LMFDB - Embedded newform 4508.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4508,2,Mod(1,4508)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4508, 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("4508.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4508 = 2^{2} \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4508.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:	$35.9965612312$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.6963152.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 10x^{3} + 10x^{2} + 29x + 10 $ x^5 - 2x^4 - 10x^3 + 10x^2 + 29x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 644)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.18229$ of defining polynomial
Character	$\chi$	$=$	4508.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.18229 q^{3} -4.04332 q^{5} +1.76237 q^{9} +O(q^{10})$	$q-2.18229 q^{3} -4.04332 q^{5} +1.76237 q^{9} -3.98384 q^{11} -6.74621 q^{13} +8.82367 q^{15} +4.92849 q^{17} +1.24680 q^{19} -1.00000 q^{23} +11.3484 q^{25} +2.70086 q^{27} +6.48443 q^{29} +8.38256 q^{31} +8.69387 q^{33} -7.61137 q^{37} +14.7222 q^{39} +7.11078 q^{41} -1.11777 q^{43} -7.12582 q^{45} -8.31426 q^{47} -10.7554 q^{51} -4.77154 q^{53} +16.1079 q^{55} -2.72088 q^{57} -1.31712 q^{59} -0.838914 q^{61} +27.2771 q^{65} +12.4350 q^{67} +2.18229 q^{69} +10.0613 q^{71} -6.46827 q^{73} -24.7655 q^{75} -6.45910 q^{79} -11.1812 q^{81} +5.59303 q^{83} -19.9275 q^{85} -14.1509 q^{87} +10.2619 q^{89} -18.2931 q^{93} -5.04121 q^{95} -11.2477 q^{97} -7.02099 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{3} - 2 q^{5} + 10 q^{9}+O(q^{10}) $ 5 * q - 3 * q^3 - 2 * q^5 + 10 * q^9	$ 5 q - 3 q^{3} - 2 q^{5} + 10 q^{9} + 2 q^{11} - 13 q^{13} + 4 q^{15} - 4 q^{17} - 12 q^{19} - 5 q^{23} + 19 q^{25} - 15 q^{27} + 13 q^{29} + 3 q^{31} - 24 q^{33} - 4 q^{37} + 3 q^{39} - q^{41} - 8 q^{43} + 16 q^{45} - 5 q^{47} - 16 q^{51} - 8 q^{53} + 2 q^{55} + 12 q^{57} - 12 q^{59} - 20 q^{61} - 12 q^{65} - 12 q^{67} + 3 q^{69} + 9 q^{71} + 9 q^{73} - 35 q^{75} - 8 q^{79} - 11 q^{81} + 28 q^{83} + 16 q^{85} + 15 q^{87} - 32 q^{89} - 15 q^{93} - 36 q^{95} - 4 q^{97} + 28 q^{99}+O(q^{100}) $ 5 * q - 3 * q^3 - 2 * q^5 + 10 * q^9 + 2 * q^11 - 13 * q^13 + 4 * q^15 - 4 * q^17 - 12 * q^19 - 5 * q^23 + 19 * q^25 - 15 * q^27 + 13 * q^29 + 3 * q^31 - 24 * q^33 - 4 * q^37 + 3 * q^39 - q^41 - 8 * q^43 + 16 * q^45 - 5 * q^47 - 16 * q^51 - 8 * q^53 + 2 * q^55 + 12 * q^57 - 12 * q^59 - 20 * q^61 - 12 * q^65 - 12 * q^67 + 3 * q^69 + 9 * q^71 + 9 * q^73 - 35 * q^75 - 8 * q^79 - 11 * q^81 + 28 * q^83 + 16 * q^85 + 15 * q^87 - 32 * q^89 - 15 * q^93 - 36 * q^95 - 4 * q^97 + 28 * 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
$3$	−2.18229	−1.25994	−0.629972	−	0.776618i	$-0.716933\pi$
$3$	−2.18229	−1.25994	−0.629972	+	0.776618i	$0.716933\pi$
$4$	0	0
$5$	−4.04332	−1.80823	−0.904113	−	0.427293i	$-0.859467\pi$
$5$	−4.04332	−1.80823	−0.904113	+	0.427293i	$0.859467\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.76237	0.587456
$10$	0	0
$11$	−3.98384	−1.20117	−0.600586	−	0.799560i	$-0.705066\pi$
$11$	−3.98384	−1.20117	−0.600586	+	0.799560i	$0.705066\pi$
$12$	0	0
$13$	−6.74621	−1.87106	−0.935531	−	0.353245i	$-0.885078\pi$
$13$	−6.74621	−1.87106	−0.935531	+	0.353245i	$0.885078\pi$
$14$	0	0
$15$	8.82367	2.27826
$16$	0	0
$17$	4.92849	1.19534	0.597668	−	0.801744i	$-0.296094\pi$
$17$	4.92849	1.19534	0.597668	+	0.801744i	$0.296094\pi$
$18$	0	0
$19$	1.24680	0.286036	0.143018	−	0.989720i	$-0.454319\pi$
$19$	1.24680	0.286036	0.143018	+	0.989720i	$0.454319\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	11.3484	2.26968
$26$	0	0
$27$	2.70086	0.519781
$28$	0	0
$29$	6.48443	1.20413	0.602064	−	0.798448i	$-0.294345\pi$
$29$	6.48443	1.20413	0.602064	+	0.798448i	$0.294345\pi$
$30$	0	0
$31$	8.38256	1.50555	0.752776	−	0.658277i	$-0.228714\pi$
$31$	8.38256	1.50555	0.752776	+	0.658277i	$0.228714\pi$
$32$	0	0
$33$	8.69387	1.51341
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.61137	−1.25130	−0.625651	−	0.780103i	$-0.715167\pi$
$37$	−7.61137	−1.25130	−0.625651	+	0.780103i	$0.715167\pi$
$38$	0	0
$39$	14.7222	2.35743
$40$	0	0
$41$	7.11078	1.11052	0.555259	−	0.831678i	$-0.312619\pi$
$41$	7.11078	1.11052	0.555259	+	0.831678i	$0.312619\pi$
$42$	0	0
$43$	−1.11777	−0.170458	−0.0852291	−	0.996361i	$-0.527162\pi$
$43$	−1.11777	−0.170458	−0.0852291	+	0.996361i	$0.527162\pi$
$44$	0	0
$45$	−7.12582	−1.06225
$46$	0	0
$47$	−8.31426	−1.21276	−0.606380	−	0.795175i	$-0.707379\pi$
$47$	−8.31426	−1.21276	−0.606380	+	0.795175i	$0.707379\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−10.7554	−1.50605
$52$	0	0
$53$	−4.77154	−0.655421	−0.327711	−	0.944778i	$-0.606277\pi$
$53$	−4.77154	−0.655421	−0.327711	+	0.944778i	$0.606277\pi$
$54$	0	0
$55$	16.1079	2.17199
$56$	0	0
$57$	−2.72088	−0.360389
$58$	0	0
$59$	−1.31712	−0.171475	−0.0857373	−	0.996318i	$-0.527325\pi$
$59$	−1.31712	−0.171475	−0.0857373	+	0.996318i	$0.527325\pi$
$60$	0	0
$61$	−0.838914	−0.107412	−0.0537060	−	0.998557i	$-0.517103\pi$
$61$	−0.838914	−0.107412	−0.0537060	+	0.998557i	$0.517103\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	27.2771	3.38330
$66$	0	0
$67$	12.4350	1.51918	0.759591	−	0.650401i	$-0.225399\pi$
$67$	12.4350	1.51918	0.759591	+	0.650401i	$0.225399\pi$
$68$	0	0
$69$	2.18229	0.262716
$70$	0	0
$71$	10.0613	1.19406	0.597028	−	0.802220i	$-0.296348\pi$
$71$	10.0613	1.19406	0.597028	+	0.802220i	$0.296348\pi$
$72$	0	0
$73$	−6.46827	−0.757054	−0.378527	−	0.925590i	$-0.623569\pi$
$73$	−6.46827	−0.757054	−0.378527	+	0.925590i	$0.623569\pi$
$74$	0	0
$75$	−24.7655	−2.85967
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.45910	−0.726706	−0.363353	−	0.931652i	$-0.618368\pi$
$79$	−6.45910	−0.726706	−0.363353	+	0.931652i	$0.618368\pi$
$80$	0	0
$81$	−11.1812	−1.24235
$82$	0	0
$83$	5.59303	0.613915	0.306957	−	0.951723i	$-0.400689\pi$
$83$	5.59303	0.613915	0.306957	+	0.951723i	$0.400689\pi$
$84$	0	0
$85$	−19.9275	−2.16144
$86$	0	0
$87$	−14.1509	−1.51713
$88$	0	0
$89$	10.2619	1.08776	0.543881	−	0.839162i	$-0.316954\pi$
$89$	10.2619	1.08776	0.543881	+	0.839162i	$0.316954\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−18.2931	−1.89691
$94$	0	0
$95$	−5.04121	−0.517217
$96$	0	0
$97$	−11.2477	−1.14203	−0.571016	−	0.820939i	$-0.693451\pi$
$97$	−11.2477	−1.14203	−0.571016	+	0.820939i	$0.693451\pi$
$98$	0	0
$99$	−7.02099	−0.705636
$100$	0	0
$101$	13.5690	1.35016	0.675082	−	0.737743i	$-0.264108\pi$
$101$	13.5690	1.35016	0.675082	+	0.737743i	$0.264108\pi$
$102$	0	0
$103$	−12.4512	−1.22685	−0.613427	−	0.789752i	$-0.710209\pi$
$103$	−12.4512	−1.22685	−0.613427	+	0.789752i	$0.710209\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.11777	−0.494753	−0.247377	−	0.968919i	$-0.579569\pi$
$107$	−5.11777	−0.494753	−0.247377	+	0.968919i	$0.579569\pi$
$108$	0	0
$109$	−3.31509	−0.317528	−0.158764	−	0.987317i	$-0.550751\pi$
$109$	−3.31509	−0.317528	−0.158764	+	0.987317i	$0.550751\pi$
$110$	0	0
$111$	16.6102	1.57657
$112$	0	0
$113$	13.1721	1.23913	0.619563	−	0.784947i	$-0.287310\pi$
$113$	13.1721	1.23913	0.619563	+	0.784947i	$0.287310\pi$
$114$	0	0
$115$	4.04332	0.377041
$116$	0	0
$117$	−11.8893	−1.09917
$118$	0	0
$119$	0	0
$120$	0	0
$121$	4.87097	0.442815
$122$	0	0
$123$	−15.5177	−1.39919
$124$	0	0
$125$	−25.6686	−2.29587
$126$	0	0
$127$	−4.50059	−0.399363	−0.199682	−	0.979861i	$-0.563991\pi$
$127$	−4.50059	−0.399363	−0.199682	+	0.979861i	$0.563991\pi$
$128$	0	0
$129$	2.43929	0.214768
$130$	0	0
$131$	1.22882	0.107362	0.0536811	−	0.998558i	$-0.482905\pi$
$131$	1.22882	0.107362	0.0536811	+	0.998558i	$0.482905\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−10.9204	−0.939882
$136$	0	0
$137$	2.61018	0.223003	0.111502	−	0.993764i	$-0.464434\pi$
$137$	2.61018	0.223003	0.111502	+	0.993764i	$0.464434\pi$
$138$	0	0
$139$	11.5157	0.976751	0.488375	−	0.872634i	$-0.337590\pi$
$139$	11.5157	0.976751	0.488375	+	0.872634i	$0.337590\pi$
$140$	0	0
$141$	18.1441	1.52801
$142$	0	0
$143$	26.8758	2.24747
$144$	0	0
$145$	−26.2186	−2.17734
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.45120	0.528503	0.264252	−	0.964454i	$-0.414875\pi$
$149$	6.45120	0.528503	0.264252	+	0.964454i	$0.414875\pi$
$150$	0	0
$151$	3.22147	0.262159	0.131080	−	0.991372i	$-0.458156\pi$
$151$	3.22147	0.262159	0.131080	+	0.991372i	$0.458156\pi$
$152$	0	0
$153$	8.68582	0.702207
$154$	0	0
$155$	−33.8933	−2.72238
$156$	0	0
$157$	22.2296	1.77412	0.887058	−	0.461658i	$-0.152745\pi$
$157$	22.2296	1.77412	0.887058	+	0.461658i	$0.152745\pi$
$158$	0	0
$159$	10.4129	0.825794
$160$	0	0
$161$	0	0
$162$	0	0
$163$	13.1291	1.02835	0.514176	−	0.857685i	$-0.328098\pi$
$163$	13.1291	1.02835	0.514176	+	0.857685i	$0.328098\pi$
$164$	0	0
$165$	−35.1521	−2.73659
$166$	0	0
$167$	7.53573	0.583132	0.291566	−	0.956551i	$-0.405824\pi$
$167$	7.53573	0.583132	0.291566	+	0.956551i	$0.405824\pi$
$168$	0	0
$169$	32.5113	2.50087
$170$	0	0
$171$	2.19732	0.168034
$172$	0	0
$173$	7.24680	0.550964	0.275482	−	0.961306i	$-0.411163\pi$
$173$	7.24680	0.550964	0.275482	+	0.961306i	$0.411163\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.87433	0.216048
$178$	0	0
$179$	−8.90113	−0.665302	−0.332651	−	0.943050i	$-0.607943\pi$
$179$	−8.90113	−0.665302	−0.332651	+	0.943050i	$0.607943\pi$
$180$	0	0
$181$	−21.8974	−1.62762	−0.813809	−	0.581133i	$-0.802610\pi$
$181$	−21.8974	−1.62762	−0.813809	+	0.581133i	$0.802610\pi$
$182$	0	0
$183$	1.83075	0.135333
$184$	0	0
$185$	30.7752	2.26264
$186$	0	0
$187$	−19.6343	−1.43580
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.3405	−1.47179	−0.735894	−	0.677097i	$-0.763238\pi$
$191$	−20.3405	−1.47179	−0.735894	+	0.677097i	$0.763238\pi$
$192$	0	0
$193$	−9.38466	−0.675523	−0.337761	−	0.941232i	$-0.609670\pi$
$193$	−9.38466	−0.675523	−0.337761	+	0.941232i	$0.609670\pi$
$194$	0	0
$195$	−59.5263	−4.26277
$196$	0	0
$197$	−21.0544	−1.50006	−0.750032	−	0.661402i	$-0.769962\pi$
$197$	−21.0544	−1.50006	−0.750032	+	0.661402i	$0.769962\pi$
$198$	0	0
$199$	2.99881	0.212580	0.106290	−	0.994335i	$-0.466103\pi$
$199$	2.99881	0.212580	0.106290	+	0.994335i	$0.466103\pi$
$200$	0	0
$201$	−27.1368	−1.91408
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−28.7511	−2.00807
$206$	0	0
$207$	−1.76237	−0.122493
$208$	0	0
$209$	−4.96705	−0.343578
$210$	0	0
$211$	−20.2981	−1.39738	−0.698690	−	0.715425i	$-0.746233\pi$
$211$	−20.2981	−1.39738	−0.698690	+	0.715425i	$0.746233\pi$
$212$	0	0
$213$	−21.9566	−1.50444
$214$	0	0
$215$	4.51950	0.308227
$216$	0	0
$217$	0	0
$218$	0	0
$219$	14.1156	0.953845
$220$	0	0
$221$	−33.2486	−2.23655
$222$	0	0
$223$	−20.5075	−1.37329	−0.686643	−	0.726995i	$-0.740917\pi$
$223$	−20.5075	−1.37329	−0.686643	+	0.726995i	$0.740917\pi$
$224$	0	0
$225$	20.0001	1.33334
$226$	0	0
$227$	3.77273	0.250405	0.125202	−	0.992131i	$-0.460042\pi$
$227$	3.77273	0.250405	0.125202	+	0.992131i	$0.460042\pi$
$228$	0	0
$229$	−3.12217	−0.206319	−0.103159	−	0.994665i	$-0.532895\pi$
$229$	−3.12217	−0.206319	−0.103159	+	0.994665i	$0.532895\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.8921	1.56522	0.782610	−	0.622512i	$-0.213888\pi$
$233$	23.8921	1.56522	0.782610	+	0.622512i	$0.213888\pi$
$234$	0	0
$235$	33.6172	2.19294
$236$	0	0
$237$	14.0956	0.915608
$238$	0	0
$239$	1.54180	0.0997311	0.0498655	−	0.998756i	$-0.484121\pi$
$239$	1.54180	0.0997311	0.0498655	+	0.998756i	$0.484121\pi$
$240$	0	0
$241$	−12.5639	−0.809313	−0.404657	−	0.914469i	$-0.632609\pi$
$241$	−12.5639	−0.809313	−0.404657	+	0.914469i	$0.632609\pi$
$242$	0	0
$243$	16.2979	1.04551
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.41118	−0.535191
$248$	0	0
$249$	−12.2056	−0.773498
$250$	0	0
$251$	−14.4753	−0.913670	−0.456835	−	0.889551i	$-0.651017\pi$
$251$	−14.4753	−0.913670	−0.456835	+	0.889551i	$0.651017\pi$
$252$	0	0
$253$	3.98384	0.250462
$254$	0	0
$255$	43.4874	2.72329
$256$	0	0
$257$	−12.5125	−0.780509	−0.390254	−	0.920707i	$-0.627613\pi$
$257$	−12.5125	−0.780509	−0.390254	+	0.920707i	$0.627613\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	11.4280	0.707373
$262$	0	0
$263$	−19.7242	−1.21625	−0.608124	−	0.793842i	$-0.708078\pi$
$263$	−19.7242	−1.21625	−0.608124	+	0.793842i	$0.708078\pi$
$264$	0	0
$265$	19.2928	1.18515
$266$	0	0
$267$	−22.3945	−1.37052
$268$	0	0
$269$	−1.86398	−0.113649	−0.0568244	−	0.998384i	$-0.518098\pi$
$269$	−1.86398	−0.113649	−0.0568244	+	0.998384i	$0.518098\pi$
$270$	0	0
$271$	−6.47729	−0.393467	−0.196734	−	0.980457i	$-0.563033\pi$
$271$	−6.47729	−0.393467	−0.196734	+	0.980457i	$0.563033\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−45.2102	−2.72628
$276$	0	0
$277$	26.1077	1.56866	0.784330	−	0.620343i	$-0.213007\pi$
$277$	26.1077	1.56866	0.784330	+	0.620343i	$0.213007\pi$
$278$	0	0
$279$	14.7732	0.884446
$280$	0	0
$281$	3.12784	0.186592	0.0932958	−	0.995638i	$-0.470260\pi$
$281$	3.12784	0.186592	0.0932958	+	0.995638i	$0.470260\pi$
$282$	0	0
$283$	−21.4017	−1.27220	−0.636100	−	0.771606i	$-0.719454\pi$
$283$	−21.4017	−1.27220	−0.636100	+	0.771606i	$0.719454\pi$
$284$	0	0
$285$	11.0014	0.651664
$286$	0	0
$287$	0	0
$288$	0	0
$289$	7.29004	0.428826
$290$	0	0
$291$	24.5457	1.43890
$292$	0	0
$293$	20.7743	1.21365	0.606823	−	0.794837i	$-0.292444\pi$
$293$	20.7743	1.21365	0.606823	+	0.794837i	$0.292444\pi$
$294$	0	0
$295$	5.32554	0.310065
$296$	0	0
$297$	−10.7598	−0.624347
$298$	0	0
$299$	6.74621	0.390143
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−29.6114	−1.70113
$304$	0	0
$305$	3.39199	0.194225
$306$	0	0
$307$	−6.95787	−0.397107	−0.198553	−	0.980090i	$-0.563624\pi$
$307$	−6.95787	−0.397107	−0.198553	+	0.980090i	$0.563624\pi$
$308$	0	0
$309$	27.1721	1.54577
$310$	0	0
$311$	−31.5759	−1.79051	−0.895253	−	0.445558i	$-0.853005\pi$
$311$	−31.5759	−1.79051	−0.895253	+	0.445558i	$0.853005\pi$
$312$	0	0
$313$	7.00505	0.395949	0.197974	−	0.980207i	$-0.436564\pi$
$313$	7.00505	0.395949	0.197974	+	0.980207i	$0.436564\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.2749	1.19492	0.597458	−	0.801900i	$-0.296177\pi$
$317$	21.2749	1.19492	0.597458	+	0.801900i	$0.296177\pi$
$318$	0	0
$319$	−25.8329	−1.44637
$320$	0	0
$321$	11.1684	0.623361
$322$	0	0
$323$	6.14485	0.341909
$324$	0	0
$325$	−76.5587	−4.24671
$326$	0	0
$327$	7.23448	0.400068
$328$	0	0
$329$	0	0
$330$	0	0
$331$	22.9759	1.26287	0.631434	−	0.775430i	$-0.282467\pi$
$331$	22.9759	1.26287	0.631434	+	0.775430i	$0.282467\pi$
$332$	0	0
$333$	−13.4140	−0.735085
$334$	0	0
$335$	−50.2788	−2.74702
$336$	0	0
$337$	−1.62145	−0.0883258	−0.0441629	−	0.999024i	$-0.514062\pi$
$337$	−1.62145	−0.0883258	−0.0441629	+	0.999024i	$0.514062\pi$
$338$	0	0
$339$	−28.7452	−1.56123
$340$	0	0
$341$	−33.3947	−1.80843
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−8.82367	−0.475050
$346$	0	0
$347$	−3.01310	−0.161752	−0.0808758	−	0.996724i	$-0.525772\pi$
$347$	−3.01310	−0.161752	−0.0808758	+	0.996724i	$0.525772\pi$
$348$	0	0
$349$	−3.83992	−0.205546	−0.102773	−	0.994705i	$-0.532772\pi$
$349$	−3.83992	−0.205546	−0.102773	+	0.994705i	$0.532772\pi$
$350$	0	0
$351$	−18.2206	−0.972543
$352$	0	0
$353$	33.8459	1.80144	0.900719	−	0.434403i	$-0.143040\pi$
$353$	33.8459	1.80144	0.900719	+	0.434403i	$0.143040\pi$
$354$	0	0
$355$	−40.6810	−2.15912
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.1147	0.744946	0.372473	−	0.928043i	$-0.378510\pi$
$359$	14.1147	0.744946	0.372473	+	0.928043i	$0.378510\pi$
$360$	0	0
$361$	−17.4455	−0.918184
$362$	0	0
$363$	−10.6298	−0.557922
$364$	0	0
$365$	26.1533	1.36892
$366$	0	0
$367$	−26.7917	−1.39852	−0.699258	−	0.714869i	$-0.746486\pi$
$367$	−26.7917	−1.39852	−0.699258	+	0.714869i	$0.746486\pi$
$368$	0	0
$369$	12.5318	0.652380
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.9931	0.569201	0.284600	−	0.958646i	$-0.408139\pi$
$373$	10.9931	0.569201	0.284600	+	0.958646i	$0.408139\pi$
$374$	0	0
$375$	56.0163	2.89267
$376$	0	0
$377$	−43.7453	−2.25300
$378$	0	0
$379$	−0.280117	−0.0143886	−0.00719431	−	0.999974i	$-0.502290\pi$
$379$	−0.280117	−0.0143886	−0.00719431	+	0.999974i	$0.502290\pi$
$380$	0	0
$381$	9.82158	0.503175
$382$	0	0
$383$	11.3817	0.581579	0.290789	−	0.956787i	$-0.406082\pi$
$383$	11.3817	0.581579	0.290789	+	0.956787i	$0.406082\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.96992	−0.100137
$388$	0	0
$389$	−0.129032	−0.00654216	−0.00327108	−	0.999995i	$-0.501041\pi$
$389$	−0.129032	−0.00654216	−0.00327108	+	0.999995i	$0.501041\pi$
$390$	0	0
$391$	−4.92849	−0.249245
$392$	0	0
$393$	−2.68163	−0.135270
$394$	0	0
$395$	26.1162	1.31405
$396$	0	0
$397$	−2.48225	−0.124581	−0.0622903	−	0.998058i	$-0.519840\pi$
$397$	−2.48225	−0.124581	−0.0622903	+	0.998058i	$0.519840\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.0843	−0.603459	−0.301730	−	0.953394i	$-0.597564\pi$
$401$	−12.0843	−0.603459	−0.301730	+	0.953394i	$0.597564\pi$
$402$	0	0
$403$	−56.5505	−2.81698
$404$	0	0
$405$	45.2090	2.24645
$406$	0	0
$407$	30.3225	1.50303
$408$	0	0
$409$	4.32161	0.213690	0.106845	−	0.994276i	$-0.465925\pi$
$409$	4.32161	0.213690	0.106845	+	0.994276i	$0.465925\pi$
$410$	0	0
$411$	−5.69617	−0.280971
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−22.6144	−1.11010
$416$	0	0
$417$	−25.1306	−1.23065
$418$	0	0
$419$	20.0044	0.977277	0.488638	−	0.872486i	$-0.337494\pi$
$419$	20.0044	0.977277	0.488638	+	0.872486i	$0.337494\pi$
$420$	0	0
$421$	27.4017	1.33548	0.667739	−	0.744395i	$-0.267262\pi$
$421$	27.4017	1.33548	0.667739	+	0.744395i	$0.267262\pi$
$422$	0	0
$423$	−14.6528	−0.712444
$424$	0	0
$425$	55.9306	2.71303
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−58.6507	−2.83168
$430$	0	0
$431$	−16.4571	−0.792710	−0.396355	−	0.918097i	$-0.629725\pi$
$431$	−16.4571	−0.792710	−0.396355	+	0.918097i	$0.629725\pi$
$432$	0	0
$433$	4.63332	0.222663	0.111332	−	0.993783i	$-0.464488\pi$
$433$	4.63332	0.222663	0.111332	+	0.993783i	$0.464488\pi$
$434$	0	0
$435$	57.2165	2.74332
$436$	0	0
$437$	−1.24680	−0.0596426
$438$	0	0
$439$	−25.6359	−1.22354	−0.611769	−	0.791037i	$-0.709542\pi$
$439$	−25.6359	−1.22354	−0.611769	+	0.791037i	$0.709542\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9.07846	0.431330	0.215665	−	0.976467i	$-0.430808\pi$
$443$	9.07846	0.431330	0.215665	+	0.976467i	$0.430808\pi$
$444$	0	0
$445$	−41.4922	−1.96692
$446$	0	0
$447$	−14.0784	−0.665884
$448$	0	0
$449$	28.6206	1.35069	0.675346	−	0.737501i	$-0.263995\pi$
$449$	28.6206	1.35069	0.675346	+	0.737501i	$0.263995\pi$
$450$	0	0
$451$	−28.3282	−1.33392
$452$	0	0
$453$	−7.03016	−0.330306
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.79560	0.130773	0.0653863	−	0.997860i	$-0.479172\pi$
$457$	2.79560	0.130773	0.0653863	+	0.997860i	$0.479172\pi$
$458$	0	0
$459$	13.3112	0.621313
$460$	0	0
$461$	10.1360	0.472082	0.236041	−	0.971743i	$-0.424150\pi$
$461$	10.1360	0.472082	0.236041	+	0.971743i	$0.424150\pi$
$462$	0	0
$463$	11.6273	0.540368	0.270184	−	0.962809i	$-0.412915\pi$
$463$	11.6273	0.540368	0.270184	+	0.962809i	$0.412915\pi$
$464$	0	0
$465$	73.9649	3.43004
$466$	0	0
$467$	−36.9748	−1.71099	−0.855494	−	0.517813i	$-0.826746\pi$
$467$	−36.9748	−1.71099	−0.855494	+	0.517813i	$0.826746\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−48.5113	−2.23528
$472$	0	0
$473$	4.45301	0.204750
$474$	0	0
$475$	14.1492	0.649210
$476$	0	0
$477$	−8.40921	−0.385032
$478$	0	0
$479$	10.8039	0.493641	0.246821	−	0.969061i	$-0.420614\pi$
$479$	10.8039	0.493641	0.246821	+	0.969061i	$0.420614\pi$
$480$	0	0
$481$	51.3479	2.34126
$482$	0	0
$483$	0	0
$484$	0	0
$485$	45.4781	2.06505
$486$	0	0
$487$	−33.5054	−1.51828	−0.759138	−	0.650930i	$-0.774379\pi$
$487$	−33.5054	−1.51828	−0.759138	+	0.650930i	$0.774379\pi$
$488$	0	0
$489$	−28.6515	−1.29566
$490$	0	0
$491$	31.8118	1.43565	0.717823	−	0.696226i	$-0.245139\pi$
$491$	31.8118	1.43565	0.717823	+	0.696226i	$0.245139\pi$
$492$	0	0
$493$	31.9585	1.43934
$494$	0	0
$495$	28.3881	1.27595
$496$	0	0
$497$	0	0
$498$	0	0
$499$	8.24689	0.369181	0.184591	−	0.982815i	$-0.440904\pi$
$499$	8.24689	0.369181	0.184591	+	0.982815i	$0.440904\pi$
$500$	0	0
$501$	−16.4451	−0.734714
$502$	0	0
$503$	16.6968	0.744474	0.372237	−	0.928138i	$-0.378591\pi$
$503$	16.6968	0.744474	0.372237	+	0.928138i	$0.378591\pi$
$504$	0	0
$505$	−54.8637	−2.44140
$506$	0	0
$507$	−70.9490	−3.15095
$508$	0	0
$509$	12.0106	0.532362	0.266181	−	0.963923i	$-0.414238\pi$
$509$	12.0106	0.532362	0.266181	+	0.963923i	$0.414238\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3.36744	0.148676
$514$	0	0
$515$	50.3442	2.21843
$516$	0	0
$517$	33.1227	1.45673
$518$	0	0
$519$	−15.8146	−0.694183
$520$	0	0
$521$	3.19522	0.139985	0.0699925	−	0.997548i	$-0.477702\pi$
$521$	3.19522	0.139985	0.0699925	+	0.997548i	$0.477702\pi$
$522$	0	0
$523$	19.2569	0.842044	0.421022	−	0.907050i	$-0.361671\pi$
$523$	19.2569	0.842044	0.421022	+	0.907050i	$0.361671\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	41.3134	1.79964
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−2.32125	−0.100734
$532$	0	0
$533$	−47.9708	−2.07785
$534$	0	0
$535$	20.6928	0.894626
$536$	0	0
$537$	19.4248	0.838243
$538$	0	0
$539$	0	0
$540$	0	0
$541$	21.7131	0.933518	0.466759	−	0.884385i	$-0.345422\pi$
$541$	21.7131	0.933518	0.466759	+	0.884385i	$0.345422\pi$
$542$	0	0
$543$	47.7863	2.05071
$544$	0	0
$545$	13.4040	0.574163
$546$	0	0
$547$	−23.9565	−1.02431	−0.512153	−	0.858894i	$-0.671152\pi$
$547$	−23.9565	−1.02431	−0.512153	+	0.858894i	$0.671152\pi$
$548$	0	0
$549$	−1.47848	−0.0630998
$550$	0	0
$551$	8.08480	0.344424
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−67.1602	−2.85079
$556$	0	0
$557$	−17.7727	−0.753055	−0.376527	−	0.926406i	$-0.622882\pi$
$557$	−17.7727	−0.753055	−0.376527	+	0.926406i	$0.622882\pi$
$558$	0	0
$559$	7.54071	0.318938
$560$	0	0
$561$	42.8477	1.80903
$562$	0	0
$563$	−29.4443	−1.24093	−0.620465	−	0.784234i	$-0.713056\pi$
$563$	−29.4443	−1.24093	−0.620465	+	0.784234i	$0.713056\pi$
$564$	0	0
$565$	−53.2589	−2.24062
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.12604	0.131050	0.0655251	−	0.997851i	$-0.479128\pi$
$569$	3.12604	0.131050	0.0655251	+	0.997851i	$0.479128\pi$
$570$	0	0
$571$	15.3393	0.641931	0.320965	−	0.947091i	$-0.395993\pi$
$571$	15.3393	0.641931	0.320965	+	0.947091i	$0.395993\pi$
$572$	0	0
$573$	44.3888	1.85437
$574$	0	0
$575$	−11.3484	−0.473261
$576$	0	0
$577$	−35.9790	−1.49783	−0.748913	−	0.662668i	$-0.769424\pi$
$577$	−35.9790	−1.49783	−0.748913	+	0.662668i	$0.769424\pi$
$578$	0	0
$579$	20.4800	0.851120
$580$	0	0
$581$	0	0
$582$	0	0
$583$	19.0090	0.787274
$584$	0	0
$585$	48.0722	1.98754
$586$	0	0
$587$	−7.42319	−0.306388	−0.153194	−	0.988196i	$-0.548956\pi$
$587$	−7.42319	−0.306388	−0.153194	+	0.988196i	$0.548956\pi$
$588$	0	0
$589$	10.4514	0.430642
$590$	0	0
$591$	45.9467	1.88999
$592$	0	0
$593$	9.96768	0.409323	0.204662	−	0.978833i	$-0.434391\pi$
$593$	9.96768	0.409323	0.204662	+	0.978833i	$0.434391\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.54427	−0.267839
$598$	0	0
$599$	10.6126	0.433617	0.216809	−	0.976214i	$-0.430435\pi$
$599$	10.6126	0.433617	0.216809	+	0.976214i	$0.430435\pi$
$600$	0	0
$601$	31.2158	1.27332	0.636659	−	0.771146i	$-0.280316\pi$
$601$	31.2158	1.27332	0.636659	+	0.771146i	$0.280316\pi$
$602$	0	0
$603$	21.9151	0.892453
$604$	0	0
$605$	−19.6949	−0.800710
$606$	0	0
$607$	17.9603	0.728987	0.364494	−	0.931206i	$-0.381242\pi$
$607$	17.9603	0.728987	0.364494	+	0.931206i	$0.381242\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	56.0897	2.26915
$612$	0	0
$613$	20.1209	0.812677	0.406339	−	0.913723i	$-0.366805\pi$
$613$	20.1209	0.812677	0.406339	+	0.913723i	$0.366805\pi$
$614$	0	0
$615$	62.7432	2.53005
$616$	0	0
$617$	−26.3689	−1.06157	−0.530787	−	0.847506i	$-0.678103\pi$
$617$	−26.3689	−1.06157	−0.530787	+	0.847506i	$0.678103\pi$
$618$	0	0
$619$	35.6959	1.43474	0.717370	−	0.696692i	$-0.245346\pi$
$619$	35.6959	1.43474	0.717370	+	0.696692i	$0.245346\pi$
$620$	0	0
$621$	−2.70086	−0.108382
$622$	0	0
$623$	0	0
$624$	0	0
$625$	47.0443	1.88177
$626$	0	0
$627$	10.8395	0.432889
$628$	0	0
$629$	−37.5126	−1.49573
$630$	0	0
$631$	−8.26178	−0.328896	−0.164448	−	0.986386i	$-0.552584\pi$
$631$	−8.26178	−0.328896	−0.164448	+	0.986386i	$0.552584\pi$
$632$	0	0
$633$	44.2963	1.76062
$634$	0	0
$635$	18.1973	0.722139
$636$	0	0
$637$	0	0
$638$	0	0
$639$	17.7317	0.701456
$640$	0	0
$641$	7.85518	0.310261	0.155130	−	0.987894i	$-0.450420\pi$
$641$	7.85518	0.310261	0.155130	+	0.987894i	$0.450420\pi$
$642$	0	0
$643$	−14.9907	−0.591176	−0.295588	−	0.955315i	$-0.595516\pi$
$643$	−14.9907	−0.591176	−0.295588	+	0.955315i	$0.595516\pi$
$644$	0	0
$645$	−9.86283	−0.388349
$646$	0	0
$647$	−16.1310	−0.634177	−0.317088	−	0.948396i	$-0.602705\pi$
$647$	−16.1310	−0.634177	−0.317088	+	0.948396i	$0.602705\pi$
$648$	0	0
$649$	5.24720	0.205971
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.15080	−0.279833	−0.139916	−	0.990163i	$-0.544683\pi$
$653$	−7.15080	−0.279833	−0.139916	+	0.990163i	$0.544683\pi$
$654$	0	0
$655$	−4.96849	−0.194135
$656$	0	0
$657$	−11.3995	−0.444736
$658$	0	0
$659$	27.7646	1.08156	0.540778	−	0.841165i	$-0.318130\pi$
$659$	27.7646	1.08156	0.540778	+	0.841165i	$0.318130\pi$
$660$	0	0
$661$	0.707812	0.0275307	0.0137653	−	0.999905i	$-0.495618\pi$
$661$	0.707812	0.0275307	0.0137653	+	0.999905i	$0.495618\pi$
$662$	0	0
$663$	72.5580	2.81792
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.48443	−0.251078
$668$	0	0
$669$	44.7533	1.73026
$670$	0	0
$671$	3.34210	0.129020
$672$	0	0
$673$	−8.96378	−0.345528	−0.172764	−	0.984963i	$-0.555270\pi$
$673$	−8.96378	−0.345528	−0.172764	+	0.984963i	$0.555270\pi$
$674$	0	0
$675$	30.6505	1.17974
$676$	0	0
$677$	46.2685	1.77824	0.889122	−	0.457671i	$-0.151316\pi$
$677$	46.2685	1.77824	0.889122	+	0.457671i	$0.151316\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−8.23317	−0.315496
$682$	0	0
$683$	−18.6674	−0.714289	−0.357145	−	0.934049i	$-0.616250\pi$
$683$	−18.6674	−0.714289	−0.357145	+	0.934049i	$0.616250\pi$
$684$	0	0
$685$	−10.5538	−0.403240
$686$	0	0
$687$	6.81347	0.259950
$688$	0	0
$689$	32.1898	1.22633
$690$	0	0
$691$	−47.2398	−1.79709	−0.898543	−	0.438886i	$-0.855373\pi$
$691$	−47.2398	−1.79709	−0.898543	+	0.438886i	$0.855373\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−46.5617	−1.76619
$696$	0	0
$697$	35.0454	1.32744
$698$	0	0
$699$	−52.1393	−1.97209
$700$	0	0
$701$	−32.8765	−1.24173	−0.620864	−	0.783918i	$-0.713218\pi$
$701$	−32.8765	−1.24173	−0.620864	+	0.783918i	$0.713218\pi$
$702$	0	0
$703$	−9.48987	−0.357917
$704$	0	0
$705$	−73.3623	−2.76298
$706$	0	0
$707$	0	0
$708$	0	0
$709$	25.3234	0.951039	0.475519	−	0.879705i	$-0.342260\pi$
$709$	25.3234	0.951039	0.475519	+	0.879705i	$0.342260\pi$
$710$	0	0
$711$	−11.3833	−0.426908
$712$	0	0
$713$	−8.38256	−0.313929
$714$	0	0
$715$	−108.667	−4.06393
$716$	0	0
$717$	−3.36466	−0.125655
$718$	0	0
$719$	3.48142	0.129835	0.0649176	−	0.997891i	$-0.479322\pi$
$719$	3.48142	0.129835	0.0649176	+	0.997891i	$0.479322\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	27.4181	1.01969
$724$	0	0
$725$	73.5880	2.73299
$726$	0	0
$727$	−14.9789	−0.555538	−0.277769	−	0.960648i	$-0.589595\pi$
$727$	−14.9789	−0.555538	−0.277769	+	0.960648i	$0.589595\pi$
$728$	0	0
$729$	−2.02317	−0.0749324
$730$	0	0
$731$	−5.50892	−0.203755
$732$	0	0
$733$	−40.6230	−1.50045	−0.750223	−	0.661185i	$-0.770054\pi$
$733$	−40.6230	−1.50045	−0.750223	+	0.661185i	$0.770054\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−49.5392	−1.82480
$738$	0	0
$739$	15.5537	0.572153	0.286076	−	0.958207i	$-0.407649\pi$
$739$	15.5537	0.572153	0.286076	+	0.958207i	$0.407649\pi$
$740$	0	0
$741$	18.3556	0.674310
$742$	0	0
$743$	1.02324	0.0375390	0.0187695	−	0.999824i	$-0.494025\pi$
$743$	1.02324	0.0375390	0.0187695	+	0.999824i	$0.494025\pi$
$744$	0	0
$745$	−26.0843	−0.955653
$746$	0	0
$747$	9.85699	0.360648
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−20.1696	−0.735997	−0.367999	−	0.929826i	$-0.619957\pi$
$751$	−20.1696	−0.735997	−0.367999	+	0.929826i	$0.619957\pi$
$752$	0	0
$753$	31.5892	1.15117
$754$	0	0
$755$	−13.0254	−0.474044
$756$	0	0
$757$	−10.4310	−0.379122	−0.189561	−	0.981869i	$-0.560706\pi$
$757$	−10.4310	−0.379122	−0.189561	+	0.981869i	$0.560706\pi$
$758$	0	0
$759$	−8.69387	−0.315568
$760$	0	0
$761$	−5.80967	−0.210600	−0.105300	−	0.994440i	$-0.533580\pi$
$761$	−5.80967	−0.210600	−0.105300	+	0.994440i	$0.533580\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−35.1195	−1.26975
$766$	0	0
$767$	8.88557	0.320839
$768$	0	0
$769$	−35.3898	−1.27619	−0.638094	−	0.769959i	$-0.720277\pi$
$769$	−35.3898	−1.27619	−0.638094	+	0.769959i	$0.720277\pi$
$770$	0	0
$771$	27.3059	0.983396
$772$	0	0
$773$	28.5275	1.02606	0.513032	−	0.858369i	$-0.328522\pi$
$773$	28.5275	1.02606	0.513032	+	0.858369i	$0.328522\pi$
$774$	0	0
$775$	95.1287	3.41712
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.86573	0.317648
$780$	0	0
$781$	−40.0826	−1.43427
$782$	0	0
$783$	17.5136	0.625884
$784$	0	0
$785$	−89.8813	−3.20800
$786$	0	0
$787$	25.8412	0.921139	0.460570	−	0.887624i	$-0.347645\pi$
$787$	25.8412	0.921139	0.460570	+	0.887624i	$0.347645\pi$
$788$	0	0
$789$	43.0439	1.53240
$790$	0	0
$791$	0	0
$792$	0	0
$793$	5.65949	0.200974
$794$	0	0
$795$	−42.1025	−1.49322
$796$	0	0
$797$	32.9288	1.16640	0.583199	−	0.812329i	$-0.301801\pi$
$797$	32.9288	1.16640	0.583199	+	0.812329i	$0.301801\pi$
$798$	0	0
$799$	−40.9768	−1.44965
$800$	0	0
$801$	18.0853	0.639013
$802$	0	0
$803$	25.7685	0.909352
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.06773	0.143191
$808$	0	0
$809$	−22.7087	−0.798396	−0.399198	−	0.916865i	$-0.630711\pi$
$809$	−22.7087	−0.798396	−0.399198	+	0.916865i	$0.630711\pi$
$810$	0	0
$811$	−53.2167	−1.86869	−0.934345	−	0.356370i	$-0.884014\pi$
$811$	−53.2167	−1.86869	−0.934345	+	0.356370i	$0.884014\pi$
$812$	0	0
$813$	14.1353	0.495746
$814$	0	0
$815$	−53.0852	−1.85949
$816$	0	0
$817$	−1.39364	−0.0487572
$818$	0	0
$819$	0	0
$820$	0	0
$821$	9.09572	0.317443	0.158721	−	0.987323i	$-0.449263\pi$
$821$	9.09572	0.317443	0.158721	+	0.987323i	$0.449263\pi$
$822$	0	0
$823$	18.7581	0.653867	0.326933	−	0.945047i	$-0.393985\pi$
$823$	18.7581	0.653867	0.326933	+	0.945047i	$0.393985\pi$
$824$	0	0
$825$	98.6616	3.43496
$826$	0	0
$827$	−18.9647	−0.659467	−0.329733	−	0.944074i	$-0.606959\pi$
$827$	−18.9647	−0.659467	−0.329733	+	0.944074i	$0.606959\pi$
$828$	0	0
$829$	−34.1733	−1.18689	−0.593443	−	0.804876i	$-0.702232\pi$
$829$	−34.1733	−1.18689	−0.593443	+	0.804876i	$0.702232\pi$
$830$	0	0
$831$	−56.9745	−1.97642
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−30.4694	−1.05444
$836$	0	0
$837$	22.6401	0.782558
$838$	0	0
$839$	43.0050	1.48470	0.742349	−	0.670014i	$-0.233712\pi$
$839$	43.0050	1.48470	0.742349	+	0.670014i	$0.233712\pi$
$840$	0	0
$841$	13.0479	0.449926
$842$	0	0
$843$	−6.82585	−0.235095
$844$	0	0
$845$	−131.454	−4.52214
$846$	0	0
$847$	0	0
$848$	0	0
$849$	46.7047	1.60290
$850$	0	0
$851$	7.61137	0.260914
$852$	0	0
$853$	40.3790	1.38255	0.691275	−	0.722591i	$-0.257049\pi$
$853$	40.3790	1.38255	0.691275	+	0.722591i	$0.257049\pi$
$854$	0	0
$855$	−8.88448	−0.303843
$856$	0	0
$857$	29.6044	1.01127	0.505633	−	0.862749i	$-0.331259\pi$
$857$	29.6044	1.01127	0.505633	+	0.862749i	$0.331259\pi$
$858$	0	0
$859$	15.6912	0.535378	0.267689	−	0.963505i	$-0.413740\pi$
$859$	15.6912	0.535378	0.267689	+	0.963505i	$0.413740\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18.2405	−0.620912	−0.310456	−	0.950588i	$-0.600482\pi$
$863$	−18.2405	−0.620912	−0.310456	+	0.950588i	$0.600482\pi$
$864$	0	0
$865$	−29.3011	−0.996268
$866$	0	0
$867$	−15.9090	−0.540296
$868$	0	0
$869$	25.7320	0.872899
$870$	0	0
$871$	−83.8894	−2.84248
$872$	0	0
$873$	−19.8226	−0.670895
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−30.2155	−1.02030	−0.510152	−	0.860084i	$-0.670411\pi$
$877$	−30.2155	−1.02030	−0.510152	+	0.860084i	$0.670411\pi$
$878$	0	0
$879$	−45.3354	−1.52912
$880$	0	0
$881$	−0.443129	−0.0149294	−0.00746471	−	0.999972i	$-0.502376\pi$
$881$	−0.443129	−0.0149294	−0.00746471	+	0.999972i	$0.502376\pi$
$882$	0	0
$883$	−33.9979	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$883$	−33.9979	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$884$	0	0
$885$	−11.6218	−0.390664
$886$	0	0
$887$	−55.1449	−1.85158	−0.925792	−	0.378033i	$-0.876601\pi$
$887$	−55.1449	−1.85158	−0.925792	+	0.378033i	$0.876601\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	44.5439	1.49228
$892$	0	0
$893$	−10.3662	−0.346893
$894$	0	0
$895$	35.9901	1.20302
$896$	0	0
$897$	−14.7222	−0.491558
$898$	0	0
$899$	54.3561	1.81288
$900$	0	0
$901$	−23.5165	−0.783448
$902$	0	0
$903$	0	0
$904$	0	0
$905$	88.5379	2.94310
$906$	0	0
$907$	−13.0471	−0.433222	−0.216611	−	0.976258i	$-0.569500\pi$
$907$	−13.0471	−0.433222	−0.216611	+	0.976258i	$0.569500\pi$
$908$	0	0
$909$	23.9135	0.793162
$910$	0	0
$911$	4.86610	0.161221	0.0806105	−	0.996746i	$-0.474313\pi$
$911$	4.86610	0.161221	0.0806105	+	0.996746i	$0.474313\pi$
$912$	0	0
$913$	−22.2817	−0.737418
$914$	0	0
$915$	−7.40230	−0.244712
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−0.0102695	−0.000338761 0	−0.000169381	−	1.00000i	$-0.500054\pi$
$919$	−0.0102695	−0.000338761 0	−0.000169381		1.00000i	$0.500054\pi$
$920$	0	0
$921$	15.1841	0.500332
$922$	0	0
$923$	−67.8756	−2.23415
$924$	0	0
$925$	−86.3770	−2.84006
$926$	0	0
$927$	−21.9436	−0.720723
$928$	0	0
$929$	−13.4689	−0.441901	−0.220950	−	0.975285i	$-0.570916\pi$
$929$	−13.4689	−0.441901	−0.220950	+	0.975285i	$0.570916\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	68.9077	2.25594
$934$	0	0
$935$	79.3878	2.59626
$936$	0	0
$937$	−23.3740	−0.763595	−0.381797	−	0.924246i	$-0.624695\pi$
$937$	−23.3740	−0.763595	−0.381797	+	0.924246i	$0.624695\pi$
$938$	0	0
$939$	−15.2870	−0.498873
$940$	0	0
$941$	−34.1588	−1.11354	−0.556772	−	0.830665i	$-0.687960\pi$
$941$	−34.1588	−1.11354	−0.556772	+	0.830665i	$0.687960\pi$
$942$	0	0
$943$	−7.11078	−0.231559
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−25.1775	−0.818160	−0.409080	−	0.912499i	$-0.634150\pi$
$947$	−25.1775	−0.818160	−0.409080	+	0.912499i	$0.634150\pi$
$948$	0	0
$949$	43.6363	1.41649
$950$	0	0
$951$	−46.4278	−1.50553
$952$	0	0
$953$	−26.3000	−0.851939	−0.425969	−	0.904738i	$-0.640067\pi$
$953$	−26.3000	−0.851939	−0.425969	+	0.904738i	$0.640067\pi$
$954$	0	0
$955$	82.2431	2.66133
$956$	0	0
$957$	56.3748	1.82234
$958$	0	0
$959$	0	0
$960$	0	0
$961$	39.2672	1.26668
$962$	0	0
$963$	−9.01940	−0.290646
$964$	0	0
$965$	37.9452	1.22150
$966$	0	0
$967$	40.9316	1.31627	0.658135	−	0.752900i	$-0.271346\pi$
$967$	40.9316	1.31627	0.658135	+	0.752900i	$0.271346\pi$
$968$	0	0
$969$	−13.4098	−0.430785
$970$	0	0
$971$	−5.65729	−0.181551	−0.0907755	−	0.995871i	$-0.528935\pi$
$971$	−5.65729	−0.181551	−0.0907755	+	0.995871i	$0.528935\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	167.073	5.35062
$976$	0	0
$977$	42.5639	1.36174	0.680870	−	0.732405i	$-0.261602\pi$
$977$	42.5639	1.36174	0.680870	+	0.732405i	$0.261602\pi$
$978$	0	0
$979$	−40.8819	−1.30659
$980$	0	0
$981$	−5.84242	−0.186534
$982$	0	0
$983$	−31.7904	−1.01395	−0.506977	−	0.861959i	$-0.669237\pi$
$983$	−31.7904	−1.01395	−0.506977	+	0.861959i	$0.669237\pi$
$984$	0	0
$985$	85.1296	2.71245
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.11777	0.0355430
$990$	0	0
$991$	−56.0238	−1.77966	−0.889828	−	0.456297i	$-0.849176\pi$
$991$	−56.0238	−1.77966	−0.889828	+	0.456297i	$0.849176\pi$
$992$	0	0
$993$	−50.1399	−1.59114
$994$	0	0
$995$	−12.1252	−0.384393
$996$	0	0
$997$	−50.3201	−1.59365	−0.796827	−	0.604208i	$-0.793490\pi$
$997$	−50.3201	−1.59365	−0.796827	+	0.604208i	$0.793490\pi$
$998$	0	0
$999$	−20.5573	−0.650403

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4508.2.a.f.1.2		5
7.6	odd	2		644.2.a.d.1.4	✓	5
21.20	even	2		5796.2.a.t.1.1		5
28.27	even	2		2576.2.a.bb.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
644.2.a.d.1.4	✓	5	7.6	odd	2
2576.2.a.bb.1.2		5	28.27	even	2
4508.2.a.f.1.2		5	1.1	even	1	trivial
5796.2.a.t.1.1		5	21.20	even	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:	\( N \)	\(=\)	\( 4508 = 2^{2} \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4508.a (trivial)

\(f(q)\)	\(=\)	\(q-2.18229 q^{3} -4.04332 q^{5} +1.76237 q^{9} +O(q^{10})\)	\(q-2.18229 q^{3} -4.04332 q^{5} +1.76237 q^{9} -3.98384 q^{11} -6.74621 q^{13} +8.82367 q^{15} +4.92849 q^{17} +1.24680 q^{19} -1.00000 q^{23} +11.3484 q^{25} +2.70086 q^{27} +6.48443 q^{29} +8.38256 q^{31} +8.69387 q^{33} -7.61137 q^{37} +14.7222 q^{39} +7.11078 q^{41} -1.11777 q^{43} -7.12582 q^{45} -8.31426 q^{47} -10.7554 q^{51} -4.77154 q^{53} +16.1079 q^{55} -2.72088 q^{57} -1.31712 q^{59} -0.838914 q^{61} +27.2771 q^{65} +12.4350 q^{67} +2.18229 q^{69} +10.0613 q^{71} -6.46827 q^{73} -24.7655 q^{75} -6.45910 q^{79} -11.1812 q^{81} +5.59303 q^{83} -19.9275 q^{85} -14.1509 q^{87} +10.2619 q^{89} -18.2931 q^{93} -5.04121 q^{95} -11.2477 q^{97} -7.02099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} - 2 q^{5} + 10 q^{9}+O(q^{10}) \) 5 * q - 3 * q^3 - 2 * q^5 + 10 * q^9	\( 5 q - 3 q^{3} - 2 q^{5} + 10 q^{9} + 2 q^{11} - 13 q^{13} + 4 q^{15} - 4 q^{17} - 12 q^{19} - 5 q^{23} + 19 q^{25} - 15 q^{27} + 13 q^{29} + 3 q^{31} - 24 q^{33} - 4 q^{37} + 3 q^{39} - q^{41} - 8 q^{43} + 16 q^{45} - 5 q^{47} - 16 q^{51} - 8 q^{53} + 2 q^{55} + 12 q^{57} - 12 q^{59} - 20 q^{61} - 12 q^{65} - 12 q^{67} + 3 q^{69} + 9 q^{71} + 9 q^{73} - 35 q^{75} - 8 q^{79} - 11 q^{81} + 28 q^{83} + 16 q^{85} + 15 q^{87} - 32 q^{89} - 15 q^{93} - 36 q^{95} - 4 q^{97} + 28 q^{99}+O(q^{100}) \) 5 * q - 3 * q^3 - 2 * q^5 + 10 * q^9 + 2 * q^11 - 13 * q^13 + 4 * q^15 - 4 * q^17 - 12 * q^19 - 5 * q^23 + 19 * q^25 - 15 * q^27 + 13 * q^29 + 3 * q^31 - 24 * q^33 - 4 * q^37 + 3 * q^39 - q^41 - 8 * q^43 + 16 * q^45 - 5 * q^47 - 16 * q^51 - 8 * q^53 + 2 * q^55 + 12 * q^57 - 12 * q^59 - 20 * q^61 - 12 * q^65 - 12 * q^67 + 3 * q^69 + 9 * q^71 + 9 * q^73 - 35 * q^75 - 8 * q^79 - 11 * q^81 + 28 * q^83 + 16 * q^85 + 15 * q^87 - 32 * q^89 - 15 * q^93 - 36 * q^95 - 4 * q^97 + 28 * q^99

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