LMFDB - Embedded newform 896.2.b.h.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [896,2,Mod(449,896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(896, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("896.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 896 = 2^{7} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	896.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$7.15459602111$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$0.765367i$ of defining polynomial
Character	$\chi$	$=$	896.449
Dual form			896.2.b.h.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/896\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$645$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	2.61313i	1.50869i	0.656479	+	0.754344i	$0.272045\pi$
$3$	2.61313i	1.50869i	−0.656479	+	0.754344i	$0.727955\pi$
$4$	0	0
$5$	2.61313i	1.16863i	0.811529	+	0.584313i	$0.198636\pi$
$5$	2.61313i	1.16863i	−0.811529	+	0.584313i	$0.801364\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−3.82843	−1.27614
$10$	0	0
$11$	2.16478i	0.652707i	0.945248	+	0.326354i	$0.105820\pi$
$11$	2.16478i	0.652707i	−0.945248	+	0.326354i	$0.894180\pi$
$12$	0	0
$13$	− 0.448342i	− 0.124348i	−0.998065	−	0.0621738i	$-0.980197\pi$
$13$	− 0.448342i	− 0.124348i	0.998065	−	0.0621738i	$-0.0198033\pi$
$14$	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
$19$	4.77791i	1.09613i	0.836436	+	0.548064i	$0.184635\pi$
$19$	4.77791i	1.09613i	−0.836436	+	0.548064i	$0.815365\pi$
$20$	0	0
$21$	2.61313i	0.570231i
$22$	0	0
$23$	6.82843	1.42383	0.711913	−	0.702268i	$-0.247829\pi$
$23$	6.82843	1.42383	0.711913	+	0.702268i	$0.247829\pi$
$24$	0	0
$25$	−1.82843	−0.365685
$26$	0	0
$27$	− 2.16478i	− 0.416613i
$28$	0	0
$29$	− 9.55582i	− 1.77447i	−0.461316	−	0.887236i	$-0.652623\pi$
$29$	− 9.55582i	− 1.77447i	0.461316	−	0.887236i	$-0.347377\pi$
$30$	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
$35$	2.61313i	0.441699i
$36$	0	0
$37$	5.22625i	0.859191i	0.903022	+	0.429595i	$0.141344\pi$
$37$	5.22625i	0.859191i	−0.903022	+	0.429595i	$0.858656\pi$
$38$	0	0
$39$	1.17157	0.187602
$40$	0	0
$41$	−3.65685	−0.571105	−0.285552	−	0.958363i	$-0.592177\pi$
$41$	−3.65685	−0.571105	−0.285552	+	0.958363i	$0.592177\pi$
$42$	0	0
$43$	− 2.16478i	− 0.330127i	−0.986283	−	0.165063i	$-0.947217\pi$
$43$	− 2.16478i	− 0.330127i	0.986283	−	0.165063i	$-0.0527828\pi$
$44$	0	0
$45$	− 10.0042i	− 1.49133i
$46$	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
$51$	− 20.0083i	− 2.80173i
$52$	0	0
$53$	− 10.4525i	− 1.43576i	−0.696166	−	0.717881i	$-0.745112\pi$
$53$	− 10.4525i	− 1.43576i	0.696166	−	0.717881i	$-0.254888\pi$
$54$	0	0
$55$	−5.65685	−0.762770
$56$	0	0
$57$	−12.4853	−1.65372
$58$	0	0
$59$	− 0.448342i	− 0.0583691i	−0.999574	−	0.0291845i	$-0.990709\pi$
$59$	− 0.448342i	− 0.0583691i	0.999574	−	0.0291845i	$-0.00929105\pi$
$60$	0	0
$61$	12.1689i	1.55807i	0.626978	+	0.779037i	$0.284292\pi$
$61$	12.1689i	1.55807i	−0.626978	+	0.779037i	$0.715708\pi$
$62$	0	0
$63$	−3.82843	−0.482336
$64$	0	0
$65$	1.17157	0.145316
$66$	0	0
$67$	3.06147i	0.374018i	0.982358	+	0.187009i	$0.0598793\pi$
$67$	3.06147i	0.374018i	−0.982358	+	0.187009i	$0.940121\pi$
$68$	0	0
$69$	17.8435i	2.14811i
$70$	0	0
$71$	2.34315	0.278080	0.139040	−	0.990287i	$-0.455598\pi$
$71$	2.34315	0.278080	0.139040	+	0.990287i	$0.455598\pi$
$72$	0	0
$73$	11.6569	1.36433	0.682166	−	0.731198i	$-0.261038\pi$
$73$	11.6569	1.36433	0.682166	+	0.731198i	$0.261038\pi$
$74$	0	0
$75$	− 4.77791i	− 0.551706i
$76$	0	0
$77$	2.16478i	0.246700i
$78$	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
$83$	− 13.0656i	− 1.43414i	−0.697002	−	0.717070i	$-0.745483\pi$
$83$	− 13.0656i	− 1.43414i	0.697002	−	0.717070i	$-0.254517\pi$
$84$	0	0
$85$	− 20.0083i	− 2.17021i
$86$	0	0
$87$	24.9706	2.67713
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	− 0.448342i	− 0.0469990i
$92$	0	0
$93$	14.7821i	1.53283i
$94$	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
$99$	− 8.28772i	− 0.832947i
$100$	0	0
$101$	− 4.77791i	− 0.475420i	−0.971336	−	0.237710i	$-0.923603\pi$
$101$	− 4.77791i	− 0.475420i	0.971336	−	0.237710i	$-0.0763968\pi$
$102$	0	0
$103$	13.6569	1.34565	0.672825	−	0.739802i	$-0.265081\pi$
$103$	13.6569	1.34565	0.672825	+	0.739802i	$0.265081\pi$
$104$	0	0
$105$	−6.82843	−0.666386
$106$	0	0
$107$	13.5140i	1.30644i	0.757166	+	0.653222i	$0.226583\pi$
$107$	13.5140i	1.30644i	−0.757166	+	0.653222i	$0.773417\pi$
$108$	0	0
$109$	11.3492i	1.08705i	0.839391	+	0.543527i	$0.182912\pi$
$109$	11.3492i	1.08705i	−0.839391	+	0.543527i	$0.817088\pi$
$110$	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
$115$	17.8435i	1.66392i
$116$	0	0
$117$	1.71644i	0.158685i
$118$	0	0
$119$	−7.65685	−0.701903
$120$	0	0
$121$	6.31371	0.573973
$122$	0	0
$123$	− 9.55582i	− 0.861619i
$124$	0	0
$125$	8.28772i	0.741276i
$126$	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
$131$	18.2919i	1.59817i	0.601219	+	0.799085i	$0.294682\pi$
$131$	18.2919i	1.59817i	−0.601219	+	0.799085i	$0.705318\pi$
$132$	0	0
$133$	4.77791i	0.414297i
$134$	0	0
$135$	5.65685	0.486864
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	19.5600i	1.65906i	0.558465	+	0.829528i	$0.311390\pi$
$139$	19.5600i	1.65906i	−0.558465	+	0.829528i	$0.688610\pi$
$140$	0	0
$141$	− 20.9050i	− 1.76052i
$142$	0	0
$143$	0.970563	0.0811625
$144$	0	0
$145$	24.9706	2.07369
$146$	0	0
$147$	2.61313i	0.215527i
$148$	0	0
$149$	− 16.5754i	− 1.35791i	−0.734179	−	0.678956i	$-0.762433\pi$
$149$	− 16.5754i	− 1.35791i	0.734179	−	0.678956i	$-0.237567\pi$
$150$	0	0
$151$	−18.1421	−1.47639	−0.738193	−	0.674590i	$-0.764321\pi$
$151$	−18.1421	−1.47639	−0.738193	+	0.674590i	$0.764321\pi$
$152$	0	0
$153$	29.3137	2.36987
$154$	0	0
$155$	14.7821i	1.18732i
$156$	0	0
$157$	− 4.77791i	− 0.381319i	−0.981656	−	0.190659i	$-0.938937\pi$
$157$	− 4.77791i	− 0.381319i	0.981656	−	0.190659i	$-0.0610626\pi$
$158$	0	0
$159$	27.3137	2.16612
$160$	0	0
$161$	6.82843	0.538155
$162$	0	0
$163$	− 3.95815i	− 0.310026i	−0.987912	−	0.155013i	$-0.950458\pi$
$163$	− 3.95815i	− 0.310026i	0.987912	−	0.155013i	$-0.0495420\pi$
$164$	0	0
$165$	− 14.7821i	− 1.15078i
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	12.7990	0.984538
$170$	0	0
$171$	− 18.2919i	− 1.39882i
$172$	0	0
$173$	16.1271i	1.22612i	0.790036	+	0.613060i	$0.210062\pi$
$173$	16.1271i	1.22612i	−0.790036	+	0.613060i	$0.789938\pi$
$174$	0	0
$175$	−1.82843	−0.138216
$176$	0	0
$177$	1.17157	0.0880608
$178$	0	0
$179$	22.1731i	1.65730i	0.559770	+	0.828648i	$0.310889\pi$
$179$	22.1731i	1.65730i	−0.559770	+	0.828648i	$0.689111\pi$
$180$	0	0
$181$	− 2.61313i	− 0.194232i	−0.995273	−	0.0971161i	$-0.969038\pi$
$181$	− 2.61313i	− 0.194232i	0.995273	−	0.0971161i	$-0.0309618\pi$
$182$	0	0
$183$	−31.7990	−2.35065
$184$	0	0
$185$	−13.6569	−1.00407
$186$	0	0
$187$	− 16.5754i	− 1.21212i
$188$	0	0
$189$	− 2.16478i	− 0.157465i
$190$	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
$195$	3.06147i	0.219236i
$196$	0	0
$197$	6.12293i	0.436241i	0.975922	+	0.218121i	$0.0699926\pi$
$197$	6.12293i	0.436241i	−0.975922	+	0.218121i	$0.930007\pi$
$198$	0	0
$199$	8.97056	0.635906	0.317953	−	0.948106i	$-0.397005\pi$
$199$	8.97056	0.635906	0.317953	+	0.948106i	$0.397005\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	− 9.55582i	− 0.670687i
$204$	0	0
$205$	− 9.55582i	− 0.667407i
$206$	0	0
$207$	−26.1421	−1.81700
$208$	0	0
$209$	−10.3431	−0.715450
$210$	0	0
$211$	9.18440i	0.632280i	0.948712	+	0.316140i	$0.102387\pi$
$211$	9.18440i	0.632280i	−0.948712	+	0.316140i	$0.897613\pi$
$212$	0	0
$213$	6.12293i	0.419537i
$214$	0	0
$215$	5.65685	0.385794
$216$	0	0
$217$	5.65685	0.384012
$218$	0	0
$219$	30.4608i	2.05835i
$220$	0	0
$221$	3.43289i	0.230921i
$222$	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
$227$	− 14.3337i	− 0.951363i	−0.879618	−	0.475682i	$-0.842201\pi$
$227$	− 14.3337i	− 0.951363i	0.879618	−	0.475682i	$-0.157799\pi$
$228$	0	0
$229$	− 5.67459i	− 0.374988i	−0.982266	−	0.187494i	$-0.939964\pi$
$229$	− 5.67459i	− 0.374988i	0.982266	−	0.187494i	$-0.0600365\pi$
$230$	0	0
$231$	−5.65685	−0.372194
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	− 20.9050i	− 1.36369i
$236$	0	0
$237$	− 6.12293i	− 0.397727i
$238$	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
$243$	− 21.7248i	− 1.39364i
$244$	0	0
$245$	2.61313i	0.166946i
$246$	0	0
$247$	2.14214	0.136301
$248$	0	0
$249$	34.1421	2.16367
$250$	0	0
$251$	− 0.819760i	− 0.0517428i	−0.999665	−	0.0258714i	$-0.991764\pi$
$251$	− 0.819760i	− 0.0517428i	0.999665	−	0.0258714i	$-0.00823604\pi$
$252$	0	0
$253$	14.7821i	0.929341i
$254$	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
$259$	5.22625i	0.324743i
$260$	0	0
$261$	36.5838i	2.26448i
$262$	0	0
$263$	2.34315	0.144485	0.0722423	−	0.997387i	$-0.476985\pi$
$263$	2.34315	0.144485	0.0722423	+	0.997387i	$0.476985\pi$
$264$	0	0
$265$	27.3137	1.67787
$266$	0	0
$267$	− 5.22625i	− 0.319841i
$268$	0	0
$269$	8.21080i	0.500621i	0.968166	+	0.250311i	$0.0805327\pi$
$269$	8.21080i	0.500621i	−0.968166	+	0.250311i	$0.919467\pi$
$270$	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
$275$	− 3.95815i	− 0.238685i
$276$	0	0
$277$	14.7821i	0.888169i	0.895985	+	0.444084i	$0.146471\pi$
$277$	14.7821i	0.888169i	−0.895985	+	0.444084i	$0.853529\pi$
$278$	0	0
$279$	−21.6569	−1.29656
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	− 20.8281i	− 1.23810i	−0.785351	−	0.619051i	$-0.787518\pi$
$283$	− 20.8281i	− 1.23810i	0.785351	−	0.619051i	$-0.212482\pi$
$284$	0	0
$285$	− 32.6256i	− 1.93257i
$286$	0	0
$287$	−3.65685	−0.215857
$288$	0	0
$289$	41.6274	2.44867
$290$	0	0
$291$	− 26.1313i	− 1.53184i
$292$	0	0
$293$	− 10.9008i	− 0.636834i	−0.947951	−	0.318417i	$-0.896849\pi$
$293$	− 10.9008i	− 0.636834i	0.947951	−	0.318417i	$-0.103151\pi$
$294$	0	0
$295$	1.17157	0.0682116
$296$	0	0
$297$	4.68629	0.271926
$298$	0	0
$299$	− 3.06147i	− 0.177049i
$300$	0	0
$301$	− 2.16478i	− 0.124776i
$302$	0	0
$303$	12.4853	0.717261
$304$	0	0
$305$	−31.7990	−1.82080
$306$	0	0
$307$	− 8.73606i	− 0.498593i	−0.968427	−	0.249297i	$-0.919801\pi$
$307$	− 8.73606i	− 0.498593i	0.968427	−	0.249297i	$-0.0801994\pi$
$308$	0	0
$309$	35.6871i	2.03017i
$310$	0	0
$311$	14.6274	0.829445	0.414722	−	0.909948i	$-0.363879\pi$
$311$	14.6274	0.829445	0.414722	+	0.909948i	$0.363879\pi$
$312$	0	0
$313$	−8.34315	−0.471582	−0.235791	−	0.971804i	$-0.575768\pi$
$313$	−8.34315	−0.471582	−0.235791	+	0.971804i	$0.575768\pi$
$314$	0	0
$315$	− 10.0042i	− 0.563671i
$316$	0	0
$317$	− 8.65914i	− 0.486346i	−0.969983	−	0.243173i	$-0.921812\pi$
$317$	− 8.65914i	− 0.486346i	0.969983	−	0.243173i	$-0.0781882\pi$
$318$	0	0
$319$	20.6863	1.15821
$320$	0	0
$321$	−35.3137	−1.97102
$322$	0	0
$323$	− 36.5838i	− 2.03558i
$324$	0	0
$325$	0.819760i	0.0454721i
$326$	0	0
$327$	−29.6569	−1.64003
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	18.7402i	1.03006i	0.857173	+	0.515028i	$0.172218\pi$
$331$	18.7402i	1.03006i	−0.857173	+	0.515028i	$0.827782\pi$
$332$	0	0
$333$	− 20.0083i	− 1.09645i
$334$	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
$339$	23.0698i	1.25298i
$340$	0	0
$341$	12.2459i	0.663151i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−46.6274	−2.51034
$346$	0	0
$347$	12.6173i	0.677332i	0.940907	+	0.338666i	$0.109976\pi$
$347$	12.6173i	0.677332i	−0.940907	+	0.338666i	$0.890024\pi$
$348$	0	0
$349$	− 22.6215i	− 1.21090i	−0.795884	−	0.605449i	$-0.792993\pi$
$349$	− 22.6215i	− 1.21090i	0.795884	−	0.605449i	$-0.207007\pi$
$350$	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
$355$	6.12293i	0.324972i
$356$	0	0
$357$	− 20.0083i	− 1.05895i
$358$	0	0
$359$	6.82843	0.360391	0.180195	−	0.983631i	$-0.442327\pi$
$359$	6.82843	0.360391	0.180195	+	0.983631i	$0.442327\pi$
$360$	0	0
$361$	−3.82843	−0.201496
$362$	0	0
$363$	16.4985i	0.865947i
$364$	0	0
$365$	30.4608i	1.59439i
$366$	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
$371$	− 10.4525i	− 0.542667i
$372$	0	0
$373$	− 33.8937i	− 1.75495i	−0.479622	−	0.877475i	$-0.659226\pi$
$373$	− 33.8937i	− 1.75495i	0.479622	−	0.877475i	$-0.340774\pi$
$374$	0	0
$375$	−21.6569	−1.11836
$376$	0	0
$377$	−4.28427	−0.220651
$378$	0	0
$379$	23.0698i	1.18502i	0.805565	+	0.592508i	$0.201862\pi$
$379$	23.0698i	1.18502i	−0.805565	+	0.592508i	$0.798138\pi$
$380$	0	0
$381$	− 11.7206i	− 0.600465i
$382$	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
$387$	8.28772i	0.421288i
$388$	0	0
$389$	20.0083i	1.01446i	0.861810	+	0.507231i	$0.169331\pi$
$389$	20.0083i	1.01446i	−0.861810	+	0.507231i	$0.830669\pi$
$390$	0	0
$391$	−52.2843	−2.64413
$392$	0	0
$393$	−47.7990	−2.41114
$394$	0	0
$395$	− 6.12293i	− 0.308078i
$396$	0	0
$397$	− 18.2919i	− 0.918043i	−0.888425	−	0.459022i	$-0.848200\pi$
$397$	− 18.2919i	− 0.918043i	0.888425	−	0.459022i	$-0.151800\pi$
$398$	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
$403$	− 2.53620i	− 0.126337i
$404$	0	0
$405$	− 15.2304i	− 0.756805i
$406$	0	0
$407$	−11.3137	−0.560800
$408$	0	0
$409$	5.31371	0.262746	0.131373	−	0.991333i	$-0.458061\pi$
$409$	5.31371	0.262746	0.131373	+	0.991333i	$0.458061\pi$
$410$	0	0
$411$	5.22625i	0.257792i
$412$	0	0
$413$	− 0.448342i	− 0.0220614i
$414$	0	0
$415$	34.1421	1.67597
$416$	0	0
$417$	−51.1127	−2.50300
$418$	0	0
$419$	13.0656i	0.638298i	0.947705	+	0.319149i	$0.103397\pi$
$419$	13.0656i	0.638298i	−0.947705	+	0.319149i	$0.896603\pi$
$420$	0	0
$421$	2.53620i	0.123607i	0.998088	+	0.0618035i	$0.0196852\pi$
$421$	2.53620i	0.123607i	−0.998088	+	0.0618035i	$0.980315\pi$
$422$	0	0
$423$	30.6274	1.48916
$424$	0	0
$425$	14.0000	0.679100
$426$	0	0
$427$	12.1689i	0.588897i
$428$	0	0
$429$	2.53620i	0.122449i
$430$	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
$435$	65.2512i	3.12856i
$436$	0	0
$437$	32.6256i	1.56069i
$438$	0	0
$439$	0.970563	0.0463224	0.0231612	−	0.999732i	$-0.492627\pi$
$439$	0.970563	0.0463224	0.0231612	+	0.999732i	$0.492627\pi$
$440$	0	0
$441$	−3.82843	−0.182306
$442$	0	0
$443$	− 9.18440i	− 0.436364i	−0.975908	−	0.218182i	$-0.929987\pi$
$443$	− 9.18440i	− 0.436364i	0.975908	−	0.218182i	$-0.0700127\pi$
$444$	0	0
$445$	− 5.22625i	− 0.247748i
$446$	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
$451$	− 7.91630i	− 0.372764i
$452$	0	0
$453$	− 47.4077i	− 2.22741i
$454$	0	0
$455$	1.17157	0.0549242
$456$	0	0
$457$	−10.4853	−0.490481	−0.245240	−	0.969462i	$-0.578867\pi$
$457$	−10.4853	−0.490481	−0.245240	+	0.969462i	$0.578867\pi$
$458$	0	0
$459$	16.5754i	0.773675i
$460$	0	0
$461$	− 10.3756i	− 0.483239i	−0.970371	−	0.241619i	$-0.922321\pi$
$461$	− 10.3756i	− 0.483239i	0.970371	−	0.241619i	$-0.0776786\pi$
$462$	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
$467$	− 1.34502i	− 0.0622403i	−0.999516	−	0.0311202i	$-0.990093\pi$
$467$	− 1.34502i	− 0.0622403i	0.999516	−	0.0311202i	$-0.00990745\pi$
$468$	0	0
$469$	3.06147i	0.141365i
$470$	0	0
$471$	12.4853	0.575291
$472$	0	0
$473$	4.68629	0.215476
$474$	0	0
$475$	− 8.73606i	− 0.400838i
$476$	0	0
$477$	40.0166i	1.83224i
$478$	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
$483$	17.8435i	0.811909i
$484$	0	0
$485$	− 26.1313i	− 1.18656i
$486$	0	0
$487$	−31.7990	−1.44095	−0.720475	−	0.693481i	$-0.756076\pi$
$487$	−31.7990	−1.44095	−0.720475	+	0.693481i	$0.756076\pi$
$488$	0	0
$489$	10.3431	0.467733
$490$	0	0
$491$	− 32.6256i	− 1.47237i	−0.676779	−	0.736187i	$-0.736625\pi$
$491$	− 32.6256i	− 1.47237i	0.676779	−	0.736187i	$-0.263375\pi$
$492$	0	0
$493$	73.1675i	3.29530i
$494$	0	0
$495$	21.6569	0.973403
$496$	0	0
$497$	2.34315	0.105104
$498$	0	0
$499$	− 11.7206i	− 0.524686i	−0.964975	−	0.262343i	$-0.915505\pi$
$499$	− 11.7206i	− 0.524686i	0.964975	−	0.262343i	$-0.0844952\pi$
$500$	0	0
$501$	41.8100i	1.86793i
$502$	0	0
$503$	5.65685	0.252227	0.126113	−	0.992016i	$-0.459750\pi$
$503$	5.65685	0.252227	0.126113	+	0.992016i	$0.459750\pi$
$504$	0	0
$505$	12.4853	0.555588
$506$	0	0
$507$	33.4454i	1.48536i
$508$	0	0
$509$	− 3.50981i	− 0.155570i	−0.996970	−	0.0777848i	$-0.975215\pi$
$509$	− 3.50981i	− 0.155570i	0.996970	−	0.0777848i	$-0.0247847\pi$
$510$	0	0
$511$	11.6569	0.515669
$512$	0	0
$513$	10.3431	0.456661
$514$	0	0
$515$	35.6871i	1.57256i
$516$	0	0
$517$	− 17.3183i	− 0.761657i
$518$	0	0
$519$	−42.1421	−1.84983
$520$	0	0
$521$	14.6863	0.643418	0.321709	−	0.946839i	$-0.395743\pi$
$521$	14.6863	0.643418	0.321709	+	0.946839i	$0.395743\pi$
$522$	0	0
$523$	7.46796i	0.326551i	0.986581	+	0.163276i	$0.0522060\pi$
$523$	7.46796i	0.326551i	−0.986581	+	0.163276i	$0.947794\pi$
$524$	0	0
$525$	− 4.77791i	− 0.208525i
$526$	0	0
$527$	−43.3137	−1.88677
$528$	0	0
$529$	23.6274	1.02728
$530$	0	0
$531$	1.71644i	0.0744873i
$532$	0	0
$533$	1.63952i	0.0710155i
$534$	0	0
$535$	−35.3137	−1.52674
$536$	0	0
$537$	−57.9411	−2.50034
$538$	0	0
$539$	2.16478i	0.0932439i
$540$	0	0
$541$	− 31.3575i	− 1.34816i	−0.738656	−	0.674082i	$-0.764539\pi$
$541$	− 31.3575i	− 1.34816i	0.738656	−	0.674082i	$-0.235461\pi$
$542$	0	0
$543$	6.82843	0.293036
$544$	0	0
$545$	−29.6569	−1.27036
$546$	0	0
$547$	− 43.9748i	− 1.88023i	−0.340862	−	0.940113i	$-0.610719\pi$
$547$	− 43.9748i	− 1.88023i	0.340862	−	0.940113i	$-0.389281\pi$
$548$	0	0
$549$	− 46.5879i	− 1.98832i
$550$	0	0
$551$	45.6569	1.94505
$552$	0	0
$553$	−2.34315	−0.0996407
$554$	0	0
$555$	− 35.6871i	− 1.51483i
$556$	0	0
$557$	14.7821i	0.626337i	0.949698	+	0.313168i	$0.101390\pi$
$557$	14.7821i	0.626337i	−0.949698	+	0.313168i	$0.898610\pi$
$558$	0	0
$559$	−0.970563	−0.0410504
$560$	0	0
$561$	43.3137	1.82871
$562$	0	0
$563$	− 17.7666i	− 0.748774i	−0.927273	−	0.374387i	$-0.877853\pi$
$563$	− 17.7666i	− 0.748774i	0.927273	−	0.374387i	$-0.122147\pi$
$564$	0	0
$565$	23.0698i	0.970553i
$566$	0	0
$567$	−5.82843	−0.244771
$568$	0	0
$569$	−17.5147	−0.734255	−0.367128	−	0.930171i	$-0.619659\pi$
$569$	−17.5147	−0.734255	−0.367128	+	0.930171i	$0.619659\pi$
$570$	0	0
$571$	− 18.7402i	− 0.784254i	−0.919911	−	0.392127i	$-0.871739\pi$
$571$	− 18.7402i	− 0.784254i	0.919911	−	0.392127i	$-0.128261\pi$
$572$	0	0
$573$	50.4692i	2.10838i
$574$	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
$579$	36.0585i	1.49854i
$580$	0	0
$581$	− 13.0656i	− 0.542054i
$582$	0	0
$583$	22.6274	0.937132
$584$	0	0
$585$	−4.48528	−0.185444
$586$	0	0
$587$	− 33.4454i	− 1.38044i	−0.723600	−	0.690219i	$-0.757514\pi$
$587$	− 33.4454i	− 1.38044i	0.723600	−	0.690219i	$-0.242486\pi$
$588$	0	0
$589$	27.0279i	1.11367i
$590$	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
$595$	− 20.0083i	− 0.820261i
$596$	0	0
$597$	23.4412i	0.959385i
$598$	0	0
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	41.3137	1.68522	0.842611	−	0.538523i	$-0.181018\pi$
$601$	41.3137	1.68522	0.842611	+	0.538523i	$0.181018\pi$
$602$	0	0
$603$	− 11.7206i	− 0.477300i
$604$	0	0
$605$	16.4985i	0.670760i
$606$	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
$611$	3.58673i	0.145104i
$612$	0	0
$613$	− 26.1313i	− 1.05543i	−0.849421	−	0.527716i	$-0.823049\pi$
$613$	− 26.1313i	− 1.05543i	0.849421	−	0.527716i	$-0.176951\pi$
$614$	0	0
$615$	24.9706	1.00691
$616$	0	0
$617$	−41.7990	−1.68276	−0.841382	−	0.540441i	$-0.818257\pi$
$617$	−41.7990	−1.68276	−0.841382	+	0.540441i	$0.818257\pi$
$618$	0	0
$619$	33.9706i	1.36540i	0.730701	+	0.682698i	$0.239193\pi$
$619$	33.9706i	1.36540i	−0.730701	+	0.682698i	$0.760807\pi$
$620$	0	0
$621$	− 14.7821i	− 0.593184i
$622$	0	0
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	−30.7990	−1.23196
$626$	0	0
$627$	− 27.0279i	− 1.07939i
$628$	0	0
$629$	− 40.0166i	− 1.59557i
$630$	0	0
$631$	30.6274	1.21926	0.609629	−	0.792687i	$-0.291318\pi$
$631$	30.6274	1.21926	0.609629	+	0.792687i	$0.291318\pi$
$632$	0	0
$633$	−24.0000	−0.953914
$634$	0	0
$635$	− 11.7206i	− 0.465118i
$636$	0	0
$637$	− 0.448342i	− 0.0177639i
$638$	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
$643$	− 2.98454i	− 0.117699i	−0.998267	−	0.0588495i	$-0.981257\pi$
$643$	− 2.98454i	− 0.117699i	0.998267	−	0.0588495i	$-0.0187432\pi$
$644$	0	0
$645$	14.7821i	0.582044i
$646$	0	0
$647$	29.6569	1.16593	0.582966	−	0.812497i	$-0.301892\pi$
$647$	29.6569	1.16593	0.582966	+	0.812497i	$0.301892\pi$
$648$	0	0
$649$	0.970563	0.0380979
$650$	0	0
$651$	14.7821i	0.579355i
$652$	0	0
$653$	− 22.5445i	− 0.882236i	−0.897449	−	0.441118i	$-0.854582\pi$
$653$	− 22.5445i	− 0.882236i	0.897449	−	0.441118i	$-0.145418\pi$
$654$	0	0
$655$	−47.7990	−1.86766
$656$	0	0
$657$	−44.6274	−1.74108
$658$	0	0
$659$	10.0811i	0.392703i	0.980534	+	0.196352i	$0.0629094\pi$
$659$	10.0811i	0.392703i	−0.980534	+	0.196352i	$0.937091\pi$
$660$	0	0
$661$	− 0.448342i	− 0.0174385i	−0.999962	−	0.00871923i	$-0.997225\pi$
$661$	− 0.448342i	− 0.0174385i	0.999962	−	0.00871923i	$-0.00277545\pi$
$662$	0	0
$663$	−8.97056	−0.348388
$664$	0	0
$665$	−12.4853	−0.484158
$666$	0	0
$667$	− 65.2512i	− 2.52654i
$668$	0	0
$669$	− 23.4412i	− 0.906290i
$670$	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
$675$	3.95815i	0.152349i
$676$	0	0
$677$	− 8.73606i	− 0.335754i	−0.985808	−	0.167877i	$-0.946309\pi$
$677$	− 8.73606i	− 0.335754i	0.985808	−	0.167877i	$-0.0536912\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	37.4558	1.43531
$682$	0	0
$683$	5.59767i	0.214189i	0.994249	+	0.107094i	$0.0341547\pi$
$683$	5.59767i	0.214189i	−0.994249	+	0.107094i	$0.965845\pi$
$684$	0	0
$685$	5.22625i	0.199685i
$686$	0	0
$687$	14.8284	0.565740
$688$	0	0
$689$	−4.68629	−0.178533
$690$	0	0
$691$	− 35.7640i	− 1.36053i	−0.732968	−	0.680263i	$-0.761865\pi$
$691$	− 35.7640i	− 1.36053i	0.732968	−	0.680263i	$-0.238135\pi$
$692$	0	0
$693$	− 8.28772i	− 0.314824i
$694$	0	0
$695$	−51.1127	−1.93882
$696$	0	0
$697$	28.0000	1.06058
$698$	0	0
$699$	− 26.1313i	− 0.988375i
$700$	0	0
$701$	− 40.9133i	− 1.54528i	−0.634847	−	0.772638i	$-0.718937\pi$
$701$	− 40.9133i	− 1.54528i	0.634847	−	0.772638i	$-0.281063\pi$
$702$	0	0
$703$	−24.9706	−0.941783
$704$	0	0
$705$	54.6274	2.05739
$706$	0	0
$707$	− 4.77791i	− 0.179692i
$708$	0	0
$709$	17.4721i	0.656179i	0.944647	+	0.328090i	$0.106405\pi$
$709$	17.4721i	0.656179i	−0.944647	+	0.328090i	$0.893595\pi$
$710$	0	0
$711$	8.97056	0.336422
$712$	0	0
$713$	38.6274	1.44661
$714$	0	0
$715$	2.53620i	0.0948486i
$716$	0	0
$717$	− 47.4077i	− 1.77047i
$718$	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
$723$	3.43289i	0.127670i
$724$	0	0
$725$	17.4721i	0.648898i
$726$	0	0
$727$	20.6863	0.767212	0.383606	−	0.923497i	$-0.374682\pi$
$727$	20.6863	0.767212	0.383606	+	0.923497i	$0.374682\pi$
$728$	0	0
$729$	39.2843	1.45497
$730$	0	0
$731$	16.5754i	0.613065i
$732$	0	0
$733$	− 21.7248i	− 0.802423i	−0.915986	−	0.401211i	$-0.868589\pi$
$733$	− 21.7248i	− 0.802423i	0.915986	−	0.401211i	$-0.131411\pi$
$734$	0	0
$735$	−6.82843	−0.251870
$736$	0	0
$737$	−6.62742	−0.244124
$738$	0	0
$739$	− 24.8632i	− 0.914606i	−0.889311	−	0.457303i	$-0.848815\pi$
$739$	− 24.8632i	− 0.914606i	0.889311	−	0.457303i	$-0.151185\pi$
$740$	0	0
$741$	5.59767i	0.205636i
$742$	0	0
$743$	27.5147	1.00942	0.504709	−	0.863290i	$-0.331600\pi$
$743$	27.5147	1.00942	0.504709	+	0.863290i	$0.331600\pi$
$744$	0	0
$745$	43.3137	1.58689
$746$	0	0
$747$	50.0208i	1.83017i
$748$	0	0
$749$	13.5140i	0.493790i
$750$	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
$755$	− 47.4077i	− 1.72534i
$756$	0	0
$757$	− 22.5445i	− 0.819395i	−0.912221	−	0.409697i	$-0.865634\pi$
$757$	− 22.5445i	− 0.819395i	0.912221	−	0.409697i	$-0.134366\pi$
$758$	0	0
$759$	−38.6274	−1.40209
$760$	0	0
$761$	18.9706	0.687682	0.343841	−	0.939028i	$-0.388272\pi$
$761$	18.9706	0.687682	0.343841	+	0.939028i	$0.388272\pi$
$762$	0	0
$763$	11.3492i	0.410868i
$764$	0	0
$765$	76.6004i	2.76949i
$766$	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
$771$	− 24.3379i	− 0.876508i
$772$	0	0
$773$	8.73606i	0.314214i	0.987582	+	0.157107i	$0.0502168\pi$
$773$	8.73606i	0.314214i	−0.987582	+	0.157107i	$0.949783\pi$
$774$	0	0
$775$	−10.3431	−0.371537
$776$	0	0
$777$	−13.6569	−0.489937
$778$	0	0
$779$	− 17.4721i	− 0.626004i
$780$	0	0
$781$	5.07241i	0.181505i
$782$	0	0
$783$	−20.6863	−0.739268
$784$	0	0
$785$	12.4853	0.445619
$786$	0	0
$787$	− 7.46796i	− 0.266204i	−0.991102	−	0.133102i	$-0.957506\pi$
$787$	− 7.46796i	− 0.266204i	0.991102	−	0.133102i	$-0.0424938\pi$
$788$	0	0
$789$	6.12293i	0.217982i
$790$	0	0
$791$	8.82843	0.313903
$792$	0	0
$793$	5.45584	0.193743
$794$	0	0
$795$	71.3742i	2.53138i
$796$	0	0
$797$	− 27.4763i	− 0.973260i	−0.873608	−	0.486630i	$-0.838226\pi$
$797$	− 27.4763i	− 0.973260i	0.873608	−	0.486630i	$-0.161774\pi$
$798$	0	0
$799$	61.2548	2.16704
$800$	0	0
$801$	7.65685	0.270542
$802$	0	0
$803$	25.2346i	0.890509i
$804$	0	0
$805$	17.8435i	0.628902i
$806$	0	0
$807$	−21.4558	−0.755281
$808$	0	0
$809$	−31.1716	−1.09593	−0.547967	−	0.836500i	$-0.684598\pi$
$809$	−31.1716	−1.09593	−0.547967	+	0.836500i	$0.684598\pi$
$810$	0	0
$811$	20.9819i	0.736775i	0.929672	+	0.368388i	$0.120090\pi$
$811$	20.9819i	0.736775i	−0.929672	+	0.368388i	$0.879910\pi$
$812$	0	0
$813$	− 6.12293i	− 0.214741i
$814$	0	0
$815$	10.3431	0.362305
$816$	0	0
$817$	10.3431	0.361861
$818$	0	0
$819$	1.71644i	0.0599774i
$820$	0	0
$821$	− 8.65914i	− 0.302206i	−0.988518	−	0.151103i	$-0.951717\pi$
$821$	− 8.65914i	− 0.302206i	0.988518	−	0.151103i	$-0.0482825\pi$
$822$	0	0
$823$	40.9706	1.42814	0.714072	−	0.700072i	$-0.246849\pi$
$823$	40.9706	1.42814	0.714072	+	0.700072i	$0.246849\pi$
$824$	0	0
$825$	10.3431	0.360102
$826$	0	0
$827$	10.9778i	0.381734i	0.981616	+	0.190867i	$0.0611300\pi$
$827$	10.9778i	0.381734i	−0.981616	+	0.190867i	$0.938870\pi$
$828$	0	0
$829$	− 6.19986i	− 0.215330i	−0.994187	−	0.107665i	$-0.965663\pi$
$829$	− 6.19986i	− 0.215330i	0.994187	−	0.107665i	$-0.0343374\pi$
$830$	0	0
$831$	−38.6274	−1.33997
$832$	0	0
$833$	−7.65685	−0.265294
$834$	0	0
$835$	41.8100i	1.44690i
$836$	0	0
$837$	− 12.2459i	− 0.423279i
$838$	0	0
$839$	36.2843	1.25267	0.626336	−	0.779553i	$-0.284554\pi$
$839$	36.2843	1.25267	0.626336	+	0.779553i	$0.284554\pi$
$840$	0	0
$841$	−62.3137	−2.14875
$842$	0	0
$843$	− 5.22625i	− 0.180002i
$844$	0	0
$845$	33.4454i	1.15056i
$846$	0	0
$847$	6.31371	0.216942
$848$	0	0
$849$	54.4264	1.86791
$850$	0	0
$851$	35.6871i	1.22334i
$852$	0	0
$853$	47.3308i	1.62057i	0.586033	+	0.810287i	$0.300689\pi$
$853$	47.3308i	1.62057i	−0.586033	+	0.810287i	$0.699311\pi$
$854$	0	0
$855$	47.7990	1.63469
$856$	0	0
$857$	0.627417	0.0214322	0.0107161	−	0.999943i	$-0.496589\pi$
$857$	0.627417	0.0214322	0.0107161	+	0.999943i	$0.496589\pi$
$858$	0	0
$859$	3.50981i	0.119753i	0.998206	+	0.0598766i	$0.0190707\pi$
$859$	3.50981i	0.119753i	−0.998206	+	0.0598766i	$0.980929\pi$
$860$	0	0
$861$	− 9.55582i	− 0.325661i
$862$	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
$867$	108.778i	3.69428i
$868$	0	0
$869$	− 5.07241i	− 0.172070i
$870$	0	0
$871$	1.37258	0.0465082
$872$	0	0
$873$	38.2843	1.29573
$874$	0	0
$875$	8.28772i	0.280176i
$876$	0	0
$877$	− 32.2542i	− 1.08915i	−0.838713	−	0.544573i	$-0.816692\pi$
$877$	− 32.2542i	− 1.08915i	0.838713	−	0.544573i	$-0.183308\pi$
$878$	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
$883$	51.7373i	1.74110i	0.492082	+	0.870549i	$0.336236\pi$
$883$	51.7373i	1.74110i	−0.492082	+	0.870549i	$0.663764\pi$
$884$	0	0
$885$	3.06147i	0.102910i
$886$	0	0
$887$	−22.6274	−0.759754	−0.379877	−	0.925037i	$-0.624034\pi$
$887$	−22.6274	−0.759754	−0.379877	+	0.925037i	$0.624034\pi$
$888$	0	0
$889$	−4.48528	−0.150432
$890$	0	0
$891$	− 12.6173i	− 0.422695i
$892$	0	0
$893$	− 38.2233i	− 1.27909i
$894$	0	0
$895$	−57.9411	−1.93676
$896$	0	0
$897$	8.00000	0.267112
$898$	0	0
$899$	− 54.0559i	− 1.80286i
$900$	0	0
$901$	80.0333i	2.66630i
$902$	0	0
$903$	5.65685	0.188248
$904$	0	0
$905$	6.82843	0.226985
$906$	0	0
$907$	− 1.26810i	− 0.0421066i	−0.999778	−	0.0210533i	$-0.993298\pi$
$907$	− 1.26810i	− 0.0421066i	0.999778	−	0.0210533i	$-0.00670197\pi$
$908$	0	0
$909$	18.2919i	0.606703i
$910$	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
$915$	− 83.0948i	− 2.74703i
$916$	0	0
$917$	18.2919i	0.604051i
$918$	0	0
$919$	2.34315	0.0772932	0.0386466	−	0.999253i	$-0.487695\pi$
$919$	2.34315	0.0772932	0.0386466	+	0.999253i	$0.487695\pi$
$920$	0	0
$921$	22.8284	0.752222
$922$	0	0
$923$	− 1.05053i	− 0.0345786i
$924$	0	0
$925$	− 9.55582i	− 0.314193i
$926$	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
$931$	4.77791i	0.156590i
$932$	0	0
$933$	38.2233i	1.25137i
$934$	0	0
$935$	43.3137	1.41651
$936$	0	0
$937$	−38.2843	−1.25069	−0.625346	−	0.780347i	$-0.715042\pi$
$937$	−38.2843	−1.25069	−0.625346	+	0.780347i	$0.715042\pi$
$938$	0	0
$939$	− 21.8017i	− 0.711471i
$940$	0	0
$941$	− 37.7749i	− 1.23143i	−0.787970	−	0.615714i	$-0.788868\pi$
$941$	− 37.7749i	− 1.23143i	0.787970	−	0.615714i	$-0.211132\pi$
$942$	0	0
$943$	−24.9706	−0.813153
$944$	0	0
$945$	5.65685	0.184017
$946$	0	0
$947$	48.3044i	1.56968i	0.619698	+	0.784841i	$0.287255\pi$
$947$	48.3044i	1.56968i	−0.619698	+	0.784841i	$0.712745\pi$
$948$	0	0
$949$	− 5.22625i	− 0.169651i
$950$	0	0
$951$	22.6274	0.733744
$952$	0	0
$953$	37.3137	1.20871	0.604355	−	0.796715i	$-0.293431\pi$
$953$	37.3137	1.20871	0.604355	+	0.796715i	$0.293431\pi$
$954$	0	0
$955$	50.4692i	1.63314i
$956$	0	0
$957$	54.0559i	1.74738i
$958$	0	0
$959$	2.00000	0.0645834
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	− 51.7373i	− 1.66721i
$964$	0	0
$965$	36.0585i	1.16076i
$966$	0	0
$967$	−29.8579	−0.960164	−0.480082	−	0.877224i	$-0.659393\pi$
$967$	−29.8579	−0.960164	−0.480082	+	0.877224i	$0.659393\pi$
$968$	0	0
$969$	95.5980	3.07105
$970$	0	0
$971$	52.5570i	1.68663i	0.537416	+	0.843317i	$0.319401\pi$
$971$	52.5570i	1.68663i	−0.537416	+	0.843317i	$0.680599\pi$
$972$	0	0
$973$	19.5600i	0.627064i
$974$	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
$979$	− 4.32957i	− 0.138374i
$980$	0	0
$981$	− 43.4495i	− 1.38724i
$982$	0	0
$983$	23.0294	0.734525	0.367262	−	0.930117i	$-0.380295\pi$
$983$	23.0294	0.734525	0.367262	+	0.930117i	$0.380295\pi$
$984$	0	0
$985$	−16.0000	−0.509802
$986$	0	0
$987$	− 20.9050i	− 0.665414i
$988$	0	0
$989$	− 14.7821i	− 0.470043i
$990$	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
$995$	23.4412i	0.743136i
$996$	0	0
$997$	0.0769232i	0.00243618i	0.999999	+	0.00121809i	$0.000387731\pi$
$997$	0.0769232i	0.00243618i	−0.999999	+	0.00121809i	$0.999612\pi$
$998$	0	0
$999$	11.3137	0.357950

Twists

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

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

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:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.b (of order \(2\), degree \(1\), minimal)

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

Introduction

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