LMFDB - Embedded newform 1216.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.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:	$9.70980888579$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.961.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 10x + 8 $ x^3 - x^2 - 10*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.29707$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.29707 q^{3} +3.08387 q^{5} +1.78680 q^{7} +7.87067 q^{9} +O(q^{10})$	$q-3.29707 q^{3} +3.08387 q^{5} +1.78680 q^{7} +7.87067 q^{9} +5.08387 q^{11} +1.29707 q^{13} -10.1677 q^{15} +0.213198 q^{17} +1.00000 q^{19} -5.89121 q^{21} -3.72347 q^{23} +4.51027 q^{25} -16.0590 q^{27} -0.870674 q^{29} -16.7619 q^{33} +5.51027 q^{35} +2.00000 q^{37} -4.27653 q^{39} +8.59414 q^{41} +3.67801 q^{43} +24.2722 q^{45} -4.65748 q^{47} -3.80734 q^{49} -0.702929 q^{51} -11.0384 q^{53} +15.6780 q^{55} -3.29707 q^{57} +4.70293 q^{59} -3.51027 q^{61} +14.0633 q^{63} +4.00000 q^{65} -1.12933 q^{67} +12.2765 q^{69} +8.76189 q^{71} +6.80734 q^{73} -14.8707 q^{75} +9.08387 q^{77} +14.5941 q^{79} +29.3355 q^{81} +9.74135 q^{83} +0.657476 q^{85} +2.87067 q^{87} -6.76189 q^{89} +2.31761 q^{91} +3.08387 q^{95} +4.16774 q^{97} +40.0135 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) $ 3 * q - q^3 - q^5 + 4 * q^7 + 12 * q^9	$ 3 q - q^{3} - q^{5} + 4 q^{7} + 12 q^{9} + 5 q^{11} - 5 q^{13} - 10 q^{15} + 2 q^{17} + 3 q^{19} + 9 q^{21} - 5 q^{23} + 6 q^{25} - q^{27} + 9 q^{29} - 12 q^{33} + 9 q^{35} + 6 q^{37} - 19 q^{39} + 8 q^{41} - 17 q^{43} + 27 q^{45} - q^{47} + 5 q^{49} - 11 q^{51} - q^{53} + 19 q^{55} - q^{57} + 23 q^{59} - 3 q^{61} + 47 q^{63} + 12 q^{65} - 15 q^{67} + 43 q^{69} - 12 q^{71} + 4 q^{73} - 33 q^{75} + 17 q^{77} + 26 q^{79} + 47 q^{81} + 6 q^{83} - 11 q^{85} - 3 q^{87} + 18 q^{89} - 17 q^{91} - q^{95} - 8 q^{97} + 51 q^{99}+O(q^{100}) $ 3 * q - q^3 - q^5 + 4 * q^7 + 12 * q^9 + 5 * q^11 - 5 * q^13 - 10 * q^15 + 2 * q^17 + 3 * q^19 + 9 * q^21 - 5 * q^23 + 6 * q^25 - q^27 + 9 * q^29 - 12 * q^33 + 9 * q^35 + 6 * q^37 - 19 * q^39 + 8 * q^41 - 17 * q^43 + 27 * q^45 - q^47 + 5 * q^49 - 11 * q^51 - q^53 + 19 * q^55 - q^57 + 23 * q^59 - 3 * q^61 + 47 * q^63 + 12 * q^65 - 15 * q^67 + 43 * q^69 - 12 * q^71 + 4 * q^73 - 33 * q^75 + 17 * q^77 + 26 * q^79 + 47 * q^81 + 6 * q^83 - 11 * q^85 - 3 * q^87 + 18 * q^89 - 17 * q^91 - q^95 - 8 * q^97 + 51 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.29707	−1.90356	−0.951782	−	0.306774i	$-0.900750\pi$
$3$	−3.29707	−1.90356	−0.951782	+	0.306774i	$0.900750\pi$
$4$	0	0
$5$	3.08387	1.37915	0.689575	−	0.724214i	$-0.257797\pi$
$5$	3.08387	1.37915	0.689575	+	0.724214i	$0.257797\pi$
$6$	0	0
$7$	1.78680	0.675348	0.337674	−	0.941263i	$-0.390360\pi$
$7$	1.78680	0.675348	0.337674	+	0.941263i	$0.390360\pi$
$8$	0	0
$9$	7.87067	2.62356
$10$	0	0
$11$	5.08387	1.53285	0.766423	−	0.642337i	$-0.222035\pi$
$11$	5.08387	1.53285	0.766423	+	0.642337i	$0.222035\pi$
$12$	0	0
$13$	1.29707	0.359743	0.179871	−	0.983690i	$-0.442432\pi$
$13$	1.29707	0.359743	0.179871	+	0.983690i	$0.442432\pi$
$14$	0	0
$15$	−10.1677	−2.62530
$16$	0	0
$17$	0.213198	0.0517082	0.0258541	−	0.999666i	$-0.491769\pi$
$17$	0.213198	0.0517082	0.0258541	+	0.999666i	$0.491769\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−5.89121	−1.28557
$22$	0	0
$23$	−3.72347	−0.776397	−0.388198	−	0.921576i	$-0.626902\pi$
$23$	−3.72347	−0.776397	−0.388198	+	0.921576i	$0.626902\pi$
$24$	0	0
$25$	4.51027	0.902054
$26$	0	0
$27$	−16.0590	−3.09055
$28$	0	0
$29$	−0.870674	−0.161680	−0.0808401	−	0.996727i	$-0.525760\pi$
$29$	−0.870674	−0.161680	−0.0808401	+	0.996727i	$0.525760\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−16.7619	−2.91787
$34$	0	0
$35$	5.51027	0.931405
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	−4.27653	−0.684793
$40$	0	0
$41$	8.59414	1.34218	0.671090	−	0.741376i	$-0.265827\pi$
$41$	8.59414	1.34218	0.671090	+	0.741376i	$0.265827\pi$
$42$	0	0
$43$	3.67801	0.560892	0.280446	−	0.959870i	$-0.409518\pi$
$43$	3.67801	0.560892	0.280446	+	0.959870i	$0.409518\pi$
$44$	0	0
$45$	24.2722	3.61828
$46$	0	0
$47$	−4.65748	−0.679363	−0.339681	−	0.940541i	$-0.610319\pi$
$47$	−4.65748	−0.679363	−0.339681	+	0.940541i	$0.610319\pi$
$48$	0	0
$49$	−3.80734	−0.543906
$50$	0	0
$51$	−0.702929	−0.0984298
$52$	0	0
$53$	−11.0384	−1.51624	−0.758122	−	0.652113i	$-0.773883\pi$
$53$	−11.0384	−1.51624	−0.758122	+	0.652113i	$0.773883\pi$
$54$	0	0
$55$	15.6780	2.11402
$56$	0	0
$57$	−3.29707	−0.436708
$58$	0	0
$59$	4.70293	0.612269	0.306135	−	0.951988i	$-0.400964\pi$
$59$	4.70293	0.612269	0.306135	+	0.951988i	$0.400964\pi$
$60$	0	0
$61$	−3.51027	−0.449444	−0.224722	−	0.974423i	$-0.572147\pi$
$61$	−3.51027	−0.449444	−0.224722	+	0.974423i	$0.572147\pi$
$62$	0	0
$63$	14.0633	1.77181
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	−1.12933	−0.137969	−0.0689846	−	0.997618i	$-0.521976\pi$
$67$	−1.12933	−0.137969	−0.0689846	+	0.997618i	$0.521976\pi$
$68$	0	0
$69$	12.2765	1.47792
$70$	0	0
$71$	8.76189	1.03984	0.519922	−	0.854214i	$-0.325961\pi$
$71$	8.76189	1.03984	0.519922	+	0.854214i	$0.325961\pi$
$72$	0	0
$73$	6.80734	0.796739	0.398369	−	0.917225i	$-0.369576\pi$
$73$	6.80734	0.796739	0.398369	+	0.917225i	$0.369576\pi$
$74$	0	0
$75$	−14.8707	−1.71712
$76$	0	0
$77$	9.08387	1.03520
$78$	0	0
$79$	14.5941	1.64197	0.820985	−	0.570950i	$-0.193425\pi$
$79$	14.5941	1.64197	0.820985	+	0.570950i	$0.193425\pi$
$80$	0	0
$81$	29.3355	3.25950
$82$	0	0
$83$	9.74135	1.06925	0.534626	−	0.845089i	$-0.320453\pi$
$83$	9.74135	1.06925	0.534626	+	0.845089i	$0.320453\pi$
$84$	0	0
$85$	0.657476	0.0713133
$86$	0	0
$87$	2.87067	0.307769
$88$	0	0
$89$	−6.76189	−0.716758	−0.358379	−	0.933576i	$-0.616671\pi$
$89$	−6.76189	−0.716758	−0.358379	+	0.933576i	$0.616671\pi$
$90$	0	0
$91$	2.31761	0.242951
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.08387	0.316399
$96$	0	0
$97$	4.16774	0.423170	0.211585	−	0.977360i	$-0.432137\pi$
$97$	4.16774	0.423170	0.211585	+	0.977360i	$0.432137\pi$
$98$	0	0
$99$	40.0135	4.02151
$100$	0	0
$101$	−19.1883	−1.90931	−0.954653	−	0.297722i	$-0.903773\pi$
$101$	−19.1883	−1.90931	−0.954653	+	0.297722i	$0.903773\pi$
$102$	0	0
$103$	−15.9091	−1.56757	−0.783785	−	0.621033i	$-0.786713\pi$
$103$	−15.9091	−1.56757	−0.783785	+	0.621033i	$0.786713\pi$
$104$	0	0
$105$	−18.1677	−1.77299
$106$	0	0
$107$	3.29707	0.318740	0.159370	−	0.987219i	$-0.449054\pi$
$107$	3.29707	0.318740	0.159370	+	0.987219i	$0.449054\pi$
$108$	0	0
$109$	−4.44428	−0.425685	−0.212842	−	0.977087i	$-0.568272\pi$
$109$	−4.44428	−0.425685	−0.212842	+	0.977087i	$0.568272\pi$
$110$	0	0
$111$	−6.59414	−0.625888
$112$	0	0
$113$	−11.1883	−1.05251	−0.526253	−	0.850328i	$-0.676403\pi$
$113$	−11.1883	−1.05251	−0.526253	+	0.850328i	$0.676403\pi$
$114$	0	0
$115$	−11.4827	−1.07077
$116$	0	0
$117$	10.2088	0.943806
$118$	0	0
$119$	0.380943	0.0349210
$120$	0	0
$121$	14.8458	1.34961
$122$	0	0
$123$	−28.3355	−2.55493
$124$	0	0
$125$	−1.51027	−0.135083
$126$	0	0
$127$	16.7619	1.48738	0.743688	−	0.668526i	$-0.233075\pi$
$127$	16.7619	1.48738	0.743688	+	0.668526i	$0.233075\pi$
$128$	0	0
$129$	−12.1267	−1.06769
$130$	0	0
$131$	−3.67801	−0.321350	−0.160675	−	0.987007i	$-0.551367\pi$
$131$	−3.67801	−0.321350	−0.160675	+	0.987007i	$0.551367\pi$
$132$	0	0
$133$	1.78680	0.154935
$134$	0	0
$135$	−49.5238	−4.26233
$136$	0	0
$137$	21.4015	1.82845	0.914226	−	0.405205i	$-0.132800\pi$
$137$	21.4015	1.82845	0.914226	+	0.405205i	$0.132800\pi$
$138$	0	0
$139$	10.8252	0.918183	0.459092	−	0.888389i	$-0.348175\pi$
$139$	10.8252	0.918183	0.459092	+	0.888389i	$0.348175\pi$
$140$	0	0
$141$	15.3560	1.29321
$142$	0	0
$143$	6.59414	0.551430
$144$	0	0
$145$	−2.68505	−0.222981
$146$	0	0
$147$	12.5531	1.03536
$148$	0	0
$149$	21.2516	1.74100	0.870500	−	0.492168i	$-0.163795\pi$
$149$	21.2516	1.74100	0.870500	+	0.492168i	$0.163795\pi$
$150$	0	0
$151$	−15.9091	−1.29466	−0.647332	−	0.762208i	$-0.724115\pi$
$151$	−15.9091	−1.29466	−0.647332	+	0.762208i	$0.724115\pi$
$152$	0	0
$153$	1.67801	0.135659
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	36.3944	2.88627
$160$	0	0
$161$	−6.65310	−0.524338
$162$	0	0
$163$	−9.18828	−0.719682	−0.359841	−	0.933014i	$-0.617169\pi$
$163$	−9.18828	−0.719682	−0.359841	+	0.933014i	$0.617169\pi$
$164$	0	0
$165$	−51.6915	−4.02418
$166$	0	0
$167$	−5.83226	−0.451313	−0.225657	−	0.974207i	$-0.572453\pi$
$167$	−5.83226	−0.451313	−0.225657	+	0.974207i	$0.572453\pi$
$168$	0	0
$169$	−11.3176	−0.870585
$170$	0	0
$171$	7.87067	0.601885
$172$	0	0
$173$	7.18828	0.546515	0.273257	−	0.961941i	$-0.411899\pi$
$173$	7.18828	0.546515	0.273257	+	0.961941i	$0.411899\pi$
$174$	0	0
$175$	8.05896	0.609200
$176$	0	0
$177$	−15.5059	−1.16549
$178$	0	0
$179$	7.87333	0.588480	0.294240	−	0.955732i	$-0.404933\pi$
$179$	7.87333	0.588480	0.294240	+	0.955732i	$0.404933\pi$
$180$	0	0
$181$	22.3355	1.66018	0.830092	−	0.557627i	$-0.188288\pi$
$181$	22.3355	1.66018	0.830092	+	0.557627i	$0.188288\pi$
$182$	0	0
$183$	11.5736	0.855545
$184$	0	0
$185$	6.16774	0.453462
$186$	0	0
$187$	1.08387	0.0792606
$188$	0	0
$189$	−28.6942	−2.08719
$190$	0	0
$191$	−23.7370	−1.71755	−0.858773	−	0.512356i	$-0.828773\pi$
$191$	−23.7370	−1.71755	−0.858773	+	0.512356i	$0.828773\pi$
$192$	0	0
$193$	−7.31495	−0.526542	−0.263271	−	0.964722i	$-0.584801\pi$
$193$	−7.31495	−0.526542	−0.263271	+	0.964722i	$0.584801\pi$
$194$	0	0
$195$	−13.1883	−0.944433
$196$	0	0
$197$	8.59414	0.612307	0.306154	−	0.951982i	$-0.400958\pi$
$197$	8.59414	0.612307	0.306154	+	0.951982i	$0.400958\pi$
$198$	0	0
$199$	−7.06599	−0.500895	−0.250447	−	0.968130i	$-0.580578\pi$
$199$	−7.06599	−0.500895	−0.250447	+	0.968130i	$0.580578\pi$
$200$	0	0
$201$	3.72347	0.262633
$202$	0	0
$203$	−1.55572	−0.109190
$204$	0	0
$205$	26.5032	1.85107
$206$	0	0
$207$	−29.3062	−2.03692
$208$	0	0
$209$	5.08387	0.351659
$210$	0	0
$211$	5.55572	0.382472	0.191236	−	0.981544i	$-0.438750\pi$
$211$	5.55572	0.382472	0.191236	+	0.981544i	$0.438750\pi$
$212$	0	0
$213$	−28.8886	−1.97941
$214$	0	0
$215$	11.3425	0.773554
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−22.4443	−1.51664
$220$	0	0
$221$	0.276533	0.0186016
$222$	0	0
$223$	−16.7619	−1.12246	−0.561229	−	0.827660i	$-0.689671\pi$
$223$	−16.7619	−1.12246	−0.561229	+	0.827660i	$0.689671\pi$
$224$	0	0
$225$	35.4989	2.36659
$226$	0	0
$227$	−8.27653	−0.549333	−0.274666	−	0.961540i	$-0.588567\pi$
$227$	−8.27653	−0.549333	−0.274666	+	0.961540i	$0.588567\pi$
$228$	0	0
$229$	12.6986	0.839144	0.419572	−	0.907722i	$-0.362180\pi$
$229$	12.6986	0.839144	0.419572	+	0.907722i	$0.362180\pi$
$230$	0	0
$231$	−29.9502	−1.97058
$232$	0	0
$233$	8.36306	0.547882	0.273941	−	0.961746i	$-0.411673\pi$
$233$	8.36306	0.547882	0.273941	+	0.961746i	$0.411673\pi$
$234$	0	0
$235$	−14.3631	−0.936943
$236$	0	0
$237$	−48.1179	−3.12559
$238$	0	0
$239$	6.97508	0.451181	0.225590	−	0.974222i	$-0.427569\pi$
$239$	6.97508	0.451181	0.225590	+	0.974222i	$0.427569\pi$
$240$	0	0
$241$	−19.7413	−1.27165	−0.635826	−	0.771832i	$-0.719340\pi$
$241$	−19.7413	−1.27165	−0.635826	+	0.771832i	$0.719340\pi$
$242$	0	0
$243$	−48.5443	−3.11412
$244$	0	0
$245$	−11.7413	−0.750127
$246$	0	0
$247$	1.29707	0.0825306
$248$	0	0
$249$	−32.1179	−2.03539
$250$	0	0
$251$	−10.2722	−0.648373	−0.324186	−	0.945993i	$-0.605090\pi$
$251$	−10.2722	−0.648373	−0.324186	+	0.945993i	$0.605090\pi$
$252$	0	0
$253$	−18.9296	−1.19010
$254$	0	0
$255$	−2.16774	−0.135749
$256$	0	0
$257$	9.90909	0.618112	0.309056	−	0.951044i	$-0.399987\pi$
$257$	9.90909	0.618112	0.309056	+	0.951044i	$0.399987\pi$
$258$	0	0
$259$	3.57360	0.222053
$260$	0	0
$261$	−6.85279	−0.424177
$262$	0	0
$263$	23.6780	1.46005	0.730024	−	0.683421i	$-0.239509\pi$
$263$	23.6780	1.46005	0.730024	+	0.683421i	$0.239509\pi$
$264$	0	0
$265$	−34.0411	−2.09113
$266$	0	0
$267$	22.2944	1.36440
$268$	0	0
$269$	5.02054	0.306108	0.153054	−	0.988218i	$-0.451089\pi$
$269$	5.02054	0.306108	0.153054	+	0.988218i	$0.451089\pi$
$270$	0	0
$271$	4.27653	0.259781	0.129890	−	0.991528i	$-0.458537\pi$
$271$	4.27653	0.259781	0.129890	+	0.991528i	$0.458537\pi$
$272$	0	0
$273$	−7.64132	−0.462474
$274$	0	0
$275$	22.9296	1.38271
$276$	0	0
$277$	−7.93667	−0.476868	−0.238434	−	0.971159i	$-0.576634\pi$
$277$	−7.93667	−0.476868	−0.238434	+	0.971159i	$0.576634\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	22.8886	1.36542	0.682708	−	0.730691i	$-0.260802\pi$
$281$	22.8886	1.36542	0.682708	+	0.730691i	$0.260802\pi$
$282$	0	0
$283$	5.08387	0.302205	0.151102	−	0.988518i	$-0.451718\pi$
$283$	5.08387	0.302205	0.151102	+	0.988518i	$0.451718\pi$
$284$	0	0
$285$	−10.1677	−0.602285
$286$	0	0
$287$	15.3560	0.906438
$288$	0	0
$289$	−16.9545	−0.997326
$290$	0	0
$291$	−13.7413	−0.805532
$292$	0	0
$293$	3.55572	0.207728	0.103864	−	0.994592i	$-0.466879\pi$
$293$	3.55572	0.207728	0.103864	+	0.994592i	$0.466879\pi$
$294$	0	0
$295$	14.5032	0.844411
$296$	0	0
$297$	−81.6417	−4.73733
$298$	0	0
$299$	−4.82960	−0.279303
$300$	0	0
$301$	6.57188	0.378797
$302$	0	0
$303$	63.2651	3.63449
$304$	0	0
$305$	−10.8252	−0.619850
$306$	0	0
$307$	−16.3355	−0.932316	−0.466158	−	0.884702i	$-0.654362\pi$
$307$	−16.3355	−0.932316	−0.466158	+	0.884702i	$0.654362\pi$
$308$	0	0
$309$	52.4534	2.98397
$310$	0	0
$311$	−16.3809	−0.928878	−0.464439	−	0.885605i	$-0.653744\pi$
$311$	−16.3809	−0.928878	−0.464439	+	0.885605i	$0.653744\pi$
$312$	0	0
$313$	−30.0590	−1.69903	−0.849516	−	0.527562i	$-0.823106\pi$
$313$	−30.0590	−1.69903	−0.849516	+	0.527562i	$0.823106\pi$
$314$	0	0
$315$	43.3695	2.44360
$316$	0	0
$317$	12.0179	0.674991	0.337496	−	0.941327i	$-0.390420\pi$
$317$	12.0179	0.674991	0.337496	+	0.941327i	$0.390420\pi$
$318$	0	0
$319$	−4.42640	−0.247831
$320$	0	0
$321$	−10.8707	−0.606742
$322$	0	0
$323$	0.213198	0.0118627
$324$	0	0
$325$	5.85014	0.324507
$326$	0	0
$327$	14.6531	0.810318
$328$	0	0
$329$	−8.32199	−0.458806
$330$	0	0
$331$	14.3176	0.786967	0.393483	−	0.919332i	$-0.371270\pi$
$331$	14.3176	0.786967	0.393483	+	0.919332i	$0.371270\pi$
$332$	0	0
$333$	15.7413	0.862621
$334$	0	0
$335$	−3.48270	−0.190280
$336$	0	0
$337$	11.6147	0.632692	0.316346	−	0.948644i	$-0.397544\pi$
$337$	11.6147	0.632692	0.316346	+	0.948644i	$0.397544\pi$
$338$	0	0
$339$	36.8886	2.00351
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−19.3106	−1.04267
$344$	0	0
$345$	37.8593	2.03827
$346$	0	0
$347$	2.91613	0.156546	0.0782730	−	0.996932i	$-0.475059\pi$
$347$	2.91613	0.156546	0.0782730	+	0.996932i	$0.475059\pi$
$348$	0	0
$349$	−12.8252	−0.686518	−0.343259	−	0.939241i	$-0.611531\pi$
$349$	−12.8252	−0.686518	−0.343259	+	0.939241i	$0.611531\pi$
$350$	0	0
$351$	−20.8296	−1.11180
$352$	0	0
$353$	25.7592	1.37103	0.685513	−	0.728061i	$-0.259578\pi$
$353$	25.7592	1.37103	0.685513	+	0.728061i	$0.259578\pi$
$354$	0	0
$355$	27.0205	1.43410
$356$	0	0
$357$	−1.25600	−0.0664743
$358$	0	0
$359$	22.4220	1.18339	0.591694	−	0.806162i	$-0.298459\pi$
$359$	22.4220	1.18339	0.591694	+	0.806162i	$0.298459\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−48.9475	−2.56908
$364$	0	0
$365$	20.9930	1.09882
$366$	0	0
$367$	5.18828	0.270826	0.135413	−	0.990789i	$-0.456764\pi$
$367$	5.18828	0.270826	0.135413	+	0.990789i	$0.456764\pi$
$368$	0	0
$369$	67.6417	3.52129
$370$	0	0
$371$	−19.7235	−1.02399
$372$	0	0
$373$	−36.0091	−1.86448	−0.932241	−	0.361838i	$-0.882149\pi$
$373$	−36.0091	−1.86448	−0.932241	+	0.361838i	$0.882149\pi$
$374$	0	0
$375$	4.97946	0.257138
$376$	0	0
$377$	−1.12933	−0.0581632
$378$	0	0
$379$	25.0384	1.28614	0.643069	−	0.765809i	$-0.277661\pi$
$379$	25.0384	1.28614	0.643069	+	0.765809i	$0.277661\pi$
$380$	0	0
$381$	−55.2651	−2.83132
$382$	0	0
$383$	−21.7413	−1.11093	−0.555466	−	0.831540i	$-0.687460\pi$
$383$	−21.7413	−1.11093	−0.555466	+	0.831540i	$0.687460\pi$
$384$	0	0
$385$	28.0135	1.42770
$386$	0	0
$387$	28.9484	1.47153
$388$	0	0
$389$	−11.5103	−0.583594	−0.291797	−	0.956480i	$-0.594253\pi$
$389$	−11.5103	−0.583594	−0.291797	+	0.956480i	$0.594253\pi$
$390$	0	0
$391$	−0.793836	−0.0401460
$392$	0	0
$393$	12.1267	0.611710
$394$	0	0
$395$	45.0065	2.26452
$396$	0	0
$397$	7.21054	0.361887	0.180943	−	0.983494i	$-0.442085\pi$
$397$	7.21054	0.361887	0.180943	+	0.983494i	$0.442085\pi$
$398$	0	0
$399$	−5.89121	−0.294929
$400$	0	0
$401$	1.23811	0.0618285	0.0309142	−	0.999522i	$-0.490158\pi$
$401$	1.23811	0.0618285	0.0309142	+	0.999522i	$0.490158\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	90.4669	4.49534
$406$	0	0
$407$	10.1677	0.503996
$408$	0	0
$409$	32.8030	1.62200	0.811001	−	0.585045i	$-0.198923\pi$
$409$	32.8030	1.62200	0.811001	+	0.585045i	$0.198923\pi$
$410$	0	0
$411$	−70.5622	−3.48058
$412$	0	0
$413$	8.40320	0.413495
$414$	0	0
$415$	30.0411	1.47466
$416$	0	0
$417$	−35.6915	−1.74782
$418$	0	0
$419$	17.1883	0.839703	0.419851	−	0.907593i	$-0.362082\pi$
$419$	17.1883	0.839703	0.419851	+	0.907593i	$0.362082\pi$
$420$	0	0
$421$	−19.8912	−0.969438	−0.484719	−	0.874670i	$-0.661078\pi$
$421$	−19.8912	−0.969438	−0.484719	+	0.874670i	$0.661078\pi$
$422$	0	0
$423$	−36.6575	−1.78235
$424$	0	0
$425$	0.961581	0.0466435
$426$	0	0
$427$	−6.27215	−0.303531
$428$	0	0
$429$	−21.7413	−1.04968
$430$	0	0
$431$	8.55307	0.411987	0.205993	−	0.978553i	$-0.433957\pi$
$431$	8.55307	0.411987	0.205993	+	0.978553i	$0.433957\pi$
$432$	0	0
$433$	−31.5238	−1.51494	−0.757468	−	0.652872i	$-0.773564\pi$
$433$	−31.5238	−1.51494	−0.757468	+	0.652872i	$0.773564\pi$
$434$	0	0
$435$	8.85279	0.424459
$436$	0	0
$437$	−3.72347	−0.178118
$438$	0	0
$439$	−4.12667	−0.196955	−0.0984776	−	0.995139i	$-0.531397\pi$
$439$	−4.12667	−0.196955	−0.0984776	+	0.995139i	$0.531397\pi$
$440$	0	0
$441$	−29.9663	−1.42697
$442$	0	0
$443$	−37.7549	−1.79379	−0.896894	−	0.442246i	$-0.854182\pi$
$443$	−37.7549	−1.79379	−0.896894	+	0.442246i	$0.854182\pi$
$444$	0	0
$445$	−20.8528	−0.988517
$446$	0	0
$447$	−70.0681	−3.31411
$448$	0	0
$449$	33.3560	1.57417	0.787084	−	0.616846i	$-0.211590\pi$
$449$	33.3560	1.57417	0.787084	+	0.616846i	$0.211590\pi$
$450$	0	0
$451$	43.6915	2.05735
$452$	0	0
$453$	52.4534	2.46448
$454$	0	0
$455$	7.14721	0.335066
$456$	0	0
$457$	32.1223	1.50262	0.751309	−	0.659951i	$-0.229423\pi$
$457$	32.1223	1.50262	0.751309	+	0.659951i	$0.229423\pi$
$458$	0	0
$459$	−3.42374	−0.159807
$460$	0	0
$461$	7.81000	0.363748	0.181874	−	0.983322i	$-0.441784\pi$
$461$	7.81000	0.363748	0.181874	+	0.983322i	$0.441784\pi$
$462$	0	0
$463$	18.4897	0.859291	0.429645	−	0.902998i	$-0.358639\pi$
$463$	18.4897	0.859291	0.429645	+	0.902998i	$0.358639\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−21.0839	−0.975645	−0.487823	−	0.872943i	$-0.662209\pi$
$467$	−21.0839	−0.975645	−0.487823	+	0.872943i	$0.662209\pi$
$468$	0	0
$469$	−2.01788	−0.0931771
$470$	0	0
$471$	46.1590	2.12689
$472$	0	0
$473$	18.6986	0.859760
$474$	0	0
$475$	4.51027	0.206945
$476$	0	0
$477$	−86.8798	−3.97795
$478$	0	0
$479$	−25.5238	−1.16621	−0.583105	−	0.812396i	$-0.698163\pi$
$479$	−25.5238	−1.16621	−0.583105	+	0.812396i	$0.698163\pi$
$480$	0	0
$481$	2.59414	0.118283
$482$	0	0
$483$	21.9357	0.998110
$484$	0	0
$485$	12.8528	0.583615
$486$	0	0
$487$	−10.4675	−0.474326	−0.237163	−	0.971470i	$-0.576218\pi$
$487$	−10.4675	−0.474326	−0.237163	+	0.971470i	$0.576218\pi$
$488$	0	0
$489$	30.2944	1.36996
$490$	0	0
$491$	−12.6710	−0.571833	−0.285917	−	0.958254i	$-0.592298\pi$
$491$	−12.6710	−0.571833	−0.285917	+	0.958254i	$0.592298\pi$
$492$	0	0
$493$	−0.185626	−0.00836018
$494$	0	0
$495$	123.397	5.54626
$496$	0	0
$497$	15.6558	0.702257
$498$	0	0
$499$	13.8458	0.619821	0.309911	−	0.950766i	$-0.399701\pi$
$499$	13.8458	0.619821	0.309911	+	0.950766i	$0.399701\pi$
$500$	0	0
$501$	19.2294	0.859104
$502$	0	0
$503$	−9.76454	−0.435379	−0.217690	−	0.976018i	$-0.569852\pi$
$503$	−9.76454	−0.435379	−0.217690	+	0.976018i	$0.569852\pi$
$504$	0	0
$505$	−59.1742	−2.63322
$506$	0	0
$507$	37.3150	1.65722
$508$	0	0
$509$	28.1677	1.24851	0.624257	−	0.781219i	$-0.285402\pi$
$509$	28.1677	1.24851	0.624257	+	0.781219i	$0.285402\pi$
$510$	0	0
$511$	12.1634	0.538076
$512$	0	0
$513$	−16.0590	−0.709020
$514$	0	0
$515$	−49.0616	−2.16191
$516$	0	0
$517$	−23.6780	−1.04136
$518$	0	0
$519$	−23.7003	−1.04033
$520$	0	0
$521$	−28.2035	−1.23562	−0.617809	−	0.786328i	$-0.711980\pi$
$521$	−28.2035	−1.23562	−0.617809	+	0.786328i	$0.711980\pi$
$522$	0	0
$523$	−1.22023	−0.0533571	−0.0266785	−	0.999644i	$-0.508493\pi$
$523$	−1.22023	−0.0533571	−0.0266785	+	0.999644i	$0.508493\pi$
$524$	0	0
$525$	−26.5709	−1.15965
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−9.13579	−0.397208
$530$	0	0
$531$	37.0152	1.60632
$532$	0	0
$533$	11.1472	0.482839
$534$	0	0
$535$	10.1677	0.439590
$536$	0	0
$537$	−25.9589	−1.12021
$538$	0	0
$539$	−19.3560	−0.833723
$540$	0	0
$541$	−12.1812	−0.523713	−0.261856	−	0.965107i	$-0.584335\pi$
$541$	−12.1812	−0.523713	−0.261856	+	0.965107i	$0.584335\pi$
$542$	0	0
$543$	−73.6417	−3.16027
$544$	0	0
$545$	−13.7056	−0.587083
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	0	0
$549$	−27.6282	−1.17914
$550$	0	0
$551$	−0.870674	−0.0370920
$552$	0	0
$553$	26.0768	1.10890
$554$	0	0
$555$	−20.3355	−0.863193
$556$	0	0
$557$	−8.60764	−0.364718	−0.182359	−	0.983232i	$-0.558373\pi$
$557$	−8.60764	−0.364718	−0.182359	+	0.983232i	$0.558373\pi$
$558$	0	0
$559$	4.77064	0.201777
$560$	0	0
$561$	−3.57360	−0.150878
$562$	0	0
$563$	19.3560	0.815759	0.407880	−	0.913036i	$-0.366268\pi$
$563$	19.3560	0.815759	0.407880	+	0.913036i	$0.366268\pi$
$564$	0	0
$565$	−34.5032	−1.45156
$566$	0	0
$567$	52.4167	2.20129
$568$	0	0
$569$	9.23811	0.387282	0.193641	−	0.981072i	$-0.437970\pi$
$569$	9.23811	0.387282	0.193641	+	0.981072i	$0.437970\pi$
$570$	0	0
$571$	−14.3766	−0.601640	−0.300820	−	0.953681i	$-0.597260\pi$
$571$	−14.3766	−0.601640	−0.300820	+	0.953681i	$0.597260\pi$
$572$	0	0
$573$	78.2625	3.26946
$574$	0	0
$575$	−16.7938	−0.700351
$576$	0	0
$577$	5.19266	0.216173	0.108087	−	0.994141i	$-0.465528\pi$
$577$	5.19266	0.216173	0.108087	+	0.994141i	$0.465528\pi$
$578$	0	0
$579$	24.1179	1.00231
$580$	0	0
$581$	17.4059	0.722117
$582$	0	0
$583$	−56.1179	−2.32417
$584$	0	0
$585$	31.4827	1.30165
$586$	0	0
$587$	−30.3079	−1.25094	−0.625471	−	0.780248i	$-0.715093\pi$
$587$	−30.3079	−1.25094	−0.625471	+	0.780248i	$0.715093\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−28.3355	−1.16557
$592$	0	0
$593$	−4.04107	−0.165947	−0.0829735	−	0.996552i	$-0.526442\pi$
$593$	−4.04107	−0.165947	−0.0829735	+	0.996552i	$0.526442\pi$
$594$	0	0
$595$	1.17478	0.0481613
$596$	0	0
$597$	23.2971	0.953486
$598$	0	0
$599$	−6.29441	−0.257183	−0.128591	−	0.991698i	$-0.541046\pi$
$599$	−6.29441	−0.257183	−0.128591	+	0.991698i	$0.541046\pi$
$600$	0	0
$601$	−10.0358	−0.409367	−0.204684	−	0.978828i	$-0.565617\pi$
$601$	−10.0358	−0.409367	−0.204684	+	0.978828i	$0.565617\pi$
$602$	0	0
$603$	−8.88856	−0.361970
$604$	0	0
$605$	45.7824	1.86132
$606$	0	0
$607$	−45.3062	−1.83892	−0.919461	−	0.393182i	$-0.871374\pi$
$607$	−45.3062	−1.83892	−0.919461	+	0.393182i	$0.871374\pi$
$608$	0	0
$609$	5.12933	0.207851
$610$	0	0
$611$	−6.04107	−0.244396
$612$	0	0
$613$	−25.1607	−1.01623	−0.508116	−	0.861289i	$-0.669658\pi$
$613$	−25.1607	−1.01623	−0.508116	+	0.861289i	$0.669658\pi$
$614$	0	0
$615$	−87.3830	−3.52362
$616$	0	0
$617$	6.01350	0.242095	0.121047	−	0.992647i	$-0.461375\pi$
$617$	6.01350	0.242095	0.121047	+	0.992647i	$0.461375\pi$
$618$	0	0
$619$	−37.2240	−1.49616	−0.748080	−	0.663608i	$-0.769024\pi$
$619$	−37.2240	−1.49616	−0.748080	+	0.663608i	$0.769024\pi$
$620$	0	0
$621$	59.7950	2.39949
$622$	0	0
$623$	−12.0821	−0.484061
$624$	0	0
$625$	−27.2088	−1.08835
$626$	0	0
$627$	−16.7619	−0.669405
$628$	0	0
$629$	0.426396	0.0170015
$630$	0	0
$631$	−12.0135	−0.478250	−0.239125	−	0.970989i	$-0.576861\pi$
$631$	−12.0135	−0.478250	−0.239125	+	0.970989i	$0.576861\pi$
$632$	0	0
$633$	−18.3176	−0.728060
$634$	0	0
$635$	51.6915	2.05132
$636$	0	0
$637$	−4.93839	−0.195666
$638$	0	0
$639$	68.9619	2.72809
$640$	0	0
$641$	6.97946	0.275672	0.137836	−	0.990455i	$-0.455985\pi$
$641$	6.97946	0.275672	0.137836	+	0.990455i	$0.455985\pi$
$642$	0	0
$643$	−40.7754	−1.60802	−0.804012	−	0.594613i	$-0.797305\pi$
$643$	−40.7754	−1.60802	−0.804012	+	0.594613i	$0.797305\pi$
$644$	0	0
$645$	−37.3971	−1.47251
$646$	0	0
$647$	24.9340	0.980257	0.490129	−	0.871650i	$-0.336950\pi$
$647$	24.9340	0.980257	0.490129	+	0.871650i	$0.336950\pi$
$648$	0	0
$649$	23.9091	0.938514
$650$	0	0
$651$	0	0
$652$	0	0
$653$	36.3988	1.42440	0.712198	−	0.701979i	$-0.247700\pi$
$653$	36.3988	1.42440	0.712198	+	0.701979i	$0.247700\pi$
$654$	0	0
$655$	−11.3425	−0.443189
$656$	0	0
$657$	53.5783	2.09029
$658$	0	0
$659$	−11.8501	−0.461616	−0.230808	−	0.972999i	$-0.574137\pi$
$659$	−11.8501	−0.461616	−0.230808	+	0.972999i	$0.574137\pi$
$660$	0	0
$661$	−34.4854	−1.34132	−0.670662	−	0.741763i	$-0.733990\pi$
$661$	−34.4854	−1.34132	−0.670662	+	0.741763i	$0.733990\pi$
$662$	0	0
$663$	−0.911749	−0.0354094
$664$	0	0
$665$	5.51027	0.213679
$666$	0	0
$667$	3.24193	0.125528
$668$	0	0
$669$	55.2651	2.13667
$670$	0	0
$671$	−17.8458	−0.688928
$672$	0	0
$673$	17.7003	0.682295	0.341148	−	0.940010i	$-0.389184\pi$
$673$	17.7003	0.682295	0.341148	+	0.940010i	$0.389184\pi$
$674$	0	0
$675$	−72.4302	−2.78784
$676$	0	0
$677$	26.0590	1.00153	0.500764	−	0.865584i	$-0.333053\pi$
$677$	26.0590	1.00153	0.500764	+	0.865584i	$0.333053\pi$
$678$	0	0
$679$	7.44693	0.285787
$680$	0	0
$681$	27.2883	1.04569
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	65.9994	2.52171
$686$	0	0
$687$	−41.8680	−1.59736
$688$	0	0
$689$	−14.3176	−0.545457
$690$	0	0
$691$	−13.9367	−0.530176	−0.265088	−	0.964224i	$-0.585401\pi$
$691$	−13.9367	−0.530176	−0.265088	+	0.964224i	$0.585401\pi$
$692$	0	0
$693$	71.4962	2.71592
$694$	0	0
$695$	33.3836	1.26631
$696$	0	0
$697$	1.83226	0.0694016
$698$	0	0
$699$	−27.5736	−1.04293
$700$	0	0
$701$	−6.55307	−0.247506	−0.123753	−	0.992313i	$-0.539493\pi$
$701$	−6.55307	−0.247506	−0.123753	+	0.992313i	$0.539493\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	47.3560	1.78353
$706$	0	0
$707$	−34.2857	−1.28944
$708$	0	0
$709$	−23.5238	−0.883454	−0.441727	−	0.897150i	$-0.645634\pi$
$709$	−23.5238	−0.883454	−0.441727	+	0.897150i	$0.645634\pi$
$710$	0	0
$711$	114.866	4.30780
$712$	0	0
$713$	0	0
$714$	0	0
$715$	20.3355	0.760504
$716$	0	0
$717$	−22.9973	−0.858852
$718$	0	0
$719$	−14.8842	−0.555086	−0.277543	−	0.960713i	$-0.589520\pi$
$719$	−14.8842	−0.555086	−0.277543	+	0.960713i	$0.589520\pi$
$720$	0	0
$721$	−28.4264	−1.05865
$722$	0	0
$723$	65.0886	2.42067
$724$	0	0
$725$	−3.92697	−0.145844
$726$	0	0
$727$	1.32464	0.0491283	0.0245641	−	0.999698i	$-0.492180\pi$
$727$	1.32464	0.0491283	0.0245641	+	0.999698i	$0.492180\pi$
$728$	0	0
$729$	72.0475	2.66843
$730$	0	0
$731$	0.784146	0.0290027
$732$	0	0
$733$	13.7824	0.509065	0.254533	−	0.967064i	$-0.418078\pi$
$733$	13.7824	0.509065	0.254533	+	0.967064i	$0.418078\pi$
$734$	0	0
$735$	38.7121	1.42792
$736$	0	0
$737$	−5.74135	−0.211485
$738$	0	0
$739$	−23.5513	−0.866350	−0.433175	−	0.901310i	$-0.642607\pi$
$739$	−23.5513	−0.866350	−0.433175	+	0.901310i	$0.642607\pi$
$740$	0	0
$741$	−4.27653	−0.157102
$742$	0	0
$743$	−14.0411	−0.515117	−0.257559	−	0.966263i	$-0.582918\pi$
$743$	−14.0411	−0.515117	−0.257559	+	0.966263i	$0.582918\pi$
$744$	0	0
$745$	65.5373	2.40110
$746$	0	0
$747$	76.6710	2.80524
$748$	0	0
$749$	5.89121	0.215260
$750$	0	0
$751$	−17.7056	−0.646086	−0.323043	−	0.946384i	$-0.604706\pi$
$751$	−17.7056	−0.646086	−0.323043	+	0.946384i	$0.604706\pi$
$752$	0	0
$753$	33.8680	1.23422
$754$	0	0
$755$	−49.0616	−1.78553
$756$	0	0
$757$	−30.3220	−1.10207	−0.551036	−	0.834482i	$-0.685767\pi$
$757$	−30.3220	−1.10207	−0.551036	+	0.834482i	$0.685767\pi$
$758$	0	0
$759$	62.4123	2.26542
$760$	0	0
$761$	12.5487	0.454890	0.227445	−	0.973791i	$-0.426963\pi$
$761$	12.5487	0.454890	0.227445	+	0.973791i	$0.426963\pi$
$762$	0	0
$763$	−7.94104	−0.287485
$764$	0	0
$765$	5.17478	0.187095
$766$	0	0
$767$	6.10003	0.220259
$768$	0	0
$769$	−50.0725	−1.80566	−0.902830	−	0.429999i	$-0.858514\pi$
$769$	−50.0725	−1.80566	−0.902830	+	0.429999i	$0.858514\pi$
$770$	0	0
$771$	−32.6710	−1.17662
$772$	0	0
$773$	−11.8003	−0.424427	−0.212214	−	0.977223i	$-0.568067\pi$
$773$	−11.8003	−0.424427	−0.212214	+	0.977223i	$0.568067\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−11.7824	−0.422692
$778$	0	0
$779$	8.59414	0.307917
$780$	0	0
$781$	44.5443	1.59392
$782$	0	0
$783$	13.9821	0.499680
$784$	0	0
$785$	−43.1742	−1.54095
$786$	0	0
$787$	−32.9475	−1.17445	−0.587226	−	0.809423i	$-0.699780\pi$
$787$	−32.9475	−1.17445	−0.587226	+	0.809423i	$0.699780\pi$
$788$	0	0
$789$	−78.0681	−2.77930
$790$	0	0
$791$	−19.9912	−0.710807
$792$	0	0
$793$	−4.55307	−0.161684
$794$	0	0
$795$	112.236	3.98059
$796$	0	0
$797$	18.8208	0.666668	0.333334	−	0.942809i	$-0.391826\pi$
$797$	18.8208	0.666668	0.333334	+	0.942809i	$0.391826\pi$
$798$	0	0
$799$	−0.992965	−0.0351286
$800$	0	0
$801$	−53.2206	−1.88046
$802$	0	0
$803$	34.6076	1.22128
$804$	0	0
$805$	−20.5173	−0.723140
$806$	0	0
$807$	−16.5531	−0.582696
$808$	0	0
$809$	−39.2607	−1.38033	−0.690167	−	0.723650i	$-0.742463\pi$
$809$	−39.2607	−1.38033	−0.690167	+	0.723650i	$0.742463\pi$
$810$	0	0
$811$	−33.9182	−1.19103	−0.595515	−	0.803344i	$-0.703052\pi$
$811$	−33.9182	−1.19103	−0.595515	+	0.803344i	$0.703052\pi$
$812$	0	0
$813$	−14.1000	−0.494510
$814$	0	0
$815$	−28.3355	−0.992549
$816$	0	0
$817$	3.67801	0.128677
$818$	0	0
$819$	18.2411	0.637397
$820$	0	0
$821$	−38.6487	−1.34885	−0.674425	−	0.738344i	$-0.735608\pi$
$821$	−38.6487	−1.34885	−0.674425	+	0.738344i	$0.735608\pi$
$822$	0	0
$823$	−8.47185	−0.295310	−0.147655	−	0.989039i	$-0.547173\pi$
$823$	−8.47185	−0.295310	−0.147655	+	0.989039i	$0.547173\pi$
$824$	0	0
$825$	−75.6006	−2.63208
$826$	0	0
$827$	44.7299	1.55541	0.777706	−	0.628628i	$-0.216383\pi$
$827$	44.7299	1.55541	0.777706	+	0.628628i	$0.216383\pi$
$828$	0	0
$829$	38.8566	1.34955	0.674773	−	0.738025i	$-0.264242\pi$
$829$	38.8566	1.34955	0.674773	+	0.738025i	$0.264242\pi$
$830$	0	0
$831$	26.1677	0.907749
$832$	0	0
$833$	−0.811718	−0.0281244
$834$	0	0
$835$	−17.9859	−0.622429
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.53784	−0.260235	−0.130118	−	0.991499i	$-0.541535\pi$
$839$	−7.53784	−0.260235	−0.130118	+	0.991499i	$0.541535\pi$
$840$	0	0
$841$	−28.2419	−0.973860
$842$	0	0
$843$	−75.4652	−2.59916
$844$	0	0
$845$	−34.9021	−1.20067
$846$	0	0
$847$	26.5264	0.911459
$848$	0	0
$849$	−16.7619	−0.575266
$850$	0	0
$851$	−7.44693	−0.255278
$852$	0	0
$853$	43.5238	1.49023	0.745113	−	0.666938i	$-0.232396\pi$
$853$	43.5238	1.49023	0.745113	+	0.666938i	$0.232396\pi$
$854$	0	0
$855$	24.2722	0.830090
$856$	0	0
$857$	−14.5531	−0.497123	−0.248562	−	0.968616i	$-0.579958\pi$
$857$	−14.5531	−0.497123	−0.248562	+	0.968616i	$0.579958\pi$
$858$	0	0
$859$	24.3132	0.829557	0.414778	−	0.909922i	$-0.363859\pi$
$859$	24.3132	0.829557	0.414778	+	0.909922i	$0.363859\pi$
$860$	0	0
$861$	−50.6299	−1.72546
$862$	0	0
$863$	−12.4264	−0.422999	−0.211500	−	0.977378i	$-0.567835\pi$
$863$	−12.4264	−0.422999	−0.211500	+	0.977378i	$0.567835\pi$
$864$	0	0
$865$	22.1677	0.753726
$866$	0	0
$867$	55.9003	1.89847
$868$	0	0
$869$	74.1948	2.51688
$870$	0	0
$871$	−1.46482	−0.0496334
$872$	0	0
$873$	32.8030	1.11021
$874$	0	0
$875$	−2.69855	−0.0912277
$876$	0	0
$877$	−54.6120	−1.84412	−0.922058	−	0.387051i	$-0.873494\pi$
$877$	−54.6120	−1.84412	−0.922058	+	0.387051i	$0.873494\pi$
$878$	0	0
$879$	−11.7235	−0.395423
$880$	0	0
$881$	35.4551	1.19451	0.597257	−	0.802050i	$-0.296257\pi$
$881$	35.4551	1.19451	0.597257	+	0.802050i	$0.296257\pi$
$882$	0	0
$883$	−21.5460	−0.725082	−0.362541	−	0.931968i	$-0.618091\pi$
$883$	−21.5460	−0.725082	−0.362541	+	0.931968i	$0.618091\pi$
$884$	0	0
$885$	−47.8182	−1.60739
$886$	0	0
$887$	49.6147	1.66590	0.832949	−	0.553350i	$-0.186651\pi$
$887$	49.6147	1.66590	0.832949	+	0.553350i	$0.186651\pi$
$888$	0	0
$889$	29.9502	1.00450
$890$	0	0
$891$	149.138	4.99631
$892$	0	0
$893$	−4.65748	−0.155856
$894$	0	0
$895$	24.2803	0.811602
$896$	0	0
$897$	15.9235	0.531671
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−2.35337	−0.0784022
$902$	0	0
$903$	−21.6680	−0.721064
$904$	0	0
$905$	68.8798	2.28964
$906$	0	0
$907$	11.0883	0.368179	0.184090	−	0.982909i	$-0.441066\pi$
$907$	11.0883	0.368179	0.184090	+	0.982909i	$0.441066\pi$
$908$	0	0
$909$	−151.025	−5.00917
$910$	0	0
$911$	12.5443	0.415612	0.207806	−	0.978170i	$-0.433368\pi$
$911$	12.5443	0.415612	0.207806	+	0.978170i	$0.433368\pi$
$912$	0	0
$913$	49.5238	1.63900
$914$	0	0
$915$	35.6915	1.17993
$916$	0	0
$917$	−6.57188	−0.217023
$918$	0	0
$919$	42.8707	1.41417	0.707087	−	0.707127i	$-0.250009\pi$
$919$	42.8707	1.41417	0.707087	+	0.707127i	$0.250009\pi$
$920$	0	0
$921$	53.8593	1.77472
$922$	0	0
$923$	11.3648	0.374076
$924$	0	0
$925$	9.02054	0.296593
$926$	0	0
$927$	−125.215	−4.11261
$928$	0	0
$929$	−27.9182	−0.915967	−0.457984	−	0.888961i	$-0.651428\pi$
$929$	−27.9182	−0.915967	−0.457984	+	0.888961i	$0.651428\pi$
$930$	0	0
$931$	−3.80734	−0.124781
$932$	0	0
$933$	54.0091	1.76818
$934$	0	0
$935$	3.34252	0.109312
$936$	0	0
$937$	−10.8982	−0.356030	−0.178015	−	0.984028i	$-0.556968\pi$
$937$	−10.8982	−0.356030	−0.178015	+	0.984028i	$0.556968\pi$
$938$	0	0
$939$	99.1065	3.23422
$940$	0	0
$941$	30.9475	1.00886	0.504430	−	0.863453i	$-0.331703\pi$
$941$	30.9475	1.00886	0.504430	+	0.863453i	$0.331703\pi$
$942$	0	0
$943$	−32.0000	−1.04206
$944$	0	0
$945$	−88.4892	−2.87855
$946$	0	0
$947$	18.2944	0.594489	0.297244	−	0.954801i	$-0.403932\pi$
$947$	18.2944	0.594489	0.297244	+	0.954801i	$0.403932\pi$
$948$	0	0
$949$	8.82960	0.286621
$950$	0	0
$951$	−39.6238	−1.28489
$952$	0	0
$953$	−23.0616	−0.747039	−0.373519	−	0.927622i	$-0.621849\pi$
$953$	−23.0616	−0.747039	−0.373519	+	0.927622i	$0.621849\pi$
$954$	0	0
$955$	−73.2018	−2.36875
$956$	0	0
$957$	14.5941	0.471762
$958$	0	0
$959$	38.2402	1.23484
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	25.9502	0.836232
$964$	0	0
$965$	−22.5584	−0.726180
$966$	0	0
$967$	26.3766	0.848213	0.424107	−	0.905612i	$-0.360588\pi$
$967$	26.3766	0.848213	0.424107	+	0.905612i	$0.360588\pi$
$968$	0	0
$969$	−0.702929	−0.0225813
$970$	0	0
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	19.3425	0.620093
$974$	0	0
$975$	−19.2883	−0.617720
$976$	0	0
$977$	−26.4264	−0.845455	−0.422728	−	0.906257i	$-0.638927\pi$
$977$	−26.4264	−0.845455	−0.422728	+	0.906257i	$0.638927\pi$
$978$	0	0
$979$	−34.3766	−1.09868
$980$	0	0
$981$	−34.9795	−1.11681
$982$	0	0
$983$	25.4059	0.810321	0.405161	−	0.914246i	$-0.367216\pi$
$983$	25.4059	0.810321	0.405161	+	0.914246i	$0.367216\pi$
$984$	0	0
$985$	26.5032	0.844463
$986$	0	0
$987$	27.4382	0.873367
$988$	0	0
$989$	−13.6950	−0.435474
$990$	0	0
$991$	29.6504	0.941877	0.470939	−	0.882166i	$-0.343915\pi$
$991$	29.6504	0.941877	0.470939	+	0.882166i	$0.343915\pi$
$992$	0	0
$993$	−47.2062	−1.49804
$994$	0	0
$995$	−21.7906	−0.690809
$996$	0	0
$997$	38.7754	1.22803	0.614014	−	0.789295i	$-0.289554\pi$
$997$	38.7754	1.22803	0.614014	+	0.789295i	$0.289554\pi$
$998$	0	0
$999$	−32.1179	−1.01617

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.a.u.1.1		3
4.3	odd	2		1216.2.a.v.1.3		3
8.3	odd	2		304.2.a.g.1.1		3
8.5	even	2		152.2.a.c.1.3	✓	3
24.5	odd	2		1368.2.a.n.1.3		3
24.11	even	2		2736.2.a.bd.1.3		3
40.13	odd	4		3800.2.d.j.3649.6		6
40.19	odd	2		7600.2.a.bv.1.3		3
40.29	even	2		3800.2.a.r.1.1		3
40.37	odd	4		3800.2.d.j.3649.1		6
56.13	odd	2		7448.2.a.bf.1.1		3
152.37	odd	2		2888.2.a.o.1.1		3
152.75	even	2		5776.2.a.bp.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.a.c.1.3	✓	3	8.5	even	2
304.2.a.g.1.1		3	8.3	odd	2
1216.2.a.u.1.1		3	1.1	even	1	trivial
1216.2.a.v.1.3		3	4.3	odd	2
1368.2.a.n.1.3		3	24.5	odd	2
2736.2.a.bd.1.3		3	24.11	even	2
2888.2.a.o.1.1		3	152.37	odd	2
3800.2.a.r.1.1		3	40.29	even	2
3800.2.d.j.3649.1		6	40.37	odd	4
3800.2.d.j.3649.6		6	40.13	odd	4
5776.2.a.bp.1.3		3	152.75	even	2
7448.2.a.bf.1.1		3	56.13	odd	2
7600.2.a.bv.1.3		3	40.19	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q-3.29707 q^{3} +3.08387 q^{5} +1.78680 q^{7} +7.87067 q^{9} +O(q^{10})\)	\(q-3.29707 q^{3} +3.08387 q^{5} +1.78680 q^{7} +7.87067 q^{9} +5.08387 q^{11} +1.29707 q^{13} -10.1677 q^{15} +0.213198 q^{17} +1.00000 q^{19} -5.89121 q^{21} -3.72347 q^{23} +4.51027 q^{25} -16.0590 q^{27} -0.870674 q^{29} -16.7619 q^{33} +5.51027 q^{35} +2.00000 q^{37} -4.27653 q^{39} +8.59414 q^{41} +3.67801 q^{43} +24.2722 q^{45} -4.65748 q^{47} -3.80734 q^{49} -0.702929 q^{51} -11.0384 q^{53} +15.6780 q^{55} -3.29707 q^{57} +4.70293 q^{59} -3.51027 q^{61} +14.0633 q^{63} +4.00000 q^{65} -1.12933 q^{67} +12.2765 q^{69} +8.76189 q^{71} +6.80734 q^{73} -14.8707 q^{75} +9.08387 q^{77} +14.5941 q^{79} +29.3355 q^{81} +9.74135 q^{83} +0.657476 q^{85} +2.87067 q^{87} -6.76189 q^{89} +2.31761 q^{91} +3.08387 q^{95} +4.16774 q^{97} +40.0135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 3 * q - q^3 - q^5 + 4 * q^7 + 12 * q^9	\( 3 q - q^{3} - q^{5} + 4 q^{7} + 12 q^{9} + 5 q^{11} - 5 q^{13} - 10 q^{15} + 2 q^{17} + 3 q^{19} + 9 q^{21} - 5 q^{23} + 6 q^{25} - q^{27} + 9 q^{29} - 12 q^{33} + 9 q^{35} + 6 q^{37} - 19 q^{39} + 8 q^{41} - 17 q^{43} + 27 q^{45} - q^{47} + 5 q^{49} - 11 q^{51} - q^{53} + 19 q^{55} - q^{57} + 23 q^{59} - 3 q^{61} + 47 q^{63} + 12 q^{65} - 15 q^{67} + 43 q^{69} - 12 q^{71} + 4 q^{73} - 33 q^{75} + 17 q^{77} + 26 q^{79} + 47 q^{81} + 6 q^{83} - 11 q^{85} - 3 q^{87} + 18 q^{89} - 17 q^{91} - q^{95} - 8 q^{97} + 51 q^{99}+O(q^{100}) \) 3 * q - q^3 - q^5 + 4 * q^7 + 12 * q^9 + 5 * q^11 - 5 * q^13 - 10 * q^15 + 2 * q^17 + 3 * q^19 + 9 * q^21 - 5 * q^23 + 6 * q^25 - q^27 + 9 * q^29 - 12 * q^33 + 9 * q^35 + 6 * q^37 - 19 * q^39 + 8 * q^41 - 17 * q^43 + 27 * q^45 - q^47 + 5 * q^49 - 11 * q^51 - q^53 + 19 * q^55 - q^57 + 23 * q^59 - 3 * q^61 + 47 * q^63 + 12 * q^65 - 15 * q^67 + 43 * q^69 - 12 * q^71 + 4 * q^73 - 33 * q^75 + 17 * q^77 + 26 * q^79 + 47 * q^81 + 6 * q^83 - 11 * q^85 - 3 * q^87 + 18 * q^89 - 17 * q^91 - q^95 - 8 * q^97 + 51 * q^99

Introduction

L-functions

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