LMFDB - Embedded newform 912.2.d.a.191.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(191,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("912.191");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.d (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:	$7.28235666434$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.2593100598870016.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2x^{10} + x^{8} + 4x^{6} + 4x^{4} - 32x^{2} + 64 $ x^12 - 2x^10 + x^8 + 4x^6 + 4x^4 - 32x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.8
Root			$1.19877 - 0.750295i$ of defining polynomial
Character	$\chi$	$=$	912.191
Dual form			912.2.d.a.191.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.448478 + 1.67298i) q^{3} -2.39755i q^{5} +3.59774i q^{7} +(-2.59774 + 1.50059i) q^{9} -5.96012 q^{11} -3.34596 q^{13} +(4.01105 - 1.07525i) q^{15} -6.52151i q^{17} +1.00000i q^{19} +(-6.01894 + 1.61350i) q^{21} -3.33674 q^{23} -0.748228 q^{25} +(-3.67549 - 3.67298i) q^{27} +5.66680i q^{29} -1.34596i q^{31} +(-2.67298 - 9.97117i) q^{33} +8.62574 q^{35} -6.84242 q^{37} +(-1.50059 - 5.59774i) q^{39} +7.68654i q^{41} +0.402265i q^{43} +(3.59774 + 6.22819i) q^{45} +1.83615 q^{47} -5.94370 q^{49} +(10.9104 - 2.92475i) q^{51} +1.76866i q^{53} +14.2897i q^{55} +(-1.67298 + 0.448478i) q^{57} +6.22819 q^{59} -12.1392 q^{61} +(-5.39873 - 9.34596i) q^{63} +8.02210i q^{65} +0.300986i q^{67} +(-1.49646 - 5.58231i) q^{69} -7.35097 q^{71} +15.6356 q^{73} +(-0.335564 - 1.25177i) q^{75} -21.4429i q^{77} +2.80453i q^{79} +(4.49646 - 7.79627i) q^{81} -15.7086 q^{83} -15.6356 q^{85} +(-9.48045 + 2.54143i) q^{87} -10.6877i q^{89} -12.0379i q^{91} +(2.25177 - 0.603635i) q^{93} +2.39755 q^{95} -7.04498 q^{97} +(15.4828 - 8.94370i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{9} - 12 q^{21} + 4 q^{25} - 12 q^{33} - 16 q^{37} + 16 q^{45} - 4 q^{49} - 24 q^{61} + 8 q^{69} + 40 q^{73} + 28 q^{81} - 40 q^{85} + 40 q^{93} - 16 q^{97}+O(q^{100}) $ 12 * q - 4 * q^9 - 12 * q^21 + 4 * q^25 - 12 * q^33 - 16 * q^37 + 16 * q^45 - 4 * q^49 - 24 * q^61 + 8 * q^69 + 40 * q^73 + 28 * q^81 - 40 * q^85 + 40 * q^93 - 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	0.448478	+	1.67298i	0.258929	+	0.965896i
$3$	0.448478	+	1.67298i	0.258929	+	0.965896i
$4$		0			0
$5$		−	2.39755i		−	1.07222i	−0.844150	−	0.536108i	$-0.819894\pi$
$5$		−	2.39755i		−	1.07222i	0.844150	−	0.536108i	$-0.180106\pi$
$6$		0			0
$7$			3.59774i			1.35982i	0.733297	+	0.679908i	$0.237980\pi$
$7$			3.59774i			1.35982i	−0.733297	+	0.679908i	$0.762020\pi$
$8$		0			0
$9$	−2.59774	+	1.50059i	−0.865912	+	0.500197i
$10$		0			0
$11$	−5.96012			−1.79704			−0.898522	−	0.438929i	$-0.855358\pi$
$11$	−5.96012			−1.79704			−0.898522	+	0.438929i	$0.855358\pi$
$12$		0			0
$13$	−3.34596			−0.928003			−0.464002	−	0.885834i	$-0.653587\pi$
$13$	−3.34596			−0.928003			−0.464002	+	0.885834i	$0.653587\pi$
$14$		0			0
$15$	4.01105	−	1.07525i	1.03565	−	0.277627i
$16$		0			0
$17$		−	6.52151i		−	1.58170i	−0.612011	−	0.790849i	$-0.709639\pi$
$17$		−	6.52151i		−	1.58170i	0.612011	−	0.790849i	$-0.290361\pi$
$18$		0			0
$19$			1.00000i			0.229416i
$19$			1.00000i			0.229416i
$20$		0			0
$21$	−6.01894	+	1.61350i	−1.31344	+	0.352096i
$22$		0			0
$23$	−3.33674			−0.695759			−0.347880	−	0.937539i	$-0.613098\pi$
$23$	−3.33674			−0.695759			−0.347880	+	0.937539i	$0.613098\pi$
$24$		0			0
$25$	−0.748228			−0.149646
$26$		0			0
$27$	−3.67549	−	3.67298i	−0.707348	−	0.706866i
$28$		0			0
$29$			5.66680i			1.05230i	0.850392	+	0.526149i	$0.176365\pi$
$29$			5.66680i			1.05230i	−0.850392	+	0.526149i	$0.823635\pi$
$30$		0			0
$31$		−	1.34596i		−	0.241742i	−0.992668	−	0.120871i	$-0.961431\pi$
$31$		−	1.34596i		−	0.241742i	0.992668	−	0.120871i	$-0.0385688\pi$
$32$		0			0
$33$	−2.67298	−	9.97117i	−0.465306	−	1.73576i
$34$		0			0
$35$	8.62574			1.45802
$36$		0			0
$37$	−6.84242			−1.12489			−0.562443	−	0.826836i	$-0.690139\pi$
$37$	−6.84242			−1.12489			−0.562443	+	0.826836i	$0.690139\pi$
$38$		0			0
$39$	−1.50059	−	5.59774i	−0.240287	−	0.896355i
$40$		0			0
$41$			7.68654i			1.20044i	0.799837	+	0.600218i	$0.204919\pi$
$41$			7.68654i			1.20044i	−0.799837	+	0.600218i	$0.795081\pi$
$42$		0			0
$43$			0.402265i			0.0613448i	0.999529	+	0.0306724i	$0.00976486\pi$
$43$			0.402265i			0.0613448i	−0.999529	+	0.0306724i	$0.990235\pi$
$44$		0			0
$45$	3.59774	+	6.22819i	0.536319	+	0.928444i
$46$		0			0
$47$	1.83615			0.267831			0.133915	−	0.990993i	$-0.457245\pi$
$47$	1.83615			0.267831			0.133915	+	0.990993i	$0.457245\pi$
$48$		0			0
$49$	−5.94370			−0.849100
$50$		0			0
$51$	10.9104	−	2.92475i	1.52776	−	0.409547i
$52$		0			0
$53$			1.76866i			0.242944i	0.992595	+	0.121472i	$0.0387615\pi$
$53$			1.76866i			0.242944i	−0.992595	+	0.121472i	$0.961238\pi$
$54$		0			0
$55$			14.2897i			1.92682i
$56$		0			0
$57$	−1.67298	+	0.448478i	−0.221592	+	0.0594023i
$58$		0			0
$59$	6.22819			0.810841			0.405421	−	0.914130i	$-0.367125\pi$
$59$	6.22819			0.810841			0.405421	+	0.914130i	$0.367125\pi$
$60$		0			0
$61$	−12.1392			−1.55426			−0.777131	−	0.629339i	$-0.783326\pi$
$61$	−12.1392			−1.55426			−0.777131	+	0.629339i	$0.783326\pi$
$62$		0			0
$63$	−5.39873	−	9.34596i	−0.680176	−	1.17748i
$64$		0			0
$65$			8.02210i			0.995019i
$66$		0			0
$67$			0.300986i			0.0367713i	0.999831	+	0.0183856i	$0.00585266\pi$
$67$			0.300986i			0.0367713i	−0.999831	+	0.0183856i	$0.994147\pi$
$68$		0			0
$69$	−1.49646	−	5.58231i	−0.180152	−	0.672031i
$70$		0			0
$71$	−7.35097			−0.872400			−0.436200	−	0.899850i	$-0.643676\pi$
$71$	−7.35097			−0.872400			−0.436200	+	0.899850i	$0.643676\pi$
$72$		0			0
$73$	15.6356			1.83001			0.915006	−	0.403441i	$-0.132186\pi$
$73$	15.6356			1.83001			0.915006	+	0.403441i	$0.132186\pi$
$74$		0			0
$75$	−0.335564	−	1.25177i	−0.0387476	−	0.144542i
$76$		0			0
$77$		−	21.4429i		−	2.44365i
$78$		0			0
$79$			2.80453i			0.315534i	0.987476	+	0.157767i	$0.0504295\pi$
$79$			2.80453i			0.315534i	−0.987476	+	0.157767i	$0.949570\pi$
$80$		0			0
$81$	4.49646	−	7.79627i	0.499606	−	0.866253i
$82$		0			0
$83$	−15.7086			−1.72425			−0.862124	−	0.506698i	$-0.830866\pi$
$83$	−15.7086			−1.72425			−0.862124	+	0.506698i	$0.830866\pi$
$84$		0			0
$85$	−15.6356			−1.69592
$86$		0			0
$87$	−9.48045	+	2.54143i	−1.01641	+	0.272470i
$88$		0			0
$89$		−	10.6877i		−	1.13290i	−0.824097	−	0.566448i	$-0.808317\pi$
$89$		−	10.6877i		−	1.13290i	0.824097	−	0.566448i	$-0.191683\pi$
$90$		0			0
$91$		−	12.0379i		−	1.26191i
$92$		0			0
$93$	2.25177	−	0.603635i	0.233498	−	0.0625940i
$94$		0			0
$95$	2.39755			0.245983
$96$		0			0
$97$	−7.04498			−0.715309			−0.357655	−	0.933854i	$-0.616423\pi$
$97$	−7.04498			−0.715309			−0.357655	+	0.933854i	$0.616423\pi$
$98$		0			0
$99$	15.4828	−	8.94370i	1.55608	−	0.898876i
$100$		0			0
$101$		−	1.12278i		−	0.111721i	−0.998439	−	0.0558606i	$-0.982210\pi$
$101$		−	1.12278i		−	0.111721i	0.998439	−	0.0558606i	$-0.0177902\pi$
$102$		0			0
$103$			5.69901i			0.561541i	0.959775	+	0.280770i	$0.0905899\pi$
$103$			5.69901i			0.561541i	−0.959775	+	0.280770i	$0.909410\pi$
$104$		0			0
$105$	3.86845	+	14.4307i	0.377522	+	1.40829i
$106$		0			0
$107$	14.8369			1.43434			0.717170	−	0.696898i	$-0.245437\pi$
$107$	14.8369			1.43434			0.717170	+	0.696898i	$0.245437\pi$
$108$		0			0
$109$	−1.19547			−0.114505			−0.0572526	−	0.998360i	$-0.518234\pi$
$109$	−1.19547			−0.114505			−0.0572526	+	0.998360i	$0.518234\pi$
$110$		0			0
$111$	−3.06867	−	11.4472i	−0.291266	−	1.08652i
$112$		0			0
$113$			15.9281i			1.49839i	0.662349	+	0.749195i	$0.269560\pi$
$113$			15.9281i			1.49839i	−0.662349	+	0.749195i	$0.730440\pi$
$114$		0			0
$115$			8.00000i			0.746004i
$116$		0			0
$117$	8.69193	−	5.02092i	0.803569	−	0.464184i
$118$		0			0
$119$	23.4627			2.15082
$120$		0			0
$121$	24.5230			2.22937
$122$		0			0
$123$	−12.8594	+	3.44724i	−1.15950	+	0.310827i
$124$		0			0
$125$		−	10.1938i		−	0.911763i
$126$		0			0
$127$			11.4965i			1.02015i	0.860131	+	0.510073i	$0.170382\pi$
$127$			11.4965i			1.02015i	−0.860131	+	0.510073i	$0.829618\pi$
$128$		0			0
$129$	−0.672982	+	0.180407i	−0.0592527	+	0.0158839i
$130$		0			0
$131$	−9.54794			−0.834208			−0.417104	−	0.908859i	$-0.636955\pi$
$131$	−9.54794			−0.834208			−0.417104	+	0.908859i	$0.636955\pi$
$132$		0			0
$133$	−3.59774			−0.311963
$134$		0			0
$135$	−8.80614	+	8.81215i	−0.757912	+	0.758429i
$136$		0			0
$137$			2.31306i			0.197618i	0.995106	+	0.0988091i	$0.0315033\pi$
$137$			2.31306i			0.197618i	−0.995106	+	0.0988091i	$0.968497\pi$
$138$		0			0
$139$		−	14.0871i		−	1.19485i	−0.801924	−	0.597426i	$-0.796190\pi$
$139$		−	14.0871i		−	1.19485i	0.801924	−	0.597426i	$-0.203810\pi$
$140$		0			0
$141$	0.823475	+	3.07185i	0.0693491	+	0.258697i
$142$		0			0
$143$	19.9423			1.66766
$144$		0			0
$145$	13.5864			1.12829
$146$		0			0
$147$	−2.66562	−	9.94370i	−0.219856	−	0.820142i
$148$		0			0
$149$			12.9755i			1.06300i	0.847059	+	0.531498i	$0.178371\pi$
$149$			12.9755i			1.06300i	−0.847059	+	0.531498i	$0.821629\pi$
$150$		0			0
$151$			22.0379i			1.79342i	0.442621	+	0.896709i	$0.354049\pi$
$151$			22.0379i			1.79342i	−0.442621	+	0.896709i	$0.645951\pi$
$152$		0			0
$153$	9.78612	+	16.9412i	0.791161	+	1.36961i
$154$		0			0
$155$	−3.22701			−0.259200
$156$		0			0
$157$	2.99291			0.238860			0.119430	−	0.992843i	$-0.461893\pi$
$157$	2.99291			0.238860			0.119430	+	0.992843i	$0.461893\pi$
$158$		0			0
$159$	−2.95894	+	0.793206i	−0.234659	+	0.0629053i
$160$		0			0
$161$		−	12.0047i		−	0.946105i
$162$		0			0
$163$			13.0829i			1.02473i	0.858768	+	0.512365i	$0.171230\pi$
$163$			13.0829i			1.02473i	−0.858768	+	0.512365i	$0.828770\pi$
$164$		0			0
$165$	−23.9063	+	6.40860i	−1.86111	+	0.498909i
$166$		0			0
$167$	−25.0477			−1.93825			−0.969126	−	0.246564i	$-0.920698\pi$
$167$	−25.0477			−1.93825			−0.969126	+	0.246564i	$0.920698\pi$
$168$		0			0
$169$	−1.80453			−0.138810
$170$		0			0
$171$	−1.50059	−	2.59774i	−0.114753	−	0.198654i
$172$		0			0
$173$		−	3.20176i		−	0.243425i	−0.992565	−	0.121713i	$-0.961161\pi$
$173$		−	3.20176i		−	0.243425i	0.992565	−	0.121713i	$-0.0388386\pi$
$174$		0			0
$175$		−	2.69193i		−	0.203491i
$176$		0			0
$177$	2.79321	+	10.4196i	0.209950	+	0.783189i
$178$		0			0
$179$	−15.4576			−1.15535			−0.577676	−	0.816266i	$-0.696040\pi$
$179$	−15.4576			−1.15535			−0.577676	+	0.816266i	$0.696040\pi$
$180$		0			0
$181$	26.7819			1.99068			0.995341	−	0.0964195i	$-0.0307390\pi$
$181$	26.7819			1.99068			0.995341	+	0.0964195i	$0.0307390\pi$
$182$		0			0
$183$	−5.44415	−	20.3086i	−0.402443	−	1.50126i
$184$		0			0
$185$			16.4050i			1.20612i
$186$		0			0
$187$			38.8690i			2.84238i
$188$		0			0
$189$	13.2144	−	13.2234i	0.961207	−	0.961863i
$190$		0			0
$191$	10.7552			0.778220			0.389110	−	0.921191i	$-0.372783\pi$
$191$	10.7552			0.778220			0.389110	+	0.921191i	$0.372783\pi$
$192$		0			0
$193$	8.18838			0.589413			0.294706	−	0.955588i	$-0.404778\pi$
$193$	8.18838			0.589413			0.294706	+	0.955588i	$0.404778\pi$
$194$		0			0
$195$	−13.4208	+	3.59774i	−0.961086	+	0.257639i
$196$		0			0
$197$			13.7986i			0.983112i	0.870846	+	0.491556i	$0.163572\pi$
$197$			13.7986i			0.983112i	−0.870846	+	0.491556i	$0.836428\pi$
$198$		0			0
$199$		−	6.79321i		−	0.481558i	−0.970580	−	0.240779i	$-0.922597\pi$
$199$		−	6.79321i		−	0.481558i	0.970580	−	0.240779i	$-0.0774028\pi$
$200$		0			0
$201$	−0.503544	+	0.134985i	−0.0355172	+	0.00952114i
$202$		0			0
$203$	−20.3876			−1.43093
$204$		0			0
$205$	18.4288			1.28713
$206$		0			0
$207$	8.66798	−	5.00709i	0.602466	−	0.348017i
$208$		0			0
$209$		−	5.96012i		−	0.412270i
$210$		0			0
$211$		−	2.85659i		−	0.196656i	−0.995154	−	0.0983280i	$-0.968651\pi$
$211$		−	2.85659i		−	0.196656i	0.995154	−	0.0983280i	$-0.0313494\pi$
$212$		0			0
$213$	−3.29675	−	12.2980i	−0.225890	−	0.842648i
$214$		0			0
$215$	0.964448			0.0657748
$216$		0			0
$217$	4.84242			0.328725
$218$		0			0
$219$	7.01223	+	26.1581i	0.473843	+	1.76760i
$220$		0			0
$221$			21.8207i			1.46782i
$222$		0			0
$223$			8.35305i			0.559362i	0.960093	+	0.279681i	$0.0902286\pi$
$223$			8.35305i			0.559362i	−0.960093	+	0.279681i	$0.909771\pi$
$224$		0			0
$225$	1.94370	−	1.12278i	0.129580	−	0.0748523i
$226$		0			0
$227$	14.5606			0.966421			0.483211	−	0.875504i	$-0.339471\pi$
$227$	14.5606			0.966421			0.483211	+	0.875504i	$0.339471\pi$
$228$		0			0
$229$	−16.8311			−1.11223			−0.556115	−	0.831105i	$-0.687709\pi$
$229$	−16.8311			−1.11223			−0.556115	+	0.831105i	$0.687709\pi$
$230$		0			0
$231$	35.8736	−	9.61668i	2.36031	−	0.632731i
$232$		0			0
$233$			4.77810i			0.313024i	0.987676	+	0.156512i	$0.0500249\pi$
$233$			4.77810i			0.313024i	−0.987676	+	0.156512i	$0.949975\pi$
$234$		0			0
$235$		−	4.40226i		−	0.287172i
$236$		0			0
$237$	−4.69193	+	1.25777i	−0.304773	+	0.0817009i
$238$		0			0
$239$	−2.33831			−0.151253			−0.0756264	−	0.997136i	$-0.524096\pi$
$239$	−2.33831			−0.151253			−0.0756264	+	0.997136i	$0.524096\pi$
$240$		0			0
$241$	21.5343			1.38715			0.693575	−	0.720385i	$-0.256035\pi$
$241$	21.5343			1.38715			0.693575	+	0.720385i	$0.256035\pi$
$242$		0			0
$243$	15.0596	+	4.02603i	0.966073	+	0.258270i
$244$		0			0
$245$			14.2503i			0.910418i
$246$		0			0
$247$		−	3.34596i		−	0.212899i
$248$		0			0
$249$	−7.04498	−	26.2803i	−0.446457	−	1.66544i
$250$		0			0
$251$	6.63125			0.418561			0.209280	−	0.977856i	$-0.432888\pi$
$251$	6.63125			0.418561			0.209280	+	0.977856i	$0.432888\pi$
$252$		0			0
$253$	19.8874			1.25031
$254$		0			0
$255$	−7.01223	−	26.1581i	−0.439123	−	1.63808i
$256$		0			0
$257$		−	19.8326i		−	1.23712i	−0.785736	−	0.618562i	$-0.787716\pi$
$257$		−	19.8326i		−	1.23712i	0.785736	−	0.618562i	$-0.212284\pi$
$258$		0			0
$259$		−	24.6172i		−	1.52964i
$260$		0			0
$261$	−8.50354	−	14.7208i	−0.526356	−	0.911197i
$262$		0			0
$263$	−2.42280			−0.149396			−0.0746980	−	0.997206i	$-0.523799\pi$
$263$	−2.42280			−0.149396			−0.0746980	+	0.997206i	$0.523799\pi$
$264$		0			0
$265$	4.24045			0.260489
$266$		0			0
$267$	17.8804	−	4.79321i	1.09426	−	0.293339i
$268$		0			0
$269$		−	9.56494i		−	0.583184i	−0.956543	−	0.291592i	$-0.905815\pi$
$269$		−	9.56494i		−	0.583184i	0.956543	−	0.291592i	$-0.0941850\pi$
$270$		0			0
$271$		−	23.4738i		−	1.42593i	−0.701198	−	0.712966i	$-0.747351\pi$
$271$		−	23.4738i		−	1.42593i	0.701198	−	0.712966i	$-0.252649\pi$
$272$		0			0
$273$	20.1392	−	5.39873i	1.21888	−	0.326746i
$274$		0			0
$275$	4.45953			0.268920
$276$		0			0
$277$	−10.1392			−0.609204			−0.304602	−	0.952480i	$-0.598523\pi$
$277$	−10.1392			−0.609204			−0.304602	+	0.952480i	$0.598523\pi$
$278$		0			0
$279$	2.01974	+	3.49646i	0.120919	+	0.209327i
$280$		0			0
$281$		−	21.7110i		−	1.29517i	−0.761993	−	0.647585i	$-0.775779\pi$
$281$		−	21.7110i		−	1.29517i	0.761993	−	0.647585i	$-0.224221\pi$
$282$		0			0
$283$			22.3654i			1.32949i	0.747071	+	0.664744i	$0.231459\pi$
$283$			22.3654i			1.32949i	−0.747071	+	0.664744i	$0.768541\pi$
$284$		0			0
$285$	1.07525	+	4.01105i	0.0636921	+	0.237594i
$286$		0			0
$287$	−27.6541			−1.63237
$288$		0			0
$289$	−25.5301			−1.50177
$290$		0			0
$291$	−3.15952	−	11.7861i	−0.185214	−	0.690914i
$292$		0			0
$293$		−	26.5061i		−	1.54850i	−0.632878	−	0.774251i	$-0.718127\pi$
$293$		−	26.5061i		−	1.54850i	0.632878	−	0.774251i	$-0.281873\pi$
$294$		0			0
$295$		−	14.9324i		−	0.869396i
$296$		0			0
$297$	21.9063	+	21.8914i	1.27113	+	1.27027i
$298$		0			0
$299$	11.1646			0.645667
$300$		0			0
$301$	−1.44724			−0.0834177
$302$		0			0
$303$	1.87840	−	0.503544i	0.107911	−	0.0289278i
$304$		0			0
$305$			29.1042i			1.66650i
$306$		0			0
$307$		−	13.1955i		−	0.753105i	−0.926395	−	0.376553i	$-0.877109\pi$
$307$		−	13.1955i		−	0.753105i	0.926395	−	0.376553i	$-0.122891\pi$
$308$		0			0
$309$	−9.53435	+	2.55588i	−0.542390	+	0.145399i
$310$		0			0
$311$	1.03004			0.0584083			0.0292041	−	0.999573i	$-0.490703\pi$
$311$	1.03004			0.0584083			0.0292041	+	0.999573i	$0.490703\pi$
$312$		0			0
$313$	9.98582			0.564432			0.282216	−	0.959351i	$-0.408930\pi$
$313$	9.98582			0.564432			0.282216	+	0.959351i	$0.408930\pi$
$314$		0			0
$315$	−22.4074	+	12.9437i	−1.26251	+	0.729295i
$316$		0			0
$317$			14.8117i			0.831907i	0.909386	+	0.415954i	$0.136552\pi$
$317$			14.8117i			0.831907i	−0.909386	+	0.415954i	$0.863448\pi$
$318$		0			0
$319$		−	33.7748i		−	1.89103i
$320$		0			0
$321$	6.65404	+	24.8219i	0.371392	+	1.38542i
$322$		0			0
$323$	6.52151			0.362867
$324$		0			0
$325$	2.50354			0.138872
$326$		0			0
$327$	−0.536142	−	2.00000i	−0.0296487	−	0.110600i
$328$		0			0
$329$			6.60600i			0.364200i
$330$		0			0
$331$		−	25.5722i		−	1.40558i	−0.711399	−	0.702789i	$-0.751938\pi$
$331$		−	25.5722i		−	1.40558i	0.711399	−	0.702789i	$-0.248062\pi$
$332$		0			0
$333$	17.7748	−	10.2677i	0.974053	−	0.562665i
$334$		0			0
$335$	0.721627			0.0394267
$336$		0			0
$337$	−6.03789			−0.328905			−0.164452	−	0.986385i	$-0.552586\pi$
$337$	−6.03789			−0.328905			−0.164452	+	0.986385i	$0.552586\pi$
$338$		0			0
$339$	−26.6474	+	7.14341i	−1.44729	+	0.387977i
$340$		0			0
$341$			8.02210i			0.434421i
$342$		0			0
$343$			3.80029i			0.205197i
$344$		0			0
$345$	−13.3839	+	3.58782i	−0.720562	+	0.193162i
$346$		0			0
$347$	6.17959			0.331738			0.165869	−	0.986148i	$-0.446957\pi$
$347$	6.17959			0.331738			0.165869	+	0.986148i	$0.446957\pi$
$348$		0			0
$349$	−24.9295			−1.33445			−0.667223	−	0.744858i	$-0.732517\pi$
$349$	−24.9295			−1.33445			−0.667223	+	0.744858i	$0.732517\pi$
$350$		0			0
$351$	12.2980	+	12.2897i	0.656421	+	0.655974i
$352$		0			0
$353$		−	0.219471i		−	0.0116813i	−0.999983	−	0.00584063i	$-0.998141\pi$
$353$		−	0.219471i		−	0.0116813i	0.999983	−	0.00584063i	$-0.00185914\pi$
$354$		0			0
$355$			17.6243i			0.935401i
$356$		0			0
$357$	10.5225	+	39.2526i	0.556909	+	2.07747i
$358$		0			0
$359$	−17.9309			−0.946354			−0.473177	−	0.880967i	$-0.656893\pi$
$359$	−17.9309			−0.946354			−0.473177	+	0.880967i	$0.656893\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$	10.9980	+	41.0266i	0.577247	+	2.15334i
$364$		0			0
$365$		−	37.4871i		−	1.96217i
$366$		0			0
$367$			20.0000i			1.04399i	0.852948	+	0.521996i	$0.174812\pi$
$367$			20.0000i			1.04399i	−0.852948	+	0.521996i	$0.825188\pi$
$368$		0			0
$369$	−11.5343	−	19.9676i	−0.600454	−	1.03947i
$370$		0			0
$371$	−6.36318			−0.330360
$372$		0			0
$373$	0.894485			0.0463147			0.0231573	−	0.999732i	$-0.492628\pi$
$373$	0.894485			0.0463147			0.0231573	+	0.999732i	$0.492628\pi$
$374$		0			0
$375$	17.0541	−	4.57170i	0.880669	−	0.236082i
$376$		0			0
$377$		−	18.9609i		−	0.976536i
$378$		0			0
$379$			2.07578i			0.106626i	0.998578	+	0.0533128i	$0.0169780\pi$
$379$			2.07578i			0.106626i	−0.998578	+	0.0533128i	$0.983022\pi$
$380$		0			0
$381$	−19.2334	+	5.15591i	−0.985355	+	0.264145i
$382$		0			0
$383$	−7.65493			−0.391149			−0.195574	−	0.980689i	$-0.562657\pi$
$383$	−7.65493			−0.391149			−0.195574	+	0.980689i	$0.562657\pi$
$384$		0			0
$385$	−51.4104			−2.62012
$386$		0			0
$387$	−0.603635	−	1.04498i	−0.0306845	−	0.0531192i
$388$		0			0
$389$		−	22.6162i		−	1.14669i	−0.819315	−	0.573344i	$-0.805646\pi$
$389$		−	22.6162i		−	1.14669i	0.819315	−	0.573344i	$-0.194354\pi$
$390$		0			0
$391$			21.7606i			1.10048i
$392$		0			0
$393$	−4.28204	−	15.9735i	−0.216000	−	0.805758i
$394$		0			0
$395$	6.72399			0.338321
$396$		0			0
$397$	−5.55985			−0.279041			−0.139520	−	0.990219i	$-0.544556\pi$
$397$	−5.55985			−0.279041			−0.139520	+	0.990219i	$0.544556\pi$
$398$		0			0
$399$	−1.61350	−	6.01894i	−0.0807763	−	0.301324i
$400$		0			0
$401$			6.70510i			0.334837i	0.985886	+	0.167418i	$0.0535430\pi$
$401$			6.70510i			0.334837i	−0.985886	+	0.167418i	$0.946457\pi$
$402$		0			0
$403$			4.50354i			0.224337i
$404$		0			0
$405$	−18.6919	−	10.7805i	−0.928809	−	0.535685i
$406$		0			0
$407$	40.7816			2.02147
$408$		0			0
$409$	1.96211			0.0970201			0.0485101	−	0.998823i	$-0.484553\pi$
$409$	1.96211			0.0970201			0.0485101	+	0.998823i	$0.484553\pi$
$410$		0			0
$411$	−3.86971	+	1.03736i	−0.190879	+	0.0511690i
$412$		0			0
$413$			22.4074i			1.10260i
$414$		0			0
$415$			37.6622i			1.84876i
$416$		0			0
$417$	23.5675	−	6.31775i	1.15410	−	0.309382i
$418$		0			0
$419$	−16.4643			−0.804331			−0.402166	−	0.915567i	$-0.631742\pi$
$419$	−16.4643			−0.804331			−0.402166	+	0.915567i	$0.631742\pi$
$420$		0			0
$421$	−23.7227			−1.15618			−0.578088	−	0.815975i	$-0.696201\pi$
$421$	−23.7227			−1.15618			−0.578088	+	0.815975i	$0.696201\pi$
$422$		0			0
$423$	−4.76984	+	2.75532i	−0.231918	+	0.133968i
$424$		0			0
$425$			4.87958i			0.236694i
$426$		0			0
$427$		−	43.6735i		−	2.11351i
$428$		0			0
$429$	8.94370	+	33.3632i	0.431806	+	1.61079i
$430$		0			0
$431$	31.4954			1.51708			0.758540	−	0.651626i	$-0.225913\pi$
$431$	31.4954			1.51708			0.758540	+	0.651626i	$0.225913\pi$
$432$		0			0
$433$	−36.4667			−1.75248			−0.876239	−	0.481876i	$-0.839956\pi$
$433$	−36.4667			−1.75248			−0.876239	+	0.481876i	$0.839956\pi$
$434$		0			0
$435$	6.09320	+	22.7298i	0.292147	+	1.08981i
$436$		0			0
$437$		−	3.33674i		−	0.159618i
$438$		0			0
$439$			39.1586i			1.86894i	0.356042	+	0.934470i	$0.384126\pi$
$439$			39.1586i			1.86894i	−0.356042	+	0.934470i	$0.615874\pi$
$440$		0			0
$441$	15.4402	−	8.91906i	0.735245	−	0.424717i
$442$		0			0
$443$	−22.0383			−1.04707			−0.523536	−	0.852004i	$-0.675387\pi$
$443$	−22.0383			−1.04707			−0.523536	+	0.852004i	$0.675387\pi$
$444$		0			0
$445$	−25.6243			−1.21471
$446$		0			0
$447$	−21.7078	+	5.81924i	−1.02674	+	0.275241i
$448$		0			0
$449$		−	7.42672i		−	0.350489i	−0.984525	−	0.175244i	$-0.943928\pi$
$449$		−	7.42672i		−	0.350489i	0.984525	−	0.175244i	$-0.0560716\pi$
$450$		0			0
$451$		−	45.8127i		−	2.15724i
$452$		0			0
$453$	−36.8690	+	9.88351i	−1.73226	+	0.464368i
$454$		0			0
$455$	−28.8614			−1.35304
$456$		0			0
$457$	5.44724			0.254811			0.127406	−	0.991851i	$-0.459335\pi$
$457$	5.44724			0.254811			0.127406	+	0.991851i	$0.459335\pi$
$458$		0			0
$459$	−23.9534	+	23.9697i	−1.11805	+	1.11881i
$460$		0			0
$461$			26.9092i			1.25328i	0.779307	+	0.626642i	$0.215571\pi$
$461$			26.9092i			1.25328i	−0.779307	+	0.626642i	$0.784429\pi$
$462$		0			0
$463$		−	1.40935i		−	0.0654982i	−0.999464	−	0.0327491i	$-0.989574\pi$
$463$		−	1.40935i		−	0.0654982i	0.999464	−	0.0327491i	$-0.0104262\pi$
$464$		0			0
$465$	−1.44724	−	5.39873i	−0.0671143	−	0.250360i
$466$		0			0
$467$	20.7466			0.960036			0.480018	−	0.877259i	$-0.340630\pi$
$467$	20.7466			0.960036			0.480018	+	0.877259i	$0.340630\pi$
$468$		0			0
$469$	−1.08287			−0.0500022
$470$		0			0
$471$	1.34226	+	5.00709i	0.0618478	+	0.230714i
$472$		0			0
$473$		−	2.39755i		−	0.110239i
$474$		0			0
$475$		−	0.748228i		−	0.0343311i
$476$		0			0
$477$	−2.65404	−	4.59451i	−0.121520	−	0.210368i
$478$		0			0
$479$	−30.4951			−1.39336			−0.696678	−	0.717384i	$-0.745339\pi$
$479$	−30.4951			−1.39336			−0.696678	+	0.717384i	$0.745339\pi$
$480$		0			0
$481$	22.8945			1.04390
$482$		0			0
$483$	20.0837	−	5.38385i	0.913839	−	0.244974i
$484$		0			0
$485$			16.8907i			0.766965i
$486$		0			0
$487$		−	35.9111i		−	1.62729i	−0.581364	−	0.813644i	$-0.697481\pi$
$487$		−	35.9111i		−	1.62729i	0.581364	−	0.813644i	$-0.302519\pi$
$488$		0			0
$489$	−21.8874	+	5.86738i	−0.989782	+	0.265332i
$490$		0			0
$491$	−12.0868			−0.545471			−0.272736	−	0.962089i	$-0.587928\pi$
$491$	−12.0868			−0.545471			−0.272736	+	0.962089i	$0.587928\pi$
$492$		0			0
$493$	36.9561			1.66442
$494$		0			0
$495$	−21.4429	−	37.1208i	−0.963788	−	1.66845i
$496$		0			0
$497$		−	26.4469i		−	1.18630i
$498$		0			0
$499$		−	13.5078i		−	0.604691i	−0.953198	−	0.302346i	$-0.902230\pi$
$499$		−	13.5078i		−	0.604691i	0.953198	−	0.302346i	$-0.0977697\pi$
$500$		0			0
$501$	−11.2334	−	41.9044i	−0.501870	−	1.87215i
$502$		0			0
$503$	−1.09118			−0.0486532			−0.0243266	−	0.999704i	$-0.507744\pi$
$503$	−1.09118			−0.0486532			−0.0243266	+	0.999704i	$0.507744\pi$
$504$		0			0
$505$	−2.69193			−0.119789
$506$		0			0
$507$	−0.809292	−	3.01894i	−0.0359419	−	0.134076i
$508$		0			0
$509$			7.55155i			0.334717i	0.985896	+	0.167358i	$0.0535237\pi$
$509$			7.55155i			0.334717i	−0.985896	+	0.167358i	$0.946476\pi$
$510$		0			0
$511$			56.2528i			2.48848i
$512$		0			0
$513$	3.67298	−	3.67549i	0.162166	−	0.162277i
$514$		0			0
$515$	13.6637			0.602092
$516$		0			0
$517$	−10.9437			−0.481303
$518$		0			0
$519$	5.35648	−	1.43592i	0.235124	−	0.0630298i
$520$		0			0
$521$			0.470549i			0.0206151i	0.999947	+	0.0103076i	$0.00328106\pi$
$521$			0.470549i			0.0206151i	−0.999947	+	0.0103076i	$0.996719\pi$
$522$		0			0
$523$		−	7.42068i		−	0.324484i	−0.986751	−	0.162242i	$-0.948128\pi$
$523$		−	7.42068i		−	0.324484i	0.986751	−	0.162242i	$-0.0518724\pi$
$524$		0			0
$525$	4.50354	−	1.20727i	0.196551	−	0.0526896i
$526$		0			0
$527$	−8.77771			−0.382363
$528$		0			0
$529$	−11.8661			−0.515919
$530$		0			0
$531$	−16.1792	+	9.34596i	−0.702117	+	0.405580i
$532$		0			0
$533$		−	25.7189i		−	1.11401i
$534$		0			0
$535$		−	35.5722i		−	1.53792i
$536$		0			0
$537$	−6.93237	−	25.8602i	−0.299154	−	1.11595i
$538$		0			0
$539$	35.4252			1.52587
$540$		0			0
$541$	0.642713			0.0276324			0.0138162	−	0.999905i	$-0.495602\pi$
$541$	0.642713			0.0276324			0.0138162	+	0.999905i	$0.495602\pi$
$542$		0			0
$543$	12.0111	+	44.8056i	0.515445	+	1.92279i
$544$		0			0
$545$			2.86620i			0.122774i
$546$		0			0
$547$		−	29.0082i		−	1.24030i	−0.784484	−	0.620150i	$-0.787072\pi$
$547$		−	29.0082i		−	1.24030i	0.784484	−	0.620150i	$-0.212928\pi$
$548$		0			0
$549$	31.5343	−	18.2159i	1.34585	−	0.777437i
$550$		0			0
$551$	−5.66680			−0.241414
$552$		0			0
$553$	−10.0900			−0.429069
$554$		0			0
$555$	−27.4453	+	7.35729i	−1.16499	+	0.312299i
$556$		0			0
$557$			14.1488i			0.599504i	0.954017	+	0.299752i	$0.0969040\pi$
$557$			14.1488i			0.599504i	−0.954017	+	0.299752i	$0.903096\pi$
$558$		0			0
$559$		−	1.34596i		−	0.0569282i
$560$		0			0
$561$	−65.0271	+	17.4319i	−2.74545	+	0.735975i
$562$		0			0
$563$	39.2577			1.65451			0.827257	−	0.561823i	$-0.189900\pi$
$563$	39.2577			1.65451			0.827257	+	0.561823i	$0.189900\pi$
$564$		0			0
$565$	38.1884			1.60660
$566$		0			0
$567$	28.0489	+	16.1771i	1.17794	+	0.679373i
$568$		0			0
$569$			18.0323i			0.755955i	0.925815	+	0.377977i	$0.123380\pi$
$569$			18.0323i			0.755955i	−0.925815	+	0.377977i	$0.876620\pi$
$570$		0			0
$571$		−	19.6990i		−	0.824378i	−0.911098	−	0.412189i	$-0.864764\pi$
$571$		−	19.6990i		−	0.824378i	0.911098	−	0.412189i	$-0.135236\pi$
$572$		0			0
$573$	4.82347	+	17.9933i	0.201504	+	0.751680i
$574$		0			0
$575$	2.49665			0.104117
$576$		0			0
$577$	−24.1392			−1.00493			−0.502463	−	0.864598i	$-0.667573\pi$
$577$	−24.1392			−1.00493			−0.502463	+	0.864598i	$0.667573\pi$
$578$		0			0
$579$	3.67231	+	13.6990i	0.152616	+	0.569312i
$580$		0			0
$581$		−	56.5155i		−	2.34466i
$582$		0			0
$583$		−	10.5414i		−	0.436582i
$584$		0			0
$585$	−12.0379	−	20.8393i	−0.497705	−	0.861599i
$586$		0			0
$587$	−1.13104			−0.0466831			−0.0233415	−	0.999728i	$-0.507431\pi$
$587$	−1.13104			−0.0466831			−0.0233415	+	0.999728i	$0.507431\pi$
$588$		0			0
$589$	1.34596			0.0554595
$590$		0			0
$591$	−23.0849	+	6.18838i	−0.949584	+	0.254556i
$592$		0			0
$593$			35.0896i			1.44096i	0.693477	+	0.720478i	$0.256078\pi$
$593$			35.0896i			1.44096i	−0.693477	+	0.720478i	$0.743922\pi$
$594$		0			0
$595$		−	56.2528i		−	2.30614i
$596$		0			0
$597$	11.3649	−	3.04660i	0.465135	−	0.124689i
$598$		0			0
$599$	−14.7019			−0.600705			−0.300353	−	0.953828i	$-0.597104\pi$
$599$	−14.7019			−0.600705			−0.300353	+	0.953828i	$0.597104\pi$
$600$		0			0
$601$	24.2263			0.988210			0.494105	−	0.869402i	$-0.335496\pi$
$601$	24.2263			0.988210			0.494105	+	0.869402i	$0.335496\pi$
$602$		0			0
$603$	−0.451656	−	0.781882i	−0.0183929	−	0.0318407i
$604$		0			0
$605$		−	58.7951i		−	2.39036i
$606$		0			0
$607$			18.4288i			0.748003i	0.927428	+	0.374002i	$0.122015\pi$
$607$			18.4288i			0.748003i	−0.927428	+	0.374002i	$0.877985\pi$
$608$		0			0
$609$	−9.14341	−	34.1081i	−0.370509	−	1.38213i
$610$		0			0
$611$	−6.14370			−0.248548
$612$		0			0
$613$	−6.04921			−0.244325			−0.122163	−	0.992510i	$-0.538983\pi$
$613$	−6.04921			−0.244325			−0.122163	+	0.992510i	$0.538983\pi$
$614$		0			0
$615$	8.26492	+	30.8311i	0.333274	+	1.24323i
$616$		0			0
$617$			19.8812i			0.800387i	0.916431	+	0.400194i	$0.131057\pi$
$617$			19.8812i			0.800387i	−0.916431	+	0.400194i	$0.868943\pi$
$618$		0			0
$619$			6.79035i			0.272927i	0.990645	+	0.136464i	$0.0435737\pi$
$619$			6.79035i			0.272927i	−0.990645	+	0.136464i	$0.956426\pi$
$620$		0			0
$621$	12.2642	+	12.2558i	0.492144	+	0.491808i
$622$		0			0
$623$	38.4516			1.54053
$624$		0			0
$625$	−28.1813			−1.12725
$626$		0			0
$627$	9.97117	−	2.67298i	0.398210	−	0.106749i
$628$		0			0
$629$			44.6229i			1.77923i
$630$		0			0
$631$		−	28.9816i		−	1.15374i	−0.816836	−	0.576869i	$-0.804274\pi$
$631$		−	28.9816i		−	1.15374i	0.816836	−	0.576869i	$-0.195726\pi$
$632$		0			0
$633$	4.77903	−	1.28112i	0.189949	−	0.0509199i
$634$		0			0
$635$	27.5633			1.09382
$636$		0			0
$637$	19.8874			0.787967
$638$		0			0
$639$	19.0959	−	11.0308i	0.755422	−	0.436372i
$640$		0			0
$641$		−	27.1836i		−	1.07369i	−0.843682	−	0.536843i	$-0.819617\pi$
$641$		−	27.1836i		−	1.07369i	0.843682	−	0.536843i	$-0.180383\pi$
$642$		0			0
$643$			21.0574i			0.830422i	0.909725	+	0.415211i	$0.136292\pi$
$643$			21.0574i			0.830422i	−0.909725	+	0.415211i	$0.863708\pi$
$644$		0			0
$645$	0.432534	+	1.61350i	0.0170310	+	0.0635317i
$646$		0			0
$647$	−14.2925			−0.561898			−0.280949	−	0.959723i	$-0.590649\pi$
$647$	−14.2925			−0.561898			−0.280949	+	0.959723i	$0.590649\pi$
$648$		0			0
$649$	−37.1208			−1.45712
$650$		0			0
$651$	2.17172	+	8.10128i	0.0851163	+	0.317514i
$652$		0			0
$653$			35.8787i			1.40404i	0.712156	+	0.702021i	$0.247719\pi$
$653$			35.8787i			1.40404i	−0.712156	+	0.702021i	$0.752281\pi$
$654$		0			0
$655$			22.8916i			0.894450i
$656$		0			0
$657$	−40.6172	+	23.4627i	−1.58463	+	0.915366i
$658$		0			0
$659$	−27.3438			−1.06516			−0.532582	−	0.846378i	$-0.678778\pi$
$659$	−27.3438			−1.06516			−0.532582	+	0.846378i	$0.678778\pi$
$660$		0			0
$661$	17.2713			0.671774			0.335887	−	0.941902i	$-0.390964\pi$
$661$	17.2713			0.671774			0.335887	+	0.941902i	$0.390964\pi$
$662$		0			0
$663$	−36.5057	+	9.78612i	−1.41776	+	0.380061i
$664$		0			0
$665$			8.62574i			0.334492i
$666$		0			0
$667$		−	18.9087i		−	0.732146i
$668$		0			0
$669$	−13.9745	+	3.74616i	−0.540285	+	0.144835i
$670$		0			0
$671$	72.3509			2.79308
$672$		0			0
$673$	−7.44439			−0.286960			−0.143480	−	0.989653i	$-0.545829\pi$
$673$	−7.44439			−0.286960			−0.143480	+	0.989653i	$0.545829\pi$
$674$		0			0
$675$	2.75010	+	2.74823i	0.105852	+	0.105779i
$676$		0			0
$677$		−	9.42359i		−	0.362178i	−0.983467	−	0.181089i	$-0.942038\pi$
$677$		−	9.42359i		−	0.362178i	0.983467	−	0.181089i	$-0.0579622\pi$
$678$		0			0
$679$		−	25.3460i		−	0.972689i
$680$		0			0
$681$	6.53011	+	24.3596i	0.250234	+	0.933463i
$682$		0			0
$683$	−25.9952			−0.994679			−0.497339	−	0.867556i	$-0.665690\pi$
$683$	−25.9952			−0.994679			−0.497339	+	0.867556i	$0.665690\pi$
$684$		0			0
$685$	5.54567			0.211889
$686$		0			0
$687$	−7.54837	−	28.1581i	−0.287988	−	1.07430i
$688$		0			0
$689$		−	5.91788i		−	0.225453i
$690$		0			0
$691$			37.7861i			1.43745i	0.695294	+	0.718726i	$0.255274\pi$
$691$			37.7861i			1.43745i	−0.695294	+	0.718726i	$0.744726\pi$
$692$		0			0
$693$	32.1771	+	55.7031i	1.22231	+	2.11598i
$694$		0			0
$695$	−33.7745			−1.28114
$696$		0			0
$697$	50.1278			1.89873
$698$		0			0
$699$	−7.99367	+	2.14287i	−0.302348	+	0.0810509i
$700$		0			0
$701$		−	19.1299i		−	0.722525i	−0.932464	−	0.361263i	$-0.882346\pi$
$701$		−	19.1299i		−	0.722525i	0.932464	−	0.361263i	$-0.117654\pi$
$702$		0			0
$703$		−	6.84242i		−	0.258067i
$704$		0			0
$705$	7.36491	−	1.97432i	0.277378	−	0.0743571i
$706$		0			0
$707$	4.03948			0.151920
$708$		0			0
$709$	8.30099			0.311750			0.155875	−	0.987777i	$-0.450180\pi$
$709$	8.30099			0.311750			0.155875	+	0.987777i	$0.450180\pi$
$710$		0			0
$711$	−4.20845	−	7.28543i	−0.157829	−	0.273225i
$712$		0			0
$713$			4.49114i			0.168194i
$714$		0			0
$715$		−	47.8127i		−	1.78809i
$716$		0			0
$717$	−1.04868	−	3.91195i	−0.0391637	−	0.146094i
$718$		0			0
$719$	−35.8535			−1.33711			−0.668554	−	0.743663i	$-0.733087\pi$
$719$	−35.8535			−1.33711			−0.668554	+	0.743663i	$0.733087\pi$
$720$		0			0
$721$	−20.5035			−0.763592
$722$		0			0
$723$	9.65768	+	36.0266i	0.359173	+	1.33984i
$724$		0			0
$725$		−	4.24006i		−	0.157472i
$726$		0			0
$727$		−	11.4710i		−	0.425434i	−0.977114	−	0.212717i	$-0.931769\pi$
$727$		−	11.4710i		−	0.425434i	0.977114	−	0.212717i	$-0.0682313\pi$
$728$		0			0
$729$	0.0184116	+	27.0000i	0.000681912	+	1.00000i
$730$		0			0
$731$	2.62337			0.0970290
$732$		0			0
$733$	46.5425			1.71909			0.859543	−	0.511063i	$-0.170748\pi$
$733$	46.5425			1.71909			0.859543	+	0.511063i	$0.170748\pi$
$734$		0			0
$735$	−23.8405	+	6.39094i	−0.879369	+	0.235733i
$736$		0			0
$737$		−	1.79391i		−	0.0660796i
$738$		0			0
$739$			24.2528i			0.892155i	0.894994	+	0.446078i	$0.147179\pi$
$739$			24.2528i			0.892155i	−0.894994	+	0.446078i	$0.852821\pi$
$740$		0			0
$741$	5.59774	−	1.50059i	0.205638	−	0.0551256i
$742$		0			0
$743$	11.9542			0.438558			0.219279	−	0.975662i	$-0.429630\pi$
$743$	11.9542			0.438558			0.219279	+	0.975662i	$0.429630\pi$
$744$		0			0
$745$	31.1094			1.13976
$746$		0			0
$747$	40.8069	−	23.5722i	1.49305	−	0.862463i
$748$		0			0
$749$			53.3794i			1.95044i
$750$		0			0
$751$		−	45.8732i		−	1.67394i	−0.547251	−	0.836969i	$-0.684326\pi$
$751$		−	45.8732i		−	1.67394i	0.547251	−	0.836969i	$-0.315674\pi$
$752$		0			0
$753$	2.97397	+	11.0940i	0.108377	+	0.404286i
$754$		0			0
$755$	52.8369			1.92293
$756$		0			0
$757$	−10.0407			−0.364937			−0.182468	−	0.983212i	$-0.558409\pi$
$757$	−10.0407			−0.364937			−0.182468	+	0.983212i	$0.558409\pi$
$758$		0			0
$759$	8.91906	+	33.2713i	0.323741	+	1.20767i
$760$		0			0
$761$		−	5.93487i		−	0.215139i	−0.994198	−	0.107569i	$-0.965693\pi$
$761$		−	5.93487i		−	0.215139i	0.994198	−	0.107569i	$-0.0343068\pi$
$762$		0			0
$763$		−	4.30099i		−	0.155706i
$764$		0			0
$765$	40.6172	−	23.4627i	1.46852	−	0.848295i
$766$		0			0
$767$	−20.8393			−0.752463
$768$		0			0
$769$	−25.8240			−0.931238			−0.465619	−	0.884985i	$-0.654168\pi$
$769$	−25.8240			−0.931238			−0.465619	+	0.884985i	$0.654168\pi$
$770$		0			0
$771$	33.1796	−	8.89448i	1.19493	−	0.320327i
$772$		0			0
$773$		−	38.6017i		−	1.38841i	−0.719780	−	0.694203i	$-0.755757\pi$
$773$		−	38.6017i		−	1.38841i	0.719780	−	0.694203i	$-0.244243\pi$
$774$		0			0
$775$			1.00709i			0.0361757i
$776$		0			0
$777$	41.1841	−	11.0403i	1.47747	−	0.396068i
$778$		0			0
$779$	−7.68654			−0.275399
$780$		0			0
$781$	43.8127			1.56774
$782$		0			0
$783$	20.8140	−	20.8282i	0.743833	−	0.744341i
$784$		0			0
$785$		−	7.17565i		−	0.256110i
$786$		0			0
$787$		−	43.5496i		−	1.55238i	−0.630502	−	0.776188i	$-0.717151\pi$
$787$		−	43.5496i		−	1.55238i	0.630502	−	0.776188i	$-0.282849\pi$
$788$		0			0
$789$	−1.08657	−	4.05329i	−0.0386829	−	0.144301i
$790$		0			0
$791$	−57.3051			−2.03754
$792$		0			0
$793$	40.6172			1.44236
$794$		0			0
$795$	1.90175	+	7.09419i	0.0674480	+	0.251605i
$796$		0			0
$797$			10.9640i			0.388366i	0.980965	+	0.194183i	$0.0622056\pi$
$797$			10.9640i			0.388366i	−0.980965	+	0.194183i	$0.937794\pi$
$798$		0			0
$799$		−	11.9745i		−	0.423627i
$800$		0			0
$801$	16.0379	+	27.7639i	0.566671	+	0.980988i
$802$		0			0
$803$	−93.1902			−3.28861
$804$		0			0
$805$	−28.7819			−1.01443
$806$		0			0
$807$	16.0020	−	4.28966i	0.563296	−	0.151003i
$808$		0			0
$809$		−	11.2321i		−	0.394900i	−0.980313	−	0.197450i	$-0.936734\pi$
$809$		−	11.2321i		−	0.394900i	0.980313	−	0.197450i	$-0.0632660\pi$
$810$		0			0
$811$		−	0.375699i		−	0.0131926i	−0.999978	−	0.00659629i	$-0.997900\pi$
$811$		−	0.375699i		−	0.0131926i	0.999978	−	0.00659629i	$-0.00209968\pi$
$812$		0			0
$813$	39.2713	−	10.5275i	1.37730	−	0.369215i
$814$		0			0
$815$	31.3668			1.09873
$816$		0			0
$817$	−0.402265			−0.0140735
$818$		0			0
$819$	18.0639	+	31.2713i	0.631205	+	1.09271i
$820$		0			0
$821$			51.3363i			1.79165i	0.444409	+	0.895824i	$0.353414\pi$
$821$			51.3363i			1.79165i	−0.444409	+	0.895824i	$0.646586\pi$
$822$		0			0
$823$		−	27.5751i		−	0.961207i	−0.876938	−	0.480604i	$-0.840418\pi$
$823$		−	27.5751i		−	0.961207i	0.876938	−	0.480604i	$-0.159582\pi$
$824$		0			0
$825$	2.00000	+	7.46071i	0.0696311	+	0.259749i
$826$		0			0
$827$	−55.3083			−1.92326			−0.961628	−	0.274356i	$-0.911535\pi$
$827$	−55.3083			−1.92326			−0.961628	+	0.274356i	$0.911535\pi$
$828$		0			0
$829$	−31.2854			−1.08659			−0.543294	−	0.839543i	$-0.682823\pi$
$829$	−31.2854			−1.08659			−0.543294	+	0.839543i	$0.682823\pi$
$830$		0			0
$831$	−4.54719	−	16.9626i	−0.157740	−	0.588428i
$832$		0			0
$833$			38.7619i			1.34302i
$834$		0			0
$835$			60.0531i			2.07822i
$836$		0			0
$837$	−4.94370	+	4.94707i	−0.170879	+	0.170996i
$838$		0			0
$839$	26.4405			0.912827			0.456414	−	0.889768i	$-0.349134\pi$
$839$	26.4405			0.912827			0.456414	+	0.889768i	$0.349134\pi$
$840$		0			0
$841$	−3.11260			−0.107331
$842$		0			0
$843$	36.3221	−	9.73690i	1.25100	−	0.335357i
$844$		0			0
$845$			4.32644i			0.148834i
$846$		0			0
$847$			88.2273i			3.03153i
$848$		0			0
$849$	−37.4170	+	10.0304i	−1.28415	+	0.344243i
$850$		0			0
$851$	22.8314			0.782651
$852$		0			0
$853$	−19.5106			−0.668031			−0.334016	−	0.942568i	$-0.608404\pi$
$853$	−19.5106			−0.668031			−0.334016	+	0.942568i	$0.608404\pi$
$854$		0			0
$855$	−6.22819	+	3.59774i	−0.213000	+	0.123040i
$856$		0			0
$857$		−	5.39047i		−	0.184135i	−0.995753	−	0.0920675i	$-0.970652\pi$
$857$		−	5.39047i		−	0.184135i	0.995753	−	0.0920675i	$-0.0293475\pi$
$858$		0			0
$859$		−	19.3952i		−	0.661755i	−0.943674	−	0.330877i	$-0.892655\pi$
$859$		−	19.3952i		−	0.661755i	0.943674	−	0.330877i	$-0.107345\pi$
$860$		0			0
$861$	−12.4023	−	46.2648i	−0.422668	−	1.57670i
$862$		0			0
$863$	−44.5384			−1.51611			−0.758053	−	0.652193i	$-0.773849\pi$
$863$	−44.5384			−1.51611			−0.758053	+	0.652193i	$0.773849\pi$
$864$		0			0
$865$	−7.67637			−0.261004
$866$		0			0
$867$	−11.4497	−	42.7114i	−0.388852	−	1.45056i
$868$		0			0
$869$		−	16.7153i		−	0.567029i
$870$		0			0
$871$		−	1.00709i		−	0.0341239i
$872$		0			0
$873$	18.3010	−	10.5716i	0.619395	−	0.357795i
$874$		0			0
$875$	36.6747			1.23983
$876$		0			0
$877$	26.1516			0.883075			0.441538	−	0.897243i	$-0.354433\pi$
$877$	26.1516			0.883075			0.441538	+	0.897243i	$0.354433\pi$
$878$		0			0
$879$	44.3442	−	11.8874i	1.49569	−	0.400952i
$880$		0			0
$881$		−	30.6659i		−	1.03316i	−0.856238	−	0.516581i	$-0.827205\pi$
$881$		−	30.6659i		−	1.03316i	0.856238	−	0.516581i	$-0.172795\pi$
$882$		0			0
$883$			13.1841i			0.443682i	0.975083	+	0.221841i	$0.0712066\pi$
$883$			13.1841i			0.443682i	−0.975083	+	0.221841i	$0.928793\pi$
$884$		0			0
$885$	24.9816	−	6.69684i	0.839747	−	0.225112i
$886$		0			0
$887$	−15.9873			−0.536803			−0.268401	−	0.963307i	$-0.586495\pi$
$887$	−15.9873			−0.536803			−0.268401	+	0.963307i	$0.586495\pi$
$888$		0			0
$889$	−41.3612			−1.38721
$890$		0			0
$891$	−26.7994	+	46.4667i	−0.897814	+	1.55669i
$892$		0			0
$893$			1.83615i			0.0614446i
$894$		0			0
$895$			37.0602i			1.23879i
$896$		0			0
$897$	5.00709	+	18.6782i	0.167182	+	0.623647i
$898$		0			0
$899$	7.62730			0.254385
$900$		0			0
$901$	11.5343			0.384265
$902$		0			0
$903$	−0.649056	−	2.42121i	−0.0215992	−	0.0805728i
$904$		0			0
$905$		−	64.2108i		−	2.13444i
$906$		0			0
$907$			51.7369i			1.71790i	0.512063	+	0.858948i	$0.328881\pi$
$907$			51.7369i			1.71790i	−0.512063	+	0.858948i	$0.671119\pi$
$908$		0			0
$909$	1.68484	+	2.91670i	0.0558826	+	0.0967407i
$910$		0			0
$911$	13.6637			0.452697			0.226348	−	0.974046i	$-0.427321\pi$
$911$	13.6637			0.452697			0.226348	+	0.974046i	$0.427321\pi$
$912$		0			0
$913$	93.6254			3.09855
$914$		0			0
$915$	−48.6908	+	13.0526i	−1.60967	+	0.431506i
$916$		0			0
$917$		−	34.3510i		−	1.13437i
$918$		0			0
$919$		−	9.71319i		−	0.320409i	−0.987084	−	0.160204i	$-0.948785\pi$
$919$		−	9.71319i		−	0.320409i	0.987084	−	0.160204i	$-0.0512153\pi$
$920$		0			0
$921$	22.0758	−	5.91788i	0.727422	−	0.195001i
$922$		0			0
$923$	24.5961			0.809590
$924$		0			0
$925$	5.11969			0.168334
$926$		0			0
$927$	−8.55189	−	14.8045i	−0.280881	−	0.486245i
$928$		0			0
$929$		−	21.4259i		−	0.702962i	−0.936195	−	0.351481i	$-0.885678\pi$
$929$		−	21.4259i		−	0.702962i	0.936195	−	0.351481i	$-0.114322\pi$
$930$		0			0
$931$		−	5.94370i		−	0.194797i
$932$		0			0
$933$	0.461951	+	1.72324i	0.0151236	+	0.0564163i
$934$		0			0
$935$	93.1902			3.04765
$936$		0			0
$937$	−8.02657			−0.262216			−0.131108	−	0.991368i	$-0.541854\pi$
$937$	−8.02657			−0.262216			−0.131108	+	0.991368i	$0.541854\pi$
$938$		0			0
$939$	4.47842	+	16.7061i	0.146148	+	0.545183i
$940$		0			0
$941$			48.7381i			1.58882i	0.607383	+	0.794409i	$0.292219\pi$
$941$			48.7381i			1.58882i	−0.607383	+	0.794409i	$0.707781\pi$
$942$		0			0
$943$		−	25.6480i		−	0.835214i
$944$		0			0
$945$	−31.7038	−	31.6822i	−1.03132	−	1.03062i
$946$		0			0
$947$	−19.6509			−0.638569			−0.319285	−	0.947659i	$-0.603443\pi$
$947$	−19.6509			−0.638569			−0.319285	+	0.947659i	$0.603443\pi$
$948$		0			0
$949$	−52.3162			−1.69826
$950$		0			0
$951$	−24.7797	+	6.64271i	−0.803536	+	0.215405i
$952$		0			0
$953$		−	4.62850i		−	0.149932i	−0.997186	−	0.0749659i	$-0.976115\pi$
$953$		−	4.62850i		−	0.149932i	0.997186	−	0.0749659i	$-0.0238848\pi$
$954$		0			0
$955$		−	25.7861i		−	0.834419i
$956$		0			0
$957$	56.5046	−	15.1472i	1.82653	−	0.489641i
$958$		0			0
$959$	−8.32178			−0.268724
$960$		0			0
$961$	29.1884			0.941561
$962$		0			0
$963$	−38.5424	+	22.2642i	−1.24201	+	0.717453i
$964$		0			0
$965$		−	19.6320i		−	0.631977i
$966$		0			0
$967$			24.1516i			0.776662i	0.921520	+	0.388331i	$0.126948\pi$
$967$			24.1516i			0.776662i	−0.921520	+	0.388331i	$0.873052\pi$
$968$		0			0
$969$	2.92475	+	10.9104i	0.0939566	+	0.350492i
$970$		0			0
$971$	−45.0342			−1.44522			−0.722609	−	0.691257i	$-0.757057\pi$
$971$	−45.0342			−1.44522			−0.722609	+	0.691257i	$0.757057\pi$
$972$		0			0
$973$	50.6817			1.62478
$974$		0			0
$975$	1.12278	+	4.18838i	0.0359579	+	0.134136i
$976$		0			0
$977$			33.5744i			1.07414i	0.843538	+	0.537070i	$0.180469\pi$
$977$			33.5744i			1.07414i	−0.843538	+	0.537070i	$0.819531\pi$
$978$		0			0
$979$			63.7001i			2.03586i
$980$		0			0
$981$	3.10552	−	1.79391i	0.0991515	−	0.0572752i
$982$		0			0
$983$	−47.2420			−1.50679			−0.753393	−	0.657570i	$-0.771584\pi$
$983$	−47.2420			−1.50679			−0.753393	+	0.657570i	$0.771584\pi$
$984$		0			0
$985$	33.0829			1.05411
$986$		0			0
$987$	−11.0517	+	2.96264i	−0.351780	+	0.0943020i
$988$		0			0
$989$		−	1.34226i		−	0.0426812i
$990$		0			0
$991$			29.2475i			0.929079i	0.885553	+	0.464539i	$0.153780\pi$
$991$			29.2475i			0.929079i	−0.885553	+	0.464539i	$0.846220\pi$
$992$		0			0
$993$	42.7819	−	11.4686i	1.35764	−	0.363944i
$994$		0			0
$995$	−16.2870			−0.516333
$996$		0			0
$997$	24.4175			0.773310			0.386655	−	0.922224i	$-0.373630\pi$
$997$	24.4175			0.773310			0.386655	+	0.922224i	$0.373630\pi$
$998$		0			0
$999$	25.1492	+	25.1321i	0.795686	+	0.795144i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.d.a.191.8	yes	12
3.2	odd	2	inner	912.2.d.a.191.6	yes	12
4.3	odd	2	inner	912.2.d.a.191.5	✓	12
12.11	even	2	inner	912.2.d.a.191.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.d.a.191.5	✓	12	4.3	odd	2	inner
912.2.d.a.191.6	yes	12	3.2	odd	2	inner
912.2.d.a.191.7	yes	12	12.11	even	2	inner
912.2.d.a.191.8	yes	12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.448478 + 1.67298i) q^{3} -2.39755i q^{5} +3.59774i q^{7} +(-2.59774 + 1.50059i) q^{9} -5.96012 q^{11} -3.34596 q^{13} +(4.01105 - 1.07525i) q^{15} -6.52151i q^{17} +1.00000i q^{19} +(-6.01894 + 1.61350i) q^{21} -3.33674 q^{23} -0.748228 q^{25} +(-3.67549 - 3.67298i) q^{27} +5.66680i q^{29} -1.34596i q^{31} +(-2.67298 - 9.97117i) q^{33} +8.62574 q^{35} -6.84242 q^{37} +(-1.50059 - 5.59774i) q^{39} +7.68654i q^{41} +0.402265i q^{43} +(3.59774 + 6.22819i) q^{45} +1.83615 q^{47} -5.94370 q^{49} +(10.9104 - 2.92475i) q^{51} +1.76866i q^{53} +14.2897i q^{55} +(-1.67298 + 0.448478i) q^{57} +6.22819 q^{59} -12.1392 q^{61} +(-5.39873 - 9.34596i) q^{63} +8.02210i q^{65} +0.300986i q^{67} +(-1.49646 - 5.58231i) q^{69} -7.35097 q^{71} +15.6356 q^{73} +(-0.335564 - 1.25177i) q^{75} -21.4429i q^{77} +2.80453i q^{79} +(4.49646 - 7.79627i) q^{81} -15.7086 q^{83} -15.6356 q^{85} +(-9.48045 + 2.54143i) q^{87} -10.6877i q^{89} -12.0379i q^{91} +(2.25177 - 0.603635i) q^{93} +2.39755 q^{95} -7.04498 q^{97} +(15.4828 - 8.94370i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{9} - 12 q^{21} + 4 q^{25} - 12 q^{33} - 16 q^{37} + 16 q^{45} - 4 q^{49} - 24 q^{61} + 8 q^{69} + 40 q^{73} + 28 q^{81} - 40 q^{85} + 40 q^{93} - 16 q^{97}+O(q^{100}) \) 12 * q - 4 * q^9 - 12 * q^21 + 4 * q^25 - 12 * q^33 - 16 * q^37 + 16 * q^45 - 4 * q^49 - 24 * q^61 + 8 * q^69 + 40 * q^73 + 28 * q^81 - 40 * q^85 + 40 * q^93 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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