LMFDB - Embedded newform 7168.2.a.bk.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7168,2,Mod(1,7168)] 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("7168.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7168, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7168 = 2^{10} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7168.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,0,8,0,0,0,-12,0,12,0,16,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$57.2367681689$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 14 x^{10} + 60 x^{9} + 71 x^{8} - 312 x^{7} - 164 x^{6} + 648 x^{5} + 167 x^{4} + \cdots + 1 $ x^12 - 4x^11 - 14x^10 + 60x^9 + 71x^8 - 312x^7 - 164x^6 + 648x^5 + 167x^4 - 484x^3 - 30x^2 + 92*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 3584)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.0108380$ of defining polynomial
Character	$\chi$	$=$	7168.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.01084 q^{3} -0.303370 q^{5} -1.00000 q^{7} -1.97821 q^{9} -0.882882 q^{11} -5.59992 q^{13} -0.306658 q^{15} +1.54776 q^{17} -0.0998752 q^{19} -1.01084 q^{21} -8.23807 q^{23} -4.90797 q^{25} -5.03216 q^{27} +1.69722 q^{29} +4.52772 q^{31} -0.892451 q^{33} +0.303370 q^{35} +0.883991 q^{37} -5.66061 q^{39} +9.92605 q^{41} +3.30486 q^{43} +0.600128 q^{45} -6.00904 q^{47} +1.00000 q^{49} +1.56453 q^{51} +11.1794 q^{53} +0.267840 q^{55} -0.100958 q^{57} +8.71281 q^{59} -2.57260 q^{61} +1.97821 q^{63} +1.69885 q^{65} +11.5232 q^{67} -8.32736 q^{69} +6.56548 q^{71} +14.4035 q^{73} -4.96116 q^{75} +0.882882 q^{77} +13.1905 q^{79} +0.847921 q^{81} -4.91904 q^{83} -0.469544 q^{85} +1.71561 q^{87} -4.10561 q^{89} +5.59992 q^{91} +4.57679 q^{93} +0.0302991 q^{95} +5.05748 q^{97} +1.74652 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 8 q^{3} - 12 q^{7} + 12 q^{9} + 16 q^{11} + 8 q^{19} - 8 q^{21} + 12 q^{25} + 32 q^{27} - 24 q^{29} + 16 q^{33} - 8 q^{37} + 32 q^{39} + 16 q^{41} + 16 q^{43} + 12 q^{49} + 16 q^{51} - 24 q^{53}+ \cdots + 48 q^{99}+O(q^{100}) $ 12 * q + 8 * q^3 - 12 * q^7 + 12 * q^9 + 16 * q^11 + 8 * q^19 - 8 * q^21 + 12 * q^25 + 32 * q^27 - 24 * q^29 + 16 * q^33 - 8 * q^37 + 32 * q^39 + 16 * q^41 + 16 * q^43 + 12 * q^49 + 16 * q^51 - 24 * q^53 + 16 * q^57 + 40 * q^59 - 12 * q^63 - 8 * q^65 + 32 * q^67 - 8 * q^73 + 40 * q^75 - 16 * q^77 + 28 * q^81 + 56 * q^83 - 8 * q^89 + 32 * q^93 + 32 * q^95 + 48 * 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
$3$	1.01084	0.583608	0.291804	−	0.956478i	$-0.405745\pi$
$3$	1.01084	0.583608	0.291804	+	0.956478i	$0.405745\pi$
$4$	0	0
$5$	−0.303370	−0.135671	−0.0678356	−	0.997697i	$-0.521609\pi$
$5$	−0.303370	−0.135671	−0.0678356	+	0.997697i	$0.521609\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.97821	−0.659402
$10$	0	0
$11$	−0.882882	−0.266199	−0.133099	−	0.991103i	$-0.542493\pi$
$11$	−0.882882	−0.266199	−0.133099	+	0.991103i	$0.542493\pi$
$12$	0	0
$13$	−5.59992	−1.55314	−0.776569	−	0.630033i	$-0.783041\pi$
$13$	−5.59992	−1.55314	−0.776569	+	0.630033i	$0.783041\pi$
$14$	0	0
$15$	−0.306658	−0.0791787
$16$	0	0
$17$	1.54776	0.375387	0.187693	−	0.982228i	$-0.439899\pi$
$17$	1.54776	0.375387	0.187693	+	0.982228i	$0.439899\pi$
$18$	0	0
$19$	−0.0998752	−0.0229129	−0.0114565	−	0.999934i	$-0.503647\pi$
$19$	−0.0998752	−0.0229129	−0.0114565	+	0.999934i	$0.503647\pi$
$20$	0	0
$21$	−1.01084	−0.220583
$22$	0	0
$23$	−8.23807	−1.71776	−0.858879	−	0.512179i	$-0.828838\pi$
$23$	−8.23807	−1.71776	−0.858879	+	0.512179i	$0.828838\pi$
$24$	0	0
$25$	−4.90797	−0.981593
$26$	0	0
$27$	−5.03216	−0.968440
$28$	0	0
$29$	1.69722	0.315165	0.157583	−	0.987506i	$-0.449630\pi$
$29$	1.69722	0.315165	0.157583	+	0.987506i	$0.449630\pi$
$30$	0	0
$31$	4.52772	0.813202	0.406601	−	0.913606i	$-0.366714\pi$
$31$	4.52772	0.813202	0.406601	+	0.913606i	$0.366714\pi$
$32$	0	0
$33$	−0.892451	−0.155356
$34$	0	0
$35$	0.303370	0.0512789
$36$	0	0
$37$	0.883991	0.145327	0.0726636	−	0.997357i	$-0.476850\pi$
$37$	0.883991	0.145327	0.0726636	+	0.997357i	$0.476850\pi$
$38$	0	0
$39$	−5.66061	−0.906423
$40$	0	0
$41$	9.92605	1.55019	0.775094	−	0.631845i	$-0.217702\pi$
$41$	9.92605	1.55019	0.775094	+	0.631845i	$0.217702\pi$
$42$	0	0
$43$	3.30486	0.503986	0.251993	−	0.967729i	$-0.418914\pi$
$43$	3.30486	0.503986	0.251993	+	0.967729i	$0.418914\pi$
$44$	0	0
$45$	0.600128	0.0894618
$46$	0	0
$47$	−6.00904	−0.876508	−0.438254	−	0.898851i	$-0.644403\pi$
$47$	−6.00904	−0.876508	−0.438254	+	0.898851i	$0.644403\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.56453	0.219079
$52$	0	0
$53$	11.1794	1.53561	0.767807	−	0.640682i	$-0.221348\pi$
$53$	11.1794	1.53561	0.767807	+	0.640682i	$0.221348\pi$
$54$	0	0
$55$	0.267840	0.0361155
$56$	0	0
$57$	−0.100958	−0.0133722
$58$	0	0
$59$	8.71281	1.13431	0.567156	−	0.823611i	$-0.308044\pi$
$59$	8.71281	1.13431	0.567156	+	0.823611i	$0.308044\pi$
$60$	0	0
$61$	−2.57260	−0.329387	−0.164694	−	0.986345i	$-0.552664\pi$
$61$	−2.57260	−0.329387	−0.164694	+	0.986345i	$0.552664\pi$
$62$	0	0
$63$	1.97821	0.249231
$64$	0	0
$65$	1.69885	0.210716
$66$	0	0
$67$	11.5232	1.40778	0.703890	−	0.710309i	$-0.251445\pi$
$67$	11.5232	1.40778	0.703890	+	0.710309i	$0.251445\pi$
$68$	0	0
$69$	−8.32736	−1.00250
$70$	0	0
$71$	6.56548	0.779179	0.389589	−	0.920989i	$-0.372617\pi$
$71$	6.56548	0.779179	0.389589	+	0.920989i	$0.372617\pi$
$72$	0	0
$73$	14.4035	1.68581	0.842904	−	0.538064i	$-0.180844\pi$
$73$	14.4035	1.68581	0.842904	+	0.538064i	$0.180844\pi$
$74$	0	0
$75$	−4.96116	−0.572865
$76$	0	0
$77$	0.882882	0.100614
$78$	0	0
$79$	13.1905	1.48405	0.742023	−	0.670374i	$-0.233866\pi$
$79$	13.1905	1.48405	0.742023	+	0.670374i	$0.233866\pi$
$80$	0	0
$81$	0.847921	0.0942134
$82$	0	0
$83$	−4.91904	−0.539935	−0.269968	−	0.962869i	$-0.587013\pi$
$83$	−4.91904	−0.539935	−0.269968	+	0.962869i	$0.587013\pi$
$84$	0	0
$85$	−0.469544	−0.0509292
$86$	0	0
$87$	1.71561	0.183933
$88$	0	0
$89$	−4.10561	−0.435194	−0.217597	−	0.976039i	$-0.569822\pi$
$89$	−4.10561	−0.435194	−0.217597	+	0.976039i	$0.569822\pi$
$90$	0	0
$91$	5.59992	0.587031
$92$	0	0
$93$	4.57679	0.474591
$94$	0	0
$95$	0.0302991	0.00310863
$96$	0	0
$97$	5.05748	0.513510	0.256755	−	0.966477i	$-0.417347\pi$
$97$	5.05748	0.513510	0.256755	+	0.966477i	$0.417347\pi$
$98$	0	0
$99$	1.74652	0.175532

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	7168.2.a.bk.1.7		12
4.3	odd	2		7168.2.a.bh.1.6		12
8.3	odd	2		7168.2.a.bl.1.7		12
8.5	even	2		7168.2.a.bg.1.6		12
32.3	odd	8		3584.2.m.bm.2689.5	yes	24
32.5	even	8		3584.2.m.bk.897.5	✓	24
32.11	odd	8		3584.2.m.bm.897.5	yes	24
32.13	even	8		3584.2.m.bk.2689.5	yes	24
32.19	odd	8		3584.2.m.bl.2689.8	yes	24
32.21	even	8		3584.2.m.bn.897.8	yes	24
32.27	odd	8		3584.2.m.bl.897.8	yes	24
32.29	even	8		3584.2.m.bn.2689.8	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.m.bk.897.5	✓	24	32.5	even	8
3584.2.m.bk.2689.5	yes	24	32.13	even	8
3584.2.m.bl.897.8	yes	24	32.27	odd	8
3584.2.m.bl.2689.8	yes	24	32.19	odd	8
3584.2.m.bm.897.5	yes	24	32.11	odd	8
3584.2.m.bm.2689.5	yes	24	32.3	odd	8
3584.2.m.bn.897.8	yes	24	32.21	even	8
3584.2.m.bn.2689.8	yes	24	32.29	even	8
7168.2.a.bg.1.6		12	8.5	even	2
7168.2.a.bh.1.6		12	4.3	odd	2
7168.2.a.bk.1.7		12	1.1	even	1	trivial
7168.2.a.bl.1.7		12	8.3	odd	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:	\( N \)	\(=\)	\( 7168 = 2^{10} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7168.a (trivial)

\(f(q)\)	\(=\)	\(q+1.01084 q^{3} -0.303370 q^{5} -1.00000 q^{7} -1.97821 q^{9} -0.882882 q^{11} -5.59992 q^{13} -0.306658 q^{15} +1.54776 q^{17} -0.0998752 q^{19} -1.01084 q^{21} -8.23807 q^{23} -4.90797 q^{25} -5.03216 q^{27} +1.69722 q^{29} +4.52772 q^{31} -0.892451 q^{33} +0.303370 q^{35} +0.883991 q^{37} -5.66061 q^{39} +9.92605 q^{41} +3.30486 q^{43} +0.600128 q^{45} -6.00904 q^{47} +1.00000 q^{49} +1.56453 q^{51} +11.1794 q^{53} +0.267840 q^{55} -0.100958 q^{57} +8.71281 q^{59} -2.57260 q^{61} +1.97821 q^{63} +1.69885 q^{65} +11.5232 q^{67} -8.32736 q^{69} +6.56548 q^{71} +14.4035 q^{73} -4.96116 q^{75} +0.882882 q^{77} +13.1905 q^{79} +0.847921 q^{81} -4.91904 q^{83} -0.469544 q^{85} +1.71561 q^{87} -4.10561 q^{89} +5.59992 q^{91} +4.57679 q^{93} +0.0302991 q^{95} +5.05748 q^{97} +1.74652 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 8 q^{3} - 12 q^{7} + 12 q^{9} + 16 q^{11} + 8 q^{19} - 8 q^{21} + 12 q^{25} + 32 q^{27} - 24 q^{29} + 16 q^{33} - 8 q^{37} + 32 q^{39} + 16 q^{41} + 16 q^{43} + 12 q^{49} + 16 q^{51} - 24 q^{53}+ \cdots + 48 q^{99}+O(q^{100}) \) 12 * q + 8 * q^3 - 12 * q^7 + 12 * q^9 + 16 * q^11 + 8 * q^19 - 8 * q^21 + 12 * q^25 + 32 * q^27 - 24 * q^29 + 16 * q^33 - 8 * q^37 + 32 * q^39 + 16 * q^41 + 16 * q^43 + 12 * q^49 + 16 * q^51 - 24 * q^53 + 16 * q^57 + 40 * q^59 - 12 * q^63 - 8 * q^65 + 32 * q^67 - 8 * q^73 + 40 * q^75 - 16 * q^77 + 28 * q^81 + 56 * q^83 - 8 * q^89 + 32 * q^93 + 32 * q^95 + 48 * q^99

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