LMFDB - Embedded newform 8032.2.a.j.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, 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("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.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:	$64.1358429035$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73095 q^{3} +0.367690 q^{5} +0.998919 q^{7} -0.00381480 q^{9} +O(q^{10})$	$q-1.73095 q^{3} +0.367690 q^{5} +0.998919 q^{7} -0.00381480 q^{9} -2.58707 q^{11} +3.65095 q^{13} -0.636452 q^{15} +1.48909 q^{17} +5.32379 q^{19} -1.72908 q^{21} +7.18344 q^{23} -4.86480 q^{25} +5.19945 q^{27} +2.32154 q^{29} +8.43045 q^{31} +4.47809 q^{33} +0.367292 q^{35} +7.73080 q^{37} -6.31961 q^{39} -8.54986 q^{41} -2.60382 q^{43} -0.00140266 q^{45} +3.71525 q^{47} -6.00216 q^{49} -2.57753 q^{51} -7.53258 q^{53} -0.951240 q^{55} -9.21521 q^{57} +7.82229 q^{59} -7.23169 q^{61} -0.00381068 q^{63} +1.34242 q^{65} -16.1674 q^{67} -12.4342 q^{69} -7.75414 q^{71} -6.59417 q^{73} +8.42073 q^{75} -2.58427 q^{77} +3.28244 q^{79} -8.98854 q^{81} +2.84893 q^{83} +0.547522 q^{85} -4.01847 q^{87} +12.5460 q^{89} +3.64700 q^{91} -14.5927 q^{93} +1.95750 q^{95} -7.84949 q^{97} +0.00986917 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9	$ 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9} + 31 q^{11} + 5 q^{13} + 8 q^{15} - q^{17} + 29 q^{19} + 6 q^{21} + 27 q^{23} + 34 q^{25} + 36 q^{27} + q^{29} - 9 q^{31} - 8 q^{33} + 51 q^{35} + 5 q^{37} - 8 q^{39} - 3 q^{41} + 43 q^{43} + 42 q^{47} + 41 q^{49} + 54 q^{51} - 11 q^{53} - 10 q^{55} + 2 q^{57} + 49 q^{59} + 21 q^{61} + 15 q^{63} - 14 q^{65} + 68 q^{67} - 10 q^{69} + 44 q^{71} + 25 q^{73} + 51 q^{75} - 20 q^{77} - 49 q^{79} + 6 q^{81} + 88 q^{83} - 5 q^{85} + 16 q^{87} - 16 q^{89} + 46 q^{91} - 4 q^{93} + 28 q^{95} - 4 q^{97} + 71 q^{99}+O(q^{100}) $ 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9 + 31 * q^11 + 5 * q^13 + 8 * q^15 - q^17 + 29 * q^19 + 6 * q^21 + 27 * q^23 + 34 * q^25 + 36 * q^27 + q^29 - 9 * q^31 - 8 * q^33 + 51 * q^35 + 5 * q^37 - 8 * q^39 - 3 * q^41 + 43 * q^43 + 42 * q^47 + 41 * q^49 + 54 * q^51 - 11 * q^53 - 10 * q^55 + 2 * q^57 + 49 * q^59 + 21 * q^61 + 15 * q^63 - 14 * q^65 + 68 * q^67 - 10 * q^69 + 44 * q^71 + 25 * q^73 + 51 * q^75 - 20 * q^77 - 49 * q^79 + 6 * q^81 + 88 * q^83 - 5 * q^85 + 16 * q^87 - 16 * q^89 + 46 * q^91 - 4 * q^93 + 28 * q^95 - 4 * q^97 + 71 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.73095	−0.999364	−0.499682	−	0.866209i	$-0.666550\pi$
$3$	−1.73095	−0.999364	−0.499682	+	0.866209i	$0.666550\pi$
$4$	0	0
$5$	0.367690	0.164436	0.0822179	−	0.996614i	$-0.473800\pi$
$5$	0.367690	0.164436	0.0822179	+	0.996614i	$0.473800\pi$
$6$	0	0
$7$	0.998919	0.377556	0.188778	−	0.982020i	$-0.439547\pi$
$7$	0.998919	0.377556	0.188778	+	0.982020i	$0.439547\pi$
$8$	0	0
$9$	−0.00381480	−0.00127160
$10$	0	0
$11$	−2.58707	−0.780032	−0.390016	−	0.920808i	$-0.627530\pi$
$11$	−2.58707	−0.780032	−0.390016	+	0.920808i	$0.627530\pi$
$12$	0	0
$13$	3.65095	1.01259	0.506296	−	0.862360i	$-0.331014\pi$
$13$	3.65095	1.01259	0.506296	+	0.862360i	$0.331014\pi$
$14$	0	0
$15$	−0.636452	−0.164331
$16$	0	0
$17$	1.48909	0.361157	0.180578	−	0.983561i	$-0.442203\pi$
$17$	1.48909	0.361157	0.180578	+	0.983561i	$0.442203\pi$
$18$	0	0
$19$	5.32379	1.22136	0.610680	−	0.791877i	$-0.290896\pi$
$19$	5.32379	1.22136	0.610680	+	0.791877i	$0.290896\pi$
$20$	0	0
$21$	−1.72908	−0.377316
$22$	0	0
$23$	7.18344	1.49785	0.748925	−	0.662654i	$-0.230570\pi$
$23$	7.18344	1.49785	0.748925	+	0.662654i	$0.230570\pi$
$24$	0	0
$25$	−4.86480	−0.972961
$26$	0	0
$27$	5.19945	1.00063
$28$	0	0
$29$	2.32154	0.431099	0.215549	−	0.976493i	$-0.430846\pi$
$29$	2.32154	0.431099	0.215549	+	0.976493i	$0.430846\pi$
$30$	0	0
$31$	8.43045	1.51415	0.757077	−	0.653326i	$-0.226627\pi$
$31$	8.43045	1.51415	0.757077	+	0.653326i	$0.226627\pi$
$32$	0	0
$33$	4.47809	0.779535
$34$	0	0
$35$	0.367292	0.0620837
$36$	0	0
$37$	7.73080	1.27094	0.635468	−	0.772128i	$-0.280807\pi$
$37$	7.73080	1.27094	0.635468	+	0.772128i	$0.280807\pi$
$38$	0	0
$39$	−6.31961	−1.01195
$40$	0	0
$41$	−8.54986	−1.33526	−0.667632	−	0.744491i	$-0.732692\pi$
$41$	−8.54986	−1.33526	−0.667632	+	0.744491i	$0.732692\pi$
$42$	0	0
$43$	−2.60382	−0.397079	−0.198540	−	0.980093i	$-0.563620\pi$
$43$	−2.60382	−0.397079	−0.198540	+	0.980093i	$0.563620\pi$
$44$	0	0
$45$	−0.00140266	−0.000209097 0
$46$	0	0
$47$	3.71525	0.541925	0.270963	−	0.962590i	$-0.412658\pi$
$47$	3.71525	0.541925	0.270963	+	0.962590i	$0.412658\pi$
$48$	0	0
$49$	−6.00216	−0.857452
$50$	0	0
$51$	−2.57753	−0.360927
$52$	0	0
$53$	−7.53258	−1.03468	−0.517339	−	0.855780i	$-0.673078\pi$
$53$	−7.53258	−1.03468	−0.517339	+	0.855780i	$0.673078\pi$
$54$	0	0
$55$	−0.951240	−0.128265
$56$	0	0
$57$	−9.21521	−1.22058
$58$	0	0
$59$	7.82229	1.01837	0.509187	−	0.860656i	$-0.329946\pi$
$59$	7.82229	1.01837	0.509187	+	0.860656i	$0.329946\pi$
$60$	0	0
$61$	−7.23169	−0.925922	−0.462961	−	0.886379i	$-0.653213\pi$
$61$	−7.23169	−0.925922	−0.462961	+	0.886379i	$0.653213\pi$
$62$	0	0
$63$	−0.00381068	−0.000480100 0
$64$	0	0
$65$	1.34242	0.166506
$66$	0	0
$67$	−16.1674	−1.97516	−0.987581	−	0.157108i	$-0.949783\pi$
$67$	−16.1674	−1.97516	−0.987581	+	0.157108i	$0.949783\pi$
$68$	0	0
$69$	−12.4342	−1.49690
$70$	0	0
$71$	−7.75414	−0.920247	−0.460124	−	0.887855i	$-0.652195\pi$
$71$	−7.75414	−0.920247	−0.460124	+	0.887855i	$0.652195\pi$
$72$	0	0
$73$	−6.59417	−0.771789	−0.385895	−	0.922543i	$-0.626107\pi$
$73$	−6.59417	−0.771789	−0.385895	+	0.922543i	$0.626107\pi$
$74$	0	0
$75$	8.42073	0.972342
$76$	0	0
$77$	−2.58427	−0.294505
$78$	0	0
$79$	3.28244	0.369304	0.184652	−	0.982804i	$-0.440884\pi$
$79$	3.28244	0.369304	0.184652	+	0.982804i	$0.440884\pi$
$80$	0	0
$81$	−8.98854	−0.998727
$82$	0	0
$83$	2.84893	0.312711	0.156356	−	0.987701i	$-0.450025\pi$
$83$	2.84893	0.312711	0.156356	+	0.987701i	$0.450025\pi$
$84$	0	0
$85$	0.547522	0.0593871
$86$	0	0
$87$	−4.01847	−0.430825
$88$	0	0
$89$	12.5460	1.32987	0.664935	−	0.746901i	$-0.268459\pi$
$89$	12.5460	1.32987	0.664935	+	0.746901i	$0.268459\pi$
$90$	0	0
$91$	3.64700	0.382310
$92$	0	0
$93$	−14.5927	−1.51319
$94$	0	0
$95$	1.95750	0.200836
$96$	0	0
$97$	−7.84949	−0.796995	−0.398497	−	0.917170i	$-0.630468\pi$
$97$	−7.84949	−0.796995	−0.398497	+	0.917170i	$0.630468\pi$
$98$	0	0
$99$	0.00986917	0.000991889 0
$100$	0	0
$101$	17.8554	1.77668	0.888341	−	0.459183i	$-0.151858\pi$
$101$	17.8554	1.77668	0.888341	+	0.459183i	$0.151858\pi$
$102$	0	0
$103$	10.0957	0.994761	0.497380	−	0.867533i	$-0.334295\pi$
$103$	10.0957	0.994761	0.497380	+	0.867533i	$0.334295\pi$
$104$	0	0
$105$	−0.635764	−0.0620442
$106$	0	0
$107$	11.4542	1.10732	0.553661	−	0.832742i	$-0.313230\pi$
$107$	11.4542	1.10732	0.553661	+	0.832742i	$0.313230\pi$
$108$	0	0
$109$	19.6184	1.87911	0.939553	−	0.342405i	$-0.111241\pi$
$109$	19.6184	1.87911	0.939553	+	0.342405i	$0.111241\pi$
$110$	0	0
$111$	−13.3816	−1.27013
$112$	0	0
$113$	−14.3555	−1.35045	−0.675227	−	0.737610i	$-0.735955\pi$
$113$	−14.3555	−1.35045	−0.675227	+	0.737610i	$0.735955\pi$
$114$	0	0
$115$	2.64128	0.246300
$116$	0	0
$117$	−0.0139277	−0.00128761
$118$	0	0
$119$	1.48748	0.136357
$120$	0	0
$121$	−4.30706	−0.391551
$122$	0	0
$123$	14.7994	1.33441
$124$	0	0
$125$	−3.62719	−0.324426
$126$	0	0
$127$	14.3437	1.27279	0.636397	−	0.771361i	$-0.280424\pi$
$127$	14.3437	1.27279	0.636397	+	0.771361i	$0.280424\pi$
$128$	0	0
$129$	4.50709	0.396827
$130$	0	0
$131$	7.91033	0.691129	0.345564	−	0.938395i	$-0.387688\pi$
$131$	7.91033	0.691129	0.345564	+	0.938395i	$0.387688\pi$
$132$	0	0
$133$	5.31803	0.461132
$134$	0	0
$135$	1.91178	0.164540
$136$	0	0
$137$	5.33305	0.455633	0.227817	−	0.973704i	$-0.426841\pi$
$137$	5.33305	0.455633	0.227817	+	0.973704i	$0.426841\pi$
$138$	0	0
$139$	4.07040	0.345247	0.172623	−	0.984988i	$-0.444776\pi$
$139$	4.07040	0.345247	0.172623	+	0.984988i	$0.444776\pi$
$140$	0	0
$141$	−6.43091	−0.541580
$142$	0	0
$143$	−9.44527	−0.789854
$144$	0	0
$145$	0.853606	0.0708881
$146$	0	0
$147$	10.3894	0.856906
$148$	0	0
$149$	4.27101	0.349895	0.174947	−	0.984578i	$-0.444024\pi$
$149$	4.27101	0.349895	0.174947	+	0.984578i	$0.444024\pi$
$150$	0	0
$151$	−14.8084	−1.20509	−0.602546	−	0.798084i	$-0.705847\pi$
$151$	−14.8084	−1.20509	−0.602546	+	0.798084i	$0.705847\pi$
$152$	0	0
$153$	−0.00568058	−0.000459247 0
$154$	0	0
$155$	3.09979	0.248981
$156$	0	0
$157$	2.75260	0.219681	0.109841	−	0.993949i	$-0.464966\pi$
$157$	2.75260	0.219681	0.109841	+	0.993949i	$0.464966\pi$
$158$	0	0
$159$	13.0385	1.03402
$160$	0	0
$161$	7.17567	0.565522
$162$	0	0
$163$	−21.0564	−1.64926	−0.824631	−	0.565672i	$-0.808617\pi$
$163$	−21.0564	−1.64926	−0.824631	+	0.565672i	$0.808617\pi$
$164$	0	0
$165$	1.64655	0.128184
$166$	0	0
$167$	−9.11199	−0.705107	−0.352553	−	0.935792i	$-0.614686\pi$
$167$	−9.11199	−0.705107	−0.352553	+	0.935792i	$0.614686\pi$
$168$	0	0
$169$	0.329448	0.0253421
$170$	0	0
$171$	−0.0203092	−0.00155308
$172$	0	0
$173$	20.0332	1.52310	0.761549	−	0.648108i	$-0.224439\pi$
$173$	20.0332	1.52310	0.761549	+	0.648108i	$0.224439\pi$
$174$	0	0
$175$	−4.85954	−0.367347
$176$	0	0
$177$	−13.5400	−1.01773
$178$	0	0
$179$	−4.68562	−0.350220	−0.175110	−	0.984549i	$-0.556028\pi$
$179$	−4.68562	−0.350220	−0.175110	+	0.984549i	$0.556028\pi$
$180$	0	0
$181$	−7.17156	−0.533058	−0.266529	−	0.963827i	$-0.585877\pi$
$181$	−7.17156	−0.533058	−0.266529	+	0.963827i	$0.585877\pi$
$182$	0	0
$183$	12.5177	0.925334
$184$	0	0
$185$	2.84253	0.208987
$186$	0	0
$187$	−3.85238	−0.281714
$188$	0	0
$189$	5.19383	0.377795
$190$	0	0
$191$	9.19557	0.665368	0.332684	−	0.943038i	$-0.392046\pi$
$191$	9.19557	0.665368	0.332684	+	0.943038i	$0.392046\pi$
$192$	0	0
$193$	−10.8554	−0.781390	−0.390695	−	0.920520i	$-0.627765\pi$
$193$	−10.8554	−0.781390	−0.390695	+	0.920520i	$0.627765\pi$
$194$	0	0
$195$	−2.32366	−0.166401
$196$	0	0
$197$	10.1578	0.723715	0.361858	−	0.932233i	$-0.382143\pi$
$197$	10.1578	0.723715	0.361858	+	0.932233i	$0.382143\pi$
$198$	0	0
$199$	1.54338	0.109407	0.0547035	−	0.998503i	$-0.482579\pi$
$199$	1.54338	0.109407	0.0547035	+	0.998503i	$0.482579\pi$
$200$	0	0
$201$	27.9850	1.97391
$202$	0	0
$203$	2.31903	0.162764
$204$	0	0
$205$	−3.14370	−0.219565
$206$	0	0
$207$	−0.0274034	−0.00190467
$208$	0	0
$209$	−13.7730	−0.952700
$210$	0	0
$211$	0.760829	0.0523776	0.0261888	−	0.999657i	$-0.491663\pi$
$211$	0.760829	0.0523776	0.0261888	+	0.999657i	$0.491663\pi$
$212$	0	0
$213$	13.4220	0.919662
$214$	0	0
$215$	−0.957399	−0.0652941
$216$	0	0
$217$	8.42134	0.571678
$218$	0	0
$219$	11.4142	0.771299
$220$	0	0
$221$	5.43659	0.365704
$222$	0	0
$223$	−26.9793	−1.80667	−0.903335	−	0.428937i	$-0.858888\pi$
$223$	−26.9793	−1.80667	−0.903335	+	0.428937i	$0.858888\pi$
$224$	0	0
$225$	0.0185583	0.00123722
$226$	0	0
$227$	−0.782188	−0.0519157	−0.0259578	−	0.999663i	$-0.508264\pi$
$227$	−0.782188	−0.0519157	−0.0259578	+	0.999663i	$0.508264\pi$
$228$	0	0
$229$	−14.1685	−0.936279	−0.468139	−	0.883655i	$-0.655076\pi$
$229$	−14.1685	−0.936279	−0.468139	+	0.883655i	$0.655076\pi$
$230$	0	0
$231$	4.47325	0.294318
$232$	0	0
$233$	−23.1918	−1.51935	−0.759673	−	0.650305i	$-0.774641\pi$
$233$	−23.1918	−1.51935	−0.759673	+	0.650305i	$0.774641\pi$
$234$	0	0
$235$	1.36606	0.0891119
$236$	0	0
$237$	−5.68174	−0.369069
$238$	0	0
$239$	14.9135	0.964672	0.482336	−	0.875986i	$-0.339788\pi$
$239$	14.9135	0.964672	0.482336	+	0.875986i	$0.339788\pi$
$240$	0	0
$241$	4.85314	0.312618	0.156309	−	0.987708i	$-0.450040\pi$
$241$	4.85314	0.312618	0.156309	+	0.987708i	$0.450040\pi$
$242$	0	0
$243$	−0.0396446	−0.00254320
$244$	0	0
$245$	−2.20693	−0.140996
$246$	0	0
$247$	19.4369	1.23674
$248$	0	0
$249$	−4.93136	−0.312512
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−18.5841	−1.16837
$254$	0	0
$255$	−0.947733	−0.0593494
$256$	0	0
$257$	−2.73244	−0.170445	−0.0852225	−	0.996362i	$-0.527160\pi$
$257$	−2.73244	−0.170445	−0.0852225	+	0.996362i	$0.527160\pi$
$258$	0	0
$259$	7.72244	0.479849
$260$	0	0
$261$	−0.00885621	−0.000548186 0
$262$	0	0
$263$	1.95928	0.120815	0.0604073	−	0.998174i	$-0.480760\pi$
$263$	1.95928	0.120815	0.0604073	+	0.998174i	$0.480760\pi$
$264$	0	0
$265$	−2.76965	−0.170138
$266$	0	0
$267$	−21.7164	−1.32902
$268$	0	0
$269$	9.22733	0.562600	0.281300	−	0.959620i	$-0.409234\pi$
$269$	9.22733	0.562600	0.281300	+	0.959620i	$0.409234\pi$
$270$	0	0
$271$	30.2845	1.83965	0.919827	−	0.392325i	$-0.128329\pi$
$271$	30.2845	1.83965	0.919827	+	0.392325i	$0.128329\pi$
$272$	0	0
$273$	−6.31278	−0.382067
$274$	0	0
$275$	12.5856	0.758940
$276$	0	0
$277$	−1.63031	−0.0979556	−0.0489778	−	0.998800i	$-0.515596\pi$
$277$	−1.63031	−0.0979556	−0.0489778	+	0.998800i	$0.515596\pi$
$278$	0	0
$279$	−0.0321605	−0.00192540
$280$	0	0
$281$	−9.14141	−0.545331	−0.272665	−	0.962109i	$-0.587905\pi$
$281$	−9.14141	−0.545331	−0.272665	+	0.962109i	$0.587905\pi$
$282$	0	0
$283$	16.8709	1.00287	0.501435	−	0.865196i	$-0.332806\pi$
$283$	16.8709	1.00287	0.501435	+	0.865196i	$0.332806\pi$
$284$	0	0
$285$	−3.38834	−0.200708
$286$	0	0
$287$	−8.54062	−0.504137
$288$	0	0
$289$	−14.7826	−0.869566
$290$	0	0
$291$	13.5871	0.796488
$292$	0	0
$293$	14.1753	0.828128	0.414064	−	0.910248i	$-0.364109\pi$
$293$	14.1753	0.828128	0.414064	+	0.910248i	$0.364109\pi$
$294$	0	0
$295$	2.87617	0.167457
$296$	0	0
$297$	−13.4514	−0.780527
$298$	0	0
$299$	26.2264	1.51671
$300$	0	0
$301$	−2.60101	−0.149920
$302$	0	0
$303$	−30.9069	−1.77555
$304$	0	0
$305$	−2.65902	−0.152255
$306$	0	0
$307$	33.6693	1.92161	0.960805	−	0.277224i	$-0.0894144\pi$
$307$	33.6693	1.92161	0.960805	+	0.277224i	$0.0894144\pi$
$308$	0	0
$309$	−17.4752	−0.994128
$310$	0	0
$311$	24.3875	1.38289	0.691445	−	0.722429i	$-0.256974\pi$
$311$	24.3875	1.38289	0.691445	+	0.722429i	$0.256974\pi$
$312$	0	0
$313$	23.9757	1.35519	0.677594	−	0.735436i	$-0.263023\pi$
$313$	23.9757	1.35519	0.677594	+	0.735436i	$0.263023\pi$
$314$	0	0
$315$	−0.00140115	−7.89457e−5 0
$316$	0	0
$317$	31.6681	1.77866	0.889330	−	0.457266i	$-0.151171\pi$
$317$	31.6681	1.77866	0.889330	+	0.457266i	$0.151171\pi$
$318$	0	0
$319$	−6.00599	−0.336271
$320$	0	0
$321$	−19.8267	−1.10662
$322$	0	0
$323$	7.92759	0.441103
$324$	0	0
$325$	−17.7612	−0.985212
$326$	0	0
$327$	−33.9585	−1.87791
$328$	0	0
$329$	3.71123	0.204607
$330$	0	0
$331$	9.34467	0.513629	0.256815	−	0.966461i	$-0.417327\pi$
$331$	9.34467	0.513629	0.256815	+	0.966461i	$0.417327\pi$
$332$	0	0
$333$	−0.0294915	−0.00161612
$334$	0	0
$335$	−5.94459	−0.324788
$336$	0	0
$337$	−29.8623	−1.62670	−0.813350	−	0.581774i	$-0.802359\pi$
$337$	−29.8623	−1.62670	−0.813350	+	0.581774i	$0.802359\pi$
$338$	0	0
$339$	24.8487	1.34960
$340$	0	0
$341$	−21.8102	−1.18109
$342$	0	0
$343$	−12.9881	−0.701292
$344$	0	0
$345$	−4.57192	−0.246144
$346$	0	0
$347$	27.9150	1.49855	0.749277	−	0.662256i	$-0.230401\pi$
$347$	27.9150	1.49855	0.749277	+	0.662256i	$0.230401\pi$
$348$	0	0
$349$	−12.4805	−0.668065	−0.334033	−	0.942562i	$-0.608410\pi$
$349$	−12.4805	−0.668065	−0.334033	+	0.942562i	$0.608410\pi$
$350$	0	0
$351$	18.9829	1.01323
$352$	0	0
$353$	24.2802	1.29231	0.646153	−	0.763208i	$-0.276377\pi$
$353$	24.2802	1.29231	0.646153	+	0.763208i	$0.276377\pi$
$354$	0	0
$355$	−2.85112	−0.151322
$356$	0	0
$357$	−2.57475	−0.136270
$358$	0	0
$359$	17.1955	0.907545	0.453773	−	0.891118i	$-0.350078\pi$
$359$	17.1955	0.907545	0.453773	+	0.891118i	$0.350078\pi$
$360$	0	0
$361$	9.34272	0.491722
$362$	0	0
$363$	7.45530	0.391302
$364$	0	0
$365$	−2.42461	−0.126910
$366$	0	0
$367$	−7.49086	−0.391020	−0.195510	−	0.980702i	$-0.562636\pi$
$367$	−7.49086	−0.391020	−0.195510	+	0.980702i	$0.562636\pi$
$368$	0	0
$369$	0.0326160	0.00169792
$370$	0	0
$371$	−7.52443	−0.390649
$372$	0	0
$373$	−11.0842	−0.573918	−0.286959	−	0.957943i	$-0.592644\pi$
$373$	−11.0842	−0.573918	−0.286959	+	0.957943i	$0.592644\pi$
$374$	0	0
$375$	6.27848	0.324219
$376$	0	0
$377$	8.47583	0.436527
$378$	0	0
$379$	15.6004	0.801339	0.400669	−	0.916223i	$-0.368778\pi$
$379$	15.6004	0.801339	0.400669	+	0.916223i	$0.368778\pi$
$380$	0	0
$381$	−24.8282	−1.27199
$382$	0	0
$383$	20.9817	1.07211	0.536057	−	0.844182i	$-0.319913\pi$
$383$	20.9817	1.07211	0.536057	+	0.844182i	$0.319913\pi$
$384$	0	0
$385$	−0.950211	−0.0484273
$386$	0	0
$387$	0.00993307	0.000504926 0
$388$	0	0
$389$	−7.59663	−0.385165	−0.192582	−	0.981281i	$-0.561686\pi$
$389$	−7.59663	−0.385165	−0.192582	+	0.981281i	$0.561686\pi$
$390$	0	0
$391$	10.6968	0.540959
$392$	0	0
$393$	−13.6924	−0.690689
$394$	0	0
$395$	1.20692	0.0607268
$396$	0	0
$397$	36.3497	1.82434	0.912169	−	0.409813i	$-0.134406\pi$
$397$	36.3497	1.82434	0.912169	+	0.409813i	$0.134406\pi$
$398$	0	0
$399$	−9.20524	−0.460839
$400$	0	0
$401$	9.20634	0.459743	0.229871	−	0.973221i	$-0.426169\pi$
$401$	9.20634	0.459743	0.229871	+	0.973221i	$0.426169\pi$
$402$	0	0
$403$	30.7792	1.53322
$404$	0	0
$405$	−3.30499	−0.164226
$406$	0	0
$407$	−20.0001	−0.991369
$408$	0	0
$409$	29.5407	1.46069	0.730346	−	0.683077i	$-0.239359\pi$
$409$	29.5407	1.46069	0.730346	+	0.683077i	$0.239359\pi$
$410$	0	0
$411$	−9.23124	−0.455344
$412$	0	0
$413$	7.81383	0.384493
$414$	0	0
$415$	1.04752	0.0514209
$416$	0	0
$417$	−7.04566	−0.345027
$418$	0	0
$419$	9.58526	0.468271	0.234135	−	0.972204i	$-0.424774\pi$
$419$	9.58526	0.468271	0.234135	+	0.972204i	$0.424774\pi$
$420$	0	0
$421$	−8.66328	−0.422222	−0.211111	−	0.977462i	$-0.567708\pi$
$421$	−8.66328	−0.422222	−0.211111	+	0.977462i	$0.567708\pi$
$422$	0	0
$423$	−0.0141729	−0.000689112 0
$424$	0	0
$425$	−7.24412	−0.351391
$426$	0	0
$427$	−7.22387	−0.349587
$428$	0	0
$429$	16.3493	0.789351
$430$	0	0
$431$	23.9105	1.15173	0.575864	−	0.817546i	$-0.304666\pi$
$431$	23.9105	1.15173	0.575864	+	0.817546i	$0.304666\pi$
$432$	0	0
$433$	33.8549	1.62696	0.813480	−	0.581593i	$-0.197570\pi$
$433$	33.8549	1.62696	0.813480	+	0.581593i	$0.197570\pi$
$434$	0	0
$435$	−1.47755	−0.0708430
$436$	0	0
$437$	38.2431	1.82942
$438$	0	0
$439$	1.13273	0.0540622	0.0270311	−	0.999635i	$-0.491395\pi$
$439$	1.13273	0.0540622	0.0270311	+	0.999635i	$0.491395\pi$
$440$	0	0
$441$	0.0228971	0.00109034
$442$	0	0
$443$	−36.6314	−1.74041	−0.870206	−	0.492689i	$-0.836014\pi$
$443$	−36.6314	−1.74041	−0.870206	+	0.492689i	$0.836014\pi$
$444$	0	0
$445$	4.61302	0.218678
$446$	0	0
$447$	−7.39290	−0.349672
$448$	0	0
$449$	13.9143	0.656655	0.328327	−	0.944564i	$-0.393515\pi$
$449$	13.9143	0.656655	0.328327	+	0.944564i	$0.393515\pi$
$450$	0	0
$451$	22.1191	1.04155
$452$	0	0
$453$	25.6326	1.20433
$454$	0	0
$455$	1.34097	0.0628655
$456$	0	0
$457$	−38.8855	−1.81899	−0.909494	−	0.415717i	$-0.863531\pi$
$457$	−38.8855	−1.81899	−0.909494	+	0.415717i	$0.863531\pi$
$458$	0	0
$459$	7.74244	0.361386
$460$	0	0
$461$	33.9588	1.58162	0.790808	−	0.612064i	$-0.209660\pi$
$461$	33.9588	1.58162	0.790808	+	0.612064i	$0.209660\pi$
$462$	0	0
$463$	−27.8297	−1.29336	−0.646678	−	0.762763i	$-0.723842\pi$
$463$	−27.8297	−1.29336	−0.646678	+	0.762763i	$0.723842\pi$
$464$	0	0
$465$	−5.36558	−0.248823
$466$	0	0
$467$	−14.9168	−0.690269	−0.345135	−	0.938553i	$-0.612167\pi$
$467$	−14.9168	−0.690269	−0.345135	+	0.938553i	$0.612167\pi$
$468$	0	0
$469$	−16.1499	−0.745734
$470$	0	0
$471$	−4.76461	−0.219542
$472$	0	0
$473$	6.73628	0.309734
$474$	0	0
$475$	−25.8992	−1.18834
$476$	0	0
$477$	0.0287353	0.00131570
$478$	0	0
$479$	−35.5123	−1.62260	−0.811300	−	0.584630i	$-0.801240\pi$
$479$	−35.5123	−1.62260	−0.811300	+	0.584630i	$0.801240\pi$
$480$	0	0
$481$	28.2248	1.28694
$482$	0	0
$483$	−12.4207	−0.565163
$484$	0	0
$485$	−2.88618	−0.131054
$486$	0	0
$487$	24.4411	1.10753	0.553765	−	0.832673i	$-0.313190\pi$
$487$	24.4411	1.10753	0.553765	+	0.832673i	$0.313190\pi$
$488$	0	0
$489$	36.4475	1.64821
$490$	0	0
$491$	41.1777	1.85832	0.929161	−	0.369675i	$-0.120531\pi$
$491$	41.1777	1.85832	0.929161	+	0.369675i	$0.120531\pi$
$492$	0	0
$493$	3.45697	0.155694
$494$	0	0
$495$	0.00362879	0.000163102 0
$496$	0	0
$497$	−7.74576	−0.347445
$498$	0	0
$499$	−32.8955	−1.47261	−0.736303	−	0.676652i	$-0.763430\pi$
$499$	−32.8955	−1.47261	−0.736303	+	0.676652i	$0.763430\pi$
$500$	0	0
$501$	15.7724	0.704658
$502$	0	0
$503$	−8.49155	−0.378620	−0.189310	−	0.981917i	$-0.560625\pi$
$503$	−8.49155	−0.378620	−0.189310	+	0.981917i	$0.560625\pi$
$504$	0	0
$505$	6.56526	0.292150
$506$	0	0
$507$	−0.570257	−0.0253260
$508$	0	0
$509$	−11.0786	−0.491051	−0.245525	−	0.969390i	$-0.578960\pi$
$509$	−11.0786	−0.491051	−0.245525	+	0.969390i	$0.578960\pi$
$510$	0	0
$511$	−6.58704	−0.291394
$512$	0	0
$513$	27.6808	1.22214
$514$	0	0
$515$	3.71209	0.163574
$516$	0	0
$517$	−9.61162	−0.422719
$518$	0	0
$519$	−34.6765	−1.52213
$520$	0	0
$521$	−37.3406	−1.63592	−0.817961	−	0.575273i	$-0.804896\pi$
$521$	−37.3406	−1.63592	−0.817961	+	0.575273i	$0.804896\pi$
$522$	0	0
$523$	20.2694	0.886317	0.443159	−	0.896443i	$-0.353858\pi$
$523$	20.2694	0.886317	0.443159	+	0.896443i	$0.353858\pi$
$524$	0	0
$525$	8.41162	0.367113
$526$	0	0
$527$	12.5537	0.546847
$528$	0	0
$529$	28.6018	1.24356
$530$	0	0
$531$	−0.0298405	−0.00129497
$532$	0	0
$533$	−31.2151	−1.35208
$534$	0	0
$535$	4.21160	0.182083
$536$	0	0
$537$	8.11057	0.349997
$538$	0	0
$539$	15.5280	0.668839
$540$	0	0
$541$	−21.8309	−0.938584	−0.469292	−	0.883043i	$-0.655491\pi$
$541$	−21.8309	−0.938584	−0.469292	+	0.883043i	$0.655491\pi$
$542$	0	0
$543$	12.4136	0.532719
$544$	0	0
$545$	7.21350	0.308992
$546$	0	0
$547$	−1.60634	−0.0686821	−0.0343410	−	0.999410i	$-0.510933\pi$
$547$	−1.60634	−0.0686821	−0.0343410	+	0.999410i	$0.510933\pi$
$548$	0	0
$549$	0.0275875	0.00117740
$550$	0	0
$551$	12.3594	0.526527
$552$	0	0
$553$	3.27889	0.139433
$554$	0	0
$555$	−4.92028	−0.208854
$556$	0	0
$557$	−8.39322	−0.355632	−0.177816	−	0.984064i	$-0.556903\pi$
$557$	−8.39322	−0.355632	−0.177816	+	0.984064i	$0.556903\pi$
$558$	0	0
$559$	−9.50643	−0.402079
$560$	0	0
$561$	6.66827	0.281534
$562$	0	0
$563$	−2.41151	−0.101633	−0.0508166	−	0.998708i	$-0.516182\pi$
$563$	−2.41151	−0.101633	−0.0508166	+	0.998708i	$0.516182\pi$
$564$	0	0
$565$	−5.27838	−0.222063
$566$	0	0
$567$	−8.97882	−0.377075
$568$	0	0
$569$	−12.4734	−0.522911	−0.261456	−	0.965215i	$-0.584203\pi$
$569$	−12.4734	−0.522911	−0.261456	+	0.965215i	$0.584203\pi$
$570$	0	0
$571$	−5.73064	−0.239820	−0.119910	−	0.992785i	$-0.538261\pi$
$571$	−5.73064	−0.239820	−0.119910	+	0.992785i	$0.538261\pi$
$572$	0	0
$573$	−15.9171	−0.664945
$574$	0	0
$575$	−34.9460	−1.45735
$576$	0	0
$577$	2.13122	0.0887237	0.0443619	−	0.999016i	$-0.485875\pi$
$577$	2.13122	0.0887237	0.0443619	+	0.999016i	$0.485875\pi$
$578$	0	0
$579$	18.7902	0.780893
$580$	0	0
$581$	2.84585	0.118066
$582$	0	0
$583$	19.4873	0.807082
$584$	0	0
$585$	−0.00512106	−0.000211730 0
$586$	0	0
$587$	4.49984	0.185728	0.0928642	−	0.995679i	$-0.470398\pi$
$587$	4.49984	0.185728	0.0928642	+	0.995679i	$0.470398\pi$
$588$	0	0
$589$	44.8819	1.84933
$590$	0	0
$591$	−17.5827	−0.723255
$592$	0	0
$593$	43.0798	1.76908	0.884538	−	0.466468i	$-0.154474\pi$
$593$	43.0798	1.76908	0.884538	+	0.466468i	$0.154474\pi$
$594$	0	0
$595$	0.546930	0.0224220
$596$	0	0
$597$	−2.67151	−0.109337
$598$	0	0
$599$	−25.7694	−1.05291	−0.526454	−	0.850203i	$-0.676479\pi$
$599$	−25.7694	−1.05291	−0.526454	+	0.850203i	$0.676479\pi$
$600$	0	0
$601$	16.0617	0.655169	0.327584	−	0.944822i	$-0.393765\pi$
$601$	16.0617	0.655169	0.327584	+	0.944822i	$0.393765\pi$
$602$	0	0
$603$	0.0616755	0.00251162
$604$	0	0
$605$	−1.58366	−0.0643850
$606$	0	0
$607$	25.4225	1.03187	0.515934	−	0.856628i	$-0.327445\pi$
$607$	25.4225	1.03187	0.515934	+	0.856628i	$0.327445\pi$
$608$	0	0
$609$	−4.01412	−0.162660
$610$	0	0
$611$	13.5642	0.548749
$612$	0	0
$613$	−6.66768	−0.269305	−0.134653	−	0.990893i	$-0.542992\pi$
$613$	−6.66768	−0.269305	−0.134653	+	0.990893i	$0.542992\pi$
$614$	0	0
$615$	5.44158	0.219426
$616$	0	0
$617$	−42.8810	−1.72632	−0.863162	−	0.504928i	$-0.831519\pi$
$617$	−42.8810	−1.72632	−0.863162	+	0.504928i	$0.831519\pi$
$618$	0	0
$619$	20.8499	0.838029	0.419015	−	0.907979i	$-0.362376\pi$
$619$	20.8499	0.838029	0.419015	+	0.907979i	$0.362376\pi$
$620$	0	0
$621$	37.3499	1.49880
$622$	0	0
$623$	12.5324	0.502100
$624$	0	0
$625$	22.9903	0.919614
$626$	0	0
$627$	23.8404	0.952094
$628$	0	0
$629$	11.5118	0.459007
$630$	0	0
$631$	−0.633132	−0.0252046	−0.0126023	−	0.999921i	$-0.504012\pi$
$631$	−0.633132	−0.0252046	−0.0126023	+	0.999921i	$0.504012\pi$
$632$	0	0
$633$	−1.31696	−0.0523443
$634$	0	0
$635$	5.27402	0.209293
$636$	0	0
$637$	−21.9136	−0.868248
$638$	0	0
$639$	0.0295805	0.00117019
$640$	0	0
$641$	−18.3682	−0.725499	−0.362750	−	0.931887i	$-0.618162\pi$
$641$	−18.3682	−0.725499	−0.362750	+	0.931887i	$0.618162\pi$
$642$	0	0
$643$	−45.1176	−1.77926	−0.889632	−	0.456678i	$-0.849039\pi$
$643$	−45.1176	−1.77926	−0.889632	+	0.456678i	$0.849039\pi$
$644$	0	0
$645$	1.65721	0.0652526
$646$	0	0
$647$	2.04504	0.0803988	0.0401994	−	0.999192i	$-0.487201\pi$
$647$	2.04504	0.0803988	0.0401994	+	0.999192i	$0.487201\pi$
$648$	0	0
$649$	−20.2368	−0.794365
$650$	0	0
$651$	−14.5769	−0.571314
$652$	0	0
$653$	−43.3606	−1.69683	−0.848415	−	0.529332i	$-0.822443\pi$
$653$	−43.3606	−1.69683	−0.848415	+	0.529332i	$0.822443\pi$
$654$	0	0
$655$	2.90855	0.113646
$656$	0	0
$657$	0.0251555	0.000981408 0
$658$	0	0
$659$	6.28376	0.244781	0.122390	−	0.992482i	$-0.460944\pi$
$659$	6.28376	0.244781	0.122390	+	0.992482i	$0.460944\pi$
$660$	0	0
$661$	30.0839	1.17013	0.585064	−	0.810987i	$-0.301069\pi$
$661$	30.0839	1.17013	0.585064	+	0.810987i	$0.301069\pi$
$662$	0	0
$663$	−9.41045	−0.365472
$664$	0	0
$665$	1.95539	0.0758266
$666$	0	0
$667$	16.6766	0.645722
$668$	0	0
$669$	46.6998	1.80552
$670$	0	0
$671$	18.7089	0.722249
$672$	0	0
$673$	30.6718	1.18231	0.591156	−	0.806558i	$-0.298672\pi$
$673$	30.6718	1.18231	0.591156	+	0.806558i	$0.298672\pi$
$674$	0	0
$675$	−25.2943	−0.973578
$676$	0	0
$677$	30.5287	1.17331	0.586657	−	0.809835i	$-0.300444\pi$
$677$	30.5287	1.17331	0.586657	+	0.809835i	$0.300444\pi$
$678$	0	0
$679$	−7.84100	−0.300910
$680$	0	0
$681$	1.35393	0.0518826
$682$	0	0
$683$	18.0101	0.689139	0.344569	−	0.938761i	$-0.388025\pi$
$683$	18.0101	0.689139	0.344569	+	0.938761i	$0.388025\pi$
$684$	0	0
$685$	1.96091	0.0749225
$686$	0	0
$687$	24.5249	0.935683
$688$	0	0
$689$	−27.5011	−1.04771
$690$	0	0
$691$	0.852184	0.0324186	0.0162093	−	0.999869i	$-0.494840\pi$
$691$	0.852184	0.0324186	0.0162093	+	0.999869i	$0.494840\pi$
$692$	0	0
$693$	0.00985850	0.000374493 0
$694$	0	0
$695$	1.49664	0.0567710
$696$	0	0
$697$	−12.7315	−0.482240
$698$	0	0
$699$	40.1439	1.51838
$700$	0	0
$701$	3.47557	0.131270	0.0656352	−	0.997844i	$-0.479093\pi$
$701$	3.47557	0.131270	0.0656352	+	0.997844i	$0.479093\pi$
$702$	0	0
$703$	41.1571	1.55227
$704$	0	0
$705$	−2.36458	−0.0890552
$706$	0	0
$707$	17.8361	0.670797
$708$	0	0
$709$	−8.38329	−0.314841	−0.157421	−	0.987532i	$-0.550318\pi$
$709$	−8.38329	−0.314841	−0.157421	+	0.987532i	$0.550318\pi$
$710$	0	0
$711$	−0.0125219	−0.000469607 0
$712$	0	0
$713$	60.5597	2.26798
$714$	0	0
$715$	−3.47293	−0.129880
$716$	0	0
$717$	−25.8145	−0.964059
$718$	0	0
$719$	37.8627	1.41204	0.706020	−	0.708192i	$-0.250489\pi$
$719$	37.8627	1.41204	0.706020	+	0.708192i	$0.250489\pi$
$720$	0	0
$721$	10.0848	0.375578
$722$	0	0
$723$	−8.40054	−0.312419
$724$	0	0
$725$	−11.2938	−0.419442
$726$	0	0
$727$	−52.9525	−1.96390	−0.981950	−	0.189139i	$-0.939430\pi$
$727$	−52.9525	−1.96390	−0.981950	+	0.189139i	$0.939430\pi$
$728$	0	0
$729$	27.0342	1.00127
$730$	0	0
$731$	−3.87732	−0.143408
$732$	0	0
$733$	11.0299	0.407397	0.203699	−	0.979034i	$-0.434704\pi$
$733$	11.0299	0.407397	0.203699	+	0.979034i	$0.434704\pi$
$734$	0	0
$735$	3.82009	0.140906
$736$	0	0
$737$	41.8262	1.54069
$738$	0	0
$739$	11.4891	0.422634	0.211317	−	0.977418i	$-0.432225\pi$
$739$	11.4891	0.422634	0.211317	+	0.977418i	$0.432225\pi$
$740$	0	0
$741$	−33.6443	−1.23595
$742$	0	0
$743$	−13.6277	−0.499954	−0.249977	−	0.968252i	$-0.580423\pi$
$743$	−13.6277	−0.499954	−0.249977	+	0.968252i	$0.580423\pi$
$744$	0	0
$745$	1.57041	0.0575352
$746$	0	0
$747$	−0.0108681	−0.000397644 0
$748$	0	0
$749$	11.4418	0.418076
$750$	0	0
$751$	−18.0192	−0.657530	−0.328765	−	0.944412i	$-0.606632\pi$
$751$	−18.0192	−0.657530	−0.328765	+	0.944412i	$0.606632\pi$
$752$	0	0
$753$	1.73095	0.0630793
$754$	0	0
$755$	−5.44491	−0.198160
$756$	0	0
$757$	38.1689	1.38727	0.693636	−	0.720326i	$-0.256008\pi$
$757$	38.1689	1.38727	0.693636	+	0.720326i	$0.256008\pi$
$758$	0	0
$759$	32.1681	1.16763
$760$	0	0
$761$	−38.4247	−1.39290	−0.696448	−	0.717607i	$-0.745237\pi$
$761$	−38.4247	−1.39290	−0.696448	+	0.717607i	$0.745237\pi$
$762$	0	0
$763$	19.5972	0.709467
$764$	0	0
$765$	−0.00208869	−7.55167e−5 0
$766$	0	0
$767$	28.5588	1.03120
$768$	0	0
$769$	5.42411	0.195599	0.0977993	−	0.995206i	$-0.468820\pi$
$769$	5.42411	0.195599	0.0977993	+	0.995206i	$0.468820\pi$
$770$	0	0
$771$	4.72972	0.170337
$772$	0	0
$773$	39.5883	1.42389	0.711947	−	0.702234i	$-0.247814\pi$
$773$	39.5883	1.42389	0.711947	+	0.702234i	$0.247814\pi$
$774$	0	0
$775$	−41.0125	−1.47321
$776$	0	0
$777$	−13.3671	−0.479544
$778$	0	0
$779$	−45.5177	−1.63084
$780$	0	0
$781$	20.0605	0.717822
$782$	0	0
$783$	12.0707	0.431373
$784$	0	0
$785$	1.01210	0.0361235
$786$	0	0
$787$	0.335907	0.0119738	0.00598690	−	0.999982i	$-0.498094\pi$
$787$	0.335907	0.0119738	0.00598690	+	0.999982i	$0.498094\pi$
$788$	0	0
$789$	−3.39142	−0.120738
$790$	0	0
$791$	−14.3400	−0.509872
$792$	0	0
$793$	−26.4025	−0.937581
$794$	0	0
$795$	4.79412	0.170030
$796$	0	0
$797$	−16.5704	−0.586955	−0.293478	−	0.955966i	$-0.594813\pi$
$797$	−16.5704	−0.586955	−0.293478	+	0.955966i	$0.594813\pi$
$798$	0	0
$799$	5.53233	0.195720
$800$	0	0
$801$	−0.0478604	−0.00169106
$802$	0	0
$803$	17.0596	0.602020
$804$	0	0
$805$	2.63842	0.0929921
$806$	0	0
$807$	−15.9720	−0.562243
$808$	0	0
$809$	53.8636	1.89374	0.946872	−	0.321610i	$-0.104224\pi$
$809$	53.8636	1.89374	0.946872	+	0.321610i	$0.104224\pi$
$810$	0	0
$811$	23.3991	0.821653	0.410827	−	0.911714i	$-0.365240\pi$
$811$	23.3991	0.821653	0.410827	+	0.911714i	$0.365240\pi$
$812$	0	0
$813$	−52.4210	−1.83848
$814$	0	0
$815$	−7.74221	−0.271198
$816$	0	0
$817$	−13.8622	−0.484977
$818$	0	0
$819$	−0.0139126	−0.000486146 0
$820$	0	0
$821$	−28.0985	−0.980646	−0.490323	−	0.871541i	$-0.663121\pi$
$821$	−28.0985	−0.980646	−0.490323	+	0.871541i	$0.663121\pi$
$822$	0	0
$823$	−8.65358	−0.301645	−0.150822	−	0.988561i	$-0.548192\pi$
$823$	−8.65358	−0.301645	−0.150822	+	0.988561i	$0.548192\pi$
$824$	0	0
$825$	−21.7850	−0.758457
$826$	0	0
$827$	35.9634	1.25057	0.625284	−	0.780397i	$-0.284983\pi$
$827$	35.9634	1.25057	0.625284	+	0.780397i	$0.284983\pi$
$828$	0	0
$829$	39.7918	1.38203	0.691013	−	0.722842i	$-0.257165\pi$
$829$	39.7918	1.38203	0.691013	+	0.722842i	$0.257165\pi$
$830$	0	0
$831$	2.82198	0.0978933
$832$	0	0
$833$	−8.93774	−0.309674
$834$	0	0
$835$	−3.35038	−0.115945
$836$	0	0
$837$	43.8337	1.51512
$838$	0	0
$839$	47.7188	1.64743	0.823717	−	0.567001i	$-0.191896\pi$
$839$	47.7188	1.64743	0.823717	+	0.567001i	$0.191896\pi$
$840$	0	0
$841$	−23.6105	−0.814154
$842$	0	0
$843$	15.8233	0.544984
$844$	0	0
$845$	0.121135	0.00416716
$846$	0	0
$847$	−4.30240	−0.147832
$848$	0	0
$849$	−29.2026	−1.00223
$850$	0	0
$851$	55.5337	1.90367
$852$	0	0
$853$	−50.7345	−1.73712	−0.868558	−	0.495587i	$-0.834953\pi$
$853$	−50.7345	−1.73712	−0.868558	+	0.495587i	$0.834953\pi$
$854$	0	0
$855$	−0.00746749	−0.000255383 0
$856$	0	0
$857$	3.68495	0.125876	0.0629378	−	0.998017i	$-0.479953\pi$
$857$	3.68495	0.125876	0.0629378	+	0.998017i	$0.479953\pi$
$858$	0	0
$859$	−35.5883	−1.21426	−0.607129	−	0.794604i	$-0.707679\pi$
$859$	−35.5883	−1.21426	−0.607129	+	0.794604i	$0.707679\pi$
$860$	0	0
$861$	14.7834	0.503816
$862$	0	0
$863$	−30.4730	−1.03731	−0.518657	−	0.854982i	$-0.673568\pi$
$863$	−30.4730	−1.03731	−0.518657	+	0.854982i	$0.673568\pi$
$864$	0	0
$865$	7.36601	0.250452
$866$	0	0
$867$	25.5880	0.869013
$868$	0	0
$869$	−8.49191	−0.288068
$870$	0	0
$871$	−59.0264	−2.00003
$872$	0	0
$873$	0.0299442	0.00101346
$874$	0	0
$875$	−3.62327	−0.122489
$876$	0	0
$877$	34.8133	1.17556	0.587780	−	0.809021i	$-0.300002\pi$
$877$	34.8133	1.17556	0.587780	+	0.809021i	$0.300002\pi$
$878$	0	0
$879$	−24.5367	−0.827601
$880$	0	0
$881$	27.4064	0.923344	0.461672	−	0.887051i	$-0.347250\pi$
$881$	27.4064	0.923344	0.461672	+	0.887051i	$0.347250\pi$
$882$	0	0
$883$	−15.1709	−0.510541	−0.255271	−	0.966870i	$-0.582165\pi$
$883$	−15.1709	−0.510541	−0.255271	+	0.966870i	$0.582165\pi$
$884$	0	0
$885$	−4.97851	−0.167351
$886$	0	0
$887$	−17.9899	−0.604042	−0.302021	−	0.953301i	$-0.597661\pi$
$887$	−17.9899	−0.604042	−0.302021	+	0.953301i	$0.597661\pi$
$888$	0	0
$889$	14.3282	0.480551
$890$	0	0
$891$	23.2540	0.779038
$892$	0	0
$893$	19.7792	0.661886
$894$	0	0
$895$	−1.72286	−0.0575887
$896$	0	0
$897$	−45.3966	−1.51575
$898$	0	0
$899$	19.5716	0.652750
$900$	0	0
$901$	−11.2167	−0.373681
$902$	0	0
$903$	4.50221	0.149824
$904$	0	0
$905$	−2.63691	−0.0876538
$906$	0	0
$907$	2.30055	0.0763886	0.0381943	−	0.999270i	$-0.487839\pi$
$907$	2.30055	0.0763886	0.0381943	+	0.999270i	$0.487839\pi$
$908$	0	0
$909$	−0.0681150	−0.00225923
$910$	0	0
$911$	53.8022	1.78255	0.891274	−	0.453466i	$-0.149813\pi$
$911$	53.8022	1.78255	0.891274	+	0.453466i	$0.149813\pi$
$912$	0	0
$913$	−7.37040	−0.243925
$914$	0	0
$915$	4.60262	0.152158
$916$	0	0
$917$	7.90178	0.260940
$918$	0	0
$919$	−33.5787	−1.10766	−0.553829	−	0.832630i	$-0.686834\pi$
$919$	−33.5787	−1.10766	−0.553829	+	0.832630i	$0.686834\pi$
$920$	0	0
$921$	−58.2799	−1.92039
$922$	0	0
$923$	−28.3100	−0.931835
$924$	0	0
$925$	−37.6088	−1.23657
$926$	0	0
$927$	−0.0385132	−0.00126494
$928$	0	0
$929$	57.0666	1.87229	0.936146	−	0.351610i	$-0.114366\pi$
$929$	57.0666	1.87229	0.936146	+	0.351610i	$0.114366\pi$
$930$	0	0
$931$	−31.9542	−1.04726
$932$	0	0
$933$	−42.2136	−1.38201
$934$	0	0
$935$	−1.41648	−0.0463238
$936$	0	0
$937$	−13.1168	−0.428509	−0.214254	−	0.976778i	$-0.568732\pi$
$937$	−13.1168	−0.428509	−0.214254	+	0.976778i	$0.568732\pi$
$938$	0	0
$939$	−41.5008	−1.35433
$940$	0	0
$941$	18.3385	0.597818	0.298909	−	0.954282i	$-0.403377\pi$
$941$	18.3385	0.597818	0.298909	+	0.954282i	$0.403377\pi$
$942$	0	0
$943$	−61.4174	−2.00003
$944$	0	0
$945$	1.90972	0.0621231
$946$	0	0
$947$	34.0462	1.10635	0.553177	−	0.833064i	$-0.313415\pi$
$947$	34.0462	1.10635	0.553177	+	0.833064i	$0.313415\pi$
$948$	0	0
$949$	−24.0750	−0.781508
$950$	0	0
$951$	−54.8160	−1.77753
$952$	0	0
$953$	−52.3107	−1.69451	−0.847255	−	0.531186i	$-0.821746\pi$
$953$	−52.3107	−1.69451	−0.847255	+	0.531186i	$0.821746\pi$
$954$	0	0
$955$	3.38112	0.109410
$956$	0	0
$957$	10.3961	0.336057
$958$	0	0
$959$	5.32729	0.172027
$960$	0	0
$961$	40.0725	1.29266
$962$	0	0
$963$	−0.0436956	−0.00140807
$964$	0	0
$965$	−3.99142	−0.128489
$966$	0	0
$967$	30.1294	0.968897	0.484449	−	0.874820i	$-0.339020\pi$
$967$	30.1294	0.968897	0.484449	+	0.874820i	$0.339020\pi$
$968$	0	0
$969$	−13.7222	−0.440822
$970$	0	0
$971$	−21.9092	−0.703099	−0.351550	−	0.936169i	$-0.614345\pi$
$971$	−21.9092	−0.703099	−0.351550	+	0.936169i	$0.614345\pi$
$972$	0	0
$973$	4.06600	0.130350
$974$	0	0
$975$	30.7437	0.984586
$976$	0	0
$977$	−50.4405	−1.61374	−0.806868	−	0.590732i	$-0.798839\pi$
$977$	−50.4405	−1.61374	−0.806868	+	0.590732i	$0.798839\pi$
$978$	0	0
$979$	−32.4573	−1.03734
$980$	0	0
$981$	−0.0748405	−0.00238947
$982$	0	0
$983$	41.7989	1.33318	0.666589	−	0.745425i	$-0.267754\pi$
$983$	41.7989	1.33318	0.666589	+	0.745425i	$0.267754\pi$
$984$	0	0
$985$	3.73493	0.119005
$986$	0	0
$987$	−6.42396	−0.204477
$988$	0	0
$989$	−18.7044	−0.594766
$990$	0	0
$991$	−25.6835	−0.815862	−0.407931	−	0.913013i	$-0.633750\pi$
$991$	−25.6835	−0.815862	−0.407931	+	0.913013i	$0.633750\pi$
$992$	0	0
$993$	−16.1751	−0.513303
$994$	0	0
$995$	0.567484	0.0179904
$996$	0	0
$997$	−31.6501	−1.00237	−0.501185	−	0.865340i	$-0.667102\pi$
$997$	−31.6501	−1.00237	−0.501185	+	0.865340i	$0.667102\pi$
$998$	0	0
$999$	40.1959	1.27174

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.j.1.7	yes	30
4.3	odd	2		8032.2.a.g.1.24	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.g.1.24	✓	30	4.3	odd	2
8032.2.a.j.1.7	yes	30	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73095 q^{3} +0.367690 q^{5} +0.998919 q^{7} -0.00381480 q^{9} +O(q^{10})\)	\(q-1.73095 q^{3} +0.367690 q^{5} +0.998919 q^{7} -0.00381480 q^{9} -2.58707 q^{11} +3.65095 q^{13} -0.636452 q^{15} +1.48909 q^{17} +5.32379 q^{19} -1.72908 q^{21} +7.18344 q^{23} -4.86480 q^{25} +5.19945 q^{27} +2.32154 q^{29} +8.43045 q^{31} +4.47809 q^{33} +0.367292 q^{35} +7.73080 q^{37} -6.31961 q^{39} -8.54986 q^{41} -2.60382 q^{43} -0.00140266 q^{45} +3.71525 q^{47} -6.00216 q^{49} -2.57753 q^{51} -7.53258 q^{53} -0.951240 q^{55} -9.21521 q^{57} +7.82229 q^{59} -7.23169 q^{61} -0.00381068 q^{63} +1.34242 q^{65} -16.1674 q^{67} -12.4342 q^{69} -7.75414 q^{71} -6.59417 q^{73} +8.42073 q^{75} -2.58427 q^{77} +3.28244 q^{79} -8.98854 q^{81} +2.84893 q^{83} +0.547522 q^{85} -4.01847 q^{87} +12.5460 q^{89} +3.64700 q^{91} -14.5927 q^{93} +1.95750 q^{95} -7.84949 q^{97} +0.00986917 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9} + 31 q^{11} + 5 q^{13} + 8 q^{15} - q^{17} + 29 q^{19} + 6 q^{21} + 27 q^{23} + 34 q^{25} + 36 q^{27} + q^{29} - 9 q^{31} - 8 q^{33} + 51 q^{35} + 5 q^{37} - 8 q^{39} - 3 q^{41} + 43 q^{43} + 42 q^{47} + 41 q^{49} + 54 q^{51} - 11 q^{53} - 10 q^{55} + 2 q^{57} + 49 q^{59} + 21 q^{61} + 15 q^{63} - 14 q^{65} + 68 q^{67} - 10 q^{69} + 44 q^{71} + 25 q^{73} + 51 q^{75} - 20 q^{77} - 49 q^{79} + 6 q^{81} + 88 q^{83} - 5 q^{85} + 16 q^{87} - 16 q^{89} + 46 q^{91} - 4 q^{93} + 28 q^{95} - 4 q^{97} + 71 q^{99}+O(q^{100}) \) 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9 + 31 * q^11 + 5 * q^13 + 8 * q^15 - q^17 + 29 * q^19 + 6 * q^21 + 27 * q^23 + 34 * q^25 + 36 * q^27 + q^29 - 9 * q^31 - 8 * q^33 + 51 * q^35 + 5 * q^37 - 8 * q^39 - 3 * q^41 + 43 * q^43 + 42 * q^47 + 41 * q^49 + 54 * q^51 - 11 * q^53 - 10 * q^55 + 2 * q^57 + 49 * q^59 + 21 * q^61 + 15 * q^63 - 14 * q^65 + 68 * q^67 - 10 * q^69 + 44 * q^71 + 25 * q^73 + 51 * q^75 - 20 * q^77 - 49 * q^79 + 6 * q^81 + 88 * q^83 - 5 * q^85 + 16 * q^87 - 16 * q^89 + 46 * q^91 - 4 * q^93 + 28 * q^95 - 4 * q^97 + 71 * q^99

Introduction

L-functions

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