LMFDB - Embedded newform 4024.2.a.g.1.6

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			1.6
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)

$f(q)$	$=$	$q-2.15460 q^{3} -0.0166844 q^{5} -4.51155 q^{7} +1.64231 q^{9} +O(q^{10})$	$q-2.15460 q^{3} -0.0166844 q^{5} -4.51155 q^{7} +1.64231 q^{9} +1.00082 q^{11} -5.74436 q^{13} +0.0359482 q^{15} -6.04961 q^{17} +5.55323 q^{19} +9.72059 q^{21} +0.430080 q^{23} -4.99972 q^{25} +2.92529 q^{27} +0.293283 q^{29} -9.26322 q^{31} -2.15636 q^{33} +0.0752725 q^{35} -8.67185 q^{37} +12.3768 q^{39} -11.9433 q^{41} -5.90376 q^{43} -0.0274009 q^{45} -13.2132 q^{47} +13.3541 q^{49} +13.0345 q^{51} -0.840404 q^{53} -0.0166980 q^{55} -11.9650 q^{57} +13.3362 q^{59} -3.84837 q^{61} -7.40935 q^{63} +0.0958412 q^{65} +3.29996 q^{67} -0.926651 q^{69} +1.77661 q^{71} -1.32799 q^{73} +10.7724 q^{75} -4.51523 q^{77} +15.9162 q^{79} -11.2297 q^{81} -14.8071 q^{83} +0.100934 q^{85} -0.631907 q^{87} -4.18265 q^{89} +25.9160 q^{91} +19.9585 q^{93} -0.0926523 q^{95} +17.1027 q^{97} +1.64365 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.15460	−1.24396	−0.621980	−	0.783033i	$-0.713671\pi$
$3$	−2.15460	−1.24396	−0.621980	+	0.783033i	$0.713671\pi$
$4$	0	0
$5$	−0.0166844	−0.00746149	−0.00373074	−	0.999993i	$-0.501188\pi$
$5$	−0.0166844	−0.00746149	−0.00373074	+	0.999993i	$0.501188\pi$
$6$	0	0
$7$	−4.51155	−1.70521	−0.852603	−	0.522560i	$-0.824977\pi$
$7$	−4.51155	−1.70521	−0.852603	+	0.522560i	$0.824977\pi$
$8$	0	0
$9$	1.64231	0.547436
$10$	0	0
$11$	1.00082	0.301758	0.150879	−	0.988552i	$-0.451790\pi$
$11$	1.00082	0.301758	0.150879	+	0.988552i	$0.451790\pi$
$12$	0	0
$13$	−5.74436	−1.59320	−0.796600	−	0.604507i	$-0.793370\pi$
$13$	−5.74436	−1.59320	−0.796600	+	0.604507i	$0.793370\pi$
$14$	0	0
$15$	0.0359482	0.00928179
$16$	0	0
$17$	−6.04961	−1.46725	−0.733623	−	0.679556i	$-0.762172\pi$
$17$	−6.04961	−1.46725	−0.733623	+	0.679556i	$0.762172\pi$
$18$	0	0
$19$	5.55323	1.27400	0.636999	−	0.770865i	$-0.280176\pi$
$19$	5.55323	1.27400	0.636999	+	0.770865i	$0.280176\pi$
$20$	0	0
$21$	9.72059	2.12121
$22$	0	0
$23$	0.430080	0.0896779	0.0448389	−	0.998994i	$-0.485723\pi$
$23$	0.430080	0.0896779	0.0448389	+	0.998994i	$0.485723\pi$
$24$	0	0
$25$	−4.99972	−0.999944
$26$	0	0
$27$	2.92529	0.562972
$28$	0	0
$29$	0.293283	0.0544612	0.0272306	−	0.999629i	$-0.491331\pi$
$29$	0.293283	0.0544612	0.0272306	+	0.999629i	$0.491331\pi$
$30$	0	0
$31$	−9.26322	−1.66372	−0.831861	−	0.554983i	$-0.812725\pi$
$31$	−9.26322	−1.66372	−0.831861	+	0.554983i	$0.812725\pi$
$32$	0	0
$33$	−2.15636	−0.375374
$34$	0	0
$35$	0.0752725	0.0127234
$36$	0	0
$37$	−8.67185	−1.42564	−0.712822	−	0.701345i	$-0.752583\pi$
$37$	−8.67185	−1.42564	−0.712822	+	0.701345i	$0.752583\pi$
$38$	0	0
$39$	12.3768	1.98188
$40$	0	0
$41$	−11.9433	−1.86523	−0.932617	−	0.360867i	$-0.882481\pi$
$41$	−11.9433	−1.86523	−0.932617	+	0.360867i	$0.882481\pi$
$42$	0	0
$43$	−5.90376	−0.900315	−0.450157	−	0.892949i	$-0.648632\pi$
$43$	−5.90376	−0.900315	−0.450157	+	0.892949i	$0.648632\pi$
$44$	0	0
$45$	−0.0274009	−0.00408468
$46$	0	0
$47$	−13.2132	−1.92735	−0.963675	−	0.267079i	$-0.913942\pi$
$47$	−13.2132	−1.92735	−0.963675	+	0.267079i	$0.913942\pi$
$48$	0	0
$49$	13.3541	1.90773
$50$	0	0
$51$	13.0345	1.82520
$52$	0	0
$53$	−0.840404	−0.115438	−0.0577192	−	0.998333i	$-0.518383\pi$
$53$	−0.840404	−0.115438	−0.0577192	+	0.998333i	$0.518383\pi$
$54$	0	0
$55$	−0.0166980	−0.00225156
$56$	0	0
$57$	−11.9650	−1.58480
$58$	0	0
$59$	13.3362	1.73623	0.868116	−	0.496361i	$-0.165331\pi$
$59$	13.3362	1.73623	0.868116	+	0.496361i	$0.165331\pi$
$60$	0	0
$61$	−3.84837	−0.492733	−0.246367	−	0.969177i	$-0.579237\pi$
$61$	−3.84837	−0.492733	−0.246367	+	0.969177i	$0.579237\pi$
$62$	0	0
$63$	−7.40935	−0.933490
$64$	0	0
$65$	0.0958412	0.0118876
$66$	0	0
$67$	3.29996	0.403155	0.201577	−	0.979473i	$-0.435393\pi$
$67$	3.29996	0.403155	0.201577	+	0.979473i	$0.435393\pi$
$68$	0	0
$69$	−0.926651	−0.111556
$70$	0	0
$71$	1.77661	0.210845	0.105423	−	0.994428i	$-0.466380\pi$
$71$	1.77661	0.210845	0.105423	+	0.994428i	$0.466380\pi$
$72$	0	0
$73$	−1.32799	−0.155429	−0.0777146	−	0.996976i	$-0.524762\pi$
$73$	−1.32799	−0.155429	−0.0777146	+	0.996976i	$0.524762\pi$
$74$	0	0
$75$	10.7724	1.24389
$76$	0	0
$77$	−4.51523	−0.514559
$78$	0	0
$79$	15.9162	1.79072	0.895358	−	0.445347i	$-0.146920\pi$
$79$	15.9162	1.79072	0.895358	+	0.445347i	$0.146920\pi$
$80$	0	0
$81$	−11.2297	−1.24775
$82$	0	0
$83$	−14.8071	−1.62529	−0.812646	−	0.582757i	$-0.801974\pi$
$83$	−14.8071	−1.62529	−0.812646	+	0.582757i	$0.801974\pi$
$84$	0	0
$85$	0.100934	0.0109478
$86$	0	0
$87$	−0.631907	−0.0677476
$88$	0	0
$89$	−4.18265	−0.443360	−0.221680	−	0.975120i	$-0.571154\pi$
$89$	−4.18265	−0.443360	−0.221680	+	0.975120i	$0.571154\pi$
$90$	0	0
$91$	25.9160	2.71673
$92$	0	0
$93$	19.9585	2.06960
$94$	0	0
$95$	−0.0926523	−0.00950592
$96$	0	0
$97$	17.1027	1.73652	0.868259	−	0.496111i	$-0.165239\pi$
$97$	17.1027	1.73652	0.868259	+	0.496111i	$0.165239\pi$
$98$	0	0
$99$	1.64365	0.165193
$100$	0	0
$101$	−0.889014	−0.0884602	−0.0442301	−	0.999021i	$-0.514083\pi$
$101$	−0.889014	−0.0884602	−0.0442301	+	0.999021i	$0.514083\pi$
$102$	0	0
$103$	3.38074	0.333115	0.166557	−	0.986032i	$-0.446735\pi$
$103$	3.38074	0.333115	0.166557	+	0.986032i	$0.446735\pi$
$104$	0	0
$105$	−0.162182	−0.0158274
$106$	0	0
$107$	−13.0677	−1.26330	−0.631651	−	0.775253i	$-0.717622\pi$
$107$	−13.0677	−1.26330	−0.631651	+	0.775253i	$0.717622\pi$
$108$	0	0
$109$	−14.4658	−1.38557	−0.692787	−	0.721143i	$-0.743617\pi$
$109$	−14.4658	−1.38557	−0.692787	+	0.721143i	$0.743617\pi$
$110$	0	0
$111$	18.6844	1.77344
$112$	0	0
$113$	6.05237	0.569359	0.284680	−	0.958623i	$-0.408113\pi$
$113$	6.05237	0.569359	0.284680	+	0.958623i	$0.408113\pi$
$114$	0	0
$115$	−0.00717563	−0.000669130 0
$116$	0	0
$117$	−9.43400	−0.872174
$118$	0	0
$119$	27.2931	2.50196
$120$	0	0
$121$	−9.99837	−0.908942
$122$	0	0
$123$	25.7331	2.32028
$124$	0	0
$125$	0.166839	0.0149226
$126$	0	0
$127$	4.46474	0.396182	0.198091	−	0.980184i	$-0.436526\pi$
$127$	4.46474	0.396182	0.198091	+	0.980184i	$0.436526\pi$
$128$	0	0
$129$	12.7202	1.11996
$130$	0	0
$131$	−5.06316	−0.442370	−0.221185	−	0.975232i	$-0.570993\pi$
$131$	−5.06316	−0.442370	−0.221185	+	0.975232i	$0.570993\pi$
$132$	0	0
$133$	−25.0537	−2.17243
$134$	0	0
$135$	−0.0488067	−0.00420061
$136$	0	0
$137$	−0.252596	−0.0215807	−0.0107904	−	0.999942i	$-0.503435\pi$
$137$	−0.252596	−0.0215807	−0.0107904	+	0.999942i	$0.503435\pi$
$138$	0	0
$139$	7.09932	0.602156	0.301078	−	0.953599i	$-0.402653\pi$
$139$	7.09932	0.602156	0.301078	+	0.953599i	$0.402653\pi$
$140$	0	0
$141$	28.4693	2.39754
$142$	0	0
$143$	−5.74905	−0.480760
$144$	0	0
$145$	−0.00489324	−0.000406362 0
$146$	0	0
$147$	−28.7727	−2.37313
$148$	0	0
$149$	−10.0817	−0.825924	−0.412962	−	0.910748i	$-0.635506\pi$
$149$	−10.0817	−0.825924	−0.412962	+	0.910748i	$0.635506\pi$
$150$	0	0
$151$	10.0551	0.818272	0.409136	−	0.912474i	$-0.365830\pi$
$151$	10.0551	0.818272	0.409136	+	0.912474i	$0.365830\pi$
$152$	0	0
$153$	−9.93532	−0.803223
$154$	0	0
$155$	0.154551	0.0124139
$156$	0	0
$157$	−7.83109	−0.624989	−0.312494	−	0.949920i	$-0.601165\pi$
$157$	−7.83109	−0.624989	−0.312494	+	0.949920i	$0.601165\pi$
$158$	0	0
$159$	1.81074	0.143601
$160$	0	0
$161$	−1.94033	−0.152919
$162$	0	0
$163$	0.915465	0.0717047	0.0358524	−	0.999357i	$-0.488585\pi$
$163$	0.915465	0.0717047	0.0358524	+	0.999357i	$0.488585\pi$
$164$	0	0
$165$	0.0359776	0.00280085
$166$	0	0
$167$	−18.9741	−1.46826	−0.734131	−	0.679008i	$-0.762410\pi$
$167$	−18.9741	−1.46826	−0.734131	+	0.679008i	$0.762410\pi$
$168$	0	0
$169$	19.9977	1.53828
$170$	0	0
$171$	9.12010	0.697432
$172$	0	0
$173$	3.27924	0.249316	0.124658	−	0.992200i	$-0.460217\pi$
$173$	3.27924	0.249316	0.124658	+	0.992200i	$0.460217\pi$
$174$	0	0
$175$	22.5565	1.70511
$176$	0	0
$177$	−28.7343	−2.15980
$178$	0	0
$179$	11.4539	0.856105	0.428052	−	0.903754i	$-0.359200\pi$
$179$	11.4539	0.856105	0.428052	+	0.903754i	$0.359200\pi$
$180$	0	0
$181$	−6.88044	−0.511419	−0.255709	−	0.966754i	$-0.582309\pi$
$181$	−6.88044	−0.511419	−0.255709	+	0.966754i	$0.582309\pi$
$182$	0	0
$183$	8.29170	0.612940
$184$	0	0
$185$	0.144685	0.0106374
$186$	0	0
$187$	−6.05455	−0.442753
$188$	0	0
$189$	−13.1976	−0.959983
$190$	0	0
$191$	−8.64272	−0.625365	−0.312683	−	0.949858i	$-0.601228\pi$
$191$	−8.64272	−0.625365	−0.312683	+	0.949858i	$0.601228\pi$
$192$	0	0
$193$	17.7687	1.27902	0.639508	−	0.768785i	$-0.279138\pi$
$193$	17.7687	1.27902	0.639508	+	0.768785i	$0.279138\pi$
$194$	0	0
$195$	−0.206500	−0.0147877
$196$	0	0
$197$	17.9915	1.28184	0.640921	−	0.767606i	$-0.278552\pi$
$197$	17.9915	1.28184	0.640921	+	0.767606i	$0.278552\pi$
$198$	0	0
$199$	−18.3546	−1.30112	−0.650560	−	0.759454i	$-0.725466\pi$
$199$	−18.3546	−1.30112	−0.650560	+	0.759454i	$0.725466\pi$
$200$	0	0
$201$	−7.11011	−0.501508
$202$	0	0
$203$	−1.32316	−0.0928676
$204$	0	0
$205$	0.199267	0.0139174
$206$	0	0
$207$	0.706323	0.0490929
$208$	0	0
$209$	5.55776	0.384439
$210$	0	0
$211$	−6.81395	−0.469092	−0.234546	−	0.972105i	$-0.575360\pi$
$211$	−6.81395	−0.469092	−0.234546	+	0.972105i	$0.575360\pi$
$212$	0	0
$213$	−3.82789	−0.262283
$214$	0	0
$215$	0.0985007	0.00671769
$216$	0	0
$217$	41.7915	2.83699
$218$	0	0
$219$	2.86128	0.193348
$220$	0	0
$221$	34.7512	2.33762
$222$	0	0
$223$	14.7424	0.987226	0.493613	−	0.869682i	$-0.335676\pi$
$223$	14.7424	0.987226	0.493613	+	0.869682i	$0.335676\pi$
$224$	0	0
$225$	−8.21108	−0.547405
$226$	0	0
$227$	22.8474	1.51644	0.758218	−	0.652001i	$-0.226070\pi$
$227$	22.8474	1.51644	0.758218	+	0.652001i	$0.226070\pi$
$228$	0	0
$229$	−26.4454	−1.74756	−0.873781	−	0.486319i	$-0.838339\pi$
$229$	−26.4454	−1.74756	−0.873781	+	0.486319i	$0.838339\pi$
$230$	0	0
$231$	9.72853	0.640090
$232$	0	0
$233$	10.1986	0.668129	0.334065	−	0.942550i	$-0.391580\pi$
$233$	10.1986	0.668129	0.334065	+	0.942550i	$0.391580\pi$
$234$	0	0
$235$	0.220455	0.0143809
$236$	0	0
$237$	−34.2931	−2.22758
$238$	0	0
$239$	14.4807	0.936681	0.468341	−	0.883548i	$-0.344852\pi$
$239$	14.4807	0.936681	0.468341	+	0.883548i	$0.344852\pi$
$240$	0	0
$241$	27.3387	1.76104	0.880520	−	0.474009i	$-0.157193\pi$
$241$	27.3387	1.76104	0.880520	+	0.474009i	$0.157193\pi$
$242$	0	0
$243$	15.4198	0.989179
$244$	0	0
$245$	−0.222805	−0.0142345
$246$	0	0
$247$	−31.8998	−2.02973
$248$	0	0
$249$	31.9034	2.02180
$250$	0	0
$251$	14.2635	0.900304	0.450152	−	0.892952i	$-0.351370\pi$
$251$	14.2635	0.900304	0.450152	+	0.892952i	$0.351370\pi$
$252$	0	0
$253$	0.430431	0.0270610
$254$	0	0
$255$	−0.217473	−0.0136187
$256$	0	0
$257$	13.1228	0.818579	0.409290	−	0.912405i	$-0.365777\pi$
$257$	13.1228	0.818579	0.409290	+	0.912405i	$0.365777\pi$
$258$	0	0
$259$	39.1235	2.43101
$260$	0	0
$261$	0.481660	0.0298140
$262$	0	0
$263$	−12.5305	−0.772666	−0.386333	−	0.922359i	$-0.626259\pi$
$263$	−12.5305	−0.772666	−0.386333	+	0.922359i	$0.626259\pi$
$264$	0	0
$265$	0.0140216	0.000861342 0
$266$	0	0
$267$	9.01194	0.551521
$268$	0	0
$269$	−2.87026	−0.175003	−0.0875013	−	0.996164i	$-0.527888\pi$
$269$	−2.87026	−0.175003	−0.0875013	+	0.996164i	$0.527888\pi$
$270$	0	0
$271$	−1.33273	−0.0809577	−0.0404789	−	0.999180i	$-0.512888\pi$
$271$	−1.33273	−0.0809577	−0.0404789	+	0.999180i	$0.512888\pi$
$272$	0	0
$273$	−55.8386	−3.37950
$274$	0	0
$275$	−5.00381	−0.301741
$276$	0	0
$277$	17.6580	1.06097	0.530484	−	0.847695i	$-0.322010\pi$
$277$	17.6580	1.06097	0.530484	+	0.847695i	$0.322010\pi$
$278$	0	0
$279$	−15.2130	−0.910781
$280$	0	0
$281$	−0.131212	−0.00782745	−0.00391372	−	0.999992i	$-0.501246\pi$
$281$	−0.131212	−0.00782745	−0.00391372	+	0.999992i	$0.501246\pi$
$282$	0	0
$283$	−3.21278	−0.190980	−0.0954901	−	0.995430i	$-0.530442\pi$
$283$	−3.21278	−0.190980	−0.0954901	+	0.995430i	$0.530442\pi$
$284$	0	0
$285$	0.199629	0.0118250
$286$	0	0
$287$	53.8829	3.18061
$288$	0	0
$289$	19.5978	1.15281
$290$	0	0
$291$	−36.8495	−2.16016
$292$	0	0
$293$	14.5252	0.848571	0.424286	−	0.905528i	$-0.360525\pi$
$293$	14.5252	0.848571	0.424286	+	0.905528i	$0.360525\pi$
$294$	0	0
$295$	−0.222507	−0.0129549
$296$	0	0
$297$	2.92768	0.169881
$298$	0	0
$299$	−2.47054	−0.142875
$300$	0	0
$301$	26.6351	1.53522
$302$	0	0
$303$	1.91547	0.110041
$304$	0	0
$305$	0.0642077	0.00367652
$306$	0	0
$307$	−7.28980	−0.416051	−0.208025	−	0.978123i	$-0.566704\pi$
$307$	−7.28980	−0.416051	−0.208025	+	0.978123i	$0.566704\pi$
$308$	0	0
$309$	−7.28415	−0.414381
$310$	0	0
$311$	−4.17803	−0.236914	−0.118457	−	0.992959i	$-0.537795\pi$
$311$	−4.17803	−0.236914	−0.118457	+	0.992959i	$0.537795\pi$
$312$	0	0
$313$	−25.2004	−1.42441	−0.712204	−	0.701972i	$-0.752303\pi$
$313$	−25.2004	−1.42441	−0.712204	+	0.701972i	$0.752303\pi$
$314$	0	0
$315$	0.123620	0.00696523
$316$	0	0
$317$	7.44536	0.418173	0.209086	−	0.977897i	$-0.432951\pi$
$317$	7.44536	0.418173	0.209086	+	0.977897i	$0.432951\pi$
$318$	0	0
$319$	0.293522	0.0164341
$320$	0	0
$321$	28.1557	1.57150
$322$	0	0
$323$	−33.5949	−1.86927
$324$	0	0
$325$	28.7202	1.59311
$326$	0	0
$327$	31.1681	1.72360
$328$	0	0
$329$	59.6122	3.28653
$330$	0	0
$331$	−9.72443	−0.534503	−0.267251	−	0.963627i	$-0.586115\pi$
$331$	−9.72443	−0.534503	−0.267251	+	0.963627i	$0.586115\pi$
$332$	0	0
$333$	−14.2418	−0.780448
$334$	0	0
$335$	−0.0550579	−0.00300813
$336$	0	0
$337$	21.9644	1.19647	0.598237	−	0.801319i	$-0.295868\pi$
$337$	21.9644	1.19647	0.598237	+	0.801319i	$0.295868\pi$
$338$	0	0
$339$	−13.0404	−0.708260
$340$	0	0
$341$	−9.27078	−0.502041
$342$	0	0
$343$	−28.6667	−1.54786
$344$	0	0
$345$	0.0154606	0.000832371 0
$346$	0	0
$347$	−2.50896	−0.134688	−0.0673440	−	0.997730i	$-0.521452\pi$
$347$	−2.50896	−0.134688	−0.0673440	+	0.997730i	$0.521452\pi$
$348$	0	0
$349$	8.06328	0.431618	0.215809	−	0.976436i	$-0.430761\pi$
$349$	8.06328	0.431618	0.215809	+	0.976436i	$0.430761\pi$
$350$	0	0
$351$	−16.8039	−0.896926
$352$	0	0
$353$	8.44698	0.449587	0.224794	−	0.974406i	$-0.427829\pi$
$353$	8.44698	0.449587	0.224794	+	0.974406i	$0.427829\pi$
$354$	0	0
$355$	−0.0296417	−0.00157322
$356$	0	0
$357$	−58.8058	−3.11233
$358$	0	0
$359$	−6.93251	−0.365884	−0.182942	−	0.983124i	$-0.558562\pi$
$359$	−6.93251	−0.365884	−0.182942	+	0.983124i	$0.558562\pi$
$360$	0	0
$361$	11.8383	0.623071
$362$	0	0
$363$	21.5425	1.13069
$364$	0	0
$365$	0.0221567	0.00115973
$366$	0	0
$367$	3.04580	0.158990	0.0794949	−	0.996835i	$-0.474669\pi$
$367$	3.04580	0.158990	0.0794949	+	0.996835i	$0.474669\pi$
$368$	0	0
$369$	−19.6146	−1.02110
$370$	0	0
$371$	3.79152	0.196846
$372$	0	0
$373$	−2.42901	−0.125770	−0.0628848	−	0.998021i	$-0.520030\pi$
$373$	−2.42901	−0.125770	−0.0628848	+	0.998021i	$0.520030\pi$
$374$	0	0
$375$	−0.359472	−0.0185631
$376$	0	0
$377$	−1.68472	−0.0867676
$378$	0	0
$379$	−31.8038	−1.63365	−0.816826	−	0.576885i	$-0.804268\pi$
$379$	−31.8038	−1.63365	−0.816826	+	0.576885i	$0.804268\pi$
$380$	0	0
$381$	−9.61973	−0.492834
$382$	0	0
$383$	−25.2349	−1.28944	−0.644721	−	0.764418i	$-0.723026\pi$
$383$	−25.2349	−1.28944	−0.644721	+	0.764418i	$0.723026\pi$
$384$	0	0
$385$	0.0753340	0.00383937
$386$	0	0
$387$	−9.69578	−0.492864
$388$	0	0
$389$	−12.5643	−0.637035	−0.318517	−	0.947917i	$-0.603185\pi$
$389$	−12.5643	−0.637035	−0.318517	+	0.947917i	$0.603185\pi$
$390$	0	0
$391$	−2.60182	−0.131580
$392$	0	0
$393$	10.9091	0.550291
$394$	0	0
$395$	−0.265553	−0.0133614
$396$	0	0
$397$	−33.5997	−1.68632	−0.843160	−	0.537663i	$-0.819307\pi$
$397$	−33.5997	−1.68632	−0.843160	+	0.537663i	$0.819307\pi$
$398$	0	0
$399$	53.9807	2.70241
$400$	0	0
$401$	−13.3145	−0.664896	−0.332448	−	0.943122i	$-0.607875\pi$
$401$	−13.3145	−0.664896	−0.332448	+	0.943122i	$0.607875\pi$
$402$	0	0
$403$	53.2113	2.65064
$404$	0	0
$405$	0.187362	0.00931007
$406$	0	0
$407$	−8.67893	−0.430199
$408$	0	0
$409$	−15.7622	−0.779390	−0.389695	−	0.920944i	$-0.627420\pi$
$409$	−15.7622	−0.779390	−0.389695	+	0.920944i	$0.627420\pi$
$410$	0	0
$411$	0.544243	0.0268455
$412$	0	0
$413$	−60.1671	−2.96063
$414$	0	0
$415$	0.247048	0.0121271
$416$	0	0
$417$	−15.2962	−0.749058
$418$	0	0
$419$	1.80836	0.0883441	0.0441721	−	0.999024i	$-0.485935\pi$
$419$	1.80836	0.0883441	0.0441721	+	0.999024i	$0.485935\pi$
$420$	0	0
$421$	5.73478	0.279496	0.139748	−	0.990187i	$-0.455371\pi$
$421$	5.73478	0.279496	0.139748	+	0.990187i	$0.455371\pi$
$422$	0	0
$423$	−21.7002	−1.05510
$424$	0	0
$425$	30.2464	1.46717
$426$	0	0
$427$	17.3621	0.840212
$428$	0	0
$429$	12.3869	0.598046
$430$	0	0
$431$	30.9922	1.49284	0.746420	−	0.665475i	$-0.231771\pi$
$431$	30.9922	1.49284	0.746420	+	0.665475i	$0.231771\pi$
$432$	0	0
$433$	−19.3114	−0.928046	−0.464023	−	0.885823i	$-0.653595\pi$
$433$	−19.3114	−0.928046	−0.464023	+	0.885823i	$0.653595\pi$
$434$	0	0
$435$	0.0105430	0.000505498 0
$436$	0	0
$437$	2.38833	0.114249
$438$	0	0
$439$	23.2567	1.10998	0.554991	−	0.831857i	$-0.312722\pi$
$439$	23.2567	1.10998	0.554991	+	0.831857i	$0.312722\pi$
$440$	0	0
$441$	21.9315	1.04436
$442$	0	0
$443$	−8.77339	−0.416837	−0.208418	−	0.978040i	$-0.566832\pi$
$443$	−8.77339	−0.416837	−0.208418	+	0.978040i	$0.566832\pi$
$444$	0	0
$445$	0.0697849	0.00330812
$446$	0	0
$447$	21.7220	1.02742
$448$	0	0
$449$	−21.9814	−1.03736	−0.518682	−	0.854967i	$-0.673577\pi$
$449$	−21.9814	−1.03736	−0.518682	+	0.854967i	$0.673577\pi$
$450$	0	0
$451$	−11.9531	−0.562849
$452$	0	0
$453$	−21.6647	−1.01790
$454$	0	0
$455$	−0.432392	−0.0202709
$456$	0	0
$457$	−23.8890	−1.11748	−0.558741	−	0.829342i	$-0.688716\pi$
$457$	−23.8890	−1.11748	−0.558741	+	0.829342i	$0.688716\pi$
$458$	0	0
$459$	−17.6969	−0.826019
$460$	0	0
$461$	−29.4137	−1.36993	−0.684967	−	0.728574i	$-0.740184\pi$
$461$	−29.4137	−1.36993	−0.684967	+	0.728574i	$0.740184\pi$
$462$	0	0
$463$	3.98010	0.184971	0.0924855	−	0.995714i	$-0.470519\pi$
$463$	3.98010	0.184971	0.0924855	+	0.995714i	$0.470519\pi$
$464$	0	0
$465$	−0.332996	−0.0154423
$466$	0	0
$467$	−10.3164	−0.477386	−0.238693	−	0.971095i	$-0.576719\pi$
$467$	−10.3164	−0.477386	−0.238693	+	0.971095i	$0.576719\pi$
$468$	0	0
$469$	−14.8879	−0.687462
$470$	0	0
$471$	16.8729	0.777461
$472$	0	0
$473$	−5.90858	−0.271677
$474$	0	0
$475$	−27.7646	−1.27393
$476$	0	0
$477$	−1.38020	−0.0631951
$478$	0	0
$479$	−22.4988	−1.02800	−0.513999	−	0.857791i	$-0.671837\pi$
$479$	−22.4988	−1.02800	−0.513999	+	0.857791i	$0.671837\pi$
$480$	0	0
$481$	49.8142	2.27133
$482$	0	0
$483$	4.18063	0.190225
$484$	0	0
$485$	−0.285349	−0.0129570
$486$	0	0
$487$	36.8675	1.67063	0.835314	−	0.549774i	$-0.185286\pi$
$487$	36.8675	1.67063	0.835314	+	0.549774i	$0.185286\pi$
$488$	0	0
$489$	−1.97246	−0.0891978
$490$	0	0
$491$	−22.1263	−0.998546	−0.499273	−	0.866445i	$-0.666399\pi$
$491$	−22.1263	−0.998546	−0.499273	+	0.866445i	$0.666399\pi$
$492$	0	0
$493$	−1.77425	−0.0799081
$494$	0	0
$495$	−0.0274233	−0.00123258
$496$	0	0
$497$	−8.01527	−0.359534
$498$	0	0
$499$	−22.7483	−1.01836	−0.509178	−	0.860661i	$-0.670050\pi$
$499$	−22.7483	−1.01836	−0.509178	+	0.860661i	$0.670050\pi$
$500$	0	0
$501$	40.8817	1.82646
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	0.0148327	0.000660045 0
$506$	0	0
$507$	−43.0871	−1.91356
$508$	0	0
$509$	−11.7543	−0.521000	−0.260500	−	0.965474i	$-0.583887\pi$
$509$	−11.7543	−0.521000	−0.260500	+	0.965474i	$0.583887\pi$
$510$	0	0
$511$	5.99128	0.265039
$512$	0	0
$513$	16.2448	0.717225
$514$	0	0
$515$	−0.0564057	−0.00248553
$516$	0	0
$517$	−13.2240	−0.581592
$518$	0	0
$519$	−7.06545	−0.310139
$520$	0	0
$521$	−40.2578	−1.76373	−0.881864	−	0.471505i	$-0.843711\pi$
$521$	−40.2578	−1.76373	−0.881864	+	0.471505i	$0.843711\pi$
$522$	0	0
$523$	0.964213	0.0421621	0.0210810	−	0.999778i	$-0.493289\pi$
$523$	0.964213	0.0421621	0.0210810	+	0.999778i	$0.493289\pi$
$524$	0	0
$525$	−48.6002	−2.12109
$526$	0	0
$527$	56.0389	2.44109
$528$	0	0
$529$	−22.8150	−0.991958
$530$	0	0
$531$	21.9022	0.950475
$532$	0	0
$533$	68.6068	2.97169
$534$	0	0
$535$	0.218027	0.00942611
$536$	0	0
$537$	−24.6786	−1.06496
$538$	0	0
$539$	13.3650	0.575671
$540$	0	0
$541$	28.5395	1.22701	0.613505	−	0.789691i	$-0.289759\pi$
$541$	28.5395	1.22701	0.613505	+	0.789691i	$0.289759\pi$
$542$	0	0
$543$	14.8246	0.636184
$544$	0	0
$545$	0.241353	0.0103384
$546$	0	0
$547$	43.7993	1.87272	0.936361	−	0.351037i	$-0.114171\pi$
$547$	43.7993	1.87272	0.936361	+	0.351037i	$0.114171\pi$
$548$	0	0
$549$	−6.32021	−0.269740
$550$	0	0
$551$	1.62867	0.0693835
$552$	0	0
$553$	−71.8069	−3.05354
$554$	0	0
$555$	−0.311738	−0.0132325
$556$	0	0
$557$	21.4929	0.910685	0.455342	−	0.890316i	$-0.349517\pi$
$557$	21.4929	0.910685	0.455342	+	0.890316i	$0.349517\pi$
$558$	0	0
$559$	33.9133	1.43438
$560$	0	0
$561$	13.0452	0.550767
$562$	0	0
$563$	−27.6181	−1.16396	−0.581982	−	0.813202i	$-0.697722\pi$
$563$	−27.6181	−1.16396	−0.581982	+	0.813202i	$0.697722\pi$
$564$	0	0
$565$	−0.100980	−0.00424827
$566$	0	0
$567$	50.6636	2.12767
$568$	0	0
$569$	16.3501	0.685433	0.342717	−	0.939439i	$-0.388653\pi$
$569$	16.3501	0.685433	0.342717	+	0.939439i	$0.388653\pi$
$570$	0	0
$571$	34.9691	1.46341	0.731706	−	0.681620i	$-0.238724\pi$
$571$	34.9691	1.46341	0.731706	+	0.681620i	$0.238724\pi$
$572$	0	0
$573$	18.6216	0.777929
$574$	0	0
$575$	−2.15028	−0.0896729
$576$	0	0
$577$	−27.0921	−1.12786	−0.563929	−	0.825823i	$-0.690711\pi$
$577$	−27.0921	−1.12786	−0.563929	+	0.825823i	$0.690711\pi$
$578$	0	0
$579$	−38.2844	−1.59104
$580$	0	0
$581$	66.8031	2.77146
$582$	0	0
$583$	−0.841090	−0.0348344
$584$	0	0
$585$	0.157401	0.00650772
$586$	0	0
$587$	40.4433	1.66928	0.834638	−	0.550799i	$-0.185677\pi$
$587$	40.4433	1.66928	0.834638	+	0.550799i	$0.185677\pi$
$588$	0	0
$589$	−51.4408	−2.11958
$590$	0	0
$591$	−38.7646	−1.59456
$592$	0	0
$593$	−16.2758	−0.668366	−0.334183	−	0.942508i	$-0.608460\pi$
$593$	−16.2758	−0.668366	−0.334183	+	0.942508i	$0.608460\pi$
$594$	0	0
$595$	−0.455369	−0.0186683
$596$	0	0
$597$	39.5468	1.61854
$598$	0	0
$599$	18.7534	0.766245	0.383123	−	0.923698i	$-0.374849\pi$
$599$	18.7534	0.766245	0.383123	+	0.923698i	$0.374849\pi$
$600$	0	0
$601$	20.9038	0.852683	0.426341	−	0.904562i	$-0.359802\pi$
$601$	20.9038	0.852683	0.426341	+	0.904562i	$0.359802\pi$
$602$	0	0
$603$	5.41955	0.220701
$604$	0	0
$605$	0.166817	0.00678206
$606$	0	0
$607$	−40.2726	−1.63462	−0.817308	−	0.576202i	$-0.804534\pi$
$607$	−40.2726	−1.63462	−0.817308	+	0.576202i	$0.804534\pi$
$608$	0	0
$609$	2.85088	0.115524
$610$	0	0
$611$	75.9016	3.07065
$612$	0	0
$613$	24.6695	0.996393	0.498197	−	0.867064i	$-0.333996\pi$
$613$	24.6695	0.996393	0.498197	+	0.867064i	$0.333996\pi$
$614$	0	0
$615$	−0.429342	−0.0173127
$616$	0	0
$617$	−17.0828	−0.687729	−0.343864	−	0.939019i	$-0.611736\pi$
$617$	−17.0828	−0.687729	−0.343864	+	0.939019i	$0.611736\pi$
$618$	0	0
$619$	29.7344	1.19513	0.597564	−	0.801821i	$-0.296135\pi$
$619$	29.7344	1.19513	0.597564	+	0.801821i	$0.296135\pi$
$620$	0	0
$621$	1.25811	0.0504861
$622$	0	0
$623$	18.8702	0.756019
$624$	0	0
$625$	24.9958	0.999833
$626$	0	0
$627$	−11.9748	−0.478226
$628$	0	0
$629$	52.4613	2.09177
$630$	0	0
$631$	4.66941	0.185886	0.0929431	−	0.995671i	$-0.470373\pi$
$631$	4.66941	0.185886	0.0929431	+	0.995671i	$0.470373\pi$
$632$	0	0
$633$	14.6813	0.583531
$634$	0	0
$635$	−0.0744915	−0.00295610
$636$	0	0
$637$	−76.7106	−3.03939
$638$	0	0
$639$	2.91774	0.115424
$640$	0	0
$641$	−26.0159	−1.02757	−0.513784	−	0.857920i	$-0.671757\pi$
$641$	−26.0159	−1.02757	−0.513784	+	0.857920i	$0.671757\pi$
$642$	0	0
$643$	−9.87315	−0.389359	−0.194680	−	0.980867i	$-0.562367\pi$
$643$	−9.87315	−0.389359	−0.194680	+	0.980867i	$0.562367\pi$
$644$	0	0
$645$	−0.212230	−0.00835653
$646$	0	0
$647$	−25.3906	−0.998207	−0.499103	−	0.866543i	$-0.666337\pi$
$647$	−25.3906	−0.998207	−0.499103	+	0.866543i	$0.666337\pi$
$648$	0	0
$649$	13.3471	0.523921
$650$	0	0
$651$	−90.0439	−3.52910
$652$	0	0
$653$	−1.57791	−0.0617485	−0.0308742	−	0.999523i	$-0.509829\pi$
$653$	−1.57791	−0.0617485	−0.0308742	+	0.999523i	$0.509829\pi$
$654$	0	0
$655$	0.0844758	0.00330074
$656$	0	0
$657$	−2.18096	−0.0850874
$658$	0	0
$659$	−47.8125	−1.86251	−0.931256	−	0.364367i	$-0.881286\pi$
$659$	−47.8125	−1.86251	−0.931256	+	0.364367i	$0.881286\pi$
$660$	0	0
$661$	−26.1048	−1.01536	−0.507680	−	0.861546i	$-0.669497\pi$
$661$	−26.1048	−1.01536	−0.507680	+	0.861546i	$0.669497\pi$
$662$	0	0
$663$	−74.8749	−2.90790
$664$	0	0
$665$	0.418005	0.0162095
$666$	0	0
$667$	0.126135	0.00488397
$668$	0	0
$669$	−31.7641	−1.22807
$670$	0	0
$671$	−3.85151	−0.148686
$672$	0	0
$673$	−10.6083	−0.408918	−0.204459	−	0.978875i	$-0.565544\pi$
$673$	−10.6083	−0.408918	−0.204459	+	0.978875i	$0.565544\pi$
$674$	0	0
$675$	−14.6256	−0.562941
$676$	0	0
$677$	23.3016	0.895552	0.447776	−	0.894146i	$-0.352216\pi$
$677$	23.3016	0.895552	0.447776	+	0.894146i	$0.352216\pi$
$678$	0	0
$679$	−77.1598	−2.96112
$680$	0	0
$681$	−49.2270	−1.88638
$682$	0	0
$683$	−9.77758	−0.374129	−0.187064	−	0.982348i	$-0.559897\pi$
$683$	−9.77758	−0.374129	−0.187064	+	0.982348i	$0.559897\pi$
$684$	0	0
$685$	0.00421441	0.000161024 0
$686$	0	0
$687$	56.9793	2.17390
$688$	0	0
$689$	4.82758	0.183916
$690$	0	0
$691$	−17.5693	−0.668367	−0.334184	−	0.942508i	$-0.608461\pi$
$691$	−17.5693	−0.668367	−0.334184	+	0.942508i	$0.608461\pi$
$692$	0	0
$693$	−7.41540	−0.281688
$694$	0	0
$695$	−0.118448	−0.00449298
$696$	0	0
$697$	72.2525	2.73676
$698$	0	0
$699$	−21.9738	−0.831126
$700$	0	0
$701$	−39.4931	−1.49163	−0.745817	−	0.666151i	$-0.767940\pi$
$701$	−39.4931	−1.49163	−0.745817	+	0.666151i	$0.767940\pi$
$702$	0	0
$703$	−48.1568	−1.81627
$704$	0	0
$705$	−0.474993	−0.0178893
$706$	0	0
$707$	4.01083	0.150843
$708$	0	0
$709$	−31.8474	−1.19606	−0.598028	−	0.801475i	$-0.704049\pi$
$709$	−31.8474	−1.19606	−0.598028	+	0.801475i	$0.704049\pi$
$710$	0	0
$711$	26.1393	0.980302
$712$	0	0
$713$	−3.98392	−0.149199
$714$	0	0
$715$	0.0959195	0.00358719
$716$	0	0
$717$	−31.2002	−1.16519
$718$	0	0
$719$	22.5015	0.839165	0.419582	−	0.907717i	$-0.362177\pi$
$719$	22.5015	0.839165	0.419582	+	0.907717i	$0.362177\pi$
$720$	0	0
$721$	−15.2524	−0.568029
$722$	0	0
$723$	−58.9040	−2.19066
$724$	0	0
$725$	−1.46633	−0.0544582
$726$	0	0
$727$	31.8285	1.18045	0.590227	−	0.807237i	$-0.299038\pi$
$727$	31.8285	1.18045	0.590227	+	0.807237i	$0.299038\pi$
$728$	0	0
$729$	0.465794	0.0172516
$730$	0	0
$731$	35.7155	1.32098
$732$	0	0
$733$	−15.1502	−0.559586	−0.279793	−	0.960060i	$-0.590266\pi$
$733$	−15.1502	−0.559586	−0.279793	+	0.960060i	$0.590266\pi$
$734$	0	0
$735$	0.480055	0.0177071
$736$	0	0
$737$	3.30266	0.121655
$738$	0	0
$739$	8.40561	0.309205	0.154603	−	0.987977i	$-0.450590\pi$
$739$	8.40561	0.309205	0.154603	+	0.987977i	$0.450590\pi$
$740$	0	0
$741$	68.7312	2.52491
$742$	0	0
$743$	−21.3511	−0.783295	−0.391647	−	0.920115i	$-0.628095\pi$
$743$	−21.3511	−0.783295	−0.391647	+	0.920115i	$0.628095\pi$
$744$	0	0
$745$	0.168207	0.00616262
$746$	0	0
$747$	−24.3178	−0.889743
$748$	0	0
$749$	58.9555	2.15419
$750$	0	0
$751$	−32.2516	−1.17688	−0.588438	−	0.808542i	$-0.700257\pi$
$751$	−32.2516	−1.17688	−0.588438	+	0.808542i	$0.700257\pi$
$752$	0	0
$753$	−30.7321	−1.11994
$754$	0	0
$755$	−0.167763	−0.00610552
$756$	0	0
$757$	−49.5145	−1.79963	−0.899817	−	0.436268i	$-0.856300\pi$
$757$	−49.5145	−1.79963	−0.899817	+	0.436268i	$0.856300\pi$
$758$	0	0
$759$	−0.927408	−0.0336628
$760$	0	0
$761$	48.7518	1.76725	0.883625	−	0.468195i	$-0.155095\pi$
$761$	48.7518	1.76725	0.883625	+	0.468195i	$0.155095\pi$
$762$	0	0
$763$	65.2632	2.36269
$764$	0	0
$765$	0.165765	0.00599324
$766$	0	0
$767$	−76.6082	−2.76616
$768$	0	0
$769$	−21.4827	−0.774684	−0.387342	−	0.921936i	$-0.626607\pi$
$769$	−21.4827	−0.774684	−0.387342	+	0.921936i	$0.626607\pi$
$770$	0	0
$771$	−28.2744	−1.01828
$772$	0	0
$773$	12.0050	0.431790	0.215895	−	0.976417i	$-0.430733\pi$
$773$	12.0050	0.431790	0.215895	+	0.976417i	$0.430733\pi$
$774$	0	0
$775$	46.3135	1.66363
$776$	0	0
$777$	−84.2955	−3.02408
$778$	0	0
$779$	−66.3240	−2.37631
$780$	0	0
$781$	1.77806	0.0636241
$782$	0	0
$783$	0.857936	0.0306601
$784$	0	0
$785$	0.130657	0.00466335
$786$	0	0
$787$	−29.3534	−1.04634	−0.523168	−	0.852230i	$-0.675250\pi$
$787$	−29.3534	−1.04634	−0.523168	+	0.852230i	$0.675250\pi$
$788$	0	0
$789$	26.9983	0.961166
$790$	0	0
$791$	−27.3056	−0.970874
$792$	0	0
$793$	22.1064	0.785023
$794$	0	0
$795$	−0.0302110	−0.00107147
$796$	0	0
$797$	10.8538	0.384460	0.192230	−	0.981350i	$-0.438428\pi$
$797$	10.8538	0.384460	0.192230	+	0.981350i	$0.438428\pi$
$798$	0	0
$799$	79.9350	2.82790
$800$	0	0
$801$	−6.86919	−0.242711
$802$	0	0
$803$	−1.32907	−0.0469019
$804$	0	0
$805$	0.0323732	0.00114100
$806$	0	0
$807$	6.18426	0.217696
$808$	0	0
$809$	−11.1823	−0.393148	−0.196574	−	0.980489i	$-0.562982\pi$
$809$	−11.1823	−0.393148	−0.196574	+	0.980489i	$0.562982\pi$
$810$	0	0
$811$	23.9683	0.841641	0.420821	−	0.907144i	$-0.361742\pi$
$811$	23.9683	0.841641	0.420821	+	0.907144i	$0.361742\pi$
$812$	0	0
$813$	2.87151	0.100708
$814$	0	0
$815$	−0.0152740	−0.000535024 0
$816$	0	0
$817$	−32.7849	−1.14700
$818$	0	0
$819$	42.5620	1.48724
$820$	0	0
$821$	0.881870	0.0307775	0.0153887	−	0.999882i	$-0.495101\pi$
$821$	0.881870	0.0307775	0.0153887	+	0.999882i	$0.495101\pi$
$822$	0	0
$823$	−43.0432	−1.50039	−0.750196	−	0.661216i	$-0.770041\pi$
$823$	−43.0432	−1.50039	−0.750196	+	0.661216i	$0.770041\pi$
$824$	0	0
$825$	10.7812	0.375353
$826$	0	0
$827$	−13.4157	−0.466508	−0.233254	−	0.972416i	$-0.574937\pi$
$827$	−13.4157	−0.466508	−0.233254	+	0.972416i	$0.574937\pi$
$828$	0	0
$829$	−42.3903	−1.47228	−0.736139	−	0.676831i	$-0.763353\pi$
$829$	−42.3903	−1.47228	−0.736139	+	0.676831i	$0.763353\pi$
$830$	0	0
$831$	−38.0460	−1.31980
$832$	0	0
$833$	−80.7870	−2.79910
$834$	0	0
$835$	0.316572	0.0109554
$836$	0	0
$837$	−27.0976	−0.936629
$838$	0	0
$839$	6.35574	0.219424	0.109712	−	0.993963i	$-0.465007\pi$
$839$	6.35574	0.219424	0.109712	+	0.993963i	$0.465007\pi$
$840$	0	0
$841$	−28.9140	−0.997034
$842$	0	0
$843$	0.282709	0.00973703
$844$	0	0
$845$	−0.333649	−0.0114779
$846$	0	0
$847$	45.1081	1.54993
$848$	0	0
$849$	6.92227	0.237572
$850$	0	0
$851$	−3.72959	−0.127849
$852$	0	0
$853$	35.9790	1.23190	0.615949	−	0.787786i	$-0.288773\pi$
$853$	35.9790	1.23190	0.615949	+	0.787786i	$0.288773\pi$
$854$	0	0
$855$	−0.152163	−0.00520388
$856$	0	0
$857$	−13.0478	−0.445704	−0.222852	−	0.974852i	$-0.571537\pi$
$857$	−13.0478	−0.445704	−0.222852	+	0.974852i	$0.571537\pi$
$858$	0	0
$859$	−46.8140	−1.59727	−0.798636	−	0.601814i	$-0.794445\pi$
$859$	−46.8140	−1.59727	−0.798636	+	0.601814i	$0.794445\pi$
$860$	0	0
$861$	−116.096	−3.95655
$862$	0	0
$863$	17.9173	0.609913	0.304956	−	0.952366i	$-0.401358\pi$
$863$	17.9173	0.609913	0.304956	+	0.952366i	$0.401358\pi$
$864$	0	0
$865$	−0.0547121	−0.00186027
$866$	0	0
$867$	−42.2255	−1.43405
$868$	0	0
$869$	15.9292	0.540362
$870$	0	0
$871$	−18.9562	−0.642306
$872$	0	0
$873$	28.0879	0.950632
$874$	0	0
$875$	−0.752704	−0.0254460
$876$	0	0
$877$	−2.15230	−0.0726779	−0.0363390	−	0.999340i	$-0.511570\pi$
$877$	−2.15230	−0.0726779	−0.0363390	+	0.999340i	$0.511570\pi$
$878$	0	0
$879$	−31.2960	−1.05559
$880$	0	0
$881$	2.39051	0.0805384	0.0402692	−	0.999189i	$-0.487178\pi$
$881$	2.39051	0.0805384	0.0402692	+	0.999189i	$0.487178\pi$
$882$	0	0
$883$	13.8390	0.465719	0.232859	−	0.972510i	$-0.425192\pi$
$883$	13.8390	0.465719	0.232859	+	0.972510i	$0.425192\pi$
$884$	0	0
$885$	0.479414	0.0161153
$886$	0	0
$887$	−2.04793	−0.0687629	−0.0343814	−	0.999409i	$-0.510946\pi$
$887$	−2.04793	−0.0687629	−0.0343814	+	0.999409i	$0.510946\pi$
$888$	0	0
$889$	−20.1429	−0.675571
$890$	0	0
$891$	−11.2389	−0.376518
$892$	0	0
$893$	−73.3761	−2.45544
$894$	0	0
$895$	−0.191101	−0.00638782
$896$	0	0
$897$	5.32302	0.177730
$898$	0	0
$899$	−2.71674	−0.0906084
$900$	0	0
$901$	5.08412	0.169377
$902$	0	0
$903$	−57.3880	−1.90975
$904$	0	0
$905$	0.114796	0.00381595
$906$	0	0
$907$	−25.2202	−0.837422	−0.418711	−	0.908119i	$-0.637518\pi$
$907$	−25.2202	−0.837422	−0.418711	+	0.908119i	$0.637518\pi$
$908$	0	0
$909$	−1.46003	−0.0484263
$910$	0	0
$911$	−7.10295	−0.235331	−0.117666	−	0.993053i	$-0.537541\pi$
$911$	−7.10295	−0.235331	−0.117666	+	0.993053i	$0.537541\pi$
$912$	0	0
$913$	−14.8192	−0.490444
$914$	0	0
$915$	−0.138342	−0.00457345
$916$	0	0
$917$	22.8427	0.754332
$918$	0	0
$919$	−47.0729	−1.55279	−0.776396	−	0.630246i	$-0.782954\pi$
$919$	−47.0729	−1.55279	−0.776396	+	0.630246i	$0.782954\pi$
$920$	0	0
$921$	15.7066	0.517550
$922$	0	0
$923$	−10.2055	−0.335918
$924$	0	0
$925$	43.3568	1.42556
$926$	0	0
$927$	5.55222	0.182359
$928$	0	0
$929$	5.12702	0.168212	0.0841060	−	0.996457i	$-0.473197\pi$
$929$	5.12702	0.168212	0.0841060	+	0.996457i	$0.473197\pi$
$930$	0	0
$931$	74.1582	2.43044
$932$	0	0
$933$	9.00198	0.294712
$934$	0	0
$935$	0.101017	0.00330360
$936$	0	0
$937$	−3.51646	−0.114878	−0.0574389	−	0.998349i	$-0.518293\pi$
$937$	−3.51646	−0.114878	−0.0574389	+	0.998349i	$0.518293\pi$
$938$	0	0
$939$	54.2967	1.77191
$940$	0	0
$941$	60.1482	1.96077	0.980387	−	0.197082i	$-0.0631464\pi$
$941$	60.1482	1.96077	0.980387	+	0.197082i	$0.0631464\pi$
$942$	0	0
$943$	−5.13659	−0.167270
$944$	0	0
$945$	0.220194	0.00716290
$946$	0	0
$947$	31.7588	1.03202	0.516012	−	0.856582i	$-0.327416\pi$
$947$	31.7588	1.03202	0.516012	+	0.856582i	$0.327416\pi$
$948$	0	0
$949$	7.62844	0.247630
$950$	0	0
$951$	−16.0418	−0.520190
$952$	0	0
$953$	51.7281	1.67564	0.837818	−	0.545949i	$-0.183831\pi$
$953$	51.7281	1.67564	0.837818	+	0.545949i	$0.183831\pi$
$954$	0	0
$955$	0.144199	0.00466616
$956$	0	0
$957$	−0.632423	−0.0204433
$958$	0	0
$959$	1.13960	0.0367995
$960$	0	0
$961$	54.8072	1.76797
$962$	0	0
$963$	−21.4612	−0.691576
$964$	0	0
$965$	−0.296459	−0.00954336
$966$	0	0
$967$	3.73236	0.120025	0.0600123	−	0.998198i	$-0.480886\pi$
$967$	3.73236	0.120025	0.0600123	+	0.998198i	$0.480886\pi$
$968$	0	0
$969$	72.3836	2.32530
$970$	0	0
$971$	23.3178	0.748304	0.374152	−	0.927367i	$-0.377934\pi$
$971$	23.3178	0.748304	0.374152	+	0.927367i	$0.377934\pi$
$972$	0	0
$973$	−32.0289	−1.02680
$974$	0	0
$975$	−61.8806	−1.98177
$976$	0	0
$977$	−16.0457	−0.513348	−0.256674	−	0.966498i	$-0.582627\pi$
$977$	−16.0457	−0.513348	−0.256674	+	0.966498i	$0.582627\pi$
$978$	0	0
$979$	−4.18606	−0.133787
$980$	0	0
$981$	−23.7573	−0.758512
$982$	0	0
$983$	−33.4969	−1.06838	−0.534192	−	0.845363i	$-0.679384\pi$
$983$	−33.4969	−1.06838	−0.534192	+	0.845363i	$0.679384\pi$
$984$	0	0
$985$	−0.300178	−0.00956446
$986$	0	0
$987$	−128.441	−4.08831
$988$	0	0
$989$	−2.53909	−0.0807383
$990$	0	0
$991$	22.9762	0.729862	0.364931	−	0.931035i	$-0.381093\pi$
$991$	22.9762	0.729862	0.364931	+	0.931035i	$0.381093\pi$
$992$	0	0
$993$	20.9523	0.664900
$994$	0	0
$995$	0.306235	0.00970830
$996$	0	0
$997$	−41.9893	−1.32981	−0.664907	−	0.746926i	$-0.731529\pi$
$997$	−41.9893	−1.32981	−0.664907	+	0.746926i	$0.731529\pi$
$998$	0	0
$999$	−25.3677	−0.802597

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.g.1.6	✓	33
4.3	odd	2		8048.2.a.x.1.28		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.6	✓	33	1.1	even	1	trivial
8048.2.a.x.1.28		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)

\(f(q)\)	\(=\)	\(q-2.15460 q^{3} -0.0166844 q^{5} -4.51155 q^{7} +1.64231 q^{9} +O(q^{10})\)	\(q-2.15460 q^{3} -0.0166844 q^{5} -4.51155 q^{7} +1.64231 q^{9} +1.00082 q^{11} -5.74436 q^{13} +0.0359482 q^{15} -6.04961 q^{17} +5.55323 q^{19} +9.72059 q^{21} +0.430080 q^{23} -4.99972 q^{25} +2.92529 q^{27} +0.293283 q^{29} -9.26322 q^{31} -2.15636 q^{33} +0.0752725 q^{35} -8.67185 q^{37} +12.3768 q^{39} -11.9433 q^{41} -5.90376 q^{43} -0.0274009 q^{45} -13.2132 q^{47} +13.3541 q^{49} +13.0345 q^{51} -0.840404 q^{53} -0.0166980 q^{55} -11.9650 q^{57} +13.3362 q^{59} -3.84837 q^{61} -7.40935 q^{63} +0.0958412 q^{65} +3.29996 q^{67} -0.926651 q^{69} +1.77661 q^{71} -1.32799 q^{73} +10.7724 q^{75} -4.51523 q^{77} +15.9162 q^{79} -11.2297 q^{81} -14.8071 q^{83} +0.100934 q^{85} -0.631907 q^{87} -4.18265 q^{89} +25.9160 q^{91} +19.9585 q^{93} -0.0926523 q^{95} +17.1027 q^{97} +1.64365 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) 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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