Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.56506787962\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
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

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{3} + \zeta_{12}^{4} q^{5} +O(q^{10})\) \( q + \zeta_{12}^{5} q^{3} + \zeta_{12}^{4} q^{5} + \zeta_{12}^{5} q^{11} -\zeta_{12}^{3} q^{15} + \zeta_{12}^{2} q^{17} + \zeta_{12} q^{19} -\zeta_{12} q^{23} -\zeta_{12}^{3} q^{27} + \zeta_{12}^{5} q^{31} -\zeta_{12}^{4} q^{33} + \zeta_{12}^{4} q^{37} -\zeta_{12} q^{47} -\zeta_{12} q^{51} -\zeta_{12}^{2} q^{53} -\zeta_{12}^{3} q^{55} - q^{57} + \zeta_{12}^{5} q^{59} -\zeta_{12}^{4} q^{61} -\zeta_{12}^{5} q^{67} + q^{69} -\zeta_{12}^{2} q^{73} -\zeta_{12} q^{79} + \zeta_{12}^{2} q^{81} - q^{85} + \zeta_{12}^{4} q^{89} -\zeta_{12}^{4} q^{93} + \zeta_{12}^{5} q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} + O(q^{10}) \) \( 4q - 2q^{5} + 2q^{17} + 2q^{33} - 2q^{37} - 2q^{53} - 4q^{57} + 2q^{61} + 4q^{69} - 2q^{73} + 2q^{81} - 4q^{85} - 2q^{89} + 2q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{12}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2431.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −0.866025 + 0.500000i 0 −0.500000 + 0.866025i 0 0 0 0 0
2431.2 0 0.866025 0.500000i 0 −0.500000 + 0.866025i 0 0 0 0 0
3007.1 0 −0.866025 0.500000i 0 −0.500000 0.866025i 0 0 0 0 0
3007.2 0 0.866025 + 0.500000i 0 −0.500000 0.866025i 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
7.c even 3 1 inner
28.g odd 6 1 inner

Twists

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

Hecke kernels

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

\( T_{3}^{4} - T_{3}^{2} + 1 \)
\( T_{5}^{2} + T_{5} + 1 \)

Hecke characteristic polynomials

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