LMFDB - Embedded newform 6016.2.a.p.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6016,2,Mod(1,6016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6016, 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("6016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6016 = 2^{7} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6016.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:	$48.0380018560$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 163 x^{9} - 840 x^{8} - 438 x^{7} + 3358 x^{6} + 5 x^{5} + \cdots - 832 $ x^13 - 4x^12 - 22x^11 + 96x^10 + 163x^9 - 840x^8 - 438x^7 + 3358x^6 + 5x^5 - 6230x^4 + 1488x^3 + 4720x^2 - 1440x - 832
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$1.76446$ of defining polynomial
Character	$\chi$	$=$	6016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.76446 q^{3} +4.26056 q^{5} +2.37797 q^{7} +0.113336 q^{9} +O(q^{10})$	$q+1.76446 q^{3} +4.26056 q^{5} +2.37797 q^{7} +0.113336 q^{9} +6.24560 q^{11} +2.70287 q^{13} +7.51761 q^{15} -4.14632 q^{17} -2.80080 q^{19} +4.19584 q^{21} +3.25870 q^{23} +13.1524 q^{25} -5.09342 q^{27} -6.91112 q^{29} +2.09232 q^{31} +11.0201 q^{33} +10.1315 q^{35} -1.19439 q^{37} +4.76912 q^{39} -2.25223 q^{41} -4.81440 q^{43} +0.482874 q^{45} +1.00000 q^{47} -1.34527 q^{49} -7.31604 q^{51} +8.16169 q^{53} +26.6097 q^{55} -4.94192 q^{57} -1.89371 q^{59} -7.83670 q^{61} +0.269509 q^{63} +11.5157 q^{65} +1.97756 q^{67} +5.74987 q^{69} -13.7046 q^{71} +8.99761 q^{73} +23.2069 q^{75} +14.8518 q^{77} -7.04445 q^{79} -9.32716 q^{81} -11.3022 q^{83} -17.6656 q^{85} -12.1944 q^{87} +16.1015 q^{89} +6.42734 q^{91} +3.69183 q^{93} -11.9330 q^{95} +0.560299 q^{97} +0.707850 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 4 q^{3} + 6 q^{5} + 2 q^{7} + 21 q^{9}+O(q^{10}) $ 13 * q + 4 * q^3 + 6 * q^5 + 2 * q^7 + 21 * q^9	$ 13 q + 4 q^{3} + 6 q^{5} + 2 q^{7} + 21 q^{9} + 10 q^{11} + 4 q^{13} - 14 q^{15} + 10 q^{17} + 8 q^{19} + 10 q^{21} - 18 q^{23} + 23 q^{25} + 16 q^{27} + 14 q^{29} - 4 q^{31} + 14 q^{33} + 14 q^{35} + 16 q^{37} - 12 q^{39} + 10 q^{41} + 12 q^{43} + 10 q^{45} + 13 q^{47} + 9 q^{49} + 22 q^{51} + 26 q^{53} + 2 q^{55} + 20 q^{57} + 30 q^{59} + 18 q^{61} - 12 q^{63} - 4 q^{65} + 4 q^{67} + 2 q^{69} - 36 q^{71} + 10 q^{73} + 38 q^{75} + 42 q^{77} + 21 q^{81} + 12 q^{83} + 4 q^{85} - 6 q^{87} + 50 q^{89} - 4 q^{91} + 52 q^{93} - 8 q^{95} - 10 q^{97} + 34 q^{99}+O(q^{100}) $ 13 * q + 4 * q^3 + 6 * q^5 + 2 * q^7 + 21 * q^9 + 10 * q^11 + 4 * q^13 - 14 * q^15 + 10 * q^17 + 8 * q^19 + 10 * q^21 - 18 * q^23 + 23 * q^25 + 16 * q^27 + 14 * q^29 - 4 * q^31 + 14 * q^33 + 14 * q^35 + 16 * q^37 - 12 * q^39 + 10 * q^41 + 12 * q^43 + 10 * q^45 + 13 * q^47 + 9 * q^49 + 22 * q^51 + 26 * q^53 + 2 * q^55 + 20 * q^57 + 30 * q^59 + 18 * q^61 - 12 * q^63 - 4 * q^65 + 4 * q^67 + 2 * q^69 - 36 * q^71 + 10 * q^73 + 38 * q^75 + 42 * q^77 + 21 * q^81 + 12 * q^83 + 4 * q^85 - 6 * q^87 + 50 * q^89 - 4 * q^91 + 52 * q^93 - 8 * q^95 - 10 * q^97 + 34 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.76446	1.01871	0.509357	−	0.860555i	$-0.329883\pi$
$3$	1.76446	1.01871	0.509357	+	0.860555i	$0.329883\pi$
$4$	0	0
$5$	4.26056	1.90538	0.952690	−	0.303944i	$-0.0983036\pi$
$5$	4.26056	1.90538	0.952690	+	0.303944i	$0.0983036\pi$
$6$	0	0
$7$	2.37797	0.898787	0.449394	−	0.893334i	$-0.351640\pi$
$7$	2.37797	0.898787	0.449394	+	0.893334i	$0.351640\pi$
$8$	0	0
$9$	0.113336	0.0377786
$10$	0	0
$11$	6.24560	1.88312	0.941560	−	0.336846i	$-0.109360\pi$
$11$	6.24560	1.88312	0.941560	+	0.336846i	$0.109360\pi$
$12$	0	0
$13$	2.70287	0.749642	0.374821	−	0.927097i	$-0.377704\pi$
$13$	2.70287	0.749642	0.374821	+	0.927097i	$0.377704\pi$
$14$	0	0
$15$	7.51761	1.94104
$16$	0	0
$17$	−4.14632	−1.00563	−0.502815	−	0.864394i	$-0.667702\pi$
$17$	−4.14632	−1.00563	−0.502815	+	0.864394i	$0.667702\pi$
$18$	0	0
$19$	−2.80080	−0.642548	−0.321274	−	0.946986i	$-0.604111\pi$
$19$	−2.80080	−0.642548	−0.321274	+	0.946986i	$0.604111\pi$
$20$	0	0
$21$	4.19584	0.915607
$22$	0	0
$23$	3.25870	0.679487	0.339743	−	0.940518i	$-0.389660\pi$
$23$	3.25870	0.679487	0.339743	+	0.940518i	$0.389660\pi$
$24$	0	0
$25$	13.1524	2.63047
$26$	0	0
$27$	−5.09342	−0.980229
$28$	0	0
$29$	−6.91112	−1.28336	−0.641681	−	0.766971i	$-0.721763\pi$
$29$	−6.91112	−1.28336	−0.641681	+	0.766971i	$0.721763\pi$
$30$	0	0
$31$	2.09232	0.375792	0.187896	−	0.982189i	$-0.439833\pi$
$31$	2.09232	0.375792	0.187896	+	0.982189i	$0.439833\pi$
$32$	0	0
$33$	11.0201	1.91836
$34$	0	0
$35$	10.1315	1.71253
$36$	0	0
$37$	−1.19439	−0.196356	−0.0981781	−	0.995169i	$-0.531301\pi$
$37$	−1.19439	−0.196356	−0.0981781	+	0.995169i	$0.531301\pi$
$38$	0	0
$39$	4.76912	0.763671
$40$	0	0
$41$	−2.25223	−0.351739	−0.175870	−	0.984413i	$-0.556274\pi$
$41$	−2.25223	−0.351739	−0.175870	+	0.984413i	$0.556274\pi$
$42$	0	0
$43$	−4.81440	−0.734189	−0.367095	−	0.930184i	$-0.619647\pi$
$43$	−4.81440	−0.734189	−0.367095	+	0.930184i	$0.619647\pi$
$44$	0	0
$45$	0.482874	0.0719826
$46$	0	0
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−1.34527	−0.192182
$50$	0	0
$51$	−7.31604	−1.02445
$52$	0	0
$53$	8.16169	1.12109	0.560547	−	0.828123i	$-0.310591\pi$
$53$	8.16169	1.12109	0.560547	+	0.828123i	$0.310591\pi$
$54$	0	0
$55$	26.6097	3.58806
$56$	0	0
$57$	−4.94192	−0.654573
$58$	0	0
$59$	−1.89371	−0.246540	−0.123270	−	0.992373i	$-0.539338\pi$
$59$	−1.89371	−0.246540	−0.123270	+	0.992373i	$0.539338\pi$
$60$	0	0
$61$	−7.83670	−1.00339	−0.501693	−	0.865045i	$-0.667289\pi$
$61$	−7.83670	−1.00339	−0.501693	+	0.865045i	$0.667289\pi$
$62$	0	0
$63$	0.269509	0.0339549
$64$	0	0
$65$	11.5157	1.42835
$66$	0	0
$67$	1.97756	0.241598	0.120799	−	0.992677i	$-0.461454\pi$
$67$	1.97756	0.241598	0.120799	+	0.992677i	$0.461454\pi$
$68$	0	0
$69$	5.74987	0.692203
$70$	0	0
$71$	−13.7046	−1.62644	−0.813218	−	0.581960i	$-0.802286\pi$
$71$	−13.7046	−1.62644	−0.813218	+	0.581960i	$0.802286\pi$
$72$	0	0
$73$	8.99761	1.05309	0.526545	−	0.850147i	$-0.323487\pi$
$73$	8.99761	1.05309	0.526545	+	0.850147i	$0.323487\pi$
$74$	0	0
$75$	23.2069	2.67970
$76$	0	0
$77$	14.8518	1.69252
$78$	0	0
$79$	−7.04445	−0.792562	−0.396281	−	0.918129i	$-0.629699\pi$
$79$	−7.04445	−0.792562	−0.396281	+	0.918129i	$0.629699\pi$
$80$	0	0
$81$	−9.32716	−1.03635
$82$	0	0
$83$	−11.3022	−1.24058	−0.620288	−	0.784375i	$-0.712984\pi$
$83$	−11.3022	−1.24058	−0.620288	+	0.784375i	$0.712984\pi$
$84$	0	0
$85$	−17.6656	−1.91611
$86$	0	0
$87$	−12.1944	−1.30738
$88$	0	0
$89$	16.1015	1.70676	0.853379	−	0.521290i	$-0.174549\pi$
$89$	16.1015	1.70676	0.853379	+	0.521290i	$0.174549\pi$
$90$	0	0
$91$	6.42734	0.673769
$92$	0	0
$93$	3.69183	0.382825
$94$	0	0
$95$	−11.9330	−1.22430
$96$	0	0
$97$	0.560299	0.0568898	0.0284449	−	0.999595i	$-0.490944\pi$
$97$	0.560299	0.0568898	0.0284449	+	0.999595i	$0.490944\pi$
$98$	0	0
$99$	0.707850	0.0711416
$100$	0	0
$101$	9.40426	0.935759	0.467879	−	0.883792i	$-0.345018\pi$
$101$	9.40426	0.935759	0.467879	+	0.883792i	$0.345018\pi$
$102$	0	0
$103$	−8.70473	−0.857703	−0.428851	−	0.903375i	$-0.641082\pi$
$103$	−8.70473	−0.857703	−0.428851	+	0.903375i	$0.641082\pi$
$104$	0	0
$105$	17.8766	1.74458
$106$	0	0
$107$	−1.94184	−0.187725	−0.0938623	−	0.995585i	$-0.529921\pi$
$107$	−1.94184	−0.187725	−0.0938623	+	0.995585i	$0.529921\pi$
$108$	0	0
$109$	−19.5490	−1.87246	−0.936229	−	0.351391i	$-0.885709\pi$
$109$	−19.5490	−1.87246	−0.936229	+	0.351391i	$0.885709\pi$
$110$	0	0
$111$	−2.10745	−0.200031
$112$	0	0
$113$	1.94537	0.183005	0.0915025	−	0.995805i	$-0.470833\pi$
$113$	1.94537	0.183005	0.0915025	+	0.995805i	$0.470833\pi$
$114$	0	0
$115$	13.8839	1.29468
$116$	0	0
$117$	0.306332	0.0283204
$118$	0	0
$119$	−9.85981	−0.903848
$120$	0	0
$121$	28.0075	2.54614
$122$	0	0
$123$	−3.97398	−0.358322
$124$	0	0
$125$	34.7336	3.10667
$126$	0	0
$127$	1.74057	0.154451	0.0772253	−	0.997014i	$-0.475394\pi$
$127$	1.74057	0.154451	0.0772253	+	0.997014i	$0.475394\pi$
$128$	0	0
$129$	−8.49484	−0.747929
$130$	0	0
$131$	5.08028	0.443866	0.221933	−	0.975062i	$-0.428763\pi$
$131$	5.08028	0.443866	0.221933	+	0.975062i	$0.428763\pi$
$132$	0	0
$133$	−6.66021	−0.577514
$134$	0	0
$135$	−21.7008	−1.86771
$136$	0	0
$137$	−1.58677	−0.135567	−0.0677833	−	0.997700i	$-0.521593\pi$
$137$	−1.58677	−0.135567	−0.0677833	+	0.997700i	$0.521593\pi$
$138$	0	0
$139$	−8.73467	−0.740865	−0.370433	−	0.928859i	$-0.620791\pi$
$139$	−8.73467	−0.740865	−0.370433	+	0.928859i	$0.620791\pi$
$140$	0	0
$141$	1.76446	0.148595
$142$	0	0
$143$	16.8811	1.41167
$144$	0	0
$145$	−29.4452	−2.44529
$146$	0	0
$147$	−2.37368	−0.195778
$148$	0	0
$149$	−14.4739	−1.18575	−0.592874	−	0.805295i	$-0.702007\pi$
$149$	−14.4739	−1.18575	−0.592874	+	0.805295i	$0.702007\pi$
$150$	0	0
$151$	−23.4194	−1.90584	−0.952920	−	0.303221i	$-0.901938\pi$
$151$	−23.4194	−1.90584	−0.952920	+	0.303221i	$0.901938\pi$
$152$	0	0
$153$	−0.469927	−0.0379913
$154$	0	0
$155$	8.91446	0.716027
$156$	0	0
$157$	3.56003	0.284121	0.142060	−	0.989858i	$-0.454627\pi$
$157$	3.56003	0.284121	0.142060	+	0.989858i	$0.454627\pi$
$158$	0	0
$159$	14.4010	1.14207
$160$	0	0
$161$	7.74909	0.610714
$162$	0	0
$163$	18.4346	1.44391	0.721954	−	0.691941i	$-0.243244\pi$
$163$	18.4346	1.44391	0.721954	+	0.691941i	$0.243244\pi$
$164$	0	0
$165$	46.9520	3.65521
$166$	0	0
$167$	−7.01901	−0.543148	−0.271574	−	0.962418i	$-0.587544\pi$
$167$	−7.01901	−0.543148	−0.271574	+	0.962418i	$0.587544\pi$
$168$	0	0
$169$	−5.69448	−0.438037
$170$	0	0
$171$	−0.317431	−0.0242746
$172$	0	0
$173$	7.46974	0.567913	0.283957	−	0.958837i	$-0.408353\pi$
$173$	7.46974	0.567913	0.283957	+	0.958837i	$0.408353\pi$
$174$	0	0
$175$	31.2759	2.36423
$176$	0	0
$177$	−3.34138	−0.251154
$178$	0	0
$179$	−17.0226	−1.27233	−0.636166	−	0.771553i	$-0.719480\pi$
$179$	−17.0226	−1.27233	−0.636166	+	0.771553i	$0.719480\pi$
$180$	0	0
$181$	3.67333	0.273036	0.136518	−	0.990638i	$-0.456409\pi$
$181$	3.67333	0.273036	0.136518	+	0.990638i	$0.456409\pi$
$182$	0	0
$183$	−13.8276	−1.02216
$184$	0	0
$185$	−5.08876	−0.374133
$186$	0	0
$187$	−25.8963	−1.89372
$188$	0	0
$189$	−12.1120	−0.881017
$190$	0	0
$191$	−16.5313	−1.19616	−0.598080	−	0.801436i	$-0.704070\pi$
$191$	−16.5313	−1.19616	−0.598080	+	0.801436i	$0.704070\pi$
$192$	0	0
$193$	−18.2049	−1.31042	−0.655210	−	0.755447i	$-0.727420\pi$
$193$	−18.2049	−1.31042	−0.655210	+	0.755447i	$0.727420\pi$
$194$	0	0
$195$	20.3191	1.45508
$196$	0	0
$197$	−22.1711	−1.57962	−0.789812	−	0.613348i	$-0.789822\pi$
$197$	−22.1711	−1.57962	−0.789812	+	0.613348i	$0.789822\pi$
$198$	0	0
$199$	21.5971	1.53098	0.765488	−	0.643450i	$-0.222497\pi$
$199$	21.5971	1.53098	0.765488	+	0.643450i	$0.222497\pi$
$200$	0	0
$201$	3.48934	0.246119
$202$	0	0
$203$	−16.4344	−1.15347
$204$	0	0
$205$	−9.59576	−0.670197
$206$	0	0
$207$	0.369328	0.0256701
$208$	0	0
$209$	−17.4927	−1.20999
$210$	0	0
$211$	14.8343	1.02124	0.510618	−	0.859808i	$-0.329417\pi$
$211$	14.8343	1.02124	0.510618	+	0.859808i	$0.329417\pi$
$212$	0	0
$213$	−24.1813	−1.65687
$214$	0	0
$215$	−20.5120	−1.39891
$216$	0	0
$217$	4.97548	0.337757
$218$	0	0
$219$	15.8760	1.07280
$220$	0	0
$221$	−11.2070	−0.753863
$222$	0	0
$223$	13.0468	0.873679	0.436840	−	0.899539i	$-0.356098\pi$
$223$	13.0468	0.873679	0.436840	+	0.899539i	$0.356098\pi$
$224$	0	0
$225$	1.49063	0.0993756
$226$	0	0
$227$	8.46786	0.562031	0.281016	−	0.959703i	$-0.409329\pi$
$227$	8.46786	0.562031	0.281016	+	0.959703i	$0.409329\pi$
$228$	0	0
$229$	0.146187	0.00966028	0.00483014	−	0.999988i	$-0.498463\pi$
$229$	0.146187	0.00966028	0.00483014	+	0.999988i	$0.498463\pi$
$230$	0	0
$231$	26.2055	1.72420
$232$	0	0
$233$	27.3788	1.79365	0.896823	−	0.442389i	$-0.145869\pi$
$233$	27.3788	1.79365	0.896823	+	0.442389i	$0.145869\pi$
$234$	0	0
$235$	4.26056	0.277928
$236$	0	0
$237$	−12.4297	−0.807395
$238$	0	0
$239$	−14.0144	−0.906516	−0.453258	−	0.891379i	$-0.649738\pi$
$239$	−14.0144	−0.906516	−0.453258	+	0.891379i	$0.649738\pi$
$240$	0	0
$241$	21.3913	1.37793	0.688966	−	0.724793i	$-0.258065\pi$
$241$	21.3913	1.37793	0.688966	+	0.724793i	$0.258065\pi$
$242$	0	0
$243$	−1.17720	−0.0755173
$244$	0	0
$245$	−5.73161	−0.366179
$246$	0	0
$247$	−7.57021	−0.481681
$248$	0	0
$249$	−19.9423	−1.26379
$250$	0	0
$251$	−5.78185	−0.364947	−0.182473	−	0.983211i	$-0.558410\pi$
$251$	−5.78185	−0.364947	−0.182473	+	0.983211i	$0.558410\pi$
$252$	0	0
$253$	20.3526	1.27956
$254$	0	0
$255$	−31.1704	−1.95197
$256$	0	0
$257$	29.9115	1.86583	0.932914	−	0.360098i	$-0.117257\pi$
$257$	29.9115	1.86583	0.932914	+	0.360098i	$0.117257\pi$
$258$	0	0
$259$	−2.84021	−0.176482
$260$	0	0
$261$	−0.783277	−0.0484836
$262$	0	0
$263$	−19.1048	−1.17805	−0.589026	−	0.808114i	$-0.700488\pi$
$263$	−19.1048	−1.17805	−0.589026	+	0.808114i	$0.700488\pi$
$264$	0	0
$265$	34.7734	2.13611
$266$	0	0
$267$	28.4106	1.73870
$268$	0	0
$269$	−4.49124	−0.273836	−0.136918	−	0.990582i	$-0.543720\pi$
$269$	−4.49124	−0.273836	−0.136918	+	0.990582i	$0.543720\pi$
$270$	0	0
$271$	17.2850	1.04999	0.524994	−	0.851106i	$-0.324068\pi$
$271$	17.2850	1.04999	0.524994	+	0.851106i	$0.324068\pi$
$272$	0	0
$273$	11.3408	0.686378
$274$	0	0
$275$	82.1444	4.95349
$276$	0	0
$277$	15.7184	0.944425	0.472212	−	0.881485i	$-0.343456\pi$
$277$	15.7184	0.944425	0.472212	+	0.881485i	$0.343456\pi$
$278$	0	0
$279$	0.237135	0.0141969
$280$	0	0
$281$	−15.4132	−0.919475	−0.459738	−	0.888055i	$-0.652057\pi$
$281$	−15.4132	−0.919475	−0.459738	+	0.888055i	$0.652057\pi$
$282$	0	0
$283$	18.2010	1.08194	0.540970	−	0.841042i	$-0.318057\pi$
$283$	18.2010	1.08194	0.540970	+	0.841042i	$0.318057\pi$
$284$	0	0
$285$	−21.0553	−1.24721
$286$	0	0
$287$	−5.35573	−0.316139
$288$	0	0
$289$	0.191970	0.0112923
$290$	0	0
$291$	0.988628	0.0579544
$292$	0	0
$293$	−7.54964	−0.441055	−0.220527	−	0.975381i	$-0.570778\pi$
$293$	−7.54964	−0.441055	−0.220527	+	0.975381i	$0.570778\pi$
$294$	0	0
$295$	−8.06826	−0.469752
$296$	0	0
$297$	−31.8115	−1.84589
$298$	0	0
$299$	8.80786	0.509372
$300$	0	0
$301$	−11.4485	−0.659880
$302$	0	0
$303$	16.5935	0.953271
$304$	0	0
$305$	−33.3887	−1.91183
$306$	0	0
$307$	−3.32738	−0.189903	−0.0949517	−	0.995482i	$-0.530270\pi$
$307$	−3.32738	−0.189903	−0.0949517	+	0.995482i	$0.530270\pi$
$308$	0	0
$309$	−15.3592	−0.873754
$310$	0	0
$311$	−30.3682	−1.72202	−0.861011	−	0.508587i	$-0.830168\pi$
$311$	−30.3682	−1.72202	−0.861011	+	0.508587i	$0.830168\pi$
$312$	0	0
$313$	29.6502	1.67593	0.837963	−	0.545727i	$-0.183746\pi$
$313$	29.6502	1.67593	0.837963	+	0.545727i	$0.183746\pi$
$314$	0	0
$315$	1.14826	0.0646970
$316$	0	0
$317$	19.3385	1.08616	0.543078	−	0.839682i	$-0.317259\pi$
$317$	19.3385	1.08616	0.543078	+	0.839682i	$0.317259\pi$
$318$	0	0
$319$	−43.1641	−2.41673
$320$	0	0
$321$	−3.42630	−0.191238
$322$	0	0
$323$	11.6130	0.646166
$324$	0	0
$325$	35.5492	1.97191
$326$	0	0
$327$	−34.4936	−1.90750
$328$	0	0
$329$	2.37797	0.131102
$330$	0	0
$331$	−15.2954	−0.840711	−0.420356	−	0.907359i	$-0.638095\pi$
$331$	−15.2954	−0.840711	−0.420356	+	0.907359i	$0.638095\pi$
$332$	0	0
$333$	−0.135367	−0.00741806
$334$	0	0
$335$	8.42552	0.460335
$336$	0	0
$337$	−31.8066	−1.73262	−0.866309	−	0.499509i	$-0.833514\pi$
$337$	−31.8066	−1.73262	−0.866309	+	0.499509i	$0.833514\pi$
$338$	0	0
$339$	3.43254	0.186430
$340$	0	0
$341$	13.0678	0.707662
$342$	0	0
$343$	−19.8448	−1.07152
$344$	0	0
$345$	24.4977	1.31891
$346$	0	0
$347$	−13.3097	−0.714503	−0.357252	−	0.934008i	$-0.616286\pi$
$347$	−13.3097	−0.714503	−0.357252	+	0.934008i	$0.616286\pi$
$348$	0	0
$349$	20.9411	1.12095	0.560476	−	0.828171i	$-0.310618\pi$
$349$	20.9411	1.12095	0.560476	+	0.828171i	$0.310618\pi$
$350$	0	0
$351$	−13.7669	−0.734820
$352$	0	0
$353$	−10.5398	−0.560979	−0.280489	−	0.959857i	$-0.590497\pi$
$353$	−10.5398	−0.560979	−0.280489	+	0.959857i	$0.590497\pi$
$354$	0	0
$355$	−58.3892	−3.09898
$356$	0	0
$357$	−17.3973	−0.920762
$358$	0	0
$359$	20.3309	1.07302	0.536512	−	0.843893i	$-0.319742\pi$
$359$	20.3309	1.07302	0.536512	+	0.843893i	$0.319742\pi$
$360$	0	0
$361$	−11.1555	−0.587132
$362$	0	0
$363$	49.4183	2.59379
$364$	0	0
$365$	38.3348	2.00654
$366$	0	0
$367$	−17.8098	−0.929666	−0.464833	−	0.885398i	$-0.653886\pi$
$367$	−17.8098	−0.929666	−0.464833	+	0.885398i	$0.653886\pi$
$368$	0	0
$369$	−0.255258	−0.0132882
$370$	0	0
$371$	19.4082	1.00763
$372$	0	0
$373$	29.6347	1.53443	0.767213	−	0.641392i	$-0.221643\pi$
$373$	29.6347	1.53443	0.767213	+	0.641392i	$0.221643\pi$
$374$	0	0
$375$	61.2862	3.16481
$376$	0	0
$377$	−18.6799	−0.962062
$378$	0	0
$379$	−9.38553	−0.482102	−0.241051	−	0.970512i	$-0.577492\pi$
$379$	−9.38553	−0.482102	−0.241051	+	0.970512i	$0.577492\pi$
$380$	0	0
$381$	3.07118	0.157341
$382$	0	0
$383$	−18.7118	−0.956129	−0.478064	−	0.878325i	$-0.658661\pi$
$383$	−18.7118	−0.956129	−0.478064	+	0.878325i	$0.658661\pi$
$384$	0	0
$385$	63.2771	3.22490
$386$	0	0
$387$	−0.545644	−0.0277366
$388$	0	0
$389$	−5.36935	−0.272237	−0.136118	−	0.990693i	$-0.543463\pi$
$389$	−5.36935	−0.272237	−0.136118	+	0.990693i	$0.543463\pi$
$390$	0	0
$391$	−13.5116	−0.683313
$392$	0	0
$393$	8.96397	0.452172
$394$	0	0
$395$	−30.0133	−1.51013
$396$	0	0
$397$	10.1101	0.507411	0.253705	−	0.967282i	$-0.418351\pi$
$397$	10.1101	0.507411	0.253705	+	0.967282i	$0.418351\pi$
$398$	0	0
$399$	−11.7517	−0.588322
$400$	0	0
$401$	25.3618	1.26651	0.633253	−	0.773945i	$-0.281719\pi$
$401$	25.3618	1.26651	0.633253	+	0.773945i	$0.281719\pi$
$402$	0	0
$403$	5.65528	0.281710
$404$	0	0
$405$	−39.7389	−1.97464
$406$	0	0
$407$	−7.45967	−0.369762
$408$	0	0
$409$	12.4427	0.615254	0.307627	−	0.951507i	$-0.400465\pi$
$409$	12.4427	0.615254	0.307627	+	0.951507i	$0.400465\pi$
$410$	0	0
$411$	−2.79979	−0.138104
$412$	0	0
$413$	−4.50318	−0.221587
$414$	0	0
$415$	−48.1536	−2.36377
$416$	0	0
$417$	−15.4120	−0.754730
$418$	0	0
$419$	25.9548	1.26797	0.633987	−	0.773344i	$-0.281417\pi$
$419$	25.9548	1.26797	0.633987	+	0.773344i	$0.281417\pi$
$420$	0	0
$421$	−24.7181	−1.20469	−0.602343	−	0.798238i	$-0.705766\pi$
$421$	−24.7181	−1.20469	−0.602343	+	0.798238i	$0.705766\pi$
$422$	0	0
$423$	0.113336	0.00551058
$424$	0	0
$425$	−54.5339	−2.64528
$426$	0	0
$427$	−18.6354	−0.901831
$428$	0	0
$429$	29.7860	1.43808
$430$	0	0
$431$	8.81741	0.424720	0.212360	−	0.977192i	$-0.431885\pi$
$431$	8.81741	0.424720	0.212360	+	0.977192i	$0.431885\pi$
$432$	0	0
$433$	22.9316	1.10202	0.551010	−	0.834498i	$-0.314242\pi$
$433$	22.9316	1.10202	0.551010	+	0.834498i	$0.314242\pi$
$434$	0	0
$435$	−51.9551	−2.49105
$436$	0	0
$437$	−9.12699	−0.436603
$438$	0	0
$439$	−6.04838	−0.288673	−0.144337	−	0.989529i	$-0.546105\pi$
$439$	−6.04838	−0.288673	−0.144337	+	0.989529i	$0.546105\pi$
$440$	0	0
$441$	−0.152467	−0.00726036
$442$	0	0
$443$	−27.7989	−1.32076	−0.660382	−	0.750930i	$-0.729606\pi$
$443$	−27.7989	−1.32076	−0.660382	+	0.750930i	$0.729606\pi$
$444$	0	0
$445$	68.6015	3.25202
$446$	0	0
$447$	−25.5387	−1.20794
$448$	0	0
$449$	−5.33168	−0.251617	−0.125809	−	0.992055i	$-0.540153\pi$
$449$	−5.33168	−0.251617	−0.125809	+	0.992055i	$0.540153\pi$
$450$	0	0
$451$	−14.0665	−0.662367
$452$	0	0
$453$	−41.3226	−1.94151
$454$	0	0
$455$	27.3841	1.28378
$456$	0	0
$457$	−42.2225	−1.97508	−0.987542	−	0.157356i	$-0.949703\pi$
$457$	−42.2225	−1.97508	−0.987542	+	0.157356i	$0.949703\pi$
$458$	0	0
$459$	21.1189	0.985748
$460$	0	0
$461$	30.0952	1.40168	0.700838	−	0.713321i	$-0.252810\pi$
$461$	30.0952	1.40168	0.700838	+	0.713321i	$0.252810\pi$
$462$	0	0
$463$	12.2243	0.568110	0.284055	−	0.958808i	$-0.408320\pi$
$463$	12.2243	0.568110	0.284055	+	0.958808i	$0.408320\pi$
$464$	0	0
$465$	15.7293	0.729427
$466$	0	0
$467$	10.0547	0.465278	0.232639	−	0.972563i	$-0.425264\pi$
$467$	10.0547	0.465278	0.232639	+	0.972563i	$0.425264\pi$
$468$	0	0
$469$	4.70258	0.217145
$470$	0	0
$471$	6.28154	0.289438
$472$	0	0
$473$	−30.0688	−1.38257
$474$	0	0
$475$	−36.8371	−1.69020
$476$	0	0
$477$	0.925012	0.0423534
$478$	0	0
$479$	−16.7130	−0.763635	−0.381817	−	0.924238i	$-0.624702\pi$
$479$	−16.7130	−0.763635	−0.381817	+	0.924238i	$0.624702\pi$
$480$	0	0
$481$	−3.22828	−0.147197
$482$	0	0
$483$	13.6730	0.622143
$484$	0	0
$485$	2.38719	0.108397
$486$	0	0
$487$	34.7146	1.57307	0.786535	−	0.617545i	$-0.211873\pi$
$487$	34.7146	1.57307	0.786535	+	0.617545i	$0.211873\pi$
$488$	0	0
$489$	32.5272	1.47093
$490$	0	0
$491$	18.8558	0.850951	0.425475	−	0.904970i	$-0.360107\pi$
$491$	18.8558	0.850951	0.425475	+	0.904970i	$0.360107\pi$
$492$	0	0
$493$	28.6557	1.29059
$494$	0	0
$495$	3.01584	0.135552
$496$	0	0
$497$	−32.5891	−1.46182
$498$	0	0
$499$	36.7289	1.64421	0.822106	−	0.569334i	$-0.192799\pi$
$499$	36.7289	1.64421	0.822106	+	0.569334i	$0.192799\pi$
$500$	0	0
$501$	−12.3848	−0.553312
$502$	0	0
$503$	14.5495	0.648731	0.324366	−	0.945932i	$-0.394849\pi$
$503$	14.5495	0.648731	0.324366	+	0.945932i	$0.394849\pi$
$504$	0	0
$505$	40.0674	1.78298
$506$	0	0
$507$	−10.0477	−0.446234
$508$	0	0
$509$	26.4560	1.17264	0.586322	−	0.810078i	$-0.300575\pi$
$509$	26.4560	1.17264	0.586322	+	0.810078i	$0.300575\pi$
$510$	0	0
$511$	21.3960	0.946504
$512$	0	0
$513$	14.2657	0.629844
$514$	0	0
$515$	−37.0870	−1.63425
$516$	0	0
$517$	6.24560	0.274681
$518$	0	0
$519$	13.1801	0.578541
$520$	0	0
$521$	14.1423	0.619587	0.309793	−	0.950804i	$-0.399740\pi$
$521$	14.1423	0.619587	0.309793	+	0.950804i	$0.399740\pi$
$522$	0	0
$523$	−19.2853	−0.843286	−0.421643	−	0.906762i	$-0.638546\pi$
$523$	−19.2853	−0.843286	−0.421643	+	0.906762i	$0.638546\pi$
$524$	0	0
$525$	55.1852	2.40848
$526$	0	0
$527$	−8.67544	−0.377908
$528$	0	0
$529$	−12.3808	−0.538298
$530$	0	0
$531$	−0.214625	−0.00931394
$532$	0	0
$533$	−6.08749	−0.263679
$534$	0	0
$535$	−8.27331	−0.357687
$536$	0	0
$537$	−30.0358	−1.29614
$538$	0	0
$539$	−8.40203	−0.361901
$540$	0	0
$541$	−7.55878	−0.324977	−0.162489	−	0.986710i	$-0.551952\pi$
$541$	−7.55878	−0.324977	−0.162489	+	0.986710i	$0.551952\pi$
$542$	0	0
$543$	6.48145	0.278146
$544$	0	0
$545$	−83.2898	−3.56774
$546$	0	0
$547$	16.1587	0.690897	0.345448	−	0.938438i	$-0.387727\pi$
$547$	16.1587	0.690897	0.345448	+	0.938438i	$0.387727\pi$
$548$	0	0
$549$	−0.888179	−0.0379066
$550$	0	0
$551$	19.3567	0.824622
$552$	0	0
$553$	−16.7515	−0.712345
$554$	0	0
$555$	−8.97893	−0.381135
$556$	0	0
$557$	−19.2361	−0.815058	−0.407529	−	0.913192i	$-0.633609\pi$
$557$	−19.2361	−0.815058	−0.407529	+	0.913192i	$0.633609\pi$
$558$	0	0
$559$	−13.0127	−0.550379
$560$	0	0
$561$	−45.6930	−1.92916
$562$	0	0
$563$	10.6672	0.449569	0.224785	−	0.974408i	$-0.427832\pi$
$563$	10.6672	0.449569	0.224785	+	0.974408i	$0.427832\pi$
$564$	0	0
$565$	8.28836	0.348694
$566$	0	0
$567$	−22.1797	−0.931459
$568$	0	0
$569$	35.4436	1.48587	0.742936	−	0.669362i	$-0.233432\pi$
$569$	35.4436	1.48587	0.742936	+	0.669362i	$0.233432\pi$
$570$	0	0
$571$	35.2430	1.47487	0.737437	−	0.675416i	$-0.236036\pi$
$571$	35.2430	1.47487	0.737437	+	0.675416i	$0.236036\pi$
$572$	0	0
$573$	−29.1688	−1.21855
$574$	0	0
$575$	42.8597	1.78737
$576$	0	0
$577$	−36.7515	−1.52999	−0.764993	−	0.644039i	$-0.777257\pi$
$577$	−36.7515	−1.52999	−0.764993	+	0.644039i	$0.777257\pi$
$578$	0	0
$579$	−32.1220	−1.33494
$580$	0	0
$581$	−26.8762	−1.11501
$582$	0	0
$583$	50.9747	2.11115
$584$	0	0
$585$	1.30515	0.0539612
$586$	0	0
$587$	2.17547	0.0897914	0.0448957	−	0.998992i	$-0.485704\pi$
$587$	2.17547	0.0897914	0.0448957	+	0.998992i	$0.485704\pi$
$588$	0	0
$589$	−5.86018	−0.241465
$590$	0	0
$591$	−39.1201	−1.60919
$592$	0	0
$593$	32.7052	1.34304	0.671521	−	0.740986i	$-0.265641\pi$
$593$	32.7052	1.34304	0.671521	+	0.740986i	$0.265641\pi$
$594$	0	0
$595$	−42.0083	−1.72217
$596$	0	0
$597$	38.1073	1.55963
$598$	0	0
$599$	−32.9034	−1.34440	−0.672198	−	0.740371i	$-0.734650\pi$
$599$	−32.9034	−1.34440	−0.672198	+	0.740371i	$0.734650\pi$
$600$	0	0
$601$	37.2315	1.51871	0.759353	−	0.650679i	$-0.225515\pi$
$601$	37.2315	1.51871	0.759353	+	0.650679i	$0.225515\pi$
$602$	0	0
$603$	0.224129	0.00912722
$604$	0	0
$605$	119.328	4.85136
$606$	0	0
$607$	28.2602	1.14705	0.573524	−	0.819189i	$-0.305576\pi$
$607$	28.2602	1.14705	0.573524	+	0.819189i	$0.305576\pi$
$608$	0	0
$609$	−28.9979	−1.17506
$610$	0	0
$611$	2.70287	0.109347
$612$	0	0
$613$	0.804668	0.0325002	0.0162501	−	0.999868i	$-0.494827\pi$
$613$	0.804668	0.0325002	0.0162501	+	0.999868i	$0.494827\pi$
$614$	0	0
$615$	−16.9314	−0.682739
$616$	0	0
$617$	−13.6905	−0.551158	−0.275579	−	0.961278i	$-0.588870\pi$
$617$	−13.6905	−0.551158	−0.275579	+	0.961278i	$0.588870\pi$
$618$	0	0
$619$	−20.7258	−0.833041	−0.416520	−	0.909126i	$-0.636751\pi$
$619$	−20.7258	−0.833041	−0.416520	+	0.909126i	$0.636751\pi$
$620$	0	0
$621$	−16.5979	−0.666053
$622$	0	0
$623$	38.2889	1.53401
$624$	0	0
$625$	82.2228	3.28891
$626$	0	0
$627$	−30.8652	−1.23264
$628$	0	0
$629$	4.95231	0.197462
$630$	0	0
$631$	17.6836	0.703972	0.351986	−	0.936005i	$-0.385506\pi$
$631$	17.6836	0.703972	0.351986	+	0.936005i	$0.385506\pi$
$632$	0	0
$633$	26.1746	1.04035
$634$	0	0
$635$	7.41580	0.294287
$636$	0	0
$637$	−3.63610	−0.144067
$638$	0	0
$639$	−1.55322	−0.0614445
$640$	0	0
$641$	43.9749	1.73690	0.868452	−	0.495774i	$-0.165115\pi$
$641$	43.9749	1.73690	0.868452	+	0.495774i	$0.165115\pi$
$642$	0	0
$643$	3.65325	0.144070	0.0720350	−	0.997402i	$-0.477051\pi$
$643$	3.65325	0.144070	0.0720350	+	0.997402i	$0.477051\pi$
$644$	0	0
$645$	−36.1928	−1.42509
$646$	0	0
$647$	−10.2990	−0.404896	−0.202448	−	0.979293i	$-0.564890\pi$
$647$	−10.2990	−0.404896	−0.202448	+	0.979293i	$0.564890\pi$
$648$	0	0
$649$	−11.8274	−0.464264
$650$	0	0
$651$	8.77905	0.344078
$652$	0	0
$653$	−16.0598	−0.628470	−0.314235	−	0.949345i	$-0.601748\pi$
$653$	−16.0598	−0.628470	−0.314235	+	0.949345i	$0.601748\pi$
$654$	0	0
$655$	21.6448	0.845733
$656$	0	0
$657$	1.01975	0.0397843
$658$	0	0
$659$	−3.55955	−0.138660	−0.0693302	−	0.997594i	$-0.522086\pi$
$659$	−3.55955	−0.138660	−0.0693302	+	0.997594i	$0.522086\pi$
$660$	0	0
$661$	39.3796	1.53169	0.765845	−	0.643025i	$-0.222321\pi$
$661$	39.3796	1.53169	0.765845	+	0.643025i	$0.222321\pi$
$662$	0	0
$663$	−19.7743	−0.767971
$664$	0	0
$665$	−28.3762	−1.10038
$666$	0	0
$667$	−22.5213	−0.872028
$668$	0	0
$669$	23.0206	0.890029
$670$	0	0
$671$	−48.9449	−1.88950
$672$	0	0
$673$	16.6727	0.642686	0.321343	−	0.946963i	$-0.395866\pi$
$673$	16.6727	0.642686	0.321343	+	0.946963i	$0.395866\pi$
$674$	0	0
$675$	−66.9905	−2.57846
$676$	0	0
$677$	2.55182	0.0980743	0.0490371	−	0.998797i	$-0.484385\pi$
$677$	2.55182	0.0980743	0.0490371	+	0.998797i	$0.484385\pi$
$678$	0	0
$679$	1.33237	0.0511318
$680$	0	0
$681$	14.9412	0.572549
$682$	0	0
$683$	33.2587	1.27261	0.636304	−	0.771439i	$-0.280463\pi$
$683$	33.2587	1.27261	0.636304	+	0.771439i	$0.280463\pi$
$684$	0	0
$685$	−6.76051	−0.258306
$686$	0	0
$687$	0.257941	0.00984106
$688$	0	0
$689$	22.0600	0.840419
$690$	0	0
$691$	−49.0025	−1.86414	−0.932071	−	0.362275i	$-0.882000\pi$
$691$	−49.0025	−1.86414	−0.932071	+	0.362275i	$0.882000\pi$
$692$	0	0
$693$	1.68324	0.0639412
$694$	0	0
$695$	−37.2146	−1.41163
$696$	0	0
$697$	9.33847	0.353720
$698$	0	0
$699$	48.3090	1.82721
$700$	0	0
$701$	18.8163	0.710680	0.355340	−	0.934737i	$-0.384365\pi$
$701$	18.8163	0.710680	0.355340	+	0.934737i	$0.384365\pi$
$702$	0	0
$703$	3.34524	0.126168
$704$	0	0
$705$	7.51761	0.283129
$706$	0	0
$707$	22.3630	0.841048
$708$	0	0
$709$	32.8159	1.23243	0.616213	−	0.787580i	$-0.288666\pi$
$709$	32.8159	1.23243	0.616213	+	0.787580i	$0.288666\pi$
$710$	0	0
$711$	−0.798388	−0.0299419
$712$	0	0
$713$	6.81826	0.255346
$714$	0	0
$715$	71.9228	2.68976
$716$	0	0
$717$	−24.7279	−0.923481
$718$	0	0
$719$	16.1088	0.600757	0.300378	−	0.953820i	$-0.402887\pi$
$719$	16.1088	0.600757	0.300378	+	0.953820i	$0.402887\pi$
$720$	0	0
$721$	−20.6996	−0.770892
$722$	0	0
$723$	37.7441	1.40372
$724$	0	0
$725$	−90.8975	−3.37585
$726$	0	0
$727$	18.9281	0.702004	0.351002	−	0.936375i	$-0.385841\pi$
$727$	18.9281	0.702004	0.351002	+	0.936375i	$0.385841\pi$
$728$	0	0
$729$	25.9044	0.959421
$730$	0	0
$731$	19.9620	0.738323
$732$	0	0
$733$	0.878130	0.0324345	0.0162172	−	0.999868i	$-0.494838\pi$
$733$	0.878130	0.0324345	0.0162172	+	0.999868i	$0.494838\pi$
$734$	0	0
$735$	−10.1132	−0.373032
$736$	0	0
$737$	12.3511	0.454957
$738$	0	0
$739$	−14.5836	−0.536466	−0.268233	−	0.963354i	$-0.586440\pi$
$739$	−14.5836	−0.536466	−0.268233	+	0.963354i	$0.586440\pi$
$740$	0	0
$741$	−13.3574	−0.490695
$742$	0	0
$743$	18.1869	0.667213	0.333606	−	0.942712i	$-0.391734\pi$
$743$	18.1869	0.667213	0.333606	+	0.942712i	$0.391734\pi$
$744$	0	0
$745$	−61.6669	−2.25930
$746$	0	0
$747$	−1.28094	−0.0468672
$748$	0	0
$749$	−4.61763	−0.168724
$750$	0	0
$751$	−42.6802	−1.55742	−0.778712	−	0.627382i	$-0.784127\pi$
$751$	−42.6802	−1.55742	−0.778712	+	0.627382i	$0.784127\pi$
$752$	0	0
$753$	−10.2019	−0.371777
$754$	0	0
$755$	−99.7795	−3.63135
$756$	0	0
$757$	38.9714	1.41644	0.708219	−	0.705992i	$-0.249499\pi$
$757$	38.9714	1.41644	0.708219	+	0.705992i	$0.249499\pi$
$758$	0	0
$759$	35.9114	1.30350
$760$	0	0
$761$	−10.9616	−0.397357	−0.198678	−	0.980065i	$-0.563665\pi$
$761$	−10.9616	−0.397357	−0.198678	+	0.980065i	$0.563665\pi$
$762$	0	0
$763$	−46.4870	−1.68294
$764$	0	0
$765$	−2.00215	−0.0723879
$766$	0	0
$767$	−5.11846	−0.184817
$768$	0	0
$769$	21.9772	0.792517	0.396258	−	0.918139i	$-0.370308\pi$
$769$	21.9772	0.792517	0.396258	+	0.918139i	$0.370308\pi$
$770$	0	0
$771$	52.7778	1.90075
$772$	0	0
$773$	28.0702	1.00962	0.504808	−	0.863232i	$-0.331563\pi$
$773$	28.0702	1.00962	0.504808	+	0.863232i	$0.331563\pi$
$774$	0	0
$775$	27.5190	0.988511
$776$	0	0
$777$	−5.01146	−0.179785
$778$	0	0
$779$	6.30805	0.226009
$780$	0	0
$781$	−85.5934	−3.06277
$782$	0	0
$783$	35.2012	1.25799
$784$	0	0
$785$	15.1677	0.541358
$786$	0	0
$787$	−33.9866	−1.21149	−0.605746	−	0.795658i	$-0.707125\pi$
$787$	−33.9866	−1.21149	−0.605746	+	0.795658i	$0.707125\pi$
$788$	0	0
$789$	−33.7097	−1.20010
$790$	0	0
$791$	4.62602	0.164483
$792$	0	0
$793$	−21.1816	−0.752181
$794$	0	0
$795$	61.3564	2.17609
$796$	0	0
$797$	−20.1964	−0.715392	−0.357696	−	0.933838i	$-0.616438\pi$
$797$	−20.1964	−0.715392	−0.357696	+	0.933838i	$0.616438\pi$
$798$	0	0
$799$	−4.14632	−0.146686
$800$	0	0
$801$	1.82488	0.0644790
$802$	0	0
$803$	56.1955	1.98309
$804$	0	0
$805$	33.0155	1.16364
$806$	0	0
$807$	−7.92463	−0.278960
$808$	0	0
$809$	−0.677947	−0.0238353	−0.0119177	−	0.999929i	$-0.503794\pi$
$809$	−0.677947	−0.0238353	−0.0119177	+	0.999929i	$0.503794\pi$
$810$	0	0
$811$	−46.0868	−1.61833	−0.809164	−	0.587583i	$-0.800080\pi$
$811$	−46.0868	−1.61833	−0.809164	+	0.587583i	$0.800080\pi$
$812$	0	0
$813$	30.4987	1.06964
$814$	0	0
$815$	78.5416	2.75119
$816$	0	0
$817$	13.4842	0.471752
$818$	0	0
$819$	0.728448	0.0254540
$820$	0	0
$821$	50.8290	1.77394	0.886971	−	0.461824i	$-0.152805\pi$
$821$	50.8290	1.77394	0.886971	+	0.461824i	$0.152805\pi$
$822$	0	0
$823$	42.3198	1.47518	0.737588	−	0.675251i	$-0.235965\pi$
$823$	42.3198	1.47518	0.737588	+	0.675251i	$0.235965\pi$
$824$	0	0
$825$	144.941	5.04619
$826$	0	0
$827$	−12.3461	−0.429317	−0.214659	−	0.976689i	$-0.568864\pi$
$827$	−12.3461	−0.429317	−0.214659	+	0.976689i	$0.568864\pi$
$828$	0	0
$829$	−13.9943	−0.486044	−0.243022	−	0.970021i	$-0.578139\pi$
$829$	−13.9943	−0.486044	−0.243022	+	0.970021i	$0.578139\pi$
$830$	0	0
$831$	27.7345	0.962099
$832$	0	0
$833$	5.57793	0.193264
$834$	0	0
$835$	−29.9049	−1.03490
$836$	0	0
$837$	−10.6571	−0.368362
$838$	0	0
$839$	30.0995	1.03915	0.519575	−	0.854425i	$-0.326090\pi$
$839$	30.0995	1.03915	0.519575	+	0.854425i	$0.326090\pi$
$840$	0	0
$841$	18.7636	0.647019
$842$	0	0
$843$	−27.1961	−0.936683
$844$	0	0
$845$	−24.2617	−0.834627
$846$	0	0
$847$	66.6010	2.28844
$848$	0	0
$849$	32.1151	1.10219
$850$	0	0
$851$	−3.89216	−0.133421
$852$	0	0
$853$	−49.0354	−1.67894	−0.839470	−	0.543407i	$-0.817134\pi$
$853$	−49.0354	−1.67894	−0.839470	+	0.543407i	$0.817134\pi$
$854$	0	0
$855$	−1.35243	−0.0462523
$856$	0	0
$857$	57.6866	1.97054	0.985268	−	0.171020i	$-0.0547062\pi$
$857$	57.6866	1.97054	0.985268	+	0.171020i	$0.0547062\pi$
$858$	0	0
$859$	24.2025	0.825778	0.412889	−	0.910781i	$-0.364520\pi$
$859$	24.2025	0.825778	0.412889	+	0.910781i	$0.364520\pi$
$860$	0	0
$861$	−9.45000	−0.322055
$862$	0	0
$863$	14.5927	0.496740	0.248370	−	0.968665i	$-0.420105\pi$
$863$	14.5927	0.496740	0.248370	+	0.968665i	$0.420105\pi$
$864$	0	0
$865$	31.8252	1.08209
$866$	0	0
$867$	0.338724	0.0115037
$868$	0	0
$869$	−43.9968	−1.49249
$870$	0	0
$871$	5.34510	0.181112
$872$	0	0
$873$	0.0635020	0.00214922
$874$	0	0
$875$	82.5954	2.79223
$876$	0	0
$877$	−1.14166	−0.0385510	−0.0192755	−	0.999814i	$-0.506136\pi$
$877$	−1.14166	−0.0385510	−0.0192755	+	0.999814i	$0.506136\pi$
$878$	0	0
$879$	−13.3211	−0.449309
$880$	0	0
$881$	16.0514	0.540785	0.270392	−	0.962750i	$-0.412847\pi$
$881$	16.0514	0.540785	0.270392	+	0.962750i	$0.412847\pi$
$882$	0	0
$883$	31.1101	1.04694	0.523469	−	0.852045i	$-0.324638\pi$
$883$	31.1101	1.04694	0.523469	+	0.852045i	$0.324638\pi$
$884$	0	0
$885$	−14.2362	−0.478543
$886$	0	0
$887$	−37.1430	−1.24714	−0.623571	−	0.781767i	$-0.714319\pi$
$887$	−37.1430	−1.24714	−0.623571	+	0.781767i	$0.714319\pi$
$888$	0	0
$889$	4.13902	0.138818
$890$	0	0
$891$	−58.2537	−1.95157
$892$	0	0
$893$	−2.80080	−0.0937253
$894$	0	0
$895$	−72.5259	−2.42427
$896$	0	0
$897$	15.5412	0.518904
$898$	0	0
$899$	−14.4603	−0.482278
$900$	0	0
$901$	−33.8410	−1.12741
$902$	0	0
$903$	−20.2005	−0.672229
$904$	0	0
$905$	15.6504	0.520237
$906$	0	0
$907$	15.2385	0.505986	0.252993	−	0.967468i	$-0.418585\pi$
$907$	15.2385	0.505986	0.252993	+	0.967468i	$0.418585\pi$
$908$	0	0
$909$	1.06584	0.0353517
$910$	0	0
$911$	−16.5106	−0.547019	−0.273509	−	0.961869i	$-0.588184\pi$
$911$	−16.5106	−0.547019	−0.273509	+	0.961869i	$0.588184\pi$
$912$	0	0
$913$	−70.5889	−2.33615
$914$	0	0
$915$	−58.9132	−1.94761
$916$	0	0
$917$	12.0807	0.398941
$918$	0	0
$919$	46.7659	1.54267	0.771333	−	0.636432i	$-0.219591\pi$
$919$	46.7659	1.54267	0.771333	+	0.636432i	$0.219591\pi$
$920$	0	0
$921$	−5.87104	−0.193457
$922$	0	0
$923$	−37.0418	−1.21924
$924$	0	0
$925$	−15.7090	−0.516509
$926$	0	0
$927$	−0.986558	−0.0324028
$928$	0	0
$929$	−43.0921	−1.41381	−0.706903	−	0.707310i	$-0.749908\pi$
$929$	−43.0921	−1.41381	−0.706903	+	0.707310i	$0.749908\pi$
$930$	0	0
$931$	3.76784	0.123486
$932$	0	0
$933$	−53.5836	−1.75425
$934$	0	0
$935$	−110.333	−3.60826
$936$	0	0
$937$	18.5102	0.604700	0.302350	−	0.953197i	$-0.402229\pi$
$937$	18.5102	0.604700	0.302350	+	0.953197i	$0.402229\pi$
$938$	0	0
$939$	52.3167	1.70729
$940$	0	0
$941$	−34.8720	−1.13679	−0.568397	−	0.822754i	$-0.692436\pi$
$941$	−34.8720	−1.13679	−0.568397	+	0.822754i	$0.692436\pi$
$942$	0	0
$943$	−7.33935	−0.239002
$944$	0	0
$945$	−51.6038	−1.67867
$946$	0	0
$947$	−21.3358	−0.693320	−0.346660	−	0.937991i	$-0.612684\pi$
$947$	−21.3358	−0.693320	−0.346660	+	0.937991i	$0.612684\pi$
$948$	0	0
$949$	24.3194	0.789441
$950$	0	0
$951$	34.1220	1.10648
$952$	0	0
$953$	−12.2393	−0.396471	−0.198235	−	0.980154i	$-0.563521\pi$
$953$	−12.2393	−0.396471	−0.198235	+	0.980154i	$0.563521\pi$
$954$	0	0
$955$	−70.4324	−2.27914
$956$	0	0
$957$	−76.1615	−2.46195
$958$	0	0
$959$	−3.77328	−0.121846
$960$	0	0
$961$	−26.6222	−0.858780
$962$	0	0
$963$	−0.220080	−0.00709197
$964$	0	0
$965$	−77.5632	−2.49685
$966$	0	0
$967$	2.50931	0.0806938	0.0403469	−	0.999186i	$-0.487154\pi$
$967$	2.50931	0.0806938	0.0403469	+	0.999186i	$0.487154\pi$
$968$	0	0
$969$	20.4908	0.658258
$970$	0	0
$971$	0.710803	0.0228108	0.0114054	−	0.999935i	$-0.496369\pi$
$971$	0.710803	0.0228108	0.0114054	+	0.999935i	$0.496369\pi$
$972$	0	0
$973$	−20.7708	−0.665880
$974$	0	0
$975$	62.7252	2.00881
$976$	0	0
$977$	4.75828	0.152231	0.0761154	−	0.997099i	$-0.475748\pi$
$977$	4.75828	0.152231	0.0761154	+	0.997099i	$0.475748\pi$
$978$	0	0
$979$	100.564	3.21403
$980$	0	0
$981$	−2.21561	−0.0707388
$982$	0	0
$983$	−38.3277	−1.22246	−0.611232	−	0.791451i	$-0.709326\pi$
$983$	−38.3277	−1.22246	−0.611232	+	0.791451i	$0.709326\pi$
$984$	0	0
$985$	−94.4612	−3.00979
$986$	0	0
$987$	4.19584	0.133555
$988$	0	0
$989$	−15.6887	−0.498872
$990$	0	0
$991$	−44.9133	−1.42672	−0.713359	−	0.700799i	$-0.752827\pi$
$991$	−44.9133	−1.42672	−0.713359	+	0.700799i	$0.752827\pi$
$992$	0	0
$993$	−26.9882	−0.856445
$994$	0	0
$995$	92.0157	2.91709
$996$	0	0
$997$	10.0843	0.319375	0.159687	−	0.987168i	$-0.448951\pi$
$997$	10.0843	0.319375	0.159687	+	0.987168i	$0.448951\pi$
$998$	0	0
$999$	6.08351	0.192474

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6016.2.a.p.1.10	yes	13
4.3	odd	2		6016.2.a.n.1.4	yes	13
8.3	odd	2		6016.2.a.o.1.10	yes	13
8.5	even	2		6016.2.a.m.1.4	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6016.2.a.m.1.4	✓	13	8.5	even	2
6016.2.a.n.1.4	yes	13	4.3	odd	2
6016.2.a.o.1.10	yes	13	8.3	odd	2
6016.2.a.p.1.10	yes	13	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:	\( N \)	\(=\)	\( 6016 = 2^{7} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.76446 q^{3} +4.26056 q^{5} +2.37797 q^{7} +0.113336 q^{9} +O(q^{10})\)	\(q+1.76446 q^{3} +4.26056 q^{5} +2.37797 q^{7} +0.113336 q^{9} +6.24560 q^{11} +2.70287 q^{13} +7.51761 q^{15} -4.14632 q^{17} -2.80080 q^{19} +4.19584 q^{21} +3.25870 q^{23} +13.1524 q^{25} -5.09342 q^{27} -6.91112 q^{29} +2.09232 q^{31} +11.0201 q^{33} +10.1315 q^{35} -1.19439 q^{37} +4.76912 q^{39} -2.25223 q^{41} -4.81440 q^{43} +0.482874 q^{45} +1.00000 q^{47} -1.34527 q^{49} -7.31604 q^{51} +8.16169 q^{53} +26.6097 q^{55} -4.94192 q^{57} -1.89371 q^{59} -7.83670 q^{61} +0.269509 q^{63} +11.5157 q^{65} +1.97756 q^{67} +5.74987 q^{69} -13.7046 q^{71} +8.99761 q^{73} +23.2069 q^{75} +14.8518 q^{77} -7.04445 q^{79} -9.32716 q^{81} -11.3022 q^{83} -17.6656 q^{85} -12.1944 q^{87} +16.1015 q^{89} +6.42734 q^{91} +3.69183 q^{93} -11.9330 q^{95} +0.560299 q^{97} +0.707850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 4 q^{3} + 6 q^{5} + 2 q^{7} + 21 q^{9}+O(q^{10}) \) 13 * q + 4 * q^3 + 6 * q^5 + 2 * q^7 + 21 * q^9	\( 13 q + 4 q^{3} + 6 q^{5} + 2 q^{7} + 21 q^{9} + 10 q^{11} + 4 q^{13} - 14 q^{15} + 10 q^{17} + 8 q^{19} + 10 q^{21} - 18 q^{23} + 23 q^{25} + 16 q^{27} + 14 q^{29} - 4 q^{31} + 14 q^{33} + 14 q^{35} + 16 q^{37} - 12 q^{39} + 10 q^{41} + 12 q^{43} + 10 q^{45} + 13 q^{47} + 9 q^{49} + 22 q^{51} + 26 q^{53} + 2 q^{55} + 20 q^{57} + 30 q^{59} + 18 q^{61} - 12 q^{63} - 4 q^{65} + 4 q^{67} + 2 q^{69} - 36 q^{71} + 10 q^{73} + 38 q^{75} + 42 q^{77} + 21 q^{81} + 12 q^{83} + 4 q^{85} - 6 q^{87} + 50 q^{89} - 4 q^{91} + 52 q^{93} - 8 q^{95} - 10 q^{97} + 34 q^{99}+O(q^{100}) \) 13 * q + 4 * q^3 + 6 * q^5 + 2 * q^7 + 21 * q^9 + 10 * q^11 + 4 * q^13 - 14 * q^15 + 10 * q^17 + 8 * q^19 + 10 * q^21 - 18 * q^23 + 23 * q^25 + 16 * q^27 + 14 * q^29 - 4 * q^31 + 14 * q^33 + 14 * q^35 + 16 * q^37 - 12 * q^39 + 10 * q^41 + 12 * q^43 + 10 * q^45 + 13 * q^47 + 9 * q^49 + 22 * q^51 + 26 * q^53 + 2 * q^55 + 20 * q^57 + 30 * q^59 + 18 * q^61 - 12 * q^63 - 4 * q^65 + 4 * q^67 + 2 * q^69 - 36 * q^71 + 10 * q^73 + 38 * q^75 + 42 * q^77 + 21 * q^81 + 12 * q^83 + 4 * q^85 - 6 * q^87 + 50 * q^89 - 4 * q^91 + 52 * q^93 - 8 * q^95 - 10 * q^97 + 34 * q^99

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