Properties

Label 3150.2.a.bp
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 210)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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:

Atkin-Lehner signs

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

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 3150.2.a.bp 1
3.b odd 2 1 1050.2.a.c 1
5.b even 2 1 630.2.a.a 1
5.c odd 4 2 3150.2.g.q 2
12.b even 2 1 8400.2.a.ce 1
15.d odd 2 1 210.2.a.e 1
15.e even 4 2 1050.2.g.g 2
20.d odd 2 1 5040.2.a.k 1
21.c even 2 1 7350.2.a.w 1
35.c odd 2 1 4410.2.a.t 1
60.h even 2 1 1680.2.a.j 1
105.g even 2 1 1470.2.a.j 1
105.o odd 6 2 1470.2.i.a 2
105.p even 6 2 1470.2.i.j 2
120.i odd 2 1 6720.2.a.j 1
120.m even 2 1 6720.2.a.bq 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.a.e 1 15.d odd 2 1
630.2.a.a 1 5.b even 2 1
1050.2.a.c 1 3.b odd 2 1
1050.2.g.g 2 15.e even 4 2
1470.2.a.j 1 105.g even 2 1
1470.2.i.a 2 105.o odd 6 2
1470.2.i.j 2 105.p even 6 2
1680.2.a.j 1 60.h even 2 1
3150.2.a.bp 1 1.a even 1 1 trivial
3150.2.g.q 2 5.c odd 4 2
4410.2.a.t 1 35.c odd 2 1
5040.2.a.k 1 20.d odd 2 1
6720.2.a.j 1 120.i odd 2 1
6720.2.a.bq 1 120.m even 2 1
7350.2.a.w 1 21.c even 2 1
8400.2.a.ce 1 12.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3150))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -1 + T \)
$11$ \( -4 + T \)
$13$ \( -2 + T \)
$17$ \( -2 + T \)
$19$ \( -4 + T \)
$23$ \( 8 + T \)
$29$ \( -2 + T \)
$31$ \( T \)
$37$ \( 6 + T \)
$41$ \( -6 + T \)
$43$ \( -4 + T \)
$47$ \( T \)
$53$ \( 10 + T \)
$59$ \( 12 + T \)
$61$ \( -14 + T \)
$67$ \( -12 + T \)
$71$ \( -8 + T \)
$73$ \( 10 + T \)
$79$ \( -16 + T \)
$83$ \( 12 + T \)
$89$ \( 10 + T \)
$97$ \( 2 + T \)
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