Properties

Label 735.2.a.f
Level $735$
Weight $2$
Character orbit 735.a
Self dual yes
Analytic conductor $5.869$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - 3q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - 3q^{8} + q^{9} - q^{10} + q^{12} + 6q^{13} + q^{15} - q^{16} - 2q^{17} + q^{18} + 8q^{19} + q^{20} + 8q^{23} + 3q^{24} + q^{25} + 6q^{26} - q^{27} - 2q^{29} + q^{30} - 4q^{31} + 5q^{32} - 2q^{34} - q^{36} - 2q^{37} + 8q^{38} - 6q^{39} + 3q^{40} + 6q^{41} + 4q^{43} - q^{45} + 8q^{46} - 8q^{47} + q^{48} + q^{50} + 2q^{51} - 6q^{52} + 10q^{53} - q^{54} - 8q^{57} - 2q^{58} - 4q^{59} - q^{60} + 2q^{61} - 4q^{62} + 7q^{64} - 6q^{65} + 4q^{67} + 2q^{68} - 8q^{69} - 12q^{71} - 3q^{72} + 2q^{73} - 2q^{74} - q^{75} - 8q^{76} - 6q^{78} + 8q^{79} + q^{80} + q^{81} + 6q^{82} + 4q^{83} + 2q^{85} + 4q^{86} + 2q^{87} + 6q^{89} - q^{90} - 8q^{92} + 4q^{93} - 8q^{94} - 8q^{95} - 5q^{96} + 18q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
1.00000 −1.00000 −1.00000 −1.00000 −1.00000 0 −3.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.a.f 1
3.b odd 2 1 2205.2.a.b 1
5.b even 2 1 3675.2.a.f 1
7.b odd 2 1 105.2.a.a 1
7.c even 3 2 735.2.i.b 2
7.d odd 6 2 735.2.i.a 2
21.c even 2 1 315.2.a.a 1
28.d even 2 1 1680.2.a.f 1
35.c odd 2 1 525.2.a.a 1
35.f even 4 2 525.2.d.b 2
56.e even 2 1 6720.2.a.bk 1
56.h odd 2 1 6720.2.a.p 1
84.h odd 2 1 5040.2.a.d 1
105.g even 2 1 1575.2.a.h 1
105.k odd 4 2 1575.2.d.b 2
140.c even 2 1 8400.2.a.co 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.a 1 7.b odd 2 1
315.2.a.a 1 21.c even 2 1
525.2.a.a 1 35.c odd 2 1
525.2.d.b 2 35.f even 4 2
735.2.a.f 1 1.a even 1 1 trivial
735.2.i.a 2 7.d odd 6 2
735.2.i.b 2 7.c even 3 2
1575.2.a.h 1 105.g even 2 1
1575.2.d.b 2 105.k odd 4 2
1680.2.a.f 1 28.d even 2 1
2205.2.a.b 1 3.b odd 2 1
3675.2.a.f 1 5.b even 2 1
5040.2.a.d 1 84.h odd 2 1
6720.2.a.p 1 56.h odd 2 1
6720.2.a.bk 1 56.e even 2 1
8400.2.a.co 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(735))\):

\( T_{2} - 1 \)
\( T_{13} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 1 + T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( -6 + T \)
$17$ \( 2 + T \)
$19$ \( -8 + T \)
$23$ \( -8 + T \)
$29$ \( 2 + T \)
$31$ \( 4 + T \)
$37$ \( 2 + T \)
$41$ \( -6 + T \)
$43$ \( -4 + T \)
$47$ \( 8 + T \)
$53$ \( -10 + T \)
$59$ \( 4 + T \)
$61$ \( -2 + T \)
$67$ \( -4 + T \)
$71$ \( 12 + T \)
$73$ \( -2 + T \)
$79$ \( -8 + T \)
$83$ \( -4 + T \)
$89$ \( -6 + T \)
$97$ \( -18 + T \)
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