LMFDB - Embedded newform 1575.2.d.l.1324.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1324,1575)]

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([0, 1, 0]))

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

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

chi := DirichletCharacter("1575.1324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.d (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:	8.0.30599805184.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 13x^{4} + 27x^{2} + 81 $ x^8 + 3x^6 + 13x^4 + 27*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1324.6
Root			$0.672201 - 1.59629i$ of defining polynomial
Character	$\chi$	$=$	1575.1324
Dual form			1575.2.d.l.1324.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.34440i q^{2} +0.192582 q^{4} +1.00000i q^{7} +2.94771i q^{8} +O(q^{10})$	$q+1.34440i q^{2} +0.192582 q^{4} +1.00000i q^{7} +2.94771i q^{8} -5.63652 q^{11} -4.38516i q^{13} -1.34440 q^{14} -3.57775 q^{16} +2.68880i q^{17} -8.38516 q^{19} -7.57775i q^{22} -5.63652i q^{23} +5.89543 q^{26} +0.192582i q^{28} -8.32532 q^{29} +6.00000 q^{31} +1.08549i q^{32} -3.61484 q^{34} -3.00000i q^{37} -11.2730i q^{38} -5.37761 q^{41} +1.38516i q^{43} -1.08549 q^{44} +7.57775 q^{46} +8.58423i q^{47} -1.00000 q^{49} -0.844506i q^{52} +5.37761i q^{53} -2.94771 q^{56} -11.1926i q^{58} +8.58423 q^{59} +8.06641i q^{62} -8.61484 q^{64} -1.38516i q^{67} +0.517816i q^{68} +0.258908 q^{71} -6.38516i q^{73} +4.03321 q^{74} -1.61484 q^{76} -5.63652i q^{77} -5.38516 q^{79} -7.22967i q^{82} +16.6506i q^{83} -1.86222 q^{86} -16.6148i q^{88} -13.9618 q^{89} +4.38516 q^{91} -1.08549i q^{92} -11.5407 q^{94} -10.7703i q^{97} -1.34440i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 20 q^{4}+O(q^{10}) $ 8 * q - 20 * q^4	$ 8 q - 20 q^{4} + 36 q^{16} - 24 q^{19} + 48 q^{31} - 72 q^{34} - 4 q^{46} - 8 q^{49} - 112 q^{64} - 56 q^{76} - 8 q^{91} + 80 q^{94}+O(q^{100}) $ 8 * q - 20 * q^4 + 36 * q^16 - 24 * q^19 + 48 * q^31 - 72 * q^34 - 4 * q^46 - 8 * q^49 - 112 * q^64 - 56 * q^76 - 8 * q^91 + 80 * q^94

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 $

$2$	1.34440i	0.950636i	0.879814	+	0.475318i	$0.157667\pi$
$2$	1.34440i	0.950636i	−0.879814	+	0.475318i	$0.842333\pi$
$3$	0	0
$3$	0	0
$4$	0.192582	0.0962912
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	2.94771i	1.04217i
$9$	0	0
$10$	0	0
$11$	−5.63652	−1.69947	−0.849737	−	0.527207i	$-0.823239\pi$
$11$	−5.63652	−1.69947	−0.849737	+	0.527207i	$0.823239\pi$
$12$	0	0
$13$	− 4.38516i	− 1.21623i	−0.793851	−	0.608113i	$-0.791927\pi$
$13$	− 4.38516i	− 1.21623i	0.793851	−	0.608113i	$-0.208073\pi$
$14$	−1.34440	−0.359307
$15$	0	0
$16$	−3.57775	−0.894437
$17$	2.68880i	0.652131i	0.945347	+	0.326065i	$0.105723\pi$
$17$	2.68880i	0.652131i	−0.945347	+	0.326065i	$0.894277\pi$
$18$	0	0
$19$	−8.38516	−1.92369	−0.961844	−	0.273597i	$-0.911786\pi$
$19$	−8.38516	−1.92369	−0.961844	+	0.273597i	$0.911786\pi$
$20$	0	0
$21$	0	0
$22$	− 7.57775i	− 1.61558i
$23$	− 5.63652i	− 1.17530i	−0.809117	−	0.587648i	$-0.800054\pi$
$23$	− 5.63652i	− 1.17530i	0.809117	−	0.587648i	$-0.199946\pi$
$24$	0	0
$25$	0	0
$26$	5.89543	1.15619
$27$	0	0
$28$	0.192582i	0.0363947i
$29$	−8.32532	−1.54597	−0.772987	−	0.634422i	$-0.781238\pi$
$29$	−8.32532	−1.54597	−0.772987	+	0.634422i	$0.781238\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	1.08549i	0.191890i
$33$	0	0
$34$	−3.61484	−0.619939
$35$	0	0
$36$	0	0
$37$	− 3.00000i	− 0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$	− 3.00000i	− 0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$	− 11.2730i	− 1.82873i
$39$	0	0
$40$	0	0
$41$	−5.37761	−0.839841	−0.419921	−	0.907561i	$-0.637942\pi$
$41$	−5.37761	−0.839841	−0.419921	+	0.907561i	$0.637942\pi$
$42$	0	0
$43$	1.38516i	0.211236i	0.994407	+	0.105618i	$0.0336820\pi$
$43$	1.38516i	0.211236i	−0.994407	+	0.105618i	$0.966318\pi$
$44$	−1.08549	−0.163644
$45$	0	0
$46$	7.57775	1.11728
$47$	8.58423i	1.25214i	0.779767	+	0.626069i	$0.215337\pi$
$47$	8.58423i	1.25214i	−0.779767	+	0.626069i	$0.784663\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 0.844506i	− 0.117112i
$53$	5.37761i	0.738671i	0.929296	+	0.369336i	$0.120415\pi$
$53$	5.37761i	0.738671i	−0.929296	+	0.369336i	$0.879585\pi$
$54$	0	0
$55$	0	0
$56$	−2.94771	−0.393905
$57$	0	0
$58$	− 11.1926i	− 1.46966i
$59$	8.58423	1.11757	0.558786	−	0.829312i	$-0.311267\pi$
$59$	8.58423	1.11757	0.558786	+	0.829312i	$0.311267\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	8.06641i	1.02444i
$63$	0	0
$64$	−8.61484	−1.07685
$65$	0	0
$66$	0	0
$67$	− 1.38516i	− 0.169225i	−0.996414	−	0.0846124i	$-0.973035\pi$
$67$	− 1.38516i	− 0.169225i	0.996414	−	0.0846124i	$-0.0269652\pi$
$68$	0.517816i	0.0627945i
$69$	0	0
$70$	0	0
$71$	0.258908	0.0307268	0.0153634	−	0.999882i	$-0.495109\pi$
$71$	0.258908	0.0307268	0.0153634	+	0.999882i	$0.495109\pi$
$72$	0	0
$73$	− 6.38516i	− 0.747327i	−0.927564	−	0.373664i	$-0.878101\pi$
$73$	− 6.38516i	− 0.747327i	0.927564	−	0.373664i	$-0.121899\pi$
$74$	4.03321	0.468851
$75$	0	0
$76$	−1.61484	−0.185234
$77$	− 5.63652i	− 0.642341i
$78$	0	0
$79$	−5.38516	−0.605878	−0.302939	−	0.953010i	$-0.597968\pi$
$79$	−5.38516	−0.605878	−0.302939	+	0.953010i	$0.597968\pi$
$80$	0	0
$81$	0	0
$82$	− 7.22967i	− 0.798384i
$83$	16.6506i	1.82765i	0.406113	+	0.913823i	$0.366884\pi$
$83$	16.6506i	1.82765i	−0.406113	+	0.913823i	$0.633116\pi$
$84$	0	0
$85$	0	0
$86$	−1.86222	−0.200808
$87$	0	0
$88$	− 16.6148i	− 1.77115i
$89$	−13.9618	−1.47995	−0.739976	−	0.672633i	$-0.765163\pi$
$89$	−13.9618	−1.47995	−0.739976	+	0.672633i	$0.765163\pi$
$90$	0	0
$91$	4.38516	0.459690
$92$	− 1.08549i	− 0.113171i
$93$	0	0
$94$	−11.5407	−1.19033
$95$	0	0
$96$	0	0
$97$	− 10.7703i	− 1.09356i	−0.837276	−	0.546781i	$-0.815853\pi$
$97$	− 10.7703i	− 1.09356i	0.837276	−	0.546781i	$-0.184147\pi$
$98$	− 1.34440i	− 0.135805i
$99$	0	0
$100$	0	0
$101$	2.68880	0.267546	0.133773	−	0.991012i	$-0.457291\pi$
$101$	2.68880	0.267546	0.133773	+	0.991012i	$0.457291\pi$
$102$	0	0
$103$	16.3852i	1.61448i	0.590225	+	0.807239i	$0.299039\pi$
$103$	16.3852i	1.61448i	−0.590225	+	0.807239i	$0.700961\pi$
$104$	12.9262	1.26752
$105$	0	0
$106$	−7.22967	−0.702208
$107$	− 16.6506i	− 1.60968i	−0.593493	−	0.804839i	$-0.702251\pi$
$107$	− 16.6506i	− 1.60968i	0.593493	−	0.804839i	$-0.297749\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	0	0
$111$	0	0
$112$	− 3.57775i	− 0.338065i
$113$	− 8.32532i	− 0.783180i	−0.920140	−	0.391590i	$-0.871925\pi$
$113$	− 8.32532i	− 0.783180i	0.920140	−	0.391590i	$-0.128075\pi$
$114$	0	0
$115$	0	0
$116$	−1.60331	−0.148864
$117$	0	0
$118$	11.5407i	1.06240i
$119$	−2.68880	−0.246482
$120$	0	0
$121$	20.7703	1.88821
$122$	0	0
$123$	0	0
$124$	1.15549	0.103766
$125$	0	0
$126$	0	0
$127$	15.3852i	1.36521i	0.730786	+	0.682606i	$0.239154\pi$
$127$	15.3852i	1.36521i	−0.730786	+	0.682606i	$0.760846\pi$
$128$	− 9.41082i	− 0.831806i
$129$	0	0
$130$	0	0
$131$	2.68880	0.234922	0.117461	−	0.993077i	$-0.462524\pi$
$131$	2.68880	0.234922	0.117461	+	0.993077i	$0.462524\pi$
$132$	0	0
$133$	− 8.38516i	− 0.727086i
$134$	1.86222	0.160871
$135$	0	0
$136$	−7.92582	−0.679634
$137$	− 5.37761i	− 0.459440i	−0.973257	−	0.229720i	$-0.926219\pi$
$137$	− 5.37761i	− 0.459440i	0.973257	−	0.229720i	$-0.0737811\pi$
$138$	0	0
$139$	−6.38516	−0.541583	−0.270791	−	0.962638i	$-0.587285\pi$
$139$	−6.38516	−0.541583	−0.270791	+	0.962638i	$0.587285\pi$
$140$	0	0
$141$	0	0
$142$	0.348077i	0.0292100i
$143$	24.7171i	2.06694i
$144$	0	0
$145$	0	0
$146$	8.58423	0.710436
$147$	0	0
$148$	− 0.577747i	− 0.0474905i
$149$	−13.7029	−1.12259	−0.561294	−	0.827617i	$-0.689696\pi$
$149$	−13.7029	−1.12259	−0.561294	+	0.827617i	$0.689696\pi$
$150$	0	0
$151$	0.614835	0.0500346	0.0250173	−	0.999687i	$-0.492036\pi$
$151$	0.614835	0.0500346	0.0250173	+	0.999687i	$0.492036\pi$
$152$	− 24.7171i	− 2.00482i
$153$	0	0
$154$	7.57775	0.610632
$155$	0	0
$156$	0	0
$157$	4.77033i	0.380714i	0.981715	+	0.190357i	$0.0609645\pi$
$157$	4.77033i	0.380714i	−0.981715	+	0.190357i	$0.939035\pi$
$158$	− 7.23983i	− 0.575970i
$159$	0	0
$160$	0	0
$161$	5.63652	0.444220
$162$	0	0
$163$	− 4.77033i	− 0.373641i	−0.982394	−	0.186821i	$-0.940182\pi$
$163$	− 4.77033i	− 0.373641i	0.982394	−	0.186821i	$-0.0598183\pi$
$164$	−1.03563	−0.0808693
$165$	0	0
$166$	−22.3852	−1.73743
$167$	13.9618i	1.08040i	0.841537	+	0.540200i	$0.181651\pi$
$167$	13.9618i	1.08040i	−0.841537	+	0.540200i	$0.818349\pi$
$168$	0	0
$169$	−6.22967	−0.479205
$170$	0	0
$171$	0	0
$172$	0.266758i	0.0203401i
$173$	− 3.20662i	− 0.243795i	−0.992543	−	0.121897i	$-0.961102\pi$
$173$	− 3.20662i	− 0.243795i	0.992543	−	0.121897i	$-0.0388979\pi$
$174$	0	0
$175$	0	0
$176$	20.1660	1.52007
$177$	0	0
$178$	− 18.7703i	− 1.40690i
$179$	16.6506	1.24453	0.622264	−	0.782808i	$-0.286213\pi$
$179$	16.6506	1.24453	0.622264	+	0.782808i	$0.286213\pi$
$180$	0	0
$181$	9.61484	0.714665	0.357333	−	0.933977i	$-0.383686\pi$
$181$	9.61484	0.714665	0.357333	+	0.933977i	$0.383686\pi$
$182$	5.89543i	0.436998i
$183$	0	0
$184$	16.6148	1.22486
$185$	0	0
$186$	0	0
$187$	− 15.1555i	− 1.10828i
$188$	1.65317i	0.120570i
$189$	0	0
$190$	0	0
$191$	11.2730	0.815688	0.407844	−	0.913052i	$-0.366281\pi$
$191$	11.2730	0.815688	0.407844	+	0.913052i	$0.366281\pi$
$192$	0	0
$193$	1.00000i	0.0719816i	0.999352	+	0.0359908i	$0.0114587\pi$
$193$	1.00000i	0.0719816i	−0.999352	+	0.0359908i	$0.988541\pi$
$194$	14.4797	1.03958
$195$	0	0
$196$	−0.192582	−0.0137559
$197$	− 19.0805i	− 1.35943i	−0.733475	−	0.679716i	$-0.762103\pi$
$197$	− 19.0805i	− 1.35943i	0.733475	−	0.679716i	$-0.237897\pi$
$198$	0	0
$199$	20.7703	1.47237	0.736185	−	0.676781i	$-0.236625\pi$
$199$	20.7703	1.47237	0.736185	+	0.676781i	$0.236625\pi$
$200$	0	0
$201$	0	0
$202$	3.61484i	0.254339i
$203$	− 8.32532i	− 0.584323i
$204$	0	0
$205$	0	0
$206$	−22.0283	−1.53478
$207$	0	0
$208$	15.6890i	1.08784i
$209$	47.2631	3.26926
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	1.03563i	0.0711276i
$213$	0	0
$214$	22.3852	1.53022
$215$	0	0
$216$	0	0
$217$	6.00000i	0.407307i
$218$	− 9.41082i	− 0.637381i
$219$	0	0
$220$	0	0
$221$	11.7909	0.793139
$222$	0	0
$223$	14.3852i	0.963302i	0.876363	+	0.481651i	$0.159963\pi$
$223$	14.3852i	0.963302i	−0.876363	+	0.481651i	$0.840037\pi$
$224$	−1.08549	−0.0725276
$225$	0	0
$226$	11.1926	0.744520
$227$	− 8.58423i	− 0.569755i	−0.958564	−	0.284878i	$-0.908047\pi$
$227$	− 8.58423i	− 0.569755i	0.958564	−	0.284878i	$-0.0919530\pi$
$228$	0	0
$229$	−27.5407	−1.81994	−0.909969	−	0.414676i	$-0.863895\pi$
$229$	−27.5407	−1.81994	−0.909969	+	0.414676i	$0.863895\pi$
$230$	0	0
$231$	0	0
$232$	− 24.5407i	− 1.61117i
$233$	19.5984i	1.28393i	0.766733	+	0.641966i	$0.221881\pi$
$233$	19.5984i	1.28393i	−0.766733	+	0.641966i	$0.778119\pi$
$234$	0	0
$235$	0	0
$236$	1.65317	0.107612
$237$	0	0
$238$	− 3.61484i	− 0.234315i
$239$	−11.2730	−0.729192	−0.364596	−	0.931166i	$-0.618793\pi$
$239$	−11.2730	−0.729192	−0.364596	+	0.931166i	$0.618793\pi$
$240$	0	0
$241$	7.61484	0.490515	0.245257	−	0.969458i	$-0.421128\pi$
$241$	7.61484	0.490515	0.245257	+	0.969458i	$0.421128\pi$
$242$	27.9237i	1.79500i
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	36.7703i	2.33964i
$248$	17.6863i	1.12308i
$249$	0	0
$250$	0	0
$251$	−30.6125	−1.93224	−0.966121	−	0.258088i	$-0.916908\pi$
$251$	−30.6125	−1.93224	−0.966121	+	0.258088i	$0.916908\pi$
$252$	0	0
$253$	31.7703i	1.99738i
$254$	−20.6839	−1.29782
$255$	0	0
$256$	−4.57775	−0.286109
$257$	− 5.89543i	− 0.367747i	−0.982950	−	0.183873i	$-0.941136\pi$
$257$	− 5.89543i	− 0.367747i	0.982950	−	0.183873i	$-0.0588636\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	0	0
$261$	0	0
$262$	3.61484i	0.223325i
$263$	11.5319i	0.711090i	0.934659	+	0.355545i	$0.115705\pi$
$263$	11.5319i	0.711090i	−0.934659	+	0.355545i	$0.884295\pi$
$264$	0	0
$265$	0	0
$266$	11.2730	0.691194
$267$	0	0
$268$	− 0.266758i	− 0.0162949i
$269$	−16.6506	−1.01521	−0.507604	−	0.861591i	$-0.669469\pi$
$269$	−16.6506	−1.01521	−0.507604	+	0.861591i	$0.669469\pi$
$270$	0	0
$271$	−17.1555	−1.04212	−0.521061	−	0.853519i	$-0.674464\pi$
$271$	−17.1555	−1.04212	−0.521061	+	0.853519i	$0.674464\pi$
$272$	− 9.61986i	− 0.583290i
$273$	0	0
$274$	7.22967	0.436760
$275$	0	0
$276$	0	0
$277$	19.5407i	1.17408i	0.809556	+	0.587042i	$0.199708\pi$
$277$	19.5407i	1.17408i	−0.809556	+	0.587042i	$0.800292\pi$
$278$	− 8.58423i	− 0.514848i
$279$	0	0
$280$	0	0
$281$	−14.2207	−0.848339	−0.424169	−	0.905583i	$-0.639434\pi$
$281$	−14.2207	−0.848339	−0.424169	+	0.905583i	$0.639434\pi$
$282$	0	0
$283$	14.0000i	0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$	14.0000i	0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$	0.0498612	0.00295872
$285$	0	0
$286$	−33.2297	−1.96491
$287$	− 5.37761i	− 0.317430i
$288$	0	0
$289$	9.77033	0.574725
$290$	0	0
$291$	0	0
$292$	− 1.22967i	− 0.0719610i
$293$	− 8.58423i	− 0.501496i	−0.968052	−	0.250748i	$-0.919323\pi$
$293$	− 8.58423i	− 0.501496i	0.968052	−	0.250748i	$-0.0806766\pi$
$294$	0	0
$295$	0	0
$296$	8.84314	0.513997
$297$	0	0
$298$	− 18.4223i	− 1.06717i
$299$	−24.7171	−1.42942
$300$	0	0
$301$	−1.38516	−0.0798396
$302$	0.826586i	0.0475647i
$303$	0	0
$304$	30.0000	1.72062
$305$	0	0
$306$	0	0
$307$	19.1555i	1.09326i	0.837374	+	0.546631i	$0.184090\pi$
$307$	19.1555i	1.09326i	−0.837374	+	0.546631i	$0.815910\pi$
$308$	− 1.08549i	− 0.0618518i
$309$	0	0
$310$	0	0
$311$	13.4440	0.762341	0.381170	−	0.924505i	$-0.375521\pi$
$311$	13.4440	0.762341	0.381170	+	0.924505i	$0.375521\pi$
$312$	0	0
$313$	29.9258i	1.69151i	0.533573	+	0.845754i	$0.320849\pi$
$313$	29.9258i	1.69151i	−0.533573	+	0.845754i	$0.679151\pi$
$314$	−6.41324	−0.361920
$315$	0	0
$316$	−1.03709	−0.0583408
$317$	− 2.94771i	− 0.165560i	−0.996568	−	0.0827800i	$-0.973620\pi$
$317$	− 2.94771i	− 0.165560i	0.996568	−	0.0827800i	$-0.0263799\pi$
$318$	0	0
$319$	46.9258	2.62734
$320$	0	0
$321$	0	0
$322$	7.57775i	0.422291i
$323$	− 22.5461i	− 1.25450i
$324$	0	0
$325$	0	0
$326$	6.41324	0.355197
$327$	0	0
$328$	− 15.8516i	− 0.875261i
$329$	−8.58423	−0.473264
$330$	0	0
$331$	−22.1555	−1.21778	−0.608888	−	0.793256i	$-0.708384\pi$
$331$	−22.1555	−1.21778	−0.608888	+	0.793256i	$0.708384\pi$
$332$	3.20662i	0.175986i
$333$	0	0
$334$	−18.7703	−1.02707
$335$	0	0
$336$	0	0
$337$	− 10.7703i	− 0.586697i	−0.956006	−	0.293349i	$-0.905230\pi$
$337$	− 10.7703i	− 0.586697i	0.956006	−	0.293349i	$-0.0947697\pi$
$338$	− 8.37518i	− 0.455550i
$339$	0	0
$340$	0	0
$341$	−33.8191	−1.83141
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	−4.08307	−0.220144
$345$	0	0
$346$	4.31099	0.231760
$347$	5.11870i	0.274786i	0.990517	+	0.137393i	$0.0438724\pi$
$347$	5.11870i	0.274786i	−0.990517	+	0.137393i	$0.956128\pi$
$348$	0	0
$349$	8.38516	0.448848	0.224424	−	0.974492i	$-0.427950\pi$
$349$	8.38516	0.448848	0.224424	+	0.974492i	$0.427950\pi$
$350$	0	0
$351$	0	0
$352$	− 6.11841i	− 0.326112i
$353$	− 5.89543i	− 0.313782i	−0.987616	−	0.156891i	$-0.949853\pi$
$353$	− 5.89543i	− 0.313782i	0.987616	−	0.156891i	$-0.0501471\pi$
$354$	0	0
$355$	0	0
$356$	−2.68880	−0.142506
$357$	0	0
$358$	22.3852i	1.18309i
$359$	−16.3917	−0.865123	−0.432561	−	0.901604i	$-0.642390\pi$
$359$	−16.3917	−0.865123	−0.432561	+	0.901604i	$0.642390\pi$
$360$	0	0
$361$	51.3110	2.70058
$362$	12.9262i	0.679386i
$363$	0	0
$364$	0.844506	0.0442641
$365$	0	0
$366$	0	0
$367$	4.00000i	0.208798i	0.994535	+	0.104399i	$0.0332919\pi$
$367$	4.00000i	0.208798i	−0.994535	+	0.104399i	$0.966708\pi$
$368$	20.1660i	1.05123i
$369$	0	0
$370$	0	0
$371$	−5.37761	−0.279192
$372$	0	0
$373$	− 25.0000i	− 1.29445i	−0.762299	−	0.647225i	$-0.775929\pi$
$373$	− 25.0000i	− 1.29445i	0.762299	−	0.647225i	$-0.224071\pi$
$374$	20.3751	1.05357
$375$	0	0
$376$	−25.3038	−1.30495
$377$	36.5079i	1.88025i
$378$	0	0
$379$	0.614835	0.0315820	0.0157910	−	0.999875i	$-0.494973\pi$
$379$	0.614835	0.0315820	0.0157910	+	0.999875i	$0.494973\pi$
$380$	0	0
$381$	0	0
$382$	15.1555i	0.775423i
$383$	− 19.3394i	− 0.988200i	−0.869405	−	0.494100i	$-0.835498\pi$
$383$	− 19.3394i	− 0.988200i	0.869405	−	0.494100i	$-0.164502\pi$
$384$	0	0
$385$	0	0
$386$	−1.34440	−0.0684283
$387$	0	0
$388$	− 2.07418i	− 0.105300i
$389$	30.8714	1.56524	0.782621	−	0.622499i	$-0.213882\pi$
$389$	30.8714	1.56524	0.782621	+	0.622499i	$0.213882\pi$
$390$	0	0
$391$	15.1555	0.766446
$392$	− 2.94771i	− 0.148882i
$393$	0	0
$394$	25.6519	1.29233
$395$	0	0
$396$	0	0
$397$	22.7703i	1.14281i	0.820668	+	0.571405i	$0.193601\pi$
$397$	22.7703i	1.14281i	−0.820668	+	0.571405i	$0.806399\pi$
$398$	27.9237i	1.39969i
$399$	0	0
$400$	0	0
$401$	13.7029	0.684292	0.342146	−	0.939647i	$-0.388846\pi$
$401$	13.7029	0.684292	0.342146	+	0.939647i	$0.388846\pi$
$402$	0	0
$403$	− 26.3110i	− 1.31064i
$404$	0.517816	0.0257623
$405$	0	0
$406$	11.1926	0.555479
$407$	16.9096i	0.838175i
$408$	0	0
$409$	−11.6148	−0.574317	−0.287158	−	0.957883i	$-0.592711\pi$
$409$	−11.6148	−0.574317	−0.287158	+	0.957883i	$0.592711\pi$
$410$	0	0
$411$	0	0
$412$	3.15549i	0.155460i
$413$	8.58423i	0.422402i
$414$	0	0
$415$	0	0
$416$	4.76007	0.233382
$417$	0	0
$418$	63.5407i	3.10788i
$419$	−5.89543	−0.288010	−0.144005	−	0.989577i	$-0.545998\pi$
$419$	−5.89543	−0.288010	−0.144005	+	0.989577i	$0.545998\pi$
$420$	0	0
$421$	−39.3110	−1.91590	−0.957950	−	0.286935i	$-0.907364\pi$
$421$	−39.3110	−1.91590	−0.957950	+	0.286935i	$0.907364\pi$
$422$	0	0
$423$	0	0
$424$	−15.8516	−0.769824
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 3.20662i	− 0.154998i
$429$	0	0
$430$	0	0
$431$	−27.4059	−1.32009	−0.660047	−	0.751224i	$-0.729464\pi$
$431$	−27.4059	−1.32009	−0.660047	+	0.751224i	$0.729464\pi$
$432$	0	0
$433$	33.5407i	1.61186i	0.592010	+	0.805931i	$0.298335\pi$
$433$	33.5407i	1.61186i	−0.592010	+	0.805931i	$0.701665\pi$
$434$	−8.06641	−0.387200
$435$	0	0
$436$	−1.34808	−0.0645612
$437$	47.2631i	2.26090i
$438$	0	0
$439$	13.1555	0.627877	0.313939	−	0.949443i	$-0.398351\pi$
$439$	13.1555	0.627877	0.313939	+	0.949443i	$0.398351\pi$
$440$	0	0
$441$	0	0
$442$	15.8516i	0.753986i
$443$	− 33.8191i	− 1.60679i	−0.595444	−	0.803397i	$-0.703024\pi$
$443$	− 33.8191i	− 1.60679i	0.595444	−	0.803397i	$-0.296976\pi$
$444$	0	0
$445$	0	0
$446$	−19.3394	−0.915749
$447$	0	0
$448$	− 8.61484i	− 0.407013i
$449$	−13.7029	−0.646681	−0.323341	−	0.946283i	$-0.604806\pi$
$449$	−13.7029	−0.646681	−0.323341	+	0.946283i	$0.604806\pi$
$450$	0	0
$451$	30.3110	1.42729
$452$	− 1.60331i	− 0.0754134i
$453$	0	0
$454$	11.5407	0.541630
$455$	0	0
$456$	0	0
$457$	− 14.5407i	− 0.680183i	−0.940392	−	0.340092i	$-0.889542\pi$
$457$	− 14.5407i	− 0.680183i	0.940392	−	0.340092i	$-0.110458\pi$
$458$	− 37.0257i	− 1.73010i
$459$	0	0
$460$	0	0
$461$	13.9618	0.650268	0.325134	−	0.945668i	$-0.394591\pi$
$461$	13.9618	0.650268	0.325134	+	0.945668i	$0.394591\pi$
$462$	0	0
$463$	12.7703i	0.593488i	0.954957	+	0.296744i	$0.0959007\pi$
$463$	12.7703i	0.593488i	−0.954957	+	0.296744i	$0.904099\pi$
$464$	29.7859	1.38278
$465$	0	0
$466$	−26.3481	−1.22055
$467$	− 11.2730i	− 0.521654i	−0.965386	−	0.260827i	$-0.916005\pi$
$467$	− 11.2730i	− 0.521654i	0.965386	−	0.260827i	$-0.0839952\pi$
$468$	0	0
$469$	1.38516	0.0639610
$470$	0	0
$471$	0	0
$472$	25.3038i	1.16470i
$473$	− 7.80751i	− 0.358989i
$474$	0	0
$475$	0	0
$476$	−0.517816	−0.0237341
$477$	0	0
$478$	− 15.1555i	− 0.693196i
$479$	2.17099	0.0991950	0.0495975	−	0.998769i	$-0.484206\pi$
$479$	2.17099	0.0991950	0.0495975	+	0.998769i	$0.484206\pi$
$480$	0	0
$481$	−13.1555	−0.599839
$482$	10.2374i	0.466301i
$483$	0	0
$484$	4.00000	0.181818
$485$	0	0
$486$	0	0
$487$	2.61484i	0.118489i	0.998243	+	0.0592447i	$0.0188692\pi$
$487$	2.61484i	0.118489i	−0.998243	+	0.0592447i	$0.981131\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10.4963	−0.473692	−0.236846	−	0.971547i	$-0.576114\pi$
$491$	−10.4963	−0.473692	−0.236846	+	0.971547i	$0.576114\pi$
$492$	0	0
$493$	− 22.3852i	− 1.00818i
$494$	−49.4341	−2.22415
$495$	0	0
$496$	−21.4665	−0.963874
$497$	0.258908i	0.0116136i
$498$	0	0
$499$	13.5407	0.606163	0.303082	−	0.952965i	$-0.401985\pi$
$499$	13.5407	0.606163	0.303082	+	0.952965i	$0.401985\pi$
$500$	0	0
$501$	0	0
$502$	− 41.1555i	− 1.83686i
$503$	23.0639i	1.02837i	0.857680	+	0.514184i	$0.171905\pi$
$503$	23.0639i	1.02837i	−0.857680	+	0.514184i	$0.828095\pi$
$504$	0	0
$505$	0	0
$506$	−42.7121	−1.89878
$507$	0	0
$508$	2.96291i	0.131458i
$509$	19.3394	0.857206	0.428603	−	0.903493i	$-0.359006\pi$
$509$	19.3394	0.857206	0.428603	+	0.903493i	$0.359006\pi$
$510$	0	0
$511$	6.38516	0.282463
$512$	− 24.9760i	− 1.10379i
$513$	0	0
$514$	7.92582	0.349593
$515$	0	0
$516$	0	0
$517$	− 48.3852i	− 2.12798i
$518$	4.03321i	0.177209i
$519$	0	0
$520$	0	0
$521$	32.7835	1.43627	0.718135	−	0.695904i	$-0.244996\pi$
$521$	32.7835	1.43627	0.718135	+	0.695904i	$0.244996\pi$
$522$	0	0
$523$	40.7703i	1.78276i	0.453255	+	0.891381i	$0.350263\pi$
$523$	40.7703i	1.78276i	−0.453255	+	0.891381i	$0.649737\pi$
$524$	0.517816	0.0226209
$525$	0	0
$526$	−15.5036	−0.675988
$527$	16.1328i	0.702757i
$528$	0	0
$529$	−8.77033	−0.381319
$530$	0	0
$531$	0	0
$532$	− 1.61484i	− 0.0700120i
$533$	23.5817i	1.02144i
$534$	0	0
$535$	0	0
$536$	4.08307	0.176362
$537$	0	0
$538$	− 22.3852i	− 0.965093i
$539$	5.63652	0.242782
$540$	0	0
$541$	28.5407	1.22706	0.613529	−	0.789672i	$-0.289749\pi$
$541$	28.5407	1.22706	0.613529	+	0.789672i	$0.289749\pi$
$542$	− 23.0639i	− 0.990679i
$543$	0	0
$544$	−2.91868	−0.125137
$545$	0	0
$546$	0	0
$547$	23.3852i	0.999877i	0.866061	+	0.499939i	$0.166644\pi$
$547$	23.3852i	0.999877i	−0.866061	+	0.499939i	$0.833356\pi$
$548$	− 1.03563i	− 0.0442400i
$549$	0	0
$550$	0	0
$551$	69.8092	2.97397
$552$	0	0
$553$	− 5.38516i	− 0.229001i
$554$	−26.2705	−1.11613
$555$	0	0
$556$	−1.22967	−0.0521496
$557$	2.94771i	0.124899i	0.998048	+	0.0624493i	$0.0198912\pi$
$557$	2.94771i	0.124899i	−0.998048	+	0.0624493i	$0.980109\pi$
$558$	0	0
$559$	6.07418	0.256910
$560$	0	0
$561$	0	0
$562$	− 19.1184i	− 0.806461i
$563$	− 16.6506i	− 0.701741i	−0.936424	−	0.350870i	$-0.885886\pi$
$563$	− 16.6506i	− 0.701741i	0.936424	−	0.350870i	$-0.114114\pi$
$564$	0	0
$565$	0	0
$566$	−18.8216	−0.791132
$567$	0	0
$568$	0.763187i	0.0320226i
$569$	−20.1162	−0.843314	−0.421657	−	0.906755i	$-0.638551\pi$
$569$	−20.1162	−0.843314	−0.421657	+	0.906755i	$0.638551\pi$
$570$	0	0
$571$	24.6148	1.03010	0.515049	−	0.857160i	$-0.327774\pi$
$571$	24.6148	1.03010	0.515049	+	0.857160i	$0.327774\pi$
$572$	4.76007i	0.199029i
$573$	0	0
$574$	7.22967	0.301761
$575$	0	0
$576$	0	0
$577$	14.8445i	0.617985i	0.951064	+	0.308992i	$0.0999918\pi$
$577$	14.8445i	0.617985i	−0.951064	+	0.308992i	$0.900008\pi$
$578$	13.1353i	0.546355i
$579$	0	0
$580$	0	0
$581$	−16.6506	−0.690785
$582$	0	0
$583$	− 30.3110i	− 1.25535i
$584$	18.8216	0.778845
$585$	0	0
$586$	11.5407	0.476740
$587$	− 35.9901i	− 1.48547i	−0.669585	−	0.742735i	$-0.733528\pi$
$587$	− 35.9901i	− 1.48547i	0.669585	−	0.742735i	$-0.266472\pi$
$588$	0	0
$589$	−50.3110	−2.07303
$590$	0	0
$591$	0	0
$592$	10.7332i	0.441134i
$593$	11.7909i	0.484192i	0.970252	+	0.242096i	$0.0778349\pi$
$593$	11.7909i	0.484192i	−0.970252	+	0.242096i	$0.922165\pi$
$594$	0	0
$595$	0	0
$596$	−2.63894	−0.108095
$597$	0	0
$598$	− 33.2297i	− 1.35886i
$599$	−22.8050	−0.931786	−0.465893	−	0.884841i	$-0.654267\pi$
$599$	−22.8050	−0.931786	−0.465893	+	0.884841i	$0.654267\pi$
$600$	0	0
$601$	−1.54066	−0.0628448	−0.0314224	−	0.999506i	$-0.510004\pi$
$601$	−1.54066	−0.0628448	−0.0314224	+	0.999506i	$0.510004\pi$
$602$	− 1.86222i	− 0.0758984i
$603$	0	0
$604$	0.118406	0.00481789
$605$	0	0
$606$	0	0
$607$	− 9.15549i	− 0.371610i	−0.982587	−	0.185805i	$-0.940511\pi$
$607$	− 9.15549i	− 0.371610i	0.982587	−	0.185805i	$-0.0594893\pi$
$608$	− 9.10205i	− 0.369137i
$609$	0	0
$610$	0	0
$611$	37.6433	1.52288
$612$	0	0
$613$	− 23.0000i	− 0.928961i	−0.885583	−	0.464481i	$-0.846241\pi$
$613$	− 23.0000i	− 0.928961i	0.885583	−	0.464481i	$-0.153759\pi$
$614$	−25.7527	−1.03929
$615$	0	0
$616$	16.6148	0.669431
$617$	− 30.8714i	− 1.24284i	−0.783479	−	0.621418i	$-0.786557\pi$
$617$	− 30.8714i	− 1.24284i	0.783479	−	0.621418i	$-0.213443\pi$
$618$	0	0
$619$	−5.15549	−0.207217	−0.103608	−	0.994618i	$-0.533039\pi$
$619$	−5.15549	−0.207217	−0.103608	+	0.994618i	$0.533039\pi$
$620$	0	0
$621$	0	0
$622$	18.0742i	0.724708i
$623$	− 13.9618i	− 0.559369i
$624$	0	0
$625$	0	0
$626$	−40.2323	−1.60801
$627$	0	0
$628$	0.918682i	0.0366594i
$629$	8.06641	0.321629
$630$	0	0
$631$	5.38516	0.214380	0.107190	−	0.994239i	$-0.465815\pi$
$631$	5.38516	0.214380	0.107190	+	0.994239i	$0.465815\pi$
$632$	− 15.8739i	− 0.631431i
$633$	0	0
$634$	3.96291	0.157387
$635$	0	0
$636$	0	0
$637$	4.38516i	0.173747i
$638$	63.0872i	2.49765i
$639$	0	0
$640$	0	0
$641$	−25.4938	−1.00694	−0.503472	−	0.864012i	$-0.667944\pi$
$641$	−25.4938	−1.00694	−0.503472	+	0.864012i	$0.667944\pi$
$642$	0	0
$643$	49.5407i	1.95369i	0.213942	+	0.976846i	$0.431370\pi$
$643$	49.5407i	1.95369i	−0.213942	+	0.976846i	$0.568630\pi$
$644$	1.08549	0.0427745
$645$	0	0
$646$	30.3110	1.19257
$647$	4.85979i	0.191058i	0.995427	+	0.0955291i	$0.0304543\pi$
$647$	4.85979i	0.191058i	−0.995427	+	0.0955291i	$0.969546\pi$
$648$	0	0
$649$	−48.3852	−1.89928
$650$	0	0
$651$	0	0
$652$	− 0.918682i	− 0.0359783i
$653$	5.37761i	0.210442i	0.994449	+	0.105221i	$0.0335550\pi$
$653$	5.37761i	0.210442i	−0.994449	+	0.105221i	$0.966445\pi$
$654$	0	0
$655$	0	0
$656$	19.2397	0.751185
$657$	0	0
$658$	− 11.5407i	− 0.449902i
$659$	38.1611	1.48654	0.743272	−	0.668989i	$-0.233273\pi$
$659$	38.1611	1.48654	0.743272	+	0.668989i	$0.233273\pi$
$660$	0	0
$661$	−21.9258	−0.852816	−0.426408	−	0.904531i	$-0.640221\pi$
$661$	−21.9258	−0.852816	−0.426408	+	0.904531i	$0.640221\pi$
$662$	− 29.7859i	− 1.15766i
$663$	0	0
$664$	−49.0813	−1.90472
$665$	0	0
$666$	0	0
$667$	46.9258i	1.81698i
$668$	2.68880i	0.104033i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 39.5407i	− 1.52418i	−0.647471	−	0.762090i	$-0.724173\pi$
$673$	− 39.5407i	− 1.52418i	0.647471	−	0.762090i	$-0.275827\pi$
$674$	14.4797	0.557736
$675$	0	0
$676$	−1.19972	−0.0461433
$677$	11.2730i	0.433258i	0.976254	+	0.216629i	$0.0695062\pi$
$677$	11.2730i	0.433258i	−0.976254	+	0.216629i	$0.930494\pi$
$678$	0	0
$679$	10.7703	0.413327
$680$	0	0
$681$	0	0
$682$	− 45.4665i	− 1.74100i
$683$	− 16.9096i	− 0.647026i	−0.946224	−	0.323513i	$-0.895136\pi$
$683$	− 16.9096i	− 0.647026i	0.946224	−	0.323513i	$-0.104864\pi$
$684$	0	0
$685$	0	0
$686$	1.34440	0.0513295
$687$	0	0
$688$	− 4.95577i	− 0.188937i
$689$	23.5817	0.898391
$690$	0	0
$691$	−36.7703	−1.39881	−0.699405	−	0.714726i	$-0.746551\pi$
$691$	−36.7703	−1.39881	−0.699405	+	0.714726i	$0.746551\pi$
$692$	− 0.617539i	− 0.0234753i
$693$	0	0
$694$	−6.88159	−0.261222
$695$	0	0
$696$	0	0
$697$	− 14.4593i	− 0.547687i
$698$	11.2730i	0.426691i
$699$	0	0
$700$	0	0
$701$	6.41324	0.242225	0.121112	−	0.992639i	$-0.461354\pi$
$701$	6.41324	0.242225	0.121112	+	0.992639i	$0.461354\pi$
$702$	0	0
$703$	25.1555i	0.948757i
$704$	48.5577	1.83009
$705$	0	0
$706$	7.92582	0.298292
$707$	2.68880i	0.101123i
$708$	0	0
$709$	3.54066	0.132972	0.0664861	−	0.997787i	$-0.478821\pi$
$709$	3.54066	0.132972	0.0664861	+	0.997787i	$0.478821\pi$
$710$	0	0
$711$	0	0
$712$	− 41.1555i	− 1.54237i
$713$	− 33.8191i	− 1.26654i
$714$	0	0
$715$	0	0
$716$	3.20662	0.119837
$717$	0	0
$718$	− 22.0371i	− 0.822417i
$719$	38.6789	1.44248	0.721240	−	0.692686i	$-0.243573\pi$
$719$	38.6789	1.44248	0.721240	+	0.692686i	$0.243573\pi$
$720$	0	0
$721$	−16.3852	−0.610215
$722$	68.9826i	2.56727i
$723$	0	0
$724$	1.85165	0.0688160
$725$	0	0
$726$	0	0
$727$	− 21.6148i	− 0.801650i	−0.916155	−	0.400825i	$-0.868724\pi$
$727$	− 21.6148i	− 0.801650i	0.916155	−	0.400825i	$-0.131276\pi$
$728$	12.9262i	0.479077i
$729$	0	0
$730$	0	0
$731$	−3.72444	−0.137753
$732$	0	0
$733$	− 28.0000i	− 1.03420i	−0.855924	−	0.517102i	$-0.827011\pi$
$733$	− 28.0000i	− 1.03420i	0.855924	−	0.517102i	$-0.172989\pi$
$734$	−5.37761	−0.198491
$735$	0	0
$736$	6.11841	0.225527
$737$	7.80751i	0.287593i
$738$	0	0
$739$	32.6148	1.19976	0.599878	−	0.800091i	$-0.295216\pi$
$739$	32.6148	1.19976	0.599878	+	0.800091i	$0.295216\pi$
$740$	0	0
$741$	0	0
$742$	− 7.22967i	− 0.265410i
$743$	− 22.0283i	− 0.808138i	−0.914728	−	0.404069i	$-0.867596\pi$
$743$	− 22.0283i	− 0.808138i	0.914728	−	0.404069i	$-0.132404\pi$
$744$	0	0
$745$	0	0
$746$	33.6101	1.23055
$747$	0	0
$748$	− 2.91868i	− 0.106718i
$749$	16.6506	0.608401
$750$	0	0
$751$	−7.22967	−0.263814	−0.131907	−	0.991262i	$-0.542110\pi$
$751$	−7.22967	−0.263814	−0.131907	+	0.991262i	$0.542110\pi$
$752$	− 30.7122i	− 1.11996i
$753$	0	0
$754$	−49.0813	−1.78744
$755$	0	0
$756$	0	0
$757$	− 53.3110i	− 1.93762i	−0.247803	−	0.968810i	$-0.579709\pi$
$757$	− 53.3110i	− 1.93762i	0.247803	−	0.968810i	$-0.420291\pi$
$758$	0.826586i	0.0300230i
$759$	0	0
$760$	0	0
$761$	22.5461	0.817294	0.408647	−	0.912692i	$-0.366001\pi$
$761$	22.5461	0.817294	0.408647	+	0.912692i	$0.366001\pi$
$762$	0	0
$763$	− 7.00000i	− 0.253417i
$764$	2.17099	0.0785436
$765$	0	0
$766$	26.0000	0.939418
$767$	− 37.6433i	− 1.35922i
$768$	0	0
$769$	−23.2297	−0.837683	−0.418842	−	0.908059i	$-0.637564\pi$
$769$	−23.2297	−0.837683	−0.418842	+	0.908059i	$0.637564\pi$
$770$	0	0
$771$	0	0
$772$	0.192582i	0.00693119i
$773$	22.0283i	0.792301i	0.918186	+	0.396151i	$0.129654\pi$
$773$	22.0283i	0.792301i	−0.918186	+	0.396151i	$0.870346\pi$
$774$	0	0
$775$	0	0
$776$	31.7478	1.13968
$777$	0	0
$778$	41.5036i	1.48798i
$779$	45.0921	1.61559
$780$	0	0
$781$	−1.45934	−0.0522193
$782$	20.3751i	0.728611i
$783$	0	0
$784$	3.57775	0.127777
$785$	0	0
$786$	0	0
$787$	− 10.3852i	− 0.370191i	−0.982721	−	0.185096i	$-0.940741\pi$
$787$	− 10.3852i	− 0.370191i	0.982721	−	0.185096i	$-0.0592595\pi$
$788$	− 3.67458i	− 0.130901i
$789$	0	0
$790$	0	0
$791$	8.32532	0.296014
$792$	0	0
$793$	0	0
$794$	−30.6125	−1.08640
$795$	0	0
$796$	4.00000	0.141776
$797$	− 42.4033i	− 1.50200i	−0.660300	−	0.751002i	$-0.729571\pi$
$797$	− 42.4033i	− 1.50200i	0.660300	−	0.751002i	$-0.270429\pi$
$798$	0	0
$799$	−23.0813	−0.816558
$800$	0	0
$801$	0	0
$802$	18.4223i	0.650512i
$803$	35.9901i	1.27006i
$804$	0	0
$805$	0	0
$806$	35.3726	1.24595
$807$	0	0
$808$	7.92582i	0.278830i
$809$	−3.46553	−0.121842	−0.0609208	−	0.998143i	$-0.519404\pi$
$809$	−3.46553	−0.121842	−0.0609208	+	0.998143i	$0.519404\pi$
$810$	0	0
$811$	9.22967	0.324098	0.162049	−	0.986783i	$-0.448190\pi$
$811$	9.22967	0.324098	0.162049	+	0.986783i	$0.448190\pi$
$812$	− 1.60331i	− 0.0562652i
$813$	0	0
$814$	−22.7332	−0.796800
$815$	0	0
$816$	0	0
$817$	− 11.6148i	− 0.406352i
$818$	− 15.6150i	− 0.545966i
$819$	0	0
$820$	0	0
$821$	−34.3369	−1.19837	−0.599183	−	0.800612i	$-0.704508\pi$
$821$	−34.3369	−1.19837	−0.599183	+	0.800612i	$0.704508\pi$
$822$	0	0
$823$	− 28.9258i	− 1.00829i	−0.863619	−	0.504145i	$-0.831808\pi$
$823$	− 28.9258i	− 1.00829i	0.863619	−	0.504145i	$-0.168192\pi$
$824$	−48.2988	−1.68257
$825$	0	0
$826$	−11.5407	−0.401551
$827$	0.258908i	0.00900312i	0.999990	+	0.00450156i	$0.00143290\pi$
$827$	0.258908i	0.00900312i	−0.999990	+	0.00450156i	$0.998567\pi$
$828$	0	0
$829$	−2.38516	−0.0828402	−0.0414201	−	0.999142i	$-0.513188\pi$
$829$	−2.38516	−0.0828402	−0.0414201	+	0.999142i	$0.513188\pi$
$830$	0	0
$831$	0	0
$832$	37.7775i	1.30970i
$833$	− 2.68880i	− 0.0931616i
$834$	0	0
$835$	0	0
$836$	9.10205	0.314801
$837$	0	0
$838$	− 7.92582i	− 0.273793i
$839$	−35.9901	−1.24252	−0.621258	−	0.783606i	$-0.713378\pi$
$839$	−35.9901	−1.24252	−0.621258	+	0.783606i	$0.713378\pi$
$840$	0	0
$841$	40.3110	1.39003
$842$	− 52.8498i	− 1.82132i
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	20.7703i	0.713677i
$848$	− 19.2397i	− 0.660695i
$849$	0	0
$850$	0	0
$851$	−16.9096	−0.579652
$852$	0	0
$853$	− 34.3110i	− 1.17479i	−0.809302	−	0.587393i	$-0.800154\pi$
$853$	− 34.3110i	− 1.17479i	0.809302	−	0.587393i	$-0.199846\pi$
$854$	0	0
$855$	0	0
$856$	49.0813	1.67756
$857$	− 39.7145i	− 1.35662i	−0.734775	−	0.678311i	$-0.762712\pi$
$857$	− 39.7145i	− 1.35662i	0.734775	−	0.678311i	$-0.237288\pi$
$858$	0	0
$859$	−15.5407	−0.530240	−0.265120	−	0.964215i	$-0.585412\pi$
$859$	−15.5407	−0.530240	−0.265120	+	0.964215i	$0.585412\pi$
$860$	0	0
$861$	0	0
$862$	− 36.8445i	− 1.25493i
$863$	22.2872i	0.758664i	0.925261	+	0.379332i	$0.123846\pi$
$863$	22.2872i	0.758664i	−0.925261	+	0.379332i	$0.876154\pi$
$864$	0	0
$865$	0	0
$866$	−45.0921	−1.53229
$867$	0	0
$868$	1.15549i	0.0392200i
$869$	30.3536	1.02967
$870$	0	0
$871$	−6.07418	−0.205816
$872$	− 20.6340i	− 0.698755i
$873$	0	0
$874$	−63.5407	−2.14929
$875$	0	0
$876$	0	0
$877$	23.5407i	0.794912i	0.917621	+	0.397456i	$0.130107\pi$
$877$	23.5407i	0.794912i	−0.917621	+	0.397456i	$0.869893\pi$
$878$	17.6863i	0.596883i
$879$	0	0
$880$	0	0
$881$	23.0639	0.777042	0.388521	−	0.921440i	$-0.372986\pi$
$881$	23.0639	0.777042	0.388521	+	0.921440i	$0.372986\pi$
$882$	0	0
$883$	19.3852i	0.652363i	0.945307	+	0.326181i	$0.105762\pi$
$883$	19.3852i	0.652363i	−0.945307	+	0.326181i	$0.894238\pi$
$884$	2.27071	0.0763723
$885$	0	0
$886$	45.4665	1.52748
$887$	5.37761i	0.180562i	0.995916	+	0.0902812i	$0.0287766\pi$
$887$	5.37761i	0.180562i	−0.995916	+	0.0902812i	$0.971223\pi$
$888$	0	0
$889$	−15.3852	−0.516002
$890$	0	0
$891$	0	0
$892$	2.77033i	0.0927575i
$893$	− 71.9802i	− 2.40873i
$894$	0	0
$895$	0	0
$896$	9.41082	0.314393
$897$	0	0
$898$	− 18.4223i	− 0.614759i
$899$	−49.9519	−1.66599
$900$	0	0
$901$	−14.4593	−0.481710
$902$	40.7502i	1.35683i
$903$	0	0
$904$	24.5407	0.816210
$905$	0	0
$906$	0	0
$907$	− 42.3110i	− 1.40491i	−0.711727	−	0.702457i	$-0.752086\pi$
$907$	− 42.3110i	− 1.40491i	0.711727	−	0.702457i	$-0.247914\pi$
$908$	− 1.65317i	− 0.0548624i
$909$	0	0
$910$	0	0
$911$	38.9378	1.29007	0.645034	−	0.764154i	$-0.276843\pi$
$911$	38.9378	1.29007	0.645034	+	0.764154i	$0.276843\pi$
$912$	0	0
$913$	− 93.8516i	− 3.10604i
$914$	19.5485	0.646607
$915$	0	0
$916$	−5.30385	−0.175244
$917$	2.68880i	0.0887922i
$918$	0	0
$919$	−25.6962	−0.847638	−0.423819	−	0.905747i	$-0.639311\pi$
$919$	−25.6962	−0.847638	−0.423819	+	0.905747i	$0.639311\pi$
$920$	0	0
$921$	0	0
$922$	18.7703i	0.618168i
$923$	− 1.13536i	− 0.0373707i
$924$	0	0
$925$	0	0
$926$	−17.1685	−0.564191
$927$	0	0
$928$	− 9.03709i	− 0.296657i
$929$	8.06641	0.264650	0.132325	−	0.991206i	$-0.457756\pi$
$929$	8.06641	0.264650	0.132325	+	0.991206i	$0.457756\pi$
$930$	0	0
$931$	8.38516	0.274813
$932$	3.77430i	0.123631i
$933$	0	0
$934$	15.1555	0.495903
$935$	0	0
$936$	0	0
$937$	− 40.7703i	− 1.33191i	−0.745993	−	0.665954i	$-0.768025\pi$
$937$	− 40.7703i	− 1.33191i	0.745993	−	0.665954i	$-0.231975\pi$
$938$	1.86222i	0.0608036i
$939$	0	0
$940$	0	0
$941$	10.2374	0.333730	0.166865	−	0.985980i	$-0.446636\pi$
$941$	10.2374	0.333730	0.166865	+	0.985980i	$0.446636\pi$
$942$	0	0
$943$	30.3110i	0.987062i
$944$	−30.7122	−0.999597
$945$	0	0
$946$	10.4964	0.341268
$947$	50.9876i	1.65687i	0.560083	+	0.828437i	$0.310769\pi$
$947$	50.9876i	1.65687i	−0.560083	+	0.828437i	$0.689231\pi$
$948$	0	0
$949$	−28.0000	−0.908918
$950$	0	0
$951$	0	0
$952$	− 7.92582i	− 0.256877i
$953$	− 36.2490i	− 1.17422i	−0.809507	−	0.587110i	$-0.800266\pi$
$953$	− 36.2490i	− 1.17422i	0.809507	−	0.587110i	$-0.199734\pi$
$954$	0	0
$955$	0	0
$956$	−2.17099	−0.0702148
$957$	0	0
$958$	2.91868i	0.0942983i
$959$	5.37761	0.173652
$960$	0	0
$961$	5.00000	0.161290
$962$	− 17.6863i	− 0.570228i
$963$	0	0
$964$	1.46648	0.0472322
$965$	0	0
$966$	0	0
$967$	− 5.54066i	− 0.178176i	−0.996024	−	0.0890878i	$-0.971605\pi$
$967$	− 5.54066i	− 0.178176i	0.996024	−	0.0890878i	$-0.0283952\pi$
$968$	61.2250i	1.96784i
$969$	0	0
$970$	0	0
$971$	−41.8855	−1.34417	−0.672085	−	0.740474i	$-0.734601\pi$
$971$	−41.8855	−1.34417	−0.672085	+	0.740474i	$0.734601\pi$
$972$	0	0
$973$	− 6.38516i	− 0.204699i
$974$	−3.51539	−0.112640
$975$	0	0
$976$	0	0
$977$	48.0399i	1.53693i	0.639891	+	0.768466i	$0.278979\pi$
$977$	48.0399i	1.53693i	−0.639891	+	0.768466i	$0.721021\pi$
$978$	0	0
$979$	78.6962	2.51514
$980$	0	0
$981$	0	0
$982$	− 14.1113i	− 0.450309i
$983$	32.7835i	1.04563i	0.852446	+	0.522815i	$0.175118\pi$
$983$	32.7835i	1.04563i	−0.852446	+	0.522815i	$0.824882\pi$
$984$	0	0
$985$	0	0
$986$	30.0947	0.958409
$987$	0	0
$988$	7.08132i	0.225287i
$989$	7.80751	0.248264
$990$	0	0
$991$	−35.6962	−1.13393	−0.566963	−	0.823743i	$-0.691882\pi$
$991$	−35.6962	−1.13393	−0.566963	+	0.823743i	$0.691882\pi$
$992$	6.51296i	0.206787i
$993$	0	0
$994$	−0.348077	−0.0110403
$995$	0	0
$996$	0	0
$997$	− 57.1555i	− 1.81013i	−0.425270	−	0.905066i	$-0.639821\pi$
$997$	− 57.1555i	− 1.81013i	0.425270	−	0.905066i	$-0.360179\pi$
$998$	18.2041i	0.576241i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.d.l.1324.6		8
3.2	odd	2	inner	1575.2.d.l.1324.4		8
5.2	odd	4		1575.2.a.y.1.2	✓	4
5.3	odd	4		1575.2.a.z.1.3	yes	4
5.4	even	2	inner	1575.2.d.l.1324.3		8
15.2	even	4		1575.2.a.y.1.3	yes	4
15.8	even	4		1575.2.a.z.1.2	yes	4
15.14	odd	2	inner	1575.2.d.l.1324.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1575.2.a.y.1.2	✓	4	5.2	odd	4
1575.2.a.y.1.3	yes	4	15.2	even	4
1575.2.a.z.1.2	yes	4	15.8	even	4
1575.2.a.z.1.3	yes	4	5.3	odd	4
1575.2.d.l.1324.3		8	5.4	even	2	inner
1575.2.d.l.1324.4		8	3.2	odd	2	inner
1575.2.d.l.1324.5		8	15.14	odd	2	inner
1575.2.d.l.1324.6		8	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.34440i q^{2} +0.192582 q^{4} +1.00000i q^{7} +2.94771i q^{8} +O(q^{10})\)	\(q+1.34440i q^{2} +0.192582 q^{4} +1.00000i q^{7} +2.94771i q^{8} -5.63652 q^{11} -4.38516i q^{13} -1.34440 q^{14} -3.57775 q^{16} +2.68880i q^{17} -8.38516 q^{19} -7.57775i q^{22} -5.63652i q^{23} +5.89543 q^{26} +0.192582i q^{28} -8.32532 q^{29} +6.00000 q^{31} +1.08549i q^{32} -3.61484 q^{34} -3.00000i q^{37} -11.2730i q^{38} -5.37761 q^{41} +1.38516i q^{43} -1.08549 q^{44} +7.57775 q^{46} +8.58423i q^{47} -1.00000 q^{49} -0.844506i q^{52} +5.37761i q^{53} -2.94771 q^{56} -11.1926i q^{58} +8.58423 q^{59} +8.06641i q^{62} -8.61484 q^{64} -1.38516i q^{67} +0.517816i q^{68} +0.258908 q^{71} -6.38516i q^{73} +4.03321 q^{74} -1.61484 q^{76} -5.63652i q^{77} -5.38516 q^{79} -7.22967i q^{82} +16.6506i q^{83} -1.86222 q^{86} -16.6148i q^{88} -13.9618 q^{89} +4.38516 q^{91} -1.08549i q^{92} -11.5407 q^{94} -10.7703i q^{97} -1.34440i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 20 q^{4}+O(q^{10}) \) 8 * q - 20 * q^4	\( 8 q - 20 q^{4} + 36 q^{16} - 24 q^{19} + 48 q^{31} - 72 q^{34} - 4 q^{46} - 8 q^{49} - 112 q^{64} - 56 q^{76} - 8 q^{91} + 80 q^{94}+O(q^{100}) \) 8 * q - 20 * q^4 + 36 * q^16 - 24 * q^19 + 48 * q^31 - 72 * q^34 - 4 * q^46 - 8 * q^49 - 112 * q^64 - 56 * q^76 - 8 * q^91 + 80 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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