LMFDB - Embedded newform 5070.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, 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("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} +1.00000 q^{22} -3.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{28} -1.00000 q^{29} +1.00000 q^{30} +3.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +5.00000 q^{37} +6.00000 q^{38} -1.00000 q^{40} -10.0000 q^{41} -2.00000 q^{42} +5.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} +3.00000 q^{46} -3.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +14.0000 q^{53} +1.00000 q^{54} -1.00000 q^{55} +2.00000 q^{56} +6.00000 q^{57} +1.00000 q^{58} +5.00000 q^{59} -1.00000 q^{60} -10.0000 q^{61} -3.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} +2.00000 q^{68} +3.00000 q^{69} +2.00000 q^{70} -4.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} -5.00000 q^{74} -1.00000 q^{75} -6.00000 q^{76} +2.00000 q^{77} +5.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +6.00000 q^{83} +2.00000 q^{84} +2.00000 q^{85} -5.00000 q^{86} +1.00000 q^{87} +1.00000 q^{88} -10.0000 q^{89} -1.00000 q^{90} -3.00000 q^{92} -3.00000 q^{93} +3.00000 q^{94} -6.00000 q^{95} +1.00000 q^{96} +10.0000 q^{97} +3.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	2.00000	0.534522
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−1.00000	−0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	1.00000	0.223607
$21$	2.00000	0.436436
$22$	1.00000	0.213201
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	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
$34$	−2.00000	−0.342997
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	−2.00000	−0.308607
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	−1.00000	−0.150756
$45$	1.00000	0.149071
$46$	3.00000	0.442326
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	−2.00000	−0.280056
$52$	0	0
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	1.00000	0.136083
$55$	−1.00000	−0.134840
$56$	2.00000	0.267261
$57$	6.00000	0.794719
$58$	1.00000	0.131306
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	−1.00000	−0.129099
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−3.00000	−0.381000
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	2.00000	0.242536
$69$	3.00000	0.361158
$70$	2.00000	0.239046
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	−1.00000	−0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−5.00000	−0.581238
$75$	−1.00000	−0.115470
$76$	−6.00000	−0.688247
$77$	2.00000	0.227921
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	2.00000	0.218218
$85$	2.00000	0.216930
$86$	−5.00000	−0.539164
$87$	1.00000	0.107211
$88$	1.00000	0.106600
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−3.00000	−0.312772
$93$	−3.00000	−0.311086
$94$	3.00000	0.309426
$95$	−6.00000	−0.615587
$96$	1.00000	0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	3.00000	0.303046
$99$	−1.00000	−0.100504
$100$	1.00000	0.100000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	2.00000	0.198030
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	−14.0000	−1.35980
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−1.00000	−0.0962250
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	1.00000	0.0953463
$111$	−5.00000	−0.474579
$112$	−2.00000	−0.188982
$113$	17.0000	1.59923	0.799613	−	0.600516i	$-0.205038\pi$
$113$	17.0000	1.59923	0.799613	+	0.600516i	$0.205038\pi$
$114$	−6.00000	−0.561951
$115$	−3.00000	−0.279751
$116$	−1.00000	−0.0928477
$117$	0	0
$118$	−5.00000	−0.460287
$119$	−4.00000	−0.366679
$120$	1.00000	0.0912871
$121$	−10.0000	−0.909091
$122$	10.0000	0.905357
$123$	10.0000	0.901670
$124$	3.00000	0.269408
$125$	1.00000	0.0894427
$126$	2.00000	0.178174
$127$	−14.0000	−1.24230	−0.621150	−	0.783692i	$-0.713334\pi$
$127$	−14.0000	−1.24230	−0.621150	+	0.783692i	$0.713334\pi$
$128$	−1.00000	−0.0883883
$129$	−5.00000	−0.440225
$130$	0	0
$131$	13.0000	1.13582	0.567908	−	0.823092i	$-0.307753\pi$
$131$	13.0000	1.13582	0.567908	+	0.823092i	$0.307753\pi$
$132$	1.00000	0.0870388
$133$	12.0000	1.04053
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	−2.00000	−0.171499
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	−3.00000	−0.255377
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	−2.00000	−0.169031
$141$	3.00000	0.252646
$142$	4.00000	0.335673
$143$	0	0
$144$	1.00000	0.0833333
$145$	−1.00000	−0.0830455
$146$	−2.00000	−0.165521
$147$	3.00000	0.247436
$148$	5.00000	0.410997
$149$	−11.0000	−0.901155	−0.450578	−	0.892737i	$-0.648782\pi$
$149$	−11.0000	−0.901155	−0.450578	+	0.892737i	$0.648782\pi$
$150$	1.00000	0.0816497
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	6.00000	0.486664
$153$	2.00000	0.161690
$154$	−2.00000	−0.161165
$155$	3.00000	0.240966
$156$	0	0
$157$	25.0000	1.99522	0.997609	−	0.0691164i	$-0.0220180\pi$
$157$	25.0000	1.99522	0.997609	+	0.0691164i	$0.0220180\pi$
$158$	−5.00000	−0.397779
$159$	−14.0000	−1.11027
$160$	−1.00000	−0.0790569
$161$	6.00000	0.472866
$162$	−1.00000	−0.0785674
$163$	−17.0000	−1.33154	−0.665771	−	0.746156i	$-0.731897\pi$
$163$	−17.0000	−1.33154	−0.665771	+	0.746156i	$0.731897\pi$
$164$	−10.0000	−0.780869
$165$	1.00000	0.0778499
$166$	−6.00000	−0.465690
$167$	7.00000	0.541676	0.270838	−	0.962625i	$-0.412699\pi$
$167$	7.00000	0.541676	0.270838	+	0.962625i	$0.412699\pi$
$168$	−2.00000	−0.154303
$169$	0	0
$170$	−2.00000	−0.153393
$171$	−6.00000	−0.458831
$172$	5.00000	0.381246
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	−1.00000	−0.0758098
$175$	−2.00000	−0.151186
$176$	−1.00000	−0.0753778
$177$	−5.00000	−0.375823
$178$	10.0000	0.749532
$179$	−7.00000	−0.523205	−0.261602	−	0.965176i	$-0.584251\pi$
$179$	−7.00000	−0.523205	−0.261602	+	0.965176i	$0.584251\pi$
$180$	1.00000	0.0745356
$181$	16.0000	1.18927	0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000	1.18927	0.594635	+	0.803996i	$0.297296\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	3.00000	0.221163
$185$	5.00000	0.367607
$186$	3.00000	0.219971
$187$	−2.00000	−0.146254
$188$	−3.00000	−0.218797
$189$	2.00000	0.145479
$190$	6.00000	0.435286
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	−1.00000	−0.0721688
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	−3.00000	−0.214286
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	1.00000	0.0710669
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−14.0000	−0.985037
$203$	2.00000	0.140372
$204$	−2.00000	−0.140028
$205$	−10.0000	−0.698430
$206$	6.00000	0.418040
$207$	−3.00000	−0.208514
$208$	0	0
$209$	6.00000	0.415029
$210$	−2.00000	−0.138013
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	14.0000	0.961524
$213$	4.00000	0.274075
$214$	6.00000	0.410152
$215$	5.00000	0.340997
$216$	1.00000	0.0680414
$217$	−6.00000	−0.407307
$218$	−6.00000	−0.406371
$219$	−2.00000	−0.135147
$220$	−1.00000	−0.0674200
$221$	0	0
$222$	5.00000	0.335578
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	2.00000	0.133631
$225$	1.00000	0.0666667
$226$	−17.0000	−1.13082
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	6.00000	0.397360
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	3.00000	0.197814
$231$	−2.00000	−0.131590
$232$	1.00000	0.0656532
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	0	0
$235$	−3.00000	−0.195698
$236$	5.00000	0.325472
$237$	−5.00000	−0.324785
$238$	4.00000	0.259281
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	−1.00000	−0.0645497
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	10.0000	0.642824
$243$	−1.00000	−0.0641500
$244$	−10.0000	−0.640184
$245$	−3.00000	−0.191663
$246$	−10.0000	−0.637577
$247$	0	0
$248$	−3.00000	−0.190500
$249$	−6.00000	−0.380235
$250$	−1.00000	−0.0632456
$251$	3.00000	0.189358	0.0946792	−	0.995508i	$-0.469817\pi$
$251$	3.00000	0.189358	0.0946792	+	0.995508i	$0.469817\pi$
$252$	−2.00000	−0.125988
$253$	3.00000	0.188608
$254$	14.0000	0.878438
$255$	−2.00000	−0.125245
$256$	1.00000	0.0625000
$257$	21.0000	1.30994	0.654972	−	0.755653i	$-0.272680\pi$
$257$	21.0000	1.30994	0.654972	+	0.755653i	$0.272680\pi$
$258$	5.00000	0.311286
$259$	−10.0000	−0.621370
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	−13.0000	−0.803143
$263$	23.0000	1.41824	0.709120	−	0.705087i	$-0.249092\pi$
$263$	23.0000	1.41824	0.709120	+	0.705087i	$0.249092\pi$
$264$	−1.00000	−0.0615457
$265$	14.0000	0.860013
$266$	−12.0000	−0.735767
$267$	10.0000	0.611990
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	1.00000	0.0608581
$271$	29.0000	1.76162	0.880812	−	0.473466i	$-0.156997\pi$
$271$	29.0000	1.76162	0.880812	+	0.473466i	$0.156997\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−9.00000	−0.543710
$275$	−1.00000	−0.0603023
$276$	3.00000	0.180579
$277$	−19.0000	−1.14160	−0.570800	−	0.821089i	$-0.693367\pi$
$277$	−19.0000	−1.14160	−0.570800	+	0.821089i	$0.693367\pi$
$278$	−10.0000	−0.599760
$279$	3.00000	0.179605
$280$	2.00000	0.119523
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	−3.00000	−0.178647
$283$	−13.0000	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$284$	−4.00000	−0.237356
$285$	6.00000	0.355409
$286$	0	0
$287$	20.0000	1.18056
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	1.00000	0.0587220
$291$	−10.0000	−0.586210
$292$	2.00000	0.117041
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	−3.00000	−0.174964
$295$	5.00000	0.291111
$296$	−5.00000	−0.290619
$297$	1.00000	0.0580259
$298$	11.0000	0.637213
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−10.0000	−0.576390
$302$	24.0000	1.38104
$303$	−14.0000	−0.804279
$304$	−6.00000	−0.344124
$305$	−10.0000	−0.572598
$306$	−2.00000	−0.114332
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	2.00000	0.113961
$309$	6.00000	0.341328
$310$	−3.00000	−0.170389
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−12.0000	−0.678280	−0.339140	−	0.940736i	$-0.610136\pi$
$313$	−12.0000	−0.678280	−0.339140	+	0.940736i	$0.610136\pi$
$314$	−25.0000	−1.41083
$315$	−2.00000	−0.112687
$316$	5.00000	0.281272
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	14.0000	0.785081
$319$	1.00000	0.0559893
$320$	1.00000	0.0559017
$321$	6.00000	0.334887
$322$	−6.00000	−0.334367
$323$	−12.0000	−0.667698
$324$	1.00000	0.0555556
$325$	0	0
$326$	17.0000	0.941543
$327$	−6.00000	−0.331801
$328$	10.0000	0.552158
$329$	6.00000	0.330791
$330$	−1.00000	−0.0550482
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	6.00000	0.329293
$333$	5.00000	0.273998
$334$	−7.00000	−0.383023
$335$	0	0
$336$	2.00000	0.109109
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	−17.0000	−0.923313
$340$	2.00000	0.108465
$341$	−3.00000	−0.162459
$342$	6.00000	0.324443
$343$	20.0000	1.07990
$344$	−5.00000	−0.269582
$345$	3.00000	0.161515
$346$	−4.00000	−0.215041
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	1.00000	0.0536056
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	1.00000	0.0533002
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	5.00000	0.265747
$355$	−4.00000	−0.212298
$356$	−10.0000	−0.529999
$357$	4.00000	0.211702
$358$	7.00000	0.369961
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	−1.00000	−0.0527046
$361$	17.0000	0.894737
$362$	−16.0000	−0.840941
$363$	10.0000	0.524864
$364$	0	0
$365$	2.00000	0.104685
$366$	−10.0000	−0.522708
$367$	36.0000	1.87918	0.939592	−	0.342296i	$-0.111204\pi$
$367$	36.0000	1.87918	0.939592	+	0.342296i	$0.111204\pi$
$368$	−3.00000	−0.156386
$369$	−10.0000	−0.520579
$370$	−5.00000	−0.259938
$371$	−28.0000	−1.45369
$372$	−3.00000	−0.155543
$373$	37.0000	1.91579	0.957894	−	0.287123i	$-0.0926989\pi$
$373$	37.0000	1.91579	0.957894	+	0.287123i	$0.0926989\pi$
$374$	2.00000	0.103418
$375$	−1.00000	−0.0516398
$376$	3.00000	0.154713
$377$	0	0
$378$	−2.00000	−0.102869
$379$	30.0000	1.54100	0.770498	−	0.637442i	$-0.220007\pi$
$379$	30.0000	1.54100	0.770498	+	0.637442i	$0.220007\pi$
$380$	−6.00000	−0.307794
$381$	14.0000	0.717242
$382$	−24.0000	−1.22795
$383$	27.0000	1.37964	0.689818	−	0.723983i	$-0.257691\pi$
$383$	27.0000	1.37964	0.689818	+	0.723983i	$0.257691\pi$
$384$	1.00000	0.0510310
$385$	2.00000	0.101929
$386$	−4.00000	−0.203595
$387$	5.00000	0.254164
$388$	10.0000	0.507673
$389$	−1.00000	−0.0507020	−0.0253510	−	0.999679i	$-0.508070\pi$
$389$	−1.00000	−0.0507020	−0.0253510	+	0.999679i	$0.508070\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	3.00000	0.151523
$393$	−13.0000	−0.655763
$394$	−12.0000	−0.604551
$395$	5.00000	0.251577
$396$	−1.00000	−0.0502519
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	0	0
$399$	−12.0000	−0.600751
$400$	1.00000	0.0500000
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	0	0
$404$	14.0000	0.696526
$405$	1.00000	0.0496904
$406$	−2.00000	−0.0992583
$407$	−5.00000	−0.247841
$408$	2.00000	0.0990148
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	10.0000	0.493865
$411$	−9.00000	−0.443937
$412$	−6.00000	−0.295599
$413$	−10.0000	−0.492068
$414$	3.00000	0.147442
$415$	6.00000	0.294528
$416$	0	0
$417$	−10.0000	−0.489702
$418$	−6.00000	−0.293470
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	2.00000	0.0975900
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	8.00000	0.389434
$423$	−3.00000	−0.145865
$424$	−14.0000	−0.679900
$425$	2.00000	0.0970143
$426$	−4.00000	−0.193801
$427$	20.0000	0.967868
$428$	−6.00000	−0.290021
$429$	0	0
$430$	−5.00000	−0.241121
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	−1.00000	−0.0481125
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	6.00000	0.288009
$435$	1.00000	0.0479463
$436$	6.00000	0.287348
$437$	18.0000	0.861057
$438$	2.00000	0.0955637
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	1.00000	0.0476731
$441$	−3.00000	−0.142857
$442$	0	0
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	−5.00000	−0.237289
$445$	−10.0000	−0.474045
$446$	10.0000	0.473514
$447$	11.0000	0.520282
$448$	−2.00000	−0.0944911
$449$	36.0000	1.69895	0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000	1.69895	0.849473	+	0.527633i	$0.176920\pi$
$450$	−1.00000	−0.0471405
$451$	10.0000	0.470882
$452$	17.0000	0.799613
$453$	24.0000	1.12762
$454$	20.0000	0.938647
$455$	0	0
$456$	−6.00000	−0.280976
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	22.0000	1.02799
$459$	−2.00000	−0.0933520
$460$	−3.00000	−0.139876
$461$	−27.0000	−1.25752	−0.628758	−	0.777601i	$-0.716436\pi$
$461$	−27.0000	−1.25752	−0.628758	+	0.777601i	$0.716436\pi$
$462$	2.00000	0.0930484
$463$	−10.0000	−0.464739	−0.232370	−	0.972628i	$-0.574648\pi$
$463$	−10.0000	−0.464739	−0.232370	+	0.972628i	$0.574648\pi$
$464$	−1.00000	−0.0464238
$465$	−3.00000	−0.139122
$466$	−3.00000	−0.138972
$467$	−2.00000	−0.0925490	−0.0462745	−	0.998929i	$-0.514735\pi$
$467$	−2.00000	−0.0925490	−0.0462745	+	0.998929i	$0.514735\pi$
$468$	0	0
$469$	0	0
$470$	3.00000	0.138380
$471$	−25.0000	−1.15194
$472$	−5.00000	−0.230144
$473$	−5.00000	−0.229900
$474$	5.00000	0.229658
$475$	−6.00000	−0.275299
$476$	−4.00000	−0.183340
$477$	14.0000	0.641016
$478$	−8.00000	−0.365911
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−7.00000	−0.318841
$483$	−6.00000	−0.273009
$484$	−10.0000	−0.454545
$485$	10.0000	0.454077
$486$	1.00000	0.0453609
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	10.0000	0.452679
$489$	17.0000	0.768767
$490$	3.00000	0.135526
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	10.0000	0.450835
$493$	−2.00000	−0.0900755
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	3.00000	0.134704
$497$	8.00000	0.358849
$498$	6.00000	0.268866
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	1.00000	0.0447214
$501$	−7.00000	−0.312737
$502$	−3.00000	−0.133897
$503$	40.0000	1.78351	0.891756	−	0.452517i	$-0.149474\pi$
$503$	40.0000	1.78351	0.891756	+	0.452517i	$0.149474\pi$
$504$	2.00000	0.0890871
$505$	14.0000	0.622992
$506$	−3.00000	−0.133366
$507$	0	0
$508$	−14.0000	−0.621150
$509$	9.00000	0.398918	0.199459	−	0.979906i	$-0.436082\pi$
$509$	9.00000	0.398918	0.199459	+	0.979906i	$0.436082\pi$
$510$	2.00000	0.0885615
$511$	−4.00000	−0.176950
$512$	−1.00000	−0.0441942
$513$	6.00000	0.264906
$514$	−21.0000	−0.926270
$515$	−6.00000	−0.264392
$516$	−5.00000	−0.220113
$517$	3.00000	0.131940
$518$	10.0000	0.439375
$519$	−4.00000	−0.175581
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	1.00000	0.0437688
$523$	33.0000	1.44299	0.721495	−	0.692420i	$-0.243455\pi$
$523$	33.0000	1.44299	0.721495	+	0.692420i	$0.243455\pi$
$524$	13.0000	0.567908
$525$	2.00000	0.0872872
$526$	−23.0000	−1.00285
$527$	6.00000	0.261364
$528$	1.00000	0.0435194
$529$	−14.0000	−0.608696
$530$	−14.0000	−0.608121
$531$	5.00000	0.216982
$532$	12.0000	0.520266
$533$	0	0
$534$	−10.0000	−0.432742
$535$	−6.00000	−0.259403
$536$	0	0
$537$	7.00000	0.302072
$538$	−18.0000	−0.776035
$539$	3.00000	0.129219
$540$	−1.00000	−0.0430331
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	−29.0000	−1.24566
$543$	−16.0000	−0.686626
$544$	−2.00000	−0.0857493
$545$	6.00000	0.257012
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	9.00000	0.384461
$549$	−10.0000	−0.426790
$550$	1.00000	0.0426401
$551$	6.00000	0.255609
$552$	−3.00000	−0.127688
$553$	−10.0000	−0.425243
$554$	19.0000	0.807233
$555$	−5.00000	−0.212238
$556$	10.0000	0.424094
$557$	34.0000	1.44063	0.720313	−	0.693649i	$-0.243998\pi$
$557$	34.0000	1.44063	0.720313	+	0.693649i	$0.243998\pi$
$558$	−3.00000	−0.127000
$559$	0	0
$560$	−2.00000	−0.0845154
$561$	2.00000	0.0844401
$562$	−30.0000	−1.26547
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	3.00000	0.126323
$565$	17.0000	0.715195
$566$	13.0000	0.546431
$567$	−2.00000	−0.0839921
$568$	4.00000	0.167836
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	−6.00000	−0.251312
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	−24.0000	−1.00261
$574$	−20.0000	−0.834784
$575$	−3.00000	−0.125109
$576$	1.00000	0.0416667
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	13.0000	0.540729
$579$	−4.00000	−0.166234
$580$	−1.00000	−0.0415227
$581$	−12.0000	−0.497844
$582$	10.0000	0.414513
$583$	−14.0000	−0.579821
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−14.0000	−0.578335
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	3.00000	0.123718
$589$	−18.0000	−0.741677
$590$	−5.00000	−0.205847
$591$	−12.0000	−0.493614
$592$	5.00000	0.205499
$593$	35.0000	1.43728	0.718639	−	0.695383i	$-0.244765\pi$
$593$	35.0000	1.43728	0.718639	+	0.695383i	$0.244765\pi$
$594$	−1.00000	−0.0410305
$595$	−4.00000	−0.163984
$596$	−11.0000	−0.450578
$597$	0	0
$598$	0	0
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	1.00000	0.0408248
$601$	−11.0000	−0.448699	−0.224350	−	0.974509i	$-0.572026\pi$
$601$	−11.0000	−0.448699	−0.224350	+	0.974509i	$0.572026\pi$
$602$	10.0000	0.407570
$603$	0	0
$604$	−24.0000	−0.976546
$605$	−10.0000	−0.406558
$606$	14.0000	0.568711
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	6.00000	0.243332
$609$	−2.00000	−0.0810441
$610$	10.0000	0.404888
$611$	0	0
$612$	2.00000	0.0808452
$613$	37.0000	1.49442	0.747208	−	0.664590i	$-0.231394\pi$
$613$	37.0000	1.49442	0.747208	+	0.664590i	$0.231394\pi$
$614$	0	0
$615$	10.0000	0.403239
$616$	−2.00000	−0.0805823
$617$	−3.00000	−0.120775	−0.0603877	−	0.998175i	$-0.519234\pi$
$617$	−3.00000	−0.120775	−0.0603877	+	0.998175i	$0.519234\pi$
$618$	−6.00000	−0.241355
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	3.00000	0.120483
$621$	3.00000	0.120386
$622$	0	0
$623$	20.0000	0.801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	12.0000	0.479616
$627$	−6.00000	−0.239617
$628$	25.0000	0.997609
$629$	10.0000	0.398726
$630$	2.00000	0.0796819
$631$	−48.0000	−1.91085	−0.955425	−	0.295234i	$-0.904602\pi$
$631$	−48.0000	−1.91085	−0.955425	+	0.295234i	$0.904602\pi$
$632$	−5.00000	−0.198889
$633$	8.00000	0.317971
$634$	12.0000	0.476581
$635$	−14.0000	−0.555573
$636$	−14.0000	−0.555136
$637$	0	0
$638$	−1.00000	−0.0395904
$639$	−4.00000	−0.158238
$640$	−1.00000	−0.0395285
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−6.00000	−0.236801
$643$	40.0000	1.57745	0.788723	−	0.614749i	$-0.210743\pi$
$643$	40.0000	1.57745	0.788723	+	0.614749i	$0.210743\pi$
$644$	6.00000	0.236433
$645$	−5.00000	−0.196875
$646$	12.0000	0.472134
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	−1.00000	−0.0392837
$649$	−5.00000	−0.196267
$650$	0	0
$651$	6.00000	0.235159
$652$	−17.0000	−0.665771
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	6.00000	0.234619
$655$	13.0000	0.507952
$656$	−10.0000	−0.390434
$657$	2.00000	0.0780274
$658$	−6.00000	−0.233904
$659$	−13.0000	−0.506408	−0.253204	−	0.967413i	$-0.581484\pi$
$659$	−13.0000	−0.506408	−0.253204	+	0.967413i	$0.581484\pi$
$660$	1.00000	0.0389249
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	−6.00000	−0.232845
$665$	12.0000	0.465340
$666$	−5.00000	−0.193746
$667$	3.00000	0.116160
$668$	7.00000	0.270838
$669$	10.0000	0.386622
$670$	0	0
$671$	10.0000	0.386046
$672$	−2.00000	−0.0771517
$673$	−16.0000	−0.616755	−0.308377	−	0.951264i	$-0.599786\pi$
$673$	−16.0000	−0.616755	−0.308377	+	0.951264i	$0.599786\pi$
$674$	22.0000	0.847408
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	17.0000	0.652881
$679$	−20.0000	−0.767530
$680$	−2.00000	−0.0766965
$681$	20.0000	0.766402
$682$	3.00000	0.114876
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	−6.00000	−0.229416
$685$	9.00000	0.343872
$686$	−20.0000	−0.763604
$687$	22.0000	0.839352
$688$	5.00000	0.190623
$689$	0	0
$690$	−3.00000	−0.114208
$691$	14.0000	0.532585	0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000	0.532585	0.266293	+	0.963892i	$0.414201\pi$
$692$	4.00000	0.152057
$693$	2.00000	0.0759737
$694$	18.0000	0.683271
$695$	10.0000	0.379322
$696$	−1.00000	−0.0379049
$697$	−20.0000	−0.757554
$698$	8.00000	0.302804
$699$	−3.00000	−0.113470
$700$	−2.00000	−0.0755929
$701$	23.0000	0.868698	0.434349	−	0.900745i	$-0.356978\pi$
$701$	23.0000	0.868698	0.434349	+	0.900745i	$0.356978\pi$
$702$	0	0
$703$	−30.0000	−1.13147
$704$	−1.00000	−0.0376889
$705$	3.00000	0.112987
$706$	−14.0000	−0.526897
$707$	−28.0000	−1.05305
$708$	−5.00000	−0.187912
$709$	16.0000	0.600893	0.300446	−	0.953799i	$-0.402864\pi$
$709$	16.0000	0.600893	0.300446	+	0.953799i	$0.402864\pi$
$710$	4.00000	0.150117
$711$	5.00000	0.187515
$712$	10.0000	0.374766
$713$	−9.00000	−0.337053
$714$	−4.00000	−0.149696
$715$	0	0
$716$	−7.00000	−0.261602
$717$	−8.00000	−0.298765
$718$	12.0000	0.447836
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	1.00000	0.0372678
$721$	12.0000	0.446903
$722$	−17.0000	−0.632674
$723$	−7.00000	−0.260333
$724$	16.0000	0.594635
$725$	−1.00000	−0.0371391
$726$	−10.0000	−0.371135
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−2.00000	−0.0740233
$731$	10.0000	0.369863
$732$	10.0000	0.369611
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	−36.0000	−1.32878
$735$	3.00000	0.110657
$736$	3.00000	0.110581
$737$	0	0
$738$	10.0000	0.368105
$739$	−24.0000	−0.882854	−0.441427	−	0.897297i	$-0.645528\pi$
$739$	−24.0000	−0.882854	−0.441427	+	0.897297i	$0.645528\pi$
$740$	5.00000	0.183804
$741$	0	0
$742$	28.0000	1.02791
$743$	15.0000	0.550297	0.275148	−	0.961402i	$-0.411273\pi$
$743$	15.0000	0.550297	0.275148	+	0.961402i	$0.411273\pi$
$744$	3.00000	0.109985
$745$	−11.0000	−0.403009
$746$	−37.0000	−1.35467
$747$	6.00000	0.219529
$748$	−2.00000	−0.0731272
$749$	12.0000	0.438470
$750$	1.00000	0.0365148
$751$	41.0000	1.49611	0.748056	−	0.663636i	$-0.230988\pi$
$751$	41.0000	1.49611	0.748056	+	0.663636i	$0.230988\pi$
$752$	−3.00000	−0.109399
$753$	−3.00000	−0.109326
$754$	0	0
$755$	−24.0000	−0.873449
$756$	2.00000	0.0727393
$757$	−54.0000	−1.96266	−0.981332	−	0.192323i	$-0.938398\pi$
$757$	−54.0000	−1.96266	−0.981332	+	0.192323i	$0.938398\pi$
$758$	−30.0000	−1.08965
$759$	−3.00000	−0.108893
$760$	6.00000	0.217643
$761$	20.0000	0.724999	0.362500	−	0.931984i	$-0.381923\pi$
$761$	20.0000	0.724999	0.362500	+	0.931984i	$0.381923\pi$
$762$	−14.0000	−0.507166
$763$	−12.0000	−0.434429
$764$	24.0000	0.868290
$765$	2.00000	0.0723102
$766$	−27.0000	−0.975550
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	29.0000	1.04577	0.522883	−	0.852404i	$-0.324856\pi$
$769$	29.0000	1.04577	0.522883	+	0.852404i	$0.324856\pi$
$770$	−2.00000	−0.0720750
$771$	−21.0000	−0.756297
$772$	4.00000	0.143963
$773$	16.0000	0.575480	0.287740	−	0.957709i	$-0.407096\pi$
$773$	16.0000	0.575480	0.287740	+	0.957709i	$0.407096\pi$
$774$	−5.00000	−0.179721
$775$	3.00000	0.107763
$776$	−10.0000	−0.358979
$777$	10.0000	0.358748
$778$	1.00000	0.0358517
$779$	60.0000	2.14972
$780$	0	0
$781$	4.00000	0.143131
$782$	6.00000	0.214560
$783$	1.00000	0.0357371
$784$	−3.00000	−0.107143
$785$	25.0000	0.892288
$786$	13.0000	0.463695
$787$	−25.0000	−0.891154	−0.445577	−	0.895244i	$-0.647001\pi$
$787$	−25.0000	−0.891154	−0.445577	+	0.895244i	$0.647001\pi$
$788$	12.0000	0.427482
$789$	−23.0000	−0.818822
$790$	−5.00000	−0.177892
$791$	−34.0000	−1.20890
$792$	1.00000	0.0355335
$793$	0	0
$794$	−13.0000	−0.461353
$795$	−14.0000	−0.496529
$796$	0	0
$797$	−52.0000	−1.84193	−0.920967	−	0.389640i	$-0.872599\pi$
$797$	−52.0000	−1.84193	−0.920967	+	0.389640i	$0.872599\pi$
$798$	12.0000	0.424795
$799$	−6.00000	−0.212265
$800$	−1.00000	−0.0353553
$801$	−10.0000	−0.353333
$802$	−12.0000	−0.423735
$803$	−2.00000	−0.0705785
$804$	0	0
$805$	6.00000	0.211472
$806$	0	0
$807$	−18.0000	−0.633630
$808$	−14.0000	−0.492518
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	−1.00000	−0.0351364
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	2.00000	0.0701862
$813$	−29.0000	−1.01707
$814$	5.00000	0.175250
$815$	−17.0000	−0.595484
$816$	−2.00000	−0.0700140
$817$	−30.0000	−1.04957
$818$	−26.0000	−0.909069
$819$	0	0
$820$	−10.0000	−0.349215
$821$	−23.0000	−0.802706	−0.401353	−	0.915924i	$-0.631460\pi$
$821$	−23.0000	−0.802706	−0.401353	+	0.915924i	$0.631460\pi$
$822$	9.00000	0.313911
$823$	−22.0000	−0.766872	−0.383436	−	0.923567i	$-0.625259\pi$
$823$	−22.0000	−0.766872	−0.383436	+	0.923567i	$0.625259\pi$
$824$	6.00000	0.209020
$825$	1.00000	0.0348155
$826$	10.0000	0.347945
$827$	18.0000	0.625921	0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000	0.625921	0.312961	+	0.949766i	$0.398679\pi$
$828$	−3.00000	−0.104257
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	−6.00000	−0.208263
$831$	19.0000	0.659103
$832$	0	0
$833$	−6.00000	−0.207888
$834$	10.0000	0.346272
$835$	7.00000	0.242245
$836$	6.00000	0.207514
$837$	−3.00000	−0.103695
$838$	−4.00000	−0.138178
$839$	−38.0000	−1.31191	−0.655953	−	0.754802i	$-0.727733\pi$
$839$	−38.0000	−1.31191	−0.655953	+	0.754802i	$0.727733\pi$
$840$	−2.00000	−0.0690066
$841$	−28.0000	−0.965517
$842$	28.0000	0.964944
$843$	−30.0000	−1.03325
$844$	−8.00000	−0.275371
$845$	0	0
$846$	3.00000	0.103142
$847$	20.0000	0.687208
$848$	14.0000	0.480762
$849$	13.0000	0.446159
$850$	−2.00000	−0.0685994
$851$	−15.0000	−0.514193
$852$	4.00000	0.137038
$853$	−7.00000	−0.239675	−0.119838	−	0.992793i	$-0.538237\pi$
$853$	−7.00000	−0.239675	−0.119838	+	0.992793i	$0.538237\pi$
$854$	−20.0000	−0.684386
$855$	−6.00000	−0.205196
$856$	6.00000	0.205076
$857$	−35.0000	−1.19558	−0.597789	−	0.801654i	$-0.703954\pi$
$857$	−35.0000	−1.19558	−0.597789	+	0.801654i	$0.703954\pi$
$858$	0	0
$859$	18.0000	0.614152	0.307076	−	0.951685i	$-0.400649\pi$
$859$	18.0000	0.614152	0.307076	+	0.951685i	$0.400649\pi$
$860$	5.00000	0.170499
$861$	−20.0000	−0.681598
$862$	12.0000	0.408722
$863$	51.0000	1.73606	0.868030	−	0.496512i	$-0.165386\pi$
$863$	51.0000	1.73606	0.868030	+	0.496512i	$0.165386\pi$
$864$	1.00000	0.0340207
$865$	4.00000	0.136004
$866$	−16.0000	−0.543702
$867$	13.0000	0.441503
$868$	−6.00000	−0.203653
$869$	−5.00000	−0.169613
$870$	−1.00000	−0.0339032
$871$	0	0
$872$	−6.00000	−0.203186
$873$	10.0000	0.338449
$874$	−18.0000	−0.608859
$875$	−2.00000	−0.0676123
$876$	−2.00000	−0.0675737
$877$	−13.0000	−0.438979	−0.219489	−	0.975615i	$-0.570439\pi$
$877$	−13.0000	−0.438979	−0.219489	+	0.975615i	$0.570439\pi$
$878$	−28.0000	−0.944954
$879$	−14.0000	−0.472208
$880$	−1.00000	−0.0337100
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	3.00000	0.101015
$883$	−1.00000	−0.0336527	−0.0168263	−	0.999858i	$-0.505356\pi$
$883$	−1.00000	−0.0336527	−0.0168263	+	0.999858i	$0.505356\pi$
$884$	0	0
$885$	−5.00000	−0.168073
$886$	−16.0000	−0.537531
$887$	1.00000	0.0335767	0.0167884	−	0.999859i	$-0.494656\pi$
$887$	1.00000	0.0335767	0.0167884	+	0.999859i	$0.494656\pi$
$888$	5.00000	0.167789
$889$	28.0000	0.939090
$890$	10.0000	0.335201
$891$	−1.00000	−0.0335013
$892$	−10.0000	−0.334825
$893$	18.0000	0.602347
$894$	−11.0000	−0.367895
$895$	−7.00000	−0.233984
$896$	2.00000	0.0668153
$897$	0	0
$898$	−36.0000	−1.20134
$899$	−3.00000	−0.100056
$900$	1.00000	0.0333333
$901$	28.0000	0.932815
$902$	−10.0000	−0.332964
$903$	10.0000	0.332779
$904$	−17.0000	−0.565412
$905$	16.0000	0.531858
$906$	−24.0000	−0.797347
$907$	−53.0000	−1.75984	−0.879918	−	0.475125i	$-0.842403\pi$
$907$	−53.0000	−1.75984	−0.879918	+	0.475125i	$0.842403\pi$
$908$	−20.0000	−0.663723
$909$	14.0000	0.464351
$910$	0	0
$911$	34.0000	1.12647	0.563235	−	0.826297i	$-0.309557\pi$
$911$	34.0000	1.12647	0.563235	+	0.826297i	$0.309557\pi$
$912$	6.00000	0.198680
$913$	−6.00000	−0.198571
$914$	14.0000	0.463079
$915$	10.0000	0.330590
$916$	−22.0000	−0.726900
$917$	−26.0000	−0.858596
$918$	2.00000	0.0660098
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	3.00000	0.0989071
$921$	0	0
$922$	27.0000	0.889198
$923$	0	0
$924$	−2.00000	−0.0657952
$925$	5.00000	0.164399
$926$	10.0000	0.328620
$927$	−6.00000	−0.197066
$928$	1.00000	0.0328266
$929$	−24.0000	−0.787414	−0.393707	−	0.919236i	$-0.628808\pi$
$929$	−24.0000	−0.787414	−0.393707	+	0.919236i	$0.628808\pi$
$930$	3.00000	0.0983739
$931$	18.0000	0.589926
$932$	3.00000	0.0982683
$933$	0	0
$934$	2.00000	0.0654420
$935$	−2.00000	−0.0654070
$936$	0	0
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	0	0
$939$	12.0000	0.391605
$940$	−3.00000	−0.0978492
$941$	−26.0000	−0.847576	−0.423788	−	0.905761i	$-0.639300\pi$
$941$	−26.0000	−0.847576	−0.423788	+	0.905761i	$0.639300\pi$
$942$	25.0000	0.814544
$943$	30.0000	0.976934
$944$	5.00000	0.162736
$945$	2.00000	0.0650600
$946$	5.00000	0.162564
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	−5.00000	−0.162392
$949$	0	0
$950$	6.00000	0.194666
$951$	12.0000	0.389127
$952$	4.00000	0.129641
$953$	15.0000	0.485898	0.242949	−	0.970039i	$-0.421885\pi$
$953$	15.0000	0.485898	0.242949	+	0.970039i	$0.421885\pi$
$954$	−14.0000	−0.453267
$955$	24.0000	0.776622
$956$	8.00000	0.258738
$957$	−1.00000	−0.0323254
$958$	−4.00000	−0.129234
$959$	−18.0000	−0.581250
$960$	−1.00000	−0.0322749
$961$	−22.0000	−0.709677
$962$	0	0
$963$	−6.00000	−0.193347
$964$	7.00000	0.225455
$965$	4.00000	0.128765
$966$	6.00000	0.193047
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	10.0000	0.321412
$969$	12.0000	0.385496
$970$	−10.0000	−0.321081
$971$	−40.0000	−1.28366	−0.641831	−	0.766846i	$-0.721825\pi$
$971$	−40.0000	−1.28366	−0.641831	+	0.766846i	$0.721825\pi$
$972$	−1.00000	−0.0320750
$973$	−20.0000	−0.641171
$974$	2.00000	0.0640841
$975$	0	0
$976$	−10.0000	−0.320092
$977$	39.0000	1.24772	0.623860	−	0.781536i	$-0.285563\pi$
$977$	39.0000	1.24772	0.623860	+	0.781536i	$0.285563\pi$
$978$	−17.0000	−0.543600
$979$	10.0000	0.319601
$980$	−3.00000	−0.0958315
$981$	6.00000	0.191565
$982$	−24.0000	−0.765871
$983$	25.0000	0.797376	0.398688	−	0.917087i	$-0.369466\pi$
$983$	25.0000	0.797376	0.398688	+	0.917087i	$0.369466\pi$
$984$	−10.0000	−0.318788
$985$	12.0000	0.382352
$986$	2.00000	0.0636930
$987$	−6.00000	−0.190982
$988$	0	0
$989$	−15.0000	−0.476972
$990$	1.00000	0.0317821
$991$	39.0000	1.23888	0.619438	−	0.785046i	$-0.287361\pi$
$991$	39.0000	1.23888	0.619438	+	0.785046i	$0.287361\pi$
$992$	−3.00000	−0.0952501
$993$	4.00000	0.126936
$994$	−8.00000	−0.253745
$995$	0	0
$996$	−6.00000	−0.190117
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	−40.0000	−1.26618
$999$	−5.00000	−0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.f.1.1		1
13.4	even	6		390.2.i.a.211.1	yes	2
13.5	odd	4		5070.2.b.g.1351.2		2
13.8	odd	4		5070.2.b.g.1351.1		2
13.10	even	6		390.2.i.a.61.1	✓	2
13.12	even	2		5070.2.a.o.1.1		1
39.17	odd	6		1170.2.i.k.991.1		2
39.23	odd	6		1170.2.i.k.451.1		2
65.4	even	6		1950.2.i.s.601.1		2
65.17	odd	12		1950.2.z.h.1849.2		4
65.23	odd	12		1950.2.z.h.1699.2		4
65.43	odd	12		1950.2.z.h.1849.1		4
65.49	even	6		1950.2.i.s.451.1		2
65.62	odd	12		1950.2.z.h.1699.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.i.a.61.1	✓	2	13.10	even	6
390.2.i.a.211.1	yes	2	13.4	even	6
1170.2.i.k.451.1		2	39.23	odd	6
1170.2.i.k.991.1		2	39.17	odd	6
1950.2.i.s.451.1		2	65.49	even	6
1950.2.i.s.601.1		2	65.4	even	6
1950.2.z.h.1699.1		4	65.62	odd	12
1950.2.z.h.1699.2		4	65.23	odd	12
1950.2.z.h.1849.1		4	65.43	odd	12
1950.2.z.h.1849.2		4	65.17	odd	12
5070.2.a.f.1.1		1	1.1	even	1	trivial
5070.2.a.o.1.1		1	13.12	even	2
5070.2.b.g.1351.1		2	13.8	odd	4
5070.2.b.g.1351.2		2	13.5	odd	4

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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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