Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( -1 + 2 \zeta_{6} ) q^{5} -2 q^{9} +O(q^{10})\) \( q + q^{3} + ( -1 + 2 \zeta_{6} ) q^{5} -2 q^{9} + ( 1 - 2 \zeta_{6} ) q^{11} + ( -1 + 2 \zeta_{6} ) q^{15} + ( 3 - 6 \zeta_{6} ) q^{17} + 7 q^{19} + ( -5 + 10 \zeta_{6} ) q^{23} + 2 q^{25} -5 q^{27} + 6 q^{29} -5 q^{31} + ( 1 - 2 \zeta_{6} ) q^{33} + 5 q^{37} + ( -4 + 8 \zeta_{6} ) q^{41} + ( 2 - 4 \zeta_{6} ) q^{43} + ( 2 - 4 \zeta_{6} ) q^{45} -3 q^{47} + ( 3 - 6 \zeta_{6} ) q^{51} + 9 q^{53} + 3 q^{55} + 7 q^{57} + 9 q^{59} + ( -5 + 10 \zeta_{6} ) q^{61} + ( -3 + 6 \zeta_{6} ) q^{67} + ( -5 + 10 \zeta_{6} ) q^{69} + ( 2 - 4 \zeta_{6} ) q^{71} + ( -1 + 2 \zeta_{6} ) q^{73} + 2 q^{75} + ( 3 - 6 \zeta_{6} ) q^{79} + q^{81} + 12 q^{83} + 9 q^{85} + 6 q^{87} + ( -7 + 14 \zeta_{6} ) q^{89} -5 q^{93} + ( -7 + 14 \zeta_{6} ) q^{95} + ( -4 + 8 \zeta_{6} ) q^{97} + ( -2 + 4 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{9} + 14q^{19} + 4q^{25} - 10q^{27} + 12q^{29} - 10q^{31} + 10q^{37} - 6q^{47} + 18q^{53} + 6q^{55} + 14q^{57} + 18q^{59} + 4q^{75} + 2q^{81} + 24q^{83} + 18q^{85} + 12q^{87} - 10q^{93} + O(q^{100}) \)

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
0.500000 0.866025i
0.500000 + 0.866025i
0 1.00000 0 1.73205i 0 0 0 −2.00000 0
3135.2 0 1.00000 0 1.73205i 0 0 0 −2.00000 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
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.b 2
4.b odd 2 1 3136.2.f.a 2
7.b odd 2 1 3136.2.f.a 2
7.c even 3 1 448.2.p.a 2
7.d odd 6 1 448.2.p.b 2
8.b even 2 1 784.2.f.a 2
8.d odd 2 1 784.2.f.b 2
24.f even 2 1 7056.2.b.m 2
24.h odd 2 1 7056.2.b.b 2
28.d even 2 1 inner 3136.2.f.b 2
28.f even 6 1 448.2.p.a 2
28.g odd 6 1 448.2.p.b 2
56.e even 2 1 784.2.f.a 2
56.h odd 2 1 784.2.f.b 2
56.j odd 6 1 112.2.p.a 2
56.j odd 6 1 784.2.p.c 2
56.k odd 6 1 112.2.p.a 2
56.k odd 6 1 784.2.p.c 2
56.m even 6 1 112.2.p.b yes 2
56.m even 6 1 784.2.p.d 2
56.p even 6 1 112.2.p.b yes 2
56.p even 6 1 784.2.p.d 2
168.e odd 2 1 7056.2.b.b 2
168.i even 2 1 7056.2.b.m 2
168.s odd 6 1 1008.2.cs.c 2
168.v even 6 1 1008.2.cs.f 2
168.ba even 6 1 1008.2.cs.f 2
168.be odd 6 1 1008.2.cs.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.a 2 56.j odd 6 1
112.2.p.a 2 56.k odd 6 1
112.2.p.b yes 2 56.m even 6 1
112.2.p.b yes 2 56.p even 6 1
448.2.p.a 2 7.c even 3 1
448.2.p.a 2 28.f even 6 1
448.2.p.b 2 7.d odd 6 1
448.2.p.b 2 28.g odd 6 1
784.2.f.a 2 8.b even 2 1
784.2.f.a 2 56.e even 2 1
784.2.f.b 2 8.d odd 2 1
784.2.f.b 2 56.h odd 2 1
784.2.p.c 2 56.j odd 6 1
784.2.p.c 2 56.k odd 6 1
784.2.p.d 2 56.m even 6 1
784.2.p.d 2 56.p even 6 1
1008.2.cs.c 2 168.s odd 6 1
1008.2.cs.c 2 168.be odd 6 1
1008.2.cs.f 2 168.v even 6 1
1008.2.cs.f 2 168.ba even 6 1
3136.2.f.a 2 4.b odd 2 1
3136.2.f.a 2 7.b odd 2 1
3136.2.f.b 2 1.a even 1 1 trivial
3136.2.f.b 2 28.d even 2 1 inner
7056.2.b.b 2 24.h odd 2 1
7056.2.b.b 2 168.e odd 2 1
7056.2.b.m 2 24.f even 2 1
7056.2.b.m 2 168.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 \(S_{2}^{\mathrm{new}}(3136, [\chi])\):

\( T_{3} - 1 \)
\( T_{5}^{2} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T + 3 T^{2} )^{2} \)
$5$ \( 1 - 7 T^{2} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - 19 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - 13 T^{2} )^{2} \)
$17$ \( 1 - 7 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 29 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 5 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 5 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 34 T^{2} + 1681 T^{4} \)
$43$ \( 1 - 74 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 + 3 T + 47 T^{2} )^{2} \)
$53$ \( ( 1 - 9 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 - 9 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 13 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \)
$67$ \( 1 - 107 T^{2} + 4489 T^{4} \)
$71$ \( 1 - 130 T^{2} + 5041 T^{4} \)
$73$ \( ( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \)
$79$ \( ( 1 - 17 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \)
$83$ \( ( 1 - 12 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 31 T^{2} + 7921 T^{4} \)
$97$ \( 1 - 146 T^{2} + 9409 T^{4} \)
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