LMFDB - Embedded newform 6075.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6075,2,Mod(1,6075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6075, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6075 = 3^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6075.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:	$48.5091192279$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 243)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	6075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{2} +4.00000 q^{4} -2.00000 q^{7} +4.89898 q^{8} -2.44949 q^{11} +1.00000 q^{13} -4.89898 q^{14} +4.00000 q^{16} -7.34847 q^{17} -1.00000 q^{19} -6.00000 q^{22} -2.44949 q^{23} +2.44949 q^{26} -8.00000 q^{28} +4.89898 q^{29} -1.00000 q^{31} -18.0000 q^{34} -8.00000 q^{37} -2.44949 q^{38} -4.89898 q^{41} -11.0000 q^{43} -9.79796 q^{44} -6.00000 q^{46} +9.79796 q^{47} -3.00000 q^{49} +4.00000 q^{52} +7.34847 q^{53} -9.79796 q^{56} +12.0000 q^{58} +2.44949 q^{59} +5.00000 q^{61} -2.44949 q^{62} -8.00000 q^{64} +7.00000 q^{67} -29.3939 q^{68} -7.34847 q^{71} -11.0000 q^{73} -19.5959 q^{74} -4.00000 q^{76} +4.89898 q^{77} -7.00000 q^{79} -12.0000 q^{82} -12.2474 q^{83} -26.9444 q^{86} -12.0000 q^{88} -2.00000 q^{91} -9.79796 q^{92} +24.0000 q^{94} +7.00000 q^{97} -7.34847 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{4} - 4 q^{7} + 2 q^{13} + 8 q^{16} - 2 q^{19} - 12 q^{22} - 16 q^{28} - 2 q^{31} - 36 q^{34} - 16 q^{37} - 22 q^{43} - 12 q^{46} - 6 q^{49} + 8 q^{52} + 24 q^{58} + 10 q^{61} - 16 q^{64} + 14 q^{67}+ \cdots + 14 q^{97}+O(q^{100}) $ 2 * q + 8 * q^4 - 4 * q^7 + 2 * q^13 + 8 * q^16 - 2 * q^19 - 12 * q^22 - 16 * q^28 - 2 * q^31 - 36 * q^34 - 16 * q^37 - 22 * q^43 - 12 * q^46 - 6 * q^49 + 8 * q^52 + 24 * q^58 + 10 * q^61 - 16 * q^64 + 14 * q^67 - 22 * q^73 - 8 * q^76 - 14 * q^79 - 24 * q^82 - 24 * q^88 - 4 * q^91 + 48 * q^94 + 14 * q^97

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 $

$2$	2.44949	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$2$	2.44949	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$3$	0	0
$3$	0	0
$4$	4.00000	2.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	4.89898	1.73205
$9$	0	0
$10$	0	0
$11$	−2.44949	−0.738549	−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949	−0.738549	−0.369274	+	0.929320i	$0.620394\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−4.89898	−1.30931
$15$	0	0
$16$	4.00000	1.00000
$17$	−7.34847	−1.78227	−0.891133	−	0.453743i	$-0.850089\pi$
$17$	−7.34847	−1.78227	−0.891133	+	0.453743i	$0.850089\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	−6.00000	−1.27920
$23$	−2.44949	−0.510754	−0.255377	−	0.966842i	$-0.582200\pi$
$23$	−2.44949	−0.510754	−0.255377	+	0.966842i	$0.582200\pi$
$24$	0	0
$25$	0	0
$26$	2.44949	0.480384
$27$	0	0
$28$	−8.00000	−1.51186
$29$	4.89898	0.909718	0.454859	−	0.890564i	$-0.349690\pi$
$29$	4.89898	0.909718	0.454859	+	0.890564i	$0.349690\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	0	0
$33$	0	0
$34$	−18.0000	−3.08697
$35$	0	0
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−2.44949	−0.397360
$39$	0	0
$40$	0	0
$41$	−4.89898	−0.765092	−0.382546	−	0.923936i	$-0.624953\pi$
$41$	−4.89898	−0.765092	−0.382546	+	0.923936i	$0.624953\pi$
$42$	0	0
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	−9.79796	−1.47710
$45$	0	0
$46$	−6.00000	−0.884652
$47$	9.79796	1.42918	0.714590	−	0.699544i	$-0.246613\pi$
$47$	9.79796	1.42918	0.714590	+	0.699544i	$0.246613\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	4.00000	0.554700
$53$	7.34847	1.00939	0.504695	−	0.863298i	$-0.331605\pi$
$53$	7.34847	1.00939	0.504695	+	0.863298i	$0.331605\pi$
$54$	0	0
$55$	0	0
$56$	−9.79796	−1.30931
$57$	0	0
$58$	12.0000	1.57568
$59$	2.44949	0.318896	0.159448	−	0.987206i	$-0.449029\pi$
$59$	2.44949	0.318896	0.159448	+	0.987206i	$0.449029\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	−2.44949	−0.311086
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	−29.3939	−3.56453
$69$	0	0
$70$	0	0
$71$	−7.34847	−0.872103	−0.436051	−	0.899922i	$-0.643623\pi$
$71$	−7.34847	−0.872103	−0.436051	+	0.899922i	$0.643623\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	−19.5959	−2.27798
$75$	0	0
$76$	−4.00000	−0.458831
$77$	4.89898	0.558291
$78$	0	0
$79$	−7.00000	−0.787562	−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000	−0.787562	−0.393781	+	0.919204i	$0.628833\pi$
$80$	0	0
$81$	0	0
$82$	−12.0000	−1.32518
$83$	−12.2474	−1.34433	−0.672166	−	0.740400i	$-0.734636\pi$
$83$	−12.2474	−1.34433	−0.672166	+	0.740400i	$0.734636\pi$
$84$	0	0
$85$	0	0
$86$	−26.9444	−2.90549
$87$	0	0
$88$	−12.0000	−1.27920
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	−9.79796	−1.02151
$93$	0	0
$94$	24.0000	2.47541
$95$	0	0
$96$	0	0
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	−7.34847	−0.742307
$99$	0	0
$100$	0	0
$101$	4.89898	0.487467	0.243733	−	0.969842i	$-0.421628\pi$
$101$	4.89898	0.487467	0.243733	+	0.969842i	$0.421628\pi$
$102$	0	0
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	4.89898	0.480384
$105$	0	0
$106$	18.0000	1.74831
$107$	14.6969	1.42081	0.710403	−	0.703795i	$-0.248513\pi$
$107$	14.6969	1.42081	0.710403	+	0.703795i	$0.248513\pi$
$108$	0	0
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	0	0
$111$	0	0
$112$	−8.00000	−0.755929
$113$	−9.79796	−0.921714	−0.460857	−	0.887474i	$-0.652458\pi$
$113$	−9.79796	−0.921714	−0.460857	+	0.887474i	$0.652458\pi$
$114$	0	0
$115$	0	0
$116$	19.5959	1.81944
$117$	0	0
$118$	6.00000	0.552345
$119$	14.6969	1.34727
$120$	0	0
$121$	−5.00000	−0.454545
$122$	12.2474	1.10883
$123$	0	0
$124$	−4.00000	−0.359211
$125$	0	0
$126$	0	0
$127$	19.0000	1.68598	0.842989	−	0.537931i	$-0.180794\pi$
$127$	19.0000	1.68598	0.842989	+	0.537931i	$0.180794\pi$
$128$	−19.5959	−1.73205
$129$	0	0
$130$	0	0
$131$	−12.2474	−1.07006	−0.535032	−	0.844832i	$-0.679701\pi$
$131$	−12.2474	−1.07006	−0.535032	+	0.844832i	$0.679701\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	17.1464	1.48123
$135$	0	0
$136$	−36.0000	−3.08697
$137$	9.79796	0.837096	0.418548	−	0.908195i	$-0.362539\pi$
$137$	9.79796	0.837096	0.418548	+	0.908195i	$0.362539\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	0	0
$142$	−18.0000	−1.51053
$143$	−2.44949	−0.204837
$144$	0	0
$145$	0	0
$146$	−26.9444	−2.22993
$147$	0	0
$148$	−32.0000	−2.63038
$149$	−12.2474	−1.00335	−0.501675	−	0.865056i	$-0.667283\pi$
$149$	−12.2474	−1.00335	−0.501675	+	0.865056i	$0.667283\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	−4.89898	−0.397360
$153$	0	0
$154$	12.0000	0.966988
$155$	0	0
$156$	0	0
$157$	−17.0000	−1.35675	−0.678374	−	0.734717i	$-0.737315\pi$
$157$	−17.0000	−1.35675	−0.678374	+	0.734717i	$0.737315\pi$
$158$	−17.1464	−1.36410
$159$	0	0
$160$	0	0
$161$	4.89898	0.386094
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	−19.5959	−1.53018
$165$	0	0
$166$	−30.0000	−2.32845
$167$	4.89898	0.379094	0.189547	−	0.981872i	$-0.439298\pi$
$167$	4.89898	0.379094	0.189547	+	0.981872i	$0.439298\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	−44.0000	−3.35497
$173$	9.79796	0.744925	0.372463	−	0.928047i	$-0.378514\pi$
$173$	9.79796	0.744925	0.372463	+	0.928047i	$0.378514\pi$
$174$	0	0
$175$	0	0
$176$	−9.79796	−0.738549
$177$	0	0
$178$	0	0
$179$	14.6969	1.09850	0.549250	−	0.835658i	$-0.314913\pi$
$179$	14.6969	1.09850	0.549250	+	0.835658i	$0.314913\pi$
$180$	0	0
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	−4.89898	−0.363137
$183$	0	0
$184$	−12.0000	−0.884652
$185$	0	0
$186$	0	0
$187$	18.0000	1.31629
$188$	39.1918	2.85836
$189$	0	0
$190$	0	0
$191$	−9.79796	−0.708955	−0.354478	−	0.935064i	$-0.615341\pi$
$191$	−9.79796	−0.708955	−0.354478	+	0.935064i	$0.615341\pi$
$192$	0	0
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	17.1464	1.23104
$195$	0	0
$196$	−12.0000	−0.857143
$197$	14.6969	1.04711	0.523557	−	0.851991i	$-0.324605\pi$
$197$	14.6969	1.04711	0.523557	+	0.851991i	$0.324605\pi$
$198$	0	0
$199$	−1.00000	−0.0708881	−0.0354441	−	0.999372i	$-0.511285\pi$
$199$	−1.00000	−0.0708881	−0.0354441	+	0.999372i	$0.511285\pi$
$200$	0	0
$201$	0	0
$202$	12.0000	0.844317
$203$	−9.79796	−0.687682
$204$	0	0
$205$	0	0
$206$	17.1464	1.19465
$207$	0	0
$208$	4.00000	0.277350
$209$	2.44949	0.169435
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	29.3939	2.01878
$213$	0	0
$214$	36.0000	2.46091
$215$	0	0
$216$	0	0
$217$	2.00000	0.135769
$218$	−2.44949	−0.165900
$219$	0	0
$220$	0	0
$221$	−7.34847	−0.494312
$222$	0	0
$223$	7.00000	0.468755	0.234377	−	0.972146i	$-0.424695\pi$
$223$	7.00000	0.468755	0.234377	+	0.972146i	$0.424695\pi$
$224$	0	0
$225$	0	0
$226$	−24.0000	−1.59646
$227$	9.79796	0.650313	0.325157	−	0.945660i	$-0.394583\pi$
$227$	9.79796	0.650313	0.325157	+	0.945660i	$0.394583\pi$
$228$	0	0
$229$	−1.00000	−0.0660819	−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000	−0.0660819	−0.0330409	+	0.999454i	$0.510519\pi$
$230$	0	0
$231$	0	0
$232$	24.0000	1.57568
$233$	−7.34847	−0.481414	−0.240707	−	0.970598i	$-0.577379\pi$
$233$	−7.34847	−0.481414	−0.240707	+	0.970598i	$0.577379\pi$
$234$	0	0
$235$	0	0
$236$	9.79796	0.637793
$237$	0	0
$238$	36.0000	2.33353
$239$	2.44949	0.158444	0.0792222	−	0.996857i	$-0.474756\pi$
$239$	2.44949	0.158444	0.0792222	+	0.996857i	$0.474756\pi$
$240$	0	0
$241$	−16.0000	−1.03065	−0.515325	−	0.856995i	$-0.672329\pi$
$241$	−16.0000	−1.03065	−0.515325	+	0.856995i	$0.672329\pi$
$242$	−12.2474	−0.787296
$243$	0	0
$244$	20.0000	1.28037
$245$	0	0
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	−4.89898	−0.311086
$249$	0	0
$250$	0	0
$251$	7.34847	0.463831	0.231916	−	0.972736i	$-0.425501\pi$
$251$	7.34847	0.463831	0.231916	+	0.972736i	$0.425501\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	46.5403	2.92020
$255$	0	0
$256$	−32.0000	−2.00000
$257$	−17.1464	−1.06956	−0.534782	−	0.844990i	$-0.679606\pi$
$257$	−17.1464	−1.06956	−0.534782	+	0.844990i	$0.679606\pi$
$258$	0	0
$259$	16.0000	0.994192
$260$	0	0
$261$	0	0
$262$	−30.0000	−1.85341
$263$	−26.9444	−1.66146	−0.830731	−	0.556674i	$-0.812077\pi$
$263$	−26.9444	−1.66146	−0.830731	+	0.556674i	$0.812077\pi$
$264$	0	0
$265$	0	0
$266$	4.89898	0.300376
$267$	0	0
$268$	28.0000	1.71037
$269$	22.0454	1.34413	0.672066	−	0.740491i	$-0.265407\pi$
$269$	22.0454	1.34413	0.672066	+	0.740491i	$0.265407\pi$
$270$	0	0
$271$	−7.00000	−0.425220	−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000	−0.425220	−0.212610	+	0.977137i	$0.568196\pi$
$272$	−29.3939	−1.78227
$273$	0	0
$274$	24.0000	1.44989
$275$	0	0
$276$	0	0
$277$	−11.0000	−0.660926	−0.330463	−	0.943819i	$-0.607205\pi$
$277$	−11.0000	−0.660926	−0.330463	+	0.943819i	$0.607205\pi$
$278$	−24.4949	−1.46911
$279$	0	0
$280$	0	0
$281$	12.2474	0.730622	0.365311	−	0.930886i	$-0.380963\pi$
$281$	12.2474	0.730622	0.365311	+	0.930886i	$0.380963\pi$
$282$	0	0
$283$	−17.0000	−1.01055	−0.505273	−	0.862960i	$-0.668608\pi$
$283$	−17.0000	−1.01055	−0.505273	+	0.862960i	$0.668608\pi$
$284$	−29.3939	−1.74421
$285$	0	0
$286$	−6.00000	−0.354787
$287$	9.79796	0.578355
$288$	0	0
$289$	37.0000	2.17647
$290$	0	0
$291$	0	0
$292$	−44.0000	−2.57491
$293$	4.89898	0.286201	0.143101	−	0.989708i	$-0.454293\pi$
$293$	4.89898	0.286201	0.143101	+	0.989708i	$0.454293\pi$
$294$	0	0
$295$	0	0
$296$	−39.1918	−2.27798
$297$	0	0
$298$	−30.0000	−1.73785
$299$	−2.44949	−0.141658
$300$	0	0
$301$	22.0000	1.26806
$302$	12.2474	0.704761
$303$	0	0
$304$	−4.00000	−0.229416
$305$	0	0
$306$	0	0
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	19.5959	1.11658
$309$	0	0
$310$	0	0
$311$	24.4949	1.38898	0.694489	−	0.719503i	$-0.255630\pi$
$311$	24.4949	1.38898	0.694489	+	0.719503i	$0.255630\pi$
$312$	0	0
$313$	16.0000	0.904373	0.452187	−	0.891923i	$-0.350644\pi$
$313$	16.0000	0.904373	0.452187	+	0.891923i	$0.350644\pi$
$314$	−41.6413	−2.34996
$315$	0	0
$316$	−28.0000	−1.57512
$317$	9.79796	0.550308	0.275154	−	0.961400i	$-0.411271\pi$
$317$	9.79796	0.550308	0.275154	+	0.961400i	$0.411271\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	0	0
$322$	12.0000	0.668734
$323$	7.34847	0.408880
$324$	0	0
$325$	0	0
$326$	24.4949	1.35665
$327$	0	0
$328$	−24.0000	−1.32518
$329$	−19.5959	−1.08036
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	−48.9898	−2.68866
$333$	0	0
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	28.0000	1.52526	0.762629	−	0.646837i	$-0.223908\pi$
$337$	28.0000	1.52526	0.762629	+	0.646837i	$0.223908\pi$
$338$	−29.3939	−1.59882
$339$	0	0
$340$	0	0
$341$	2.44949	0.132647
$342$	0	0
$343$	20.0000	1.07990
$344$	−53.8888	−2.90549
$345$	0	0
$346$	24.0000	1.29025
$347$	−24.4949	−1.31495	−0.657477	−	0.753474i	$-0.728377\pi$
$347$	−24.4949	−1.31495	−0.657477	+	0.753474i	$0.728377\pi$
$348$	0	0
$349$	20.0000	1.07058	0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000	1.07058	0.535288	+	0.844670i	$0.320203\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.44949	0.130373	0.0651866	−	0.997873i	$-0.479236\pi$
$353$	2.44949	0.130373	0.0651866	+	0.997873i	$0.479236\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	36.0000	1.90266
$359$	−29.3939	−1.55135	−0.775675	−	0.631133i	$-0.782590\pi$
$359$	−29.3939	−1.55135	−0.775675	+	0.631133i	$0.782590\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	19.5959	1.02994
$363$	0	0
$364$	−8.00000	−0.419314
$365$	0	0
$366$	0	0
$367$	−5.00000	−0.260998	−0.130499	−	0.991448i	$-0.541658\pi$
$367$	−5.00000	−0.260998	−0.130499	+	0.991448i	$0.541658\pi$
$368$	−9.79796	−0.510754
$369$	0	0
$370$	0	0
$371$	−14.6969	−0.763027
$372$	0	0
$373$	−35.0000	−1.81223	−0.906116	−	0.423030i	$-0.860966\pi$
$373$	−35.0000	−1.81223	−0.906116	+	0.423030i	$0.860966\pi$
$374$	44.0908	2.27988
$375$	0	0
$376$	48.0000	2.47541
$377$	4.89898	0.252310
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	0	0
$382$	−24.0000	−1.22795
$383$	34.2929	1.75228	0.876142	−	0.482054i	$-0.160109\pi$
$383$	34.2929	1.75228	0.876142	+	0.482054i	$0.160109\pi$
$384$	0	0
$385$	0	0
$386$	−26.9444	−1.37143
$387$	0	0
$388$	28.0000	1.42148
$389$	26.9444	1.36613	0.683067	−	0.730355i	$-0.260646\pi$
$389$	26.9444	1.36613	0.683067	+	0.730355i	$0.260646\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	−14.6969	−0.742307
$393$	0	0
$394$	36.0000	1.81365
$395$	0	0
$396$	0	0
$397$	1.00000	0.0501886	0.0250943	−	0.999685i	$-0.492011\pi$
$397$	1.00000	0.0501886	0.0250943	+	0.999685i	$0.492011\pi$
$398$	−2.44949	−0.122782
$399$	0	0
$400$	0	0
$401$	−34.2929	−1.71250	−0.856252	−	0.516559i	$-0.827213\pi$
$401$	−34.2929	−1.71250	−0.856252	+	0.516559i	$0.827213\pi$
$402$	0	0
$403$	−1.00000	−0.0498135
$404$	19.5959	0.974933
$405$	0	0
$406$	−24.0000	−1.19110
$407$	19.5959	0.971334
$408$	0	0
$409$	−28.0000	−1.38451	−0.692255	−	0.721653i	$-0.743383\pi$
$409$	−28.0000	−1.38451	−0.692255	+	0.721653i	$0.743383\pi$
$410$	0	0
$411$	0	0
$412$	28.0000	1.37946
$413$	−4.89898	−0.241063
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	6.00000	0.293470
$419$	−34.2929	−1.67532	−0.837658	−	0.546195i	$-0.816076\pi$
$419$	−34.2929	−1.67532	−0.837658	+	0.546195i	$0.816076\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	−2.44949	−0.119239
$423$	0	0
$424$	36.0000	1.74831
$425$	0	0
$426$	0	0
$427$	−10.0000	−0.483934
$428$	58.7878	2.84161
$429$	0	0
$430$	0	0
$431$	7.34847	0.353963	0.176982	−	0.984214i	$-0.443367\pi$
$431$	7.34847	0.353963	0.176982	+	0.984214i	$0.443367\pi$
$432$	0	0
$433$	−17.0000	−0.816968	−0.408484	−	0.912766i	$-0.633942\pi$
$433$	−17.0000	−0.816968	−0.408484	+	0.912766i	$0.633942\pi$
$434$	4.89898	0.235159
$435$	0	0
$436$	−4.00000	−0.191565
$437$	2.44949	0.117175
$438$	0	0
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	0	0
$441$	0	0
$442$	−18.0000	−0.856173
$443$	−12.2474	−0.581894	−0.290947	−	0.956739i	$-0.593970\pi$
$443$	−12.2474	−0.581894	−0.290947	+	0.956739i	$0.593970\pi$
$444$	0	0
$445$	0	0
$446$	17.1464	0.811907
$447$	0	0
$448$	16.0000	0.755929
$449$	−22.0454	−1.04039	−0.520194	−	0.854048i	$-0.674140\pi$
$449$	−22.0454	−1.04039	−0.520194	+	0.854048i	$0.674140\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	−39.1918	−1.84343
$453$	0	0
$454$	24.0000	1.12638
$455$	0	0
$456$	0	0
$457$	−29.0000	−1.35656	−0.678281	−	0.734802i	$-0.737275\pi$
$457$	−29.0000	−1.35656	−0.678281	+	0.734802i	$0.737275\pi$
$458$	−2.44949	−0.114457
$459$	0	0
$460$	0	0
$461$	26.9444	1.25493	0.627463	−	0.778647i	$-0.284094\pi$
$461$	26.9444	1.25493	0.627463	+	0.778647i	$0.284094\pi$
$462$	0	0
$463$	19.0000	0.883005	0.441502	−	0.897260i	$-0.354446\pi$
$463$	19.0000	0.883005	0.441502	+	0.897260i	$0.354446\pi$
$464$	19.5959	0.909718
$465$	0	0
$466$	−18.0000	−0.833834
$467$	14.6969	0.680093	0.340047	−	0.940409i	$-0.389557\pi$
$467$	14.6969	0.680093	0.340047	+	0.940409i	$0.389557\pi$
$468$	0	0
$469$	−14.0000	−0.646460
$470$	0	0
$471$	0	0
$472$	12.0000	0.552345
$473$	26.9444	1.23890
$474$	0	0
$475$	0	0
$476$	58.7878	2.69453
$477$	0	0
$478$	6.00000	0.274434
$479$	26.9444	1.23112	0.615560	−	0.788090i	$-0.288930\pi$
$479$	26.9444	1.23112	0.615560	+	0.788090i	$0.288930\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	−39.1918	−1.78514
$483$	0	0
$484$	−20.0000	−0.909091
$485$	0	0
$486$	0	0
$487$	−35.0000	−1.58600	−0.793001	−	0.609221i	$-0.791482\pi$
$487$	−35.0000	−1.58600	−0.793001	+	0.609221i	$0.791482\pi$
$488$	24.4949	1.10883
$489$	0	0
$490$	0	0
$491$	39.1918	1.76870	0.884351	−	0.466822i	$-0.154601\pi$
$491$	39.1918	1.76870	0.884351	+	0.466822i	$0.154601\pi$
$492$	0	0
$493$	−36.0000	−1.62136
$494$	−2.44949	−0.110208
$495$	0	0
$496$	−4.00000	−0.179605
$497$	14.6969	0.659248
$498$	0	0
$499$	2.00000	0.0895323	0.0447661	−	0.998997i	$-0.485746\pi$
$499$	2.00000	0.0895323	0.0447661	+	0.998997i	$0.485746\pi$
$500$	0	0
$501$	0	0
$502$	18.0000	0.803379
$503$	−14.6969	−0.655304	−0.327652	−	0.944798i	$-0.606257\pi$
$503$	−14.6969	−0.655304	−0.327652	+	0.944798i	$0.606257\pi$
$504$	0	0
$505$	0	0
$506$	14.6969	0.653359
$507$	0	0
$508$	76.0000	3.37195
$509$	9.79796	0.434287	0.217143	−	0.976140i	$-0.430326\pi$
$509$	9.79796	0.434287	0.217143	+	0.976140i	$0.430326\pi$
$510$	0	0
$511$	22.0000	0.973223
$512$	−39.1918	−1.73205
$513$	0	0
$514$	−42.0000	−1.85254
$515$	0	0
$516$	0	0
$517$	−24.0000	−1.05552
$518$	39.1918	1.72199
$519$	0	0
$520$	0	0
$521$	−22.0454	−0.965827	−0.482913	−	0.875668i	$-0.660421\pi$
$521$	−22.0454	−0.965827	−0.482913	+	0.875668i	$0.660421\pi$
$522$	0	0
$523$	25.0000	1.09317	0.546587	−	0.837402i	$-0.315927\pi$
$523$	25.0000	1.09317	0.546587	+	0.837402i	$0.315927\pi$
$524$	−48.9898	−2.14013
$525$	0	0
$526$	−66.0000	−2.87774
$527$	7.34847	0.320104
$528$	0	0
$529$	−17.0000	−0.739130
$530$	0	0
$531$	0	0
$532$	8.00000	0.346844
$533$	−4.89898	−0.212198
$534$	0	0
$535$	0	0
$536$	34.2929	1.48123
$537$	0	0
$538$	54.0000	2.32811
$539$	7.34847	0.316521
$540$	0	0
$541$	−28.0000	−1.20381	−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000	−1.20381	−0.601907	+	0.798566i	$0.705592\pi$
$542$	−17.1464	−0.736502
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	13.0000	0.555840	0.277920	−	0.960604i	$-0.410355\pi$
$547$	13.0000	0.555840	0.277920	+	0.960604i	$0.410355\pi$
$548$	39.1918	1.67419
$549$	0	0
$550$	0	0
$551$	−4.89898	−0.208704
$552$	0	0
$553$	14.0000	0.595341
$554$	−26.9444	−1.14476
$555$	0	0
$556$	−40.0000	−1.69638
$557$	7.34847	0.311365	0.155682	−	0.987807i	$-0.450242\pi$
$557$	7.34847	0.311365	0.155682	+	0.987807i	$0.450242\pi$
$558$	0	0
$559$	−11.0000	−0.465250
$560$	0	0
$561$	0	0
$562$	30.0000	1.26547
$563$	12.2474	0.516168	0.258084	−	0.966122i	$-0.416909\pi$
$563$	12.2474	0.516168	0.258084	+	0.966122i	$0.416909\pi$
$564$	0	0
$565$	0	0
$566$	−41.6413	−1.75032
$567$	0	0
$568$	−36.0000	−1.51053
$569$	12.2474	0.513440	0.256720	−	0.966486i	$-0.417358\pi$
$569$	12.2474	0.513440	0.256720	+	0.966486i	$0.417358\pi$
$570$	0	0
$571$	−10.0000	−0.418487	−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000	−0.418487	−0.209243	+	0.977864i	$0.567100\pi$
$572$	−9.79796	−0.409673
$573$	0	0
$574$	24.0000	1.00174
$575$	0	0
$576$	0	0
$577$	25.0000	1.04076	0.520382	−	0.853934i	$-0.325790\pi$
$577$	25.0000	1.04076	0.520382	+	0.853934i	$0.325790\pi$
$578$	90.6311	3.76976
$579$	0	0
$580$	0	0
$581$	24.4949	1.01622
$582$	0	0
$583$	−18.0000	−0.745484
$584$	−53.8888	−2.22993
$585$	0	0
$586$	12.0000	0.495715
$587$	2.44949	0.101101	0.0505506	−	0.998721i	$-0.483902\pi$
$587$	2.44949	0.101101	0.0505506	+	0.998721i	$0.483902\pi$
$588$	0	0
$589$	1.00000	0.0412043
$590$	0	0
$591$	0	0
$592$	−32.0000	−1.31519
$593$	7.34847	0.301765	0.150883	−	0.988552i	$-0.451788\pi$
$593$	7.34847	0.301765	0.150883	+	0.988552i	$0.451788\pi$
$594$	0	0
$595$	0	0
$596$	−48.9898	−2.00670
$597$	0	0
$598$	−6.00000	−0.245358
$599$	39.1918	1.60134	0.800668	−	0.599109i	$-0.204478\pi$
$599$	39.1918	1.60134	0.800668	+	0.599109i	$0.204478\pi$
$600$	0	0
$601$	−7.00000	−0.285536	−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000	−0.285536	−0.142768	+	0.989756i	$0.545600\pi$
$602$	53.8888	2.19634
$603$	0	0
$604$	20.0000	0.813788
$605$	0	0
$606$	0	0
$607$	−44.0000	−1.78590	−0.892952	−	0.450151i	$-0.851370\pi$
$607$	−44.0000	−1.78590	−0.892952	+	0.450151i	$0.851370\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.79796	0.396383
$612$	0	0
$613$	−11.0000	−0.444286	−0.222143	−	0.975014i	$-0.571305\pi$
$613$	−11.0000	−0.444286	−0.222143	+	0.975014i	$0.571305\pi$
$614$	−4.89898	−0.197707
$615$	0	0
$616$	24.0000	0.966988
$617$	−24.4949	−0.986127	−0.493064	−	0.869993i	$-0.664123\pi$
$617$	−24.4949	−0.986127	−0.493064	+	0.869993i	$0.664123\pi$
$618$	0	0
$619$	−49.0000	−1.96948	−0.984738	−	0.174042i	$-0.944317\pi$
$619$	−49.0000	−1.96948	−0.984738	+	0.174042i	$0.944317\pi$
$620$	0	0
$621$	0	0
$622$	60.0000	2.40578
$623$	0	0
$624$	0	0
$625$	0	0
$626$	39.1918	1.56642
$627$	0	0
$628$	−68.0000	−2.71350
$629$	58.7878	2.34402
$630$	0	0
$631$	44.0000	1.75161	0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000	1.75161	0.875806	+	0.482663i	$0.160330\pi$
$632$	−34.2929	−1.36410
$633$	0	0
$634$	24.0000	0.953162
$635$	0	0
$636$	0	0
$637$	−3.00000	−0.118864
$638$	−29.3939	−1.16371
$639$	0	0
$640$	0	0
$641$	−17.1464	−0.677243	−0.338622	−	0.940923i	$-0.609961\pi$
$641$	−17.1464	−0.677243	−0.338622	+	0.940923i	$0.609961\pi$
$642$	0	0
$643$	−38.0000	−1.49857	−0.749287	−	0.662246i	$-0.769604\pi$
$643$	−38.0000	−1.49857	−0.749287	+	0.662246i	$0.769604\pi$
$644$	19.5959	0.772187
$645$	0	0
$646$	18.0000	0.708201
$647$	36.7423	1.44449	0.722245	−	0.691637i	$-0.243110\pi$
$647$	36.7423	1.44449	0.722245	+	0.691637i	$0.243110\pi$
$648$	0	0
$649$	−6.00000	−0.235521
$650$	0	0
$651$	0	0
$652$	40.0000	1.56652
$653$	−9.79796	−0.383424	−0.191712	−	0.981451i	$-0.561404\pi$
$653$	−9.79796	−0.383424	−0.191712	+	0.981451i	$0.561404\pi$
$654$	0	0
$655$	0	0
$656$	−19.5959	−0.765092
$657$	0	0
$658$	−48.0000	−1.87123
$659$	19.5959	0.763349	0.381674	−	0.924297i	$-0.375348\pi$
$659$	19.5959	0.763349	0.381674	+	0.924297i	$0.375348\pi$
$660$	0	0
$661$	11.0000	0.427850	0.213925	−	0.976850i	$-0.431375\pi$
$661$	11.0000	0.427850	0.213925	+	0.976850i	$0.431375\pi$
$662$	−17.1464	−0.666415
$663$	0	0
$664$	−60.0000	−2.32845
$665$	0	0
$666$	0	0
$667$	−12.0000	−0.464642
$668$	19.5959	0.758189
$669$	0	0
$670$	0	0
$671$	−12.2474	−0.472808
$672$	0	0
$673$	−29.0000	−1.11787	−0.558934	−	0.829212i	$-0.688789\pi$
$673$	−29.0000	−1.11787	−0.558934	+	0.829212i	$0.688789\pi$
$674$	68.5857	2.64182
$675$	0	0
$676$	−48.0000	−1.84615
$677$	46.5403	1.78869	0.894345	−	0.447379i	$-0.147642\pi$
$677$	46.5403	1.78869	0.894345	+	0.447379i	$0.147642\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	0	0
$682$	6.00000	0.229752
$683$	−22.0454	−0.843544	−0.421772	−	0.906702i	$-0.638592\pi$
$683$	−22.0454	−0.843544	−0.421772	+	0.906702i	$0.638592\pi$
$684$	0	0
$685$	0	0
$686$	48.9898	1.87044
$687$	0	0
$688$	−44.0000	−1.67748
$689$	7.34847	0.279954
$690$	0	0
$691$	47.0000	1.78796	0.893982	−	0.448103i	$-0.147900\pi$
$691$	47.0000	1.78796	0.893982	+	0.448103i	$0.147900\pi$
$692$	39.1918	1.48985
$693$	0	0
$694$	−60.0000	−2.27757
$695$	0	0
$696$	0	0
$697$	36.0000	1.36360
$698$	48.9898	1.85429
$699$	0	0
$700$	0	0
$701$	14.6969	0.555096	0.277548	−	0.960712i	$-0.410478\pi$
$701$	14.6969	0.555096	0.277548	+	0.960712i	$0.410478\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	19.5959	0.738549
$705$	0	0
$706$	6.00000	0.225813
$707$	−9.79796	−0.368490
$708$	0	0
$709$	−7.00000	−0.262891	−0.131445	−	0.991323i	$-0.541962\pi$
$709$	−7.00000	−0.262891	−0.131445	+	0.991323i	$0.541962\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.44949	0.0917341
$714$	0	0
$715$	0	0
$716$	58.7878	2.19700
$717$	0	0
$718$	−72.0000	−2.68702
$719$	36.7423	1.37026	0.685129	−	0.728422i	$-0.259746\pi$
$719$	36.7423	1.37026	0.685129	+	0.728422i	$0.259746\pi$
$720$	0	0
$721$	−14.0000	−0.521387
$722$	−44.0908	−1.64089
$723$	0	0
$724$	32.0000	1.18927
$725$	0	0
$726$	0	0
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	−9.79796	−0.363137
$729$	0	0
$730$	0	0
$731$	80.8332	2.98972
$732$	0	0
$733$	−17.0000	−0.627909	−0.313955	−	0.949438i	$-0.601654\pi$
$733$	−17.0000	−0.627909	−0.313955	+	0.949438i	$0.601654\pi$
$734$	−12.2474	−0.452062
$735$	0	0
$736$	0	0
$737$	−17.1464	−0.631597
$738$	0	0
$739$	−1.00000	−0.0367856	−0.0183928	−	0.999831i	$-0.505855\pi$
$739$	−1.00000	−0.0367856	−0.0183928	+	0.999831i	$0.505855\pi$
$740$	0	0
$741$	0	0
$742$	−36.0000	−1.32160
$743$	−31.8434	−1.16822	−0.584110	−	0.811675i	$-0.698556\pi$
$743$	−31.8434	−1.16822	−0.584110	+	0.811675i	$0.698556\pi$
$744$	0	0
$745$	0	0
$746$	−85.7321	−3.13888
$747$	0	0
$748$	72.0000	2.63258
$749$	−29.3939	−1.07403
$750$	0	0
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	39.1918	1.42918
$753$	0	0
$754$	12.0000	0.437014
$755$	0	0
$756$	0	0
$757$	7.00000	0.254419	0.127210	−	0.991876i	$-0.459398\pi$
$757$	7.00000	0.254419	0.127210	+	0.991876i	$0.459398\pi$
$758$	19.5959	0.711756
$759$	0	0
$760$	0	0
$761$	2.44949	0.0887939	0.0443970	−	0.999014i	$-0.485863\pi$
$761$	2.44949	0.0887939	0.0443970	+	0.999014i	$0.485863\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	−39.1918	−1.41791
$765$	0	0
$766$	84.0000	3.03504
$767$	2.44949	0.0884459
$768$	0	0
$769$	−37.0000	−1.33425	−0.667127	−	0.744944i	$-0.732476\pi$
$769$	−37.0000	−1.33425	−0.667127	+	0.744944i	$0.732476\pi$
$770$	0	0
$771$	0	0
$772$	−44.0000	−1.58359
$773$	−44.0908	−1.58584	−0.792918	−	0.609328i	$-0.791439\pi$
$773$	−44.0908	−1.58584	−0.792918	+	0.609328i	$0.791439\pi$
$774$	0	0
$775$	0	0
$776$	34.2929	1.23104
$777$	0	0
$778$	66.0000	2.36621
$779$	4.89898	0.175524
$780$	0	0
$781$	18.0000	0.644091
$782$	44.0908	1.57668
$783$	0	0
$784$	−12.0000	−0.428571
$785$	0	0
$786$	0	0
$787$	25.0000	0.891154	0.445577	−	0.895244i	$-0.352999\pi$
$787$	25.0000	0.891154	0.445577	+	0.895244i	$0.352999\pi$
$788$	58.7878	2.09423
$789$	0	0
$790$	0	0
$791$	19.5959	0.696751
$792$	0	0
$793$	5.00000	0.177555
$794$	2.44949	0.0869291
$795$	0	0
$796$	−4.00000	−0.141776
$797$	41.6413	1.47501	0.737506	−	0.675341i	$-0.236003\pi$
$797$	41.6413	1.47501	0.737506	+	0.675341i	$0.236003\pi$
$798$	0	0
$799$	−72.0000	−2.54718
$800$	0	0
$801$	0	0
$802$	−84.0000	−2.96614
$803$	26.9444	0.950847
$804$	0	0
$805$	0	0
$806$	−2.44949	−0.0862796
$807$	0	0
$808$	24.0000	0.844317
$809$	−22.0454	−0.775075	−0.387538	−	0.921854i	$-0.626674\pi$
$809$	−22.0454	−0.775075	−0.387538	+	0.921854i	$0.626674\pi$
$810$	0	0
$811$	35.0000	1.22902	0.614508	−	0.788911i	$-0.289355\pi$
$811$	35.0000	1.22902	0.614508	+	0.788911i	$0.289355\pi$
$812$	−39.1918	−1.37536
$813$	0	0
$814$	48.0000	1.68240
$815$	0	0
$816$	0	0
$817$	11.0000	0.384841
$818$	−68.5857	−2.39804
$819$	0	0
$820$	0	0
$821$	−39.1918	−1.36780	−0.683902	−	0.729574i	$-0.739719\pi$
$821$	−39.1918	−1.36780	−0.683902	+	0.729574i	$0.739719\pi$
$822$	0	0
$823$	−35.0000	−1.22002	−0.610012	−	0.792392i	$-0.708835\pi$
$823$	−35.0000	−1.22002	−0.610012	+	0.792392i	$0.708835\pi$
$824$	34.2929	1.19465
$825$	0	0
$826$	−12.0000	−0.417533
$827$	−22.0454	−0.766594	−0.383297	−	0.923625i	$-0.625211\pi$
$827$	−22.0454	−0.766594	−0.383297	+	0.923625i	$0.625211\pi$
$828$	0	0
$829$	−37.0000	−1.28506	−0.642532	−	0.766259i	$-0.722116\pi$
$829$	−37.0000	−1.28506	−0.642532	+	0.766259i	$0.722116\pi$
$830$	0	0
$831$	0	0
$832$	−8.00000	−0.277350
$833$	22.0454	0.763828
$834$	0	0
$835$	0	0
$836$	9.79796	0.338869
$837$	0	0
$838$	−84.0000	−2.90173
$839$	4.89898	0.169132	0.0845658	−	0.996418i	$-0.473050\pi$
$839$	4.89898	0.169132	0.0845658	+	0.996418i	$0.473050\pi$
$840$	0	0
$841$	−5.00000	−0.172414
$842$	4.89898	0.168830
$843$	0	0
$844$	−4.00000	−0.137686
$845$	0	0
$846$	0	0
$847$	10.0000	0.343604
$848$	29.3939	1.00939
$849$	0	0
$850$	0	0
$851$	19.5959	0.671739
$852$	0	0
$853$	13.0000	0.445112	0.222556	−	0.974920i	$-0.428560\pi$
$853$	13.0000	0.445112	0.222556	+	0.974920i	$0.428560\pi$
$854$	−24.4949	−0.838198
$855$	0	0
$856$	72.0000	2.46091
$857$	24.4949	0.836730	0.418365	−	0.908279i	$-0.362603\pi$
$857$	24.4949	0.836730	0.418365	+	0.908279i	$0.362603\pi$
$858$	0	0
$859$	26.0000	0.887109	0.443554	−	0.896248i	$-0.353717\pi$
$859$	26.0000	0.887109	0.443554	+	0.896248i	$0.353717\pi$
$860$	0	0
$861$	0	0
$862$	18.0000	0.613082
$863$	−7.34847	−0.250145	−0.125072	−	0.992148i	$-0.539916\pi$
$863$	−7.34847	−0.250145	−0.125072	+	0.992148i	$0.539916\pi$
$864$	0	0
$865$	0	0
$866$	−41.6413	−1.41503
$867$	0	0
$868$	8.00000	0.271538
$869$	17.1464	0.581653
$870$	0	0
$871$	7.00000	0.237186
$872$	−4.89898	−0.165900
$873$	0	0
$874$	6.00000	0.202953
$875$	0	0
$876$	0	0
$877$	−8.00000	−0.270141	−0.135070	−	0.990836i	$-0.543126\pi$
$877$	−8.00000	−0.270141	−0.135070	+	0.990836i	$0.543126\pi$
$878$	34.2929	1.15733
$879$	0	0
$880$	0	0
$881$	−36.7423	−1.23788	−0.618941	−	0.785438i	$-0.712438\pi$
$881$	−36.7423	−1.23788	−0.618941	+	0.785438i	$0.712438\pi$
$882$	0	0
$883$	−17.0000	−0.572096	−0.286048	−	0.958215i	$-0.592342\pi$
$883$	−17.0000	−0.572096	−0.286048	+	0.958215i	$0.592342\pi$
$884$	−29.3939	−0.988623
$885$	0	0
$886$	−30.0000	−1.00787
$887$	−17.1464	−0.575721	−0.287860	−	0.957672i	$-0.592944\pi$
$887$	−17.1464	−0.575721	−0.287860	+	0.957672i	$0.592944\pi$
$888$	0	0
$889$	−38.0000	−1.27448
$890$	0	0
$891$	0	0
$892$	28.0000	0.937509
$893$	−9.79796	−0.327876
$894$	0	0
$895$	0	0
$896$	39.1918	1.30931
$897$	0	0
$898$	−54.0000	−1.80200
$899$	−4.89898	−0.163390
$900$	0	0
$901$	−54.0000	−1.79900
$902$	29.3939	0.978709
$903$	0	0
$904$	−48.0000	−1.59646
$905$	0	0
$906$	0	0
$907$	7.00000	0.232431	0.116216	−	0.993224i	$-0.462924\pi$
$907$	7.00000	0.232431	0.116216	+	0.993224i	$0.462924\pi$
$908$	39.1918	1.30063
$909$	0	0
$910$	0	0
$911$	12.2474	0.405776	0.202888	−	0.979202i	$-0.434967\pi$
$911$	12.2474	0.405776	0.202888	+	0.979202i	$0.434967\pi$
$912$	0	0
$913$	30.0000	0.992855
$914$	−71.0352	−2.34964
$915$	0	0
$916$	−4.00000	−0.132164
$917$	24.4949	0.808893
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	0	0
$922$	66.0000	2.17359
$923$	−7.34847	−0.241878
$924$	0	0
$925$	0	0
$926$	46.5403	1.52941
$927$	0	0
$928$	0	0
$929$	26.9444	0.884017	0.442008	−	0.897011i	$-0.354266\pi$
$929$	26.9444	0.884017	0.442008	+	0.897011i	$0.354266\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	−29.3939	−0.962828
$933$	0	0
$934$	36.0000	1.17796
$935$	0	0
$936$	0	0
$937$	−8.00000	−0.261349	−0.130674	−	0.991425i	$-0.541714\pi$
$937$	−8.00000	−0.261349	−0.130674	+	0.991425i	$0.541714\pi$
$938$	−34.2929	−1.11970
$939$	0	0
$940$	0	0
$941$	9.79796	0.319404	0.159702	−	0.987165i	$-0.448947\pi$
$941$	9.79796	0.319404	0.159702	+	0.987165i	$0.448947\pi$
$942$	0	0
$943$	12.0000	0.390774
$944$	9.79796	0.318896
$945$	0	0
$946$	66.0000	2.14585
$947$	24.4949	0.795977	0.397989	−	0.917390i	$-0.369708\pi$
$947$	24.4949	0.795977	0.397989	+	0.917390i	$0.369708\pi$
$948$	0	0
$949$	−11.0000	−0.357075
$950$	0	0
$951$	0	0
$952$	72.0000	2.33353
$953$	29.3939	0.952161	0.476081	−	0.879402i	$-0.342057\pi$
$953$	29.3939	0.952161	0.476081	+	0.879402i	$0.342057\pi$
$954$	0	0
$955$	0	0
$956$	9.79796	0.316889
$957$	0	0
$958$	66.0000	2.13236
$959$	−19.5959	−0.632785
$960$	0	0
$961$	−30.0000	−0.967742
$962$	−19.5959	−0.631798
$963$	0	0
$964$	−64.0000	−2.06130
$965$	0	0
$966$	0	0
$967$	7.00000	0.225105	0.112552	−	0.993646i	$-0.464097\pi$
$967$	7.00000	0.225105	0.112552	+	0.993646i	$0.464097\pi$
$968$	−24.4949	−0.787296
$969$	0	0
$970$	0	0
$971$	−29.3939	−0.943294	−0.471647	−	0.881787i	$-0.656340\pi$
$971$	−29.3939	−0.943294	−0.471647	+	0.881787i	$0.656340\pi$
$972$	0	0
$973$	20.0000	0.641171
$974$	−85.7321	−2.74703
$975$	0	0
$976$	20.0000	0.640184
$977$	−24.4949	−0.783661	−0.391831	−	0.920037i	$-0.628158\pi$
$977$	−24.4949	−0.783661	−0.391831	+	0.920037i	$0.628158\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	96.0000	3.06348
$983$	−4.89898	−0.156253	−0.0781266	−	0.996943i	$-0.524894\pi$
$983$	−4.89898	−0.156253	−0.0781266	+	0.996943i	$0.524894\pi$
$984$	0	0
$985$	0	0
$986$	−88.1816	−2.80828
$987$	0	0
$988$	−4.00000	−0.127257
$989$	26.9444	0.856782
$990$	0	0
$991$	−7.00000	−0.222362	−0.111181	−	0.993800i	$-0.535463\pi$
$991$	−7.00000	−0.222362	−0.111181	+	0.993800i	$0.535463\pi$
$992$	0	0
$993$	0	0
$994$	36.0000	1.14185
$995$	0	0
$996$	0	0
$997$	−50.0000	−1.58352	−0.791758	−	0.610835i	$-0.790834\pi$
$997$	−50.0000	−1.58352	−0.791758	+	0.610835i	$0.790834\pi$
$998$	4.89898	0.155074
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6075.2.a.bn.1.2		2
3.2	odd	2	inner	6075.2.a.bn.1.1		2
5.4	even	2		243.2.a.d.1.1	✓	2
15.14	odd	2		243.2.a.d.1.2	yes	2
20.19	odd	2		3888.2.a.z.1.2		2
45.4	even	6		243.2.c.c.163.2		4
45.14	odd	6		243.2.c.c.163.1		4
45.29	odd	6		243.2.c.c.82.1		4
45.34	even	6		243.2.c.c.82.2		4
60.59	even	2		3888.2.a.z.1.1		2
135.4	even	18		729.2.e.p.406.1		12
135.14	odd	18		729.2.e.p.163.1		12
135.29	odd	18		729.2.e.p.568.1		12
135.34	even	18		729.2.e.p.325.1		12
135.49	even	18		729.2.e.p.649.1		12
135.59	odd	18		729.2.e.p.649.2		12
135.74	odd	18		729.2.e.p.325.2		12
135.79	even	18		729.2.e.p.568.2		12
135.94	even	18		729.2.e.p.163.2		12
135.104	odd	18		729.2.e.p.406.2		12
135.119	odd	18		729.2.e.p.82.2		12
135.124	even	18		729.2.e.p.82.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.a.d.1.1	✓	2	5.4	even	2
243.2.a.d.1.2	yes	2	15.14	odd	2
243.2.c.c.82.1		4	45.29	odd	6
243.2.c.c.82.2		4	45.34	even	6
243.2.c.c.163.1		4	45.14	odd	6
243.2.c.c.163.2		4	45.4	even	6
729.2.e.p.82.1		12	135.124	even	18
729.2.e.p.82.2		12	135.119	odd	18
729.2.e.p.163.1		12	135.14	odd	18
729.2.e.p.163.2		12	135.94	even	18
729.2.e.p.325.1		12	135.34	even	18
729.2.e.p.325.2		12	135.74	odd	18
729.2.e.p.406.1		12	135.4	even	18
729.2.e.p.406.2		12	135.104	odd	18
729.2.e.p.568.1		12	135.29	odd	18
729.2.e.p.568.2		12	135.79	even	18
729.2.e.p.649.1		12	135.49	even	18
729.2.e.p.649.2		12	135.59	odd	18
3888.2.a.z.1.1		2	60.59	even	2
3888.2.a.z.1.2		2	20.19	odd	2
6075.2.a.bn.1.1		2	3.2	odd	2	inner
6075.2.a.bn.1.2		2	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:	\( N \)	\(=\)	\( 6075 = 3^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6075.a (trivial)

\(f(q)\)	\(=\)	\(q+2.44949 q^{2} +4.00000 q^{4} -2.00000 q^{7} +4.89898 q^{8} -2.44949 q^{11} +1.00000 q^{13} -4.89898 q^{14} +4.00000 q^{16} -7.34847 q^{17} -1.00000 q^{19} -6.00000 q^{22} -2.44949 q^{23} +2.44949 q^{26} -8.00000 q^{28} +4.89898 q^{29} -1.00000 q^{31} -18.0000 q^{34} -8.00000 q^{37} -2.44949 q^{38} -4.89898 q^{41} -11.0000 q^{43} -9.79796 q^{44} -6.00000 q^{46} +9.79796 q^{47} -3.00000 q^{49} +4.00000 q^{52} +7.34847 q^{53} -9.79796 q^{56} +12.0000 q^{58} +2.44949 q^{59} +5.00000 q^{61} -2.44949 q^{62} -8.00000 q^{64} +7.00000 q^{67} -29.3939 q^{68} -7.34847 q^{71} -11.0000 q^{73} -19.5959 q^{74} -4.00000 q^{76} +4.89898 q^{77} -7.00000 q^{79} -12.0000 q^{82} -12.2474 q^{83} -26.9444 q^{86} -12.0000 q^{88} -2.00000 q^{91} -9.79796 q^{92} +24.0000 q^{94} +7.00000 q^{97} -7.34847 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{4} - 4 q^{7} + 2 q^{13} + 8 q^{16} - 2 q^{19} - 12 q^{22} - 16 q^{28} - 2 q^{31} - 36 q^{34} - 16 q^{37} - 22 q^{43} - 12 q^{46} - 6 q^{49} + 8 q^{52} + 24 q^{58} + 10 q^{61} - 16 q^{64} + 14 q^{67}+ \cdots + 14 q^{97}+O(q^{100}) \) 2 * q + 8 * q^4 - 4 * q^7 + 2 * q^13 + 8 * q^16 - 2 * q^19 - 12 * q^22 - 16 * q^28 - 2 * q^31 - 36 * q^34 - 16 * q^37 - 22 * q^43 - 12 * q^46 - 6 * q^49 + 8 * q^52 + 24 * q^58 + 10 * q^61 - 16 * q^64 + 14 * q^67 - 22 * q^73 - 8 * q^76 - 14 * q^79 - 24 * q^82 - 24 * q^88 - 4 * q^91 + 48 * q^94 + 14 * q^97

Introduction

L-functions

Modular forms

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