Properties

Label 3136.2.b.b
Level $3136$
Weight $2$
Character orbit 3136.b
Analytic conductor $25.041$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3136.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.0410860739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 64)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{3} - q^{9} +O(q^{10})\) \( q + 2 i q^{3} - q^{9} + 6 i q^{11} + 6 q^{17} + 2 i q^{19} + 5 q^{25} + 4 i q^{27} -12 q^{33} + 6 q^{41} -10 i q^{43} + 12 i q^{51} -4 q^{57} -6 i q^{59} + 14 i q^{67} -2 q^{73} + 10 i q^{75} -11 q^{81} + 18 i q^{83} -18 q^{89} -10 q^{97} -6 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 12q^{17} + 10q^{25} - 24q^{33} + 12q^{41} - 8q^{57} - 4q^{73} - 22q^{81} - 36q^{89} - 20q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1569.1
1.00000i
1.00000i
0 2.00000i 0 0 0 0 0 −1.00000 0
1569.2 0 2.00000i 0 0 0 0 0 −1.00000 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b 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.b.b 2
4.b odd 2 1 inner 3136.2.b.b 2
7.b odd 2 1 64.2.b.a 2
8.b even 2 1 inner 3136.2.b.b 2
8.d odd 2 1 CM 3136.2.b.b 2
21.c even 2 1 576.2.d.a 2
28.d even 2 1 64.2.b.a 2
35.c odd 2 1 1600.2.d.a 2
35.f even 4 1 1600.2.f.a 2
35.f even 4 1 1600.2.f.b 2
56.e even 2 1 64.2.b.a 2
56.h odd 2 1 64.2.b.a 2
84.h odd 2 1 576.2.d.a 2
112.j even 4 1 256.2.a.a 1
112.j even 4 1 256.2.a.d 1
112.l odd 4 1 256.2.a.a 1
112.l odd 4 1 256.2.a.d 1
140.c even 2 1 1600.2.d.a 2
140.j odd 4 1 1600.2.f.a 2
140.j odd 4 1 1600.2.f.b 2
168.e odd 2 1 576.2.d.a 2
168.i even 2 1 576.2.d.a 2
224.v odd 8 4 1024.2.e.l 4
224.x even 8 4 1024.2.e.l 4
280.c odd 2 1 1600.2.d.a 2
280.n even 2 1 1600.2.d.a 2
280.s even 4 1 1600.2.f.a 2
280.s even 4 1 1600.2.f.b 2
280.y odd 4 1 1600.2.f.a 2
280.y odd 4 1 1600.2.f.b 2
336.v odd 4 1 2304.2.a.h 1
336.v odd 4 1 2304.2.a.i 1
336.y even 4 1 2304.2.a.h 1
336.y even 4 1 2304.2.a.i 1
560.be even 4 1 6400.2.a.a 1
560.be even 4 1 6400.2.a.x 1
560.bf odd 4 1 6400.2.a.a 1
560.bf odd 4 1 6400.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.b.a 2 7.b odd 2 1
64.2.b.a 2 28.d even 2 1
64.2.b.a 2 56.e even 2 1
64.2.b.a 2 56.h odd 2 1
256.2.a.a 1 112.j even 4 1
256.2.a.a 1 112.l odd 4 1
256.2.a.d 1 112.j even 4 1
256.2.a.d 1 112.l odd 4 1
576.2.d.a 2 21.c even 2 1
576.2.d.a 2 84.h odd 2 1
576.2.d.a 2 168.e odd 2 1
576.2.d.a 2 168.i even 2 1
1024.2.e.l 4 224.v odd 8 4
1024.2.e.l 4 224.x even 8 4
1600.2.d.a 2 35.c odd 2 1
1600.2.d.a 2 140.c even 2 1
1600.2.d.a 2 280.c odd 2 1
1600.2.d.a 2 280.n even 2 1
1600.2.f.a 2 35.f even 4 1
1600.2.f.a 2 140.j odd 4 1
1600.2.f.a 2 280.s even 4 1
1600.2.f.a 2 280.y odd 4 1
1600.2.f.b 2 35.f even 4 1
1600.2.f.b 2 140.j odd 4 1
1600.2.f.b 2 280.s even 4 1
1600.2.f.b 2 280.y odd 4 1
2304.2.a.h 1 336.v odd 4 1
2304.2.a.h 1 336.y even 4 1
2304.2.a.i 1 336.v odd 4 1
2304.2.a.i 1 336.y even 4 1
3136.2.b.b 2 1.a even 1 1 trivial
3136.2.b.b 2 4.b odd 2 1 inner
3136.2.b.b 2 8.b even 2 1 inner
3136.2.b.b 2 8.d odd 2 1 CM
6400.2.a.a 1 560.be even 4 1
6400.2.a.a 1 560.bf odd 4 1
6400.2.a.x 1 560.be even 4 1
6400.2.a.x 1 560.bf odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 4 \)
\( T_{5} \)
\( T_{17} - 6 \)
\( T_{31} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( 36 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -6 + T )^{2} \)
$19$ \( 4 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 100 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( 36 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( 196 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 2 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( 324 + T^{2} \)
$89$ \( ( 18 + T )^{2} \)
$97$ \( ( 10 + T )^{2} \)
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