LMFDB - Embedded newform 1650.2.c.i.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1650,2,Mod(199,1650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1650, 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("1650.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1650.c (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:	$13.1753163335$
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 330)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1650.199
Dual form			1650.2.c.i.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} +1.00000i q^{22} -1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +2.00000 q^{29} -1.00000i q^{32} -1.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} -4.00000i q^{38} +2.00000 q^{39} +2.00000 q^{41} -12.0000i q^{43} +1.00000 q^{44} -8.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +2.00000 q^{51} +2.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} +4.00000i q^{57} -2.00000i q^{58} +12.0000 q^{59} +6.00000 q^{61} -1.00000 q^{64} -1.00000 q^{66} -4.00000i q^{67} +2.00000i q^{68} -1.00000i q^{72} -6.00000i q^{73} +2.00000 q^{74} -4.00000 q^{76} -2.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} -2.00000i q^{82} +4.00000i q^{83} -12.0000 q^{86} +2.00000i q^{87} -1.00000i q^{88} -10.0000 q^{89} -8.00000 q^{94} +1.00000 q^{96} -2.00000i q^{97} -7.00000i q^{98} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{11} + 2 q^{16} + 8 q^{19} - 2 q^{24} - 4 q^{26} + 4 q^{29} - 4 q^{34} + 2 q^{36} + 4 q^{39} + 4 q^{41} + 2 q^{44} + 14 q^{49} + 4 q^{51} - 2 q^{54} + 24 q^{59} + 12 q^{61}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^11 + 2 * q^16 + 8 * q^19 - 2 * q^24 - 4 * q^26 + 4 * q^29 - 4 * q^34 + 2 * q^36 + 4 * q^39 + 4 * q^41 + 2 * q^44 + 14 * q^49 + 4 * q^51 - 2 * q^54 + 24 * q^59 + 12 * q^61 - 2 * q^64 - 2 * q^66 + 4 * q^74 - 8 * q^76 + 32 * q^79 + 2 * q^81 - 24 * q^86 - 20 * q^89 - 16 * q^94 + 2 * q^96 + 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1650\mathbb{Z}\right)^\times$.

$n$	$551$	$727$	$1201$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	− 1.00000i	− 0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	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	$-0.0779791\pi$
$18$	1.00000i	0.235702i
$19$	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
$22$	1.00000i	0.213201i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 1.00000i	− 0.174078i
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	− 4.00000i	− 0.648886i
$39$	2.00000	0.320256
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	− 12.0000i	− 1.82998i	−0.403473	−	0.914991i	$-0.632197\pi$
$43$	− 12.0000i	− 1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	0	0
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	1.00000i	0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	2.00000	0.280056
$52$	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	$0.864802\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	4.00000i	0.529813i
$58$	− 2.00000i	− 0.262613i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	2.00000i	0.242536i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	− 2.00000i	− 0.226455i
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 2.00000i	− 0.220863i
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	−12.0000	−1.29399
$87$	2.00000i	0.214423i
$88$	− 1.00000i	− 0.106600i
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−8.00000	−0.825137
$95$	0	0
$96$	1.00000	0.102062
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	− 7.00000i	− 0.707107i
$99$	1.00000	0.100504
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	− 2.00000i	− 0.198030i
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	1.00000i	0.0962250i
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	10.0000i	0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$	10.0000i	0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	−2.00000	−0.185695
$117$	2.00000i	0.184900i
$118$	− 12.0000i	− 1.10469i
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 6.00000i	− 0.543214i
$123$	2.00000i	0.180334i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	1.00000i	0.0883883i
$129$	12.0000	1.05654
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	1.00000i	0.0870388i
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	2.00000	0.171499
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	2.00000i	0.167248i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−6.00000	−0.496564
$147$	7.00000i	0.577350i
$148$	− 2.00000i	− 0.164399i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	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
$153$	2.00000i	0.161690i
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	− 22.0000i	− 1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	− 22.0000i	− 1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$	− 16.0000i	− 1.27289i
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	4.00000	0.310460
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	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	$0.821365\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	12.0000i	0.901975i
$178$	10.0000i	0.749532i
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	6.00000i	0.443533i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.00000i	0.146254i
$188$	8.00000i	0.583460i
$189$	0	0
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	18.0000i	1.29567i	0.761781	+	0.647834i	$0.224325\pi$
$193$	18.0000i	1.29567i	−0.761781	+	0.647834i	$0.775675\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	− 1.00000i	− 0.0710669i
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	− 6.00000i	− 0.422159i
$203$	0	0
$204$	−2.00000	−0.140028
$205$	0	0
$206$	0	0
$207$	0	0
$208$	− 2.00000i	− 0.138675i
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	− 6.00000i	− 0.412082i
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	− 10.0000i	− 0.677285i
$219$	6.00000	0.405442
$220$	0	0
$221$	−4.00000	−0.269069
$222$	2.00000i	0.134231i
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	0	0
$225$	0	0
$226$	10.0000	0.665190
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	− 4.00000i	− 0.264906i
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	2.00000i	0.131306i
$233$	10.0000i	0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$	10.0000i	0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−12.0000	−0.781133
$237$	16.0000i	1.03931i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	1.00000i	0.0641500i
$244$	−6.00000	−0.384111
$245$	0	0
$246$	2.00000	0.127515
$247$	− 8.00000i	− 0.509028i
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.0000i	1.37232i	0.727450	+	0.686161i	$0.240706\pi$
$257$	22.0000i	1.37232i	−0.727450	+	0.686161i	$0.759294\pi$
$258$	− 12.0000i	− 0.747087i
$259$	0	0
$260$	0	0
$261$	−2.00000	−0.123797
$262$	20.0000i	1.23560i
$263$	− 8.00000i	− 0.493301i	−0.969104	−	0.246651i	$-0.920670\pi$
$263$	− 8.00000i	− 0.493301i	0.969104	−	0.246651i	$-0.0793300\pi$
$264$	1.00000	0.0615457
$265$	0	0
$266$	0	0
$267$	− 10.0000i	− 0.611990i
$268$	4.00000i	0.244339i
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	−18.0000	−1.08742
$275$	0	0
$276$	0	0
$277$	26.0000i	1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$	26.0000i	1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$	4.00000i	0.239904i
$279$	0	0
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	− 8.00000i	− 0.476393i
$283$	− 12.0000i	− 0.713326i	−0.934233	−	0.356663i	$-0.883914\pi$
$283$	− 12.0000i	− 0.713326i	0.934233	−	0.356663i	$-0.116086\pi$
$284$	0	0
$285$	0	0
$286$	2.00000	0.118262
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	2.00000	0.117242
$292$	6.00000i	0.351123i
$293$	− 26.0000i	− 1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$	− 26.0000i	− 1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	−2.00000	−0.116248
$297$	1.00000i	0.0580259i
$298$	6.00000i	0.347571i
$299$	0	0
$300$	0	0
$301$	0	0
$302$	− 8.00000i	− 0.460348i
$303$	6.00000i	0.344691i
$304$	4.00000	0.229416
$305$	0	0
$306$	2.00000	0.114332
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−32.0000	−1.81455	−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000	−1.81455	−0.907277	+	0.420534i	$0.861843\pi$
$312$	2.00000i	0.113228i
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	−16.0000	−0.900070
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	6.00000i	0.336463i
$319$	−2.00000	−0.111979
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	− 8.00000i	− 0.445132i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	10.0000i	0.553001i
$328$	2.00000i	0.110432i
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	− 4.00000i	− 0.219529i
$333$	− 2.00000i	− 0.109599i
$334$	−8.00000	−0.437741
$335$	0	0
$336$	0	0
$337$	30.0000i	1.63420i	0.576493	+	0.817102i	$0.304421\pi$
$337$	30.0000i	1.63420i	−0.576493	+	0.817102i	$0.695579\pi$
$338$	− 9.00000i	− 0.489535i
$339$	−10.0000	−0.543125
$340$	0	0
$341$	0	0
$342$	4.00000i	0.216295i
$343$	0	0
$344$	12.0000	0.646997
$345$	0	0
$346$	14.0000	0.752645
$347$	36.0000i	1.93258i	0.257454	+	0.966291i	$0.417117\pi$
$347$	36.0000i	1.93258i	−0.257454	+	0.966291i	$0.582883\pi$
$348$	− 2.00000i	− 0.107211i
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	1.00000i	0.0533002i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	10.0000	0.529999
$357$	0	0
$358$	− 4.00000i	− 0.211407i
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 22.0000i	− 1.15629i
$363$	1.00000i	0.0524864i
$364$	0	0
$365$	0	0
$366$	6.00000	0.313625
$367$	− 24.0000i	− 1.25279i	−0.779506	−	0.626395i	$-0.784530\pi$
$367$	− 24.0000i	− 1.25279i	0.779506	−	0.626395i	$-0.215470\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 10.0000i	− 0.517780i	−0.965907	−	0.258890i	$-0.916643\pi$
$373$	− 10.0000i	− 0.517780i	0.965907	−	0.258890i	$-0.0833568\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	8.00000	0.412568
$377$	− 4.00000i	− 0.206010i
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	24.0000i	1.22795i
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	18.0000	0.916176
$387$	12.0000i	0.609994i
$388$	2.00000i	0.101535i
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	0	0
$392$	7.00000i	0.353553i
$393$	− 20.0000i	− 1.00887i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	10.0000i	0.501886i	0.968002	+	0.250943i	$0.0807406\pi$
$397$	10.0000i	0.501886i	−0.968002	+	0.250943i	$0.919259\pi$
$398$	8.00000i	0.401004i
$399$	0	0
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	− 4.00000i	− 0.199502i
$403$	0	0
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	− 2.00000i	− 0.0991363i
$408$	2.00000i	0.0990148i
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	− 4.00000i	− 0.195881i
$418$	4.00000i	0.195646i
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	4.00000i	0.194717i
$423$	8.00000i	0.388973i
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	0	0
$428$	12.0000i	0.580042i
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	0	0
$438$	− 6.00000i	− 0.286691i
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	4.00000i	0.190261i
$443$	− 36.0000i	− 1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$	− 36.0000i	− 1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	8.00000	0.378811
$447$	− 6.00000i	− 0.283790i
$448$	0	0
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	− 10.0000i	− 0.470360i
$453$	8.00000i	0.375873i
$454$	12.0000	0.563188
$455$	0	0
$456$	−4.00000	−0.187317
$457$	38.0000i	1.77757i	0.458329	+	0.888783i	$0.348448\pi$
$457$	38.0000i	1.77757i	−0.458329	+	0.888783i	$0.651552\pi$
$458$	22.0000i	1.02799i
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	− 8.00000i	− 0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$	− 8.00000i	− 0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	10.0000	0.463241
$467$	− 20.0000i	− 0.925490i	−0.886492	−	0.462745i	$-0.846865\pi$
$467$	− 20.0000i	− 0.925490i	0.886492	−	0.462745i	$-0.153135\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	0	0
$470$	0	0
$471$	22.0000	1.01371
$472$	12.0000i	0.552345i
$473$	12.0000i	0.551761i
$474$	16.0000	0.734904
$475$	0	0
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	14.0000i	0.637683i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 32.0000i	− 1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$487$	− 32.0000i	− 1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$488$	6.00000i	0.271607i
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	− 2.00000i	− 0.0901670i
$493$	− 4.00000i	− 0.180151i
$494$	−8.00000	−0.359937
$495$	0	0
$496$	0	0
$497$	0	0
$498$	4.00000i	0.179244i
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	12.0000i	0.535586i
$503$	40.0000i	1.78351i	0.452517	+	0.891756i	$0.350526\pi$
$503$	40.0000i	1.78351i	−0.452517	+	0.891756i	$0.649474\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.00000i	0.399704i
$508$	8.00000i	0.354943i
$509$	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
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	− 4.00000i	− 0.176604i
$514$	22.0000	0.970378
$515$	0	0
$516$	−12.0000	−0.528271
$517$	8.00000i	0.351840i
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	2.00000i	0.0875376i
$523$	36.0000i	1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$	36.0000i	1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$	20.0000	0.873704
$525$	0	0
$526$	−8.00000	−0.348817
$527$	0	0
$528$	− 1.00000i	− 0.0435194i
$529$	23.0000	1.00000
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	− 4.00000i	− 0.173259i
$534$	−10.0000	−0.432742
$535$	0	0
$536$	4.00000	0.172774
$537$	4.00000i	0.172613i
$538$	14.0000i	0.603583i
$539$	−7.00000	−0.301511
$540$	0	0
$541$	−26.0000	−1.11783	−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000	−1.11783	−0.558914	+	0.829226i	$0.688782\pi$
$542$	− 32.0000i	− 1.37452i
$543$	22.0000i	0.944110i
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	18.0000i	0.768922i
$549$	−6.00000	−0.256074
$550$	0	0
$551$	8.00000	0.340811
$552$	0	0
$553$	0	0
$554$	26.0000	1.10463
$555$	0	0
$556$	4.00000	0.169638
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	− 2.00000i	− 0.0843649i
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	−12.0000	−0.504398
$567$	0	0
$568$	0	0
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	− 2.00000i	− 0.0836242i
$573$	− 24.0000i	− 1.00261i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	30.0000i	1.24892i	0.781058	+	0.624458i	$0.214680\pi$
$577$	30.0000i	1.24892i	−0.781058	+	0.624458i	$0.785320\pi$
$578$	− 13.0000i	− 0.540729i
$579$	−18.0000	−0.748054
$580$	0	0
$581$	0	0
$582$	− 2.00000i	− 0.0829027i
$583$	− 6.00000i	− 0.248495i
$584$	6.00000	0.248282
$585$	0	0
$586$	−26.0000	−1.07405
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	− 7.00000i	− 0.288675i
$589$	0	0
$590$	0	0
$591$	6.00000	0.246807
$592$	2.00000i	0.0821995i
$593$	2.00000i	0.0821302i	0.999156	+	0.0410651i	$0.0130751\pi$
$593$	2.00000i	0.0821302i	−0.999156	+	0.0410651i	$0.986925\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	6.00000	0.245770
$597$	− 8.00000i	− 0.327418i
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	6.00000	0.243733
$607$	− 24.0000i	− 0.974130i	−0.873366	−	0.487065i	$-0.838067\pi$
$607$	− 24.0000i	− 0.974130i	0.873366	−	0.487065i	$-0.161933\pi$
$608$	− 4.00000i	− 0.162221i
$609$	0	0
$610$	0	0
$611$	−16.0000	−0.647291
$612$	− 2.00000i	− 0.0808452i
$613$	6.00000i	0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$	6.00000i	0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	0	0
$622$	32.0000i	1.28308i
$623$	0	0
$624$	2.00000	0.0800641
$625$	0	0
$626$	10.0000	0.399680
$627$	− 4.00000i	− 0.159745i
$628$	22.0000i	0.877896i
$629$	4.00000	0.159490
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	16.0000i	0.636446i
$633$	− 4.00000i	− 0.158986i
$634$	18.0000	0.714871
$635$	0	0
$636$	6.00000	0.237915
$637$	− 14.0000i	− 0.554700i
$638$	2.00000i	0.0791808i
$639$	0	0
$640$	0	0
$641$	−14.0000	−0.552967	−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000	−0.552967	−0.276483	+	0.961019i	$0.589169\pi$
$642$	− 12.0000i	− 0.473602i
$643$	− 44.0000i	− 1.73519i	−0.497271	−	0.867595i	$-0.665665\pi$
$643$	− 44.0000i	− 1.73519i	0.497271	−	0.867595i	$-0.334335\pi$
$644$	0	0
$645$	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	$0.783454\pi$
$648$	1.00000i	0.0392837i
$649$	−12.0000	−0.471041
$650$	0	0
$651$	0	0
$652$	− 4.00000i	− 0.156652i
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	2.00000	0.0780869
$657$	6.00000i	0.234082i
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	20.0000i	0.777322i
$663$	− 4.00000i	− 0.155347i
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	8.00000i	0.309529i
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−6.00000	−0.231627
$672$	0	0
$673$	− 14.0000i	− 0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$	− 14.0000i	− 0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$	30.0000	1.15556
$675$	0	0
$676$	−9.00000	−0.346154
$677$	26.0000i	0.999261i	0.866239	+	0.499631i	$0.166531\pi$
$677$	26.0000i	0.999261i	−0.866239	+	0.499631i	$0.833469\pi$
$678$	10.0000i	0.384048i
$679$	0	0
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	− 20.0000i	− 0.765279i	−0.923898	−	0.382639i	$-0.875015\pi$
$683$	− 20.0000i	− 0.765279i	0.923898	−	0.382639i	$-0.124985\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	0	0
$687$	− 22.0000i	− 0.839352i
$688$	− 12.0000i	− 0.457496i
$689$	12.0000	0.457164
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	− 14.0000i	− 0.532200i
$693$	0	0
$694$	36.0000	1.36654
$695$	0	0
$696$	−2.00000	−0.0758098
$697$	− 4.00000i	− 0.151511i
$698$	− 10.0000i	− 0.378506i
$699$	−10.0000	−0.378235
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	2.00000i	0.0754851i
$703$	8.00000i	0.301726i
$704$	1.00000	0.0376889
$705$	0	0
$706$	−6.00000	−0.225813
$707$	0	0
$708$	− 12.0000i	− 0.450988i
$709$	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
$711$	−16.0000	−0.600047
$712$	− 10.0000i	− 0.374766i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−4.00000	−0.149487
$717$	0	0
$718$	24.0000i	0.895672i
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	0	0
$722$	3.00000i	0.111648i
$723$	− 14.0000i	− 0.520666i
$724$	−22.0000	−0.817624
$725$	0	0
$726$	1.00000	0.0371135
$727$	− 16.0000i	− 0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$	− 16.0000i	− 0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	− 6.00000i	− 0.221766i
$733$	− 2.00000i	− 0.0738717i	−0.999318	−	0.0369358i	$-0.988240\pi$
$733$	− 2.00000i	− 0.0738717i	0.999318	−	0.0369358i	$-0.0117597\pi$
$734$	−24.0000	−0.885856
$735$	0	0
$736$	0	0
$737$	4.00000i	0.147342i
$738$	2.00000i	0.0736210i
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	8.00000	0.293887
$742$	0	0
$743$	8.00000i	0.293492i	0.989174	+	0.146746i	$0.0468799\pi$
$743$	8.00000i	0.293492i	−0.989174	+	0.146746i	$0.953120\pi$
$744$	0	0
$745$	0	0
$746$	−10.0000	−0.366126
$747$	− 4.00000i	− 0.146352i
$748$	− 2.00000i	− 0.0731272i
$749$	0	0
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	− 8.00000i	− 0.291730i
$753$	− 12.0000i	− 0.437304i
$754$	−4.00000	−0.145671
$755$	0	0
$756$	0	0
$757$	− 46.0000i	− 1.67190i	−0.548807	−	0.835949i	$-0.684918\pi$
$757$	− 46.0000i	− 1.67190i	0.548807	−	0.835949i	$-0.315082\pi$
$758$	− 20.0000i	− 0.726433i
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	− 8.00000i	− 0.289809i
$763$	0	0
$764$	24.0000	0.868290
$765$	0	0
$766$	24.0000	0.867155
$767$	− 24.0000i	− 0.866590i
$768$	1.00000i	0.0360844i
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	−22.0000	−0.792311
$772$	− 18.0000i	− 0.647834i
$773$	− 42.0000i	− 1.51064i	−0.655359	−	0.755318i	$-0.727483\pi$
$773$	− 42.0000i	− 1.51064i	0.655359	−	0.755318i	$-0.272517\pi$
$774$	12.0000	0.431331
$775$	0	0
$776$	2.00000	0.0717958
$777$	0	0
$778$	− 26.0000i	− 0.932145i
$779$	8.00000	0.286630
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 2.00000i	− 0.0714742i
$784$	7.00000	0.250000
$785$	0	0
$786$	−20.0000	−0.713376
$787$	20.0000i	0.712923i	0.934310	+	0.356462i	$0.116017\pi$
$787$	20.0000i	0.712923i	−0.934310	+	0.356462i	$0.883983\pi$
$788$	6.00000i	0.213741i
$789$	8.00000	0.284808
$790$	0	0
$791$	0	0
$792$	1.00000i	0.0355335i
$793$	− 12.0000i	− 0.426132i
$794$	10.0000	0.354887
$795$	0	0
$796$	8.00000	0.283552
$797$	2.00000i	0.0708436i	0.999372	+	0.0354218i	$0.0112775\pi$
$797$	2.00000i	0.0708436i	−0.999372	+	0.0354218i	$0.988723\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	10.0000	0.353333
$802$	− 2.00000i	− 0.0706225i
$803$	6.00000i	0.211735i
$804$	−4.00000	−0.141069
$805$	0	0
$806$	0	0
$807$	− 14.0000i	− 0.492823i
$808$	6.00000i	0.211079i
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	32.0000i	1.12229i
$814$	−2.00000	−0.0701000
$815$	0	0
$816$	2.00000	0.0700140
$817$	− 48.0000i	− 1.67931i
$818$	− 6.00000i	− 0.209785i
$819$	0	0
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	− 18.0000i	− 0.627822i
$823$	− 16.0000i	− 0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$	− 16.0000i	− 0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.00000i	0.139094i	0.997579	+	0.0695468i	$0.0221553\pi$
$827$	4.00000i	0.139094i	−0.997579	+	0.0695468i	$0.977845\pi$
$828$	0	0
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0	0
$831$	−26.0000	−0.901930
$832$	2.00000i	0.0693375i
$833$	− 14.0000i	− 0.485071i
$834$	−4.00000	−0.138509
$835$	0	0
$836$	4.00000	0.138343
$837$	0	0
$838$	28.0000i	0.967244i
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	− 6.00000i	− 0.206774i
$843$	2.00000i	0.0688837i
$844$	4.00000	0.137686
$845$	0	0
$846$	8.00000	0.275046
$847$	0	0
$848$	6.00000i	0.206041i
$849$	12.0000	0.411839
$850$	0	0
$851$	0	0
$852$	0	0
$853$	22.0000i	0.753266i	0.926363	+	0.376633i	$0.122918\pi$
$853$	22.0000i	0.753266i	−0.926363	+	0.376633i	$0.877082\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	− 42.0000i	− 1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$	− 42.0000i	− 1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$	2.00000i	0.0682789i
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	0	0
$861$	0	0
$862$	− 32.0000i	− 1.08992i
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	2.00000	0.0679628
$867$	13.0000i	0.441503i
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	−8.00000	−0.271070
$872$	10.0000i	0.338643i
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	0	0
$876$	−6.00000	−0.202721
$877$	− 46.0000i	− 1.55331i	−0.629926	−	0.776655i	$-0.716915\pi$
$877$	− 46.0000i	− 1.55331i	0.629926	−	0.776655i	$-0.283085\pi$
$878$	− 8.00000i	− 0.269987i
$879$	26.0000	0.876958
$880$	0	0
$881$	50.0000	1.68454	0.842271	−	0.539054i	$-0.181218\pi$
$881$	50.0000	1.68454	0.842271	+	0.539054i	$0.181218\pi$
$882$	7.00000i	0.235702i
$883$	− 44.0000i	− 1.48072i	−0.672212	−	0.740359i	$-0.734656\pi$
$883$	− 44.0000i	− 1.48072i	0.672212	−	0.740359i	$-0.265344\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	−36.0000	−1.20944
$887$	− 8.00000i	− 0.268614i	−0.990940	−	0.134307i	$-0.957119\pi$
$887$	− 8.00000i	− 0.268614i	0.990940	−	0.134307i	$-0.0428808\pi$
$888$	− 2.00000i	− 0.0671156i
$889$	0	0
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	− 8.00000i	− 0.267860i
$893$	− 32.0000i	− 1.07084i
$894$	−6.00000	−0.200670
$895$	0	0
$896$	0	0
$897$	0	0
$898$	34.0000i	1.13459i
$899$	0	0
$900$	0	0
$901$	12.0000	0.399778
$902$	2.00000i	0.0665927i
$903$	0	0
$904$	−10.0000	−0.332595
$905$	0	0
$906$	8.00000	0.265782
$907$	− 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
$909$	−6.00000	−0.199007
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	4.00000i	0.132453i
$913$	− 4.00000i	− 0.132381i
$914$	38.0000	1.25693
$915$	0	0
$916$	22.0000	0.726900
$917$	0	0
$918$	2.00000i	0.0660098i
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	− 30.0000i	− 0.987997i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−8.00000	−0.262896
$927$	0	0
$928$	− 2.00000i	− 0.0656532i
$929$	46.0000	1.50921	0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000	1.50921	0.754606	+	0.656179i	$0.227828\pi$
$930$	0	0
$931$	28.0000	0.917663
$932$	− 10.0000i	− 0.327561i
$933$	− 32.0000i	− 1.04763i
$934$	−20.0000	−0.654420
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	6.00000i	0.196011i	0.995186	+	0.0980057i	$0.0312463\pi$
$937$	6.00000i	0.196011i	−0.995186	+	0.0980057i	$0.968754\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	− 22.0000i	− 0.716799i
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	12.0000	0.390154
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	− 16.0000i	− 0.519656i
$949$	−12.0000	−0.389536
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	0	0
$957$	− 2.00000i	− 0.0646508i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	− 4.00000i	− 0.128965i
$963$	12.0000i	0.386695i
$964$	14.0000	0.450910
$965$	0	0
$966$	0	0
$967$	− 16.0000i	− 0.514525i	−0.966342	−	0.257263i	$-0.917179\pi$
$967$	− 16.0000i	− 0.514525i	0.966342	−	0.257263i	$-0.0828206\pi$
$968$	1.00000i	0.0321412i
$969$	8.00000	0.256997
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	0	0
$974$	−32.0000	−1.02535
$975$	0	0
$976$	6.00000	0.192055
$977$	54.0000i	1.72761i	0.503824	+	0.863807i	$0.331926\pi$
$977$	54.0000i	1.72761i	−0.503824	+	0.863807i	$0.668074\pi$
$978$	4.00000i	0.127906i
$979$	10.0000	0.319601
$980$	0	0
$981$	−10.0000	−0.319275
$982$	12.0000i	0.382935i
$983$	16.0000i	0.510321i	0.966899	+	0.255160i	$0.0821283\pi$
$983$	16.0000i	0.510321i	−0.966899	+	0.255160i	$0.917872\pi$
$984$	−2.00000	−0.0637577
$985$	0	0
$986$	−4.00000	−0.127386
$987$	0	0
$988$	8.00000i	0.254514i
$989$	0	0
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	− 20.0000i	− 0.634681i
$994$	0	0
$995$	0	0
$996$	4.00000	0.126745
$997$	26.0000i	0.823428i	0.911313	+	0.411714i	$0.135070\pi$
$997$	26.0000i	0.823428i	−0.911313	+	0.411714i	$0.864930\pi$
$998$	36.0000i	1.13956i
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1650.2.c.i.199.1		2
3.2	odd	2		4950.2.c.v.199.2		2
5.2	odd	4		330.2.a.e.1.1	✓	1
5.3	odd	4		1650.2.a.b.1.1		1
5.4	even	2	inner	1650.2.c.i.199.2		2
15.2	even	4		990.2.a.c.1.1		1
15.8	even	4		4950.2.a.bk.1.1		1
15.14	odd	2		4950.2.c.v.199.1		2
20.7	even	4		2640.2.a.h.1.1		1
55.32	even	4		3630.2.a.k.1.1		1
60.47	odd	4		7920.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
330.2.a.e.1.1	✓	1	5.2	odd	4
990.2.a.c.1.1		1	15.2	even	4
1650.2.a.b.1.1		1	5.3	odd	4
1650.2.c.i.199.1		2	1.1	even	1	trivial
1650.2.c.i.199.2		2	5.4	even	2	inner
2640.2.a.h.1.1		1	20.7	even	4
3630.2.a.k.1.1		1	55.32	even	4
4950.2.a.bk.1.1		1	15.8	even	4
4950.2.c.v.199.1		2	15.14	odd	2
4950.2.c.v.199.2		2	3.2	odd	2
7920.2.a.g.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:	\( N \)	\(=\)	\( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1650.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} +1.00000i q^{22} -1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +2.00000 q^{29} -1.00000i q^{32} -1.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} -4.00000i q^{38} +2.00000 q^{39} +2.00000 q^{41} -12.0000i q^{43} +1.00000 q^{44} -8.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +2.00000 q^{51} +2.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} +4.00000i q^{57} -2.00000i q^{58} +12.0000 q^{59} +6.00000 q^{61} -1.00000 q^{64} -1.00000 q^{66} -4.00000i q^{67} +2.00000i q^{68} -1.00000i q^{72} -6.00000i q^{73} +2.00000 q^{74} -4.00000 q^{76} -2.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} -2.00000i q^{82} +4.00000i q^{83} -12.0000 q^{86} +2.00000i q^{87} -1.00000i q^{88} -10.0000 q^{89} -8.00000 q^{94} +1.00000 q^{96} -2.00000i q^{97} -7.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{11} + 2 q^{16} + 8 q^{19} - 2 q^{24} - 4 q^{26} + 4 q^{29} - 4 q^{34} + 2 q^{36} + 4 q^{39} + 4 q^{41} + 2 q^{44} + 14 q^{49} + 4 q^{51} - 2 q^{54} + 24 q^{59} + 12 q^{61}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^11 + 2 * q^16 + 8 * q^19 - 2 * q^24 - 4 * q^26 + 4 * q^29 - 4 * q^34 + 2 * q^36 + 4 * q^39 + 4 * q^41 + 2 * q^44 + 14 * q^49 + 4 * q^51 - 2 * q^54 + 24 * q^59 + 12 * q^61 - 2 * q^64 - 2 * q^66 + 4 * q^74 - 8 * q^76 + 32 * q^79 + 2 * q^81 - 24 * q^86 - 20 * q^89 - 16 * q^94 + 2 * q^96 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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