LMFDB - Embedded newform 5472.2.g.b.2737.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5472,2,Mod(2737,5472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5472, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5472.2737");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5472 = 2^{5} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5472.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:	$43.6941399860$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 4 x^{12} + 4 x^{11} - 10 x^{10} + 24 x^{9} - 40 x^{8} + \cdots + 256 $ x^16 - 2x^15 + 3x^14 - 4x^13 + 4x^12 + 4x^11 - 10x^10 + 24x^9 - 40x^8 + 48x^7 - 40x^6 + 32x^5 + 64x^4 - 128x^3 + 192x^2 - 256*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2737.9
Root			$1.14052 + 0.836196i$ of defining polynomial
Character	$\chi$	$=$	5472.2737
Dual form			5472.2.g.b.2737.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.594041i q^{5} +3.48756 q^{7} +O(q^{10})$	$q+0.594041i q^{5} +3.48756 q^{7} +4.83520i q^{11} +0.215597i q^{13} -1.29720 q^{17} -1.00000i q^{19} +4.52815 q^{23} +4.64712 q^{25} -9.41093i q^{29} +1.22031 q^{31} +2.07176i q^{35} -5.62653i q^{37} +0.450021 q^{41} +0.794359i q^{43} +12.1986 q^{47} +5.16310 q^{49} +2.56409i q^{53} -2.87231 q^{55} -2.75191i q^{59} +7.76665i q^{61} -0.128073 q^{65} +4.11631i q^{67} -7.82788 q^{71} +3.08931 q^{73} +16.8631i q^{77} +10.0731 q^{79} +11.6296i q^{83} -0.770591i q^{85} -13.7091 q^{89} +0.751907i q^{91} +0.594041 q^{95} +2.08846 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{7}+O(q^{10}) $ 16 * q + 8 * q^7	$ 16 q + 8 q^{7} + 8 q^{17} - 24 q^{25} - 16 q^{31} - 16 q^{41} + 24 q^{47} + 24 q^{49} - 16 q^{55} - 16 q^{65} + 48 q^{71} + 48 q^{79} + 16 q^{89} + 16 q^{95} + 32 q^{97}+O(q^{100}) $ 16 * q + 8 * q^7 + 8 * q^17 - 24 * q^25 - 16 * q^31 - 16 * q^41 + 24 * q^47 + 24 * q^49 - 16 * q^55 - 16 * q^65 + 48 * q^71 + 48 * q^79 + 16 * q^89 + 16 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times$.

$n$	$1217$	$2053$	$3745$	$4447$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	0.594041i	0.265663i	0.991139	+	0.132832i	$0.0424069\pi$
$5$	0.594041i	0.265663i	−0.991139	+	0.132832i	$0.957593\pi$
$6$	0	0
$7$	3.48756	1.31818	0.659088	−	0.752066i	$-0.270943\pi$
$7$	3.48756	1.31818	0.659088	+	0.752066i	$0.270943\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.83520i	1.45787i	0.684585	+	0.728933i	$0.259984\pi$
$11$	4.83520i	1.45787i	−0.684585	+	0.728933i	$0.740016\pi$
$12$	0	0
$13$	0.215597i	0.0597957i	0.999553	+	0.0298979i	$0.00951821\pi$
$13$	0.215597i	0.0597957i	−0.999553	+	0.0298979i	$0.990482\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.29720	−0.314618	−0.157309	−	0.987549i	$-0.550282\pi$
$17$	−1.29720	−0.314618	−0.157309	+	0.987549i	$0.550282\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.52815	0.944184	0.472092	−	0.881549i	$-0.343499\pi$
$23$	4.52815	0.944184	0.472092	+	0.881549i	$0.343499\pi$
$24$	0	0
$25$	4.64712	0.929423
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 9.41093i	− 1.74757i	−0.486316	−	0.873783i	$-0.661660\pi$
$29$	− 9.41093i	− 1.74757i	0.486316	−	0.873783i	$-0.338340\pi$
$30$	0	0
$31$	1.22031	0.219174	0.109587	−	0.993977i	$-0.465047\pi$
$31$	1.22031	0.219174	0.109587	+	0.993977i	$0.465047\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.07176i	0.350191i
$36$	0	0
$37$	− 5.62653i	− 0.924995i	−0.886621	−	0.462498i	$-0.846953\pi$
$37$	− 5.62653i	− 0.924995i	0.886621	−	0.462498i	$-0.153047\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.450021	0.0702815	0.0351407	−	0.999382i	$-0.488812\pi$
$41$	0.450021	0.0702815	0.0351407	+	0.999382i	$0.488812\pi$
$42$	0	0
$43$	0.794359i	0.121139i	0.998164	+	0.0605693i	$0.0192916\pi$
$43$	0.794359i	0.121139i	−0.998164	+	0.0605693i	$0.980708\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.1986	1.77935	0.889676	−	0.456593i	$-0.150930\pi$
$47$	12.1986	1.77935	0.889676	+	0.456593i	$0.150930\pi$
$48$	0	0
$49$	5.16310	0.737586
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.56409i	0.352205i	0.984372	+	0.176103i	$0.0563490\pi$
$53$	2.56409i	0.352205i	−0.984372	+	0.176103i	$0.943651\pi$
$54$	0	0
$55$	−2.87231	−0.387302
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 2.75191i	− 0.358268i	−0.983825	−	0.179134i	$-0.942670\pi$
$59$	− 2.75191i	− 0.358268i	0.983825	−	0.179134i	$-0.0573295\pi$
$60$	0	0
$61$	7.76665i	0.994417i	0.867631	+	0.497209i	$0.165642\pi$
$61$	7.76665i	0.994417i	−0.867631	+	0.497209i	$0.834358\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.128073	−0.0158855
$66$	0	0
$67$	4.11631i	0.502887i	0.967872	+	0.251443i	$0.0809053\pi$
$67$	4.11631i	0.502887i	−0.967872	+	0.251443i	$0.919095\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.82788	−0.928999	−0.464499	−	0.885573i	$-0.653766\pi$
$71$	−7.82788	−0.928999	−0.464499	+	0.885573i	$0.653766\pi$
$72$	0	0
$73$	3.08931	0.361577	0.180788	−	0.983522i	$-0.442135\pi$
$73$	3.08931	0.361577	0.180788	+	0.983522i	$0.442135\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.8631i	1.92172i
$78$	0	0
$79$	10.0731	1.13331	0.566654	−	0.823956i	$-0.308237\pi$
$79$	10.0731	1.13331	0.566654	+	0.823956i	$0.308237\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.6296i	1.27651i	0.769825	+	0.638255i	$0.220343\pi$
$83$	11.6296i	1.27651i	−0.769825	+	0.638255i	$0.779657\pi$
$84$	0	0
$85$	− 0.770591i	− 0.0835823i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13.7091	−1.45317	−0.726583	−	0.687079i	$-0.758893\pi$
$89$	−13.7091	−1.45317	−0.726583	+	0.687079i	$0.758893\pi$
$90$	0	0
$91$	0.751907i	0.0788213i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.594041	0.0609473
$96$	0	0
$97$	2.08846	0.212051	0.106026	−	0.994363i	$-0.466187\pi$
$97$	2.08846	0.212051	0.106026	+	0.994363i	$0.466187\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.77074i	0.275699i	0.990453	+	0.137849i	$0.0440190\pi$
$101$	2.77074i	0.275699i	−0.990453	+	0.137849i	$0.955981\pi$
$102$	0	0
$103$	14.3363	1.41260	0.706301	−	0.707912i	$-0.250363\pi$
$103$	14.3363	1.41260	0.706301	+	0.707912i	$0.250363\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.42388i	− 0.234325i	−0.993113	−	0.117163i	$-0.962620\pi$
$107$	− 2.42388i	− 0.234325i	0.993113	−	0.117163i	$-0.0373799\pi$
$108$	0	0
$109$	0.00123810i	0 0.000118589i	1.00000		5.92945e-5i	$1.88740e-5\pi$
$109$	0.00123810i	0 0.000118589i	−1.00000		5.92945e-5i	$0.999981\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.81614	−0.170848	−0.0854242	−	0.996345i	$-0.527225\pi$
$113$	−1.81614	−0.170848	−0.0854242	+	0.996345i	$0.527225\pi$
$114$	0	0
$115$	2.68990i	0.250835i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.52407	−0.414721
$120$	0	0
$121$	−12.3791	−1.12538
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.73078i	0.512577i
$126$	0	0
$127$	16.1269	1.43103	0.715517	−	0.698596i	$-0.246191\pi$
$127$	16.1269	1.43103	0.715517	+	0.698596i	$0.246191\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.1477i	0.973983i	0.873407	+	0.486992i	$0.161906\pi$
$131$	11.1477i	0.973983i	−0.873407	+	0.486992i	$0.838094\pi$
$132$	0	0
$133$	− 3.48756i	− 0.302410i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.1666	−0.954025	−0.477013	−	0.878896i	$-0.658280\pi$
$137$	−11.1666	−0.954025	−0.477013	+	0.878896i	$0.658280\pi$
$138$	0	0
$139$	− 13.0534i	− 1.10718i	−0.832790	−	0.553589i	$-0.813258\pi$
$139$	− 13.0534i	− 1.10718i	0.832790	−	0.553589i	$-0.186742\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.04245	−0.0871742
$144$	0	0
$145$	5.59048	0.464264
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.3811i	1.17814i	0.808080	+	0.589072i	$0.200507\pi$
$149$	14.3811i	1.17814i	−0.808080	+	0.589072i	$0.799493\pi$
$150$	0	0
$151$	−17.3489	−1.41184	−0.705918	−	0.708293i	$-0.749465\pi$
$151$	−17.3489	−1.41184	−0.705918	+	0.708293i	$0.749465\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.724915i	0.0582266i
$156$	0	0
$157$	− 9.63293i	− 0.768791i	−0.923168	−	0.384396i	$-0.874410\pi$
$157$	− 9.63293i	− 0.768791i	0.923168	−	0.384396i	$-0.125590\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	15.7922	1.24460
$162$	0	0
$163$	22.5365i	1.76520i	0.470129	+	0.882598i	$0.344207\pi$
$163$	22.5365i	1.76520i	−0.470129	+	0.882598i	$0.655793\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.5108	1.27765	0.638823	−	0.769353i	$-0.279421\pi$
$167$	16.5108	1.27765	0.638823	+	0.769353i	$0.279421\pi$
$168$	0	0
$169$	12.9535	0.996424
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 5.14911i	− 0.391479i	−0.980656	−	0.195740i	$-0.937289\pi$
$173$	− 5.14911i	− 0.391479i	0.980656	−	0.195740i	$-0.0627108\pi$
$174$	0	0
$175$	16.2071	1.22514
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 9.31580i	− 0.696295i	−0.937440	−	0.348148i	$-0.886811\pi$
$179$	− 9.31580i	− 0.696295i	0.937440	−	0.348148i	$-0.113189\pi$
$180$	0	0
$181$	5.15517i	0.383180i	0.981475	+	0.191590i	$0.0613645\pi$
$181$	5.15517i	0.383180i	−0.981475	+	0.191590i	$0.938636\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.34239	0.245737
$186$	0	0
$187$	− 6.27223i	− 0.458671i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.8805	−1.07672	−0.538359	−	0.842715i	$-0.680956\pi$
$191$	−14.8805	−1.07672	−0.538359	+	0.842715i	$0.680956\pi$
$192$	0	0
$193$	−21.2754	−1.53144	−0.765719	−	0.643175i	$-0.777617\pi$
$193$	−21.2754	−1.53144	−0.765719	+	0.643175i	$0.777617\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.69169i	− 0.263022i	−0.991315	−	0.131511i	$-0.958017\pi$
$197$	− 3.69169i	− 0.263022i	0.991315	−	0.131511i	$-0.0419829\pi$
$198$	0	0
$199$	−11.6857	−0.828377	−0.414188	−	0.910191i	$-0.635935\pi$
$199$	−11.6857	−0.828377	−0.414188	+	0.910191i	$0.635935\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 32.8212i	− 2.30360i
$204$	0	0
$205$	0.267331i	0.0186712i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.83520	0.334458
$210$	0	0
$211$	7.61611i	0.524315i	0.965025	+	0.262157i	$0.0844340\pi$
$211$	7.61611i	0.524315i	−0.965025	+	0.262157i	$0.915566\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.471882	−0.0321821
$216$	0	0
$217$	4.25591	0.288910
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 0.279672i	− 0.0188128i
$222$	0	0
$223$	1.95477	0.130901	0.0654505	−	0.997856i	$-0.479152\pi$
$223$	1.95477	0.130901	0.0654505	+	0.997856i	$0.479152\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 13.3709i	− 0.887461i	−0.896160	−	0.443730i	$-0.853655\pi$
$227$	− 13.3709i	− 0.887461i	0.896160	−	0.443730i	$-0.146345\pi$
$228$	0	0
$229$	11.5800i	0.765225i	0.923909	+	0.382613i	$0.124976\pi$
$229$	11.5800i	0.765225i	−0.923909	+	0.382613i	$0.875024\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.58872	−0.104080	−0.0520401	−	0.998645i	$-0.516572\pi$
$233$	−1.58872	−0.104080	−0.0520401	+	0.998645i	$0.516572\pi$
$234$	0	0
$235$	7.24648i	0.472708i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.8219	1.54091	0.770455	−	0.637494i	$-0.220029\pi$
$239$	23.8219	1.54091	0.770455	+	0.637494i	$0.220029\pi$
$240$	0	0
$241$	22.8554	1.47225	0.736123	−	0.676848i	$-0.236654\pi$
$241$	22.8554	1.47225	0.736123	+	0.676848i	$0.236654\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.06710i	0.195950i
$246$	0	0
$247$	0.215597	0.0137181
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 2.63524i	− 0.166335i	−0.996536	−	0.0831676i	$-0.973496\pi$
$251$	− 2.63524i	− 0.166335i	0.996536	−	0.0831676i	$-0.0265037\pi$
$252$	0	0
$253$	21.8945i	1.37649i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.0579	−1.25118	−0.625588	−	0.780153i	$-0.715141\pi$
$257$	−20.0579	−1.25118	−0.625588	+	0.780153i	$0.715141\pi$
$258$	0	0
$259$	− 19.6229i	− 1.21931i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.03667	0.248912	0.124456	−	0.992225i	$-0.460282\pi$
$263$	4.03667	0.248912	0.124456	+	0.992225i	$0.460282\pi$
$264$	0	0
$265$	−1.52318	−0.0935679
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 14.3095i	− 0.872467i	−0.899834	−	0.436233i	$-0.856312\pi$
$269$	− 14.3095i	− 0.872467i	0.899834	−	0.436233i	$-0.143688\pi$
$270$	0	0
$271$	−9.85034	−0.598366	−0.299183	−	0.954196i	$-0.596714\pi$
$271$	−9.85034	−0.598366	−0.299183	+	0.954196i	$0.596714\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	22.4697i	1.35498i
$276$	0	0
$277$	16.9641i	1.01928i	0.860389	+	0.509638i	$0.170221\pi$
$277$	16.9641i	1.01928i	−0.860389	+	0.509638i	$0.829779\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	10.2078	0.608944	0.304472	−	0.952521i	$-0.401520\pi$
$281$	10.2078	0.608944	0.304472	+	0.952521i	$0.401520\pi$
$282$	0	0
$283$	− 18.7709i	− 1.11581i	−0.829903	−	0.557907i	$-0.811604\pi$
$283$	− 18.7709i	− 1.11581i	0.829903	−	0.557907i	$-0.188396\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.56948	0.0926433
$288$	0	0
$289$	−15.3173	−0.901016
$290$	0	0
$291$	0	0
$292$	0	0
$293$	19.8271i	1.15831i	0.815217	+	0.579155i	$0.196618\pi$
$293$	19.8271i	1.15831i	−0.815217	+	0.579155i	$0.803382\pi$
$294$	0	0
$295$	1.63475	0.0951786
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.976253i	0.0564582i
$300$	0	0
$301$	2.77038i	0.159682i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4.61371	−0.264180
$306$	0	0
$307$	18.3935i	1.04977i	0.851173	+	0.524886i	$0.175892\pi$
$307$	18.3935i	1.04977i	−0.851173	+	0.524886i	$0.824108\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	19.1747	1.08730	0.543648	−	0.839313i	$-0.317043\pi$
$311$	19.1747	1.08730	0.543648	+	0.839313i	$0.317043\pi$
$312$	0	0
$313$	14.2274	0.804183	0.402091	−	0.915600i	$-0.368283\pi$
$313$	14.2274	0.804183	0.402091	+	0.915600i	$0.368283\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 0.777938i	− 0.0436934i	−0.999761	−	0.0218467i	$-0.993045\pi$
$317$	− 0.777938i	− 0.0436934i	0.999761	−	0.0218467i	$-0.00695457\pi$
$318$	0	0
$319$	45.5037	2.54772
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.29720i	0.0721782i
$324$	0	0
$325$	1.00190i	0.0555755i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	42.5435	2.34550
$330$	0	0
$331$	16.3988i	0.901359i	0.892686	+	0.450680i	$0.148818\pi$
$331$	16.3988i	0.901359i	−0.892686	+	0.450680i	$0.851182\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.44525	−0.133599
$336$	0	0
$337$	24.6109	1.34064	0.670320	−	0.742072i	$-0.266157\pi$
$337$	24.6109	1.34064	0.670320	+	0.742072i	$0.266157\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.90045i	0.319527i
$342$	0	0
$343$	−6.40629	−0.345907
$344$	0	0
$345$	0	0
$346$	0	0
$347$	13.4635i	0.722760i	0.932419	+	0.361380i	$0.117694\pi$
$347$	13.4635i	0.722760i	−0.932419	+	0.361380i	$0.882306\pi$
$348$	0	0
$349$	− 2.83422i	− 0.151712i	−0.997119	−	0.0758560i	$-0.975831\pi$
$349$	− 2.83422i	− 0.151712i	0.997119	−	0.0758560i	$-0.0241689\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.02227	0.267308	0.133654	−	0.991028i	$-0.457329\pi$
$353$	5.02227	0.267308	0.133654	+	0.991028i	$0.457329\pi$
$354$	0	0
$355$	− 4.65008i	− 0.246801i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	23.6898	1.25030	0.625149	−	0.780505i	$-0.285038\pi$
$359$	23.6898	1.25030	0.625149	+	0.780505i	$0.285038\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.83518i	0.0960577i
$366$	0	0
$367$	16.9338	0.883938	0.441969	−	0.897030i	$-0.354280\pi$
$367$	16.9338	0.883938	0.441969	+	0.897030i	$0.354280\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.94244i	0.464268i
$372$	0	0
$373$	12.8599i	0.665860i	0.942952	+	0.332930i	$0.108037\pi$
$373$	12.8599i	0.665860i	−0.942952	+	0.332930i	$0.891963\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.02896	0.104497
$378$	0	0
$379$	20.7810i	1.06745i	0.845659	+	0.533723i	$0.179207\pi$
$379$	20.7810i	1.06745i	−0.845659	+	0.533723i	$0.820793\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.6170	−0.797993	−0.398996	−	0.916953i	$-0.630641\pi$
$383$	−15.6170	−0.797993	−0.398996	+	0.916953i	$0.630641\pi$
$384$	0	0
$385$	−10.0173	−0.510531
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 10.1133i	− 0.512764i	−0.966576	−	0.256382i	$-0.917470\pi$
$389$	− 10.1133i	− 0.512764i	0.966576	−	0.256382i	$-0.0825304\pi$
$390$	0	0
$391$	−5.87392	−0.297057
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.98381i	0.301078i
$396$	0	0
$397$	4.60677i	0.231207i	0.993295	+	0.115604i	$0.0368802\pi$
$397$	4.60677i	0.231207i	−0.993295	+	0.115604i	$0.963120\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.00112	0.199806	0.0999032	−	0.994997i	$-0.468147\pi$
$401$	4.00112	0.199806	0.0999032	+	0.994997i	$0.468147\pi$
$402$	0	0
$403$	0.263095i	0.0131057i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	27.2054	1.34852
$408$	0	0
$409$	−14.5111	−0.717529	−0.358765	−	0.933428i	$-0.616802\pi$
$409$	−14.5111	−0.717529	−0.358765	+	0.933428i	$0.616802\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 9.59745i	− 0.472260i
$414$	0	0
$415$	−6.90843	−0.339122
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 5.02078i	− 0.245281i	−0.992451	−	0.122641i	$-0.960864\pi$
$419$	− 5.02078i	− 0.245281i	0.992451	−	0.122641i	$-0.0391362\pi$
$420$	0	0
$421$	4.10152i	0.199896i	0.994993	+	0.0999479i	$0.0318676\pi$
$421$	4.10152i	0.199896i	−0.994993	+	0.0999479i	$0.968132\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.02825	−0.292413
$426$	0	0
$427$	27.0867i	1.31082i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.2920	−0.784760	−0.392380	−	0.919803i	$-0.628348\pi$
$431$	−16.2920	−0.784760	−0.392380	+	0.919803i	$0.628348\pi$
$432$	0	0
$433$	7.44858	0.357956	0.178978	−	0.983853i	$-0.442721\pi$
$433$	7.44858	0.357956	0.178978	+	0.983853i	$0.442721\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.52815i	− 0.216611i
$438$	0	0
$439$	−15.8898	−0.758379	−0.379189	−	0.925319i	$-0.623797\pi$
$439$	−15.8898	−0.758379	−0.379189	+	0.925319i	$0.623797\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15.1742i	0.720949i	0.932769	+	0.360474i	$0.117385\pi$
$443$	15.1742i	0.720949i	−0.932769	+	0.360474i	$0.882615\pi$
$444$	0	0
$445$	− 8.14379i	− 0.386053i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	24.3909	1.15108	0.575538	−	0.817775i	$-0.304793\pi$
$449$	24.3909	1.15108	0.575538	+	0.817775i	$0.304793\pi$
$450$	0	0
$451$	2.17594i	0.102461i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.446664	−0.0209399
$456$	0	0
$457$	−10.6860	−0.499869	−0.249934	−	0.968263i	$-0.580409\pi$
$457$	−10.6860	−0.499869	−0.249934	+	0.968263i	$0.580409\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 21.8232i	− 1.01641i	−0.861236	−	0.508205i	$-0.830309\pi$
$461$	− 21.8232i	− 1.01641i	0.861236	−	0.508205i	$-0.169691\pi$
$462$	0	0
$463$	−0.0258442	−0.00120108	−0.000600541	−	1.00000i	$-0.500191\pi$
$463$	−0.0258442	−0.00120108	−0.000600541		1.00000i	$0.500191\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.66985i	0.216095i	0.994146	+	0.108047i	$0.0344598\pi$
$467$	4.66985i	0.216095i	−0.994146	+	0.108047i	$0.965540\pi$
$468$	0	0
$469$	14.3559i	0.662893i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.84088	−0.176604
$474$	0	0
$475$	− 4.64712i	− 0.213224i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−30.5428	−1.39554	−0.697768	−	0.716324i	$-0.745823\pi$
$479$	−30.5428	−1.39554	−0.697768	+	0.716324i	$0.745823\pi$
$480$	0	0
$481$	1.21306	0.0553108
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.24063i	0.0563342i
$486$	0	0
$487$	2.37133	0.107455	0.0537277	−	0.998556i	$-0.482890\pi$
$487$	2.37133	0.107455	0.0537277	+	0.998556i	$0.482890\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 41.3206i	− 1.86477i	−0.361466	−	0.932385i	$-0.617724\pi$
$491$	− 41.3206i	− 1.86477i	0.361466	−	0.932385i	$-0.382276\pi$
$492$	0	0
$493$	12.2079i	0.549815i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−27.3002	−1.22458
$498$	0	0
$499$	− 15.8610i	− 0.710035i	−0.934860	−	0.355018i	$-0.884475\pi$
$499$	− 15.8610i	− 0.710035i	0.934860	−	0.355018i	$-0.115525\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18.6879	−0.833254	−0.416627	−	0.909078i	$-0.636788\pi$
$503$	−18.6879	−0.833254	−0.416627	+	0.909078i	$0.636788\pi$
$504$	0	0
$505$	−1.64593	−0.0732431
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 33.2172i	− 1.47233i	−0.676804	−	0.736163i	$-0.736636\pi$
$509$	− 33.2172i	− 1.47233i	0.676804	−	0.736163i	$-0.263364\pi$
$510$	0	0
$511$	10.7742	0.476622
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.51637i	0.375276i
$516$	0	0
$517$	58.9827i	2.59406i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.9831	0.481178	0.240589	−	0.970627i	$-0.422659\pi$
$521$	10.9831	0.481178	0.240589	+	0.970627i	$0.422659\pi$
$522$	0	0
$523$	8.00433i	0.350005i	0.984568	+	0.175002i	$0.0559933\pi$
$523$	8.00433i	0.350005i	−0.984568	+	0.175002i	$0.944007\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.58299	−0.0689561
$528$	0	0
$529$	−2.49589	−0.108517
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.0970230i	0.00420253i
$534$	0	0
$535$	1.43988	0.0622516
$536$	0	0
$537$	0	0
$538$	0	0
$539$	24.9646i	1.07530i
$540$	0	0
$541$	− 24.4359i	− 1.05058i	−0.850923	−	0.525291i	$-0.823956\pi$
$541$	− 24.4359i	− 1.05058i	0.850923	−	0.525291i	$-0.176044\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−0.000735485 0	−3.15047e−5 0
$546$	0	0
$547$	− 5.25284i	− 0.224595i	−0.993675	−	0.112298i	$-0.964179\pi$
$547$	− 5.25284i	− 0.224595i	0.993675	−	0.112298i	$-0.0358210\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.41093	−0.400919
$552$	0	0
$553$	35.1305	1.49390
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.56791i	0.151177i	0.997139	+	0.0755886i	$0.0240836\pi$
$557$	3.56791i	0.151177i	−0.997139	+	0.0755886i	$0.975916\pi$
$558$	0	0
$559$	−0.171261	−0.00724357
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17.0555i	0.718805i	0.933183	+	0.359403i	$0.117020\pi$
$563$	17.0555i	0.718805i	−0.933183	+	0.359403i	$0.882980\pi$
$564$	0	0
$565$	− 1.07886i	− 0.0453881i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0.858816	0.0360034	0.0180017	−	0.999838i	$-0.494270\pi$
$569$	0.858816	0.0360034	0.0180017	+	0.999838i	$0.494270\pi$
$570$	0	0
$571$	40.5440i	1.69671i	0.529426	+	0.848356i	$0.322407\pi$
$571$	40.5440i	1.69671i	−0.529426	+	0.848356i	$0.677593\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	21.0428	0.877546
$576$	0	0
$577$	36.5405	1.52120	0.760600	−	0.649220i	$-0.224905\pi$
$577$	36.5405	1.52120	0.760600	+	0.649220i	$0.224905\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	40.5588i	1.68266i
$582$	0	0
$583$	−12.3979	−0.513468
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 30.4022i	− 1.25483i	−0.778684	−	0.627416i	$-0.784113\pi$
$587$	− 30.4022i	− 1.25483i	0.778684	−	0.627416i	$-0.215887\pi$
$588$	0	0
$589$	− 1.22031i	− 0.0502821i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	16.3814	0.672703	0.336351	−	0.941737i	$-0.390807\pi$
$593$	16.3814	0.672703	0.336351	+	0.941737i	$0.390807\pi$
$594$	0	0
$595$	− 2.68749i	− 0.110176i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2.24749	−0.0918298	−0.0459149	−	0.998945i	$-0.514620\pi$
$599$	−2.24749	−0.0918298	−0.0459149	+	0.998945i	$0.514620\pi$
$600$	0	0
$601$	−33.1805	−1.35346	−0.676730	−	0.736231i	$-0.736604\pi$
$601$	−33.1805	−1.35346	−0.676730	+	0.736231i	$0.736604\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 7.35371i	− 0.298971i
$606$	0	0
$607$	−35.7847	−1.45246	−0.726228	−	0.687454i	$-0.758728\pi$
$607$	−35.7847	−1.45246	−0.726228	+	0.687454i	$0.758728\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.62998i	0.106398i
$612$	0	0
$613$	− 42.2856i	− 1.70790i	−0.520355	−	0.853950i	$-0.674200\pi$
$613$	− 42.2856i	− 1.70790i	0.520355	−	0.853950i	$-0.325800\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26.2079	1.05509	0.527545	−	0.849527i	$-0.323113\pi$
$617$	26.2079	1.05509	0.527545	+	0.849527i	$0.323113\pi$
$618$	0	0
$619$	− 40.7494i	− 1.63786i	−0.573896	−	0.818929i	$-0.694568\pi$
$619$	− 40.7494i	− 1.63786i	0.573896	−	0.818929i	$-0.305432\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−47.8115	−1.91553
$624$	0	0
$625$	19.8313	0.793250
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7.29874i	0.291020i
$630$	0	0
$631$	−42.3804	−1.68714	−0.843569	−	0.537021i	$-0.819550\pi$
$631$	−42.3804	−1.68714	−0.843569	+	0.537021i	$0.819550\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9.58005i	0.380173i
$636$	0	0
$637$	1.11315i	0.0441045i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	40.0356	1.58131	0.790656	−	0.612261i	$-0.209740\pi$
$641$	40.0356	1.58131	0.790656	+	0.612261i	$0.209740\pi$
$642$	0	0
$643$	− 2.25369i	− 0.0888767i	−0.999012	−	0.0444384i	$-0.985850\pi$
$643$	− 2.25369i	− 0.0888767i	0.999012	−	0.0444384i	$-0.0141498\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.89272	−0.270981	−0.135490	−	0.990779i	$-0.543261\pi$
$647$	−6.89272	−0.270981	−0.135490	+	0.990779i	$0.543261\pi$
$648$	0	0
$649$	13.3060	0.522307
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.75398i	0.225170i	0.993642	+	0.112585i	$0.0359131\pi$
$653$	5.75398i	0.225170i	−0.993642	+	0.112585i	$0.964087\pi$
$654$	0	0
$655$	−6.62222	−0.258751
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 34.1024i	− 1.32844i	−0.747537	−	0.664220i	$-0.768764\pi$
$659$	− 34.1024i	− 1.32844i	0.747537	−	0.664220i	$-0.231236\pi$
$660$	0	0
$661$	− 46.3767i	− 1.80384i	−0.431900	−	0.901922i	$-0.642157\pi$
$661$	− 46.3767i	− 1.80384i	0.431900	−	0.901922i	$-0.357843\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.07176	0.0803392
$666$	0	0
$667$	− 42.6141i	− 1.65002i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−37.5533	−1.44973
$672$	0	0
$673$	−29.1579	−1.12395	−0.561977	−	0.827153i	$-0.689959\pi$
$673$	−29.1579	−1.12395	−0.561977	+	0.827153i	$0.689959\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 10.0436i	− 0.386006i	−0.981198	−	0.193003i	$-0.938177\pi$
$677$	− 10.0436i	− 0.386006i	0.981198	−	0.193003i	$-0.0618228\pi$
$678$	0	0
$679$	7.28365	0.279521
$680$	0	0
$681$	0	0
$682$	0	0
$683$	25.5866i	0.979042i	0.871991	+	0.489521i	$0.162828\pi$
$683$	25.5866i	0.979042i	−0.871991	+	0.489521i	$0.837172\pi$
$684$	0	0
$685$	− 6.63341i	− 0.253449i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−0.552809	−0.0210604
$690$	0	0
$691$	42.2386i	1.60683i	0.595417	+	0.803417i	$0.296987\pi$
$691$	42.2386i	1.60683i	−0.595417	+	0.803417i	$0.703013\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.75428	0.294136
$696$	0	0
$697$	−0.583768	−0.0221118
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 43.9811i	− 1.66114i	−0.556911	−	0.830572i	$-0.688014\pi$
$701$	− 43.9811i	− 1.66114i	0.556911	−	0.830572i	$-0.311986\pi$
$702$	0	0
$703$	−5.62653	−0.212208
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.66314i	0.363420i
$708$	0	0
$709$	11.1307i	0.418023i	0.977913	+	0.209012i	$0.0670246\pi$
$709$	11.1307i	0.418023i	−0.977913	+	0.209012i	$0.932975\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.52575	0.206941
$714$	0	0
$715$	− 0.619259i	− 0.0231590i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	17.8961	0.667413	0.333706	−	0.942677i	$-0.391701\pi$
$719$	17.8961	0.667413	0.333706	+	0.942677i	$0.391701\pi$
$720$	0	0
$721$	49.9989	1.86206
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 43.7337i	− 1.62423i
$726$	0	0
$727$	35.8772	1.33061	0.665306	−	0.746571i	$-0.268301\pi$
$727$	35.8772	1.33061	0.665306	+	0.746571i	$0.268301\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1.03044i	− 0.0381123i
$732$	0	0
$733$	− 24.4618i	− 0.903516i	−0.892141	−	0.451758i	$-0.850797\pi$
$733$	− 24.4618i	− 0.903516i	0.892141	−	0.451758i	$-0.149203\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−19.9032	−0.733142
$738$	0	0
$739$	26.7820i	0.985193i	0.870258	+	0.492596i	$0.163952\pi$
$739$	26.7820i	0.985193i	−0.870258	+	0.492596i	$0.836048\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	23.7835	0.872534	0.436267	−	0.899817i	$-0.356300\pi$
$743$	23.7835	0.872534	0.436267	+	0.899817i	$0.356300\pi$
$744$	0	0
$745$	−8.54295	−0.312990
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 8.45344i	− 0.308882i
$750$	0	0
$751$	−22.4303	−0.818494	−0.409247	−	0.912424i	$-0.634209\pi$
$751$	−22.4303	−0.818494	−0.409247	+	0.912424i	$0.634209\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 10.3060i	− 0.375073i
$756$	0	0
$757$	− 32.2421i	− 1.17186i	−0.810362	−	0.585930i	$-0.800729\pi$
$757$	− 32.2421i	− 1.17186i	0.810362	−	0.585930i	$-0.199271\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−16.3559	−0.592902	−0.296451	−	0.955048i	$-0.595803\pi$
$761$	−16.3559	−0.592902	−0.296451	+	0.955048i	$0.595803\pi$
$762$	0	0
$763$	0.00431797i	0 0.000156321i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.593302	0.0214229
$768$	0	0
$769$	−19.6832	−0.709794	−0.354897	−	0.934905i	$-0.615484\pi$
$769$	−19.6832	−0.709794	−0.354897	+	0.934905i	$0.615484\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	33.0819i	1.18987i	0.803773	+	0.594936i	$0.202823\pi$
$773$	33.0819i	1.18987i	−0.803773	+	0.594936i	$0.797177\pi$
$774$	0	0
$775$	5.67093	0.203706
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 0.450021i	− 0.0161237i
$780$	0	0
$781$	− 37.8494i	− 1.35436i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	5.72235	0.204240
$786$	0	0
$787$	− 35.1617i	− 1.25338i	−0.779269	−	0.626690i	$-0.784409\pi$
$787$	− 35.1617i	− 1.25338i	0.779269	−	0.626690i	$-0.215591\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.33392	−0.225208
$792$	0	0
$793$	−1.67446	−0.0594619
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 3.75101i	− 0.132868i	−0.997791	−	0.0664338i	$-0.978838\pi$
$797$	− 3.75101i	− 0.132868i	0.997791	−	0.0664338i	$-0.0211621\pi$
$798$	0	0
$799$	−15.8241	−0.559815
$800$	0	0
$801$	0	0
$802$	0	0
$803$	14.9374i	0.527131i
$804$	0	0
$805$	9.38121i	0.330644i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−23.9158	−0.840836	−0.420418	−	0.907331i	$-0.638117\pi$
$809$	−23.9158	−0.840836	−0.420418	+	0.907331i	$0.638117\pi$
$810$	0	0
$811$	− 52.0538i	− 1.82785i	−0.405878	−	0.913927i	$-0.633034\pi$
$811$	− 52.0538i	− 1.82785i	0.405878	−	0.913927i	$-0.366966\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−13.3876	−0.468947
$816$	0	0
$817$	0.794359	0.0277911
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 0.703590i	− 0.0245555i	−0.999925	−	0.0122777i	$-0.996092\pi$
$821$	− 0.703590i	− 0.0245555i	0.999925	−	0.0122777i	$-0.00390822\pi$
$822$	0	0
$823$	−23.4559	−0.817622	−0.408811	−	0.912619i	$-0.634057\pi$
$823$	−23.4559	−0.817622	−0.408811	+	0.912619i	$0.634057\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 29.5282i	− 1.02680i	−0.858150	−	0.513399i	$-0.828386\pi$
$827$	− 29.5282i	− 1.02680i	0.858150	−	0.513399i	$-0.171614\pi$
$828$	0	0
$829$	− 4.42946i	− 0.153841i	−0.997037	−	0.0769207i	$-0.975491\pi$
$829$	− 4.42946i	− 0.153841i	0.997037	−	0.0769207i	$-0.0245088\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.69759	−0.232058
$834$	0	0
$835$	9.80811i	0.339424i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−36.4506	−1.25841	−0.629207	−	0.777237i	$-0.716620\pi$
$839$	−36.4506	−1.25841	−0.629207	+	0.777237i	$0.716620\pi$
$840$	0	0
$841$	−59.5656	−2.05399
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7.69492i	0.264713i
$846$	0	0
$847$	−43.1730	−1.48344
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 25.4777i	− 0.873366i
$852$	0	0
$853$	− 52.7043i	− 1.80456i	−0.431149	−	0.902281i	$-0.641892\pi$
$853$	− 52.7043i	− 1.80456i	0.431149	−	0.902281i	$-0.358108\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−13.7199	−0.468662	−0.234331	−	0.972157i	$-0.575290\pi$
$857$	−13.7199	−0.468662	−0.234331	+	0.972157i	$0.575290\pi$
$858$	0	0
$859$	55.0757i	1.87916i	0.342330	+	0.939580i	$0.388784\pi$
$859$	55.0757i	1.87916i	−0.342330	+	0.939580i	$0.611216\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38.3384	1.30505	0.652527	−	0.757766i	$-0.273709\pi$
$863$	38.3384	1.30505	0.652527	+	0.757766i	$0.273709\pi$
$864$	0	0
$865$	3.05878	0.104002
$866$	0	0
$867$	0	0
$868$	0	0
$869$	48.7053i	1.65221i
$870$	0	0
$871$	−0.887462	−0.0300705
$872$	0	0
$873$	0	0
$874$	0	0
$875$	19.9865i	0.675666i
$876$	0	0
$877$	21.4788i	0.725288i	0.931928	+	0.362644i	$0.118126\pi$
$877$	21.4788i	0.725288i	−0.931928	+	0.362644i	$0.881874\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	11.2383	0.378629	0.189314	−	0.981917i	$-0.439373\pi$
$881$	11.2383	0.378629	0.189314	+	0.981917i	$0.439373\pi$
$882$	0	0
$883$	− 15.7137i	− 0.528809i	−0.964412	−	0.264405i	$-0.914825\pi$
$883$	− 15.7137i	− 0.528809i	0.964412	−	0.264405i	$-0.0851755\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.48008	0.150426	0.0752131	−	0.997167i	$-0.476036\pi$
$887$	4.48008	0.150426	0.0752131	+	0.997167i	$0.476036\pi$
$888$	0	0
$889$	56.2437	1.88635
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 12.1986i	− 0.408211i
$894$	0	0
$895$	5.53396	0.184980
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 11.4843i	− 0.383022i
$900$	0	0
$901$	− 3.32614i	− 0.110810i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.06238	−0.101797
$906$	0	0
$907$	6.37021i	0.211519i	0.994392	+	0.105760i	$0.0337274\pi$
$907$	6.37021i	0.211519i	−0.994392	+	0.105760i	$0.966273\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−30.5275	−1.01142	−0.505711	−	0.862703i	$-0.668770\pi$
$911$	−30.5275	−1.01142	−0.505711	+	0.862703i	$0.668770\pi$
$912$	0	0
$913$	−56.2312	−1.86098
$914$	0	0
$915$	0	0
$916$	0	0
$917$	38.8785i	1.28388i
$918$	0	0
$919$	48.0650	1.58552	0.792758	−	0.609536i	$-0.208644\pi$
$919$	48.0650	1.58552	0.792758	+	0.609536i	$0.208644\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1.68767i	− 0.0555502i
$924$	0	0
$925$	− 26.1471i	− 0.859712i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−47.2953	−1.55171	−0.775854	−	0.630912i	$-0.782681\pi$
$929$	−47.2953	−1.55171	−0.775854	+	0.630912i	$0.782681\pi$
$930$	0	0
$931$	− 5.16310i	− 0.169214i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.72596	0.121852
$936$	0	0
$937$	−8.30237	−0.271227	−0.135613	−	0.990762i	$-0.543301\pi$
$937$	−8.30237	−0.271227	−0.135613	+	0.990762i	$0.543301\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	31.3955i	1.02347i	0.859145	+	0.511733i	$0.170996\pi$
$941$	31.3955i	1.02347i	−0.859145	+	0.511733i	$0.829004\pi$
$942$	0	0
$943$	2.03776	0.0663586
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.14805i	0.0698024i	0.999391	+	0.0349012i	$0.0111117\pi$
$947$	2.14805i	0.0698024i	−0.999391	+	0.0349012i	$0.988888\pi$
$948$	0	0
$949$	0.666046i	0.0216208i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−25.3661	−0.821689	−0.410844	−	0.911705i	$-0.634766\pi$
$953$	−25.3661	−0.821689	−0.410844	+	0.911705i	$0.634766\pi$
$954$	0	0
$955$	− 8.83965i	− 0.286044i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−38.9442	−1.25757
$960$	0	0
$961$	−29.5108	−0.951963
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 12.6385i	− 0.406847i
$966$	0	0
$967$	12.8395	0.412892	0.206446	−	0.978458i	$-0.433810\pi$
$967$	12.8395	0.412892	0.206446	+	0.978458i	$0.433810\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8.77909i	0.281734i	0.990028	+	0.140867i	$0.0449891\pi$
$971$	8.77909i	0.281734i	−0.990028	+	0.140867i	$0.955011\pi$
$972$	0	0
$973$	− 45.5247i	− 1.45945i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	45.9701	1.47071	0.735357	−	0.677680i	$-0.237014\pi$
$977$	45.9701	1.47071	0.735357	+	0.677680i	$0.237014\pi$
$978$	0	0
$979$	− 66.2864i	− 2.11852i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−51.4992	−1.64257	−0.821284	−	0.570519i	$-0.806742\pi$
$983$	−51.4992	−1.64257	−0.821284	+	0.570519i	$0.806742\pi$
$984$	0	0
$985$	2.19302	0.0698753
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.59697i	0.114377i
$990$	0	0
$991$	29.4762	0.936343	0.468171	−	0.883638i	$-0.344913\pi$
$991$	29.4762	0.936343	0.468171	+	0.883638i	$0.344913\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 6.94178i	− 0.220069i
$996$	0	0
$997$	54.3591i	1.72157i	0.508968	+	0.860786i	$0.330027\pi$
$997$	54.3591i	1.72157i	−0.508968	+	0.860786i	$0.669973\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.g.b.2737.9		16
3.2	odd	2		608.2.c.b.305.1		16
4.3	odd	2		1368.2.g.b.685.12		16
8.3	odd	2		1368.2.g.b.685.11		16
8.5	even	2	inner	5472.2.g.b.2737.8		16
12.11	even	2		152.2.c.b.77.5	✓	16
24.5	odd	2		608.2.c.b.305.16		16
24.11	even	2		152.2.c.b.77.6	yes	16
48.5	odd	4		4864.2.a.bp.1.1		8
48.11	even	4		4864.2.a.bq.1.8		8
48.29	odd	4		4864.2.a.bn.1.8		8
48.35	even	4		4864.2.a.bo.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.c.b.77.5	✓	16	12.11	even	2
152.2.c.b.77.6	yes	16	24.11	even	2
608.2.c.b.305.1		16	3.2	odd	2
608.2.c.b.305.16		16	24.5	odd	2
1368.2.g.b.685.11		16	8.3	odd	2
1368.2.g.b.685.12		16	4.3	odd	2
4864.2.a.bn.1.8		8	48.29	odd	4
4864.2.a.bo.1.1		8	48.35	even	4
4864.2.a.bp.1.1		8	48.5	odd	4
4864.2.a.bq.1.8		8	48.11	even	4
5472.2.g.b.2737.8		16	8.5	even	2	inner
5472.2.g.b.2737.9		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.594041i q^{5} +3.48756 q^{7} +O(q^{10})\)	\(q+0.594041i q^{5} +3.48756 q^{7} +4.83520i q^{11} +0.215597i q^{13} -1.29720 q^{17} -1.00000i q^{19} +4.52815 q^{23} +4.64712 q^{25} -9.41093i q^{29} +1.22031 q^{31} +2.07176i q^{35} -5.62653i q^{37} +0.450021 q^{41} +0.794359i q^{43} +12.1986 q^{47} +5.16310 q^{49} +2.56409i q^{53} -2.87231 q^{55} -2.75191i q^{59} +7.76665i q^{61} -0.128073 q^{65} +4.11631i q^{67} -7.82788 q^{71} +3.08931 q^{73} +16.8631i q^{77} +10.0731 q^{79} +11.6296i q^{83} -0.770591i q^{85} -13.7091 q^{89} +0.751907i q^{91} +0.594041 q^{95} +2.08846 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{7}+O(q^{10}) \) 16 * q + 8 * q^7	\( 16 q + 8 q^{7} + 8 q^{17} - 24 q^{25} - 16 q^{31} - 16 q^{41} + 24 q^{47} + 24 q^{49} - 16 q^{55} - 16 q^{65} + 48 q^{71} + 48 q^{79} + 16 q^{89} + 16 q^{95} + 32 q^{97}+O(q^{100}) \) 16 * q + 8 * q^7 + 8 * q^17 - 24 * q^25 - 16 * q^31 - 16 * q^41 + 24 * q^47 + 24 * q^49 - 16 * q^55 - 16 * q^65 + 48 * q^71 + 48 * q^79 + 16 * q^89 + 16 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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