Properties

Label 35.9.c.b
Level $35$
Weight $9$
Character orbit 35.c
Self dual yes
Analytic conductor $14.258$
Analytic rank $0$
Dimension $1$
CM discriminant -35
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 35.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(14.2582513521\)
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

\(f(q)\) \(=\) \( q + 127q^{3} + 256q^{4} + 625q^{5} + 2401q^{7} + 9568q^{9} + O(q^{10}) \) \( q + 127q^{3} + 256q^{4} + 625q^{5} + 2401q^{7} + 9568q^{9} - 23953q^{11} + 32512q^{12} - 56593q^{13} + 79375q^{15} + 65536q^{16} - 97873q^{17} + 160000q^{20} + 304927q^{21} + 390625q^{25} + 381889q^{27} + 614656q^{28} - 85153q^{29} - 3042031q^{33} + 1500625q^{35} + 2449408q^{36} - 7187311q^{39} - 6131968q^{44} + 5980000q^{45} + 2191487q^{47} + 8323072q^{48} + 5764801q^{49} - 12429871q^{51} - 14487808q^{52} - 14970625q^{55} + 20320000q^{60} + 22972768q^{63} + 16777216q^{64} - 35370625q^{65} - 25055488q^{68} + 50742722q^{71} + 33491522q^{73} + 49609375q^{75} - 57511153q^{77} + 70135727q^{79} + 40960000q^{80} - 14275745q^{81} - 54186718q^{83} + 78061312q^{84} - 61170625q^{85} - 10814431q^{87} - 135879793q^{91} - 165613873q^{97} - 229182304q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1
0
0 127.000 256.000 625.000 0 2401.00 0 9568.00 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
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.9.c.b yes 1
5.b even 2 1 35.9.c.a 1
5.c odd 4 2 175.9.d.c 2
7.b odd 2 1 35.9.c.a 1
35.c odd 2 1 CM 35.9.c.b yes 1
35.f even 4 2 175.9.d.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.9.c.a 1 5.b even 2 1
35.9.c.a 1 7.b odd 2 1
35.9.c.b yes 1 1.a even 1 1 trivial
35.9.c.b yes 1 35.c odd 2 1 CM
175.9.d.c 2 5.c odd 4 2
175.9.d.c 2 35.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(35, [\chi])\):

\( T_{2} \)
\( T_{3} - 127 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -127 + T \)
$5$ \( -625 + T \)
$7$ \( -2401 + T \)
$11$ \( 23953 + T \)
$13$ \( 56593 + T \)
$17$ \( 97873 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( 85153 + T \)
$31$ \( T \)
$37$ \( T \)
$41$ \( T \)
$43$ \( T \)
$47$ \( -2191487 + T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( T \)
$67$ \( T \)
$71$ \( -50742722 + T \)
$73$ \( -33491522 + T \)
$79$ \( -70135727 + T \)
$83$ \( 54186718 + T \)
$89$ \( T \)
$97$ \( 165613873 + T \)
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