LMFDB - Embedded newform 1024.2.b.g.513.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,2,Mod(513,1024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1024.513");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1024.b (of order $2$, degree $1$, not 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:	$8.17668116698$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			513.4
Root			$-0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	1024.513
Dual form			1024.2.b.g.513.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.08239i q^{3} +0.585786i q^{5} -3.69552 q^{7} +1.82843 q^{9} +4.14386i q^{11} -3.41421i q^{13} +0.634051 q^{15} +2.82843 q^{17} +6.30864i q^{19} +4.00000i q^{21} +6.75699 q^{23} +4.65685 q^{25} -5.22625i q^{27} -7.41421i q^{29} +3.06147 q^{31} +4.48528 q^{33} -2.16478i q^{35} +9.07107i q^{37} -3.69552 q^{39} +4.00000 q^{41} -1.08239i q^{43} +1.07107i q^{45} +3.06147 q^{47} +6.65685 q^{49} -3.06147i q^{51} +4.58579i q^{53} -2.42742 q^{55} +6.82843 q^{57} +1.08239i q^{59} +1.07107i q^{61} -6.75699 q^{63} +2.00000 q^{65} -1.97908i q^{67} -7.31371i q^{69} +8.02509 q^{71} +6.48528 q^{73} -5.04054i q^{75} -15.3137i q^{77} -14.7821 q^{79} -0.171573 q^{81} -13.6997i q^{83} +1.65685i q^{85} -8.02509 q^{87} -4.82843 q^{89} +12.6173i q^{91} -3.31371i q^{93} -3.69552 q^{95} +5.17157 q^{97} +7.57675i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9} - 8 q^{25} - 32 q^{33} + 32 q^{41} + 8 q^{49} + 32 q^{57} + 16 q^{65} - 16 q^{73} - 24 q^{81} - 16 q^{89} + 64 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 - 8 * q^25 - 32 * q^33 + 32 * q^41 + 8 * q^49 + 32 * q^57 + 16 * q^65 - 16 * q^73 - 24 * q^81 - 16 * q^89 + 64 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$.

$n$	$5$	$1023$
$\chi(n)$	$-1$	$1$

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	$-0.898847\pi$
$3$	− 1.08239i	− 0.624919i	0.949931	−	0.312460i	$-0.101153\pi$
$4$	0	0
$5$	0.585786i	0.261972i	0.991384	+	0.130986i	$0.0418142\pi$
$5$	0.585786i	0.261972i	−0.991384	+	0.130986i	$0.958186\pi$
$6$	0	0
$7$	−3.69552	−1.39677	−0.698387	−	0.715720i	$-0.746099\pi$
$7$	−3.69552	−1.39677	−0.698387	+	0.715720i	$0.746099\pi$
$8$	0	0
$9$	1.82843	0.609476
$10$	0	0
$11$	4.14386i	1.24942i	0.780857	+	0.624710i	$0.214783\pi$
$11$	4.14386i	1.24942i	−0.780857	+	0.624710i	$0.785217\pi$
$12$	0	0
$13$	− 3.41421i	− 0.946932i	−0.880812	−	0.473466i	$-0.843003\pi$
$13$	− 3.41421i	− 0.946932i	0.880812	−	0.473466i	$-0.156997\pi$
$14$	0	0
$15$	0.634051	0.163711
$16$	0	0
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	0	0
$19$	6.30864i	1.44730i	0.690166	+	0.723651i	$0.257538\pi$
$19$	6.30864i	1.44730i	−0.690166	+	0.723651i	$0.742462\pi$
$20$	0	0
$21$	4.00000i	0.872872i
$22$	0	0
$23$	6.75699	1.40893	0.704464	−	0.709739i	$-0.251187\pi$
$23$	6.75699	1.40893	0.704464	+	0.709739i	$0.251187\pi$
$24$	0	0
$25$	4.65685	0.931371
$26$	0	0
$27$	− 5.22625i	− 1.00579i
$28$	0	0
$29$	− 7.41421i	− 1.37678i	−0.725338	−	0.688392i	$-0.758317\pi$
$29$	− 7.41421i	− 1.37678i	0.725338	−	0.688392i	$-0.241683\pi$
$30$	0	0
$31$	3.06147	0.549856	0.274928	−	0.961465i	$-0.411346\pi$
$31$	3.06147	0.549856	0.274928	+	0.961465i	$0.411346\pi$
$32$	0	0
$33$	4.48528	0.780787
$34$	0	0
$35$	− 2.16478i	− 0.365915i
$36$	0	0
$37$	9.07107i	1.49127i	0.666352	+	0.745637i	$0.267855\pi$
$37$	9.07107i	1.49127i	−0.666352	+	0.745637i	$0.732145\pi$
$38$	0	0
$39$	−3.69552	−0.591756
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	− 1.08239i	− 0.165063i	−0.996588	−	0.0825316i	$-0.973699\pi$
$43$	− 1.08239i	− 0.165063i	0.996588	−	0.0825316i	$-0.0263005\pi$
$44$	0	0
$45$	1.07107i	0.159665i
$46$	0	0
$47$	3.06147	0.446561	0.223280	−	0.974754i	$-0.428323\pi$
$47$	3.06147	0.446561	0.223280	+	0.974754i	$0.428323\pi$
$48$	0	0
$49$	6.65685	0.950979
$50$	0	0
$51$	− 3.06147i	− 0.428691i
$52$	0	0
$53$	4.58579i	0.629906i	0.949107	+	0.314953i	$0.101989\pi$
$53$	4.58579i	0.629906i	−0.949107	+	0.314953i	$0.898011\pi$
$54$	0	0
$55$	−2.42742	−0.327313
$56$	0	0
$57$	6.82843	0.904447
$58$	0	0
$59$	1.08239i	0.140915i	0.997515	+	0.0704577i	$0.0224460\pi$
$59$	1.08239i	0.140915i	−0.997515	+	0.0704577i	$0.977554\pi$
$60$	0	0
$61$	1.07107i	0.137136i	0.997646	+	0.0685681i	$0.0218430\pi$
$61$	1.07107i	0.137136i	−0.997646	+	0.0685681i	$0.978157\pi$
$62$	0	0
$63$	−6.75699	−0.851300
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	− 1.97908i	− 0.241783i	−0.992666	−	0.120891i	$-0.961425\pi$
$67$	− 1.97908i	− 0.241783i	0.992666	−	0.120891i	$-0.0385752\pi$
$68$	0	0
$69$	− 7.31371i	− 0.880467i
$70$	0	0
$71$	8.02509	0.952403	0.476201	−	0.879336i	$-0.342013\pi$
$71$	8.02509	0.952403	0.476201	+	0.879336i	$0.342013\pi$
$72$	0	0
$73$	6.48528	0.759045	0.379522	−	0.925183i	$-0.376088\pi$
$73$	6.48528	0.759045	0.379522	+	0.925183i	$0.376088\pi$
$74$	0	0
$75$	− 5.04054i	− 0.582032i
$76$	0	0
$77$	− 15.3137i	− 1.74516i
$78$	0	0
$79$	−14.7821	−1.66311	−0.831557	−	0.555440i	$-0.812550\pi$
$79$	−14.7821	−1.66311	−0.831557	+	0.555440i	$0.812550\pi$
$80$	0	0
$81$	−0.171573	−0.0190637
$82$	0	0
$83$	− 13.6997i	− 1.50374i	−0.659314	−	0.751868i	$-0.729153\pi$
$83$	− 13.6997i	− 1.50374i	0.659314	−	0.751868i	$-0.270847\pi$
$84$	0	0
$85$	1.65685i	0.179711i
$86$	0	0
$87$	−8.02509	−0.860380
$88$	0	0
$89$	−4.82843	−0.511812	−0.255906	−	0.966702i	$-0.582374\pi$
$89$	−4.82843	−0.511812	−0.255906	+	0.966702i	$0.582374\pi$
$90$	0	0
$91$	12.6173i	1.32265i
$92$	0	0
$93$	− 3.31371i	− 0.343616i
$94$	0	0
$95$	−3.69552	−0.379152
$96$	0	0
$97$	5.17157	0.525094	0.262547	−	0.964919i	$-0.415438\pi$
$97$	5.17157	0.525094	0.262547	+	0.964919i	$0.415438\pi$
$98$	0	0
$99$	7.57675i	0.761492i
$100$	0	0
$101$	17.0711i	1.69863i	0.527883	+	0.849317i	$0.322986\pi$
$101$	17.0711i	1.69863i	−0.527883	+	0.849317i	$0.677014\pi$
$102$	0	0
$103$	14.1480	1.39405	0.697023	−	0.717049i	$-0.254508\pi$
$103$	14.1480	1.39405	0.697023	+	0.717049i	$0.254508\pi$
$104$	0	0
$105$	−2.34315	−0.228668
$106$	0	0
$107$	− 1.97908i	− 0.191324i	−0.995414	−	0.0956622i	$-0.969503\pi$
$107$	− 1.97908i	− 0.191324i	0.995414	−	0.0956622i	$-0.0304969\pi$
$108$	0	0
$109$	13.0711i	1.25198i	0.779831	+	0.625991i	$0.215305\pi$
$109$	13.0711i	1.25198i	−0.779831	+	0.625991i	$0.784695\pi$
$110$	0	0
$111$	9.81845	0.931926
$112$	0	0
$113$	−7.65685	−0.720296	−0.360148	−	0.932895i	$-0.617274\pi$
$113$	−7.65685	−0.720296	−0.360148	+	0.932895i	$0.617274\pi$
$114$	0	0
$115$	3.95815i	0.369099i
$116$	0	0
$117$	− 6.24264i	− 0.577132i
$118$	0	0
$119$	−10.4525	−0.958179
$120$	0	0
$121$	−6.17157	−0.561052
$122$	0	0
$123$	− 4.32957i	− 0.390384i
$124$	0	0
$125$	5.65685i	0.505964i
$126$	0	0
$127$	11.7206	1.04004	0.520018	−	0.854155i	$-0.325925\pi$
$127$	11.7206	1.04004	0.520018	+	0.854155i	$0.325925\pi$
$128$	0	0
$129$	−1.17157	−0.103151
$130$	0	0
$131$	14.5964i	1.27529i	0.770330	+	0.637645i	$0.220091\pi$
$131$	14.5964i	1.27529i	−0.770330	+	0.637645i	$0.779909\pi$
$132$	0	0
$133$	− 23.3137i	− 2.02155i
$134$	0	0
$135$	3.06147	0.263489
$136$	0	0
$137$	−12.9706	−1.10815	−0.554075	−	0.832467i	$-0.686928\pi$
$137$	−12.9706	−1.10815	−0.554075	+	0.832467i	$0.686928\pi$
$138$	0	0
$139$	− 20.1940i	− 1.71284i	−0.516283	−	0.856418i	$-0.672685\pi$
$139$	− 20.1940i	− 1.71284i	0.516283	−	0.856418i	$-0.327315\pi$
$140$	0	0
$141$	− 3.31371i	− 0.279065i
$142$	0	0
$143$	14.1480	1.18312
$144$	0	0
$145$	4.34315	0.360679
$146$	0	0
$147$	− 7.20533i	− 0.594285i
$148$	0	0
$149$	1.75736i	0.143968i	0.997406	+	0.0719842i	$0.0229331\pi$
$149$	1.75736i	0.143968i	−0.997406	+	0.0719842i	$0.977067\pi$
$150$	0	0
$151$	0.634051	0.0515983	0.0257992	−	0.999667i	$-0.491787\pi$
$151$	0.634051	0.0515983	0.0257992	+	0.999667i	$0.491787\pi$
$152$	0	0
$153$	5.17157	0.418097
$154$	0	0
$155$	1.79337i	0.144047i
$156$	0	0
$157$	− 2.24264i	− 0.178982i	−0.995988	−	0.0894911i	$-0.971476\pi$
$157$	− 2.24264i	− 0.178982i	0.995988	−	0.0894911i	$-0.0285241\pi$
$158$	0	0
$159$	4.96362	0.393641
$160$	0	0
$161$	−24.9706	−1.96796
$162$	0	0
$163$	− 5.41196i	− 0.423898i	−0.977281	−	0.211949i	$-0.932019\pi$
$163$	− 5.41196i	− 0.423898i	0.977281	−	0.211949i	$-0.0679810\pi$
$164$	0	0
$165$	2.62742i	0.204544i
$166$	0	0
$167$	−3.69552	−0.285968	−0.142984	−	0.989725i	$-0.545670\pi$
$167$	−3.69552	−0.285968	−0.142984	+	0.989725i	$0.545670\pi$
$168$	0	0
$169$	1.34315	0.103319
$170$	0	0
$171$	11.5349i	0.882096i
$172$	0	0
$173$	19.2132i	1.46075i	0.683045	+	0.730376i	$0.260655\pi$
$173$	19.2132i	1.46075i	−0.683045	+	0.730376i	$0.739345\pi$
$174$	0	0
$175$	−17.2095	−1.30092
$176$	0	0
$177$	1.17157	0.0880608
$178$	0	0
$179$	11.5349i	0.862159i	0.902314	+	0.431079i	$0.141867\pi$
$179$	11.5349i	0.862159i	−0.902314	+	0.431079i	$0.858133\pi$
$180$	0	0
$181$	− 14.2426i	− 1.05865i	−0.848420	−	0.529324i	$-0.822446\pi$
$181$	− 14.2426i	− 1.05865i	0.848420	−	0.529324i	$-0.177554\pi$
$182$	0	0
$183$	1.15932	0.0856991
$184$	0	0
$185$	−5.31371	−0.390672
$186$	0	0
$187$	11.7206i	0.857096i
$188$	0	0
$189$	19.3137i	1.40487i
$190$	0	0
$191$	−11.7206	−0.848073	−0.424037	−	0.905645i	$-0.639387\pi$
$191$	−11.7206	−0.848073	−0.424037	+	0.905645i	$0.639387\pi$
$192$	0	0
$193$	0.485281	0.0349313	0.0174657	−	0.999847i	$-0.494440\pi$
$193$	0.485281	0.0349313	0.0174657	+	0.999847i	$0.494440\pi$
$194$	0	0
$195$	− 2.16478i	− 0.155023i
$196$	0	0
$197$	− 10.7279i	− 0.764333i	−0.924094	−	0.382166i	$-0.875178\pi$
$197$	− 10.7279i	− 0.764333i	0.924094	−	0.382166i	$-0.124822\pi$
$198$	0	0
$199$	−12.3547	−0.875798	−0.437899	−	0.899024i	$-0.644277\pi$
$199$	−12.3547	−0.875798	−0.437899	+	0.899024i	$0.644277\pi$
$200$	0	0
$201$	−2.14214	−0.151095
$202$	0	0
$203$	27.3994i	1.92306i
$204$	0	0
$205$	2.34315i	0.163652i
$206$	0	0
$207$	12.3547	0.858708
$208$	0	0
$209$	−26.1421	−1.80829
$210$	0	0
$211$	13.3283i	0.917555i	0.888551	+	0.458778i	$0.151713\pi$
$211$	13.3283i	0.917555i	−0.888551	+	0.458778i	$0.848287\pi$
$212$	0	0
$213$	− 8.68629i	− 0.595175i
$214$	0	0
$215$	0.634051	0.0432419
$216$	0	0
$217$	−11.3137	−0.768025
$218$	0	0
$219$	− 7.01962i	− 0.474342i
$220$	0	0
$221$	− 9.65685i	− 0.649590i
$222$	0	0
$223$	3.06147	0.205011	0.102506	−	0.994732i	$-0.467314\pi$
$223$	3.06147	0.205011	0.102506	+	0.994732i	$0.467314\pi$
$224$	0	0
$225$	8.51472	0.567648
$226$	0	0
$227$	− 1.08239i	− 0.0718409i	−0.999355	−	0.0359204i	$-0.988564\pi$
$227$	− 1.08239i	− 0.0718409i	0.999355	−	0.0359204i	$-0.0114363\pi$
$228$	0	0
$229$	3.89949i	0.257686i	0.991665	+	0.128843i	$0.0411263\pi$
$229$	3.89949i	0.257686i	−0.991665	+	0.128843i	$0.958874\pi$
$230$	0	0
$231$	−16.5754	−1.09058
$232$	0	0
$233$	−20.8284	−1.36452	−0.682258	−	0.731112i	$-0.739002\pi$
$233$	−20.8284	−1.36452	−0.682258	+	0.731112i	$0.739002\pi$
$234$	0	0
$235$	1.79337i	0.116986i
$236$	0	0
$237$	16.0000i	1.03931i
$238$	0	0
$239$	−12.2459	−0.792119	−0.396060	−	0.918225i	$-0.629623\pi$
$239$	−12.2459	−0.792119	−0.396060	+	0.918225i	$0.629623\pi$
$240$	0	0
$241$	−13.1716	−0.848456	−0.424228	−	0.905556i	$-0.639454\pi$
$241$	−13.1716	−0.848456	−0.424228	+	0.905556i	$0.639454\pi$
$242$	0	0
$243$	− 15.4930i	− 0.993879i
$244$	0	0
$245$	3.89949i	0.249130i
$246$	0	0
$247$	21.5391	1.37050
$248$	0	0
$249$	−14.8284	−0.939713
$250$	0	0
$251$	− 18.0292i	− 1.13800i	−0.822339	−	0.568998i	$-0.807331\pi$
$251$	− 18.0292i	− 1.13800i	0.822339	−	0.568998i	$-0.192669\pi$
$252$	0	0
$253$	28.0000i	1.76034i
$254$	0	0
$255$	1.79337	0.112305
$256$	0	0
$257$	−14.9706	−0.933838	−0.466919	−	0.884300i	$-0.654636\pi$
$257$	−14.9706	−0.933838	−0.466919	+	0.884300i	$0.654636\pi$
$258$	0	0
$259$	− 33.5223i	− 2.08297i
$260$	0	0
$261$	− 13.5563i	− 0.839117i
$262$	0	0
$263$	−12.8799	−0.794210	−0.397105	−	0.917773i	$-0.629985\pi$
$263$	−12.8799	−0.794210	−0.397105	+	0.917773i	$0.629985\pi$
$264$	0	0
$265$	−2.68629	−0.165018
$266$	0	0
$267$	5.22625i	0.319841i
$268$	0	0
$269$	− 25.5563i	− 1.55820i	−0.626901	−	0.779099i	$-0.715677\pi$
$269$	− 25.5563i	− 1.55820i	0.626901	−	0.779099i	$-0.284323\pi$
$270$	0	0
$271$	−5.59767	−0.340034	−0.170017	−	0.985441i	$-0.554382\pi$
$271$	−5.59767	−0.340034	−0.170017	+	0.985441i	$0.554382\pi$
$272$	0	0
$273$	13.6569	0.826550
$274$	0	0
$275$	19.2974i	1.16367i
$276$	0	0
$277$	− 3.41421i	− 0.205140i	−0.994726	−	0.102570i	$-0.967293\pi$
$277$	− 3.41421i	− 0.205140i	0.994726	−	0.102570i	$-0.0327066\pi$
$278$	0	0
$279$	5.59767	0.335124
$280$	0	0
$281$	−17.7990	−1.06180	−0.530899	−	0.847435i	$-0.678146\pi$
$281$	−17.7990	−1.06180	−0.530899	+	0.847435i	$0.678146\pi$
$282$	0	0
$283$	− 14.5964i	− 0.867664i	−0.900994	−	0.433832i	$-0.857161\pi$
$283$	− 14.5964i	− 0.867664i	0.900994	−	0.433832i	$-0.142839\pi$
$284$	0	0
$285$	4.00000i	0.236940i
$286$	0	0
$287$	−14.7821	−0.872558
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	− 5.59767i	− 0.328141i
$292$	0	0
$293$	0.585786i	0.0342220i	0.999854	+	0.0171110i	$0.00544687\pi$
$293$	0.585786i	0.0342220i	−0.999854	+	0.0171110i	$0.994553\pi$
$294$	0	0
$295$	−0.634051	−0.0369159
$296$	0	0
$297$	21.6569	1.25666
$298$	0	0
$299$	− 23.0698i	− 1.33416i
$300$	0	0
$301$	4.00000i	0.230556i
$302$	0	0
$303$	18.4776	1.06151
$304$	0	0
$305$	−0.627417	−0.0359258
$306$	0	0
$307$	4.51528i	0.257701i	0.991664	+	0.128850i	$0.0411287\pi$
$307$	4.51528i	0.257701i	−0.991664	+	0.128850i	$0.958871\pi$
$308$	0	0
$309$	− 15.3137i	− 0.871166i
$310$	0	0
$311$	24.6005	1.39497	0.697484	−	0.716600i	$-0.254303\pi$
$311$	24.6005	1.39497	0.697484	+	0.716600i	$0.254303\pi$
$312$	0	0
$313$	31.3137	1.76996	0.884978	−	0.465633i	$-0.154173\pi$
$313$	31.3137	1.76996	0.884978	+	0.465633i	$0.154173\pi$
$314$	0	0
$315$	− 3.95815i	− 0.223017i
$316$	0	0
$317$	− 15.4142i	− 0.865748i	−0.901454	−	0.432874i	$-0.857499\pi$
$317$	− 15.4142i	− 0.865748i	0.901454	−	0.432874i	$-0.142501\pi$
$318$	0	0
$319$	30.7235	1.72018
$320$	0	0
$321$	−2.14214	−0.119562
$322$	0	0
$323$	17.8435i	0.992841i
$324$	0	0
$325$	− 15.8995i	− 0.881945i
$326$	0	0
$327$	14.1480	0.782387
$328$	0	0
$329$	−11.3137	−0.623745
$330$	0	0
$331$	8.47343i	0.465742i	0.972508	+	0.232871i	$0.0748120\pi$
$331$	8.47343i	0.465742i	−0.972508	+	0.232871i	$0.925188\pi$
$332$	0	0
$333$	16.5858i	0.908895i
$334$	0	0
$335$	1.15932	0.0633402
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	0	0
$339$	8.28772i	0.450127i
$340$	0	0
$341$	12.6863i	0.687001i
$342$	0	0
$343$	1.26810	0.0684710
$344$	0	0
$345$	4.28427	0.230657
$346$	0	0
$347$	− 15.4930i	− 0.831710i	−0.909431	−	0.415855i	$-0.863482\pi$
$347$	− 15.4930i	− 0.831710i	0.909431	−	0.415855i	$-0.136518\pi$
$348$	0	0
$349$	− 2.72792i	− 0.146022i	−0.997331	−	0.0730112i	$-0.976739\pi$
$349$	− 2.72792i	− 0.146022i	0.997331	−	0.0730112i	$-0.0232609\pi$
$350$	0	0
$351$	−17.8435	−0.952418
$352$	0	0
$353$	22.9706	1.22260	0.611300	−	0.791399i	$-0.290647\pi$
$353$	22.9706	1.22260	0.611300	+	0.791399i	$0.290647\pi$
$354$	0	0
$355$	4.70099i	0.249502i
$356$	0	0
$357$	11.3137i	0.598785i
$358$	0	0
$359$	−0.634051	−0.0334639	−0.0167320	−	0.999860i	$-0.505326\pi$
$359$	−0.634051	−0.0334639	−0.0167320	+	0.999860i	$0.505326\pi$
$360$	0	0
$361$	−20.7990	−1.09468
$362$	0	0
$363$	6.68006i	0.350612i
$364$	0	0
$365$	3.79899i	0.198848i
$366$	0	0
$367$	−9.18440	−0.479422	−0.239711	−	0.970844i	$-0.577053\pi$
$367$	−9.18440	−0.479422	−0.239711	+	0.970844i	$0.577053\pi$
$368$	0	0
$369$	7.31371	0.380736
$370$	0	0
$371$	− 16.9469i	− 0.879837i
$372$	0	0
$373$	− 6.72792i	− 0.348359i	−0.984714	−	0.174179i	$-0.944273\pi$
$373$	− 6.72792i	− 0.348359i	0.984714	−	0.174179i	$-0.0557272\pi$
$374$	0	0
$375$	6.12293	0.316187
$376$	0	0
$377$	−25.3137	−1.30372
$378$	0	0
$379$	26.3170i	1.35181i	0.736988	+	0.675906i	$0.236247\pi$
$379$	26.3170i	1.35181i	−0.736988	+	0.675906i	$0.763753\pi$
$380$	0	0
$381$	− 12.6863i	− 0.649938i
$382$	0	0
$383$	−20.9050	−1.06820	−0.534098	−	0.845423i	$-0.679349\pi$
$383$	−20.9050	−1.06820	−0.534098	+	0.845423i	$0.679349\pi$
$384$	0	0
$385$	8.97056	0.457182
$386$	0	0
$387$	− 1.97908i	− 0.100602i
$388$	0	0
$389$	20.3848i	1.03355i	0.856121	+	0.516775i	$0.172867\pi$
$389$	20.3848i	1.03355i	−0.856121	+	0.516775i	$0.827133\pi$
$390$	0	0
$391$	19.1116	0.966517
$392$	0	0
$393$	15.7990	0.796954
$394$	0	0
$395$	− 8.65914i	− 0.435688i
$396$	0	0
$397$	5.07107i	0.254510i	0.991870	+	0.127255i	$0.0406166\pi$
$397$	5.07107i	0.254510i	−0.991870	+	0.127255i	$0.959383\pi$
$398$	0	0
$399$	−25.2346	−1.26331
$400$	0	0
$401$	7.51472	0.375267	0.187634	−	0.982239i	$-0.439918\pi$
$401$	7.51472	0.375267	0.187634	+	0.982239i	$0.439918\pi$
$402$	0	0
$403$	− 10.4525i	− 0.520676i
$404$	0	0
$405$	− 0.100505i	− 0.00499414i
$406$	0	0
$407$	−37.5892	−1.86323
$408$	0	0
$409$	31.3137	1.54836	0.774182	−	0.632964i	$-0.218162\pi$
$409$	31.3137	1.54836	0.774182	+	0.632964i	$0.218162\pi$
$410$	0	0
$411$	14.0392i	0.692504i
$412$	0	0
$413$	− 4.00000i	− 0.196827i
$414$	0	0
$415$	8.02509	0.393936
$416$	0	0
$417$	−21.8579	−1.07038
$418$	0	0
$419$	− 12.4316i	− 0.607322i	−0.952780	−	0.303661i	$-0.901791\pi$
$419$	− 12.4316i	− 0.607322i	0.952780	−	0.303661i	$-0.0982091\pi$
$420$	0	0
$421$	8.58579i	0.418446i	0.977868	+	0.209223i	$0.0670934\pi$
$421$	8.58579i	0.418446i	−0.977868	+	0.209223i	$0.932907\pi$
$422$	0	0
$423$	5.59767	0.272168
$424$	0	0
$425$	13.1716	0.638915
$426$	0	0
$427$	− 3.95815i	− 0.191548i
$428$	0	0
$429$	− 15.3137i	− 0.739353i
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	25.4558	1.22333	0.611665	−	0.791117i	$-0.290500\pi$
$433$	25.4558	1.22333	0.611665	+	0.791117i	$0.290500\pi$
$434$	0	0
$435$	− 4.70099i	− 0.225395i
$436$	0	0
$437$	42.6274i	2.03915i
$438$	0	0
$439$	−38.1145	−1.81911	−0.909553	−	0.415588i	$-0.863576\pi$
$439$	−38.1145	−1.81911	−0.909553	+	0.415588i	$0.863576\pi$
$440$	0	0
$441$	12.1716	0.579599
$442$	0	0
$443$	1.97908i	0.0940287i	0.998894	+	0.0470144i	$0.0149707\pi$
$443$	1.97908i	0.0940287i	−0.998894	+	0.0470144i	$0.985029\pi$
$444$	0	0
$445$	− 2.82843i	− 0.134080i
$446$	0	0
$447$	1.90215	0.0899687
$448$	0	0
$449$	−6.14214	−0.289865	−0.144933	−	0.989442i	$-0.546297\pi$
$449$	−6.14214	−0.289865	−0.144933	+	0.989442i	$0.546297\pi$
$450$	0	0
$451$	16.5754i	0.780507i
$452$	0	0
$453$	− 0.686292i	− 0.0322448i
$454$	0	0
$455$	−7.39104	−0.346497
$456$	0	0
$457$	26.6274	1.24558	0.622789	−	0.782390i	$-0.285999\pi$
$457$	26.6274	1.24558	0.622789	+	0.782390i	$0.285999\pi$
$458$	0	0
$459$	− 14.7821i	− 0.689968i
$460$	0	0
$461$	− 38.2426i	− 1.78114i	−0.454849	−	0.890569i	$-0.650307\pi$
$461$	− 38.2426i	− 1.78114i	0.454849	−	0.890569i	$-0.349693\pi$
$462$	0	0
$463$	35.6871	1.65852	0.829260	−	0.558864i	$-0.188762\pi$
$463$	35.6871	1.65852	0.829260	+	0.558864i	$0.188762\pi$
$464$	0	0
$465$	1.94113	0.0900175
$466$	0	0
$467$	− 38.0376i	− 1.76017i	−0.474817	−	0.880084i	$-0.657486\pi$
$467$	− 38.0376i	− 1.76017i	0.474817	−	0.880084i	$-0.342514\pi$
$468$	0	0
$469$	7.31371i	0.337716i
$470$	0	0
$471$	−2.42742	−0.111849
$472$	0	0
$473$	4.48528	0.206233
$474$	0	0
$475$	29.3784i	1.34798i
$476$	0	0
$477$	8.38478i	0.383913i
$478$	0	0
$479$	35.1618	1.60658	0.803292	−	0.595585i	$-0.203080\pi$
$479$	35.1618	1.60658	0.803292	+	0.595585i	$0.203080\pi$
$480$	0	0
$481$	30.9706	1.41214
$482$	0	0
$483$	27.0279i	1.22981i
$484$	0	0
$485$	3.02944i	0.137560i
$486$	0	0
$487$	−6.23172	−0.282386	−0.141193	−	0.989982i	$-0.545094\pi$
$487$	−6.23172	−0.282386	−0.141193	+	0.989982i	$0.545094\pi$
$488$	0	0
$489$	−5.85786	−0.264902
$490$	0	0
$491$	5.04054i	0.227477i	0.993511	+	0.113738i	$0.0362825\pi$
$491$	5.04054i	0.227477i	−0.993511	+	0.113738i	$0.963717\pi$
$492$	0	0
$493$	− 20.9706i	− 0.944467i
$494$	0	0
$495$	−4.43835	−0.199489
$496$	0	0
$497$	−29.6569	−1.33029
$498$	0	0
$499$	17.6578i	0.790473i	0.918579	+	0.395237i	$0.129337\pi$
$499$	17.6578i	0.790473i	−0.918579	+	0.395237i	$0.870663\pi$
$500$	0	0
$501$	4.00000i	0.178707i
$502$	0	0
$503$	−22.8072	−1.01692	−0.508460	−	0.861085i	$-0.669785\pi$
$503$	−22.8072	−1.01692	−0.508460	+	0.861085i	$0.669785\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	0	0
$507$	− 1.45381i	− 0.0645660i
$508$	0	0
$509$	− 26.2426i	− 1.16318i	−0.813480	−	0.581592i	$-0.802430\pi$
$509$	− 26.2426i	− 1.16318i	0.813480	−	0.581592i	$-0.197570\pi$
$510$	0	0
$511$	−23.9665	−1.06021
$512$	0	0
$513$	32.9706	1.45569
$514$	0	0
$515$	8.28772i	0.365201i
$516$	0	0
$517$	12.6863i	0.557942i
$518$	0	0
$519$	20.7962	0.912853
$520$	0	0
$521$	−6.34315	−0.277898	−0.138949	−	0.990300i	$-0.544372\pi$
$521$	−6.34315	−0.277898	−0.138949	+	0.990300i	$0.544372\pi$
$522$	0	0
$523$	− 14.2249i	− 0.622013i	−0.950408	−	0.311007i	$-0.899334\pi$
$523$	− 14.2249i	− 0.622013i	0.950408	−	0.311007i	$-0.100666\pi$
$524$	0	0
$525$	18.6274i	0.812967i
$526$	0	0
$527$	8.65914	0.377198
$528$	0	0
$529$	22.6569	0.985081
$530$	0	0
$531$	1.97908i	0.0858846i
$532$	0	0
$533$	− 13.6569i	− 0.591544i
$534$	0	0
$535$	1.15932	0.0501216
$536$	0	0
$537$	12.4853	0.538780
$538$	0	0
$539$	27.5851i	1.18817i
$540$	0	0
$541$	− 10.2426i	− 0.440366i	−0.975459	−	0.220183i	$-0.929335\pi$
$541$	− 10.2426i	− 0.440366i	0.975459	−	0.220183i	$-0.0706654\pi$
$542$	0	0
$543$	−15.4161	−0.661569
$544$	0	0
$545$	−7.65685	−0.327984
$546$	0	0
$547$	18.9259i	0.809214i	0.914491	+	0.404607i	$0.132592\pi$
$547$	18.9259i	0.809214i	−0.914491	+	0.404607i	$0.867408\pi$
$548$	0	0
$549$	1.95837i	0.0835812i
$550$	0	0
$551$	46.7736	1.99262
$552$	0	0
$553$	54.6274	2.32299
$554$	0	0
$555$	5.75152i	0.244138i
$556$	0	0
$557$	− 11.4142i	− 0.483636i	−0.970322	−	0.241818i	$-0.922256\pi$
$557$	− 11.4142i	− 0.483636i	0.970322	−	0.241818i	$-0.0777437\pi$
$558$	0	0
$559$	−3.69552	−0.156304
$560$	0	0
$561$	12.6863	0.535616
$562$	0	0
$563$	14.9678i	0.630817i	0.948956	+	0.315408i	$0.102141\pi$
$563$	14.9678i	0.630817i	−0.948956	+	0.315408i	$0.897859\pi$
$564$	0	0
$565$	− 4.48528i	− 0.188697i
$566$	0	0
$567$	0.634051	0.0266276
$568$	0	0
$569$	25.6569	1.07559	0.537796	−	0.843075i	$-0.319257\pi$
$569$	25.6569	1.07559	0.537796	+	0.843075i	$0.319257\pi$
$570$	0	0
$571$	− 27.5851i	− 1.15440i	−0.816603	−	0.577200i	$-0.804145\pi$
$571$	− 27.5851i	− 1.15440i	0.816603	−	0.577200i	$-0.195855\pi$
$572$	0	0
$573$	12.6863i	0.529977i
$574$	0	0
$575$	31.4663	1.31224
$576$	0	0
$577$	17.3137	0.720779	0.360390	−	0.932802i	$-0.382644\pi$
$577$	17.3137	0.720779	0.360390	+	0.932802i	$0.382644\pi$
$578$	0	0
$579$	− 0.525265i	− 0.0218293i
$580$	0	0
$581$	50.6274i	2.10038i
$582$	0	0
$583$	−19.0029	−0.787018
$584$	0	0
$585$	3.65685	0.151192
$586$	0	0
$587$	− 18.5545i	− 0.765827i	−0.923784	−	0.382913i	$-0.874921\pi$
$587$	− 18.5545i	− 0.765827i	0.923784	−	0.382913i	$-0.125079\pi$
$588$	0	0
$589$	19.3137i	0.795807i
$590$	0	0
$591$	−11.6118	−0.477646
$592$	0	0
$593$	9.31371	0.382468	0.191234	−	0.981544i	$-0.438751\pi$
$593$	9.31371	0.382468	0.191234	+	0.981544i	$0.438751\pi$
$594$	0	0
$595$	− 6.12293i	− 0.251016i
$596$	0	0
$597$	13.3726i	0.547303i
$598$	0	0
$599$	45.5055	1.85931	0.929653	−	0.368437i	$-0.120107\pi$
$599$	45.5055	1.85931	0.929653	+	0.368437i	$0.120107\pi$
$600$	0	0
$601$	−20.8284	−0.849609	−0.424805	−	0.905285i	$-0.639657\pi$
$601$	−20.8284	−0.849609	−0.424805	+	0.905285i	$0.639657\pi$
$602$	0	0
$603$	− 3.61859i	− 0.147361i
$604$	0	0
$605$	− 3.61522i	− 0.146980i
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	29.6569	1.20176
$610$	0	0
$611$	− 10.4525i	− 0.422863i
$612$	0	0
$613$	− 39.4142i	− 1.59193i	−0.605346	−	0.795963i	$-0.706965\pi$
$613$	− 39.4142i	− 1.59193i	0.605346	−	0.795963i	$-0.293035\pi$
$614$	0	0
$615$	2.53620	0.102270
$616$	0	0
$617$	−38.4853	−1.54936	−0.774680	−	0.632354i	$-0.782089\pi$
$617$	−38.4853	−1.54936	−0.774680	+	0.632354i	$0.782089\pi$
$618$	0	0
$619$	4.14386i	0.166556i	0.996526	+	0.0832779i	$0.0265389\pi$
$619$	4.14386i	0.166556i	−0.996526	+	0.0832779i	$0.973461\pi$
$620$	0	0
$621$	− 35.3137i	− 1.41709i
$622$	0	0
$623$	17.8435	0.714886
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	28.2960i	1.13004i
$628$	0	0
$629$	25.6569i	1.02301i
$630$	0	0
$631$	−14.1480	−0.563224	−0.281612	−	0.959528i	$-0.590869\pi$
$631$	−14.1480	−0.563224	−0.281612	+	0.959528i	$0.590869\pi$
$632$	0	0
$633$	14.4264	0.573398
$634$	0	0
$635$	6.86577i	0.272460i
$636$	0	0
$637$	− 22.7279i	− 0.900513i
$638$	0	0
$639$	14.6733	0.580466
$640$	0	0
$641$	−38.1421	−1.50652	−0.753262	−	0.657721i	$-0.771521\pi$
$641$	−38.1421	−1.50652	−0.753262	+	0.657721i	$0.771521\pi$
$642$	0	0
$643$	18.9259i	0.746366i	0.927758	+	0.373183i	$0.121734\pi$
$643$	18.9259i	0.746366i	−0.927758	+	0.373183i	$0.878266\pi$
$644$	0	0
$645$	− 0.686292i	− 0.0270227i
$646$	0	0
$647$	−21.5391	−0.846788	−0.423394	−	0.905946i	$-0.639161\pi$
$647$	−21.5391	−0.846788	−0.423394	+	0.905946i	$0.639161\pi$
$648$	0	0
$649$	−4.48528	−0.176063
$650$	0	0
$651$	12.2459i	0.479953i
$652$	0	0
$653$	6.44365i	0.252160i	0.992020	+	0.126080i	$0.0402395\pi$
$653$	6.44365i	0.252160i	−0.992020	+	0.126080i	$0.959760\pi$
$654$	0	0
$655$	−8.55035	−0.334090
$656$	0	0
$657$	11.8579	0.462619
$658$	0	0
$659$	− 1.45381i	− 0.0566324i	−0.999599	−	0.0283162i	$-0.990985\pi$
$659$	− 1.45381i	− 0.0566324i	0.999599	−	0.0283162i	$-0.00901453\pi$
$660$	0	0
$661$	29.0711i	1.13073i	0.824840	+	0.565367i	$0.191265\pi$
$661$	29.0711i	1.13073i	−0.824840	+	0.565367i	$0.808735\pi$
$662$	0	0
$663$	−10.4525	−0.405942
$664$	0	0
$665$	13.6569	0.529590
$666$	0	0
$667$	− 50.0977i	− 1.93979i
$668$	0	0
$669$	− 3.31371i	− 0.128115i
$670$	0	0
$671$	−4.43835	−0.171341
$672$	0	0
$673$	−26.8284	−1.03416	−0.517080	−	0.855937i	$-0.672981\pi$
$673$	−26.8284	−1.03416	−0.517080	+	0.855937i	$0.672981\pi$
$674$	0	0
$675$	− 24.3379i	− 0.936766i
$676$	0	0
$677$	− 13.5563i	− 0.521013i	−0.965472	−	0.260506i	$-0.916110\pi$
$677$	− 13.5563i	− 0.521013i	0.965472	−	0.260506i	$-0.0838895\pi$
$678$	0	0
$679$	−19.1116	−0.733437
$680$	0	0
$681$	−1.17157	−0.0448948
$682$	0	0
$683$	32.8113i	1.25549i	0.778419	+	0.627745i	$0.216022\pi$
$683$	32.8113i	1.25549i	−0.778419	+	0.627745i	$0.783978\pi$
$684$	0	0
$685$	− 7.59798i	− 0.290304i
$686$	0	0
$687$	4.22078	0.161033
$688$	0	0
$689$	15.6569	0.596479
$690$	0	0
$691$	− 40.7276i	− 1.54935i	−0.632358	−	0.774676i	$-0.717913\pi$
$691$	− 40.7276i	− 1.54935i	0.632358	−	0.774676i	$-0.282087\pi$
$692$	0	0
$693$	− 28.0000i	− 1.06363i
$694$	0	0
$695$	11.8294	0.448714
$696$	0	0
$697$	11.3137	0.428537
$698$	0	0
$699$	22.5445i	0.852712i
$700$	0	0
$701$	− 24.8701i	− 0.939329i	−0.882845	−	0.469665i	$-0.844375\pi$
$701$	− 24.8701i	− 0.939329i	0.882845	−	0.469665i	$-0.155625\pi$
$702$	0	0
$703$	−57.2261	−2.15832
$704$	0	0
$705$	1.94113	0.0731070
$706$	0	0
$707$	− 63.0864i	− 2.37261i
$708$	0	0
$709$	23.6985i	0.890015i	0.895527	+	0.445008i	$0.146799\pi$
$709$	23.6985i	0.890015i	−0.895527	+	0.445008i	$0.853201\pi$
$710$	0	0
$711$	−27.0279	−1.01363
$712$	0	0
$713$	20.6863	0.774708
$714$	0	0
$715$	8.28772i	0.309943i
$716$	0	0
$717$	13.2548i	0.495011i
$718$	0	0
$719$	−29.5641	−1.10256	−0.551278	−	0.834321i	$-0.685860\pi$
$719$	−29.5641	−1.10256	−0.551278	+	0.834321i	$0.685860\pi$
$720$	0	0
$721$	−52.2843	−1.94717
$722$	0	0
$723$	14.2568i	0.530216i
$724$	0	0
$725$	− 34.5269i	− 1.28230i
$726$	0	0
$727$	22.0643	0.818320	0.409160	−	0.912463i	$-0.365822\pi$
$727$	22.0643	0.818320	0.409160	+	0.912463i	$0.365822\pi$
$728$	0	0
$729$	−17.2843	−0.640158
$730$	0	0
$731$	− 3.06147i	− 0.113232i
$732$	0	0
$733$	18.5269i	0.684307i	0.939644	+	0.342154i	$0.111156\pi$
$733$	18.5269i	0.684307i	−0.939644	+	0.342154i	$0.888844\pi$
$734$	0	0
$735$	4.22078	0.155686
$736$	0	0
$737$	8.20101	0.302088
$738$	0	0
$739$	− 49.7582i	− 1.83038i	−0.403018	−	0.915192i	$-0.632039\pi$
$739$	− 49.7582i	− 1.83038i	0.403018	−	0.915192i	$-0.367961\pi$
$740$	0	0
$741$	− 23.3137i	− 0.856450i
$742$	0	0
$743$	−27.1367	−0.995550	−0.497775	−	0.867306i	$-0.665849\pi$
$743$	−27.1367	−0.995550	−0.497775	+	0.867306i	$0.665849\pi$
$744$	0	0
$745$	−1.02944	−0.0377157
$746$	0	0
$747$	− 25.0489i	− 0.916490i
$748$	0	0
$749$	7.31371i	0.267237i
$750$	0	0
$751$	14.7821	0.539405	0.269703	−	0.962944i	$-0.413075\pi$
$751$	14.7821	0.539405	0.269703	+	0.962944i	$0.413075\pi$
$752$	0	0
$753$	−19.5147	−0.711156
$754$	0	0
$755$	0.371418i	0.0135173i
$756$	0	0
$757$	− 18.9289i	− 0.687984i	−0.938973	−	0.343992i	$-0.888221\pi$
$757$	− 18.9289i	− 0.687984i	0.938973	−	0.343992i	$-0.111779\pi$
$758$	0	0
$759$	30.3070	1.10007
$760$	0	0
$761$	−34.6274	−1.25524	−0.627621	−	0.778519i	$-0.715971\pi$
$761$	−34.6274	−1.25524	−0.627621	+	0.778519i	$0.715971\pi$
$762$	0	0
$763$	− 48.3044i	− 1.74874i
$764$	0	0
$765$	3.02944i	0.109530i
$766$	0	0
$767$	3.69552	0.133437
$768$	0	0
$769$	5.17157	0.186492	0.0932458	−	0.995643i	$-0.470276\pi$
$769$	5.17157	0.186492	0.0932458	+	0.995643i	$0.470276\pi$
$770$	0	0
$771$	16.2040i	0.583574i
$772$	0	0
$773$	39.6985i	1.42786i	0.700219	+	0.713928i	$0.253085\pi$
$773$	39.6985i	1.42786i	−0.700219	+	0.713928i	$0.746915\pi$
$774$	0	0
$775$	14.2568	0.512120
$776$	0	0
$777$	−36.2843	−1.30169
$778$	0	0
$779$	25.2346i	0.904123i
$780$	0	0
$781$	33.2548i	1.18995i
$782$	0	0
$783$	−38.7485	−1.38476
$784$	0	0
$785$	1.31371	0.0468883
$786$	0	0
$787$	7.94816i	0.283321i	0.989915	+	0.141661i	$0.0452442\pi$
$787$	7.94816i	0.283321i	−0.989915	+	0.141661i	$0.954756\pi$
$788$	0	0
$789$	13.9411i	0.496317i
$790$	0	0
$791$	28.2960	1.00609
$792$	0	0
$793$	3.65685	0.129859
$794$	0	0
$795$	2.90762i	0.103123i
$796$	0	0
$797$	39.6985i	1.40619i	0.711095	+	0.703096i	$0.248200\pi$
$797$	39.6985i	1.40619i	−0.711095	+	0.703096i	$0.751800\pi$
$798$	0	0
$799$	8.65914	0.306338
$800$	0	0
$801$	−8.82843	−0.311937
$802$	0	0
$803$	26.8741i	0.948366i
$804$	0	0
$805$	− 14.6274i	− 0.515549i
$806$	0	0
$807$	−27.6620	−0.973748
$808$	0	0
$809$	24.6863	0.867924	0.433962	−	0.900931i	$-0.357115\pi$
$809$	24.6863	0.867924	0.433962	+	0.900931i	$0.357115\pi$
$810$	0	0
$811$	2.35049i	0.0825370i	0.999148	+	0.0412685i	$0.0131399\pi$
$811$	2.35049i	0.0825370i	−0.999148	+	0.0412685i	$0.986860\pi$
$812$	0	0
$813$	6.05887i	0.212494i
$814$	0	0
$815$	3.17025	0.111049
$816$	0	0
$817$	6.82843	0.238896
$818$	0	0
$819$	23.0698i	0.806124i
$820$	0	0
$821$	38.5269i	1.34460i	0.740279	+	0.672299i	$0.234693\pi$
$821$	38.5269i	1.34460i	−0.740279	+	0.672299i	$0.765307\pi$
$822$	0	0
$823$	22.0643	0.769114	0.384557	−	0.923101i	$-0.374354\pi$
$823$	22.0643	0.769114	0.384557	+	0.923101i	$0.374354\pi$
$824$	0	0
$825$	20.8873	0.727203
$826$	0	0
$827$	18.5545i	0.645204i	0.946535	+	0.322602i	$0.104557\pi$
$827$	18.5545i	0.645204i	−0.946535	+	0.322602i	$0.895443\pi$
$828$	0	0
$829$	12.3848i	0.430141i	0.976599	+	0.215071i	$0.0689981\pi$
$829$	12.3848i	0.430141i	−0.976599	+	0.215071i	$0.931002\pi$
$830$	0	0
$831$	−3.69552	−0.128196
$832$	0	0
$833$	18.8284	0.652366
$834$	0	0
$835$	− 2.16478i	− 0.0749155i
$836$	0	0
$837$	− 16.0000i	− 0.553041i
$838$	0	0
$839$	29.4554	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$839$	29.4554	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$840$	0	0
$841$	−25.9706	−0.895537
$842$	0	0
$843$	19.2655i	0.663539i
$844$	0	0
$845$	0.786797i	0.0270666i
$846$	0	0
$847$	22.8072	0.783663
$848$	0	0
$849$	−15.7990	−0.542220
$850$	0	0
$851$	61.2931i	2.10110i
$852$	0	0
$853$	− 32.1005i	− 1.09910i	−0.835461	−	0.549550i	$-0.814799\pi$
$853$	− 32.1005i	− 1.09910i	0.835461	−	0.549550i	$-0.185201\pi$
$854$	0	0
$855$	−6.75699	−0.231084
$856$	0	0
$857$	−24.2843	−0.829535	−0.414767	−	0.909927i	$-0.636137\pi$
$857$	−24.2843	−0.829535	−0.414767	+	0.909927i	$0.636137\pi$
$858$	0	0
$859$	− 11.9063i	− 0.406238i	−0.979154	−	0.203119i	$-0.934892\pi$
$859$	− 11.9063i	− 0.406238i	0.979154	−	0.203119i	$-0.0651079\pi$
$860$	0	0
$861$	16.0000i	0.545279i
$862$	0	0
$863$	0.525265	0.0178802	0.00894011	−	0.999960i	$-0.497154\pi$
$863$	0.525265	0.0178802	0.00894011	+	0.999960i	$0.497154\pi$
$864$	0	0
$865$	−11.2548	−0.382676
$866$	0	0
$867$	9.74153i	0.330840i
$868$	0	0
$869$	− 61.2548i	− 2.07793i
$870$	0	0
$871$	−6.75699	−0.228952
$872$	0	0
$873$	9.45584	0.320032
$874$	0	0
$875$	− 20.9050i	− 0.706718i
$876$	0	0
$877$	− 49.5563i	− 1.67340i	−0.547662	−	0.836700i	$-0.684482\pi$
$877$	− 49.5563i	− 1.67340i	0.547662	−	0.836700i	$-0.315518\pi$
$878$	0	0
$879$	0.634051	0.0213860
$880$	0	0
$881$	−39.9411	−1.34565	−0.672825	−	0.739801i	$-0.734919\pi$
$881$	−39.9411	−1.34565	−0.672825	+	0.739801i	$0.734919\pi$
$882$	0	0
$883$	− 29.3784i	− 0.988663i	−0.869273	−	0.494332i	$-0.835413\pi$
$883$	− 29.3784i	− 0.988663i	0.869273	−	0.494332i	$-0.164587\pi$
$884$	0	0
$885$	0.686292i	0.0230694i
$886$	0	0
$887$	−17.2095	−0.577838	−0.288919	−	0.957354i	$-0.593296\pi$
$887$	−17.2095	−0.577838	−0.288919	+	0.957354i	$0.593296\pi$
$888$	0	0
$889$	−43.3137	−1.45270
$890$	0	0
$891$	− 0.710974i	− 0.0238185i
$892$	0	0
$893$	19.3137i	0.646309i
$894$	0	0
$895$	−6.75699	−0.225861
$896$	0	0
$897$	−24.9706	−0.833743
$898$	0	0
$899$	− 22.6984i	− 0.757033i
$900$	0	0
$901$	12.9706i	0.432112i
$902$	0	0
$903$	4.32957	0.144079
$904$	0	0
$905$	8.34315	0.277336
$906$	0	0
$907$	45.0572i	1.49610i	0.663643	+	0.748050i	$0.269010\pi$
$907$	45.0572i	1.49610i	−0.663643	+	0.748050i	$0.730990\pi$
$908$	0	0
$909$	31.2132i	1.03528i
$910$	0	0
$911$	23.9665	0.794045	0.397022	−	0.917809i	$-0.370044\pi$
$911$	23.9665	0.794045	0.397022	+	0.917809i	$0.370044\pi$
$912$	0	0
$913$	56.7696	1.87880
$914$	0	0
$915$	0.679111i	0.0224507i
$916$	0	0
$917$	− 53.9411i	− 1.78129i
$918$	0	0
$919$	19.0029	0.626846	0.313423	−	0.949614i	$-0.398524\pi$
$919$	19.0029	0.626846	0.313423	+	0.949614i	$0.398524\pi$
$920$	0	0
$921$	4.88730	0.161042
$922$	0	0
$923$	− 27.3994i	− 0.901861i
$924$	0	0
$925$	42.2426i	1.38893i
$926$	0	0
$927$	25.8686	0.849637
$928$	0	0
$929$	−42.8284	−1.40516	−0.702578	−	0.711607i	$-0.747968\pi$
$929$	−42.8284	−1.40516	−0.702578	+	0.711607i	$0.747968\pi$
$930$	0	0
$931$	41.9957i	1.37635i
$932$	0	0
$933$	− 26.6274i	− 0.871742i
$934$	0	0
$935$	−6.86577	−0.224535
$936$	0	0
$937$	−43.4558	−1.41964	−0.709820	−	0.704383i	$-0.751224\pi$
$937$	−43.4558	−1.41964	−0.709820	+	0.704383i	$0.751224\pi$
$938$	0	0
$939$	− 33.8937i	− 1.10608i
$940$	0	0
$941$	− 2.92893i	− 0.0954805i	−0.998860	−	0.0477402i	$-0.984798\pi$
$941$	− 2.92893i	− 0.0954805i	0.998860	−	0.0477402i	$-0.0152020\pi$
$942$	0	0
$943$	27.0279	0.880151
$944$	0	0
$945$	−11.3137	−0.368035
$946$	0	0
$947$	− 21.4621i	− 0.697426i	−0.937230	−	0.348713i	$-0.886619\pi$
$947$	− 21.4621i	− 0.697426i	0.937230	−	0.348713i	$-0.113381\pi$
$948$	0	0
$949$	− 22.1421i	− 0.718764i
$950$	0	0
$951$	−16.6842	−0.541023
$952$	0	0
$953$	−11.0294	−0.357279	−0.178639	−	0.983915i	$-0.557170\pi$
$953$	−11.0294	−0.357279	−0.178639	+	0.983915i	$0.557170\pi$
$954$	0	0
$955$	− 6.86577i	− 0.222171i
$956$	0	0
$957$	− 33.2548i	− 1.07498i
$958$	0	0
$959$	47.9329	1.54784
$960$	0	0
$961$	−21.6274	−0.697659
$962$	0	0
$963$	− 3.61859i	− 0.116608i
$964$	0	0
$965$	0.284271i	0.00915102i
$966$	0	0
$967$	11.0866	0.356520	0.178260	−	0.983983i	$-0.442953\pi$
$967$	11.0866	0.356520	0.178260	+	0.983983i	$0.442953\pi$
$968$	0	0
$969$	19.3137	0.620446
$970$	0	0
$971$	− 57.6745i	− 1.85086i	−0.378916	−	0.925431i	$-0.623703\pi$
$971$	− 57.6745i	− 1.85086i	0.378916	−	0.925431i	$-0.376297\pi$
$972$	0	0
$973$	74.6274i	2.39245i
$974$	0	0
$975$	−17.2095	−0.551145
$976$	0	0
$977$	−47.1127	−1.50727	−0.753634	−	0.657294i	$-0.771701\pi$
$977$	−47.1127	−1.50727	−0.753634	+	0.657294i	$0.771701\pi$
$978$	0	0
$979$	− 20.0083i	− 0.639469i
$980$	0	0
$981$	23.8995i	0.763052i
$982$	0	0
$983$	−46.7736	−1.49185	−0.745924	−	0.666031i	$-0.767992\pi$
$983$	−46.7736	−1.49185	−0.745924	+	0.666031i	$0.767992\pi$
$984$	0	0
$985$	6.28427	0.200234
$986$	0	0
$987$	12.2459i	0.389790i
$988$	0	0
$989$	− 7.31371i	− 0.232562i
$990$	0	0
$991$	−27.0279	−0.858571	−0.429285	−	0.903169i	$-0.641235\pi$
$991$	−27.0279	−0.858571	−0.429285	+	0.903169i	$0.641235\pi$
$992$	0	0
$993$	9.17157	0.291051
$994$	0	0
$995$	− 7.23719i	− 0.229434i
$996$	0	0
$997$	28.3848i	0.898955i	0.893292	+	0.449477i	$0.148390\pi$
$997$	28.3848i	0.898955i	−0.893292	+	0.449477i	$0.851610\pi$
$998$	0	0
$999$	47.4077	1.49991

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.2.b.g.513.4		8
4.3	odd	2	inner	1024.2.b.g.513.6		8
8.3	odd	2	inner	1024.2.b.g.513.3		8
8.5	even	2	inner	1024.2.b.g.513.5		8
16.3	odd	4		1024.2.a.i.1.2		4
16.5	even	4		1024.2.a.h.1.2		4
16.11	odd	4		1024.2.a.h.1.3		4
16.13	even	4		1024.2.a.i.1.3		4
32.3	odd	8		512.2.e.j.385.2	yes	8
32.5	even	8		512.2.e.j.129.3	yes	8
32.11	odd	8		512.2.e.i.129.3	yes	8
32.13	even	8		512.2.e.i.385.2	yes	8
32.19	odd	8		512.2.e.i.385.3	yes	8
32.21	even	8		512.2.e.i.129.2	✓	8
32.27	odd	8		512.2.e.j.129.2	yes	8
32.29	even	8		512.2.e.j.385.3	yes	8
48.5	odd	4		9216.2.a.bp.1.2		4
48.11	even	4		9216.2.a.bp.1.1		4
48.29	odd	4		9216.2.a.w.1.4		4
48.35	even	4		9216.2.a.w.1.3		4
96.5	odd	8		4608.2.k.bd.1153.3		8
96.11	even	8		4608.2.k.bi.1153.2		8
96.29	odd	8		4608.2.k.bd.3457.4		8
96.35	even	8		4608.2.k.bd.3457.3		8
96.53	odd	8		4608.2.k.bi.1153.1		8
96.59	even	8		4608.2.k.bd.1153.4		8
96.77	odd	8		4608.2.k.bi.3457.2		8
96.83	even	8		4608.2.k.bi.3457.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.e.i.129.2	✓	8	32.21	even	8
512.2.e.i.129.3	yes	8	32.11	odd	8
512.2.e.i.385.2	yes	8	32.13	even	8
512.2.e.i.385.3	yes	8	32.19	odd	8
512.2.e.j.129.2	yes	8	32.27	odd	8
512.2.e.j.129.3	yes	8	32.5	even	8
512.2.e.j.385.2	yes	8	32.3	odd	8
512.2.e.j.385.3	yes	8	32.29	even	8
1024.2.a.h.1.2		4	16.5	even	4
1024.2.a.h.1.3		4	16.11	odd	4
1024.2.a.i.1.2		4	16.3	odd	4
1024.2.a.i.1.3		4	16.13	even	4
1024.2.b.g.513.3		8	8.3	odd	2	inner
1024.2.b.g.513.4		8	1.1	even	1	trivial
1024.2.b.g.513.5		8	8.5	even	2	inner
1024.2.b.g.513.6		8	4.3	odd	2	inner
4608.2.k.bd.1153.3		8	96.5	odd	8
4608.2.k.bd.1153.4		8	96.59	even	8
4608.2.k.bd.3457.3		8	96.35	even	8
4608.2.k.bd.3457.4		8	96.29	odd	8
4608.2.k.bi.1153.1		8	96.53	odd	8
4608.2.k.bi.1153.2		8	96.11	even	8
4608.2.k.bi.3457.1		8	96.83	even	8
4608.2.k.bi.3457.2		8	96.77	odd	8
9216.2.a.w.1.3		4	48.35	even	4
9216.2.a.w.1.4		4	48.29	odd	4
9216.2.a.bp.1.1		4	48.11	even	4
9216.2.a.bp.1.2		4	48.5	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:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1024.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.08239i q^{3} +0.585786i q^{5} -3.69552 q^{7} +1.82843 q^{9} +4.14386i q^{11} -3.41421i q^{13} +0.634051 q^{15} +2.82843 q^{17} +6.30864i q^{19} +4.00000i q^{21} +6.75699 q^{23} +4.65685 q^{25} -5.22625i q^{27} -7.41421i q^{29} +3.06147 q^{31} +4.48528 q^{33} -2.16478i q^{35} +9.07107i q^{37} -3.69552 q^{39} +4.00000 q^{41} -1.08239i q^{43} +1.07107i q^{45} +3.06147 q^{47} +6.65685 q^{49} -3.06147i q^{51} +4.58579i q^{53} -2.42742 q^{55} +6.82843 q^{57} +1.08239i q^{59} +1.07107i q^{61} -6.75699 q^{63} +2.00000 q^{65} -1.97908i q^{67} -7.31371i q^{69} +8.02509 q^{71} +6.48528 q^{73} -5.04054i q^{75} -15.3137i q^{77} -14.7821 q^{79} -0.171573 q^{81} -13.6997i q^{83} +1.65685i q^{85} -8.02509 q^{87} -4.82843 q^{89} +12.6173i q^{91} -3.31371i q^{93} -3.69552 q^{95} +5.17157 q^{97} +7.57675i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} - 8 q^{25} - 32 q^{33} + 32 q^{41} + 8 q^{49} + 32 q^{57} + 16 q^{65} - 16 q^{73} - 24 q^{81} - 16 q^{89} + 64 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 - 8 * q^25 - 32 * q^33 + 32 * q^41 + 8 * q^49 + 32 * q^57 + 16 * q^65 - 16 * q^73 - 24 * q^81 - 16 * q^89 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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