LMFDB - Embedded newform 2312.2.b.j.577.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2312,2,Mod(577,2312)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2312.577"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2312.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,4,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$18.4614129473$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.1
Root			$-1.84776i$ of defining polynomial
Character	$\chi$	$=$	2312.577
Dual form			2312.2.b.j.577.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.84776i q^{3} -0.765367i q^{5} -4.46088i q^{7} -0.414214 q^{9} +1.84776i q^{11} -1.41421 q^{15} +6.24264 q^{19} -8.24264 q^{21} -4.90923i q^{23} +4.41421 q^{25} -4.77791i q^{27} -2.29610i q^{29} -5.54328i q^{31} +3.41421 q^{33} -3.41421 q^{35} +1.84776i q^{37} -9.68714i q^{41} +1.75736 q^{43} +0.317025i q^{45} -7.17157 q^{47} -12.8995 q^{49} -11.0711 q^{53} +1.41421 q^{55} -11.5349i q^{57} +11.8995 q^{59} +12.7486i q^{61} +1.84776i q^{63} +1.65685 q^{67} -9.07107 q^{69} +2.48181i q^{71} +14.0167i q^{73} -8.15640i q^{75} +8.24264 q^{77} +13.5684i q^{79} -10.0711 q^{81} +1.75736 q^{83} -4.24264 q^{87} -9.65685 q^{89} -10.2426 q^{93} -4.77791i q^{95} +2.03347i q^{97} -0.765367i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9} + 8 q^{19} - 16 q^{21} + 12 q^{25} + 8 q^{33} - 8 q^{35} + 24 q^{43} - 40 q^{47} - 12 q^{49} - 16 q^{53} + 8 q^{59} - 16 q^{67} - 8 q^{69} + 16 q^{77} - 12 q^{81} + 24 q^{83} - 16 q^{89} - 24 q^{93}+O(q^{100}) $ 4 * q + 4 * q^9 + 8 * q^19 - 16 * q^21 + 12 * q^25 + 8 * q^33 - 8 * q^35 + 24 * q^43 - 40 * q^47 - 12 * q^49 - 16 * q^53 + 8 * q^59 - 16 * q^67 - 8 * q^69 + 16 * q^77 - 12 * q^81 + 24 * q^83 - 16 * q^89 - 24 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times$.

$n$	$1157$	$1735$	$1737$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.84776i	− 1.06680i	−0.845862	−	0.533402i	$-0.820913\pi$
$3$	− 1.84776i	− 1.06680i	0.845862	−	0.533402i	$-0.179087\pi$
$4$	0	0
$5$	− 0.765367i	− 0.342282i	−0.985247	−	0.171141i	$-0.945255\pi$
$5$	− 0.765367i	− 0.342282i	0.985247	−	0.171141i	$-0.0547454\pi$
$6$	0	0
$7$	− 4.46088i	− 1.68606i	−0.537870	−	0.843028i	$-0.680771\pi$
$7$	− 4.46088i	− 1.68606i	0.537870	−	0.843028i	$-0.319229\pi$
$8$	0	0
$9$	−0.414214	−0.138071
$10$	0	0
$11$	1.84776i	0.557120i	0.960419	+	0.278560i	$0.0898572\pi$
$11$	1.84776i	0.557120i	−0.960419	+	0.278560i	$0.910143\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−1.41421	−0.365148
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	6.24264	1.43216	0.716080	−	0.698018i	$-0.245935\pi$
$19$	6.24264	1.43216	0.716080	+	0.698018i	$0.245935\pi$
$20$	0	0
$21$	−8.24264	−1.79869
$22$	0	0
$23$	− 4.90923i	− 1.02364i	−0.859091	−	0.511822i	$-0.828971\pi$
$23$	− 4.90923i	− 1.02364i	0.859091	−	0.511822i	$-0.171029\pi$
$24$	0	0
$25$	4.41421	0.882843
$26$	0	0
$27$	− 4.77791i	− 0.919509i
$28$	0	0
$29$	− 2.29610i	− 0.426375i	−0.977011	−	0.213188i	$-0.931615\pi$
$29$	− 2.29610i	− 0.426375i	0.977011	−	0.213188i	$-0.0683845\pi$
$30$	0	0
$31$	− 5.54328i	− 0.995602i	−0.867291	−	0.497801i	$-0.834141\pi$
$31$	− 5.54328i	− 0.995602i	0.867291	−	0.497801i	$-0.165859\pi$
$32$	0	0
$33$	3.41421	0.594338
$34$	0	0
$35$	−3.41421	−0.577107
$36$	0	0
$37$	1.84776i	0.303770i	0.988398	+	0.151885i	$0.0485343\pi$
$37$	1.84776i	0.303770i	−0.988398	+	0.151885i	$0.951466\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 9.68714i	− 1.51288i	−0.654065	−	0.756438i	$-0.726938\pi$
$41$	− 9.68714i	− 1.51288i	0.654065	−	0.756438i	$-0.273062\pi$
$42$	0	0
$43$	1.75736	0.267995	0.133997	−	0.990982i	$-0.457219\pi$
$43$	1.75736	0.267995	0.133997	+	0.990982i	$0.457219\pi$
$44$	0	0
$45$	0.317025i	0.0472593i
$46$	0	0
$47$	−7.17157	−1.04608	−0.523041	−	0.852308i	$-0.675202\pi$
$47$	−7.17157	−1.04608	−0.523041	+	0.852308i	$0.675202\pi$
$48$	0	0
$49$	−12.8995	−1.84278
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−11.0711	−1.52073	−0.760364	−	0.649497i	$-0.774979\pi$
$53$	−11.0711	−1.52073	−0.760364	+	0.649497i	$0.774979\pi$
$54$	0	0
$55$	1.41421	0.190693
$56$	0	0
$57$	− 11.5349i	− 1.52783i
$58$	0	0
$59$	11.8995	1.54918	0.774591	−	0.632462i	$-0.217956\pi$
$59$	11.8995	1.54918	0.774591	+	0.632462i	$0.217956\pi$
$60$	0	0
$61$	12.7486i	1.63229i	0.577846	+	0.816146i	$0.303893\pi$
$61$	12.7486i	1.63229i	−0.577846	+	0.816146i	$0.696107\pi$
$62$	0	0
$63$	1.84776i	0.232796i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.65685	0.202417	0.101208	−	0.994865i	$-0.467729\pi$
$67$	1.65685	0.202417	0.101208	+	0.994865i	$0.467729\pi$
$68$	0	0
$69$	−9.07107	−1.09203
$70$	0	0
$71$	2.48181i	0.294537i	0.989097	+	0.147268i	$0.0470481\pi$
$71$	2.48181i	0.294537i	−0.989097	+	0.147268i	$0.952952\pi$
$72$	0	0
$73$	14.0167i	1.64053i	0.571983	+	0.820266i	$0.306174\pi$
$73$	14.0167i	1.64053i	−0.571983	+	0.820266i	$0.693826\pi$
$74$	0	0
$75$	− 8.15640i	− 0.941820i
$76$	0	0
$77$	8.24264	0.939336
$78$	0	0
$79$	13.5684i	1.52656i	0.646068	+	0.763280i	$0.276412\pi$
$79$	13.5684i	1.52656i	−0.646068	+	0.763280i	$0.723588\pi$
$80$	0	0
$81$	−10.0711	−1.11901
$82$	0	0
$83$	1.75736	0.192895	0.0964476	−	0.995338i	$-0.469252\pi$
$83$	1.75736	0.192895	0.0964476	+	0.995338i	$0.469252\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.24264	−0.454859
$88$	0	0
$89$	−9.65685	−1.02362	−0.511812	−	0.859097i	$-0.671026\pi$
$89$	−9.65685	−1.02362	−0.511812	+	0.859097i	$0.671026\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−10.2426	−1.06211
$94$	0	0
$95$	− 4.77791i	− 0.490203i
$96$	0	0
$97$	2.03347i	0.206467i	0.994657	+	0.103234i	$0.0329190\pi$
$97$	2.03347i	0.206467i	−0.994657	+	0.103234i	$0.967081\pi$
$98$	0	0
$99$	− 0.765367i	− 0.0769223i
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	−13.6569	−1.34565	−0.672825	−	0.739802i	$-0.734919\pi$
$103$	−13.6569	−1.34565	−0.672825	+	0.739802i	$0.734919\pi$
$104$	0	0
$105$	6.30864i	0.615661i
$106$	0	0
$107$	1.66205i	0.160676i	0.996768	+	0.0803382i	$0.0256000\pi$
$107$	1.66205i	0.160676i	−0.996768	+	0.0803382i	$0.974400\pi$
$108$	0	0
$109$	12.4860i	1.19594i	0.801519	+	0.597970i	$0.204026\pi$
$109$	12.4860i	1.19594i	−0.801519	+	0.597970i	$0.795974\pi$
$110$	0	0
$111$	3.41421	0.324063
$112$	0	0
$113$	− 1.21371i	− 0.114176i	−0.998369	−	0.0570880i	$-0.981818\pi$
$113$	− 1.21371i	− 0.114176i	0.998369	−	0.0570880i	$-0.0181816\pi$
$114$	0	0
$115$	−3.75736	−0.350376
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.58579	0.689617
$122$	0	0
$123$	−17.8995	−1.61394
$124$	0	0
$125$	− 7.20533i	− 0.644464i
$126$	0	0
$127$	10.2426	0.908887	0.454444	−	0.890775i	$-0.349838\pi$
$127$	10.2426	0.908887	0.454444	+	0.890775i	$0.349838\pi$
$128$	0	0
$129$	− 3.24718i	− 0.285898i
$130$	0	0
$131$	− 1.66205i	− 0.145214i	−0.997361	−	0.0726070i	$-0.976868\pi$
$131$	− 1.66205i	− 0.145214i	0.997361	−	0.0726070i	$-0.0231319\pi$
$132$	0	0
$133$	− 27.8477i	− 2.41470i
$134$	0	0
$135$	−3.65685	−0.314732
$136$	0	0
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	− 19.6913i	− 1.67019i	−0.550103	−	0.835097i	$-0.685411\pi$
$139$	− 19.6913i	− 1.67019i	0.550103	−	0.835097i	$-0.314589\pi$
$140$	0	0
$141$	13.2513i	1.11596i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1.75736	−0.145941
$146$	0	0
$147$	23.8352i	1.96589i
$148$	0	0
$149$	17.6569	1.44651	0.723253	−	0.690583i	$-0.242646\pi$
$149$	17.6569	1.44651	0.723253	+	0.690583i	$0.242646\pi$
$150$	0	0
$151$	0.100505	0.00817899	0.00408949	−	0.999992i	$-0.498698\pi$
$151$	0.100505	0.00817899	0.00408949	+	0.999992i	$0.498698\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.24264	−0.340777
$156$	0	0
$157$	−7.31371	−0.583697	−0.291849	−	0.956464i	$-0.594270\pi$
$157$	−7.31371	−0.583697	−0.291849	+	0.956464i	$0.594270\pi$
$158$	0	0
$159$	20.4567i	1.62232i
$160$	0	0
$161$	−21.8995	−1.72592
$162$	0	0
$163$	14.9134i	1.16811i	0.811715	+	0.584053i	$0.198534\pi$
$163$	14.9134i	1.16811i	−0.811715	+	0.584053i	$0.801466\pi$
$164$	0	0
$165$	− 2.61313i	− 0.203432i
$166$	0	0
$167$	1.84776i	0.142984i	0.997441	+	0.0714919i	$0.0227760\pi$
$167$	1.84776i	0.142984i	−0.997441	+	0.0714919i	$0.977224\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	−2.58579	−0.197740
$172$	0	0
$173$	− 24.6549i	− 1.87448i	−0.348687	−	0.937239i	$-0.613372\pi$
$173$	− 24.6549i	− 1.87448i	0.348687	−	0.937239i	$-0.386628\pi$
$174$	0	0
$175$	− 19.6913i	− 1.48852i
$176$	0	0
$177$	− 21.9874i	− 1.65267i
$178$	0	0
$179$	9.75736	0.729299	0.364650	−	0.931145i	$-0.381189\pi$
$179$	9.75736	0.729299	0.364650	+	0.931145i	$0.381189\pi$
$180$	0	0
$181$	− 1.21371i	− 0.0902142i	−0.998982	−	0.0451071i	$-0.985637\pi$
$181$	− 1.21371i	− 0.0902142i	0.998982	−	0.0451071i	$-0.0143629\pi$
$182$	0	0
$183$	23.5563	1.74134
$184$	0	0
$185$	1.41421	0.103975
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−21.3137	−1.55034
$190$	0	0
$191$	−3.17157	−0.229487	−0.114743	−	0.993395i	$-0.536605\pi$
$191$	−3.17157	−0.229487	−0.114743	+	0.993395i	$0.536605\pi$
$192$	0	0
$193$	− 1.84776i	− 0.133005i	−0.997786	−	0.0665023i	$-0.978816\pi$
$193$	− 1.84776i	− 0.133005i	0.997786	−	0.0665023i	$-0.0211840\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.56420i	0.253939i	0.991907	+	0.126969i	$0.0405250\pi$
$197$	3.56420i	0.253939i	−0.991907	+	0.126969i	$0.959475\pi$
$198$	0	0
$199$	13.6453i	0.967289i	0.875265	+	0.483644i	$0.160687\pi$
$199$	13.6453i	0.967289i	−0.875265	+	0.483644i	$0.839313\pi$
$200$	0	0
$201$	− 3.06147i	− 0.215939i
$202$	0	0
$203$	−10.2426	−0.718892
$204$	0	0
$205$	−7.41421	−0.517831
$206$	0	0
$207$	2.03347i	0.141336i
$208$	0	0
$209$	11.5349i	0.797885i
$210$	0	0
$211$	15.1760i	1.04476i	0.852713	+	0.522380i	$0.174956\pi$
$211$	15.1760i	1.04476i	−0.852713	+	0.522380i	$0.825044\pi$
$212$	0	0
$213$	4.58579	0.314213
$214$	0	0
$215$	− 1.34502i	− 0.0917299i
$216$	0	0
$217$	−24.7279	−1.67864
$218$	0	0
$219$	25.8995	1.75013
$220$	0	0
$221$	0	0
$222$	0	0
$223$	13.5563	0.907800	0.453900	−	0.891053i	$-0.350032\pi$
$223$	13.5563	0.907800	0.453900	+	0.891053i	$0.350032\pi$
$224$	0	0
$225$	−1.82843	−0.121895
$226$	0	0
$227$	− 12.3003i	− 0.816397i	−0.912893	−	0.408199i	$-0.866157\pi$
$227$	− 12.3003i	− 0.816397i	0.912893	−	0.408199i	$-0.133843\pi$
$228$	0	0
$229$	−9.89949	−0.654177	−0.327089	−	0.944994i	$-0.606068\pi$
$229$	−9.89949	−0.654177	−0.327089	+	0.944994i	$0.606068\pi$
$230$	0	0
$231$	− 15.2304i	− 1.00209i
$232$	0	0
$233$	− 17.0782i	− 1.11883i	−0.828888	−	0.559414i	$-0.811026\pi$
$233$	− 17.0782i	− 1.11883i	0.828888	−	0.559414i	$-0.188974\pi$
$234$	0	0
$235$	5.48888i	0.358055i
$236$	0	0
$237$	25.0711	1.62854
$238$	0	0
$239$	−11.3137	−0.731823	−0.365911	−	0.930650i	$-0.619243\pi$
$239$	−11.3137	−0.731823	−0.365911	+	0.930650i	$0.619243\pi$
$240$	0	0
$241$	7.97069i	0.513438i	0.966486	+	0.256719i	$0.0826414\pi$
$241$	7.97069i	0.513438i	−0.966486	+	0.256719i	$0.917359\pi$
$242$	0	0
$243$	4.27518i	0.274253i
$244$	0	0
$245$	9.87285i	0.630753i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 3.24718i	− 0.205781i
$250$	0	0
$251$	4.82843	0.304768	0.152384	−	0.988321i	$-0.451305\pi$
$251$	4.82843	0.304768	0.152384	+	0.988321i	$0.451305\pi$
$252$	0	0
$253$	9.07107	0.570293
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.242641	0.0151355	0.00756776	−	0.999971i	$-0.497591\pi$
$257$	0.242641	0.0151355	0.00756776	+	0.999971i	$0.497591\pi$
$258$	0	0
$259$	8.24264	0.512173
$260$	0	0
$261$	0.951076i	0.0588701i
$262$	0	0
$263$	15.8995	0.980405	0.490202	−	0.871609i	$-0.336923\pi$
$263$	15.8995	0.980405	0.490202	+	0.871609i	$0.336923\pi$
$264$	0	0
$265$	8.47343i	0.520519i
$266$	0	0
$267$	17.8435i	1.09201i
$268$	0	0
$269$	− 7.97069i	− 0.485982i	−0.970029	−	0.242991i	$-0.921871\pi$
$269$	− 7.97069i	− 0.485982i	0.970029	−	0.242991i	$-0.0781285\pi$
$270$	0	0
$271$	−11.3137	−0.687259	−0.343629	−	0.939105i	$-0.611656\pi$
$271$	−11.3137	−0.687259	−0.343629	+	0.939105i	$0.611656\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.15640i	0.491850i
$276$	0	0
$277$	− 3.82683i	− 0.229932i	−0.993369	−	0.114966i	$-0.963324\pi$
$277$	− 3.82683i	− 0.229932i	0.993369	−	0.114966i	$-0.0366759\pi$
$278$	0	0
$279$	2.29610i	0.137464i
$280$	0	0
$281$	−28.7279	−1.71376	−0.856882	−	0.515512i	$-0.827602\pi$
$281$	−28.7279	−1.71376	−0.856882	+	0.515512i	$0.827602\pi$
$282$	0	0
$283$	− 21.5935i	− 1.28360i	−0.766873	−	0.641799i	$-0.778188\pi$
$283$	− 21.5935i	− 1.28360i	0.766873	−	0.641799i	$-0.221812\pi$
$284$	0	0
$285$	−8.82843	−0.522951
$286$	0	0
$287$	−43.2132	−2.55079
$288$	0	0
$289$	0	0
$290$	0	0
$291$	3.75736	0.220260
$292$	0	0
$293$	15.3137	0.894636	0.447318	−	0.894375i	$-0.352379\pi$
$293$	15.3137	0.894636	0.447318	+	0.894375i	$0.352379\pi$
$294$	0	0
$295$	− 9.10748i	− 0.530258i
$296$	0	0
$297$	8.82843	0.512277
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 7.83938i	− 0.451854i
$302$	0	0
$303$	18.4776i	1.06151i
$304$	0	0
$305$	9.75736	0.558705
$306$	0	0
$307$	18.6274	1.06312	0.531561	−	0.847020i	$-0.321605\pi$
$307$	18.6274	1.06312	0.531561	+	0.847020i	$0.321605\pi$
$308$	0	0
$309$	25.2346i	1.43554i
$310$	0	0
$311$	5.99162i	0.339754i	0.985465	+	0.169877i	$0.0543370\pi$
$311$	5.99162i	0.339754i	−0.985465	+	0.169877i	$0.945663\pi$
$312$	0	0
$313$	15.5474i	0.878793i	0.898293	+	0.439397i	$0.144808\pi$
$313$	15.5474i	0.878793i	−0.898293	+	0.439397i	$0.855192\pi$
$314$	0	0
$315$	1.41421	0.0796819
$316$	0	0
$317$	− 17.2639i	− 0.969636i	−0.874615	−	0.484818i	$-0.838886\pi$
$317$	− 17.2639i	− 0.969636i	0.874615	−	0.484818i	$-0.161114\pi$
$318$	0	0
$319$	4.24264	0.237542
$320$	0	0
$321$	3.07107	0.171410
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	23.0711	1.27583
$328$	0	0
$329$	31.9916i	1.76375i
$330$	0	0
$331$	22.2426	1.22257	0.611283	−	0.791412i	$-0.290654\pi$
$331$	22.2426	1.22257	0.611283	+	0.791412i	$0.290654\pi$
$332$	0	0
$333$	− 0.765367i	− 0.0419418i
$334$	0	0
$335$	− 1.26810i	− 0.0692838i
$336$	0	0
$337$	4.27518i	0.232884i	0.993198	+	0.116442i	$0.0371489\pi$
$337$	4.27518i	0.232884i	−0.993198	+	0.116442i	$0.962851\pi$
$338$	0	0
$339$	−2.24264	−0.121804
$340$	0	0
$341$	10.2426	0.554670
$342$	0	0
$343$	26.3170i	1.42098i
$344$	0	0
$345$	6.94269i	0.373782i
$346$	0	0
$347$	− 19.2430i	− 1.03302i	−0.856282	−	0.516508i	$-0.827232\pi$
$347$	− 19.2430i	− 1.03302i	0.856282	−	0.516508i	$-0.172768\pi$
$348$	0	0
$349$	−2.10051	−0.112437	−0.0562187	−	0.998418i	$-0.517904\pi$
$349$	−2.10051	−0.112437	−0.0562187	+	0.998418i	$0.517904\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.97056	−0.477455	−0.238727	−	0.971087i	$-0.576730\pi$
$353$	−8.97056	−0.477455	−0.238727	+	0.971087i	$0.576730\pi$
$354$	0	0
$355$	1.89949	0.100815
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.7574	−0.726086	−0.363043	−	0.931772i	$-0.618262\pi$
$359$	−13.7574	−0.726086	−0.363043	+	0.931772i	$0.618262\pi$
$360$	0	0
$361$	19.9706	1.05108
$362$	0	0
$363$	− 14.0167i	− 0.735686i
$364$	0	0
$365$	10.7279	0.561525
$366$	0	0
$367$	− 20.7737i	− 1.08438i	−0.840257	−	0.542189i	$-0.817596\pi$
$367$	− 20.7737i	− 1.08438i	0.840257	−	0.542189i	$-0.182404\pi$
$368$	0	0
$369$	4.01254i	0.208885i
$370$	0	0
$371$	49.3868i	2.56403i
$372$	0	0
$373$	31.9411	1.65385	0.826924	−	0.562313i	$-0.190088\pi$
$373$	31.9411	1.65385	0.826924	+	0.562313i	$0.190088\pi$
$374$	0	0
$375$	−13.3137	−0.687517
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 10.5838i	− 0.543654i	−0.962346	−	0.271827i	$-0.912372\pi$
$379$	− 10.5838i	− 0.543654i	0.962346	−	0.271827i	$-0.0876279\pi$
$380$	0	0
$381$	− 18.9259i	− 0.969605i
$382$	0	0
$383$	14.7279	0.752562	0.376281	−	0.926506i	$-0.377203\pi$
$383$	14.7279	0.752562	0.376281	+	0.926506i	$0.377203\pi$
$384$	0	0
$385$	− 6.30864i	− 0.321518i
$386$	0	0
$387$	−0.727922	−0.0370024
$388$	0	0
$389$	−3.27208	−0.165901	−0.0829505	−	0.996554i	$-0.526434\pi$
$389$	−3.27208	−0.165901	−0.0829505	+	0.996554i	$0.526434\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−3.07107	−0.154915
$394$	0	0
$395$	10.3848	0.522515
$396$	0	0
$397$	− 7.44543i	− 0.373675i	−0.982391	−	0.186838i	$-0.940176\pi$
$397$	− 7.44543i	− 0.373675i	0.982391	−	0.186838i	$-0.0598238\pi$
$398$	0	0
$399$	−51.4558	−2.57601
$400$	0	0
$401$	− 2.55873i	− 0.127777i	−0.997957	−	0.0638885i	$-0.979650\pi$
$401$	− 2.55873i	− 0.127777i	0.997957	−	0.0638885i	$-0.0203502\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	7.70806i	0.383017i
$406$	0	0
$407$	−3.41421	−0.169236
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	− 25.8686i	− 1.27601i
$412$	0	0
$413$	− 53.0823i	− 2.61201i
$414$	0	0
$415$	− 1.34502i	− 0.0660247i
$416$	0	0
$417$	−36.3848	−1.78177
$418$	0	0
$419$	4.90923i	0.239831i	0.992784	+	0.119916i	$0.0382624\pi$
$419$	4.90923i	0.239831i	−0.992784	+	0.119916i	$0.961738\pi$
$420$	0	0
$421$	−11.3137	−0.551396	−0.275698	−	0.961244i	$-0.588909\pi$
$421$	−11.3137	−0.551396	−0.275698	+	0.961244i	$0.588909\pi$
$422$	0	0
$423$	2.97056	0.144434
$424$	0	0
$425$	0	0
$426$	0	0
$427$	56.8701	2.75213
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.54328i	0.267010i	0.991048	+	0.133505i	$0.0426233\pi$
$431$	5.54328i	0.267010i	−0.991048	+	0.133505i	$0.957377\pi$
$432$	0	0
$433$	−2.58579	−0.124265	−0.0621325	−	0.998068i	$-0.519790\pi$
$433$	−2.58579	−0.124265	−0.0621325	+	0.998068i	$0.519790\pi$
$434$	0	0
$435$	3.24718i	0.155690i
$436$	0	0
$437$	− 30.6465i	− 1.46602i
$438$	0	0
$439$	− 19.0572i	− 0.909553i	−0.890606	−	0.454776i	$-0.849719\pi$
$439$	− 19.0572i	− 0.909553i	0.890606	−	0.454776i	$-0.150281\pi$
$440$	0	0
$441$	5.34315	0.254436
$442$	0	0
$443$	16.2843	0.773689	0.386845	−	0.922145i	$-0.373565\pi$
$443$	16.2843	0.773689	0.386845	+	0.922145i	$0.373565\pi$
$444$	0	0
$445$	7.39104i	0.350369i
$446$	0	0
$447$	− 32.6256i	− 1.54314i
$448$	0	0
$449$	24.7318i	1.16717i	0.812053	+	0.583584i	$0.198350\pi$
$449$	24.7318i	1.16717i	−0.812053	+	0.583584i	$0.801650\pi$
$450$	0	0
$451$	17.8995	0.842854
$452$	0	0
$453$	− 0.185709i	− 0.00872538i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.0711	0.892107	0.446053	−	0.895006i	$-0.352829\pi$
$457$	19.0711	0.892107	0.446053	+	0.895006i	$0.352829\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	27.5563	1.28343	0.641714	−	0.766944i	$-0.278224\pi$
$461$	27.5563	1.28343	0.641714	+	0.766944i	$0.278224\pi$
$462$	0	0
$463$	−4.14214	−0.192501	−0.0962507	−	0.995357i	$-0.530685\pi$
$463$	−4.14214	−0.192501	−0.0962507	+	0.995357i	$0.530685\pi$
$464$	0	0
$465$	7.83938i	0.363542i
$466$	0	0
$467$	17.5563	0.812411	0.406205	−	0.913782i	$-0.366852\pi$
$467$	17.5563	0.812411	0.406205	+	0.913782i	$0.366852\pi$
$468$	0	0
$469$	− 7.39104i	− 0.341286i
$470$	0	0
$471$	13.5140i	0.622691i
$472$	0	0
$473$	3.24718i	0.149305i
$474$	0	0
$475$	27.5563	1.26437
$476$	0	0
$477$	4.58579	0.209969
$478$	0	0
$479$	8.07948i	0.369161i	0.982817	+	0.184580i	$0.0590926\pi$
$479$	8.07948i	0.369161i	−0.982817	+	0.184580i	$0.940907\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	40.4650i	1.84122i
$484$	0	0
$485$	1.55635	0.0706702
$486$	0	0
$487$	− 42.4985i	− 1.92579i	−0.269878	−	0.962894i	$-0.586983\pi$
$487$	− 42.4985i	− 1.92579i	0.269878	−	0.962894i	$-0.413017\pi$
$488$	0	0
$489$	27.5563	1.24614
$490$	0	0
$491$	16.3848	0.739435	0.369717	−	0.929144i	$-0.379455\pi$
$491$	16.3848	0.739435	0.369717	+	0.929144i	$0.379455\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−0.585786	−0.0263291
$496$	0	0
$497$	11.0711	0.496605
$498$	0	0
$499$	22.1187i	0.990170i	0.868845	+	0.495085i	$0.164863\pi$
$499$	22.1187i	0.990170i	−0.868845	+	0.495085i	$0.835137\pi$
$500$	0	0
$501$	3.41421	0.152536
$502$	0	0
$503$	4.46088i	0.198901i	0.995043	+	0.0994505i	$0.0317085\pi$
$503$	4.46088i	0.198901i	−0.995043	+	0.0994505i	$0.968292\pi$
$504$	0	0
$505$	7.65367i	0.340584i
$506$	0	0
$507$	24.0209i	1.06680i
$508$	0	0
$509$	−18.2843	−0.810436	−0.405218	−	0.914220i	$-0.632804\pi$
$509$	−18.2843	−0.810436	−0.405218	+	0.914220i	$0.632804\pi$
$510$	0	0
$511$	62.5269	2.76603
$512$	0	0
$513$	− 29.8268i	− 1.31688i
$514$	0	0
$515$	10.4525i	0.460592i
$516$	0	0
$517$	− 13.2513i	− 0.582793i
$518$	0	0
$519$	−45.5563	−1.99970
$520$	0	0
$521$	9.76406i	0.427771i	0.976859	+	0.213886i	$0.0686120\pi$
$521$	9.76406i	0.427771i	−0.976859	+	0.213886i	$0.931388\pi$
$522$	0	0
$523$	24.1421	1.05566	0.527831	−	0.849349i	$-0.323005\pi$
$523$	24.1421	1.05566	0.527831	+	0.849349i	$0.323005\pi$
$524$	0	0
$525$	−36.3848	−1.58796
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−1.10051	−0.0478480
$530$	0	0
$531$	−4.92893	−0.213897
$532$	0	0
$533$	0	0
$534$	0	0
$535$	1.27208	0.0549967
$536$	0	0
$537$	− 18.0292i	− 0.778020i
$538$	0	0
$539$	− 23.8352i	− 1.02665i
$540$	0	0
$541$	− 0.0543929i	− 0.00233854i	−0.999999	−	0.00116927i	$-0.999628\pi$
$541$	− 0.0543929i	− 0.00233854i	0.999999	−	0.00116927i	$-0.000372190\pi$
$542$	0	0
$543$	−2.24264	−0.0962409
$544$	0	0
$545$	9.55635	0.409349
$546$	0	0
$547$	16.5210i	0.706389i	0.935550	+	0.353194i	$0.114905\pi$
$547$	16.5210i	0.706389i	−0.935550	+	0.353194i	$0.885095\pi$
$548$	0	0
$549$	− 5.28064i	− 0.225372i
$550$	0	0
$551$	− 14.3337i	− 0.610637i
$552$	0	0
$553$	60.5269	2.57387
$554$	0	0
$555$	− 2.61313i	− 0.110921i
$556$	0	0
$557$	16.0000	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−22.0416	−0.928944	−0.464472	−	0.885588i	$-0.653756\pi$
$563$	−22.0416	−0.928944	−0.464472	+	0.885588i	$0.653756\pi$
$564$	0	0
$565$	−0.928932	−0.0390805
$566$	0	0
$567$	44.9259i	1.88671i
$568$	0	0
$569$	−46.1838	−1.93612	−0.968062	−	0.250711i	$-0.919336\pi$
$569$	−46.1838	−1.93612	−0.968062	+	0.250711i	$0.919336\pi$
$570$	0	0
$571$	− 0.393949i	− 0.0164862i	−0.999966	−	0.00824312i	$-0.997376\pi$
$571$	− 0.393949i	− 0.0164862i	0.999966	−	0.00824312i	$-0.00262390\pi$
$572$	0	0
$573$	5.86030i	0.244818i
$574$	0	0
$575$	− 21.6704i	− 0.903717i
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	0	0
$579$	−3.41421	−0.141890
$580$	0	0
$581$	− 7.83938i	− 0.325232i
$582$	0	0
$583$	− 20.4567i	− 0.847229i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−44.8701	−1.85199	−0.925993	−	0.377541i	$-0.876770\pi$
$587$	−44.8701	−1.85199	−0.925993	+	0.377541i	$0.876770\pi$
$588$	0	0
$589$	− 34.6047i	− 1.42586i
$590$	0	0
$591$	6.58579	0.270903
$592$	0	0
$593$	37.6985	1.54809	0.774046	−	0.633130i	$-0.218230\pi$
$593$	37.6985	1.54809	0.774046	+	0.633130i	$0.218230\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	25.2132	1.03191
$598$	0	0
$599$	39.1716	1.60051	0.800254	−	0.599662i	$-0.204698\pi$
$599$	39.1716	1.60051	0.800254	+	0.599662i	$0.204698\pi$
$600$	0	0
$601$	10.3981i	0.424148i	0.977254	+	0.212074i	$0.0680218\pi$
$601$	10.3981i	0.424148i	−0.977254	+	0.212074i	$0.931978\pi$
$602$	0	0
$603$	−0.686292	−0.0279480
$604$	0	0
$605$	− 5.80591i	− 0.236044i
$606$	0	0
$607$	− 5.46635i	− 0.221872i	−0.993828	−	0.110936i	$-0.964615\pi$
$607$	− 5.46635i	− 0.221872i	0.993828	−	0.110936i	$-0.0353849\pi$
$608$	0	0
$609$	18.9259i	0.766917i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−6.68629	−0.270057	−0.135028	−	0.990842i	$-0.543113\pi$
$613$	−6.68629	−0.270057	−0.135028	+	0.990842i	$0.543113\pi$
$614$	0	0
$615$	13.6997i	0.552424i
$616$	0	0
$617$	18.8715i	0.759740i	0.925040	+	0.379870i	$0.124031\pi$
$617$	18.8715i	0.759740i	−0.925040	+	0.379870i	$0.875969\pi$
$618$	0	0
$619$	20.7737i	0.834965i	0.908685	+	0.417483i	$0.137088\pi$
$619$	20.7737i	0.834965i	−0.908685	+	0.417483i	$0.862912\pi$
$620$	0	0
$621$	−23.4558	−0.941251
$622$	0	0
$623$	43.0781i	1.72589i
$624$	0	0
$625$	16.5563	0.662254
$626$	0	0
$627$	21.3137	0.851188
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−27.4142	−1.09134	−0.545671	−	0.837999i	$-0.683725\pi$
$631$	−27.4142	−1.09134	−0.545671	+	0.837999i	$0.683725\pi$
$632$	0	0
$633$	28.0416	1.11455
$634$	0	0
$635$	− 7.83938i	− 0.311096i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 1.02800i	− 0.0406670i
$640$	0	0
$641$	39.5139i	1.56071i	0.625339	+	0.780353i	$0.284961\pi$
$641$	39.5139i	1.56071i	−0.625339	+	0.780353i	$0.715039\pi$
$642$	0	0
$643$	9.76406i	0.385057i	0.981291	+	0.192528i	$0.0616688\pi$
$643$	9.76406i	0.385057i	−0.981291	+	0.192528i	$0.938331\pi$
$644$	0	0
$645$	−2.48528	−0.0978579
$646$	0	0
$647$	−45.2548	−1.77915	−0.889576	−	0.456788i	$-0.849000\pi$
$647$	−45.2548	−1.77915	−0.889576	+	0.456788i	$0.849000\pi$
$648$	0	0
$649$	21.9874i	0.863081i
$650$	0	0
$651$	45.6912i	1.79078i
$652$	0	0
$653$	27.7933i	1.08764i	0.839203	+	0.543818i	$0.183022\pi$
$653$	27.7933i	1.08764i	−0.839203	+	0.543818i	$0.816978\pi$
$654$	0	0
$655$	−1.27208	−0.0497042
$656$	0	0
$657$	− 5.80591i	− 0.226510i
$658$	0	0
$659$	−50.7696	−1.97770	−0.988850	−	0.148912i	$-0.952423\pi$
$659$	−50.7696	−1.97770	−0.988850	+	0.148912i	$0.952423\pi$
$660$	0	0
$661$	29.4142	1.14408	0.572040	−	0.820226i	$-0.306152\pi$
$661$	29.4142	1.14408	0.572040	+	0.820226i	$0.306152\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−21.3137	−0.826510
$666$	0	0
$667$	−11.2721	−0.436457
$668$	0	0
$669$	− 25.0489i	− 0.968445i
$670$	0	0
$671$	−23.5563	−0.909383
$672$	0	0
$673$	10.9552i	0.422293i	0.977454	+	0.211147i	$0.0677198\pi$
$673$	10.9552i	0.422293i	−0.977454	+	0.211147i	$0.932280\pi$
$674$	0	0
$675$	− 21.0907i	− 0.811782i
$676$	0	0
$677$	45.5599i	1.75101i	0.483209	+	0.875505i	$0.339471\pi$
$677$	45.5599i	1.75101i	−0.483209	+	0.875505i	$0.660529\pi$
$678$	0	0
$679$	9.07107	0.348116
$680$	0	0
$681$	−22.7279	−0.870936
$682$	0	0
$683$	− 40.0710i	− 1.53328i	−0.642080	−	0.766638i	$-0.721928\pi$
$683$	− 40.0710i	− 1.53328i	0.642080	−	0.766638i	$-0.278072\pi$
$684$	0	0
$685$	− 10.7151i	− 0.409404i
$686$	0	0
$687$	18.2919i	0.697879i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 36.3755i	− 1.38379i	−0.721998	−	0.691895i	$-0.756776\pi$
$691$	− 36.3755i	− 1.38379i	0.721998	−	0.691895i	$-0.243224\pi$
$692$	0	0
$693$	−3.41421	−0.129695
$694$	0	0
$695$	−15.0711	−0.571678
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−31.5563	−1.19357
$700$	0	0
$701$	51.3137	1.93809	0.969046	−	0.246880i	$-0.0794054\pi$
$701$	51.3137	1.93809	0.969046	+	0.246880i	$0.0794054\pi$
$702$	0	0
$703$	11.5349i	0.435047i
$704$	0	0
$705$	10.1421	0.381975
$706$	0	0
$707$	44.6088i	1.67769i
$708$	0	0
$709$	41.3073i	1.55133i	0.631146	+	0.775664i	$0.282585\pi$
$709$	41.3073i	1.55133i	−0.631146	+	0.775664i	$0.717415\pi$
$710$	0	0
$711$	− 5.62020i	− 0.210774i
$712$	0	0
$713$	−27.2132	−1.01914
$714$	0	0
$715$	0	0
$716$	0	0
$717$	20.9050i	0.780712i
$718$	0	0
$719$	− 21.0363i	− 0.784522i	−0.919854	−	0.392261i	$-0.871693\pi$
$719$	− 21.0363i	− 0.784522i	0.919854	−	0.392261i	$-0.128307\pi$
$720$	0	0
$721$	60.9217i	2.26884i
$722$	0	0
$723$	14.7279	0.547737
$724$	0	0
$725$	− 10.1355i	− 0.376422i
$726$	0	0
$727$	−21.7990	−0.808480	−0.404240	−	0.914653i	$-0.632464\pi$
$727$	−21.7990	−0.808480	−0.404240	+	0.914653i	$0.632464\pi$
$728$	0	0
$729$	−22.3137	−0.826434
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−0.727922	−0.0268864	−0.0134432	−	0.999910i	$-0.504279\pi$
$733$	−0.727922	−0.0268864	−0.0134432	+	0.999910i	$0.504279\pi$
$734$	0	0
$735$	18.2426	0.672890
$736$	0	0
$737$	3.06147i	0.112771i
$738$	0	0
$739$	−5.07107	−0.186542	−0.0932711	−	0.995641i	$-0.529732\pi$
$739$	−5.07107	−0.186542	−0.0932711	+	0.995641i	$0.529732\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 0.393949i	− 0.0144526i	−0.999974	−	0.00722629i	$-0.997700\pi$
$743$	− 0.393949i	− 0.0144526i	0.999974	−	0.00722629i	$-0.00230022\pi$
$744$	0	0
$745$	− 13.5140i	− 0.495114i
$746$	0	0
$747$	−0.727922	−0.0266333
$748$	0	0
$749$	7.41421	0.270909
$750$	0	0
$751$	25.1802i	0.918838i	0.888220	+	0.459419i	$0.151942\pi$
$751$	25.1802i	0.918838i	−0.888220	+	0.459419i	$0.848058\pi$
$752$	0	0
$753$	− 8.92177i	− 0.325127i
$754$	0	0
$755$	− 0.0769232i	− 0.00279952i
$756$	0	0
$757$	33.8995	1.23210	0.616049	−	0.787708i	$-0.288732\pi$
$757$	33.8995	1.23210	0.616049	+	0.787708i	$0.288732\pi$
$758$	0	0
$759$	− 16.7611i	− 0.608391i
$760$	0	0
$761$	36.9706	1.34018	0.670091	−	0.742279i	$-0.266255\pi$
$761$	36.9706	1.34018	0.670091	+	0.742279i	$0.266255\pi$
$762$	0	0
$763$	55.6985	2.01642
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−35.5980	−1.28370	−0.641848	−	0.766832i	$-0.721832\pi$
$769$	−35.5980	−1.28370	−0.641848	+	0.766832i	$0.721832\pi$
$770$	0	0
$771$	− 0.448342i	− 0.0161466i
$772$	0	0
$773$	47.0711	1.69303	0.846514	−	0.532366i	$-0.178697\pi$
$773$	47.0711	1.69303	0.846514	+	0.532366i	$0.178697\pi$
$774$	0	0
$775$	− 24.4692i	− 0.878960i
$776$	0	0
$777$	− 15.2304i	− 0.546388i
$778$	0	0
$779$	− 60.4733i	− 2.16668i
$780$	0	0
$781$	−4.58579	−0.164092
$782$	0	0
$783$	−10.9706	−0.392056
$784$	0	0
$785$	5.59767i	0.199789i
$786$	0	0
$787$	42.9468i	1.53089i	0.643502	+	0.765444i	$0.277481\pi$
$787$	42.9468i	1.53089i	−0.643502	+	0.765444i	$0.722519\pi$
$788$	0	0
$789$	− 29.3784i	− 1.04590i
$790$	0	0
$791$	−5.41421	−0.192507
$792$	0	0
$793$	0	0
$794$	0	0
$795$	15.6569	0.555291
$796$	0	0
$797$	−10.5858	−0.374968	−0.187484	−	0.982268i	$-0.560033\pi$
$797$	−10.5858	−0.374968	−0.187484	+	0.982268i	$0.560033\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	4.00000	0.141333
$802$	0	0
$803$	−25.8995	−0.913973
$804$	0	0
$805$	16.7611i	0.590753i
$806$	0	0
$807$	−14.7279	−0.518447
$808$	0	0
$809$	− 2.03347i	− 0.0714929i	−0.999361	−	0.0357465i	$-0.988619\pi$
$809$	− 2.03347i	− 0.0714929i	0.999361	−	0.0357465i	$-0.0113809\pi$
$810$	0	0
$811$	3.93562i	0.138198i	0.997610	+	0.0690992i	$0.0220125\pi$
$811$	3.93562i	0.138198i	−0.997610	+	0.0690992i	$0.977988\pi$
$812$	0	0
$813$	20.9050i	0.733171i
$814$	0	0
$815$	11.4142	0.399822
$816$	0	0
$817$	10.9706	0.383811
$818$	0	0
$819$	0	0
$820$	0	0
$821$	45.8995i	1.60190i	0.598729	+	0.800951i	$0.295673\pi$
$821$	45.8995i	1.60190i	−0.598729	+	0.800951i	$0.704327\pi$
$822$	0	0
$823$	17.7122i	0.617409i	0.951158	+	0.308705i	$0.0998955\pi$
$823$	17.7122i	0.617409i	−0.951158	+	0.308705i	$0.900105\pi$
$824$	0	0
$825$	15.0711	0.524707
$826$	0	0
$827$	− 1.21371i	− 0.0422048i	−0.999777	−	0.0211024i	$-0.993282\pi$
$827$	− 1.21371i	− 0.0422048i	0.999777	−	0.0211024i	$-0.00671760\pi$
$828$	0	0
$829$	13.9411	0.484195	0.242098	−	0.970252i	$-0.422165\pi$
$829$	13.9411	0.484195	0.242098	+	0.970252i	$0.422165\pi$
$830$	0	0
$831$	−7.07107	−0.245293
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.41421	0.0489409
$836$	0	0
$837$	−26.4853	−0.915465
$838$	0	0
$839$	0.688444i	0.0237677i	0.999929	+	0.0118839i	$0.00378284\pi$
$839$	0.688444i	0.0237677i	−0.999929	+	0.0118839i	$0.996217\pi$
$840$	0	0
$841$	23.7279	0.818204
$842$	0	0
$843$	53.0823i	1.82825i
$844$	0	0
$845$	9.94977i	0.342282i
$846$	0	0
$847$	− 33.8393i	− 1.16273i
$848$	0	0
$849$	−39.8995	−1.36935
$850$	0	0
$851$	9.07107	0.310952
$852$	0	0
$853$	− 17.2639i	− 0.591104i	−0.955327	−	0.295552i	$-0.904496\pi$
$853$	− 17.2639i	− 0.591104i	0.955327	−	0.295552i	$-0.0955036\pi$
$854$	0	0
$855$	1.97908i	0.0676829i
$856$	0	0
$857$	− 14.0167i	− 0.478802i	−0.970921	−	0.239401i	$-0.923049\pi$
$857$	− 14.0167i	− 0.478802i	0.970921	−	0.239401i	$-0.0769510\pi$
$858$	0	0
$859$	6.04163	0.206138	0.103069	−	0.994674i	$-0.467134\pi$
$859$	6.04163	0.206138	0.103069	+	0.994674i	$0.467134\pi$
$860$	0	0
$861$	79.8476i	2.72120i
$862$	0	0
$863$	44.1421	1.50262	0.751308	−	0.659952i	$-0.229423\pi$
$863$	44.1421	1.50262	0.751308	+	0.659952i	$0.229423\pi$
$864$	0	0
$865$	−18.8701	−0.641601
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−25.0711	−0.850478
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 0.842290i	− 0.0285072i
$874$	0	0
$875$	−32.1421	−1.08660
$876$	0	0
$877$	− 22.9385i	− 0.774577i	−0.921959	−	0.387289i	$-0.873412\pi$
$877$	− 22.9385i	− 0.774577i	0.921959	−	0.387289i	$-0.126588\pi$
$878$	0	0
$879$	− 28.2960i	− 0.954402i
$880$	0	0
$881$	25.1802i	0.848342i	0.905582	+	0.424171i	$0.139434\pi$
$881$	25.1802i	0.848342i	−0.905582	+	0.424171i	$0.860566\pi$
$882$	0	0
$883$	49.2548	1.65756	0.828779	−	0.559577i	$-0.189036\pi$
$883$	49.2548	1.65756	0.828779	+	0.559577i	$0.189036\pi$
$884$	0	0
$885$	−16.8284	−0.565681
$886$	0	0
$887$	− 25.8142i	− 0.866757i	−0.901212	−	0.433379i	$-0.857321\pi$
$887$	− 25.8142i	− 0.866757i	0.901212	−	0.433379i	$-0.142679\pi$
$888$	0	0
$889$	− 45.6912i	− 1.53244i
$890$	0	0
$891$	− 18.6089i	− 0.623422i
$892$	0	0
$893$	−44.7696	−1.49816
$894$	0	0
$895$	− 7.46796i	− 0.249626i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.7279	−0.424500
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−14.4853	−0.482040
$904$	0	0
$905$	−0.928932	−0.0308788
$906$	0	0
$907$	− 35.7415i	− 1.18678i	−0.804917	−	0.593388i	$-0.797790\pi$
$907$	− 35.7415i	− 1.18678i	0.804917	−	0.593388i	$-0.202210\pi$
$908$	0	0
$909$	4.14214	0.137386
$910$	0	0
$911$	− 44.2149i	− 1.46491i	−0.680818	−	0.732453i	$-0.738376\pi$
$911$	− 44.2149i	− 1.46491i	0.680818	−	0.732453i	$-0.261624\pi$
$912$	0	0
$913$	3.24718i	0.107466i
$914$	0	0
$915$	− 18.0292i	− 0.596029i
$916$	0	0
$917$	−7.41421	−0.244839
$918$	0	0
$919$	−12.2843	−0.405221	−0.202610	−	0.979259i	$-0.564942\pi$
$919$	−12.2843	−0.405221	−0.202610	+	0.979259i	$0.564942\pi$
$920$	0	0
$921$	− 34.4190i	− 1.13414i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	8.15640i	0.268181i
$926$	0	0
$927$	5.65685	0.185795
$928$	0	0
$929$	25.2890i	0.829704i	0.909889	+	0.414852i	$0.136167\pi$
$929$	25.2890i	0.829704i	−0.909889	+	0.414852i	$0.863833\pi$
$930$	0	0
$931$	−80.5269	−2.63916
$932$	0	0
$933$	11.0711	0.362450
$934$	0	0
$935$	0	0
$936$	0	0
$937$	31.5563	1.03090	0.515450	−	0.856919i	$-0.327625\pi$
$937$	31.5563	1.03090	0.515450	+	0.856919i	$0.327625\pi$
$938$	0	0
$939$	28.7279	0.937500
$940$	0	0
$941$	31.3031i	1.02045i	0.860040	+	0.510226i	$0.170438\pi$
$941$	31.3031i	1.02045i	−0.860040	+	0.510226i	$0.829562\pi$
$942$	0	0
$943$	−47.5563	−1.54865
$944$	0	0
$945$	16.3128i	0.530656i
$946$	0	0
$947$	− 16.9694i	− 0.551431i	−0.961239	−	0.275715i	$-0.911085\pi$
$947$	− 16.9694i	− 0.551431i	0.961239	−	0.275715i	$-0.0889148\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−31.8995	−1.03441
$952$	0	0
$953$	−50.0000	−1.61966	−0.809829	−	0.586665i	$-0.800440\pi$
$953$	−50.0000	−1.61966	−0.809829	+	0.586665i	$0.800440\pi$
$954$	0	0
$955$	2.42742i	0.0785494i
$956$	0	0
$957$	− 7.83938i	− 0.253411i
$958$	0	0
$959$	− 62.4524i	− 2.01669i
$960$	0	0
$961$	0.272078	0.00877671
$962$	0	0
$963$	− 0.688444i	− 0.0221848i
$964$	0	0
$965$	−1.41421	−0.0455251
$966$	0	0
$967$	−30.5269	−0.981679	−0.490840	−	0.871250i	$-0.663310\pi$
$967$	−30.5269	−0.981679	−0.490840	+	0.871250i	$0.663310\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	29.2721	0.939386	0.469693	−	0.882830i	$-0.344365\pi$
$971$	29.2721	0.939386	0.469693	+	0.882830i	$0.344365\pi$
$972$	0	0
$973$	−87.8406	−2.81604
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.38478	0.0762958	0.0381479	−	0.999272i	$-0.487854\pi$
$977$	2.38478	0.0762958	0.0381479	+	0.999272i	$0.487854\pi$
$978$	0	0
$979$	− 17.8435i	− 0.570282i
$980$	0	0
$981$	− 5.17186i	− 0.165125i
$982$	0	0
$983$	− 31.3031i	− 0.998414i	−0.866483	−	0.499207i	$-0.833625\pi$
$983$	− 31.3031i	− 0.998414i	0.866483	−	0.499207i	$-0.166375\pi$
$984$	0	0
$985$	2.72792	0.0869188
$986$	0	0
$987$	59.1127	1.88158
$988$	0	0
$989$	− 8.62727i	− 0.274331i
$990$	0	0
$991$	− 8.04762i	− 0.255641i	−0.991797	−	0.127820i	$-0.959202\pi$
$991$	− 8.04762i	− 0.255641i	0.991797	−	0.127820i	$-0.0407981\pi$
$992$	0	0
$993$	− 41.0990i	− 1.30424i
$994$	0	0
$995$	10.4437	0.331086
$996$	0	0
$997$	41.8644i	1.32586i	0.748682	+	0.662930i	$0.230687\pi$
$997$	41.8644i	1.32586i	−0.748682	+	0.662930i	$0.769313\pi$
$998$	0	0
$999$	8.82843	0.279319

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.b.j.577.1		4
17.4	even	4		2312.2.a.s.1.1		4
17.7	odd	16		136.2.n.a.121.1	yes	4
17.12	odd	16		136.2.n.a.9.1	✓	4
17.13	even	4		2312.2.a.s.1.4		4
17.16	even	2	inner	2312.2.b.j.577.4		4
51.29	even	16		1224.2.bq.a.145.1		4
51.41	even	16		1224.2.bq.a.937.1		4
68.7	even	16		272.2.v.e.257.1		4
68.47	odd	4		4624.2.a.bm.1.1		4
68.55	odd	4		4624.2.a.bm.1.4		4
68.63	even	16		272.2.v.e.145.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.n.a.9.1	✓	4	17.12	odd	16
136.2.n.a.121.1	yes	4	17.7	odd	16
272.2.v.e.145.1		4	68.63	even	16
272.2.v.e.257.1		4	68.7	even	16
1224.2.bq.a.145.1		4	51.29	even	16
1224.2.bq.a.937.1		4	51.41	even	16
2312.2.a.s.1.1		4	17.4	even	4
2312.2.a.s.1.4		4	17.13	even	4
2312.2.b.j.577.1		4	1.1	even	1	trivial
2312.2.b.j.577.4		4	17.16	even	2	inner
4624.2.a.bm.1.1		4	68.47	odd	4
4624.2.a.bm.1.4		4	68.55	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2312.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.84776i q^{3} -0.765367i q^{5} -4.46088i q^{7} -0.414214 q^{9} +1.84776i q^{11} -1.41421 q^{15} +6.24264 q^{19} -8.24264 q^{21} -4.90923i q^{23} +4.41421 q^{25} -4.77791i q^{27} -2.29610i q^{29} -5.54328i q^{31} +3.41421 q^{33} -3.41421 q^{35} +1.84776i q^{37} -9.68714i q^{41} +1.75736 q^{43} +0.317025i q^{45} -7.17157 q^{47} -12.8995 q^{49} -11.0711 q^{53} +1.41421 q^{55} -11.5349i q^{57} +11.8995 q^{59} +12.7486i q^{61} +1.84776i q^{63} +1.65685 q^{67} -9.07107 q^{69} +2.48181i q^{71} +14.0167i q^{73} -8.15640i q^{75} +8.24264 q^{77} +13.5684i q^{79} -10.0711 q^{81} +1.75736 q^{83} -4.24264 q^{87} -9.65685 q^{89} -10.2426 q^{93} -4.77791i q^{95} +2.03347i q^{97} -0.765367i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9} + 8 q^{19} - 16 q^{21} + 12 q^{25} + 8 q^{33} - 8 q^{35} + 24 q^{43} - 40 q^{47} - 12 q^{49} - 16 q^{53} + 8 q^{59} - 16 q^{67} - 8 q^{69} + 16 q^{77} - 12 q^{81} + 24 q^{83} - 16 q^{89} - 24 q^{93}+O(q^{100}) \) 4 * q + 4 * q^9 + 8 * q^19 - 16 * q^21 + 12 * q^25 + 8 * q^33 - 8 * q^35 + 24 * q^43 - 40 * q^47 - 12 * q^49 - 16 * q^53 + 8 * q^59 - 16 * q^67 - 8 * q^69 + 16 * q^77 - 12 * q^81 + 24 * q^83 - 16 * q^89 - 24 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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