Properties

Label 35.3.c.a
Level $35$
Weight $3$
Character orbit 35.c
Self dual yes
Analytic conductor $0.954$
Analytic rank $0$
Dimension $1$
CM discriminant -35
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 35.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.953680925261\)
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

\(f(q)\) \(=\) \( q - q^{3} + 4q^{4} + 5q^{5} - 7q^{7} - 8q^{9} + O(q^{10}) \) \( q - q^{3} + 4q^{4} + 5q^{5} - 7q^{7} - 8q^{9} - 13q^{11} - 4q^{12} + 19q^{13} - 5q^{15} + 16q^{16} - 29q^{17} + 20q^{20} + 7q^{21} + 25q^{25} + 17q^{27} - 28q^{28} + 23q^{29} + 13q^{33} - 35q^{35} - 32q^{36} - 19q^{39} - 52q^{44} - 40q^{45} + 31q^{47} - 16q^{48} + 49q^{49} + 29q^{51} + 76q^{52} - 65q^{55} - 20q^{60} + 56q^{63} + 64q^{64} + 95q^{65} - 116q^{68} + 2q^{71} + 34q^{73} - 25q^{75} + 91q^{77} - 157q^{79} + 80q^{80} + 55q^{81} - 86q^{83} + 28q^{84} - 145q^{85} - 23q^{87} - 133q^{91} - 149q^{97} + 104q^{99} + O(q^{100}) \)

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 −1.00000 4.00000 5.00000 0 −7.00000 0 −8.00000 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 35.3.c.a 1
3.b odd 2 1 315.3.e.a 1
4.b odd 2 1 560.3.p.b 1
5.b even 2 1 35.3.c.b yes 1
5.c odd 4 2 175.3.d.e 2
7.b odd 2 1 35.3.c.b yes 1
7.c even 3 2 245.3.i.b 2
7.d odd 6 2 245.3.i.a 2
15.d odd 2 1 315.3.e.b 1
20.d odd 2 1 560.3.p.a 1
21.c even 2 1 315.3.e.b 1
28.d even 2 1 560.3.p.a 1
35.c odd 2 1 CM 35.3.c.a 1
35.f even 4 2 175.3.d.e 2
35.i odd 6 2 245.3.i.b 2
35.j even 6 2 245.3.i.a 2
105.g even 2 1 315.3.e.a 1
140.c even 2 1 560.3.p.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.c.a 1 1.a even 1 1 trivial
35.3.c.a 1 35.c odd 2 1 CM
35.3.c.b yes 1 5.b even 2 1
35.3.c.b yes 1 7.b odd 2 1
175.3.d.e 2 5.c odd 4 2
175.3.d.e 2 35.f even 4 2
245.3.i.a 2 7.d odd 6 2
245.3.i.a 2 35.j even 6 2
245.3.i.b 2 7.c even 3 2
245.3.i.b 2 35.i odd 6 2
315.3.e.a 1 3.b odd 2 1
315.3.e.a 1 105.g even 2 1
315.3.e.b 1 15.d odd 2 1
315.3.e.b 1 21.c even 2 1
560.3.p.a 1 20.d odd 2 1
560.3.p.a 1 28.d even 2 1
560.3.p.b 1 4.b odd 2 1
560.3.p.b 1 140.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(35, [\chi])\):

\( T_{2} \)
\( T_{3} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T )( 1 + 2 T ) \)
$3$ \( 1 + T + 9 T^{2} \)
$5$ \( 1 - 5 T \)
$7$ \( 1 + 7 T \)
$11$ \( 1 + 13 T + 121 T^{2} \)
$13$ \( 1 - 19 T + 169 T^{2} \)
$17$ \( 1 + 29 T + 289 T^{2} \)
$19$ \( ( 1 - 19 T )( 1 + 19 T ) \)
$23$ \( ( 1 - 23 T )( 1 + 23 T ) \)
$29$ \( 1 - 23 T + 841 T^{2} \)
$31$ \( ( 1 - 31 T )( 1 + 31 T ) \)
$37$ \( ( 1 - 37 T )( 1 + 37 T ) \)
$41$ \( ( 1 - 41 T )( 1 + 41 T ) \)
$43$ \( ( 1 - 43 T )( 1 + 43 T ) \)
$47$ \( 1 - 31 T + 2209 T^{2} \)
$53$ \( ( 1 - 53 T )( 1 + 53 T ) \)
$59$ \( ( 1 - 59 T )( 1 + 59 T ) \)
$61$ \( ( 1 - 61 T )( 1 + 61 T ) \)
$67$ \( ( 1 - 67 T )( 1 + 67 T ) \)
$71$ \( 1 - 2 T + 5041 T^{2} \)
$73$ \( 1 - 34 T + 5329 T^{2} \)
$79$ \( 1 + 157 T + 6241 T^{2} \)
$83$ \( 1 + 86 T + 6889 T^{2} \)
$89$ \( ( 1 - 89 T )( 1 + 89 T ) \)
$97$ \( 1 + 149 T + 9409 T^{2} \)
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