LMFDB - Embedded newform 1150.2.b.a.599.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,2,Mod(599,1150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1150.599");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1150.b (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:	$9.18279623245$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1150.599
Dual form			1150.2.b.a.599.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -6.00000 q^{9} +3.00000 q^{11} +3.00000i q^{12} +6.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +5.00000i q^{17} +6.00000i q^{18} +1.00000 q^{19} +12.0000 q^{21} -3.00000i q^{22} -1.00000i q^{23} +3.00000 q^{24} +6.00000 q^{26} +9.00000i q^{27} -4.00000i q^{28} +8.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -9.00000i q^{33} +5.00000 q^{34} +6.00000 q^{36} +2.00000i q^{37} -1.00000i q^{38} +18.0000 q^{39} -7.00000 q^{41} -12.0000i q^{42} -4.00000i q^{43} -3.00000 q^{44} -1.00000 q^{46} +10.0000i q^{47} -3.00000i q^{48} -9.00000 q^{49} +15.0000 q^{51} -6.00000i q^{52} +12.0000i q^{53} +9.00000 q^{54} -4.00000 q^{56} -3.00000i q^{57} -8.00000i q^{58} -4.00000 q^{59} -8.00000 q^{61} +8.00000i q^{62} -24.0000i q^{63} -1.00000 q^{64} -9.00000 q^{66} +3.00000i q^{67} -5.00000i q^{68} -3.00000 q^{69} +4.00000 q^{71} -6.00000i q^{72} +7.00000i q^{73} +2.00000 q^{74} -1.00000 q^{76} +12.0000i q^{77} -18.0000i q^{78} +6.00000 q^{79} +9.00000 q^{81} +7.00000i q^{82} -11.0000i q^{83} -12.0000 q^{84} -4.00000 q^{86} -24.0000i q^{87} +3.00000i q^{88} +3.00000 q^{89} -24.0000 q^{91} +1.00000i q^{92} +24.0000i q^{93} +10.0000 q^{94} -3.00000 q^{96} -14.0000i q^{97} +9.00000i q^{98} -18.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 6 q^{6} - 12 q^{9} + 6 q^{11} + 8 q^{14} + 2 q^{16} + 2 q^{19} + 24 q^{21} + 6 q^{24} + 12 q^{26} + 16 q^{29} - 16 q^{31} + 10 q^{34} + 12 q^{36} + 36 q^{39} - 14 q^{41} - 6 q^{44} - 2 q^{46}+ \cdots - 36 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 6 * q^6 - 12 * q^9 + 6 * q^11 + 8 * q^14 + 2 * q^16 + 2 * q^19 + 24 * q^21 + 6 * q^24 + 12 * q^26 + 16 * q^29 - 16 * q^31 + 10 * q^34 + 12 * q^36 + 36 * q^39 - 14 * q^41 - 6 * q^44 - 2 * q^46 - 18 * q^49 + 30 * q^51 + 18 * q^54 - 8 * q^56 - 8 * q^59 - 16 * q^61 - 2 * q^64 - 18 * q^66 - 6 * q^69 + 8 * q^71 + 4 * q^74 - 2 * q^76 + 12 * q^79 + 18 * q^81 - 24 * q^84 - 8 * q^86 + 6 * q^89 - 48 * q^91 + 20 * q^94 - 6 * q^96 - 36 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

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
$3$	− 3.00000i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	− 3.00000i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−3.00000	−1.22474
$7$	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
$9$	−6.00000	−2.00000
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	3.00000i	0.866025i
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	5.00000i	1.21268i	0.795206	+	0.606339i	$0.207363\pi$
$17$	5.00000i	1.21268i	−0.795206	+	0.606339i	$0.792637\pi$
$18$	6.00000i	1.41421i
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	12.0000	2.61861
$22$	− 3.00000i	− 0.639602i
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	3.00000	0.612372
$25$	0	0
$26$	6.00000	1.17670
$27$	9.00000i	1.73205i
$28$	− 4.00000i	− 0.755929i
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 9.00000i	− 1.56670i
$34$	5.00000	0.857493
$35$	0	0
$36$	6.00000	1.00000
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	− 1.00000i	− 0.162221i
$39$	18.0000	2.88231
$40$	0	0
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	− 12.0000i	− 1.85164i
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−1.00000	−0.147442
$47$	10.0000i	1.45865i	0.684167	+	0.729325i	$0.260166\pi$
$47$	10.0000i	1.45865i	−0.684167	+	0.729325i	$0.739834\pi$
$48$	− 3.00000i	− 0.433013i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	15.0000	2.10042
$52$	− 6.00000i	− 0.832050i
$53$	12.0000i	1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$	12.0000i	1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$	9.00000	1.22474
$55$	0	0
$56$	−4.00000	−0.534522
$57$	− 3.00000i	− 0.397360i
$58$	− 8.00000i	− 1.05045i
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	8.00000i	1.01600i
$63$	− 24.0000i	− 3.02372i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−9.00000	−1.10782
$67$	3.00000i	0.366508i	0.983066	+	0.183254i	$0.0586631\pi$
$67$	3.00000i	0.366508i	−0.983066	+	0.183254i	$0.941337\pi$
$68$	− 5.00000i	− 0.606339i
$69$	−3.00000	−0.361158
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	− 6.00000i	− 0.707107i
$73$	7.00000i	0.819288i	0.912245	+	0.409644i	$0.134347\pi$
$73$	7.00000i	0.819288i	−0.912245	+	0.409644i	$0.865653\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−1.00000	−0.114708
$77$	12.0000i	1.36753i
$78$	− 18.0000i	− 2.03810i
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	7.00000i	0.773021i
$83$	− 11.0000i	− 1.20741i	−0.797209	−	0.603703i	$-0.793691\pi$
$83$	− 11.0000i	− 1.20741i	0.797209	−	0.603703i	$-0.206309\pi$
$84$	−12.0000	−1.30931
$85$	0	0
$86$	−4.00000	−0.431331
$87$	− 24.0000i	− 2.57307i
$88$	3.00000i	0.319801i
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	−24.0000	−2.51588
$92$	1.00000i	0.104257i
$93$	24.0000i	2.48868i
$94$	10.0000	1.03142
$95$	0	0
$96$	−3.00000	−0.306186
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	9.00000i	0.909137i
$99$	−18.0000	−1.80907
$100$	0	0
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	− 15.0000i	− 1.48522i
$103$	− 10.0000i	− 0.985329i	−0.870219	−	0.492665i	$-0.836023\pi$
$103$	− 10.0000i	− 0.985329i	0.870219	−	0.492665i	$-0.163977\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	12.0000	1.16554
$107$	5.00000i	0.483368i	0.970355	+	0.241684i	$0.0776998\pi$
$107$	5.00000i	0.483368i	−0.970355	+	0.241684i	$0.922300\pi$
$108$	− 9.00000i	− 0.866025i
$109$	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
$111$	6.00000	0.569495
$112$	4.00000i	0.377964i
$113$	− 15.0000i	− 1.41108i	−0.708669	−	0.705541i	$-0.750704\pi$
$113$	− 15.0000i	− 1.41108i	0.708669	−	0.705541i	$-0.249296\pi$
$114$	−3.00000	−0.280976
$115$	0	0
$116$	−8.00000	−0.742781
$117$	− 36.0000i	− 3.32820i
$118$	4.00000i	0.368230i
$119$	−20.0000	−1.83340
$120$	0	0
$121$	−2.00000	−0.181818
$122$	8.00000i	0.724286i
$123$	21.0000i	1.89351i
$124$	8.00000	0.718421
$125$	0	0
$126$	−24.0000	−2.13809
$127$	− 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
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	9.00000i	0.783349i
$133$	4.00000i	0.346844i
$134$	3.00000	0.259161
$135$	0	0
$136$	−5.00000	−0.428746
$137$	− 3.00000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 3.00000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	3.00000i	0.255377i
$139$	19.0000	1.61156	0.805779	−	0.592216i	$-0.201747\pi$
$139$	19.0000	1.61156	0.805779	+	0.592216i	$0.201747\pi$
$140$	0	0
$141$	30.0000	2.52646
$142$	− 4.00000i	− 0.335673i
$143$	18.0000i	1.50524i
$144$	−6.00000	−0.500000
$145$	0	0
$146$	7.00000	0.579324
$147$	27.0000i	2.22692i
$148$	− 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
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	1.00000i	0.0811107i
$153$	− 30.0000i	− 2.42536i
$154$	12.0000	0.966988
$155$	0	0
$156$	−18.0000	−1.44115
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	− 6.00000i	− 0.477334i
$159$	36.0000	2.85499
$160$	0	0
$161$	4.00000	0.315244
$162$	− 9.00000i	− 0.707107i
$163$	− 5.00000i	− 0.391630i	−0.980641	−	0.195815i	$-0.937265\pi$
$163$	− 5.00000i	− 0.391630i	0.980641	−	0.195815i	$-0.0627352\pi$
$164$	7.00000	0.546608
$165$	0	0
$166$	−11.0000	−0.853766
$167$	22.0000i	1.70241i	0.524832	+	0.851206i	$0.324128\pi$
$167$	22.0000i	1.70241i	−0.524832	+	0.851206i	$0.675872\pi$
$168$	12.0000i	0.925820i
$169$	−23.0000	−1.76923
$170$	0	0
$171$	−6.00000	−0.458831
$172$	4.00000i	0.304997i
$173$	− 4.00000i	− 0.304114i	−0.988372	−	0.152057i	$-0.951410\pi$
$173$	− 4.00000i	− 0.304114i	0.988372	−	0.152057i	$-0.0485898\pi$
$174$	−24.0000	−1.81944
$175$	0	0
$176$	3.00000	0.226134
$177$	12.0000i	0.901975i
$178$	− 3.00000i	− 0.224860i
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	24.0000i	1.77900i
$183$	24.0000i	1.77413i
$184$	1.00000	0.0737210
$185$	0	0
$186$	24.0000	1.75977
$187$	15.0000i	1.09691i
$188$	− 10.0000i	− 0.729325i
$189$	−36.0000	−2.61861
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	3.00000i	0.216506i
$193$	9.00000i	0.647834i	0.946085	+	0.323917i	$0.105000\pi$
$193$	9.00000i	0.647834i	−0.946085	+	0.323917i	$0.895000\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	9.00000	0.642857
$197$	− 8.00000i	− 0.569976i	−0.958531	−	0.284988i	$-0.908010\pi$
$197$	− 8.00000i	− 0.569976i	0.958531	−	0.284988i	$-0.0919897\pi$
$198$	18.0000i	1.27920i
$199$	−18.0000	−1.27599	−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000	−1.27599	−0.637993	+	0.770042i	$0.720235\pi$
$200$	0	0
$201$	9.00000	0.634811
$202$	− 4.00000i	− 0.281439i
$203$	32.0000i	2.24596i
$204$	−15.0000	−1.05021
$205$	0	0
$206$	−10.0000	−0.696733
$207$	6.00000i	0.417029i
$208$	6.00000i	0.416025i
$209$	3.00000	0.207514
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	− 12.0000i	− 0.824163i
$213$	− 12.0000i	− 0.822226i
$214$	5.00000	0.341793
$215$	0	0
$216$	−9.00000	−0.612372
$217$	− 32.0000i	− 2.17230i
$218$	− 10.0000i	− 0.677285i
$219$	21.0000	1.41905
$220$	0	0
$221$	−30.0000	−2.01802
$222$	− 6.00000i	− 0.402694i
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	−15.0000	−0.997785
$227$	− 8.00000i	− 0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$	− 8.00000i	− 0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$	3.00000i	0.198680i
$229$	30.0000	1.98246	0.991228	−	0.132164i	$-0.0421925\pi$
$229$	30.0000	1.98246	0.991228	+	0.132164i	$0.0421925\pi$
$230$	0	0
$231$	36.0000	2.36863
$232$	8.00000i	0.525226i
$233$	14.0000i	0.917170i	0.888650	+	0.458585i	$0.151644\pi$
$233$	14.0000i	0.917170i	−0.888650	+	0.458585i	$0.848356\pi$
$234$	−36.0000	−2.35339
$235$	0	0
$236$	4.00000	0.260378
$237$	− 18.0000i	− 1.16923i
$238$	20.0000i	1.29641i
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	23.0000	1.48156	0.740780	−	0.671748i	$-0.234456\pi$
$241$	23.0000	1.48156	0.740780	+	0.671748i	$0.234456\pi$
$242$	2.00000i	0.128565i
$243$	0	0
$244$	8.00000	0.512148
$245$	0	0
$246$	21.0000	1.33891
$247$	6.00000i	0.381771i
$248$	− 8.00000i	− 0.508001i
$249$	−33.0000	−2.09129
$250$	0	0
$251$	−7.00000	−0.441836	−0.220918	−	0.975292i	$-0.570905\pi$
$251$	−7.00000	−0.441836	−0.220918	+	0.975292i	$0.570905\pi$
$252$	24.0000i	1.51186i
$253$	− 3.00000i	− 0.188608i
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000i	0.124757i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$	2.00000i	0.124757i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$	12.0000i	0.747087i
$259$	−8.00000	−0.497096
$260$	0	0
$261$	−48.0000	−2.97113
$262$	− 12.0000i	− 0.741362i
$263$	4.00000i	0.246651i	0.992366	+	0.123325i	$0.0393559\pi$
$263$	4.00000i	0.246651i	−0.992366	+	0.123325i	$0.960644\pi$
$264$	9.00000	0.553912
$265$	0	0
$266$	4.00000	0.245256
$267$	− 9.00000i	− 0.550791i
$268$	− 3.00000i	− 0.183254i
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	10.0000	0.607457	0.303728	−	0.952759i	$-0.401768\pi$
$271$	10.0000	0.607457	0.303728	+	0.952759i	$0.401768\pi$
$272$	5.00000i	0.303170i
$273$	72.0000i	4.35764i
$274$	−3.00000	−0.181237
$275$	0	0
$276$	3.00000	0.180579
$277$	− 28.0000i	− 1.68236i	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	− 28.0000i	− 1.68236i	0.540758	−	0.841178i	$-0.318138\pi$
$278$	− 19.0000i	− 1.13954i
$279$	48.0000	2.87368
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	− 30.0000i	− 1.78647i
$283$	33.0000i	1.96165i	0.194900	+	0.980823i	$0.437562\pi$
$283$	33.0000i	1.96165i	−0.194900	+	0.980823i	$0.562438\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	18.0000	1.06436
$287$	− 28.0000i	− 1.65279i
$288$	6.00000i	0.353553i
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−42.0000	−2.46208
$292$	− 7.00000i	− 0.409644i
$293$	− 24.0000i	− 1.40209i	−0.713115	−	0.701047i	$-0.752716\pi$
$293$	− 24.0000i	− 1.40209i	0.713115	−	0.701047i	$-0.247284\pi$
$294$	27.0000	1.57467
$295$	0	0
$296$	−2.00000	−0.116248
$297$	27.0000i	1.56670i
$298$	6.00000i	0.347571i
$299$	6.00000	0.346989
$300$	0	0
$301$	16.0000	0.922225
$302$	− 20.0000i	− 1.15087i
$303$	− 12.0000i	− 0.689382i
$304$	1.00000	0.0573539
$305$	0	0
$306$	−30.0000	−1.71499
$307$	− 9.00000i	− 0.513657i	−0.966457	−	0.256829i	$-0.917322\pi$
$307$	− 9.00000i	− 0.513657i	0.966457	−	0.256829i	$-0.0826776\pi$
$308$	− 12.0000i	− 0.683763i
$309$	−30.0000	−1.70664
$310$	0	0
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	18.0000i	1.01905i
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−6.00000	−0.337526
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	− 36.0000i	− 2.01878i
$319$	24.0000	1.34374
$320$	0	0
$321$	15.0000	0.837218
$322$	− 4.00000i	− 0.222911i
$323$	5.00000i	0.278207i
$324$	−9.00000	−0.500000
$325$	0	0
$326$	−5.00000	−0.276924
$327$	− 30.0000i	− 1.65900i
$328$	− 7.00000i	− 0.386510i
$329$	−40.0000	−2.20527
$330$	0	0
$331$	25.0000	1.37412	0.687062	−	0.726599i	$-0.258900\pi$
$331$	25.0000	1.37412	0.687062	+	0.726599i	$0.258900\pi$
$332$	11.0000i	0.603703i
$333$	− 12.0000i	− 0.657596i
$334$	22.0000	1.20379
$335$	0	0
$336$	12.0000	0.654654
$337$	3.00000i	0.163420i	0.996656	+	0.0817102i	$0.0260382\pi$
$337$	3.00000i	0.163420i	−0.996656	+	0.0817102i	$0.973962\pi$
$338$	23.0000i	1.25104i
$339$	−45.0000	−2.44406
$340$	0	0
$341$	−24.0000	−1.29967
$342$	6.00000i	0.324443i
$343$	− 8.00000i	− 0.431959i
$344$	4.00000	0.215666
$345$	0	0
$346$	−4.00000	−0.215041
$347$	7.00000i	0.375780i	0.982190	+	0.187890i	$0.0601648\pi$
$347$	7.00000i	0.375780i	−0.982190	+	0.187890i	$0.939835\pi$
$348$	24.0000i	1.28654i
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	−54.0000	−2.88231
$352$	− 3.00000i	− 0.159901i
$353$	− 18.0000i	− 0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$	− 18.0000i	− 0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−3.00000	−0.159000
$357$	60.0000i	3.17554i
$358$	− 3.00000i	− 0.158555i
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	22.0000i	1.15629i
$363$	6.00000i	0.314918i
$364$	24.0000	1.25794
$365$	0	0
$366$	24.0000	1.25450
$367$	2.00000i	0.104399i	0.998637	+	0.0521996i	$0.0166232\pi$
$367$	2.00000i	0.104399i	−0.998637	+	0.0521996i	$0.983377\pi$
$368$	− 1.00000i	− 0.0521286i
$369$	42.0000	2.18643
$370$	0	0
$371$	−48.0000	−2.49204
$372$	− 24.0000i	− 1.24434i
$373$	12.0000i	0.621336i	0.950518	+	0.310668i	$0.100553\pi$
$373$	12.0000i	0.621336i	−0.950518	+	0.310668i	$0.899447\pi$
$374$	15.0000	0.775632
$375$	0	0
$376$	−10.0000	−0.515711
$377$	48.0000i	2.47213i
$378$	36.0000i	1.85164i
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	0	0
$381$	−24.0000	−1.22956
$382$	− 12.0000i	− 0.613973i
$383$	6.00000i	0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$	6.00000i	0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	9.00000	0.458088
$387$	24.0000i	1.21999i
$388$	14.0000i	0.710742i
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	5.00000	0.252861
$392$	− 9.00000i	− 0.454569i
$393$	− 36.0000i	− 1.81596i
$394$	−8.00000	−0.403034
$395$	0	0
$396$	18.0000	0.904534
$397$	− 12.0000i	− 0.602263i	−0.953583	−	0.301131i	$-0.902636\pi$
$397$	− 12.0000i	− 0.602263i	0.953583	−	0.301131i	$-0.0973643\pi$
$398$	18.0000i	0.902258i
$399$	12.0000	0.600751
$400$	0	0
$401$	−5.00000	−0.249688	−0.124844	−	0.992176i	$-0.539843\pi$
$401$	−5.00000	−0.249688	−0.124844	+	0.992176i	$0.539843\pi$
$402$	− 9.00000i	− 0.448879i
$403$	− 48.0000i	− 2.39105i
$404$	−4.00000	−0.199007
$405$	0	0
$406$	32.0000	1.58813
$407$	6.00000i	0.297409i
$408$	15.0000i	0.742611i
$409$	−21.0000	−1.03838	−0.519192	−	0.854658i	$-0.673767\pi$
$409$	−21.0000	−1.03838	−0.519192	+	0.854658i	$0.673767\pi$
$410$	0	0
$411$	−9.00000	−0.443937
$412$	10.0000i	0.492665i
$413$	− 16.0000i	− 0.787309i
$414$	6.00000	0.294884
$415$	0	0
$416$	6.00000	0.294174
$417$	− 57.0000i	− 2.79130i
$418$	− 3.00000i	− 0.146735i
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	1.00000i	0.0486792i
$423$	− 60.0000i	− 2.91730i
$424$	−12.0000	−0.582772
$425$	0	0
$426$	−12.0000	−0.581402
$427$	− 32.0000i	− 1.54859i
$428$	− 5.00000i	− 0.241684i
$429$	54.0000	2.60714
$430$	0	0
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	9.00000i	0.433013i
$433$	− 29.0000i	− 1.39365i	−0.717241	−	0.696826i	$-0.754595\pi$
$433$	− 29.0000i	− 1.39365i	0.717241	−	0.696826i	$-0.245405\pi$
$434$	−32.0000	−1.53605
$435$	0	0
$436$	−10.0000	−0.478913
$437$	− 1.00000i	− 0.0478365i
$438$	− 21.0000i	− 1.00342i
$439$	6.00000	0.286364	0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000	0.286364	0.143182	+	0.989696i	$0.454267\pi$
$440$	0	0
$441$	54.0000	2.57143
$442$	30.0000i	1.42695i
$443$	39.0000i	1.85295i	0.376361	+	0.926473i	$0.377175\pi$
$443$	39.0000i	1.85295i	−0.376361	+	0.926473i	$0.622825\pi$
$444$	−6.00000	−0.284747
$445$	0	0
$446$	0	0
$447$	18.0000i	0.851371i
$448$	− 4.00000i	− 0.188982i
$449$	−5.00000	−0.235965	−0.117982	−	0.993016i	$-0.537643\pi$
$449$	−5.00000	−0.235965	−0.117982	+	0.993016i	$0.537643\pi$
$450$	0	0
$451$	−21.0000	−0.988851
$452$	15.0000i	0.705541i
$453$	− 60.0000i	− 2.81905i
$454$	−8.00000	−0.375459
$455$	0	0
$456$	3.00000	0.140488
$457$	− 7.00000i	− 0.327446i	−0.986506	−	0.163723i	$-0.947650\pi$
$457$	− 7.00000i	− 0.327446i	0.986506	−	0.163723i	$-0.0523504\pi$
$458$	− 30.0000i	− 1.40181i
$459$	−45.0000	−2.10042
$460$	0	0
$461$	−24.0000	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$462$	− 36.0000i	− 1.67487i
$463$	− 26.0000i	− 1.20832i	−0.796862	−	0.604161i	$-0.793508\pi$
$463$	− 26.0000i	− 1.20832i	0.796862	−	0.604161i	$-0.206492\pi$
$464$	8.00000	0.371391
$465$	0	0
$466$	14.0000	0.648537
$467$	28.0000i	1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$	28.0000i	1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$	36.0000i	1.66410i
$469$	−12.0000	−0.554109
$470$	0	0
$471$	42.0000	1.93526
$472$	− 4.00000i	− 0.184115i
$473$	− 12.0000i	− 0.551761i
$474$	−18.0000	−0.826767
$475$	0	0
$476$	20.0000	0.916698
$477$	− 72.0000i	− 3.29665i
$478$	− 6.00000i	− 0.274434i
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	− 23.0000i	− 1.04762i
$483$	− 12.0000i	− 0.546019i
$484$	2.00000	0.0909091
$485$	0	0
$486$	0	0
$487$	− 30.0000i	− 1.35943i	−0.733476	−	0.679715i	$-0.762104\pi$
$487$	− 30.0000i	− 1.35943i	0.733476	−	0.679715i	$-0.237896\pi$
$488$	− 8.00000i	− 0.362143i
$489$	−15.0000	−0.678323
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	− 21.0000i	− 0.946753i
$493$	40.0000i	1.80151i
$494$	6.00000	0.269953
$495$	0	0
$496$	−8.00000	−0.359211
$497$	16.0000i	0.717698i
$498$	33.0000i	1.47877i
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	0	0
$501$	66.0000	2.94866
$502$	7.00000i	0.312425i
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	24.0000	1.06904
$505$	0	0
$506$	−3.00000	−0.133366
$507$	69.0000i	3.06440i
$508$	8.00000i	0.354943i
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	0	0
$511$	−28.0000	−1.23865
$512$	− 1.00000i	− 0.0441942i
$513$	9.00000i	0.397360i
$514$	2.00000	0.0882162
$515$	0	0
$516$	12.0000	0.528271
$517$	30.0000i	1.31940i
$518$	8.00000i	0.351500i
$519$	−12.0000	−0.526742
$520$	0	0
$521$	31.0000	1.35813	0.679067	−	0.734076i	$-0.262384\pi$
$521$	31.0000	1.35813	0.679067	+	0.734076i	$0.262384\pi$
$522$	48.0000i	2.10090i
$523$	− 37.0000i	− 1.61790i	−0.587879	−	0.808949i	$-0.700037\pi$
$523$	− 37.0000i	− 1.61790i	0.587879	−	0.808949i	$-0.299963\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	4.00000	0.174408
$527$	− 40.0000i	− 1.74243i
$528$	− 9.00000i	− 0.391675i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	24.0000	1.04151
$532$	− 4.00000i	− 0.173422i
$533$	− 42.0000i	− 1.81922i
$534$	−9.00000	−0.389468
$535$	0	0
$536$	−3.00000	−0.129580
$537$	− 9.00000i	− 0.388379i
$538$	− 6.00000i	− 0.258678i
$539$	−27.0000	−1.16297
$540$	0	0
$541$	16.0000	0.687894	0.343947	−	0.938989i	$-0.388236\pi$
$541$	16.0000	0.687894	0.343947	+	0.938989i	$0.388236\pi$
$542$	− 10.0000i	− 0.429537i
$543$	66.0000i	2.83233i
$544$	5.00000	0.214373
$545$	0	0
$546$	72.0000	3.08132
$547$	− 17.0000i	− 0.726868i	−0.931620	−	0.363434i	$-0.881604\pi$
$547$	− 17.0000i	− 0.726868i	0.931620	−	0.363434i	$-0.118396\pi$
$548$	3.00000i	0.128154i
$549$	48.0000	2.04859
$550$	0	0
$551$	8.00000	0.340811
$552$	− 3.00000i	− 0.127688i
$553$	24.0000i	1.02058i
$554$	−28.0000	−1.18961
$555$	0	0
$556$	−19.0000	−0.805779
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	− 48.0000i	− 2.03200i
$559$	24.0000	1.01509
$560$	0	0
$561$	45.0000	1.89990
$562$	30.0000i	1.26547i
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	−30.0000	−1.26323
$565$	0	0
$566$	33.0000	1.38709
$567$	36.0000i	1.51186i
$568$	4.00000i	0.167836i
$569$	9.00000	0.377300	0.188650	−	0.982044i	$-0.439589\pi$
$569$	9.00000	0.377300	0.188650	+	0.982044i	$0.439589\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	− 18.0000i	− 0.752618i
$573$	− 36.0000i	− 1.50392i
$574$	−28.0000	−1.16870
$575$	0	0
$576$	6.00000	0.250000
$577$	17.0000i	0.707719i	0.935299	+	0.353860i	$0.115131\pi$
$577$	17.0000i	0.707719i	−0.935299	+	0.353860i	$0.884869\pi$
$578$	8.00000i	0.332756i
$579$	27.0000	1.12208
$580$	0	0
$581$	44.0000	1.82543
$582$	42.0000i	1.74096i
$583$	36.0000i	1.49097i
$584$	−7.00000	−0.289662
$585$	0	0
$586$	−24.0000	−0.991431
$587$	17.0000i	0.701665i	0.936438	+	0.350833i	$0.114101\pi$
$587$	17.0000i	0.701665i	−0.936438	+	0.350833i	$0.885899\pi$
$588$	− 27.0000i	− 1.11346i
$589$	−8.00000	−0.329634
$590$	0	0
$591$	−24.0000	−0.987228
$592$	2.00000i	0.0821995i
$593$	35.0000i	1.43728i	0.695383	+	0.718639i	$0.255235\pi$
$593$	35.0000i	1.43728i	−0.695383	+	0.718639i	$0.744765\pi$
$594$	27.0000	1.10782
$595$	0	0
$596$	6.00000	0.245770
$597$	54.0000i	2.21007i
$598$	− 6.00000i	− 0.245358i
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	31.0000	1.26452	0.632258	−	0.774758i	$-0.282128\pi$
$601$	31.0000	1.26452	0.632258	+	0.774758i	$0.282128\pi$
$602$	− 16.0000i	− 0.652111i
$603$	− 18.0000i	− 0.733017i
$604$	−20.0000	−0.813788
$605$	0	0
$606$	−12.0000	−0.487467
$607$	10.0000i	0.405887i	0.979190	+	0.202944i	$0.0650509\pi$
$607$	10.0000i	0.405887i	−0.979190	+	0.202944i	$0.934949\pi$
$608$	− 1.00000i	− 0.0405554i
$609$	96.0000	3.89012
$610$	0	0
$611$	−60.0000	−2.42734
$612$	30.0000i	1.21268i
$613$	8.00000i	0.323117i	0.986863	+	0.161558i	$0.0516520\pi$
$613$	8.00000i	0.323117i	−0.986863	+	0.161558i	$0.948348\pi$
$614$	−9.00000	−0.363210
$615$	0	0
$616$	−12.0000	−0.483494
$617$	30.0000i	1.20775i	0.797077	+	0.603877i	$0.206378\pi$
$617$	30.0000i	1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$618$	30.0000i	1.20678i
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	9.00000	0.361158
$622$	− 2.00000i	− 0.0801927i
$623$	12.0000i	0.480770i
$624$	18.0000	0.720577
$625$	0	0
$626$	10.0000	0.399680
$627$	− 9.00000i	− 0.359425i
$628$	− 14.0000i	− 0.558661i
$629$	−10.0000	−0.398726
$630$	0	0
$631$	30.0000	1.19428	0.597141	−	0.802137i	$-0.296303\pi$
$631$	30.0000	1.19428	0.597141	+	0.802137i	$0.296303\pi$
$632$	6.00000i	0.238667i
$633$	3.00000i	0.119239i
$634$	12.0000	0.476581
$635$	0	0
$636$	−36.0000	−1.42749
$637$	− 54.0000i	− 2.13956i
$638$	− 24.0000i	− 0.950169i
$639$	−24.0000	−0.949425
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	− 15.0000i	− 0.592003i
$643$	− 32.0000i	− 1.26196i	−0.775800	−	0.630978i	$-0.782654\pi$
$643$	− 32.0000i	− 1.26196i	0.775800	−	0.630978i	$-0.217346\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	5.00000	0.196722
$647$	28.0000i	1.10079i	0.834903	+	0.550397i	$0.185524\pi$
$647$	28.0000i	1.10079i	−0.834903	+	0.550397i	$0.814476\pi$
$648$	9.00000i	0.353553i
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−96.0000	−3.76254
$652$	5.00000i	0.195815i
$653$	− 14.0000i	− 0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$	− 14.0000i	− 0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$	−30.0000	−1.17309
$655$	0	0
$656$	−7.00000	−0.273304
$657$	− 42.0000i	− 1.63858i
$658$	40.0000i	1.55936i
$659$	−5.00000	−0.194772	−0.0973862	−	0.995247i	$-0.531048\pi$
$659$	−5.00000	−0.194772	−0.0973862	+	0.995247i	$0.531048\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	− 25.0000i	− 0.971653i
$663$	90.0000i	3.49531i
$664$	11.0000	0.426883
$665$	0	0
$666$	−12.0000	−0.464991
$667$	− 8.00000i	− 0.309761i
$668$	− 22.0000i	− 0.851206i
$669$	0	0
$670$	0	0
$671$	−24.0000	−0.926510
$672$	− 12.0000i	− 0.462910i
$673$	− 34.0000i	− 1.31060i	−0.755367	−	0.655302i	$-0.772541\pi$
$673$	− 34.0000i	− 1.31060i	0.755367	−	0.655302i	$-0.227459\pi$
$674$	3.00000	0.115556
$675$	0	0
$676$	23.0000	0.884615
$677$	42.0000i	1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$	42.0000i	1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$	45.0000i	1.72821i
$679$	56.0000	2.14908
$680$	0	0
$681$	−24.0000	−0.919682
$682$	24.0000i	0.919007i
$683$	− 35.0000i	− 1.33924i	−0.742705	−	0.669619i	$-0.766457\pi$
$683$	− 35.0000i	− 1.33924i	0.742705	−	0.669619i	$-0.233543\pi$
$684$	6.00000	0.229416
$685$	0	0
$686$	−8.00000	−0.305441
$687$	− 90.0000i	− 3.43371i
$688$	− 4.00000i	− 0.152499i
$689$	−72.0000	−2.74298
$690$	0	0
$691$	29.0000	1.10321	0.551606	−	0.834105i	$-0.314015\pi$
$691$	29.0000	1.10321	0.551606	+	0.834105i	$0.314015\pi$
$692$	4.00000i	0.152057i
$693$	− 72.0000i	− 2.73505i
$694$	7.00000	0.265716
$695$	0	0
$696$	24.0000	0.909718
$697$	− 35.0000i	− 1.32572i
$698$	26.0000i	0.984115i
$699$	42.0000	1.58859
$700$	0	0
$701$	−8.00000	−0.302156	−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000	−0.302156	−0.151078	+	0.988522i	$0.548274\pi$
$702$	54.0000i	2.03810i
$703$	2.00000i	0.0754314i
$704$	−3.00000	−0.113067
$705$	0	0
$706$	−18.0000	−0.677439
$707$	16.0000i	0.601742i
$708$	− 12.0000i	− 0.450988i
$709$	−16.0000	−0.600893	−0.300446	−	0.953799i	$-0.597136\pi$
$709$	−16.0000	−0.600893	−0.300446	+	0.953799i	$0.597136\pi$
$710$	0	0
$711$	−36.0000	−1.35011
$712$	3.00000i	0.112430i
$713$	8.00000i	0.299602i
$714$	60.0000	2.24544
$715$	0	0
$716$	−3.00000	−0.112115
$717$	− 18.0000i	− 0.672222i
$718$	− 20.0000i	− 0.746393i
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	40.0000	1.48968
$722$	18.0000i	0.669891i
$723$	− 69.0000i	− 2.56614i
$724$	22.0000	0.817624
$725$	0	0
$726$	6.00000	0.222681
$727$	− 28.0000i	− 1.03846i	−0.854634	−	0.519231i	$-0.826218\pi$
$727$	− 28.0000i	− 1.03846i	0.854634	−	0.519231i	$-0.173782\pi$
$728$	− 24.0000i	− 0.889499i
$729$	27.0000	1.00000
$730$	0	0
$731$	20.0000	0.739727
$732$	− 24.0000i	− 0.887066i
$733$	− 38.0000i	− 1.40356i	−0.712393	−	0.701781i	$-0.752388\pi$
$733$	− 38.0000i	− 1.40356i	0.712393	−	0.701781i	$-0.247612\pi$
$734$	2.00000	0.0738213
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	9.00000i	0.331519i
$738$	− 42.0000i	− 1.54604i
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	18.0000	0.661247
$742$	48.0000i	1.76214i
$743$	− 26.0000i	− 0.953847i	−0.878945	−	0.476924i	$-0.841752\pi$
$743$	− 26.0000i	− 0.953847i	0.878945	−	0.476924i	$-0.158248\pi$
$744$	−24.0000	−0.879883
$745$	0	0
$746$	12.0000	0.439351
$747$	66.0000i	2.41481i
$748$	− 15.0000i	− 0.548454i
$749$	−20.0000	−0.730784
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	10.0000i	0.364662i
$753$	21.0000i	0.765283i
$754$	48.0000	1.74806
$755$	0	0
$756$	36.0000	1.30931
$757$	16.0000i	0.581530i	0.956795	+	0.290765i	$0.0939098\pi$
$757$	16.0000i	0.581530i	−0.956795	+	0.290765i	$0.906090\pi$
$758$	5.00000i	0.181608i
$759$	−9.00000	−0.326679
$760$	0	0
$761$	−15.0000	−0.543750	−0.271875	−	0.962333i	$-0.587644\pi$
$761$	−15.0000	−0.543750	−0.271875	+	0.962333i	$0.587644\pi$
$762$	24.0000i	0.869428i
$763$	40.0000i	1.44810i
$764$	−12.0000	−0.434145
$765$	0	0
$766$	6.00000	0.216789
$767$	− 24.0000i	− 0.866590i
$768$	− 3.00000i	− 0.108253i
$769$	21.0000	0.757279	0.378640	−	0.925544i	$-0.376392\pi$
$769$	21.0000	0.757279	0.378640	+	0.925544i	$0.376392\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	− 9.00000i	− 0.323917i
$773$	− 4.00000i	− 0.143870i	−0.997409	−	0.0719350i	$-0.977083\pi$
$773$	− 4.00000i	− 0.143870i	0.997409	−	0.0719350i	$-0.0229174\pi$
$774$	24.0000	0.862662
$775$	0	0
$776$	14.0000	0.502571
$777$	24.0000i	0.860995i
$778$	− 36.0000i	− 1.29066i
$779$	−7.00000	−0.250801
$780$	0	0
$781$	12.0000	0.429394
$782$	− 5.00000i	− 0.178800i
$783$	72.0000i	2.57307i
$784$	−9.00000	−0.321429
$785$	0	0
$786$	−36.0000	−1.28408
$787$	− 52.0000i	− 1.85360i	−0.375555	−	0.926800i	$-0.622548\pi$
$787$	− 52.0000i	− 1.85360i	0.375555	−	0.926800i	$-0.377452\pi$
$788$	8.00000i	0.284988i
$789$	12.0000	0.427211
$790$	0	0
$791$	60.0000	2.13335
$792$	− 18.0000i	− 0.639602i
$793$	− 48.0000i	− 1.70453i
$794$	−12.0000	−0.425864
$795$	0	0
$796$	18.0000	0.637993
$797$	− 18.0000i	− 0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$	− 18.0000i	− 0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$	− 12.0000i	− 0.424795i
$799$	−50.0000	−1.76887
$800$	0	0
$801$	−18.0000	−0.635999
$802$	5.00000i	0.176556i
$803$	21.0000i	0.741074i
$804$	−9.00000	−0.317406
$805$	0	0
$806$	−48.0000	−1.69073
$807$	− 18.0000i	− 0.633630i
$808$	4.00000i	0.140720i
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	− 32.0000i	− 1.12298i
$813$	− 30.0000i	− 1.05215i
$814$	6.00000	0.210300
$815$	0	0
$816$	15.0000	0.525105
$817$	− 4.00000i	− 0.139942i
$818$	21.0000i	0.734248i
$819$	144.000	5.03177
$820$	0	0
$821$	16.0000	0.558404	0.279202	−	0.960232i	$-0.409930\pi$
$821$	16.0000	0.558404	0.279202	+	0.960232i	$0.409930\pi$
$822$	9.00000i	0.313911i
$823$	26.0000i	0.906303i	0.891434	+	0.453152i	$0.149700\pi$
$823$	26.0000i	0.906303i	−0.891434	+	0.453152i	$0.850300\pi$
$824$	10.0000	0.348367
$825$	0	0
$826$	−16.0000	−0.556711
$827$	− 17.0000i	− 0.591148i	−0.955320	−	0.295574i	$-0.904489\pi$
$827$	− 17.0000i	− 0.591148i	0.955320	−	0.295574i	$-0.0955109\pi$
$828$	− 6.00000i	− 0.208514i
$829$	−24.0000	−0.833554	−0.416777	−	0.909009i	$-0.636840\pi$
$829$	−24.0000	−0.833554	−0.416777	+	0.909009i	$0.636840\pi$
$830$	0	0
$831$	−84.0000	−2.91393
$832$	− 6.00000i	− 0.208013i
$833$	− 45.0000i	− 1.55916i
$834$	−57.0000	−1.97375
$835$	0	0
$836$	−3.00000	−0.103757
$837$	− 72.0000i	− 2.48868i
$838$	3.00000i	0.103633i
$839$	−26.0000	−0.897620	−0.448810	−	0.893627i	$-0.648152\pi$
$839$	−26.0000	−0.897620	−0.448810	+	0.893627i	$0.648152\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	14.0000i	0.482472i
$843$	90.0000i	3.09976i
$844$	1.00000	0.0344214
$845$	0	0
$846$	−60.0000	−2.06284
$847$	− 8.00000i	− 0.274883i
$848$	12.0000i	0.412082i
$849$	99.0000	3.39767
$850$	0	0
$851$	2.00000	0.0685591
$852$	12.0000i	0.411113i
$853$	− 12.0000i	− 0.410872i	−0.978671	−	0.205436i	$-0.934139\pi$
$853$	− 12.0000i	− 0.410872i	0.978671	−	0.205436i	$-0.0658613\pi$
$854$	−32.0000	−1.09502
$855$	0	0
$856$	−5.00000	−0.170896
$857$	− 21.0000i	− 0.717346i	−0.933463	−	0.358673i	$-0.883229\pi$
$857$	− 21.0000i	− 0.717346i	0.933463	−	0.358673i	$-0.116771\pi$
$858$	− 54.0000i	− 1.84353i
$859$	11.0000	0.375315	0.187658	−	0.982235i	$-0.439910\pi$
$859$	11.0000	0.375315	0.187658	+	0.982235i	$0.439910\pi$
$860$	0	0
$861$	−84.0000	−2.86271
$862$	4.00000i	0.136241i
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	9.00000	0.306186
$865$	0	0
$866$	−29.0000	−0.985460
$867$	24.0000i	0.815083i
$868$	32.0000i	1.08615i
$869$	18.0000	0.610608
$870$	0	0
$871$	−18.0000	−0.609907
$872$	10.0000i	0.338643i
$873$	84.0000i	2.84297i
$874$	−1.00000	−0.0338255
$875$	0	0
$876$	−21.0000	−0.709524
$877$	38.0000i	1.28317i	0.767052	+	0.641584i	$0.221723\pi$
$877$	38.0000i	1.28317i	−0.767052	+	0.641584i	$0.778277\pi$
$878$	− 6.00000i	− 0.202490i
$879$	−72.0000	−2.42850
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	− 54.0000i	− 1.81827i
$883$	37.0000i	1.24515i	0.782560	+	0.622575i	$0.213913\pi$
$883$	37.0000i	1.24515i	−0.782560	+	0.622575i	$0.786087\pi$
$884$	30.0000	1.00901
$885$	0	0
$886$	39.0000	1.31023
$887$	12.0000i	0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$	12.0000i	0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$	6.00000i	0.201347i
$889$	32.0000	1.07325
$890$	0	0
$891$	27.0000	0.904534
$892$	0	0
$893$	10.0000i	0.334637i
$894$	18.0000	0.602010
$895$	0	0
$896$	−4.00000	−0.133631
$897$	− 18.0000i	− 0.601003i
$898$	5.00000i	0.166852i
$899$	−64.0000	−2.13452
$900$	0	0
$901$	−60.0000	−1.99889
$902$	21.0000i	0.699224i
$903$	− 48.0000i	− 1.59734i
$904$	15.0000	0.498893
$905$	0	0
$906$	−60.0000	−1.99337
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	8.00000i	0.265489i
$909$	−24.0000	−0.796030
$910$	0	0
$911$	−10.0000	−0.331315	−0.165657	−	0.986183i	$-0.552975\pi$
$911$	−10.0000	−0.331315	−0.165657	+	0.986183i	$0.552975\pi$
$912$	− 3.00000i	− 0.0993399i
$913$	− 33.0000i	− 1.09214i
$914$	−7.00000	−0.231539
$915$	0	0
$916$	−30.0000	−0.991228
$917$	48.0000i	1.58510i
$918$	45.0000i	1.48522i
$919$	−52.0000	−1.71532	−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000	−1.71532	−0.857661	+	0.514216i	$0.828083\pi$
$920$	0	0
$921$	−27.0000	−0.889680
$922$	24.0000i	0.790398i
$923$	24.0000i	0.789970i
$924$	−36.0000	−1.18431
$925$	0	0
$926$	−26.0000	−0.854413
$927$	60.0000i	1.97066i
$928$	− 8.00000i	− 0.262613i
$929$	22.0000	0.721797	0.360898	−	0.932605i	$-0.382470\pi$
$929$	22.0000	0.721797	0.360898	+	0.932605i	$0.382470\pi$
$930$	0	0
$931$	−9.00000	−0.294963
$932$	− 14.0000i	− 0.458585i
$933$	− 6.00000i	− 0.196431i
$934$	28.0000	0.916188
$935$	0	0
$936$	36.0000	1.17670
$937$	− 9.00000i	− 0.294017i	−0.989135	−	0.147009i	$-0.953036\pi$
$937$	− 9.00000i	− 0.294017i	0.989135	−	0.147009i	$-0.0469645\pi$
$938$	12.0000i	0.391814i
$939$	30.0000	0.979013
$940$	0	0
$941$	4.00000	0.130396	0.0651981	−	0.997872i	$-0.479232\pi$
$941$	4.00000	0.130396	0.0651981	+	0.997872i	$0.479232\pi$
$942$	− 42.0000i	− 1.36843i
$943$	7.00000i	0.227951i
$944$	−4.00000	−0.130189
$945$	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	$0.937538\pi$
$948$	18.0000i	0.584613i
$949$	−42.0000	−1.36338
$950$	0	0
$951$	36.0000	1.16738
$952$	− 20.0000i	− 0.648204i
$953$	27.0000i	0.874616i	0.899312	+	0.437308i	$0.144068\pi$
$953$	27.0000i	0.874616i	−0.899312	+	0.437308i	$0.855932\pi$
$954$	−72.0000	−2.33109
$955$	0	0
$956$	−6.00000	−0.194054
$957$	− 72.0000i	− 2.32743i
$958$	− 4.00000i	− 0.129234i
$959$	12.0000	0.387500
$960$	0	0
$961$	33.0000	1.06452
$962$	12.0000i	0.386896i
$963$	− 30.0000i	− 0.966736i
$964$	−23.0000	−0.740780
$965$	0	0
$966$	−12.0000	−0.386094
$967$	− 28.0000i	− 0.900419i	−0.892923	−	0.450210i	$-0.851349\pi$
$967$	− 28.0000i	− 0.900419i	0.892923	−	0.450210i	$-0.148651\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	15.0000	0.481869
$970$	0	0
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	0	0
$973$	76.0000i	2.43645i
$974$	−30.0000	−0.961262
$975$	0	0
$976$	−8.00000	−0.256074
$977$	17.0000i	0.543878i	0.962314	+	0.271939i	$0.0876649\pi$
$977$	17.0000i	0.543878i	−0.962314	+	0.271939i	$0.912335\pi$
$978$	15.0000i	0.479647i
$979$	9.00000	0.287641
$980$	0	0
$981$	−60.0000	−1.91565
$982$	0	0
$983$	− 42.0000i	− 1.33959i	−0.742545	−	0.669796i	$-0.766382\pi$
$983$	− 42.0000i	− 1.33959i	0.742545	−	0.669796i	$-0.233618\pi$
$984$	−21.0000	−0.669456
$985$	0	0
$986$	40.0000	1.27386
$987$	120.000i	3.81964i
$988$	− 6.00000i	− 0.190885i
$989$	−4.00000	−0.127193
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	8.00000i	0.254000i
$993$	− 75.0000i	− 2.38005i
$994$	16.0000	0.507489
$995$	0	0
$996$	33.0000	1.04565
$997$	22.0000i	0.696747i	0.937356	+	0.348373i	$0.113266\pi$
$997$	22.0000i	0.696747i	−0.937356	+	0.348373i	$0.886734\pi$
$998$	− 12.0000i	− 0.379853i
$999$	−18.0000	−0.569495

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.2.b.a.599.1		2
5.2	odd	4		1150.2.a.e.1.1	yes	1
5.3	odd	4		1150.2.a.d.1.1	✓	1
5.4	even	2	inner	1150.2.b.a.599.2		2
20.3	even	4		9200.2.a.a.1.1		1
20.7	even	4		9200.2.a.bl.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1150.2.a.d.1.1	✓	1	5.3	odd	4
1150.2.a.e.1.1	yes	1	5.2	odd	4
1150.2.b.a.599.1		2	1.1	even	1	trivial
1150.2.b.a.599.2		2	5.4	even	2	inner
9200.2.a.a.1.1		1	20.3	even	4
9200.2.a.bl.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:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1150.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -6.00000 q^{9} +3.00000 q^{11} +3.00000i q^{12} +6.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +5.00000i q^{17} +6.00000i q^{18} +1.00000 q^{19} +12.0000 q^{21} -3.00000i q^{22} -1.00000i q^{23} +3.00000 q^{24} +6.00000 q^{26} +9.00000i q^{27} -4.00000i q^{28} +8.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -9.00000i q^{33} +5.00000 q^{34} +6.00000 q^{36} +2.00000i q^{37} -1.00000i q^{38} +18.0000 q^{39} -7.00000 q^{41} -12.0000i q^{42} -4.00000i q^{43} -3.00000 q^{44} -1.00000 q^{46} +10.0000i q^{47} -3.00000i q^{48} -9.00000 q^{49} +15.0000 q^{51} -6.00000i q^{52} +12.0000i q^{53} +9.00000 q^{54} -4.00000 q^{56} -3.00000i q^{57} -8.00000i q^{58} -4.00000 q^{59} -8.00000 q^{61} +8.00000i q^{62} -24.0000i q^{63} -1.00000 q^{64} -9.00000 q^{66} +3.00000i q^{67} -5.00000i q^{68} -3.00000 q^{69} +4.00000 q^{71} -6.00000i q^{72} +7.00000i q^{73} +2.00000 q^{74} -1.00000 q^{76} +12.0000i q^{77} -18.0000i q^{78} +6.00000 q^{79} +9.00000 q^{81} +7.00000i q^{82} -11.0000i q^{83} -12.0000 q^{84} -4.00000 q^{86} -24.0000i q^{87} +3.00000i q^{88} +3.00000 q^{89} -24.0000 q^{91} +1.00000i q^{92} +24.0000i q^{93} +10.0000 q^{94} -3.00000 q^{96} -14.0000i q^{97} +9.00000i q^{98} -18.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 6 q^{6} - 12 q^{9} + 6 q^{11} + 8 q^{14} + 2 q^{16} + 2 q^{19} + 24 q^{21} + 6 q^{24} + 12 q^{26} + 16 q^{29} - 16 q^{31} + 10 q^{34} + 12 q^{36} + 36 q^{39} - 14 q^{41} - 6 q^{44} - 2 q^{46}+ \cdots - 36 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 6 * q^6 - 12 * q^9 + 6 * q^11 + 8 * q^14 + 2 * q^16 + 2 * q^19 + 24 * q^21 + 6 * q^24 + 12 * q^26 + 16 * q^29 - 16 * q^31 + 10 * q^34 + 12 * q^36 + 36 * q^39 - 14 * q^41 - 6 * q^44 - 2 * q^46 - 18 * q^49 + 30 * q^51 + 18 * q^54 - 8 * q^56 - 8 * q^59 - 16 * q^61 - 2 * q^64 - 18 * q^66 - 6 * q^69 + 8 * q^71 + 4 * q^74 - 2 * q^76 + 12 * q^79 + 18 * q^81 - 24 * q^84 - 8 * q^86 + 6 * q^89 - 48 * q^91 + 20 * q^94 - 6 * q^96 - 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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