LMFDB - Embedded newform 3856.2.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3856,2,Mod(1,3856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3856, 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("3856.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3856 = 2^{4} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3856.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:	$30.7903150194$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} + \cdots - 1 $ x^12 - 3x^11 - 14x^10 + 44x^9 + 65x^8 - 219x^7 - 123x^6 + 444x^5 + 105x^4 - 328x^3 - 45x^2 + 18*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 241)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.0822506$ of defining polynomial
Character	$\chi$	$=$	3856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.81824 q^{3} +4.31963 q^{5} -0.690569 q^{7} +0.306010 q^{9} +O(q^{10})$	$q-1.81824 q^{3} +4.31963 q^{5} -0.690569 q^{7} +0.306010 q^{9} +2.95833 q^{11} -1.93470 q^{13} -7.85414 q^{15} -2.07477 q^{17} -3.74689 q^{19} +1.25562 q^{21} -4.34832 q^{23} +13.6592 q^{25} +4.89833 q^{27} -8.10772 q^{29} +2.80197 q^{31} -5.37896 q^{33} -2.98300 q^{35} -9.72312 q^{37} +3.51777 q^{39} -4.09401 q^{41} -3.02779 q^{43} +1.32185 q^{45} -6.71240 q^{47} -6.52312 q^{49} +3.77244 q^{51} -0.0484656 q^{53} +12.7789 q^{55} +6.81275 q^{57} -4.50676 q^{59} -9.62232 q^{61} -0.211321 q^{63} -8.35721 q^{65} +0.964160 q^{67} +7.90631 q^{69} -7.76289 q^{71} +16.4250 q^{73} -24.8358 q^{75} -2.04293 q^{77} +6.83310 q^{79} -9.82439 q^{81} +9.15477 q^{83} -8.96225 q^{85} +14.7418 q^{87} -2.36597 q^{89} +1.33605 q^{91} -5.09467 q^{93} -16.1852 q^{95} -5.25335 q^{97} +0.905280 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9	$ 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9} - 22 q^{11} - 5 q^{13} - 13 q^{15} - 4 q^{17} + 6 q^{19} - 14 q^{21} - 32 q^{23} + 4 q^{25} + 5 q^{27} + 6 q^{29} - 8 q^{31} - 24 q^{33} - 15 q^{35} - 8 q^{37} - 31 q^{39} - q^{41} + 2 q^{43} - 15 q^{45} - 34 q^{47} - 9 q^{49} + 3 q^{51} + 5 q^{53} + 3 q^{55} - 22 q^{57} - 26 q^{59} - 26 q^{61} + 4 q^{63} - 25 q^{65} - 6 q^{67} - 2 q^{69} - 94 q^{71} - 22 q^{73} - 7 q^{77} - 9 q^{79} + 4 q^{81} + 8 q^{83} + 4 q^{85} - 4 q^{87} - 3 q^{89} + 20 q^{91} + 12 q^{93} - 33 q^{95} - 29 q^{97} - 36 q^{99}+O(q^{100}) $ 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9 - 22 * q^11 - 5 * q^13 - 13 * q^15 - 4 * q^17 + 6 * q^19 - 14 * q^21 - 32 * q^23 + 4 * q^25 + 5 * q^27 + 6 * q^29 - 8 * q^31 - 24 * q^33 - 15 * q^35 - 8 * q^37 - 31 * q^39 - q^41 + 2 * q^43 - 15 * q^45 - 34 * q^47 - 9 * q^49 + 3 * q^51 + 5 * q^53 + 3 * q^55 - 22 * q^57 - 26 * q^59 - 26 * q^61 + 4 * q^63 - 25 * q^65 - 6 * q^67 - 2 * q^69 - 94 * q^71 - 22 * q^73 - 7 * q^77 - 9 * q^79 + 4 * q^81 + 8 * q^83 + 4 * q^85 - 4 * q^87 - 3 * q^89 + 20 * q^91 + 12 * q^93 - 33 * q^95 - 29 * q^97 - 36 * 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
$3$	−1.81824	−1.04976	−0.524882	−	0.851175i	$-0.675890\pi$
$3$	−1.81824	−1.04976	−0.524882	+	0.851175i	$0.675890\pi$
$4$	0	0
$5$	4.31963	1.93180	0.965899	−	0.258920i	$-0.0833667\pi$
$5$	4.31963	1.93180	0.965899	+	0.258920i	$0.0833667\pi$
$6$	0	0
$7$	−0.690569	−0.261010	−0.130505	−	0.991448i	$-0.541660\pi$
$7$	−0.690569	−0.261010	−0.130505	+	0.991448i	$0.541660\pi$
$8$	0	0
$9$	0.306010	0.102003
$10$	0	0
$11$	2.95833	0.891970	0.445985	−	0.895040i	$-0.352853\pi$
$11$	2.95833	0.891970	0.445985	+	0.895040i	$0.352853\pi$
$12$	0	0
$13$	−1.93470	−0.536591	−0.268295	−	0.963337i	$-0.586460\pi$
$13$	−1.93470	−0.536591	−0.268295	+	0.963337i	$0.586460\pi$
$14$	0	0
$15$	−7.85414	−2.02793
$16$	0	0
$17$	−2.07477	−0.503206	−0.251603	−	0.967830i	$-0.580958\pi$
$17$	−2.07477	−0.503206	−0.251603	+	0.967830i	$0.580958\pi$
$18$	0	0
$19$	−3.74689	−0.859595	−0.429797	−	0.902925i	$-0.641415\pi$
$19$	−3.74689	−0.859595	−0.429797	+	0.902925i	$0.641415\pi$
$20$	0	0
$21$	1.25562	0.273999
$22$	0	0
$23$	−4.34832	−0.906688	−0.453344	−	0.891336i	$-0.649769\pi$
$23$	−4.34832	−0.906688	−0.453344	+	0.891336i	$0.649769\pi$
$24$	0	0
$25$	13.6592	2.73184
$26$	0	0
$27$	4.89833	0.942684
$28$	0	0
$29$	−8.10772	−1.50557	−0.752783	−	0.658269i	$-0.771289\pi$
$29$	−8.10772	−1.50557	−0.752783	+	0.658269i	$0.771289\pi$
$30$	0	0
$31$	2.80197	0.503249	0.251624	−	0.967825i	$-0.419035\pi$
$31$	2.80197	0.503249	0.251624	+	0.967825i	$0.419035\pi$
$32$	0	0
$33$	−5.37896	−0.936358
$34$	0	0
$35$	−2.98300	−0.504219
$36$	0	0
$37$	−9.72312	−1.59847	−0.799236	−	0.601018i	$-0.794762\pi$
$37$	−9.72312	−1.59847	−0.799236	+	0.601018i	$0.794762\pi$
$38$	0	0
$39$	3.51777	0.563293
$40$	0	0
$41$	−4.09401	−0.639378	−0.319689	−	0.947523i	$-0.603578\pi$
$41$	−4.09401	−0.639378	−0.319689	+	0.947523i	$0.603578\pi$
$42$	0	0
$43$	−3.02779	−0.461733	−0.230867	−	0.972985i	$-0.574156\pi$
$43$	−3.02779	−0.461733	−0.230867	+	0.972985i	$0.574156\pi$
$44$	0	0
$45$	1.32185	0.197050
$46$	0	0
$47$	−6.71240	−0.979104	−0.489552	−	0.871974i	$-0.662840\pi$
$47$	−6.71240	−0.979104	−0.489552	+	0.871974i	$0.662840\pi$
$48$	0	0
$49$	−6.52312	−0.931874
$50$	0	0
$51$	3.77244	0.528248
$52$	0	0
$53$	−0.0484656	−0.00665726	−0.00332863	−	0.999994i	$-0.501060\pi$
$53$	−0.0484656	−0.00665726	−0.00332863	+	0.999994i	$0.501060\pi$
$54$	0	0
$55$	12.7789	1.72311
$56$	0	0
$57$	6.81275	0.902371
$58$	0	0
$59$	−4.50676	−0.586730	−0.293365	−	0.956000i	$-0.594775\pi$
$59$	−4.50676	−0.586730	−0.293365	+	0.956000i	$0.594775\pi$
$60$	0	0
$61$	−9.62232	−1.23201	−0.616006	−	0.787742i	$-0.711250\pi$
$61$	−9.62232	−1.23201	−0.616006	+	0.787742i	$0.711250\pi$
$62$	0	0
$63$	−0.211321	−0.0266240
$64$	0	0
$65$	−8.35721	−1.03658
$66$	0	0
$67$	0.964160	0.117791	0.0588954	−	0.998264i	$-0.481242\pi$
$67$	0.964160	0.117791	0.0588954	+	0.998264i	$0.481242\pi$
$68$	0	0
$69$	7.90631	0.951808
$70$	0	0
$71$	−7.76289	−0.921286	−0.460643	−	0.887586i	$-0.652381\pi$
$71$	−7.76289	−0.921286	−0.460643	+	0.887586i	$0.652381\pi$
$72$	0	0
$73$	16.4250	1.92241	0.961203	−	0.275841i	$-0.0889563\pi$
$73$	16.4250	1.92241	0.961203	+	0.275841i	$0.0889563\pi$
$74$	0	0
$75$	−24.8358	−2.86779
$76$	0	0
$77$	−2.04293	−0.232813
$78$	0	0
$79$	6.83310	0.768783	0.384392	−	0.923170i	$-0.374411\pi$
$79$	6.83310	0.768783	0.384392	+	0.923170i	$0.374411\pi$
$80$	0	0
$81$	−9.82439	−1.09160
$82$	0	0
$83$	9.15477	1.00487	0.502433	−	0.864616i	$-0.332438\pi$
$83$	9.15477	1.00487	0.502433	+	0.864616i	$0.332438\pi$
$84$	0	0
$85$	−8.96225	−0.972092
$86$	0	0
$87$	14.7418	1.58049
$88$	0	0
$89$	−2.36597	−0.250793	−0.125396	−	0.992107i	$-0.540020\pi$
$89$	−2.36597	−0.250793	−0.125396	+	0.992107i	$0.540020\pi$
$90$	0	0
$91$	1.33605	0.140056
$92$	0	0
$93$	−5.09467	−0.528292
$94$	0	0
$95$	−16.1852	−1.66056
$96$	0	0
$97$	−5.25335	−0.533396	−0.266698	−	0.963780i	$-0.585933\pi$
$97$	−5.25335	−0.533396	−0.266698	+	0.963780i	$0.585933\pi$
$98$	0	0
$99$	0.905280	0.0909840
$100$	0	0
$101$	9.93584	0.988653	0.494327	−	0.869276i	$-0.335415\pi$
$101$	9.93584	0.988653	0.494327	+	0.869276i	$0.335415\pi$
$102$	0	0
$103$	−19.2644	−1.89818	−0.949090	−	0.315005i	$-0.897994\pi$
$103$	−19.2644	−1.89818	−0.949090	+	0.315005i	$0.897994\pi$
$104$	0	0
$105$	5.42382	0.529311
$106$	0	0
$107$	7.88030	0.761817	0.380909	−	0.924613i	$-0.375611\pi$
$107$	7.88030	0.761817	0.380909	+	0.924613i	$0.375611\pi$
$108$	0	0
$109$	−1.72728	−0.165443	−0.0827215	−	0.996573i	$-0.526361\pi$
$109$	−1.72728	−0.165443	−0.0827215	+	0.996573i	$0.526361\pi$
$110$	0	0
$111$	17.6790	1.67802
$112$	0	0
$113$	14.0431	1.32107	0.660533	−	0.750797i	$-0.270330\pi$
$113$	14.0431	1.32107	0.660533	+	0.750797i	$0.270330\pi$
$114$	0	0
$115$	−18.7831	−1.75154
$116$	0	0
$117$	−0.592040	−0.0547341
$118$	0	0
$119$	1.43277	0.131342
$120$	0	0
$121$	−2.24829	−0.204390
$122$	0	0
$123$	7.44392	0.671195
$124$	0	0
$125$	37.4046	3.34557
$126$	0	0
$127$	1.65498	0.146856	0.0734278	−	0.997301i	$-0.476606\pi$
$127$	1.65498	0.146856	0.0734278	+	0.997301i	$0.476606\pi$
$128$	0	0
$129$	5.50525	0.484711
$130$	0	0
$131$	−17.1541	−1.49876	−0.749381	−	0.662139i	$-0.769649\pi$
$131$	−17.1541	−1.49876	−0.749381	+	0.662139i	$0.769649\pi$
$132$	0	0
$133$	2.58748	0.224363
$134$	0	0
$135$	21.1590	1.82107
$136$	0	0
$137$	14.7651	1.26147	0.630733	−	0.776000i	$-0.282755\pi$
$137$	14.7651	1.26147	0.630733	+	0.776000i	$0.282755\pi$
$138$	0	0
$139$	−20.3570	−1.72666	−0.863329	−	0.504641i	$-0.831625\pi$
$139$	−20.3570	−1.72666	−0.863329	+	0.504641i	$0.831625\pi$
$140$	0	0
$141$	12.2048	1.02783
$142$	0	0
$143$	−5.72349	−0.478623
$144$	0	0
$145$	−35.0223	−2.90845
$146$	0	0
$147$	11.8606	0.978247
$148$	0	0
$149$	8.37820	0.686369	0.343185	−	0.939268i	$-0.388494\pi$
$149$	8.37820	0.686369	0.343185	+	0.939268i	$0.388494\pi$
$150$	0	0
$151$	22.0074	1.79093	0.895467	−	0.445128i	$-0.146842\pi$
$151$	22.0074	1.79093	0.895467	+	0.445128i	$0.146842\pi$
$152$	0	0
$153$	−0.634902	−0.0513288
$154$	0	0
$155$	12.1035	0.972175
$156$	0	0
$157$	−0.208187	−0.0166151	−0.00830757	−	0.999965i	$-0.502644\pi$
$157$	−0.208187	−0.0166151	−0.00830757	+	0.999965i	$0.502644\pi$
$158$	0	0
$159$	0.0881222	0.00698855
$160$	0	0
$161$	3.00281	0.236655
$162$	0	0
$163$	−7.85060	−0.614906	−0.307453	−	0.951563i	$-0.599477\pi$
$163$	−7.85060	−0.614906	−0.307453	+	0.951563i	$0.599477\pi$
$164$	0	0
$165$	−23.2351	−1.80885
$166$	0	0
$167$	13.8766	1.07380	0.536900	−	0.843646i	$-0.319595\pi$
$167$	13.8766	1.07380	0.536900	+	0.843646i	$0.319595\pi$
$168$	0	0
$169$	−9.25692	−0.712071
$170$	0	0
$171$	−1.14659	−0.0876816
$172$	0	0
$173$	19.3500	1.47115	0.735577	−	0.677441i	$-0.236911\pi$
$173$	19.3500	1.47115	0.735577	+	0.677441i	$0.236911\pi$
$174$	0	0
$175$	−9.43262	−0.713039
$176$	0	0
$177$	8.19439	0.615928
$178$	0	0
$179$	7.02479	0.525057	0.262529	−	0.964924i	$-0.415444\pi$
$179$	7.02479	0.525057	0.262529	+	0.964924i	$0.415444\pi$
$180$	0	0
$181$	7.22067	0.536708	0.268354	−	0.963320i	$-0.413520\pi$
$181$	7.22067	0.536708	0.268354	+	0.963320i	$0.413520\pi$
$182$	0	0
$183$	17.4957	1.29332
$184$	0	0
$185$	−42.0003	−3.08792
$186$	0	0
$187$	−6.13786	−0.448845
$188$	0	0
$189$	−3.38263	−0.246050
$190$	0	0
$191$	−20.7694	−1.50282	−0.751412	−	0.659834i	$-0.770627\pi$
$191$	−20.7694	−1.50282	−0.751412	+	0.659834i	$0.770627\pi$
$192$	0	0
$193$	−19.7206	−1.41952	−0.709762	−	0.704442i	$-0.751197\pi$
$193$	−19.7206	−1.41952	−0.709762	+	0.704442i	$0.751197\pi$
$194$	0	0
$195$	15.1954	1.08817
$196$	0	0
$197$	−16.2977	−1.16116	−0.580582	−	0.814201i	$-0.697175\pi$
$197$	−16.2977	−1.16116	−0.580582	+	0.814201i	$0.697175\pi$
$198$	0	0
$199$	−4.79014	−0.339564	−0.169782	−	0.985482i	$-0.554306\pi$
$199$	−4.79014	−0.339564	−0.169782	+	0.985482i	$0.554306\pi$
$200$	0	0
$201$	−1.75308	−0.123653
$202$	0	0
$203$	5.59893	0.392968
$204$	0	0
$205$	−17.6846	−1.23515
$206$	0	0
$207$	−1.33063	−0.0924853
$208$	0	0
$209$	−11.0845	−0.766733
$210$	0	0
$211$	4.30473	0.296350	0.148175	−	0.988961i	$-0.452660\pi$
$211$	4.30473	0.296350	0.148175	+	0.988961i	$0.452660\pi$
$212$	0	0
$213$	14.1148	0.967132
$214$	0	0
$215$	−13.0789	−0.891975
$216$	0	0
$217$	−1.93495	−0.131353
$218$	0	0
$219$	−29.8647	−2.01807
$220$	0	0
$221$	4.01407	0.270016
$222$	0	0
$223$	−14.8068	−0.991540	−0.495770	−	0.868454i	$-0.665114\pi$
$223$	−14.8068	−0.991540	−0.495770	+	0.868454i	$0.665114\pi$
$224$	0	0
$225$	4.17986	0.278657
$226$	0	0
$227$	11.9026	0.790000	0.395000	−	0.918681i	$-0.370745\pi$
$227$	11.9026	0.790000	0.395000	+	0.918681i	$0.370745\pi$
$228$	0	0
$229$	−5.11465	−0.337985	−0.168993	−	0.985617i	$-0.554051\pi$
$229$	−5.11465	−0.337985	−0.168993	+	0.985617i	$0.554051\pi$
$230$	0	0
$231$	3.71454	0.244399
$232$	0	0
$233$	−14.9300	−0.978097	−0.489048	−	0.872257i	$-0.662656\pi$
$233$	−14.9300	−0.978097	−0.489048	+	0.872257i	$0.662656\pi$
$234$	0	0
$235$	−28.9951	−1.89143
$236$	0	0
$237$	−12.4242	−0.807041
$238$	0	0
$239$	−29.8624	−1.93164	−0.965820	−	0.259215i	$-0.916536\pi$
$239$	−29.8624	−1.93164	−0.965820	+	0.259215i	$0.916536\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	0	0
$243$	3.16814	0.203237
$244$	0	0
$245$	−28.1774	−1.80019
$246$	0	0
$247$	7.24912	0.461250
$248$	0	0
$249$	−16.6456	−1.05487
$250$	0	0
$251$	−13.0620	−0.824464	−0.412232	−	0.911079i	$-0.635251\pi$
$251$	−13.0620	−0.824464	−0.412232	+	0.911079i	$0.635251\pi$
$252$	0	0
$253$	−12.8638	−0.808738
$254$	0	0
$255$	16.2956	1.02047
$256$	0	0
$257$	6.35830	0.396620	0.198310	−	0.980139i	$-0.436455\pi$
$257$	6.35830	0.396620	0.198310	+	0.980139i	$0.436455\pi$
$258$	0	0
$259$	6.71448	0.417218
$260$	0	0
$261$	−2.48105	−0.153573
$262$	0	0
$263$	−4.66189	−0.287465	−0.143732	−	0.989617i	$-0.545910\pi$
$263$	−4.66189	−0.287465	−0.143732	+	0.989617i	$0.545910\pi$
$264$	0	0
$265$	−0.209353	−0.0128605
$266$	0	0
$267$	4.30192	0.263273
$268$	0	0
$269$	−10.5251	−0.641728	−0.320864	−	0.947125i	$-0.603973\pi$
$269$	−10.5251	−0.641728	−0.320864	+	0.947125i	$0.603973\pi$
$270$	0	0
$271$	−10.3212	−0.626968	−0.313484	−	0.949593i	$-0.601496\pi$
$271$	−10.3212	−0.626968	−0.313484	+	0.949593i	$0.601496\pi$
$272$	0	0
$273$	−2.42926	−0.147025
$274$	0	0
$275$	40.4084	2.43672
$276$	0	0
$277$	22.1023	1.32800	0.663999	−	0.747734i	$-0.268858\pi$
$277$	22.1023	1.32800	0.663999	+	0.747734i	$0.268858\pi$
$278$	0	0
$279$	0.857432	0.0513331
$280$	0	0
$281$	4.22803	0.252223	0.126111	−	0.992016i	$-0.459750\pi$
$281$	4.22803	0.252223	0.126111	+	0.992016i	$0.459750\pi$
$282$	0	0
$283$	3.06769	0.182355	0.0911776	−	0.995835i	$-0.470937\pi$
$283$	3.06769	0.182355	0.0911776	+	0.995835i	$0.470937\pi$
$284$	0	0
$285$	29.4286	1.74320
$286$	0	0
$287$	2.82720	0.166884
$288$	0	0
$289$	−12.6953	−0.746783
$290$	0	0
$291$	9.55186	0.559940
$292$	0	0
$293$	−15.4694	−0.903735	−0.451867	−	0.892085i	$-0.649242\pi$
$293$	−15.4694	−0.903735	−0.451867	+	0.892085i	$0.649242\pi$
$294$	0	0
$295$	−19.4675	−1.13344
$296$	0	0
$297$	14.4909	0.840846
$298$	0	0
$299$	8.41272	0.486520
$300$	0	0
$301$	2.09089	0.120517
$302$	0	0
$303$	−18.0658	−1.03785
$304$	0	0
$305$	−41.5648	−2.38000
$306$	0	0
$307$	20.2412	1.15523	0.577614	−	0.816310i	$-0.303984\pi$
$307$	20.2412	1.15523	0.577614	+	0.816310i	$0.303984\pi$
$308$	0	0
$309$	35.0274	1.99264
$310$	0	0
$311$	−1.49434	−0.0847363	−0.0423681	−	0.999102i	$-0.513490\pi$
$311$	−1.49434	−0.0847363	−0.0423681	+	0.999102i	$0.513490\pi$
$312$	0	0
$313$	−21.6470	−1.22356	−0.611780	−	0.791028i	$-0.709546\pi$
$313$	−21.6470	−1.22356	−0.611780	+	0.791028i	$0.709546\pi$
$314$	0	0
$315$	−0.912829	−0.0514321
$316$	0	0
$317$	28.9329	1.62504	0.812518	−	0.582936i	$-0.198096\pi$
$317$	28.9329	1.62504	0.812518	+	0.582936i	$0.198096\pi$
$318$	0	0
$319$	−23.9853	−1.34292
$320$	0	0
$321$	−14.3283	−0.799728
$322$	0	0
$323$	7.77394	0.432553
$324$	0	0
$325$	−26.4265	−1.46588
$326$	0	0
$327$	3.14061	0.173676
$328$	0	0
$329$	4.63537	0.255556
$330$	0	0
$331$	27.5701	1.51539	0.757695	−	0.652609i	$-0.226325\pi$
$331$	27.5701	1.51539	0.757695	+	0.652609i	$0.226325\pi$
$332$	0	0
$333$	−2.97538	−0.163050
$334$	0	0
$335$	4.16481	0.227548
$336$	0	0
$337$	−8.25450	−0.449651	−0.224826	−	0.974399i	$-0.572181\pi$
$337$	−8.25450	−0.449651	−0.224826	+	0.974399i	$0.572181\pi$
$338$	0	0
$339$	−25.5338	−1.38681
$340$	0	0
$341$	8.28915	0.448883
$342$	0	0
$343$	9.33864	0.504239
$344$	0	0
$345$	34.1523	1.83870
$346$	0	0
$347$	−21.2018	−1.13817	−0.569085	−	0.822278i	$-0.692703\pi$
$347$	−21.2018	−1.13817	−0.569085	+	0.822278i	$0.692703\pi$
$348$	0	0
$349$	21.3257	1.14154	0.570770	−	0.821110i	$-0.306645\pi$
$349$	21.3257	1.14154	0.570770	+	0.821110i	$0.306645\pi$
$350$	0	0
$351$	−9.47682	−0.505835
$352$	0	0
$353$	2.80184	0.149127	0.0745635	−	0.997216i	$-0.476244\pi$
$353$	2.80184	0.149127	0.0745635	+	0.997216i	$0.476244\pi$
$354$	0	0
$355$	−33.5328	−1.77974
$356$	0	0
$357$	−2.60513	−0.137878
$358$	0	0
$359$	−4.62788	−0.244250	−0.122125	−	0.992515i	$-0.538971\pi$
$359$	−4.62788	−0.244250	−0.122125	+	0.992515i	$0.538971\pi$
$360$	0	0
$361$	−4.96085	−0.261097
$362$	0	0
$363$	4.08793	0.214561
$364$	0	0
$365$	70.9501	3.71370
$366$	0	0
$367$	0.106183	0.00554270	0.00277135	−	0.999996i	$-0.499118\pi$
$367$	0.106183	0.00554270	0.00277135	+	0.999996i	$0.499118\pi$
$368$	0	0
$369$	−1.25281	−0.0652187
$370$	0	0
$371$	0.0334688	0.00173761
$372$	0	0
$373$	−16.6703	−0.863157	−0.431579	−	0.902075i	$-0.642043\pi$
$373$	−16.6703	−0.863157	−0.431579	+	0.902075i	$0.642043\pi$
$374$	0	0
$375$	−68.0106	−3.51205
$376$	0	0
$377$	15.6860	0.807872
$378$	0	0
$379$	16.5660	0.850939	0.425469	−	0.904973i	$-0.360109\pi$
$379$	16.5660	0.850939	0.425469	+	0.904973i	$0.360109\pi$
$380$	0	0
$381$	−3.00915	−0.154164
$382$	0	0
$383$	25.0198	1.27845	0.639226	−	0.769019i	$-0.279255\pi$
$383$	25.0198	1.27845	0.639226	+	0.769019i	$0.279255\pi$
$384$	0	0
$385$	−8.82470	−0.449748
$386$	0	0
$387$	−0.926534	−0.0470984
$388$	0	0
$389$	−32.6696	−1.65641	−0.828207	−	0.560423i	$-0.810639\pi$
$389$	−32.6696	−1.65641	−0.828207	+	0.560423i	$0.810639\pi$
$390$	0	0
$391$	9.02178	0.456251
$392$	0	0
$393$	31.1904	1.57335
$394$	0	0
$395$	29.5165	1.48513
$396$	0	0
$397$	−21.9103	−1.09964	−0.549822	−	0.835282i	$-0.685305\pi$
$397$	−21.9103	−1.09964	−0.549822	+	0.835282i	$0.685305\pi$
$398$	0	0
$399$	−4.70467	−0.235528
$400$	0	0
$401$	10.2229	0.510508	0.255254	−	0.966874i	$-0.417841\pi$
$401$	10.2229	0.510508	0.255254	+	0.966874i	$0.417841\pi$
$402$	0	0
$403$	−5.42099	−0.270039
$404$	0	0
$405$	−42.4377	−2.10875
$406$	0	0
$407$	−28.7642	−1.42579
$408$	0	0
$409$	−16.7739	−0.829417	−0.414709	−	0.909954i	$-0.636117\pi$
$409$	−16.7739	−0.829417	−0.414709	+	0.909954i	$0.636117\pi$
$410$	0	0
$411$	−26.8465	−1.32424
$412$	0	0
$413$	3.11223	0.153143
$414$	0	0
$415$	39.5452	1.94120
$416$	0	0
$417$	37.0140	1.81258
$418$	0	0
$419$	−32.5790	−1.59159	−0.795793	−	0.605569i	$-0.792946\pi$
$419$	−32.5790	−1.59159	−0.795793	+	0.605569i	$0.792946\pi$
$420$	0	0
$421$	11.5346	0.562163	0.281082	−	0.959684i	$-0.409307\pi$
$421$	11.5346	0.562163	0.281082	+	0.959684i	$0.409307\pi$
$422$	0	0
$423$	−2.05406	−0.0998721
$424$	0	0
$425$	−28.3397	−1.37468
$426$	0	0
$427$	6.64487	0.321568
$428$	0	0
$429$	10.4067	0.502441
$430$	0	0
$431$	−15.2713	−0.735590	−0.367795	−	0.929907i	$-0.619887\pi$
$431$	−15.2713	−0.735590	−0.367795	+	0.929907i	$0.619887\pi$
$432$	0	0
$433$	8.69941	0.418067	0.209034	−	0.977908i	$-0.432968\pi$
$433$	8.69941	0.418067	0.209034	+	0.977908i	$0.432968\pi$
$434$	0	0
$435$	63.6792	3.05318
$436$	0	0
$437$	16.2927	0.779384
$438$	0	0
$439$	−29.4040	−1.40338	−0.701689	−	0.712484i	$-0.747570\pi$
$439$	−29.4040	−1.40338	−0.701689	+	0.712484i	$0.747570\pi$
$440$	0	0
$441$	−1.99614	−0.0950543
$442$	0	0
$443$	35.7878	1.70033	0.850164	−	0.526518i	$-0.176503\pi$
$443$	35.7878	1.70033	0.850164	+	0.526518i	$0.176503\pi$
$444$	0	0
$445$	−10.2201	−0.484481
$446$	0	0
$447$	−15.2336	−0.720525
$448$	0	0
$449$	−7.50777	−0.354314	−0.177157	−	0.984183i	$-0.556690\pi$
$449$	−7.50777	−0.354314	−0.177157	+	0.984183i	$0.556690\pi$
$450$	0	0
$451$	−12.1114	−0.570306
$452$	0	0
$453$	−40.0147	−1.88006
$454$	0	0
$455$	5.77123	0.270559
$456$	0	0
$457$	−31.7902	−1.48708	−0.743540	−	0.668691i	$-0.766855\pi$
$457$	−31.7902	−1.48708	−0.743540	+	0.668691i	$0.766855\pi$
$458$	0	0
$459$	−10.1629	−0.474364
$460$	0	0
$461$	−12.8715	−0.599487	−0.299743	−	0.954020i	$-0.596901\pi$
$461$	−12.8715	−0.599487	−0.299743	+	0.954020i	$0.596901\pi$
$462$	0	0
$463$	11.8687	0.551587	0.275794	−	0.961217i	$-0.411059\pi$
$463$	11.8687	0.551587	0.275794	+	0.961217i	$0.411059\pi$
$464$	0	0
$465$	−22.0071	−1.02055
$466$	0	0
$467$	18.9191	0.875473	0.437736	−	0.899103i	$-0.355780\pi$
$467$	18.9191	0.875473	0.437736	+	0.899103i	$0.355780\pi$
$468$	0	0
$469$	−0.665818	−0.0307446
$470$	0	0
$471$	0.378535	0.0174420
$472$	0	0
$473$	−8.95719	−0.411852
$474$	0	0
$475$	−51.1795	−2.34828
$476$	0	0
$477$	−0.0148310	−0.000679063 0
$478$	0	0
$479$	−23.7911	−1.08704	−0.543522	−	0.839395i	$-0.682909\pi$
$479$	−23.7911	−1.08704	−0.543522	+	0.839395i	$0.682909\pi$
$480$	0	0
$481$	18.8114	0.857725
$482$	0	0
$483$	−5.45985	−0.248432
$484$	0	0
$485$	−22.6925	−1.03041
$486$	0	0
$487$	−1.59589	−0.0723165	−0.0361583	−	0.999346i	$-0.511512\pi$
$487$	−1.59589	−0.0723165	−0.0361583	+	0.999346i	$0.511512\pi$
$488$	0	0
$489$	14.2743	0.645506
$490$	0	0
$491$	31.8604	1.43784	0.718920	−	0.695093i	$-0.244637\pi$
$491$	31.8604	1.43784	0.718920	+	0.695093i	$0.244637\pi$
$492$	0	0
$493$	16.8217	0.757610
$494$	0	0
$495$	3.91047	0.175763
$496$	0	0
$497$	5.36081	0.240465
$498$	0	0
$499$	−17.3389	−0.776194	−0.388097	−	0.921619i	$-0.626867\pi$
$499$	−17.3389	−0.776194	−0.388097	+	0.921619i	$0.626867\pi$
$500$	0	0
$501$	−25.2310	−1.12724
$502$	0	0
$503$	13.8066	0.615607	0.307803	−	0.951450i	$-0.400406\pi$
$503$	13.8066	0.615607	0.307803	+	0.951450i	$0.400406\pi$
$504$	0	0
$505$	42.9192	1.90988
$506$	0	0
$507$	16.8313	0.747506
$508$	0	0
$509$	37.2876	1.65274	0.826372	−	0.563124i	$-0.190401\pi$
$509$	37.2876	1.65274	0.826372	+	0.563124i	$0.190401\pi$
$510$	0	0
$511$	−11.3426	−0.501768
$512$	0	0
$513$	−18.3535	−0.810326
$514$	0	0
$515$	−83.2152	−3.66690
$516$	0	0
$517$	−19.8575	−0.873332
$518$	0	0
$519$	−35.1830	−1.54436
$520$	0	0
$521$	35.4359	1.55248	0.776238	−	0.630439i	$-0.217125\pi$
$521$	35.4359	1.55248	0.776238	+	0.630439i	$0.217125\pi$
$522$	0	0
$523$	22.2983	0.975036	0.487518	−	0.873113i	$-0.337902\pi$
$523$	22.2983	0.975036	0.487518	+	0.873113i	$0.337902\pi$
$524$	0	0
$525$	17.1508	0.748522
$526$	0	0
$527$	−5.81345	−0.253238
$528$	0	0
$529$	−4.09210	−0.177917
$530$	0	0
$531$	−1.37912	−0.0598485
$532$	0	0
$533$	7.92071	0.343084
$534$	0	0
$535$	34.0400	1.47168
$536$	0	0
$537$	−12.7728	−0.551186
$538$	0	0
$539$	−19.2975	−0.831203
$540$	0	0
$541$	−28.0367	−1.20539	−0.602697	−	0.797970i	$-0.705907\pi$
$541$	−28.0367	−1.20539	−0.602697	+	0.797970i	$0.705907\pi$
$542$	0	0
$543$	−13.1289	−0.563416
$544$	0	0
$545$	−7.46119	−0.319602
$546$	0	0
$547$	13.1556	0.562491	0.281245	−	0.959636i	$-0.409253\pi$
$547$	13.1556	0.562491	0.281245	+	0.959636i	$0.409253\pi$
$548$	0	0
$549$	−2.94453	−0.125669
$550$	0	0
$551$	30.3787	1.29418
$552$	0	0
$553$	−4.71872	−0.200660
$554$	0	0
$555$	76.3668	3.24159
$556$	0	0
$557$	−20.1003	−0.851678	−0.425839	−	0.904799i	$-0.640021\pi$
$557$	−20.1003	−0.851678	−0.425839	+	0.904799i	$0.640021\pi$
$558$	0	0
$559$	5.85787	0.247762
$560$	0	0
$561$	11.1601	0.471181
$562$	0	0
$563$	1.97240	0.0831268	0.0415634	−	0.999136i	$-0.486766\pi$
$563$	1.97240	0.0831268	0.0415634	+	0.999136i	$0.486766\pi$
$564$	0	0
$565$	60.6611	2.55203
$566$	0	0
$567$	6.78441	0.284919
$568$	0	0
$569$	19.0863	0.800139	0.400070	−	0.916485i	$-0.368986\pi$
$569$	19.0863	0.800139	0.400070	+	0.916485i	$0.368986\pi$
$570$	0	0
$571$	−10.1484	−0.424697	−0.212348	−	0.977194i	$-0.568111\pi$
$571$	−10.1484	−0.424697	−0.212348	+	0.977194i	$0.568111\pi$
$572$	0	0
$573$	37.7639	1.57761
$574$	0	0
$575$	−59.3946	−2.47693
$576$	0	0
$577$	11.2993	0.470395	0.235198	−	0.971948i	$-0.424426\pi$
$577$	11.2993	0.470395	0.235198	+	0.971948i	$0.424426\pi$
$578$	0	0
$579$	35.8569	1.49016
$580$	0	0
$581$	−6.32200	−0.262281
$582$	0	0
$583$	−0.143377	−0.00593807
$584$	0	0
$585$	−2.55739	−0.105735
$586$	0	0
$587$	−36.1850	−1.49352	−0.746758	−	0.665095i	$-0.768391\pi$
$587$	−36.1850	−1.49352	−0.746758	+	0.665095i	$0.768391\pi$
$588$	0	0
$589$	−10.4987	−0.432590
$590$	0	0
$591$	29.6332	1.21895
$592$	0	0
$593$	−28.2897	−1.16172	−0.580858	−	0.814005i	$-0.697283\pi$
$593$	−28.2897	−1.16172	−0.580858	+	0.814005i	$0.697283\pi$
$594$	0	0
$595$	6.18905	0.253726
$596$	0	0
$597$	8.70965	0.356462
$598$	0	0
$599$	10.9989	0.449404	0.224702	−	0.974427i	$-0.427859\pi$
$599$	10.9989	0.449404	0.224702	+	0.974427i	$0.427859\pi$
$600$	0	0
$601$	−10.3224	−0.421058	−0.210529	−	0.977588i	$-0.567519\pi$
$601$	−10.3224	−0.421058	−0.210529	+	0.977588i	$0.567519\pi$
$602$	0	0
$603$	0.295043	0.0120151
$604$	0	0
$605$	−9.71176	−0.394839
$606$	0	0
$607$	26.4559	1.07381	0.536906	−	0.843642i	$-0.319593\pi$
$607$	26.4559	1.07381	0.536906	+	0.843642i	$0.319593\pi$
$608$	0	0
$609$	−10.1802	−0.412524
$610$	0	0
$611$	12.9865	0.525378
$612$	0	0
$613$	33.8484	1.36712	0.683561	−	0.729893i	$-0.260430\pi$
$613$	33.8484	1.36712	0.683561	+	0.729893i	$0.260430\pi$
$614$	0	0
$615$	32.1550	1.29661
$616$	0	0
$617$	−0.659865	−0.0265651	−0.0132826	−	0.999912i	$-0.504228\pi$
$617$	−0.659865	−0.0265651	−0.0132826	+	0.999912i	$0.504228\pi$
$618$	0	0
$619$	34.8866	1.40221	0.701106	−	0.713057i	$-0.252690\pi$
$619$	34.8866	1.40221	0.701106	+	0.713057i	$0.252690\pi$
$620$	0	0
$621$	−21.2995	−0.854720
$622$	0	0
$623$	1.63387	0.0654595
$624$	0	0
$625$	93.2779	3.73111
$626$	0	0
$627$	20.1544	0.804888
$628$	0	0
$629$	20.1733	0.804361
$630$	0	0
$631$	6.39474	0.254571	0.127285	−	0.991866i	$-0.459374\pi$
$631$	6.39474	0.254571	0.127285	+	0.991866i	$0.459374\pi$
$632$	0	0
$633$	−7.82705	−0.311097
$634$	0	0
$635$	7.14890	0.283695
$636$	0	0
$637$	12.6203	0.500035
$638$	0	0
$639$	−2.37553	−0.0939743
$640$	0	0
$641$	16.9466	0.669352	0.334676	−	0.942333i	$-0.391373\pi$
$641$	16.9466	0.669352	0.334676	+	0.942333i	$0.391373\pi$
$642$	0	0
$643$	−14.3993	−0.567852	−0.283926	−	0.958846i	$-0.591637\pi$
$643$	−14.3993	−0.567852	−0.283926	+	0.958846i	$0.591637\pi$
$644$	0	0
$645$	23.7807	0.936363
$646$	0	0
$647$	−4.38744	−0.172488	−0.0862441	−	0.996274i	$-0.527486\pi$
$647$	−4.38744	−0.172488	−0.0862441	+	0.996274i	$0.527486\pi$
$648$	0	0
$649$	−13.3325	−0.523346
$650$	0	0
$651$	3.51822	0.137890
$652$	0	0
$653$	−4.56970	−0.178826	−0.0894132	−	0.995995i	$-0.528499\pi$
$653$	−4.56970	−0.178826	−0.0894132	+	0.995995i	$0.528499\pi$
$654$	0	0
$655$	−74.0994	−2.89530
$656$	0	0
$657$	5.02624	0.196092
$658$	0	0
$659$	13.6506	0.531751	0.265875	−	0.964007i	$-0.414339\pi$
$659$	13.6506	0.531751	0.265875	+	0.964007i	$0.414339\pi$
$660$	0	0
$661$	32.4151	1.26080	0.630401	−	0.776270i	$-0.282891\pi$
$661$	32.4151	1.26080	0.630401	+	0.776270i	$0.282891\pi$
$662$	0	0
$663$	−7.29856	−0.283453
$664$	0	0
$665$	11.1770	0.433424
$666$	0	0
$667$	35.2550	1.36508
$668$	0	0
$669$	26.9225	1.04088
$670$	0	0
$671$	−28.4660	−1.09892
$672$	0	0
$673$	0.237992	0.00917391	0.00458696	−	0.999989i	$-0.498540\pi$
$673$	0.237992	0.00917391	0.00458696	+	0.999989i	$0.498540\pi$
$674$	0	0
$675$	66.9073	2.57526
$676$	0	0
$677$	−30.9534	−1.18964	−0.594818	−	0.803860i	$-0.702776\pi$
$677$	−30.9534	−1.18964	−0.594818	+	0.803860i	$0.702776\pi$
$678$	0	0
$679$	3.62780	0.139222
$680$	0	0
$681$	−21.6417	−0.829313
$682$	0	0
$683$	28.5801	1.09359	0.546794	−	0.837267i	$-0.315848\pi$
$683$	28.5801	1.09359	0.546794	+	0.837267i	$0.315848\pi$
$684$	0	0
$685$	63.7796	2.43689
$686$	0	0
$687$	9.29968	0.354805
$688$	0	0
$689$	0.0937666	0.00357222
$690$	0	0
$691$	6.89241	0.262200	0.131100	−	0.991369i	$-0.458149\pi$
$691$	6.89241	0.262200	0.131100	+	0.991369i	$0.458149\pi$
$692$	0	0
$693$	−0.625158	−0.0237478
$694$	0	0
$695$	−87.9347	−3.33555
$696$	0	0
$697$	8.49415	0.321739
$698$	0	0
$699$	27.1464	1.02677
$700$	0	0
$701$	−29.3972	−1.11032	−0.555158	−	0.831745i	$-0.687342\pi$
$701$	−29.3972	−1.11032	−0.555158	+	0.831745i	$0.687342\pi$
$702$	0	0
$703$	36.4314	1.37404
$704$	0	0
$705$	52.7202	1.98556
$706$	0	0
$707$	−6.86138	−0.258049
$708$	0	0
$709$	−10.8690	−0.408195	−0.204098	−	0.978951i	$-0.565426\pi$
$709$	−10.8690	−0.408195	−0.204098	+	0.978951i	$0.565426\pi$
$710$	0	0
$711$	2.09100	0.0784186
$712$	0	0
$713$	−12.1839	−0.456290
$714$	0	0
$715$	−24.7234	−0.924602
$716$	0	0
$717$	54.2972	2.02776
$718$	0	0
$719$	30.8283	1.14970	0.574850	−	0.818259i	$-0.305060\pi$
$719$	30.8283	1.14970	0.574850	+	0.818259i	$0.305060\pi$
$720$	0	0
$721$	13.3034	0.495445
$722$	0	0
$723$	−1.81824	−0.0676212
$724$	0	0
$725$	−110.745	−4.11297
$726$	0	0
$727$	−28.0322	−1.03966	−0.519828	−	0.854271i	$-0.674004\pi$
$727$	−28.0322	−1.03966	−0.519828	+	0.854271i	$0.674004\pi$
$728$	0	0
$729$	23.7127	0.878248
$730$	0	0
$731$	6.28197	0.232347
$732$	0	0
$733$	25.8442	0.954578	0.477289	−	0.878746i	$-0.341619\pi$
$733$	25.8442	0.954578	0.477289	+	0.878746i	$0.341619\pi$
$734$	0	0
$735$	51.2335	1.88977
$736$	0	0
$737$	2.85230	0.105066
$738$	0	0
$739$	43.0752	1.58455	0.792273	−	0.610166i	$-0.208897\pi$
$739$	43.0752	1.58455	0.792273	+	0.610166i	$0.208897\pi$
$740$	0	0
$741$	−13.1807	−0.484204
$742$	0	0
$743$	−13.1212	−0.481372	−0.240686	−	0.970603i	$-0.577372\pi$
$743$	−13.1212	−0.481372	−0.240686	+	0.970603i	$0.577372\pi$
$744$	0	0
$745$	36.1907	1.32593
$746$	0	0
$747$	2.80146	0.102500
$748$	0	0
$749$	−5.44189	−0.198842
$750$	0	0
$751$	−33.6870	−1.22926	−0.614629	−	0.788817i	$-0.710694\pi$
$751$	−33.6870	−1.22926	−0.614629	+	0.788817i	$0.710694\pi$
$752$	0	0
$753$	23.7498	0.865492
$754$	0	0
$755$	95.0637	3.45972
$756$	0	0
$757$	−46.0911	−1.67521	−0.837604	−	0.546278i	$-0.816044\pi$
$757$	−46.0911	−1.67521	−0.837604	+	0.546278i	$0.816044\pi$
$758$	0	0
$759$	23.3895	0.848984
$760$	0	0
$761$	14.2727	0.517387	0.258693	−	0.965960i	$-0.416708\pi$
$761$	14.2727	0.517387	0.258693	+	0.965960i	$0.416708\pi$
$762$	0	0
$763$	1.19280	0.0431823
$764$	0	0
$765$	−2.74254	−0.0991568
$766$	0	0
$767$	8.71925	0.314834
$768$	0	0
$769$	13.0154	0.469349	0.234674	−	0.972074i	$-0.424598\pi$
$769$	13.0154	0.469349	0.234674	+	0.972074i	$0.424598\pi$
$770$	0	0
$771$	−11.5609	−0.416357
$772$	0	0
$773$	−29.9438	−1.07700	−0.538502	−	0.842624i	$-0.681010\pi$
$773$	−29.9438	−1.07700	−0.538502	+	0.842624i	$0.681010\pi$
$774$	0	0
$775$	38.2727	1.37480
$776$	0	0
$777$	−12.2086	−0.437980
$778$	0	0
$779$	15.3398	0.549606
$780$	0	0
$781$	−22.9652	−0.821759
$782$	0	0
$783$	−39.7143	−1.41927
$784$	0	0
$785$	−0.899291	−0.0320971
$786$	0	0
$787$	−13.7932	−0.491676	−0.245838	−	0.969311i	$-0.579063\pi$
$787$	−13.7932	−0.491676	−0.245838	+	0.969311i	$0.579063\pi$
$788$	0	0
$789$	8.47646	0.301770
$790$	0	0
$791$	−9.69774	−0.344812
$792$	0	0
$793$	18.6163	0.661086
$794$	0	0
$795$	0.380655	0.0135005
$796$	0	0
$797$	18.4152	0.652300	0.326150	−	0.945318i	$-0.394249\pi$
$797$	18.4152	0.652300	0.326150	+	0.945318i	$0.394249\pi$
$798$	0	0
$799$	13.9267	0.492691
$800$	0	0
$801$	−0.724013	−0.0255817
$802$	0	0
$803$	48.5907	1.71473
$804$	0	0
$805$	12.9710	0.457169
$806$	0	0
$807$	19.1372	0.673663
$808$	0	0
$809$	−14.7007	−0.516849	−0.258425	−	0.966031i	$-0.583203\pi$
$809$	−14.7007	−0.516849	−0.258425	+	0.966031i	$0.583203\pi$
$810$	0	0
$811$	46.3555	1.62776	0.813882	−	0.581031i	$-0.197350\pi$
$811$	46.3555	1.62776	0.813882	+	0.581031i	$0.197350\pi$
$812$	0	0
$813$	18.7665	0.658168
$814$	0	0
$815$	−33.9117	−1.18787
$816$	0	0
$817$	11.3448	0.396903
$818$	0	0
$819$	0.408844	0.0142862
$820$	0	0
$821$	39.2626	1.37027	0.685137	−	0.728414i	$-0.259742\pi$
$821$	39.2626	1.37027	0.685137	+	0.728414i	$0.259742\pi$
$822$	0	0
$823$	29.0804	1.01368	0.506839	−	0.862041i	$-0.330814\pi$
$823$	29.0804	1.01368	0.506839	+	0.862041i	$0.330814\pi$
$824$	0	0
$825$	−73.4724	−2.55798
$826$	0	0
$827$	−30.7270	−1.06848	−0.534242	−	0.845332i	$-0.679403\pi$
$827$	−30.7270	−1.06848	−0.534242	+	0.845332i	$0.679403\pi$
$828$	0	0
$829$	−16.4599	−0.571675	−0.285837	−	0.958278i	$-0.592272\pi$
$829$	−16.4599	−0.571675	−0.285837	+	0.958278i	$0.592272\pi$
$830$	0	0
$831$	−40.1873	−1.39408
$832$	0	0
$833$	13.5340	0.468925
$834$	0	0
$835$	59.9416	2.07436
$836$	0	0
$837$	13.7250	0.474405
$838$	0	0
$839$	−43.2285	−1.49241	−0.746206	−	0.665715i	$-0.768127\pi$
$839$	−43.2285	−1.49241	−0.746206	+	0.665715i	$0.768127\pi$
$840$	0	0
$841$	36.7351	1.26673
$842$	0	0
$843$	−7.68758	−0.264774
$844$	0	0
$845$	−39.9865	−1.37558
$846$	0	0
$847$	1.55260	0.0533478
$848$	0	0
$849$	−5.57781	−0.191430
$850$	0	0
$851$	42.2793	1.44931
$852$	0	0
$853$	8.36996	0.286582	0.143291	−	0.989681i	$-0.454232\pi$
$853$	8.36996	0.286582	0.143291	+	0.989681i	$0.454232\pi$
$854$	0	0
$855$	−4.95283	−0.169383
$856$	0	0
$857$	18.3641	0.627305	0.313653	−	0.949538i	$-0.398447\pi$
$857$	18.3641	0.627305	0.313653	+	0.949538i	$0.398447\pi$
$858$	0	0
$859$	−2.47499	−0.0844457	−0.0422228	−	0.999108i	$-0.513444\pi$
$859$	−2.47499	−0.0844457	−0.0422228	+	0.999108i	$0.513444\pi$
$860$	0	0
$861$	−5.14053	−0.175189
$862$	0	0
$863$	10.0917	0.343525	0.171762	−	0.985138i	$-0.445054\pi$
$863$	10.0917	0.343525	0.171762	+	0.985138i	$0.445054\pi$
$864$	0	0
$865$	83.5849	2.84197
$866$	0	0
$867$	23.0832	0.783946
$868$	0	0
$869$	20.2146	0.685732
$870$	0	0
$871$	−1.86536	−0.0632055
$872$	0	0
$873$	−1.60758	−0.0544083
$874$	0	0
$875$	−25.8304	−0.873227
$876$	0	0
$877$	53.6453	1.81147	0.905736	−	0.423843i	$-0.139319\pi$
$877$	53.6453	1.81147	0.905736	+	0.423843i	$0.139319\pi$
$878$	0	0
$879$	28.1272	0.948708
$880$	0	0
$881$	30.6985	1.03426	0.517130	−	0.855907i	$-0.327001\pi$
$881$	30.6985	1.03426	0.517130	+	0.855907i	$0.327001\pi$
$882$	0	0
$883$	21.8831	0.736424	0.368212	−	0.929742i	$-0.379970\pi$
$883$	21.8831	0.736424	0.368212	+	0.929742i	$0.379970\pi$
$884$	0	0
$885$	35.3967	1.18985
$886$	0	0
$887$	23.3921	0.785431	0.392715	−	0.919660i	$-0.371536\pi$
$887$	23.3921	0.785431	0.392715	+	0.919660i	$0.371536\pi$
$888$	0	0
$889$	−1.14288	−0.0383308
$890$	0	0
$891$	−29.0638	−0.973673
$892$	0	0
$893$	25.1506	0.841633
$894$	0	0
$895$	30.3445	1.01430
$896$	0	0
$897$	−15.2964	−0.510731
$898$	0	0
$899$	−22.7176	−0.757674
$900$	0	0
$901$	0.100555	0.00334997
$902$	0	0
$903$	−3.80175	−0.126514
$904$	0	0
$905$	31.1906	1.03681
$906$	0	0
$907$	2.98723	0.0991893	0.0495947	−	0.998769i	$-0.484207\pi$
$907$	2.98723	0.0991893	0.0495947	+	0.998769i	$0.484207\pi$
$908$	0	0
$909$	3.04047	0.100846
$910$	0	0
$911$	−42.1883	−1.39776	−0.698880	−	0.715239i	$-0.746318\pi$
$911$	−42.1883	−1.39776	−0.698880	+	0.715239i	$0.746318\pi$
$912$	0	0
$913$	27.0828	0.896311
$914$	0	0
$915$	75.5750	2.49843
$916$	0	0
$917$	11.8461	0.391192
$918$	0	0
$919$	53.2428	1.75632	0.878160	−	0.478367i	$-0.158771\pi$
$919$	53.2428	1.75632	0.878160	+	0.478367i	$0.158771\pi$
$920$	0	0
$921$	−36.8035	−1.21272
$922$	0	0
$923$	15.0189	0.494353
$924$	0	0
$925$	−132.810	−4.36677
$926$	0	0
$927$	−5.89511	−0.193621
$928$	0	0
$929$	−27.2213	−0.893103	−0.446552	−	0.894758i	$-0.647348\pi$
$929$	−27.2213	−0.893103	−0.446552	+	0.894758i	$0.647348\pi$
$930$	0	0
$931$	24.4414	0.801033
$932$	0	0
$933$	2.71708	0.0889531
$934$	0	0
$935$	−26.5133	−0.867077
$936$	0	0
$937$	7.54293	0.246417	0.123208	−	0.992381i	$-0.460682\pi$
$937$	7.54293	0.246417	0.123208	+	0.992381i	$0.460682\pi$
$938$	0	0
$939$	39.3595	1.28445
$940$	0	0
$941$	35.4553	1.15581	0.577905	−	0.816104i	$-0.303870\pi$
$941$	35.4553	1.15581	0.577905	+	0.816104i	$0.303870\pi$
$942$	0	0
$943$	17.8021	0.579716
$944$	0	0
$945$	−14.6117	−0.475319
$946$	0	0
$947$	−33.0630	−1.07440	−0.537202	−	0.843454i	$-0.680519\pi$
$947$	−33.0630	−1.07440	−0.537202	+	0.843454i	$0.680519\pi$
$948$	0	0
$949$	−31.7776	−1.03155
$950$	0	0
$951$	−52.6071	−1.70590
$952$	0	0
$953$	−5.14501	−0.166663	−0.0833316	−	0.996522i	$-0.526556\pi$
$953$	−5.14501	−0.166663	−0.0833316	+	0.996522i	$0.526556\pi$
$954$	0	0
$955$	−89.7162	−2.90315
$956$	0	0
$957$	43.6111	1.40975
$958$	0	0
$959$	−10.1963	−0.329255
$960$	0	0
$961$	−23.1490	−0.746741
$962$	0	0
$963$	2.41145	0.0777080
$964$	0	0
$965$	−85.1859	−2.74223
$966$	0	0
$967$	−2.82789	−0.0909390	−0.0454695	−	0.998966i	$-0.514478\pi$
$967$	−2.82789	−0.0909390	−0.0454695	+	0.998966i	$0.514478\pi$
$968$	0	0
$969$	−14.1349	−0.454079
$970$	0	0
$971$	43.0152	1.38042	0.690211	−	0.723608i	$-0.257518\pi$
$971$	43.0152	1.38042	0.690211	+	0.723608i	$0.257518\pi$
$972$	0	0
$973$	14.0579	0.450676
$974$	0	0
$975$	48.0499	1.53883
$976$	0	0
$977$	−5.45876	−0.174641	−0.0873207	−	0.996180i	$-0.527830\pi$
$977$	−5.45876	−0.174641	−0.0873207	+	0.996180i	$0.527830\pi$
$978$	0	0
$979$	−6.99933	−0.223700
$980$	0	0
$981$	−0.528564	−0.0168758
$982$	0	0
$983$	−16.8518	−0.537489	−0.268745	−	0.963211i	$-0.586609\pi$
$983$	−16.8518	−0.537489	−0.268745	+	0.963211i	$0.586609\pi$
$984$	0	0
$985$	−70.4002	−2.24314
$986$	0	0
$987$	−8.42824	−0.268274
$988$	0	0
$989$	13.1658	0.418648
$990$	0	0
$991$	−57.3144	−1.82065	−0.910326	−	0.413892i	$-0.864169\pi$
$991$	−57.3144	−1.82065	−0.910326	+	0.413892i	$0.864169\pi$
$992$	0	0
$993$	−50.1292	−1.59080
$994$	0	0
$995$	−20.6916	−0.655969
$996$	0	0
$997$	−39.3642	−1.24668	−0.623339	−	0.781952i	$-0.714224\pi$
$997$	−39.3642	−1.24668	−0.623339	+	0.781952i	$0.714224\pi$
$998$	0	0
$999$	−47.6271	−1.50685

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3856.2.a.n.1.4		12
4.3	odd	2		241.2.a.b.1.6	✓	12
12.11	even	2		2169.2.a.h.1.7		12
20.19	odd	2		6025.2.a.h.1.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.6	✓	12	4.3	odd	2
2169.2.a.h.1.7		12	12.11	even	2
3856.2.a.n.1.4		12	1.1	even	1	trivial
6025.2.a.h.1.7		12	20.19	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:	\( N \)	\(=\)	\( 3856 = 2^{4} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3856.a (trivial)

\(f(q)\)	\(=\)	\(q-1.81824 q^{3} +4.31963 q^{5} -0.690569 q^{7} +0.306010 q^{9} +O(q^{10})\)	\(q-1.81824 q^{3} +4.31963 q^{5} -0.690569 q^{7} +0.306010 q^{9} +2.95833 q^{11} -1.93470 q^{13} -7.85414 q^{15} -2.07477 q^{17} -3.74689 q^{19} +1.25562 q^{21} -4.34832 q^{23} +13.6592 q^{25} +4.89833 q^{27} -8.10772 q^{29} +2.80197 q^{31} -5.37896 q^{33} -2.98300 q^{35} -9.72312 q^{37} +3.51777 q^{39} -4.09401 q^{41} -3.02779 q^{43} +1.32185 q^{45} -6.71240 q^{47} -6.52312 q^{49} +3.77244 q^{51} -0.0484656 q^{53} +12.7789 q^{55} +6.81275 q^{57} -4.50676 q^{59} -9.62232 q^{61} -0.211321 q^{63} -8.35721 q^{65} +0.964160 q^{67} +7.90631 q^{69} -7.76289 q^{71} +16.4250 q^{73} -24.8358 q^{75} -2.04293 q^{77} +6.83310 q^{79} -9.82439 q^{81} +9.15477 q^{83} -8.96225 q^{85} +14.7418 q^{87} -2.36597 q^{89} +1.33605 q^{91} -5.09467 q^{93} -16.1852 q^{95} -5.25335 q^{97} +0.905280 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9	\( 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9} - 22 q^{11} - 5 q^{13} - 13 q^{15} - 4 q^{17} + 6 q^{19} - 14 q^{21} - 32 q^{23} + 4 q^{25} + 5 q^{27} + 6 q^{29} - 8 q^{31} - 24 q^{33} - 15 q^{35} - 8 q^{37} - 31 q^{39} - q^{41} + 2 q^{43} - 15 q^{45} - 34 q^{47} - 9 q^{49} + 3 q^{51} + 5 q^{53} + 3 q^{55} - 22 q^{57} - 26 q^{59} - 26 q^{61} + 4 q^{63} - 25 q^{65} - 6 q^{67} - 2 q^{69} - 94 q^{71} - 22 q^{73} - 7 q^{77} - 9 q^{79} + 4 q^{81} + 8 q^{83} + 4 q^{85} - 4 q^{87} - 3 q^{89} + 20 q^{91} + 12 q^{93} - 33 q^{95} - 29 q^{97} - 36 q^{99}+O(q^{100}) \) 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9 - 22 * q^11 - 5 * q^13 - 13 * q^15 - 4 * q^17 + 6 * q^19 - 14 * q^21 - 32 * q^23 + 4 * q^25 + 5 * q^27 + 6 * q^29 - 8 * q^31 - 24 * q^33 - 15 * q^35 - 8 * q^37 - 31 * q^39 - q^41 + 2 * q^43 - 15 * q^45 - 34 * q^47 - 9 * q^49 + 3 * q^51 + 5 * q^53 + 3 * q^55 - 22 * q^57 - 26 * q^59 - 26 * q^61 + 4 * q^63 - 25 * q^65 - 6 * q^67 - 2 * q^69 - 94 * q^71 - 22 * q^73 - 7 * q^77 - 9 * q^79 + 4 * q^81 + 8 * q^83 + 4 * q^85 - 4 * q^87 - 3 * q^89 + 20 * q^91 + 12 * q^93 - 33 * q^95 - 29 * q^97 - 36 * q^99

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