LMFDB - Embedded newform 2496.2.d.q.1535.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,2,Mod(1535,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, 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("2496.1535");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2496.d (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:	$19.9306603445$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{17} + 18 x^{16} + 8 x^{14} - 8 x^{13} + 241 x^{12} - 44 x^{11} - 112 x^{10} - 132 x^{9} + \cdots + 59049 $ x^20 - 4x^17 + 18x^16 + 8x^14 - 8x^13 + 241x^12 - 44x^11 - 112x^10 - 132x^9 + 2169x^8 - 216x^7 + 648x^6 + 13122x^4 - 8748*x^3 + 59049
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{23}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1248)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1535.5
Root			$0.587478 + 1.62938i$ of defining polynomial
Character	$\chi$	$=$	2496.1535
Dual form			2496.2.d.q.1535.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.736734 - 1.56755i) q^{3} +2.65543i q^{5} -4.30836i q^{7} +(-1.91445 + 2.30974i) q^{9} +2.82843 q^{11} -1.00000 q^{13} +(4.16253 - 1.95635i) q^{15} +1.67254i q^{17} +0.395673i q^{19} +(-6.75359 + 3.17412i) q^{21} -7.57972 q^{23} -2.05131 q^{25} +(5.03108 + 1.29933i) q^{27} -2.82843i q^{29} -11.1150i q^{31} +(-2.08380 - 4.43371i) q^{33} +11.4406 q^{35} +5.06937 q^{37} +(0.736734 + 1.56755i) q^{39} -8.00304i q^{41} +3.30270i q^{43} +(-6.13335 - 5.08368i) q^{45} -2.64266 q^{47} -11.5620 q^{49} +(2.62180 - 1.23222i) q^{51} +7.02015i q^{53} +7.51069i q^{55} +(0.620239 - 0.291506i) q^{57} +2.82843 q^{59} -9.31801 q^{61} +(9.95120 + 8.24813i) q^{63} -2.65543i q^{65} -7.60433i q^{67} +(5.58424 + 11.8816i) q^{69} -10.2908 q^{71} +11.4081 q^{73} +(1.51127 + 3.21553i) q^{75} -12.1859i q^{77} +12.7374i q^{79} +(-1.66979 - 8.84374i) q^{81} -9.35747 q^{83} -4.44132 q^{85} +(-4.43371 + 2.08380i) q^{87} -13.6854i q^{89} +4.30836i q^{91} +(-17.4234 + 8.18881i) q^{93} -1.05068 q^{95} +0.628049 q^{97} +(-5.41487 + 6.53293i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 8 q^{9} - 20 q^{13} + 12 q^{21} - 36 q^{25} - 16 q^{37} - 4 q^{45} - 76 q^{49} - 16 q^{57} + 56 q^{61} - 24 q^{69} + 88 q^{73} + 72 q^{81} - 56 q^{85} - 96 q^{93} - 72 q^{97}+O(q^{100}) $ 20 * q - 8 * q^9 - 20 * q^13 + 12 * q^21 - 36 * q^25 - 16 * q^37 - 4 * q^45 - 76 * q^49 - 16 * q^57 + 56 * q^61 - 24 * q^69 + 88 * q^73 + 72 * q^81 - 56 * q^85 - 96 * q^93 - 72 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	−0.736734	−	1.56755i	−0.425354	−	0.905027i
$3$	−0.736734	−	1.56755i	−0.425354	−	0.905027i
$4$		0			0
$5$			2.65543i			1.18754i	0.804633	+	0.593772i	$0.202362\pi$
$5$			2.65543i			1.18754i	−0.804633	+	0.593772i	$0.797638\pi$
$6$		0			0
$7$		−	4.30836i		−	1.62841i	−0.580579	−	0.814204i	$-0.697174\pi$
$7$		−	4.30836i		−	1.62841i	0.580579	−	0.814204i	$-0.302826\pi$
$8$		0			0
$9$	−1.91445	+	2.30974i	−0.638149	+	0.769913i
$10$		0			0
$11$	2.82843			0.852803			0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843			0.852803			0.426401	+	0.904534i	$0.359781\pi$
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$	4.16253	−	1.95635i	1.07476	−	0.505126i
$16$		0			0
$17$			1.67254i			0.405651i	0.979215	+	0.202826i	$0.0650125\pi$
$17$			1.67254i			0.405651i	−0.979215	+	0.202826i	$0.934988\pi$
$18$		0			0
$19$			0.395673i			0.0907736i	0.998969	+	0.0453868i	$0.0144520\pi$
$19$			0.395673i			0.0907736i	−0.998969	+	0.0453868i	$0.985548\pi$
$20$		0			0
$21$	−6.75359	+	3.17412i	−1.47375	+	0.692649i
$22$		0			0
$23$	−7.57972			−1.58048			−0.790241	−	0.612797i	$-0.790044\pi$
$23$	−7.57972			−1.58048			−0.790241	+	0.612797i	$0.790044\pi$
$24$		0			0
$25$	−2.05131			−0.410262
$26$		0			0
$27$	5.03108	+	1.29933i	0.968231	+	0.250057i
$28$		0			0
$29$		−	2.82843i		−	0.525226i	−0.964901	−	0.262613i	$-0.915416\pi$
$29$		−	2.82843i		−	0.525226i	0.964901	−	0.262613i	$-0.0845842\pi$
$30$		0			0
$31$		−	11.1150i		−	1.99632i	−0.0606681	−	0.998158i	$-0.519323\pi$
$31$		−	11.1150i		−	1.99632i	0.0606681	−	0.998158i	$-0.480677\pi$
$32$		0			0
$33$	−2.08380	−	4.43371i	−0.362743	−	0.771810i
$34$		0			0
$35$	11.4406			1.93381
$36$		0			0
$37$	5.06937			0.833399			0.416700	−	0.909044i	$-0.363187\pi$
$37$	5.06937			0.833399			0.416700	+	0.909044i	$0.363187\pi$
$38$		0			0
$39$	0.736734	+	1.56755i	0.117972	+	0.251009i
$40$		0			0
$41$		−	8.00304i		−	1.24987i	−0.780679	−	0.624933i	$-0.785126\pi$
$41$		−	8.00304i		−	1.24987i	0.780679	−	0.624933i	$-0.214874\pi$
$42$		0			0
$43$			3.30270i			0.503657i	0.967772	+	0.251829i	$0.0810320\pi$
$43$			3.30270i			0.503657i	−0.967772	+	0.251829i	$0.918968\pi$
$44$		0			0
$45$	−6.13335	−	5.08368i	−0.914306	−	0.757830i
$46$		0			0
$47$	−2.64266			−0.385471			−0.192736	−	0.981251i	$-0.561736\pi$
$47$	−2.64266			−0.385471			−0.192736	+	0.981251i	$0.561736\pi$
$48$		0			0
$49$	−11.5620			−1.65171
$50$		0			0
$51$	2.62180	−	1.23222i	0.367125	−	0.172545i
$52$		0			0
$53$			7.02015i			0.964292i	0.876091	+	0.482146i	$0.160143\pi$
$53$			7.02015i			0.964292i	−0.876091	+	0.482146i	$0.839857\pi$
$54$		0			0
$55$			7.51069i			1.01274i
$56$		0			0
$57$	0.620239	−	0.291506i	0.0821526	−	0.0386109i
$58$		0			0
$59$	2.82843			0.368230			0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843			0.368230			0.184115	+	0.982905i	$0.441058\pi$
$60$		0			0
$61$	−9.31801			−1.19305			−0.596524	−	0.802595i	$-0.703452\pi$
$61$	−9.31801			−1.19305			−0.596524	+	0.802595i	$0.703452\pi$
$62$		0			0
$63$	9.95120	+	8.24813i	1.25373	+	1.03917i
$64$		0			0
$65$		−	2.65543i		−	0.329366i
$66$		0			0
$67$		−	7.60433i		−	0.929016i	−0.885569	−	0.464508i	$-0.846231\pi$
$67$		−	7.60433i		−	0.929016i	0.885569	−	0.464508i	$-0.153769\pi$
$68$		0			0
$69$	5.58424	+	11.8816i	0.672263	+	1.43038i
$70$		0			0
$71$	−10.2908			−1.22130			−0.610648	−	0.791902i	$-0.709091\pi$
$71$	−10.2908			−1.22130			−0.610648	+	0.791902i	$0.709091\pi$
$72$		0			0
$73$	11.4081			1.33521			0.667607	−	0.744514i	$-0.267319\pi$
$73$	11.4081			1.33521			0.667607	+	0.744514i	$0.267319\pi$
$74$		0			0
$75$	1.51127	+	3.21553i	0.174506	+	0.371298i
$76$		0			0
$77$		−	12.1859i		−	1.38871i
$78$		0			0
$79$			12.7374i			1.43307i	0.697551	+	0.716535i	$0.254273\pi$
$79$			12.7374i			1.43307i	−0.697551	+	0.716535i	$0.745727\pi$
$80$		0			0
$81$	−1.66979	−	8.84374i	−0.185532	−	0.982638i
$82$		0			0
$83$	−9.35747			−1.02712			−0.513558	−	0.858055i	$-0.671673\pi$
$83$	−9.35747			−1.02712			−0.513558	+	0.858055i	$0.671673\pi$
$84$		0			0
$85$	−4.44132			−0.481729
$86$		0			0
$87$	−4.43371	+	2.08380i	−0.475344	+	0.223407i
$88$		0			0
$89$		−	13.6854i		−	1.45065i	−0.688405	−	0.725327i	$-0.741689\pi$
$89$		−	13.6854i		−	1.45065i	0.688405	−	0.725327i	$-0.258311\pi$
$90$		0			0
$91$			4.30836i			0.451639i
$92$		0			0
$93$	−17.4234	+	8.18881i	−1.80672	+	0.849140i
$94$		0			0
$95$	−1.05068			−0.107798
$96$		0			0
$97$	0.628049			0.0637687			0.0318844	−	0.999492i	$-0.489849\pi$
$97$	0.628049			0.0637687			0.0318844	+	0.999492i	$0.489849\pi$
$98$		0			0
$99$	−5.41487	+	6.53293i	−0.544215	+	0.656584i
$100$		0			0
$101$		−	15.0143i		−	1.49398i	−0.664835	−	0.746990i	$-0.731498\pi$
$101$		−	15.0143i		−	1.49398i	0.664835	−	0.746990i	$-0.268502\pi$
$102$		0			0
$103$		−	16.7374i		−	1.64919i	−0.565727	−	0.824593i	$-0.691404\pi$
$103$		−	16.7374i		−	1.64919i	0.565727	−	0.824593i	$-0.308596\pi$
$104$		0			0
$105$	−8.42865	−	17.9337i	−0.822552	−	1.75015i
$106$		0			0
$107$	−14.2968			−1.38212			−0.691061	−	0.722796i	$-0.742857\pi$
$107$	−14.2968			−1.38212			−0.691061	+	0.722796i	$0.742857\pi$
$108$		0			0
$109$	−9.06937			−0.868688			−0.434344	−	0.900747i	$-0.643020\pi$
$109$	−9.06937			−0.868688			−0.434344	+	0.900747i	$0.643020\pi$
$110$		0			0
$111$	−3.73478	−	7.94651i	−0.354489	−	0.754249i
$112$		0			0
$113$		−	0.846642i		−	0.0796454i	−0.999207	−	0.0398227i	$-0.987321\pi$
$113$		−	0.846642i		−	0.0796454i	0.999207	−	0.0398227i	$-0.0126793\pi$
$114$		0			0
$115$		−	20.1274i		−	1.87689i
$116$		0			0
$117$	1.91445	−	2.30974i	0.176991	−	0.213535i
$118$		0			0
$119$	7.20592			0.660566
$120$		0			0
$121$	−3.00000			−0.272727
$122$		0			0
$123$	−12.5452	+	5.89611i	−1.13116	+	0.531635i
$124$		0			0
$125$			7.83004i			0.700340i
$126$		0			0
$127$		−	7.39001i		−	0.655758i	−0.944720	−	0.327879i	$-0.893666\pi$
$127$		−	7.39001i		−	0.655758i	0.944720	−	0.327879i	$-0.106334\pi$
$128$		0			0
$129$	5.17716	−	2.43321i	0.455824	−	0.214232i
$130$		0			0
$131$	−3.53085			−0.308492			−0.154246	−	0.988032i	$-0.549295\pi$
$131$	−3.53085			−0.308492			−0.154246	+	0.988032i	$0.549295\pi$
$132$		0			0
$133$	1.70470			0.147817
$134$		0			0
$135$	−3.45029	+	13.3597i	−0.296953	+	1.14982i
$136$		0			0
$137$			11.6941i			0.999096i	0.866286	+	0.499548i	$0.166501\pi$
$137$			11.6941i			0.999096i	−0.866286	+	0.499548i	$0.833499\pi$
$138$		0			0
$139$			14.0220i			1.18933i	0.803972	+	0.594667i	$0.202716\pi$
$139$			14.0220i			1.18933i	−0.803972	+	0.594667i	$0.797284\pi$
$140$		0			0
$141$	1.94694	+	4.14251i	0.163962	+	0.348862i
$142$		0			0
$143$	−2.82843			−0.236525
$144$		0			0
$145$	7.51069			0.623729
$146$		0			0
$147$	8.51812	+	18.1240i	0.702562	+	1.49485i
$148$		0			0
$149$		−	18.1081i		−	1.48347i	−0.670691	−	0.741737i	$-0.734002\pi$
$149$		−	18.1081i		−	1.48347i	0.670691	−	0.741737i	$-0.265998\pi$
$150$		0			0
$151$		−	2.99709i		−	0.243900i	−0.992536	−	0.121950i	$-0.961085\pi$
$151$		−	2.99709i		−	0.243900i	0.992536	−	0.121950i	$-0.0389148\pi$
$152$		0			0
$153$	−3.86314	−	3.20199i	−0.312316	−	0.258866i
$154$		0			0
$155$	29.5151			2.37071
$156$		0			0
$157$	−11.2154			−0.895086			−0.447543	−	0.894262i	$-0.647701\pi$
$157$	−11.2154			−0.895086			−0.447543	+	0.894262i	$0.647701\pi$
$158$		0			0
$159$	11.0045	−	5.17199i	0.872711	−	0.410165i
$160$		0			0
$161$			32.6562i			2.57367i
$162$		0			0
$163$		−	9.89504i		−	0.775039i	−0.921861	−	0.387520i	$-0.873332\pi$
$163$		−	9.89504i		−	0.775039i	0.921861	−	0.387520i	$-0.126668\pi$
$164$		0			0
$165$	11.7734	−	5.53338i	0.916558	−	0.430773i
$166$		0			0
$167$	−10.2201			−0.790857			−0.395428	−	0.918497i	$-0.629404\pi$
$167$	−10.2201			−0.790857			−0.395428	+	0.918497i	$0.629404\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−0.913902	−	0.757495i	−0.0698878	−	0.0579271i
$172$		0			0
$173$		−	10.0495i		−	0.764046i	−0.924153	−	0.382023i	$-0.875228\pi$
$173$		−	10.0495i		−	0.764046i	0.924153	−	0.382023i	$-0.124772\pi$
$174$		0			0
$175$			8.83778i			0.668074i
$176$		0			0
$177$	−2.08380	−	4.43371i	−0.156628	−	0.333258i
$178$		0			0
$179$	5.84262			0.436698			0.218349	−	0.975871i	$-0.429933\pi$
$179$	5.84262			0.436698			0.218349	+	0.975871i	$0.429933\pi$
$180$		0			0
$181$	−18.7261			−1.39190			−0.695949	−	0.718091i	$-0.745016\pi$
$181$	−18.7261			−1.39190			−0.695949	+	0.718091i	$0.745016\pi$
$182$		0			0
$183$	6.86489	+	14.6065i	0.507468	+	1.07974i
$184$		0			0
$185$			13.4614i			0.989699i
$186$		0			0
$187$			4.73067i			0.345941i
$188$		0			0
$189$	5.59800	−	21.6757i	0.407195	−	1.57668i
$190$		0			0
$191$	−12.4890			−0.903671			−0.451835	−	0.892101i	$-0.649231\pi$
$191$	−12.4890			−0.903671			−0.451835	+	0.892101i	$0.649231\pi$
$192$		0			0
$193$	−14.9188			−1.07388			−0.536938	−	0.843622i	$-0.680419\pi$
$193$	−14.9188			−1.07388			−0.536938	+	0.843622i	$0.680419\pi$
$194$		0			0
$195$	−4.16253	+	1.95635i	−0.298085	+	0.140097i
$196$		0			0
$197$			19.6515i			1.40011i	0.714087	+	0.700057i	$0.246842\pi$
$197$			19.6515i			1.40011i	−0.714087	+	0.700057i	$0.753158\pi$
$198$		0			0
$199$			2.90070i			0.205625i	0.994701	+	0.102813i	$0.0327842\pi$
$199$			2.90070i			0.205625i	−0.994701	+	0.102813i	$0.967216\pi$
$200$		0			0
$201$	−11.9202	+	5.60237i	−0.840785	+	0.395160i
$202$		0			0
$203$	−12.1859			−0.855282
$204$		0			0
$205$	21.2515			1.48427
$206$		0			0
$207$	14.5110	−	17.5072i	1.00858	−	1.21683i
$208$		0			0
$209$			1.11913i			0.0774120i
$210$		0			0
$211$		−	15.3027i		−	1.05348i	−0.850026	−	0.526741i	$-0.823414\pi$
$211$		−	15.3027i		−	1.05348i	0.850026	−	0.526741i	$-0.176586\pi$
$212$		0			0
$213$	7.58160	+	16.1314i	0.519483	+	1.10531i
$214$		0			0
$215$	−8.77010			−0.598116
$216$		0			0
$217$	−47.8875			−3.25082
$218$		0			0
$219$	−8.40471	−	17.8828i	−0.567938	−	1.20841i
$220$		0			0
$221$		−	1.67254i		−	0.112507i
$222$		0			0
$223$		−	5.61963i		−	0.376319i	−0.982139	−	0.188159i	$-0.939748\pi$
$223$		−	5.61963i		−	0.376319i	0.982139	−	0.188159i	$-0.0602521\pi$
$224$		0			0
$225$	3.92712	−	4.73799i	0.261808	−	0.315866i
$226$		0			0
$227$	−11.9595			−0.793778			−0.396889	−	0.917867i	$-0.629910\pi$
$227$	−11.9595			−0.793778			−0.396889	+	0.917867i	$0.629910\pi$
$228$		0			0
$229$	−1.55868			−0.103000			−0.0515002	−	0.998673i	$-0.516400\pi$
$229$	−1.55868			−0.103000			−0.0515002	+	0.998673i	$0.516400\pi$
$230$		0			0
$231$	−19.1020	+	8.97776i	−1.25682	+	0.590693i
$232$		0			0
$233$		−	10.1674i		−	0.666086i	−0.942912	−	0.333043i	$-0.891925\pi$
$233$		−	10.1674i		−	0.666086i	0.942912	−	0.333043i	$-0.108075\pi$
$234$		0			0
$235$		−	7.01739i		−	0.457764i
$236$		0			0
$237$	19.9666	−	9.38408i	1.29697	−	0.609561i
$238$		0			0
$239$	29.6165			1.91573			0.957865	−	0.287220i	$-0.0927311\pi$
$239$	29.6165			1.91573			0.957865	+	0.287220i	$0.0927311\pi$
$240$		0			0
$241$	−4.41731			−0.284544			−0.142272	−	0.989828i	$-0.545441\pi$
$241$	−4.41731			−0.284544			−0.142272	+	0.989828i	$0.545441\pi$
$242$		0			0
$243$	−12.6328	+	9.13297i	−0.810397	+	0.585880i
$244$		0			0
$245$		−	30.7021i		−	1.96148i
$246$		0			0
$247$		−	0.395673i		−	0.0251761i
$248$		0			0
$249$	6.89396	+	14.6683i	0.436887	+	0.929567i
$250$		0			0
$251$	14.5999			0.921536			0.460768	−	0.887521i	$-0.347574\pi$
$251$	14.5999			0.921536			0.460768	+	0.887521i	$0.347574\pi$
$252$		0			0
$253$	−21.4387			−1.34784
$254$		0			0
$255$	3.27207	+	6.96201i	0.204905	+	0.435978i
$256$		0			0
$257$			17.1780i			1.07153i	0.844366	+	0.535766i	$0.179977\pi$
$257$			17.1780i			1.07153i	−0.844366	+	0.535766i	$0.820023\pi$
$258$		0			0
$259$		−	21.8407i		−	1.35711i
$260$		0			0
$261$	6.53293	+	5.41487i	0.404378	+	0.335172i
$262$		0			0
$263$	10.0032			0.616826			0.308413	−	0.951253i	$-0.400202\pi$
$263$	10.0032			0.616826			0.308413	+	0.951253i	$0.400202\pi$
$264$		0			0
$265$	−18.6415			−1.14514
$266$		0			0
$267$	−21.4527	+	10.0825i	−1.31288	+	0.617041i
$268$		0			0
$269$			16.5050i			1.00633i	0.864191	+	0.503163i	$0.167831\pi$
$269$			16.5050i			1.00633i	−0.864191	+	0.503163i	$0.832169\pi$
$270$		0			0
$271$		−	20.6410i		−	1.25385i	−0.779079	−	0.626926i	$-0.784313\pi$
$271$		−	20.6410i		−	1.25385i	0.779079	−	0.626926i	$-0.215687\pi$
$272$		0			0
$273$	6.75359	−	3.17412i	0.408746	−	0.192106i
$274$		0			0
$275$	−5.80198			−0.349872
$276$		0			0
$277$	−5.51069			−0.331105			−0.165553	−	0.986201i	$-0.552941\pi$
$277$	−5.51069			−0.331105			−0.165553	+	0.986201i	$0.552941\pi$
$278$		0			0
$279$	25.6728	+	21.2791i	1.53699	+	1.27395i
$280$		0			0
$281$		−	16.6334i		−	0.992268i	−0.868246	−	0.496134i	$-0.834752\pi$
$281$		−	16.6334i		−	0.992268i	0.868246	−	0.496134i	$-0.165248\pi$
$282$		0			0
$283$			3.63137i			0.215862i	0.994158	+	0.107931i	$0.0344226\pi$
$283$			3.63137i			0.215862i	−0.994158	+	0.107931i	$0.965577\pi$
$284$		0			0
$285$	0.774073	+	1.64700i	0.0458521	+	0.0975599i
$286$		0			0
$287$	−34.4800			−2.03529
$288$		0			0
$289$	14.2026			0.835447
$290$		0			0
$291$	−0.462705	−	0.984501i	−0.0271243	−	0.0577124i
$292$		0			0
$293$			15.9605i			0.932420i	0.884674	+	0.466210i	$0.154381\pi$
$293$			15.9605i			0.932420i	−0.884674	+	0.466210i	$0.845619\pi$
$294$		0			0
$295$			7.51069i			0.437289i
$296$		0			0
$297$	14.2300	+	3.67507i	0.825710	+	0.213249i
$298$		0			0
$299$	7.57972			0.438347
$300$		0			0
$301$	14.2292			0.820160
$302$		0			0
$303$	−23.5357	+	11.0616i	−1.35209	+	0.635470i
$304$		0			0
$305$		−	24.7433i		−	1.41680i
$306$		0			0
$307$			18.9404i			1.08099i	0.841349	+	0.540493i	$0.181762\pi$
$307$			18.9404i			1.08099i	−0.841349	+	0.540493i	$0.818238\pi$
$308$		0			0
$309$	−26.2368	+	12.3310i	−1.49256	+	0.701487i
$310$		0			0
$311$	27.5749			1.56363			0.781815	−	0.623510i	$-0.214294\pi$
$311$	27.5749			1.56363			0.781815	+	0.623510i	$0.214294\pi$
$312$		0			0
$313$	31.1855			1.76271			0.881353	−	0.472458i	$-0.156633\pi$
$313$	31.1855			1.76271			0.881353	+	0.472458i	$0.156633\pi$
$314$		0			0
$315$	−21.9023	+	26.4247i	−1.23406	+	1.48886i
$316$		0			0
$317$		−	29.1493i		−	1.63719i	−0.574372	−	0.818594i	$-0.694754\pi$
$317$		−	29.1493i		−	1.63719i	0.574372	−	0.818594i	$-0.305246\pi$
$318$		0			0
$319$		−	8.00000i		−	0.447914i
$320$		0			0
$321$	10.5329	+	22.4110i	0.587891	+	1.25086i
$322$		0			0
$323$	−0.661780			−0.0368224
$324$		0			0
$325$	2.05131			0.113786
$326$		0			0
$327$	6.68171	+	14.2167i	0.369500	+	0.786186i
$328$		0			0
$329$			11.3855i			0.627705i
$330$		0			0
$331$			4.42629i			0.243291i	0.992574	+	0.121645i	$0.0388171\pi$
$331$			4.42629i			0.243291i	−0.992574	+	0.121645i	$0.961183\pi$
$332$		0			0
$333$	−9.70504	+	11.7089i	−0.531833	+	0.641645i
$334$		0			0
$335$	20.1928			1.10325
$336$		0			0
$337$	−18.2667			−0.995051			−0.497525	−	0.867449i	$-0.665758\pi$
$337$	−18.2667			−0.995051			−0.497525	+	0.867449i	$0.665758\pi$
$338$		0			0
$339$	−1.32716	+	0.623750i	−0.0720812	+	0.0338774i
$340$		0			0
$341$		−	31.4380i		−	1.70246i
$342$		0			0
$343$			19.6547i			1.06126i
$344$		0			0
$345$	−31.5508	+	14.8286i	−1.69864	+	0.798342i
$346$		0			0
$347$	0.915999			0.0491734			0.0245867	−	0.999698i	$-0.492173\pi$
$347$	0.915999			0.0491734			0.0245867	+	0.999698i	$0.492173\pi$
$348$		0			0
$349$	30.9747			1.65804			0.829019	−	0.559220i	$-0.188899\pi$
$349$	30.9747			1.65804			0.829019	+	0.559220i	$0.188899\pi$
$350$		0			0
$351$	−5.03108	−	1.29933i	−0.268539	−	0.0693533i
$352$		0			0
$353$			34.2416i			1.82249i	0.411860	+	0.911247i	$0.364879\pi$
$353$			34.2416i			1.82249i	−0.411860	+	0.911247i	$0.635121\pi$
$354$		0			0
$355$		−	27.3266i		−	1.45034i
$356$		0			0
$357$	−5.30885	−	11.2957i	−0.280974	−	0.597830i
$358$		0			0
$359$	−24.7734			−1.30749			−0.653745	−	0.756715i	$-0.726803\pi$
$359$	−24.7734			−1.30749			−0.653745	+	0.756715i	$0.726803\pi$
$360$		0			0
$361$	18.8434			0.991760
$362$		0			0
$363$	2.21020	+	4.70266i	0.116006	+	0.246826i
$364$		0			0
$365$			30.2933i			1.58563i
$366$		0			0
$367$			7.89280i			0.412001i	0.978552	+	0.206000i	$0.0660448\pi$
$367$			7.89280i			0.412001i	−0.978552	+	0.206000i	$0.933955\pi$
$368$		0			0
$369$	18.4849	+	15.3214i	0.962288	+	0.797600i
$370$		0			0
$371$	30.2454			1.57026
$372$		0			0
$373$	7.12400			0.368867			0.184433	−	0.982845i	$-0.440955\pi$
$373$	7.12400			0.368867			0.184433	+	0.982845i	$0.440955\pi$
$374$		0			0
$375$	12.2740	−	5.76866i	0.633827	−	0.297892i
$376$		0			0
$377$			2.82843i			0.145671i
$378$		0			0
$379$		−	27.9858i		−	1.43753i	−0.695252	−	0.718766i	$-0.744707\pi$
$379$		−	27.9858i		−	1.43753i	0.695252	−	0.718766i	$-0.255293\pi$
$380$		0			0
$381$	−11.5842	+	5.44447i	−0.593478	+	0.278929i
$382$		0			0
$383$	25.5238			1.30420			0.652102	−	0.758131i	$-0.273887\pi$
$383$	25.5238			1.30420			0.652102	+	0.758131i	$0.273887\pi$
$384$		0			0
$385$	32.3588			1.64916
$386$		0			0
$387$	−7.62838	−	6.32285i	−0.387772	−	0.321408i
$388$		0			0
$389$			7.52080i			0.381320i	0.981656	+	0.190660i	$0.0610628\pi$
$389$			7.52080i			0.381320i	−0.981656	+	0.190660i	$0.938937\pi$
$390$		0			0
$391$		−	12.6774i		−	0.641124i
$392$		0			0
$393$	2.60130	+	5.53480i	0.131218	+	0.279194i
$394$		0			0
$395$	−33.8233			−1.70183
$396$		0			0
$397$	6.99210			0.350923			0.175462	−	0.984486i	$-0.443858\pi$
$397$	6.99210			0.350923			0.175462	+	0.984486i	$0.443858\pi$
$398$		0			0
$399$	−1.25591	−	2.67221i	−0.0628743	−	0.133778i
$400$		0			0
$401$		−	11.9666i		−	0.597584i	−0.954318	−	0.298792i	$-0.903416\pi$
$401$		−	11.9666i		−	0.597584i	0.954318	−	0.298792i	$-0.0965837\pi$
$402$		0			0
$403$			11.1150i			0.553678i
$404$		0			0
$405$	23.4839	−	4.43401i	1.16693	−	0.220328i
$406$		0			0
$407$	14.3383			0.710725
$408$		0			0
$409$	−6.70802			−0.331690			−0.165845	−	0.986152i	$-0.553035\pi$
$409$	−6.70802			−0.331690			−0.165845	+	0.986152i	$0.553035\pi$
$410$		0			0
$411$	18.3312	−	8.61546i	0.904209	−	0.424969i
$412$		0			0
$413$		−	12.1859i		−	0.599629i
$414$		0			0
$415$		−	24.8481i		−	1.21975i
$416$		0			0
$417$	21.9803	−	10.3305i	1.07638	−	0.505888i
$418$		0			0
$419$	29.5224			1.44226			0.721132	−	0.692797i	$-0.243622\pi$
$419$	29.5224			1.44226			0.721132	+	0.692797i	$0.243622\pi$
$420$		0			0
$421$	13.4627			0.656132			0.328066	−	0.944655i	$-0.393603\pi$
$421$	13.4627			0.656132			0.328066	+	0.944655i	$0.393603\pi$
$422$		0			0
$423$	5.05923	−	6.10385i	0.245988	−	0.296779i
$424$		0			0
$425$		−	3.43090i		−	0.166423i
$426$		0			0
$427$			40.1454i			1.94277i
$428$		0			0
$429$	2.08380	+	4.43371i	0.100607	+	0.214062i
$430$		0			0
$431$	15.8921			0.765494			0.382747	−	0.923853i	$-0.374978\pi$
$431$	15.8921			0.765494			0.382747	+	0.923853i	$0.374978\pi$
$432$		0			0
$433$	−0.899333			−0.0432192			−0.0216096	−	0.999766i	$-0.506879\pi$
$433$	−0.899333			−0.0432192			−0.0216096	+	0.999766i	$0.506879\pi$
$434$		0			0
$435$	−5.53338	−	11.7734i	−0.265305	−	0.564492i
$436$		0			0
$437$		−	2.99909i		−	0.143466i
$438$		0			0
$439$		−	21.4681i		−	1.02462i	−0.858802	−	0.512308i	$-0.828791\pi$
$439$		−	21.4681i		−	1.02462i	0.858802	−	0.512308i	$-0.171209\pi$
$440$		0			0
$441$	22.1348	−	26.7052i	1.05404	−	1.27168i
$442$		0			0
$443$	−32.8675			−1.56158			−0.780791	−	0.624792i	$-0.785184\pi$
$443$	−32.8675			−1.56158			−0.780791	+	0.624792i	$0.785184\pi$
$444$		0			0
$445$	36.3407			1.72272
$446$		0			0
$447$	−28.3854	+	13.3409i	−1.34258	+	0.631001i
$448$		0			0
$449$		−	6.38326i		−	0.301245i	−0.988591	−	0.150622i	$-0.951872\pi$
$449$		−	6.38326i		−	0.301245i	0.988591	−	0.150622i	$-0.0481278\pi$
$450$		0			0
$451$		−	22.6360i		−	1.06589i
$452$		0			0
$453$	−4.69810	+	2.20806i	−0.220736	+	0.103744i
$454$		0			0
$455$	−11.4406			−0.536342
$456$		0			0
$457$	−3.82405			−0.178881			−0.0894407	−	0.995992i	$-0.528508\pi$
$457$	−3.82405			−0.178881			−0.0894407	+	0.995992i	$0.528508\pi$
$458$		0			0
$459$	−2.17319	+	8.41469i	−0.101436	+	0.392764i
$460$		0			0
$461$			18.6871i			0.870343i	0.900348	+	0.435171i	$0.143312\pi$
$461$			18.6871i			0.870343i	−0.900348	+	0.435171i	$0.856688\pi$
$462$		0			0
$463$		−	13.8589i		−	0.644079i	−0.946726	−	0.322039i	$-0.895632\pi$
$463$		−	13.8589i		−	0.644079i	0.946726	−	0.322039i	$-0.104368\pi$
$464$		0			0
$465$	−21.7448	−	46.2666i	−1.00839	−	2.14556i
$466$		0			0
$467$	9.63501			0.445855			0.222928	−	0.974835i	$-0.428439\pi$
$467$	9.63501			0.445855			0.222928	+	0.974835i	$0.428439\pi$
$468$		0			0
$469$	−32.7622			−1.51282
$470$		0			0
$471$	8.26276	+	17.5807i	0.380728	+	0.810077i
$472$		0			0
$473$			9.34145i			0.429521i
$474$		0			0
$475$		−	0.811647i		−	0.0372409i
$476$		0			0
$477$	−16.2147	−	13.4397i	−0.742421	−	0.615362i
$478$		0			0
$479$	6.48839			0.296462			0.148231	−	0.988953i	$-0.452642\pi$
$479$	6.48839			0.296462			0.148231	+	0.988953i	$0.452642\pi$
$480$		0			0
$481$	−5.06937			−0.231143
$482$		0			0
$483$	51.1903	−	24.0589i	2.32924	−	1.09472i
$484$		0			0
$485$			1.66774i			0.0757282i
$486$		0			0
$487$			33.8950i			1.53593i	0.640491	+	0.767965i	$0.278731\pi$
$487$			33.8950i			1.53593i	−0.640491	+	0.767965i	$0.721269\pi$
$488$		0			0
$489$	−15.5110	+	7.29001i	−0.701432	+	0.329666i
$490$		0			0
$491$	−7.84678			−0.354120			−0.177060	−	0.984200i	$-0.556659\pi$
$491$	−7.84678			−0.354120			−0.177060	+	0.984200i	$0.556659\pi$
$492$		0			0
$493$	4.73067			0.213058
$494$		0			0
$495$	−17.3477	−	14.3788i	−0.779723	−	0.646280i
$496$		0			0
$497$			44.3366i			1.98877i
$498$		0			0
$499$		−	23.7011i		−	1.06101i	−0.847682	−	0.530504i	$-0.822003\pi$
$499$		−	23.7011i		−	1.06101i	0.847682	−	0.530504i	$-0.177997\pi$
$500$		0			0
$501$	7.52951	+	16.0206i	0.336394	+	0.715747i
$502$		0			0
$503$	22.3070			0.994618			0.497309	−	0.867573i	$-0.334321\pi$
$503$	22.3070			0.994618			0.497309	+	0.867573i	$0.334321\pi$
$504$		0			0
$505$	39.8695			1.77417
$506$		0			0
$507$	−0.736734	−	1.56755i	−0.0327195	−	0.0696175i
$508$		0			0
$509$			32.5502i			1.44276i	0.692539	+	0.721380i	$0.256492\pi$
$509$			32.5502i			1.44276i	−0.692539	+	0.721380i	$0.743508\pi$
$510$		0			0
$511$		−	49.1501i		−	2.17427i
$512$		0			0
$513$	−0.514111	+	1.99066i	−0.0226986	+	0.0878899i
$514$		0			0
$515$	44.4450			1.95848
$516$		0			0
$517$	−7.47457			−0.328731
$518$		0			0
$519$	−15.7531	+	7.40377i	−0.691482	+	0.324990i
$520$		0			0
$521$		−	27.9288i		−	1.22358i	−0.791019	−	0.611792i	$-0.790449\pi$
$521$		−	27.9288i		−	1.22358i	0.791019	−	0.611792i	$-0.209551\pi$
$522$		0			0
$523$		−	11.4261i		−	0.499630i	−0.968294	−	0.249815i	$-0.919630\pi$
$523$		−	11.4261i		−	0.499630i	0.968294	−	0.249815i	$-0.0803698\pi$
$524$		0			0
$525$	13.8537	−	6.51109i	0.604625	−	0.284167i
$526$		0			0
$527$	18.5903			0.809808
$528$		0			0
$529$	34.4522			1.49792
$530$		0			0
$531$	−5.41487	+	6.53293i	−0.234985	+	0.283505i
$532$		0			0
$533$			8.00304i			0.346650i
$534$		0			0
$535$		−	37.9641i		−	1.64133i
$536$		0			0
$537$	−4.30446	−	9.15862i	−0.185751	−	0.395224i
$538$		0			0
$539$	−32.7023			−1.40859
$540$		0			0
$541$	37.8306			1.62647			0.813233	−	0.581938i	$-0.197705\pi$
$541$	37.8306			1.62647			0.813233	+	0.581938i	$0.197705\pi$
$542$		0			0
$543$	13.7961	+	29.3541i	0.592049	+	1.25971i
$544$		0			0
$545$		−	24.0831i		−	1.03161i
$546$		0			0
$547$		−	7.77385i		−	0.332386i	−0.986093	−	0.166193i	$-0.946853\pi$
$547$		−	7.77385i		−	0.332386i	0.986093	−	0.166193i	$-0.0531474\pi$
$548$		0			0
$549$	17.8388	−	21.5222i	0.761343	−	0.918544i
$550$		0			0
$551$	1.11913			0.0476766
$552$		0			0
$553$	54.8774			2.33362
$554$		0			0
$555$	21.1014	−	9.91744i	0.895704	−	0.420972i
$556$		0			0
$557$		−	6.44556i		−	0.273107i	−0.990633	−	0.136553i	$-0.956397\pi$
$557$		−	6.44556i		−	0.273107i	0.990633	−	0.136553i	$-0.0436026\pi$
$558$		0			0
$559$		−	3.30270i		−	0.139689i
$560$		0			0
$561$	7.41557	−	3.48524i	0.313086	−	0.147147i
$562$		0			0
$563$	10.4360			0.439823			0.219911	−	0.975520i	$-0.429423\pi$
$563$	10.4360			0.439823			0.219911	+	0.975520i	$0.429423\pi$
$564$		0			0
$565$	2.24820			0.0945824
$566$		0			0
$567$	−38.1021	+	7.19407i	−1.60014	+	0.302122i
$568$		0			0
$569$		−	20.5421i		−	0.861171i	−0.902550	−	0.430585i	$-0.858307\pi$
$569$		−	20.5421i		−	0.861171i	0.902550	−	0.430585i	$-0.141693\pi$
$570$		0			0
$571$		−	27.6580i		−	1.15745i	−0.815522	−	0.578726i	$-0.803550\pi$
$571$		−	27.6580i		−	1.15745i	0.815522	−	0.578726i	$-0.196450\pi$
$572$		0			0
$573$	9.20106	+	19.5771i	0.384380	+	0.817847i
$574$		0			0
$575$	15.5483			0.648411
$576$		0			0
$577$	−33.8389			−1.40873			−0.704365	−	0.709838i	$-0.748768\pi$
$577$	−33.8389			−1.40873			−0.704365	+	0.709838i	$0.748768\pi$
$578$		0			0
$579$	10.9912	+	23.3860i	0.456777	+	0.971887i
$580$		0			0
$581$			40.3154i			1.67256i
$582$		0			0
$583$			19.8560i			0.822351i
$584$		0			0
$585$	6.13335	+	5.08368i	0.253583	+	0.210184i
$586$		0			0
$587$	21.0860			0.870313			0.435156	−	0.900355i	$-0.356693\pi$
$587$	21.0860			0.870313			0.435156	+	0.900355i	$0.356693\pi$
$588$		0			0
$589$	4.39791			0.181213
$590$		0			0
$591$	30.8048	−	14.4780i	1.26714	−	0.595543i
$592$		0			0
$593$		−	0.255824i		−	0.0105054i	−0.999986	−	0.00525272i	$-0.998328\pi$
$593$		−	0.255824i		−	0.0105054i	0.999986	−	0.00525272i	$-0.00167200\pi$
$594$		0			0
$595$			19.1348i			0.784451i
$596$		0			0
$597$	4.54701	−	2.13705i	0.186097	−	0.0874635i
$598$		0			0
$599$	−25.9103			−1.05867			−0.529333	−	0.848414i	$-0.677558\pi$
$599$	−25.9103			−1.05867			−0.529333	+	0.848414i	$0.677558\pi$
$600$		0			0
$601$	28.7626			1.17325			0.586626	−	0.809858i	$-0.300456\pi$
$601$	28.7626			1.17325			0.586626	+	0.809858i	$0.300456\pi$
$602$		0			0
$603$	17.5640	+	14.5581i	0.715262	+	0.592851i
$604$		0			0
$605$		−	7.96629i		−	0.323876i
$606$		0			0
$607$		−	23.2515i		−	0.943750i	−0.881665	−	0.471875i	$-0.843577\pi$
$607$		−	23.2515i		−	0.943750i	0.881665	−	0.471875i	$-0.156423\pi$
$608$		0			0
$609$	8.97776	+	19.1020i	0.363797	+	0.774054i
$610$		0			0
$611$	2.64266			0.106911
$612$		0			0
$613$	36.6495			1.48026			0.740130	−	0.672464i	$-0.234764\pi$
$613$	36.6495			1.48026			0.740130	+	0.672464i	$0.234764\pi$
$614$		0			0
$615$	−15.6567	−	33.3129i	−0.631340	−	1.34331i
$616$		0			0
$617$			3.28046i			0.132066i	0.997817	+	0.0660331i	$0.0210343\pi$
$617$			3.28046i			0.132066i	−0.997817	+	0.0660331i	$0.978966\pi$
$618$		0			0
$619$		−	4.33499i		−	0.174238i	−0.996198	−	0.0871190i	$-0.972234\pi$
$619$		−	4.33499i		−	0.174238i	0.996198	−	0.0871190i	$-0.0277660\pi$
$620$		0			0
$621$	−38.1342	−	9.84858i	−1.53027	−	0.395210i
$622$		0			0
$623$	−58.9618			−2.36226
$624$		0			0
$625$	−31.0487			−1.24195
$626$		0			0
$627$	1.75430	−	0.824503i	0.0700600	−	0.0329275i
$628$		0			0
$629$			8.47874i			0.338069i
$630$		0			0
$631$			39.2691i			1.56328i	0.623732	+	0.781638i	$0.285616\pi$
$631$			39.2691i			1.56328i	−0.623732	+	0.781638i	$0.714384\pi$
$632$		0			0
$633$	−23.9878	+	11.2740i	−0.953430	+	0.448102i
$634$		0			0
$635$	19.6237			0.778741
$636$		0			0
$637$	11.5620			0.458103
$638$		0			0
$639$	19.7012	−	23.7691i	0.779369	−	0.940292i
$640$		0			0
$641$		−	49.4099i		−	1.95157i	−0.218726	−	0.975786i	$-0.570190\pi$
$641$		−	49.4099i		−	1.95157i	0.218726	−	0.975786i	$-0.429810\pi$
$642$		0			0
$643$		−	12.6144i		−	0.497463i	−0.968573	−	0.248731i	$-0.919986\pi$
$643$		−	12.6144i		−	0.497463i	0.968573	−	0.248731i	$-0.0800136\pi$
$644$		0			0
$645$	6.46123	+	13.7476i	0.254411	+	0.541311i
$646$		0			0
$647$	−5.95489			−0.234111			−0.117055	−	0.993125i	$-0.537346\pi$
$647$	−5.95489			−0.234111			−0.117055	+	0.993125i	$0.537346\pi$
$648$		0			0
$649$	8.00000			0.314027
$650$		0			0
$651$	35.2804	+	75.0663i	1.38275	+	2.94208i
$652$		0			0
$653$		−	6.09222i		−	0.238407i	−0.992870	−	0.119203i	$-0.961966\pi$
$653$		−	6.09222i		−	0.238407i	0.992870	−	0.119203i	$-0.0380341\pi$
$654$		0			0
$655$		−	9.37594i		−	0.366348i
$656$		0			0
$657$	−21.8401	+	26.3497i	−0.852065	+	1.02800i
$658$		0			0
$659$	−6.82259			−0.265771			−0.132885	−	0.991131i	$-0.542424\pi$
$659$	−6.82259			−0.265771			−0.132885	+	0.991131i	$0.542424\pi$
$660$		0			0
$661$	−19.2561			−0.748975			−0.374488	−	0.927232i	$-0.622181\pi$
$661$	−19.2561			−0.748975			−0.374488	+	0.927232i	$0.622181\pi$
$662$		0			0
$663$	−2.62180	+	1.23222i	−0.101822	+	0.0478554i
$664$		0			0
$665$			4.52672i			0.175539i
$666$		0			0
$667$			21.4387i			0.830109i
$668$		0			0
$669$	−8.80908	+	4.14018i	−0.340579	+	0.160068i
$670$		0			0
$671$	−26.3553			−1.01744
$672$		0			0
$673$	6.77958			0.261334			0.130667	−	0.991426i	$-0.458288\pi$
$673$	6.77958			0.261334			0.130667	+	0.991426i	$0.458288\pi$
$674$		0			0
$675$	−10.3203	−	2.66533i	−0.397228	−	0.102589i
$676$		0			0
$677$		−	27.0889i		−	1.04111i	−0.853828	−	0.520555i	$-0.825725\pi$
$677$		−	27.0889i		−	1.04111i	0.853828	−	0.520555i	$-0.174275\pi$
$678$		0			0
$679$		−	2.70586i		−	0.103842i
$680$		0			0
$681$	8.81095	+	18.7471i	0.337636	+	0.718391i
$682$		0			0
$683$	−50.4159			−1.92911			−0.964556	−	0.263880i	$-0.914998\pi$
$683$	−50.4159			−1.92911			−0.964556	+	0.263880i	$0.914998\pi$
$684$		0			0
$685$	−31.0529			−1.18647
$686$		0			0
$687$	1.14833	+	2.44331i	0.0438116	+	0.0932182i
$688$		0			0
$689$		−	7.02015i		−	0.267447i
$690$		0			0
$691$			37.2311i			1.41634i	0.706043	+	0.708169i	$0.250479\pi$
$691$			37.2311i			1.41634i	−0.706043	+	0.708169i	$0.749521\pi$
$692$		0			0
$693$	28.1462	+	23.3292i	1.06919	+	0.886204i
$694$		0			0
$695$	−37.2346			−1.41239
$696$		0			0
$697$	13.3854			0.507009
$698$		0			0
$699$	−15.9379	+	7.49064i	−0.602826	+	0.283322i
$700$		0			0
$701$			27.8365i			1.05137i	0.850680	+	0.525684i	$0.176191\pi$
$701$			27.8365i			1.05137i	−0.850680	+	0.525684i	$0.823809\pi$
$702$		0			0
$703$			2.00581i			0.0756507i
$704$		0			0
$705$	−11.0001	+	5.16995i	−0.414289	+	0.194712i
$706$		0			0
$707$	−64.6872			−2.43281
$708$		0			0
$709$	31.4387			1.18070			0.590352	−	0.807146i	$-0.298989\pi$
$709$	31.4387			1.18070			0.590352	+	0.807146i	$0.298989\pi$
$710$		0			0
$711$	−29.4201	−	24.3851i	−1.10334	−	0.914512i
$712$		0			0
$713$			84.2487i			3.15514i
$714$		0			0
$715$		−	7.51069i		−	0.280884i
$716$		0			0
$717$	−21.8194	−	46.4254i	−0.814862	−	1.73379i
$718$		0			0
$719$	0.760439			0.0283596			0.0141798	−	0.999899i	$-0.495486\pi$
$719$	0.760439			0.0283596			0.0141798	+	0.999899i	$0.495486\pi$
$720$		0			0
$721$	−72.1108			−2.68555
$722$		0			0
$723$	3.25438	+	6.92436i	0.121032	+	0.257520i
$724$		0			0
$725$			5.80198i			0.215480i
$726$		0			0
$727$			36.8954i			1.36838i	0.729306	+	0.684188i	$0.239843\pi$
$727$			36.8954i			1.36838i	−0.729306	+	0.684188i	$0.760157\pi$
$728$		0			0
$729$	23.6235	+	13.0741i	0.874943	+	0.484226i
$730$		0			0
$731$	−5.52391			−0.204309
$732$		0			0
$733$	15.1052			0.557925			0.278963	−	0.960302i	$-0.410009\pi$
$733$	15.1052			0.557925			0.278963	+	0.960302i	$0.410009\pi$
$734$		0			0
$735$	−48.1271	+	22.6193i	−1.77520	+	0.834324i
$736$		0			0
$737$		−	21.5083i		−	0.792268i
$738$		0			0
$739$			12.1659i			0.447529i	0.974643	+	0.223765i	$0.0718347\pi$
$739$			12.1659i			0.447529i	−0.974643	+	0.223765i	$0.928165\pi$
$740$		0			0
$741$	−0.620239	+	0.291506i	−0.0227850	+	0.0107087i
$742$		0			0
$743$	−24.5113			−0.899234			−0.449617	−	0.893222i	$-0.648439\pi$
$743$	−24.5113			−0.899234			−0.449617	+	0.893222i	$0.648439\pi$
$744$		0			0
$745$	48.0848			1.76169
$746$		0			0
$747$	17.9144	−	21.6133i	0.655452	−	0.790790i
$748$		0			0
$749$			61.5958i			2.25066i
$750$		0			0
$751$		−	4.18136i		−	0.152580i	−0.997086	−	0.0762900i	$-0.975693\pi$
$751$		−	4.18136i		−	0.152580i	0.997086	−	0.0762900i	$-0.0243075\pi$
$752$		0			0
$753$	−10.7562	−	22.8861i	−0.391979	−	0.834015i
$754$		0			0
$755$	7.95857			0.289642
$756$		0			0
$757$	0.447272			0.0162564			0.00812818	−	0.999967i	$-0.497413\pi$
$757$	0.447272			0.0162564			0.00812818	+	0.999967i	$0.497413\pi$
$758$		0			0
$759$	15.7946	+	33.6063i	0.573308	+	1.21983i
$760$		0			0
$761$		−	19.6627i		−	0.712774i	−0.934338	−	0.356387i	$-0.884008\pi$
$761$		−	19.6627i		−	0.712774i	0.934338	−	0.356387i	$-0.115992\pi$
$762$		0			0
$763$			39.0741i			1.41458i
$764$		0			0
$765$	8.50267	−	10.2583i	0.307415	−	0.370889i
$766$		0			0
$767$	−2.82843			−0.102129
$768$		0			0
$769$	48.5400			1.75040			0.875199	−	0.483762i	$-0.160730\pi$
$769$	48.5400			1.75040			0.875199	+	0.483762i	$0.160730\pi$
$770$		0			0
$771$	26.9274	−	12.6556i	0.969766	−	0.455780i
$772$		0			0
$773$		−	3.31540i		−	0.119247i	−0.998221	−	0.0596233i	$-0.981010\pi$
$773$		−	3.31540i		−	0.119247i	0.998221	−	0.0596233i	$-0.0189899\pi$
$774$		0			0
$775$			22.8003i			0.819012i
$776$		0			0
$777$	−34.2364	+	16.0908i	−1.22823	+	0.577253i
$778$		0			0
$779$	3.16659			0.113455
$780$		0			0
$781$	−29.1069			−1.04153
$782$		0			0
$783$	3.67507	−	14.2300i	0.131336	−	0.508540i
$784$		0			0
$785$		−	29.7817i		−	1.06295i
$786$		0			0
$787$			9.53232i			0.339791i	0.985462	+	0.169895i	$0.0543430\pi$
$787$			9.53232i			0.339791i	−0.985462	+	0.169895i	$0.945657\pi$
$788$		0			0
$789$	−7.36972	−	15.6806i	−0.262369	−	0.558244i
$790$		0			0
$791$	−3.64764			−0.129695
$792$		0			0
$793$	9.31801			0.330892
$794$		0			0
$795$	13.7338	+	29.2216i	0.487089	+	1.03638i
$796$		0			0
$797$			30.5198i			1.08107i	0.841323	+	0.540533i	$0.181777\pi$
$797$			30.5198i			1.08107i	−0.841323	+	0.540533i	$0.818223\pi$
$798$		0			0
$799$		−	4.41996i		−	0.156367i
$800$		0			0
$801$	31.6098	+	26.2000i	1.11688	+	0.925733i
$802$		0			0
$803$	32.2669			1.13867
$804$		0			0
$805$	−86.7162			−3.05635
$806$		0			0
$807$	25.8724	−	12.1598i	0.910753	−	0.428045i
$808$		0			0
$809$			37.2270i			1.30883i	0.756135	+	0.654415i	$0.227085\pi$
$809$			37.2270i			1.30883i	−0.756135	+	0.654415i	$0.772915\pi$
$810$		0			0
$811$		−	1.56239i		−	0.0548630i	−0.999624	−	0.0274315i	$-0.991267\pi$
$811$		−	1.56239i		−	0.0548630i	0.999624	−	0.0274315i	$-0.00873281\pi$
$812$		0			0
$813$	−32.3559	+	15.2069i	−1.13477	+	0.533331i
$814$		0			0
$815$	26.2756			0.920394
$816$		0			0
$817$	−1.30679			−0.0457188
$818$		0			0
$819$	−9.95120	−	8.24813i	−0.347723	−	0.288213i
$820$		0			0
$821$			45.2205i			1.57821i	0.614261	+	0.789103i	$0.289454\pi$
$821$			45.2205i			1.57821i	−0.614261	+	0.789103i	$0.710546\pi$
$822$		0			0
$823$			3.01464i			0.105084i	0.998619	+	0.0525419i	$0.0167323\pi$
$823$			3.01464i			0.105084i	−0.998619	+	0.0525419i	$0.983268\pi$
$824$		0			0
$825$	4.27451	+	9.09491i	0.148819	+	0.316644i
$826$		0			0
$827$	−29.9711			−1.04220			−0.521099	−	0.853496i	$-0.674478\pi$
$827$	−29.9711			−1.04220			−0.521099	+	0.853496i	$0.674478\pi$
$828$		0			0
$829$	−6.01255			−0.208825			−0.104412	−	0.994534i	$-0.533296\pi$
$829$	−6.01255			−0.208825			−0.104412	+	0.994534i	$0.533296\pi$
$830$		0			0
$831$	4.05991	+	8.63830i	0.140837	+	0.299659i
$832$		0			0
$833$		−	19.3379i		−	0.670020i
$834$		0			0
$835$		−	27.1388i		−	0.939177i
$836$		0			0
$837$	14.4421	−	55.9205i	0.499192	−	1.93290i
$838$		0			0
$839$	4.85351			0.167562			0.0837808	−	0.996484i	$-0.473300\pi$
$839$	4.85351			0.167562			0.0837808	+	0.996484i	$0.473300\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		0			0
$843$	−26.0738	+	12.2544i	−0.898030	+	0.422065i
$844$		0			0
$845$			2.65543i			0.0913496i
$846$		0			0
$847$			12.9251i			0.444111i
$848$		0			0
$849$	5.69236	−	2.67535i	0.195361	−	0.0918178i
$850$		0			0
$851$	−38.4244			−1.31717
$852$		0			0
$853$	33.4762			1.14620			0.573101	−	0.819485i	$-0.305740\pi$
$853$	33.4762			1.14620			0.573101	+	0.819485i	$0.305740\pi$
$854$		0			0
$855$	2.01147	−	2.42680i	0.0687910	−	0.0829949i
$856$		0			0
$857$		−	4.40213i		−	0.150374i	−0.997169	−	0.0751870i	$-0.976045\pi$
$857$		−	4.40213i		−	0.150374i	0.997169	−	0.0751870i	$-0.0239554\pi$
$858$		0			0
$859$			31.0587i			1.05971i	0.848088	+	0.529855i	$0.177754\pi$
$859$			31.0587i			1.05971i	−0.848088	+	0.529855i	$0.822246\pi$
$860$		0			0
$861$	25.4026	+	54.0493i	0.865718	+	1.84199i
$862$		0			0
$863$	5.27488			0.179559			0.0897795	−	0.995962i	$-0.471384\pi$
$863$	5.27488			0.179559			0.0897795	+	0.995962i	$0.471384\pi$
$864$		0			0
$865$	26.6856			0.907338
$866$		0			0
$867$	−10.4635	−	22.2633i	−0.355360	−	0.756102i
$868$		0			0
$869$			36.0268i			1.22213i
$870$		0			0
$871$			7.60433i			0.257663i
$872$		0			0
$873$	−1.20237	+	1.45063i	−0.0406939	+	0.0490964i
$874$		0			0
$875$	33.7347			1.14044
$876$		0			0
$877$	−12.0491			−0.406870			−0.203435	−	0.979088i	$-0.565211\pi$
$877$	−12.0491			−0.406870			−0.203435	+	0.979088i	$0.565211\pi$
$878$		0			0
$879$	25.0189	−	11.7586i	0.843865	−	0.396608i
$880$		0			0
$881$		−	40.3919i		−	1.36084i	−0.732824	−	0.680418i	$-0.761798\pi$
$881$		−	40.3919i		−	1.36084i	0.732824	−	0.680418i	$-0.238202\pi$
$882$		0			0
$883$			40.1020i			1.34954i	0.738028	+	0.674770i	$0.235757\pi$
$883$			40.1020i			1.34954i	−0.738028	+	0.674770i	$0.764243\pi$
$884$		0			0
$885$	11.7734	−	5.53338i	0.395759	−	0.186003i
$886$		0			0
$887$	51.6230			1.73333			0.866665	−	0.498890i	$-0.166259\pi$
$887$	51.6230			1.73333			0.866665	+	0.498890i	$0.166259\pi$
$888$		0			0
$889$	−31.8389			−1.06784
$890$		0			0
$891$	−4.72288	−	25.0139i	−0.158223	−	0.837997i
$892$		0			0
$893$		−	1.04563i		−	0.0349906i
$894$		0			0
$895$			15.5147i			0.518598i
$896$		0			0
$897$	−5.58424	−	11.8816i	−0.186452	−	0.396716i
$898$		0			0
$899$	−31.4380			−1.04852
$900$		0			0
$901$	−11.7415			−0.391166
$902$		0			0
$903$	−10.4832	−	22.3051i	−0.348858	−	0.742267i
$904$		0			0
$905$		−	49.7258i		−	1.65294i
$906$		0			0
$907$		−	31.9862i		−	1.06208i	−0.847346	−	0.531042i	$-0.821801\pi$
$907$		−	31.9862i		−	1.06208i	0.847346	−	0.531042i	$-0.178199\pi$
$908$		0			0
$909$	34.6792	+	28.7441i	1.15024	+	0.953382i
$910$		0			0
$911$	−38.3560			−1.27079			−0.635395	−	0.772187i	$-0.719163\pi$
$911$	−38.3560			−1.27079			−0.635395	+	0.772187i	$0.719163\pi$
$912$		0			0
$913$	−26.4669			−0.875927
$914$		0			0
$915$	−38.7865	+	18.2292i	−1.28224	+	0.602640i
$916$		0			0
$917$			15.2122i			0.502351i
$918$		0			0
$919$			2.48941i			0.0821181i	0.999157	+	0.0410591i	$0.0130732\pi$
$919$			2.48941i			0.0821181i	−0.999157	+	0.0410591i	$0.986927\pi$
$920$		0			0
$921$	29.6901	−	13.9540i	0.978321	−	0.459801i
$922$		0			0
$923$	10.2908			0.338727
$924$		0			0
$925$	−10.3988			−0.341912
$926$		0			0
$927$	38.6590	+	32.0429i	1.26973	+	1.05243i
$928$		0			0
$929$			41.6415i			1.36621i	0.730319	+	0.683106i	$0.239371\pi$
$929$			41.6415i			1.36621i	−0.730319	+	0.683106i	$0.760629\pi$
$930$		0			0
$931$		−	4.57477i		−	0.149932i
$932$		0			0
$933$	−20.3154	−	43.2252i	−0.665096	−	1.41513i
$934$		0			0
$935$	−12.5620			−0.410820
$936$		0			0
$937$	−8.82870			−0.288421			−0.144211	−	0.989547i	$-0.546064\pi$
$937$	−8.82870			−0.288421			−0.144211	+	0.989547i	$0.546064\pi$
$938$		0			0
$939$	−22.9754	−	48.8849i	−0.749774	−	1.59530i
$940$		0			0
$941$			4.16400i			0.135742i	0.997694	+	0.0678712i	$0.0216207\pi$
$941$			4.16400i			0.135742i	−0.997694	+	0.0678712i	$0.978379\pi$
$942$		0			0
$943$			60.6608i			1.97539i
$944$		0			0
$945$	57.5583	+	14.8651i	1.87237	+	0.483562i
$946$		0			0
$947$	−21.2274			−0.689799			−0.344899	−	0.938640i	$-0.612087\pi$
$947$	−21.2274			−0.689799			−0.344899	+	0.938640i	$0.612087\pi$
$948$		0			0
$949$	−11.4081			−0.370322
$950$		0			0
$951$	−45.6931	+	21.4753i	−1.48170	+	0.696384i
$952$		0			0
$953$			10.4143i			0.337352i	0.985672	+	0.168676i	$0.0539492\pi$
$953$			10.4143i			0.337352i	−0.985672	+	0.168676i	$0.946051\pi$
$954$		0			0
$955$		−	33.1636i		−	1.07315i
$956$		0			0
$957$	−12.5404	+	5.89387i	−0.405374	+	0.190522i
$958$		0			0
$959$	50.3825			1.62694
$960$		0			0
$961$	−92.5436			−2.98528
$962$		0			0
$963$	27.3704	−	33.0219i	0.882000	−	1.06411i
$964$		0			0
$965$		−	39.6157i		−	1.27528i
$966$		0			0
$967$			5.30046i			0.170451i	0.996362	+	0.0852257i	$0.0271611\pi$
$967$			5.30046i			0.170451i	−0.996362	+	0.0852257i	$0.972839\pi$
$968$		0			0
$969$	0.487556	+	1.03738i	0.0156626	+	0.0333253i
$970$		0			0
$971$	−1.71170			−0.0549311			−0.0274655	−	0.999623i	$-0.508744\pi$
$971$	−1.71170			−0.0549311			−0.0274655	+	0.999623i	$0.508744\pi$
$972$		0			0
$973$	60.4121			1.93672
$974$		0			0
$975$	−1.51127	−	3.21553i	−0.0483993	−	0.102980i
$976$		0			0
$977$		−	39.0217i		−	1.24841i	−0.781259	−	0.624207i	$-0.785422\pi$
$977$		−	39.0217i		−	1.24841i	0.781259	−	0.624207i	$-0.214578\pi$
$978$		0			0
$979$		−	38.7083i		−	1.23712i
$980$		0			0
$981$	17.3628	−	20.9479i	0.554352	−	0.668814i
$982$		0			0
$983$	−23.6712			−0.754993			−0.377497	−	0.926011i	$-0.623215\pi$
$983$	−23.6712			−0.754993			−0.377497	+	0.926011i	$0.623215\pi$
$984$		0			0
$985$	−52.1833			−1.66270
$986$		0			0
$987$	17.8474	−	8.38811i	0.568090	−	0.266996i
$988$		0			0
$989$		−	25.0336i		−	0.796021i
$990$		0			0
$991$		−	1.11278i		−	0.0353485i	−0.999844	−	0.0176743i	$-0.994374\pi$
$991$		−	1.11278i		−	0.0353485i	0.999844	−	0.0176743i	$-0.00562619\pi$
$992$		0			0
$993$	6.93844	−	3.26100i	0.220185	−	0.103485i
$994$		0			0
$995$	−7.70261			−0.244189
$996$		0			0
$997$	−51.7107			−1.63769			−0.818846	−	0.574013i	$-0.805386\pi$
$997$	−51.7107			−1.63769			−0.818846	+	0.574013i	$0.805386\pi$
$998$		0			0
$999$	25.5044	+	6.58680i	0.806923	+	0.208397i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.2.d.q.1535.5		20
3.2	odd	2	inner	2496.2.d.q.1535.15		20
4.3	odd	2	inner	2496.2.d.q.1535.16		20
8.3	odd	2		1248.2.d.d.287.5	✓	20
8.5	even	2		1248.2.d.d.287.16	yes	20
12.11	even	2	inner	2496.2.d.q.1535.6		20
24.5	odd	2		1248.2.d.d.287.6	yes	20
24.11	even	2		1248.2.d.d.287.15	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.2.d.d.287.5	✓	20	8.3	odd	2
1248.2.d.d.287.6	yes	20	24.5	odd	2
1248.2.d.d.287.15	yes	20	24.11	even	2
1248.2.d.d.287.16	yes	20	8.5	even	2
2496.2.d.q.1535.5		20	1.1	even	1	trivial
2496.2.d.q.1535.6		20	12.11	even	2	inner
2496.2.d.q.1535.15		20	3.2	odd	2	inner
2496.2.d.q.1535.16		20	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2496.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.736734 - 1.56755i) q^{3} +2.65543i q^{5} -4.30836i q^{7} +(-1.91445 + 2.30974i) q^{9} +2.82843 q^{11} -1.00000 q^{13} +(4.16253 - 1.95635i) q^{15} +1.67254i q^{17} +0.395673i q^{19} +(-6.75359 + 3.17412i) q^{21} -7.57972 q^{23} -2.05131 q^{25} +(5.03108 + 1.29933i) q^{27} -2.82843i q^{29} -11.1150i q^{31} +(-2.08380 - 4.43371i) q^{33} +11.4406 q^{35} +5.06937 q^{37} +(0.736734 + 1.56755i) q^{39} -8.00304i q^{41} +3.30270i q^{43} +(-6.13335 - 5.08368i) q^{45} -2.64266 q^{47} -11.5620 q^{49} +(2.62180 - 1.23222i) q^{51} +7.02015i q^{53} +7.51069i q^{55} +(0.620239 - 0.291506i) q^{57} +2.82843 q^{59} -9.31801 q^{61} +(9.95120 + 8.24813i) q^{63} -2.65543i q^{65} -7.60433i q^{67} +(5.58424 + 11.8816i) q^{69} -10.2908 q^{71} +11.4081 q^{73} +(1.51127 + 3.21553i) q^{75} -12.1859i q^{77} +12.7374i q^{79} +(-1.66979 - 8.84374i) q^{81} -9.35747 q^{83} -4.44132 q^{85} +(-4.43371 + 2.08380i) q^{87} -13.6854i q^{89} +4.30836i q^{91} +(-17.4234 + 8.18881i) q^{93} -1.05068 q^{95} +0.628049 q^{97} +(-5.41487 + 6.53293i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{9} - 20 q^{13} + 12 q^{21} - 36 q^{25} - 16 q^{37} - 4 q^{45} - 76 q^{49} - 16 q^{57} + 56 q^{61} - 24 q^{69} + 88 q^{73} + 72 q^{81} - 56 q^{85} - 96 q^{93} - 72 q^{97}+O(q^{100}) \) 20 * q - 8 * q^9 - 20 * q^13 + 12 * q^21 - 36 * q^25 - 16 * q^37 - 4 * q^45 - 76 * q^49 - 16 * q^57 + 56 * q^61 - 24 * q^69 + 88 * q^73 + 72 * q^81 - 56 * q^85 - 96 * q^93 - 72 * q^97

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