LMFDB - Embedded newform 4012.2.a.g.1.1

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			1.1
Root			$3.27119$ 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)

$f(q)$	$=$	$q-3.27119 q^{3} +3.82141 q^{5} -1.10063 q^{7} +7.70067 q^{9} +O(q^{10})$	$q-3.27119 q^{3} +3.82141 q^{5} -1.10063 q^{7} +7.70067 q^{9} +4.05597 q^{11} +2.22527 q^{13} -12.5006 q^{15} -1.00000 q^{17} -5.58725 q^{19} +3.60037 q^{21} -8.79238 q^{23} +9.60321 q^{25} -15.3768 q^{27} -7.67625 q^{29} +4.75500 q^{31} -13.2678 q^{33} -4.20597 q^{35} +1.29636 q^{37} -7.27928 q^{39} -6.85530 q^{41} -2.65701 q^{43} +29.4274 q^{45} -11.7214 q^{47} -5.78861 q^{49} +3.27119 q^{51} -9.59673 q^{53} +15.4995 q^{55} +18.2769 q^{57} +1.00000 q^{59} -11.2376 q^{61} -8.47560 q^{63} +8.50368 q^{65} +2.30373 q^{67} +28.7615 q^{69} +6.69275 q^{71} +9.58317 q^{73} -31.4139 q^{75} -4.46413 q^{77} +5.58062 q^{79} +27.1983 q^{81} -8.10186 q^{83} -3.82141 q^{85} +25.1105 q^{87} +8.45021 q^{89} -2.44920 q^{91} -15.5545 q^{93} -21.3512 q^{95} -3.32787 q^{97} +31.2337 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.27119	−1.88862	−0.944310	−	0.329056i	$-0.893270\pi$
$3$	−3.27119	−1.88862	−0.944310	+	0.329056i	$0.893270\pi$
$4$	0	0
$5$	3.82141	1.70899	0.854494	−	0.519461i	$-0.173867\pi$
$5$	3.82141	1.70899	0.854494	+	0.519461i	$0.173867\pi$
$6$	0	0
$7$	−1.10063	−0.416000	−0.208000	−	0.978129i	$-0.566695\pi$
$7$	−1.10063	−0.416000	−0.208000	+	0.978129i	$0.566695\pi$
$8$	0	0
$9$	7.70067	2.56689
$10$	0	0
$11$	4.05597	1.22292	0.611461	−	0.791275i	$-0.290582\pi$
$11$	4.05597	1.22292	0.611461	+	0.791275i	$0.290582\pi$
$12$	0	0
$13$	2.22527	0.617179	0.308589	−	0.951195i	$-0.400143\pi$
$13$	2.22527	0.617179	0.308589	+	0.951195i	$0.400143\pi$
$14$	0	0
$15$	−12.5006	−3.22763
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−5.58725	−1.28180	−0.640901	−	0.767623i	$-0.721439\pi$
$19$	−5.58725	−1.28180	−0.640901	+	0.767623i	$0.721439\pi$
$20$	0	0
$21$	3.60037	0.785666
$22$	0	0
$23$	−8.79238	−1.83334	−0.916669	−	0.399648i	$-0.869132\pi$
$23$	−8.79238	−1.83334	−0.916669	+	0.399648i	$0.869132\pi$
$24$	0	0
$25$	9.60321	1.92064
$26$	0	0
$27$	−15.3768	−2.95926
$28$	0	0
$29$	−7.67625	−1.42544	−0.712722	−	0.701447i	$-0.752538\pi$
$29$	−7.67625	−1.42544	−0.712722	+	0.701447i	$0.752538\pi$
$30$	0	0
$31$	4.75500	0.854023	0.427011	−	0.904246i	$-0.359566\pi$
$31$	4.75500	0.854023	0.427011	+	0.904246i	$0.359566\pi$
$32$	0	0
$33$	−13.2678	−2.30963
$34$	0	0
$35$	−4.20597	−0.710939
$36$	0	0
$37$	1.29636	0.213120	0.106560	−	0.994306i	$-0.466016\pi$
$37$	1.29636	0.213120	0.106560	+	0.994306i	$0.466016\pi$
$38$	0	0
$39$	−7.27928	−1.16562
$40$	0	0
$41$	−6.85530	−1.07062	−0.535309	−	0.844656i	$-0.679805\pi$
$41$	−6.85530	−1.07062	−0.535309	+	0.844656i	$0.679805\pi$
$42$	0	0
$43$	−2.65701	−0.405191	−0.202595	−	0.979263i	$-0.564938\pi$
$43$	−2.65701	−0.405191	−0.202595	+	0.979263i	$0.564938\pi$
$44$	0	0
$45$	29.4274	4.38678
$46$	0	0
$47$	−11.7214	−1.70974	−0.854871	−	0.518841i	$-0.826364\pi$
$47$	−11.7214	−1.70974	−0.854871	+	0.518841i	$0.826364\pi$
$48$	0	0
$49$	−5.78861	−0.826944
$50$	0	0
$51$	3.27119	0.458058
$52$	0	0
$53$	−9.59673	−1.31821	−0.659106	−	0.752050i	$-0.729065\pi$
$53$	−9.59673	−1.31821	−0.659106	+	0.752050i	$0.729065\pi$
$54$	0	0
$55$	15.4995	2.08996
$56$	0	0
$57$	18.2769	2.42084
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−11.2376	−1.43883	−0.719414	−	0.694582i	$-0.755590\pi$
$61$	−11.2376	−1.43883	−0.719414	+	0.694582i	$0.755590\pi$
$62$	0	0
$63$	−8.47560	−1.06783
$64$	0	0
$65$	8.50368	1.05475
$66$	0	0
$67$	2.30373	0.281445	0.140722	−	0.990049i	$-0.455057\pi$
$67$	2.30373	0.281445	0.140722	+	0.990049i	$0.455057\pi$
$68$	0	0
$69$	28.7615	3.46248
$70$	0	0
$71$	6.69275	0.794284	0.397142	−	0.917757i	$-0.370002\pi$
$71$	6.69275	0.794284	0.397142	+	0.917757i	$0.370002\pi$
$72$	0	0
$73$	9.58317	1.12163	0.560813	−	0.827943i	$-0.310489\pi$
$73$	9.58317	1.12163	0.560813	+	0.827943i	$0.310489\pi$
$74$	0	0
$75$	−31.4139	−3.62736
$76$	0	0
$77$	−4.46413	−0.508735
$78$	0	0
$79$	5.58062	0.627869	0.313934	−	0.949445i	$-0.398353\pi$
$79$	5.58062	0.627869	0.313934	+	0.949445i	$0.398353\pi$
$80$	0	0
$81$	27.1983	3.02203
$82$	0	0
$83$	−8.10186	−0.889295	−0.444648	−	0.895706i	$-0.646671\pi$
$83$	−8.10186	−0.889295	−0.444648	+	0.895706i	$0.646671\pi$
$84$	0	0
$85$	−3.82141	−0.414491
$86$	0	0
$87$	25.1105	2.69212
$88$	0	0
$89$	8.45021	0.895720	0.447860	−	0.894104i	$-0.352186\pi$
$89$	8.45021	0.895720	0.447860	+	0.894104i	$0.352186\pi$
$90$	0	0
$91$	−2.44920	−0.256746
$92$	0	0
$93$	−15.5545	−1.61293
$94$	0	0
$95$	−21.3512	−2.19059
$96$	0	0
$97$	−3.32787	−0.337894	−0.168947	−	0.985625i	$-0.554037\pi$
$97$	−3.32787	−0.337894	−0.168947	+	0.985625i	$0.554037\pi$
$98$	0	0
$99$	31.2337	3.13910
$100$	0	0
$101$	5.24482	0.521879	0.260940	−	0.965355i	$-0.415968\pi$
$101$	5.24482	0.521879	0.260940	+	0.965355i	$0.415968\pi$
$102$	0	0
$103$	−18.9660	−1.86877	−0.934387	−	0.356259i	$-0.884052\pi$
$103$	−18.9660	−1.86877	−0.934387	+	0.356259i	$0.884052\pi$
$104$	0	0
$105$	13.7585	1.34269
$106$	0	0
$107$	12.4038	1.19912	0.599558	−	0.800331i	$-0.295343\pi$
$107$	12.4038	1.19912	0.599558	+	0.800331i	$0.295343\pi$
$108$	0	0
$109$	−1.00930	−0.0966734	−0.0483367	−	0.998831i	$-0.515392\pi$
$109$	−1.00930	−0.0966734	−0.0483367	+	0.998831i	$0.515392\pi$
$110$	0	0
$111$	−4.24063	−0.402502
$112$	0	0
$113$	−7.76908	−0.730853	−0.365427	−	0.930840i	$-0.619077\pi$
$113$	−7.76908	−0.730853	−0.365427	+	0.930840i	$0.619077\pi$
$114$	0	0
$115$	−33.5993	−3.13315
$116$	0	0
$117$	17.1361	1.58423
$118$	0	0
$119$	1.10063	0.100895
$120$	0	0
$121$	5.45090	0.495536
$122$	0	0
$123$	22.4250	2.02199
$124$	0	0
$125$	17.5908	1.57337
$126$	0	0
$127$	3.32657	0.295186	0.147593	−	0.989048i	$-0.452848\pi$
$127$	3.32657	0.295186	0.147593	+	0.989048i	$0.452848\pi$
$128$	0	0
$129$	8.69159	0.765252
$130$	0	0
$131$	7.21932	0.630755	0.315378	−	0.948966i	$-0.397869\pi$
$131$	7.21932	0.630755	0.315378	+	0.948966i	$0.397869\pi$
$132$	0	0
$133$	6.14950	0.533230
$134$	0	0
$135$	−58.7610	−5.05734
$136$	0	0
$137$	−15.9196	−1.36010	−0.680050	−	0.733166i	$-0.738042\pi$
$137$	−15.9196	−1.36010	−0.680050	+	0.733166i	$0.738042\pi$
$138$	0	0
$139$	−10.5516	−0.894977	−0.447489	−	0.894290i	$-0.647681\pi$
$139$	−10.5516	−0.894977	−0.447489	+	0.894290i	$0.647681\pi$
$140$	0	0
$141$	38.3429	3.22905
$142$	0	0
$143$	9.02563	0.754761
$144$	0	0
$145$	−29.3341	−2.43607
$146$	0	0
$147$	18.9356	1.56178
$148$	0	0
$149$	−11.1086	−0.910048	−0.455024	−	0.890479i	$-0.650369\pi$
$149$	−11.1086	−0.910048	−0.455024	+	0.890479i	$0.650369\pi$
$150$	0	0
$151$	12.1732	0.990642	0.495321	−	0.868710i	$-0.335050\pi$
$151$	12.1732	0.990642	0.495321	+	0.868710i	$0.335050\pi$
$152$	0	0
$153$	−7.70067	−0.622562
$154$	0	0
$155$	18.1708	1.45952
$156$	0	0
$157$	−5.52818	−0.441197	−0.220598	−	0.975365i	$-0.570801\pi$
$157$	−5.52818	−0.441197	−0.220598	+	0.975365i	$0.570801\pi$
$158$	0	0
$159$	31.3927	2.48960
$160$	0	0
$161$	9.67717	0.762668
$162$	0	0
$163$	−8.07857	−0.632762	−0.316381	−	0.948632i	$-0.602468\pi$
$163$	−8.07857	−0.632762	−0.316381	+	0.948632i	$0.602468\pi$
$164$	0	0
$165$	−50.7019	−3.94714
$166$	0	0
$167$	−6.57513	−0.508799	−0.254399	−	0.967099i	$-0.581878\pi$
$167$	−6.57513	−0.508799	−0.254399	+	0.967099i	$0.581878\pi$
$168$	0	0
$169$	−8.04817	−0.619090
$170$	0	0
$171$	−43.0255	−3.29024
$172$	0	0
$173$	9.85196	0.749031	0.374515	−	0.927221i	$-0.377809\pi$
$173$	9.85196	0.749031	0.374515	+	0.927221i	$0.377809\pi$
$174$	0	0
$175$	−10.5696	−0.798987
$176$	0	0
$177$	−3.27119	−0.245877
$178$	0	0
$179$	16.4294	1.22799	0.613997	−	0.789308i	$-0.289561\pi$
$179$	16.4294	1.22799	0.613997	+	0.789308i	$0.289561\pi$
$180$	0	0
$181$	1.87598	0.139441	0.0697203	−	0.997567i	$-0.477789\pi$
$181$	1.87598	0.139441	0.0697203	+	0.997567i	$0.477789\pi$
$182$	0	0
$183$	36.7603	2.71740
$184$	0	0
$185$	4.95392	0.364219
$186$	0	0
$187$	−4.05597	−0.296602
$188$	0	0
$189$	16.9242	1.23105
$190$	0	0
$191$	−12.5411	−0.907439	−0.453719	−	0.891145i	$-0.649903\pi$
$191$	−12.5411	−0.907439	−0.453719	+	0.891145i	$0.649903\pi$
$192$	0	0
$193$	−12.2994	−0.885327	−0.442663	−	0.896688i	$-0.645966\pi$
$193$	−12.2994	−0.885327	−0.442663	+	0.896688i	$0.645966\pi$
$194$	0	0
$195$	−27.8171	−1.99203
$196$	0	0
$197$	23.9912	1.70930	0.854652	−	0.519201i	$-0.173770\pi$
$197$	23.9912	1.70930	0.854652	+	0.519201i	$0.173770\pi$
$198$	0	0
$199$	19.2957	1.36784	0.683919	−	0.729558i	$-0.260274\pi$
$199$	19.2957	1.36784	0.683919	+	0.729558i	$0.260274\pi$
$200$	0	0
$201$	−7.53592	−0.531543
$202$	0	0
$203$	8.44873	0.592985
$204$	0	0
$205$	−26.1970	−1.82967
$206$	0	0
$207$	−67.7072	−4.70597
$208$	0	0
$209$	−22.6617	−1.56754
$210$	0	0
$211$	5.90494	0.406513	0.203257	−	0.979126i	$-0.434847\pi$
$211$	5.90494	0.406513	0.203257	+	0.979126i	$0.434847\pi$
$212$	0	0
$213$	−21.8932	−1.50010
$214$	0	0
$215$	−10.1536	−0.692467
$216$	0	0
$217$	−5.23350	−0.355273
$218$	0	0
$219$	−31.3483	−2.11832
$220$	0	0
$221$	−2.22527	−0.149688
$222$	0	0
$223$	18.0199	1.20670	0.603352	−	0.797475i	$-0.293831\pi$
$223$	18.0199	1.20670	0.603352	+	0.797475i	$0.293831\pi$
$224$	0	0
$225$	73.9511	4.93007
$226$	0	0
$227$	−15.8257	−1.05039	−0.525194	−	0.850983i	$-0.676007\pi$
$227$	−15.8257	−1.05039	−0.525194	+	0.850983i	$0.676007\pi$
$228$	0	0
$229$	−5.02899	−0.332325	−0.166163	−	0.986098i	$-0.553138\pi$
$229$	−5.02899	−0.332325	−0.166163	+	0.986098i	$0.553138\pi$
$230$	0	0
$231$	14.6030	0.960808
$232$	0	0
$233$	7.10234	0.465290	0.232645	−	0.972562i	$-0.425262\pi$
$233$	7.10234	0.465290	0.232645	+	0.972562i	$0.425262\pi$
$234$	0	0
$235$	−44.7923	−2.92193
$236$	0	0
$237$	−18.2552	−1.18581
$238$	0	0
$239$	5.04925	0.326609	0.163304	−	0.986576i	$-0.447785\pi$
$239$	5.04925	0.326609	0.163304	+	0.986576i	$0.447785\pi$
$240$	0	0
$241$	−13.8701	−0.893453	−0.446727	−	0.894671i	$-0.647410\pi$
$241$	−13.8701	−0.893453	−0.446727	+	0.894671i	$0.647410\pi$
$242$	0	0
$243$	−42.8403	−2.74821
$244$	0	0
$245$	−22.1207	−1.41324
$246$	0	0
$247$	−12.4331	−0.791101
$248$	0	0
$249$	26.5027	1.67954
$250$	0	0
$251$	−10.5787	−0.667724	−0.333862	−	0.942622i	$-0.608352\pi$
$251$	−10.5787	−0.667724	−0.333862	+	0.942622i	$0.608352\pi$
$252$	0	0
$253$	−35.6616	−2.24203
$254$	0	0
$255$	12.5006	0.782816
$256$	0	0
$257$	19.2234	1.19913	0.599563	−	0.800328i	$-0.295341\pi$
$257$	19.2234	1.19913	0.599563	+	0.800328i	$0.295341\pi$
$258$	0	0
$259$	−1.42681	−0.0886578
$260$	0	0
$261$	−59.1123	−3.65896
$262$	0	0
$263$	3.33295	0.205518	0.102759	−	0.994706i	$-0.467233\pi$
$263$	3.33295	0.205518	0.102759	+	0.994706i	$0.467233\pi$
$264$	0	0
$265$	−36.6731	−2.25281
$266$	0	0
$267$	−27.6422	−1.69168
$268$	0	0
$269$	−21.6195	−1.31817	−0.659083	−	0.752071i	$-0.729055\pi$
$269$	−21.6195	−1.31817	−0.659083	+	0.752071i	$0.729055\pi$
$270$	0	0
$271$	17.5371	1.06530	0.532651	−	0.846335i	$-0.321196\pi$
$271$	17.5371	1.06530	0.532651	+	0.846335i	$0.321196\pi$
$272$	0	0
$273$	8.01181	0.484897
$274$	0	0
$275$	38.9503	2.34879
$276$	0	0
$277$	−20.4180	−1.22680	−0.613400	−	0.789773i	$-0.710198\pi$
$277$	−20.4180	−1.22680	−0.613400	+	0.789773i	$0.710198\pi$
$278$	0	0
$279$	36.6167	2.19218
$280$	0	0
$281$	−9.57349	−0.571106	−0.285553	−	0.958363i	$-0.592177\pi$
$281$	−9.57349	−0.571106	−0.285553	+	0.958363i	$0.592177\pi$
$282$	0	0
$283$	−20.0460	−1.19161	−0.595806	−	0.803129i	$-0.703167\pi$
$283$	−20.0460	−1.19161	−0.595806	+	0.803129i	$0.703167\pi$
$284$	0	0
$285$	69.8437	4.13719
$286$	0	0
$287$	7.54517	0.445377
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	10.8861	0.638153
$292$	0	0
$293$	−17.7665	−1.03793	−0.518965	−	0.854795i	$-0.673683\pi$
$293$	−17.7665	−1.03793	−0.518965	+	0.854795i	$0.673683\pi$
$294$	0	0
$295$	3.82141	0.222491
$296$	0	0
$297$	−62.3677	−3.61894
$298$	0	0
$299$	−19.5654	−1.13150
$300$	0	0
$301$	2.92440	0.168559
$302$	0	0
$303$	−17.1568	−0.985632
$304$	0	0
$305$	−42.9435	−2.45894
$306$	0	0
$307$	12.0091	0.685396	0.342698	−	0.939446i	$-0.388659\pi$
$307$	12.0091	0.685396	0.342698	+	0.939446i	$0.388659\pi$
$308$	0	0
$309$	62.0413	3.52941
$310$	0	0
$311$	−20.1162	−1.14068	−0.570342	−	0.821407i	$-0.693189\pi$
$311$	−20.1162	−1.14068	−0.570342	+	0.821407i	$0.693189\pi$
$312$	0	0
$313$	13.4115	0.758062	0.379031	−	0.925384i	$-0.376257\pi$
$313$	13.4115	0.758062	0.379031	+	0.925384i	$0.376257\pi$
$314$	0	0
$315$	−32.3888	−1.82490
$316$	0	0
$317$	17.0862	0.959656	0.479828	−	0.877362i	$-0.340699\pi$
$317$	17.0862	0.959656	0.479828	+	0.877362i	$0.340699\pi$
$318$	0	0
$319$	−31.1347	−1.74321
$320$	0	0
$321$	−40.5750	−2.26468
$322$	0	0
$323$	5.58725	0.310883
$324$	0	0
$325$	21.3697	1.18538
$326$	0	0
$327$	3.30161	0.182579
$328$	0	0
$329$	12.9009	0.711252
$330$	0	0
$331$	−25.9892	−1.42850	−0.714249	−	0.699892i	$-0.753232\pi$
$331$	−25.9892	−1.42850	−0.714249	+	0.699892i	$0.753232\pi$
$332$	0	0
$333$	9.98281	0.547055
$334$	0	0
$335$	8.80349	0.480986
$336$	0	0
$337$	−6.10967	−0.332815	−0.166408	−	0.986057i	$-0.553217\pi$
$337$	−6.10967	−0.332815	−0.166408	+	0.986057i	$0.553217\pi$
$338$	0	0
$339$	25.4141	1.38030
$340$	0	0
$341$	19.2861	1.04440
$342$	0	0
$343$	14.0756	0.760009
$344$	0	0
$345$	109.910	5.91734
$346$	0	0
$347$	−16.8679	−0.905517	−0.452758	−	0.891633i	$-0.649560\pi$
$347$	−16.8679	−0.905517	−0.452758	+	0.891633i	$0.649560\pi$
$348$	0	0
$349$	8.48831	0.454369	0.227184	−	0.973852i	$-0.427048\pi$
$349$	8.48831	0.454369	0.227184	+	0.973852i	$0.427048\pi$
$350$	0	0
$351$	−34.2175	−1.82639
$352$	0	0
$353$	4.53541	0.241396	0.120698	−	0.992689i	$-0.461487\pi$
$353$	4.53541	0.241396	0.120698	+	0.992689i	$0.461487\pi$
$354$	0	0
$355$	25.5758	1.35742
$356$	0	0
$357$	−3.60037	−0.190552
$358$	0	0
$359$	−1.06421	−0.0561670	−0.0280835	−	0.999606i	$-0.508940\pi$
$359$	−1.06421	−0.0561670	−0.0280835	+	0.999606i	$0.508940\pi$
$360$	0	0
$361$	12.2173	0.643017
$362$	0	0
$363$	−17.8309	−0.935880
$364$	0	0
$365$	36.6213	1.91684
$366$	0	0
$367$	23.9923	1.25239	0.626193	−	0.779668i	$-0.284612\pi$
$367$	23.9923	1.25239	0.626193	+	0.779668i	$0.284612\pi$
$368$	0	0
$369$	−52.7904	−2.74816
$370$	0	0
$371$	10.5625	0.548376
$372$	0	0
$373$	28.7593	1.48910	0.744549	−	0.667567i	$-0.232664\pi$
$373$	28.7593	1.48910	0.744549	+	0.667567i	$0.232664\pi$
$374$	0	0
$375$	−57.5427	−2.97149
$376$	0	0
$377$	−17.0817	−0.879754
$378$	0	0
$379$	−24.7869	−1.27322	−0.636609	−	0.771187i	$-0.719663\pi$
$379$	−24.7869	−1.27322	−0.636609	+	0.771187i	$0.719663\pi$
$380$	0	0
$381$	−10.8818	−0.557494
$382$	0	0
$383$	−6.84679	−0.349855	−0.174927	−	0.984581i	$-0.555969\pi$
$383$	−6.84679	−0.349855	−0.174927	+	0.984581i	$0.555969\pi$
$384$	0	0
$385$	−17.0593	−0.869422
$386$	0	0
$387$	−20.4608	−1.04008
$388$	0	0
$389$	19.0487	0.965809	0.482904	−	0.875673i	$-0.339582\pi$
$389$	19.0487	0.965809	0.482904	+	0.875673i	$0.339582\pi$
$390$	0	0
$391$	8.79238	0.444650
$392$	0	0
$393$	−23.6158	−1.19126
$394$	0	0
$395$	21.3259	1.07302
$396$	0	0
$397$	1.73345	0.0869992	0.0434996	−	0.999053i	$-0.486149\pi$
$397$	1.73345	0.0869992	0.0434996	+	0.999053i	$0.486149\pi$
$398$	0	0
$399$	−20.1162	−1.00707
$400$	0	0
$401$	27.5134	1.37395	0.686976	−	0.726680i	$-0.258938\pi$
$401$	27.5134	1.37395	0.686976	+	0.726680i	$0.258938\pi$
$402$	0	0
$403$	10.5812	0.527085
$404$	0	0
$405$	103.936	5.16461
$406$	0	0
$407$	5.25798	0.260629
$408$	0	0
$409$	−0.234273	−0.0115840	−0.00579202	−	0.999983i	$-0.501844\pi$
$409$	−0.234273	−0.0115840	−0.00579202	+	0.999983i	$0.501844\pi$
$410$	0	0
$411$	52.0759	2.56871
$412$	0	0
$413$	−1.10063	−0.0541586
$414$	0	0
$415$	−30.9606	−1.51980
$416$	0	0
$417$	34.5163	1.69027
$418$	0	0
$419$	9.31142	0.454893	0.227446	−	0.973791i	$-0.426962\pi$
$419$	9.31142	0.454893	0.227446	+	0.973791i	$0.426962\pi$
$420$	0	0
$421$	28.3807	1.38319	0.691596	−	0.722285i	$-0.256908\pi$
$421$	28.3807	1.38319	0.691596	+	0.722285i	$0.256908\pi$
$422$	0	0
$423$	−90.2625	−4.38872
$424$	0	0
$425$	−9.60321	−0.465824
$426$	0	0
$427$	12.3685	0.598552
$428$	0	0
$429$	−29.5245	−1.42546
$430$	0	0
$431$	−14.4066	−0.693940	−0.346970	−	0.937876i	$-0.612789\pi$
$431$	−14.4066	−0.693940	−0.346970	+	0.937876i	$0.612789\pi$
$432$	0	0
$433$	−36.5511	−1.75653	−0.878267	−	0.478170i	$-0.841300\pi$
$433$	−36.5511	−1.75653	−0.878267	+	0.478170i	$0.841300\pi$
$434$	0	0
$435$	95.9575	4.60081
$436$	0	0
$437$	49.1252	2.34998
$438$	0	0
$439$	36.2178	1.72858	0.864292	−	0.502991i	$-0.167767\pi$
$439$	36.2178	1.72858	0.864292	+	0.502991i	$0.167767\pi$
$440$	0	0
$441$	−44.5761	−2.12267
$442$	0	0
$443$	30.6620	1.45679	0.728397	−	0.685155i	$-0.240266\pi$
$443$	30.6620	1.45679	0.728397	+	0.685155i	$0.240266\pi$
$444$	0	0
$445$	32.2917	1.53078
$446$	0	0
$447$	36.3382	1.71874
$448$	0	0
$449$	−22.7514	−1.07370	−0.536852	−	0.843676i	$-0.680387\pi$
$449$	−22.7514	−1.07370	−0.536852	+	0.843676i	$0.680387\pi$
$450$	0	0
$451$	−27.8049	−1.30928
$452$	0	0
$453$	−39.8209	−1.87095
$454$	0	0
$455$	−9.35943	−0.438777
$456$	0	0
$457$	−41.9599	−1.96280	−0.981401	−	0.191967i	$-0.938513\pi$
$457$	−41.9599	−1.96280	−0.981401	+	0.191967i	$0.938513\pi$
$458$	0	0
$459$	15.3768	0.717726
$460$	0	0
$461$	28.6681	1.33521	0.667603	−	0.744518i	$-0.267320\pi$
$461$	28.6681	1.33521	0.667603	+	0.744518i	$0.267320\pi$
$462$	0	0
$463$	−33.6589	−1.56426	−0.782131	−	0.623114i	$-0.785867\pi$
$463$	−33.6589	−1.56426	−0.782131	+	0.623114i	$0.785867\pi$
$464$	0	0
$465$	−59.4401	−2.75647
$466$	0	0
$467$	−37.3425	−1.72801	−0.864003	−	0.503487i	$-0.832050\pi$
$467$	−37.3425	−1.72801	−0.864003	+	0.503487i	$0.832050\pi$
$468$	0	0
$469$	−2.53556	−0.117081
$470$	0	0
$471$	18.0837	0.833253
$472$	0	0
$473$	−10.7768	−0.495517
$474$	0	0
$475$	−53.6555	−2.46188
$476$	0	0
$477$	−73.9012	−3.38371
$478$	0	0
$479$	−33.0170	−1.50859	−0.754293	−	0.656538i	$-0.772020\pi$
$479$	−33.0170	−1.50859	−0.754293	+	0.656538i	$0.772020\pi$
$480$	0	0
$481$	2.88474	0.131533
$482$	0	0
$483$	−31.6559	−1.44039
$484$	0	0
$485$	−12.7172	−0.577456
$486$	0	0
$487$	27.2360	1.23418	0.617090	−	0.786893i	$-0.288311\pi$
$487$	27.2360	1.23418	0.617090	+	0.786893i	$0.288311\pi$
$488$	0	0
$489$	26.4265	1.19505
$490$	0	0
$491$	−25.6768	−1.15878	−0.579388	−	0.815052i	$-0.696708\pi$
$491$	−25.6768	−1.15878	−0.579388	+	0.815052i	$0.696708\pi$
$492$	0	0
$493$	7.67625	0.345721
$494$	0	0
$495$	119.357	5.36469
$496$	0	0
$497$	−7.36626	−0.330422
$498$	0	0
$499$	36.6075	1.63878	0.819389	−	0.573237i	$-0.194313\pi$
$499$	36.6075	1.63878	0.819389	+	0.573237i	$0.194313\pi$
$500$	0	0
$501$	21.5085	0.960928
$502$	0	0
$503$	−12.0013	−0.535113	−0.267556	−	0.963542i	$-0.586216\pi$
$503$	−12.0013	−0.535113	−0.267556	+	0.963542i	$0.586216\pi$
$504$	0	0
$505$	20.0426	0.891885
$506$	0	0
$507$	26.3271	1.16923
$508$	0	0
$509$	−19.1208	−0.847513	−0.423757	−	0.905776i	$-0.639289\pi$
$509$	−19.1208	−0.847513	−0.423757	+	0.905776i	$0.639289\pi$
$510$	0	0
$511$	−10.5475	−0.466596
$512$	0	0
$513$	85.9138	3.79318
$514$	0	0
$515$	−72.4769	−3.19371
$516$	0	0
$517$	−47.5416	−2.09088
$518$	0	0
$519$	−32.2276	−1.41464
$520$	0	0
$521$	17.9633	0.786985	0.393493	−	0.919328i	$-0.371267\pi$
$521$	17.9633	0.786985	0.393493	+	0.919328i	$0.371267\pi$
$522$	0	0
$523$	12.6040	0.551134	0.275567	−	0.961282i	$-0.411134\pi$
$523$	12.6040	0.551134	0.275567	+	0.961282i	$0.411134\pi$
$524$	0	0
$525$	34.5751	1.50898
$526$	0	0
$527$	−4.75500	−0.207131
$528$	0	0
$529$	54.3059	2.36113
$530$	0	0
$531$	7.70067	0.334180
$532$	0	0
$533$	−15.2549	−0.660763
$534$	0	0
$535$	47.3999	2.04928
$536$	0	0
$537$	−53.7438	−2.31922
$538$	0	0
$539$	−23.4784	−1.01129
$540$	0	0
$541$	39.1769	1.68435	0.842174	−	0.539206i	$-0.181276\pi$
$541$	39.1769	1.68435	0.842174	+	0.539206i	$0.181276\pi$
$542$	0	0
$543$	−6.13669	−0.263351
$544$	0	0
$545$	−3.85695	−0.165214
$546$	0	0
$547$	21.5389	0.920936	0.460468	−	0.887676i	$-0.347682\pi$
$547$	21.5389	0.920936	0.460468	+	0.887676i	$0.347682\pi$
$548$	0	0
$549$	−86.5371	−3.69331
$550$	0	0
$551$	42.8891	1.82714
$552$	0	0
$553$	−6.14221	−0.261193
$554$	0	0
$555$	−16.2052	−0.687872
$556$	0	0
$557$	−4.62692	−0.196049	−0.0980244	−	0.995184i	$-0.531252\pi$
$557$	−4.62692	−0.196049	−0.0980244	+	0.995184i	$0.531252\pi$
$558$	0	0
$559$	−5.91258	−0.250075
$560$	0	0
$561$	13.2678	0.560169
$562$	0	0
$563$	−1.01676	−0.0428513	−0.0214257	−	0.999770i	$-0.506821\pi$
$563$	−1.01676	−0.0428513	−0.0214257	+	0.999770i	$0.506821\pi$
$564$	0	0
$565$	−29.6889	−1.24902
$566$	0	0
$567$	−29.9353	−1.25716
$568$	0	0
$569$	18.4491	0.773425	0.386713	−	0.922200i	$-0.373611\pi$
$569$	18.4491	0.773425	0.386713	+	0.922200i	$0.373611\pi$
$570$	0	0
$571$	29.4338	1.23177	0.615883	−	0.787837i	$-0.288799\pi$
$571$	29.4338	1.23177	0.615883	+	0.787837i	$0.288799\pi$
$572$	0	0
$573$	41.0241	1.71381
$574$	0	0
$575$	−84.4350	−3.52118
$576$	0	0
$577$	−32.4556	−1.35115	−0.675573	−	0.737293i	$-0.736104\pi$
$577$	−32.4556	−1.35115	−0.675573	+	0.737293i	$0.736104\pi$
$578$	0	0
$579$	40.2335	1.67205
$580$	0	0
$581$	8.91717	0.369947
$582$	0	0
$583$	−38.9241	−1.61207
$584$	0	0
$585$	65.4840	2.70743
$586$	0	0
$587$	13.7984	0.569520	0.284760	−	0.958599i	$-0.408086\pi$
$587$	13.7984	0.569520	0.284760	+	0.958599i	$0.408086\pi$
$588$	0	0
$589$	−26.5673	−1.09469
$590$	0	0
$591$	−78.4798	−3.22823
$592$	0	0
$593$	−47.6437	−1.95649	−0.978245	−	0.207452i	$-0.933483\pi$
$593$	−47.6437	−1.95649	−0.978245	+	0.207452i	$0.933483\pi$
$594$	0	0
$595$	4.20597	0.172428
$596$	0	0
$597$	−63.1199	−2.58333
$598$	0	0
$599$	−11.9684	−0.489017	−0.244508	−	0.969647i	$-0.578627\pi$
$599$	−11.9684	−0.489017	−0.244508	+	0.969647i	$0.578627\pi$
$600$	0	0
$601$	38.1249	1.55515	0.777574	−	0.628792i	$-0.216450\pi$
$601$	38.1249	1.55515	0.777574	+	0.628792i	$0.216450\pi$
$602$	0	0
$603$	17.7402	0.722438
$604$	0	0
$605$	20.8301	0.846866
$606$	0	0
$607$	−48.3892	−1.96406	−0.982029	−	0.188732i	$-0.939562\pi$
$607$	−48.3892	−1.96406	−0.982029	+	0.188732i	$0.939562\pi$
$608$	0	0
$609$	−27.6374	−1.11992
$610$	0	0
$611$	−26.0833	−1.05522
$612$	0	0
$613$	−41.0223	−1.65688	−0.828438	−	0.560080i	$-0.810770\pi$
$613$	−41.0223	−1.65688	−0.828438	+	0.560080i	$0.810770\pi$
$614$	0	0
$615$	85.6951	3.45556
$616$	0	0
$617$	−1.83156	−0.0737360	−0.0368680	−	0.999320i	$-0.511738\pi$
$617$	−1.83156	−0.0737360	−0.0368680	+	0.999320i	$0.511738\pi$
$618$	0	0
$619$	13.0869	0.526005	0.263003	−	0.964795i	$-0.415287\pi$
$619$	13.0869	0.526005	0.263003	+	0.964795i	$0.415287\pi$
$620$	0	0
$621$	135.198	5.42532
$622$	0	0
$623$	−9.30057	−0.372620
$624$	0	0
$625$	19.2056	0.768222
$626$	0	0
$627$	74.1307	2.96049
$628$	0	0
$629$	−1.29636	−0.0516891
$630$	0	0
$631$	−20.7142	−0.824618	−0.412309	−	0.911044i	$-0.635278\pi$
$631$	−20.7142	−0.824618	−0.412309	+	0.911044i	$0.635278\pi$
$632$	0	0
$633$	−19.3162	−0.767749
$634$	0	0
$635$	12.7122	0.504469
$636$	0	0
$637$	−12.8812	−0.510372
$638$	0	0
$639$	51.5387	2.03884
$640$	0	0
$641$	25.0404	0.989035	0.494518	−	0.869168i	$-0.335345\pi$
$641$	25.0404	0.989035	0.494518	+	0.869168i	$0.335345\pi$
$642$	0	0
$643$	0.573937	0.0226339	0.0113169	−	0.999936i	$-0.496398\pi$
$643$	0.573937	0.0226339	0.0113169	+	0.999936i	$0.496398\pi$
$644$	0	0
$645$	33.2142	1.30781
$646$	0	0
$647$	13.0674	0.513731	0.256865	−	0.966447i	$-0.417310\pi$
$647$	13.0674	0.513731	0.256865	+	0.966447i	$0.417310\pi$
$648$	0	0
$649$	4.05597	0.159211
$650$	0	0
$651$	17.1198	0.670977
$652$	0	0
$653$	−19.5129	−0.763598	−0.381799	−	0.924245i	$-0.624695\pi$
$653$	−19.5129	−0.763598	−0.381799	+	0.924245i	$0.624695\pi$
$654$	0	0
$655$	27.5880	1.07795
$656$	0	0
$657$	73.7968	2.87909
$658$	0	0
$659$	25.7407	1.00271	0.501357	−	0.865240i	$-0.332834\pi$
$659$	25.7407	1.00271	0.501357	+	0.865240i	$0.332834\pi$
$660$	0	0
$661$	24.6236	0.957746	0.478873	−	0.877884i	$-0.341045\pi$
$661$	24.6236	0.957746	0.478873	+	0.877884i	$0.341045\pi$
$662$	0	0
$663$	7.27928	0.282704
$664$	0	0
$665$	23.4998	0.911283
$666$	0	0
$667$	67.4925	2.61332
$668$	0	0
$669$	−58.9466	−2.27901
$670$	0	0
$671$	−45.5794	−1.75957
$672$	0	0
$673$	−20.2603	−0.780977	−0.390488	−	0.920608i	$-0.627694\pi$
$673$	−20.2603	−0.780977	−0.390488	+	0.920608i	$0.627694\pi$
$674$	0	0
$675$	−147.666	−5.68368
$676$	0	0
$677$	16.2029	0.622729	0.311364	−	0.950291i	$-0.399214\pi$
$677$	16.2029	0.622729	0.311364	+	0.950291i	$0.399214\pi$
$678$	0	0
$679$	3.66276	0.140564
$680$	0	0
$681$	51.7688	1.98378
$682$	0	0
$683$	5.17044	0.197841	0.0989206	−	0.995095i	$-0.468461\pi$
$683$	5.17044	0.197841	0.0989206	+	0.995095i	$0.468461\pi$
$684$	0	0
$685$	−60.8352	−2.32440
$686$	0	0
$687$	16.4508	0.627636
$688$	0	0
$689$	−21.3553	−0.813573
$690$	0	0
$691$	38.6586	1.47064	0.735321	−	0.677719i	$-0.237031\pi$
$691$	38.6586	1.47064	0.735321	+	0.677719i	$0.237031\pi$
$692$	0	0
$693$	−34.3768	−1.30587
$694$	0	0
$695$	−40.3221	−1.52951
$696$	0	0
$697$	6.85530	0.259663
$698$	0	0
$699$	−23.2331	−0.878756
$700$	0	0
$701$	14.5356	0.549003	0.274501	−	0.961587i	$-0.411487\pi$
$701$	14.5356	0.549003	0.274501	+	0.961587i	$0.411487\pi$
$702$	0	0
$703$	−7.24306	−0.273177
$704$	0	0
$705$	146.524	5.51841
$706$	0	0
$707$	−5.77262	−0.217102
$708$	0	0
$709$	−10.7826	−0.404948	−0.202474	−	0.979288i	$-0.564898\pi$
$709$	−10.7826	−0.404948	−0.202474	+	0.979288i	$0.564898\pi$
$710$	0	0
$711$	42.9745	1.61167
$712$	0	0
$713$	−41.8077	−1.56571
$714$	0	0
$715$	34.4907	1.28988
$716$	0	0
$717$	−16.5170	−0.616840
$718$	0	0
$719$	−39.8258	−1.48525	−0.742625	−	0.669707i	$-0.766420\pi$
$719$	−39.8258	−1.48525	−0.742625	+	0.669707i	$0.766420\pi$
$720$	0	0
$721$	20.8746	0.777410
$722$	0	0
$723$	45.3718	1.68739
$724$	0	0
$725$	−73.7166	−2.73777
$726$	0	0
$727$	−2.38628	−0.0885022	−0.0442511	−	0.999020i	$-0.514090\pi$
$727$	−2.38628	−0.0885022	−0.0442511	+	0.999020i	$0.514090\pi$
$728$	0	0
$729$	58.5440	2.16830
$730$	0	0
$731$	2.65701	0.0982732
$732$	0	0
$733$	32.2166	1.18995	0.594973	−	0.803746i	$-0.297163\pi$
$733$	32.2166	1.18995	0.594973	+	0.803746i	$0.297163\pi$
$734$	0	0
$735$	72.3609	2.66907
$736$	0	0
$737$	9.34385	0.344185
$738$	0	0
$739$	51.6588	1.90030	0.950150	−	0.311792i	$-0.100929\pi$
$739$	51.6588	1.90030	0.950150	+	0.311792i	$0.100929\pi$
$740$	0	0
$741$	40.6711	1.49409
$742$	0	0
$743$	51.4392	1.88712	0.943561	−	0.331199i	$-0.107453\pi$
$743$	51.4392	1.88712	0.943561	+	0.331199i	$0.107453\pi$
$744$	0	0
$745$	−42.4504	−1.55526
$746$	0	0
$747$	−62.3898	−2.28272
$748$	0	0
$749$	−13.6520	−0.498832
$750$	0	0
$751$	−12.1121	−0.441976	−0.220988	−	0.975277i	$-0.570928\pi$
$751$	−12.1121	−0.441976	−0.220988	+	0.975277i	$0.570928\pi$
$752$	0	0
$753$	34.6050	1.26108
$754$	0	0
$755$	46.5189	1.69300
$756$	0	0
$757$	29.0001	1.05403	0.527014	−	0.849857i	$-0.323312\pi$
$757$	29.0001	1.05403	0.527014	+	0.849857i	$0.323312\pi$
$758$	0	0
$759$	116.656	4.23434
$760$	0	0
$761$	−6.67684	−0.242035	−0.121018	−	0.992650i	$-0.538616\pi$
$761$	−6.67684	−0.242035	−0.121018	+	0.992650i	$0.538616\pi$
$762$	0	0
$763$	1.11087	0.0402161
$764$	0	0
$765$	−29.4274	−1.06395
$766$	0	0
$767$	2.22527	0.0803499
$768$	0	0
$769$	−37.9377	−1.36807	−0.684034	−	0.729450i	$-0.739776\pi$
$769$	−37.9377	−1.36807	−0.684034	+	0.729450i	$0.739776\pi$
$770$	0	0
$771$	−62.8835	−2.26469
$772$	0	0
$773$	9.39476	0.337906	0.168953	−	0.985624i	$-0.445961\pi$
$773$	9.39476	0.337906	0.168953	+	0.985624i	$0.445961\pi$
$774$	0	0
$775$	45.6632	1.64027
$776$	0	0
$777$	4.66737	0.167441
$778$	0	0
$779$	38.3023	1.37232
$780$	0	0
$781$	27.1456	0.971346
$782$	0	0
$783$	118.036	4.21826
$784$	0	0
$785$	−21.1255	−0.754000
$786$	0	0
$787$	48.5118	1.72926	0.864630	−	0.502410i	$-0.167553\pi$
$787$	48.5118	1.72926	0.864630	+	0.502410i	$0.167553\pi$
$788$	0	0
$789$	−10.9027	−0.388146
$790$	0	0
$791$	8.55090	0.304035
$792$	0	0
$793$	−25.0067	−0.888014
$794$	0	0
$795$	119.965	4.25470
$796$	0	0
$797$	17.8000	0.630509	0.315254	−	0.949007i	$-0.397910\pi$
$797$	17.8000	0.630509	0.315254	+	0.949007i	$0.397910\pi$
$798$	0	0
$799$	11.7214	0.414673
$800$	0	0
$801$	65.0722	2.29921
$802$	0	0
$803$	38.8691	1.37166
$804$	0	0
$805$	36.9805	1.30339
$806$	0	0
$807$	70.7215	2.48951
$808$	0	0
$809$	−43.7199	−1.53711	−0.768555	−	0.639784i	$-0.779024\pi$
$809$	−43.7199	−1.53711	−0.768555	+	0.639784i	$0.779024\pi$
$810$	0	0
$811$	−10.2920	−0.361400	−0.180700	−	0.983538i	$-0.557836\pi$
$811$	−10.2920	−0.361400	−0.180700	+	0.983538i	$0.557836\pi$
$812$	0	0
$813$	−57.3671	−2.01195
$814$	0	0
$815$	−30.8715	−1.08138
$816$	0	0
$817$	14.8454	0.519375
$818$	0	0
$819$	−18.8605	−0.659040
$820$	0	0
$821$	−2.85285	−0.0995651	−0.0497826	−	0.998760i	$-0.515853\pi$
$821$	−2.85285	−0.0995651	−0.0497826	+	0.998760i	$0.515853\pi$
$822$	0	0
$823$	39.7632	1.38606	0.693030	−	0.720909i	$-0.256275\pi$
$823$	39.7632	1.38606	0.693030	+	0.720909i	$0.256275\pi$
$824$	0	0
$825$	−127.414	−4.43598
$826$	0	0
$827$	44.2553	1.53891	0.769454	−	0.638702i	$-0.220529\pi$
$827$	44.2553	1.53891	0.769454	+	0.638702i	$0.220529\pi$
$828$	0	0
$829$	−21.8192	−0.757811	−0.378905	−	0.925435i	$-0.623699\pi$
$829$	−21.8192	−0.757811	−0.378905	+	0.925435i	$0.623699\pi$
$830$	0	0
$831$	66.7911	2.31696
$832$	0	0
$833$	5.78861	0.200563
$834$	0	0
$835$	−25.1263	−0.869531
$836$	0	0
$837$	−73.1165	−2.52727
$838$	0	0
$839$	−9.77123	−0.337340	−0.168670	−	0.985673i	$-0.553947\pi$
$839$	−9.77123	−0.337340	−0.168670	+	0.985673i	$0.553947\pi$
$840$	0	0
$841$	29.9248	1.03189
$842$	0	0
$843$	31.3167	1.07860
$844$	0	0
$845$	−30.7554	−1.05802
$846$	0	0
$847$	−5.99944	−0.206143
$848$	0	0
$849$	65.5742	2.25050
$850$	0	0
$851$	−11.3981	−0.390720
$852$	0	0
$853$	33.6353	1.15165	0.575826	−	0.817573i	$-0.304681\pi$
$853$	33.6353	1.15165	0.575826	+	0.817573i	$0.304681\pi$
$854$	0	0
$855$	−164.418	−5.62299
$856$	0	0
$857$	2.09687	0.0716278	0.0358139	−	0.999358i	$-0.488598\pi$
$857$	2.09687	0.0716278	0.0358139	+	0.999358i	$0.488598\pi$
$858$	0	0
$859$	−40.5154	−1.38237	−0.691183	−	0.722680i	$-0.742910\pi$
$859$	−40.5154	−1.38237	−0.691183	+	0.722680i	$0.742910\pi$
$860$	0	0
$861$	−24.6817	−0.841149
$862$	0	0
$863$	−34.0458	−1.15893	−0.579466	−	0.814996i	$-0.696739\pi$
$863$	−34.0458	−1.15893	−0.579466	+	0.814996i	$0.696739\pi$
$864$	0	0
$865$	37.6484	1.28009
$866$	0	0
$867$	−3.27119	−0.111095
$868$	0	0
$869$	22.6348	0.767834
$870$	0	0
$871$	5.12641	0.173702
$872$	0	0
$873$	−25.6268	−0.867335
$874$	0	0
$875$	−19.3610	−0.654520
$876$	0	0
$877$	30.9700	1.04578	0.522892	−	0.852399i	$-0.324853\pi$
$877$	30.9700	1.04578	0.522892	+	0.852399i	$0.324853\pi$
$878$	0	0
$879$	58.1176	1.96026
$880$	0	0
$881$	16.3507	0.550868	0.275434	−	0.961320i	$-0.411178\pi$
$881$	16.3507	0.550868	0.275434	+	0.961320i	$0.411178\pi$
$882$	0	0
$883$	−51.3129	−1.72682	−0.863409	−	0.504505i	$-0.831675\pi$
$883$	−51.3129	−1.72682	−0.863409	+	0.504505i	$0.831675\pi$
$884$	0	0
$885$	−12.5006	−0.420202
$886$	0	0
$887$	−5.76280	−0.193496	−0.0967479	−	0.995309i	$-0.530844\pi$
$887$	−5.76280	−0.193496	−0.0967479	+	0.995309i	$0.530844\pi$
$888$	0	0
$889$	−3.66133	−0.122797
$890$	0	0
$891$	110.315	3.69570
$892$	0	0
$893$	65.4903	2.19155
$894$	0	0
$895$	62.7837	2.09863
$896$	0	0
$897$	64.0021	2.13697
$898$	0	0
$899$	−36.5006	−1.21736
$900$	0	0
$901$	9.59673	0.319714
$902$	0	0
$903$	−9.56625	−0.318345
$904$	0	0
$905$	7.16891	0.238303
$906$	0	0
$907$	−21.6904	−0.720218	−0.360109	−	0.932910i	$-0.617261\pi$
$907$	−21.6904	−0.720218	−0.360109	+	0.932910i	$0.617261\pi$
$908$	0	0
$909$	40.3886	1.33961
$910$	0	0
$911$	17.7764	0.588957	0.294479	−	0.955658i	$-0.404854\pi$
$911$	17.7764	0.588957	0.294479	+	0.955658i	$0.404854\pi$
$912$	0	0
$913$	−32.8609	−1.08754
$914$	0	0
$915$	140.476	4.64401
$916$	0	0
$917$	−7.94582	−0.262394
$918$	0	0
$919$	−29.0756	−0.959117	−0.479558	−	0.877510i	$-0.659203\pi$
$919$	−29.0756	−0.959117	−0.479558	+	0.877510i	$0.659203\pi$
$920$	0	0
$921$	−39.2841	−1.29445
$922$	0	0
$923$	14.8932	0.490215
$924$	0	0
$925$	12.4492	0.409327
$926$	0	0
$927$	−146.051	−4.79694
$928$	0	0
$929$	−13.6276	−0.447108	−0.223554	−	0.974692i	$-0.571766\pi$
$929$	−13.6276	−0.447108	−0.223554	+	0.974692i	$0.571766\pi$
$930$	0	0
$931$	32.3424	1.05998
$932$	0	0
$933$	65.8038	2.15432
$934$	0	0
$935$	−15.4995	−0.506889
$936$	0	0
$937$	−29.3366	−0.958384	−0.479192	−	0.877710i	$-0.659070\pi$
$937$	−29.3366	−0.958384	−0.479192	+	0.877710i	$0.659070\pi$
$938$	0	0
$939$	−43.8715	−1.43169
$940$	0	0
$941$	−55.3831	−1.80544	−0.902718	−	0.430232i	$-0.858432\pi$
$941$	−55.3831	−1.80544	−0.902718	+	0.430232i	$0.858432\pi$
$942$	0	0
$943$	60.2744	1.96280
$944$	0	0
$945$	64.6742	2.10385
$946$	0	0
$947$	25.8521	0.840080	0.420040	−	0.907505i	$-0.362016\pi$
$947$	25.8521	0.840080	0.420040	+	0.907505i	$0.362016\pi$
$948$	0	0
$949$	21.3251	0.692243
$950$	0	0
$951$	−55.8922	−1.81243
$952$	0	0
$953$	−28.0028	−0.907100	−0.453550	−	0.891231i	$-0.649843\pi$
$953$	−28.0028	−0.907100	−0.453550	+	0.891231i	$0.649843\pi$
$954$	0	0
$955$	−47.9246	−1.55080
$956$	0	0
$957$	101.847	3.29225
$958$	0	0
$959$	17.5216	0.565802
$960$	0	0
$961$	−8.39000	−0.270645
$962$	0	0
$963$	95.5172	3.07800
$964$	0	0
$965$	−47.0009	−1.51301
$966$	0	0
$967$	−42.3369	−1.36146	−0.680732	−	0.732533i	$-0.738338\pi$
$967$	−42.3369	−1.36146	−0.680732	+	0.732533i	$0.738338\pi$
$968$	0	0
$969$	−18.2769	−0.587140
$970$	0	0
$971$	−33.4135	−1.07229	−0.536145	−	0.844126i	$-0.680120\pi$
$971$	−33.4135	−1.07229	−0.536145	+	0.844126i	$0.680120\pi$
$972$	0	0
$973$	11.6135	0.372310
$974$	0	0
$975$	−69.9044	−2.23873
$976$	0	0
$977$	24.3795	0.779969	0.389984	−	0.920821i	$-0.372480\pi$
$977$	24.3795	0.779969	0.389984	+	0.920821i	$0.372480\pi$
$978$	0	0
$979$	34.2738	1.09540
$980$	0	0
$981$	−7.77228	−0.248150
$982$	0	0
$983$	−49.0524	−1.56453	−0.782265	−	0.622946i	$-0.785936\pi$
$983$	−49.0524	−1.56453	−0.782265	+	0.622946i	$0.785936\pi$
$984$	0	0
$985$	91.6804	2.92118
$986$	0	0
$987$	−42.2014	−1.34329
$988$	0	0
$989$	23.3615	0.742852
$990$	0	0
$991$	−50.6874	−1.61014	−0.805068	−	0.593182i	$-0.797872\pi$
$991$	−50.6874	−1.61014	−0.805068	+	0.593182i	$0.797872\pi$
$992$	0	0
$993$	85.0157	2.69789
$994$	0	0
$995$	73.7369	2.33762
$996$	0	0
$997$	−33.7304	−1.06825	−0.534126	−	0.845405i	$-0.679359\pi$
$997$	−33.7304	−1.06825	−0.534126	+	0.845405i	$0.679359\pi$
$998$	0	0
$999$	−19.9338	−0.630676

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.g.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-3.27119 q^{3} +3.82141 q^{5} -1.10063 q^{7} +7.70067 q^{9} +O(q^{10})\)	\(q-3.27119 q^{3} +3.82141 q^{5} -1.10063 q^{7} +7.70067 q^{9} +4.05597 q^{11} +2.22527 q^{13} -12.5006 q^{15} -1.00000 q^{17} -5.58725 q^{19} +3.60037 q^{21} -8.79238 q^{23} +9.60321 q^{25} -15.3768 q^{27} -7.67625 q^{29} +4.75500 q^{31} -13.2678 q^{33} -4.20597 q^{35} +1.29636 q^{37} -7.27928 q^{39} -6.85530 q^{41} -2.65701 q^{43} +29.4274 q^{45} -11.7214 q^{47} -5.78861 q^{49} +3.27119 q^{51} -9.59673 q^{53} +15.4995 q^{55} +18.2769 q^{57} +1.00000 q^{59} -11.2376 q^{61} -8.47560 q^{63} +8.50368 q^{65} +2.30373 q^{67} +28.7615 q^{69} +6.69275 q^{71} +9.58317 q^{73} -31.4139 q^{75} -4.46413 q^{77} +5.58062 q^{79} +27.1983 q^{81} -8.10186 q^{83} -3.82141 q^{85} +25.1105 q^{87} +8.45021 q^{89} -2.44920 q^{91} -15.5545 q^{93} -21.3512 q^{95} -3.32787 q^{97} +31.2337 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 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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