Properties

Label 35.17.c.b
Level $35$
Weight $17$
Character orbit 35.c
Self dual yes
Analytic conductor $56.814$
Analytic rank $0$
Dimension $1$
CM discriminant -35
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 35.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(56.8135903498\)
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

\(f(q)\) \(=\) \( q + 3007q^{3} + 65536q^{4} + 390625q^{5} + 5764801q^{7} - 34004672q^{9} + O(q^{10}) \) \( q + 3007q^{3} + 65536q^{4} + 390625q^{5} + 5764801q^{7} - 34004672q^{9} + 145028447q^{11} + 197066752q^{12} + 1571306207q^{13} + 1174609375q^{15} + 4294967296q^{16} - 4372390753q^{17} + 25600000000q^{20} + 17334756607q^{21} + 152587890625q^{25} - 231693538751q^{27} + 377801998336q^{28} - 993241792513q^{29} + 436100540129q^{33} + 2251875390625q^{35} - 2228530184192q^{36} + 4724917764449q^{39} + 9504584302592q^{44} - 13283075000000q^{45} - 42819958052353q^{47} + 12914966659072q^{48} + 33232930569601q^{49} - 13147778994271q^{51} + 102977123581952q^{52} + 56651737109375q^{55} + 76979200000000q^{60} - 196030167150272q^{63} + 281474976710656q^{64} + 613791487109375q^{65} - 286549000388608q^{68} + 1283316773477762q^{71} - 491238137911678q^{73} + 458831787109375q^{75} + 836060136294047q^{77} + 1884802582005407q^{79} + 1677721600000000q^{80} + 767087157256255q^{81} - 1568384056666558q^{83} + 1136050608996352q^{84} - 1707965137890625q^{85} - 2986678070086591q^{87} + 9058267593419807q^{91} + 11753087741306207q^{97} - 4931644770904384q^{99} + O(q^{100}) \)

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 3007.00 65536.0 390625. 0 5.76480e6 0 −3.40047e7 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 35.17.c.b yes 1
5.b even 2 1 35.17.c.a 1
7.b odd 2 1 35.17.c.a 1
35.c odd 2 1 CM 35.17.c.b yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.17.c.a 1 5.b even 2 1
35.17.c.a 1 7.b odd 2 1
35.17.c.b yes 1 1.a even 1 1 trivial
35.17.c.b yes 1 35.c odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{17}^{\mathrm{new}}(35, [\chi])\):

\( T_{2} \)
\( T_{3} - 3007 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3007 + T \)
$5$ \( -390625 + T \)
$7$ \( -5764801 + T \)
$11$ \( -145028447 + T \)
$13$ \( -1571306207 + T \)
$17$ \( 4372390753 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( 993241792513 + T \)
$31$ \( T \)
$37$ \( T \)
$41$ \( T \)
$43$ \( T \)
$47$ \( 42819958052353 + T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( T \)
$67$ \( T \)
$71$ \( -1283316773477762 + T \)
$73$ \( 491238137911678 + T \)
$79$ \( -1884802582005407 + T \)
$83$ \( 1568384056666558 + T \)
$89$ \( T \)
$97$ \( -11753087741306207 + T \)
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