LMFDB - Embedded newform 4012.2.a.g.1.6

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			1.6
Root			$0.859886$ 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)

$f(q)$	$=$	$q-0.859886 q^{3} +2.49803 q^{5} -3.53399 q^{7} -2.26060 q^{9} +O(q^{10})$	$q-0.859886 q^{3} +2.49803 q^{5} -3.53399 q^{7} -2.26060 q^{9} +1.54144 q^{11} +3.17355 q^{13} -2.14802 q^{15} -1.00000 q^{17} +0.523938 q^{19} +3.03883 q^{21} -5.23538 q^{23} +1.24016 q^{25} +4.52351 q^{27} +4.96421 q^{29} +3.14090 q^{31} -1.32546 q^{33} -8.82803 q^{35} -7.80477 q^{37} -2.72889 q^{39} -1.58833 q^{41} -1.96140 q^{43} -5.64704 q^{45} +2.57853 q^{47} +5.48912 q^{49} +0.859886 q^{51} +2.58777 q^{53} +3.85056 q^{55} -0.450527 q^{57} +1.00000 q^{59} +6.54735 q^{61} +7.98893 q^{63} +7.92762 q^{65} -5.68694 q^{67} +4.50183 q^{69} -13.9886 q^{71} -1.37301 q^{73} -1.06639 q^{75} -5.44743 q^{77} -6.64369 q^{79} +2.89208 q^{81} -16.8533 q^{83} -2.49803 q^{85} -4.26866 q^{87} +2.73484 q^{89} -11.2153 q^{91} -2.70082 q^{93} +1.30881 q^{95} +3.59905 q^{97} -3.48457 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.859886	−0.496455	−0.248228	−	0.968702i	$-0.579848\pi$
$3$	−0.859886	−0.496455	−0.248228	+	0.968702i	$0.579848\pi$
$4$	0	0
$5$	2.49803	1.11715	0.558577	−	0.829453i	$-0.311348\pi$
$5$	2.49803	1.11715	0.558577	+	0.829453i	$0.311348\pi$
$6$	0	0
$7$	−3.53399	−1.33572	−0.667862	−	0.744285i	$-0.732790\pi$
$7$	−3.53399	−1.33572	−0.667862	+	0.744285i	$0.732790\pi$
$8$	0	0
$9$	−2.26060	−0.753532
$10$	0	0
$11$	1.54144	0.464761	0.232380	−	0.972625i	$-0.425349\pi$
$11$	1.54144	0.464761	0.232380	+	0.972625i	$0.425349\pi$
$12$	0	0
$13$	3.17355	0.880184	0.440092	−	0.897953i	$-0.354946\pi$
$13$	3.17355	0.880184	0.440092	+	0.897953i	$0.354946\pi$
$14$	0	0
$15$	−2.14802	−0.554617
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	0.523938	0.120200	0.0600998	−	0.998192i	$-0.480858\pi$
$19$	0.523938	0.120200	0.0600998	+	0.998192i	$0.480858\pi$
$20$	0	0
$21$	3.03883	0.663128
$22$	0	0
$23$	−5.23538	−1.09165	−0.545826	−	0.837898i	$-0.683784\pi$
$23$	−5.23538	−1.09165	−0.545826	+	0.837898i	$0.683784\pi$
$24$	0	0
$25$	1.24016	0.248031
$26$	0	0
$27$	4.52351	0.870550
$28$	0	0
$29$	4.96421	0.921831	0.460916	−	0.887444i	$-0.347521\pi$
$29$	4.96421	0.921831	0.460916	+	0.887444i	$0.347521\pi$
$30$	0	0
$31$	3.14090	0.564123	0.282061	−	0.959396i	$-0.408982\pi$
$31$	3.14090	0.564123	0.282061	+	0.959396i	$0.408982\pi$
$32$	0	0
$33$	−1.32546	−0.230733
$34$	0	0
$35$	−8.82803	−1.49221
$36$	0	0
$37$	−7.80477	−1.28310	−0.641548	−	0.767083i	$-0.721708\pi$
$37$	−7.80477	−1.28310	−0.641548	+	0.767083i	$0.721708\pi$
$38$	0	0
$39$	−2.72889	−0.436972
$40$	0	0
$41$	−1.58833	−0.248055	−0.124028	−	0.992279i	$-0.539581\pi$
$41$	−1.58833	−0.248055	−0.124028	+	0.992279i	$0.539581\pi$
$42$	0	0
$43$	−1.96140	−0.299111	−0.149556	−	0.988753i	$-0.547784\pi$
$43$	−1.96140	−0.299111	−0.149556	+	0.988753i	$0.547784\pi$
$44$	0	0
$45$	−5.64704	−0.841811
$46$	0	0
$47$	2.57853	0.376117	0.188059	−	0.982158i	$-0.439780\pi$
$47$	2.57853	0.376117	0.188059	+	0.982158i	$0.439780\pi$
$48$	0	0
$49$	5.48912	0.784160
$50$	0	0
$51$	0.859886	0.120408
$52$	0	0
$53$	2.58777	0.355457	0.177729	−	0.984080i	$-0.443125\pi$
$53$	2.58777	0.355457	0.177729	+	0.984080i	$0.443125\pi$
$54$	0	0
$55$	3.85056	0.519209
$56$	0	0
$57$	−0.450527	−0.0596737
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	6.54735	0.838302	0.419151	−	0.907917i	$-0.362328\pi$
$61$	6.54735	0.838302	0.419151	+	0.907917i	$0.362328\pi$
$62$	0	0
$63$	7.98893	1.00651
$64$	0	0
$65$	7.92762	0.983300
$66$	0	0
$67$	−5.68694	−0.694770	−0.347385	−	0.937723i	$-0.612930\pi$
$67$	−5.68694	−0.694770	−0.347385	+	0.937723i	$0.612930\pi$
$68$	0	0
$69$	4.50183	0.541957
$70$	0	0
$71$	−13.9886	−1.66014	−0.830072	−	0.557656i	$-0.811701\pi$
$71$	−13.9886	−1.66014	−0.830072	+	0.557656i	$0.811701\pi$
$72$	0	0
$73$	−1.37301	−0.160699	−0.0803494	−	0.996767i	$-0.525604\pi$
$73$	−1.37301	−0.160699	−0.0803494	+	0.996767i	$0.525604\pi$
$74$	0	0
$75$	−1.06639	−0.123137
$76$	0	0
$77$	−5.44743	−0.620792
$78$	0	0
$79$	−6.64369	−0.747474	−0.373737	−	0.927535i	$-0.621924\pi$
$79$	−6.64369	−0.747474	−0.373737	+	0.927535i	$0.621924\pi$
$80$	0	0
$81$	2.89208	0.321343
$82$	0	0
$83$	−16.8533	−1.84990	−0.924948	−	0.380094i	$-0.875891\pi$
$83$	−16.8533	−1.84990	−0.924948	+	0.380094i	$0.875891\pi$
$84$	0	0
$85$	−2.49803	−0.270949
$86$	0	0
$87$	−4.26866	−0.457648
$88$	0	0
$89$	2.73484	0.289893	0.144946	−	0.989439i	$-0.453699\pi$
$89$	2.73484	0.289893	0.144946	+	0.989439i	$0.453699\pi$
$90$	0	0
$91$	−11.2153	−1.17568
$92$	0	0
$93$	−2.70082	−0.280062
$94$	0	0
$95$	1.30881	0.134281
$96$	0	0
$97$	3.59905	0.365428	0.182714	−	0.983166i	$-0.441512\pi$
$97$	3.59905	0.365428	0.182714	+	0.983166i	$0.441512\pi$
$98$	0	0
$99$	−3.48457	−0.350212
$100$	0	0
$101$	−13.0337	−1.29690	−0.648451	−	0.761256i	$-0.724583\pi$
$101$	−13.0337	−1.29690	−0.648451	+	0.761256i	$0.724583\pi$
$102$	0	0
$103$	−8.90735	−0.877667	−0.438834	−	0.898568i	$-0.644608\pi$
$103$	−8.90735	−0.877667	−0.438834	+	0.898568i	$0.644608\pi$
$104$	0	0
$105$	7.59110	0.740815
$106$	0	0
$107$	−9.92583	−0.959566	−0.479783	−	0.877387i	$-0.659285\pi$
$107$	−9.92583	−0.959566	−0.479783	+	0.877387i	$0.659285\pi$
$108$	0	0
$109$	1.21141	0.116032	0.0580161	−	0.998316i	$-0.481523\pi$
$109$	1.21141	0.116032	0.0580161	+	0.998316i	$0.481523\pi$
$110$	0	0
$111$	6.71121	0.637000
$112$	0	0
$113$	8.03970	0.756312	0.378156	−	0.925742i	$-0.376558\pi$
$113$	8.03970	0.756312	0.378156	+	0.925742i	$0.376558\pi$
$114$	0	0
$115$	−13.0781	−1.21954
$116$	0	0
$117$	−7.17411	−0.663247
$118$	0	0
$119$	3.53399	0.323961
$120$	0	0
$121$	−8.62397	−0.783997
$122$	0	0
$123$	1.36578	0.123148
$124$	0	0
$125$	−9.39220	−0.840064
$126$	0	0
$127$	−17.7870	−1.57834	−0.789171	−	0.614173i	$-0.789490\pi$
$127$	−17.7870	−1.57834	−0.789171	+	0.614173i	$0.789490\pi$
$128$	0	0
$129$	1.68658	0.148495
$130$	0	0
$131$	11.1752	0.976385	0.488192	−	0.872736i	$-0.337656\pi$
$131$	11.1752	0.976385	0.488192	+	0.872736i	$0.337656\pi$
$132$	0	0
$133$	−1.85159	−0.160554
$134$	0	0
$135$	11.2999	0.972538
$136$	0	0
$137$	5.64695	0.482451	0.241226	−	0.970469i	$-0.422451\pi$
$137$	5.64695	0.482451	0.241226	+	0.970469i	$0.422451\pi$
$138$	0	0
$139$	−4.13706	−0.350901	−0.175450	−	0.984488i	$-0.556138\pi$
$139$	−4.13706	−0.350901	−0.175450	+	0.984488i	$0.556138\pi$
$140$	0	0
$141$	−2.21724	−0.186726
$142$	0	0
$143$	4.89182	0.409075
$144$	0	0
$145$	12.4008	1.02983
$146$	0	0
$147$	−4.72001	−0.389300
$148$	0	0
$149$	13.7421	1.12580	0.562898	−	0.826527i	$-0.309687\pi$
$149$	13.7421	1.12580	0.562898	+	0.826527i	$0.309687\pi$
$150$	0	0
$151$	−0.425241	−0.0346056	−0.0173028	−	0.999850i	$-0.505508\pi$
$151$	−0.425241	−0.0346056	−0.0173028	+	0.999850i	$0.505508\pi$
$152$	0	0
$153$	2.26060	0.182758
$154$	0	0
$155$	7.84607	0.630212
$156$	0	0
$157$	2.00538	0.160047	0.0800234	−	0.996793i	$-0.474500\pi$
$157$	2.00538	0.160047	0.0800234	+	0.996793i	$0.474500\pi$
$158$	0	0
$159$	−2.22518	−0.176469
$160$	0	0
$161$	18.5018	1.45815
$162$	0	0
$163$	−9.86513	−0.772697	−0.386348	−	0.922353i	$-0.626264\pi$
$163$	−9.86513	−0.772697	−0.386348	+	0.922353i	$0.626264\pi$
$164$	0	0
$165$	−3.31104	−0.257764
$166$	0	0
$167$	15.4041	1.19201	0.596003	−	0.802982i	$-0.296755\pi$
$167$	15.4041	1.19201	0.596003	+	0.802982i	$0.296755\pi$
$168$	0	0
$169$	−2.92860	−0.225277
$170$	0	0
$171$	−1.18441	−0.0905742
$172$	0	0
$173$	−2.50616	−0.190539	−0.0952697	−	0.995451i	$-0.530371\pi$
$173$	−2.50616	−0.190539	−0.0952697	+	0.995451i	$0.530371\pi$
$174$	0	0
$175$	−4.38271	−0.331302
$176$	0	0
$177$	−0.859886	−0.0646330
$178$	0	0
$179$	−10.6318	−0.794657	−0.397328	−	0.917676i	$-0.630063\pi$
$179$	−10.6318	−0.794657	−0.397328	+	0.917676i	$0.630063\pi$
$180$	0	0
$181$	−4.52625	−0.336434	−0.168217	−	0.985750i	$-0.553801\pi$
$181$	−4.52625	−0.336434	−0.168217	+	0.985750i	$0.553801\pi$
$182$	0	0
$183$	−5.62997	−0.416179
$184$	0	0
$185$	−19.4966	−1.43342
$186$	0	0
$187$	−1.54144	−0.112721
$188$	0	0
$189$	−15.9861	−1.16282
$190$	0	0
$191$	−1.62563	−0.117626	−0.0588132	−	0.998269i	$-0.518732\pi$
$191$	−1.62563	−0.117626	−0.0588132	+	0.998269i	$0.518732\pi$
$192$	0	0
$193$	11.2000	0.806193	0.403096	−	0.915158i	$-0.367934\pi$
$193$	11.2000	0.806193	0.403096	+	0.915158i	$0.367934\pi$
$194$	0	0
$195$	−6.81685	−0.488165
$196$	0	0
$197$	−24.3579	−1.73543	−0.867715	−	0.497061i	$-0.834412\pi$
$197$	−24.3579	−1.73543	−0.867715	+	0.497061i	$0.834412\pi$
$198$	0	0
$199$	10.6308	0.753595	0.376798	−	0.926296i	$-0.377025\pi$
$199$	10.6308	0.753595	0.376798	+	0.926296i	$0.377025\pi$
$200$	0	0
$201$	4.89012	0.344922
$202$	0	0
$203$	−17.5435	−1.23131
$204$	0	0
$205$	−3.96769	−0.277116
$206$	0	0
$207$	11.8351	0.822595
$208$	0	0
$209$	0.807617	0.0558641
$210$	0	0
$211$	−24.1799	−1.66461	−0.832307	−	0.554315i	$-0.812980\pi$
$211$	−24.1799	−1.66461	−0.832307	+	0.554315i	$0.812980\pi$
$212$	0	0
$213$	12.0286	0.824187
$214$	0	0
$215$	−4.89965	−0.334153
$216$	0	0
$217$	−11.0999	−0.753513
$218$	0	0
$219$	1.18063	0.0797798
$220$	0	0
$221$	−3.17355	−0.213476
$222$	0	0
$223$	−10.4427	−0.699298	−0.349649	−	0.936881i	$-0.613699\pi$
$223$	−10.4427	−0.699298	−0.349649	+	0.936881i	$0.613699\pi$
$224$	0	0
$225$	−2.80349	−0.186900
$226$	0	0
$227$	10.7140	0.711115	0.355558	−	0.934654i	$-0.384291\pi$
$227$	10.7140	0.711115	0.355558	+	0.934654i	$0.384291\pi$
$228$	0	0
$229$	−4.09820	−0.270817	−0.135408	−	0.990790i	$-0.543235\pi$
$229$	−4.09820	−0.270817	−0.135408	+	0.990790i	$0.543235\pi$
$230$	0	0
$231$	4.68417	0.308196
$232$	0	0
$233$	−26.4695	−1.73407	−0.867036	−	0.498245i	$-0.833978\pi$
$233$	−26.4695	−1.73407	−0.867036	+	0.498245i	$0.833978\pi$
$234$	0	0
$235$	6.44125	0.420181
$236$	0	0
$237$	5.71282	0.371087
$238$	0	0
$239$	12.5621	0.812576	0.406288	−	0.913745i	$-0.366823\pi$
$239$	12.5621	0.812576	0.406288	+	0.913745i	$0.366823\pi$
$240$	0	0
$241$	6.48443	0.417699	0.208849	−	0.977948i	$-0.433028\pi$
$241$	6.48443	0.417699	0.208849	+	0.977948i	$0.433028\pi$
$242$	0	0
$243$	−16.0574	−1.03008
$244$	0	0
$245$	13.7120	0.876026
$246$	0	0
$247$	1.66274	0.105798
$248$	0	0
$249$	14.4920	0.918391
$250$	0	0
$251$	0.613499	0.0387237	0.0193618	−	0.999813i	$-0.493837\pi$
$251$	0.613499	0.0387237	0.0193618	+	0.999813i	$0.493837\pi$
$252$	0	0
$253$	−8.07001	−0.507357
$254$	0	0
$255$	2.14802	0.134514
$256$	0	0
$257$	27.3070	1.70336	0.851681	−	0.524060i	$-0.175583\pi$
$257$	27.3070	1.70336	0.851681	+	0.524060i	$0.175583\pi$
$258$	0	0
$259$	27.5820	1.71386
$260$	0	0
$261$	−11.2221	−0.694629
$262$	0	0
$263$	−25.6118	−1.57929	−0.789645	−	0.613564i	$-0.789735\pi$
$263$	−25.6118	−1.57929	−0.789645	+	0.613564i	$0.789735\pi$
$264$	0	0
$265$	6.46432	0.397100
$266$	0	0
$267$	−2.35165	−0.143919
$268$	0	0
$269$	18.9356	1.15453	0.577263	−	0.816558i	$-0.304121\pi$
$269$	18.9356	1.15453	0.577263	+	0.816558i	$0.304121\pi$
$270$	0	0
$271$	−22.8234	−1.38642	−0.693212	−	0.720734i	$-0.743805\pi$
$271$	−22.8234	−1.38642	−0.693212	+	0.720734i	$0.743805\pi$
$272$	0	0
$273$	9.64388	0.583674
$274$	0	0
$275$	1.91162	0.115275
$276$	0	0
$277$	11.9504	0.718030	0.359015	−	0.933332i	$-0.383113\pi$
$277$	11.9504	0.718030	0.359015	+	0.933332i	$0.383113\pi$
$278$	0	0
$279$	−7.10031	−0.425085
$280$	0	0
$281$	−8.81255	−0.525713	−0.262856	−	0.964835i	$-0.584665\pi$
$281$	−8.81255	−0.525713	−0.262856	+	0.964835i	$0.584665\pi$
$282$	0	0
$283$	−21.4080	−1.27257	−0.636286	−	0.771453i	$-0.719530\pi$
$283$	−21.4080	−1.27257	−0.636286	+	0.771453i	$0.719530\pi$
$284$	0	0
$285$	−1.12543	−0.0666647
$286$	0	0
$287$	5.61314	0.331333
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−3.09477	−0.181419
$292$	0	0
$293$	17.9472	1.04849	0.524245	−	0.851568i	$-0.324348\pi$
$293$	17.9472	1.04849	0.524245	+	0.851568i	$0.324348\pi$
$294$	0	0
$295$	2.49803	0.145441
$296$	0	0
$297$	6.97271	0.404598
$298$	0	0
$299$	−16.6147	−0.960855
$300$	0	0
$301$	6.93159	0.399530
$302$	0	0
$303$	11.2075	0.643854
$304$	0	0
$305$	16.3555	0.936511
$306$	0	0
$307$	−3.03015	−0.172940	−0.0864699	−	0.996254i	$-0.527559\pi$
$307$	−3.03015	−0.172940	−0.0864699	+	0.996254i	$0.527559\pi$
$308$	0	0
$309$	7.65930	0.435723
$310$	0	0
$311$	10.5182	0.596432	0.298216	−	0.954498i	$-0.403608\pi$
$311$	10.5182	0.596432	0.298216	+	0.954498i	$0.403608\pi$
$312$	0	0
$313$	10.8453	0.613014	0.306507	−	0.951868i	$-0.400840\pi$
$313$	10.8453	0.613014	0.306507	+	0.951868i	$0.400840\pi$
$314$	0	0
$315$	19.9566	1.12443
$316$	0	0
$317$	−28.1582	−1.58152	−0.790760	−	0.612127i	$-0.790314\pi$
$317$	−28.1582	−1.58152	−0.790760	+	0.612127i	$0.790314\pi$
$318$	0	0
$319$	7.65202	0.428431
$320$	0	0
$321$	8.53508	0.476382
$322$	0	0
$323$	−0.523938	−0.0291527
$324$	0	0
$325$	3.93570	0.218313
$326$	0	0
$327$	−1.04168	−0.0576048
$328$	0	0
$329$	−9.11251	−0.502389
$330$	0	0
$331$	−14.3683	−0.789751	−0.394875	−	0.918735i	$-0.629212\pi$
$331$	−14.3683	−0.789751	−0.394875	+	0.918735i	$0.629212\pi$
$332$	0	0
$333$	17.6434	0.966854
$334$	0	0
$335$	−14.2061	−0.776164
$336$	0	0
$337$	0.225511	0.0122843	0.00614217	−	0.999981i	$-0.498045\pi$
$337$	0.225511	0.0122843	0.00614217	+	0.999981i	$0.498045\pi$
$338$	0	0
$339$	−6.91323	−0.375475
$340$	0	0
$341$	4.84151	0.262182
$342$	0	0
$343$	5.33945	0.288303
$344$	0	0
$345$	11.2457	0.605449
$346$	0	0
$347$	−14.8161	−0.795372	−0.397686	−	0.917522i	$-0.630187\pi$
$347$	−14.8161	−0.795372	−0.397686	+	0.917522i	$0.630187\pi$
$348$	0	0
$349$	−0.379178	−0.0202969	−0.0101485	−	0.999949i	$-0.503230\pi$
$349$	−0.379178	−0.0202969	−0.0101485	+	0.999949i	$0.503230\pi$
$350$	0	0
$351$	14.3556	0.766244
$352$	0	0
$353$	2.37860	0.126600	0.0633000	−	0.997995i	$-0.479838\pi$
$353$	2.37860	0.126600	0.0633000	+	0.997995i	$0.479838\pi$
$354$	0	0
$355$	−34.9440	−1.85464
$356$	0	0
$357$	−3.03883	−0.160832
$358$	0	0
$359$	12.4888	0.659133	0.329566	−	0.944132i	$-0.393097\pi$
$359$	12.4888	0.659133	0.329566	+	0.944132i	$0.393097\pi$
$360$	0	0
$361$	−18.7255	−0.985552
$362$	0	0
$363$	7.41563	0.389220
$364$	0	0
$365$	−3.42982	−0.179525
$366$	0	0
$367$	−24.3802	−1.27264	−0.636319	−	0.771426i	$-0.719544\pi$
$367$	−24.3802	−1.27264	−0.636319	+	0.771426i	$0.719544\pi$
$368$	0	0
$369$	3.59057	0.186918
$370$	0	0
$371$	−9.14516	−0.474793
$372$	0	0
$373$	−33.6638	−1.74304	−0.871522	−	0.490357i	$-0.836867\pi$
$373$	−33.6638	−1.74304	−0.871522	+	0.490357i	$0.836867\pi$
$374$	0	0
$375$	8.07622	0.417054
$376$	0	0
$377$	15.7542	0.811381
$378$	0	0
$379$	−7.27325	−0.373602	−0.186801	−	0.982398i	$-0.559812\pi$
$379$	−7.27325	−0.373602	−0.186801	+	0.982398i	$0.559812\pi$
$380$	0	0
$381$	15.2948	0.783576
$382$	0	0
$383$	24.0268	1.22771	0.613857	−	0.789417i	$-0.289617\pi$
$383$	24.0268	1.22771	0.613857	+	0.789417i	$0.289617\pi$
$384$	0	0
$385$	−13.6079	−0.693520
$386$	0	0
$387$	4.43394	0.225390
$388$	0	0
$389$	4.37457	0.221799	0.110900	−	0.993832i	$-0.464627\pi$
$389$	4.37457	0.221799	0.110900	+	0.993832i	$0.464627\pi$
$390$	0	0
$391$	5.23538	0.264765
$392$	0	0
$393$	−9.60943	−0.484731
$394$	0	0
$395$	−16.5961	−0.835043
$396$	0	0
$397$	−35.7166	−1.79257	−0.896283	−	0.443482i	$-0.853743\pi$
$397$	−35.7166	−1.79257	−0.896283	+	0.443482i	$0.853743\pi$
$398$	0	0
$399$	1.59216	0.0797077
$400$	0	0
$401$	−10.3711	−0.517906	−0.258953	−	0.965890i	$-0.583378\pi$
$401$	−10.3711	−0.517906	−0.258953	+	0.965890i	$0.583378\pi$
$402$	0	0
$403$	9.96781	0.496532
$404$	0	0
$405$	7.22451	0.358989
$406$	0	0
$407$	−12.0306	−0.596333
$408$	0	0
$409$	25.6922	1.27040	0.635198	−	0.772349i	$-0.280918\pi$
$409$	25.6922	1.27040	0.635198	+	0.772349i	$0.280918\pi$
$410$	0	0
$411$	−4.85573	−0.239516
$412$	0	0
$413$	−3.53399	−0.173897
$414$	0	0
$415$	−42.1002	−2.06662
$416$	0	0
$417$	3.55740	0.174206
$418$	0	0
$419$	23.3221	1.13936	0.569679	−	0.821868i	$-0.307068\pi$
$419$	23.3221	1.13936	0.569679	+	0.821868i	$0.307068\pi$
$420$	0	0
$421$	−27.2315	−1.32718	−0.663591	−	0.748096i	$-0.730968\pi$
$421$	−27.2315	−1.32718	−0.663591	+	0.748096i	$0.730968\pi$
$422$	0	0
$423$	−5.82902	−0.283417
$424$	0	0
$425$	−1.24016	−0.0601565
$426$	0	0
$427$	−23.1383	−1.11974
$428$	0	0
$429$	−4.20641	−0.203087
$430$	0	0
$431$	−31.6188	−1.52303	−0.761513	−	0.648150i	$-0.775543\pi$
$431$	−31.6188	−1.52303	−0.761513	+	0.648150i	$0.775543\pi$
$432$	0	0
$433$	−2.91444	−0.140059	−0.0700295	−	0.997545i	$-0.522309\pi$
$433$	−2.91444	−0.140059	−0.0700295	+	0.997545i	$0.522309\pi$
$434$	0	0
$435$	−10.6632	−0.511263
$436$	0	0
$437$	−2.74301	−0.131216
$438$	0	0
$439$	25.7288	1.22797	0.613985	−	0.789318i	$-0.289566\pi$
$439$	25.7288	1.22797	0.613985	+	0.789318i	$0.289566\pi$
$440$	0	0
$441$	−12.4087	−0.590889
$442$	0	0
$443$	9.86400	0.468653	0.234326	−	0.972158i	$-0.424712\pi$
$443$	9.86400	0.468653	0.234326	+	0.972158i	$0.424712\pi$
$444$	0	0
$445$	6.83173	0.323855
$446$	0	0
$447$	−11.8166	−0.558907
$448$	0	0
$449$	−4.43929	−0.209503	−0.104751	−	0.994498i	$-0.533405\pi$
$449$	−4.43929	−0.209503	−0.104751	+	0.994498i	$0.533405\pi$
$450$	0	0
$451$	−2.44831	−0.115286
$452$	0	0
$453$	0.365659	0.0171801
$454$	0	0
$455$	−28.0162	−1.31342
$456$	0	0
$457$	−16.8780	−0.789521	−0.394761	−	0.918784i	$-0.629172\pi$
$457$	−16.8780	−0.789521	−0.394761	+	0.918784i	$0.629172\pi$
$458$	0	0
$459$	−4.52351	−0.211139
$460$	0	0
$461$	1.44300	0.0672073	0.0336037	−	0.999435i	$-0.489302\pi$
$461$	1.44300	0.0672073	0.0336037	+	0.999435i	$0.489302\pi$
$462$	0	0
$463$	13.3709	0.621400	0.310700	−	0.950508i	$-0.399437\pi$
$463$	13.3709	0.621400	0.310700	+	0.950508i	$0.399437\pi$
$464$	0	0
$465$	−6.74673	−0.312872
$466$	0	0
$467$	1.24377	0.0575548	0.0287774	−	0.999586i	$-0.490839\pi$
$467$	1.24377	0.0575548	0.0287774	+	0.999586i	$0.490839\pi$
$468$	0	0
$469$	20.0976	0.928021
$470$	0	0
$471$	−1.72440	−0.0794561
$472$	0	0
$473$	−3.02338	−0.139015
$474$	0	0
$475$	0.649765	0.0298133
$476$	0	0
$477$	−5.84990	−0.267848
$478$	0	0
$479$	11.0190	0.503471	0.251735	−	0.967796i	$-0.418999\pi$
$479$	11.0190	0.503471	0.251735	+	0.967796i	$0.418999\pi$
$480$	0	0
$481$	−24.7688	−1.12936
$482$	0	0
$483$	−15.9094	−0.723905
$484$	0	0
$485$	8.99054	0.408239
$486$	0	0
$487$	15.9006	0.720525	0.360262	−	0.932851i	$-0.382687\pi$
$487$	15.9006	0.720525	0.360262	+	0.932851i	$0.382687\pi$
$488$	0	0
$489$	8.48289	0.383609
$490$	0	0
$491$	14.1416	0.638202	0.319101	−	0.947721i	$-0.396619\pi$
$491$	14.1416	0.638202	0.319101	+	0.947721i	$0.396619\pi$
$492$	0	0
$493$	−4.96421	−0.223577
$494$	0	0
$495$	−8.70456	−0.391241
$496$	0	0
$497$	49.4357	2.21749
$498$	0	0
$499$	6.92123	0.309837	0.154918	−	0.987927i	$-0.450489\pi$
$499$	6.92123	0.309837	0.154918	+	0.987927i	$0.450489\pi$
$500$	0	0
$501$	−13.2458	−0.591778
$502$	0	0
$503$	−13.0842	−0.583396	−0.291698	−	0.956510i	$-0.594220\pi$
$503$	−13.0842	−0.583396	−0.291698	+	0.956510i	$0.594220\pi$
$504$	0	0
$505$	−32.5586	−1.44884
$506$	0	0
$507$	2.51826	0.111840
$508$	0	0
$509$	−16.6478	−0.737903	−0.368951	−	0.929449i	$-0.620283\pi$
$509$	−16.6478	−0.737903	−0.368951	+	0.929449i	$0.620283\pi$
$510$	0	0
$511$	4.85221	0.214649
$512$	0	0
$513$	2.37004	0.104640
$514$	0	0
$515$	−22.2508	−0.980489
$516$	0	0
$517$	3.97464	0.174805
$518$	0	0
$519$	2.15501	0.0945944
$520$	0	0
$521$	9.64096	0.422378	0.211189	−	0.977445i	$-0.432266\pi$
$521$	9.64096	0.422378	0.211189	+	0.977445i	$0.432266\pi$
$522$	0	0
$523$	20.9852	0.917617	0.458809	−	0.888535i	$-0.348276\pi$
$523$	20.9852	0.917617	0.458809	+	0.888535i	$0.348276\pi$
$524$	0	0
$525$	3.76863	0.164477
$526$	0	0
$527$	−3.14090	−0.136820
$528$	0	0
$529$	4.40921	0.191705
$530$	0	0
$531$	−2.26060	−0.0981015
$532$	0	0
$533$	−5.04063	−0.218334
$534$	0	0
$535$	−24.7950	−1.07198
$536$	0	0
$537$	9.14212	0.394512
$538$	0	0
$539$	8.46113	0.364447
$540$	0	0
$541$	22.6895	0.975496	0.487748	−	0.872984i	$-0.337819\pi$
$541$	22.6895	0.975496	0.487748	+	0.872984i	$0.337819\pi$
$542$	0	0
$543$	3.89206	0.167024
$544$	0	0
$545$	3.02614	0.129626
$546$	0	0
$547$	42.1363	1.80162	0.900808	−	0.434217i	$-0.142975\pi$
$547$	42.1363	1.80162	0.900808	+	0.434217i	$0.142975\pi$
$548$	0	0
$549$	−14.8009	−0.631687
$550$	0	0
$551$	2.60094	0.110804
$552$	0	0
$553$	23.4788	0.998419
$554$	0	0
$555$	16.7648	0.711627
$556$	0	0
$557$	−10.3277	−0.437601	−0.218800	−	0.975770i	$-0.570214\pi$
$557$	−10.3277	−0.437601	−0.218800	+	0.975770i	$0.570214\pi$
$558$	0	0
$559$	−6.22461	−0.263273
$560$	0	0
$561$	1.32546	0.0559610
$562$	0	0
$563$	−8.23744	−0.347167	−0.173583	−	0.984819i	$-0.555535\pi$
$563$	−8.23744	−0.347167	−0.173583	+	0.984819i	$0.555535\pi$
$564$	0	0
$565$	20.0834	0.844916
$566$	0	0
$567$	−10.2206	−0.429225
$568$	0	0
$569$	−26.6695	−1.11805	−0.559023	−	0.829152i	$-0.688824\pi$
$569$	−26.6695	−1.11805	−0.559023	+	0.829152i	$0.688824\pi$
$570$	0	0
$571$	28.8644	1.20794	0.603968	−	0.797009i	$-0.293586\pi$
$571$	28.8644	1.20794	0.603968	+	0.797009i	$0.293586\pi$
$572$	0	0
$573$	1.39785	0.0583962
$574$	0	0
$575$	−6.49270	−0.270764
$576$	0	0
$577$	28.2153	1.17462	0.587310	−	0.809362i	$-0.300187\pi$
$577$	28.2153	1.17462	0.587310	+	0.809362i	$0.300187\pi$
$578$	0	0
$579$	−9.63071	−0.400239
$580$	0	0
$581$	59.5596	2.47095
$582$	0	0
$583$	3.98888	0.165203
$584$	0	0
$585$	−17.9211	−0.740948
$586$	0	0
$587$	0.367332	0.0151614	0.00758070	−	0.999971i	$-0.497587\pi$
$587$	0.367332	0.0151614	0.00758070	+	0.999971i	$0.497587\pi$
$588$	0	0
$589$	1.64564	0.0678073
$590$	0	0
$591$	20.9450	0.861564
$592$	0	0
$593$	2.06664	0.0848669	0.0424335	−	0.999099i	$-0.486489\pi$
$593$	2.06664	0.0848669	0.0424335	+	0.999099i	$0.486489\pi$
$594$	0	0
$595$	8.82803	0.361914
$596$	0	0
$597$	−9.14125	−0.374126
$598$	0	0
$599$	−18.0232	−0.736409	−0.368205	−	0.929745i	$-0.620027\pi$
$599$	−18.0232	−0.736409	−0.368205	+	0.929745i	$0.620027\pi$
$600$	0	0
$601$	−2.24935	−0.0917530	−0.0458765	−	0.998947i	$-0.514608\pi$
$601$	−2.24935	−0.0917530	−0.0458765	+	0.998947i	$0.514608\pi$
$602$	0	0
$603$	12.8559	0.523531
$604$	0	0
$605$	−21.5429	−0.875845
$606$	0	0
$607$	39.7027	1.61148	0.805742	−	0.592266i	$-0.201767\pi$
$607$	39.7027	1.61148	0.805742	+	0.592266i	$0.201767\pi$
$608$	0	0
$609$	15.0854	0.611292
$610$	0	0
$611$	8.18309	0.331052
$612$	0	0
$613$	−5.25236	−0.212141	−0.106070	−	0.994359i	$-0.533827\pi$
$613$	−5.25236	−0.212141	−0.106070	+	0.994359i	$0.533827\pi$
$614$	0	0
$615$	3.41176	0.137576
$616$	0	0
$617$	−44.4029	−1.78759	−0.893797	−	0.448472i	$-0.851968\pi$
$617$	−44.4029	−1.78759	−0.893797	+	0.448472i	$0.851968\pi$
$618$	0	0
$619$	21.7679	0.874928	0.437464	−	0.899236i	$-0.355877\pi$
$619$	21.7679	0.874928	0.437464	+	0.899236i	$0.355877\pi$
$620$	0	0
$621$	−23.6823	−0.950338
$622$	0	0
$623$	−9.66493	−0.387217
$624$	0	0
$625$	−29.6628	−1.18651
$626$	0	0
$627$	−0.694459	−0.0277340
$628$	0	0
$629$	7.80477	0.311197
$630$	0	0
$631$	−17.2043	−0.684894	−0.342447	−	0.939537i	$-0.611256\pi$
$631$	−17.2043	−0.684894	−0.342447	+	0.939537i	$0.611256\pi$
$632$	0	0
$633$	20.7920	0.826407
$634$	0	0
$635$	−44.4325	−1.76325
$636$	0	0
$637$	17.4200	0.690205
$638$	0	0
$639$	31.6226	1.25097
$640$	0	0
$641$	38.1210	1.50569	0.752844	−	0.658198i	$-0.228681\pi$
$641$	38.1210	1.50569	0.752844	+	0.658198i	$0.228681\pi$
$642$	0	0
$643$	−23.4962	−0.926598	−0.463299	−	0.886202i	$-0.653334\pi$
$643$	−23.4962	−0.926598	−0.463299	+	0.886202i	$0.653334\pi$
$644$	0	0
$645$	4.21314	0.165892
$646$	0	0
$647$	−6.58999	−0.259079	−0.129540	−	0.991574i	$-0.541350\pi$
$647$	−6.58999	−0.259079	−0.129540	+	0.991574i	$0.541350\pi$
$648$	0	0
$649$	1.54144	0.0605067
$650$	0	0
$651$	9.54468	0.374085
$652$	0	0
$653$	34.8504	1.36380	0.681900	−	0.731445i	$-0.261154\pi$
$653$	34.8504	1.36380	0.681900	+	0.731445i	$0.261154\pi$
$654$	0	0
$655$	27.9161	1.09077
$656$	0	0
$657$	3.10382	0.121092
$658$	0	0
$659$	−25.6375	−0.998693	−0.499347	−	0.866402i	$-0.666427\pi$
$659$	−25.6375	−0.998693	−0.499347	+	0.866402i	$0.666427\pi$
$660$	0	0
$661$	−31.2665	−1.21613	−0.608064	−	0.793888i	$-0.708053\pi$
$661$	−31.2665	−1.21613	−0.608064	+	0.793888i	$0.708053\pi$
$662$	0	0
$663$	2.72889	0.105981
$664$	0	0
$665$	−4.62534	−0.179363
$666$	0	0
$667$	−25.9895	−1.00632
$668$	0	0
$669$	8.97957	0.347170
$670$	0	0
$671$	10.0923	0.389610
$672$	0	0
$673$	−1.99467	−0.0768889	−0.0384445	−	0.999261i	$-0.512240\pi$
$673$	−1.99467	−0.0768889	−0.0384445	+	0.999261i	$0.512240\pi$
$674$	0	0
$675$	5.60987	0.215924
$676$	0	0
$677$	−0.331632	−0.0127457	−0.00637283	−	0.999980i	$-0.502029\pi$
$677$	−0.331632	−0.0127457	−0.00637283	+	0.999980i	$0.502029\pi$
$678$	0	0
$679$	−12.7190	−0.488111
$680$	0	0
$681$	−9.21285	−0.353037
$682$	0	0
$683$	19.2037	0.734810	0.367405	−	0.930061i	$-0.380246\pi$
$683$	19.2037	0.734810	0.367405	+	0.930061i	$0.380246\pi$
$684$	0	0
$685$	14.1063	0.538972
$686$	0	0
$687$	3.52399	0.134449
$688$	0	0
$689$	8.21240	0.312868
$690$	0	0
$691$	30.6610	1.16640	0.583200	−	0.812329i	$-0.301801\pi$
$691$	30.6610	1.16640	0.583200	+	0.812329i	$0.301801\pi$
$692$	0	0
$693$	12.3144	0.467787
$694$	0	0
$695$	−10.3345	−0.392010
$696$	0	0
$697$	1.58833	0.0601622
$698$	0	0
$699$	22.7607	0.860889
$700$	0	0
$701$	10.0165	0.378317	0.189158	−	0.981947i	$-0.439424\pi$
$701$	10.0165	0.378317	0.189158	+	0.981947i	$0.439424\pi$
$702$	0	0
$703$	−4.08921	−0.154228
$704$	0	0
$705$	−5.53874	−0.208601
$706$	0	0
$707$	46.0610	1.73230
$708$	0	0
$709$	−13.5734	−0.509759	−0.254880	−	0.966973i	$-0.582036\pi$
$709$	−13.5734	−0.509759	−0.254880	+	0.966973i	$0.582036\pi$
$710$	0	0
$711$	15.0187	0.563245
$712$	0	0
$713$	−16.4438	−0.615826
$714$	0	0
$715$	12.2199	0.456999
$716$	0	0
$717$	−10.8020	−0.403408
$718$	0	0
$719$	8.75380	0.326461	0.163231	−	0.986588i	$-0.447809\pi$
$719$	8.75380	0.326461	0.163231	+	0.986588i	$0.447809\pi$
$720$	0	0
$721$	31.4785	1.17232
$722$	0	0
$723$	−5.57587	−0.207369
$724$	0	0
$725$	6.15640	0.228643
$726$	0	0
$727$	18.0212	0.668370	0.334185	−	0.942508i	$-0.391539\pi$
$727$	18.0212	0.668370	0.334185	+	0.942508i	$0.391539\pi$
$728$	0	0
$729$	5.13128	0.190048
$730$	0	0
$731$	1.96140	0.0725452
$732$	0	0
$733$	23.1797	0.856163	0.428082	−	0.903740i	$-0.359190\pi$
$733$	23.1797	0.856163	0.428082	+	0.903740i	$0.359190\pi$
$734$	0	0
$735$	−11.7907	−0.434908
$736$	0	0
$737$	−8.76606	−0.322902
$738$	0	0
$739$	24.3936	0.897335	0.448667	−	0.893699i	$-0.351899\pi$
$739$	24.3936	0.897335	0.448667	+	0.893699i	$0.351899\pi$
$740$	0	0
$741$	−1.42977	−0.0525238
$742$	0	0
$743$	26.6662	0.978287	0.489143	−	0.872203i	$-0.337309\pi$
$743$	26.6662	0.978287	0.489143	+	0.872203i	$0.337309\pi$
$744$	0	0
$745$	34.3281	1.25769
$746$	0	0
$747$	38.0986	1.39396
$748$	0	0
$749$	35.0778	1.28172
$750$	0	0
$751$	46.7858	1.70724	0.853618	−	0.520899i	$-0.174403\pi$
$751$	46.7858	1.70724	0.853618	+	0.520899i	$0.174403\pi$
$752$	0	0
$753$	−0.527539	−0.0192246
$754$	0	0
$755$	−1.06226	−0.0386598
$756$	0	0
$757$	−13.8514	−0.503438	−0.251719	−	0.967800i	$-0.580996\pi$
$757$	−13.8514	−0.503438	−0.251719	+	0.967800i	$0.580996\pi$
$758$	0	0
$759$	6.93929	0.251880
$760$	0	0
$761$	9.87791	0.358074	0.179037	−	0.983842i	$-0.442702\pi$
$761$	9.87791	0.358074	0.179037	+	0.983842i	$0.442702\pi$
$762$	0	0
$763$	−4.28112	−0.154987
$764$	0	0
$765$	5.64704	0.204169
$766$	0	0
$767$	3.17355	0.114590
$768$	0	0
$769$	5.78278	0.208532	0.104266	−	0.994549i	$-0.466751\pi$
$769$	5.78278	0.208532	0.104266	+	0.994549i	$0.466751\pi$
$770$	0	0
$771$	−23.4809	−0.845644
$772$	0	0
$773$	33.2286	1.19515	0.597574	−	0.801814i	$-0.296131\pi$
$773$	33.2286	1.19515	0.597574	+	0.801814i	$0.296131\pi$
$774$	0	0
$775$	3.89521	0.139920
$776$	0	0
$777$	−23.7174	−0.850857
$778$	0	0
$779$	−0.832185	−0.0298161
$780$	0	0
$781$	−21.5626	−0.771570
$782$	0	0
$783$	22.4557	0.802501
$784$	0	0
$785$	5.00950	0.178797
$786$	0	0
$787$	−27.1882	−0.969155	−0.484577	−	0.874748i	$-0.661027\pi$
$787$	−27.1882	−0.969155	−0.484577	+	0.874748i	$0.661027\pi$
$788$	0	0
$789$	22.0232	0.784047
$790$	0	0
$791$	−28.4123	−1.01022
$792$	0	0
$793$	20.7783	0.737859
$794$	0	0
$795$	−5.55858	−0.197143
$796$	0	0
$797$	52.3247	1.85343	0.926717	−	0.375759i	$-0.122618\pi$
$797$	52.3247	1.85343	0.926717	+	0.375759i	$0.122618\pi$
$798$	0	0
$799$	−2.57853	−0.0912219
$800$	0	0
$801$	−6.18238	−0.218444
$802$	0	0
$803$	−2.11641	−0.0746865
$804$	0	0
$805$	46.2181	1.62897
$806$	0	0
$807$	−16.2825	−0.573171
$808$	0	0
$809$	36.9217	1.29810	0.649048	−	0.760747i	$-0.275167\pi$
$809$	36.9217	1.29810	0.649048	+	0.760747i	$0.275167\pi$
$810$	0	0
$811$	7.03514	0.247037	0.123519	−	0.992342i	$-0.460582\pi$
$811$	7.03514	0.247037	0.123519	+	0.992342i	$0.460582\pi$
$812$	0	0
$813$	19.6255	0.688297
$814$	0	0
$815$	−24.6434	−0.863221
$816$	0	0
$817$	−1.02765	−0.0359531
$818$	0	0
$819$	25.3533	0.885915
$820$	0	0
$821$	−18.7702	−0.655085	−0.327542	−	0.944836i	$-0.606220\pi$
$821$	−18.7702	−0.655085	−0.327542	+	0.944836i	$0.606220\pi$
$822$	0	0
$823$	52.7145	1.83751	0.918756	−	0.394825i	$-0.129195\pi$
$823$	52.7145	1.83751	0.918756	+	0.394825i	$0.129195\pi$
$824$	0	0
$825$	−1.64378	−0.0572291
$826$	0	0
$827$	−17.6427	−0.613498	−0.306749	−	0.951790i	$-0.599241\pi$
$827$	−17.6427	−0.613498	−0.306749	+	0.951790i	$0.599241\pi$
$828$	0	0
$829$	34.9409	1.21355	0.606775	−	0.794874i	$-0.292463\pi$
$829$	34.9409	1.21355	0.606775	+	0.794874i	$0.292463\pi$
$830$	0	0
$831$	−10.2760	−0.356470
$832$	0	0
$833$	−5.48912	−0.190187
$834$	0	0
$835$	38.4799	1.33165
$836$	0	0
$837$	14.2079	0.491097
$838$	0	0
$839$	12.2284	0.422170	0.211085	−	0.977468i	$-0.432300\pi$
$839$	12.2284	0.422170	0.211085	+	0.977468i	$0.432300\pi$
$840$	0	0
$841$	−4.35659	−0.150227
$842$	0	0
$843$	7.57779	0.260993
$844$	0	0
$845$	−7.31572	−0.251669
$846$	0	0
$847$	30.4771	1.04720
$848$	0	0
$849$	18.4084	0.631775
$850$	0	0
$851$	40.8609	1.40069
$852$	0	0
$853$	−14.9504	−0.511891	−0.255945	−	0.966691i	$-0.582387\pi$
$853$	−14.9504	−0.511891	−0.255945	+	0.966691i	$0.582387\pi$
$854$	0	0
$855$	−2.95870	−0.101185
$856$	0	0
$857$	−28.8802	−0.986530	−0.493265	−	0.869879i	$-0.664197\pi$
$857$	−28.8802	−0.986530	−0.493265	+	0.869879i	$0.664197\pi$
$858$	0	0
$859$	18.3304	0.625424	0.312712	−	0.949848i	$-0.398763\pi$
$859$	18.3304	0.625424	0.312712	+	0.949848i	$0.398763\pi$
$860$	0	0
$861$	−4.82666	−0.164492
$862$	0	0
$863$	−30.9280	−1.05280	−0.526401	−	0.850237i	$-0.676459\pi$
$863$	−30.9280	−1.05280	−0.526401	+	0.850237i	$0.676459\pi$
$864$	0	0
$865$	−6.26045	−0.212862
$866$	0	0
$867$	−0.859886	−0.0292033
$868$	0	0
$869$	−10.2408	−0.347397
$870$	0	0
$871$	−18.0478	−0.611525
$872$	0	0
$873$	−8.13600	−0.275362
$874$	0	0
$875$	33.1920	1.12209
$876$	0	0
$877$	42.0300	1.41925	0.709627	−	0.704578i	$-0.248864\pi$
$877$	42.0300	1.41925	0.709627	+	0.704578i	$0.248864\pi$
$878$	0	0
$879$	−15.4326	−0.520528
$880$	0	0
$881$	−58.8585	−1.98299	−0.991497	−	0.130127i	$-0.958461\pi$
$881$	−58.8585	−1.98299	−0.991497	+	0.130127i	$0.958461\pi$
$882$	0	0
$883$	48.8906	1.64530	0.822650	−	0.568548i	$-0.192495\pi$
$883$	48.8906	1.64530	0.822650	+	0.568548i	$0.192495\pi$
$884$	0	0
$885$	−2.14802	−0.0722050
$886$	0	0
$887$	21.3749	0.717698	0.358849	−	0.933396i	$-0.383169\pi$
$887$	21.3749	0.717698	0.358849	+	0.933396i	$0.383169\pi$
$888$	0	0
$889$	62.8592	2.10823
$890$	0	0
$891$	4.45797	0.149347
$892$	0	0
$893$	1.35099	0.0452092
$894$	0	0
$895$	−26.5585	−0.887753
$896$	0	0
$897$	14.2868	0.477021
$898$	0	0
$899$	15.5921	0.520026
$900$	0	0
$901$	−2.58777	−0.0862110
$902$	0	0
$903$	−5.96038	−0.198349
$904$	0	0
$905$	−11.3067	−0.375848
$906$	0	0
$907$	53.1059	1.76335	0.881676	−	0.471854i	$-0.156415\pi$
$907$	53.1059	1.76335	0.881676	+	0.471854i	$0.156415\pi$
$908$	0	0
$909$	29.4639	0.977257
$910$	0	0
$911$	−41.3354	−1.36950	−0.684752	−	0.728776i	$-0.740090\pi$
$911$	−41.3354	−1.36950	−0.684752	+	0.728776i	$0.740090\pi$
$912$	0	0
$913$	−25.9784	−0.859759
$914$	0	0
$915$	−14.0638	−0.464936
$916$	0	0
$917$	−39.4932	−1.30418
$918$	0	0
$919$	21.9600	0.724395	0.362197	−	0.932101i	$-0.382027\pi$
$919$	21.9600	0.724395	0.362197	+	0.932101i	$0.382027\pi$
$920$	0	0
$921$	2.60558	0.0858569
$922$	0	0
$923$	−44.3936	−1.46123
$924$	0	0
$925$	−9.67914	−0.318248
$926$	0	0
$927$	20.1359	0.661350
$928$	0	0
$929$	−9.76489	−0.320376	−0.160188	−	0.987087i	$-0.551210\pi$
$929$	−9.76489	−0.320376	−0.160188	+	0.987087i	$0.551210\pi$
$930$	0	0
$931$	2.87596	0.0942557
$932$	0	0
$933$	−9.04445	−0.296102
$934$	0	0
$935$	−3.85056	−0.125927
$936$	0	0
$937$	6.07167	0.198353	0.0991765	−	0.995070i	$-0.468379\pi$
$937$	6.07167	0.198353	0.0991765	+	0.995070i	$0.468379\pi$
$938$	0	0
$939$	−9.32574	−0.304334
$940$	0	0
$941$	−2.95743	−0.0964095	−0.0482048	−	0.998837i	$-0.515350\pi$
$941$	−2.95743	−0.0964095	−0.0482048	+	0.998837i	$0.515350\pi$
$942$	0	0
$943$	8.31550	0.270790
$944$	0	0
$945$	−39.9337	−1.29904
$946$	0	0
$947$	−3.09886	−0.100699	−0.0503497	−	0.998732i	$-0.516034\pi$
$947$	−3.09886	−0.100699	−0.0503497	+	0.998732i	$0.516034\pi$
$948$	0	0
$949$	−4.35732	−0.141444
$950$	0	0
$951$	24.2128	0.785154
$952$	0	0
$953$	16.4416	0.532596	0.266298	−	0.963891i	$-0.414199\pi$
$953$	16.4416	0.532596	0.266298	+	0.963891i	$0.414199\pi$
$954$	0	0
$955$	−4.06087	−0.131407
$956$	0	0
$957$	−6.57987	−0.212697
$958$	0	0
$959$	−19.9563	−0.644422
$960$	0	0
$961$	−21.1347	−0.681765
$962$	0	0
$963$	22.4383	0.723064
$964$	0	0
$965$	27.9779	0.900641
$966$	0	0
$967$	−26.0967	−0.839213	−0.419606	−	0.907706i	$-0.637832\pi$
$967$	−26.0967	−0.839213	−0.419606	+	0.907706i	$0.637832\pi$
$968$	0	0
$969$	0.450527	0.0144730
$970$	0	0
$971$	14.5210	0.466000	0.233000	−	0.972477i	$-0.425146\pi$
$971$	14.5210	0.466000	0.233000	+	0.972477i	$0.425146\pi$
$972$	0	0
$973$	14.6203	0.468706
$974$	0	0
$975$	−3.38425	−0.108383
$976$	0	0
$977$	−39.9161	−1.27703	−0.638515	−	0.769609i	$-0.720451\pi$
$977$	−39.9161	−1.27703	−0.638515	+	0.769609i	$0.720451\pi$
$978$	0	0
$979$	4.21559	0.134731
$980$	0	0
$981$	−2.73851	−0.0874340
$982$	0	0
$983$	−12.7747	−0.407449	−0.203725	−	0.979028i	$-0.565305\pi$
$983$	−12.7747	−0.407449	−0.203725	+	0.979028i	$0.565305\pi$
$984$	0	0
$985$	−60.8469	−1.93874
$986$	0	0
$987$	7.83572	0.249414
$988$	0	0
$989$	10.2687	0.326526
$990$	0	0
$991$	−31.3208	−0.994939	−0.497470	−	0.867481i	$-0.665737\pi$
$991$	−31.3208	−0.994939	−0.497470	+	0.867481i	$0.665737\pi$
$992$	0	0
$993$	12.3551	0.392076
$994$	0	0
$995$	26.5560	0.841881
$996$	0	0
$997$	4.01525	0.127164	0.0635821	−	0.997977i	$-0.479748\pi$
$997$	4.01525	0.127164	0.0635821	+	0.997977i	$0.479748\pi$
$998$	0	0
$999$	−35.3050	−1.11700

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.g.1.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.g.1.6	✓	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)

\(f(q)\)	\(=\)	\(q-0.859886 q^{3} +2.49803 q^{5} -3.53399 q^{7} -2.26060 q^{9} +O(q^{10})\)	\(q-0.859886 q^{3} +2.49803 q^{5} -3.53399 q^{7} -2.26060 q^{9} +1.54144 q^{11} +3.17355 q^{13} -2.14802 q^{15} -1.00000 q^{17} +0.523938 q^{19} +3.03883 q^{21} -5.23538 q^{23} +1.24016 q^{25} +4.52351 q^{27} +4.96421 q^{29} +3.14090 q^{31} -1.32546 q^{33} -8.82803 q^{35} -7.80477 q^{37} -2.72889 q^{39} -1.58833 q^{41} -1.96140 q^{43} -5.64704 q^{45} +2.57853 q^{47} +5.48912 q^{49} +0.859886 q^{51} +2.58777 q^{53} +3.85056 q^{55} -0.450527 q^{57} +1.00000 q^{59} +6.54735 q^{61} +7.98893 q^{63} +7.92762 q^{65} -5.68694 q^{67} +4.50183 q^{69} -13.9886 q^{71} -1.37301 q^{73} -1.06639 q^{75} -5.44743 q^{77} -6.64369 q^{79} +2.89208 q^{81} -16.8533 q^{83} -2.49803 q^{85} -4.26866 q^{87} +2.73484 q^{89} -11.2153 q^{91} -2.70082 q^{93} +1.30881 q^{95} +3.59905 q^{97} -3.48457 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 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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