LMFDB - Embedded newform 53.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [53,2,Mod(1,53)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(53, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("53.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	53.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:	$0.423207130713$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -3.00000 q^{3} -1.00000 q^{4} +3.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -3.00000 q^{3} -1.00000 q^{4} +3.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} +6.00000 q^{9} +3.00000 q^{12} -3.00000 q^{13} +4.00000 q^{14} -1.00000 q^{16} -3.00000 q^{17} -6.00000 q^{18} -5.00000 q^{19} +12.0000 q^{21} +7.00000 q^{23} -9.00000 q^{24} -5.00000 q^{25} +3.00000 q^{26} -9.00000 q^{27} +4.00000 q^{28} -7.00000 q^{29} +4.00000 q^{31} -5.00000 q^{32} +3.00000 q^{34} -6.00000 q^{36} +5.00000 q^{37} +5.00000 q^{38} +9.00000 q^{39} +6.00000 q^{41} -12.0000 q^{42} -2.00000 q^{43} -7.00000 q^{46} -2.00000 q^{47} +3.00000 q^{48} +9.00000 q^{49} +5.00000 q^{50} +9.00000 q^{51} +3.00000 q^{52} -1.00000 q^{53} +9.00000 q^{54} -12.0000 q^{56} +15.0000 q^{57} +7.00000 q^{58} -2.00000 q^{59} -8.00000 q^{61} -4.00000 q^{62} -24.0000 q^{63} +7.00000 q^{64} -12.0000 q^{67} +3.00000 q^{68} -21.0000 q^{69} +1.00000 q^{71} +18.0000 q^{72} -4.00000 q^{73} -5.00000 q^{74} +15.0000 q^{75} +5.00000 q^{76} -9.00000 q^{78} -1.00000 q^{79} +9.00000 q^{81} -6.00000 q^{82} -1.00000 q^{83} -12.0000 q^{84} +2.00000 q^{86} +21.0000 q^{87} -14.0000 q^{89} +12.0000 q^{91} -7.00000 q^{92} -12.0000 q^{93} +2.00000 q^{94} +15.0000 q^{96} +1.00000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	−1.00000	−0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	3.00000	1.22474
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	3.00000	1.06066
$9$	6.00000	2.00000
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	3.00000	0.866025
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	−6.00000	−1.41421
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	12.0000	2.61861
$22$	0	0
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	−9.00000	−1.83712
$25$	−5.00000	−1.00000
$26$	3.00000	0.588348
$27$	−9.00000	−1.73205
$28$	4.00000	0.755929
$29$	−7.00000	−1.29987	−0.649934	−	0.759991i	$-0.725203\pi$
$29$	−7.00000	−1.29987	−0.649934	+	0.759991i	$0.725203\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	3.00000	0.514496
$35$	0	0
$36$	−6.00000	−1.00000
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	5.00000	0.811107
$39$	9.00000	1.44115
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−12.0000	−1.85164
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	0	0
$46$	−7.00000	−1.03209
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	3.00000	0.433013
$49$	9.00000	1.28571
$50$	5.00000	0.707107
$51$	9.00000	1.26025
$52$	3.00000	0.416025
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	9.00000	1.22474
$55$	0	0
$56$	−12.0000	−1.60357
$57$	15.0000	1.98680
$58$	7.00000	0.919145
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	−4.00000	−0.508001
$63$	−24.0000	−3.02372
$64$	7.00000	0.875000
$65$	0	0
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	3.00000	0.363803
$69$	−21.0000	−2.52810
$70$	0	0
$71$	1.00000	0.118678	0.0593391	−	0.998238i	$-0.481101\pi$
$71$	1.00000	0.118678	0.0593391	+	0.998238i	$0.481101\pi$
$72$	18.0000	2.12132
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−5.00000	−0.581238
$75$	15.0000	1.73205
$76$	5.00000	0.573539
$77$	0	0
$78$	−9.00000	−1.01905
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	−6.00000	−0.662589
$83$	−1.00000	−0.109764	−0.0548821	−	0.998493i	$-0.517478\pi$
$83$	−1.00000	−0.109764	−0.0548821	+	0.998493i	$0.517478\pi$
$84$	−12.0000	−1.30931
$85$	0	0
$86$	2.00000	0.215666
$87$	21.0000	2.25144
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	−7.00000	−0.729800
$93$	−12.0000	−1.24434
$94$	2.00000	0.206284
$95$	0	0
$96$	15.0000	1.53093
$97$	1.00000	0.101535	0.0507673	−	0.998711i	$-0.483833\pi$
$97$	1.00000	0.101535	0.0507673	+	0.998711i	$0.483833\pi$
$98$	−9.00000	−0.909137
$99$	0	0
$100$	5.00000	0.500000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	−9.00000	−0.891133
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	−9.00000	−0.882523
$105$	0	0
$106$	1.00000	0.0971286
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	9.00000	0.866025
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	−15.0000	−1.42374
$112$	4.00000	0.377964
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	−15.0000	−1.40488
$115$	0	0
$116$	7.00000	0.649934
$117$	−18.0000	−1.66410
$118$	2.00000	0.184115
$119$	12.0000	1.10004
$120$	0	0
$121$	−11.0000	−1.00000
$122$	8.00000	0.724286
$123$	−18.0000	−1.62301
$124$	−4.00000	−0.359211
$125$	0	0
$126$	24.0000	2.13809
$127$	13.0000	1.15356	0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000	1.15356	0.576782	+	0.816898i	$0.304308\pi$
$128$	3.00000	0.265165
$129$	6.00000	0.528271
$130$	0	0
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	0	0
$133$	20.0000	1.73422
$134$	12.0000	1.03664
$135$	0	0
$136$	−9.00000	−0.771744
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	21.0000	1.78764
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	−1.00000	−0.0839181
$143$	0	0
$144$	−6.00000	−0.500000
$145$	0	0
$146$	4.00000	0.331042
$147$	−27.0000	−2.22692
$148$	−5.00000	−0.410997
$149$	−5.00000	−0.409616	−0.204808	−	0.978802i	$-0.565657\pi$
$149$	−5.00000	−0.409616	−0.204808	+	0.978802i	$0.565657\pi$
$150$	−15.0000	−1.22474
$151$	−3.00000	−0.244137	−0.122068	−	0.992522i	$-0.538953\pi$
$151$	−3.00000	−0.244137	−0.122068	+	0.992522i	$0.538953\pi$
$152$	−15.0000	−1.21666
$153$	−18.0000	−1.45521
$154$	0	0
$155$	0	0
$156$	−9.00000	−0.720577
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	1.00000	0.0795557
$159$	3.00000	0.237915
$160$	0	0
$161$	−28.0000	−2.20671
$162$	−9.00000	−0.707107
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	1.00000	0.0776151
$167$	21.0000	1.62503	0.812514	−	0.582941i	$-0.198098\pi$
$167$	21.0000	1.62503	0.812514	+	0.582941i	$0.198098\pi$
$168$	36.0000	2.77746
$169$	−4.00000	−0.307692
$170$	0	0
$171$	−30.0000	−2.29416
$172$	2.00000	0.152499
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	−21.0000	−1.59201
$175$	20.0000	1.51186
$176$	0	0
$177$	6.00000	0.450988
$178$	14.0000	1.04934
$179$	11.0000	0.822179	0.411089	−	0.911595i	$-0.365148\pi$
$179$	11.0000	0.822179	0.411089	+	0.911595i	$0.365148\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	−12.0000	−0.889499
$183$	24.0000	1.77413
$184$	21.0000	1.54814
$185$	0	0
$186$	12.0000	0.879883
$187$	0	0
$188$	2.00000	0.145865
$189$	36.0000	2.61861
$190$	0	0
$191$	−21.0000	−1.51951	−0.759753	−	0.650211i	$-0.774680\pi$
$191$	−21.0000	−1.51951	−0.759753	+	0.650211i	$0.774680\pi$
$192$	−21.0000	−1.51554
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	−1.00000	−0.0717958
$195$	0	0
$196$	−9.00000	−0.642857
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	−15.0000	−1.06066
$201$	36.0000	2.53924
$202$	2.00000	0.140720
$203$	28.0000	1.96521
$204$	−9.00000	−0.630126
$205$	0	0
$206$	1.00000	0.0696733
$207$	42.0000	2.91920
$208$	3.00000	0.208013
$209$	0	0
$210$	0	0
$211$	−2.00000	−0.137686	−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000	−0.137686	−0.0688428	+	0.997628i	$0.521931\pi$
$212$	1.00000	0.0686803
$213$	−3.00000	−0.205557
$214$	−6.00000	−0.410152
$215$	0	0
$216$	−27.0000	−1.83712
$217$	−16.0000	−1.08615
$218$	−16.0000	−1.08366
$219$	12.0000	0.810885
$220$	0	0
$221$	9.00000	0.605406
$222$	15.0000	1.00673
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	20.0000	1.33631
$225$	−30.0000	−2.00000
$226$	−15.0000	−0.997785
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	−15.0000	−0.993399
$229$	21.0000	1.38772	0.693860	−	0.720110i	$-0.255909\pi$
$229$	21.0000	1.38772	0.693860	+	0.720110i	$0.255909\pi$
$230$	0	0
$231$	0	0
$232$	−21.0000	−1.37872
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	18.0000	1.17670
$235$	0	0
$236$	2.00000	0.130189
$237$	3.00000	0.194871
$238$	−12.0000	−0.777844
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	8.00000	0.512148
$245$	0	0
$246$	18.0000	1.14764
$247$	15.0000	0.954427
$248$	12.0000	0.762001
$249$	3.00000	0.190117
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	24.0000	1.51186
$253$	0	0
$254$	−13.0000	−0.815693
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−28.0000	−1.74659	−0.873296	−	0.487190i	$-0.838022\pi$
$257$	−28.0000	−1.74659	−0.873296	+	0.487190i	$0.838022\pi$
$258$	−6.00000	−0.373544
$259$	−20.0000	−1.24274
$260$	0	0
$261$	−42.0000	−2.59973
$262$	2.00000	0.123560
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	0	0
$265$	0	0
$266$	−20.0000	−1.22628
$267$	42.0000	2.57036
$268$	12.0000	0.733017
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	0	0
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	3.00000	0.181902
$273$	−36.0000	−2.17882
$274$	−12.0000	−0.724947
$275$	0	0
$276$	21.0000	1.26405
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	20.0000	1.19952
$279$	24.0000	1.43684
$280$	0	0
$281$	17.0000	1.01413	0.507067	−	0.861906i	$-0.330729\pi$
$281$	17.0000	1.01413	0.507067	+	0.861906i	$0.330729\pi$
$282$	−6.00000	−0.357295
$283$	−9.00000	−0.534994	−0.267497	−	0.963559i	$-0.586197\pi$
$283$	−9.00000	−0.534994	−0.267497	+	0.963559i	$0.586197\pi$
$284$	−1.00000	−0.0593391
$285$	0	0
$286$	0	0
$287$	−24.0000	−1.41668
$288$	−30.0000	−1.76777
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−3.00000	−0.175863
$292$	4.00000	0.234082
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	27.0000	1.57467
$295$	0	0
$296$	15.0000	0.871857
$297$	0	0
$298$	5.00000	0.289642
$299$	−21.0000	−1.21446
$300$	−15.0000	−0.866025
$301$	8.00000	0.461112
$302$	3.00000	0.172631
$303$	6.00000	0.344691
$304$	5.00000	0.286770
$305$	0	0
$306$	18.0000	1.02899
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	3.00000	0.170664
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	27.0000	1.52857
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	1.00000	0.0562544
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	−3.00000	−0.168232
$319$	0	0
$320$	0	0
$321$	−18.0000	−1.00466
$322$	28.0000	1.56038
$323$	15.0000	0.834622
$324$	−9.00000	−0.500000
$325$	15.0000	0.832050
$326$	6.00000	0.332309
$327$	−48.0000	−2.65441
$328$	18.0000	0.993884
$329$	8.00000	0.441054
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	1.00000	0.0548821
$333$	30.0000	1.64399
$334$	−21.0000	−1.14907
$335$	0	0
$336$	−12.0000	−0.654654
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	4.00000	0.217571
$339$	−45.0000	−2.44406
$340$	0	0
$341$	0	0
$342$	30.0000	1.62221
$343$	−8.00000	−0.431959
$344$	−6.00000	−0.323498
$345$	0	0
$346$	−10.0000	−0.537603
$347$	−30.0000	−1.61048	−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000	−1.61048	−0.805242	+	0.592946i	$0.797965\pi$
$348$	−21.0000	−1.12572
$349$	24.0000	1.28469	0.642345	−	0.766415i	$-0.277962\pi$
$349$	24.0000	1.28469	0.642345	+	0.766415i	$0.277962\pi$
$350$	−20.0000	−1.06904
$351$	27.0000	1.44115
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	14.0000	0.741999
$357$	−36.0000	−1.90532
$358$	−11.0000	−0.581368
$359$	9.00000	0.475002	0.237501	−	0.971387i	$-0.423672\pi$
$359$	9.00000	0.475002	0.237501	+	0.971387i	$0.423672\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	2.00000	0.105118
$363$	33.0000	1.73205
$364$	−12.0000	−0.628971
$365$	0	0
$366$	−24.0000	−1.25450
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	−7.00000	−0.364900
$369$	36.0000	1.87409
$370$	0	0
$371$	4.00000	0.207670
$372$	12.0000	0.622171
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	0	0
$376$	−6.00000	−0.309426
$377$	21.0000	1.08156
$378$	−36.0000	−1.85164
$379$	11.0000	0.565032	0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000	0.565032	0.282516	+	0.959263i	$0.408831\pi$
$380$	0	0
$381$	−39.0000	−1.99803
$382$	21.0000	1.07445
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	−9.00000	−0.459279
$385$	0	0
$386$	16.0000	0.814379
$387$	−12.0000	−0.609994
$388$	−1.00000	−0.0507673
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	−21.0000	−1.06202
$392$	27.0000	1.36371
$393$	6.00000	0.302660
$394$	18.0000	0.906827
$395$	0	0
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	−4.00000	−0.200502
$399$	−60.0000	−3.00376
$400$	5.00000	0.250000
$401$	−32.0000	−1.59800	−0.799002	−	0.601329i	$-0.794638\pi$
$401$	−32.0000	−1.59800	−0.799002	+	0.601329i	$0.794638\pi$
$402$	−36.0000	−1.79552
$403$	−12.0000	−0.597763
$404$	2.00000	0.0995037
$405$	0	0
$406$	−28.0000	−1.38962
$407$	0	0
$408$	27.0000	1.33670
$409$	19.0000	0.939490	0.469745	−	0.882802i	$-0.344346\pi$
$409$	19.0000	0.939490	0.469745	+	0.882802i	$0.344346\pi$
$410$	0	0
$411$	−36.0000	−1.77575
$412$	1.00000	0.0492665
$413$	8.00000	0.393654
$414$	−42.0000	−2.06419
$415$	0	0
$416$	15.0000	0.735436
$417$	60.0000	2.93821
$418$	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
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	2.00000	0.0973585
$423$	−12.0000	−0.583460
$424$	−3.00000	−0.145693
$425$	15.0000	0.727607
$426$	3.00000	0.145350
$427$	32.0000	1.54859
$428$	−6.00000	−0.290021
$429$	0	0
$430$	0	0
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	9.00000	0.433013
$433$	−7.00000	−0.336399	−0.168199	−	0.985753i	$-0.553795\pi$
$433$	−7.00000	−0.336399	−0.168199	+	0.985753i	$0.553795\pi$
$434$	16.0000	0.768025
$435$	0	0
$436$	−16.0000	−0.766261
$437$	−35.0000	−1.67428
$438$	−12.0000	−0.573382
$439$	−34.0000	−1.62273	−0.811366	−	0.584539i	$-0.801275\pi$
$439$	−34.0000	−1.62273	−0.811366	+	0.584539i	$0.801275\pi$
$440$	0	0
$441$	54.0000	2.57143
$442$	−9.00000	−0.428086
$443$	−33.0000	−1.56788	−0.783939	−	0.620838i	$-0.786792\pi$
$443$	−33.0000	−1.56788	−0.783939	+	0.620838i	$0.786792\pi$
$444$	15.0000	0.711868
$445$	0	0
$446$	14.0000	0.662919
$447$	15.0000	0.709476
$448$	−28.0000	−1.32288
$449$	−11.0000	−0.519122	−0.259561	−	0.965727i	$-0.583578\pi$
$449$	−11.0000	−0.519122	−0.259561	+	0.965727i	$0.583578\pi$
$450$	30.0000	1.41421
$451$	0	0
$452$	−15.0000	−0.705541
$453$	9.00000	0.422857
$454$	−6.00000	−0.281594
$455$	0	0
$456$	45.0000	2.10732
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	−21.0000	−0.981266
$459$	27.0000	1.26025
$460$	0	0
$461$	−9.00000	−0.419172	−0.209586	−	0.977790i	$-0.567212\pi$
$461$	−9.00000	−0.419172	−0.209586	+	0.977790i	$0.567212\pi$
$462$	0	0
$463$	−23.0000	−1.06890	−0.534450	−	0.845200i	$-0.679481\pi$
$463$	−23.0000	−1.06890	−0.534450	+	0.845200i	$0.679481\pi$
$464$	7.00000	0.324967
$465$	0	0
$466$	8.00000	0.370593
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	18.0000	0.832050
$469$	48.0000	2.21643
$470$	0	0
$471$	12.0000	0.552931
$472$	−6.00000	−0.276172
$473$	0	0
$474$	−3.00000	−0.137795
$475$	25.0000	1.14708
$476$	−12.0000	−0.550019
$477$	−6.00000	−0.274721
$478$	0	0
$479$	3.00000	0.137073	0.0685367	−	0.997649i	$-0.478167\pi$
$479$	3.00000	0.137073	0.0685367	+	0.997649i	$0.478167\pi$
$480$	0	0
$481$	−15.0000	−0.683941
$482$	11.0000	0.501036
$483$	84.0000	3.82213
$484$	11.0000	0.500000
$485$	0	0
$486$	0	0
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	−24.0000	−1.08643
$489$	18.0000	0.813988
$490$	0	0
$491$	27.0000	1.21849	0.609246	−	0.792981i	$-0.291472\pi$
$491$	27.0000	1.21849	0.609246	+	0.792981i	$0.291472\pi$
$492$	18.0000	0.811503
$493$	21.0000	0.945792
$494$	−15.0000	−0.674882
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−4.00000	−0.179425
$498$	−3.00000	−0.134433
$499$	−23.0000	−1.02962	−0.514811	−	0.857304i	$-0.672138\pi$
$499$	−23.0000	−1.02962	−0.514811	+	0.857304i	$0.672138\pi$
$500$	0	0
$501$	−63.0000	−2.81463
$502$	−20.0000	−0.892644
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	−72.0000	−3.20713
$505$	0	0
$506$	0	0
$507$	12.0000	0.532939
$508$	−13.0000	−0.576782
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	11.0000	0.486136
$513$	45.0000	1.98680
$514$	28.0000	1.23503
$515$	0	0
$516$	−6.00000	−0.264135
$517$	0	0
$518$	20.0000	0.878750
$519$	−30.0000	−1.31685
$520$	0	0
$521$	−45.0000	−1.97149	−0.985743	−	0.168259i	$-0.946186\pi$
$521$	−45.0000	−1.97149	−0.985743	+	0.168259i	$0.946186\pi$
$522$	42.0000	1.83829
$523$	−42.0000	−1.83653	−0.918266	−	0.395964i	$-0.870410\pi$
$523$	−42.0000	−1.83653	−0.918266	+	0.395964i	$0.870410\pi$
$524$	2.00000	0.0873704
$525$	−60.0000	−2.61861
$526$	28.0000	1.22086
$527$	−12.0000	−0.522728
$528$	0	0
$529$	26.0000	1.13043
$530$	0	0
$531$	−12.0000	−0.520756
$532$	−20.0000	−0.867110
$533$	−18.0000	−0.779667
$534$	−42.0000	−1.81752
$535$	0	0
$536$	−36.0000	−1.55496
$537$	−33.0000	−1.42406
$538$	−9.00000	−0.388018
$539$	0	0
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	14.0000	0.601351
$543$	6.00000	0.257485
$544$	15.0000	0.643120
$545$	0	0
$546$	36.0000	1.54066
$547$	−38.0000	−1.62476	−0.812381	−	0.583127i	$-0.801829\pi$
$547$	−38.0000	−1.62476	−0.812381	+	0.583127i	$0.801829\pi$
$548$	−12.0000	−0.512615
$549$	−48.0000	−2.04859
$550$	0	0
$551$	35.0000	1.49105
$552$	−63.0000	−2.68146
$553$	4.00000	0.170097
$554$	8.00000	0.339887
$555$	0	0
$556$	20.0000	0.848189
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	−24.0000	−1.01600
$559$	6.00000	0.253773
$560$	0	0
$561$	0	0
$562$	−17.0000	−0.717102
$563$	−1.00000	−0.0421450	−0.0210725	−	0.999778i	$-0.506708\pi$
$563$	−1.00000	−0.0421450	−0.0210725	+	0.999778i	$0.506708\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	9.00000	0.378298
$567$	−36.0000	−1.51186
$568$	3.00000	0.125877
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	63.0000	2.63186
$574$	24.0000	1.00174
$575$	−35.0000	−1.45960
$576$	42.0000	1.75000
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	8.00000	0.332756
$579$	48.0000	1.99481
$580$	0	0
$581$	4.00000	0.165948
$582$	3.00000	0.124354
$583$	0	0
$584$	−12.0000	−0.496564
$585$	0	0
$586$	−26.0000	−1.07405
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	27.0000	1.11346
$589$	−20.0000	−0.824086
$590$	0	0
$591$	54.0000	2.22126
$592$	−5.00000	−0.205499
$593$	25.0000	1.02663	0.513313	−	0.858201i	$-0.328418\pi$
$593$	25.0000	1.02663	0.513313	+	0.858201i	$0.328418\pi$
$594$	0	0
$595$	0	0
$596$	5.00000	0.204808
$597$	−12.0000	−0.491127
$598$	21.0000	0.858754
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	45.0000	1.83712
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	−8.00000	−0.326056
$603$	−72.0000	−2.93207
$604$	3.00000	0.122068
$605$	0	0
$606$	−6.00000	−0.243733
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	25.0000	1.01388
$609$	−84.0000	−3.40385
$610$	0	0
$611$	6.00000	0.242734
$612$	18.0000	0.727607
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	0	0
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	−3.00000	−0.120678
$619$	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
$621$	−63.0000	−2.52810
$622$	−16.0000	−0.641542
$623$	56.0000	2.24359
$624$	−9.00000	−0.360288
$625$	25.0000	1.00000
$626$	6.00000	0.239808
$627$	0	0
$628$	4.00000	0.159617
$629$	−15.0000	−0.598089
$630$	0	0
$631$	7.00000	0.278666	0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000	0.278666	0.139333	+	0.990246i	$0.455504\pi$
$632$	−3.00000	−0.119334
$633$	6.00000	0.238479
$634$	−3.00000	−0.119145
$635$	0	0
$636$	−3.00000	−0.118958
$637$	−27.0000	−1.06978
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	46.0000	1.81689	0.908445	−	0.418004i	$-0.137270\pi$
$641$	46.0000	1.81689	0.908445	+	0.418004i	$0.137270\pi$
$642$	18.0000	0.710403
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	28.0000	1.10335
$645$	0	0
$646$	−15.0000	−0.590167
$647$	38.0000	1.49393	0.746967	−	0.664861i	$-0.231509\pi$
$647$	38.0000	1.49393	0.746967	+	0.664861i	$0.231509\pi$
$648$	27.0000	1.06066
$649$	0	0
$650$	−15.0000	−0.588348
$651$	48.0000	1.88127
$652$	6.00000	0.234978
$653$	−38.0000	−1.48705	−0.743527	−	0.668705i	$-0.766849\pi$
$653$	−38.0000	−1.48705	−0.743527	+	0.668705i	$0.766849\pi$
$654$	48.0000	1.87695
$655$	0	0
$656$	−6.00000	−0.234261
$657$	−24.0000	−0.936329
$658$	−8.00000	−0.311872
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−1.00000	−0.0388955	−0.0194477	−	0.999811i	$-0.506191\pi$
$661$	−1.00000	−0.0388955	−0.0194477	+	0.999811i	$0.506191\pi$
$662$	−28.0000	−1.08825
$663$	−27.0000	−1.04859
$664$	−3.00000	−0.116423
$665$	0	0
$666$	−30.0000	−1.16248
$667$	−49.0000	−1.89729
$668$	−21.0000	−0.812514
$669$	42.0000	1.62381
$670$	0	0
$671$	0	0
$672$	−60.0000	−2.31455
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	−10.0000	−0.385186
$675$	45.0000	1.73205
$676$	4.00000	0.153846
$677$	40.0000	1.53732	0.768662	−	0.639655i	$-0.220923\pi$
$677$	40.0000	1.53732	0.768662	+	0.639655i	$0.220923\pi$
$678$	45.0000	1.72821
$679$	−4.00000	−0.153506
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	2.00000	0.0765279	0.0382639	−	0.999268i	$-0.487817\pi$
$683$	2.00000	0.0765279	0.0382639	+	0.999268i	$0.487817\pi$
$684$	30.0000	1.14708
$685$	0	0
$686$	8.00000	0.305441
$687$	−63.0000	−2.40360
$688$	2.00000	0.0762493
$689$	3.00000	0.114291
$690$	0	0
$691$	41.0000	1.55971	0.779857	−	0.625958i	$-0.215292\pi$
$691$	41.0000	1.55971	0.779857	+	0.625958i	$0.215292\pi$
$692$	−10.0000	−0.380143
$693$	0	0
$694$	30.0000	1.13878
$695$	0	0
$696$	63.0000	2.38801
$697$	−18.0000	−0.681799
$698$	−24.0000	−0.908413
$699$	24.0000	0.907763
$700$	−20.0000	−0.755929
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	−27.0000	−1.01905
$703$	−25.0000	−0.942893
$704$	0	0
$705$	0	0
$706$	18.0000	0.677439
$707$	8.00000	0.300871
$708$	−6.00000	−0.225494
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	−42.0000	−1.57402
$713$	28.0000	1.04861
$714$	36.0000	1.34727
$715$	0	0
$716$	−11.0000	−0.411089
$717$	0	0
$718$	−9.00000	−0.335877
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	−6.00000	−0.223297
$723$	33.0000	1.22728
$724$	2.00000	0.0743294
$725$	35.0000	1.29987
$726$	−33.0000	−1.22474
$727$	−30.0000	−1.11264	−0.556319	−	0.830969i	$-0.687787\pi$
$727$	−30.0000	−1.11264	−0.556319	+	0.830969i	$0.687787\pi$
$728$	36.0000	1.33425
$729$	−27.0000	−1.00000
$730$	0	0
$731$	6.00000	0.221918
$732$	−24.0000	−0.887066
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−22.0000	−0.812035
$735$	0	0
$736$	−35.0000	−1.29012
$737$	0	0
$738$	−36.0000	−1.32518
$739$	15.0000	0.551784	0.275892	−	0.961189i	$-0.411027\pi$
$739$	15.0000	0.551784	0.275892	+	0.961189i	$0.411027\pi$
$740$	0	0
$741$	−45.0000	−1.65312
$742$	−4.00000	−0.146845
$743$	−26.0000	−0.953847	−0.476924	−	0.878945i	$-0.658248\pi$
$743$	−26.0000	−0.953847	−0.476924	+	0.878945i	$0.658248\pi$
$744$	−36.0000	−1.31982
$745$	0	0
$746$	10.0000	0.366126
$747$	−6.00000	−0.219529
$748$	0	0
$749$	−24.0000	−0.876941
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	2.00000	0.0729325
$753$	−60.0000	−2.18652
$754$	−21.0000	−0.764775
$755$	0	0
$756$	−36.0000	−1.30931
$757$	−43.0000	−1.56286	−0.781431	−	0.623992i	$-0.785510\pi$
$757$	−43.0000	−1.56286	−0.781431	+	0.623992i	$0.785510\pi$
$758$	−11.0000	−0.399538
$759$	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	$0.669430\pi$
$762$	39.0000	1.41282
$763$	−64.0000	−2.31696
$764$	21.0000	0.759753
$765$	0	0
$766$	−16.0000	−0.578103
$767$	6.00000	0.216647
$768$	51.0000	1.84030
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	84.0000	3.02519
$772$	16.0000	0.575853
$773$	40.0000	1.43870	0.719350	−	0.694648i	$-0.244440\pi$
$773$	40.0000	1.43870	0.719350	+	0.694648i	$0.244440\pi$
$774$	12.0000	0.431331
$775$	−20.0000	−0.718421
$776$	3.00000	0.107694
$777$	60.0000	2.15249
$778$	−12.0000	−0.430221
$779$	−30.0000	−1.07486
$780$	0	0
$781$	0	0
$782$	21.0000	0.750958
$783$	63.0000	2.25144
$784$	−9.00000	−0.321429
$785$	0	0
$786$	−6.00000	−0.214013
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	18.0000	0.641223
$789$	84.0000	2.99048
$790$	0	0
$791$	−60.0000	−2.13335
$792$	0	0
$793$	24.0000	0.852265
$794$	−18.0000	−0.638796
$795$	0	0
$796$	−4.00000	−0.141776
$797$	16.0000	0.566749	0.283375	−	0.959009i	$-0.408546\pi$
$797$	16.0000	0.566749	0.283375	+	0.959009i	$0.408546\pi$
$798$	60.0000	2.12398
$799$	6.00000	0.212265
$800$	25.0000	0.883883
$801$	−84.0000	−2.96799
$802$	32.0000	1.12996
$803$	0	0
$804$	−36.0000	−1.26962
$805$	0	0
$806$	12.0000	0.422682
$807$	−27.0000	−0.950445
$808$	−6.00000	−0.211079
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	0	0
$811$	42.0000	1.47482	0.737410	−	0.675446i	$-0.236049\pi$
$811$	42.0000	1.47482	0.737410	+	0.675446i	$0.236049\pi$
$812$	−28.0000	−0.982607
$813$	42.0000	1.47300
$814$	0	0
$815$	0	0
$816$	−9.00000	−0.315063
$817$	10.0000	0.349856
$818$	−19.0000	−0.664319
$819$	72.0000	2.51588
$820$	0	0
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	36.0000	1.25564
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	−3.00000	−0.104510
$825$	0	0
$826$	−8.00000	−0.278356
$827$	47.0000	1.63435	0.817175	−	0.576390i	$-0.195539\pi$
$827$	47.0000	1.63435	0.817175	+	0.576390i	$0.195539\pi$
$828$	−42.0000	−1.45960
$829$	−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
$831$	24.0000	0.832551
$832$	−21.0000	−0.728044
$833$	−27.0000	−0.935495
$834$	−60.0000	−2.07763
$835$	0	0
$836$	0	0
$837$	−36.0000	−1.24434
$838$	12.0000	0.414533
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	−14.0000	−0.482472
$843$	−51.0000	−1.75653
$844$	2.00000	0.0688428
$845$	0	0
$846$	12.0000	0.412568
$847$	44.0000	1.51186
$848$	1.00000	0.0343401
$849$	27.0000	0.926638
$850$	−15.0000	−0.514496
$851$	35.0000	1.19978
$852$	3.00000	0.102778
$853$	−56.0000	−1.91740	−0.958702	−	0.284413i	$-0.908201\pi$
$853$	−56.0000	−1.91740	−0.958702	+	0.284413i	$0.908201\pi$
$854$	−32.0000	−1.09502
$855$	0	0
$856$	18.0000	0.615227
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	−58.0000	−1.97893	−0.989467	−	0.144757i	$-0.953760\pi$
$859$	−58.0000	−1.97893	−0.989467	+	0.144757i	$0.953760\pi$
$860$	0	0
$861$	72.0000	2.45375
$862$	36.0000	1.22616
$863$	−30.0000	−1.02121	−0.510606	−	0.859815i	$-0.670579\pi$
$863$	−30.0000	−1.02121	−0.510606	+	0.859815i	$0.670579\pi$
$864$	45.0000	1.53093
$865$	0	0
$866$	7.00000	0.237870
$867$	24.0000	0.815083
$868$	16.0000	0.543075
$869$	0	0
$870$	0	0
$871$	36.0000	1.21981
$872$	48.0000	1.62549
$873$	6.00000	0.203069
$874$	35.0000	1.18389
$875$	0	0
$876$	−12.0000	−0.405442
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	34.0000	1.14744
$879$	−78.0000	−2.63087
$880$	0	0
$881$	16.0000	0.539054	0.269527	−	0.962993i	$-0.413133\pi$
$881$	16.0000	0.539054	0.269527	+	0.962993i	$0.413133\pi$
$882$	−54.0000	−1.81827
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	−9.00000	−0.302703
$885$	0	0
$886$	33.0000	1.10866
$887$	−7.00000	−0.235037	−0.117518	−	0.993071i	$-0.537494\pi$
$887$	−7.00000	−0.235037	−0.117518	+	0.993071i	$0.537494\pi$
$888$	−45.0000	−1.51010
$889$	−52.0000	−1.74402
$890$	0	0
$891$	0	0
$892$	14.0000	0.468755
$893$	10.0000	0.334637
$894$	−15.0000	−0.501675
$895$	0	0
$896$	−12.0000	−0.400892
$897$	63.0000	2.10351
$898$	11.0000	0.367075
$899$	−28.0000	−0.933852
$900$	30.0000	1.00000
$901$	3.00000	0.0999445
$902$	0	0
$903$	−24.0000	−0.798670
$904$	45.0000	1.49668
$905$	0	0
$906$	−9.00000	−0.299005
$907$	46.0000	1.52740	0.763702	−	0.645568i	$-0.223379\pi$
$907$	46.0000	1.52740	0.763702	+	0.645568i	$0.223379\pi$
$908$	−6.00000	−0.199117
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	−15.0000	−0.496700
$913$	0	0
$914$	−38.0000	−1.25693
$915$	0	0
$916$	−21.0000	−0.693860
$917$	8.00000	0.264183
$918$	−27.0000	−0.891133
$919$	15.0000	0.494804	0.247402	−	0.968913i	$-0.420423\pi$
$919$	15.0000	0.494804	0.247402	+	0.968913i	$0.420423\pi$
$920$	0	0
$921$	48.0000	1.58165
$922$	9.00000	0.296399
$923$	−3.00000	−0.0987462
$924$	0	0
$925$	−25.0000	−0.821995
$926$	23.0000	0.755827
$927$	−6.00000	−0.197066
$928$	35.0000	1.14893
$929$	38.0000	1.24674	0.623370	−	0.781927i	$-0.285763\pi$
$929$	38.0000	1.24674	0.623370	+	0.781927i	$0.285763\pi$
$930$	0	0
$931$	−45.0000	−1.47482
$932$	8.00000	0.262049
$933$	−48.0000	−1.57145
$934$	28.0000	0.916188
$935$	0	0
$936$	−54.0000	−1.76505
$937$	−55.0000	−1.79677	−0.898386	−	0.439207i	$-0.855259\pi$
$937$	−55.0000	−1.79677	−0.898386	+	0.439207i	$0.855259\pi$
$938$	−48.0000	−1.56726
$939$	18.0000	0.587408
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	−12.0000	−0.390981
$943$	42.0000	1.36771
$944$	2.00000	0.0650945
$945$	0	0
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−3.00000	−0.0974355
$949$	12.0000	0.389536
$950$	−25.0000	−0.811107
$951$	−9.00000	−0.291845
$952$	36.0000	1.16677
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	0	0
$957$	0	0
$958$	−3.00000	−0.0969256
$959$	−48.0000	−1.55000
$960$	0	0
$961$	−15.0000	−0.483871
$962$	15.0000	0.483619
$963$	36.0000	1.16008
$964$	11.0000	0.354286
$965$	0	0
$966$	−84.0000	−2.70266
$967$	52.0000	1.67221	0.836104	−	0.548572i	$-0.184828\pi$
$967$	52.0000	1.67221	0.836104	+	0.548572i	$0.184828\pi$
$968$	−33.0000	−1.06066
$969$	−45.0000	−1.44561
$970$	0	0
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	0	0
$973$	80.0000	2.56468
$974$	−20.0000	−0.640841
$975$	−45.0000	−1.44115
$976$	8.00000	0.256074
$977$	−20.0000	−0.639857	−0.319928	−	0.947442i	$-0.603659\pi$
$977$	−20.0000	−0.639857	−0.319928	+	0.947442i	$0.603659\pi$
$978$	−18.0000	−0.575577
$979$	0	0
$980$	0	0
$981$	96.0000	3.06504
$982$	−27.0000	−0.861605
$983$	46.0000	1.46717	0.733586	−	0.679597i	$-0.237845\pi$
$983$	46.0000	1.46717	0.733586	+	0.679597i	$0.237845\pi$
$984$	−54.0000	−1.72146
$985$	0	0
$986$	−21.0000	−0.668776
$987$	−24.0000	−0.763928
$988$	−15.0000	−0.477214
$989$	−14.0000	−0.445174
$990$	0	0
$991$	34.0000	1.08005	0.540023	−	0.841650i	$-0.318416\pi$
$991$	34.0000	1.08005	0.540023	+	0.841650i	$0.318416\pi$
$992$	−20.0000	−0.635001
$993$	−84.0000	−2.66566
$994$	4.00000	0.126872
$995$	0	0
$996$	−3.00000	−0.0950586
$997$	−13.0000	−0.411714	−0.205857	−	0.978582i	$-0.565998\pi$
$997$	−13.0000	−0.411714	−0.205857	+	0.978582i	$0.565998\pi$
$998$	23.0000	0.728052
$999$	−45.0000	−1.42374

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	53.2.a.a.1.1	✓	1
3.2	odd	2		477.2.a.a.1.1		1
4.3	odd	2		848.2.a.g.1.1		1
5.2	odd	4		1325.2.b.c.849.1		2
5.3	odd	4		1325.2.b.c.849.2		2
5.4	even	2		1325.2.a.e.1.1		1
7.6	odd	2		2597.2.a.a.1.1		1
8.3	odd	2		3392.2.a.a.1.1		1
8.5	even	2		3392.2.a.s.1.1		1
11.10	odd	2		6413.2.a.h.1.1		1
12.11	even	2		7632.2.a.j.1.1		1
13.12	even	2		8957.2.a.b.1.1		1
53.52	even	2		2809.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
53.2.a.a.1.1	✓	1	1.1	even	1	trivial
477.2.a.a.1.1		1	3.2	odd	2
848.2.a.g.1.1		1	4.3	odd	2
1325.2.a.e.1.1		1	5.4	even	2
1325.2.b.c.849.1		2	5.2	odd	4
1325.2.b.c.849.2		2	5.3	odd	4
2597.2.a.a.1.1		1	7.6	odd	2
2809.2.a.a.1.1		1	53.52	even	2
3392.2.a.a.1.1		1	8.3	odd	2
3392.2.a.s.1.1		1	8.5	even	2
6413.2.a.h.1.1		1	11.10	odd	2
7632.2.a.j.1.1		1	12.11	even	2
8957.2.a.b.1.1		1	13.12	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:	\( N \)	\(=\)	\( 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	53.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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