LMFDB - Embedded newform 1728.3.g.j.703.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(703,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1728.703");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.g (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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.56070144.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 $ x^8 - 4x^7 + 16x^6 - 34x^5 + 63x^4 - 74x^3 + 70x^2 - 38*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 864)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.1
Root			$0.500000 + 0.564882i$ of defining polynomial
Character	$\chi$	$=$	1728.703
Dual form			1728.3.g.j.703.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.42091 q^{5} -13.8102i q^{7} +O(q^{10})$	$q-6.42091 q^{5} -13.8102i q^{7} -13.0350i q^{11} -7.16454 q^{13} +31.5918 q^{17} -16.4875i q^{19} -16.9778i q^{23} +16.2281 q^{25} +42.4562 q^{29} -29.6836i q^{31} +88.6741i q^{35} -39.3037 q^{37} +39.8856 q^{41} -16.3291i q^{43} -57.8888i q^{47} -141.722 q^{49} -46.4110 q^{53} +83.6964i q^{55} +14.2179i q^{59} +63.7479 q^{61} +46.0028 q^{65} +32.5634i q^{67} -22.4738i q^{71} +24.9252 q^{73} -180.016 q^{77} +61.9501i q^{79} +44.7901i q^{83} -202.848 q^{85} -1.95333 q^{89} +98.9437i q^{91} +105.865i q^{95} -44.5813 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^5	$ 8 q - 8 q^{5} - 8 q^{13} + 24 q^{17} + 24 q^{25} + 128 q^{29} - 24 q^{37} + 160 q^{41} - 144 q^{49} + 48 q^{53} + 136 q^{61} - 280 q^{65} + 72 q^{73} - 520 q^{77} - 96 q^{85} - 168 q^{89} + 104 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 - 8 * q^13 + 24 * q^17 + 24 * q^25 + 128 * q^29 - 24 * q^37 + 160 * q^41 - 144 * q^49 + 48 * q^53 + 136 * q^61 - 280 * q^65 + 72 * q^73 - 520 * q^77 - 96 * q^85 - 168 * q^89 + 104 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−6.42091	−1.28418	−0.642091	−	0.766628i	$-0.721933\pi$
$5$	−6.42091	−1.28418	−0.642091	+	0.766628i	$0.721933\pi$
$6$	0	0
$7$	− 13.8102i	− 1.97289i	−0.164102	−	0.986443i	$-0.552473\pi$
$7$	− 13.8102i	− 1.97289i	0.164102	−	0.986443i	$-0.447527\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 13.0350i	− 1.18500i	−0.805572	−	0.592498i	$-0.798142\pi$
$11$	− 13.0350i	− 1.18500i	0.805572	−	0.592498i	$-0.201858\pi$
$12$	0	0
$13$	−7.16454	−0.551118	−0.275559	−	0.961284i	$-0.588863\pi$
$13$	−7.16454	−0.551118	−0.275559	+	0.961284i	$0.588863\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	31.5918	1.85834	0.929171	−	0.369651i	$-0.120523\pi$
$17$	31.5918	1.85834	0.929171	+	0.369651i	$0.120523\pi$
$18$	0	0
$19$	− 16.4875i	− 0.867763i	−0.900970	−	0.433881i	$-0.857144\pi$
$19$	− 16.4875i	− 0.867763i	0.900970	−	0.433881i	$-0.142856\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 16.9778i	− 0.738163i	−0.929397	−	0.369082i	$-0.879672\pi$
$23$	− 16.9778i	− 0.738163i	0.929397	−	0.369082i	$-0.120328\pi$
$24$	0	0
$25$	16.2281	0.649124
$26$	0	0
$27$	0	0
$28$	0	0
$29$	42.4562	1.46401	0.732004	−	0.681301i	$-0.238585\pi$
$29$	42.4562	1.46401	0.732004	+	0.681301i	$0.238585\pi$
$30$	0	0
$31$	− 29.6836i	− 0.957537i	−0.877941	−	0.478769i	$-0.841083\pi$
$31$	− 29.6836i	− 0.957537i	0.877941	−	0.478769i	$-0.158917\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	88.6741i	2.53355i
$36$	0	0
$37$	−39.3037	−1.06226	−0.531131	−	0.847289i	$-0.678233\pi$
$37$	−39.3037	−1.06226	−0.531131	+	0.847289i	$0.678233\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	39.8856	0.972819	0.486409	−	0.873731i	$-0.338306\pi$
$41$	39.8856	0.972819	0.486409	+	0.873731i	$0.338306\pi$
$42$	0	0
$43$	− 16.3291i	− 0.379746i	−0.981809	−	0.189873i	$-0.939192\pi$
$43$	− 16.3291i	− 0.379746i	0.981809	−	0.189873i	$-0.0608076\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 57.8888i	− 1.23168i	−0.787872	−	0.615839i	$-0.788817\pi$
$47$	− 57.8888i	− 1.23168i	0.787872	−	0.615839i	$-0.211183\pi$
$48$	0	0
$49$	−141.722	−2.89228
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−46.4110	−0.875680	−0.437840	−	0.899053i	$-0.644256\pi$
$53$	−46.4110	−0.875680	−0.437840	+	0.899053i	$0.644256\pi$
$54$	0	0
$55$	83.6964i	1.52175i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	14.2179i	0.240981i	0.992714	+	0.120491i	$0.0384468\pi$
$59$	14.2179i	0.240981i	−0.992714	+	0.120491i	$0.961553\pi$
$60$	0	0
$61$	63.7479	1.04505	0.522524	−	0.852625i	$-0.324991\pi$
$61$	63.7479	1.04505	0.522524	+	0.852625i	$0.324991\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	46.0028	0.707736
$66$	0	0
$67$	32.5634i	0.486022i	0.970024	+	0.243011i	$0.0781350\pi$
$67$	32.5634i	0.486022i	−0.970024	+	0.243011i	$0.921865\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 22.4738i	− 0.316532i	−0.987397	−	0.158266i	$-0.949410\pi$
$71$	− 22.4738i	− 0.316532i	0.987397	−	0.158266i	$-0.0505904\pi$
$72$	0	0
$73$	24.9252	0.341441	0.170721	−	0.985319i	$-0.445390\pi$
$73$	24.9252	0.341441	0.170721	+	0.985319i	$0.445390\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−180.016	−2.33786
$78$	0	0
$79$	61.9501i	0.784178i	0.919927	+	0.392089i	$0.128247\pi$
$79$	61.9501i	0.784178i	−0.919927	+	0.392089i	$0.871753\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	44.7901i	0.539640i	0.962911	+	0.269820i	$0.0869642\pi$
$83$	44.7901i	0.539640i	−0.962911	+	0.269820i	$0.913036\pi$
$84$	0	0
$85$	−202.848	−2.38645
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.95333	−0.0219475	−0.0109738	−	0.999940i	$-0.503493\pi$
$89$	−1.95333	−0.0219475	−0.0109738	+	0.999940i	$0.503493\pi$
$90$	0	0
$91$	98.9437i	1.08729i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	105.865i	1.11437i
$96$	0	0
$97$	−44.5813	−0.459601	−0.229800	−	0.973238i	$-0.573807\pi$
$97$	−44.5813	−0.459601	−0.229800	+	0.973238i	$0.573807\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−138.393	−1.37022	−0.685112	−	0.728438i	$-0.740247\pi$
$101$	−138.393	−1.37022	−0.685112	+	0.728438i	$0.740247\pi$
$102$	0	0
$103$	19.5698i	0.189998i	0.995477	+	0.0949991i	$0.0302848\pi$
$103$	19.5698i	0.189998i	−0.995477	+	0.0949991i	$0.969715\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 39.6245i	− 0.370323i	−0.982708	−	0.185161i	$-0.940719\pi$
$107$	− 39.6245i	− 0.370323i	0.982708	−	0.185161i	$-0.0592808\pi$
$108$	0	0
$109$	−45.2012	−0.414690	−0.207345	−	0.978268i	$-0.566482\pi$
$109$	−45.2012	−0.414690	−0.207345	+	0.978268i	$0.566482\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	111.845	0.989776	0.494888	−	0.868957i	$-0.335209\pi$
$113$	111.845	0.989776	0.494888	+	0.868957i	$0.335209\pi$
$114$	0	0
$115$	109.013i	0.947936i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 436.289i	− 3.66630i
$120$	0	0
$121$	−48.9103	−0.404218
$122$	0	0
$123$	0	0
$124$	0	0
$125$	56.3235	0.450588
$126$	0	0
$127$	− 110.064i	− 0.866642i	−0.901240	−	0.433321i	$-0.857342\pi$
$127$	− 110.064i	− 0.866642i	0.901240	−	0.433321i	$-0.142658\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.3672i	0.109673i	0.998495	+	0.0548367i	$0.0174638\pi$
$131$	14.3672i	0.109673i	−0.998495	+	0.0548367i	$0.982536\pi$
$132$	0	0
$133$	−227.696	−1.71200
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−170.048	−1.24123	−0.620613	−	0.784117i	$-0.713116\pi$
$137$	−170.048	−1.24123	−0.620613	+	0.784117i	$0.713116\pi$
$138$	0	0
$139$	− 213.171i	− 1.53360i	−0.641884	−	0.766802i	$-0.721847\pi$
$139$	− 213.171i	− 1.53360i	0.641884	−	0.766802i	$-0.278153\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	93.3895i	0.653073i
$144$	0	0
$145$	−272.608	−1.88005
$146$	0	0
$147$	0	0
$148$	0	0
$149$	84.5143	0.567210	0.283605	−	0.958941i	$-0.408470\pi$
$149$	84.5143	0.567210	0.283605	+	0.958941i	$0.408470\pi$
$150$	0	0
$151$	− 180.407i	− 1.19475i	−0.801962	−	0.597376i	$-0.796210\pi$
$151$	− 180.407i	− 1.19475i	0.801962	−	0.597376i	$-0.203790\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	190.596i	1.22965i
$156$	0	0
$157$	−49.7071	−0.316605	−0.158303	−	0.987391i	$-0.550602\pi$
$157$	−49.7071	−0.316605	−0.158303	+	0.987391i	$0.550602\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−234.466	−1.45631
$162$	0	0
$163$	49.3718i	0.302894i	0.988465	+	0.151447i	$0.0483933\pi$
$163$	49.3718i	0.302894i	−0.988465	+	0.151447i	$0.951607\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	161.617i	0.967766i	0.875133	+	0.483883i	$0.160774\pi$
$167$	161.617i	0.967766i	−0.875133	+	0.483883i	$0.839226\pi$
$168$	0	0
$169$	−117.669	−0.696269
$170$	0	0
$171$	0	0
$172$	0	0
$173$	63.6058	0.367664	0.183832	−	0.982958i	$-0.441150\pi$
$173$	63.6058	0.367664	0.183832	+	0.982958i	$0.441150\pi$
$174$	0	0
$175$	− 224.114i	− 1.28065i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	203.780i	1.13844i	0.822186	+	0.569219i	$0.192754\pi$
$179$	203.780i	1.13844i	−0.822186	+	0.569219i	$0.807246\pi$
$180$	0	0
$181$	−23.1886	−0.128114	−0.0640570	−	0.997946i	$-0.520404\pi$
$181$	−23.1886	−0.128114	−0.0640570	+	0.997946i	$0.520404\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	252.366	1.36414
$186$	0	0
$187$	− 411.798i	− 2.20213i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	247.017i	1.29328i	0.762793	+	0.646642i	$0.223827\pi$
$191$	247.017i	1.29328i	−0.762793	+	0.646642i	$0.776173\pi$
$192$	0	0
$193$	126.813	0.657063	0.328531	−	0.944493i	$-0.393446\pi$
$193$	126.813	0.657063	0.328531	+	0.944493i	$0.393446\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−47.1824	−0.239505	−0.119752	−	0.992804i	$-0.538210\pi$
$197$	−47.1824	−0.239505	−0.119752	+	0.992804i	$0.538210\pi$
$198$	0	0
$199$	298.835i	1.50168i	0.660482	+	0.750842i	$0.270352\pi$
$199$	298.835i	1.50168i	−0.660482	+	0.750842i	$0.729648\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 586.329i	− 2.88832i
$204$	0	0
$205$	−256.102	−1.24928
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−214.914	−1.02830
$210$	0	0
$211$	− 270.034i	− 1.27978i	−0.768466	−	0.639891i	$-0.778980\pi$
$211$	− 270.034i	− 1.27978i	0.768466	−	0.639891i	$-0.221020\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	104.848i	0.487663i
$216$	0	0
$217$	−409.937	−1.88911
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−226.341	−1.02417
$222$	0	0
$223$	171.763i	0.770237i	0.922867	+	0.385118i	$0.125839\pi$
$223$	171.763i	0.770237i	−0.922867	+	0.385118i	$0.874161\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 146.057i	− 0.643424i	−0.946838	−	0.321712i	$-0.895742\pi$
$227$	− 146.057i	− 0.643424i	0.946838	−	0.321712i	$-0.104258\pi$
$228$	0	0
$229$	325.016	1.41928	0.709641	−	0.704563i	$-0.248857\pi$
$229$	325.016	1.41928	0.709641	+	0.704563i	$0.248857\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	405.093	1.73860	0.869299	−	0.494286i	$-0.164570\pi$
$233$	405.093	1.73860	0.869299	+	0.494286i	$0.164570\pi$
$234$	0	0
$235$	371.699i	1.58170i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	85.3071i	0.356933i	0.983946	+	0.178467i	$0.0571137\pi$
$239$	85.3071i	0.356933i	−0.983946	+	0.178467i	$0.942886\pi$
$240$	0	0
$241$	−357.900	−1.48506	−0.742532	−	0.669811i	$-0.766375\pi$
$241$	−357.900	−1.48506	−0.742532	+	0.669811i	$0.766375\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	909.983	3.71422
$246$	0	0
$247$	118.125i	0.478240i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 88.8097i	− 0.353823i	−0.984227	−	0.176912i	$-0.943389\pi$
$251$	− 88.8097i	− 0.353823i	0.984227	−	0.176912i	$-0.0566107\pi$
$252$	0	0
$253$	−221.304	−0.874721
$254$	0	0
$255$	0	0
$256$	0	0
$257$	237.843	0.925460	0.462730	−	0.886499i	$-0.346870\pi$
$257$	237.843	0.925460	0.462730	+	0.886499i	$0.346870\pi$
$258$	0	0
$259$	542.793i	2.09572i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	309.702i	1.17757i	0.808289	+	0.588786i	$0.200394\pi$
$263$	309.702i	1.17757i	−0.808289	+	0.588786i	$0.799606\pi$
$264$	0	0
$265$	298.001	1.12453
$266$	0	0
$267$	0	0
$268$	0	0
$269$	490.940	1.82505	0.912527	−	0.409016i	$-0.134128\pi$
$269$	490.940	1.82505	0.912527	+	0.409016i	$0.134128\pi$
$270$	0	0
$271$	− 0.493459i	− 0.00182088i	−1.00000		0.000910442i	$-0.999710\pi$
$271$	− 0.493459i	− 0.00182088i	1.00000		0.000910442i	$-0.000289803\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 211.533i	− 0.769210i
$276$	0	0
$277$	10.8497	0.0391686	0.0195843	−	0.999808i	$-0.493766\pi$
$277$	10.8497	0.0391686	0.0195843	+	0.999808i	$0.493766\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−351.417	−1.25059	−0.625297	−	0.780387i	$-0.715022\pi$
$281$	−351.417	−1.25059	−0.625297	+	0.780387i	$0.715022\pi$
$282$	0	0
$283$	62.5357i	0.220974i	0.993878	+	0.110487i	$0.0352411\pi$
$283$	62.5357i	0.220974i	−0.993878	+	0.110487i	$0.964759\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 550.828i	− 1.91926i
$288$	0	0
$289$	709.042	2.45343
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−74.3035	−0.253596	−0.126798	−	0.991929i	$-0.540470\pi$
$293$	−74.3035	−0.253596	−0.126798	+	0.991929i	$0.540470\pi$
$294$	0	0
$295$	− 91.2919i	− 0.309464i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	121.638i	0.406815i
$300$	0	0
$301$	−225.508	−0.749195
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−409.320	−1.34203
$306$	0	0
$307$	460.555i	1.50018i	0.661336	+	0.750089i	$0.269990\pi$
$307$	460.555i	1.50018i	−0.661336	+	0.750089i	$0.730010\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	433.687i	1.39449i	0.716832	+	0.697246i	$0.245591\pi$
$311$	433.687i	1.39449i	−0.716832	+	0.697246i	$0.754409\pi$
$312$	0	0
$313$	−434.686	−1.38877	−0.694387	−	0.719602i	$-0.744324\pi$
$313$	−434.686	−1.38877	−0.694387	+	0.719602i	$0.744324\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	212.120	0.669149	0.334574	−	0.942369i	$-0.391407\pi$
$317$	212.120	0.669149	0.334574	+	0.942369i	$0.391407\pi$
$318$	0	0
$319$	− 553.415i	− 1.73484i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 520.870i	− 1.61260i
$324$	0	0
$325$	−116.267	−0.357744
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−799.457	−2.42996
$330$	0	0
$331$	− 578.614i	− 1.74808i	−0.485854	−	0.874040i	$-0.661491\pi$
$331$	− 578.614i	− 1.74808i	0.485854	−	0.874040i	$-0.338509\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 209.087i	− 0.624140i
$336$	0	0
$337$	−214.736	−0.637198	−0.318599	−	0.947890i	$-0.603212\pi$
$337$	−214.736	−0.637198	−0.318599	+	0.947890i	$0.603212\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−386.925	−1.13468
$342$	0	0
$343$	1280.51i	3.73326i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	280.616i	0.808692i	0.914606	+	0.404346i	$0.132501\pi$
$347$	280.616i	0.808692i	−0.914606	+	0.404346i	$0.867499\pi$
$348$	0	0
$349$	−232.447	−0.666036	−0.333018	−	0.942920i	$-0.608067\pi$
$349$	−232.447	−0.666036	−0.333018	+	0.942920i	$0.608067\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−97.0279	−0.274866	−0.137433	−	0.990511i	$-0.543885\pi$
$353$	−97.0279	−0.274866	−0.137433	+	0.990511i	$0.543885\pi$
$354$	0	0
$355$	144.302i	0.406485i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 37.7190i	− 0.105067i	−0.998619	−	0.0525334i	$-0.983270\pi$
$359$	− 37.7190i	− 0.105067i	0.998619	−	0.0525334i	$-0.0167296\pi$
$360$	0	0
$361$	89.1625	0.246988
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−160.043	−0.438473
$366$	0	0
$367$	− 303.506i	− 0.826991i	−0.910506	−	0.413496i	$-0.864308\pi$
$367$	− 303.506i	− 0.826991i	0.910506	−	0.413496i	$-0.135692\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	640.946i	1.72762i
$372$	0	0
$373$	576.215	1.54481	0.772406	−	0.635129i	$-0.219053\pi$
$373$	576.215	1.54481	0.772406	+	0.635129i	$0.219053\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−304.179	−0.806841
$378$	0	0
$379$	− 36.0808i	− 0.0952001i	−0.998866	−	0.0476000i	$-0.984843\pi$
$379$	− 36.0808i	− 0.0952001i	0.998866	−	0.0476000i	$-0.0151573\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	521.053i	1.36045i	0.733003	+	0.680225i	$0.238118\pi$
$383$	521.053i	1.36045i	−0.733003	+	0.680225i	$0.761882\pi$
$384$	0	0
$385$	1155.86	3.00224
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−472.918	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$389$	−472.918	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$390$	0	0
$391$	− 536.358i	− 1.37176i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 397.776i	− 1.00703i
$396$	0	0
$397$	209.784	0.528423	0.264212	−	0.964465i	$-0.414888\pi$
$397$	209.784	0.528423	0.264212	+	0.964465i	$0.414888\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	664.907	1.65812	0.829062	−	0.559157i	$-0.188875\pi$
$401$	664.907	1.65812	0.829062	+	0.559157i	$0.188875\pi$
$402$	0	0
$403$	212.670i	0.527716i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	512.323i	1.25878i
$408$	0	0
$409$	−573.373	−1.40189	−0.700945	−	0.713215i	$-0.747238\pi$
$409$	−573.373	−1.40189	−0.700945	+	0.713215i	$0.747238\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	196.352	0.475429
$414$	0	0
$415$	− 287.594i	− 0.692996i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 672.621i	− 1.60530i	−0.596450	−	0.802651i	$-0.703422\pi$
$419$	− 672.621i	− 1.60530i	0.596450	−	0.802651i	$-0.296578\pi$
$420$	0	0
$421$	659.810	1.56724	0.783622	−	0.621238i	$-0.213370\pi$
$421$	659.810	1.56724	0.783622	+	0.621238i	$0.213370\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	512.675	1.20629
$426$	0	0
$427$	− 880.372i	− 2.06176i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	360.903i	0.837361i	0.908134	+	0.418681i	$0.137507\pi$
$431$	360.903i	0.837361i	−0.908134	+	0.418681i	$0.862493\pi$
$432$	0	0
$433$	−319.790	−0.738546	−0.369273	−	0.929321i	$-0.620393\pi$
$433$	−319.790	−0.738546	−0.369273	+	0.929321i	$0.620393\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−279.921	−0.640551
$438$	0	0
$439$	80.6635i	0.183744i	0.995771	+	0.0918719i	$0.0292850\pi$
$439$	80.6635i	0.183744i	−0.995771	+	0.0918719i	$0.970715\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	551.362i	1.24461i	0.782775	+	0.622305i	$0.213803\pi$
$443$	551.362i	1.24461i	−0.782775	+	0.622305i	$0.786197\pi$
$444$	0	0
$445$	12.5422	0.0281847
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−43.0735	−0.0959321	−0.0479661	−	0.998849i	$-0.515274\pi$
$449$	−43.0735	−0.0959321	−0.0479661	+	0.998849i	$0.515274\pi$
$450$	0	0
$451$	− 519.907i	− 1.15279i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 635.309i	− 1.39628i
$456$	0	0
$457$	205.936	0.450626	0.225313	−	0.974286i	$-0.427659\pi$
$457$	205.936	0.450626	0.225313	+	0.974286i	$0.427659\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−650.729	−1.41156	−0.705780	−	0.708431i	$-0.749403\pi$
$461$	−650.729	−1.41156	−0.705780	+	0.708431i	$0.749403\pi$
$462$	0	0
$463$	− 333.871i	− 0.721103i	−0.932739	−	0.360552i	$-0.882588\pi$
$463$	− 333.871i	− 0.721103i	0.932739	−	0.360552i	$-0.117412\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 359.357i	− 0.769502i	−0.923020	−	0.384751i	$-0.874287\pi$
$467$	− 359.357i	− 0.769502i	0.923020	−	0.384751i	$-0.125713\pi$
$468$	0	0
$469$	449.708	0.958865
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−212.849	−0.449998
$474$	0	0
$475$	− 267.561i	− 0.563286i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 635.502i	− 1.32673i	−0.748297	−	0.663363i	$-0.769128\pi$
$479$	− 635.502i	− 1.32673i	0.748297	−	0.663363i	$-0.230872\pi$
$480$	0	0
$481$	281.593	0.585432
$482$	0	0
$483$	0	0
$484$	0	0
$485$	286.252	0.590211
$486$	0	0
$487$	233.959i	0.480409i	0.970722	+	0.240205i	$0.0772146\pi$
$487$	233.959i	0.480409i	−0.970722	+	0.240205i	$0.922785\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	494.173i	1.00646i	0.864152	+	0.503231i	$0.167855\pi$
$491$	494.173i	1.00646i	−0.864152	+	0.503231i	$0.832145\pi$
$492$	0	0
$493$	1341.27	2.72063
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−310.368	−0.624482
$498$	0	0
$499$	− 468.260i	− 0.938396i	−0.883093	−	0.469198i	$-0.844543\pi$
$499$	− 468.260i	− 0.938396i	0.883093	−	0.469198i	$-0.155457\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 59.2449i	− 0.117783i	−0.998264	−	0.0588916i	$-0.981243\pi$
$503$	− 59.2449i	− 0.117783i	0.998264	−	0.0588916i	$-0.0187566\pi$
$504$	0	0
$505$	888.607	1.75962
$506$	0	0
$507$	0	0
$508$	0	0
$509$	161.243	0.316784	0.158392	−	0.987376i	$-0.449369\pi$
$509$	161.243	0.316784	0.158392	+	0.987376i	$0.449369\pi$
$510$	0	0
$511$	− 344.222i	− 0.673625i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 125.656i	− 0.243992i
$516$	0	0
$517$	−754.579	−1.45953
$518$	0	0
$519$	0	0
$520$	0	0
$521$	350.872	0.673458	0.336729	−	0.941602i	$-0.390679\pi$
$521$	350.872	0.673458	0.336729	+	0.941602i	$0.390679\pi$
$522$	0	0
$523$	729.342i	1.39454i	0.716811	+	0.697268i	$0.245601\pi$
$523$	729.342i	1.39454i	−0.716811	+	0.697268i	$0.754399\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 937.760i	− 1.77943i
$528$	0	0
$529$	240.756	0.455115
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−285.762	−0.536138
$534$	0	0
$535$	254.426i	0.475562i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1847.34i	3.42734i
$540$	0	0
$541$	365.728	0.676021	0.338011	−	0.941142i	$-0.390246\pi$
$541$	365.728	0.676021	0.338011	+	0.941142i	$0.390246\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	290.233	0.532537
$546$	0	0
$547$	468.636i	0.856738i	0.903604	+	0.428369i	$0.140912\pi$
$547$	468.636i	0.856738i	−0.903604	+	0.428369i	$0.859088\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 699.997i	− 1.27041i
$552$	0	0
$553$	855.543	1.54709
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−288.006	−0.517066	−0.258533	−	0.966002i	$-0.583239\pi$
$557$	−288.006	−0.517066	−0.258533	+	0.966002i	$0.583239\pi$
$558$	0	0
$559$	116.990i	0.209285i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	158.644i	0.281783i	0.990025	+	0.140892i	$0.0449969\pi$
$563$	158.644i	0.281783i	−0.990025	+	0.140892i	$0.955003\pi$
$564$	0	0
$565$	−718.145	−1.27105
$566$	0	0
$567$	0	0
$568$	0	0
$569$	438.309	0.770315	0.385157	−	0.922851i	$-0.374147\pi$
$569$	438.309	0.770315	0.385157	+	0.922851i	$0.374147\pi$
$570$	0	0
$571$	− 683.282i	− 1.19664i	−0.801257	−	0.598321i	$-0.795835\pi$
$571$	− 683.282i	− 1.19664i	0.801257	−	0.598321i	$-0.204165\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 275.517i	− 0.479160i
$576$	0	0
$577$	−166.370	−0.288337	−0.144168	−	0.989553i	$-0.546051\pi$
$577$	−166.370	−0.288337	−0.144168	+	0.989553i	$0.546051\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	618.561	1.06465
$582$	0	0
$583$	604.966i	1.03768i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1095.17i	− 1.86570i	−0.360265	−	0.932850i	$-0.617314\pi$
$587$	− 1095.17i	− 1.86570i	0.360265	−	0.932850i	$-0.382686\pi$
$588$	0	0
$589$	−489.409	−0.830915
$590$	0	0
$591$	0	0
$592$	0	0
$593$	131.905	0.222437	0.111218	−	0.993796i	$-0.464525\pi$
$593$	131.905	0.222437	0.111218	+	0.993796i	$0.464525\pi$
$594$	0	0
$595$	2801.38i	4.70819i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	228.590i	0.381619i	0.981627	+	0.190809i	$0.0611112\pi$
$599$	228.590i	0.381619i	−0.981627	+	0.190809i	$0.938889\pi$
$600$	0	0
$601$	945.987	1.57402	0.787010	−	0.616940i	$-0.211628\pi$
$601$	945.987	1.57402	0.787010	+	0.616940i	$0.211628\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	314.049	0.519089
$606$	0	0
$607$	− 539.641i	− 0.889029i	−0.895772	−	0.444514i	$-0.853376\pi$
$607$	− 539.641i	− 0.889029i	0.895772	−	0.444514i	$-0.146624\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	414.747i	0.678800i
$612$	0	0
$613$	305.962	0.499123	0.249561	−	0.968359i	$-0.419714\pi$
$613$	305.962	0.499123	0.249561	+	0.968359i	$0.419714\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	409.095	0.663038	0.331519	−	0.943449i	$-0.392439\pi$
$617$	409.095	0.663038	0.331519	+	0.943449i	$0.392439\pi$
$618$	0	0
$619$	− 453.938i	− 0.733341i	−0.930351	−	0.366670i	$-0.880498\pi$
$619$	− 453.938i	− 0.733341i	0.930351	−	0.366670i	$-0.119502\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	26.9759i	0.0433000i
$624$	0	0
$625$	−767.351	−1.22776
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1241.68	−1.97405
$630$	0	0
$631$	393.383i	0.623428i	0.950176	+	0.311714i	$0.100903\pi$
$631$	393.383i	0.623428i	−0.950176	+	0.311714i	$0.899097\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	706.709i	1.11293i
$636$	0	0
$637$	1015.37	1.59399
$638$	0	0
$639$	0	0
$640$	0	0
$641$	982.724	1.53311	0.766555	−	0.642179i	$-0.221969\pi$
$641$	982.724	1.53311	0.766555	+	0.642179i	$0.221969\pi$
$642$	0	0
$643$	1078.21i	1.67684i	0.545025	+	0.838420i	$0.316520\pi$
$643$	1078.21i	1.67684i	−0.545025	+	0.838420i	$0.683480\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.4217i	0.0593844i	0.999559	+	0.0296922i	$0.00945271\pi$
$647$	38.4217i	0.0593844i	−0.999559	+	0.0296922i	$0.990547\pi$
$648$	0	0
$649$	185.330	0.285562
$650$	0	0
$651$	0	0
$652$	0	0
$653$	317.953	0.486911	0.243456	−	0.969912i	$-0.421719\pi$
$653$	317.953	0.486911	0.243456	+	0.969912i	$0.421719\pi$
$654$	0	0
$655$	− 92.2507i	− 0.140841i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 792.921i	− 1.20322i	−0.798790	−	0.601609i	$-0.794526\pi$
$659$	− 792.921i	− 1.20322i	0.798790	−	0.601609i	$-0.205474\pi$
$660$	0	0
$661$	1029.55	1.55757	0.778786	−	0.627290i	$-0.215836\pi$
$661$	1029.55	1.55757	0.778786	+	0.627290i	$0.215836\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1462.01	2.19852
$666$	0	0
$667$	− 720.811i	− 1.08068i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 830.952i	− 1.23838i
$672$	0	0
$673$	−6.46119	−0.00960059	−0.00480029	−	0.999988i	$-0.501528\pi$
$673$	−6.46119	−0.00960059	−0.00480029	+	0.999988i	$0.501528\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−21.6650	−0.0320015	−0.0160008	−	0.999872i	$-0.505093\pi$
$677$	−21.6650	−0.0320015	−0.0160008	+	0.999872i	$0.505093\pi$
$678$	0	0
$679$	615.676i	0.906740i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 503.008i	− 0.736469i	−0.929733	−	0.368234i	$-0.879962\pi$
$683$	− 503.008i	− 0.736469i	0.929733	−	0.368234i	$-0.120038\pi$
$684$	0	0
$685$	1091.86	1.59396
$686$	0	0
$687$	0	0
$688$	0	0
$689$	332.514	0.482603
$690$	0	0
$691$	− 514.362i	− 0.744374i	−0.928158	−	0.372187i	$-0.878608\pi$
$691$	− 514.362i	− 0.744374i	0.928158	−	0.372187i	$-0.121392\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1368.75i	1.96943i
$696$	0	0
$697$	1260.06	1.80783
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−457.404	−0.652502	−0.326251	−	0.945283i	$-0.605786\pi$
$701$	−457.404	−0.652502	−0.326251	+	0.945283i	$0.605786\pi$
$702$	0	0
$703$	648.020i	0.921792i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1911.23i	2.70330i
$708$	0	0
$709$	−390.207	−0.550363	−0.275182	−	0.961392i	$-0.588738\pi$
$709$	−390.207	−0.550363	−0.275182	+	0.961392i	$0.588738\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−503.962	−0.706819
$714$	0	0
$715$	− 599.646i	− 0.838665i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	979.508i	1.36232i	0.732134	+	0.681160i	$0.238524\pi$
$719$	979.508i	1.36232i	−0.732134	+	0.681160i	$0.761476\pi$
$720$	0	0
$721$	270.263	0.374845
$722$	0	0
$723$	0	0
$724$	0	0
$725$	688.984	0.950323
$726$	0	0
$727$	553.802i	0.761764i	0.924624	+	0.380882i	$0.124380\pi$
$727$	553.802i	0.761764i	−0.924624	+	0.380882i	$0.875620\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 515.865i	− 0.705697i
$732$	0	0
$733$	−250.392	−0.341599	−0.170800	−	0.985306i	$-0.554635\pi$
$733$	−250.392	−0.341599	−0.170800	+	0.985306i	$0.554635\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	424.463	0.575934
$738$	0	0
$739$	− 403.959i	− 0.546630i	−0.961925	−	0.273315i	$-0.911880\pi$
$739$	− 403.959i	− 0.546630i	0.961925	−	0.273315i	$-0.0881201\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	142.091i	0.191240i	0.995418	+	0.0956201i	$0.0304834\pi$
$743$	142.091i	0.191240i	−0.995418	+	0.0956201i	$0.969517\pi$
$744$	0	0
$745$	−542.659	−0.728401
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−547.223	−0.730605
$750$	0	0
$751$	− 667.242i	− 0.888471i	−0.895910	−	0.444235i	$-0.853475\pi$
$751$	− 667.242i	− 0.888471i	0.895910	−	0.444235i	$-0.146525\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1158.38i	1.53428i
$756$	0	0
$757$	−970.866	−1.28252	−0.641259	−	0.767325i	$-0.721587\pi$
$757$	−970.866	−1.28252	−0.641259	+	0.767325i	$0.721587\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−295.323	−0.388072	−0.194036	−	0.980994i	$-0.562158\pi$
$761$	−295.323	−0.388072	−0.194036	+	0.980994i	$0.562158\pi$
$762$	0	0
$763$	624.238i	0.818136i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 101.865i	− 0.132809i
$768$	0	0
$769$	−552.240	−0.718128	−0.359064	−	0.933313i	$-0.616904\pi$
$769$	−552.240	−0.718128	−0.359064	+	0.933313i	$0.616904\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	797.819	1.03211	0.516053	−	0.856556i	$-0.327401\pi$
$773$	797.819	1.03211	0.516053	+	0.856556i	$0.327401\pi$
$774$	0	0
$775$	− 481.709i	− 0.621561i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 657.613i	− 0.844176i
$780$	0	0
$781$	−292.945	−0.375090
$782$	0	0
$783$	0	0
$784$	0	0
$785$	319.165	0.406579
$786$	0	0
$787$	− 1140.72i	− 1.44946i	−0.689033	−	0.724730i	$-0.741965\pi$
$787$	− 1140.72i	− 1.44946i	0.689033	−	0.724730i	$-0.258035\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1544.60i	− 1.95272i
$792$	0	0
$793$	−456.724	−0.575945
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1453.75	−1.82403	−0.912016	−	0.410154i	$-0.865475\pi$
$797$	−1453.75	−1.82403	−0.912016	+	0.410154i	$0.865475\pi$
$798$	0	0
$799$	− 1828.81i	− 2.28888i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 324.899i	− 0.404607i
$804$	0	0
$805$	1505.49	1.87017
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−981.664	−1.21343	−0.606714	−	0.794920i	$-0.707513\pi$
$809$	−981.664	−1.21343	−0.606714	+	0.794920i	$0.707513\pi$
$810$	0	0
$811$	− 1090.81i	− 1.34502i	−0.740087	−	0.672511i	$-0.765216\pi$
$811$	− 1090.81i	− 1.34502i	0.740087	−	0.672511i	$-0.234784\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 317.012i	− 0.388972i
$816$	0	0
$817$	−269.225	−0.329529
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−282.456	−0.344039	−0.172019	−	0.985094i	$-0.555029\pi$
$821$	−282.456	−0.344039	−0.172019	+	0.985094i	$0.555029\pi$
$822$	0	0
$823$	497.382i	0.604352i	0.953252	+	0.302176i	$0.0977131\pi$
$823$	497.382i	0.604352i	−0.953252	+	0.302176i	$0.902287\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1159.09i	− 1.40156i	−0.713379	−	0.700778i	$-0.752836\pi$
$827$	− 1159.09i	− 1.40156i	0.713379	−	0.700778i	$-0.247164\pi$
$828$	0	0
$829$	806.908	0.973351	0.486675	−	0.873583i	$-0.338209\pi$
$829$	806.908	0.973351	0.486675	+	0.873583i	$0.338209\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4477.25	−5.37485
$834$	0	0
$835$	− 1037.73i	− 1.24279i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1045.71i	− 1.24638i	−0.782071	−	0.623189i	$-0.785837\pi$
$839$	− 1045.71i	− 1.24638i	0.782071	−	0.623189i	$-0.214163\pi$
$840$	0	0
$841$	961.530	1.14332
$842$	0	0
$843$	0	0
$844$	0	0
$845$	755.545	0.894136
$846$	0	0
$847$	675.462i	0.797476i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	667.289i	0.784123i
$852$	0	0
$853$	−1126.10	−1.32017	−0.660083	−	0.751193i	$-0.729479\pi$
$853$	−1126.10	−1.32017	−0.660083	+	0.751193i	$0.729479\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−112.853	−0.131684	−0.0658419	−	0.997830i	$-0.520973\pi$
$857$	−112.853	−0.131684	−0.0658419	+	0.997830i	$0.520973\pi$
$858$	0	0
$859$	− 679.360i	− 0.790873i	−0.918493	−	0.395437i	$-0.870593\pi$
$859$	− 679.360i	− 0.790873i	0.918493	−	0.395437i	$-0.129407\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	560.002i	0.648901i	0.945903	+	0.324451i	$0.105179\pi$
$863$	560.002i	0.648901i	−0.945903	+	0.324451i	$0.894821\pi$
$864$	0	0
$865$	−408.408	−0.472147
$866$	0	0
$867$	0	0
$868$	0	0
$869$	807.517	0.929249
$870$	0	0
$871$	− 233.302i	− 0.267855i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 777.840i	− 0.888960i
$876$	0	0
$877$	775.431	0.884186	0.442093	−	0.896969i	$-0.354236\pi$
$877$	775.431	0.884186	0.442093	+	0.896969i	$0.354236\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1024.25	−1.16260	−0.581298	−	0.813691i	$-0.697455\pi$
$881$	−1024.25	−1.16260	−0.581298	+	0.813691i	$0.697455\pi$
$882$	0	0
$883$	1301.28i	1.47370i	0.676057	+	0.736850i	$0.263687\pi$
$883$	1301.28i	1.47370i	−0.676057	+	0.736850i	$0.736313\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1284.02i	− 1.44760i	−0.690010	−	0.723800i	$-0.742394\pi$
$887$	− 1284.02i	− 1.44760i	0.690010	−	0.723800i	$-0.257606\pi$
$888$	0	0
$889$	−1520.00	−1.70979
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−954.442	−1.06880
$894$	0	0
$895$	− 1308.46i	− 1.46196i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1260.26i	− 1.40184i
$900$	0	0
$901$	−1466.21	−1.62731
$902$	0	0
$903$	0	0
$904$	0	0
$905$	148.892	0.164522
$906$	0	0
$907$	1042.84i	1.14976i	0.818236	+	0.574882i	$0.194952\pi$
$907$	1042.84i	1.14976i	−0.818236	+	0.574882i	$0.805048\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	857.109i	0.940844i	0.882442	+	0.470422i	$0.155898\pi$
$911$	857.109i	0.940844i	−0.882442	+	0.470422i	$0.844102\pi$
$912$	0	0
$913$	583.838	0.639472
$914$	0	0
$915$	0	0
$916$	0	0
$917$	198.414	0.216373
$918$	0	0
$919$	451.721i	0.491536i	0.969329	+	0.245768i	$0.0790401\pi$
$919$	451.721i	0.491536i	−0.969329	+	0.245768i	$0.920960\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	161.014i	0.174447i
$924$	0	0
$925$	−637.825	−0.689541
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1826.21	1.96578	0.982890	−	0.184191i	$-0.0589666\pi$
$929$	1826.21	1.96578	0.982890	+	0.184191i	$0.0589666\pi$
$930$	0	0
$931$	2336.64i	2.50981i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2644.12i	2.82794i
$936$	0	0
$937$	−264.729	−0.282529	−0.141264	−	0.989972i	$-0.545117\pi$
$937$	−264.729	−0.282529	−0.141264	+	0.989972i	$0.545117\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1401.16	1.48901	0.744505	−	0.667616i	$-0.232685\pi$
$941$	1401.16	1.48901	0.744505	+	0.667616i	$0.232685\pi$
$942$	0	0
$943$	− 677.167i	− 0.718099i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	389.952i	0.411776i	0.978576	+	0.205888i	$0.0660082\pi$
$947$	389.952i	0.411776i	−0.978576	+	0.205888i	$0.933992\pi$
$948$	0	0
$949$	−178.578	−0.188175
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1291.61	−1.35531	−0.677656	−	0.735379i	$-0.737004\pi$
$953$	−1291.61	−1.35531	−0.677656	+	0.735379i	$0.737004\pi$
$954$	0	0
$955$	− 1586.08i	− 1.66081i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2348.40i	2.44880i
$960$	0	0
$961$	79.8811	0.0831229
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−814.256	−0.843789
$966$	0	0
$967$	− 1802.57i	− 1.86408i	−0.362349	−	0.932042i	$-0.618025\pi$
$967$	− 1802.57i	− 1.86408i	0.362349	−	0.932042i	$-0.381975\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	473.260i	0.487395i	0.969851	+	0.243697i	$0.0783603\pi$
$971$	473.260i	0.487395i	−0.969851	+	0.243697i	$0.921640\pi$
$972$	0	0
$973$	−2943.93	−3.02563
$974$	0	0
$975$	0	0
$976$	0	0
$977$	656.147	0.671594	0.335797	−	0.941934i	$-0.390994\pi$
$977$	656.147	0.671594	0.335797	+	0.941934i	$0.390994\pi$
$978$	0	0
$979$	25.4616i	0.0260078i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 267.000i	− 0.271617i	−0.990735	−	0.135809i	$-0.956637\pi$
$983$	− 267.000i	− 0.271617i	0.990735	−	0.135809i	$-0.0433632\pi$
$984$	0	0
$985$	302.954	0.307568
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−277.231	−0.280314
$990$	0	0
$991$	− 549.499i	− 0.554489i	−0.960799	−	0.277245i	$-0.910579\pi$
$991$	− 549.499i	− 0.554489i	0.960799	−	0.277245i	$-0.0894213\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1918.79i	− 1.92844i
$996$	0	0
$997$	−546.254	−0.547898	−0.273949	−	0.961744i	$-0.588330\pi$
$997$	−546.254	−0.547898	−0.273949	+	0.961744i	$0.588330\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.g.j.703.1		8
3.2	odd	2		1728.3.g.m.703.7		8
4.3	odd	2	inner	1728.3.g.j.703.2		8
8.3	odd	2		864.3.g.d.703.8	yes	8
8.5	even	2		864.3.g.d.703.7	yes	8
12.11	even	2		1728.3.g.m.703.8		8
24.5	odd	2		864.3.g.b.703.1	✓	8
24.11	even	2		864.3.g.b.703.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.3.g.b.703.1	✓	8	24.5	odd	2
864.3.g.b.703.2	yes	8	24.11	even	2
864.3.g.d.703.7	yes	8	8.5	even	2
864.3.g.d.703.8	yes	8	8.3	odd	2
1728.3.g.j.703.1		8	1.1	even	1	trivial
1728.3.g.j.703.2		8	4.3	odd	2	inner
1728.3.g.m.703.7		8	3.2	odd	2
1728.3.g.m.703.8		8	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-6.42091 q^{5} -13.8102i q^{7} +O(q^{10})\)	\(q-6.42091 q^{5} -13.8102i q^{7} -13.0350i q^{11} -7.16454 q^{13} +31.5918 q^{17} -16.4875i q^{19} -16.9778i q^{23} +16.2281 q^{25} +42.4562 q^{29} -29.6836i q^{31} +88.6741i q^{35} -39.3037 q^{37} +39.8856 q^{41} -16.3291i q^{43} -57.8888i q^{47} -141.722 q^{49} -46.4110 q^{53} +83.6964i q^{55} +14.2179i q^{59} +63.7479 q^{61} +46.0028 q^{65} +32.5634i q^{67} -22.4738i q^{71} +24.9252 q^{73} -180.016 q^{77} +61.9501i q^{79} +44.7901i q^{83} -202.848 q^{85} -1.95333 q^{89} +98.9437i q^{91} +105.865i q^{95} -44.5813 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^5	\( 8 q - 8 q^{5} - 8 q^{13} + 24 q^{17} + 24 q^{25} + 128 q^{29} - 24 q^{37} + 160 q^{41} - 144 q^{49} + 48 q^{53} + 136 q^{61} - 280 q^{65} + 72 q^{73} - 520 q^{77} - 96 q^{85} - 168 q^{89} + 104 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 - 8 * q^13 + 24 * q^17 + 24 * q^25 + 128 * q^29 - 24 * q^37 + 160 * q^41 - 144 * q^49 + 48 * q^53 + 136 * q^61 - 280 * q^65 + 72 * q^73 - 520 * q^77 - 96 * q^85 - 168 * q^89 + 104 * q^97

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