LMFDB - Embedded newform 630.2.g.f.379.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [630,2,Mod(379,630)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(630, 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("630.379");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	630.g (of order $2$, degree $1$, 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:	$5.03057532734$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			379.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	630.379
Dual form			630.2.g.f.379.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} -1.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} -1.00000i q^{7} -1.00000i q^{8} +(1.00000 + 2.00000i) q^{10} +4.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +8.00000 q^{19} +(-2.00000 + 1.00000i) q^{20} -8.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -4.00000 q^{26} +1.00000i q^{28} +8.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} +2.00000 q^{34} +(-1.00000 - 2.00000i) q^{35} +8.00000i q^{37} +8.00000i q^{38} +(-1.00000 - 2.00000i) q^{40} -12.0000 q^{41} +8.00000i q^{43} +8.00000 q^{46} -4.00000i q^{47} -1.00000 q^{49} +(4.00000 + 3.00000i) q^{50} -4.00000i q^{52} +6.00000i q^{53} -1.00000 q^{56} +8.00000i q^{58} +8.00000 q^{59} -6.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +(4.00000 + 8.00000i) q^{65} -8.00000i q^{67} +2.00000i q^{68} +(2.00000 - 1.00000i) q^{70} -4.00000i q^{73} -8.00000 q^{74} -8.00000 q^{76} -8.00000 q^{79} +(2.00000 - 1.00000i) q^{80} -12.0000i q^{82} +(-2.00000 - 4.00000i) q^{85} -8.00000 q^{86} -4.00000 q^{89} +4.00000 q^{91} +8.00000i q^{92} +4.00000 q^{94} +(16.0000 - 8.00000i) q^{95} +12.0000i q^{97} -1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 4 q^{5}+O(q^{10}) $ 2 * q - 2 * q^4 + 4 * q^5	$ 2 q - 2 q^{4} + 4 q^{5} + 2 q^{10} + 2 q^{14} + 2 q^{16} + 16 q^{19} - 4 q^{20} + 6 q^{25} - 8 q^{26} + 16 q^{29} + 8 q^{31} + 4 q^{34} - 2 q^{35} - 2 q^{40} - 24 q^{41} + 16 q^{46} - 2 q^{49} + 8 q^{50} - 2 q^{56} + 16 q^{59} - 12 q^{61} - 2 q^{64} + 8 q^{65} + 4 q^{70} - 16 q^{74} - 16 q^{76} - 16 q^{79} + 4 q^{80} - 4 q^{85} - 16 q^{86} - 8 q^{89} + 8 q^{91} + 8 q^{94} + 32 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 + 4 * q^5 + 2 * q^10 + 2 * q^14 + 2 * q^16 + 16 * q^19 - 4 * q^20 + 6 * q^25 - 8 * q^26 + 16 * q^29 + 8 * q^31 + 4 * q^34 - 2 * q^35 - 2 * q^40 - 24 * q^41 + 16 * q^46 - 2 * q^49 + 8 * q^50 - 2 * q^56 + 16 * q^59 - 12 * q^61 - 2 * q^64 + 8 * q^65 + 4 * q^70 - 16 * q^74 - 16 * q^76 - 16 * q^79 + 4 * q^80 - 4 * q^85 - 16 * q^86 - 8 * q^89 + 8 * q^91 + 8 * q^94 + 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/630\mathbb{Z}\right)^\times$.

$n$	$127$	$281$	$451$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	1.00000	+	2.00000i	0.316228	+	0.632456i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	1.00000			0.267261
$15$		0			0
$16$	1.00000			0.250000
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$	8.00000			1.83533			0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000			1.83533			0.917663	+	0.397360i	$0.130073\pi$
$20$	−2.00000	+	1.00000i	−0.447214	+	0.223607i
$21$		0			0
$22$		0			0
$23$		−	8.00000i		−	1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$		−	8.00000i		−	1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$	−4.00000			−0.784465
$27$		0			0
$28$			1.00000i			0.188982i
$29$	8.00000			1.48556			0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000			1.48556			0.742781	+	0.669534i	$0.233506\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	2.00000			0.342997
$35$	−1.00000	−	2.00000i	−0.169031	−	0.338062i
$36$		0			0
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$			8.00000i			1.29777i
$39$		0			0
$40$	−1.00000	−	2.00000i	−0.158114	−	0.316228i
$41$	−12.0000			−1.87409			−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000			−1.87409			−0.937043	+	0.349215i	$0.886448\pi$
$42$		0			0
$43$			8.00000i			1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$			8.00000i			1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$		0			0
$45$		0			0
$46$	8.00000			1.17954
$47$		−	4.00000i		−	0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$		−	4.00000i		−	0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$	4.00000	+	3.00000i	0.565685	+	0.424264i
$51$		0			0
$52$		−	4.00000i		−	0.554700i
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$		0			0
$56$	−1.00000			−0.133631
$57$		0			0
$58$			8.00000i			1.05045i
$59$	8.00000			1.04151			0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000			1.04151			0.520756	+	0.853706i	$0.325650\pi$
$60$		0			0
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$			4.00000i			0.508001i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	4.00000	+	8.00000i	0.496139	+	0.992278i
$66$		0			0
$67$		−	8.00000i		−	0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$		−	8.00000i		−	0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$			2.00000i			0.242536i
$69$		0			0
$70$	2.00000	−	1.00000i	0.239046	−	0.119523i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−8.00000			−0.929981
$75$		0			0
$76$	−8.00000			−0.917663
$77$		0			0
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$	2.00000	−	1.00000i	0.223607	−	0.111803i
$81$		0			0
$82$		−	12.0000i		−	1.32518i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	−2.00000	−	4.00000i	−0.216930	−	0.433861i
$86$	−8.00000			−0.862662
$87$		0			0
$88$		0			0
$89$	−4.00000			−0.423999			−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000			−0.423999			−0.212000	+	0.977270i	$0.567998\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$			8.00000i			0.834058i
$93$		0			0
$94$	4.00000			0.412568
$95$	16.0000	−	8.00000i	1.64157	−	0.820783i
$96$		0			0
$97$			12.0000i			1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$			12.0000i			1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$		−	1.00000i		−	0.101015i
$99$		0			0
$100$	−3.00000	+	4.00000i	−0.300000	+	0.400000i
$101$	−12.0000			−1.19404			−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000			−1.19404			−0.597022	+	0.802225i	$0.703650\pi$
$102$		0			0
$103$		−	8.00000i		−	0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$		−	8.00000i		−	0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	4.00000			0.392232
$105$		0			0
$106$	−6.00000			−0.582772
$107$		−	4.00000i		−	0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$		−	4.00000i		−	0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$		0			0
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$		0			0
$112$		−	1.00000i		−	0.0944911i
$113$		−	2.00000i		−	0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$		−	2.00000i		−	0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$		0			0
$115$	−8.00000	−	16.0000i	−0.746004	−	1.49201i
$116$	−8.00000			−0.742781
$117$		0			0
$118$			8.00000i			0.736460i
$119$	−2.00000			−0.183340
$120$		0			0
$121$	−11.0000			−1.00000
$122$		−	6.00000i		−	0.543214i
$123$		0			0
$124$	−4.00000			−0.359211
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		0			0
$127$			8.00000i			0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$			8.00000i			0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−8.00000	+	4.00000i	−0.701646	+	0.350823i
$131$	−8.00000			−0.698963			−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000			−0.698963			−0.349482	+	0.936943i	$0.613642\pi$
$132$		0			0
$133$		−	8.00000i		−	0.693688i
$134$	8.00000			0.691095
$135$		0			0
$136$	−2.00000			−0.171499
$137$			10.0000i			0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$			10.0000i			0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$		0			0
$139$	−16.0000			−1.35710			−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000			−1.35710			−0.678551	+	0.734553i	$0.737392\pi$
$140$	1.00000	+	2.00000i	0.0845154	+	0.169031i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	16.0000	−	8.00000i	1.32873	−	0.664364i
$146$	4.00000			0.331042
$147$		0			0
$148$		−	8.00000i		−	0.657596i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		−	8.00000i		−	0.648886i
$153$		0			0
$154$		0			0
$155$	8.00000	−	4.00000i	0.642575	−	0.321288i
$156$		0			0
$157$		−	12.0000i		−	0.957704i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$		−	12.0000i		−	0.957704i	0.877896	−	0.478852i	$-0.158947\pi$
$158$		−	8.00000i		−	0.636446i
$159$		0			0
$160$	1.00000	+	2.00000i	0.0790569	+	0.158114i
$161$	−8.00000			−0.630488
$162$		0			0
$163$			24.0000i			1.87983i	0.341415	+	0.939913i	$0.389094\pi$
$163$			24.0000i			1.87983i	−0.341415	+	0.939913i	$0.610906\pi$
$164$	12.0000			0.937043
$165$		0			0
$166$		0			0
$167$		−	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
$169$	−3.00000			−0.230769
$170$	4.00000	−	2.00000i	0.306786	−	0.153393i
$171$		0			0
$172$		−	8.00000i		−	0.609994i
$173$		−	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
$175$	−4.00000	−	3.00000i	−0.302372	−	0.226779i
$176$		0			0
$177$		0			0
$178$		−	4.00000i		−	0.299813i
$179$	16.0000			1.19590			0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000			1.19590			0.597948	+	0.801535i	$0.295983\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$			4.00000i			0.296500i
$183$		0			0
$184$	−8.00000			−0.589768
$185$	8.00000	+	16.0000i	0.588172	+	1.17634i
$186$		0			0
$187$		0			0
$188$			4.00000i			0.291730i
$189$		0			0
$190$	8.00000	+	16.0000i	0.580381	+	1.16076i
$191$	−16.0000			−1.15772			−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000			−1.15772			−0.578860	+	0.815427i	$0.696502\pi$
$192$		0			0
$193$		−	16.0000i		−	1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$		−	16.0000i		−	1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$	−12.0000			−0.861550
$195$		0			0
$196$	1.00000			0.0714286
$197$			22.0000i			1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$			22.0000i			1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$		0			0
$199$	12.0000			0.850657			0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000			0.850657			0.425329	+	0.905039i	$0.360158\pi$
$200$	−4.00000	−	3.00000i	−0.282843	−	0.212132i
$201$		0			0
$202$		−	12.0000i		−	0.844317i
$203$		−	8.00000i		−	0.561490i
$204$		0			0
$205$	−24.0000	+	12.0000i	−1.67623	+	0.838116i
$206$	8.00000			0.557386
$207$		0			0
$208$			4.00000i			0.277350i
$209$		0			0
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$		−	6.00000i		−	0.412082i
$213$		0			0
$214$	4.00000			0.273434
$215$	8.00000	+	16.0000i	0.545595	+	1.09119i
$216$		0			0
$217$		−	4.00000i		−	0.271538i
$218$		−	2.00000i		−	0.135457i
$219$		0			0
$220$		0			0
$221$	8.00000			0.538138
$222$		0			0
$223$		−	8.00000i		−	0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$		−	8.00000i		−	0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$	2.00000			0.133038
$227$		−	24.0000i		−	1.59294i	−0.604681	−	0.796468i	$-0.706699\pi$
$227$		−	24.0000i		−	1.59294i	0.604681	−	0.796468i	$-0.293301\pi$
$228$		0			0
$229$	2.00000			0.132164			0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000			0.132164			0.0660819	+	0.997814i	$0.478950\pi$
$230$	16.0000	−	8.00000i	1.05501	−	0.527504i
$231$		0			0
$232$		−	8.00000i		−	0.525226i
$233$			26.0000i			1.70332i	0.524097	+	0.851658i	$0.324403\pi$
$233$			26.0000i			1.70332i	−0.524097	+	0.851658i	$0.675597\pi$
$234$		0			0
$235$	−4.00000	−	8.00000i	−0.260931	−	0.521862i
$236$	−8.00000			−0.520756
$237$		0			0
$238$		−	2.00000i		−	0.129641i
$239$	−16.0000			−1.03495			−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000			−1.03495			−0.517477	+	0.855697i	$0.673129\pi$
$240$		0			0
$241$	18.0000			1.15948			0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000			1.15948			0.579741	+	0.814801i	$0.303154\pi$
$242$		−	11.0000i		−	0.707107i
$243$		0			0
$244$	6.00000			0.384111
$245$	−2.00000	+	1.00000i	−0.127775	+	0.0638877i
$246$		0			0
$247$			32.0000i			2.03611i
$248$		−	4.00000i		−	0.254000i
$249$		0			0
$250$	11.0000	+	2.00000i	0.695701	+	0.126491i
$251$	−8.00000			−0.504956			−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000			−0.504956			−0.252478	+	0.967603i	$0.581245\pi$
$252$		0			0
$253$		0			0
$254$	−8.00000			−0.501965
$255$		0			0
$256$	1.00000			0.0625000
$257$			18.0000i			1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$			18.0000i			1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$		0			0
$259$	8.00000			0.497096
$260$	−4.00000	−	8.00000i	−0.248069	−	0.496139i
$261$		0			0
$262$		−	8.00000i		−	0.494242i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	6.00000	+	12.0000i	0.368577	+	0.737154i
$266$	8.00000			0.490511
$267$		0			0
$268$			8.00000i			0.488678i
$269$	−28.0000			−1.70719			−0.853595	−	0.520937i	$-0.825583\pi$
$269$	−28.0000			−1.70719			−0.853595	+	0.520937i	$0.825583\pi$
$270$		0			0
$271$	−12.0000			−0.728948			−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000			−0.728948			−0.364474	+	0.931214i	$0.618751\pi$
$272$		−	2.00000i		−	0.121268i
$273$		0			0
$274$	−10.0000			−0.604122
$275$		0			0
$276$		0			0
$277$			32.0000i			1.92269i	0.275340	+	0.961347i	$0.411209\pi$
$277$			32.0000i			1.92269i	−0.275340	+	0.961347i	$0.588791\pi$
$278$		−	16.0000i		−	0.959616i
$279$		0			0
$280$	−2.00000	+	1.00000i	−0.119523	+	0.0597614i
$281$	−24.0000			−1.43172			−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000			−1.43172			−0.715860	+	0.698244i	$0.753965\pi$
$282$		0			0
$283$		−	16.0000i		−	0.951101i	−0.879688	−	0.475551i	$-0.842249\pi$
$283$		−	16.0000i		−	0.951101i	0.879688	−	0.475551i	$-0.157751\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			12.0000i			0.708338i
$288$		0			0
$289$	13.0000			0.764706
$290$	8.00000	+	16.0000i	0.469776	+	0.939552i
$291$		0			0
$292$			4.00000i			0.234082i
$293$			18.0000i			1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$			18.0000i			1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$		0			0
$295$	16.0000	−	8.00000i	0.931556	−	0.465778i
$296$	8.00000			0.464991
$297$		0			0
$298$		0			0
$299$	32.0000			1.85061
$300$		0			0
$301$	8.00000			0.461112
$302$		0			0
$303$		0			0
$304$	8.00000			0.458831
$305$	−12.0000	+	6.00000i	−0.687118	+	0.343559i
$306$		0			0
$307$			16.0000i			0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$			16.0000i			0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$		0			0
$309$		0			0
$310$	4.00000	+	8.00000i	0.227185	+	0.454369i
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	28.0000i		−	1.58265i	−0.611393	−	0.791327i	$-0.709391\pi$
$313$		−	28.0000i		−	1.58265i	0.611393	−	0.791327i	$-0.290609\pi$
$314$	12.0000			0.677199
$315$		0			0
$316$	8.00000			0.450035
$317$		−	14.0000i		−	0.786318i	−0.919470	−	0.393159i	$-0.871382\pi$
$317$		−	14.0000i		−	0.786318i	0.919470	−	0.393159i	$-0.128618\pi$
$318$		0			0
$319$		0			0
$320$	−2.00000	+	1.00000i	−0.111803	+	0.0559017i
$321$		0			0
$322$		−	8.00000i		−	0.445823i
$323$		−	16.0000i		−	0.890264i
$324$		0			0
$325$	16.0000	+	12.0000i	0.887520	+	0.665640i
$326$	−24.0000			−1.32924
$327$		0			0
$328$			12.0000i			0.662589i
$329$	−4.00000			−0.220527
$330$		0			0
$331$	20.0000			1.09930			0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000			1.09930			0.549650	+	0.835395i	$0.314761\pi$
$332$		0			0
$333$		0			0
$334$	12.0000			0.656611
$335$	−8.00000	−	16.0000i	−0.437087	−	0.874173i
$336$		0			0
$337$			8.00000i			0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$			8.00000i			0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$		−	3.00000i		−	0.163178i
$339$		0			0
$340$	2.00000	+	4.00000i	0.108465	+	0.216930i
$341$		0			0
$342$		0			0
$343$			1.00000i			0.0539949i
$344$	8.00000			0.431331
$345$		0			0
$346$	6.00000			0.322562
$347$			36.0000i			1.93258i	0.257454	+	0.966291i	$0.417117\pi$
$347$			36.0000i			1.93258i	−0.257454	+	0.966291i	$0.582883\pi$
$348$		0			0
$349$	6.00000			0.321173			0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000			0.321173			0.160586	+	0.987022i	$0.448662\pi$
$350$	3.00000	−	4.00000i	0.160357	−	0.213809i
$351$		0			0
$352$		0			0
$353$			18.0000i			0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$			18.0000i			0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$		0			0
$355$		0			0
$356$	4.00000			0.212000
$357$		0			0
$358$			16.0000i			0.845626i
$359$	16.0000			0.844448			0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000			0.844448			0.422224	+	0.906492i	$0.361250\pi$
$360$		0			0
$361$	45.0000			2.36842
$362$			2.00000i			0.105118i
$363$		0			0
$364$	−4.00000			−0.209657
$365$	−4.00000	−	8.00000i	−0.209370	−	0.418739i
$366$		0			0
$367$			8.00000i			0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$			8.00000i			0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$		−	8.00000i		−	0.417029i
$369$		0			0
$370$	−16.0000	+	8.00000i	−0.831800	+	0.415900i
$371$	6.00000			0.311504
$372$		0			0
$373$		−	16.0000i		−	0.828449i	−0.910175	−	0.414224i	$-0.864053\pi$
$373$		−	16.0000i		−	0.828449i	0.910175	−	0.414224i	$-0.135947\pi$
$374$		0			0
$375$		0			0
$376$	−4.00000			−0.206284
$377$			32.0000i			1.64808i
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$	−16.0000	+	8.00000i	−0.820783	+	0.410391i
$381$		0			0
$382$		−	16.0000i		−	0.818631i
$383$			12.0000i			0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$			12.0000i			0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$		0			0
$385$		0			0
$386$	16.0000			0.814379
$387$		0			0
$388$		−	12.0000i		−	0.609208i
$389$	24.0000			1.21685			0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000			1.21685			0.608424	+	0.793612i	$0.291802\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$			1.00000i			0.0505076i
$393$		0			0
$394$	−22.0000			−1.10834
$395$	−16.0000	+	8.00000i	−0.805047	+	0.402524i
$396$		0			0
$397$		−	4.00000i		−	0.200754i	−0.994949	−	0.100377i	$-0.967995\pi$
$397$		−	4.00000i		−	0.200754i	0.994949	−	0.100377i	$-0.0320049\pi$
$398$			12.0000i			0.601506i
$399$		0			0
$400$	3.00000	−	4.00000i	0.150000	−	0.200000i
$401$	24.0000			1.19850			0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000			1.19850			0.599251	+	0.800561i	$0.295465\pi$
$402$		0			0
$403$			16.0000i			0.797017i
$404$	12.0000			0.597022
$405$		0			0
$406$	8.00000			0.397033
$407$		0			0
$408$		0			0
$409$	−22.0000			−1.08783			−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000			−1.08783			−0.543915	+	0.839140i	$0.683059\pi$
$410$	−12.0000	−	24.0000i	−0.592638	−	1.18528i
$411$		0			0
$412$			8.00000i			0.394132i
$413$		−	8.00000i		−	0.393654i
$414$		0			0
$415$		0			0
$416$	−4.00000			−0.196116
$417$		0			0
$418$		0			0
$419$	−40.0000			−1.95413			−0.977064	−	0.212946i	$-0.931694\pi$
$419$	−40.0000			−1.95413			−0.977064	+	0.212946i	$0.931694\pi$
$420$		0			0
$421$	−26.0000			−1.26716			−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$		−	12.0000i		−	0.584151i
$423$		0			0
$424$	6.00000			0.291386
$425$	−8.00000	−	6.00000i	−0.388057	−	0.291043i
$426$		0			0
$427$			6.00000i			0.290360i
$428$			4.00000i			0.193347i
$429$		0			0
$430$	−16.0000	+	8.00000i	−0.771589	+	0.385794i
$431$	16.0000			0.770693			0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000			0.770693			0.385346	+	0.922772i	$0.374082\pi$
$432$		0			0
$433$			4.00000i			0.192228i	0.995370	+	0.0961139i	$0.0306413\pi$
$433$			4.00000i			0.192228i	−0.995370	+	0.0961139i	$0.969359\pi$
$434$	4.00000			0.192006
$435$		0			0
$436$	2.00000			0.0957826
$437$		−	64.0000i		−	3.06154i
$438$		0			0
$439$	28.0000			1.33637			0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000			1.33637			0.668184	+	0.743996i	$0.267072\pi$
$440$		0			0
$441$		0			0
$442$			8.00000i			0.380521i
$443$		−	12.0000i		−	0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$		−	12.0000i		−	0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$		0			0
$445$	−8.00000	+	4.00000i	−0.379236	+	0.189618i
$446$	8.00000			0.378811
$447$		0			0
$448$			1.00000i			0.0472456i
$449$	−16.0000			−0.755087			−0.377543	−	0.925992i	$-0.623231\pi$
$449$	−16.0000			−0.755087			−0.377543	+	0.925992i	$0.623231\pi$
$450$		0			0
$451$		0			0
$452$			2.00000i			0.0940721i
$453$		0			0
$454$	24.0000			1.12638
$455$	8.00000	−	4.00000i	0.375046	−	0.187523i
$456$		0			0
$457$			16.0000i			0.748448i	0.927338	+	0.374224i	$0.122091\pi$
$457$			16.0000i			0.748448i	−0.927338	+	0.374224i	$0.877909\pi$
$458$			2.00000i			0.0934539i
$459$		0			0
$460$	8.00000	+	16.0000i	0.373002	+	0.746004i
$461$	20.0000			0.931493			0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000			0.931493			0.465746	+	0.884918i	$0.345786\pi$
$462$		0			0
$463$		−	8.00000i		−	0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$		−	8.00000i		−	0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$	8.00000			0.371391
$465$		0			0
$466$	−26.0000			−1.20443
$467$		−	32.0000i		−	1.48078i	−0.672176	−	0.740392i	$-0.734640\pi$
$467$		−	32.0000i		−	1.48078i	0.672176	−	0.740392i	$-0.265360\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$	8.00000	−	4.00000i	0.369012	−	0.184506i
$471$		0			0
$472$		−	8.00000i		−	0.368230i
$473$		0			0
$474$		0			0
$475$	24.0000	−	32.0000i	1.10120	−	1.46826i
$476$	2.00000			0.0916698
$477$		0			0
$478$		−	16.0000i		−	0.731823i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−32.0000			−1.45907
$482$			18.0000i			0.819878i
$483$		0			0
$484$	11.0000			0.500000
$485$	12.0000	+	24.0000i	0.544892	+	1.08978i
$486$		0			0
$487$		−	40.0000i		−	1.81257i	−0.422664	−	0.906287i	$-0.638905\pi$
$487$		−	40.0000i		−	1.81257i	0.422664	−	0.906287i	$-0.361095\pi$
$488$			6.00000i			0.271607i
$489$		0			0
$490$	−1.00000	−	2.00000i	−0.0451754	−	0.0903508i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		−	16.0000i		−	0.720604i
$494$	−32.0000			−1.43975
$495$		0			0
$496$	4.00000			0.179605
$497$		0			0
$498$		0			0
$499$	−28.0000			−1.25345			−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000			−1.25345			−0.626726	+	0.779240i	$0.715605\pi$
$500$	−2.00000	+	11.0000i	−0.0894427	+	0.491935i
$501$		0			0
$502$		−	8.00000i		−	0.357057i
$503$		−	28.0000i		−	1.24846i	−0.781241	−	0.624229i	$-0.785413\pi$
$503$		−	28.0000i		−	1.24846i	0.781241	−	0.624229i	$-0.214587\pi$
$504$		0			0
$505$	−24.0000	+	12.0000i	−1.06799	+	0.533993i
$506$		0			0
$507$		0			0
$508$		−	8.00000i		−	0.354943i
$509$	−4.00000			−0.177297			−0.0886484	−	0.996063i	$-0.528255\pi$
$509$	−4.00000			−0.177297			−0.0886484	+	0.996063i	$0.528255\pi$
$510$		0			0
$511$	−4.00000			−0.176950
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−18.0000			−0.793946
$515$	−8.00000	−	16.0000i	−0.352522	−	0.705044i
$516$		0			0
$517$		0			0
$518$			8.00000i			0.351500i
$519$		0			0
$520$	8.00000	−	4.00000i	0.350823	−	0.175412i
$521$	−12.0000			−0.525730			−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000			−0.525730			−0.262865	+	0.964833i	$0.584667\pi$
$522$		0			0
$523$			16.0000i			0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$			16.0000i			0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	8.00000			0.349482
$525$		0			0
$526$		0			0
$527$		−	8.00000i		−	0.348485i
$528$		0			0
$529$	−41.0000			−1.78261
$530$	−12.0000	+	6.00000i	−0.521247	+	0.260623i
$531$		0			0
$532$			8.00000i			0.346844i
$533$		−	48.0000i		−	2.07911i
$534$		0			0
$535$	−4.00000	−	8.00000i	−0.172935	−	0.345870i
$536$	−8.00000			−0.345547
$537$		0			0
$538$		−	28.0000i		−	1.20717i
$539$		0			0
$540$		0			0
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$		−	12.0000i		−	0.515444i
$543$		0			0
$544$	2.00000			0.0857493
$545$	−4.00000	+	2.00000i	−0.171341	+	0.0856706i
$546$		0			0
$547$			8.00000i			0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$			8.00000i			0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$		−	10.0000i		−	0.427179i
$549$		0			0
$550$		0			0
$551$	64.0000			2.72649
$552$		0			0
$553$			8.00000i			0.340195i
$554$	−32.0000			−1.35955
$555$		0			0
$556$	16.0000			0.678551
$557$			2.00000i			0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$			2.00000i			0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$		0			0
$559$	−32.0000			−1.35346
$560$	−1.00000	−	2.00000i	−0.0422577	−	0.0845154i
$561$		0			0
$562$		−	24.0000i		−	1.01238i
$563$			40.0000i			1.68580i	0.538071	+	0.842900i	$0.319153\pi$
$563$			40.0000i			1.68580i	−0.538071	+	0.842900i	$0.680847\pi$
$564$		0			0
$565$	−2.00000	−	4.00000i	−0.0841406	−	0.168281i
$566$	16.0000			0.672530
$567$		0			0
$568$		0			0
$569$	24.0000			1.00613			0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000			1.00613			0.503066	+	0.864248i	$0.332205\pi$
$570$		0			0
$571$	20.0000			0.836974			0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000			0.836974			0.418487	+	0.908223i	$0.362561\pi$
$572$		0			0
$573$		0			0
$574$	−12.0000			−0.500870
$575$	−32.0000	−	24.0000i	−1.33449	−	1.00087i
$576$		0			0
$577$			28.0000i			1.16566i	0.812596	+	0.582828i	$0.198054\pi$
$577$			28.0000i			1.16566i	−0.812596	+	0.582828i	$0.801946\pi$
$578$			13.0000i			0.540729i
$579$		0			0
$580$	−16.0000	+	8.00000i	−0.664364	+	0.332182i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−4.00000			−0.165521
$585$		0			0
$586$	−18.0000			−0.743573
$587$			16.0000i			0.660391i	0.943913	+	0.330195i	$0.107115\pi$
$587$			16.0000i			0.660391i	−0.943913	+	0.330195i	$0.892885\pi$
$588$		0			0
$589$	32.0000			1.31854
$590$	8.00000	+	16.0000i	0.329355	+	0.658710i
$591$		0			0
$592$			8.00000i			0.328798i
$593$			30.0000i			1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$			30.0000i			1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$		0			0
$595$	−4.00000	+	2.00000i	−0.163984	+	0.0819920i
$596$		0			0
$597$		0			0
$598$			32.0000i			1.30858i
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$			8.00000i			0.326056i
$603$		0			0
$604$		0			0
$605$	−22.0000	+	11.0000i	−0.894427	+	0.447214i
$606$		0			0
$607$		−	8.00000i		−	0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$		−	8.00000i		−	0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$			8.00000i			0.324443i
$609$		0			0
$610$	−6.00000	−	12.0000i	−0.242933	−	0.485866i
$611$	16.0000			0.647291
$612$		0			0
$613$		−	24.0000i		−	0.969351i	−0.874694	−	0.484675i	$-0.838938\pi$
$613$		−	24.0000i		−	0.969351i	0.874694	−	0.484675i	$-0.161062\pi$
$614$	−16.0000			−0.645707
$615$		0			0
$616$		0			0
$617$			38.0000i			1.52982i	0.644136	+	0.764911i	$0.277217\pi$
$617$			38.0000i			1.52982i	−0.644136	+	0.764911i	$0.722783\pi$
$618$		0			0
$619$	−16.0000			−0.643094			−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000			−0.643094			−0.321547	+	0.946894i	$0.604203\pi$
$620$	−8.00000	+	4.00000i	−0.321288	+	0.160644i
$621$		0			0
$622$		0			0
$623$			4.00000i			0.160257i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	28.0000			1.11911
$627$		0			0
$628$			12.0000i			0.478852i
$629$	16.0000			0.637962
$630$		0			0
$631$	−24.0000			−0.955425			−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000			−0.955425			−0.477712	+	0.878516i	$0.658534\pi$
$632$			8.00000i			0.318223i
$633$		0			0
$634$	14.0000			0.556011
$635$	8.00000	+	16.0000i	0.317470	+	0.634941i
$636$		0			0
$637$		−	4.00000i		−	0.158486i
$638$		0			0
$639$		0			0
$640$	−1.00000	−	2.00000i	−0.0395285	−	0.0790569i
$641$	8.00000			0.315981			0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000			0.315981			0.157991	+	0.987441i	$0.449498\pi$
$642$		0			0
$643$			16.0000i			0.630978i	0.948929	+	0.315489i	$0.102169\pi$
$643$			16.0000i			0.630978i	−0.948929	+	0.315489i	$0.897831\pi$
$644$	8.00000			0.315244
$645$		0			0
$646$	16.0000			0.629512
$647$		−	36.0000i		−	1.41531i	−0.706560	−	0.707653i	$-0.749754\pi$
$647$		−	36.0000i		−	1.41531i	0.706560	−	0.707653i	$-0.250246\pi$
$648$		0			0
$649$		0			0
$650$	−12.0000	+	16.0000i	−0.470679	+	0.627572i
$651$		0			0
$652$		−	24.0000i		−	0.939913i
$653$		−	18.0000i		−	0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$		−	18.0000i		−	0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$		0			0
$655$	−16.0000	+	8.00000i	−0.625172	+	0.312586i
$656$	−12.0000			−0.468521
$657$		0			0
$658$		−	4.00000i		−	0.155936i
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−14.0000			−0.544537			−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000			−0.544537			−0.272268	+	0.962221i	$0.587774\pi$
$662$			20.0000i			0.777322i
$663$		0			0
$664$		0			0
$665$	−8.00000	−	16.0000i	−0.310227	−	0.620453i
$666$		0			0
$667$		−	64.0000i		−	2.47809i
$668$			12.0000i			0.464294i
$669$		0			0
$670$	16.0000	−	8.00000i	0.618134	−	0.309067i
$671$		0			0
$672$		0			0
$673$		−	40.0000i		−	1.54189i	−0.636904	−	0.770943i	$-0.719785\pi$
$673$		−	40.0000i		−	1.54189i	0.636904	−	0.770943i	$-0.280215\pi$
$674$	−8.00000			−0.308148
$675$		0			0
$676$	3.00000			0.115385
$677$		−	14.0000i		−	0.538064i	−0.963131	−	0.269032i	$-0.913296\pi$
$677$		−	14.0000i		−	0.538064i	0.963131	−	0.269032i	$-0.0867037\pi$
$678$		0			0
$679$	12.0000			0.460518
$680$	−4.00000	+	2.00000i	−0.153393	+	0.0766965i
$681$		0			0
$682$		0			0
$683$		−	4.00000i		−	0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$		−	4.00000i		−	0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$		0			0
$685$	10.0000	+	20.0000i	0.382080	+	0.764161i
$686$	−1.00000			−0.0381802
$687$		0			0
$688$			8.00000i			0.304997i
$689$	−24.0000			−0.914327
$690$		0			0
$691$	−32.0000			−1.21734			−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000			−1.21734			−0.608669	+	0.793424i	$0.708296\pi$
$692$			6.00000i			0.228086i
$693$		0			0
$694$	−36.0000			−1.36654
$695$	−32.0000	+	16.0000i	−1.21383	+	0.606915i
$696$		0			0
$697$			24.0000i			0.909065i
$698$			6.00000i			0.227103i
$699$		0			0
$700$	4.00000	+	3.00000i	0.151186	+	0.113389i
$701$	40.0000			1.51078			0.755390	−	0.655276i	$-0.227448\pi$
$701$	40.0000			1.51078			0.755390	+	0.655276i	$0.227448\pi$
$702$		0			0
$703$			64.0000i			2.41381i
$704$		0			0
$705$		0			0
$706$	−18.0000			−0.677439
$707$			12.0000i			0.451306i
$708$		0			0
$709$	42.0000			1.57734			0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000			1.57734			0.788672	+	0.614815i	$0.210769\pi$
$710$		0			0
$711$		0			0
$712$			4.00000i			0.149906i
$713$		−	32.0000i		−	1.19841i
$714$		0			0
$715$		0			0
$716$	−16.0000			−0.597948
$717$		0			0
$718$			16.0000i			0.597115i
$719$	−16.0000			−0.596699			−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000			−0.596699			−0.298350	+	0.954457i	$0.596436\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$			45.0000i			1.67473i
$723$		0			0
$724$	−2.00000			−0.0743294
$725$	24.0000	−	32.0000i	0.891338	−	1.18845i
$726$		0			0
$727$		−	40.0000i		−	1.48352i	−0.670667	−	0.741759i	$-0.733992\pi$
$727$		−	40.0000i		−	1.48352i	0.670667	−	0.741759i	$-0.266008\pi$
$728$		−	4.00000i		−	0.148250i
$729$		0			0
$730$	8.00000	−	4.00000i	0.296093	−	0.148047i
$731$	16.0000			0.591781
$732$		0			0
$733$			20.0000i			0.738717i	0.929287	+	0.369358i	$0.120423\pi$
$733$			20.0000i			0.738717i	−0.929287	+	0.369358i	$0.879577\pi$
$734$	−8.00000			−0.295285
$735$		0			0
$736$	8.00000			0.294884
$737$		0			0
$738$		0			0
$739$	−4.00000			−0.147142			−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000			−0.147142			−0.0735712	+	0.997290i	$0.523440\pi$
$740$	−8.00000	−	16.0000i	−0.294086	−	0.588172i
$741$		0			0
$742$			6.00000i			0.220267i
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$	16.0000			0.585802
$747$		0			0
$748$		0			0
$749$	−4.00000			−0.146157
$750$		0			0
$751$	40.0000			1.45962			0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000			1.45962			0.729810	+	0.683650i	$0.239608\pi$
$752$		−	4.00000i		−	0.145865i
$753$		0			0
$754$	−32.0000			−1.16537
$755$		0			0
$756$		0			0
$757$		−	24.0000i		−	0.872295i	−0.899875	−	0.436147i	$-0.856343\pi$
$757$		−	24.0000i		−	0.872295i	0.899875	−	0.436147i	$-0.143657\pi$
$758$			20.0000i			0.726433i
$759$		0			0
$760$	−8.00000	−	16.0000i	−0.290191	−	0.580381i
$761$	−4.00000			−0.145000			−0.0724999	−	0.997368i	$-0.523098\pi$
$761$	−4.00000			−0.145000			−0.0724999	+	0.997368i	$0.523098\pi$
$762$		0			0
$763$			2.00000i			0.0724049i
$764$	16.0000			0.578860
$765$		0			0
$766$	−12.0000			−0.433578
$767$			32.0000i			1.15545i
$768$		0			0
$769$	2.00000			0.0721218			0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000			0.0721218			0.0360609	+	0.999350i	$0.488519\pi$
$770$		0			0
$771$		0			0
$772$			16.0000i			0.575853i
$773$		−	18.0000i		−	0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$		−	18.0000i		−	0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$		0			0
$775$	12.0000	−	16.0000i	0.431053	−	0.574737i
$776$	12.0000			0.430775
$777$		0			0
$778$			24.0000i			0.860442i
$779$	−96.0000			−3.43956
$780$		0			0
$781$		0			0
$782$		−	16.0000i		−	0.572159i
$783$		0			0
$784$	−1.00000			−0.0357143
$785$	−12.0000	−	24.0000i	−0.428298	−	0.856597i
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		−	22.0000i		−	0.783718i
$789$		0			0
$790$	−8.00000	−	16.0000i	−0.284627	−	0.569254i
$791$	−2.00000			−0.0711118
$792$		0			0
$793$		−	24.0000i		−	0.852265i
$794$	4.00000			0.141955
$795$		0			0
$796$	−12.0000			−0.425329
$797$		−	26.0000i		−	0.920967i	−0.887668	−	0.460484i	$-0.847676\pi$
$797$		−	26.0000i		−	0.920967i	0.887668	−	0.460484i	$-0.152324\pi$
$798$		0			0
$799$	−8.00000			−0.283020
$800$	4.00000	+	3.00000i	0.141421	+	0.106066i
$801$		0			0
$802$			24.0000i			0.847469i
$803$		0			0
$804$		0			0
$805$	−16.0000	+	8.00000i	−0.563926	+	0.281963i
$806$	−16.0000			−0.563576
$807$		0			0
$808$			12.0000i			0.422159i
$809$	−32.0000			−1.12506			−0.562530	−	0.826777i	$-0.690172\pi$
$809$	−32.0000			−1.12506			−0.562530	+	0.826777i	$0.690172\pi$
$810$		0			0
$811$	24.0000			0.842754			0.421377	−	0.906886i	$-0.361547\pi$
$811$	24.0000			0.842754			0.421377	+	0.906886i	$0.361547\pi$
$812$			8.00000i			0.280745i
$813$		0			0
$814$		0			0
$815$	24.0000	+	48.0000i	0.840683	+	1.68137i
$816$		0			0
$817$			64.0000i			2.23908i
$818$		−	22.0000i		−	0.769212i
$819$		0			0
$820$	24.0000	−	12.0000i	0.838116	−	0.419058i
$821$	48.0000			1.67521			0.837606	−	0.546275i	$-0.183955\pi$
$821$	48.0000			1.67521			0.837606	+	0.546275i	$0.183955\pi$
$822$		0			0
$823$			24.0000i			0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$			24.0000i			0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$	−8.00000			−0.278693
$825$		0			0
$826$	8.00000			0.278356
$827$			20.0000i			0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$			20.0000i			0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$		0			0
$829$	−10.0000			−0.347314			−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000			−0.347314			−0.173657	+	0.984806i	$0.555558\pi$
$830$		0			0
$831$		0			0
$832$		−	4.00000i		−	0.138675i
$833$			2.00000i			0.0692959i
$834$		0			0
$835$	−12.0000	−	24.0000i	−0.415277	−	0.830554i
$836$		0			0
$837$		0			0
$838$		−	40.0000i		−	1.38178i
$839$	32.0000			1.10476			0.552381	−	0.833592i	$-0.313719\pi$
$839$	32.0000			1.10476			0.552381	+	0.833592i	$0.313719\pi$
$840$		0			0
$841$	35.0000			1.20690
$842$		−	26.0000i		−	0.896019i
$843$		0			0
$844$	12.0000			0.413057
$845$	−6.00000	+	3.00000i	−0.206406	+	0.103203i
$846$		0			0
$847$			11.0000i			0.377964i
$848$			6.00000i			0.206041i
$849$		0			0
$850$	6.00000	−	8.00000i	0.205798	−	0.274398i
$851$	64.0000			2.19389
$852$		0			0
$853$		−	28.0000i		−	0.958702i	−0.877623	−	0.479351i	$-0.840872\pi$
$853$		−	28.0000i		−	0.958702i	0.877623	−	0.479351i	$-0.159128\pi$
$854$	−6.00000			−0.205316
$855$		0			0
$856$	−4.00000			−0.136717
$857$		−	58.0000i		−	1.98124i	−0.136637	−	0.990621i	$-0.543630\pi$
$857$		−	58.0000i		−	1.98124i	0.136637	−	0.990621i	$-0.456370\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$	−8.00000	−	16.0000i	−0.272798	−	0.545595i
$861$		0			0
$862$			16.0000i			0.544962i
$863$			32.0000i			1.08929i	0.838666	+	0.544646i	$0.183336\pi$
$863$			32.0000i			1.08929i	−0.838666	+	0.544646i	$0.816664\pi$
$864$		0			0
$865$	−6.00000	−	12.0000i	−0.204006	−	0.408012i
$866$	−4.00000			−0.135926
$867$		0			0
$868$			4.00000i			0.135769i
$869$		0			0
$870$		0			0
$871$	32.0000			1.08428
$872$			2.00000i			0.0677285i
$873$		0			0
$874$	64.0000			2.16483
$875$	−11.0000	−	2.00000i	−0.371868	−	0.0676123i
$876$		0			0
$877$		−	40.0000i		−	1.35070i	−0.737496	−	0.675352i	$-0.763992\pi$
$877$		−	40.0000i		−	1.35070i	0.737496	−	0.675352i	$-0.236008\pi$
$878$			28.0000i			0.944954i
$879$		0			0
$880$		0			0
$881$	12.0000			0.404290			0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000			0.404290			0.202145	+	0.979356i	$0.435209\pi$
$882$		0			0
$883$			40.0000i			1.34611i	0.739594	+	0.673054i	$0.235018\pi$
$883$			40.0000i			1.34611i	−0.739594	+	0.673054i	$0.764982\pi$
$884$	−8.00000			−0.269069
$885$		0			0
$886$	12.0000			0.403148
$887$			12.0000i			0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$			12.0000i			0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$		0			0
$889$	8.00000			0.268311
$890$	−4.00000	−	8.00000i	−0.134080	−	0.268161i
$891$		0			0
$892$			8.00000i			0.267860i
$893$		−	32.0000i		−	1.07084i
$894$		0			0
$895$	32.0000	−	16.0000i	1.06964	−	0.534821i
$896$	−1.00000			−0.0334077
$897$		0			0
$898$		−	16.0000i		−	0.533927i
$899$	32.0000			1.06726
$900$		0			0
$901$	12.0000			0.399778
$902$		0			0
$903$		0			0
$904$	−2.00000			−0.0665190
$905$	4.00000	−	2.00000i	0.132964	−	0.0664822i
$906$		0			0
$907$			40.0000i			1.32818i	0.747653	+	0.664089i	$0.231180\pi$
$907$			40.0000i			1.32818i	−0.747653	+	0.664089i	$0.768820\pi$
$908$			24.0000i			0.796468i
$909$		0			0
$910$	4.00000	+	8.00000i	0.132599	+	0.265197i
$911$	−32.0000			−1.06021			−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000			−1.06021			−0.530104	+	0.847933i	$0.677847\pi$
$912$		0			0
$913$		0			0
$914$	−16.0000			−0.529233
$915$		0			0
$916$	−2.00000			−0.0660819
$917$			8.00000i			0.264183i
$918$		0			0
$919$	40.0000			1.31948			0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000			1.31948			0.659739	+	0.751495i	$0.270667\pi$
$920$	−16.0000	+	8.00000i	−0.527504	+	0.263752i
$921$		0			0
$922$			20.0000i			0.658665i
$923$		0			0
$924$		0			0
$925$	32.0000	+	24.0000i	1.05215	+	0.789115i
$926$	8.00000			0.262896
$927$		0			0
$928$			8.00000i			0.262613i
$929$	−12.0000			−0.393707			−0.196854	−	0.980433i	$-0.563072\pi$
$929$	−12.0000			−0.393707			−0.196854	+	0.980433i	$0.563072\pi$
$930$		0			0
$931$	−8.00000			−0.262189
$932$		−	26.0000i		−	0.851658i
$933$		0			0
$934$	32.0000			1.04707
$935$		0			0
$936$		0			0
$937$		−	28.0000i		−	0.914720i	−0.889282	−	0.457360i	$-0.848795\pi$
$937$		−	28.0000i		−	0.914720i	0.889282	−	0.457360i	$-0.151205\pi$
$938$		−	8.00000i		−	0.261209i
$939$		0			0
$940$	4.00000	+	8.00000i	0.130466	+	0.260931i
$941$	28.0000			0.912774			0.456387	−	0.889781i	$-0.349143\pi$
$941$	28.0000			0.912774			0.456387	+	0.889781i	$0.349143\pi$
$942$		0			0
$943$			96.0000i			3.12619i
$944$	8.00000			0.260378
$945$		0			0
$946$		0			0
$947$		−	60.0000i		−	1.94974i	−0.222779	−	0.974869i	$-0.571513\pi$
$947$		−	60.0000i		−	1.94974i	0.222779	−	0.974869i	$-0.428487\pi$
$948$		0			0
$949$	16.0000			0.519382
$950$	32.0000	+	24.0000i	1.03822	+	0.778663i
$951$		0			0
$952$			2.00000i			0.0648204i
$953$			54.0000i			1.74923i	0.484817	+	0.874616i	$0.338886\pi$
$953$			54.0000i			1.74923i	−0.484817	+	0.874616i	$0.661114\pi$
$954$		0			0
$955$	−32.0000	+	16.0000i	−1.03550	+	0.517748i
$956$	16.0000			0.517477
$957$		0			0
$958$		0			0
$959$	10.0000			0.322917
$960$		0			0
$961$	−15.0000			−0.483871
$962$		−	32.0000i		−	1.03172i
$963$		0			0
$964$	−18.0000			−0.579741
$965$	−16.0000	−	32.0000i	−0.515058	−	1.03012i
$966$		0			0
$967$		−	8.00000i		−	0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$		−	8.00000i		−	0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$			11.0000i			0.353553i
$969$		0			0
$970$	−24.0000	+	12.0000i	−0.770594	+	0.385297i
$971$	−24.0000			−0.770197			−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000			−0.770197			−0.385098	+	0.922876i	$0.625832\pi$
$972$		0			0
$973$			16.0000i			0.512936i
$974$	40.0000			1.28168
$975$		0			0
$976$	−6.00000			−0.192055
$977$			30.0000i			0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$			30.0000i			0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$		0			0
$979$		0			0
$980$	2.00000	−	1.00000i	0.0638877	−	0.0319438i
$981$		0			0
$982$		0			0
$983$		−	4.00000i		−	0.127580i	−0.997963	−	0.0637901i	$-0.979681\pi$
$983$		−	4.00000i		−	0.127580i	0.997963	−	0.0637901i	$-0.0203188\pi$
$984$		0			0
$985$	22.0000	+	44.0000i	0.700978	+	1.40196i
$986$	16.0000			0.509544
$987$		0			0
$988$		−	32.0000i		−	1.01806i
$989$	64.0000			2.03508
$990$		0			0
$991$	−32.0000			−1.01651			−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000			−1.01651			−0.508257	+	0.861206i	$0.669710\pi$
$992$			4.00000i			0.127000i
$993$		0			0
$994$		0			0
$995$	24.0000	−	12.0000i	0.760851	−	0.380426i
$996$		0			0
$997$		−	4.00000i		−	0.126681i	−0.997992	−	0.0633406i	$-0.979825\pi$
$997$		−	4.00000i		−	0.126681i	0.997992	−	0.0633406i	$-0.0201755\pi$
$998$		−	28.0000i		−	0.886325i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	630.2.g.f.379.2	yes	2
3.2	odd	2		630.2.g.a.379.1	✓	2
4.3	odd	2		5040.2.t.q.1009.1		2
5.2	odd	4		3150.2.a.p.1.1		1
5.3	odd	4		3150.2.a.z.1.1		1
5.4	even	2	inner	630.2.g.f.379.1	yes	2
12.11	even	2		5040.2.t.b.1009.2		2
15.2	even	4		3150.2.a.bn.1.1		1
15.8	even	4		3150.2.a.e.1.1		1
15.14	odd	2		630.2.g.a.379.2	yes	2
20.19	odd	2		5040.2.t.q.1009.2		2
60.59	even	2		5040.2.t.b.1009.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.g.a.379.1	✓	2	3.2	odd	2
630.2.g.a.379.2	yes	2	15.14	odd	2
630.2.g.f.379.1	yes	2	5.4	even	2	inner
630.2.g.f.379.2	yes	2	1.1	even	1	trivial
3150.2.a.e.1.1		1	15.8	even	4
3150.2.a.p.1.1		1	5.2	odd	4
3150.2.a.z.1.1		1	5.3	odd	4
3150.2.a.bn.1.1		1	15.2	even	4
5040.2.t.b.1009.1		2	60.59	even	2
5040.2.t.b.1009.2		2	12.11	even	2
5040.2.t.q.1009.1		2	4.3	odd	2
5040.2.t.q.1009.2		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	630.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} -1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} -1.00000i q^{7} -1.00000i q^{8} +(1.00000 + 2.00000i) q^{10} +4.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +8.00000 q^{19} +(-2.00000 + 1.00000i) q^{20} -8.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -4.00000 q^{26} +1.00000i q^{28} +8.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} +2.00000 q^{34} +(-1.00000 - 2.00000i) q^{35} +8.00000i q^{37} +8.00000i q^{38} +(-1.00000 - 2.00000i) q^{40} -12.0000 q^{41} +8.00000i q^{43} +8.00000 q^{46} -4.00000i q^{47} -1.00000 q^{49} +(4.00000 + 3.00000i) q^{50} -4.00000i q^{52} +6.00000i q^{53} -1.00000 q^{56} +8.00000i q^{58} +8.00000 q^{59} -6.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +(4.00000 + 8.00000i) q^{65} -8.00000i q^{67} +2.00000i q^{68} +(2.00000 - 1.00000i) q^{70} -4.00000i q^{73} -8.00000 q^{74} -8.00000 q^{76} -8.00000 q^{79} +(2.00000 - 1.00000i) q^{80} -12.0000i q^{82} +(-2.00000 - 4.00000i) q^{85} -8.00000 q^{86} -4.00000 q^{89} +4.00000 q^{91} +8.00000i q^{92} +4.00000 q^{94} +(16.0000 - 8.00000i) q^{95} +12.0000i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 4 q^{5}+O(q^{10}) \) 2 * q - 2 * q^4 + 4 * q^5	\( 2 q - 2 q^{4} + 4 q^{5} + 2 q^{10} + 2 q^{14} + 2 q^{16} + 16 q^{19} - 4 q^{20} + 6 q^{25} - 8 q^{26} + 16 q^{29} + 8 q^{31} + 4 q^{34} - 2 q^{35} - 2 q^{40} - 24 q^{41} + 16 q^{46} - 2 q^{49} + 8 q^{50} - 2 q^{56} + 16 q^{59} - 12 q^{61} - 2 q^{64} + 8 q^{65} + 4 q^{70} - 16 q^{74} - 16 q^{76} - 16 q^{79} + 4 q^{80} - 4 q^{85} - 16 q^{86} - 8 q^{89} + 8 q^{91} + 8 q^{94} + 32 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^5 + 2 * q^10 + 2 * q^14 + 2 * q^16 + 16 * q^19 - 4 * q^20 + 6 * q^25 - 8 * q^26 + 16 * q^29 + 8 * q^31 + 4 * q^34 - 2 * q^35 - 2 * q^40 - 24 * q^41 + 16 * q^46 - 2 * q^49 + 8 * q^50 - 2 * q^56 + 16 * q^59 - 12 * q^61 - 2 * q^64 + 8 * q^65 + 4 * q^70 - 16 * q^74 - 16 * q^76 - 16 * q^79 + 4 * q^80 - 4 * q^85 - 16 * q^86 - 8 * q^89 + 8 * q^91 + 8 * q^94 + 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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