LMFDB - Embedded newform 4016.2.a.j.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, 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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 27 x^{12} + 79 x^{11} + 274 x^{10} - 747 x^{9} - 1422 x^{8} + 3287 x^{7} + \cdots - 196 $ x^14 - 3x^13 - 27x^12 + 79x^11 + 274x^10 - 747x^9 - 1422x^8 + 3287x^7 + 4161x^6 - 6861x^5 - 6676x^4 + 5599x^3 + 4627x^2 - 359*x - 196
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1004)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$-3.43087$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.43087 q^{3} -0.723735 q^{5} -3.63900 q^{7} +8.77086 q^{9} +O(q^{10})$	$q+3.43087 q^{3} -0.723735 q^{5} -3.63900 q^{7} +8.77086 q^{9} +2.52672 q^{11} -5.63831 q^{13} -2.48304 q^{15} -6.45157 q^{17} -6.03003 q^{19} -12.4849 q^{21} -6.51176 q^{23} -4.47621 q^{25} +19.7991 q^{27} +1.42186 q^{29} -1.46844 q^{31} +8.66884 q^{33} +2.63367 q^{35} +9.25058 q^{37} -19.3443 q^{39} -3.54493 q^{41} -2.04958 q^{43} -6.34778 q^{45} +1.56997 q^{47} +6.24233 q^{49} -22.1345 q^{51} +0.205679 q^{53} -1.82868 q^{55} -20.6882 q^{57} -13.1714 q^{59} +3.53160 q^{61} -31.9172 q^{63} +4.08064 q^{65} -3.41564 q^{67} -22.3410 q^{69} -8.33065 q^{71} +10.3377 q^{73} -15.3573 q^{75} -9.19473 q^{77} -15.1079 q^{79} +41.6154 q^{81} +10.2212 q^{83} +4.66923 q^{85} +4.87821 q^{87} +8.98950 q^{89} +20.5178 q^{91} -5.03804 q^{93} +4.36414 q^{95} +12.6010 q^{97} +22.1615 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9}+O(q^{10}) $ 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9	$ 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9} - 9 q^{11} - q^{13} - 14 q^{15} - 27 q^{19} - 3 q^{21} - 13 q^{23} + 26 q^{25} - 15 q^{27} - 25 q^{31} + 16 q^{33} - 21 q^{35} - q^{37} - 33 q^{39} + 10 q^{41} - 35 q^{43} - 4 q^{45} - 6 q^{47} + 36 q^{49} - 48 q^{51} - q^{53} - 41 q^{55} + 14 q^{57} - 30 q^{59} + 3 q^{61} - 31 q^{63} + 7 q^{65} - 22 q^{67} - 17 q^{69} - 6 q^{71} + 5 q^{73} - 4 q^{75} - 14 q^{77} - 56 q^{79} + 26 q^{81} + 28 q^{83} - 23 q^{85} - 11 q^{87} - 24 q^{89} - 38 q^{91} - 55 q^{93} + 4 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) $ 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9 - 9 * q^11 - q^13 - 14 * q^15 - 27 * q^19 - 3 * q^21 - 13 * q^23 + 26 * q^25 - 15 * q^27 - 25 * q^31 + 16 * q^33 - 21 * q^35 - q^37 - 33 * q^39 + 10 * q^41 - 35 * q^43 - 4 * q^45 - 6 * q^47 + 36 * q^49 - 48 * q^51 - q^53 - 41 * q^55 + 14 * q^57 - 30 * q^59 + 3 * q^61 - 31 * q^63 + 7 * q^65 - 22 * q^67 - 17 * q^69 - 6 * q^71 + 5 * q^73 - 4 * q^75 - 14 * q^77 - 56 * q^79 + 26 * q^81 + 28 * q^83 - 23 * q^85 - 11 * q^87 - 24 * q^89 - 38 * q^91 - 55 * q^93 + 4 * q^95 + 6 * q^97 - 12 * 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
$3$	3.43087	1.98081	0.990407	−	0.138184i	$-0.0441267\pi$
$3$	3.43087	1.98081	0.990407	+	0.138184i	$0.0441267\pi$
$4$	0	0
$5$	−0.723735	−0.323664	−0.161832	−	0.986818i	$-0.551740\pi$
$5$	−0.723735	−0.323664	−0.161832	+	0.986818i	$0.551740\pi$
$6$	0	0
$7$	−3.63900	−1.37541	−0.687707	−	0.725989i	$-0.741382\pi$
$7$	−3.63900	−1.37541	−0.687707	+	0.725989i	$0.741382\pi$
$8$	0	0
$9$	8.77086	2.92362
$10$	0	0
$11$	2.52672	0.761834	0.380917	−	0.924609i	$-0.375608\pi$
$11$	2.52672	0.761834	0.380917	+	0.924609i	$0.375608\pi$
$12$	0	0
$13$	−5.63831	−1.56379	−0.781893	−	0.623413i	$-0.785746\pi$
$13$	−5.63831	−1.56379	−0.781893	+	0.623413i	$0.785746\pi$
$14$	0	0
$15$	−2.48304	−0.641118
$16$	0	0
$17$	−6.45157	−1.56474	−0.782368	−	0.622816i	$-0.785989\pi$
$17$	−6.45157	−1.56474	−0.782368	+	0.622816i	$0.785989\pi$
$18$	0	0
$19$	−6.03003	−1.38338	−0.691691	−	0.722193i	$-0.743134\pi$
$19$	−6.03003	−1.38338	−0.691691	+	0.722193i	$0.743134\pi$
$20$	0	0
$21$	−12.4849	−2.72444
$22$	0	0
$23$	−6.51176	−1.35780	−0.678898	−	0.734232i	$-0.737542\pi$
$23$	−6.51176	−1.35780	−0.678898	+	0.734232i	$0.737542\pi$
$24$	0	0
$25$	−4.47621	−0.895241
$26$	0	0
$27$	19.7991	3.81033
$28$	0	0
$29$	1.42186	0.264032	0.132016	−	0.991248i	$-0.457855\pi$
$29$	1.42186	0.264032	0.132016	+	0.991248i	$0.457855\pi$
$30$	0	0
$31$	−1.46844	−0.263740	−0.131870	−	0.991267i	$-0.542098\pi$
$31$	−1.46844	−0.263740	−0.131870	+	0.991267i	$0.542098\pi$
$32$	0	0
$33$	8.66884	1.50905
$34$	0	0
$35$	2.63367	0.445172
$36$	0	0
$37$	9.25058	1.52079	0.760393	−	0.649464i	$-0.225007\pi$
$37$	9.25058	1.52079	0.760393	+	0.649464i	$0.225007\pi$
$38$	0	0
$39$	−19.3443	−3.09757
$40$	0	0
$41$	−3.54493	−0.553625	−0.276812	−	0.960924i	$-0.589278\pi$
$41$	−3.54493	−0.553625	−0.276812	+	0.960924i	$0.589278\pi$
$42$	0	0
$43$	−2.04958	−0.312557	−0.156279	−	0.987713i	$-0.549950\pi$
$43$	−2.04958	−0.312557	−0.156279	+	0.987713i	$0.549950\pi$
$44$	0	0
$45$	−6.34778	−0.946271
$46$	0	0
$47$	1.56997	0.229004	0.114502	−	0.993423i	$-0.463473\pi$
$47$	1.56997	0.229004	0.114502	+	0.993423i	$0.463473\pi$
$48$	0	0
$49$	6.24233	0.891761
$50$	0	0
$51$	−22.1345	−3.09945
$52$	0	0
$53$	0.205679	0.0282522	0.0141261	−	0.999900i	$-0.495503\pi$
$53$	0.205679	0.0282522	0.0141261	+	0.999900i	$0.495503\pi$
$54$	0	0
$55$	−1.82868	−0.246579
$56$	0	0
$57$	−20.6882	−2.74022
$58$	0	0
$59$	−13.1714	−1.71476	−0.857382	−	0.514680i	$-0.827911\pi$
$59$	−13.1714	−1.71476	−0.857382	+	0.514680i	$0.827911\pi$
$60$	0	0
$61$	3.53160	0.452175	0.226088	−	0.974107i	$-0.427406\pi$
$61$	3.53160	0.452175	0.226088	+	0.974107i	$0.427406\pi$
$62$	0	0
$63$	−31.9172	−4.02119
$64$	0	0
$65$	4.08064	0.506142
$66$	0	0
$67$	−3.41564	−0.417287	−0.208644	−	0.977992i	$-0.566905\pi$
$67$	−3.41564	−0.417287	−0.208644	+	0.977992i	$0.566905\pi$
$68$	0	0
$69$	−22.3410	−2.68954
$70$	0	0
$71$	−8.33065	−0.988666	−0.494333	−	0.869273i	$-0.664588\pi$
$71$	−8.33065	−0.988666	−0.494333	+	0.869273i	$0.664588\pi$
$72$	0	0
$73$	10.3377	1.20993	0.604965	−	0.796252i	$-0.293187\pi$
$73$	10.3377	1.20993	0.604965	+	0.796252i	$0.293187\pi$
$74$	0	0
$75$	−15.3573	−1.77331
$76$	0	0
$77$	−9.19473	−1.04784
$78$	0	0
$79$	−15.1079	−1.69978	−0.849888	−	0.526963i	$-0.823331\pi$
$79$	−15.1079	−1.69978	−0.849888	+	0.526963i	$0.823331\pi$
$80$	0	0
$81$	41.6154	4.62394
$82$	0	0
$83$	10.2212	1.12192	0.560960	−	0.827843i	$-0.310432\pi$
$83$	10.2212	1.12192	0.560960	+	0.827843i	$0.310432\pi$
$84$	0	0
$85$	4.66923	0.506449
$86$	0	0
$87$	4.87821	0.522999
$88$	0	0
$89$	8.98950	0.952885	0.476443	−	0.879206i	$-0.341926\pi$
$89$	8.98950	0.952885	0.476443	+	0.879206i	$0.341926\pi$
$90$	0	0
$91$	20.5178	2.15085
$92$	0	0
$93$	−5.03804	−0.522420
$94$	0	0
$95$	4.36414	0.447752
$96$	0	0
$97$	12.6010	1.27943	0.639717	−	0.768610i	$-0.279051\pi$
$97$	12.6010	1.27943	0.639717	+	0.768610i	$0.279051\pi$
$98$	0	0
$99$	22.1615	2.22731
$100$	0	0
$101$	−5.59354	−0.556578	−0.278289	−	0.960497i	$-0.589767\pi$
$101$	−5.59354	−0.556578	−0.278289	+	0.960497i	$0.589767\pi$
$102$	0	0
$103$	1.91655	0.188844	0.0944219	−	0.995532i	$-0.469900\pi$
$103$	1.91655	0.188844	0.0944219	+	0.995532i	$0.469900\pi$
$104$	0	0
$105$	9.03579	0.881803
$106$	0	0
$107$	5.74368	0.555262	0.277631	−	0.960688i	$-0.410451\pi$
$107$	5.74368	0.555262	0.277631	+	0.960688i	$0.410451\pi$
$108$	0	0
$109$	−4.29875	−0.411746	−0.205873	−	0.978579i	$-0.566003\pi$
$109$	−4.29875	−0.411746	−0.205873	+	0.978579i	$0.566003\pi$
$110$	0	0
$111$	31.7375	3.01239
$112$	0	0
$113$	4.64136	0.436622	0.218311	−	0.975879i	$-0.429945\pi$
$113$	4.64136	0.436622	0.218311	+	0.975879i	$0.429945\pi$
$114$	0	0
$115$	4.71279	0.439470
$116$	0	0
$117$	−49.4528	−4.57192
$118$	0	0
$119$	23.4773	2.15216
$120$	0	0
$121$	−4.61569	−0.419608
$122$	0	0
$123$	−12.1622	−1.09663
$124$	0	0
$125$	6.85827	0.613422
$126$	0	0
$127$	−8.94460	−0.793705	−0.396853	−	0.917882i	$-0.629898\pi$
$127$	−8.94460	−0.793705	−0.396853	+	0.917882i	$0.629898\pi$
$128$	0	0
$129$	−7.03183	−0.619118
$130$	0	0
$131$	−4.54442	−0.397048	−0.198524	−	0.980096i	$-0.563615\pi$
$131$	−4.54442	−0.397048	−0.198524	+	0.980096i	$0.563615\pi$
$132$	0	0
$133$	21.9433	1.90272
$134$	0	0
$135$	−14.3293	−1.23327
$136$	0	0
$137$	−20.1604	−1.72242	−0.861210	−	0.508249i	$-0.830293\pi$
$137$	−20.1604	−1.72242	−0.861210	+	0.508249i	$0.830293\pi$
$138$	0	0
$139$	5.30311	0.449804	0.224902	−	0.974381i	$-0.427794\pi$
$139$	5.30311	0.449804	0.224902	+	0.974381i	$0.427794\pi$
$140$	0	0
$141$	5.38636	0.453613
$142$	0	0
$143$	−14.2464	−1.19135
$144$	0	0
$145$	−1.02905	−0.0854579
$146$	0	0
$147$	21.4166	1.76641
$148$	0	0
$149$	3.61985	0.296550	0.148275	−	0.988946i	$-0.452628\pi$
$149$	3.61985	0.296550	0.148275	+	0.988946i	$0.452628\pi$
$150$	0	0
$151$	−0.0261224	−0.00212581	−0.00106290	−	0.999999i	$-0.500338\pi$
$151$	−0.0261224	−0.00212581	−0.00106290	+	0.999999i	$0.500338\pi$
$152$	0	0
$153$	−56.5859	−4.57470
$154$	0	0
$155$	1.06276	0.0853633
$156$	0	0
$157$	12.6819	1.01212	0.506062	−	0.862497i	$-0.331101\pi$
$157$	12.6819	1.01212	0.506062	+	0.862497i	$0.331101\pi$
$158$	0	0
$159$	0.705659	0.0559624
$160$	0	0
$161$	23.6963	1.86753
$162$	0	0
$163$	13.0905	1.02533	0.512663	−	0.858590i	$-0.328659\pi$
$163$	13.0905	1.02533	0.512663	+	0.858590i	$0.328659\pi$
$164$	0	0
$165$	−6.27395	−0.488426
$166$	0	0
$167$	−2.36543	−0.183043	−0.0915213	−	0.995803i	$-0.529173\pi$
$167$	−2.36543	−0.183043	−0.0915213	+	0.995803i	$0.529173\pi$
$168$	0	0
$169$	18.7905	1.44543
$170$	0	0
$171$	−52.8885	−4.04449
$172$	0	0
$173$	0.261656	0.0198934	0.00994668	−	0.999951i	$-0.496834\pi$
$173$	0.261656	0.0198934	0.00994668	+	0.999951i	$0.496834\pi$
$174$	0	0
$175$	16.2889	1.23133
$176$	0	0
$177$	−45.1892	−3.39663
$178$	0	0
$179$	7.87847	0.588865	0.294432	−	0.955672i	$-0.404869\pi$
$179$	7.87847	0.588865	0.294432	+	0.955672i	$0.404869\pi$
$180$	0	0
$181$	10.0023	0.743462	0.371731	−	0.928340i	$-0.378764\pi$
$181$	10.0023	0.743462	0.371731	+	0.928340i	$0.378764\pi$
$182$	0	0
$183$	12.1165	0.895674
$184$	0	0
$185$	−6.69497	−0.492224
$186$	0	0
$187$	−16.3013	−1.19207
$188$	0	0
$189$	−72.0488	−5.24078
$190$	0	0
$191$	−23.9403	−1.73226	−0.866131	−	0.499817i	$-0.833401\pi$
$191$	−23.9403	−1.73226	−0.866131	+	0.499817i	$0.833401\pi$
$192$	0	0
$193$	14.0727	1.01297	0.506486	−	0.862248i	$-0.330944\pi$
$193$	14.0727	1.01297	0.506486	+	0.862248i	$0.330944\pi$
$194$	0	0
$195$	14.0002	1.00257
$196$	0	0
$197$	19.9341	1.42024	0.710122	−	0.704079i	$-0.248640\pi$
$197$	19.9341	1.42024	0.710122	+	0.704079i	$0.248640\pi$
$198$	0	0
$199$	18.7250	1.32738	0.663690	−	0.748007i	$-0.268989\pi$
$199$	18.7250	1.32738	0.663690	+	0.748007i	$0.268989\pi$
$200$	0	0
$201$	−11.7186	−0.826568
$202$	0	0
$203$	−5.17414	−0.363154
$204$	0	0
$205$	2.56559	0.179189
$206$	0	0
$207$	−57.1138	−3.96968
$208$	0	0
$209$	−15.2362	−1.05391
$210$	0	0
$211$	−12.6468	−0.870644	−0.435322	−	0.900275i	$-0.643365\pi$
$211$	−12.6468	−0.870644	−0.435322	+	0.900275i	$0.643365\pi$
$212$	0	0
$213$	−28.5814	−1.95836
$214$	0	0
$215$	1.48335	0.101164
$216$	0	0
$217$	5.34367	0.362752
$218$	0	0
$219$	35.4671	2.39665
$220$	0	0
$221$	36.3760	2.44691
$222$	0	0
$223$	−18.1325	−1.21424	−0.607120	−	0.794611i	$-0.707675\pi$
$223$	−18.1325	−1.21424	−0.607120	+	0.794611i	$0.707675\pi$
$224$	0	0
$225$	−39.2602	−2.61735
$226$	0	0
$227$	−14.1136	−0.936755	−0.468378	−	0.883528i	$-0.655161\pi$
$227$	−14.1136	−0.936755	−0.468378	+	0.883528i	$0.655161\pi$
$228$	0	0
$229$	20.4737	1.35294	0.676471	−	0.736470i	$-0.263509\pi$
$229$	20.4737	1.35294	0.676471	+	0.736470i	$0.263509\pi$
$230$	0	0
$231$	−31.5459	−2.07557
$232$	0	0
$233$	−15.8633	−1.03924	−0.519621	−	0.854397i	$-0.673927\pi$
$233$	−15.8633	−1.03924	−0.519621	+	0.854397i	$0.673927\pi$
$234$	0	0
$235$	−1.13624	−0.0741203
$236$	0	0
$237$	−51.8334	−3.36694
$238$	0	0
$239$	−7.37849	−0.477275	−0.238638	−	0.971109i	$-0.576701\pi$
$239$	−7.37849	−0.477275	−0.238638	+	0.971109i	$0.576701\pi$
$240$	0	0
$241$	−20.6724	−1.33163	−0.665813	−	0.746119i	$-0.731915\pi$
$241$	−20.6724	−1.33163	−0.665813	+	0.746119i	$0.731915\pi$
$242$	0	0
$243$	83.3798	5.34882
$244$	0	0
$245$	−4.51779	−0.288631
$246$	0	0
$247$	33.9992	2.16331
$248$	0	0
$249$	35.0675	2.22231
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−16.4534	−1.03442
$254$	0	0
$255$	16.0195	1.00318
$256$	0	0
$257$	−2.61549	−0.163150	−0.0815749	−	0.996667i	$-0.525995\pi$
$257$	−2.61549	−0.163150	−0.0815749	+	0.996667i	$0.525995\pi$
$258$	0	0
$259$	−33.6629	−2.09171
$260$	0	0
$261$	12.4709	0.771931
$262$	0	0
$263$	0.904492	0.0557734	0.0278867	−	0.999611i	$-0.491122\pi$
$263$	0.904492	0.0557734	0.0278867	+	0.999611i	$0.491122\pi$
$264$	0	0
$265$	−0.148857	−0.00914424
$266$	0	0
$267$	30.8418	1.88749
$268$	0	0
$269$	−12.4453	−0.758804	−0.379402	−	0.925232i	$-0.623870\pi$
$269$	−12.4453	−0.758804	−0.379402	+	0.925232i	$0.623870\pi$
$270$	0	0
$271$	−10.9431	−0.664743	−0.332372	−	0.943149i	$-0.607849\pi$
$271$	−10.9431	−0.664743	−0.332372	+	0.943149i	$0.607849\pi$
$272$	0	0
$273$	70.3939	4.26044
$274$	0	0
$275$	−11.3101	−0.682026
$276$	0	0
$277$	−32.8452	−1.97347	−0.986737	−	0.162325i	$-0.948101\pi$
$277$	−32.8452	−1.97347	−0.986737	+	0.162325i	$0.948101\pi$
$278$	0	0
$279$	−12.8795	−0.771077
$280$	0	0
$281$	−4.46149	−0.266150	−0.133075	−	0.991106i	$-0.542485\pi$
$281$	−4.46149	−0.266150	−0.133075	+	0.991106i	$0.542485\pi$
$282$	0	0
$283$	−1.70939	−0.101613	−0.0508064	−	0.998709i	$-0.516179\pi$
$283$	−1.70939	−0.101613	−0.0508064	+	0.998709i	$0.516179\pi$
$284$	0	0
$285$	14.9728	0.886912
$286$	0	0
$287$	12.9000	0.761463
$288$	0	0
$289$	24.6228	1.44840
$290$	0	0
$291$	43.2323	2.53432
$292$	0	0
$293$	30.4549	1.77919	0.889597	−	0.456746i	$-0.150985\pi$
$293$	30.4549	1.77919	0.889597	+	0.456746i	$0.150985\pi$
$294$	0	0
$295$	9.53258	0.555008
$296$	0	0
$297$	50.0267	2.90284
$298$	0	0
$299$	36.7153	2.12330
$300$	0	0
$301$	7.45841	0.429896
$302$	0	0
$303$	−19.1907	−1.10248
$304$	0	0
$305$	−2.55594	−0.146353
$306$	0	0
$307$	−28.0628	−1.60163	−0.800814	−	0.598914i	$-0.795599\pi$
$307$	−28.0628	−1.60163	−0.800814	+	0.598914i	$0.795599\pi$
$308$	0	0
$309$	6.57545	0.374064
$310$	0	0
$311$	7.85762	0.445565	0.222782	−	0.974868i	$-0.428486\pi$
$311$	7.85762	0.445565	0.222782	+	0.974868i	$0.428486\pi$
$312$	0	0
$313$	−27.1857	−1.53663	−0.768313	−	0.640074i	$-0.778904\pi$
$313$	−27.1857	−1.53663	−0.768313	+	0.640074i	$0.778904\pi$
$314$	0	0
$315$	23.0996	1.30151
$316$	0	0
$317$	3.18003	0.178608	0.0893042	−	0.996004i	$-0.471536\pi$
$317$	3.18003	0.178608	0.0893042	+	0.996004i	$0.471536\pi$
$318$	0	0
$319$	3.59264	0.201149
$320$	0	0
$321$	19.7058	1.09987
$322$	0	0
$323$	38.9032	2.16463
$324$	0	0
$325$	25.2382	1.39997
$326$	0	0
$327$	−14.7484	−0.815591
$328$	0	0
$329$	−5.71312	−0.314975
$330$	0	0
$331$	−25.0412	−1.37639	−0.688195	−	0.725526i	$-0.741597\pi$
$331$	−25.0412	−1.37639	−0.688195	+	0.725526i	$0.741597\pi$
$332$	0	0
$333$	81.1355	4.44620
$334$	0	0
$335$	2.47202	0.135061
$336$	0	0
$337$	−9.10140	−0.495785	−0.247893	−	0.968788i	$-0.579738\pi$
$337$	−9.10140	−0.495785	−0.247893	+	0.968788i	$0.579738\pi$
$338$	0	0
$339$	15.9239	0.864867
$340$	0	0
$341$	−3.71035	−0.200927
$342$	0	0
$343$	2.75716	0.148873
$344$	0	0
$345$	16.1690	0.870508
$346$	0	0
$347$	8.63401	0.463498	0.231749	−	0.972776i	$-0.425555\pi$
$347$	8.63401	0.463498	0.231749	+	0.972776i	$0.425555\pi$
$348$	0	0
$349$	1.10069	0.0589185	0.0294592	−	0.999566i	$-0.490621\pi$
$349$	1.10069	0.0589185	0.0294592	+	0.999566i	$0.490621\pi$
$350$	0	0
$351$	−111.633	−5.95854
$352$	0	0
$353$	−20.2146	−1.07592	−0.537958	−	0.842972i	$-0.680804\pi$
$353$	−20.2146	−1.07592	−0.537958	+	0.842972i	$0.680804\pi$
$354$	0	0
$355$	6.02919	0.319996
$356$	0	0
$357$	80.5475	4.26303
$358$	0	0
$359$	−8.16004	−0.430670	−0.215335	−	0.976540i	$-0.569084\pi$
$359$	−8.16004	−0.430670	−0.215335	+	0.976540i	$0.569084\pi$
$360$	0	0
$361$	17.3612	0.913748
$362$	0	0
$363$	−15.8358	−0.831165
$364$	0	0
$365$	−7.48173	−0.391611
$366$	0	0
$367$	7.69201	0.401520	0.200760	−	0.979641i	$-0.435659\pi$
$367$	7.69201	0.401520	0.200760	+	0.979641i	$0.435659\pi$
$368$	0	0
$369$	−31.0921	−1.61859
$370$	0	0
$371$	−0.748468	−0.0388585
$372$	0	0
$373$	9.95031	0.515208	0.257604	−	0.966251i	$-0.417067\pi$
$373$	9.95031	0.515208	0.257604	+	0.966251i	$0.417067\pi$
$374$	0	0
$375$	23.5298	1.21507
$376$	0	0
$377$	−8.01688	−0.412890
$378$	0	0
$379$	12.4432	0.639163	0.319581	−	0.947559i	$-0.396458\pi$
$379$	12.4432	0.639163	0.319581	+	0.947559i	$0.396458\pi$
$380$	0	0
$381$	−30.6878	−1.57218
$382$	0	0
$383$	5.03410	0.257231	0.128615	−	0.991695i	$-0.458947\pi$
$383$	5.03410	0.257231	0.128615	+	0.991695i	$0.458947\pi$
$384$	0	0
$385$	6.65455	0.339147
$386$	0	0
$387$	−17.9765	−0.913799
$388$	0	0
$389$	−29.0902	−1.47493	−0.737465	−	0.675386i	$-0.763977\pi$
$389$	−29.0902	−1.47493	−0.737465	+	0.675386i	$0.763977\pi$
$390$	0	0
$391$	42.0111	2.12459
$392$	0	0
$393$	−15.5913	−0.786477
$394$	0	0
$395$	10.9342	0.550157
$396$	0	0
$397$	−27.1949	−1.36487	−0.682436	−	0.730945i	$-0.739080\pi$
$397$	−27.1949	−1.36487	−0.682436	+	0.730945i	$0.739080\pi$
$398$	0	0
$399$	75.2845	3.76894
$400$	0	0
$401$	−19.0641	−0.952014	−0.476007	−	0.879441i	$-0.657916\pi$
$401$	−19.0641	−0.952014	−0.476007	+	0.879441i	$0.657916\pi$
$402$	0	0
$403$	8.27954	0.412433
$404$	0	0
$405$	−30.1185	−1.49660
$406$	0	0
$407$	23.3736	1.15859
$408$	0	0
$409$	32.9101	1.62730	0.813650	−	0.581355i	$-0.197477\pi$
$409$	32.9101	1.62730	0.813650	+	0.581355i	$0.197477\pi$
$410$	0	0
$411$	−69.1677	−3.41179
$412$	0	0
$413$	47.9306	2.35851
$414$	0	0
$415$	−7.39743	−0.363125
$416$	0	0
$417$	18.1943	0.890978
$418$	0	0
$419$	−36.2983	−1.77329	−0.886643	−	0.462455i	$-0.846969\pi$
$419$	−36.2983	−1.77329	−0.886643	+	0.462455i	$0.846969\pi$
$420$	0	0
$421$	−12.6401	−0.616043	−0.308022	−	0.951379i	$-0.599667\pi$
$421$	−12.6401	−0.616043	−0.308022	+	0.951379i	$0.599667\pi$
$422$	0	0
$423$	13.7700	0.669520
$424$	0	0
$425$	28.8786	1.40082
$426$	0	0
$427$	−12.8515	−0.621928
$428$	0	0
$429$	−48.8776	−2.35983
$430$	0	0
$431$	16.6760	0.803254	0.401627	−	0.915803i	$-0.368445\pi$
$431$	16.6760	0.803254	0.401627	+	0.915803i	$0.368445\pi$
$432$	0	0
$433$	−31.4938	−1.51349	−0.756747	−	0.653708i	$-0.773213\pi$
$433$	−31.4938	−1.51349	−0.756747	+	0.653708i	$0.773213\pi$
$434$	0	0
$435$	−3.53053	−0.169276
$436$	0	0
$437$	39.2661	1.87835
$438$	0	0
$439$	−5.87833	−0.280557	−0.140279	−	0.990112i	$-0.544800\pi$
$439$	−5.87833	−0.280557	−0.140279	+	0.990112i	$0.544800\pi$
$440$	0	0
$441$	54.7506	2.60717
$442$	0	0
$443$	−2.95913	−0.140593	−0.0702963	−	0.997526i	$-0.522394\pi$
$443$	−2.95913	−0.140593	−0.0702963	+	0.997526i	$0.522394\pi$
$444$	0	0
$445$	−6.50602	−0.308415
$446$	0	0
$447$	12.4192	0.587409
$448$	0	0
$449$	−21.1681	−0.998984	−0.499492	−	0.866318i	$-0.666480\pi$
$449$	−21.1681	−0.998984	−0.499492	+	0.866318i	$0.666480\pi$
$450$	0	0
$451$	−8.95704	−0.421770
$452$	0	0
$453$	−0.0896225	−0.00421083
$454$	0	0
$455$	−14.8495	−0.696154
$456$	0	0
$457$	−4.61220	−0.215750	−0.107875	−	0.994164i	$-0.534405\pi$
$457$	−4.61220	−0.215750	−0.107875	+	0.994164i	$0.534405\pi$
$458$	0	0
$459$	−127.735	−5.96217
$460$	0	0
$461$	13.3346	0.621052	0.310526	−	0.950565i	$-0.399495\pi$
$461$	13.3346	0.621052	0.310526	+	0.950565i	$0.399495\pi$
$462$	0	0
$463$	20.7591	0.964759	0.482379	−	0.875962i	$-0.339773\pi$
$463$	20.7591	0.964759	0.482379	+	0.875962i	$0.339773\pi$
$464$	0	0
$465$	3.64621	0.169089
$466$	0	0
$467$	24.8016	1.14768	0.573841	−	0.818967i	$-0.305453\pi$
$467$	24.8016	1.14768	0.573841	+	0.818967i	$0.305453\pi$
$468$	0	0
$469$	12.4295	0.573942
$470$	0	0
$471$	43.5098	2.00483
$472$	0	0
$473$	−5.17870	−0.238117
$474$	0	0
$475$	26.9916	1.23846
$476$	0	0
$477$	1.80399	0.0825988
$478$	0	0
$479$	27.4260	1.25312	0.626562	−	0.779372i	$-0.284462\pi$
$479$	27.4260	1.25312	0.626562	+	0.779372i	$0.284462\pi$
$480$	0	0
$481$	−52.1576	−2.37818
$482$	0	0
$483$	81.2990	3.69923
$484$	0	0
$485$	−9.11977	−0.414107
$486$	0	0
$487$	3.89273	0.176396	0.0881982	−	0.996103i	$-0.471889\pi$
$487$	3.89273	0.176396	0.0881982	+	0.996103i	$0.471889\pi$
$488$	0	0
$489$	44.9118	2.03098
$490$	0	0
$491$	−21.8949	−0.988102	−0.494051	−	0.869433i	$-0.664484\pi$
$491$	−21.8949	−0.988102	−0.494051	+	0.869433i	$0.664484\pi$
$492$	0	0
$493$	−9.17323	−0.413141
$494$	0	0
$495$	−16.0391	−0.720902
$496$	0	0
$497$	30.3152	1.35982
$498$	0	0
$499$	−10.3263	−0.462268	−0.231134	−	0.972922i	$-0.574244\pi$
$499$	−10.3263	−0.462268	−0.231134	+	0.972922i	$0.574244\pi$
$500$	0	0
$501$	−8.11548	−0.362573
$502$	0	0
$503$	1.51173	0.0674047	0.0337024	−	0.999432i	$-0.489270\pi$
$503$	1.51173	0.0674047	0.0337024	+	0.999432i	$0.489270\pi$
$504$	0	0
$505$	4.04824	0.180144
$506$	0	0
$507$	64.4679	2.86312
$508$	0	0
$509$	−6.45143	−0.285955	−0.142977	−	0.989726i	$-0.545668\pi$
$509$	−6.45143	−0.285955	−0.142977	+	0.989726i	$0.545668\pi$
$510$	0	0
$511$	−37.6187	−1.66415
$512$	0	0
$513$	−119.389	−5.27115
$514$	0	0
$515$	−1.38708	−0.0611220
$516$	0	0
$517$	3.96687	0.174463
$518$	0	0
$519$	0.897709	0.0394050
$520$	0	0
$521$	−37.7331	−1.65312	−0.826559	−	0.562851i	$-0.809705\pi$
$521$	−37.7331	−1.65312	−0.826559	+	0.562851i	$0.809705\pi$
$522$	0	0
$523$	1.68283	0.0735850	0.0367925	−	0.999323i	$-0.488286\pi$
$523$	1.68283	0.0735850	0.0367925	+	0.999323i	$0.488286\pi$
$524$	0	0
$525$	55.8852	2.43903
$526$	0	0
$527$	9.47378	0.412684
$528$	0	0
$529$	19.4031	0.843612
$530$	0	0
$531$	−115.524	−5.01332
$532$	0	0
$533$	19.9874	0.865751
$534$	0	0
$535$	−4.15690	−0.179718
$536$	0	0
$537$	27.0300	1.16643
$538$	0	0
$539$	15.7726	0.679375
$540$	0	0
$541$	−11.5811	−0.497911	−0.248955	−	0.968515i	$-0.580087\pi$
$541$	−11.5811	−0.497911	−0.248955	+	0.968515i	$0.580087\pi$
$542$	0	0
$543$	34.3164	1.47266
$544$	0	0
$545$	3.11116	0.133267
$546$	0	0
$547$	16.7657	0.716851	0.358426	−	0.933558i	$-0.383314\pi$
$547$	16.7657	0.716851	0.358426	+	0.933558i	$0.383314\pi$
$548$	0	0
$549$	30.9752	1.32199
$550$	0	0
$551$	−8.57384	−0.365258
$552$	0	0
$553$	54.9778	2.33789
$554$	0	0
$555$	−22.9696	−0.975003
$556$	0	0
$557$	10.1292	0.429187	0.214594	−	0.976703i	$-0.431157\pi$
$557$	10.1292	0.429187	0.214594	+	0.976703i	$0.431157\pi$
$558$	0	0
$559$	11.5561	0.488773
$560$	0	0
$561$	−55.9277	−2.36127
$562$	0	0
$563$	−3.16995	−0.133597	−0.0667986	−	0.997766i	$-0.521279\pi$
$563$	−3.16995	−0.133597	−0.0667986	+	0.997766i	$0.521279\pi$
$564$	0	0
$565$	−3.35911	−0.141319
$566$	0	0
$567$	−151.439	−6.35982
$568$	0	0
$569$	13.6663	0.572922	0.286461	−	0.958092i	$-0.407521\pi$
$569$	13.6663	0.572922	0.286461	+	0.958092i	$0.407521\pi$
$570$	0	0
$571$	−1.56169	−0.0653548	−0.0326774	−	0.999466i	$-0.510403\pi$
$571$	−1.56169	−0.0653548	−0.0326774	+	0.999466i	$0.510403\pi$
$572$	0	0
$573$	−82.1362	−3.43129
$574$	0	0
$575$	29.1480	1.21556
$576$	0	0
$577$	24.9922	1.04044	0.520219	−	0.854033i	$-0.325850\pi$
$577$	24.9922	1.04044	0.520219	+	0.854033i	$0.325850\pi$
$578$	0	0
$579$	48.2815	2.00651
$580$	0	0
$581$	−37.1949	−1.54310
$582$	0	0
$583$	0.519694	0.0215235
$584$	0	0
$585$	35.7908	1.47977
$586$	0	0
$587$	35.2462	1.45476	0.727382	−	0.686232i	$-0.240737\pi$
$587$	35.2462	1.45476	0.727382	+	0.686232i	$0.240737\pi$
$588$	0	0
$589$	8.85476	0.364854
$590$	0	0
$591$	68.3912	2.81324
$592$	0	0
$593$	29.0161	1.19155	0.595775	−	0.803152i	$-0.296845\pi$
$593$	29.0161	1.19155	0.595775	+	0.803152i	$0.296845\pi$
$594$	0	0
$595$	−16.9913	−0.696577
$596$	0	0
$597$	64.2431	2.62929
$598$	0	0
$599$	−16.9321	−0.691826	−0.345913	−	0.938267i	$-0.612431\pi$
$599$	−16.9321	−0.691826	−0.345913	+	0.938267i	$0.612431\pi$
$600$	0	0
$601$	−1.63518	−0.0667002	−0.0333501	−	0.999444i	$-0.510618\pi$
$601$	−1.63518	−0.0667002	−0.0333501	+	0.999444i	$0.510618\pi$
$602$	0	0
$603$	−29.9581	−1.21999
$604$	0	0
$605$	3.34054	0.135812
$606$	0	0
$607$	−25.1440	−1.02056	−0.510282	−	0.860007i	$-0.670459\pi$
$607$	−25.1440	−1.02056	−0.510282	+	0.860007i	$0.670459\pi$
$608$	0	0
$609$	−17.7518	−0.719340
$610$	0	0
$611$	−8.85198	−0.358113
$612$	0	0
$613$	6.65330	0.268724	0.134362	−	0.990932i	$-0.457101\pi$
$613$	6.65330	0.268724	0.134362	+	0.990932i	$0.457101\pi$
$614$	0	0
$615$	8.80220	0.354939
$616$	0	0
$617$	29.8300	1.20091	0.600455	−	0.799658i	$-0.294986\pi$
$617$	29.8300	1.20091	0.600455	+	0.799658i	$0.294986\pi$
$618$	0	0
$619$	−12.8717	−0.517357	−0.258679	−	0.965963i	$-0.583287\pi$
$619$	−12.8717	−0.517357	−0.258679	+	0.965963i	$0.583287\pi$
$620$	0	0
$621$	−128.927	−5.17366
$622$	0	0
$623$	−32.7128	−1.31061
$624$	0	0
$625$	17.4175	0.696699
$626$	0	0
$627$	−52.2733	−2.08760
$628$	0	0
$629$	−59.6808	−2.37963
$630$	0	0
$631$	−20.0313	−0.797432	−0.398716	−	0.917074i	$-0.630544\pi$
$631$	−20.0313	−0.797432	−0.398716	+	0.917074i	$0.630544\pi$
$632$	0	0
$633$	−43.3896	−1.72458
$634$	0	0
$635$	6.47352	0.256894
$636$	0	0
$637$	−35.1962	−1.39452
$638$	0	0
$639$	−73.0670	−2.89048
$640$	0	0
$641$	−17.9163	−0.707651	−0.353826	−	0.935311i	$-0.615119\pi$
$641$	−17.9163	−0.707651	−0.353826	+	0.935311i	$0.615119\pi$
$642$	0	0
$643$	−28.3687	−1.11875	−0.559376	−	0.828914i	$-0.688959\pi$
$643$	−28.3687	−1.11875	−0.559376	+	0.828914i	$0.688959\pi$
$644$	0	0
$645$	5.08918	0.200386
$646$	0	0
$647$	40.8144	1.60458	0.802289	−	0.596936i	$-0.203615\pi$
$647$	40.8144	1.60458	0.802289	+	0.596936i	$0.203615\pi$
$648$	0	0
$649$	−33.2803	−1.30637
$650$	0	0
$651$	18.3334	0.718544
$652$	0	0
$653$	−19.2986	−0.755214	−0.377607	−	0.925966i	$-0.623253\pi$
$653$	−19.2986	−0.755214	−0.377607	+	0.925966i	$0.623253\pi$
$654$	0	0
$655$	3.28896	0.128510
$656$	0	0
$657$	90.6701	3.53738
$658$	0	0
$659$	13.1073	0.510588	0.255294	−	0.966863i	$-0.417828\pi$
$659$	13.1073	0.510588	0.255294	+	0.966863i	$0.417828\pi$
$660$	0	0
$661$	−1.91986	−0.0746738	−0.0373369	−	0.999303i	$-0.511887\pi$
$661$	−1.91986	−0.0746738	−0.0373369	+	0.999303i	$0.511887\pi$
$662$	0	0
$663$	124.801	4.84688
$664$	0	0
$665$	−15.8811	−0.615843
$666$	0	0
$667$	−9.25881	−0.358502
$668$	0	0
$669$	−62.2101	−2.40518
$670$	0	0
$671$	8.92336	0.344483
$672$	0	0
$673$	−6.08921	−0.234722	−0.117361	−	0.993089i	$-0.537443\pi$
$673$	−6.08921	−0.234722	−0.117361	+	0.993089i	$0.537443\pi$
$674$	0	0
$675$	−88.6247	−3.41117
$676$	0	0
$677$	33.9307	1.30406	0.652031	−	0.758192i	$-0.273917\pi$
$677$	33.9307	1.30406	0.652031	+	0.758192i	$0.273917\pi$
$678$	0	0
$679$	−45.8549	−1.75975
$680$	0	0
$681$	−48.4220	−1.85554
$682$	0	0
$683$	−5.85648	−0.224092	−0.112046	−	0.993703i	$-0.535740\pi$
$683$	−5.85648	−0.224092	−0.112046	+	0.993703i	$0.535740\pi$
$684$	0	0
$685$	14.5908	0.557486
$686$	0	0
$687$	70.2426	2.67992
$688$	0	0
$689$	−1.15968	−0.0441805
$690$	0	0
$691$	22.2030	0.844642	0.422321	−	0.906446i	$-0.361216\pi$
$691$	22.2030	0.844642	0.422321	+	0.906446i	$0.361216\pi$
$692$	0	0
$693$	−80.6457	−3.06348
$694$	0	0
$695$	−3.83805	−0.145586
$696$	0	0
$697$	22.8704	0.866277
$698$	0	0
$699$	−54.4251	−2.05854
$700$	0	0
$701$	38.5848	1.45733	0.728664	−	0.684871i	$-0.240141\pi$
$701$	38.5848	1.45733	0.728664	+	0.684871i	$0.240141\pi$
$702$	0	0
$703$	−55.7812	−2.10383
$704$	0	0
$705$	−3.89830	−0.146818
$706$	0	0
$707$	20.3549	0.765525
$708$	0	0
$709$	11.2496	0.422487	0.211244	−	0.977433i	$-0.432249\pi$
$709$	11.2496	0.422487	0.211244	+	0.977433i	$0.432249\pi$
$710$	0	0
$711$	−132.510	−4.96950
$712$	0	0
$713$	9.56216	0.358106
$714$	0	0
$715$	10.3106	0.385596
$716$	0	0
$717$	−25.3146	−0.945393
$718$	0	0
$719$	41.3583	1.54241	0.771203	−	0.636589i	$-0.219655\pi$
$719$	41.3583	1.54241	0.771203	+	0.636589i	$0.219655\pi$
$720$	0	0
$721$	−6.97434	−0.259738
$722$	0	0
$723$	−70.9243	−2.63770
$724$	0	0
$725$	−6.36453	−0.236373
$726$	0	0
$727$	−14.5116	−0.538206	−0.269103	−	0.963111i	$-0.586727\pi$
$727$	−14.5116	−0.538206	−0.269103	+	0.963111i	$0.586727\pi$
$728$	0	0
$729$	161.219	5.97108
$730$	0	0
$731$	13.2230	0.489070
$732$	0	0
$733$	−1.29193	−0.0477186	−0.0238593	−	0.999715i	$-0.507595\pi$
$733$	−1.29193	−0.0477186	−0.0238593	+	0.999715i	$0.507595\pi$
$734$	0	0
$735$	−15.5000	−0.571725
$736$	0	0
$737$	−8.63037	−0.317904
$738$	0	0
$739$	−20.7227	−0.762297	−0.381148	−	0.924514i	$-0.624471\pi$
$739$	−20.7227	−0.762297	−0.381148	+	0.924514i	$0.624471\pi$
$740$	0	0
$741$	116.647	4.28512
$742$	0	0
$743$	−23.7398	−0.870928	−0.435464	−	0.900206i	$-0.643416\pi$
$743$	−23.7398	−0.870928	−0.435464	+	0.900206i	$0.643416\pi$
$744$	0	0
$745$	−2.61981	−0.0959825
$746$	0	0
$747$	89.6485	3.28007
$748$	0	0
$749$	−20.9012	−0.763715
$750$	0	0
$751$	−38.5970	−1.40843	−0.704213	−	0.709989i	$-0.748700\pi$
$751$	−38.5970	−1.40843	−0.704213	+	0.709989i	$0.748700\pi$
$752$	0	0
$753$	3.43087	0.125028
$754$	0	0
$755$	0.0189057	0.000688049 0
$756$	0	0
$757$	8.13606	0.295710	0.147855	−	0.989009i	$-0.452763\pi$
$757$	8.13606	0.295710	0.147855	+	0.989009i	$0.452763\pi$
$758$	0	0
$759$	−56.4495	−2.04899
$760$	0	0
$761$	33.9969	1.23239	0.616193	−	0.787595i	$-0.288674\pi$
$761$	33.9969	1.23239	0.616193	+	0.787595i	$0.288674\pi$
$762$	0	0
$763$	15.6432	0.566320
$764$	0	0
$765$	40.9532	1.48067
$766$	0	0
$767$	74.2642	2.68152
$768$	0	0
$769$	16.7579	0.604304	0.302152	−	0.953260i	$-0.402295\pi$
$769$	16.7579	0.604304	0.302152	+	0.953260i	$0.402295\pi$
$770$	0	0
$771$	−8.97340	−0.323169
$772$	0	0
$773$	26.4671	0.951954	0.475977	−	0.879458i	$-0.342094\pi$
$773$	26.4671	0.951954	0.475977	+	0.879458i	$0.342094\pi$
$774$	0	0
$775$	6.57306	0.236111
$776$	0	0
$777$	−115.493	−4.14328
$778$	0	0
$779$	21.3760	0.765875
$780$	0	0
$781$	−21.0492	−0.753200
$782$	0	0
$783$	28.1515	1.00605
$784$	0	0
$785$	−9.17831	−0.327588
$786$	0	0
$787$	−18.0762	−0.644347	−0.322173	−	0.946681i	$-0.604413\pi$
$787$	−18.0762	−0.644347	−0.322173	+	0.946681i	$0.604413\pi$
$788$	0	0
$789$	3.10319	0.110477
$790$	0	0
$791$	−16.8899	−0.600536
$792$	0	0
$793$	−19.9123	−0.707105
$794$	0	0
$795$	−0.510711	−0.0181130
$796$	0	0
$797$	−14.4221	−0.510857	−0.255428	−	0.966828i	$-0.582217\pi$
$797$	−14.4221	−0.510857	−0.255428	+	0.966828i	$0.582217\pi$
$798$	0	0
$799$	−10.1288	−0.358330
$800$	0	0
$801$	78.8456	2.78587
$802$	0	0
$803$	26.1203	0.921767
$804$	0	0
$805$	−17.1499	−0.604453
$806$	0	0
$807$	−42.6982	−1.50305
$808$	0	0
$809$	−4.60706	−0.161976	−0.0809878	−	0.996715i	$-0.525807\pi$
$809$	−4.60706	−0.161976	−0.0809878	+	0.996715i	$0.525807\pi$
$810$	0	0
$811$	−22.1427	−0.777537	−0.388768	−	0.921335i	$-0.627099\pi$
$811$	−22.1427	−0.777537	−0.388768	+	0.921335i	$0.627099\pi$
$812$	0	0
$813$	−37.5442	−1.31673
$814$	0	0
$815$	−9.47405	−0.331862
$816$	0	0
$817$	12.3590	0.432387
$818$	0	0
$819$	179.959	6.28827
$820$	0	0
$821$	24.0919	0.840814	0.420407	−	0.907336i	$-0.361887\pi$
$821$	24.0919	0.840814	0.420407	+	0.907336i	$0.361887\pi$
$822$	0	0
$823$	45.0243	1.56945	0.784724	−	0.619846i	$-0.212805\pi$
$823$	45.0243	1.56945	0.784724	+	0.619846i	$0.212805\pi$
$824$	0	0
$825$	−38.8035	−1.35097
$826$	0	0
$827$	−7.99846	−0.278134	−0.139067	−	0.990283i	$-0.544410\pi$
$827$	−7.99846	−0.278134	−0.139067	+	0.990283i	$0.544410\pi$
$828$	0	0
$829$	5.76449	0.200209	0.100104	−	0.994977i	$-0.468082\pi$
$829$	5.76449	0.200209	0.100104	+	0.994977i	$0.468082\pi$
$830$	0	0
$831$	−112.687	−3.90908
$832$	0	0
$833$	−40.2729	−1.39537
$834$	0	0
$835$	1.71195	0.0592443
$836$	0	0
$837$	−29.0738	−1.00494
$838$	0	0
$839$	34.2824	1.18356	0.591780	−	0.806099i	$-0.298425\pi$
$839$	34.2824	1.18356	0.591780	+	0.806099i	$0.298425\pi$
$840$	0	0
$841$	−26.9783	−0.930287
$842$	0	0
$843$	−15.3068	−0.527193
$844$	0	0
$845$	−13.5994	−0.467833
$846$	0	0
$847$	16.7965	0.577135
$848$	0	0
$849$	−5.86470	−0.201276
$850$	0	0
$851$	−60.2376	−2.06492
$852$	0	0
$853$	26.3752	0.903071	0.451536	−	0.892253i	$-0.350876\pi$
$853$	26.3752	0.903071	0.451536	+	0.892253i	$0.350876\pi$
$854$	0	0
$855$	38.2773	1.30906
$856$	0	0
$857$	−56.0512	−1.91467	−0.957337	−	0.288975i	$-0.906685\pi$
$857$	−56.0512	−1.91467	−0.957337	+	0.288975i	$0.906685\pi$
$858$	0	0
$859$	3.81995	0.130335	0.0651675	−	0.997874i	$-0.479242\pi$
$859$	3.81995	0.130335	0.0651675	+	0.997874i	$0.479242\pi$
$860$	0	0
$861$	44.2582	1.50832
$862$	0	0
$863$	−12.6539	−0.430743	−0.215372	−	0.976532i	$-0.569096\pi$
$863$	−12.6539	−0.430743	−0.215372	+	0.976532i	$0.569096\pi$
$864$	0	0
$865$	−0.189370	−0.00643877
$866$	0	0
$867$	84.4777	2.86901
$868$	0	0
$869$	−38.1735	−1.29495
$870$	0	0
$871$	19.2585	0.652548
$872$	0	0
$873$	110.521	3.74058
$874$	0	0
$875$	−24.9572	−0.843709
$876$	0	0
$877$	−21.4012	−0.722669	−0.361334	−	0.932436i	$-0.617679\pi$
$877$	−21.4012	−0.722669	−0.361334	+	0.932436i	$0.617679\pi$
$878$	0	0
$879$	104.487	3.52425
$880$	0	0
$881$	−23.6793	−0.797776	−0.398888	−	0.917000i	$-0.630604\pi$
$881$	−23.6793	−0.797776	−0.398888	+	0.917000i	$0.630604\pi$
$882$	0	0
$883$	9.46676	0.318582	0.159291	−	0.987232i	$-0.449079\pi$
$883$	9.46676	0.318582	0.159291	+	0.987232i	$0.449079\pi$
$884$	0	0
$885$	32.7050	1.09937
$886$	0	0
$887$	40.2162	1.35033	0.675163	−	0.737668i	$-0.264073\pi$
$887$	40.2162	1.35033	0.675163	+	0.737668i	$0.264073\pi$
$888$	0	0
$889$	32.5494	1.09167
$890$	0	0
$891$	105.150	3.52267
$892$	0	0
$893$	−9.46696	−0.316800
$894$	0	0
$895$	−5.70193	−0.190594
$896$	0	0
$897$	125.966	4.20587
$898$	0	0
$899$	−2.08792	−0.0696360
$900$	0	0
$901$	−1.32696	−0.0442073
$902$	0	0
$903$	25.5888	0.851543
$904$	0	0
$905$	−7.23899	−0.240632
$906$	0	0
$907$	−11.1114	−0.368949	−0.184474	−	0.982837i	$-0.559058\pi$
$907$	−11.1114	−0.368949	−0.184474	+	0.982837i	$0.559058\pi$
$908$	0	0
$909$	−49.0602	−1.62722
$910$	0	0
$911$	38.2838	1.26840	0.634199	−	0.773170i	$-0.281330\pi$
$911$	38.2838	1.26840	0.634199	+	0.773170i	$0.281330\pi$
$912$	0	0
$913$	25.8260	0.854717
$914$	0	0
$915$	−8.76911	−0.289898
$916$	0	0
$917$	16.5371	0.546104
$918$	0	0
$919$	−25.3922	−0.837611	−0.418805	−	0.908076i	$-0.637551\pi$
$919$	−25.3922	−0.837611	−0.418805	+	0.908076i	$0.637551\pi$
$920$	0	0
$921$	−96.2797	−3.17252
$922$	0	0
$923$	46.9708	1.54606
$924$	0	0
$925$	−41.4075	−1.36147
$926$	0	0
$927$	16.8098	0.552107
$928$	0	0
$929$	8.21129	0.269404	0.134702	−	0.990886i	$-0.456992\pi$
$929$	8.21129	0.269404	0.134702	+	0.990886i	$0.456992\pi$
$930$	0	0
$931$	−37.6414	−1.23365
$932$	0	0
$933$	26.9585	0.882580
$934$	0	0
$935$	11.7978	0.385831
$936$	0	0
$937$	8.03467	0.262481	0.131241	−	0.991351i	$-0.458104\pi$
$937$	8.03467	0.262481	0.131241	+	0.991351i	$0.458104\pi$
$938$	0	0
$939$	−93.2705	−3.04377
$940$	0	0
$941$	59.1566	1.92845	0.964225	−	0.265086i	$-0.0854004\pi$
$941$	59.1566	1.92845	0.964225	+	0.265086i	$0.0854004\pi$
$942$	0	0
$943$	23.0837	0.751710
$944$	0	0
$945$	52.1443	1.69625
$946$	0	0
$947$	−50.9047	−1.65418	−0.827090	−	0.562070i	$-0.810005\pi$
$947$	−50.9047	−1.65418	−0.827090	+	0.562070i	$0.810005\pi$
$948$	0	0
$949$	−58.2869	−1.89207
$950$	0	0
$951$	10.9103	0.353790
$952$	0	0
$953$	41.8185	1.35463	0.677317	−	0.735691i	$-0.263143\pi$
$953$	41.8185	1.35463	0.677317	+	0.735691i	$0.263143\pi$
$954$	0	0
$955$	17.3265	0.560671
$956$	0	0
$957$	12.3259	0.398439
$958$	0	0
$959$	73.3637	2.36904
$960$	0	0
$961$	−28.8437	−0.930441
$962$	0	0
$963$	50.3770	1.62338
$964$	0	0
$965$	−10.1849	−0.327863
$966$	0	0
$967$	15.7805	0.507466	0.253733	−	0.967274i	$-0.418342\pi$
$967$	15.7805	0.507466	0.253733	+	0.967274i	$0.418342\pi$
$968$	0	0
$969$	133.472	4.28773
$970$	0	0
$971$	41.7555	1.34000	0.669999	−	0.742362i	$-0.266295\pi$
$971$	41.7555	1.34000	0.669999	+	0.742362i	$0.266295\pi$
$972$	0	0
$973$	−19.2980	−0.618667
$974$	0	0
$975$	86.5891	2.77307
$976$	0	0
$977$	5.41034	0.173092	0.0865461	−	0.996248i	$-0.472417\pi$
$977$	5.41034	0.173092	0.0865461	+	0.996248i	$0.472417\pi$
$978$	0	0
$979$	22.7139	0.725941
$980$	0	0
$981$	−37.7037	−1.20379
$982$	0	0
$983$	−8.25791	−0.263387	−0.131693	−	0.991291i	$-0.542041\pi$
$983$	−8.25791	−0.263387	−0.131693	+	0.991291i	$0.542041\pi$
$984$	0	0
$985$	−14.4270	−0.459682
$986$	0	0
$987$	−19.6010	−0.623906
$988$	0	0
$989$	13.3464	0.424389
$990$	0	0
$991$	0.261438	0.00830487	0.00415243	−	0.999991i	$-0.498678\pi$
$991$	0.261438	0.00830487	0.00415243	+	0.999991i	$0.498678\pi$
$992$	0	0
$993$	−85.9131	−2.72637
$994$	0	0
$995$	−13.5520	−0.429626
$996$	0	0
$997$	−24.3734	−0.771913	−0.385956	−	0.922517i	$-0.626128\pi$
$997$	−24.3734	−0.771913	−0.385956	+	0.922517i	$0.626128\pi$
$998$	0	0
$999$	183.153	5.79470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.j.1.14		14
4.3	odd	2		1004.2.a.b.1.1	✓	14
12.11	even	2		9036.2.a.m.1.8		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1004.2.a.b.1.1	✓	14	4.3	odd	2
4016.2.a.j.1.14		14	1.1	even	1	trivial
9036.2.a.m.1.8		14	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+3.43087 q^{3} -0.723735 q^{5} -3.63900 q^{7} +8.77086 q^{9} +O(q^{10})\)	\(q+3.43087 q^{3} -0.723735 q^{5} -3.63900 q^{7} +8.77086 q^{9} +2.52672 q^{11} -5.63831 q^{13} -2.48304 q^{15} -6.45157 q^{17} -6.03003 q^{19} -12.4849 q^{21} -6.51176 q^{23} -4.47621 q^{25} +19.7991 q^{27} +1.42186 q^{29} -1.46844 q^{31} +8.66884 q^{33} +2.63367 q^{35} +9.25058 q^{37} -19.3443 q^{39} -3.54493 q^{41} -2.04958 q^{43} -6.34778 q^{45} +1.56997 q^{47} +6.24233 q^{49} -22.1345 q^{51} +0.205679 q^{53} -1.82868 q^{55} -20.6882 q^{57} -13.1714 q^{59} +3.53160 q^{61} -31.9172 q^{63} +4.08064 q^{65} -3.41564 q^{67} -22.3410 q^{69} -8.33065 q^{71} +10.3377 q^{73} -15.3573 q^{75} -9.19473 q^{77} -15.1079 q^{79} +41.6154 q^{81} +10.2212 q^{83} +4.66923 q^{85} +4.87821 q^{87} +8.98950 q^{89} +20.5178 q^{91} -5.03804 q^{93} +4.36414 q^{95} +12.6010 q^{97} +22.1615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9}+O(q^{10}) \) 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9	\( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9} - 9 q^{11} - q^{13} - 14 q^{15} - 27 q^{19} - 3 q^{21} - 13 q^{23} + 26 q^{25} - 15 q^{27} - 25 q^{31} + 16 q^{33} - 21 q^{35} - q^{37} - 33 q^{39} + 10 q^{41} - 35 q^{43} - 4 q^{45} - 6 q^{47} + 36 q^{49} - 48 q^{51} - q^{53} - 41 q^{55} + 14 q^{57} - 30 q^{59} + 3 q^{61} - 31 q^{63} + 7 q^{65} - 22 q^{67} - 17 q^{69} - 6 q^{71} + 5 q^{73} - 4 q^{75} - 14 q^{77} - 56 q^{79} + 26 q^{81} + 28 q^{83} - 23 q^{85} - 11 q^{87} - 24 q^{89} - 38 q^{91} - 55 q^{93} + 4 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) \) 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9 - 9 * q^11 - q^13 - 14 * q^15 - 27 * q^19 - 3 * q^21 - 13 * q^23 + 26 * q^25 - 15 * q^27 - 25 * q^31 + 16 * q^33 - 21 * q^35 - q^37 - 33 * q^39 + 10 * q^41 - 35 * q^43 - 4 * q^45 - 6 * q^47 + 36 * q^49 - 48 * q^51 - q^53 - 41 * q^55 + 14 * q^57 - 30 * q^59 + 3 * q^61 - 31 * q^63 + 7 * q^65 - 22 * q^67 - 17 * q^69 - 6 * q^71 + 5 * q^73 - 4 * q^75 - 14 * q^77 - 56 * q^79 + 26 * q^81 + 28 * q^83 - 23 * q^85 - 11 * q^87 - 24 * q^89 - 38 * q^91 - 55 * q^93 + 4 * q^95 + 6 * q^97 - 12 * q^99

Introduction

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