LMFDB - Embedded newform 896.2.b.f.449.3

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			449.3
Root			$1.84776i$ of defining polynomial
Character	$\chi$	$=$	896.449
Dual form			896.2.b.f.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.08239i q^{3} -1.08239i q^{5} -1.00000 q^{7} +1.82843 q^{9} +O(q^{10})$	$q+1.08239i q^{3} -1.08239i q^{5} -1.00000 q^{7} +1.82843 q^{9} -5.22625i q^{11} +6.30864i q^{13} +1.17157 q^{15} +3.65685 q^{17} -4.14386i q^{19} -1.08239i q^{21} -1.17157 q^{23} +3.82843 q^{25} +5.22625i q^{27} -8.28772i q^{29} +5.65685 q^{31} +5.65685 q^{33} +1.08239i q^{35} -2.16478i q^{37} -6.82843 q^{39} +7.65685 q^{41} +5.22625i q^{43} -1.97908i q^{45} +8.00000 q^{47} +1.00000 q^{49} +3.95815i q^{51} +4.32957i q^{53} -5.65685 q^{55} +4.48528 q^{57} -6.30864i q^{59} +7.20533i q^{61} -1.82843 q^{63} +6.82843 q^{65} +7.39104i q^{67} -1.26810i q^{69} -13.6569 q^{71} +0.343146 q^{73} +4.14386i q^{75} +5.22625i q^{77} +13.6569 q^{79} -0.171573 q^{81} -5.41196i q^{83} -3.95815i q^{85} +8.97056 q^{87} -2.00000 q^{89} -6.30864i q^{91} +6.12293i q^{93} -4.48528 q^{95} -10.0000 q^{97} -9.55582i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.08239i	0.624919i	0.949931	+	0.312460i	$0.101153\pi$
$3$	1.08239i	0.624919i	−0.949931	+	0.312460i	$0.898847\pi$
$4$	0	0
$5$	− 1.08239i	− 0.484061i	−0.970269	−	0.242030i	$-0.922187\pi$
$5$	− 1.08239i	− 0.484061i	0.970269	−	0.242030i	$-0.0778133\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.82843	0.609476
$10$	0	0
$11$	− 5.22625i	− 1.57577i	−0.615820	−	0.787887i	$-0.711175\pi$
$11$	− 5.22625i	− 1.57577i	0.615820	−	0.787887i	$-0.288825\pi$
$12$	0	0
$13$	6.30864i	1.74970i	0.484391	+	0.874852i	$0.339041\pi$
$13$	6.30864i	1.74970i	−0.484391	+	0.874852i	$0.660959\pi$
$14$	0	0
$15$	1.17157	0.302499
$16$	0	0
$17$	3.65685	0.886917	0.443459	−	0.896295i	$-0.353751\pi$
$17$	3.65685	0.886917	0.443459	+	0.896295i	$0.353751\pi$
$18$	0	0
$19$	− 4.14386i	− 0.950667i	−0.879806	−	0.475333i	$-0.842327\pi$
$19$	− 4.14386i	− 0.950667i	0.879806	−	0.475333i	$-0.157673\pi$
$20$	0	0
$21$	− 1.08239i	− 0.236197i
$22$	0	0
$23$	−1.17157	−0.244290	−0.122145	−	0.992512i	$-0.538977\pi$
$23$	−1.17157	−0.244290	−0.122145	+	0.992512i	$0.538977\pi$
$24$	0	0
$25$	3.82843	0.765685
$26$	0	0
$27$	5.22625i	1.00579i
$28$	0	0
$29$	− 8.28772i	− 1.53899i	−0.638652	−	0.769495i	$-0.720508\pi$
$29$	− 8.28772i	− 1.53899i	0.638652	−	0.769495i	$-0.279492\pi$
$30$	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
$35$	1.08239i	0.182958i
$36$	0	0
$37$	− 2.16478i	− 0.355888i	−0.984041	−	0.177944i	$-0.943055\pi$
$37$	− 2.16478i	− 0.355888i	0.984041	−	0.177944i	$-0.0569447\pi$
$38$	0	0
$39$	−6.82843	−1.09342
$40$	0	0
$41$	7.65685	1.19580	0.597900	−	0.801571i	$-0.296002\pi$
$41$	7.65685	1.19580	0.597900	+	0.801571i	$0.296002\pi$
$42$	0	0
$43$	5.22625i	0.796996i	0.917169	+	0.398498i	$0.130468\pi$
$43$	5.22625i	0.796996i	−0.917169	+	0.398498i	$0.869532\pi$
$44$	0	0
$45$	− 1.97908i	− 0.295023i
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.95815i	0.554252i
$52$	0	0
$53$	4.32957i	0.594712i	0.954767	+	0.297356i	$0.0961048\pi$
$53$	4.32957i	0.594712i	−0.954767	+	0.297356i	$0.903895\pi$
$54$	0	0
$55$	−5.65685	−0.762770
$56$	0	0
$57$	4.48528	0.594090
$58$	0	0
$59$	− 6.30864i	− 0.821315i	−0.911790	−	0.410658i	$-0.865299\pi$
$59$	− 6.30864i	− 0.821315i	0.911790	−	0.410658i	$-0.134701\pi$
$60$	0	0
$61$	7.20533i	0.922548i	0.887258	+	0.461274i	$0.152607\pi$
$61$	7.20533i	0.922548i	−0.887258	+	0.461274i	$0.847393\pi$
$62$	0	0
$63$	−1.82843	−0.230360
$64$	0	0
$65$	6.82843	0.846962
$66$	0	0
$67$	7.39104i	0.902959i	0.892282	+	0.451479i	$0.149104\pi$
$67$	7.39104i	0.902959i	−0.892282	+	0.451479i	$0.850896\pi$
$68$	0	0
$69$	− 1.26810i	− 0.152661i
$70$	0	0
$71$	−13.6569	−1.62077	−0.810385	−	0.585897i	$-0.800742\pi$
$71$	−13.6569	−1.62077	−0.810385	+	0.585897i	$0.800742\pi$
$72$	0	0
$73$	0.343146	0.0401622	0.0200811	−	0.999798i	$-0.493608\pi$
$73$	0.343146	0.0401622	0.0200811	+	0.999798i	$0.493608\pi$
$74$	0	0
$75$	4.14386i	0.478492i
$76$	0	0
$77$	5.22625i	0.595587i
$78$	0	0
$79$	13.6569	1.53652	0.768258	−	0.640140i	$-0.221124\pi$
$79$	13.6569	1.53652	0.768258	+	0.640140i	$0.221124\pi$
$80$	0	0
$81$	−0.171573	−0.0190637
$82$	0	0
$83$	− 5.41196i	− 0.594040i	−0.954871	−	0.297020i	$-0.904007\pi$
$83$	− 5.41196i	− 0.594040i	0.954871	−	0.297020i	$-0.0959928\pi$
$84$	0	0
$85$	− 3.95815i	− 0.429322i
$86$	0	0
$87$	8.97056	0.961745
$88$	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
$91$	− 6.30864i	− 0.661326i
$92$	0	0
$93$	6.12293i	0.634919i
$94$	0	0
$95$	−4.48528	−0.460180
$96$	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
$99$	− 9.55582i	− 0.960396i
$100$	0	0
$101$	− 4.14386i	− 0.412329i	−0.978517	−	0.206165i	$-0.933902\pi$
$101$	− 4.14386i	− 0.412329i	0.978517	−	0.206165i	$-0.0660983\pi$
$102$	0	0
$103$	−2.34315	−0.230877	−0.115439	−	0.993315i	$-0.536827\pi$
$103$	−2.34315	−0.230877	−0.115439	+	0.993315i	$0.536827\pi$
$104$	0	0
$105$	−1.17157	−0.114334
$106$	0	0
$107$	11.7206i	1.13307i	0.824036	+	0.566537i	$0.191717\pi$
$107$	11.7206i	1.13307i	−0.824036	+	0.566537i	$0.808283\pi$
$108$	0	0
$109$	− 16.9469i	− 1.62321i	−0.584203	−	0.811607i	$-0.698593\pi$
$109$	− 16.9469i	− 1.62321i	0.584203	−	0.811607i	$-0.301407\pi$
$110$	0	0
$111$	2.34315	0.222402
$112$	0	0
$113$	3.17157	0.298356	0.149178	−	0.988810i	$-0.452337\pi$
$113$	3.17157	0.298356	0.149178	+	0.988810i	$0.452337\pi$
$114$	0	0
$115$	1.26810i	0.118251i
$116$	0	0
$117$	11.5349i	1.06640i
$118$	0	0
$119$	−3.65685	−0.335223
$120$	0	0
$121$	−16.3137	−1.48306
$122$	0	0
$123$	8.28772i	0.747278i
$124$	0	0
$125$	− 9.55582i	− 0.854699i
$126$	0	0
$127$	−12.4853	−1.10789	−0.553945	−	0.832553i	$-0.686878\pi$
$127$	−12.4853	−1.10789	−0.553945	+	0.832553i	$0.686878\pi$
$128$	0	0
$129$	−5.65685	−0.498058
$130$	0	0
$131$	7.57675i	0.661983i	0.943634	+	0.330992i	$0.107383\pi$
$131$	7.57675i	0.661983i	−0.943634	+	0.330992i	$0.892617\pi$
$132$	0	0
$133$	4.14386i	0.359318i
$134$	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
$139$	− 10.2668i	− 0.870818i	−0.900233	−	0.435409i	$-0.856604\pi$
$139$	− 10.2668i	− 0.870818i	0.900233	−	0.435409i	$-0.143396\pi$
$140$	0	0
$141$	8.65914i	0.729231i
$142$	0	0
$143$	32.9706	2.75714
$144$	0	0
$145$	−8.97056	−0.744965
$146$	0	0
$147$	1.08239i	0.0892742i
$148$	0	0
$149$	19.1116i	1.56569i	0.622219	+	0.782843i	$0.286231\pi$
$149$	19.1116i	1.56569i	−0.622219	+	0.782843i	$0.713769\pi$
$150$	0	0
$151$	−10.1421	−0.825355	−0.412678	−	0.910877i	$-0.635406\pi$
$151$	−10.1421	−0.825355	−0.412678	+	0.910877i	$0.635406\pi$
$152$	0	0
$153$	6.68629	0.540555
$154$	0	0
$155$	− 6.12293i	− 0.491806i
$156$	0	0
$157$	− 4.14386i	− 0.330716i	−0.986234	−	0.165358i	$-0.947122\pi$
$157$	− 4.14386i	− 0.330716i	0.986234	−	0.165358i	$-0.0528780\pi$
$158$	0	0
$159$	−4.68629	−0.371647
$160$	0	0
$161$	1.17157	0.0923329
$162$	0	0
$163$	− 20.0083i	− 1.56717i	−0.621283	−	0.783586i	$-0.713388\pi$
$163$	− 20.0083i	− 1.56717i	0.621283	−	0.783586i	$-0.286612\pi$
$164$	0	0
$165$	− 6.12293i	− 0.476670i
$166$	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
$169$	−26.7990	−2.06146
$170$	0	0
$171$	− 7.57675i	− 0.579408i
$172$	0	0
$173$	− 12.8030i	− 0.973394i	−0.873571	−	0.486697i	$-0.838202\pi$
$173$	− 12.8030i	− 0.973394i	0.873571	−	0.486697i	$-0.161798\pi$
$174$	0	0
$175$	−3.82843	−0.289402
$176$	0	0
$177$	6.82843	0.513256
$178$	0	0
$179$	− 9.18440i	− 0.686474i	−0.939249	−	0.343237i	$-0.888477\pi$
$179$	− 9.18440i	− 0.686474i	0.939249	−	0.343237i	$-0.111523\pi$
$180$	0	0
$181$	1.08239i	0.0804536i	0.999191	+	0.0402268i	$0.0128080\pi$
$181$	1.08239i	0.0804536i	−0.999191	+	0.0402268i	$0.987192\pi$
$182$	0	0
$183$	−7.79899	−0.576518
$184$	0	0
$185$	−2.34315	−0.172272
$186$	0	0
$187$	− 19.1116i	− 1.39758i
$188$	0	0
$189$	− 5.22625i	− 0.380154i
$190$	0	0
$191$	3.31371	0.239772	0.119886	−	0.992788i	$-0.461747\pi$
$191$	3.31371	0.239772	0.119886	+	0.992788i	$0.461747\pi$
$192$	0	0
$193$	−25.7990	−1.85705	−0.928526	−	0.371267i	$-0.878923\pi$
$193$	−25.7990	−1.85705	−0.928526	+	0.371267i	$0.878923\pi$
$194$	0	0
$195$	7.39104i	0.529283i
$196$	0	0
$197$	− 14.7821i	− 1.05318i	−0.850120	−	0.526590i	$-0.823470\pi$
$197$	− 14.7821i	− 1.05318i	0.850120	−	0.526590i	$-0.176530\pi$
$198$	0	0
$199$	24.9706	1.77012	0.885058	−	0.465481i	$-0.154118\pi$
$199$	24.9706	1.77012	0.885058	+	0.465481i	$0.154118\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	8.28772i	0.581684i
$204$	0	0
$205$	− 8.28772i	− 0.578839i
$206$	0	0
$207$	−2.14214	−0.148889
$208$	0	0
$209$	−21.6569	−1.49804
$210$	0	0
$211$	22.1731i	1.52646i	0.646127	+	0.763230i	$0.276388\pi$
$211$	22.1731i	1.52646i	−0.646127	+	0.763230i	$0.723612\pi$
$212$	0	0
$213$	− 14.7821i	− 1.01285i
$214$	0	0
$215$	5.65685	0.385794
$216$	0	0
$217$	−5.65685	−0.384012
$218$	0	0
$219$	0.371418i	0.0250981i
$220$	0	0
$221$	23.0698i	1.55184i
$222$	0	0
$223$	−24.9706	−1.67215	−0.836076	−	0.548613i	$-0.815156\pi$
$223$	−24.9706	−1.67215	−0.836076	+	0.548613i	$0.815156\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	12.4316i	0.825113i	0.910932	+	0.412556i	$0.135364\pi$
$227$	12.4316i	0.825113i	−0.910932	+	0.412556i	$0.864636\pi$
$228$	0	0
$229$	8.47343i	0.559940i	0.960009	+	0.279970i	$0.0903245\pi$
$229$	8.47343i	0.559940i	−0.960009	+	0.279970i	$0.909675\pi$
$230$	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
$235$	− 8.65914i	− 0.564860i
$236$	0	0
$237$	14.7821i	0.960199i
$238$	0	0
$239$	−10.1421	−0.656040	−0.328020	−	0.944671i	$-0.606381\pi$
$239$	−10.1421	−0.656040	−0.328020	+	0.944671i	$0.606381\pi$
$240$	0	0
$241$	−21.3137	−1.37294	−0.686468	−	0.727160i	$-0.740840\pi$
$241$	−21.3137	−1.37294	−0.686468	+	0.727160i	$0.740840\pi$
$242$	0	0
$243$	15.4930i	0.993879i
$244$	0	0
$245$	− 1.08239i	− 0.0691515i
$246$	0	0
$247$	26.1421	1.66338
$248$	0	0
$249$	5.85786	0.371227
$250$	0	0
$251$	24.1522i	1.52447i	0.647299	+	0.762236i	$0.275899\pi$
$251$	24.1522i	1.52447i	−0.647299	+	0.762236i	$0.724101\pi$
$252$	0	0
$253$	6.12293i	0.384946i
$254$	0	0
$255$	4.28427	0.268291
$256$	0	0
$257$	13.3137	0.830486	0.415243	−	0.909710i	$-0.363696\pi$
$257$	13.3137	0.830486	0.415243	+	0.909710i	$0.363696\pi$
$258$	0	0
$259$	2.16478i	0.134513i
$260$	0	0
$261$	− 15.1535i	− 0.937978i
$262$	0	0
$263$	−13.6569	−0.842118	−0.421059	−	0.907033i	$-0.638341\pi$
$263$	−13.6569	−0.842118	−0.421059	+	0.907033i	$0.638341\pi$
$264$	0	0
$265$	4.68629	0.287877
$266$	0	0
$267$	− 2.16478i	− 0.132483i
$268$	0	0
$269$	27.2137i	1.65925i	0.558324	+	0.829623i	$0.311445\pi$
$269$	27.2137i	1.65925i	−0.558324	+	0.829623i	$0.688555\pi$
$270$	0	0
$271$	13.6569	0.829595	0.414797	−	0.909914i	$-0.363852\pi$
$271$	13.6569	0.829595	0.414797	+	0.909914i	$0.363852\pi$
$272$	0	0
$273$	6.82843	0.413275
$274$	0	0
$275$	− 20.0083i	− 1.20655i
$276$	0	0
$277$	6.12293i	0.367892i	0.982936	+	0.183946i	$0.0588871\pi$
$277$	6.12293i	0.367892i	−0.982936	+	0.183946i	$0.941113\pi$
$278$	0	0
$279$	10.3431	0.619228
$280$	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
$283$	28.1103i	1.67099i	0.549501	+	0.835493i	$0.314818\pi$
$283$	28.1103i	1.67099i	−0.549501	+	0.835493i	$0.685182\pi$
$284$	0	0
$285$	− 4.85483i	− 0.287576i
$286$	0	0
$287$	−7.65685	−0.451970
$288$	0	0
$289$	−3.62742	−0.213377
$290$	0	0
$291$	− 10.8239i	− 0.634510i
$292$	0	0
$293$	10.6382i	0.621491i	0.950493	+	0.310746i	$0.100579\pi$
$293$	10.6382i	0.621491i	−0.950493	+	0.310746i	$0.899421\pi$
$294$	0	0
$295$	−6.82843	−0.397566
$296$	0	0
$297$	27.3137	1.58490
$298$	0	0
$299$	− 7.39104i	− 0.427435i
$300$	0	0
$301$	− 5.22625i	− 0.301236i
$302$	0	0
$303$	4.48528	0.257673
$304$	0	0
$305$	7.79899	0.446569
$306$	0	0
$307$	− 15.8645i	− 0.905433i	−0.891655	−	0.452716i	$-0.850455\pi$
$307$	− 15.8645i	− 0.905433i	0.891655	−	0.452716i	$-0.149545\pi$
$308$	0	0
$309$	− 2.53620i	− 0.144280i
$310$	0	0
$311$	30.6274	1.73672	0.868361	−	0.495933i	$-0.165174\pi$
$311$	30.6274	1.73672	0.868361	+	0.495933i	$0.165174\pi$
$312$	0	0
$313$	−19.6569	−1.11107	−0.555536	−	0.831493i	$-0.687487\pi$
$313$	−19.6569	−1.11107	−0.555536	+	0.831493i	$0.687487\pi$
$314$	0	0
$315$	1.97908i	0.111508i
$316$	0	0
$317$	− 20.9050i	− 1.17414i	−0.809535	−	0.587071i	$-0.800281\pi$
$317$	− 20.9050i	− 1.17414i	0.809535	−	0.587071i	$-0.199719\pi$
$318$	0	0
$319$	−43.3137	−2.42510
$320$	0	0
$321$	−12.6863	−0.708080
$322$	0	0
$323$	− 15.1535i	− 0.843163i
$324$	0	0
$325$	24.1522i	1.33972i
$326$	0	0
$327$	18.3431	1.01438
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	13.8854i	0.763210i	0.924325	+	0.381605i	$0.124629\pi$
$331$	13.8854i	0.763210i	−0.924325	+	0.381605i	$0.875371\pi$
$332$	0	0
$333$	− 3.95815i	− 0.216905i
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	16.8284	0.916703	0.458351	−	0.888771i	$-0.348440\pi$
$337$	16.8284	0.916703	0.458351	+	0.888771i	$0.348440\pi$
$338$	0	0
$339$	3.43289i	0.186449i
$340$	0	0
$341$	− 29.5641i	− 1.60099i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.37258	−0.0738974
$346$	0	0
$347$	− 0.896683i	− 0.0481365i	−0.999710	−	0.0240682i	$-0.992338\pi$
$347$	− 0.896683i	− 0.0481365i	0.999710	−	0.0240682i	$-0.00766190\pi$
$348$	0	0
$349$	− 2.87576i	− 0.153936i	−0.997034	−	0.0769679i	$-0.975476\pi$
$349$	− 2.87576i	− 0.153936i	0.997034	−	0.0769679i	$-0.0245239\pi$
$350$	0	0
$351$	−32.9706	−1.75984
$352$	0	0
$353$	4.34315	0.231162	0.115581	−	0.993298i	$-0.463127\pi$
$353$	4.34315	0.231162	0.115581	+	0.993298i	$0.463127\pi$
$354$	0	0
$355$	14.7821i	0.784551i
$356$	0	0
$357$	− 3.95815i	− 0.209488i
$358$	0	0
$359$	−1.17157	−0.0618333	−0.0309166	−	0.999522i	$-0.509843\pi$
$359$	−1.17157	−0.0618333	−0.0309166	+	0.999522i	$0.509843\pi$
$360$	0	0
$361$	1.82843	0.0962330
$362$	0	0
$363$	− 17.6578i	− 0.926796i
$364$	0	0
$365$	− 0.371418i	− 0.0194409i
$366$	0	0
$367$	8.97056	0.468260	0.234130	−	0.972205i	$-0.424776\pi$
$367$	8.97056	0.468260	0.234130	+	0.972205i	$0.424776\pi$
$368$	0	0
$369$	14.0000	0.728811
$370$	0	0
$371$	− 4.32957i	− 0.224780i
$372$	0	0
$373$	− 22.6984i	− 1.17528i	−0.809124	−	0.587639i	$-0.800058\pi$
$373$	− 22.6984i	− 1.17528i	0.809124	−	0.587639i	$-0.199942\pi$
$374$	0	0
$375$	10.3431	0.534118
$376$	0	0
$377$	52.2843	2.69278
$378$	0	0
$379$	3.43289i	0.176335i	0.996106	+	0.0881677i	$0.0281012\pi$
$379$	3.43289i	0.176335i	−0.996106	+	0.0881677i	$0.971899\pi$
$380$	0	0
$381$	− 13.5140i	− 0.692342i
$382$	0	0
$383$	−3.31371	−0.169323	−0.0846613	−	0.996410i	$-0.526981\pi$
$383$	−3.31371	−0.169323	−0.0846613	+	0.996410i	$0.526981\pi$
$384$	0	0
$385$	5.65685	0.288300
$386$	0	0
$387$	9.55582i	0.485750i
$388$	0	0
$389$	3.95815i	0.200686i	0.994953	+	0.100343i	$0.0319940\pi$
$389$	3.95815i	0.200686i	−0.994953	+	0.100343i	$0.968006\pi$
$390$	0	0
$391$	−4.28427	−0.216665
$392$	0	0
$393$	−8.20101	−0.413686
$394$	0	0
$395$	− 14.7821i	− 0.743767i
$396$	0	0
$397$	7.57675i	0.380266i	0.981758	+	0.190133i	$0.0608919\pi$
$397$	7.57675i	0.380266i	−0.981758	+	0.190133i	$0.939108\pi$
$398$	0	0
$399$	−4.48528	−0.224545
$400$	0	0
$401$	26.4853	1.32261	0.661306	−	0.750116i	$-0.270003\pi$
$401$	26.4853	1.32261	0.661306	+	0.750116i	$0.270003\pi$
$402$	0	0
$403$	35.6871i	1.77770i
$404$	0	0
$405$	0.185709i	0.00922796i
$406$	0	0
$407$	−11.3137	−0.560800
$408$	0	0
$409$	−17.3137	−0.856108	−0.428054	−	0.903753i	$-0.640801\pi$
$409$	−17.3137	−0.856108	−0.428054	+	0.903753i	$0.640801\pi$
$410$	0	0
$411$	2.16478i	0.106781i
$412$	0	0
$413$	6.30864i	0.310428i
$414$	0	0
$415$	−5.85786	−0.287551
$416$	0	0
$417$	11.1127	0.544191
$418$	0	0
$419$	5.41196i	0.264392i	0.991224	+	0.132196i	$0.0422028\pi$
$419$	5.41196i	0.264392i	−0.991224	+	0.132196i	$0.957797\pi$
$420$	0	0
$421$	35.6871i	1.73928i	0.493685	+	0.869641i	$0.335650\pi$
$421$	35.6871i	1.73928i	−0.493685	+	0.869641i	$0.664350\pi$
$422$	0	0
$423$	14.6274	0.711209
$424$	0	0
$425$	14.0000	0.679100
$426$	0	0
$427$	− 7.20533i	− 0.348690i
$428$	0	0
$429$	35.6871i	1.72299i
$430$	0	0
$431$	−1.17157	−0.0564327	−0.0282163	−	0.999602i	$-0.508983\pi$
$431$	−1.17157	−0.0564327	−0.0282163	+	0.999602i	$0.508983\pi$
$432$	0	0
$433$	−23.6569	−1.13688	−0.568438	−	0.822726i	$-0.692452\pi$
$433$	−23.6569	−1.13688	−0.568438	+	0.822726i	$0.692452\pi$
$434$	0	0
$435$	− 9.70967i	− 0.465543i
$436$	0	0
$437$	4.85483i	0.232238i
$438$	0	0
$439$	32.9706	1.57360	0.786800	−	0.617209i	$-0.211737\pi$
$439$	32.9706	1.57360	0.786800	+	0.617209i	$0.211737\pi$
$440$	0	0
$441$	1.82843	0.0870680
$442$	0	0
$443$	− 22.1731i	− 1.05348i	−0.850028	−	0.526738i	$-0.823415\pi$
$443$	− 22.1731i	− 1.05348i	0.850028	−	0.526738i	$-0.176585\pi$
$444$	0	0
$445$	2.16478i	0.102621i
$446$	0	0
$447$	−20.6863	−0.978428
$448$	0	0
$449$	20.6274	0.973468	0.486734	−	0.873550i	$-0.338188\pi$
$449$	20.6274	0.973468	0.486734	+	0.873550i	$0.338188\pi$
$450$	0	0
$451$	− 40.0166i	− 1.88431i
$452$	0	0
$453$	− 10.9778i	− 0.515781i
$454$	0	0
$455$	−6.82843	−0.320122
$456$	0	0
$457$	6.48528	0.303369	0.151684	−	0.988429i	$-0.451530\pi$
$457$	6.48528	0.303369	0.151684	+	0.988429i	$0.451530\pi$
$458$	0	0
$459$	19.1116i	0.892055i
$460$	0	0
$461$	− 32.4399i	− 1.51088i	−0.655220	−	0.755438i	$-0.727424\pi$
$461$	− 32.4399i	− 1.51088i	0.655220	−	0.755438i	$-0.272576\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	6.62742	0.307339
$466$	0	0
$467$	− 18.9259i	− 0.875788i	−0.899027	−	0.437894i	$-0.855725\pi$
$467$	− 18.9259i	− 0.875788i	0.899027	−	0.437894i	$-0.144275\pi$
$468$	0	0
$469$	− 7.39104i	− 0.341286i
$470$	0	0
$471$	4.48528	0.206671
$472$	0	0
$473$	27.3137	1.25589
$474$	0	0
$475$	− 15.8645i	− 0.727912i
$476$	0	0
$477$	7.91630i	0.362463i
$478$	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
$481$	13.6569	0.622699
$482$	0	0
$483$	1.26810i	0.0577006i
$484$	0	0
$485$	10.8239i	0.491489i
$486$	0	0
$487$	−7.79899	−0.353406	−0.176703	−	0.984264i	$-0.556543\pi$
$487$	−7.79899	−0.353406	−0.176703	+	0.984264i	$0.556543\pi$
$488$	0	0
$489$	21.6569	0.979357
$490$	0	0
$491$	4.85483i	0.219096i	0.993982	+	0.109548i	$0.0349403\pi$
$491$	4.85483i	0.219096i	−0.993982	+	0.109548i	$0.965060\pi$
$492$	0	0
$493$	− 30.3070i	− 1.36496i
$494$	0	0
$495$	−10.3431	−0.464890
$496$	0	0
$497$	13.6569	0.612594
$498$	0	0
$499$	13.5140i	0.604968i	0.953154	+	0.302484i	$0.0978160\pi$
$499$	13.5140i	0.604968i	−0.953154	+	0.302484i	$0.902184\pi$
$500$	0	0
$501$	− 17.3183i	− 0.773723i
$502$	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
$505$	−4.48528	−0.199592
$506$	0	0
$507$	− 29.0070i	− 1.28825i
$508$	0	0
$509$	13.6997i	0.607228i	0.952795	+	0.303614i	$0.0981933\pi$
$509$	13.6997i	0.607228i	−0.952795	+	0.303614i	$0.901807\pi$
$510$	0	0
$511$	−0.343146	−0.0151799
$512$	0	0
$513$	21.6569	0.956173
$514$	0	0
$515$	2.53620i	0.111758i
$516$	0	0
$517$	− 41.8100i	− 1.83880i
$518$	0	0
$519$	13.8579	0.608293
$520$	0	0
$521$	37.3137	1.63474	0.817372	−	0.576111i	$-0.195430\pi$
$521$	37.3137	1.63474	0.817372	+	0.576111i	$0.195430\pi$
$522$	0	0
$523$	33.7080i	1.47395i	0.675921	+	0.736974i	$0.263746\pi$
$523$	33.7080i	1.47395i	−0.675921	+	0.736974i	$0.736254\pi$
$524$	0	0
$525$	− 4.14386i	− 0.180853i
$526$	0	0
$527$	20.6863	0.901109
$528$	0	0
$529$	−21.6274	−0.940322
$530$	0	0
$531$	− 11.5349i	− 0.500572i
$532$	0	0
$533$	48.3044i	2.09229i
$534$	0	0
$535$	12.6863	0.548476
$536$	0	0
$537$	9.94113	0.428991
$538$	0	0
$539$	− 5.22625i	− 0.225111i
$540$	0	0
$541$	12.9887i	0.558428i	0.960229	+	0.279214i	$0.0900739\pi$
$541$	12.9887i	0.558428i	−0.960229	+	0.279214i	$0.909926\pi$
$542$	0	0
$543$	−1.17157	−0.0502770
$544$	0	0
$545$	−18.3431	−0.785734
$546$	0	0
$547$	− 12.0920i	− 0.517018i	−0.966009	−	0.258509i	$-0.916769\pi$
$547$	− 12.0920i	− 0.517018i	0.966009	−	0.258509i	$-0.0832311\pi$
$548$	0	0
$549$	13.1744i	0.562270i
$550$	0	0
$551$	−34.3431	−1.46307
$552$	0	0
$553$	−13.6569	−0.580749
$554$	0	0
$555$	− 2.53620i	− 0.107656i
$556$	0	0
$557$	6.12293i	0.259437i	0.991551	+	0.129719i	$0.0414074\pi$
$557$	6.12293i	0.259437i	−0.991551	+	0.129719i	$0.958593\pi$
$558$	0	0
$559$	−32.9706	−1.39451
$560$	0	0
$561$	20.6863	0.873376
$562$	0	0
$563$	35.5014i	1.49620i	0.663584	+	0.748102i	$0.269035\pi$
$563$	35.5014i	1.49620i	−0.663584	+	0.748102i	$0.730965\pi$
$564$	0	0
$565$	− 3.43289i	− 0.144423i
$566$	0	0
$567$	0.171573	0.00720538
$568$	0	0
$569$	−34.4853	−1.44570	−0.722849	−	0.691006i	$-0.757168\pi$
$569$	−34.4853	−1.44570	−0.722849	+	0.691006i	$0.757168\pi$
$570$	0	0
$571$	− 13.8854i	− 0.581085i	−0.956862	−	0.290543i	$-0.906164\pi$
$571$	− 13.8854i	− 0.581085i	0.956862	−	0.290543i	$-0.0938358\pi$
$572$	0	0
$573$	3.58673i	0.149838i
$574$	0	0
$575$	−4.48528	−0.187049
$576$	0	0
$577$	10.9706	0.456711	0.228355	−	0.973578i	$-0.426665\pi$
$577$	10.9706	0.456711	0.228355	+	0.973578i	$0.426665\pi$
$578$	0	0
$579$	− 27.9246i	− 1.16051i
$580$	0	0
$581$	5.41196i	0.224526i
$582$	0	0
$583$	22.6274	0.937132
$584$	0	0
$585$	12.4853	0.516203
$586$	0	0
$587$	29.0070i	1.19725i	0.801030	+	0.598624i	$0.204286\pi$
$587$	29.0070i	1.19725i	−0.801030	+	0.598624i	$0.795714\pi$
$588$	0	0
$589$	− 23.4412i	− 0.965878i
$590$	0	0
$591$	16.0000	0.658152
$592$	0	0
$593$	−2.68629	−0.110313	−0.0551564	−	0.998478i	$-0.517566\pi$
$593$	−2.68629	−0.110313	−0.0551564	+	0.998478i	$0.517566\pi$
$594$	0	0
$595$	3.95815i	0.162268i
$596$	0	0
$597$	27.0279i	1.10618i
$598$	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
$601$	18.6863	0.762230	0.381115	−	0.924528i	$-0.375540\pi$
$601$	18.6863	0.762230	0.381115	+	0.924528i	$0.375540\pi$
$602$	0	0
$603$	13.5140i	0.550331i
$604$	0	0
$605$	17.6578i	0.717893i
$606$	0	0
$607$	−4.68629	−0.190211	−0.0951054	−	0.995467i	$-0.530319\pi$
$607$	−4.68629	−0.190211	−0.0951054	+	0.995467i	$0.530319\pi$
$608$	0	0
$609$	−8.97056	−0.363506
$610$	0	0
$611$	50.4692i	2.04176i
$612$	0	0
$613$	10.8239i	0.437174i	0.975817	+	0.218587i	$0.0701448\pi$
$613$	10.8239i	0.437174i	−0.975817	+	0.218587i	$0.929855\pi$
$614$	0	0
$615$	8.97056	0.361728
$616$	0	0
$617$	−2.20101	−0.0886093	−0.0443047	−	0.999018i	$-0.514107\pi$
$617$	−2.20101	−0.0886093	−0.0443047	+	0.999018i	$0.514107\pi$
$618$	0	0
$619$	14.0711i	0.565565i	0.959184	+	0.282783i	$0.0912575\pi$
$619$	14.0711i	0.565565i	−0.959184	+	0.282783i	$0.908743\pi$
$620$	0	0
$621$	− 6.12293i	− 0.245705i
$622$	0	0
$623$	2.00000	0.0801283
$624$	0	0
$625$	8.79899	0.351960
$626$	0	0
$627$	− 23.4412i	− 0.936152i
$628$	0	0
$629$	− 7.91630i	− 0.315644i
$630$	0	0
$631$	14.6274	0.582308	0.291154	−	0.956676i	$-0.405961\pi$
$631$	14.6274	0.582308	0.291154	+	0.956676i	$0.405961\pi$
$632$	0	0
$633$	−24.0000	−0.953914
$634$	0	0
$635$	13.5140i	0.536286i
$636$	0	0
$637$	6.30864i	0.249958i
$638$	0	0
$639$	−24.9706	−0.987820
$640$	0	0
$641$	−23.4558	−0.926450	−0.463225	−	0.886241i	$-0.653308\pi$
$641$	−23.4558	−0.926450	−0.463225	+	0.886241i	$0.653308\pi$
$642$	0	0
$643$	29.3784i	1.15857i	0.815124	+	0.579286i	$0.196669\pi$
$643$	29.3784i	1.15857i	−0.815124	+	0.579286i	$0.803331\pi$
$644$	0	0
$645$	6.12293i	0.241090i
$646$	0	0
$647$	−18.3431	−0.721143	−0.360572	−	0.932731i	$-0.617418\pi$
$647$	−18.3431	−0.721143	−0.360572	+	0.932731i	$0.617418\pi$
$648$	0	0
$649$	−32.9706	−1.29421
$650$	0	0
$651$	− 6.12293i	− 0.239977i
$652$	0	0
$653$	− 39.6452i	− 1.55144i	−0.631079	−	0.775719i	$-0.717387\pi$
$653$	− 39.6452i	− 1.55144i	0.631079	−	0.775719i	$-0.282613\pi$
$654$	0	0
$655$	8.20101	0.320440
$656$	0	0
$657$	0.627417	0.0244779
$658$	0	0
$659$	34.7904i	1.35524i	0.735412	+	0.677621i	$0.236989\pi$
$659$	34.7904i	1.35524i	−0.735412	+	0.677621i	$0.763011\pi$
$660$	0	0
$661$	6.30864i	0.245378i	0.992445	+	0.122689i	$0.0391517\pi$
$661$	6.30864i	0.245378i	−0.992445	+	0.122689i	$0.960848\pi$
$662$	0	0
$663$	−24.9706	−0.969776
$664$	0	0
$665$	4.48528	0.173932
$666$	0	0
$667$	9.70967i	0.375960i
$668$	0	0
$669$	− 27.0279i	− 1.04496i
$670$	0	0
$671$	37.6569	1.45373
$672$	0	0
$673$	−16.6274	−0.640940	−0.320470	−	0.947259i	$-0.603841\pi$
$673$	−16.6274	−0.640940	−0.320470	+	0.947259i	$0.603841\pi$
$674$	0	0
$675$	20.0083i	0.770121i
$676$	0	0
$677$	15.8645i	0.609721i	0.952397	+	0.304860i	$0.0986098\pi$
$677$	15.8645i	0.609721i	−0.952397	+	0.304860i	$0.901390\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	0	0
$681$	−13.4558	−0.515629
$682$	0	0
$683$	− 28.2960i	− 1.08272i	−0.840792	−	0.541359i	$-0.817910\pi$
$683$	− 28.2960i	− 1.08272i	0.840792	−	0.541359i	$-0.182090\pi$
$684$	0	0
$685$	− 2.16478i	− 0.0827122i
$686$	0	0
$687$	−9.17157	−0.349917
$688$	0	0
$689$	−27.3137	−1.04057
$690$	0	0
$691$	− 39.3057i	− 1.49526i	−0.664116	−	0.747629i	$-0.731192\pi$
$691$	− 39.3057i	− 1.49526i	0.664116	−	0.747629i	$-0.268808\pi$
$692$	0	0
$693$	9.55582i	0.362996i
$694$	0	0
$695$	−11.1127	−0.421529
$696$	0	0
$697$	28.0000	1.06058
$698$	0	0
$699$	− 10.8239i	− 0.409398i
$700$	0	0
$701$	4.70099i	0.177554i	0.996052	+	0.0887769i	$0.0282958\pi$
$701$	4.70099i	0.177554i	−0.996052	+	0.0887769i	$0.971704\pi$
$702$	0	0
$703$	−8.97056	−0.338331
$704$	0	0
$705$	9.37258	0.352992
$706$	0	0
$707$	4.14386i	0.155846i
$708$	0	0
$709$	− 31.7289i	− 1.19160i	−0.803131	−	0.595802i	$-0.796834\pi$
$709$	− 31.7289i	− 1.19160i	0.803131	−	0.595802i	$-0.203166\pi$
$710$	0	0
$711$	24.9706	0.936469
$712$	0	0
$713$	−6.62742	−0.248199
$714$	0	0
$715$	− 35.6871i	− 1.33462i
$716$	0	0
$717$	− 10.9778i	− 0.409972i
$718$	0	0
$719$	19.3137	0.720280	0.360140	−	0.932898i	$-0.382729\pi$
$719$	19.3137	0.720280	0.360140	+	0.932898i	$0.382729\pi$
$720$	0	0
$721$	2.34315	0.0872633
$722$	0	0
$723$	− 23.0698i	− 0.857975i
$724$	0	0
$725$	− 31.7289i	− 1.17838i
$726$	0	0
$727$	−43.3137	−1.60642	−0.803208	−	0.595698i	$-0.796875\pi$
$727$	−43.3137	−1.60642	−0.803208	+	0.595698i	$0.796875\pi$
$728$	0	0
$729$	−17.2843	−0.640158
$730$	0	0
$731$	19.1116i	0.706870i
$732$	0	0
$733$	− 15.4930i	− 0.572249i	−0.958192	−	0.286124i	$-0.907633\pi$
$733$	− 15.4930i	− 0.572249i	0.958192	−	0.286124i	$-0.0923670\pi$
$734$	0	0
$735$	1.17157	0.0432141
$736$	0	0
$737$	38.6274	1.42286
$738$	0	0
$739$	− 28.6675i	− 1.05455i	−0.849695	−	0.527275i	$-0.823214\pi$
$739$	− 28.6675i	− 1.05455i	0.849695	−	0.527275i	$-0.176786\pi$
$740$	0	0
$741$	28.2960i	1.03948i
$742$	0	0
$743$	−44.4853	−1.63201	−0.816003	−	0.578047i	$-0.803815\pi$
$743$	−44.4853	−1.63201	−0.816003	+	0.578047i	$0.803815\pi$
$744$	0	0
$745$	20.6863	0.757887
$746$	0	0
$747$	− 9.89538i	− 0.362053i
$748$	0	0
$749$	− 11.7206i	− 0.428262i
$750$	0	0
$751$	5.45584	0.199087	0.0995433	−	0.995033i	$-0.468262\pi$
$751$	5.45584	0.199087	0.0995433	+	0.995033i	$0.468262\pi$
$752$	0	0
$753$	−26.1421	−0.952672
$754$	0	0
$755$	10.9778i	0.399522i
$756$	0	0
$757$	− 39.6452i	− 1.44093i	−0.693491	−	0.720465i	$-0.743928\pi$
$757$	− 39.6452i	− 1.44093i	0.693491	−	0.720465i	$-0.256072\pi$
$758$	0	0
$759$	−6.62742	−0.240560
$760$	0	0
$761$	−14.9706	−0.542682	−0.271341	−	0.962483i	$-0.587467\pi$
$761$	−14.9706	−0.542682	−0.271341	+	0.962483i	$0.587467\pi$
$762$	0	0
$763$	16.9469i	0.613517i
$764$	0	0
$765$	− 7.23719i	− 0.261661i
$766$	0	0
$767$	39.7990	1.43706
$768$	0	0
$769$	−55.2548	−1.99254	−0.996270	−	0.0862891i	$-0.972499\pi$
$769$	−55.2548	−1.99254	−0.996270	+	0.0862891i	$0.972499\pi$
$770$	0	0
$771$	14.4107i	0.518987i
$772$	0	0
$773$	− 15.8645i	− 0.570605i	−0.958438	−	0.285303i	$-0.907906\pi$
$773$	− 15.8645i	− 0.570605i	0.958438	−	0.285303i	$-0.0920941\pi$
$774$	0	0
$775$	21.6569	0.777937
$776$	0	0
$777$	−2.34315	−0.0840599
$778$	0	0
$779$	− 31.7289i	− 1.13681i
$780$	0	0
$781$	71.3742i	2.55397i
$782$	0	0
$783$	43.3137	1.54791
$784$	0	0
$785$	−4.48528	−0.160087
$786$	0	0
$787$	− 33.7080i	− 1.20156i	−0.799414	−	0.600780i	$-0.794857\pi$
$787$	− 33.7080i	− 1.20156i	0.799414	−	0.600780i	$-0.205143\pi$
$788$	0	0
$789$	− 14.7821i	− 0.526256i
$790$	0	0
$791$	−3.17157	−0.112768
$792$	0	0
$793$	−45.4558	−1.61418
$794$	0	0
$795$	5.07241i	0.179900i
$796$	0	0
$797$	29.7499i	1.05379i	0.849929	+	0.526897i	$0.176645\pi$
$797$	29.7499i	1.05379i	−0.849929	+	0.526897i	$0.823355\pi$
$798$	0	0
$799$	29.2548	1.03496
$800$	0	0
$801$	−3.65685	−0.129209
$802$	0	0
$803$	− 1.79337i	− 0.0632865i
$804$	0	0
$805$	− 1.26810i	− 0.0446947i
$806$	0	0
$807$	−29.4558	−1.03689
$808$	0	0
$809$	−36.8284	−1.29482	−0.647409	−	0.762143i	$-0.724148\pi$
$809$	−36.8284	−1.29482	−0.647409	+	0.762143i	$0.724148\pi$
$810$	0	0
$811$	45.4286i	1.59521i	0.603177	+	0.797607i	$0.293901\pi$
$811$	45.4286i	1.59521i	−0.603177	+	0.797607i	$0.706099\pi$
$812$	0	0
$813$	14.7821i	0.518430i
$814$	0	0
$815$	−21.6569	−0.758607
$816$	0	0
$817$	21.6569	0.757677
$818$	0	0
$819$	− 11.5349i	− 0.403062i
$820$	0	0
$821$	− 20.9050i	− 0.729590i	−0.931088	−	0.364795i	$-0.881139\pi$
$821$	− 20.9050i	− 0.729590i	0.931088	−	0.364795i	$-0.118861\pi$
$822$	0	0
$823$	−7.02944	−0.245031	−0.122515	−	0.992467i	$-0.539096\pi$
$823$	−7.02944	−0.245031	−0.122515	+	0.992467i	$0.539096\pi$
$824$	0	0
$825$	21.6569	0.753995
$826$	0	0
$827$	47.4077i	1.64853i	0.566207	+	0.824263i	$0.308410\pi$
$827$	47.4077i	1.64853i	−0.566207	+	0.824263i	$0.691590\pi$
$828$	0	0
$829$	51.5515i	1.79046i	0.445605	+	0.895230i	$0.352989\pi$
$829$	51.5515i	1.79046i	−0.445605	+	0.895230i	$0.647011\pi$
$830$	0	0
$831$	−6.62742	−0.229903
$832$	0	0
$833$	3.65685	0.126702
$834$	0	0
$835$	17.3183i	0.599324i
$836$	0	0
$837$	29.5641i	1.02189i
$838$	0	0
$839$	20.2843	0.700291	0.350145	−	0.936695i	$-0.386132\pi$
$839$	20.2843	0.700291	0.350145	+	0.936695i	$0.386132\pi$
$840$	0	0
$841$	−39.6863	−1.36849
$842$	0	0
$843$	− 2.16478i	− 0.0745591i
$844$	0	0
$845$	29.0070i	0.997872i
$846$	0	0
$847$	16.3137	0.560546
$848$	0	0
$849$	−30.4264	−1.04423
$850$	0	0
$851$	2.53620i	0.0869399i
$852$	0	0
$853$	47.7472i	1.63483i	0.576046	+	0.817417i	$0.304595\pi$
$853$	47.7472i	1.63483i	−0.576046	+	0.817417i	$0.695405\pi$
$854$	0	0
$855$	−8.20101	−0.280469
$856$	0	0
$857$	−44.6274	−1.52444	−0.762222	−	0.647316i	$-0.775891\pi$
$857$	−44.6274	−1.52444	−0.762222	+	0.647316i	$0.775891\pi$
$858$	0	0
$859$	13.6997i	0.467427i	0.972306	+	0.233714i	$0.0750878\pi$
$859$	13.6997i	0.467427i	−0.972306	+	0.233714i	$0.924912\pi$
$860$	0	0
$861$	− 8.28772i	− 0.282445i
$862$	0	0
$863$	28.6863	0.976493	0.488246	−	0.872706i	$-0.337637\pi$
$863$	28.6863	0.976493	0.488246	+	0.872706i	$0.337637\pi$
$864$	0	0
$865$	−13.8579	−0.471182
$866$	0	0
$867$	− 3.92629i	− 0.133344i
$868$	0	0
$869$	− 71.3742i	− 2.42120i
$870$	0	0
$871$	−46.6274	−1.57991
$872$	0	0
$873$	−18.2843	−0.618829
$874$	0	0
$875$	9.55582i	0.323046i
$876$	0	0
$877$	25.6060i	0.864653i	0.901717	+	0.432326i	$0.142307\pi$
$877$	25.6060i	0.864653i	−0.901717	+	0.432326i	$0.857693\pi$
$878$	0	0
$879$	−11.5147	−0.388382
$880$	0	0
$881$	−22.9706	−0.773898	−0.386949	−	0.922101i	$-0.626471\pi$
$881$	−22.9706	−0.773898	−0.386949	+	0.922101i	$0.626471\pi$
$882$	0	0
$883$	− 21.4303i	− 0.721186i	−0.932723	−	0.360593i	$-0.882574\pi$
$883$	− 21.4303i	− 0.721186i	0.932723	−	0.360593i	$-0.117426\pi$
$884$	0	0
$885$	− 7.39104i	− 0.248447i
$886$	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
$889$	12.4853	0.418743
$890$	0	0
$891$	0.896683i	0.0300400i
$892$	0	0
$893$	− 33.1509i	− 1.10935i
$894$	0	0
$895$	−9.94113	−0.332295
$896$	0	0
$897$	8.00000	0.267112
$898$	0	0
$899$	− 46.8824i	− 1.56362i
$900$	0	0
$901$	15.8326i	0.527460i
$902$	0	0
$903$	5.65685	0.188248
$904$	0	0
$905$	1.17157	0.0389444
$906$	0	0
$907$	17.8435i	0.592485i	0.955113	+	0.296243i	$0.0957337\pi$
$907$	17.8435i	0.592485i	−0.955113	+	0.296243i	$0.904266\pi$
$908$	0	0
$909$	− 7.57675i	− 0.251305i
$910$	0	0
$911$	33.1716	1.09902	0.549512	−	0.835486i	$-0.314814\pi$
$911$	33.1716	1.09902	0.549512	+	0.835486i	$0.314814\pi$
$912$	0	0
$913$	−28.2843	−0.936073
$914$	0	0
$915$	8.44157i	0.279070i
$916$	0	0
$917$	− 7.57675i	− 0.250206i
$918$	0	0
$919$	−13.6569	−0.450498	−0.225249	−	0.974301i	$-0.572320\pi$
$919$	−13.6569	−0.450498	−0.225249	+	0.974301i	$0.572320\pi$
$920$	0	0
$921$	17.1716	0.565823
$922$	0	0
$923$	− 86.1562i	− 2.83587i
$924$	0	0
$925$	− 8.28772i	− 0.272499i
$926$	0	0
$927$	−4.28427	−0.140714
$928$	0	0
$929$	−46.2843	−1.51854	−0.759269	−	0.650777i	$-0.774443\pi$
$929$	−46.2843	−1.51854	−0.759269	+	0.650777i	$0.774443\pi$
$930$	0	0
$931$	− 4.14386i	− 0.135810i
$932$	0	0
$933$	33.1509i	1.08531i
$934$	0	0
$935$	−20.6863	−0.676514
$936$	0	0
$937$	18.2843	0.597321	0.298661	−	0.954359i	$-0.403460\pi$
$937$	18.2843	0.597321	0.298661	+	0.954359i	$0.403460\pi$
$938$	0	0
$939$	− 21.2764i	− 0.694330i
$940$	0	0
$941$	− 39.4595i	− 1.28634i	−0.765722	−	0.643172i	$-0.777618\pi$
$941$	− 39.4595i	− 1.28634i	0.765722	−	0.643172i	$-0.222382\pi$
$942$	0	0
$943$	−8.97056	−0.292122
$944$	0	0
$945$	−5.65685	−0.184017
$946$	0	0
$947$	1.63952i	0.0532772i	0.999645	+	0.0266386i	$0.00848034\pi$
$947$	1.63952i	0.0532772i	−0.999645	+	0.0266386i	$0.991520\pi$
$948$	0	0
$949$	2.16478i	0.0702719i
$950$	0	0
$951$	22.6274	0.733744
$952$	0	0
$953$	14.6863	0.475736	0.237868	−	0.971298i	$-0.423551\pi$
$953$	14.6863	0.475736	0.237868	+	0.971298i	$0.423551\pi$
$954$	0	0
$955$	− 3.58673i	− 0.116064i
$956$	0	0
$957$	− 46.8824i	− 1.51549i
$958$	0	0
$959$	−2.00000	−0.0645834
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	21.4303i	0.690581i
$964$	0	0
$965$	27.9246i	0.898925i
$966$	0	0
$967$	58.1421	1.86973	0.934863	−	0.355010i	$-0.115523\pi$
$967$	58.1421	1.86973	0.934863	+	0.355010i	$0.115523\pi$
$968$	0	0
$969$	16.4020	0.526909
$970$	0	0
$971$	− 45.5825i	− 1.46281i	−0.681943	−	0.731405i	$-0.738865\pi$
$971$	− 45.5825i	− 1.46281i	0.681943	−	0.731405i	$-0.261135\pi$
$972$	0	0
$973$	10.2668i	0.329138i
$974$	0	0
$975$	−26.1421	−0.837218
$976$	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
$979$	10.4525i	0.334063i
$980$	0	0
$981$	− 30.9861i	− 0.989310i
$982$	0	0
$983$	−56.9706	−1.81708	−0.908539	−	0.417799i	$-0.862802\pi$
$983$	−56.9706	−1.81708	−0.908539	+	0.417799i	$0.862802\pi$
$984$	0	0
$985$	−16.0000	−0.509802
$986$	0	0
$987$	− 8.65914i	− 0.275623i
$988$	0	0
$989$	− 6.12293i	− 0.194698i
$990$	0	0
$991$	18.3431	0.582689	0.291345	−	0.956618i	$-0.405897\pi$
$991$	18.3431	0.582689	0.291345	+	0.956618i	$0.405897\pi$
$992$	0	0
$993$	−15.0294	−0.476945
$994$	0	0
$995$	− 27.0279i	− 0.856843i
$996$	0	0
$997$	− 36.7695i	− 1.16450i	−0.813009	−	0.582250i	$-0.802172\pi$
$997$	− 36.7695i	− 1.16450i	0.813009	−	0.582250i	$-0.197828\pi$
$998$	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.f.449.3	yes	4
4.3	odd	2		896.2.b.h.449.2	yes	4
8.3	odd	2		896.2.b.h.449.3	yes	4
8.5	even	2	inner	896.2.b.f.449.2	✓	4
16.3	odd	4		1792.2.a.u.1.3		4
16.5	even	4		1792.2.a.w.1.3		4
16.11	odd	4		1792.2.a.u.1.2		4
16.13	even	4		1792.2.a.w.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.b.f.449.2	✓	4	8.5	even	2	inner
896.2.b.f.449.3	yes	4	1.1	even	1	trivial
896.2.b.h.449.2	yes	4	4.3	odd	2
896.2.b.h.449.3	yes	4	8.3	odd	2
1792.2.a.u.1.2		4	16.11	odd	4
1792.2.a.u.1.3		4	16.3	odd	4
1792.2.a.w.1.2		4	16.13	even	4
1792.2.a.w.1.3		4	16.5	even	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)

\(f(q)\)	\(=\)	\(q+1.08239i q^{3} -1.08239i q^{5} -1.00000 q^{7} +1.82843 q^{9} +O(q^{10})\)	\(q+1.08239i q^{3} -1.08239i q^{5} -1.00000 q^{7} +1.82843 q^{9} -5.22625i q^{11} +6.30864i q^{13} +1.17157 q^{15} +3.65685 q^{17} -4.14386i q^{19} -1.08239i q^{21} -1.17157 q^{23} +3.82843 q^{25} +5.22625i q^{27} -8.28772i q^{29} +5.65685 q^{31} +5.65685 q^{33} +1.08239i q^{35} -2.16478i q^{37} -6.82843 q^{39} +7.65685 q^{41} +5.22625i q^{43} -1.97908i q^{45} +8.00000 q^{47} +1.00000 q^{49} +3.95815i q^{51} +4.32957i q^{53} -5.65685 q^{55} +4.48528 q^{57} -6.30864i q^{59} +7.20533i q^{61} -1.82843 q^{63} +6.82843 q^{65} +7.39104i q^{67} -1.26810i q^{69} -13.6569 q^{71} +0.343146 q^{73} +4.14386i q^{75} +5.22625i q^{77} +13.6569 q^{79} -0.171573 q^{81} -5.41196i q^{83} -3.95815i q^{85} +8.97056 q^{87} -2.00000 q^{89} -6.30864i q^{91} +6.12293i q^{93} -4.48528 q^{95} -10.0000 q^{97} -9.55582i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} - 4 q^{9}+O(q^{10}) \) 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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Modular forms

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