Properties

Label 3025.2.a.bf
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: \(\Q(\zeta_{36})^+\)
Defining polynomial: \( x^{6} - 6x^{4} + 9x^{2} - 3 \) Copy content Toggle raw display
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of \(\nu = \zeta_{36} + \zeta_{36}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 5\nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 5\nu^{3} + 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 5\beta_{3} + 11\beta_1 \) 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
−1.96962
−1.28558
−0.684040
0.684040
1.28558
1.96962
−1.96962 −0.652704 1.87939 0 1.28558 −0.0825054 0.237565 −2.57398 0
1.2 −1.28558 0.532089 −0.347296 0 −0.684040 −4.62327 3.01763 −2.71688 0
1.3 −0.684040 −2.87939 −1.53209 0 1.96962 4.54077 2.41609 5.29086 0
1.4 0.684040 −2.87939 −1.53209 0 −1.96962 −4.54077 −2.41609 5.29086 0
1.5 1.28558 0.532089 −0.347296 0 0.684040 4.62327 −3.01763 −2.71688 0
1.6 1.96962 −0.652704 1.87939 0 −1.28558 0.0825054 −0.237565 −2.57398 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.bf 6
5.b even 2 1 3025.2.a.bh yes 6
11.b odd 2 1 inner 3025.2.a.bf 6
55.d odd 2 1 3025.2.a.bh yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3025.2.a.bf 6 1.a even 1 1 trivial
3025.2.a.bf 6 11.b odd 2 1 inner
3025.2.a.bh yes 6 5.b even 2 1
3025.2.a.bh yes 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3025))\):

\( T_{2}^{6} - 6T_{2}^{4} + 9T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{3}^{3} + 3T_{3}^{2} - 1 \) Copy content Toggle raw display
\( T_{19}^{6} - 87T_{19}^{4} + 1242T_{19}^{2} - 1083 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 6 T^{4} + 9 T^{2} - 3 \) Copy content Toggle raw display
$3$ \( (T^{3} + 3 T^{2} - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 42 T^{4} + 441 T^{2} - 3 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 27 T^{4} + 162 T^{2} + \cdots - 243 \) Copy content Toggle raw display
$17$ \( T^{6} - 42 T^{4} + 441 T^{2} - 3 \) Copy content Toggle raw display
$19$ \( T^{6} - 87 T^{4} + 1242 T^{2} + \cdots - 1083 \) Copy content Toggle raw display
$23$ \( (T^{3} + 12 T^{2} + 27 T - 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} - 135 T^{4} + 4914 T^{2} + \cdots - 36963 \) Copy content Toggle raw display
$31$ \( (T^{3} - 12 T^{2} + 12 T + 152)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 93 T + 289)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 177 T^{4} + 6939 T^{2} + \cdots - 15987 \) Copy content Toggle raw display
$43$ \( T^{6} - 105 T^{4} + 1899 T^{2} + \cdots - 8427 \) Copy content Toggle raw display
$47$ \( (T^{3} + 12 T^{2} + 45 T + 51)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 3 T^{2} - 54 T - 219)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 3 T^{2} - 90 T + 381)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} - 15 T^{4} + 54 T^{2} - 3 \) Copy content Toggle raw display
$67$ \( (T^{3} + 6 T^{2} - 9 T - 17)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 12 T^{2} - 45 T - 597)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 267 T^{4} + 17226 T^{2} + \cdots - 220323 \) Copy content Toggle raw display
$79$ \( T^{6} - 330 T^{4} + 23373 T^{2} + \cdots - 282747 \) Copy content Toggle raw display
$83$ \( T^{6} - 411 T^{4} + 44946 T^{2} + \cdots - 604803 \) Copy content Toggle raw display
$89$ \( (T^{3} + 3 T^{2} - 72 T + 51)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 6 T^{2} - 159 T + 323)^{2} \) Copy content Toggle raw display
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