LMFDB - Embedded newform 1792.2.a.u.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(1,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, 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("1792.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.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:	$14.3091920422$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 896)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.84776$ of defining polynomial
Character	$\chi$	$=$	1792.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61313 q^{3} -2.61313 q^{5} -1.00000 q^{7} +3.82843 q^{9} +O(q^{10})$	$q+2.61313 q^{3} -2.61313 q^{5} -1.00000 q^{7} +3.82843 q^{9} -2.16478 q^{11} -0.448342 q^{13} -6.82843 q^{15} -7.65685 q^{17} +4.77791 q^{19} -2.61313 q^{21} -6.82843 q^{23} +1.82843 q^{25} +2.16478 q^{27} -9.55582 q^{29} +5.65685 q^{31} -5.65685 q^{33} +2.61313 q^{35} -5.22625 q^{37} -1.17157 q^{39} +3.65685 q^{41} +2.16478 q^{43} -10.0042 q^{45} -8.00000 q^{47} +1.00000 q^{49} -20.0083 q^{51} +10.4525 q^{53} +5.65685 q^{55} +12.4853 q^{57} +0.448342 q^{59} +12.1689 q^{61} -3.82843 q^{63} +1.17157 q^{65} +3.06147 q^{67} -17.8435 q^{69} -2.34315 q^{71} -11.6569 q^{73} +4.77791 q^{75} +2.16478 q^{77} -2.34315 q^{79} -5.82843 q^{81} -13.0656 q^{83} +20.0083 q^{85} -24.9706 q^{87} +2.00000 q^{89} +0.448342 q^{91} +14.7821 q^{93} -12.4853 q^{95} -10.0000 q^{97} -8.28772 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{7} + 4 q^{9} - 16 q^{15} - 8 q^{17} - 16 q^{23} - 4 q^{25} - 16 q^{39} - 8 q^{41} - 32 q^{47} + 4 q^{49} + 16 q^{57} - 4 q^{63} + 16 q^{65} - 32 q^{71} - 24 q^{73} - 32 q^{79} - 12 q^{81} - 32 q^{87} + 8 q^{89} - 16 q^{95} - 40 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 + 4 * q^9 - 16 * q^15 - 8 * q^17 - 16 * q^23 - 4 * q^25 - 16 * q^39 - 8 * q^41 - 32 * q^47 + 4 * q^49 + 16 * q^57 - 4 * q^63 + 16 * q^65 - 32 * q^71 - 24 * q^73 - 32 * q^79 - 12 * q^81 - 32 * q^87 + 8 * q^89 - 16 * q^95 - 40 * q^97

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
$3$	2.61313	1.50869	0.754344	−	0.656479i	$-0.227955\pi$
$3$	2.61313	1.50869	0.754344	+	0.656479i	$0.227955\pi$
$4$	0	0
$5$	−2.61313	−1.16863	−0.584313	−	0.811529i	$-0.698636\pi$
$5$	−2.61313	−1.16863	−0.584313	+	0.811529i	$0.698636\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.82843	1.27614
$10$	0	0
$11$	−2.16478	−0.652707	−0.326354	−	0.945248i	$-0.605820\pi$
$11$	−2.16478	−0.652707	−0.326354	+	0.945248i	$0.605820\pi$
$12$	0	0
$13$	−0.448342	−0.124348	−0.0621738	−	0.998065i	$-0.519803\pi$
$13$	−0.448342	−0.124348	−0.0621738	+	0.998065i	$0.519803\pi$
$14$	0	0
$15$	−6.82843	−1.76309
$16$	0	0
$17$	−7.65685	−1.85706	−0.928530	−	0.371257i	$-0.878927\pi$
$17$	−7.65685	−1.85706	−0.928530	+	0.371257i	$0.878927\pi$
$18$	0	0
$19$	4.77791	1.09613	0.548064	−	0.836436i	$-0.315365\pi$
$19$	4.77791	1.09613	0.548064	+	0.836436i	$0.315365\pi$
$20$	0	0
$21$	−2.61313	−0.570231
$22$	0	0
$23$	−6.82843	−1.42383	−0.711913	−	0.702268i	$-0.752171\pi$
$23$	−6.82843	−1.42383	−0.711913	+	0.702268i	$0.752171\pi$
$24$	0	0
$25$	1.82843	0.365685
$26$	0	0
$27$	2.16478	0.416613
$28$	0	0
$29$	−9.55582	−1.77447	−0.887236	−	0.461316i	$-0.847377\pi$
$29$	−9.55582	−1.77447	−0.887236	+	0.461316i	$0.847377\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	0	0
$33$	−5.65685	−0.984732
$34$	0	0
$35$	2.61313	0.441699
$36$	0	0
$37$	−5.22625	−0.859191	−0.429595	−	0.903022i	$-0.641344\pi$
$37$	−5.22625	−0.859191	−0.429595	+	0.903022i	$0.641344\pi$
$38$	0	0
$39$	−1.17157	−0.187602
$40$	0	0
$41$	3.65685	0.571105	0.285552	−	0.958363i	$-0.407823\pi$
$41$	3.65685	0.571105	0.285552	+	0.958363i	$0.407823\pi$
$42$	0	0
$43$	2.16478	0.330127	0.165063	−	0.986283i	$-0.447217\pi$
$43$	2.16478	0.330127	0.165063	+	0.986283i	$0.447217\pi$
$44$	0	0
$45$	−10.0042	−1.49133
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−20.0083	−2.80173
$52$	0	0
$53$	10.4525	1.43576	0.717881	−	0.696166i	$-0.245112\pi$
$53$	10.4525	1.43576	0.717881	+	0.696166i	$0.245112\pi$
$54$	0	0
$55$	5.65685	0.762770
$56$	0	0
$57$	12.4853	1.65372
$58$	0	0
$59$	0.448342	0.0583691	0.0291845	−	0.999574i	$-0.490709\pi$
$59$	0.448342	0.0583691	0.0291845	+	0.999574i	$0.490709\pi$
$60$	0	0
$61$	12.1689	1.55807	0.779037	−	0.626978i	$-0.215708\pi$
$61$	12.1689	1.55807	0.779037	+	0.626978i	$0.215708\pi$
$62$	0	0
$63$	−3.82843	−0.482336
$64$	0	0
$65$	1.17157	0.145316
$66$	0	0
$67$	3.06147	0.374018	0.187009	−	0.982358i	$-0.440121\pi$
$67$	3.06147	0.374018	0.187009	+	0.982358i	$0.440121\pi$
$68$	0	0
$69$	−17.8435	−2.14811
$70$	0	0
$71$	−2.34315	−0.278080	−0.139040	−	0.990287i	$-0.544402\pi$
$71$	−2.34315	−0.278080	−0.139040	+	0.990287i	$0.544402\pi$
$72$	0	0
$73$	−11.6569	−1.36433	−0.682166	−	0.731198i	$-0.738962\pi$
$73$	−11.6569	−1.36433	−0.682166	+	0.731198i	$0.738962\pi$
$74$	0	0
$75$	4.77791	0.551706
$76$	0	0
$77$	2.16478	0.246700
$78$	0	0
$79$	−2.34315	−0.263624	−0.131812	−	0.991275i	$-0.542080\pi$
$79$	−2.34315	−0.263624	−0.131812	+	0.991275i	$0.542080\pi$
$80$	0	0
$81$	−5.82843	−0.647603
$82$	0	0
$83$	−13.0656	−1.43414	−0.717070	−	0.697002i	$-0.754517\pi$
$83$	−13.0656	−1.43414	−0.717070	+	0.697002i	$0.754517\pi$
$84$	0	0
$85$	20.0083	2.17021
$86$	0	0
$87$	−24.9706	−2.67713
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	0.448342	0.0469990
$92$	0	0
$93$	14.7821	1.53283
$94$	0	0
$95$	−12.4853	−1.28096
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	−8.28772	−0.832947
$100$	0	0
$101$	4.77791	0.475420	0.237710	−	0.971336i	$-0.423603\pi$
$101$	4.77791	0.475420	0.237710	+	0.971336i	$0.423603\pi$
$102$	0	0
$103$	−13.6569	−1.34565	−0.672825	−	0.739802i	$-0.734919\pi$
$103$	−13.6569	−1.34565	−0.672825	+	0.739802i	$0.734919\pi$
$104$	0	0
$105$	6.82843	0.666386
$106$	0	0
$107$	−13.5140	−1.30644	−0.653222	−	0.757166i	$-0.726583\pi$
$107$	−13.5140	−1.30644	−0.653222	+	0.757166i	$0.726583\pi$
$108$	0	0
$109$	11.3492	1.08705	0.543527	−	0.839391i	$-0.317088\pi$
$109$	11.3492	1.08705	0.543527	+	0.839391i	$0.317088\pi$
$110$	0	0
$111$	−13.6569	−1.29625
$112$	0	0
$113$	8.82843	0.830509	0.415254	−	0.909705i	$-0.363693\pi$
$113$	8.82843	0.830509	0.415254	+	0.909705i	$0.363693\pi$
$114$	0	0
$115$	17.8435	1.66392
$116$	0	0
$117$	−1.71644	−0.158685
$118$	0	0
$119$	7.65685	0.701903
$120$	0	0
$121$	−6.31371	−0.573973
$122$	0	0
$123$	9.55582	0.861619
$124$	0	0
$125$	8.28772	0.741276
$126$	0	0
$127$	−4.48528	−0.398004	−0.199002	−	0.979999i	$-0.563770\pi$
$127$	−4.48528	−0.398004	−0.199002	+	0.979999i	$0.563770\pi$
$128$	0	0
$129$	5.65685	0.498058
$130$	0	0
$131$	18.2919	1.59817	0.799085	−	0.601219i	$-0.205318\pi$
$131$	18.2919	1.59817	0.799085	+	0.601219i	$0.205318\pi$
$132$	0	0
$133$	−4.77791	−0.414297
$134$	0	0
$135$	−5.65685	−0.486864
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	−19.5600	−1.65906	−0.829528	−	0.558465i	$-0.811390\pi$
$139$	−19.5600	−1.65906	−0.829528	+	0.558465i	$0.811390\pi$
$140$	0	0
$141$	−20.9050	−1.76052
$142$	0	0
$143$	0.970563	0.0811625
$144$	0	0
$145$	24.9706	2.07369
$146$	0	0
$147$	2.61313	0.215527
$148$	0	0
$149$	16.5754	1.35791	0.678956	−	0.734179i	$-0.262433\pi$
$149$	16.5754	1.35791	0.678956	+	0.734179i	$0.262433\pi$
$150$	0	0
$151$	18.1421	1.47639	0.738193	−	0.674590i	$-0.235679\pi$
$151$	18.1421	1.47639	0.738193	+	0.674590i	$0.235679\pi$
$152$	0	0
$153$	−29.3137	−2.36987
$154$	0	0
$155$	−14.7821	−1.18732
$156$	0	0
$157$	−4.77791	−0.381319	−0.190659	−	0.981656i	$-0.561063\pi$
$157$	−4.77791	−0.381319	−0.190659	+	0.981656i	$0.561063\pi$
$158$	0	0
$159$	27.3137	2.16612
$160$	0	0
$161$	6.82843	0.538155
$162$	0	0
$163$	−3.95815	−0.310026	−0.155013	−	0.987912i	$-0.549542\pi$
$163$	−3.95815	−0.310026	−0.155013	+	0.987912i	$0.549542\pi$
$164$	0	0
$165$	14.7821	1.15078
$166$	0	0
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	−12.7990	−0.984538
$170$	0	0
$171$	18.2919	1.39882
$172$	0	0
$173$	16.1271	1.22612	0.613060	−	0.790036i	$-0.289938\pi$
$173$	16.1271	1.22612	0.613060	+	0.790036i	$0.289938\pi$
$174$	0	0
$175$	−1.82843	−0.138216
$176$	0	0
$177$	1.17157	0.0880608
$178$	0	0
$179$	22.1731	1.65730	0.828648	−	0.559770i	$-0.189111\pi$
$179$	22.1731	1.65730	0.828648	+	0.559770i	$0.189111\pi$
$180$	0	0
$181$	2.61313	0.194232	0.0971161	−	0.995273i	$-0.469038\pi$
$181$	2.61313	0.194232	0.0971161	+	0.995273i	$0.469038\pi$
$182$	0	0
$183$	31.7990	2.35065
$184$	0	0
$185$	13.6569	1.00407
$186$	0	0
$187$	16.5754	1.21212
$188$	0	0
$189$	−2.16478	−0.157465
$190$	0	0
$191$	19.3137	1.39749	0.698745	−	0.715370i	$-0.253742\pi$
$191$	19.3137	1.39749	0.698745	+	0.715370i	$0.253742\pi$
$192$	0	0
$193$	13.7990	0.993273	0.496637	−	0.867959i	$-0.334568\pi$
$193$	13.7990	0.993273	0.496637	+	0.867959i	$0.334568\pi$
$194$	0	0
$195$	3.06147	0.219236
$196$	0	0
$197$	−6.12293	−0.436241	−0.218121	−	0.975922i	$-0.569993\pi$
$197$	−6.12293	−0.436241	−0.218121	+	0.975922i	$0.569993\pi$
$198$	0	0
$199$	−8.97056	−0.635906	−0.317953	−	0.948106i	$-0.602995\pi$
$199$	−8.97056	−0.635906	−0.317953	+	0.948106i	$0.602995\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	9.55582	0.670687
$204$	0	0
$205$	−9.55582	−0.667407
$206$	0	0
$207$	−26.1421	−1.81700
$208$	0	0
$209$	−10.3431	−0.715450
$210$	0	0
$211$	9.18440	0.632280	0.316140	−	0.948712i	$-0.397613\pi$
$211$	9.18440	0.632280	0.316140	+	0.948712i	$0.397613\pi$
$212$	0	0
$213$	−6.12293	−0.419537
$214$	0	0
$215$	−5.65685	−0.385794
$216$	0	0
$217$	−5.65685	−0.384012
$218$	0	0
$219$	−30.4608	−2.05835
$220$	0	0
$221$	3.43289	0.230921
$222$	0	0
$223$	−8.97056	−0.600713	−0.300357	−	0.953827i	$-0.597106\pi$
$223$	−8.97056	−0.600713	−0.300357	+	0.953827i	$0.597106\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	−14.3337	−0.951363	−0.475682	−	0.879618i	$-0.657799\pi$
$227$	−14.3337	−0.951363	−0.475682	+	0.879618i	$0.657799\pi$
$228$	0	0
$229$	5.67459	0.374988	0.187494	−	0.982266i	$-0.439964\pi$
$229$	5.67459	0.374988	0.187494	+	0.982266i	$0.439964\pi$
$230$	0	0
$231$	5.65685	0.372194
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	20.9050	1.36369
$236$	0	0
$237$	−6.12293	−0.397727
$238$	0	0
$239$	−18.1421	−1.17352	−0.586759	−	0.809762i	$-0.699596\pi$
$239$	−18.1421	−1.17352	−0.586759	+	0.809762i	$0.699596\pi$
$240$	0	0
$241$	1.31371	0.0846234	0.0423117	−	0.999104i	$-0.486528\pi$
$241$	1.31371	0.0846234	0.0423117	+	0.999104i	$0.486528\pi$
$242$	0	0
$243$	−21.7248	−1.39364
$244$	0	0
$245$	−2.61313	−0.166946
$246$	0	0
$247$	−2.14214	−0.136301
$248$	0	0
$249$	−34.1421	−2.16367
$250$	0	0
$251$	0.819760	0.0517428	0.0258714	−	0.999665i	$-0.491764\pi$
$251$	0.819760	0.0517428	0.0258714	+	0.999665i	$0.491764\pi$
$252$	0	0
$253$	14.7821	0.929341
$254$	0	0
$255$	52.2843	3.27417
$256$	0	0
$257$	−9.31371	−0.580973	−0.290487	−	0.956879i	$-0.593817\pi$
$257$	−9.31371	−0.580973	−0.290487	+	0.956879i	$0.593817\pi$
$258$	0	0
$259$	5.22625	0.324743
$260$	0	0
$261$	−36.5838	−2.26448
$262$	0	0
$263$	−2.34315	−0.144485	−0.0722423	−	0.997387i	$-0.523015\pi$
$263$	−2.34315	−0.144485	−0.0722423	+	0.997387i	$0.523015\pi$
$264$	0	0
$265$	−27.3137	−1.67787
$266$	0	0
$267$	5.22625	0.319841
$268$	0	0
$269$	8.21080	0.500621	0.250311	−	0.968166i	$-0.419467\pi$
$269$	8.21080	0.500621	0.250311	+	0.968166i	$0.419467\pi$
$270$	0	0
$271$	−2.34315	−0.142336	−0.0711680	−	0.997464i	$-0.522673\pi$
$271$	−2.34315	−0.142336	−0.0711680	+	0.997464i	$0.522673\pi$
$272$	0	0
$273$	1.17157	0.0709068
$274$	0	0
$275$	−3.95815	−0.238685
$276$	0	0
$277$	−14.7821	−0.888169	−0.444084	−	0.895985i	$-0.646471\pi$
$277$	−14.7821	−0.888169	−0.444084	+	0.895985i	$0.646471\pi$
$278$	0	0
$279$	21.6569	1.29656
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	20.8281	1.23810	0.619051	−	0.785351i	$-0.287518\pi$
$283$	20.8281	1.23810	0.619051	+	0.785351i	$0.287518\pi$
$284$	0	0
$285$	−32.6256	−1.93257
$286$	0	0
$287$	−3.65685	−0.215857
$288$	0	0
$289$	41.6274	2.44867
$290$	0	0
$291$	−26.1313	−1.53184
$292$	0	0
$293$	10.9008	0.636834	0.318417	−	0.947951i	$-0.396849\pi$
$293$	10.9008	0.636834	0.318417	+	0.947951i	$0.396849\pi$
$294$	0	0
$295$	−1.17157	−0.0682116
$296$	0	0
$297$	−4.68629	−0.271926
$298$	0	0
$299$	3.06147	0.177049
$300$	0	0
$301$	−2.16478	−0.124776
$302$	0	0
$303$	12.4853	0.717261
$304$	0	0
$305$	−31.7990	−1.82080
$306$	0	0
$307$	−8.73606	−0.498593	−0.249297	−	0.968427i	$-0.580199\pi$
$307$	−8.73606	−0.498593	−0.249297	+	0.968427i	$0.580199\pi$
$308$	0	0
$309$	−35.6871	−2.03017
$310$	0	0
$311$	−14.6274	−0.829445	−0.414722	−	0.909948i	$-0.636121\pi$
$311$	−14.6274	−0.829445	−0.414722	+	0.909948i	$0.636121\pi$
$312$	0	0
$313$	8.34315	0.471582	0.235791	−	0.971804i	$-0.424232\pi$
$313$	8.34315	0.471582	0.235791	+	0.971804i	$0.424232\pi$
$314$	0	0
$315$	10.0042	0.563671
$316$	0	0
$317$	−8.65914	−0.486346	−0.243173	−	0.969983i	$-0.578188\pi$
$317$	−8.65914	−0.486346	−0.243173	+	0.969983i	$0.578188\pi$
$318$	0	0
$319$	20.6863	1.15821
$320$	0	0
$321$	−35.3137	−1.97102
$322$	0	0
$323$	−36.5838	−2.03558
$324$	0	0
$325$	−0.819760	−0.0454721
$326$	0	0
$327$	29.6569	1.64003
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	−18.7402	−1.03006	−0.515028	−	0.857173i	$-0.672218\pi$
$331$	−18.7402	−1.03006	−0.515028	+	0.857173i	$0.672218\pi$
$332$	0	0
$333$	−20.0083	−1.09645
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	11.1716	0.608554	0.304277	−	0.952584i	$-0.401585\pi$
$337$	11.1716	0.608554	0.304277	+	0.952584i	$0.401585\pi$
$338$	0	0
$339$	23.0698	1.25298
$340$	0	0
$341$	−12.2459	−0.663151
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	46.6274	2.51034
$346$	0	0
$347$	−12.6173	−0.677332	−0.338666	−	0.940907i	$-0.609976\pi$
$347$	−12.6173	−0.677332	−0.338666	+	0.940907i	$0.609976\pi$
$348$	0	0
$349$	−22.6215	−1.21090	−0.605449	−	0.795884i	$-0.707007\pi$
$349$	−22.6215	−1.21090	−0.605449	+	0.795884i	$0.707007\pi$
$350$	0	0
$351$	−0.970563	−0.0518048
$352$	0	0
$353$	15.6569	0.833330	0.416665	−	0.909060i	$-0.363199\pi$
$353$	15.6569	0.833330	0.416665	+	0.909060i	$0.363199\pi$
$354$	0	0
$355$	6.12293	0.324972
$356$	0	0
$357$	20.0083	1.05895
$358$	0	0
$359$	−6.82843	−0.360391	−0.180195	−	0.983631i	$-0.557673\pi$
$359$	−6.82843	−0.360391	−0.180195	+	0.983631i	$0.557673\pi$
$360$	0	0
$361$	3.82843	0.201496
$362$	0	0
$363$	−16.4985	−0.865947
$364$	0	0
$365$	30.4608	1.59439
$366$	0	0
$367$	24.9706	1.30345	0.651726	−	0.758454i	$-0.274045\pi$
$367$	24.9706	1.30345	0.651726	+	0.758454i	$0.274045\pi$
$368$	0	0
$369$	14.0000	0.728811
$370$	0	0
$371$	−10.4525	−0.542667
$372$	0	0
$373$	33.8937	1.75495	0.877475	−	0.479622i	$-0.159226\pi$
$373$	33.8937	1.75495	0.877475	+	0.479622i	$0.159226\pi$
$374$	0	0
$375$	21.6569	1.11836
$376$	0	0
$377$	4.28427	0.220651
$378$	0	0
$379$	−23.0698	−1.18502	−0.592508	−	0.805565i	$-0.701862\pi$
$379$	−23.0698	−1.18502	−0.592508	+	0.805565i	$0.701862\pi$
$380$	0	0
$381$	−11.7206	−0.600465
$382$	0	0
$383$	−19.3137	−0.986884	−0.493442	−	0.869779i	$-0.664262\pi$
$383$	−19.3137	−0.986884	−0.493442	+	0.869779i	$0.664262\pi$
$384$	0	0
$385$	−5.65685	−0.288300
$386$	0	0
$387$	8.28772	0.421288
$388$	0	0
$389$	−20.0083	−1.01446	−0.507231	−	0.861810i	$-0.669331\pi$
$389$	−20.0083	−1.01446	−0.507231	+	0.861810i	$0.669331\pi$
$390$	0	0
$391$	52.2843	2.64413
$392$	0	0
$393$	47.7990	2.41114
$394$	0	0
$395$	6.12293	0.308078
$396$	0	0
$397$	−18.2919	−0.918043	−0.459022	−	0.888425i	$-0.651800\pi$
$397$	−18.2919	−0.918043	−0.459022	+	0.888425i	$0.651800\pi$
$398$	0	0
$399$	−12.4853	−0.625046
$400$	0	0
$401$	9.51472	0.475142	0.237571	−	0.971370i	$-0.423649\pi$
$401$	9.51472	0.475142	0.237571	+	0.971370i	$0.423649\pi$
$402$	0	0
$403$	−2.53620	−0.126337
$404$	0	0
$405$	15.2304	0.756805
$406$	0	0
$407$	11.3137	0.560800
$408$	0	0
$409$	−5.31371	−0.262746	−0.131373	−	0.991333i	$-0.541939\pi$
$409$	−5.31371	−0.262746	−0.131373	+	0.991333i	$0.541939\pi$
$410$	0	0
$411$	−5.22625	−0.257792
$412$	0	0
$413$	−0.448342	−0.0220614
$414$	0	0
$415$	34.1421	1.67597
$416$	0	0
$417$	−51.1127	−2.50300
$418$	0	0
$419$	13.0656	0.638298	0.319149	−	0.947705i	$-0.396603\pi$
$419$	13.0656	0.638298	0.319149	+	0.947705i	$0.396603\pi$
$420$	0	0
$421$	−2.53620	−0.123607	−0.0618035	−	0.998088i	$-0.519685\pi$
$421$	−2.53620	−0.123607	−0.0618035	+	0.998088i	$0.519685\pi$
$422$	0	0
$423$	−30.6274	−1.48916
$424$	0	0
$425$	−14.0000	−0.679100
$426$	0	0
$427$	−12.1689	−0.588897
$428$	0	0
$429$	2.53620	0.122449
$430$	0	0
$431$	6.82843	0.328914	0.164457	−	0.986384i	$-0.447413\pi$
$431$	6.82843	0.328914	0.164457	+	0.986384i	$0.447413\pi$
$432$	0	0
$433$	−12.3431	−0.593174	−0.296587	−	0.955006i	$-0.595848\pi$
$433$	−12.3431	−0.593174	−0.296587	+	0.955006i	$0.595848\pi$
$434$	0	0
$435$	65.2512	3.12856
$436$	0	0
$437$	−32.6256	−1.56069
$438$	0	0
$439$	−0.970563	−0.0463224	−0.0231612	−	0.999732i	$-0.507373\pi$
$439$	−0.970563	−0.0463224	−0.0231612	+	0.999732i	$0.507373\pi$
$440$	0	0
$441$	3.82843	0.182306
$442$	0	0
$443$	9.18440	0.436364	0.218182	−	0.975908i	$-0.429987\pi$
$443$	9.18440	0.436364	0.218182	+	0.975908i	$0.429987\pi$
$444$	0	0
$445$	−5.22625	−0.247748
$446$	0	0
$447$	43.3137	2.04867
$448$	0	0
$449$	−24.6274	−1.16224	−0.581120	−	0.813818i	$-0.697385\pi$
$449$	−24.6274	−1.16224	−0.581120	+	0.813818i	$0.697385\pi$
$450$	0	0
$451$	−7.91630	−0.372764
$452$	0	0
$453$	47.4077	2.22741
$454$	0	0
$455$	−1.17157	−0.0549242
$456$	0	0
$457$	10.4853	0.490481	0.245240	−	0.969462i	$-0.421133\pi$
$457$	10.4853	0.490481	0.245240	+	0.969462i	$0.421133\pi$
$458$	0	0
$459$	−16.5754	−0.773675
$460$	0	0
$461$	−10.3756	−0.483239	−0.241619	−	0.970371i	$-0.577679\pi$
$461$	−10.3756	−0.483239	−0.241619	+	0.970371i	$0.577679\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	−38.6274	−1.79130
$466$	0	0
$467$	−1.34502	−0.0622403	−0.0311202	−	0.999516i	$-0.509907\pi$
$467$	−1.34502	−0.0622403	−0.0311202	+	0.999516i	$0.509907\pi$
$468$	0	0
$469$	−3.06147	−0.141365
$470$	0	0
$471$	−12.4853	−0.575291
$472$	0	0
$473$	−4.68629	−0.215476
$474$	0	0
$475$	8.73606	0.400838
$476$	0	0
$477$	40.0166	1.83224
$478$	0	0
$479$	5.65685	0.258468	0.129234	−	0.991614i	$-0.458748\pi$
$479$	5.65685	0.258468	0.129234	+	0.991614i	$0.458748\pi$
$480$	0	0
$481$	2.34315	0.106838
$482$	0	0
$483$	17.8435	0.811909
$484$	0	0
$485$	26.1313	1.18656
$486$	0	0
$487$	31.7990	1.44095	0.720475	−	0.693481i	$-0.243924\pi$
$487$	31.7990	1.44095	0.720475	+	0.693481i	$0.243924\pi$
$488$	0	0
$489$	−10.3431	−0.467733
$490$	0	0
$491$	32.6256	1.47237	0.736187	−	0.676779i	$-0.236625\pi$
$491$	32.6256	1.47237	0.736187	+	0.676779i	$0.236625\pi$
$492$	0	0
$493$	73.1675	3.29530
$494$	0	0
$495$	21.6569	0.973403
$496$	0	0
$497$	2.34315	0.105104
$498$	0	0
$499$	−11.7206	−0.524686	−0.262343	−	0.964975i	$-0.584495\pi$
$499$	−11.7206	−0.524686	−0.262343	+	0.964975i	$0.584495\pi$
$500$	0	0
$501$	−41.8100	−1.86793
$502$	0	0
$503$	−5.65685	−0.252227	−0.126113	−	0.992016i	$-0.540250\pi$
$503$	−5.65685	−0.252227	−0.126113	+	0.992016i	$0.540250\pi$
$504$	0	0
$505$	−12.4853	−0.555588
$506$	0	0
$507$	−33.4454	−1.48536
$508$	0	0
$509$	−3.50981	−0.155570	−0.0777848	−	0.996970i	$-0.524785\pi$
$509$	−3.50981	−0.155570	−0.0777848	+	0.996970i	$0.524785\pi$
$510$	0	0
$511$	11.6569	0.515669
$512$	0	0
$513$	10.3431	0.456661
$514$	0	0
$515$	35.6871	1.57256
$516$	0	0
$517$	17.3183	0.761657
$518$	0	0
$519$	42.1421	1.84983
$520$	0	0
$521$	−14.6863	−0.643418	−0.321709	−	0.946839i	$-0.604257\pi$
$521$	−14.6863	−0.643418	−0.321709	+	0.946839i	$0.604257\pi$
$522$	0	0
$523$	−7.46796	−0.326551	−0.163276	−	0.986581i	$-0.552206\pi$
$523$	−7.46796	−0.326551	−0.163276	+	0.986581i	$0.552206\pi$
$524$	0	0
$525$	−4.77791	−0.208525
$526$	0	0
$527$	−43.3137	−1.88677
$528$	0	0
$529$	23.6274	1.02728
$530$	0	0
$531$	1.71644	0.0744873
$532$	0	0
$533$	−1.63952	−0.0710155
$534$	0	0
$535$	35.3137	1.52674
$536$	0	0
$537$	57.9411	2.50034
$538$	0	0
$539$	−2.16478	−0.0932439
$540$	0	0
$541$	−31.3575	−1.34816	−0.674082	−	0.738656i	$-0.735461\pi$
$541$	−31.3575	−1.34816	−0.674082	+	0.738656i	$0.735461\pi$
$542$	0	0
$543$	6.82843	0.293036
$544$	0	0
$545$	−29.6569	−1.27036
$546$	0	0
$547$	−43.9748	−1.88023	−0.940113	−	0.340862i	$-0.889281\pi$
$547$	−43.9748	−1.88023	−0.940113	+	0.340862i	$0.889281\pi$
$548$	0	0
$549$	46.5879	1.98832
$550$	0	0
$551$	−45.6569	−1.94505
$552$	0	0
$553$	2.34315	0.0996407
$554$	0	0
$555$	35.6871	1.51483
$556$	0	0
$557$	14.7821	0.626337	0.313168	−	0.949698i	$-0.398610\pi$
$557$	14.7821	0.626337	0.313168	+	0.949698i	$0.398610\pi$
$558$	0	0
$559$	−0.970563	−0.0410504
$560$	0	0
$561$	43.3137	1.82871
$562$	0	0
$563$	−17.7666	−0.748774	−0.374387	−	0.927273i	$-0.622147\pi$
$563$	−17.7666	−0.748774	−0.374387	+	0.927273i	$0.622147\pi$
$564$	0	0
$565$	−23.0698	−0.970553
$566$	0	0
$567$	5.82843	0.244771
$568$	0	0
$569$	17.5147	0.734255	0.367128	−	0.930171i	$-0.380341\pi$
$569$	17.5147	0.734255	0.367128	+	0.930171i	$0.380341\pi$
$570$	0	0
$571$	18.7402	0.784254	0.392127	−	0.919911i	$-0.371739\pi$
$571$	18.7402	0.784254	0.392127	+	0.919911i	$0.371739\pi$
$572$	0	0
$573$	50.4692	2.10838
$574$	0	0
$575$	−12.4853	−0.520672
$576$	0	0
$577$	−22.9706	−0.956277	−0.478139	−	0.878284i	$-0.658688\pi$
$577$	−22.9706	−0.956277	−0.478139	+	0.878284i	$0.658688\pi$
$578$	0	0
$579$	36.0585	1.49854
$580$	0	0
$581$	13.0656	0.542054
$582$	0	0
$583$	−22.6274	−0.937132
$584$	0	0
$585$	4.48528	0.185444
$586$	0	0
$587$	33.4454	1.38044	0.690219	−	0.723600i	$-0.257514\pi$
$587$	33.4454	1.38044	0.690219	+	0.723600i	$0.257514\pi$
$588$	0	0
$589$	27.0279	1.11367
$590$	0	0
$591$	−16.0000	−0.658152
$592$	0	0
$593$	−25.3137	−1.03951	−0.519755	−	0.854316i	$-0.673977\pi$
$593$	−25.3137	−1.03951	−0.519755	+	0.854316i	$0.673977\pi$
$594$	0	0
$595$	−20.0083	−0.820261
$596$	0	0
$597$	−23.4412	−0.959385
$598$	0	0
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	0	0
$601$	−41.3137	−1.68522	−0.842611	−	0.538523i	$-0.818982\pi$
$601$	−41.3137	−1.68522	−0.842611	+	0.538523i	$0.818982\pi$
$602$	0	0
$603$	11.7206	0.477300
$604$	0	0
$605$	16.4985	0.670760
$606$	0	0
$607$	27.3137	1.10863	0.554315	−	0.832307i	$-0.312980\pi$
$607$	27.3137	1.10863	0.554315	+	0.832307i	$0.312980\pi$
$608$	0	0
$609$	24.9706	1.01186
$610$	0	0
$611$	3.58673	0.145104
$612$	0	0
$613$	26.1313	1.05543	0.527716	−	0.849421i	$-0.323049\pi$
$613$	26.1313	1.05543	0.527716	+	0.849421i	$0.323049\pi$
$614$	0	0
$615$	−24.9706	−1.00691
$616$	0	0
$617$	41.7990	1.68276	0.841382	−	0.540441i	$-0.181743\pi$
$617$	41.7990	1.68276	0.841382	+	0.540441i	$0.181743\pi$
$618$	0	0
$619$	−33.9706	−1.36540	−0.682698	−	0.730701i	$-0.739193\pi$
$619$	−33.9706	−1.36540	−0.682698	+	0.730701i	$0.739193\pi$
$620$	0	0
$621$	−14.7821	−0.593184
$622$	0	0
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	−30.7990	−1.23196
$626$	0	0
$627$	−27.0279	−1.07939
$628$	0	0
$629$	40.0166	1.59557
$630$	0	0
$631$	−30.6274	−1.21926	−0.609629	−	0.792687i	$-0.708682\pi$
$631$	−30.6274	−1.21926	−0.609629	+	0.792687i	$0.708682\pi$
$632$	0	0
$633$	24.0000	0.953914
$634$	0	0
$635$	11.7206	0.465118
$636$	0	0
$637$	−0.448342	−0.0177639
$638$	0	0
$639$	−8.97056	−0.354870
$640$	0	0
$641$	27.4558	1.08444	0.542220	−	0.840236i	$-0.317584\pi$
$641$	27.4558	1.08444	0.542220	+	0.840236i	$0.317584\pi$
$642$	0	0
$643$	−2.98454	−0.117699	−0.0588495	−	0.998267i	$-0.518743\pi$
$643$	−2.98454	−0.117699	−0.0588495	+	0.998267i	$0.518743\pi$
$644$	0	0
$645$	−14.7821	−0.582044
$646$	0	0
$647$	−29.6569	−1.16593	−0.582966	−	0.812497i	$-0.698108\pi$
$647$	−29.6569	−1.16593	−0.582966	+	0.812497i	$0.698108\pi$
$648$	0	0
$649$	−0.970563	−0.0380979
$650$	0	0
$651$	−14.7821	−0.579355
$652$	0	0
$653$	−22.5445	−0.882236	−0.441118	−	0.897449i	$-0.645418\pi$
$653$	−22.5445	−0.882236	−0.441118	+	0.897449i	$0.645418\pi$
$654$	0	0
$655$	−47.7990	−1.86766
$656$	0	0
$657$	−44.6274	−1.74108
$658$	0	0
$659$	10.0811	0.392703	0.196352	−	0.980534i	$-0.437091\pi$
$659$	10.0811	0.392703	0.196352	+	0.980534i	$0.437091\pi$
$660$	0	0
$661$	0.448342	0.0174385	0.00871923	−	0.999962i	$-0.497225\pi$
$661$	0.448342	0.0174385	0.00871923	+	0.999962i	$0.497225\pi$
$662$	0	0
$663$	8.97056	0.348388
$664$	0	0
$665$	12.4853	0.484158
$666$	0	0
$667$	65.2512	2.52654
$668$	0	0
$669$	−23.4412	−0.906290
$670$	0	0
$671$	−26.3431	−1.01697
$672$	0	0
$673$	28.6274	1.10351	0.551753	−	0.834008i	$-0.313959\pi$
$673$	28.6274	1.10351	0.551753	+	0.834008i	$0.313959\pi$
$674$	0	0
$675$	3.95815	0.152349
$676$	0	0
$677$	8.73606	0.335754	0.167877	−	0.985808i	$-0.446309\pi$
$677$	8.73606	0.335754	0.167877	+	0.985808i	$0.446309\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	0	0
$681$	−37.4558	−1.43531
$682$	0	0
$683$	−5.59767	−0.214189	−0.107094	−	0.994249i	$-0.534155\pi$
$683$	−5.59767	−0.214189	−0.107094	+	0.994249i	$0.534155\pi$
$684$	0	0
$685$	5.22625	0.199685
$686$	0	0
$687$	14.8284	0.565740
$688$	0	0
$689$	−4.68629	−0.178533
$690$	0	0
$691$	−35.7640	−1.36053	−0.680263	−	0.732968i	$-0.738135\pi$
$691$	−35.7640	−1.36053	−0.680263	+	0.732968i	$0.738135\pi$
$692$	0	0
$693$	8.28772	0.314824
$694$	0	0
$695$	51.1127	1.93882
$696$	0	0
$697$	−28.0000	−1.06058
$698$	0	0
$699$	26.1313	0.988375
$700$	0	0
$701$	−40.9133	−1.54528	−0.772638	−	0.634847i	$-0.781063\pi$
$701$	−40.9133	−1.54528	−0.772638	+	0.634847i	$0.781063\pi$
$702$	0	0
$703$	−24.9706	−0.941783
$704$	0	0
$705$	54.6274	2.05739
$706$	0	0
$707$	−4.77791	−0.179692
$708$	0	0
$709$	−17.4721	−0.656179	−0.328090	−	0.944647i	$-0.606405\pi$
$709$	−17.4721	−0.656179	−0.328090	+	0.944647i	$0.606405\pi$
$710$	0	0
$711$	−8.97056	−0.336422
$712$	0	0
$713$	−38.6274	−1.44661
$714$	0	0
$715$	−2.53620	−0.0948486
$716$	0	0
$717$	−47.4077	−1.77047
$718$	0	0
$719$	3.31371	0.123580	0.0617902	−	0.998089i	$-0.480319\pi$
$719$	3.31371	0.123580	0.0617902	+	0.998089i	$0.480319\pi$
$720$	0	0
$721$	13.6569	0.508608
$722$	0	0
$723$	3.43289	0.127670
$724$	0	0
$725$	−17.4721	−0.648898
$726$	0	0
$727$	−20.6863	−0.767212	−0.383606	−	0.923497i	$-0.625318\pi$
$727$	−20.6863	−0.767212	−0.383606	+	0.923497i	$0.625318\pi$
$728$	0	0
$729$	−39.2843	−1.45497
$730$	0	0
$731$	−16.5754	−0.613065
$732$	0	0
$733$	−21.7248	−0.802423	−0.401211	−	0.915986i	$-0.631411\pi$
$733$	−21.7248	−0.802423	−0.401211	+	0.915986i	$0.631411\pi$
$734$	0	0
$735$	−6.82843	−0.251870
$736$	0	0
$737$	−6.62742	−0.244124
$738$	0	0
$739$	−24.8632	−0.914606	−0.457303	−	0.889311i	$-0.651185\pi$
$739$	−24.8632	−0.914606	−0.457303	+	0.889311i	$0.651185\pi$
$740$	0	0
$741$	−5.59767	−0.205636
$742$	0	0
$743$	−27.5147	−1.00942	−0.504709	−	0.863290i	$-0.668400\pi$
$743$	−27.5147	−1.00942	−0.504709	+	0.863290i	$0.668400\pi$
$744$	0	0
$745$	−43.3137	−1.58689
$746$	0	0
$747$	−50.0208	−1.83017
$748$	0	0
$749$	13.5140	0.493790
$750$	0	0
$751$	45.4558	1.65871	0.829354	−	0.558724i	$-0.188709\pi$
$751$	45.4558	1.65871	0.829354	+	0.558724i	$0.188709\pi$
$752$	0	0
$753$	2.14214	0.0780638
$754$	0	0
$755$	−47.4077	−1.72534
$756$	0	0
$757$	22.5445	0.819395	0.409697	−	0.912221i	$-0.365634\pi$
$757$	22.5445	0.819395	0.409697	+	0.912221i	$0.365634\pi$
$758$	0	0
$759$	38.6274	1.40209
$760$	0	0
$761$	−18.9706	−0.687682	−0.343841	−	0.939028i	$-0.611728\pi$
$761$	−18.9706	−0.687682	−0.343841	+	0.939028i	$0.611728\pi$
$762$	0	0
$763$	−11.3492	−0.410868
$764$	0	0
$765$	76.6004	2.76949
$766$	0	0
$767$	−0.201010	−0.00725806
$768$	0	0
$769$	35.2548	1.27132	0.635661	−	0.771968i	$-0.280728\pi$
$769$	35.2548	1.27132	0.635661	+	0.771968i	$0.280728\pi$
$770$	0	0
$771$	−24.3379	−0.876508
$772$	0	0
$773$	−8.73606	−0.314214	−0.157107	−	0.987582i	$-0.550217\pi$
$773$	−8.73606	−0.314214	−0.157107	+	0.987582i	$0.550217\pi$
$774$	0	0
$775$	10.3431	0.371537
$776$	0	0
$777$	13.6569	0.489937
$778$	0	0
$779$	17.4721	0.626004
$780$	0	0
$781$	5.07241	0.181505
$782$	0	0
$783$	−20.6863	−0.739268
$784$	0	0
$785$	12.4853	0.445619
$786$	0	0
$787$	−7.46796	−0.266204	−0.133102	−	0.991102i	$-0.542494\pi$
$787$	−7.46796	−0.266204	−0.133102	+	0.991102i	$0.542494\pi$
$788$	0	0
$789$	−6.12293	−0.217982
$790$	0	0
$791$	−8.82843	−0.313903
$792$	0	0
$793$	−5.45584	−0.193743
$794$	0	0
$795$	−71.3742	−2.53138
$796$	0	0
$797$	−27.4763	−0.973260	−0.486630	−	0.873608i	$-0.661774\pi$
$797$	−27.4763	−0.973260	−0.486630	+	0.873608i	$0.661774\pi$
$798$	0	0
$799$	61.2548	2.16704
$800$	0	0
$801$	7.65685	0.270542
$802$	0	0
$803$	25.2346	0.890509
$804$	0	0
$805$	−17.8435	−0.628902
$806$	0	0
$807$	21.4558	0.755281
$808$	0	0
$809$	31.1716	1.09593	0.547967	−	0.836500i	$-0.315402\pi$
$809$	31.1716	1.09593	0.547967	+	0.836500i	$0.315402\pi$
$810$	0	0
$811$	−20.9819	−0.736775	−0.368388	−	0.929672i	$-0.620090\pi$
$811$	−20.9819	−0.736775	−0.368388	+	0.929672i	$0.620090\pi$
$812$	0	0
$813$	−6.12293	−0.214741
$814$	0	0
$815$	10.3431	0.362305
$816$	0	0
$817$	10.3431	0.361861
$818$	0	0
$819$	1.71644	0.0599774
$820$	0	0
$821$	8.65914	0.302206	0.151103	−	0.988518i	$-0.451717\pi$
$821$	8.65914	0.302206	0.151103	+	0.988518i	$0.451717\pi$
$822$	0	0
$823$	−40.9706	−1.42814	−0.714072	−	0.700072i	$-0.753151\pi$
$823$	−40.9706	−1.42814	−0.714072	+	0.700072i	$0.753151\pi$
$824$	0	0
$825$	−10.3431	−0.360102
$826$	0	0
$827$	−10.9778	−0.381734	−0.190867	−	0.981616i	$-0.561130\pi$
$827$	−10.9778	−0.381734	−0.190867	+	0.981616i	$0.561130\pi$
$828$	0	0
$829$	−6.19986	−0.215330	−0.107665	−	0.994187i	$-0.534337\pi$
$829$	−6.19986	−0.215330	−0.107665	+	0.994187i	$0.534337\pi$
$830$	0	0
$831$	−38.6274	−1.33997
$832$	0	0
$833$	−7.65685	−0.265294
$834$	0	0
$835$	41.8100	1.44690
$836$	0	0
$837$	12.2459	0.423279
$838$	0	0
$839$	−36.2843	−1.25267	−0.626336	−	0.779553i	$-0.715446\pi$
$839$	−36.2843	−1.25267	−0.626336	+	0.779553i	$0.715446\pi$
$840$	0	0
$841$	62.3137	2.14875
$842$	0	0
$843$	5.22625	0.180002
$844$	0	0
$845$	33.4454	1.15056
$846$	0	0
$847$	6.31371	0.216942
$848$	0	0
$849$	54.4264	1.86791
$850$	0	0
$851$	35.6871	1.22334
$852$	0	0
$853$	−47.3308	−1.62057	−0.810287	−	0.586033i	$-0.800689\pi$
$853$	−47.3308	−1.62057	−0.810287	+	0.586033i	$0.800689\pi$
$854$	0	0
$855$	−47.7990	−1.63469
$856$	0	0
$857$	−0.627417	−0.0214322	−0.0107161	−	0.999943i	$-0.503411\pi$
$857$	−0.627417	−0.0214322	−0.0107161	+	0.999943i	$0.503411\pi$
$858$	0	0
$859$	−3.50981	−0.119753	−0.0598766	−	0.998206i	$-0.519071\pi$
$859$	−3.50981	−0.119753	−0.0598766	+	0.998206i	$0.519071\pi$
$860$	0	0
$861$	−9.55582	−0.325661
$862$	0	0
$863$	−51.3137	−1.74674	−0.873369	−	0.487058i	$-0.838070\pi$
$863$	−51.3137	−1.74674	−0.873369	+	0.487058i	$0.838070\pi$
$864$	0	0
$865$	−42.1421	−1.43288
$866$	0	0
$867$	108.778	3.69428
$868$	0	0
$869$	5.07241	0.172070
$870$	0	0
$871$	−1.37258	−0.0465082
$872$	0	0
$873$	−38.2843	−1.29573
$874$	0	0
$875$	−8.28772	−0.280176
$876$	0	0
$877$	−32.2542	−1.08915	−0.544573	−	0.838713i	$-0.683308\pi$
$877$	−32.2542	−1.08915	−0.544573	+	0.838713i	$0.683308\pi$
$878$	0	0
$879$	28.4853	0.960785
$880$	0	0
$881$	10.9706	0.369608	0.184804	−	0.982775i	$-0.440835\pi$
$881$	10.9706	0.369608	0.184804	+	0.982775i	$0.440835\pi$
$882$	0	0
$883$	51.7373	1.74110	0.870549	−	0.492082i	$-0.163764\pi$
$883$	51.7373	1.74110	0.870549	+	0.492082i	$0.163764\pi$
$884$	0	0
$885$	−3.06147	−0.102910
$886$	0	0
$887$	22.6274	0.759754	0.379877	−	0.925037i	$-0.375966\pi$
$887$	22.6274	0.759754	0.379877	+	0.925037i	$0.375966\pi$
$888$	0	0
$889$	4.48528	0.150432
$890$	0	0
$891$	12.6173	0.422695
$892$	0	0
$893$	−38.2233	−1.27909
$894$	0	0
$895$	−57.9411	−1.93676
$896$	0	0
$897$	8.00000	0.267112
$898$	0	0
$899$	−54.0559	−1.80286
$900$	0	0
$901$	−80.0333	−2.66630
$902$	0	0
$903$	−5.65685	−0.188248
$904$	0	0
$905$	−6.82843	−0.226985
$906$	0	0
$907$	1.26810	0.0421066	0.0210533	−	0.999778i	$-0.493298\pi$
$907$	1.26810	0.0421066	0.0210533	+	0.999778i	$0.493298\pi$
$908$	0	0
$909$	18.2919	0.606703
$910$	0	0
$911$	−38.8284	−1.28644	−0.643222	−	0.765680i	$-0.722403\pi$
$911$	−38.8284	−1.28644	−0.643222	+	0.765680i	$0.722403\pi$
$912$	0	0
$913$	28.2843	0.936073
$914$	0	0
$915$	−83.0948	−2.74703
$916$	0	0
$917$	−18.2919	−0.604051
$918$	0	0
$919$	−2.34315	−0.0772932	−0.0386466	−	0.999253i	$-0.512305\pi$
$919$	−2.34315	−0.0772932	−0.0386466	+	0.999253i	$0.512305\pi$
$920$	0	0
$921$	−22.8284	−0.752222
$922$	0	0
$923$	1.05053	0.0345786
$924$	0	0
$925$	−9.55582	−0.314193
$926$	0	0
$927$	−52.2843	−1.71724
$928$	0	0
$929$	10.2843	0.337416	0.168708	−	0.985666i	$-0.446041\pi$
$929$	10.2843	0.337416	0.168708	+	0.985666i	$0.446041\pi$
$930$	0	0
$931$	4.77791	0.156590
$932$	0	0
$933$	−38.2233	−1.25137
$934$	0	0
$935$	−43.3137	−1.41651
$936$	0	0
$937$	38.2843	1.25069	0.625346	−	0.780347i	$-0.284958\pi$
$937$	38.2843	1.25069	0.625346	+	0.780347i	$0.284958\pi$
$938$	0	0
$939$	21.8017	0.711471
$940$	0	0
$941$	−37.7749	−1.23143	−0.615714	−	0.787970i	$-0.711132\pi$
$941$	−37.7749	−1.23143	−0.615714	+	0.787970i	$0.711132\pi$
$942$	0	0
$943$	−24.9706	−0.813153
$944$	0	0
$945$	5.65685	0.184017
$946$	0	0
$947$	48.3044	1.56968	0.784841	−	0.619698i	$-0.212745\pi$
$947$	48.3044	1.56968	0.784841	+	0.619698i	$0.212745\pi$
$948$	0	0
$949$	5.22625	0.169651
$950$	0	0
$951$	−22.6274	−0.733744
$952$	0	0
$953$	−37.3137	−1.20871	−0.604355	−	0.796715i	$-0.706569\pi$
$953$	−37.3137	−1.20871	−0.604355	+	0.796715i	$0.706569\pi$
$954$	0	0
$955$	−50.4692	−1.63314
$956$	0	0
$957$	54.0559	1.74738
$958$	0	0
$959$	2.00000	0.0645834
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−51.7373	−1.66721
$964$	0	0
$965$	−36.0585	−1.16076
$966$	0	0
$967$	29.8579	0.960164	0.480082	−	0.877224i	$-0.340607\pi$
$967$	29.8579	0.960164	0.480082	+	0.877224i	$0.340607\pi$
$968$	0	0
$969$	−95.5980	−3.07105
$970$	0	0
$971$	−52.5570	−1.68663	−0.843317	−	0.537416i	$-0.819401\pi$
$971$	−52.5570	−1.68663	−0.843317	+	0.537416i	$0.819401\pi$
$972$	0	0
$973$	19.5600	0.627064
$974$	0	0
$975$	−2.14214	−0.0686032
$976$	0	0
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	−4.32957	−0.138374
$980$	0	0
$981$	43.4495	1.38724
$982$	0	0
$983$	−23.0294	−0.734525	−0.367262	−	0.930117i	$-0.619705\pi$
$983$	−23.0294	−0.734525	−0.367262	+	0.930117i	$0.619705\pi$
$984$	0	0
$985$	16.0000	0.509802
$986$	0	0
$987$	20.9050	0.665414
$988$	0	0
$989$	−14.7821	−0.470043
$990$	0	0
$991$	−29.6569	−0.942081	−0.471041	−	0.882112i	$-0.656121\pi$
$991$	−29.6569	−0.942081	−0.471041	+	0.882112i	$0.656121\pi$
$992$	0	0
$993$	−48.9706	−1.55403
$994$	0	0
$995$	23.4412	0.743136
$996$	0	0
$997$	−0.0769232	−0.00243618	−0.00121809	−	0.999999i	$-0.500388\pi$
$997$	−0.0769232	−0.00243618	−0.00121809	+	0.999999i	$0.500388\pi$
$998$	0	0
$999$	−11.3137	−0.357950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.a.u.1.4		4
4.3	odd	2		1792.2.a.w.1.1		4
8.3	odd	2		1792.2.a.w.1.4		4
8.5	even	2	inner	1792.2.a.u.1.1		4
16.3	odd	4		896.2.b.f.449.1	✓	4
16.5	even	4		896.2.b.h.449.1	yes	4
16.11	odd	4		896.2.b.f.449.4	yes	4
16.13	even	4		896.2.b.h.449.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.b.f.449.1	✓	4	16.3	odd	4
896.2.b.f.449.4	yes	4	16.11	odd	4
896.2.b.h.449.1	yes	4	16.5	even	4
896.2.b.h.449.4	yes	4	16.13	even	4
1792.2.a.u.1.1		4	8.5	even	2	inner
1792.2.a.u.1.4		4	1.1	even	1	trivial
1792.2.a.w.1.1		4	4.3	odd	2
1792.2.a.w.1.4		4	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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.a (trivial)

\(f(q)\)	\(=\)	\(q+2.61313 q^{3} -2.61313 q^{5} -1.00000 q^{7} +3.82843 q^{9} +O(q^{10})\)	\(q+2.61313 q^{3} -2.61313 q^{5} -1.00000 q^{7} +3.82843 q^{9} -2.16478 q^{11} -0.448342 q^{13} -6.82843 q^{15} -7.65685 q^{17} +4.77791 q^{19} -2.61313 q^{21} -6.82843 q^{23} +1.82843 q^{25} +2.16478 q^{27} -9.55582 q^{29} +5.65685 q^{31} -5.65685 q^{33} +2.61313 q^{35} -5.22625 q^{37} -1.17157 q^{39} +3.65685 q^{41} +2.16478 q^{43} -10.0042 q^{45} -8.00000 q^{47} +1.00000 q^{49} -20.0083 q^{51} +10.4525 q^{53} +5.65685 q^{55} +12.4853 q^{57} +0.448342 q^{59} +12.1689 q^{61} -3.82843 q^{63} +1.17157 q^{65} +3.06147 q^{67} -17.8435 q^{69} -2.34315 q^{71} -11.6569 q^{73} +4.77791 q^{75} +2.16478 q^{77} -2.34315 q^{79} -5.82843 q^{81} -13.0656 q^{83} +20.0083 q^{85} -24.9706 q^{87} +2.00000 q^{89} +0.448342 q^{91} +14.7821 q^{93} -12.4853 q^{95} -10.0000 q^{97} -8.28772 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{7} + 4 q^{9} - 16 q^{15} - 8 q^{17} - 16 q^{23} - 4 q^{25} - 16 q^{39} - 8 q^{41} - 32 q^{47} + 4 q^{49} + 16 q^{57} - 4 q^{63} + 16 q^{65} - 32 q^{71} - 24 q^{73} - 32 q^{79} - 12 q^{81} - 32 q^{87} + 8 q^{89} - 16 q^{95} - 40 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 + 4 * q^9 - 16 * q^15 - 8 * q^17 - 16 * q^23 - 4 * q^25 - 16 * q^39 - 8 * q^41 - 32 * q^47 + 4 * q^49 + 16 * q^57 - 4 * q^63 + 16 * q^65 - 32 * q^71 - 24 * q^73 - 32 * q^79 - 12 * q^81 - 32 * q^87 + 8 * q^89 - 16 * q^95 - 40 * q^97

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