LMFDB - Embedded newform 6016.2.a.o.1.13

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			1.13
Root			$3.29402$ 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)

$f(q)$	$=$	$q+3.29402 q^{3} +3.40346 q^{5} -1.01019 q^{7} +7.85059 q^{9} +O(q^{10})$	$q+3.29402 q^{3} +3.40346 q^{5} -1.01019 q^{7} +7.85059 q^{9} +5.72520 q^{11} -2.46556 q^{13} +11.2111 q^{15} +5.23012 q^{17} -4.81048 q^{19} -3.32760 q^{21} +5.25423 q^{23} +6.58356 q^{25} +15.9780 q^{27} -5.28804 q^{29} -8.08188 q^{31} +18.8589 q^{33} -3.43816 q^{35} +4.26903 q^{37} -8.12160 q^{39} -12.5727 q^{41} +3.79762 q^{43} +26.7192 q^{45} -1.00000 q^{47} -5.97951 q^{49} +17.2281 q^{51} -4.27708 q^{53} +19.4855 q^{55} -15.8458 q^{57} -4.14200 q^{59} -5.15979 q^{61} -7.93062 q^{63} -8.39143 q^{65} -5.15333 q^{67} +17.3075 q^{69} +10.5001 q^{71} -5.30202 q^{73} +21.6864 q^{75} -5.78356 q^{77} -14.3486 q^{79} +29.0800 q^{81} +9.84381 q^{83} +17.8005 q^{85} -17.4189 q^{87} -7.70282 q^{89} +2.49069 q^{91} -26.6219 q^{93} -16.3723 q^{95} -10.2647 q^{97} +44.9462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.29402	1.90181	0.950903	−	0.309490i	$-0.100158\pi$
$3$	3.29402	1.90181	0.950903	+	0.309490i	$0.100158\pi$
$4$	0	0
$5$	3.40346	1.52207	0.761037	−	0.648708i	$-0.224690\pi$
$5$	3.40346	1.52207	0.761037	+	0.648708i	$0.224690\pi$
$6$	0	0
$7$	−1.01019	−0.381817	−0.190909	−	0.981608i	$-0.561143\pi$
$7$	−1.01019	−0.381817	−0.190909	+	0.981608i	$0.561143\pi$
$8$	0	0
$9$	7.85059	2.61686
$10$	0	0
$11$	5.72520	1.72621	0.863106	−	0.505023i	$-0.168516\pi$
$11$	5.72520	1.72621	0.863106	+	0.505023i	$0.168516\pi$
$12$	0	0
$13$	−2.46556	−0.683822	−0.341911	−	0.939732i	$-0.611074\pi$
$13$	−2.46556	−0.683822	−0.341911	+	0.939732i	$0.611074\pi$
$14$	0	0
$15$	11.2111	2.89469
$16$	0	0
$17$	5.23012	1.26849	0.634246	−	0.773132i	$-0.281311\pi$
$17$	5.23012	1.26849	0.634246	+	0.773132i	$0.281311\pi$
$18$	0	0
$19$	−4.81048	−1.10360	−0.551799	−	0.833977i	$-0.686059\pi$
$19$	−4.81048	−1.10360	−0.551799	+	0.833977i	$0.686059\pi$
$20$	0	0
$21$	−3.32760	−0.726142
$22$	0	0
$23$	5.25423	1.09558	0.547791	−	0.836615i	$-0.315469\pi$
$23$	5.25423	1.09558	0.547791	+	0.836615i	$0.315469\pi$
$24$	0	0
$25$	6.58356	1.31671
$26$	0	0
$27$	15.9780	3.07496
$28$	0	0
$29$	−5.28804	−0.981964	−0.490982	−	0.871170i	$-0.663362\pi$
$29$	−5.28804	−0.981964	−0.490982	+	0.871170i	$0.663362\pi$
$30$	0	0
$31$	−8.08188	−1.45155	−0.725774	−	0.687933i	$-0.758518\pi$
$31$	−8.08188	−1.45155	−0.725774	+	0.687933i	$0.758518\pi$
$32$	0	0
$33$	18.8589	3.28292
$34$	0	0
$35$	−3.43816	−0.581154
$36$	0	0
$37$	4.26903	0.701824	0.350912	−	0.936408i	$-0.385872\pi$
$37$	4.26903	0.701824	0.350912	+	0.936408i	$0.385872\pi$
$38$	0	0
$39$	−8.12160	−1.30050
$40$	0	0
$41$	−12.5727	−1.96352	−0.981760	−	0.190123i	$-0.939111\pi$
$41$	−12.5727	−1.96352	−0.981760	+	0.190123i	$0.939111\pi$
$42$	0	0
$43$	3.79762	0.579131	0.289566	−	0.957158i	$-0.406489\pi$
$43$	3.79762	0.579131	0.289566	+	0.957158i	$0.406489\pi$
$44$	0	0
$45$	26.7192	3.98306
$46$	0	0
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−5.97951	−0.854216
$50$	0	0
$51$	17.2281	2.41242
$52$	0	0
$53$	−4.27708	−0.587502	−0.293751	−	0.955882i	$-0.594904\pi$
$53$	−4.27708	−0.587502	−0.293751	+	0.955882i	$0.594904\pi$
$54$	0	0
$55$	19.4855	2.62742
$56$	0	0
$57$	−15.8458	−2.09883
$58$	0	0
$59$	−4.14200	−0.539243	−0.269621	−	0.962966i	$-0.586899\pi$
$59$	−4.14200	−0.539243	−0.269621	+	0.962966i	$0.586899\pi$
$60$	0	0
$61$	−5.15979	−0.660643	−0.330322	−	0.943868i	$-0.607157\pi$
$61$	−5.15979	−0.660643	−0.330322	+	0.943868i	$0.607157\pi$
$62$	0	0
$63$	−7.93062	−0.999164
$64$	0	0
$65$	−8.39143	−1.04083
$66$	0	0
$67$	−5.15333	−0.629579	−0.314789	−	0.949162i	$-0.601934\pi$
$67$	−5.15333	−0.629579	−0.314789	+	0.949162i	$0.601934\pi$
$68$	0	0
$69$	17.3075	2.08358
$70$	0	0
$71$	10.5001	1.24613	0.623065	−	0.782170i	$-0.285887\pi$
$71$	10.5001	1.24613	0.623065	+	0.782170i	$0.285887\pi$
$72$	0	0
$73$	−5.30202	−0.620554	−0.310277	−	0.950646i	$-0.600422\pi$
$73$	−5.30202	−0.620554	−0.310277	+	0.950646i	$0.600422\pi$
$74$	0	0
$75$	21.6864	2.50413
$76$	0	0
$77$	−5.78356	−0.659097
$78$	0	0
$79$	−14.3486	−1.61435	−0.807173	−	0.590315i	$-0.799003\pi$
$79$	−14.3486	−1.61435	−0.807173	+	0.590315i	$0.799003\pi$
$80$	0	0
$81$	29.0800	3.23111
$82$	0	0
$83$	9.84381	1.08050	0.540250	−	0.841505i	$-0.318330\pi$
$83$	9.84381	1.08050	0.540250	+	0.841505i	$0.318330\pi$
$84$	0	0
$85$	17.8005	1.93074
$86$	0	0
$87$	−17.4189	−1.86750
$88$	0	0
$89$	−7.70282	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$89$	−7.70282	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$90$	0	0
$91$	2.49069	0.261095
$92$	0	0
$93$	−26.6219	−2.76056
$94$	0	0
$95$	−16.3723	−1.67976
$96$	0	0
$97$	−10.2647	−1.04222	−0.521110	−	0.853490i	$-0.674482\pi$
$97$	−10.2647	−1.04222	−0.521110	+	0.853490i	$0.674482\pi$
$98$	0	0
$99$	44.9462	4.51726
$100$	0	0
$101$	−19.7066	−1.96088	−0.980438	−	0.196828i	$-0.936936\pi$
$101$	−19.7066	−1.96088	−0.980438	+	0.196828i	$0.936936\pi$
$102$	0	0
$103$	−6.98446	−0.688199	−0.344100	−	0.938933i	$-0.611816\pi$
$103$	−6.98446	−0.688199	−0.344100	+	0.938933i	$0.611816\pi$
$104$	0	0
$105$	−11.3254	−1.10524
$106$	0	0
$107$	6.50926	0.629273	0.314637	−	0.949212i	$-0.398117\pi$
$107$	6.50926	0.629273	0.314637	+	0.949212i	$0.398117\pi$
$108$	0	0
$109$	16.5829	1.58835	0.794177	−	0.607686i	$-0.207902\pi$
$109$	16.5829	1.58835	0.794177	+	0.607686i	$0.207902\pi$
$110$	0	0
$111$	14.0623	1.33473
$112$	0	0
$113$	5.91614	0.556543	0.278272	−	0.960502i	$-0.410238\pi$
$113$	5.91614	0.556543	0.278272	+	0.960502i	$0.410238\pi$
$114$	0	0
$115$	17.8826	1.66756
$116$	0	0
$117$	−19.3561	−1.78947
$118$	0	0
$119$	−5.28344	−0.484332
$120$	0	0
$121$	21.7779	1.97981
$122$	0	0
$123$	−41.4147	−3.73423
$124$	0	0
$125$	5.38958	0.482058
$126$	0	0
$127$	−1.84506	−0.163722	−0.0818612	−	0.996644i	$-0.526086\pi$
$127$	−1.84506	−0.163722	−0.0818612	+	0.996644i	$0.526086\pi$
$128$	0	0
$129$	12.5094	1.10140
$130$	0	0
$131$	−10.5001	−0.917396	−0.458698	−	0.888592i	$-0.651684\pi$
$131$	−10.5001	−0.917396	−0.458698	+	0.888592i	$0.651684\pi$
$132$	0	0
$133$	4.85951	0.421373
$134$	0	0
$135$	54.3804	4.68032
$136$	0	0
$137$	11.4238	0.976004	0.488002	−	0.872843i	$-0.337726\pi$
$137$	11.4238	0.976004	0.488002	+	0.872843i	$0.337726\pi$
$138$	0	0
$139$	−7.91429	−0.671281	−0.335641	−	0.941990i	$-0.608953\pi$
$139$	−7.91429	−0.671281	−0.335641	+	0.941990i	$0.608953\pi$
$140$	0	0
$141$	−3.29402	−0.277407
$142$	0	0
$143$	−14.1158	−1.18042
$144$	0	0
$145$	−17.9976	−1.49462
$146$	0	0
$147$	−19.6966	−1.62455
$148$	0	0
$149$	−18.0805	−1.48121	−0.740605	−	0.671940i	$-0.765461\pi$
$149$	−18.0805	−1.48121	−0.740605	+	0.671940i	$0.765461\pi$
$150$	0	0
$151$	0.949002	0.0772287	0.0386144	−	0.999254i	$-0.487706\pi$
$151$	0.949002	0.0772287	0.0386144	+	0.999254i	$0.487706\pi$
$152$	0	0
$153$	41.0596	3.31947
$154$	0	0
$155$	−27.5064	−2.20936
$156$	0	0
$157$	10.0579	0.802710	0.401355	−	0.915923i	$-0.368539\pi$
$157$	10.0579	0.802710	0.401355	+	0.915923i	$0.368539\pi$
$158$	0	0
$159$	−14.0888	−1.11731
$160$	0	0
$161$	−5.30779	−0.418312
$162$	0	0
$163$	7.61522	0.596470	0.298235	−	0.954492i	$-0.403602\pi$
$163$	7.61522	0.596470	0.298235	+	0.954492i	$0.403602\pi$
$164$	0	0
$165$	64.1857	4.99685
$166$	0	0
$167$	−7.71566	−0.597056	−0.298528	−	0.954401i	$-0.596496\pi$
$167$	−7.71566	−0.597056	−0.298528	+	0.954401i	$0.596496\pi$
$168$	0	0
$169$	−6.92103	−0.532387
$170$	0	0
$171$	−37.7651	−2.88797
$172$	0	0
$173$	9.15546	0.696077	0.348038	−	0.937480i	$-0.386848\pi$
$173$	9.15546	0.696077	0.348038	+	0.937480i	$0.386848\pi$
$174$	0	0
$175$	−6.65067	−0.502743
$176$	0	0
$177$	−13.6439	−1.02554
$178$	0	0
$179$	12.9473	0.967727	0.483863	−	0.875144i	$-0.339233\pi$
$179$	12.9473	0.967727	0.483863	+	0.875144i	$0.339233\pi$
$180$	0	0
$181$	0.263246	0.0195669	0.00978347	−	0.999952i	$-0.496886\pi$
$181$	0.263246	0.0195669	0.00978347	+	0.999952i	$0.496886\pi$
$182$	0	0
$183$	−16.9965	−1.25641
$184$	0	0
$185$	14.5295	1.06823
$186$	0	0
$187$	29.9435	2.18968
$188$	0	0
$189$	−16.1408	−1.17407
$190$	0	0
$191$	−7.74895	−0.560695	−0.280347	−	0.959899i	$-0.590450\pi$
$191$	−7.74895	−0.560695	−0.280347	+	0.959899i	$0.590450\pi$
$192$	0	0
$193$	20.2678	1.45891	0.729455	−	0.684028i	$-0.239774\pi$
$193$	20.2678	1.45891	0.729455	+	0.684028i	$0.239774\pi$
$194$	0	0
$195$	−27.6416	−1.97945
$196$	0	0
$197$	1.16480	0.0829889	0.0414944	−	0.999139i	$-0.486788\pi$
$197$	1.16480	0.0829889	0.0414944	+	0.999139i	$0.486788\pi$
$198$	0	0
$199$	21.5395	1.52689	0.763447	−	0.645871i	$-0.223506\pi$
$199$	21.5395	1.52689	0.763447	+	0.645871i	$0.223506\pi$
$200$	0	0
$201$	−16.9752	−1.19734
$202$	0	0
$203$	5.34194	0.374931
$204$	0	0
$205$	−42.7906	−2.98863
$206$	0	0
$207$	41.2488	2.86699
$208$	0	0
$209$	−27.5409	−1.90505
$210$	0	0
$211$	11.1401	0.766916	0.383458	−	0.923558i	$-0.374733\pi$
$211$	11.1401	0.766916	0.383458	+	0.923558i	$0.374733\pi$
$212$	0	0
$213$	34.5875	2.36990
$214$	0	0
$215$	12.9251	0.881481
$216$	0	0
$217$	8.16426	0.554226
$218$	0	0
$219$	−17.4650	−1.18017
$220$	0	0
$221$	−12.8952	−0.867423
$222$	0	0
$223$	2.86516	0.191865	0.0959327	−	0.995388i	$-0.469417\pi$
$223$	2.86516	0.191865	0.0959327	+	0.995388i	$0.469417\pi$
$224$	0	0
$225$	51.6848	3.44565
$226$	0	0
$227$	−7.92238	−0.525827	−0.262913	−	0.964819i	$-0.584683\pi$
$227$	−7.92238	−0.525827	−0.262913	+	0.964819i	$0.584683\pi$
$228$	0	0
$229$	10.1286	0.669319	0.334659	−	0.942339i	$-0.391379\pi$
$229$	10.1286	0.669319	0.334659	+	0.942339i	$0.391379\pi$
$230$	0	0
$231$	−19.0512	−1.25347
$232$	0	0
$233$	−5.45696	−0.357497	−0.178749	−	0.983895i	$-0.557205\pi$
$233$	−5.45696	−0.357497	−0.178749	+	0.983895i	$0.557205\pi$
$234$	0	0
$235$	−3.40346	−0.222017
$236$	0	0
$237$	−47.2647	−3.07017
$238$	0	0
$239$	−2.47493	−0.160090	−0.0800450	−	0.996791i	$-0.525506\pi$
$239$	−2.47493	−0.160090	−0.0800450	+	0.996791i	$0.525506\pi$
$240$	0	0
$241$	−17.7997	−1.14658	−0.573291	−	0.819352i	$-0.694333\pi$
$241$	−17.7997	−1.14658	−0.573291	+	0.819352i	$0.694333\pi$
$242$	0	0
$243$	47.8563	3.06998
$244$	0	0
$245$	−20.3510	−1.30018
$246$	0	0
$247$	11.8605	0.754666
$248$	0	0
$249$	32.4258	2.05490
$250$	0	0
$251$	16.3045	1.02913	0.514566	−	0.857451i	$-0.327953\pi$
$251$	16.3045	1.02913	0.514566	+	0.857451i	$0.327953\pi$
$252$	0	0
$253$	30.0815	1.89121
$254$	0	0
$255$	58.6354	3.67189
$256$	0	0
$257$	0.476521	0.0297246	0.0148623	−	0.999890i	$-0.495269\pi$
$257$	0.476521	0.0297246	0.0148623	+	0.999890i	$0.495269\pi$
$258$	0	0
$259$	−4.31255	−0.267969
$260$	0	0
$261$	−41.5142	−2.56967
$262$	0	0
$263$	0.982319	0.0605724	0.0302862	−	0.999541i	$-0.490358\pi$
$263$	0.982319	0.0605724	0.0302862	+	0.999541i	$0.490358\pi$
$264$	0	0
$265$	−14.5569	−0.894221
$266$	0	0
$267$	−25.3733	−1.55282
$268$	0	0
$269$	−10.2190	−0.623064	−0.311532	−	0.950236i	$-0.600842\pi$
$269$	−10.2190	−0.623064	−0.311532	+	0.950236i	$0.600842\pi$
$270$	0	0
$271$	5.62542	0.341720	0.170860	−	0.985295i	$-0.445345\pi$
$271$	5.62542	0.341720	0.170860	+	0.985295i	$0.445345\pi$
$272$	0	0
$273$	8.20439	0.496552
$274$	0	0
$275$	37.6922	2.27292
$276$	0	0
$277$	12.2346	0.735107	0.367553	−	0.930002i	$-0.380195\pi$
$277$	12.2346	0.735107	0.367553	+	0.930002i	$0.380195\pi$
$278$	0	0
$279$	−63.4475	−3.79850
$280$	0	0
$281$	−14.7699	−0.881096	−0.440548	−	0.897729i	$-0.645216\pi$
$281$	−14.7699	−0.881096	−0.440548	+	0.897729i	$0.645216\pi$
$282$	0	0
$283$	19.4910	1.15862	0.579311	−	0.815107i	$-0.303322\pi$
$283$	19.4910	1.15862	0.579311	+	0.815107i	$0.303322\pi$
$284$	0	0
$285$	−53.9307	−3.19458
$286$	0	0
$287$	12.7008	0.749706
$288$	0	0
$289$	10.3542	0.609070
$290$	0	0
$291$	−33.8121	−1.98210
$292$	0	0
$293$	11.5024	0.671975	0.335988	−	0.941866i	$-0.390930\pi$
$293$	11.5024	0.671975	0.335988	+	0.941866i	$0.390930\pi$
$294$	0	0
$295$	−14.0972	−0.820768
$296$	0	0
$297$	91.4770	5.30803
$298$	0	0
$299$	−12.9546	−0.749184
$300$	0	0
$301$	−3.83633	−0.221122
$302$	0	0
$303$	−64.9139	−3.72920
$304$	0	0
$305$	−17.5611	−1.00555
$306$	0	0
$307$	27.1455	1.54927	0.774637	−	0.632407i	$-0.217933\pi$
$307$	27.1455	1.54927	0.774637	+	0.632407i	$0.217933\pi$
$308$	0	0
$309$	−23.0070	−1.30882
$310$	0	0
$311$	29.2076	1.65621	0.828105	−	0.560573i	$-0.189419\pi$
$311$	29.2076	1.65621	0.828105	+	0.560573i	$0.189419\pi$
$312$	0	0
$313$	17.3355	0.979862	0.489931	−	0.871761i	$-0.337022\pi$
$313$	17.3355	0.979862	0.489931	+	0.871761i	$0.337022\pi$
$314$	0	0
$315$	−26.9916	−1.52080
$316$	0	0
$317$	−8.16893	−0.458813	−0.229406	−	0.973331i	$-0.573678\pi$
$317$	−8.16893	−0.458813	−0.229406	+	0.973331i	$0.573678\pi$
$318$	0	0
$319$	−30.2750	−1.69508
$320$	0	0
$321$	21.4416	1.19676
$322$	0	0
$323$	−25.1594	−1.39991
$324$	0	0
$325$	−16.2321	−0.900397
$326$	0	0
$327$	54.6245	3.02074
$328$	0	0
$329$	1.01019	0.0556938
$330$	0	0
$331$	8.54399	0.469620	0.234810	−	0.972041i	$-0.424553\pi$
$331$	8.54399	0.469620	0.234810	+	0.972041i	$0.424553\pi$
$332$	0	0
$333$	33.5144	1.83658
$334$	0	0
$335$	−17.5392	−0.958266
$336$	0	0
$337$	14.5188	0.790888	0.395444	−	0.918490i	$-0.370591\pi$
$337$	14.5188	0.790888	0.395444	+	0.918490i	$0.370591\pi$
$338$	0	0
$339$	19.4879	1.05844
$340$	0	0
$341$	−46.2703	−2.50568
$342$	0	0
$343$	13.1118	0.707972
$344$	0	0
$345$	58.9056	3.17137
$346$	0	0
$347$	9.43033	0.506247	0.253123	−	0.967434i	$-0.418542\pi$
$347$	9.43033	0.506247	0.253123	+	0.967434i	$0.418542\pi$
$348$	0	0
$349$	7.36249	0.394105	0.197052	−	0.980393i	$-0.436863\pi$
$349$	7.36249	0.394105	0.197052	+	0.980393i	$0.436863\pi$
$350$	0	0
$351$	−39.3946	−2.10273
$352$	0	0
$353$	−13.0551	−0.694853	−0.347426	−	0.937707i	$-0.612944\pi$
$353$	−13.0551	−0.694853	−0.347426	+	0.937707i	$0.612944\pi$
$354$	0	0
$355$	35.7366	1.89670
$356$	0	0
$357$	−17.4038	−0.921105
$358$	0	0
$359$	27.7361	1.46386	0.731929	−	0.681381i	$-0.238620\pi$
$359$	27.7361	1.46386	0.731929	+	0.681381i	$0.238620\pi$
$360$	0	0
$361$	4.14068	0.217931
$362$	0	0
$363$	71.7368	3.76521
$364$	0	0
$365$	−18.0452	−0.944530
$366$	0	0
$367$	36.6642	1.91386	0.956929	−	0.290323i	$-0.0937627\pi$
$367$	36.6642	1.91386	0.956929	+	0.290323i	$0.0937627\pi$
$368$	0	0
$369$	−98.7029	−5.13827
$370$	0	0
$371$	4.32067	0.224318
$372$	0	0
$373$	−9.01404	−0.466729	−0.233365	−	0.972389i	$-0.574974\pi$
$373$	−9.01404	−0.466729	−0.233365	+	0.972389i	$0.574974\pi$
$374$	0	0
$375$	17.7534	0.916781
$376$	0	0
$377$	13.0380	0.671489
$378$	0	0
$379$	−11.2137	−0.576010	−0.288005	−	0.957629i	$-0.592992\pi$
$379$	−11.2137	−0.576010	−0.288005	+	0.957629i	$0.592992\pi$
$380$	0	0
$381$	−6.07766	−0.311368
$382$	0	0
$383$	−12.3247	−0.629760	−0.314880	−	0.949131i	$-0.601964\pi$
$383$	−12.3247	−0.629760	−0.314880	+	0.949131i	$0.601964\pi$
$384$	0	0
$385$	−19.6841	−1.00320
$386$	0	0
$387$	29.8135	1.51551
$388$	0	0
$389$	−11.6693	−0.591658	−0.295829	−	0.955241i	$-0.595596\pi$
$389$	−11.6693	−0.591658	−0.295829	+	0.955241i	$0.595596\pi$
$390$	0	0
$391$	27.4803	1.38974
$392$	0	0
$393$	−34.5875	−1.74471
$394$	0	0
$395$	−48.8350	−2.45715
$396$	0	0
$397$	−5.65277	−0.283705	−0.141852	−	0.989888i	$-0.545306\pi$
$397$	−5.65277	−0.283705	−0.141852	+	0.989888i	$0.545306\pi$
$398$	0	0
$399$	16.0073	0.801370
$400$	0	0
$401$	39.7695	1.98599	0.992997	−	0.118139i	$-0.0376929\pi$
$401$	39.7695	1.98599	0.992997	+	0.118139i	$0.0376929\pi$
$402$	0	0
$403$	19.9263	0.992601
$404$	0	0
$405$	98.9727	4.91799
$406$	0	0
$407$	24.4410	1.21150
$408$	0	0
$409$	−14.0369	−0.694078	−0.347039	−	0.937851i	$-0.612813\pi$
$409$	−14.0369	−0.694078	−0.347039	+	0.937851i	$0.612813\pi$
$410$	0	0
$411$	37.6304	1.85617
$412$	0	0
$413$	4.18423	0.205892
$414$	0	0
$415$	33.5031	1.64460
$416$	0	0
$417$	−26.0699	−1.27665
$418$	0	0
$419$	−38.8002	−1.89551	−0.947756	−	0.318996i	$-0.896654\pi$
$419$	−38.8002	−1.89551	−0.947756	+	0.318996i	$0.896654\pi$
$420$	0	0
$421$	−8.76782	−0.427317	−0.213659	−	0.976908i	$-0.568538\pi$
$421$	−8.76782	−0.427317	−0.213659	+	0.976908i	$0.568538\pi$
$422$	0	0
$423$	−7.85059	−0.381709
$424$	0	0
$425$	34.4328	1.67024
$426$	0	0
$427$	5.21239	0.252245
$428$	0	0
$429$	−46.4978	−2.24493
$430$	0	0
$431$	15.9651	0.769011	0.384505	−	0.923123i	$-0.374372\pi$
$431$	15.9651	0.769011	0.384505	+	0.923123i	$0.374372\pi$
$432$	0	0
$433$	−35.6916	−1.71523	−0.857615	−	0.514292i	$-0.828055\pi$
$433$	−35.6916	−1.71523	−0.857615	+	0.514292i	$0.828055\pi$
$434$	0	0
$435$	−59.2846	−2.84248
$436$	0	0
$437$	−25.2753	−1.20908
$438$	0	0
$439$	−27.1930	−1.29785	−0.648925	−	0.760852i	$-0.724781\pi$
$439$	−27.1930	−1.29785	−0.648925	+	0.760852i	$0.724781\pi$
$440$	0	0
$441$	−46.9427	−2.23537
$442$	0	0
$443$	−16.9119	−0.803507	−0.401753	−	0.915748i	$-0.631599\pi$
$443$	−16.9119	−0.803507	−0.401753	+	0.915748i	$0.631599\pi$
$444$	0	0
$445$	−26.2162	−1.24277
$446$	0	0
$447$	−59.5575	−2.81697
$448$	0	0
$449$	−5.54470	−0.261671	−0.130835	−	0.991404i	$-0.541766\pi$
$449$	−5.54470	−0.261671	−0.130835	+	0.991404i	$0.541766\pi$
$450$	0	0
$451$	−71.9810	−3.38945
$452$	0	0
$453$	3.12604	0.146874
$454$	0	0
$455$	8.47697	0.397406
$456$	0	0
$457$	−16.4027	−0.767286	−0.383643	−	0.923481i	$-0.625331\pi$
$457$	−16.4027	−0.767286	−0.383643	+	0.923481i	$0.625331\pi$
$458$	0	0
$459$	83.5667	3.90056
$460$	0	0
$461$	−8.06731	−0.375732	−0.187866	−	0.982195i	$-0.560157\pi$
$461$	−8.06731	−0.375732	−0.187866	+	0.982195i	$0.560157\pi$
$462$	0	0
$463$	35.3959	1.64499	0.822493	−	0.568776i	$-0.192583\pi$
$463$	35.3959	1.64499	0.822493	+	0.568776i	$0.192583\pi$
$464$	0	0
$465$	−90.6066	−4.20178
$466$	0	0
$467$	−0.348500	−0.0161267	−0.00806334	−	0.999967i	$-0.502567\pi$
$467$	−0.348500	−0.0161267	−0.00806334	+	0.999967i	$0.502567\pi$
$468$	0	0
$469$	5.20586	0.240384
$470$	0	0
$471$	33.1311	1.52660
$472$	0	0
$473$	21.7421	0.999703
$474$	0	0
$475$	−31.6700	−1.45312
$476$	0	0
$477$	−33.5776	−1.53741
$478$	0	0
$479$	−17.7205	−0.809669	−0.404835	−	0.914390i	$-0.632671\pi$
$479$	−17.7205	−0.809669	−0.404835	+	0.914390i	$0.632671\pi$
$480$	0	0
$481$	−10.5255	−0.479923
$482$	0	0
$483$	−17.4840	−0.795548
$484$	0	0
$485$	−34.9354	−1.58634
$486$	0	0
$487$	−1.44876	−0.0656495	−0.0328247	−	0.999461i	$-0.510450\pi$
$487$	−1.44876	−0.0656495	−0.0328247	+	0.999461i	$0.510450\pi$
$488$	0	0
$489$	25.0847	1.13437
$490$	0	0
$491$	2.15520	0.0972627	0.0486313	−	0.998817i	$-0.484514\pi$
$491$	2.15520	0.0972627	0.0486313	+	0.998817i	$0.484514\pi$
$492$	0	0
$493$	−27.6571	−1.24561
$494$	0	0
$495$	152.973	6.87561
$496$	0	0
$497$	−10.6071	−0.475794
$498$	0	0
$499$	−22.5468	−1.00933	−0.504666	−	0.863314i	$-0.668384\pi$
$499$	−22.5468	−1.00933	−0.504666	+	0.863314i	$0.668384\pi$
$500$	0	0
$501$	−25.4156	−1.13548
$502$	0	0
$503$	12.1442	0.541483	0.270742	−	0.962652i	$-0.412731\pi$
$503$	12.1442	0.541483	0.270742	+	0.962652i	$0.412731\pi$
$504$	0	0
$505$	−67.0705	−2.98460
$506$	0	0
$507$	−22.7980	−1.01250
$508$	0	0
$509$	−1.45073	−0.0643027	−0.0321513	−	0.999483i	$-0.510236\pi$
$509$	−1.45073	−0.0643027	−0.0321513	+	0.999483i	$0.510236\pi$
$510$	0	0
$511$	5.35606	0.236938
$512$	0	0
$513$	−76.8616	−3.39352
$514$	0	0
$515$	−23.7713	−1.04749
$516$	0	0
$517$	−5.72520	−0.251794
$518$	0	0
$519$	30.1583	1.32380
$520$	0	0
$521$	−14.2519	−0.624388	−0.312194	−	0.950018i	$-0.601064\pi$
$521$	−14.2519	−0.624388	−0.312194	+	0.950018i	$0.601064\pi$
$522$	0	0
$523$	−1.78280	−0.0779565	−0.0389782	−	0.999240i	$-0.512410\pi$
$523$	−1.78280	−0.0779565	−0.0389782	+	0.999240i	$0.512410\pi$
$524$	0	0
$525$	−21.9075	−0.956120
$526$	0	0
$527$	−42.2692	−1.84128
$528$	0	0
$529$	4.60691	0.200300
$530$	0	0
$531$	−32.5172	−1.41113
$532$	0	0
$533$	30.9986	1.34270
$534$	0	0
$535$	22.1540	0.957801
$536$	0	0
$537$	42.6487	1.84043
$538$	0	0
$539$	−34.2339	−1.47456
$540$	0	0
$541$	1.55369	0.0667982	0.0333991	−	0.999442i	$-0.489367\pi$
$541$	1.55369	0.0667982	0.0333991	+	0.999442i	$0.489367\pi$
$542$	0	0
$543$	0.867140	0.0372125
$544$	0	0
$545$	56.4393	2.41759
$546$	0	0
$547$	23.8059	1.01787	0.508933	−	0.860806i	$-0.330040\pi$
$547$	23.8059	1.01787	0.508933	+	0.860806i	$0.330040\pi$
$548$	0	0
$549$	−40.5074	−1.72881
$550$	0	0
$551$	25.4380	1.08369
$552$	0	0
$553$	14.4949	0.616385
$554$	0	0
$555$	47.8605	2.03156
$556$	0	0
$557$	−24.3308	−1.03093	−0.515465	−	0.856911i	$-0.672381\pi$
$557$	−24.3308	−1.03093	−0.515465	+	0.856911i	$0.672381\pi$
$558$	0	0
$559$	−9.36324	−0.396023
$560$	0	0
$561$	98.6345	4.16435
$562$	0	0
$563$	−6.88333	−0.290098	−0.145049	−	0.989424i	$-0.546334\pi$
$563$	−6.88333	−0.290098	−0.145049	+	0.989424i	$0.546334\pi$
$564$	0	0
$565$	20.1353	0.847101
$566$	0	0
$567$	−29.3764	−1.23369
$568$	0	0
$569$	6.32305	0.265076	0.132538	−	0.991178i	$-0.457687\pi$
$569$	6.32305	0.265076	0.132538	+	0.991178i	$0.457687\pi$
$570$	0	0
$571$	6.32641	0.264752	0.132376	−	0.991200i	$-0.457739\pi$
$571$	6.32641	0.264752	0.132376	+	0.991200i	$0.457739\pi$
$572$	0	0
$573$	−25.5252	−1.06633
$574$	0	0
$575$	34.5915	1.44257
$576$	0	0
$577$	24.5472	1.02191	0.510957	−	0.859606i	$-0.329291\pi$
$577$	24.5472	1.02191	0.510957	+	0.859606i	$0.329291\pi$
$578$	0	0
$579$	66.7627	2.77456
$580$	0	0
$581$	−9.94416	−0.412553
$582$	0	0
$583$	−24.4871	−1.01415
$584$	0	0
$585$	−65.8777	−2.72371
$586$	0	0
$587$	34.4747	1.42293	0.711463	−	0.702724i	$-0.248033\pi$
$587$	34.4747	1.42293	0.711463	+	0.702724i	$0.248033\pi$
$588$	0	0
$589$	38.8777	1.60193
$590$	0	0
$591$	3.83689	0.157829
$592$	0	0
$593$	−16.5677	−0.680354	−0.340177	−	0.940361i	$-0.610487\pi$
$593$	−16.5677	−0.680354	−0.340177	+	0.940361i	$0.610487\pi$
$594$	0	0
$595$	−17.9820	−0.737189
$596$	0	0
$597$	70.9515	2.90385
$598$	0	0
$599$	3.33391	0.136220	0.0681099	−	0.997678i	$-0.478303\pi$
$599$	3.33391	0.136220	0.0681099	+	0.997678i	$0.478303\pi$
$600$	0	0
$601$	−22.6444	−0.923684	−0.461842	−	0.886962i	$-0.652811\pi$
$601$	−22.6444	−0.923684	−0.461842	+	0.886962i	$0.652811\pi$
$602$	0	0
$603$	−40.4566	−1.64752
$604$	0	0
$605$	74.1202	3.01341
$606$	0	0
$607$	39.5991	1.60728	0.803639	−	0.595117i	$-0.202894\pi$
$607$	39.5991	1.60728	0.803639	+	0.595117i	$0.202894\pi$
$608$	0	0
$609$	17.5965	0.713045
$610$	0	0
$611$	2.46556	0.0997457
$612$	0	0
$613$	5.24650	0.211904	0.105952	−	0.994371i	$-0.466211\pi$
$613$	5.24650	0.211904	0.105952	+	0.994371i	$0.466211\pi$
$614$	0	0
$615$	−140.953	−5.68378
$616$	0	0
$617$	3.90461	0.157194	0.0785969	−	0.996906i	$-0.474956\pi$
$617$	3.90461	0.157194	0.0785969	+	0.996906i	$0.474956\pi$
$618$	0	0
$619$	−0.752490	−0.0302451	−0.0151226	−	0.999886i	$-0.504814\pi$
$619$	−0.752490	−0.0302451	−0.0151226	+	0.999886i	$0.504814\pi$
$620$	0	0
$621$	83.9518	3.36887
$622$	0	0
$623$	7.78134	0.311753
$624$	0	0
$625$	−14.5746	−0.582982
$626$	0	0
$627$	−90.7204	−3.62303
$628$	0	0
$629$	22.3276	0.890258
$630$	0	0
$631$	16.1111	0.641373	0.320686	−	0.947185i	$-0.396086\pi$
$631$	16.1111	0.641373	0.320686	+	0.947185i	$0.396086\pi$
$632$	0	0
$633$	36.6958	1.45853
$634$	0	0
$635$	−6.27958	−0.249198
$636$	0	0
$637$	14.7428	0.584132
$638$	0	0
$639$	82.4318	3.26095
$640$	0	0
$641$	3.57148	0.141065	0.0705325	−	0.997509i	$-0.477530\pi$
$641$	3.57148	0.141065	0.0705325	+	0.997509i	$0.477530\pi$
$642$	0	0
$643$	17.1521	0.676411	0.338206	−	0.941072i	$-0.390180\pi$
$643$	17.1521	0.676411	0.338206	+	0.941072i	$0.390180\pi$
$644$	0	0
$645$	42.5754	1.67641
$646$	0	0
$647$	16.7145	0.657114	0.328557	−	0.944484i	$-0.393438\pi$
$647$	16.7145	0.657114	0.328557	+	0.944484i	$0.393438\pi$
$648$	0	0
$649$	−23.7138	−0.930847
$650$	0	0
$651$	26.8933	1.05403
$652$	0	0
$653$	40.0870	1.56873	0.784364	−	0.620301i	$-0.212990\pi$
$653$	40.0870	1.56873	0.784364	+	0.620301i	$0.212990\pi$
$654$	0	0
$655$	−35.7366	−1.39635
$656$	0	0
$657$	−41.6240	−1.62391
$658$	0	0
$659$	18.5765	0.723638	0.361819	−	0.932248i	$-0.382156\pi$
$659$	18.5765	0.723638	0.361819	+	0.932248i	$0.382156\pi$
$660$	0	0
$661$	−25.8030	−1.00362	−0.501811	−	0.864977i	$-0.667333\pi$
$661$	−25.8030	−1.00362	−0.501811	+	0.864977i	$0.667333\pi$
$662$	0	0
$663$	−42.4770	−1.64967
$664$	0	0
$665$	16.5392	0.641361
$666$	0	0
$667$	−27.7845	−1.07582
$668$	0	0
$669$	9.43791	0.364891
$670$	0	0
$671$	−29.5408	−1.14041
$672$	0	0
$673$	−10.0161	−0.386092	−0.193046	−	0.981190i	$-0.561837\pi$
$673$	−10.0161	−0.386092	−0.193046	+	0.981190i	$0.561837\pi$
$674$	0	0
$675$	105.192	4.04883
$676$	0	0
$677$	48.0177	1.84547	0.922736	−	0.385433i	$-0.125948\pi$
$677$	48.0177	1.84547	0.922736	+	0.385433i	$0.125948\pi$
$678$	0	0
$679$	10.3693	0.397937
$680$	0	0
$681$	−26.0965	−1.00002
$682$	0	0
$683$	−20.3988	−0.780537	−0.390268	−	0.920701i	$-0.627618\pi$
$683$	−20.3988	−0.780537	−0.390268	+	0.920701i	$0.627618\pi$
$684$	0	0
$685$	38.8806	1.48555
$686$	0	0
$687$	33.3639	1.27291
$688$	0	0
$689$	10.5454	0.401747
$690$	0	0
$691$	−3.07747	−0.117072	−0.0585362	−	0.998285i	$-0.518643\pi$
$691$	−3.07747	−0.117072	−0.0585362	+	0.998285i	$0.518643\pi$
$692$	0	0
$693$	−45.4043	−1.72477
$694$	0	0
$695$	−26.9360	−1.02174
$696$	0	0
$697$	−65.7566	−2.49071
$698$	0	0
$699$	−17.9754	−0.679890
$700$	0	0
$701$	−11.7014	−0.441956	−0.220978	−	0.975279i	$-0.570925\pi$
$701$	−11.7014	−0.441956	−0.220978	+	0.975279i	$0.570925\pi$
$702$	0	0
$703$	−20.5361	−0.774533
$704$	0	0
$705$	−11.2111	−0.422234
$706$	0	0
$707$	19.9074	0.748696
$708$	0	0
$709$	−39.4191	−1.48042	−0.740209	−	0.672377i	$-0.765273\pi$
$709$	−39.4191	−1.48042	−0.740209	+	0.672377i	$0.765273\pi$
$710$	0	0
$711$	−112.645	−4.22452
$712$	0	0
$713$	−42.4640	−1.59029
$714$	0	0
$715$	−48.0426	−1.79669
$716$	0	0
$717$	−8.15248	−0.304460
$718$	0	0
$719$	41.1251	1.53371	0.766854	−	0.641822i	$-0.221821\pi$
$719$	41.1251	1.53371	0.766854	+	0.641822i	$0.221821\pi$
$720$	0	0
$721$	7.05566	0.262766
$722$	0	0
$723$	−58.6327	−2.18057
$724$	0	0
$725$	−34.8141	−1.29296
$726$	0	0
$727$	−29.8097	−1.10558	−0.552790	−	0.833321i	$-0.686437\pi$
$727$	−29.8097	−1.10558	−0.552790	+	0.833321i	$0.686437\pi$
$728$	0	0
$729$	70.3999	2.60740
$730$	0	0
$731$	19.8620	0.734623
$732$	0	0
$733$	2.39218	0.0883573	0.0441786	−	0.999024i	$-0.485933\pi$
$733$	2.39218	0.0883573	0.0441786	+	0.999024i	$0.485933\pi$
$734$	0	0
$735$	−67.0368	−2.47269
$736$	0	0
$737$	−29.5038	−1.08679
$738$	0	0
$739$	−29.4914	−1.08486	−0.542428	−	0.840102i	$-0.682495\pi$
$739$	−29.4914	−1.08486	−0.542428	+	0.840102i	$0.682495\pi$
$740$	0	0
$741$	39.0688	1.43523
$742$	0	0
$743$	18.1362	0.665353	0.332677	−	0.943041i	$-0.392048\pi$
$743$	18.1362	0.665353	0.332677	+	0.943041i	$0.392048\pi$
$744$	0	0
$745$	−61.5362	−2.25451
$746$	0	0
$747$	77.2798	2.82752
$748$	0	0
$749$	−6.57561	−0.240267
$750$	0	0
$751$	36.7238	1.34007	0.670035	−	0.742329i	$-0.266279\pi$
$751$	36.7238	1.34007	0.670035	+	0.742329i	$0.266279\pi$
$752$	0	0
$753$	53.7075	1.95721
$754$	0	0
$755$	3.22989	0.117548
$756$	0	0
$757$	16.3203	0.593173	0.296587	−	0.955006i	$-0.404152\pi$
$757$	16.3203	0.593173	0.296587	+	0.955006i	$0.404152\pi$
$758$	0	0
$759$	99.0891	3.59671
$760$	0	0
$761$	34.0552	1.23450	0.617250	−	0.786767i	$-0.288247\pi$
$761$	34.0552	1.23450	0.617250	+	0.786767i	$0.288247\pi$
$762$	0	0
$763$	−16.7519	−0.606461
$764$	0	0
$765$	139.745	5.05248
$766$	0	0
$767$	10.2123	0.368746
$768$	0	0
$769$	−1.76745	−0.0637360	−0.0318680	−	0.999492i	$-0.510146\pi$
$769$	−1.76745	−0.0637360	−0.0318680	+	0.999492i	$0.510146\pi$
$770$	0	0
$771$	1.56967	0.0565303
$772$	0	0
$773$	15.3189	0.550984	0.275492	−	0.961303i	$-0.411159\pi$
$773$	15.3189	0.550984	0.275492	+	0.961303i	$0.411159\pi$
$774$	0	0
$775$	−53.2075	−1.91127
$776$	0	0
$777$	−14.2056	−0.509624
$778$	0	0
$779$	60.4805	2.16694
$780$	0	0
$781$	60.1150	2.15108
$782$	0	0
$783$	−84.4920	−3.01950
$784$	0	0
$785$	34.2318	1.22178
$786$	0	0
$787$	−49.6113	−1.76845	−0.884226	−	0.467059i	$-0.845314\pi$
$787$	−49.6113	−1.76845	−0.884226	+	0.467059i	$0.845314\pi$
$788$	0	0
$789$	3.23578	0.115197
$790$	0	0
$791$	−5.97644	−0.212498
$792$	0	0
$793$	12.7218	0.451763
$794$	0	0
$795$	−47.9507	−1.70063
$796$	0	0
$797$	−28.4624	−1.00819	−0.504096	−	0.863648i	$-0.668174\pi$
$797$	−28.4624	−1.00819	−0.504096	+	0.863648i	$0.668174\pi$
$798$	0	0
$799$	−5.23012	−0.185028
$800$	0	0
$801$	−60.4717	−2.13666
$802$	0	0
$803$	−30.3551	−1.07121
$804$	0	0
$805$	−18.0649	−0.636702
$806$	0	0
$807$	−33.6617	−1.18495
$808$	0	0
$809$	55.0741	1.93630	0.968151	−	0.250366i	$-0.0805509\pi$
$809$	55.0741	1.93630	0.968151	+	0.250366i	$0.0805509\pi$
$810$	0	0
$811$	14.1223	0.495902	0.247951	−	0.968773i	$-0.420243\pi$
$811$	14.1223	0.495902	0.247951	+	0.968773i	$0.420243\pi$
$812$	0	0
$813$	18.5303	0.649885
$814$	0	0
$815$	25.9181	0.907872
$816$	0	0
$817$	−18.2684	−0.639129
$818$	0	0
$819$	19.5534	0.683250
$820$	0	0
$821$	42.8827	1.49662	0.748308	−	0.663352i	$-0.230867\pi$
$821$	42.8827	1.49662	0.748308	+	0.663352i	$0.230867\pi$
$822$	0	0
$823$	39.9210	1.39156	0.695779	−	0.718256i	$-0.255059\pi$
$823$	39.9210	1.39156	0.695779	+	0.718256i	$0.255059\pi$
$824$	0	0
$825$	124.159	4.32266
$826$	0	0
$827$	48.2874	1.67912	0.839558	−	0.543270i	$-0.182814\pi$
$827$	48.2874	1.67912	0.839558	+	0.543270i	$0.182814\pi$
$828$	0	0
$829$	7.12920	0.247607	0.123804	−	0.992307i	$-0.460491\pi$
$829$	7.12920	0.247607	0.123804	+	0.992307i	$0.460491\pi$
$830$	0	0
$831$	40.3011	1.39803
$832$	0	0
$833$	−31.2736	−1.08356
$834$	0	0
$835$	−26.2600	−0.908764
$836$	0	0
$837$	−129.132	−4.46345
$838$	0	0
$839$	−33.3109	−1.15002	−0.575010	−	0.818146i	$-0.695002\pi$
$839$	−33.3109	−1.15002	−0.575010	+	0.818146i	$0.695002\pi$
$840$	0	0
$841$	−1.03666	−0.0357471
$842$	0	0
$843$	−48.6523	−1.67567
$844$	0	0
$845$	−23.5555	−0.810333
$846$	0	0
$847$	−21.9999	−0.755924
$848$	0	0
$849$	64.2039	2.20347
$850$	0	0
$851$	22.4305	0.768906
$852$	0	0
$853$	9.10908	0.311889	0.155944	−	0.987766i	$-0.450158\pi$
$853$	9.10908	0.311889	0.155944	+	0.987766i	$0.450158\pi$
$854$	0	0
$855$	−128.532	−4.39570
$856$	0	0
$857$	−7.22096	−0.246663	−0.123332	−	0.992366i	$-0.539358\pi$
$857$	−7.22096	−0.246663	−0.123332	+	0.992366i	$0.539358\pi$
$858$	0	0
$859$	15.4990	0.528818	0.264409	−	0.964411i	$-0.414823\pi$
$859$	15.4990	0.528818	0.264409	+	0.964411i	$0.414823\pi$
$860$	0	0
$861$	41.8368	1.42580
$862$	0	0
$863$	−8.22895	−0.280117	−0.140058	−	0.990143i	$-0.544729\pi$
$863$	−8.22895	−0.280117	−0.140058	+	0.990143i	$0.544729\pi$
$864$	0	0
$865$	31.1603	1.05948
$866$	0	0
$867$	34.1069	1.15833
$868$	0	0
$869$	−82.1486	−2.78670
$870$	0	0
$871$	12.7058	0.430520
$872$	0	0
$873$	−80.5837	−2.72735
$874$	0	0
$875$	−5.44452	−0.184058
$876$	0	0
$877$	27.2276	0.919411	0.459706	−	0.888071i	$-0.347955\pi$
$877$	27.2276	0.919411	0.459706	+	0.888071i	$0.347955\pi$
$878$	0	0
$879$	37.8890	1.27797
$880$	0	0
$881$	35.2731	1.18838	0.594191	−	0.804324i	$-0.297472\pi$
$881$	35.2731	1.18838	0.594191	+	0.804324i	$0.297472\pi$
$882$	0	0
$883$	−26.3181	−0.885674	−0.442837	−	0.896602i	$-0.646028\pi$
$883$	−26.3181	−0.885674	−0.442837	+	0.896602i	$0.646028\pi$
$884$	0	0
$885$	−46.4364	−1.56094
$886$	0	0
$887$	−52.0204	−1.74667	−0.873337	−	0.487116i	$-0.838049\pi$
$887$	−52.0204	−1.74667	−0.873337	+	0.487116i	$0.838049\pi$
$888$	0	0
$889$	1.86386	0.0625120
$890$	0	0
$891$	166.489	5.57758
$892$	0	0
$893$	4.81048	0.160976
$894$	0	0
$895$	44.0657	1.47295
$896$	0	0
$897$	−42.6727	−1.42480
$898$	0	0
$899$	42.7373	1.42537
$900$	0	0
$901$	−22.3696	−0.745241
$902$	0	0
$903$	−12.6370	−0.420532
$904$	0	0
$905$	0.895949	0.0297824
$906$	0	0
$907$	−46.2955	−1.53722	−0.768608	−	0.639720i	$-0.779050\pi$
$907$	−46.2955	−1.53722	−0.768608	+	0.639720i	$0.779050\pi$
$908$	0	0
$909$	−154.708	−5.13135
$910$	0	0
$911$	34.5338	1.14416	0.572079	−	0.820199i	$-0.306137\pi$
$911$	34.5338	1.14416	0.572079	+	0.820199i	$0.306137\pi$
$912$	0	0
$913$	56.3578	1.86517
$914$	0	0
$915$	−57.8468	−1.91236
$916$	0	0
$917$	10.6071	0.350278
$918$	0	0
$919$	35.4876	1.17063	0.585315	−	0.810806i	$-0.300971\pi$
$919$	35.4876	1.17063	0.585315	+	0.810806i	$0.300971\pi$
$920$	0	0
$921$	89.4178	2.94642
$922$	0	0
$923$	−25.8885	−0.852131
$924$	0	0
$925$	28.1054	0.924100
$926$	0	0
$927$	−54.8321	−1.80092
$928$	0	0
$929$	−27.0523	−0.887557	−0.443779	−	0.896136i	$-0.646362\pi$
$929$	−27.0523	−0.887557	−0.443779	+	0.896136i	$0.646362\pi$
$930$	0	0
$931$	28.7643	0.942711
$932$	0	0
$933$	96.2105	3.14979
$934$	0	0
$935$	101.912	3.33286
$936$	0	0
$937$	−51.6534	−1.68744	−0.843722	−	0.536780i	$-0.819640\pi$
$937$	−51.6534	−1.68744	−0.843722	+	0.536780i	$0.819640\pi$
$938$	0	0
$939$	57.1036	1.86351
$940$	0	0
$941$	−45.5605	−1.48523	−0.742615	−	0.669718i	$-0.766415\pi$
$941$	−45.5605	−1.48523	−0.742615	+	0.669718i	$0.766415\pi$
$942$	0	0
$943$	−66.0596	−2.15120
$944$	0	0
$945$	−54.9347	−1.78703
$946$	0	0
$947$	16.6145	0.539898	0.269949	−	0.962875i	$-0.412993\pi$
$947$	16.6145	0.539898	0.269949	+	0.962875i	$0.412993\pi$
$948$	0	0
$949$	13.0724	0.424349
$950$	0	0
$951$	−26.9086	−0.872572
$952$	0	0
$953$	4.18754	0.135648	0.0678239	−	0.997697i	$-0.478394\pi$
$953$	4.18754	0.135648	0.0678239	+	0.997697i	$0.478394\pi$
$954$	0	0
$955$	−26.3733	−0.853419
$956$	0	0
$957$	−99.7267	−3.22371
$958$	0	0
$959$	−11.5403	−0.372655
$960$	0	0
$961$	34.3167	1.10699
$962$	0	0
$963$	51.1015	1.64672
$964$	0	0
$965$	68.9808	2.22057
$966$	0	0
$967$	39.4921	1.26998	0.634990	−	0.772520i	$-0.281004\pi$
$967$	39.4921	1.26998	0.634990	+	0.772520i	$0.281004\pi$
$968$	0	0
$969$	−82.8756	−2.66235
$970$	0	0
$971$	−10.5070	−0.337185	−0.168592	−	0.985686i	$-0.553922\pi$
$971$	−10.5070	−0.337185	−0.168592	+	0.985686i	$0.553922\pi$
$972$	0	0
$973$	7.99496	0.256307
$974$	0	0
$975$	−53.4690	−1.71238
$976$	0	0
$977$	−43.7558	−1.39987	−0.699937	−	0.714205i	$-0.746788\pi$
$977$	−43.7558	−1.39987	−0.699937	+	0.714205i	$0.746788\pi$
$978$	0	0
$979$	−44.1001	−1.40945
$980$	0	0
$981$	130.186	4.15651
$982$	0	0
$983$	−61.6919	−1.96767	−0.983833	−	0.179089i	$-0.942685\pi$
$983$	−61.6919	−1.96767	−0.983833	+	0.179089i	$0.942685\pi$
$984$	0	0
$985$	3.96437	0.126315
$986$	0	0
$987$	3.32760	0.105919
$988$	0	0
$989$	19.9536	0.634486
$990$	0	0
$991$	52.9225	1.68114	0.840569	−	0.541705i	$-0.182221\pi$
$991$	52.9225	1.68114	0.840569	+	0.541705i	$0.182221\pi$
$992$	0	0
$993$	28.1441	0.893127
$994$	0	0
$995$	73.3088	2.32405
$996$	0	0
$997$	58.6863	1.85861	0.929306	−	0.369310i	$-0.120406\pi$
$997$	58.6863	1.85861	0.929306	+	0.369310i	$0.120406\pi$
$998$	0	0
$999$	68.2104	2.15808

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6016.2.a.o.1.13	yes	13
4.3	odd	2		6016.2.a.m.1.1	✓	13
8.3	odd	2		6016.2.a.p.1.13	yes	13
8.5	even	2		6016.2.a.n.1.1	yes	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6016.2.a.m.1.1	✓	13	4.3	odd	2
6016.2.a.n.1.1	yes	13	8.5	even	2
6016.2.a.o.1.13	yes	13	1.1	even	1	trivial
6016.2.a.p.1.13	yes	13	8.3	odd	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:	\( N \)	\(=\)	\( 6016 = 2^{7} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6016.a (trivial)

\(f(q)\)	\(=\)	\(q+3.29402 q^{3} +3.40346 q^{5} -1.01019 q^{7} +7.85059 q^{9} +O(q^{10})\)	\(q+3.29402 q^{3} +3.40346 q^{5} -1.01019 q^{7} +7.85059 q^{9} +5.72520 q^{11} -2.46556 q^{13} +11.2111 q^{15} +5.23012 q^{17} -4.81048 q^{19} -3.32760 q^{21} +5.25423 q^{23} +6.58356 q^{25} +15.9780 q^{27} -5.28804 q^{29} -8.08188 q^{31} +18.8589 q^{33} -3.43816 q^{35} +4.26903 q^{37} -8.12160 q^{39} -12.5727 q^{41} +3.79762 q^{43} +26.7192 q^{45} -1.00000 q^{47} -5.97951 q^{49} +17.2281 q^{51} -4.27708 q^{53} +19.4855 q^{55} -15.8458 q^{57} -4.14200 q^{59} -5.15979 q^{61} -7.93062 q^{63} -8.39143 q^{65} -5.15333 q^{67} +17.3075 q^{69} +10.5001 q^{71} -5.30202 q^{73} +21.6864 q^{75} -5.78356 q^{77} -14.3486 q^{79} +29.0800 q^{81} +9.84381 q^{83} +17.8005 q^{85} -17.4189 q^{87} -7.70282 q^{89} +2.49069 q^{91} -26.6219 q^{93} -16.3723 q^{95} -10.2647 q^{97} +44.9462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 4 q^{3} - 6 q^{5} - 2 q^{7} + 21 q^{9}+O(q^{10}) \) 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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