Properties

Label 3610.2.a.b
Level $3610$
Weight $2$
Character orbit 3610.a
Self dual yes
Analytic conductor $28.826$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3610 = 2 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3610.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.8259951297\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} - q^{8} - 2q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} - q^{8} - 2q^{9} - q^{10} - q^{12} + q^{13} + q^{14} - q^{15} + q^{16} - 3q^{17} + 2q^{18} + q^{20} + q^{21} + 3q^{23} + q^{24} + q^{25} - q^{26} + 5q^{27} - q^{28} + 3q^{29} + q^{30} - 2q^{31} - q^{32} + 3q^{34} - q^{35} - 2q^{36} + 10q^{37} - q^{39} - q^{40} - 6q^{41} - q^{42} + 2q^{43} - 2q^{45} - 3q^{46} - q^{48} - 6q^{49} - q^{50} + 3q^{51} + q^{52} - 3q^{53} - 5q^{54} + q^{56} - 3q^{58} - 3q^{59} - q^{60} + 8q^{61} + 2q^{62} + 2q^{63} + q^{64} + q^{65} + 7q^{67} - 3q^{68} - 3q^{69} + q^{70} - 12q^{71} + 2q^{72} - 13q^{73} - 10q^{74} - q^{75} + q^{78} - 14q^{79} + q^{80} + q^{81} + 6q^{82} + 6q^{83} + q^{84} - 3q^{85} - 2q^{86} - 3q^{87} - 6q^{89} + 2q^{90} - q^{91} + 3q^{92} + 2q^{93} + q^{96} + 10q^{97} + 6q^{98} + O(q^{100}) \)

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 −1.00000 −1.00000 −2.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(19\) \(-1\)

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 3610.2.a.b 1
19.b odd 2 1 190.2.a.c 1
57.d even 2 1 1710.2.a.d 1
76.d even 2 1 1520.2.a.d 1
95.d odd 2 1 950.2.a.a 1
95.g even 4 2 950.2.b.e 2
133.c even 2 1 9310.2.a.o 1
152.b even 2 1 6080.2.a.p 1
152.g odd 2 1 6080.2.a.h 1
285.b even 2 1 8550.2.a.bd 1
380.d even 2 1 7600.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.c 1 19.b odd 2 1
950.2.a.a 1 95.d odd 2 1
950.2.b.e 2 95.g even 4 2
1520.2.a.d 1 76.d even 2 1
1710.2.a.d 1 57.d even 2 1
3610.2.a.b 1 1.a even 1 1 trivial
6080.2.a.h 1 152.g odd 2 1
6080.2.a.p 1 152.b even 2 1
7600.2.a.m 1 380.d even 2 1
8550.2.a.bd 1 285.b even 2 1
9310.2.a.o 1 133.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3610))\):

\( T_{3} + 1 \)
\( T_{7} + 1 \)
\( T_{13} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + T \)
$5$ \( -1 + T \)
$7$ \( 1 + T \)
$11$ \( T \)
$13$ \( -1 + T \)
$17$ \( 3 + T \)
$19$ \( T \)
$23$ \( -3 + T \)
$29$ \( -3 + T \)
$31$ \( 2 + T \)
$37$ \( -10 + T \)
$41$ \( 6 + T \)
$43$ \( -2 + T \)
$47$ \( T \)
$53$ \( 3 + T \)
$59$ \( 3 + T \)
$61$ \( -8 + T \)
$67$ \( -7 + T \)
$71$ \( 12 + T \)
$73$ \( 13 + T \)
$79$ \( 14 + T \)
$83$ \( -6 + T \)
$89$ \( 6 + T \)
$97$ \( -10 + T \)
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