Properties

Label 3025.2.a.q
Level $3025$
Weight $2$
Character orbit 3025.a
Self dual yes
Analytic conductor $24.155$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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: \(3\)
Coefficient field: 3.3.473.1
Defining polynomial: \(x^{3} - 5 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -1 - \beta_{1} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -3 - \beta_{1} - \beta_{2} ) q^{6} + ( 1 - \beta_{1} + \beta_{2} ) q^{7} + ( 1 + \beta_{1} ) q^{8} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -1 - \beta_{1} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -3 - \beta_{1} - \beta_{2} ) q^{6} + ( 1 - \beta_{1} + \beta_{2} ) q^{7} + ( 1 + \beta_{1} ) q^{8} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{9} + ( -2 - 3 \beta_{1} - \beta_{2} ) q^{12} + ( -3 - \beta_{1} ) q^{13} + ( -2 + 3 \beta_{1} - \beta_{2} ) q^{14} + ( 1 + \beta_{1} - \beta_{2} ) q^{16} + ( \beta_{1} - 2 \beta_{2} ) q^{17} + ( 7 + 3 \beta_{1} + 2 \beta_{2} ) q^{18} + ( 2 - \beta_{1} + \beta_{2} ) q^{19} + ( 1 - 2 \beta_{1} ) q^{21} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{23} + ( -4 - 2 \beta_{1} - \beta_{2} ) q^{24} + ( -3 - 3 \beta_{1} - \beta_{2} ) q^{26} + ( -5 - 2 \beta_{1} - 3 \beta_{2} ) q^{27} + ( 6 - 2 \beta_{1} + \beta_{2} ) q^{28} + ( 4 - \beta_{2} ) q^{29} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{31} + ( -3 \beta_{1} + \beta_{2} ) q^{32} + ( 1 - 4 \beta_{1} + \beta_{2} ) q^{34} + ( 9 + 7 \beta_{1} + \beta_{2} ) q^{36} + ( 2 + \beta_{1} ) q^{37} + ( -2 + 4 \beta_{1} - \beta_{2} ) q^{38} + ( 6 + 4 \beta_{1} + \beta_{2} ) q^{39} + ( -1 + 2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -6 + \beta_{1} - 2 \beta_{2} ) q^{42} + ( -5 + 4 \beta_{1} + 2 \beta_{2} ) q^{43} + ( 1 - 6 \beta_{1} + \beta_{2} ) q^{46} + ( -8 + \beta_{1} ) q^{47} -3 q^{48} + ( 1 - 5 \beta_{1} + 2 \beta_{2} ) q^{49} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{51} + ( -4 - 3 \beta_{1} - 3 \beta_{2} ) q^{52} + ( -8 + \beta_{1} + \beta_{2} ) q^{53} + ( -9 - 11 \beta_{1} - 2 \beta_{2} ) q^{54} + ( -1 + 2 \beta_{1} ) q^{56} -3 \beta_{1} q^{57} + ( -1 + 2 \beta_{1} ) q^{58} + ( -4 - \beta_{2} ) q^{59} + ( -7 + \beta_{1} ) q^{61} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{62} + ( 2 + 4 \beta_{1} - \beta_{2} ) q^{63} + ( -10 - \beta_{2} ) q^{64} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{67} + ( -11 + \beta_{1} ) q^{68} + ( 1 + 5 \beta_{1} + \beta_{2} ) q^{69} + ( -1 - 4 \beta_{1} - \beta_{2} ) q^{71} + ( 8 + 5 \beta_{1} + 3 \beta_{2} ) q^{72} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{73} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{74} + ( 7 - 2 \beta_{1} + 2 \beta_{2} ) q^{76} + ( 13 + 8 \beta_{1} + 4 \beta_{2} ) q^{78} + ( -6 - 5 \beta_{1} + 2 \beta_{2} ) q^{79} + ( 11 + 7 \beta_{1} + 2 \beta_{2} ) q^{81} + ( 4 - 5 \beta_{1} + 2 \beta_{2} ) q^{82} + ( 2 - \beta_{1} - 5 \beta_{2} ) q^{83} + ( -1 - 6 \beta_{1} + \beta_{2} ) q^{84} + ( 14 - \beta_{1} + 4 \beta_{2} ) q^{86} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{87} + ( 6 - \beta_{1} - 3 \beta_{2} ) q^{89} + ( -1 - 2 \beta_{2} ) q^{91} + ( -13 + \beta_{1} - 2 \beta_{2} ) q^{92} -6 q^{93} + ( 3 - 8 \beta_{1} + \beta_{2} ) q^{94} + ( 8 + \beta_{1} + 2 \beta_{2} ) q^{96} + ( -3 + 2 \beta_{1} + 3 \beta_{2} ) q^{97} + ( -13 + 5 \beta_{1} - 5 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 4 q^{4} - 10 q^{6} + 4 q^{7} + 3 q^{8} + 4 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + 4 q^{4} - 10 q^{6} + 4 q^{7} + 3 q^{8} + 4 q^{9} - 7 q^{12} - 9 q^{13} - 7 q^{14} + 2 q^{16} - 2 q^{17} + 23 q^{18} + 7 q^{19} + 3 q^{21} - 8 q^{23} - 13 q^{24} - 10 q^{26} - 18 q^{27} + 19 q^{28} + 11 q^{29} + 4 q^{31} + q^{32} + 4 q^{34} + 28 q^{36} + 6 q^{37} - 7 q^{38} + 19 q^{39} - 5 q^{41} - 20 q^{42} - 13 q^{43} + 4 q^{46} - 24 q^{47} - 9 q^{48} + 5 q^{49} - 2 q^{51} - 15 q^{52} - 23 q^{53} - 29 q^{54} - 3 q^{56} - 3 q^{58} - 13 q^{59} - 21 q^{61} + 14 q^{62} + 5 q^{63} - 31 q^{64} + 10 q^{67} - 33 q^{68} + 4 q^{69} - 4 q^{71} + 27 q^{72} - 3 q^{73} + 10 q^{74} + 23 q^{76} + 43 q^{78} - 16 q^{79} + 35 q^{81} + 14 q^{82} + q^{83} - 2 q^{84} + 46 q^{86} - 8 q^{87} + 15 q^{89} - 5 q^{91} - 41 q^{92} - 18 q^{93} + 10 q^{94} + 26 q^{96} - 6 q^{97} - 44 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 5 x - 1\):

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

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
−2.12842
−0.201640
2.33006
−2.12842 1.12842 2.53017 0 −2.40175 4.65859 −1.12842 −1.72667 0
1.2 −0.201640 −0.798360 −1.95934 0 0.160981 −1.75770 0.798360 −2.36262 0
1.3 2.33006 −3.33006 3.42917 0 −7.75923 1.09911 3.33006 8.08929 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(-1\)

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 3025.2.a.q 3
5.b even 2 1 3025.2.a.s yes 3
11.b odd 2 1 3025.2.a.r yes 3
55.d odd 2 1 3025.2.a.t yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3025.2.a.q 3 1.a even 1 1 trivial
3025.2.a.r yes 3 11.b odd 2 1
3025.2.a.s yes 3 5.b even 2 1
3025.2.a.t yes 3 55.d odd 2 1

Hecke kernels

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

\( T_{2}^{3} - 5 T_{2} - 1 \)
\( T_{3}^{3} + 3 T_{3}^{2} - 2 T_{3} - 3 \)
\( T_{19}^{3} - 7 T_{19}^{2} + 6 T_{19} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 5 T + T^{3} \)
$3$ \( -3 - 2 T + 3 T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( 9 - 5 T - 4 T^{2} + T^{3} \)
$11$ \( T^{3} \)
$13$ \( 13 + 22 T + 9 T^{2} + T^{3} \)
$17$ \( -75 - 31 T + 2 T^{2} + T^{3} \)
$19$ \( 9 + 6 T - 7 T^{2} + T^{3} \)
$23$ \( -121 - 11 T + 8 T^{2} + T^{3} \)
$29$ \( -27 + 32 T - 11 T^{2} + T^{3} \)
$31$ \( 72 - 36 T - 4 T^{2} + T^{3} \)
$37$ \( 1 + 7 T - 6 T^{2} + T^{3} \)
$41$ \( -45 - 33 T + 5 T^{2} + T^{3} \)
$43$ \( -1125 - 81 T + 13 T^{2} + T^{3} \)
$47$ \( 471 + 187 T + 24 T^{2} + T^{3} \)
$53$ \( 311 + 160 T + 23 T^{2} + T^{3} \)
$59$ \( 37 + 48 T + 13 T^{2} + T^{3} \)
$61$ \( 307 + 142 T + 21 T^{2} + T^{3} \)
$67$ \( 1 - T - 10 T^{2} + T^{3} \)
$71$ \( 211 - 95 T + 4 T^{2} + T^{3} \)
$73$ \( -197 - 74 T + 3 T^{2} + T^{3} \)
$79$ \( -1075 - 43 T + 16 T^{2} + T^{3} \)
$83$ \( -747 - 228 T - T^{2} + T^{3} \)
$89$ \( 193 - 14 T - 15 T^{2} + T^{3} \)
$97$ \( -293 - 101 T + 6 T^{2} + T^{3} \)
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