LMFDB - Embedded newform 3360.2.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3360,2,Mod(1,3360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("3360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.a (trivial)

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:	yes
Analytic conductor:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	3360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.41855 q^{11} +3.26180 q^{13} +1.00000 q^{15} -6.68035 q^{17} +7.41855 q^{19} +1.00000 q^{21} -2.15676 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.68035 q^{29} +9.57531 q^{31} -3.41855 q^{33} +1.00000 q^{35} +6.68035 q^{37} +3.26180 q^{39} +8.83710 q^{41} +8.68035 q^{43} +1.00000 q^{45} +2.15676 q^{47} +1.00000 q^{49} -6.68035 q^{51} +1.41855 q^{53} -3.41855 q^{55} +7.41855 q^{57} +8.00000 q^{59} +0.156755 q^{61} +1.00000 q^{63} +3.26180 q^{65} -15.5174 q^{67} -2.15676 q^{69} -1.26180 q^{71} +0.738205 q^{73} +1.00000 q^{75} -3.41855 q^{77} +4.00000 q^{79} +1.00000 q^{81} +10.8371 q^{83} -6.68035 q^{85} -6.68035 q^{87} +4.52359 q^{89} +3.26180 q^{91} +9.57531 q^{93} +7.41855 q^{95} +3.26180 q^{97} -3.41855 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9} + 4 q^{11} + 2 q^{13} + 3 q^{15} + 2 q^{17} + 8 q^{19} + 3 q^{21} + 3 q^{25} + 3 q^{27} + 2 q^{29} + 8 q^{31} + 4 q^{33} + 3 q^{35} - 2 q^{37} + 2 q^{39} - 2 q^{41} + 4 q^{43} + 3 q^{45} + 3 q^{49} + 2 q^{51} - 10 q^{53} + 4 q^{55} + 8 q^{57} + 24 q^{59} - 6 q^{61} + 3 q^{63} + 2 q^{65} + 4 q^{67} + 4 q^{71} + 10 q^{73} + 3 q^{75} + 4 q^{77} + 12 q^{79} + 3 q^{81} + 4 q^{83} + 2 q^{85} + 2 q^{87} - 2 q^{89} + 2 q^{91} + 8 q^{93} + 8 q^{95} + 2 q^{97} + 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^7 + 3 * q^9 + 4 * q^11 + 2 * q^13 + 3 * q^15 + 2 * q^17 + 8 * q^19 + 3 * q^21 + 3 * q^25 + 3 * q^27 + 2 * q^29 + 8 * q^31 + 4 * q^33 + 3 * q^35 - 2 * q^37 + 2 * q^39 - 2 * q^41 + 4 * q^43 + 3 * q^45 + 3 * q^49 + 2 * q^51 - 10 * q^53 + 4 * q^55 + 8 * q^57 + 24 * q^59 - 6 * q^61 + 3 * q^63 + 2 * q^65 + 4 * q^67 + 4 * q^71 + 10 * q^73 + 3 * q^75 + 4 * q^77 + 12 * q^79 + 3 * q^81 + 4 * q^83 + 2 * q^85 + 2 * q^87 - 2 * q^89 + 2 * q^91 + 8 * q^93 + 8 * q^95 + 2 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.41855	−1.03073	−0.515366	−	0.856970i	$-0.672344\pi$
$11$	−3.41855	−1.03073	−0.515366	+	0.856970i	$0.672344\pi$
$12$	0	0
$13$	3.26180	0.904659	0.452330	−	0.891851i	$-0.350593\pi$
$13$	3.26180	0.904659	0.452330	+	0.891851i	$0.350593\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−6.68035	−1.62022	−0.810111	−	0.586277i	$-0.800593\pi$
$17$	−6.68035	−1.62022	−0.810111	+	0.586277i	$0.800593\pi$
$18$	0	0
$19$	7.41855	1.70193	0.850966	−	0.525221i	$-0.176017\pi$
$19$	7.41855	1.70193	0.850966	+	0.525221i	$0.176017\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−2.15676	−0.449715	−0.224857	−	0.974392i	$-0.572192\pi$
$23$	−2.15676	−0.449715	−0.224857	+	0.974392i	$0.572192\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.68035	−1.24051	−0.620255	−	0.784401i	$-0.712971\pi$
$29$	−6.68035	−1.24051	−0.620255	+	0.784401i	$0.712971\pi$
$30$	0	0
$31$	9.57531	1.71978	0.859888	−	0.510483i	$-0.170533\pi$
$31$	9.57531	1.71978	0.859888	+	0.510483i	$0.170533\pi$
$32$	0	0
$33$	−3.41855	−0.595093
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	6.68035	1.09824	0.549121	−	0.835743i	$-0.314963\pi$
$37$	6.68035	1.09824	0.549121	+	0.835743i	$0.314963\pi$
$38$	0	0
$39$	3.26180	0.522305
$40$	0	0
$41$	8.83710	1.38012	0.690062	−	0.723751i	$-0.257583\pi$
$41$	8.83710	1.38012	0.690062	+	0.723751i	$0.257583\pi$
$42$	0	0
$43$	8.68035	1.32374	0.661870	−	0.749618i	$-0.269763\pi$
$43$	8.68035	1.32374	0.661870	+	0.749618i	$0.269763\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	2.15676	0.314595	0.157298	−	0.987551i	$-0.449722\pi$
$47$	2.15676	0.314595	0.157298	+	0.987551i	$0.449722\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.68035	−0.935436
$52$	0	0
$53$	1.41855	0.194853	0.0974265	−	0.995243i	$-0.468939\pi$
$53$	1.41855	0.194853	0.0974265	+	0.995243i	$0.468939\pi$
$54$	0	0
$55$	−3.41855	−0.460957
$56$	0	0
$57$	7.41855	0.982611
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	0.156755	0.0200705	0.0100352	−	0.999950i	$-0.496806\pi$
$61$	0.156755	0.0200705	0.0100352	+	0.999950i	$0.496806\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	3.26180	0.404576
$66$	0	0
$67$	−15.5174	−1.89576	−0.947879	−	0.318631i	$-0.896777\pi$
$67$	−15.5174	−1.89576	−0.947879	+	0.318631i	$0.896777\pi$
$68$	0	0
$69$	−2.15676	−0.259643
$70$	0	0
$71$	−1.26180	−0.149748	−0.0748738	−	0.997193i	$-0.523855\pi$
$71$	−1.26180	−0.149748	−0.0748738	+	0.997193i	$0.523855\pi$
$72$	0	0
$73$	0.738205	0.0864003	0.0432002	−	0.999066i	$-0.486245\pi$
$73$	0.738205	0.0864003	0.0432002	+	0.999066i	$0.486245\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−3.41855	−0.389580
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.8371	1.18953	0.594763	−	0.803901i	$-0.297246\pi$
$83$	10.8371	1.18953	0.594763	+	0.803901i	$0.297246\pi$
$84$	0	0
$85$	−6.68035	−0.724585
$86$	0	0
$87$	−6.68035	−0.716208
$88$	0	0
$89$	4.52359	0.479500	0.239750	−	0.970835i	$-0.422935\pi$
$89$	4.52359	0.479500	0.239750	+	0.970835i	$0.422935\pi$
$90$	0	0
$91$	3.26180	0.341929
$92$	0	0
$93$	9.57531	0.992913
$94$	0	0
$95$	7.41855	0.761127
$96$	0	0
$97$	3.26180	0.331185	0.165593	−	0.986194i	$-0.447046\pi$
$97$	3.26180	0.331185	0.165593	+	0.986194i	$0.447046\pi$
$98$	0	0
$99$	−3.41855	−0.343577
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	6.15676	0.595196	0.297598	−	0.954691i	$-0.403814\pi$
$107$	6.15676	0.595196	0.297598	+	0.954691i	$0.403814\pi$
$108$	0	0
$109$	−19.3607	−1.85442	−0.927209	−	0.374544i	$-0.877799\pi$
$109$	−19.3607	−1.85442	−0.927209	+	0.374544i	$0.877799\pi$
$110$	0	0
$111$	6.68035	0.634070
$112$	0	0
$113$	16.2557	1.52920	0.764602	−	0.644503i	$-0.222936\pi$
$113$	16.2557	1.52920	0.764602	+	0.644503i	$0.222936\pi$
$114$	0	0
$115$	−2.15676	−0.201118
$116$	0	0
$117$	3.26180	0.301553
$118$	0	0
$119$	−6.68035	−0.612386
$120$	0	0
$121$	0.686489	0.0624081
$122$	0	0
$123$	8.83710	0.796815
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.5236	0.933818	0.466909	−	0.884305i	$-0.345368\pi$
$127$	10.5236	0.933818	0.466909	+	0.884305i	$0.345368\pi$
$128$	0	0
$129$	8.68035	0.764262
$130$	0	0
$131$	10.5236	0.919450	0.459725	−	0.888061i	$-0.347948\pi$
$131$	10.5236	0.919450	0.459725	+	0.888061i	$0.347948\pi$
$132$	0	0
$133$	7.41855	0.643270
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−15.9421	−1.36203	−0.681015	−	0.732270i	$-0.738461\pi$
$137$	−15.9421	−1.36203	−0.681015	+	0.732270i	$0.738461\pi$
$138$	0	0
$139$	15.4186	1.30778	0.653892	−	0.756588i	$-0.273135\pi$
$139$	15.4186	1.30778	0.653892	+	0.756588i	$0.273135\pi$
$140$	0	0
$141$	2.15676	0.181632
$142$	0	0
$143$	−11.1506	−0.932461
$144$	0	0
$145$	−6.68035	−0.554773
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−12.1568	−0.995920	−0.497960	−	0.867200i	$-0.665917\pi$
$149$	−12.1568	−0.995920	−0.497960	+	0.867200i	$0.665917\pi$
$150$	0	0
$151$	−2.83710	−0.230880	−0.115440	−	0.993314i	$-0.536828\pi$
$151$	−2.83710	−0.230880	−0.115440	+	0.993314i	$0.536828\pi$
$152$	0	0
$153$	−6.68035	−0.540074
$154$	0	0
$155$	9.57531	0.769107
$156$	0	0
$157$	−7.26180	−0.579554	−0.289777	−	0.957094i	$-0.593581\pi$
$157$	−7.26180	−0.579554	−0.289777	+	0.957094i	$0.593581\pi$
$158$	0	0
$159$	1.41855	0.112498
$160$	0	0
$161$	−2.15676	−0.169976
$162$	0	0
$163$	−16.6803	−1.30651	−0.653253	−	0.757140i	$-0.726596\pi$
$163$	−16.6803	−1.30651	−0.653253	+	0.757140i	$0.726596\pi$
$164$	0	0
$165$	−3.41855	−0.266134
$166$	0	0
$167$	−3.31965	−0.256883	−0.128441	−	0.991717i	$-0.540997\pi$
$167$	−3.31965	−0.256883	−0.128441	+	0.991717i	$0.540997\pi$
$168$	0	0
$169$	−2.36069	−0.181592
$170$	0	0
$171$	7.41855	0.567311
$172$	0	0
$173$	−8.83710	−0.671872	−0.335936	−	0.941885i	$-0.609053\pi$
$173$	−8.83710	−0.671872	−0.335936	+	0.941885i	$0.609053\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	12.5814	0.940382	0.470191	−	0.882565i	$-0.344185\pi$
$179$	12.5814	0.940382	0.470191	+	0.882565i	$0.344185\pi$
$180$	0	0
$181$	−22.6803	−1.68582	−0.842908	−	0.538057i	$-0.819158\pi$
$181$	−22.6803	−1.68582	−0.842908	+	0.538057i	$0.819158\pi$
$182$	0	0
$183$	0.156755	0.0115877
$184$	0	0
$185$	6.68035	0.491149
$186$	0	0
$187$	22.8371	1.67001
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	1.26180	0.0913003	0.0456501	−	0.998957i	$-0.485464\pi$
$191$	1.26180	0.0913003	0.0456501	+	0.998957i	$0.485464\pi$
$192$	0	0
$193$	19.3607	1.39361	0.696807	−	0.717259i	$-0.254604\pi$
$193$	19.3607	1.39361	0.696807	+	0.717259i	$0.254604\pi$
$194$	0	0
$195$	3.26180	0.233582
$196$	0	0
$197$	3.94214	0.280866	0.140433	−	0.990090i	$-0.455151\pi$
$197$	3.94214	0.280866	0.140433	+	0.990090i	$0.455151\pi$
$198$	0	0
$199$	3.90110	0.276542	0.138271	−	0.990394i	$-0.455846\pi$
$199$	3.90110	0.276542	0.138271	+	0.990394i	$0.455846\pi$
$200$	0	0
$201$	−15.5174	−1.09452
$202$	0	0
$203$	−6.68035	−0.468868
$204$	0	0
$205$	8.83710	0.617210
$206$	0	0
$207$	−2.15676	−0.149905
$208$	0	0
$209$	−25.3607	−1.75424
$210$	0	0
$211$	0.313511	0.0215830	0.0107915	−	0.999942i	$-0.496565\pi$
$211$	0.313511	0.0215830	0.0107915	+	0.999942i	$0.496565\pi$
$212$	0	0
$213$	−1.26180	−0.0864568
$214$	0	0
$215$	8.68035	0.591995
$216$	0	0
$217$	9.57531	0.650014
$218$	0	0
$219$	0.738205	0.0498833
$220$	0	0
$221$	−21.7899	−1.46575
$222$	0	0
$223$	−19.8843	−1.33155	−0.665775	−	0.746153i	$-0.731899\pi$
$223$	−19.8843	−1.33155	−0.665775	+	0.746153i	$0.731899\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	23.8843	1.58526	0.792628	−	0.609706i	$-0.208712\pi$
$227$	23.8843	1.58526	0.792628	+	0.609706i	$0.208712\pi$
$228$	0	0
$229$	26.8781	1.77616	0.888079	−	0.459691i	$-0.152040\pi$
$229$	26.8781	1.77616	0.888079	+	0.459691i	$0.152040\pi$
$230$	0	0
$231$	−3.41855	−0.224924
$232$	0	0
$233$	−22.7792	−1.49232	−0.746159	−	0.665768i	$-0.768104\pi$
$233$	−22.7792	−1.49232	−0.746159	+	0.665768i	$0.768104\pi$
$234$	0	0
$235$	2.15676	0.140691
$236$	0	0
$237$	4.00000	0.259828
$238$	0	0
$239$	−1.26180	−0.0816187	−0.0408094	−	0.999167i	$-0.512994\pi$
$239$	−1.26180	−0.0816187	−0.0408094	+	0.999167i	$0.512994\pi$
$240$	0	0
$241$	−4.83710	−0.311585	−0.155793	−	0.987790i	$-0.549793\pi$
$241$	−4.83710	−0.311585	−0.155793	+	0.987790i	$0.549793\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	24.1978	1.53967
$248$	0	0
$249$	10.8371	0.686773
$250$	0	0
$251$	−17.3607	−1.09580	−0.547899	−	0.836545i	$-0.684572\pi$
$251$	−17.3607	−1.09580	−0.547899	+	0.836545i	$0.684572\pi$
$252$	0	0
$253$	7.37298	0.463535
$254$	0	0
$255$	−6.68035	−0.418339
$256$	0	0
$257$	25.5174	1.59173	0.795867	−	0.605471i	$-0.207015\pi$
$257$	25.5174	1.59173	0.795867	+	0.605471i	$0.207015\pi$
$258$	0	0
$259$	6.68035	0.415096
$260$	0	0
$261$	−6.68035	−0.413503
$262$	0	0
$263$	21.3074	1.31387	0.656934	−	0.753948i	$-0.271853\pi$
$263$	21.3074	1.31387	0.656934	+	0.753948i	$0.271853\pi$
$264$	0	0
$265$	1.41855	0.0871409
$266$	0	0
$267$	4.52359	0.276839
$268$	0	0
$269$	21.0349	1.28252	0.641260	−	0.767324i	$-0.278412\pi$
$269$	21.0349	1.28252	0.641260	+	0.767324i	$0.278412\pi$
$270$	0	0
$271$	−1.57531	−0.0956930	−0.0478465	−	0.998855i	$-0.515236\pi$
$271$	−1.57531	−0.0956930	−0.0478465	+	0.998855i	$0.515236\pi$
$272$	0	0
$273$	3.26180	0.197413
$274$	0	0
$275$	−3.41855	−0.206146
$276$	0	0
$277$	−26.6803	−1.60307	−0.801533	−	0.597950i	$-0.795982\pi$
$277$	−26.6803	−1.60307	−0.801533	+	0.597950i	$0.795982\pi$
$278$	0	0
$279$	9.57531	0.573259
$280$	0	0
$281$	8.83710	0.527177	0.263589	−	0.964635i	$-0.415094\pi$
$281$	8.83710	0.527177	0.263589	+	0.964635i	$0.415094\pi$
$282$	0	0
$283$	−25.6742	−1.52617	−0.763086	−	0.646296i	$-0.776317\pi$
$283$	−25.6742	−1.52617	−0.763086	+	0.646296i	$0.776317\pi$
$284$	0	0
$285$	7.41855	0.439437
$286$	0	0
$287$	8.83710	0.521638
$288$	0	0
$289$	27.6270	1.62512
$290$	0	0
$291$	3.26180	0.191210
$292$	0	0
$293$	−20.5236	−1.19900	−0.599500	−	0.800374i	$-0.704634\pi$
$293$	−20.5236	−1.19900	−0.599500	+	0.800374i	$0.704634\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	−3.41855	−0.198364
$298$	0	0
$299$	−7.03489	−0.406838
$300$	0	0
$301$	8.68035	0.500327
$302$	0	0
$303$	−10.0000	−0.574485
$304$	0	0
$305$	0.156755	0.00897579
$306$	0	0
$307$	−28.1978	−1.60933	−0.804667	−	0.593727i	$-0.797656\pi$
$307$	−28.1978	−1.60933	−0.804667	+	0.593727i	$0.797656\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.6742	0.775393	0.387696	−	0.921787i	$-0.373271\pi$
$311$	13.6742	0.775393	0.387696	+	0.921787i	$0.373271\pi$
$312$	0	0
$313$	23.5753	1.33256	0.666278	−	0.745704i	$-0.267887\pi$
$313$	23.5753	1.33256	0.666278	+	0.745704i	$0.267887\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	−11.6286	−0.653129	−0.326564	−	0.945175i	$-0.605891\pi$
$317$	−11.6286	−0.653129	−0.326564	+	0.945175i	$0.605891\pi$
$318$	0	0
$319$	22.8371	1.27863
$320$	0	0
$321$	6.15676	0.343637
$322$	0	0
$323$	−49.5585	−2.75751
$324$	0	0
$325$	3.26180	0.180932
$326$	0	0
$327$	−19.3607	−1.07065
$328$	0	0
$329$	2.15676	0.118906
$330$	0	0
$331$	−24.3135	−1.33639	−0.668196	−	0.743986i	$-0.732933\pi$
$331$	−24.3135	−1.33639	−0.668196	+	0.743986i	$0.732933\pi$
$332$	0	0
$333$	6.68035	0.366081
$334$	0	0
$335$	−15.5174	−0.847809
$336$	0	0
$337$	5.68649	0.309763	0.154881	−	0.987933i	$-0.450500\pi$
$337$	5.68649	0.309763	0.154881	+	0.987933i	$0.450500\pi$
$338$	0	0
$339$	16.2557	0.882886
$340$	0	0
$341$	−32.7337	−1.77263
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.15676	−0.116116
$346$	0	0
$347$	8.68035	0.465985	0.232993	−	0.972478i	$-0.425148\pi$
$347$	8.68035	0.465985	0.232993	+	0.972478i	$0.425148\pi$
$348$	0	0
$349$	4.47027	0.239288	0.119644	−	0.992817i	$-0.461825\pi$
$349$	4.47027	0.239288	0.119644	+	0.992817i	$0.461825\pi$
$350$	0	0
$351$	3.26180	0.174102
$352$	0	0
$353$	−32.6681	−1.73875	−0.869373	−	0.494157i	$-0.835477\pi$
$353$	−32.6681	−1.73875	−0.869373	+	0.494157i	$0.835477\pi$
$354$	0	0
$355$	−1.26180	−0.0669691
$356$	0	0
$357$	−6.68035	−0.353561
$358$	0	0
$359$	4.94828	0.261160	0.130580	−	0.991438i	$-0.458316\pi$
$359$	4.94828	0.261160	0.130580	+	0.991438i	$0.458316\pi$
$360$	0	0
$361$	36.0349	1.89657
$362$	0	0
$363$	0.686489	0.0360313
$364$	0	0
$365$	0.738205	0.0386394
$366$	0	0
$367$	13.4764	0.703463	0.351731	−	0.936101i	$-0.385593\pi$
$367$	13.4764	0.703463	0.351731	+	0.936101i	$0.385593\pi$
$368$	0	0
$369$	8.83710	0.460041
$370$	0	0
$371$	1.41855	0.0736475
$372$	0	0
$373$	−31.1917	−1.61504	−0.807521	−	0.589839i	$-0.799191\pi$
$373$	−31.1917	−1.61504	−0.807521	+	0.589839i	$0.799191\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−21.7899	−1.12224
$378$	0	0
$379$	10.2101	0.524457	0.262228	−	0.965006i	$-0.415543\pi$
$379$	10.2101	0.524457	0.262228	+	0.965006i	$0.415543\pi$
$380$	0	0
$381$	10.5236	0.539140
$382$	0	0
$383$	10.3545	0.529093	0.264546	−	0.964373i	$-0.414778\pi$
$383$	10.3545	0.529093	0.264546	+	0.964373i	$0.414778\pi$
$384$	0	0
$385$	−3.41855	−0.174225
$386$	0	0
$387$	8.68035	0.441247
$388$	0	0
$389$	−23.8432	−1.20890	−0.604450	−	0.796643i	$-0.706607\pi$
$389$	−23.8432	−1.20890	−0.604450	+	0.796643i	$0.706607\pi$
$390$	0	0
$391$	14.4079	0.728637
$392$	0	0
$393$	10.5236	0.530845
$394$	0	0
$395$	4.00000	0.201262
$396$	0	0
$397$	−30.7259	−1.54209	−0.771045	−	0.636781i	$-0.780266\pi$
$397$	−30.7259	−1.54209	−0.771045	+	0.636781i	$0.780266\pi$
$398$	0	0
$399$	7.41855	0.371392
$400$	0	0
$401$	8.63931	0.431426	0.215713	−	0.976457i	$-0.430792\pi$
$401$	8.63931	0.431426	0.215713	+	0.976457i	$0.430792\pi$
$402$	0	0
$403$	31.2327	1.55581
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−22.8371	−1.13199
$408$	0	0
$409$	10.7337	0.530746	0.265373	−	0.964146i	$-0.414505\pi$
$409$	10.7337	0.530746	0.265373	+	0.964146i	$0.414505\pi$
$410$	0	0
$411$	−15.9421	−0.786368
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	10.8371	0.531972
$416$	0	0
$417$	15.4186	0.755050
$418$	0	0
$419$	32.1978	1.57297	0.786483	−	0.617612i	$-0.211900\pi$
$419$	32.1978	1.57297	0.786483	+	0.617612i	$0.211900\pi$
$420$	0	0
$421$	−7.67420	−0.374018	−0.187009	−	0.982358i	$-0.559879\pi$
$421$	−7.67420	−0.374018	−0.187009	+	0.982358i	$0.559879\pi$
$422$	0	0
$423$	2.15676	0.104865
$424$	0	0
$425$	−6.68035	−0.324044
$426$	0	0
$427$	0.156755	0.00758593
$428$	0	0
$429$	−11.1506	−0.538357
$430$	0	0
$431$	1.06400	0.0512512	0.0256256	−	0.999672i	$-0.491842\pi$
$431$	1.06400	0.0512512	0.0256256	+	0.999672i	$0.491842\pi$
$432$	0	0
$433$	4.62249	0.222143	0.111071	−	0.993812i	$-0.464572\pi$
$433$	4.62249	0.222143	0.111071	+	0.993812i	$0.464572\pi$
$434$	0	0
$435$	−6.68035	−0.320298
$436$	0	0
$437$	−16.0000	−0.765384
$438$	0	0
$439$	13.2618	0.632951	0.316475	−	0.948601i	$-0.397501\pi$
$439$	13.2618	0.632951	0.316475	+	0.948601i	$0.397501\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	6.35455	0.301914	0.150957	−	0.988540i	$-0.451765\pi$
$443$	6.35455	0.301914	0.150957	+	0.988540i	$0.451765\pi$
$444$	0	0
$445$	4.52359	0.214439
$446$	0	0
$447$	−12.1568	−0.574995
$448$	0	0
$449$	−2.31351	−0.109181	−0.0545907	−	0.998509i	$-0.517385\pi$
$449$	−2.31351	−0.109181	−0.0545907	+	0.998509i	$0.517385\pi$
$450$	0	0
$451$	−30.2101	−1.42254
$452$	0	0
$453$	−2.83710	−0.133299
$454$	0	0
$455$	3.26180	0.152915
$456$	0	0
$457$	7.47641	0.349732	0.174866	−	0.984592i	$-0.444051\pi$
$457$	7.47641	0.349732	0.174866	+	0.984592i	$0.444051\pi$
$458$	0	0
$459$	−6.68035	−0.311812
$460$	0	0
$461$	−22.3135	−1.03924	−0.519622	−	0.854396i	$-0.673927\pi$
$461$	−22.3135	−1.03924	−0.519622	+	0.854396i	$0.673927\pi$
$462$	0	0
$463$	−1.78992	−0.0831847	−0.0415923	−	0.999135i	$-0.513243\pi$
$463$	−1.78992	−0.0831847	−0.0415923	+	0.999135i	$0.513243\pi$
$464$	0	0
$465$	9.57531	0.444044
$466$	0	0
$467$	22.5236	1.04227	0.521134	−	0.853475i	$-0.325509\pi$
$467$	22.5236	1.04227	0.521134	+	0.853475i	$0.325509\pi$
$468$	0	0
$469$	−15.5174	−0.716529
$470$	0	0
$471$	−7.26180	−0.334606
$472$	0	0
$473$	−29.6742	−1.36442
$474$	0	0
$475$	7.41855	0.340386
$476$	0	0
$477$	1.41855	0.0649510
$478$	0	0
$479$	34.7214	1.58646	0.793230	−	0.608922i	$-0.208398\pi$
$479$	34.7214	1.58646	0.793230	+	0.608922i	$0.208398\pi$
$480$	0	0
$481$	21.7899	0.993535
$482$	0	0
$483$	−2.15676	−0.0981358
$484$	0	0
$485$	3.26180	0.148110
$486$	0	0
$487$	−22.8371	−1.03485	−0.517424	−	0.855729i	$-0.673109\pi$
$487$	−22.8371	−1.03485	−0.517424	+	0.855729i	$0.673109\pi$
$488$	0	0
$489$	−16.6803	−0.754311
$490$	0	0
$491$	15.3028	0.690607	0.345304	−	0.938491i	$-0.387776\pi$
$491$	15.3028	0.690607	0.345304	+	0.938491i	$0.387776\pi$
$492$	0	0
$493$	44.6270	2.00990
$494$	0	0
$495$	−3.41855	−0.153652
$496$	0	0
$497$	−1.26180	−0.0565993
$498$	0	0
$499$	−39.8843	−1.78547	−0.892733	−	0.450586i	$-0.851215\pi$
$499$	−39.8843	−1.78547	−0.892733	+	0.450586i	$0.851215\pi$
$500$	0	0
$501$	−3.31965	−0.148311
$502$	0	0
$503$	−37.3074	−1.66345	−0.831727	−	0.555185i	$-0.812647\pi$
$503$	−37.3074	−1.66345	−0.831727	+	0.555185i	$0.812647\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	0	0
$507$	−2.36069	−0.104842
$508$	0	0
$509$	−25.0349	−1.10965	−0.554826	−	0.831966i	$-0.687215\pi$
$509$	−25.0349	−1.10965	−0.554826	+	0.831966i	$0.687215\pi$
$510$	0	0
$511$	0.738205	0.0326563
$512$	0	0
$513$	7.41855	0.327537
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−7.37298	−0.324263
$518$	0	0
$519$	−8.83710	−0.387906
$520$	0	0
$521$	5.15061	0.225652	0.112826	−	0.993615i	$-0.464010\pi$
$521$	5.15061	0.225652	0.112826	+	0.993615i	$0.464010\pi$
$522$	0	0
$523$	−6.52359	−0.285257	−0.142628	−	0.989776i	$-0.545555\pi$
$523$	−6.52359	−0.285257	−0.142628	+	0.989776i	$0.545555\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	−63.9664	−2.78642
$528$	0	0
$529$	−18.3484	−0.797757
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	28.8248	1.24854
$534$	0	0
$535$	6.15676	0.266180
$536$	0	0
$537$	12.5814	0.542930
$538$	0	0
$539$	−3.41855	−0.147247
$540$	0	0
$541$	45.0349	1.93620	0.968101	−	0.250562i	$-0.0806152\pi$
$541$	45.0349	1.93620	0.968101	+	0.250562i	$0.0806152\pi$
$542$	0	0
$543$	−22.6803	−0.973307
$544$	0	0
$545$	−19.3607	−0.829321
$546$	0	0
$547$	−22.8904	−0.978724	−0.489362	−	0.872081i	$-0.662770\pi$
$547$	−22.8904	−0.978724	−0.489362	+	0.872081i	$0.662770\pi$
$548$	0	0
$549$	0.156755	0.00669016
$550$	0	0
$551$	−49.5585	−2.11126
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	6.68035	0.283565
$556$	0	0
$557$	15.0928	0.639500	0.319750	−	0.947502i	$-0.396401\pi$
$557$	15.0928	0.639500	0.319750	+	0.947502i	$0.396401\pi$
$558$	0	0
$559$	28.3135	1.19753
$560$	0	0
$561$	22.8371	0.964183
$562$	0	0
$563$	−25.0472	−1.05561	−0.527806	−	0.849365i	$-0.676985\pi$
$563$	−25.0472	−1.05561	−0.527806	+	0.849365i	$0.676985\pi$
$564$	0	0
$565$	16.2557	0.683880
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	33.0349	1.38489	0.692447	−	0.721468i	$-0.256532\pi$
$569$	33.0349	1.38489	0.692447	+	0.721468i	$0.256532\pi$
$570$	0	0
$571$	−42.8371	−1.79268	−0.896338	−	0.443370i	$-0.853783\pi$
$571$	−42.8371	−1.79268	−0.896338	+	0.443370i	$0.853783\pi$
$572$	0	0
$573$	1.26180	0.0527123
$574$	0	0
$575$	−2.15676	−0.0899429
$576$	0	0
$577$	−8.42469	−0.350725	−0.175362	−	0.984504i	$-0.556110\pi$
$577$	−8.42469	−0.350725	−0.175362	+	0.984504i	$0.556110\pi$
$578$	0	0
$579$	19.3607	0.804603
$580$	0	0
$581$	10.8371	0.449599
$582$	0	0
$583$	−4.84939	−0.200841
$584$	0	0
$585$	3.26180	0.134859
$586$	0	0
$587$	−33.0472	−1.36400	−0.682002	−	0.731351i	$-0.738890\pi$
$587$	−33.0472	−1.36400	−0.682002	+	0.731351i	$0.738890\pi$
$588$	0	0
$589$	71.0349	2.92694
$590$	0	0
$591$	3.94214	0.162158
$592$	0	0
$593$	41.7152	1.71304	0.856520	−	0.516114i	$-0.172622\pi$
$593$	41.7152	1.71304	0.856520	+	0.516114i	$0.172622\pi$
$594$	0	0
$595$	−6.68035	−0.273867
$596$	0	0
$597$	3.90110	0.159662
$598$	0	0
$599$	4.21461	0.172204	0.0861022	−	0.996286i	$-0.472559\pi$
$599$	4.21461	0.172204	0.0861022	+	0.996286i	$0.472559\pi$
$600$	0	0
$601$	−22.1978	−0.905467	−0.452733	−	0.891646i	$-0.649551\pi$
$601$	−22.1978	−0.905467	−0.452733	+	0.891646i	$0.649551\pi$
$602$	0	0
$603$	−15.5174	−0.631919
$604$	0	0
$605$	0.686489	0.0279097
$606$	0	0
$607$	−36.5113	−1.48195	−0.740974	−	0.671534i	$-0.765636\pi$
$607$	−36.5113	−1.48195	−0.740974	+	0.671534i	$0.765636\pi$
$608$	0	0
$609$	−6.68035	−0.270701
$610$	0	0
$611$	7.03489	0.284601
$612$	0	0
$613$	1.63317	0.0659629	0.0329815	−	0.999456i	$-0.489500\pi$
$613$	1.63317	0.0659629	0.0329815	+	0.999456i	$0.489500\pi$
$614$	0	0
$615$	8.83710	0.356346
$616$	0	0
$617$	−1.73206	−0.0697302	−0.0348651	−	0.999392i	$-0.511100\pi$
$617$	−1.73206	−0.0697302	−0.0348651	+	0.999392i	$0.511100\pi$
$618$	0	0
$619$	19.7321	0.793099	0.396549	−	0.918013i	$-0.370208\pi$
$619$	19.7321	0.793099	0.396549	+	0.918013i	$0.370208\pi$
$620$	0	0
$621$	−2.15676	−0.0865476
$622$	0	0
$623$	4.52359	0.181234
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−25.3607	−1.01281
$628$	0	0
$629$	−44.6270	−1.77940
$630$	0	0
$631$	−4.73367	−0.188444	−0.0942222	−	0.995551i	$-0.530036\pi$
$631$	−4.73367	−0.188444	−0.0942222	+	0.995551i	$0.530036\pi$
$632$	0	0
$633$	0.313511	0.0124609
$634$	0	0
$635$	10.5236	0.417616
$636$	0	0
$637$	3.26180	0.129237
$638$	0	0
$639$	−1.26180	−0.0499158
$640$	0	0
$641$	−17.1506	−0.677408	−0.338704	−	0.940893i	$-0.609989\pi$
$641$	−17.1506	−0.677408	−0.338704	+	0.940893i	$0.609989\pi$
$642$	0	0
$643$	−13.7899	−0.543821	−0.271911	−	0.962322i	$-0.587656\pi$
$643$	−13.7899	−0.543821	−0.271911	+	0.962322i	$0.587656\pi$
$644$	0	0
$645$	8.68035	0.341788
$646$	0	0
$647$	0.993857	0.0390725	0.0195363	−	0.999809i	$-0.493781\pi$
$647$	0.993857	0.0390725	0.0195363	+	0.999809i	$0.493781\pi$
$648$	0	0
$649$	−27.3484	−1.07352
$650$	0	0
$651$	9.57531	0.375286
$652$	0	0
$653$	29.3028	1.14671	0.573354	−	0.819308i	$-0.305642\pi$
$653$	29.3028	1.14671	0.573354	+	0.819308i	$0.305642\pi$
$654$	0	0
$655$	10.5236	0.411191
$656$	0	0
$657$	0.738205	0.0288001
$658$	0	0
$659$	20.7792	0.809444	0.404722	−	0.914440i	$-0.367368\pi$
$659$	20.7792	0.809444	0.404722	+	0.914440i	$0.367368\pi$
$660$	0	0
$661$	−46.1445	−1.79481	−0.897406	−	0.441206i	$-0.854551\pi$
$661$	−46.1445	−1.79481	−0.897406	+	0.441206i	$0.854551\pi$
$662$	0	0
$663$	−21.7899	−0.846250
$664$	0	0
$665$	7.41855	0.287679
$666$	0	0
$667$	14.4079	0.557875
$668$	0	0
$669$	−19.8843	−0.768771
$670$	0	0
$671$	−0.535877	−0.0206873
$672$	0	0
$673$	−17.8843	−0.689388	−0.344694	−	0.938715i	$-0.612017\pi$
$673$	−17.8843	−0.689388	−0.344694	+	0.938715i	$0.612017\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	4.21008	0.161806	0.0809032	−	0.996722i	$-0.474220\pi$
$677$	4.21008	0.161806	0.0809032	+	0.996722i	$0.474220\pi$
$678$	0	0
$679$	3.26180	0.125176
$680$	0	0
$681$	23.8843	0.915248
$682$	0	0
$683$	−33.3074	−1.27447	−0.637236	−	0.770669i	$-0.719922\pi$
$683$	−33.3074	−1.27447	−0.637236	+	0.770669i	$0.719922\pi$
$684$	0	0
$685$	−15.9421	−0.609118
$686$	0	0
$687$	26.8781	1.02546
$688$	0	0
$689$	4.62702	0.176275
$690$	0	0
$691$	−24.0456	−0.914737	−0.457368	−	0.889277i	$-0.651208\pi$
$691$	−24.0456	−0.914737	−0.457368	+	0.889277i	$0.651208\pi$
$692$	0	0
$693$	−3.41855	−0.129860
$694$	0	0
$695$	15.4186	0.584859
$696$	0	0
$697$	−59.0349	−2.23611
$698$	0	0
$699$	−22.7792	−0.861590
$700$	0	0
$701$	−20.3545	−0.768781	−0.384390	−	0.923171i	$-0.625588\pi$
$701$	−20.3545	−0.768781	−0.384390	+	0.923171i	$0.625588\pi$
$702$	0	0
$703$	49.5585	1.86913
$704$	0	0
$705$	2.15676	0.0812281
$706$	0	0
$707$	−10.0000	−0.376089
$708$	0	0
$709$	47.3607	1.77867	0.889334	−	0.457258i	$-0.151168\pi$
$709$	47.3607	1.77867	0.889334	+	0.457258i	$0.151168\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	−20.6516	−0.773408
$714$	0	0
$715$	−11.1506	−0.417009
$716$	0	0
$717$	−1.26180	−0.0471226
$718$	0	0
$719$	−30.8371	−1.15003	−0.575015	−	0.818143i	$-0.695004\pi$
$719$	−30.8371	−1.15003	−0.575015	+	0.818143i	$0.695004\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−4.83710	−0.179894
$724$	0	0
$725$	−6.68035	−0.248102
$726$	0	0
$727$	16.1978	0.600743	0.300371	−	0.953822i	$-0.402889\pi$
$727$	16.1978	0.600743	0.300371	+	0.953822i	$0.402889\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−57.9877	−2.14475
$732$	0	0
$733$	22.2146	0.820516	0.410258	−	0.911970i	$-0.365439\pi$
$733$	22.2146	0.820516	0.410258	+	0.911970i	$0.365439\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	53.0472	1.95402
$738$	0	0
$739$	−9.47641	−0.348595	−0.174298	−	0.984693i	$-0.555766\pi$
$739$	−9.47641	−0.348595	−0.174298	+	0.984693i	$0.555766\pi$
$740$	0	0
$741$	24.1978	0.888928
$742$	0	0
$743$	−27.3197	−1.00226	−0.501130	−	0.865372i	$-0.667082\pi$
$743$	−27.3197	−1.00226	−0.501130	+	0.865372i	$0.667082\pi$
$744$	0	0
$745$	−12.1568	−0.445389
$746$	0	0
$747$	10.8371	0.396509
$748$	0	0
$749$	6.15676	0.224963
$750$	0	0
$751$	−30.7214	−1.12104	−0.560520	−	0.828141i	$-0.689399\pi$
$751$	−30.7214	−1.12104	−0.560520	+	0.828141i	$0.689399\pi$
$752$	0	0
$753$	−17.3607	−0.632659
$754$	0	0
$755$	−2.83710	−0.103253
$756$	0	0
$757$	9.83096	0.357312	0.178656	−	0.983912i	$-0.442825\pi$
$757$	9.83096	0.357312	0.178656	+	0.983912i	$0.442825\pi$
$758$	0	0
$759$	7.37298	0.267622
$760$	0	0
$761$	−26.5113	−0.961034	−0.480517	−	0.876985i	$-0.659551\pi$
$761$	−26.5113	−0.961034	−0.480517	+	0.876985i	$0.659551\pi$
$762$	0	0
$763$	−19.3607	−0.700904
$764$	0	0
$765$	−6.68035	−0.241528
$766$	0	0
$767$	26.0944	0.942213
$768$	0	0
$769$	43.5585	1.57076	0.785380	−	0.619014i	$-0.212468\pi$
$769$	43.5585	1.57076	0.785380	+	0.619014i	$0.212468\pi$
$770$	0	0
$771$	25.5174	0.918988
$772$	0	0
$773$	−26.8248	−0.964822	−0.482411	−	0.875945i	$-0.660239\pi$
$773$	−26.8248	−0.964822	−0.482411	+	0.875945i	$0.660239\pi$
$774$	0	0
$775$	9.57531	0.343955
$776$	0	0
$777$	6.68035	0.239656
$778$	0	0
$779$	65.5585	2.34888
$780$	0	0
$781$	4.31351	0.154350
$782$	0	0
$783$	−6.68035	−0.238736
$784$	0	0
$785$	−7.26180	−0.259185
$786$	0	0
$787$	−0.115718	−0.00412489	−0.00206244	−	0.999998i	$-0.500656\pi$
$787$	−0.115718	−0.00412489	−0.00206244	+	0.999998i	$0.500656\pi$
$788$	0	0
$789$	21.3074	0.758562
$790$	0	0
$791$	16.2557	0.577984
$792$	0	0
$793$	0.511304	0.0181569
$794$	0	0
$795$	1.41855	0.0503108
$796$	0	0
$797$	−8.83710	−0.313026	−0.156513	−	0.987676i	$-0.550025\pi$
$797$	−8.83710	−0.313026	−0.156513	+	0.987676i	$0.550025\pi$
$798$	0	0
$799$	−14.4079	−0.509714
$800$	0	0
$801$	4.52359	0.159833
$802$	0	0
$803$	−2.52359	−0.0890556
$804$	0	0
$805$	−2.15676	−0.0760156
$806$	0	0
$807$	21.0349	0.740463
$808$	0	0
$809$	26.8248	0.943110	0.471555	−	0.881837i	$-0.343693\pi$
$809$	26.8248	0.943110	0.471555	+	0.881837i	$0.343693\pi$
$810$	0	0
$811$	16.0456	0.563436	0.281718	−	0.959497i	$-0.409096\pi$
$811$	16.0456	0.563436	0.281718	+	0.959497i	$0.409096\pi$
$812$	0	0
$813$	−1.57531	−0.0552484
$814$	0	0
$815$	−16.6803	−0.584287
$816$	0	0
$817$	64.3956	2.25292
$818$	0	0
$819$	3.26180	0.113976
$820$	0	0
$821$	45.4017	1.58453	0.792266	−	0.610176i	$-0.208901\pi$
$821$	45.4017	1.58453	0.792266	+	0.610176i	$0.208901\pi$
$822$	0	0
$823$	−52.5113	−1.83043	−0.915215	−	0.402967i	$-0.867979\pi$
$823$	−52.5113	−1.83043	−0.915215	+	0.402967i	$0.867979\pi$
$824$	0	0
$825$	−3.41855	−0.119019
$826$	0	0
$827$	2.47027	0.0858996	0.0429498	−	0.999077i	$-0.486324\pi$
$827$	2.47027	0.0858996	0.0429498	+	0.999077i	$0.486324\pi$
$828$	0	0
$829$	−21.7152	−0.754201	−0.377101	−	0.926172i	$-0.623079\pi$
$829$	−21.7152	−0.754201	−0.377101	+	0.926172i	$0.623079\pi$
$830$	0	0
$831$	−26.6803	−0.925531
$832$	0	0
$833$	−6.68035	−0.231460
$834$	0	0
$835$	−3.31965	−0.114881
$836$	0	0
$837$	9.57531	0.330971
$838$	0	0
$839$	−38.8371	−1.34081	−0.670403	−	0.741997i	$-0.733879\pi$
$839$	−38.8371	−1.34081	−0.670403	+	0.741997i	$0.733879\pi$
$840$	0	0
$841$	15.6270	0.538863
$842$	0	0
$843$	8.83710	0.304366
$844$	0	0
$845$	−2.36069	−0.0812103
$846$	0	0
$847$	0.686489	0.0235880
$848$	0	0
$849$	−25.6742	−0.881136
$850$	0	0
$851$	−14.4079	−0.493896
$852$	0	0
$853$	−25.0517	−0.857754	−0.428877	−	0.903363i	$-0.641091\pi$
$853$	−25.0517	−0.857754	−0.428877	+	0.903363i	$0.641091\pi$
$854$	0	0
$855$	7.41855	0.253709
$856$	0	0
$857$	−3.52973	−0.120573	−0.0602867	−	0.998181i	$-0.519201\pi$
$857$	−3.52973	−0.120573	−0.0602867	+	0.998181i	$0.519201\pi$
$858$	0	0
$859$	20.8950	0.712927	0.356463	−	0.934309i	$-0.383982\pi$
$859$	20.8950	0.712927	0.356463	+	0.934309i	$0.383982\pi$
$860$	0	0
$861$	8.83710	0.301168
$862$	0	0
$863$	20.8781	0.710700	0.355350	−	0.934733i	$-0.384362\pi$
$863$	20.8781	0.710700	0.355350	+	0.934733i	$0.384362\pi$
$864$	0	0
$865$	−8.83710	−0.300470
$866$	0	0
$867$	27.6270	0.938263
$868$	0	0
$869$	−13.6742	−0.463866
$870$	0	0
$871$	−50.6147	−1.71501
$872$	0	0
$873$	3.26180	0.110395
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−32.3545	−1.09254	−0.546268	−	0.837611i	$-0.683952\pi$
$877$	−32.3545	−1.09254	−0.546268	+	0.837611i	$0.683952\pi$
$878$	0	0
$879$	−20.5236	−0.692244
$880$	0	0
$881$	−38.1978	−1.28692	−0.643458	−	0.765481i	$-0.722501\pi$
$881$	−38.1978	−1.28692	−0.643458	+	0.765481i	$0.722501\pi$
$882$	0	0
$883$	6.89043	0.231881	0.115941	−	0.993256i	$-0.463012\pi$
$883$	6.89043	0.231881	0.115941	+	0.993256i	$0.463012\pi$
$884$	0	0
$885$	8.00000	0.268917
$886$	0	0
$887$	8.56463	0.287572	0.143786	−	0.989609i	$-0.454072\pi$
$887$	8.56463	0.287572	0.143786	+	0.989609i	$0.454072\pi$
$888$	0	0
$889$	10.5236	0.352950
$890$	0	0
$891$	−3.41855	−0.114526
$892$	0	0
$893$	16.0000	0.535420
$894$	0	0
$895$	12.5814	0.420551
$896$	0	0
$897$	−7.03489	−0.234888
$898$	0	0
$899$	−63.9664	−2.13340
$900$	0	0
$901$	−9.47641	−0.315705
$902$	0	0
$903$	8.68035	0.288864
$904$	0	0
$905$	−22.6803	−0.753920
$906$	0	0
$907$	−35.2039	−1.16893	−0.584464	−	0.811420i	$-0.698695\pi$
$907$	−35.2039	−1.16893	−0.584464	+	0.811420i	$0.698695\pi$
$908$	0	0
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−1.69102	−0.0560261	−0.0280131	−	0.999608i	$-0.508918\pi$
$911$	−1.69102	−0.0560261	−0.0280131	+	0.999608i	$0.508918\pi$
$912$	0	0
$913$	−37.0472	−1.22608
$914$	0	0
$915$	0.156755	0.00518218
$916$	0	0
$917$	10.5236	0.347520
$918$	0	0
$919$	14.3258	0.472564	0.236282	−	0.971684i	$-0.424071\pi$
$919$	14.3258	0.472564	0.236282	+	0.971684i	$0.424071\pi$
$920$	0	0
$921$	−28.1978	−0.929149
$922$	0	0
$923$	−4.11572	−0.135470
$924$	0	0
$925$	6.68035	0.219648
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−34.5113	−1.13228	−0.566140	−	0.824309i	$-0.691564\pi$
$929$	−34.5113	−1.13228	−0.566140	+	0.824309i	$0.691564\pi$
$930$	0	0
$931$	7.41855	0.243133
$932$	0	0
$933$	13.6742	0.447673
$934$	0	0
$935$	22.8371	0.746853
$936$	0	0
$937$	−29.9011	−0.976826	−0.488413	−	0.872612i	$-0.662424\pi$
$937$	−29.9011	−0.976826	−0.488413	+	0.872612i	$0.662424\pi$
$938$	0	0
$939$	23.5753	0.769351
$940$	0	0
$941$	−28.0944	−0.915850	−0.457925	−	0.888991i	$-0.651407\pi$
$941$	−28.0944	−0.915850	−0.457925	+	0.888991i	$0.651407\pi$
$942$	0	0
$943$	−19.0595	−0.620662
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	−48.3423	−1.57091	−0.785456	−	0.618917i	$-0.787572\pi$
$947$	−48.3423	−1.57091	−0.785456	+	0.618917i	$0.787572\pi$
$948$	0	0
$949$	2.40787	0.0781629
$950$	0	0
$951$	−11.6286	−0.377084
$952$	0	0
$953$	−43.0928	−1.39591	−0.697956	−	0.716141i	$-0.745907\pi$
$953$	−43.0928	−1.39591	−0.697956	+	0.716141i	$0.745907\pi$
$954$	0	0
$955$	1.26180	0.0408307
$956$	0	0
$957$	22.8371	0.738219
$958$	0	0
$959$	−15.9421	−0.514799
$960$	0	0
$961$	60.6865	1.95763
$962$	0	0
$963$	6.15676	0.198399
$964$	0	0
$965$	19.3607	0.623243
$966$	0	0
$967$	−16.7337	−0.538119	−0.269059	−	0.963124i	$-0.586713\pi$
$967$	−16.7337	−0.538119	−0.269059	+	0.963124i	$0.586713\pi$
$968$	0	0
$969$	−49.5585	−1.59205
$970$	0	0
$971$	30.3012	0.972413	0.486206	−	0.873844i	$-0.338380\pi$
$971$	30.3012	0.972413	0.486206	+	0.873844i	$0.338380\pi$
$972$	0	0
$973$	15.4186	0.494296
$974$	0	0
$975$	3.26180	0.104461
$976$	0	0
$977$	41.6163	1.33142	0.665712	−	0.746208i	$-0.268128\pi$
$977$	41.6163	1.33142	0.665712	+	0.746208i	$0.268128\pi$
$978$	0	0
$979$	−15.4641	−0.494236
$980$	0	0
$981$	−19.3607	−0.618139
$982$	0	0
$983$	−28.2511	−0.901071	−0.450535	−	0.892759i	$-0.648767\pi$
$983$	−28.2511	−0.901071	−0.450535	+	0.892759i	$0.648767\pi$
$984$	0	0
$985$	3.94214	0.125607
$986$	0	0
$987$	2.15676	0.0686503
$988$	0	0
$989$	−18.7214	−0.595305
$990$	0	0
$991$	41.6742	1.32382	0.661912	−	0.749581i	$-0.269745\pi$
$991$	41.6742	1.32382	0.661912	+	0.749581i	$0.269745\pi$
$992$	0	0
$993$	−24.3135	−0.771566
$994$	0	0
$995$	3.90110	0.123673
$996$	0	0
$997$	12.6225	0.399758	0.199879	−	0.979821i	$-0.435945\pi$
$997$	12.6225	0.399758	0.199879	+	0.979821i	$0.435945\pi$
$998$	0	0
$999$	6.68035	0.211357

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.a.bj.1.1	yes	3
4.3	odd	2		3360.2.a.bi.1.3	✓	3
8.3	odd	2		6720.2.a.db.1.1		3
8.5	even	2		6720.2.a.da.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3360.2.a.bi.1.3	✓	3	4.3	odd	2
3360.2.a.bj.1.1	yes	3	1.1	even	1	trivial
6720.2.a.da.1.3		3	8.5	even	2
6720.2.a.db.1.1		3	8.3	odd	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:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.41855 q^{11} +3.26180 q^{13} +1.00000 q^{15} -6.68035 q^{17} +7.41855 q^{19} +1.00000 q^{21} -2.15676 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.68035 q^{29} +9.57531 q^{31} -3.41855 q^{33} +1.00000 q^{35} +6.68035 q^{37} +3.26180 q^{39} +8.83710 q^{41} +8.68035 q^{43} +1.00000 q^{45} +2.15676 q^{47} +1.00000 q^{49} -6.68035 q^{51} +1.41855 q^{53} -3.41855 q^{55} +7.41855 q^{57} +8.00000 q^{59} +0.156755 q^{61} +1.00000 q^{63} +3.26180 q^{65} -15.5174 q^{67} -2.15676 q^{69} -1.26180 q^{71} +0.738205 q^{73} +1.00000 q^{75} -3.41855 q^{77} +4.00000 q^{79} +1.00000 q^{81} +10.8371 q^{83} -6.68035 q^{85} -6.68035 q^{87} +4.52359 q^{89} +3.26180 q^{91} +9.57531 q^{93} +7.41855 q^{95} +3.26180 q^{97} -3.41855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9} + 4 q^{11} + 2 q^{13} + 3 q^{15} + 2 q^{17} + 8 q^{19} + 3 q^{21} + 3 q^{25} + 3 q^{27} + 2 q^{29} + 8 q^{31} + 4 q^{33} + 3 q^{35} - 2 q^{37} + 2 q^{39} - 2 q^{41} + 4 q^{43} + 3 q^{45} + 3 q^{49} + 2 q^{51} - 10 q^{53} + 4 q^{55} + 8 q^{57} + 24 q^{59} - 6 q^{61} + 3 q^{63} + 2 q^{65} + 4 q^{67} + 4 q^{71} + 10 q^{73} + 3 q^{75} + 4 q^{77} + 12 q^{79} + 3 q^{81} + 4 q^{83} + 2 q^{85} + 2 q^{87} - 2 q^{89} + 2 q^{91} + 8 q^{93} + 8 q^{95} + 2 q^{97} + 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^7 + 3 * q^9 + 4 * q^11 + 2 * q^13 + 3 * q^15 + 2 * q^17 + 8 * q^19 + 3 * q^21 + 3 * q^25 + 3 * q^27 + 2 * q^29 + 8 * q^31 + 4 * q^33 + 3 * q^35 - 2 * q^37 + 2 * q^39 - 2 * q^41 + 4 * q^43 + 3 * q^45 + 3 * q^49 + 2 * q^51 - 10 * q^53 + 4 * q^55 + 8 * q^57 + 24 * q^59 - 6 * q^61 + 3 * q^63 + 2 * q^65 + 4 * q^67 + 4 * q^71 + 10 * q^73 + 3 * q^75 + 4 * q^77 + 12 * q^79 + 3 * q^81 + 4 * q^83 + 2 * q^85 + 2 * q^87 - 2 * q^89 + 2 * q^91 + 8 * q^93 + 8 * q^95 + 2 * q^97 + 4 * q^99

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