Properties

Label 3025.2.a.bk
Level $3025$
Weight $2$
Character orbit 3025.a
Self dual yes
Analytic conductor $24.155$
Analytic rank $1$
Dimension $8$
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: \(8\)
Coefficient field: 8.8.1480160000.1
Defining polynomial: \( x^{8} - 9x^{6} + 27x^{4} - 31x^{2} + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{7} - \beta_{6} - \beta_1) q^{3} + \beta_{2} q^{4} + ( - \beta_{5} - 2 \beta_{3} - \beta_{2} - 2) q^{6} - \beta_{4} q^{7} + (\beta_{6} + \beta_{4} - \beta_1) q^{8} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{7} - \beta_{6} - \beta_1) q^{3} + \beta_{2} q^{4} + ( - \beta_{5} - 2 \beta_{3} - \beta_{2} - 2) q^{6} - \beta_{4} q^{7} + (\beta_{6} + \beta_{4} - \beta_1) q^{8} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2}) q^{9} + ( - 2 \beta_{6} - \beta_{4} - \beta_1) q^{12} + (2 \beta_{6} + \beta_1) q^{13} + (2 \beta_{5} - 2 \beta_{2} + 1) q^{14} + (\beta_{3} - \beta_{2} - 2) q^{16} + (3 \beta_{7} + \beta_{6} + 2 \beta_1) q^{17} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{4} + 2 \beta_1) q^{18} + (\beta_{5} + \beta_{3} - 2 \beta_{2}) q^{19} + (2 \beta_{5} + \beta_{3} + 1) q^{21} + ( - \beta_{6} + 2 \beta_{4} - 2 \beta_1) q^{23} + (2 \beta_{3} - \beta_{2} + 1) q^{24} + (4 \beta_{5} + 2 \beta_{3} + \beta_{2} + 4) q^{26} + (2 \beta_{7} - \beta_{4}) q^{27} - \beta_1 q^{28} + ( - \beta_{5} - \beta_{3} - 4) q^{29} + ( - \beta_{5} + \beta_{3} - \beta_{2} + 2) q^{31} + (\beta_{7} - 2 \beta_{6} - 3 \beta_{4} - \beta_1) q^{32} + ( - \beta_{5} + 4 \beta_{3} + 2 \beta_{2} + 2) q^{34} + (\beta_{5} + \beta_{3} + 2 \beta_{2} + 3) q^{36} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{4}) q^{37} + (\beta_{7} - 2 \beta_{4} - 2 \beta_1) q^{38} + (\beta_{5} - 5 \beta_{2} - 2) q^{39} + (5 \beta_{5} - 2 \beta_{3} - 3 \beta_{2} - 2) q^{41} + (\beta_{7} + 3 \beta_{6} + \beta_1) q^{42} + (2 \beta_{7} - 2 \beta_{6} + 3 \beta_1) q^{43} + ( - 6 \beta_{5} - \beta_{3} + 2 \beta_{2} - 7) q^{46} + ( - \beta_{7} - \beta_{6} + 3 \beta_{4} + \beta_1) q^{47} + (2 \beta_{7} + 5 \beta_{6} + \beta_{4} + 2 \beta_1) q^{48} + (\beta_{5} - 2 \beta_{3} - \beta_{2} - 4) q^{49} + (2 \beta_{5} - 3 \beta_{3} - \beta_{2} - 7) q^{51} + (2 \beta_{7} + 3 \beta_{6} + \beta_{4} + 3 \beta_1) q^{52} + (2 \beta_{7} + 3 \beta_{6} + \beta_{4} - 3 \beta_1) q^{53} + (2 \beta_{3} - 2 \beta_{2} - 1) q^{54} + ( - 4 \beta_{5} + 3 \beta_{2} - 4) q^{56} + (\beta_{7} + 5 \beta_{6} + \beta_{4} + \beta_1) q^{57} + ( - \beta_{7} - 2 \beta_{6} - 4 \beta_1) q^{58} + (2 \beta_{5} - 3 \beta_{3} - 5 \beta_{2}) q^{59} + ( - 5 \beta_{5} + 4 \beta_{3} + 4 \beta_{2} - 4) q^{61} + (\beta_{7} - \beta_{6} - \beta_{4} + \beta_1) q^{62} + (\beta_{7} + \beta_{4} - 2 \beta_1) q^{63} + (\beta_{5} - 3 \beta_{3} - 5 \beta_{2} + 2) q^{64} + ( - 3 \beta_{7} - 2 \beta_{6} - \beta_{4} - 4 \beta_1) q^{67} + ( - 2 \beta_{7} + 3 \beta_{6} + 2 \beta_{4}) q^{68} + ( - 3 \beta_{5} + \beta_{3} + 4 \beta_{2} + 2) q^{69} + ( - 3 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} - 9) q^{71} + ( - \beta_{7} - 2 \beta_{4} + \beta_1) q^{72} + (\beta_{7} - 5 \beta_{4} + 2 \beta_1) q^{73} + (3 \beta_{5} + 2 \beta_{3} + 4 \beta_{2} + 2) q^{74} + (\beta_{5} - \beta_{3} - 2 \beta_{2} - 3) q^{76} + ( - 4 \beta_{6} - 5 \beta_{4} - 7 \beta_1) q^{78} + ( - 5 \beta_{5} - \beta_{3} + 4 \beta_{2} - 6) q^{79} + (7 \beta_{5} - 2 \beta_{3} - 4 \beta_{2} - 1) q^{81} + ( - 2 \beta_{7} - 3 \beta_{4} - 5 \beta_1) q^{82} + (\beta_{7} - 3 \beta_{6} - \beta_{4} - \beta_1) q^{83} + (\beta_{5} + 2 \beta_{3} + \beta_{2} + 2) q^{84} + ( - 6 \beta_{5} + 3 \beta_{2} + 2) q^{86} + (3 \beta_{7} + 3 \beta_{6} + \beta_{4} + 5 \beta_1) q^{87} + ( - 2 \beta_{5} - 6 \beta_{3} - 2 \beta_{2} - 5) q^{89} + ( - 4 \beta_{3} - 2 \beta_{2} + 1) q^{91} + ( - \beta_{7} - 3 \beta_{6} - 2 \beta_{4} - \beta_1) q^{92} + ( - 3 \beta_{7} + \beta_{6} + 2 \beta_{4} - 2 \beta_1) q^{93} + ( - 7 \beta_{5} - 2 \beta_{3} + 7 \beta_{2} - 1) q^{94} + (6 \beta_{5} + 3 \beta_{3} + 6 \beta_{2} + 4) q^{96} + (3 \beta_{7} + \beta_{4} - \beta_1) q^{97} + ( - 2 \beta_{7} - 2 \beta_{6} - \beta_{4} - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} - 6 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} - 6 q^{6} + 4 q^{9} - 4 q^{14} - 22 q^{16} - 12 q^{19} - 4 q^{21} - 2 q^{24} + 10 q^{26} - 24 q^{29} + 14 q^{31} + 8 q^{34} + 20 q^{36} - 30 q^{39} - 34 q^{41} - 24 q^{46} - 30 q^{49} - 54 q^{51} - 20 q^{54} - 10 q^{56} - 6 q^{59} - 20 q^{61} + 14 q^{64} + 32 q^{69} - 42 q^{71} + 4 q^{74} - 28 q^{76} - 16 q^{79} - 36 q^{81} + 6 q^{84} + 46 q^{86} - 12 q^{89} + 20 q^{91} + 42 q^{94} + 8 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 9x^{6} + 27x^{4} - 31x^{2} + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 5\nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{7} + 7\nu^{5} - 13\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} - 7\nu^{4} + 14\nu^{2} - 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} - 7\nu^{5} + 14\nu^{3} - 8\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{7} + 8\nu^{5} - 19\nu^{3} + 12\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{4} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 5\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 6\beta_{6} + 5\beta_{4} + 11\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{5} + 7\beta_{3} + 21\beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7\beta_{7} + 29\beta_{6} + 21\beta_{4} + 43\beta_1 \) 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
−2.02368
−1.65458
−1.23399
−0.802699
0.802699
1.23399
1.65458
2.02368
−2.02368 2.62059 2.09529 0 −5.30325 0.965823 −0.192845 3.86752 0
1.2 −1.65458 −1.97479 0.737640 0 3.26745 2.24307 2.08868 0.899788 0
1.3 −1.23399 −0.363982 −0.477260 0 0.449152 −2.58558 3.05692 −2.86752 0
1.4 −0.802699 1.76074 −1.35567 0 −1.41335 −0.592103 2.69360 0.100212 0
1.5 0.802699 −1.76074 −1.35567 0 −1.41335 0.592103 −2.69360 0.100212 0
1.6 1.23399 0.363982 −0.477260 0 0.449152 2.58558 −3.05692 −2.86752 0
1.7 1.65458 1.97479 0.737640 0 3.26745 −2.24307 −2.08868 0.899788 0
1.8 2.02368 −2.62059 2.09529 0 −5.30325 −0.965823 0.192845 3.86752 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.bk 8
5.b even 2 1 inner 3025.2.a.bk 8
5.c odd 4 2 605.2.b.f 8
11.b odd 2 1 3025.2.a.bl 8
11.d odd 10 2 275.2.h.d 16
55.d odd 2 1 3025.2.a.bl 8
55.e even 4 2 605.2.b.g 8
55.h odd 10 2 275.2.h.d 16
55.k odd 20 4 605.2.j.d 16
55.k odd 20 4 605.2.j.g 16
55.l even 20 4 55.2.j.a 16
55.l even 20 4 605.2.j.h 16
165.u odd 20 4 495.2.ba.a 16
220.w odd 20 4 880.2.cd.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.j.a 16 55.l even 20 4
275.2.h.d 16 11.d odd 10 2
275.2.h.d 16 55.h odd 10 2
495.2.ba.a 16 165.u odd 20 4
605.2.b.f 8 5.c odd 4 2
605.2.b.g 8 55.e even 4 2
605.2.j.d 16 55.k odd 20 4
605.2.j.g 16 55.k odd 20 4
605.2.j.h 16 55.l even 20 4
880.2.cd.c 16 220.w odd 20 4
3025.2.a.bk 8 1.a even 1 1 trivial
3025.2.a.bk 8 5.b even 2 1 inner
3025.2.a.bl 8 11.b odd 2 1
3025.2.a.bl 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3025))\):

\( T_{2}^{8} - 9T_{2}^{6} + 27T_{2}^{4} - 31T_{2}^{2} + 11 \) Copy content Toggle raw display
\( T_{3}^{8} - 14T_{3}^{6} + 62T_{3}^{4} - 91T_{3}^{2} + 11 \) Copy content Toggle raw display
\( T_{19}^{4} + 6T_{19}^{3} - 4T_{19}^{2} - 39T_{19} + 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 9 T^{6} + 27 T^{4} - 31 T^{2} + \cdots + 11 \) Copy content Toggle raw display
$3$ \( T^{8} - 14 T^{6} + 62 T^{4} - 91 T^{2} + \cdots + 11 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 13 T^{6} + 49 T^{4} - 47 T^{2} + \cdots + 11 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 45 T^{6} + 675 T^{4} + \cdots + 6875 \) Copy content Toggle raw display
$17$ \( T^{8} - 81 T^{6} + 1842 T^{4} + \cdots + 40931 \) Copy content Toggle raw display
$19$ \( (T^{4} + 6 T^{3} - 4 T^{2} - 39 T + 11)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 99 T^{6} + 2232 T^{4} + \cdots + 26411 \) Copy content Toggle raw display
$29$ \( (T^{4} + 12 T^{3} + 48 T^{2} + 67 T + 11)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 7 T^{3} + 5 T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 101 T^{6} + 2822 T^{4} + \cdots + 3971 \) Copy content Toggle raw display
$41$ \( (T^{4} + 17 T^{3} + 48 T^{2} - 438 T - 1969)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 173 T^{6} + 8319 T^{4} + \cdots + 212531 \) Copy content Toggle raw display
$47$ \( T^{8} - 191 T^{6} + 7351 T^{4} + \cdots + 244211 \) Copy content Toggle raw display
$53$ \( T^{8} - 249 T^{6} + 15237 T^{4} + \cdots + 489731 \) Copy content Toggle raw display
$59$ \( (T^{4} + 3 T^{3} - 75 T^{2} - 29 T - 1)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 10 T^{3} - 61 T^{2} - 10 T + 209)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 149 T^{6} + 5736 T^{4} + \cdots + 18491 \) Copy content Toggle raw display
$71$ \( (T^{4} + 21 T^{3} + 90 T^{2} - 432 T - 2511)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 297 T^{6} + 25504 T^{4} + \cdots + 1279091 \) Copy content Toggle raw display
$79$ \( (T^{4} + 8 T^{3} - 52 T^{2} - 377 T - 319)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 111 T^{6} + 3931 T^{4} + \cdots + 212531 \) Copy content Toggle raw display
$89$ \( (T^{4} + 6 T^{3} - 128 T^{2} - 486 T + 1871)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 162 T^{6} + 5588 T^{4} + \cdots + 161051 \) Copy content Toggle raw display
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