LMFDB - Embedded newform 2940.2.a.q.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2940,2,Mod(1,2940)] 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("2940.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2940, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2940.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,-2,0,0,0,2,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$23.4760181943$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2940.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} -2.00000 q^{11} +2.82843 q^{13} -1.00000 q^{15} -6.82843 q^{17} +2.58579 q^{19} -4.58579 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.65685 q^{29} -4.24264 q^{31} -2.00000 q^{33} +6.48528 q^{37} +2.82843 q^{39} -2.00000 q^{41} -9.65685 q^{43} -1.00000 q^{45} +6.48528 q^{47} -6.82843 q^{51} -7.41421 q^{53} +2.00000 q^{55} +2.58579 q^{57} -2.82843 q^{59} +9.89949 q^{61} -2.82843 q^{65} -1.17157 q^{67} -4.58579 q^{69} +6.48528 q^{71} -12.4853 q^{73} +1.00000 q^{75} -10.0000 q^{79} +1.00000 q^{81} +0.828427 q^{83} +6.82843 q^{85} -7.65685 q^{87} +0.828427 q^{89} -4.24264 q^{93} -2.58579 q^{95} +4.00000 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{11} - 2 q^{15} - 8 q^{17} + 8 q^{19} - 12 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 4 q^{33} - 4 q^{37} - 4 q^{41} - 8 q^{43} - 2 q^{45} - 4 q^{47} - 8 q^{51} - 12 q^{53}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9 - 4 * q^11 - 2 * q^15 - 8 * q^17 + 8 * q^19 - 12 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 - 4 * q^33 - 4 * q^37 - 4 * q^41 - 8 * q^43 - 2 * q^45 - 4 * q^47 - 8 * q^51 - 12 * q^53 + 4 * q^55 + 8 * q^57 - 8 * q^67 - 12 * q^69 - 4 * q^71 - 8 * q^73 + 2 * q^75 - 20 * q^79 + 2 * q^81 - 4 * q^83 + 8 * q^85 - 4 * q^87 - 4 * q^89 - 8 * q^95 + 8 * q^97 - 4 * q^99

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	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
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−6.82843	−1.65614	−0.828068	−	0.560627i	$-0.810560\pi$
$17$	−6.82843	−1.65614	−0.828068	+	0.560627i	$0.810560\pi$
$18$	0	0
$19$	2.58579	0.593220	0.296610	−	0.954999i	$-0.404144\pi$
$19$	2.58579	0.593220	0.296610	+	0.954999i	$0.404144\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.58579	−0.956203	−0.478101	−	0.878305i	$-0.658675\pi$
$23$	−4.58579	−0.956203	−0.478101	+	0.878305i	$0.658675\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.65685	−1.42184	−0.710921	−	0.703272i	$-0.751722\pi$
$29$	−7.65685	−1.42184	−0.710921	+	0.703272i	$0.751722\pi$
$30$	0	0
$31$	−4.24264	−0.762001	−0.381000	−	0.924575i	$-0.624420\pi$
$31$	−4.24264	−0.762001	−0.381000	+	0.924575i	$0.624420\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.48528	1.06617	0.533087	−	0.846061i	$-0.321032\pi$
$37$	6.48528	1.06617	0.533087	+	0.846061i	$0.321032\pi$
$38$	0	0
$39$	2.82843	0.452911
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−9.65685	−1.47266	−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685	−1.47266	−0.736328	+	0.676625i	$0.763442\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	6.48528	0.945976	0.472988	−	0.881069i	$-0.343175\pi$
$47$	6.48528	0.945976	0.472988	+	0.881069i	$0.343175\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−6.82843	−0.956171
$52$	0	0
$53$	−7.41421	−1.01842	−0.509210	−	0.860642i	$-0.670062\pi$
$53$	−7.41421	−1.01842	−0.509210	+	0.860642i	$0.670062\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	2.58579	0.342496
$58$	0	0
$59$	−2.82843	−0.368230	−0.184115	−	0.982905i	$-0.558942\pi$
$59$	−2.82843	−0.368230	−0.184115	+	0.982905i	$0.558942\pi$
$60$	0	0
$61$	9.89949	1.26750	0.633750	−	0.773538i	$-0.281515\pi$
$61$	9.89949	1.26750	0.633750	+	0.773538i	$0.281515\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.82843	−0.350823
$66$	0	0
$67$	−1.17157	−0.143130	−0.0715652	−	0.997436i	$-0.522799\pi$
$67$	−1.17157	−0.143130	−0.0715652	+	0.997436i	$0.522799\pi$
$68$	0	0
$69$	−4.58579	−0.552064
$70$	0	0
$71$	6.48528	0.769661	0.384831	−	0.922987i	$-0.374260\pi$
$71$	6.48528	0.769661	0.384831	+	0.922987i	$0.374260\pi$
$72$	0	0
$73$	−12.4853	−1.46129	−0.730646	−	0.682757i	$-0.760781\pi$
$73$	−12.4853	−1.46129	−0.730646	+	0.682757i	$0.760781\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.828427	0.0909317	0.0454658	−	0.998966i	$-0.485523\pi$
$83$	0.828427	0.0909317	0.0454658	+	0.998966i	$0.485523\pi$
$84$	0	0
$85$	6.82843	0.740647
$86$	0	0
$87$	−7.65685	−0.820901
$88$	0	0
$89$	0.828427	0.0878131	0.0439065	−	0.999036i	$-0.486020\pi$
$89$	0.828427	0.0878131	0.0439065	+	0.999036i	$0.486020\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−4.24264	−0.439941
$94$	0	0
$95$	−2.58579	−0.265296
$96$	0	0
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	0	0
$99$	−2.00000	−0.201008

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	2940.2.a.q.1.2	yes	2
3.2	odd	2		8820.2.a.bm.1.2		2
7.2	even	3		2940.2.q.p.361.2		4
7.3	odd	6		2940.2.q.r.961.1		4
7.4	even	3		2940.2.q.p.961.2		4
7.5	odd	6		2940.2.q.r.361.1		4
7.6	odd	2		2940.2.a.o.1.1	✓	2
21.20	even	2		8820.2.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2940.2.a.o.1.1	✓	2	7.6	odd	2
2940.2.a.q.1.2	yes	2	1.1	even	1	trivial
2940.2.q.p.361.2		4	7.2	even	3
2940.2.q.p.961.2		4	7.4	even	3
2940.2.q.r.361.1		4	7.5	odd	6
2940.2.q.r.961.1		4	7.3	odd	6
8820.2.a.bh.1.1		2	21.20	even	2
8820.2.a.bm.1.2		2	3.2	odd	2

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Level:	\( N \)	\(=\)	\( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2940.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} -2.00000 q^{11} +2.82843 q^{13} -1.00000 q^{15} -6.82843 q^{17} +2.58579 q^{19} -4.58579 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.65685 q^{29} -4.24264 q^{31} -2.00000 q^{33} +6.48528 q^{37} +2.82843 q^{39} -2.00000 q^{41} -9.65685 q^{43} -1.00000 q^{45} +6.48528 q^{47} -6.82843 q^{51} -7.41421 q^{53} +2.00000 q^{55} +2.58579 q^{57} -2.82843 q^{59} +9.89949 q^{61} -2.82843 q^{65} -1.17157 q^{67} -4.58579 q^{69} +6.48528 q^{71} -12.4853 q^{73} +1.00000 q^{75} -10.0000 q^{79} +1.00000 q^{81} +0.828427 q^{83} +6.82843 q^{85} -7.65685 q^{87} +0.828427 q^{89} -4.24264 q^{93} -2.58579 q^{95} +4.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{11} - 2 q^{15} - 8 q^{17} + 8 q^{19} - 12 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 4 q^{33} - 4 q^{37} - 4 q^{41} - 8 q^{43} - 2 q^{45} - 4 q^{47} - 8 q^{51} - 12 q^{53}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9 - 4 * q^11 - 2 * q^15 - 8 * q^17 + 8 * q^19 - 12 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 - 4 * q^33 - 4 * q^37 - 4 * q^41 - 8 * q^43 - 2 * q^45 - 4 * q^47 - 8 * q^51 - 12 * q^53 + 4 * q^55 + 8 * q^57 - 8 * q^67 - 12 * q^69 - 4 * q^71 - 8 * q^73 + 2 * q^75 - 20 * q^79 + 2 * q^81 - 4 * q^83 + 8 * q^85 - 4 * q^87 - 4 * q^89 - 8 * q^95 + 8 * q^97 - 4 * q^99

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