Properties

Label 3136.1.d.b
Level 3136
Weight 1
Character orbit 3136.d
Analytic conductor 1.565
Analytic rank 0
Dimension 2
Projective image \(A_{4}\)
CM/RM no
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.56506787962\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Projective image \(A_{4}\)
Projective field Galois closure of 4.0.3136.1
Artin image $\SL(2,3):C_2$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + i q^{3} - q^{5} +O(q^{10})\) \( q + i q^{3} - q^{5} -i q^{11} -i q^{15} + q^{17} + i q^{19} + i q^{23} + i q^{27} + i q^{31} + q^{33} + q^{37} -i q^{47} + i q^{51} + q^{53} + i q^{55} - q^{57} + i q^{59} + q^{61} + i q^{67} - q^{69} - q^{73} + i q^{79} - q^{81} - q^{85} - q^{89} - q^{93} -i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{5} + 2q^{17} + 2q^{33} + 2q^{37} + 2q^{53} - 2q^{57} + 2q^{61} - 2q^{69} - 2q^{73} - 2q^{81} - 2q^{85} - 2q^{89} - 2q^{93} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1471.1
1.00000i
1.00000i
0 1.00000i 0 −1.00000 0 0 0 0 0
1471.2 0 1.00000i 0 −1.00000 0 0 0 0 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.1.d.b 2
4.b odd 2 1 inner 3136.1.d.b 2
7.b odd 2 1 3136.1.d.d 2
7.c even 3 2 448.1.r.a 4
7.d odd 6 2 3136.1.r.b 4
8.b even 2 1 1568.1.d.b 2
8.d odd 2 1 1568.1.d.b 2
28.d even 2 1 3136.1.d.d 2
28.f even 6 2 3136.1.r.b 4
28.g odd 6 2 448.1.r.a 4
56.e even 2 1 1568.1.d.a 2
56.h odd 2 1 1568.1.d.a 2
56.j odd 6 2 1568.1.r.a 4
56.k odd 6 2 224.1.r.a 4
56.m even 6 2 1568.1.r.a 4
56.p even 6 2 224.1.r.a 4
112.u odd 12 2 1792.1.o.a 4
112.u odd 12 2 1792.1.o.b 4
112.w even 12 2 1792.1.o.a 4
112.w even 12 2 1792.1.o.b 4
168.s odd 6 2 2016.1.cd.a 4
168.v even 6 2 2016.1.cd.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.1.r.a 4 56.k odd 6 2
224.1.r.a 4 56.p even 6 2
448.1.r.a 4 7.c even 3 2
448.1.r.a 4 28.g odd 6 2
1568.1.d.a 2 56.e even 2 1
1568.1.d.a 2 56.h odd 2 1
1568.1.d.b 2 8.b even 2 1
1568.1.d.b 2 8.d odd 2 1
1568.1.r.a 4 56.j odd 6 2
1568.1.r.a 4 56.m even 6 2
1792.1.o.a 4 112.u odd 12 2
1792.1.o.a 4 112.w even 12 2
1792.1.o.b 4 112.u odd 12 2
1792.1.o.b 4 112.w even 12 2
2016.1.cd.a 4 168.s odd 6 2
2016.1.cd.a 4 168.v even 6 2
3136.1.d.b 2 1.a even 1 1 trivial
3136.1.d.b 2 4.b odd 2 1 inner
3136.1.d.d 2 7.b odd 2 1
3136.1.d.d 2 28.d even 2 1
3136.1.r.b 4 7.d odd 6 2
3136.1.r.b 4 28.f even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3136, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ 1
$11$ \( 1 - T^{2} + T^{4} \)
$13$ \( ( 1 + T^{2} )^{2} \)
$17$ \( ( 1 - T + T^{2} )^{2} \)
$19$ \( 1 - T^{2} + T^{4} \)
$23$ \( 1 - T^{2} + T^{4} \)
$29$ \( ( 1 + T^{2} )^{2} \)
$31$ \( 1 - T^{2} + T^{4} \)
$37$ \( ( 1 - T + T^{2} )^{2} \)
$41$ \( ( 1 + T^{2} )^{2} \)
$43$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$47$ \( 1 - T^{2} + T^{4} \)
$53$ \( ( 1 - T + T^{2} )^{2} \)
$59$ \( 1 - T^{2} + T^{4} \)
$61$ \( ( 1 - T + T^{2} )^{2} \)
$67$ \( 1 - T^{2} + T^{4} \)
$71$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$73$ \( ( 1 + T + T^{2} )^{2} \)
$79$ \( 1 - T^{2} + T^{4} \)
$83$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$89$ \( ( 1 + T + T^{2} )^{2} \)
$97$ \( ( 1 + T^{2} )^{2} \)
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