Properties

Label 35.21.c.a
Level $35$
Weight $21$
Character orbit 35.c
Self dual yes
Analytic conductor $88.730$
Analytic rank $0$
Dimension $1$
CM discriminant -35
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 21 \)
Character orbit: \([\chi]\) \(=\) 35.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(88.7298177859\)
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

\(f(q)\) \(=\) \( q - 12223q^{3} + 1048576q^{4} - 9765625q^{5} - 282475249q^{7} - 3337382672q^{9} + O(q^{10}) \) \( q - 12223q^{3} + 1048576q^{4} - 9765625q^{5} - 282475249q^{7} - 3337382672q^{9} - 51836073673q^{11} - 12816744448q^{12} - 92077960823q^{13} + 119365234375q^{15} + 1099511627776q^{16} - 2834566008023q^{17} - 10240000000000q^{20} + 3452694968527q^{21} + 95367431640625q^{25} + 83411794133279q^{27} - 296196766695424q^{28} + 498981827484527q^{29} + 633592328505079q^{33} + 2758547353515625q^{35} - 3499499372675072q^{36} + 1125468915139529q^{39} - 54354062787739648q^{44} + 32591627656250000q^{45} - 102681296590588223q^{47} - 13439330626306048q^{48} + 79792266297612001q^{49} + 34646900316065129q^{51} - 96550739847938048q^{52} + 506211656962890625q^{55} + 125163520000000000q^{60} + 942728001281485328q^{63} + 1152921504606846976q^{64} + 899198836162109375q^{65} - 2972257886428725248q^{68} - 6446014172213061598q^{71} - 6042116169407251298q^{73} - 1165676116943359375q^{75} + 14642407817963019577q^{77} + 8153339628309741527q^{79} - 10737418240000000000q^{80} + 10617191481206230255q^{81} + 20776721187438397102q^{83} + 3620413079318167552q^{84} + 27681308672099609375q^{85} - 6099054877343373521q^{87} + 26009744910889169927q^{91} - 115875187308982861223q^{97} + 172996814060785594256q^{99} + O(q^{100}) \)

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 −12223.0 1.04858e6 −9.76562e6 0 −2.82475e8 0 −3.33738e9 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 35.21.c.a 1
5.b even 2 1 35.21.c.b yes 1
7.b odd 2 1 35.21.c.b yes 1
35.c odd 2 1 CM 35.21.c.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.21.c.a 1 1.a even 1 1 trivial
35.21.c.a 1 35.c odd 2 1 CM
35.21.c.b yes 1 5.b even 2 1
35.21.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 \(S_{21}^{\mathrm{new}}(35, [\chi])\):

\( T_{2} \)
\( T_{3} + 12223 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 12223 + T \)
$5$ \( 9765625 + T \)
$7$ \( 282475249 + T \)
$11$ \( 51836073673 + T \)
$13$ \( 92077960823 + T \)
$17$ \( 2834566008023 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( -498981827484527 + T \)
$31$ \( T \)
$37$ \( T \)
$41$ \( T \)
$43$ \( T \)
$47$ \( 102681296590588223 + T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( T \)
$67$ \( T \)
$71$ \( 6446014172213061598 + T \)
$73$ \( 6042116169407251298 + T \)
$79$ \( -8153339628309741527 + T \)
$83$ \( -20776721187438397102 + T \)
$89$ \( T \)
$97$ \( \)\(11\!\cdots\!23\)\( + T \)
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