Properties

Label 3025.2.a.bg
Level $3025$
Weight $2$
Character orbit 3025.a
Self dual yes
Analytic conductor $24.155$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.1547466114\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.27433728.1
Defining polynomial: \( x^{6} - 9x^{4} + 15x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 605)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{4} - 2 \beta_1) q^{6} - \beta_1 q^{7} + (\beta_{5} + \beta_{4} + \beta_1) q^{8} + ( - 2 \beta_{3} - \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - 1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{4} - 2 \beta_1) q^{6} - \beta_1 q^{7} + (\beta_{5} + \beta_{4} + \beta_1) q^{8} + ( - 2 \beta_{3} - \beta_{2} + 2) q^{9} + ( - \beta_{3} - \beta_{2} - 3) q^{12} + ( - \beta_{5} + \beta_{4} - 2 \beta_1) q^{13} + ( - \beta_{2} - 3) q^{14} + (2 \beta_{3} + 2 \beta_{2} + 3) q^{16} + ( - 2 \beta_{4} + 2 \beta_1) q^{17} + ( - \beta_{5} - 3 \beta_{4} + 2 \beta_1) q^{18} + (\beta_{5} - \beta_{4}) q^{19} + ( - \beta_{4} + 2 \beta_1) q^{21} + ( - 2 \beta_{3} - \beta_{2} - 2) q^{23} + ( - \beta_{5} - 4 \beta_{4}) q^{24} + ( - 3 \beta_{2} - 6) q^{26} + (3 \beta_{3} + 3 \beta_{2} - 5) q^{27} + ( - \beta_{5} - \beta_{4} - 3 \beta_1) q^{28} + ( - 2 \beta_{5} + 2 \beta_1) q^{29} + (2 \beta_{3} - \beta_{2}) q^{31} + (2 \beta_{4} + 3 \beta_1) q^{32} + ( - 2 \beta_{3} + 4) q^{34} + ( - \beta_{2} - 2) q^{36} - 2 \beta_{3} q^{37} + \beta_{2} q^{38} + ( - \beta_{5} + \beta_{4} + 4 \beta_1) q^{39} + (\beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{41} + ( - \beta_{3} + \beta_{2} + 5) q^{42} + (2 \beta_{5} - \beta_1) q^{43} + ( - \beta_{5} - 3 \beta_{4} - 2 \beta_1) q^{46} + ( - \beta_{3} - 7) q^{47} + ( - 3 \beta_{3} - 4 \beta_{2} + 1) q^{48} + (\beta_{2} - 4) q^{49} + (2 \beta_{5} + 4 \beta_{4} - 6 \beta_1) q^{51} + ( - \beta_{5} - 5 \beta_{4} - 8 \beta_1) q^{52} + ( - 4 \beta_{3} - 3 \beta_{2} - 4) q^{53} + (3 \beta_{5} + 6 \beta_{4} - 2 \beta_1) q^{54} + ( - 2 \beta_{3} - 4 \beta_{2} - 5) q^{56} + (\beta_{5} - 3 \beta_{4}) q^{57} + ( - 2 \beta_{3} - 2 \beta_{2} + 4) q^{58} - \beta_{2} q^{59} - \beta_{5} q^{61} + ( - \beta_{5} + \beta_{4} - 4 \beta_1) q^{62} + (\beta_{5} + 3 \beta_{4} - 2 \beta_1) q^{63} + ( - 2 \beta_{3} + \beta_{2} + 5) q^{64} + ( - \beta_{3} - 5) q^{67} + (2 \beta_{4} + 2 \beta_1) q^{68} + (2 \beta_{3} + 3 \beta_{2} - 4) q^{69} + \beta_{2} q^{71} + (\beta_{5} + 5 \beta_{4} - 8 \beta_1) q^{72} + (4 \beta_{4} - 4 \beta_1) q^{73} + ( - 2 \beta_{4} + 2 \beta_1) q^{74} + ( - \beta_{5} + 3 \beta_{4} + 2 \beta_1) q^{76} + (3 \beta_{2} + 12) q^{78} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_1) q^{79} + ( - 8 \beta_{3} - 3 \beta_{2} + 5) q^{81} + ( - \beta_{3} - 2 \beta_{2} - 7) q^{82} + ( - \beta_{5} - 3 \beta_{4} + 6 \beta_1) q^{83} + (\beta_{5} + 2 \beta_{4} + 4 \beta_1) q^{84} + (2 \beta_{3} + 3 \beta_{2} - 1) q^{86} + (10 \beta_{4} - 6 \beta_1) q^{87} + ( - 2 \beta_{3} + 2 \beta_{2} - 5) q^{89} + (3 \beta_{2} + 6) q^{91} + ( - 5 \beta_{2} - 6) q^{92} + ( - \beta_{2} + 10) q^{93} + ( - \beta_{4} - 6 \beta_1) q^{94} + ( - 2 \beta_{5} + \beta_{4} - 4 \beta_1) q^{96} + (4 \beta_{3} + \beta_{2} - 4) q^{97} + (\beta_{5} + \beta_{4} - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 6 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{3} + 6 q^{4} + 12 q^{9} - 18 q^{12} - 18 q^{14} + 18 q^{16} - 12 q^{23} - 36 q^{26} - 30 q^{27} + 24 q^{34} - 12 q^{36} + 30 q^{42} - 42 q^{47} + 6 q^{48} - 24 q^{49} - 24 q^{53} - 30 q^{56} + 24 q^{58} + 30 q^{64} - 30 q^{67} - 24 q^{69} + 72 q^{78} + 30 q^{81} - 42 q^{82} - 6 q^{86} - 30 q^{89} + 36 q^{91} - 36 q^{92} + 60 q^{93} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 9x^{4} + 15x^{2} - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 8\nu^{2} + 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 8\nu^{3} + 9\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 10\nu^{3} - 19\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{3} + 8\beta_{2} + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{5} + 10\beta_{4} + 31\beta_1 \) 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
−2.62383
−1.37268
−0.480901
0.480901
1.37268
2.62383
−2.62383 −1.33988 4.88448 0 3.51561 2.62383 −7.56839 −1.20473 0
1.2 −1.37268 −3.26180 −0.115749 0 4.47741 1.37268 2.90425 7.63935 0
1.3 −0.480901 1.60168 −1.76873 0 −0.770249 0.480901 1.81239 −0.434624 0
1.4 0.480901 1.60168 −1.76873 0 0.770249 −0.480901 −1.81239 −0.434624 0
1.5 1.37268 −3.26180 −0.115749 0 −4.47741 −1.37268 −2.90425 7.63935 0
1.6 2.62383 −1.33988 4.88448 0 −3.51561 −2.62383 7.56839 −1.20473 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
\(5\) \(1\)
\(11\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3025.2.a.bg 6
5.b even 2 1 605.2.a.m 6
11.b odd 2 1 inner 3025.2.a.bg 6
15.d odd 2 1 5445.2.a.bx 6
20.d odd 2 1 9680.2.a.cw 6
55.d odd 2 1 605.2.a.m 6
55.h odd 10 4 605.2.g.q 24
55.j even 10 4 605.2.g.q 24
165.d even 2 1 5445.2.a.bx 6
220.g even 2 1 9680.2.a.cw 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.m 6 5.b even 2 1
605.2.a.m 6 55.d odd 2 1
605.2.g.q 24 55.h odd 10 4
605.2.g.q 24 55.j even 10 4
3025.2.a.bg 6 1.a even 1 1 trivial
3025.2.a.bg 6 11.b odd 2 1 inner
5445.2.a.bx 6 15.d odd 2 1
5445.2.a.bx 6 165.d even 2 1
9680.2.a.cw 6 20.d odd 2 1
9680.2.a.cw 6 220.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3025))\):

\( T_{2}^{6} - 9T_{2}^{4} + 15T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{3}^{3} + 3T_{3}^{2} - 3T_{3} - 7 \) Copy content Toggle raw display
\( T_{19}^{6} - 36T_{19}^{4} + 96T_{19}^{2} - 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 9 T^{4} + 15 T^{2} - 3 \) Copy content Toggle raw display
$3$ \( (T^{3} + 3 T^{2} - 3 T - 7)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 9 T^{4} + 15 T^{2} - 3 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 72 T^{4} + 1296 T^{2} + \cdots - 3888 \) Copy content Toggle raw display
$17$ \( T^{6} - 48 T^{4} + 384 T^{2} + \cdots - 768 \) Copy content Toggle raw display
$19$ \( T^{6} - 36 T^{4} + 96 T^{2} - 48 \) Copy content Toggle raw display
$23$ \( (T^{3} + 6 T^{2} - 12 T - 84)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} - 144 T^{4} + 5184 T^{2} + \cdots - 6912 \) Copy content Toggle raw display
$31$ \( (T^{3} - 48 T - 124)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 24 T + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 105 T^{4} + 2499 T^{2} + \cdots - 7203 \) Copy content Toggle raw display
$43$ \( T^{6} - 129 T^{4} + 4695 T^{2} + \cdots - 43923 \) Copy content Toggle raw display
$47$ \( (T^{3} + 21 T^{2} + 141 T + 303)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 12 T^{2} - 84 T - 732)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 12 T + 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} - 33 T^{4} + 339 T^{2} + \cdots - 1083 \) Copy content Toggle raw display
$67$ \( (T^{3} + 15 T^{2} + 69 T + 97)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 12 T - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 192 T^{4} + 6144 T^{2} + \cdots - 49152 \) Copy content Toggle raw display
$79$ \( T^{6} - 276 T^{4} + 11184 T^{2} + \cdots - 101568 \) Copy content Toggle raw display
$83$ \( T^{6} - 312 T^{4} + 14640 T^{2} + \cdots - 40368 \) Copy content Toggle raw display
$89$ \( (T^{3} + 15 T^{2} - 21 T - 147)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 12 T^{2} - 36 T - 76)^{2} \) Copy content Toggle raw display
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