LMFDB - Embedded newform 4012.2.a.j.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.90340 q^{3} +1.03137 q^{5} +0.216287 q^{7} +0.622916 q^{9} +O(q^{10})$	$q+1.90340 q^{3} +1.03137 q^{5} +0.216287 q^{7} +0.622916 q^{9} +3.51567 q^{11} +3.04742 q^{13} +1.96310 q^{15} +1.00000 q^{17} +4.47014 q^{19} +0.411679 q^{21} -5.89924 q^{23} -3.93628 q^{25} -4.52453 q^{27} +0.947662 q^{29} +6.79764 q^{31} +6.69172 q^{33} +0.223071 q^{35} +0.950560 q^{37} +5.80045 q^{39} +11.2734 q^{41} +0.955438 q^{43} +0.642454 q^{45} +5.04222 q^{47} -6.95322 q^{49} +1.90340 q^{51} +3.63947 q^{53} +3.62594 q^{55} +8.50844 q^{57} +1.00000 q^{59} -2.00757 q^{61} +0.134729 q^{63} +3.14301 q^{65} +2.97464 q^{67} -11.2286 q^{69} -8.63937 q^{71} +5.98475 q^{73} -7.49231 q^{75} +0.760393 q^{77} -0.883128 q^{79} -10.4807 q^{81} -17.0632 q^{83} +1.03137 q^{85} +1.80378 q^{87} +5.52883 q^{89} +0.659117 q^{91} +12.9386 q^{93} +4.61035 q^{95} -11.1821 q^{97} +2.18997 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * 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
$3$	1.90340	1.09893	0.549463	−	0.835518i	$-0.314832\pi$
$3$	1.90340	1.09893	0.549463	+	0.835518i	$0.314832\pi$
$4$	0	0
$5$	1.03137	0.461241	0.230620	−	0.973044i	$-0.425924\pi$
$5$	1.03137	0.461241	0.230620	+	0.973044i	$0.425924\pi$
$6$	0	0
$7$	0.216287	0.0817487	0.0408744	−	0.999164i	$-0.486986\pi$
$7$	0.216287	0.0817487	0.0408744	+	0.999164i	$0.486986\pi$
$8$	0	0
$9$	0.622916	0.207639
$10$	0	0
$11$	3.51567	1.06002	0.530008	−	0.847993i	$-0.322189\pi$
$11$	3.51567	1.06002	0.530008	+	0.847993i	$0.322189\pi$
$12$	0	0
$13$	3.04742	0.845202	0.422601	−	0.906316i	$-0.361117\pi$
$13$	3.04742	0.845202	0.422601	+	0.906316i	$0.361117\pi$
$14$	0	0
$15$	1.96310	0.506870
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	4.47014	1.02552	0.512760	−	0.858532i	$-0.328623\pi$
$19$	4.47014	1.02552	0.512760	+	0.858532i	$0.328623\pi$
$20$	0	0
$21$	0.411679	0.0898358
$22$	0	0
$23$	−5.89924	−1.23008	−0.615038	−	0.788497i	$-0.710859\pi$
$23$	−5.89924	−1.23008	−0.615038	+	0.788497i	$0.710859\pi$
$24$	0	0
$25$	−3.93628	−0.787257
$26$	0	0
$27$	−4.52453	−0.870747
$28$	0	0
$29$	0.947662	0.175976	0.0879882	−	0.996122i	$-0.471956\pi$
$29$	0.947662	0.175976	0.0879882	+	0.996122i	$0.471956\pi$
$30$	0	0
$31$	6.79764	1.22089	0.610446	−	0.792058i	$-0.290990\pi$
$31$	6.79764	1.22089	0.610446	+	0.792058i	$0.290990\pi$
$32$	0	0
$33$	6.69172	1.16488
$34$	0	0
$35$	0.223071	0.0377058
$36$	0	0
$37$	0.950560	0.156271	0.0781356	−	0.996943i	$-0.475103\pi$
$37$	0.950560	0.156271	0.0781356	+	0.996943i	$0.475103\pi$
$38$	0	0
$39$	5.80045	0.928815
$40$	0	0
$41$	11.2734	1.76061	0.880306	−	0.474407i	$-0.157338\pi$
$41$	11.2734	1.76061	0.880306	+	0.474407i	$0.157338\pi$
$42$	0	0
$43$	0.955438	0.145703	0.0728515	−	0.997343i	$-0.476790\pi$
$43$	0.955438	0.145703	0.0728515	+	0.997343i	$0.476790\pi$
$44$	0	0
$45$	0.642454	0.0957715
$46$	0	0
$47$	5.04222	0.735484	0.367742	−	0.929928i	$-0.380131\pi$
$47$	5.04222	0.735484	0.367742	+	0.929928i	$0.380131\pi$
$48$	0	0
$49$	−6.95322	−0.993317
$50$	0	0
$51$	1.90340	0.266529
$52$	0	0
$53$	3.63947	0.499920	0.249960	−	0.968256i	$-0.419583\pi$
$53$	3.63947	0.499920	0.249960	+	0.968256i	$0.419583\pi$
$54$	0	0
$55$	3.62594	0.488922
$56$	0	0
$57$	8.50844	1.12697
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−2.00757	−0.257043	−0.128522	−	0.991707i	$-0.541023\pi$
$61$	−2.00757	−0.257043	−0.128522	+	0.991707i	$0.541023\pi$
$62$	0	0
$63$	0.134729	0.0169742
$64$	0	0
$65$	3.14301	0.389842
$66$	0	0
$67$	2.97464	0.363411	0.181705	−	0.983353i	$-0.441838\pi$
$67$	2.97464	0.363411	0.181705	+	0.983353i	$0.441838\pi$
$68$	0	0
$69$	−11.2286	−1.35176
$70$	0	0
$71$	−8.63937	−1.02530	−0.512652	−	0.858596i	$-0.671337\pi$
$71$	−8.63937	−1.02530	−0.512652	+	0.858596i	$0.671337\pi$
$72$	0	0
$73$	5.98475	0.700462	0.350231	−	0.936663i	$-0.386103\pi$
$73$	5.98475	0.700462	0.350231	+	0.936663i	$0.386103\pi$
$74$	0	0
$75$	−7.49231	−0.865137
$76$	0	0
$77$	0.760393	0.0866549
$78$	0	0
$79$	−0.883128	−0.0993597	−0.0496798	−	0.998765i	$-0.515820\pi$
$79$	−0.883128	−0.0993597	−0.0496798	+	0.998765i	$0.515820\pi$
$80$	0	0
$81$	−10.4807	−1.16452
$82$	0	0
$83$	−17.0632	−1.87293	−0.936467	−	0.350757i	$-0.885924\pi$
$83$	−17.0632	−1.87293	−0.936467	+	0.350757i	$0.885924\pi$
$84$	0	0
$85$	1.03137	0.111867
$86$	0	0
$87$	1.80378	0.193385
$88$	0	0
$89$	5.52883	0.586055	0.293027	−	0.956104i	$-0.405337\pi$
$89$	5.52883	0.586055	0.293027	+	0.956104i	$0.405337\pi$
$90$	0	0
$91$	0.659117	0.0690942
$92$	0	0
$93$	12.9386	1.34167
$94$	0	0
$95$	4.61035	0.473012
$96$	0	0
$97$	−11.1821	−1.13537	−0.567683	−	0.823247i	$-0.692160\pi$
$97$	−11.1821	−1.13537	−0.567683	+	0.823247i	$0.692160\pi$
$98$	0	0
$99$	2.18997	0.220100
$100$	0	0
$101$	16.2446	1.61640	0.808198	−	0.588911i	$-0.200443\pi$
$101$	16.2446	1.61640	0.808198	+	0.588911i	$0.200443\pi$
$102$	0	0
$103$	0.903193	0.0889943	0.0444971	−	0.999010i	$-0.485831\pi$
$103$	0.903193	0.0889943	0.0444971	+	0.999010i	$0.485831\pi$
$104$	0	0
$105$	0.424592	0.0414359
$106$	0	0
$107$	6.87112	0.664256	0.332128	−	0.943234i	$-0.392233\pi$
$107$	6.87112	0.664256	0.332128	+	0.943234i	$0.392233\pi$
$108$	0	0
$109$	−8.93814	−0.856119	−0.428059	−	0.903751i	$-0.640803\pi$
$109$	−8.93814	−0.856119	−0.428059	+	0.903751i	$0.640803\pi$
$110$	0	0
$111$	1.80929	0.171730
$112$	0	0
$113$	−6.81848	−0.641429	−0.320714	−	0.947176i	$-0.603923\pi$
$113$	−6.81848	−0.641429	−0.320714	+	0.947176i	$0.603923\pi$
$114$	0	0
$115$	−6.08427	−0.567361
$116$	0	0
$117$	1.89829	0.175497
$118$	0	0
$119$	0.216287	0.0198270
$120$	0	0
$121$	1.35995	0.123632
$122$	0	0
$123$	21.4578	1.93478
$124$	0	0
$125$	−9.21658	−0.824356
$126$	0	0
$127$	10.2391	0.908571	0.454285	−	0.890856i	$-0.349895\pi$
$127$	10.2391	0.908571	0.454285	+	0.890856i	$0.349895\pi$
$128$	0	0
$129$	1.81858	0.160117
$130$	0	0
$131$	13.4597	1.17598	0.587991	−	0.808867i	$-0.299919\pi$
$131$	13.4597	1.17598	0.587991	+	0.808867i	$0.299919\pi$
$132$	0	0
$133$	0.966832	0.0838349
$134$	0	0
$135$	−4.66645	−0.401624
$136$	0	0
$137$	22.0325	1.88236	0.941181	−	0.337903i	$-0.109718\pi$
$137$	22.0325	1.88236	0.941181	+	0.337903i	$0.109718\pi$
$138$	0	0
$139$	13.5661	1.15066	0.575331	−	0.817921i	$-0.304873\pi$
$139$	13.5661	1.15066	0.575331	+	0.817921i	$0.304873\pi$
$140$	0	0
$141$	9.59735	0.808242
$142$	0	0
$143$	10.7137	0.895927
$144$	0	0
$145$	0.977386	0.0811675
$146$	0	0
$147$	−13.2347	−1.09158
$148$	0	0
$149$	−18.3530	−1.50353	−0.751767	−	0.659428i	$-0.770798\pi$
$149$	−18.3530	−1.50353	−0.751767	+	0.659428i	$0.770798\pi$
$150$	0	0
$151$	2.77805	0.226074	0.113037	−	0.993591i	$-0.463942\pi$
$151$	2.77805	0.226074	0.113037	+	0.993591i	$0.463942\pi$
$152$	0	0
$153$	0.622916	0.0503598
$154$	0	0
$155$	7.01086	0.563125
$156$	0	0
$157$	11.4695	0.915365	0.457682	−	0.889116i	$-0.348680\pi$
$157$	11.4695	0.915365	0.457682	+	0.889116i	$0.348680\pi$
$158$	0	0
$159$	6.92736	0.549375
$160$	0	0
$161$	−1.27593	−0.100557
$162$	0	0
$163$	3.51227	0.275102	0.137551	−	0.990495i	$-0.456077\pi$
$163$	3.51227	0.275102	0.137551	+	0.990495i	$0.456077\pi$
$164$	0	0
$165$	6.90161	0.537289
$166$	0	0
$167$	−24.2189	−1.87412	−0.937058	−	0.349174i	$-0.886462\pi$
$167$	−24.2189	−1.87412	−0.937058	+	0.349174i	$0.886462\pi$
$168$	0	0
$169$	−3.71323	−0.285633
$170$	0	0
$171$	2.78452	0.212938
$172$	0	0
$173$	23.9358	1.81980	0.909902	−	0.414824i	$-0.136157\pi$
$173$	23.9358	1.81980	0.909902	+	0.414824i	$0.136157\pi$
$174$	0	0
$175$	−0.851366	−0.0643572
$176$	0	0
$177$	1.90340	0.143068
$178$	0	0
$179$	13.1496	0.982845	0.491423	−	0.870921i	$-0.336477\pi$
$179$	13.1496	0.982845	0.491423	+	0.870921i	$0.336477\pi$
$180$	0	0
$181$	−6.81980	−0.506912	−0.253456	−	0.967347i	$-0.581567\pi$
$181$	−6.81980	−0.506912	−0.253456	+	0.967347i	$0.581567\pi$
$182$	0	0
$183$	−3.82120	−0.282471
$184$	0	0
$185$	0.980375	0.0720786
$186$	0	0
$187$	3.51567	0.257091
$188$	0	0
$189$	−0.978596	−0.0711824
$190$	0	0
$191$	−26.9619	−1.95090	−0.975448	−	0.220232i	$-0.929319\pi$
$191$	−26.9619	−1.95090	−0.975448	+	0.220232i	$0.929319\pi$
$192$	0	0
$193$	21.8976	1.57623	0.788113	−	0.615530i	$-0.211058\pi$
$193$	21.8976	1.57623	0.788113	+	0.615530i	$0.211058\pi$
$194$	0	0
$195$	5.98238	0.428407
$196$	0	0
$197$	9.71099	0.691880	0.345940	−	0.938257i	$-0.387560\pi$
$197$	9.71099	0.691880	0.345940	+	0.938257i	$0.387560\pi$
$198$	0	0
$199$	−22.0667	−1.56427	−0.782135	−	0.623109i	$-0.785869\pi$
$199$	−22.0667	−1.56427	−0.782135	+	0.623109i	$0.785869\pi$
$200$	0	0
$201$	5.66192	0.399361
$202$	0	0
$203$	0.204967	0.0143859
$204$	0	0
$205$	11.6270	0.812066
$206$	0	0
$207$	−3.67473	−0.255411
$208$	0	0
$209$	15.7155	1.08707
$210$	0	0
$211$	10.6344	0.732100	0.366050	−	0.930595i	$-0.380710\pi$
$211$	10.6344	0.732100	0.366050	+	0.930595i	$0.380710\pi$
$212$	0	0
$213$	−16.4441	−1.12673
$214$	0	0
$215$	0.985406	0.0672041
$216$	0	0
$217$	1.47024	0.0998064
$218$	0	0
$219$	11.3913	0.769756
$220$	0	0
$221$	3.04742	0.204992
$222$	0	0
$223$	−21.6665	−1.45090	−0.725448	−	0.688277i	$-0.758368\pi$
$223$	−21.6665	−1.45090	−0.725448	+	0.688277i	$0.758368\pi$
$224$	0	0
$225$	−2.45198	−0.163465
$226$	0	0
$227$	4.13190	0.274244	0.137122	−	0.990554i	$-0.456215\pi$
$227$	4.13190	0.274244	0.137122	+	0.990554i	$0.456215\pi$
$228$	0	0
$229$	−21.9592	−1.45111	−0.725554	−	0.688166i	$-0.758416\pi$
$229$	−21.9592	−1.45111	−0.725554	+	0.688166i	$0.758416\pi$
$230$	0	0
$231$	1.44733	0.0952273
$232$	0	0
$233$	−18.1071	−1.18624	−0.593119	−	0.805115i	$-0.702104\pi$
$233$	−18.1071	−1.18624	−0.593119	+	0.805115i	$0.702104\pi$
$234$	0	0
$235$	5.20037	0.339235
$236$	0	0
$237$	−1.68094	−0.109189
$238$	0	0
$239$	16.2966	1.05414	0.527070	−	0.849822i	$-0.323290\pi$
$239$	16.2966	1.05414	0.527070	+	0.849822i	$0.323290\pi$
$240$	0	0
$241$	−22.3729	−1.44117	−0.720583	−	0.693369i	$-0.756126\pi$
$241$	−22.3729	−1.44117	−0.720583	+	0.693369i	$0.756126\pi$
$242$	0	0
$243$	−6.37537	−0.408980
$244$	0	0
$245$	−7.17131	−0.458158
$246$	0	0
$247$	13.6224	0.866772
$248$	0	0
$249$	−32.4781	−2.05822
$250$	0	0
$251$	0.476860	0.0300991	0.0150496	−	0.999887i	$-0.495209\pi$
$251$	0.476860	0.0300991	0.0150496	+	0.999887i	$0.495209\pi$
$252$	0	0
$253$	−20.7398	−1.30390
$254$	0	0
$255$	1.96310	0.122934
$256$	0	0
$257$	−18.7475	−1.16944	−0.584718	−	0.811237i	$-0.698795\pi$
$257$	−18.7475	−1.16944	−0.584718	+	0.811237i	$0.698795\pi$
$258$	0	0
$259$	0.205594	0.0127750
$260$	0	0
$261$	0.590314	0.0365395
$262$	0	0
$263$	−13.0087	−0.802153	−0.401076	−	0.916045i	$-0.631364\pi$
$263$	−13.0087	−0.802153	−0.401076	+	0.916045i	$0.631364\pi$
$264$	0	0
$265$	3.75363	0.230583
$266$	0	0
$267$	10.5236	0.644031
$268$	0	0
$269$	−14.9849	−0.913647	−0.456824	−	0.889557i	$-0.651013\pi$
$269$	−14.9849	−0.913647	−0.456824	+	0.889557i	$0.651013\pi$
$270$	0	0
$271$	−16.3433	−0.992784	−0.496392	−	0.868098i	$-0.665342\pi$
$271$	−16.3433	−0.992784	−0.496392	+	0.868098i	$0.665342\pi$
$272$	0	0
$273$	1.25456	0.0759294
$274$	0	0
$275$	−13.8387	−0.834504
$276$	0	0
$277$	9.04593	0.543517	0.271759	−	0.962365i	$-0.412395\pi$
$277$	9.04593	0.543517	0.271759	+	0.962365i	$0.412395\pi$
$278$	0	0
$279$	4.23436	0.253505
$280$	0	0
$281$	−15.8721	−0.946852	−0.473426	−	0.880834i	$-0.656983\pi$
$281$	−15.8721	−0.946852	−0.473426	+	0.880834i	$0.656983\pi$
$282$	0	0
$283$	13.9891	0.831565	0.415782	−	0.909464i	$-0.363508\pi$
$283$	13.9891	0.831565	0.415782	+	0.909464i	$0.363508\pi$
$284$	0	0
$285$	8.77532	0.519805
$286$	0	0
$287$	2.43829	0.143928
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−21.2839	−1.24768
$292$	0	0
$293$	20.8954	1.22072	0.610362	−	0.792122i	$-0.291024\pi$
$293$	20.8954	1.22072	0.610362	+	0.792122i	$0.291024\pi$
$294$	0	0
$295$	1.03137	0.0600484
$296$	0	0
$297$	−15.9068	−0.923005
$298$	0	0
$299$	−17.9775	−1.03966
$300$	0	0
$301$	0.206649	0.0119110
$302$	0	0
$303$	30.9199	1.77630
$304$	0	0
$305$	−2.07054	−0.118559
$306$	0	0
$307$	31.8522	1.81790	0.908950	−	0.416905i	$-0.136885\pi$
$307$	31.8522	1.81790	0.908950	+	0.416905i	$0.136885\pi$
$308$	0	0
$309$	1.71913	0.0977981
$310$	0	0
$311$	17.4639	0.990289	0.495145	−	0.868811i	$-0.335115\pi$
$311$	17.4639	0.990289	0.495145	+	0.868811i	$0.335115\pi$
$312$	0	0
$313$	−6.53646	−0.369463	−0.184731	−	0.982789i	$-0.559142\pi$
$313$	−6.53646	−0.369463	−0.184731	+	0.982789i	$0.559142\pi$
$314$	0	0
$315$	0.138954	0.00782919
$316$	0	0
$317$	31.9319	1.79347	0.896737	−	0.442563i	$-0.145931\pi$
$317$	31.9319	1.79347	0.896737	+	0.442563i	$0.145931\pi$
$318$	0	0
$319$	3.33167	0.186538
$320$	0	0
$321$	13.0785	0.729969
$322$	0	0
$323$	4.47014	0.248725
$324$	0	0
$325$	−11.9955	−0.665392
$326$	0	0
$327$	−17.0128	−0.940811
$328$	0	0
$329$	1.09057	0.0601248
$330$	0	0
$331$	15.6969	0.862778	0.431389	−	0.902166i	$-0.358024\pi$
$331$	15.6969	0.862778	0.431389	+	0.902166i	$0.358024\pi$
$332$	0	0
$333$	0.592119	0.0324479
$334$	0	0
$335$	3.06795	0.167620
$336$	0	0
$337$	34.5969	1.88461	0.942307	−	0.334750i	$-0.108652\pi$
$337$	34.5969	1.88461	0.942307	+	0.334750i	$0.108652\pi$
$338$	0	0
$339$	−12.9783	−0.704883
$340$	0	0
$341$	23.8983	1.29416
$342$	0	0
$343$	−3.01790	−0.162951
$344$	0	0
$345$	−11.5808	−0.623488
$346$	0	0
$347$	−7.29460	−0.391595	−0.195797	−	0.980644i	$-0.562729\pi$
$347$	−7.29460	−0.391595	−0.195797	+	0.980644i	$0.562729\pi$
$348$	0	0
$349$	−1.05988	−0.0567341	−0.0283670	−	0.999598i	$-0.509031\pi$
$349$	−1.05988	−0.0567341	−0.0283670	+	0.999598i	$0.509031\pi$
$350$	0	0
$351$	−13.7882	−0.735957
$352$	0	0
$353$	−28.9126	−1.53886	−0.769430	−	0.638731i	$-0.779460\pi$
$353$	−28.9126	−1.53886	−0.769430	+	0.638731i	$0.779460\pi$
$354$	0	0
$355$	−8.91035	−0.472912
$356$	0	0
$357$	0.411679	0.0217884
$358$	0	0
$359$	35.9880	1.89938	0.949688	−	0.313198i	$-0.101400\pi$
$359$	35.9880	1.89938	0.949688	+	0.313198i	$0.101400\pi$
$360$	0	0
$361$	0.982138	0.0516915
$362$	0	0
$363$	2.58853	0.135863
$364$	0	0
$365$	6.17246	0.323081
$366$	0	0
$367$	5.94955	0.310564	0.155282	−	0.987870i	$-0.450371\pi$
$367$	5.94955	0.310564	0.155282	+	0.987870i	$0.450371\pi$
$368$	0	0
$369$	7.02239	0.365571
$370$	0	0
$371$	0.787170	0.0408678
$372$	0	0
$373$	−26.0844	−1.35060	−0.675300	−	0.737543i	$-0.735986\pi$
$373$	−26.0844	−1.35060	−0.675300	+	0.737543i	$0.735986\pi$
$374$	0	0
$375$	−17.5428	−0.905906
$376$	0	0
$377$	2.88793	0.148736
$378$	0	0
$379$	−26.6223	−1.36749	−0.683747	−	0.729719i	$-0.739651\pi$
$379$	−26.6223	−1.36749	−0.683747	+	0.729719i	$0.739651\pi$
$380$	0	0
$381$	19.4890	0.998452
$382$	0	0
$383$	−7.28262	−0.372124	−0.186062	−	0.982538i	$-0.559573\pi$
$383$	−7.28262	−0.372124	−0.186062	+	0.982538i	$0.559573\pi$
$384$	0	0
$385$	0.784244	0.0399688
$386$	0	0
$387$	0.595158	0.0302536
$388$	0	0
$389$	−25.9516	−1.31580	−0.657898	−	0.753107i	$-0.728554\pi$
$389$	−25.9516	−1.31580	−0.657898	+	0.753107i	$0.728554\pi$
$390$	0	0
$391$	−5.89924	−0.298337
$392$	0	0
$393$	25.6192	1.29232
$394$	0	0
$395$	−0.910828	−0.0458287
$396$	0	0
$397$	−6.30347	−0.316362	−0.158181	−	0.987410i	$-0.550563\pi$
$397$	−6.30347	−0.316362	−0.158181	+	0.987410i	$0.550563\pi$
$398$	0	0
$399$	1.84026	0.0921284
$400$	0	0
$401$	−25.3855	−1.26769	−0.633846	−	0.773459i	$-0.718525\pi$
$401$	−25.3855	−1.26769	−0.633846	+	0.773459i	$0.718525\pi$
$402$	0	0
$403$	20.7153	1.03190
$404$	0	0
$405$	−10.8095	−0.537126
$406$	0	0
$407$	3.34186	0.165650
$408$	0	0
$409$	−28.6845	−1.41836	−0.709179	−	0.705029i	$-0.750934\pi$
$409$	−28.6845	−1.41836	−0.709179	+	0.705029i	$0.750934\pi$
$410$	0	0
$411$	41.9365	2.06858
$412$	0	0
$413$	0.216287	0.0106428
$414$	0	0
$415$	−17.5984	−0.863873
$416$	0	0
$417$	25.8217	1.26449
$418$	0	0
$419$	6.66762	0.325735	0.162867	−	0.986648i	$-0.447926\pi$
$419$	6.66762	0.325735	0.162867	+	0.986648i	$0.447926\pi$
$420$	0	0
$421$	34.8127	1.69667	0.848335	−	0.529460i	$-0.177606\pi$
$421$	34.8127	1.69667	0.848335	+	0.529460i	$0.177606\pi$
$422$	0	0
$423$	3.14088	0.152715
$424$	0	0
$425$	−3.93628	−0.190938
$426$	0	0
$427$	−0.434211	−0.0210129
$428$	0	0
$429$	20.3925	0.984558
$430$	0	0
$431$	−3.80346	−0.183206	−0.0916031	−	0.995796i	$-0.529199\pi$
$431$	−3.80346	−0.183206	−0.0916031	+	0.995796i	$0.529199\pi$
$432$	0	0
$433$	5.69532	0.273700	0.136850	−	0.990592i	$-0.456302\pi$
$433$	5.69532	0.273700	0.136850	+	0.990592i	$0.456302\pi$
$434$	0	0
$435$	1.86035	0.0891971
$436$	0	0
$437$	−26.3704	−1.26147
$438$	0	0
$439$	−20.3384	−0.970699	−0.485349	−	0.874320i	$-0.661308\pi$
$439$	−20.3384	−0.970699	−0.485349	+	0.874320i	$0.661308\pi$
$440$	0	0
$441$	−4.33127	−0.206251
$442$	0	0
$443$	9.45305	0.449128	0.224564	−	0.974459i	$-0.427904\pi$
$443$	9.45305	0.449128	0.224564	+	0.974459i	$0.427904\pi$
$444$	0	0
$445$	5.70225	0.270312
$446$	0	0
$447$	−34.9330	−1.65227
$448$	0	0
$449$	−33.9709	−1.60319	−0.801593	−	0.597869i	$-0.796014\pi$
$449$	−33.9709	−1.60319	−0.801593	+	0.597869i	$0.796014\pi$
$450$	0	0
$451$	39.6336	1.86627
$452$	0	0
$453$	5.28772	0.248439
$454$	0	0
$455$	0.679790	0.0318691
$456$	0	0
$457$	26.8841	1.25758	0.628792	−	0.777574i	$-0.283550\pi$
$457$	26.8841	1.25758	0.628792	+	0.777574i	$0.283550\pi$
$458$	0	0
$459$	−4.52453	−0.211187
$460$	0	0
$461$	−29.2675	−1.36312	−0.681561	−	0.731762i	$-0.738698\pi$
$461$	−29.2675	−1.36312	−0.681561	+	0.731762i	$0.738698\pi$
$462$	0	0
$463$	−25.4318	−1.18192	−0.590959	−	0.806702i	$-0.701251\pi$
$463$	−25.4318	−1.18192	−0.590959	+	0.806702i	$0.701251\pi$
$464$	0	0
$465$	13.3444	0.618833
$466$	0	0
$467$	−12.2513	−0.566923	−0.283461	−	0.958984i	$-0.591483\pi$
$467$	−12.2513	−0.566923	−0.283461	+	0.958984i	$0.591483\pi$
$468$	0	0
$469$	0.643376	0.0297083
$470$	0	0
$471$	21.8310	1.00592
$472$	0	0
$473$	3.35901	0.154447
$474$	0	0
$475$	−17.5957	−0.807348
$476$	0	0
$477$	2.26709	0.103803
$478$	0	0
$479$	−23.0877	−1.05490	−0.527451	−	0.849586i	$-0.676852\pi$
$479$	−23.0877	−1.05490	−0.527451	+	0.849586i	$0.676852\pi$
$480$	0	0
$481$	2.89676	0.132081
$482$	0	0
$483$	−2.42859	−0.110505
$484$	0	0
$485$	−11.5328	−0.523677
$486$	0	0
$487$	−8.12176	−0.368032	−0.184016	−	0.982923i	$-0.558910\pi$
$487$	−8.12176	−0.368032	−0.184016	+	0.982923i	$0.558910\pi$
$488$	0	0
$489$	6.68524	0.302317
$490$	0	0
$491$	−0.346773	−0.0156497	−0.00782483	−	0.999969i	$-0.502491\pi$
$491$	−0.346773	−0.0156497	−0.00782483	+	0.999969i	$0.502491\pi$
$492$	0	0
$493$	0.947662	0.0426806
$494$	0	0
$495$	2.25866	0.101519
$496$	0	0
$497$	−1.86858	−0.0838173
$498$	0	0
$499$	33.8956	1.51738	0.758688	−	0.651454i	$-0.225841\pi$
$499$	33.8956	1.51738	0.758688	+	0.651454i	$0.225841\pi$
$500$	0	0
$501$	−46.0982	−2.05951
$502$	0	0
$503$	−23.1846	−1.03375	−0.516874	−	0.856061i	$-0.672905\pi$
$503$	−23.1846	−1.03375	−0.516874	+	0.856061i	$0.672905\pi$
$504$	0	0
$505$	16.7541	0.745548
$506$	0	0
$507$	−7.06774	−0.313889
$508$	0	0
$509$	20.7139	0.918126	0.459063	−	0.888404i	$-0.348185\pi$
$509$	20.7139	0.918126	0.459063	+	0.888404i	$0.348185\pi$
$510$	0	0
$511$	1.29442	0.0572618
$512$	0	0
$513$	−20.2253	−0.892968
$514$	0	0
$515$	0.931523	0.0410478
$516$	0	0
$517$	17.7268	0.779624
$518$	0	0
$519$	45.5593	1.99983
$520$	0	0
$521$	−4.30747	−0.188714	−0.0943569	−	0.995538i	$-0.530079\pi$
$521$	−4.30747	−0.188714	−0.0943569	+	0.995538i	$0.530079\pi$
$522$	0	0
$523$	−22.8261	−0.998115	−0.499057	−	0.866569i	$-0.666320\pi$
$523$	−22.8261	−0.998115	−0.499057	+	0.866569i	$0.666320\pi$
$524$	0	0
$525$	−1.62049	−0.0707239
$526$	0	0
$527$	6.79764	0.296110
$528$	0	0
$529$	11.8010	0.513087
$530$	0	0
$531$	0.622916	0.0270323
$532$	0	0
$533$	34.3548	1.48807
$534$	0	0
$535$	7.08664	0.306382
$536$	0	0
$537$	25.0288	1.08007
$538$	0	0
$539$	−24.4452	−1.05293
$540$	0	0
$541$	−35.8634	−1.54189	−0.770943	−	0.636904i	$-0.780215\pi$
$541$	−35.8634	−1.54189	−0.770943	+	0.636904i	$0.780215\pi$
$542$	0	0
$543$	−12.9808	−0.557058
$544$	0	0
$545$	−9.21850	−0.394877
$546$	0	0
$547$	11.7913	0.504162	0.252081	−	0.967706i	$-0.418885\pi$
$547$	11.7913	0.504162	0.252081	+	0.967706i	$0.418885\pi$
$548$	0	0
$549$	−1.25055	−0.0533721
$550$	0	0
$551$	4.23618	0.180467
$552$	0	0
$553$	−0.191009	−0.00812253
$554$	0	0
$555$	1.86604	0.0792091
$556$	0	0
$557$	−6.70819	−0.284235	−0.142118	−	0.989850i	$-0.545391\pi$
$557$	−6.70819	−0.284235	−0.142118	+	0.989850i	$0.545391\pi$
$558$	0	0
$559$	2.91162	0.123148
$560$	0	0
$561$	6.69172	0.282525
$562$	0	0
$563$	31.8125	1.34074	0.670368	−	0.742029i	$-0.266136\pi$
$563$	31.8125	1.34074	0.670368	+	0.742029i	$0.266136\pi$
$564$	0	0
$565$	−7.03235	−0.295853
$566$	0	0
$567$	−2.26684	−0.0951984
$568$	0	0
$569$	17.4193	0.730257	0.365128	−	0.930957i	$-0.381025\pi$
$569$	17.4193	0.730257	0.365128	+	0.930957i	$0.381025\pi$
$570$	0	0
$571$	−24.1259	−1.00964	−0.504818	−	0.863226i	$-0.668440\pi$
$571$	−24.1259	−1.00964	−0.504818	+	0.863226i	$0.668440\pi$
$572$	0	0
$573$	−51.3192	−2.14389
$574$	0	0
$575$	23.2211	0.968386
$576$	0	0
$577$	14.2509	0.593271	0.296635	−	0.954991i	$-0.404135\pi$
$577$	14.2509	0.593271	0.296635	+	0.954991i	$0.404135\pi$
$578$	0	0
$579$	41.6799	1.73216
$580$	0	0
$581$	−3.69055	−0.153110
$582$	0	0
$583$	12.7952	0.529923
$584$	0	0
$585$	1.95783	0.0809463
$586$	0	0
$587$	5.11482	0.211111	0.105556	−	0.994413i	$-0.466338\pi$
$587$	5.11482	0.211111	0.105556	+	0.994413i	$0.466338\pi$
$588$	0	0
$589$	30.3864	1.25205
$590$	0	0
$591$	18.4839	0.760325
$592$	0	0
$593$	25.8217	1.06037	0.530185	−	0.847882i	$-0.322122\pi$
$593$	25.8217	1.06037	0.530185	+	0.847882i	$0.322122\pi$
$594$	0	0
$595$	0.223071	0.00914501
$596$	0	0
$597$	−42.0017	−1.71902
$598$	0	0
$599$	−3.58462	−0.146464	−0.0732318	−	0.997315i	$-0.523331\pi$
$599$	−3.58462	−0.146464	−0.0732318	+	0.997315i	$0.523331\pi$
$600$	0	0
$601$	−46.7003	−1.90494	−0.952472	−	0.304625i	$-0.901469\pi$
$601$	−46.7003	−1.90494	−0.952472	+	0.304625i	$0.901469\pi$
$602$	0	0
$603$	1.85295	0.0754581
$604$	0	0
$605$	1.40261	0.0570242
$606$	0	0
$607$	−37.9940	−1.54213	−0.771065	−	0.636756i	$-0.780276\pi$
$607$	−37.9940	−1.54213	−0.771065	+	0.636756i	$0.780276\pi$
$608$	0	0
$609$	0.390133	0.0158090
$610$	0	0
$611$	15.3658	0.621633
$612$	0	0
$613$	−22.6355	−0.914240	−0.457120	−	0.889405i	$-0.651119\pi$
$613$	−22.6355	−0.914240	−0.457120	+	0.889405i	$0.651119\pi$
$614$	0	0
$615$	22.1308	0.892400
$616$	0	0
$617$	−36.1369	−1.45482	−0.727408	−	0.686205i	$-0.759275\pi$
$617$	−36.1369	−1.45482	−0.727408	+	0.686205i	$0.759275\pi$
$618$	0	0
$619$	−37.0516	−1.48923	−0.744614	−	0.667495i	$-0.767366\pi$
$619$	−37.0516	−1.48923	−0.744614	+	0.667495i	$0.767366\pi$
$620$	0	0
$621$	26.6913	1.07108
$622$	0	0
$623$	1.19581	0.0479092
$624$	0	0
$625$	10.1758	0.407030
$626$	0	0
$627$	29.9129	1.19461
$628$	0	0
$629$	0.950560	0.0379013
$630$	0	0
$631$	−7.90350	−0.314633	−0.157317	−	0.987548i	$-0.550284\pi$
$631$	−7.90350	−0.314633	−0.157317	+	0.987548i	$0.550284\pi$
$632$	0	0
$633$	20.2414	0.804524
$634$	0	0
$635$	10.5602	0.419070
$636$	0	0
$637$	−21.1894	−0.839554
$638$	0	0
$639$	−5.38160	−0.212893
$640$	0	0
$641$	20.5973	0.813545	0.406772	−	0.913530i	$-0.366654\pi$
$641$	20.5973	0.813545	0.406772	+	0.913530i	$0.366654\pi$
$642$	0	0
$643$	−23.5878	−0.930214	−0.465107	−	0.885255i	$-0.653984\pi$
$643$	−23.5878	−0.930214	−0.465107	+	0.885255i	$0.653984\pi$
$644$	0	0
$645$	1.87562	0.0738524
$646$	0	0
$647$	24.5770	0.966221	0.483110	−	0.875559i	$-0.339507\pi$
$647$	24.5770	0.966221	0.483110	+	0.875559i	$0.339507\pi$
$648$	0	0
$649$	3.51567	0.138002
$650$	0	0
$651$	2.79845	0.109680
$652$	0	0
$653$	20.9951	0.821600	0.410800	−	0.911725i	$-0.365249\pi$
$653$	20.9951	0.821600	0.410800	+	0.911725i	$0.365249\pi$
$654$	0	0
$655$	13.8819	0.542411
$656$	0	0
$657$	3.72800	0.145443
$658$	0	0
$659$	11.1293	0.433535	0.216768	−	0.976223i	$-0.430449\pi$
$659$	11.1293	0.433535	0.216768	+	0.976223i	$0.430449\pi$
$660$	0	0
$661$	20.3000	0.789580	0.394790	−	0.918771i	$-0.370817\pi$
$661$	20.3000	0.789580	0.394790	+	0.918771i	$0.370817\pi$
$662$	0	0
$663$	5.80045	0.225271
$664$	0	0
$665$	0.997157	0.0386681
$666$	0	0
$667$	−5.59049	−0.216464
$668$	0	0
$669$	−41.2399	−1.59443
$670$	0	0
$671$	−7.05796	−0.272470
$672$	0	0
$673$	−19.2879	−0.743494	−0.371747	−	0.928334i	$-0.621241\pi$
$673$	−19.2879	−0.743494	−0.371747	+	0.928334i	$0.621241\pi$
$674$	0	0
$675$	17.8098	0.685501
$676$	0	0
$677$	−30.6794	−1.17911	−0.589553	−	0.807730i	$-0.700696\pi$
$677$	−30.6794	−1.17911	−0.589553	+	0.807730i	$0.700696\pi$
$678$	0	0
$679$	−2.41853	−0.0928148
$680$	0	0
$681$	7.86464	0.301374
$682$	0	0
$683$	15.7043	0.600908	0.300454	−	0.953796i	$-0.402862\pi$
$683$	15.7043	0.600908	0.300454	+	0.953796i	$0.402862\pi$
$684$	0	0
$685$	22.7235	0.868222
$686$	0	0
$687$	−41.7971	−1.59466
$688$	0	0
$689$	11.0910	0.422534
$690$	0	0
$691$	−45.3035	−1.72343	−0.861713	−	0.507396i	$-0.830608\pi$
$691$	−45.3035	−1.72343	−0.861713	+	0.507396i	$0.830608\pi$
$692$	0	0
$693$	0.473661	0.0179929
$694$	0	0
$695$	13.9916	0.530732
$696$	0	0
$697$	11.2734	0.427011
$698$	0	0
$699$	−34.4650	−1.30359
$700$	0	0
$701$	38.5405	1.45565	0.727827	−	0.685761i	$-0.240531\pi$
$701$	38.5405	1.45565	0.727827	+	0.685761i	$0.240531\pi$
$702$	0	0
$703$	4.24914	0.160259
$704$	0	0
$705$	9.89837	0.372794
$706$	0	0
$707$	3.51349	0.132138
$708$	0	0
$709$	−43.6024	−1.63752	−0.818762	−	0.574133i	$-0.805339\pi$
$709$	−43.6024	−1.63752	−0.818762	+	0.574133i	$0.805339\pi$
$710$	0	0
$711$	−0.550115	−0.0206309
$712$	0	0
$713$	−40.1009	−1.50179
$714$	0	0
$715$	11.0498	0.413238
$716$	0	0
$717$	31.0189	1.15842
$718$	0	0
$719$	33.2205	1.23892	0.619459	−	0.785029i	$-0.287352\pi$
$719$	33.2205	1.23892	0.619459	+	0.785029i	$0.287352\pi$
$720$	0	0
$721$	0.195349	0.00727517
$722$	0	0
$723$	−42.5845	−1.58373
$724$	0	0
$725$	−3.73027	−0.138539
$726$	0	0
$727$	−0.438430	−0.0162605	−0.00813023	−	0.999967i	$-0.502588\pi$
$727$	−0.438430	−0.0162605	−0.00813023	+	0.999967i	$0.502588\pi$
$728$	0	0
$729$	19.3073	0.715086
$730$	0	0
$731$	0.955438	0.0353382
$732$	0	0
$733$	−7.33979	−0.271101	−0.135551	−	0.990770i	$-0.543280\pi$
$733$	−7.33979	−0.271101	−0.135551	+	0.990770i	$0.543280\pi$
$734$	0	0
$735$	−13.6498	−0.503482
$736$	0	0
$737$	10.4579	0.385221
$738$	0	0
$739$	19.3336	0.711197	0.355598	−	0.934639i	$-0.384277\pi$
$739$	19.3336	0.711197	0.355598	+	0.934639i	$0.384277\pi$
$740$	0	0
$741$	25.9288	0.952519
$742$	0	0
$743$	23.1009	0.847491	0.423745	−	0.905781i	$-0.360715\pi$
$743$	23.1009	0.847491	0.423745	+	0.905781i	$0.360715\pi$
$744$	0	0
$745$	−18.9286	−0.693492
$746$	0	0
$747$	−10.6290	−0.388894
$748$	0	0
$749$	1.48613	0.0543021
$750$	0	0
$751$	13.5011	0.492663	0.246331	−	0.969186i	$-0.420775\pi$
$751$	13.5011	0.492663	0.246331	+	0.969186i	$0.420775\pi$
$752$	0	0
$753$	0.907653	0.0330767
$754$	0	0
$755$	2.86518	0.104275
$756$	0	0
$757$	−41.2000	−1.49744	−0.748721	−	0.662886i	$-0.769331\pi$
$757$	−41.2000	−1.49744	−0.748721	+	0.662886i	$0.769331\pi$
$758$	0	0
$759$	−39.4760	−1.43289
$760$	0	0
$761$	33.2904	1.20678	0.603388	−	0.797447i	$-0.293817\pi$
$761$	33.2904	1.20678	0.603388	+	0.797447i	$0.293817\pi$
$762$	0	0
$763$	−1.93320	−0.0699866
$764$	0	0
$765$	0.642454	0.0232280
$766$	0	0
$767$	3.04742	0.110036
$768$	0	0
$769$	−37.4058	−1.34889	−0.674444	−	0.738326i	$-0.735617\pi$
$769$	−37.4058	−1.34889	−0.674444	+	0.738326i	$0.735617\pi$
$770$	0	0
$771$	−35.6839	−1.28512
$772$	0	0
$773$	−33.8236	−1.21655	−0.608274	−	0.793727i	$-0.708138\pi$
$773$	−33.8236	−1.21655	−0.608274	+	0.793727i	$0.708138\pi$
$774$	0	0
$775$	−26.7575	−0.961156
$776$	0	0
$777$	0.391326	0.0140387
$778$	0	0
$779$	50.3937	1.80554
$780$	0	0
$781$	−30.3732	−1.08684
$782$	0	0
$783$	−4.28773	−0.153231
$784$	0	0
$785$	11.8292	0.422204
$786$	0	0
$787$	35.9617	1.28189	0.640947	−	0.767585i	$-0.278542\pi$
$787$	35.9617	1.28189	0.640947	+	0.767585i	$0.278542\pi$
$788$	0	0
$789$	−24.7608	−0.881507
$790$	0	0
$791$	−1.47475	−0.0524360
$792$	0	0
$793$	−6.11791	−0.217253
$794$	0	0
$795$	7.14464	0.253394
$796$	0	0
$797$	−28.5948	−1.01288	−0.506440	−	0.862275i	$-0.669039\pi$
$797$	−28.5948	−1.01288	−0.506440	+	0.862275i	$0.669039\pi$
$798$	0	0
$799$	5.04222	0.178381
$800$	0	0
$801$	3.44400	0.121688
$802$	0	0
$803$	21.0404	0.742500
$804$	0	0
$805$	−1.31595	−0.0463810
$806$	0	0
$807$	−28.5223	−1.00403
$808$	0	0
$809$	−44.5418	−1.56601	−0.783003	−	0.622017i	$-0.786313\pi$
$809$	−44.5418	−1.56601	−0.783003	+	0.622017i	$0.786313\pi$
$810$	0	0
$811$	1.57945	0.0554621	0.0277310	−	0.999615i	$-0.491172\pi$
$811$	1.57945	0.0554621	0.0277310	+	0.999615i	$0.491172\pi$
$812$	0	0
$813$	−31.1077	−1.09100
$814$	0	0
$815$	3.62243	0.126888
$816$	0	0
$817$	4.27094	0.149421
$818$	0	0
$819$	0.410574	0.0143466
$820$	0	0
$821$	13.5026	0.471246	0.235623	−	0.971845i	$-0.424287\pi$
$821$	13.5026	0.471246	0.235623	+	0.971845i	$0.424287\pi$
$822$	0	0
$823$	−2.55785	−0.0891609	−0.0445804	−	0.999006i	$-0.514195\pi$
$823$	−2.55785	−0.0891609	−0.0445804	+	0.999006i	$0.514195\pi$
$824$	0	0
$825$	−26.3405	−0.917059
$826$	0	0
$827$	−1.80034	−0.0626039	−0.0313019	−	0.999510i	$-0.509965\pi$
$827$	−1.80034	−0.0626039	−0.0313019	+	0.999510i	$0.509965\pi$
$828$	0	0
$829$	−22.9560	−0.797296	−0.398648	−	0.917104i	$-0.630521\pi$
$829$	−22.9560	−0.797296	−0.398648	+	0.917104i	$0.630521\pi$
$830$	0	0
$831$	17.2180	0.597285
$832$	0	0
$833$	−6.95322	−0.240915
$834$	0	0
$835$	−24.9786	−0.864418
$836$	0	0
$837$	−30.7561	−1.06309
$838$	0	0
$839$	11.0934	0.382986	0.191493	−	0.981494i	$-0.438667\pi$
$839$	11.0934	0.382986	0.191493	+	0.981494i	$0.438667\pi$
$840$	0	0
$841$	−28.1019	−0.969032
$842$	0	0
$843$	−30.2110	−1.04052
$844$	0	0
$845$	−3.82969	−0.131745
$846$	0	0
$847$	0.294140	0.0101068
$848$	0	0
$849$	26.6268	0.913828
$850$	0	0
$851$	−5.60758	−0.192225
$852$	0	0
$853$	−28.4742	−0.974938	−0.487469	−	0.873140i	$-0.662080\pi$
$853$	−28.4742	−0.974938	−0.487469	+	0.873140i	$0.662080\pi$
$854$	0	0
$855$	2.87186	0.0982155
$856$	0	0
$857$	27.1037	0.925845	0.462922	−	0.886399i	$-0.346801\pi$
$857$	27.1037	0.925845	0.462922	+	0.886399i	$0.346801\pi$
$858$	0	0
$859$	−9.74070	−0.332349	−0.166174	−	0.986096i	$-0.553141\pi$
$859$	−9.74070	−0.332349	−0.166174	+	0.986096i	$0.553141\pi$
$860$	0	0
$861$	4.64103	0.158166
$862$	0	0
$863$	−35.3579	−1.20360	−0.601799	−	0.798647i	$-0.705549\pi$
$863$	−35.3579	−1.20360	−0.601799	+	0.798647i	$0.705549\pi$
$864$	0	0
$865$	24.6865	0.839368
$866$	0	0
$867$	1.90340	0.0646427
$868$	0	0
$869$	−3.10479	−0.105323
$870$	0	0
$871$	9.06499	0.307155
$872$	0	0
$873$	−6.96549	−0.235746
$874$	0	0
$875$	−1.99342	−0.0673900
$876$	0	0
$877$	−19.0490	−0.643240	−0.321620	−	0.946869i	$-0.604227\pi$
$877$	−19.0490	−0.643240	−0.321620	+	0.946869i	$0.604227\pi$
$878$	0	0
$879$	39.7723	1.34149
$880$	0	0
$881$	−35.6999	−1.20276	−0.601380	−	0.798963i	$-0.705382\pi$
$881$	−35.6999	−1.20276	−0.601380	+	0.798963i	$0.705382\pi$
$882$	0	0
$883$	15.8339	0.532854	0.266427	−	0.963855i	$-0.414157\pi$
$883$	15.8339	0.532854	0.266427	+	0.963855i	$0.414157\pi$
$884$	0	0
$885$	1.96310	0.0659888
$886$	0	0
$887$	−7.77405	−0.261027	−0.130514	−	0.991447i	$-0.541663\pi$
$887$	−7.77405	−0.261027	−0.130514	+	0.991447i	$0.541663\pi$
$888$	0	0
$889$	2.21458	0.0742745
$890$	0	0
$891$	−36.8468	−1.23441
$892$	0	0
$893$	22.5394	0.754253
$894$	0	0
$895$	13.5620	0.453328
$896$	0	0
$897$	−34.2182	−1.14251
$898$	0	0
$899$	6.44187	0.214848
$900$	0	0
$901$	3.63947	0.121248
$902$	0	0
$903$	0.393334	0.0130893
$904$	0	0
$905$	−7.03371	−0.233808
$906$	0	0
$907$	0.421904	0.0140091	0.00700455	−	0.999975i	$-0.497770\pi$
$907$	0.421904	0.0140091	0.00700455	+	0.999975i	$0.497770\pi$
$908$	0	0
$909$	10.1190	0.335626
$910$	0	0
$911$	11.4786	0.380301	0.190151	−	0.981755i	$-0.439102\pi$
$911$	11.4786	0.380301	0.190151	+	0.981755i	$0.439102\pi$
$912$	0	0
$913$	−59.9887	−1.98534
$914$	0	0
$915$	−3.94106	−0.130287
$916$	0	0
$917$	2.91116	0.0961351
$918$	0	0
$919$	14.3864	0.474563	0.237281	−	0.971441i	$-0.423744\pi$
$919$	14.3864	0.474563	0.237281	+	0.971441i	$0.423744\pi$
$920$	0	0
$921$	60.6273	1.99774
$922$	0	0
$923$	−26.3278	−0.866590
$924$	0	0
$925$	−3.74168	−0.123026
$926$	0	0
$927$	0.562614	0.0184787
$928$	0	0
$929$	21.3578	0.700728	0.350364	−	0.936614i	$-0.386058\pi$
$929$	21.3578	0.700728	0.350364	+	0.936614i	$0.386058\pi$
$930$	0	0
$931$	−31.0819	−1.01867
$932$	0	0
$933$	33.2408	1.08825
$934$	0	0
$935$	3.62594	0.118581
$936$	0	0
$937$	22.2560	0.727073	0.363537	−	0.931580i	$-0.381569\pi$
$937$	22.2560	0.727073	0.363537	+	0.931580i	$0.381569\pi$
$938$	0	0
$939$	−12.4415	−0.406012
$940$	0	0
$941$	−55.4567	−1.80784	−0.903918	−	0.427706i	$-0.859322\pi$
$941$	−55.4567	−1.80784	−0.903918	+	0.427706i	$0.859322\pi$
$942$	0	0
$943$	−66.5045	−2.16569
$944$	0	0
$945$	−1.00929	−0.0328322
$946$	0	0
$947$	40.3650	1.31169	0.655843	−	0.754897i	$-0.272313\pi$
$947$	40.3650	1.31169	0.655843	+	0.754897i	$0.272313\pi$
$948$	0	0
$949$	18.2380	0.592032
$950$	0	0
$951$	60.7791	1.97090
$952$	0	0
$953$	47.4916	1.53840	0.769202	−	0.639005i	$-0.220654\pi$
$953$	47.4916	1.53840	0.769202	+	0.639005i	$0.220654\pi$
$954$	0	0
$955$	−27.8076	−0.899832
$956$	0	0
$957$	6.34149	0.204991
$958$	0	0
$959$	4.76533	0.153881
$960$	0	0
$961$	15.2079	0.490579
$962$	0	0
$963$	4.28013	0.137925
$964$	0	0
$965$	22.5845	0.727020
$966$	0	0
$967$	−19.8685	−0.638929	−0.319465	−	0.947598i	$-0.603503\pi$
$967$	−19.8685	−0.638929	−0.319465	+	0.947598i	$0.603503\pi$
$968$	0	0
$969$	8.50844	0.273331
$970$	0	0
$971$	58.8734	1.88934	0.944669	−	0.328026i	$-0.106383\pi$
$971$	58.8734	1.88934	0.944669	+	0.328026i	$0.106383\pi$
$972$	0	0
$973$	2.93417	0.0940652
$974$	0	0
$975$	−22.8322	−0.731216
$976$	0	0
$977$	−61.7211	−1.97463	−0.987317	−	0.158764i	$-0.949249\pi$
$977$	−61.7211	−1.97463	−0.987317	+	0.158764i	$0.949249\pi$
$978$	0	0
$979$	19.4376	0.621227
$980$	0	0
$981$	−5.56771	−0.177763
$982$	0	0
$983$	20.6908	0.659934	0.329967	−	0.943992i	$-0.392962\pi$
$983$	20.6908	0.659934	0.329967	+	0.943992i	$0.392962\pi$
$984$	0	0
$985$	10.0156	0.319123
$986$	0	0
$987$	2.07578	0.0660728
$988$	0	0
$989$	−5.63636	−0.179226
$990$	0	0
$991$	38.6996	1.22933	0.614667	−	0.788787i	$-0.289291\pi$
$991$	38.6996	1.22933	0.614667	+	0.788787i	$0.289291\pi$
$992$	0	0
$993$	29.8773	0.948129
$994$	0	0
$995$	−22.7589	−0.721505
$996$	0	0
$997$	15.0770	0.477493	0.238746	−	0.971082i	$-0.423264\pi$
$997$	15.0770	0.477493	0.238746	+	0.971082i	$0.423264\pi$
$998$	0	0
$999$	−4.30084	−0.136073

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.j.1.14	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.j.1.14	✓	21	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q+1.90340 q^{3} +1.03137 q^{5} +0.216287 q^{7} +0.622916 q^{9} +O(q^{10})\)	\(q+1.90340 q^{3} +1.03137 q^{5} +0.216287 q^{7} +0.622916 q^{9} +3.51567 q^{11} +3.04742 q^{13} +1.96310 q^{15} +1.00000 q^{17} +4.47014 q^{19} +0.411679 q^{21} -5.89924 q^{23} -3.93628 q^{25} -4.52453 q^{27} +0.947662 q^{29} +6.79764 q^{31} +6.69172 q^{33} +0.223071 q^{35} +0.950560 q^{37} +5.80045 q^{39} +11.2734 q^{41} +0.955438 q^{43} +0.642454 q^{45} +5.04222 q^{47} -6.95322 q^{49} +1.90340 q^{51} +3.63947 q^{53} +3.62594 q^{55} +8.50844 q^{57} +1.00000 q^{59} -2.00757 q^{61} +0.134729 q^{63} +3.14301 q^{65} +2.97464 q^{67} -11.2286 q^{69} -8.63937 q^{71} +5.98475 q^{73} -7.49231 q^{75} +0.760393 q^{77} -0.883128 q^{79} -10.4807 q^{81} -17.0632 q^{83} +1.03137 q^{85} +1.80378 q^{87} +5.52883 q^{89} +0.659117 q^{91} +12.9386 q^{93} +4.61035 q^{95} -11.1821 q^{97} +2.18997 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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