LMFDB - Embedded newform 3328.2.a.p.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] 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("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,-2,0,6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 832)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{3} +1.82843 q^{5} +4.41421 q^{7} +2.82843 q^{9} +0.828427 q^{11} -1.00000 q^{13} -4.41421 q^{15} +1.00000 q^{17} +5.65685 q^{19} -10.6569 q^{21} +8.82843 q^{23} -1.65685 q^{25} +0.414214 q^{27} -3.65685 q^{29} +3.65685 q^{31} -2.00000 q^{33} +8.07107 q^{35} -7.00000 q^{37} +2.41421 q^{39} +9.65685 q^{41} +8.41421 q^{43} +5.17157 q^{45} +0.757359 q^{47} +12.4853 q^{49} -2.41421 q^{51} -3.65685 q^{53} +1.51472 q^{55} -13.6569 q^{57} -8.00000 q^{59} +12.4853 q^{63} -1.82843 q^{65} -8.82843 q^{67} -21.3137 q^{69} -11.7279 q^{71} +1.65685 q^{73} +4.00000 q^{75} +3.65685 q^{77} -16.1421 q^{79} -9.48528 q^{81} -12.1421 q^{83} +1.82843 q^{85} +8.82843 q^{87} +13.6569 q^{89} -4.41421 q^{91} -8.82843 q^{93} +10.3431 q^{95} -7.65685 q^{97} +2.34315 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 6 q^{7} - 4 q^{11} - 2 q^{13} - 6 q^{15} + 2 q^{17} - 10 q^{21} + 12 q^{23} + 8 q^{25} - 2 q^{27} + 4 q^{29} - 4 q^{31} - 4 q^{33} + 2 q^{35} - 14 q^{37} + 2 q^{39} + 8 q^{41}+ \cdots + 16 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 6 * q^7 - 4 * q^11 - 2 * q^13 - 6 * q^15 + 2 * q^17 - 10 * q^21 + 12 * q^23 + 8 * q^25 - 2 * q^27 + 4 * q^29 - 4 * q^31 - 4 * q^33 + 2 * q^35 - 14 * q^37 + 2 * q^39 + 8 * q^41 + 14 * q^43 + 16 * q^45 + 10 * q^47 + 8 * q^49 - 2 * q^51 + 4 * q^53 + 20 * q^55 - 16 * q^57 - 16 * q^59 + 8 * q^63 + 2 * q^65 - 12 * q^67 - 20 * q^69 + 2 * q^71 - 8 * q^73 + 8 * q^75 - 4 * q^77 - 4 * q^79 - 2 * q^81 + 4 * q^83 - 2 * q^85 + 12 * q^87 + 16 * q^89 - 6 * q^91 - 12 * q^93 + 32 * q^95 - 4 * q^97 + 16 * 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
$3$	−2.41421	−1.39385	−0.696923	−	0.717146i	$-0.745448\pi$
$3$	−2.41421	−1.39385	−0.696923	+	0.717146i	$0.745448\pi$
$4$	0	0
$5$	1.82843	0.817697	0.408849	−	0.912602i	$-0.365930\pi$
$5$	1.82843	0.817697	0.408849	+	0.912602i	$0.365930\pi$
$6$	0	0
$7$	4.41421	1.66842	0.834208	−	0.551450i	$-0.185925\pi$
$7$	4.41421	1.66842	0.834208	+	0.551450i	$0.185925\pi$
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−4.41421	−1.13975
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	5.65685	1.29777	0.648886	−	0.760886i	$-0.275235\pi$
$19$	5.65685	1.29777	0.648886	+	0.760886i	$0.275235\pi$
$20$	0	0
$21$	−10.6569	−2.32552
$22$	0	0
$23$	8.82843	1.84085	0.920427	−	0.390914i	$-0.127841\pi$
$23$	8.82843	1.84085	0.920427	+	0.390914i	$0.127841\pi$
$24$	0	0
$25$	−1.65685	−0.331371
$26$	0	0
$27$	0.414214	0.0797154
$28$	0	0
$29$	−3.65685	−0.679061	−0.339530	−	0.940595i	$-0.610268\pi$
$29$	−3.65685	−0.679061	−0.339530	+	0.940595i	$0.610268\pi$
$30$	0	0
$31$	3.65685	0.656790	0.328395	−	0.944540i	$-0.393492\pi$
$31$	3.65685	0.656790	0.328395	+	0.944540i	$0.393492\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	8.07107	1.36426
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	2.41421	0.386584
$40$	0	0
$41$	9.65685	1.50815	0.754074	−	0.656790i	$-0.228086\pi$
$41$	9.65685	1.50815	0.754074	+	0.656790i	$0.228086\pi$
$42$	0	0
$43$	8.41421	1.28316	0.641578	−	0.767058i	$-0.278280\pi$
$43$	8.41421	1.28316	0.641578	+	0.767058i	$0.278280\pi$
$44$	0	0
$45$	5.17157	0.770933
$46$	0	0
$47$	0.757359	0.110472	0.0552361	−	0.998473i	$-0.482409\pi$
$47$	0.757359	0.110472	0.0552361	+	0.998473i	$0.482409\pi$
$48$	0	0
$49$	12.4853	1.78361
$50$	0	0
$51$	−2.41421	−0.338058
$52$	0	0
$53$	−3.65685	−0.502308	−0.251154	−	0.967947i	$-0.580810\pi$
$53$	−3.65685	−0.502308	−0.251154	+	0.967947i	$0.580810\pi$
$54$	0	0
$55$	1.51472	0.204245
$56$	0	0
$57$	−13.6569	−1.80889
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	12.4853	1.57300
$64$	0	0
$65$	−1.82843	−0.226788
$66$	0	0
$67$	−8.82843	−1.07856	−0.539282	−	0.842125i	$-0.681304\pi$
$67$	−8.82843	−1.07856	−0.539282	+	0.842125i	$0.681304\pi$
$68$	0	0
$69$	−21.3137	−2.56587
$70$	0	0
$71$	−11.7279	−1.39185	−0.695924	−	0.718115i	$-0.745005\pi$
$71$	−11.7279	−1.39185	−0.695924	+	0.718115i	$0.745005\pi$
$72$	0	0
$73$	1.65685	0.193920	0.0969601	−	0.995288i	$-0.469088\pi$
$73$	1.65685	0.193920	0.0969601	+	0.995288i	$0.469088\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	3.65685	0.416737
$78$	0	0
$79$	−16.1421	−1.81613	−0.908066	−	0.418827i	$-0.862441\pi$
$79$	−16.1421	−1.81613	−0.908066	+	0.418827i	$0.862441\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	−12.1421	−1.33277	−0.666386	−	0.745607i	$-0.732160\pi$
$83$	−12.1421	−1.33277	−0.666386	+	0.745607i	$0.732160\pi$
$84$	0	0
$85$	1.82843	0.198321
$86$	0	0
$87$	8.82843	0.946507
$88$	0	0
$89$	13.6569	1.44762	0.723812	−	0.689997i	$-0.242388\pi$
$89$	13.6569	1.44762	0.723812	+	0.689997i	$0.242388\pi$
$90$	0	0
$91$	−4.41421	−0.462735
$92$	0	0
$93$	−8.82843	−0.915465
$94$	0	0
$95$	10.3431	1.06118
$96$	0	0
$97$	−7.65685	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$97$	−7.65685	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$98$	0	0
$99$	2.34315	0.235495

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	3328.2.a.p.1.1		2
4.3	odd	2		3328.2.a.y.1.2		2
8.3	odd	2		3328.2.a.q.1.1		2
8.5	even	2		3328.2.a.bb.1.2		2
16.3	odd	4		832.2.b.b.417.4	yes	4
16.5	even	4		832.2.b.a.417.4	yes	4
16.11	odd	4		832.2.b.b.417.1	yes	4
16.13	even	4		832.2.b.a.417.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
832.2.b.a.417.1	✓	4	16.13	even	4
832.2.b.a.417.4	yes	4	16.5	even	4
832.2.b.b.417.1	yes	4	16.11	odd	4
832.2.b.b.417.4	yes	4	16.3	odd	4
3328.2.a.p.1.1		2	1.1	even	1	trivial
3328.2.a.q.1.1		2	8.3	odd	2
3328.2.a.y.1.2		2	4.3	odd	2
3328.2.a.bb.1.2		2	8.5	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{3} +1.82843 q^{5} +4.41421 q^{7} +2.82843 q^{9} +0.828427 q^{11} -1.00000 q^{13} -4.41421 q^{15} +1.00000 q^{17} +5.65685 q^{19} -10.6569 q^{21} +8.82843 q^{23} -1.65685 q^{25} +0.414214 q^{27} -3.65685 q^{29} +3.65685 q^{31} -2.00000 q^{33} +8.07107 q^{35} -7.00000 q^{37} +2.41421 q^{39} +9.65685 q^{41} +8.41421 q^{43} +5.17157 q^{45} +0.757359 q^{47} +12.4853 q^{49} -2.41421 q^{51} -3.65685 q^{53} +1.51472 q^{55} -13.6569 q^{57} -8.00000 q^{59} +12.4853 q^{63} -1.82843 q^{65} -8.82843 q^{67} -21.3137 q^{69} -11.7279 q^{71} +1.65685 q^{73} +4.00000 q^{75} +3.65685 q^{77} -16.1421 q^{79} -9.48528 q^{81} -12.1421 q^{83} +1.82843 q^{85} +8.82843 q^{87} +13.6569 q^{89} -4.41421 q^{91} -8.82843 q^{93} +10.3431 q^{95} -7.65685 q^{97} +2.34315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 6 q^{7} - 4 q^{11} - 2 q^{13} - 6 q^{15} + 2 q^{17} - 10 q^{21} + 12 q^{23} + 8 q^{25} - 2 q^{27} + 4 q^{29} - 4 q^{31} - 4 q^{33} + 2 q^{35} - 14 q^{37} + 2 q^{39} + 8 q^{41}+ \cdots + 16 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 6 * q^7 - 4 * q^11 - 2 * q^13 - 6 * q^15 + 2 * q^17 - 10 * q^21 + 12 * q^23 + 8 * q^25 - 2 * q^27 + 4 * q^29 - 4 * q^31 - 4 * q^33 + 2 * q^35 - 14 * q^37 + 2 * q^39 + 8 * q^41 + 14 * q^43 + 16 * q^45 + 10 * q^47 + 8 * q^49 - 2 * q^51 + 4 * q^53 + 20 * q^55 - 16 * q^57 - 16 * q^59 + 8 * q^63 + 2 * q^65 - 12 * q^67 - 20 * q^69 + 2 * q^71 - 8 * q^73 + 8 * q^75 - 4 * q^77 - 4 * q^79 - 2 * q^81 + 4 * q^83 - 2 * q^85 + 12 * q^87 + 16 * q^89 - 6 * q^91 - 12 * q^93 + 32 * q^95 - 4 * q^97 + 16 * q^99

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