LMFDB - Embedded newform 1575.2.g.c.1574.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1574,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

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

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

chi := DirichletCharacter("1575.1574");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.g (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$12.5764383184$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 315)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1574.6
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1575.1574
Dual form			1575.2.g.c.1574.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.517638 q^{2} -1.73205 q^{4} +(2.44949 + 1.00000i) q^{7} -1.93185 q^{8} +O(q^{10})$	$q+0.517638 q^{2} -1.73205 q^{4} +(2.44949 + 1.00000i) q^{7} -1.93185 q^{8} +0.378937i q^{11} -1.79315 q^{13} +(1.26795 + 0.517638i) q^{14} +2.46410 q^{16} +3.46410i q^{17} -1.79315i q^{19} +0.196152i q^{22} -1.41421 q^{23} -0.928203 q^{26} +(-4.24264 - 1.73205i) q^{28} +1.41421i q^{29} +6.69213i q^{31} +5.13922 q^{32} +1.79315i q^{34} -1.46410i q^{37} -0.928203i q^{38} -10.3923 q^{41} +4.92820i q^{43} -0.656339i q^{44} -0.732051 q^{46} +9.46410i q^{47} +(5.00000 + 4.89898i) q^{49} +3.10583 q^{52} +10.1769 q^{53} +(-4.73205 - 1.93185i) q^{56} +0.732051i q^{58} -9.46410 q^{59} +13.3843i q^{61} +3.46410i q^{62} -2.26795 q^{64} -10.9282i q^{67} -6.00000i q^{68} +15.0759i q^{71} +1.79315 q^{73} -0.757875i q^{74} +3.10583i q^{76} +(-0.378937 + 0.928203i) q^{77} +10.9282 q^{79} -5.37945 q^{82} +9.46410i q^{83} +2.55103i q^{86} -0.732051i q^{88} -12.9282 q^{89} +(-4.39230 - 1.79315i) q^{91} +2.44949 q^{92} +4.89898i q^{94} +15.1774 q^{97} +(2.58819 + 2.53590i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 24 q^{14} - 8 q^{16} + 48 q^{26} + 8 q^{46} + 40 q^{49} - 24 q^{56} - 48 q^{59} - 32 q^{64} + 32 q^{79} - 48 q^{89} + 48 q^{91}+O(q^{100}) $ 8 * q + 24 * q^14 - 8 * q^16 + 48 * q^26 + 8 * q^46 + 40 * q^49 - 24 * q^56 - 48 * q^59 - 32 * q^64 + 32 * q^79 - 48 * q^89 + 48 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$1226$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	0.517638			0.366025			0.183013	−	0.983111i	$-0.441415\pi$
$2$	0.517638			0.366025			0.183013	+	0.983111i	$0.441415\pi$
$3$		0			0
$3$		0			0
$4$	−1.73205			−0.866025
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.44949	+	1.00000i	0.925820	+	0.377964i
$7$	2.44949	+	1.00000i	0.925820	+	0.377964i
$8$	−1.93185			−0.683013
$9$		0			0
$10$		0			0
$11$			0.378937i			0.114254i	0.998367	+	0.0571270i	$0.0181940\pi$
$11$			0.378937i			0.114254i	−0.998367	+	0.0571270i	$0.981806\pi$
$12$		0			0
$13$	−1.79315			−0.497331			−0.248665	−	0.968589i	$-0.579992\pi$
$13$	−1.79315			−0.497331			−0.248665	+	0.968589i	$0.579992\pi$
$14$	1.26795	+	0.517638i	0.338874	+	0.138345i
$15$		0			0
$16$	2.46410			0.616025
$17$			3.46410i			0.840168i	0.907485	+	0.420084i	$0.137999\pi$
$17$			3.46410i			0.840168i	−0.907485	+	0.420084i	$0.862001\pi$
$18$		0			0
$19$		−	1.79315i		−	0.411377i	−0.978618	−	0.205689i	$-0.934057\pi$
$19$		−	1.79315i		−	0.411377i	0.978618	−	0.205689i	$-0.0659434\pi$
$20$		0			0
$21$		0			0
$22$			0.196152i			0.0418198i
$23$	−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
$25$		0			0
$26$	−0.928203			−0.182036
$27$		0			0
$28$	−4.24264	−	1.73205i	−0.801784	−	0.327327i
$29$			1.41421i			0.262613i	0.991342	+	0.131306i	$0.0419172\pi$
$29$			1.41421i			0.262613i	−0.991342	+	0.131306i	$0.958083\pi$
$30$		0			0
$31$			6.69213i			1.20194i	0.799271	+	0.600971i	$0.205219\pi$
$31$			6.69213i			1.20194i	−0.799271	+	0.600971i	$0.794781\pi$
$32$	5.13922			0.908494
$33$		0			0
$34$			1.79315i			0.307523i
$35$		0			0
$36$		0			0
$37$		−	1.46410i		−	0.240697i	−0.992732	−	0.120348i	$-0.961599\pi$
$37$		−	1.46410i		−	0.240697i	0.992732	−	0.120348i	$-0.0384012\pi$
$38$		−	0.928203i		−	0.150574i
$39$		0			0
$40$		0			0
$41$	−10.3923			−1.62301			−0.811503	−	0.584349i	$-0.801350\pi$
$41$	−10.3923			−1.62301			−0.811503	+	0.584349i	$0.801350\pi$
$42$		0			0
$43$			4.92820i			0.751544i	0.926712	+	0.375772i	$0.122622\pi$
$43$			4.92820i			0.751544i	−0.926712	+	0.375772i	$0.877378\pi$
$44$		−	0.656339i		−	0.0989468i
$45$		0			0
$46$	−0.732051			−0.107935
$47$			9.46410i			1.38048i	0.723580	+	0.690241i	$0.242495\pi$
$47$			9.46410i			1.38048i	−0.723580	+	0.690241i	$0.757505\pi$
$48$		0			0
$49$	5.00000	+	4.89898i	0.714286	+	0.699854i
$50$		0			0
$51$		0			0
$52$	3.10583			0.430701
$53$	10.1769			1.39790			0.698952	−	0.715168i	$-0.253650\pi$
$53$	10.1769			1.39790			0.698952	+	0.715168i	$0.253650\pi$
$54$		0			0
$55$		0			0
$56$	−4.73205	−	1.93185i	−0.632347	−	0.258155i
$57$		0			0
$58$			0.732051i			0.0961230i
$59$	−9.46410			−1.23212			−0.616061	−	0.787699i	$-0.711272\pi$
$59$	−9.46410			−1.23212			−0.616061	+	0.787699i	$0.711272\pi$
$60$		0			0
$61$			13.3843i			1.71368i	0.515583	+	0.856840i	$0.327575\pi$
$61$			13.3843i			1.71368i	−0.515583	+	0.856840i	$0.672425\pi$
$62$			3.46410i			0.439941i
$63$		0			0
$64$	−2.26795			−0.283494
$65$		0			0
$66$		0			0
$67$		−	10.9282i		−	1.33509i	−0.744568	−	0.667546i	$-0.767345\pi$
$67$		−	10.9282i		−	1.33509i	0.744568	−	0.667546i	$-0.232655\pi$
$68$		−	6.00000i		−	0.727607i
$69$		0			0
$70$		0			0
$71$			15.0759i			1.78918i	0.446891	+	0.894589i	$0.352531\pi$
$71$			15.0759i			1.78918i	−0.446891	+	0.894589i	$0.647469\pi$
$72$		0			0
$73$	1.79315			0.209872			0.104936	−	0.994479i	$-0.466536\pi$
$73$	1.79315			0.209872			0.104936	+	0.994479i	$0.466536\pi$
$74$		−	0.757875i		−	0.0881012i
$75$		0			0
$76$			3.10583i			0.356263i
$77$	−0.378937	+	0.928203i	−0.0431839	+	0.105779i
$78$		0			0
$79$	10.9282			1.22952			0.614759	−	0.788715i	$-0.289253\pi$
$79$	10.9282			1.22952			0.614759	+	0.788715i	$0.289253\pi$
$80$		0			0
$81$		0			0
$82$	−5.37945			−0.594061
$83$			9.46410i			1.03882i	0.854525	+	0.519410i	$0.173848\pi$
$83$			9.46410i			1.03882i	−0.854525	+	0.519410i	$0.826152\pi$
$84$		0			0
$85$		0			0
$86$			2.55103i			0.275084i
$87$		0			0
$88$		−	0.732051i		−	0.0780369i
$89$	−12.9282			−1.37039			−0.685193	−	0.728361i	$-0.740282\pi$
$89$	−12.9282			−1.37039			−0.685193	+	0.728361i	$0.740282\pi$
$90$		0			0
$91$	−4.39230	−	1.79315i	−0.460439	−	0.187973i
$92$	2.44949			0.255377
$93$		0			0
$94$			4.89898i			0.505291i
$95$		0			0
$96$		0			0
$97$	15.1774			1.54103			0.770516	−	0.637420i	$-0.219998\pi$
$97$	15.1774			1.54103			0.770516	+	0.637420i	$0.219998\pi$
$98$	2.58819	+	2.53590i	0.261447	+	0.256164i
$99$		0			0
$100$		0			0
$101$	−10.3923			−1.03407			−0.517036	−	0.855963i	$-0.672965\pi$
$101$	−10.3923			−1.03407			−0.517036	+	0.855963i	$0.672965\pi$
$102$		0			0
$103$	13.3843			1.31879			0.659395	−	0.751796i	$-0.270812\pi$
$103$	13.3843			1.31879			0.659395	+	0.751796i	$0.270812\pi$
$104$	3.46410			0.339683
$105$		0			0
$106$	5.26795			0.511668
$107$	−13.2827			−1.28409			−0.642045	−	0.766667i	$-0.721914\pi$
$107$	−13.2827			−1.28409			−0.642045	+	0.766667i	$0.721914\pi$
$108$		0			0
$109$	−8.00000			−0.766261			−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000			−0.766261			−0.383131	+	0.923694i	$0.625154\pi$
$110$		0			0
$111$		0			0
$112$	6.03579	+	2.46410i	0.570329	+	0.232836i
$113$	−9.41902			−0.886067			−0.443034	−	0.896505i	$-0.646098\pi$
$113$	−9.41902			−0.886067			−0.443034	+	0.896505i	$0.646098\pi$
$114$		0			0
$115$		0			0
$116$		−	2.44949i		−	0.227429i
$117$		0			0
$118$	−4.89898			−0.450988
$119$	−3.46410	+	8.48528i	−0.317554	+	0.777844i
$120$		0			0
$121$	10.8564			0.986946
$122$			6.92820i			0.627250i
$123$		0			0
$124$		−	11.5911i		−	1.04091i
$125$		0			0
$126$		0			0
$127$			2.92820i			0.259836i	0.991525	+	0.129918i	$0.0414714\pi$
$127$			2.92820i			0.259836i	−0.991525	+	0.129918i	$0.958529\pi$
$128$	−11.4524			−1.01226
$129$		0			0
$130$		0			0
$131$	−2.53590			−0.221562			−0.110781	−	0.993845i	$-0.535335\pi$
$131$	−2.53590			−0.221562			−0.110781	+	0.993845i	$0.535335\pi$
$132$		0			0
$133$	1.79315	−	4.39230i	0.155486	−	0.380861i
$134$		−	5.65685i		−	0.488678i
$135$		0			0
$136$		−	6.69213i		−	0.573845i
$137$	21.4906			1.83607			0.918033	−	0.396504i	$-0.129777\pi$
$137$	21.4906			1.83607			0.918033	+	0.396504i	$0.129777\pi$
$138$		0			0
$139$			16.4901i			1.39867i	0.714794	+	0.699336i	$0.246521\pi$
$139$			16.4901i			1.39867i	−0.714794	+	0.699336i	$0.753479\pi$
$140$		0			0
$141$		0			0
$142$			7.80385i			0.654884i
$143$		−	0.679492i		−	0.0568220i
$144$		0			0
$145$		0			0
$146$	0.928203			0.0768186
$147$		0			0
$148$			2.53590i			0.208450i
$149$			14.7985i			1.21234i	0.795336	+	0.606169i	$0.207295\pi$
$149$			14.7985i			1.21234i	−0.795336	+	0.606169i	$0.792705\pi$
$150$		0			0
$151$	−0.535898			−0.0436108			−0.0218054	−	0.999762i	$-0.506941\pi$
$151$	−0.535898			−0.0436108			−0.0218054	+	0.999762i	$0.506941\pi$
$152$			3.46410i			0.280976i
$153$		0			0
$154$	−0.196152	+	0.480473i	−0.0158064	+	0.0387176i
$155$		0			0
$156$		0			0
$157$	−3.10583			−0.247872			−0.123936	−	0.992290i	$-0.539552\pi$
$157$	−3.10583			−0.247872			−0.123936	+	0.992290i	$0.539552\pi$
$158$	5.65685			0.450035
$159$		0			0
$160$		0			0
$161$	−3.46410	−	1.41421i	−0.273009	−	0.111456i
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	18.0000			1.40556
$165$		0			0
$166$			4.89898i			0.380235i
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$		0			0
$169$	−9.78461			−0.752662
$170$		0			0
$171$		0			0
$172$		−	8.53590i		−	0.650856i
$173$		−	6.00000i		−	0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$		−	6.00000i		−	0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$		0			0
$175$		0			0
$176$			0.933740i			0.0703833i
$177$		0			0
$178$	−6.69213			−0.501596
$179$		−	18.6622i		−	1.39488i	−0.716645	−	0.697438i	$-0.754323\pi$
$179$		−	18.6622i		−	1.39488i	0.716645	−	0.697438i	$-0.245677\pi$
$180$		0			0
$181$		−	12.0716i		−	0.897274i	−0.893714	−	0.448637i	$-0.851910\pi$
$181$		−	12.0716i		−	0.897274i	0.893714	−	0.448637i	$-0.148090\pi$
$182$	−2.27362	−	0.928203i	−0.168532	−	0.0688030i
$183$		0			0
$184$	2.73205			0.201409
$185$		0			0
$186$		0			0
$187$	−1.31268			−0.0959925
$188$		−	16.3923i		−	1.19553i
$189$		0			0
$190$		0			0
$191$		−	13.0053i		−	0.941032i	−0.882391	−	0.470516i	$-0.844068\pi$
$191$		−	13.0053i		−	0.941032i	0.882391	−	0.470516i	$-0.155932\pi$
$192$		0			0
$193$		−	24.3923i		−	1.75580i	−0.478847	−	0.877898i	$-0.658945\pi$
$193$		−	24.3923i		−	1.75580i	0.478847	−	0.877898i	$-0.341055\pi$
$194$	7.85641			0.564057
$195$		0			0
$196$	−8.66025	−	8.48528i	−0.618590	−	0.606092i
$197$	8.10634			0.577553			0.288777	−	0.957397i	$-0.406752\pi$
$197$	8.10634			0.577553			0.288777	+	0.957397i	$0.406752\pi$
$198$		0			0
$199$		−	15.1774i		−	1.07590i	−0.842977	−	0.537949i	$-0.819199\pi$
$199$		−	15.1774i		−	1.07590i	0.842977	−	0.537949i	$-0.180801\pi$
$200$		0			0
$201$		0			0
$202$	−5.37945			−0.378497
$203$	−1.41421	+	3.46410i	−0.0992583	+	0.243132i
$204$		0			0
$205$		0			0
$206$	6.92820			0.482711
$207$		0			0
$208$	−4.41851			−0.306368
$209$	0.679492			0.0470014
$210$		0			0
$211$	−2.39230			−0.164693			−0.0823465	−	0.996604i	$-0.526241\pi$
$211$	−2.39230			−0.164693			−0.0823465	+	0.996604i	$0.526241\pi$
$212$	−17.6269			−1.21062
$213$		0			0
$214$	−6.87564			−0.470009
$215$		0			0
$216$		0			0
$217$	−6.69213	+	16.3923i	−0.454291	+	1.11278i
$218$	−4.14110			−0.280471
$219$		0			0
$220$		0			0
$221$		−	6.21166i		−	0.417841i
$222$		0			0
$223$	−4.89898			−0.328060			−0.164030	−	0.986455i	$-0.552449\pi$
$223$	−4.89898			−0.328060			−0.164030	+	0.986455i	$0.552449\pi$
$224$	12.5885	+	5.13922i	0.841102	+	0.343378i
$225$		0			0
$226$	−4.87564			−0.324323
$227$		−	11.3205i		−	0.751369i	−0.926748	−	0.375684i	$-0.877408\pi$
$227$		−	11.3205i		−	0.751369i	0.926748	−	0.375684i	$-0.122592\pi$
$228$		0			0
$229$			14.6969i			0.971201i	0.874181	+	0.485601i	$0.161399\pi$
$229$			14.6969i			0.971201i	−0.874181	+	0.485601i	$0.838601\pi$
$230$		0			0
$231$		0			0
$232$		−	2.73205i		−	0.179368i
$233$	3.96524			0.259771			0.129886	−	0.991529i	$-0.458539\pi$
$233$	3.96524			0.259771			0.129886	+	0.991529i	$0.458539\pi$
$234$		0			0
$235$		0			0
$236$	16.3923			1.06705
$237$		0			0
$238$	−1.79315	+	4.39230i	−0.116233	+	0.284711i
$239$		−	3.96524i		−	0.256490i	−0.991743	−	0.128245i	$-0.959066\pi$
$239$		−	3.96524i		−	0.256490i	0.991743	−	0.128245i	$-0.0409344\pi$
$240$		0			0
$241$			1.31268i			0.0845570i	0.999106	+	0.0422785i	$0.0134617\pi$
$241$			1.31268i			0.0845570i	−0.999106	+	0.0422785i	$0.986538\pi$
$242$	5.61969			0.361247
$243$		0			0
$244$		−	23.1822i		−	1.48409i
$245$		0			0
$246$		0			0
$247$			3.21539i			0.204590i
$248$		−	12.9282i		−	0.820942i
$249$		0			0
$250$		0			0
$251$	11.3205			0.714544			0.357272	−	0.934000i	$-0.383707\pi$
$251$	11.3205			0.714544			0.357272	+	0.934000i	$0.383707\pi$
$252$		0			0
$253$		−	0.535898i		−	0.0336916i
$254$			1.51575i			0.0951066i
$255$		0			0
$256$	−1.39230			−0.0870191
$257$		−	15.4641i		−	0.964624i	−0.875999	−	0.482312i	$-0.839797\pi$
$257$		−	15.4641i		−	0.964624i	0.875999	−	0.482312i	$-0.160203\pi$
$258$		0			0
$259$	1.46410	−	3.58630i	0.0909748	−	0.222842i
$260$		0			0
$261$		0			0
$262$	−1.31268			−0.0810975
$263$	11.9700			0.738105			0.369052	−	0.929409i	$-0.379682\pi$
$263$	11.9700			0.738105			0.369052	+	0.929409i	$0.379682\pi$
$264$		0			0
$265$		0			0
$266$	0.928203	−	2.27362i	0.0569118	−	0.139405i
$267$		0			0
$268$			18.9282i			1.15622i
$269$	−8.53590			−0.520443			−0.260221	−	0.965549i	$-0.583796\pi$
$269$	−8.53590			−0.520443			−0.260221	+	0.965549i	$0.583796\pi$
$270$		0			0
$271$		−	26.2880i		−	1.59689i	−0.602071	−	0.798443i	$-0.705658\pi$
$271$		−	26.2880i		−	1.59689i	0.602071	−	0.798443i	$-0.294342\pi$
$272$			8.53590i			0.517565i
$273$		0			0
$274$	11.1244			0.672047
$275$		0			0
$276$		0			0
$277$		−	22.0000i		−	1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$		−	22.0000i		−	1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$			8.53590i			0.511949i
$279$		0			0
$280$		0			0
$281$			3.48477i			0.207884i	0.994583	+	0.103942i	$0.0331456\pi$
$281$			3.48477i			0.207884i	−0.994583	+	0.103942i	$0.966854\pi$
$282$		0			0
$283$	24.4949			1.45607			0.728035	−	0.685540i	$-0.240434\pi$
$283$	24.4949			1.45607			0.728035	+	0.685540i	$0.240434\pi$
$284$		−	26.1122i		−	1.54947i
$285$		0			0
$286$		−	0.351731i		−	0.0207983i
$287$	−25.4558	−	10.3923i	−1.50261	−	0.613438i
$288$		0			0
$289$	5.00000			0.294118
$290$		0			0
$291$		0			0
$292$	−3.10583			−0.181755
$293$			10.3923i			0.607125i	0.952812	+	0.303562i	$0.0981761\pi$
$293$			10.3923i			0.607125i	−0.952812	+	0.303562i	$0.901824\pi$
$294$		0			0
$295$		0			0
$296$			2.82843i			0.164399i
$297$		0			0
$298$			7.66025i			0.443747i
$299$	2.53590			0.146655
$300$		0			0
$301$	−4.92820	+	12.0716i	−0.284057	+	0.695794i
$302$	−0.277401			−0.0159627
$303$		0			0
$304$		−	4.41851i		−	0.253419i
$305$		0			0
$306$		0			0
$307$	−14.6969			−0.838799			−0.419399	−	0.907802i	$-0.637759\pi$
$307$	−14.6969			−0.838799			−0.419399	+	0.907802i	$0.637759\pi$
$308$	0.656339	−	1.60770i	0.0373984	−	0.0916069i
$309$		0			0
$310$		0			0
$311$	28.3923			1.60998			0.804990	−	0.593288i	$-0.202171\pi$
$311$	28.3923			1.60998			0.804990	+	0.593288i	$0.202171\pi$
$312$		0			0
$313$	−10.2784			−0.580971			−0.290486	−	0.956879i	$-0.593817\pi$
$313$	−10.2784			−0.580971			−0.290486	+	0.956879i	$0.593817\pi$
$314$	−1.60770			−0.0907275
$315$		0			0
$316$	−18.9282			−1.06479
$317$	−27.1475			−1.52475			−0.762377	−	0.647134i	$-0.775968\pi$
$317$	−27.1475			−1.52475			−0.762377	+	0.647134i	$0.775968\pi$
$318$		0			0
$319$	−0.535898			−0.0300045
$320$		0			0
$321$		0			0
$322$	−1.79315	−	0.732051i	−0.0999284	−	0.0407956i
$323$	6.21166			0.345626
$324$		0			0
$325$		0			0
$326$			2.07055i			0.114677i
$327$		0			0
$328$	20.0764			1.10853
$329$	−9.46410	+	23.1822i	−0.521773	+	1.27808i
$330$		0			0
$331$	−5.60770			−0.308227			−0.154113	−	0.988053i	$-0.549252\pi$
$331$	−5.60770			−0.308227			−0.154113	+	0.988053i	$0.549252\pi$
$332$		−	16.3923i		−	0.899645i
$333$		0			0
$334$		−	6.21166i		−	0.339887i
$335$		0			0
$336$		0			0
$337$			12.3923i			0.675052i	0.941316	+	0.337526i	$0.109590\pi$
$337$			12.3923i			0.675052i	−0.941316	+	0.337526i	$0.890410\pi$
$338$	−5.06489			−0.275494
$339$		0			0
$340$		0			0
$341$	−2.53590			−0.137327
$342$		0			0
$343$	7.34847	+	17.0000i	0.396780	+	0.917914i
$344$		−	9.52056i		−	0.513314i
$345$		0			0
$346$		−	3.10583i		−	0.166970i
$347$	28.1827			1.51293			0.756464	−	0.654035i	$-0.226925\pi$
$347$	28.1827			1.51293			0.756464	+	0.654035i	$0.226925\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$			1.94744i			0.103799i
$353$			30.0000i			1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$			30.0000i			1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$		0			0
$355$		0			0
$356$	22.3923			1.18679
$357$		0			0
$358$		−	9.66025i		−	0.510560i
$359$			20.1779i			1.06495i	0.846446	+	0.532475i	$0.178738\pi$
$359$			20.1779i			1.06495i	−0.846446	+	0.532475i	$0.821262\pi$
$360$		0			0
$361$	15.7846			0.830769
$362$		−	6.24871i		−	0.328425i
$363$		0			0
$364$	7.60770	+	3.10583i	0.398752	+	0.162790i
$365$		0			0
$366$		0			0
$367$	−21.8695			−1.14158			−0.570790	−	0.821096i	$-0.693363\pi$
$367$	−21.8695			−1.14158			−0.570790	+	0.821096i	$0.693363\pi$
$368$	−3.48477			−0.181656
$369$		0			0
$370$		0			0
$371$	24.9282	+	10.1769i	1.29421	+	0.528358i
$372$		0			0
$373$		−	7.07180i		−	0.366164i	−0.983098	−	0.183082i	$-0.941393\pi$
$373$		−	7.07180i		−	0.366164i	0.983098	−	0.183082i	$-0.0586073\pi$
$374$	−0.679492			−0.0351357
$375$		0			0
$376$		−	18.2832i		−	0.942886i
$377$		−	2.53590i		−	0.130605i
$378$		0			0
$379$	−11.4641			−0.588871			−0.294436	−	0.955671i	$-0.595132\pi$
$379$	−11.4641			−0.588871			−0.294436	+	0.955671i	$0.595132\pi$
$380$		0			0
$381$		0			0
$382$		−	6.73205i		−	0.344442i
$383$			30.9282i			1.58036i	0.612877	+	0.790179i	$0.290012\pi$
$383$			30.9282i			1.58036i	−0.612877	+	0.790179i	$0.709988\pi$
$384$		0			0
$385$		0			0
$386$		−	12.6264i		−	0.642666i
$387$		0			0
$388$	−26.2880			−1.33457
$389$		−	30.2533i		−	1.53390i	−0.641705	−	0.766951i	$-0.721773\pi$
$389$		−	30.2533i		−	1.53390i	0.641705	−	0.766951i	$-0.278227\pi$
$390$		0			0
$391$		−	4.89898i		−	0.247752i
$392$	−9.65926	−	9.46410i	−0.487866	−	0.478009i
$393$		0			0
$394$	4.19615			0.211399
$395$		0			0
$396$		0			0
$397$	12.9038			0.647623			0.323811	−	0.946122i	$-0.395036\pi$
$397$	12.9038			0.647623			0.323811	+	0.946122i	$0.395036\pi$
$398$		−	7.85641i		−	0.393806i
$399$		0			0
$400$		0			0
$401$		−	21.0101i		−	1.04920i	−0.851350	−	0.524598i	$-0.824216\pi$
$401$		−	21.0101i		−	1.04920i	0.851350	−	0.524598i	$-0.175784\pi$
$402$		0			0
$403$		−	12.0000i		−	0.597763i
$404$	18.0000			0.895533
$405$		0			0
$406$	−0.732051	+	1.79315i	−0.0363311	+	0.0889926i
$407$	0.554803			0.0275006
$408$		0			0
$409$			31.6675i			1.56586i	0.622112	+	0.782929i	$0.286275\pi$
$409$			31.6675i			1.56586i	−0.622112	+	0.782929i	$0.713725\pi$
$410$		0			0
$411$		0			0
$412$	−23.1822			−1.14211
$413$	−23.1822	−	9.46410i	−1.14072	−	0.465698i
$414$		0			0
$415$		0			0
$416$	−9.21539			−0.451822
$417$		0			0
$418$	0.351731			0.0172037
$419$	−10.1436			−0.495547			−0.247773	−	0.968818i	$-0.579699\pi$
$419$	−10.1436			−0.495547			−0.247773	+	0.968818i	$0.579699\pi$
$420$		0			0
$421$	0.143594			0.00699832			0.00349916	−	0.999994i	$-0.498886\pi$
$421$	0.143594			0.00699832			0.00349916	+	0.999994i	$0.498886\pi$
$422$	−1.23835			−0.0602818
$423$		0			0
$424$	−19.6603			−0.954786
$425$		0			0
$426$		0			0
$427$	−13.3843	+	32.7846i	−0.647710	+	1.58656i
$428$	23.0064			1.11205
$429$		0			0
$430$		0			0
$431$		−	6.79367i		−	0.327239i	−0.986523	−	0.163620i	$-0.947683\pi$
$431$		−	6.79367i		−	0.327239i	0.986523	−	0.163620i	$-0.0523170\pi$
$432$		0			0
$433$	−0.480473			−0.0230901			−0.0115450	−	0.999933i	$-0.503675\pi$
$433$	−0.480473			−0.0230901			−0.0115450	+	0.999933i	$0.503675\pi$
$434$	−3.46410	+	8.48528i	−0.166282	+	0.407307i
$435$		0			0
$436$	13.8564			0.663602
$437$			2.53590i			0.121308i
$438$		0			0
$439$			8.96575i			0.427912i	0.976843	+	0.213956i	$0.0686349\pi$
$439$			8.96575i			0.427912i	−0.976843	+	0.213956i	$0.931365\pi$
$440$		0			0
$441$		0			0
$442$		−	3.21539i		−	0.152941i
$443$	15.5563			0.739104			0.369552	−	0.929210i	$-0.379511\pi$
$443$	15.5563			0.739104			0.369552	+	0.929210i	$0.379511\pi$
$444$		0			0
$445$		0			0
$446$	−2.53590			−0.120078
$447$		0			0
$448$	−5.55532	−	2.26795i	−0.262464	−	0.107151i
$449$			12.5249i			0.591084i	0.955330	+	0.295542i	$0.0955003\pi$
$449$			12.5249i			0.591084i	−0.955330	+	0.295542i	$0.904500\pi$
$450$		0			0
$451$		−	3.93803i		−	0.185435i
$452$	16.3142			0.767357
$453$		0			0
$454$		−	5.85993i		−	0.275020i
$455$		0			0
$456$		0			0
$457$			26.2487i			1.22786i	0.789359	+	0.613931i	$0.210413\pi$
$457$			26.2487i			1.22786i	−0.789359	+	0.613931i	$0.789587\pi$
$458$			7.60770i			0.355484i
$459$		0			0
$460$		0			0
$461$	−7.85641			−0.365909			−0.182955	−	0.983121i	$-0.558566\pi$
$461$	−7.85641			−0.365909			−0.182955	+	0.983121i	$0.558566\pi$
$462$		0			0
$463$		−	8.00000i		−	0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$		−	8.00000i		−	0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$			3.48477i			0.161776i
$465$		0			0
$466$	2.05256			0.0950830
$467$		−	28.3923i		−	1.31384i	−0.753961	−	0.656920i	$-0.771859\pi$
$467$		−	28.3923i		−	1.31384i	0.753961	−	0.656920i	$-0.228141\pi$
$468$		0			0
$469$	10.9282	−	26.7685i	0.504618	−	1.23606i
$470$		0			0
$471$		0			0
$472$	18.2832			0.841554
$473$	−1.86748			−0.0858668
$474$		0			0
$475$		0			0
$476$	6.00000	−	14.6969i	0.275010	−	0.673633i
$477$		0			0
$478$		−	2.05256i		−	0.0938819i
$479$	14.5359			0.664162			0.332081	−	0.943251i	$-0.392249\pi$
$479$	14.5359			0.664162			0.332081	+	0.943251i	$0.392249\pi$
$480$		0			0
$481$			2.62536i			0.119706i
$482$			0.679492i			0.0309500i
$483$		0			0
$484$	−18.8038			−0.854720
$485$		0			0
$486$		0			0
$487$		−	11.8564i		−	0.537265i	−0.963243	−	0.268633i	$-0.913428\pi$
$487$		−	11.8564i		−	0.537265i	0.963243	−	0.268633i	$-0.0865717\pi$
$488$		−	25.8564i		−	1.17046i
$489$		0			0
$490$		0			0
$491$		−	13.0053i		−	0.586922i	−0.955971	−	0.293461i	$-0.905193\pi$
$491$		−	13.0053i		−	0.586922i	0.955971	−	0.293461i	$-0.0948071\pi$
$492$		0			0
$493$	−4.89898			−0.220639
$494$			1.66441i			0.0748853i
$495$		0			0
$496$			16.4901i			0.740427i
$497$	−15.0759	+	36.9282i	−0.676245	+	1.65646i
$498$		0			0
$499$	17.6077			0.788229			0.394114	−	0.919061i	$-0.371051\pi$
$499$	17.6077			0.788229			0.394114	+	0.919061i	$0.371051\pi$
$500$		0			0
$501$		0			0
$502$	5.85993			0.261541
$503$			6.92820i			0.308913i	0.988000	+	0.154457i	$0.0493627\pi$
$503$			6.92820i			0.308913i	−0.988000	+	0.154457i	$0.950637\pi$
$504$		0			0
$505$		0			0
$506$		−	0.277401i		−	0.0123320i
$507$		0			0
$508$		−	5.07180i		−	0.225025i
$509$	13.6077			0.603150			0.301575	−	0.953442i	$-0.402488\pi$
$509$	13.6077			0.603150			0.301575	+	0.953442i	$0.402488\pi$
$510$		0			0
$511$	4.39230	+	1.79315i	0.194304	+	0.0793243i
$512$	22.1841			0.980408
$513$		0			0
$514$		−	8.00481i		−	0.353077i
$515$		0			0
$516$		0			0
$517$	−3.58630			−0.157725
$518$	0.757875	−	1.85641i	0.0332991	−	0.0815658i
$519$		0			0
$520$		0			0
$521$	−10.3923			−0.455295			−0.227648	−	0.973744i	$-0.573103\pi$
$521$	−10.3923			−0.455295			−0.227648	+	0.973744i	$0.573103\pi$
$522$		0			0
$523$	−40.1528			−1.75576			−0.877879	−	0.478882i	$-0.841042\pi$
$523$	−40.1528			−1.75576			−0.877879	+	0.478882i	$0.841042\pi$
$524$	4.39230			0.191879
$525$		0			0
$526$	6.19615			0.270165
$527$	−23.1822			−1.00983
$528$		0			0
$529$	−21.0000			−0.913043
$530$		0			0
$531$		0			0
$532$	−3.10583	+	7.60770i	−0.134655	+	0.329835i
$533$	18.6350			0.807170
$534$		0			0
$535$		0			0
$536$			21.1117i			0.911885i
$537$		0			0
$538$	−4.41851			−0.190495
$539$	−1.85641	+	1.89469i	−0.0799611	+	0.0816099i
$540$		0			0
$541$	−11.8564			−0.509747			−0.254873	−	0.966974i	$-0.582034\pi$
$541$	−11.8564			−0.509747			−0.254873	+	0.966974i	$0.582034\pi$
$542$		−	13.6077i		−	0.584501i
$543$		0			0
$544$			17.8028i			0.763287i
$545$		0			0
$546$		0			0
$547$		−	29.8564i		−	1.27657i	−0.769801	−	0.638284i	$-0.779645\pi$
$547$		−	29.8564i		−	1.27657i	0.769801	−	0.638284i	$-0.220355\pi$
$548$	−37.2228			−1.59008
$549$		0			0
$550$		0			0
$551$	2.53590			0.108033
$552$		0			0
$553$	26.7685	+	10.9282i	1.13831	+	0.464714i
$554$		−	11.3880i		−	0.483831i
$555$		0			0
$556$		−	28.5617i		−	1.21128i
$557$	28.6632			1.21450			0.607250	−	0.794511i	$-0.292273\pi$
$557$	28.6632			1.21450			0.607250	+	0.794511i	$0.292273\pi$
$558$		0			0
$559$		−	8.83701i		−	0.373766i
$560$		0			0
$561$		0			0
$562$			1.80385i			0.0760907i
$563$		−	30.9282i		−	1.30347i	−0.758447	−	0.651734i	$-0.774042\pi$
$563$		−	30.9282i		−	1.30347i	0.758447	−	0.651734i	$-0.225958\pi$
$564$		0			0
$565$		0			0
$566$	12.6795			0.532959
$567$		0			0
$568$		−	29.1244i		−	1.22203i
$569$			17.0721i			0.715700i	0.933779	+	0.357850i	$0.116490\pi$
$569$			17.0721i			0.715700i	−0.933779	+	0.357850i	$0.883510\pi$
$570$		0			0
$571$	−4.00000			−0.167395			−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000			−0.167395			−0.0836974	+	0.996491i	$0.526673\pi$
$572$			1.17691i			0.0492093i
$573$		0			0
$574$	−13.1769	−	5.37945i	−0.549994	−	0.224534i
$575$		0			0
$576$		0			0
$577$	26.2880			1.09439			0.547193	−	0.837007i	$-0.315696\pi$
$577$	26.2880			1.09439			0.547193	+	0.837007i	$0.315696\pi$
$578$	2.58819			0.107655
$579$		0			0
$580$		0			0
$581$	−9.46410	+	23.1822i	−0.392637	+	0.961761i
$582$		0			0
$583$			3.85641i			0.159716i
$584$	−3.46410			−0.143346
$585$		0			0
$586$			5.37945i			0.222223i
$587$		−	15.7128i		−	0.648537i	−0.945965	−	0.324269i	$-0.894882\pi$
$587$		−	15.7128i		−	0.648537i	0.945965	−	0.324269i	$-0.105118\pi$
$588$		0			0
$589$	12.0000			0.494451
$590$		0			0
$591$		0			0
$592$		−	3.60770i		−	0.148275i
$593$			9.71281i			0.398857i	0.979912	+	0.199429i	$0.0639086\pi$
$593$			9.71281i			0.398857i	−0.979912	+	0.199429i	$0.936091\pi$
$594$		0			0
$595$		0			0
$596$		−	25.6317i		−	1.04992i
$597$		0			0
$598$	1.31268			0.0536794
$599$		−	26.1865i		−	1.06995i	−0.844867	−	0.534976i	$-0.820321\pi$
$599$		−	26.1865i		−	1.06995i	0.844867	−	0.534976i	$-0.179679\pi$
$600$		0			0
$601$			19.5959i			0.799334i	0.916660	+	0.399667i	$0.130874\pi$
$601$			19.5959i			0.799334i	−0.916660	+	0.399667i	$0.869126\pi$
$602$	−2.55103	+	6.24871i	−0.103972	+	0.254678i
$603$		0			0
$604$	0.928203			0.0377681
$605$		0			0
$606$		0			0
$607$	−1.31268			−0.0532799			−0.0266400	−	0.999645i	$-0.508481\pi$
$607$	−1.31268			−0.0532799			−0.0266400	+	0.999645i	$0.508481\pi$
$608$		−	9.21539i		−	0.373733i
$609$		0			0
$610$		0			0
$611$		−	16.9706i		−	0.686555i
$612$		0			0
$613$			28.9282i			1.16840i	0.811610	+	0.584200i	$0.198591\pi$
$613$			28.9282i			1.16840i	−0.811610	+	0.584200i	$0.801409\pi$
$614$	−7.60770			−0.307022
$615$		0			0
$616$	0.732051	−	1.79315i	0.0294952	−	0.0722481i
$617$	13.0053			0.523575			0.261787	−	0.965126i	$-0.415688\pi$
$617$	13.0053			0.523575			0.261787	+	0.965126i	$0.415688\pi$
$618$		0			0
$619$		−	17.4510i		−	0.701416i	−0.936485	−	0.350708i	$-0.885941\pi$
$619$		−	17.4510i		−	0.701416i	0.936485	−	0.350708i	$-0.114059\pi$
$620$		0			0
$621$		0			0
$622$	14.6969			0.589294
$623$	−31.6675	−	12.9282i	−1.26873	−	0.517958i
$624$		0			0
$625$		0			0
$626$	−5.32051			−0.212650
$627$		0			0
$628$	5.37945			0.214664
$629$	5.07180			0.202226
$630$		0			0
$631$	−24.7846			−0.986660			−0.493330	−	0.869842i	$-0.664220\pi$
$631$	−24.7846			−0.986660			−0.493330	+	0.869842i	$0.664220\pi$
$632$	−21.1117			−0.839777
$633$		0			0
$634$	−14.0526			−0.558098
$635$		0			0
$636$		0			0
$637$	−8.96575	−	8.78461i	−0.355236	−	0.348059i
$638$	−0.277401			−0.0109824
$639$		0			0
$640$		0			0
$641$			40.0512i			1.58193i	0.611862	+	0.790965i	$0.290421\pi$
$641$			40.0512i			1.58193i	−0.611862	+	0.790965i	$0.709579\pi$
$642$		0			0
$643$	28.0812			1.10741			0.553707	−	0.832711i	$-0.313213\pi$
$643$	28.0812			1.10741			0.553707	+	0.832711i	$0.313213\pi$
$644$	6.00000	+	2.44949i	0.236433	+	0.0965234i
$645$		0			0
$646$	3.21539			0.126508
$647$			32.7846i			1.28890i	0.764648	+	0.644448i	$0.222913\pi$
$647$			32.7846i			1.28890i	−0.764648	+	0.644448i	$0.777087\pi$
$648$		0			0
$649$		−	3.58630i		−	0.140775i
$650$		0			0
$651$		0			0
$652$		−	6.92820i		−	0.271329i
$653$	7.90327			0.309279			0.154639	−	0.987971i	$-0.450578\pi$
$653$	7.90327			0.309279			0.154639	+	0.987971i	$0.450578\pi$
$654$		0			0
$655$		0			0
$656$	−25.6077			−0.999813
$657$		0			0
$658$	−4.89898	+	12.0000i	−0.190982	+	0.467809i
$659$		−	18.6622i		−	0.726975i	−0.931599	−	0.363488i	$-0.881586\pi$
$659$		−	18.6622i		−	0.726975i	0.931599	−	0.363488i	$-0.118414\pi$
$660$		0			0
$661$			16.0096i			0.622702i	0.950295	+	0.311351i	$0.100781\pi$
$661$			16.0096i			0.622702i	−0.950295	+	0.311351i	$0.899219\pi$
$662$	−2.90276			−0.112819
$663$		0			0
$664$		−	18.2832i		−	0.709527i
$665$		0			0
$666$		0			0
$667$		−	2.00000i		−	0.0774403i
$668$			20.7846i			0.804181i
$669$		0			0
$670$		0			0
$671$	−5.07180			−0.195795
$672$		0			0
$673$			23.6077i			0.910010i	0.890489	+	0.455005i	$0.150362\pi$
$673$			23.6077i			0.910010i	−0.890489	+	0.455005i	$0.849638\pi$
$674$			6.41473i			0.247086i
$675$		0			0
$676$	16.9474			0.651825
$677$			41.3205i			1.58808i	0.607868	+	0.794038i	$0.292025\pi$
$677$			41.3205i			1.58808i	−0.607868	+	0.794038i	$0.707975\pi$
$678$		0			0
$679$	37.1769	+	15.1774i	1.42672	+	0.582456i
$680$		0			0
$681$		0			0
$682$	−1.31268			−0.0502650
$683$	14.2437			0.545019			0.272509	−	0.962153i	$-0.412146\pi$
$683$	14.2437			0.545019			0.272509	+	0.962153i	$0.412146\pi$
$684$		0			0
$685$		0			0
$686$	3.80385	+	8.79985i	0.145232	+	0.335980i
$687$		0			0
$688$			12.1436i			0.462970i
$689$	−18.2487			−0.695221
$690$		0			0
$691$		−	17.4510i		−	0.663869i	−0.943302	−	0.331934i	$-0.892299\pi$
$691$		−	17.4510i		−	0.663869i	0.943302	−	0.331934i	$-0.107701\pi$
$692$			10.3923i			0.395056i
$693$		0			0
$694$	14.5885			0.553770
$695$		0			0
$696$		0			0
$697$		−	36.0000i		−	1.36360i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			32.5269i			1.22852i	0.789102	+	0.614262i	$0.210546\pi$
$701$			32.5269i			1.22852i	−0.789102	+	0.614262i	$0.789454\pi$
$702$		0			0
$703$	−2.62536			−0.0990171
$704$		−	0.859411i		−	0.0323903i
$705$		0			0
$706$			15.5291i			0.584447i
$707$	−25.4558	−	10.3923i	−0.957366	−	0.390843i
$708$		0			0
$709$	45.5692			1.71139			0.855694	−	0.517482i	$-0.173131\pi$
$709$	45.5692			1.71139			0.855694	+	0.517482i	$0.173131\pi$
$710$		0			0
$711$		0			0
$712$	24.9754			0.935992
$713$		−	9.46410i		−	0.354433i
$714$		0			0
$715$		0			0
$716$			32.3238i			1.20800i
$717$		0			0
$718$			10.4449i			0.389799i
$719$	4.39230			0.163805			0.0819027	−	0.996640i	$-0.473900\pi$
$719$	4.39230			0.163805			0.0819027	+	0.996640i	$0.473900\pi$
$720$		0			0
$721$	32.7846	+	13.3843i	1.22096	+	0.498456i
$722$	8.17072			0.304083
$723$		0			0
$724$			20.9086i			0.777062i
$725$		0			0
$726$		0			0
$727$	19.5959			0.726772			0.363386	−	0.931639i	$-0.381621\pi$
$727$	19.5959			0.726772			0.363386	+	0.931639i	$0.381621\pi$
$728$	8.48528	+	3.46410i	0.314485	+	0.128388i
$729$		0			0
$730$		0			0
$731$	−17.0718			−0.631423
$732$		0			0
$733$	32.1480			1.18741			0.593706	−	0.804682i	$-0.297664\pi$
$733$	32.1480			1.18741			0.593706	+	0.804682i	$0.297664\pi$
$734$	−11.3205			−0.417848
$735$		0			0
$736$	−7.26795			−0.267900
$737$	4.14110			0.152540
$738$		0			0
$739$	−25.0718			−0.922281			−0.461140	−	0.887327i	$-0.652560\pi$
$739$	−25.0718			−0.922281			−0.461140	+	0.887327i	$0.652560\pi$
$740$		0			0
$741$		0			0
$742$	12.9038	+	5.26795i	0.473713	+	0.193392i
$743$	13.2827			0.487296			0.243648	−	0.969864i	$-0.421656\pi$
$743$	13.2827			0.487296			0.243648	+	0.969864i	$0.421656\pi$
$744$		0			0
$745$		0			0
$746$		−	3.66063i		−	0.134025i
$747$		0			0
$748$	2.27362			0.0831319
$749$	−32.5359	−	13.2827i	−1.18884	−	0.485340i
$750$		0			0
$751$	−26.3923			−0.963069			−0.481534	−	0.876427i	$-0.659920\pi$
$751$	−26.3923			−0.963069			−0.481534	+	0.876427i	$0.659920\pi$
$752$			23.3205i			0.850411i
$753$		0			0
$754$		−	1.31268i		−	0.0478049i
$755$		0			0
$756$		0			0
$757$		−	17.1769i		−	0.624306i	−0.950032	−	0.312153i	$-0.898950\pi$
$757$		−	17.1769i		−	0.624306i	0.950032	−	0.312153i	$-0.101050\pi$
$758$	−5.93426			−0.215542
$759$		0			0
$760$		0			0
$761$	−35.5692			−1.28938			−0.644692	−	0.764443i	$-0.723014\pi$
$761$	−35.5692			−1.28938			−0.644692	+	0.764443i	$0.723014\pi$
$762$		0			0
$763$	−19.5959	−	8.00000i	−0.709420	−	0.289619i
$764$			22.5259i			0.814958i
$765$		0			0
$766$			16.0096i			0.578451i
$767$	16.9706			0.612772
$768$		0			0
$769$			39.1918i			1.41329i	0.707566	+	0.706647i	$0.249793\pi$
$769$			39.1918i			1.41329i	−0.707566	+	0.706647i	$0.750207\pi$
$770$		0			0
$771$		0			0
$772$			42.2487i			1.52056i
$773$		−	33.7128i		−	1.21257i	−0.795249	−	0.606283i	$-0.792660\pi$
$773$		−	33.7128i		−	1.21257i	0.795249	−	0.606283i	$-0.207340\pi$
$774$		0			0
$775$		0			0
$776$	−29.3205			−1.05254
$777$		0			0
$778$		−	15.6603i		−	0.561447i
$779$			18.6350i			0.667667i
$780$		0			0
$781$	−5.71281			−0.204421
$782$		−	2.53590i		−	0.0906835i
$783$		0			0
$784$	12.3205	+	12.0716i	0.440018	+	0.431128i
$785$		0			0
$786$		0			0
$787$	−36.5665			−1.30345			−0.651727	−	0.758454i	$-0.725955\pi$
$787$	−36.5665			−1.30345			−0.651727	+	0.758454i	$0.725955\pi$
$788$	−14.0406			−0.500176
$789$		0			0
$790$		0			0
$791$	−23.0718	−	9.41902i	−0.820339	−	0.334902i
$792$		0			0
$793$		−	24.0000i		−	0.852265i
$794$	6.67949			0.237046
$795$		0			0
$796$			26.2880i			0.931755i
$797$			54.4974i			1.93040i	0.261517	+	0.965199i	$0.415777\pi$
$797$			54.4974i			1.93040i	−0.261517	+	0.965199i	$0.584223\pi$
$798$		0			0
$799$	−32.7846			−1.15984
$800$		0			0
$801$		0			0
$802$		−	10.8756i		−	0.384032i
$803$			0.679492i			0.0239787i
$804$		0			0
$805$		0			0
$806$		−	6.21166i		−	0.218796i
$807$		0			0
$808$	20.0764			0.706285
$809$		−	11.9700i		−	0.420844i	−0.977611	−	0.210422i	$-0.932516\pi$
$809$		−	11.9700i		−	0.420844i	0.977611	−	0.210422i	$-0.0674839\pi$
$810$		0			0
$811$			0.480473i			0.0168717i	0.999964	+	0.00843585i	$0.00268525\pi$
$811$			0.480473i			0.0168717i	−0.999964	+	0.00843585i	$0.997315\pi$
$812$	2.44949	−	6.00000i	0.0859602	−	0.210559i
$813$		0			0
$814$	0.287187			0.0100659
$815$		0			0
$816$		0			0
$817$	8.83701			0.309168
$818$			16.3923i			0.573143i
$819$		0			0
$820$		0			0
$821$			11.0091i			0.384220i	0.981373	+	0.192110i	$0.0615331\pi$
$821$			11.0091i			0.384220i	−0.981373	+	0.192110i	$0.938467\pi$
$822$		0			0
$823$		−	16.7846i		−	0.585075i	−0.956254	−	0.292537i	$-0.905500\pi$
$823$		−	16.7846i		−	0.585075i	0.956254	−	0.292537i	$-0.0944996\pi$
$824$	−25.8564			−0.900751
$825$		0			0
$826$	−12.0000	−	4.89898i	−0.417533	−	0.170457i
$827$	21.9711			0.764009			0.382005	−	0.924160i	$-0.375234\pi$
$827$	21.9711			0.764009			0.382005	+	0.924160i	$0.375234\pi$
$828$		0			0
$829$			5.85993i			0.203524i	0.994809	+	0.101762i	$0.0324480\pi$
$829$			5.85993i			0.203524i	−0.994809	+	0.101762i	$0.967552\pi$
$830$		0			0
$831$		0			0
$832$	4.06678			0.140990
$833$	−16.9706	+	17.3205i	−0.587995	+	0.600120i
$834$		0			0
$835$		0			0
$836$	−1.17691			−0.0407044
$837$		0			0
$838$	−5.25071			−0.181383
$839$	6.92820			0.239188			0.119594	−	0.992823i	$-0.461841\pi$
$839$	6.92820			0.239188			0.119594	+	0.992823i	$0.461841\pi$
$840$		0			0
$841$	27.0000			0.931034
$842$	0.0743295			0.00256156
$843$		0			0
$844$	4.14359			0.142628
$845$		0			0
$846$		0			0
$847$	26.5927	+	10.8564i	0.913734	+	0.373031i
$848$	25.0769			0.861145
$849$		0			0
$850$		0			0
$851$			2.07055i			0.0709776i
$852$		0			0
$853$	−33.4607			−1.14567			−0.572835	−	0.819670i	$-0.694157\pi$
$853$	−33.4607			−1.14567			−0.572835	+	0.819670i	$0.694157\pi$
$854$	−6.92820	+	16.9706i	−0.237078	+	0.580721i
$855$		0			0
$856$	25.6603			0.877049
$857$		−	9.21539i		−	0.314792i	−0.987536	−	0.157396i	$-0.949690\pi$
$857$		−	9.21539i		−	0.314792i	0.987536	−	0.157396i	$-0.0503099\pi$
$858$		0			0
$859$			22.3500i			0.762573i	0.924457	+	0.381286i	$0.124519\pi$
$859$			22.3500i			0.762573i	−0.924457	+	0.381286i	$0.875481\pi$
$860$		0			0
$861$		0			0
$862$		−	3.51666i		−	0.119778i
$863$	5.75839			0.196018			0.0980089	−	0.995186i	$-0.468753\pi$
$863$	5.75839			0.196018			0.0980089	+	0.995186i	$0.468753\pi$
$864$		0			0
$865$		0			0
$866$	−0.248711			−0.00845155
$867$		0			0
$868$	11.5911	−	28.3923i	0.393428	−	0.963698i
$869$			4.14110i			0.140477i
$870$		0			0
$871$			19.5959i			0.663982i
$872$	15.4548			0.523366
$873$		0			0
$874$			1.31268i			0.0444020i
$875$		0			0
$876$		0			0
$877$		−	22.0000i		−	0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$		−	22.0000i		−	0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$			4.64102i			0.156627i
$879$		0			0
$880$		0			0
$881$	−50.7846			−1.71098			−0.855488	−	0.517822i	$-0.826743\pi$
$881$	−50.7846			−1.71098			−0.855488	+	0.517822i	$0.826743\pi$
$882$		0			0
$883$		−	12.1436i		−	0.408664i	−0.978902	−	0.204332i	$-0.934498\pi$
$883$		−	12.1436i		−	0.408664i	0.978902	−	0.204332i	$-0.0655023\pi$
$884$			10.7589i			0.361861i
$885$		0			0
$886$	8.05256			0.270531
$887$		−	18.2487i		−	0.612732i	−0.951914	−	0.306366i	$-0.900887\pi$
$887$		−	18.2487i		−	0.612732i	0.951914	−	0.306366i	$-0.0991131\pi$
$888$		0			0
$889$	−2.92820	+	7.17260i	−0.0982088	+	0.240561i
$890$		0			0
$891$		0			0
$892$	8.48528			0.284108
$893$	16.9706			0.567898
$894$		0			0
$895$		0			0
$896$	−28.0526	−	11.4524i	−0.937170	−	0.382598i
$897$		0			0
$898$			6.48334i			0.216352i
$899$	−9.46410			−0.315645
$900$		0			0
$901$			35.2538i			1.17447i
$902$		−	2.03848i		−	0.0678738i
$903$		0			0
$904$	18.1962			0.605195
$905$		0			0
$906$		0			0
$907$			10.7846i			0.358097i	0.983840	+	0.179049i	$0.0573019\pi$
$907$			10.7846i			0.358097i	−0.983840	+	0.179049i	$0.942698\pi$
$908$			19.6077i			0.650704i
$909$		0			0
$910$		0			0
$911$		−	10.7317i		−	0.355557i	−0.984071	−	0.177779i	$-0.943109\pi$
$911$		−	10.7317i		−	0.355557i	0.984071	−	0.177779i	$-0.0568911\pi$
$912$		0			0
$913$	−3.58630			−0.118689
$914$			13.5873i			0.449429i
$915$		0			0
$916$		−	25.4558i		−	0.841085i
$917$	−6.21166	−	2.53590i	−0.205127	−	0.0837427i
$918$		0			0
$919$	−34.1051			−1.12502			−0.562512	−	0.826789i	$-0.690165\pi$
$919$	−34.1051			−1.12502			−0.562512	+	0.826789i	$0.690165\pi$
$920$		0			0
$921$		0			0
$922$	−4.06678			−0.133932
$923$		−	27.0333i		−	0.889813i
$924$		0			0
$925$		0			0
$926$		−	4.14110i		−	0.136085i
$927$		0			0
$928$			7.26795i			0.238582i
$929$	−36.9282			−1.21158			−0.605788	−	0.795626i	$-0.707142\pi$
$929$	−36.9282			−1.21158			−0.605788	+	0.795626i	$0.707142\pi$
$930$		0			0
$931$	8.78461	−	8.96575i	0.287904	−	0.293841i
$932$	−6.86800			−0.224969
$933$		0			0
$934$		−	14.6969i		−	0.480899i
$935$		0			0
$936$		0			0
$937$	10.2784			0.335782			0.167891	−	0.985806i	$-0.446304\pi$
$937$	10.2784			0.335782			0.167891	+	0.985806i	$0.446304\pi$
$938$	5.65685	−	13.8564i	0.184703	−	0.452428i
$939$		0			0
$940$		0			0
$941$	−12.9282			−0.421447			−0.210724	−	0.977546i	$-0.567582\pi$
$941$	−12.9282			−0.421447			−0.210724	+	0.977546i	$0.567582\pi$
$942$		0			0
$943$	14.6969			0.478598
$944$	−23.3205			−0.759018
$945$		0			0
$946$	−0.966679			−0.0314294
$947$	39.2934			1.27686			0.638432	−	0.769679i	$-0.279584\pi$
$947$	39.2934			1.27686			0.638432	+	0.769679i	$0.279584\pi$
$948$		0			0
$949$	−3.21539			−0.104376
$950$		0			0
$951$		0			0
$952$	6.69213	−	16.3923i	0.216893	−	0.531278i
$953$	−21.4906			−0.696149			−0.348074	−	0.937467i	$-0.613164\pi$
$953$	−21.4906			−0.696149			−0.348074	+	0.937467i	$0.613164\pi$
$954$		0			0
$955$		0			0
$956$			6.86800i			0.222127i
$957$		0			0
$958$	7.52433			0.243100
$959$	52.6410	+	21.4906i	1.69987	+	0.693968i
$960$		0			0
$961$	−13.7846			−0.444665
$962$			1.35898i			0.0438154i
$963$		0			0
$964$		−	2.27362i		−	0.0732285i
$965$		0			0
$966$		0			0
$967$			28.7846i			0.925651i	0.886450	+	0.462825i	$0.153164\pi$
$967$			28.7846i			0.925651i	−0.886450	+	0.462825i	$0.846836\pi$
$968$	−20.9730			−0.674097
$969$		0			0
$970$		0			0
$971$	18.9282			0.607435			0.303717	−	0.952762i	$-0.401772\pi$
$971$	18.9282			0.607435			0.303717	+	0.952762i	$0.401772\pi$
$972$		0			0
$973$	−16.4901	+	40.3923i	−0.528648	+	1.29492i
$974$		−	6.13733i		−	0.196653i
$975$		0			0
$976$			32.9802i			1.05567i
$977$	42.3992			1.35647			0.678235	−	0.734845i	$-0.262745\pi$
$977$	42.3992			1.35647			0.678235	+	0.734845i	$0.262745\pi$
$978$		0			0
$979$		−	4.89898i		−	0.156572i
$980$		0			0
$981$		0			0
$982$		−	6.73205i		−	0.214828i
$983$			9.46410i			0.301858i	0.988545	+	0.150929i	$0.0482265\pi$
$983$			9.46410i			0.301858i	−0.988545	+	0.150929i	$0.951774\pi$
$984$		0			0
$985$		0			0
$986$	−2.53590			−0.0807595
$987$		0			0
$988$		−	5.56922i		−	0.177180i
$989$		−	6.96953i		−	0.221618i
$990$		0			0
$991$	13.3205			0.423140			0.211570	−	0.977363i	$-0.432142\pi$
$991$	13.3205			0.423140			0.211570	+	0.977363i	$0.432142\pi$
$992$			34.3923i			1.09196i
$993$		0			0
$994$	−7.80385	+	19.1154i	−0.247523	+	0.606305i
$995$		0			0
$996$		0			0
$997$	59.2682			1.87704			0.938522	−	0.345220i	$-0.112196\pi$
$997$	59.2682			1.87704			0.938522	+	0.345220i	$0.112196\pi$
$998$	9.11441			0.288512
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.g.c.1574.6		8
3.2	odd	2		1575.2.g.a.1574.4		8
5.2	odd	4		1575.2.b.b.251.3		4
5.3	odd	4		315.2.b.a.251.2	✓	4
5.4	even	2	inner	1575.2.g.c.1574.3		8
7.6	odd	2		1575.2.g.a.1574.6		8
15.2	even	4		1575.2.b.c.251.2		4
15.8	even	4		315.2.b.b.251.3	yes	4
15.14	odd	2		1575.2.g.a.1574.5		8
20.3	even	4		5040.2.f.a.881.3		4
21.20	even	2	inner	1575.2.g.c.1574.4		8
35.13	even	4		315.2.b.b.251.2	yes	4
35.27	even	4		1575.2.b.c.251.3		4
35.34	odd	2		1575.2.g.a.1574.3		8
60.23	odd	4		5040.2.f.c.881.4		4
105.62	odd	4		1575.2.b.b.251.2		4
105.83	odd	4		315.2.b.a.251.3	yes	4
105.104	even	2	inner	1575.2.g.c.1574.5		8
140.83	odd	4		5040.2.f.c.881.1		4
420.83	even	4		5040.2.f.a.881.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.b.a.251.2	✓	4	5.3	odd	4
315.2.b.a.251.3	yes	4	105.83	odd	4
315.2.b.b.251.2	yes	4	35.13	even	4
315.2.b.b.251.3	yes	4	15.8	even	4
1575.2.b.b.251.2		4	105.62	odd	4
1575.2.b.b.251.3		4	5.2	odd	4
1575.2.b.c.251.2		4	15.2	even	4
1575.2.b.c.251.3		4	35.27	even	4
1575.2.g.a.1574.3		8	35.34	odd	2
1575.2.g.a.1574.4		8	3.2	odd	2
1575.2.g.a.1574.5		8	15.14	odd	2
1575.2.g.a.1574.6		8	7.6	odd	2
1575.2.g.c.1574.3		8	5.4	even	2	inner
1575.2.g.c.1574.4		8	21.20	even	2	inner
1575.2.g.c.1574.5		8	105.104	even	2	inner
1575.2.g.c.1574.6		8	1.1	even	1	trivial
5040.2.f.a.881.2		4	420.83	even	4
5040.2.f.a.881.3		4	20.3	even	4
5040.2.f.c.881.1		4	140.83	odd	4
5040.2.f.c.881.4		4	60.23	odd	4

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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.517638 q^{2} -1.73205 q^{4} +(2.44949 + 1.00000i) q^{7} -1.93185 q^{8} +O(q^{10})\)	\(q+0.517638 q^{2} -1.73205 q^{4} +(2.44949 + 1.00000i) q^{7} -1.93185 q^{8} +0.378937i q^{11} -1.79315 q^{13} +(1.26795 + 0.517638i) q^{14} +2.46410 q^{16} +3.46410i q^{17} -1.79315i q^{19} +0.196152i q^{22} -1.41421 q^{23} -0.928203 q^{26} +(-4.24264 - 1.73205i) q^{28} +1.41421i q^{29} +6.69213i q^{31} +5.13922 q^{32} +1.79315i q^{34} -1.46410i q^{37} -0.928203i q^{38} -10.3923 q^{41} +4.92820i q^{43} -0.656339i q^{44} -0.732051 q^{46} +9.46410i q^{47} +(5.00000 + 4.89898i) q^{49} +3.10583 q^{52} +10.1769 q^{53} +(-4.73205 - 1.93185i) q^{56} +0.732051i q^{58} -9.46410 q^{59} +13.3843i q^{61} +3.46410i q^{62} -2.26795 q^{64} -10.9282i q^{67} -6.00000i q^{68} +15.0759i q^{71} +1.79315 q^{73} -0.757875i q^{74} +3.10583i q^{76} +(-0.378937 + 0.928203i) q^{77} +10.9282 q^{79} -5.37945 q^{82} +9.46410i q^{83} +2.55103i q^{86} -0.732051i q^{88} -12.9282 q^{89} +(-4.39230 - 1.79315i) q^{91} +2.44949 q^{92} +4.89898i q^{94} +15.1774 q^{97} +(2.58819 + 2.53590i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 24 q^{14} - 8 q^{16} + 48 q^{26} + 8 q^{46} + 40 q^{49} - 24 q^{56} - 48 q^{59} - 32 q^{64} + 32 q^{79} - 48 q^{89} + 48 q^{91}+O(q^{100}) \) 8 * q + 24 * q^14 - 8 * q^16 + 48 * q^26 + 8 * q^46 + 40 * q^49 - 24 * q^56 - 48 * q^59 - 32 * q^64 + 32 * q^79 - 48 * q^89 + 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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