407i	− 1.46007i	0.683408	−	0.730036i	$-0.260497\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	−4.49242	−0.198345
$514$	0	0
$515$	47.5130i	2.09367i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	16.3845	0.719198
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	− 35.9300i	− 1.57111i	−0.618791	−	0.785555i	$-0.712377\pi$
$523$	− 35.9300i	− 1.57111i	0.618791	−	0.785555i	$-0.287623\pi$
$524$	0	0
$525$	5.73384i	0.250245i
$526$	0	0
$527$	−12.4924	−0.544178
$528$	0	0
$529$	−13.2462	−0.575922
$530$	0	0
$531$	− 16.1498i	− 0.700842i
$532$	0	0
$533$	40.8427i	1.76910i
$534$	0	0
$535$	−24.0000	−1.03761
$536$	0	0
$537$	−2.24621	−0.0969312
$538$	0	0
$539$	4.27156i	0.183989i
$540$	0	0
$541$	42.7156i	1.83649i	0.396017	+	0.918243i	$0.370392\pi$
$541$	42.7156i	1.83649i	−0.396017	+	0.918243i	$0.629608\pi$
$542$	0	0
$543$	12.8769	0.552600
$544$	0	0
$545$	−18.7386	−0.802675
$546$	0	0
$547$	− 39.4957i	− 1.68872i	−0.535780	−	0.844358i	$-0.679982\pi$
$547$	− 39.4957i	− 1.68872i	0.535780	−	0.844358i	$-0.320018\pi$
$548$	0	0
$549$	− 7.08084i	− 0.302203i
$550$	0	0
$551$	−1.75379	−0.0747139
$552$	0	0
$553$	0	0
$554$	0	0
$555$	5.84912i	0.248281i
$556$	0	0
$557$	8.54312i	0.361983i	0.983485	+	0.180992i	$0.0579307\pi$
$557$	8.54312i	0.361983i	−0.983485	+	0.180992i	$0.942069\pi$
$558$	0	0
$559$	14.2462	0.602551
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	16.9710i	0.715240i	0.933867	+	0.357620i	$0.116412\pi$
$563$	16.9710i	0.715240i	−0.933867	+	0.357620i	$0.883588\pi$
$564$	0	0
$565$	43.7673i	1.84130i
$566$	0	0
$567$	1.87689	0.0788222
$568$	0	0
$569$	11.3693	0.476627	0.238313	−	0.971188i	$-0.423405\pi$
$569$	11.3693	0.476627	0.238313	+	0.971188i	$0.423405\pi$
$570$	0	0
$571$	− 29.9009i	− 1.25131i	−0.780098	−	0.625657i	$-0.784831\pi$
$571$	− 29.9009i	− 1.25131i	0.780098	−	0.625657i	$-0.215169\pi$
$572$	0	0
$573$	14.9828i	0.625916i
$574$	0	0
$575$	−19.1231	−0.797489
$576$	0	0
$577$	24.2462	1.00938	0.504691	−	0.863300i	$-0.331606\pi$
$577$	24.2462	1.00938	0.504691	+	0.863300i	$0.331606\pi$
$578$	0	0
$579$	2.69400i	0.111959i
$580$	0	0
$581$	9.47954i	0.393278i
$582$	0	0
$583$	36.4924	1.51136
$584$	0	0
$585$	23.6155	0.976382
$586$	0	0
$587$	8.65840i	0.357370i	0.983906	+	0.178685i	$0.0571843\pi$
$587$	8.65840i	0.357370i	−0.983906	+	0.178685i	$0.942816\pi$
$588$	0	0
$589$	5.84912i	0.241009i
$590$	0	0
$591$	−0.984845	−0.0405111
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	6.67026i	0.273454i
$596$	0	0
$597$	− 1.64229i	− 0.0672146i
$598$	0	0
$599$	−20.4924	−0.837298	−0.418649	−	0.908148i	$-0.637496\pi$
$599$	−20.4924	−0.837298	−0.418649	+	0.908148i	$0.637496\pi$
$600$	0	0
$601$	0.246211	0.0100432	0.00502158	−	0.999987i	$-0.498402\pi$
$601$	0.246211	0.0100432	0.00502158	+	0.999987i	$0.498402\pi$
$602$	0	0
$603$	− 33.4161i	− 1.36081i
$604$	0	0
$605$	− 24.1671i	− 0.982531i
$606$	0	0
$607$	24.9848	1.01410	0.507052	−	0.861916i	$-0.330735\pi$
$607$	24.9848	1.01410	0.507052	+	0.861916i	$0.330735\pi$
$608$	0	0
$609$	−1.75379	−0.0710671
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 45.6401i	− 1.84339i	−0.387918	−	0.921694i	$-0.626806\pi$
$613$	− 45.6401i	− 1.84339i	0.387918	−	0.921694i	$-0.373194\pi$
$614$	0	0
$615$	−38.2462	−1.54224
$616$	0	0
$617$	−4.63068	−0.186424	−0.0932121	−	0.995646i	$-0.529713\pi$
$617$	−4.63068	−0.186424	−0.0932121	+	0.995646i	$0.529713\pi$
$618$	0	0
$619$	36.9817i	1.48642i	0.669057	+	0.743211i	$0.266698\pi$
$619$	36.9817i	1.48642i	−0.669057	+	0.743211i	$0.733302\pi$
$620$	0	0
$621$	− 14.9828i	− 0.601240i
$622$	0	0
$623$	0.246211	0.00986425
$624$	0	0
$625$	−18.1231	−0.724924
$626$	0	0
$627$	− 3.74571i	− 0.149589i
$628$	0	0
$629$	3.74571i	0.149351i
$630$	0	0
$631$	36.4924	1.45274	0.726370	−	0.687304i	$-0.241206\pi$
$631$	36.4924	1.45274	0.726370	+	0.687304i	$0.241206\pi$
$632$	0	0
$633$	−2.24621	−0.0892789
$634$	0	0
$635$	− 16.2651i	− 0.645460i
$636$	0	0
$637$	− 3.33513i	− 0.132143i
$638$	0	0
$639$	16.9848	0.671910
$640$	0	0
$641$	19.3693	0.765042	0.382521	−	0.923947i	$-0.375056\pi$
$641$	19.3693	0.765042	0.382521	+	0.923947i	$0.375056\pi$
$642$	0	0
$643$	9.47954i	0.373837i	0.982375	+	0.186918i	$0.0598500\pi$
$643$	9.47954i	0.373837i	−0.982375	+	0.186918i	$0.940150\pi$
$644$	0	0
$645$	13.3405i	0.525283i
$646$	0	0
$647$	39.2311	1.54233	0.771166	−	0.636634i	$-0.219674\pi$
$647$	39.2311	1.54233	0.771166	+	0.636634i	$0.219674\pi$
$648$	0	0
$649$	32.4924	1.27544
$650$	0	0
$651$	5.84912i	0.229245i
$652$	0	0
$653$	− 21.0625i	− 0.824239i	−0.911130	−	0.412120i	$-0.864789\pi$
$653$	− 21.0625i	− 0.824239i	0.911130	−	0.412120i	$-0.135211\pi$
$654$	0	0
$655$	12.8769	0.503142
$656$	0	0
$657$	−12.7386	−0.496981
$658$	0	0
$659$	21.3578i	0.831981i	0.909369	+	0.415991i	$0.136565\pi$
$659$	21.3578i	0.831981i	−0.909369	+	0.415991i	$0.863435\pi$
$660$	0	0
$661$	− 5.43854i	− 0.211535i	−0.994391	−	0.105767i	$-0.966270\pi$
$661$	− 5.43854i	− 0.211535i	0.994391	−	0.105767i	$-0.0337299\pi$
$662$	0	0
$663$	−6.24621	−0.242583
$664$	0	0
$665$	3.12311	0.121109
$666$	0	0
$667$	− 5.84912i	− 0.226479i
$668$	0	0
$669$	− 20.8319i	− 0.805409i
$670$	0	0
$671$	14.2462	0.549969
$672$	0	0
$673$	26.9848	1.04019	0.520095	−	0.854109i	$-0.325897\pi$
$673$	26.9848	1.04019	0.520095	+	0.854109i	$0.325897\pi$
$674$	0	0
$675$	29.3751i	1.13065i
$676$	0	0
$677$	6.25969i	0.240579i	0.992739	+	0.120290i	$0.0383824\pi$
$677$	6.25969i	0.240579i	−0.992739	+	0.120290i	$0.961618\pi$
$678$	0	0
$679$	−4.24621	−0.162955
$680$	0	0
$681$	11.6155	0.445108
$682$	0	0
$683$	− 19.4849i	− 0.745570i	−0.927918	−	0.372785i	$-0.878403\pi$
$683$	− 19.4849i	− 0.745570i	0.927918	−	0.372785i	$-0.121597\pi$
$684$	0	0
$685$	0.821147i	0.0313744i
$686$	0	0
$687$	−3.89205	−0.148491
$688$	0	0
$689$	−28.4924	−1.08547
$690$	0	0
$691$	42.0097i	1.59812i	0.601248	+	0.799062i	$0.294670\pi$
$691$	42.0097i	1.59812i	−0.601248	+	0.799062i	$0.705330\pi$
$692$	0	0
$693$	9.06897i	0.344502i
$694$	0	0
$695$	60.1080	2.28002
$696$	0	0
$697$	−24.4924	−0.927717
$698$	0	0
$699$	− 0.230559i	− 0.00872053i
$700$	0	0
$701$	45.6401i	1.72380i	0.507075	+	0.861902i	$0.330727\pi$
$701$	45.6401i	1.72380i	−0.507075	+	0.861902i	$0.669273\pi$
$702$	0	0
$703$	1.75379	0.0661454
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 7.08084i	− 0.266302i
$708$	0	0
$709$	− 39.7910i	− 1.49438i	−0.664609	−	0.747192i	$-0.731402\pi$
$709$	− 39.7910i	− 1.49438i	0.664609	−	0.747192i	$-0.268598\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−19.5076	−0.730565
$714$	0	0
$715$	47.5130i	1.77689i
$716$	0	0
$717$	22.1142i	0.825870i
$718$	0	0
$719$	19.5076	0.727510	0.363755	−	0.931495i	$-0.381495\pi$
$719$	19.5076	0.727510	0.363755	+	0.931495i	$0.381495\pi$
$720$	0	0
$721$	14.2462	0.530557
$722$	0	0
$723$	18.9591i	0.705096i
$724$	0	0
$725$	11.4677i	0.425899i
$726$	0	0
$727$	−48.9848	−1.81675	−0.908374	−	0.418159i	$-0.862675\pi$
$727$	−48.9848	−1.81675	−0.908374	+	0.418159i	$0.862675\pi$
$728$	0	0
$729$	−9.49242	−0.351571
$730$	0	0
$731$	8.54312i	0.315979i
$732$	0	0
$733$	2.51398i	0.0928562i	0.998922	+	0.0464281i	$0.0147838\pi$
$733$	2.51398i	0.0928562i	−0.998922	+	0.0464281i	$0.985216\pi$
$734$	0	0
$735$	3.12311	0.115197
$736$	0	0
$737$	67.2311	2.47649
$738$	0	0
$739$	− 33.6466i	− 1.23771i	−0.785505	−	0.618855i	$-0.787597\pi$
$739$	− 33.6466i	− 1.23771i	0.785505	−	0.618855i	$-0.212403\pi$
$740$	0	0
$741$	2.92456i	0.107436i
$742$	0	0
$743$	−25.3693	−0.930710	−0.465355	−	0.885124i	$-0.654073\pi$
$743$	−25.3693	−0.930710	−0.465355	+	0.885124i	$0.654073\pi$
$744$	0	0
$745$	−40.9848	−1.50157
$746$	0	0
$747$	20.1261i	0.736374i
$748$	0	0
$749$	7.19612i	0.262940i
$750$	0	0
$751$	−33.3693	−1.21766	−0.608832	−	0.793299i	$-0.708362\pi$
$751$	−33.3693	−1.21766	−0.608832	+	0.793299i	$0.708362\pi$
$752$	0	0
$753$	24.8769	0.906564
$754$	0	0
$755$	− 63.7781i	− 2.32112i
$756$	0	0
$757$	− 38.1487i	− 1.38654i	−0.720678	−	0.693270i	$-0.756170\pi$
$757$	− 38.1487i	− 1.38654i	0.720678	−	0.693270i	$-0.243830\pi$
$758$	0	0
$759$	12.4924	0.453446
$760$	0	0
$761$	−40.7386	−1.47677	−0.738387	−	0.674377i	$-0.764412\pi$
$761$	−40.7386	−1.47677	−0.738387	+	0.674377i	$0.764412\pi$
$762$	0	0
$763$	5.61856i	0.203405i
$764$	0	0
$765$	14.1617i	0.512016i
$766$	0	0
$767$	−25.3693	−0.916033
$768$	0	0
$769$	−23.7538	−0.856584	−0.428292	−	0.903641i	$-0.640884\pi$
$769$	−23.7538	−0.856584	−0.428292	+	0.903641i	$0.640884\pi$
$770$	0	0
$771$	9.82538i	0.353852i
$772$	0	0
$773$	3.33513i	0.119956i	0.998200	+	0.0599782i	$0.0191031\pi$
$773$	3.33513i	0.119956i	−0.998200	+	0.0599782i	$0.980897\pi$
$774$	0	0
$775$	38.2462	1.37384
$776$	0	0
$777$	1.75379	0.0629168
$778$	0	0
$779$	11.4677i	0.410872i
$780$	0	0
$781$	34.1725i	1.22279i
$782$	0	0
$783$	−8.98485	−0.321092
$784$	0	0
$785$	55.6155	1.98500
$786$	0	0
$787$	− 27.6175i	− 0.984457i	−0.870466	−	0.492228i	$-0.836182\pi$
$787$	− 27.6175i	− 0.984457i	0.870466	−	0.492228i	$-0.163818\pi$
$788$	0	0
$789$	− 19.1896i	− 0.683169i
$790$	0	0
$791$	13.1231	0.466604
$792$	0	0
$793$	−11.1231	−0.394993
$794$	0	0
$795$	− 26.6811i	− 0.946280i
$796$	0	0
$797$	4.15628i	0.147223i	0.997287	+	0.0736115i	$0.0234525\pi$
$797$	4.15628i	0.147223i	−0.997287	+	0.0736115i	$0.976548\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0.522732	0.0184698
$802$	0	0
$803$	− 25.6294i	− 0.904440i
$804$	0	0
$805$	10.4160i	0.367115i
$806$	0	0
$807$	21.8617	0.769570
$808$	0	0
$809$	−19.8617	−0.698302	−0.349151	−	0.937067i	$-0.613530\pi$
$809$	−19.8617	−0.698302	−0.349151	+	0.937067i	$0.613530\pi$
$810$	0	0
$811$	− 10.5312i	− 0.369802i	−0.982757	−	0.184901i	$-0.940804\pi$
$811$	− 10.5312i	− 0.369802i	0.982757	−	0.184901i	$-0.0591965\pi$
$812$	0	0
$813$	− 5.84912i	− 0.205137i
$814$	0	0
$815$	−26.7386	−0.936613
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	− 7.08084i	− 0.247424i
$820$	0	0
$821$	− 2.69400i	− 0.0940212i	−0.998894	−	0.0470106i	$-0.985031\pi$
$821$	− 2.69400i	− 0.0940212i	0.998894	−	0.0470106i	$-0.0149695\pi$
$822$	0	0
$823$	−32.9848	−1.14978	−0.574890	−	0.818231i	$-0.694955\pi$
$823$	−32.9848	−1.14978	−0.574890	+	0.818231i	$0.694955\pi$
$824$	0	0
$825$	−24.4924	−0.852717
$826$	0	0
$827$	− 32.8255i	− 1.14145i	−0.821140	−	0.570727i	$-0.806662\pi$
$827$	− 32.8255i	− 1.14145i	0.821140	−	0.570727i	$-0.193338\pi$
$828$	0	0
$829$	− 18.3180i	− 0.636209i	−0.948056	−	0.318104i	$-0.896954\pi$
$829$	− 18.3180i	− 0.636209i	0.948056	−	0.318104i	$-0.103046\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	− 26.6811i	− 0.923336i
$836$	0	0
$837$	29.9656i	1.03576i
$838$	0	0
$839$	−42.7386	−1.47550	−0.737751	−	0.675073i	$-0.764112\pi$
$839$	−42.7386	−1.47550	−0.737751	+	0.675073i	$0.764112\pi$
$840$	0	0
$841$	25.4924	0.879049
$842$	0	0
$843$	5.61856i	0.193513i
$844$	0	0
$845$	6.25969i	0.215340i
$846$	0	0
$847$	−7.24621	−0.248983
$848$	0	0
$849$	8.10795	0.278264
$850$	0	0
$851$	5.84912i	0.200505i
$852$	0	0
$853$	− 50.0270i	− 1.71289i	−0.516237	−	0.856446i	$-0.672668\pi$
$853$	− 50.0270i	− 1.71289i	0.516237	−	0.856446i	$-0.327332\pi$
$854$	0	0
$855$	6.63068	0.226765
$856$	0	0
$857$	18.9848	0.648510	0.324255	−	0.945970i	$-0.394886\pi$
$857$	18.9848	0.648510	0.324255	+	0.945970i	$0.394886\pi$
$858$	0	0
$859$	47.3977i	1.61719i	0.588366	+	0.808595i	$0.299771\pi$
$859$	47.3977i	1.61719i	−0.588366	+	0.808595i	$0.700229\pi$
$860$	0	0
$861$	11.4677i	0.390817i
$862$	0	0
$863$	−3.50758	−0.119399	−0.0596997	−	0.998216i	$-0.519014\pi$
$863$	−3.50758	−0.119399	−0.0596997	+	0.998216i	$0.519014\pi$
$864$	0	0
$865$	−58.3542	−1.98410
$866$	0	0
$867$	12.1735i	0.413435i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−52.4924	−1.77864
$872$	0	0
$873$	−9.01515	−0.305117
$874$	0	0
$875$	− 3.74571i	− 0.126628i
$876$	0	0
$877$	26.4505i	0.893170i	0.894741	+	0.446585i	$0.147360\pi$
$877$	26.4505i	0.893170i	−0.894741	+	0.446585i	$0.852640\pi$
$878$	0	0
$879$	−6.63068	−0.223647
$880$	0	0
$881$	27.7538	0.935049	0.467524	−	0.883980i	$-0.345146\pi$
$881$	27.7538	0.935049	0.467524	+	0.883980i	$0.345146\pi$
$882$	0	0
$883$	− 2.39871i	− 0.0807229i	−0.999185	−	0.0403614i	$-0.987149\pi$
$883$	− 2.39871i	− 0.0807229i	0.999185	−	0.0403614i	$-0.0128509\pi$
$884$	0	0
$885$	− 23.7565i	− 0.798566i
$886$	0	0
$887$	−4.49242	−0.150841	−0.0754204	−	0.997152i	$-0.524030\pi$
$887$	−4.49242	−0.150841	−0.0754204	+	0.997152i	$0.524030\pi$
$888$	0	0
$889$	−4.87689	−0.163566
$890$	0	0
$891$	8.01726i	0.268588i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	8.00000	0.267411
$896$	0	0
$897$	−9.75379	−0.325670
$898$	0	0
$899$	11.6982i	0.390158i
$900$	0	0
$901$	− 17.0862i	− 0.569225i
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	−45.8617	−1.52450
$906$	0	0
$907$	43.0109i	1.42815i	0.700068	+	0.714076i	$0.253153\pi$
$907$	43.0109i	1.42815i	−0.700068	+	0.714076i	$0.746847\pi$
$908$	0	0
$909$	− 15.0334i	− 0.498625i
$910$	0	0
$911$	45.8617	1.51947	0.759734	−	0.650234i	$-0.225329\pi$
$911$	45.8617	1.51947	0.759734	+	0.650234i	$0.225329\pi$
$912$	0	0
$913$	−40.4924	−1.34010
$914$	0	0
$915$	− 10.4160i	− 0.344341i
$916$	0	0
$917$	− 3.86098i	− 0.127501i
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	14.3542	0.472986
$922$	0	0
$923$	− 26.6811i	− 0.878218i
$924$	0	0
$925$	− 11.4677i	− 0.377055i
$926$	0	0
$927$	30.2462	0.993416
$928$	0	0
$929$	30.4924	1.00042	0.500212	−	0.865903i	$-0.333255\pi$
$929$	30.4924	1.00042	0.500212	+	0.865903i	$0.333255\pi$
$930$	0	0
$931$	− 0.936426i	− 0.0306901i
$932$	0	0
$933$	19.1896i	0.628241i
$934$	0	0
$935$	−28.4924	−0.931802
$936$	0	0
$937$	−30.9848	−1.01223	−0.506115	−	0.862466i	$-0.668919\pi$
$937$	−30.9848	−1.01223	−0.506115	+	0.862466i	$0.668919\pi$
$938$	0	0
$939$	− 26.9116i	− 0.878227i
$940$	0	0
$941$	− 32.9407i	− 1.07384i	−0.843634	−	0.536919i	$-0.819588\pi$
$941$	− 32.9407i	− 1.07384i	0.843634	−	0.536919i	$-0.180412\pi$
$942$	0	0
$943$	−38.2462	−1.24547
$944$	0	0
$945$	16.0000	0.520480
$946$	0	0
$947$	6.37497i	0.207159i	0.994621	+	0.103579i	$0.0330296\pi$
$947$	6.37497i	0.207159i	−0.994621	+	0.103579i	$0.966970\pi$
$948$	0	0
$949$	20.0108i	0.649578i
$950$	0	0
$951$	20.4924	0.664512
$952$	0	0
$953$	−50.4924	−1.63561	−0.817805	−	0.575495i	$-0.804809\pi$
$953$	−50.4924	−1.63561	−0.817805	+	0.575495i	$0.804809\pi$
$954$	0	0
$955$	− 53.3621i	− 1.72676i
$956$	0	0
$957$	− 7.49141i	− 0.242163i
$958$	0	0
$959$	0.246211	0.00795058
$960$	0	0
$961$	8.01515	0.258553
$962$	0	0
$963$	15.2781i	0.492330i
$964$	0	0
$965$	− 9.59482i	− 0.308868i
$966$	0	0
$967$	19.1231	0.614958	0.307479	−	0.951555i	$-0.400515\pi$
$967$	19.1231	0.614958	0.307479	+	0.951555i	$0.400515\pi$
$968$	0	0
$969$	−1.75379	−0.0563398
$970$	0	0
$971$	26.5658i	0.852536i	0.904597	+	0.426268i	$0.140172\pi$
$971$	26.5658i	0.852536i	−0.904597	+	0.426268i	$0.859828\pi$
$972$	0	0
$973$	− 18.0227i	− 0.577780i
$974$	0	0
$975$	19.1231	0.612430
$976$	0	0
$977$	40.2462	1.28759	0.643795	−	0.765198i	$-0.277359\pi$
$977$	40.2462	1.28759	0.643795	+	0.765198i	$0.277359\pi$
$978$	0	0
$979$	1.05171i	0.0336127i
$980$	0	0
$981$	11.9288i	0.380857i
$982$	0	0
$983$	17.7538	0.566258	0.283129	−	0.959082i	$-0.408628\pi$
$983$	17.7538	0.566258	0.283129	+	0.959082i	$0.408628\pi$
$984$	0	0
$985$	3.50758	0.111761
$986$	0	0
$987$	0	0
$988$	0	0
$989$	13.3405i	0.424204i
$990$	0	0
$991$	3.50758	0.111422	0.0557109	−	0.998447i	$-0.482257\pi$
$991$	3.50758	0.111422	0.0557109	+	0.998447i	$0.482257\pi$
$992$	0	0
$993$	−20.9848	−0.665934
$994$	0	0
$995$	5.84912i	0.185429i
$996$	0	0
$997$	44.9990i	1.42513i	0.701605	+	0.712566i	$0.252467\pi$
$997$	44.9990i	1.42513i	−0.701605	+	0.712566i	$0.747533\pi$
$998$	0	0
$999$	8.98485	0.284268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.2.b.b.113.2		4
3.2	odd	2		2016.2.c.c.1009.1		4
4.3	odd	2		56.2.b.b.29.2	yes	4
7.2	even	3		1568.2.t.d.753.3		8
7.3	odd	6		1568.2.t.e.177.3		8
7.4	even	3		1568.2.t.d.177.2		8
7.5	odd	6		1568.2.t.e.753.2		8
7.6	odd	2		1568.2.b.d.785.3		4
8.3	odd	2		56.2.b.b.29.1	✓	4
8.5	even	2	inner	224.2.b.b.113.3		4
12.11	even	2		504.2.c.d.253.3		4
16.3	odd	4		1792.2.a.x.1.2		4
16.5	even	4		1792.2.a.v.1.2		4
16.11	odd	4		1792.2.a.x.1.3		4
16.13	even	4		1792.2.a.v.1.3		4
24.5	odd	2		2016.2.c.c.1009.4		4
24.11	even	2		504.2.c.d.253.4		4
28.3	even	6		392.2.p.e.373.2		8
28.11	odd	6		392.2.p.f.373.2		8
28.19	even	6		392.2.p.e.165.4		8
28.23	odd	6		392.2.p.f.165.4		8
28.27	even	2		392.2.b.c.197.2		4
56.3	even	6		392.2.p.e.373.4		8
56.5	odd	6		1568.2.t.e.753.3		8
56.11	odd	6		392.2.p.f.373.4		8
56.13	odd	2		1568.2.b.d.785.2		4
56.19	even	6		392.2.p.e.165.2		8
56.27	even	2		392.2.b.c.197.1		4
56.37	even	6		1568.2.t.d.753.2		8
56.45	odd	6		1568.2.t.e.177.2		8
56.51	odd	6		392.2.p.f.165.2		8
56.53	even	6		1568.2.t.d.177.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.b.b.29.1	✓	4	8.3	odd	2
56.2.b.b.29.2	yes	4	4.3	odd	2
224.2.b.b.113.2		4	1.1	even	1	trivial
224.2.b.b.113.3		4	8.5	even	2	inner
392.2.b.c.197.1		4	56.27	even	2
392.2.b.c.197.2		4	28.27	even	2
392.2.p.e.165.2		8	56.19	even	6
392.2.p.e.165.4		8	28.19	even	6
392.2.p.e.373.2		8	28.3	even	6
392.2.p.e.373.4		8	56.3	even	6
392.2.p.f.165.2		8	56.51	odd	6
392.2.p.f.165.4		8	28.23	odd	6
392.2.p.f.373.2		8	28.11	odd	6
392.2.p.f.373.4		8	56.11	odd	6
504.2.c.d.253.3		4	12.11	even	2
504.2.c.d.253.4		4	24.11	even	2
1568.2.b.d.785.2		4	56.13	odd	2
1568.2.b.d.785.3		4	7.6	odd	2
1568.2.t.d.177.2		8	7.4	even	3
1568.2.t.d.177.3		8	56.53	even	6
1568.2.t.d.753.2		8	56.37	even	6
1568.2.t.d.753.3		8	7.2	even	3
1568.2.t.e.177.2		8	56.45	odd	6
1568.2.t.e.177.3		8	7.3	odd	6
1568.2.t.e.753.2		8	7.5	odd	6
1568.2.t.e.753.3		8	56.5	odd	6
1792.2.a.v.1.2		4	16.5	even	4
1792.2.a.v.1.3		4	16.13	even	4
1792.2.a.x.1.2		4	16.3	odd	4
1792.2.a.x.1.3		4	16.11	odd	4
2016.2.c.c.1009.1		4	3.2	odd	2
2016.2.c.c.1009.4		4	24.5	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:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	224.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.936426i q^{3} +3.33513i q^{5} +1.00000 q^{7} +2.12311 q^{9} +O(q^{10})\)	\(q-0.936426i q^{3} +3.33513i q^{5} +1.00000 q^{7} +2.12311 q^{9} +4.27156i q^{11} -3.33513i q^{13} +3.12311 q^{15} +2.00000 q^{17} -0.936426i q^{19} -0.936426i q^{21} +3.12311 q^{23} -6.12311 q^{25} -4.79741i q^{27} -1.87285i q^{29} -6.24621 q^{31} +4.00000 q^{33} +3.33513i q^{35} +1.87285i q^{37} -3.12311 q^{39} -12.2462 q^{41} +4.27156i q^{43} +7.08084i q^{45} +1.00000 q^{49} -1.87285i q^{51} -8.54312i q^{53} -14.2462 q^{55} -0.876894 q^{57} -7.60669i q^{59} -3.33513i q^{61} +2.12311 q^{63} +11.1231 q^{65} -15.7392i q^{67} -2.92456i q^{69} +8.00000 q^{71} -6.00000 q^{73} +5.73384i q^{75} +4.27156i q^{77} +1.87689 q^{81} +9.47954i q^{83} +6.67026i q^{85} -1.75379 q^{87} +0.246211 q^{89} -3.33513i q^{91} +5.84912i q^{93} +3.12311 q^{95} -4.24621 q^{97} +9.06897i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7} - 8 q^{9}+O(q^{10}) \) 4 * q + 4 * q^7 - 8 * q^9	\( 4 q + 4 q^{7} - 8 q^{9} - 4 q^{15} + 8 q^{17} - 4 q^{23} - 8 q^{25} + 8 q^{31} + 16 q^{33} + 4 q^{39} - 16 q^{41} + 4 q^{49} - 24 q^{55} - 20 q^{57} - 8 q^{63} + 28 q^{65} + 32 q^{71} - 24 q^{73} + 24 q^{81} - 40 q^{87} - 32 q^{89} - 4 q^{95} + 16 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 - 8 * q^9 - 4 * q^15 + 8 * q^17 - 4 * q^23 - 8 * q^25 + 8 * q^31 + 16 * q^33 + 4 * q^39 - 16 * q^41 + 4 * q^49 - 24 * q^55 - 20 * q^57 - 8 * q^63 + 28 * q^65 + 32 * q^71 - 24 * q^73 + 24 * q^81 - 40 * q^87 - 32 * q^89 - 4 * q^95 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more