LMFDB - Embedded newform 4800.2.f.bd.3649.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4800,2,Mod(3649,4800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4800.3649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4800 = 2^{6} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4800.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-2,0,8,0,0,0,0,0,0,0,-2,0,-2,0,0,0,0,0,0,0,-16, 0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$38.3281929702$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2400)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4800.3649
Dual form			4800.2.f.bd.3649.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +4.00000 q^{11} +3.00000i q^{13} +4.00000i q^{17} -1.00000 q^{19} -1.00000 q^{21} +1.00000i q^{27} -8.00000 q^{29} +1.00000 q^{31} -4.00000i q^{33} +2.00000i q^{37} +3.00000 q^{39} +2.00000 q^{41} +11.0000i q^{43} +2.00000i q^{47} +6.00000 q^{49} +4.00000 q^{51} +10.0000i q^{53} +1.00000i q^{57} -6.00000 q^{59} -11.0000 q^{61} +1.00000i q^{63} +9.00000i q^{67} +6.00000 q^{71} -14.0000i q^{73} -4.00000i q^{77} -16.0000 q^{79} +1.00000 q^{81} -2.00000i q^{83} +8.00000i q^{87} +3.00000 q^{91} -1.00000i q^{93} +11.0000i q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} + 8 q^{11} - 2 q^{19} - 2 q^{21} - 16 q^{29} + 2 q^{31} + 6 q^{39} + 4 q^{41} + 12 q^{49} + 8 q^{51} - 12 q^{59} - 22 q^{61} + 12 q^{71} - 32 q^{79} + 2 q^{81} + 6 q^{91} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 + 8 * q^11 - 2 * q^19 - 2 * q^21 - 16 * q^29 + 2 * q^31 + 6 * q^39 + 4 * q^41 + 12 * q^49 + 8 * q^51 - 12 * q^59 - 22 * q^61 + 12 * q^71 - 32 * q^79 + 2 * q^81 + 6 * q^91 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$4351$
$\chi(n)$	$-1$	$1$	$1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	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
$13$	3.00000i	0.832050i	0.909353	+	0.416025i	$0.136577\pi$
$13$	3.00000i	0.832050i	−0.909353	+	0.416025i	$0.863423\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	− 4.00000i	− 0.696311i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	3.00000	0.480384
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	11.0000i	1.67748i	0.544529	+	0.838742i	$0.316708\pi$
$43$	11.0000i	1.67748i	−0.544529	+	0.838742i	$0.683292\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.00000i	0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$	2.00000i	0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	10.0000i	1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$	10.0000i	1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.00000i	0.132453i
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−11.0000	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$62$	0	0
$63$	1.00000i	0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.00000i	1.09952i	0.835321	+	0.549762i	$0.185282\pi$
$67$	9.00000i	1.09952i	−0.835321	+	0.549762i	$0.814718\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	− 14.0000i	− 1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$	− 14.0000i	− 1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 2.00000i	− 0.219529i	−0.993958	−	0.109764i	$-0.964990\pi$
$83$	− 2.00000i	− 0.219529i	0.993958	−	0.109764i	$-0.0350096\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.00000i	0.857690i
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	0	0
$93$	− 1.00000i	− 0.103695i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.0000i	1.11688i	0.829545	+	0.558440i	$0.188600\pi$
$97$	11.0000i	1.11688i	−0.829545	+	0.558440i	$0.811400\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	− 12.0000i	− 1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$	− 12.0000i	− 1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.00000i	− 0.277350i
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	− 2.00000i	− 0.180334i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	11.0000	0.968496
$130$	0	0
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	0	0
$133$	1.00000i	0.0867110i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	0	0
$143$	12.0000i	1.00349i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 6.00000i	− 0.494872i
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	23.0000	1.87171	0.935857	−	0.352381i	$-0.114628\pi$
$151$	23.0000	1.87171	0.935857	+	0.352381i	$0.114628\pi$
$152$	0	0
$153$	− 4.00000i	− 0.323381i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 5.00000i	− 0.399043i	−0.979893	−	0.199522i	$-0.936061\pi$
$157$	− 5.00000i	− 0.399043i	0.979893	−	0.199522i	$-0.0639388\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.00000i	0.704934i	0.935824	+	0.352467i	$0.114657\pi$
$163$	9.00000i	0.704934i	−0.935824	+	0.352467i	$0.885343\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.0000i	1.70241i	0.524832	+	0.851206i	$0.324128\pi$
$167$	22.0000i	1.70241i	−0.524832	+	0.851206i	$0.675872\pi$
$168$	0	0
$169$	4.00000	0.307692
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	24.0000i	1.82469i	0.409426	+	0.912343i	$0.365729\pi$
$173$	24.0000i	1.82469i	−0.409426	+	0.912343i	$0.634271\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.00000i	0.450988i
$178$	0	0
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	0	0
$181$	15.0000	1.11494	0.557471	−	0.830197i	$-0.311772\pi$
$181$	15.0000	1.11494	0.557471	+	0.830197i	$0.311772\pi$
$182$	0	0
$183$	11.0000i	0.813143i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	16.0000i	1.17004i
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	− 13.0000i	− 0.935760i	−0.883792	−	0.467880i	$-0.845018\pi$
$193$	− 13.0000i	− 0.935760i	0.883792	−	0.467880i	$-0.154982\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	9.00000	0.634811
$202$	0	0
$203$	8.00000i	0.561490i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	0	0
$213$	− 6.00000i	− 0.411113i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1.00000i	− 0.0678844i
$218$	0	0
$219$	−14.0000	−0.946032
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	− 19.0000i	− 1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$	− 19.0000i	− 1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 18.0000i	− 1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$	− 18.0000i	− 1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$	0	0
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	26.0000i	1.70332i	0.524097	+	0.851658i	$0.324403\pi$
$233$	26.0000i	1.70332i	−0.524097	+	0.851658i	$0.675597\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	16.0000i	1.03931i
$238$	0	0
$239$	−10.0000	−0.646846	−0.323423	−	0.946254i	$-0.604834\pi$
$239$	−10.0000	−0.646846	−0.323423	+	0.946254i	$0.604834\pi$
$240$	0	0
$241$	21.0000	1.35273	0.676364	−	0.736567i	$-0.263554\pi$
$241$	21.0000	1.35273	0.676364	+	0.736567i	$0.263554\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 3.00000i	− 0.190885i
$248$	0	0
$249$	−2.00000	−0.126745
$250$	0	0
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 8.00000i	− 0.499026i	−0.968371	−	0.249513i	$-0.919729\pi$
$257$	− 8.00000i	− 0.499026i	0.968371	−	0.249513i	$-0.0802706\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	8.00000	0.495188
$262$	0	0
$263$	26.0000i	1.60323i	0.597841	+	0.801614i	$0.296025\pi$
$263$	26.0000i	1.60323i	−0.597841	+	0.801614i	$0.703975\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	28.0000	1.70088	0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000	1.70088	0.850439	+	0.526073i	$0.176336\pi$
$272$	0	0
$273$	− 3.00000i	− 0.181568i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 7.00000i	− 0.420589i	−0.977638	−	0.210295i	$-0.932558\pi$
$277$	− 7.00000i	− 0.420589i	0.977638	−	0.210295i	$-0.0674423\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	0	0
$283$	17.0000i	1.01055i	0.862960	+	0.505273i	$0.168608\pi$
$283$	17.0000i	1.01055i	−0.862960	+	0.505273i	$0.831392\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.00000i	− 0.118056i
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	11.0000	0.644831
$292$	0	0
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.00000i	0.232104i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	11.0000	0.634029
$302$	0	0
$303$	14.0000i	0.804279i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	23.0000i	1.31268i	0.754466	+	0.656340i	$0.227896\pi$
$307$	23.0000i	1.31268i	−0.754466	+	0.656340i	$0.772104\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−26.0000	−1.47432	−0.737162	−	0.675716i	$-0.763835\pi$
$311$	−26.0000	−1.47432	−0.737162	+	0.675716i	$0.763835\pi$
$312$	0	0
$313$	15.0000i	0.847850i	0.905697	+	0.423925i	$0.139348\pi$
$313$	15.0000i	0.847850i	−0.905697	+	0.423925i	$0.860652\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 28.0000i	− 1.57264i	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	− 28.0000i	− 1.57264i	0.617822	−	0.786318i	$-0.288015\pi$
$318$	0	0
$319$	−32.0000	−1.79166
$320$	0	0
$321$	6.00000	0.334887
$322$	0	0
$323$	− 4.00000i	− 0.222566i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11.0000i	0.608301i
$328$	0	0
$329$	2.00000	0.110264
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 13.0000i	− 0.708155i	−0.935216	−	0.354078i	$-0.884795\pi$
$337$	− 13.0000i	− 0.708155i	0.935216	−	0.354078i	$-0.115205\pi$
$338$	0	0
$339$	−12.0000	−0.651751
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	− 13.0000i	− 0.701934i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.0000i	1.61048i	0.592946	+	0.805242i	$0.297965\pi$
$347$	30.0000i	1.61048i	−0.592946	+	0.805242i	$0.702035\pi$
$348$	0	0
$349$	34.0000	1.81998	0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	0	0
$351$	−3.00000	−0.160128
$352$	0	0
$353$	24.0000i	1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$	24.0000i	1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 4.00000i	− 0.211702i
$358$	0	0
$359$	26.0000	1.37223	0.686114	−	0.727494i	$-0.259315\pi$
$359$	26.0000	1.37223	0.686114	+	0.727494i	$0.259315\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 3.00000i	− 0.156599i	−0.996930	−	0.0782994i	$-0.975051\pi$
$367$	− 3.00000i	− 0.156599i	0.996930	−	0.0782994i	$-0.0249490\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	10.0000	0.519174
$372$	0	0
$373$	− 19.0000i	− 0.983783i	−0.870657	−	0.491891i	$-0.836306\pi$
$373$	− 19.0000i	− 0.983783i	0.870657	−	0.491891i	$-0.163694\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 24.0000i	− 1.23606i
$378$	0	0
$379$	33.0000	1.69510	0.847548	−	0.530719i	$-0.178078\pi$
$379$	33.0000	1.69510	0.847548	+	0.530719i	$0.178078\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 8.00000i	− 0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$	− 8.00000i	− 0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 11.0000i	− 0.559161i
$388$	0	0
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	− 14.0000i	− 0.706207i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	23.0000i	1.15434i	0.816625	+	0.577168i	$0.195842\pi$
$397$	23.0000i	1.15434i	−0.816625	+	0.577168i	$0.804158\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	3.00000i	0.149441i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.00000i	0.396545i
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	6.00000i	0.295241i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 20.0000i	− 0.979404i
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	− 2.00000i	− 0.0972433i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11.0000i	0.532327i
$428$	0	0
$429$	12.0000	0.579365
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	− 1.00000i	− 0.0480569i	−0.999711	−	0.0240285i	$-0.992351\pi$
$433$	− 1.00000i	− 0.0480569i	0.999711	−	0.0240285i	$-0.00764923\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	7.00000	0.334092	0.167046	−	0.985949i	$-0.446577\pi$
$439$	7.00000	0.334092	0.167046	+	0.985949i	$0.446577\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.00000i	0.0945968i
$448$	0	0
$449$	−40.0000	−1.88772	−0.943858	−	0.330350i	$-0.892833\pi$
$449$	−40.0000	−1.88772	−0.943858	+	0.330350i	$0.892833\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	− 23.0000i	− 1.08063i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.00000i	0.280668i	0.990104	+	0.140334i	$0.0448177\pi$
$457$	6.00000i	0.280668i	−0.990104	+	0.140334i	$0.955182\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	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	$-0.846148\pi$
$463$	− 20.0000i	− 0.929479i	0.885448	−	0.464739i	$-0.153852\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	0	0
$469$	9.00000	0.415581
$470$	0	0
$471$	−5.00000	−0.230388
$472$	0	0
$473$	44.0000i	2.02312i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 10.0000i	− 0.457869i
$478$	0	0
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 11.0000i	− 0.498458i	−0.968445	−	0.249229i	$-0.919823\pi$
$487$	− 11.0000i	− 0.498458i	0.968445	−	0.249229i	$-0.0801771\pi$
$488$	0	0
$489$	9.00000	0.406994
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	− 32.0000i	− 1.44121i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6.00000i	− 0.269137i
$498$	0	0
$499$	−13.0000	−0.581960	−0.290980	−	0.956729i	$-0.593981\pi$
$499$	−13.0000	−0.581960	−0.290980	+	0.956729i	$0.593981\pi$
$500$	0	0
$501$	22.0000	0.982888
$502$	0	0
$503$	10.0000i	0.445878i	0.974832	+	0.222939i	$0.0715651\pi$
$503$	10.0000i	0.445878i	−0.974832	+	0.222939i	$0.928435\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 4.00000i	− 0.177646i
$508$	0	0
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	0	0
$513$	− 1.00000i	− 0.0441511i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.00000i	0.351840i
$518$	0	0
$519$	24.0000	1.05348
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	17.0000i	0.743358i	0.928361	+	0.371679i	$0.121218\pi$
$523$	17.0000i	0.743358i	−0.928361	+	0.371679i	$0.878782\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.00000i	0.174243i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	6.00000i	0.259889i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	18.0000i	0.776757i
$538$	0	0
$539$	24.0000	1.03375
$540$	0	0
$541$	7.00000	0.300954	0.150477	−	0.988614i	$-0.451919\pi$
$541$	7.00000	0.300954	0.150477	+	0.988614i	$0.451919\pi$
$542$	0	0
$543$	− 15.0000i	− 0.643712i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	0	0
$549$	11.0000	0.469469
$550$	0	0
$551$	8.00000	0.340811
$552$	0	0
$553$	16.0000i	0.680389i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	0	0
$559$	−33.0000	−1.39575
$560$	0	0
$561$	16.0000	0.675521
$562$	0	0
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.00000i	− 0.0419961i
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	0	0
$573$	− 20.0000i	− 0.835512i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 7.00000i	− 0.291414i	−0.989328	−	0.145707i	$-0.953454\pi$
$577$	− 7.00000i	− 0.291414i	0.989328	−	0.145707i	$-0.0465456\pi$
$578$	0	0
$579$	−13.0000	−0.540262
$580$	0	0
$581$	−2.00000	−0.0829740
$582$	0	0
$583$	40.0000i	1.65663i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 26.0000i	− 1.07313i	−0.843857	−	0.536567i	$-0.819721\pi$
$587$	− 26.0000i	− 1.07313i	0.843857	−	0.536567i	$-0.180279\pi$
$588$	0	0
$589$	−1.00000	−0.0412043
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	11.0000i	0.450200i
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	0	0
$603$	− 9.00000i	− 0.366508i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 40.0000i	− 1.62355i	−0.583970	−	0.811775i	$-0.698502\pi$
$607$	− 40.0000i	− 1.62355i	0.583970	−	0.811775i	$-0.301498\pi$
$608$	0	0
$609$	8.00000	0.324176
$610$	0	0
$611$	−6.00000	−0.242734
$612$	0	0
$613$	− 38.0000i	− 1.53481i	−0.641165	−	0.767403i	$-0.721549\pi$
$613$	− 38.0000i	− 1.53481i	0.641165	−	0.767403i	$-0.278451\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	0	0
$619$	23.0000	0.924448	0.462224	−	0.886763i	$-0.347052\pi$
$619$	23.0000	0.924448	0.462224	+	0.886763i	$0.347052\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.00000i	0.159745i
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−15.0000	−0.597141	−0.298570	−	0.954388i	$-0.596510\pi$
$631$	−15.0000	−0.597141	−0.298570	+	0.954388i	$0.596510\pi$
$632$	0	0
$633$	− 5.00000i	− 0.198732i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18.0000i	0.713186i
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	− 12.0000i	− 0.473234i	−0.971603	−	0.236617i	$-0.923961\pi$
$643$	− 12.0000i	− 0.473234i	0.971603	−	0.236617i	$-0.0760386\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0000i	0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$	12.0000i	0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	−1.00000	−0.0391931
$652$	0	0
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14.0000i	0.546192i
$658$	0	0
$659$	14.0000	0.545363	0.272681	−	0.962104i	$-0.412090\pi$
$659$	14.0000	0.545363	0.272681	+	0.962104i	$0.412090\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	0	0
$663$	12.0000i	0.466041i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−19.0000	−0.734582
$670$	0	0
$671$	−44.0000	−1.69860
$672$	0	0
$673$	− 2.00000i	− 0.0770943i	−0.999257	−	0.0385472i	$-0.987727\pi$
$673$	− 2.00000i	− 0.0770943i	0.999257	−	0.0385472i	$-0.0122730\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	44.0000i	1.69106i	0.533930	+	0.845529i	$0.320715\pi$
$677$	44.0000i	1.69106i	−0.533930	+	0.845529i	$0.679285\pi$
$678$	0	0
$679$	11.0000	0.422141
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	− 8.00000i	− 0.306111i	−0.988218	−	0.153056i	$-0.951089\pi$
$683$	− 8.00000i	− 0.306111i	0.988218	−	0.153056i	$-0.0489114\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 5.00000i	− 0.190762i
$688$	0	0
$689$	−30.0000	−1.14291
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	0	0
$693$	4.00000i	0.151947i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8.00000i	0.303022i
$698$	0	0
$699$	26.0000	0.983410
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	− 2.00000i	− 0.0754314i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.0000i	0.526524i
$708$	0	0
$709$	17.0000	0.638448	0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000	0.638448	0.319224	+	0.947679i	$0.396578\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	10.0000i	0.373457i
$718$	0	0
$719$	−32.0000	−1.19340	−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000	−1.19340	−0.596699	+	0.802465i	$0.703521\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	− 21.0000i	− 0.780998i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23.0000i	0.853023i	0.904482	+	0.426511i	$0.140258\pi$
$727$	23.0000i	0.853023i	−0.904482	+	0.426511i	$0.859742\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−44.0000	−1.62740
$732$	0	0
$733$	− 6.00000i	− 0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$	− 6.00000i	− 0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	36.0000i	1.32608i
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−3.00000	−0.110208
$742$	0	0
$743$	− 36.0000i	− 1.32071i	−0.750953	−	0.660356i	$-0.770405\pi$
$743$	− 36.0000i	− 1.32071i	0.750953	−	0.660356i	$-0.229595\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.00000i	0.0731762i
$748$	0	0
$749$	6.00000	0.219235
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	− 8.00000i	− 0.291536i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 11.0000i	− 0.399802i	−0.979816	−	0.199901i	$-0.935938\pi$
$757$	− 11.0000i	− 0.399802i	0.979816	−	0.199901i	$-0.0640620\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20.0000	−0.724999	−0.362500	−	0.931984i	$-0.618077\pi$
$761$	−20.0000	−0.724999	−0.362500	+	0.931984i	$0.618077\pi$
$762$	0	0
$763$	11.0000i	0.398227i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 18.0000i	− 0.649942i
$768$	0	0
$769$	9.00000	0.324548	0.162274	−	0.986746i	$-0.448117\pi$
$769$	9.00000	0.324548	0.162274	+	0.986746i	$0.448117\pi$
$770$	0	0
$771$	−8.00000	−0.288113
$772$	0	0
$773$	2.00000i	0.0719350i	0.999353	+	0.0359675i	$0.0114513\pi$
$773$	2.00000i	0.0719350i	−0.999353	+	0.0359675i	$0.988549\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2.00000i	− 0.0717496i
$778$	0	0
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	24.0000	0.858788
$782$	0	0
$783$	− 8.00000i	− 0.285897i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 53.0000i	− 1.88925i	−0.328158	−	0.944623i	$-0.606428\pi$
$787$	− 53.0000i	− 1.88925i	0.328158	−	0.944623i	$-0.393572\pi$
$788$	0	0
$789$	26.0000	0.925625
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	− 33.0000i	− 1.17186i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 24.0000i	− 0.850124i	−0.905164	−	0.425062i	$-0.860252\pi$
$797$	− 24.0000i	− 0.850124i	0.905164	−	0.425062i	$-0.139748\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 56.0000i	− 1.97620i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.0000i	0.422420i
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−43.0000	−1.50993	−0.754967	−	0.655763i	$-0.772347\pi$
$811$	−43.0000	−1.50993	−0.754967	+	0.655763i	$0.772347\pi$
$812$	0	0
$813$	− 28.0000i	− 0.982003i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 11.0000i	− 0.384841i
$818$	0	0
$819$	−3.00000	−0.104828
$820$	0	0
$821$	−46.0000	−1.60541	−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000	−1.60541	−0.802706	+	0.596376i	$0.796607\pi$
$822$	0	0
$823$	− 39.0000i	− 1.35945i	−0.733465	−	0.679727i	$-0.762098\pi$
$823$	− 39.0000i	− 1.35945i	0.733465	−	0.679727i	$-0.237902\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 10.0000i	− 0.347734i	−0.984769	−	0.173867i	$-0.944374\pi$
$827$	− 10.0000i	− 0.347734i	0.984769	−	0.173867i	$-0.0556263\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	−7.00000	−0.242827
$832$	0	0
$833$	24.0000i	0.831551i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.00000i	0.0345651i
$838$	0	0
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	16.0000i	0.551069i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 5.00000i	− 0.171802i
$848$	0	0
$849$	17.0000	0.583438
$850$	0	0
$851$	0	0
$852$	0	0
$853$	17.0000i	0.582069i	0.956713	+	0.291034i	$0.0939994\pi$
$853$	17.0000i	0.582069i	−0.956713	+	0.291034i	$0.906001\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24.0000i	− 0.819824i	−0.912125	−	0.409912i	$-0.865559\pi$
$857$	− 24.0000i	− 0.819824i	0.912125	−	0.409912i	$-0.134441\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	−2.00000	−0.0681598
$862$	0	0
$863$	− 28.0000i	− 0.953131i	−0.879139	−	0.476566i	$-0.841881\pi$
$863$	− 28.0000i	− 0.953131i	0.879139	−	0.476566i	$-0.158119\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.00000i	− 0.0339618i
$868$	0	0
$869$	−64.0000	−2.17105
$870$	0	0
$871$	−27.0000	−0.914860
$872$	0	0
$873$	− 11.0000i	− 0.372294i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 17.0000i	− 0.574049i	−0.957923	−	0.287025i	$-0.907334\pi$
$877$	− 17.0000i	− 0.574049i	0.957923	−	0.287025i	$-0.0926662\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−52.0000	−1.75192	−0.875962	−	0.482380i	$-0.839773\pi$
$881$	−52.0000	−1.75192	−0.875962	+	0.482380i	$0.839773\pi$
$882$	0	0
$883$	− 25.0000i	− 0.841317i	−0.907219	−	0.420658i	$-0.861799\pi$
$883$	− 25.0000i	− 0.841317i	0.907219	−	0.420658i	$-0.138201\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 34.0000i	− 1.14161i	−0.821086	−	0.570804i	$-0.806632\pi$
$887$	− 34.0000i	− 1.14161i	0.821086	−	0.570804i	$-0.193368\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	− 2.00000i	− 0.0669274i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8.00000	−0.266815
$900$	0	0
$901$	−40.0000	−1.33259
$902$	0	0
$903$	− 11.0000i	− 0.366057i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	20.0000i	0.664089i	0.943264	+	0.332045i	$0.107738\pi$
$907$	20.0000i	0.664089i	−0.943264	+	0.332045i	$0.892262\pi$
$908$	0	0
$909$	14.0000	0.464351
$910$	0	0
$911$	50.0000	1.65657	0.828287	−	0.560304i	$-0.189316\pi$
$911$	50.0000	1.65657	0.828287	+	0.560304i	$0.189316\pi$
$912$	0	0
$913$	− 8.00000i	− 0.264761i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 14.0000i	− 0.462321i
$918$	0	0
$919$	49.0000	1.61636	0.808180	−	0.588935i	$-0.200453\pi$
$919$	49.0000	1.61636	0.808180	+	0.588935i	$0.200453\pi$
$920$	0	0
$921$	23.0000	0.757876
$922$	0	0
$923$	18.0000i	0.592477i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 8.00000i	− 0.262754i
$928$	0	0
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	26.0000i	0.851202i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	53.0000i	1.73143i	0.500533	+	0.865717i	$0.333137\pi$
$937$	53.0000i	1.73143i	−0.500533	+	0.865717i	$0.666863\pi$
$938$	0	0
$939$	15.0000	0.489506
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 32.0000i	− 1.03986i	−0.854209	−	0.519930i	$-0.825958\pi$
$947$	− 32.0000i	− 1.03986i	0.854209	−	0.519930i	$-0.174042\pi$
$948$	0	0
$949$	42.0000	1.36338
$950$	0	0
$951$	−28.0000	−0.907962
$952$	0	0
$953$	− 8.00000i	− 0.259145i	−0.991570	−	0.129573i	$-0.958639\pi$
$953$	− 8.00000i	− 0.259145i	0.991570	−	0.129573i	$-0.0413606\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	32.0000i	1.03441i
$958$	0	0
$959$	−2.00000	−0.0645834
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	− 6.00000i	− 0.193347i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	0	0
$969$	−4.00000	−0.128499
$970$	0	0
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	0	0
$973$	− 20.0000i	− 0.641171i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	11.0000	0.351203
$982$	0	0
$983$	44.0000i	1.40338i	0.712481	+	0.701691i	$0.247571\pi$
$983$	44.0000i	1.40338i	−0.712481	+	0.701691i	$0.752429\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.00000i	− 0.0636607i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−47.0000	−1.49300	−0.746502	−	0.665383i	$-0.768268\pi$
$991$	−47.0000	−1.49300	−0.746502	+	0.665383i	$0.768268\pi$
$992$	0	0
$993$	− 8.00000i	− 0.253872i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	18.0000i	0.570066i	0.958518	+	0.285033i	$0.0920045\pi$
$997$	18.0000i	0.570066i	−0.958518	+	0.285033i	$0.907995\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4800.2.f.bd.3649.1		2
4.3	odd	2		4800.2.f.g.3649.2		2
5.2	odd	4		4800.2.a.y.1.1		1
5.3	odd	4		4800.2.a.by.1.1		1
5.4	even	2	inner	4800.2.f.bd.3649.2		2
8.3	odd	2		2400.2.f.p.1249.1		2
8.5	even	2		2400.2.f.c.1249.2		2
20.3	even	4		4800.2.a.v.1.1		1
20.7	even	4		4800.2.a.bv.1.1		1
20.19	odd	2		4800.2.f.g.3649.1		2
24.5	odd	2		7200.2.f.z.6049.1		2
24.11	even	2		7200.2.f.d.6049.2		2
40.3	even	4		2400.2.a.bd.1.1	yes	1
40.13	odd	4		2400.2.a.e.1.1	✓	1
40.19	odd	2		2400.2.f.p.1249.2		2
40.27	even	4		2400.2.a.h.1.1	yes	1
40.29	even	2		2400.2.f.c.1249.1		2
40.37	odd	4		2400.2.a.ba.1.1	yes	1
120.29	odd	2		7200.2.f.z.6049.2		2
120.53	even	4		7200.2.a.t.1.1		1
120.59	even	2		7200.2.f.d.6049.1		2
120.77	even	4		7200.2.a.bk.1.1		1
120.83	odd	4		7200.2.a.bh.1.1		1
120.107	odd	4		7200.2.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2400.2.a.e.1.1	✓	1	40.13	odd	4
2400.2.a.h.1.1	yes	1	40.27	even	4
2400.2.a.ba.1.1	yes	1	40.37	odd	4
2400.2.a.bd.1.1	yes	1	40.3	even	4
2400.2.f.c.1249.1		2	40.29	even	2
2400.2.f.c.1249.2		2	8.5	even	2
2400.2.f.p.1249.1		2	8.3	odd	2
2400.2.f.p.1249.2		2	40.19	odd	2
4800.2.a.v.1.1		1	20.3	even	4
4800.2.a.y.1.1		1	5.2	odd	4
4800.2.a.bv.1.1		1	20.7	even	4
4800.2.a.by.1.1		1	5.3	odd	4
4800.2.f.g.3649.1		2	20.19	odd	2
4800.2.f.g.3649.2		2	4.3	odd	2
4800.2.f.bd.3649.1		2	1.1	even	1	trivial
4800.2.f.bd.3649.2		2	5.4	even	2	inner
7200.2.a.q.1.1		1	120.107	odd	4
7200.2.a.t.1.1		1	120.53	even	4
7200.2.a.bh.1.1		1	120.83	odd	4
7200.2.a.bk.1.1		1	120.77	even	4
7200.2.f.d.6049.1		2	120.59	even	2
7200.2.f.d.6049.2		2	24.11	even	2
7200.2.f.z.6049.1		2	24.5	odd	2
7200.2.f.z.6049.2		2	120.29	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +4.00000 q^{11} +3.00000i q^{13} +4.00000i q^{17} -1.00000 q^{19} -1.00000 q^{21} +1.00000i q^{27} -8.00000 q^{29} +1.00000 q^{31} -4.00000i q^{33} +2.00000i q^{37} +3.00000 q^{39} +2.00000 q^{41} +11.0000i q^{43} +2.00000i q^{47} +6.00000 q^{49} +4.00000 q^{51} +10.0000i q^{53} +1.00000i q^{57} -6.00000 q^{59} -11.0000 q^{61} +1.00000i q^{63} +9.00000i q^{67} +6.00000 q^{71} -14.0000i q^{73} -4.00000i q^{77} -16.0000 q^{79} +1.00000 q^{81} -2.00000i q^{83} +8.00000i q^{87} +3.00000 q^{91} -1.00000i q^{93} +11.0000i q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} + 8 q^{11} - 2 q^{19} - 2 q^{21} - 16 q^{29} + 2 q^{31} + 6 q^{39} + 4 q^{41} + 12 q^{49} + 8 q^{51} - 12 q^{59} - 22 q^{61} + 12 q^{71} - 32 q^{79} + 2 q^{81} + 6 q^{91} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 + 8 * q^11 - 2 * q^19 - 2 * q^21 - 16 * q^29 + 2 * q^31 + 6 * q^39 + 4 * q^41 + 12 * q^49 + 8 * q^51 - 12 * q^59 - 22 * q^61 + 12 * q^71 - 32 * q^79 + 2 * q^81 + 6 * q^91 - 8 * q^99

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