Properties

Label 4225.2.a.bk
Level $4225$
Weight $2$
Character orbit 4225.a
Self dual yes
Analytic conductor $33.737$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4225 = 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.7367948540\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + q^{3} -\beta_{2} q^{4} + \beta_{3} q^{6} + ( -\beta_{1} - \beta_{3} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} -2 q^{9} +O(q^{10})\) \( q + \beta_{3} q^{2} + q^{3} -\beta_{2} q^{4} + \beta_{3} q^{6} + ( -\beta_{1} - \beta_{3} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} -2 q^{9} + ( \beta_{1} - \beta_{3} ) q^{11} -\beta_{2} q^{12} + ( -1 + \beta_{2} ) q^{14} + ( 1 + \beta_{2} ) q^{16} + ( 1 + 2 \beta_{2} ) q^{17} -2 \beta_{3} q^{18} + ( -\beta_{1} - \beta_{3} ) q^{19} + ( -\beta_{1} - \beta_{3} ) q^{21} + ( -3 + \beta_{2} ) q^{22} + ( -1 - 2 \beta_{2} ) q^{23} + ( \beta_{1} + \beta_{3} ) q^{24} -5 q^{27} + ( \beta_{1} - 2 \beta_{3} ) q^{28} + ( 1 + 2 \beta_{2} ) q^{29} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{31} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{32} + ( \beta_{1} - \beta_{3} ) q^{33} + ( -2 \beta_{1} - 5 \beta_{3} ) q^{34} + 2 \beta_{2} q^{36} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{37} + ( -1 + \beta_{2} ) q^{38} + ( \beta_{1} - \beta_{3} ) q^{41} + ( -1 + \beta_{2} ) q^{42} + ( -7 - 2 \beta_{2} ) q^{43} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{44} + ( 2 \beta_{1} + 5 \beta_{3} ) q^{46} + 4 \beta_{3} q^{47} + ( 1 + \beta_{2} ) q^{48} -4 q^{49} + ( 1 + 2 \beta_{2} ) q^{51} + ( 4 + 2 \beta_{2} ) q^{53} -5 \beta_{3} q^{54} -3 q^{56} + ( -\beta_{1} - \beta_{3} ) q^{57} + ( -2 \beta_{1} - 5 \beta_{3} ) q^{58} + ( 7 \beta_{1} + 3 \beta_{3} ) q^{59} + ( -5 + 2 \beta_{2} ) q^{61} + ( 10 - 4 \beta_{2} ) q^{62} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{63} + ( -7 + 2 \beta_{2} ) q^{64} + ( -3 + \beta_{2} ) q^{66} + ( -\beta_{1} - 7 \beta_{3} ) q^{67} + ( -10 + \beta_{2} ) q^{68} + ( -1 - 2 \beta_{2} ) q^{69} + ( -3 \beta_{1} + \beta_{3} ) q^{71} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{72} + ( -9 + 3 \beta_{2} ) q^{74} + ( \beta_{1} - 2 \beta_{3} ) q^{76} + ( -1 - 2 \beta_{2} ) q^{77} -6 q^{79} + q^{81} + ( -3 + \beta_{2} ) q^{82} + ( 4 \beta_{1} + 6 \beta_{3} ) q^{83} + ( \beta_{1} - 2 \beta_{3} ) q^{84} + ( 2 \beta_{1} - \beta_{3} ) q^{86} + ( 1 + 2 \beta_{2} ) q^{87} + ( 1 + 2 \beta_{2} ) q^{88} + ( 5 \beta_{1} + 3 \beta_{3} ) q^{89} + ( 10 - \beta_{2} ) q^{92} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{93} + ( 8 - 4 \beta_{2} ) q^{94} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{96} + ( 5 \beta_{1} - \beta_{3} ) q^{97} -4 \beta_{3} q^{98} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 2q^{4} - 8q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 2q^{4} - 8q^{9} + 2q^{12} - 6q^{14} + 2q^{16} - 14q^{22} - 20q^{27} - 4q^{36} - 6q^{38} - 6q^{42} - 24q^{43} + 2q^{48} - 16q^{49} + 12q^{53} - 12q^{56} - 24q^{61} + 48q^{62} - 32q^{64} - 14q^{66} - 42q^{68} - 42q^{74} - 24q^{79} + 4q^{81} - 14q^{82} + 42q^{92} + 40q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 1\):

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

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
0.456850
2.18890
−2.18890
−0.456850
−2.18890 1.00000 2.79129 0 −2.18890 1.73205 −1.73205 −2.00000 0
1.2 −0.456850 1.00000 −1.79129 0 −0.456850 −1.73205 1.73205 −2.00000 0
1.3 0.456850 1.00000 −1.79129 0 0.456850 1.73205 −1.73205 −2.00000 0
1.4 2.18890 1.00000 2.79129 0 2.18890 −1.73205 1.73205 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(13\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4225.2.a.bk 4
5.b even 2 1 4225.2.a.bj 4
5.c odd 4 2 845.2.b.f 8
13.b even 2 1 inner 4225.2.a.bk 4
13.f odd 12 2 325.2.n.c 4
65.d even 2 1 4225.2.a.bj 4
65.f even 4 2 845.2.d.c 8
65.h odd 4 2 845.2.b.f 8
65.k even 4 2 845.2.d.c 8
65.o even 12 2 65.2.l.a 8
65.o even 12 2 845.2.l.c 8
65.q odd 12 2 845.2.n.c 8
65.q odd 12 2 845.2.n.d 8
65.r odd 12 2 845.2.n.c 8
65.r odd 12 2 845.2.n.d 8
65.s odd 12 2 325.2.n.b 4
65.t even 12 2 65.2.l.a 8
65.t even 12 2 845.2.l.c 8
195.bc odd 12 2 585.2.bf.a 8
195.bn odd 12 2 585.2.bf.a 8
260.be odd 12 2 1040.2.df.b 8
260.bl odd 12 2 1040.2.df.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.l.a 8 65.o even 12 2
65.2.l.a 8 65.t even 12 2
325.2.n.b 4 65.s odd 12 2
325.2.n.c 4 13.f odd 12 2
585.2.bf.a 8 195.bc odd 12 2
585.2.bf.a 8 195.bn odd 12 2
845.2.b.f 8 5.c odd 4 2
845.2.b.f 8 65.h odd 4 2
845.2.d.c 8 65.f even 4 2
845.2.d.c 8 65.k even 4 2
845.2.l.c 8 65.o even 12 2
845.2.l.c 8 65.t even 12 2
845.2.n.c 8 65.q odd 12 2
845.2.n.c 8 65.r odd 12 2
845.2.n.d 8 65.q odd 12 2
845.2.n.d 8 65.r odd 12 2
1040.2.df.b 8 260.be odd 12 2
1040.2.df.b 8 260.bl odd 12 2
4225.2.a.bj 4 5.b even 2 1
4225.2.a.bj 4 65.d even 2 1
4225.2.a.bk 4 1.a even 1 1 trivial
4225.2.a.bk 4 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4225))\):

\( T_{2}^{4} - 5 T_{2}^{2} + 1 \)
\( T_{3} - 1 \)
\( T_{7}^{2} - 3 \)
\( T_{11}^{2} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5 T^{2} + T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -3 + T^{2} )^{2} \)
$11$ \( ( -7 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( -21 + T^{2} )^{2} \)
$19$ \( ( -3 + T^{2} )^{2} \)
$23$ \( ( -21 + T^{2} )^{2} \)
$29$ \( ( -21 + T^{2} )^{2} \)
$31$ \( 3600 - 132 T^{2} + T^{4} \)
$37$ \( ( -63 + T^{2} )^{2} \)
$41$ \( ( -7 + T^{2} )^{2} \)
$43$ \( ( 15 + 12 T + T^{2} )^{2} \)
$47$ \( 256 - 80 T^{2} + T^{4} \)
$53$ \( ( -12 - 6 T + T^{2} )^{2} \)
$59$ \( 2209 - 206 T^{2} + T^{4} \)
$61$ \( ( 15 + 12 T + T^{2} )^{2} \)
$67$ \( 225 - 222 T^{2} + T^{4} \)
$71$ \( 625 - 62 T^{2} + T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 6 + T )^{4} \)
$83$ \( 4624 - 164 T^{2} + T^{4} \)
$89$ \( 1681 - 110 T^{2} + T^{4} \)
$97$ \( 2601 - 150 T^{2} + T^{4} \)
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