Properties

Label 3150.2.a.bt
Level $3150$
Weight $2$
Character orbit 3150.a
Self dual yes
Analytic conductor $25.153$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} - q^{7} + q^{8} + 2 \beta q^{11} + ( 2 + \beta ) q^{13} - q^{14} + q^{16} -2 q^{17} + ( 4 - \beta ) q^{19} + 2 \beta q^{22} + ( 2 - 2 \beta ) q^{23} + ( 2 + \beta ) q^{26} - q^{28} + ( -2 - 2 \beta ) q^{29} + ( 4 + 2 \beta ) q^{31} + q^{32} -2 q^{34} + 2 q^{37} + ( 4 - \beta ) q^{38} + ( 6 - 2 \beta ) q^{41} + ( 4 - 2 \beta ) q^{43} + 2 \beta q^{44} + ( 2 - 2 \beta ) q^{46} + ( -4 - 2 \beta ) q^{47} + q^{49} + ( 2 + \beta ) q^{52} + ( 6 + 2 \beta ) q^{53} - q^{56} + ( -2 - 2 \beta ) q^{58} + ( 4 - \beta ) q^{59} + ( 6 - \beta ) q^{61} + ( 4 + 2 \beta ) q^{62} + q^{64} -8 q^{67} -2 q^{68} + ( 6 - 2 \beta ) q^{71} + ( -2 + 2 \beta ) q^{73} + 2 q^{74} + ( 4 - \beta ) q^{76} -2 \beta q^{77} + ( 2 + 2 \beta ) q^{79} + ( 6 - 2 \beta ) q^{82} + \beta q^{83} + ( 4 - 2 \beta ) q^{86} + 2 \beta q^{88} + 10 q^{89} + ( -2 - \beta ) q^{91} + ( 2 - 2 \beta ) q^{92} + ( -4 - 2 \beta ) q^{94} + ( 6 + 4 \beta ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 4 q^{13} - 2 q^{14} + 2 q^{16} - 4 q^{17} + 8 q^{19} + 4 q^{23} + 4 q^{26} - 2 q^{28} - 4 q^{29} + 8 q^{31} + 2 q^{32} - 4 q^{34} + 4 q^{37} + 8 q^{38} + 12 q^{41} + 8 q^{43} + 4 q^{46} - 8 q^{47} + 2 q^{49} + 4 q^{52} + 12 q^{53} - 2 q^{56} - 4 q^{58} + 8 q^{59} + 12 q^{61} + 8 q^{62} + 2 q^{64} - 16 q^{67} - 4 q^{68} + 12 q^{71} - 4 q^{73} + 4 q^{74} + 8 q^{76} + 4 q^{79} + 12 q^{82} + 8 q^{86} + 20 q^{89} - 4 q^{91} + 4 q^{92} - 8 q^{94} + 12 q^{97} + 2 q^{98} + O(q^{100}) \)

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
−2.44949
2.44949
1.00000 0 1.00000 0 0 −1.00000 1.00000 0 0
1.2 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.bt 2
3.b odd 2 1 350.2.a.g 2
5.b even 2 1 3150.2.a.bs 2
5.c odd 4 2 630.2.g.g 4
12.b even 2 1 2800.2.a.bm 2
15.d odd 2 1 350.2.a.h 2
15.e even 4 2 70.2.c.a 4
20.e even 4 2 5040.2.t.t 4
21.c even 2 1 2450.2.a.bl 2
60.h even 2 1 2800.2.a.bl 2
60.l odd 4 2 560.2.g.e 4
105.g even 2 1 2450.2.a.bq 2
105.k odd 4 2 490.2.c.e 4
105.w odd 12 4 490.2.i.f 8
105.x even 12 4 490.2.i.c 8
120.q odd 4 2 2240.2.g.i 4
120.w even 4 2 2240.2.g.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 15.e even 4 2
350.2.a.g 2 3.b odd 2 1
350.2.a.h 2 15.d odd 2 1
490.2.c.e 4 105.k odd 4 2
490.2.i.c 8 105.x even 12 4
490.2.i.f 8 105.w odd 12 4
560.2.g.e 4 60.l odd 4 2
630.2.g.g 4 5.c odd 4 2
2240.2.g.i 4 120.q odd 4 2
2240.2.g.j 4 120.w even 4 2
2450.2.a.bl 2 21.c even 2 1
2450.2.a.bq 2 105.g even 2 1
2800.2.a.bl 2 60.h even 2 1
2800.2.a.bm 2 12.b even 2 1
3150.2.a.bs 2 5.b even 2 1
3150.2.a.bt 2 1.a even 1 1 trivial
5040.2.t.t 4 20.e even 4 2

Hecke kernels

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

\( T_{11}^{2} - 24 \)
\( T_{13}^{2} - 4 T_{13} - 2 \)
\( T_{17} + 2 \)
\( T_{19}^{2} - 8 T_{19} + 10 \)
\( T_{29}^{2} + 4 T_{29} - 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -24 + T^{2} \)
$13$ \( -2 - 4 T + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( 10 - 8 T + T^{2} \)
$23$ \( -20 - 4 T + T^{2} \)
$29$ \( -20 + 4 T + T^{2} \)
$31$ \( -8 - 8 T + T^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( 12 - 12 T + T^{2} \)
$43$ \( -8 - 8 T + T^{2} \)
$47$ \( -8 + 8 T + T^{2} \)
$53$ \( 12 - 12 T + T^{2} \)
$59$ \( 10 - 8 T + T^{2} \)
$61$ \( 30 - 12 T + T^{2} \)
$67$ \( ( 8 + T )^{2} \)
$71$ \( 12 - 12 T + T^{2} \)
$73$ \( -20 + 4 T + T^{2} \)
$79$ \( -20 - 4 T + T^{2} \)
$83$ \( -6 + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( -60 - 12 T + T^{2} \)
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