Properties

Label 2450.2.a.bu
Level $2450$
Weight $2$
Character orbit 2450.a
Self dual yes
Analytic conductor $19.563$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{11})\)
Defining polynomial: \(x^{4} - 12 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{1} q^{3} + q^{4} + \beta_{1} q^{6} + q^{8} + ( 3 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + q^{2} + \beta_{1} q^{3} + q^{4} + \beta_{1} q^{6} + q^{8} + ( 3 + \beta_{3} ) q^{9} + ( 2 + \beta_{3} ) q^{11} + \beta_{1} q^{12} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{13} + q^{16} + ( -\beta_{1} - 3 \beta_{2} ) q^{17} + ( 3 + \beta_{3} ) q^{18} + ( -\beta_{1} + \beta_{2} ) q^{19} + ( 2 + \beta_{3} ) q^{22} + ( -2 - 2 \beta_{3} ) q^{23} + \beta_{1} q^{24} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{26} + ( \beta_{1} + 5 \beta_{2} ) q^{27} + ( 4 - 2 \beta_{3} ) q^{29} + 4 \beta_{2} q^{31} + q^{32} + ( 3 \beta_{1} + 5 \beta_{2} ) q^{33} + ( -\beta_{1} - 3 \beta_{2} ) q^{34} + ( 3 + \beta_{3} ) q^{36} + ( 2 - 2 \beta_{3} ) q^{37} + ( -\beta_{1} + \beta_{2} ) q^{38} + 10 q^{39} + ( \beta_{1} - 4 \beta_{2} ) q^{41} + 2 \beta_{3} q^{43} + ( 2 + \beta_{3} ) q^{44} + ( -2 - 2 \beta_{3} ) q^{46} -2 \beta_{1} q^{47} + \beta_{1} q^{48} + ( -9 - 4 \beta_{3} ) q^{51} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{52} + 8 q^{53} + ( \beta_{1} + 5 \beta_{2} ) q^{54} -5 q^{57} + ( 4 - 2 \beta_{3} ) q^{58} + ( -2 \beta_{1} + 3 \beta_{2} ) q^{59} -2 \beta_{1} q^{61} + 4 \beta_{2} q^{62} + q^{64} + ( 3 \beta_{1} + 5 \beta_{2} ) q^{66} + ( -5 + 2 \beta_{3} ) q^{67} + ( -\beta_{1} - 3 \beta_{2} ) q^{68} + ( -4 \beta_{1} - 10 \beta_{2} ) q^{69} -2 q^{71} + ( 3 + \beta_{3} ) q^{72} + ( -\beta_{1} - 3 \beta_{2} ) q^{73} + ( 2 - 2 \beta_{3} ) q^{74} + ( -\beta_{1} + \beta_{2} ) q^{76} + 10 q^{78} + ( 8 - 2 \beta_{3} ) q^{79} + ( 2 + 3 \beta_{3} ) q^{81} + ( \beta_{1} - 4 \beta_{2} ) q^{82} + ( \beta_{1} - 8 \beta_{2} ) q^{83} + 2 \beta_{3} q^{86} + ( 2 \beta_{1} - 10 \beta_{2} ) q^{87} + ( 2 + \beta_{3} ) q^{88} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{89} + ( -2 - 2 \beta_{3} ) q^{92} + ( 4 + 4 \beta_{3} ) q^{93} -2 \beta_{1} q^{94} + \beta_{1} q^{96} + 3 \beta_{2} q^{97} + ( 17 + 5 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{4} + 4q^{8} + 12q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{4} + 4q^{8} + 12q^{9} + 8q^{11} + 4q^{16} + 12q^{18} + 8q^{22} - 8q^{23} + 16q^{29} + 4q^{32} + 12q^{36} + 8q^{37} + 40q^{39} + 8q^{44} - 8q^{46} - 36q^{51} + 32q^{53} - 20q^{57} + 16q^{58} + 4q^{64} - 20q^{67} - 8q^{71} + 12q^{72} + 8q^{74} + 40q^{78} + 32q^{79} + 8q^{81} + 8q^{88} - 8q^{92} + 16q^{93} + 68q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 12 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 7 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 6\)
\(\nu^{3}\)\(=\)\(5 \beta_{2} + 7 \beta_{1}\)

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
−3.05231
−1.63810
1.63810
3.05231
1.00000 −3.05231 1.00000 0 −3.05231 0 1.00000 6.31662 0
1.2 1.00000 −1.63810 1.00000 0 −1.63810 0 1.00000 −0.316625 0
1.3 1.00000 1.63810 1.00000 0 1.63810 0 1.00000 −0.316625 0
1.4 1.00000 3.05231 1.00000 0 3.05231 0 1.00000 6.31662 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2450.2.a.bu yes 4
5.b even 2 1 2450.2.a.bt 4
5.c odd 4 2 2450.2.c.x 8
7.b odd 2 1 inner 2450.2.a.bu yes 4
35.c odd 2 1 2450.2.a.bt 4
35.f even 4 2 2450.2.c.x 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2450.2.a.bt 4 5.b even 2 1
2450.2.a.bt 4 35.c odd 2 1
2450.2.a.bu yes 4 1.a even 1 1 trivial
2450.2.a.bu yes 4 7.b odd 2 1 inner
2450.2.c.x 8 5.c odd 4 2
2450.2.c.x 8 35.f even 4 2

Hecke kernels

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

\( T_{3}^{4} - 12 T_{3}^{2} + 25 \)
\( T_{11}^{2} - 4 T_{11} - 7 \)
\( T_{13}^{4} - 48 T_{13}^{2} + 400 \)
\( T_{17}^{4} - 60 T_{17}^{2} + 361 \)
\( T_{19}^{4} - 12 T_{19}^{2} + 25 \)
\( T_{23}^{2} + 4 T_{23} - 40 \)
\( T_{37}^{2} - 4 T_{37} - 40 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( 25 - 12 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -7 - 4 T + T^{2} )^{2} \)
$13$ \( 400 - 48 T^{2} + T^{4} \)
$17$ \( 361 - 60 T^{2} + T^{4} \)
$19$ \( 25 - 12 T^{2} + T^{4} \)
$23$ \( ( -40 + 4 T + T^{2} )^{2} \)
$29$ \( ( -28 - 8 T + T^{2} )^{2} \)
$31$ \( ( -32 + T^{2} )^{2} \)
$37$ \( ( -40 - 4 T + T^{2} )^{2} \)
$41$ \( 361 - 60 T^{2} + T^{4} \)
$43$ \( ( -44 + T^{2} )^{2} \)
$47$ \( 400 - 48 T^{2} + T^{4} \)
$53$ \( ( -8 + T )^{4} \)
$59$ \( 196 - 60 T^{2} + T^{4} \)
$61$ \( 400 - 48 T^{2} + T^{4} \)
$67$ \( ( -19 + 10 T + T^{2} )^{2} \)
$71$ \( ( 2 + T )^{4} \)
$73$ \( 361 - 60 T^{2} + T^{4} \)
$79$ \( ( 20 - 16 T + T^{2} )^{2} \)
$83$ \( 11449 - 236 T^{2} + T^{4} \)
$89$ \( 2025 - 108 T^{2} + T^{4} \)
$97$ \( ( -18 + T^{2} )^{2} \)
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