LMFDB - Embedded newform 630.2.g.d.379.1

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:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} +1.00000i q^{7} +1.00000i q^{8} +(-2.00000 - 1.00000i) q^{10} +2.00000 q^{11} -2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -8.00000i q^{17} +2.00000 q^{19} +(-1.00000 + 2.00000i) q^{20} -2.00000i q^{22} +(-3.00000 - 4.00000i) q^{25} -2.00000 q^{26} -1.00000i q^{28} -6.00000 q^{29} +6.00000 q^{31} -1.00000i q^{32} -8.00000 q^{34} +(2.00000 + 1.00000i) q^{35} +8.00000i q^{37} -2.00000i q^{38} +(2.00000 + 1.00000i) q^{40} -6.00000 q^{41} -8.00000i q^{43} -2.00000 q^{44} -4.00000i q^{47} -1.00000 q^{49} +(-4.00000 + 3.00000i) q^{50} +2.00000i q^{52} -2.00000i q^{53} +(2.00000 - 4.00000i) q^{55} -1.00000 q^{56} +6.00000i q^{58} -8.00000 q^{59} +10.0000 q^{61} -6.00000i q^{62} -1.00000 q^{64} +(-4.00000 - 2.00000i) q^{65} -12.0000i q^{67} +8.00000i q^{68} +(1.00000 - 2.00000i) q^{70} +14.0000 q^{71} +10.0000i q^{73} +8.00000 q^{74} -2.00000 q^{76} +2.00000i q^{77} -4.00000 q^{79} +(1.00000 - 2.00000i) q^{80} +6.00000i q^{82} +16.0000i q^{83} +(-16.0000 - 8.00000i) q^{85} -8.00000 q^{86} +2.00000i q^{88} +10.0000 q^{89} +2.00000 q^{91} -4.00000 q^{94} +(2.00000 - 4.00000i) q^{95} +10.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^5	$ 2 q - 2 q^{4} + 2 q^{5} - 4 q^{10} + 4 q^{11} + 2 q^{14} + 2 q^{16} + 4 q^{19} - 2 q^{20} - 6 q^{25} - 4 q^{26} - 12 q^{29} + 12 q^{31} - 16 q^{34} + 4 q^{35} + 4 q^{40} - 12 q^{41} - 4 q^{44} - 2 q^{49} - 8 q^{50} + 4 q^{55} - 2 q^{56} - 16 q^{59} + 20 q^{61} - 2 q^{64} - 8 q^{65} + 2 q^{70} + 28 q^{71} + 16 q^{74} - 4 q^{76} - 8 q^{79} + 2 q^{80} - 32 q^{85} - 16 q^{86} + 20 q^{89} + 4 q^{91} - 8 q^{94} + 4 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^5 - 4 * q^10 + 4 * q^11 + 2 * q^14 + 2 * q^16 + 4 * q^19 - 2 * q^20 - 6 * q^25 - 4 * q^26 - 12 * q^29 + 12 * q^31 - 16 * q^34 + 4 * q^35 + 4 * q^40 - 12 * q^41 - 4 * q^44 - 2 * q^49 - 8 * q^50 + 4 * q^55 - 2 * q^56 - 16 * q^59 + 20 * q^61 - 2 * q^64 - 8 * q^65 + 2 * q^70 + 28 * q^71 + 16 * q^74 - 4 * q^76 - 8 * q^79 + 2 * q^80 - 32 * q^85 - 16 * q^86 + 20 * q^89 + 4 * q^91 - 8 * q^94 + 4 * 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
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$			1.00000i			0.353553i
$9$		0			0
$10$	−2.00000	−	1.00000i	−0.632456	−	0.316228i
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$		−	2.00000i		−	0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$		−	2.00000i		−	0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	1.00000			0.267261
$15$		0			0
$16$	1.00000			0.250000
$17$		−	8.00000i		−	1.94029i	−0.242536	−	0.970143i	$-0.577979\pi$
$17$		−	8.00000i		−	1.94029i	0.242536	−	0.970143i	$-0.422021\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$	−1.00000	+	2.00000i	−0.223607	+	0.447214i
$21$		0			0
$22$		−	2.00000i		−	0.426401i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$	−2.00000			−0.392232
$27$		0			0
$28$		−	1.00000i		−	0.188982i
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	6.00000			1.07763			0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000			1.07763			0.538816	+	0.842424i	$0.318872\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	−8.00000			−1.37199
$35$	2.00000	+	1.00000i	0.338062	+	0.169031i
$36$		0			0
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$		−	2.00000i		−	0.324443i
$39$		0			0
$40$	2.00000	+	1.00000i	0.316228	+	0.158114i
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$		−	8.00000i		−	1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$		−	8.00000i		−	1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	−2.00000			−0.301511
$45$		0			0
$46$		0			0
$47$		−	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
$52$			2.00000i			0.277350i
$53$		−	2.00000i		−	0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$		−	2.00000i		−	0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$		0			0
$55$	2.00000	−	4.00000i	0.269680	−	0.539360i
$56$	−1.00000			−0.133631
$57$		0			0
$58$			6.00000i			0.787839i
$59$	−8.00000			−1.04151			−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000			−1.04151			−0.520756	+	0.853706i	$0.674350\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		−	6.00000i		−	0.762001i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−4.00000	−	2.00000i	−0.496139	−	0.248069i
$66$		0			0
$67$		−	12.0000i		−	1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$		−	12.0000i		−	1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$			8.00000i			0.970143i
$69$		0			0
$70$	1.00000	−	2.00000i	0.119523	−	0.239046i
$71$	14.0000			1.66149			0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000			1.66149			0.830747	+	0.556650i	$0.187914\pi$
$72$		0			0
$73$			10.0000i			1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$			10.0000i			1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	8.00000			0.929981
$75$		0			0
$76$	−2.00000			−0.229416
$77$			2.00000i			0.227921i
$78$		0			0
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$	1.00000	−	2.00000i	0.111803	−	0.223607i
$81$		0			0
$82$			6.00000i			0.662589i
$83$			16.0000i			1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$			16.0000i			1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$		0			0
$85$	−16.0000	−	8.00000i	−1.73544	−	0.867722i
$86$	−8.00000			−0.862662
$87$		0			0
$88$			2.00000i			0.213201i
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$	2.00000			0.209657
$92$		0			0
$93$		0			0
$94$	−4.00000			−0.412568
$95$	2.00000	−	4.00000i	0.205196	−	0.410391i
$96$		0			0
$97$			10.0000i			1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$			10.0000i			1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$			1.00000i			0.101015i
$99$		0			0
$100$	3.00000	+	4.00000i	0.300000	+	0.400000i
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	2.00000			0.196116
$105$		0			0
$106$	−2.00000			−0.194257
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$	6.00000			0.574696			0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000			0.574696			0.287348	+	0.957826i	$0.407226\pi$
$110$	−4.00000	−	2.00000i	−0.381385	−	0.190693i
$111$		0			0
$112$			1.00000i			0.0944911i
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$		0			0
$115$		0			0
$116$	6.00000			0.557086
$117$		0			0
$118$			8.00000i			0.736460i
$119$	8.00000			0.733359
$120$		0			0
$121$	−7.00000			−0.636364
$122$		−	10.0000i		−	0.905357i
$123$		0			0
$124$	−6.00000			−0.538816
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$			12.0000i			1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$			12.0000i			1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−2.00000	+	4.00000i	−0.175412	+	0.350823i
$131$	20.0000			1.74741			0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000			1.74741			0.873704	+	0.486458i	$0.161711\pi$
$132$		0			0
$133$			2.00000i			0.173422i
$134$	−12.0000			−1.03664
$135$		0			0
$136$	8.00000			0.685994
$137$			14.0000i			1.19610i	0.801459	+	0.598050i	$0.204058\pi$
$137$			14.0000i			1.19610i	−0.801459	+	0.598050i	$0.795942\pi$
$138$		0			0
$139$	2.00000			0.169638			0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000			0.169638			0.0848189	+	0.996396i	$0.472969\pi$
$140$	−2.00000	−	1.00000i	−0.169031	−	0.0845154i
$141$		0			0
$142$		−	14.0000i		−	1.17485i
$143$		−	4.00000i		−	0.334497i
$144$		0			0
$145$	−6.00000	+	12.0000i	−0.498273	+	0.996546i
$146$	10.0000			0.827606
$147$		0			0
$148$		−	8.00000i		−	0.657596i
$149$	−18.0000			−1.47462			−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000			−1.47462			−0.737309	+	0.675556i	$0.763904\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$			2.00000i			0.162221i
$153$		0			0
$154$	2.00000			0.161165
$155$	6.00000	−	12.0000i	0.481932	−	0.963863i
$156$		0			0
$157$		−	10.0000i		−	0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$		−	10.0000i		−	0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$			4.00000i			0.318223i
$159$		0			0
$160$	−2.00000	−	1.00000i	−0.158114	−	0.0790569i
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$	6.00000			0.468521
$165$		0			0
$166$	16.0000			1.24184
$167$			8.00000i			0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$			8.00000i			0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$	−8.00000	+	16.0000i	−0.613572	+	1.22714i
$171$		0			0
$172$			8.00000i			0.609994i
$173$		−	16.0000i		−	1.21646i	−0.793762	−	0.608229i	$-0.791880\pi$
$173$		−	16.0000i		−	1.21646i	0.793762	−	0.608229i	$-0.208120\pi$
$174$		0			0
$175$	4.00000	−	3.00000i	0.302372	−	0.226779i
$176$	2.00000			0.150756
$177$		0			0
$178$		−	10.0000i		−	0.749532i
$179$	−2.00000			−0.149487			−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000			−0.149487			−0.0747435	+	0.997203i	$0.523814\pi$
$180$		0			0
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		−	2.00000i		−	0.148250i
$183$		0			0
$184$		0			0
$185$	16.0000	+	8.00000i	1.17634	+	0.588172i
$186$		0			0
$187$		−	16.0000i		−	1.17004i
$188$			4.00000i			0.291730i
$189$		0			0
$190$	−4.00000	−	2.00000i	−0.290191	−	0.145095i
$191$	−14.0000			−1.01300			−0.506502	−	0.862239i	$-0.669062\pi$
$191$	−14.0000			−1.01300			−0.506502	+	0.862239i	$0.669062\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$	10.0000			0.717958
$195$		0			0
$196$	1.00000			0.0714286
$197$			6.00000i			0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$			6.00000i			0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$		0			0
$199$	26.0000			1.84309			0.921546	−	0.388270i	$-0.126927\pi$
$199$	26.0000			1.84309			0.921546	+	0.388270i	$0.126927\pi$
$200$	4.00000	−	3.00000i	0.282843	−	0.212132i
$201$		0			0
$202$		−	10.0000i		−	0.703598i
$203$		−	6.00000i		−	0.421117i
$204$		0			0
$205$	−6.00000	+	12.0000i	−0.419058	+	0.838116i
$206$		0			0
$207$		0			0
$208$		−	2.00000i		−	0.138675i
$209$	4.00000			0.276686
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$			2.00000i			0.137361i
$213$		0			0
$214$	12.0000			0.820303
$215$	−16.0000	−	8.00000i	−1.09119	−	0.545595i
$216$		0			0
$217$			6.00000i			0.407307i
$218$		−	6.00000i		−	0.406371i
$219$		0			0
$220$	−2.00000	+	4.00000i	−0.134840	+	0.269680i
$221$	−16.0000			−1.07628
$222$		0			0
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$	6.00000			0.399114
$227$			8.00000i			0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$			8.00000i			0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$		0			0
$229$	−26.0000			−1.71813			−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000			−1.71813			−0.859064	+	0.511868i	$0.828954\pi$
$230$		0			0
$231$		0			0
$232$		−	6.00000i		−	0.393919i
$233$			6.00000i			0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$			6.00000i			0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$		0			0
$235$	−8.00000	−	4.00000i	−0.521862	−	0.260931i
$236$	8.00000			0.520756
$237$		0			0
$238$		−	8.00000i		−	0.518563i
$239$	−22.0000			−1.42306			−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000			−1.42306			−0.711531	+	0.702655i	$0.751998\pi$
$240$		0			0
$241$	−10.0000			−0.644157			−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000			−0.644157			−0.322078	+	0.946713i	$0.604381\pi$
$242$			7.00000i			0.449977i
$243$		0			0
$244$	−10.0000			−0.640184
$245$	−1.00000	+	2.00000i	−0.0638877	+	0.127775i
$246$		0			0
$247$		−	4.00000i		−	0.254514i
$248$			6.00000i			0.381000i
$249$		0			0
$250$	2.00000	+	11.0000i	0.126491	+	0.695701i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$	12.0000			0.752947
$255$		0			0
$256$	1.00000			0.0625000
$257$			20.0000i			1.24757i	0.781598	+	0.623783i	$0.214405\pi$
$257$			20.0000i			1.24757i	−0.781598	+	0.623783i	$0.785595\pi$
$258$		0			0
$259$	−8.00000			−0.497096
$260$	4.00000	+	2.00000i	0.248069	+	0.124035i
$261$		0			0
$262$		−	20.0000i		−	1.23560i
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$	−4.00000	−	2.00000i	−0.245718	−	0.122859i
$266$	2.00000			0.122628
$267$		0			0
$268$			12.0000i			0.733017i
$269$	14.0000			0.853595			0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000			0.853595			0.426798	+	0.904347i	$0.359642\pi$
$270$		0			0
$271$	2.00000			0.121491			0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000			0.121491			0.0607457	+	0.998153i	$0.480652\pi$
$272$		−	8.00000i		−	0.485071i
$273$		0			0
$274$	14.0000			0.845771
$275$	−6.00000	−	8.00000i	−0.361814	−	0.482418i
$276$		0			0
$277$		−	8.00000i		−	0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$		−	8.00000i		−	0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$		−	2.00000i		−	0.119952i
$279$		0			0
$280$	−1.00000	+	2.00000i	−0.0597614	+	0.119523i
$281$	−22.0000			−1.31241			−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000			−1.31241			−0.656205	+	0.754583i	$0.727839\pi$
$282$		0			0
$283$		−	4.00000i		−	0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$		−	4.00000i		−	0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	−14.0000			−0.830747
$285$		0			0
$286$	−4.00000			−0.236525
$287$		−	6.00000i		−	0.354169i
$288$		0			0
$289$	−47.0000			−2.76471
$290$	12.0000	+	6.00000i	0.704664	+	0.352332i
$291$		0			0
$292$		−	10.0000i		−	0.585206i
$293$		−	16.0000i		−	0.934730i	−0.884064	−	0.467365i	$-0.845203\pi$
$293$		−	16.0000i		−	0.934730i	0.884064	−	0.467365i	$-0.154797\pi$
$294$		0			0
$295$	−8.00000	+	16.0000i	−0.465778	+	0.931556i
$296$	−8.00000			−0.464991
$297$		0			0
$298$			18.0000i			1.04271i
$299$		0			0
$300$		0			0
$301$	8.00000			0.461112
$302$		−	8.00000i		−	0.460348i
$303$		0			0
$304$	2.00000			0.114708
$305$	10.0000	−	20.0000i	0.572598	−	1.14520i
$306$		0			0
$307$			28.0000i			1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$			28.0000i			1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$		−	2.00000i		−	0.113961i
$309$		0			0
$310$	−12.0000	−	6.00000i	−0.681554	−	0.340777i
$311$	8.00000			0.453638			0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000			0.453638			0.226819	+	0.973937i	$0.427167\pi$
$312$		0			0
$313$		−	10.0000i		−	0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$		−	10.0000i		−	0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	−10.0000			−0.564333
$315$		0			0
$316$	4.00000			0.225018
$317$			2.00000i			0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$			2.00000i			0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$		0			0
$319$	−12.0000			−0.671871
$320$	−1.00000	+	2.00000i	−0.0559017	+	0.111803i
$321$		0			0
$322$		0			0
$323$		−	16.0000i		−	0.890264i
$324$		0			0
$325$	−8.00000	+	6.00000i	−0.443760	+	0.332820i
$326$		0			0
$327$		0			0
$328$		−	6.00000i		−	0.331295i
$329$	4.00000			0.220527
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		−	16.0000i		−	0.878114i
$333$		0			0
$334$	8.00000			0.437741
$335$	−24.0000	−	12.0000i	−1.31126	−	0.655630i
$336$		0			0
$337$			32.0000i			1.74315i	0.490261	+	0.871576i	$0.336901\pi$
$337$			32.0000i			1.74315i	−0.490261	+	0.871576i	$0.663099\pi$
$338$		−	9.00000i		−	0.489535i
$339$		0			0
$340$	16.0000	+	8.00000i	0.867722	+	0.433861i
$341$	12.0000			0.649836
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$	8.00000			0.431331
$345$		0			0
$346$	−16.0000			−0.860165
$347$		−	4.00000i		−	0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$		−	4.00000i		−	0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$		0			0
$349$	−18.0000			−0.963518			−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000			−0.963518			−0.481759	+	0.876304i	$0.660002\pi$
$350$	−3.00000	−	4.00000i	−0.160357	−	0.213809i
$351$		0			0
$352$		−	2.00000i		−	0.106600i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$	14.0000	−	28.0000i	0.743043	−	1.48609i
$356$	−10.0000			−0.529999
$357$		0			0
$358$			2.00000i			0.105703i
$359$	14.0000			0.738892			0.369446	−	0.929252i	$-0.379548\pi$
$359$	14.0000			0.738892			0.369446	+	0.929252i	$0.379548\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		−	22.0000i		−	1.15629i
$363$		0			0
$364$	−2.00000			−0.104828
$365$	20.0000	+	10.0000i	1.04685	+	0.523424i
$366$		0			0
$367$		−	32.0000i		−	1.67039i	−0.549957	−	0.835193i	$-0.685356\pi$
$367$		−	32.0000i		−	1.67039i	0.549957	−	0.835193i	$-0.314644\pi$
$368$		0			0
$369$		0			0
$370$	8.00000	−	16.0000i	0.415900	−	0.831800i
$371$	2.00000			0.103835
$372$		0			0
$373$		−	36.0000i		−	1.86401i	−0.362446	−	0.932005i	$-0.618058\pi$
$373$		−	36.0000i		−	1.86401i	0.362446	−	0.932005i	$-0.381942\pi$
$374$	−16.0000			−0.827340
$375$		0			0
$376$	4.00000			0.206284
$377$			12.0000i			0.618031i
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$	−2.00000	+	4.00000i	−0.102598	+	0.205196i
$381$		0			0
$382$			14.0000i			0.716302i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$	4.00000	+	2.00000i	0.203859	+	0.101929i
$386$		0			0
$387$		0			0
$388$		−	10.0000i		−	0.507673i
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$		0			0
$391$		0			0
$392$		−	1.00000i		−	0.0505076i
$393$		0			0
$394$	6.00000			0.302276
$395$	−4.00000	+	8.00000i	−0.201262	+	0.402524i
$396$		0			0
$397$			2.00000i			0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$			2.00000i			0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$		−	26.0000i		−	1.30326i
$399$		0			0
$400$	−3.00000	−	4.00000i	−0.150000	−	0.200000i
$401$	−18.0000			−0.898877			−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000			−0.898877			−0.449439	+	0.893311i	$0.648376\pi$
$402$		0			0
$403$		−	12.0000i		−	0.597763i
$404$	−10.0000			−0.497519
$405$		0			0
$406$	−6.00000			−0.297775
$407$			16.0000i			0.793091i
$408$		0			0
$409$	−10.0000			−0.494468			−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000			−0.494468			−0.247234	+	0.968956i	$0.579522\pi$
$410$	12.0000	+	6.00000i	0.592638	+	0.296319i
$411$		0			0
$412$		0			0
$413$		−	8.00000i		−	0.393654i
$414$		0			0
$415$	32.0000	+	16.0000i	1.57082	+	0.785409i
$416$	−2.00000			−0.0980581
$417$		0			0
$418$		−	4.00000i		−	0.195646i
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$			8.00000i			0.389434i
$423$		0			0
$424$	2.00000			0.0971286
$425$	−32.0000	+	24.0000i	−1.55223	+	1.16417i
$426$		0			0
$427$			10.0000i			0.483934i
$428$		−	12.0000i		−	0.580042i
$429$		0			0
$430$	−8.00000	+	16.0000i	−0.385794	+	0.771589i
$431$	6.00000			0.289010			0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000			0.289010			0.144505	+	0.989504i	$0.453841\pi$
$432$		0			0
$433$		−	26.0000i		−	1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$		−	26.0000i		−	1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$	6.00000			0.288009
$435$		0			0
$436$	−6.00000			−0.287348
$437$		0			0
$438$		0			0
$439$	−6.00000			−0.286364			−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000			−0.286364			−0.143182	+	0.989696i	$0.545733\pi$
$440$	4.00000	+	2.00000i	0.190693	+	0.0953463i
$441$		0			0
$442$			16.0000i			0.761042i
$443$			36.0000i			1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$			36.0000i			1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$		0			0
$445$	10.0000	−	20.0000i	0.474045	−	0.948091i
$446$	16.0000			0.757622
$447$		0			0
$448$		−	1.00000i		−	0.0472456i
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$	−12.0000			−0.565058
$452$		−	6.00000i		−	0.282216i
$453$		0			0
$454$	8.00000			0.375459
$455$	2.00000	−	4.00000i	0.0937614	−	0.187523i
$456$		0			0
$457$		−	28.0000i		−	1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$		−	28.0000i		−	1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$			26.0000i			1.21490i
$459$		0			0
$460$		0			0
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		0			0
$463$			12.0000i			0.557687i	0.960337	+	0.278844i	$0.0899511\pi$
$463$			12.0000i			0.557687i	−0.960337	+	0.278844i	$0.910049\pi$
$464$	−6.00000			−0.278543
$465$		0			0
$466$	6.00000			0.277945
$467$		−	16.0000i		−	0.740392i	−0.928954	−	0.370196i	$-0.879291\pi$
$467$		−	16.0000i		−	0.740392i	0.928954	−	0.370196i	$-0.120709\pi$
$468$		0			0
$469$	12.0000			0.554109
$470$	−4.00000	+	8.00000i	−0.184506	+	0.369012i
$471$		0			0
$472$		−	8.00000i		−	0.368230i
$473$		−	16.0000i		−	0.735681i
$474$		0			0
$475$	−6.00000	−	8.00000i	−0.275299	−	0.367065i
$476$	−8.00000			−0.366679
$477$		0			0
$478$			22.0000i			1.00626i
$479$	8.00000			0.365529			0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000			0.365529			0.182765	+	0.983157i	$0.441495\pi$
$480$		0			0
$481$	16.0000			0.729537
$482$			10.0000i			0.455488i
$483$		0			0
$484$	7.00000			0.318182
$485$	20.0000	+	10.0000i	0.908153	+	0.454077i
$486$		0			0
$487$			36.0000i			1.63132i	0.578535	+	0.815658i	$0.303625\pi$
$487$			36.0000i			1.63132i	−0.578535	+	0.815658i	$0.696375\pi$
$488$			10.0000i			0.452679i
$489$		0			0
$490$	2.00000	+	1.00000i	0.0903508	+	0.0451754i
$491$	42.0000			1.89543			0.947717	−	0.319113i	$-0.103385\pi$
$491$	42.0000			1.89543			0.947717	+	0.319113i	$0.103385\pi$
$492$		0			0
$493$			48.0000i			2.16181i
$494$	−4.00000			−0.179969
$495$		0			0
$496$	6.00000			0.269408
$497$			14.0000i			0.627986i
$498$		0			0
$499$	28.0000			1.25345			0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000			1.25345			0.626726	+	0.779240i	$0.284395\pi$
$500$	11.0000	−	2.00000i	0.491935	−	0.0894427i
$501$		0			0
$502$		0			0
$503$			24.0000i			1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$			24.0000i			1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$		0			0
$505$	10.0000	−	20.0000i	0.444994	−	0.889988i
$506$		0			0
$507$		0			0
$508$		−	12.0000i		−	0.532414i
$509$	2.00000			0.0886484			0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000			0.0886484			0.0443242	+	0.999017i	$0.485887\pi$
$510$		0			0
$511$	−10.0000			−0.442374
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	20.0000			0.882162
$515$		0			0
$516$		0			0
$517$		−	8.00000i		−	0.351840i
$518$			8.00000i			0.351500i
$519$		0			0
$520$	2.00000	−	4.00000i	0.0877058	−	0.175412i
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$		0			0
$523$			12.0000i			0.524723i	0.964970	+	0.262362i	$0.0845013\pi$
$523$			12.0000i			0.524723i	−0.964970	+	0.262362i	$0.915499\pi$
$524$	−20.0000			−0.873704
$525$		0			0
$526$	−24.0000			−1.04645
$527$		−	48.0000i		−	2.09091i
$528$		0			0
$529$	23.0000			1.00000
$530$	−2.00000	+	4.00000i	−0.0868744	+	0.173749i
$531$		0			0
$532$		−	2.00000i		−	0.0867110i
$533$			12.0000i			0.519778i
$534$		0			0
$535$	24.0000	+	12.0000i	1.03761	+	0.518805i
$536$	12.0000			0.518321
$537$		0			0
$538$		−	14.0000i		−	0.603583i
$539$	−2.00000			−0.0861461
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		−	2.00000i		−	0.0859074i
$543$		0			0
$544$	−8.00000			−0.342997
$545$	6.00000	−	12.0000i	0.257012	−	0.514024i
$546$		0			0
$547$			20.0000i			0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$			20.0000i			0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$		−	14.0000i		−	0.598050i
$549$		0			0
$550$	−8.00000	+	6.00000i	−0.341121	+	0.255841i
$551$	−12.0000			−0.511217
$552$		0			0
$553$		−	4.00000i		−	0.170097i
$554$	−8.00000			−0.339887
$555$		0			0
$556$	−2.00000			−0.0848189
$557$		−	6.00000i		−	0.254228i	−0.991888	−	0.127114i	$-0.959429\pi$
$557$		−	6.00000i		−	0.254228i	0.991888	−	0.127114i	$-0.0405714\pi$
$558$		0			0
$559$	−16.0000			−0.676728
$560$	2.00000	+	1.00000i	0.0845154	+	0.0422577i
$561$		0			0
$562$			22.0000i			0.928014i
$563$			12.0000i			0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$			12.0000i			0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$		0			0
$565$	12.0000	+	6.00000i	0.504844	+	0.252422i
$566$	−4.00000			−0.168133
$567$		0			0
$568$			14.0000i			0.587427i
$569$	18.0000			0.754599			0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000			0.754599			0.377300	+	0.926091i	$0.376853\pi$
$570$		0			0
$571$	−44.0000			−1.84134			−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000			−1.84134			−0.920671	+	0.390339i	$0.872358\pi$
$572$			4.00000i			0.167248i
$573$		0			0
$574$	−6.00000			−0.250435
$575$		0			0
$576$		0			0
$577$			18.0000i			0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$			18.0000i			0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$			47.0000i			1.95494i
$579$		0			0
$580$	6.00000	−	12.0000i	0.249136	−	0.498273i
$581$	−16.0000			−0.663792
$582$		0			0
$583$		−	4.00000i		−	0.165663i
$584$	−10.0000			−0.413803
$585$		0			0
$586$	−16.0000			−0.660954
$587$		−	4.00000i		−	0.165098i	−0.996587	−	0.0825488i	$-0.973694\pi$
$587$		−	4.00000i		−	0.165098i	0.996587	−	0.0825488i	$-0.0263060\pi$
$588$		0			0
$589$	12.0000			0.494451
$590$	16.0000	+	8.00000i	0.658710	+	0.329355i
$591$		0			0
$592$			8.00000i			0.328798i
$593$		−	32.0000i		−	1.31408i	−0.753855	−	0.657041i	$-0.771808\pi$
$593$		−	32.0000i		−	1.31408i	0.753855	−	0.657041i	$-0.228192\pi$
$594$		0			0
$595$	8.00000	−	16.0000i	0.327968	−	0.655936i
$596$	18.0000			0.737309
$597$		0			0
$598$		0			0
$599$	14.0000			0.572024			0.286012	−	0.958226i	$-0.407670\pi$
$599$	14.0000			0.572024			0.286012	+	0.958226i	$0.407670\pi$
$600$		0			0
$601$	−42.0000			−1.71322			−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000			−1.71322			−0.856608	+	0.515968i	$0.827432\pi$
$602$		−	8.00000i		−	0.326056i
$603$		0			0
$604$	−8.00000			−0.325515
$605$	−7.00000	+	14.0000i	−0.284590	+	0.569181i
$606$		0			0
$607$		−	16.0000i		−	0.649420i	−0.945814	−	0.324710i	$-0.894733\pi$
$607$		−	16.0000i		−	0.649420i	0.945814	−	0.324710i	$-0.105267\pi$
$608$		−	2.00000i		−	0.0811107i
$609$		0			0
$610$	−20.0000	−	10.0000i	−0.809776	−	0.404888i
$611$	−8.00000			−0.323645
$612$		0			0
$613$		−	32.0000i		−	1.29247i	−0.763139	−	0.646234i	$-0.776343\pi$
$613$		−	32.0000i		−	1.29247i	0.763139	−	0.646234i	$-0.223657\pi$
$614$	28.0000			1.12999
$615$		0			0
$616$	−2.00000			−0.0805823
$617$			18.0000i			0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$			18.0000i			0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$		0			0
$619$	18.0000			0.723481			0.361741	−	0.932279i	$-0.382183\pi$
$619$	18.0000			0.723481			0.361741	+	0.932279i	$0.382183\pi$
$620$	−6.00000	+	12.0000i	−0.240966	+	0.481932i
$621$		0			0
$622$		−	8.00000i		−	0.320771i
$623$			10.0000i			0.400642i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	−10.0000			−0.399680
$627$		0			0
$628$			10.0000i			0.399043i
$629$	64.0000			2.55185
$630$		0			0
$631$	12.0000			0.477712			0.238856	−	0.971055i	$-0.423228\pi$
$631$	12.0000			0.477712			0.238856	+	0.971055i	$0.423228\pi$
$632$		−	4.00000i		−	0.159111i
$633$		0			0
$634$	2.00000			0.0794301
$635$	24.0000	+	12.0000i	0.952411	+	0.476205i
$636$		0			0
$637$			2.00000i			0.0792429i
$638$			12.0000i			0.475085i
$639$		0			0
$640$	2.00000	+	1.00000i	0.0790569	+	0.0395285i
$641$	−30.0000			−1.18493			−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000			−1.18493			−0.592464	+	0.805597i	$0.701845\pi$
$642$		0			0
$643$			20.0000i			0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$			20.0000i			0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$		0			0
$645$		0			0
$646$	−16.0000			−0.629512
$647$		−	16.0000i		−	0.629025i	−0.949253	−	0.314512i	$-0.898159\pi$
$647$		−	16.0000i		−	0.629025i	0.949253	−	0.314512i	$-0.101841\pi$
$648$		0			0
$649$	−16.0000			−0.628055
$650$	6.00000	+	8.00000i	0.235339	+	0.313786i
$651$		0			0
$652$		0			0
$653$		−	14.0000i		−	0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$		−	14.0000i		−	0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$		0			0
$655$	20.0000	−	40.0000i	0.781465	−	1.56293i
$656$	−6.00000			−0.234261
$657$		0			0
$658$		−	4.00000i		−	0.155936i
$659$	−6.00000			−0.233727			−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000			−0.233727			−0.116863	+	0.993148i	$0.537284\pi$
$660$		0			0
$661$	10.0000			0.388955			0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000			0.388955			0.194477	+	0.980907i	$0.437699\pi$
$662$		0			0
$663$		0			0
$664$	−16.0000			−0.620920
$665$	4.00000	+	2.00000i	0.155113	+	0.0775567i
$666$		0			0
$667$		0			0
$668$		−	8.00000i		−	0.309529i
$669$		0			0
$670$	−12.0000	+	24.0000i	−0.463600	+	0.927201i
$671$	20.0000			0.772091
$672$		0			0
$673$		−	12.0000i		−	0.462566i	−0.972887	−	0.231283i	$-0.925708\pi$
$673$		−	12.0000i		−	0.462566i	0.972887	−	0.231283i	$-0.0742923\pi$
$674$	32.0000			1.23259
$675$		0			0
$676$	−9.00000			−0.346154
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	−10.0000			−0.383765
$680$	8.00000	−	16.0000i	0.306786	−	0.613572i
$681$		0			0
$682$		−	12.0000i		−	0.459504i
$683$			28.0000i			1.07139i	0.844411	+	0.535695i	$0.179950\pi$
$683$			28.0000i			1.07139i	−0.844411	+	0.535695i	$0.820050\pi$
$684$		0			0
$685$	28.0000	+	14.0000i	1.06983	+	0.534913i
$686$	−1.00000			−0.0381802
$687$		0			0
$688$		−	8.00000i		−	0.304997i
$689$	−4.00000			−0.152388
$690$		0			0
$691$	−14.0000			−0.532585			−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000			−0.532585			−0.266293	+	0.963892i	$0.585799\pi$
$692$			16.0000i			0.608229i
$693$		0			0
$694$	−4.00000			−0.151838
$695$	2.00000	−	4.00000i	0.0758643	−	0.151729i
$696$		0			0
$697$			48.0000i			1.81813i
$698$			18.0000i			0.681310i
$699$		0			0
$700$	−4.00000	+	3.00000i	−0.151186	+	0.113389i
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$			16.0000i			0.603451i
$704$	−2.00000			−0.0753778
$705$		0			0
$706$		0			0
$707$			10.0000i			0.376089i
$708$		0			0
$709$	38.0000			1.42712			0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000			1.42712			0.713560	+	0.700594i	$0.247082\pi$
$710$	−28.0000	−	14.0000i	−1.05082	−	0.525411i
$711$		0			0
$712$			10.0000i			0.374766i
$713$		0			0
$714$		0			0
$715$	−8.00000	−	4.00000i	−0.299183	−	0.149592i
$716$	2.00000			0.0747435
$717$		0			0
$718$		−	14.0000i		−	0.522475i
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$		0			0
$721$		0			0
$722$			15.0000i			0.558242i
$723$		0			0
$724$	−22.0000			−0.817624
$725$	18.0000	+	24.0000i	0.668503	+	0.891338i
$726$		0			0
$727$			16.0000i			0.593407i	0.954970	+	0.296704i	$0.0958873\pi$
$727$			16.0000i			0.593407i	−0.954970	+	0.296704i	$0.904113\pi$
$728$			2.00000i			0.0741249i
$729$		0			0
$730$	10.0000	−	20.0000i	0.370117	−	0.740233i
$731$	−64.0000			−2.36713
$732$		0			0
$733$			30.0000i			1.10808i	0.832492	+	0.554038i	$0.186914\pi$
$733$			30.0000i			1.10808i	−0.832492	+	0.554038i	$0.813086\pi$
$734$	−32.0000			−1.18114
$735$		0			0
$736$		0			0
$737$		−	24.0000i		−	0.884051i
$738$		0			0
$739$	−20.0000			−0.735712			−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000			−0.735712			−0.367856	+	0.929883i	$0.619908\pi$
$740$	−16.0000	−	8.00000i	−0.588172	−	0.294086i
$741$		0			0
$742$		−	2.00000i		−	0.0734223i
$743$			16.0000i			0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$			16.0000i			0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$		0			0
$745$	−18.0000	+	36.0000i	−0.659469	+	1.31894i
$746$	−36.0000			−1.31805
$747$		0			0
$748$			16.0000i			0.585018i
$749$	−12.0000			−0.438470
$750$		0			0
$751$	4.00000			0.145962			0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000			0.145962			0.0729810	+	0.997333i	$0.476749\pi$
$752$		−	4.00000i		−	0.145865i
$753$		0			0
$754$	12.0000			0.437014
$755$	8.00000	−	16.0000i	0.291150	−	0.582300i
$756$		0			0
$757$		−	4.00000i		−	0.145382i	−0.997354	−	0.0726912i	$-0.976841\pi$
$757$		−	4.00000i		−	0.145382i	0.997354	−	0.0726912i	$-0.0231588\pi$
$758$		−	28.0000i		−	1.01701i
$759$		0			0
$760$	4.00000	+	2.00000i	0.145095	+	0.0725476i
$761$	−6.00000			−0.217500			−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000			−0.217500			−0.108750	+	0.994069i	$0.534685\pi$
$762$		0			0
$763$			6.00000i			0.217215i
$764$	14.0000			0.506502
$765$		0			0
$766$		0			0
$767$			16.0000i			0.577727i
$768$		0			0
$769$	42.0000			1.51456			0.757279	−	0.653091i	$-0.226528\pi$
$769$	42.0000			1.51456			0.757279	+	0.653091i	$0.226528\pi$
$770$	2.00000	−	4.00000i	0.0720750	−	0.144150i
$771$		0			0
$772$		0			0
$773$		−	32.0000i		−	1.15096i	−0.817816	−	0.575480i	$-0.804815\pi$
$773$		−	32.0000i		−	1.15096i	0.817816	−	0.575480i	$-0.195185\pi$
$774$		0			0
$775$	−18.0000	−	24.0000i	−0.646579	−	0.862105i
$776$	−10.0000			−0.358979
$777$		0			0
$778$		−	6.00000i		−	0.215110i
$779$	−12.0000			−0.429945
$780$		0			0
$781$	28.0000			1.00192
$782$		0			0
$783$		0			0
$784$	−1.00000			−0.0357143
$785$	−20.0000	−	10.0000i	−0.713831	−	0.356915i
$786$		0			0
$787$		−	44.0000i		−	1.56843i	−0.620489	−	0.784215i	$-0.713066\pi$
$787$		−	44.0000i		−	1.56843i	0.620489	−	0.784215i	$-0.286934\pi$
$788$		−	6.00000i		−	0.213741i
$789$		0			0
$790$	8.00000	+	4.00000i	0.284627	+	0.142314i
$791$	−6.00000			−0.213335
$792$		0			0
$793$		−	20.0000i		−	0.710221i
$794$	2.00000			0.0709773
$795$		0			0
$796$	−26.0000			−0.921546
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$	−32.0000			−1.13208
$800$	−4.00000	+	3.00000i	−0.141421	+	0.106066i
$801$		0			0
$802$			18.0000i			0.635602i
$803$			20.0000i			0.705785i
$804$		0			0
$805$		0			0
$806$	−12.0000			−0.422682
$807$		0			0
$808$			10.0000i			0.351799i
$809$	6.00000			0.210949			0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000			0.210949			0.105474	+	0.994422i	$0.466364\pi$
$810$		0			0
$811$	14.0000			0.491606			0.245803	−	0.969320i	$-0.420948\pi$
$811$	14.0000			0.491606			0.245803	+	0.969320i	$0.420948\pi$
$812$			6.00000i			0.210559i
$813$		0			0
$814$	16.0000			0.560800
$815$		0			0
$816$		0			0
$817$		−	16.0000i		−	0.559769i
$818$			10.0000i			0.349642i
$819$		0			0
$820$	6.00000	−	12.0000i	0.209529	−	0.419058i
$821$	6.00000			0.209401			0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000			0.209401			0.104701	+	0.994504i	$0.466612\pi$
$822$		0			0
$823$			4.00000i			0.139431i	0.997567	+	0.0697156i	$0.0222092\pi$
$823$			4.00000i			0.139431i	−0.997567	+	0.0697156i	$0.977791\pi$
$824$		0			0
$825$		0			0
$826$	−8.00000			−0.278356
$827$			36.0000i			1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$			36.0000i			1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$		0			0
$829$	34.0000			1.18087			0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000			1.18087			0.590434	+	0.807086i	$0.298956\pi$
$830$	16.0000	−	32.0000i	0.555368	−	1.11074i
$831$		0			0
$832$			2.00000i			0.0693375i
$833$			8.00000i			0.277184i
$834$		0			0
$835$	16.0000	+	8.00000i	0.553703	+	0.276851i
$836$	−4.00000			−0.138343
$837$		0			0
$838$		−	12.0000i		−	0.414533i
$839$	44.0000			1.51905			0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000			1.51905			0.759524	+	0.650479i	$0.225432\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		−	10.0000i		−	0.344623i
$843$		0			0
$844$	8.00000			0.275371
$845$	9.00000	−	18.0000i	0.309609	−	0.619219i
$846$		0			0
$847$		−	7.00000i		−	0.240523i
$848$		−	2.00000i		−	0.0686803i
$849$		0			0
$850$	24.0000	+	32.0000i	0.823193	+	1.09759i
$851$		0			0
$852$		0			0
$853$		−	10.0000i		−	0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$		−	10.0000i		−	0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	10.0000			0.342193
$855$		0			0
$856$	−12.0000			−0.410152
$857$		−	20.0000i		−	0.683187i	−0.939848	−	0.341593i	$-0.889033\pi$
$857$		−	20.0000i		−	0.683187i	0.939848	−	0.341593i	$-0.110967\pi$
$858$		0			0
$859$	6.00000			0.204717			0.102359	−	0.994748i	$-0.467361\pi$
$859$	6.00000			0.204717			0.102359	+	0.994748i	$0.467361\pi$
$860$	16.0000	+	8.00000i	0.545595	+	0.272798i
$861$		0			0
$862$		−	6.00000i		−	0.204361i
$863$			48.0000i			1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$			48.0000i			1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$		0			0
$865$	−32.0000	−	16.0000i	−1.08803	−	0.544016i
$866$	−26.0000			−0.883516
$867$		0			0
$868$		−	6.00000i		−	0.203653i
$869$	−8.00000			−0.271381
$870$		0			0
$871$	−24.0000			−0.813209
$872$			6.00000i			0.203186i
$873$		0			0
$874$		0			0
$875$	−2.00000	−	11.0000i	−0.0676123	−	0.371868i
$876$		0			0
$877$		−	48.0000i		−	1.62084i	−0.585846	−	0.810422i	$-0.699238\pi$
$877$		−	48.0000i		−	1.62084i	0.585846	−	0.810422i	$-0.300762\pi$
$878$			6.00000i			0.202490i
$879$		0			0
$880$	2.00000	−	4.00000i	0.0674200	−	0.134840i
$881$	6.00000			0.202145			0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000			0.202145			0.101073	+	0.994879i	$0.467773\pi$
$882$		0			0
$883$		−	52.0000i		−	1.74994i	−0.484178	−	0.874970i	$-0.660881\pi$
$883$		−	52.0000i		−	1.74994i	0.484178	−	0.874970i	$-0.339119\pi$
$884$	16.0000			0.538138
$885$		0			0
$886$	36.0000			1.20944
$887$		−	36.0000i		−	1.20876i	−0.796696	−	0.604381i	$-0.793421\pi$
$887$		−	36.0000i		−	1.20876i	0.796696	−	0.604381i	$-0.206579\pi$
$888$		0			0
$889$	−12.0000			−0.402467
$890$	−20.0000	−	10.0000i	−0.670402	−	0.335201i
$891$		0			0
$892$		−	16.0000i		−	0.535720i
$893$		−	8.00000i		−	0.267710i
$894$		0			0
$895$	−2.00000	+	4.00000i	−0.0668526	+	0.133705i
$896$	−1.00000			−0.0334077
$897$		0			0
$898$			30.0000i			1.00111i
$899$	−36.0000			−1.20067
$900$		0			0
$901$	−16.0000			−0.533037
$902$			12.0000i			0.399556i
$903$		0			0
$904$	−6.00000			−0.199557
$905$	22.0000	−	44.0000i	0.731305	−	1.46261i
$906$		0			0
$907$		−	44.0000i		−	1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$		−	44.0000i		−	1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$		−	8.00000i		−	0.265489i
$909$		0			0
$910$	−4.00000	−	2.00000i	−0.132599	−	0.0662994i
$911$	−6.00000			−0.198789			−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000			−0.198789			−0.0993944	+	0.995048i	$0.531691\pi$
$912$		0			0
$913$			32.0000i			1.05905i
$914$	−28.0000			−0.926158
$915$		0			0
$916$	26.0000			0.859064
$917$			20.0000i			0.660458i
$918$		0			0
$919$	4.00000			0.131948			0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000			0.131948			0.0659739	+	0.997821i	$0.478985\pi$
$920$		0			0
$921$		0			0
$922$			18.0000i			0.592798i
$923$		−	28.0000i		−	0.921631i
$924$		0			0
$925$	32.0000	−	24.0000i	1.05215	−	0.789115i
$926$	12.0000			0.394344
$927$		0			0
$928$			6.00000i			0.196960i
$929$	−18.0000			−0.590561			−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000			−0.590561			−0.295280	+	0.955411i	$0.595413\pi$
$930$		0			0
$931$	−2.00000			−0.0655474
$932$		−	6.00000i		−	0.196537i
$933$		0			0
$934$	−16.0000			−0.523536
$935$	−32.0000	−	16.0000i	−1.04651	−	0.523256i
$936$		0			0
$937$		−	26.0000i		−	0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$		−	26.0000i		−	0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$		−	12.0000i		−	0.391814i
$939$		0			0
$940$	8.00000	+	4.00000i	0.260931	+	0.130466i
$941$	6.00000			0.195594			0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000			0.195594			0.0977972	+	0.995206i	$0.468820\pi$
$942$		0			0
$943$		0			0
$944$	−8.00000			−0.260378
$945$		0			0
$946$	−16.0000			−0.520205
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$		0			0
$949$	20.0000			0.649227
$950$	−8.00000	+	6.00000i	−0.259554	+	0.194666i
$951$		0			0
$952$			8.00000i			0.259281i
$953$		−	18.0000i		−	0.583077i	−0.956559	−	0.291539i	$-0.905833\pi$
$953$		−	18.0000i		−	0.583077i	0.956559	−	0.291539i	$-0.0941672\pi$
$954$		0			0
$955$	−14.0000	+	28.0000i	−0.453029	+	0.906059i
$956$	22.0000			0.711531
$957$		0			0
$958$		−	8.00000i		−	0.258468i
$959$	−14.0000			−0.452084
$960$		0			0
$961$	5.00000			0.161290
$962$		−	16.0000i		−	0.515861i
$963$		0			0
$964$	10.0000			0.322078
$965$		0			0
$966$		0			0
$967$		−	24.0000i		−	0.771788i	−0.922543	−	0.385894i	$-0.873893\pi$
$967$		−	24.0000i		−	0.771788i	0.922543	−	0.385894i	$-0.126107\pi$
$968$		−	7.00000i		−	0.224989i
$969$		0			0
$970$	10.0000	−	20.0000i	0.321081	−	0.642161i
$971$	−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
$973$			2.00000i			0.0641171i
$974$	36.0000			1.15351
$975$		0			0
$976$	10.0000			0.320092
$977$			18.0000i			0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$			18.0000i			0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$		0			0
$979$	20.0000			0.639203
$980$	1.00000	−	2.00000i	0.0319438	−	0.0638877i
$981$		0			0
$982$		−	42.0000i		−	1.34027i
$983$			12.0000i			0.382741i	0.981518	+	0.191370i	$0.0612931\pi$
$983$			12.0000i			0.382741i	−0.981518	+	0.191370i	$0.938707\pi$
$984$		0			0
$985$	12.0000	+	6.00000i	0.382352	+	0.191176i
$986$	48.0000			1.52863
$987$		0			0
$988$			4.00000i			0.127257i
$989$		0			0
$990$		0			0
$991$	−60.0000			−1.90596			−0.952981	−	0.303029i	$-0.902002\pi$
$991$	−60.0000			−1.90596			−0.952981	+	0.303029i	$0.902002\pi$
$992$		−	6.00000i		−	0.190500i
$993$		0			0
$994$	14.0000			0.444053
$995$	26.0000	−	52.0000i	0.824255	−	1.64851i
$996$		0			0
$997$		−	38.0000i		−	1.20347i	−0.798695	−	0.601736i	$-0.794476\pi$
$997$		−	38.0000i		−	1.20347i	0.798695	−	0.601736i	$-0.205524\pi$
$998$		−	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.d.379.1		2
3.2	odd	2		210.2.g.a.169.2	yes	2
4.3	odd	2		5040.2.t.k.1009.1		2
5.2	odd	4		3150.2.a.be.1.1		1
5.3	odd	4		3150.2.a.q.1.1		1
5.4	even	2	inner	630.2.g.d.379.2		2
12.11	even	2		1680.2.t.d.1009.1		2
15.2	even	4		1050.2.a.g.1.1		1
15.8	even	4		1050.2.a.m.1.1		1
15.14	odd	2		210.2.g.a.169.1	✓	2
20.19	odd	2		5040.2.t.k.1009.2		2
21.2	odd	6		1470.2.n.g.949.2		4
21.5	even	6		1470.2.n.c.949.2		4
21.11	odd	6		1470.2.n.g.79.1		4
21.17	even	6		1470.2.n.c.79.1		4
21.20	even	2		1470.2.g.e.589.2		2
60.23	odd	4		8400.2.a.ca.1.1		1
60.47	odd	4		8400.2.a.bd.1.1		1
60.59	even	2		1680.2.t.d.1009.2		2
105.44	odd	6		1470.2.n.g.949.1		4
105.59	even	6		1470.2.n.c.79.2		4
105.62	odd	4		7350.2.a.g.1.1		1
105.74	odd	6		1470.2.n.g.79.2		4
105.83	odd	4		7350.2.a.co.1.1		1
105.89	even	6		1470.2.n.c.949.1		4
105.104	even	2		1470.2.g.e.589.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.g.a.169.1	✓	2	15.14	odd	2
210.2.g.a.169.2	yes	2	3.2	odd	2
630.2.g.d.379.1		2	1.1	even	1	trivial
630.2.g.d.379.2		2	5.4	even	2	inner
1050.2.a.g.1.1		1	15.2	even	4
1050.2.a.m.1.1		1	15.8	even	4
1470.2.g.e.589.1		2	105.104	even	2
1470.2.g.e.589.2		2	21.20	even	2
1470.2.n.c.79.1		4	21.17	even	6
1470.2.n.c.79.2		4	105.59	even	6
1470.2.n.c.949.1		4	105.89	even	6
1470.2.n.c.949.2		4	21.5	even	6
1470.2.n.g.79.1		4	21.11	odd	6
1470.2.n.g.79.2		4	105.74	odd	6
1470.2.n.g.949.1		4	105.44	odd	6
1470.2.n.g.949.2		4	21.2	odd	6
1680.2.t.d.1009.1		2	12.11	even	2
1680.2.t.d.1009.2		2	60.59	even	2
3150.2.a.q.1.1		1	5.3	odd	4
3150.2.a.be.1.1		1	5.2	odd	4
5040.2.t.k.1009.1		2	4.3	odd	2
5040.2.t.k.1009.2		2	20.19	odd	2
7350.2.a.g.1.1		1	105.62	odd	4
7350.2.a.co.1.1		1	105.83	odd	4
8400.2.a.bd.1.1		1	60.47	odd	4
8400.2.a.ca.1.1		1	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:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	630.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} +1.00000i q^{7} +1.00000i q^{8} +(-2.00000 - 1.00000i) q^{10} +2.00000 q^{11} -2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -8.00000i q^{17} +2.00000 q^{19} +(-1.00000 + 2.00000i) q^{20} -2.00000i q^{22} +(-3.00000 - 4.00000i) q^{25} -2.00000 q^{26} -1.00000i q^{28} -6.00000 q^{29} +6.00000 q^{31} -1.00000i q^{32} -8.00000 q^{34} +(2.00000 + 1.00000i) q^{35} +8.00000i q^{37} -2.00000i q^{38} +(2.00000 + 1.00000i) q^{40} -6.00000 q^{41} -8.00000i q^{43} -2.00000 q^{44} -4.00000i q^{47} -1.00000 q^{49} +(-4.00000 + 3.00000i) q^{50} +2.00000i q^{52} -2.00000i q^{53} +(2.00000 - 4.00000i) q^{55} -1.00000 q^{56} +6.00000i q^{58} -8.00000 q^{59} +10.0000 q^{61} -6.00000i q^{62} -1.00000 q^{64} +(-4.00000 - 2.00000i) q^{65} -12.0000i q^{67} +8.00000i q^{68} +(1.00000 - 2.00000i) q^{70} +14.0000 q^{71} +10.0000i q^{73} +8.00000 q^{74} -2.00000 q^{76} +2.00000i q^{77} -4.00000 q^{79} +(1.00000 - 2.00000i) q^{80} +6.00000i q^{82} +16.0000i q^{83} +(-16.0000 - 8.00000i) q^{85} -8.00000 q^{86} +2.00000i q^{88} +10.0000 q^{89} +2.00000 q^{91} -4.00000 q^{94} +(2.00000 - 4.00000i) q^{95} +10.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^5	\( 2 q - 2 q^{4} + 2 q^{5} - 4 q^{10} + 4 q^{11} + 2 q^{14} + 2 q^{16} + 4 q^{19} - 2 q^{20} - 6 q^{25} - 4 q^{26} - 12 q^{29} + 12 q^{31} - 16 q^{34} + 4 q^{35} + 4 q^{40} - 12 q^{41} - 4 q^{44} - 2 q^{49} - 8 q^{50} + 4 q^{55} - 2 q^{56} - 16 q^{59} + 20 q^{61} - 2 q^{64} - 8 q^{65} + 2 q^{70} + 28 q^{71} + 16 q^{74} - 4 q^{76} - 8 q^{79} + 2 q^{80} - 32 q^{85} - 16 q^{86} + 20 q^{89} + 4 q^{91} - 8 q^{94} + 4 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^5 - 4 * q^10 + 4 * q^11 + 2 * q^14 + 2 * q^16 + 4 * q^19 - 2 * q^20 - 6 * q^25 - 4 * q^26 - 12 * q^29 + 12 * q^31 - 16 * q^34 + 4 * q^35 + 4 * q^40 - 12 * q^41 - 4 * q^44 - 2 * q^49 - 8 * q^50 + 4 * q^55 - 2 * q^56 - 16 * q^59 + 20 * q^61 - 2 * q^64 - 8 * q^65 + 2 * q^70 + 28 * q^71 + 16 * q^74 - 4 * q^76 - 8 * q^79 + 2 * q^80 - 32 * q^85 - 16 * q^86 + 20 * q^89 + 4 * q^91 - 8 * q^94 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more