Properties

Label 3150.2.a.bl
Level 3150
Weight 2
Character orbit 3150.a
Self dual yes
Analytic conductor 25.153
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1050)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + q^{7} + q^{8} - 2q^{11} - q^{13} + q^{14} + q^{16} - q^{17} + 4q^{19} - 2q^{22} + 7q^{23} - q^{26} + q^{28} - q^{29} + 3q^{31} + q^{32} - q^{34} + 6q^{37} + 4q^{38} + 3q^{41} + q^{43} - 2q^{44} + 7q^{46} - 12q^{47} + q^{49} - q^{52} + 11q^{53} + q^{56} - q^{58} + 3q^{59} + 5q^{61} + 3q^{62} + q^{64} + 12q^{67} - q^{68} - 4q^{71} + 14q^{73} + 6q^{74} + 4q^{76} - 2q^{77} - 2q^{79} + 3q^{82} - 3q^{83} + q^{86} - 2q^{88} - 10q^{89} - q^{91} + 7q^{92} - 12q^{94} + 10q^{97} + q^{98} + O(q^{100}) \)

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 0 1.00000 0 0 1.00000 1.00000 0 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 3150.2.a.bl 1
3.b odd 2 1 1050.2.a.e 1
5.b even 2 1 3150.2.a.c 1
5.c odd 4 2 3150.2.g.g 2
12.b even 2 1 8400.2.a.bt 1
15.d odd 2 1 1050.2.a.o yes 1
15.e even 4 2 1050.2.g.j 2
21.c even 2 1 7350.2.a.bj 1
60.h even 2 1 8400.2.a.t 1
105.g even 2 1 7350.2.a.ca 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.a.e 1 3.b odd 2 1
1050.2.a.o yes 1 15.d odd 2 1
1050.2.g.j 2 15.e even 4 2
3150.2.a.c 1 5.b even 2 1
3150.2.a.bl 1 1.a even 1 1 trivial
3150.2.g.g 2 5.c odd 4 2
7350.2.a.bj 1 21.c even 2 1
7350.2.a.ca 1 105.g even 2 1
8400.2.a.t 1 60.h even 2 1
8400.2.a.bt 1 12.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3150))\):

\( T_{11} + 2 \)
\( T_{13} + 1 \)
\( T_{17} + 1 \)
\( T_{19} - 4 \)
\( T_{29} + 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ 1
$5$ 1
$7$ \( 1 - T \)
$11$ \( 1 + 2 T + 11 T^{2} \)
$13$ \( 1 + T + 13 T^{2} \)
$17$ \( 1 + T + 17 T^{2} \)
$19$ \( 1 - 4 T + 19 T^{2} \)
$23$ \( 1 - 7 T + 23 T^{2} \)
$29$ \( 1 + T + 29 T^{2} \)
$31$ \( 1 - 3 T + 31 T^{2} \)
$37$ \( 1 - 6 T + 37 T^{2} \)
$41$ \( 1 - 3 T + 41 T^{2} \)
$43$ \( 1 - T + 43 T^{2} \)
$47$ \( 1 + 12 T + 47 T^{2} \)
$53$ \( 1 - 11 T + 53 T^{2} \)
$59$ \( 1 - 3 T + 59 T^{2} \)
$61$ \( 1 - 5 T + 61 T^{2} \)
$67$ \( 1 - 12 T + 67 T^{2} \)
$71$ \( 1 + 4 T + 71 T^{2} \)
$73$ \( 1 - 14 T + 73 T^{2} \)
$79$ \( 1 + 2 T + 79 T^{2} \)
$83$ \( 1 + 3 T + 83 T^{2} \)
$89$ \( 1 + 10 T + 89 T^{2} \)
$97$ \( 1 - 10 T + 97 T^{2} \)
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