Properties

Label 35.7.c.b
Level $35$
Weight $7$
Character orbit 35.c
Self dual yes
Analytic conductor $8.052$
Analytic rank $0$
Dimension $1$
CM discriminant -35
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(8.05189292669\)
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

\(f(q)\) \(=\) \( q + 26q^{3} + 64q^{4} + 125q^{5} - 343q^{7} - 53q^{9} + O(q^{10}) \) \( q + 26q^{3} + 64q^{4} + 125q^{5} - 343q^{7} - 53q^{9} + 2522q^{11} + 1664q^{12} - 2774q^{13} + 3250q^{15} + 4096q^{16} + 754q^{17} + 8000q^{20} - 8918q^{21} + 15625q^{25} - 20332q^{27} - 21952q^{28} - 45862q^{29} + 65572q^{33} - 42875q^{35} - 3392q^{36} - 72124q^{39} + 161408q^{44} - 6625q^{45} - 175646q^{47} + 106496q^{48} + 117649q^{49} + 19604q^{51} - 177536q^{52} + 315250q^{55} + 208000q^{60} + 18179q^{63} + 262144q^{64} - 346750q^{65} + 48256q^{68} - 30238q^{71} - 504254q^{73} + 406250q^{75} - 865046q^{77} - 930382q^{79} + 512000q^{80} - 489995q^{81} + 1141306q^{83} - 570752q^{84} + 94250q^{85} - 1192412q^{87} + 951482q^{91} + 897874q^{97} - 133666q^{99} + O(q^{100}) \)

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 26.0000 64.0000 125.000 0 −343.000 0 −53.0000 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 35.7.c.b yes 1
5.b even 2 1 35.7.c.a 1
5.c odd 4 2 175.7.d.c 2
7.b odd 2 1 35.7.c.a 1
35.c odd 2 1 CM 35.7.c.b yes 1
35.f even 4 2 175.7.d.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.7.c.a 1 5.b even 2 1
35.7.c.a 1 7.b odd 2 1
35.7.c.b yes 1 1.a even 1 1 trivial
35.7.c.b yes 1 35.c odd 2 1 CM
175.7.d.c 2 5.c odd 4 2
175.7.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 \(S_{7}^{\mathrm{new}}(35, [\chi])\):

\( T_{2} \)
\( T_{3} - 26 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -26 + T \)
$5$ \( -125 + T \)
$7$ \( 343 + T \)
$11$ \( -2522 + T \)
$13$ \( 2774 + T \)
$17$ \( -754 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( 45862 + T \)
$31$ \( T \)
$37$ \( T \)
$41$ \( T \)
$43$ \( T \)
$47$ \( 175646 + T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( T \)
$67$ \( T \)
$71$ \( 30238 + T \)
$73$ \( 504254 + T \)
$79$ \( 930382 + T \)
$83$ \( -1141306 + T \)
$89$ \( T \)
$97$ \( -897874 + T \)
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