Properties

Label 2450.2.a.bq
Level $2450$
Weight $2$
Character orbit 2450.a
Self dual yes
Analytic conductor $19.563$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.5633484952\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + q^{8} + 3 q^{9} + 2 \beta q^{11} + \beta q^{12} + ( - \beta + 2) q^{13} + q^{16} + 2 q^{17} + 3 q^{18} + ( - \beta - 4) q^{19} + 2 \beta q^{22} + (2 \beta + 2) q^{23} + \beta q^{24} + ( - \beta + 2) q^{26} + ( - 2 \beta + 2) q^{29} + (2 \beta - 4) q^{31} + q^{32} + 12 q^{33} + 2 q^{34} + 3 q^{36} - 2 q^{37} + ( - \beta - 4) q^{38} + (2 \beta - 6) q^{39} + (2 \beta + 6) q^{41} + ( - 2 \beta - 4) q^{43} + 2 \beta q^{44} + (2 \beta + 2) q^{46} + ( - 2 \beta + 4) q^{47} + \beta q^{48} + 2 \beta q^{51} + ( - \beta + 2) q^{52} + ( - 2 \beta + 6) q^{53} + ( - 4 \beta - 6) q^{57} + ( - 2 \beta + 2) q^{58} + (\beta + 4) q^{59} + ( - \beta - 6) q^{61} + (2 \beta - 4) q^{62} + q^{64} + 12 q^{66} + 8 q^{67} + 2 q^{68} + (2 \beta + 12) q^{69} + ( - 2 \beta - 6) q^{71} + 3 q^{72} + ( - 2 \beta - 2) q^{73} - 2 q^{74} + ( - \beta - 4) q^{76} + (2 \beta - 6) q^{78} + ( - 2 \beta + 2) q^{79} - 9 q^{81} + (2 \beta + 6) q^{82} + \beta q^{83} + ( - 2 \beta - 4) q^{86} + (2 \beta - 12) q^{87} + 2 \beta q^{88} + 10 q^{89} + (2 \beta + 2) q^{92} + ( - 4 \beta + 12) q^{93} + ( - 2 \beta + 4) q^{94} + \beta q^{96} + ( - 4 \beta + 6) q^{97} + 6 \beta q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 6 q^{9} + 4 q^{13} + 2 q^{16} + 4 q^{17} + 6 q^{18} - 8 q^{19} + 4 q^{23} + 4 q^{26} + 4 q^{29} - 8 q^{31} + 2 q^{32} + 24 q^{33} + 4 q^{34} + 6 q^{36} - 4 q^{37} - 8 q^{38} - 12 q^{39} + 12 q^{41} - 8 q^{43} + 4 q^{46} + 8 q^{47} + 4 q^{52} + 12 q^{53} - 12 q^{57} + 4 q^{58} + 8 q^{59} - 12 q^{61} - 8 q^{62} + 2 q^{64} + 24 q^{66} + 16 q^{67} + 4 q^{68} + 24 q^{69} - 12 q^{71} + 6 q^{72} - 4 q^{73} - 4 q^{74} - 8 q^{76} - 12 q^{78} + 4 q^{79} - 18 q^{81} + 12 q^{82} - 8 q^{86} - 24 q^{87} + 20 q^{89} + 4 q^{92} + 24 q^{93} + 8 q^{94} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 −2.44949 1.00000 0 −2.44949 0 1.00000 3.00000 0
1.2 1.00000 2.44949 1.00000 0 2.44949 0 1.00000 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 2450.2.a.bq 2
5.b even 2 1 2450.2.a.bl 2
5.c odd 4 2 490.2.c.e 4
7.b odd 2 1 350.2.a.h 2
21.c even 2 1 3150.2.a.bs 2
28.d even 2 1 2800.2.a.bl 2
35.c odd 2 1 350.2.a.g 2
35.f even 4 2 70.2.c.a 4
35.k even 12 4 490.2.i.c 8
35.l odd 12 4 490.2.i.f 8
105.g even 2 1 3150.2.a.bt 2
105.k odd 4 2 630.2.g.g 4
140.c even 2 1 2800.2.a.bm 2
140.j odd 4 2 560.2.g.e 4
280.s even 4 2 2240.2.g.j 4
280.y odd 4 2 2240.2.g.i 4
420.w even 4 2 5040.2.t.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 35.f even 4 2
350.2.a.g 2 35.c odd 2 1
350.2.a.h 2 7.b odd 2 1
490.2.c.e 4 5.c odd 4 2
490.2.i.c 8 35.k even 12 4
490.2.i.f 8 35.l odd 12 4
560.2.g.e 4 140.j odd 4 2
630.2.g.g 4 105.k odd 4 2
2240.2.g.i 4 280.y odd 4 2
2240.2.g.j 4 280.s even 4 2
2450.2.a.bl 2 5.b even 2 1
2450.2.a.bq 2 1.a even 1 1 trivial
2800.2.a.bl 2 28.d even 2 1
2800.2.a.bm 2 140.c even 2 1
3150.2.a.bs 2 21.c even 2 1
3150.2.a.bt 2 105.g even 2 1
5040.2.t.t 4 420.w even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2450))\):

\( T_{3}^{2} - 6 \) Copy content Toggle raw display
\( T_{11}^{2} - 24 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} - 2 \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display
\( T_{19}^{2} + 8T_{19} + 10 \) Copy content Toggle raw display
\( T_{23}^{2} - 4T_{23} - 20 \) Copy content Toggle raw display
\( T_{37} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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