LMFDB - Embedded newform 3120.2.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,2,Mod(1,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, 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("3120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.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:	$24.9133254306$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1849.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 14x - 8 $ x^3 - x^2 - 14*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.615072$ of defining polynomial
Character	$\chi$	$=$	3120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -2.61507 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} -2.61507 q^{7} +1.00000 q^{9} +6.39154 q^{11} +1.00000 q^{13} +1.00000 q^{15} -0.615072 q^{17} -7.77647 q^{19} +2.61507 q^{21} -2.61507 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.00662 q^{29} -5.23014 q^{31} -6.39154 q^{33} +2.61507 q^{35} +7.16140 q^{37} -1.00000 q^{39} +7.16140 q^{41} -4.00000 q^{43} -1.00000 q^{45} +1.23014 q^{47} -0.161400 q^{49} +0.615072 q^{51} -13.6217 q^{53} -6.39154 q^{55} +7.77647 q^{57} -0.391544 q^{61} -2.61507 q^{63} -1.00000 q^{65} +14.0132 q^{67} +2.61507 q^{69} -5.16140 q^{71} +0.993385 q^{73} -1.00000 q^{75} -16.7143 q^{77} +14.3915 q^{79} +1.00000 q^{81} -18.0132 q^{83} +0.615072 q^{85} -7.00662 q^{87} +14.8518 q^{89} -2.61507 q^{91} +5.23014 q^{93} +7.77647 q^{95} +7.38493 q^{97} +6.39154 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + 3 q^{15} + q^{17} - 4 q^{19} + 5 q^{21} - 5 q^{23} + 3 q^{25} - 3 q^{27} - 4 q^{29} - 10 q^{31} + 3 q^{33} + 5 q^{35} + 5 q^{37} - 3 q^{39} + 5 q^{41} - 12 q^{43} - 3 q^{45} - 2 q^{47} + 16 q^{49} - q^{51} - 13 q^{53} + 3 q^{55} + 4 q^{57} + 21 q^{61} - 5 q^{63} - 3 q^{65} - 8 q^{67} + 5 q^{69} + q^{71} + 28 q^{73} - 3 q^{75} + 5 q^{77} + 21 q^{79} + 3 q^{81} - 4 q^{83} - q^{85} + 4 q^{87} + 11 q^{89} - 5 q^{91} + 10 q^{93} + 4 q^{95} + 25 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + 3 * q^15 + q^17 - 4 * q^19 + 5 * q^21 - 5 * q^23 + 3 * q^25 - 3 * q^27 - 4 * q^29 - 10 * q^31 + 3 * q^33 + 5 * q^35 + 5 * q^37 - 3 * q^39 + 5 * q^41 - 12 * q^43 - 3 * q^45 - 2 * q^47 + 16 * q^49 - q^51 - 13 * q^53 + 3 * q^55 + 4 * q^57 + 21 * q^61 - 5 * q^63 - 3 * q^65 - 8 * q^67 + 5 * q^69 + q^71 + 28 * q^73 - 3 * q^75 + 5 * q^77 + 21 * q^79 + 3 * q^81 - 4 * q^83 - q^85 + 4 * q^87 + 11 * q^89 - 5 * q^91 + 10 * q^93 + 4 * q^95 + 25 * q^97 - 3 * 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
$7$	−2.61507	−0.988404	−0.494202	−	0.869347i	$-0.664540\pi$
$7$	−2.61507	−0.988404	−0.494202	+	0.869347i	$0.664540\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.39154	1.92712	0.963561	−	0.267487i	$-0.0861933\pi$
$11$	6.39154	1.92712	0.963561	+	0.267487i	$0.0861933\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−0.615072	−0.149177	−0.0745884	−	0.997214i	$-0.523764\pi$
$17$	−0.615072	−0.149177	−0.0745884	+	0.997214i	$0.523764\pi$
$18$	0	0
$19$	−7.77647	−1.78405	−0.892023	−	0.451991i	$-0.850714\pi$
$19$	−7.77647	−1.78405	−0.892023	+	0.451991i	$0.850714\pi$
$20$	0	0
$21$	2.61507	0.570655
$22$	0	0
$23$	−2.61507	−0.545280	−0.272640	−	0.962116i	$-0.587897\pi$
$23$	−2.61507	−0.545280	−0.272640	+	0.962116i	$0.587897\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.00662	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$29$	7.00662	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$30$	0	0
$31$	−5.23014	−0.939361	−0.469681	−	0.882836i	$-0.655631\pi$
$31$	−5.23014	−0.939361	−0.469681	+	0.882836i	$0.655631\pi$
$32$	0	0
$33$	−6.39154	−1.11262
$34$	0	0
$35$	2.61507	0.442028
$36$	0	0
$37$	7.16140	1.17733	0.588663	−	0.808378i	$-0.299654\pi$
$37$	7.16140	1.17733	0.588663	+	0.808378i	$0.299654\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	7.16140	1.11842	0.559211	−	0.829025i	$-0.311104\pi$
$41$	7.16140	1.11842	0.559211	+	0.829025i	$0.311104\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	1.23014	0.179435	0.0897174	−	0.995967i	$-0.471404\pi$
$47$	1.23014	0.179435	0.0897174	+	0.995967i	$0.471404\pi$
$48$	0	0
$49$	−0.161400	−0.0230572
$50$	0	0
$51$	0.615072	0.0861273
$52$	0	0
$53$	−13.6217	−1.87108	−0.935541	−	0.353217i	$-0.885088\pi$
$53$	−13.6217	−1.87108	−0.935541	+	0.353217i	$0.885088\pi$
$54$	0	0
$55$	−6.39154	−0.861836
$56$	0	0
$57$	7.77647	1.03002
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−0.391544	−0.0501320	−0.0250660	−	0.999686i	$-0.507980\pi$
$61$	−0.391544	−0.0501320	−0.0250660	+	0.999686i	$0.507980\pi$
$62$	0	0
$63$	−2.61507	−0.329468
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	14.0132	1.71199	0.855994	−	0.516985i	$-0.172946\pi$
$67$	14.0132	1.71199	0.855994	+	0.516985i	$0.172946\pi$
$68$	0	0
$69$	2.61507	0.314818
$70$	0	0
$71$	−5.16140	−0.612546	−0.306273	−	0.951944i	$-0.599082\pi$
$71$	−5.16140	−0.612546	−0.306273	+	0.951944i	$0.599082\pi$
$72$	0	0
$73$	0.993385	0.116267	0.0581334	−	0.998309i	$-0.481485\pi$
$73$	0.993385	0.116267	0.0581334	+	0.998309i	$0.481485\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−16.7143	−1.90478
$78$	0	0
$79$	14.3915	1.61918	0.809588	−	0.586999i	$-0.199691\pi$
$79$	14.3915	1.61918	0.809588	+	0.586999i	$0.199691\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−18.0132	−1.97721	−0.988604	−	0.150536i	$-0.951900\pi$
$83$	−18.0132	−1.97721	−0.988604	+	0.150536i	$0.951900\pi$
$84$	0	0
$85$	0.615072	0.0667139
$86$	0	0
$87$	−7.00662	−0.751188
$88$	0	0
$89$	14.8518	1.57429	0.787145	−	0.616767i	$-0.211558\pi$
$89$	14.8518	1.57429	0.787145	+	0.616767i	$0.211558\pi$
$90$	0	0
$91$	−2.61507	−0.274134
$92$	0	0
$93$	5.23014	0.542341
$94$	0	0
$95$	7.77647	0.797849
$96$	0	0
$97$	7.38493	0.749826	0.374913	−	0.927060i	$-0.377673\pi$
$97$	7.38493	0.749826	0.374913	+	0.927060i	$0.377673\pi$
$98$	0	0
$99$	6.39154	0.642374
$100$	0	0
$101$	9.77647	0.972795	0.486398	−	0.873738i	$-0.338311\pi$
$101$	9.77647	0.972795	0.486398	+	0.873738i	$0.338311\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	−2.61507	−0.255205
$106$	0	0
$107$	10.3915	1.00459	0.502294	−	0.864697i	$-0.332489\pi$
$107$	10.3915	1.00459	0.502294	+	0.864697i	$0.332489\pi$
$108$	0	0
$109$	−8.23676	−0.788938	−0.394469	−	0.918909i	$-0.629072\pi$
$109$	−8.23676	−0.788938	−0.394469	+	0.918909i	$0.629072\pi$
$110$	0	0
$111$	−7.16140	−0.679730
$112$	0	0
$113$	11.0066	1.03542	0.517708	−	0.855558i	$-0.326786\pi$
$113$	11.0066	1.03542	0.517708	+	0.855558i	$0.326786\pi$
$114$	0	0
$115$	2.61507	0.243857
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	1.60846	0.147447
$120$	0	0
$121$	29.8518	2.71380
$122$	0	0
$123$	−7.16140	−0.645722
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.7831	−1.13432	−0.567158	−	0.823609i	$-0.691957\pi$
$127$	−12.7831	−1.13432	−0.567158	+	0.823609i	$0.691957\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	15.7765	1.37840	0.689198	−	0.724573i	$-0.257963\pi$
$131$	15.7765	1.37840	0.689198	+	0.724573i	$0.257963\pi$
$132$	0	0
$133$	20.3360	1.76336
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	16.0132	1.36810	0.684051	−	0.729434i	$-0.260216\pi$
$137$	16.0132	1.36810	0.684051	+	0.729434i	$0.260216\pi$
$138$	0	0
$139$	21.1614	1.79489	0.897443	−	0.441130i	$-0.145422\pi$
$139$	21.1614	1.79489	0.897443	+	0.441130i	$0.145422\pi$
$140$	0	0
$141$	−1.23014	−0.103597
$142$	0	0
$143$	6.39154	0.534488
$144$	0	0
$145$	−7.00662	−0.581868
$146$	0	0
$147$	0.161400	0.0133121
$148$	0	0
$149$	−17.6217	−1.44362	−0.721812	−	0.692089i	$-0.756691\pi$
$149$	−17.6217	−1.44362	−0.721812	+	0.692089i	$0.756691\pi$
$150$	0	0
$151$	15.5529	1.26568	0.632840	−	0.774282i	$-0.281889\pi$
$151$	15.5529	1.26568	0.632840	+	0.774282i	$0.281889\pi$
$152$	0	0
$153$	−0.615072	−0.0497256
$154$	0	0
$155$	5.23014	0.420095
$156$	0	0
$157$	5.55294	0.443173	0.221587	−	0.975141i	$-0.428876\pi$
$157$	5.55294	0.443173	0.221587	+	0.975141i	$0.428876\pi$
$158$	0	0
$159$	13.6217	1.08027
$160$	0	0
$161$	6.83860	0.538957
$162$	0	0
$163$	7.93126	0.621224	0.310612	−	0.950537i	$-0.399466\pi$
$163$	7.93126	0.621224	0.310612	+	0.950537i	$0.399466\pi$
$164$	0	0
$165$	6.39154	0.497581
$166$	0	0
$167$	4.00000	0.309529	0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000	0.309529	0.154765	+	0.987951i	$0.450538\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.77647	−0.594682
$172$	0	0
$173$	−17.5529	−1.33453	−0.667263	−	0.744822i	$-0.732534\pi$
$173$	−17.5529	−1.33453	−0.667263	+	0.744822i	$0.732534\pi$
$174$	0	0
$175$	−2.61507	−0.197681
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0.223528	0.0167073	0.00835363	−	0.999965i	$-0.497341\pi$
$179$	0.223528	0.0167073	0.00835363	+	0.999965i	$0.497341\pi$
$180$	0	0
$181$	15.1614	1.12694	0.563469	−	0.826137i	$-0.309466\pi$
$181$	15.1614	1.12694	0.563469	+	0.826137i	$0.309466\pi$
$182$	0	0
$183$	0.391544	0.0289437
$184$	0	0
$185$	−7.16140	−0.526517
$186$	0	0
$187$	−3.93126	−0.287482
$188$	0	0
$189$	2.61507	0.190218
$190$	0	0
$191$	5.53971	0.400840	0.200420	−	0.979710i	$-0.435769\pi$
$191$	5.53971	0.400840	0.200420	+	0.979710i	$0.435769\pi$
$192$	0	0
$193$	4.61507	0.332200	0.166100	−	0.986109i	$-0.446883\pi$
$193$	4.61507	0.332200	0.166100	+	0.986109i	$0.446883\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	−0.769857	−0.0548500	−0.0274250	−	0.999624i	$-0.508731\pi$
$197$	−0.769857	−0.0548500	−0.0274250	+	0.999624i	$0.508731\pi$
$198$	0	0
$199$	18.4603	1.30862	0.654308	−	0.756229i	$-0.272960\pi$
$199$	18.4603	1.30862	0.654308	+	0.756229i	$0.272960\pi$
$200$	0	0
$201$	−14.0132	−0.988417
$202$	0	0
$203$	−18.3228	−1.28601
$204$	0	0
$205$	−7.16140	−0.500174
$206$	0	0
$207$	−2.61507	−0.181760
$208$	0	0
$209$	−49.7037	−3.43807
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	0	0
$213$	5.16140	0.353653
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	13.6772	0.928469
$218$	0	0
$219$	−0.993385	−0.0671267
$220$	0	0
$221$	−0.615072	−0.0413742
$222$	0	0
$223$	11.4669	0.767881	0.383940	−	0.923358i	$-0.374567\pi$
$223$	11.4669	0.767881	0.383940	+	0.923358i	$0.374567\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−7.69043	−0.510432	−0.255216	−	0.966884i	$-0.582147\pi$
$227$	−7.69043	−0.510432	−0.255216	+	0.966884i	$0.582147\pi$
$228$	0	0
$229$	7.00662	0.463010	0.231505	−	0.972834i	$-0.425635\pi$
$229$	7.00662	0.463010	0.231505	+	0.972834i	$0.425635\pi$
$230$	0	0
$231$	16.7143	1.09972
$232$	0	0
$233$	17.7077	1.16007	0.580036	−	0.814591i	$-0.303038\pi$
$233$	17.7077	1.16007	0.580036	+	0.814591i	$0.303038\pi$
$234$	0	0
$235$	−1.23014	−0.0802457
$236$	0	0
$237$	−14.3915	−0.934831
$238$	0	0
$239$	17.9445	1.16073	0.580366	−	0.814356i	$-0.302909\pi$
$239$	17.9445	1.16073	0.580366	+	0.814356i	$0.302909\pi$
$240$	0	0
$241$	−0.460287	−0.0296497	−0.0148248	−	0.999890i	$-0.504719\pi$
$241$	−0.460287	−0.0296497	−0.0148248	+	0.999890i	$0.504719\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0.161400	0.0103115
$246$	0	0
$247$	−7.77647	−0.494805
$248$	0	0
$249$	18.0132	1.14154
$250$	0	0
$251$	−12.5596	−0.792752	−0.396376	−	0.918088i	$-0.629732\pi$
$251$	−12.5596	−0.792752	−0.396376	+	0.918088i	$0.629732\pi$
$252$	0	0
$253$	−16.7143	−1.05082
$254$	0	0
$255$	−0.615072	−0.0385173
$256$	0	0
$257$	−25.0198	−1.56070	−0.780348	−	0.625346i	$-0.784958\pi$
$257$	−25.0198	−1.56070	−0.780348	+	0.625346i	$0.784958\pi$
$258$	0	0
$259$	−18.7276	−1.16367
$260$	0	0
$261$	7.00662	0.433699
$262$	0	0
$263$	6.54633	0.403664	0.201832	−	0.979420i	$-0.435311\pi$
$263$	6.54633	0.403664	0.201832	+	0.979420i	$0.435311\pi$
$264$	0	0
$265$	13.6217	0.836774
$266$	0	0
$267$	−14.8518	−0.908917
$268$	0	0
$269$	1.77647	0.108313	0.0541567	−	0.998532i	$-0.482753\pi$
$269$	1.77647	0.108313	0.0541567	+	0.998532i	$0.482753\pi$
$270$	0	0
$271$	−28.3360	−1.72129	−0.860646	−	0.509204i	$-0.829940\pi$
$271$	−28.3360	−1.72129	−0.860646	+	0.509204i	$0.829940\pi$
$272$	0	0
$273$	2.61507	0.158271
$274$	0	0
$275$	6.39154	0.385425
$276$	0	0
$277$	−12.0132	−0.721805	−0.360903	−	0.932604i	$-0.617531\pi$
$277$	−12.0132	−0.721805	−0.360903	+	0.932604i	$0.617531\pi$
$278$	0	0
$279$	−5.23014	−0.313120
$280$	0	0
$281$	3.53971	0.211162	0.105581	−	0.994411i	$-0.466330\pi$
$281$	3.53971	0.211162	0.105581	+	0.994411i	$0.466330\pi$
$282$	0	0
$283$	−1.53971	−0.0915265	−0.0457632	−	0.998952i	$-0.514572\pi$
$283$	−1.53971	−0.0915265	−0.0457632	+	0.998952i	$0.514572\pi$
$284$	0	0
$285$	−7.77647	−0.460638
$286$	0	0
$287$	−18.7276	−1.10545
$288$	0	0
$289$	−16.6217	−0.977746
$290$	0	0
$291$	−7.38493	−0.432912
$292$	0	0
$293$	−19.2301	−1.12344	−0.561718	−	0.827328i	$-0.689860\pi$
$293$	−19.2301	−1.12344	−0.561718	+	0.827328i	$0.689860\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−6.39154	−0.370875
$298$	0	0
$299$	−2.61507	−0.151233
$300$	0	0
$301$	10.4603	0.602921
$302$	0	0
$303$	−9.77647	−0.561644
$304$	0	0
$305$	0.391544	0.0224197
$306$	0	0
$307$	−18.0820	−1.03199	−0.515996	−	0.856591i	$-0.672578\pi$
$307$	−18.0820	−1.03199	−0.515996	+	0.856591i	$0.672578\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	12.0132	0.679028	0.339514	−	0.940601i	$-0.389737\pi$
$313$	12.0132	0.679028	0.339514	+	0.940601i	$0.389737\pi$
$314$	0	0
$315$	2.61507	0.147343
$316$	0	0
$317$	14.7831	0.830301	0.415150	−	0.909753i	$-0.363729\pi$
$317$	14.7831	0.830301	0.415150	+	0.909753i	$0.363729\pi$
$318$	0	0
$319$	44.7831	2.50737
$320$	0	0
$321$	−10.3915	−0.579999
$322$	0	0
$323$	4.78309	0.266138
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	8.23676	0.455494
$328$	0	0
$329$	−3.21691	−0.177354
$330$	0	0
$331$	15.4669	0.850138	0.425069	−	0.905161i	$-0.360250\pi$
$331$	15.4669	0.850138	0.425069	+	0.905161i	$0.360250\pi$
$332$	0	0
$333$	7.16140	0.392442
$334$	0	0
$335$	−14.0132	−0.765625
$336$	0	0
$337$	17.6904	0.963659	0.481830	−	0.876265i	$-0.339972\pi$
$337$	17.6904	0.963659	0.481830	+	0.876265i	$0.339972\pi$
$338$	0	0
$339$	−11.0066	−0.597797
$340$	0	0
$341$	−33.4287	−1.81026
$342$	0	0
$343$	18.7276	1.01119
$344$	0	0
$345$	−2.61507	−0.140791
$346$	0	0
$347$	25.9445	1.39277	0.696387	−	0.717667i	$-0.254790\pi$
$347$	25.9445	1.39277	0.696387	+	0.717667i	$0.254790\pi$
$348$	0	0
$349$	−21.7765	−1.16567	−0.582834	−	0.812591i	$-0.698056\pi$
$349$	−21.7765	−1.16567	−0.582834	+	0.812591i	$0.698056\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	18.4735	0.983246	0.491623	−	0.870808i	$-0.336404\pi$
$353$	18.4735	0.983246	0.491623	+	0.870808i	$0.336404\pi$
$354$	0	0
$355$	5.16140	0.273939
$356$	0	0
$357$	−1.60846	−0.0851285
$358$	0	0
$359$	11.5529	0.609741	0.304871	−	0.952394i	$-0.401387\pi$
$359$	11.5529	0.609741	0.304871	+	0.952394i	$0.401387\pi$
$360$	0	0
$361$	41.4735	2.18282
$362$	0	0
$363$	−29.8518	−1.56681
$364$	0	0
$365$	−0.993385	−0.0519961
$366$	0	0
$367$	33.5662	1.75214	0.876070	−	0.482184i	$-0.160156\pi$
$367$	33.5662	1.75214	0.876070	+	0.482184i	$0.160156\pi$
$368$	0	0
$369$	7.16140	0.372808
$370$	0	0
$371$	35.6217	1.84939
$372$	0	0
$373$	0.769857	0.0398617	0.0199308	−	0.999801i	$-0.493655\pi$
$373$	0.769857	0.0398617	0.0199308	+	0.999801i	$0.493655\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	7.00662	0.360859
$378$	0	0
$379$	−28.2500	−1.45110	−0.725552	−	0.688167i	$-0.758416\pi$
$379$	−28.2500	−1.45110	−0.725552	+	0.688167i	$0.758416\pi$
$380$	0	0
$381$	12.7831	0.654897
$382$	0	0
$383$	0.783087	0.0400139	0.0200069	−	0.999800i	$-0.493631\pi$
$383$	0.783087	0.0400139	0.0200069	+	0.999800i	$0.493631\pi$
$384$	0	0
$385$	16.7143	0.851842
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	−14.2235	−0.721161	−0.360581	−	0.932728i	$-0.617421\pi$
$389$	−14.2235	−0.721161	−0.360581	+	0.932728i	$0.617421\pi$
$390$	0	0
$391$	1.60846	0.0813431
$392$	0	0
$393$	−15.7765	−0.795818
$394$	0	0
$395$	−14.3915	−0.724117
$396$	0	0
$397$	−0.838600	−0.0420881	−0.0210441	−	0.999779i	$-0.506699\pi$
$397$	−0.838600	−0.0420881	−0.0210441	+	0.999779i	$0.506699\pi$
$398$	0	0
$399$	−20.3360	−1.01807
$400$	0	0
$401$	−14.4735	−0.722773	−0.361386	−	0.932416i	$-0.617696\pi$
$401$	−14.4735	−0.722773	−0.361386	+	0.932416i	$0.617696\pi$
$402$	0	0
$403$	−5.23014	−0.260532
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	45.7724	2.26885
$408$	0	0
$409$	9.55294	0.472363	0.236181	−	0.971709i	$-0.424104\pi$
$409$	9.55294	0.472363	0.236181	+	0.971709i	$0.424104\pi$
$410$	0	0
$411$	−16.0132	−0.789874
$412$	0	0
$413$	0	0
$414$	0	0
$415$	18.0132	0.884235
$416$	0	0
$417$	−21.1614	−1.03628
$418$	0	0
$419$	−25.7897	−1.25991	−0.629955	−	0.776632i	$-0.716927\pi$
$419$	−25.7897	−1.25991	−0.629955	+	0.776632i	$0.716927\pi$
$420$	0	0
$421$	−23.7897	−1.15944	−0.579720	−	0.814816i	$-0.696838\pi$
$421$	−23.7897	−1.15944	−0.579720	+	0.814816i	$0.696838\pi$
$422$	0	0
$423$	1.23014	0.0598116
$424$	0	0
$425$	−0.615072	−0.0298354
$426$	0	0
$427$	1.02391	0.0495507
$428$	0	0
$429$	−6.39154	−0.308587
$430$	0	0
$431$	32.4735	1.56419	0.782097	−	0.623157i	$-0.214150\pi$
$431$	32.4735	1.56419	0.782097	+	0.623157i	$0.214150\pi$
$432$	0	0
$433$	−3.23014	−0.155231	−0.0776154	−	0.996983i	$-0.524731\pi$
$433$	−3.23014	−0.155231	−0.0776154	+	0.996983i	$0.524731\pi$
$434$	0	0
$435$	7.00662	0.335941
$436$	0	0
$437$	20.3360	0.972804
$438$	0	0
$439$	12.0687	0.576010	0.288005	−	0.957629i	$-0.407008\pi$
$439$	12.0687	0.576010	0.288005	+	0.957629i	$0.407008\pi$
$440$	0	0
$441$	−0.161400	−0.00768573
$442$	0	0
$443$	12.4048	0.589369	0.294684	−	0.955595i	$-0.404786\pi$
$443$	12.4048	0.589369	0.294684	+	0.955595i	$0.404786\pi$
$444$	0	0
$445$	−14.8518	−0.704044
$446$	0	0
$447$	17.6217	0.833477
$448$	0	0
$449$	−21.9313	−1.03500	−0.517500	−	0.855683i	$-0.673137\pi$
$449$	−21.9313	−1.03500	−0.517500	+	0.855683i	$0.673137\pi$
$450$	0	0
$451$	45.7724	2.15534
$452$	0	0
$453$	−15.5529	−0.730741
$454$	0	0
$455$	2.61507	0.122596
$456$	0	0
$457$	−10.1812	−0.476259	−0.238129	−	0.971233i	$-0.576534\pi$
$457$	−10.1812	−0.476259	−0.238129	+	0.971233i	$0.576534\pi$
$458$	0	0
$459$	0.615072	0.0287091
$460$	0	0
$461$	7.63492	0.355594	0.177797	−	0.984067i	$-0.443103\pi$
$461$	7.63492	0.355594	0.177797	+	0.984067i	$0.443103\pi$
$462$	0	0
$463$	−28.6283	−1.33047	−0.665235	−	0.746634i	$-0.731669\pi$
$463$	−28.6283	−1.33047	−0.665235	+	0.746634i	$0.731669\pi$
$464$	0	0
$465$	−5.23014	−0.242542
$466$	0	0
$467$	23.3121	1.07876	0.539378	−	0.842064i	$-0.318659\pi$
$467$	23.3121	1.07876	0.539378	+	0.842064i	$0.318659\pi$
$468$	0	0
$469$	−36.6456	−1.69214
$470$	0	0
$471$	−5.55294	−0.255866
$472$	0	0
$473$	−25.5662	−1.17553
$474$	0	0
$475$	−7.77647	−0.356809
$476$	0	0
$477$	−13.6217	−0.623694
$478$	0	0
$479$	−8.37831	−0.382815	−0.191407	−	0.981511i	$-0.561305\pi$
$479$	−8.37831	−0.382815	−0.191407	+	0.981511i	$0.561305\pi$
$480$	0	0
$481$	7.16140	0.326532
$482$	0	0
$483$	−6.83860	−0.311167
$484$	0	0
$485$	−7.38493	−0.335332
$486$	0	0
$487$	−28.9379	−1.31130	−0.655650	−	0.755065i	$-0.727605\pi$
$487$	−28.9379	−1.31130	−0.655650	+	0.755065i	$0.727605\pi$
$488$	0	0
$489$	−7.93126	−0.358664
$490$	0	0
$491$	−21.0066	−0.948015	−0.474008	−	0.880521i	$-0.657193\pi$
$491$	−21.0066	−0.948015	−0.474008	+	0.880521i	$0.657193\pi$
$492$	0	0
$493$	−4.30957	−0.194093
$494$	0	0
$495$	−6.39154	−0.287279
$496$	0	0
$497$	13.4974	0.605443
$498$	0	0
$499$	23.7765	1.06438	0.532191	−	0.846625i	$-0.321369\pi$
$499$	23.7765	1.06438	0.532191	+	0.846625i	$0.321369\pi$
$500$	0	0
$501$	−4.00000	−0.178707
$502$	0	0
$503$	−14.0993	−0.628656	−0.314328	−	0.949315i	$-0.601779\pi$
$503$	−14.0993	−0.628656	−0.314328	+	0.949315i	$0.601779\pi$
$504$	0	0
$505$	−9.77647	−0.435047
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−10.0687	−0.446289	−0.223145	−	0.974785i	$-0.571632\pi$
$509$	−10.0687	−0.446289	−0.223145	+	0.974785i	$0.571632\pi$
$510$	0	0
$511$	−2.59777	−0.114919
$512$	0	0
$513$	7.77647	0.343340
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.86251	0.345793
$518$	0	0
$519$	17.5529	0.770489
$520$	0	0
$521$	14.4735	0.634096	0.317048	−	0.948409i	$-0.397308\pi$
$521$	14.4735	0.634096	0.317048	+	0.948409i	$0.397308\pi$
$522$	0	0
$523$	6.46029	0.282489	0.141244	−	0.989975i	$-0.454890\pi$
$523$	6.46029	0.282489	0.141244	+	0.989975i	$0.454890\pi$
$524$	0	0
$525$	2.61507	0.114131
$526$	0	0
$527$	3.21691	0.140131
$528$	0	0
$529$	−16.1614	−0.702670
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.16140	0.310195
$534$	0	0
$535$	−10.3915	−0.449266
$536$	0	0
$537$	−0.223528	−0.00964594
$538$	0	0
$539$	−1.03160	−0.0444341
$540$	0	0
$541$	−3.00662	−0.129264	−0.0646322	−	0.997909i	$-0.520587\pi$
$541$	−3.00662	−0.129264	−0.0646322	+	0.997909i	$0.520587\pi$
$542$	0	0
$543$	−15.1614	−0.650638
$544$	0	0
$545$	8.23676	0.352824
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	−0.391544	−0.0167107
$550$	0	0
$551$	−54.4867	−2.32121
$552$	0	0
$553$	−37.6349	−1.60040
$554$	0	0
$555$	7.16140	0.303985
$556$	0	0
$557$	−1.07943	−0.0457368	−0.0228684	−	0.999738i	$-0.507280\pi$
$557$	−1.07943	−0.0457368	−0.0228684	+	0.999738i	$0.507280\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	3.93126	0.165978
$562$	0	0
$563$	−33.1879	−1.39870	−0.699351	−	0.714779i	$-0.746527\pi$
$563$	−33.1879	−1.39870	−0.699351	+	0.714779i	$0.746527\pi$
$564$	0	0
$565$	−11.0066	−0.463052
$566$	0	0
$567$	−2.61507	−0.109823
$568$	0	0
$569$	−12.7963	−0.536450	−0.268225	−	0.963356i	$-0.586437\pi$
$569$	−12.7963	−0.536450	−0.268225	+	0.963356i	$0.586437\pi$
$570$	0	0
$571$	41.9445	1.75532	0.877661	−	0.479282i	$-0.159103\pi$
$571$	41.9445	1.75532	0.877661	+	0.479282i	$0.159103\pi$
$572$	0	0
$573$	−5.53971	−0.231425
$574$	0	0
$575$	−2.61507	−0.109056
$576$	0	0
$577$	38.9379	1.62100	0.810502	−	0.585735i	$-0.199194\pi$
$577$	38.9379	1.62100	0.810502	+	0.585735i	$0.199194\pi$
$578$	0	0
$579$	−4.61507	−0.191796
$580$	0	0
$581$	47.1059	1.95428
$582$	0	0
$583$	−87.0636	−3.60581
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	−29.2301	−1.20646	−0.603229	−	0.797568i	$-0.706119\pi$
$587$	−29.2301	−1.20646	−0.603229	+	0.797568i	$0.706119\pi$
$588$	0	0
$589$	40.6721	1.67586
$590$	0	0
$591$	0.769857	0.0316677
$592$	0	0
$593$	11.6772	0.479525	0.239763	−	0.970832i	$-0.422930\pi$
$593$	11.6772	0.479525	0.239763	+	0.970832i	$0.422930\pi$
$594$	0	0
$595$	−1.60846	−0.0659403
$596$	0	0
$597$	−18.4603	−0.755529
$598$	0	0
$599$	−36.0265	−1.47200	−0.736001	−	0.676981i	$-0.763288\pi$
$599$	−36.0265	−1.47200	−0.736001	+	0.676981i	$0.763288\pi$
$600$	0	0
$601$	−0.0555123	−0.00226439	−0.00113220	−	0.999999i	$-0.500360\pi$
$601$	−0.0555123	−0.00226439	−0.00113220	+	0.999999i	$0.500360\pi$
$602$	0	0
$603$	14.0132	0.570663
$604$	0	0
$605$	−29.8518	−1.21365
$606$	0	0
$607$	−33.5662	−1.36241	−0.681204	−	0.732093i	$-0.738544\pi$
$607$	−33.5662	−1.36241	−0.681204	+	0.732093i	$0.738544\pi$
$608$	0	0
$609$	18.3228	0.742477
$610$	0	0
$611$	1.23014	0.0497663
$612$	0	0
$613$	25.4842	1.02930	0.514649	−	0.857401i	$-0.327922\pi$
$613$	25.4842	1.02930	0.514649	+	0.857401i	$0.327922\pi$
$614$	0	0
$615$	7.16140	0.288776
$616$	0	0
$617$	−9.55294	−0.384587	−0.192294	−	0.981337i	$-0.561593\pi$
$617$	−9.55294	−0.384587	−0.192294	+	0.981337i	$0.561593\pi$
$618$	0	0
$619$	26.2368	1.05454	0.527272	−	0.849696i	$-0.323215\pi$
$619$	26.2368	1.05454	0.527272	+	0.849696i	$0.323215\pi$
$620$	0	0
$621$	2.61507	0.104939
$622$	0	0
$623$	−38.8386	−1.55604
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	49.7037	1.98497
$628$	0	0
$629$	−4.40477	−0.175630
$630$	0	0
$631$	18.4603	0.734892	0.367446	−	0.930045i	$-0.380232\pi$
$631$	18.4603	0.734892	0.367446	+	0.930045i	$0.380232\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	0	0
$635$	12.7831	0.507281
$636$	0	0
$637$	−0.161400	−0.00639492
$638$	0	0
$639$	−5.16140	−0.204182
$640$	0	0
$641$	−41.8890	−1.65452	−0.827258	−	0.561823i	$-0.810100\pi$
$641$	−41.8890	−1.65452	−0.827258	+	0.561823i	$0.810100\pi$
$642$	0	0
$643$	−23.6217	−0.931548	−0.465774	−	0.884904i	$-0.654224\pi$
$643$	−23.6217	−0.931548	−0.465774	+	0.884904i	$0.654224\pi$
$644$	0	0
$645$	−4.00000	−0.157500
$646$	0	0
$647$	15.2607	0.599959	0.299979	−	0.953946i	$-0.403020\pi$
$647$	15.2607	0.599959	0.299979	+	0.953946i	$0.403020\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−13.6772	−0.536052
$652$	0	0
$653$	22.0000	0.860927	0.430463	−	0.902608i	$-0.358350\pi$
$653$	22.0000	0.860927	0.430463	+	0.902608i	$0.358350\pi$
$654$	0	0
$655$	−15.7765	−0.616438
$656$	0	0
$657$	0.993385	0.0387556
$658$	0	0
$659$	15.0198	0.585090	0.292545	−	0.956252i	$-0.405498\pi$
$659$	15.0198	0.585090	0.292545	+	0.956252i	$0.405498\pi$
$660$	0	0
$661$	−31.7897	−1.23648	−0.618238	−	0.785991i	$-0.712153\pi$
$661$	−31.7897	−1.23648	−0.618238	+	0.785991i	$0.712153\pi$
$662$	0	0
$663$	0.615072	0.0238874
$664$	0	0
$665$	−20.3360	−0.788597
$666$	0	0
$667$	−18.3228	−0.709462
$668$	0	0
$669$	−11.4669	−0.443336
$670$	0	0
$671$	−2.50257	−0.0966106
$672$	0	0
$673$	38.0265	1.46581	0.732906	−	0.680330i	$-0.238163\pi$
$673$	38.0265	1.46581	0.732906	+	0.680330i	$0.238163\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−42.7143	−1.64165	−0.820823	−	0.571183i	$-0.806485\pi$
$677$	−42.7143	−1.64165	−0.820823	+	0.571183i	$0.806485\pi$
$678$	0	0
$679$	−19.3121	−0.741131
$680$	0	0
$681$	7.69043	0.294698
$682$	0	0
$683$	21.0927	0.807088	0.403544	−	0.914960i	$-0.367778\pi$
$683$	21.0927	0.807088	0.403544	+	0.914960i	$0.367778\pi$
$684$	0	0
$685$	−16.0132	−0.611834
$686$	0	0
$687$	−7.00662	−0.267319
$688$	0	0
$689$	−13.6217	−0.518945
$690$	0	0
$691$	−31.1573	−1.18528	−0.592640	−	0.805467i	$-0.701914\pi$
$691$	−31.1573	−1.18528	−0.592640	+	0.805467i	$0.701914\pi$
$692$	0	0
$693$	−16.7143	−0.634925
$694$	0	0
$695$	−21.1614	−0.802698
$696$	0	0
$697$	−4.40477	−0.166843
$698$	0	0
$699$	−17.7077	−0.669768
$700$	0	0
$701$	40.8824	1.54411	0.772053	−	0.635558i	$-0.219230\pi$
$701$	40.8824	1.54411	0.772053	+	0.635558i	$0.219230\pi$
$702$	0	0
$703$	−55.6904	−2.10040
$704$	0	0
$705$	1.23014	0.0463299
$706$	0	0
$707$	−25.5662	−0.961515
$708$	0	0
$709$	4.68381	0.175904	0.0879522	−	0.996125i	$-0.471968\pi$
$709$	4.68381	0.175904	0.0879522	+	0.996125i	$0.471968\pi$
$710$	0	0
$711$	14.3915	0.539725
$712$	0	0
$713$	13.6772	0.512215
$714$	0	0
$715$	−6.39154	−0.239030
$716$	0	0
$717$	−17.9445	−0.670149
$718$	0	0
$719$	0.447056	0.0166724	0.00833619	−	0.999965i	$-0.497346\pi$
$719$	0.447056	0.0166724	0.00833619	+	0.999965i	$0.497346\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0.460287	0.0171182
$724$	0	0
$725$	7.00662	0.260219
$726$	0	0
$727$	33.2566	1.23342	0.616710	−	0.787191i	$-0.288465\pi$
$727$	33.2566	1.23342	0.616710	+	0.787191i	$0.288465\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.46029	0.0909970
$732$	0	0
$733$	−14.3783	−0.531075	−0.265538	−	0.964101i	$-0.585549\pi$
$733$	−14.3783	−0.531075	−0.265538	+	0.964101i	$0.585549\pi$
$734$	0	0
$735$	−0.161400	−0.00595334
$736$	0	0
$737$	89.5662	3.29921
$738$	0	0
$739$	17.3426	0.637960	0.318980	−	0.947762i	$-0.396660\pi$
$739$	17.3426	0.637960	0.318980	+	0.947762i	$0.396660\pi$
$740$	0	0
$741$	7.77647	0.285676
$742$	0	0
$743$	−35.2434	−1.29295	−0.646477	−	0.762933i	$-0.723758\pi$
$743$	−35.2434	−1.29295	−0.646477	+	0.762933i	$0.723758\pi$
$744$	0	0
$745$	17.6217	0.645609
$746$	0	0
$747$	−18.0132	−0.659070
$748$	0	0
$749$	−27.1746	−0.992939
$750$	0	0
$751$	13.6349	0.497545	0.248773	−	0.968562i	$-0.419973\pi$
$751$	13.6349	0.497545	0.248773	+	0.968562i	$0.419973\pi$
$752$	0	0
$753$	12.5596	0.457696
$754$	0	0
$755$	−15.5529	−0.566030
$756$	0	0
$757$	−28.3228	−1.02941	−0.514705	−	0.857367i	$-0.672098\pi$
$757$	−28.3228	−1.02941	−0.514705	+	0.857367i	$0.672098\pi$
$758$	0	0
$759$	16.7143	0.606692
$760$	0	0
$761$	11.0927	0.402109	0.201054	−	0.979580i	$-0.435563\pi$
$761$	11.0927	0.402109	0.201054	+	0.979580i	$0.435563\pi$
$762$	0	0
$763$	21.5397	0.779790
$764$	0	0
$765$	0.615072	0.0222380
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−32.3228	−1.16559	−0.582795	−	0.812619i	$-0.698041\pi$
$769$	−32.3228	−1.16559	−0.582795	+	0.812619i	$0.698041\pi$
$770$	0	0
$771$	25.0198	0.901068
$772$	0	0
$773$	6.33603	0.227891	0.113946	−	0.993487i	$-0.463651\pi$
$773$	6.33603	0.227891	0.113946	+	0.993487i	$0.463651\pi$
$774$	0	0
$775$	−5.23014	−0.187872
$776$	0	0
$777$	18.7276	0.671848
$778$	0	0
$779$	−55.6904	−1.99532
$780$	0	0
$781$	−32.9893	−1.18045
$782$	0	0
$783$	−7.00662	−0.250396
$784$	0	0
$785$	−5.55294	−0.198193
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	0	0
$789$	−6.54633	−0.233055
$790$	0	0
$791$	−28.7831	−1.02341
$792$	0	0
$793$	−0.391544	−0.0139041
$794$	0	0
$795$	−13.6217	−0.483111
$796$	0	0
$797$	−50.4048	−1.78543	−0.892714	−	0.450623i	$-0.851202\pi$
$797$	−50.4048	−1.78543	−0.892714	+	0.450623i	$0.851202\pi$
$798$	0	0
$799$	−0.756626	−0.0267675
$800$	0	0
$801$	14.8518	0.524764
$802$	0	0
$803$	6.34926	0.224061
$804$	0	0
$805$	−6.83860	−0.241029
$806$	0	0
$807$	−1.77647	−0.0625348
$808$	0	0
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	−28.5596	−1.00286	−0.501431	−	0.865198i	$-0.667193\pi$
$811$	−28.5596	−1.00286	−0.501431	+	0.865198i	$0.667193\pi$
$812$	0	0
$813$	28.3360	0.993788
$814$	0	0
$815$	−7.93126	−0.277820
$816$	0	0
$817$	31.1059	1.08826
$818$	0	0
$819$	−2.61507	−0.0913780
$820$	0	0
$821$	−3.94449	−0.137664	−0.0688318	−	0.997628i	$-0.521927\pi$
$821$	−3.94449	−0.137664	−0.0688318	+	0.997628i	$0.521927\pi$
$822$	0	0
$823$	20.9206	0.729245	0.364623	−	0.931155i	$-0.381198\pi$
$823$	20.9206	0.729245	0.364623	+	0.931155i	$0.381198\pi$
$824$	0	0
$825$	−6.39154	−0.222525
$826$	0	0
$827$	−49.8757	−1.73435	−0.867175	−	0.498004i	$-0.834067\pi$
$827$	−49.8757	−1.73435	−0.867175	+	0.498004i	$0.834067\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	0	0
$831$	12.0132	0.416734
$832$	0	0
$833$	0.0992728	0.00343960
$834$	0	0
$835$	−4.00000	−0.138426
$836$	0	0
$837$	5.23014	0.180780
$838$	0	0
$839$	54.8651	1.89415	0.947076	−	0.321009i	$-0.104022\pi$
$839$	54.8651	1.89415	0.947076	+	0.321009i	$0.104022\pi$
$840$	0	0
$841$	20.0927	0.692850
$842$	0	0
$843$	−3.53971	−0.121914
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−78.0647	−2.68233
$848$	0	0
$849$	1.53971	0.0528428
$850$	0	0
$851$	−18.7276	−0.641973
$852$	0	0
$853$	−31.8070	−1.08905	−0.544526	−	0.838744i	$-0.683290\pi$
$853$	−31.8070	−1.08905	−0.544526	+	0.838744i	$0.683290\pi$
$854$	0	0
$855$	7.77647	0.265950
$856$	0	0
$857$	−31.4114	−1.07299	−0.536496	−	0.843903i	$-0.680252\pi$
$857$	−31.4114	−1.07299	−0.536496	+	0.843903i	$0.680252\pi$
$858$	0	0
$859$	57.6349	1.96648	0.983239	−	0.182321i	$-0.0583611\pi$
$859$	57.6349	1.96648	0.983239	+	0.182321i	$0.0583611\pi$
$860$	0	0
$861$	18.7276	0.638234
$862$	0	0
$863$	−9.40223	−0.320056	−0.160028	−	0.987113i	$-0.551158\pi$
$863$	−9.40223	−0.320056	−0.160028	+	0.987113i	$0.551158\pi$
$864$	0	0
$865$	17.5529	0.596818
$866$	0	0
$867$	16.6217	0.564502
$868$	0	0
$869$	91.9842	3.12035
$870$	0	0
$871$	14.0132	0.474820
$872$	0	0
$873$	7.38493	0.249942
$874$	0	0
$875$	2.61507	0.0884056
$876$	0	0
$877$	46.9470	1.58529	0.792644	−	0.609684i	$-0.208704\pi$
$877$	46.9470	1.58529	0.792644	+	0.609684i	$0.208704\pi$
$878$	0	0
$879$	19.2301	0.648617
$880$	0	0
$881$	44.0132	1.48284	0.741422	−	0.671039i	$-0.234152\pi$
$881$	44.0132	1.48284	0.741422	+	0.671039i	$0.234152\pi$
$882$	0	0
$883$	22.4603	0.755849	0.377924	−	0.925836i	$-0.376638\pi$
$883$	22.4603	0.755849	0.377924	+	0.925836i	$0.376638\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−12.3187	−0.413623	−0.206811	−	0.978381i	$-0.566309\pi$
$887$	−12.3187	−0.413623	−0.206811	+	0.978381i	$0.566309\pi$
$888$	0	0
$889$	33.4287	1.12116
$890$	0	0
$891$	6.39154	0.214125
$892$	0	0
$893$	−9.56617	−0.320120
$894$	0	0
$895$	−0.223528	−0.00747172
$896$	0	0
$897$	2.61507	0.0873147
$898$	0	0
$899$	−36.6456	−1.22220
$900$	0	0
$901$	8.37831	0.279122
$902$	0	0
$903$	−10.4603	−0.348097
$904$	0	0
$905$	−15.1614	−0.503982
$906$	0	0
$907$	6.32280	0.209945	0.104973	−	0.994475i	$-0.466525\pi$
$907$	6.32280	0.209945	0.104973	+	0.994475i	$0.466525\pi$
$908$	0	0
$909$	9.77647	0.324265
$910$	0	0
$911$	11.5794	0.383643	0.191821	−	0.981430i	$-0.438561\pi$
$911$	11.5794	0.383643	0.191821	+	0.981430i	$0.438561\pi$
$912$	0	0
$913$	−115.132	−3.81032
$914$	0	0
$915$	−0.391544	−0.0129440
$916$	0	0
$917$	−41.2566	−1.36241
$918$	0	0
$919$	17.4710	0.576314	0.288157	−	0.957583i	$-0.406957\pi$
$919$	17.4710	0.576314	0.288157	+	0.957583i	$0.406957\pi$
$920$	0	0
$921$	18.0820	0.595821
$922$	0	0
$923$	−5.16140	−0.169890
$924$	0	0
$925$	7.16140	0.235465
$926$	0	0
$927$	0	0
$928$	0	0
$929$	34.8783	1.14432	0.572160	−	0.820142i	$-0.306106\pi$
$929$	34.8783	1.14432	0.572160	+	0.820142i	$0.306106\pi$
$930$	0	0
$931$	1.25513	0.0411351
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.93126	0.128566
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	−12.0132	−0.392037
$940$	0	0
$941$	47.3253	1.54276	0.771381	−	0.636373i	$-0.219566\pi$
$941$	47.3253	1.54276	0.771381	+	0.636373i	$0.219566\pi$
$942$	0	0
$943$	−18.7276	−0.609854
$944$	0	0
$945$	−2.61507	−0.0850683
$946$	0	0
$947$	−41.1191	−1.33619	−0.668096	−	0.744075i	$-0.732890\pi$
$947$	−41.1191	−1.33619	−0.668096	+	0.744075i	$0.732890\pi$
$948$	0	0
$949$	0.993385	0.0322466
$950$	0	0
$951$	−14.7831	−0.479374
$952$	0	0
$953$	−5.53564	−0.179317	−0.0896586	−	0.995973i	$-0.528578\pi$
$953$	−5.53564	−0.179317	−0.0896586	+	0.995973i	$0.528578\pi$
$954$	0	0
$955$	−5.53971	−0.179261
$956$	0	0
$957$	−44.7831	−1.44763
$958$	0	0
$959$	−41.8757	−1.35224
$960$	0	0
$961$	−3.64560	−0.117600
$962$	0	0
$963$	10.3915	0.334863
$964$	0	0
$965$	−4.61507	−0.148564
$966$	0	0
$967$	21.4801	0.690754	0.345377	−	0.938464i	$-0.387751\pi$
$967$	21.4801	0.690754	0.345377	+	0.938464i	$0.387751\pi$
$968$	0	0
$969$	−4.78309	−0.153655
$970$	0	0
$971$	31.4669	1.00982	0.504910	−	0.863172i	$-0.331526\pi$
$971$	31.4669	1.00982	0.504910	+	0.863172i	$0.331526\pi$
$972$	0	0
$973$	−55.3386	−1.77407
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	16.3228	0.522213	0.261106	−	0.965310i	$-0.415913\pi$
$977$	16.3228	0.522213	0.261106	+	0.965310i	$0.415913\pi$
$978$	0	0
$979$	94.9261	3.03385
$980$	0	0
$981$	−8.23676	−0.262979
$982$	0	0
$983$	−24.9206	−0.794843	−0.397421	−	0.917636i	$-0.630095\pi$
$983$	−24.9206	−0.794843	−0.397421	+	0.917636i	$0.630095\pi$
$984$	0	0
$985$	0.769857	0.0245297
$986$	0	0
$987$	3.21691	0.102395
$988$	0	0
$989$	10.4603	0.332618
$990$	0	0
$991$	−52.8783	−1.67973	−0.839867	−	0.542792i	$-0.817367\pi$
$991$	−52.8783	−1.67973	−0.839867	+	0.542792i	$0.817367\pi$
$992$	0	0
$993$	−15.4669	−0.490827
$994$	0	0
$995$	−18.4603	−0.585230
$996$	0	0
$997$	32.1507	1.01822	0.509112	−	0.860700i	$-0.329974\pi$
$997$	32.1507	1.01822	0.509112	+	0.860700i	$0.329974\pi$
$998$	0	0
$999$	−7.16140	−0.226577

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.a.bh.1.2		3
3.2	odd	2		9360.2.a.db.1.2		3
4.3	odd	2		1560.2.a.p.1.2	✓	3
12.11	even	2		4680.2.a.bl.1.2		3
20.19	odd	2		7800.2.a.bf.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.a.p.1.2	✓	3	4.3	odd	2
3120.2.a.bh.1.2		3	1.1	even	1	trivial
4680.2.a.bl.1.2		3	12.11	even	2
7800.2.a.bf.1.2		3	20.19	odd	2
9360.2.a.db.1.2		3	3.2	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:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -2.61507 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} -2.61507 q^{7} +1.00000 q^{9} +6.39154 q^{11} +1.00000 q^{13} +1.00000 q^{15} -0.615072 q^{17} -7.77647 q^{19} +2.61507 q^{21} -2.61507 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.00662 q^{29} -5.23014 q^{31} -6.39154 q^{33} +2.61507 q^{35} +7.16140 q^{37} -1.00000 q^{39} +7.16140 q^{41} -4.00000 q^{43} -1.00000 q^{45} +1.23014 q^{47} -0.161400 q^{49} +0.615072 q^{51} -13.6217 q^{53} -6.39154 q^{55} +7.77647 q^{57} -0.391544 q^{61} -2.61507 q^{63} -1.00000 q^{65} +14.0132 q^{67} +2.61507 q^{69} -5.16140 q^{71} +0.993385 q^{73} -1.00000 q^{75} -16.7143 q^{77} +14.3915 q^{79} +1.00000 q^{81} -18.0132 q^{83} +0.615072 q^{85} -7.00662 q^{87} +14.8518 q^{89} -2.61507 q^{91} +5.23014 q^{93} +7.77647 q^{95} +7.38493 q^{97} +6.39154 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + 3 q^{15} + q^{17} - 4 q^{19} + 5 q^{21} - 5 q^{23} + 3 q^{25} - 3 q^{27} - 4 q^{29} - 10 q^{31} + 3 q^{33} + 5 q^{35} + 5 q^{37} - 3 q^{39} + 5 q^{41} - 12 q^{43} - 3 q^{45} - 2 q^{47} + 16 q^{49} - q^{51} - 13 q^{53} + 3 q^{55} + 4 q^{57} + 21 q^{61} - 5 q^{63} - 3 q^{65} - 8 q^{67} + 5 q^{69} + q^{71} + 28 q^{73} - 3 q^{75} + 5 q^{77} + 21 q^{79} + 3 q^{81} - 4 q^{83} - q^{85} + 4 q^{87} + 11 q^{89} - 5 q^{91} + 10 q^{93} + 4 q^{95} + 25 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + 3 * q^15 + q^17 - 4 * q^19 + 5 * q^21 - 5 * q^23 + 3 * q^25 - 3 * q^27 - 4 * q^29 - 10 * q^31 + 3 * q^33 + 5 * q^35 + 5 * q^37 - 3 * q^39 + 5 * q^41 - 12 * q^43 - 3 * q^45 - 2 * q^47 + 16 * q^49 - q^51 - 13 * q^53 + 3 * q^55 + 4 * q^57 + 21 * q^61 - 5 * q^63 - 3 * q^65 - 8 * q^67 + 5 * q^69 + q^71 + 28 * q^73 - 3 * q^75 + 5 * q^77 + 21 * q^79 + 3 * q^81 - 4 * q^83 - q^85 + 4 * q^87 + 11 * q^89 - 5 * q^91 + 10 * q^93 + 4 * q^95 + 25 * q^97 - 3 * q^99

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