LMFDB - Embedded newform 7872.2.a.cf.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7872,2,Mod(1,7872)] 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("7872.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7872, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7872 = 2^{6} \cdot 3 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7872.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,4,0,-6,0,4,0,-3,0,2] 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:	$62.8582364712$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.15188.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + x + 2 $ x^4 - x^3 - 7*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3936)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.599159$ of defining polynomial
Character	$\chi$	$=$	7872.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.25066 q^{5} -4.98951 q^{7} +1.00000 q^{9} -0.651498 q^{11} -2.98951 q^{13} -1.25066 q^{15} -0.150184 q^{17} -4.18783 q^{19} +4.98951 q^{21} +6.72836 q^{23} -3.43586 q^{25} -1.00000 q^{27} +5.88904 q^{29} +8.31703 q^{31} +0.651498 q^{33} -6.24017 q^{35} +9.27518 q^{37} +2.98951 q^{39} -1.00000 q^{41} +8.57818 q^{43} +1.25066 q^{45} -5.08735 q^{47} +17.8952 q^{49} +0.150184 q^{51} +6.54053 q^{53} -0.814801 q^{55} +4.18783 q^{57} -2.69700 q^{59} +4.44269 q^{61} -4.98951 q^{63} -3.73885 q^{65} +2.39664 q^{67} -6.72836 q^{69} -14.7660 q^{71} +9.11871 q^{73} +3.43586 q^{75} +3.25066 q^{77} -9.17734 q^{79} +1.00000 q^{81} -12.8246 q^{83} -0.187828 q^{85} -5.88904 q^{87} -5.39927 q^{89} +14.9162 q^{91} -8.31703 q^{93} -5.23754 q^{95} -13.4698 q^{97} -0.651498 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} - 6 q^{7} + 4 q^{9} - 3 q^{11} + 2 q^{13} - 4 q^{15} - 3 q^{17} + 6 q^{21} - 4 q^{27} + 13 q^{29} - 9 q^{31} + 3 q^{33} - 10 q^{35} + 7 q^{37} - 2 q^{39} - 4 q^{41} + 5 q^{43} + 4 q^{45}+ \cdots - 3 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 - 6 * q^7 + 4 * q^9 - 3 * q^11 + 2 * q^13 - 4 * q^15 - 3 * q^17 + 6 * q^21 - 4 * q^27 + 13 * q^29 - 9 * q^31 + 3 * q^33 - 10 * q^35 + 7 * q^37 - 2 * q^39 - 4 * q^41 + 5 * q^43 + 4 * q^45 - 7 * q^47 + 3 * q^51 + 16 * q^53 - 16 * q^55 - 10 * q^59 + 7 * q^61 - 6 * q^63 - 2 * q^65 + 4 * q^67 - 13 * q^71 - 3 * q^73 + 12 * q^77 - 6 * q^79 + 4 * q^81 - 14 * q^83 + 16 * q^85 - 13 * q^87 - 12 * q^89 + 16 * q^91 + 9 * q^93 - 10 * q^95 - 10 * q^97 - 3 * 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
$5$	1.25066	0.559311	0.279655	−	0.960100i	$-0.409780\pi$
$5$	1.25066	0.559311	0.279655	+	0.960100i	$0.409780\pi$
$6$	0	0
$7$	−4.98951	−1.88586	−0.942929	−	0.332994i	$-0.891941\pi$
$7$	−4.98951	−1.88586	−0.942929	+	0.332994i	$0.891941\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.651498	−0.196434	−0.0982171	−	0.995165i	$-0.531314\pi$
$11$	−0.651498	−0.196434	−0.0982171	+	0.995165i	$0.531314\pi$
$12$	0	0
$13$	−2.98951	−0.829141	−0.414570	−	0.910017i	$-0.636068\pi$
$13$	−2.98951	−0.829141	−0.414570	+	0.910017i	$0.636068\pi$
$14$	0	0
$15$	−1.25066	−0.322918
$16$	0	0
$17$	−0.150184	−0.0364249	−0.0182124	−	0.999834i	$-0.505798\pi$
$17$	−0.150184	−0.0364249	−0.0182124	+	0.999834i	$0.505798\pi$
$18$	0	0
$19$	−4.18783	−0.960754	−0.480377	−	0.877062i	$-0.659500\pi$
$19$	−4.18783	−0.960754	−0.480377	+	0.877062i	$0.659500\pi$
$20$	0	0
$21$	4.98951	1.08880
$22$	0	0
$23$	6.72836	1.40296	0.701480	−	0.712689i	$-0.252523\pi$
$23$	6.72836	1.40296	0.701480	+	0.712689i	$0.252523\pi$
$24$	0	0
$25$	−3.43586	−0.687171
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.88904	1.09357	0.546783	−	0.837274i	$-0.315852\pi$
$29$	5.88904	1.09357	0.546783	+	0.837274i	$0.315852\pi$
$30$	0	0
$31$	8.31703	1.49378	0.746892	−	0.664946i	$-0.231545\pi$
$31$	8.31703	1.49378	0.746892	+	0.664946i	$0.231545\pi$
$32$	0	0
$33$	0.651498	0.113411
$34$	0	0
$35$	−6.24017	−1.05478
$36$	0	0
$37$	9.27518	1.52483	0.762415	−	0.647088i	$-0.224013\pi$
$37$	9.27518	1.52483	0.762415	+	0.647088i	$0.224013\pi$
$38$	0	0
$39$	2.98951	0.478705
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	8.57818	1.30816	0.654080	−	0.756425i	$-0.273056\pi$
$43$	8.57818	1.30816	0.654080	+	0.756425i	$0.273056\pi$
$44$	0	0
$45$	1.25066	0.186437
$46$	0	0
$47$	−5.08735	−0.742067	−0.371033	−	0.928619i	$-0.620996\pi$
$47$	−5.08735	−0.742067	−0.371033	+	0.928619i	$0.620996\pi$
$48$	0	0
$49$	17.8952	2.55646
$50$	0	0
$51$	0.150184	0.0210299
$52$	0	0
$53$	6.54053	0.898412	0.449206	−	0.893428i	$-0.351707\pi$
$53$	6.54053	0.898412	0.449206	+	0.893428i	$0.351707\pi$
$54$	0	0
$55$	−0.814801	−0.109868
$56$	0	0
$57$	4.18783	0.554691
$58$	0	0
$59$	−2.69700	−0.351120	−0.175560	−	0.984469i	$-0.556174\pi$
$59$	−2.69700	−0.351120	−0.175560	+	0.984469i	$0.556174\pi$
$60$	0	0
$61$	4.44269	0.568828	0.284414	−	0.958702i	$-0.408201\pi$
$61$	4.44269	0.568828	0.284414	+	0.958702i	$0.408201\pi$
$62$	0	0
$63$	−4.98951	−0.628619
$64$	0	0
$65$	−3.73885	−0.463748
$66$	0	0
$67$	2.39664	0.292796	0.146398	−	0.989226i	$-0.453232\pi$
$67$	2.39664	0.292796	0.146398	+	0.989226i	$0.453232\pi$
$68$	0	0
$69$	−6.72836	−0.810000
$70$	0	0
$71$	−14.7660	−1.75240	−0.876201	−	0.481945i	$-0.839930\pi$
$71$	−14.7660	−1.75240	−0.876201	+	0.481945i	$0.839930\pi$
$72$	0	0
$73$	9.11871	1.06726	0.533632	−	0.845717i	$-0.320827\pi$
$73$	9.11871	1.06726	0.533632	+	0.845717i	$0.320827\pi$
$74$	0	0
$75$	3.43586	0.396739
$76$	0	0
$77$	3.25066	0.370447
$78$	0	0
$79$	−9.17734	−1.03253	−0.516266	−	0.856429i	$-0.672678\pi$
$79$	−9.17734	−1.03253	−0.516266	+	0.856429i	$0.672678\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.8246	−1.40769	−0.703843	−	0.710355i	$-0.748534\pi$
$83$	−12.8246	−1.40769	−0.703843	+	0.710355i	$0.748534\pi$
$84$	0	0
$85$	−0.187828	−0.0203728
$86$	0	0
$87$	−5.88904	−0.631371
$88$	0	0
$89$	−5.39927	−0.572321	−0.286161	−	0.958182i	$-0.592379\pi$
$89$	−5.39927	−0.572321	−0.286161	+	0.958182i	$0.592379\pi$
$90$	0	0
$91$	14.9162	1.56364
$92$	0	0
$93$	−8.31703	−0.862436
$94$	0	0
$95$	−5.23754	−0.537360
$96$	0	0
$97$	−13.4698	−1.36766	−0.683828	−	0.729643i	$-0.739686\pi$
$97$	−13.4698	−1.36766	−0.683828	+	0.729643i	$0.739686\pi$
$98$	0	0
$99$	−0.651498	−0.0654780

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	7872.2.a.cf.1.3		4
4.3	odd	2		7872.2.a.cj.1.3		4
8.3	odd	2		3936.2.a.i.1.2	✓	4
8.5	even	2		3936.2.a.m.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3936.2.a.i.1.2	✓	4	8.3	odd	2
3936.2.a.m.1.2	yes	4	8.5	even	2
7872.2.a.cf.1.3		4	1.1	even	1	trivial
7872.2.a.cj.1.3		4	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7872 = 2^{6} \cdot 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7872.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.25066 q^{5} -4.98951 q^{7} +1.00000 q^{9} -0.651498 q^{11} -2.98951 q^{13} -1.25066 q^{15} -0.150184 q^{17} -4.18783 q^{19} +4.98951 q^{21} +6.72836 q^{23} -3.43586 q^{25} -1.00000 q^{27} +5.88904 q^{29} +8.31703 q^{31} +0.651498 q^{33} -6.24017 q^{35} +9.27518 q^{37} +2.98951 q^{39} -1.00000 q^{41} +8.57818 q^{43} +1.25066 q^{45} -5.08735 q^{47} +17.8952 q^{49} +0.150184 q^{51} +6.54053 q^{53} -0.814801 q^{55} +4.18783 q^{57} -2.69700 q^{59} +4.44269 q^{61} -4.98951 q^{63} -3.73885 q^{65} +2.39664 q^{67} -6.72836 q^{69} -14.7660 q^{71} +9.11871 q^{73} +3.43586 q^{75} +3.25066 q^{77} -9.17734 q^{79} +1.00000 q^{81} -12.8246 q^{83} -0.187828 q^{85} -5.88904 q^{87} -5.39927 q^{89} +14.9162 q^{91} -8.31703 q^{93} -5.23754 q^{95} -13.4698 q^{97} -0.651498 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - 6 q^{7} + 4 q^{9} - 3 q^{11} + 2 q^{13} - 4 q^{15} - 3 q^{17} + 6 q^{21} - 4 q^{27} + 13 q^{29} - 9 q^{31} + 3 q^{33} - 10 q^{35} + 7 q^{37} - 2 q^{39} - 4 q^{41} + 5 q^{43} + 4 q^{45}+ \cdots - 3 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 - 6 * q^7 + 4 * q^9 - 3 * q^11 + 2 * q^13 - 4 * q^15 - 3 * q^17 + 6 * q^21 - 4 * q^27 + 13 * q^29 - 9 * q^31 + 3 * q^33 - 10 * q^35 + 7 * q^37 - 2 * q^39 - 4 * q^41 + 5 * q^43 + 4 * q^45 - 7 * q^47 + 3 * q^51 + 16 * q^53 - 16 * q^55 - 10 * q^59 + 7 * q^61 - 6 * q^63 - 2 * q^65 + 4 * q^67 - 13 * q^71 - 3 * q^73 + 12 * q^77 - 6 * q^79 + 4 * q^81 - 14 * q^83 + 16 * q^85 - 13 * q^87 - 12 * q^89 + 16 * q^91 + 9 * q^93 - 10 * q^95 - 10 * q^97 - 3 * q^99

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