LMFDB - Embedded newform 325.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(1,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, 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("325.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	325.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:	$2.59513806569$
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	$\chi$	$=$	325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} +2.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} -2.00000 q^{12} +1.00000 q^{13} +4.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -6.00000 q^{19} +8.00000 q^{21} +2.00000 q^{22} +6.00000 q^{23} -6.00000 q^{24} +1.00000 q^{26} -4.00000 q^{27} -4.00000 q^{28} +2.00000 q^{29} -10.0000 q^{31} +5.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} -1.00000 q^{36} +2.00000 q^{37} -6.00000 q^{38} +2.00000 q^{39} -6.00000 q^{41} +8.00000 q^{42} -10.0000 q^{43} -2.00000 q^{44} +6.00000 q^{46} -4.00000 q^{47} -2.00000 q^{48} +9.00000 q^{49} -4.00000 q^{51} -1.00000 q^{52} -2.00000 q^{53} -4.00000 q^{54} -12.0000 q^{56} -12.0000 q^{57} +2.00000 q^{58} +6.00000 q^{59} +2.00000 q^{61} -10.0000 q^{62} +4.00000 q^{63} +7.00000 q^{64} +4.00000 q^{66} +4.00000 q^{67} +2.00000 q^{68} +12.0000 q^{69} +6.00000 q^{71} -3.00000 q^{72} +6.00000 q^{73} +2.00000 q^{74} +6.00000 q^{76} +8.00000 q^{77} +2.00000 q^{78} -12.0000 q^{79} -11.0000 q^{81} -6.00000 q^{82} +16.0000 q^{83} -8.00000 q^{84} -10.0000 q^{86} +4.00000 q^{87} -6.00000 q^{88} +2.00000 q^{89} +4.00000 q^{91} -6.00000 q^{92} -20.0000 q^{93} -4.00000 q^{94} +10.0000 q^{96} +2.00000 q^{97} +9.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−2.00000	−0.577350
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	4.00000	1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	8.00000	1.74574
$22$	2.00000	0.426401
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	−6.00000	−1.22474
$25$	0	0
$26$	1.00000	0.196116
$27$	−4.00000	−0.769800
$28$	−4.00000	−0.755929
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	5.00000	0.883883
$33$	4.00000	0.696311
$34$	−2.00000	−0.342997
$35$	0	0
$36$	−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
$39$	2.00000	0.320256
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	8.00000	1.23443
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	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
$50$	0	0
$51$	−4.00000	−0.560112
$52$	−1.00000	−0.138675
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	−12.0000	−1.60357
$57$	−12.0000	−1.58944
$58$	2.00000	0.262613
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	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
$70$	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
$73$	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
$75$	0	0
$76$	6.00000	0.688247
$77$	8.00000	0.911685
$78$	2.00000	0.226455
$79$	−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
$81$	−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
$85$	0	0
$86$	−10.0000	−1.07833
$87$	4.00000	0.428845
$88$	−6.00000	−0.639602
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	−6.00000	−0.625543
$93$	−20.0000	−2.07390
$94$	−4.00000	−0.412568
$95$	0	0
$96$	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
$100$	0	0
$101$	−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	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−2.00000	−0.194257
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	4.00000	0.384900
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	−4.00000	−0.377964
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	−12.0000	−1.12390
$115$	0	0
$116$	−2.00000	−0.185695
$117$	1.00000	0.0924500
$118$	6.00000	0.552345
$119$	−8.00000	−0.733359
$120$	0	0
$121$	−7.00000	−0.636364
$122$	2.00000	0.181071
$123$	−12.0000	−1.08200
$124$	10.0000	0.898027
$125$	0	0
$126$	4.00000	0.356348
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−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
$135$	0	0
$136$	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	$0.5\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	6.00000	0.503509
$143$	2.00000	0.167248
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	6.00000	0.496564
$147$	18.0000	1.48461
$148$	−2.00000	−0.164399
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	18.0000	1.45999
$153$	−2.00000	−0.161690
$154$	8.00000	0.644658
$155$	0	0
$156$	−2.00000	−0.160128
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	−12.0000	−0.954669
$159$	−4.00000	−0.317221
$160$	0	0
$161$	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
$165$	0	0
$166$	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
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.00000	−0.458831
$172$	10.0000	0.762493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	4.00000	0.303239
$175$	0	0
$176$	−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	$0.351972\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	4.00000	0.296500
$183$	4.00000	0.295689
$184$	−18.0000	−1.32698
$185$	0	0
$186$	−20.0000	−1.46647
$187$	−4.00000	−0.292509
$188$	4.00000	0.291730
$189$	−16.0000	−1.16383
$190$	0	0
$191$	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	$0.691937\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	−18.0000	−1.26648
$203$	8.00000	0.561490
$204$	4.00000	0.280056
$205$	0	0
$206$	−2.00000	−0.139347
$207$	6.00000	0.417029
$208$	−1.00000	−0.0693375
$209$	−12.0000	−0.830057
$210$	0	0
$211$	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
$215$	0	0
$216$	12.0000	0.816497
$217$	−40.0000	−2.71538
$218$	10.0000	0.677285
$219$	12.0000	0.810885
$220$	0	0
$221$	−2.00000	−0.134535
$222$	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
$225$	0	0
$226$	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	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	−6.00000	−0.393919
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	−6.00000	−0.390567
$237$	−24.0000	−1.55897
$238$	−8.00000	−0.518563
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−7.00000	−0.449977
$243$	−10.0000	−0.641500
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−12.0000	−0.765092
$247$	−6.00000	−0.381771
$248$	30.0000	1.90500
$249$	32.0000	2.02792
$250$	0	0
$251$	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
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	−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	$-0.642064\pi$
$263$	−14.0000	−0.863277	−0.431638	+	0.902047i	$0.642064\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	−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	$0.441447\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	2.00000	0.121268
$273$	8.00000	0.484182
$274$	2.00000	0.120824
$275$	0	0
$276$	−12.0000	−0.722315
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	0	0
$279$	−10.0000	−0.598684
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	−8.00000	−0.476393
$283$	−2.00000	−0.118888	−0.0594438	−	0.998232i	$-0.518933\pi$
$283$	−2.00000	−0.118888	−0.0594438	+	0.998232i	$0.518933\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	2.00000	0.118262
$287$	−24.0000	−1.41668
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	0	0
$291$	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
$295$	0	0
$296$	−6.00000	−0.348743
$297$	−8.00000	−0.464207
$298$	18.0000	1.04271
$299$	6.00000	0.346989
$300$	0	0
$301$	−40.0000	−2.30556
$302$	10.0000	0.575435
$303$	−36.0000	−2.06815
$304$	6.00000	0.344124
$305$	0	0
$306$	−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
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	−6.00000	−0.339683
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	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
$320$	0	0
$321$	−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
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	−16.0000	−0.878114
$333$	2.00000	0.109599
$334$	12.0000	0.656611
$335$	0	0
$336$	−8.00000	−0.436436
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	1.00000	0.0543928
$339$	28.0000	1.52075
$340$	0	0
$341$	−20.0000	−1.08306
$342$	−6.00000	−0.324443
$343$	8.00000	0.431959
$344$	30.0000	1.61749
$345$	0	0
$346$	6.00000	0.322562
$347$	22.0000	1.18102	0.590511	−	0.807030i	$-0.298926\pi$
$347$	22.0000	1.18102	0.590511	+	0.807030i	$0.298926\pi$
$348$	−4.00000	−0.214423
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	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
$355$	0	0
$356$	−2.00000	−0.106000
$357$	−16.0000	−0.846810
$358$	12.0000	0.634220
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−22.0000	−1.15629
$363$	−14.0000	−0.734809
$364$	−4.00000	−0.209657
$365$	0	0
$366$	4.00000	0.209083
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	−6.00000	−0.312772
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−8.00000	−0.415339
$372$	20.0000	1.03695
$373$	−34.0000	−1.76045	−0.880227	−	0.474554i	$-0.842610\pi$
$373$	−34.0000	−1.76045	−0.880227	+	0.474554i	$0.842610\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	12.0000	0.618853
$377$	2.00000	0.103005
$378$	−16.0000	−0.822951
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	0	0
$381$	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
$385$	0	0
$386$	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
$395$	0	0
$396$	−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
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	8.00000	0.399004
$403$	−10.0000	−0.498135
$404$	18.0000	0.895533
$405$	0	0
$406$	8.00000	0.397033
$407$	4.00000	0.198273
$408$	12.0000	0.594089
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	4.00000	0.197305
$412$	2.00000	0.0985329
$413$	24.0000	1.18096
$414$	6.00000	0.294884
$415$	0	0
$416$	5.00000	0.245145
$417$	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	$0.931694\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	12.0000	0.584151
$423$	−4.00000	−0.194487
$424$	6.00000	0.291386
$425$	0	0
$426$	12.0000	0.581402
$427$	8.00000	0.387147
$428$	10.0000	0.483368
$429$	4.00000	0.193122
$430$	0	0
$431$	14.0000	0.674356	0.337178	−	0.941441i	$-0.390528\pi$
$431$	14.0000	0.674356	0.337178	+	0.941441i	$0.390528\pi$
$432$	4.00000	0.192450
$433$	−10.0000	−0.480569	−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000	−0.480569	−0.240285	+	0.970702i	$0.577241\pi$
$434$	−40.0000	−1.92006
$435$	0	0
$436$	−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	$0.5\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	−2.00000	−0.0951303
$443$	−14.0000	−0.665160	−0.332580	−	0.943075i	$-0.607919\pi$
$443$	−14.0000	−0.665160	−0.332580	+	0.943075i	$0.607919\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	−4.00000	−0.189405
$447$	36.0000	1.70274
$448$	28.0000	1.32288
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	−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
$460$	0	0
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	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
$465$	0	0
$466$	−10.0000	−0.463241
$467$	10.0000	0.462745	0.231372	−	0.972865i	$-0.425678\pi$
$467$	10.0000	0.462745	0.231372	+	0.972865i	$0.425678\pi$
$468$	−1.00000	−0.0462250
$469$	16.0000	0.738811
$470$	0	0
$471$	12.0000	0.552931
$472$	−18.0000	−0.828517
$473$	−20.0000	−0.919601
$474$	−24.0000	−1.10236
$475$	0	0
$476$	8.00000	0.366679
$477$	−2.00000	−0.0915737
$478$	−6.00000	−0.274434
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	10.0000	0.455488
$483$	48.0000	2.18408
$484$	7.00000	0.318182
$485$	0	0
$486$	−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
$490$	0	0
$491$	−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
$494$	−6.00000	−0.269953
$495$	0	0
$496$	10.0000	0.449013
$497$	24.0000	1.07655
$498$	32.0000	1.43395
$499$	10.0000	0.447661	0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000	0.447661	0.223831	+	0.974628i	$0.428144\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	24.0000	1.07117
$503$	18.0000	0.802580	0.401290	−	0.915951i	$-0.368562\pi$
$503$	18.0000	0.802580	0.401290	+	0.915951i	$0.368562\pi$
$504$	−12.0000	−0.534522
$505$	0	0
$506$	12.0000	0.533465
$507$	2.00000	0.0888231
$508$	−2.00000	−0.0887357
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	−11.0000	−0.486136
$513$	24.0000	1.05963
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	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	$-0.541877\pi$
$523$	−6.00000	−0.262362	−0.131181	+	0.991358i	$0.541877\pi$
$524$	−20.0000	−0.873704
$525$	0	0
$526$	−14.0000	−0.610429
$527$	20.0000	0.871214
$528$	−4.00000	−0.174078
$529$	13.0000	0.565217
$530$	0	0
$531$	6.00000	0.260378
$532$	24.0000	1.04053
$533$	−6.00000	−0.259889
$534$	4.00000	0.173097
$535$	0	0
$536$	−12.0000	−0.518321
$537$	24.0000	1.03568
$538$	6.00000	0.258678
$539$	18.0000	0.775315
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	2.00000	0.0859074
$543$	−44.0000	−1.88822
$544$	−10.0000	−0.428746
$545$	0	0
$546$	8.00000	0.342368
$547$	6.00000	0.256541	0.128271	−	0.991739i	$-0.459057\pi$
$547$	6.00000	0.256541	0.128271	+	0.991739i	$0.459057\pi$
$548$	−2.00000	−0.0854358
$549$	2.00000	0.0853579
$550$	0	0
$551$	−12.0000	−0.511217
$552$	−36.0000	−1.53226
$553$	−48.0000	−2.04117
$554$	14.0000	0.594803
$555$	0	0
$556$	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
$559$	−10.0000	−0.422955
$560$	0	0
$561$	−8.00000	−0.337760
$562$	−6.00000	−0.253095
$563$	22.0000	0.927189	0.463595	−	0.886047i	$-0.346559\pi$
$563$	22.0000	0.927189	0.463595	+	0.886047i	$0.346559\pi$
$564$	8.00000	0.336861
$565$	0	0
$566$	−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	$0.683466\pi$
$570$	0	0
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	−2.00000	−0.0836242
$573$	0	0
$574$	−24.0000	−1.00174
$575$	0	0
$576$	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
$580$	0	0
$581$	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
$590$	0	0
$591$	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
$595$	0	0
$596$	−18.0000	−0.737309
$597$	−32.0000	−1.30967
$598$	6.00000	0.245358
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	0	0
$601$	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
$605$	0	0
$606$	−36.0000	−1.46240
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	−30.0000	−1.21666
$609$	16.0000	0.648353
$610$	0	0
$611$	−4.00000	−0.161823
$612$	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
$615$	0	0
$616$	−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	$-0.512797\pi$
$619$	−2.00000	−0.0803868	−0.0401934	+	0.999192i	$0.512797\pi$
$620$	0	0
$621$	−24.0000	−0.963087
$622$	−4.00000	−0.160385
$623$	8.00000	0.320513
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	22.0000	0.879297
$627$	−24.0000	−0.958468
$628$	−6.00000	−0.239426
$629$	−4.00000	−0.159490
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	36.0000	1.43200
$633$	24.0000	0.953914
$634$	18.0000	0.714871
$635$	0	0
$636$	4.00000	0.158610
$637$	9.00000	0.356593
$638$	4.00000	0.158362
$639$	6.00000	0.237356
$640$	0	0
$641$	−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
$645$	0	0
$646$	12.0000	0.472134
$647$	−38.0000	−1.49393	−0.746967	−	0.664861i	$-0.768491\pi$
$647$	−38.0000	−1.49393	−0.746967	+	0.664861i	$0.768491\pi$
$648$	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	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	20.0000	0.782062
$655$	0	0
$656$	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	$0.5\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	−18.0000	−0.699590
$663$	−4.00000	−0.155347
$664$	−48.0000	−1.86276
$665$	0	0
$666$	2.00000	0.0774984
$667$	12.0000	0.464642
$668$	−12.0000	−0.464294
$669$	−8.00000	−0.309298
$670$	0	0
$671$	4.00000	0.154418
$672$	40.0000	1.54303
$673$	30.0000	1.15642	0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000	1.15642	0.578208	+	0.815890i	$0.303752\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	28.0000	1.07533
$679$	8.00000	0.307012
$680$	0	0
$681$	−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
$685$	0	0
$686$	8.00000	0.305441
$687$	−44.0000	−1.67870
$688$	10.0000	0.381246
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	−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
$700$	0	0
$701$	−26.0000	−0.982006	−0.491003	−	0.871158i	$-0.663370\pi$
$701$	−26.0000	−0.982006	−0.491003	+	0.871158i	$0.663370\pi$
$702$	−4.00000	−0.150970
$703$	−12.0000	−0.452589
$704$	14.0000	0.527645
$705$	0	0
$706$	−18.0000	−0.677439
$707$	−72.0000	−2.70784
$708$	−12.0000	−0.450988
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	−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	$0.703521\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	17.0000	0.632674
$723$	20.0000	0.743808
$724$	22.0000	0.817624
$725$	0	0
$726$	−14.0000	−0.519589
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	−12.0000	−0.444750
$729$	13.0000	0.481481
$730$	0	0
$731$	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
$735$	0	0
$736$	30.0000	1.10581
$737$	8.00000	0.294684
$738$	−6.00000	−0.220863
$739$	6.00000	0.220714	0.110357	−	0.993892i	$-0.464801\pi$
$739$	6.00000	0.220714	0.110357	+	0.993892i	$0.464801\pi$
$740$	0	0
$741$	−12.0000	−0.440831
$742$	−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
$745$	0	0
$746$	−34.0000	−1.24483
$747$	16.0000	0.585409
$748$	4.00000	0.146254
$749$	−40.0000	−1.46157
$750$	0	0
$751$	−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
$754$	2.00000	0.0728357
$755$	0	0
$756$	16.0000	0.581914
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	10.0000	0.363216
$759$	24.0000	0.871145
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	4.00000	0.144905
$763$	40.0000	1.44810
$764$	0	0
$765$	0	0
$766$	−12.0000	−0.433578
$767$	6.00000	0.216647
$768$	−34.0000	−1.22687
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	−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
$775$	0	0
$776$	−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
$785$	0	0
$786$	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
$790$	0	0
$791$	56.0000	1.99113
$792$	−6.00000	−0.213201
$793$	2.00000	0.0710221
$794$	−6.00000	−0.212932
$795$	0	0
$796$	16.0000	0.567105
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	−48.0000	−1.69918
$799$	8.00000	0.283020
$800$	0	0
$801$	2.00000	0.0706665
$802$	10.0000	0.353112
$803$	12.0000	0.423471
$804$	−8.00000	−0.282138
$805$	0	0
$806$	−10.0000	−0.352235
$807$	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	$0.511193\pi$
$810$	0	0
$811$	−42.0000	−1.47482	−0.737410	−	0.675446i	$-0.763951\pi$
$811$	−42.0000	−1.47482	−0.737410	+	0.675446i	$0.763951\pi$
$812$	−8.00000	−0.280745
$813$	4.00000	0.140286
$814$	4.00000	0.140200
$815$	0	0
$816$	4.00000	0.140028
$817$	60.0000	2.09913
$818$	18.0000	0.629355
$819$	4.00000	0.139771
$820$	0	0
$821$	50.0000	1.74501	0.872506	−	0.488603i	$-0.162493\pi$
$821$	50.0000	1.74501	0.872506	+	0.488603i	$0.162493\pi$
$822$	4.00000	0.139516
$823$	46.0000	1.60346	0.801730	−	0.597687i	$-0.203913\pi$
$823$	46.0000	1.60346	0.801730	+	0.597687i	$0.203913\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	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	$0.298956\pi$
$830$	0	0
$831$	28.0000	0.971309
$832$	7.00000	0.242681
$833$	−18.0000	−0.623663
$834$	0	0
$835$	0	0
$836$	12.0000	0.415029
$837$	40.0000	1.38260
$838$	−40.0000	−1.38178
$839$	38.0000	1.31191	0.655953	−	0.754802i	$-0.272267\pi$
$839$	38.0000	1.31191	0.655953	+	0.754802i	$0.272267\pi$
$840$	0	0
$841$	−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
$850$	0	0
$851$	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
$855$	0	0
$856$	30.0000	1.02538
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	4.00000	0.136558
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	−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
$865$	0	0
$866$	−10.0000	−0.339814
$867$	−26.0000	−0.883006
$868$	40.0000	1.35769
$869$	−24.0000	−0.814144
$870$	0	0
$871$	4.00000	0.135535
$872$	−30.0000	−1.01593
$873$	2.00000	0.0676897
$874$	−36.0000	−1.21772
$875$	0	0
$876$	−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
$880$	0	0
$881$	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	$-0.379296\pi$
$883$	22.0000	0.740359	0.370179	+	0.928960i	$0.379296\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	−14.0000	−0.470339
$887$	−58.0000	−1.94745	−0.973725	−	0.227728i	$-0.926870\pi$
$887$	−58.0000	−1.94745	−0.973725	+	0.227728i	$0.926870\pi$
$888$	−12.0000	−0.402694
$889$	8.00000	0.268311
$890$	0	0
$891$	−22.0000	−0.737028
$892$	4.00000	0.133930
$893$	24.0000	0.803129
$894$	36.0000	1.20402
$895$	0	0
$896$	−12.0000	−0.400892
$897$	12.0000	0.400668
$898$	−6.00000	−0.200223
$899$	−20.0000	−0.667037
$900$	0	0
$901$	4.00000	0.133259
$902$	−12.0000	−0.399556
$903$	−80.0000	−2.66223
$904$	−42.0000	−1.39690
$905$	0	0
$906$	20.0000	0.664455
$907$	34.0000	1.12895	0.564476	−	0.825450i	$-0.309078\pi$
$907$	34.0000	1.12895	0.564476	+	0.825450i	$0.309078\pi$
$908$	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
$915$	0	0
$916$	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	$0.828083\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	10.0000	0.329332
$923$	6.00000	0.197492
$924$	−16.0000	−0.526361
$925$	0	0
$926$	24.0000	0.788689
$927$	−2.00000	−0.0656886
$928$	10.0000	0.328266
$929$	−38.0000	−1.24674	−0.623370	−	0.781927i	$-0.714237\pi$
$929$	−38.0000	−1.24674	−0.623370	+	0.781927i	$0.714237\pi$
$930$	0	0
$931$	−54.0000	−1.76978
$932$	10.0000	0.327561
$933$	−8.00000	−0.261908
$934$	10.0000	0.327210
$935$	0	0
$936$	−3.00000	−0.0980581
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	16.0000	0.522419
$939$	44.0000	1.43589
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	12.0000	0.390981
$943$	−36.0000	−1.17232
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−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
$949$	6.00000	0.194768
$950$	0	0
$951$	36.0000	1.16738
$952$	24.0000	0.777844
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	−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
$960$	0	0
$961$	69.0000	2.22581
$962$	2.00000	0.0644826
$963$	−10.0000	−0.322245
$964$	−10.0000	−0.322078
$965$	0	0
$966$	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
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	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
$980$	0	0
$981$	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
$985$	0	0
$986$	−4.00000	−0.127386
$987$	−32.0000	−1.01857
$988$	6.00000	0.190885
$989$	−60.0000	−1.90789
$990$	0	0
$991$	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
$995$	0	0
$996$	−32.0000	−1.01396
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	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	325.2.a.d.1.1		1
3.2	odd	2		2925.2.a.f.1.1		1
4.3	odd	2		5200.2.a.d.1.1		1
5.2	odd	4		325.2.b.b.274.2		2
5.3	odd	4		325.2.b.b.274.1		2
5.4	even	2		65.2.a.a.1.1	✓	1
13.12	even	2		4225.2.a.g.1.1		1
15.2	even	4		2925.2.c.h.2224.1		2
15.8	even	4		2925.2.c.h.2224.2		2
15.14	odd	2		585.2.a.h.1.1		1
20.19	odd	2		1040.2.a.f.1.1		1
35.34	odd	2		3185.2.a.e.1.1		1
40.19	odd	2		4160.2.a.f.1.1		1
40.29	even	2		4160.2.a.q.1.1		1
55.54	odd	2		7865.2.a.c.1.1		1
60.59	even	2		9360.2.a.ca.1.1		1
65.4	even	6		845.2.e.a.146.1		2
65.9	even	6		845.2.e.b.146.1		2
65.19	odd	12		845.2.m.b.361.2		4
65.24	odd	12		845.2.m.b.316.2		4
65.29	even	6		845.2.e.b.191.1		2
65.34	odd	4		845.2.c.a.506.1		2
65.44	odd	4		845.2.c.a.506.2		2
65.49	even	6		845.2.e.a.191.1		2
65.54	odd	12		845.2.m.b.316.1		4
65.59	odd	12		845.2.m.b.361.1		4
65.64	even	2		845.2.a.a.1.1		1
195.194	odd	2		7605.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.a.1.1	✓	1	5.4	even	2
325.2.a.d.1.1		1	1.1	even	1	trivial
325.2.b.b.274.1		2	5.3	odd	4
325.2.b.b.274.2		2	5.2	odd	4
585.2.a.h.1.1		1	15.14	odd	2
845.2.a.a.1.1		1	65.64	even	2
845.2.c.a.506.1		2	65.34	odd	4
845.2.c.a.506.2		2	65.44	odd	4
845.2.e.a.146.1		2	65.4	even	6
845.2.e.a.191.1		2	65.49	even	6
845.2.e.b.146.1		2	65.9	even	6
845.2.e.b.191.1		2	65.29	even	6
845.2.m.b.316.1		4	65.54	odd	12
845.2.m.b.316.2		4	65.24	odd	12
845.2.m.b.361.1		4	65.59	odd	12
845.2.m.b.361.2		4	65.19	odd	12
1040.2.a.f.1.1		1	20.19	odd	2
2925.2.a.f.1.1		1	3.2	odd	2
2925.2.c.h.2224.1		2	15.2	even	4
2925.2.c.h.2224.2		2	15.8	even	4
3185.2.a.e.1.1		1	35.34	odd	2
4160.2.a.f.1.1		1	40.19	odd	2
4160.2.a.q.1.1		1	40.29	even	2
4225.2.a.g.1.1		1	13.12	even	2
5200.2.a.d.1.1		1	4.3	odd	2
7605.2.a.f.1.1		1	195.194	odd	2
7865.2.a.c.1.1		1	55.54	odd	2
9360.2.a.ca.1.1		1	60.59	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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