LMFDB - Embedded newform 7605.2.a.bk.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7605,2,Mod(1,7605)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7605.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7605, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,0,4,2,0,2,12,0,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$60.7262307372$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051 q^{2} -1.46410 q^{4} +1.00000 q^{5} +4.46410 q^{7} +2.53590 q^{8} -0.732051 q^{10} +3.46410 q^{11} -3.26795 q^{14} +1.07180 q^{16} +6.73205 q^{17} -5.46410 q^{19} -1.46410 q^{20} -2.53590 q^{22} +0.535898 q^{23} +1.00000 q^{25} -6.53590 q^{28} +2.73205 q^{29} -3.19615 q^{31} -5.85641 q^{32} -4.92820 q^{34} +4.46410 q^{35} +4.00000 q^{37} +4.00000 q^{38} +2.53590 q^{40} +5.26795 q^{41} -0.267949 q^{43} -5.07180 q^{44} -0.392305 q^{46} +0.196152 q^{47} +12.9282 q^{49} -0.732051 q^{50} +6.92820 q^{53} +3.46410 q^{55} +11.3205 q^{56} -2.00000 q^{58} +7.26795 q^{59} +4.46410 q^{61} +2.33975 q^{62} +2.14359 q^{64} +12.4641 q^{67} -9.85641 q^{68} -3.26795 q^{70} +12.7321 q^{71} -15.3923 q^{73} -2.92820 q^{74} +8.00000 q^{76} +15.4641 q^{77} +1.92820 q^{79} +1.07180 q^{80} -3.85641 q^{82} +2.53590 q^{83} +6.73205 q^{85} +0.196152 q^{86} +8.78461 q^{88} +1.26795 q^{89} -0.784610 q^{92} -0.143594 q^{94} -5.46410 q^{95} -16.4641 q^{97} -9.46410 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} + 12 q^{8} + 2 q^{10} - 10 q^{14} + 16 q^{16} + 10 q^{17} - 4 q^{19} + 4 q^{20} - 12 q^{22} + 8 q^{23} + 2 q^{25} - 20 q^{28} + 2 q^{29} + 4 q^{31} + 16 q^{32}+ \cdots - 12 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 + 12 * q^8 + 2 * q^10 - 10 * q^14 + 16 * q^16 + 10 * q^17 - 4 * q^19 + 4 * q^20 - 12 * q^22 + 8 * q^23 + 2 * q^25 - 20 * q^28 + 2 * q^29 + 4 * q^31 + 16 * q^32 + 4 * q^34 + 2 * q^35 + 8 * q^37 + 8 * q^38 + 12 * q^40 + 14 * q^41 - 4 * q^43 - 24 * q^44 + 20 * q^46 - 10 * q^47 + 12 * q^49 + 2 * q^50 - 12 * q^56 - 4 * q^58 + 18 * q^59 + 2 * q^61 + 22 * q^62 + 32 * q^64 + 18 * q^67 + 8 * q^68 - 10 * q^70 + 22 * q^71 - 10 * q^73 + 8 * q^74 + 16 * q^76 + 24 * q^77 - 10 * q^79 + 16 * q^80 + 20 * q^82 + 12 * q^83 + 10 * q^85 - 10 * q^86 - 24 * q^88 + 6 * q^89 + 40 * q^92 - 28 * q^94 - 4 * q^95 - 26 * q^97 - 12 * q^98

Coefficient data

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

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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 $

$2$	−0.732051	−0.517638	−0.258819	−	0.965926i	$-0.583333\pi$
$2$	−0.732051	−0.517638	−0.258819	+	0.965926i	$0.583333\pi$
$3$	0	0
$3$	0	0
$4$	−1.46410	−0.732051
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.46410	1.68727	0.843636	−	0.536916i	$-0.180411\pi$
$7$	4.46410	1.68727	0.843636	+	0.536916i	$0.180411\pi$
$8$	2.53590	0.896575
$9$	0	0
$10$	−0.732051	−0.231495
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−3.26795	−0.873396
$15$	0	0
$16$	1.07180	0.267949
$17$	6.73205	1.63276	0.816381	−	0.577514i	$-0.195977\pi$
$17$	6.73205	1.63276	0.816381	+	0.577514i	$0.195977\pi$
$18$	0	0
$19$	−5.46410	−1.25355	−0.626775	−	0.779200i	$-0.715626\pi$
$19$	−5.46410	−1.25355	−0.626775	+	0.779200i	$0.715626\pi$
$20$	−1.46410	−0.327383
$21$	0	0
$22$	−2.53590	−0.540655
$23$	0.535898	0.111743	0.0558713	−	0.998438i	$-0.482206\pi$
$23$	0.535898	0.111743	0.0558713	+	0.998438i	$0.482206\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−6.53590	−1.23517
$29$	2.73205	0.507329	0.253665	−	0.967292i	$-0.418364\pi$
$29$	2.73205	0.507329	0.253665	+	0.967292i	$0.418364\pi$
$30$	0	0
$31$	−3.19615	−0.574046	−0.287023	−	0.957924i	$-0.592666\pi$
$31$	−3.19615	−0.574046	−0.287023	+	0.957924i	$0.592666\pi$
$32$	−5.85641	−1.03528
$33$	0	0
$34$	−4.92820	−0.845180
$35$	4.46410	0.754571
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	2.53590	0.400961
$41$	5.26795	0.822715	0.411358	−	0.911474i	$-0.365055\pi$
$41$	5.26795	0.822715	0.411358	+	0.911474i	$0.365055\pi$
$42$	0	0
$43$	−0.267949	−0.0408619	−0.0204309	−	0.999791i	$-0.506504\pi$
$43$	−0.267949	−0.0408619	−0.0204309	+	0.999791i	$0.506504\pi$
$44$	−5.07180	−0.764602
$45$	0	0
$46$	−0.392305	−0.0578422
$47$	0.196152	0.0286118	0.0143059	−	0.999898i	$-0.495446\pi$
$47$	0.196152	0.0286118	0.0143059	+	0.999898i	$0.495446\pi$
$48$	0	0
$49$	12.9282	1.84689
$50$	−0.732051	−0.103528
$51$	0	0
$52$	0	0
$53$	6.92820	0.951662	0.475831	−	0.879537i	$-0.342147\pi$
$53$	6.92820	0.951662	0.475831	+	0.879537i	$0.342147\pi$
$54$	0	0
$55$	3.46410	0.467099
$56$	11.3205	1.51277
$57$	0	0
$58$	−2.00000	−0.262613
$59$	7.26795	0.946206	0.473103	−	0.881007i	$-0.343134\pi$
$59$	7.26795	0.946206	0.473103	+	0.881007i	$0.343134\pi$
$60$	0	0
$61$	4.46410	0.571570	0.285785	−	0.958294i	$-0.407746\pi$
$61$	4.46410	0.571570	0.285785	+	0.958294i	$0.407746\pi$
$62$	2.33975	0.297148
$63$	0	0
$64$	2.14359	0.267949
$65$	0	0
$66$	0	0
$67$	12.4641	1.52273	0.761366	−	0.648322i	$-0.224529\pi$
$67$	12.4641	1.52273	0.761366	+	0.648322i	$0.224529\pi$
$68$	−9.85641	−1.19526
$69$	0	0
$70$	−3.26795	−0.390595
$71$	12.7321	1.51102	0.755508	−	0.655139i	$-0.227390\pi$
$71$	12.7321	1.51102	0.755508	+	0.655139i	$0.227390\pi$
$72$	0	0
$73$	−15.3923	−1.80153	−0.900767	−	0.434304i	$-0.856994\pi$
$73$	−15.3923	−1.80153	−0.900767	+	0.434304i	$0.856994\pi$
$74$	−2.92820	−0.340397
$75$	0	0
$76$	8.00000	0.917663
$77$	15.4641	1.76230
$78$	0	0
$79$	1.92820	0.216940	0.108470	−	0.994100i	$-0.465405\pi$
$79$	1.92820	0.216940	0.108470	+	0.994100i	$0.465405\pi$
$80$	1.07180	0.119831
$81$	0	0
$82$	−3.85641	−0.425869
$83$	2.53590	0.278351	0.139176	−	0.990268i	$-0.455555\pi$
$83$	2.53590	0.278351	0.139176	+	0.990268i	$0.455555\pi$
$84$	0	0
$85$	6.73205	0.730193
$86$	0.196152	0.0211517
$87$	0	0
$88$	8.78461	0.936443
$89$	1.26795	0.134402	0.0672012	−	0.997739i	$-0.478593\pi$
$89$	1.26795	0.134402	0.0672012	+	0.997739i	$0.478593\pi$
$90$	0	0
$91$	0	0
$92$	−0.784610	−0.0818012
$93$	0	0
$94$	−0.143594	−0.0148105
$95$	−5.46410	−0.560605
$96$	0	0
$97$	−16.4641	−1.67168	−0.835838	−	0.548976i	$-0.815018\pi$
$97$	−16.4641	−1.67168	−0.835838	+	0.548976i	$0.815018\pi$
$98$	−9.46410	−0.956019
$99$	0	0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.bk.1.1		2
3.2	odd	2		2535.2.a.n.1.2		2
13.6	odd	12		585.2.bu.a.361.1		4
13.11	odd	12		585.2.bu.a.316.1		4
13.12	even	2		7605.2.a.y.1.2		2
39.11	even	12		195.2.bb.a.121.2	✓	4
39.32	even	12		195.2.bb.a.166.2	yes	4
39.38	odd	2		2535.2.a.s.1.1		2
195.32	odd	12		975.2.w.f.49.1		4
195.89	even	12		975.2.bc.h.901.1		4
195.128	odd	12		975.2.w.f.199.1		4
195.149	even	12		975.2.bc.h.751.1		4
195.167	odd	12		975.2.w.a.199.2		4
195.188	odd	12		975.2.w.a.49.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.bb.a.121.2	✓	4	39.11	even	12
195.2.bb.a.166.2	yes	4	39.32	even	12
585.2.bu.a.316.1		4	13.11	odd	12
585.2.bu.a.361.1		4	13.6	odd	12
975.2.w.a.49.2		4	195.188	odd	12
975.2.w.a.199.2		4	195.167	odd	12
975.2.w.f.49.1		4	195.32	odd	12
975.2.w.f.199.1		4	195.128	odd	12
975.2.bc.h.751.1		4	195.149	even	12
975.2.bc.h.901.1		4	195.89	even	12
2535.2.a.n.1.2		2	3.2	odd	2
2535.2.a.s.1.1		2	39.38	odd	2
7605.2.a.y.1.2		2	13.12	even	2
7605.2.a.bk.1.1		2	1.1	even	1	trivial

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Level:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q-0.732051 q^{2} -1.46410 q^{4} +1.00000 q^{5} +4.46410 q^{7} +2.53590 q^{8} -0.732051 q^{10} +3.46410 q^{11} -3.26795 q^{14} +1.07180 q^{16} +6.73205 q^{17} -5.46410 q^{19} -1.46410 q^{20} -2.53590 q^{22} +0.535898 q^{23} +1.00000 q^{25} -6.53590 q^{28} +2.73205 q^{29} -3.19615 q^{31} -5.85641 q^{32} -4.92820 q^{34} +4.46410 q^{35} +4.00000 q^{37} +4.00000 q^{38} +2.53590 q^{40} +5.26795 q^{41} -0.267949 q^{43} -5.07180 q^{44} -0.392305 q^{46} +0.196152 q^{47} +12.9282 q^{49} -0.732051 q^{50} +6.92820 q^{53} +3.46410 q^{55} +11.3205 q^{56} -2.00000 q^{58} +7.26795 q^{59} +4.46410 q^{61} +2.33975 q^{62} +2.14359 q^{64} +12.4641 q^{67} -9.85641 q^{68} -3.26795 q^{70} +12.7321 q^{71} -15.3923 q^{73} -2.92820 q^{74} +8.00000 q^{76} +15.4641 q^{77} +1.92820 q^{79} +1.07180 q^{80} -3.85641 q^{82} +2.53590 q^{83} +6.73205 q^{85} +0.196152 q^{86} +8.78461 q^{88} +1.26795 q^{89} -0.784610 q^{92} -0.143594 q^{94} -5.46410 q^{95} -16.4641 q^{97} -9.46410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} + 12 q^{8} + 2 q^{10} - 10 q^{14} + 16 q^{16} + 10 q^{17} - 4 q^{19} + 4 q^{20} - 12 q^{22} + 8 q^{23} + 2 q^{25} - 20 q^{28} + 2 q^{29} + 4 q^{31} + 16 q^{32}+ \cdots - 12 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 + 12 * q^8 + 2 * q^10 - 10 * q^14 + 16 * q^16 + 10 * q^17 - 4 * q^19 + 4 * q^20 - 12 * q^22 + 8 * q^23 + 2 * q^25 - 20 * q^28 + 2 * q^29 + 4 * q^31 + 16 * q^32 + 4 * q^34 + 2 * q^35 + 8 * q^37 + 8 * q^38 + 12 * q^40 + 14 * q^41 - 4 * q^43 - 24 * q^44 + 20 * q^46 - 10 * q^47 + 12 * q^49 + 2 * q^50 - 12 * q^56 - 4 * q^58 + 18 * q^59 + 2 * q^61 + 22 * q^62 + 32 * q^64 + 18 * q^67 + 8 * q^68 - 10 * q^70 + 22 * q^71 - 10 * q^73 + 8 * q^74 + 16 * q^76 + 24 * q^77 - 10 * q^79 + 16 * q^80 + 20 * q^82 + 12 * q^83 + 10 * q^85 - 10 * q^86 - 24 * q^88 + 6 * q^89 + 40 * q^92 - 28 * q^94 - 4 * q^95 - 26 * q^97 - 12 * q^98

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