Properties

Label 4225.2.a.bq
Level $4225$
Weight $2$
Character orbit 4225.a
Self dual yes
Analytic conductor $33.737$
Analytic rank $0$
Dimension $6$
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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.199374400.1
Defining polynomial: \(x^{6} - 8 x^{4} + 10 x^{2} - 1\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{5} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -2 - \beta_{2} ) q^{6} + ( \beta_{1} + \beta_{5} ) q^{7} + ( 2 \beta_{1} + \beta_{4} - \beta_{5} ) q^{8} + ( 1 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{5} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -2 - \beta_{2} ) q^{6} + ( \beta_{1} + \beta_{5} ) q^{7} + ( 2 \beta_{1} + \beta_{4} - \beta_{5} ) q^{8} + ( 1 + \beta_{3} ) q^{9} + \beta_{3} q^{11} + ( -3 \beta_{1} - \beta_{4} - \beta_{5} ) q^{12} + ( 4 + \beta_{2} ) q^{14} + ( 3 + \beta_{2} + \beta_{3} ) q^{16} + ( -\beta_{4} + \beta_{5} ) q^{17} + ( \beta_{1} + \beta_{4} ) q^{18} + ( -2 + \beta_{3} ) q^{19} + ( -2 \beta_{2} + \beta_{3} ) q^{21} + \beta_{4} q^{22} + ( \beta_{1} - \beta_{5} ) q^{23} + ( -6 - 2 \beta_{2} - \beta_{3} ) q^{24} + ( \beta_{1} + \beta_{5} ) q^{27} + ( 5 \beta_{1} + \beta_{4} - 3 \beta_{5} ) q^{28} + 3 q^{29} + ( 2 + 2 \beta_{2} ) q^{31} + ( 2 \beta_{1} + \beta_{5} ) q^{32} + ( -\beta_{1} + 3 \beta_{5} ) q^{33} + ( 1 - \beta_{2} - \beta_{3} ) q^{34} + ( 1 + 2 \beta_{2} - \beta_{3} ) q^{36} + ( -\beta_{4} + \beta_{5} ) q^{37} + ( -2 \beta_{1} + \beta_{4} ) q^{38} + ( 3 + 2 \beta_{2} ) q^{41} + ( -6 \beta_{1} - \beta_{4} + 2 \beta_{5} ) q^{42} + ( -\beta_{1} + 3 \beta_{5} ) q^{43} + ( \beta_{2} - \beta_{3} ) q^{44} + ( 2 + \beta_{2} ) q^{46} + ( 4 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{47} + ( -6 \beta_{1} - \beta_{4} + 4 \beta_{5} ) q^{48} + ( 1 + \beta_{3} ) q^{49} + ( 2 + \beta_{3} ) q^{51} + ( -\beta_{1} + \beta_{4} + 4 \beta_{5} ) q^{53} + ( 4 + \beta_{2} ) q^{54} + ( 4 + 4 \beta_{2} + \beta_{3} ) q^{56} + ( \beta_{1} + \beta_{5} ) q^{57} + 3 \beta_{1} q^{58} + ( -2 \beta_{2} - \beta_{3} ) q^{59} + ( -1 - 2 \beta_{3} ) q^{61} + ( 8 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} ) q^{62} + ( 2 \beta_{4} + 4 \beta_{5} ) q^{63} + ( 1 - 2 \beta_{3} ) q^{64} -\beta_{2} q^{66} + ( -3 \beta_{1} - \beta_{5} ) q^{67} + ( -2 \beta_{1} - \beta_{5} ) q^{68} + ( -4 - \beta_{3} ) q^{69} + ( -2 - \beta_{3} ) q^{71} + ( 5 \beta_{1} - \beta_{4} - 2 \beta_{5} ) q^{72} + ( 5 \beta_{1} - \beta_{4} - 2 \beta_{5} ) q^{73} + ( 1 - \beta_{2} - \beta_{3} ) q^{74} + ( -2 - \beta_{2} - \beta_{3} ) q^{76} + ( -\beta_{1} + 2 \beta_{4} + 3 \beta_{5} ) q^{77} + ( 8 - 2 \beta_{2} ) q^{79} + ( -3 - 2 \beta_{2} - 2 \beta_{3} ) q^{81} + ( 9 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} ) q^{82} + ( -4 \beta_{1} - 2 \beta_{4} + 4 \beta_{5} ) q^{83} + ( -16 - 3 \beta_{2} - 3 \beta_{3} ) q^{84} -\beta_{2} q^{86} + ( -3 \beta_{1} + 3 \beta_{5} ) q^{87} + ( 3 \beta_{1} - 2 \beta_{4} - \beta_{5} ) q^{88} + ( -2 + 4 \beta_{2} + \beta_{3} ) q^{89} + ( 3 \beta_{1} + \beta_{4} + \beta_{5} ) q^{92} + ( -6 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{93} + ( 10 + 2 \beta_{2} - 2 \beta_{3} ) q^{94} + ( -2 - 3 \beta_{2} + \beta_{3} ) q^{96} + ( 5 \beta_{1} + 2 \beta_{4} - 3 \beta_{5} ) q^{97} + ( \beta_{1} + \beta_{4} ) q^{98} + ( 8 - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 4q^{4} - 10q^{6} + 6q^{9} + O(q^{10}) \) \( 6q + 4q^{4} - 10q^{6} + 6q^{9} + 22q^{14} + 16q^{16} - 12q^{19} + 4q^{21} - 32q^{24} + 18q^{29} + 8q^{31} + 8q^{34} + 2q^{36} + 14q^{41} - 2q^{44} + 10q^{46} + 6q^{49} + 12q^{51} + 22q^{54} + 16q^{56} + 4q^{59} - 6q^{61} + 6q^{64} + 2q^{66} - 24q^{69} - 12q^{71} + 8q^{74} - 10q^{76} + 52q^{79} - 14q^{81} - 90q^{84} + 2q^{86} - 20q^{89} + 56q^{94} - 6q^{96} + 52q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 8 x^{4} + 10 x^{2} - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 7 \nu^{2} + 4 \)
\(\beta_{4}\)\(=\)\( \nu^{5} - 7 \nu^{3} + 4 \nu \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 8 \nu^{3} + 10 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{5} + \beta_{4} + 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{3} + 7 \beta_{2} + 17\)
\(\nu^{5}\)\(=\)\(-7 \beta_{5} + 8 \beta_{4} + 38 \beta_{1}\)

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
−2.54574
−1.18733
−0.330837
0.330837
1.18733
2.54574
−2.54574 2.15293 4.48079 0 −5.48079 −2.93855 −6.31544 1.63509 0
1.2 −1.18733 0.345110 −0.590239 0 −0.409761 −2.02956 3.07548 −2.88090 0
1.3 −0.330837 −2.69180 −1.89055 0 0.890547 −3.35348 1.28714 4.24581 0
1.4 0.330837 2.69180 −1.89055 0 0.890547 3.35348 −1.28714 4.24581 0
1.5 1.18733 −0.345110 −0.590239 0 −0.409761 2.02956 −3.07548 −2.88090 0
1.6 2.54574 −2.15293 4.48079 0 −5.48079 2.93855 6.31544 1.63509 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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
5.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.bq 6
5.b even 2 1 inner 4225.2.a.bq 6
5.c odd 4 2 845.2.b.e 6
13.b even 2 1 4225.2.a.br 6
13.e even 6 2 325.2.e.e 12
65.d even 2 1 4225.2.a.br 6
65.f even 4 2 845.2.d.d 12
65.h odd 4 2 845.2.b.d 6
65.k even 4 2 845.2.d.d 12
65.l even 6 2 325.2.e.e 12
65.o even 12 4 845.2.l.f 24
65.q odd 12 4 845.2.n.e 12
65.r odd 12 4 65.2.n.a 12
65.t even 12 4 845.2.l.f 24
195.bf even 12 4 585.2.bs.a 12
260.bg even 12 4 1040.2.dh.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.n.a 12 65.r odd 12 4
325.2.e.e 12 13.e even 6 2
325.2.e.e 12 65.l even 6 2
585.2.bs.a 12 195.bf even 12 4
845.2.b.d 6 65.h odd 4 2
845.2.b.e 6 5.c odd 4 2
845.2.d.d 12 65.f even 4 2
845.2.d.d 12 65.k even 4 2
845.2.l.f 24 65.o even 12 4
845.2.l.f 24 65.t even 12 4
845.2.n.e 12 65.q odd 12 4
1040.2.dh.a 12 260.bg even 12 4
4225.2.a.bq 6 1.a even 1 1 trivial
4225.2.a.bq 6 5.b even 2 1 inner
4225.2.a.br 6 13.b even 2 1
4225.2.a.br 6 65.d even 2 1

Hecke kernels

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

\( T_{2}^{6} - 8 T_{2}^{4} + 10 T_{2}^{2} - 1 \)
\( T_{3}^{6} - 12 T_{3}^{4} + 35 T_{3}^{2} - 4 \)
\( T_{7}^{6} - 24 T_{7}^{4} + 179 T_{7}^{2} - 400 \)
\( T_{11}^{3} - 13 T_{11} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 10 T^{2} - 8 T^{4} + T^{6} \)
$3$ \( -4 + 35 T^{2} - 12 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( -400 + 179 T^{2} - 24 T^{4} + T^{6} \)
$11$ \( ( 8 - 13 T + T^{3} )^{2} \)
$13$ \( T^{6} \)
$17$ \( -169 + 163 T^{2} - 35 T^{4} + T^{6} \)
$19$ \( ( -10 - T + 6 T^{2} + T^{3} )^{2} \)
$23$ \( -4 + 35 T^{2} - 12 T^{4} + T^{6} \)
$29$ \( ( -3 + T )^{6} \)
$31$ \( ( -40 - 40 T - 4 T^{2} + T^{3} )^{2} \)
$37$ \( -169 + 163 T^{2} - 35 T^{4} + T^{6} \)
$41$ \( ( -5 - 29 T - 7 T^{2} + T^{3} )^{2} \)
$43$ \( -256 + 283 T^{2} - 80 T^{4} + T^{6} \)
$47$ \( -270400 + 14640 T^{2} - 236 T^{4} + T^{6} \)
$53$ \( -400 + 1040 T^{2} - 171 T^{4} + T^{6} \)
$59$ \( ( 136 - 55 T - 2 T^{2} + T^{3} )^{2} \)
$61$ \( ( -115 - 49 T + 3 T^{2} + T^{3} )^{2} \)
$67$ \( -20164 + 2603 T^{2} - 100 T^{4} + T^{6} \)
$71$ \( ( -26 - T + 6 T^{2} + T^{3} )^{2} \)
$73$ \( -250000 + 13900 T^{2} - 215 T^{4} + T^{6} \)
$79$ \( ( -160 + 180 T - 26 T^{2} + T^{3} )^{2} \)
$83$ \( -640000 + 23600 T^{2} - 276 T^{4} + T^{6} \)
$89$ \( ( -1586 - 157 T + 10 T^{2} + T^{3} )^{2} \)
$97$ \( -204304 + 14363 T^{2} - 280 T^{4} + T^{6} \)
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