Properties

Label 3675.2.a.b
Level $3675$
Weight $2$
Character orbit 3675.a
Self dual yes
Analytic conductor $29.345$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} - q^{3} + 2q^{4} + 2q^{6} + q^{9} + O(q^{10}) \) \( q - 2q^{2} - q^{3} + 2q^{4} + 2q^{6} + q^{9} + 2q^{11} - 2q^{12} + q^{13} - 4q^{16} + 2q^{17} - 2q^{18} + 5q^{19} - 4q^{22} - 6q^{23} - 2q^{26} - q^{27} + 10q^{29} + 3q^{31} + 8q^{32} - 2q^{33} - 4q^{34} + 2q^{36} - 2q^{37} - 10q^{38} - q^{39} + 8q^{41} - q^{43} + 4q^{44} + 12q^{46} + 2q^{47} + 4q^{48} - 2q^{51} + 2q^{52} + 4q^{53} + 2q^{54} - 5q^{57} - 20q^{58} + 10q^{59} - 7q^{61} - 6q^{62} - 8q^{64} + 4q^{66} + 3q^{67} + 4q^{68} + 6q^{69} - 8q^{71} - 14q^{73} + 4q^{74} + 10q^{76} + 2q^{78} + q^{81} - 16q^{82} + 6q^{83} + 2q^{86} - 10q^{87} - 12q^{92} - 3q^{93} - 4q^{94} - 8q^{96} + 17q^{97} + 2q^{99} + O(q^{100}) \)

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.b 1
5.b even 2 1 3675.2.a.q 1
7.b odd 2 1 75.2.a.a 1
21.c even 2 1 225.2.a.e 1
28.d even 2 1 1200.2.a.c 1
35.c odd 2 1 75.2.a.c yes 1
35.f even 4 2 75.2.b.a 2
56.e even 2 1 4800.2.a.br 1
56.h odd 2 1 4800.2.a.bb 1
77.b even 2 1 9075.2.a.s 1
84.h odd 2 1 3600.2.a.j 1
105.g even 2 1 225.2.a.a 1
105.k odd 4 2 225.2.b.a 2
140.c even 2 1 1200.2.a.p 1
140.j odd 4 2 1200.2.f.d 2
280.c odd 2 1 4800.2.a.bq 1
280.n even 2 1 4800.2.a.be 1
280.s even 4 2 4800.2.f.l 2
280.y odd 4 2 4800.2.f.y 2
385.h even 2 1 9075.2.a.a 1
420.o odd 2 1 3600.2.a.bk 1
420.w even 4 2 3600.2.f.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.a.a 1 7.b odd 2 1
75.2.a.c yes 1 35.c odd 2 1
75.2.b.a 2 35.f even 4 2
225.2.a.a 1 105.g even 2 1
225.2.a.e 1 21.c even 2 1
225.2.b.a 2 105.k odd 4 2
1200.2.a.c 1 28.d even 2 1
1200.2.a.p 1 140.c even 2 1
1200.2.f.d 2 140.j odd 4 2
3600.2.a.j 1 84.h odd 2 1
3600.2.a.bk 1 420.o odd 2 1
3600.2.f.p 2 420.w even 4 2
3675.2.a.b 1 1.a even 1 1 trivial
3675.2.a.q 1 5.b even 2 1
4800.2.a.bb 1 56.h odd 2 1
4800.2.a.be 1 280.n even 2 1
4800.2.a.bq 1 280.c odd 2 1
4800.2.a.br 1 56.e even 2 1
4800.2.f.l 2 280.s even 4 2
4800.2.f.y 2 280.y odd 4 2
9075.2.a.a 1 385.h even 2 1
9075.2.a.s 1 77.b even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( 1 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( -2 + T \)
$13$ \( -1 + T \)
$17$ \( -2 + T \)
$19$ \( -5 + T \)
$23$ \( 6 + T \)
$29$ \( -10 + T \)
$31$ \( -3 + T \)
$37$ \( 2 + T \)
$41$ \( -8 + T \)
$43$ \( 1 + T \)
$47$ \( -2 + T \)
$53$ \( -4 + T \)
$59$ \( -10 + T \)
$61$ \( 7 + T \)
$67$ \( -3 + T \)
$71$ \( 8 + T \)
$73$ \( 14 + T \)
$79$ \( T \)
$83$ \( -6 + T \)
$89$ \( T \)
$97$ \( -17 + T \)
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