LMFDB - Embedded newform 1336.2.a.e.1.8

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 15 x^{10} + 100 x^{9} + 36 x^{8} - 641 x^{7} + 129 x^{6} + 1804 x^{5} - 433 x^{4} + \cdots + 224 $ x^12 - 5x^11 - 15x^10 + 100x^9 + 36x^8 - 641x^7 + 129x^6 + 1804x^5 - 433x^4 - 2399x^3 + 148x^2 + 1224*x + 224
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.85311$ 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)

$f(q)$	$=$	$q+1.85311 q^{3} +3.00198 q^{5} +0.497697 q^{7} +0.434028 q^{9} +O(q^{10})$	$q+1.85311 q^{3} +3.00198 q^{5} +0.497697 q^{7} +0.434028 q^{9} +5.10991 q^{11} +1.20475 q^{13} +5.56301 q^{15} +1.51222 q^{17} -4.66674 q^{19} +0.922290 q^{21} -1.17497 q^{23} +4.01187 q^{25} -4.75504 q^{27} -8.43956 q^{29} +4.49118 q^{31} +9.46924 q^{33} +1.49408 q^{35} -0.519618 q^{37} +2.23254 q^{39} -6.36901 q^{41} +11.7452 q^{43} +1.30294 q^{45} -7.73171 q^{47} -6.75230 q^{49} +2.80231 q^{51} +8.72907 q^{53} +15.3398 q^{55} -8.64799 q^{57} +8.02160 q^{59} -0.243358 q^{61} +0.216015 q^{63} +3.61664 q^{65} -0.980848 q^{67} -2.17735 q^{69} -13.8181 q^{71} -0.526178 q^{73} +7.43446 q^{75} +2.54319 q^{77} +1.17133 q^{79} -10.1137 q^{81} -2.26900 q^{83} +4.53965 q^{85} -15.6395 q^{87} -4.82221 q^{89} +0.599602 q^{91} +8.32266 q^{93} -14.0094 q^{95} -9.80151 q^{97} +2.21784 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9}+O(q^{10}) $ 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9	$ 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9} + 14 q^{11} - 9 q^{13} + 8 q^{15} + 4 q^{17} + 9 q^{19} + 7 q^{21} + 15 q^{23} + 18 q^{25} + 20 q^{27} + 17 q^{29} + 11 q^{31} + 8 q^{33} + 19 q^{35} - 29 q^{37} + 26 q^{39} + 14 q^{41} + 15 q^{43} - 26 q^{45} + 17 q^{47} + 20 q^{49} + 36 q^{51} - 7 q^{53} + 13 q^{55} + 3 q^{57} + 32 q^{59} - 8 q^{61} + 50 q^{63} + 37 q^{65} + 39 q^{67} + q^{69} + 35 q^{71} - 12 q^{73} + 29 q^{75} + 13 q^{77} + 36 q^{79} + 44 q^{81} + 25 q^{83} - 38 q^{85} + 14 q^{87} + 23 q^{89} - 2 q^{91} - 25 q^{93} + 38 q^{95} - 2 q^{97} + 43 q^{99}+O(q^{100}) $ 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9 + 14 * q^11 - 9 * q^13 + 8 * q^15 + 4 * q^17 + 9 * q^19 + 7 * q^21 + 15 * q^23 + 18 * q^25 + 20 * q^27 + 17 * q^29 + 11 * q^31 + 8 * q^33 + 19 * q^35 - 29 * q^37 + 26 * q^39 + 14 * q^41 + 15 * q^43 - 26 * q^45 + 17 * q^47 + 20 * q^49 + 36 * q^51 - 7 * q^53 + 13 * q^55 + 3 * q^57 + 32 * q^59 - 8 * q^61 + 50 * q^63 + 37 * q^65 + 39 * q^67 + q^69 + 35 * q^71 - 12 * q^73 + 29 * q^75 + 13 * q^77 + 36 * q^79 + 44 * q^81 + 25 * q^83 - 38 * q^85 + 14 * q^87 + 23 * q^89 - 2 * q^91 - 25 * q^93 + 38 * q^95 - 2 * q^97 + 43 * 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
$3$	1.85311	1.06990	0.534948	−	0.844885i	$-0.320331\pi$
$3$	1.85311	1.06990	0.534948	+	0.844885i	$0.320331\pi$
$4$	0	0
$5$	3.00198	1.34253	0.671263	−	0.741219i	$-0.265752\pi$
$5$	3.00198	1.34253	0.671263	+	0.741219i	$0.265752\pi$
$6$	0	0
$7$	0.497697	0.188112	0.0940560	−	0.995567i	$-0.470017\pi$
$7$	0.497697	0.188112	0.0940560	+	0.995567i	$0.470017\pi$
$8$	0	0
$9$	0.434028	0.144676
$10$	0	0
$11$	5.10991	1.54070	0.770348	−	0.637624i	$-0.220083\pi$
$11$	5.10991	1.54070	0.770348	+	0.637624i	$0.220083\pi$
$12$	0	0
$13$	1.20475	0.334138	0.167069	−	0.985945i	$-0.446570\pi$
$13$	1.20475	0.334138	0.167069	+	0.985945i	$0.446570\pi$
$14$	0	0
$15$	5.56301	1.43636
$16$	0	0
$17$	1.51222	0.366767	0.183383	−	0.983041i	$-0.441295\pi$
$17$	1.51222	0.366767	0.183383	+	0.983041i	$0.441295\pi$
$18$	0	0
$19$	−4.66674	−1.07062	−0.535312	−	0.844655i	$-0.679806\pi$
$19$	−4.66674	−1.07062	−0.535312	+	0.844655i	$0.679806\pi$
$20$	0	0
$21$	0.922290	0.201260
$22$	0	0
$23$	−1.17497	−0.244998	−0.122499	−	0.992469i	$-0.539091\pi$
$23$	−1.17497	−0.244998	−0.122499	+	0.992469i	$0.539091\pi$
$24$	0	0
$25$	4.01187	0.802375
$26$	0	0
$27$	−4.75504	−0.915107
$28$	0	0
$29$	−8.43956	−1.56719	−0.783594	−	0.621273i	$-0.786616\pi$
$29$	−8.43956	−1.56719	−0.783594	+	0.621273i	$0.786616\pi$
$30$	0	0
$31$	4.49118	0.806639	0.403319	−	0.915059i	$-0.367856\pi$
$31$	4.49118	0.806639	0.403319	+	0.915059i	$0.367856\pi$
$32$	0	0
$33$	9.46924	1.64838
$34$	0	0
$35$	1.49408	0.252545
$36$	0	0
$37$	−0.519618	−0.0854247	−0.0427124	−	0.999087i	$-0.513600\pi$
$37$	−0.519618	−0.0854247	−0.0427124	+	0.999087i	$0.513600\pi$
$38$	0	0
$39$	2.23254	0.357493
$40$	0	0
$41$	−6.36901	−0.994673	−0.497336	−	0.867558i	$-0.665689\pi$
$41$	−6.36901	−0.994673	−0.497336	+	0.867558i	$0.665689\pi$
$42$	0	0
$43$	11.7452	1.79113	0.895564	−	0.444932i	$-0.146772\pi$
$43$	11.7452	1.79113	0.895564	+	0.444932i	$0.146772\pi$
$44$	0	0
$45$	1.30294	0.194231
$46$	0	0
$47$	−7.73171	−1.12779	−0.563893	−	0.825848i	$-0.690697\pi$
$47$	−7.73171	−1.12779	−0.563893	+	0.825848i	$0.690697\pi$
$48$	0	0
$49$	−6.75230	−0.964614
$50$	0	0
$51$	2.80231	0.392402
$52$	0	0
$53$	8.72907	1.19903	0.599515	−	0.800363i	$-0.295360\pi$
$53$	8.72907	1.19903	0.599515	+	0.800363i	$0.295360\pi$
$54$	0	0
$55$	15.3398	2.06842
$56$	0	0
$57$	−8.64799	−1.14545
$58$	0	0
$59$	8.02160	1.04432	0.522162	−	0.852846i	$-0.325126\pi$
$59$	8.02160	1.04432	0.522162	+	0.852846i	$0.325126\pi$
$60$	0	0
$61$	−0.243358	−0.0311588	−0.0155794	−	0.999879i	$-0.504959\pi$
$61$	−0.243358	−0.0311588	−0.0155794	+	0.999879i	$0.504959\pi$
$62$	0	0
$63$	0.216015	0.0272153
$64$	0	0
$65$	3.61664	0.448589
$66$	0	0
$67$	−0.980848	−0.119830	−0.0599148	−	0.998203i	$-0.519083\pi$
$67$	−0.980848	−0.119830	−0.0599148	+	0.998203i	$0.519083\pi$
$68$	0	0
$69$	−2.17735	−0.262122
$70$	0	0
$71$	−13.8181	−1.63990	−0.819951	−	0.572434i	$-0.805999\pi$
$71$	−13.8181	−1.63990	−0.819951	+	0.572434i	$0.805999\pi$
$72$	0	0
$73$	−0.526178	−0.0615845	−0.0307923	−	0.999526i	$-0.509803\pi$
$73$	−0.526178	−0.0615845	−0.0307923	+	0.999526i	$0.509803\pi$
$74$	0	0
$75$	7.43446	0.858457
$76$	0	0
$77$	2.54319	0.289823
$78$	0	0
$79$	1.17133	0.131784	0.0658922	−	0.997827i	$-0.479011\pi$
$79$	1.17133	0.131784	0.0658922	+	0.997827i	$0.479011\pi$
$80$	0	0
$81$	−10.1137	−1.12374
$82$	0	0
$83$	−2.26900	−0.249055	−0.124528	−	0.992216i	$-0.539742\pi$
$83$	−2.26900	−0.249055	−0.124528	+	0.992216i	$0.539742\pi$
$84$	0	0
$85$	4.53965	0.492394
$86$	0	0
$87$	−15.6395	−1.67673
$88$	0	0
$89$	−4.82221	−0.511153	−0.255577	−	0.966789i	$-0.582265\pi$
$89$	−4.82221	−0.511153	−0.255577	+	0.966789i	$0.582265\pi$
$90$	0	0
$91$	0.599602	0.0628554
$92$	0	0
$93$	8.32266	0.863019
$94$	0	0
$95$	−14.0094	−1.43734
$96$	0	0
$97$	−9.80151	−0.995193	−0.497596	−	0.867409i	$-0.665784\pi$
$97$	−9.80151	−0.995193	−0.497596	+	0.867409i	$0.665784\pi$
$98$	0	0
$99$	2.21784	0.222902
$100$	0	0
$101$	0.425894	0.0423780	0.0211890	−	0.999775i	$-0.493255\pi$
$101$	0.425894	0.0423780	0.0211890	+	0.999775i	$0.493255\pi$
$102$	0	0
$103$	7.97539	0.785839	0.392919	−	0.919573i	$-0.371465\pi$
$103$	7.97539	0.785839	0.392919	+	0.919573i	$0.371465\pi$
$104$	0	0
$105$	2.76869	0.270197
$106$	0	0
$107$	14.1186	1.36489	0.682446	−	0.730936i	$-0.260916\pi$
$107$	14.1186	1.36489	0.682446	+	0.730936i	$0.260916\pi$
$108$	0	0
$109$	5.15668	0.493920	0.246960	−	0.969026i	$-0.420568\pi$
$109$	5.15668	0.493920	0.246960	+	0.969026i	$0.420568\pi$
$110$	0	0
$111$	−0.962912	−0.0913955
$112$	0	0
$113$	−9.30765	−0.875590	−0.437795	−	0.899075i	$-0.644240\pi$
$113$	−9.30765	−0.875590	−0.437795	+	0.899075i	$0.644240\pi$
$114$	0	0
$115$	−3.52723	−0.328916
$116$	0	0
$117$	0.522896	0.0483418
$118$	0	0
$119$	0.752627	0.0689932
$120$	0	0
$121$	15.1112	1.37374
$122$	0	0
$123$	−11.8025	−1.06420
$124$	0	0
$125$	−2.96633	−0.265317
$126$	0	0
$127$	−0.542796	−0.0481654	−0.0240827	−	0.999710i	$-0.507666\pi$
$127$	−0.542796	−0.0481654	−0.0240827	+	0.999710i	$0.507666\pi$
$128$	0	0
$129$	21.7652	1.91632
$130$	0	0
$131$	9.66726	0.844632	0.422316	−	0.906449i	$-0.361217\pi$
$131$	9.66726	0.844632	0.422316	+	0.906449i	$0.361217\pi$
$132$	0	0
$133$	−2.32262	−0.201397
$134$	0	0
$135$	−14.2745	−1.22855
$136$	0	0
$137$	13.8344	1.18195	0.590975	−	0.806690i	$-0.298743\pi$
$137$	13.8344	1.18195	0.590975	+	0.806690i	$0.298743\pi$
$138$	0	0
$139$	16.9576	1.43833	0.719164	−	0.694840i	$-0.244525\pi$
$139$	16.9576	1.43833	0.719164	+	0.694840i	$0.244525\pi$
$140$	0	0
$141$	−14.3277	−1.20661
$142$	0	0
$143$	6.15617	0.514805
$144$	0	0
$145$	−25.3354	−2.10399
$146$	0	0
$147$	−12.5128	−1.03204
$148$	0	0
$149$	5.55615	0.455177	0.227589	−	0.973757i	$-0.426916\pi$
$149$	5.55615	0.455177	0.227589	+	0.973757i	$0.426916\pi$
$150$	0	0
$151$	0.620765	0.0505172	0.0252586	−	0.999681i	$-0.491959\pi$
$151$	0.620765	0.0505172	0.0252586	+	0.999681i	$0.491959\pi$
$152$	0	0
$153$	0.656346	0.0530624
$154$	0	0
$155$	13.4824	1.08293
$156$	0	0
$157$	−23.1285	−1.84586	−0.922928	−	0.384973i	$-0.874211\pi$
$157$	−23.1285	−1.84586	−0.922928	+	0.384973i	$0.874211\pi$
$158$	0	0
$159$	16.1760	1.28284
$160$	0	0
$161$	−0.584778	−0.0460870
$162$	0	0
$163$	3.52342	0.275975	0.137988	−	0.990434i	$-0.455937\pi$
$163$	3.52342	0.275975	0.137988	+	0.990434i	$0.455937\pi$
$164$	0	0
$165$	28.4264	2.21300
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−11.5486	−0.888352
$170$	0	0
$171$	−2.02550	−0.154894
$172$	0	0
$173$	−8.66666	−0.658914	−0.329457	−	0.944171i	$-0.606866\pi$
$173$	−8.66666	−0.658914	−0.329457	+	0.944171i	$0.606866\pi$
$174$	0	0
$175$	1.99670	0.150936
$176$	0	0
$177$	14.8649	1.11732
$178$	0	0
$179$	23.8203	1.78041	0.890207	−	0.455556i	$-0.150560\pi$
$179$	23.8203	1.78041	0.890207	+	0.455556i	$0.150560\pi$
$180$	0	0
$181$	−11.3423	−0.843069	−0.421534	−	0.906812i	$-0.638508\pi$
$181$	−11.3423	−0.843069	−0.421534	+	0.906812i	$0.638508\pi$
$182$	0	0
$183$	−0.450969	−0.0333366
$184$	0	0
$185$	−1.55988	−0.114685
$186$	0	0
$187$	7.72730	0.565076
$188$	0	0
$189$	−2.36657	−0.172143
$190$	0	0
$191$	12.0570	0.872416	0.436208	−	0.899846i	$-0.356321\pi$
$191$	12.0570	0.872416	0.436208	+	0.899846i	$0.356321\pi$
$192$	0	0
$193$	18.5308	1.33388	0.666939	−	0.745112i	$-0.267604\pi$
$193$	18.5308	1.33388	0.666939	+	0.745112i	$0.267604\pi$
$194$	0	0
$195$	6.70204	0.479943
$196$	0	0
$197$	−22.1225	−1.57616	−0.788080	−	0.615573i	$-0.788925\pi$
$197$	−22.1225	−1.57616	−0.788080	+	0.615573i	$0.788925\pi$
$198$	0	0
$199$	−24.2524	−1.71921	−0.859605	−	0.510959i	$-0.829291\pi$
$199$	−24.2524	−1.71921	−0.859605	+	0.510959i	$0.829291\pi$
$200$	0	0
$201$	−1.81762	−0.128205
$202$	0	0
$203$	−4.20035	−0.294807
$204$	0	0
$205$	−19.1196	−1.33537
$206$	0	0
$207$	−0.509969	−0.0354453
$208$	0	0
$209$	−23.8466	−1.64950
$210$	0	0
$211$	−21.3641	−1.47076	−0.735381	−	0.677654i	$-0.762997\pi$
$211$	−21.3641	−1.47076	−0.735381	+	0.677654i	$0.762997\pi$
$212$	0	0
$213$	−25.6064	−1.75452
$214$	0	0
$215$	35.2589	2.40464
$216$	0	0
$217$	2.23525	0.151738
$218$	0	0
$219$	−0.975068	−0.0658890
$220$	0	0
$221$	1.82185	0.122551
$222$	0	0
$223$	−0.565896	−0.0378952	−0.0189476	−	0.999820i	$-0.506032\pi$
$223$	−0.565896	−0.0378952	−0.0189476	+	0.999820i	$0.506032\pi$
$224$	0	0
$225$	1.74127	0.116084
$226$	0	0
$227$	16.4318	1.09061	0.545307	−	0.838236i	$-0.316413\pi$
$227$	16.4318	1.09061	0.545307	+	0.838236i	$0.316413\pi$
$228$	0	0
$229$	−14.2192	−0.939629	−0.469815	−	0.882765i	$-0.655679\pi$
$229$	−14.2192	−0.939629	−0.469815	+	0.882765i	$0.655679\pi$
$230$	0	0
$231$	4.71281	0.310080
$232$	0	0
$233$	5.61451	0.367818	0.183909	−	0.982943i	$-0.441125\pi$
$233$	5.61451	0.367818	0.183909	+	0.982943i	$0.441125\pi$
$234$	0	0
$235$	−23.2104	−1.51408
$236$	0	0
$237$	2.17060	0.140996
$238$	0	0
$239$	4.23918	0.274210	0.137105	−	0.990557i	$-0.456220\pi$
$239$	4.23918	0.274210	0.137105	+	0.990557i	$0.456220\pi$
$240$	0	0
$241$	12.8872	0.830138	0.415069	−	0.909790i	$-0.363757\pi$
$241$	12.8872	0.830138	0.415069	+	0.909790i	$0.363757\pi$
$242$	0	0
$243$	−4.47673	−0.287182
$244$	0	0
$245$	−20.2703	−1.29502
$246$	0	0
$247$	−5.62226	−0.357736
$248$	0	0
$249$	−4.20472	−0.266463
$250$	0	0
$251$	−21.6808	−1.36848	−0.684239	−	0.729258i	$-0.739865\pi$
$251$	−21.6808	−1.36848	−0.684239	+	0.729258i	$0.739865\pi$
$252$	0	0
$253$	−6.00398	−0.377467
$254$	0	0
$255$	8.41248	0.526810
$256$	0	0
$257$	−14.5966	−0.910511	−0.455255	−	0.890361i	$-0.650452\pi$
$257$	−14.5966	−0.910511	−0.455255	+	0.890361i	$0.650452\pi$
$258$	0	0
$259$	−0.258613	−0.0160694
$260$	0	0
$261$	−3.66301	−0.226735
$262$	0	0
$263$	−8.24735	−0.508553	−0.254277	−	0.967132i	$-0.581837\pi$
$263$	−8.24735	−0.508553	−0.254277	+	0.967132i	$0.581837\pi$
$264$	0	0
$265$	26.2045	1.60973
$266$	0	0
$267$	−8.93610	−0.546881
$268$	0	0
$269$	−3.05734	−0.186409	−0.0932046	−	0.995647i	$-0.529711\pi$
$269$	−3.05734	−0.186409	−0.0932046	+	0.995647i	$0.529711\pi$
$270$	0	0
$271$	−4.76742	−0.289600	−0.144800	−	0.989461i	$-0.546254\pi$
$271$	−4.76742	−0.289600	−0.144800	+	0.989461i	$0.546254\pi$
$272$	0	0
$273$	1.11113	0.0672487
$274$	0	0
$275$	20.5003	1.23621
$276$	0	0
$277$	3.07895	0.184996	0.0924982	−	0.995713i	$-0.470515\pi$
$277$	3.07895	0.184996	0.0924982	+	0.995713i	$0.470515\pi$
$278$	0	0
$279$	1.94930	0.116701
$280$	0	0
$281$	17.4668	1.04198	0.520991	−	0.853562i	$-0.325562\pi$
$281$	17.4668	1.04198	0.520991	+	0.853562i	$0.325562\pi$
$282$	0	0
$283$	12.6412	0.751444	0.375722	−	0.926732i	$-0.377395\pi$
$283$	12.6412	0.751444	0.375722	+	0.926732i	$0.377395\pi$
$284$	0	0
$285$	−25.9611	−1.53780
$286$	0	0
$287$	−3.16984	−0.187110
$288$	0	0
$289$	−14.7132	−0.865482
$290$	0	0
$291$	−18.1633	−1.06475
$292$	0	0
$293$	−1.08460	−0.0633628	−0.0316814	−	0.999498i	$-0.510086\pi$
$293$	−1.08460	−0.0633628	−0.0316814	+	0.999498i	$0.510086\pi$
$294$	0	0
$295$	24.0807	1.40203
$296$	0	0
$297$	−24.2978	−1.40990
$298$	0	0
$299$	−1.41554	−0.0818631
$300$	0	0
$301$	5.84556	0.336933
$302$	0	0
$303$	0.789230	0.0453401
$304$	0	0
$305$	−0.730555	−0.0418314
$306$	0	0
$307$	−3.26851	−0.186544	−0.0932720	−	0.995641i	$-0.529733\pi$
$307$	−3.26851	−0.186544	−0.0932720	+	0.995641i	$0.529733\pi$
$308$	0	0
$309$	14.7793	0.840765
$310$	0	0
$311$	1.58826	0.0900622	0.0450311	−	0.998986i	$-0.485661\pi$
$311$	1.58826	0.0900622	0.0450311	+	0.998986i	$0.485661\pi$
$312$	0	0
$313$	−24.3415	−1.37586	−0.687931	−	0.725776i	$-0.741481\pi$
$313$	−24.3415	−1.37586	−0.687931	+	0.725776i	$0.741481\pi$
$314$	0	0
$315$	0.648471	0.0365372
$316$	0	0
$317$	24.2276	1.36076	0.680378	−	0.732861i	$-0.261816\pi$
$317$	24.2276	1.36076	0.680378	+	0.732861i	$0.261816\pi$
$318$	0	0
$319$	−43.1254	−2.41456
$320$	0	0
$321$	26.1633	1.46029
$322$	0	0
$323$	−7.05713	−0.392669
$324$	0	0
$325$	4.83331	0.268104
$326$	0	0
$327$	9.55591	0.528443
$328$	0	0
$329$	−3.84805	−0.212150
$330$	0	0
$331$	9.93092	0.545853	0.272926	−	0.962035i	$-0.412008\pi$
$331$	9.93092	0.545853	0.272926	+	0.962035i	$0.412008\pi$
$332$	0	0
$333$	−0.225529	−0.0123589
$334$	0	0
$335$	−2.94448	−0.160874
$336$	0	0
$337$	−23.6467	−1.28812	−0.644058	−	0.764976i	$-0.722751\pi$
$337$	−23.6467	−1.28812	−0.644058	+	0.764976i	$0.722751\pi$
$338$	0	0
$339$	−17.2481	−0.936790
$340$	0	0
$341$	22.9495	1.24278
$342$	0	0
$343$	−6.84448	−0.369567
$344$	0	0
$345$	−6.53635	−0.351905
$346$	0	0
$347$	−21.8624	−1.17364	−0.586818	−	0.809719i	$-0.699620\pi$
$347$	−21.8624	−1.17364	−0.586818	+	0.809719i	$0.699620\pi$
$348$	0	0
$349$	7.12642	0.381468	0.190734	−	0.981642i	$-0.438913\pi$
$349$	7.12642	0.381468	0.190734	+	0.981642i	$0.438913\pi$
$350$	0	0
$351$	−5.72864	−0.305772
$352$	0	0
$353$	3.97154	0.211384	0.105692	−	0.994399i	$-0.466294\pi$
$353$	3.97154	0.211384	0.105692	+	0.994399i	$0.466294\pi$
$354$	0	0
$355$	−41.4815	−2.20161
$356$	0	0
$357$	1.39470	0.0738155
$358$	0	0
$359$	−33.1772	−1.75102	−0.875512	−	0.483197i	$-0.839475\pi$
$359$	−33.1772	−1.75102	−0.875512	+	0.483197i	$0.839475\pi$
$360$	0	0
$361$	2.77844	0.146234
$362$	0	0
$363$	28.0027	1.46976
$364$	0	0
$365$	−1.57958	−0.0826788
$366$	0	0
$367$	18.8929	0.986201	0.493100	−	0.869972i	$-0.335864\pi$
$367$	18.8929	0.986201	0.493100	+	0.869972i	$0.335864\pi$
$368$	0	0
$369$	−2.76433	−0.143905
$370$	0	0
$371$	4.34444	0.225552
$372$	0	0
$373$	−30.0134	−1.55404	−0.777018	−	0.629478i	$-0.783269\pi$
$373$	−30.0134	−1.55404	−0.777018	+	0.629478i	$0.783269\pi$
$374$	0	0
$375$	−5.49695	−0.283861
$376$	0	0
$377$	−10.1676	−0.523657
$378$	0	0
$379$	25.8908	1.32992	0.664961	−	0.746878i	$-0.268448\pi$
$379$	25.8908	1.32992	0.664961	+	0.746878i	$0.268448\pi$
$380$	0	0
$381$	−1.00586	−0.0515319
$382$	0	0
$383$	−16.8716	−0.862099	−0.431050	−	0.902328i	$-0.641857\pi$
$383$	−16.8716	−0.862099	−0.431050	+	0.902328i	$0.641857\pi$
$384$	0	0
$385$	7.63459	0.389095
$386$	0	0
$387$	5.09776	0.259133
$388$	0	0
$389$	−21.1954	−1.07465	−0.537326	−	0.843375i	$-0.680565\pi$
$389$	−21.1954	−1.07465	−0.537326	+	0.843375i	$0.680565\pi$
$390$	0	0
$391$	−1.77681	−0.0898570
$392$	0	0
$393$	17.9145	0.903668
$394$	0	0
$395$	3.51629	0.176924
$396$	0	0
$397$	−1.05601	−0.0529997	−0.0264999	−	0.999649i	$-0.508436\pi$
$397$	−1.05601	−0.0529997	−0.0264999	+	0.999649i	$0.508436\pi$
$398$	0	0
$399$	−4.30408	−0.215474
$400$	0	0
$401$	20.3497	1.01622	0.508109	−	0.861293i	$-0.330345\pi$
$401$	20.3497	1.01622	0.508109	+	0.861293i	$0.330345\pi$
$402$	0	0
$403$	5.41075	0.269529
$404$	0	0
$405$	−30.3611	−1.50866
$406$	0	0
$407$	−2.65520	−0.131613
$408$	0	0
$409$	39.2942	1.94297	0.971486	−	0.237097i	$-0.0761959\pi$
$409$	39.2942	1.94297	0.971486	+	0.237097i	$0.0761959\pi$
$410$	0	0
$411$	25.6367	1.26456
$412$	0	0
$413$	3.99233	0.196450
$414$	0	0
$415$	−6.81149	−0.334363
$416$	0	0
$417$	31.4244	1.53886
$418$	0	0
$419$	18.4942	0.903503	0.451751	−	0.892144i	$-0.350799\pi$
$419$	18.4942	0.903503	0.451751	+	0.892144i	$0.350799\pi$
$420$	0	0
$421$	−7.92041	−0.386017	−0.193009	−	0.981197i	$-0.561825\pi$
$421$	−7.92041	−0.386017	−0.193009	+	0.981197i	$0.561825\pi$
$422$	0	0
$423$	−3.35578	−0.163164
$424$	0	0
$425$	6.06683	0.294285
$426$	0	0
$427$	−0.121119	−0.00586134
$428$	0	0
$429$	11.4081	0.550788
$430$	0	0
$431$	17.6624	0.850766	0.425383	−	0.905014i	$-0.360139\pi$
$431$	17.6624	0.850766	0.425383	+	0.905014i	$0.360139\pi$
$432$	0	0
$433$	18.5094	0.889505	0.444753	−	0.895653i	$-0.353292\pi$
$433$	18.5094	0.889505	0.444753	+	0.895653i	$0.353292\pi$
$434$	0	0
$435$	−46.9493	−2.25105
$436$	0	0
$437$	5.48326	0.262300
$438$	0	0
$439$	−13.1950	−0.629765	−0.314882	−	0.949131i	$-0.601965\pi$
$439$	−13.1950	−0.629765	−0.314882	+	0.949131i	$0.601965\pi$
$440$	0	0
$441$	−2.93069	−0.139557
$442$	0	0
$443$	31.5363	1.49834	0.749168	−	0.662380i	$-0.230453\pi$
$443$	31.5363	1.49834	0.749168	+	0.662380i	$0.230453\pi$
$444$	0	0
$445$	−14.4762	−0.686236
$446$	0	0
$447$	10.2962	0.486992
$448$	0	0
$449$	20.2646	0.956346	0.478173	−	0.878266i	$-0.341299\pi$
$449$	20.2646	0.956346	0.478173	+	0.878266i	$0.341299\pi$
$450$	0	0
$451$	−32.5451	−1.53249
$452$	0	0
$453$	1.15035	0.0540481
$454$	0	0
$455$	1.79999	0.0843849
$456$	0	0
$457$	−14.3305	−0.670352	−0.335176	−	0.942155i	$-0.608796\pi$
$457$	−14.3305	−0.670352	−0.335176	+	0.942155i	$0.608796\pi$
$458$	0	0
$459$	−7.19065	−0.335631
$460$	0	0
$461$	15.8846	0.739821	0.369911	−	0.929067i	$-0.379388\pi$
$461$	15.8846	0.739821	0.369911	+	0.929067i	$0.379388\pi$
$462$	0	0
$463$	6.13068	0.284917	0.142458	−	0.989801i	$-0.454499\pi$
$463$	6.13068	0.284917	0.142458	+	0.989801i	$0.454499\pi$
$464$	0	0
$465$	24.9844	1.15863
$466$	0	0
$467$	−37.2918	−1.72566	−0.862830	−	0.505494i	$-0.831310\pi$
$467$	−37.2918	−1.72566	−0.862830	+	0.505494i	$0.831310\pi$
$468$	0	0
$469$	−0.488165	−0.0225414
$470$	0	0
$471$	−42.8597	−1.97487
$472$	0	0
$473$	60.0170	2.75958
$474$	0	0
$475$	−18.7224	−0.859041
$476$	0	0
$477$	3.78866	0.173471
$478$	0	0
$479$	−36.7675	−1.67995	−0.839975	−	0.542626i	$-0.817430\pi$
$479$	−36.7675	−1.67995	−0.839975	+	0.542626i	$0.817430\pi$
$480$	0	0
$481$	−0.626011	−0.0285437
$482$	0	0
$483$	−1.08366	−0.0493082
$484$	0	0
$485$	−29.4239	−1.33607
$486$	0	0
$487$	−28.9705	−1.31278	−0.656390	−	0.754421i	$-0.727918\pi$
$487$	−28.9705	−1.31278	−0.656390	+	0.754421i	$0.727918\pi$
$488$	0	0
$489$	6.52929	0.295265
$490$	0	0
$491$	−19.4947	−0.879784	−0.439892	−	0.898051i	$-0.644983\pi$
$491$	−19.4947	−0.879784	−0.439892	+	0.898051i	$0.644983\pi$
$492$	0	0
$493$	−12.7625	−0.574793
$494$	0	0
$495$	6.65792	0.299251
$496$	0	0
$497$	−6.87721	−0.308485
$498$	0	0
$499$	−29.8417	−1.33590	−0.667948	−	0.744208i	$-0.732827\pi$
$499$	−29.8417	−1.33590	−0.667948	+	0.744208i	$0.732827\pi$
$500$	0	0
$501$	1.85311	0.0827910
$502$	0	0
$503$	10.6274	0.473853	0.236927	−	0.971528i	$-0.423860\pi$
$503$	10.6274	0.473853	0.236927	+	0.971528i	$0.423860\pi$
$504$	0	0
$505$	1.27852	0.0568936
$506$	0	0
$507$	−21.4008	−0.950443
$508$	0	0
$509$	3.90778	0.173209	0.0866046	−	0.996243i	$-0.472398\pi$
$509$	3.90778	0.173209	0.0866046	+	0.996243i	$0.472398\pi$
$510$	0	0
$511$	−0.261878	−0.0115848
$512$	0	0
$513$	22.1905	0.979735
$514$	0	0
$515$	23.9420	1.05501
$516$	0	0
$517$	−39.5083	−1.73757
$518$	0	0
$519$	−16.0603	−0.704969
$520$	0	0
$521$	−37.6443	−1.64923	−0.824613	−	0.565698i	$-0.808607\pi$
$521$	−37.6443	−1.64923	−0.824613	+	0.565698i	$0.808607\pi$
$522$	0	0
$523$	−5.82944	−0.254904	−0.127452	−	0.991845i	$-0.540680\pi$
$523$	−5.82944	−0.254904	−0.127452	+	0.991845i	$0.540680\pi$
$524$	0	0
$525$	3.70011	0.161486
$526$	0	0
$527$	6.79164	0.295848
$528$	0	0
$529$	−21.6195	−0.939976
$530$	0	0
$531$	3.48160	0.151089
$532$	0	0
$533$	−7.67308	−0.332358
$534$	0	0
$535$	42.3836	1.83240
$536$	0	0
$537$	44.1417	1.90486
$538$	0	0
$539$	−34.5036	−1.48618
$540$	0	0
$541$	27.9697	1.20251	0.601257	−	0.799056i	$-0.294667\pi$
$541$	27.9697	1.20251	0.601257	+	0.799056i	$0.294667\pi$
$542$	0	0
$543$	−21.0186	−0.901996
$544$	0	0
$545$	15.4802	0.663101
$546$	0	0
$547$	2.42478	0.103676	0.0518380	−	0.998656i	$-0.483492\pi$
$547$	2.42478	0.103676	0.0518380	+	0.998656i	$0.483492\pi$
$548$	0	0
$549$	−0.105624	−0.00450793
$550$	0	0
$551$	39.3852	1.67787
$552$	0	0
$553$	0.582966	0.0247902
$554$	0	0
$555$	−2.89064	−0.122701
$556$	0	0
$557$	28.9556	1.22689	0.613444	−	0.789739i	$-0.289784\pi$
$557$	28.9556	1.22689	0.613444	+	0.789739i	$0.289784\pi$
$558$	0	0
$559$	14.1501	0.598484
$560$	0	0
$561$	14.3196	0.604572
$562$	0	0
$563$	24.9644	1.05213	0.526063	−	0.850446i	$-0.323668\pi$
$563$	24.9644	1.05213	0.526063	+	0.850446i	$0.323668\pi$
$564$	0	0
$565$	−27.9414	−1.17550
$566$	0	0
$567$	−5.03356	−0.211390
$568$	0	0
$569$	29.5380	1.23830	0.619148	−	0.785275i	$-0.287478\pi$
$569$	29.5380	1.23830	0.619148	+	0.785275i	$0.287478\pi$
$570$	0	0
$571$	34.5437	1.44561	0.722805	−	0.691052i	$-0.242853\pi$
$571$	34.5437	1.44561	0.722805	+	0.691052i	$0.242853\pi$
$572$	0	0
$573$	22.3430	0.933394
$574$	0	0
$575$	−4.71382	−0.196580
$576$	0	0
$577$	35.8513	1.49251	0.746254	−	0.665662i	$-0.231851\pi$
$577$	35.8513	1.49251	0.746254	+	0.665662i	$0.231851\pi$
$578$	0	0
$579$	34.3397	1.42711
$580$	0	0
$581$	−1.12928	−0.0468503
$582$	0	0
$583$	44.6047	1.84734
$584$	0	0
$585$	1.56972	0.0649001
$586$	0	0
$587$	18.7057	0.772069	0.386034	−	0.922484i	$-0.373845\pi$
$587$	18.7057	0.772069	0.386034	+	0.922484i	$0.373845\pi$
$588$	0	0
$589$	−20.9591	−0.863606
$590$	0	0
$591$	−40.9954	−1.68633
$592$	0	0
$593$	−3.20184	−0.131484	−0.0657420	−	0.997837i	$-0.520941\pi$
$593$	−3.20184	−0.131484	−0.0657420	+	0.997837i	$0.520941\pi$
$594$	0	0
$595$	2.25937	0.0926252
$596$	0	0
$597$	−44.9425	−1.83938
$598$	0	0
$599$	−33.2377	−1.35806	−0.679028	−	0.734112i	$-0.737599\pi$
$599$	−33.2377	−1.35806	−0.679028	+	0.734112i	$0.737599\pi$
$600$	0	0
$601$	35.0600	1.43013	0.715064	−	0.699059i	$-0.246398\pi$
$601$	35.0600	1.43013	0.715064	+	0.699059i	$0.246398\pi$
$602$	0	0
$603$	−0.425716	−0.0173365
$604$	0	0
$605$	45.3634	1.84428
$606$	0	0
$607$	7.63852	0.310038	0.155019	−	0.987912i	$-0.450456\pi$
$607$	7.63852	0.310038	0.155019	+	0.987912i	$0.450456\pi$
$608$	0	0
$609$	−7.78372	−0.315412
$610$	0	0
$611$	−9.31479	−0.376836
$612$	0	0
$613$	−10.5781	−0.427246	−0.213623	−	0.976916i	$-0.568526\pi$
$613$	−10.5781	−0.427246	−0.213623	+	0.976916i	$0.568526\pi$
$614$	0	0
$615$	−35.4309	−1.42871
$616$	0	0
$617$	19.6393	0.790650	0.395325	−	0.918541i	$-0.370632\pi$
$617$	19.6393	0.790650	0.395325	+	0.918541i	$0.370632\pi$
$618$	0	0
$619$	8.39452	0.337404	0.168702	−	0.985667i	$-0.446042\pi$
$619$	8.39452	0.337404	0.168702	+	0.985667i	$0.446042\pi$
$620$	0	0
$621$	5.58701	0.224199
$622$	0	0
$623$	−2.40000	−0.0961540
$624$	0	0
$625$	−28.9642	−1.15857
$626$	0	0
$627$	−44.1904	−1.76480
$628$	0	0
$629$	−0.785777	−0.0313310
$630$	0	0
$631$	3.99991	0.159234	0.0796170	−	0.996826i	$-0.474630\pi$
$631$	3.99991	0.159234	0.0796170	+	0.996826i	$0.474630\pi$
$632$	0	0
$633$	−39.5900	−1.57356
$634$	0	0
$635$	−1.62946	−0.0646632
$636$	0	0
$637$	−8.13485	−0.322314
$638$	0	0
$639$	−5.99743	−0.237255
$640$	0	0
$641$	32.3851	1.27913	0.639567	−	0.768735i	$-0.279114\pi$
$641$	32.3851	1.27913	0.639567	+	0.768735i	$0.279114\pi$
$642$	0	0
$643$	−26.2848	−1.03657	−0.518286	−	0.855208i	$-0.673430\pi$
$643$	−26.2848	−1.03657	−0.518286	+	0.855208i	$0.673430\pi$
$644$	0	0
$645$	65.3387	2.57271
$646$	0	0
$647$	46.4858	1.82755	0.913773	−	0.406226i	$-0.133155\pi$
$647$	46.4858	1.82755	0.913773	+	0.406226i	$0.133155\pi$
$648$	0	0
$649$	40.9896	1.60898
$650$	0	0
$651$	4.14216	0.162344
$652$	0	0
$653$	23.4486	0.917613	0.458807	−	0.888536i	$-0.348277\pi$
$653$	23.4486	0.917613	0.458807	+	0.888536i	$0.348277\pi$
$654$	0	0
$655$	29.0209	1.13394
$656$	0	0
$657$	−0.228376	−0.00890981
$658$	0	0
$659$	7.46279	0.290709	0.145354	−	0.989380i	$-0.453568\pi$
$659$	7.46279	0.290709	0.145354	+	0.989380i	$0.453568\pi$
$660$	0	0
$661$	−36.3331	−1.41319	−0.706597	−	0.707616i	$-0.749771\pi$
$661$	−36.3331	−1.41319	−0.706597	+	0.707616i	$0.749771\pi$
$662$	0	0
$663$	3.37609	0.131117
$664$	0	0
$665$	−6.97246	−0.270381
$666$	0	0
$667$	9.91621	0.383957
$668$	0	0
$669$	−1.04867	−0.0405439
$670$	0	0
$671$	−1.24354	−0.0480062
$672$	0	0
$673$	6.05116	0.233255	0.116628	−	0.993176i	$-0.462792\pi$
$673$	6.05116	0.233255	0.116628	+	0.993176i	$0.462792\pi$
$674$	0	0
$675$	−19.0766	−0.734259
$676$	0	0
$677$	45.7377	1.75784	0.878921	−	0.476968i	$-0.158264\pi$
$677$	45.7377	1.75784	0.878921	+	0.476968i	$0.158264\pi$
$678$	0	0
$679$	−4.87819	−0.187208
$680$	0	0
$681$	30.4499	1.16684
$682$	0	0
$683$	42.4609	1.62472	0.812360	−	0.583156i	$-0.198183\pi$
$683$	42.4609	1.62472	0.812360	+	0.583156i	$0.198183\pi$
$684$	0	0
$685$	41.5305	1.58680
$686$	0	0
$687$	−26.3497	−1.00530
$688$	0	0
$689$	10.5164	0.400642
$690$	0	0
$691$	−5.15598	−0.196143	−0.0980714	−	0.995179i	$-0.531267\pi$
$691$	−5.15598	−0.196143	−0.0980714	+	0.995179i	$0.531267\pi$
$692$	0	0
$693$	1.10382	0.0419305
$694$	0	0
$695$	50.9065	1.93099
$696$	0	0
$697$	−9.63134	−0.364813
$698$	0	0
$699$	10.4043	0.393527
$700$	0	0
$701$	−8.65116	−0.326750	−0.163375	−	0.986564i	$-0.552238\pi$
$701$	−8.65116	−0.326750	−0.163375	+	0.986564i	$0.552238\pi$
$702$	0	0
$703$	2.42492	0.0914577
$704$	0	0
$705$	−43.0115	−1.61991
$706$	0	0
$707$	0.211966	0.00797181
$708$	0	0
$709$	−5.63523	−0.211636	−0.105818	−	0.994386i	$-0.533746\pi$
$709$	−5.63523	−0.211636	−0.105818	+	0.994386i	$0.533746\pi$
$710$	0	0
$711$	0.508388	0.0190661
$712$	0	0
$713$	−5.27698	−0.197625
$714$	0	0
$715$	18.4807	0.691139
$716$	0	0
$717$	7.85568	0.293376
$718$	0	0
$719$	41.4088	1.54429	0.772144	−	0.635448i	$-0.219184\pi$
$719$	41.4088	1.54429	0.772144	+	0.635448i	$0.219184\pi$
$720$	0	0
$721$	3.96933	0.147826
$722$	0	0
$723$	23.8815	0.888161
$724$	0	0
$725$	−33.8585	−1.25747
$726$	0	0
$727$	−20.9423	−0.776705	−0.388353	−	0.921511i	$-0.626956\pi$
$727$	−20.9423	−0.776705	−0.388353	+	0.921511i	$0.626956\pi$
$728$	0	0
$729$	22.0452	0.816490
$730$	0	0
$731$	17.7613	0.656927
$732$	0	0
$733$	−13.4118	−0.495377	−0.247688	−	0.968840i	$-0.579671\pi$
$733$	−13.4118	−0.495377	−0.247688	+	0.968840i	$0.579671\pi$
$734$	0	0
$735$	−37.5631	−1.38553
$736$	0	0
$737$	−5.01204	−0.184621
$738$	0	0
$739$	−14.1872	−0.521886	−0.260943	−	0.965354i	$-0.584033\pi$
$739$	−14.1872	−0.521886	−0.260943	+	0.965354i	$0.584033\pi$
$740$	0	0
$741$	−10.4187	−0.382740
$742$	0	0
$743$	−15.2552	−0.559661	−0.279830	−	0.960049i	$-0.590278\pi$
$743$	−15.2552	−0.559661	−0.279830	+	0.960049i	$0.590278\pi$
$744$	0	0
$745$	16.6794	0.611087
$746$	0	0
$747$	−0.984811	−0.0360323
$748$	0	0
$749$	7.02677	0.256753
$750$	0	0
$751$	38.5201	1.40562	0.702808	−	0.711379i	$-0.251929\pi$
$751$	38.5201	1.40562	0.702808	+	0.711379i	$0.251929\pi$
$752$	0	0
$753$	−40.1769	−1.46413
$754$	0	0
$755$	1.86352	0.0678206
$756$	0	0
$757$	0.272510	0.00990456	0.00495228	−	0.999988i	$-0.498424\pi$
$757$	0.272510	0.00990456	0.00495228	+	0.999988i	$0.498424\pi$
$758$	0	0
$759$	−11.1260	−0.403850
$760$	0	0
$761$	10.2453	0.371391	0.185696	−	0.982607i	$-0.440546\pi$
$761$	10.2453	0.371391	0.185696	+	0.982607i	$0.440546\pi$
$762$	0	0
$763$	2.56647	0.0929123
$764$	0	0
$765$	1.97034	0.0712376
$766$	0	0
$767$	9.66404	0.348948
$768$	0	0
$769$	30.7252	1.10798	0.553990	−	0.832523i	$-0.313104\pi$
$769$	30.7252	1.10798	0.553990	+	0.832523i	$0.313104\pi$
$770$	0	0
$771$	−27.0491	−0.974151
$772$	0	0
$773$	51.2608	1.84372	0.921861	−	0.387521i	$-0.126669\pi$
$773$	51.2608	1.84372	0.921861	+	0.387521i	$0.126669\pi$
$774$	0	0
$775$	18.0180	0.647227
$776$	0	0
$777$	−0.479239	−0.0171926
$778$	0	0
$779$	29.7225	1.06492
$780$	0	0
$781$	−70.6090	−2.52659
$782$	0	0
$783$	40.1304	1.43414
$784$	0	0
$785$	−69.4313	−2.47811
$786$	0	0
$787$	31.1384	1.10997	0.554983	−	0.831862i	$-0.312725\pi$
$787$	31.1384	1.10997	0.554983	+	0.831862i	$0.312725\pi$
$788$	0	0
$789$	−15.2833	−0.544099
$790$	0	0
$791$	−4.63239	−0.164709
$792$	0	0
$793$	−0.293186	−0.0104113
$794$	0	0
$795$	48.5599	1.72224
$796$	0	0
$797$	21.9560	0.777720	0.388860	−	0.921297i	$-0.372869\pi$
$797$	21.9560	0.777720	0.388860	+	0.921297i	$0.372869\pi$
$798$	0	0
$799$	−11.6920	−0.413634
$800$	0	0
$801$	−2.09298	−0.0739517
$802$	0	0
$803$	−2.68872	−0.0948830
$804$	0	0
$805$	−1.75549	−0.0618729
$806$	0	0
$807$	−5.66559	−0.199438
$808$	0	0
$809$	17.3887	0.611355	0.305678	−	0.952135i	$-0.401117\pi$
$809$	17.3887	0.611355	0.305678	+	0.952135i	$0.401117\pi$
$810$	0	0
$811$	−26.1628	−0.918699	−0.459349	−	0.888256i	$-0.651917\pi$
$811$	−26.1628	−0.918699	−0.459349	+	0.888256i	$0.651917\pi$
$812$	0	0
$813$	−8.83457	−0.309842
$814$	0	0
$815$	10.5772	0.370504
$816$	0	0
$817$	−54.8118	−1.91762
$818$	0	0
$819$	0.260244	0.00909367
$820$	0	0
$821$	−3.67022	−0.128092	−0.0640458	−	0.997947i	$-0.520400\pi$
$821$	−3.67022	−0.128092	−0.0640458	+	0.997947i	$0.520400\pi$
$822$	0	0
$823$	13.2160	0.460679	0.230340	−	0.973110i	$-0.426016\pi$
$823$	13.2160	0.460679	0.230340	+	0.973110i	$0.426016\pi$
$824$	0	0
$825$	37.9894	1.32262
$826$	0	0
$827$	−30.0794	−1.04596	−0.522981	−	0.852344i	$-0.675180\pi$
$827$	−30.0794	−1.04596	−0.522981	+	0.852344i	$0.675180\pi$
$828$	0	0
$829$	51.0787	1.77404	0.887018	−	0.461734i	$-0.152773\pi$
$829$	51.0787	1.77404	0.887018	+	0.461734i	$0.152773\pi$
$830$	0	0
$831$	5.70565	0.197927
$832$	0	0
$833$	−10.2110	−0.353788
$834$	0	0
$835$	3.00198	0.103888
$836$	0	0
$837$	−21.3557	−0.738161
$838$	0	0
$839$	44.6087	1.54006	0.770031	−	0.638007i	$-0.220241\pi$
$839$	44.6087	1.54006	0.770031	+	0.638007i	$0.220241\pi$
$840$	0	0
$841$	42.2263	1.45608
$842$	0	0
$843$	32.3680	1.11481
$844$	0	0
$845$	−34.6686	−1.19263
$846$	0	0
$847$	7.52078	0.258417
$848$	0	0
$849$	23.4256	0.803966
$850$	0	0
$851$	0.610535	0.0209289
$852$	0	0
$853$	−48.3841	−1.65664	−0.828320	−	0.560256i	$-0.810703\pi$
$853$	−48.3841	−1.65664	−0.828320	+	0.560256i	$0.810703\pi$
$854$	0	0
$855$	−6.08049	−0.207949
$856$	0	0
$857$	39.4758	1.34847	0.674234	−	0.738518i	$-0.264474\pi$
$857$	39.4758	1.34847	0.674234	+	0.738518i	$0.264474\pi$
$858$	0	0
$859$	5.62522	0.191930	0.0959650	−	0.995385i	$-0.469406\pi$
$859$	5.62522	0.191930	0.0959650	+	0.995385i	$0.469406\pi$
$860$	0	0
$861$	−5.87408	−0.200188
$862$	0	0
$863$	−2.73707	−0.0931710	−0.0465855	−	0.998914i	$-0.514834\pi$
$863$	−2.73707	−0.0931710	−0.0465855	+	0.998914i	$0.514834\pi$
$864$	0	0
$865$	−26.0171	−0.884609
$866$	0	0
$867$	−27.2652	−0.925975
$868$	0	0
$869$	5.98537	0.203040
$870$	0	0
$871$	−1.18168	−0.0400397
$872$	0	0
$873$	−4.25413	−0.143981
$874$	0	0
$875$	−1.47634	−0.0499093
$876$	0	0
$877$	−3.91277	−0.132125	−0.0660625	−	0.997815i	$-0.521044\pi$
$877$	−3.91277	−0.132125	−0.0660625	+	0.997815i	$0.521044\pi$
$878$	0	0
$879$	−2.00988	−0.0677916
$880$	0	0
$881$	−4.74009	−0.159698	−0.0798488	−	0.996807i	$-0.525444\pi$
$881$	−4.74009	−0.159698	−0.0798488	+	0.996807i	$0.525444\pi$
$882$	0	0
$883$	−21.8335	−0.734754	−0.367377	−	0.930072i	$-0.619744\pi$
$883$	−21.8335	−0.734754	−0.367377	+	0.930072i	$0.619744\pi$
$884$	0	0
$885$	44.6242	1.50003
$886$	0	0
$887$	−2.38041	−0.0799264	−0.0399632	−	0.999201i	$-0.512724\pi$
$887$	−2.38041	−0.0799264	−0.0399632	+	0.999201i	$0.512724\pi$
$888$	0	0
$889$	−0.270148	−0.00906048
$890$	0	0
$891$	−51.6801	−1.73135
$892$	0	0
$893$	36.0818	1.20743
$894$	0	0
$895$	71.5081	2.39025
$896$	0	0
$897$	−2.62316	−0.0875849
$898$	0	0
$899$	−37.9036	−1.26415
$900$	0	0
$901$	13.2003	0.439765
$902$	0	0
$903$	10.8325	0.360483
$904$	0	0
$905$	−34.0494	−1.13184
$906$	0	0
$907$	−15.3653	−0.510195	−0.255098	−	0.966915i	$-0.582108\pi$
$907$	−15.3653	−0.510195	−0.255098	+	0.966915i	$0.582108\pi$
$908$	0	0
$909$	0.184850	0.00613109
$910$	0	0
$911$	2.59038	0.0858231	0.0429115	−	0.999079i	$-0.486337\pi$
$911$	2.59038	0.0858231	0.0429115	+	0.999079i	$0.486337\pi$
$912$	0	0
$913$	−11.5944	−0.383718
$914$	0	0
$915$	−1.35380	−0.0447553
$916$	0	0
$917$	4.81137	0.158885
$918$	0	0
$919$	9.54700	0.314926	0.157463	−	0.987525i	$-0.449668\pi$
$919$	9.54700	0.314926	0.157463	+	0.987525i	$0.449668\pi$
$920$	0	0
$921$	−6.05693	−0.199582
$922$	0	0
$923$	−16.6473	−0.547954
$924$	0	0
$925$	−2.08464	−0.0685427
$926$	0	0
$927$	3.46155	0.113692
$928$	0	0
$929$	−18.0650	−0.592692	−0.296346	−	0.955081i	$-0.595768\pi$
$929$	−18.0650	−0.592692	−0.296346	+	0.955081i	$0.595768\pi$
$930$	0	0
$931$	31.5112	1.03274
$932$	0	0
$933$	2.94323	0.0963571
$934$	0	0
$935$	23.1972	0.758629
$936$	0	0
$937$	5.39036	0.176095	0.0880476	−	0.996116i	$-0.471937\pi$
$937$	5.39036	0.176095	0.0880476	+	0.996116i	$0.471937\pi$
$938$	0	0
$939$	−45.1075	−1.47203
$940$	0	0
$941$	14.2930	0.465938	0.232969	−	0.972484i	$-0.425156\pi$
$941$	14.2930	0.465938	0.232969	+	0.972484i	$0.425156\pi$
$942$	0	0
$943$	7.48338	0.243693
$944$	0	0
$945$	−7.10439	−0.231106
$946$	0	0
$947$	−30.3027	−0.984707	−0.492353	−	0.870395i	$-0.663863\pi$
$947$	−30.3027	−0.984707	−0.492353	+	0.870395i	$0.663863\pi$
$948$	0	0
$949$	−0.633914	−0.0205777
$950$	0	0
$951$	44.8964	1.45587
$952$	0	0
$953$	22.0748	0.715073	0.357536	−	0.933899i	$-0.383617\pi$
$953$	22.0748	0.715073	0.357536	+	0.933899i	$0.383617\pi$
$954$	0	0
$955$	36.1949	1.17124
$956$	0	0
$957$	−79.9162	−2.58333
$958$	0	0
$959$	6.88533	0.222339
$960$	0	0
$961$	−10.8293	−0.349334
$962$	0	0
$963$	6.12785	0.197467
$964$	0	0
$965$	55.6292	1.79077
$966$	0	0
$967$	34.3522	1.10469	0.552345	−	0.833615i	$-0.313733\pi$
$967$	34.3522	1.10469	0.552345	+	0.833615i	$0.313733\pi$
$968$	0	0
$969$	−13.0777	−0.420115
$970$	0	0
$971$	−45.3338	−1.45483	−0.727415	−	0.686198i	$-0.759278\pi$
$971$	−45.3338	−1.45483	−0.727415	+	0.686198i	$0.759278\pi$
$972$	0	0
$973$	8.43977	0.270567
$974$	0	0
$975$	8.95668	0.286843
$976$	0	0
$977$	−0.905536	−0.0289707	−0.0144853	−	0.999895i	$-0.504611\pi$
$977$	−0.905536	−0.0289707	−0.0144853	+	0.999895i	$0.504611\pi$
$978$	0	0
$979$	−24.6411	−0.787531
$980$	0	0
$981$	2.23815	0.0714585
$982$	0	0
$983$	−15.0683	−0.480605	−0.240302	−	0.970698i	$-0.577247\pi$
$983$	−15.0683	−0.480605	−0.240302	+	0.970698i	$0.577247\pi$
$984$	0	0
$985$	−66.4111	−2.11603
$986$	0	0
$987$	−7.13087	−0.226978
$988$	0	0
$989$	−13.8002	−0.438822
$990$	0	0
$991$	27.6975	0.879839	0.439920	−	0.898037i	$-0.355007\pi$
$991$	27.6975	0.879839	0.439920	+	0.898037i	$0.355007\pi$
$992$	0	0
$993$	18.4031	0.584005
$994$	0	0
$995$	−72.8053	−2.30808
$996$	0	0
$997$	−21.8659	−0.692501	−0.346251	−	0.938142i	$-0.612545\pi$
$997$	−21.8659	−0.692501	−0.346251	+	0.938142i	$0.612545\pi$
$998$	0	0
$999$	2.47080	0.0781728

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1336.2.a.e.1.8	✓	12
4.3	odd	2		2672.2.a.n.1.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.e.1.8	✓	12	1.1	even	1	trivial
2672.2.a.n.1.5		12	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)

\(f(q)\)	\(=\)	\(q+1.85311 q^{3} +3.00198 q^{5} +0.497697 q^{7} +0.434028 q^{9} +O(q^{10})\)	\(q+1.85311 q^{3} +3.00198 q^{5} +0.497697 q^{7} +0.434028 q^{9} +5.10991 q^{11} +1.20475 q^{13} +5.56301 q^{15} +1.51222 q^{17} -4.66674 q^{19} +0.922290 q^{21} -1.17497 q^{23} +4.01187 q^{25} -4.75504 q^{27} -8.43956 q^{29} +4.49118 q^{31} +9.46924 q^{33} +1.49408 q^{35} -0.519618 q^{37} +2.23254 q^{39} -6.36901 q^{41} +11.7452 q^{43} +1.30294 q^{45} -7.73171 q^{47} -6.75230 q^{49} +2.80231 q^{51} +8.72907 q^{53} +15.3398 q^{55} -8.64799 q^{57} +8.02160 q^{59} -0.243358 q^{61} +0.216015 q^{63} +3.61664 q^{65} -0.980848 q^{67} -2.17735 q^{69} -13.8181 q^{71} -0.526178 q^{73} +7.43446 q^{75} +2.54319 q^{77} +1.17133 q^{79} -10.1137 q^{81} -2.26900 q^{83} +4.53965 q^{85} -15.6395 q^{87} -4.82221 q^{89} +0.599602 q^{91} +8.32266 q^{93} -14.0094 q^{95} -9.80151 q^{97} +2.21784 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9}+O(q^{10}) \) 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9	\( 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9} + 14 q^{11} - 9 q^{13} + 8 q^{15} + 4 q^{17} + 9 q^{19} + 7 q^{21} + 15 q^{23} + 18 q^{25} + 20 q^{27} + 17 q^{29} + 11 q^{31} + 8 q^{33} + 19 q^{35} - 29 q^{37} + 26 q^{39} + 14 q^{41} + 15 q^{43} - 26 q^{45} + 17 q^{47} + 20 q^{49} + 36 q^{51} - 7 q^{53} + 13 q^{55} + 3 q^{57} + 32 q^{59} - 8 q^{61} + 50 q^{63} + 37 q^{65} + 39 q^{67} + q^{69} + 35 q^{71} - 12 q^{73} + 29 q^{75} + 13 q^{77} + 36 q^{79} + 44 q^{81} + 25 q^{83} - 38 q^{85} + 14 q^{87} + 23 q^{89} - 2 q^{91} - 25 q^{93} + 38 q^{95} - 2 q^{97} + 43 q^{99}+O(q^{100}) \) 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9 + 14 * q^11 - 9 * q^13 + 8 * q^15 + 4 * q^17 + 9 * q^19 + 7 * q^21 + 15 * q^23 + 18 * q^25 + 20 * q^27 + 17 * q^29 + 11 * q^31 + 8 * q^33 + 19 * q^35 - 29 * q^37 + 26 * q^39 + 14 * q^41 + 15 * q^43 - 26 * q^45 + 17 * q^47 + 20 * q^49 + 36 * q^51 - 7 * q^53 + 13 * q^55 + 3 * q^57 + 32 * q^59 - 8 * q^61 + 50 * q^63 + 37 * q^65 + 39 * q^67 + q^69 + 35 * q^71 - 12 * q^73 + 29 * q^75 + 13 * q^77 + 36 * q^79 + 44 * q^81 + 25 * q^83 - 38 * q^85 + 14 * q^87 + 23 * q^89 - 2 * q^91 - 25 * q^93 + 38 * q^95 - 2 * q^97 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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