Properties

Label 3675.2.a.p
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

\(f(q)\) \(=\) \( q + 2q^{2} + q^{3} + 2q^{4} + 2q^{6} + q^{9} + O(q^{10}) \) \( q + 2q^{2} + q^{3} + 2q^{4} + 2q^{6} + q^{9} - 6q^{11} + 2q^{12} - 3q^{13} - 4q^{16} - 4q^{17} + 2q^{18} - q^{19} - 12q^{22} + 4q^{23} - 6q^{26} + q^{27} - 8q^{29} - q^{31} - 8q^{32} - 6q^{33} - 8q^{34} + 2q^{36} - 7q^{37} - 2q^{38} - 3q^{39} + 6q^{41} - q^{43} - 12q^{44} + 8q^{46} + 2q^{47} - 4q^{48} - 4q^{51} - 6q^{52} - 4q^{53} + 2q^{54} - q^{57} - 16q^{58} + 8q^{59} + 14q^{61} - 2q^{62} - 8q^{64} - 12q^{66} - 7q^{67} - 8q^{68} + 4q^{69} + 6q^{71} + q^{73} - 14q^{74} - 2q^{76} - 6q^{78} - q^{79} + q^{81} + 12q^{82} + 2q^{83} - 2q^{86} - 8q^{87} + 12q^{89} + 8q^{92} - q^{93} + 4q^{94} - 8q^{96} - 6q^{97} - 6q^{99} + O(q^{100}) \)

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
2.00000 1.00000 2.00000 0 2.00000 0 0 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.p 1
5.b even 2 1 735.2.a.a 1
7.b odd 2 1 3675.2.a.o 1
7.d odd 6 2 525.2.i.a 2
15.d odd 2 1 2205.2.a.m 1
35.c odd 2 1 735.2.a.b 1
35.i odd 6 2 105.2.i.b 2
35.j even 6 2 735.2.i.f 2
35.k even 12 4 525.2.r.d 4
105.g even 2 1 2205.2.a.k 1
105.p even 6 2 315.2.j.a 2
140.s even 6 2 1680.2.bg.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.b 2 35.i odd 6 2
315.2.j.a 2 105.p even 6 2
525.2.i.a 2 7.d odd 6 2
525.2.r.d 4 35.k even 12 4
735.2.a.a 1 5.b even 2 1
735.2.a.b 1 35.c odd 2 1
735.2.i.f 2 35.j even 6 2
1680.2.bg.l 2 140.s even 6 2
2205.2.a.k 1 105.g even 2 1
2205.2.a.m 1 15.d odd 2 1
3675.2.a.o 1 7.b odd 2 1
3675.2.a.p 1 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2} - 2 \)
\( T_{11} + 6 \)
\( T_{13} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( 6 + T \)
$13$ \( 3 + T \)
$17$ \( 4 + T \)
$19$ \( 1 + T \)
$23$ \( -4 + T \)
$29$ \( 8 + T \)
$31$ \( 1 + T \)
$37$ \( 7 + T \)
$41$ \( -6 + T \)
$43$ \( 1 + T \)
$47$ \( -2 + T \)
$53$ \( 4 + T \)
$59$ \( -8 + T \)
$61$ \( -14 + T \)
$67$ \( 7 + T \)
$71$ \( -6 + T \)
$73$ \( -1 + T \)
$79$ \( 1 + T \)
$83$ \( -2 + T \)
$89$ \( -12 + T \)
$97$ \( 6 + T \)
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