LMFDB - Embedded newform 3856.2.a.n.1.5

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			1.5
Root			$2.49073$ 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)

$f(q)$	$=$	$q-1.22208 q^{3} -3.14843 q^{5} -0.136122 q^{7} -1.50653 q^{9} +O(q^{10})$	$q-1.22208 q^{3} -3.14843 q^{5} -0.136122 q^{7} -1.50653 q^{9} +0.905365 q^{11} -0.123706 q^{13} +3.84762 q^{15} +1.26034 q^{17} +2.13460 q^{19} +0.166352 q^{21} -6.64978 q^{23} +4.91264 q^{25} +5.50732 q^{27} +5.36862 q^{29} +9.78467 q^{31} -1.10642 q^{33} +0.428573 q^{35} +5.76688 q^{37} +0.151178 q^{39} +6.43642 q^{41} +3.18712 q^{43} +4.74322 q^{45} -12.9849 q^{47} -6.98147 q^{49} -1.54023 q^{51} +3.90862 q^{53} -2.85048 q^{55} -2.60864 q^{57} -8.15085 q^{59} -14.3712 q^{61} +0.205073 q^{63} +0.389481 q^{65} +4.89534 q^{67} +8.12653 q^{69} -4.32869 q^{71} +5.64935 q^{73} -6.00362 q^{75} -0.123241 q^{77} -1.43490 q^{79} -2.21077 q^{81} +11.7625 q^{83} -3.96811 q^{85} -6.56086 q^{87} -13.7381 q^{89} +0.0168392 q^{91} -11.9576 q^{93} -6.72063 q^{95} +13.6204 q^{97} -1.36396 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.22208	−0.705566	−0.352783	−	0.935705i	$-0.614765\pi$
$3$	−1.22208	−0.705566	−0.352783	+	0.935705i	$0.614765\pi$
$4$	0	0
$5$	−3.14843	−1.40802	−0.704011	−	0.710189i	$-0.748610\pi$
$5$	−3.14843	−1.40802	−0.704011	+	0.710189i	$0.748610\pi$
$6$	0	0
$7$	−0.136122	−0.0514495	−0.0257247	−	0.999669i	$-0.508189\pi$
$7$	−0.136122	−0.0514495	−0.0257247	+	0.999669i	$0.508189\pi$
$8$	0	0
$9$	−1.50653	−0.502177
$10$	0	0
$11$	0.905365	0.272978	0.136489	−	0.990642i	$-0.456418\pi$
$11$	0.905365	0.272978	0.136489	+	0.990642i	$0.456418\pi$
$12$	0	0
$13$	−0.123706	−0.0343099	−0.0171549	−	0.999853i	$-0.505461\pi$
$13$	−0.123706	−0.0343099	−0.0171549	+	0.999853i	$0.505461\pi$
$14$	0	0
$15$	3.84762	0.993452
$16$	0	0
$17$	1.26034	0.305678	0.152839	−	0.988251i	$-0.451158\pi$
$17$	1.26034	0.305678	0.152839	+	0.988251i	$0.451158\pi$
$18$	0	0
$19$	2.13460	0.489710	0.244855	−	0.969560i	$-0.421260\pi$
$19$	2.13460	0.489710	0.244855	+	0.969560i	$0.421260\pi$
$20$	0	0
$21$	0.166352	0.0363010
$22$	0	0
$23$	−6.64978	−1.38657	−0.693287	−	0.720661i	$-0.743838\pi$
$23$	−6.64978	−1.38657	−0.693287	+	0.720661i	$0.743838\pi$
$24$	0	0
$25$	4.91264	0.982528
$26$	0	0
$27$	5.50732	1.05988
$28$	0	0
$29$	5.36862	0.996928	0.498464	−	0.866911i	$-0.333898\pi$
$29$	5.36862	0.996928	0.498464	+	0.866911i	$0.333898\pi$
$30$	0	0
$31$	9.78467	1.75738	0.878689	−	0.477395i	$-0.158419\pi$
$31$	9.78467	1.75738	0.878689	+	0.477395i	$0.158419\pi$
$32$	0	0
$33$	−1.10642	−0.192604
$34$	0	0
$35$	0.428573	0.0724420
$36$	0	0
$37$	5.76688	0.948070	0.474035	−	0.880506i	$-0.342797\pi$
$37$	5.76688	0.948070	0.474035	+	0.880506i	$0.342797\pi$
$38$	0	0
$39$	0.151178	0.0242079
$40$	0	0
$41$	6.43642	1.00520	0.502600	−	0.864519i	$-0.332377\pi$
$41$	6.43642	1.00520	0.502600	+	0.864519i	$0.332377\pi$
$42$	0	0
$43$	3.18712	0.486032	0.243016	−	0.970022i	$-0.421863\pi$
$43$	3.18712	0.486032	0.243016	+	0.970022i	$0.421863\pi$
$44$	0	0
$45$	4.74322	0.707077
$46$	0	0
$47$	−12.9849	−1.89404	−0.947022	−	0.321168i	$-0.895925\pi$
$47$	−12.9849	−1.89404	−0.947022	+	0.321168i	$0.895925\pi$
$48$	0	0
$49$	−6.98147	−0.997353
$50$	0	0
$51$	−1.54023	−0.215676
$52$	0	0
$53$	3.90862	0.536890	0.268445	−	0.963295i	$-0.413490\pi$
$53$	3.90862	0.536890	0.268445	+	0.963295i	$0.413490\pi$
$54$	0	0
$55$	−2.85048	−0.384359
$56$	0	0
$57$	−2.60864	−0.345522
$58$	0	0
$59$	−8.15085	−1.06115	−0.530575	−	0.847638i	$-0.678024\pi$
$59$	−8.15085	−1.06115	−0.530575	+	0.847638i	$0.678024\pi$
$60$	0	0
$61$	−14.3712	−1.84004	−0.920021	−	0.391870i	$-0.871828\pi$
$61$	−14.3712	−1.84004	−0.920021	+	0.391870i	$0.871828\pi$
$62$	0	0
$63$	0.205073	0.0258368
$64$	0	0
$65$	0.389481	0.0483091
$66$	0	0
$67$	4.89534	0.598062	0.299031	−	0.954243i	$-0.403337\pi$
$67$	4.89534	0.598062	0.299031	+	0.954243i	$0.403337\pi$
$68$	0	0
$69$	8.12653	0.978319
$70$	0	0
$71$	−4.32869	−0.513720	−0.256860	−	0.966449i	$-0.582688\pi$
$71$	−4.32869	−0.513720	−0.256860	+	0.966449i	$0.582688\pi$
$72$	0	0
$73$	5.64935	0.661206	0.330603	−	0.943770i	$-0.392748\pi$
$73$	5.64935	0.661206	0.330603	+	0.943770i	$0.392748\pi$
$74$	0	0
$75$	−6.00362	−0.693238
$76$	0	0
$77$	−0.123241	−0.0140446
$78$	0	0
$79$	−1.43490	−0.161439	−0.0807195	−	0.996737i	$-0.525722\pi$
$79$	−1.43490	−0.161439	−0.0807195	+	0.996737i	$0.525722\pi$
$80$	0	0
$81$	−2.21077	−0.245641
$82$	0	0
$83$	11.7625	1.29110	0.645549	−	0.763718i	$-0.276628\pi$
$83$	11.7625	1.29110	0.645549	+	0.763718i	$0.276628\pi$
$84$	0	0
$85$	−3.96811	−0.430402
$86$	0	0
$87$	−6.56086	−0.703398
$88$	0	0
$89$	−13.7381	−1.45624	−0.728118	−	0.685452i	$-0.759605\pi$
$89$	−13.7381	−1.45624	−0.728118	+	0.685452i	$0.759605\pi$
$90$	0	0
$91$	0.0168392	0.00176523
$92$	0	0
$93$	−11.9576	−1.23995
$94$	0	0
$95$	−6.72063	−0.689522
$96$	0	0
$97$	13.6204	1.38294	0.691472	−	0.722404i	$-0.256963\pi$
$97$	13.6204	1.38294	0.691472	+	0.722404i	$0.256963\pi$
$98$	0	0
$99$	−1.36396	−0.137083
$100$	0	0
$101$	−0.0787660	−0.00783751	−0.00391876	−	0.999992i	$-0.501247\pi$
$101$	−0.0787660	−0.00783751	−0.00391876	+	0.999992i	$0.501247\pi$
$102$	0	0
$103$	6.36701	0.627360	0.313680	−	0.949529i	$-0.398438\pi$
$103$	6.36701	0.627360	0.313680	+	0.949529i	$0.398438\pi$
$104$	0	0
$105$	−0.523748	−0.0511126
$106$	0	0
$107$	1.99570	0.192932	0.0964660	−	0.995336i	$-0.469246\pi$
$107$	1.99570	0.192932	0.0964660	+	0.995336i	$0.469246\pi$
$108$	0	0
$109$	14.8223	1.41972	0.709861	−	0.704341i	$-0.248758\pi$
$109$	14.8223	1.41972	0.709861	+	0.704341i	$0.248758\pi$
$110$	0	0
$111$	−7.04756	−0.668925
$112$	0	0
$113$	−12.3669	−1.16338	−0.581690	−	0.813410i	$-0.697608\pi$
$113$	−12.3669	−1.16338	−0.581690	+	0.813410i	$0.697608\pi$
$114$	0	0
$115$	20.9364	1.95233
$116$	0	0
$117$	0.186367	0.0172296
$118$	0	0
$119$	−0.171561	−0.0157270
$120$	0	0
$121$	−10.1803	−0.925483
$122$	0	0
$123$	−7.86579	−0.709234
$124$	0	0
$125$	0.275046	0.0246009
$126$	0	0
$127$	−15.9678	−1.41691	−0.708456	−	0.705754i	$-0.750608\pi$
$127$	−15.9678	−1.41691	−0.708456	+	0.705754i	$0.750608\pi$
$128$	0	0
$129$	−3.89491	−0.342927
$130$	0	0
$131$	12.1390	1.06059	0.530293	−	0.847814i	$-0.322082\pi$
$131$	12.1390	1.06059	0.530293	+	0.847814i	$0.322082\pi$
$132$	0	0
$133$	−0.290566	−0.0251953
$134$	0	0
$135$	−17.3394	−1.49234
$136$	0	0
$137$	0.846052	0.0722831	0.0361416	−	0.999347i	$-0.488493\pi$
$137$	0.846052	0.0722831	0.0361416	+	0.999347i	$0.488493\pi$
$138$	0	0
$139$	−15.6472	−1.32717	−0.663587	−	0.748099i	$-0.730967\pi$
$139$	−15.6472	−1.32717	−0.663587	+	0.748099i	$0.730967\pi$
$140$	0	0
$141$	15.8685	1.33637
$142$	0	0
$143$	−0.111999	−0.00936584
$144$	0	0
$145$	−16.9027	−1.40370
$146$	0	0
$147$	8.53188	0.703698
$148$	0	0
$149$	−0.542212	−0.0444198	−0.0222099	−	0.999753i	$-0.507070\pi$
$149$	−0.542212	−0.0444198	−0.0222099	+	0.999753i	$0.507070\pi$
$150$	0	0
$151$	−8.53046	−0.694199	−0.347100	−	0.937828i	$-0.612833\pi$
$151$	−8.53046	−0.694199	−0.347100	+	0.937828i	$0.612833\pi$
$152$	0	0
$153$	−1.89875	−0.153505
$154$	0	0
$155$	−30.8064	−2.47443
$156$	0	0
$157$	−16.2204	−1.29453	−0.647263	−	0.762267i	$-0.724086\pi$
$157$	−16.2204	−1.29453	−0.647263	+	0.762267i	$0.724086\pi$
$158$	0	0
$159$	−4.77662	−0.378811
$160$	0	0
$161$	0.905184	0.0713385
$162$	0	0
$163$	−1.36363	−0.106808	−0.0534039	−	0.998573i	$-0.517007\pi$
$163$	−1.36363	−0.106808	−0.0534039	+	0.998573i	$0.517007\pi$
$164$	0	0
$165$	3.48351	0.271191
$166$	0	0
$167$	−13.4045	−1.03727	−0.518636	−	0.854995i	$-0.673560\pi$
$167$	−13.4045	−1.03727	−0.518636	+	0.854995i	$0.673560\pi$
$168$	0	0
$169$	−12.9847	−0.998823
$170$	0	0
$171$	−3.21584	−0.245921
$172$	0	0
$173$	−13.6007	−1.03404	−0.517022	−	0.855972i	$-0.672959\pi$
$173$	−13.6007	−1.03404	−0.517022	+	0.855972i	$0.672959\pi$
$174$	0	0
$175$	−0.668721	−0.0505505
$176$	0	0
$177$	9.96096	0.748711
$178$	0	0
$179$	−23.9070	−1.78689	−0.893445	−	0.449172i	$-0.851719\pi$
$179$	−23.9070	−1.78689	−0.893445	+	0.449172i	$0.851719\pi$
$180$	0	0
$181$	−11.6044	−0.862545	−0.431273	−	0.902222i	$-0.641935\pi$
$181$	−11.6044	−0.862545	−0.431273	+	0.902222i	$0.641935\pi$
$182$	0	0
$183$	17.5627	1.29827
$184$	0	0
$185$	−18.1567	−1.33490
$186$	0	0
$187$	1.14107	0.0834433
$188$	0	0
$189$	−0.749670	−0.0545305
$190$	0	0
$191$	4.85929	0.351606	0.175803	−	0.984425i	$-0.443748\pi$
$191$	4.85929	0.351606	0.175803	+	0.984425i	$0.443748\pi$
$192$	0	0
$193$	25.6451	1.84597	0.922987	−	0.384831i	$-0.125740\pi$
$193$	25.6451	1.84597	0.922987	+	0.384831i	$0.125740\pi$
$194$	0	0
$195$	−0.475975	−0.0340852
$196$	0	0
$197$	1.46990	0.104726	0.0523629	−	0.998628i	$-0.483325\pi$
$197$	1.46990	0.104726	0.0523629	+	0.998628i	$0.483325\pi$
$198$	0	0
$199$	−7.46714	−0.529332	−0.264666	−	0.964340i	$-0.585262\pi$
$199$	−7.46714	−0.529332	−0.264666	+	0.964340i	$0.585262\pi$
$200$	0	0
$201$	−5.98248	−0.421972
$202$	0	0
$203$	−0.730790	−0.0512914
$204$	0	0
$205$	−20.2646	−1.41534
$206$	0	0
$207$	10.0181	0.696306
$208$	0	0
$209$	1.93259	0.133680
$210$	0	0
$211$	20.1549	1.38752	0.693759	−	0.720208i	$-0.255953\pi$
$211$	20.1549	1.38752	0.693759	+	0.720208i	$0.255953\pi$
$212$	0	0
$213$	5.28998	0.362463
$214$	0	0
$215$	−10.0345	−0.684344
$216$	0	0
$217$	−1.33191	−0.0904162
$218$	0	0
$219$	−6.90393	−0.466524
$220$	0	0
$221$	−0.155912	−0.0104878
$222$	0	0
$223$	8.91317	0.596870	0.298435	−	0.954430i	$-0.403535\pi$
$223$	8.91317	0.596870	0.298435	+	0.954430i	$0.403535\pi$
$224$	0	0
$225$	−7.40105	−0.493403
$226$	0	0
$227$	17.2725	1.14642	0.573209	−	0.819409i	$-0.305698\pi$
$227$	17.2725	1.14642	0.573209	+	0.819409i	$0.305698\pi$
$228$	0	0
$229$	−12.9392	−0.855044	−0.427522	−	0.904005i	$-0.640613\pi$
$229$	−12.9392	−0.855044	−0.427522	+	0.904005i	$0.640613\pi$
$230$	0	0
$231$	0.150609	0.00990936
$232$	0	0
$233$	−24.8301	−1.62668	−0.813338	−	0.581791i	$-0.802352\pi$
$233$	−24.8301	−1.62668	−0.813338	+	0.581791i	$0.802352\pi$
$234$	0	0
$235$	40.8822	2.66686
$236$	0	0
$237$	1.75356	0.113906
$238$	0	0
$239$	−7.52122	−0.486507	−0.243254	−	0.969963i	$-0.578215\pi$
$239$	−7.52122	−0.486507	−0.243254	+	0.969963i	$0.578215\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	0	0
$243$	−13.8202	−0.886569
$244$	0	0
$245$	21.9807	1.40430
$246$	0	0
$247$	−0.264062	−0.0168019
$248$	0	0
$249$	−14.3746	−0.910955
$250$	0	0
$251$	30.0279	1.89534	0.947671	−	0.319250i	$-0.103431\pi$
$251$	30.0279	1.89534	0.947671	+	0.319250i	$0.103431\pi$
$252$	0	0
$253$	−6.02048	−0.378504
$254$	0	0
$255$	4.84932	0.303677
$256$	0	0
$257$	−9.93458	−0.619702	−0.309851	−	0.950785i	$-0.600279\pi$
$257$	−9.93458	−0.619702	−0.309851	+	0.950785i	$0.600279\pi$
$258$	0	0
$259$	−0.785002	−0.0487777
$260$	0	0
$261$	−8.08800	−0.500634
$262$	0	0
$263$	−18.4575	−1.13814	−0.569069	−	0.822290i	$-0.692696\pi$
$263$	−18.4575	−1.13814	−0.569069	+	0.822290i	$0.692696\pi$
$264$	0	0
$265$	−12.3060	−0.755953
$266$	0	0
$267$	16.7890	1.02747
$268$	0	0
$269$	−1.14902	−0.0700569	−0.0350285	−	0.999386i	$-0.511152\pi$
$269$	−1.14902	−0.0700569	−0.0350285	+	0.999386i	$0.511152\pi$
$270$	0	0
$271$	−12.2034	−0.741303	−0.370651	−	0.928772i	$-0.620866\pi$
$271$	−12.2034	−0.741303	−0.370651	+	0.928772i	$0.620866\pi$
$272$	0	0
$273$	−0.0205787	−0.00124548
$274$	0	0
$275$	4.44773	0.268208
$276$	0	0
$277$	−4.31327	−0.259159	−0.129580	−	0.991569i	$-0.541363\pi$
$277$	−4.31327	−0.259159	−0.129580	+	0.991569i	$0.541363\pi$
$278$	0	0
$279$	−14.7409	−0.882515
$280$	0	0
$281$	−17.1399	−1.02248	−0.511240	−	0.859438i	$-0.670814\pi$
$281$	−17.1399	−1.02248	−0.511240	+	0.859438i	$0.670814\pi$
$282$	0	0
$283$	−13.8905	−0.825704	−0.412852	−	0.910798i	$-0.635467\pi$
$283$	−13.8905	−0.825704	−0.412852	+	0.910798i	$0.635467\pi$
$284$	0	0
$285$	8.21312	0.486503
$286$	0	0
$287$	−0.876142	−0.0517170
$288$	0	0
$289$	−15.4115	−0.906561
$290$	0	0
$291$	−16.6452	−0.975757
$292$	0	0
$293$	−0.325090	−0.0189920	−0.00949598	−	0.999955i	$-0.503023\pi$
$293$	−0.325090	−0.0189920	−0.00949598	+	0.999955i	$0.503023\pi$
$294$	0	0
$295$	25.6624	1.49412
$296$	0	0
$297$	4.98614	0.289325
$298$	0	0
$299$	0.822618	0.0475732
$300$	0	0
$301$	−0.433839	−0.0250061
$302$	0	0
$303$	0.0962580	0.00552988
$304$	0	0
$305$	45.2467	2.59082
$306$	0	0
$307$	14.3600	0.819566	0.409783	−	0.912183i	$-0.365604\pi$
$307$	14.3600	0.819566	0.409783	+	0.912183i	$0.365604\pi$
$308$	0	0
$309$	−7.78096	−0.442644
$310$	0	0
$311$	11.2072	0.635502	0.317751	−	0.948174i	$-0.397072\pi$
$311$	11.2072	0.635502	0.317751	+	0.948174i	$0.397072\pi$
$312$	0	0
$313$	19.3860	1.09576	0.547879	−	0.836557i	$-0.315435\pi$
$313$	19.3860	1.09576	0.547879	+	0.836557i	$0.315435\pi$
$314$	0	0
$315$	−0.645658	−0.0363787
$316$	0	0
$317$	5.19578	0.291824	0.145912	−	0.989298i	$-0.453388\pi$
$317$	5.19578	0.291824	0.145912	+	0.989298i	$0.453388\pi$
$318$	0	0
$319$	4.86056	0.272139
$320$	0	0
$321$	−2.43890	−0.136126
$322$	0	0
$323$	2.69032	0.149693
$324$	0	0
$325$	−0.607724	−0.0337104
$326$	0	0
$327$	−18.1140	−1.00171
$328$	0	0
$329$	1.76754	0.0974476
$330$	0	0
$331$	−0.0173250	−0.000952268 0	−0.000476134	−	1.00000i	$-0.500152\pi$
$331$	−0.0173250	−0.000952268 0	−0.000476134		1.00000i	$0.500152\pi$
$332$	0	0
$333$	−8.68799	−0.476099
$334$	0	0
$335$	−15.4127	−0.842084
$336$	0	0
$337$	0.347899	0.0189513	0.00947563	−	0.999955i	$-0.496984\pi$
$337$	0.347899	0.0189513	0.00947563	+	0.999955i	$0.496984\pi$
$338$	0	0
$339$	15.1133	0.820841
$340$	0	0
$341$	8.85870	0.479725
$342$	0	0
$343$	1.90319	0.102763
$344$	0	0
$345$	−25.5858	−1.37750
$346$	0	0
$347$	−6.37601	−0.342282	−0.171141	−	0.985247i	$-0.554745\pi$
$347$	−6.37601	−0.342282	−0.171141	+	0.985247i	$0.554745\pi$
$348$	0	0
$349$	−0.970720	−0.0519614	−0.0259807	−	0.999662i	$-0.508271\pi$
$349$	−0.970720	−0.0519614	−0.0259807	+	0.999662i	$0.508271\pi$
$350$	0	0
$351$	−0.681289	−0.0363645
$352$	0	0
$353$	30.5488	1.62595	0.812975	−	0.582299i	$-0.197847\pi$
$353$	30.5488	1.62595	0.812975	+	0.582299i	$0.197847\pi$
$354$	0	0
$355$	13.6286	0.723330
$356$	0	0
$357$	0.209660	0.0110964
$358$	0	0
$359$	−8.22124	−0.433901	−0.216950	−	0.976183i	$-0.569611\pi$
$359$	−8.22124	−0.433901	−0.216950	+	0.976183i	$0.569611\pi$
$360$	0	0
$361$	−14.4435	−0.760184
$362$	0	0
$363$	12.4411	0.652989
$364$	0	0
$365$	−17.7866	−0.930994
$366$	0	0
$367$	−25.1108	−1.31077	−0.655387	−	0.755294i	$-0.727494\pi$
$367$	−25.1108	−1.31077	−0.655387	+	0.755294i	$0.727494\pi$
$368$	0	0
$369$	−9.69667	−0.504789
$370$	0	0
$371$	−0.532051	−0.0276227
$372$	0	0
$373$	−27.0231	−1.39920	−0.699601	−	0.714534i	$-0.746639\pi$
$373$	−27.0231	−1.39920	−0.699601	+	0.714534i	$0.746639\pi$
$374$	0	0
$375$	−0.336127	−0.0173575
$376$	0	0
$377$	−0.664131	−0.0342045
$378$	0	0
$379$	−22.9961	−1.18123	−0.590616	−	0.806953i	$-0.701115\pi$
$379$	−22.9961	−1.18123	−0.590616	+	0.806953i	$0.701115\pi$
$380$	0	0
$381$	19.5139	0.999725
$382$	0	0
$383$	−3.72670	−0.190425	−0.0952126	−	0.995457i	$-0.530353\pi$
$383$	−3.72670	−0.190425	−0.0952126	+	0.995457i	$0.530353\pi$
$384$	0	0
$385$	0.388015	0.0197751
$386$	0	0
$387$	−4.80151	−0.244074
$388$	0	0
$389$	4.72525	0.239580	0.119790	−	0.992799i	$-0.461778\pi$
$389$	4.72525	0.239580	0.119790	+	0.992799i	$0.461778\pi$
$390$	0	0
$391$	−8.38100	−0.423845
$392$	0	0
$393$	−14.8347	−0.748313
$394$	0	0
$395$	4.51769	0.227310
$396$	0	0
$397$	32.2227	1.61721	0.808606	−	0.588350i	$-0.200222\pi$
$397$	32.2227	1.61721	0.808606	+	0.588350i	$0.200222\pi$
$398$	0	0
$399$	0.355094	0.0177769
$400$	0	0
$401$	−6.02370	−0.300809	−0.150405	−	0.988625i	$-0.548058\pi$
$401$	−6.02370	−0.300809	−0.150405	+	0.988625i	$0.548058\pi$
$402$	0	0
$403$	−1.21042	−0.0602955
$404$	0	0
$405$	6.96045	0.345868
$406$	0	0
$407$	5.22113	0.258802
$408$	0	0
$409$	18.5405	0.916767	0.458384	−	0.888754i	$-0.348429\pi$
$409$	18.5405	0.916767	0.458384	+	0.888754i	$0.348429\pi$
$410$	0	0
$411$	−1.03394	−0.0510005
$412$	0	0
$413$	1.10951	0.0545956
$414$	0	0
$415$	−37.0334	−1.81790
$416$	0	0
$417$	19.1220	0.936409
$418$	0	0
$419$	−29.7394	−1.45286	−0.726432	−	0.687238i	$-0.758823\pi$
$419$	−29.7394	−1.45286	−0.726432	+	0.687238i	$0.758823\pi$
$420$	0	0
$421$	2.65516	0.129405	0.0647023	−	0.997905i	$-0.479390\pi$
$421$	2.65516	0.129405	0.0647023	+	0.997905i	$0.479390\pi$
$422$	0	0
$423$	19.5622	0.951146
$424$	0	0
$425$	6.19161	0.300337
$426$	0	0
$427$	1.95624	0.0946691
$428$	0	0
$429$	0.136871	0.00660822
$430$	0	0
$431$	−27.9512	−1.34636	−0.673181	−	0.739478i	$-0.735073\pi$
$431$	−27.9512	−1.34636	−0.673181	+	0.739478i	$0.735073\pi$
$432$	0	0
$433$	13.5815	0.652687	0.326344	−	0.945251i	$-0.394183\pi$
$433$	13.5815	0.652687	0.326344	+	0.945251i	$0.394183\pi$
$434$	0	0
$435$	20.6564	0.990400
$436$	0	0
$437$	−14.1946	−0.679019
$438$	0	0
$439$	−27.3113	−1.30350	−0.651748	−	0.758435i	$-0.725964\pi$
$439$	−27.3113	−1.30350	−0.651748	+	0.758435i	$0.725964\pi$
$440$	0	0
$441$	10.5178	0.500848
$442$	0	0
$443$	−4.06001	−0.192897	−0.0964484	−	0.995338i	$-0.530748\pi$
$443$	−4.06001	−0.192897	−0.0964484	+	0.995338i	$0.530748\pi$
$444$	0	0
$445$	43.2535	2.05041
$446$	0	0
$447$	0.662624	0.0313410
$448$	0	0
$449$	−1.53571	−0.0724746	−0.0362373	−	0.999343i	$-0.511537\pi$
$449$	−1.53571	−0.0724746	−0.0362373	+	0.999343i	$0.511537\pi$
$450$	0	0
$451$	5.82731	0.274397
$452$	0	0
$453$	10.4249	0.489803
$454$	0	0
$455$	−0.0530171	−0.00248548
$456$	0	0
$457$	−4.50705	−0.210831	−0.105415	−	0.994428i	$-0.533617\pi$
$457$	−4.50705	−0.210831	−0.105415	+	0.994428i	$0.533617\pi$
$458$	0	0
$459$	6.94111	0.323983
$460$	0	0
$461$	17.5820	0.818875	0.409438	−	0.912338i	$-0.365725\pi$
$461$	17.5820	0.818875	0.409438	+	0.912338i	$0.365725\pi$
$462$	0	0
$463$	42.1082	1.95693	0.978466	−	0.206407i	$-0.0661772\pi$
$463$	42.1082	1.95693	0.978466	+	0.206407i	$0.0661772\pi$
$464$	0	0
$465$	37.6477	1.74587
$466$	0	0
$467$	−10.6368	−0.492210	−0.246105	−	0.969243i	$-0.579151\pi$
$467$	−10.6368	−0.492210	−0.246105	+	0.969243i	$0.579151\pi$
$468$	0	0
$469$	−0.666367	−0.0307699
$470$	0	0
$471$	19.8225	0.913373
$472$	0	0
$473$	2.88551	0.132676
$474$	0	0
$475$	10.4865	0.481154
$476$	0	0
$477$	−5.88845	−0.269614
$478$	0	0
$479$	−24.3303	−1.11168	−0.555839	−	0.831290i	$-0.687603\pi$
$479$	−24.3303	−1.11168	−0.555839	+	0.831290i	$0.687603\pi$
$480$	0	0
$481$	−0.713398	−0.0325282
$482$	0	0
$483$	−1.10620	−0.0503340
$484$	0	0
$485$	−42.8830	−1.94722
$486$	0	0
$487$	−15.1777	−0.687768	−0.343884	−	0.939012i	$-0.611743\pi$
$487$	−15.1777	−0.687768	−0.343884	+	0.939012i	$0.611743\pi$
$488$	0	0
$489$	1.66646	0.0753598
$490$	0	0
$491$	−10.0194	−0.452169	−0.226084	−	0.974108i	$-0.572592\pi$
$491$	−10.0194	−0.452169	−0.226084	+	0.974108i	$0.572592\pi$
$492$	0	0
$493$	6.76630	0.304739
$494$	0	0
$495$	4.29434	0.193016
$496$	0	0
$497$	0.589231	0.0264306
$498$	0	0
$499$	7.82989	0.350514	0.175257	−	0.984523i	$-0.443924\pi$
$499$	7.82989	0.350514	0.175257	+	0.984523i	$0.443924\pi$
$500$	0	0
$501$	16.3813	0.731863
$502$	0	0
$503$	7.39455	0.329707	0.164853	−	0.986318i	$-0.447285\pi$
$503$	7.39455	0.329707	0.164853	+	0.986318i	$0.447285\pi$
$504$	0	0
$505$	0.247990	0.0110354
$506$	0	0
$507$	15.8683	0.704735
$508$	0	0
$509$	13.7526	0.609572	0.304786	−	0.952421i	$-0.401415\pi$
$509$	13.7526	0.609572	0.304786	+	0.952421i	$0.401415\pi$
$510$	0	0
$511$	−0.769003	−0.0340187
$512$	0	0
$513$	11.7559	0.519036
$514$	0	0
$515$	−20.0461	−0.883337
$516$	0	0
$517$	−11.7561	−0.517032
$518$	0	0
$519$	16.6211	0.729585
$520$	0	0
$521$	−14.2825	−0.625730	−0.312865	−	0.949798i	$-0.601289\pi$
$521$	−14.2825	−0.625730	−0.312865	+	0.949798i	$0.601289\pi$
$522$	0	0
$523$	29.4674	1.28852	0.644260	−	0.764806i	$-0.277165\pi$
$523$	29.4674	1.28852	0.644260	+	0.764806i	$0.277165\pi$
$524$	0	0
$525$	0.817227	0.0356667
$526$	0	0
$527$	12.3320	0.537192
$528$	0	0
$529$	21.2195	0.922589
$530$	0	0
$531$	12.2795	0.532886
$532$	0	0
$533$	−0.796224	−0.0344883
$534$	0	0
$535$	−6.28334	−0.271653
$536$	0	0
$537$	29.2161	1.26077
$538$	0	0
$539$	−6.32078	−0.272255
$540$	0	0
$541$	31.6583	1.36110	0.680549	−	0.732703i	$-0.261741\pi$
$541$	31.6583	1.36110	0.680549	+	0.732703i	$0.261741\pi$
$542$	0	0
$543$	14.1814	0.608582
$544$	0	0
$545$	−46.6672	−1.99900
$546$	0	0
$547$	31.5186	1.34764	0.673820	−	0.738896i	$-0.264652\pi$
$547$	31.5186	1.34764	0.673820	+	0.738896i	$0.264652\pi$
$548$	0	0
$549$	21.6506	0.924027
$550$	0	0
$551$	11.4598	0.488205
$552$	0	0
$553$	0.195322	0.00830595
$554$	0	0
$555$	22.1888	0.941862
$556$	0	0
$557$	−32.3589	−1.37109	−0.685545	−	0.728030i	$-0.740436\pi$
$557$	−32.3589	−1.37109	−0.685545	+	0.728030i	$0.740436\pi$
$558$	0	0
$559$	−0.394267	−0.0166757
$560$	0	0
$561$	−1.39447	−0.0588747
$562$	0	0
$563$	−20.8428	−0.878419	−0.439209	−	0.898385i	$-0.644741\pi$
$563$	−20.8428	−0.878419	−0.439209	+	0.898385i	$0.644741\pi$
$564$	0	0
$565$	38.9364	1.63807
$566$	0	0
$567$	0.300935	0.0126381
$568$	0	0
$569$	26.6787	1.11843	0.559215	−	0.829022i	$-0.311103\pi$
$569$	26.6787	1.11843	0.559215	+	0.829022i	$0.311103\pi$
$570$	0	0
$571$	−0.500659	−0.0209519	−0.0104760	−	0.999945i	$-0.503335\pi$
$571$	−0.500659	−0.0209519	−0.0104760	+	0.999945i	$0.503335\pi$
$572$	0	0
$573$	−5.93842	−0.248081
$574$	0	0
$575$	−32.6680	−1.36235
$576$	0	0
$577$	−9.59281	−0.399354	−0.199677	−	0.979862i	$-0.563989\pi$
$577$	−9.59281	−0.399354	−0.199677	+	0.979862i	$0.563989\pi$
$578$	0	0
$579$	−31.3402	−1.30246
$580$	0	0
$581$	−1.60114	−0.0664263
$582$	0	0
$583$	3.53873	0.146559
$584$	0	0
$585$	−0.586765	−0.0242597
$586$	0	0
$587$	−19.6676	−0.811770	−0.405885	−	0.913924i	$-0.633037\pi$
$587$	−19.6676	−0.811770	−0.405885	+	0.913924i	$0.633037\pi$
$588$	0	0
$589$	20.8863	0.860605
$590$	0	0
$591$	−1.79632	−0.0738909
$592$	0	0
$593$	35.9622	1.47679	0.738394	−	0.674369i	$-0.235584\pi$
$593$	35.9622	1.47679	0.738394	+	0.674369i	$0.235584\pi$
$594$	0	0
$595$	0.540148	0.0221439
$596$	0	0
$597$	9.12541	0.373478
$598$	0	0
$599$	8.60123	0.351437	0.175718	−	0.984440i	$-0.443775\pi$
$599$	8.60123	0.351437	0.175718	+	0.984440i	$0.443775\pi$
$600$	0	0
$601$	36.8567	1.50342	0.751708	−	0.659496i	$-0.229230\pi$
$601$	36.8567	1.50342	0.751708	+	0.659496i	$0.229230\pi$
$602$	0	0
$603$	−7.37499	−0.300333
$604$	0	0
$605$	32.0521	1.30310
$606$	0	0
$607$	38.6665	1.56942	0.784712	−	0.619861i	$-0.212811\pi$
$607$	38.6665	1.56942	0.784712	+	0.619861i	$0.212811\pi$
$608$	0	0
$609$	0.893080	0.0361894
$610$	0	0
$611$	1.60631	0.0649845
$612$	0	0
$613$	−38.3341	−1.54830	−0.774150	−	0.633002i	$-0.781822\pi$
$613$	−38.3341	−1.54830	−0.774150	+	0.633002i	$0.781822\pi$
$614$	0	0
$615$	24.7649	0.998618
$616$	0	0
$617$	−44.3829	−1.78679	−0.893395	−	0.449272i	$-0.851683\pi$
$617$	−44.3829	−1.78679	−0.893395	+	0.449272i	$0.851683\pi$
$618$	0	0
$619$	15.6150	0.627619	0.313810	−	0.949486i	$-0.398395\pi$
$619$	15.6150	0.627619	0.313810	+	0.949486i	$0.398395\pi$
$620$	0	0
$621$	−36.6225	−1.46961
$622$	0	0
$623$	1.87007	0.0749226
$624$	0	0
$625$	−25.4292	−1.01717
$626$	0	0
$627$	−2.36177	−0.0943200
$628$	0	0
$629$	7.26825	0.289804
$630$	0	0
$631$	7.52472	0.299555	0.149777	−	0.988720i	$-0.452144\pi$
$631$	7.52472	0.299555	0.149777	+	0.988720i	$0.452144\pi$
$632$	0	0
$633$	−24.6307	−0.978984
$634$	0	0
$635$	50.2736	1.99505
$636$	0	0
$637$	0.863650	0.0342191
$638$	0	0
$639$	6.52130	0.257979
$640$	0	0
$641$	2.42509	0.0957851	0.0478926	−	0.998852i	$-0.484749\pi$
$641$	2.42509	0.0957851	0.0478926	+	0.998852i	$0.484749\pi$
$642$	0	0
$643$	−12.1224	−0.478061	−0.239030	−	0.971012i	$-0.576830\pi$
$643$	−12.1224	−0.478061	−0.239030	+	0.971012i	$0.576830\pi$
$644$	0	0
$645$	12.2629	0.482850
$646$	0	0
$647$	−12.4628	−0.489964	−0.244982	−	0.969528i	$-0.578782\pi$
$647$	−12.4628	−0.489964	−0.244982	+	0.969528i	$0.578782\pi$
$648$	0	0
$649$	−7.37950	−0.289671
$650$	0	0
$651$	1.62770	0.0637945
$652$	0	0
$653$	−49.4695	−1.93589	−0.967945	−	0.251163i	$-0.919187\pi$
$653$	−49.4695	−1.93589	−0.967945	+	0.251163i	$0.919187\pi$
$654$	0	0
$655$	−38.2187	−1.49333
$656$	0	0
$657$	−8.51092	−0.332043
$658$	0	0
$659$	−19.5067	−0.759874	−0.379937	−	0.925012i	$-0.624054\pi$
$659$	−19.5067	−0.759874	−0.379937	+	0.925012i	$0.624054\pi$
$660$	0	0
$661$	33.2398	1.29288	0.646439	−	0.762966i	$-0.276258\pi$
$661$	33.2398	1.29288	0.646439	+	0.762966i	$0.276258\pi$
$662$	0	0
$663$	0.190536	0.00739982
$664$	0	0
$665$	0.914829	0.0354756
$666$	0	0
$667$	−35.7001	−1.38231
$668$	0	0
$669$	−10.8926	−0.421131
$670$	0	0
$671$	−13.0112	−0.502291
$672$	0	0
$673$	28.8764	1.11310	0.556552	−	0.830813i	$-0.312124\pi$
$673$	28.8764	1.11310	0.556552	+	0.830813i	$0.312124\pi$
$674$	0	0
$675$	27.0555	1.04137
$676$	0	0
$677$	29.4480	1.13178	0.565889	−	0.824482i	$-0.308533\pi$
$677$	29.4480	1.13178	0.565889	+	0.824482i	$0.308533\pi$
$678$	0	0
$679$	−1.85404	−0.0711517
$680$	0	0
$681$	−21.1083	−0.808873
$682$	0	0
$683$	−19.6959	−0.753643	−0.376821	−	0.926286i	$-0.622983\pi$
$683$	−19.6959	−0.753643	−0.376821	+	0.926286i	$0.622983\pi$
$684$	0	0
$685$	−2.66374	−0.101776
$686$	0	0
$687$	15.8126	0.603289
$688$	0	0
$689$	−0.483520	−0.0184206
$690$	0	0
$691$	−15.6587	−0.595683	−0.297842	−	0.954615i	$-0.596267\pi$
$691$	−15.6587	−0.595683	−0.297842	+	0.954615i	$0.596267\pi$
$692$	0	0
$693$	0.185666	0.00705286
$694$	0	0
$695$	49.2641	1.86869
$696$	0	0
$697$	8.11209	0.307267
$698$	0	0
$699$	30.3443	1.14773
$700$	0	0
$701$	32.6726	1.23403	0.617014	−	0.786952i	$-0.288342\pi$
$701$	32.6726	1.23403	0.617014	+	0.786952i	$0.288342\pi$
$702$	0	0
$703$	12.3100	0.464279
$704$	0	0
$705$	−49.9611	−1.88164
$706$	0	0
$707$	0.0107218	0.000403236 0
$708$	0	0
$709$	−5.63690	−0.211698	−0.105849	−	0.994382i	$-0.533756\pi$
$709$	−5.63690	−0.211698	−0.105849	+	0.994382i	$0.533756\pi$
$710$	0	0
$711$	2.16172	0.0810710
$712$	0	0
$713$	−65.0659	−2.43674
$714$	0	0
$715$	0.352622	0.0131873
$716$	0	0
$717$	9.19150	0.343263
$718$	0	0
$719$	−13.2196	−0.493006	−0.246503	−	0.969142i	$-0.579282\pi$
$719$	−13.2196	−0.493006	−0.246503	+	0.969142i	$0.579282\pi$
$720$	0	0
$721$	−0.866693	−0.0322773
$722$	0	0
$723$	−1.22208	−0.0454495
$724$	0	0
$725$	26.3741	0.979509
$726$	0	0
$727$	−21.6517	−0.803018	−0.401509	−	0.915855i	$-0.631514\pi$
$727$	−21.6517	−0.803018	−0.401509	+	0.915855i	$0.631514\pi$
$728$	0	0
$729$	23.5217	0.871173
$730$	0	0
$731$	4.01687	0.148569
$732$	0	0
$733$	−37.5385	−1.38651	−0.693257	−	0.720690i	$-0.743825\pi$
$733$	−37.5385	−1.38651	−0.693257	+	0.720690i	$0.743825\pi$
$734$	0	0
$735$	−26.8621	−0.990823
$736$	0	0
$737$	4.43208	0.163258
$738$	0	0
$739$	28.8751	1.06219	0.531093	−	0.847313i	$-0.321781\pi$
$739$	28.8751	1.06219	0.531093	+	0.847313i	$0.321781\pi$
$740$	0	0
$741$	0.322704	0.0118548
$742$	0	0
$743$	24.5272	0.899817	0.449909	−	0.893075i	$-0.351457\pi$
$743$	24.5272	0.899817	0.449909	+	0.893075i	$0.351457\pi$
$744$	0	0
$745$	1.70712	0.0625440
$746$	0	0
$747$	−17.7205	−0.648361
$748$	0	0
$749$	−0.271660	−0.00992624
$750$	0	0
$751$	−38.3355	−1.39888	−0.699442	−	0.714689i	$-0.746568\pi$
$751$	−38.3355	−1.39888	−0.699442	+	0.714689i	$0.746568\pi$
$752$	0	0
$753$	−36.6963	−1.33729
$754$	0	0
$755$	26.8576	0.977448
$756$	0	0
$757$	31.6239	1.14939	0.574694	−	0.818368i	$-0.305121\pi$
$757$	31.6239	1.14939	0.574694	+	0.818368i	$0.305121\pi$
$758$	0	0
$759$	7.35748	0.267060
$760$	0	0
$761$	−28.4924	−1.03285	−0.516424	−	0.856333i	$-0.672737\pi$
$761$	−28.4924	−1.03285	−0.516424	+	0.856333i	$0.672737\pi$
$762$	0	0
$763$	−2.01765	−0.0730440
$764$	0	0
$765$	5.97808	0.216138
$766$	0	0
$767$	1.00831	0.0364080
$768$	0	0
$769$	−8.72245	−0.314539	−0.157270	−	0.987556i	$-0.550269\pi$
$769$	−8.72245	−0.314539	−0.157270	+	0.987556i	$0.550269\pi$
$770$	0	0
$771$	12.1408	0.437241
$772$	0	0
$773$	−42.2655	−1.52019	−0.760093	−	0.649814i	$-0.774847\pi$
$773$	−42.2655	−1.52019	−0.760093	+	0.649814i	$0.774847\pi$
$774$	0	0
$775$	48.0685	1.72667
$776$	0	0
$777$	0.959332	0.0344158
$778$	0	0
$779$	13.7392	0.492256
$780$	0	0
$781$	−3.91904	−0.140234
$782$	0	0
$783$	29.5667	1.05663
$784$	0	0
$785$	51.0687	1.82272
$786$	0	0
$787$	−17.5127	−0.624261	−0.312130	−	0.950039i	$-0.601043\pi$
$787$	−17.5127	−0.624261	−0.312130	+	0.950039i	$0.601043\pi$
$788$	0	0
$789$	22.5565	0.803031
$790$	0	0
$791$	1.68341	0.0598553
$792$	0	0
$793$	1.77780	0.0631316
$794$	0	0
$795$	15.0389	0.533374
$796$	0	0
$797$	−44.6851	−1.58283	−0.791413	−	0.611281i	$-0.790654\pi$
$797$	−44.6851	−1.58283	−0.791413	+	0.611281i	$0.790654\pi$
$798$	0	0
$799$	−16.3654	−0.578968
$800$	0	0
$801$	20.6969	0.731289
$802$	0	0
$803$	5.11472	0.180495
$804$	0	0
$805$	−2.84991	−0.100446
$806$	0	0
$807$	1.40419	0.0494297
$808$	0	0
$809$	37.7898	1.32862	0.664309	−	0.747458i	$-0.268726\pi$
$809$	37.7898	1.32862	0.664309	+	0.747458i	$0.268726\pi$
$810$	0	0
$811$	−17.5453	−0.616097	−0.308049	−	0.951371i	$-0.599676\pi$
$811$	−17.5453	−0.616097	−0.308049	+	0.951371i	$0.599676\pi$
$812$	0	0
$813$	14.9135	0.523038
$814$	0	0
$815$	4.29330	0.150388
$816$	0	0
$817$	6.80322	0.238015
$818$	0	0
$819$	−0.0253688	−0.000886456 0
$820$	0	0
$821$	−21.1741	−0.738980	−0.369490	−	0.929235i	$-0.620468\pi$
$821$	−21.1741	−0.738980	−0.369490	+	0.929235i	$0.620468\pi$
$822$	0	0
$823$	47.1268	1.64274	0.821369	−	0.570397i	$-0.193211\pi$
$823$	47.1268	1.64274	0.821369	+	0.570397i	$0.193211\pi$
$824$	0	0
$825$	−5.43547	−0.189239
$826$	0	0
$827$	13.3177	0.463103	0.231552	−	0.972823i	$-0.425620\pi$
$827$	13.3177	0.463103	0.231552	+	0.972823i	$0.425620\pi$
$828$	0	0
$829$	−41.6063	−1.44505	−0.722523	−	0.691347i	$-0.757017\pi$
$829$	−41.6063	−1.44505	−0.722523	+	0.691347i	$0.757017\pi$
$830$	0	0
$831$	5.27114	0.182854
$832$	0	0
$833$	−8.79904	−0.304869
$834$	0	0
$835$	42.2032	1.46050
$836$	0	0
$837$	53.8873	1.86262
$838$	0	0
$839$	−20.2863	−0.700359	−0.350180	−	0.936683i	$-0.613879\pi$
$839$	−20.2863	−0.700359	−0.350180	+	0.936683i	$0.613879\pi$
$840$	0	0
$841$	−0.177928	−0.00613546
$842$	0	0
$843$	20.9462	0.721426
$844$	0	0
$845$	40.8815	1.40637
$846$	0	0
$847$	1.38577	0.0476156
$848$	0	0
$849$	16.9752	0.582588
$850$	0	0
$851$	−38.3485	−1.31457
$852$	0	0
$853$	−26.7153	−0.914715	−0.457358	−	0.889283i	$-0.651204\pi$
$853$	−26.7153	−0.914715	−0.457358	+	0.889283i	$0.651204\pi$
$854$	0	0
$855$	10.1248	0.346263
$856$	0	0
$857$	−1.26642	−0.0432602	−0.0216301	−	0.999766i	$-0.506886\pi$
$857$	−1.26642	−0.0432602	−0.0216301	+	0.999766i	$0.506886\pi$
$858$	0	0
$859$	45.0089	1.53569	0.767843	−	0.640638i	$-0.221330\pi$
$859$	45.0089	1.53569	0.767843	+	0.640638i	$0.221330\pi$
$860$	0	0
$861$	1.07071	0.0364897
$862$	0	0
$863$	−26.1118	−0.888855	−0.444428	−	0.895815i	$-0.646593\pi$
$863$	−26.1118	−0.888855	−0.444428	+	0.895815i	$0.646593\pi$
$864$	0	0
$865$	42.8210	1.45596
$866$	0	0
$867$	18.8341	0.639638
$868$	0	0
$869$	−1.29911	−0.0440693
$870$	0	0
$871$	−0.605584	−0.0205194
$872$	0	0
$873$	−20.5196	−0.694483
$874$	0	0
$875$	−0.0374399	−0.00126570
$876$	0	0
$877$	18.2107	0.614931	0.307466	−	0.951559i	$-0.400519\pi$
$877$	18.2107	0.614931	0.307466	+	0.951559i	$0.400519\pi$
$878$	0	0
$879$	0.397285	0.0134001
$880$	0	0
$881$	7.40050	0.249329	0.124665	−	0.992199i	$-0.460214\pi$
$881$	7.40050	0.249329	0.124665	+	0.992199i	$0.460214\pi$
$882$	0	0
$883$	1.57266	0.0529243	0.0264621	−	0.999650i	$-0.491576\pi$
$883$	1.57266	0.0529243	0.0264621	+	0.999650i	$0.491576\pi$
$884$	0	0
$885$	−31.3614	−1.05420
$886$	0	0
$887$	1.08367	0.0363861	0.0181930	−	0.999834i	$-0.494209\pi$
$887$	1.08367	0.0363861	0.0181930	+	0.999834i	$0.494209\pi$
$888$	0	0
$889$	2.17358	0.0728994
$890$	0	0
$891$	−2.00155	−0.0670545
$892$	0	0
$893$	−27.7175	−0.927532
$894$	0	0
$895$	75.2695	2.51598
$896$	0	0
$897$	−1.00530	−0.0335660
$898$	0	0
$899$	52.5301	1.75198
$900$	0	0
$901$	4.92620	0.164115
$902$	0	0
$903$	0.530184	0.0176434
$904$	0	0
$905$	36.5356	1.21448
$906$	0	0
$907$	−39.1868	−1.30118	−0.650588	−	0.759431i	$-0.725478\pi$
$907$	−39.1868	−1.30118	−0.650588	+	0.759431i	$0.725478\pi$
$908$	0	0
$909$	0.118664	0.00393582
$910$	0	0
$911$	−11.8818	−0.393662	−0.196831	−	0.980437i	$-0.563065\pi$
$911$	−11.8818	−0.393662	−0.196831	+	0.980437i	$0.563065\pi$
$912$	0	0
$913$	10.6493	0.352441
$914$	0	0
$915$	−55.2949	−1.82799
$916$	0	0
$917$	−1.65239	−0.0545666
$918$	0	0
$919$	28.1347	0.928077	0.464039	−	0.885815i	$-0.346400\pi$
$919$	28.1347	0.928077	0.464039	+	0.885815i	$0.346400\pi$
$920$	0	0
$921$	−17.5490	−0.578258
$922$	0	0
$923$	0.535485	0.0176257
$924$	0	0
$925$	28.3306	0.931505
$926$	0	0
$927$	−9.59210	−0.315046
$928$	0	0
$929$	59.5689	1.95439	0.977197	−	0.212335i	$-0.0681068\pi$
$929$	59.5689	1.95439	0.977197	+	0.212335i	$0.0681068\pi$
$930$	0	0
$931$	−14.9026	−0.488413
$932$	0	0
$933$	−13.6961	−0.448389
$934$	0	0
$935$	−3.59259	−0.117490
$936$	0	0
$937$	25.8722	0.845209	0.422605	−	0.906314i	$-0.361116\pi$
$937$	25.8722	0.845209	0.422605	+	0.906314i	$0.361116\pi$
$938$	0	0
$939$	−23.6911	−0.773130
$940$	0	0
$941$	−43.2143	−1.40875	−0.704373	−	0.709830i	$-0.748772\pi$
$941$	−43.2143	−1.40875	−0.704373	+	0.709830i	$0.748772\pi$
$942$	0	0
$943$	−42.8008	−1.39378
$944$	0	0
$945$	2.36029	0.0767802
$946$	0	0
$947$	53.0980	1.72545	0.862726	−	0.505672i	$-0.168755\pi$
$947$	53.0980	1.72545	0.862726	+	0.505672i	$0.168755\pi$
$948$	0	0
$949$	−0.698859	−0.0226859
$950$	0	0
$951$	−6.34963	−0.205901
$952$	0	0
$953$	−28.8397	−0.934210	−0.467105	−	0.884202i	$-0.654703\pi$
$953$	−28.8397	−0.934210	−0.467105	+	0.884202i	$0.654703\pi$
$954$	0	0
$955$	−15.2992	−0.495069
$956$	0	0
$957$	−5.93997	−0.192012
$958$	0	0
$959$	−0.115167	−0.00371893
$960$	0	0
$961$	64.7397	2.08838
$962$	0	0
$963$	−3.00659	−0.0968860
$964$	0	0
$965$	−80.7419	−2.59917
$966$	0	0
$967$	−6.08196	−0.195583	−0.0977914	−	0.995207i	$-0.531178\pi$
$967$	−6.08196	−0.195583	−0.0977914	+	0.995207i	$0.531178\pi$
$968$	0	0
$969$	−3.28778	−0.105619
$970$	0	0
$971$	−2.57155	−0.0825251	−0.0412626	−	0.999148i	$-0.513138\pi$
$971$	−2.57155	−0.0825251	−0.0412626	+	0.999148i	$0.513138\pi$
$972$	0	0
$973$	2.12993	0.0682824
$974$	0	0
$975$	0.742684	0.0237849
$976$	0	0
$977$	52.2605	1.67196	0.835980	−	0.548760i	$-0.184900\pi$
$977$	52.2605	1.67196	0.835980	+	0.548760i	$0.184900\pi$
$978$	0	0
$979$	−12.4380	−0.397520
$980$	0	0
$981$	−22.3303	−0.712953
$982$	0	0
$983$	−5.96528	−0.190263	−0.0951314	−	0.995465i	$-0.530327\pi$
$983$	−5.96528	−0.190263	−0.0951314	+	0.995465i	$0.530327\pi$
$984$	0	0
$985$	−4.62787	−0.147456
$986$	0	0
$987$	−2.16007	−0.0687557
$988$	0	0
$989$	−21.1937	−0.673920
$990$	0	0
$991$	−1.36771	−0.0434466	−0.0217233	−	0.999764i	$-0.506915\pi$
$991$	−1.36771	−0.0434466	−0.0217233	+	0.999764i	$0.506915\pi$
$992$	0	0
$993$	0.0211725	0.000671888 0
$994$	0	0
$995$	23.5098	0.745311
$996$	0	0
$997$	−17.2016	−0.544780	−0.272390	−	0.962187i	$-0.587814\pi$
$997$	−17.2016	−0.544780	−0.272390	+	0.962187i	$0.587814\pi$
$998$	0	0
$999$	31.7601	1.00484

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3856.2.a.n.1.5		12
4.3	odd	2		241.2.a.b.1.11	✓	12
12.11	even	2		2169.2.a.h.1.2		12
20.19	odd	2		6025.2.a.h.1.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.11	✓	12	4.3	odd	2
2169.2.a.h.1.2		12	12.11	even	2
3856.2.a.n.1.5		12	1.1	even	1	trivial
6025.2.a.h.1.2		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)

\(f(q)\)	\(=\)	\(q-1.22208 q^{3} -3.14843 q^{5} -0.136122 q^{7} -1.50653 q^{9} +O(q^{10})\)	\(q-1.22208 q^{3} -3.14843 q^{5} -0.136122 q^{7} -1.50653 q^{9} +0.905365 q^{11} -0.123706 q^{13} +3.84762 q^{15} +1.26034 q^{17} +2.13460 q^{19} +0.166352 q^{21} -6.64978 q^{23} +4.91264 q^{25} +5.50732 q^{27} +5.36862 q^{29} +9.78467 q^{31} -1.10642 q^{33} +0.428573 q^{35} +5.76688 q^{37} +0.151178 q^{39} +6.43642 q^{41} +3.18712 q^{43} +4.74322 q^{45} -12.9849 q^{47} -6.98147 q^{49} -1.54023 q^{51} +3.90862 q^{53} -2.85048 q^{55} -2.60864 q^{57} -8.15085 q^{59} -14.3712 q^{61} +0.205073 q^{63} +0.389481 q^{65} +4.89534 q^{67} +8.12653 q^{69} -4.32869 q^{71} +5.64935 q^{73} -6.00362 q^{75} -0.123241 q^{77} -1.43490 q^{79} -2.21077 q^{81} +11.7625 q^{83} -3.96811 q^{85} -6.56086 q^{87} -13.7381 q^{89} +0.0168392 q^{91} -11.9576 q^{93} -6.72063 q^{95} +13.6204 q^{97} -1.36396 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9}+O(q^{10}) \) 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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