LMFDB - Embedded newform 3150.2.g.e.2899.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(2899,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, 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("3150.2899");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$25.1528766367$
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			2899.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3150.2899
Dual form			3150.2.g.e.2899.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} -4.00000 q^{11} +2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +4.00000 q^{19} +4.00000i q^{22} -8.00000i q^{23} +2.00000 q^{26} -1.00000i q^{28} +6.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -2.00000 q^{34} -2.00000i q^{37} -4.00000i q^{38} -2.00000 q^{41} +12.0000i q^{43} +4.00000 q^{44} -8.00000 q^{46} +8.00000i q^{47} -1.00000 q^{49} -2.00000i q^{52} +6.00000i q^{53} -1.00000 q^{56} -6.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} +12.0000i q^{67} +2.00000i q^{68} -8.00000 q^{71} +14.0000i q^{73} -2.00000 q^{74} -4.00000 q^{76} -4.00000i q^{77} +2.00000i q^{82} +12.0000i q^{83} +12.0000 q^{86} -4.00000i q^{88} +2.00000 q^{89} -2.00000 q^{91} +8.00000i q^{92} +8.00000 q^{94} +10.0000i q^{97} +1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 8 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{19} + 4 q^{26} + 12 q^{29} - 16 q^{31} - 4 q^{34} - 4 q^{41} + 8 q^{44} - 16 q^{46} - 2 q^{49} - 2 q^{56} + 8 q^{59} - 4 q^{61} - 2 q^{64} - 16 q^{71} - 4 q^{74} - 8 q^{76} + 24 q^{86} + 4 q^{89} - 4 q^{91} + 16 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 8 * q^11 + 2 * q^14 + 2 * q^16 + 8 * q^19 + 4 * q^26 + 12 * q^29 - 16 * q^31 - 4 * q^34 - 4 * q^41 + 8 * q^44 - 16 * q^46 - 2 * q^49 - 2 * q^56 + 8 * q^59 - 4 * q^61 - 2 * q^64 - 16 * q^71 - 4 * q^74 - 8 * q^76 + 24 * q^86 + 4 * q^89 - 4 * q^91 + 16 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	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
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	4.00000i	0.852803i
$23$	− 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
$25$	0	0
$26$	2.00000	0.392232
$27$	0	0
$28$	− 1.00000i	− 0.188982i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	− 4.00000i	− 0.648886i
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	12.0000i	1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$	12.0000i	1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−8.00000	−1.17954
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	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
$58$	− 6.00000i	− 0.787839i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	8.00000i	1.01600i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	2.00000i	0.242536i
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−4.00000	−0.458831
$77$	− 4.00000i	− 0.455842i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	2.00000i	0.220863i
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	0	0
$86$	12.0000	1.29399
$87$	0	0
$88$	− 4.00000i	− 0.426401i
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	8.00000i	0.834058i
$93$	0	0
$94$	8.00000	0.825137
$95$	0	0
$96$	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
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	20.0000i	1.93347i	0.255774	+	0.966736i	$0.417670\pi$
$107$	20.0000i	1.93347i	−0.255774	+	0.966736i	$0.582330\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	1.00000i	0.0944911i
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	− 4.00000i	− 0.368230i
$119$	2.00000	0.183340
$120$	0	0
$121$	5.00000	0.454545
$122$	2.00000i	0.181071i
$123$	0	0
$124$	8.00000	0.718421
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	4.00000i	0.346844i
$134$	12.0000	1.03664
$135$	0	0
$136$	2.00000	0.171499
$137$	− 10.0000i	− 0.854358i	−0.904167	−	0.427179i	$-0.859507\pi$
$137$	− 10.0000i	− 0.854358i	0.904167	−	0.427179i	$-0.140493\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	0	0
$142$	8.00000i	0.671345i
$143$	− 8.00000i	− 0.668994i
$144$	0	0
$145$	0	0
$146$	14.0000	1.15865
$147$	0	0
$148$	2.00000i	0.164399i
$149$	−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	$0.394462\pi$
$152$	4.00000i	0.324443i
$153$	0	0
$154$	−4.00000	−0.322329
$155$	0	0
$156$	0	0
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	12.0000	0.931381
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	− 12.0000i	− 0.914991i
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	− 2.00000i	− 0.149906i
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	2.00000i	0.148250i
$183$	0	0
$184$	8.00000	0.589768
$185$	0	0
$186$	0	0
$187$	8.00000i	0.585018i
$188$	− 8.00000i	− 0.583460i
$189$	0	0
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	0	0
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	6.00000i	0.422159i
$203$	6.00000i	0.421117i
$204$	0	0
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	2.00000i	0.138675i
$209$	−16.0000	−1.10674
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	− 6.00000i	− 0.412082i
$213$	0	0
$214$	20.0000	1.36717
$215$	0	0
$216$	0	0
$217$	− 8.00000i	− 0.543075i
$218$	− 2.00000i	− 0.135457i
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−14.0000	−0.931266
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	10.0000i	0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$	10.0000i	0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	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	$0.303154\pi$
$242$	− 5.00000i	− 0.321412i
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	8.00000i	0.509028i
$248$	− 8.00000i	− 0.508001i
$249$	0	0
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	32.0000i	2.01182i
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	0	0
$262$	12.0000i	0.741362i
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	0	0
$265$	0	0
$266$	4.00000	0.245256
$267$	0	0
$268$	− 12.0000i	− 0.733017i
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	−10.0000	−0.604122
$275$	0	0
$276$	0	0
$277$	14.0000i	0.841178i	0.907251	+	0.420589i	$0.138177\pi$
$277$	14.0000i	0.841178i	−0.907251	+	0.420589i	$0.861823\pi$
$278$	20.0000i	1.19952i
$279$	0	0
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	− 20.0000i	− 1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$	− 20.0000i	− 1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−8.00000	−0.473050
$287$	− 2.00000i	− 0.118056i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	− 14.0000i	− 0.819288i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	0	0
$296$	2.00000	0.116248
$297$	0	0
$298$	18.0000i	1.04271i
$299$	16.0000	0.925304
$300$	0	0
$301$	−12.0000	−0.691669
$302$	− 8.00000i	− 0.460348i
$303$	0	0
$304$	4.00000	0.229416
$305$	0	0
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	4.00000i	0.227921i
$309$	0	0
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	− 34.0000i	− 1.92179i	−0.276907	−	0.960897i	$-0.589309\pi$
$313$	− 34.0000i	− 1.92179i	0.276907	−	0.960897i	$-0.410691\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	0	0
$317$	− 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
$319$	−24.0000	−1.34374
$320$	0	0
$321$	0	0
$322$	− 8.00000i	− 0.445823i
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	0	0
$326$	−12.0000	−0.664619
$327$	0	0
$328$	− 2.00000i	− 0.110432i
$329$	−8.00000	−0.441054
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	− 12.0000i	− 0.658586i
$333$	0	0
$334$	16.0000	0.875481
$335$	0	0
$336$	0	0
$337$	2.00000i	0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$	2.00000i	0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	− 9.00000i	− 0.489535i
$339$	0	0
$340$	0	0
$341$	32.0000	1.73290
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	−12.0000	−0.646997
$345$	0	0
$346$	14.0000	0.752645
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	0	0
$349$	34.0000	1.81998	0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	0	0
$351$	0	0
$352$	4.00000i	0.213201i
$353$	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
$356$	−2.00000	−0.106000
$357$	0	0
$358$	20.0000i	1.05703i
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	10.0000i	0.525588i
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	0	0
$367$	− 32.0000i	− 1.67039i	−0.549957	−	0.835193i	$-0.685356\pi$
$367$	− 32.0000i	− 1.67039i	0.549957	−	0.835193i	$-0.314644\pi$
$368$	− 8.00000i	− 0.417029i
$369$	0	0
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	− 14.0000i	− 0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	− 14.0000i	− 0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	−8.00000	−0.412568
$377$	12.0000i	0.618031i
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	0	0
$382$	− 16.0000i	− 0.818631i
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	0	0
$385$	0	0
$386$	14.0000	0.712581
$387$	0	0
$388$	− 10.0000i	− 0.507673i
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	− 1.00000i	− 0.0505076i
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	− 16.0000i	− 0.802008i
$399$	0	0
$400$	0	0
$401$	−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
$403$	− 16.0000i	− 0.797017i
$404$	6.00000	0.298511
$405$	0	0
$406$	6.00000	0.297775
$407$	8.00000i	0.396545i
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	0	0
$412$	8.00000i	0.394132i
$413$	4.00000i	0.196827i
$414$	0	0
$415$	0	0
$416$	2.00000	0.0980581
$417$	0	0
$418$	16.0000i	0.782586i
$419$	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
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	− 20.0000i	− 0.973585i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	− 2.00000i	− 0.0967868i
$428$	− 20.0000i	− 0.966736i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	6.00000i	0.288342i	0.989553	+	0.144171i	$0.0460515\pi$
$433$	6.00000i	0.288342i	−0.989553	+	0.144171i	$0.953949\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	− 32.0000i	− 1.53077i
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	0	0
$442$	− 4.00000i	− 0.190261i
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	− 1.00000i	− 0.0472456i
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	14.0000i	0.658505i
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	− 38.0000i	− 1.77757i	−0.458329	−	0.888783i	$-0.651552\pi$
$457$	− 38.0000i	− 1.77757i	0.458329	−	0.888783i	$-0.348448\pi$
$458$	− 26.0000i	− 1.21490i
$459$	0	0
$460$	0	0
$461$	34.0000	1.58354	0.791769	−	0.610821i	$-0.209160\pi$
$461$	34.0000	1.58354	0.791769	+	0.610821i	$0.209160\pi$
$462$	0	0
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	10.0000	0.463241
$467$	20.0000i	0.925490i	0.886492	+	0.462745i	$0.153135\pi$
$467$	20.0000i	0.925490i	−0.886492	+	0.462745i	$0.846865\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	0	0
$471$	0	0
$472$	4.00000i	0.184115i
$473$	− 48.0000i	− 2.20704i
$474$	0	0
$475$	0	0
$476$	−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
$481$	4.00000	0.182384
$482$	− 18.0000i	− 0.819878i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	0	0
$487$	− 24.0000i	− 1.08754i	−0.839233	−	0.543772i	$-0.816996\pi$
$487$	− 24.0000i	− 1.08754i	0.839233	−	0.543772i	$-0.183004\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	0	0
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	8.00000	0.359937
$495$	0	0
$496$	−8.00000	−0.359211
$497$	− 8.00000i	− 0.358849i
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	0	0
$502$	4.00000i	0.178529i
$503$	16.0000i	0.713405i	0.934218	+	0.356702i	$0.116099\pi$
$503$	16.0000i	0.713405i	−0.934218	+	0.356702i	$0.883901\pi$
$504$	0	0
$505$	0	0
$506$	32.0000	1.42257
$507$	0	0
$508$	0	0
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	0	0
$517$	− 32.0000i	− 1.40736i
$518$	− 2.00000i	− 0.0878750i
$519$	0	0
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	28.0000i	1.22435i	0.790721	+	0.612177i	$0.209706\pi$
$523$	28.0000i	1.22435i	−0.790721	+	0.612177i	$0.790294\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	24.0000	1.04645
$527$	16.0000i	0.696971i
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	0	0
$532$	− 4.00000i	− 0.173422i
$533$	− 4.00000i	− 0.173259i
$534$	0	0
$535$	0	0
$536$	−12.0000	−0.518321
$537$	0	0
$538$	− 30.0000i	− 1.29339i
$539$	4.00000	0.172292
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	24.0000i	1.03089i
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	10.0000i	0.427179i
$549$	0	0
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	0	0
$554$	14.0000	0.594803
$555$	0	0
$556$	20.0000	0.848189
$557$	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
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	10.0000i	0.421825i
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	0	0
$565$	0	0
$566$	−20.0000	−0.840663
$567$	0	0
$568$	− 8.00000i	− 0.335673i
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	8.00000i	0.334497i
$573$	0	0
$574$	−2.00000	−0.0834784
$575$	0	0
$576$	0	0
$577$	26.0000i	1.08239i	0.840896	+	0.541197i	$0.182029\pi$
$577$	26.0000i	1.08239i	−0.840896	+	0.541197i	$0.817971\pi$
$578$	− 13.0000i	− 0.540729i
$579$	0	0
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	− 24.0000i	− 0.993978i
$584$	−14.0000	−0.579324
$585$	0	0
$586$	6.00000	0.247858
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	0	0
$592$	− 2.00000i	− 0.0821995i
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	0	0
$595$	0	0
$596$	18.0000	0.737309
$597$	0	0
$598$	− 16.0000i	− 0.654289i
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	12.0000i	0.489083i
$603$	0	0
$604$	−8.00000	−0.325515
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	− 4.00000i	− 0.162221i
$609$	0	0
$610$	0	0
$611$	−16.0000	−0.647291
$612$	0	0
$613$	34.0000i	1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$	34.0000i	1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	4.00000	0.161165
$617$	− 10.0000i	− 0.402585i	−0.979531	−	0.201292i	$-0.935486\pi$
$617$	− 10.0000i	− 0.402585i	0.979531	−	0.201292i	$-0.0645141\pi$
$618$	0	0
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	0	0
$621$	0	0
$622$	8.00000i	0.320771i
$623$	2.00000i	0.0801283i
$624$	0	0
$625$	0	0
$626$	−34.0000	−1.35891
$627$	0	0
$628$	− 14.0000i	− 0.558661i
$629$	−4.00000	−0.159490
$630$	0	0
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	0	0
$633$	0	0
$634$	−14.0000	−0.556011
$635$	0	0
$636$	0	0
$637$	− 2.00000i	− 0.0792429i
$638$	24.0000i	0.950169i
$639$	0	0
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	20.0000i	0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$	20.0000i	0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	−8.00000	−0.314756
$647$	32.0000i	1.25805i	0.777385	+	0.629025i	$0.216546\pi$
$647$	32.0000i	1.25805i	−0.777385	+	0.629025i	$0.783454\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	12.0000i	0.469956i
$653$	− 34.0000i	− 1.33052i	−0.746611	−	0.665261i	$-0.768320\pi$
$653$	− 34.0000i	− 1.33052i	0.746611	−	0.665261i	$-0.231680\pi$
$654$	0	0
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	8.00000i	0.311872i
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	− 28.0000i	− 1.08825i
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	− 48.0000i	− 1.85857i
$668$	− 16.0000i	− 0.619059i
$669$	0	0
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	− 2.00000i	− 0.0770943i	−0.999257	−	0.0385472i	$-0.987727\pi$
$673$	− 2.00000i	− 0.0770943i	0.999257	−	0.0385472i	$-0.0122730\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	0	0
$682$	− 32.0000i	− 1.22534i
$683$	− 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
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	12.0000i	0.457496i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	− 14.0000i	− 0.532200i
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	0	0
$697$	4.00000i	0.151511i
$698$	− 34.0000i	− 1.28692i
$699$	0	0
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	− 8.00000i	− 0.301726i
$704$	4.00000	0.150756
$705$	0	0
$706$	18.0000	0.677439
$707$	− 6.00000i	− 0.225653i
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	0	0
$712$	2.00000i	0.0749532i
$713$	64.0000i	2.39682i
$714$	0	0
$715$	0	0
$716$	20.0000	0.747435
$717$	0	0
$718$	8.00000i	0.298557i
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	3.00000i	0.111648i
$723$	0	0
$724$	10.0000	0.371647
$725$	0	0
$726$	0	0
$727$	− 24.0000i	− 0.890111i	−0.895503	−	0.445055i	$-0.853184\pi$
$727$	− 24.0000i	− 0.890111i	0.895503	−	0.445055i	$-0.146816\pi$
$728$	− 2.00000i	− 0.0741249i
$729$	0	0
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	− 30.0000i	− 1.10808i	−0.832492	−	0.554038i	$-0.813086\pi$
$733$	− 30.0000i	− 1.10808i	0.832492	−	0.554038i	$-0.186914\pi$
$734$	−32.0000	−1.18114
$735$	0	0
$736$	−8.00000	−0.294884
$737$	− 48.0000i	− 1.76810i
$738$	0	0
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	0	0
$741$	0	0
$742$	6.00000i	0.220267i
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	0	0
$745$	0	0
$746$	−14.0000	−0.512576
$747$	0	0
$748$	− 8.00000i	− 0.292509i
$749$	−20.0000	−0.730784
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	8.00000i	0.291730i
$753$	0	0
$754$	12.0000	0.437014
$755$	0	0
$756$	0	0
$757$	− 34.0000i	− 1.23575i	−0.786276	−	0.617876i	$-0.787994\pi$
$757$	− 34.0000i	− 1.23575i	0.786276	−	0.617876i	$-0.212006\pi$
$758$	28.0000i	1.01701i
$759$	0	0
$760$	0	0
$761$	−2.00000	−0.0724999	−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000	−0.0724999	−0.0362500	+	0.999343i	$0.511541\pi$
$762$	0	0
$763$	2.00000i	0.0724049i
$764$	−16.0000	−0.578860
$765$	0	0
$766$	24.0000	0.867155
$767$	8.00000i	0.288863i
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	− 14.0000i	− 0.503871i
$773$	− 26.0000i	− 0.935155i	−0.883952	−	0.467578i	$-0.845127\pi$
$773$	− 26.0000i	− 0.935155i	0.883952	−	0.467578i	$-0.154873\pi$
$774$	0	0
$775$	0	0
$776$	−10.0000	−0.358979
$777$	0	0
$778$	18.0000i	0.645331i
$779$	−8.00000	−0.286630
$780$	0	0
$781$	32.0000	1.14505
$782$	16.0000i	0.572159i
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	6.00000i	0.213741i
$789$	0	0
$790$	0	0
$791$	14.0000	0.497783
$792$	0	0
$793$	− 4.00000i	− 0.142044i
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−16.0000	−0.567105
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	0	0
$802$	18.0000i	0.635602i
$803$	− 56.0000i	− 1.97620i
$804$	0	0
$805$	0	0
$806$	−16.0000	−0.563576
$807$	0	0
$808$	− 6.00000i	− 0.211079i
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	0	0
$811$	36.0000	1.26413	0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000	1.26413	0.632065	+	0.774915i	$0.282207\pi$
$812$	− 6.00000i	− 0.210559i
$813$	0	0
$814$	8.00000	0.280400
$815$	0	0
$816$	0	0
$817$	48.0000i	1.67931i
$818$	10.0000i	0.349642i
$819$	0	0
$820$	0	0
$821$	−46.0000	−1.60541	−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000	−1.60541	−0.802706	+	0.596376i	$0.796607\pi$
$822$	0	0
$823$	8.00000i	0.278862i	0.990232	+	0.139431i	$0.0445274\pi$
$823$	8.00000i	0.278862i	−0.990232	+	0.139431i	$0.955473\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	4.00000	0.139178
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	0	0
$832$	− 2.00000i	− 0.0693375i
$833$	2.00000i	0.0692959i
$834$	0	0
$835$	0	0
$836$	16.0000	0.553372
$837$	0	0
$838$	− 12.0000i	− 0.414533i
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	26.0000i	0.896019i
$843$	0	0
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	5.00000i	0.171802i
$848$	6.00000i	0.206041i
$849$	0	0
$850$	0	0
$851$	−16.0000	−0.548473
$852$	0	0
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	−20.0000	−0.683586
$857$	22.0000i	0.751506i	0.926720	+	0.375753i	$0.122616\pi$
$857$	22.0000i	0.751506i	−0.926720	+	0.375753i	$0.877384\pi$
$858$	0	0
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	6.00000	0.203888
$867$	0	0
$868$	8.00000i	0.271538i
$869$	0	0
$870$	0	0
$871$	−24.0000	−0.813209
$872$	2.00000i	0.0677285i
$873$	0	0
$874$	−32.0000	−1.08242
$875$	0	0
$876$	0	0
$877$	− 10.0000i	− 0.337676i	−0.985644	−	0.168838i	$-0.945999\pi$
$877$	− 10.0000i	− 0.337676i	0.985644	−	0.168838i	$-0.0540015\pi$
$878$	32.0000i	1.07995i
$879$	0	0
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	− 28.0000i	− 0.942275i	−0.882060	−	0.471138i	$-0.843844\pi$
$883$	− 28.0000i	− 0.942275i	0.882060	−	0.471138i	$-0.156156\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−4.00000	−0.134383
$887$	− 16.0000i	− 0.537227i	−0.963248	−	0.268614i	$-0.913434\pi$
$887$	− 16.0000i	− 0.537227i	0.963248	−	0.268614i	$-0.0865655\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	32.0000i	1.07084i
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	14.0000i	0.467186i
$899$	−48.0000	−1.60089
$900$	0	0
$901$	12.0000	0.399778
$902$	− 8.00000i	− 0.266371i
$903$	0	0
$904$	14.0000	0.465633
$905$	0	0
$906$	0	0
$907$	− 12.0000i	− 0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$	− 12.0000i	− 0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$	12.0000i	0.398234i
$909$	0	0
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	− 48.0000i	− 1.58857i
$914$	−38.0000	−1.25693
$915$	0	0
$916$	−26.0000	−0.859064
$917$	− 12.0000i	− 0.396275i
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	0	0
$922$	− 34.0000i	− 1.11973i
$923$	− 16.0000i	− 0.526646i
$924$	0	0
$925$	0	0
$926$	16.0000	0.525793
$927$	0	0
$928$	− 6.00000i	− 0.196960i
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	− 10.0000i	− 0.327561i
$933$	0	0
$934$	20.0000	0.654420
$935$	0	0
$936$	0	0
$937$	18.0000i	0.588034i	0.955800	+	0.294017i	$0.0949923\pi$
$937$	18.0000i	0.588034i	−0.955800	+	0.294017i	$0.905008\pi$
$938$	12.0000i	0.391814i
$939$	0	0
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	16.0000i	0.521032i
$944$	4.00000	0.130189
$945$	0	0
$946$	−48.0000	−1.56061
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	0	0
$949$	−28.0000	−0.908918
$950$	0	0
$951$	0	0
$952$	2.00000i	0.0648204i
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	0	0
$955$	0	0
$956$	16.0000	0.517477
$957$	0	0
$958$	0	0
$959$	10.0000	0.322917
$960$	0	0
$961$	33.0000	1.06452
$962$	− 4.00000i	− 0.128965i
$963$	0	0
$964$	−18.0000	−0.579741
$965$	0	0
$966$	0	0
$967$	− 8.00000i	− 0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$	− 8.00000i	− 0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$	5.00000i	0.160706i
$969$	0	0
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	− 20.0000i	− 0.641171i
$974$	−24.0000	−0.769010
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	0	0
$982$	36.0000i	1.14881i
$983$	32.0000i	1.02064i	0.859984	+	0.510321i	$0.170473\pi$
$983$	32.0000i	1.02064i	−0.859984	+	0.510321i	$0.829527\pi$
$984$	0	0
$985$	0	0
$986$	−12.0000	−0.382158
$987$	0	0
$988$	− 8.00000i	− 0.254514i
$989$	96.0000	3.05262
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	8.00000i	0.254000i
$993$	0	0
$994$	−8.00000	−0.253745
$995$	0	0
$996$	0	0
$997$	38.0000i	1.20347i	0.798695	+	0.601736i	$0.205524\pi$
$997$	38.0000i	1.20347i	−0.798695	+	0.601736i	$0.794476\pi$
$998$	4.00000i	0.126618i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.g.e.2899.1		2
3.2	odd	2		1050.2.g.d.799.2		2
5.2	odd	4		3150.2.a.w.1.1		1
5.3	odd	4		630.2.a.b.1.1		1
5.4	even	2	inner	3150.2.g.e.2899.2		2
15.2	even	4		1050.2.a.h.1.1		1
15.8	even	4		210.2.a.c.1.1	✓	1
15.14	odd	2		1050.2.g.d.799.1		2
20.3	even	4		5040.2.a.i.1.1		1
35.13	even	4		4410.2.a.l.1.1		1
60.23	odd	4		1680.2.a.q.1.1		1
60.47	odd	4		8400.2.a.p.1.1		1
105.23	even	12		1470.2.i.f.361.1		2
105.38	odd	12		1470.2.i.b.961.1		2
105.53	even	12		1470.2.i.f.961.1		2
105.62	odd	4		7350.2.a.p.1.1		1
105.68	odd	12		1470.2.i.b.361.1		2
105.83	odd	4		1470.2.a.q.1.1		1
120.53	even	4		6720.2.a.bp.1.1		1
120.83	odd	4		6720.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.a.c.1.1	✓	1	15.8	even	4
630.2.a.b.1.1		1	5.3	odd	4
1050.2.a.h.1.1		1	15.2	even	4
1050.2.g.d.799.1		2	15.14	odd	2
1050.2.g.d.799.2		2	3.2	odd	2
1470.2.a.q.1.1		1	105.83	odd	4
1470.2.i.b.361.1		2	105.68	odd	12
1470.2.i.b.961.1		2	105.38	odd	12
1470.2.i.f.361.1		2	105.23	even	12
1470.2.i.f.961.1		2	105.53	even	12
1680.2.a.q.1.1		1	60.23	odd	4
3150.2.a.w.1.1		1	5.2	odd	4
3150.2.g.e.2899.1		2	1.1	even	1	trivial
3150.2.g.e.2899.2		2	5.4	even	2	inner
4410.2.a.l.1.1		1	35.13	even	4
5040.2.a.i.1.1		1	20.3	even	4
6720.2.a.k.1.1		1	120.83	odd	4
6720.2.a.bp.1.1		1	120.53	even	4
7350.2.a.p.1.1		1	105.62	odd	4
8400.2.a.p.1.1		1	60.47	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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} -4.00000 q^{11} +2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +4.00000 q^{19} +4.00000i q^{22} -8.00000i q^{23} +2.00000 q^{26} -1.00000i q^{28} +6.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -2.00000 q^{34} -2.00000i q^{37} -4.00000i q^{38} -2.00000 q^{41} +12.0000i q^{43} +4.00000 q^{44} -8.00000 q^{46} +8.00000i q^{47} -1.00000 q^{49} -2.00000i q^{52} +6.00000i q^{53} -1.00000 q^{56} -6.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} +12.0000i q^{67} +2.00000i q^{68} -8.00000 q^{71} +14.0000i q^{73} -2.00000 q^{74} -4.00000 q^{76} -4.00000i q^{77} +2.00000i q^{82} +12.0000i q^{83} +12.0000 q^{86} -4.00000i q^{88} +2.00000 q^{89} -2.00000 q^{91} +8.00000i q^{92} +8.00000 q^{94} +10.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 8 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{19} + 4 q^{26} + 12 q^{29} - 16 q^{31} - 4 q^{34} - 4 q^{41} + 8 q^{44} - 16 q^{46} - 2 q^{49} - 2 q^{56} + 8 q^{59} - 4 q^{61} - 2 q^{64} - 16 q^{71} - 4 q^{74} - 8 q^{76} + 24 q^{86} + 4 q^{89} - 4 q^{91} + 16 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 8 * q^11 + 2 * q^14 + 2 * q^16 + 8 * q^19 + 4 * q^26 + 12 * q^29 - 16 * q^31 - 4 * q^34 - 4 * q^41 + 8 * q^44 - 16 * q^46 - 2 * q^49 - 2 * q^56 + 8 * q^59 - 4 * q^61 - 2 * q^64 - 16 * q^71 - 4 * q^74 - 8 * q^76 + 24 * q^86 + 4 * q^89 - 4 * q^91 + 16 * q^94

Introduction

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