LMFDB - Embedded newform 5850.2.e.bb.5149.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5850,2,Mod(5149,5850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5850, 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("5850.5149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5850.e (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:	$46.7124851824$
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 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5149.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	5850.5149
Dual form			5850.2.e.bb.5149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +4.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +4.00000i q^{7} -1.00000i q^{8} +4.00000 q^{11} -1.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +8.00000 q^{19} +4.00000i q^{22} +1.00000 q^{26} -4.00000i q^{28} +6.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} +2.00000 q^{34} -2.00000i q^{37} +8.00000i q^{38} +10.0000 q^{41} -4.00000i q^{43} -4.00000 q^{44} -8.00000i q^{47} -9.00000 q^{49} +1.00000i q^{52} -10.0000i q^{53} +4.00000 q^{56} +6.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} -16.0000i q^{67} +2.00000i q^{68} +8.00000 q^{71} -2.00000i q^{73} +2.00000 q^{74} -8.00000 q^{76} +16.0000i q^{77} -8.00000 q^{79} +10.0000i q^{82} +12.0000i q^{83} +4.00000 q^{86} -4.00000i q^{88} +14.0000 q^{89} +4.00000 q^{91} +8.00000 q^{94} +10.0000i q^{97} -9.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 8 q^{11} - 8 q^{14} + 2 q^{16} + 16 q^{19} + 2 q^{26} + 12 q^{29} - 8 q^{31} + 4 q^{34} + 20 q^{41} - 8 q^{44} - 18 q^{49} + 8 q^{56} + 8 q^{59} - 4 q^{61} - 2 q^{64} + 16 q^{71} + 4 q^{74} - 16 q^{76} - 16 q^{79} + 8 q^{86} + 28 q^{89} + 8 q^{91} + 16 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 8 * q^11 - 8 * q^14 + 2 * q^16 + 16 * q^19 + 2 * q^26 + 12 * q^29 - 8 * q^31 + 4 * q^34 + 20 * q^41 - 8 * q^44 - 18 * q^49 + 8 * q^56 + 8 * q^59 - 4 * q^61 - 2 * q^64 + 16 * q^71 + 4 * q^74 - 16 * q^76 - 16 * q^79 + 8 * q^86 + 28 * q^89 + 8 * q^91 + 16 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times$.

$n$	$2251$	$3251$	$3277$
$\chi(n)$	$1$	$1$	$-1$

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
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	0	0
$22$	4.00000i	0.852803i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	1.00000	0.196116
$27$	0	0
$28$	− 4.00000i	− 0.755929i
$29$	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
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	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	$-0.0525685\pi$
$38$	8.00000i	1.29777i
$39$	0	0
$40$	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
$46$	0	0
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	1.00000i	0.138675i
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	0	0
$55$	0	0
$56$	4.00000	0.534522
$57$	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	$0.540868\pi$
$62$	− 4.00000i	− 0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 16.0000i	− 1.95471i	−0.211604	−	0.977356i	$-0.567869\pi$
$67$	− 16.0000i	− 1.95471i	0.211604	−	0.977356i	$-0.432131\pi$
$68$	2.00000i	0.242536i
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−8.00000	−0.917663
$77$	16.0000i	1.82337i
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	0	0
$82$	10.0000i	1.10432i
$83$	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
$86$	4.00000	0.431331
$87$	0	0
$88$	− 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
$91$	4.00000	0.419314
$92$	0	0
$93$	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	$0.830506\pi$
$98$	− 9.00000i	− 0.909137i
$99$	0	0
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	10.0000	0.971286
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	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
$112$	4.00000i	0.377964i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	4.00000i	0.368230i
$119$	8.00000	0.733359
$120$	0	0
$121$	5.00000	0.454545
$122$	− 2.00000i	− 0.181071i
$123$	0	0
$124$	4.00000	0.359211
$125$	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
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	32.0000i	2.77475i
$134$	16.0000	1.38219
$135$	0	0
$136$	−2.00000	−0.171499
$137$	10.0000i	0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$	10.0000i	0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	8.00000i	0.671345i
$143$	− 4.00000i	− 0.334497i
$144$	0	0
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	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
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	− 8.00000i	− 0.648886i
$153$	0	0
$154$	−16.0000	−1.28932
$155$	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	$0.811315\pi$
$158$	− 8.00000i	− 0.636446i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	−12.0000	−0.931381
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	4.00000i	0.304997i
$173$	− 10.0000i	− 0.760286i	−0.924928	−	0.380143i	$-0.875875\pi$
$173$	− 10.0000i	− 0.760286i	0.924928	−	0.380143i	$-0.124125\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	14.0000i	1.04934i
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	4.00000i	0.296500i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	8.00000i	0.583460i
$189$	0	0
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	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
$196$	9.00000	0.642857
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	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
$201$	0	0
$202$	2.00000i	0.140720i
$203$	24.0000i	1.68447i
$204$	0	0
$205$	0	0
$206$	16.0000	1.11477
$207$	0	0
$208$	− 1.00000i	− 0.0693375i
$209$	32.0000	2.21349
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	10.0000i	0.686803i
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	0	0
$217$	− 16.0000i	− 1.08615i
$218$	2.00000i	0.135457i
$219$	0	0
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	4.00000i	0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$	4.00000i	0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	6.00000	0.399114
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	0	0
$229$	−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
$232$	− 6.00000i	− 0.393919i
$233$	18.0000i	1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$	18.0000i	1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	0	0
$238$	8.00000i	0.518563i
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	5.00000i	0.321412i
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	− 8.00000i	− 0.509028i
$248$	4.00000i	0.254000i
$249$	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
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000i	0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$	6.00000i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	0	0
$262$	− 4.00000i	− 0.247121i
$263$	8.00000i	0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$	8.00000i	0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	0	0
$265$	0	0
$266$	−32.0000	−1.96205
$267$	0	0
$268$	16.0000i	0.977356i
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	−10.0000	−0.604122
$275$	0	0
$276$	0	0
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	− 12.0000i	− 0.719712i
$279$	0	0
$280$	0	0
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	4.00000	0.236525
$287$	40.0000i	2.36113i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	2.00000i	0.117041i
$293$	26.0000i	1.51894i	0.650545	+	0.759468i	$0.274541\pi$
$293$	26.0000i	1.51894i	−0.650545	+	0.759468i	$0.725459\pi$
$294$	0	0
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	− 6.00000i	− 0.347571i
$299$	0	0
$300$	0	0
$301$	16.0000	0.922225
$302$	12.0000i	0.690522i
$303$	0	0
$304$	8.00000	0.458831
$305$	0	0
$306$	0	0
$307$	− 8.00000i	− 0.456584i	−0.973593	−	0.228292i	$-0.926686\pi$
$307$	− 8.00000i	− 0.456584i	0.973593	−	0.228292i	$-0.0733141\pi$
$308$	− 16.0000i	− 0.911685i
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	8.00000	0.450035
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 16.0000i	− 0.890264i
$324$	0	0
$325$	0	0
$326$	−16.0000	−0.886158
$327$	0	0
$328$	− 10.0000i	− 0.552158i
$329$	32.0000	1.76422
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	− 12.0000i	− 0.658586i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.0000i	0.980522i	0.871576	+	0.490261i	$0.163099\pi$
$337$	18.0000i	0.980522i	−0.871576	+	0.490261i	$0.836901\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	0	0
$340$	0	0
$341$	−16.0000	−0.866449
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	10.0000	0.537603
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	0	0
$352$	4.00000i	0.213201i
$353$	14.0000i	0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$	14.0000i	0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	0	0
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	− 10.0000i	− 0.525588i
$363$	0	0
$364$	−4.00000	−0.209657
$365$	0	0
$366$	0	0
$367$	16.0000i	0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$	16.0000i	0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	40.0000	2.07670
$372$	0	0
$373$	− 6.00000i	− 0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$	− 6.00000i	− 0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	−8.00000	−0.412568
$377$	− 6.00000i	− 0.309016i
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	8.00000i	0.409316i
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	0	0
$385$	0	0
$386$	−14.0000	−0.712581
$387$	0	0
$388$	− 10.0000i	− 0.507673i
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	0	0
$392$	9.00000i	0.454569i
$393$	0	0
$394$	18.0000	0.906827
$395$	0	0
$396$	0	0
$397$	6.00000i	0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$	6.00000i	0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$	8.00000i	0.401004i
$399$	0	0
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	−24.0000	−1.19110
$407$	− 8.00000i	− 0.396545i
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	0	0
$412$	16.0000i	0.788263i
$413$	16.0000i	0.787309i
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	32.0000i	1.56517i
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	12.0000i	0.584151i
$423$	0	0
$424$	−10.0000	−0.485643
$425$	0	0
$426$	0	0
$427$	− 8.00000i	− 0.387147i
$428$	12.0000i	0.580042i
$429$	0	0
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	30.0000i	1.44171i	0.693087	+	0.720854i	$0.256250\pi$
$433$	30.0000i	1.44171i	−0.693087	+	0.720854i	$0.743750\pi$
$434$	16.0000	0.768025
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	0	0
$442$	− 2.00000i	− 0.0951303i
$443$	− 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
$446$	−4.00000	−0.189405
$447$	0	0
$448$	− 4.00000i	− 0.188982i
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	40.0000	1.88353
$452$	6.00000i	0.282216i
$453$	0	0
$454$	20.0000	0.938647
$455$	0	0
$456$	0	0
$457$	− 30.0000i	− 1.40334i	−0.712502	−	0.701670i	$-0.752438\pi$
$457$	− 30.0000i	− 1.40334i	0.712502	−	0.701670i	$-0.247562\pi$
$458$	− 22.0000i	− 1.02799i
$459$	0	0
$460$	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
$463$	20.0000i	0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$	20.0000i	0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−18.0000	−0.833834
$467$	4.00000i	0.185098i	0.995708	+	0.0925490i	$0.0295015\pi$
$467$	4.00000i	0.185098i	−0.995708	+	0.0925490i	$0.970499\pi$
$468$	0	0
$469$	64.0000	2.95525
$470$	0	0
$471$	0	0
$472$	− 4.00000i	− 0.184115i
$473$	− 16.0000i	− 0.735681i
$474$	0	0
$475$	0	0
$476$	−8.00000	−0.366679
$477$	0	0
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	10.0000i	0.455488i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	0	0
$487$	4.00000i	0.181257i	0.995885	+	0.0906287i	$0.0288876\pi$
$487$	4.00000i	0.181257i	−0.995885	+	0.0906287i	$0.971112\pi$
$488$	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
$496$	−4.00000	−0.179605
$497$	32.0000i	1.43540i
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	0	0
$502$	− 4.00000i	− 0.178529i
$503$	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
$507$	0	0
$508$	0	0
$509$	42.0000	1.86162	0.930809	−	0.365507i	$-0.119104\pi$
$509$	42.0000	1.86162	0.930809	+	0.365507i	$0.119104\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−6.00000	−0.264649
$515$	0	0
$516$	0	0
$517$	− 32.0000i	− 1.40736i
$518$	8.00000i	0.351500i
$519$	0	0
$520$	0	0
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	−8.00000	−0.348817
$527$	8.00000i	0.348485i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	− 32.0000i	− 1.38738i
$533$	− 10.0000i	− 0.433148i
$534$	0	0
$535$	0	0
$536$	−16.0000	−0.691095
$537$	0	0
$538$	− 26.0000i	− 1.12094i
$539$	−36.0000	−1.55063
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	− 4.00000i	− 0.171815i
$543$	0	0
$544$	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	$-0.204275\pi$
$548$	− 10.0000i	− 0.427179i
$549$	0	0
$550$	0	0
$551$	48.0000	2.04487
$552$	0	0
$553$	− 32.0000i	− 1.36078i
$554$	−22.0000	−0.934690
$555$	0	0
$556$	12.0000	0.508913
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	26.0000i	1.09674i
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	0	0
$565$	0	0
$566$	−4.00000	−0.168133
$567$	0	0
$568$	− 8.00000i	− 0.335673i
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	4.00000i	0.167248i
$573$	0	0
$574$	−40.0000	−1.66957
$575$	0	0
$576$	0	0
$577$	18.0000i	0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$	18.0000i	0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$	13.0000i	0.540729i
$579$	0	0
$580$	0	0
$581$	−48.0000	−1.99138
$582$	0	0
$583$	− 40.0000i	− 1.65663i
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−26.0000	−1.07405
$587$	− 4.00000i	− 0.165098i	−0.996587	−	0.0825488i	$-0.973694\pi$
$587$	− 4.00000i	− 0.165098i	0.996587	−	0.0825488i	$-0.0263060\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	0	0
$592$	− 2.00000i	− 0.0821995i
$593$	− 42.0000i	− 1.72473i	−0.506284	−	0.862367i	$-0.668981\pi$
$593$	− 42.0000i	− 1.72473i	0.506284	−	0.862367i	$-0.331019\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	16.0000i	0.652111i
$603$	0	0
$604$	−12.0000	−0.488273
$605$	0	0
$606$	0	0
$607$	− 16.0000i	− 0.649420i	−0.945814	−	0.324710i	$-0.894733\pi$
$607$	− 16.0000i	− 0.649420i	0.945814	−	0.324710i	$-0.105267\pi$
$608$	8.00000i	0.324443i
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	16.0000	0.644658
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	0	0
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	56.0000i	2.24359i
$624$	0	0
$625$	0	0
$626$	−6.00000	−0.239808
$627$	0	0
$628$	− 14.0000i	− 0.558661i
$629$	−4.00000	−0.159490
$630$	0	0
$631$	−36.0000	−1.43314	−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000	−1.43314	−0.716569	+	0.697517i	$0.754288\pi$
$632$	8.00000i	0.318223i
$633$	0	0
$634$	−6.00000	−0.238290
$635$	0	0
$636$	0	0
$637$	9.00000i	0.356593i
$638$	24.0000i	0.950169i
$639$	0	0
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	16.0000i	0.630978i	0.948929	+	0.315489i	$0.102169\pi$
$643$	16.0000i	0.630978i	−0.948929	+	0.315489i	$0.897831\pi$
$644$	0	0
$645$	0	0
$646$	16.0000	0.629512
$647$	− 24.0000i	− 0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$	− 24.0000i	− 0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	− 16.0000i	− 0.626608i
$653$	− 10.0000i	− 0.391330i	−0.980671	−	0.195665i	$-0.937313\pi$
$653$	− 10.0000i	− 0.391330i	0.980671	−	0.195665i	$-0.0626866\pi$
$654$	0	0
$655$	0	0
$656$	10.0000	0.390434
$657$	0	0
$658$	32.0000i	1.24749i
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	8.00000i	0.310929i
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	14.0000i	0.539660i	0.962908	+	0.269830i	$0.0869676\pi$
$673$	14.0000i	0.539660i	−0.962908	+	0.269830i	$0.913032\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	1.00000	0.0384615
$677$	− 38.0000i	− 1.46046i	−0.683202	−	0.730229i	$-0.739413\pi$
$677$	− 38.0000i	− 1.46046i	0.683202	−	0.730229i	$-0.260587\pi$
$678$	0	0
$679$	−40.0000	−1.53506
$680$	0	0
$681$	0	0
$682$	− 16.0000i	− 0.612672i
$683$	44.0000i	1.68361i	0.539779	+	0.841807i	$0.318508\pi$
$683$	44.0000i	1.68361i	−0.539779	+	0.841807i	$0.681492\pi$
$684$	0	0
$685$	0	0
$686$	8.00000	0.305441
$687$	0	0
$688$	− 4.00000i	− 0.152499i
$689$	−10.0000	−0.380970
$690$	0	0
$691$	−32.0000	−1.21734	−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000	−1.21734	−0.608669	+	0.793424i	$0.708296\pi$
$692$	10.0000i	0.380143i
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	0	0
$697$	− 20.0000i	− 0.757554i
$698$	− 6.00000i	− 0.227103i
$699$	0	0
$700$	0	0
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	0	0
$703$	− 16.0000i	− 0.603451i
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−14.0000	−0.526897
$707$	8.00000i	0.300871i
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	0	0
$712$	− 14.0000i	− 0.524672i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	0	0
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	64.0000	2.38348
$722$	45.0000i	1.67473i
$723$	0	0
$724$	10.0000	0.371647
$725$	0	0
$726$	0	0
$727$	− 40.0000i	− 1.48352i	−0.670667	−	0.741759i	$-0.733992\pi$
$727$	− 40.0000i	− 1.48352i	0.670667	−	0.741759i	$-0.266008\pi$
$728$	− 4.00000i	− 0.148250i
$729$	0	0
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	2.00000i	0.0738717i	0.999318	+	0.0369358i	$0.0117597\pi$
$733$	2.00000i	0.0738717i	−0.999318	+	0.0369358i	$0.988240\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	0	0
$737$	− 64.0000i	− 2.35747i
$738$	0	0
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	0	0
$741$	0	0
$742$	40.0000i	1.46845i
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	0	0
$746$	6.00000	0.219676
$747$	0	0
$748$	8.00000i	0.292509i
$749$	48.0000	1.75388
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	− 8.00000i	− 0.291730i
$753$	0	0
$754$	6.00000	0.218507
$755$	0	0
$756$	0	0
$757$	54.0000i	1.96266i	0.192323	+	0.981332i	$0.438398\pi$
$757$	54.0000i	1.96266i	−0.192323	+	0.981332i	$0.561602\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	26.0000	0.942499	0.471250	−	0.882000i	$-0.343803\pi$
$761$	26.0000	0.942499	0.471250	+	0.882000i	$0.343803\pi$
$762$	0	0
$763$	8.00000i	0.289619i
$764$	−8.00000	−0.289430
$765$	0	0
$766$	24.0000	0.867155
$767$	− 4.00000i	− 0.144432i
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	0	0
$772$	− 14.0000i	− 0.503871i
$773$	− 54.0000i	− 1.94225i	−0.238581	−	0.971123i	$-0.576682\pi$
$773$	− 54.0000i	− 1.94225i	0.238581	−	0.971123i	$-0.423318\pi$
$774$	0	0
$775$	0	0
$776$	10.0000	0.358979
$777$	0	0
$778$	− 26.0000i	− 0.932145i
$779$	80.0000	2.86630
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	0	0
$784$	−9.00000	−0.321429
$785$	0	0
$786$	0	0
$787$	40.0000i	1.42585i	0.701242	+	0.712923i	$0.252629\pi$
$787$	40.0000i	1.42585i	−0.701242	+	0.712923i	$0.747371\pi$
$788$	18.0000i	0.641223i
$789$	0	0
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	2.00000i	0.0710221i
$794$	−6.00000	−0.212932
$795$	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
$801$	0	0
$802$	− 6.00000i	− 0.211867i
$803$	− 8.00000i	− 0.282314i
$804$	0	0
$805$	0	0
$806$	−4.00000	−0.140894
$807$	0	0
$808$	− 2.00000i	− 0.0703598i
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	− 24.0000i	− 0.842235i
$813$	0	0
$814$	8.00000	0.280400
$815$	0	0
$816$	0	0
$817$	− 32.0000i	− 1.11954i
$818$	− 2.00000i	− 0.0699284i
$819$	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	$0.761836\pi$
$822$	0	0
$823$	16.0000i	0.557725i	0.960331	+	0.278862i	$0.0899574\pi$
$823$	16.0000i	0.557725i	−0.960331	+	0.278862i	$0.910043\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	−16.0000	−0.556711
$827$	− 28.0000i	− 0.973655i	−0.873498	−	0.486828i	$-0.838154\pi$
$827$	− 28.0000i	− 0.973655i	0.873498	−	0.486828i	$-0.161846\pi$
$828$	0	0
$829$	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
$831$	0	0
$832$	1.00000i	0.0346688i
$833$	18.0000i	0.623663i
$834$	0	0
$835$	0	0
$836$	−32.0000	−1.10674
$837$	0	0
$838$	4.00000i	0.138178i
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	22.0000i	0.758170i
$843$	0	0
$844$	−12.0000	−0.413057
$845$	0	0
$846$	0	0
$847$	20.0000i	0.687208i
$848$	− 10.0000i	− 0.343401i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	2.00000i	0.0684787i	0.999414	+	0.0342393i	$0.0109009\pi$
$853$	2.00000i	0.0684787i	−0.999414	+	0.0342393i	$0.989099\pi$
$854$	8.00000	0.273754
$855$	0	0
$856$	−12.0000	−0.410152
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	0	0
$861$	0	0
$862$	8.00000i	0.272481i
$863$	40.0000i	1.36162i	0.732462	+	0.680808i	$0.238371\pi$
$863$	40.0000i	1.36162i	−0.732462	+	0.680808i	$0.761629\pi$
$864$	0	0
$865$	0	0
$866$	−30.0000	−1.01944
$867$	0	0
$868$	16.0000i	0.543075i
$869$	−32.0000	−1.08553
$870$	0	0
$871$	−16.0000	−0.542139
$872$	− 2.00000i	− 0.0677285i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	− 16.0000i	− 0.539974i
$879$	0	0
$880$	0	0
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	0	0
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	4.00000	0.134383
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	− 4.00000i	− 0.133930i
$893$	− 64.0000i	− 2.14168i
$894$	0	0
$895$	0	0
$896$	4.00000	0.133631
$897$	0	0
$898$	6.00000i	0.200223i
$899$	−24.0000	−0.800445
$900$	0	0
$901$	−20.0000	−0.666297
$902$	40.0000i	1.33185i
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	20.0000i	0.663723i
$909$	0	0
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	48.0000i	1.58857i
$914$	30.0000	0.992312
$915$	0	0
$916$	22.0000	0.726900
$917$	− 16.0000i	− 0.528367i
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	0	0
$922$	6.00000i	0.197599i
$923$	− 8.00000i	− 0.263323i
$924$	0	0
$925$	0	0
$926$	−20.0000	−0.657241
$927$	0	0
$928$	6.00000i	0.196960i
$929$	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
$931$	−72.0000	−2.35970
$932$	− 18.0000i	− 0.589610i
$933$	0	0
$934$	−4.00000	−0.130884
$935$	0	0
$936$	0	0
$937$	26.0000i	0.849383i	0.905338	+	0.424691i	$0.139617\pi$
$937$	26.0000i	0.849383i	−0.905338	+	0.424691i	$0.860383\pi$
$938$	64.0000i	2.08967i
$939$	0	0
$940$	0	0
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	0	0
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	16.0000	0.520205
$947$	4.00000i	0.129983i	0.997886	+	0.0649913i	$0.0207020\pi$
$947$	4.00000i	0.129983i	−0.997886	+	0.0649913i	$0.979298\pi$
$948$	0	0
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	0	0
$952$	− 8.00000i	− 0.259281i
$953$	− 30.0000i	− 0.971795i	−0.874016	−	0.485898i	$-0.838493\pi$
$953$	− 30.0000i	− 0.971795i	0.874016	−	0.485898i	$-0.161507\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	− 16.0000i	− 0.516937i
$959$	−40.0000	−1.29167
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 2.00000i	− 0.0644826i
$963$	0	0
$964$	−10.0000	−0.322078
$965$	0	0
$966$	0	0
$967$	4.00000i	0.128631i	0.997930	+	0.0643157i	$0.0204865\pi$
$967$	4.00000i	0.128631i	−0.997930	+	0.0643157i	$0.979514\pi$
$968$	− 5.00000i	− 0.160706i
$969$	0	0
$970$	0	0
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	− 48.0000i	− 1.53881i
$974$	−4.00000	−0.128168
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	− 6.00000i	− 0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	− 6.00000i	− 0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$	0	0
$979$	56.0000	1.78977
$980$	0	0
$981$	0	0
$982$	− 36.0000i	− 1.14881i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	0	0
$985$	0	0
$986$	12.0000	0.382158
$987$	0	0
$988$	8.00000i	0.254514i
$989$	0	0
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	− 4.00000i	− 0.127000i
$993$	0	0
$994$	−32.0000	−1.01498
$995$	0	0
$996$	0	0
$997$	− 26.0000i	− 0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$	− 26.0000i	− 0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.e.bb.5149.2		2
3.2	odd	2		1950.2.e.i.1249.1		2
5.2	odd	4		5850.2.a.d.1.1		1
5.3	odd	4		234.2.a.c.1.1		1
5.4	even	2	inner	5850.2.e.bb.5149.1		2
15.2	even	4		1950.2.a.w.1.1		1
15.8	even	4		78.2.a.a.1.1	✓	1
15.14	odd	2		1950.2.e.i.1249.2		2
20.3	even	4		1872.2.a.c.1.1		1
40.3	even	4		7488.2.a.bk.1.1		1
40.13	odd	4		7488.2.a.bz.1.1		1
45.13	odd	12		2106.2.e.j.703.1		2
45.23	even	12		2106.2.e.q.703.1		2
45.38	even	12		2106.2.e.q.1405.1		2
45.43	odd	12		2106.2.e.j.1405.1		2
60.23	odd	4		624.2.a.h.1.1		1
65.8	even	4		3042.2.b.g.1351.2		2
65.18	even	4		3042.2.b.g.1351.1		2
65.38	odd	4		3042.2.a.f.1.1		1
105.83	odd	4		3822.2.a.j.1.1		1
120.53	even	4		2496.2.a.t.1.1		1
120.83	odd	4		2496.2.a.b.1.1		1
165.98	odd	4		9438.2.a.t.1.1		1
195.8	odd	4		1014.2.b.b.337.1		2
195.23	even	12		1014.2.e.c.529.1		2
195.38	even	4		1014.2.a.d.1.1		1
195.68	even	12		1014.2.e.f.529.1		2
195.83	odd	4		1014.2.b.b.337.2		2
195.98	odd	12		1014.2.i.d.361.1		4
195.113	even	12		1014.2.e.f.991.1		2
195.128	odd	12		1014.2.i.d.823.2		4
195.158	odd	12		1014.2.i.d.823.1		4
195.173	even	12		1014.2.e.c.991.1		2
195.188	odd	12		1014.2.i.d.361.2		4
780.623	odd	4		8112.2.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.a.a.1.1	✓	1	15.8	even	4
234.2.a.c.1.1		1	5.3	odd	4
624.2.a.h.1.1		1	60.23	odd	4
1014.2.a.d.1.1		1	195.38	even	4
1014.2.b.b.337.1		2	195.8	odd	4
1014.2.b.b.337.2		2	195.83	odd	4
1014.2.e.c.529.1		2	195.23	even	12
1014.2.e.c.991.1		2	195.173	even	12
1014.2.e.f.529.1		2	195.68	even	12
1014.2.e.f.991.1		2	195.113	even	12
1014.2.i.d.361.1		4	195.98	odd	12
1014.2.i.d.361.2		4	195.188	odd	12
1014.2.i.d.823.1		4	195.158	odd	12
1014.2.i.d.823.2		4	195.128	odd	12
1872.2.a.c.1.1		1	20.3	even	4
1950.2.a.w.1.1		1	15.2	even	4
1950.2.e.i.1249.1		2	3.2	odd	2
1950.2.e.i.1249.2		2	15.14	odd	2
2106.2.e.j.703.1		2	45.13	odd	12
2106.2.e.j.1405.1		2	45.43	odd	12
2106.2.e.q.703.1		2	45.23	even	12
2106.2.e.q.1405.1		2	45.38	even	12
2496.2.a.b.1.1		1	120.83	odd	4
2496.2.a.t.1.1		1	120.53	even	4
3042.2.a.f.1.1		1	65.38	odd	4
3042.2.b.g.1351.1		2	65.18	even	4
3042.2.b.g.1351.2		2	65.8	even	4
3822.2.a.j.1.1		1	105.83	odd	4
5850.2.a.d.1.1		1	5.2	odd	4
5850.2.e.bb.5149.1		2	5.4	even	2	inner
5850.2.e.bb.5149.2		2	1.1	even	1	trivial
7488.2.a.bk.1.1		1	40.3	even	4
7488.2.a.bz.1.1		1	40.13	odd	4
8112.2.a.v.1.1		1	780.623	odd	4
9438.2.a.t.1.1		1	165.98	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:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +4.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +4.00000i q^{7} -1.00000i q^{8} +4.00000 q^{11} -1.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +8.00000 q^{19} +4.00000i q^{22} +1.00000 q^{26} -4.00000i q^{28} +6.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} +2.00000 q^{34} -2.00000i q^{37} +8.00000i q^{38} +10.0000 q^{41} -4.00000i q^{43} -4.00000 q^{44} -8.00000i q^{47} -9.00000 q^{49} +1.00000i q^{52} -10.0000i q^{53} +4.00000 q^{56} +6.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} -16.0000i q^{67} +2.00000i q^{68} +8.00000 q^{71} -2.00000i q^{73} +2.00000 q^{74} -8.00000 q^{76} +16.0000i q^{77} -8.00000 q^{79} +10.0000i q^{82} +12.0000i q^{83} +4.00000 q^{86} -4.00000i q^{88} +14.0000 q^{89} +4.00000 q^{91} +8.00000 q^{94} +10.0000i q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 8 q^{11} - 8 q^{14} + 2 q^{16} + 16 q^{19} + 2 q^{26} + 12 q^{29} - 8 q^{31} + 4 q^{34} + 20 q^{41} - 8 q^{44} - 18 q^{49} + 8 q^{56} + 8 q^{59} - 4 q^{61} - 2 q^{64} + 16 q^{71} + 4 q^{74} - 16 q^{76} - 16 q^{79} + 8 q^{86} + 28 q^{89} + 8 q^{91} + 16 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 8 * q^11 - 8 * q^14 + 2 * q^16 + 16 * q^19 + 2 * q^26 + 12 * q^29 - 8 * q^31 + 4 * q^34 + 20 * q^41 - 8 * q^44 - 18 * q^49 + 8 * q^56 + 8 * q^59 - 4 * q^61 - 2 * q^64 + 16 * q^71 + 4 * q^74 - 16 * q^76 - 16 * q^79 + 8 * q^86 + 28 * q^89 + 8 * q^91 + 16 * q^94

Introduction

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