LMFDB - Embedded newform 8032.2.a.i.1.3

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			1.3
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)

$f(q)$	$=$	$q-2.87266 q^{3} +1.81575 q^{5} +4.67512 q^{7} +5.25220 q^{9} +O(q^{10})$	$q-2.87266 q^{3} +1.81575 q^{5} +4.67512 q^{7} +5.25220 q^{9} +5.05963 q^{11} -2.90118 q^{13} -5.21605 q^{15} +4.51155 q^{17} -0.989496 q^{19} -13.4300 q^{21} -3.68921 q^{23} -1.70304 q^{25} -6.46982 q^{27} +2.64753 q^{29} +6.99338 q^{31} -14.5346 q^{33} +8.48886 q^{35} +7.54712 q^{37} +8.33411 q^{39} -4.10588 q^{41} -3.93817 q^{43} +9.53670 q^{45} +12.1010 q^{47} +14.8567 q^{49} -12.9602 q^{51} -12.7083 q^{53} +9.18703 q^{55} +2.84249 q^{57} +4.97103 q^{59} -6.41226 q^{61} +24.5546 q^{63} -5.26782 q^{65} +4.76200 q^{67} +10.5979 q^{69} +3.33693 q^{71} +10.9388 q^{73} +4.89226 q^{75} +23.6543 q^{77} +14.0018 q^{79} +2.82901 q^{81} +16.1166 q^{83} +8.19185 q^{85} -7.60545 q^{87} -14.2070 q^{89} -13.5633 q^{91} -20.0896 q^{93} -1.79668 q^{95} -4.74155 q^{97} +26.5742 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.87266	−1.65853	−0.829267	−	0.558853i	$-0.811242\pi$
$3$	−2.87266	−1.65853	−0.829267	+	0.558853i	$0.811242\pi$
$4$	0	0
$5$	1.81575	0.812030	0.406015	−	0.913866i	$-0.366918\pi$
$5$	1.81575	0.812030	0.406015	+	0.913866i	$0.366918\pi$
$6$	0	0
$7$	4.67512	1.76703	0.883514	−	0.468405i	$-0.155171\pi$
$7$	4.67512	1.76703	0.883514	+	0.468405i	$0.155171\pi$
$8$	0	0
$9$	5.25220	1.75073
$10$	0	0
$11$	5.05963	1.52553	0.762767	−	0.646673i	$-0.223840\pi$
$11$	5.05963	1.52553	0.762767	+	0.646673i	$0.223840\pi$
$12$	0	0
$13$	−2.90118	−0.804642	−0.402321	−	0.915499i	$-0.631796\pi$
$13$	−2.90118	−0.804642	−0.402321	+	0.915499i	$0.631796\pi$
$14$	0	0
$15$	−5.21605	−1.34678
$16$	0	0
$17$	4.51155	1.09421	0.547105	−	0.837064i	$-0.315730\pi$
$17$	4.51155	1.09421	0.547105	+	0.837064i	$0.315730\pi$
$18$	0	0
$19$	−0.989496	−0.227006	−0.113503	−	0.993538i	$-0.536207\pi$
$19$	−0.989496	−0.227006	−0.113503	+	0.993538i	$0.536207\pi$
$20$	0	0
$21$	−13.4300	−2.93068
$22$	0	0
$23$	−3.68921	−0.769254	−0.384627	−	0.923072i	$-0.625670\pi$
$23$	−3.68921	−0.769254	−0.384627	+	0.923072i	$0.625670\pi$
$24$	0	0
$25$	−1.70304	−0.340608
$26$	0	0
$27$	−6.46982	−1.24512
$28$	0	0
$29$	2.64753	0.491633	0.245817	−	0.969316i	$-0.420944\pi$
$29$	2.64753	0.491633	0.245817	+	0.969316i	$0.420944\pi$
$30$	0	0
$31$	6.99338	1.25605	0.628024	−	0.778194i	$-0.283864\pi$
$31$	6.99338	1.25605	0.628024	+	0.778194i	$0.283864\pi$
$32$	0	0
$33$	−14.5346	−2.53015
$34$	0	0
$35$	8.48886	1.43488
$36$	0	0
$37$	7.54712	1.24074	0.620370	−	0.784310i	$-0.286983\pi$
$37$	7.54712	1.24074	0.620370	+	0.784310i	$0.286983\pi$
$38$	0	0
$39$	8.33411	1.33453
$40$	0	0
$41$	−4.10588	−0.641231	−0.320615	−	0.947209i	$-0.603890\pi$
$41$	−4.10588	−0.641231	−0.320615	+	0.947209i	$0.603890\pi$
$42$	0	0
$43$	−3.93817	−0.600566	−0.300283	−	0.953850i	$-0.597081\pi$
$43$	−3.93817	−0.600566	−0.300283	+	0.953850i	$0.597081\pi$
$44$	0	0
$45$	9.53670	1.42165
$46$	0	0
$47$	12.1010	1.76511	0.882557	−	0.470205i	$-0.155820\pi$
$47$	12.1010	1.76511	0.882557	+	0.470205i	$0.155820\pi$
$48$	0	0
$49$	14.8567	2.12239
$50$	0	0
$51$	−12.9602	−1.81478
$52$	0	0
$53$	−12.7083	−1.74561	−0.872807	−	0.488065i	$-0.837703\pi$
$53$	−12.7083	−1.74561	−0.872807	+	0.488065i	$0.837703\pi$
$54$	0	0
$55$	9.18703	1.23878
$56$	0	0
$57$	2.84249	0.376497
$58$	0	0
$59$	4.97103	0.647173	0.323587	−	0.946199i	$-0.395111\pi$
$59$	4.97103	0.647173	0.323587	+	0.946199i	$0.395111\pi$
$60$	0	0
$61$	−6.41226	−0.821006	−0.410503	−	0.911859i	$-0.634647\pi$
$61$	−6.41226	−0.821006	−0.410503	+	0.911859i	$0.634647\pi$
$62$	0	0
$63$	24.5546	3.09359
$64$	0	0
$65$	−5.26782	−0.653393
$66$	0	0
$67$	4.76200	0.581771	0.290886	−	0.956758i	$-0.406050\pi$
$67$	4.76200	0.581771	0.290886	+	0.956758i	$0.406050\pi$
$68$	0	0
$69$	10.5979	1.27583
$70$	0	0
$71$	3.33693	0.396021	0.198010	−	0.980200i	$-0.436552\pi$
$71$	3.33693	0.396021	0.198010	+	0.980200i	$0.436552\pi$
$72$	0	0
$73$	10.9388	1.28029	0.640143	−	0.768256i	$-0.278875\pi$
$73$	10.9388	1.28029	0.640143	+	0.768256i	$0.278875\pi$
$74$	0	0
$75$	4.89226	0.564909
$76$	0	0
$77$	23.6543	2.69566
$78$	0	0
$79$	14.0018	1.57533	0.787663	−	0.616106i	$-0.211291\pi$
$79$	14.0018	1.57533	0.787663	+	0.616106i	$0.211291\pi$
$80$	0	0
$81$	2.82901	0.314334
$82$	0	0
$83$	16.1166	1.76902	0.884511	−	0.466518i	$-0.154492\pi$
$83$	16.1166	1.76902	0.884511	+	0.466518i	$0.154492\pi$
$84$	0	0
$85$	8.19185	0.888531
$86$	0	0
$87$	−7.60545	−0.815390
$88$	0	0
$89$	−14.2070	−1.50594	−0.752968	−	0.658057i	$-0.771378\pi$
$89$	−14.2070	−1.50594	−0.752968	+	0.658057i	$0.771378\pi$
$90$	0	0
$91$	−13.5633	−1.42182
$92$	0	0
$93$	−20.0896	−2.08320
$94$	0	0
$95$	−1.79668	−0.184336
$96$	0	0
$97$	−4.74155	−0.481431	−0.240716	−	0.970596i	$-0.577382\pi$
$97$	−4.74155	−0.481431	−0.240716	+	0.970596i	$0.577382\pi$
$98$	0	0
$99$	26.5742	2.67080
$100$	0	0
$101$	12.9069	1.28429	0.642143	−	0.766585i	$-0.278046\pi$
$101$	12.9069	1.28429	0.642143	+	0.766585i	$0.278046\pi$
$102$	0	0
$103$	−3.68515	−0.363109	−0.181554	−	0.983381i	$-0.558113\pi$
$103$	−3.68515	−0.363109	−0.181554	+	0.983381i	$0.558113\pi$
$104$	0	0
$105$	−24.3856	−2.37980
$106$	0	0
$107$	−20.4013	−1.97227	−0.986135	−	0.165944i	$-0.946933\pi$
$107$	−20.4013	−1.97227	−0.986135	+	0.165944i	$0.946933\pi$
$108$	0	0
$109$	−17.5968	−1.68547	−0.842735	−	0.538329i	$-0.819056\pi$
$109$	−17.5968	−1.68547	−0.842735	+	0.538329i	$0.819056\pi$
$110$	0	0
$111$	−21.6804	−2.05781
$112$	0	0
$113$	7.29472	0.686229	0.343115	−	0.939294i	$-0.388518\pi$
$113$	7.29472	0.686229	0.343115	+	0.939294i	$0.388518\pi$
$114$	0	0
$115$	−6.69870	−0.624657
$116$	0	0
$117$	−15.2376	−1.40871
$118$	0	0
$119$	21.0920	1.93350
$120$	0	0
$121$	14.5998	1.32726
$122$	0	0
$123$	11.7948	1.06350
$124$	0	0
$125$	−12.1711	−1.08861
$126$	0	0
$127$	18.4857	1.64034	0.820169	−	0.572122i	$-0.193880\pi$
$127$	18.4857	1.64034	0.820169	+	0.572122i	$0.193880\pi$
$128$	0	0
$129$	11.3130	0.996058
$130$	0	0
$131$	−2.49024	−0.217574	−0.108787	−	0.994065i	$-0.534697\pi$
$131$	−2.49024	−0.217574	−0.108787	+	0.994065i	$0.534697\pi$
$132$	0	0
$133$	−4.62601	−0.401126
$134$	0	0
$135$	−11.7476	−1.01107
$136$	0	0
$137$	12.5997	1.07646	0.538231	−	0.842798i	$-0.319093\pi$
$137$	12.5997	1.07646	0.538231	+	0.842798i	$0.319093\pi$
$138$	0	0
$139$	1.89465	0.160702	0.0803509	−	0.996767i	$-0.474396\pi$
$139$	1.89465	0.160702	0.0803509	+	0.996767i	$0.474396\pi$
$140$	0	0
$141$	−34.7621	−2.92750
$142$	0	0
$143$	−14.6789	−1.22751
$144$	0	0
$145$	4.80726	0.399221
$146$	0	0
$147$	−42.6784	−3.52005
$148$	0	0
$149$	13.7638	1.12757	0.563786	−	0.825921i	$-0.309344\pi$
$149$	13.7638	1.12757	0.563786	+	0.825921i	$0.309344\pi$
$150$	0	0
$151$	−7.58452	−0.617220	−0.308610	−	0.951189i	$-0.599864\pi$
$151$	−7.58452	−0.617220	−0.308610	+	0.951189i	$0.599864\pi$
$152$	0	0
$153$	23.6955	1.91567
$154$	0	0
$155$	12.6982	1.01995
$156$	0	0
$157$	−13.9306	−1.11179	−0.555893	−	0.831254i	$-0.687623\pi$
$157$	−13.9306	−1.11179	−0.555893	+	0.831254i	$0.687623\pi$
$158$	0	0
$159$	36.5066	2.89516
$160$	0	0
$161$	−17.2475	−1.35929
$162$	0	0
$163$	18.7547	1.46898	0.734492	−	0.678618i	$-0.237421\pi$
$163$	18.7547	1.46898	0.734492	+	0.678618i	$0.237421\pi$
$164$	0	0
$165$	−26.3913	−2.05456
$166$	0	0
$167$	−12.8639	−0.995437	−0.497719	−	0.867339i	$-0.665829\pi$
$167$	−12.8639	−0.995437	−0.497719	+	0.867339i	$0.665829\pi$
$168$	0	0
$169$	−4.58317	−0.352551
$170$	0	0
$171$	−5.19703	−0.397427
$172$	0	0
$173$	−8.32288	−0.632777	−0.316389	−	0.948630i	$-0.602470\pi$
$173$	−8.32288	−0.632777	−0.316389	+	0.948630i	$0.602470\pi$
$174$	0	0
$175$	−7.96190	−0.601863
$176$	0	0
$177$	−14.2801	−1.07336
$178$	0	0
$179$	−16.3111	−1.21915	−0.609575	−	0.792729i	$-0.708660\pi$
$179$	−16.3111	−1.21915	−0.609575	+	0.792729i	$0.708660\pi$
$180$	0	0
$181$	−24.1199	−1.79282	−0.896410	−	0.443227i	$-0.853834\pi$
$181$	−24.1199	−1.79282	−0.896410	+	0.443227i	$0.853834\pi$
$182$	0	0
$183$	18.4203	1.36167
$184$	0	0
$185$	13.7037	1.00752
$186$	0	0
$187$	22.8267	1.66926
$188$	0	0
$189$	−30.2471	−2.20016
$190$	0	0
$191$	−4.99338	−0.361308	−0.180654	−	0.983547i	$-0.557821\pi$
$191$	−4.99338	−0.361308	−0.180654	+	0.983547i	$0.557821\pi$
$192$	0	0
$193$	11.1976	0.806017	0.403009	−	0.915196i	$-0.367964\pi$
$193$	11.1976	0.806017	0.403009	+	0.915196i	$0.367964\pi$
$194$	0	0
$195$	15.1327	1.08367
$196$	0	0
$197$	5.00892	0.356871	0.178435	−	0.983952i	$-0.442896\pi$
$197$	5.00892	0.356871	0.178435	+	0.983952i	$0.442896\pi$
$198$	0	0
$199$	−12.5833	−0.892009	−0.446004	−	0.895031i	$-0.647153\pi$
$199$	−12.5833	−0.892009	−0.446004	+	0.895031i	$0.647153\pi$
$200$	0	0
$201$	−13.6796	−0.964887
$202$	0	0
$203$	12.3775	0.868730
$204$	0	0
$205$	−7.45527	−0.520698
$206$	0	0
$207$	−19.3765	−1.34676
$208$	0	0
$209$	−5.00648	−0.346305
$210$	0	0
$211$	−16.5945	−1.14241	−0.571205	−	0.820808i	$-0.693524\pi$
$211$	−16.5945	−1.14241	−0.571205	+	0.820808i	$0.693524\pi$
$212$	0	0
$213$	−9.58588	−0.656814
$214$	0	0
$215$	−7.15075	−0.487677
$216$	0	0
$217$	32.6948	2.21947
$218$	0	0
$219$	−31.4234	−2.12340
$220$	0	0
$221$	−13.0888	−0.880448
$222$	0	0
$223$	−1.03756	−0.0694803	−0.0347402	−	0.999396i	$-0.511060\pi$
$223$	−1.03756	−0.0694803	−0.0347402	+	0.999396i	$0.511060\pi$
$224$	0	0
$225$	−8.94470	−0.596313
$226$	0	0
$227$	−1.57694	−0.104665	−0.0523326	−	0.998630i	$-0.516666\pi$
$227$	−1.57694	−0.104665	−0.0523326	+	0.998630i	$0.516666\pi$
$228$	0	0
$229$	−1.93630	−0.127954	−0.0639770	−	0.997951i	$-0.520378\pi$
$229$	−1.93630	−0.127954	−0.0639770	+	0.997951i	$0.520378\pi$
$230$	0	0
$231$	−67.9510	−4.47085
$232$	0	0
$233$	−15.6854	−1.02759	−0.513794	−	0.857914i	$-0.671760\pi$
$233$	−15.6854	−1.02759	−0.513794	+	0.857914i	$0.671760\pi$
$234$	0	0
$235$	21.9725	1.43333
$236$	0	0
$237$	−40.2225	−2.61273
$238$	0	0
$239$	3.42958	0.221841	0.110920	−	0.993829i	$-0.464620\pi$
$239$	3.42958	0.221841	0.110920	+	0.993829i	$0.464620\pi$
$240$	0	0
$241$	−0.200575	−0.0129202	−0.00646009	−	0.999979i	$-0.502056\pi$
$241$	−0.200575	−0.0129202	−0.00646009	+	0.999979i	$0.502056\pi$
$242$	0	0
$243$	11.2827	0.723783
$244$	0	0
$245$	26.9761	1.72344
$246$	0	0
$247$	2.87070	0.182659
$248$	0	0
$249$	−46.2975	−2.93398
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−18.6660	−1.17352
$254$	0	0
$255$	−23.5324	−1.47366
$256$	0	0
$257$	−12.7334	−0.794290	−0.397145	−	0.917756i	$-0.629999\pi$
$257$	−12.7334	−0.794290	−0.397145	+	0.917756i	$0.629999\pi$
$258$	0	0
$259$	35.2837	2.19242
$260$	0	0
$261$	13.9053	0.860719
$262$	0	0
$263$	17.6539	1.08859	0.544294	−	0.838894i	$-0.316797\pi$
$263$	17.6539	1.08859	0.544294	+	0.838894i	$0.316797\pi$
$264$	0	0
$265$	−23.0751	−1.41749
$266$	0	0
$267$	40.8119	2.49765
$268$	0	0
$269$	−29.8350	−1.81907	−0.909535	−	0.415627i	$-0.863562\pi$
$269$	−29.8350	−1.81907	−0.909535	+	0.415627i	$0.863562\pi$
$270$	0	0
$271$	12.5114	0.760011	0.380005	−	0.924984i	$-0.375922\pi$
$271$	12.5114	0.760011	0.380005	+	0.924984i	$0.375922\pi$
$272$	0	0
$273$	38.9629	2.35814
$274$	0	0
$275$	−8.61674	−0.519609
$276$	0	0
$277$	−2.45010	−0.147212	−0.0736060	−	0.997287i	$-0.523451\pi$
$277$	−2.45010	−0.147212	−0.0736060	+	0.997287i	$0.523451\pi$
$278$	0	0
$279$	36.7306	2.19900
$280$	0	0
$281$	15.9705	0.952721	0.476361	−	0.879250i	$-0.341956\pi$
$281$	15.9705	0.952721	0.476361	+	0.879250i	$0.341956\pi$
$282$	0	0
$283$	8.23392	0.489456	0.244728	−	0.969592i	$-0.421301\pi$
$283$	8.23392	0.489456	0.244728	+	0.969592i	$0.421301\pi$
$284$	0	0
$285$	5.16126	0.305727
$286$	0	0
$287$	−19.1955	−1.13307
$288$	0	0
$289$	3.35404	0.197296
$290$	0	0
$291$	13.6209	0.798470
$292$	0	0
$293$	27.0487	1.58020	0.790101	−	0.612977i	$-0.210028\pi$
$293$	27.0487	1.58020	0.790101	+	0.612977i	$0.210028\pi$
$294$	0	0
$295$	9.02617	0.525524
$296$	0	0
$297$	−32.7348	−1.89947
$298$	0	0
$299$	10.7031	0.618974
$300$	0	0
$301$	−18.4114	−1.06122
$302$	0	0
$303$	−37.0772	−2.13003
$304$	0	0
$305$	−11.6431	−0.666681
$306$	0	0
$307$	29.4892	1.68304	0.841518	−	0.540229i	$-0.181662\pi$
$307$	29.4892	1.68304	0.841518	+	0.540229i	$0.181662\pi$
$308$	0	0
$309$	10.5862	0.602228
$310$	0	0
$311$	−19.7995	−1.12273	−0.561364	−	0.827569i	$-0.689723\pi$
$311$	−19.7995	−1.12273	−0.561364	+	0.827569i	$0.689723\pi$
$312$	0	0
$313$	18.8179	1.06365	0.531826	−	0.846854i	$-0.321506\pi$
$313$	18.8179	1.06365	0.531826	+	0.846854i	$0.321506\pi$
$314$	0	0
$315$	44.5852	2.51209
$316$	0	0
$317$	−3.75954	−0.211157	−0.105578	−	0.994411i	$-0.533669\pi$
$317$	−3.75954	−0.211157	−0.105578	+	0.994411i	$0.533669\pi$
$318$	0	0
$319$	13.3955	0.750004
$320$	0	0
$321$	58.6062	3.27108
$322$	0	0
$323$	−4.46416	−0.248392
$324$	0	0
$325$	4.94082	0.274067
$326$	0	0
$327$	50.5498	2.79541
$328$	0	0
$329$	56.5736	3.11901
$330$	0	0
$331$	−9.24412	−0.508103	−0.254051	−	0.967191i	$-0.581763\pi$
$331$	−9.24412	−0.508103	−0.254051	+	0.967191i	$0.581763\pi$
$332$	0	0
$333$	39.6390	2.17220
$334$	0	0
$335$	8.64662	0.472415
$336$	0	0
$337$	−6.64871	−0.362179	−0.181089	−	0.983467i	$-0.557962\pi$
$337$	−6.64871	−0.362179	−0.181089	+	0.983467i	$0.557962\pi$
$338$	0	0
$339$	−20.9553	−1.13813
$340$	0	0
$341$	35.3839	1.91614
$342$	0	0
$343$	36.7311	1.98329
$344$	0	0
$345$	19.2431	1.03601
$346$	0	0
$347$	−22.4275	−1.20397	−0.601985	−	0.798507i	$-0.705623\pi$
$347$	−22.4275	−1.20397	−0.601985	+	0.798507i	$0.705623\pi$
$348$	0	0
$349$	25.6845	1.37486	0.687429	−	0.726251i	$-0.258739\pi$
$349$	25.6845	1.37486	0.687429	+	0.726251i	$0.258739\pi$
$350$	0	0
$351$	18.7701	1.00187
$352$	0	0
$353$	14.0398	0.747265	0.373632	−	0.927577i	$-0.378112\pi$
$353$	14.0398	0.747265	0.373632	+	0.927577i	$0.378112\pi$
$354$	0	0
$355$	6.05904	0.321581
$356$	0	0
$357$	−60.5902	−3.20678
$358$	0	0
$359$	−13.2586	−0.699763	−0.349881	−	0.936794i	$-0.613778\pi$
$359$	−13.2586	−0.699763	−0.349881	+	0.936794i	$0.613778\pi$
$360$	0	0
$361$	−18.0209	−0.948468
$362$	0	0
$363$	−41.9404	−2.20130
$364$	0	0
$365$	19.8621	1.03963
$366$	0	0
$367$	−7.60236	−0.396840	−0.198420	−	0.980117i	$-0.563581\pi$
$367$	−7.60236	−0.396840	−0.198420	+	0.980117i	$0.563581\pi$
$368$	0	0
$369$	−21.5649	−1.12262
$370$	0	0
$371$	−59.4126	−3.08455
$372$	0	0
$373$	29.9391	1.55019	0.775093	−	0.631847i	$-0.217703\pi$
$373$	29.9391	1.55019	0.775093	+	0.631847i	$0.217703\pi$
$374$	0	0
$375$	34.9634	1.80550
$376$	0	0
$377$	−7.68094	−0.395589
$378$	0	0
$379$	6.77355	0.347934	0.173967	−	0.984751i	$-0.444341\pi$
$379$	6.77355	0.347934	0.173967	+	0.984751i	$0.444341\pi$
$380$	0	0
$381$	−53.1031	−2.72055
$382$	0	0
$383$	−26.0716	−1.33220	−0.666098	−	0.745865i	$-0.732037\pi$
$383$	−26.0716	−1.33220	−0.666098	+	0.745865i	$0.732037\pi$
$384$	0	0
$385$	42.9505	2.18896
$386$	0	0
$387$	−20.6841	−1.05143
$388$	0	0
$389$	−1.75574	−0.0890196	−0.0445098	−	0.999009i	$-0.514173\pi$
$389$	−1.75574	−0.0890196	−0.0445098	+	0.999009i	$0.514173\pi$
$390$	0	0
$391$	−16.6440	−0.841726
$392$	0	0
$393$	7.15363	0.360853
$394$	0	0
$395$	25.4238	1.27921
$396$	0	0
$397$	−6.34362	−0.318377	−0.159189	−	0.987248i	$-0.550888\pi$
$397$	−6.34362	−0.318377	−0.159189	+	0.987248i	$0.550888\pi$
$398$	0	0
$399$	13.2890	0.665281
$400$	0	0
$401$	−11.2456	−0.561579	−0.280790	−	0.959769i	$-0.590596\pi$
$401$	−11.2456	−0.561579	−0.280790	+	0.959769i	$0.590596\pi$
$402$	0	0
$403$	−20.2890	−1.01067
$404$	0	0
$405$	5.13678	0.255249
$406$	0	0
$407$	38.1856	1.89279
$408$	0	0
$409$	−14.8109	−0.732352	−0.366176	−	0.930546i	$-0.619333\pi$
$409$	−14.8109	−0.732352	−0.366176	+	0.930546i	$0.619333\pi$
$410$	0	0
$411$	−36.1946	−1.78535
$412$	0	0
$413$	23.2402	1.14357
$414$	0	0
$415$	29.2637	1.43650
$416$	0	0
$417$	−5.44268	−0.266529
$418$	0	0
$419$	4.16284	0.203368	0.101684	−	0.994817i	$-0.467577\pi$
$419$	4.16284	0.203368	0.101684	+	0.994817i	$0.467577\pi$
$420$	0	0
$421$	10.5040	0.511935	0.255968	−	0.966685i	$-0.417606\pi$
$421$	10.5040	0.511935	0.255968	+	0.966685i	$0.417606\pi$
$422$	0	0
$423$	63.5569	3.09024
$424$	0	0
$425$	−7.68334	−0.372696
$426$	0	0
$427$	−29.9781	−1.45074
$428$	0	0
$429$	42.1675	2.03586
$430$	0	0
$431$	28.3778	1.36691	0.683455	−	0.729992i	$-0.260476\pi$
$431$	28.3778	1.36691	0.683455	+	0.729992i	$0.260476\pi$
$432$	0	0
$433$	19.3881	0.931731	0.465866	−	0.884855i	$-0.345743\pi$
$433$	19.3881	0.931731	0.465866	+	0.884855i	$0.345743\pi$
$434$	0	0
$435$	−13.8096	−0.662121
$436$	0	0
$437$	3.65046	0.174625
$438$	0	0
$439$	22.7003	1.08343	0.541713	−	0.840564i	$-0.317776\pi$
$439$	22.7003	1.08343	0.541713	+	0.840564i	$0.317776\pi$
$440$	0	0
$441$	78.0304	3.71574
$442$	0	0
$443$	29.6313	1.40783	0.703914	−	0.710285i	$-0.251434\pi$
$443$	29.6313	1.40783	0.703914	+	0.710285i	$0.251434\pi$
$444$	0	0
$445$	−25.7964	−1.22286
$446$	0	0
$447$	−39.5387	−1.87012
$448$	0	0
$449$	14.0777	0.664369	0.332184	−	0.943214i	$-0.392214\pi$
$449$	14.0777	0.664369	0.332184	+	0.943214i	$0.392214\pi$
$450$	0	0
$451$	−20.7742	−0.978220
$452$	0	0
$453$	21.7878	1.02368
$454$	0	0
$455$	−24.6277	−1.15456
$456$	0	0
$457$	12.2640	0.573684	0.286842	−	0.957978i	$-0.407394\pi$
$457$	12.2640	0.573684	0.286842	+	0.957978i	$0.407394\pi$
$458$	0	0
$459$	−29.1889	−1.36242
$460$	0	0
$461$	21.4353	0.998342	0.499171	−	0.866503i	$-0.333638\pi$
$461$	21.4353	0.998342	0.499171	+	0.866503i	$0.333638\pi$
$462$	0	0
$463$	−33.9641	−1.57844	−0.789222	−	0.614108i	$-0.789516\pi$
$463$	−33.9641	−1.57844	−0.789222	+	0.614108i	$0.789516\pi$
$464$	0	0
$465$	−36.4778	−1.69162
$466$	0	0
$467$	−23.7415	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$467$	−23.7415	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$468$	0	0
$469$	22.2629	1.02801
$470$	0	0
$471$	40.0180	1.84393
$472$	0	0
$473$	−19.9257	−0.916184
$474$	0	0
$475$	1.68515	0.0773200
$476$	0	0
$477$	−66.7464	−3.05611
$478$	0	0
$479$	27.6792	1.26470	0.632348	−	0.774684i	$-0.282091\pi$
$479$	27.6792	1.26470	0.632348	+	0.774684i	$0.282091\pi$
$480$	0	0
$481$	−21.8955	−0.998351
$482$	0	0
$483$	49.5463	2.25443
$484$	0	0
$485$	−8.60949	−0.390937
$486$	0	0
$487$	−4.76376	−0.215867	−0.107933	−	0.994158i	$-0.534423\pi$
$487$	−4.76376	−0.215867	−0.107933	+	0.994158i	$0.534423\pi$
$488$	0	0
$489$	−53.8760	−2.43636
$490$	0	0
$491$	−1.85951	−0.0839185	−0.0419593	−	0.999119i	$-0.513360\pi$
$491$	−1.85951	−0.0839185	−0.0419593	+	0.999119i	$0.513360\pi$
$492$	0	0
$493$	11.9444	0.537950
$494$	0	0
$495$	48.2521	2.16877
$496$	0	0
$497$	15.6005	0.699780
$498$	0	0
$499$	23.3689	1.04613	0.523067	−	0.852292i	$-0.324788\pi$
$499$	23.3689	1.04613	0.523067	+	0.852292i	$0.324788\pi$
$500$	0	0
$501$	36.9536	1.65097
$502$	0	0
$503$	−22.0081	−0.981291	−0.490646	−	0.871359i	$-0.663239\pi$
$503$	−22.0081	−0.981291	−0.490646	+	0.871359i	$0.663239\pi$
$504$	0	0
$505$	23.4358	1.04288
$506$	0	0
$507$	13.1659	0.584718
$508$	0	0
$509$	6.47511	0.287004	0.143502	−	0.989650i	$-0.454164\pi$
$509$	6.47511	0.287004	0.143502	+	0.989650i	$0.454164\pi$
$510$	0	0
$511$	51.1400	2.26230
$512$	0	0
$513$	6.40186	0.282649
$514$	0	0
$515$	−6.69133	−0.294855
$516$	0	0
$517$	61.2266	2.69274
$518$	0	0
$519$	23.9089	1.04948
$520$	0	0
$521$	9.45572	0.414263	0.207131	−	0.978313i	$-0.433587\pi$
$521$	9.45572	0.414263	0.207131	+	0.978313i	$0.433587\pi$
$522$	0	0
$523$	−10.6959	−0.467700	−0.233850	−	0.972273i	$-0.575133\pi$
$523$	−10.6959	−0.467700	−0.233850	+	0.972273i	$0.575133\pi$
$524$	0	0
$525$	22.8719	0.998210
$526$	0	0
$527$	31.5509	1.37438
$528$	0	0
$529$	−9.38972	−0.408249
$530$	0	0
$531$	26.1089	1.13303
$532$	0	0
$533$	11.9119	0.515961
$534$	0	0
$535$	−37.0438	−1.60154
$536$	0	0
$537$	46.8563	2.02200
$538$	0	0
$539$	75.1694	3.23778
$540$	0	0
$541$	−2.48559	−0.106864	−0.0534320	−	0.998571i	$-0.517016\pi$
$541$	−2.48559	−0.106864	−0.0534320	+	0.998571i	$0.517016\pi$
$542$	0	0
$543$	69.2884	2.97345
$544$	0	0
$545$	−31.9515	−1.36865
$546$	0	0
$547$	−5.84521	−0.249923	−0.124962	−	0.992162i	$-0.539881\pi$
$547$	−5.84521	−0.249923	−0.124962	+	0.992162i	$0.539881\pi$
$548$	0	0
$549$	−33.6785	−1.43736
$550$	0	0
$551$	−2.61972	−0.111604
$552$	0	0
$553$	65.4601	2.78365
$554$	0	0
$555$	−39.3662	−1.67100
$556$	0	0
$557$	−10.2895	−0.435980	−0.217990	−	0.975951i	$-0.569950\pi$
$557$	−10.2895	−0.435980	−0.217990	+	0.975951i	$0.569950\pi$
$558$	0	0
$559$	11.4253	0.483240
$560$	0	0
$561$	−65.5735	−2.76852
$562$	0	0
$563$	−39.5695	−1.66766	−0.833828	−	0.552024i	$-0.813855\pi$
$563$	−39.5695	−1.66766	−0.833828	+	0.552024i	$0.813855\pi$
$564$	0	0
$565$	13.2454	0.557239
$566$	0	0
$567$	13.2259	0.555437
$568$	0	0
$569$	−17.6349	−0.739293	−0.369647	−	0.929172i	$-0.620521\pi$
$569$	−17.6349	−0.739293	−0.369647	+	0.929172i	$0.620521\pi$
$570$	0	0
$571$	7.69870	0.322181	0.161090	−	0.986940i	$-0.448499\pi$
$571$	7.69870	0.322181	0.161090	+	0.986940i	$0.448499\pi$
$572$	0	0
$573$	14.3443	0.599242
$574$	0	0
$575$	6.28287	0.262014
$576$	0	0
$577$	−33.2437	−1.38395	−0.691977	−	0.721920i	$-0.743260\pi$
$577$	−33.2437	−1.38395	−0.691977	+	0.721920i	$0.743260\pi$
$578$	0	0
$579$	−32.1668	−1.33681
$580$	0	0
$581$	75.3468	3.12591
$582$	0	0
$583$	−64.2991	−2.66300
$584$	0	0
$585$	−27.6677	−1.14392
$586$	0	0
$587$	−7.37181	−0.304267	−0.152134	−	0.988360i	$-0.548614\pi$
$587$	−7.37181	−0.304267	−0.152134	+	0.988360i	$0.548614\pi$
$588$	0	0
$589$	−6.91992	−0.285130
$590$	0	0
$591$	−14.3890	−0.591882
$592$	0	0
$593$	5.23742	0.215075	0.107537	−	0.994201i	$-0.465703\pi$
$593$	5.23742	0.215075	0.107537	+	0.994201i	$0.465703\pi$
$594$	0	0
$595$	38.2979	1.57006
$596$	0	0
$597$	36.1477	1.47943
$598$	0	0
$599$	39.1935	1.60140	0.800701	−	0.599065i	$-0.204461\pi$
$599$	39.1935	1.60140	0.800701	+	0.599065i	$0.204461\pi$
$600$	0	0
$601$	15.9273	0.649690	0.324845	−	0.945767i	$-0.394688\pi$
$601$	15.9273	0.649690	0.324845	+	0.945767i	$0.394688\pi$
$602$	0	0
$603$	25.0110	1.01853
$604$	0	0
$605$	26.5097	1.07777
$606$	0	0
$607$	−29.2907	−1.18887	−0.594437	−	0.804142i	$-0.702625\pi$
$607$	−29.2907	−1.18887	−0.594437	+	0.804142i	$0.702625\pi$
$608$	0	0
$609$	−35.5564	−1.44082
$610$	0	0
$611$	−35.1072	−1.42028
$612$	0	0
$613$	−27.5459	−1.11257	−0.556285	−	0.830992i	$-0.687774\pi$
$613$	−27.5459	−1.11257	−0.556285	+	0.830992i	$0.687774\pi$
$614$	0	0
$615$	21.4165	0.863596
$616$	0	0
$617$	21.2705	0.856319	0.428160	−	0.903703i	$-0.359162\pi$
$617$	21.2705	0.856319	0.428160	+	0.903703i	$0.359162\pi$
$618$	0	0
$619$	9.00454	0.361923	0.180962	−	0.983490i	$-0.442079\pi$
$619$	9.00454	0.361923	0.180962	+	0.983490i	$0.442079\pi$
$620$	0	0
$621$	23.8685	0.957811
$622$	0	0
$623$	−66.4192	−2.66103
$624$	0	0
$625$	−13.5845	−0.543379
$626$	0	0
$627$	14.3819	0.574359
$628$	0	0
$629$	34.0492	1.35763
$630$	0	0
$631$	−20.9592	−0.834373	−0.417186	−	0.908821i	$-0.636984\pi$
$631$	−20.9592	−0.834373	−0.417186	+	0.908821i	$0.636984\pi$
$632$	0	0
$633$	47.6703	1.89472
$634$	0	0
$635$	33.5654	1.33200
$636$	0	0
$637$	−43.1020	−1.70776
$638$	0	0
$639$	17.5262	0.693327
$640$	0	0
$641$	4.70090	0.185674	0.0928372	−	0.995681i	$-0.470406\pi$
$641$	4.70090	0.185674	0.0928372	+	0.995681i	$0.470406\pi$
$642$	0	0
$643$	−17.2442	−0.680047	−0.340023	−	0.940417i	$-0.610435\pi$
$643$	−17.2442	−0.680047	−0.340023	+	0.940417i	$0.610435\pi$
$644$	0	0
$645$	20.5417	0.808829
$646$	0	0
$647$	−16.1341	−0.634298	−0.317149	−	0.948376i	$-0.602725\pi$
$647$	−16.1341	−0.634298	−0.317149	+	0.948376i	$0.602725\pi$
$648$	0	0
$649$	25.1516	0.987285
$650$	0	0
$651$	−93.9213	−3.68107
$652$	0	0
$653$	7.05473	0.276073	0.138036	−	0.990427i	$-0.455921\pi$
$653$	7.05473	0.276073	0.138036	+	0.990427i	$0.455921\pi$
$654$	0	0
$655$	−4.52167	−0.176676
$656$	0	0
$657$	57.4526	2.24144
$658$	0	0
$659$	−13.9805	−0.544604	−0.272302	−	0.962212i	$-0.587785\pi$
$659$	−13.9805	−0.544604	−0.272302	+	0.962212i	$0.587785\pi$
$660$	0	0
$661$	−15.2429	−0.592880	−0.296440	−	0.955051i	$-0.595800\pi$
$661$	−15.2429	−0.592880	−0.296440	+	0.955051i	$0.595800\pi$
$662$	0	0
$663$	37.5997	1.46025
$664$	0	0
$665$	−8.39969	−0.325726
$666$	0	0
$667$	−9.76728	−0.378191
$668$	0	0
$669$	2.98057	0.115235
$670$	0	0
$671$	−32.4436	−1.25247
$672$	0	0
$673$	7.68714	0.296318	0.148159	−	0.988964i	$-0.452665\pi$
$673$	7.68714	0.296318	0.148159	+	0.988964i	$0.452665\pi$
$674$	0	0
$675$	11.0183	0.424096
$676$	0	0
$677$	41.9267	1.61138	0.805688	−	0.592340i	$-0.201796\pi$
$677$	41.9267	1.61138	0.805688	+	0.592340i	$0.201796\pi$
$678$	0	0
$679$	−22.1673	−0.850703
$680$	0	0
$681$	4.53002	0.173591
$682$	0	0
$683$	51.6478	1.97625	0.988124	−	0.153656i	$-0.0491048\pi$
$683$	51.6478	1.97625	0.988124	+	0.153656i	$0.0491048\pi$
$684$	0	0
$685$	22.8779	0.874119
$686$	0	0
$687$	5.56233	0.212216
$688$	0	0
$689$	36.8689	1.40459
$690$	0	0
$691$	−19.1329	−0.727850	−0.363925	−	0.931428i	$-0.618564\pi$
$691$	−19.1329	−0.727850	−0.363925	+	0.931428i	$0.618564\pi$
$692$	0	0
$693$	124.237	4.71939
$694$	0	0
$695$	3.44021	0.130495
$696$	0	0
$697$	−18.5239	−0.701641
$698$	0	0
$699$	45.0590	1.70429
$700$	0	0
$701$	2.27895	0.0860748	0.0430374	−	0.999073i	$-0.486297\pi$
$701$	2.27895	0.0860748	0.0430374	+	0.999073i	$0.486297\pi$
$702$	0	0
$703$	−7.46785	−0.281655
$704$	0	0
$705$	−63.1195	−2.37722
$706$	0	0
$707$	60.3413	2.26937
$708$	0	0
$709$	−7.09774	−0.266561	−0.133281	−	0.991078i	$-0.542551\pi$
$709$	−7.09774	−0.266561	−0.133281	+	0.991078i	$0.542551\pi$
$710$	0	0
$711$	73.5403	2.75798
$712$	0	0
$713$	−25.8000	−0.966219
$714$	0	0
$715$	−26.6532	−0.996774
$716$	0	0
$717$	−9.85202	−0.367931
$718$	0	0
$719$	−50.3548	−1.87792	−0.938959	−	0.344030i	$-0.888208\pi$
$719$	−50.3548	−1.87792	−0.938959	+	0.344030i	$0.888208\pi$
$720$	0	0
$721$	−17.2285	−0.641623
$722$	0	0
$723$	0.576185	0.0214285
$724$	0	0
$725$	−4.50884	−0.167454
$726$	0	0
$727$	−43.6201	−1.61778	−0.808890	−	0.587961i	$-0.799931\pi$
$727$	−43.6201	−1.61778	−0.808890	+	0.587961i	$0.799931\pi$
$728$	0	0
$729$	−40.8983	−1.51475
$730$	0	0
$731$	−17.7672	−0.657145
$732$	0	0
$733$	3.41803	0.126248	0.0631239	−	0.998006i	$-0.479894\pi$
$733$	3.41803	0.126248	0.0631239	+	0.998006i	$0.479894\pi$
$734$	0	0
$735$	−77.4934	−2.85839
$736$	0	0
$737$	24.0939	0.887512
$738$	0	0
$739$	−42.3615	−1.55829	−0.779147	−	0.626841i	$-0.784347\pi$
$739$	−42.3615	−1.55829	−0.779147	+	0.626841i	$0.784347\pi$
$740$	0	0
$741$	−8.24657	−0.302945
$742$	0	0
$743$	23.9988	0.880431	0.440216	−	0.897892i	$-0.354902\pi$
$743$	23.9988	0.880431	0.440216	+	0.897892i	$0.354902\pi$
$744$	0	0
$745$	24.9916	0.915622
$746$	0	0
$747$	84.6474	3.09709
$748$	0	0
$749$	−95.3786	−3.48506
$750$	0	0
$751$	37.2428	1.35901	0.679504	−	0.733672i	$-0.262195\pi$
$751$	37.2428	1.35901	0.679504	+	0.733672i	$0.262195\pi$
$752$	0	0
$753$	−2.87266	−0.104686
$754$	0	0
$755$	−13.7716	−0.501201
$756$	0	0
$757$	43.5458	1.58270	0.791350	−	0.611363i	$-0.209379\pi$
$757$	43.5458	1.58270	0.791350	+	0.611363i	$0.209379\pi$
$758$	0	0
$759$	53.6212	1.94633
$760$	0	0
$761$	6.64205	0.240774	0.120387	−	0.992727i	$-0.461586\pi$
$761$	6.64205	0.240774	0.120387	+	0.992727i	$0.461586\pi$
$762$	0	0
$763$	−82.2672	−2.97827
$764$	0	0
$765$	43.0253	1.55558
$766$	0	0
$767$	−14.4218	−0.520743
$768$	0	0
$769$	−1.29794	−0.0468048	−0.0234024	−	0.999726i	$-0.507450\pi$
$769$	−1.29794	−0.0468048	−0.0234024	+	0.999726i	$0.507450\pi$
$770$	0	0
$771$	36.5789	1.31736
$772$	0	0
$773$	0.630760	0.0226869	0.0113434	−	0.999936i	$-0.496389\pi$
$773$	0.630760	0.0226869	0.0113434	+	0.999936i	$0.496389\pi$
$774$	0	0
$775$	−11.9100	−0.427819
$776$	0	0
$777$	−101.358	−3.63620
$778$	0	0
$779$	4.06275	0.145563
$780$	0	0
$781$	16.8836	0.604143
$782$	0	0
$783$	−17.1290	−0.612141
$784$	0	0
$785$	−25.2946	−0.902803
$786$	0	0
$787$	35.6524	1.27087	0.635435	−	0.772154i	$-0.280821\pi$
$787$	35.6524	1.27087	0.635435	+	0.772154i	$0.280821\pi$
$788$	0	0
$789$	−50.7139	−1.80546
$790$	0	0
$791$	34.1037	1.21259
$792$	0	0
$793$	18.6031	0.660616
$794$	0	0
$795$	66.2870	2.35096
$796$	0	0
$797$	−26.8409	−0.950752	−0.475376	−	0.879783i	$-0.657688\pi$
$797$	−26.8409	−0.950752	−0.475376	+	0.879783i	$0.657688\pi$
$798$	0	0
$799$	54.5943	1.93141
$800$	0	0
$801$	−74.6179	−2.63649
$802$	0	0
$803$	55.3461	1.95312
$804$	0	0
$805$	−31.3172	−1.10379
$806$	0	0
$807$	85.7059	3.01699
$808$	0	0
$809$	33.5709	1.18029	0.590146	−	0.807297i	$-0.299070\pi$
$809$	33.5709	1.18029	0.590146	+	0.807297i	$0.299070\pi$
$810$	0	0
$811$	−14.5521	−0.510993	−0.255496	−	0.966810i	$-0.582239\pi$
$811$	−14.5521	−0.510993	−0.255496	+	0.966810i	$0.582239\pi$
$812$	0	0
$813$	−35.9409	−1.26050
$814$	0	0
$815$	34.0540	1.19286
$816$	0	0
$817$	3.89681	0.136332
$818$	0	0
$819$	−71.2374	−2.48924
$820$	0	0
$821$	−7.65458	−0.267147	−0.133573	−	0.991039i	$-0.542645\pi$
$821$	−7.65458	−0.267147	−0.133573	+	0.991039i	$0.542645\pi$
$822$	0	0
$823$	−15.7298	−0.548308	−0.274154	−	0.961686i	$-0.588398\pi$
$823$	−15.7298	−0.548308	−0.274154	+	0.961686i	$0.588398\pi$
$824$	0	0
$825$	24.7530	0.861789
$826$	0	0
$827$	12.9986	0.452006	0.226003	−	0.974127i	$-0.427434\pi$
$827$	12.9986	0.452006	0.226003	+	0.974127i	$0.427434\pi$
$828$	0	0
$829$	−16.9179	−0.587582	−0.293791	−	0.955870i	$-0.594917\pi$
$829$	−16.9179	−0.587582	−0.293791	+	0.955870i	$0.594917\pi$
$830$	0	0
$831$	7.03831	0.244156
$832$	0	0
$833$	67.0267	2.32234
$834$	0	0
$835$	−23.3576	−0.808325
$836$	0	0
$837$	−45.2459	−1.56393
$838$	0	0
$839$	−19.7891	−0.683195	−0.341598	−	0.939846i	$-0.610968\pi$
$839$	−19.7891	−0.683195	−0.341598	+	0.939846i	$0.610968\pi$
$840$	0	0
$841$	−21.9906	−0.758297
$842$	0	0
$843$	−45.8779	−1.58012
$844$	0	0
$845$	−8.32190	−0.286282
$846$	0	0
$847$	68.2558	2.34530
$848$	0	0
$849$	−23.6533	−0.811779
$850$	0	0
$851$	−27.8429	−0.954443
$852$	0	0
$853$	−30.0724	−1.02966	−0.514830	−	0.857293i	$-0.672145\pi$
$853$	−30.0724	−1.02966	−0.514830	+	0.857293i	$0.672145\pi$
$854$	0	0
$855$	−9.43653	−0.322723
$856$	0	0
$857$	−17.3011	−0.590993	−0.295496	−	0.955344i	$-0.595485\pi$
$857$	−17.3011	−0.590993	−0.295496	+	0.955344i	$0.595485\pi$
$858$	0	0
$859$	−6.99912	−0.238807	−0.119404	−	0.992846i	$-0.538098\pi$
$859$	−6.99912	−0.238807	−0.119404	+	0.992846i	$0.538098\pi$
$860$	0	0
$861$	55.1421	1.87924
$862$	0	0
$863$	0.258179	0.00878852	0.00439426	−	0.999990i	$-0.498601\pi$
$863$	0.258179	0.00878852	0.00439426	+	0.999990i	$0.498601\pi$
$864$	0	0
$865$	−15.1123	−0.513834
$866$	0	0
$867$	−9.63503	−0.327223
$868$	0	0
$869$	70.8439	2.40322
$870$	0	0
$871$	−13.8154	−0.468117
$872$	0	0
$873$	−24.9036	−0.842858
$874$	0	0
$875$	−56.9012	−1.92361
$876$	0	0
$877$	54.3625	1.83569	0.917846	−	0.396937i	$-0.129927\pi$
$877$	54.3625	1.83569	0.917846	+	0.396937i	$0.129927\pi$
$878$	0	0
$879$	−77.7018	−2.62082
$880$	0	0
$881$	41.3117	1.39183	0.695913	−	0.718126i	$-0.255000\pi$
$881$	41.3117	1.39183	0.695913	+	0.718126i	$0.255000\pi$
$882$	0	0
$883$	−19.7050	−0.663127	−0.331563	−	0.943433i	$-0.607576\pi$
$883$	−19.7050	−0.663127	−0.331563	+	0.943433i	$0.607576\pi$
$884$	0	0
$885$	−25.9292	−0.871599
$886$	0	0
$887$	−8.75605	−0.293999	−0.147000	−	0.989137i	$-0.546962\pi$
$887$	−8.75605	−0.293999	−0.147000	+	0.989137i	$0.546962\pi$
$888$	0	0
$889$	86.4226	2.89852
$890$	0	0
$891$	14.3137	0.479528
$892$	0	0
$893$	−11.9739	−0.400691
$894$	0	0
$895$	−29.6170	−0.989986
$896$	0	0
$897$	−30.7463	−1.02659
$898$	0	0
$899$	18.5151	0.617515
$900$	0	0
$901$	−57.3339	−1.91007
$902$	0	0
$903$	52.8898	1.76006
$904$	0	0
$905$	−43.7958	−1.45582
$906$	0	0
$907$	3.30854	0.109858	0.0549291	−	0.998490i	$-0.482507\pi$
$907$	3.30854	0.109858	0.0549291	+	0.998490i	$0.482507\pi$
$908$	0	0
$909$	67.7897	2.24844
$910$	0	0
$911$	58.8790	1.95075	0.975374	−	0.220556i	$-0.0707871\pi$
$911$	58.8790	1.95075	0.975374	+	0.220556i	$0.0707871\pi$
$912$	0	0
$913$	81.5438	2.69871
$914$	0	0
$915$	33.4467	1.10571
$916$	0	0
$917$	−11.6422	−0.384458
$918$	0	0
$919$	10.9742	0.362004	0.181002	−	0.983483i	$-0.442066\pi$
$919$	10.9742	0.362004	0.181002	+	0.983483i	$0.442066\pi$
$920$	0	0
$921$	−84.7125	−2.79137
$922$	0	0
$923$	−9.68103	−0.318655
$924$	0	0
$925$	−12.8530	−0.422605
$926$	0	0
$927$	−19.3552	−0.635707
$928$	0	0
$929$	−22.9648	−0.753452	−0.376726	−	0.926325i	$-0.622950\pi$
$929$	−22.9648	−0.753452	−0.376726	+	0.926325i	$0.622950\pi$
$930$	0	0
$931$	−14.7007	−0.481795
$932$	0	0
$933$	56.8773	1.86208
$934$	0	0
$935$	41.4477	1.35549
$936$	0	0
$937$	3.09239	0.101024	0.0505120	−	0.998723i	$-0.483915\pi$
$937$	3.09239	0.101024	0.0505120	+	0.998723i	$0.483915\pi$
$938$	0	0
$939$	−54.0575	−1.76410
$940$	0	0
$941$	11.1410	0.363186	0.181593	−	0.983374i	$-0.441875\pi$
$941$	11.1410	0.363186	0.181593	+	0.983374i	$0.441875\pi$
$942$	0	0
$943$	15.1475	0.493269
$944$	0	0
$945$	−54.9214	−1.78659
$946$	0	0
$947$	56.5858	1.83879	0.919395	−	0.393335i	$-0.128679\pi$
$947$	56.5858	1.83879	0.919395	+	0.393335i	$0.128679\pi$
$948$	0	0
$949$	−31.7353	−1.03017
$950$	0	0
$951$	10.7999	0.350211
$952$	0	0
$953$	−43.0707	−1.39520	−0.697598	−	0.716489i	$-0.745748\pi$
$953$	−43.0707	−1.39520	−0.697598	+	0.716489i	$0.745748\pi$
$954$	0	0
$955$	−9.06674	−0.293393
$956$	0	0
$957$	−38.4808	−1.24391
$958$	0	0
$959$	58.9049	1.90214
$960$	0	0
$961$	17.9073	0.577655
$962$	0	0
$963$	−107.152	−3.45292
$964$	0	0
$965$	20.3320	0.654510
$966$	0	0
$967$	42.0847	1.35335	0.676676	−	0.736281i	$-0.263420\pi$
$967$	42.0847	1.35335	0.676676	+	0.736281i	$0.263420\pi$
$968$	0	0
$969$	12.8240	0.411967
$970$	0	0
$971$	18.2262	0.584905	0.292453	−	0.956280i	$-0.405529\pi$
$971$	18.2262	0.584905	0.292453	+	0.956280i	$0.405529\pi$
$972$	0	0
$973$	8.85769	0.283965
$974$	0	0
$975$	−14.1933	−0.454550
$976$	0	0
$977$	54.2715	1.73630	0.868149	−	0.496303i	$-0.165310\pi$
$977$	54.2715	1.73630	0.868149	+	0.496303i	$0.165310\pi$
$978$	0	0
$979$	−71.8819	−2.29736
$980$	0	0
$981$	−92.4220	−2.95081
$982$	0	0
$983$	41.6051	1.32700	0.663498	−	0.748178i	$-0.269071\pi$
$983$	41.6051	1.32700	0.663498	+	0.748178i	$0.269071\pi$
$984$	0	0
$985$	9.09497	0.289790
$986$	0	0
$987$	−162.517	−5.17298
$988$	0	0
$989$	14.5288	0.461988
$990$	0	0
$991$	−38.7863	−1.23209	−0.616044	−	0.787712i	$-0.711266\pi$
$991$	−38.7863	−1.23209	−0.616044	+	0.787712i	$0.711266\pi$
$992$	0	0
$993$	26.5552	0.842705
$994$	0	0
$995$	−22.8482	−0.724338
$996$	0	0
$997$	14.9471	0.473379	0.236690	−	0.971585i	$-0.423938\pi$
$997$	14.9471	0.473379	0.236690	+	0.971585i	$0.423938\pi$
$998$	0	0
$999$	−48.8285	−1.54487

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.i.1.3	yes	30
4.3	odd	2		8032.2.a.h.1.28	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.h.1.28	✓	30	4.3	odd	2
8032.2.a.i.1.3	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)

\(f(q)\)	\(=\)	\(q-2.87266 q^{3} +1.81575 q^{5} +4.67512 q^{7} +5.25220 q^{9} +O(q^{10})\)	\(q-2.87266 q^{3} +1.81575 q^{5} +4.67512 q^{7} +5.25220 q^{9} +5.05963 q^{11} -2.90118 q^{13} -5.21605 q^{15} +4.51155 q^{17} -0.989496 q^{19} -13.4300 q^{21} -3.68921 q^{23} -1.70304 q^{25} -6.46982 q^{27} +2.64753 q^{29} +6.99338 q^{31} -14.5346 q^{33} +8.48886 q^{35} +7.54712 q^{37} +8.33411 q^{39} -4.10588 q^{41} -3.93817 q^{43} +9.53670 q^{45} +12.1010 q^{47} +14.8567 q^{49} -12.9602 q^{51} -12.7083 q^{53} +9.18703 q^{55} +2.84249 q^{57} +4.97103 q^{59} -6.41226 q^{61} +24.5546 q^{63} -5.26782 q^{65} +4.76200 q^{67} +10.5979 q^{69} +3.33693 q^{71} +10.9388 q^{73} +4.89226 q^{75} +23.6543 q^{77} +14.0018 q^{79} +2.82901 q^{81} +16.1166 q^{83} +8.19185 q^{85} -7.60545 q^{87} -14.2070 q^{89} -13.5633 q^{91} -20.0896 q^{93} -1.79668 q^{95} -4.74155 q^{97} +26.5742 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) \) 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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