LMFDB - Embedded newform 1075.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1075,2,Mod(1,1075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1075, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1075.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:	$8.58391821729$
Analytic rank:	$0$
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	$\chi$	$=$	1075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} -2.00000 q^{6} -4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} -2.00000 q^{6} -4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} +2.00000 q^{12} -1.00000 q^{13} -4.00000 q^{14} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +8.00000 q^{21} +3.00000 q^{22} -3.00000 q^{23} +6.00000 q^{24} -1.00000 q^{26} +4.00000 q^{27} +4.00000 q^{28} +1.00000 q^{31} +5.00000 q^{32} -6.00000 q^{33} +2.00000 q^{34} -1.00000 q^{36} +10.0000 q^{37} +2.00000 q^{39} +9.00000 q^{41} +8.00000 q^{42} -1.00000 q^{43} -3.00000 q^{44} -3.00000 q^{46} -5.00000 q^{47} +2.00000 q^{48} +9.00000 q^{49} -4.00000 q^{51} +1.00000 q^{52} -1.00000 q^{53} +4.00000 q^{54} +12.0000 q^{56} +1.00000 q^{59} +1.00000 q^{62} -4.00000 q^{63} +7.00000 q^{64} -6.00000 q^{66} -12.0000 q^{67} -2.00000 q^{68} +6.00000 q^{69} +6.00000 q^{71} -3.00000 q^{72} +14.0000 q^{73} +10.0000 q^{74} -12.0000 q^{77} +2.00000 q^{78} +13.0000 q^{79} -11.0000 q^{81} +9.00000 q^{82} -9.00000 q^{83} -8.00000 q^{84} -1.00000 q^{86} -9.00000 q^{88} -8.00000 q^{89} +4.00000 q^{91} +3.00000 q^{92} -2.00000 q^{93} -5.00000 q^{94} -10.0000 q^{96} -9.00000 q^{97} +9.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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	$0.384973\pi$
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	2.00000	0.577350
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	1.00000	0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	8.00000	1.74574
$22$	3.00000	0.639602
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	6.00000	1.22474
$25$	0	0
$26$	−1.00000	−0.196116
$27$	4.00000	0.769800
$28$	4.00000	0.755929
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	5.00000	0.883883
$33$	−6.00000	−1.04447
$34$	2.00000	0.342997
$35$	0	0
$36$	−1.00000	−0.166667
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	8.00000	1.23443
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−3.00000	−0.442326
$47$	−5.00000	−0.729325	−0.364662	−	0.931140i	$-0.618816\pi$
$47$	−5.00000	−0.729325	−0.364662	+	0.931140i	$0.618816\pi$
$48$	2.00000	0.288675
$49$	9.00000	1.28571
$50$	0	0
$51$	−4.00000	−0.560112
$52$	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	$0.521879\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	12.0000	1.60357
$57$	0	0
$58$	0	0
$59$	1.00000	0.130189	0.0650945	−	0.997879i	$-0.479265\pi$
$59$	1.00000	0.130189	0.0650945	+	0.997879i	$0.479265\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	1.00000	0.127000
$63$	−4.00000	−0.503953
$64$	7.00000	0.875000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−2.00000	−0.242536
$69$	6.00000	0.722315
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	−3.00000	−0.353553
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	0	0
$77$	−12.0000	−1.36753
$78$	2.00000	0.226455
$79$	13.0000	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	9.00000	0.993884
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	−8.00000	−0.872872
$85$	0	0
$86$	−1.00000	−0.107833
$87$	0	0
$88$	−9.00000	−0.959403
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	3.00000	0.312772
$93$	−2.00000	−0.207390
$94$	−5.00000	−0.515711
$95$	0	0
$96$	−10.0000	−1.02062
$97$	−9.00000	−0.913812	−0.456906	−	0.889515i	$-0.651042\pi$
$97$	−9.00000	−0.913812	−0.456906	+	0.889515i	$0.651042\pi$
$98$	9.00000	0.909137
$99$	3.00000	0.301511
$100$	0	0
$101$	−13.0000	−1.29355	−0.646774	−	0.762682i	$-0.723882\pi$
$101$	−13.0000	−1.29355	−0.646774	+	0.762682i	$0.723882\pi$
$102$	−4.00000	−0.396059
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	−9.00000	−0.870063	−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000	−0.870063	−0.435031	+	0.900415i	$0.643263\pi$
$108$	−4.00000	−0.384900
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	0	0
$111$	−20.0000	−1.89832
$112$	4.00000	0.377964
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	1.00000	0.0920575
$119$	−8.00000	−0.733359
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−18.0000	−1.62301
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	−4.00000	−0.356348
$127$	13.0000	1.15356	0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000	1.15356	0.576782	+	0.816898i	$0.304308\pi$
$128$	−3.00000	−0.265165
$129$	2.00000	0.176090
$130$	0	0
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	6.00000	0.522233
$133$	0	0
$134$	−12.0000	−1.03664
$135$	0	0
$136$	−6.00000	−0.514496
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	6.00000	0.510754
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	10.0000	0.842152
$142$	6.00000	0.503509
$143$	−3.00000	−0.250873
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	14.0000	1.15865
$147$	−18.0000	−1.48461
$148$	−10.0000	−0.821995
$149$	24.0000	1.96616	0.983078	−	0.183186i	$-0.0586410\pi$
$149$	24.0000	1.96616	0.983078	+	0.183186i	$0.0586410\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	−12.0000	−0.966988
$155$	0	0
$156$	−2.00000	−0.160128
$157$	20.0000	1.59617	0.798087	−	0.602542i	$-0.205846\pi$
$157$	20.0000	1.59617	0.798087	+	0.602542i	$0.205846\pi$
$158$	13.0000	1.03422
$159$	2.00000	0.158610
$160$	0	0
$161$	12.0000	0.945732
$162$	−11.0000	−0.864242
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	−9.00000	−0.702782
$165$	0	0
$166$	−9.00000	−0.698535
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	−24.0000	−1.85164
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	1.00000	0.0762493
$173$	17.0000	1.29249	0.646243	−	0.763132i	$-0.276339\pi$
$173$	17.0000	1.29249	0.646243	+	0.763132i	$0.276339\pi$
$174$	0	0
$175$	0	0
$176$	−3.00000	−0.226134
$177$	−2.00000	−0.150329
$178$	−8.00000	−0.599625
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	4.00000	0.296500
$183$	0	0
$184$	9.00000	0.663489
$185$	0	0
$186$	−2.00000	−0.146647
$187$	6.00000	0.438763
$188$	5.00000	0.364662
$189$	−16.0000	−1.16383
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−14.0000	−1.01036
$193$	7.00000	0.503871	0.251936	−	0.967744i	$-0.418933\pi$
$193$	7.00000	0.503871	0.251936	+	0.967744i	$0.418933\pi$
$194$	−9.00000	−0.646162
$195$	0	0
$196$	−9.00000	−0.642857
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	3.00000	0.213201
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	24.0000	1.69283
$202$	−13.0000	−0.914677
$203$	0	0
$204$	4.00000	0.280056
$205$	0	0
$206$	16.0000	1.11477
$207$	−3.00000	−0.208514
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	−26.0000	−1.78991	−0.894957	−	0.446153i	$-0.852794\pi$
$211$	−26.0000	−1.78991	−0.894957	+	0.446153i	$0.852794\pi$
$212$	1.00000	0.0686803
$213$	−12.0000	−0.822226
$214$	−9.00000	−0.615227
$215$	0	0
$216$	−12.0000	−0.816497
$217$	−4.00000	−0.271538
$218$	−1.00000	−0.0677285
$219$	−28.0000	−1.89206
$220$	0	0
$221$	−2.00000	−0.134535
$222$	−20.0000	−1.34231
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	−20.0000	−1.33631
$225$	0	0
$226$	12.0000	0.798228
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	−11.0000	−0.726900	−0.363450	−	0.931614i	$-0.618401\pi$
$229$	−11.0000	−0.726900	−0.363450	+	0.931614i	$0.618401\pi$
$230$	0	0
$231$	24.0000	1.57908
$232$	0	0
$233$	−4.00000	−0.262049	−0.131024	−	0.991379i	$-0.541827\pi$
$233$	−4.00000	−0.262049	−0.131024	+	0.991379i	$0.541827\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	−1.00000	−0.0650945
$237$	−26.0000	−1.68888
$238$	−8.00000	−0.518563
$239$	−5.00000	−0.323423	−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000	−0.323423	−0.161712	+	0.986838i	$0.551701\pi$
$240$	0	0
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	−2.00000	−0.128565
$243$	10.0000	0.641500
$244$	0	0
$245$	0	0
$246$	−18.0000	−1.14764
$247$	0	0
$248$	−3.00000	−0.190500
$249$	18.0000	1.14070
$250$	0	0
$251$	11.0000	0.694314	0.347157	−	0.937807i	$-0.387147\pi$
$251$	11.0000	0.694314	0.347157	+	0.937807i	$0.387147\pi$
$252$	4.00000	0.251976
$253$	−9.00000	−0.565825
$254$	13.0000	0.815693
$255$	0	0
$256$	−17.0000	−1.06250
$257$	20.0000	1.24757	0.623783	−	0.781598i	$-0.285595\pi$
$257$	20.0000	1.24757	0.623783	+	0.781598i	$0.285595\pi$
$258$	2.00000	0.124515
$259$	−40.0000	−2.48548
$260$	0	0
$261$	0	0
$262$	20.0000	1.23560
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	18.0000	1.10782
$265$	0	0
$266$	0	0
$267$	16.0000	0.979184
$268$	12.0000	0.733017
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	−2.00000	−0.121268
$273$	−8.00000	−0.484182
$274$	12.0000	0.724947
$275$	0	0
$276$	−6.00000	−0.361158
$277$	30.0000	1.80253	0.901263	−	0.433273i	$-0.142641\pi$
$277$	30.0000	1.80253	0.901263	+	0.433273i	$0.142641\pi$
$278$	5.00000	0.299880
$279$	1.00000	0.0598684
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	10.0000	0.595491
$283$	11.0000	0.653882	0.326941	−	0.945045i	$-0.393982\pi$
$283$	11.0000	0.653882	0.326941	+	0.945045i	$0.393982\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−3.00000	−0.177394
$287$	−36.0000	−2.12501
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	0	0
$291$	18.0000	1.05518
$292$	−14.0000	−0.819288
$293$	31.0000	1.81104	0.905520	−	0.424304i	$-0.139481\pi$
$293$	31.0000	1.81104	0.905520	+	0.424304i	$0.139481\pi$
$294$	−18.0000	−1.04978
$295$	0	0
$296$	−30.0000	−1.74371
$297$	12.0000	0.696311
$298$	24.0000	1.39028
$299$	3.00000	0.173494
$300$	0	0
$301$	4.00000	0.230556
$302$	2.00000	0.115087
$303$	26.0000	1.49366
$304$	0	0
$305$	0	0
$306$	2.00000	0.114332
$307$	9.00000	0.513657	0.256829	−	0.966457i	$-0.417322\pi$
$307$	9.00000	0.513657	0.256829	+	0.966457i	$0.417322\pi$
$308$	12.0000	0.683763
$309$	−32.0000	−1.82042
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	−6.00000	−0.339683
$313$	28.0000	1.58265	0.791327	−	0.611393i	$-0.209391\pi$
$313$	28.0000	1.58265	0.791327	+	0.611393i	$0.209391\pi$
$314$	20.0000	1.12867
$315$	0	0
$316$	−13.0000	−0.731307
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	2.00000	0.112154
$319$	0	0
$320$	0	0
$321$	18.0000	1.00466
$322$	12.0000	0.668734
$323$	0	0
$324$	11.0000	0.611111
$325$	0	0
$326$	−14.0000	−0.775388
$327$	2.00000	0.110600
$328$	−27.0000	−1.49083
$329$	20.0000	1.10264
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	9.00000	0.493939
$333$	10.0000	0.547997
$334$	−24.0000	−1.31322
$335$	0	0
$336$	−8.00000	−0.436436
$337$	−27.0000	−1.47078	−0.735392	−	0.677642i	$-0.763002\pi$
$337$	−27.0000	−1.47078	−0.735392	+	0.677642i	$0.763002\pi$
$338$	−12.0000	−0.652714
$339$	−24.0000	−1.30350
$340$	0	0
$341$	3.00000	0.162459
$342$	0	0
$343$	−8.00000	−0.431959
$344$	3.00000	0.161749
$345$	0	0
$346$	17.0000	0.913926
$347$	32.0000	1.71785	0.858925	−	0.512101i	$-0.171133\pi$
$347$	32.0000	1.71785	0.858925	+	0.512101i	$0.171133\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	15.0000	0.799503
$353$	−3.00000	−0.159674	−0.0798369	−	0.996808i	$-0.525440\pi$
$353$	−3.00000	−0.159674	−0.0798369	+	0.996808i	$0.525440\pi$
$354$	−2.00000	−0.106299
$355$	0	0
$356$	8.00000	0.423999
$357$	16.0000	0.846810
$358$	−16.0000	−0.845626
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	5.00000	0.262794
$363$	4.00000	0.209946
$364$	−4.00000	−0.209657
$365$	0	0
$366$	0	0
$367$	−11.0000	−0.574195	−0.287098	−	0.957901i	$-0.592690\pi$
$367$	−11.0000	−0.574195	−0.287098	+	0.957901i	$0.592690\pi$
$368$	3.00000	0.156386
$369$	9.00000	0.468521
$370$	0	0
$371$	4.00000	0.207670
$372$	2.00000	0.103695
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	15.0000	0.773566
$377$	0	0
$378$	−16.0000	−0.822951
$379$	−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
$381$	−26.0000	−1.33202
$382$	−12.0000	−0.613973
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	6.00000	0.306186
$385$	0	0
$386$	7.00000	0.356291
$387$	−1.00000	−0.0508329
$388$	9.00000	0.456906
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	−27.0000	−1.36371
$393$	−40.0000	−2.01773
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−3.00000	−0.150756
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	24.0000	1.19701
$403$	−1.00000	−0.0498135
$404$	13.0000	0.646774
$405$	0	0
$406$	0	0
$407$	30.0000	1.48704
$408$	12.0000	0.594089
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	−24.0000	−1.18383
$412$	−16.0000	−0.788263
$413$	−4.00000	−0.196827
$414$	−3.00000	−0.147442
$415$	0	0
$416$	−5.00000	−0.245145
$417$	−10.0000	−0.489702
$418$	0	0
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	−26.0000	−1.26566
$423$	−5.00000	−0.243108
$424$	3.00000	0.145693
$425$	0	0
$426$	−12.0000	−0.581402
$427$	0	0
$428$	9.00000	0.435031
$429$	6.00000	0.289683
$430$	0	0
$431$	15.0000	0.722525	0.361262	−	0.932464i	$-0.382346\pi$
$431$	15.0000	0.722525	0.361262	+	0.932464i	$0.382346\pi$
$432$	−4.00000	−0.192450
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	1.00000	0.0478913
$437$	0	0
$438$	−28.0000	−1.33789
$439$	−11.0000	−0.525001	−0.262501	−	0.964932i	$-0.584547\pi$
$439$	−11.0000	−0.525001	−0.262501	+	0.964932i	$0.584547\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	−2.00000	−0.0951303
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	20.0000	0.949158
$445$	0	0
$446$	14.0000	0.662919
$447$	−48.0000	−2.27032
$448$	−28.0000	−1.32288
$449$	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
$451$	27.0000	1.27138
$452$	−12.0000	−0.564433
$453$	−4.00000	−0.187936
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	−11.0000	−0.513996
$459$	8.00000	0.373408
$460$	0	0
$461$	25.0000	1.16437	0.582183	−	0.813058i	$-0.302199\pi$
$461$	25.0000	1.16437	0.582183	+	0.813058i	$0.302199\pi$
$462$	24.0000	1.11658
$463$	30.0000	1.39422	0.697109	−	0.716965i	$-0.254469\pi$
$463$	30.0000	1.39422	0.697109	+	0.716965i	$0.254469\pi$
$464$	0	0
$465$	0	0
$466$	−4.00000	−0.185296
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	1.00000	0.0462250
$469$	48.0000	2.21643
$470$	0	0
$471$	−40.0000	−1.84310
$472$	−3.00000	−0.138086
$473$	−3.00000	−0.137940
$474$	−26.0000	−1.19422
$475$	0	0
$476$	8.00000	0.366679
$477$	−1.00000	−0.0457869
$478$	−5.00000	−0.228695
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−10.0000	−0.455961
$482$	−12.0000	−0.546585
$483$	−24.0000	−1.09204
$484$	2.00000	0.0909091
$485$	0	0
$486$	10.0000	0.453609
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	0	0
$489$	28.0000	1.26620
$490$	0	0
$491$	26.0000	1.17336	0.586682	−	0.809818i	$-0.300434\pi$
$491$	26.0000	1.17336	0.586682	+	0.809818i	$0.300434\pi$
$492$	18.0000	0.811503
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	−24.0000	−1.07655
$498$	18.0000	0.806599
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	0	0
$501$	48.0000	2.14448
$502$	11.0000	0.490954
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	12.0000	0.534522
$505$	0	0
$506$	−9.00000	−0.400099
$507$	24.0000	1.06588
$508$	−13.0000	−0.576782
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	−56.0000	−2.47729
$512$	−11.0000	−0.486136
$513$	0	0
$514$	20.0000	0.882162
$515$	0	0
$516$	−2.00000	−0.0880451
$517$	−15.0000	−0.659699
$518$	−40.0000	−1.75750
$519$	−34.0000	−1.49243
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	−20.0000	−0.873704
$525$	0	0
$526$	18.0000	0.784837
$527$	2.00000	0.0871214
$528$	6.00000	0.261116
$529$	−14.0000	−0.608696
$530$	0	0
$531$	1.00000	0.0433963
$532$	0	0
$533$	−9.00000	−0.389833
$534$	16.0000	0.692388
$535$	0	0
$536$	36.0000	1.55496
$537$	32.0000	1.38090
$538$	18.0000	0.776035
$539$	27.0000	1.16297
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	16.0000	0.687259
$543$	−10.0000	−0.429141
$544$	10.0000	0.428746
$545$	0	0
$546$	−8.00000	−0.342368
$547$	−31.0000	−1.32546	−0.662732	−	0.748857i	$-0.730603\pi$
$547$	−31.0000	−1.32546	−0.662732	+	0.748857i	$0.730603\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	0	0
$551$	0	0
$552$	−18.0000	−0.766131
$553$	−52.0000	−2.21126
$554$	30.0000	1.27458
$555$	0	0
$556$	−5.00000	−0.212047
$557$	1.00000	0.0423714	0.0211857	−	0.999776i	$-0.493256\pi$
$557$	1.00000	0.0423714	0.0211857	+	0.999776i	$0.493256\pi$
$558$	1.00000	0.0423334
$559$	1.00000	0.0422955
$560$	0	0
$561$	−12.0000	−0.506640
$562$	−6.00000	−0.253095
$563$	39.0000	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$564$	−10.0000	−0.421076
$565$	0	0
$566$	11.0000	0.462364
$567$	44.0000	1.84783
$568$	−18.0000	−0.755263
$569$	3.00000	0.125767	0.0628833	−	0.998021i	$-0.479970\pi$
$569$	3.00000	0.125767	0.0628833	+	0.998021i	$0.479970\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	3.00000	0.125436
$573$	24.0000	1.00261
$574$	−36.0000	−1.50261
$575$	0	0
$576$	7.00000	0.291667
$577$	4.00000	0.166522	0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000	0.166522	0.0832611	+	0.996528i	$0.473466\pi$
$578$	−13.0000	−0.540729
$579$	−14.0000	−0.581820
$580$	0	0
$581$	36.0000	1.49353
$582$	18.0000	0.746124
$583$	−3.00000	−0.124247
$584$	−42.0000	−1.73797
$585$	0	0
$586$	31.0000	1.28060
$587$	−8.00000	−0.330195	−0.165098	−	0.986277i	$-0.552794\pi$
$587$	−8.00000	−0.330195	−0.165098	+	0.986277i	$0.552794\pi$
$588$	18.0000	0.742307
$589$	0	0
$590$	0	0
$591$	12.0000	0.493614
$592$	−10.0000	−0.410997
$593$	−8.00000	−0.328521	−0.164260	−	0.986417i	$-0.552524\pi$
$593$	−8.00000	−0.328521	−0.164260	+	0.986417i	$0.552524\pi$
$594$	12.0000	0.492366
$595$	0	0
$596$	−24.0000	−0.983078
$597$	−8.00000	−0.327418
$598$	3.00000	0.122679
$599$	29.0000	1.18491	0.592454	−	0.805604i	$-0.298159\pi$
$599$	29.0000	1.18491	0.592454	+	0.805604i	$0.298159\pi$
$600$	0	0
$601$	−20.0000	−0.815817	−0.407909	−	0.913023i	$-0.633742\pi$
$601$	−20.0000	−0.815817	−0.407909	+	0.913023i	$0.633742\pi$
$602$	4.00000	0.163028
$603$	−12.0000	−0.488678
$604$	−2.00000	−0.0813788
$605$	0	0
$606$	26.0000	1.05618
$607$	−46.0000	−1.86708	−0.933541	−	0.358470i	$-0.883298\pi$
$607$	−46.0000	−1.86708	−0.933541	+	0.358470i	$0.883298\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.00000	0.202278
$612$	−2.00000	−0.0808452
$613$	−31.0000	−1.25208	−0.626039	−	0.779792i	$-0.715325\pi$
$613$	−31.0000	−1.25208	−0.626039	+	0.779792i	$0.715325\pi$
$614$	9.00000	0.363210
$615$	0	0
$616$	36.0000	1.45048
$617$	19.0000	0.764911	0.382456	−	0.923974i	$-0.375078\pi$
$617$	19.0000	0.764911	0.382456	+	0.923974i	$0.375078\pi$
$618$	−32.0000	−1.28723
$619$	−7.00000	−0.281354	−0.140677	−	0.990056i	$-0.544928\pi$
$619$	−7.00000	−0.281354	−0.140677	+	0.990056i	$0.544928\pi$
$620$	0	0
$621$	−12.0000	−0.481543
$622$	16.0000	0.641542
$623$	32.0000	1.28205
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	28.0000	1.11911
$627$	0	0
$628$	−20.0000	−0.798087
$629$	20.0000	0.797452
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	−39.0000	−1.55134
$633$	52.0000	2.06681
$634$	2.00000	0.0794301
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	−9.00000	−0.356593
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	18.0000	0.710403
$643$	−23.0000	−0.907031	−0.453516	−	0.891248i	$-0.649830\pi$
$643$	−23.0000	−0.907031	−0.453516	+	0.891248i	$0.649830\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	0	0
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	33.0000	1.29636
$649$	3.00000	0.117760
$650$	0	0
$651$	8.00000	0.313545
$652$	14.0000	0.548282
$653$	−12.0000	−0.469596	−0.234798	−	0.972044i	$-0.575443\pi$
$653$	−12.0000	−0.469596	−0.234798	+	0.972044i	$0.575443\pi$
$654$	2.00000	0.0782062
$655$	0	0
$656$	−9.00000	−0.351391
$657$	14.0000	0.546192
$658$	20.0000	0.779681
$659$	−45.0000	−1.75295	−0.876476	−	0.481446i	$-0.840112\pi$
$659$	−45.0000	−1.75295	−0.876476	+	0.481446i	$0.840112\pi$
$660$	0	0
$661$	−5.00000	−0.194477	−0.0972387	−	0.995261i	$-0.531001\pi$
$661$	−5.00000	−0.194477	−0.0972387	+	0.995261i	$0.531001\pi$
$662$	8.00000	0.310929
$663$	4.00000	0.155347
$664$	27.0000	1.04780
$665$	0	0
$666$	10.0000	0.387492
$667$	0	0
$668$	24.0000	0.928588
$669$	−28.0000	−1.08254
$670$	0	0
$671$	0	0
$672$	40.0000	1.54303
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	−27.0000	−1.04000
$675$	0	0
$676$	12.0000	0.461538
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	−24.0000	−0.921714
$679$	36.0000	1.38155
$680$	0	0
$681$	24.0000	0.919682
$682$	3.00000	0.114876
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	0	0
$686$	−8.00000	−0.305441
$687$	22.0000	0.839352
$688$	1.00000	0.0381246
$689$	1.00000	0.0380970
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	−17.0000	−0.646243
$693$	−12.0000	−0.455842
$694$	32.0000	1.21470
$695$	0	0
$696$	0	0
$697$	18.0000	0.681799
$698$	26.0000	0.984115
$699$	8.00000	0.302588
$700$	0	0
$701$	11.0000	0.415464	0.207732	−	0.978186i	$-0.433392\pi$
$701$	11.0000	0.415464	0.207732	+	0.978186i	$0.433392\pi$
$702$	−4.00000	−0.150970
$703$	0	0
$704$	21.0000	0.791467
$705$	0	0
$706$	−3.00000	−0.112906
$707$	52.0000	1.95566
$708$	2.00000	0.0751646
$709$	27.0000	1.01401	0.507003	−	0.861944i	$-0.330753\pi$
$709$	27.0000	1.01401	0.507003	+	0.861944i	$0.330753\pi$
$710$	0	0
$711$	13.0000	0.487538
$712$	24.0000	0.899438
$713$	−3.00000	−0.112351
$714$	16.0000	0.598785
$715$	0	0
$716$	16.0000	0.597948
$717$	10.0000	0.373457
$718$	0	0
$719$	27.0000	1.00693	0.503465	−	0.864016i	$-0.332058\pi$
$719$	27.0000	1.00693	0.503465	+	0.864016i	$0.332058\pi$
$720$	0	0
$721$	−64.0000	−2.38348
$722$	−19.0000	−0.707107
$723$	24.0000	0.892570
$724$	−5.00000	−0.185824
$725$	0	0
$726$	4.00000	0.148454
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	−12.0000	−0.444750
$729$	13.0000	0.481481
$730$	0	0
$731$	−2.00000	−0.0739727
$732$	0	0
$733$	10.0000	0.369358	0.184679	−	0.982799i	$-0.440875\pi$
$733$	10.0000	0.369358	0.184679	+	0.982799i	$0.440875\pi$
$734$	−11.0000	−0.406017
$735$	0	0
$736$	−15.0000	−0.552907
$737$	−36.0000	−1.32608
$738$	9.00000	0.331295
$739$	−14.0000	−0.514998	−0.257499	−	0.966279i	$-0.582898\pi$
$739$	−14.0000	−0.514998	−0.257499	+	0.966279i	$0.582898\pi$
$740$	0	0
$741$	0	0
$742$	4.00000	0.146845
$743$	−40.0000	−1.46746	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−40.0000	−1.46746	−0.733729	+	0.679442i	$0.762222\pi$
$744$	6.00000	0.219971
$745$	0	0
$746$	16.0000	0.585802
$747$	−9.00000	−0.329293
$748$	−6.00000	−0.219382
$749$	36.0000	1.31541
$750$	0	0
$751$	−38.0000	−1.38664	−0.693320	−	0.720630i	$-0.743853\pi$
$751$	−38.0000	−1.38664	−0.693320	+	0.720630i	$0.743853\pi$
$752$	5.00000	0.182331
$753$	−22.0000	−0.801725
$754$	0	0
$755$	0	0
$756$	16.0000	0.581914
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	−20.0000	−0.726433
$759$	18.0000	0.653359
$760$	0	0
$761$	−36.0000	−1.30500	−0.652499	−	0.757789i	$-0.726280\pi$
$761$	−36.0000	−1.30500	−0.652499	+	0.757789i	$0.726280\pi$
$762$	−26.0000	−0.941881
$763$	4.00000	0.144810
$764$	12.0000	0.434145
$765$	0	0
$766$	−16.0000	−0.578103
$767$	−1.00000	−0.0361079
$768$	34.0000	1.22687
$769$	−21.0000	−0.757279	−0.378640	−	0.925544i	$-0.623608\pi$
$769$	−21.0000	−0.757279	−0.378640	+	0.925544i	$0.623608\pi$
$770$	0	0
$771$	−40.0000	−1.44056
$772$	−7.00000	−0.251936
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	−1.00000	−0.0359443
$775$	0	0
$776$	27.0000	0.969244
$777$	80.0000	2.86998
$778$	36.0000	1.29066
$779$	0	0
$780$	0	0
$781$	18.0000	0.644091
$782$	−6.00000	−0.214560
$783$	0	0
$784$	−9.00000	−0.321429
$785$	0	0
$786$	−40.0000	−1.42675
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	6.00000	0.213741
$789$	−36.0000	−1.28163
$790$	0	0
$791$	−48.0000	−1.70668
$792$	−9.00000	−0.319801
$793$	0	0
$794$	34.0000	1.20661
$795$	0	0
$796$	−4.00000	−0.141776
$797$	−10.0000	−0.354218	−0.177109	−	0.984191i	$-0.556675\pi$
$797$	−10.0000	−0.354218	−0.177109	+	0.984191i	$0.556675\pi$
$798$	0	0
$799$	−10.0000	−0.353775
$800$	0	0
$801$	−8.00000	−0.282666
$802$	−2.00000	−0.0706225
$803$	42.0000	1.48215
$804$	−24.0000	−0.846415
$805$	0	0
$806$	−1.00000	−0.0352235
$807$	−36.0000	−1.26726
$808$	39.0000	1.37202
$809$	−55.0000	−1.93370	−0.966849	−	0.255351i	$-0.917809\pi$
$809$	−55.0000	−1.93370	−0.966849	+	0.255351i	$0.917809\pi$
$810$	0	0
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	0	0
$813$	−32.0000	−1.12229
$814$	30.0000	1.05150
$815$	0	0
$816$	4.00000	0.140028
$817$	0	0
$818$	30.0000	1.04893
$819$	4.00000	0.139771
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	−24.0000	−0.837096
$823$	23.0000	0.801730	0.400865	−	0.916137i	$-0.368710\pi$
$823$	23.0000	0.801730	0.400865	+	0.916137i	$0.368710\pi$
$824$	−48.0000	−1.67216
$825$	0	0
$826$	−4.00000	−0.139178
$827$	−21.0000	−0.730242	−0.365121	−	0.930960i	$-0.618972\pi$
$827$	−21.0000	−0.730242	−0.365121	+	0.930960i	$0.618972\pi$
$828$	3.00000	0.104257
$829$	−36.0000	−1.25033	−0.625166	−	0.780492i	$-0.714969\pi$
$829$	−36.0000	−1.25033	−0.625166	+	0.780492i	$0.714969\pi$
$830$	0	0
$831$	−60.0000	−2.08138
$832$	−7.00000	−0.242681
$833$	18.0000	0.623663
$834$	−10.0000	−0.346272
$835$	0	0
$836$	0	0
$837$	4.00000	0.138260
$838$	−26.0000	−0.898155
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−20.0000	−0.689246
$843$	12.0000	0.413302
$844$	26.0000	0.894957
$845$	0	0
$846$	−5.00000	−0.171904
$847$	8.00000	0.274883
$848$	1.00000	0.0343401
$849$	−22.0000	−0.755038
$850$	0	0
$851$	−30.0000	−1.02839
$852$	12.0000	0.411113
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	0	0
$855$	0	0
$856$	27.0000	0.922841
$857$	15.0000	0.512390	0.256195	−	0.966625i	$-0.417531\pi$
$857$	15.0000	0.512390	0.256195	+	0.966625i	$0.417531\pi$
$858$	6.00000	0.204837
$859$	6.00000	0.204717	0.102359	−	0.994748i	$-0.467361\pi$
$859$	6.00000	0.204717	0.102359	+	0.994748i	$0.467361\pi$
$860$	0	0
$861$	72.0000	2.45375
$862$	15.0000	0.510902
$863$	14.0000	0.476566	0.238283	−	0.971196i	$-0.423415\pi$
$863$	14.0000	0.476566	0.238283	+	0.971196i	$0.423415\pi$
$864$	20.0000	0.680414
$865$	0	0
$866$	−26.0000	−0.883516
$867$	26.0000	0.883006
$868$	4.00000	0.135769
$869$	39.0000	1.32298
$870$	0	0
$871$	12.0000	0.406604
$872$	3.00000	0.101593
$873$	−9.00000	−0.304604
$874$	0	0
$875$	0	0
$876$	28.0000	0.946032
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	−11.0000	−0.371232
$879$	−62.0000	−2.09121
$880$	0	0
$881$	25.0000	0.842271	0.421136	−	0.906998i	$-0.361632\pi$
$881$	25.0000	0.842271	0.421136	+	0.906998i	$0.361632\pi$
$882$	9.00000	0.303046
$883$	35.0000	1.17784	0.588922	−	0.808190i	$-0.299553\pi$
$883$	35.0000	1.17784	0.588922	+	0.808190i	$0.299553\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	0	0
$887$	30.0000	1.00730	0.503651	−	0.863907i	$-0.331990\pi$
$887$	30.0000	1.00730	0.503651	+	0.863907i	$0.331990\pi$
$888$	60.0000	2.01347
$889$	−52.0000	−1.74402
$890$	0	0
$891$	−33.0000	−1.10554
$892$	−14.0000	−0.468755
$893$	0	0
$894$	−48.0000	−1.60536
$895$	0	0
$896$	12.0000	0.400892
$897$	−6.00000	−0.200334
$898$	12.0000	0.400445
$899$	0	0
$900$	0	0
$901$	−2.00000	−0.0666297
$902$	27.0000	0.899002
$903$	−8.00000	−0.266223
$904$	−36.0000	−1.19734
$905$	0	0
$906$	−4.00000	−0.132891
$907$	3.00000	0.0996134	0.0498067	−	0.998759i	$-0.484139\pi$
$907$	3.00000	0.0996134	0.0498067	+	0.998759i	$0.484139\pi$
$908$	12.0000	0.398234
$909$	−13.0000	−0.431183
$910$	0	0
$911$	34.0000	1.12647	0.563235	−	0.826297i	$-0.309557\pi$
$911$	34.0000	1.12647	0.563235	+	0.826297i	$0.309557\pi$
$912$	0	0
$913$	−27.0000	−0.893570
$914$	6.00000	0.198462
$915$	0	0
$916$	11.0000	0.363450
$917$	−80.0000	−2.64183
$918$	8.00000	0.264039
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	−18.0000	−0.593120
$922$	25.0000	0.823331
$923$	−6.00000	−0.197492
$924$	−24.0000	−0.789542
$925$	0	0
$926$	30.0000	0.985861
$927$	16.0000	0.525509
$928$	0	0
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	0	0
$931$	0	0
$932$	4.00000	0.131024
$933$	−32.0000	−1.04763
$934$	−6.00000	−0.196326
$935$	0	0
$936$	3.00000	0.0980581
$937$	−24.0000	−0.784046	−0.392023	−	0.919955i	$-0.628225\pi$
$937$	−24.0000	−0.784046	−0.392023	+	0.919955i	$0.628225\pi$
$938$	48.0000	1.56726
$939$	−56.0000	−1.82749
$940$	0	0
$941$	21.0000	0.684580	0.342290	−	0.939594i	$-0.388797\pi$
$941$	21.0000	0.684580	0.342290	+	0.939594i	$0.388797\pi$
$942$	−40.0000	−1.30327
$943$	−27.0000	−0.879241
$944$	−1.00000	−0.0325472
$945$	0	0
$946$	−3.00000	−0.0975384
$947$	51.0000	1.65728	0.828639	−	0.559784i	$-0.189116\pi$
$947$	51.0000	1.65728	0.828639	+	0.559784i	$0.189116\pi$
$948$	26.0000	0.844441
$949$	−14.0000	−0.454459
$950$	0	0
$951$	−4.00000	−0.129709
$952$	24.0000	0.777844
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	−1.00000	−0.0323762
$955$	0	0
$956$	5.00000	0.161712
$957$	0	0
$958$	0	0
$959$	−48.0000	−1.55000
$960$	0	0
$961$	−30.0000	−0.967742
$962$	−10.0000	−0.322413
$963$	−9.00000	−0.290021
$964$	12.0000	0.386494
$965$	0	0
$966$	−24.0000	−0.772187
$967$	−41.0000	−1.31847	−0.659236	−	0.751936i	$-0.729120\pi$
$967$	−41.0000	−1.31847	−0.659236	+	0.751936i	$0.729120\pi$
$968$	6.00000	0.192847
$969$	0	0
$970$	0	0
$971$	52.0000	1.66876	0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000	1.66876	0.834380	+	0.551190i	$0.185826\pi$
$972$	−10.0000	−0.320750
$973$	−20.0000	−0.641171
$974$	−12.0000	−0.384505
$975$	0	0
$976$	0	0
$977$	−13.0000	−0.415907	−0.207953	−	0.978139i	$-0.566680\pi$
$977$	−13.0000	−0.415907	−0.207953	+	0.978139i	$0.566680\pi$
$978$	28.0000	0.895341
$979$	−24.0000	−0.767043
$980$	0	0
$981$	−1.00000	−0.0319275
$982$	26.0000	0.829693
$983$	4.00000	0.127580	0.0637901	−	0.997963i	$-0.479681\pi$
$983$	4.00000	0.127580	0.0637901	+	0.997963i	$0.479681\pi$
$984$	54.0000	1.72146
$985$	0	0
$986$	0	0
$987$	−40.0000	−1.27321
$988$	0	0
$989$	3.00000	0.0953945
$990$	0	0
$991$	46.0000	1.46124	0.730619	−	0.682785i	$-0.239232\pi$
$991$	46.0000	1.46124	0.730619	+	0.682785i	$0.239232\pi$
$992$	5.00000	0.158750
$993$	−16.0000	−0.507745
$994$	−24.0000	−0.761234
$995$	0	0
$996$	−18.0000	−0.570352
$997$	−56.0000	−1.77354	−0.886769	−	0.462213i	$-0.847056\pi$
$997$	−56.0000	−1.77354	−0.886769	+	0.462213i	$0.847056\pi$
$998$	24.0000	0.759707
$999$	40.0000	1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.2.a.f.1.1	yes	1
3.2	odd	2		9675.2.a.e.1.1		1
5.2	odd	4		1075.2.b.c.474.2		2
5.3	odd	4		1075.2.b.c.474.1		2
5.4	even	2		1075.2.a.b.1.1	✓	1
15.14	odd	2		9675.2.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1075.2.a.b.1.1	✓	1	5.4	even	2
1075.2.a.f.1.1	yes	1	1.1	even	1	trivial
1075.2.b.c.474.1		2	5.3	odd	4
1075.2.b.c.474.2		2	5.2	odd	4
9675.2.a.e.1.1		1	3.2	odd	2
9675.2.a.u.1.1		1	15.14	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:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1075.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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