Properties

Label 3136.2.f.f
Level $3136$
Weight $2$
Character orbit 3136.f
Analytic conductor $25.041$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3136.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.0410860739\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{2} q^{5} + 4 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{2} q^{5} + 4 q^{9} -\beta_{3} q^{11} + 2 \beta_{2} q^{13} + \beta_{3} q^{15} + 3 \beta_{2} q^{17} + \beta_{1} q^{19} -\beta_{3} q^{23} + 2 q^{25} + \beta_{1} q^{27} + \beta_{1} q^{31} + 7 \beta_{2} q^{33} -7 q^{37} -2 \beta_{3} q^{39} + 2 \beta_{2} q^{41} + 2 \beta_{3} q^{43} -4 \beta_{2} q^{45} + 3 \beta_{1} q^{47} -3 \beta_{3} q^{51} -3 q^{53} + 3 \beta_{1} q^{55} + 7 q^{57} -3 \beta_{1} q^{59} + \beta_{2} q^{61} + 6 q^{65} + \beta_{3} q^{67} + 7 \beta_{2} q^{69} -2 \beta_{3} q^{71} -3 \beta_{2} q^{73} + 2 \beta_{1} q^{75} -\beta_{3} q^{79} -5 q^{81} + 9 q^{85} -\beta_{2} q^{89} + 7 q^{93} + \beta_{3} q^{95} + 2 \beta_{2} q^{97} -4 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 16q^{9} + O(q^{10}) \) \( 4q + 16q^{9} + 8q^{25} - 28q^{37} - 12q^{53} + 28q^{57} + 24q^{65} - 20q^{81} + 36q^{85} + 28q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/7\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{2} + 7 \)\()/7\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 14 \nu \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(7 \beta_{2} - 7\)\()/2\)
\(\nu^{3}\)\(=\)\(7 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3135.1
1.32288 + 2.29129i
1.32288 2.29129i
−1.32288 2.29129i
−1.32288 + 2.29129i
0 −2.64575 0 1.73205i 0 0 0 4.00000 0
3135.2 0 −2.64575 0 1.73205i 0 0 0 4.00000 0
3135.3 0 2.64575 0 1.73205i 0 0 0 4.00000 0
3135.4 0 2.64575 0 1.73205i 0 0 0 4.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.2.f.f 4
4.b odd 2 1 inner 3136.2.f.f 4
7.b odd 2 1 inner 3136.2.f.f 4
7.c even 3 1 448.2.p.c 4
7.d odd 6 1 448.2.p.c 4
8.b even 2 1 784.2.f.d 4
8.d odd 2 1 784.2.f.d 4
24.f even 2 1 7056.2.b.s 4
24.h odd 2 1 7056.2.b.s 4
28.d even 2 1 inner 3136.2.f.f 4
28.f even 6 1 448.2.p.c 4
28.g odd 6 1 448.2.p.c 4
56.e even 2 1 784.2.f.d 4
56.h odd 2 1 784.2.f.d 4
56.j odd 6 1 112.2.p.c 4
56.j odd 6 1 784.2.p.g 4
56.k odd 6 1 112.2.p.c 4
56.k odd 6 1 784.2.p.g 4
56.m even 6 1 112.2.p.c 4
56.m even 6 1 784.2.p.g 4
56.p even 6 1 112.2.p.c 4
56.p even 6 1 784.2.p.g 4
168.e odd 2 1 7056.2.b.s 4
168.i even 2 1 7056.2.b.s 4
168.s odd 6 1 1008.2.cs.q 4
168.v even 6 1 1008.2.cs.q 4
168.ba even 6 1 1008.2.cs.q 4
168.be odd 6 1 1008.2.cs.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.c 4 56.j odd 6 1
112.2.p.c 4 56.k odd 6 1
112.2.p.c 4 56.m even 6 1
112.2.p.c 4 56.p even 6 1
448.2.p.c 4 7.c even 3 1
448.2.p.c 4 7.d odd 6 1
448.2.p.c 4 28.f even 6 1
448.2.p.c 4 28.g odd 6 1
784.2.f.d 4 8.b even 2 1
784.2.f.d 4 8.d odd 2 1
784.2.f.d 4 56.e even 2 1
784.2.f.d 4 56.h odd 2 1
784.2.p.g 4 56.j odd 6 1
784.2.p.g 4 56.k odd 6 1
784.2.p.g 4 56.m even 6 1
784.2.p.g 4 56.p even 6 1
1008.2.cs.q 4 168.s odd 6 1
1008.2.cs.q 4 168.v even 6 1
1008.2.cs.q 4 168.ba even 6 1
1008.2.cs.q 4 168.be odd 6 1
3136.2.f.f 4 1.a even 1 1 trivial
3136.2.f.f 4 4.b odd 2 1 inner
3136.2.f.f 4 7.b odd 2 1 inner
3136.2.f.f 4 28.d even 2 1 inner
7056.2.b.s 4 24.f even 2 1
7056.2.b.s 4 24.h odd 2 1
7056.2.b.s 4 168.e odd 2 1
7056.2.b.s 4 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3136, [\chi])\):

\( T_{3}^{2} - 7 \)
\( T_{5}^{2} + 3 \)