LMFDB - Embedded newform 1336.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1336,2,Mod(1,1336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1336, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1336.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1336 = 2^{3} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1336.a (trivial)

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:	yes
Analytic conductor:	$10.6680137100$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 9x^{5} + 5x^{4} + 25x^{3} - 4x^{2} - 17x + 1 $ x^7 - x^6 - 9x^5 + 5x^4 + 25x^3 - 4x^2 - 17*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.71785$ of defining polynomial
Character	$\chi$	$=$	1336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.71785 q^{3} +2.12216 q^{5} +1.38168 q^{7} +4.38672 q^{9} +O(q^{10})$	$q-2.71785 q^{3} +2.12216 q^{5} +1.38168 q^{7} +4.38672 q^{9} -6.32172 q^{11} +1.31355 q^{13} -5.76771 q^{15} -3.64555 q^{17} +3.20814 q^{19} -3.75521 q^{21} -5.96178 q^{23} -0.496444 q^{25} -3.76889 q^{27} +3.77852 q^{29} +3.53458 q^{31} +17.1815 q^{33} +2.93215 q^{35} +10.2797 q^{37} -3.57002 q^{39} -8.65920 q^{41} +7.34106 q^{43} +9.30931 q^{45} -12.6864 q^{47} -5.09096 q^{49} +9.90807 q^{51} -4.25549 q^{53} -13.4157 q^{55} -8.71924 q^{57} -3.30431 q^{59} -12.6049 q^{61} +6.06105 q^{63} +2.78755 q^{65} -1.49164 q^{67} +16.2032 q^{69} -9.73373 q^{71} -16.2510 q^{73} +1.34926 q^{75} -8.73460 q^{77} -5.74713 q^{79} -2.91686 q^{81} +10.1117 q^{83} -7.73644 q^{85} -10.2695 q^{87} +16.3782 q^{89} +1.81490 q^{91} -9.60647 q^{93} +6.80818 q^{95} +0.796932 q^{97} -27.7316 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 6 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) $ 7 * q - 6 * q^3 - q^7 + 3 * q^9	$ 7 q - 6 q^{3} - q^{7} + 3 q^{9} - 6 q^{11} - 2 q^{13} - 9 q^{15} - 9 q^{17} - 3 q^{19} + 7 q^{21} - 10 q^{23} + 3 q^{25} - 12 q^{27} - 5 q^{29} - 21 q^{31} + 8 q^{33} - 12 q^{35} + 19 q^{37} - 27 q^{39} - 22 q^{41} - 19 q^{43} + 13 q^{45} - 13 q^{47} - 14 q^{49} + 4 q^{51} + 5 q^{53} - 17 q^{55} + 5 q^{57} - 18 q^{59} + 26 q^{61} - 20 q^{63} - 20 q^{65} - 27 q^{67} - 3 q^{69} - 46 q^{71} - 25 q^{73} - 19 q^{75} - 19 q^{77} - 22 q^{79} - 9 q^{81} + q^{83} - 11 q^{85} + 9 q^{87} - 3 q^{89} + 33 q^{93} - 40 q^{95} + 11 q^{97} - 21 q^{99}+O(q^{100}) $ 7 * q - 6 * q^3 - q^7 + 3 * q^9 - 6 * q^11 - 2 * q^13 - 9 * q^15 - 9 * q^17 - 3 * q^19 + 7 * q^21 - 10 * q^23 + 3 * q^25 - 12 * q^27 - 5 * q^29 - 21 * q^31 + 8 * q^33 - 12 * q^35 + 19 * q^37 - 27 * q^39 - 22 * q^41 - 19 * q^43 + 13 * q^45 - 13 * q^47 - 14 * q^49 + 4 * q^51 + 5 * q^53 - 17 * q^55 + 5 * q^57 - 18 * q^59 + 26 * q^61 - 20 * q^63 - 20 * q^65 - 27 * q^67 - 3 * q^69 - 46 * q^71 - 25 * q^73 - 19 * q^75 - 19 * q^77 - 22 * q^79 - 9 * q^81 + q^83 - 11 * q^85 + 9 * q^87 - 3 * q^89 + 33 * q^93 - 40 * q^95 + 11 * q^97 - 21 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−2.71785	−1.56915	−0.784576	−	0.620032i	$-0.787119\pi$
$3$	−2.71785	−1.56915	−0.784576	+	0.620032i	$0.787119\pi$
$4$	0	0
$5$	2.12216	0.949058	0.474529	−	0.880240i	$-0.342618\pi$
$5$	2.12216	0.949058	0.474529	+	0.880240i	$0.342618\pi$
$6$	0	0
$7$	1.38168	0.522227	0.261113	−	0.965308i	$-0.415910\pi$
$7$	1.38168	0.522227	0.261113	+	0.965308i	$0.415910\pi$
$8$	0	0
$9$	4.38672	1.46224
$10$	0	0
$11$	−6.32172	−1.90607	−0.953035	−	0.302860i	$-0.902058\pi$
$11$	−6.32172	−1.90607	−0.953035	+	0.302860i	$0.902058\pi$
$12$	0	0
$13$	1.31355	0.364312	0.182156	−	0.983270i	$-0.441692\pi$
$13$	1.31355	0.364312	0.182156	+	0.983270i	$0.441692\pi$
$14$	0	0
$15$	−5.76771	−1.48922
$16$	0	0
$17$	−3.64555	−0.884177	−0.442088	−	0.896972i	$-0.645762\pi$
$17$	−3.64555	−0.884177	−0.442088	+	0.896972i	$0.645762\pi$
$18$	0	0
$19$	3.20814	0.735997	0.367999	−	0.929826i	$-0.380043\pi$
$19$	3.20814	0.735997	0.367999	+	0.929826i	$0.380043\pi$
$20$	0	0
$21$	−3.75521	−0.819453
$22$	0	0
$23$	−5.96178	−1.24312	−0.621559	−	0.783368i	$-0.713500\pi$
$23$	−5.96178	−1.24312	−0.621559	+	0.783368i	$0.713500\pi$
$24$	0	0
$25$	−0.496444	−0.0992888
$26$	0	0
$27$	−3.76889	−0.725324
$28$	0	0
$29$	3.77852	0.701653	0.350827	−	0.936440i	$-0.385901\pi$
$29$	3.77852	0.701653	0.350827	+	0.936440i	$0.385901\pi$
$30$	0	0
$31$	3.53458	0.634830	0.317415	−	0.948287i	$-0.397185\pi$
$31$	3.53458	0.634830	0.317415	+	0.948287i	$0.397185\pi$
$32$	0	0
$33$	17.1815	2.99091
$34$	0	0
$35$	2.93215	0.495623
$36$	0	0
$37$	10.2797	1.68997	0.844984	−	0.534791i	$-0.179610\pi$
$37$	10.2797	1.68997	0.844984	+	0.534791i	$0.179610\pi$
$38$	0	0
$39$	−3.57002	−0.571661
$40$	0	0
$41$	−8.65920	−1.35234	−0.676170	−	0.736746i	$-0.736362\pi$
$41$	−8.65920	−1.35234	−0.676170	+	0.736746i	$0.736362\pi$
$42$	0	0
$43$	7.34106	1.11950	0.559751	−	0.828661i	$-0.310897\pi$
$43$	7.34106	1.11950	0.559751	+	0.828661i	$0.310897\pi$
$44$	0	0
$45$	9.30931	1.38775
$46$	0	0
$47$	−12.6864	−1.85051	−0.925254	−	0.379349i	$-0.876148\pi$
$47$	−12.6864	−1.85051	−0.925254	+	0.379349i	$0.876148\pi$
$48$	0	0
$49$	−5.09096	−0.727279
$50$	0	0
$51$	9.90807	1.38741
$52$	0	0
$53$	−4.25549	−0.584536	−0.292268	−	0.956336i	$-0.594410\pi$
$53$	−4.25549	−0.584536	−0.292268	+	0.956336i	$0.594410\pi$
$54$	0	0
$55$	−13.4157	−1.80897
$56$	0	0
$57$	−8.71924	−1.15489
$58$	0	0
$59$	−3.30431	−0.430185	−0.215092	−	0.976594i	$-0.569005\pi$
$59$	−3.30431	−0.430185	−0.215092	+	0.976594i	$0.569005\pi$
$60$	0	0
$61$	−12.6049	−1.61389	−0.806944	−	0.590628i	$-0.798880\pi$
$61$	−12.6049	−1.61389	−0.806944	+	0.590628i	$0.798880\pi$
$62$	0	0
$63$	6.06105	0.763620
$64$	0	0
$65$	2.78755	0.345753
$66$	0	0
$67$	−1.49164	−0.182233	−0.0911165	−	0.995840i	$-0.529044\pi$
$67$	−1.49164	−0.182233	−0.0911165	+	0.995840i	$0.529044\pi$
$68$	0	0
$69$	16.2032	1.95064
$70$	0	0
$71$	−9.73373	−1.15518	−0.577591	−	0.816327i	$-0.696007\pi$
$71$	−9.73373	−1.15518	−0.577591	+	0.816327i	$0.696007\pi$
$72$	0	0
$73$	−16.2510	−1.90204	−0.951020	−	0.309128i	$-0.899963\pi$
$73$	−16.2510	−1.90204	−0.951020	+	0.309128i	$0.899963\pi$
$74$	0	0
$75$	1.34926	0.155799
$76$	0	0
$77$	−8.73460	−0.995400
$78$	0	0
$79$	−5.74713	−0.646602	−0.323301	−	0.946296i	$-0.604793\pi$
$79$	−5.74713	−0.646602	−0.323301	+	0.946296i	$0.604793\pi$
$80$	0	0
$81$	−2.91686	−0.324096
$82$	0	0
$83$	10.1117	1.10991	0.554953	−	0.831881i	$-0.312736\pi$
$83$	10.1117	1.10991	0.554953	+	0.831881i	$0.312736\pi$
$84$	0	0
$85$	−7.73644	−0.839135
$86$	0	0
$87$	−10.2695	−1.10100
$88$	0	0
$89$	16.3782	1.73609	0.868044	−	0.496488i	$-0.165377\pi$
$89$	16.3782	1.73609	0.868044	+	0.496488i	$0.165377\pi$
$90$	0	0
$91$	1.81490	0.190253
$92$	0	0
$93$	−9.60647	−0.996145
$94$	0	0
$95$	6.80818	0.698504
$96$	0	0
$97$	0.796932	0.0809161	0.0404581	−	0.999181i	$-0.487118\pi$
$97$	0.796932	0.0809161	0.0404581	+	0.999181i	$0.487118\pi$
$98$	0	0
$99$	−27.7316	−2.78713
$100$	0	0
$101$	−0.945643	−0.0940950	−0.0470475	−	0.998893i	$-0.514981\pi$
$101$	−0.945643	−0.0940950	−0.0470475	+	0.998893i	$0.514981\pi$
$102$	0	0
$103$	0.487569	0.0480416	0.0240208	−	0.999711i	$-0.492353\pi$
$103$	0.487569	0.0480416	0.0240208	+	0.999711i	$0.492353\pi$
$104$	0	0
$105$	−7.96914	−0.777709
$106$	0	0
$107$	−11.5833	−1.11980	−0.559898	−	0.828561i	$-0.689160\pi$
$107$	−11.5833	−1.11980	−0.559898	+	0.828561i	$0.689160\pi$
$108$	0	0
$109$	−8.96419	−0.858614	−0.429307	−	0.903159i	$-0.641242\pi$
$109$	−8.96419	−0.858614	−0.429307	+	0.903159i	$0.641242\pi$
$110$	0	0
$111$	−27.9386	−2.65182
$112$	0	0
$113$	1.05981	0.0996981	0.0498491	−	0.998757i	$-0.484126\pi$
$113$	1.05981	0.0996981	0.0498491	+	0.998757i	$0.484126\pi$
$114$	0	0
$115$	−12.6518	−1.17979
$116$	0	0
$117$	5.76215	0.532711
$118$	0	0
$119$	−5.03699	−0.461741
$120$	0	0
$121$	28.9641	2.63310
$122$	0	0
$123$	23.5344	2.12203
$124$	0	0
$125$	−11.6643	−1.04329
$126$	0	0
$127$	−4.94980	−0.439223	−0.219612	−	0.975587i	$-0.570479\pi$
$127$	−4.94980	−0.439223	−0.219612	+	0.975587i	$0.570479\pi$
$128$	0	0
$129$	−19.9519	−1.75667
$130$	0	0
$131$	−10.2008	−0.891248	−0.445624	−	0.895220i	$-0.647018\pi$
$131$	−10.2008	−0.891248	−0.445624	+	0.895220i	$0.647018\pi$
$132$	0	0
$133$	4.43263	0.384357
$134$	0	0
$135$	−7.99819	−0.688374
$136$	0	0
$137$	3.13270	0.267644	0.133822	−	0.991005i	$-0.457275\pi$
$137$	3.13270	0.267644	0.133822	+	0.991005i	$0.457275\pi$
$138$	0	0
$139$	−2.31262	−0.196154	−0.0980771	−	0.995179i	$-0.531269\pi$
$139$	−2.31262	−0.196154	−0.0980771	+	0.995179i	$0.531269\pi$
$140$	0	0
$141$	34.4799	2.90373
$142$	0	0
$143$	−8.30386	−0.694404
$144$	0	0
$145$	8.01861	0.665910
$146$	0	0
$147$	13.8365	1.14121
$148$	0	0
$149$	−5.08091	−0.416245	−0.208122	−	0.978103i	$-0.566735\pi$
$149$	−5.08091	−0.416245	−0.208122	+	0.978103i	$0.566735\pi$
$150$	0	0
$151$	8.59828	0.699718	0.349859	−	0.936802i	$-0.386229\pi$
$151$	8.59828	0.699718	0.349859	+	0.936802i	$0.386229\pi$
$152$	0	0
$153$	−15.9920	−1.29288
$154$	0	0
$155$	7.50094	0.602490
$156$	0	0
$157$	−14.5021	−1.15739	−0.578697	−	0.815543i	$-0.696439\pi$
$157$	−14.5021	−1.15739	−0.578697	+	0.815543i	$0.696439\pi$
$158$	0	0
$159$	11.5658	0.917226
$160$	0	0
$161$	−8.23728	−0.649189
$162$	0	0
$163$	14.7797	1.15763	0.578817	−	0.815457i	$-0.303514\pi$
$163$	14.7797	1.15763	0.578817	+	0.815457i	$0.303514\pi$
$164$	0	0
$165$	36.4619	2.83855
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−11.2746	−0.867277
$170$	0	0
$171$	14.0732	1.07620
$172$	0	0
$173$	19.8570	1.50970	0.754850	−	0.655897i	$-0.227710\pi$
$173$	19.8570	1.50970	0.754850	+	0.655897i	$0.227710\pi$
$174$	0	0
$175$	−0.685928	−0.0518512
$176$	0	0
$177$	8.98063	0.675026
$178$	0	0
$179$	3.11996	0.233197	0.116598	−	0.993179i	$-0.462801\pi$
$179$	3.11996	0.233197	0.116598	+	0.993179i	$0.462801\pi$
$180$	0	0
$181$	−9.59225	−0.712987	−0.356493	−	0.934298i	$-0.616028\pi$
$181$	−9.59225	−0.712987	−0.356493	+	0.934298i	$0.616028\pi$
$182$	0	0
$183$	34.2581	2.53243
$184$	0	0
$185$	21.8151	1.60388
$186$	0	0
$187$	23.0462	1.68530
$188$	0	0
$189$	−5.20741	−0.378783
$190$	0	0
$191$	−18.4833	−1.33741	−0.668704	−	0.743529i	$-0.733151\pi$
$191$	−18.4833	−1.33741	−0.668704	+	0.743529i	$0.733151\pi$
$192$	0	0
$193$	−21.1551	−1.52278	−0.761389	−	0.648296i	$-0.775482\pi$
$193$	−21.1551	−1.52278	−0.761389	+	0.648296i	$0.775482\pi$
$194$	0	0
$195$	−7.57615	−0.542539
$196$	0	0
$197$	−21.2234	−1.51210	−0.756051	−	0.654513i	$-0.772874\pi$
$197$	−21.2234	−1.51210	−0.756051	+	0.654513i	$0.772874\pi$
$198$	0	0
$199$	−2.94751	−0.208944	−0.104472	−	0.994528i	$-0.533315\pi$
$199$	−2.94751	−0.208944	−0.104472	+	0.994528i	$0.533315\pi$
$200$	0	0
$201$	4.05406	0.285951
$202$	0	0
$203$	5.22071	0.366422
$204$	0	0
$205$	−18.3762	−1.28345
$206$	0	0
$207$	−26.1527	−1.81774
$208$	0	0
$209$	−20.2809	−1.40286
$210$	0	0
$211$	−17.4438	−1.20088	−0.600441	−	0.799669i	$-0.705008\pi$
$211$	−17.4438	−1.20088	−0.600441	+	0.799669i	$0.705008\pi$
$212$	0	0
$213$	26.4548	1.81266
$214$	0	0
$215$	15.5789	1.06247
$216$	0	0
$217$	4.88367	0.331525
$218$	0	0
$219$	44.1679	2.98459
$220$	0	0
$221$	−4.78860	−0.322116
$222$	0	0
$223$	14.1235	0.945781	0.472890	−	0.881121i	$-0.343211\pi$
$223$	14.1235	0.945781	0.472890	+	0.881121i	$0.343211\pi$
$224$	0	0
$225$	−2.17776	−0.145184
$226$	0	0
$227$	14.8108	0.983029	0.491515	−	0.870869i	$-0.336443\pi$
$227$	14.8108	0.983029	0.491515	+	0.870869i	$0.336443\pi$
$228$	0	0
$229$	−9.92697	−0.655993	−0.327996	−	0.944679i	$-0.606373\pi$
$229$	−9.92697	−0.655993	−0.327996	+	0.944679i	$0.606373\pi$
$230$	0	0
$231$	23.7394	1.56193
$232$	0	0
$233$	19.2666	1.26220	0.631098	−	0.775703i	$-0.282605\pi$
$233$	19.2666	1.26220	0.631098	+	0.775703i	$0.282605\pi$
$234$	0	0
$235$	−26.9226	−1.75624
$236$	0	0
$237$	15.6198	1.01462
$238$	0	0
$239$	14.4651	0.935667	0.467834	−	0.883817i	$-0.345035\pi$
$239$	14.4651	0.935667	0.467834	+	0.883817i	$0.345035\pi$
$240$	0	0
$241$	7.33586	0.472544	0.236272	−	0.971687i	$-0.424074\pi$
$241$	7.33586	0.472544	0.236272	+	0.971687i	$0.424074\pi$
$242$	0	0
$243$	19.2343	1.23388
$244$	0	0
$245$	−10.8038	−0.690230
$246$	0	0
$247$	4.21403	0.268133
$248$	0	0
$249$	−27.4822	−1.74161
$250$	0	0
$251$	−0.560389	−0.0353714	−0.0176857	−	0.999844i	$-0.505630\pi$
$251$	−0.560389	−0.0353714	−0.0176857	+	0.999844i	$0.505630\pi$
$252$	0	0
$253$	37.6887	2.36947
$254$	0	0
$255$	21.0265	1.31673
$256$	0	0
$257$	22.3271	1.39272	0.696362	−	0.717690i	$-0.254801\pi$
$257$	22.3271	1.39272	0.696362	+	0.717690i	$0.254801\pi$
$258$	0	0
$259$	14.2032	0.882546
$260$	0	0
$261$	16.5753	1.02599
$262$	0	0
$263$	4.03965	0.249095	0.124548	−	0.992214i	$-0.460252\pi$
$263$	4.03965	0.249095	0.124548	+	0.992214i	$0.460252\pi$
$264$	0	0
$265$	−9.03082	−0.554759
$266$	0	0
$267$	−44.5136	−2.72419
$268$	0	0
$269$	−0.0467585	−0.00285091	−0.00142546	−	0.999999i	$-0.500454\pi$
$269$	−0.0467585	−0.00285091	−0.00142546	+	0.999999i	$0.500454\pi$
$270$	0	0
$271$	−8.06143	−0.489697	−0.244849	−	0.969561i	$-0.578738\pi$
$271$	−8.06143	−0.489697	−0.244849	+	0.969561i	$0.578738\pi$
$272$	0	0
$273$	−4.93263	−0.298536
$274$	0	0
$275$	3.13838	0.189251
$276$	0	0
$277$	28.3493	1.70334	0.851672	−	0.524075i	$-0.175589\pi$
$277$	28.3493	1.70334	0.851672	+	0.524075i	$0.175589\pi$
$278$	0	0
$279$	15.5052	0.928273
$280$	0	0
$281$	15.2851	0.911830	0.455915	−	0.890023i	$-0.349312\pi$
$281$	15.2851	0.911830	0.455915	+	0.890023i	$0.349312\pi$
$282$	0	0
$283$	6.43024	0.382238	0.191119	−	0.981567i	$-0.438788\pi$
$283$	6.43024	0.382238	0.191119	+	0.981567i	$0.438788\pi$
$284$	0	0
$285$	−18.5036	−1.09606
$286$	0	0
$287$	−11.9643	−0.706228
$288$	0	0
$289$	−3.70994	−0.218232
$290$	0	0
$291$	−2.16594	−0.126970
$292$	0	0
$293$	−10.2633	−0.599587	−0.299794	−	0.954004i	$-0.596918\pi$
$293$	−10.2633	−0.599587	−0.299794	+	0.954004i	$0.596918\pi$
$294$	0	0
$295$	−7.01228	−0.408270
$296$	0	0
$297$	23.8259	1.38252
$298$	0	0
$299$	−7.83107	−0.452882
$300$	0	0
$301$	10.1430	0.584634
$302$	0	0
$303$	2.57012	0.147649
$304$	0	0
$305$	−26.7495	−1.53167
$306$	0	0
$307$	−11.6759	−0.666378	−0.333189	−	0.942860i	$-0.608125\pi$
$307$	−11.6759	−0.666378	−0.333189	+	0.942860i	$0.608125\pi$
$308$	0	0
$309$	−1.32514	−0.0753846
$310$	0	0
$311$	20.1645	1.14342	0.571712	−	0.820455i	$-0.306280\pi$
$311$	20.1645	1.14342	0.571712	+	0.820455i	$0.306280\pi$
$312$	0	0
$313$	−5.41890	−0.306295	−0.153147	−	0.988203i	$-0.548941\pi$
$313$	−5.41890	−0.306295	−0.153147	+	0.988203i	$0.548941\pi$
$314$	0	0
$315$	12.8625	0.724720
$316$	0	0
$317$	−4.37800	−0.245893	−0.122947	−	0.992413i	$-0.539234\pi$
$317$	−4.37800	−0.245893	−0.122947	+	0.992413i	$0.539234\pi$
$318$	0	0
$319$	−23.8867	−1.33740
$320$	0	0
$321$	31.4816	1.75713
$322$	0	0
$323$	−11.6954	−0.650752
$324$	0	0
$325$	−0.652101	−0.0361721
$326$	0	0
$327$	24.3633	1.34730
$328$	0	0
$329$	−17.5286	−0.966384
$330$	0	0
$331$	17.4368	0.958414	0.479207	−	0.877702i	$-0.340924\pi$
$331$	17.4368	0.958414	0.479207	+	0.877702i	$0.340924\pi$
$332$	0	0
$333$	45.0940	2.47114
$334$	0	0
$335$	−3.16550	−0.172950
$336$	0	0
$337$	24.1333	1.31462	0.657311	−	0.753619i	$-0.271694\pi$
$337$	24.1333	1.31462	0.657311	+	0.753619i	$0.271694\pi$
$338$	0	0
$339$	−2.88039	−0.156442
$340$	0	0
$341$	−22.3446	−1.21003
$342$	0	0
$343$	−16.7059	−0.902031
$344$	0	0
$345$	34.3858	1.85127
$346$	0	0
$347$	11.5803	0.621665	0.310833	−	0.950465i	$-0.399392\pi$
$347$	11.5803	0.621665	0.310833	+	0.950465i	$0.399392\pi$
$348$	0	0
$349$	−8.58123	−0.459342	−0.229671	−	0.973268i	$-0.573765\pi$
$349$	−8.58123	−0.459342	−0.229671	+	0.973268i	$0.573765\pi$
$350$	0	0
$351$	−4.95061	−0.264244
$352$	0	0
$353$	−8.96077	−0.476934	−0.238467	−	0.971151i	$-0.576645\pi$
$353$	−8.96077	−0.476934	−0.238467	+	0.971151i	$0.576645\pi$
$354$	0	0
$355$	−20.6565	−1.09633
$356$	0	0
$357$	13.6898	0.724541
$358$	0	0
$359$	7.52055	0.396920	0.198460	−	0.980109i	$-0.436406\pi$
$359$	7.52055	0.396920	0.198460	+	0.980109i	$0.436406\pi$
$360$	0	0
$361$	−8.70785	−0.458308
$362$	0	0
$363$	−78.7202	−4.13174
$364$	0	0
$365$	−34.4873	−1.80515
$366$	0	0
$367$	34.9627	1.82504	0.912520	−	0.409033i	$-0.134134\pi$
$367$	34.9627	1.82504	0.912520	+	0.409033i	$0.134134\pi$
$368$	0	0
$369$	−37.9855	−1.97744
$370$	0	0
$371$	−5.87973	−0.305260
$372$	0	0
$373$	16.8856	0.874302	0.437151	−	0.899388i	$-0.355987\pi$
$373$	16.8856	0.874302	0.437151	+	0.899388i	$0.355987\pi$
$374$	0	0
$375$	31.7019	1.63708
$376$	0	0
$377$	4.96325	0.255621
$378$	0	0
$379$	−11.9437	−0.613507	−0.306753	−	0.951789i	$-0.599243\pi$
$379$	−11.9437	−0.613507	−0.306753	+	0.951789i	$0.599243\pi$
$380$	0	0
$381$	13.4528	0.689208
$382$	0	0
$383$	0.353435	0.0180597	0.00902985	−	0.999959i	$-0.497126\pi$
$383$	0.353435	0.0180597	0.00902985	+	0.999959i	$0.497126\pi$
$384$	0	0
$385$	−18.5362	−0.944693
$386$	0	0
$387$	32.2032	1.63698
$388$	0	0
$389$	4.30463	0.218254	0.109127	−	0.994028i	$-0.465195\pi$
$389$	4.30463	0.218254	0.109127	+	0.994028i	$0.465195\pi$
$390$	0	0
$391$	21.7340	1.09914
$392$	0	0
$393$	27.7243	1.39850
$394$	0	0
$395$	−12.1963	−0.613663
$396$	0	0
$397$	15.5330	0.779579	0.389789	−	0.920904i	$-0.372548\pi$
$397$	15.5330	0.779579	0.389789	+	0.920904i	$0.372548\pi$
$398$	0	0
$399$	−12.0472	−0.603115
$400$	0	0
$401$	11.4347	0.571023	0.285512	−	0.958375i	$-0.407836\pi$
$401$	11.4347	0.571023	0.285512	+	0.958375i	$0.407836\pi$
$402$	0	0
$403$	4.64283	0.231276
$404$	0	0
$405$	−6.19004	−0.307586
$406$	0	0
$407$	−64.9852	−3.22120
$408$	0	0
$409$	−1.88436	−0.0931754	−0.0465877	−	0.998914i	$-0.514835\pi$
$409$	−1.88436	−0.0931754	−0.0465877	+	0.998914i	$0.514835\pi$
$410$	0	0
$411$	−8.51420	−0.419975
$412$	0	0
$413$	−4.56551	−0.224654
$414$	0	0
$415$	21.4587	1.05337
$416$	0	0
$417$	6.28537	0.307796
$418$	0	0
$419$	18.1403	0.886209	0.443105	−	0.896470i	$-0.353877\pi$
$419$	18.1403	0.886209	0.443105	+	0.896470i	$0.353877\pi$
$420$	0	0
$421$	−15.2573	−0.743597	−0.371799	−	0.928313i	$-0.621259\pi$
$421$	−15.2573	−0.743597	−0.371799	+	0.928313i	$0.621259\pi$
$422$	0	0
$423$	−55.6518	−2.70588
$424$	0	0
$425$	1.80981	0.0877888
$426$	0	0
$427$	−17.4159	−0.842815
$428$	0	0
$429$	22.5687	1.08963
$430$	0	0
$431$	−13.4969	−0.650123	−0.325061	−	0.945693i	$-0.605385\pi$
$431$	−13.4969	−0.650123	−0.325061	+	0.945693i	$0.605385\pi$
$432$	0	0
$433$	−35.1020	−1.68689	−0.843446	−	0.537214i	$-0.819477\pi$
$433$	−35.1020	−1.68689	−0.843446	+	0.537214i	$0.819477\pi$
$434$	0	0
$435$	−21.7934	−1.04491
$436$	0	0
$437$	−19.1262	−0.914931
$438$	0	0
$439$	1.80991	0.0863822	0.0431911	−	0.999067i	$-0.486248\pi$
$439$	1.80991	0.0863822	0.0431911	+	0.999067i	$0.486248\pi$
$440$	0	0
$441$	−22.3326	−1.06346
$442$	0	0
$443$	−36.0658	−1.71354	−0.856769	−	0.515700i	$-0.827532\pi$
$443$	−36.0658	−1.71354	−0.856769	+	0.515700i	$0.827532\pi$
$444$	0	0
$445$	34.7572	1.64765
$446$	0	0
$447$	13.8092	0.653151
$448$	0	0
$449$	−27.6935	−1.30694	−0.653468	−	0.756954i	$-0.726687\pi$
$449$	−27.6935	−1.30694	−0.653468	+	0.756954i	$0.726687\pi$
$450$	0	0
$451$	54.7410	2.57765
$452$	0	0
$453$	−23.3689	−1.09796
$454$	0	0
$455$	3.85151	0.180561
$456$	0	0
$457$	13.6438	0.638230	0.319115	−	0.947716i	$-0.396614\pi$
$457$	13.6438	0.638230	0.319115	+	0.947716i	$0.396614\pi$
$458$	0	0
$459$	13.7397	0.641314
$460$	0	0
$461$	−20.4875	−0.954200	−0.477100	−	0.878849i	$-0.658312\pi$
$461$	−20.4875	−0.954200	−0.477100	+	0.878849i	$0.658312\pi$
$462$	0	0
$463$	26.8665	1.24859	0.624297	−	0.781187i	$-0.285386\pi$
$463$	26.8665	1.24859	0.624297	+	0.781187i	$0.285386\pi$
$464$	0	0
$465$	−20.3865	−0.945399
$466$	0	0
$467$	1.38093	0.0639016	0.0319508	−	0.999489i	$-0.489828\pi$
$467$	1.38093	0.0639016	0.0319508	+	0.999489i	$0.489828\pi$
$468$	0	0
$469$	−2.06097	−0.0951669
$470$	0	0
$471$	39.4146	1.81613
$472$	0	0
$473$	−46.4081	−2.13385
$474$	0	0
$475$	−1.59266	−0.0730763
$476$	0	0
$477$	−18.6676	−0.854732
$478$	0	0
$479$	−7.85710	−0.359000	−0.179500	−	0.983758i	$-0.557448\pi$
$479$	−7.85710	−0.359000	−0.179500	+	0.983758i	$0.557448\pi$
$480$	0	0
$481$	13.5028	0.615675
$482$	0	0
$483$	22.3877	1.01868
$484$	0	0
$485$	1.69122	0.0767941
$486$	0	0
$487$	−28.6566	−1.29856	−0.649278	−	0.760551i	$-0.724929\pi$
$487$	−28.6566	−1.29856	−0.649278	+	0.760551i	$0.724929\pi$
$488$	0	0
$489$	−40.1690	−1.81650
$490$	0	0
$491$	35.7503	1.61339	0.806694	−	0.590970i	$-0.201255\pi$
$491$	35.7503	1.61339	0.806694	+	0.590970i	$0.201255\pi$
$492$	0	0
$493$	−13.7748	−0.620385
$494$	0	0
$495$	−58.8508	−2.64515
$496$	0	0
$497$	−13.4489	−0.603266
$498$	0	0
$499$	−12.7973	−0.572884	−0.286442	−	0.958098i	$-0.592473\pi$
$499$	−12.7973	−0.572884	−0.286442	+	0.958098i	$0.592473\pi$
$500$	0	0
$501$	2.71785	0.121425
$502$	0	0
$503$	−12.2345	−0.545510	−0.272755	−	0.962083i	$-0.587935\pi$
$503$	−12.2345	−0.545510	−0.272755	+	0.962083i	$0.587935\pi$
$504$	0	0
$505$	−2.00680	−0.0893016
$506$	0	0
$507$	30.6427	1.36089
$508$	0	0
$509$	18.9950	0.841940	0.420970	−	0.907075i	$-0.361690\pi$
$509$	18.9950	0.841940	0.420970	+	0.907075i	$0.361690\pi$
$510$	0	0
$511$	−22.4538	−0.993296
$512$	0	0
$513$	−12.0911	−0.533836
$514$	0	0
$515$	1.03470	0.0455943
$516$	0	0
$517$	80.2001	3.52720
$518$	0	0
$519$	−53.9684	−2.36895
$520$	0	0
$521$	5.90004	0.258485	0.129243	−	0.991613i	$-0.458745\pi$
$521$	5.90004	0.258485	0.129243	+	0.991613i	$0.458745\pi$
$522$	0	0
$523$	24.5549	1.07371	0.536855	−	0.843674i	$-0.319612\pi$
$523$	24.5549	1.07371	0.536855	+	0.843674i	$0.319612\pi$
$524$	0	0
$525$	1.86425	0.0813625
$526$	0	0
$527$	−12.8855	−0.561302
$528$	0	0
$529$	12.5428	0.545341
$530$	0	0
$531$	−14.4951	−0.629033
$532$	0	0
$533$	−11.3742	−0.492673
$534$	0	0
$535$	−24.5815	−1.06275
$536$	0	0
$537$	−8.47960	−0.365922
$538$	0	0
$539$	32.1836	1.38625
$540$	0	0
$541$	16.6133	0.714262	0.357131	−	0.934054i	$-0.383755\pi$
$541$	16.6133	0.714262	0.357131	+	0.934054i	$0.383755\pi$
$542$	0	0
$543$	26.0703	1.11878
$544$	0	0
$545$	−19.0234	−0.814874
$546$	0	0
$547$	6.78464	0.290090	0.145045	−	0.989425i	$-0.453667\pi$
$547$	6.78464	0.290090	0.145045	+	0.989425i	$0.453667\pi$
$548$	0	0
$549$	−55.2940	−2.35989
$550$	0	0
$551$	12.1220	0.516415
$552$	0	0
$553$	−7.94070	−0.337673
$554$	0	0
$555$	−59.2902	−2.51673
$556$	0	0
$557$	37.0895	1.57153	0.785766	−	0.618524i	$-0.212269\pi$
$557$	37.0895	1.57153	0.785766	+	0.618524i	$0.212269\pi$
$558$	0	0
$559$	9.64282	0.407848
$560$	0	0
$561$	−62.6361	−2.64450
$562$	0	0
$563$	25.3600	1.06880	0.534399	−	0.845233i	$-0.320538\pi$
$563$	25.3600	1.06880	0.534399	+	0.845233i	$0.320538\pi$
$564$	0	0
$565$	2.24908	0.0946193
$566$	0	0
$567$	−4.03017	−0.169251
$568$	0	0
$569$	−21.2838	−0.892262	−0.446131	−	0.894968i	$-0.647198\pi$
$569$	−21.2838	−0.892262	−0.446131	+	0.894968i	$0.647198\pi$
$570$	0	0
$571$	−36.0127	−1.50709	−0.753543	−	0.657399i	$-0.771657\pi$
$571$	−36.0127	−1.50709	−0.753543	+	0.657399i	$0.771657\pi$
$572$	0	0
$573$	50.2350	2.09860
$574$	0	0
$575$	2.95969	0.123428
$576$	0	0
$577$	−15.0352	−0.625923	−0.312961	−	0.949766i	$-0.601321\pi$
$577$	−15.0352	−0.625923	−0.312961	+	0.949766i	$0.601321\pi$
$578$	0	0
$579$	57.4964	2.38947
$580$	0	0
$581$	13.9712	0.579623
$582$	0	0
$583$	26.9020	1.11417
$584$	0	0
$585$	12.2282	0.505574
$586$	0	0
$587$	19.8891	0.820911	0.410456	−	0.911881i	$-0.365370\pi$
$587$	19.8891	0.820911	0.410456	+	0.911881i	$0.365370\pi$
$588$	0	0
$589$	11.3394	0.467233
$590$	0	0
$591$	57.6819	2.37272
$592$	0	0
$593$	−25.9542	−1.06581	−0.532905	−	0.846175i	$-0.678900\pi$
$593$	−25.9542	−1.06581	−0.532905	+	0.846175i	$0.678900\pi$
$594$	0	0
$595$	−10.6893	−0.438219
$596$	0	0
$597$	8.01090	0.327864
$598$	0	0
$599$	−29.4425	−1.20299	−0.601494	−	0.798878i	$-0.705427\pi$
$599$	−29.4425	−1.20299	−0.601494	+	0.798878i	$0.705427\pi$
$600$	0	0
$601$	20.5685	0.839005	0.419503	−	0.907754i	$-0.362204\pi$
$601$	20.5685	0.839005	0.419503	+	0.907754i	$0.362204\pi$
$602$	0	0
$603$	−6.54341	−0.266468
$604$	0	0
$605$	61.4665	2.49897
$606$	0	0
$607$	14.8400	0.602338	0.301169	−	0.953571i	$-0.402623\pi$
$607$	14.8400	0.602338	0.301169	+	0.953571i	$0.402623\pi$
$608$	0	0
$609$	−14.1891	−0.574972
$610$	0	0
$611$	−16.6642	−0.674162
$612$	0	0
$613$	22.8567	0.923175	0.461587	−	0.887095i	$-0.347280\pi$
$613$	22.8567	0.923175	0.461587	+	0.887095i	$0.347280\pi$
$614$	0	0
$615$	49.9438	2.01393
$616$	0	0
$617$	43.5745	1.75424	0.877121	−	0.480270i	$-0.159461\pi$
$617$	43.5745	1.75424	0.877121	+	0.480270i	$0.159461\pi$
$618$	0	0
$619$	31.7050	1.27433	0.637166	−	0.770727i	$-0.280107\pi$
$619$	31.7050	1.27433	0.637166	+	0.770727i	$0.280107\pi$
$620$	0	0
$621$	22.4693	0.901663
$622$	0	0
$623$	22.6295	0.906631
$624$	0	0
$625$	−22.2713	−0.890853
$626$	0	0
$627$	55.1206	2.20131
$628$	0	0
$629$	−37.4751	−1.49423
$630$	0	0
$631$	35.1082	1.39764	0.698818	−	0.715300i	$-0.253710\pi$
$631$	35.1082	1.39764	0.698818	+	0.715300i	$0.253710\pi$
$632$	0	0
$633$	47.4097	1.88437
$634$	0	0
$635$	−10.5043	−0.416849
$636$	0	0
$637$	−6.68720	−0.264956
$638$	0	0
$639$	−42.6991	−1.68915
$640$	0	0
$641$	−36.5975	−1.44552	−0.722758	−	0.691101i	$-0.757126\pi$
$641$	−36.5975	−1.44552	−0.722758	+	0.691101i	$0.757126\pi$
$642$	0	0
$643$	6.28922	0.248023	0.124011	−	0.992281i	$-0.460424\pi$
$643$	6.28922	0.248023	0.124011	+	0.992281i	$0.460424\pi$
$644$	0	0
$645$	−42.3411	−1.66718
$646$	0	0
$647$	23.5370	0.925336	0.462668	−	0.886532i	$-0.346892\pi$
$647$	23.5370	0.925336	0.462668	+	0.886532i	$0.346892\pi$
$648$	0	0
$649$	20.8889	0.819963
$650$	0	0
$651$	−13.2731	−0.520213
$652$	0	0
$653$	44.9870	1.76048	0.880239	−	0.474531i	$-0.157382\pi$
$653$	44.9870	1.76048	0.880239	+	0.474531i	$0.157382\pi$
$654$	0	0
$655$	−21.6477	−0.845846
$656$	0	0
$657$	−71.2887	−2.78124
$658$	0	0
$659$	−11.0964	−0.432254	−0.216127	−	0.976365i	$-0.569342\pi$
$659$	−11.0964	−0.432254	−0.216127	+	0.976365i	$0.569342\pi$
$660$	0	0
$661$	16.7216	0.650394	0.325197	−	0.945646i	$-0.394569\pi$
$661$	16.7216	0.650394	0.325197	+	0.945646i	$0.394569\pi$
$662$	0	0
$663$	13.0147	0.505449
$664$	0	0
$665$	9.40673	0.364777
$666$	0	0
$667$	−22.5267	−0.872237
$668$	0	0
$669$	−38.3856	−1.48407
$670$	0	0
$671$	79.6844	3.07618
$672$	0	0
$673$	−41.1517	−1.58628	−0.793141	−	0.609038i	$-0.791556\pi$
$673$	−41.1517	−1.58628	−0.793141	+	0.609038i	$0.791556\pi$
$674$	0	0
$675$	1.87104	0.0720165
$676$	0	0
$677$	0.672273	0.0258375	0.0129188	−	0.999917i	$-0.495888\pi$
$677$	0.672273	0.0258375	0.0129188	+	0.999917i	$0.495888\pi$
$678$	0	0
$679$	1.10111	0.0422566
$680$	0	0
$681$	−40.2536	−1.54252
$682$	0	0
$683$	−35.0937	−1.34282	−0.671412	−	0.741085i	$-0.734312\pi$
$683$	−35.0937	−1.34282	−0.671412	+	0.741085i	$0.734312\pi$
$684$	0	0
$685$	6.64808	0.254010
$686$	0	0
$687$	26.9800	1.02935
$688$	0	0
$689$	−5.58978	−0.212953
$690$	0	0
$691$	−1.13684	−0.0432474	−0.0216237	−	0.999766i	$-0.506884\pi$
$691$	−1.13684	−0.0432474	−0.0216237	+	0.999766i	$0.506884\pi$
$692$	0	0
$693$	−38.3162	−1.45551
$694$	0	0
$695$	−4.90776	−0.186162
$696$	0	0
$697$	31.5676	1.19571
$698$	0	0
$699$	−52.3637	−1.98058
$700$	0	0
$701$	−40.7141	−1.53775	−0.768875	−	0.639399i	$-0.779183\pi$
$701$	−40.7141	−1.53775	−0.768875	+	0.639399i	$0.779183\pi$
$702$	0	0
$703$	32.9786	1.24381
$704$	0	0
$705$	73.1717	2.75581
$706$	0	0
$707$	−1.30658	−0.0491389
$708$	0	0
$709$	−42.9875	−1.61443	−0.807214	−	0.590258i	$-0.799026\pi$
$709$	−42.9875	−1.61443	−0.807214	+	0.590258i	$0.799026\pi$
$710$	0	0
$711$	−25.2110	−0.945488
$712$	0	0
$713$	−21.0724	−0.789168
$714$	0	0
$715$	−17.6221	−0.659030
$716$	0	0
$717$	−39.3139	−1.46820
$718$	0	0
$719$	8.82916	0.329272	0.164636	−	0.986354i	$-0.447355\pi$
$719$	8.82916	0.329272	0.164636	+	0.986354i	$0.447355\pi$
$720$	0	0
$721$	0.673665	0.0250886
$722$	0	0
$723$	−19.9378	−0.741494
$724$	0	0
$725$	−1.87582	−0.0696663
$726$	0	0
$727$	25.7527	0.955116	0.477558	−	0.878600i	$-0.341522\pi$
$727$	25.7527	0.955116	0.477558	+	0.878600i	$0.341522\pi$
$728$	0	0
$729$	−43.5253	−1.61205
$730$	0	0
$731$	−26.7622	−0.989837
$732$	0	0
$733$	17.3507	0.640862	0.320431	−	0.947272i	$-0.396172\pi$
$733$	17.3507	0.640862	0.320431	+	0.947272i	$0.396172\pi$
$734$	0	0
$735$	29.3632	1.08308
$736$	0	0
$737$	9.42973	0.347349
$738$	0	0
$739$	−17.6448	−0.649075	−0.324537	−	0.945873i	$-0.605209\pi$
$739$	−17.6448	−0.649075	−0.324537	+	0.945873i	$0.605209\pi$
$740$	0	0
$741$	−11.4531	−0.420741
$742$	0	0
$743$	−9.36132	−0.343433	−0.171717	−	0.985146i	$-0.554931\pi$
$743$	−9.36132	−0.343433	−0.171717	+	0.985146i	$0.554931\pi$
$744$	0	0
$745$	−10.7825	−0.395040
$746$	0	0
$747$	44.3573	1.62295
$748$	0	0
$749$	−16.0044	−0.584788
$750$	0	0
$751$	−36.4063	−1.32848	−0.664242	−	0.747518i	$-0.731245\pi$
$751$	−36.4063	−1.32848	−0.664242	+	0.747518i	$0.731245\pi$
$752$	0	0
$753$	1.52305	0.0555031
$754$	0	0
$755$	18.2469	0.664073
$756$	0	0
$757$	−1.87511	−0.0681520	−0.0340760	−	0.999419i	$-0.510849\pi$
$757$	−1.87511	−0.0681520	−0.0340760	+	0.999419i	$0.510849\pi$
$758$	0	0
$759$	−102.432	−3.71806
$760$	0	0
$761$	12.2993	0.445848	0.222924	−	0.974836i	$-0.428440\pi$
$761$	12.2993	0.445848	0.222924	+	0.974836i	$0.428440\pi$
$762$	0	0
$763$	−12.3857	−0.448391
$764$	0	0
$765$	−33.9376	−1.22702
$766$	0	0
$767$	−4.34036	−0.156721
$768$	0	0
$769$	44.6462	1.60998	0.804992	−	0.593286i	$-0.202170\pi$
$769$	44.6462	1.60998	0.804992	+	0.593286i	$0.202170\pi$
$770$	0	0
$771$	−60.6817	−2.18540
$772$	0	0
$773$	−43.8955	−1.57881	−0.789405	−	0.613873i	$-0.789611\pi$
$773$	−43.8955	−1.57881	−0.789405	+	0.613873i	$0.789611\pi$
$774$	0	0
$775$	−1.75472	−0.0630315
$776$	0	0
$777$	−38.6023	−1.38485
$778$	0	0
$779$	−27.7799	−0.995319
$780$	0	0
$781$	61.5339	2.20186
$782$	0	0
$783$	−14.2408	−0.508926
$784$	0	0
$785$	−30.7758	−1.09843
$786$	0	0
$787$	37.4506	1.33497	0.667484	−	0.744624i	$-0.267371\pi$
$787$	37.4506	1.33497	0.667484	+	0.744624i	$0.267371\pi$
$788$	0	0
$789$	−10.9792	−0.390869
$790$	0	0
$791$	1.46431	0.0520650
$792$	0	0
$793$	−16.5571	−0.587958
$794$	0	0
$795$	24.5444	0.870501
$796$	0	0
$797$	9.10276	0.322436	0.161218	−	0.986919i	$-0.448458\pi$
$797$	9.10276	0.322436	0.161218	+	0.986919i	$0.448458\pi$
$798$	0	0
$799$	46.2491	1.63618
$800$	0	0
$801$	71.8466	2.53857
$802$	0	0
$803$	102.735	3.62542
$804$	0	0
$805$	−17.4808	−0.616118
$806$	0	0
$807$	0.127083	0.00447352
$808$	0	0
$809$	−49.2269	−1.73072	−0.865362	−	0.501147i	$-0.832911\pi$
$809$	−49.2269	−1.73072	−0.865362	+	0.501147i	$0.832911\pi$
$810$	0	0
$811$	−42.8106	−1.50328	−0.751642	−	0.659572i	$-0.770738\pi$
$811$	−42.8106	−1.50328	−0.751642	+	0.659572i	$0.770738\pi$
$812$	0	0
$813$	21.9098	0.768410
$814$	0	0
$815$	31.3648	1.09866
$816$	0	0
$817$	23.5511	0.823950
$818$	0	0
$819$	7.96146	0.278196
$820$	0	0
$821$	−19.5764	−0.683221	−0.341610	−	0.939842i	$-0.610972\pi$
$821$	−19.5764	−0.683221	−0.341610	+	0.939842i	$0.610972\pi$
$822$	0	0
$823$	−41.5485	−1.44829	−0.724144	−	0.689648i	$-0.757765\pi$
$823$	−41.5485	−1.44829	−0.724144	+	0.689648i	$0.757765\pi$
$824$	0	0
$825$	−8.52965	−0.296964
$826$	0	0
$827$	−53.6318	−1.86496	−0.932479	−	0.361223i	$-0.882359\pi$
$827$	−53.6318	−1.86496	−0.932479	+	0.361223i	$0.882359\pi$
$828$	0	0
$829$	26.4078	0.917180	0.458590	−	0.888648i	$-0.348355\pi$
$829$	26.4078	0.917180	0.458590	+	0.888648i	$0.348355\pi$
$830$	0	0
$831$	−77.0492	−2.67281
$832$	0	0
$833$	18.5594	0.643043
$834$	0	0
$835$	−2.12216	−0.0734403
$836$	0	0
$837$	−13.3215	−0.460457
$838$	0	0
$839$	−40.5767	−1.40086	−0.700431	−	0.713720i	$-0.747009\pi$
$839$	−40.5767	−1.40086	−0.700431	+	0.713720i	$0.747009\pi$
$840$	0	0
$841$	−14.7228	−0.507683
$842$	0	0
$843$	−41.5425	−1.43080
$844$	0	0
$845$	−23.9265	−0.823096
$846$	0	0
$847$	40.0192	1.37508
$848$	0	0
$849$	−17.4764	−0.599790
$850$	0	0
$851$	−61.2852	−2.10083
$852$	0	0
$853$	46.2264	1.58276	0.791381	−	0.611323i	$-0.209362\pi$
$853$	46.2264	1.58276	0.791381	+	0.611323i	$0.209362\pi$
$854$	0	0
$855$	29.8656	1.02138
$856$	0	0
$857$	−15.3409	−0.524036	−0.262018	−	0.965063i	$-0.584388\pi$
$857$	−15.3409	−0.524036	−0.262018	+	0.965063i	$0.584388\pi$
$858$	0	0
$859$	−29.5131	−1.00697	−0.503486	−	0.864003i	$-0.667950\pi$
$859$	−29.5131	−1.00697	−0.503486	+	0.864003i	$0.667950\pi$
$860$	0	0
$861$	32.5171	1.10818
$862$	0	0
$863$	19.4375	0.661660	0.330830	−	0.943690i	$-0.392671\pi$
$863$	19.4375	0.661660	0.330830	+	0.943690i	$0.392671\pi$
$864$	0	0
$865$	42.1397	1.43279
$866$	0	0
$867$	10.0831	0.342439
$868$	0	0
$869$	36.3317	1.23247
$870$	0	0
$871$	−1.95934	−0.0663896
$872$	0	0
$873$	3.49591	0.118319
$874$	0	0
$875$	−16.1164	−0.544833
$876$	0	0
$877$	44.0182	1.48639	0.743194	−	0.669076i	$-0.233310\pi$
$877$	44.0182	1.48639	0.743194	+	0.669076i	$0.233310\pi$
$878$	0	0
$879$	27.8941	0.940844
$880$	0	0
$881$	−0.413062	−0.0139164	−0.00695821	−	0.999976i	$-0.502215\pi$
$881$	−0.413062	−0.0139164	−0.00695821	+	0.999976i	$0.502215\pi$
$882$	0	0
$883$	11.0661	0.372403	0.186201	−	0.982512i	$-0.440382\pi$
$883$	11.0661	0.372403	0.186201	+	0.982512i	$0.440382\pi$
$884$	0	0
$885$	19.0583	0.640639
$886$	0	0
$887$	36.6893	1.23191	0.615953	−	0.787783i	$-0.288771\pi$
$887$	36.6893	1.23191	0.615953	+	0.787783i	$0.288771\pi$
$888$	0	0
$889$	−6.83904	−0.229374
$890$	0	0
$891$	18.4396	0.617749
$892$	0	0
$893$	−40.6998	−1.36197
$894$	0	0
$895$	6.62105	0.221317
$896$	0	0
$897$	21.2837	0.710642
$898$	0	0
$899$	13.3555	0.445430
$900$	0	0
$901$	15.5136	0.516833
$902$	0	0
$903$	−27.5672	−0.917379
$904$	0	0
$905$	−20.3563	−0.676666
$906$	0	0
$907$	−27.2986	−0.906434	−0.453217	−	0.891400i	$-0.649724\pi$
$907$	−27.2986	−0.906434	−0.453217	+	0.891400i	$0.649724\pi$
$908$	0	0
$909$	−4.14827	−0.137589
$910$	0	0
$911$	26.0291	0.862381	0.431191	−	0.902261i	$-0.358094\pi$
$911$	26.0291	0.862381	0.431191	+	0.902261i	$0.358094\pi$
$912$	0	0
$913$	−63.9235	−2.11556
$914$	0	0
$915$	72.7012	2.40343
$916$	0	0
$917$	−14.0943	−0.465433
$918$	0	0
$919$	37.4159	1.23424	0.617118	−	0.786871i	$-0.288300\pi$
$919$	37.4159	1.23424	0.617118	+	0.786871i	$0.288300\pi$
$920$	0	0
$921$	31.7333	1.04565
$922$	0	0
$923$	−12.7857	−0.420846
$924$	0	0
$925$	−5.10328	−0.167795
$926$	0	0
$927$	2.13883	0.0702483
$928$	0	0
$929$	−9.23818	−0.303095	−0.151547	−	0.988450i	$-0.548426\pi$
$929$	−9.23818	−0.303095	−0.151547	+	0.988450i	$0.548426\pi$
$930$	0	0
$931$	−16.3325	−0.535276
$932$	0	0
$933$	−54.8041	−1.79421
$934$	0	0
$935$	48.9076	1.59945
$936$	0	0
$937$	−11.9710	−0.391076	−0.195538	−	0.980696i	$-0.562645\pi$
$937$	−11.9710	−0.391076	−0.195538	+	0.980696i	$0.562645\pi$
$938$	0	0
$939$	14.7278	0.480623
$940$	0	0
$941$	25.4018	0.828076	0.414038	−	0.910259i	$-0.364118\pi$
$941$	25.4018	0.828076	0.414038	+	0.910259i	$0.364118\pi$
$942$	0	0
$943$	51.6243	1.68112
$944$	0	0
$945$	−11.0510	−0.359487
$946$	0	0
$947$	24.5700	0.798419	0.399209	−	0.916860i	$-0.369285\pi$
$947$	24.5700	0.798419	0.399209	+	0.916860i	$0.369285\pi$
$948$	0	0
$949$	−21.3465	−0.692936
$950$	0	0
$951$	11.8988	0.385844
$952$	0	0
$953$	47.9115	1.55201	0.776003	−	0.630729i	$-0.217244\pi$
$953$	47.9115	1.55201	0.776003	+	0.630729i	$0.217244\pi$
$954$	0	0
$955$	−39.2246	−1.26928
$956$	0	0
$957$	64.9206	2.09858
$958$	0	0
$959$	4.32839	0.139771
$960$	0	0
$961$	−18.5067	−0.596991
$962$	0	0
$963$	−50.8125	−1.63741
$964$	0	0
$965$	−44.8945	−1.44520
$966$	0	0
$967$	−18.2254	−0.586088	−0.293044	−	0.956099i	$-0.594668\pi$
$967$	−18.2254	−0.586088	−0.293044	+	0.956099i	$0.594668\pi$
$968$	0	0
$969$	31.7865	1.02113
$970$	0	0
$971$	42.7236	1.37106	0.685532	−	0.728042i	$-0.259570\pi$
$971$	42.7236	1.37106	0.685532	+	0.728042i	$0.259570\pi$
$972$	0	0
$973$	−3.19531	−0.102437
$974$	0	0
$975$	1.77232	0.0567595
$976$	0	0
$977$	−43.0646	−1.37776	−0.688879	−	0.724876i	$-0.741897\pi$
$977$	−43.0646	−1.37776	−0.688879	+	0.724876i	$0.741897\pi$
$978$	0	0
$979$	−103.538	−3.30910
$980$	0	0
$981$	−39.3234	−1.25550
$982$	0	0
$983$	1.78011	0.0567767	0.0283884	−	0.999597i	$-0.490962\pi$
$983$	1.78011	0.0567767	0.0283884	+	0.999597i	$0.490962\pi$
$984$	0	0
$985$	−45.0393	−1.43507
$986$	0	0
$987$	47.6402	1.51640
$988$	0	0
$989$	−43.7658	−1.39167
$990$	0	0
$991$	14.5061	0.460800	0.230400	−	0.973096i	$-0.425997\pi$
$991$	14.5061	0.460800	0.230400	+	0.973096i	$0.425997\pi$
$992$	0	0
$993$	−47.3907	−1.50390
$994$	0	0
$995$	−6.25509	−0.198300
$996$	0	0
$997$	22.0698	0.698957	0.349479	−	0.936944i	$-0.386359\pi$
$997$	22.0698	0.698957	0.349479	+	0.936944i	$0.386359\pi$
$998$	0	0
$999$	−38.7430	−1.22577

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1336.2.a.b.1.2	✓	7
4.3	odd	2		2672.2.a.l.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.b.1.2	✓	7	1.1	even	1	trivial
2672.2.a.l.1.6		7	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1336 = 2^{3} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1336.a (trivial)

\(f(q)\)	\(=\)	\(q-2.71785 q^{3} +2.12216 q^{5} +1.38168 q^{7} +4.38672 q^{9} +O(q^{10})\)	\(q-2.71785 q^{3} +2.12216 q^{5} +1.38168 q^{7} +4.38672 q^{9} -6.32172 q^{11} +1.31355 q^{13} -5.76771 q^{15} -3.64555 q^{17} +3.20814 q^{19} -3.75521 q^{21} -5.96178 q^{23} -0.496444 q^{25} -3.76889 q^{27} +3.77852 q^{29} +3.53458 q^{31} +17.1815 q^{33} +2.93215 q^{35} +10.2797 q^{37} -3.57002 q^{39} -8.65920 q^{41} +7.34106 q^{43} +9.30931 q^{45} -12.6864 q^{47} -5.09096 q^{49} +9.90807 q^{51} -4.25549 q^{53} -13.4157 q^{55} -8.71924 q^{57} -3.30431 q^{59} -12.6049 q^{61} +6.06105 q^{63} +2.78755 q^{65} -1.49164 q^{67} +16.2032 q^{69} -9.73373 q^{71} -16.2510 q^{73} +1.34926 q^{75} -8.73460 q^{77} -5.74713 q^{79} -2.91686 q^{81} +10.1117 q^{83} -7.73644 q^{85} -10.2695 q^{87} +16.3782 q^{89} +1.81490 q^{91} -9.60647 q^{93} +6.80818 q^{95} +0.796932 q^{97} -27.7316 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 6 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) \) 7 * q - 6 * q^3 - q^7 + 3 * q^9	\( 7 q - 6 q^{3} - q^{7} + 3 q^{9} - 6 q^{11} - 2 q^{13} - 9 q^{15} - 9 q^{17} - 3 q^{19} + 7 q^{21} - 10 q^{23} + 3 q^{25} - 12 q^{27} - 5 q^{29} - 21 q^{31} + 8 q^{33} - 12 q^{35} + 19 q^{37} - 27 q^{39} - 22 q^{41} - 19 q^{43} + 13 q^{45} - 13 q^{47} - 14 q^{49} + 4 q^{51} + 5 q^{53} - 17 q^{55} + 5 q^{57} - 18 q^{59} + 26 q^{61} - 20 q^{63} - 20 q^{65} - 27 q^{67} - 3 q^{69} - 46 q^{71} - 25 q^{73} - 19 q^{75} - 19 q^{77} - 22 q^{79} - 9 q^{81} + q^{83} - 11 q^{85} + 9 q^{87} - 3 q^{89} + 33 q^{93} - 40 q^{95} + 11 q^{97} - 21 q^{99}+O(q^{100}) \) 7 * q - 6 * q^3 - q^7 + 3 * q^9 - 6 * q^11 - 2 * q^13 - 9 * q^15 - 9 * q^17 - 3 * q^19 + 7 * q^21 - 10 * q^23 + 3 * q^25 - 12 * q^27 - 5 * q^29 - 21 * q^31 + 8 * q^33 - 12 * q^35 + 19 * q^37 - 27 * q^39 - 22 * q^41 - 19 * q^43 + 13 * q^45 - 13 * q^47 - 14 * q^49 + 4 * q^51 + 5 * q^53 - 17 * q^55 + 5 * q^57 - 18 * q^59 + 26 * q^61 - 20 * q^63 - 20 * q^65 - 27 * q^67 - 3 * q^69 - 46 * q^71 - 25 * q^73 - 19 * q^75 - 19 * q^77 - 22 * q^79 - 9 * q^81 + q^83 - 11 * q^85 + 9 * q^87 - 3 * q^89 + 33 * q^93 - 40 * q^95 + 11 * q^97 - 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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