Properties

Label 3675.2.a.f
Level $3675$
Weight $2$
Character orbit 3675.a
Self dual yes
Analytic conductor $29.345$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.3450227428\)
Analytic rank: \(1\)
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

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

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 0 −1.00000 0 3.00000 1.00000 0
\(n\): 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 3675.2.a.f 1
5.b even 2 1 735.2.a.f 1
7.b odd 2 1 525.2.a.a 1
15.d odd 2 1 2205.2.a.b 1
21.c even 2 1 1575.2.a.h 1
28.d even 2 1 8400.2.a.co 1
35.c odd 2 1 105.2.a.a 1
35.f even 4 2 525.2.d.b 2
35.i odd 6 2 735.2.i.a 2
35.j even 6 2 735.2.i.b 2
105.g even 2 1 315.2.a.a 1
105.k odd 4 2 1575.2.d.b 2
140.c even 2 1 1680.2.a.f 1
280.c odd 2 1 6720.2.a.p 1
280.n even 2 1 6720.2.a.bk 1
420.o odd 2 1 5040.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.a 1 35.c odd 2 1
315.2.a.a 1 105.g even 2 1
525.2.a.a 1 7.b odd 2 1
525.2.d.b 2 35.f even 4 2
735.2.a.f 1 5.b even 2 1
735.2.i.a 2 35.i odd 6 2
735.2.i.b 2 35.j even 6 2
1575.2.a.h 1 21.c even 2 1
1575.2.d.b 2 105.k odd 4 2
1680.2.a.f 1 140.c even 2 1
2205.2.a.b 1 15.d odd 2 1
3675.2.a.f 1 1.a even 1 1 trivial
5040.2.a.d 1 420.o odd 2 1
6720.2.a.p 1 280.c odd 2 1
6720.2.a.bk 1 280.n even 2 1
8400.2.a.co 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3675))\):

\( T_{2} + 1 \)
\( T_{11} \)
\( T_{13} + 6 \)

Hecke characteristic polynomials

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