LMFDB - Embedded newform 8100.2.a.bd.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8100,2,Mod(1,8100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8100, 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("8100.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.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:	$64.6788256372$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.1207701504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 14x^{4} + 43x^{2} - 36 $ x^6 - 14x^4 + 43x^2 - 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 180)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.56613$ of defining polynomial
Character	$\chi$	$=$	8100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.26496 q^{7} +O(q^{10})$	$q+2.26496 q^{7} +2.54722 q^{11} +5.02510 q^{13} +5.72392 q^{17} -1.86997 q^{19} +5.39722 q^{23} +3.00000 q^{29} +9.38162 q^{31} +2.59165 q^{37} -3.96442 q^{41} -10.2084 q^{43} +1.07095 q^{47} -1.86997 q^{49} +10.7036 q^{53} +1.58280 q^{59} +0.869975 q^{61} +4.85661 q^{67} +1.86997 q^{71} -3.87652 q^{73} +5.76935 q^{77} -13.2516 q^{79} -14.7032 q^{83} +13.4172 q^{89} +11.3816 q^{91} -17.6669 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 2 q^{11} + 18 q^{29} - 6 q^{31} + 14 q^{41} + 34 q^{59} - 6 q^{61} - 6 q^{79} + 56 q^{89} + 6 q^{91}+O(q^{100}) $ 6 * q + 2 * q^11 + 18 * q^29 - 6 * q^31 + 14 * q^41 + 34 * q^59 - 6 * q^61 - 6 * q^79 + 56 * q^89 + 6 * q^91

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.26496	0.856073	0.428036	−	0.903761i	$-0.359205\pi$
$7$	2.26496	0.856073	0.428036	+	0.903761i	$0.359205\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.54722	0.768017	0.384009	−	0.923330i	$-0.374543\pi$
$11$	2.54722	0.768017	0.384009	+	0.923330i	$0.374543\pi$
$12$	0	0
$13$	5.02510	1.39371	0.696856	−	0.717212i	$-0.254582\pi$
$13$	5.02510	1.39371	0.696856	+	0.717212i	$0.254582\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.72392	1.38825	0.694127	−	0.719852i	$-0.255791\pi$
$17$	5.72392	1.38825	0.694127	+	0.719852i	$0.255791\pi$
$18$	0	0
$19$	−1.86997	−0.429002	−0.214501	−	0.976724i	$-0.568812\pi$
$19$	−1.86997	−0.429002	−0.214501	+	0.976724i	$0.568812\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.39722	1.12540	0.562699	−	0.826662i	$-0.309763\pi$
$23$	5.39722	1.12540	0.562699	+	0.826662i	$0.309763\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	9.38162	1.68499	0.842495	−	0.538705i	$-0.181086\pi$
$31$	9.38162	1.68499	0.842495	+	0.538705i	$0.181086\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.59165	0.426065	0.213032	−	0.977045i	$-0.431666\pi$
$37$	2.59165	0.426065	0.213032	+	0.977045i	$0.431666\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.96442	−0.619139	−0.309569	−	0.950877i	$-0.600185\pi$
$41$	−3.96442	−0.619139	−0.309569	+	0.950877i	$0.600185\pi$
$42$	0	0
$43$	−10.2084	−1.55677	−0.778383	−	0.627790i	$-0.783960\pi$
$43$	−10.2084	−1.55677	−0.778383	+	0.627790i	$0.783960\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.07095	0.156214	0.0781070	−	0.996945i	$-0.475112\pi$
$47$	1.07095	0.156214	0.0781070	+	0.996945i	$0.475112\pi$
$48$	0	0
$49$	−1.86997	−0.267139
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.7036	1.47025	0.735125	−	0.677931i	$-0.237123\pi$
$53$	10.7036	1.47025	0.735125	+	0.677931i	$0.237123\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.58280	0.206063	0.103032	−	0.994678i	$-0.467146\pi$
$59$	1.58280	0.206063	0.103032	+	0.994678i	$0.467146\pi$
$60$	0	0
$61$	0.869975	0.111389	0.0556944	−	0.998448i	$-0.482263\pi$
$61$	0.869975	0.111389	0.0556944	+	0.998448i	$0.482263\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.85661	0.593329	0.296664	−	0.954982i	$-0.404126\pi$
$67$	4.85661	0.593329	0.296664	+	0.954982i	$0.404126\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.86997	0.221925	0.110963	−	0.993825i	$-0.464607\pi$
$71$	1.86997	0.221925	0.110963	+	0.993825i	$0.464607\pi$
$72$	0	0
$73$	−3.87652	−0.453713	−0.226856	−	0.973928i	$-0.572845\pi$
$73$	−3.87652	−0.453713	−0.226856	+	0.973928i	$0.572845\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.76935	0.657479
$78$	0	0
$79$	−13.2516	−1.49092	−0.745461	−	0.666550i	$-0.767770\pi$
$79$	−13.2516	−1.49092	−0.745461	+	0.666550i	$0.767770\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−14.7032	−1.61388	−0.806941	−	0.590632i	$-0.798878\pi$
$83$	−14.7032	−1.61388	−0.806941	+	0.590632i	$0.798878\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.4172	1.42222	0.711110	−	0.703081i	$-0.248193\pi$
$89$	13.4172	1.42222	0.711110	+	0.703081i	$0.248193\pi$
$90$	0	0
$91$	11.3816	1.19312
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−17.6669	−1.79381	−0.896903	−	0.442227i	$-0.854188\pi$
$97$	−17.6669	−1.79381	−0.896903	+	0.442227i	$0.854188\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.32275	0.529633	0.264817	−	0.964299i	$-0.414689\pi$
$101$	5.32275	0.529633	0.264817	+	0.964299i	$0.414689\pi$
$102$	0	0
$103$	6.01547	0.592722	0.296361	−	0.955076i	$-0.404227\pi$
$103$	6.01547	0.592722	0.296361	+	0.955076i	$0.404227\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.69840	−0.454212	−0.227106	−	0.973870i	$-0.572926\pi$
$107$	−4.69840	−0.454212	−0.227106	+	0.973870i	$0.572926\pi$
$108$	0	0
$109$	−6.64167	−0.636157	−0.318078	−	0.948064i	$-0.603038\pi$
$109$	−6.64167	−0.636157	−0.318078	+	0.948064i	$0.603038\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.8000	−1.20413	−0.602064	−	0.798448i	$-0.705655\pi$
$113$	−12.8000	−1.20413	−0.602064	+	0.798448i	$0.705655\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12.9644	1.18845
$120$	0	0
$121$	−4.51165	−0.410150
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.8103	−1.13673	−0.568367	−	0.822775i	$-0.692424\pi$
$127$	−12.8103	−1.13673	−0.568367	+	0.822775i	$0.692424\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.51165	0.831037	0.415518	−	0.909585i	$-0.363600\pi$
$131$	9.51165	0.831037	0.415518	+	0.909585i	$0.363600\pi$
$132$	0	0
$133$	−4.23541	−0.367257
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.4933	−0.981936	−0.490968	−	0.871178i	$-0.663357\pi$
$137$	−11.4933	−0.981936	−0.490968	+	0.871178i	$0.663357\pi$
$138$	0	0
$139$	−11.5116	−0.976405	−0.488203	−	0.872730i	$-0.662347\pi$
$139$	−11.5116	−0.976405	−0.488203	+	0.872730i	$0.662347\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	12.8000	1.07039
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.7044	1.61425	0.807123	−	0.590384i	$-0.201024\pi$
$149$	19.7044	1.61425	0.807123	+	0.590384i	$0.201024\pi$
$150$	0	0
$151$	15.3816	1.25174	0.625869	−	0.779928i	$-0.284744\pi$
$151$	15.3816	1.25174	0.625869	+	0.779928i	$0.284744\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.4836	0.996303	0.498151	−	0.867090i	$-0.334012\pi$
$157$	12.4836	0.996303	0.498151	+	0.867090i	$0.334012\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	12.2245	0.963424
$162$	0	0
$163$	−1.93826	−0.151816	−0.0759082	−	0.997115i	$-0.524186\pi$
$163$	−1.93826	−0.151816	−0.0759082	+	0.997115i	$0.524186\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.326695	0.0252804	0.0126402	−	0.999920i	$-0.495976\pi$
$167$	0.326695	0.0252804	0.0126402	+	0.999920i	$0.495976\pi$
$168$	0	0
$169$	12.2516	0.942431
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.7992	−1.58133	−0.790667	−	0.612246i	$-0.790266\pi$
$173$	−20.7992	−1.58133	−0.790667	+	0.612246i	$0.790266\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3.79882	−0.283937	−0.141969	−	0.989871i	$-0.545343\pi$
$179$	−3.79882	−0.283937	−0.141969	+	0.989871i	$0.545343\pi$
$180$	0	0
$181$	−16.5116	−1.22730	−0.613651	−	0.789578i	$-0.710300\pi$
$181$	−16.5116	−1.22730	−0.613651	+	0.789578i	$0.710300\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	14.5801	1.06620
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.3816	−0.823545	−0.411773	−	0.911287i	$-0.635090\pi$
$191$	−11.3816	−0.823545	−0.411773	+	0.911287i	$0.635090\pi$
$192$	0	0
$193$	14.0849	1.01385	0.506927	−	0.861989i	$-0.330781\pi$
$193$	14.0849	1.01385	0.506927	+	0.861989i	$0.330781\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.90830	0.563443	0.281721	−	0.959496i	$-0.409095\pi$
$197$	7.90830	0.563443	0.281721	+	0.959496i	$0.409095\pi$
$198$	0	0
$199$	−1.86997	−0.132559	−0.0662795	−	0.997801i	$-0.521113\pi$
$199$	−1.86997	−0.132559	−0.0662795	+	0.997801i	$0.521113\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.79487	0.476906
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.76325	−0.329481
$210$	0	0
$211$	−11.3816	−0.783543	−0.391772	−	0.920063i	$-0.628138\pi$
$211$	−11.3816	−0.783543	−0.391772	+	0.920063i	$0.628138\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	21.2490	1.44247
$218$	0	0
$219$	0	0
$220$	0	0
$221$	28.7632	1.93483
$222$	0	0
$223$	−4.85661	−0.325222	−0.162611	−	0.986690i	$-0.551992\pi$
$223$	−4.85661	−0.325222	−0.162611	+	0.986690i	$0.551992\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.07613	−0.469659	−0.234830	−	0.972037i	$-0.575453\pi$
$227$	−7.07613	−0.469659	−0.234830	+	0.972037i	$0.575453\pi$
$228$	0	0
$229$	8.73995	0.577552	0.288776	−	0.957397i	$-0.406752\pi$
$229$	8.73995	0.577552	0.288776	+	0.957397i	$0.406752\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.26346	−0.606869	−0.303435	−	0.952852i	$-0.598133\pi$
$233$	−9.26346	−0.606869	−0.303435	+	0.952852i	$0.598133\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.6772	0.820023	0.410012	−	0.912080i	$-0.365525\pi$
$239$	12.6772	0.820023	0.410012	+	0.912080i	$0.365525\pi$
$240$	0	0
$241$	−10.6099	−0.683445	−0.341723	−	0.939801i	$-0.611010\pi$
$241$	−10.6099	−0.683445	−0.341723	+	0.939801i	$0.611010\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−9.39680	−0.597904
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−18.6332	−1.17612	−0.588059	−	0.808818i	$-0.700108\pi$
$251$	−18.6332	−1.17612	−0.588059	+	0.808818i	$0.700108\pi$
$252$	0	0
$253$	13.7479	0.864326
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−26.3898	−1.64615	−0.823075	−	0.567933i	$-0.807743\pi$
$257$	−26.3898	−1.64615	−0.823075	+	0.567933i	$0.807743\pi$
$258$	0	0
$259$	5.86997	0.364742
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.6776	0.843400	0.421700	−	0.906735i	$-0.361434\pi$
$263$	13.6776	0.843400	0.421700	+	0.906735i	$0.361434\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.6417	1.38049	0.690244	−	0.723577i	$-0.257503\pi$
$269$	22.6417	1.38049	0.690244	+	0.723577i	$0.257503\pi$
$270$	0	0
$271$	5.86997	0.356576	0.178288	−	0.983978i	$-0.442944\pi$
$271$	5.86997	0.356576	0.178288	+	0.983978i	$0.442944\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−0.158205	−0.00950560	−0.00475280	−	0.999989i	$-0.501513\pi$
$277$	−0.158205	−0.00950560	−0.00475280	+	0.999989i	$0.501513\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.39008	0.202235	0.101117	−	0.994874i	$-0.467758\pi$
$281$	3.39008	0.202235	0.101117	+	0.994874i	$0.467758\pi$
$282$	0	0
$283$	11.9782	0.712028	0.356014	−	0.934481i	$-0.384135\pi$
$283$	11.9782	0.712028	0.356014	+	0.934481i	$0.384135\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.97924	−0.530028
$288$	0	0
$289$	15.7632	0.927250
$290$	0	0
$291$	0	0
$292$	0	0
$293$	15.0753	0.880708	0.440354	−	0.897824i	$-0.354853\pi$
$293$	15.0753	0.880708	0.440354	+	0.897824i	$0.354853\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	27.1216	1.56848
$300$	0	0
$301$	−23.1216	−1.33271
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−6.00518	−0.342734	−0.171367	−	0.985207i	$-0.554818\pi$
$307$	−6.00518	−0.342734	−0.171367	+	0.985207i	$0.554818\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−30.4049	−1.72410	−0.862052	−	0.506819i	$-0.830821\pi$
$311$	−30.4049	−1.72410	−0.862052	+	0.506819i	$0.830821\pi$
$312$	0	0
$313$	−18.2984	−1.03429	−0.517144	−	0.855898i	$-0.673005\pi$
$313$	−18.2984	−1.03429	−0.517144	+	0.855898i	$0.673005\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.47377	0.475935	0.237967	−	0.971273i	$-0.423519\pi$
$317$	8.47377	0.475935	0.237967	+	0.971273i	$0.423519\pi$
$318$	0	0
$319$	7.64167	0.427852
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−10.7036	−0.595563
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.42565	0.133731
$330$	0	0
$331$	−15.2516	−0.838304	−0.419152	−	0.907916i	$-0.637672\pi$
$331$	−15.2516	−0.838304	−0.419152	+	0.907916i	$0.637672\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−17.9834	−0.979616	−0.489808	−	0.871830i	$-0.662933\pi$
$337$	−17.9834	−0.979616	−0.489808	+	0.871830i	$0.662933\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	23.8971	1.29410
$342$	0	0
$343$	−20.0901	−1.08476
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.3821	0.879436	0.439718	−	0.898136i	$-0.355078\pi$
$347$	16.3821	0.879436	0.439718	+	0.898136i	$0.355078\pi$
$348$	0	0
$349$	17.7632	0.950845	0.475422	−	0.879758i	$-0.342295\pi$
$349$	17.7632	0.950845	0.475422	+	0.879758i	$0.342295\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.7083	1.10219	0.551097	−	0.834441i	$-0.314209\pi$
$353$	20.7083	1.10219	0.551097	+	0.834441i	$0.314209\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	34.3732	1.81415	0.907073	−	0.420973i	$-0.138311\pi$
$359$	34.3732	1.81415	0.907073	+	0.420973i	$0.138311\pi$
$360$	0	0
$361$	−15.5032	−0.815958
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.14857	0.0599551	0.0299776	−	0.999551i	$-0.490456\pi$
$367$	1.14857	0.0599551	0.0299776	+	0.999551i	$0.490456\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	24.2431	1.25864
$372$	0	0
$373$	−21.5435	−1.11548	−0.557739	−	0.830016i	$-0.688331\pi$
$373$	−21.5435	−1.11548	−0.557739	+	0.830016i	$0.688331\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	15.0753	0.776417
$378$	0	0
$379$	24.7632	1.27200	0.636001	−	0.771688i	$-0.280587\pi$
$379$	24.7632	1.27200	0.636001	+	0.771688i	$0.280587\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.53659	0.180711	0.0903556	−	0.995910i	$-0.471200\pi$
$383$	3.53659	0.180711	0.0903556	+	0.995910i	$0.471200\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	30.7988	1.56156	0.780781	−	0.624805i	$-0.214821\pi$
$389$	30.7988	1.56156	0.780781	+	0.624805i	$0.214821\pi$
$390$	0	0
$391$	30.8933	1.56234
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−18.1196	−0.909399	−0.454700	−	0.890645i	$-0.650253\pi$
$397$	−18.1196	−0.909399	−0.454700	+	0.890645i	$0.650253\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.5828	0.678293	0.339146	−	0.940734i	$-0.389862\pi$
$401$	13.5828	0.678293	0.339146	+	0.940734i	$0.389862\pi$
$402$	0	0
$403$	47.1436	2.34839
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.60152	0.327225
$408$	0	0
$409$	28.2749	1.39810	0.699052	−	0.715071i	$-0.253606\pi$
$409$	28.2749	1.39810	0.699052	+	0.715071i	$0.253606\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.58497	0.176405
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	22.0271	1.07610	0.538048	−	0.842914i	$-0.319162\pi$
$419$	22.0271	1.07610	0.538048	+	0.842914i	$0.319162\pi$
$420$	0	0
$421$	−16.2749	−0.793190	−0.396595	−	0.917994i	$-0.629808\pi$
$421$	−16.2749	−0.793190	−0.396595	+	0.917994i	$0.629808\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.97045	0.0953570
$428$	0	0
$429$	0	0
$430$	0	0
$431$	18.6921	0.900366	0.450183	−	0.892936i	$-0.351359\pi$
$431$	18.6921	0.900366	0.450183	+	0.892936i	$0.351359\pi$
$432$	0	0
$433$	11.9885	0.576128	0.288064	−	0.957611i	$-0.406988\pi$
$433$	11.9885	0.576128	0.288064	+	0.957611i	$0.406988\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10.0927	−0.482798
$438$	0	0
$439$	−13.2516	−0.632464	−0.316232	−	0.948682i	$-0.602418\pi$
$439$	−13.2516	−0.632464	−0.316232	+	0.948682i	$0.602418\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	23.3132	1.10765	0.553823	−	0.832635i	$-0.313169\pi$
$443$	23.3132	1.10765	0.553823	+	0.832635i	$0.313169\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.7632	−0.507949	−0.253974	−	0.967211i	$-0.581738\pi$
$449$	−10.7632	−0.507949	−0.253974	+	0.967211i	$0.581738\pi$
$450$	0	0
$451$	−10.0983	−0.475509
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−9.55501	−0.446964	−0.223482	−	0.974708i	$-0.571742\pi$
$457$	−9.55501	−0.446964	−0.223482	+	0.974708i	$0.571742\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17.2601	0.803881	0.401940	−	0.915666i	$-0.368336\pi$
$461$	17.2601	0.803881	0.401940	+	0.915666i	$0.368336\pi$
$462$	0	0
$463$	−35.8291	−1.66512	−0.832559	−	0.553936i	$-0.813125\pi$
$463$	−35.8291	−1.66512	−0.832559	+	0.553936i	$0.813125\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−28.5287	−1.32015	−0.660076	−	0.751199i	$-0.729476\pi$
$467$	−28.5287	−1.32015	−0.660076	+	0.751199i	$0.729476\pi$
$468$	0	0
$469$	11.0000	0.507933
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−26.0031	−1.19562
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−25.8259	−1.18002	−0.590009	−	0.807397i	$-0.700876\pi$
$479$	−25.8259	−1.18002	−0.590009	+	0.807397i	$0.700876\pi$
$480$	0	0
$481$	13.0233	0.593811
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−4.88880	−0.221533	−0.110766	−	0.993846i	$-0.535331\pi$
$487$	−4.88880	−0.221533	−0.110766	+	0.993846i	$0.535331\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9.51165	0.429255	0.214627	−	0.976696i	$-0.431146\pi$
$491$	9.51165	0.429255	0.214627	+	0.976696i	$0.431146\pi$
$492$	0	0
$493$	17.1718	0.773377
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.23541	0.189984
$498$	0	0
$499$	9.51165	0.425800	0.212900	−	0.977074i	$-0.431709\pi$
$499$	9.51165	0.425800	0.212900	+	0.977074i	$0.431709\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−13.7801	−0.614426	−0.307213	−	0.951641i	$-0.599396\pi$
$503$	−13.7801	−0.614426	−0.307213	+	0.951641i	$0.599396\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.97670	0.0876159	0.0438079	−	0.999040i	$-0.486051\pi$
$509$	1.97670	0.0876159	0.0438079	+	0.999040i	$0.486051\pi$
$510$	0	0
$511$	−8.78015	−0.388411
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.72795	0.119975
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−18.1804	−0.796500	−0.398250	−	0.917277i	$-0.630382\pi$
$521$	−18.1804	−0.796500	−0.398250	+	0.917277i	$0.630382\pi$
$522$	0	0
$523$	15.0431	0.657789	0.328894	−	0.944367i	$-0.393324\pi$
$523$	15.0431	0.657789	0.328894	+	0.944367i	$0.393324\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	53.6996	2.33919
$528$	0	0
$529$	6.13003	0.266523
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−19.9216	−0.862901
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.76325	−0.205167
$540$	0	0
$541$	12.7717	0.549098	0.274549	−	0.961573i	$-0.411471\pi$
$541$	12.7717	0.549098	0.274549	+	0.961573i	$0.411471\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.42900	0.403155	0.201577	−	0.979473i	$-0.435393\pi$
$547$	9.42900	0.403155	0.201577	+	0.979473i	$0.435393\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.60992	−0.238991
$552$	0	0
$553$	−30.0143	−1.27634
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.85726	−0.248180	−0.124090	−	0.992271i	$-0.539601\pi$
$557$	−5.85726	−0.248180	−0.124090	+	0.992271i	$0.539601\pi$
$558$	0	0
$559$	−51.2982	−2.16968
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−31.3124	−1.31966	−0.659830	−	0.751415i	$-0.729372\pi$
$563$	−31.3124	−1.31966	−0.659830	+	0.751415i	$0.729372\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	35.4405	1.48574	0.742871	−	0.669434i	$-0.233463\pi$
$569$	35.4405	1.48574	0.742871	+	0.669434i	$0.233463\pi$
$570$	0	0
$571$	−38.7950	−1.62352	−0.811760	−	0.583991i	$-0.801490\pi$
$571$	−38.7950	−1.62352	−0.811760	+	0.583991i	$0.801490\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	19.7634	0.822761	0.411381	−	0.911464i	$-0.365047\pi$
$577$	19.7634	0.822761	0.411381	+	0.911464i	$0.365047\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−33.3020	−1.38160
$582$	0	0
$583$	27.2644	1.12918
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.0380	0.868331	0.434165	−	0.900833i	$-0.357043\pi$
$587$	21.0380	0.868331	0.434165	+	0.900833i	$0.357043\pi$
$588$	0	0
$589$	−17.5434	−0.722863
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14.9390	−0.613471	−0.306735	−	0.951795i	$-0.599237\pi$
$593$	−14.9390	−0.613471	−0.306735	+	0.951795i	$0.599237\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−21.5116	−0.878942	−0.439471	−	0.898257i	$-0.644834\pi$
$599$	−21.5116	−0.878942	−0.439471	+	0.898257i	$0.644834\pi$
$600$	0	0
$601$	5.51165	0.224825	0.112412	−	0.993662i	$-0.464142\pi$
$601$	5.51165	0.224825	0.112412	+	0.993662i	$0.464142\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9.70293	0.393830	0.196915	−	0.980421i	$-0.436908\pi$
$607$	9.70293	0.393830	0.196915	+	0.980421i	$0.436908\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.38162	0.217717
$612$	0	0
$613$	−2.25467	−0.0910653	−0.0455326	−	0.998963i	$-0.514499\pi$
$613$	−2.25467	−0.0910653	−0.0455326	+	0.998963i	$0.514499\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−40.1553	−1.61659	−0.808297	−	0.588776i	$-0.799610\pi$
$617$	−40.1553	−1.61659	−0.808297	+	0.588776i	$0.799610\pi$
$618$	0	0
$619$	5.12157	0.205853	0.102927	−	0.994689i	$-0.467179\pi$
$619$	5.12157	0.205853	0.102927	+	0.994689i	$0.467179\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	30.3894	1.21752
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.8344	0.591486
$630$	0	0
$631$	−1.28335	−0.0510892	−0.0255446	−	0.999674i	$-0.508132\pi$
$631$	−1.28335	−0.0510892	−0.0255446	+	0.999674i	$0.508132\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−9.39680	−0.372315
$638$	0	0
$639$	0	0
$640$	0	0
$641$	28.5388	1.12721	0.563607	−	0.826043i	$-0.309413\pi$
$641$	28.5388	1.12721	0.563607	+	0.826043i	$0.309413\pi$
$642$	0	0
$643$	−23.3132	−0.919384	−0.459692	−	0.888078i	$-0.652040\pi$
$643$	−23.3132	−0.919384	−0.459692	+	0.888078i	$0.652040\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.93381	−0.351224	−0.175612	−	0.984459i	$-0.556190\pi$
$647$	−8.93381	−0.351224	−0.175612	+	0.984459i	$0.556190\pi$
$648$	0	0
$649$	4.03175	0.158260
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14.3310	0.560817	0.280408	−	0.959881i	$-0.409530\pi$
$653$	14.3310	0.560817	0.280408	+	0.959881i	$0.409530\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	7.58280	0.295384	0.147692	−	0.989033i	$-0.452816\pi$
$659$	7.58280	0.295384	0.147692	+	0.989033i	$0.452816\pi$
$660$	0	0
$661$	29.5116	1.14787	0.573935	−	0.818901i	$-0.305416\pi$
$661$	29.5116	1.14787	0.573935	+	0.818901i	$0.305416\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.1917	0.626944
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.21602	0.0855485
$672$	0	0
$673$	6.98527	0.269262	0.134631	−	0.990896i	$-0.457015\pi$
$673$	6.98527	0.269262	0.134631	+	0.990896i	$0.457015\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20.1033	0.772634	0.386317	−	0.922366i	$-0.373747\pi$
$677$	20.1033	0.772634	0.386317	+	0.922366i	$0.373747\pi$
$678$	0	0
$679$	−40.0148	−1.53563
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−44.5270	−1.70378	−0.851890	−	0.523721i	$-0.824544\pi$
$683$	−44.5270	−1.70378	−0.851890	+	0.523721i	$0.824544\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	53.7865	2.04910
$690$	0	0
$691$	33.7717	1.28474	0.642368	−	0.766396i	$-0.277952\pi$
$691$	33.7717	1.28474	0.642368	+	0.766396i	$0.277952\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−22.6920	−0.859522
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.78398	0.256227	0.128114	−	0.991759i	$-0.459108\pi$
$701$	6.78398	0.256227	0.128114	+	0.991759i	$0.459108\pi$
$702$	0	0
$703$	−4.84632	−0.182782
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.0558	0.453405
$708$	0	0
$709$	−44.7865	−1.68199	−0.840997	−	0.541040i	$-0.818031\pi$
$709$	−44.7865	−1.68199	−0.840997	+	0.541040i	$0.818031\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	50.6347	1.89629
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−5.55105	−0.207019	−0.103510	−	0.994628i	$-0.533007\pi$
$719$	−5.55105	−0.207019	−0.103510	+	0.994628i	$0.533007\pi$
$720$	0	0
$721$	13.6248	0.507413
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	37.5564	1.39289	0.696444	−	0.717611i	$-0.254764\pi$
$727$	37.5564	1.39289	0.696444	+	0.717611i	$0.254764\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−58.4320	−2.16119
$732$	0	0
$733$	34.5442	1.27592	0.637959	−	0.770070i	$-0.279779\pi$
$733$	34.5442	1.27592	0.637959	+	0.770070i	$0.279779\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.3709	0.455687
$738$	0	0
$739$	−0.520101	−0.0191322	−0.00956612	−	0.999954i	$-0.503045\pi$
$739$	−0.520101	−0.0191322	−0.00956612	+	0.999954i	$0.503045\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	41.7918	1.53319	0.766596	−	0.642130i	$-0.221949\pi$
$743$	41.7918	1.53319	0.766596	+	0.642130i	$0.221949\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−10.6417	−0.388838
$750$	0	0
$751$	17.1216	0.624775	0.312388	−	0.949955i	$-0.398871\pi$
$751$	17.1216	0.624775	0.312388	+	0.949955i	$0.398871\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−6.80515	−0.247337	−0.123669	−	0.992324i	$-0.539466\pi$
$757$	−6.80515	−0.247337	−0.123669	+	0.992324i	$0.539466\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	45.2431	1.64006	0.820031	−	0.572319i	$-0.193956\pi$
$761$	45.2431	1.64006	0.820031	+	0.572319i	$0.193956\pi$
$762$	0	0
$763$	−15.0431	−0.544597
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.95373	0.287192
$768$	0	0
$769$	−15.2431	−0.549682	−0.274841	−	0.961490i	$-0.588625\pi$
$769$	−15.2431	−0.549682	−0.274841	+	0.961490i	$0.588625\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.41337	0.265612
$780$	0	0
$781$	4.76325	0.170442
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−20.2586	−0.722141	−0.361070	−	0.932539i	$-0.617589\pi$
$787$	−20.2586	−0.722141	−0.361070	+	0.932539i	$0.617589\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−28.9915	−1.03082
$792$	0	0
$793$	4.37171	0.155244
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11.2691	−0.399171	−0.199585	−	0.979880i	$-0.563960\pi$
$797$	−11.2691	−0.399171	−0.199585	+	0.979880i	$0.563960\pi$
$798$	0	0
$799$	6.13003	0.216865
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−9.87437	−0.348459
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	26.5032	0.931803	0.465901	−	0.884837i	$-0.345730\pi$
$809$	26.5032	0.931803	0.465901	+	0.884837i	$0.345730\pi$
$810$	0	0
$811$	−1.54340	−0.0541960	−0.0270980	−	0.999633i	$-0.508627\pi$
$811$	−1.54340	−0.0541960	−0.0270980	+	0.999633i	$0.508627\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	19.0894	0.667855
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−11.3189	−0.395033	−0.197517	−	0.980300i	$-0.563288\pi$
$821$	−11.3189	−0.395033	−0.197517	+	0.980300i	$0.563288\pi$
$822$	0	0
$823$	9.72350	0.338940	0.169470	−	0.985535i	$-0.445794\pi$
$823$	9.72350	0.338940	0.169470	+	0.985535i	$0.445794\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.9722	0.903143	0.451571	−	0.892235i	$-0.350864\pi$
$827$	25.9722	0.903143	0.451571	+	0.892235i	$0.350864\pi$
$828$	0	0
$829$	−9.61838	−0.334060	−0.167030	−	0.985952i	$-0.553418\pi$
$829$	−9.61838	−0.334060	−0.167030	+	0.985952i	$0.553418\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−10.7036	−0.370857
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−20.0906	−0.693605	−0.346803	−	0.937938i	$-0.612733\pi$
$839$	−20.0906	−0.693605	−0.346803	+	0.937938i	$0.612733\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−10.2187	−0.351118
$848$	0	0
$849$	0	0
$850$	0	0
$851$	13.9877	0.479493
$852$	0	0
$853$	−18.9312	−0.648193	−0.324097	−	0.946024i	$-0.605060\pi$
$853$	−18.9312	−0.648193	−0.324097	+	0.946024i	$0.605060\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−38.0618	−1.30017	−0.650084	−	0.759863i	$-0.725266\pi$
$857$	−38.0618	−1.30017	−0.650084	+	0.759863i	$0.725266\pi$
$858$	0	0
$859$	6.35833	0.216943	0.108472	−	0.994100i	$-0.465404\pi$
$859$	6.35833	0.216943	0.108472	+	0.994100i	$0.465404\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6.61609	−0.225214	−0.112607	−	0.993640i	$-0.535920\pi$
$863$	−6.61609	−0.225214	−0.112607	+	0.993640i	$0.535920\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−33.7548	−1.14505
$870$	0	0
$871$	24.4049	0.826929
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	21.5654	0.728211	0.364105	−	0.931358i	$-0.381375\pi$
$877$	21.5654	0.728211	0.364105	+	0.931358i	$0.381375\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	43.1449	1.45359	0.726794	−	0.686856i	$-0.241010\pi$
$881$	43.1449	1.45359	0.726794	+	0.686856i	$0.241010\pi$
$882$	0	0
$883$	−18.3306	−0.616874	−0.308437	−	0.951245i	$-0.599806\pi$
$883$	−18.3306	−0.616874	−0.308437	+	0.951245i	$0.599806\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38.0618	−1.27799	−0.638996	−	0.769210i	$-0.720650\pi$
$887$	−38.0618	−1.27799	−0.638996	+	0.769210i	$0.720650\pi$
$888$	0	0
$889$	−29.0148	−0.973127
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.00265	−0.0670160
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	28.1449	0.938684
$900$	0	0
$901$	61.2664	2.04108
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	29.1705	0.968590	0.484295	−	0.874905i	$-0.339076\pi$
$907$	29.1705	0.968590	0.484295	+	0.874905i	$0.339076\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.51165	0.116346	0.0581730	−	0.998307i	$-0.481472\pi$
$911$	3.51165	0.116346	0.0581730	+	0.998307i	$0.481472\pi$
$912$	0	0
$913$	−37.4523	−1.23949
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21.5435	0.711428
$918$	0	0
$919$	−24.2431	−0.799708	−0.399854	−	0.916579i	$-0.630939\pi$
$919$	−24.2431	−0.799708	−0.399854	+	0.916579i	$0.630939\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.39680	0.309300
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8.73150	0.286471	0.143236	−	0.989689i	$-0.454249\pi$
$929$	8.73150	0.286471	0.143236	+	0.989689i	$0.454249\pi$
$930$	0	0
$931$	3.49681	0.114603
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35.9873	1.17565	0.587826	−	0.808987i	$-0.299984\pi$
$937$	35.9873	1.17565	0.587826	+	0.808987i	$0.299984\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	26.6688	0.869378	0.434689	−	0.900581i	$-0.356858\pi$
$941$	26.6688	0.869378	0.434689	+	0.900581i	$0.356858\pi$
$942$	0	0
$943$	−21.3969	−0.696778
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−8.05917	−0.261888	−0.130944	−	0.991390i	$-0.541801\pi$
$947$	−8.05917	−0.261888	−0.130944	+	0.991390i	$0.541801\pi$
$948$	0	0
$949$	−19.4799	−0.632344
$950$	0	0
$951$	0	0
$952$	0	0
$953$	23.7733	0.770092	0.385046	−	0.922897i	$-0.374186\pi$
$953$	23.7733	0.770092	0.385046	+	0.922897i	$0.374186\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−26.0317	−0.840609
$960$	0	0
$961$	57.0148	1.83919
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−33.7223	−1.08444	−0.542218	−	0.840238i	$-0.682415\pi$
$967$	−33.7223	−1.08444	−0.542218	+	0.840238i	$0.682415\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−36.1766	−1.16096	−0.580481	−	0.814273i	$-0.697136\pi$
$971$	−36.1766	−1.16096	−0.580481	+	0.814273i	$0.697136\pi$
$972$	0	0
$973$	−26.0734	−0.835874
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50.0722	1.60195	0.800976	−	0.598697i	$-0.204315\pi$
$977$	50.0722	1.60195	0.800976	+	0.598697i	$0.204315\pi$
$978$	0	0
$979$	34.1766	1.09229
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−40.3941	−1.28837	−0.644186	−	0.764869i	$-0.722804\pi$
$983$	−40.3941	−1.28837	−0.644186	+	0.764869i	$0.722804\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−55.0970	−1.75198
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−42.9506	−1.36026	−0.680130	−	0.733091i	$-0.738077\pi$
$997$	−42.9506	−1.36026	−0.680130	+	0.733091i	$0.738077\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.a.bd.1.5		6
3.2	odd	2		8100.2.a.bc.1.5		6
5.2	odd	4		1620.2.d.d.649.5		6
5.3	odd	4		1620.2.d.d.649.6		6
5.4	even	2	inner	8100.2.a.bd.1.2		6
9.2	odd	6		900.2.i.f.301.2		12
9.4	even	3		2700.2.i.f.1801.2		12
9.5	odd	6		900.2.i.f.601.2		12
9.7	even	3		2700.2.i.f.901.2		12
15.2	even	4		1620.2.d.c.649.2		6
15.8	even	4		1620.2.d.c.649.1		6
15.14	odd	2		8100.2.a.bc.1.2		6
45.2	even	12		180.2.r.a.49.2	✓	12
45.4	even	6		2700.2.i.f.1801.5		12
45.7	odd	12		540.2.r.a.469.3		12
45.13	odd	12		540.2.r.a.289.3		12
45.14	odd	6		900.2.i.f.601.5		12
45.22	odd	12		540.2.r.a.289.2		12
45.23	even	12		180.2.r.a.169.2	yes	12
45.29	odd	6		900.2.i.f.301.5		12
45.32	even	12		180.2.r.a.169.5	yes	12
45.34	even	6		2700.2.i.f.901.5		12
45.38	even	12		180.2.r.a.49.5	yes	12
45.43	odd	12		540.2.r.a.469.2		12
180.7	even	12		2160.2.by.e.1009.3		12
180.23	odd	12		720.2.by.e.529.5		12
180.43	even	12		2160.2.by.e.1009.2		12
180.47	odd	12		720.2.by.e.49.5		12
180.67	even	12		2160.2.by.e.289.2		12
180.83	odd	12		720.2.by.e.49.2		12
180.103	even	12		2160.2.by.e.289.3		12
180.167	odd	12		720.2.by.e.529.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.r.a.49.2	✓	12	45.2	even	12
180.2.r.a.49.5	yes	12	45.38	even	12
180.2.r.a.169.2	yes	12	45.23	even	12
180.2.r.a.169.5	yes	12	45.32	even	12
540.2.r.a.289.2		12	45.22	odd	12
540.2.r.a.289.3		12	45.13	odd	12
540.2.r.a.469.2		12	45.43	odd	12
540.2.r.a.469.3		12	45.7	odd	12
720.2.by.e.49.2		12	180.83	odd	12
720.2.by.e.49.5		12	180.47	odd	12
720.2.by.e.529.2		12	180.167	odd	12
720.2.by.e.529.5		12	180.23	odd	12
900.2.i.f.301.2		12	9.2	odd	6
900.2.i.f.301.5		12	45.29	odd	6
900.2.i.f.601.2		12	9.5	odd	6
900.2.i.f.601.5		12	45.14	odd	6
1620.2.d.c.649.1		6	15.8	even	4
1620.2.d.c.649.2		6	15.2	even	4
1620.2.d.d.649.5		6	5.2	odd	4
1620.2.d.d.649.6		6	5.3	odd	4
2160.2.by.e.289.2		12	180.67	even	12
2160.2.by.e.289.3		12	180.103	even	12
2160.2.by.e.1009.2		12	180.43	even	12
2160.2.by.e.1009.3		12	180.7	even	12
2700.2.i.f.901.2		12	9.7	even	3
2700.2.i.f.901.5		12	45.34	even	6
2700.2.i.f.1801.2		12	9.4	even	3
2700.2.i.f.1801.5		12	45.4	even	6
8100.2.a.bc.1.2		6	15.14	odd	2
8100.2.a.bc.1.5		6	3.2	odd	2
8100.2.a.bd.1.2		6	5.4	even	2	inner
8100.2.a.bd.1.5		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.a (trivial)

\(f(q)\)	\(=\)	\(q+2.26496 q^{7} +O(q^{10})\)	\(q+2.26496 q^{7} +2.54722 q^{11} +5.02510 q^{13} +5.72392 q^{17} -1.86997 q^{19} +5.39722 q^{23} +3.00000 q^{29} +9.38162 q^{31} +2.59165 q^{37} -3.96442 q^{41} -10.2084 q^{43} +1.07095 q^{47} -1.86997 q^{49} +10.7036 q^{53} +1.58280 q^{59} +0.869975 q^{61} +4.85661 q^{67} +1.86997 q^{71} -3.87652 q^{73} +5.76935 q^{77} -13.2516 q^{79} -14.7032 q^{83} +13.4172 q^{89} +11.3816 q^{91} -17.6669 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 2 q^{11} + 18 q^{29} - 6 q^{31} + 14 q^{41} + 34 q^{59} - 6 q^{61} - 6 q^{79} + 56 q^{89} + 6 q^{91}+O(q^{100}) \) 6 * q + 2 * q^11 + 18 * q^29 - 6 * q^31 + 14 * q^41 + 34 * q^59 - 6 * q^61 - 6 * q^79 + 56 * q^89 + 6 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more