LMFDB - Embedded newform 8024.2.a.ba.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.974333 q^{3} -2.75294 q^{5} -1.75498 q^{7} -2.05067 q^{9} +O(q^{10})$	$q+0.974333 q^{3} -2.75294 q^{5} -1.75498 q^{7} -2.05067 q^{9} -3.54130 q^{11} -5.69309 q^{13} -2.68228 q^{15} -1.00000 q^{17} -3.25913 q^{19} -1.70994 q^{21} -4.46974 q^{23} +2.57868 q^{25} -4.92104 q^{27} +10.4018 q^{29} +0.905387 q^{31} -3.45040 q^{33} +4.83136 q^{35} -10.1827 q^{37} -5.54697 q^{39} -8.68776 q^{41} +8.29401 q^{43} +5.64539 q^{45} -5.19995 q^{47} -3.92004 q^{49} -0.974333 q^{51} -1.69997 q^{53} +9.74898 q^{55} -3.17548 q^{57} +1.00000 q^{59} -10.1075 q^{61} +3.59889 q^{63} +15.6728 q^{65} -5.28344 q^{67} -4.35501 q^{69} +4.49720 q^{71} +3.04578 q^{73} +2.51249 q^{75} +6.21491 q^{77} -13.7931 q^{79} +1.35729 q^{81} -16.5642 q^{83} +2.75294 q^{85} +10.1349 q^{87} +3.97324 q^{89} +9.99127 q^{91} +0.882149 q^{93} +8.97220 q^{95} -15.8875 q^{97} +7.26205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9}+O(q^{10}) $ 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9	$ 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9} + 3 q^{11} + 9 q^{13} + 14 q^{15} - 30 q^{17} + 24 q^{19} + 7 q^{21} + 9 q^{23} + 40 q^{25} + 19 q^{27} + 9 q^{29} + 11 q^{31} - 14 q^{33} + 30 q^{35} - 13 q^{37} + 16 q^{39} - 13 q^{41} + 23 q^{43} + 12 q^{45} + 43 q^{47} + 35 q^{49} - 4 q^{51} - 4 q^{53} + 43 q^{55} + 3 q^{57} + 30 q^{59} + 43 q^{61} + 38 q^{63} + 3 q^{65} + 50 q^{67} + 34 q^{69} + 3 q^{71} - 16 q^{73} + 21 q^{75} + 18 q^{77} + 45 q^{79} + 6 q^{81} + 63 q^{83} - 2 q^{85} + 42 q^{87} + 6 q^{89} + 22 q^{91} - 2 q^{93} + 19 q^{95} - 28 q^{97} + 51 q^{99}+O(q^{100}) $ 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9 + 3 * q^11 + 9 * q^13 + 14 * q^15 - 30 * q^17 + 24 * q^19 + 7 * q^21 + 9 * q^23 + 40 * q^25 + 19 * q^27 + 9 * q^29 + 11 * q^31 - 14 * q^33 + 30 * q^35 - 13 * q^37 + 16 * q^39 - 13 * q^41 + 23 * q^43 + 12 * q^45 + 43 * q^47 + 35 * q^49 - 4 * q^51 - 4 * q^53 + 43 * q^55 + 3 * q^57 + 30 * q^59 + 43 * q^61 + 38 * q^63 + 3 * q^65 + 50 * q^67 + 34 * q^69 + 3 * q^71 - 16 * q^73 + 21 * q^75 + 18 * q^77 + 45 * q^79 + 6 * q^81 + 63 * q^83 - 2 * q^85 + 42 * q^87 + 6 * q^89 + 22 * q^91 - 2 * q^93 + 19 * q^95 - 28 * q^97 + 51 * 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
$3$	0.974333	0.562531	0.281266	−	0.959630i	$-0.409246\pi$
$3$	0.974333	0.562531	0.281266	+	0.959630i	$0.409246\pi$
$4$	0	0
$5$	−2.75294	−1.23115	−0.615576	−	0.788077i	$-0.711077\pi$
$5$	−2.75294	−1.23115	−0.615576	+	0.788077i	$0.711077\pi$
$6$	0	0
$7$	−1.75498	−0.663320	−0.331660	−	0.943399i	$-0.607609\pi$
$7$	−1.75498	−0.663320	−0.331660	+	0.943399i	$0.607609\pi$
$8$	0	0
$9$	−2.05067	−0.683558
$10$	0	0
$11$	−3.54130	−1.06774	−0.533871	−	0.845566i	$-0.679263\pi$
$11$	−3.54130	−1.06774	−0.533871	+	0.845566i	$0.679263\pi$
$12$	0	0
$13$	−5.69309	−1.57898	−0.789490	−	0.613763i	$-0.789655\pi$
$13$	−5.69309	−1.57898	−0.789490	+	0.613763i	$0.789655\pi$
$14$	0	0
$15$	−2.68228	−0.692562
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−3.25913	−0.747697	−0.373848	−	0.927490i	$-0.621962\pi$
$19$	−3.25913	−0.747697	−0.373848	+	0.927490i	$0.621962\pi$
$20$	0	0
$21$	−1.70994	−0.373139
$22$	0	0
$23$	−4.46974	−0.932005	−0.466002	−	0.884783i	$-0.654306\pi$
$23$	−4.46974	−0.932005	−0.466002	+	0.884783i	$0.654306\pi$
$24$	0	0
$25$	2.57868	0.515736
$26$	0	0
$27$	−4.92104	−0.947055
$28$	0	0
$29$	10.4018	1.93157	0.965787	−	0.259338i	$-0.0835042\pi$
$29$	10.4018	1.93157	0.965787	+	0.259338i	$0.0835042\pi$
$30$	0	0
$31$	0.905387	0.162612	0.0813062	−	0.996689i	$-0.474091\pi$
$31$	0.905387	0.162612	0.0813062	+	0.996689i	$0.474091\pi$
$32$	0	0
$33$	−3.45040	−0.600638
$34$	0	0
$35$	4.83136	0.816648
$36$	0	0
$37$	−10.1827	−1.67402	−0.837009	−	0.547188i	$-0.815698\pi$
$37$	−10.1827	−1.67402	−0.837009	+	0.547188i	$0.815698\pi$
$38$	0	0
$39$	−5.54697	−0.888226
$40$	0	0
$41$	−8.68776	−1.35680	−0.678400	−	0.734692i	$-0.737327\pi$
$41$	−8.68776	−1.35680	−0.678400	+	0.734692i	$0.737327\pi$
$42$	0	0
$43$	8.29401	1.26482	0.632412	−	0.774632i	$-0.282065\pi$
$43$	8.29401	1.26482	0.632412	+	0.774632i	$0.282065\pi$
$44$	0	0
$45$	5.64539	0.841564
$46$	0	0
$47$	−5.19995	−0.758491	−0.379246	−	0.925296i	$-0.623816\pi$
$47$	−5.19995	−0.758491	−0.379246	+	0.925296i	$0.623816\pi$
$48$	0	0
$49$	−3.92004	−0.560006
$50$	0	0
$51$	−0.974333	−0.136434
$52$	0	0
$53$	−1.69997	−0.233509	−0.116755	−	0.993161i	$-0.537249\pi$
$53$	−1.69997	−0.233509	−0.116755	+	0.993161i	$0.537249\pi$
$54$	0	0
$55$	9.74898	1.31455
$56$	0	0
$57$	−3.17548	−0.420603
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−10.1075	−1.29413	−0.647065	−	0.762435i	$-0.724004\pi$
$61$	−10.1075	−1.29413	−0.647065	+	0.762435i	$0.724004\pi$
$62$	0	0
$63$	3.59889	0.453418
$64$	0	0
$65$	15.6728	1.94397
$66$	0	0
$67$	−5.28344	−0.645474	−0.322737	−	0.946489i	$-0.604603\pi$
$67$	−5.28344	−0.645474	−0.322737	+	0.946489i	$0.604603\pi$
$68$	0	0
$69$	−4.35501	−0.524282
$70$	0	0
$71$	4.49720	0.533719	0.266860	−	0.963735i	$-0.414014\pi$
$71$	4.49720	0.533719	0.266860	+	0.963735i	$0.414014\pi$
$72$	0	0
$73$	3.04578	0.356482	0.178241	−	0.983987i	$-0.442959\pi$
$73$	3.04578	0.356482	0.178241	+	0.983987i	$0.442959\pi$
$74$	0	0
$75$	2.51249	0.290118
$76$	0	0
$77$	6.21491	0.708255
$78$	0	0
$79$	−13.7931	−1.55184	−0.775921	−	0.630830i	$-0.782715\pi$
$79$	−13.7931	−1.55184	−0.775921	+	0.630830i	$0.782715\pi$
$80$	0	0
$81$	1.35729	0.150810
$82$	0	0
$83$	−16.5642	−1.81816	−0.909078	−	0.416625i	$-0.863213\pi$
$83$	−16.5642	−1.81816	−0.909078	+	0.416625i	$0.863213\pi$
$84$	0	0
$85$	2.75294	0.298598
$86$	0	0
$87$	10.1349	1.08657
$88$	0	0
$89$	3.97324	0.421163	0.210581	−	0.977576i	$-0.432464\pi$
$89$	3.97324	0.421163	0.210581	+	0.977576i	$0.432464\pi$
$90$	0	0
$91$	9.99127	1.04737
$92$	0	0
$93$	0.882149	0.0914745
$94$	0	0
$95$	8.97220	0.920529
$96$	0	0
$97$	−15.8875	−1.61313	−0.806564	−	0.591147i	$-0.798675\pi$
$97$	−15.8875	−1.61313	−0.806564	+	0.591147i	$0.798675\pi$
$98$	0	0
$99$	7.26205	0.729863
$100$	0	0
$101$	10.5967	1.05441	0.527206	−	0.849738i	$-0.323240\pi$
$101$	10.5967	1.05441	0.527206	+	0.849738i	$0.323240\pi$
$102$	0	0
$103$	17.4108	1.71554	0.857769	−	0.514036i	$-0.171850\pi$
$103$	17.4108	1.71554	0.857769	+	0.514036i	$0.171850\pi$
$104$	0	0
$105$	4.70735	0.459390
$106$	0	0
$107$	−18.6769	−1.80556	−0.902780	−	0.430103i	$-0.858477\pi$
$107$	−18.6769	−1.80556	−0.902780	+	0.430103i	$0.858477\pi$
$108$	0	0
$109$	19.4925	1.86704	0.933521	−	0.358523i	$-0.116719\pi$
$109$	19.4925	1.86704	0.933521	+	0.358523i	$0.116719\pi$
$110$	0	0
$111$	−9.92130	−0.941688
$112$	0	0
$113$	−7.83410	−0.736970	−0.368485	−	0.929634i	$-0.620123\pi$
$113$	−7.83410	−0.736970	−0.368485	+	0.929634i	$0.620123\pi$
$114$	0	0
$115$	12.3049	1.14744
$116$	0	0
$117$	11.6747	1.07933
$118$	0	0
$119$	1.75498	0.160879
$120$	0	0
$121$	1.54079	0.140072
$122$	0	0
$123$	−8.46478	−0.763243
$124$	0	0
$125$	6.66575	0.596203
$126$	0	0
$127$	−3.37236	−0.299248	−0.149624	−	0.988743i	$-0.547806\pi$
$127$	−3.37236	−0.299248	−0.149624	+	0.988743i	$0.547806\pi$
$128$	0	0
$129$	8.08113	0.711503
$130$	0	0
$131$	3.64354	0.318337	0.159169	−	0.987251i	$-0.449119\pi$
$131$	3.64354	0.318337	0.159169	+	0.987251i	$0.449119\pi$
$132$	0	0
$133$	5.71972	0.495962
$134$	0	0
$135$	13.5473	1.16597
$136$	0	0
$137$	15.7723	1.34752	0.673758	−	0.738952i	$-0.264679\pi$
$137$	15.7723	1.34752	0.673758	+	0.738952i	$0.264679\pi$
$138$	0	0
$139$	9.50672	0.806350	0.403175	−	0.915123i	$-0.367907\pi$
$139$	9.50672	0.806350	0.403175	+	0.915123i	$0.367907\pi$
$140$	0	0
$141$	−5.06649	−0.426675
$142$	0	0
$143$	20.1609	1.68594
$144$	0	0
$145$	−28.6356	−2.37806
$146$	0	0
$147$	−3.81943	−0.315021
$148$	0	0
$149$	6.50289	0.532738	0.266369	−	0.963871i	$-0.414176\pi$
$149$	6.50289	0.532738	0.266369	+	0.963871i	$0.414176\pi$
$150$	0	0
$151$	−2.80510	−0.228276	−0.114138	−	0.993465i	$-0.536411\pi$
$151$	−2.80510	−0.228276	−0.114138	+	0.993465i	$0.536411\pi$
$152$	0	0
$153$	2.05067	0.165787
$154$	0	0
$155$	−2.49248	−0.200201
$156$	0	0
$157$	16.4403	1.31208	0.656040	−	0.754726i	$-0.272230\pi$
$157$	16.4403	1.31208	0.656040	+	0.754726i	$0.272230\pi$
$158$	0	0
$159$	−1.65634	−0.131356
$160$	0	0
$161$	7.84430	0.618218
$162$	0	0
$163$	−16.2519	−1.27295	−0.636475	−	0.771297i	$-0.719608\pi$
$163$	−16.2519	−1.27295	−0.636475	+	0.771297i	$0.719608\pi$
$164$	0	0
$165$	9.49875	0.739477
$166$	0	0
$167$	14.0796	1.08951	0.544755	−	0.838595i	$-0.316623\pi$
$167$	14.0796	1.08951	0.544755	+	0.838595i	$0.316623\pi$
$168$	0	0
$169$	19.4113	1.49318
$170$	0	0
$171$	6.68342	0.511094
$172$	0	0
$173$	−21.1937	−1.61133	−0.805663	−	0.592374i	$-0.798191\pi$
$173$	−21.1937	−1.61133	−0.805663	+	0.592374i	$0.798191\pi$
$174$	0	0
$175$	−4.52554	−0.342098
$176$	0	0
$177$	0.974333	0.0732354
$178$	0	0
$179$	−14.9633	−1.11841	−0.559205	−	0.829029i	$-0.688894\pi$
$179$	−14.9633	−1.11841	−0.559205	+	0.829029i	$0.688894\pi$
$180$	0	0
$181$	14.1163	1.04926	0.524628	−	0.851331i	$-0.324204\pi$
$181$	14.1163	1.04926	0.524628	+	0.851331i	$0.324204\pi$
$182$	0	0
$183$	−9.84806	−0.727989
$184$	0	0
$185$	28.0323	2.06097
$186$	0	0
$187$	3.54130	0.258965
$188$	0	0
$189$	8.63633	0.628201
$190$	0	0
$191$	−2.70045	−0.195398	−0.0976990	−	0.995216i	$-0.531148\pi$
$191$	−2.70045	−0.195398	−0.0976990	+	0.995216i	$0.531148\pi$
$192$	0	0
$193$	17.3251	1.24709	0.623543	−	0.781789i	$-0.285693\pi$
$193$	17.3251	1.24709	0.623543	+	0.781789i	$0.285693\pi$
$194$	0	0
$195$	15.2705	1.09354
$196$	0	0
$197$	9.54915	0.680349	0.340174	−	0.940362i	$-0.389514\pi$
$197$	9.54915	0.680349	0.340174	+	0.940362i	$0.389514\pi$
$198$	0	0
$199$	5.33145	0.377936	0.188968	−	0.981983i	$-0.439486\pi$
$199$	5.33145	0.377936	0.188968	+	0.981983i	$0.439486\pi$
$200$	0	0
$201$	−5.14783	−0.363100
$202$	0	0
$203$	−18.2550	−1.28125
$204$	0	0
$205$	23.9169	1.67043
$206$	0	0
$207$	9.16598	0.637080
$208$	0	0
$209$	11.5416	0.798347
$210$	0	0
$211$	0.908450	0.0625403	0.0312702	−	0.999511i	$-0.490045\pi$
$211$	0.908450	0.0625403	0.0312702	+	0.999511i	$0.490045\pi$
$212$	0	0
$213$	4.38177	0.300234
$214$	0	0
$215$	−22.8329	−1.55719
$216$	0	0
$217$	−1.58894	−0.107864
$218$	0	0
$219$	2.96761	0.200532
$220$	0	0
$221$	5.69309	0.382959
$222$	0	0
$223$	−7.91055	−0.529730	−0.264865	−	0.964286i	$-0.585327\pi$
$223$	−7.91055	−0.529730	−0.264865	+	0.964286i	$0.585327\pi$
$224$	0	0
$225$	−5.28804	−0.352536
$226$	0	0
$227$	8.57972	0.569456	0.284728	−	0.958608i	$-0.408097\pi$
$227$	8.57972	0.569456	0.284728	+	0.958608i	$0.408097\pi$
$228$	0	0
$229$	−22.3760	−1.47865	−0.739323	−	0.673351i	$-0.764854\pi$
$229$	−22.3760	−1.47865	−0.739323	+	0.673351i	$0.764854\pi$
$230$	0	0
$231$	6.05539	0.398415
$232$	0	0
$233$	−17.0102	−1.11438	−0.557189	−	0.830386i	$-0.688120\pi$
$233$	−17.0102	−1.11438	−0.557189	+	0.830386i	$0.688120\pi$
$234$	0	0
$235$	14.3152	0.933818
$236$	0	0
$237$	−13.4390	−0.872960
$238$	0	0
$239$	12.7459	0.824464	0.412232	−	0.911079i	$-0.364749\pi$
$239$	12.7459	0.824464	0.412232	+	0.911079i	$0.364749\pi$
$240$	0	0
$241$	−6.91531	−0.445454	−0.222727	−	0.974881i	$-0.571496\pi$
$241$	−6.91531	−0.445454	−0.222727	+	0.974881i	$0.571496\pi$
$242$	0	0
$243$	16.0856	1.03189
$244$	0	0
$245$	10.7916	0.689453
$246$	0	0
$247$	18.5546	1.18060
$248$	0	0
$249$	−16.1390	−1.02277
$250$	0	0
$251$	22.9899	1.45111	0.725556	−	0.688163i	$-0.241583\pi$
$251$	22.9899	1.45111	0.725556	+	0.688163i	$0.241583\pi$
$252$	0	0
$253$	15.8287	0.995140
$254$	0	0
$255$	2.68228	0.167971
$256$	0	0
$257$	−19.8252	−1.23666	−0.618330	−	0.785918i	$-0.712191\pi$
$257$	−19.8252	−1.23666	−0.618330	+	0.785918i	$0.712191\pi$
$258$	0	0
$259$	17.8704	1.11041
$260$	0	0
$261$	−21.3308	−1.32034
$262$	0	0
$263$	−26.6281	−1.64196	−0.820979	−	0.570959i	$-0.806572\pi$
$263$	−26.6281	−1.64196	−0.820979	+	0.570959i	$0.806572\pi$
$264$	0	0
$265$	4.67992	0.287485
$266$	0	0
$267$	3.87126	0.236917
$268$	0	0
$269$	−0.695933	−0.0424318	−0.0212159	−	0.999775i	$-0.506754\pi$
$269$	−0.695933	−0.0424318	−0.0212159	+	0.999775i	$0.506754\pi$
$270$	0	0
$271$	−17.1595	−1.04237	−0.521184	−	0.853444i	$-0.674509\pi$
$271$	−17.1595	−1.04237	−0.521184	+	0.853444i	$0.674509\pi$
$272$	0	0
$273$	9.73483	0.589179
$274$	0	0
$275$	−9.13188	−0.550673
$276$	0	0
$277$	−17.1587	−1.03097	−0.515483	−	0.856900i	$-0.672388\pi$
$277$	−17.1587	−1.03097	−0.515483	+	0.856900i	$0.672388\pi$
$278$	0	0
$279$	−1.85665	−0.111155
$280$	0	0
$281$	−32.4223	−1.93415	−0.967077	−	0.254486i	$-0.918094\pi$
$281$	−32.4223	−1.93415	−0.967077	+	0.254486i	$0.918094\pi$
$282$	0	0
$283$	16.5292	0.982557	0.491278	−	0.871003i	$-0.336530\pi$
$283$	16.5292	0.982557	0.491278	+	0.871003i	$0.336530\pi$
$284$	0	0
$285$	8.74191	0.517826
$286$	0	0
$287$	15.2469	0.899994
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−15.4797	−0.907435
$292$	0	0
$293$	10.3777	0.606274	0.303137	−	0.952947i	$-0.401966\pi$
$293$	10.3777	0.606274	0.303137	+	0.952947i	$0.401966\pi$
$294$	0	0
$295$	−2.75294	−0.160282
$296$	0	0
$297$	17.4269	1.01121
$298$	0	0
$299$	25.4466	1.47162
$300$	0	0
$301$	−14.5558	−0.838984
$302$	0	0
$303$	10.3247	0.593140
$304$	0	0
$305$	27.8253	1.59327
$306$	0	0
$307$	−11.3367	−0.647020	−0.323510	−	0.946225i	$-0.604863\pi$
$307$	−11.3367	−0.647020	−0.323510	+	0.946225i	$0.604863\pi$
$308$	0	0
$309$	16.9639	0.965044
$310$	0	0
$311$	−10.0860	−0.571926	−0.285963	−	0.958241i	$-0.592313\pi$
$311$	−10.0860	−0.571926	−0.285963	+	0.958241i	$0.592313\pi$
$312$	0	0
$313$	−7.91798	−0.447551	−0.223775	−	0.974641i	$-0.571838\pi$
$313$	−7.91798	−0.447551	−0.223775	+	0.974641i	$0.571838\pi$
$314$	0	0
$315$	−9.90754	−0.558227
$316$	0	0
$317$	−26.0685	−1.46415	−0.732075	−	0.681224i	$-0.761448\pi$
$317$	−26.0685	−1.46415	−0.732075	+	0.681224i	$0.761448\pi$
$318$	0	0
$319$	−36.8360	−2.06242
$320$	0	0
$321$	−18.1975	−1.01568
$322$	0	0
$323$	3.25913	0.181343
$324$	0	0
$325$	−14.6807	−0.814337
$326$	0	0
$327$	18.9922	1.05027
$328$	0	0
$329$	9.12582	0.503123
$330$	0	0
$331$	19.6899	1.08225	0.541127	−	0.840941i	$-0.317998\pi$
$331$	19.6899	1.08225	0.541127	+	0.840941i	$0.317998\pi$
$332$	0	0
$333$	20.8813	1.14429
$334$	0	0
$335$	14.5450	0.794677
$336$	0	0
$337$	−3.04200	−0.165708	−0.0828542	−	0.996562i	$-0.526404\pi$
$337$	−3.04200	−0.165708	−0.0828542	+	0.996562i	$0.526404\pi$
$338$	0	0
$339$	−7.63302	−0.414569
$340$	0	0
$341$	−3.20624	−0.173628
$342$	0	0
$343$	19.1645	1.03478
$344$	0	0
$345$	11.9891	0.645471
$346$	0	0
$347$	−8.79881	−0.472345	−0.236173	−	0.971711i	$-0.575893\pi$
$347$	−8.79881	−0.472345	−0.236173	+	0.971711i	$0.575893\pi$
$348$	0	0
$349$	21.7654	1.16507	0.582537	−	0.812804i	$-0.302060\pi$
$349$	21.7654	1.16507	0.582537	+	0.812804i	$0.302060\pi$
$350$	0	0
$351$	28.0159	1.49538
$352$	0	0
$353$	−18.8175	−1.00156	−0.500778	−	0.865576i	$-0.666953\pi$
$353$	−18.8175	−1.00156	−0.500778	+	0.865576i	$0.666953\pi$
$354$	0	0
$355$	−12.3805	−0.657090
$356$	0	0
$357$	1.70994	0.0904994
$358$	0	0
$359$	−0.158295	−0.00835448	−0.00417724	−	0.999991i	$-0.501330\pi$
$359$	−0.158295	−0.00835448	−0.00417724	+	0.999991i	$0.501330\pi$
$360$	0	0
$361$	−8.37804	−0.440950
$362$	0	0
$363$	1.50124	0.0787947
$364$	0	0
$365$	−8.38486	−0.438884
$366$	0	0
$367$	11.6955	0.610498	0.305249	−	0.952273i	$-0.401260\pi$
$367$	11.6955	0.610498	0.305249	+	0.952273i	$0.401260\pi$
$368$	0	0
$369$	17.8158	0.927453
$370$	0	0
$371$	2.98342	0.154891
$372$	0	0
$373$	−4.42201	−0.228963	−0.114481	−	0.993425i	$-0.536521\pi$
$373$	−4.42201	−0.228963	−0.114481	+	0.993425i	$0.536521\pi$
$374$	0	0
$375$	6.49466	0.335383
$376$	0	0
$377$	−59.2187	−3.04992
$378$	0	0
$379$	18.3172	0.940891	0.470445	−	0.882429i	$-0.344093\pi$
$379$	18.3172	0.940891	0.470445	+	0.882429i	$0.344093\pi$
$380$	0	0
$381$	−3.28580	−0.168337
$382$	0	0
$383$	−2.97421	−0.151975	−0.0759874	−	0.997109i	$-0.524211\pi$
$383$	−2.97421	−0.151975	−0.0759874	+	0.997109i	$0.524211\pi$
$384$	0	0
$385$	−17.1093	−0.871969
$386$	0	0
$387$	−17.0083	−0.864581
$388$	0	0
$389$	2.20434	0.111765	0.0558823	−	0.998437i	$-0.482203\pi$
$389$	2.20434	0.111765	0.0558823	+	0.998437i	$0.482203\pi$
$390$	0	0
$391$	4.46974	0.226044
$392$	0	0
$393$	3.55002	0.179075
$394$	0	0
$395$	37.9715	1.91055
$396$	0	0
$397$	8.58423	0.430830	0.215415	−	0.976523i	$-0.430890\pi$
$397$	8.58423	0.430830	0.215415	+	0.976523i	$0.430890\pi$
$398$	0	0
$399$	5.57291	0.278994
$400$	0	0
$401$	−17.6752	−0.882656	−0.441328	−	0.897346i	$-0.645493\pi$
$401$	−17.6752	−0.882656	−0.441328	+	0.897346i	$0.645493\pi$
$402$	0	0
$403$	−5.15445	−0.256762
$404$	0	0
$405$	−3.73655	−0.185670
$406$	0	0
$407$	36.0598	1.78742
$408$	0	0
$409$	8.35058	0.412910	0.206455	−	0.978456i	$-0.433807\pi$
$409$	8.35058	0.412910	0.206455	+	0.978456i	$0.433807\pi$
$410$	0	0
$411$	15.3675	0.758021
$412$	0	0
$413$	−1.75498	−0.0863570
$414$	0	0
$415$	45.6002	2.23843
$416$	0	0
$417$	9.26271	0.453597
$418$	0	0
$419$	−2.97706	−0.145439	−0.0727194	−	0.997352i	$-0.523168\pi$
$419$	−2.97706	−0.145439	−0.0727194	+	0.997352i	$0.523168\pi$
$420$	0	0
$421$	18.6392	0.908419	0.454210	−	0.890895i	$-0.349922\pi$
$421$	18.6392	0.908419	0.454210	+	0.890895i	$0.349922\pi$
$422$	0	0
$423$	10.6634	0.518473
$424$	0	0
$425$	−2.57868	−0.125084
$426$	0	0
$427$	17.7384	0.858423
$428$	0	0
$429$	19.6435	0.948396
$430$	0	0
$431$	−21.6395	−1.04234	−0.521169	−	0.853454i	$-0.674504\pi$
$431$	−21.6395	−1.04234	−0.521169	+	0.853454i	$0.674504\pi$
$432$	0	0
$433$	2.18728	0.105114	0.0525569	−	0.998618i	$-0.483263\pi$
$433$	2.18728	0.105114	0.0525569	+	0.998618i	$0.483263\pi$
$434$	0	0
$435$	−27.9007	−1.33773
$436$	0	0
$437$	14.5675	0.696857
$438$	0	0
$439$	−16.7785	−0.800794	−0.400397	−	0.916342i	$-0.631128\pi$
$439$	−16.7785	−0.800794	−0.400397	+	0.916342i	$0.631128\pi$
$440$	0	0
$441$	8.03873	0.382797
$442$	0	0
$443$	22.8814	1.08713	0.543563	−	0.839368i	$-0.317075\pi$
$443$	22.8814	1.08713	0.543563	+	0.839368i	$0.317075\pi$
$444$	0	0
$445$	−10.9381	−0.518515
$446$	0	0
$447$	6.33598	0.299682
$448$	0	0
$449$	−10.8781	−0.513369	−0.256684	−	0.966495i	$-0.582630\pi$
$449$	−10.8781	−0.513369	−0.256684	+	0.966495i	$0.582630\pi$
$450$	0	0
$451$	30.7660	1.44871
$452$	0	0
$453$	−2.73310	−0.128412
$454$	0	0
$455$	−27.5054	−1.28947
$456$	0	0
$457$	−13.4568	−0.629481	−0.314741	−	0.949178i	$-0.601918\pi$
$457$	−13.4568	−0.629481	−0.314741	+	0.949178i	$0.601918\pi$
$458$	0	0
$459$	4.92104	0.229694
$460$	0	0
$461$	14.1712	0.660020	0.330010	−	0.943977i	$-0.392948\pi$
$461$	14.1712	0.660020	0.330010	+	0.943977i	$0.392948\pi$
$462$	0	0
$463$	−14.5105	−0.674359	−0.337179	−	0.941440i	$-0.609473\pi$
$463$	−14.5105	−0.674359	−0.337179	+	0.941440i	$0.609473\pi$
$464$	0	0
$465$	−2.42850	−0.112619
$466$	0	0
$467$	6.12086	0.283239	0.141620	−	0.989921i	$-0.454769\pi$
$467$	6.12086	0.283239	0.141620	+	0.989921i	$0.454769\pi$
$468$	0	0
$469$	9.27233	0.428156
$470$	0	0
$471$	16.0183	0.738086
$472$	0	0
$473$	−29.3715	−1.35051
$474$	0	0
$475$	−8.40427	−0.385614
$476$	0	0
$477$	3.48609	0.159617
$478$	0	0
$479$	−3.75800	−0.171707	−0.0858536	−	0.996308i	$-0.527362\pi$
$479$	−3.75800	−0.171707	−0.0858536	+	0.996308i	$0.527362\pi$
$480$	0	0
$481$	57.9708	2.64324
$482$	0	0
$483$	7.64296	0.347767
$484$	0	0
$485$	43.7372	1.98601
$486$	0	0
$487$	−37.2088	−1.68609	−0.843045	−	0.537843i	$-0.819239\pi$
$487$	−37.2088	−1.68609	−0.843045	+	0.537843i	$0.819239\pi$
$488$	0	0
$489$	−15.8348	−0.716075
$490$	0	0
$491$	−12.0211	−0.542505	−0.271252	−	0.962508i	$-0.587438\pi$
$491$	−12.0211	−0.542505	−0.271252	+	0.962508i	$0.587438\pi$
$492$	0	0
$493$	−10.4018	−0.468475
$494$	0	0
$495$	−19.9920	−0.898573
$496$	0	0
$497$	−7.89250	−0.354027
$498$	0	0
$499$	−39.0515	−1.74819	−0.874094	−	0.485758i	$-0.838544\pi$
$499$	−39.0515	−1.74819	−0.874094	+	0.485758i	$0.838544\pi$
$500$	0	0
$501$	13.7182	0.612883
$502$	0	0
$503$	35.6655	1.59024	0.795122	−	0.606449i	$-0.207407\pi$
$503$	35.6655	1.59024	0.795122	+	0.606449i	$0.207407\pi$
$504$	0	0
$505$	−29.1721	−1.29814
$506$	0	0
$507$	18.9131	0.839960
$508$	0	0
$509$	−24.1691	−1.07128	−0.535639	−	0.844447i	$-0.679929\pi$
$509$	−24.1691	−1.07128	−0.535639	+	0.844447i	$0.679929\pi$
$510$	0	0
$511$	−5.34529	−0.236462
$512$	0	0
$513$	16.0383	0.708110
$514$	0	0
$515$	−47.9309	−2.11209
$516$	0	0
$517$	18.4146	0.809872
$518$	0	0
$519$	−20.6497	−0.906422
$520$	0	0
$521$	−17.0922	−0.748823	−0.374411	−	0.927263i	$-0.622155\pi$
$521$	−17.0922	−0.748823	−0.374411	+	0.927263i	$0.622155\pi$
$522$	0	0
$523$	−19.9510	−0.872398	−0.436199	−	0.899850i	$-0.643676\pi$
$523$	−19.9510	−0.872398	−0.436199	+	0.899850i	$0.643676\pi$
$524$	0	0
$525$	−4.40938	−0.192441
$526$	0	0
$527$	−0.905387	−0.0394393
$528$	0	0
$529$	−3.02144	−0.131367
$530$	0	0
$531$	−2.05067	−0.0889917
$532$	0	0
$533$	49.4603	2.14236
$534$	0	0
$535$	51.4163	2.22292
$536$	0	0
$537$	−14.5792	−0.629141
$538$	0	0
$539$	13.8820	0.597942
$540$	0	0
$541$	9.07585	0.390201	0.195101	−	0.980783i	$-0.437497\pi$
$541$	9.07585	0.390201	0.195101	+	0.980783i	$0.437497\pi$
$542$	0	0
$543$	13.7540	0.590240
$544$	0	0
$545$	−53.6617	−2.29861
$546$	0	0
$547$	−3.77093	−0.161234	−0.0806168	−	0.996745i	$-0.525689\pi$
$547$	−3.77093	−0.161234	−0.0806168	+	0.996745i	$0.525689\pi$
$548$	0	0
$549$	20.7272	0.884614
$550$	0	0
$551$	−33.9010	−1.44423
$552$	0	0
$553$	24.2066	1.02937
$554$	0	0
$555$	27.3128	1.15936
$556$	0	0
$557$	−41.9907	−1.77920	−0.889602	−	0.456736i	$-0.849018\pi$
$557$	−41.9907	−1.77920	−0.889602	+	0.456736i	$0.849018\pi$
$558$	0	0
$559$	−47.2186	−1.99713
$560$	0	0
$561$	3.45040	0.145676
$562$	0	0
$563$	3.00068	0.126464	0.0632318	−	0.997999i	$-0.479859\pi$
$563$	3.00068	0.126464	0.0632318	+	0.997999i	$0.479859\pi$
$564$	0	0
$565$	21.5668	0.907323
$566$	0	0
$567$	−2.38202	−0.100036
$568$	0	0
$569$	13.5827	0.569417	0.284708	−	0.958614i	$-0.408103\pi$
$569$	13.5827	0.569417	0.284708	+	0.958614i	$0.408103\pi$
$570$	0	0
$571$	2.58254	0.108076	0.0540379	−	0.998539i	$-0.482791\pi$
$571$	2.58254	0.108076	0.0540379	+	0.998539i	$0.482791\pi$
$572$	0	0
$573$	−2.63114	−0.109918
$574$	0	0
$575$	−11.5260	−0.480669
$576$	0	0
$577$	21.7131	0.903928	0.451964	−	0.892036i	$-0.350724\pi$
$577$	21.7131	0.903928	0.451964	+	0.892036i	$0.350724\pi$
$578$	0	0
$579$	16.8804	0.701525
$580$	0	0
$581$	29.0698	1.20602
$582$	0	0
$583$	6.02010	0.249327
$584$	0	0
$585$	−32.1397	−1.32881
$586$	0	0
$587$	−21.2255	−0.876072	−0.438036	−	0.898958i	$-0.644326\pi$
$587$	−21.2255	−0.876072	−0.438036	+	0.898958i	$0.644326\pi$
$588$	0	0
$589$	−2.95078	−0.121585
$590$	0	0
$591$	9.30406	0.382718
$592$	0	0
$593$	−48.4646	−1.99020	−0.995101	−	0.0988618i	$-0.968480\pi$
$593$	−48.4646	−1.99020	−0.995101	+	0.0988618i	$0.968480\pi$
$594$	0	0
$595$	−4.83136	−0.198066
$596$	0	0
$597$	5.19461	0.212601
$598$	0	0
$599$	21.6510	0.884635	0.442317	−	0.896859i	$-0.354156\pi$
$599$	21.6510	0.884635	0.442317	+	0.896859i	$0.354156\pi$
$600$	0	0
$601$	4.10572	0.167476	0.0837379	−	0.996488i	$-0.473314\pi$
$601$	4.10572	0.167476	0.0837379	+	0.996488i	$0.473314\pi$
$602$	0	0
$603$	10.8346	0.441219
$604$	0	0
$605$	−4.24170	−0.172449
$606$	0	0
$607$	30.0129	1.21819	0.609094	−	0.793098i	$-0.291533\pi$
$607$	30.0129	1.21819	0.609094	+	0.793098i	$0.291533\pi$
$608$	0	0
$609$	−17.7865	−0.720745
$610$	0	0
$611$	29.6038	1.19764
$612$	0	0
$613$	−30.9448	−1.24985	−0.624925	−	0.780685i	$-0.714870\pi$
$613$	−30.9448	−1.24985	−0.624925	+	0.780685i	$0.714870\pi$
$614$	0	0
$615$	23.3030	0.939669
$616$	0	0
$617$	10.4158	0.419324	0.209662	−	0.977774i	$-0.432764\pi$
$617$	10.4158	0.419324	0.209662	+	0.977774i	$0.432764\pi$
$618$	0	0
$619$	−15.1766	−0.610000	−0.305000	−	0.952352i	$-0.598656\pi$
$619$	−15.1766	−0.610000	−0.305000	+	0.952352i	$0.598656\pi$
$620$	0	0
$621$	21.9958	0.882659
$622$	0	0
$623$	−6.97296	−0.279366
$624$	0	0
$625$	−31.2438	−1.24975
$626$	0	0
$627$	11.2453	0.449095
$628$	0	0
$629$	10.1827	0.406009
$630$	0	0
$631$	−21.9467	−0.873685	−0.436843	−	0.899538i	$-0.643903\pi$
$631$	−21.9467	−0.873685	−0.436843	+	0.899538i	$0.643903\pi$
$632$	0	0
$633$	0.885133	0.0351809
$634$	0	0
$635$	9.28390	0.368420
$636$	0	0
$637$	22.3172	0.884239
$638$	0	0
$639$	−9.22229	−0.364828
$640$	0	0
$641$	−3.97777	−0.157113	−0.0785563	−	0.996910i	$-0.525031\pi$
$641$	−3.97777	−0.157113	−0.0785563	+	0.996910i	$0.525031\pi$
$642$	0	0
$643$	37.7794	1.48987	0.744937	−	0.667135i	$-0.232480\pi$
$643$	37.7794	1.48987	0.744937	+	0.667135i	$0.232480\pi$
$644$	0	0
$645$	−22.2469	−0.875969
$646$	0	0
$647$	−3.77436	−0.148385	−0.0741926	−	0.997244i	$-0.523638\pi$
$647$	−3.77436	−0.148385	−0.0741926	+	0.997244i	$0.523638\pi$
$648$	0	0
$649$	−3.54130	−0.139008
$650$	0	0
$651$	−1.54815	−0.0606769
$652$	0	0
$653$	−20.0178	−0.783355	−0.391678	−	0.920103i	$-0.628105\pi$
$653$	−20.0178	−0.783355	−0.391678	+	0.920103i	$0.628105\pi$
$654$	0	0
$655$	−10.0304	−0.391922
$656$	0	0
$657$	−6.24591	−0.243676
$658$	0	0
$659$	−12.9731	−0.505361	−0.252680	−	0.967550i	$-0.581312\pi$
$659$	−12.9731	−0.505361	−0.252680	+	0.967550i	$0.581312\pi$
$660$	0	0
$661$	15.3656	0.597653	0.298826	−	0.954308i	$-0.403405\pi$
$661$	15.3656	0.597653	0.298826	+	0.954308i	$0.403405\pi$
$662$	0	0
$663$	5.54697	0.215426
$664$	0	0
$665$	−15.7460	−0.610605
$666$	0	0
$667$	−46.4935	−1.80024
$668$	0	0
$669$	−7.70751	−0.297990
$670$	0	0
$671$	35.7936	1.38180
$672$	0	0
$673$	−20.2538	−0.780728	−0.390364	−	0.920661i	$-0.627651\pi$
$673$	−20.2538	−0.780728	−0.390364	+	0.920661i	$0.627651\pi$
$674$	0	0
$675$	−12.6898	−0.488430
$676$	0	0
$677$	13.8478	0.532213	0.266106	−	0.963944i	$-0.414263\pi$
$677$	13.8478	0.532213	0.266106	+	0.963944i	$0.414263\pi$
$678$	0	0
$679$	27.8822	1.07002
$680$	0	0
$681$	8.35951	0.320337
$682$	0	0
$683$	−13.2435	−0.506749	−0.253375	−	0.967368i	$-0.581541\pi$
$683$	−13.2435	−0.506749	−0.253375	+	0.967368i	$0.581541\pi$
$684$	0	0
$685$	−43.4201	−1.65900
$686$	0	0
$687$	−21.8017	−0.831785
$688$	0	0
$689$	9.67810	0.368706
$690$	0	0
$691$	16.4425	0.625504	0.312752	−	0.949835i	$-0.398749\pi$
$691$	16.4425	0.625504	0.312752	+	0.949835i	$0.398749\pi$
$692$	0	0
$693$	−12.7448	−0.484133
$694$	0	0
$695$	−26.1714	−0.992739
$696$	0	0
$697$	8.68776	0.329073
$698$	0	0
$699$	−16.5736	−0.626873
$700$	0	0
$701$	−34.3193	−1.29622	−0.648111	−	0.761546i	$-0.724441\pi$
$701$	−34.3193	−1.29622	−0.648111	+	0.761546i	$0.724441\pi$
$702$	0	0
$703$	33.1867	1.25166
$704$	0	0
$705$	13.9477	0.525302
$706$	0	0
$707$	−18.5970	−0.699413
$708$	0	0
$709$	−30.7119	−1.15341	−0.576705	−	0.816952i	$-0.695662\pi$
$709$	−30.7119	−1.15341	−0.576705	+	0.816952i	$0.695662\pi$
$710$	0	0
$711$	28.2851	1.06077
$712$	0	0
$713$	−4.04684	−0.151555
$714$	0	0
$715$	−55.5019	−2.07565
$716$	0	0
$717$	12.4188	0.463787
$718$	0	0
$719$	50.9022	1.89833	0.949167	−	0.314774i	$-0.101929\pi$
$719$	50.9022	1.89833	0.949167	+	0.314774i	$0.101929\pi$
$720$	0	0
$721$	−30.5556	−1.13795
$722$	0	0
$723$	−6.73782	−0.250582
$724$	0	0
$725$	26.8230	0.996182
$726$	0	0
$727$	−9.56627	−0.354793	−0.177397	−	0.984139i	$-0.556768\pi$
$727$	−9.56627	−0.354793	−0.177397	+	0.984139i	$0.556768\pi$
$728$	0	0
$729$	11.6008	0.429660
$730$	0	0
$731$	−8.29401	−0.306765
$732$	0	0
$733$	−20.1229	−0.743257	−0.371628	−	0.928382i	$-0.621200\pi$
$733$	−20.1229	−0.743257	−0.371628	+	0.928382i	$0.621200\pi$
$734$	0	0
$735$	10.5147	0.387839
$736$	0	0
$737$	18.7102	0.689200
$738$	0	0
$739$	19.3691	0.712504	0.356252	−	0.934390i	$-0.384054\pi$
$739$	19.3691	0.712504	0.356252	+	0.934390i	$0.384054\pi$
$740$	0	0
$741$	18.0783	0.664124
$742$	0	0
$743$	−17.7419	−0.650889	−0.325444	−	0.945561i	$-0.605514\pi$
$743$	−17.7419	−0.650889	−0.325444	+	0.945561i	$0.605514\pi$
$744$	0	0
$745$	−17.9021	−0.655881
$746$	0	0
$747$	33.9678	1.24282
$748$	0	0
$749$	32.7775	1.19766
$750$	0	0
$751$	52.3449	1.91009	0.955046	−	0.296458i	$-0.0958056\pi$
$751$	52.3449	1.91009	0.955046	+	0.296458i	$0.0958056\pi$
$752$	0	0
$753$	22.3999	0.816296
$754$	0	0
$755$	7.72226	0.281042
$756$	0	0
$757$	0.519339	0.0188757	0.00943786	−	0.999955i	$-0.496996\pi$
$757$	0.519339	0.0188757	0.00943786	+	0.999955i	$0.496996\pi$
$758$	0	0
$759$	15.4224	0.559798
$760$	0	0
$761$	−2.82848	−0.102532	−0.0512661	−	0.998685i	$-0.516326\pi$
$761$	−2.82848	−0.102532	−0.0512661	+	0.998685i	$0.516326\pi$
$762$	0	0
$763$	−34.2089	−1.23845
$764$	0	0
$765$	−5.64539	−0.204109
$766$	0	0
$767$	−5.69309	−0.205566
$768$	0	0
$769$	30.0885	1.08502	0.542510	−	0.840049i	$-0.317474\pi$
$769$	30.0885	1.08502	0.542510	+	0.840049i	$0.317474\pi$
$770$	0	0
$771$	−19.3163	−0.695661
$772$	0	0
$773$	−12.6011	−0.453231	−0.226616	−	0.973984i	$-0.572766\pi$
$773$	−12.6011	−0.453231	−0.226616	+	0.973984i	$0.572766\pi$
$774$	0	0
$775$	2.33470	0.0838651
$776$	0	0
$777$	17.4117	0.624641
$778$	0	0
$779$	28.3146	1.01448
$780$	0	0
$781$	−15.9259	−0.569874
$782$	0	0
$783$	−51.1879	−1.82931
$784$	0	0
$785$	−45.2592	−1.61537
$786$	0	0
$787$	−29.9707	−1.06834	−0.534170	−	0.845377i	$-0.679376\pi$
$787$	−29.9707	−1.06834	−0.534170	+	0.845377i	$0.679376\pi$
$788$	0	0
$789$	−25.9446	−0.923653
$790$	0	0
$791$	13.7487	0.488847
$792$	0	0
$793$	57.5429	2.04341
$794$	0	0
$795$	4.55980	0.161719
$796$	0	0
$797$	−10.6848	−0.378477	−0.189238	−	0.981931i	$-0.560602\pi$
$797$	−10.6848	−0.378477	−0.189238	+	0.981931i	$0.560602\pi$
$798$	0	0
$799$	5.19995	0.183961
$800$	0	0
$801$	−8.14782	−0.287889
$802$	0	0
$803$	−10.7860	−0.380631
$804$	0	0
$805$	−21.5949	−0.761120
$806$	0	0
$807$	−0.678070	−0.0238692
$808$	0	0
$809$	−31.6589	−1.11307	−0.556534	−	0.830825i	$-0.687869\pi$
$809$	−31.6589	−1.11307	−0.556534	+	0.830825i	$0.687869\pi$
$810$	0	0
$811$	28.7987	1.01126	0.505630	−	0.862751i	$-0.331260\pi$
$811$	28.7987	1.01126	0.505630	+	0.862751i	$0.331260\pi$
$812$	0	0
$813$	−16.7191	−0.586365
$814$	0	0
$815$	44.7406	1.56720
$816$	0	0
$817$	−27.0313	−0.945705
$818$	0	0
$819$	−20.4889	−0.715938
$820$	0	0
$821$	−13.5509	−0.472931	−0.236465	−	0.971640i	$-0.575989\pi$
$821$	−13.5509	−0.472931	−0.236465	+	0.971640i	$0.575989\pi$
$822$	0	0
$823$	50.8105	1.77114	0.885571	−	0.464505i	$-0.153768\pi$
$823$	50.8105	1.77114	0.885571	+	0.464505i	$0.153768\pi$
$824$	0	0
$825$	−8.89749	−0.309771
$826$	0	0
$827$	6.92950	0.240962	0.120481	−	0.992716i	$-0.461556\pi$
$827$	6.92950	0.240962	0.120481	+	0.992716i	$0.461556\pi$
$828$	0	0
$829$	−8.48179	−0.294585	−0.147292	−	0.989093i	$-0.547056\pi$
$829$	−8.48179	−0.294585	−0.147292	+	0.989093i	$0.547056\pi$
$830$	0	0
$831$	−16.7183	−0.579951
$832$	0	0
$833$	3.92004	0.135821
$834$	0	0
$835$	−38.7602	−1.34135
$836$	0	0
$837$	−4.45545	−0.154003
$838$	0	0
$839$	−50.6884	−1.74996	−0.874980	−	0.484160i	$-0.839125\pi$
$839$	−50.6884	−1.74996	−0.874980	+	0.484160i	$0.839125\pi$
$840$	0	0
$841$	79.1983	2.73098
$842$	0	0
$843$	−31.5901	−1.08802
$844$	0	0
$845$	−53.4382	−1.83833
$846$	0	0
$847$	−2.70405	−0.0929123
$848$	0	0
$849$	16.1049	0.552719
$850$	0	0
$851$	45.5138	1.56019
$852$	0	0
$853$	7.12236	0.243865	0.121933	−	0.992538i	$-0.461091\pi$
$853$	7.12236	0.243865	0.121933	+	0.992538i	$0.461091\pi$
$854$	0	0
$855$	−18.3991	−0.629235
$856$	0	0
$857$	−27.5301	−0.940409	−0.470204	−	0.882558i	$-0.655820\pi$
$857$	−27.5301	−0.940409	−0.470204	+	0.882558i	$0.655820\pi$
$858$	0	0
$859$	17.4725	0.596154	0.298077	−	0.954542i	$-0.403655\pi$
$859$	17.4725	0.596154	0.298077	+	0.954542i	$0.403655\pi$
$860$	0	0
$861$	14.8555	0.506275
$862$	0	0
$863$	40.7655	1.38767	0.693836	−	0.720133i	$-0.255919\pi$
$863$	40.7655	1.38767	0.693836	+	0.720133i	$0.255919\pi$
$864$	0	0
$865$	58.3450	1.98379
$866$	0	0
$867$	0.974333	0.0330901
$868$	0	0
$869$	48.8454	1.65697
$870$	0	0
$871$	30.0791	1.01919
$872$	0	0
$873$	32.5800	1.10267
$874$	0	0
$875$	−11.6983	−0.395473
$876$	0	0
$877$	−29.3881	−0.992367	−0.496184	−	0.868218i	$-0.665266\pi$
$877$	−29.3881	−0.992367	−0.496184	+	0.868218i	$0.665266\pi$
$878$	0	0
$879$	10.1114	0.341048
$880$	0	0
$881$	−13.3691	−0.450416	−0.225208	−	0.974311i	$-0.572306\pi$
$881$	−13.3691	−0.450416	−0.225208	+	0.974311i	$0.572306\pi$
$882$	0	0
$883$	40.0014	1.34615	0.673077	−	0.739573i	$-0.264972\pi$
$883$	40.0014	1.34615	0.673077	+	0.739573i	$0.264972\pi$
$884$	0	0
$885$	−2.68228	−0.0901639
$886$	0	0
$887$	23.4670	0.787946	0.393973	−	0.919122i	$-0.371100\pi$
$887$	23.4670	0.787946	0.393973	+	0.919122i	$0.371100\pi$
$888$	0	0
$889$	5.91842	0.198498
$890$	0	0
$891$	−4.80658	−0.161026
$892$	0	0
$893$	16.9473	0.567121
$894$	0	0
$895$	41.1931	1.37693
$896$	0	0
$897$	24.7935	0.827831
$898$	0	0
$899$	9.41769	0.314098
$900$	0	0
$901$	1.69997	0.0566343
$902$	0	0
$903$	−14.1822	−0.471955
$904$	0	0
$905$	−38.8614	−1.29180
$906$	0	0
$907$	−1.46667	−0.0486999	−0.0243500	−	0.999703i	$-0.507752\pi$
$907$	−1.46667	−0.0486999	−0.0243500	+	0.999703i	$0.507752\pi$
$908$	0	0
$909$	−21.7304	−0.720752
$910$	0	0
$911$	−7.57758	−0.251056	−0.125528	−	0.992090i	$-0.540063\pi$
$911$	−7.57758	−0.251056	−0.125528	+	0.992090i	$0.540063\pi$
$912$	0	0
$913$	58.6587	1.94132
$914$	0	0
$915$	27.1111	0.896266
$916$	0	0
$917$	−6.39434	−0.211160
$918$	0	0
$919$	12.5144	0.412812	0.206406	−	0.978466i	$-0.433823\pi$
$919$	12.5144	0.412812	0.206406	+	0.978466i	$0.433823\pi$
$920$	0	0
$921$	−11.0457	−0.363969
$922$	0	0
$923$	−25.6030	−0.842732
$924$	0	0
$925$	−26.2578	−0.863352
$926$	0	0
$927$	−35.7039	−1.17267
$928$	0	0
$929$	4.03651	0.132434	0.0662168	−	0.997805i	$-0.478907\pi$
$929$	4.03651	0.132434	0.0662168	+	0.997805i	$0.478907\pi$
$930$	0	0
$931$	12.7759	0.418715
$932$	0	0
$933$	−9.82715	−0.321726
$934$	0	0
$935$	−9.74898	−0.318826
$936$	0	0
$937$	−29.7876	−0.973119	−0.486560	−	0.873647i	$-0.661748\pi$
$937$	−29.7876	−0.973119	−0.486560	+	0.873647i	$0.661748\pi$
$938$	0	0
$939$	−7.71475	−0.251761
$940$	0	0
$941$	−5.59510	−0.182395	−0.0911975	−	0.995833i	$-0.529069\pi$
$941$	−5.59510	−0.182395	−0.0911975	+	0.995833i	$0.529069\pi$
$942$	0	0
$943$	38.8320	1.26454
$944$	0	0
$945$	−23.7753	−0.773411
$946$	0	0
$947$	−9.80501	−0.318620	−0.159310	−	0.987229i	$-0.550927\pi$
$947$	−9.80501	−0.318620	−0.159310	+	0.987229i	$0.550927\pi$
$948$	0	0
$949$	−17.3399	−0.562878
$950$	0	0
$951$	−25.3994	−0.823631
$952$	0	0
$953$	−51.2530	−1.66025	−0.830124	−	0.557578i	$-0.811731\pi$
$953$	−51.2530	−1.66025	−0.830124	+	0.557578i	$0.811731\pi$
$954$	0	0
$955$	7.43419	0.240565
$956$	0	0
$957$	−35.8905	−1.16018
$958$	0	0
$959$	−27.6800	−0.893835
$960$	0	0
$961$	−30.1803	−0.973557
$962$	0	0
$963$	38.3002	1.23421
$964$	0	0
$965$	−47.6949	−1.53535
$966$	0	0
$967$	11.8882	0.382299	0.191150	−	0.981561i	$-0.438778\pi$
$967$	11.8882	0.382299	0.191150	+	0.981561i	$0.438778\pi$
$968$	0	0
$969$	3.17548	0.102011
$970$	0	0
$971$	−31.9223	−1.02443	−0.512217	−	0.858856i	$-0.671176\pi$
$971$	−31.9223	−1.02443	−0.512217	+	0.858856i	$0.671176\pi$
$972$	0	0
$973$	−16.6841	−0.534868
$974$	0	0
$975$	−14.3039	−0.458090
$976$	0	0
$977$	−61.0754	−1.95397	−0.976987	−	0.213298i	$-0.931580\pi$
$977$	−61.0754	−1.95397	−0.976987	+	0.213298i	$0.931580\pi$
$978$	0	0
$979$	−14.0704	−0.449693
$980$	0	0
$981$	−39.9728	−1.27623
$982$	0	0
$983$	−10.7120	−0.341659	−0.170829	−	0.985301i	$-0.554645\pi$
$983$	−10.7120	−0.341659	−0.170829	+	0.985301i	$0.554645\pi$
$984$	0	0
$985$	−26.2883	−0.837613
$986$	0	0
$987$	8.89159	0.283022
$988$	0	0
$989$	−37.0720	−1.17882
$990$	0	0
$991$	−16.9125	−0.537244	−0.268622	−	0.963246i	$-0.586568\pi$
$991$	−16.9125	−0.537244	−0.268622	+	0.963246i	$0.586568\pi$
$992$	0	0
$993$	19.1845	0.608802
$994$	0	0
$995$	−14.6772	−0.465297
$996$	0	0
$997$	−7.52845	−0.238428	−0.119214	−	0.992869i	$-0.538038\pi$
$997$	−7.52845	−0.238428	−0.119214	+	0.992869i	$0.538038\pi$
$998$	0	0
$999$	50.1093	1.58539

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.ba.1.19	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.ba.1.19	✓	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q+0.974333 q^{3} -2.75294 q^{5} -1.75498 q^{7} -2.05067 q^{9} +O(q^{10})\)	\(q+0.974333 q^{3} -2.75294 q^{5} -1.75498 q^{7} -2.05067 q^{9} -3.54130 q^{11} -5.69309 q^{13} -2.68228 q^{15} -1.00000 q^{17} -3.25913 q^{19} -1.70994 q^{21} -4.46974 q^{23} +2.57868 q^{25} -4.92104 q^{27} +10.4018 q^{29} +0.905387 q^{31} -3.45040 q^{33} +4.83136 q^{35} -10.1827 q^{37} -5.54697 q^{39} -8.68776 q^{41} +8.29401 q^{43} +5.64539 q^{45} -5.19995 q^{47} -3.92004 q^{49} -0.974333 q^{51} -1.69997 q^{53} +9.74898 q^{55} -3.17548 q^{57} +1.00000 q^{59} -10.1075 q^{61} +3.59889 q^{63} +15.6728 q^{65} -5.28344 q^{67} -4.35501 q^{69} +4.49720 q^{71} +3.04578 q^{73} +2.51249 q^{75} +6.21491 q^{77} -13.7931 q^{79} +1.35729 q^{81} -16.5642 q^{83} +2.75294 q^{85} +10.1349 q^{87} +3.97324 q^{89} +9.99127 q^{91} +0.882149 q^{93} +8.97220 q^{95} -15.8875 q^{97} +7.26205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9}+O(q^{10}) \) 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9	\( 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9} + 3 q^{11} + 9 q^{13} + 14 q^{15} - 30 q^{17} + 24 q^{19} + 7 q^{21} + 9 q^{23} + 40 q^{25} + 19 q^{27} + 9 q^{29} + 11 q^{31} - 14 q^{33} + 30 q^{35} - 13 q^{37} + 16 q^{39} - 13 q^{41} + 23 q^{43} + 12 q^{45} + 43 q^{47} + 35 q^{49} - 4 q^{51} - 4 q^{53} + 43 q^{55} + 3 q^{57} + 30 q^{59} + 43 q^{61} + 38 q^{63} + 3 q^{65} + 50 q^{67} + 34 q^{69} + 3 q^{71} - 16 q^{73} + 21 q^{75} + 18 q^{77} + 45 q^{79} + 6 q^{81} + 63 q^{83} - 2 q^{85} + 42 q^{87} + 6 q^{89} + 22 q^{91} - 2 q^{93} + 19 q^{95} - 28 q^{97} + 51 q^{99}+O(q^{100}) \) 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9 + 3 * q^11 + 9 * q^13 + 14 * q^15 - 30 * q^17 + 24 * q^19 + 7 * q^21 + 9 * q^23 + 40 * q^25 + 19 * q^27 + 9 * q^29 + 11 * q^31 - 14 * q^33 + 30 * q^35 - 13 * q^37 + 16 * q^39 - 13 * q^41 + 23 * q^43 + 12 * q^45 + 43 * q^47 + 35 * q^49 - 4 * q^51 - 4 * q^53 + 43 * q^55 + 3 * q^57 + 30 * q^59 + 43 * q^61 + 38 * q^63 + 3 * q^65 + 50 * q^67 + 34 * q^69 + 3 * q^71 - 16 * q^73 + 21 * q^75 + 18 * q^77 + 45 * q^79 + 6 * q^81 + 63 * q^83 - 2 * q^85 + 42 * q^87 + 6 * q^89 + 22 * q^91 - 2 * q^93 + 19 * q^95 - 28 * q^97 + 51 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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