LMFDB - Embedded newform 5775.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5775,2,Mod(1,5775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5775, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5775.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5775.a (trivial)

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:	yes
Analytic conductor:	$46.1136071673$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1155)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5775.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

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

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	−1.00000	−0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	1.00000	0.213201
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	−3.00000	−0.612372
$25$	0	0
$26$	2.00000	0.392232
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	5.00000	0.883883
$33$	1.00000	0.174078
$34$	6.00000	1.02899
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−4.00000	−0.648886
$39$	2.00000	0.320256
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	1.00000	0.154303
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−8.00000	−1.17954
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	0	0
$51$	6.00000	0.840168
$52$	−2.00000	−0.277350
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−3.00000	−0.400892
$57$	−4.00000	−0.529813
$58$	6.00000	0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	8.00000	1.01600
$63$	1.00000	0.125988
$64$	7.00000	0.875000
$65$	0	0
$66$	1.00000	0.123091
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−6.00000	−0.727607
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−3.00000	−0.353553
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	1.00000	0.113961
$78$	2.00000	0.226455
$79$	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
$82$	−6.00000	−0.662589
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	4.00000	0.431331
$87$	6.00000	0.643268
$88$	−3.00000	−0.319801
$89$	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
$92$	8.00000	0.834058
$93$	8.00000	0.829561
$94$	8.00000	0.825137
$95$	0	0
$96$	5.00000	0.510310
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	1.00000	0.101015
$99$	1.00000	0.100504
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	6.00000	0.594089
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	10.0000	0.971286
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	−1.00000	−0.0944911
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	−6.00000	−0.557086
$117$	2.00000	0.184900
$118$	−12.0000	−1.10469
$119$	6.00000	0.550019
$120$	0	0
$121$	1.00000	0.0909091
$122$	−10.0000	−0.905357
$123$	−6.00000	−0.541002
$124$	−8.00000	−0.718421
$125$	0	0
$126$	1.00000	0.0890871
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−3.00000	−0.265165
$129$	4.00000	0.352180
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	−1.00000	−0.0870388
$133$	−4.00000	−0.346844
$134$	12.0000	1.03664
$135$	0	0
$136$	−18.0000	−1.54349
$137$	22.0000	1.87959	0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000	1.87959	0.939793	+	0.341743i	$0.111017\pi$
$138$	−8.00000	−0.681005
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	−8.00000	−0.671345
$143$	2.00000	0.167248
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	6.00000	0.496564
$147$	1.00000	0.0824786
$148$	6.00000	0.493197
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	12.0000	0.973329
$153$	6.00000	0.485071
$154$	1.00000	0.0805823
$155$	0	0
$156$	−2.00000	−0.160128
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	8.00000	0.636446
$159$	10.0000	0.793052
$160$	0	0
$161$	−8.00000	−0.630488
$162$	1.00000	0.0785674
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	4.00000	0.310460
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	−3.00000	−0.231455
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	−4.00000	−0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−12.0000	−0.901975
$178$	10.0000	0.749532
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	2.00000	0.148250
$183$	−10.0000	−0.739221
$184$	24.0000	1.76930
$185$	0	0
$186$	8.00000	0.586588
$187$	6.00000	0.438763
$188$	−8.00000	−0.583460
$189$	1.00000	0.0727393
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	7.00000	0.505181
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	1.00000	0.0710669
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	6.00000	0.422159
$203$	6.00000	0.421117
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−16.0000	−1.11477
$207$	−8.00000	−0.556038
$208$	−2.00000	−0.138675
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−10.0000	−0.686803
$213$	−8.00000	−0.548151
$214$	12.0000	0.820303
$215$	0	0
$216$	−3.00000	−0.204124
$217$	8.00000	0.543075
$218$	−2.00000	−0.135457
$219$	6.00000	0.405442
$220$	0	0
$221$	12.0000	0.807207
$222$	−6.00000	−0.402694
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	5.00000	0.334077
$225$	0	0
$226$	14.0000	0.931266
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	4.00000	0.264906
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	−18.0000	−1.18176
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	12.0000	0.781133
$237$	8.00000	0.519656
$238$	6.00000	0.388922
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	10.0000	0.640184
$245$	0	0
$246$	−6.00000	−0.382546
$247$	−8.00000	−0.509028
$248$	−24.0000	−1.52400
$249$	4.00000	0.253490
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	−1.00000	−0.0629941
$253$	−8.00000	−0.502956
$254$	0	0
$255$	0	0
$256$	−17.0000	−1.06250
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	4.00000	0.249029
$259$	−6.00000	−0.372822
$260$	0	0
$261$	6.00000	0.371391
$262$	−12.0000	−0.741362
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	−4.00000	−0.245256
$267$	10.0000	0.611990
$268$	−12.0000	−0.733017
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−6.00000	−0.363803
$273$	2.00000	0.121046
$274$	22.0000	1.32907
$275$	0	0
$276$	8.00000	0.481543
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	4.00000	0.239904
$279$	8.00000	0.478947
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	8.00000	0.476393
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	2.00000	0.118262
$287$	−6.00000	−0.354169
$288$	5.00000	0.294628
$289$	19.0000	1.11765
$290$	0	0
$291$	14.0000	0.820695
$292$	−6.00000	−0.351123
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	1.00000	0.0583212
$295$	0	0
$296$	18.0000	1.04623
$297$	1.00000	0.0580259
$298$	−18.0000	−1.04271
$299$	−16.0000	−0.925304
$300$	0	0
$301$	4.00000	0.230556
$302$	16.0000	0.920697
$303$	6.00000	0.344691
$304$	4.00000	0.229416
$305$	0	0
$306$	6.00000	0.342997
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	−1.00000	−0.0569803
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	−6.00000	−0.339683
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	−8.00000	−0.450035
$317$	34.0000	1.90963	0.954815	−	0.297200i	$-0.0960529\pi$
$317$	34.0000	1.90963	0.954815	+	0.297200i	$0.0960529\pi$
$318$	10.0000	0.560772
$319$	6.00000	0.335936
$320$	0	0
$321$	12.0000	0.669775
$322$	−8.00000	−0.445823
$323$	−24.0000	−1.33540
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	−2.00000	−0.110600
$328$	18.0000	0.993884
$329$	8.00000	0.441054
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	−4.00000	−0.219529
$333$	−6.00000	−0.328798
$334$	16.0000	0.875481
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	−9.00000	−0.489535
$339$	14.0000	0.760376
$340$	0	0
$341$	8.00000	0.433224
$342$	−4.00000	−0.216295
$343$	1.00000	0.0539949
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	−6.00000	−0.321634
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	5.00000	0.266501
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	−10.0000	−0.529999
$357$	6.00000	0.317554
$358$	4.00000	0.211407
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	6.00000	0.315353
$363$	1.00000	0.0524864
$364$	−2.00000	−0.104828
$365$	0	0
$366$	−10.0000	−0.522708
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	8.00000	0.417029
$369$	−6.00000	−0.312348
$370$	0	0
$371$	10.0000	0.519174
$372$	−8.00000	−0.414781
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	−24.0000	−1.23771
$377$	12.0000	0.618031
$378$	1.00000	0.0514344
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	0	0
$382$	16.0000	0.818631
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	22.0000	1.11977
$387$	4.00000	0.203331
$388$	−14.0000	−0.710742
$389$	−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
$391$	−48.0000	−2.42746
$392$	−3.00000	−0.151523
$393$	−12.0000	−0.605320
$394$	−22.0000	−1.10834
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	16.0000	0.802008
$399$	−4.00000	−0.200250
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	12.0000	0.598506
$403$	16.0000	0.797017
$404$	−6.00000	−0.298511
$405$	0	0
$406$	6.00000	0.297775
$407$	−6.00000	−0.297409
$408$	−18.0000	−0.891133
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	22.0000	1.08518
$412$	16.0000	0.788263
$413$	−12.0000	−0.590481
$414$	−8.00000	−0.393179
$415$	0	0
$416$	10.0000	0.490290
$417$	4.00000	0.195881
$418$	−4.00000	−0.195646
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	−4.00000	−0.194717
$423$	8.00000	0.388973
$424$	−30.0000	−1.45693
$425$	0	0
$426$	−8.00000	−0.387601
$427$	−10.0000	−0.483934
$428$	−12.0000	−0.580042
$429$	2.00000	0.0965609
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	−1.00000	−0.0481125
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	2.00000	0.0957826
$437$	32.0000	1.53077
$438$	6.00000	0.286691
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	12.0000	0.570782
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	8.00000	0.378811
$447$	−18.0000	−0.851371
$448$	7.00000	0.330719
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	−14.0000	−0.658505
$453$	16.0000	0.751746
$454$	−12.0000	−0.563188
$455$	0	0
$456$	12.0000	0.561951
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	−10.0000	−0.467269
$459$	6.00000	0.280056
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	1.00000	0.0465242
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	22.0000	1.01913
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	−2.00000	−0.0924500
$469$	12.0000	0.554109
$470$	0	0
$471$	2.00000	0.0921551
$472$	36.0000	1.65703
$473$	4.00000	0.183920
$474$	8.00000	0.367452
$475$	0	0
$476$	−6.00000	−0.275010
$477$	10.0000	0.457869
$478$	−16.0000	−0.731823
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	−22.0000	−1.00207
$483$	−8.00000	−0.364013
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	1.00000	0.0453609
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	30.0000	1.35804
$489$	−20.0000	−0.904431
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	6.00000	0.270501
$493$	36.0000	1.62136
$494$	−8.00000	−0.359937
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−8.00000	−0.358849
$498$	4.00000	0.179244
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	20.0000	0.892644
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	−8.00000	−0.355643
$507$	−9.00000	−0.399704
$508$	0	0
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	−11.0000	−0.486136
$513$	−4.00000	−0.176604
$514$	30.0000	1.32324
$515$	0	0
$516$	−4.00000	−0.176090
$517$	8.00000	0.351840
$518$	−6.00000	−0.263625
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	6.00000	0.262613
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−16.0000	−0.697633
$527$	48.0000	2.09091
$528$	−1.00000	−0.0435194
$529$	41.0000	1.78261
$530$	0	0
$531$	−12.0000	−0.520756
$532$	4.00000	0.173422
$533$	−12.0000	−0.519778
$534$	10.0000	0.432742
$535$	0	0
$536$	−36.0000	−1.55496
$537$	4.00000	0.172613
$538$	14.0000	0.603583
$539$	1.00000	0.0430730
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	8.00000	0.343629
$543$	6.00000	0.257485
$544$	30.0000	1.28624
$545$	0	0
$546$	2.00000	0.0855921
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	−22.0000	−0.939793
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−24.0000	−1.02243
$552$	24.0000	1.02151
$553$	8.00000	0.340195
$554$	18.0000	0.764747
$555$	0	0
$556$	−4.00000	−0.169638
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	8.00000	0.338667
$559$	8.00000	0.338364
$560$	0	0
$561$	6.00000	0.253320
$562$	18.0000	0.759284
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	4.00000	0.168133
$567$	1.00000	0.0419961
$568$	24.0000	1.00702
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	−2.00000	−0.0836242
$573$	16.0000	0.668410
$574$	−6.00000	−0.250435
$575$	0	0
$576$	7.00000	0.291667
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	19.0000	0.790296
$579$	22.0000	0.914289
$580$	0	0
$581$	4.00000	0.165948
$582$	14.0000	0.580319
$583$	10.0000	0.414158
$584$	−18.0000	−0.744845
$585$	0	0
$586$	18.0000	0.743573
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	−1.00000	−0.0412393
$589$	−32.0000	−1.31854
$590$	0	0
$591$	−22.0000	−0.904959
$592$	6.00000	0.246598
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	18.0000	0.737309
$597$	16.0000	0.654836
$598$	−16.0000	−0.654289
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	4.00000	0.163028
$603$	12.0000	0.488678
$604$	−16.0000	−0.651031
$605$	0	0
$606$	6.00000	0.243733
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	−20.0000	−0.811107
$609$	6.00000	0.243132
$610$	0	0
$611$	16.0000	0.647291
$612$	−6.00000	−0.242536
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	−3.00000	−0.120873
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	−16.0000	−0.643614
$619$	−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
$621$	−8.00000	−0.321029
$622$	−16.0000	−0.641542
$623$	10.0000	0.400642
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	−10.0000	−0.399680
$627$	−4.00000	−0.159745
$628$	−2.00000	−0.0798087
$629$	−36.0000	−1.43541
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	−24.0000	−0.954669
$633$	−4.00000	−0.158986
$634$	34.0000	1.35031
$635$	0	0
$636$	−10.0000	−0.396526
$637$	2.00000	0.0792429
$638$	6.00000	0.237542
$639$	−8.00000	−0.316475
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	12.0000	0.473602
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	−24.0000	−0.944267
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	−3.00000	−0.117851
$649$	−12.0000	−0.471041
$650$	0	0
$651$	8.00000	0.313545
$652$	20.0000	0.783260
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	6.00000	0.234261
$657$	6.00000	0.234082
$658$	8.00000	0.311872
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	28.0000	1.08825
$663$	12.0000	0.466041
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−6.00000	−0.232495
$667$	−48.0000	−1.85857
$668$	−16.0000	−0.619059
$669$	8.00000	0.309298
$670$	0	0
$671$	−10.0000	−0.386046
$672$	5.00000	0.192879
$673$	38.0000	1.46479	0.732396	−	0.680879i	$-0.238402\pi$
$673$	38.0000	1.46479	0.732396	+	0.680879i	$0.238402\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	9.00000	0.346154
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	14.0000	0.537667
$679$	14.0000	0.537271
$680$	0	0
$681$	−12.0000	−0.459841
$682$	8.00000	0.306336
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	1.00000	0.0381802
$687$	−10.0000	−0.381524
$688$	−4.00000	−0.152499
$689$	20.0000	0.761939
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	6.00000	0.228086
$693$	1.00000	0.0379869
$694$	−20.0000	−0.759190
$695$	0	0
$696$	−18.0000	−0.682288
$697$	−36.0000	−1.36360
$698$	−10.0000	−0.378506
$699$	22.0000	0.832116
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	2.00000	0.0754851
$703$	24.0000	0.905177
$704$	7.00000	0.263822
$705$	0	0
$706$	−34.0000	−1.27961
$707$	6.00000	0.225653
$708$	12.0000	0.450988
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	−30.0000	−1.12430
$713$	−64.0000	−2.39682
$714$	6.00000	0.224544
$715$	0	0
$716$	−4.00000	−0.149487
$717$	−16.0000	−0.597531
$718$	8.00000	0.298557
$719$	−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
$721$	−16.0000	−0.595871
$722$	−3.00000	−0.111648
$723$	−22.0000	−0.818189
$724$	−6.00000	−0.222988
$725$	0	0
$726$	1.00000	0.0371135
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	−6.00000	−0.222375
$729$	1.00000	0.0370370
$730$	0	0
$731$	24.0000	0.887672
$732$	10.0000	0.369611
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	−40.0000	−1.47442
$737$	12.0000	0.442026
$738$	−6.00000	−0.220863
$739$	−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
$741$	−8.00000	−0.293887
$742$	10.0000	0.367112
$743$	32.0000	1.17397	0.586983	−	0.809599i	$-0.300316\pi$
$743$	32.0000	1.17397	0.586983	+	0.809599i	$0.300316\pi$
$744$	−24.0000	−0.879883
$745$	0	0
$746$	−14.0000	−0.512576
$747$	4.00000	0.146352
$748$	−6.00000	−0.219382
$749$	12.0000	0.438470
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	−8.00000	−0.291730
$753$	20.0000	0.728841
$754$	12.0000	0.437014
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	−20.0000	−0.726433
$759$	−8.00000	−0.290382
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	−16.0000	−0.578860
$765$	0	0
$766$	−24.0000	−0.867155
$767$	−24.0000	−0.866590
$768$	−17.0000	−0.613435
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	−22.0000	−0.791797
$773$	10.0000	0.359675	0.179838	−	0.983696i	$-0.442443\pi$
$773$	10.0000	0.359675	0.179838	+	0.983696i	$0.442443\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−42.0000	−1.50771
$777$	−6.00000	−0.215249
$778$	−10.0000	−0.358517
$779$	24.0000	0.859889
$780$	0	0
$781$	−8.00000	−0.286263
$782$	−48.0000	−1.71648
$783$	6.00000	0.214423
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	−12.0000	−0.428026
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	22.0000	0.783718
$789$	−16.0000	−0.569615
$790$	0	0
$791$	14.0000	0.497783
$792$	−3.00000	−0.106600
$793$	−20.0000	−0.710221
$794$	18.0000	0.638796
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	−4.00000	−0.141598
$799$	48.0000	1.69812
$800$	0	0
$801$	10.0000	0.353333
$802$	2.00000	0.0706225
$803$	6.00000	0.211735
$804$	−12.0000	−0.423207
$805$	0	0
$806$	16.0000	0.563576
$807$	14.0000	0.492823
$808$	−18.0000	−0.633238
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	−6.00000	−0.210559
$813$	8.00000	0.280572
$814$	−6.00000	−0.210300
$815$	0	0
$816$	−6.00000	−0.210042
$817$	−16.0000	−0.559769
$818$	−14.0000	−0.489499
$819$	2.00000	0.0698857
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	22.0000	0.767338
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	48.0000	1.67216
$825$	0	0
$826$	−12.0000	−0.417533
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	8.00000	0.278019
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	14.0000	0.485363
$833$	6.00000	0.207888
$834$	4.00000	0.138509
$835$	0	0
$836$	4.00000	0.138343
$837$	8.00000	0.276520
$838$	−36.0000	−1.24360
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	6.00000	0.206774
$843$	18.0000	0.619953
$844$	4.00000	0.137686
$845$	0	0
$846$	8.00000	0.275046
$847$	1.00000	0.0343604
$848$	−10.0000	−0.343401
$849$	4.00000	0.137280
$850$	0	0
$851$	48.0000	1.64542
$852$	8.00000	0.274075
$853$	−38.0000	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$854$	−10.0000	−0.342193
$855$	0	0
$856$	−36.0000	−1.23045
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	2.00000	0.0682789
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	−16.0000	−0.544962
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	19.0000	0.645274
$868$	−8.00000	−0.271538
$869$	8.00000	0.271381
$870$	0	0
$871$	24.0000	0.813209
$872$	6.00000	0.203186
$873$	14.0000	0.473828
$874$	32.0000	1.08242
$875$	0	0
$876$	−6.00000	−0.202721
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	32.0000	1.07995
$879$	18.0000	0.607125
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	1.00000	0.0336718
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−12.0000	−0.403148
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	18.0000	0.604040
$889$	0	0
$890$	0	0
$891$	1.00000	0.0335013
$892$	−8.00000	−0.267860
$893$	−32.0000	−1.07084
$894$	−18.0000	−0.602010
$895$	0	0
$896$	−3.00000	−0.100223
$897$	−16.0000	−0.534224
$898$	2.00000	0.0667409
$899$	48.0000	1.60089
$900$	0	0
$901$	60.0000	1.99889
$902$	−6.00000	−0.199778
$903$	4.00000	0.133112
$904$	−42.0000	−1.39690
$905$	0	0
$906$	16.0000	0.531564
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	12.0000	0.398234
$909$	6.00000	0.199007
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	4.00000	0.132453
$913$	4.00000	0.132381
$914$	−18.0000	−0.595387
$915$	0	0
$916$	10.0000	0.330409
$917$	−12.0000	−0.396275
$918$	6.00000	0.198030
$919$	−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
$921$	−4.00000	−0.131804
$922$	14.0000	0.461065
$923$	−16.0000	−0.526646
$924$	−1.00000	−0.0328976
$925$	0	0
$926$	−16.0000	−0.525793
$927$	−16.0000	−0.525509
$928$	30.0000	0.984798
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	−22.0000	−0.720634
$933$	−16.0000	−0.523816
$934$	−28.0000	−0.916188
$935$	0	0
$936$	−6.00000	−0.196116
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	12.0000	0.391814
$939$	−10.0000	−0.326338
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	2.00000	0.0651635
$943$	48.0000	1.56310
$944$	12.0000	0.390567
$945$	0	0
$946$	4.00000	0.130051
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	−8.00000	−0.259828
$949$	12.0000	0.389536
$950$	0	0
$951$	34.0000	1.10253
$952$	−18.0000	−0.583383
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	10.0000	0.323762
$955$	0	0
$956$	16.0000	0.517477
$957$	6.00000	0.193952
$958$	−16.0000	−0.516937
$959$	22.0000	0.710417
$960$	0	0
$961$	33.0000	1.06452
$962$	−12.0000	−0.386896
$963$	12.0000	0.386695
$964$	22.0000	0.708572
$965$	0	0
$966$	−8.00000	−0.257396
$967$	−56.0000	−1.80084	−0.900419	−	0.435023i	$-0.856740\pi$
$967$	−56.0000	−1.80084	−0.900419	+	0.435023i	$0.856740\pi$
$968$	−3.00000	−0.0964237
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	−1.00000	−0.0320750
$973$	4.00000	0.128234
$974$	−24.0000	−0.769010
$975$	0	0
$976$	10.0000	0.320092
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	−20.0000	−0.639529
$979$	10.0000	0.319601
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	−20.0000	−0.638226
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	18.0000	0.573819
$985$	0	0
$986$	36.0000	1.14647
$987$	8.00000	0.254643
$988$	8.00000	0.254514
$989$	−32.0000	−1.01754
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	40.0000	1.27000
$993$	28.0000	0.888553
$994$	−8.00000	−0.253745
$995$	0	0
$996$	−4.00000	−0.126745
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	−12.0000	−0.379853
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5775.2.a.v.1.1		1
5.4	even	2		1155.2.a.e.1.1	✓	1
15.14	odd	2		3465.2.a.l.1.1		1
35.34	odd	2		8085.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.e.1.1	✓	1	5.4	even	2
3465.2.a.l.1.1		1	15.14	odd	2
5775.2.a.v.1.1		1	1.1	even	1	trivial
8085.2.a.g.1.1		1	35.34	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5775.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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