Properties

Label 35.15.c.a
Level $35$
Weight $15$
Character orbit 35.c
Self dual yes
Analytic conductor $43.515$
Analytic rank $0$
Dimension $1$
CM discriminant -35
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 35.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(43.5151388532\)
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

\(f(q)\) \(=\) \( q - 4031q^{3} + 16384q^{4} - 78125q^{5} + 823543q^{7} + 11465992q^{9} + O(q^{10}) \) \( q - 4031q^{3} + 16384q^{4} - 78125q^{5} + 823543q^{7} + 11465992q^{9} - 37379173q^{11} - 66043904q^{12} - 65279611q^{13} + 314921875q^{15} + 268435456q^{16} - 626193259q^{17} - 1280000000q^{20} - 3319701833q^{21} + 6103515625q^{25} - 26939265713q^{27} + 13492928512q^{28} - 9775649497q^{29} + 150675446363q^{33} - 64339296875q^{35} + 187858812928q^{36} + 263142111941q^{39} - 612420370432q^{44} - 895780625000q^{45} + 719081600801q^{47} - 1082063323136q^{48} + 678223072849q^{49} + 2524185027029q^{51} - 1069541146624q^{52} + 2920247890625q^{55} + 5159680000000q^{60} + 9442737449656q^{63} + 4398046511104q^{64} + 5099969609375q^{65} - 10259550355456q^{68} - 1790558995678q^{71} + 22033597628414q^{73} - 24603271484375q^{75} - 30783356269939q^{77} - 27088287440917q^{79} - 20971520000000q^{80} + 53750695798855q^{81} + 33726754263974q^{83} - 54389994831872q^{84} + 48921348359375q^{85} + 39405643122407q^{87} - 53760566681773q^{91} + 24587561871581q^{97} - 428589298584616q^{99} + O(q^{100}) \)

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 −4031.00 16384.0 −78125.0 0 823543. 0 1.14660e7 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 35.15.c.a 1
5.b even 2 1 35.15.c.b yes 1
7.b odd 2 1 35.15.c.b yes 1
35.c odd 2 1 CM 35.15.c.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.15.c.a 1 1.a even 1 1 trivial
35.15.c.a 1 35.c odd 2 1 CM
35.15.c.b yes 1 5.b even 2 1
35.15.c.b yes 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{15}^{\mathrm{new}}(35, [\chi])\):

\( T_{2} \)
\( T_{3} + 4031 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 4031 + T \)
$5$ \( 78125 + T \)
$7$ \( -823543 + T \)
$11$ \( 37379173 + T \)
$13$ \( 65279611 + T \)
$17$ \( 626193259 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( 9775649497 + T \)
$31$ \( T \)
$37$ \( T \)
$41$ \( T \)
$43$ \( T \)
$47$ \( -719081600801 + T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( T \)
$67$ \( T \)
$71$ \( 1790558995678 + T \)
$73$ \( -22033597628414 + T \)
$79$ \( 27088287440917 + T \)
$83$ \( -33726754263974 + T \)
$89$ \( T \)
$97$ \( -24587561871581 + T \)
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