Properties

Label 3675.2.a.o
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

Related objects

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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:

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.o 1
5.b even 2 1 735.2.a.b 1
7.b odd 2 1 3675.2.a.p 1
7.c even 3 2 525.2.i.a 2
15.d odd 2 1 2205.2.a.k 1
35.c odd 2 1 735.2.a.a 1
35.i odd 6 2 735.2.i.f 2
35.j even 6 2 105.2.i.b 2
35.l odd 12 4 525.2.r.d 4
105.g even 2 1 2205.2.a.m 1
105.o odd 6 2 315.2.j.a 2
140.p odd 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.j even 6 2
315.2.j.a 2 105.o odd 6 2
525.2.i.a 2 7.c even 3 2
525.2.r.d 4 35.l odd 12 4
735.2.a.a 1 35.c odd 2 1
735.2.a.b 1 5.b even 2 1
735.2.i.f 2 35.i odd 6 2
1680.2.bg.l 2 140.p odd 6 2
2205.2.a.k 1 15.d odd 2 1
2205.2.a.m 1 105.g even 2 1
3675.2.a.o 1 1.a even 1 1 trivial
3675.2.a.p 1 7.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

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

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

Hecke Characteristic Polynomials

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