LMFDB - Embedded newform 5390.2.a.bf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5390,2,Mod(1,5390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5390, 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("5390.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5390.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:	$43.0393666895$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5390.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +7.00000 q^{17} -2.00000 q^{18} -5.00000 q^{19} -1.00000 q^{20} +1.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} -5.00000 q^{27} +5.00000 q^{29} -1.00000 q^{30} +3.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} +7.00000 q^{34} -2.00000 q^{36} +3.00000 q^{37} -5.00000 q^{38} +6.00000 q^{39} -1.00000 q^{40} -2.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +2.00000 q^{45} -6.00000 q^{46} +2.00000 q^{47} +1.00000 q^{48} +1.00000 q^{50} +7.00000 q^{51} +6.00000 q^{52} -1.00000 q^{53} -5.00000 q^{54} -1.00000 q^{55} -5.00000 q^{57} +5.00000 q^{58} +10.0000 q^{59} -1.00000 q^{60} -7.00000 q^{61} +3.00000 q^{62} +1.00000 q^{64} -6.00000 q^{65} +1.00000 q^{66} +8.00000 q^{67} +7.00000 q^{68} -6.00000 q^{69} +7.00000 q^{71} -2.00000 q^{72} -14.0000 q^{73} +3.00000 q^{74} +1.00000 q^{75} -5.00000 q^{76} +6.00000 q^{78} +10.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +6.00000 q^{83} -7.00000 q^{85} +4.00000 q^{86} +5.00000 q^{87} +1.00000 q^{88} +15.0000 q^{89} +2.00000 q^{90} -6.00000 q^{92} +3.00000 q^{93} +2.00000 q^{94} +5.00000 q^{95} +1.00000 q^{96} +12.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.00000	0.288675
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	−2.00000	−0.471405
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	6.00000	1.17670
$27$	−5.00000	−0.962250
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	−1.00000	−0.182574
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	7.00000	1.20049
$35$	0	0
$36$	−2.00000	−0.333333
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	−5.00000	−0.811107
$39$	6.00000	0.960769
$40$	−1.00000	−0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	1.00000	0.150756
$45$	2.00000	0.298142
$46$	−6.00000	−0.884652
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	1.00000	0.141421
$51$	7.00000	0.980196
$52$	6.00000	0.832050
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	−5.00000	−0.680414
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−5.00000	−0.662266
$58$	5.00000	0.656532
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	−1.00000	−0.129099
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	3.00000	0.381000
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.00000	−0.744208
$66$	1.00000	0.123091
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	7.00000	0.848875
$69$	−6.00000	−0.722315
$70$	0	0
$71$	7.00000	0.830747	0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000	0.830747	0.415374	+	0.909651i	$0.363651\pi$
$72$	−2.00000	−0.235702
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	3.00000	0.348743
$75$	1.00000	0.115470
$76$	−5.00000	−0.573539
$77$	0	0
$78$	6.00000	0.679366
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−7.00000	−0.759257
$86$	4.00000	0.431331
$87$	5.00000	0.536056
$88$	1.00000	0.106600
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	−6.00000	−0.625543
$93$	3.00000	0.311086
$94$	2.00000	0.206284
$95$	5.00000	0.512989
$96$	1.00000	0.102062
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	1.00000	0.100000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	7.00000	0.693103
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	−5.00000	−0.481125
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−1.00000	−0.0953463
$111$	3.00000	0.284747
$112$	0	0
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	−5.00000	−0.468293
$115$	6.00000	0.559503
$116$	5.00000	0.464238
$117$	−12.0000	−1.10940
$118$	10.0000	0.920575
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	1.00000	0.0909091
$122$	−7.00000	−0.633750
$123$	−2.00000	−0.180334
$124$	3.00000	0.269408
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	−6.00000	−0.526235
$131$	−17.0000	−1.48530	−0.742648	−	0.669681i	$-0.766431\pi$
$131$	−17.0000	−1.48530	−0.742648	+	0.669681i	$0.766431\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	8.00000	0.691095
$135$	5.00000	0.430331
$136$	7.00000	0.600245
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	−6.00000	−0.510754
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	7.00000	0.587427
$143$	6.00000	0.501745
$144$	−2.00000	−0.166667
$145$	−5.00000	−0.415227
$146$	−14.0000	−1.15865
$147$	0	0
$148$	3.00000	0.246598
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	1.00000	0.0816497
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	−5.00000	−0.405554
$153$	−14.0000	−1.13183
$154$	0	0
$155$	−3.00000	−0.240966
$156$	6.00000	0.480384
$157$	−3.00000	−0.239426	−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000	−0.239426	−0.119713	+	0.992809i	$0.538197\pi$
$158$	10.0000	0.795557
$159$	−1.00000	−0.0793052
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	19.0000	1.48819	0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000	1.48819	0.744097	+	0.668071i	$0.232880\pi$
$164$	−2.00000	−0.156174
$165$	−1.00000	−0.0778499
$166$	6.00000	0.465690
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−7.00000	−0.536875
$171$	10.0000	0.764719
$172$	4.00000	0.304997
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	5.00000	0.379049
$175$	0	0
$176$	1.00000	0.0753778
$177$	10.0000	0.751646
$178$	15.0000	1.12430
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	2.00000	0.149071
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−7.00000	−0.517455
$184$	−6.00000	−0.442326
$185$	−3.00000	−0.220564
$186$	3.00000	0.219971
$187$	7.00000	0.511891
$188$	2.00000	0.145865
$189$	0	0
$190$	5.00000	0.362738
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	1.00000	0.0721688
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	12.0000	0.861550
$195$	−6.00000	−0.429669
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	−2.00000	−0.142134
$199$	25.0000	1.77220	0.886102	−	0.463491i	$-0.153403\pi$
$199$	25.0000	1.77220	0.886102	+	0.463491i	$0.153403\pi$
$200$	1.00000	0.0707107
$201$	8.00000	0.564276
$202$	−2.00000	−0.140720
$203$	0	0
$204$	7.00000	0.490098
$205$	2.00000	0.139686
$206$	−4.00000	−0.278693
$207$	12.0000	0.834058
$208$	6.00000	0.416025
$209$	−5.00000	−0.345857
$210$	0	0
$211$	−23.0000	−1.58339	−0.791693	−	0.610920i	$-0.790800\pi$
$211$	−23.0000	−1.58339	−0.791693	+	0.610920i	$0.790800\pi$
$212$	−1.00000	−0.0686803
$213$	7.00000	0.479632
$214$	8.00000	0.546869
$215$	−4.00000	−0.272798
$216$	−5.00000	−0.340207
$217$	0	0
$218$	−10.0000	−0.677285
$219$	−14.0000	−0.946032
$220$	−1.00000	−0.0674200
$221$	42.0000	2.82523
$222$	3.00000	0.201347
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	−16.0000	−1.06430
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	−5.00000	−0.331133
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	5.00000	0.328266
$233$	9.00000	0.589610	0.294805	−	0.955557i	$-0.404745\pi$
$233$	9.00000	0.589610	0.294805	+	0.955557i	$0.404745\pi$
$234$	−12.0000	−0.784465
$235$	−2.00000	−0.130466
$236$	10.0000	0.650945
$237$	10.0000	0.649570
$238$	0	0
$239$	10.0000	0.646846	0.323423	−	0.946254i	$-0.395166\pi$
$239$	10.0000	0.646846	0.323423	+	0.946254i	$0.395166\pi$
$240$	−1.00000	−0.0645497
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	1.00000	0.0642824
$243$	16.0000	1.02640
$244$	−7.00000	−0.448129
$245$	0	0
$246$	−2.00000	−0.127515
$247$	−30.0000	−1.90885
$248$	3.00000	0.190500
$249$	6.00000	0.380235
$250$	−1.00000	−0.0632456
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	8.00000	0.501965
$255$	−7.00000	−0.438357
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	4.00000	0.249029
$259$	0	0
$260$	−6.00000	−0.372104
$261$	−10.0000	−0.618984
$262$	−17.0000	−1.05026
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	1.00000	0.0615457
$265$	1.00000	0.0614295
$266$	0	0
$267$	15.0000	0.917985
$268$	8.00000	0.488678
$269$	20.0000	1.21942	0.609711	−	0.792624i	$-0.291286\pi$
$269$	20.0000	1.21942	0.609711	+	0.792624i	$0.291286\pi$
$270$	5.00000	0.304290
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	7.00000	0.424437
$273$	0	0
$274$	−12.0000	−0.724947
$275$	1.00000	0.0603023
$276$	−6.00000	−0.361158
$277$	−12.0000	−0.721010	−0.360505	−	0.932757i	$-0.617396\pi$
$277$	−12.0000	−0.721010	−0.360505	+	0.932757i	$0.617396\pi$
$278$	20.0000	1.19952
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	2.00000	0.119098
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	7.00000	0.415374
$285$	5.00000	0.296174
$286$	6.00000	0.354787
$287$	0	0
$288$	−2.00000	−0.117851
$289$	32.0000	1.88235
$290$	−5.00000	−0.293610
$291$	12.0000	0.703452
$292$	−14.0000	−0.819288
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−10.0000	−0.582223
$296$	3.00000	0.174371
$297$	−5.00000	−0.290129
$298$	15.0000	0.868927
$299$	−36.0000	−2.08193
$300$	1.00000	0.0577350
$301$	0	0
$302$	2.00000	0.115087
$303$	−2.00000	−0.114897
$304$	−5.00000	−0.286770
$305$	7.00000	0.400819
$306$	−14.0000	−0.800327
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	−3.00000	−0.170389
$311$	3.00000	0.170114	0.0850572	−	0.996376i	$-0.472893\pi$
$311$	3.00000	0.170114	0.0850572	+	0.996376i	$0.472893\pi$
$312$	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	$0.445762\pi$
$314$	−3.00000	−0.169300
$315$	0	0
$316$	10.0000	0.562544
$317$	−7.00000	−0.393159	−0.196580	−	0.980488i	$-0.562983\pi$
$317$	−7.00000	−0.393159	−0.196580	+	0.980488i	$0.562983\pi$
$318$	−1.00000	−0.0560772
$319$	5.00000	0.279946
$320$	−1.00000	−0.0559017
$321$	8.00000	0.446516
$322$	0	0
$323$	−35.0000	−1.94745
$324$	1.00000	0.0555556
$325$	6.00000	0.332820
$326$	19.0000	1.05231
$327$	−10.0000	−0.553001
$328$	−2.00000	−0.110432
$329$	0	0
$330$	−1.00000	−0.0550482
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	6.00000	0.329293
$333$	−6.00000	−0.328798
$334$	−3.00000	−0.164153
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−17.0000	−0.926049	−0.463025	−	0.886345i	$-0.653236\pi$
$337$	−17.0000	−0.926049	−0.463025	+	0.886345i	$0.653236\pi$
$338$	23.0000	1.25104
$339$	−16.0000	−0.869001
$340$	−7.00000	−0.379628
$341$	3.00000	0.162459
$342$	10.0000	0.540738
$343$	0	0
$344$	4.00000	0.215666
$345$	6.00000	0.323029
$346$	−14.0000	−0.752645
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	5.00000	0.268028
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	−30.0000	−1.60128
$352$	1.00000	0.0533002
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	10.0000	0.531494
$355$	−7.00000	−0.371521
$356$	15.0000	0.794998
$357$	0	0
$358$	0	0
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	2.00000	0.105409
$361$	6.00000	0.315789
$362$	−2.00000	−0.105118
$363$	1.00000	0.0524864
$364$	0	0
$365$	14.0000	0.732793
$366$	−7.00000	−0.365896
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	−6.00000	−0.312772
$369$	4.00000	0.208232
$370$	−3.00000	−0.155963
$371$	0	0
$372$	3.00000	0.155543
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	7.00000	0.361961
$375$	−1.00000	−0.0516398
$376$	2.00000	0.103142
$377$	30.0000	1.54508
$378$	0	0
$379$	−30.0000	−1.54100	−0.770498	−	0.637442i	$-0.779993\pi$
$379$	−30.0000	−1.54100	−0.770498	+	0.637442i	$0.779993\pi$
$380$	5.00000	0.256495
$381$	8.00000	0.409852
$382$	12.0000	0.613973
$383$	−34.0000	−1.73732	−0.868659	−	0.495410i	$-0.835018\pi$
$383$	−34.0000	−1.73732	−0.868659	+	0.495410i	$0.835018\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−11.0000	−0.559885
$387$	−8.00000	−0.406663
$388$	12.0000	0.609208
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	−6.00000	−0.303822
$391$	−42.0000	−2.12403
$392$	0	0
$393$	−17.0000	−0.857537
$394$	−12.0000	−0.604551
$395$	−10.0000	−0.503155
$396$	−2.00000	−0.100504
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	25.0000	1.25314
$399$	0	0
$400$	1.00000	0.0500000
$401$	−13.0000	−0.649189	−0.324595	−	0.945853i	$-0.605228\pi$
$401$	−13.0000	−0.649189	−0.324595	+	0.945853i	$0.605228\pi$
$402$	8.00000	0.399004
$403$	18.0000	0.896644
$404$	−2.00000	−0.0995037
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	3.00000	0.148704
$408$	7.00000	0.346552
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	2.00000	0.0987730
$411$	−12.0000	−0.591916
$412$	−4.00000	−0.197066
$413$	0	0
$414$	12.0000	0.589768
$415$	−6.00000	−0.294528
$416$	6.00000	0.294174
$417$	20.0000	0.979404
$418$	−5.00000	−0.244558
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	−23.0000	−1.11962
$423$	−4.00000	−0.194487
$424$	−1.00000	−0.0485643
$425$	7.00000	0.339550
$426$	7.00000	0.339151
$427$	0	0
$428$	8.00000	0.386695
$429$	6.00000	0.289683
$430$	−4.00000	−0.192897
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	−5.00000	−0.240563
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	−5.00000	−0.239732
$436$	−10.0000	−0.478913
$437$	30.0000	1.43509
$438$	−14.0000	−0.668946
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	42.0000	1.99774
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	3.00000	0.142374
$445$	−15.0000	−0.711068
$446$	6.00000	0.284108
$447$	15.0000	0.709476
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	−2.00000	−0.0942809
$451$	−2.00000	−0.0941763
$452$	−16.0000	−0.752577
$453$	2.00000	0.0939682
$454$	2.00000	0.0938647
$455$	0	0
$456$	−5.00000	−0.234146
$457$	3.00000	0.140334	0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000	0.140334	0.0701670	+	0.997535i	$0.477647\pi$
$458$	−10.0000	−0.467269
$459$	−35.0000	−1.63366
$460$	6.00000	0.279751
$461$	−27.0000	−1.25752	−0.628758	−	0.777601i	$-0.716436\pi$
$461$	−27.0000	−1.25752	−0.628758	+	0.777601i	$0.716436\pi$
$462$	0	0
$463$	34.0000	1.58011	0.790057	−	0.613033i	$-0.210051\pi$
$463$	34.0000	1.58011	0.790057	+	0.613033i	$0.210051\pi$
$464$	5.00000	0.232119
$465$	−3.00000	−0.139122
$466$	9.00000	0.416917
$467$	−23.0000	−1.06431	−0.532157	−	0.846646i	$-0.678618\pi$
$467$	−23.0000	−1.06431	−0.532157	+	0.846646i	$0.678618\pi$
$468$	−12.0000	−0.554700
$469$	0	0
$470$	−2.00000	−0.0922531
$471$	−3.00000	−0.138233
$472$	10.0000	0.460287
$473$	4.00000	0.183920
$474$	10.0000	0.459315
$475$	−5.00000	−0.229416
$476$	0	0
$477$	2.00000	0.0915737
$478$	10.0000	0.457389
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−1.00000	−0.0456435
$481$	18.0000	0.820729
$482$	18.0000	0.819878
$483$	0	0
$484$	1.00000	0.0454545
$485$	−12.0000	−0.544892
$486$	16.0000	0.725775
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	−7.00000	−0.316875
$489$	19.0000	0.859210
$490$	0	0
$491$	−3.00000	−0.135388	−0.0676941	−	0.997706i	$-0.521564\pi$
$491$	−3.00000	−0.135388	−0.0676941	+	0.997706i	$0.521564\pi$
$492$	−2.00000	−0.0901670
$493$	35.0000	1.57632
$494$	−30.0000	−1.34976
$495$	2.00000	0.0898933
$496$	3.00000	0.134704
$497$	0	0
$498$	6.00000	0.268866
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	−1.00000	−0.0447214
$501$	−3.00000	−0.134030
$502$	−2.00000	−0.0892644
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	−6.00000	−0.266733
$507$	23.0000	1.02147
$508$	8.00000	0.354943
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	−7.00000	−0.309965
$511$	0	0
$512$	1.00000	0.0441942
$513$	25.0000	1.10378
$514$	2.00000	0.0882162
$515$	4.00000	0.176261
$516$	4.00000	0.176090
$517$	2.00000	0.0879599
$518$	0	0
$519$	−14.0000	−0.614532
$520$	−6.00000	−0.263117
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	−10.0000	−0.437688
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	−17.0000	−0.742648
$525$	0	0
$526$	9.00000	0.392419
$527$	21.0000	0.914774
$528$	1.00000	0.0435194
$529$	13.0000	0.565217
$530$	1.00000	0.0434372
$531$	−20.0000	−0.867926
$532$	0	0
$533$	−12.0000	−0.519778
$534$	15.0000	0.649113
$535$	−8.00000	−0.345870
$536$	8.00000	0.345547
$537$	0	0
$538$	20.0000	0.862261
$539$	0	0
$540$	5.00000	0.215166
$541$	−23.0000	−0.988847	−0.494424	−	0.869221i	$-0.664621\pi$
$541$	−23.0000	−0.988847	−0.494424	+	0.869221i	$0.664621\pi$
$542$	8.00000	0.343629
$543$	−2.00000	−0.0858282
$544$	7.00000	0.300123
$545$	10.0000	0.428353
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−12.0000	−0.512615
$549$	14.0000	0.597505
$550$	1.00000	0.0426401
$551$	−25.0000	−1.06504
$552$	−6.00000	−0.255377
$553$	0	0
$554$	−12.0000	−0.509831
$555$	−3.00000	−0.127343
$556$	20.0000	0.848189
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	−6.00000	−0.254000
$559$	24.0000	1.01509
$560$	0	0
$561$	7.00000	0.295540
$562$	−18.0000	−0.759284
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	2.00000	0.0842152
$565$	16.0000	0.673125
$566$	6.00000	0.252199
$567$	0	0
$568$	7.00000	0.293713
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	5.00000	0.209427
$571$	27.0000	1.12991	0.564957	−	0.825120i	$-0.308893\pi$
$571$	27.0000	1.12991	0.564957	+	0.825120i	$0.308893\pi$
$572$	6.00000	0.250873
$573$	12.0000	0.501307
$574$	0	0
$575$	−6.00000	−0.250217
$576$	−2.00000	−0.0833333
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	32.0000	1.33102
$579$	−11.0000	−0.457144
$580$	−5.00000	−0.207614
$581$	0	0
$582$	12.0000	0.497416
$583$	−1.00000	−0.0414158
$584$	−14.0000	−0.579324
$585$	12.0000	0.496139
$586$	6.00000	0.247858
$587$	27.0000	1.11441	0.557205	−	0.830375i	$-0.311874\pi$
$587$	27.0000	1.11441	0.557205	+	0.830375i	$0.311874\pi$
$588$	0	0
$589$	−15.0000	−0.618064
$590$	−10.0000	−0.411693
$591$	−12.0000	−0.493614
$592$	3.00000	0.123299
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	−5.00000	−0.205152
$595$	0	0
$596$	15.0000	0.614424
$597$	25.0000	1.02318
$598$	−36.0000	−1.47215
$599$	45.0000	1.83865	0.919325	−	0.393499i	$-0.128735\pi$
$599$	45.0000	1.83865	0.919325	+	0.393499i	$0.128735\pi$
$600$	1.00000	0.0408248
$601$	−42.0000	−1.71322	−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000	−1.71322	−0.856608	+	0.515968i	$0.827432\pi$
$602$	0	0
$603$	−16.0000	−0.651570
$604$	2.00000	0.0813788
$605$	−1.00000	−0.0406558
$606$	−2.00000	−0.0812444
$607$	47.0000	1.90767	0.953836	−	0.300329i	$-0.0970966\pi$
$607$	47.0000	1.90767	0.953836	+	0.300329i	$0.0970966\pi$
$608$	−5.00000	−0.202777
$609$	0	0
$610$	7.00000	0.283422
$611$	12.0000	0.485468
$612$	−14.0000	−0.565916
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	2.00000	0.0807134
$615$	2.00000	0.0806478
$616$	0	0
$617$	8.00000	0.322068	0.161034	−	0.986949i	$-0.448517\pi$
$617$	8.00000	0.322068	0.161034	+	0.986949i	$0.448517\pi$
$618$	−4.00000	−0.160904
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	−3.00000	−0.120483
$621$	30.0000	1.20386
$622$	3.00000	0.120289
$623$	0	0
$624$	6.00000	0.240192
$625$	1.00000	0.0400000
$626$	6.00000	0.239808
$627$	−5.00000	−0.199681
$628$	−3.00000	−0.119713
$629$	21.0000	0.837325
$630$	0	0
$631$	−33.0000	−1.31371	−0.656855	−	0.754017i	$-0.728113\pi$
$631$	−33.0000	−1.31371	−0.656855	+	0.754017i	$0.728113\pi$
$632$	10.0000	0.397779
$633$	−23.0000	−0.914168
$634$	−7.00000	−0.278006
$635$	−8.00000	−0.317470
$636$	−1.00000	−0.0396526
$637$	0	0
$638$	5.00000	0.197952
$639$	−14.0000	−0.553831
$640$	−1.00000	−0.0395285
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	8.00000	0.315735
$643$	−19.0000	−0.749287	−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	0	0
$645$	−4.00000	−0.157500
$646$	−35.0000	−1.37706
$647$	42.0000	1.65119	0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000	1.65119	0.825595	+	0.564263i	$0.190840\pi$
$648$	1.00000	0.0392837
$649$	10.0000	0.392534
$650$	6.00000	0.235339
$651$	0	0
$652$	19.0000	0.744097
$653$	−31.0000	−1.21312	−0.606562	−	0.795036i	$-0.707452\pi$
$653$	−31.0000	−1.21312	−0.606562	+	0.795036i	$0.707452\pi$
$654$	−10.0000	−0.391031
$655$	17.0000	0.664245
$656$	−2.00000	−0.0780869
$657$	28.0000	1.09238
$658$	0	0
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	−1.00000	−0.0389249
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	−28.0000	−1.08825
$663$	42.0000	1.63114
$664$	6.00000	0.232845
$665$	0	0
$666$	−6.00000	−0.232495
$667$	−30.0000	−1.16160
$668$	−3.00000	−0.116073
$669$	6.00000	0.231973
$670$	−8.00000	−0.309067
$671$	−7.00000	−0.270232
$672$	0	0
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	−17.0000	−0.654816
$675$	−5.00000	−0.192450
$676$	23.0000	0.884615
$677$	−28.0000	−1.07613	−0.538064	−	0.842904i	$-0.680844\pi$
$677$	−28.0000	−1.07613	−0.538064	+	0.842904i	$0.680844\pi$
$678$	−16.0000	−0.614476
$679$	0	0
$680$	−7.00000	−0.268438
$681$	2.00000	0.0766402
$682$	3.00000	0.114876
$683$	−31.0000	−1.18618	−0.593091	−	0.805135i	$-0.702093\pi$
$683$	−31.0000	−1.18618	−0.593091	+	0.805135i	$0.702093\pi$
$684$	10.0000	0.382360
$685$	12.0000	0.458496
$686$	0	0
$687$	−10.0000	−0.381524
$688$	4.00000	0.152499
$689$	−6.00000	−0.228582
$690$	6.00000	0.228416
$691$	38.0000	1.44559	0.722794	−	0.691063i	$-0.242858\pi$
$691$	38.0000	1.44559	0.722794	+	0.691063i	$0.242858\pi$
$692$	−14.0000	−0.532200
$693$	0	0
$694$	18.0000	0.683271
$695$	−20.0000	−0.758643
$696$	5.00000	0.189525
$697$	−14.0000	−0.530288
$698$	−30.0000	−1.13552
$699$	9.00000	0.340411
$700$	0	0
$701$	7.00000	0.264386	0.132193	−	0.991224i	$-0.457798\pi$
$701$	7.00000	0.264386	0.132193	+	0.991224i	$0.457798\pi$
$702$	−30.0000	−1.13228
$703$	−15.0000	−0.565736
$704$	1.00000	0.0376889
$705$	−2.00000	−0.0753244
$706$	−34.0000	−1.27961
$707$	0	0
$708$	10.0000	0.375823
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	−7.00000	−0.262705
$711$	−20.0000	−0.750059
$712$	15.0000	0.562149
$713$	−18.0000	−0.674105
$714$	0	0
$715$	−6.00000	−0.224387
$716$	0	0
$717$	10.0000	0.373457
$718$	−20.0000	−0.746393
$719$	−25.0000	−0.932343	−0.466171	−	0.884694i	$-0.654367\pi$
$719$	−25.0000	−0.932343	−0.466171	+	0.884694i	$0.654367\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	6.00000	0.223297
$723$	18.0000	0.669427
$724$	−2.00000	−0.0743294
$725$	5.00000	0.185695
$726$	1.00000	0.0371135
$727$	22.0000	0.815935	0.407967	−	0.912996i	$-0.366238\pi$
$727$	22.0000	0.815935	0.407967	+	0.912996i	$0.366238\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	14.0000	0.518163
$731$	28.0000	1.03562
$732$	−7.00000	−0.258727
$733$	−24.0000	−0.886460	−0.443230	−	0.896408i	$-0.646168\pi$
$733$	−24.0000	−0.886460	−0.443230	+	0.896408i	$0.646168\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	−6.00000	−0.221163
$737$	8.00000	0.294684
$738$	4.00000	0.147242
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	−3.00000	−0.110282
$741$	−30.0000	−1.10208
$742$	0	0
$743$	−21.0000	−0.770415	−0.385208	−	0.922830i	$-0.625870\pi$
$743$	−21.0000	−0.770415	−0.385208	+	0.922830i	$0.625870\pi$
$744$	3.00000	0.109985
$745$	−15.0000	−0.549557
$746$	−6.00000	−0.219676
$747$	−12.0000	−0.439057
$748$	7.00000	0.255945
$749$	0	0
$750$	−1.00000	−0.0365148
$751$	17.0000	0.620339	0.310169	−	0.950681i	$-0.399614\pi$
$751$	17.0000	0.620339	0.310169	+	0.950681i	$0.399614\pi$
$752$	2.00000	0.0729325
$753$	−2.00000	−0.0728841
$754$	30.0000	1.09254
$755$	−2.00000	−0.0727875
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−30.0000	−1.08965
$759$	−6.00000	−0.217786
$760$	5.00000	0.181369
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	8.00000	0.289809
$763$	0	0
$764$	12.0000	0.434145
$765$	14.0000	0.506171
$766$	−34.0000	−1.22847
$767$	60.0000	2.16647
$768$	1.00000	0.0360844
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	−11.0000	−0.395899
$773$	−19.0000	−0.683383	−0.341691	−	0.939812i	$-0.611000\pi$
$773$	−19.0000	−0.683383	−0.341691	+	0.939812i	$0.611000\pi$
$774$	−8.00000	−0.287554
$775$	3.00000	0.107763
$776$	12.0000	0.430775
$777$	0	0
$778$	−30.0000	−1.07555
$779$	10.0000	0.358287
$780$	−6.00000	−0.214834
$781$	7.00000	0.250480
$782$	−42.0000	−1.50192
$783$	−25.0000	−0.893427
$784$	0	0
$785$	3.00000	0.107075
$786$	−17.0000	−0.606370
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	−12.0000	−0.427482
$789$	9.00000	0.320408
$790$	−10.0000	−0.355784
$791$	0	0
$792$	−2.00000	−0.0710669
$793$	−42.0000	−1.49146
$794$	2.00000	0.0709773
$795$	1.00000	0.0354663
$796$	25.0000	0.886102
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	14.0000	0.495284
$800$	1.00000	0.0353553
$801$	−30.0000	−1.06000
$802$	−13.0000	−0.459046
$803$	−14.0000	−0.494049
$804$	8.00000	0.282138
$805$	0	0
$806$	18.0000	0.634023
$807$	20.0000	0.704033
$808$	−2.00000	−0.0703598
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	−1.00000	−0.0351364
$811$	−7.00000	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$811$	−7.00000	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	3.00000	0.105150
$815$	−19.0000	−0.665541
$816$	7.00000	0.245049
$817$	−20.0000	−0.699711
$818$	20.0000	0.699284
$819$	0	0
$820$	2.00000	0.0698430
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	−12.0000	−0.418548
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	−4.00000	−0.139347
$825$	1.00000	0.0348155
$826$	0	0
$827$	8.00000	0.278187	0.139094	−	0.990279i	$-0.455581\pi$
$827$	8.00000	0.278187	0.139094	+	0.990279i	$0.455581\pi$
$828$	12.0000	0.417029
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	−6.00000	−0.208263
$831$	−12.0000	−0.416275
$832$	6.00000	0.208013
$833$	0	0
$834$	20.0000	0.692543
$835$	3.00000	0.103819
$836$	−5.00000	−0.172929
$837$	−15.0000	−0.518476
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	32.0000	1.10279
$843$	−18.0000	−0.619953
$844$	−23.0000	−0.791693
$845$	−23.0000	−0.791224
$846$	−4.00000	−0.137523
$847$	0	0
$848$	−1.00000	−0.0343401
$849$	6.00000	0.205919
$850$	7.00000	0.240098
$851$	−18.0000	−0.617032
$852$	7.00000	0.239816
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	0	0
$855$	−10.0000	−0.341993
$856$	8.00000	0.273434
$857$	7.00000	0.239115	0.119558	−	0.992827i	$-0.461852\pi$
$857$	7.00000	0.239115	0.119558	+	0.992827i	$0.461852\pi$
$858$	6.00000	0.204837
$859$	−30.0000	−1.02359	−0.511793	−	0.859109i	$-0.671019\pi$
$859$	−30.0000	−1.02359	−0.511793	+	0.859109i	$0.671019\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	−8.00000	−0.272481
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	−5.00000	−0.170103
$865$	14.0000	0.476014
$866$	16.0000	0.543702
$867$	32.0000	1.08678
$868$	0	0
$869$	10.0000	0.339227
$870$	−5.00000	−0.169516
$871$	48.0000	1.62642
$872$	−10.0000	−0.338643
$873$	−24.0000	−0.812277
$874$	30.0000	1.01477
$875$	0	0
$876$	−14.0000	−0.473016
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	−20.0000	−0.674967
$879$	6.00000	0.202375
$880$	−1.00000	−0.0337100
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	9.00000	0.302874	0.151437	−	0.988467i	$-0.451610\pi$
$883$	9.00000	0.302874	0.151437	+	0.988467i	$0.451610\pi$
$884$	42.0000	1.41261
$885$	−10.0000	−0.336146
$886$	4.00000	0.134383
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	3.00000	0.100673
$889$	0	0
$890$	−15.0000	−0.502801
$891$	1.00000	0.0335013
$892$	6.00000	0.200895
$893$	−10.0000	−0.334637
$894$	15.0000	0.501675
$895$	0	0
$896$	0	0
$897$	−36.0000	−1.20201
$898$	−30.0000	−1.00111
$899$	15.0000	0.500278
$900$	−2.00000	−0.0666667
$901$	−7.00000	−0.233204
$902$	−2.00000	−0.0665927
$903$	0	0
$904$	−16.0000	−0.532152
$905$	2.00000	0.0664822
$906$	2.00000	0.0664455
$907$	−57.0000	−1.89265	−0.946327	−	0.323211i	$-0.895238\pi$
$907$	−57.0000	−1.89265	−0.946327	+	0.323211i	$0.895238\pi$
$908$	2.00000	0.0663723
$909$	4.00000	0.132672
$910$	0	0
$911$	27.0000	0.894550	0.447275	−	0.894397i	$-0.352395\pi$
$911$	27.0000	0.894550	0.447275	+	0.894397i	$0.352395\pi$
$912$	−5.00000	−0.165567
$913$	6.00000	0.198571
$914$	3.00000	0.0992312
$915$	7.00000	0.231413
$916$	−10.0000	−0.330409
$917$	0	0
$918$	−35.0000	−1.15517
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	6.00000	0.197814
$921$	2.00000	0.0659022
$922$	−27.0000	−0.889198
$923$	42.0000	1.38245
$924$	0	0
$925$	3.00000	0.0986394
$926$	34.0000	1.11731
$927$	8.00000	0.262754
$928$	5.00000	0.164133
$929$	−35.0000	−1.14831	−0.574156	−	0.818746i	$-0.694670\pi$
$929$	−35.0000	−1.14831	−0.574156	+	0.818746i	$0.694670\pi$
$930$	−3.00000	−0.0983739
$931$	0	0
$932$	9.00000	0.294805
$933$	3.00000	0.0982156
$934$	−23.0000	−0.752583
$935$	−7.00000	−0.228924
$936$	−12.0000	−0.392232
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	0	0
$939$	6.00000	0.195803
$940$	−2.00000	−0.0652328
$941$	−17.0000	−0.554184	−0.277092	−	0.960843i	$-0.589371\pi$
$941$	−17.0000	−0.554184	−0.277092	+	0.960843i	$0.589371\pi$
$942$	−3.00000	−0.0977453
$943$	12.0000	0.390774
$944$	10.0000	0.325472
$945$	0	0
$946$	4.00000	0.130051
$947$	−27.0000	−0.877382	−0.438691	−	0.898638i	$-0.644558\pi$
$947$	−27.0000	−0.877382	−0.438691	+	0.898638i	$0.644558\pi$
$948$	10.0000	0.324785
$949$	−84.0000	−2.72676
$950$	−5.00000	−0.162221
$951$	−7.00000	−0.226991
$952$	0	0
$953$	39.0000	1.26333	0.631667	−	0.775240i	$-0.282371\pi$
$953$	39.0000	1.26333	0.631667	+	0.775240i	$0.282371\pi$
$954$	2.00000	0.0647524
$955$	−12.0000	−0.388311
$956$	10.0000	0.323423
$957$	5.00000	0.161627
$958$	0	0
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	−22.0000	−0.709677
$962$	18.0000	0.580343
$963$	−16.0000	−0.515593
$964$	18.0000	0.579741
$965$	11.0000	0.354103
$966$	0	0
$967$	−27.0000	−0.868261	−0.434131	−	0.900850i	$-0.642944\pi$
$967$	−27.0000	−0.868261	−0.434131	+	0.900850i	$0.642944\pi$
$968$	1.00000	0.0321412
$969$	−35.0000	−1.12436
$970$	−12.0000	−0.385297
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	16.0000	0.513200
$973$	0	0
$974$	−12.0000	−0.384505
$975$	6.00000	0.192154
$976$	−7.00000	−0.224065
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	19.0000	0.607553
$979$	15.0000	0.479402
$980$	0	0
$981$	20.0000	0.638551
$982$	−3.00000	−0.0957338
$983$	−54.0000	−1.72233	−0.861166	−	0.508323i	$-0.830265\pi$
$983$	−54.0000	−1.72233	−0.861166	+	0.508323i	$0.830265\pi$
$984$	−2.00000	−0.0637577
$985$	12.0000	0.382352
$986$	35.0000	1.11463
$987$	0	0
$988$	−30.0000	−0.954427
$989$	−24.0000	−0.763156
$990$	2.00000	0.0635642
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	3.00000	0.0952501
$993$	−28.0000	−0.888553
$994$	0	0
$995$	−25.0000	−0.792553
$996$	6.00000	0.190117
$997$	32.0000	1.01345	0.506725	−	0.862108i	$-0.330856\pi$
$997$	32.0000	1.01345	0.506725	+	0.862108i	$0.330856\pi$
$998$	20.0000	0.633089
$999$	−15.0000	−0.474579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5390.2.a.bf.1.1		1
7.6	odd	2		110.2.a.b.1.1	✓	1
21.20	even	2		990.2.a.d.1.1		1
28.27	even	2		880.2.a.i.1.1		1
35.13	even	4		550.2.b.a.199.1		2
35.27	even	4		550.2.b.a.199.2		2
35.34	odd	2		550.2.a.f.1.1		1
56.13	odd	2		3520.2.a.y.1.1		1
56.27	even	2		3520.2.a.h.1.1		1
77.76	even	2		1210.2.a.b.1.1		1
84.83	odd	2		7920.2.a.d.1.1		1
105.62	odd	4		4950.2.c.m.199.1		2
105.83	odd	4		4950.2.c.m.199.2		2
105.104	even	2		4950.2.a.bc.1.1		1
140.27	odd	4		4400.2.b.i.4049.1		2
140.83	odd	4		4400.2.b.i.4049.2		2
140.139	even	2		4400.2.a.l.1.1		1
308.307	odd	2		9680.2.a.x.1.1		1
385.384	even	2		6050.2.a.bj.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.b.1.1	✓	1	7.6	odd	2
550.2.a.f.1.1		1	35.34	odd	2
550.2.b.a.199.1		2	35.13	even	4
550.2.b.a.199.2		2	35.27	even	4
880.2.a.i.1.1		1	28.27	even	2
990.2.a.d.1.1		1	21.20	even	2
1210.2.a.b.1.1		1	77.76	even	2
3520.2.a.h.1.1		1	56.27	even	2
3520.2.a.y.1.1		1	56.13	odd	2
4400.2.a.l.1.1		1	140.139	even	2
4400.2.b.i.4049.1		2	140.27	odd	4
4400.2.b.i.4049.2		2	140.83	odd	4
4950.2.a.bc.1.1		1	105.104	even	2
4950.2.c.m.199.1		2	105.62	odd	4
4950.2.c.m.199.2		2	105.83	odd	4
5390.2.a.bf.1.1		1	1.1	even	1	trivial
6050.2.a.bj.1.1		1	385.384	even	2
7920.2.a.d.1.1		1	84.83	odd	2
9680.2.a.x.1.1		1	308.307	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5390.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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