Properties

Label 3136.2.f.i
Level $3136$
Weight $2$
Character orbit 3136.f
Analytic conductor $25.041$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.1212153856.10
Defining polynomial: \(x^{8} - 4 x^{6} + 10 x^{4} - 16 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 196)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + ( \beta_{1} - \beta_{2} ) q^{5} + ( 3 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{4} q^{3} + ( \beta_{1} - \beta_{2} ) q^{5} + ( 3 + \beta_{3} ) q^{9} -\beta_{5} q^{11} + ( -\beta_{1} - \beta_{2} ) q^{13} + ( -\beta_{5} + \beta_{6} ) q^{15} + ( 2 \beta_{1} + \beta_{2} ) q^{17} + ( \beta_{4} - \beta_{7} ) q^{19} + ( \beta_{5} + \beta_{6} ) q^{23} + ( 1 + 2 \beta_{3} ) q^{25} + \beta_{7} q^{27} + ( 4 - 2 \beta_{3} ) q^{29} + 2 \beta_{7} q^{31} + ( -\beta_{1} - 5 \beta_{2} ) q^{33} + 4 q^{37} + ( -\beta_{5} - \beta_{6} ) q^{39} + 3 \beta_{1} q^{41} + ( -\beta_{5} + 2 \beta_{6} ) q^{43} + ( 3 \beta_{1} - \beta_{2} ) q^{45} + 2 \beta_{4} q^{47} + ( \beta_{5} + 2 \beta_{6} ) q^{51} + ( -2 + 6 \beta_{3} ) q^{53} + ( -2 \beta_{4} + 2 \beta_{7} ) q^{55} + ( 4 - 5 \beta_{3} ) q^{57} + ( -\beta_{4} + 3 \beta_{7} ) q^{59} + ( -5 \beta_{1} - 5 \beta_{2} ) q^{61} + 2 \beta_{3} q^{65} -2 \beta_{6} q^{67} + ( 8 \beta_{1} + 6 \beta_{2} ) q^{69} + ( -\beta_{1} - 8 \beta_{2} ) q^{73} + ( \beta_{4} + 2 \beta_{7} ) q^{75} -2 \beta_{5} q^{79} + ( -7 + 3 \beta_{3} ) q^{81} + ( -3 \beta_{4} + \beta_{7} ) q^{83} + ( -2 - 2 \beta_{3} ) q^{85} + ( 4 \beta_{4} - 2 \beta_{7} ) q^{87} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 4 + 12 \beta_{3} ) q^{93} + ( -3 \beta_{5} + \beta_{6} ) q^{95} + 5 \beta_{2} q^{97} + ( -2 \beta_{5} - \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 24q^{9} + O(q^{10}) \) \( 8q + 24q^{9} + 8q^{25} + 32q^{29} + 32q^{37} - 16q^{53} + 32q^{57} - 56q^{81} - 16q^{85} + 32q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{6} + 10 x^{4} - 16 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{3} \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - 4 \nu^{5} + 10 \nu^{3} - 8 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + 4 \nu^{4} - 6 \nu^{2} + 8 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} - 10 \nu^{3} + 24 \nu \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} - 4 \nu^{4} + 14 \nu^{2} - 16 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{6} - 4 \nu^{4} + 10 \nu^{2} - 8 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + 3 \nu^{5} - 4 \nu^{3} + 6 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + \beta_{3} + 2\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 3 \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{6} + \beta_{5} + 4 \beta_{3} - 2\)\()/2\)
\(\nu^{5}\)\(=\)\(\beta_{7} - \beta_{4} + 3 \beta_{1}\)
\(\nu^{6}\)\(=\)\(2 \beta_{6} - \beta_{5} + \beta_{3} - 2\)
\(\nu^{7}\)\(=\)\(-\beta_{7} - 3 \beta_{2} + 7 \beta_{1}\)

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
−1.36145 0.382683i
−1.36145 + 0.382683i
−1.07072 + 0.923880i
−1.07072 0.923880i
1.07072 + 0.923880i
1.07072 0.923880i
1.36145 0.382683i
1.36145 + 0.382683i
0 −2.72291 0 1.08239i 0 0 0 4.41421 0
3135.2 0 −2.72291 0 1.08239i 0 0 0 4.41421 0
3135.3 0 −2.14144 0 2.61313i 0 0 0 1.58579 0
3135.4 0 −2.14144 0 2.61313i 0 0 0 1.58579 0
3135.5 0 2.14144 0 2.61313i 0 0 0 1.58579 0
3135.6 0 2.14144 0 2.61313i 0 0 0 1.58579 0
3135.7 0 2.72291 0 1.08239i 0 0 0 4.41421 0
3135.8 0 2.72291 0 1.08239i 0 0 0 4.41421 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3135.8
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.i 8
4.b odd 2 1 inner 3136.2.f.i 8
7.b odd 2 1 inner 3136.2.f.i 8
8.b even 2 1 196.2.d.c 8
8.d odd 2 1 196.2.d.c 8
24.f even 2 1 1764.2.b.k 8
24.h odd 2 1 1764.2.b.k 8
28.d even 2 1 inner 3136.2.f.i 8
56.e even 2 1 196.2.d.c 8
56.h odd 2 1 196.2.d.c 8
56.j odd 6 2 196.2.f.d 16
56.k odd 6 2 196.2.f.d 16
56.m even 6 2 196.2.f.d 16
56.p even 6 2 196.2.f.d 16
168.e odd 2 1 1764.2.b.k 8
168.i even 2 1 1764.2.b.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.2.d.c 8 8.b even 2 1
196.2.d.c 8 8.d odd 2 1
196.2.d.c 8 56.e even 2 1
196.2.d.c 8 56.h odd 2 1
196.2.f.d 16 56.j odd 6 2
196.2.f.d 16 56.k odd 6 2
196.2.f.d 16 56.m even 6 2
196.2.f.d 16 56.p even 6 2
1764.2.b.k 8 24.f even 2 1
1764.2.b.k 8 24.h odd 2 1
1764.2.b.k 8 168.e odd 2 1
1764.2.b.k 8 168.i even 2 1
3136.2.f.i 8 1.a even 1 1 trivial
3136.2.f.i 8 4.b odd 2 1 inner
3136.2.f.i 8 7.b odd 2 1 inner
3136.2.f.i 8 28.d even 2 1 inner

Hecke kernels

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

\( T_{3}^{4} - 12 T_{3}^{2} + 34 \)
\( T_{5}^{4} + 8 T_{5}^{2} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 34 - 12 T^{2} + T^{4} )^{2} \)
$5$ \( ( 8 + 8 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( 68 + 20 T^{2} + T^{4} )^{2} \)
$13$ \( ( 8 + 8 T^{2} + T^{4} )^{2} \)
$17$ \( ( 2 + 20 T^{2} + T^{4} )^{2} \)
$19$ \( ( 34 - 28 T^{2} + T^{4} )^{2} \)
$23$ \( ( 272 + 56 T^{2} + T^{4} )^{2} \)
$29$ \( ( 8 - 8 T + T^{2} )^{4} \)
$31$ \( ( 2176 - 96 T^{2} + T^{4} )^{2} \)
$37$ \( ( -4 + T )^{8} \)
$41$ \( ( 162 + 36 T^{2} + T^{4} )^{2} \)
$43$ \( ( 3332 + 116 T^{2} + T^{4} )^{2} \)
$47$ \( ( 544 - 48 T^{2} + T^{4} )^{2} \)
$53$ \( ( -68 + 4 T + T^{2} )^{4} \)
$59$ \( ( 9826 - 204 T^{2} + T^{4} )^{2} \)
$61$ \( ( 5000 + 200 T^{2} + T^{4} )^{2} \)
$67$ \( ( 1088 + 112 T^{2} + T^{4} )^{2} \)
$71$ \( T^{8} \)
$73$ \( ( 12482 + 260 T^{2} + T^{4} )^{2} \)
$79$ \( ( 1088 + 80 T^{2} + T^{4} )^{2} \)
$83$ \( ( 1666 - 108 T^{2} + T^{4} )^{2} \)
$89$ \( ( 578 + 52 T^{2} + T^{4} )^{2} \)
$97$ \( ( 1250 + 100 T^{2} + T^{4} )^{2} \)
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