LMFDB - Embedded newform 4650.2.d.n.3349.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(3349,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, 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("4650.3349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$37.1304369399$
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 930)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4650.3349
Dual form			4650.2.d.n.3349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} +6.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} +4.00000i q^{22} -8.00000i q^{23} -1.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} -6.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} -6.00000 q^{39} +10.0000 q^{41} -4.00000i q^{43} +4.00000 q^{44} -8.00000 q^{46} +1.00000i q^{48} +7.00000 q^{49} +2.00000 q^{51} -6.00000i q^{52} -10.0000i q^{53} -1.00000 q^{54} -4.00000i q^{57} +6.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} +1.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} +4.00000i q^{67} +2.00000i q^{68} +8.00000 q^{69} -1.00000i q^{72} +2.00000i q^{73} +2.00000 q^{74} +4.00000 q^{76} +6.00000i q^{78} +1.00000 q^{81} -10.0000i q^{82} +4.00000i q^{83} -4.00000 q^{86} -6.00000i q^{87} -4.00000i q^{88} +14.0000 q^{89} +8.00000i q^{92} -1.00000i q^{93} +1.00000 q^{96} -18.0000i q^{97} -7.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 2 q^{16} - 8 q^{19} - 2 q^{24} + 12 q^{26} - 12 q^{29} - 2 q^{31} - 4 q^{34} + 2 q^{36} - 12 q^{39} + 20 q^{41} + 8 q^{44} - 16 q^{46} + 14 q^{49} + 4 q^{51}+ \cdots + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 + 2 * q^16 - 8 * q^19 - 2 * q^24 + 12 * q^26 - 12 * q^29 - 2 * q^31 - 4 * q^34 + 2 * q^36 - 12 * q^39 + 20 * q^41 + 8 * q^44 - 16 * q^46 + 14 * q^49 + 4 * q^51 - 2 * q^54 + 24 * q^59 - 4 * q^61 - 2 * q^64 - 8 * q^66 + 16 * q^69 + 4 * q^74 + 8 * q^76 + 2 * q^81 - 8 * q^86 + 28 * q^89 + 2 * q^96 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$.

$n$	$1801$	$2977$	$3101$
$\chi(n)$	$1$	$-1$	$1$

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
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	− 1.00000i	− 0.288675i
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	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	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	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	$-0.313988\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	6.00000	1.17670
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−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
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	− 1.00000i	− 0.176777i
$33$	− 4.00000i	− 0.696311i
$34$	−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
$39$	−6.00000	−0.960769
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−8.00000	−1.17954
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.00000i	0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	2.00000	0.280056
$52$	− 6.00000i	− 0.832050i
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	− 4.00000i	− 0.529813i
$58$	6.00000i	0.787839i
$59$	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
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	1.00000i	0.127000i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	2.00000i	0.242536i
$69$	8.00000	0.963087
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 1.00000i	− 0.117851i
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	6.00000i	0.679366i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 10.0000i	− 1.10432i
$83$	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
$86$	−4.00000	−0.431331
$87$	− 6.00000i	− 0.643268i
$88$	− 4.00000i	− 0.426401i
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	0	0
$92$	8.00000i	0.834058i
$93$	− 1.00000i	− 0.103695i
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	− 18.0000i	− 1.82762i	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	− 18.0000i	− 1.82762i	0.406138	−	0.913812i	$-0.366875\pi$
$98$	− 7.00000i	− 0.707107i
$99$	4.00000	0.402015
$100$	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
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−10.0000	−0.971286
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	1.00000i	0.0962250i
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	− 6.00000i	− 0.554700i
$118$	− 12.0000i	− 1.10469i
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	2.00000i	0.181071i
$123$	10.0000i	0.901670i
$124$	1.00000	0.0898027
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000i	0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	4.00000i	0.348155i
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	2.00000	0.171499
$137$	22.0000i	1.87959i	0.341743	+	0.939793i	$0.388983\pi$
$137$	22.0000i	1.87959i	−0.341743	+	0.939793i	$0.611017\pi$
$138$	− 8.00000i	− 0.681005i
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 24.0000i	− 2.00698i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	7.00000i	0.577350i
$148$	− 2.00000i	− 0.164399i
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	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
$156$	6.00000	0.480384
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	4.00000	0.310460
$167$	− 24.0000i	− 1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$	− 24.0000i	− 1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	4.00000	0.305888
$172$	4.00000i	0.304997i
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	−4.00000	−0.301511
$177$	12.0000i	0.901975i
$178$	− 14.0000i	− 1.04934i
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	− 2.00000i	− 0.147844i
$184$	8.00000	0.589768
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	8.00000i	0.585018i
$188$	0	0
$189$	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
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	−18.0000	−1.29232
$195$	0	0
$196$	−7.00000	−0.500000
$197$	10.0000i	0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$	10.0000i	0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$	− 4.00000i	− 0.284268i
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	− 6.00000i	− 0.422159i
$203$	0	0
$204$	−2.00000	−0.140028
$205$	0	0
$206$	16.0000	1.11477
$207$	8.00000i	0.556038i
$208$	6.00000i	0.416025i
$209$	16.0000	1.10674
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	10.0000i	0.686803i
$213$	0	0
$214$	4.00000	0.273434
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	− 18.0000i	− 1.21911i
$219$	−2.00000	−0.135147
$220$	0	0
$221$	12.0000	0.807207
$222$	2.00000i	0.134231i
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	0	0
$225$	0	0
$226$	−14.0000	−0.931266
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	4.00000i	0.264906i
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	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	$0.893773\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	−12.0000	−0.781133
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	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
$243$	1.00000i	0.0641500i
$244$	2.00000	0.128037
$245$	0	0
$246$	10.0000	0.637577
$247$	− 24.0000i	− 1.52708i
$248$	− 1.00000i	− 0.0635001i
$249$	−4.00000	−0.253490
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	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	$-0.0198685\pi$
$258$	− 4.00000i	− 0.249029i
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	4.00000i	0.247121i
$263$	− 8.00000i	− 0.493301i	−0.969104	−	0.246651i	$-0.920670\pi$
$263$	− 8.00000i	− 0.493301i	0.969104	−	0.246651i	$-0.0793300\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	0	0
$267$	14.0000i	0.856786i
$268$	− 4.00000i	− 0.244339i
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	22.0000	1.32907
$275$	0	0
$276$	−8.00000	−0.481543
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	− 4.00000i	− 0.239904i
$279$	1.00000	0.0598684
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	0	0
$285$	0	0
$286$	−24.0000	−1.41915
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	18.0000	1.05518
$292$	− 2.00000i	− 0.117041i
$293$	− 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
$297$	4.00000i	0.232104i
$298$	− 10.0000i	− 0.579284i
$299$	48.0000	2.77591
$300$	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
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	− 6.00000i	− 0.339683i
$313$	18.0000i	1.01742i	0.860938	+	0.508710i	$0.169877\pi$
$313$	18.0000i	1.01742i	−0.860938	+	0.508710i	$0.830123\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	0	0
$317$	− 30.0000i	− 1.68497i	−0.538721	−	0.842484i	$-0.681092\pi$
$317$	− 30.0000i	− 1.68497i	0.538721	−	0.842484i	$-0.318908\pi$
$318$	− 10.0000i	− 0.560772i
$319$	24.0000	1.34374
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	8.00000i	0.445132i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	18.0000i	0.995402i
$328$	10.0000i	0.552158i
$329$	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
$334$	−24.0000	−1.31322
$335$	0	0
$336$	0	0
$337$	− 26.0000i	− 1.41631i	−0.706057	−	0.708155i	$-0.749528\pi$
$337$	− 26.0000i	− 1.41631i	0.706057	−	0.708155i	$-0.250472\pi$
$338$	23.0000i	1.25104i
$339$	14.0000	0.760376
$340$	0	0
$341$	4.00000	0.216612
$342$	− 4.00000i	− 0.216295i
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	− 28.0000i	− 1.50312i	−0.659665	−	0.751559i	$-0.729302\pi$
$347$	− 28.0000i	− 1.50312i	0.659665	−	0.751559i	$-0.270698\pi$
$348$	6.00000i	0.321634i
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	4.00000i	0.213201i
$353$	− 14.0000i	− 0.745145i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$	− 14.0000i	− 0.745145i	0.928003	−	0.372572i	$-0.121524\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	4.00000i	0.211407i
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	10.0000i	0.525588i
$363$	5.00000i	0.262432i
$364$	0	0
$365$	0	0
$366$	−2.00000	−0.104542
$367$	− 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
$369$	−10.0000	−0.520579
$370$	0	0
$371$	0	0
$372$	1.00000i	0.0518476i
$373$	38.0000i	1.96757i	0.179364	+	0.983783i	$0.442596\pi$
$373$	38.0000i	1.96757i	−0.179364	+	0.983783i	$0.557404\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	0	0
$377$	− 36.0000i	− 1.85409i
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	0	0
$382$	24.0000i	1.22795i
$383$	− 32.0000i	− 1.63512i	−0.575841	−	0.817562i	$-0.695325\pi$
$383$	− 32.0000i	− 1.63512i	0.575841	−	0.817562i	$-0.304675\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	4.00000i	0.203331i
$388$	18.0000i	0.913812i
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	7.00000i	0.353553i
$393$	− 4.00000i	− 0.201773i
$394$	10.0000	0.503793
$395$	0	0
$396$	−4.00000	−0.201008
$397$	18.0000i	0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$	18.0000i	0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	− 8.00000i	− 0.401004i
$399$	0	0
$400$	0	0
$401$	26.0000	1.29838	0.649189	−	0.760627i	$-0.275108\pi$
$401$	26.0000	1.29838	0.649189	+	0.760627i	$0.275108\pi$
$402$	4.00000i	0.199502i
$403$	− 6.00000i	− 0.298881i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	− 8.00000i	− 0.396545i
$408$	2.00000i	0.0990148i
$409$	−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
$411$	−22.0000	−1.08518
$412$	− 16.0000i	− 0.788263i
$413$	0	0
$414$	8.00000	0.393179
$415$	0	0
$416$	6.00000	0.294174
$417$	4.00000i	0.195881i
$418$	− 16.0000i	− 0.782586i
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	− 4.00000i	− 0.194717i
$423$	0	0
$424$	10.0000	0.485643
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 4.00000i	− 0.193347i
$429$	24.0000	1.15873
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 22.0000i	− 1.05725i	−0.848855	−	0.528626i	$-0.822707\pi$
$433$	− 22.0000i	− 1.05725i	0.848855	−	0.528626i	$-0.177293\pi$
$434$	0	0
$435$	0	0
$436$	−18.0000	−0.862044
$437$	32.0000i	1.53077i
$438$	2.00000i	0.0955637i
$439$	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
$442$	− 12.0000i	− 0.570782i
$443$	28.0000i	1.33032i	0.746701	+	0.665160i	$0.231637\pi$
$443$	28.0000i	1.33032i	−0.746701	+	0.665160i	$0.768363\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	−16.0000	−0.757622
$447$	10.0000i	0.472984i
$448$	0	0
$449$	−42.0000	−1.98210	−0.991051	−	0.133482i	$-0.957384\pi$
$449$	−42.0000	−1.98210	−0.991051	+	0.133482i	$0.957384\pi$
$450$	0	0
$451$	−40.0000	−1.88353
$452$	14.0000i	0.658505i
$453$	8.00000i	0.375873i
$454$	−4.00000	−0.187729
$455$	0	0
$456$	4.00000	0.187317
$457$	− 2.00000i	− 0.0935561i	−0.998905	−	0.0467780i	$-0.985105\pi$
$457$	− 2.00000i	− 0.0935561i	0.998905	−	0.0467780i	$-0.0148953\pi$
$458$	− 10.0000i	− 0.467269i
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	10.0000	0.463241
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	6.00000i	0.277350i
$469$	0	0
$470$	0	0
$471$	14.0000	0.645086
$472$	12.0000i	0.552345i
$473$	16.0000i	0.735681i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.0000i	0.457869i
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	− 18.0000i	− 0.819878i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	4.00000	0.180886
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	− 10.0000i	− 0.450835i
$493$	12.0000i	0.540453i
$494$	−24.0000	−1.07981
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	0	0
$498$	4.00000i	0.179244i
$499$	−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
$501$	24.0000	1.07224
$502$	− 12.0000i	− 0.535586i
$503$	− 24.0000i	− 1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$	− 24.0000i	− 1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$	0	0
$505$	0	0
$506$	32.0000	1.42257
$507$	− 23.0000i	− 1.02147i
$508$	0	0
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	4.00000i	0.176604i
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	− 6.00000i	− 0.262613i
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	−8.00000	−0.348817
$527$	2.00000i	0.0871214i
$528$	− 4.00000i	− 0.174078i
$529$	−41.0000	−1.78261
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	60.0000i	2.59889i
$534$	14.0000	0.605839
$535$	0	0
$536$	−4.00000	−0.172774
$537$	− 4.00000i	− 0.172613i
$538$	− 10.0000i	− 0.431131i
$539$	−28.0000	−1.20605
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	16.0000i	0.687259i
$543$	− 10.0000i	− 0.429141i
$544$	−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	$-0.204275\pi$
$548$	− 22.0000i	− 0.939793i
$549$	2.00000	0.0853579
$550$	0	0
$551$	24.0000	1.02243
$552$	8.00000i	0.340503i
$553$	0	0
$554$	2.00000	0.0849719
$555$	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	$0.875462\pi$
$558$	− 1.00000i	− 0.0423334i
$559$	24.0000	1.01509
$560$	0	0
$561$	−8.00000	−0.337760
$562$	− 10.0000i	− 0.421825i
$563$	36.0000i	1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$	36.0000i	1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$	0	0
$565$	0	0
$566$	20.0000	0.840663
$567$	0	0
$568$	0	0
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	24.0000i	1.00349i
$573$	− 24.0000i	− 1.00261i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	− 13.0000i	− 0.540729i
$579$	14.0000	0.581820
$580$	0	0
$581$	0	0
$582$	− 18.0000i	− 0.746124i
$583$	40.0000i	1.65663i
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−26.0000	−1.07405
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	− 7.00000i	− 0.288675i
$589$	4.00000	0.164817
$590$	0	0
$591$	−10.0000	−0.411345
$592$	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	$0.879500\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−10.0000	−0.409616
$597$	8.00000i	0.327418i
$598$	− 48.0000i	− 1.96287i
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	6.00000	0.243733
$607$	− 8.00000i	− 0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$	− 8.00000i	− 0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$	4.00000i	0.162221i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	− 2.00000i	− 0.0808452i
$613$	14.0000i	0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$	14.0000i	0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	0	0
$617$	22.0000i	0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$	22.0000i	0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$	16.0000i	0.643614i
$619$	−12.0000	−0.482321	−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000	−0.482321	−0.241160	+	0.970485i	$0.577528\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	0	0
$623$	0	0
$624$	−6.00000	−0.240192
$625$	0	0
$626$	18.0000	0.719425
$627$	16.0000i	0.638978i
$628$	14.0000i	0.558661i
$629$	4.00000	0.159490
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	4.00000i	0.158986i
$634$	−30.0000	−1.19145
$635$	0	0
$636$	−10.0000	−0.396526
$637$	42.0000i	1.66410i
$638$	− 24.0000i	− 0.950169i
$639$	0	0
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	4.00000i	0.157867i
$643$	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
$646$	8.00000	0.314756
$647$	− 24.0000i	− 0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$	− 24.0000i	− 0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	1.00000i	0.0392837i
$649$	−48.0000	−1.88416
$650$	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	$-0.114565\pi$
$654$	18.0000	0.703856
$655$	0	0
$656$	10.0000	0.390434
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	20.0000i	0.777322i
$663$	12.0000i	0.466041i
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	48.0000i	1.85857i
$668$	24.0000i	0.928588i
$669$	16.0000	0.618596
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	10.0000i	0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$	10.0000i	0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	23.0000	0.884615
$677$	26.0000i	0.999261i	0.866239	+	0.499631i	$0.166531\pi$
$677$	26.0000i	0.999261i	−0.866239	+	0.499631i	$0.833469\pi$
$678$	− 14.0000i	− 0.537667i
$679$	0	0
$680$	0	0
$681$	4.00000	0.153280
$682$	− 4.00000i	− 0.153168i
$683$	44.0000i	1.68361i	0.539779	+	0.841807i	$0.318508\pi$
$683$	44.0000i	1.68361i	−0.539779	+	0.841807i	$0.681492\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	0	0
$687$	10.0000i	0.381524i
$688$	− 4.00000i	− 0.152499i
$689$	60.0000	2.28582
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	− 14.0000i	− 0.532200i
$693$	0	0
$694$	−28.0000	−1.06287
$695$	0	0
$696$	6.00000	0.227429
$697$	− 20.0000i	− 0.757554i
$698$	30.0000i	1.13552i
$699$	−10.0000	−0.378235
$700$	0	0
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	− 6.00000i	− 0.226455i
$703$	− 8.00000i	− 0.301726i
$704$	4.00000	0.150756
$705$	0	0
$706$	−14.0000	−0.526897
$707$	0	0
$708$	− 12.0000i	− 0.450988i
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	0	0
$712$	14.0000i	0.524672i
$713$	8.00000i	0.299602i
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	0	0
$718$	− 16.0000i	− 0.597115i
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	0	0
$722$	3.00000i	0.111648i
$723$	18.0000i	0.669427i
$724$	10.0000	0.371647
$725$	0	0
$726$	5.00000	0.185567
$727$	48.0000i	1.78022i	0.455744	+	0.890111i	$0.349373\pi$
$727$	48.0000i	1.78022i	−0.455744	+	0.890111i	$0.650627\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−8.00000	−0.295891
$732$	2.00000i	0.0739221i
$733$	− 2.00000i	− 0.0738717i	−0.999318	−	0.0369358i	$-0.988240\pi$
$733$	− 2.00000i	− 0.0738717i	0.999318	−	0.0369358i	$-0.0117597\pi$
$734$	−32.0000	−1.18114
$735$	0	0
$736$	−8.00000	−0.294884
$737$	− 16.0000i	− 0.589368i
$738$	10.0000i	0.368105i
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	24.0000	0.881662
$742$	0	0
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	38.0000	1.39128
$747$	− 4.00000i	− 0.146352i
$748$	− 8.00000i	− 0.292509i
$749$	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	$0.594295\pi$
$752$	0	0
$753$	12.0000i	0.437304i
$754$	−36.0000	−1.31104
$755$	0	0
$756$	0	0
$757$	− 14.0000i	− 0.508839i	−0.967094	−	0.254419i	$-0.918116\pi$
$757$	− 14.0000i	− 0.508839i	0.967094	−	0.254419i	$-0.0818843\pi$
$758$	12.0000i	0.435860i
$759$	−32.0000	−1.16153
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	0	0
$763$	0	0
$764$	24.0000	0.868290
$765$	0	0
$766$	−32.0000	−1.15621
$767$	72.0000i	2.59977i
$768$	1.00000i	0.0360844i
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	14.0000i	0.503871i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	18.0000	0.646162
$777$	0	0
$778$	− 18.0000i	− 0.645331i
$779$	−40.0000	−1.43315
$780$	0	0
$781$	0	0
$782$	16.0000i	0.572159i
$783$	6.00000i	0.214423i
$784$	7.00000	0.250000
$785$	0	0
$786$	−4.00000	−0.142675
$787$	12.0000i	0.427754i	0.976861	+	0.213877i	$0.0686091\pi$
$787$	12.0000i	0.427754i	−0.976861	+	0.213877i	$0.931391\pi$
$788$	− 10.0000i	− 0.356235i
$789$	8.00000	0.284808
$790$	0	0
$791$	0	0
$792$	4.00000i	0.142134i
$793$	− 12.0000i	− 0.426132i
$794$	18.0000	0.638796
$795$	0	0
$796$	−8.00000	−0.283552
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−14.0000	−0.494666
$802$	− 26.0000i	− 0.918092i
$803$	− 8.00000i	− 0.282314i
$804$	4.00000	0.141069
$805$	0	0
$806$	−6.00000	−0.211341
$807$	10.0000i	0.352017i
$808$	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
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	0	0
$813$	− 16.0000i	− 0.561144i
$814$	−8.00000	−0.280400
$815$	0	0
$816$	2.00000	0.0700140
$817$	16.0000i	0.559769i
$818$	10.0000i	0.349642i
$819$	0	0
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	22.0000i	0.767338i
$823$	− 40.0000i	− 1.39431i	−0.716919	−	0.697156i	$-0.754448\pi$
$823$	− 40.0000i	− 1.39431i	0.716919	−	0.697156i	$-0.245552\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	0	0
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	− 8.00000i	− 0.278019i
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	− 6.00000i	− 0.208013i
$833$	− 14.0000i	− 0.485071i
$834$	4.00000	0.138509
$835$	0	0
$836$	−16.0000	−0.553372
$837$	1.00000i	0.0345651i
$838$	− 20.0000i	− 0.690889i
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 38.0000i	− 1.30957i
$843$	10.0000i	0.344418i
$844$	−4.00000	−0.137686
$845$	0	0
$846$	0	0
$847$	0	0
$848$	− 10.0000i	− 0.343401i
$849$	−20.0000	−0.686398
$850$	0	0
$851$	16.0000	0.548473
$852$	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
$856$	−4.00000	−0.136717
$857$	6.00000i	0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$	6.00000i	0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$	− 24.0000i	− 0.819346i
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	0	0
$861$	0	0
$862$	− 24.0000i	− 0.817443i
$863$	− 16.0000i	− 0.544646i	−0.962206	−	0.272323i	$-0.912208\pi$
$863$	− 16.0000i	− 0.544646i	0.962206	−	0.272323i	$-0.0877920\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−22.0000	−0.747590
$867$	13.0000i	0.441503i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−24.0000	−0.813209
$872$	18.0000i	0.609557i
$873$	18.0000i	0.609208i
$874$	32.0000	1.08242
$875$	0	0
$876$	2.00000	0.0675737
$877$	− 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
$881$	−22.0000	−0.741199	−0.370599	−	0.928793i	$-0.620848\pi$
$881$	−22.0000	−0.741199	−0.370599	+	0.928793i	$0.620848\pi$
$882$	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	$-0.265344\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	28.0000	0.940678
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	− 2.00000i	− 0.0671156i
$889$	0	0
$890$	0	0
$891$	−4.00000	−0.134005
$892$	16.0000i	0.535720i
$893$	0	0
$894$	10.0000	0.334450
$895$	0	0
$896$	0	0
$897$	48.0000i	1.60267i
$898$	42.0000i	1.40156i
$899$	6.00000	0.200111
$900$	0	0
$901$	−20.0000	−0.666297
$902$	40.0000i	1.33185i
$903$	0	0
$904$	14.0000	0.465633
$905$	0	0
$906$	8.00000	0.265782
$907$	12.0000i	0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$	12.0000i	0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$	4.00000i	0.132745i
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	− 4.00000i	− 0.132453i
$913$	− 16.0000i	− 0.529523i
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	−10.0000	−0.330409
$917$	0	0
$918$	2.00000i	0.0660098i
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	− 6.00000i	− 0.197599i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−16.0000	−0.525793
$927$	− 16.0000i	− 0.525509i
$928$	6.00000i	0.196960i
$929$	38.0000	1.24674	0.623370	−	0.781927i	$-0.285763\pi$
$929$	38.0000	1.24674	0.623370	+	0.781927i	$0.285763\pi$
$930$	0	0
$931$	−28.0000	−0.917663
$932$	− 10.0000i	− 0.327561i
$933$	0	0
$934$	−36.0000	−1.17796
$935$	0	0
$936$	6.00000	0.196116
$937$	38.0000i	1.24141i	0.784046	+	0.620703i	$0.213153\pi$
$937$	38.0000i	1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$	0	0
$939$	−18.0000	−0.587408
$940$	0	0
$941$	22.0000	0.717180	0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000	0.717180	0.358590	+	0.933495i	$0.383258\pi$
$942$	− 14.0000i	− 0.456145i
$943$	− 80.0000i	− 2.60516i
$944$	12.0000	0.390567
$945$	0	0
$946$	16.0000	0.520205
$947$	− 36.0000i	− 1.16984i	−0.811090	−	0.584921i	$-0.801125\pi$
$947$	− 36.0000i	− 1.16984i	0.811090	−	0.584921i	$-0.198875\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	30.0000	0.972817
$952$	0	0
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	10.0000	0.323762
$955$	0	0
$956$	0	0
$957$	24.0000i	0.775810i
$958$	24.0000i	0.775405i
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	12.0000i	0.386896i
$963$	− 4.00000i	− 0.128898i
$964$	−18.0000	−0.579741
$965$	0	0
$966$	0	0
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	5.00000i	0.160706i
$969$	−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
$974$	−8.00000	−0.256337
$975$	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	$-0.0929689\pi$
$978$	− 4.00000i	− 0.127906i
$979$	−56.0000	−1.78977
$980$	0	0
$981$	−18.0000	−0.574696
$982$	20.0000i	0.638226i
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	−10.0000	−0.318788
$985$	0	0
$986$	12.0000	0.382158
$987$	0	0
$988$	24.0000i	0.763542i
$989$	−32.0000	−1.01754
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	1.00000i	0.0317500i
$993$	− 20.0000i	− 0.634681i
$994$	0	0
$995$	0	0
$996$	4.00000	0.126745
$997$	42.0000i	1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$	42.0000i	1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$	4.00000i	0.126618i
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.d.n.3349.1		2
5.2	odd	4		930.2.a.o.1.1	✓	1
5.3	odd	4		4650.2.a.h.1.1		1
5.4	even	2	inner	4650.2.d.n.3349.2		2
15.2	even	4		2790.2.a.c.1.1		1
20.7	even	4		7440.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.o.1.1	✓	1	5.2	odd	4
2790.2.a.c.1.1		1	15.2	even	4
4650.2.a.h.1.1		1	5.3	odd	4
4650.2.d.n.3349.1		2	1.1	even	1	trivial
4650.2.d.n.3349.2		2	5.4	even	2	inner
7440.2.a.j.1.1		1	20.7	even	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:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} +6.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} +4.00000i q^{22} -8.00000i q^{23} -1.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} -6.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} -6.00000 q^{39} +10.0000 q^{41} -4.00000i q^{43} +4.00000 q^{44} -8.00000 q^{46} +1.00000i q^{48} +7.00000 q^{49} +2.00000 q^{51} -6.00000i q^{52} -10.0000i q^{53} -1.00000 q^{54} -4.00000i q^{57} +6.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} +1.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} +4.00000i q^{67} +2.00000i q^{68} +8.00000 q^{69} -1.00000i q^{72} +2.00000i q^{73} +2.00000 q^{74} +4.00000 q^{76} +6.00000i q^{78} +1.00000 q^{81} -10.0000i q^{82} +4.00000i q^{83} -4.00000 q^{86} -6.00000i q^{87} -4.00000i q^{88} +14.0000 q^{89} +8.00000i q^{92} -1.00000i q^{93} +1.00000 q^{96} -18.0000i q^{97} -7.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 2 q^{16} - 8 q^{19} - 2 q^{24} + 12 q^{26} - 12 q^{29} - 2 q^{31} - 4 q^{34} + 2 q^{36} - 12 q^{39} + 20 q^{41} + 8 q^{44} - 16 q^{46} + 14 q^{49} + 4 q^{51}+ \cdots + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 + 2 * q^16 - 8 * q^19 - 2 * q^24 + 12 * q^26 - 12 * q^29 - 2 * q^31 - 4 * q^34 + 2 * q^36 - 12 * q^39 + 20 * q^41 + 8 * q^44 - 16 * q^46 + 14 * q^49 + 4 * q^51 - 2 * q^54 + 24 * q^59 - 4 * q^61 - 2 * q^64 - 8 * q^66 + 16 * q^69 + 4 * q^74 + 8 * q^76 + 2 * q^81 - 8 * q^86 + 28 * q^89 + 2 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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