LMFDB - Embedded newform 4012.2.a.g.1.7

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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 13 x^{10} + 60 x^{9} + 48 x^{8} - 289 x^{7} - 89 x^{6} + 602 x^{5} + 161 x^{4} + \cdots + 82 $ x^12 - 4x^11 - 13x^10 + 60x^9 + 48x^8 - 289x^7 - 89x^6 + 602x^5 + 161x^4 - 555x^3 - 197x^2 + 181*x + 82
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.660935$ of defining polynomial
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)

$f(q)$	$=$	$q+0.660935 q^{3} +2.45345 q^{5} +3.10518 q^{7} -2.56317 q^{9} +O(q^{10})$	$q+0.660935 q^{3} +2.45345 q^{5} +3.10518 q^{7} -2.56317 q^{9} -3.33884 q^{11} -5.08477 q^{13} +1.62157 q^{15} -1.00000 q^{17} -5.38873 q^{19} +2.05232 q^{21} -3.60332 q^{23} +1.01943 q^{25} -3.67689 q^{27} -0.718872 q^{29} +8.53934 q^{31} -2.20676 q^{33} +7.61841 q^{35} +2.73317 q^{37} -3.36070 q^{39} -1.77689 q^{41} -10.0004 q^{43} -6.28861 q^{45} -9.54752 q^{47} +2.64214 q^{49} -0.660935 q^{51} -8.25925 q^{53} -8.19169 q^{55} -3.56160 q^{57} +1.00000 q^{59} +0.304416 q^{61} -7.95909 q^{63} -12.4753 q^{65} +2.16849 q^{67} -2.38156 q^{69} -15.4011 q^{71} -16.3540 q^{73} +0.673778 q^{75} -10.3677 q^{77} +14.7037 q^{79} +5.25931 q^{81} +9.25266 q^{83} -2.45345 q^{85} -0.475128 q^{87} +1.00738 q^{89} -15.7891 q^{91} +5.64394 q^{93} -13.2210 q^{95} +11.7259 q^{97} +8.55800 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) $ 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9	$ 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9} - 5 q^{11} + 9 q^{13} - 16 q^{15} - 12 q^{17} - 15 q^{19} + 5 q^{21} - 22 q^{23} + 5 q^{25} - 16 q^{27} - 17 q^{29} + 5 q^{31} - 33 q^{33} - 9 q^{35} - 2 q^{37} - 24 q^{39} - 18 q^{41} - 16 q^{43} + 30 q^{45} - 60 q^{47} + 10 q^{49} + 4 q^{51} - 24 q^{53} - q^{55} + 41 q^{57} + 12 q^{59} - 17 q^{61} - 23 q^{63} + 2 q^{65} - 22 q^{67} + 6 q^{69} - 10 q^{71} + 4 q^{73} - 62 q^{75} - 6 q^{77} + 48 q^{79} + 4 q^{81} - 20 q^{83} - 3 q^{85} + 13 q^{87} - 29 q^{89} - 28 q^{91} - 31 q^{93} - 48 q^{95} - 26 q^{97} - 3 q^{99}+O(q^{100}) $ 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9 - 5 * q^11 + 9 * q^13 - 16 * q^15 - 12 * q^17 - 15 * q^19 + 5 * q^21 - 22 * q^23 + 5 * q^25 - 16 * q^27 - 17 * q^29 + 5 * q^31 - 33 * q^33 - 9 * q^35 - 2 * q^37 - 24 * q^39 - 18 * q^41 - 16 * q^43 + 30 * q^45 - 60 * q^47 + 10 * q^49 + 4 * q^51 - 24 * q^53 - q^55 + 41 * q^57 + 12 * q^59 - 17 * q^61 - 23 * q^63 + 2 * q^65 - 22 * q^67 + 6 * q^69 - 10 * q^71 + 4 * q^73 - 62 * q^75 - 6 * q^77 + 48 * q^79 + 4 * q^81 - 20 * q^83 - 3 * q^85 + 13 * q^87 - 29 * q^89 - 28 * q^91 - 31 * q^93 - 48 * q^95 - 26 * q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.660935	0.381591	0.190795	−	0.981630i	$-0.438893\pi$
$3$	0.660935	0.381591	0.190795	+	0.981630i	$0.438893\pi$
$4$	0	0
$5$	2.45345	1.09722	0.548609	−	0.836079i	$-0.315158\pi$
$5$	2.45345	1.09722	0.548609	+	0.836079i	$0.315158\pi$
$6$	0	0
$7$	3.10518	1.17365	0.586824	−	0.809715i	$-0.300378\pi$
$7$	3.10518	1.17365	0.586824	+	0.809715i	$0.300378\pi$
$8$	0	0
$9$	−2.56317	−0.854388
$10$	0	0
$11$	−3.33884	−1.00670	−0.503349	−	0.864083i	$-0.667899\pi$
$11$	−3.33884	−1.00670	−0.503349	+	0.864083i	$0.667899\pi$
$12$	0	0
$13$	−5.08477	−1.41026	−0.705131	−	0.709077i	$-0.749112\pi$
$13$	−5.08477	−1.41026	−0.705131	+	0.709077i	$0.749112\pi$
$14$	0	0
$15$	1.62157	0.418688
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−5.38873	−1.23626	−0.618130	−	0.786076i	$-0.712109\pi$
$19$	−5.38873	−1.23626	−0.618130	+	0.786076i	$0.712109\pi$
$20$	0	0
$21$	2.05232	0.447853
$22$	0	0
$23$	−3.60332	−0.751344	−0.375672	−	0.926753i	$-0.622588\pi$
$23$	−3.60332	−0.751344	−0.375672	+	0.926753i	$0.622588\pi$
$24$	0	0
$25$	1.01943	0.203886
$26$	0	0
$27$	−3.67689	−0.707618
$28$	0	0
$29$	−0.718872	−0.133491	−0.0667456	−	0.997770i	$-0.521262\pi$
$29$	−0.718872	−0.133491	−0.0667456	+	0.997770i	$0.521262\pi$
$30$	0	0
$31$	8.53934	1.53371	0.766855	−	0.641820i	$-0.221821\pi$
$31$	8.53934	1.53371	0.766855	+	0.641820i	$0.221821\pi$
$32$	0	0
$33$	−2.20676	−0.384147
$34$	0	0
$35$	7.61841	1.28775
$36$	0	0
$37$	2.73317	0.449331	0.224665	−	0.974436i	$-0.427871\pi$
$37$	2.73317	0.449331	0.224665	+	0.974436i	$0.427871\pi$
$38$	0	0
$39$	−3.36070	−0.538143
$40$	0	0
$41$	−1.77689	−0.277503	−0.138752	−	0.990327i	$-0.544309\pi$
$41$	−1.77689	−0.277503	−0.138752	+	0.990327i	$0.544309\pi$
$42$	0	0
$43$	−10.0004	−1.52505	−0.762526	−	0.646958i	$-0.776041\pi$
$43$	−10.0004	−1.52505	−0.762526	+	0.646958i	$0.776041\pi$
$44$	0	0
$45$	−6.28861	−0.937450
$46$	0	0
$47$	−9.54752	−1.39265	−0.696325	−	0.717727i	$-0.745183\pi$
$47$	−9.54752	−1.39265	−0.696325	+	0.717727i	$0.745183\pi$
$48$	0	0
$49$	2.64214	0.377448
$50$	0	0
$51$	−0.660935	−0.0925494
$52$	0	0
$53$	−8.25925	−1.13450	−0.567248	−	0.823547i	$-0.691992\pi$
$53$	−8.25925	−1.13450	−0.567248	+	0.823547i	$0.691992\pi$
$54$	0	0
$55$	−8.19169	−1.10457
$56$	0	0
$57$	−3.56160	−0.471745
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	0.304416	0.0389765	0.0194882	−	0.999810i	$-0.493796\pi$
$61$	0.304416	0.0389765	0.0194882	+	0.999810i	$0.493796\pi$
$62$	0	0
$63$	−7.95909	−1.00275
$64$	0	0
$65$	−12.4753	−1.54736
$66$	0	0
$67$	2.16849	0.264923	0.132462	−	0.991188i	$-0.457712\pi$
$67$	2.16849	0.264923	0.132462	+	0.991188i	$0.457712\pi$
$68$	0	0
$69$	−2.38156	−0.286706
$70$	0	0
$71$	−15.4011	−1.82777	−0.913885	−	0.405973i	$-0.866933\pi$
$71$	−15.4011	−1.82777	−0.913885	+	0.405973i	$0.866933\pi$
$72$	0	0
$73$	−16.3540	−1.91410	−0.957048	−	0.289931i	$-0.906368\pi$
$73$	−16.3540	−1.91410	−0.957048	+	0.289931i	$0.906368\pi$
$74$	0	0
$75$	0.673778	0.0778012
$76$	0	0
$77$	−10.3677	−1.18151
$78$	0	0
$79$	14.7037	1.65429	0.827146	−	0.561987i	$-0.189963\pi$
$79$	14.7037	1.65429	0.827146	+	0.561987i	$0.189963\pi$
$80$	0	0
$81$	5.25931	0.584368
$82$	0	0
$83$	9.25266	1.01561	0.507806	−	0.861472i	$-0.330457\pi$
$83$	9.25266	1.01561	0.507806	+	0.861472i	$0.330457\pi$
$84$	0	0
$85$	−2.45345	−0.266114
$86$	0	0
$87$	−0.475128	−0.0509390
$88$	0	0
$89$	1.00738	0.106782	0.0533912	−	0.998574i	$-0.482997\pi$
$89$	1.00738	0.106782	0.0533912	+	0.998574i	$0.482997\pi$
$90$	0	0
$91$	−15.7891	−1.65515
$92$	0	0
$93$	5.64394	0.585250
$94$	0	0
$95$	−13.2210	−1.35645
$96$	0	0
$97$	11.7259	1.19058	0.595292	−	0.803509i	$-0.297036\pi$
$97$	11.7259	1.19058	0.595292	+	0.803509i	$0.297036\pi$
$98$	0	0
$99$	8.55800	0.860111
$100$	0	0
$101$	11.6963	1.16382	0.581910	−	0.813253i	$-0.302306\pi$
$101$	11.6963	1.16382	0.581910	+	0.813253i	$0.302306\pi$
$102$	0	0
$103$	8.56606	0.844039	0.422019	−	0.906587i	$-0.361321\pi$
$103$	8.56606	0.844039	0.422019	+	0.906587i	$0.361321\pi$
$104$	0	0
$105$	5.03527	0.491392
$106$	0	0
$107$	9.87146	0.954310	0.477155	−	0.878819i	$-0.341668\pi$
$107$	9.87146	0.954310	0.477155	+	0.878819i	$0.341668\pi$
$108$	0	0
$109$	−11.1260	−1.06567	−0.532837	−	0.846218i	$-0.678874\pi$
$109$	−11.1260	−1.06567	−0.532837	+	0.846218i	$0.678874\pi$
$110$	0	0
$111$	1.80645	0.171460
$112$	0	0
$113$	8.91916	0.839044	0.419522	−	0.907745i	$-0.362198\pi$
$113$	8.91916	0.839044	0.419522	+	0.907745i	$0.362198\pi$
$114$	0	0
$115$	−8.84058	−0.824388
$116$	0	0
$117$	13.0331	1.20491
$118$	0	0
$119$	−3.10518	−0.284651
$120$	0	0
$121$	0.147848	0.0134407
$122$	0	0
$123$	−1.17441	−0.105893
$124$	0	0
$125$	−9.76614	−0.873510
$126$	0	0
$127$	−19.2947	−1.71213	−0.856064	−	0.516871i	$-0.827097\pi$
$127$	−19.2947	−1.71213	−0.856064	+	0.516871i	$0.827097\pi$
$128$	0	0
$129$	−6.60963	−0.581946
$130$	0	0
$131$	17.0054	1.48577	0.742885	−	0.669419i	$-0.233457\pi$
$131$	17.0054	1.48577	0.742885	+	0.669419i	$0.233457\pi$
$132$	0	0
$133$	−16.7330	−1.45093
$134$	0	0
$135$	−9.02108	−0.776411
$136$	0	0
$137$	5.48969	0.469016	0.234508	−	0.972114i	$-0.424652\pi$
$137$	5.48969	0.469016	0.234508	+	0.972114i	$0.424652\pi$
$138$	0	0
$139$	22.8288	1.93631	0.968157	−	0.250343i	$-0.0805433\pi$
$139$	22.8288	1.93631	0.968157	+	0.250343i	$0.0805433\pi$
$140$	0	0
$141$	−6.31029	−0.531422
$142$	0	0
$143$	16.9772	1.41971
$144$	0	0
$145$	−1.76372	−0.146469
$146$	0	0
$147$	1.74628	0.144031
$148$	0	0
$149$	10.4895	0.859337	0.429668	−	0.902987i	$-0.358630\pi$
$149$	10.4895	0.859337	0.429668	+	0.902987i	$0.358630\pi$
$150$	0	0
$151$	4.35434	0.354352	0.177176	−	0.984179i	$-0.443304\pi$
$151$	4.35434	0.354352	0.177176	+	0.984179i	$0.443304\pi$
$152$	0	0
$153$	2.56317	0.207220
$154$	0	0
$155$	20.9509	1.68281
$156$	0	0
$157$	18.2072	1.45310	0.726548	−	0.687116i	$-0.241124\pi$
$157$	18.2072	1.45310	0.726548	+	0.687116i	$0.241124\pi$
$158$	0	0
$159$	−5.45883	−0.432913
$160$	0	0
$161$	−11.1890	−0.881813
$162$	0	0
$163$	−13.1400	−1.02920	−0.514602	−	0.857429i	$-0.672060\pi$
$163$	−13.1400	−1.02920	−0.514602	+	0.857429i	$0.672060\pi$
$164$	0	0
$165$	−5.41417	−0.421493
$166$	0	0
$167$	−16.2523	−1.25764	−0.628821	−	0.777550i	$-0.716462\pi$
$167$	−16.2523	−1.25764	−0.628821	+	0.777550i	$0.716462\pi$
$168$	0	0
$169$	12.8549	0.988840
$170$	0	0
$171$	13.8122	1.05625
$172$	0	0
$173$	−9.91123	−0.753537	−0.376768	−	0.926308i	$-0.622965\pi$
$173$	−9.91123	−0.753537	−0.376768	+	0.926308i	$0.622965\pi$
$174$	0	0
$175$	3.16552	0.239291
$176$	0	0
$177$	0.660935	0.0496789
$178$	0	0
$179$	−8.24666	−0.616384	−0.308192	−	0.951324i	$-0.599724\pi$
$179$	−8.24666	−0.616384	−0.308192	+	0.951324i	$0.599724\pi$
$180$	0	0
$181$	4.93880	0.367098	0.183549	−	0.983011i	$-0.441241\pi$
$181$	4.93880	0.367098	0.183549	+	0.983011i	$0.441241\pi$
$182$	0	0
$183$	0.201199	0.0148731
$184$	0	0
$185$	6.70571	0.493013
$186$	0	0
$187$	3.33884	0.244160
$188$	0	0
$189$	−11.4174	−0.830494
$190$	0	0
$191$	−8.32799	−0.602592	−0.301296	−	0.953531i	$-0.597419\pi$
$191$	−8.32799	−0.602592	−0.301296	+	0.953531i	$0.597419\pi$
$192$	0	0
$193$	−22.4847	−1.61848	−0.809241	−	0.587477i	$-0.800121\pi$
$193$	−22.4847	−1.61848	−0.809241	+	0.587477i	$0.800121\pi$
$194$	0	0
$195$	−8.24533	−0.590460
$196$	0	0
$197$	11.5325	0.821653	0.410827	−	0.911714i	$-0.365240\pi$
$197$	11.5325	0.821653	0.410827	+	0.911714i	$0.365240\pi$
$198$	0	0
$199$	−21.2531	−1.50659	−0.753296	−	0.657682i	$-0.771537\pi$
$199$	−21.2531	−1.50659	−0.753296	+	0.657682i	$0.771537\pi$
$200$	0	0
$201$	1.43323	0.101092
$202$	0	0
$203$	−2.23223	−0.156672
$204$	0	0
$205$	−4.35951	−0.304482
$206$	0	0
$207$	9.23590	0.641940
$208$	0	0
$209$	17.9921	1.24454
$210$	0	0
$211$	17.2682	1.18879	0.594395	−	0.804173i	$-0.297392\pi$
$211$	17.2682	1.18879	0.594395	+	0.804173i	$0.297392\pi$
$212$	0	0
$213$	−10.1791	−0.697460
$214$	0	0
$215$	−24.5356	−1.67331
$216$	0	0
$217$	26.5162	1.80003
$218$	0	0
$219$	−10.8090	−0.730401
$220$	0	0
$221$	5.08477	0.342039
$222$	0	0
$223$	−4.23045	−0.283292	−0.141646	−	0.989917i	$-0.545239\pi$
$223$	−4.23045	−0.283292	−0.141646	+	0.989917i	$0.545239\pi$
$224$	0	0
$225$	−2.61297	−0.174198
$226$	0	0
$227$	−13.2076	−0.876616	−0.438308	−	0.898825i	$-0.644422\pi$
$227$	−13.2076	−0.876616	−0.438308	+	0.898825i	$0.644422\pi$
$228$	0	0
$229$	14.1591	0.935658	0.467829	−	0.883819i	$-0.345036\pi$
$229$	14.1591	0.935658	0.467829	+	0.883819i	$0.345036\pi$
$230$	0	0
$231$	−6.85237	−0.450853
$232$	0	0
$233$	−2.12997	−0.139539	−0.0697695	−	0.997563i	$-0.522226\pi$
$233$	−2.12997	−0.139539	−0.0697695	+	0.997563i	$0.522226\pi$
$234$	0	0
$235$	−23.4244	−1.52804
$236$	0	0
$237$	9.71817	0.631263
$238$	0	0
$239$	18.4130	1.19104	0.595520	−	0.803341i	$-0.296946\pi$
$239$	18.4130	1.19104	0.595520	+	0.803341i	$0.296946\pi$
$240$	0	0
$241$	1.73250	0.111600	0.0558001	−	0.998442i	$-0.482229\pi$
$241$	1.73250	0.111600	0.0558001	+	0.998442i	$0.482229\pi$
$242$	0	0
$243$	14.5067	0.930607
$244$	0	0
$245$	6.48236	0.414143
$246$	0	0
$247$	27.4005	1.74345
$248$	0	0
$249$	6.11540	0.387548
$250$	0	0
$251$	−5.08637	−0.321049	−0.160524	−	0.987032i	$-0.551319\pi$
$251$	−5.08637	−0.321049	−0.160524	+	0.987032i	$0.551319\pi$
$252$	0	0
$253$	12.0309	0.756377
$254$	0	0
$255$	−1.62157	−0.101547
$256$	0	0
$257$	−14.7808	−0.922002	−0.461001	−	0.887400i	$-0.652509\pi$
$257$	−14.7808	−0.922002	−0.461001	+	0.887400i	$0.652509\pi$
$258$	0	0
$259$	8.48699	0.527356
$260$	0	0
$261$	1.84259	0.114053
$262$	0	0
$263$	−6.68659	−0.412313	−0.206156	−	0.978519i	$-0.566096\pi$
$263$	−6.68659	−0.412313	−0.206156	+	0.978519i	$0.566096\pi$
$264$	0	0
$265$	−20.2637	−1.24479
$266$	0	0
$267$	0.665815	0.0407472
$268$	0	0
$269$	18.8193	1.14743	0.573715	−	0.819055i	$-0.305502\pi$
$269$	18.8193	1.14743	0.573715	+	0.819055i	$0.305502\pi$
$270$	0	0
$271$	−14.5928	−0.886448	−0.443224	−	0.896411i	$-0.646165\pi$
$271$	−14.5928	−0.886448	−0.443224	+	0.896411i	$0.646165\pi$
$272$	0	0
$273$	−10.4356	−0.631590
$274$	0	0
$275$	−3.40372	−0.205252
$276$	0	0
$277$	27.8797	1.67513	0.837563	−	0.546340i	$-0.183979\pi$
$277$	27.8797	1.67513	0.837563	+	0.546340i	$0.183979\pi$
$278$	0	0
$279$	−21.8877	−1.31038
$280$	0	0
$281$	−20.7462	−1.23762	−0.618808	−	0.785543i	$-0.712384\pi$
$281$	−20.7462	−1.23762	−0.618808	+	0.785543i	$0.712384\pi$
$282$	0	0
$283$	−17.7038	−1.05238	−0.526191	−	0.850366i	$-0.676380\pi$
$283$	−17.7038	−1.05238	−0.526191	+	0.850366i	$0.676380\pi$
$284$	0	0
$285$	−8.73822	−0.517607
$286$	0	0
$287$	−5.51756	−0.325691
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	7.75005	0.454316
$292$	0	0
$293$	−11.6136	−0.678474	−0.339237	−	0.940701i	$-0.610169\pi$
$293$	−11.6136	−0.678474	−0.339237	+	0.940701i	$0.610169\pi$
$294$	0	0
$295$	2.45345	0.142846
$296$	0	0
$297$	12.2765	0.712357
$298$	0	0
$299$	18.3221	1.05959
$300$	0	0
$301$	−31.0531	−1.78987
$302$	0	0
$303$	7.73046	0.444103
$304$	0	0
$305$	0.746871	0.0427657
$306$	0	0
$307$	19.2632	1.09941	0.549705	−	0.835359i	$-0.314740\pi$
$307$	19.2632	1.09941	0.549705	+	0.835359i	$0.314740\pi$
$308$	0	0
$309$	5.66161	0.322078
$310$	0	0
$311$	−33.7242	−1.91232	−0.956162	−	0.292837i	$-0.905401\pi$
$311$	−33.7242	−1.91232	−0.956162	+	0.292837i	$0.905401\pi$
$312$	0	0
$313$	25.3089	1.43055	0.715273	−	0.698845i	$-0.246302\pi$
$313$	25.3089	1.43055	0.715273	+	0.698845i	$0.246302\pi$
$314$	0	0
$315$	−19.5272	−1.10024
$316$	0	0
$317$	−17.8815	−1.00433	−0.502164	−	0.864772i	$-0.667463\pi$
$317$	−17.8815	−1.00433	−0.502164	+	0.864772i	$0.667463\pi$
$318$	0	0
$319$	2.40020	0.134385
$320$	0	0
$321$	6.52439	0.364156
$322$	0	0
$323$	5.38873	0.299837
$324$	0	0
$325$	−5.18358	−0.287533
$326$	0	0
$327$	−7.35354	−0.406652
$328$	0	0
$329$	−29.6468	−1.63448
$330$	0	0
$331$	6.11436	0.336076	0.168038	−	0.985781i	$-0.446257\pi$
$331$	6.11436	0.336076	0.168038	+	0.985781i	$0.446257\pi$
$332$	0	0
$333$	−7.00557	−0.383903
$334$	0	0
$335$	5.32029	0.290679
$336$	0	0
$337$	0.478656	0.0260741	0.0130370	−	0.999915i	$-0.495850\pi$
$337$	0.478656	0.0260741	0.0130370	+	0.999915i	$0.495850\pi$
$338$	0	0
$339$	5.89498	0.320172
$340$	0	0
$341$	−28.5115	−1.54398
$342$	0	0
$343$	−13.5319	−0.730656
$344$	0	0
$345$	−5.84304	−0.314579
$346$	0	0
$347$	2.47317	0.132767	0.0663834	−	0.997794i	$-0.478854\pi$
$347$	2.47317	0.132767	0.0663834	+	0.997794i	$0.478854\pi$
$348$	0	0
$349$	11.5110	0.616168	0.308084	−	0.951359i	$-0.400312\pi$
$349$	11.5110	0.616168	0.308084	+	0.951359i	$0.400312\pi$
$350$	0	0
$351$	18.6962	0.997927
$352$	0	0
$353$	0.0790507	0.00420745	0.00210372	−	0.999998i	$-0.499330\pi$
$353$	0.0790507	0.00420745	0.00210372	+	0.999998i	$0.499330\pi$
$354$	0	0
$355$	−37.7858	−2.00546
$356$	0	0
$357$	−2.05232	−0.108620
$358$	0	0
$359$	−16.6318	−0.877795	−0.438897	−	0.898537i	$-0.644631\pi$
$359$	−16.6318	−0.877795	−0.438897	+	0.898537i	$0.644631\pi$
$360$	0	0
$361$	10.0384	0.528337
$362$	0	0
$363$	0.0977180	0.00512886
$364$	0	0
$365$	−40.1239	−2.10018
$366$	0	0
$367$	−31.0009	−1.61823	−0.809116	−	0.587649i	$-0.800054\pi$
$367$	−31.0009	−1.61823	−0.809116	+	0.587649i	$0.800054\pi$
$368$	0	0
$369$	4.55446	0.237096
$370$	0	0
$371$	−25.6465	−1.33150
$372$	0	0
$373$	8.28208	0.428830	0.214415	−	0.976743i	$-0.431216\pi$
$373$	8.28208	0.428830	0.214415	+	0.976743i	$0.431216\pi$
$374$	0	0
$375$	−6.45478	−0.333323
$376$	0	0
$377$	3.65530	0.188258
$378$	0	0
$379$	−35.2827	−1.81235	−0.906176	−	0.422900i	$-0.861012\pi$
$379$	−35.2827	−1.81235	−0.906176	+	0.422900i	$0.861012\pi$
$380$	0	0
$381$	−12.7525	−0.653332
$382$	0	0
$383$	−31.5494	−1.61210	−0.806049	−	0.591849i	$-0.798398\pi$
$383$	−31.5494	−1.61210	−0.806049	+	0.591849i	$0.798398\pi$
$384$	0	0
$385$	−25.4367	−1.29637
$386$	0	0
$387$	25.6328	1.30299
$388$	0	0
$389$	14.0519	0.712459	0.356230	−	0.934398i	$-0.384062\pi$
$389$	14.0519	0.712459	0.356230	+	0.934398i	$0.384062\pi$
$390$	0	0
$391$	3.60332	0.182228
$392$	0	0
$393$	11.2395	0.566956
$394$	0	0
$395$	36.0748	1.81512
$396$	0	0
$397$	3.60826	0.181094	0.0905468	−	0.995892i	$-0.471139\pi$
$397$	3.60826	0.181094	0.0905468	+	0.995892i	$0.471139\pi$
$398$	0	0
$399$	−11.0594	−0.553663
$400$	0	0
$401$	15.7980	0.788917	0.394458	−	0.918914i	$-0.370932\pi$
$401$	15.7980	0.788917	0.394458	+	0.918914i	$0.370932\pi$
$402$	0	0
$403$	−43.4206	−2.16293
$404$	0	0
$405$	12.9035	0.641179
$406$	0	0
$407$	−9.12562	−0.452340
$408$	0	0
$409$	1.36757	0.0676221	0.0338111	−	0.999428i	$-0.489236\pi$
$409$	1.36757	0.0676221	0.0338111	+	0.999428i	$0.489236\pi$
$410$	0	0
$411$	3.62833	0.178972
$412$	0	0
$413$	3.10518	0.152796
$414$	0	0
$415$	22.7010	1.11435
$416$	0	0
$417$	15.0884	0.738880
$418$	0	0
$419$	14.7484	0.720506	0.360253	−	0.932855i	$-0.382690\pi$
$419$	14.7484	0.720506	0.360253	+	0.932855i	$0.382690\pi$
$420$	0	0
$421$	−11.9744	−0.583596	−0.291798	−	0.956480i	$-0.594253\pi$
$421$	−11.9744	−0.583596	−0.291798	+	0.956480i	$0.594253\pi$
$422$	0	0
$423$	24.4719	1.18986
$424$	0	0
$425$	−1.01943	−0.0494497
$426$	0	0
$427$	0.945266	0.0457446
$428$	0	0
$429$	11.2208	0.541748
$430$	0	0
$431$	−8.85550	−0.426555	−0.213277	−	0.976992i	$-0.568414\pi$
$431$	−8.85550	−0.426555	−0.213277	+	0.976992i	$0.568414\pi$
$432$	0	0
$433$	28.4148	1.36553	0.682764	−	0.730639i	$-0.260778\pi$
$433$	28.4148	1.36553	0.682764	+	0.730639i	$0.260778\pi$
$434$	0	0
$435$	−1.16570	−0.0558912
$436$	0	0
$437$	19.4173	0.928856
$438$	0	0
$439$	−27.7491	−1.32439	−0.662196	−	0.749331i	$-0.730375\pi$
$439$	−27.7491	−1.32439	−0.662196	+	0.749331i	$0.730375\pi$
$440$	0	0
$441$	−6.77223	−0.322487
$442$	0	0
$443$	2.18906	0.104005	0.0520026	−	0.998647i	$-0.483440\pi$
$443$	2.18906	0.104005	0.0520026	+	0.998647i	$0.483440\pi$
$444$	0	0
$445$	2.47157	0.117164
$446$	0	0
$447$	6.93290	0.327915
$448$	0	0
$449$	38.0122	1.79391	0.896953	−	0.442126i	$-0.145775\pi$
$449$	38.0122	1.79391	0.896953	+	0.442126i	$0.145775\pi$
$450$	0	0
$451$	5.93275	0.279362
$452$	0	0
$453$	2.87794	0.135217
$454$	0	0
$455$	−38.7379	−1.81606
$456$	0	0
$457$	18.1792	0.850388	0.425194	−	0.905102i	$-0.360206\pi$
$457$	18.1792	0.850388	0.425194	+	0.905102i	$0.360206\pi$
$458$	0	0
$459$	3.67689	0.171622
$460$	0	0
$461$	32.6942	1.52272	0.761360	−	0.648330i	$-0.224532\pi$
$461$	32.6942	1.52272	0.761360	+	0.648330i	$0.224532\pi$
$462$	0	0
$463$	13.1884	0.612918	0.306459	−	0.951884i	$-0.400856\pi$
$463$	13.1884	0.612918	0.306459	+	0.951884i	$0.400856\pi$
$464$	0	0
$465$	13.8472	0.642146
$466$	0	0
$467$	−17.9191	−0.829197	−0.414598	−	0.910005i	$-0.636078\pi$
$467$	−17.9191	−0.829197	−0.414598	+	0.910005i	$0.636078\pi$
$468$	0	0
$469$	6.73355	0.310927
$470$	0	0
$471$	12.0338	0.554488
$472$	0	0
$473$	33.3898	1.53527
$474$	0	0
$475$	−5.49345	−0.252057
$476$	0	0
$477$	21.1698	0.969300
$478$	0	0
$479$	−1.73745	−0.0793863	−0.0396932	−	0.999212i	$-0.512638\pi$
$479$	−1.73745	−0.0793863	−0.0396932	+	0.999212i	$0.512638\pi$
$480$	0	0
$481$	−13.8976	−0.633674
$482$	0	0
$483$	−7.39517	−0.336492
$484$	0	0
$485$	28.7689	1.30633
$486$	0	0
$487$	2.90227	0.131515	0.0657573	−	0.997836i	$-0.479054\pi$
$487$	2.90227	0.131515	0.0657573	+	0.997836i	$0.479054\pi$
$488$	0	0
$489$	−8.68468	−0.392735
$490$	0	0
$491$	−3.23157	−0.145839	−0.0729193	−	0.997338i	$-0.523232\pi$
$491$	−3.23157	−0.145839	−0.0729193	+	0.997338i	$0.523232\pi$
$492$	0	0
$493$	0.718872	0.0323764
$494$	0	0
$495$	20.9966	0.943729
$496$	0	0
$497$	−47.8231	−2.14516
$498$	0	0
$499$	−36.2953	−1.62480	−0.812400	−	0.583100i	$-0.801839\pi$
$499$	−36.2953	−1.62480	−0.812400	+	0.583100i	$0.801839\pi$
$500$	0	0
$501$	−10.7417	−0.479905
$502$	0	0
$503$	9.92907	0.442715	0.221358	−	0.975193i	$-0.428951\pi$
$503$	9.92907	0.442715	0.221358	+	0.975193i	$0.428951\pi$
$504$	0	0
$505$	28.6962	1.27696
$506$	0	0
$507$	8.49627	0.377332
$508$	0	0
$509$	−42.3483	−1.87705	−0.938527	−	0.345206i	$-0.887809\pi$
$509$	−42.3483	−1.87705	−0.938527	+	0.345206i	$0.887809\pi$
$510$	0	0
$511$	−50.7822	−2.24647
$512$	0	0
$513$	19.8138	0.874799
$514$	0	0
$515$	21.0164	0.926094
$516$	0	0
$517$	31.8776	1.40198
$518$	0	0
$519$	−6.55067	−0.287543
$520$	0	0
$521$	−20.6643	−0.905321	−0.452660	−	0.891683i	$-0.649525\pi$
$521$	−20.6643	−0.905321	−0.452660	+	0.891683i	$0.649525\pi$
$522$	0	0
$523$	−17.7628	−0.776711	−0.388356	−	0.921510i	$-0.626957\pi$
$523$	−17.7628	−0.776711	−0.388356	+	0.921510i	$0.626957\pi$
$524$	0	0
$525$	2.09220	0.0913112
$526$	0	0
$527$	−8.53934	−0.371979
$528$	0	0
$529$	−10.0161	−0.435482
$530$	0	0
$531$	−2.56317	−0.111232
$532$	0	0
$533$	9.03508	0.391353
$534$	0	0
$535$	24.2192	1.04709
$536$	0	0
$537$	−5.45050	−0.235207
$538$	0	0
$539$	−8.82167	−0.379976
$540$	0	0
$541$	−5.63706	−0.242356	−0.121178	−	0.992631i	$-0.538667\pi$
$541$	−5.63706	−0.242356	−0.121178	+	0.992631i	$0.538667\pi$
$542$	0	0
$543$	3.26423	0.140081
$544$	0	0
$545$	−27.2970	−1.16928
$546$	0	0
$547$	0.384934	0.0164586	0.00822930	−	0.999966i	$-0.497381\pi$
$547$	0.384934	0.0164586	0.00822930	+	0.999966i	$0.497381\pi$
$548$	0	0
$549$	−0.780269	−0.0333011
$550$	0	0
$551$	3.87381	0.165030
$552$	0	0
$553$	45.6575	1.94156
$554$	0	0
$555$	4.43204	0.188129
$556$	0	0
$557$	11.5820	0.490743	0.245372	−	0.969429i	$-0.421090\pi$
$557$	11.5820	0.490743	0.245372	+	0.969429i	$0.421090\pi$
$558$	0	0
$559$	50.8499	2.15072
$560$	0	0
$561$	2.20676	0.0931693
$562$	0	0
$563$	43.6941	1.84149	0.920743	−	0.390170i	$-0.127584\pi$
$563$	43.6941	1.84149	0.920743	+	0.390170i	$0.127584\pi$
$564$	0	0
$565$	21.8827	0.920614
$566$	0	0
$567$	16.3311	0.685842
$568$	0	0
$569$	−12.6480	−0.530233	−0.265116	−	0.964216i	$-0.585410\pi$
$569$	−12.6480	−0.530233	−0.265116	+	0.964216i	$0.585410\pi$
$570$	0	0
$571$	−11.7061	−0.489887	−0.244943	−	0.969537i	$-0.578769\pi$
$571$	−11.7061	−0.489887	−0.244943	+	0.969537i	$0.578769\pi$
$572$	0	0
$573$	−5.50426	−0.229944
$574$	0	0
$575$	−3.67334	−0.153189
$576$	0	0
$577$	17.1630	0.714507	0.357253	−	0.934008i	$-0.383713\pi$
$577$	17.1630	0.714507	0.357253	+	0.934008i	$0.383713\pi$
$578$	0	0
$579$	−14.8609	−0.617598
$580$	0	0
$581$	28.7312	1.19197
$582$	0	0
$583$	27.5763	1.14209
$584$	0	0
$585$	31.9761	1.32205
$586$	0	0
$587$	−1.03040	−0.0425292	−0.0212646	−	0.999774i	$-0.506769\pi$
$587$	−1.03040	−0.0425292	−0.0212646	+	0.999774i	$0.506769\pi$
$588$	0	0
$589$	−46.0162	−1.89606
$590$	0	0
$591$	7.62220	0.313535
$592$	0	0
$593$	−12.9978	−0.533754	−0.266877	−	0.963731i	$-0.585992\pi$
$593$	−12.9978	−0.533754	−0.266877	+	0.963731i	$0.585992\pi$
$594$	0	0
$595$	−7.61841	−0.312324
$596$	0	0
$597$	−14.0469	−0.574902
$598$	0	0
$599$	−23.3677	−0.954779	−0.477390	−	0.878692i	$-0.658417\pi$
$599$	−23.3677	−0.954779	−0.477390	+	0.878692i	$0.658417\pi$
$600$	0	0
$601$	14.1721	0.578091	0.289045	−	0.957315i	$-0.406662\pi$
$601$	14.1721	0.578091	0.289045	+	0.957315i	$0.406662\pi$
$602$	0	0
$603$	−5.55820	−0.226347
$604$	0	0
$605$	0.362738	0.0147474
$606$	0	0
$607$	20.0134	0.812320	0.406160	−	0.913802i	$-0.366868\pi$
$607$	20.0134	0.812320	0.406160	+	0.913802i	$0.366868\pi$
$608$	0	0
$609$	−1.47536	−0.0597845
$610$	0	0
$611$	48.5470	1.96400
$612$	0	0
$613$	17.1274	0.691770	0.345885	−	0.938277i	$-0.387579\pi$
$613$	17.1274	0.691770	0.345885	+	0.938277i	$0.387579\pi$
$614$	0	0
$615$	−2.88135	−0.116187
$616$	0	0
$617$	−17.9852	−0.724056	−0.362028	−	0.932167i	$-0.617915\pi$
$617$	−17.9852	−0.724056	−0.362028	+	0.932167i	$0.617915\pi$
$618$	0	0
$619$	−28.9080	−1.16191	−0.580955	−	0.813935i	$-0.697321\pi$
$619$	−28.9080	−1.16191	−0.580955	+	0.813935i	$0.697321\pi$
$620$	0	0
$621$	13.2490	0.531664
$622$	0	0
$623$	3.12811	0.125325
$624$	0	0
$625$	−29.0579	−1.16232
$626$	0	0
$627$	11.8916	0.474905
$628$	0	0
$629$	−2.73317	−0.108979
$630$	0	0
$631$	15.7580	0.627317	0.313659	−	0.949536i	$-0.398445\pi$
$631$	15.7580	0.627317	0.313659	+	0.949536i	$0.398445\pi$
$632$	0	0
$633$	11.4131	0.453631
$634$	0	0
$635$	−47.3386	−1.87858
$636$	0	0
$637$	−13.4347	−0.532301
$638$	0	0
$639$	39.4755	1.56163
$640$	0	0
$641$	−2.90815	−0.114865	−0.0574325	−	0.998349i	$-0.518291\pi$
$641$	−2.90815	−0.114865	−0.0574325	+	0.998349i	$0.518291\pi$
$642$	0	0
$643$	−1.20261	−0.0474262	−0.0237131	−	0.999719i	$-0.507549\pi$
$643$	−1.20261	−0.0474262	−0.0237131	+	0.999719i	$0.507549\pi$
$644$	0	0
$645$	−16.2164	−0.638521
$646$	0	0
$647$	40.5464	1.59404	0.797021	−	0.603951i	$-0.206408\pi$
$647$	40.5464	1.59404	0.797021	+	0.603951i	$0.206408\pi$
$648$	0	0
$649$	−3.33884	−0.131061
$650$	0	0
$651$	17.5255	0.686877
$652$	0	0
$653$	−10.4233	−0.407894	−0.203947	−	0.978982i	$-0.565377\pi$
$653$	−10.4233	−0.407894	−0.203947	+	0.978982i	$0.565377\pi$
$654$	0	0
$655$	41.7220	1.63021
$656$	0	0
$657$	41.9181	1.63538
$658$	0	0
$659$	0.825957	0.0321747	0.0160874	−	0.999871i	$-0.494879\pi$
$659$	0.825957	0.0321747	0.0160874	+	0.999871i	$0.494879\pi$
$660$	0	0
$661$	8.75766	0.340634	0.170317	−	0.985389i	$-0.445521\pi$
$661$	8.75766	0.340634	0.170317	+	0.985389i	$0.445521\pi$
$662$	0	0
$663$	3.36070	0.130519
$664$	0	0
$665$	−41.0536	−1.59199
$666$	0	0
$667$	2.59033	0.100298
$668$	0	0
$669$	−2.79605	−0.108102
$670$	0	0
$671$	−1.01640	−0.0392375
$672$	0	0
$673$	−10.1027	−0.389431	−0.194715	−	0.980860i	$-0.562378\pi$
$673$	−10.1027	−0.389431	−0.194715	+	0.980860i	$0.562378\pi$
$674$	0	0
$675$	−3.74834	−0.144274
$676$	0	0
$677$	−24.5662	−0.944155	−0.472077	−	0.881557i	$-0.656496\pi$
$677$	−24.5662	−0.944155	−0.472077	+	0.881557i	$0.656496\pi$
$678$	0	0
$679$	36.4110	1.39733
$680$	0	0
$681$	−8.72933	−0.334509
$682$	0	0
$683$	−26.9497	−1.03120	−0.515600	−	0.856829i	$-0.672431\pi$
$683$	−26.9497	−1.03120	−0.515600	+	0.856829i	$0.672431\pi$
$684$	0	0
$685$	13.4687	0.514613
$686$	0	0
$687$	9.35823	0.357039
$688$	0	0
$689$	41.9964	1.59994
$690$	0	0
$691$	−4.96807	−0.188994	−0.0944972	−	0.995525i	$-0.530124\pi$
$691$	−4.96807	−0.188994	−0.0944972	+	0.995525i	$0.530124\pi$
$692$	0	0
$693$	26.5741	1.00947
$694$	0	0
$695$	56.0094	2.12456
$696$	0	0
$697$	1.77689	0.0673045
$698$	0	0
$699$	−1.40777	−0.0532468
$700$	0	0
$701$	−38.7570	−1.46383	−0.731915	−	0.681395i	$-0.761373\pi$
$701$	−38.7570	−1.46383	−0.731915	+	0.681395i	$0.761373\pi$
$702$	0	0
$703$	−14.7283	−0.555489
$704$	0	0
$705$	−15.4820	−0.583086
$706$	0	0
$707$	36.3190	1.36591
$708$	0	0
$709$	−49.7918	−1.86997	−0.934985	−	0.354688i	$-0.884587\pi$
$709$	−49.7918	−1.86997	−0.934985	+	0.354688i	$0.884587\pi$
$710$	0	0
$711$	−37.6879	−1.41341
$712$	0	0
$713$	−30.7700	−1.15234
$714$	0	0
$715$	41.6529	1.55773
$716$	0	0
$717$	12.1698	0.454490
$718$	0	0
$719$	−33.5082	−1.24965	−0.624823	−	0.780766i	$-0.714829\pi$
$719$	−33.5082	−1.24965	−0.624823	+	0.780766i	$0.714829\pi$
$720$	0	0
$721$	26.5991	0.990604
$722$	0	0
$723$	1.14507	0.0425856
$724$	0	0
$725$	−0.732842	−0.0272171
$726$	0	0
$727$	−20.5111	−0.760714	−0.380357	−	0.924840i	$-0.624199\pi$
$727$	−20.5111	−0.760714	−0.380357	+	0.924840i	$0.624199\pi$
$728$	0	0
$729$	−6.18993	−0.229257
$730$	0	0
$731$	10.0004	0.369879
$732$	0	0
$733$	39.7312	1.46750	0.733752	−	0.679418i	$-0.237768\pi$
$733$	39.7312	1.46750	0.733752	+	0.679418i	$0.237768\pi$
$734$	0	0
$735$	4.28442	0.158033
$736$	0	0
$737$	−7.24024	−0.266698
$738$	0	0
$739$	−3.29003	−0.121026	−0.0605128	−	0.998167i	$-0.519274\pi$
$739$	−3.29003	−0.121026	−0.0605128	+	0.998167i	$0.519274\pi$
$740$	0	0
$741$	18.1099	0.665285
$742$	0	0
$743$	−45.6042	−1.67306	−0.836529	−	0.547923i	$-0.815419\pi$
$743$	−45.6042	−1.67306	−0.836529	+	0.547923i	$0.815419\pi$
$744$	0	0
$745$	25.7356	0.942880
$746$	0	0
$747$	−23.7161	−0.867726
$748$	0	0
$749$	30.6526	1.12002
$750$	0	0
$751$	26.4517	0.965237	0.482618	−	0.875831i	$-0.339686\pi$
$751$	26.4517	0.965237	0.482618	+	0.875831i	$0.339686\pi$
$752$	0	0
$753$	−3.36176	−0.122509
$754$	0	0
$755$	10.6832	0.388801
$756$	0	0
$757$	−48.1468	−1.74992	−0.874962	−	0.484191i	$-0.839114\pi$
$757$	−48.1468	−1.74992	−0.874962	+	0.484191i	$0.839114\pi$
$758$	0	0
$759$	7.95164	0.288626
$760$	0	0
$761$	20.1841	0.731673	0.365836	−	0.930679i	$-0.380783\pi$
$761$	20.1841	0.731673	0.365836	+	0.930679i	$0.380783\pi$
$762$	0	0
$763$	−34.5481	−1.25073
$764$	0	0
$765$	6.28861	0.227365
$766$	0	0
$767$	−5.08477	−0.183601
$768$	0	0
$769$	10.1158	0.364784	0.182392	−	0.983226i	$-0.441616\pi$
$769$	10.1158	0.364784	0.182392	+	0.983226i	$0.441616\pi$
$770$	0	0
$771$	−9.76915	−0.351827
$772$	0	0
$773$	−23.0470	−0.828941	−0.414471	−	0.910063i	$-0.636033\pi$
$773$	−23.0470	−0.828941	−0.414471	+	0.910063i	$0.636033\pi$
$774$	0	0
$775$	8.70527	0.312703
$776$	0	0
$777$	5.60934	0.201234
$778$	0	0
$779$	9.57517	0.343066
$780$	0	0
$781$	51.4217	1.84001
$782$	0	0
$783$	2.64321	0.0944608
$784$	0	0
$785$	44.6706	1.59436
$786$	0	0
$787$	34.6384	1.23472	0.617362	−	0.786679i	$-0.288201\pi$
$787$	34.6384	1.23472	0.617362	+	0.786679i	$0.288201\pi$
$788$	0	0
$789$	−4.41940	−0.157335
$790$	0	0
$791$	27.6956	0.984742
$792$	0	0
$793$	−1.54789	−0.0549671
$794$	0	0
$795$	−13.3930	−0.475000
$796$	0	0
$797$	5.95250	0.210848	0.105424	−	0.994427i	$-0.466380\pi$
$797$	5.95250	0.210848	0.105424	+	0.994427i	$0.466380\pi$
$798$	0	0
$799$	9.54752	0.337767
$800$	0	0
$801$	−2.58209	−0.0912337
$802$	0	0
$803$	54.6035	1.92692
$804$	0	0
$805$	−27.4516	−0.967541
$806$	0	0
$807$	12.4383	0.437849
$808$	0	0
$809$	−51.5140	−1.81114	−0.905568	−	0.424201i	$-0.860555\pi$
$809$	−51.5140	−1.81114	−0.905568	+	0.424201i	$0.860555\pi$
$810$	0	0
$811$	−3.18270	−0.111760	−0.0558799	−	0.998438i	$-0.517796\pi$
$811$	−3.18270	−0.111760	−0.0558799	+	0.998438i	$0.517796\pi$
$812$	0	0
$813$	−9.64487	−0.338260
$814$	0	0
$815$	−32.2384	−1.12926
$816$	0	0
$817$	53.8896	1.88536
$818$	0	0
$819$	40.4702	1.41414
$820$	0	0
$821$	−19.2377	−0.671402	−0.335701	−	0.941969i	$-0.608973\pi$
$821$	−19.2377	−0.671402	−0.335701	+	0.941969i	$0.608973\pi$
$822$	0	0
$823$	−54.4984	−1.89969	−0.949847	−	0.312716i	$-0.898761\pi$
$823$	−54.4984	−1.89969	−0.949847	+	0.312716i	$0.898761\pi$
$824$	0	0
$825$	−2.24964	−0.0783223
$826$	0	0
$827$	−24.7763	−0.861557	−0.430778	−	0.902458i	$-0.641761\pi$
$827$	−24.7763	−0.861557	−0.430778	+	0.902458i	$0.641761\pi$
$828$	0	0
$829$	52.2277	1.81394	0.906971	−	0.421192i	$-0.138388\pi$
$829$	52.2277	1.81394	0.906971	+	0.421192i	$0.138388\pi$
$830$	0	0
$831$	18.4266	0.639213
$832$	0	0
$833$	−2.64214	−0.0915446
$834$	0	0
$835$	−39.8743	−1.37991
$836$	0	0
$837$	−31.3982	−1.08528
$838$	0	0
$839$	25.0058	0.863296	0.431648	−	0.902042i	$-0.357932\pi$
$839$	25.0058	0.863296	0.431648	+	0.902042i	$0.357932\pi$
$840$	0	0
$841$	−28.4832	−0.982180
$842$	0	0
$843$	−13.7119	−0.472263
$844$	0	0
$845$	31.5390	1.08497
$846$	0	0
$847$	0.459095	0.0157747
$848$	0	0
$849$	−11.7011	−0.401580
$850$	0	0
$851$	−9.84849	−0.337602
$852$	0	0
$853$	26.9546	0.922908	0.461454	−	0.887164i	$-0.347328\pi$
$853$	26.9546	0.922908	0.461454	+	0.887164i	$0.347328\pi$
$854$	0	0
$855$	33.8876	1.15893
$856$	0	0
$857$	−32.0787	−1.09579	−0.547894	−	0.836548i	$-0.684570\pi$
$857$	−32.0787	−1.09579	−0.547894	+	0.836548i	$0.684570\pi$
$858$	0	0
$859$	−28.8505	−0.984366	−0.492183	−	0.870492i	$-0.663801\pi$
$859$	−28.8505	−0.984366	−0.492183	+	0.870492i	$0.663801\pi$
$860$	0	0
$861$	−3.64675	−0.124281
$862$	0	0
$863$	7.39512	0.251733	0.125866	−	0.992047i	$-0.459829\pi$
$863$	7.39512	0.251733	0.125866	+	0.992047i	$0.459829\pi$
$864$	0	0
$865$	−24.3167	−0.826794
$866$	0	0
$867$	0.660935	0.0224465
$868$	0	0
$869$	−49.0932	−1.66537
$870$	0	0
$871$	−11.0263	−0.373612
$872$	0	0
$873$	−30.0554	−1.01722
$874$	0	0
$875$	−30.3256	−1.02519
$876$	0	0
$877$	−8.74842	−0.295413	−0.147707	−	0.989031i	$-0.547189\pi$
$877$	−8.74842	−0.295413	−0.147707	+	0.989031i	$0.547189\pi$
$878$	0	0
$879$	−7.67584	−0.258900
$880$	0	0
$881$	−36.6843	−1.23592	−0.617962	−	0.786208i	$-0.712041\pi$
$881$	−36.6843	−1.23592	−0.617962	+	0.786208i	$0.712041\pi$
$882$	0	0
$883$	12.8854	0.433629	0.216815	−	0.976213i	$-0.430433\pi$
$883$	12.8854	0.433629	0.216815	+	0.976213i	$0.430433\pi$
$884$	0	0
$885$	1.62157	0.0545086
$886$	0	0
$887$	−21.6305	−0.726282	−0.363141	−	0.931734i	$-0.618296\pi$
$887$	−21.6305	−0.726282	−0.363141	+	0.931734i	$0.618296\pi$
$888$	0	0
$889$	−59.9135	−2.00943
$890$	0	0
$891$	−17.5600	−0.588282
$892$	0	0
$893$	51.4490	1.72168
$894$	0	0
$895$	−20.2328	−0.676307
$896$	0	0
$897$	12.1097	0.404331
$898$	0	0
$899$	−6.13869	−0.204737
$900$	0	0
$901$	8.25925	0.275156
$902$	0	0
$903$	−20.5241	−0.682999
$904$	0	0
$905$	12.1171	0.402787
$906$	0	0
$907$	26.5017	0.879975	0.439988	−	0.898004i	$-0.354983\pi$
$907$	26.5017	0.879975	0.439988	+	0.898004i	$0.354983\pi$
$908$	0	0
$909$	−29.9794	−0.994355
$910$	0	0
$911$	15.8224	0.524220	0.262110	−	0.965038i	$-0.415582\pi$
$911$	15.8224	0.524220	0.262110	+	0.965038i	$0.415582\pi$
$912$	0	0
$913$	−30.8931	−1.02241
$914$	0	0
$915$	0.493633	0.0163190
$916$	0	0
$917$	52.8049	1.74377
$918$	0	0
$919$	16.0356	0.528966	0.264483	−	0.964390i	$-0.414799\pi$
$919$	16.0356	0.528966	0.264483	+	0.964390i	$0.414799\pi$
$920$	0	0
$921$	12.7317	0.419525
$922$	0	0
$923$	78.3109	2.57764
$924$	0	0
$925$	2.78628	0.0916124
$926$	0	0
$927$	−21.9562	−0.721137
$928$	0	0
$929$	29.6096	0.971461	0.485730	−	0.874109i	$-0.338554\pi$
$929$	29.6096	0.971461	0.485730	+	0.874109i	$0.338554\pi$
$930$	0	0
$931$	−14.2378	−0.466624
$932$	0	0
$933$	−22.2895	−0.729726
$934$	0	0
$935$	8.19169	0.267897
$936$	0	0
$937$	52.4643	1.71393	0.856967	−	0.515371i	$-0.172346\pi$
$937$	52.4643	1.71393	0.856967	+	0.515371i	$0.172346\pi$
$938$	0	0
$939$	16.7276	0.545883
$940$	0	0
$941$	−27.4776	−0.895744	−0.447872	−	0.894098i	$-0.647818\pi$
$941$	−27.4776	−0.895744	−0.447872	+	0.894098i	$0.647818\pi$
$942$	0	0
$943$	6.40270	0.208501
$944$	0	0
$945$	−28.0121	−0.911232
$946$	0	0
$947$	32.6627	1.06139	0.530697	−	0.847561i	$-0.321930\pi$
$947$	32.6627	1.06139	0.530697	+	0.847561i	$0.321930\pi$
$948$	0	0
$949$	83.1566	2.69938
$950$	0	0
$951$	−11.8185	−0.383242
$952$	0	0
$953$	38.0028	1.23103	0.615515	−	0.788125i	$-0.288948\pi$
$953$	38.0028	1.23103	0.615515	+	0.788125i	$0.288948\pi$
$954$	0	0
$955$	−20.4323	−0.661175
$956$	0	0
$957$	1.58638	0.0512802
$958$	0	0
$959$	17.0465	0.550460
$960$	0	0
$961$	41.9203	1.35227
$962$	0	0
$963$	−25.3022	−0.815351
$964$	0	0
$965$	−55.1651	−1.77583
$966$	0	0
$967$	−53.6660	−1.72578	−0.862891	−	0.505390i	$-0.831349\pi$
$967$	−53.6660	−1.72578	−0.862891	+	0.505390i	$0.831349\pi$
$968$	0	0
$969$	3.56160	0.114415
$970$	0	0
$971$	20.6172	0.661638	0.330819	−	0.943694i	$-0.392675\pi$
$971$	20.6172	0.661638	0.330819	+	0.943694i	$0.392675\pi$
$972$	0	0
$973$	70.8876	2.27255
$974$	0	0
$975$	−3.42601	−0.109720
$976$	0	0
$977$	−17.5439	−0.561279	−0.280640	−	0.959813i	$-0.590547\pi$
$977$	−17.5439	−0.561279	−0.280640	+	0.959813i	$0.590547\pi$
$978$	0	0
$979$	−3.36349	−0.107498
$980$	0	0
$981$	28.5177	0.910500
$982$	0	0
$983$	−20.0456	−0.639354	−0.319677	−	0.947527i	$-0.603574\pi$
$983$	−20.0456	−0.639354	−0.319677	+	0.947527i	$0.603574\pi$
$984$	0	0
$985$	28.2943	0.901532
$986$	0	0
$987$	−19.5946	−0.623702
$988$	0	0
$989$	36.0348	1.14584
$990$	0	0
$991$	22.7829	0.723723	0.361862	−	0.932232i	$-0.382141\pi$
$991$	22.7829	0.723723	0.361862	+	0.932232i	$0.382141\pi$
$992$	0	0
$993$	4.04119	0.128243
$994$	0	0
$995$	−52.1435	−1.65306
$996$	0	0
$997$	11.4479	0.362558	0.181279	−	0.983432i	$-0.441976\pi$
$997$	11.4479	0.362558	0.181279	+	0.983432i	$0.441976\pi$
$998$	0	0
$999$	−10.0496	−0.317954

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.g.1.7	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.g.1.7	✓	12	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q+0.660935 q^{3} +2.45345 q^{5} +3.10518 q^{7} -2.56317 q^{9} +O(q^{10})\)	\(q+0.660935 q^{3} +2.45345 q^{5} +3.10518 q^{7} -2.56317 q^{9} -3.33884 q^{11} -5.08477 q^{13} +1.62157 q^{15} -1.00000 q^{17} -5.38873 q^{19} +2.05232 q^{21} -3.60332 q^{23} +1.01943 q^{25} -3.67689 q^{27} -0.718872 q^{29} +8.53934 q^{31} -2.20676 q^{33} +7.61841 q^{35} +2.73317 q^{37} -3.36070 q^{39} -1.77689 q^{41} -10.0004 q^{43} -6.28861 q^{45} -9.54752 q^{47} +2.64214 q^{49} -0.660935 q^{51} -8.25925 q^{53} -8.19169 q^{55} -3.56160 q^{57} +1.00000 q^{59} +0.304416 q^{61} -7.95909 q^{63} -12.4753 q^{65} +2.16849 q^{67} -2.38156 q^{69} -15.4011 q^{71} -16.3540 q^{73} +0.673778 q^{75} -10.3677 q^{77} +14.7037 q^{79} +5.25931 q^{81} +9.25266 q^{83} -2.45345 q^{85} -0.475128 q^{87} +1.00738 q^{89} -15.7891 q^{91} +5.64394 q^{93} -13.2210 q^{95} +11.7259 q^{97} +8.55800 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9	\( 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9} - 5 q^{11} + 9 q^{13} - 16 q^{15} - 12 q^{17} - 15 q^{19} + 5 q^{21} - 22 q^{23} + 5 q^{25} - 16 q^{27} - 17 q^{29} + 5 q^{31} - 33 q^{33} - 9 q^{35} - 2 q^{37} - 24 q^{39} - 18 q^{41} - 16 q^{43} + 30 q^{45} - 60 q^{47} + 10 q^{49} + 4 q^{51} - 24 q^{53} - q^{55} + 41 q^{57} + 12 q^{59} - 17 q^{61} - 23 q^{63} + 2 q^{65} - 22 q^{67} + 6 q^{69} - 10 q^{71} + 4 q^{73} - 62 q^{75} - 6 q^{77} + 48 q^{79} + 4 q^{81} - 20 q^{83} - 3 q^{85} + 13 q^{87} - 29 q^{89} - 28 q^{91} - 31 q^{93} - 48 q^{95} - 26 q^{97} - 3 q^{99}+O(q^{100}) \) 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9 - 5 * q^11 + 9 * q^13 - 16 * q^15 - 12 * q^17 - 15 * q^19 + 5 * q^21 - 22 * q^23 + 5 * q^25 - 16 * q^27 - 17 * q^29 + 5 * q^31 - 33 * q^33 - 9 * q^35 - 2 * q^37 - 24 * q^39 - 18 * q^41 - 16 * q^43 + 30 * q^45 - 60 * q^47 + 10 * q^49 + 4 * q^51 - 24 * q^53 - q^55 + 41 * q^57 + 12 * q^59 - 17 * q^61 - 23 * q^63 + 2 * q^65 - 22 * q^67 + 6 * q^69 - 10 * q^71 + 4 * q^73 - 62 * q^75 - 6 * q^77 + 48 * q^79 + 4 * q^81 - 20 * q^83 - 3 * q^85 + 13 * q^87 - 29 * q^89 - 28 * q^91 - 31 * q^93 - 48 * q^95 - 26 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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