LMFDB - Embedded newform 845.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(1,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, 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("845.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.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:	$6.74735897080$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	845.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{11} +2.00000 q^{12} +4.00000 q^{14} -2.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} -8.00000 q^{21} -2.00000 q^{22} -6.00000 q^{23} +6.00000 q^{24} +1.00000 q^{25} +4.00000 q^{27} -4.00000 q^{28} +2.00000 q^{29} -2.00000 q^{30} +10.0000 q^{31} +5.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} +4.00000 q^{35} -1.00000 q^{36} +2.00000 q^{37} +6.00000 q^{38} -3.00000 q^{40} +6.00000 q^{41} -8.00000 q^{42} +10.0000 q^{43} +2.00000 q^{44} +1.00000 q^{45} -6.00000 q^{46} -4.00000 q^{47} +2.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -4.00000 q^{51} +2.00000 q^{53} +4.00000 q^{54} -2.00000 q^{55} -12.0000 q^{56} -12.0000 q^{57} +2.00000 q^{58} -6.00000 q^{59} +2.00000 q^{60} +2.00000 q^{61} +10.0000 q^{62} +4.00000 q^{63} +7.00000 q^{64} +4.00000 q^{66} +4.00000 q^{67} -2.00000 q^{68} +12.0000 q^{69} +4.00000 q^{70} -6.00000 q^{71} -3.00000 q^{72} +6.00000 q^{73} +2.00000 q^{74} -2.00000 q^{75} -6.00000 q^{76} -8.00000 q^{77} -12.0000 q^{79} -1.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} +16.0000 q^{83} +8.00000 q^{84} +2.00000 q^{85} +10.0000 q^{86} -4.00000 q^{87} +6.00000 q^{88} -2.00000 q^{89} +1.00000 q^{90} +6.00000 q^{92} -20.0000 q^{93} -4.00000 q^{94} +6.00000 q^{95} -10.0000 q^{96} +2.00000 q^{97} +9.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.00000	−0.816497
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	2.00000	0.577350
$13$	0	0
$13$	0	0
$14$	4.00000	1.06904
$15$	−2.00000	−0.516398
$16$	−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
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−1.00000	−0.223607
$21$	−8.00000	−1.74574
$22$	−2.00000	−0.426401
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	6.00000	1.22474
$25$	1.00000	0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	−4.00000	−0.755929
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−2.00000	−0.365148
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	5.00000	0.883883
$33$	4.00000	0.696311
$34$	2.00000	0.342997
$35$	4.00000	0.676123
$36$	−1.00000	−0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	−3.00000	−0.474342
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−8.00000	−1.23443
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	2.00000	0.301511
$45$	1.00000	0.149071
$46$	−6.00000	−0.884652
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	2.00000	0.288675
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	−4.00000	−0.560112
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	4.00000	0.544331
$55$	−2.00000	−0.269680
$56$	−12.0000	−1.60357
$57$	−12.0000	−1.58944
$58$	2.00000	0.262613
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	2.00000	0.258199
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	10.0000	1.27000
$63$	4.00000	0.503953
$64$	7.00000	0.875000
$65$	0	0
$66$	4.00000	0.492366
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−2.00000	−0.242536
$69$	12.0000	1.44463
$70$	4.00000	0.478091
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−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	$0.385800\pi$
$74$	2.00000	0.232495
$75$	−2.00000	−0.230940
$76$	−6.00000	−0.688247
$77$	−8.00000	−0.911685
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	−1.00000	−0.111803
$81$	−11.0000	−1.22222
$82$	6.00000	0.662589
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	8.00000	0.872872
$85$	2.00000	0.216930
$86$	10.0000	1.07833
$87$	−4.00000	−0.428845
$88$	6.00000	0.639602
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	6.00000	0.625543
$93$	−20.0000	−2.07390
$94$	−4.00000	−0.412568
$95$	6.00000	0.615587
$96$	−10.0000	−1.02062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	9.00000	0.909137
$99$	−2.00000	−0.201008
$100$	−1.00000	−0.100000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	−4.00000	−0.396059
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	2.00000	0.194257
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	−4.00000	−0.384900
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−2.00000	−0.190693
$111$	−4.00000	−0.379663
$112$	−4.00000	−0.377964
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	−12.0000	−1.12390
$115$	−6.00000	−0.559503
$116$	−2.00000	−0.185695
$117$	0	0
$118$	−6.00000	−0.552345
$119$	8.00000	0.733359
$120$	6.00000	0.547723
$121$	−7.00000	−0.636364
$122$	2.00000	0.181071
$123$	−12.0000	−1.08200
$124$	−10.0000	−0.898027
$125$	1.00000	0.0894427
$126$	4.00000	0.356348
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	−3.00000	−0.265165
$129$	−20.0000	−1.76090
$130$	0	0
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	−4.00000	−0.348155
$133$	24.0000	2.08106
$134$	4.00000	0.345547
$135$	4.00000	0.344265
$136$	−6.00000	−0.514496
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	12.0000	1.02151
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−4.00000	−0.338062
$141$	8.00000	0.673722
$142$	−6.00000	−0.503509
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	2.00000	0.166091
$146$	6.00000	0.496564
$147$	−18.0000	−1.48461
$148$	−2.00000	−0.164399
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	−2.00000	−0.163299
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	−18.0000	−1.45999
$153$	2.00000	0.161690
$154$	−8.00000	−0.644658
$155$	10.0000	0.803219
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	−12.0000	−0.954669
$159$	−4.00000	−0.317221
$160$	5.00000	0.395285
$161$	−24.0000	−1.89146
$162$	−11.0000	−0.864242
$163$	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
$165$	4.00000	0.311400
$166$	16.0000	1.24184
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	24.0000	1.85164
$169$	0	0
$170$	2.00000	0.153393
$171$	6.00000	0.458831
$172$	−10.0000	−0.762493
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−4.00000	−0.303239
$175$	4.00000	0.302372
$176$	2.00000	0.150756
$177$	12.0000	0.901975
$178$	−2.00000	−0.149906
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	−1.00000	−0.0745356
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	18.0000	1.32698
$185$	2.00000	0.147043
$186$	−20.0000	−1.46647
$187$	−4.00000	−0.292509
$188$	4.00000	0.291730
$189$	16.0000	1.16383
$190$	6.00000	0.435286
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−14.0000	−1.01036
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−9.00000	−0.642857
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−2.00000	−0.142134
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	−3.00000	−0.212132
$201$	−8.00000	−0.564276
$202$	−18.0000	−1.26648
$203$	8.00000	0.561490
$204$	4.00000	0.280056
$205$	6.00000	0.419058
$206$	2.00000	0.139347
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−12.0000	−0.830057
$210$	−8.00000	−0.552052
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−2.00000	−0.137361
$213$	12.0000	0.822226
$214$	10.0000	0.683586
$215$	10.0000	0.681994
$216$	−12.0000	−0.816497
$217$	40.0000	2.71538
$218$	−10.0000	−0.677285
$219$	−12.0000	−0.810885
$220$	2.00000	0.134840
$221$	0	0
$222$	−4.00000	−0.268462
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	20.0000	1.33631
$225$	1.00000	0.0666667
$226$	−14.0000	−0.931266
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	12.0000	0.794719
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	−6.00000	−0.395628
$231$	16.0000	1.05272
$232$	−6.00000	−0.393919
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	6.00000	0.390567
$237$	24.0000	1.55897
$238$	8.00000	0.518563
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	2.00000	0.129099
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−7.00000	−0.449977
$243$	10.0000	0.641500
$244$	−2.00000	−0.128037
$245$	9.00000	0.574989
$246$	−12.0000	−0.765092
$247$	0	0
$248$	−30.0000	−1.90500
$249$	−32.0000	−2.02792
$250$	1.00000	0.0632456
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	−4.00000	−0.251976
$253$	12.0000	0.754434
$254$	−2.00000	−0.125491
$255$	−4.00000	−0.250490
$256$	−17.0000	−1.06250
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	−20.0000	−1.24515
$259$	8.00000	0.497096
$260$	0	0
$261$	2.00000	0.123797
$262$	20.0000	1.23560
$263$	14.0000	0.863277	0.431638	−	0.902047i	$-0.357936\pi$
$263$	14.0000	0.863277	0.431638	+	0.902047i	$0.357936\pi$
$264$	−12.0000	−0.738549
$265$	2.00000	0.122859
$266$	24.0000	1.47153
$267$	4.00000	0.244796
$268$	−4.00000	−0.244339
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	4.00000	0.243432
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	2.00000	0.120824
$275$	−2.00000	−0.120605
$276$	−12.0000	−0.722315
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	−12.0000	−0.717137
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	8.00000	0.476393
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	6.00000	0.356034
$285$	−12.0000	−0.710819
$286$	0	0
$287$	24.0000	1.41668
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	2.00000	0.117444
$291$	−4.00000	−0.234484
$292$	−6.00000	−0.351123
$293$	−22.0000	−1.28525	−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000	−1.28525	−0.642627	+	0.766179i	$0.722155\pi$
$294$	−18.0000	−1.04978
$295$	−6.00000	−0.349334
$296$	−6.00000	−0.348743
$297$	−8.00000	−0.464207
$298$	−18.0000	−1.04271
$299$	0	0
$300$	2.00000	0.115470
$301$	40.0000	2.30556
$302$	−10.0000	−0.575435
$303$	36.0000	2.06815
$304$	−6.00000	−0.344124
$305$	2.00000	0.114520
$306$	2.00000	0.114332
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	8.00000	0.455842
$309$	−4.00000	−0.227552
$310$	10.0000	0.567962
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	−6.00000	−0.338600
$315$	4.00000	0.225374
$316$	12.0000	0.675053
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	−4.00000	−0.224309
$319$	−4.00000	−0.223957
$320$	7.00000	0.391312
$321$	−20.0000	−1.11629
$322$	−24.0000	−1.33747
$323$	12.0000	0.667698
$324$	11.0000	0.611111
$325$	0	0
$326$	12.0000	0.664619
$327$	20.0000	1.10600
$328$	−18.0000	−0.993884
$329$	−16.0000	−0.882109
$330$	4.00000	0.220193
$331$	18.0000	0.989369	0.494685	−	0.869072i	$-0.335284\pi$
$331$	18.0000	0.989369	0.494685	+	0.869072i	$0.335284\pi$
$332$	−16.0000	−0.878114
$333$	2.00000	0.109599
$334$	12.0000	0.656611
$335$	4.00000	0.218543
$336$	8.00000	0.436436
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	0	0
$339$	28.0000	1.52075
$340$	−2.00000	−0.108465
$341$	−20.0000	−1.08306
$342$	6.00000	0.324443
$343$	8.00000	0.431959
$344$	−30.0000	−1.61749
$345$	12.0000	0.646058
$346$	−6.00000	−0.322562
$347$	−22.0000	−1.18102	−0.590511	−	0.807030i	$-0.701074\pi$
$347$	−22.0000	−1.18102	−0.590511	+	0.807030i	$0.701074\pi$
$348$	4.00000	0.214423
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	4.00000	0.213809
$351$	0	0
$352$	−10.0000	−0.533002
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	12.0000	0.637793
$355$	−6.00000	−0.318447
$356$	2.00000	0.106000
$357$	−16.0000	−0.846810
$358$	12.0000	0.634220
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	−3.00000	−0.158114
$361$	17.0000	0.894737
$362$	−22.0000	−1.15629
$363$	14.0000	0.734809
$364$	0	0
$365$	6.00000	0.314054
$366$	−4.00000	−0.209083
$367$	−14.0000	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$367$	−14.0000	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$368$	6.00000	0.312772
$369$	6.00000	0.312348
$370$	2.00000	0.103975
$371$	8.00000	0.415339
$372$	20.0000	1.03695
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	−4.00000	−0.206835
$375$	−2.00000	−0.103280
$376$	12.0000	0.618853
$377$	0	0
$378$	16.0000	0.822951
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	−6.00000	−0.307794
$381$	4.00000	0.204926
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	6.00000	0.306186
$385$	−8.00000	−0.407718
$386$	2.00000	0.101797
$387$	10.0000	0.508329
$388$	−2.00000	−0.101535
$389$	−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
$391$	−12.0000	−0.606866
$392$	−27.0000	−1.36371
$393$	−40.0000	−2.01773
$394$	6.00000	0.302276
$395$	−12.0000	−0.603786
$396$	2.00000	0.100504
$397$	−6.00000	−0.301131	−0.150566	−	0.988600i	$-0.548110\pi$
$397$	−6.00000	−0.301131	−0.150566	+	0.988600i	$0.548110\pi$
$398$	−16.0000	−0.802008
$399$	−48.0000	−2.40301
$400$	−1.00000	−0.0500000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	−8.00000	−0.399004
$403$	0	0
$404$	18.0000	0.895533
$405$	−11.0000	−0.546594
$406$	8.00000	0.397033
$407$	−4.00000	−0.198273
$408$	12.0000	0.594089
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	6.00000	0.296319
$411$	−4.00000	−0.197305
$412$	−2.00000	−0.0985329
$413$	−24.0000	−1.18096
$414$	−6.00000	−0.294884
$415$	16.0000	0.785409
$416$	0	0
$417$	0	0
$418$	−12.0000	−0.586939
$419$	−40.0000	−1.95413	−0.977064	−	0.212946i	$-0.931694\pi$
$419$	−40.0000	−1.95413	−0.977064	+	0.212946i	$0.931694\pi$
$420$	8.00000	0.390360
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	12.0000	0.584151
$423$	−4.00000	−0.194487
$424$	−6.00000	−0.291386
$425$	2.00000	0.0970143
$426$	12.0000	0.581402
$427$	8.00000	0.387147
$428$	−10.0000	−0.483368
$429$	0	0
$430$	10.0000	0.482243
$431$	−14.0000	−0.674356	−0.337178	−	0.941441i	$-0.609472\pi$
$431$	−14.0000	−0.674356	−0.337178	+	0.941441i	$0.609472\pi$
$432$	−4.00000	−0.192450
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	40.0000	1.92006
$435$	−4.00000	−0.191785
$436$	10.0000	0.478913
$437$	−36.0000	−1.72211
$438$	−12.0000	−0.573382
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	6.00000	0.286039
$441$	9.00000	0.428571
$442$	0	0
$443$	14.0000	0.665160	0.332580	−	0.943075i	$-0.392081\pi$
$443$	14.0000	0.665160	0.332580	+	0.943075i	$0.392081\pi$
$444$	4.00000	0.189832
$445$	−2.00000	−0.0948091
$446$	−4.00000	−0.189405
$447$	36.0000	1.70274
$448$	28.0000	1.32288
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	1.00000	0.0471405
$451$	−12.0000	−0.565058
$452$	14.0000	0.658505
$453$	20.0000	0.939682
$454$	−4.00000	−0.187729
$455$	0	0
$456$	36.0000	1.68585
$457$	−38.0000	−1.77757	−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000	−1.77757	−0.888783	+	0.458329i	$0.848448\pi$
$458$	22.0000	1.02799
$459$	8.00000	0.373408
$460$	6.00000	0.279751
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	16.0000	0.744387
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	−2.00000	−0.0928477
$465$	−20.0000	−0.927478
$466$	10.0000	0.463241
$467$	−10.0000	−0.462745	−0.231372	−	0.972865i	$-0.574322\pi$
$467$	−10.0000	−0.462745	−0.231372	+	0.972865i	$0.574322\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	−4.00000	−0.184506
$471$	12.0000	0.552931
$472$	18.0000	0.828517
$473$	−20.0000	−0.919601
$474$	24.0000	1.10236
$475$	6.00000	0.275299
$476$	−8.00000	−0.366679
$477$	2.00000	0.0915737
$478$	6.00000	0.274434
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	−10.0000	−0.456435
$481$	0	0
$482$	−10.0000	−0.455488
$483$	48.0000	2.18408
$484$	7.00000	0.318182
$485$	2.00000	0.0908153
$486$	10.0000	0.453609
$487$	40.0000	1.81257	0.906287	−	0.422664i	$-0.138905\pi$
$487$	40.0000	1.81257	0.906287	+	0.422664i	$0.138905\pi$
$488$	−6.00000	−0.271607
$489$	−24.0000	−1.08532
$490$	9.00000	0.406579
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	12.0000	0.541002
$493$	4.00000	0.180151
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	−10.0000	−0.449013
$497$	−24.0000	−1.07655
$498$	−32.0000	−1.43395
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	−1.00000	−0.0447214
$501$	−24.0000	−1.07224
$502$	24.0000	1.07117
$503$	−18.0000	−0.802580	−0.401290	−	0.915951i	$-0.631438\pi$
$503$	−18.0000	−0.802580	−0.401290	+	0.915951i	$0.631438\pi$
$504$	−12.0000	−0.534522
$505$	−18.0000	−0.800989
$506$	12.0000	0.533465
$507$	0	0
$508$	2.00000	0.0887357
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	−4.00000	−0.177123
$511$	24.0000	1.06170
$512$	−11.0000	−0.486136
$513$	24.0000	1.05963
$514$	2.00000	0.0882162
$515$	2.00000	0.0881305
$516$	20.0000	0.880451
$517$	8.00000	0.351840
$518$	8.00000	0.351500
$519$	12.0000	0.526742
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	2.00000	0.0875376
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	−20.0000	−0.873704
$525$	−8.00000	−0.349149
$526$	14.0000	0.610429
$527$	20.0000	0.871214
$528$	−4.00000	−0.174078
$529$	13.0000	0.565217
$530$	2.00000	0.0868744
$531$	−6.00000	−0.260378
$532$	−24.0000	−1.04053
$533$	0	0
$534$	4.00000	0.173097
$535$	10.0000	0.432338
$536$	−12.0000	−0.518321
$537$	−24.0000	−1.03568
$538$	6.00000	0.258678
$539$	−18.0000	−0.775315
$540$	−4.00000	−0.172133
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	−2.00000	−0.0859074
$543$	44.0000	1.88822
$544$	10.0000	0.428746
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−6.00000	−0.256541	−0.128271	−	0.991739i	$-0.540943\pi$
$547$	−6.00000	−0.256541	−0.128271	+	0.991739i	$0.540943\pi$
$548$	−2.00000	−0.0854358
$549$	2.00000	0.0853579
$550$	−2.00000	−0.0852803
$551$	12.0000	0.511217
$552$	−36.0000	−1.53226
$553$	−48.0000	−2.04117
$554$	−14.0000	−0.594803
$555$	−4.00000	−0.169791
$556$	0	0
$557$	26.0000	1.10166	0.550828	−	0.834619i	$-0.314312\pi$
$557$	26.0000	1.10166	0.550828	+	0.834619i	$0.314312\pi$
$558$	10.0000	0.423334
$559$	0	0
$560$	−4.00000	−0.169031
$561$	8.00000	0.337760
$562$	6.00000	0.253095
$563$	−22.0000	−0.927189	−0.463595	−	0.886047i	$-0.653441\pi$
$563$	−22.0000	−0.927189	−0.463595	+	0.886047i	$0.653441\pi$
$564$	−8.00000	−0.336861
$565$	−14.0000	−0.588984
$566$	2.00000	0.0840663
$567$	−44.0000	−1.84783
$568$	18.0000	0.755263
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	−12.0000	−0.502625
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	0	0
$573$	0	0
$574$	24.0000	1.00174
$575$	−6.00000	−0.250217
$576$	7.00000	0.291667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−13.0000	−0.540729
$579$	−4.00000	−0.166234
$580$	−2.00000	−0.0830455
$581$	64.0000	2.65517
$582$	−4.00000	−0.165805
$583$	−4.00000	−0.165663
$584$	−18.0000	−0.744845
$585$	0	0
$586$	−22.0000	−0.908812
$587$	44.0000	1.81607	0.908037	−	0.418890i	$-0.137581\pi$
$587$	44.0000	1.81607	0.908037	+	0.418890i	$0.137581\pi$
$588$	18.0000	0.742307
$589$	60.0000	2.47226
$590$	−6.00000	−0.247016
$591$	−12.0000	−0.493614
$592$	−2.00000	−0.0821995
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	−8.00000	−0.328244
$595$	8.00000	0.327968
$596$	18.0000	0.737309
$597$	32.0000	1.30967
$598$	0	0
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	6.00000	0.244949
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	40.0000	1.63028
$603$	4.00000	0.162893
$604$	10.0000	0.406894
$605$	−7.00000	−0.284590
$606$	36.0000	1.46240
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	30.0000	1.21666
$609$	−16.0000	−0.648353
$610$	2.00000	0.0809776
$611$	0	0
$612$	−2.00000	−0.0808452
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	−8.00000	−0.322854
$615$	−12.0000	−0.483887
$616$	24.0000	0.966988
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	−4.00000	−0.160904
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	−10.0000	−0.401610
$621$	−24.0000	−0.963087
$622$	−4.00000	−0.160385
$623$	−8.00000	−0.320513
$624$	0	0
$625$	1.00000	0.0400000
$626$	−22.0000	−0.879297
$627$	24.0000	0.958468
$628$	6.00000	0.239426
$629$	4.00000	0.159490
$630$	4.00000	0.159364
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	36.0000	1.43200
$633$	−24.0000	−0.953914
$634$	18.0000	0.714871
$635$	−2.00000	−0.0793676
$636$	4.00000	0.158610
$637$	0	0
$638$	−4.00000	−0.158362
$639$	−6.00000	−0.237356
$640$	−3.00000	−0.118585
$641$	−46.0000	−1.81689	−0.908445	−	0.418004i	$-0.862730\pi$
$641$	−46.0000	−1.81689	−0.908445	+	0.418004i	$0.862730\pi$
$642$	−20.0000	−0.789337
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	24.0000	0.945732
$645$	−20.0000	−0.787499
$646$	12.0000	0.472134
$647$	38.0000	1.49393	0.746967	−	0.664861i	$-0.231509\pi$
$647$	38.0000	1.49393	0.746967	+	0.664861i	$0.231509\pi$
$648$	33.0000	1.29636
$649$	12.0000	0.471041
$650$	0	0
$651$	−80.0000	−3.13545
$652$	−12.0000	−0.469956
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	20.0000	0.782062
$655$	20.0000	0.781465
$656$	−6.00000	−0.234261
$657$	6.00000	0.234082
$658$	−16.0000	−0.623745
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	−4.00000	−0.155700
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	18.0000	0.699590
$663$	0	0
$664$	−48.0000	−1.86276
$665$	24.0000	0.930680
$666$	2.00000	0.0774984
$667$	−12.0000	−0.464642
$668$	−12.0000	−0.464294
$669$	8.00000	0.309298
$670$	4.00000	0.154533
$671$	−4.00000	−0.154418
$672$	−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	$0.696248\pi$
$674$	26.0000	1.00148
$675$	4.00000	0.153960
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	28.0000	1.07533
$679$	8.00000	0.307012
$680$	−6.00000	−0.230089
$681$	8.00000	0.306561
$682$	−20.0000	−0.765840
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	−6.00000	−0.229416
$685$	2.00000	0.0764161
$686$	8.00000	0.305441
$687$	−44.0000	−1.67870
$688$	−10.0000	−0.381246
$689$	0	0
$690$	12.0000	0.456832
$691$	−22.0000	−0.836919	−0.418460	−	0.908235i	$-0.637430\pi$
$691$	−22.0000	−0.836919	−0.418460	+	0.908235i	$0.637430\pi$
$692$	6.00000	0.228086
$693$	−8.00000	−0.303895
$694$	−22.0000	−0.835109
$695$	0	0
$696$	12.0000	0.454859
$697$	12.0000	0.454532
$698$	30.0000	1.13552
$699$	−20.0000	−0.756469
$700$	−4.00000	−0.151186
$701$	−26.0000	−0.982006	−0.491003	−	0.871158i	$-0.663370\pi$
$701$	−26.0000	−0.982006	−0.491003	+	0.871158i	$0.663370\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	−14.0000	−0.527645
$705$	8.00000	0.301297
$706$	−18.0000	−0.677439
$707$	−72.0000	−2.70784
$708$	−12.0000	−0.450988
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	−6.00000	−0.225176
$711$	−12.0000	−0.450035
$712$	6.00000	0.224860
$713$	−60.0000	−2.24702
$714$	−16.0000	−0.598785
$715$	0	0
$716$	−12.0000	−0.448461
$717$	−12.0000	−0.448148
$718$	10.0000	0.373197
$719$	−32.0000	−1.19340	−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000	−1.19340	−0.596699	+	0.802465i	$0.703521\pi$
$720$	−1.00000	−0.0372678
$721$	8.00000	0.297936
$722$	17.0000	0.632674
$723$	20.0000	0.743808
$724$	22.0000	0.817624
$725$	2.00000	0.0742781
$726$	14.0000	0.519589
$727$	−18.0000	−0.667583	−0.333792	−	0.942647i	$-0.608328\pi$
$727$	−18.0000	−0.667583	−0.333792	+	0.942647i	$0.608328\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	6.00000	0.222070
$731$	20.0000	0.739727
$732$	4.00000	0.147844
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	−14.0000	−0.516749
$735$	−18.0000	−0.663940
$736$	−30.0000	−1.10581
$737$	−8.00000	−0.294684
$738$	6.00000	0.220863
$739$	−6.00000	−0.220714	−0.110357	−	0.993892i	$-0.535199\pi$
$739$	−6.00000	−0.220714	−0.110357	+	0.993892i	$0.535199\pi$
$740$	−2.00000	−0.0735215
$741$	0	0
$742$	8.00000	0.293689
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	60.0000	2.19971
$745$	−18.0000	−0.659469
$746$	34.0000	1.24483
$747$	16.0000	0.585409
$748$	4.00000	0.146254
$749$	40.0000	1.46157
$750$	−2.00000	−0.0730297
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	4.00000	0.145865
$753$	−48.0000	−1.74922
$754$	0	0
$755$	−10.0000	−0.363937
$756$	−16.0000	−0.581914
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−10.0000	−0.363216
$759$	−24.0000	−0.871145
$760$	−18.0000	−0.652929
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	4.00000	0.144905
$763$	−40.0000	−1.44810
$764$	0	0
$765$	2.00000	0.0723102
$766$	−12.0000	−0.433578
$767$	0	0
$768$	34.0000	1.22687
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	−8.00000	−0.288300
$771$	−4.00000	−0.144056
$772$	−2.00000	−0.0719816
$773$	2.00000	0.0719350	0.0359675	−	0.999353i	$-0.488549\pi$
$773$	2.00000	0.0719350	0.0359675	+	0.999353i	$0.488549\pi$
$774$	10.0000	0.359443
$775$	10.0000	0.359211
$776$	−6.00000	−0.215387
$777$	−16.0000	−0.573997
$778$	−10.0000	−0.358517
$779$	36.0000	1.28983
$780$	0	0
$781$	12.0000	0.429394
$782$	−12.0000	−0.429119
$783$	8.00000	0.285897
$784$	−9.00000	−0.321429
$785$	−6.00000	−0.214149
$786$	−40.0000	−1.42675
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	−6.00000	−0.213741
$789$	−28.0000	−0.996826
$790$	−12.0000	−0.426941
$791$	−56.0000	−1.99113
$792$	6.00000	0.213201
$793$	0	0
$794$	−6.00000	−0.212932
$795$	−4.00000	−0.141865
$796$	16.0000	0.567105
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	−48.0000	−1.69918
$799$	−8.00000	−0.283020
$800$	5.00000	0.176777
$801$	−2.00000	−0.0706665
$802$	−10.0000	−0.353112
$803$	−12.0000	−0.423471
$804$	8.00000	0.282138
$805$	−24.0000	−0.845889
$806$	0	0
$807$	−12.0000	−0.422420
$808$	54.0000	1.89971
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	−11.0000	−0.386501
$811$	42.0000	1.47482	0.737410	−	0.675446i	$-0.236049\pi$
$811$	42.0000	1.47482	0.737410	+	0.675446i	$0.236049\pi$
$812$	−8.00000	−0.280745
$813$	4.00000	0.140286
$814$	−4.00000	−0.140200
$815$	12.0000	0.420342
$816$	4.00000	0.140028
$817$	60.0000	2.09913
$818$	−18.0000	−0.629355
$819$	0	0
$820$	−6.00000	−0.209529
$821$	−50.0000	−1.74501	−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000	−1.74501	−0.872506	+	0.488603i	$0.837507\pi$
$822$	−4.00000	−0.139516
$823$	−46.0000	−1.60346	−0.801730	−	0.597687i	$-0.796087\pi$
$823$	−46.0000	−1.60346	−0.801730	+	0.597687i	$0.796087\pi$
$824$	−6.00000	−0.209020
$825$	4.00000	0.139262
$826$	−24.0000	−0.835067
$827$	32.0000	1.11275	0.556375	−	0.830932i	$-0.312192\pi$
$827$	32.0000	1.11275	0.556375	+	0.830932i	$0.312192\pi$
$828$	6.00000	0.208514
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	16.0000	0.555368
$831$	28.0000	0.971309
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	12.0000	0.415277
$836$	12.0000	0.415029
$837$	40.0000	1.38260
$838$	−40.0000	−1.38178
$839$	−38.0000	−1.31191	−0.655953	−	0.754802i	$-0.727733\pi$
$839$	−38.0000	−1.31191	−0.655953	+	0.754802i	$0.727733\pi$
$840$	24.0000	0.828079
$841$	−25.0000	−0.862069
$842$	−10.0000	−0.344623
$843$	−12.0000	−0.413302
$844$	−12.0000	−0.413057
$845$	0	0
$846$	−4.00000	−0.137523
$847$	−28.0000	−0.962091
$848$	−2.00000	−0.0686803
$849$	−4.00000	−0.137280
$850$	2.00000	0.0685994
$851$	−12.0000	−0.411355
$852$	−12.0000	−0.411113
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	8.00000	0.273754
$855$	6.00000	0.205196
$856$	−30.0000	−1.02538
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	−10.0000	−0.340997
$861$	−48.0000	−1.63584
$862$	−14.0000	−0.476842
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	20.0000	0.680414
$865$	−6.00000	−0.204006
$866$	10.0000	0.339814
$867$	26.0000	0.883006
$868$	−40.0000	−1.35769
$869$	24.0000	0.814144
$870$	−4.00000	−0.135613
$871$	0	0
$872$	30.0000	1.01593
$873$	2.00000	0.0676897
$874$	−36.0000	−1.21772
$875$	4.00000	0.135225
$876$	12.0000	0.405442
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	0	0
$879$	44.0000	1.48408
$880$	2.00000	0.0674200
$881$	38.0000	1.28025	0.640126	−	0.768270i	$-0.278882\pi$
$881$	38.0000	1.28025	0.640126	+	0.768270i	$0.278882\pi$
$882$	9.00000	0.303046
$883$	−22.0000	−0.740359	−0.370179	−	0.928960i	$-0.620704\pi$
$883$	−22.0000	−0.740359	−0.370179	+	0.928960i	$0.620704\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	14.0000	0.470339
$887$	58.0000	1.94745	0.973725	−	0.227728i	$-0.0731298\pi$
$887$	58.0000	1.94745	0.973725	+	0.227728i	$0.0731298\pi$
$888$	12.0000	0.402694
$889$	−8.00000	−0.268311
$890$	−2.00000	−0.0670402
$891$	22.0000	0.737028
$892$	4.00000	0.133930
$893$	−24.0000	−0.803129
$894$	36.0000	1.20402
$895$	12.0000	0.401116
$896$	−12.0000	−0.400892
$897$	0	0
$898$	6.00000	0.200223
$899$	20.0000	0.667037
$900$	−1.00000	−0.0333333
$901$	4.00000	0.133259
$902$	−12.0000	−0.399556
$903$	−80.0000	−2.66223
$904$	42.0000	1.39690
$905$	−22.0000	−0.731305
$906$	20.0000	0.664455
$907$	−34.0000	−1.12895	−0.564476	−	0.825450i	$-0.690922\pi$
$907$	−34.0000	−1.12895	−0.564476	+	0.825450i	$0.690922\pi$
$908$	4.00000	0.132745
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	12.0000	0.397360
$913$	−32.0000	−1.05905
$914$	−38.0000	−1.25693
$915$	−4.00000	−0.132236
$916$	−22.0000	−0.726900
$917$	80.0000	2.64183
$918$	8.00000	0.264039
$919$	−52.0000	−1.71532	−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000	−1.71532	−0.857661	+	0.514216i	$0.828083\pi$
$920$	18.0000	0.593442
$921$	16.0000	0.527218
$922$	−10.0000	−0.329332
$923$	0	0
$924$	−16.0000	−0.526361
$925$	2.00000	0.0657596
$926$	24.0000	0.788689
$927$	2.00000	0.0656886
$928$	10.0000	0.328266
$929$	38.0000	1.24674	0.623370	−	0.781927i	$-0.285763\pi$
$929$	38.0000	1.24674	0.623370	+	0.781927i	$0.285763\pi$
$930$	−20.0000	−0.655826
$931$	54.0000	1.76978
$932$	−10.0000	−0.327561
$933$	8.00000	0.261908
$934$	−10.0000	−0.327210
$935$	−4.00000	−0.130814
$936$	0	0
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	16.0000	0.522419
$939$	44.0000	1.43589
$940$	4.00000	0.130466
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	12.0000	0.390981
$943$	−36.0000	−1.17232
$944$	6.00000	0.195283
$945$	16.0000	0.520480
$946$	−20.0000	−0.650256
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	−24.0000	−0.779484
$949$	0	0
$950$	6.00000	0.194666
$951$	−36.0000	−1.16738
$952$	−24.0000	−0.777844
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	−6.00000	−0.194054
$957$	8.00000	0.258603
$958$	−30.0000	−0.969256
$959$	8.00000	0.258333
$960$	−14.0000	−0.451848
$961$	69.0000	2.22581
$962$	0	0
$963$	10.0000	0.322245
$964$	10.0000	0.322078
$965$	2.00000	0.0643823
$966$	48.0000	1.54437
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	21.0000	0.674966
$969$	−24.0000	−0.770991
$970$	2.00000	0.0642161
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	−10.0000	−0.320750
$973$	0	0
$974$	40.0000	1.28168
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	−24.0000	−0.767435
$979$	4.00000	0.127841
$980$	−9.00000	−0.287494
$981$	−10.0000	−0.319275
$982$	−24.0000	−0.765871
$983$	−56.0000	−1.78612	−0.893061	−	0.449935i	$-0.851447\pi$
$983$	−56.0000	−1.78612	−0.893061	+	0.449935i	$0.851447\pi$
$984$	36.0000	1.14764
$985$	6.00000	0.191176
$986$	4.00000	0.127386
$987$	32.0000	1.01857
$988$	0	0
$989$	−60.0000	−1.90789
$990$	−2.00000	−0.0635642
$991$	48.0000	1.52477	0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000	1.52477	0.762385	+	0.647124i	$0.224028\pi$
$992$	50.0000	1.58750
$993$	−36.0000	−1.14243
$994$	−24.0000	−0.761234
$995$	−16.0000	−0.507234
$996$	32.0000	1.01396
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	−10.0000	−0.316544
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.a.a.1.1		1
3.2	odd	2		7605.2.a.f.1.1		1
5.4	even	2		4225.2.a.g.1.1		1
13.2	odd	12		845.2.m.b.316.2		4
13.3	even	3		845.2.e.a.191.1		2
13.4	even	6		845.2.e.b.146.1		2
13.5	odd	4		845.2.c.a.506.1		2
13.6	odd	12		845.2.m.b.361.1		4
13.7	odd	12		845.2.m.b.361.2		4
13.8	odd	4		845.2.c.a.506.2		2
13.9	even	3		845.2.e.a.146.1		2
13.10	even	6		845.2.e.b.191.1		2
13.11	odd	12		845.2.m.b.316.1		4
13.12	even	2		65.2.a.a.1.1	✓	1
39.38	odd	2		585.2.a.h.1.1		1
52.51	odd	2		1040.2.a.f.1.1		1
65.12	odd	4		325.2.b.b.274.1		2
65.38	odd	4		325.2.b.b.274.2		2
65.64	even	2		325.2.a.d.1.1		1
91.90	odd	2		3185.2.a.e.1.1		1
104.51	odd	2		4160.2.a.f.1.1		1
104.77	even	2		4160.2.a.q.1.1		1
143.142	odd	2		7865.2.a.c.1.1		1
156.155	even	2		9360.2.a.ca.1.1		1
195.38	even	4		2925.2.c.h.2224.1		2
195.77	even	4		2925.2.c.h.2224.2		2
195.194	odd	2		2925.2.a.f.1.1		1
260.259	odd	2		5200.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.a.1.1	✓	1	13.12	even	2
325.2.a.d.1.1		1	65.64	even	2
325.2.b.b.274.1		2	65.12	odd	4
325.2.b.b.274.2		2	65.38	odd	4
585.2.a.h.1.1		1	39.38	odd	2
845.2.a.a.1.1		1	1.1	even	1	trivial
845.2.c.a.506.1		2	13.5	odd	4
845.2.c.a.506.2		2	13.8	odd	4
845.2.e.a.146.1		2	13.9	even	3
845.2.e.a.191.1		2	13.3	even	3
845.2.e.b.146.1		2	13.4	even	6
845.2.e.b.191.1		2	13.10	even	6
845.2.m.b.316.1		4	13.11	odd	12
845.2.m.b.316.2		4	13.2	odd	12
845.2.m.b.361.1		4	13.6	odd	12
845.2.m.b.361.2		4	13.7	odd	12
1040.2.a.f.1.1		1	52.51	odd	2
2925.2.a.f.1.1		1	195.194	odd	2
2925.2.c.h.2224.1		2	195.38	even	4
2925.2.c.h.2224.2		2	195.77	even	4
3185.2.a.e.1.1		1	91.90	odd	2
4160.2.a.f.1.1		1	104.51	odd	2
4160.2.a.q.1.1		1	104.77	even	2
4225.2.a.g.1.1		1	5.4	even	2
5200.2.a.d.1.1		1	260.259	odd	2
7605.2.a.f.1.1		1	3.2	odd	2
7865.2.a.c.1.1		1	143.142	odd	2
9360.2.a.ca.1.1		1	156.155	even	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:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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