LMFDB - Embedded newform 975.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(1,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("975.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.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:	$7.78541419707$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	975.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} -1.00000 q^{16} -7.00000 q^{17} -1.00000 q^{18} -1.00000 q^{21} +1.00000 q^{22} +3.00000 q^{24} +1.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} +5.00000 q^{29} -1.00000 q^{31} -5.00000 q^{32} -1.00000 q^{33} +7.00000 q^{34} -1.00000 q^{36} -8.00000 q^{37} -1.00000 q^{39} +6.00000 q^{41} +1.00000 q^{42} -8.00000 q^{43} +1.00000 q^{44} -5.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} -7.00000 q^{51} +1.00000 q^{52} -1.00000 q^{53} -1.00000 q^{54} -3.00000 q^{56} -5.00000 q^{58} -3.00000 q^{59} -7.00000 q^{61} +1.00000 q^{62} -1.00000 q^{63} +7.00000 q^{64} +1.00000 q^{66} -7.00000 q^{67} +7.00000 q^{68} +3.00000 q^{72} -12.0000 q^{73} +8.00000 q^{74} +1.00000 q^{77} +1.00000 q^{78} -12.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +11.0000 q^{83} +1.00000 q^{84} +8.00000 q^{86} +5.00000 q^{87} -3.00000 q^{88} +8.00000 q^{89} +1.00000 q^{91} -1.00000 q^{93} +5.00000 q^{94} -5.00000 q^{96} +10.0000 q^{97} +6.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	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	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.00000	0.267261
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	1.00000	0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	1.00000	0.188982
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	−5.00000	−0.883883
$33$	−1.00000	−0.174078
$34$	7.00000	1.20049
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	1.00000	0.154303
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	0	0
$47$	−5.00000	−0.729325	−0.364662	−	0.931140i	$-0.618816\pi$
$47$	−5.00000	−0.729325	−0.364662	+	0.931140i	$0.618816\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−7.00000	−0.980196
$52$	1.00000	0.138675
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−3.00000	−0.400892
$57$	0	0
$58$	−5.00000	−0.656532
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	1.00000	0.127000
$63$	−1.00000	−0.125988
$64$	7.00000	0.875000
$65$	0	0
$66$	1.00000	0.123091
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	7.00000	0.848875
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	3.00000	0.353553
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	0	0
$77$	1.00000	0.113961
$78$	1.00000	0.113228
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	11.0000	1.20741	0.603703	−	0.797209i	$-0.293691\pi$
$83$	11.0000	1.20741	0.603703	+	0.797209i	$0.293691\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	8.00000	0.862662
$87$	5.00000	0.536056
$88$	−3.00000	−0.319801
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−1.00000	−0.103695
$94$	5.00000	0.515711
$95$	0	0
$96$	−5.00000	−0.510310
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	6.00000	0.606092
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	7.00000	0.693103
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	1.00000	0.0971286
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	−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
$111$	−8.00000	−0.759326
$112$	1.00000	0.0944911
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	−5.00000	−0.464238
$117$	−1.00000	−0.0924500
$118$	3.00000	0.276172
$119$	7.00000	0.641689
$120$	0	0
$121$	−10.0000	−0.909091
$122$	7.00000	0.633750
$123$	6.00000	0.541002
$124$	1.00000	0.0898027
$125$	0	0
$126$	1.00000	0.0890871
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	3.00000	0.265165
$129$	−8.00000	−0.704361
$130$	0	0
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	7.00000	0.604708
$135$	0	0
$136$	−21.0000	−1.80074
$137$	4.00000	0.341743	0.170872	−	0.985293i	$-0.445342\pi$
$137$	4.00000	0.341743	0.170872	+	0.985293i	$0.445342\pi$
$138$	0	0
$139$	−18.0000	−1.52674	−0.763370	−	0.645961i	$-0.776457\pi$
$139$	−18.0000	−1.52674	−0.763370	+	0.645961i	$0.776457\pi$
$140$	0	0
$141$	−5.00000	−0.421076
$142$	0	0
$143$	1.00000	0.0836242
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	12.0000	0.993127
$147$	−6.00000	−0.494872
$148$	8.00000	0.657596
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	19.0000	1.54620	0.773099	−	0.634285i	$-0.218706\pi$
$151$	19.0000	1.54620	0.773099	+	0.634285i	$0.218706\pi$
$152$	0	0
$153$	−7.00000	−0.565916
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	1.00000	0.0800641
$157$	3.00000	0.239426	0.119713	−	0.992809i	$-0.461803\pi$
$157$	3.00000	0.239426	0.119713	+	0.992809i	$0.461803\pi$
$158$	12.0000	0.954669
$159$	−1.00000	−0.0793052
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−11.0000	−0.853766
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−3.00000	−0.231455
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	8.00000	0.609994
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	−5.00000	−0.379049
$175$	0	0
$176$	1.00000	0.0753778
$177$	−3.00000	−0.225494
$178$	−8.00000	−0.599625
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	−1.00000	−0.0741249
$183$	−7.00000	−0.517455
$184$	0	0
$185$	0	0
$186$	1.00000	0.0733236
$187$	7.00000	0.511891
$188$	5.00000	0.364662
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	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	$0.209146\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	6.00000	0.428571
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	1.00000	0.0710669
$199$	−22.0000	−1.55954	−0.779769	−	0.626067i	$-0.784664\pi$
$199$	−22.0000	−1.55954	−0.779769	+	0.626067i	$0.784664\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	3.00000	0.211079
$203$	−5.00000	−0.350931
$204$	7.00000	0.490098
$205$	0	0
$206$	10.0000	0.696733
$207$	0	0
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	1.00000	0.0686803
$213$	0	0
$214$	2.00000	0.136717
$215$	0	0
$216$	3.00000	0.204124
$217$	1.00000	0.0678844
$218$	2.00000	0.135457
$219$	−12.0000	−0.810885
$220$	0	0
$221$	7.00000	0.470871
$222$	8.00000	0.536925
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	5.00000	0.334077
$225$	0	0
$226$	14.0000	0.931266
$227$	25.0000	1.65931	0.829654	−	0.558278i	$-0.188538\pi$
$227$	25.0000	1.65931	0.829654	+	0.558278i	$0.188538\pi$
$228$	0	0
$229$	−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
$232$	15.0000	0.984798
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	3.00000	0.195283
$237$	−12.0000	−0.779484
$238$	−7.00000	−0.453743
$239$	21.0000	1.35838	0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000	1.35838	0.679189	+	0.733964i	$0.262332\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	10.0000	0.642824
$243$	1.00000	0.0641500
$244$	7.00000	0.448129
$245$	0	0
$246$	−6.00000	−0.382546
$247$	0	0
$248$	−3.00000	−0.190500
$249$	11.0000	0.697097
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	1.00000	0.0629941
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−1.00000	−0.0623783	−0.0311891	−	0.999514i	$-0.509929\pi$
$257$	−1.00000	−0.0623783	−0.0311891	+	0.999514i	$0.509929\pi$
$258$	8.00000	0.498058
$259$	8.00000	0.497096
$260$	0	0
$261$	5.00000	0.309492
$262$	−14.0000	−0.864923
$263$	26.0000	1.60323	0.801614	−	0.597841i	$-0.203975\pi$
$263$	26.0000	1.60323	0.801614	+	0.597841i	$0.203975\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	0	0
$267$	8.00000	0.489592
$268$	7.00000	0.427593
$269$	−3.00000	−0.182913	−0.0914566	−	0.995809i	$-0.529152\pi$
$269$	−3.00000	−0.182913	−0.0914566	+	0.995809i	$0.529152\pi$
$270$	0	0
$271$	23.0000	1.39715	0.698575	−	0.715537i	$-0.253818\pi$
$271$	23.0000	1.39715	0.698575	+	0.715537i	$0.253818\pi$
$272$	7.00000	0.424437
$273$	1.00000	0.0605228
$274$	−4.00000	−0.241649
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	18.0000	1.07957
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	5.00000	0.297746
$283$	−22.0000	−1.30776	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000	−1.30776	−0.653882	+	0.756596i	$0.726861\pi$
$284$	0	0
$285$	0	0
$286$	−1.00000	−0.0591312
$287$	−6.00000	−0.354169
$288$	−5.00000	−0.294628
$289$	32.0000	1.88235
$290$	0	0
$291$	10.0000	0.586210
$292$	12.0000	0.702247
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	6.00000	0.349927
$295$	0	0
$296$	−24.0000	−1.39497
$297$	−1.00000	−0.0580259
$298$	6.00000	0.347571
$299$	0	0
$300$	0	0
$301$	8.00000	0.461112
$302$	−19.0000	−1.09333
$303$	−3.00000	−0.172345
$304$	0	0
$305$	0	0
$306$	7.00000	0.400163
$307$	−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
$309$	−10.0000	−0.568880
$310$	0	0
$311$	−10.0000	−0.567048	−0.283524	−	0.958965i	$-0.591504\pi$
$311$	−10.0000	−0.567048	−0.283524	+	0.958965i	$0.591504\pi$
$312$	−3.00000	−0.169842
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	−3.00000	−0.169300
$315$	0	0
$316$	12.0000	0.675053
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	1.00000	0.0560772
$319$	−5.00000	−0.279946
$320$	0	0
$321$	−2.00000	−0.111629
$322$	0	0
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−12.0000	−0.664619
$327$	−2.00000	−0.110600
$328$	18.0000	0.993884
$329$	5.00000	0.275659
$330$	0	0
$331$	−36.0000	−1.97874	−0.989369	−	0.145424i	$-0.953545\pi$
$331$	−36.0000	−1.97874	−0.989369	+	0.145424i	$0.953545\pi$
$332$	−11.0000	−0.603703
$333$	−8.00000	−0.438397
$334$	12.0000	0.656611
$335$	0	0
$336$	1.00000	0.0545545
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	−1.00000	−0.0543928
$339$	−14.0000	−0.760376
$340$	0	0
$341$	1.00000	0.0541530
$342$	0	0
$343$	13.0000	0.701934
$344$	−24.0000	−1.29399
$345$	0	0
$346$	−9.00000	−0.483843
$347$	2.00000	0.107366	0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000	0.107366	0.0536828	+	0.998558i	$0.482904\pi$
$348$	−5.00000	−0.268028
$349$	24.0000	1.28469	0.642345	−	0.766415i	$-0.277962\pi$
$349$	24.0000	1.28469	0.642345	+	0.766415i	$0.277962\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	5.00000	0.266501
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	3.00000	0.159448
$355$	0	0
$356$	−8.00000	−0.423999
$357$	7.00000	0.370479
$358$	6.00000	0.317110
$359$	−1.00000	−0.0527780	−0.0263890	−	0.999652i	$-0.508401\pi$
$359$	−1.00000	−0.0527780	−0.0263890	+	0.999652i	$0.508401\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−5.00000	−0.262794
$363$	−10.0000	−0.524864
$364$	−1.00000	−0.0524142
$365$	0	0
$366$	7.00000	0.365896
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	1.00000	0.0519174
$372$	1.00000	0.0518476
$373$	31.0000	1.60512	0.802560	−	0.596572i	$-0.203471\pi$
$373$	31.0000	1.60512	0.802560	+	0.596572i	$0.203471\pi$
$374$	−7.00000	−0.361961
$375$	0	0
$376$	−15.0000	−0.773566
$377$	−5.00000	−0.257513
$378$	1.00000	0.0514344
$379$	1.00000	0.0513665	0.0256833	−	0.999670i	$-0.491824\pi$
$379$	1.00000	0.0513665	0.0256833	+	0.999670i	$0.491824\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	−18.0000	−0.920960
$383$	−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
$387$	−8.00000	−0.406663
$388$	−10.0000	−0.507673
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	0	0
$392$	−18.0000	−0.909137
$393$	14.0000	0.706207
$394$	18.0000	0.906827
$395$	0	0
$396$	1.00000	0.0502519
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	22.0000	1.10276
$399$	0	0
$400$	0	0
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	7.00000	0.349128
$403$	1.00000	0.0498135
$404$	3.00000	0.149256
$405$	0	0
$406$	5.00000	0.248146
$407$	8.00000	0.396545
$408$	−21.0000	−1.03965
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	4.00000	0.197305
$412$	10.0000	0.492665
$413$	3.00000	0.147620
$414$	0	0
$415$	0	0
$416$	5.00000	0.245145
$417$	−18.0000	−0.881464
$418$	0	0
$419$	14.0000	0.683945	0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000	0.683945	0.341972	+	0.939710i	$0.388905\pi$
$420$	0	0
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	0	0
$423$	−5.00000	−0.243108
$424$	−3.00000	−0.145693
$425$	0	0
$426$	0	0
$427$	7.00000	0.338754
$428$	2.00000	0.0966736
$429$	1.00000	0.0482805
$430$	0	0
$431$	20.0000	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$432$	−1.00000	−0.0481125
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	−1.00000	−0.0480015
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	12.0000	0.573382
$439$	12.0000	0.572729	0.286364	−	0.958121i	$-0.407553\pi$
$439$	12.0000	0.572729	0.286364	+	0.958121i	$0.407553\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	−7.00000	−0.332956
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	−16.0000	−0.757622
$447$	−6.00000	−0.283790
$448$	−7.00000	−0.330719
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	14.0000	0.658505
$453$	19.0000	0.892698
$454$	−25.0000	−1.17331
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	10.0000	0.467269
$459$	−7.00000	−0.326732
$460$	0	0
$461$	−38.0000	−1.76984	−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−38.0000	−1.76984	−0.884918	+	0.465746i	$0.845786\pi$
$462$	−1.00000	−0.0465242
$463$	21.0000	0.975953	0.487976	−	0.872857i	$-0.337735\pi$
$463$	21.0000	0.975953	0.487976	+	0.872857i	$0.337735\pi$
$464$	−5.00000	−0.232119
$465$	0	0
$466$	26.0000	1.20443
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	1.00000	0.0462250
$469$	7.00000	0.323230
$470$	0	0
$471$	3.00000	0.138233
$472$	−9.00000	−0.414259
$473$	8.00000	0.367840
$474$	12.0000	0.551178
$475$	0	0
$476$	−7.00000	−0.320844
$477$	−1.00000	−0.0457869
$478$	−21.0000	−0.960518
$479$	−39.0000	−1.78196	−0.890978	−	0.454047i	$-0.849980\pi$
$479$	−39.0000	−1.78196	−0.890978	+	0.454047i	$0.849980\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	−10.0000	−0.455488
$483$	0	0
$484$	10.0000	0.454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	23.0000	1.04223	0.521115	−	0.853487i	$-0.325516\pi$
$487$	23.0000	1.04223	0.521115	+	0.853487i	$0.325516\pi$
$488$	−21.0000	−0.950625
$489$	12.0000	0.542659
$490$	0	0
$491$	−42.0000	−1.89543	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000	−1.89543	−0.947717	+	0.319113i	$0.896615\pi$
$492$	−6.00000	−0.270501
$493$	−35.0000	−1.57632
$494$	0	0
$495$	0	0
$496$	1.00000	0.0449013
$497$	0	0
$498$	−11.0000	−0.492922
$499$	−29.0000	−1.29822	−0.649109	−	0.760695i	$-0.724858\pi$
$499$	−29.0000	−1.29822	−0.649109	+	0.760695i	$0.724858\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	−18.0000	−0.803379
$503$	−42.0000	−1.87269	−0.936344	−	0.351085i	$-0.885813\pi$
$503$	−42.0000	−1.87269	−0.936344	+	0.351085i	$0.885813\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	−16.0000	−0.709885
$509$	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
$511$	12.0000	0.530849
$512$	11.0000	0.486136
$513$	0	0
$514$	1.00000	0.0441081
$515$	0	0
$516$	8.00000	0.352180
$517$	5.00000	0.219900
$518$	−8.00000	−0.351500
$519$	9.00000	0.395056
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	−5.00000	−0.218844
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	−14.0000	−0.611593
$525$	0	0
$526$	−26.0000	−1.13365
$527$	7.00000	0.304925
$528$	1.00000	0.0435194
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−3.00000	−0.130189
$532$	0	0
$533$	−6.00000	−0.259889
$534$	−8.00000	−0.346194
$535$	0	0
$536$	−21.0000	−0.907062
$537$	−6.00000	−0.258919
$538$	3.00000	0.129339
$539$	6.00000	0.258438
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	−23.0000	−0.987935
$543$	5.00000	0.214571
$544$	35.0000	1.50061
$545$	0	0
$546$	−1.00000	−0.0427960
$547$	6.00000	0.256541	0.128271	−	0.991739i	$-0.459057\pi$
$547$	6.00000	0.256541	0.128271	+	0.991739i	$0.459057\pi$
$548$	−4.00000	−0.170872
$549$	−7.00000	−0.298753
$550$	0	0
$551$	0	0
$552$	0	0
$553$	12.0000	0.510292
$554$	−10.0000	−0.424859
$555$	0	0
$556$	18.0000	0.763370
$557$	−38.0000	−1.61011	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000	−1.61011	−0.805056	+	0.593199i	$0.797865\pi$
$558$	1.00000	0.0423334
$559$	8.00000	0.338364
$560$	0	0
$561$	7.00000	0.295540
$562$	−30.0000	−1.26547
$563$	32.0000	1.34864	0.674320	−	0.738440i	$-0.264437\pi$
$563$	32.0000	1.34864	0.674320	+	0.738440i	$0.264437\pi$
$564$	5.00000	0.210538
$565$	0	0
$566$	22.0000	0.924729
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	19.0000	0.796521	0.398261	−	0.917272i	$-0.369614\pi$
$569$	19.0000	0.796521	0.398261	+	0.917272i	$0.369614\pi$
$570$	0	0
$571$	−24.0000	−1.00437	−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000	−1.00437	−0.502184	+	0.864761i	$0.667470\pi$
$572$	−1.00000	−0.0418121
$573$	18.0000	0.751961
$574$	6.00000	0.250435
$575$	0	0
$576$	7.00000	0.291667
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	−32.0000	−1.33102
$579$	22.0000	0.914289
$580$	0	0
$581$	−11.0000	−0.456357
$582$	−10.0000	−0.414513
$583$	1.00000	0.0414158
$584$	−36.0000	−1.48969
$585$	0	0
$586$	14.0000	0.578335
$587$	37.0000	1.52715	0.763577	−	0.645717i	$-0.223441\pi$
$587$	37.0000	1.52715	0.763577	+	0.645717i	$0.223441\pi$
$588$	6.00000	0.247436
$589$	0	0
$590$	0	0
$591$	−18.0000	−0.740421
$592$	8.00000	0.328798
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	6.00000	0.245770
$597$	−22.0000	−0.900400
$598$	0	0
$599$	−10.0000	−0.408589	−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000	−0.408589	−0.204294	+	0.978909i	$0.565490\pi$
$600$	0	0
$601$	37.0000	1.50926	0.754631	−	0.656150i	$-0.227816\pi$
$601$	37.0000	1.50926	0.754631	+	0.656150i	$0.227816\pi$
$602$	−8.00000	−0.326056
$603$	−7.00000	−0.285062
$604$	−19.0000	−0.773099
$605$	0	0
$606$	3.00000	0.121867
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	0	0
$609$	−5.00000	−0.202610
$610$	0	0
$611$	5.00000	0.202278
$612$	7.00000	0.282958
$613$	42.0000	1.69636	0.848182	−	0.529705i	$-0.177697\pi$
$613$	42.0000	1.69636	0.848182	+	0.529705i	$0.177697\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	3.00000	0.120873
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	10.0000	0.402259
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	0	0
$622$	10.0000	0.400963
$623$	−8.00000	−0.320513
$624$	1.00000	0.0400320
$625$	0	0
$626$	19.0000	0.759393
$627$	0	0
$628$	−3.00000	−0.119713
$629$	56.0000	2.23287
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	−36.0000	−1.43200
$633$	0	0
$634$	−24.0000	−0.953162
$635$	0	0
$636$	1.00000	0.0396526
$637$	6.00000	0.237729
$638$	5.00000	0.197952
$639$	0	0
$640$	0	0
$641$	−43.0000	−1.69840	−0.849199	−	0.528073i	$-0.822915\pi$
$641$	−43.0000	−1.69840	−0.849199	+	0.528073i	$0.822915\pi$
$642$	2.00000	0.0789337
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	3.00000	0.117851
$649$	3.00000	0.117760
$650$	0	0
$651$	1.00000	0.0391931
$652$	−12.0000	−0.469956
$653$	9.00000	0.352197	0.176099	−	0.984373i	$-0.443652\pi$
$653$	9.00000	0.352197	0.176099	+	0.984373i	$0.443652\pi$
$654$	2.00000	0.0782062
$655$	0	0
$656$	−6.00000	−0.234261
$657$	−12.0000	−0.468165
$658$	−5.00000	−0.194920
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	36.0000	1.39918
$663$	7.00000	0.271857
$664$	33.0000	1.28065
$665$	0	0
$666$	8.00000	0.309994
$667$	0	0
$668$	12.0000	0.464294
$669$	16.0000	0.618596
$670$	0	0
$671$	7.00000	0.270232
$672$	5.00000	0.192879
$673$	−21.0000	−0.809491	−0.404745	−	0.914429i	$-0.632640\pi$
$673$	−21.0000	−0.809491	−0.404745	+	0.914429i	$0.632640\pi$
$674$	13.0000	0.500741
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	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
$679$	−10.0000	−0.383765
$680$	0	0
$681$	25.0000	0.958002
$682$	−1.00000	−0.0382920
$683$	7.00000	0.267848	0.133924	−	0.990992i	$-0.457242\pi$
$683$	7.00000	0.267848	0.133924	+	0.990992i	$0.457242\pi$
$684$	0	0
$685$	0	0
$686$	−13.0000	−0.496342
$687$	−10.0000	−0.381524
$688$	8.00000	0.304997
$689$	1.00000	0.0380970
$690$	0	0
$691$	31.0000	1.17930	0.589648	−	0.807661i	$-0.299267\pi$
$691$	31.0000	1.17930	0.589648	+	0.807661i	$0.299267\pi$
$692$	−9.00000	−0.342129
$693$	1.00000	0.0379869
$694$	−2.00000	−0.0759190
$695$	0	0
$696$	15.0000	0.568574
$697$	−42.0000	−1.59086
$698$	−24.0000	−0.908413
$699$	−26.0000	−0.983410
$700$	0	0
$701$	−29.0000	−1.09531	−0.547657	−	0.836703i	$-0.684480\pi$
$701$	−29.0000	−1.09531	−0.547657	+	0.836703i	$0.684480\pi$
$702$	1.00000	0.0377426
$703$	0	0
$704$	−7.00000	−0.263822
$705$	0	0
$706$	18.0000	0.677439
$707$	3.00000	0.112827
$708$	3.00000	0.112747
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	−12.0000	−0.450035
$712$	24.0000	0.899438
$713$	0	0
$714$	−7.00000	−0.261968
$715$	0	0
$716$	6.00000	0.224231
$717$	21.0000	0.784259
$718$	1.00000	0.0373197
$719$	46.0000	1.71551	0.857755	−	0.514058i	$-0.171858\pi$
$719$	46.0000	1.71551	0.857755	+	0.514058i	$0.171858\pi$
$720$	0	0
$721$	10.0000	0.372419
$722$	19.0000	0.707107
$723$	10.0000	0.371904
$724$	−5.00000	−0.185824
$725$	0	0
$726$	10.0000	0.371135
$727$	42.0000	1.55769	0.778847	−	0.627214i	$-0.215805\pi$
$727$	42.0000	1.55769	0.778847	+	0.627214i	$0.215805\pi$
$728$	3.00000	0.111187
$729$	1.00000	0.0370370
$730$	0	0
$731$	56.0000	2.07123
$732$	7.00000	0.258727
$733$	−20.0000	−0.738717	−0.369358	−	0.929287i	$-0.620423\pi$
$733$	−20.0000	−0.738717	−0.369358	+	0.929287i	$0.620423\pi$
$734$	−4.00000	−0.147643
$735$	0	0
$736$	0	0
$737$	7.00000	0.257848
$738$	−6.00000	−0.220863
$739$	27.0000	0.993211	0.496606	−	0.867976i	$-0.334580\pi$
$739$	27.0000	0.993211	0.496606	+	0.867976i	$0.334580\pi$
$740$	0	0
$741$	0	0
$742$	−1.00000	−0.0367112
$743$	27.0000	0.990534	0.495267	−	0.868741i	$-0.335070\pi$
$743$	27.0000	0.990534	0.495267	+	0.868741i	$0.335070\pi$
$744$	−3.00000	−0.109985
$745$	0	0
$746$	−31.0000	−1.13499
$747$	11.0000	0.402469
$748$	−7.00000	−0.255945
$749$	2.00000	0.0730784
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	5.00000	0.182331
$753$	18.0000	0.655956
$754$	5.00000	0.182089
$755$	0	0
$756$	1.00000	0.0363696
$757$	−7.00000	−0.254419	−0.127210	−	0.991876i	$-0.540602\pi$
$757$	−7.00000	−0.254419	−0.127210	+	0.991876i	$0.540602\pi$
$758$	−1.00000	−0.0363216
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	−16.0000	−0.579619
$763$	2.00000	0.0724049
$764$	−18.0000	−0.651217
$765$	0	0
$766$	24.0000	0.867155
$767$	3.00000	0.108324
$768$	−17.0000	−0.613435
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	−1.00000	−0.0360141
$772$	−22.0000	−0.791797
$773$	−50.0000	−1.79838	−0.899188	−	0.437564i	$-0.855842\pi$
$773$	−50.0000	−1.79838	−0.899188	+	0.437564i	$0.855842\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	30.0000	1.07694
$777$	8.00000	0.286998
$778$	−2.00000	−0.0717035
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	5.00000	0.178685
$784$	6.00000	0.214286
$785$	0	0
$786$	−14.0000	−0.499363
$787$	−13.0000	−0.463400	−0.231700	−	0.972787i	$-0.574429\pi$
$787$	−13.0000	−0.463400	−0.231700	+	0.972787i	$0.574429\pi$
$788$	18.0000	0.641223
$789$	26.0000	0.925625
$790$	0	0
$791$	14.0000	0.497783
$792$	−3.00000	−0.106600
$793$	7.00000	0.248577
$794$	18.0000	0.638796
$795$	0	0
$796$	22.0000	0.779769
$797$	39.0000	1.38145	0.690725	−	0.723117i	$-0.257291\pi$
$797$	39.0000	1.38145	0.690725	+	0.723117i	$0.257291\pi$
$798$	0	0
$799$	35.0000	1.23821
$800$	0	0
$801$	8.00000	0.282666
$802$	8.00000	0.282490
$803$	12.0000	0.423471
$804$	7.00000	0.246871
$805$	0	0
$806$	−1.00000	−0.0352235
$807$	−3.00000	−0.105605
$808$	−9.00000	−0.316619
$809$	−50.0000	−1.75791	−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000	−1.75791	−0.878953	+	0.476908i	$0.841757\pi$
$810$	0	0
$811$	3.00000	0.105344	0.0526721	−	0.998612i	$-0.483226\pi$
$811$	3.00000	0.105344	0.0526721	+	0.998612i	$0.483226\pi$
$812$	5.00000	0.175466
$813$	23.0000	0.806645
$814$	−8.00000	−0.280400
$815$	0	0
$816$	7.00000	0.245049
$817$	0	0
$818$	6.00000	0.209785
$819$	1.00000	0.0349428
$820$	0	0
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	−4.00000	−0.139516
$823$	20.0000	0.697156	0.348578	−	0.937280i	$-0.386665\pi$
$823$	20.0000	0.697156	0.348578	+	0.937280i	$0.386665\pi$
$824$	−30.0000	−1.04510
$825$	0	0
$826$	−3.00000	−0.104383
$827$	31.0000	1.07798	0.538988	−	0.842314i	$-0.318807\pi$
$827$	31.0000	1.07798	0.538988	+	0.842314i	$0.318807\pi$
$828$	0	0
$829$	−29.0000	−1.00721	−0.503606	−	0.863934i	$-0.667994\pi$
$829$	−29.0000	−1.00721	−0.503606	+	0.863934i	$0.667994\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	−7.00000	−0.242681
$833$	42.0000	1.45521
$834$	18.0000	0.623289
$835$	0	0
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	−14.0000	−0.483622
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	20.0000	0.689246
$843$	30.0000	1.03325
$844$	0	0
$845$	0	0
$846$	5.00000	0.171904
$847$	10.0000	0.343604
$848$	1.00000	0.0343401
$849$	−22.0000	−0.755038
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	−7.00000	−0.239535
$855$	0	0
$856$	−6.00000	−0.205076
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	−1.00000	−0.0341394
$859$	−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
$862$	−20.0000	−0.681203
$863$	39.0000	1.32758	0.663788	−	0.747921i	$-0.268948\pi$
$863$	39.0000	1.32758	0.663788	+	0.747921i	$0.268948\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	14.0000	0.475739
$867$	32.0000	1.08678
$868$	−1.00000	−0.0339422
$869$	12.0000	0.407072
$870$	0	0
$871$	7.00000	0.237186
$872$	−6.00000	−0.203186
$873$	10.0000	0.338449
$874$	0	0
$875$	0	0
$876$	12.0000	0.405442
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	−12.0000	−0.404980
$879$	−14.0000	−0.472208
$880$	0	0
$881$	−37.0000	−1.24656	−0.623281	−	0.781998i	$-0.714201\pi$
$881$	−37.0000	−1.24656	−0.623281	+	0.781998i	$0.714201\pi$
$882$	6.00000	0.202031
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	−7.00000	−0.235435
$885$	0	0
$886$	16.0000	0.537531
$887$	28.0000	0.940148	0.470074	−	0.882627i	$-0.344227\pi$
$887$	28.0000	0.940148	0.470074	+	0.882627i	$0.344227\pi$
$888$	−24.0000	−0.805387
$889$	−16.0000	−0.536623
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	−16.0000	−0.535720
$893$	0	0
$894$	6.00000	0.200670
$895$	0	0
$896$	−3.00000	−0.100223
$897$	0	0
$898$	−12.0000	−0.400445
$899$	−5.00000	−0.166759
$900$	0	0
$901$	7.00000	0.233204
$902$	6.00000	0.199778
$903$	8.00000	0.266223
$904$	−42.0000	−1.39690
$905$	0	0
$906$	−19.0000	−0.631233
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	−25.0000	−0.829654
$909$	−3.00000	−0.0995037
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−11.0000	−0.364047
$914$	22.0000	0.727695
$915$	0	0
$916$	10.0000	0.330409
$917$	−14.0000	−0.462321
$918$	7.00000	0.231034
$919$	−58.0000	−1.91324	−0.956622	−	0.291333i	$-0.905901\pi$
$919$	−58.0000	−1.91324	−0.956622	+	0.291333i	$0.905901\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	38.0000	1.25146
$923$	0	0
$924$	−1.00000	−0.0328976
$925$	0	0
$926$	−21.0000	−0.690103
$927$	−10.0000	−0.328443
$928$	−25.0000	−0.820665
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	0	0
$932$	26.0000	0.851658
$933$	−10.0000	−0.327385
$934$	−8.00000	−0.261768
$935$	0	0
$936$	−3.00000	−0.0980581
$937$	−21.0000	−0.686040	−0.343020	−	0.939328i	$-0.611450\pi$
$937$	−21.0000	−0.686040	−0.343020	+	0.939328i	$0.611450\pi$
$938$	−7.00000	−0.228558
$939$	−19.0000	−0.620042
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	−3.00000	−0.0977453
$943$	0	0
$944$	3.00000	0.0976417
$945$	0	0
$946$	−8.00000	−0.260102
$947$	−45.0000	−1.46230	−0.731152	−	0.682215i	$-0.761017\pi$
$947$	−45.0000	−1.46230	−0.731152	+	0.682215i	$0.761017\pi$
$948$	12.0000	0.389742
$949$	12.0000	0.389536
$950$	0	0
$951$	24.0000	0.778253
$952$	21.0000	0.680614
$953$	15.0000	0.485898	0.242949	−	0.970039i	$-0.421885\pi$
$953$	15.0000	0.485898	0.242949	+	0.970039i	$0.421885\pi$
$954$	1.00000	0.0323762
$955$	0	0
$956$	−21.0000	−0.679189
$957$	−5.00000	−0.161627
$958$	39.0000	1.26003
$959$	−4.00000	−0.129167
$960$	0	0
$961$	−30.0000	−0.967742
$962$	−8.00000	−0.257930
$963$	−2.00000	−0.0644491
$964$	−10.0000	−0.322078
$965$	0	0
$966$	0	0
$967$	7.00000	0.225105	0.112552	−	0.993646i	$-0.464097\pi$
$967$	7.00000	0.225105	0.112552	+	0.993646i	$0.464097\pi$
$968$	−30.0000	−0.964237
$969$	0	0
$970$	0	0
$971$	−10.0000	−0.320915	−0.160458	−	0.987043i	$-0.551297\pi$
$971$	−10.0000	−0.320915	−0.160458	+	0.987043i	$0.551297\pi$
$972$	−1.00000	−0.0320750
$973$	18.0000	0.577054
$974$	−23.0000	−0.736968
$975$	0	0
$976$	7.00000	0.224065
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	−12.0000	−0.383718
$979$	−8.00000	−0.255681
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	42.0000	1.34027
$983$	29.0000	0.924956	0.462478	−	0.886631i	$-0.346960\pi$
$983$	29.0000	0.924956	0.462478	+	0.886631i	$0.346960\pi$
$984$	18.0000	0.573819
$985$	0	0
$986$	35.0000	1.11463
$987$	5.00000	0.159152
$988$	0	0
$989$	0	0
$990$	0	0
$991$	36.0000	1.14358	0.571789	−	0.820401i	$-0.306250\pi$
$991$	36.0000	1.14358	0.571789	+	0.820401i	$0.306250\pi$
$992$	5.00000	0.158750
$993$	−36.0000	−1.14243
$994$	0	0
$995$	0	0
$996$	−11.0000	−0.348548
$997$	−19.0000	−0.601736	−0.300868	−	0.953666i	$-0.597276\pi$
$997$	−19.0000	−0.601736	−0.300868	+	0.953666i	$0.597276\pi$
$998$	29.0000	0.917979
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.a.e.1.1	✓	1
3.2	odd	2		2925.2.a.n.1.1		1
5.2	odd	4		975.2.c.d.274.1		2
5.3	odd	4		975.2.c.d.274.2		2
5.4	even	2		975.2.a.j.1.1	yes	1
15.2	even	4		2925.2.c.j.2224.2		2
15.8	even	4		2925.2.c.j.2224.1		2
15.14	odd	2		2925.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.2.a.e.1.1	✓	1	1.1	even	1	trivial
975.2.a.j.1.1	yes	1	5.4	even	2
975.2.c.d.274.1		2	5.2	odd	4
975.2.c.d.274.2		2	5.3	odd	4
2925.2.a.e.1.1		1	15.14	odd	2
2925.2.a.n.1.1		1	3.2	odd	2
2925.2.c.j.2224.1		2	15.8	even	4
2925.2.c.j.2224.2		2	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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