Properties

Label 3025.2.a.y
Level $3025$
Weight $2$
Character orbit 3025.a
Self dual yes
Analytic conductor $24.155$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + \beta_{2} q^{4} - q^{6} + (2 \beta_{3} + \beta_1) q^{7} + \beta_{3} q^{8} + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + \beta_{2} q^{4} - q^{6} + (2 \beta_{3} + \beta_1) q^{7} + \beta_{3} q^{8} + ( - \beta_{2} - 1) q^{9} + ( - 2 \beta_{3} - \beta_1) q^{12} + ( - 3 \beta_{3} - 3 \beta_1) q^{13} + \beta_{2} q^{14} + ( - 2 \beta_{2} - 1) q^{16} - 2 \beta_{3} q^{17} + ( - \beta_{3} - 3 \beta_1) q^{18} + (3 \beta_{2} + 1) q^{19} + ( - 2 \beta_{2} + 3) q^{21} + ( - \beta_{3} + 3 \beta_1) q^{23} + ( - \beta_{2} + 2) q^{24} + ( - 3 \beta_{2} - 3) q^{26} + ( - 2 \beta_{3} + \beta_1) q^{27} - 3 \beta_{3} q^{28} - 4 \beta_{2} q^{29} + (\beta_{2} - 7) q^{31} + ( - 4 \beta_{3} - 5 \beta_1) q^{32} + 2 q^{34} + ( - \beta_{2} - 3) q^{36} + ( - 2 \beta_{3} - 4 \beta_1) q^{37} + (3 \beta_{3} + 7 \beta_1) q^{38} + (3 \beta_{2} - 3) q^{39} - \beta_{2} q^{41} + ( - 2 \beta_{3} - \beta_1) q^{42} + (2 \beta_{3} - 5 \beta_1) q^{43} + (3 \beta_{2} + 7) q^{46} + ( - 7 \beta_{3} - 4 \beta_1) q^{47} + (3 \beta_{3} + 2 \beta_1) q^{48} + ( - 3 \beta_{2} - 1) q^{49} + (2 \beta_{2} - 4) q^{51} + (3 \beta_{3} - 3 \beta_1) q^{52} + ( - 3 \beta_{3} - 7 \beta_1) q^{53} + (\beta_{2} + 4) q^{54} + ( - 2 \beta_{2} + 3) q^{56} + ( - 5 \beta_{3} - 3 \beta_1) q^{57} + ( - 4 \beta_{3} - 8 \beta_1) q^{58} + ( - \beta_{2} - 3) q^{59} + (\beta_{2} - 10) q^{61} + (\beta_{3} - 5 \beta_1) q^{62} + (\beta_{3} - \beta_1) q^{63} + ( - \beta_{2} - 4) q^{64} + (\beta_{3} + 8 \beta_1) q^{67} + (4 \beta_{3} + 2 \beta_1) q^{68} + (\beta_{2} - 5) q^{69} + (3 \beta_{2} - 3) q^{71} + (\beta_{3} + \beta_1) q^{72} + (2 \beta_{3} - 2 \beta_1) q^{73} + ( - 4 \beta_{2} - 6) q^{74} + (\beta_{2} + 9) q^{76} + (3 \beta_{3} + 3 \beta_1) q^{78} + ( - 2 \beta_{2} - 2) q^{79} + (5 \beta_{2} - 2) q^{81} + ( - \beta_{3} - 2 \beta_1) q^{82} + (7 \beta_{3} + 7 \beta_1) q^{83} + (3 \beta_{2} - 6) q^{84} + ( - 5 \beta_{2} - 12) q^{86} + (8 \beta_{3} + 4 \beta_1) q^{87} + ( - 2 \beta_{2} - 3) q^{89} + (3 \beta_{2} - 9) q^{91} + (5 \beta_{3} + 7 \beta_1) q^{92} + ( - 9 \beta_{3} - \beta_1) q^{93} + ( - 4 \beta_{2} - 1) q^{94} + (4 \beta_{2} - 3) q^{96} + (5 \beta_{3} + \beta_1) q^{97} + ( - 3 \beta_{3} - 7 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{6} - 4 q^{9} - 4 q^{16} + 4 q^{19} + 12 q^{21} + 8 q^{24} - 12 q^{26} - 28 q^{31} + 8 q^{34} - 12 q^{36} - 12 q^{39} + 28 q^{46} - 4 q^{49} - 16 q^{51} + 16 q^{54} + 12 q^{56} - 12 q^{59} - 40 q^{61} - 16 q^{64} - 20 q^{69} - 12 q^{71} - 24 q^{74} + 36 q^{76} - 8 q^{79} - 8 q^{81} - 24 q^{84} - 48 q^{86} - 12 q^{89} - 36 q^{91} - 4 q^{94} - 12 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{24} + \zeta_{24}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) 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
−1.93185
−0.517638
0.517638
1.93185
−1.93185 0.517638 1.73205 0 −1.00000 −0.896575 0.517638 −2.73205 0
1.2 −0.517638 1.93185 −1.73205 0 −1.00000 3.34607 1.93185 0.732051 0
1.3 0.517638 −1.93185 −1.73205 0 −1.00000 −3.34607 −1.93185 0.732051 0
1.4 1.93185 −0.517638 1.73205 0 −1.00000 0.896575 −0.517638 −2.73205 0
\(n\): e.g. 2-40 or 990-1000
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
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 3025.2.a.y 4
5.b even 2 1 inner 3025.2.a.y 4
5.c odd 4 2 605.2.b.d 4
11.b odd 2 1 3025.2.a.z 4
55.d odd 2 1 3025.2.a.z 4
55.e even 4 2 605.2.b.e yes 4
55.k odd 20 8 605.2.j.f 16
55.l even 20 8 605.2.j.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.d 4 5.c odd 4 2
605.2.b.e yes 4 55.e even 4 2
605.2.j.e 16 55.l even 20 8
605.2.j.f 16 55.k odd 20 8
3025.2.a.y 4 1.a even 1 1 trivial
3025.2.a.y 4 5.b even 2 1 inner
3025.2.a.z 4 11.b odd 2 1
3025.2.a.z 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3025))\):

\( T_{2}^{4} - 4T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} - 4T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{19}^{2} - 2T_{19} - 26 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 4T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 12T^{2} + 9 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 16T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 52T^{2} + 484 \) Copy content Toggle raw display
$29$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 14 T + 46)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 48T^{2} + 144 \) Copy content Toggle raw display
$41$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 156T^{2} + 4761 \) Copy content Toggle raw display
$47$ \( T^{4} - 148T^{2} + 2209 \) Copy content Toggle raw display
$53$ \( T^{4} - 148T^{2} + 676 \) Copy content Toggle raw display
$59$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 20 T + 97)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 228T^{2} + 1089 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T - 18)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 8)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 98)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T - 3)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 84T^{2} + 36 \) Copy content Toggle raw display
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