LMFDB - Embedded newform 2275.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2275,2,Mod(1,2275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2275, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2275 = 5^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2275.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:	$18.1659664598$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 455)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +6.00000 q^{11} -2.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +2.00000 q^{21} +6.00000 q^{22} -6.00000 q^{24} +1.00000 q^{26} -4.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +8.00000 q^{31} +5.00000 q^{32} +12.0000 q^{33} +2.00000 q^{34} -1.00000 q^{36} -2.00000 q^{37} +2.00000 q^{39} -4.00000 q^{41} +2.00000 q^{42} +4.00000 q^{43} -6.00000 q^{44} +12.0000 q^{47} -2.00000 q^{48} +1.00000 q^{49} +4.00000 q^{51} -1.00000 q^{52} -4.00000 q^{54} -3.00000 q^{56} -2.00000 q^{58} +4.00000 q^{59} -14.0000 q^{61} +8.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} +12.0000 q^{66} +4.00000 q^{67} -2.00000 q^{68} -2.00000 q^{71} -3.00000 q^{72} +6.00000 q^{73} -2.00000 q^{74} +6.00000 q^{77} +2.00000 q^{78} +16.0000 q^{79} -11.0000 q^{81} -4.00000 q^{82} +4.00000 q^{83} -2.00000 q^{84} +4.00000 q^{86} -4.00000 q^{87} -18.0000 q^{88} +1.00000 q^{91} +16.0000 q^{93} +12.0000 q^{94} +10.0000 q^{96} -10.0000 q^{97} +1.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	−2.00000	−0.577350
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.00000	0.267261
$15$	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
$21$	2.00000	0.436436
$22$	6.00000	1.27920
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−6.00000	−1.22474
$25$	0	0
$26$	1.00000	0.196116
$27$	−4.00000	−0.769800
$28$	−1.00000	−0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	5.00000	0.883883
$33$	12.0000	2.08893
$34$	2.00000	0.342997
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	2.00000	0.308607
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	−2.00000	−0.288675
$49$	1.00000	0.142857
$50$	0	0
$51$	4.00000	0.560112
$52$	−1.00000	−0.138675
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	−3.00000	−0.400892
$57$	0	0
$58$	−2.00000	−0.262613
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	8.00000	1.01600
$63$	1.00000	0.125988
$64$	7.00000	0.875000
$65$	0	0
$66$	12.0000	1.47710
$67$	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
$69$	0	0
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	−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
$76$	0	0
$77$	6.00000	0.683763
$78$	2.00000	0.226455
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	−4.00000	−0.441726
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	4.00000	0.431331
$87$	−4.00000	−0.428845
$88$	−18.0000	−1.91881
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	16.0000	1.65912
$94$	12.0000	1.23771
$95$	0	0
$96$	10.0000	1.02062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	1.00000	0.101015
$99$	6.00000	0.603023
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	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
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	4.00000	0.384900
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	−1.00000	−0.0944911
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	0	0
$116$	2.00000	0.185695
$117$	1.00000	0.0924500
$118$	4.00000	0.368230
$119$	2.00000	0.183340
$120$	0	0
$121$	25.0000	2.27273
$122$	−14.0000	−1.26750
$123$	−8.00000	−0.721336
$124$	−8.00000	−0.718421
$125$	0	0
$126$	1.00000	0.0890871
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−3.00000	−0.265165
$129$	8.00000	0.704361
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	−12.0000	−1.04447
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	24.0000	2.02116
$142$	−2.00000	−0.167836
$143$	6.00000	0.501745
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	6.00000	0.496564
$147$	2.00000	0.164957
$148$	2.00000	0.164399
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	6.00000	0.483494
$155$	0	0
$156$	−2.00000	−0.160128
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	16.0000	1.27289
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−11.0000	−0.864242
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	4.00000	0.310460
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−6.00000	−0.462910
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−4.00000	−0.303239
$175$	0	0
$176$	−6.00000	−0.452267
$177$	8.00000	0.601317
$178$	0	0
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	1.00000	0.0741249
$183$	−28.0000	−2.06982
$184$	0	0
$185$	0	0
$186$	16.0000	1.17318
$187$	12.0000	0.877527
$188$	−12.0000	−0.875190
$189$	−4.00000	−0.290957
$190$	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
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	6.00000	0.426401
$199$	−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
$203$	−2.00000	−0.140372
$204$	−4.00000	−0.280056
$205$	0	0
$206$	−2.00000	−0.139347
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	−4.00000	−0.274075
$214$	−4.00000	−0.273434
$215$	0	0
$216$	12.0000	0.816497
$217$	8.00000	0.543075
$218$	14.0000	0.948200
$219$	12.0000	0.810885
$220$	0	0
$221$	2.00000	0.134535
$222$	−4.00000	−0.268462
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	5.00000	0.334077
$225$	0	0
$226$	−12.0000	−0.798228
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	−28.0000	−1.85029	−0.925146	−	0.379611i	$-0.876058\pi$
$229$	−28.0000	−1.85029	−0.925146	+	0.379611i	$0.876058\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	6.00000	0.393919
$233$	−16.0000	−1.04819	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000	−1.04819	−0.524097	+	0.851658i	$0.675597\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	−4.00000	−0.260378
$237$	32.0000	2.07862
$238$	2.00000	0.129641
$239$	18.0000	1.16432	0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000	1.16432	0.582162	+	0.813073i	$0.302207\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	25.0000	1.60706
$243$	−10.0000	−0.641500
$244$	14.0000	0.896258
$245$	0	0
$246$	−8.00000	−0.510061
$247$	0	0
$248$	−24.0000	−1.52400
$249$	8.00000	0.506979
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	−1.00000	−0.0629941
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	8.00000	0.498058
$259$	−2.00000	−0.124274
$260$	0	0
$261$	−2.00000	−0.123797
$262$	−20.0000	−1.23560
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	−36.0000	−2.21565
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−4.00000	−0.244339
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	−2.00000	−0.121268
$273$	2.00000	0.121046
$274$	−18.0000	−1.08742
$275$	0	0
$276$	0	0
$277$	−20.0000	−1.20168	−0.600842	−	0.799368i	$-0.705168\pi$
$277$	−20.0000	−1.20168	−0.600842	+	0.799368i	$0.705168\pi$
$278$	−4.00000	−0.239904
$279$	8.00000	0.478947
$280$	0	0
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	24.0000	1.42918
$283$	10.0000	0.594438	0.297219	−	0.954809i	$-0.403941\pi$
$283$	10.0000	0.594438	0.297219	+	0.954809i	$0.403941\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	6.00000	0.354787
$287$	−4.00000	−0.236113
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−20.0000	−1.17242
$292$	−6.00000	−0.351123
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	2.00000	0.116642
$295$	0	0
$296$	6.00000	0.348743
$297$	−24.0000	−1.39262
$298$	−14.0000	−0.810998
$299$	0	0
$300$	0	0
$301$	4.00000	0.230556
$302$	−10.0000	−0.575435
$303$	36.0000	2.06815
$304$	0	0
$305$	0	0
$306$	2.00000	0.114332
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	−6.00000	−0.341882
$309$	−4.00000	−0.227552
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	−6.00000	−0.339683
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	−16.0000	−0.900070
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	−8.00000	−0.446516
$322$	0	0
$323$	0	0
$324$	11.0000	0.611111
$325$	0	0
$326$	20.0000	1.10770
$327$	28.0000	1.54840
$328$	12.0000	0.662589
$329$	12.0000	0.661581
$330$	0	0
$331$	6.00000	0.329790	0.164895	−	0.986311i	$-0.447272\pi$
$331$	6.00000	0.329790	0.164895	+	0.986311i	$0.447272\pi$
$332$	−4.00000	−0.219529
$333$	−2.00000	−0.109599
$334$	0	0
$335$	0	0
$336$	−2.00000	−0.109109
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	1.00000	0.0543928
$339$	−24.0000	−1.30350
$340$	0	0
$341$	48.0000	2.59935
$342$	0	0
$343$	1.00000	0.0539949
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	4.00000	0.214423
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	30.0000	1.59901
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	8.00000	0.425195
$355$	0	0
$356$	0	0
$357$	4.00000	0.211702
$358$	16.0000	0.845626
$359$	34.0000	1.79445	0.897226	−	0.441572i	$-0.145579\pi$
$359$	34.0000	1.79445	0.897226	+	0.441572i	$0.145579\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	6.00000	0.315353
$363$	50.0000	2.62432
$364$	−1.00000	−0.0524142
$365$	0	0
$366$	−28.0000	−1.46358
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	0	0
$372$	−16.0000	−0.829561
$373$	−12.0000	−0.621336	−0.310668	−	0.950518i	$-0.600553\pi$
$373$	−12.0000	−0.621336	−0.310668	+	0.950518i	$0.600553\pi$
$374$	12.0000	0.620505
$375$	0	0
$376$	−36.0000	−1.85656
$377$	−2.00000	−0.103005
$378$	−4.00000	−0.205738
$379$	−22.0000	−1.13006	−0.565032	−	0.825069i	$-0.691136\pi$
$379$	−22.0000	−1.13006	−0.565032	+	0.825069i	$0.691136\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	−12.0000	−0.613973
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	−6.00000	−0.306186
$385$	0	0
$386$	−22.0000	−1.11977
$387$	4.00000	0.203331
$388$	10.0000	0.507673
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	0	0
$392$	−3.00000	−0.151523
$393$	−40.0000	−2.01773
$394$	−2.00000	−0.100759
$395$	0	0
$396$	−6.00000	−0.301511
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	8.00000	0.399004
$403$	8.00000	0.398508
$404$	−18.0000	−0.895533
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	−12.0000	−0.594818
$408$	−12.0000	−0.594089
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	0	0
$411$	−36.0000	−1.77575
$412$	2.00000	0.0985329
$413$	4.00000	0.196827
$414$	0	0
$415$	0	0
$416$	5.00000	0.245145
$417$	−8.00000	−0.391762
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	8.00000	0.389434
$423$	12.0000	0.583460
$424$	0	0
$425$	0	0
$426$	−4.00000	−0.193801
$427$	−14.0000	−0.677507
$428$	4.00000	0.193347
$429$	12.0000	0.579365
$430$	0	0
$431$	−38.0000	−1.83040	−0.915198	−	0.403005i	$-0.867966\pi$
$431$	−38.0000	−1.83040	−0.915198	+	0.403005i	$0.867966\pi$
$432$	4.00000	0.192450
$433$	22.0000	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$433$	22.0000	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	−14.0000	−0.670478
$437$	0	0
$438$	12.0000	0.573382
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	2.00000	0.0951303
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	−24.0000	−1.13643
$447$	−28.0000	−1.32435
$448$	7.00000	0.330719
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	12.0000	0.564433
$453$	−20.0000	−0.939682
$454$	−20.0000	−0.938647
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−28.0000	−1.30835
$459$	−8.00000	−0.373408
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	12.0000	0.558291
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−16.0000	−0.741186
$467$	30.0000	1.38823	0.694117	−	0.719862i	$-0.255795\pi$
$467$	30.0000	1.38823	0.694117	+	0.719862i	$0.255795\pi$
$468$	−1.00000	−0.0462250
$469$	4.00000	0.184703
$470$	0	0
$471$	−36.0000	−1.65879
$472$	−12.0000	−0.552345
$473$	24.0000	1.10352
$474$	32.0000	1.46981
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	0	0
$478$	18.0000	0.823301
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	−8.00000	−0.364390
$483$	0	0
$484$	−25.0000	−1.13636
$485$	0	0
$486$	−10.0000	−0.453609
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	42.0000	1.90125
$489$	40.0000	1.80886
$490$	0	0
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	8.00000	0.360668
$493$	−4.00000	−0.180151
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−2.00000	−0.0897123
$498$	8.00000	0.358489
$499$	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
$501$	0	0
$502$	12.0000	0.535586
$503$	42.0000	1.87269	0.936344	−	0.351085i	$-0.114187\pi$
$503$	42.0000	1.87269	0.936344	+	0.351085i	$0.114187\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	0	0
$507$	2.00000	0.0888231
$508$	8.00000	0.354943
$509$	12.0000	0.531891	0.265945	−	0.963988i	$-0.414316\pi$
$509$	12.0000	0.531891	0.265945	+	0.963988i	$0.414316\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−22.0000	−0.970378
$515$	0	0
$516$	−8.00000	−0.352180
$517$	72.0000	3.16656
$518$	−2.00000	−0.0878750
$519$	−12.0000	−0.526742
$520$	0	0
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	−2.00000	−0.0875376
$523$	−26.0000	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$523$	−26.0000	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$524$	20.0000	0.873704
$525$	0	0
$526$	12.0000	0.523225
$527$	16.0000	0.696971
$528$	−12.0000	−0.522233
$529$	−23.0000	−1.00000
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	−4.00000	−0.173259
$534$	0	0
$535$	0	0
$536$	−12.0000	−0.518321
$537$	32.0000	1.38090
$538$	10.0000	0.431131
$539$	6.00000	0.258438
$540$	0	0
$541$	42.0000	1.80572	0.902861	−	0.429934i	$-0.141463\pi$
$541$	42.0000	1.80572	0.902861	+	0.429934i	$0.141463\pi$
$542$	4.00000	0.171815
$543$	12.0000	0.514969
$544$	10.0000	0.428746
$545$	0	0
$546$	2.00000	0.0855921
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	18.0000	0.768922
$549$	−14.0000	−0.597505
$550$	0	0
$551$	0	0
$552$	0	0
$553$	16.0000	0.680389
$554$	−20.0000	−0.849719
$555$	0	0
$556$	4.00000	0.169638
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	8.00000	0.338667
$559$	4.00000	0.169182
$560$	0	0
$561$	24.0000	1.01328
$562$	−14.0000	−0.590554
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	−24.0000	−1.01058
$565$	0	0
$566$	10.0000	0.420331
$567$	−11.0000	−0.461957
$568$	6.00000	0.251754
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	−6.00000	−0.250873
$573$	−24.0000	−1.00261
$574$	−4.00000	−0.166957
$575$	0	0
$576$	7.00000	0.291667
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	−13.0000	−0.540729
$579$	−44.0000	−1.82858
$580$	0	0
$581$	4.00000	0.165948
$582$	−20.0000	−0.829027
$583$	0	0
$584$	−18.0000	−0.744845
$585$	0	0
$586$	−18.0000	−0.743573
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	−2.00000	−0.0824786
$589$	0	0
$590$	0	0
$591$	−4.00000	−0.164538
$592$	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	$0.592809\pi$
$594$	−24.0000	−0.984732
$595$	0	0
$596$	14.0000	0.573462
$597$	−32.0000	−1.30967
$598$	0	0
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	34.0000	1.38689	0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000	1.38689	0.693444	+	0.720510i	$0.256092\pi$
$602$	4.00000	0.163028
$603$	4.00000	0.162893
$604$	10.0000	0.406894
$605$	0	0
$606$	36.0000	1.46240
$607$	18.0000	0.730597	0.365299	−	0.930890i	$-0.380967\pi$
$607$	18.0000	0.730597	0.365299	+	0.930890i	$0.380967\pi$
$608$	0	0
$609$	−4.00000	−0.162088
$610$	0	0
$611$	12.0000	0.485468
$612$	−2.00000	−0.0808452
$613$	−46.0000	−1.85792	−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000	−1.85792	−0.928961	+	0.370177i	$0.879297\pi$
$614$	32.0000	1.29141
$615$	0	0
$616$	−18.0000	−0.725241
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	−4.00000	−0.160904
$619$	16.0000	0.643094	0.321547	−	0.946894i	$-0.395797\pi$
$619$	16.0000	0.643094	0.321547	+	0.946894i	$0.395797\pi$
$620$	0	0
$621$	0	0
$622$	24.0000	0.962312
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	6.00000	0.239808
$627$	0	0
$628$	18.0000	0.718278
$629$	−4.00000	−0.159490
$630$	0	0
$631$	22.0000	0.875806	0.437903	−	0.899022i	$-0.355721\pi$
$631$	22.0000	0.875806	0.437903	+	0.899022i	$0.355721\pi$
$632$	−48.0000	−1.90934
$633$	16.0000	0.635943
$634$	18.0000	0.714871
$635$	0	0
$636$	0	0
$637$	1.00000	0.0396214
$638$	−12.0000	−0.475085
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	−14.0000	−0.552967	−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000	−0.552967	−0.276483	+	0.961019i	$0.589169\pi$
$642$	−8.00000	−0.315735
$643$	8.00000	0.315489	0.157745	−	0.987480i	$-0.449578\pi$
$643$	8.00000	0.315489	0.157745	+	0.987480i	$0.449578\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.0000	0.550397	0.275198	−	0.961387i	$-0.411256\pi$
$647$	14.0000	0.550397	0.275198	+	0.961387i	$0.411256\pi$
$648$	33.0000	1.29636
$649$	24.0000	0.942082
$650$	0	0
$651$	16.0000	0.627089
$652$	−20.0000	−0.783260
$653$	−8.00000	−0.313064	−0.156532	−	0.987673i	$-0.550031\pi$
$653$	−8.00000	−0.313064	−0.156532	+	0.987673i	$0.550031\pi$
$654$	28.0000	1.09489
$655$	0	0
$656$	4.00000	0.156174
$657$	6.00000	0.234082
$658$	12.0000	0.467809
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	−24.0000	−0.933492	−0.466746	−	0.884391i	$-0.654574\pi$
$661$	−24.0000	−0.933492	−0.466746	+	0.884391i	$0.654574\pi$
$662$	6.00000	0.233197
$663$	4.00000	0.155347
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	0	0
$669$	−48.0000	−1.85579
$670$	0	0
$671$	−84.0000	−3.24278
$672$	10.0000	0.385758
$673$	36.0000	1.38770	0.693849	−	0.720121i	$-0.255914\pi$
$673$	36.0000	1.38770	0.693849	+	0.720121i	$0.255914\pi$
$674$	−12.0000	−0.462223
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	−26.0000	−0.999261	−0.499631	−	0.866239i	$-0.666531\pi$
$677$	−26.0000	−0.999261	−0.499631	+	0.866239i	$0.666531\pi$
$678$	−24.0000	−0.921714
$679$	−10.0000	−0.383765
$680$	0	0
$681$	−40.0000	−1.53280
$682$	48.0000	1.83801
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	0	0
$686$	1.00000	0.0381802
$687$	−56.0000	−2.13653
$688$	−4.00000	−0.152499
$689$	0	0
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	6.00000	0.228086
$693$	6.00000	0.227921
$694$	−24.0000	−0.911028
$695$	0	0
$696$	12.0000	0.454859
$697$	−8.00000	−0.303022
$698$	16.0000	0.605609
$699$	−32.0000	−1.21035
$700$	0	0
$701$	38.0000	1.43524	0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000	1.43524	0.717620	+	0.696435i	$0.245231\pi$
$702$	−4.00000	−0.150970
$703$	0	0
$704$	42.0000	1.58293
$705$	0	0
$706$	−14.0000	−0.526897
$707$	18.0000	0.676960
$708$	−8.00000	−0.300658
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	0	0
$714$	4.00000	0.149696
$715$	0	0
$716$	−16.0000	−0.597948
$717$	36.0000	1.34444
$718$	34.0000	1.26887
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−2.00000	−0.0744839
$722$	−19.0000	−0.707107
$723$	−16.0000	−0.595046
$724$	−6.00000	−0.222988
$725$	0	0
$726$	50.0000	1.85567
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	−3.00000	−0.111187
$729$	13.0000	0.481481
$730$	0	0
$731$	8.00000	0.295891
$732$	28.0000	1.03491
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	18.0000	0.664392
$735$	0	0
$736$	0	0
$737$	24.0000	0.884051
$738$	−4.00000	−0.147242
$739$	−38.0000	−1.39785	−0.698926	−	0.715194i	$-0.746338\pi$
$739$	−38.0000	−1.39785	−0.698926	+	0.715194i	$0.746338\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−40.0000	−1.46746	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−40.0000	−1.46746	−0.733729	+	0.679442i	$0.762222\pi$
$744$	−48.0000	−1.75977
$745$	0	0
$746$	−12.0000	−0.439351
$747$	4.00000	0.146352
$748$	−12.0000	−0.438763
$749$	−4.00000	−0.146157
$750$	0	0
$751$	36.0000	1.31366	0.656829	−	0.754039i	$-0.271897\pi$
$751$	36.0000	1.31366	0.656829	+	0.754039i	$0.271897\pi$
$752$	−12.0000	−0.437595
$753$	24.0000	0.874609
$754$	−2.00000	−0.0728357
$755$	0	0
$756$	4.00000	0.145479
$757$	−20.0000	−0.726912	−0.363456	−	0.931611i	$-0.618403\pi$
$757$	−20.0000	−0.726912	−0.363456	+	0.931611i	$0.618403\pi$
$758$	−22.0000	−0.799076
$759$	0	0
$760$	0	0
$761$	−28.0000	−1.01500	−0.507500	−	0.861652i	$-0.669430\pi$
$761$	−28.0000	−1.01500	−0.507500	+	0.861652i	$0.669430\pi$
$762$	−16.0000	−0.579619
$763$	14.0000	0.506834
$764$	12.0000	0.434145
$765$	0	0
$766$	20.0000	0.722629
$767$	4.00000	0.144432
$768$	−34.0000	−1.22687
$769$	12.0000	0.432731	0.216366	−	0.976312i	$-0.430580\pi$
$769$	12.0000	0.432731	0.216366	+	0.976312i	$0.430580\pi$
$770$	0	0
$771$	−44.0000	−1.58462
$772$	22.0000	0.791797
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	30.0000	1.07694
$777$	−4.00000	−0.143499
$778$	2.00000	0.0717035
$779$	0	0
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	8.00000	0.285897
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	−40.0000	−1.42675
$787$	16.0000	0.570338	0.285169	−	0.958477i	$-0.407950\pi$
$787$	16.0000	0.570338	0.285169	+	0.958477i	$0.407950\pi$
$788$	2.00000	0.0712470
$789$	24.0000	0.854423
$790$	0	0
$791$	−12.0000	−0.426671
$792$	−18.0000	−0.639602
$793$	−14.0000	−0.497155
$794$	2.00000	0.0709773
$795$	0	0
$796$	16.0000	0.567105
$797$	−14.0000	−0.495905	−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000	−0.495905	−0.247953	+	0.968772i	$0.579758\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	0	0
$802$	6.00000	0.211867
$803$	36.0000	1.27041
$804$	−8.00000	−0.282138
$805$	0	0
$806$	8.00000	0.281788
$807$	20.0000	0.704033
$808$	−54.0000	−1.89971
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	2.00000	0.0701862
$813$	8.00000	0.280572
$814$	−12.0000	−0.420600
$815$	0	0
$816$	−4.00000	−0.140028
$817$	0	0
$818$	8.00000	0.279713
$819$	1.00000	0.0349428
$820$	0	0
$821$	46.0000	1.60541	0.802706	−	0.596376i	$-0.203393\pi$
$821$	46.0000	1.60541	0.802706	+	0.596376i	$0.203393\pi$
$822$	−36.0000	−1.25564
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	4.00000	0.139178
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	0	0
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	0	0
$831$	−40.0000	−1.38758
$832$	7.00000	0.242681
$833$	2.00000	0.0692959
$834$	−8.00000	−0.277017
$835$	0	0
$836$	0	0
$837$	−32.0000	−1.10608
$838$	−12.0000	−0.414533
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	6.00000	0.206774
$843$	−28.0000	−0.964371
$844$	−8.00000	−0.275371
$845$	0	0
$846$	12.0000	0.412568
$847$	25.0000	0.859010
$848$	0	0
$849$	20.0000	0.686398
$850$	0	0
$851$	0	0
$852$	4.00000	0.137038
$853$	−18.0000	−0.616308	−0.308154	−	0.951336i	$-0.599711\pi$
$853$	−18.0000	−0.616308	−0.308154	+	0.951336i	$0.599711\pi$
$854$	−14.0000	−0.479070
$855$	0	0
$856$	12.0000	0.410152
$857$	30.0000	1.02478	0.512390	−	0.858753i	$-0.328760\pi$
$857$	30.0000	1.02478	0.512390	+	0.858753i	$0.328760\pi$
$858$	12.0000	0.409673
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	−38.0000	−1.29429
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	−20.0000	−0.680414
$865$	0	0
$866$	22.0000	0.747590
$867$	−26.0000	−0.883006
$868$	−8.00000	−0.271538
$869$	96.0000	3.25658
$870$	0	0
$871$	4.00000	0.135535
$872$	−42.0000	−1.42230
$873$	−10.0000	−0.338449
$874$	0	0
$875$	0	0
$876$	−12.0000	−0.405442
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	24.0000	0.809961
$879$	−36.0000	−1.21425
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	1.00000	0.0336718
$883$	−24.0000	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$883$	−24.0000	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	−20.0000	−0.671913
$887$	46.0000	1.54453	0.772264	−	0.635301i	$-0.219124\pi$
$887$	46.0000	1.54453	0.772264	+	0.635301i	$0.219124\pi$
$888$	12.0000	0.402694
$889$	−8.00000	−0.268311
$890$	0	0
$891$	−66.0000	−2.21108
$892$	24.0000	0.803579
$893$	0	0
$894$	−28.0000	−0.936460
$895$	0	0
$896$	−3.00000	−0.100223
$897$	0	0
$898$	−14.0000	−0.467186
$899$	−16.0000	−0.533630
$900$	0	0
$901$	0	0
$902$	−24.0000	−0.799113
$903$	8.00000	0.266223
$904$	36.0000	1.19734
$905$	0	0
$906$	−20.0000	−0.664455
$907$	−16.0000	−0.531271	−0.265636	−	0.964073i	$-0.585582\pi$
$907$	−16.0000	−0.531271	−0.265636	+	0.964073i	$0.585582\pi$
$908$	20.0000	0.663723
$909$	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	$0.792611\pi$
$912$	0	0
$913$	24.0000	0.794284
$914$	−22.0000	−0.727695
$915$	0	0
$916$	28.0000	0.925146
$917$	−20.0000	−0.660458
$918$	−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
$921$	64.0000	2.10887
$922$	0	0
$923$	−2.00000	−0.0658308
$924$	−12.0000	−0.394771
$925$	0	0
$926$	−24.0000	−0.788689
$927$	−2.00000	−0.0656886
$928$	−10.0000	−0.328266
$929$	28.0000	0.918650	0.459325	−	0.888268i	$-0.348091\pi$
$929$	28.0000	0.918650	0.459325	+	0.888268i	$0.348091\pi$
$930$	0	0
$931$	0	0
$932$	16.0000	0.524097
$933$	48.0000	1.57145
$934$	30.0000	0.981630
$935$	0	0
$936$	−3.00000	−0.0980581
$937$	58.0000	1.89478	0.947389	−	0.320085i	$-0.103712\pi$
$937$	58.0000	1.89478	0.947389	+	0.320085i	$0.103712\pi$
$938$	4.00000	0.130605
$939$	12.0000	0.391605
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	−36.0000	−1.17294
$943$	0	0
$944$	−4.00000	−0.130189
$945$	0	0
$946$	24.0000	0.780307
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	−32.0000	−1.03931
$949$	6.00000	0.194768
$950$	0	0
$951$	36.0000	1.16738
$952$	−6.00000	−0.194461
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	0	0
$955$	0	0
$956$	−18.0000	−0.582162
$957$	−24.0000	−0.775810
$958$	36.0000	1.16311
$959$	−18.0000	−0.581250
$960$	0	0
$961$	33.0000	1.06452
$962$	−2.00000	−0.0644826
$963$	−4.00000	−0.128898
$964$	8.00000	0.257663
$965$	0	0
$966$	0	0
$967$	56.0000	1.80084	0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000	1.80084	0.900419	+	0.435023i	$0.143260\pi$
$968$	−75.0000	−2.41059
$969$	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
$973$	−4.00000	−0.128234
$974$	24.0000	0.769010
$975$	0	0
$976$	14.0000	0.448129
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	40.0000	1.27906
$979$	0	0
$980$	0	0
$981$	14.0000	0.446986
$982$	4.00000	0.127645
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	24.0000	0.765092
$985$	0	0
$986$	−4.00000	−0.127386
$987$	24.0000	0.763928
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−44.0000	−1.39771	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000	−1.39771	−0.698853	+	0.715265i	$0.746306\pi$
$992$	40.0000	1.27000
$993$	12.0000	0.380808
$994$	−2.00000	−0.0634361
$995$	0	0
$996$	−8.00000	−0.253490
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	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	2275.2.a.f.1.1		1
5.2	odd	4		455.2.c.a.274.2	yes	2
5.3	odd	4		455.2.c.a.274.1	✓	2
5.4	even	2		2275.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
455.2.c.a.274.1	✓	2	5.3	odd	4
455.2.c.a.274.2	yes	2	5.2	odd	4
2275.2.a.b.1.1		1	5.4	even	2
2275.2.a.f.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2275 = 5^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2275.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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