Properties

Label 7.9.b.a
Level 7
Weight 9
Character orbit 7.b
Self dual yes
Analytic conductor 2.852
Analytic rank 0
Dimension 1
CM discriminant -7
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 7.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(2.85165027043\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 31q^{2} + 705q^{4} + 2401q^{7} - 13919q^{8} + 6561q^{9} + O(q^{10}) \) \( q - 31q^{2} + 705q^{4} + 2401q^{7} - 13919q^{8} + 6561q^{9} + 13154q^{11} - 74431q^{14} + 251009q^{16} - 203391q^{18} - 407774q^{22} - 20926q^{23} + 390625q^{25} + 1692705q^{28} + 108194q^{29} - 4218015q^{32} + 4625505q^{36} - 2073886q^{37} - 6726046q^{43} + 9273570q^{44} + 648706q^{46} + 5764801q^{49} - 12109375q^{50} + 15377762q^{53} - 33419519q^{56} - 3354014q^{58} + 15752961q^{63} + 66500161q^{64} - 15839326q^{67} - 42331966q^{71} - 91322559q^{72} + 64290466q^{74} + 31582754q^{77} - 64606846q^{79} + 43046721q^{81} + 208507426q^{86} - 183090526q^{88} - 14752830q^{92} - 178708831q^{98} + 86303394q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
0
−31.0000 0 705.000 0 0 2401.00 −13919.0 6561.00 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.9.b.a 1
3.b odd 2 1 63.9.d.a 1
4.b odd 2 1 112.9.c.a 1
5.b even 2 1 175.9.d.a 1
5.c odd 4 2 175.9.c.a 2
7.b odd 2 1 CM 7.9.b.a 1
7.c even 3 2 49.9.d.a 2
7.d odd 6 2 49.9.d.a 2
21.c even 2 1 63.9.d.a 1
28.d even 2 1 112.9.c.a 1
35.c odd 2 1 175.9.d.a 1
35.f even 4 2 175.9.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.9.b.a 1 1.a even 1 1 trivial
7.9.b.a 1 7.b odd 2 1 CM
49.9.d.a 2 7.c even 3 2
49.9.d.a 2 7.d odd 6 2
63.9.d.a 1 3.b odd 2 1
63.9.d.a 1 21.c even 2 1
112.9.c.a 1 4.b odd 2 1
112.9.c.a 1 28.d even 2 1
175.9.c.a 2 5.c odd 4 2
175.9.c.a 2 35.f even 4 2
175.9.d.a 1 5.b even 2 1
175.9.d.a 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 31 \) acting on \(S_{9}^{\mathrm{new}}(7, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 31 T + 256 T^{2} \)
$3$ \( ( 1 - 81 T )( 1 + 81 T ) \)
$5$ \( ( 1 - 625 T )( 1 + 625 T ) \)
$7$ \( 1 - 2401 T \)
$11$ \( 1 - 13154 T + 214358881 T^{2} \)
$13$ \( ( 1 - 28561 T )( 1 + 28561 T ) \)
$17$ \( ( 1 - 83521 T )( 1 + 83521 T ) \)
$19$ \( ( 1 - 130321 T )( 1 + 130321 T ) \)
$23$ \( 1 + 20926 T + 78310985281 T^{2} \)
$29$ \( 1 - 108194 T + 500246412961 T^{2} \)
$31$ \( ( 1 - 923521 T )( 1 + 923521 T ) \)
$37$ \( 1 + 2073886 T + 3512479453921 T^{2} \)
$41$ \( ( 1 - 2825761 T )( 1 + 2825761 T ) \)
$43$ \( 1 + 6726046 T + 11688200277601 T^{2} \)
$47$ \( ( 1 - 4879681 T )( 1 + 4879681 T ) \)
$53$ \( 1 - 15377762 T + 62259690411361 T^{2} \)
$59$ \( ( 1 - 12117361 T )( 1 + 12117361 T ) \)
$61$ \( ( 1 - 13845841 T )( 1 + 13845841 T ) \)
$67$ \( 1 + 15839326 T + 406067677556641 T^{2} \)
$71$ \( 1 + 42331966 T + 645753531245761 T^{2} \)
$73$ \( ( 1 - 28398241 T )( 1 + 28398241 T ) \)
$79$ \( 1 + 64606846 T + 1517108809906561 T^{2} \)
$83$ \( ( 1 - 47458321 T )( 1 + 47458321 T ) \)
$89$ \( ( 1 - 62742241 T )( 1 + 62742241 T ) \)
$97$ \( ( 1 - 88529281 T )( 1 + 88529281 T ) \)
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