LMFDB - Embedded newform 3025.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3025,2,Mod(1,3025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3025, 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("3025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3025 = 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3025.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:	$24.1547466114$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 121)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} +2.00000 q^{6} +2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{12} -1.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} +5.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} -4.00000 q^{21} -2.00000 q^{23} -6.00000 q^{24} +1.00000 q^{26} +4.00000 q^{27} -2.00000 q^{28} +9.00000 q^{29} -2.00000 q^{31} -5.00000 q^{32} -5.00000 q^{34} -1.00000 q^{36} +3.00000 q^{37} -6.00000 q^{38} +2.00000 q^{39} -5.00000 q^{41} +4.00000 q^{42} +2.00000 q^{46} -2.00000 q^{47} +2.00000 q^{48} -3.00000 q^{49} -10.0000 q^{51} +1.00000 q^{52} -9.00000 q^{53} -4.00000 q^{54} +6.00000 q^{56} -12.0000 q^{57} -9.00000 q^{58} +8.00000 q^{59} +6.00000 q^{61} +2.00000 q^{62} +2.00000 q^{63} +7.00000 q^{64} -2.00000 q^{67} -5.00000 q^{68} +4.00000 q^{69} +12.0000 q^{71} +3.00000 q^{72} +2.00000 q^{73} -3.00000 q^{74} -6.00000 q^{76} -2.00000 q^{78} -10.0000 q^{79} -11.0000 q^{81} +5.00000 q^{82} -6.00000 q^{83} +4.00000 q^{84} -18.0000 q^{87} -9.00000 q^{89} -2.00000 q^{91} +2.00000 q^{92} +4.00000 q^{93} +2.00000 q^{94} +10.0000 q^{96} +13.0000 q^{97} +3.00000 q^{98} +O(q^{100})\)

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	2.00000	0.577350
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	−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	$0.258380\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	−6.00000	−1.22474
$25$	0	0
$26$	1.00000	0.196116
$27$	4.00000	0.769800
$28$	−2.00000	−0.377964
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	−5.00000	−0.857493
$35$	0	0
$36$	−1.00000	−0.166667
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	−6.00000	−0.973329
$39$	2.00000	0.320256
$40$	0	0
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	4.00000	0.617213
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	2.00000	0.294884
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	2.00000	0.288675
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−10.0000	−1.40028
$52$	1.00000	0.138675
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	6.00000	0.801784
$57$	−12.0000	−1.58944
$58$	−9.00000	−1.18176
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	2.00000	0.254000
$63$	2.00000	0.251976
$64$	7.00000	0.875000
$65$	0	0
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−5.00000	−0.606339
$69$	4.00000	0.481543
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	3.00000	0.353553
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−3.00000	−0.348743
$75$	0	0
$76$	−6.00000	−0.688247
$77$	0	0
$78$	−2.00000	−0.226455
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	5.00000	0.552158
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	0	0
$87$	−18.0000	−1.92980
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	2.00000	0.208514
$93$	4.00000	0.414781
$94$	2.00000	0.206284
$95$	0	0
$96$	10.0000	1.02062
$97$	13.0000	1.31995	0.659975	−	0.751288i	$-0.270567\pi$
$97$	13.0000	1.31995	0.659975	+	0.751288i	$0.270567\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	10.0000	0.990148
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	9.00000	0.874157
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−4.00000	−0.384900
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	−2.00000	−0.188982
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	12.0000	1.12390
$115$	0	0
$116$	−9.00000	−0.835629
$117$	−1.00000	−0.0924500
$118$	−8.00000	−0.736460
$119$	10.0000	0.916698
$120$	0	0
$121$	0	0
$122$	−6.00000	−0.543214
$123$	10.0000	0.901670
$124$	2.00000	0.179605
$125$	0	0
$126$	−2.00000	−0.178174
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	2.00000	0.172774
$135$	0	0
$136$	15.0000	1.28624
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	−4.00000	−0.340503
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	−12.0000	−1.00702
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−2.00000	−0.165521
$147$	6.00000	0.494872
$148$	−3.00000	−0.246598
$149$	17.0000	1.39269	0.696347	−	0.717705i	$-0.254807\pi$
$149$	17.0000	1.39269	0.696347	+	0.717705i	$0.254807\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	18.0000	1.45999
$153$	5.00000	0.404226
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	10.0000	0.795557
$159$	18.0000	1.42749
$160$	0	0
$161$	−4.00000	−0.315244
$162$	11.0000	0.864242
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	6.00000	0.465690
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−12.0000	−0.925820
$169$	−12.0000	−0.923077
$170$	0	0
$171$	6.00000	0.458831
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	18.0000	1.36458
$175$	0	0
$176$	0	0
$177$	−16.0000	−1.20263
$178$	9.00000	0.674579
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	1.00000	0.0743294	0.0371647	−	0.999309i	$-0.488167\pi$
$181$	1.00000	0.0743294	0.0371647	+	0.999309i	$0.488167\pi$
$182$	2.00000	0.148250
$183$	−12.0000	−0.887066
$184$	−6.00000	−0.442326
$185$	0	0
$186$	−4.00000	−0.293294
$187$	0	0
$188$	2.00000	0.145865
$189$	8.00000	0.581914
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	−14.0000	−1.01036
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	−13.0000	−0.933346
$195$	0	0
$196$	3.00000	0.214286
$197$	11.0000	0.783718	0.391859	−	0.920025i	$-0.371832\pi$
$197$	11.0000	0.783718	0.391859	+	0.920025i	$0.371832\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	10.0000	0.703598
$203$	18.0000	1.26335
$204$	10.0000	0.700140
$205$	0	0
$206$	8.00000	0.557386
$207$	−2.00000	−0.139010
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	9.00000	0.618123
$213$	−24.0000	−1.64445
$214$	6.00000	0.410152
$215$	0	0
$216$	12.0000	0.816497
$217$	−4.00000	−0.271538
$218$	11.0000	0.745014
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−5.00000	−0.336336
$222$	6.00000	0.402694
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	−10.0000	−0.668153
$225$	0	0
$226$	−9.00000	−0.598671
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	12.0000	0.794719
$229$	9.00000	0.594737	0.297368	−	0.954763i	$-0.403891\pi$
$229$	9.00000	0.594737	0.297368	+	0.954763i	$0.403891\pi$
$230$	0	0
$231$	0	0
$232$	27.0000	1.77264
$233$	21.0000	1.37576	0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000	1.37576	0.687878	+	0.725826i	$0.258542\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	−8.00000	−0.520756
$237$	20.0000	1.29914
$238$	−10.0000	−0.648204
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	−6.00000	−0.384111
$245$	0	0
$246$	−10.0000	−0.637577
$247$	−6.00000	−0.381771
$248$	−6.00000	−0.381000
$249$	12.0000	0.760469
$250$	0	0
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−19.0000	−1.18519	−0.592594	−	0.805502i	$-0.701896\pi$
$257$	−19.0000	−1.18519	−0.592594	+	0.805502i	$0.701896\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	9.00000	0.557086
$262$	0	0
$263$	22.0000	1.35658	0.678289	−	0.734795i	$-0.262722\pi$
$263$	22.0000	1.35658	0.678289	+	0.734795i	$0.262722\pi$
$264$	0	0
$265$	0	0
$266$	−12.0000	−0.735767
$267$	18.0000	1.10158
$268$	2.00000	0.122169
$269$	1.00000	0.0609711	0.0304855	−	0.999535i	$-0.490295\pi$
$269$	1.00000	0.0609711	0.0304855	+	0.999535i	$0.490295\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−5.00000	−0.303170
$273$	4.00000	0.242091
$274$	−10.0000	−0.604122
$275$	0	0
$276$	−4.00000	−0.240772
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	2.00000	0.119952
$279$	−2.00000	−0.119737
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	−4.00000	−0.238197
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	0	0
$287$	−10.0000	−0.590281
$288$	−5.00000	−0.294628
$289$	8.00000	0.470588
$290$	0	0
$291$	−26.0000	−1.52415
$292$	−2.00000	−0.117041
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	9.00000	0.523114
$297$	0	0
$298$	−17.0000	−0.984784
$299$	2.00000	0.115663
$300$	0	0
$301$	0	0
$302$	16.0000	0.920697
$303$	20.0000	1.14897
$304$	−6.00000	−0.344124
$305$	0	0
$306$	−5.00000	−0.285831
$307$	22.0000	1.25561	0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000	1.25561	0.627803	+	0.778372i	$0.283954\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	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
$313$	−23.0000	−1.30004	−0.650018	−	0.759918i	$-0.725239\pi$
$313$	−23.0000	−1.30004	−0.650018	+	0.759918i	$0.725239\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	10.0000	0.562544
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	−18.0000	−1.00939
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	4.00000	0.222911
$323$	30.0000	1.66924
$324$	11.0000	0.611111
$325$	0	0
$326$	−2.00000	−0.110770
$327$	22.0000	1.21660
$328$	−15.0000	−0.828236
$329$	−4.00000	−0.220527
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	6.00000	0.329293
$333$	3.00000	0.164399
$334$	12.0000	0.656611
$335$	0	0
$336$	4.00000	0.218218
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	12.0000	0.652714
$339$	−18.0000	−0.977626
$340$	0	0
$341$	0	0
$342$	−6.00000	−0.324443
$343$	−20.0000	−1.07990
$344$	0	0
$345$	0	0
$346$	6.00000	0.322562
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	18.0000	0.964901
$349$	−27.0000	−1.44528	−0.722638	−	0.691226i	$-0.757071\pi$
$349$	−27.0000	−1.44528	−0.722638	+	0.691226i	$0.757071\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	9.00000	0.479022	0.239511	−	0.970894i	$-0.423013\pi$
$353$	9.00000	0.479022	0.239511	+	0.970894i	$0.423013\pi$
$354$	16.0000	0.850390
$355$	0	0
$356$	9.00000	0.476999
$357$	−20.0000	−1.05851
$358$	−24.0000	−1.26844
$359$	−2.00000	−0.105556	−0.0527780	−	0.998606i	$-0.516808\pi$
$359$	−2.00000	−0.105556	−0.0527780	+	0.998606i	$0.516808\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−1.00000	−0.0525588
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	12.0000	0.627250
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	2.00000	0.104257
$369$	−5.00000	−0.260290
$370$	0	0
$371$	−18.0000	−0.934513
$372$	−4.00000	−0.207390
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	0	0
$376$	−6.00000	−0.309426
$377$	−9.00000	−0.463524
$378$	−8.00000	−0.411476
$379$	−32.0000	−1.64373	−0.821865	−	0.569683i	$-0.807066\pi$
$379$	−32.0000	−1.64373	−0.821865	+	0.569683i	$0.807066\pi$
$380$	0	0
$381$	−32.0000	−1.63941
$382$	−8.00000	−0.409316
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	−6.00000	−0.306186
$385$	0	0
$386$	−5.00000	−0.254493
$387$	0	0
$388$	−13.0000	−0.659975
$389$	−3.00000	−0.152106	−0.0760530	−	0.997104i	$-0.524232\pi$
$389$	−3.00000	−0.152106	−0.0760530	+	0.997104i	$0.524232\pi$
$390$	0	0
$391$	−10.0000	−0.505722
$392$	−9.00000	−0.454569
$393$	0	0
$394$	−11.0000	−0.554172
$395$	0	0
$396$	0	0
$397$	−13.0000	−0.652451	−0.326226	−	0.945292i	$-0.605777\pi$
$397$	−13.0000	−0.652451	−0.326226	+	0.945292i	$0.605777\pi$
$398$	−24.0000	−1.20301
$399$	−24.0000	−1.20150
$400$	0	0
$401$	23.0000	1.14857	0.574283	−	0.818657i	$-0.305281\pi$
$401$	23.0000	1.14857	0.574283	+	0.818657i	$0.305281\pi$
$402$	−4.00000	−0.199502
$403$	2.00000	0.0996271
$404$	10.0000	0.497519
$405$	0	0
$406$	−18.0000	−0.893325
$407$	0	0
$408$	−30.0000	−1.48522
$409$	−21.0000	−1.03838	−0.519192	−	0.854658i	$-0.673767\pi$
$409$	−21.0000	−1.03838	−0.519192	+	0.854658i	$0.673767\pi$
$410$	0	0
$411$	−20.0000	−0.986527
$412$	8.00000	0.394132
$413$	16.0000	0.787309
$414$	2.00000	0.0982946
$415$	0	0
$416$	5.00000	0.245145
$417$	4.00000	0.195881
$418$	0	0
$419$	2.00000	0.0977064	0.0488532	−	0.998806i	$-0.484443\pi$
$419$	2.00000	0.0977064	0.0488532	+	0.998806i	$0.484443\pi$
$420$	0	0
$421$	13.0000	0.633581	0.316791	−	0.948495i	$-0.397395\pi$
$421$	13.0000	0.633581	0.316791	+	0.948495i	$0.397395\pi$
$422$	−12.0000	−0.584151
$423$	−2.00000	−0.0972433
$424$	−27.0000	−1.31124
$425$	0	0
$426$	24.0000	1.16280
$427$	12.0000	0.580721
$428$	6.00000	0.290021
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−4.00000	−0.192450
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	11.0000	0.526804
$437$	−12.0000	−0.574038
$438$	4.00000	0.191127
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	5.00000	0.237826
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	−20.0000	−0.947027
$447$	−34.0000	−1.60814
$448$	14.0000	0.661438
$449$	−13.0000	−0.613508	−0.306754	−	0.951789i	$-0.599243\pi$
$449$	−13.0000	−0.613508	−0.306754	+	0.951789i	$0.599243\pi$
$450$	0	0
$451$	0	0
$452$	−9.00000	−0.423324
$453$	32.0000	1.50349
$454$	−24.0000	−1.12638
$455$	0	0
$456$	−36.0000	−1.68585
$457$	−39.0000	−1.82434	−0.912172	−	0.409809i	$-0.865595\pi$
$457$	−39.0000	−1.82434	−0.912172	+	0.409809i	$0.865595\pi$
$458$	−9.00000	−0.420542
$459$	20.0000	0.933520
$460$	0	0
$461$	33.0000	1.53696	0.768482	−	0.639872i	$-0.221013\pi$
$461$	33.0000	1.53696	0.768482	+	0.639872i	$0.221013\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	−9.00000	−0.417815
$465$	0	0
$466$	−21.0000	−0.972806
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	1.00000	0.0462250
$469$	−4.00000	−0.184703
$470$	0	0
$471$	4.00000	0.184310
$472$	24.0000	1.10469
$473$	0	0
$474$	−20.0000	−0.918630
$475$	0	0
$476$	−10.0000	−0.458349
$477$	−9.00000	−0.412082
$478$	−6.00000	−0.274434
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−3.00000	−0.136788
$482$	−22.0000	−1.00207
$483$	8.00000	0.364013
$484$	0	0
$485$	0	0
$486$	−10.0000	−0.453609
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	18.0000	0.814822
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	−10.0000	−0.450835
$493$	45.0000	2.02670
$494$	6.00000	0.269953
$495$	0	0
$496$	2.00000	0.0898027
$497$	24.0000	1.07655
$498$	−12.0000	−0.537733
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	2.00000	0.0892644
$503$	38.0000	1.69434	0.847168	−	0.531325i	$-0.178306\pi$
$503$	38.0000	1.69434	0.847168	+	0.531325i	$0.178306\pi$
$504$	6.00000	0.267261
$505$	0	0
$506$	0	0
$507$	24.0000	1.06588
$508$	−16.0000	−0.709885
$509$	−42.0000	−1.86162	−0.930809	−	0.365507i	$-0.880896\pi$
$509$	−42.0000	−1.86162	−0.930809	+	0.365507i	$0.880896\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	11.0000	0.486136
$513$	24.0000	1.05963
$514$	19.0000	0.838054
$515$	0	0
$516$	0	0
$517$	0	0
$518$	−6.00000	−0.263625
$519$	12.0000	0.526742
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	−9.00000	−0.393919
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	0	0
$525$	0	0
$526$	−22.0000	−0.959246
$527$	−10.0000	−0.435607
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	8.00000	0.347170
$532$	−12.0000	−0.520266
$533$	5.00000	0.216574
$534$	−18.0000	−0.778936
$535$	0	0
$536$	−6.00000	−0.259161
$537$	−48.0000	−2.07135
$538$	−1.00000	−0.0431131
$539$	0	0
$540$	0	0
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	−20.0000	−0.859074
$543$	−2.00000	−0.0858282
$544$	−25.0000	−1.07187
$545$	0	0
$546$	−4.00000	−0.171184
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	−10.0000	−0.427179
$549$	6.00000	0.256074
$550$	0	0
$551$	54.0000	2.30048
$552$	12.0000	0.510754
$553$	−20.0000	−0.850487
$554$	1.00000	0.0424859
$555$	0	0
$556$	2.00000	0.0848189
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	2.00000	0.0846668
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−6.00000	−0.253095
$563$	−34.0000	−1.43293	−0.716465	−	0.697623i	$-0.754241\pi$
$563$	−34.0000	−1.43293	−0.716465	+	0.697623i	$0.754241\pi$
$564$	−4.00000	−0.168430
$565$	0	0
$566$	28.0000	1.17693
$567$	−22.0000	−0.923913
$568$	36.0000	1.51053
$569$	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
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	0	0
$573$	−16.0000	−0.668410
$574$	10.0000	0.417392
$575$	0	0
$576$	7.00000	0.291667
$577$	21.0000	0.874241	0.437121	−	0.899403i	$-0.355998\pi$
$577$	21.0000	0.874241	0.437121	+	0.899403i	$0.355998\pi$
$578$	−8.00000	−0.332756
$579$	−10.0000	−0.415586
$580$	0	0
$581$	−12.0000	−0.497844
$582$	26.0000	1.07773
$583$	0	0
$584$	6.00000	0.248282
$585$	0	0
$586$	9.00000	0.371787
$587$	14.0000	0.577842	0.288921	−	0.957353i	$-0.406704\pi$
$587$	14.0000	0.577842	0.288921	+	0.957353i	$0.406704\pi$
$588$	−6.00000	−0.247436
$589$	−12.0000	−0.494451
$590$	0	0
$591$	−22.0000	−0.904959
$592$	−3.00000	−0.123299
$593$	−11.0000	−0.451716	−0.225858	−	0.974160i	$-0.572519\pi$
$593$	−11.0000	−0.451716	−0.225858	+	0.974160i	$0.572519\pi$
$594$	0	0
$595$	0	0
$596$	−17.0000	−0.696347
$597$	−48.0000	−1.96451
$598$	−2.00000	−0.0817861
$599$	34.0000	1.38920	0.694601	−	0.719395i	$-0.255581\pi$
$599$	34.0000	1.38920	0.694601	+	0.719395i	$0.255581\pi$
$600$	0	0
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	16.0000	0.651031
$605$	0	0
$606$	−20.0000	−0.812444
$607$	10.0000	0.405887	0.202944	−	0.979190i	$-0.434949\pi$
$607$	10.0000	0.405887	0.202944	+	0.979190i	$0.434949\pi$
$608$	−30.0000	−1.21666
$609$	−36.0000	−1.45879
$610$	0	0
$611$	2.00000	0.0809113
$612$	−5.00000	−0.202113
$613$	−17.0000	−0.686624	−0.343312	−	0.939222i	$-0.611549\pi$
$613$	−17.0000	−0.686624	−0.343312	+	0.939222i	$0.611549\pi$
$614$	−22.0000	−0.887848
$615$	0	0
$616$	0	0
$617$	9.00000	0.362326	0.181163	−	0.983453i	$-0.442014\pi$
$617$	9.00000	0.362326	0.181163	+	0.983453i	$0.442014\pi$
$618$	−16.0000	−0.643614
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	−24.0000	−0.962312
$623$	−18.0000	−0.721155
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	23.0000	0.919265
$627$	0	0
$628$	2.00000	0.0798087
$629$	15.0000	0.598089
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	−30.0000	−1.19334
$633$	−24.0000	−0.953914
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	−18.0000	−0.713746
$637$	3.00000	0.118864
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	−12.0000	−0.473602
$643$	10.0000	0.394362	0.197181	−	0.980367i	$-0.436821\pi$
$643$	10.0000	0.394362	0.197181	+	0.980367i	$0.436821\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	−30.0000	−1.18033
$647$	−20.0000	−0.786281	−0.393141	−	0.919478i	$-0.628611\pi$
$647$	−20.0000	−0.786281	−0.393141	+	0.919478i	$0.628611\pi$
$648$	−33.0000	−1.29636
$649$	0	0
$650$	0	0
$651$	8.00000	0.313545
$652$	−2.00000	−0.0783260
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	−22.0000	−0.860268
$655$	0	0
$656$	5.00000	0.195217
$657$	2.00000	0.0780274
$658$	4.00000	0.155936
$659$	22.0000	0.856998	0.428499	−	0.903542i	$-0.359042\pi$
$659$	22.0000	0.856998	0.428499	+	0.903542i	$0.359042\pi$
$660$	0	0
$661$	13.0000	0.505641	0.252821	−	0.967513i	$-0.418642\pi$
$661$	13.0000	0.505641	0.252821	+	0.967513i	$0.418642\pi$
$662$	20.0000	0.777322
$663$	10.0000	0.388368
$664$	−18.0000	−0.698535
$665$	0	0
$666$	−3.00000	−0.116248
$667$	−18.0000	−0.696963
$668$	12.0000	0.464294
$669$	−40.0000	−1.54649
$670$	0	0
$671$	0	0
$672$	20.0000	0.771517
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	12.0000	0.461538
$677$	27.0000	1.03769	0.518847	−	0.854867i	$-0.326361\pi$
$677$	27.0000	1.03769	0.518847	+	0.854867i	$0.326361\pi$
$678$	18.0000	0.691286
$679$	26.0000	0.997788
$680$	0	0
$681$	−48.0000	−1.83936
$682$	0	0
$683$	−2.00000	−0.0765279	−0.0382639	−	0.999268i	$-0.512183\pi$
$683$	−2.00000	−0.0765279	−0.0382639	+	0.999268i	$0.512183\pi$
$684$	−6.00000	−0.229416
$685$	0	0
$686$	20.0000	0.763604
$687$	−18.0000	−0.686743
$688$	0	0
$689$	9.00000	0.342873
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	28.0000	1.06287
$695$	0	0
$696$	−54.0000	−2.04686
$697$	−25.0000	−0.946943
$698$	27.0000	1.02197
$699$	−42.0000	−1.58859
$700$	0	0
$701$	17.0000	0.642081	0.321041	−	0.947065i	$-0.395967\pi$
$701$	17.0000	0.642081	0.321041	+	0.947065i	$0.395967\pi$
$702$	4.00000	0.150970
$703$	18.0000	0.678883
$704$	0	0
$705$	0	0
$706$	−9.00000	−0.338719
$707$	−20.0000	−0.752177
$708$	16.0000	0.601317
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	−27.0000	−1.01187
$713$	4.00000	0.149801
$714$	20.0000	0.748481
$715$	0	0
$716$	−24.0000	−0.896922
$717$	−12.0000	−0.448148
$718$	2.00000	0.0746393
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	−17.0000	−0.632674
$723$	−44.0000	−1.63638
$724$	−1.00000	−0.0371647
$725$	0	0
$726$	0	0
$727$	42.0000	1.55769	0.778847	−	0.627214i	$-0.215805\pi$
$727$	42.0000	1.55769	0.778847	+	0.627214i	$0.215805\pi$
$728$	−6.00000	−0.222375
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	12.0000	0.443533
$733$	−9.00000	−0.332423	−0.166211	−	0.986090i	$-0.553153\pi$
$733$	−9.00000	−0.332423	−0.166211	+	0.986090i	$0.553153\pi$
$734$	−14.0000	−0.516749
$735$	0	0
$736$	10.0000	0.368605
$737$	0	0
$738$	5.00000	0.184053
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	18.0000	0.660801
$743$	38.0000	1.39408	0.697042	−	0.717030i	$-0.254499\pi$
$743$	38.0000	1.39408	0.697042	+	0.717030i	$0.254499\pi$
$744$	12.0000	0.439941
$745$	0	0
$746$	22.0000	0.805477
$747$	−6.00000	−0.219529
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	2.00000	0.0729325
$753$	4.00000	0.145768
$754$	9.00000	0.327761
$755$	0	0
$756$	−8.00000	−0.290957
$757$	−53.0000	−1.92632	−0.963159	−	0.268933i	$-0.913329\pi$
$757$	−53.0000	−1.92632	−0.963159	+	0.268933i	$0.913329\pi$
$758$	32.0000	1.16229
$759$	0	0
$760$	0	0
$761$	−21.0000	−0.761249	−0.380625	−	0.924730i	$-0.624291\pi$
$761$	−21.0000	−0.761249	−0.380625	+	0.924730i	$0.624291\pi$
$762$	32.0000	1.15924
$763$	−22.0000	−0.796453
$764$	−8.00000	−0.289430
$765$	0	0
$766$	20.0000	0.722629
$767$	−8.00000	−0.288863
$768$	34.0000	1.22687
$769$	11.0000	0.396670	0.198335	−	0.980134i	$-0.436447\pi$
$769$	11.0000	0.396670	0.198335	+	0.980134i	$0.436447\pi$
$770$	0	0
$771$	38.0000	1.36854
$772$	−5.00000	−0.179954
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	0	0
$775$	0	0
$776$	39.0000	1.40002
$777$	−12.0000	−0.430498
$778$	3.00000	0.107555
$779$	−30.0000	−1.07486
$780$	0	0
$781$	0	0
$782$	10.0000	0.357599
$783$	36.0000	1.28654
$784$	3.00000	0.107143
$785$	0	0
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	−11.0000	−0.391859
$789$	−44.0000	−1.56644
$790$	0	0
$791$	18.0000	0.640006
$792$	0	0
$793$	−6.00000	−0.213066
$794$	13.0000	0.461353
$795$	0	0
$796$	−24.0000	−0.850657
$797$	10.0000	0.354218	0.177109	−	0.984191i	$-0.443325\pi$
$797$	10.0000	0.354218	0.177109	+	0.984191i	$0.443325\pi$
$798$	24.0000	0.849591
$799$	−10.0000	−0.353775
$800$	0	0
$801$	−9.00000	−0.317999
$802$	−23.0000	−0.812158
$803$	0	0
$804$	−4.00000	−0.141069
$805$	0	0
$806$	−2.00000	−0.0704470
$807$	−2.00000	−0.0704033
$808$	−30.0000	−1.05540
$809$	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
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	−18.0000	−0.631676
$813$	−40.0000	−1.40286
$814$	0	0
$815$	0	0
$816$	10.0000	0.350070
$817$	0	0
$818$	21.0000	0.734248
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	20.0000	0.697580
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	−24.0000	−0.836080
$825$	0	0
$826$	−16.0000	−0.556711
$827$	10.0000	0.347734	0.173867	−	0.984769i	$-0.444374\pi$
$827$	10.0000	0.347734	0.173867	+	0.984769i	$0.444374\pi$
$828$	2.00000	0.0695048
$829$	−47.0000	−1.63238	−0.816189	−	0.577785i	$-0.803917\pi$
$829$	−47.0000	−1.63238	−0.816189	+	0.577785i	$0.803917\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	−7.00000	−0.242681
$833$	−15.0000	−0.519719
$834$	−4.00000	−0.138509
$835$	0	0
$836$	0	0
$837$	−8.00000	−0.276520
$838$	−2.00000	−0.0690889
$839$	46.0000	1.58810	0.794048	−	0.607855i	$-0.207970\pi$
$839$	46.0000	1.58810	0.794048	+	0.607855i	$0.207970\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	−13.0000	−0.448010
$843$	−12.0000	−0.413302
$844$	−12.0000	−0.413057
$845$	0	0
$846$	2.00000	0.0687614
$847$	0	0
$848$	9.00000	0.309061
$849$	56.0000	1.92192
$850$	0	0
$851$	−6.00000	−0.205677
$852$	24.0000	0.822226
$853$	−17.0000	−0.582069	−0.291034	−	0.956713i	$-0.593999\pi$
$853$	−17.0000	−0.582069	−0.291034	+	0.956713i	$0.593999\pi$
$854$	−12.0000	−0.410632
$855$	0	0
$856$	−18.0000	−0.615227
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	24.0000	0.818869	0.409435	−	0.912339i	$-0.365726\pi$
$859$	24.0000	0.818869	0.409435	+	0.912339i	$0.365726\pi$
$860$	0	0
$861$	20.0000	0.681598
$862$	−12.0000	−0.408722
$863$	54.0000	1.83818	0.919091	−	0.394046i	$-0.128925\pi$
$863$	54.0000	1.83818	0.919091	+	0.394046i	$0.128925\pi$
$864$	−20.0000	−0.680414
$865$	0	0
$866$	19.0000	0.645646
$867$	−16.0000	−0.543388
$868$	4.00000	0.135769
$869$	0	0
$870$	0	0
$871$	2.00000	0.0677674
$872$	−33.0000	−1.11752
$873$	13.0000	0.439983
$874$	12.0000	0.405906
$875$	0	0
$876$	4.00000	0.135147
$877$	27.0000	0.911725	0.455863	−	0.890050i	$-0.349331\pi$
$877$	27.0000	0.911725	0.455863	+	0.890050i	$0.349331\pi$
$878$	−22.0000	−0.742464
$879$	18.0000	0.607125
$880$	0	0
$881$	35.0000	1.17918	0.589590	−	0.807703i	$-0.299289\pi$
$881$	35.0000	1.17918	0.589590	+	0.807703i	$0.299289\pi$
$882$	3.00000	0.101015
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	5.00000	0.168168
$885$	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	$0.219124\pi$
$888$	−18.0000	−0.604040
$889$	32.0000	1.07325
$890$	0	0
$891$	0	0
$892$	−20.0000	−0.669650
$893$	−12.0000	−0.401565
$894$	34.0000	1.13713
$895$	0	0
$896$	6.00000	0.200446
$897$	−4.00000	−0.133556
$898$	13.0000	0.433816
$899$	−18.0000	−0.600334
$900$	0	0
$901$	−45.0000	−1.49917
$902$	0	0
$903$	0	0
$904$	27.0000	0.898007
$905$	0	0
$906$	−32.0000	−1.06313
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	−24.0000	−0.796468
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	12.0000	0.397360
$913$	0	0
$914$	39.0000	1.29001
$915$	0	0
$916$	−9.00000	−0.297368
$917$	0	0
$918$	−20.0000	−0.660098
$919$	28.0000	0.923635	0.461817	−	0.886975i	$-0.347198\pi$
$919$	28.0000	0.923635	0.461817	+	0.886975i	$0.347198\pi$
$920$	0	0
$921$	−44.0000	−1.44985
$922$	−33.0000	−1.08680
$923$	−12.0000	−0.394985
$924$	0	0
$925$	0	0
$926$	−20.0000	−0.657241
$927$	−8.00000	−0.262754
$928$	−45.0000	−1.47720
$929$	−21.0000	−0.688988	−0.344494	−	0.938789i	$-0.611949\pi$
$929$	−21.0000	−0.688988	−0.344494	+	0.938789i	$0.611949\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	−21.0000	−0.687878
$933$	−48.0000	−1.57145
$934$	12.0000	0.392652
$935$	0	0
$936$	−3.00000	−0.0980581
$937$	−23.0000	−0.751377	−0.375689	−	0.926746i	$-0.622594\pi$
$937$	−23.0000	−0.751377	−0.375689	+	0.926746i	$0.622594\pi$
$938$	4.00000	0.130605
$939$	46.0000	1.50115
$940$	0	0
$941$	−27.0000	−0.880175	−0.440087	−	0.897955i	$-0.645053\pi$
$941$	−27.0000	−0.880175	−0.440087	+	0.897955i	$0.645053\pi$
$942$	−4.00000	−0.130327
$943$	10.0000	0.325645
$944$	−8.00000	−0.260378
$945$	0	0
$946$	0	0
$947$	42.0000	1.36482	0.682408	−	0.730971i	$-0.260933\pi$
$947$	42.0000	1.36482	0.682408	+	0.730971i	$0.260933\pi$
$948$	−20.0000	−0.649570
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	−4.00000	−0.129709
$952$	30.0000	0.972306
$953$	−31.0000	−1.00419	−0.502094	−	0.864813i	$-0.667437\pi$
$953$	−31.0000	−1.00419	−0.502094	+	0.864813i	$0.667437\pi$
$954$	9.00000	0.291386
$955$	0	0
$956$	−6.00000	−0.194054
$957$	0	0
$958$	16.0000	0.516937
$959$	20.0000	0.645834
$960$	0	0
$961$	−27.0000	−0.870968
$962$	3.00000	0.0967239
$963$	−6.00000	−0.193347
$964$	−22.0000	−0.708572
$965$	0	0
$966$	−8.00000	−0.257396
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	0	0
$969$	−60.0000	−1.92748
$970$	0	0
$971$	2.00000	0.0641831	0.0320915	−	0.999485i	$-0.489783\pi$
$971$	2.00000	0.0641831	0.0320915	+	0.999485i	$0.489783\pi$
$972$	−10.0000	−0.320750
$973$	−4.00000	−0.128234
$974$	2.00000	0.0640841
$975$	0	0
$976$	−6.00000	−0.192055
$977$	57.0000	1.82359	0.911796	−	0.410644i	$-0.134696\pi$
$977$	57.0000	1.82359	0.911796	+	0.410644i	$0.134696\pi$
$978$	4.00000	0.127906
$979$	0	0
$980$	0	0
$981$	−11.0000	−0.351203
$982$	2.00000	0.0638226
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	30.0000	0.956365
$985$	0	0
$986$	−45.0000	−1.43309
$987$	8.00000	0.254643
$988$	6.00000	0.190885
$989$	0	0
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	10.0000	0.317500
$993$	40.0000	1.26936
$994$	−24.0000	−0.761234
$995$	0	0
$996$	−12.0000	−0.380235
$997$	−53.0000	−1.67853	−0.839263	−	0.543725i	$-0.817013\pi$
$997$	−53.0000	−1.67853	−0.839263	+	0.543725i	$0.817013\pi$
$998$	−8.00000	−0.253236
$999$	12.0000	0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.b.1.1		1
5.4	even	2		121.2.a.c.1.1	yes	1
11.10	odd	2		3025.2.a.e.1.1		1
15.14	odd	2		1089.2.a.c.1.1		1
20.19	odd	2		1936.2.a.b.1.1		1
35.34	odd	2		5929.2.a.g.1.1		1
40.19	odd	2		7744.2.a.bf.1.1		1
40.29	even	2		7744.2.a.c.1.1		1
55.4	even	10		121.2.c.b.27.1		4
55.9	even	10		121.2.c.b.81.1		4
55.14	even	10		121.2.c.b.9.1		4
55.19	odd	10		121.2.c.d.9.1		4
55.24	odd	10		121.2.c.d.81.1		4
55.29	odd	10		121.2.c.d.27.1		4
55.39	odd	10		121.2.c.d.3.1		4
55.49	even	10		121.2.c.b.3.1		4
55.54	odd	2		121.2.a.a.1.1	✓	1
165.164	even	2		1089.2.a.i.1.1		1
220.219	even	2		1936.2.a.a.1.1		1
385.384	even	2		5929.2.a.a.1.1		1
440.109	odd	2		7744.2.a.f.1.1		1
440.219	even	2		7744.2.a.be.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.2.a.a.1.1	✓	1	55.54	odd	2
121.2.a.c.1.1	yes	1	5.4	even	2
121.2.c.b.3.1		4	55.49	even	10
121.2.c.b.9.1		4	55.14	even	10
121.2.c.b.27.1		4	55.4	even	10
121.2.c.b.81.1		4	55.9	even	10
121.2.c.d.3.1		4	55.39	odd	10
121.2.c.d.9.1		4	55.19	odd	10
121.2.c.d.27.1		4	55.29	odd	10
121.2.c.d.81.1		4	55.24	odd	10
1089.2.a.c.1.1		1	15.14	odd	2
1089.2.a.i.1.1		1	165.164	even	2
1936.2.a.a.1.1		1	220.219	even	2
1936.2.a.b.1.1		1	20.19	odd	2
3025.2.a.b.1.1		1	1.1	even	1	trivial
3025.2.a.e.1.1		1	11.10	odd	2
5929.2.a.a.1.1		1	385.384	even	2
5929.2.a.g.1.1		1	35.34	odd	2
7744.2.a.c.1.1		1	40.29	even	2
7744.2.a.f.1.1		1	440.109	odd	2
7744.2.a.be.1.1		1	440.219	even	2
7744.2.a.bf.1.1		1	40.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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