LMFDB - Embedded newform 8032.2.a.i.1.12

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			1.12
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)

$f(q)$	$=$	$q-0.797542 q^{3} -1.59101 q^{5} +1.45275 q^{7} -2.36393 q^{9} +O(q^{10})$	$q-0.797542 q^{3} -1.59101 q^{5} +1.45275 q^{7} -2.36393 q^{9} -5.08436 q^{11} -3.23817 q^{13} +1.26890 q^{15} -1.47535 q^{17} -4.34209 q^{19} -1.15863 q^{21} -8.66973 q^{23} -2.46870 q^{25} +4.27796 q^{27} +2.07637 q^{29} +0.368627 q^{31} +4.05499 q^{33} -2.31134 q^{35} -10.1998 q^{37} +2.58258 q^{39} +2.84460 q^{41} -0.220232 q^{43} +3.76102 q^{45} -0.444696 q^{47} -4.88950 q^{49} +1.17665 q^{51} -3.52007 q^{53} +8.08925 q^{55} +3.46300 q^{57} -1.14881 q^{59} -8.23654 q^{61} -3.43421 q^{63} +5.15195 q^{65} -4.03182 q^{67} +6.91447 q^{69} -2.27655 q^{71} +5.98502 q^{73} +1.96889 q^{75} -7.38633 q^{77} -7.87656 q^{79} +3.67993 q^{81} +5.55280 q^{83} +2.34729 q^{85} -1.65599 q^{87} +7.51285 q^{89} -4.70427 q^{91} -0.293995 q^{93} +6.90830 q^{95} +0.608887 q^{97} +12.0191 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9	$ 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9} + 13 q^{11} - 7 q^{13} + 28 q^{15} - 9 q^{17} - 17 q^{19} - 6 q^{21} + 43 q^{23} + 34 q^{25} + 12 q^{27} - q^{29} + 39 q^{31} + 17 q^{35} - q^{37} + 48 q^{39} - 3 q^{41} - 19 q^{43} + 66 q^{47} + 25 q^{49} - 14 q^{51} + 3 q^{53} + 50 q^{55} - 14 q^{57} + 27 q^{59} + 15 q^{61} + 75 q^{63} - 6 q^{65} + 8 q^{67} + 18 q^{69} + 64 q^{71} - 15 q^{73} + 9 q^{75} + 71 q^{79} + 6 q^{81} + 60 q^{83} + 15 q^{85} + 64 q^{87} - 32 q^{89} - 26 q^{91} - 4 q^{93} + 72 q^{95} - 4 q^{97} + 13 q^{99}+O(q^{100}) $ 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9 + 13 * q^11 - 7 * q^13 + 28 * q^15 - 9 * q^17 - 17 * q^19 - 6 * q^21 + 43 * q^23 + 34 * q^25 + 12 * q^27 - q^29 + 39 * q^31 + 17 * q^35 - q^37 + 48 * q^39 - 3 * q^41 - 19 * q^43 + 66 * q^47 + 25 * q^49 - 14 * q^51 + 3 * q^53 + 50 * q^55 - 14 * q^57 + 27 * q^59 + 15 * q^61 + 75 * q^63 - 6 * q^65 + 8 * q^67 + 18 * q^69 + 64 * q^71 - 15 * q^73 + 9 * q^75 + 71 * q^79 + 6 * q^81 + 60 * q^83 + 15 * q^85 + 64 * q^87 - 32 * q^89 - 26 * q^91 - 4 * q^93 + 72 * q^95 - 4 * q^97 + 13 * 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
$3$	−0.797542	−0.460461	−0.230231	−	0.973136i	$-0.573948\pi$
$3$	−0.797542	−0.460461	−0.230231	+	0.973136i	$0.573948\pi$
$4$	0	0
$5$	−1.59101	−0.711520	−0.355760	−	0.934577i	$-0.615778\pi$
$5$	−1.59101	−0.711520	−0.355760	+	0.934577i	$0.615778\pi$
$6$	0	0
$7$	1.45275	0.549090	0.274545	−	0.961574i	$-0.411473\pi$
$7$	1.45275	0.549090	0.274545	+	0.961574i	$0.411473\pi$
$8$	0	0
$9$	−2.36393	−0.787976
$10$	0	0
$11$	−5.08436	−1.53299	−0.766496	−	0.642249i	$-0.778001\pi$
$11$	−5.08436	−1.53299	−0.766496	+	0.642249i	$0.778001\pi$
$12$	0	0
$13$	−3.23817	−0.898107	−0.449053	−	0.893505i	$-0.648239\pi$
$13$	−3.23817	−0.898107	−0.449053	+	0.893505i	$0.648239\pi$
$14$	0	0
$15$	1.26890	0.327627
$16$	0	0
$17$	−1.47535	−0.357824	−0.178912	−	0.983865i	$-0.557258\pi$
$17$	−1.47535	−0.357824	−0.178912	+	0.983865i	$0.557258\pi$
$18$	0	0
$19$	−4.34209	−0.996144	−0.498072	−	0.867136i	$-0.665958\pi$
$19$	−4.34209	−0.996144	−0.498072	+	0.867136i	$0.665958\pi$
$20$	0	0
$21$	−1.15863	−0.252834
$22$	0	0
$23$	−8.66973	−1.80776	−0.903882	−	0.427782i	$-0.859295\pi$
$23$	−8.66973	−1.80776	−0.903882	+	0.427782i	$0.859295\pi$
$24$	0	0
$25$	−2.46870	−0.493739
$26$	0	0
$27$	4.27796	0.823293
$28$	0	0
$29$	2.07637	0.385572	0.192786	−	0.981241i	$-0.438248\pi$
$29$	2.07637	0.385572	0.192786	+	0.981241i	$0.438248\pi$
$30$	0	0
$31$	0.368627	0.0662073	0.0331037	−	0.999452i	$-0.489461\pi$
$31$	0.368627	0.0662073	0.0331037	+	0.999452i	$0.489461\pi$
$32$	0	0
$33$	4.05499	0.705883
$34$	0	0
$35$	−2.31134	−0.390688
$36$	0	0
$37$	−10.1998	−1.67683	−0.838416	−	0.545031i	$-0.816518\pi$
$37$	−10.1998	−1.67683	−0.838416	+	0.545031i	$0.816518\pi$
$38$	0	0
$39$	2.58258	0.413543
$40$	0	0
$41$	2.84460	0.444252	0.222126	−	0.975018i	$-0.428700\pi$
$41$	2.84460	0.444252	0.222126	+	0.975018i	$0.428700\pi$
$42$	0	0
$43$	−0.220232	−0.0335850	−0.0167925	−	0.999859i	$-0.505345\pi$
$43$	−0.220232	−0.0335850	−0.0167925	+	0.999859i	$0.505345\pi$
$44$	0	0
$45$	3.76102	0.560660
$46$	0	0
$47$	−0.444696	−0.0648656	−0.0324328	−	0.999474i	$-0.510325\pi$
$47$	−0.444696	−0.0648656	−0.0324328	+	0.999474i	$0.510325\pi$
$48$	0	0
$49$	−4.88950	−0.698500
$50$	0	0
$51$	1.17665	0.164764
$52$	0	0
$53$	−3.52007	−0.483519	−0.241760	−	0.970336i	$-0.577725\pi$
$53$	−3.52007	−0.483519	−0.241760	+	0.970336i	$0.577725\pi$
$54$	0	0
$55$	8.08925	1.09075
$56$	0	0
$57$	3.46300	0.458685
$58$	0	0
$59$	−1.14881	−0.149563	−0.0747813	−	0.997200i	$-0.523826\pi$
$59$	−1.14881	−0.149563	−0.0747813	+	0.997200i	$0.523826\pi$
$60$	0	0
$61$	−8.23654	−1.05458	−0.527291	−	0.849685i	$-0.676792\pi$
$61$	−8.23654	−1.05458	−0.527291	+	0.849685i	$0.676792\pi$
$62$	0	0
$63$	−3.43421	−0.432669
$64$	0	0
$65$	5.15195	0.639021
$66$	0	0
$67$	−4.03182	−0.492565	−0.246283	−	0.969198i	$-0.579209\pi$
$67$	−4.03182	−0.492565	−0.246283	+	0.969198i	$0.579209\pi$
$68$	0	0
$69$	6.91447	0.832405
$70$	0	0
$71$	−2.27655	−0.270176	−0.135088	−	0.990834i	$-0.543132\pi$
$71$	−2.27655	−0.270176	−0.135088	+	0.990834i	$0.543132\pi$
$72$	0	0
$73$	5.98502	0.700493	0.350247	−	0.936658i	$-0.386098\pi$
$73$	5.98502	0.700493	0.350247	+	0.936658i	$0.386098\pi$
$74$	0	0
$75$	1.96889	0.227348
$76$	0	0
$77$	−7.38633	−0.841750
$78$	0	0
$79$	−7.87656	−0.886182	−0.443091	−	0.896477i	$-0.646118\pi$
$79$	−7.87656	−0.886182	−0.443091	+	0.896477i	$0.646118\pi$
$80$	0	0
$81$	3.67993	0.408881
$82$	0	0
$83$	5.55280	0.609499	0.304750	−	0.952432i	$-0.401427\pi$
$83$	5.55280	0.609499	0.304750	+	0.952432i	$0.401427\pi$
$84$	0	0
$85$	2.34729	0.254599
$86$	0	0
$87$	−1.65599	−0.177541
$88$	0	0
$89$	7.51285	0.796361	0.398180	−	0.917307i	$-0.369642\pi$
$89$	7.51285	0.796361	0.398180	+	0.917307i	$0.369642\pi$
$90$	0	0
$91$	−4.70427	−0.493141
$92$	0	0
$93$	−0.293995	−0.0304859
$94$	0	0
$95$	6.90830	0.708776
$96$	0	0
$97$	0.608887	0.0618231	0.0309116	−	0.999522i	$-0.490159\pi$
$97$	0.608887	0.0618231	0.0309116	+	0.999522i	$0.490159\pi$
$98$	0	0
$99$	12.0191	1.20796
$100$	0	0
$101$	4.69354	0.467024	0.233512	−	0.972354i	$-0.424978\pi$
$101$	4.69354	0.467024	0.233512	+	0.972354i	$0.424978\pi$
$102$	0	0
$103$	15.4987	1.52714	0.763568	−	0.645728i	$-0.223446\pi$
$103$	15.4987	1.52714	0.763568	+	0.645728i	$0.223446\pi$
$104$	0	0
$105$	1.84339	0.179897
$106$	0	0
$107$	−11.4010	−1.10218	−0.551089	−	0.834447i	$-0.685787\pi$
$107$	−11.4010	−1.10218	−0.551089	+	0.834447i	$0.685787\pi$
$108$	0	0
$109$	−12.0275	−1.15203	−0.576013	−	0.817440i	$-0.695392\pi$
$109$	−12.0275	−1.15203	−0.576013	+	0.817440i	$0.695392\pi$
$110$	0	0
$111$	8.13474	0.772116
$112$	0	0
$113$	−0.501043	−0.0471342	−0.0235671	−	0.999722i	$-0.507502\pi$
$113$	−0.501043	−0.0471342	−0.0235671	+	0.999722i	$0.507502\pi$
$114$	0	0
$115$	13.7936	1.28626
$116$	0	0
$117$	7.65480	0.707686
$118$	0	0
$119$	−2.14332	−0.196478
$120$	0	0
$121$	14.8507	1.35006
$122$	0	0
$123$	−2.26869	−0.204561
$124$	0	0
$125$	11.8827	1.06283
$126$	0	0
$127$	13.9915	1.24155	0.620774	−	0.783990i	$-0.286819\pi$
$127$	13.9915	1.24155	0.620774	+	0.783990i	$0.286819\pi$
$128$	0	0
$129$	0.175644	0.0154646
$130$	0	0
$131$	−1.16074	−0.101414	−0.0507071	−	0.998714i	$-0.516148\pi$
$131$	−1.16074	−0.101414	−0.0507071	+	0.998714i	$0.516148\pi$
$132$	0	0
$133$	−6.30799	−0.546972
$134$	0	0
$135$	−6.80626	−0.585790
$136$	0	0
$137$	−4.04220	−0.345348	−0.172674	−	0.984979i	$-0.555241\pi$
$137$	−4.04220	−0.345348	−0.172674	+	0.984979i	$0.555241\pi$
$138$	0	0
$139$	−19.0269	−1.61384	−0.806920	−	0.590661i	$-0.798867\pi$
$139$	−19.0269	−1.61384	−0.806920	+	0.590661i	$0.798867\pi$
$140$	0	0
$141$	0.354664	0.0298681
$142$	0	0
$143$	16.4640	1.37679
$144$	0	0
$145$	−3.30352	−0.274342
$146$	0	0
$147$	3.89958	0.321632
$148$	0	0
$149$	−1.48371	−0.121551	−0.0607753	−	0.998151i	$-0.519357\pi$
$149$	−1.48371	−0.121551	−0.0607753	+	0.998151i	$0.519357\pi$
$150$	0	0
$151$	8.20708	0.667883	0.333941	−	0.942594i	$-0.391621\pi$
$151$	8.20708	0.667883	0.333941	+	0.942594i	$0.391621\pi$
$152$	0	0
$153$	3.48761	0.281957
$154$	0	0
$155$	−0.586488	−0.0471078
$156$	0	0
$157$	−5.30831	−0.423649	−0.211824	−	0.977308i	$-0.567941\pi$
$157$	−5.30831	−0.423649	−0.211824	+	0.977308i	$0.567941\pi$
$158$	0	0
$159$	2.80741	0.222642
$160$	0	0
$161$	−12.5950	−0.992624
$162$	0	0
$163$	10.1590	0.795714	0.397857	−	0.917447i	$-0.369754\pi$
$163$	10.1590	0.795714	0.397857	+	0.917447i	$0.369754\pi$
$164$	0	0
$165$	−6.45152	−0.502250
$166$	0	0
$167$	11.4519	0.886171	0.443086	−	0.896479i	$-0.353884\pi$
$167$	11.4519	0.886171	0.443086	+	0.896479i	$0.353884\pi$
$168$	0	0
$169$	−2.51425	−0.193404
$170$	0	0
$171$	10.2644	0.784937
$172$	0	0
$173$	−22.9663	−1.74610	−0.873050	−	0.487631i	$-0.837861\pi$
$173$	−22.9663	−1.74610	−0.873050	+	0.487631i	$0.837861\pi$
$174$	0	0
$175$	−3.58641	−0.271107
$176$	0	0
$177$	0.916226	0.0688677
$178$	0	0
$179$	10.0902	0.754174	0.377087	−	0.926178i	$-0.376926\pi$
$179$	10.0902	0.754174	0.377087	+	0.926178i	$0.376926\pi$
$180$	0	0
$181$	−24.2139	−1.79981	−0.899904	−	0.436088i	$-0.856364\pi$
$181$	−24.2139	−1.79981	−0.899904	+	0.436088i	$0.856364\pi$
$182$	0	0
$183$	6.56899	0.485594
$184$	0	0
$185$	16.2279	1.19310
$186$	0	0
$187$	7.50120	0.548542
$188$	0	0
$189$	6.21482	0.452062
$190$	0	0
$191$	−7.57567	−0.548156	−0.274078	−	0.961707i	$-0.588373\pi$
$191$	−7.57567	−0.548156	−0.274078	+	0.961707i	$0.588373\pi$
$192$	0	0
$193$	−12.3191	−0.886749	−0.443375	−	0.896336i	$-0.646219\pi$
$193$	−12.3191	−0.886749	−0.443375	+	0.896336i	$0.646219\pi$
$194$	0	0
$195$	−4.10890	−0.294244
$196$	0	0
$197$	10.4706	0.745997	0.372998	−	0.927832i	$-0.378330\pi$
$197$	10.4706	0.745997	0.372998	+	0.927832i	$0.378330\pi$
$198$	0	0
$199$	7.38657	0.523620	0.261810	−	0.965119i	$-0.415681\pi$
$199$	7.38657	0.523620	0.261810	+	0.965119i	$0.415681\pi$
$200$	0	0
$201$	3.21555	0.226807
$202$	0	0
$203$	3.01646	0.211714
$204$	0	0
$205$	−4.52578	−0.316094
$206$	0	0
$207$	20.4946	1.42447
$208$	0	0
$209$	22.0767	1.52708
$210$	0	0
$211$	8.20288	0.564709	0.282355	−	0.959310i	$-0.408884\pi$
$211$	8.20288	0.564709	0.282355	+	0.959310i	$0.408884\pi$
$212$	0	0
$213$	1.81564	0.124406
$214$	0	0
$215$	0.350390	0.0238964
$216$	0	0
$217$	0.535524	0.0363538
$218$	0	0
$219$	−4.77330	−0.322550
$220$	0	0
$221$	4.77743	0.321364
$222$	0	0
$223$	−12.1402	−0.812966	−0.406483	−	0.913658i	$-0.633245\pi$
$223$	−12.1402	−0.812966	−0.406483	+	0.913658i	$0.633245\pi$
$224$	0	0
$225$	5.83582	0.389055
$226$	0	0
$227$	−9.62461	−0.638808	−0.319404	−	0.947619i	$-0.603483\pi$
$227$	−9.62461	−0.638808	−0.319404	+	0.947619i	$0.603483\pi$
$228$	0	0
$229$	−14.2400	−0.941005	−0.470503	−	0.882399i	$-0.655927\pi$
$229$	−14.2400	−0.941005	−0.470503	+	0.882399i	$0.655927\pi$
$230$	0	0
$231$	5.89091	0.387593
$232$	0	0
$233$	−25.9411	−1.69946	−0.849728	−	0.527221i	$-0.823234\pi$
$233$	−25.9411	−1.69946	−0.849728	+	0.527221i	$0.823234\pi$
$234$	0	0
$235$	0.707515	0.0461532
$236$	0	0
$237$	6.28189	0.408053
$238$	0	0
$239$	4.87455	0.315308	0.157654	−	0.987494i	$-0.449607\pi$
$239$	4.87455	0.315308	0.157654	+	0.987494i	$0.449607\pi$
$240$	0	0
$241$	−16.2622	−1.04754	−0.523770	−	0.851860i	$-0.675475\pi$
$241$	−16.2622	−1.04754	−0.523770	+	0.851860i	$0.675475\pi$
$242$	0	0
$243$	−15.7688	−1.01157
$244$	0	0
$245$	7.77923	0.496997
$246$	0	0
$247$	14.0604	0.894643
$248$	0	0
$249$	−4.42859	−0.280651
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	44.0800	2.77129
$254$	0	0
$255$	−1.87206	−0.117233
$256$	0	0
$257$	−11.5663	−0.721485	−0.360743	−	0.932665i	$-0.617477\pi$
$257$	−11.5663	−0.721485	−0.360743	+	0.932665i	$0.617477\pi$
$258$	0	0
$259$	−14.8178	−0.920731
$260$	0	0
$261$	−4.90838	−0.303821
$262$	0	0
$263$	19.6584	1.21219	0.606095	−	0.795392i	$-0.292735\pi$
$263$	19.6584	1.21219	0.606095	+	0.795392i	$0.292735\pi$
$264$	0	0
$265$	5.60046	0.344034
$266$	0	0
$267$	−5.99182	−0.366693
$268$	0	0
$269$	27.5590	1.68030	0.840152	−	0.542351i	$-0.182466\pi$
$269$	27.5590	1.68030	0.840152	+	0.542351i	$0.182466\pi$
$270$	0	0
$271$	14.4953	0.880525	0.440263	−	0.897869i	$-0.354885\pi$
$271$	14.4953	0.880525	0.440263	+	0.897869i	$0.354885\pi$
$272$	0	0
$273$	3.75185	0.227072
$274$	0	0
$275$	12.5517	0.756898
$276$	0	0
$277$	−31.1876	−1.87388	−0.936942	−	0.349486i	$-0.886356\pi$
$277$	−31.1876	−1.87388	−0.936942	+	0.349486i	$0.886356\pi$
$278$	0	0
$279$	−0.871406	−0.0521697
$280$	0	0
$281$	−22.6416	−1.35068	−0.675341	−	0.737506i	$-0.736004\pi$
$281$	−22.6416	−1.35068	−0.675341	+	0.737506i	$0.736004\pi$
$282$	0	0
$283$	−15.3541	−0.912707	−0.456354	−	0.889799i	$-0.650845\pi$
$283$	−15.3541	−0.912707	−0.456354	+	0.889799i	$0.650845\pi$
$284$	0	0
$285$	−5.50966	−0.326364
$286$	0	0
$287$	4.13251	0.243934
$288$	0	0
$289$	−14.8233	−0.871962
$290$	0	0
$291$	−0.485613	−0.0284672
$292$	0	0
$293$	15.6294	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$293$	15.6294	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$294$	0	0
$295$	1.82777	0.106417
$296$	0	0
$297$	−21.7507	−1.26210
$298$	0	0
$299$	28.0741	1.62356
$300$	0	0
$301$	−0.319943	−0.0184412
$302$	0	0
$303$	−3.74329	−0.215047
$304$	0	0
$305$	13.1044	0.750356
$306$	0	0
$307$	8.79473	0.501942	0.250971	−	0.967995i	$-0.419250\pi$
$307$	8.79473	0.501942	0.250971	+	0.967995i	$0.419250\pi$
$308$	0	0
$309$	−12.3609	−0.703186
$310$	0	0
$311$	29.6368	1.68055	0.840275	−	0.542160i	$-0.182393\pi$
$311$	29.6368	1.68055	0.840275	+	0.542160i	$0.182393\pi$
$312$	0	0
$313$	1.99611	0.112827	0.0564134	−	0.998407i	$-0.482034\pi$
$313$	1.99611	0.112827	0.0564134	+	0.998407i	$0.482034\pi$
$314$	0	0
$315$	5.46385	0.307853
$316$	0	0
$317$	−20.2246	−1.13593	−0.567963	−	0.823054i	$-0.692268\pi$
$317$	−20.2246	−1.13593	−0.567963	+	0.823054i	$0.692268\pi$
$318$	0	0
$319$	−10.5570	−0.591079
$320$	0	0
$321$	9.09279	0.507510
$322$	0	0
$323$	6.40609	0.356444
$324$	0	0
$325$	7.99406	0.443431
$326$	0	0
$327$	9.59244	0.530463
$328$	0	0
$329$	−0.646034	−0.0356170
$330$	0	0
$331$	16.6830	0.916982	0.458491	−	0.888699i	$-0.348390\pi$
$331$	16.6830	0.916982	0.458491	+	0.888699i	$0.348390\pi$
$332$	0	0
$333$	24.1115	1.32130
$334$	0	0
$335$	6.41466	0.350470
$336$	0	0
$337$	−28.5413	−1.55474	−0.777371	−	0.629043i	$-0.783447\pi$
$337$	−28.5413	−1.55474	−0.777371	+	0.629043i	$0.783447\pi$
$338$	0	0
$339$	0.399603	0.0217035
$340$	0	0
$341$	−1.87423	−0.101495
$342$	0	0
$343$	−17.2725	−0.932629
$344$	0	0
$345$	−11.0010	−0.592273
$346$	0	0
$347$	−6.79212	−0.364620	−0.182310	−	0.983241i	$-0.558358\pi$
$347$	−6.79212	−0.364620	−0.182310	+	0.983241i	$0.558358\pi$
$348$	0	0
$349$	18.5327	0.992035	0.496017	−	0.868313i	$-0.334795\pi$
$349$	18.5327	0.992035	0.496017	+	0.868313i	$0.334795\pi$
$350$	0	0
$351$	−13.8528	−0.739405
$352$	0	0
$353$	28.9360	1.54011	0.770055	−	0.637977i	$-0.220229\pi$
$353$	28.9360	1.54011	0.770055	+	0.637977i	$0.220229\pi$
$354$	0	0
$355$	3.62200	0.192236
$356$	0	0
$357$	1.70939	0.0904703
$358$	0	0
$359$	11.2287	0.592628	0.296314	−	0.955091i	$-0.404243\pi$
$359$	11.2287	0.592628	0.296314	+	0.955091i	$0.404243\pi$
$360$	0	0
$361$	−0.146258	−0.00769777
$362$	0	0
$363$	−11.8441	−0.621652
$364$	0	0
$365$	−9.52220	−0.498415
$366$	0	0
$367$	33.6694	1.75753	0.878763	−	0.477258i	$-0.158369\pi$
$367$	33.6694	1.75753	0.878763	+	0.477258i	$0.158369\pi$
$368$	0	0
$369$	−6.72443	−0.350060
$370$	0	0
$371$	−5.11380	−0.265495
$372$	0	0
$373$	23.6550	1.22481	0.612405	−	0.790544i	$-0.290202\pi$
$373$	23.6550	1.22481	0.612405	+	0.790544i	$0.290202\pi$
$374$	0	0
$375$	−9.47699	−0.489390
$376$	0	0
$377$	−6.72364	−0.346285
$378$	0	0
$379$	−29.6340	−1.52219	−0.761097	−	0.648638i	$-0.775339\pi$
$379$	−29.6340	−1.52219	−0.761097	+	0.648638i	$0.775339\pi$
$380$	0	0
$381$	−11.1588	−0.571684
$382$	0	0
$383$	−17.3473	−0.886406	−0.443203	−	0.896421i	$-0.646158\pi$
$383$	−17.3473	−0.886406	−0.443203	+	0.896421i	$0.646158\pi$
$384$	0	0
$385$	11.7517	0.598922
$386$	0	0
$387$	0.520612	0.0264642
$388$	0	0
$389$	27.3614	1.38728	0.693639	−	0.720322i	$-0.256006\pi$
$389$	27.3614	1.38728	0.693639	+	0.720322i	$0.256006\pi$
$390$	0	0
$391$	12.7909	0.646862
$392$	0	0
$393$	0.925738	0.0466973
$394$	0	0
$395$	12.5317	0.630536
$396$	0	0
$397$	3.56136	0.178739	0.0893697	−	0.995999i	$-0.471515\pi$
$397$	3.56136	0.178739	0.0893697	+	0.995999i	$0.471515\pi$
$398$	0	0
$399$	5.03089	0.251859
$400$	0	0
$401$	−27.2223	−1.35942	−0.679709	−	0.733482i	$-0.737894\pi$
$401$	−27.2223	−1.35942	−0.679709	+	0.733482i	$0.737894\pi$
$402$	0	0
$403$	−1.19368	−0.0594612
$404$	0	0
$405$	−5.85479	−0.290927
$406$	0	0
$407$	51.8593	2.57057
$408$	0	0
$409$	1.31647	0.0650955	0.0325477	−	0.999470i	$-0.489638\pi$
$409$	1.31647	0.0650955	0.0325477	+	0.999470i	$0.489638\pi$
$410$	0	0
$411$	3.22382	0.159019
$412$	0	0
$413$	−1.66894	−0.0821232
$414$	0	0
$415$	−8.83455	−0.433671
$416$	0	0
$417$	15.1747	0.743111
$418$	0	0
$419$	30.8987	1.50950	0.754750	−	0.656012i	$-0.227758\pi$
$419$	30.8987	1.50950	0.754750	+	0.656012i	$0.227758\pi$
$420$	0	0
$421$	−10.3745	−0.505620	−0.252810	−	0.967516i	$-0.581355\pi$
$421$	−10.3745	−0.505620	−0.252810	+	0.967516i	$0.581355\pi$
$422$	0	0
$423$	1.05123	0.0511125
$424$	0	0
$425$	3.64219	0.176672
$426$	0	0
$427$	−11.9657	−0.579060
$428$	0	0
$429$	−13.1307	−0.633958
$430$	0	0
$431$	−10.2148	−0.492030	−0.246015	−	0.969266i	$-0.579121\pi$
$431$	−10.2148	−0.492030	−0.246015	+	0.969266i	$0.579121\pi$
$432$	0	0
$433$	13.2704	0.637733	0.318867	−	0.947800i	$-0.396698\pi$
$433$	13.2704	0.637733	0.318867	+	0.947800i	$0.396698\pi$
$434$	0	0
$435$	2.63469	0.126324
$436$	0	0
$437$	37.6447	1.80079
$438$	0	0
$439$	−0.324821	−0.0155029	−0.00775144	−	0.999970i	$-0.502467\pi$
$439$	−0.324821	−0.0155029	−0.00775144	+	0.999970i	$0.502467\pi$
$440$	0	0
$441$	11.5584	0.550401
$442$	0	0
$443$	19.5792	0.930237	0.465118	−	0.885249i	$-0.346012\pi$
$443$	19.5792	0.930237	0.465118	+	0.885249i	$0.346012\pi$
$444$	0	0
$445$	−11.9530	−0.566627
$446$	0	0
$447$	1.18332	0.0559693
$448$	0	0
$449$	12.7665	0.602487	0.301244	−	0.953547i	$-0.402598\pi$
$449$	12.7665	0.602487	0.301244	+	0.953547i	$0.402598\pi$
$450$	0	0
$451$	−14.4630	−0.681035
$452$	0	0
$453$	−6.54549	−0.307534
$454$	0	0
$455$	7.48452	0.350880
$456$	0	0
$457$	−33.7867	−1.58048	−0.790238	−	0.612799i	$-0.790043\pi$
$457$	−33.7867	−1.58048	−0.790238	+	0.612799i	$0.790043\pi$
$458$	0	0
$459$	−6.31147	−0.294594
$460$	0	0
$461$	−21.6532	−1.00849	−0.504245	−	0.863561i	$-0.668229\pi$
$461$	−21.6532	−1.00849	−0.504245	+	0.863561i	$0.668229\pi$
$462$	0	0
$463$	24.8083	1.15294	0.576471	−	0.817118i	$-0.304430\pi$
$463$	24.8083	1.15294	0.576471	+	0.817118i	$0.304430\pi$
$464$	0	0
$465$	0.467749	0.0216913
$466$	0	0
$467$	−12.1799	−0.563618	−0.281809	−	0.959470i	$-0.590935\pi$
$467$	−12.1799	−0.563618	−0.281809	+	0.959470i	$0.590935\pi$
$468$	0	0
$469$	−5.85725	−0.270463
$470$	0	0
$471$	4.23360	0.195074
$472$	0	0
$473$	1.11974	0.0514856
$474$	0	0
$475$	10.7193	0.491835
$476$	0	0
$477$	8.32119	0.381001
$478$	0	0
$479$	−15.5662	−0.711240	−0.355620	−	0.934631i	$-0.615730\pi$
$479$	−15.5662	−0.711240	−0.355620	+	0.934631i	$0.615730\pi$
$480$	0	0
$481$	33.0286	1.50597
$482$	0	0
$483$	10.0450	0.457065
$484$	0	0
$485$	−0.968744	−0.0439884
$486$	0	0
$487$	−18.2701	−0.827899	−0.413950	−	0.910300i	$-0.635851\pi$
$487$	−18.2701	−0.827899	−0.413950	+	0.910300i	$0.635851\pi$
$488$	0	0
$489$	−8.10223	−0.366395
$490$	0	0
$491$	13.8632	0.625640	0.312820	−	0.949812i	$-0.398726\pi$
$491$	13.8632	0.625640	0.312820	+	0.949812i	$0.398726\pi$
$492$	0	0
$493$	−3.06337	−0.137967
$494$	0	0
$495$	−19.1224	−0.859488
$496$	0	0
$497$	−3.30727	−0.148351
$498$	0	0
$499$	−17.3645	−0.777342	−0.388671	−	0.921377i	$-0.627066\pi$
$499$	−17.3645	−0.777342	−0.388671	+	0.921377i	$0.627066\pi$
$500$	0	0
$501$	−9.13334	−0.408047
$502$	0	0
$503$	13.8255	0.616448	0.308224	−	0.951314i	$-0.400265\pi$
$503$	13.8255	0.616448	0.308224	+	0.951314i	$0.400265\pi$
$504$	0	0
$505$	−7.46745	−0.332297
$506$	0	0
$507$	2.00522	0.0890551
$508$	0	0
$509$	−2.25942	−0.100147	−0.0500734	−	0.998746i	$-0.515946\pi$
$509$	−2.25942	−0.100147	−0.0500734	+	0.998746i	$0.515946\pi$
$510$	0	0
$511$	8.69476	0.384634
$512$	0	0
$513$	−18.5753	−0.820118
$514$	0	0
$515$	−24.6586	−1.08659
$516$	0	0
$517$	2.26099	0.0994384
$518$	0	0
$519$	18.3166	0.804011
$520$	0	0
$521$	3.22270	0.141189	0.0705945	−	0.997505i	$-0.477510\pi$
$521$	3.22270	0.141189	0.0705945	+	0.997505i	$0.477510\pi$
$522$	0	0
$523$	26.5339	1.16025	0.580123	−	0.814529i	$-0.303005\pi$
$523$	26.5339	1.16025	0.580123	+	0.814529i	$0.303005\pi$
$524$	0	0
$525$	2.86031	0.124834
$526$	0	0
$527$	−0.543852	−0.0236906
$528$	0	0
$529$	52.1642	2.26801
$530$	0	0
$531$	2.71571	0.117852
$532$	0	0
$533$	−9.21130	−0.398986
$534$	0	0
$535$	18.1391	0.784221
$536$	0	0
$537$	−8.04732	−0.347268
$538$	0	0
$539$	24.8600	1.07080
$540$	0	0
$541$	−8.55225	−0.367690	−0.183845	−	0.982955i	$-0.558854\pi$
$541$	−8.55225	−0.367690	−0.183845	+	0.982955i	$0.558854\pi$
$542$	0	0
$543$	19.3116	0.828742
$544$	0	0
$545$	19.1359	0.819690
$546$	0	0
$547$	−24.6962	−1.05593	−0.527967	−	0.849265i	$-0.677045\pi$
$547$	−24.6962	−1.05593	−0.527967	+	0.849265i	$0.677045\pi$
$548$	0	0
$549$	19.4706	0.830984
$550$	0	0
$551$	−9.01578	−0.384085
$552$	0	0
$553$	−11.4427	−0.486594
$554$	0	0
$555$	−12.9424	−0.549376
$556$	0	0
$557$	−24.1650	−1.02390	−0.511951	−	0.859015i	$-0.671077\pi$
$557$	−24.1650	−1.02390	−0.511951	+	0.859015i	$0.671077\pi$
$558$	0	0
$559$	0.713148	0.0301630
$560$	0	0
$561$	−5.98252	−0.252582
$562$	0	0
$563$	34.2975	1.44547	0.722734	−	0.691126i	$-0.242885\pi$
$563$	34.2975	1.44547	0.722734	+	0.691126i	$0.242885\pi$
$564$	0	0
$565$	0.797163	0.0335369
$566$	0	0
$567$	5.34603	0.224512
$568$	0	0
$569$	20.9009	0.876212	0.438106	−	0.898923i	$-0.355649\pi$
$569$	20.9009	0.876212	0.438106	+	0.898923i	$0.355649\pi$
$570$	0	0
$571$	−33.2036	−1.38953	−0.694764	−	0.719238i	$-0.744491\pi$
$571$	−33.2036	−1.38953	−0.694764	+	0.719238i	$0.744491\pi$
$572$	0	0
$573$	6.04192	0.252405
$574$	0	0
$575$	21.4029	0.892564
$576$	0	0
$577$	−16.3457	−0.680479	−0.340240	−	0.940339i	$-0.610508\pi$
$577$	−16.3457	−0.680479	−0.340240	+	0.940339i	$0.610508\pi$
$578$	0	0
$579$	9.82501	0.408314
$580$	0	0
$581$	8.06686	0.334670
$582$	0	0
$583$	17.8973	0.741231
$584$	0	0
$585$	−12.1788	−0.503533
$586$	0	0
$587$	−35.8472	−1.47957	−0.739786	−	0.672842i	$-0.765073\pi$
$587$	−35.8472	−1.47957	−0.739786	+	0.672842i	$0.765073\pi$
$588$	0	0
$589$	−1.60061	−0.0659520
$590$	0	0
$591$	−8.35071	−0.343502
$592$	0	0
$593$	−6.97842	−0.286569	−0.143285	−	0.989682i	$-0.545766\pi$
$593$	−6.97842	−0.286569	−0.143285	+	0.989682i	$0.545766\pi$
$594$	0	0
$595$	3.41003	0.139798
$596$	0	0
$597$	−5.89110	−0.241107
$598$	0	0
$599$	−0.642098	−0.0262354	−0.0131177	−	0.999914i	$-0.504176\pi$
$599$	−0.642098	−0.0262354	−0.0131177	+	0.999914i	$0.504176\pi$
$600$	0	0
$601$	33.1282	1.35133	0.675664	−	0.737209i	$-0.263857\pi$
$601$	33.1282	1.35133	0.675664	+	0.737209i	$0.263857\pi$
$602$	0	0
$603$	9.53093	0.388129
$604$	0	0
$605$	−23.6276	−0.960597
$606$	0	0
$607$	−0.462154	−0.0187583	−0.00937913	−	0.999956i	$-0.502986\pi$
$607$	−0.462154	−0.0187583	−0.00937913	+	0.999956i	$0.502986\pi$
$608$	0	0
$609$	−2.40575	−0.0974859
$610$	0	0
$611$	1.44000	0.0582562
$612$	0	0
$613$	3.24069	0.130890	0.0654450	−	0.997856i	$-0.479153\pi$
$613$	3.24069	0.130890	0.0654450	+	0.997856i	$0.479153\pi$
$614$	0	0
$615$	3.60950	0.145549
$616$	0	0
$617$	−1.75861	−0.0707990	−0.0353995	−	0.999373i	$-0.511270\pi$
$617$	−1.75861	−0.0707990	−0.0353995	+	0.999373i	$0.511270\pi$
$618$	0	0
$619$	−15.9414	−0.640738	−0.320369	−	0.947293i	$-0.603807\pi$
$619$	−15.9414	−0.640738	−0.320369	+	0.947293i	$0.603807\pi$
$620$	0	0
$621$	−37.0887	−1.48832
$622$	0	0
$623$	10.9143	0.437274
$624$	0	0
$625$	−6.56205	−0.262482
$626$	0	0
$627$	−17.6071	−0.703161
$628$	0	0
$629$	15.0482	0.600011
$630$	0	0
$631$	−41.4070	−1.64839	−0.824194	−	0.566308i	$-0.808371\pi$
$631$	−41.4070	−1.64839	−0.824194	+	0.566308i	$0.808371\pi$
$632$	0	0
$633$	−6.54214	−0.260027
$634$	0	0
$635$	−22.2606	−0.883386
$636$	0	0
$637$	15.8330	0.627328
$638$	0	0
$639$	5.38159	0.212892
$640$	0	0
$641$	39.0260	1.54143	0.770717	−	0.637178i	$-0.219898\pi$
$641$	39.0260	1.54143	0.770717	+	0.637178i	$0.219898\pi$
$642$	0	0
$643$	7.75645	0.305884	0.152942	−	0.988235i	$-0.451125\pi$
$643$	7.75645	0.305884	0.152942	+	0.988235i	$0.451125\pi$
$644$	0	0
$645$	−0.279451	−0.0110034
$646$	0	0
$647$	2.04119	0.0802474	0.0401237	−	0.999195i	$-0.487225\pi$
$647$	2.04119	0.0802474	0.0401237	+	0.999195i	$0.487225\pi$
$648$	0	0
$649$	5.84097	0.229278
$650$	0	0
$651$	−0.427103	−0.0167395
$652$	0	0
$653$	−9.83838	−0.385006	−0.192503	−	0.981296i	$-0.561660\pi$
$653$	−9.83838	−0.385006	−0.192503	+	0.981296i	$0.561660\pi$
$654$	0	0
$655$	1.84674	0.0721583
$656$	0	0
$657$	−14.1481	−0.551971
$658$	0	0
$659$	38.0432	1.48195	0.740976	−	0.671532i	$-0.234363\pi$
$659$	38.0432	1.48195	0.740976	+	0.671532i	$0.234363\pi$
$660$	0	0
$661$	17.6749	0.687474	0.343737	−	0.939066i	$-0.388307\pi$
$661$	17.6749	0.687474	0.343737	+	0.939066i	$0.388307\pi$
$662$	0	0
$663$	−3.81020	−0.147976
$664$	0	0
$665$	10.0361	0.389182
$666$	0	0
$667$	−18.0016	−0.697023
$668$	0	0
$669$	9.68229	0.374339
$670$	0	0
$671$	41.8775	1.61666
$672$	0	0
$673$	10.3861	0.400357	0.200178	−	0.979759i	$-0.435848\pi$
$673$	10.3861	0.400357	0.200178	+	0.979759i	$0.435848\pi$
$674$	0	0
$675$	−10.5610	−0.406492
$676$	0	0
$677$	−1.80298	−0.0692940	−0.0346470	−	0.999400i	$-0.511031\pi$
$677$	−1.80298	−0.0692940	−0.0346470	+	0.999400i	$0.511031\pi$
$678$	0	0
$679$	0.884564	0.0339464
$680$	0	0
$681$	7.67603	0.294146
$682$	0	0
$683$	−42.3787	−1.62158	−0.810789	−	0.585339i	$-0.800962\pi$
$683$	−42.3787	−1.62158	−0.810789	+	0.585339i	$0.800962\pi$
$684$	0	0
$685$	6.43116	0.245722
$686$	0	0
$687$	11.3570	0.433296
$688$	0	0
$689$	11.3986	0.434252
$690$	0	0
$691$	8.61896	0.327881	0.163940	−	0.986470i	$-0.447580\pi$
$691$	8.61896	0.327881	0.163940	+	0.986470i	$0.447580\pi$
$692$	0	0
$693$	17.4607	0.663278
$694$	0	0
$695$	30.2719	1.14828
$696$	0	0
$697$	−4.19678	−0.158964
$698$	0	0
$699$	20.6891	0.782534
$700$	0	0
$701$	32.2804	1.21922	0.609608	−	0.792703i	$-0.291327\pi$
$701$	32.2804	1.21922	0.609608	+	0.792703i	$0.291327\pi$
$702$	0	0
$703$	44.2883	1.67036
$704$	0	0
$705$	−0.564273	−0.0212517
$706$	0	0
$707$	6.81856	0.256438
$708$	0	0
$709$	26.2241	0.984867	0.492433	−	0.870350i	$-0.336107\pi$
$709$	26.2241	0.984867	0.492433	+	0.870350i	$0.336107\pi$
$710$	0	0
$711$	18.6196	0.698290
$712$	0	0
$713$	−3.19589	−0.119687
$714$	0	0
$715$	−26.1944	−0.979614
$716$	0	0
$717$	−3.88766	−0.145187
$718$	0	0
$719$	−23.9547	−0.893360	−0.446680	−	0.894694i	$-0.647394\pi$
$719$	−23.9547	−0.893360	−0.446680	+	0.894694i	$0.647394\pi$
$720$	0	0
$721$	22.5159	0.838534
$722$	0	0
$723$	12.9698	0.482352
$724$	0	0
$725$	−5.12592	−0.190372
$726$	0	0
$727$	−23.7671	−0.881474	−0.440737	−	0.897636i	$-0.645283\pi$
$727$	−23.7671	−0.881474	−0.440737	+	0.897636i	$0.645283\pi$
$728$	0	0
$729$	1.53647	0.0569064
$730$	0	0
$731$	0.324919	0.0120175
$732$	0	0
$733$	28.9899	1.07077	0.535383	−	0.844610i	$-0.320167\pi$
$733$	28.9899	1.07077	0.535383	+	0.844610i	$0.320167\pi$
$734$	0	0
$735$	−6.20427	−0.228848
$736$	0	0
$737$	20.4992	0.755099
$738$	0	0
$739$	−3.86810	−0.142290	−0.0711451	−	0.997466i	$-0.522665\pi$
$739$	−3.86810	−0.142290	−0.0711451	+	0.997466i	$0.522665\pi$
$740$	0	0
$741$	−11.2138	−0.411949
$742$	0	0
$743$	5.26145	0.193024	0.0965120	−	0.995332i	$-0.469231\pi$
$743$	5.26145	0.193024	0.0965120	+	0.995332i	$0.469231\pi$
$744$	0	0
$745$	2.36060	0.0864857
$746$	0	0
$747$	−13.1264	−0.480271
$748$	0	0
$749$	−16.5629	−0.605194
$750$	0	0
$751$	2.15809	0.0787498	0.0393749	−	0.999225i	$-0.487463\pi$
$751$	2.15809	0.0787498	0.0393749	+	0.999225i	$0.487463\pi$
$752$	0	0
$753$	−0.797542	−0.0290640
$754$	0	0
$755$	−13.0575	−0.475212
$756$	0	0
$757$	−42.9482	−1.56098	−0.780490	−	0.625168i	$-0.785030\pi$
$757$	−42.9482	−1.56098	−0.780490	+	0.625168i	$0.785030\pi$
$758$	0	0
$759$	−35.1557	−1.27607
$760$	0	0
$761$	12.8703	0.466549	0.233274	−	0.972411i	$-0.425056\pi$
$761$	12.8703	0.466549	0.233274	+	0.972411i	$0.425056\pi$
$762$	0	0
$763$	−17.4730	−0.632566
$764$	0	0
$765$	−5.54882	−0.200618
$766$	0	0
$767$	3.72005	0.134323
$768$	0	0
$769$	28.4367	1.02545	0.512727	−	0.858552i	$-0.328635\pi$
$769$	28.4367	1.02545	0.512727	+	0.858552i	$0.328635\pi$
$770$	0	0
$771$	9.22460	0.332216
$772$	0	0
$773$	11.3288	0.407467	0.203734	−	0.979026i	$-0.434692\pi$
$773$	11.3288	0.407467	0.203734	+	0.979026i	$0.434692\pi$
$774$	0	0
$775$	−0.910027	−0.0326892
$776$	0	0
$777$	11.8178	0.423961
$778$	0	0
$779$	−12.3515	−0.442539
$780$	0	0
$781$	11.5748	0.414178
$782$	0	0
$783$	8.88262	0.317439
$784$	0	0
$785$	8.44555	0.301435
$786$	0	0
$787$	9.89964	0.352884	0.176442	−	0.984311i	$-0.443541\pi$
$787$	9.89964	0.352884	0.176442	+	0.984311i	$0.443541\pi$
$788$	0	0
$789$	−15.6784	−0.558167
$790$	0	0
$791$	−0.727893	−0.0258809
$792$	0	0
$793$	26.6713	0.947127
$794$	0	0
$795$	−4.46660	−0.158414
$796$	0	0
$797$	26.4484	0.936850	0.468425	−	0.883503i	$-0.344822\pi$
$797$	26.4484	0.936850	0.468425	+	0.883503i	$0.344822\pi$
$798$	0	0
$799$	0.656081	0.0232105
$800$	0	0
$801$	−17.7598	−0.627513
$802$	0	0
$803$	−30.4300	−1.07385
$804$	0	0
$805$	20.0387	0.706272
$806$	0	0
$807$	−21.9795	−0.773715
$808$	0	0
$809$	24.6352	0.866129	0.433064	−	0.901363i	$-0.357432\pi$
$809$	24.6352	0.866129	0.433064	+	0.901363i	$0.357432\pi$
$810$	0	0
$811$	−28.9804	−1.01764	−0.508820	−	0.860873i	$-0.669918\pi$
$811$	−28.9804	−1.01764	−0.508820	+	0.860873i	$0.669918\pi$
$812$	0	0
$813$	−11.5606	−0.405448
$814$	0	0
$815$	−16.1630	−0.566166
$816$	0	0
$817$	0.956267	0.0334555
$818$	0	0
$819$	11.1205	0.388583
$820$	0	0
$821$	−11.7663	−0.410646	−0.205323	−	0.978694i	$-0.565824\pi$
$821$	−11.7663	−0.410646	−0.205323	+	0.978694i	$0.565824\pi$
$822$	0	0
$823$	7.87270	0.274425	0.137213	−	0.990542i	$-0.456186\pi$
$823$	7.87270	0.274425	0.137213	+	0.990542i	$0.456186\pi$
$824$	0	0
$825$	−10.0105	−0.348522
$826$	0	0
$827$	14.1513	0.492087	0.246044	−	0.969259i	$-0.420869\pi$
$827$	14.1513	0.492087	0.246044	+	0.969259i	$0.420869\pi$
$828$	0	0
$829$	−13.3282	−0.462907	−0.231453	−	0.972846i	$-0.574348\pi$
$829$	−13.3282	−0.462907	−0.231453	+	0.972846i	$0.574348\pi$
$830$	0	0
$831$	24.8735	0.862850
$832$	0	0
$833$	7.21372	0.249940
$834$	0	0
$835$	−18.2200	−0.630529
$836$	0	0
$837$	1.57697	0.0545080
$838$	0	0
$839$	7.59858	0.262332	0.131166	−	0.991360i	$-0.458128\pi$
$839$	7.59858	0.262332	0.131166	+	0.991360i	$0.458128\pi$
$840$	0	0
$841$	−24.6887	−0.851334
$842$	0	0
$843$	18.0576	0.621937
$844$	0	0
$845$	4.00019	0.137611
$846$	0	0
$847$	21.5744	0.741306
$848$	0	0
$849$	12.2455	0.420266
$850$	0	0
$851$	88.4292	3.03131
$852$	0	0
$853$	11.6594	0.399210	0.199605	−	0.979876i	$-0.436034\pi$
$853$	11.6594	0.399210	0.199605	+	0.979876i	$0.436034\pi$
$854$	0	0
$855$	−16.3307	−0.558498
$856$	0	0
$857$	−15.2226	−0.519994	−0.259997	−	0.965609i	$-0.583722\pi$
$857$	−15.2226	−0.519994	−0.259997	+	0.965609i	$0.583722\pi$
$858$	0	0
$859$	−5.85587	−0.199800	−0.0998998	−	0.994998i	$-0.531852\pi$
$859$	−5.85587	−0.199800	−0.0998998	+	0.994998i	$0.531852\pi$
$860$	0	0
$861$	−3.29585	−0.112322
$862$	0	0
$863$	−11.1781	−0.380509	−0.190254	−	0.981735i	$-0.560931\pi$
$863$	−11.1781	−0.380509	−0.190254	+	0.981735i	$0.560931\pi$
$864$	0	0
$865$	36.5396	1.24238
$866$	0	0
$867$	11.8222	0.401505
$868$	0	0
$869$	40.0473	1.35851
$870$	0	0
$871$	13.0557	0.442376
$872$	0	0
$873$	−1.43936	−0.0487151
$874$	0	0
$875$	17.2627	0.583586
$876$	0	0
$877$	3.91919	0.132342	0.0661709	−	0.997808i	$-0.478922\pi$
$877$	3.91919	0.132342	0.0661709	+	0.997808i	$0.478922\pi$
$878$	0	0
$879$	−12.4651	−0.420439
$880$	0	0
$881$	−23.5304	−0.792761	−0.396380	−	0.918086i	$-0.629734\pi$
$881$	−23.5304	−0.792761	−0.396380	+	0.918086i	$0.629734\pi$
$882$	0	0
$883$	30.8715	1.03891	0.519454	−	0.854499i	$-0.326135\pi$
$883$	30.8715	1.03891	0.519454	+	0.854499i	$0.326135\pi$
$884$	0	0
$885$	−1.45772	−0.0490008
$886$	0	0
$887$	−15.4426	−0.518512	−0.259256	−	0.965809i	$-0.583477\pi$
$887$	−15.4426	−0.518512	−0.259256	+	0.965809i	$0.583477\pi$
$888$	0	0
$889$	20.3263	0.681721
$890$	0	0
$891$	−18.7101	−0.626811
$892$	0	0
$893$	1.93091	0.0646155
$894$	0	0
$895$	−16.0535	−0.536610
$896$	0	0
$897$	−22.3902	−0.747589
$898$	0	0
$899$	0.765405	0.0255277
$900$	0	0
$901$	5.19333	0.173015
$902$	0	0
$903$	0.255168	0.00849146
$904$	0	0
$905$	38.5245	1.28060
$906$	0	0
$907$	−18.8410	−0.625604	−0.312802	−	0.949818i	$-0.601268\pi$
$907$	−18.8410	−0.625604	−0.312802	+	0.949818i	$0.601268\pi$
$908$	0	0
$909$	−11.0952	−0.368004
$910$	0	0
$911$	40.0658	1.32744	0.663720	−	0.747981i	$-0.268977\pi$
$911$	40.0658	1.32744	0.663720	+	0.747981i	$0.268977\pi$
$912$	0	0
$913$	−28.2324	−0.934357
$914$	0	0
$915$	−10.4513	−0.345510
$916$	0	0
$917$	−1.68627	−0.0556855
$918$	0	0
$919$	−22.1658	−0.731184	−0.365592	−	0.930775i	$-0.619133\pi$
$919$	−22.1658	−0.731184	−0.365592	+	0.930775i	$0.619133\pi$
$920$	0	0
$921$	−7.01417	−0.231125
$922$	0	0
$923$	7.37185	0.242647
$924$	0	0
$925$	25.1801	0.827918
$926$	0	0
$927$	−36.6379	−1.20335
$928$	0	0
$929$	−16.2554	−0.533322	−0.266661	−	0.963790i	$-0.585920\pi$
$929$	−16.2554	−0.533322	−0.266661	+	0.963790i	$0.585920\pi$
$930$	0	0
$931$	21.2307	0.695807
$932$	0	0
$933$	−23.6366	−0.773828
$934$	0	0
$935$	−11.9345	−0.390298
$936$	0	0
$937$	−51.2139	−1.67308	−0.836542	−	0.547903i	$-0.815426\pi$
$937$	−51.2139	−1.67308	−0.836542	+	0.547903i	$0.815426\pi$
$938$	0	0
$939$	−1.59198	−0.0519523
$940$	0	0
$941$	−5.48359	−0.178760	−0.0893799	−	0.995998i	$-0.528489\pi$
$941$	−5.48359	−0.178760	−0.0893799	+	0.995998i	$0.528489\pi$
$942$	0	0
$943$	−24.6619	−0.803103
$944$	0	0
$945$	−9.88783	−0.321651
$946$	0	0
$947$	−10.7120	−0.348094	−0.174047	−	0.984737i	$-0.555685\pi$
$947$	−10.7120	−0.348094	−0.174047	+	0.984737i	$0.555685\pi$
$948$	0	0
$949$	−19.3805	−0.629118
$950$	0	0
$951$	16.1299	0.523049
$952$	0	0
$953$	−27.9499	−0.905384	−0.452692	−	0.891667i	$-0.649536\pi$
$953$	−27.9499	−0.905384	−0.452692	+	0.891667i	$0.649536\pi$
$954$	0	0
$955$	12.0529	0.390024
$956$	0	0
$957$	8.41966	0.272169
$958$	0	0
$959$	−5.87232	−0.189627
$960$	0	0
$961$	−30.8641	−0.995617
$962$	0	0
$963$	26.9512	0.868489
$964$	0	0
$965$	19.5998	0.630940
$966$	0	0
$967$	46.9393	1.50947	0.754733	−	0.656032i	$-0.227766\pi$
$967$	46.9393	1.50947	0.754733	+	0.656032i	$0.227766\pi$
$968$	0	0
$969$	−5.10913	−0.164129
$970$	0	0
$971$	−47.0455	−1.50976	−0.754881	−	0.655862i	$-0.772305\pi$
$971$	−47.0455	−1.50976	−0.754881	+	0.655862i	$0.772305\pi$
$972$	0	0
$973$	−27.6414	−0.886143
$974$	0	0
$975$	−6.37560	−0.204183
$976$	0	0
$977$	−33.5783	−1.07427	−0.537133	−	0.843497i	$-0.680493\pi$
$977$	−33.5783	−1.07427	−0.537133	+	0.843497i	$0.680493\pi$
$978$	0	0
$979$	−38.1980	−1.22081
$980$	0	0
$981$	28.4321	0.907769
$982$	0	0
$983$	8.84733	0.282186	0.141093	−	0.989996i	$-0.454938\pi$
$983$	8.84733	0.282186	0.141093	+	0.989996i	$0.454938\pi$
$984$	0	0
$985$	−16.6587	−0.530792
$986$	0	0
$987$	0.515240	0.0164003
$988$	0	0
$989$	1.90935	0.0607138
$990$	0	0
$991$	−37.5140	−1.19167	−0.595836	−	0.803106i	$-0.703179\pi$
$991$	−37.5140	−1.19167	−0.595836	+	0.803106i	$0.703179\pi$
$992$	0	0
$993$	−13.3054	−0.422235
$994$	0	0
$995$	−11.7521	−0.372566
$996$	0	0
$997$	30.5315	0.966944	0.483472	−	0.875360i	$-0.339376\pi$
$997$	30.5315	0.966944	0.483472	+	0.875360i	$0.339376\pi$
$998$	0	0
$999$	−43.6342	−1.38052

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.i.1.12	yes	30
4.3	odd	2		8032.2.a.h.1.19	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.h.1.19	✓	30	4.3	odd	2
8032.2.a.i.1.12	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)

\(f(q)\)	\(=\)	\(q-0.797542 q^{3} -1.59101 q^{5} +1.45275 q^{7} -2.36393 q^{9} +O(q^{10})\)	\(q-0.797542 q^{3} -1.59101 q^{5} +1.45275 q^{7} -2.36393 q^{9} -5.08436 q^{11} -3.23817 q^{13} +1.26890 q^{15} -1.47535 q^{17} -4.34209 q^{19} -1.15863 q^{21} -8.66973 q^{23} -2.46870 q^{25} +4.27796 q^{27} +2.07637 q^{29} +0.368627 q^{31} +4.05499 q^{33} -2.31134 q^{35} -10.1998 q^{37} +2.58258 q^{39} +2.84460 q^{41} -0.220232 q^{43} +3.76102 q^{45} -0.444696 q^{47} -4.88950 q^{49} +1.17665 q^{51} -3.52007 q^{53} +8.08925 q^{55} +3.46300 q^{57} -1.14881 q^{59} -8.23654 q^{61} -3.43421 q^{63} +5.15195 q^{65} -4.03182 q^{67} +6.91447 q^{69} -2.27655 q^{71} +5.98502 q^{73} +1.96889 q^{75} -7.38633 q^{77} -7.87656 q^{79} +3.67993 q^{81} +5.55280 q^{83} +2.34729 q^{85} -1.65599 q^{87} +7.51285 q^{89} -4.70427 q^{91} -0.293995 q^{93} +6.90830 q^{95} +0.608887 q^{97} +12.0191 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9	\( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9} + 13 q^{11} - 7 q^{13} + 28 q^{15} - 9 q^{17} - 17 q^{19} - 6 q^{21} + 43 q^{23} + 34 q^{25} + 12 q^{27} - q^{29} + 39 q^{31} + 17 q^{35} - q^{37} + 48 q^{39} - 3 q^{41} - 19 q^{43} + 66 q^{47} + 25 q^{49} - 14 q^{51} + 3 q^{53} + 50 q^{55} - 14 q^{57} + 27 q^{59} + 15 q^{61} + 75 q^{63} - 6 q^{65} + 8 q^{67} + 18 q^{69} + 64 q^{71} - 15 q^{73} + 9 q^{75} + 71 q^{79} + 6 q^{81} + 60 q^{83} + 15 q^{85} + 64 q^{87} - 32 q^{89} - 26 q^{91} - 4 q^{93} + 72 q^{95} - 4 q^{97} + 13 q^{99}+O(q^{100}) \) 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9 + 13 * q^11 - 7 * q^13 + 28 * q^15 - 9 * q^17 - 17 * q^19 - 6 * q^21 + 43 * q^23 + 34 * q^25 + 12 * q^27 - q^29 + 39 * q^31 + 17 * q^35 - q^37 + 48 * q^39 - 3 * q^41 - 19 * q^43 + 66 * q^47 + 25 * q^49 - 14 * q^51 + 3 * q^53 + 50 * q^55 - 14 * q^57 + 27 * q^59 + 15 * q^61 + 75 * q^63 - 6 * q^65 + 8 * q^67 + 18 * q^69 + 64 * q^71 - 15 * q^73 + 9 * q^75 + 71 * q^79 + 6 * q^81 + 60 * q^83 + 15 * q^85 + 64 * q^87 - 32 * q^89 - 26 * q^91 - 4 * q^93 + 72 * q^95 - 4 * q^97 + 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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