Properties

Label 3025.2.a.z
Level $3025$
Weight $2$
Character orbit 3025.a
Self dual yes
Analytic conductor $24.155$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(x^{4} - 4 x^{2} + 1\)
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.

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

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.z 4
5.b even 2 1 inner 3025.2.a.z 4
5.c odd 4 2 605.2.b.e yes 4
11.b odd 2 1 3025.2.a.y 4
55.d odd 2 1 3025.2.a.y 4
55.e even 4 2 605.2.b.d 4
55.k odd 20 8 605.2.j.e 16
55.l even 20 8 605.2.j.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.d 4 55.e even 4 2
605.2.b.e yes 4 5.c odd 4 2
605.2.j.e 16 55.k odd 20 8
605.2.j.f 16 55.l even 20 8
3025.2.a.y 4 11.b odd 2 1
3025.2.a.y 4 55.d odd 2 1
3025.2.a.z 4 1.a even 1 1 trivial
3025.2.a.z 4 5.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3025))\):

\( T_{2}^{4} - 4 T_{2}^{2} + 1 \)
\( T_{3}^{4} - 4 T_{3}^{2} + 1 \)
\( T_{19}^{2} + 2 T_{19} - 26 \)

Hecke characteristic polynomials

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