LMFDB - Embedded newform 3456.2.d.n.1729.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3456,2,Mod(1729,3456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3456, 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("3456.1729");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3456 = 2^{7} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3456.d (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:	$27.5962989386$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1729.7
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	3456.1729
Dual form			3456.2.d.n.1729.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949i q^{5} +1.73205 q^{7} +O(q^{10})$	$q+2.44949i q^{5} +1.73205 q^{7} -1.41421i q^{11} -5.19615i q^{13} +4.24264 q^{17} +3.00000i q^{19} +7.34847 q^{23} -1.00000 q^{25} -6.92820 q^{31} +4.24264i q^{35} -5.19615i q^{37} +8.48528 q^{41} -6.00000i q^{43} -7.34847 q^{47} -4.00000 q^{49} -9.79796i q^{53} +3.46410 q^{55} +1.41421i q^{59} -5.19615i q^{61} +12.7279 q^{65} +3.00000i q^{67} +1.00000 q^{73} -2.44949i q^{77} +15.5885 q^{79} -14.1421i q^{83} +10.3923i q^{85} +12.7279 q^{89} -9.00000i q^{91} -7.34847 q^{95} +5.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{25} - 32 q^{49} + 8 q^{73} + 40 q^{97}+O(q^{100}) $ 8 * q - 8 * q^25 - 32 * q^49 + 8 * q^73 + 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times$.

$n$	$2053$	$2431$	$2945$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	2.44949i	1.09545i	0.836660	+	0.547723i	$0.184505\pi$
$5$	2.44949i	1.09545i	−0.836660	+	0.547723i	$0.815495\pi$
$6$	0	0
$7$	1.73205	0.654654	0.327327	−	0.944911i	$-0.393852\pi$
$7$	1.73205	0.654654	0.327327	+	0.944911i	$0.393852\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.41421i	− 0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$	− 1.41421i	− 0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$	0	0
$13$	− 5.19615i	− 1.44115i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	− 5.19615i	− 1.44115i	0.693375	−	0.720577i	$-0.256123\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.24264	1.02899	0.514496	−	0.857493i	$-0.327979\pi$
$17$	4.24264	1.02899	0.514496	+	0.857493i	$0.327979\pi$
$18$	0	0
$19$	3.00000i	0.688247i	0.938924	+	0.344124i	$0.111824\pi$
$19$	3.00000i	0.688247i	−0.938924	+	0.344124i	$0.888176\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.34847	1.53226	0.766131	−	0.642685i	$-0.222179\pi$
$23$	7.34847	1.53226	0.766131	+	0.642685i	$0.222179\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−6.92820	−1.24434	−0.622171	−	0.782881i	$-0.713749\pi$
$31$	−6.92820	−1.24434	−0.622171	+	0.782881i	$0.713749\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.24264i	0.717137i
$36$	0	0
$37$	− 5.19615i	− 0.854242i	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	− 5.19615i	− 0.854242i	0.904194	−	0.427121i	$-0.140472\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.48528	1.32518	0.662589	−	0.748983i	$-0.269458\pi$
$41$	8.48528	1.32518	0.662589	+	0.748983i	$0.269458\pi$
$42$	0	0
$43$	− 6.00000i	− 0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$	− 6.00000i	− 0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.34847	−1.07188	−0.535942	−	0.844255i	$-0.680044\pi$
$47$	−7.34847	−1.07188	−0.535942	+	0.844255i	$0.680044\pi$
$48$	0	0
$49$	−4.00000	−0.571429
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 9.79796i	− 1.34585i	−0.739709	−	0.672927i	$-0.765037\pi$
$53$	− 9.79796i	− 1.34585i	0.739709	−	0.672927i	$-0.234963\pi$
$54$	0	0
$55$	3.46410	0.467099
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.41421i	0.184115i	0.995754	+	0.0920575i	$0.0293443\pi$
$59$	1.41421i	0.184115i	−0.995754	+	0.0920575i	$0.970656\pi$
$60$	0	0
$61$	− 5.19615i	− 0.665299i	−0.943051	−	0.332650i	$-0.892057\pi$
$61$	− 5.19615i	− 0.665299i	0.943051	−	0.332650i	$-0.107943\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	12.7279	1.57870
$66$	0	0
$67$	3.00000i	0.366508i	0.983066	+	0.183254i	$0.0586631\pi$
$67$	3.00000i	0.366508i	−0.983066	+	0.183254i	$0.941337\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.44949i	− 0.279145i
$78$	0	0
$79$	15.5885	1.75384	0.876919	−	0.480638i	$-0.159595\pi$
$79$	15.5885	1.75384	0.876919	+	0.480638i	$0.159595\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 14.1421i	− 1.55230i	−0.630548	−	0.776151i	$-0.717170\pi$
$83$	− 14.1421i	− 1.55230i	0.630548	−	0.776151i	$-0.282830\pi$
$84$	0	0
$85$	10.3923i	1.12720i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.7279	1.34916	0.674579	−	0.738203i	$-0.264325\pi$
$89$	12.7279	1.34916	0.674579	+	0.738203i	$0.264325\pi$
$90$	0	0
$91$	− 9.00000i	− 0.943456i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.34847	−0.753937
$96$	0	0
$97$	5.00000	0.507673	0.253837	−	0.967247i	$-0.418307\pi$
$97$	5.00000	0.507673	0.253837	+	0.967247i	$0.418307\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.89898i	0.487467i	0.969842	+	0.243733i	$0.0783722\pi$
$101$	4.89898i	0.487467i	−0.969842	+	0.243733i	$0.921628\pi$
$102$	0	0
$103$	−15.5885	−1.53598	−0.767988	−	0.640464i	$-0.778742\pi$
$103$	−15.5885	−1.53598	−0.767988	+	0.640464i	$0.778742\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.89949i	0.957020i	0.878082	+	0.478510i	$0.158823\pi$
$107$	9.89949i	0.957020i	−0.878082	+	0.478510i	$0.841177\pi$
$108$	0	0
$109$	10.3923i	0.995402i	0.867349	+	0.497701i	$0.165822\pi$
$109$	10.3923i	0.995402i	−0.867349	+	0.497701i	$0.834178\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.7279	1.19734	0.598671	−	0.800995i	$-0.295696\pi$
$113$	12.7279	1.19734	0.598671	+	0.800995i	$0.295696\pi$
$114$	0	0
$115$	18.0000i	1.67851i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.34847	0.673633
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.79796i	0.876356i
$126$	0	0
$127$	−3.46410	−0.307389	−0.153695	−	0.988118i	$-0.549117\pi$
$127$	−3.46410	−0.307389	−0.153695	+	0.988118i	$0.549117\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.65685i	0.494242i	0.968985	+	0.247121i	$0.0794845\pi$
$131$	5.65685i	0.494242i	−0.968985	+	0.247121i	$0.920516\pi$
$132$	0	0
$133$	5.19615i	0.450564i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.2132	1.81237	0.906183	−	0.422885i	$-0.138983\pi$
$137$	21.2132	1.81237	0.906183	+	0.422885i	$0.138983\pi$
$138$	0	0
$139$	− 15.0000i	− 1.27228i	−0.771572	−	0.636142i	$-0.780529\pi$
$139$	− 15.0000i	− 1.27228i	0.771572	−	0.636142i	$-0.219471\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−7.34847	−0.614510
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.5959i	1.60536i	0.596410	+	0.802680i	$0.296593\pi$
$149$	19.5959i	1.60536i	−0.596410	+	0.802680i	$0.703407\pi$
$150$	0	0
$151$	−1.73205	−0.140952	−0.0704761	−	0.997513i	$-0.522452\pi$
$151$	−1.73205	−0.140952	−0.0704761	+	0.997513i	$0.522452\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 16.9706i	− 1.36311i
$156$	0	0
$157$	10.3923i	0.829396i	0.909959	+	0.414698i	$0.136113\pi$
$157$	10.3923i	0.829396i	−0.909959	+	0.414698i	$0.863887\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	12.7279	1.00310
$162$	0	0
$163$	9.00000i	0.704934i	0.935824	+	0.352467i	$0.114657\pi$
$163$	9.00000i	0.704934i	−0.935824	+	0.352467i	$0.885343\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.0454	−1.70592	−0.852962	−	0.521972i	$-0.825196\pi$
$167$	−22.0454	−1.70592	−0.852962	+	0.521972i	$0.825196\pi$
$168$	0	0
$169$	−14.0000	−1.07692
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 4.89898i	− 0.372463i	−0.982506	−	0.186231i	$-0.940373\pi$
$173$	− 4.89898i	− 0.372463i	0.982506	−	0.186231i	$-0.0596274\pi$
$174$	0	0
$175$	−1.73205	−0.130931
$176$	0	0
$177$	0	0
$178$	0	0
$179$	22.6274i	1.69125i	0.533775	+	0.845626i	$0.320773\pi$
$179$	22.6274i	1.69125i	−0.533775	+	0.845626i	$0.679227\pi$
$180$	0	0
$181$	5.19615i	0.386227i	0.981176	+	0.193113i	$0.0618586\pi$
$181$	5.19615i	0.386227i	−0.981176	+	0.193113i	$0.938141\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12.7279	0.935775
$186$	0	0
$187$	− 6.00000i	− 0.438763i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.0454	1.59515	0.797575	−	0.603220i	$-0.206116\pi$
$191$	22.0454	1.59515	0.797575	+	0.603220i	$0.206116\pi$
$192$	0	0
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 12.2474i	− 0.872595i	−0.899803	−	0.436297i	$-0.856290\pi$
$197$	− 12.2474i	− 0.872595i	0.899803	−	0.436297i	$-0.143710\pi$
$198$	0	0
$199$	12.1244	0.859473	0.429736	−	0.902954i	$-0.358606\pi$
$199$	12.1244	0.859473	0.429736	+	0.902954i	$0.358606\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	20.7846i	1.45166i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.24264	0.293470
$210$	0	0
$211$	27.0000i	1.85876i	0.369129	+	0.929378i	$0.379656\pi$
$211$	27.0000i	1.85876i	−0.369129	+	0.929378i	$0.620344\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	14.6969	1.00232
$216$	0	0
$217$	−12.0000	−0.814613
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 22.0454i	− 1.48293i
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 22.6274i	− 1.50183i	−0.660396	−	0.750917i	$-0.729612\pi$
$227$	− 22.6274i	− 1.50183i	0.660396	−	0.750917i	$-0.270388\pi$
$228$	0	0
$229$	− 20.7846i	− 1.37349i	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	− 20.7846i	− 1.37349i	0.726900	−	0.686743i	$-0.240960\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.48528	0.555889	0.277945	−	0.960597i	$-0.410347\pi$
$233$	8.48528	0.555889	0.277945	+	0.960597i	$0.410347\pi$
$234$	0	0
$235$	− 18.0000i	− 1.17419i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.6969	0.950666	0.475333	−	0.879806i	$-0.342328\pi$
$239$	14.6969	0.950666	0.475333	+	0.879806i	$0.342328\pi$
$240$	0	0
$241$	19.0000	1.22390	0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000	1.22390	0.611949	+	0.790897i	$0.290386\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 9.79796i	− 0.625969i
$246$	0	0
$247$	15.5885	0.991870
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.7990i	1.24970i	0.780744	+	0.624851i	$0.214840\pi$
$251$	19.7990i	1.24970i	−0.780744	+	0.624851i	$0.785160\pi$
$252$	0	0
$253$	− 10.3923i	− 0.653359i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.9706	1.05859	0.529297	−	0.848436i	$-0.322456\pi$
$257$	16.9706	1.05859	0.529297	+	0.848436i	$0.322456\pi$
$258$	0	0
$259$	− 9.00000i	− 0.559233i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−14.6969	−0.906252	−0.453126	−	0.891446i	$-0.649691\pi$
$263$	−14.6969	−0.906252	−0.453126	+	0.891446i	$0.649691\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.0454i	1.34413i	0.740491	+	0.672066i	$0.234593\pi$
$269$	22.0454i	1.34413i	−0.740491	+	0.672066i	$0.765407\pi$
$270$	0	0
$271$	−12.1244	−0.736502	−0.368251	−	0.929726i	$-0.620043\pi$
$271$	−12.1244	−0.736502	−0.368251	+	0.929726i	$0.620043\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.41421i	0.0852803i
$276$	0	0
$277$	− 10.3923i	− 0.624413i	−0.950014	−	0.312207i	$-0.898932\pi$
$277$	− 10.3923i	− 0.624413i	0.950014	−	0.312207i	$-0.101068\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	− 18.0000i	− 1.06999i	−0.844856	−	0.534994i	$-0.820314\pi$
$283$	− 18.0000i	− 1.06999i	0.844856	−	0.534994i	$-0.179686\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14.6969	0.867533
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 22.0454i	− 1.28791i	−0.765065	−	0.643953i	$-0.777293\pi$
$293$	− 22.0454i	− 1.28791i	0.765065	−	0.643953i	$-0.222707\pi$
$294$	0	0
$295$	−3.46410	−0.201688
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 38.1838i	− 2.20822i
$300$	0	0
$301$	− 10.3923i	− 0.599002i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	12.7279	0.728799
$306$	0	0
$307$	30.0000i	1.71219i	0.516818	+	0.856095i	$0.327116\pi$
$307$	30.0000i	1.71219i	−0.516818	+	0.856095i	$0.672884\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.34847	0.416693	0.208347	−	0.978055i	$-0.433192\pi$
$311$	7.34847	0.416693	0.208347	+	0.978055i	$0.433192\pi$
$312$	0	0
$313$	−5.00000	−0.282617	−0.141308	−	0.989966i	$-0.545131\pi$
$313$	−5.00000	−0.282617	−0.141308	+	0.989966i	$0.545131\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.89898i	0.275154i	0.990491	+	0.137577i	$0.0439315\pi$
$317$	4.89898i	0.275154i	−0.990491	+	0.137577i	$0.956069\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.7279i	0.708201i
$324$	0	0
$325$	5.19615i	0.288231i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−12.7279	−0.701713
$330$	0	0
$331$	− 27.0000i	− 1.48405i	−0.670370	−	0.742027i	$-0.733865\pi$
$331$	− 27.0000i	− 1.48405i	0.670370	−	0.742027i	$-0.266135\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.34847	−0.401490
$336$	0	0
$337$	−23.0000	−1.25289	−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000	−1.25289	−0.626445	+	0.779466i	$0.715491\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	9.79796i	0.530589i
$342$	0	0
$343$	−19.0526	−1.02874
$344$	0	0
$345$	0	0
$346$	0	0
$347$	19.7990i	1.06287i	0.847100	+	0.531433i	$0.178346\pi$
$347$	19.7990i	1.06287i	−0.847100	+	0.531433i	$0.821654\pi$
$348$	0	0
$349$	− 25.9808i	− 1.39072i	−0.718662	−	0.695359i	$-0.755245\pi$
$349$	− 25.9808i	− 1.39072i	0.718662	−	0.695359i	$-0.244755\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.48528	−0.451626	−0.225813	−	0.974171i	$-0.572504\pi$
$353$	−8.48528	−0.451626	−0.225813	+	0.974171i	$0.572504\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−36.7423	−1.93919	−0.969593	−	0.244721i	$-0.921303\pi$
$359$	−36.7423	−1.93919	−0.969593	+	0.244721i	$0.921303\pi$
$360$	0	0
$361$	10.0000	0.526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.44949i	0.128212i
$366$	0	0
$367$	−32.9090	−1.71783	−0.858917	−	0.512115i	$-0.828862\pi$
$367$	−32.9090	−1.71783	−0.858917	+	0.512115i	$0.828862\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 16.9706i	− 0.881068i
$372$	0	0
$373$	− 5.19615i	− 0.269047i	−0.990910	−	0.134523i	$-0.957050\pi$
$373$	− 5.19615i	− 0.269047i	0.990910	−	0.134523i	$-0.0429503\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	3.00000i	0.154100i	0.997027	+	0.0770498i	$0.0245501\pi$
$379$	3.00000i	0.154100i	−0.997027	+	0.0770498i	$0.975450\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	14.6969	0.750978	0.375489	−	0.926827i	$-0.377475\pi$
$383$	14.6969	0.750978	0.375489	+	0.926827i	$0.377475\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	0	0
$387$	0	0
$388$	0	0
$389$	26.9444i	1.36613i	0.730355	+	0.683067i	$0.239354\pi$
$389$	26.9444i	1.36613i	−0.730355	+	0.683067i	$0.760646\pi$
$390$	0	0
$391$	31.1769	1.57668
$392$	0	0
$393$	0	0
$394$	0	0
$395$	38.1838i	1.92123i
$396$	0	0
$397$	− 31.1769i	− 1.56472i	−0.622824	−	0.782362i	$-0.714015\pi$
$397$	− 31.1769i	− 1.56472i	0.622824	−	0.782362i	$-0.285985\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.48528	−0.423735	−0.211867	−	0.977298i	$-0.567954\pi$
$401$	−8.48528	−0.423735	−0.211867	+	0.977298i	$0.567954\pi$
$402$	0	0
$403$	36.0000i	1.79329i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.34847	−0.364250
$408$	0	0
$409$	11.0000	0.543915	0.271957	−	0.962309i	$-0.412329\pi$
$409$	11.0000	0.543915	0.271957	+	0.962309i	$0.412329\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.44949i	0.120532i
$414$	0	0
$415$	34.6410	1.70046
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 1.41421i	− 0.0690889i	−0.999403	−	0.0345444i	$-0.989002\pi$
$419$	− 1.41421i	− 0.0690889i	0.999403	−	0.0345444i	$-0.0109980\pi$
$420$	0	0
$421$	− 15.5885i	− 0.759735i	−0.925041	−	0.379867i	$-0.875970\pi$
$421$	− 15.5885i	− 0.759735i	0.925041	−	0.379867i	$-0.124030\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.24264	−0.205798
$426$	0	0
$427$	− 9.00000i	− 0.435541i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.34847	0.353963	0.176982	−	0.984214i	$-0.443367\pi$
$431$	7.34847	0.353963	0.176982	+	0.984214i	$0.443367\pi$
$432$	0	0
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	22.0454i	1.05457i
$438$	0	0
$439$	−13.8564	−0.661330	−0.330665	−	0.943748i	$-0.607273\pi$
$439$	−13.8564	−0.661330	−0.330665	+	0.943748i	$0.607273\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 22.6274i	− 1.07506i	−0.843244	−	0.537531i	$-0.819357\pi$
$443$	− 22.6274i	− 1.07506i	0.843244	−	0.537531i	$-0.180643\pi$
$444$	0	0
$445$	31.1769i	1.47793i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−29.6985	−1.40156	−0.700779	−	0.713378i	$-0.747164\pi$
$449$	−29.6985	−1.40156	−0.700779	+	0.713378i	$0.747164\pi$
$450$	0	0
$451$	− 12.0000i	− 0.565058i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	22.0454	1.03350
$456$	0	0
$457$	−40.0000	−1.87112	−0.935561	−	0.353166i	$-0.885105\pi$
$457$	−40.0000	−1.87112	−0.935561	+	0.353166i	$0.885105\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 17.1464i	− 0.798589i	−0.916823	−	0.399294i	$-0.869255\pi$
$461$	− 17.1464i	− 0.798589i	0.916823	−	0.399294i	$-0.130745\pi$
$462$	0	0
$463$	39.8372	1.85139	0.925695	−	0.378270i	$-0.123481\pi$
$463$	39.8372	1.85139	0.925695	+	0.378270i	$0.123481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.5563i	0.719862i	0.932979	+	0.359931i	$0.117200\pi$
$467$	15.5563i	0.719862i	−0.932979	+	0.359931i	$0.882800\pi$
$468$	0	0
$469$	5.19615i	0.239936i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8.48528	−0.390154
$474$	0	0
$475$	− 3.00000i	− 0.137649i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−29.3939	−1.34304	−0.671520	−	0.740986i	$-0.734358\pi$
$479$	−29.3939	−1.34304	−0.671520	+	0.740986i	$0.734358\pi$
$480$	0	0
$481$	−27.0000	−1.23109
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.2474i	0.556128i
$486$	0	0
$487$	−15.5885	−0.706380	−0.353190	−	0.935552i	$-0.614903\pi$
$487$	−15.5885	−0.706380	−0.353190	+	0.935552i	$0.614903\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 7.07107i	− 0.319113i	−0.987189	−	0.159556i	$-0.948994\pi$
$491$	− 7.07107i	− 0.319113i	0.987189	−	0.159556i	$-0.0510064\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	24.0000i	1.07439i	0.843459	+	0.537194i	$0.180516\pi$
$499$	24.0000i	1.07439i	−0.843459	+	0.537194i	$0.819484\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−7.34847	−0.327652	−0.163826	−	0.986489i	$-0.552384\pi$
$503$	−7.34847	−0.327652	−0.163826	+	0.986489i	$0.552384\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.44949i	0.108572i	0.998525	+	0.0542859i	$0.0172882\pi$
$509$	2.44949i	0.108572i	−0.998525	+	0.0542859i	$0.982712\pi$
$510$	0	0
$511$	1.73205	0.0766214
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 38.1838i	− 1.68258i
$516$	0	0
$517$	10.3923i	0.457053i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−29.6985	−1.30111	−0.650557	−	0.759457i	$-0.725465\pi$
$521$	−29.6985	−1.30111	−0.650557	+	0.759457i	$0.725465\pi$
$522$	0	0
$523$	− 27.0000i	− 1.18063i	−0.807174	−	0.590314i	$-0.799004\pi$
$523$	− 27.0000i	− 1.18063i	0.807174	−	0.590314i	$-0.200996\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−29.3939	−1.28042
$528$	0	0
$529$	31.0000	1.34783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 44.0908i	− 1.90979i
$534$	0	0
$535$	−24.2487	−1.04836
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.65685i	0.243658i
$540$	0	0
$541$	5.19615i	0.223400i	0.993742	+	0.111700i	$0.0356296\pi$
$541$	5.19615i	0.223400i	−0.993742	+	0.111700i	$0.964370\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−25.4558	−1.09041
$546$	0	0
$547$	3.00000i	0.128271i	0.997941	+	0.0641354i	$0.0204289\pi$
$547$	3.00000i	0.128271i	−0.997941	+	0.0641354i	$0.979571\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	27.0000	1.14816
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 31.8434i	− 1.34925i	−0.738162	−	0.674623i	$-0.764306\pi$
$557$	− 31.8434i	− 1.34925i	0.738162	−	0.674623i	$-0.235694\pi$
$558$	0	0
$559$	−31.1769	−1.31864
$560$	0	0
$561$	0	0
$562$	0	0
$563$	19.7990i	0.834428i	0.908808	+	0.417214i	$0.136993\pi$
$563$	19.7990i	0.834428i	−0.908808	+	0.417214i	$0.863007\pi$
$564$	0	0
$565$	31.1769i	1.31162i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.2132	−0.889304	−0.444652	−	0.895703i	$-0.646673\pi$
$569$	−21.2132	−0.889304	−0.444652	+	0.895703i	$0.646673\pi$
$570$	0	0
$571$	15.0000i	0.627730i	0.949468	+	0.313865i	$0.101624\pi$
$571$	15.0000i	0.627730i	−0.949468	+	0.313865i	$0.898376\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.34847	−0.306452
$576$	0	0
$577$	25.0000	1.04076	0.520382	−	0.853934i	$-0.325790\pi$
$577$	25.0000	1.04076	0.520382	+	0.853934i	$0.325790\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 24.4949i	− 1.01622i
$582$	0	0
$583$	−13.8564	−0.573874
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 24.0416i	− 0.992304i	−0.868236	−	0.496152i	$-0.834746\pi$
$587$	− 24.0416i	− 0.992304i	0.868236	−	0.496152i	$-0.165254\pi$
$588$	0	0
$589$	− 20.7846i	− 0.856415i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−25.4558	−1.04535	−0.522673	−	0.852533i	$-0.675065\pi$
$593$	−25.4558	−1.04535	−0.522673	+	0.852533i	$0.675065\pi$
$594$	0	0
$595$	18.0000i	0.737928i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	4.00000	0.163163	0.0815817	−	0.996667i	$-0.474003\pi$
$601$	4.00000	0.163163	0.0815817	+	0.996667i	$0.474003\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	22.0454i	0.896273i
$606$	0	0
$607$	15.5885	0.632716	0.316358	−	0.948640i	$-0.397540\pi$
$607$	15.5885	0.632716	0.316358	+	0.948640i	$0.397540\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	38.1838i	1.54475i
$612$	0	0
$613$	25.9808i	1.04935i	0.851302	+	0.524677i	$0.175814\pi$
$613$	25.9808i	1.04935i	−0.851302	+	0.524677i	$0.824186\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.24264	−0.170802	−0.0854011	−	0.996347i	$-0.527217\pi$
$617$	−4.24264	−0.170802	−0.0854011	+	0.996347i	$0.527217\pi$
$618$	0	0
$619$	− 3.00000i	− 0.120580i	−0.998181	−	0.0602901i	$-0.980797\pi$
$619$	− 3.00000i	− 0.120580i	0.998181	−	0.0602901i	$-0.0192026\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	22.0454	0.883231
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 22.0454i	− 0.879008i
$630$	0	0
$631$	12.1244	0.482663	0.241331	−	0.970443i	$-0.422416\pi$
$631$	12.1244	0.482663	0.241331	+	0.970443i	$0.422416\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 8.48528i	− 0.336728i
$636$	0	0
$637$	20.7846i	0.823516i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−16.9706	−0.670297	−0.335148	−	0.942165i	$-0.608786\pi$
$641$	−16.9706	−0.670297	−0.335148	+	0.942165i	$0.608786\pi$
$642$	0	0
$643$	− 42.0000i	− 1.65632i	−0.560493	−	0.828159i	$-0.689388\pi$
$643$	− 42.0000i	− 1.65632i	0.560493	−	0.828159i	$-0.310612\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	44.0908	1.73339	0.866694	−	0.498839i	$-0.166240\pi$
$647$	44.0908	1.73339	0.866694	+	0.498839i	$0.166240\pi$
$648$	0	0
$649$	2.00000	0.0785069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24.4949i	0.958559i	0.877662	+	0.479280i	$0.159102\pi$
$653$	24.4949i	0.958559i	−0.877662	+	0.479280i	$0.840898\pi$
$654$	0	0
$655$	−13.8564	−0.541415
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 11.3137i	− 0.440720i	−0.975419	−	0.220360i	$-0.929277\pi$
$659$	− 11.3137i	− 0.440720i	0.975419	−	0.220360i	$-0.0707231\pi$
$660$	0	0
$661$	5.19615i	0.202107i	0.994881	+	0.101053i	$0.0322213\pi$
$661$	5.19615i	0.202107i	−0.994881	+	0.101053i	$0.967779\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−12.7279	−0.493568
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.34847	−0.283685
$672$	0	0
$673$	23.0000	0.886585	0.443292	−	0.896377i	$-0.353810\pi$
$673$	23.0000	0.886585	0.443292	+	0.896377i	$0.353810\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.0454i	0.847274i	0.905832	+	0.423637i	$0.139247\pi$
$677$	22.0454i	0.847274i	−0.905832	+	0.423637i	$0.860753\pi$
$678$	0	0
$679$	8.66025	0.332350
$680$	0	0
$681$	0	0
$682$	0	0
$683$	19.7990i	0.757587i	0.925481	+	0.378794i	$0.123661\pi$
$683$	19.7990i	0.757587i	−0.925481	+	0.378794i	$0.876339\pi$
$684$	0	0
$685$	51.9615i	1.98535i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−50.9117	−1.93958
$690$	0	0
$691$	18.0000i	0.684752i	0.939563	+	0.342376i	$0.111232\pi$
$691$	18.0000i	0.684752i	−0.939563	+	0.342376i	$0.888768\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	36.7423	1.39372
$696$	0	0
$697$	36.0000	1.36360
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 12.2474i	− 0.462580i	−0.972885	−	0.231290i	$-0.925705\pi$
$701$	− 12.2474i	− 0.462580i	0.972885	−	0.231290i	$-0.0742946\pi$
$702$	0	0
$703$	15.5885	0.587930
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.48528i	0.319122i
$708$	0	0
$709$	36.3731i	1.36602i	0.730409	+	0.683010i	$0.239329\pi$
$709$	36.3731i	1.36602i	−0.730409	+	0.683010i	$0.760671\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−50.9117	−1.90666
$714$	0	0
$715$	− 18.0000i	− 0.673162i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−14.6969	−0.548103	−0.274052	−	0.961715i	$-0.588364\pi$
$719$	−14.6969	−0.548103	−0.274052	+	0.961715i	$0.588364\pi$
$720$	0	0
$721$	−27.0000	−1.00553
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	48.4974	1.79867	0.899335	−	0.437260i	$-0.144051\pi$
$727$	48.4974	1.79867	0.899335	+	0.437260i	$0.144051\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 25.4558i	− 0.941518i
$732$	0	0
$733$	− 31.1769i	− 1.15155i	−0.817610	−	0.575773i	$-0.804701\pi$
$733$	− 31.1769i	− 1.15155i	0.817610	−	0.575773i	$-0.195299\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.24264	0.156280
$738$	0	0
$739$	− 30.0000i	− 1.10357i	−0.833987	−	0.551784i	$-0.813947\pi$
$739$	− 30.0000i	− 1.10357i	0.833987	−	0.551784i	$-0.186053\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−7.34847	−0.269589	−0.134795	−	0.990874i	$-0.543037\pi$
$743$	−7.34847	−0.269589	−0.134795	+	0.990874i	$0.543037\pi$
$744$	0	0
$745$	−48.0000	−1.75858
$746$	0	0
$747$	0	0
$748$	0	0
$749$	17.1464i	0.626517i
$750$	0	0
$751$	−22.5167	−0.821645	−0.410822	−	0.911715i	$-0.634758\pi$
$751$	−22.5167	−0.821645	−0.410822	+	0.911715i	$0.634758\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 4.24264i	− 0.154406i
$756$	0	0
$757$	− 46.7654i	− 1.69972i	−0.527011	−	0.849858i	$-0.676688\pi$
$757$	− 46.7654i	− 1.69972i	0.527011	−	0.849858i	$-0.323312\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−12.7279	−0.461387	−0.230693	−	0.973026i	$-0.574099\pi$
$761$	−12.7279	−0.461387	−0.230693	+	0.973026i	$0.574099\pi$
$762$	0	0
$763$	18.0000i	0.651644i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.34847	0.265338
$768$	0	0
$769$	−29.0000	−1.04577	−0.522883	−	0.852404i	$-0.675144\pi$
$769$	−29.0000	−1.04577	−0.522883	+	0.852404i	$0.675144\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	44.0908i	1.58584i	0.609328	+	0.792918i	$0.291439\pi$
$773$	44.0908i	1.58584i	−0.609328	+	0.792918i	$0.708561\pi$
$774$	0	0
$775$	6.92820	0.248868
$776$	0	0
$777$	0	0
$778$	0	0
$779$	25.4558i	0.912050i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−25.4558	−0.908558
$786$	0	0
$787$	27.0000i	0.962446i	0.876598	+	0.481223i	$0.159807\pi$
$787$	27.0000i	0.962446i	−0.876598	+	0.481223i	$0.840193\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	22.0454	0.783844
$792$	0	0
$793$	−27.0000	−0.958798
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.4949i	0.867654i	0.900996	+	0.433827i	$0.142837\pi$
$797$	24.4949i	0.867654i	−0.900996	+	0.433827i	$0.857163\pi$
$798$	0	0
$799$	−31.1769	−1.10296
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1.41421i	− 0.0499065i
$804$	0	0
$805$	31.1769i	1.09884i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−16.9706	−0.596653	−0.298327	−	0.954464i	$-0.596428\pi$
$809$	−16.9706	−0.596653	−0.298327	+	0.954464i	$0.596428\pi$
$810$	0	0
$811$	6.00000i	0.210688i	0.994436	+	0.105344i	$0.0335944\pi$
$811$	6.00000i	0.210688i	−0.994436	+	0.105344i	$0.966406\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−22.0454	−0.772217
$816$	0	0
$817$	18.0000	0.629740
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22.0454i	0.769390i	0.923044	+	0.384695i	$0.125693\pi$
$821$	22.0454i	0.769390i	−0.923044	+	0.384695i	$0.874307\pi$
$822$	0	0
$823$	8.66025	0.301877	0.150939	−	0.988543i	$-0.451770\pi$
$823$	8.66025	0.301877	0.150939	+	0.988543i	$0.451770\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18.3848i	0.639301i	0.947535	+	0.319651i	$0.103566\pi$
$827$	18.3848i	0.639301i	−0.947535	+	0.319651i	$0.896434\pi$
$828$	0	0
$829$	15.5885i	0.541409i	0.962662	+	0.270705i	$0.0872567\pi$
$829$	15.5885i	0.541409i	−0.962662	+	0.270705i	$0.912743\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−16.9706	−0.587995
$834$	0	0
$835$	− 54.0000i	− 1.86875i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	29.3939	1.01479	0.507395	−	0.861714i	$-0.330609\pi$
$839$	29.3939	1.01479	0.507395	+	0.861714i	$0.330609\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 34.2929i	− 1.17971i
$846$	0	0
$847$	15.5885	0.535626
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 38.1838i	− 1.30892i
$852$	0	0
$853$	− 57.1577i	− 1.95704i	−0.206148	−	0.978521i	$-0.566093\pi$
$853$	− 57.1577i	− 1.95704i	0.206148	−	0.978521i	$-0.433907\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	21.0000i	0.716511i	0.933624	+	0.358255i	$0.116628\pi$
$859$	21.0000i	0.716511i	−0.933624	+	0.358255i	$0.883372\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36.7423	−1.25072	−0.625362	−	0.780335i	$-0.715049\pi$
$863$	−36.7423	−1.25072	−0.625362	+	0.780335i	$0.715049\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 22.0454i	− 0.747839i
$870$	0	0
$871$	15.5885	0.528195
$872$	0	0
$873$	0	0
$874$	0	0
$875$	16.9706i	0.573710i
$876$	0	0
$877$	25.9808i	0.877308i	0.898656	+	0.438654i	$0.144545\pi$
$877$	25.9808i	0.877308i	−0.898656	+	0.438654i	$0.855455\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−12.7279	−0.428815	−0.214407	−	0.976744i	$-0.568782\pi$
$881$	−12.7279	−0.428815	−0.214407	+	0.976744i	$0.568782\pi$
$882$	0	0
$883$	51.0000i	1.71629i	0.513410	+	0.858143i	$0.328382\pi$
$883$	51.0000i	1.71629i	−0.513410	+	0.858143i	$0.671618\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−44.0908	−1.48042	−0.740212	−	0.672373i	$-0.765275\pi$
$887$	−44.0908	−1.48042	−0.740212	+	0.672373i	$0.765275\pi$
$888$	0	0
$889$	−6.00000	−0.201234
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 22.0454i	− 0.737721i
$894$	0	0
$895$	−55.4256	−1.85267
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 41.5692i	− 1.38487i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12.7279	−0.423090
$906$	0	0
$907$	− 3.00000i	− 0.0996134i	−0.998759	−	0.0498067i	$-0.984139\pi$
$907$	− 3.00000i	− 0.0996134i	0.998759	−	0.0498067i	$-0.0158605\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	14.6969	0.486931	0.243466	−	0.969910i	$-0.421716\pi$
$911$	14.6969	0.486931	0.243466	+	0.969910i	$0.421716\pi$
$912$	0	0
$913$	−20.0000	−0.661903
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.79796i	0.323557i
$918$	0	0
$919$	−27.7128	−0.914161	−0.457081	−	0.889425i	$-0.651105\pi$
$919$	−27.7128	−0.914161	−0.457081	+	0.889425i	$0.651105\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	5.19615i	0.170848i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−25.4558	−0.835179	−0.417590	−	0.908636i	$-0.637125\pi$
$929$	−25.4558	−0.835179	−0.417590	+	0.908636i	$0.637125\pi$
$930$	0	0
$931$	− 12.0000i	− 0.393284i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	14.6969	0.480641
$936$	0	0
$937$	35.0000	1.14340	0.571700	−	0.820463i	$-0.306284\pi$
$937$	35.0000	1.14340	0.571700	+	0.820463i	$0.306284\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	41.6413i	1.35747i	0.734384	+	0.678734i	$0.237471\pi$
$941$	41.6413i	1.35747i	−0.734384	+	0.678734i	$0.762529\pi$
$942$	0	0
$943$	62.3538	2.03052
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 49.4975i	− 1.60845i	−0.594324	−	0.804226i	$-0.702580\pi$
$947$	− 49.4975i	− 1.60845i	0.594324	−	0.804226i	$-0.297420\pi$
$948$	0	0
$949$	− 5.19615i	− 0.168674i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−46.6690	−1.51176	−0.755879	−	0.654711i	$-0.772790\pi$
$953$	−46.6690	−1.51176	−0.755879	+	0.654711i	$0.772790\pi$
$954$	0	0
$955$	54.0000i	1.74740i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.7423	1.18647
$960$	0	0
$961$	17.0000	0.548387
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 26.9444i	− 0.867371i
$966$	0	0
$967$	−53.6936	−1.72667	−0.863334	−	0.504632i	$-0.831628\pi$
$967$	−53.6936	−1.72667	−0.863334	+	0.504632i	$0.831628\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 35.3553i	− 1.13461i	−0.823509	−	0.567303i	$-0.807987\pi$
$971$	− 35.3553i	− 1.13461i	0.823509	−	0.567303i	$-0.192013\pi$
$972$	0	0
$973$	− 25.9808i	− 0.832905i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	42.4264	1.35734	0.678671	−	0.734443i	$-0.262556\pi$
$977$	42.4264	1.35734	0.678671	+	0.734443i	$0.262556\pi$
$978$	0	0
$979$	− 18.0000i	− 0.575282i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−36.7423	−1.17190	−0.585949	−	0.810348i	$-0.699278\pi$
$983$	−36.7423	−1.17190	−0.585949	+	0.810348i	$0.699278\pi$
$984$	0	0
$985$	30.0000	0.955879
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 44.0908i	− 1.40201i
$990$	0	0
$991$	−19.0526	−0.605224	−0.302612	−	0.953114i	$-0.597859\pi$
$991$	−19.0526	−0.605224	−0.302612	+	0.953114i	$0.597859\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	29.6985i	0.941505i
$996$	0	0
$997$	− 31.1769i	− 0.987383i	−0.869637	−	0.493691i	$-0.835647\pi$
$997$	− 31.1769i	− 0.987383i	0.869637	−	0.493691i	$-0.164353\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3456.2.d.n.1729.7	yes	8
3.2	odd	2	inner	3456.2.d.n.1729.3	yes	8
4.3	odd	2	inner	3456.2.d.n.1729.5	yes	8
8.3	odd	2	inner	3456.2.d.n.1729.2	yes	8
8.5	even	2	inner	3456.2.d.n.1729.4	yes	8
12.11	even	2	inner	3456.2.d.n.1729.1	✓	8
16.3	odd	4		6912.2.a.ch.1.4		4
16.5	even	4		6912.2.a.ch.1.1		4
16.11	odd	4		6912.2.a.ce.1.2		4
16.13	even	4		6912.2.a.ce.1.3		4
24.5	odd	2	inner	3456.2.d.n.1729.8	yes	8
24.11	even	2	inner	3456.2.d.n.1729.6	yes	8
48.5	odd	4		6912.2.a.ch.1.3		4
48.11	even	4		6912.2.a.ce.1.4		4
48.29	odd	4		6912.2.a.ce.1.1		4
48.35	even	4		6912.2.a.ch.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3456.2.d.n.1729.1	✓	8	12.11	even	2	inner
3456.2.d.n.1729.2	yes	8	8.3	odd	2	inner
3456.2.d.n.1729.3	yes	8	3.2	odd	2	inner
3456.2.d.n.1729.4	yes	8	8.5	even	2	inner
3456.2.d.n.1729.5	yes	8	4.3	odd	2	inner
3456.2.d.n.1729.6	yes	8	24.11	even	2	inner
3456.2.d.n.1729.7	yes	8	1.1	even	1	trivial
3456.2.d.n.1729.8	yes	8	24.5	odd	2	inner
6912.2.a.ce.1.1		4	48.29	odd	4
6912.2.a.ce.1.2		4	16.11	odd	4
6912.2.a.ce.1.3		4	16.13	even	4
6912.2.a.ce.1.4		4	48.11	even	4
6912.2.a.ch.1.1		4	16.5	even	4
6912.2.a.ch.1.2		4	48.35	even	4
6912.2.a.ch.1.3		4	48.5	odd	4
6912.2.a.ch.1.4		4	16.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3456 = 2^{7} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3456.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.44949i q^{5} +1.73205 q^{7} +O(q^{10})\)	\(q+2.44949i q^{5} +1.73205 q^{7} -1.41421i q^{11} -5.19615i q^{13} +4.24264 q^{17} +3.00000i q^{19} +7.34847 q^{23} -1.00000 q^{25} -6.92820 q^{31} +4.24264i q^{35} -5.19615i q^{37} +8.48528 q^{41} -6.00000i q^{43} -7.34847 q^{47} -4.00000 q^{49} -9.79796i q^{53} +3.46410 q^{55} +1.41421i q^{59} -5.19615i q^{61} +12.7279 q^{65} +3.00000i q^{67} +1.00000 q^{73} -2.44949i q^{77} +15.5885 q^{79} -14.1421i q^{83} +10.3923i q^{85} +12.7279 q^{89} -9.00000i q^{91} -7.34847 q^{95} +5.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{25} - 32 q^{49} + 8 q^{73} + 40 q^{97}+O(q^{100}) \) 8 * q - 8 * q^25 - 32 * q^49 + 8 * q^73 + 40 * q^97

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