LMFDB - Embedded newform 1950.2.e.m.1249.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(1249,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, 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("1950.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1950.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:	$15.5708283941$
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 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1950.1249
Dual form			1950.2.e.m.1249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$.

$n$	$301$	$1301$	$1327$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−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	$0.293963\pi$
$12$	1.00000i	0.288675i
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	− 4.00000i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	− 4.00000i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$	− 1.00000i	− 0.235702i
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	4.00000i	0.852803i
$23$	2.00000i	0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$	2.00000i	0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	1.00000	0.196116
$27$	1.00000i	0.192450i
$28$	− 2.00000i	− 0.377964i
$29$	−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
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000i	0.176777i
$33$	− 4.00000i	− 0.696311i
$34$	4.00000	0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	2.00000i	0.324443i
$39$	−1.00000	−0.160128
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	2.00000i	0.308607i
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−2.00000	−0.294884
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 1.00000i	− 0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	−4.00000	−0.560112
$52$	1.00000i	0.138675i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	2.00000	0.267261
$57$	− 2.00000i	− 0.264906i
$58$	− 8.00000i	− 1.05045i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	4.00000i	0.508001i
$63$	− 2.00000i	− 0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	4.00000i	0.485071i
$69$	2.00000	0.240772
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000i	0.117851i
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	−2.00000	−0.229416
$77$	8.00000i	0.911685i
$78$	− 1.00000i	− 0.113228i
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.0000i	1.10432i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	−4.00000	−0.431331
$87$	8.00000i	0.857690i
$88$	− 4.00000i	− 0.426401i
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	− 2.00000i	− 0.208514i
$93$	− 4.00000i	− 0.414781i
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	8.00000i	0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$	8.00000i	0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	3.00000i	0.303046i
$99$	−4.00000	−0.402015
$100$	0	0
$101$	20.0000	1.99007	0.995037	−	0.0995037i	$-0.0317255\pi$
$101$	20.0000	1.99007	0.995037	+	0.0995037i	$0.0317255\pi$
$102$	− 4.00000i	− 0.396059i
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	2.00000i	0.188982i
$113$	− 16.0000i	− 1.50515i	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	− 16.0000i	− 1.50515i	0.658505	−	0.752577i	$-0.271189\pi$
$114$	2.00000	0.187317
$115$	0	0
$116$	8.00000	0.742781
$117$	1.00000i	0.0924500i
$118$	12.0000i	1.10469i
$119$	8.00000	0.733359
$120$	0	0
$121$	5.00000	0.454545
$122$	− 2.00000i	− 0.181071i
$123$	− 10.0000i	− 0.901670i
$124$	−4.00000	−0.359211
$125$	0	0
$126$	2.00000	0.178174
$127$	− 20.0000i	− 1.77471i	−0.461084	−	0.887357i	$-0.652539\pi$
$127$	− 20.0000i	− 1.77471i	0.461084	−	0.887357i	$-0.347461\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	4.00000i	0.348155i
$133$	4.00000i	0.346844i
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−4.00000	−0.342997
$137$	2.00000i	0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$	2.00000i	0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$	2.00000i	0.170251i
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 4.00000i	− 0.334497i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	0	0
$147$	− 3.00000i	− 0.247436i
$148$	6.00000i	0.493197i
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	− 2.00000i	− 0.162221i
$153$	4.00000i	0.323381i
$154$	−8.00000	−0.644658
$155$	0	0
$156$	1.00000	0.0800641
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	8.00000i	0.636446i
$159$	6.00000	0.475831
$160$	0	0
$161$	−4.00000	−0.315244
$162$	1.00000i	0.0785674i
$163$	12.0000i	0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$	12.0000i	0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	12.0000	0.931381
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	− 2.00000i	− 0.154303i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−2.00000	−0.152944
$172$	− 4.00000i	− 0.304997i
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	−8.00000	−0.606478
$175$	0	0
$176$	4.00000	0.301511
$177$	− 12.0000i	− 0.901975i
$178$	10.0000i	0.749532i
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	2.00000i	0.148250i
$183$	2.00000i	0.147844i
$184$	2.00000	0.147442
$185$	0	0
$186$	4.00000	0.293294
$187$	− 16.0000i	− 1.17004i
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	1.00000i	0.0721688i
$193$	12.0000i	0.863779i	0.901927	+	0.431889i	$0.142153\pi$
$193$	12.0000i	0.863779i	−0.901927	+	0.431889i	$0.857847\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 10.0000i	− 0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$	− 10.0000i	− 0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$	− 4.00000i	− 0.284268i
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	20.0000i	1.40720i
$203$	− 16.0000i	− 1.12298i
$204$	4.00000	0.280056
$205$	0	0
$206$	−4.00000	−0.278693
$207$	− 2.00000i	− 0.139010i
$208$	− 1.00000i	− 0.0693375i
$209$	8.00000	0.553372
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	− 6.00000i	− 0.412082i
$213$	0	0
$214$	4.00000	0.273434
$215$	0	0
$216$	1.00000	0.0680414
$217$	8.00000i	0.543075i
$218$	− 4.00000i	− 0.270914i
$219$	0	0
$220$	0	0
$221$	−4.00000	−0.269069
$222$	− 6.00000i	− 0.402694i
$223$	− 22.0000i	− 1.47323i	−0.676313	−	0.736614i	$-0.736423\pi$
$223$	− 22.0000i	− 1.47323i	0.676313	−	0.736614i	$-0.263577\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	16.0000	1.06430
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	2.00000i	0.132453i
$229$	−28.0000	−1.85029	−0.925146	−	0.379611i	$-0.876058\pi$
$229$	−28.0000	−1.85029	−0.925146	+	0.379611i	$0.876058\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	8.00000i	0.525226i
$233$	28.0000i	1.83434i	0.398495	+	0.917170i	$0.369533\pi$
$233$	28.0000i	1.83434i	−0.398495	+	0.917170i	$0.630467\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	−12.0000	−0.781133
$237$	− 8.00000i	− 0.519656i
$238$	8.00000i	0.518563i
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	5.00000i	0.321412i
$243$	− 1.00000i	− 0.0641500i
$244$	2.00000	0.128037
$245$	0	0
$246$	10.0000	0.637577
$247$	− 2.00000i	− 0.127257i
$248$	− 4.00000i	− 0.254000i
$249$	−12.0000	−0.760469
$250$	0	0
$251$	10.0000	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$252$	2.00000i	0.125988i
$253$	8.00000i	0.502956i
$254$	20.0000	1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.00000i	0.499026i	0.968371	+	0.249513i	$0.0802706\pi$
$257$	8.00000i	0.499026i	−0.968371	+	0.249513i	$0.919729\pi$
$258$	4.00000i	0.249029i
$259$	12.0000	0.745644
$260$	0	0
$261$	8.00000	0.495188
$262$	14.0000i	0.864923i
$263$	− 18.0000i	− 1.10993i	−0.831875	−	0.554964i	$-0.812732\pi$
$263$	− 18.0000i	− 1.10993i	0.831875	−	0.554964i	$-0.187268\pi$
$264$	−4.00000	−0.246183
$265$	0	0
$266$	−4.00000	−0.245256
$267$	− 10.0000i	− 0.611990i
$268$	− 8.00000i	− 0.488678i
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	− 4.00000i	− 0.242536i
$273$	− 2.00000i	− 0.121046i
$274$	−2.00000	−0.120824
$275$	0	0
$276$	−2.00000	−0.120386
$277$	− 22.0000i	− 1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$	− 22.0000i	− 1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	16.0000i	0.959616i
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	28.0000i	1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$	28.0000i	1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$	0	0
$285$	0	0
$286$	4.00000	0.236525
$287$	20.0000i	1.18056i
$288$	− 1.00000i	− 0.0589256i
$289$	1.00000	0.0588235
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	−6.00000	−0.348743
$297$	4.00000i	0.232104i
$298$	22.0000i	1.27443i
$299$	2.00000	0.115663
$300$	0	0
$301$	−8.00000	−0.461112
$302$	− 4.00000i	− 0.230174i
$303$	− 20.0000i	− 1.14897i
$304$	2.00000	0.114708
$305$	0	0
$306$	−4.00000	−0.228665
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	− 8.00000i	− 0.455842i
$309$	4.00000	0.227552
$310$	0	0
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	1.00000i	0.0566139i
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−8.00000	−0.450035
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	6.00000i	0.336463i
$319$	−32.0000	−1.79166
$320$	0	0
$321$	−4.00000	−0.223258
$322$	− 4.00000i	− 0.222911i
$323$	− 8.00000i	− 0.445132i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−12.0000	−0.664619
$327$	4.00000i	0.221201i
$328$	− 10.0000i	− 0.552158i
$329$	0	0
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	12.0000i	0.658586i
$333$	6.00000i	0.328798i
$334$	−12.0000	−0.656611
$335$	0	0
$336$	2.00000	0.109109
$337$	26.0000i	1.41631i	0.706057	+	0.708155i	$0.250472\pi$
$337$	26.0000i	1.41631i	−0.706057	+	0.708155i	$0.749528\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	−16.0000	−0.869001
$340$	0	0
$341$	16.0000	0.866449
$342$	− 2.00000i	− 0.108148i
$343$	20.0000i	1.07990i
$344$	4.00000	0.215666
$345$	0	0
$346$	−14.0000	−0.752645
$347$	− 16.0000i	− 0.858925i	−0.903085	−	0.429463i	$-0.858703\pi$
$347$	− 16.0000i	− 0.858925i	0.903085	−	0.429463i	$-0.141297\pi$
$348$	− 8.00000i	− 0.428845i
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	4.00000i	0.213201i
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−10.0000	−0.529999
$357$	− 8.00000i	− 0.423405i
$358$	− 2.00000i	− 0.105703i
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	− 22.0000i	− 1.15629i
$363$	− 5.00000i	− 0.262432i
$364$	−2.00000	−0.104828
$365$	0	0
$366$	−2.00000	−0.104542
$367$	4.00000i	0.208798i	0.994535	+	0.104399i	$0.0332919\pi$
$367$	4.00000i	0.208798i	−0.994535	+	0.104399i	$0.966708\pi$
$368$	2.00000i	0.104257i
$369$	−10.0000	−0.520579
$370$	0	0
$371$	−12.0000	−0.623009
$372$	4.00000i	0.207390i
$373$	14.0000i	0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$	14.0000i	0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	16.0000	0.827340
$375$	0	0
$376$	0	0
$377$	8.00000i	0.412021i
$378$	− 2.00000i	− 0.102869i
$379$	14.0000	0.719132	0.359566	−	0.933120i	$-0.382925\pi$
$379$	14.0000	0.719132	0.359566	+	0.933120i	$0.382925\pi$
$380$	0	0
$381$	−20.0000	−1.02463
$382$	− 8.00000i	− 0.409316i
$383$	− 20.0000i	− 1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$	− 20.0000i	− 1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−12.0000	−0.610784
$387$	− 4.00000i	− 0.203331i
$388$	− 8.00000i	− 0.406138i
$389$	−20.0000	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	−20.0000	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	− 3.00000i	− 0.151523i
$393$	− 14.0000i	− 0.706207i
$394$	10.0000	0.503793
$395$	0	0
$396$	4.00000	0.201008
$397$	− 6.00000i	− 0.301131i	−0.988600	−	0.150566i	$-0.951890\pi$
$397$	− 6.00000i	− 0.301131i	0.988600	−	0.150566i	$-0.0481095\pi$
$398$	− 24.0000i	− 1.20301i
$399$	4.00000	0.200250
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	8.00000i	0.399004i
$403$	− 4.00000i	− 0.199254i
$404$	−20.0000	−0.995037
$405$	0	0
$406$	16.0000	0.794067
$407$	− 24.0000i	− 1.18964i
$408$	4.00000i	0.198030i
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	− 4.00000i	− 0.197066i
$413$	24.0000i	1.18096i
$414$	2.00000	0.0982946
$415$	0	0
$416$	1.00000	0.0490290
$417$	− 16.0000i	− 0.783523i
$418$	8.00000i	0.391293i
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	0	0
$421$	−16.0000	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$422$	− 4.00000i	− 0.194717i
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	4.00000i	0.193347i
$429$	−4.00000	−0.193122
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	1.00000i	0.0481125i
$433$	− 26.0000i	− 1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$	− 26.0000i	− 1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	4.00000	0.191565
$437$	4.00000i	0.191346i
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	− 4.00000i	− 0.190261i
$443$	− 16.0000i	− 0.760183i	−0.924949	−	0.380091i	$-0.875893\pi$
$443$	− 16.0000i	− 0.760183i	0.924949	−	0.380091i	$-0.124107\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	22.0000	1.04173
$447$	− 22.0000i	− 1.04056i
$448$	− 2.00000i	− 0.0944911i
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	40.0000	1.88353
$452$	16.0000i	0.752577i
$453$	4.00000i	0.187936i
$454$	12.0000	0.563188
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	16.0000i	0.748448i	0.927338	+	0.374224i	$0.122091\pi$
$457$	16.0000i	0.748448i	−0.927338	+	0.374224i	$0.877909\pi$
$458$	− 28.0000i	− 1.30835i
$459$	4.00000	0.186704
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	8.00000i	0.372194i
$463$	2.00000i	0.0929479i	0.998920	+	0.0464739i	$0.0147984\pi$
$463$	2.00000i	0.0929479i	−0.998920	+	0.0464739i	$0.985202\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	−28.0000	−1.29707
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	− 1.00000i	− 0.0462250i
$469$	−16.0000	−0.738811
$470$	0	0
$471$	10.0000	0.460776
$472$	− 12.0000i	− 0.552345i
$473$	16.0000i	0.735681i
$474$	8.00000	0.367452
$475$	0	0
$476$	−8.00000	−0.366679
$477$	− 6.00000i	− 0.274721i
$478$	8.00000i	0.365911i
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	− 22.0000i	− 1.00207i
$483$	4.00000i	0.182006i
$484$	−5.00000	−0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 38.0000i	− 1.72194i	−0.508652	−	0.860972i	$-0.669856\pi$
$487$	− 38.0000i	− 1.72194i	0.508652	−	0.860972i	$-0.330144\pi$
$488$	2.00000i	0.0905357i
$489$	12.0000	0.542659
$490$	0	0
$491$	38.0000	1.71492	0.857458	−	0.514554i	$-0.172042\pi$
$491$	38.0000	1.71492	0.857458	+	0.514554i	$0.172042\pi$
$492$	10.0000i	0.450835i
$493$	32.0000i	1.44121i
$494$	2.00000	0.0899843
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	− 12.0000i	− 0.537733i
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	10.0000i	0.446322i
$503$	− 42.0000i	− 1.87269i	−0.351085	−	0.936344i	$-0.614187\pi$
$503$	− 42.0000i	− 1.87269i	0.351085	−	0.936344i	$-0.385813\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	−8.00000	−0.355643
$507$	1.00000i	0.0444116i
$508$	20.0000i	0.887357i
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	2.00000i	0.0883022i
$514$	−8.00000	−0.352865
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	12.0000i	0.527250i
$519$	14.0000	0.614532
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	8.00000i	0.350150i
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	−14.0000	−0.611593
$525$	0	0
$526$	18.0000	0.784837
$527$	− 16.0000i	− 0.696971i
$528$	− 4.00000i	− 0.174078i
$529$	19.0000	0.826087
$530$	0	0
$531$	−12.0000	−0.520756
$532$	− 4.00000i	− 0.173422i
$533$	− 10.0000i	− 0.433148i
$534$	10.0000	0.432742
$535$	0	0
$536$	8.00000	0.345547
$537$	2.00000i	0.0863064i
$538$	4.00000i	0.172452i
$539$	12.0000	0.516877
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	− 28.0000i	− 1.20270i
$543$	22.0000i	0.944110i
$544$	4.00000	0.171499
$545$	0	0
$546$	2.00000	0.0855921
$547$	12.0000i	0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$	12.0000i	0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	− 2.00000i	− 0.0854358i
$549$	2.00000	0.0853579
$550$	0	0
$551$	−16.0000	−0.681623
$552$	− 2.00000i	− 0.0851257i
$553$	16.0000i	0.680389i
$554$	22.0000	0.934690
$555$	0	0
$556$	−16.0000	−0.678551
$557$	10.0000i	0.423714i	0.977301	+	0.211857i	$0.0679510\pi$
$557$	10.0000i	0.423714i	−0.977301	+	0.211857i	$0.932049\pi$
$558$	− 4.00000i	− 0.169334i
$559$	4.00000	0.169182
$560$	0	0
$561$	−16.0000	−0.675521
$562$	− 10.0000i	− 0.421825i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	0	0
$565$	0	0
$566$	−28.0000	−1.17693
$567$	2.00000i	0.0839921i
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	4.00000i	0.167248i
$573$	8.00000i	0.334205i
$574$	−20.0000	−0.834784
$575$	0	0
$576$	1.00000	0.0416667
$577$	4.00000i	0.166522i	0.996528	+	0.0832611i	$0.0265335\pi$
$577$	4.00000i	0.166522i	−0.996528	+	0.0832611i	$0.973466\pi$
$578$	1.00000i	0.0415945i
$579$	12.0000	0.498703
$580$	0	0
$581$	24.0000	0.995688
$582$	8.00000i	0.331611i
$583$	24.0000i	0.993978i
$584$	0	0
$585$	0	0
$586$	6.00000	0.247858
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	3.00000i	0.123718i
$589$	8.00000	0.329634
$590$	0	0
$591$	−10.0000	−0.411345
$592$	− 6.00000i	− 0.246598i
$593$	34.0000i	1.39621i	0.715994	+	0.698106i	$0.245974\pi$
$593$	34.0000i	1.39621i	−0.715994	+	0.698106i	$0.754026\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	−22.0000	−0.901155
$597$	24.0000i	0.982255i
$598$	2.00000i	0.0817861i
$599$	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
$601$	−42.0000	−1.71322	−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000	−1.71322	−0.856608	+	0.515968i	$0.827432\pi$
$602$	− 8.00000i	− 0.326056i
$603$	− 8.00000i	− 0.325785i
$604$	4.00000	0.162758
$605$	0	0
$606$	20.0000	0.812444
$607$	12.0000i	0.487065i	0.969893	+	0.243532i	$0.0783062\pi$
$607$	12.0000i	0.487065i	−0.969893	+	0.243532i	$0.921694\pi$
$608$	2.00000i	0.0811107i
$609$	−16.0000	−0.648353
$610$	0	0
$611$	0	0
$612$	− 4.00000i	− 0.161690i
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	0	0
$615$	0	0
$616$	8.00000	0.322329
$617$	− 38.0000i	− 1.52982i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	− 38.0000i	− 1.52982i	0.644136	−	0.764911i	$-0.277217\pi$
$618$	4.00000i	0.160904i
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	−2.00000	−0.0802572
$622$	− 16.0000i	− 0.641542i
$623$	20.0000i	0.801283i
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	10.0000	0.399680
$627$	− 8.00000i	− 0.319489i
$628$	− 10.0000i	− 0.399043i
$629$	−24.0000	−0.956943
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	− 8.00000i	− 0.318223i
$633$	4.00000i	0.158986i
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	−6.00000	−0.237915
$637$	− 3.00000i	− 0.118864i
$638$	− 32.0000i	− 1.26689i
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	− 4.00000i	− 0.157867i
$643$	− 40.0000i	− 1.57745i	−0.614749	−	0.788723i	$-0.710743\pi$
$643$	− 40.0000i	− 1.57745i	0.614749	−	0.788723i	$-0.289257\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	8.00000	0.314756
$647$	18.0000i	0.707653i	0.935311	+	0.353827i	$0.115120\pi$
$647$	18.0000i	0.707653i	−0.935311	+	0.353827i	$0.884880\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	48.0000	1.88416
$650$	0	0
$651$	8.00000	0.313545
$652$	− 12.0000i	− 0.469956i
$653$	− 38.0000i	− 1.48705i	−0.668705	−	0.743527i	$-0.733151\pi$
$653$	− 38.0000i	− 1.48705i	0.668705	−	0.743527i	$-0.266849\pi$
$654$	−4.00000	−0.156412
$655$	0	0
$656$	10.0000	0.390434
$657$	0	0
$658$	0	0
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	− 10.0000i	− 0.388661i
$663$	4.00000i	0.155347i
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−6.00000	−0.232495
$667$	− 16.0000i	− 0.619522i
$668$	− 12.0000i	− 0.464294i
$669$	−22.0000	−0.850569
$670$	0	0
$671$	−8.00000	−0.308837
$672$	2.00000i	0.0771517i
$673$	− 46.0000i	− 1.77317i	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	− 46.0000i	− 1.77317i	0.462566	−	0.886585i	$-0.346929\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	1.00000	0.0384615
$677$	22.0000i	0.845529i	0.906240	+	0.422764i	$0.138940\pi$
$677$	22.0000i	0.845529i	−0.906240	+	0.422764i	$0.861060\pi$
$678$	− 16.0000i	− 0.614476i
$679$	−16.0000	−0.614024
$680$	0	0
$681$	−12.0000	−0.459841
$682$	16.0000i	0.612672i
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	2.00000	0.0764719
$685$	0	0
$686$	−20.0000	−0.763604
$687$	28.0000i	1.06827i
$688$	4.00000i	0.152499i
$689$	6.00000	0.228582
$690$	0	0
$691$	−18.0000	−0.684752	−0.342376	−	0.939563i	$-0.611232\pi$
$691$	−18.0000	−0.684752	−0.342376	+	0.939563i	$0.611232\pi$
$692$	− 14.0000i	− 0.532200i
$693$	− 8.00000i	− 0.303895i
$694$	16.0000	0.607352
$695$	0	0
$696$	8.00000	0.303239
$697$	− 40.0000i	− 1.51511i
$698$	28.0000i	1.05982i
$699$	28.0000	1.05906
$700$	0	0
$701$	−20.0000	−0.755390	−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000	−0.755390	−0.377695	+	0.925930i	$0.623283\pi$
$702$	1.00000i	0.0377426i
$703$	− 12.0000i	− 0.452589i
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−6.00000	−0.225813
$707$	40.0000i	1.50435i
$708$	12.0000i	0.450988i
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	− 10.0000i	− 0.374766i
$713$	8.00000i	0.299602i
$714$	8.00000	0.299392
$715$	0	0
$716$	2.00000	0.0747435
$717$	− 8.00000i	− 0.298765i
$718$	− 24.0000i	− 0.895672i
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	− 15.0000i	− 0.558242i
$723$	22.0000i	0.818189i
$724$	22.0000	0.817624
$725$	0	0
$726$	5.00000	0.185567
$727$	− 8.00000i	− 0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$	− 8.00000i	− 0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$	− 2.00000i	− 0.0741249i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	16.0000	0.591781
$732$	− 2.00000i	− 0.0739221i
$733$	6.00000i	0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$	6.00000i	0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$	−4.00000	−0.147643
$735$	0	0
$736$	−2.00000	−0.0737210
$737$	32.0000i	1.17874i
$738$	− 10.0000i	− 0.368105i
$739$	−50.0000	−1.83928	−0.919640	−	0.392763i	$-0.871519\pi$
$739$	−50.0000	−1.83928	−0.919640	+	0.392763i	$0.871519\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	− 12.0000i	− 0.440534i
$743$	36.0000i	1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$	36.0000i	1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$	−4.00000	−0.146647
$745$	0	0
$746$	−14.0000	−0.512576
$747$	12.0000i	0.439057i
$748$	16.0000i	0.585018i
$749$	8.00000	0.292314
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	− 10.0000i	− 0.364420i
$754$	−8.00000	−0.291343
$755$	0	0
$756$	2.00000	0.0727393
$757$	38.0000i	1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$	38.0000i	1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$	14.0000i	0.508503i
$759$	8.00000	0.290382
$760$	0	0
$761$	−46.0000	−1.66750	−0.833749	−	0.552143i	$-0.813810\pi$
$761$	−46.0000	−1.66750	−0.833749	+	0.552143i	$0.813810\pi$
$762$	− 20.0000i	− 0.724524i
$763$	− 8.00000i	− 0.289619i
$764$	8.00000	0.289430
$765$	0	0
$766$	20.0000	0.722629
$767$	− 12.0000i	− 0.433295i
$768$	− 1.00000i	− 0.0360844i
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	8.00000	0.288113
$772$	− 12.0000i	− 0.431889i
$773$	26.0000i	0.935155i	0.883952	+	0.467578i	$0.154873\pi$
$773$	26.0000i	0.935155i	−0.883952	+	0.467578i	$0.845127\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	8.00000	0.287183
$777$	− 12.0000i	− 0.430498i
$778$	− 20.0000i	− 0.717035i
$779$	20.0000	0.716574
$780$	0	0
$781$	0	0
$782$	8.00000i	0.286079i
$783$	− 8.00000i	− 0.285897i
$784$	3.00000	0.107143
$785$	0	0
$786$	14.0000	0.499363
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	10.0000i	0.356235i
$789$	−18.0000	−0.640817
$790$	0	0
$791$	32.0000	1.13779
$792$	4.00000i	0.142134i
$793$	2.00000i	0.0710221i
$794$	6.00000	0.212932
$795$	0	0
$796$	24.0000	0.850657
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	4.00000i	0.141598i
$799$	0	0
$800$	0	0
$801$	−10.0000	−0.353333
$802$	− 30.0000i	− 1.05934i
$803$	0	0
$804$	−8.00000	−0.282138
$805$	0	0
$806$	4.00000	0.140894
$807$	− 4.00000i	− 0.140807i
$808$	− 20.0000i	− 0.703598i
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−34.0000	−1.19390	−0.596951	−	0.802278i	$-0.703621\pi$
$811$	−34.0000	−1.19390	−0.596951	+	0.802278i	$0.703621\pi$
$812$	16.0000i	0.561490i
$813$	28.0000i	0.982003i
$814$	24.0000	0.841200
$815$	0	0
$816$	−4.00000	−0.140028
$817$	8.00000i	0.279885i
$818$	18.0000i	0.629355i
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	2.00000i	0.0697580i
$823$	− 20.0000i	− 0.697156i	−0.937280	−	0.348578i	$-0.886665\pi$
$823$	− 20.0000i	− 0.697156i	0.937280	−	0.348578i	$-0.113335\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	−24.0000	−0.835067
$827$	44.0000i	1.53003i	0.644013	+	0.765015i	$0.277268\pi$
$827$	44.0000i	1.53003i	−0.644013	+	0.765015i	$0.722732\pi$
$828$	2.00000i	0.0695048i
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	1.00000i	0.0346688i
$833$	− 12.0000i	− 0.415775i
$834$	16.0000	0.554035
$835$	0	0
$836$	−8.00000	−0.276686
$837$	4.00000i	0.138260i
$838$	− 14.0000i	− 0.483622i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	− 16.0000i	− 0.551396i
$843$	10.0000i	0.344418i
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	10.0000i	0.343604i
$848$	6.00000i	0.206041i
$849$	28.0000	0.960958
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	−4.00000	−0.136717
$857$	16.0000i	0.546550i	0.961936	+	0.273275i	$0.0881068\pi$
$857$	16.0000i	0.546550i	−0.961936	+	0.273275i	$0.911893\pi$
$858$	− 4.00000i	− 0.136558i
$859$	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
$861$	20.0000	0.681598
$862$	16.0000i	0.544962i
$863$	− 36.0000i	− 1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$	− 36.0000i	− 1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	26.0000	0.883516
$867$	− 1.00000i	− 0.0339618i
$868$	− 8.00000i	− 0.271538i
$869$	32.0000	1.08553
$870$	0	0
$871$	8.00000	0.271070
$872$	4.00000i	0.135457i
$873$	− 8.00000i	− 0.270759i
$874$	−4.00000	−0.135302
$875$	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	$0.878863\pi$
$878$	− 8.00000i	− 0.269987i
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	− 3.00000i	− 0.101015i
$883$	44.0000i	1.48072i	0.672212	+	0.740359i	$0.265344\pi$
$883$	44.0000i	1.48072i	−0.672212	+	0.740359i	$0.734656\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	16.0000	0.537531
$887$	6.00000i	0.201460i	0.994914	+	0.100730i	$0.0321179\pi$
$887$	6.00000i	0.201460i	−0.994914	+	0.100730i	$0.967882\pi$
$888$	6.00000i	0.201347i
$889$	40.0000	1.34156
$890$	0	0
$891$	4.00000	0.134005
$892$	22.0000i	0.736614i
$893$	0	0
$894$	22.0000	0.735790
$895$	0	0
$896$	2.00000	0.0668153
$897$	− 2.00000i	− 0.0667781i
$898$	18.0000i	0.600668i
$899$	−32.0000	−1.06726
$900$	0	0
$901$	24.0000	0.799556
$902$	40.0000i	1.33185i
$903$	8.00000i	0.266223i
$904$	−16.0000	−0.532152
$905$	0	0
$906$	−4.00000	−0.132891
$907$	44.0000i	1.46100i	0.682915	+	0.730498i	$0.260712\pi$
$907$	44.0000i	1.46100i	−0.682915	+	0.730498i	$0.739288\pi$
$908$	12.0000i	0.398234i
$909$	−20.0000	−0.663358
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	− 2.00000i	− 0.0662266i
$913$	− 48.0000i	− 1.58857i
$914$	−16.0000	−0.529233
$915$	0	0
$916$	28.0000	0.925146
$917$	28.0000i	0.924641i
$918$	4.00000i	0.132020i
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	0	0
$922$	30.0000i	0.987997i
$923$	0	0
$924$	−8.00000	−0.263181
$925$	0	0
$926$	−2.00000	−0.0657241
$927$	− 4.00000i	− 0.131377i
$928$	− 8.00000i	− 0.262613i
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	− 28.0000i	− 0.917170i
$933$	16.0000i	0.523816i
$934$	12.0000	0.392652
$935$	0	0
$936$	1.00000	0.0326860
$937$	− 10.0000i	− 0.326686i	−0.986569	−	0.163343i	$-0.947772\pi$
$937$	− 10.0000i	− 0.326686i	0.986569	−	0.163343i	$-0.0522277\pi$
$938$	− 16.0000i	− 0.522419i
$939$	−10.0000	−0.326338
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	10.0000i	0.325818i
$943$	20.0000i	0.651290i
$944$	12.0000	0.390567
$945$	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	$0.979298\pi$
$948$	8.00000i	0.259828i
$949$	0	0
$950$	0	0
$951$	2.00000	0.0648544
$952$	− 8.00000i	− 0.259281i
$953$	− 36.0000i	− 1.16615i	−0.812417	−	0.583077i	$-0.801849\pi$
$953$	− 36.0000i	− 1.16615i	0.812417	−	0.583077i	$-0.198151\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−8.00000	−0.258738
$957$	32.0000i	1.03441i
$958$	− 24.0000i	− 0.775405i
$959$	−4.00000	−0.129167
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 6.00000i	− 0.193448i
$963$	4.00000i	0.128898i
$964$	22.0000	0.708572
$965$	0	0
$966$	−4.00000	−0.128698
$967$	− 14.0000i	− 0.450210i	−0.974335	−	0.225105i	$-0.927728\pi$
$967$	− 14.0000i	− 0.450210i	0.974335	−	0.225105i	$-0.0722725\pi$
$968$	− 5.00000i	− 0.160706i
$969$	−8.00000	−0.256997
$970$	0	0
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	1.00000i	0.0320750i
$973$	32.0000i	1.02587i
$974$	38.0000	1.21760
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	12.0000i	0.383718i
$979$	40.0000	1.27841
$980$	0	0
$981$	4.00000	0.127710
$982$	38.0000i	1.21263i
$983$	− 12.0000i	− 0.382741i	−0.981518	−	0.191370i	$-0.938707\pi$
$983$	− 12.0000i	− 0.382741i	0.981518	−	0.191370i	$-0.0612931\pi$
$984$	−10.0000	−0.318788
$985$	0	0
$986$	−32.0000	−1.01909
$987$	0	0
$988$	2.00000i	0.0636285i
$989$	−8.00000	−0.254385
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	4.00000i	0.127000i
$993$	10.0000i	0.317340i
$994$	0	0
$995$	0	0
$996$	12.0000	0.380235
$997$	26.0000i	0.823428i	0.911313	+	0.411714i	$0.135070\pi$
$997$	26.0000i	0.823428i	−0.911313	+	0.411714i	$0.864930\pi$
$998$	14.0000i	0.443162i
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.e.m.1249.2		2
3.2	odd	2		5850.2.e.h.5149.1		2
5.2	odd	4		390.2.a.b.1.1	✓	1
5.3	odd	4		1950.2.a.ba.1.1		1
5.4	even	2	inner	1950.2.e.m.1249.1		2
15.2	even	4		1170.2.a.j.1.1		1
15.8	even	4		5850.2.a.s.1.1		1
15.14	odd	2		5850.2.e.h.5149.2		2
20.7	even	4		3120.2.a.y.1.1		1
60.47	odd	4		9360.2.a.v.1.1		1
65.12	odd	4		5070.2.a.n.1.1		1
65.47	even	4		5070.2.b.f.1351.1		2
65.57	even	4		5070.2.b.f.1351.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.b.1.1	✓	1	5.2	odd	4
1170.2.a.j.1.1		1	15.2	even	4
1950.2.a.ba.1.1		1	5.3	odd	4
1950.2.e.m.1249.1		2	5.4	even	2	inner
1950.2.e.m.1249.2		2	1.1	even	1	trivial
3120.2.a.y.1.1		1	20.7	even	4
5070.2.a.n.1.1		1	65.12	odd	4
5070.2.b.f.1351.1		2	65.47	even	4
5070.2.b.f.1351.2		2	65.57	even	4
5850.2.a.s.1.1		1	15.8	even	4
5850.2.e.h.5149.1		2	3.2	odd	2
5850.2.e.h.5149.2		2	15.14	odd	2
9360.2.a.v.1.1		1	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.e (of order \(2\), degree \(1\), not minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more