LMFDB - Embedded newform 4024.2.a.g.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.950380 q^{3} +3.01565 q^{5} +0.279316 q^{7} -2.09678 q^{9} +O(q^{10})$	$q-0.950380 q^{3} +3.01565 q^{5} +0.279316 q^{7} -2.09678 q^{9} +3.32528 q^{11} +2.71425 q^{13} -2.86601 q^{15} +6.56645 q^{17} +6.07908 q^{19} -0.265456 q^{21} +6.35649 q^{23} +4.09414 q^{25} +4.84388 q^{27} -4.96003 q^{29} -8.98530 q^{31} -3.16028 q^{33} +0.842319 q^{35} -5.88730 q^{37} -2.57957 q^{39} -5.52886 q^{41} +12.0965 q^{43} -6.32315 q^{45} +3.77650 q^{47} -6.92198 q^{49} -6.24062 q^{51} -12.2575 q^{53} +10.0279 q^{55} -5.77743 q^{57} +5.60065 q^{59} +6.29552 q^{61} -0.585664 q^{63} +8.18522 q^{65} +13.3597 q^{67} -6.04108 q^{69} -12.2422 q^{71} +0.810178 q^{73} -3.89099 q^{75} +0.928804 q^{77} -12.8938 q^{79} +1.68681 q^{81} -2.69273 q^{83} +19.8021 q^{85} +4.71391 q^{87} +3.51416 q^{89} +0.758133 q^{91} +8.53945 q^{93} +18.3324 q^{95} +0.120259 q^{97} -6.97237 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * 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
$3$	−0.950380	−0.548702	−0.274351	−	0.961630i	$-0.588463\pi$
$3$	−0.950380	−0.548702	−0.274351	+	0.961630i	$0.588463\pi$
$4$	0	0
$5$	3.01565	1.34864	0.674320	−	0.738440i	$-0.264437\pi$
$5$	3.01565	1.34864	0.674320	+	0.738440i	$0.264437\pi$
$6$	0	0
$7$	0.279316	0.105572	0.0527858	−	0.998606i	$-0.483190\pi$
$7$	0.279316	0.105572	0.0527858	+	0.998606i	$0.483190\pi$
$8$	0	0
$9$	−2.09678	−0.698926
$10$	0	0
$11$	3.32528	1.00261	0.501305	−	0.865271i	$-0.332854\pi$
$11$	3.32528	1.00261	0.501305	+	0.865271i	$0.332854\pi$
$12$	0	0
$13$	2.71425	0.752797	0.376398	−	0.926458i	$-0.377162\pi$
$13$	2.71425	0.752797	0.376398	+	0.926458i	$0.377162\pi$
$14$	0	0
$15$	−2.86601	−0.740001
$16$	0	0
$17$	6.56645	1.59260	0.796299	−	0.604903i	$-0.206788\pi$
$17$	6.56645	1.59260	0.796299	+	0.604903i	$0.206788\pi$
$18$	0	0
$19$	6.07908	1.39464	0.697318	−	0.716762i	$-0.254377\pi$
$19$	6.07908	1.39464	0.697318	+	0.716762i	$0.254377\pi$
$20$	0	0
$21$	−0.265456	−0.0579273
$22$	0	0
$23$	6.35649	1.32542	0.662710	−	0.748876i	$-0.269406\pi$
$23$	6.35649	1.32542	0.662710	+	0.748876i	$0.269406\pi$
$24$	0	0
$25$	4.09414	0.818828
$26$	0	0
$27$	4.84388	0.932204
$28$	0	0
$29$	−4.96003	−0.921054	−0.460527	−	0.887646i	$-0.652339\pi$
$29$	−4.96003	−0.921054	−0.460527	+	0.887646i	$0.652339\pi$
$30$	0	0
$31$	−8.98530	−1.61381	−0.806904	−	0.590683i	$-0.798858\pi$
$31$	−8.98530	−1.61381	−0.806904	+	0.590683i	$0.798858\pi$
$32$	0	0
$33$	−3.16028	−0.550134
$34$	0	0
$35$	0.842319	0.142378
$36$	0	0
$37$	−5.88730	−0.967866	−0.483933	−	0.875105i	$-0.660792\pi$
$37$	−5.88730	−0.967866	−0.483933	+	0.875105i	$0.660792\pi$
$38$	0	0
$39$	−2.57957	−0.413061
$40$	0	0
$41$	−5.52886	−0.863462	−0.431731	−	0.902002i	$-0.642097\pi$
$41$	−5.52886	−0.863462	−0.431731	+	0.902002i	$0.642097\pi$
$42$	0	0
$43$	12.0965	1.84470	0.922350	−	0.386356i	$-0.126266\pi$
$43$	12.0965	1.84470	0.922350	+	0.386356i	$0.126266\pi$
$44$	0	0
$45$	−6.32315	−0.942599
$46$	0	0
$47$	3.77650	0.550859	0.275430	−	0.961321i	$-0.411180\pi$
$47$	3.77650	0.550859	0.275430	+	0.961321i	$0.411180\pi$
$48$	0	0
$49$	−6.92198	−0.988855
$50$	0	0
$51$	−6.24062	−0.873862
$52$	0	0
$53$	−12.2575	−1.68369	−0.841845	−	0.539719i	$-0.818531\pi$
$53$	−12.2575	−1.68369	−0.841845	+	0.539719i	$0.818531\pi$
$54$	0	0
$55$	10.0279	1.35216
$56$	0	0
$57$	−5.77743	−0.765240
$58$	0	0
$59$	5.60065	0.729142	0.364571	−	0.931176i	$-0.381216\pi$
$59$	5.60065	0.729142	0.364571	+	0.931176i	$0.381216\pi$
$60$	0	0
$61$	6.29552	0.806059	0.403030	−	0.915187i	$-0.367957\pi$
$61$	6.29552	0.806059	0.403030	+	0.915187i	$0.367957\pi$
$62$	0	0
$63$	−0.585664	−0.0737867
$64$	0	0
$65$	8.18522	1.01525
$66$	0	0
$67$	13.3597	1.63214	0.816071	−	0.577952i	$-0.196148\pi$
$67$	13.3597	1.63214	0.816071	+	0.577952i	$0.196148\pi$
$68$	0	0
$69$	−6.04108	−0.727261
$70$	0	0
$71$	−12.2422	−1.45288	−0.726440	−	0.687229i	$-0.758827\pi$
$71$	−12.2422	−1.45288	−0.726440	+	0.687229i	$0.758827\pi$
$72$	0	0
$73$	0.810178	0.0948241	0.0474121	−	0.998875i	$-0.484903\pi$
$73$	0.810178	0.0948241	0.0474121	+	0.998875i	$0.484903\pi$
$74$	0	0
$75$	−3.89099	−0.449293
$76$	0	0
$77$	0.928804	0.105847
$78$	0	0
$79$	−12.8938	−1.45067	−0.725335	−	0.688396i	$-0.758315\pi$
$79$	−12.8938	−1.45067	−0.725335	+	0.688396i	$0.758315\pi$
$80$	0	0
$81$	1.68681	0.187423
$82$	0	0
$83$	−2.69273	−0.295566	−0.147783	−	0.989020i	$-0.547214\pi$
$83$	−2.69273	−0.295566	−0.147783	+	0.989020i	$0.547214\pi$
$84$	0	0
$85$	19.8021	2.14784
$86$	0	0
$87$	4.71391	0.505384
$88$	0	0
$89$	3.51416	0.372500	0.186250	−	0.982502i	$-0.440367\pi$
$89$	3.51416	0.372500	0.186250	+	0.982502i	$0.440367\pi$
$90$	0	0
$91$	0.758133	0.0794739
$92$	0	0
$93$	8.53945	0.885500
$94$	0	0
$95$	18.3324	1.88086
$96$	0	0
$97$	0.120259	0.0122104	0.00610521	−	0.999981i	$-0.498057\pi$
$97$	0.120259	0.0122104	0.00610521	+	0.999981i	$0.498057\pi$
$98$	0	0
$99$	−6.97237	−0.700750
$100$	0	0
$101$	13.9800	1.39106	0.695529	−	0.718498i	$-0.255170\pi$
$101$	13.9800	1.39106	0.695529	+	0.718498i	$0.255170\pi$
$102$	0	0
$103$	−3.34037	−0.329137	−0.164568	−	0.986366i	$-0.552623\pi$
$103$	−3.34037	−0.329137	−0.164568	+	0.986366i	$0.552623\pi$
$104$	0	0
$105$	−0.800523	−0.0781231
$106$	0	0
$107$	3.32358	0.321303	0.160651	−	0.987011i	$-0.448641\pi$
$107$	3.32358	0.321303	0.160651	+	0.987011i	$0.448641\pi$
$108$	0	0
$109$	−10.5163	−1.00728	−0.503642	−	0.863913i	$-0.668007\pi$
$109$	−10.5163	−1.00728	−0.503642	+	0.863913i	$0.668007\pi$
$110$	0	0
$111$	5.59517	0.531070
$112$	0	0
$113$	15.3417	1.44323	0.721614	−	0.692295i	$-0.243400\pi$
$113$	15.3417	1.44323	0.721614	+	0.692295i	$0.243400\pi$
$114$	0	0
$115$	19.1689	1.78751
$116$	0	0
$117$	−5.69117	−0.526149
$118$	0	0
$119$	1.83411	0.168133
$120$	0	0
$121$	0.0574821	0.00522565
$122$	0	0
$123$	5.25451	0.473784
$124$	0	0
$125$	−2.73176	−0.244336
$126$	0	0
$127$	5.84743	0.518876	0.259438	−	0.965760i	$-0.416463\pi$
$127$	5.84743	0.518876	0.259438	+	0.965760i	$0.416463\pi$
$128$	0	0
$129$	−11.4963	−1.01219
$130$	0	0
$131$	−6.49361	−0.567350	−0.283675	−	0.958921i	$-0.591554\pi$
$131$	−6.49361	−0.567350	−0.283675	+	0.958921i	$0.591554\pi$
$132$	0	0
$133$	1.69798	0.147234
$134$	0	0
$135$	14.6074	1.25721
$136$	0	0
$137$	−9.92065	−0.847578	−0.423789	−	0.905761i	$-0.639300\pi$
$137$	−9.92065	−0.847578	−0.423789	+	0.905761i	$0.639300\pi$
$138$	0	0
$139$	14.7835	1.25392	0.626959	−	0.779053i	$-0.284300\pi$
$139$	14.7835	1.25392	0.626959	+	0.779053i	$0.284300\pi$
$140$	0	0
$141$	−3.58911	−0.302258
$142$	0	0
$143$	9.02563	0.754761
$144$	0	0
$145$	−14.9577	−1.24217
$146$	0	0
$147$	6.57851	0.542587
$148$	0	0
$149$	−13.7535	−1.12673	−0.563366	−	0.826207i	$-0.690494\pi$
$149$	−13.7535	−1.12673	−0.563366	+	0.826207i	$0.690494\pi$
$150$	0	0
$151$	−5.31081	−0.432188	−0.216094	−	0.976373i	$-0.569332\pi$
$151$	−5.31081	−0.432188	−0.216094	+	0.976373i	$0.569332\pi$
$152$	0	0
$153$	−13.7684	−1.11311
$154$	0	0
$155$	−27.0965	−2.17644
$156$	0	0
$157$	12.2484	0.977531	0.488766	−	0.872415i	$-0.337447\pi$
$157$	12.2484	0.977531	0.488766	+	0.872415i	$0.337447\pi$
$158$	0	0
$159$	11.6492	0.923845
$160$	0	0
$161$	1.77547	0.139927
$162$	0	0
$163$	13.2357	1.03670	0.518351	−	0.855168i	$-0.326546\pi$
$163$	13.2357	1.03670	0.518351	+	0.855168i	$0.326546\pi$
$164$	0	0
$165$	−9.53029	−0.741932
$166$	0	0
$167$	−3.87061	−0.299517	−0.149758	−	0.988723i	$-0.547850\pi$
$167$	−3.87061	−0.299517	−0.149758	+	0.988723i	$0.547850\pi$
$168$	0	0
$169$	−5.63286	−0.433297
$170$	0	0
$171$	−12.7465	−0.974747
$172$	0	0
$173$	3.18082	0.241833	0.120917	−	0.992663i	$-0.461417\pi$
$173$	3.18082	0.241833	0.120917	+	0.992663i	$0.461417\pi$
$174$	0	0
$175$	1.14356	0.0864449
$176$	0	0
$177$	−5.32274	−0.400082
$178$	0	0
$179$	2.07767	0.155292	0.0776460	−	0.996981i	$-0.475260\pi$
$179$	2.07767	0.155292	0.0776460	+	0.996981i	$0.475260\pi$
$180$	0	0
$181$	17.9233	1.33223	0.666114	−	0.745850i	$-0.267956\pi$
$181$	17.9233	1.33223	0.666114	+	0.745850i	$0.267956\pi$
$182$	0	0
$183$	−5.98314	−0.442286
$184$	0	0
$185$	−17.7540	−1.30530
$186$	0	0
$187$	21.8353	1.59675
$188$	0	0
$189$	1.35297	0.0984142
$190$	0	0
$191$	−1.47243	−0.106542	−0.0532708	−	0.998580i	$-0.516965\pi$
$191$	−1.47243	−0.106542	−0.0532708	+	0.998580i	$0.516965\pi$
$192$	0	0
$193$	15.7175	1.13137	0.565686	−	0.824621i	$-0.308612\pi$
$193$	15.7175	1.13137	0.565686	+	0.824621i	$0.308612\pi$
$194$	0	0
$195$	−7.77907	−0.557071
$196$	0	0
$197$	−16.7884	−1.19612	−0.598061	−	0.801450i	$-0.704062\pi$
$197$	−16.7884	−1.19612	−0.598061	+	0.801450i	$0.704062\pi$
$198$	0	0
$199$	−1.72223	−0.122086	−0.0610428	−	0.998135i	$-0.519443\pi$
$199$	−1.72223	−0.122086	−0.0610428	+	0.998135i	$0.519443\pi$
$200$	0	0
$201$	−12.6967	−0.895560
$202$	0	0
$203$	−1.38541	−0.0972370
$204$	0	0
$205$	−16.6731	−1.16450
$206$	0	0
$207$	−13.3281	−0.926370
$208$	0	0
$209$	20.2146	1.39827
$210$	0	0
$211$	2.13360	0.146883	0.0734415	−	0.997300i	$-0.476602\pi$
$211$	2.13360	0.146883	0.0734415	+	0.997300i	$0.476602\pi$
$212$	0	0
$213$	11.6347	0.797199
$214$	0	0
$215$	36.4788	2.48783
$216$	0	0
$217$	−2.50974	−0.170372
$218$	0	0
$219$	−0.769977	−0.0520302
$220$	0	0
$221$	17.8230	1.19890
$222$	0	0
$223$	−12.2477	−0.820166	−0.410083	−	0.912048i	$-0.634500\pi$
$223$	−12.2477	−0.820166	−0.410083	+	0.912048i	$0.634500\pi$
$224$	0	0
$225$	−8.58450	−0.572300
$226$	0	0
$227$	−16.6511	−1.10517	−0.552586	−	0.833456i	$-0.686359\pi$
$227$	−16.6511	−1.10517	−0.552586	+	0.833456i	$0.686359\pi$
$228$	0	0
$229$	21.9107	1.44790	0.723950	−	0.689853i	$-0.242325\pi$
$229$	21.9107	1.44790	0.723950	+	0.689853i	$0.242325\pi$
$230$	0	0
$231$	−0.882716	−0.0580785
$232$	0	0
$233$	−18.6747	−1.22342	−0.611711	−	0.791081i	$-0.709518\pi$
$233$	−18.6747	−1.22342	−0.611711	+	0.791081i	$0.709518\pi$
$234$	0	0
$235$	11.3886	0.742910
$236$	0	0
$237$	12.2541	0.795986
$238$	0	0
$239$	6.84459	0.442740	0.221370	−	0.975190i	$-0.428947\pi$
$239$	6.84459	0.442740	0.221370	+	0.975190i	$0.428947\pi$
$240$	0	0
$241$	0.728541	0.0469295	0.0234647	−	0.999725i	$-0.492530\pi$
$241$	0.728541	0.0469295	0.0234647	+	0.999725i	$0.492530\pi$
$242$	0	0
$243$	−16.1347	−1.03504
$244$	0	0
$245$	−20.8743	−1.33361
$246$	0	0
$247$	16.5001	1.04988
$248$	0	0
$249$	2.55912	0.162178
$250$	0	0
$251$	12.0139	0.758308	0.379154	−	0.925334i	$-0.376215\pi$
$251$	12.0139	0.758308	0.379154	+	0.925334i	$0.376215\pi$
$252$	0	0
$253$	21.1371	1.32888
$254$	0	0
$255$	−18.8195	−1.17852
$256$	0	0
$257$	15.7572	0.982906	0.491453	−	0.870904i	$-0.336466\pi$
$257$	15.7572	0.982906	0.491453	+	0.870904i	$0.336466\pi$
$258$	0	0
$259$	−1.64442	−0.102179
$260$	0	0
$261$	10.4001	0.643748
$262$	0	0
$263$	22.5069	1.38783	0.693917	−	0.720055i	$-0.255883\pi$
$263$	22.5069	1.38783	0.693917	+	0.720055i	$0.255883\pi$
$264$	0	0
$265$	−36.9642	−2.27069
$266$	0	0
$267$	−3.33978	−0.204391
$268$	0	0
$269$	−4.74464	−0.289286	−0.144643	−	0.989484i	$-0.546203\pi$
$269$	−4.74464	−0.289286	−0.144643	+	0.989484i	$0.546203\pi$
$270$	0	0
$271$	13.8074	0.838741	0.419371	−	0.907815i	$-0.362251\pi$
$271$	13.8074	0.838741	0.419371	+	0.907815i	$0.362251\pi$
$272$	0	0
$273$	−0.720514	−0.0436075
$274$	0	0
$275$	13.6142	0.820964
$276$	0	0
$277$	−30.9670	−1.86063	−0.930314	−	0.366763i	$-0.880466\pi$
$277$	−30.9670	−1.86063	−0.930314	+	0.366763i	$0.880466\pi$
$278$	0	0
$279$	18.8402	1.12793
$280$	0	0
$281$	−7.04572	−0.420313	−0.210156	−	0.977668i	$-0.567397\pi$
$281$	−7.04572	−0.420313	−0.210156	+	0.977668i	$0.567397\pi$
$282$	0	0
$283$	−1.87133	−0.111239	−0.0556195	−	0.998452i	$-0.517713\pi$
$283$	−1.87133	−0.111239	−0.0556195	+	0.998452i	$0.517713\pi$
$284$	0	0
$285$	−17.4227	−1.03203
$286$	0	0
$287$	−1.54430	−0.0911570
$288$	0	0
$289$	26.1183	1.53637
$290$	0	0
$291$	−0.114291	−0.00669988
$292$	0	0
$293$	33.0509	1.93085	0.965427	−	0.260672i	$-0.0839441\pi$
$293$	33.0509	1.93085	0.965427	+	0.260672i	$0.0839441\pi$
$294$	0	0
$295$	16.8896	0.983350
$296$	0	0
$297$	16.1072	0.934637
$298$	0	0
$299$	17.2531	0.997772
$300$	0	0
$301$	3.37875	0.194748
$302$	0	0
$303$	−13.2863	−0.763276
$304$	0	0
$305$	18.9851	1.08708
$306$	0	0
$307$	−8.41748	−0.480411	−0.240206	−	0.970722i	$-0.577215\pi$
$307$	−8.41748	−0.480411	−0.240206	+	0.970722i	$0.577215\pi$
$308$	0	0
$309$	3.17462	0.180598
$310$	0	0
$311$	−16.0630	−0.910850	−0.455425	−	0.890274i	$-0.650513\pi$
$311$	−16.0630	−0.910850	−0.455425	+	0.890274i	$0.650513\pi$
$312$	0	0
$313$	−10.9158	−0.616998	−0.308499	−	0.951225i	$-0.599827\pi$
$313$	−10.9158	−0.616998	−0.308499	+	0.951225i	$0.599827\pi$
$314$	0	0
$315$	−1.76616	−0.0995116
$316$	0	0
$317$	33.2094	1.86522	0.932612	−	0.360881i	$-0.117524\pi$
$317$	33.2094	1.86522	0.932612	+	0.360881i	$0.117524\pi$
$318$	0	0
$319$	−16.4935	−0.923457
$320$	0	0
$321$	−3.15867	−0.176299
$322$	0	0
$323$	39.9179	2.22109
$324$	0	0
$325$	11.1125	0.616411
$326$	0	0
$327$	9.99453	0.552699
$328$	0	0
$329$	1.05484	0.0581550
$330$	0	0
$331$	12.6341	0.694434	0.347217	−	0.937785i	$-0.387127\pi$
$331$	12.6341	0.694434	0.347217	+	0.937785i	$0.387127\pi$
$332$	0	0
$333$	12.3444	0.676466
$334$	0	0
$335$	40.2880	2.20117
$336$	0	0
$337$	−13.0269	−0.709620	−0.354810	−	0.934938i	$-0.615454\pi$
$337$	−13.0269	−0.709620	−0.354810	+	0.934938i	$0.615454\pi$
$338$	0	0
$339$	−14.5805	−0.791903
$340$	0	0
$341$	−29.8786	−1.61802
$342$	0	0
$343$	−3.88863	−0.209966
$344$	0	0
$345$	−18.2178	−0.980812
$346$	0	0
$347$	36.2387	1.94540	0.972698	−	0.232074i	$-0.0745512\pi$
$347$	36.2387	1.94540	0.972698	+	0.232074i	$0.0745512\pi$
$348$	0	0
$349$	−5.48937	−0.293839	−0.146920	−	0.989148i	$-0.546936\pi$
$349$	−5.48937	−0.293839	−0.146920	+	0.989148i	$0.546936\pi$
$350$	0	0
$351$	13.1475	0.701760
$352$	0	0
$353$	−20.0809	−1.06880	−0.534399	−	0.845232i	$-0.679462\pi$
$353$	−20.0809	−1.06880	−0.534399	+	0.845232i	$0.679462\pi$
$354$	0	0
$355$	−36.9182	−1.95941
$356$	0	0
$357$	−1.74311	−0.0922549
$358$	0	0
$359$	−26.2886	−1.38746	−0.693729	−	0.720236i	$-0.744034\pi$
$359$	−26.2886	−1.38746	−0.693729	+	0.720236i	$0.744034\pi$
$360$	0	0
$361$	17.9552	0.945009
$362$	0	0
$363$	−0.0546299	−0.00286732
$364$	0	0
$365$	2.44321	0.127884
$366$	0	0
$367$	29.2549	1.52709	0.763547	−	0.645752i	$-0.223456\pi$
$367$	29.2549	1.52709	0.763547	+	0.645752i	$0.223456\pi$
$368$	0	0
$369$	11.5928	0.603496
$370$	0	0
$371$	−3.42370	−0.177750
$372$	0	0
$373$	−34.9194	−1.80806	−0.904030	−	0.427470i	$-0.859405\pi$
$373$	−34.9194	−1.80806	−0.904030	+	0.427470i	$0.859405\pi$
$374$	0	0
$375$	2.59621	0.134068
$376$	0	0
$377$	−13.4627	−0.693366
$378$	0	0
$379$	−21.1251	−1.08512	−0.542562	−	0.840015i	$-0.682546\pi$
$379$	−21.1251	−1.08512	−0.542562	+	0.840015i	$0.682546\pi$
$380$	0	0
$381$	−5.55728	−0.284708
$382$	0	0
$383$	−8.59403	−0.439135	−0.219567	−	0.975597i	$-0.570465\pi$
$383$	−8.59403	−0.439135	−0.219567	+	0.975597i	$0.570465\pi$
$384$	0	0
$385$	2.80095	0.142749
$386$	0	0
$387$	−25.3637	−1.28931
$388$	0	0
$389$	−10.0007	−0.507054	−0.253527	−	0.967328i	$-0.581591\pi$
$389$	−10.0007	−0.507054	−0.253527	+	0.967328i	$0.581591\pi$
$390$	0	0
$391$	41.7396	2.11086
$392$	0	0
$393$	6.17140	0.311306
$394$	0	0
$395$	−38.8833	−1.95643
$396$	0	0
$397$	11.3466	0.569469	0.284735	−	0.958606i	$-0.408095\pi$
$397$	11.3466	0.569469	0.284735	+	0.958606i	$0.408095\pi$
$398$	0	0
$399$	−1.61373	−0.0807875
$400$	0	0
$401$	−1.00441	−0.0501580	−0.0250790	−	0.999685i	$-0.507984\pi$
$401$	−1.00441	−0.0501580	−0.0250790	+	0.999685i	$0.507984\pi$
$402$	0	0
$403$	−24.3883	−1.21487
$404$	0	0
$405$	5.08683	0.252767
$406$	0	0
$407$	−19.5769	−0.970391
$408$	0	0
$409$	16.3654	0.809219	0.404609	−	0.914490i	$-0.367408\pi$
$409$	16.3654	0.809219	0.404609	+	0.914490i	$0.367408\pi$
$410$	0	0
$411$	9.42839	0.465068
$412$	0	0
$413$	1.56435	0.0769767
$414$	0	0
$415$	−8.12034	−0.398612
$416$	0	0
$417$	−14.0499	−0.688027
$418$	0	0
$419$	−2.32915	−0.113786	−0.0568931	−	0.998380i	$-0.518119\pi$
$419$	−2.32915	−0.113786	−0.0568931	+	0.998380i	$0.518119\pi$
$420$	0	0
$421$	11.2611	0.548830	0.274415	−	0.961611i	$-0.411516\pi$
$421$	11.2611	0.548830	0.274415	+	0.961611i	$0.411516\pi$
$422$	0	0
$423$	−7.91848	−0.385010
$424$	0	0
$425$	26.8840	1.30406
$426$	0	0
$427$	1.75844	0.0850969
$428$	0	0
$429$	−8.57778	−0.414139
$430$	0	0
$431$	31.8318	1.53328	0.766642	−	0.642075i	$-0.221926\pi$
$431$	31.8318	1.53328	0.766642	+	0.642075i	$0.221926\pi$
$432$	0	0
$433$	15.9886	0.768365	0.384183	−	0.923257i	$-0.374483\pi$
$433$	15.9886	0.768365	0.384183	+	0.923257i	$0.374483\pi$
$434$	0	0
$435$	14.2155	0.681581
$436$	0	0
$437$	38.6416	1.84848
$438$	0	0
$439$	−12.3182	−0.587917	−0.293959	−	0.955818i	$-0.594973\pi$
$439$	−12.3182	−0.587917	−0.293959	+	0.955818i	$0.594973\pi$
$440$	0	0
$441$	14.5139	0.691136
$442$	0	0
$443$	−36.8074	−1.74877	−0.874386	−	0.485231i	$-0.838736\pi$
$443$	−36.8074	−1.74877	−0.874386	+	0.485231i	$0.838736\pi$
$444$	0	0
$445$	10.5975	0.502368
$446$	0	0
$447$	13.0711	0.618241
$448$	0	0
$449$	−24.1335	−1.13893	−0.569465	−	0.822015i	$-0.692850\pi$
$449$	−24.1335	−1.13893	−0.569465	+	0.822015i	$0.692850\pi$
$450$	0	0
$451$	−18.3850	−0.865715
$452$	0	0
$453$	5.04729	0.237142
$454$	0	0
$455$	2.28626	0.107182
$456$	0	0
$457$	−9.87744	−0.462047	−0.231024	−	0.972948i	$-0.574207\pi$
$457$	−9.87744	−0.462047	−0.231024	+	0.972948i	$0.574207\pi$
$458$	0	0
$459$	31.8071	1.48463
$460$	0	0
$461$	0.0308211	0.00143548	0.000717741	−	1.00000i	$-0.499772\pi$
$461$	0.0308211	0.00143548	0.000717741		1.00000i	$0.499772\pi$
$462$	0	0
$463$	−26.7101	−1.24132	−0.620662	−	0.784078i	$-0.713136\pi$
$463$	−26.7101	−1.24132	−0.620662	+	0.784078i	$0.713136\pi$
$464$	0	0
$465$	25.7520	1.19422
$466$	0	0
$467$	40.6205	1.87969	0.939847	−	0.341597i	$-0.110968\pi$
$467$	40.6205	1.87969	0.939847	+	0.341597i	$0.110968\pi$
$468$	0	0
$469$	3.73157	0.172308
$470$	0	0
$471$	−11.6407	−0.536373
$472$	0	0
$473$	40.2243	1.84951
$474$	0	0
$475$	24.8886	1.14197
$476$	0	0
$477$	25.7012	1.17678
$478$	0	0
$479$	25.9930	1.18765	0.593825	−	0.804595i	$-0.297617\pi$
$479$	25.9930	1.18765	0.593825	+	0.804595i	$0.297617\pi$
$480$	0	0
$481$	−15.9796	−0.728606
$482$	0	0
$483$	−1.68737	−0.0767780
$484$	0	0
$485$	0.362658	0.0164674
$486$	0	0
$487$	28.4716	1.29017	0.645086	−	0.764110i	$-0.276821\pi$
$487$	28.4716	1.29017	0.645086	+	0.764110i	$0.276821\pi$
$488$	0	0
$489$	−12.5790	−0.568841
$490$	0	0
$491$	3.09464	0.139659	0.0698295	−	0.997559i	$-0.477754\pi$
$491$	3.09464	0.139659	0.0698295	+	0.997559i	$0.477754\pi$
$492$	0	0
$493$	−32.5698	−1.46687
$494$	0	0
$495$	−21.0262	−0.945059
$496$	0	0
$497$	−3.41944	−0.153383
$498$	0	0
$499$	9.01807	0.403704	0.201852	−	0.979416i	$-0.435304\pi$
$499$	9.01807	0.403704	0.201852	+	0.979416i	$0.435304\pi$
$500$	0	0
$501$	3.67855	0.164345
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	42.1586	1.87603
$506$	0	0
$507$	5.35336	0.237751
$508$	0	0
$509$	−3.43447	−0.152230	−0.0761150	−	0.997099i	$-0.524252\pi$
$509$	−3.43447	−0.152230	−0.0761150	+	0.997099i	$0.524252\pi$
$510$	0	0
$511$	0.226296	0.0100107
$512$	0	0
$513$	29.4463	1.30009
$514$	0	0
$515$	−10.0734	−0.443887
$516$	0	0
$517$	12.5579	0.552297
$518$	0	0
$519$	−3.02299	−0.132694
$520$	0	0
$521$	15.7399	0.689575	0.344788	−	0.938681i	$-0.387951\pi$
$521$	15.7399	0.689575	0.344788	+	0.938681i	$0.387951\pi$
$522$	0	0
$523$	3.05307	0.133501	0.0667506	−	0.997770i	$-0.478737\pi$
$523$	3.05307	0.133501	0.0667506	+	0.997770i	$0.478737\pi$
$524$	0	0
$525$	−1.08682	−0.0474325
$526$	0	0
$527$	−59.0015	−2.57015
$528$	0	0
$529$	17.4049	0.756737
$530$	0	0
$531$	−11.7433	−0.509617
$532$	0	0
$533$	−15.0067	−0.650012
$534$	0	0
$535$	10.0228	0.433321
$536$	0	0
$537$	−1.97457	−0.0852091
$538$	0	0
$539$	−23.0175	−0.991435
$540$	0	0
$541$	2.25551	0.0969718	0.0484859	−	0.998824i	$-0.484560\pi$
$541$	2.25551	0.0969718	0.0484859	+	0.998824i	$0.484560\pi$
$542$	0	0
$543$	−17.0339	−0.730996
$544$	0	0
$545$	−31.7136	−1.35846
$546$	0	0
$547$	−24.0258	−1.02727	−0.513635	−	0.858009i	$-0.671701\pi$
$547$	−24.0258	−1.02727	−0.513635	+	0.858009i	$0.671701\pi$
$548$	0	0
$549$	−13.2003	−0.563376
$550$	0	0
$551$	−30.1524	−1.28453
$552$	0	0
$553$	−3.60146	−0.153150
$554$	0	0
$555$	16.8731	0.716222
$556$	0	0
$557$	−28.1764	−1.19387	−0.596937	−	0.802288i	$-0.703616\pi$
$557$	−28.1764	−1.19387	−0.596937	+	0.802288i	$0.703616\pi$
$558$	0	0
$559$	32.8329	1.38868
$560$	0	0
$561$	−20.7518	−0.876142
$562$	0	0
$563$	−30.1041	−1.26874	−0.634369	−	0.773030i	$-0.718740\pi$
$563$	−30.1041	−1.26874	−0.634369	+	0.773030i	$0.718740\pi$
$564$	0	0
$565$	46.2653	1.94639
$566$	0	0
$567$	0.471153	0.0197866
$568$	0	0
$569$	−37.2366	−1.56104	−0.780521	−	0.625130i	$-0.785046\pi$
$569$	−37.2366	−1.56104	−0.780521	+	0.625130i	$0.785046\pi$
$570$	0	0
$571$	−29.3129	−1.22671	−0.613354	−	0.789808i	$-0.710180\pi$
$571$	−29.3129	−1.22671	−0.613354	+	0.789808i	$0.710180\pi$
$572$	0	0
$573$	1.39937	0.0584596
$574$	0	0
$575$	26.0243	1.08529
$576$	0	0
$577$	−1.14697	−0.0477490	−0.0238745	−	0.999715i	$-0.507600\pi$
$577$	−1.14697	−0.0477490	−0.0238745	+	0.999715i	$0.507600\pi$
$578$	0	0
$579$	−14.9376	−0.620786
$580$	0	0
$581$	−0.752123	−0.0312033
$582$	0	0
$583$	−40.7595	−1.68808
$584$	0	0
$585$	−17.1626	−0.709585
$586$	0	0
$587$	−1.12303	−0.0463523	−0.0231761	−	0.999731i	$-0.507378\pi$
$587$	−1.12303	−0.0463523	−0.0231761	+	0.999731i	$0.507378\pi$
$588$	0	0
$589$	−54.6223	−2.25067
$590$	0	0
$591$	15.9553	0.656315
$592$	0	0
$593$	−5.69282	−0.233776	−0.116888	−	0.993145i	$-0.537292\pi$
$593$	−5.69282	−0.233776	−0.116888	+	0.993145i	$0.537292\pi$
$594$	0	0
$595$	5.53104	0.226751
$596$	0	0
$597$	1.63677	0.0669887
$598$	0	0
$599$	−4.22811	−0.172756	−0.0863780	−	0.996262i	$-0.527529\pi$
$599$	−4.22811	−0.172756	−0.0863780	+	0.996262i	$0.527529\pi$
$600$	0	0
$601$	15.8899	0.648161	0.324081	−	0.946029i	$-0.394945\pi$
$601$	15.8899	0.648161	0.324081	+	0.946029i	$0.394945\pi$
$602$	0	0
$603$	−28.0122	−1.14075
$604$	0	0
$605$	0.173346	0.00704751
$606$	0	0
$607$	−15.7697	−0.640073	−0.320036	−	0.947405i	$-0.603695\pi$
$607$	−15.7697	−0.640073	−0.320036	+	0.947405i	$0.603695\pi$
$608$	0	0
$609$	1.31667	0.0533542
$610$	0	0
$611$	10.2504	0.414685
$612$	0	0
$613$	−1.43316	−0.0578848	−0.0289424	−	0.999581i	$-0.509214\pi$
$613$	−1.43316	−0.0578848	−0.0289424	+	0.999581i	$0.509214\pi$
$614$	0	0
$615$	15.8458	0.638963
$616$	0	0
$617$	30.4664	1.22653	0.613265	−	0.789877i	$-0.289856\pi$
$617$	30.4664	1.22653	0.613265	+	0.789877i	$0.289856\pi$
$618$	0	0
$619$	35.7764	1.43797	0.718987	−	0.695024i	$-0.244606\pi$
$619$	35.7764	1.43797	0.718987	+	0.695024i	$0.244606\pi$
$620$	0	0
$621$	30.7900	1.23556
$622$	0	0
$623$	0.981560	0.0393254
$624$	0	0
$625$	−28.7087	−1.14835
$626$	0	0
$627$	−19.2116	−0.767236
$628$	0	0
$629$	−38.6586	−1.54142
$630$	0	0
$631$	8.94606	0.356137	0.178068	−	0.984018i	$-0.443015\pi$
$631$	8.94606	0.356137	0.178068	+	0.984018i	$0.443015\pi$
$632$	0	0
$633$	−2.02773	−0.0805950
$634$	0	0
$635$	17.6338	0.699776
$636$	0	0
$637$	−18.7880	−0.744407
$638$	0	0
$639$	25.6692	1.01546
$640$	0	0
$641$	42.9609	1.69685	0.848426	−	0.529314i	$-0.177551\pi$
$641$	42.9609	1.69685	0.848426	+	0.529314i	$0.177551\pi$
$642$	0	0
$643$	−8.93023	−0.352174	−0.176087	−	0.984375i	$-0.556344\pi$
$643$	−8.93023	−0.352174	−0.176087	+	0.984375i	$0.556344\pi$
$644$	0	0
$645$	−34.6687	−1.36508
$646$	0	0
$647$	1.00563	0.0395355	0.0197677	−	0.999805i	$-0.493707\pi$
$647$	1.00563	0.0395355	0.0197677	+	0.999805i	$0.493707\pi$
$648$	0	0
$649$	18.6237	0.731045
$650$	0	0
$651$	2.38520	0.0934835
$652$	0	0
$653$	6.35700	0.248769	0.124384	−	0.992234i	$-0.460304\pi$
$653$	6.35700	0.248769	0.124384	+	0.992234i	$0.460304\pi$
$654$	0	0
$655$	−19.5825	−0.765150
$656$	0	0
$657$	−1.69876	−0.0662750
$658$	0	0
$659$	−51.0363	−1.98809	−0.994046	−	0.108962i	$-0.965247\pi$
$659$	−51.0363	−1.98809	−0.994046	+	0.108962i	$0.965247\pi$
$660$	0	0
$661$	−27.9005	−1.08520	−0.542601	−	0.839990i	$-0.682561\pi$
$661$	−27.9005	−1.08520	−0.542601	+	0.839990i	$0.682561\pi$
$662$	0	0
$663$	−16.9386	−0.657840
$664$	0	0
$665$	5.12052	0.198565
$666$	0	0
$667$	−31.5283	−1.22078
$668$	0	0
$669$	11.6400	0.450027
$670$	0	0
$671$	20.9344	0.808163
$672$	0	0
$673$	−14.7011	−0.566686	−0.283343	−	0.959019i	$-0.591443\pi$
$673$	−14.7011	−0.566686	−0.283343	+	0.959019i	$0.591443\pi$
$674$	0	0
$675$	19.8315	0.763315
$676$	0	0
$677$	−16.7319	−0.643058	−0.321529	−	0.946900i	$-0.604197\pi$
$677$	−16.7319	−0.643058	−0.321529	+	0.946900i	$0.604197\pi$
$678$	0	0
$679$	0.0335902	0.00128907
$680$	0	0
$681$	15.8249	0.606411
$682$	0	0
$683$	1.68291	0.0643948	0.0321974	−	0.999482i	$-0.489749\pi$
$683$	1.68291	0.0643948	0.0321974	+	0.999482i	$0.489749\pi$
$684$	0	0
$685$	−29.9172	−1.14308
$686$	0	0
$687$	−20.8235	−0.794465
$688$	0	0
$689$	−33.2698	−1.26748
$690$	0	0
$691$	13.2305	0.503312	0.251656	−	0.967817i	$-0.419025\pi$
$691$	13.2305	0.503312	0.251656	+	0.967817i	$0.419025\pi$
$692$	0	0
$693$	−1.94750	−0.0739792
$694$	0	0
$695$	44.5817	1.69108
$696$	0	0
$697$	−36.3049	−1.37515
$698$	0	0
$699$	17.7481	0.671294
$700$	0	0
$701$	−27.5999	−1.04243	−0.521217	−	0.853424i	$-0.674522\pi$
$701$	−27.5999	−1.04243	−0.521217	+	0.853424i	$0.674522\pi$
$702$	0	0
$703$	−35.7893	−1.34982
$704$	0	0
$705$	−10.8235	−0.407636
$706$	0	0
$707$	3.90483	0.146856
$708$	0	0
$709$	−16.7174	−0.627833	−0.313917	−	0.949451i	$-0.601641\pi$
$709$	−16.7174	−0.627833	−0.313917	+	0.949451i	$0.601641\pi$
$710$	0	0
$711$	27.0355	1.01391
$712$	0	0
$713$	−57.1150	−2.13897
$714$	0	0
$715$	27.2181	1.01790
$716$	0	0
$717$	−6.50496	−0.242932
$718$	0	0
$719$	8.92807	0.332961	0.166480	−	0.986045i	$-0.446760\pi$
$719$	8.92807	0.332961	0.166480	+	0.986045i	$0.446760\pi$
$720$	0	0
$721$	−0.933020	−0.0347475
$722$	0	0
$723$	−0.692391	−0.0257503
$724$	0	0
$725$	−20.3070	−0.754184
$726$	0	0
$727$	−26.6643	−0.988923	−0.494462	−	0.869199i	$-0.664635\pi$
$727$	−26.6643	−0.988923	−0.494462	+	0.869199i	$0.664635\pi$
$728$	0	0
$729$	10.2737	0.380507
$730$	0	0
$731$	79.4311	2.93786
$732$	0	0
$733$	−13.2620	−0.489844	−0.244922	−	0.969543i	$-0.578762\pi$
$733$	−13.2620	−0.489844	−0.244922	+	0.969543i	$0.578762\pi$
$734$	0	0
$735$	19.8385	0.731754
$736$	0	0
$737$	44.4246	1.63640
$738$	0	0
$739$	−32.7074	−1.20316	−0.601581	−	0.798812i	$-0.705462\pi$
$739$	−32.7074	−1.20316	−0.601581	+	0.798812i	$0.705462\pi$
$740$	0	0
$741$	−15.6814	−0.576070
$742$	0	0
$743$	−8.43068	−0.309292	−0.154646	−	0.987970i	$-0.549424\pi$
$743$	−8.43068	−0.309292	−0.154646	+	0.987970i	$0.549424\pi$
$744$	0	0
$745$	−41.4758	−1.51956
$746$	0	0
$747$	5.64606	0.206579
$748$	0	0
$749$	0.928329	0.0339204
$750$	0	0
$751$	−49.0358	−1.78934	−0.894672	−	0.446724i	$-0.852590\pi$
$751$	−49.0358	−1.78934	−0.894672	+	0.446724i	$0.852590\pi$
$752$	0	0
$753$	−11.4177	−0.416085
$754$	0	0
$755$	−16.0156	−0.582866
$756$	0	0
$757$	−6.93998	−0.252238	−0.126119	−	0.992015i	$-0.540252\pi$
$757$	−6.93998	−0.252238	−0.126119	+	0.992015i	$0.540252\pi$
$758$	0	0
$759$	−20.0883	−0.729158
$760$	0	0
$761$	10.5479	0.382361	0.191181	−	0.981555i	$-0.438768\pi$
$761$	10.5479	0.382361	0.191181	+	0.981555i	$0.438768\pi$
$762$	0	0
$763$	−2.93738	−0.106340
$764$	0	0
$765$	−41.5206	−1.50118
$766$	0	0
$767$	15.2015	0.548896
$768$	0	0
$769$	4.94301	0.178249	0.0891247	−	0.996020i	$-0.471593\pi$
$769$	4.94301	0.178249	0.0891247	+	0.996020i	$0.471593\pi$
$770$	0	0
$771$	−14.9753	−0.539323
$772$	0	0
$773$	−47.9574	−1.72491	−0.862454	−	0.506135i	$-0.831074\pi$
$773$	−47.9574	−1.72491	−0.862454	+	0.506135i	$0.831074\pi$
$774$	0	0
$775$	−36.7871	−1.32143
$776$	0	0
$777$	1.56282	0.0560659
$778$	0	0
$779$	−33.6103	−1.20422
$780$	0	0
$781$	−40.7087	−1.45667
$782$	0	0
$783$	−24.0257	−0.858610
$784$	0	0
$785$	36.9370	1.31834
$786$	0	0
$787$	−16.8951	−0.602246	−0.301123	−	0.953585i	$-0.597361\pi$
$787$	−16.8951	−0.602246	−0.301123	+	0.953585i	$0.597361\pi$
$788$	0	0
$789$	−21.3901	−0.761507
$790$	0	0
$791$	4.28519	0.152364
$792$	0	0
$793$	17.0876	0.606799
$794$	0	0
$795$	35.1300	1.24593
$796$	0	0
$797$	−33.5770	−1.18936	−0.594679	−	0.803963i	$-0.702721\pi$
$797$	−33.5770	−1.18936	−0.594679	+	0.803963i	$0.702721\pi$
$798$	0	0
$799$	24.7982	0.877297
$800$	0	0
$801$	−7.36840	−0.260350
$802$	0	0
$803$	2.69407	0.0950715
$804$	0	0
$805$	5.35419	0.188710
$806$	0	0
$807$	4.50921	0.158732
$808$	0	0
$809$	−4.55596	−0.160179	−0.0800895	−	0.996788i	$-0.525521\pi$
$809$	−4.55596	−0.160179	−0.0800895	+	0.996788i	$0.525521\pi$
$810$	0	0
$811$	44.3719	1.55811	0.779055	−	0.626956i	$-0.215699\pi$
$811$	44.3719	1.55811	0.779055	+	0.626956i	$0.215699\pi$
$812$	0	0
$813$	−13.1223	−0.460219
$814$	0	0
$815$	39.9143	1.39814
$816$	0	0
$817$	73.5356	2.57268
$818$	0	0
$819$	−1.58964	−0.0555464
$820$	0	0
$821$	15.3969	0.537355	0.268677	−	0.963230i	$-0.413413\pi$
$821$	15.3969	0.537355	0.268677	+	0.963230i	$0.413413\pi$
$822$	0	0
$823$	19.6051	0.683391	0.341695	−	0.939811i	$-0.388999\pi$
$823$	19.6051	0.683391	0.341695	+	0.939811i	$0.388999\pi$
$824$	0	0
$825$	−12.9386	−0.450465
$826$	0	0
$827$	−21.6339	−0.752285	−0.376142	−	0.926562i	$-0.622750\pi$
$827$	−21.6339	−0.752285	−0.376142	+	0.926562i	$0.622750\pi$
$828$	0	0
$829$	−20.1213	−0.698842	−0.349421	−	0.936966i	$-0.613622\pi$
$829$	−20.1213	−0.698842	−0.349421	+	0.936966i	$0.613622\pi$
$830$	0	0
$831$	29.4304	1.02093
$832$	0	0
$833$	−45.4528	−1.57485
$834$	0	0
$835$	−11.6724	−0.403940
$836$	0	0
$837$	−43.5237	−1.50440
$838$	0	0
$839$	13.3980	0.462551	0.231275	−	0.972888i	$-0.425710\pi$
$839$	13.3980	0.462551	0.231275	+	0.972888i	$0.425710\pi$
$840$	0	0
$841$	−4.39814	−0.151660
$842$	0	0
$843$	6.69611	0.230626
$844$	0	0
$845$	−16.9867	−0.584361
$846$	0	0
$847$	0.0160557	0.000551680 0
$848$	0	0
$849$	1.77847	0.0610371
$850$	0	0
$851$	−37.4225	−1.28283
$852$	0	0
$853$	27.5219	0.942333	0.471166	−	0.882044i	$-0.343833\pi$
$853$	27.5219	0.942333	0.471166	+	0.882044i	$0.343833\pi$
$854$	0	0
$855$	−38.4389	−1.31458
$856$	0	0
$857$	−7.77030	−0.265428	−0.132714	−	0.991154i	$-0.542369\pi$
$857$	−7.77030	−0.265428	−0.132714	+	0.991154i	$0.542369\pi$
$858$	0	0
$859$	−11.2172	−0.382725	−0.191362	−	0.981519i	$-0.561291\pi$
$859$	−11.2172	−0.382725	−0.191362	+	0.981519i	$0.561291\pi$
$860$	0	0
$861$	1.46767	0.0500180
$862$	0	0
$863$	−29.5430	−1.00565	−0.502827	−	0.864387i	$-0.667707\pi$
$863$	−29.5430	−1.00565	−0.502827	+	0.864387i	$0.667707\pi$
$864$	0	0
$865$	9.59224	0.326146
$866$	0	0
$867$	−24.8223	−0.843008
$868$	0	0
$869$	−42.8756	−1.45446
$870$	0	0
$871$	36.2614	1.22867
$872$	0	0
$873$	−0.252156	−0.00853417
$874$	0	0
$875$	−0.763024	−0.0257949
$876$	0	0
$877$	−28.4421	−0.960423	−0.480211	−	0.877153i	$-0.659440\pi$
$877$	−28.4421	−0.960423	−0.480211	+	0.877153i	$0.659440\pi$
$878$	0	0
$879$	−31.4109	−1.05946
$880$	0	0
$881$	−6.46155	−0.217695	−0.108848	−	0.994058i	$-0.534716\pi$
$881$	−6.46155	−0.217695	−0.108848	+	0.994058i	$0.534716\pi$
$882$	0	0
$883$	6.25124	0.210371	0.105185	−	0.994453i	$-0.466456\pi$
$883$	6.25124	0.210371	0.105185	+	0.994453i	$0.466456\pi$
$884$	0	0
$885$	−16.0515	−0.539566
$886$	0	0
$887$	11.2721	0.378479	0.189239	−	0.981931i	$-0.439398\pi$
$887$	11.2721	0.378479	0.189239	+	0.981931i	$0.439398\pi$
$888$	0	0
$889$	1.63328	0.0547785
$890$	0	0
$891$	5.60912	0.187913
$892$	0	0
$893$	22.9576	0.768248
$894$	0	0
$895$	6.26551	0.209433
$896$	0	0
$897$	−16.3970	−0.547479
$898$	0	0
$899$	44.5673	1.48640
$900$	0	0
$901$	−80.4880	−2.68144
$902$	0	0
$903$	−3.21109	−0.106858
$904$	0	0
$905$	54.0504	1.79670
$906$	0	0
$907$	−23.0782	−0.766299	−0.383149	−	0.923686i	$-0.625161\pi$
$907$	−23.0782	−0.766299	−0.383149	+	0.923686i	$0.625161\pi$
$908$	0	0
$909$	−29.3129	−0.972246
$910$	0	0
$911$	40.0961	1.32844	0.664221	−	0.747536i	$-0.268763\pi$
$911$	40.0961	1.32844	0.664221	+	0.747536i	$0.268763\pi$
$912$	0	0
$913$	−8.95409	−0.296337
$914$	0	0
$915$	−18.0431	−0.596485
$916$	0	0
$917$	−1.81377	−0.0598960
$918$	0	0
$919$	−15.0186	−0.495419	−0.247710	−	0.968834i	$-0.579678\pi$
$919$	−15.0186	−0.495419	−0.247710	+	0.968834i	$0.579678\pi$
$920$	0	0
$921$	7.99981	0.263603
$922$	0	0
$923$	−33.2283	−1.09372
$924$	0	0
$925$	−24.1034	−0.792515
$926$	0	0
$927$	7.00402	0.230042
$928$	0	0
$929$	41.6193	1.36549	0.682743	−	0.730659i	$-0.260787\pi$
$929$	41.6193	1.36549	0.682743	+	0.730659i	$0.260787\pi$
$930$	0	0
$931$	−42.0793	−1.37909
$932$	0	0
$933$	15.2660	0.499785
$934$	0	0
$935$	65.8475	2.15344
$936$	0	0
$937$	46.4215	1.51652	0.758262	−	0.651950i	$-0.226049\pi$
$937$	46.4215	1.51652	0.758262	+	0.651950i	$0.226049\pi$
$938$	0	0
$939$	10.3742	0.338548
$940$	0	0
$941$	−5.83243	−0.190132	−0.0950659	−	0.995471i	$-0.530306\pi$
$941$	−5.83243	−0.190132	−0.0950659	+	0.995471i	$0.530306\pi$
$942$	0	0
$943$	−35.1441	−1.14445
$944$	0	0
$945$	4.08009	0.132725
$946$	0	0
$947$	4.52852	0.147157	0.0735785	−	0.997289i	$-0.476558\pi$
$947$	4.52852	0.147157	0.0735785	+	0.997289i	$0.476558\pi$
$948$	0	0
$949$	2.19902	0.0713833
$950$	0	0
$951$	−31.5615	−1.02345
$952$	0	0
$953$	50.2071	1.62637	0.813183	−	0.582008i	$-0.197733\pi$
$953$	50.2071	1.62637	0.813183	+	0.582008i	$0.197733\pi$
$954$	0	0
$955$	−4.44034	−0.143686
$956$	0	0
$957$	15.6751	0.506703
$958$	0	0
$959$	−2.77100	−0.0894801
$960$	0	0
$961$	49.7356	1.60437
$962$	0	0
$963$	−6.96881	−0.224567
$964$	0	0
$965$	47.3985	1.52581
$966$	0	0
$967$	40.6400	1.30689	0.653447	−	0.756972i	$-0.273322\pi$
$967$	40.6400	1.30689	0.653447	+	0.756972i	$0.273322\pi$
$968$	0	0
$969$	−37.9372	−1.21872
$970$	0	0
$971$	−21.0600	−0.675848	−0.337924	−	0.941173i	$-0.609725\pi$
$971$	−21.0600	−0.675848	−0.337924	+	0.941173i	$0.609725\pi$
$972$	0	0
$973$	4.12926	0.132378
$974$	0	0
$975$	−10.5611	−0.338226
$976$	0	0
$977$	−32.8455	−1.05082	−0.525411	−	0.850849i	$-0.676088\pi$
$977$	−32.8455	−1.05082	−0.525411	+	0.850849i	$0.676088\pi$
$978$	0	0
$979$	11.6855	0.373472
$980$	0	0
$981$	22.0504	0.704017
$982$	0	0
$983$	51.2133	1.63345	0.816726	−	0.577026i	$-0.195787\pi$
$983$	51.2133	1.63345	0.816726	+	0.577026i	$0.195787\pi$
$984$	0	0
$985$	−50.6279	−1.61314
$986$	0	0
$987$	−1.00250	−0.0319098
$988$	0	0
$989$	76.8913	2.44500
$990$	0	0
$991$	−0.348340	−0.0110654	−0.00553269	−	0.999985i	$-0.501761\pi$
$991$	−0.348340	−0.0110654	−0.00553269	+	0.999985i	$0.501761\pi$
$992$	0	0
$993$	−12.0072	−0.381038
$994$	0	0
$995$	−5.19364	−0.164650
$996$	0	0
$997$	20.7317	0.656580	0.328290	−	0.944577i	$-0.393528\pi$
$997$	20.7317	0.656580	0.328290	+	0.944577i	$0.393528\pi$
$998$	0	0
$999$	−28.5173	−0.902249

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.g.1.11	✓	33
4.3	odd	2		8048.2.a.x.1.23		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.11	✓	33	1.1	even	1	trivial
8048.2.a.x.1.23		33	4.3	odd	2

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:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-0.950380 q^{3} +3.01565 q^{5} +0.279316 q^{7} -2.09678 q^{9} +O(q^{10})\)	\(q-0.950380 q^{3} +3.01565 q^{5} +0.279316 q^{7} -2.09678 q^{9} +3.32528 q^{11} +2.71425 q^{13} -2.86601 q^{15} +6.56645 q^{17} +6.07908 q^{19} -0.265456 q^{21} +6.35649 q^{23} +4.09414 q^{25} +4.84388 q^{27} -4.96003 q^{29} -8.98530 q^{31} -3.16028 q^{33} +0.842319 q^{35} -5.88730 q^{37} -2.57957 q^{39} -5.52886 q^{41} +12.0965 q^{43} -6.32315 q^{45} +3.77650 q^{47} -6.92198 q^{49} -6.24062 q^{51} -12.2575 q^{53} +10.0279 q^{55} -5.77743 q^{57} +5.60065 q^{59} +6.29552 q^{61} -0.585664 q^{63} +8.18522 q^{65} +13.3597 q^{67} -6.04108 q^{69} -12.2422 q^{71} +0.810178 q^{73} -3.89099 q^{75} +0.928804 q^{77} -12.8938 q^{79} +1.68681 q^{81} -2.69273 q^{83} +19.8021 q^{85} +4.71391 q^{87} +3.51416 q^{89} +0.758133 q^{91} +8.53945 q^{93} +18.3324 q^{95} +0.120259 q^{97} -6.97237 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * q^99

Introduction

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