LMFDB - Embedded newform 2312.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,2,Mod(1,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, 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("2312.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2312.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:	$18.4614129473$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{3} -4.24264 q^{5} -4.24264 q^{7} -1.00000 q^{9} -1.41421 q^{11} -2.00000 q^{13} -6.00000 q^{15} +6.00000 q^{19} -6.00000 q^{21} -1.41421 q^{23} +13.0000 q^{25} -5.65685 q^{27} +1.41421 q^{29} +1.41421 q^{31} -2.00000 q^{33} +18.0000 q^{35} +4.24264 q^{37} -2.82843 q^{39} +1.41421 q^{41} +2.00000 q^{43} +4.24264 q^{45} +11.0000 q^{49} -4.00000 q^{53} +6.00000 q^{55} +8.48528 q^{57} -6.00000 q^{59} +1.41421 q^{61} +4.24264 q^{63} +8.48528 q^{65} -4.00000 q^{67} -2.00000 q^{69} +7.07107 q^{71} +1.41421 q^{73} +18.3848 q^{75} +6.00000 q^{77} -12.7279 q^{79} -5.00000 q^{81} +14.0000 q^{83} +2.00000 q^{87} -6.00000 q^{89} +8.48528 q^{91} +2.00000 q^{93} -25.4558 q^{95} +18.3848 q^{97} +1.41421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} - 4 q^{13} - 12 q^{15} + 12 q^{19} - 12 q^{21} + 26 q^{25} - 4 q^{33} + 36 q^{35} + 4 q^{43} + 22 q^{49} - 8 q^{53} + 12 q^{55} - 12 q^{59} - 8 q^{67} - 4 q^{69} + 12 q^{77} - 10 q^{81} + 28 q^{83}+ \cdots + 4 q^{93}+O(q^{100}) $ 2 * q - 2 * q^9 - 4 * q^13 - 12 * q^15 + 12 * q^19 - 12 * q^21 + 26 * q^25 - 4 * q^33 + 36 * q^35 + 4 * q^43 + 22 * q^49 - 8 * q^53 + 12 * q^55 - 12 * q^59 - 8 * q^67 - 4 * q^69 + 12 * q^77 - 10 * q^81 + 28 * q^83 + 4 * q^87 - 12 * q^89 + 4 * q^93

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
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	0	0
$5$	−4.24264	−1.89737	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	−4.24264	−1.89737	−0.948683	+	0.316228i	$0.897584\pi$
$6$	0	0
$7$	−4.24264	−1.60357	−0.801784	−	0.597614i	$-0.796115\pi$
$7$	−4.24264	−1.60357	−0.801784	+	0.597614i	$0.796115\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−6.00000	−1.54919
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	−6.00000	−1.30931
$22$	0	0
$23$	−1.41421	−0.294884	−0.147442	−	0.989071i	$-0.547104\pi$
$23$	−1.41421	−0.294884	−0.147442	+	0.989071i	$0.547104\pi$
$24$	0	0
$25$	13.0000	2.60000
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	1.41421	0.262613	0.131306	−	0.991342i	$-0.458083\pi$
$29$	1.41421	0.262613	0.131306	+	0.991342i	$0.458083\pi$
$30$	0	0
$31$	1.41421	0.254000	0.127000	−	0.991903i	$-0.459465\pi$
$31$	1.41421	0.254000	0.127000	+	0.991903i	$0.459465\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	18.0000	3.04256
$36$	0	0
$37$	4.24264	0.697486	0.348743	−	0.937218i	$-0.386609\pi$
$37$	4.24264	0.697486	0.348743	+	0.937218i	$0.386609\pi$
$38$	0	0
$39$	−2.82843	−0.452911
$40$	0	0
$41$	1.41421	0.220863	0.110432	−	0.993884i	$-0.464777\pi$
$41$	1.41421	0.220863	0.110432	+	0.993884i	$0.464777\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	4.24264	0.632456
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	6.00000	0.809040
$56$	0	0
$57$	8.48528	1.12390
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	1.41421	0.181071	0.0905357	−	0.995893i	$-0.471142\pi$
$61$	1.41421	0.181071	0.0905357	+	0.995893i	$0.471142\pi$
$62$	0	0
$63$	4.24264	0.534522
$64$	0	0
$65$	8.48528	1.05247
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	7.07107	0.839181	0.419591	−	0.907713i	$-0.362174\pi$
$71$	7.07107	0.839181	0.419591	+	0.907713i	$0.362174\pi$
$72$	0	0
$73$	1.41421	0.165521	0.0827606	−	0.996569i	$-0.473626\pi$
$73$	1.41421	0.165521	0.0827606	+	0.996569i	$0.473626\pi$
$74$	0	0
$75$	18.3848	2.12289
$76$	0	0
$77$	6.00000	0.683763
$78$	0	0
$79$	−12.7279	−1.43200	−0.716002	−	0.698099i	$-0.754030\pi$
$79$	−12.7279	−1.43200	−0.716002	+	0.698099i	$0.754030\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	14.0000	1.53670	0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000	1.53670	0.768350	+	0.640030i	$0.221078\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	8.48528	0.889499
$92$	0	0
$93$	2.00000	0.207390
$94$	0	0
$95$	−25.4558	−2.61171
$96$	0	0
$97$	18.3848	1.86669	0.933346	−	0.358979i	$-0.116875\pi$
$97$	18.3848	1.86669	0.933346	+	0.358979i	$0.116875\pi$
$98$	0	0
$99$	1.41421	0.142134
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	25.4558	2.48424
$106$	0	0
$107$	9.89949	0.957020	0.478510	−	0.878082i	$-0.341177\pi$
$107$	9.89949	0.957020	0.478510	+	0.878082i	$0.341177\pi$
$108$	0	0
$109$	12.7279	1.21911	0.609557	−	0.792742i	$-0.291347\pi$
$109$	12.7279	1.21911	0.609557	+	0.792742i	$0.291347\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	−7.07107	−0.665190	−0.332595	−	0.943070i	$-0.607924\pi$
$113$	−7.07107	−0.665190	−0.332595	+	0.943070i	$0.607924\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	−33.9411	−3.03579
$126$	0	0
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	0	0
$129$	2.82843	0.249029
$130$	0	0
$131$	−12.7279	−1.11204	−0.556022	−	0.831168i	$-0.687673\pi$
$131$	−12.7279	−1.11204	−0.556022	+	0.831168i	$0.687673\pi$
$132$	0	0
$133$	−25.4558	−2.20730
$134$	0	0
$135$	24.0000	2.06559
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−15.5563	−1.31947	−0.659736	−	0.751497i	$-0.729332\pi$
$139$	−15.5563	−1.31947	−0.659736	+	0.751497i	$0.729332\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.82843	0.236525
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	15.5563	1.28307
$148$	0	0
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.00000	−0.481932
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	−5.65685	−0.448618
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	1.41421	0.110770	0.0553849	−	0.998465i	$-0.482361\pi$
$163$	1.41421	0.110770	0.0553849	+	0.998465i	$0.482361\pi$
$164$	0	0
$165$	8.48528	0.660578
$166$	0	0
$167$	18.3848	1.42266	0.711328	−	0.702860i	$-0.248094\pi$
$167$	18.3848	1.42266	0.711328	+	0.702860i	$0.248094\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	9.89949	0.752645	0.376322	−	0.926489i	$-0.377189\pi$
$173$	9.89949	0.752645	0.376322	+	0.926489i	$0.377189\pi$
$174$	0	0
$175$	−55.1543	−4.16928
$176$	0	0
$177$	−8.48528	−0.637793
$178$	0	0
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	−12.7279	−0.946059	−0.473029	−	0.881047i	$-0.656840\pi$
$181$	−12.7279	−0.946059	−0.473029	+	0.881047i	$0.656840\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	−18.0000	−1.32339
$186$	0	0
$187$	0	0
$188$	0	0
$189$	24.0000	1.74574
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	−12.7279	−0.916176	−0.458088	−	0.888907i	$-0.651466\pi$
$193$	−12.7279	−0.916176	−0.458088	+	0.888907i	$0.651466\pi$
$194$	0	0
$195$	12.0000	0.859338
$196$	0	0
$197$	−15.5563	−1.10834	−0.554172	−	0.832402i	$-0.686965\pi$
$197$	−15.5563	−1.10834	−0.554172	+	0.832402i	$0.686965\pi$
$198$	0	0
$199$	9.89949	0.701757	0.350878	−	0.936421i	$-0.385883\pi$
$199$	9.89949	0.701757	0.350878	+	0.936421i	$0.385883\pi$
$200$	0	0
$201$	−5.65685	−0.399004
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	1.41421	0.0982946
$208$	0	0
$209$	−8.48528	−0.586939
$210$	0	0
$211$	7.07107	0.486792	0.243396	−	0.969927i	$-0.421738\pi$
$211$	7.07107	0.486792	0.243396	+	0.969927i	$0.421738\pi$
$212$	0	0
$213$	10.0000	0.685189
$214$	0	0
$215$	−8.48528	−0.578691
$216$	0	0
$217$	−6.00000	−0.407307
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	0	0
$222$	0	0
$223$	26.0000	1.74109	0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000	1.74109	0.870544	+	0.492090i	$0.163767\pi$
$224$	0	0
$225$	−13.0000	−0.866667
$226$	0	0
$227$	21.2132	1.40797	0.703985	−	0.710215i	$-0.251402\pi$
$227$	21.2132	1.40797	0.703985	+	0.710215i	$0.251402\pi$
$228$	0	0
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	0	0
$231$	8.48528	0.558291
$232$	0	0
$233$	−26.8701	−1.76032	−0.880158	−	0.474681i	$-0.842563\pi$
$233$	−26.8701	−1.76032	−0.880158	+	0.474681i	$0.842563\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−18.0000	−1.16923
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−12.7279	−0.819878	−0.409939	−	0.912113i	$-0.634450\pi$
$241$	−12.7279	−0.819878	−0.409939	+	0.912113i	$0.634450\pi$
$242$	0	0
$243$	9.89949	0.635053
$244$	0	0
$245$	−46.6690	−2.98158
$246$	0	0
$247$	−12.0000	−0.763542
$248$	0	0
$249$	19.7990	1.25471
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.0000	−1.24757	−0.623783	−	0.781598i	$-0.714405\pi$
$257$	−20.0000	−1.24757	−0.623783	+	0.781598i	$0.714405\pi$
$258$	0	0
$259$	−18.0000	−1.11847
$260$	0	0
$261$	−1.41421	−0.0875376
$262$	0	0
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	16.9706	1.04249
$266$	0	0
$267$	−8.48528	−0.519291
$268$	0	0
$269$	−18.3848	−1.12094	−0.560470	−	0.828175i	$-0.689379\pi$
$269$	−18.3848	−1.12094	−0.560470	+	0.828175i	$0.689379\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	12.0000	0.726273
$274$	0	0
$275$	−18.3848	−1.10864
$276$	0	0
$277$	12.7279	0.764747	0.382373	−	0.924008i	$-0.375107\pi$
$277$	12.7279	0.764747	0.382373	+	0.924008i	$0.375107\pi$
$278$	0	0
$279$	−1.41421	−0.0846668
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	0	0
$283$	21.2132	1.26099	0.630497	−	0.776192i	$-0.282851\pi$
$283$	21.2132	1.26099	0.630497	+	0.776192i	$0.282851\pi$
$284$	0	0
$285$	−36.0000	−2.13246
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	0	0
$290$	0	0
$291$	26.0000	1.52415
$292$	0	0
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	0	0
$295$	25.4558	1.48210
$296$	0	0
$297$	8.00000	0.464207
$298$	0	0
$299$	2.82843	0.163572
$300$	0	0
$301$	−8.48528	−0.489083
$302$	0	0
$303$	−8.48528	−0.487467
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	26.8701	1.52366	0.761831	−	0.647776i	$-0.224301\pi$
$311$	26.8701	1.52366	0.761831	+	0.647776i	$0.224301\pi$
$312$	0	0
$313$	12.7279	0.719425	0.359712	−	0.933063i	$-0.382875\pi$
$313$	12.7279	0.719425	0.359712	+	0.933063i	$0.382875\pi$
$314$	0	0
$315$	−18.0000	−1.01419
$316$	0	0
$317$	−7.07107	−0.397151	−0.198575	−	0.980086i	$-0.563631\pi$
$317$	−7.07107	−0.397151	−0.198575	+	0.980086i	$0.563631\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	0	0
$321$	14.0000	0.781404
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−26.0000	−1.44222
$326$	0	0
$327$	18.0000	0.995402
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	0	0
$333$	−4.24264	−0.232495
$334$	0	0
$335$	16.9706	0.927201
$336$	0	0
$337$	21.2132	1.15556	0.577778	−	0.816194i	$-0.303920\pi$
$337$	21.2132	1.15556	0.577778	+	0.816194i	$0.303920\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	0	0
$341$	−2.00000	−0.108306
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	8.48528	0.456832
$346$	0	0
$347$	24.0416	1.29062	0.645311	−	0.763920i	$-0.276728\pi$
$347$	24.0416	1.29062	0.645311	+	0.763920i	$0.276728\pi$
$348$	0	0
$349$	12.0000	0.642345	0.321173	−	0.947021i	$-0.395923\pi$
$349$	12.0000	0.642345	0.321173	+	0.947021i	$0.395923\pi$
$350$	0	0
$351$	11.3137	0.603881
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	−30.0000	−1.59223
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.0000	−0.738892	−0.369446	−	0.929252i	$-0.620452\pi$
$359$	−14.0000	−0.738892	−0.369446	+	0.929252i	$0.620452\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−12.7279	−0.668043
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	29.6985	1.55025	0.775124	−	0.631809i	$-0.217687\pi$
$367$	29.6985	1.55025	0.775124	+	0.631809i	$0.217687\pi$
$368$	0	0
$369$	−1.41421	−0.0736210
$370$	0	0
$371$	16.9706	0.881068
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	−48.0000	−2.47871
$376$	0	0
$377$	−2.82843	−0.145671
$378$	0	0
$379$	21.2132	1.08965	0.544825	−	0.838550i	$-0.316596\pi$
$379$	21.2132	1.08965	0.544825	+	0.838550i	$0.316596\pi$
$380$	0	0
$381$	19.7990	1.01433
$382$	0	0
$383$	−26.0000	−1.32854	−0.664269	−	0.747494i	$-0.731257\pi$
$383$	−26.0000	−1.32854	−0.664269	+	0.747494i	$0.731257\pi$
$384$	0	0
$385$	−25.4558	−1.29735
$386$	0	0
$387$	−2.00000	−0.101666
$388$	0	0
$389$	16.0000	0.811232	0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000	0.811232	0.405616	+	0.914044i	$0.367057\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−18.0000	−0.907980
$394$	0	0
$395$	54.0000	2.71703
$396$	0	0
$397$	32.5269	1.63248	0.816239	−	0.577714i	$-0.196055\pi$
$397$	32.5269	1.63248	0.816239	+	0.577714i	$0.196055\pi$
$398$	0	0
$399$	−36.0000	−1.80225
$400$	0	0
$401$	12.7279	0.635602	0.317801	−	0.948157i	$-0.397056\pi$
$401$	12.7279	0.635602	0.317801	+	0.948157i	$0.397056\pi$
$402$	0	0
$403$	−2.82843	−0.140894
$404$	0	0
$405$	21.2132	1.05409
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	0	0
$411$	8.48528	0.418548
$412$	0	0
$413$	25.4558	1.25260
$414$	0	0
$415$	−59.3970	−2.91568
$416$	0	0
$417$	−22.0000	−1.07734
$418$	0	0
$419$	−7.07107	−0.345444	−0.172722	−	0.984971i	$-0.555256\pi$
$419$	−7.07107	−0.345444	−0.172722	+	0.984971i	$0.555256\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	9.89949	0.476842	0.238421	−	0.971162i	$-0.423370\pi$
$431$	9.89949	0.476842	0.238421	+	0.971162i	$0.423370\pi$
$432$	0	0
$433$	−12.0000	−0.576683	−0.288342	−	0.957528i	$-0.593104\pi$
$433$	−12.0000	−0.576683	−0.288342	+	0.957528i	$0.593104\pi$
$434$	0	0
$435$	−8.48528	−0.406838
$436$	0	0
$437$	−8.48528	−0.405906
$438$	0	0
$439$	−4.24264	−0.202490	−0.101245	−	0.994862i	$-0.532283\pi$
$439$	−4.24264	−0.202490	−0.101245	+	0.994862i	$0.532283\pi$
$440$	0	0
$441$	−11.0000	−0.523810
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	25.4558	1.20672
$446$	0	0
$447$	14.1421	0.668900
$448$	0	0
$449$	−4.24264	−0.200223	−0.100111	−	0.994976i	$-0.531920\pi$
$449$	−4.24264	−0.200223	−0.100111	+	0.994976i	$0.531920\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	0	0
$453$	2.82843	0.132891
$454$	0	0
$455$	−36.0000	−1.68771
$456$	0	0
$457$	20.0000	0.935561	0.467780	−	0.883845i	$-0.345054\pi$
$457$	20.0000	0.935561	0.467780	+	0.883845i	$0.345054\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−28.0000	−1.30409	−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000	−1.30409	−0.652045	+	0.758180i	$0.726089\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	0	0
$465$	−8.48528	−0.393496
$466$	0	0
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	0	0
$469$	16.9706	0.783628
$470$	0	0
$471$	14.1421	0.651635
$472$	0	0
$473$	−2.82843	−0.130051
$474$	0	0
$475$	78.0000	3.57889
$476$	0	0
$477$	4.00000	0.183147
$478$	0	0
$479$	12.7279	0.581554	0.290777	−	0.956791i	$-0.406086\pi$
$479$	12.7279	0.581554	0.290777	+	0.956791i	$0.406086\pi$
$480$	0	0
$481$	−8.48528	−0.386896
$482$	0	0
$483$	8.48528	0.386094
$484$	0	0
$485$	−78.0000	−3.54180
$486$	0	0
$487$	−18.3848	−0.833094	−0.416547	−	0.909114i	$-0.636760\pi$
$487$	−18.3848	−0.833094	−0.416547	+	0.909114i	$0.636760\pi$
$488$	0	0
$489$	2.00000	0.0904431
$490$	0	0
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−6.00000	−0.269680
$496$	0	0
$497$	−30.0000	−1.34568
$498$	0	0
$499$	−18.3848	−0.823016	−0.411508	−	0.911406i	$-0.634998\pi$
$499$	−18.3848	−0.823016	−0.411508	+	0.911406i	$0.634998\pi$
$500$	0	0
$501$	26.0000	1.16159
$502$	0	0
$503$	−26.8701	−1.19808	−0.599038	−	0.800720i	$-0.704450\pi$
$503$	−26.8701	−1.19808	−0.599038	+	0.800720i	$0.704450\pi$
$504$	0	0
$505$	25.4558	1.13277
$506$	0	0
$507$	−12.7279	−0.565267
$508$	0	0
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	−33.9411	−1.49854
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	32.5269	1.42503	0.712515	−	0.701657i	$-0.247556\pi$
$521$	32.5269	1.42503	0.712515	+	0.701657i	$0.247556\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	−78.0000	−3.40420
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−21.0000	−0.913043
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	−2.82843	−0.122513
$534$	0	0
$535$	−42.0000	−1.81582
$536$	0	0
$537$	25.4558	1.09850
$538$	0	0
$539$	−15.5563	−0.670059
$540$	0	0
$541$	26.8701	1.15523	0.577617	−	0.816308i	$-0.303983\pi$
$541$	26.8701	1.15523	0.577617	+	0.816308i	$0.303983\pi$
$542$	0	0
$543$	−18.0000	−0.772454
$544$	0	0
$545$	−54.0000	−2.31311
$546$	0	0
$547$	−21.2132	−0.907011	−0.453506	−	0.891253i	$-0.649827\pi$
$547$	−21.2132	−0.907011	−0.453506	+	0.891253i	$0.649827\pi$
$548$	0	0
$549$	−1.41421	−0.0603572
$550$	0	0
$551$	8.48528	0.361485
$552$	0	0
$553$	54.0000	2.29631
$554$	0	0
$555$	−25.4558	−1.08054
$556$	0	0
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22.0000	0.927189	0.463595	−	0.886047i	$-0.346559\pi$
$563$	22.0000	0.927189	0.463595	+	0.886047i	$0.346559\pi$
$564$	0	0
$565$	30.0000	1.26211
$566$	0	0
$567$	21.2132	0.890871
$568$	0	0
$569$	−16.0000	−0.670755	−0.335377	−	0.942084i	$-0.608864\pi$
$569$	−16.0000	−0.670755	−0.335377	+	0.942084i	$0.608864\pi$
$570$	0	0
$571$	18.3848	0.769379	0.384689	−	0.923046i	$-0.374309\pi$
$571$	18.3848	0.769379	0.384689	+	0.923046i	$0.374309\pi$
$572$	0	0
$573$	−33.9411	−1.41791
$574$	0	0
$575$	−18.3848	−0.766698
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	0	0
$579$	−18.0000	−0.748054
$580$	0	0
$581$	−59.3970	−2.46420
$582$	0	0
$583$	5.65685	0.234283
$584$	0	0
$585$	−8.48528	−0.350823
$586$	0	0
$587$	−38.0000	−1.56843	−0.784214	−	0.620491i	$-0.786934\pi$
$587$	−38.0000	−1.56843	−0.784214	+	0.620491i	$0.786934\pi$
$588$	0	0
$589$	8.48528	0.349630
$590$	0	0
$591$	−22.0000	−0.904959
$592$	0	0
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	14.0000	0.572982
$598$	0	0
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	−1.41421	−0.0576870	−0.0288435	−	0.999584i	$-0.509182\pi$
$601$	−1.41421	−0.0576870	−0.0288435	+	0.999584i	$0.509182\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	38.1838	1.55239
$606$	0	0
$607$	−12.7279	−0.516610	−0.258305	−	0.966063i	$-0.583164\pi$
$607$	−12.7279	−0.516610	−0.258305	+	0.966063i	$0.583164\pi$
$608$	0	0
$609$	−8.48528	−0.343841
$610$	0	0
$611$	0	0
$612$	0	0
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	0	0
$615$	−8.48528	−0.342160
$616$	0	0
$617$	7.07107	0.284670	0.142335	−	0.989819i	$-0.454539\pi$
$617$	7.07107	0.284670	0.142335	+	0.989819i	$0.454539\pi$
$618$	0	0
$619$	−21.2132	−0.852631	−0.426315	−	0.904575i	$-0.640189\pi$
$619$	−21.2132	−0.852631	−0.426315	+	0.904575i	$0.640189\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	25.4558	1.01987
$624$	0	0
$625$	79.0000	3.16000
$626$	0	0
$627$	−12.0000	−0.479234
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−22.0000	−0.875806	−0.437903	−	0.899022i	$-0.644279\pi$
$631$	−22.0000	−0.875806	−0.437903	+	0.899022i	$0.644279\pi$
$632$	0	0
$633$	10.0000	0.397464
$634$	0	0
$635$	−59.3970	−2.35710
$636$	0	0
$637$	−22.0000	−0.871672
$638$	0	0
$639$	−7.07107	−0.279727
$640$	0	0
$641$	−15.5563	−0.614439	−0.307219	−	0.951639i	$-0.599399\pi$
$641$	−15.5563	−0.614439	−0.307219	+	0.951639i	$0.599399\pi$
$642$	0	0
$643$	−43.8406	−1.72891	−0.864453	−	0.502714i	$-0.832335\pi$
$643$	−43.8406	−1.72891	−0.864453	+	0.502714i	$0.832335\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	0	0
$647$	16.0000	0.629025	0.314512	−	0.949253i	$-0.398159\pi$
$647$	16.0000	0.629025	0.314512	+	0.949253i	$0.398159\pi$
$648$	0	0
$649$	8.48528	0.333076
$650$	0	0
$651$	−8.48528	−0.332564
$652$	0	0
$653$	−15.5563	−0.608767	−0.304383	−	0.952550i	$-0.598450\pi$
$653$	−15.5563	−0.608767	−0.304383	+	0.952550i	$0.598450\pi$
$654$	0	0
$655$	54.0000	2.10995
$656$	0	0
$657$	−1.41421	−0.0551737
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	36.0000	1.40024	0.700119	−	0.714026i	$-0.253130\pi$
$661$	36.0000	1.40024	0.700119	+	0.714026i	$0.253130\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	108.000	4.18806
$666$	0	0
$667$	−2.00000	−0.0774403
$668$	0	0
$669$	36.7696	1.42159
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	−15.5563	−0.599653	−0.299827	−	0.953994i	$-0.596929\pi$
$673$	−15.5563	−0.599653	−0.299827	+	0.953994i	$0.596929\pi$
$674$	0	0
$675$	−73.5391	−2.83052
$676$	0	0
$677$	−1.41421	−0.0543526	−0.0271763	−	0.999631i	$-0.508652\pi$
$677$	−1.41421	−0.0543526	−0.0271763	+	0.999631i	$0.508652\pi$
$678$	0	0
$679$	−78.0000	−2.99337
$680$	0	0
$681$	30.0000	1.14960
$682$	0	0
$683$	1.41421	0.0541134	0.0270567	−	0.999634i	$-0.491387\pi$
$683$	1.41421	0.0541134	0.0270567	+	0.999634i	$0.491387\pi$
$684$	0	0
$685$	−25.4558	−0.972618
$686$	0	0
$687$	−22.6274	−0.863290
$688$	0	0
$689$	8.00000	0.304776
$690$	0	0
$691$	43.8406	1.66778	0.833888	−	0.551934i	$-0.186110\pi$
$691$	43.8406	1.66778	0.833888	+	0.551934i	$0.186110\pi$
$692$	0	0
$693$	−6.00000	−0.227921
$694$	0	0
$695$	66.0000	2.50352
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−38.0000	−1.43729
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	25.4558	0.960085
$704$	0	0
$705$	0	0
$706$	0	0
$707$	25.4558	0.957366
$708$	0	0
$709$	46.6690	1.75269	0.876346	−	0.481681i	$-0.159974\pi$
$709$	46.6690	1.75269	0.876346	+	0.481681i	$0.159974\pi$
$710$	0	0
$711$	12.7279	0.477334
$712$	0	0
$713$	−2.00000	−0.0749006
$714$	0	0
$715$	−12.0000	−0.448775
$716$	0	0
$717$	11.3137	0.422518
$718$	0	0
$719$	−18.3848	−0.685636	−0.342818	−	0.939402i	$-0.611381\pi$
$719$	−18.3848	−0.685636	−0.342818	+	0.939402i	$0.611381\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−18.0000	−0.669427
$724$	0	0
$725$	18.3848	0.682793
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	0	0
$732$	0	0
$733$	28.0000	1.03420	0.517102	−	0.855924i	$-0.327011\pi$
$733$	28.0000	1.03420	0.517102	+	0.855924i	$0.327011\pi$
$734$	0	0
$735$	−66.0000	−2.43445
$736$	0	0
$737$	5.65685	0.208373
$738$	0	0
$739$	−14.0000	−0.514998	−0.257499	−	0.966279i	$-0.582898\pi$
$739$	−14.0000	−0.514998	−0.257499	+	0.966279i	$0.582898\pi$
$740$	0	0
$741$	−16.9706	−0.623429
$742$	0	0
$743$	21.2132	0.778237	0.389118	−	0.921188i	$-0.372780\pi$
$743$	21.2132	0.778237	0.389118	+	0.921188i	$0.372780\pi$
$744$	0	0
$745$	−42.4264	−1.55438
$746$	0	0
$747$	−14.0000	−0.512233
$748$	0	0
$749$	−42.0000	−1.53465
$750$	0	0
$751$	−26.8701	−0.980502	−0.490251	−	0.871581i	$-0.663095\pi$
$751$	−26.8701	−0.980502	−0.490251	+	0.871581i	$0.663095\pi$
$752$	0	0
$753$	28.2843	1.03074
$754$	0	0
$755$	−8.48528	−0.308811
$756$	0	0
$757$	16.0000	0.581530	0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000	0.581530	0.290765	+	0.956795i	$0.406090\pi$
$758$	0	0
$759$	2.82843	0.102665
$760$	0	0
$761$	−38.0000	−1.37750	−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000	−1.37750	−0.688749	+	0.724999i	$0.741840\pi$
$762$	0	0
$763$	−54.0000	−1.95493
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.0000	0.433295
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	−28.2843	−1.01863
$772$	0	0
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	0	0
$775$	18.3848	0.660401
$776$	0	0
$777$	−25.4558	−0.913223
$778$	0	0
$779$	8.48528	0.304017
$780$	0	0
$781$	−10.0000	−0.357828
$782$	0	0
$783$	−8.00000	−0.285897
$784$	0	0
$785$	−42.4264	−1.51426
$786$	0	0
$787$	−7.07107	−0.252056	−0.126028	−	0.992027i	$-0.540223\pi$
$787$	−7.07107	−0.252056	−0.126028	+	0.992027i	$0.540223\pi$
$788$	0	0
$789$	−25.4558	−0.906252
$790$	0	0
$791$	30.0000	1.06668
$792$	0	0
$793$	−2.82843	−0.100440
$794$	0	0
$795$	24.0000	0.851192
$796$	0	0
$797$	−20.0000	−0.708436	−0.354218	−	0.935163i	$-0.615253\pi$
$797$	−20.0000	−0.708436	−0.354218	+	0.935163i	$0.615253\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	−2.00000	−0.0705785
$804$	0	0
$805$	−25.4558	−0.897201
$806$	0	0
$807$	−26.0000	−0.915243
$808$	0	0
$809$	−4.24264	−0.149163	−0.0745817	−	0.997215i	$-0.523762\pi$
$809$	−4.24264	−0.149163	−0.0745817	+	0.997215i	$0.523762\pi$
$810$	0	0
$811$	4.24264	0.148979	0.0744896	−	0.997222i	$-0.476267\pi$
$811$	4.24264	0.148979	0.0744896	+	0.997222i	$0.476267\pi$
$812$	0	0
$813$	11.3137	0.396789
$814$	0	0
$815$	−6.00000	−0.210171
$816$	0	0
$817$	12.0000	0.419827
$818$	0	0
$819$	−8.48528	−0.296500
$820$	0	0
$821$	1.41421	0.0493564	0.0246782	−	0.999695i	$-0.492144\pi$
$821$	1.41421	0.0493564	0.0246782	+	0.999695i	$0.492144\pi$
$822$	0	0
$823$	35.3553	1.23241	0.616205	−	0.787586i	$-0.288669\pi$
$823$	35.3553	1.23241	0.616205	+	0.787586i	$0.288669\pi$
$824$	0	0
$825$	−26.0000	−0.905204
$826$	0	0
$827$	−7.07107	−0.245885	−0.122943	−	0.992414i	$-0.539233\pi$
$827$	−7.07107	−0.245885	−0.122943	+	0.992414i	$0.539233\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−78.0000	−2.69930
$836$	0	0
$837$	−8.00000	−0.276520
$838$	0	0
$839$	38.1838	1.31825	0.659125	−	0.752033i	$-0.270927\pi$
$839$	38.1838	1.31825	0.659125	+	0.752033i	$0.270927\pi$
$840$	0	0
$841$	−27.0000	−0.931034
$842$	0	0
$843$	11.3137	0.389665
$844$	0	0
$845$	38.1838	1.31356
$846$	0	0
$847$	38.1838	1.31201
$848$	0	0
$849$	30.0000	1.02960
$850$	0	0
$851$	−6.00000	−0.205677
$852$	0	0
$853$	−7.07107	−0.242109	−0.121054	−	0.992646i	$-0.538628\pi$
$853$	−7.07107	−0.242109	−0.121054	+	0.992646i	$0.538628\pi$
$854$	0	0
$855$	25.4558	0.870572
$856$	0	0
$857$	46.6690	1.59418	0.797092	−	0.603858i	$-0.206370\pi$
$857$	46.6690	1.59418	0.797092	+	0.603858i	$0.206370\pi$
$858$	0	0
$859$	42.0000	1.43302	0.716511	−	0.697576i	$-0.245738\pi$
$859$	42.0000	1.43302	0.716511	+	0.697576i	$0.245738\pi$
$860$	0	0
$861$	−8.48528	−0.289178
$862$	0	0
$863$	8.00000	0.272323	0.136162	−	0.990687i	$-0.456523\pi$
$863$	8.00000	0.272323	0.136162	+	0.990687i	$0.456523\pi$
$864$	0	0
$865$	−42.0000	−1.42804
$866$	0	0
$867$	0	0
$868$	0	0
$869$	18.0000	0.610608
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	−18.3848	−0.622230
$874$	0	0
$875$	144.000	4.86809
$876$	0	0
$877$	−32.5269	−1.09836	−0.549178	−	0.835705i	$-0.685059\pi$
$877$	−32.5269	−1.09836	−0.549178	+	0.835705i	$0.685059\pi$
$878$	0	0
$879$	2.82843	0.0954005
$880$	0	0
$881$	−24.0416	−0.809983	−0.404992	−	0.914320i	$-0.632726\pi$
$881$	−24.0416	−0.809983	−0.404992	+	0.914320i	$0.632726\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	0	0
$885$	36.0000	1.21013
$886$	0	0
$887$	57.9828	1.94687	0.973435	−	0.228963i	$-0.0735334\pi$
$887$	57.9828	1.94687	0.973435	+	0.228963i	$0.0735334\pi$
$888$	0	0
$889$	−59.3970	−1.99211
$890$	0	0
$891$	7.07107	0.236890
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−76.3675	−2.55269
$896$	0	0
$897$	4.00000	0.133556
$898$	0	0
$899$	2.00000	0.0667037
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−12.0000	−0.399335
$904$	0	0
$905$	54.0000	1.79502
$906$	0	0
$907$	−46.6690	−1.54962	−0.774810	−	0.632194i	$-0.782155\pi$
$907$	−46.6690	−1.54962	−0.774810	+	0.632194i	$0.782155\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	29.6985	0.983955	0.491977	−	0.870608i	$-0.336274\pi$
$911$	29.6985	0.983955	0.491977	+	0.870608i	$0.336274\pi$
$912$	0	0
$913$	−19.7990	−0.655251
$914$	0	0
$915$	−8.48528	−0.280515
$916$	0	0
$917$	54.0000	1.78324
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	39.5980	1.30480
$922$	0	0
$923$	−14.1421	−0.465494
$924$	0	0
$925$	55.1543	1.81346
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9.89949	0.324792	0.162396	−	0.986726i	$-0.448078\pi$
$929$	9.89949	0.324792	0.162396	+	0.986726i	$0.448078\pi$
$930$	0	0
$931$	66.0000	2.16306
$932$	0	0
$933$	38.0000	1.24406
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−32.0000	−1.04539	−0.522697	−	0.852518i	$-0.675074\pi$
$937$	−32.0000	−1.04539	−0.522697	+	0.852518i	$0.675074\pi$
$938$	0	0
$939$	18.0000	0.587408
$940$	0	0
$941$	38.1838	1.24476	0.622378	−	0.782717i	$-0.286167\pi$
$941$	38.1838	1.24476	0.622378	+	0.782717i	$0.286167\pi$
$942$	0	0
$943$	−2.00000	−0.0651290
$944$	0	0
$945$	−101.823	−3.31231
$946$	0	0
$947$	38.1838	1.24081	0.620403	−	0.784283i	$-0.286969\pi$
$947$	38.1838	1.24081	0.620403	+	0.784283i	$0.286969\pi$
$948$	0	0
$949$	−2.82843	−0.0918146
$950$	0	0
$951$	−10.0000	−0.324272
$952$	0	0
$953$	−10.0000	−0.323932	−0.161966	−	0.986796i	$-0.551783\pi$
$953$	−10.0000	−0.323932	−0.161966	+	0.986796i	$0.551783\pi$
$954$	0	0
$955$	101.823	3.29493
$956$	0	0
$957$	−2.82843	−0.0914301
$958$	0	0
$959$	−25.4558	−0.822012
$960$	0	0
$961$	−29.0000	−0.935484
$962$	0	0
$963$	−9.89949	−0.319007
$964$	0	0
$965$	54.0000	1.73832
$966$	0	0
$967$	58.0000	1.86515	0.932577	−	0.360971i	$-0.117555\pi$
$967$	58.0000	1.86515	0.932577	+	0.360971i	$0.117555\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	14.0000	0.449281	0.224641	−	0.974442i	$-0.427879\pi$
$971$	14.0000	0.449281	0.224641	+	0.974442i	$0.427879\pi$
$972$	0	0
$973$	66.0000	2.11586
$974$	0	0
$975$	−36.7696	−1.17757
$976$	0	0
$977$	−8.00000	−0.255943	−0.127971	−	0.991778i	$-0.540847\pi$
$977$	−8.00000	−0.255943	−0.127971	+	0.991778i	$0.540847\pi$
$978$	0	0
$979$	8.48528	0.271191
$980$	0	0
$981$	−12.7279	−0.406371
$982$	0	0
$983$	−4.24264	−0.135319	−0.0676596	−	0.997708i	$-0.521553\pi$
$983$	−4.24264	−0.135319	−0.0676596	+	0.997708i	$0.521553\pi$
$984$	0	0
$985$	66.0000	2.10293
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.82843	−0.0899388
$990$	0	0
$991$	15.5563	0.494164	0.247082	−	0.968995i	$-0.420528\pi$
$991$	15.5563	0.494164	0.247082	+	0.968995i	$0.420528\pi$
$992$	0	0
$993$	2.82843	0.0897574
$994$	0	0
$995$	−42.0000	−1.33149
$996$	0	0
$997$	4.24264	0.134366	0.0671829	−	0.997741i	$-0.478599\pi$
$997$	4.24264	0.134366	0.0671829	+	0.997741i	$0.478599\pi$
$998$	0	0
$999$	−24.0000	−0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.a.h.1.2		2
4.3	odd	2		4624.2.a.q.1.1		2
17.4	even	4		2312.2.b.f.577.1		2
17.8	even	8		136.2.k.a.81.1	✓	2
17.13	even	4		2312.2.b.f.577.2		2
17.15	even	8		136.2.k.a.89.1	yes	2
17.16	even	2	inner	2312.2.a.h.1.1		2
51.8	odd	8		1224.2.w.f.217.1		2
51.32	odd	8		1224.2.w.f.361.1		2
68.15	odd	8		272.2.o.d.225.1		2
68.59	odd	8		272.2.o.d.81.1		2
68.67	odd	2		4624.2.a.q.1.2		2
136.59	odd	8		1088.2.o.g.897.1		2
136.83	odd	8		1088.2.o.g.769.1		2
136.93	even	8		1088.2.o.p.897.1		2
136.117	even	8		1088.2.o.p.769.1		2
204.59	even	8		2448.2.be.n.1441.1		2
204.83	even	8		2448.2.be.n.1585.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.k.a.81.1	✓	2	17.8	even	8
136.2.k.a.89.1	yes	2	17.15	even	8
272.2.o.d.81.1		2	68.59	odd	8
272.2.o.d.225.1		2	68.15	odd	8
1088.2.o.g.769.1		2	136.83	odd	8
1088.2.o.g.897.1		2	136.59	odd	8
1088.2.o.p.769.1		2	136.117	even	8
1088.2.o.p.897.1		2	136.93	even	8
1224.2.w.f.217.1		2	51.8	odd	8
1224.2.w.f.361.1		2	51.32	odd	8
2312.2.a.h.1.1		2	17.16	even	2	inner
2312.2.a.h.1.2		2	1.1	even	1	trivial
2312.2.b.f.577.1		2	17.4	even	4
2312.2.b.f.577.2		2	17.13	even	4
2448.2.be.n.1441.1		2	204.59	even	8
2448.2.be.n.1585.1		2	204.83	even	8
4624.2.a.q.1.1		2	4.3	odd	2
4624.2.a.q.1.2		2	68.67	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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} -4.24264 q^{5} -4.24264 q^{7} -1.00000 q^{9} -1.41421 q^{11} -2.00000 q^{13} -6.00000 q^{15} +6.00000 q^{19} -6.00000 q^{21} -1.41421 q^{23} +13.0000 q^{25} -5.65685 q^{27} +1.41421 q^{29} +1.41421 q^{31} -2.00000 q^{33} +18.0000 q^{35} +4.24264 q^{37} -2.82843 q^{39} +1.41421 q^{41} +2.00000 q^{43} +4.24264 q^{45} +11.0000 q^{49} -4.00000 q^{53} +6.00000 q^{55} +8.48528 q^{57} -6.00000 q^{59} +1.41421 q^{61} +4.24264 q^{63} +8.48528 q^{65} -4.00000 q^{67} -2.00000 q^{69} +7.07107 q^{71} +1.41421 q^{73} +18.3848 q^{75} +6.00000 q^{77} -12.7279 q^{79} -5.00000 q^{81} +14.0000 q^{83} +2.00000 q^{87} -6.00000 q^{89} +8.48528 q^{91} +2.00000 q^{93} -25.4558 q^{95} +18.3848 q^{97} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} - 4 q^{13} - 12 q^{15} + 12 q^{19} - 12 q^{21} + 26 q^{25} - 4 q^{33} + 36 q^{35} + 4 q^{43} + 22 q^{49} - 8 q^{53} + 12 q^{55} - 12 q^{59} - 8 q^{67} - 4 q^{69} + 12 q^{77} - 10 q^{81} + 28 q^{83}+ \cdots + 4 q^{93}+O(q^{100}) \) 2 * q - 2 * q^9 - 4 * q^13 - 12 * q^15 + 12 * q^19 - 12 * q^21 + 26 * q^25 - 4 * q^33 + 36 * q^35 + 4 * q^43 + 22 * q^49 - 8 * q^53 + 12 * q^55 - 12 * q^59 - 8 * q^67 - 4 * q^69 + 12 * q^77 - 10 * q^81 + 28 * q^83 + 4 * q^87 - 12 * q^89 + 4 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more