Properties

Label 1875.2.a.k
Level $1875$
Weight $2$
Character orbit 1875.a
Self dual yes
Analytic conductor $14.972$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.9719503790\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.44400625.1
Defining polynomial: \( x^{6} - 11x^{4} - x^{3} + 29x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{4} - \beta_{3} + 1) q^{4} + \beta_1 q^{6} + ( - \beta_1 + 1) q^{7} + ( - \beta_{3} + \beta_{2} + 2 \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{4} - \beta_{3} + 1) q^{4} + \beta_1 q^{6} + ( - \beta_1 + 1) q^{7} + ( - \beta_{3} + \beta_{2} + 2 \beta_1) q^{8} + q^{9} + ( - \beta_{5} + \beta_1) q^{11} + (\beta_{4} - \beta_{3} + 1) q^{12} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{13} + ( - \beta_{4} + \beta_{3} + \beta_1 - 3) q^{14} + (\beta_{5} - 3 \beta_{3} + \beta_{2} + \beta_1 + 2) q^{16} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 3) q^{17} + \beta_1 q^{18} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} + 3) q^{19} + ( - \beta_1 + 1) q^{21} + (\beta_{4} - \beta_{3} - 2 \beta_{2} + 2) q^{22} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 2) q^{23} + ( - \beta_{3} + \beta_{2} + 2 \beta_1) q^{24} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 4) q^{26} + q^{27} + (\beta_{4} - \beta_{2} - 4 \beta_1 + 1) q^{28} + (\beta_{5} + \beta_{2} + \beta_1) q^{29} + (\beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{31} + (\beta_{5} + \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{32} + ( - \beta_{5} + \beta_1) q^{33} + ( - \beta_{5} + 2 \beta_{3} + \beta_{2} + 4 \beta_1 + 3) q^{34} + (\beta_{4} - \beta_{3} + 1) q^{36} + (\beta_{5} + \beta_{2} - \beta_1 + 4) q^{37} + (\beta_{5} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{38} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{39} + ( - \beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 - 1) q^{41} + ( - \beta_{4} + \beta_{3} + \beta_1 - 3) q^{42} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{43} + (5 \beta_{3} + \beta_{2} + 3 \beta_1 + 4) q^{44} + (\beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 2) q^{46} + ( - \beta_{4} + 3 \beta_{3} + 2 \beta_1 + 4) q^{47} + (\beta_{5} - 3 \beta_{3} + \beta_{2} + \beta_1 + 2) q^{48} + (\beta_{4} - \beta_{3} - 2 \beta_1 - 3) q^{49} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 3) q^{51} + (\beta_{5} + \beta_{4} + 5 \beta_{3} - \beta_{2} - 5 \beta_1 + 5) q^{52} + ( - \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 2) q^{53} + \beta_1 q^{54} + ( - \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + \beta_1 - 4) q^{56} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} + 3) q^{57} + (\beta_{5} + \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + 2) q^{58} + (2 \beta_{5} + 3 \beta_{3} + \beta_{2} - \beta_1 + 4) q^{59} + (\beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_1 + 3) q^{61} + ( - 2 \beta_{5} - 2 \beta_{4} + 7 \beta_{3} + \beta_{2} - 1) q^{62} + ( - \beta_1 + 1) q^{63} + (\beta_{5} + \beta_{4} - 5 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 6) q^{64} + (\beta_{4} - \beta_{3} - 2 \beta_{2} + 2) q^{66} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{67} + ( - \beta_{5} + 2 \beta_{4} - 9 \beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{68} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 2) q^{69} + ( - \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + \beta_{2} - \beta_1 - 3) q^{71} + ( - \beta_{3} + \beta_{2} + 2 \beta_1) q^{72} + (\beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2) q^{73} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 4) q^{74} + ( - 2 \beta_{5} + \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 4) q^{76} + ( - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{77} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 4) q^{78} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{79} + q^{81} + ( - 3 \beta_{5} - \beta_{4} + 11 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{82} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{83} + (\beta_{4} - \beta_{2} - 4 \beta_1 + 1) q^{84} + (2 \beta_{5} - \beta_{4} - 4 \beta_{3} - \beta_{2} - 2 \beta_1 - 7) q^{86} + (\beta_{5} + \beta_{2} + \beta_1) q^{87} + (\beta_{5} + \beta_{4} - 4 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{88} + (2 \beta_{4} - 8 \beta_{3} + 3 \beta_{2} - \beta_1 - 5) q^{89} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{91} + ( - \beta_{5} + \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 11) q^{92} + (\beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{93} + (2 \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 + 6) q^{94} + (\beta_{5} + \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{96} + (2 \beta_{5} - \beta_{4} - 2 \beta_{2} - 3 \beta_1) q^{97} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} - 6) q^{98} + ( - \beta_{5} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 10 q^{4} + 6 q^{7} + 3 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} + 10 q^{4} + 6 q^{7} + 3 q^{8} + 6 q^{9} + 3 q^{11} + 10 q^{12} + 6 q^{13} - 22 q^{14} + 18 q^{16} + 13 q^{17} + 11 q^{19} + 6 q^{21} + 16 q^{22} + 13 q^{23} + 3 q^{24} - 28 q^{26} + 6 q^{27} + 7 q^{28} - 3 q^{29} - 11 q^{31} + 16 q^{32} + 3 q^{33} + 15 q^{34} + 10 q^{36} + 21 q^{37} - 9 q^{38} + 6 q^{39} - q^{41} - 22 q^{42} + 2 q^{43} + 9 q^{44} + 19 q^{46} + 14 q^{47} + 18 q^{48} - 14 q^{49} + 13 q^{51} + 13 q^{52} + 23 q^{53} - 35 q^{56} + 11 q^{57} + 22 q^{58} + 9 q^{59} + 11 q^{61} - 23 q^{62} + 6 q^{63} - 23 q^{64} + 16 q^{66} + 8 q^{67} + 50 q^{68} + 13 q^{69} - 8 q^{71} + 3 q^{72} + 13 q^{73} - 22 q^{74} - 26 q^{76} - 13 q^{77} - 28 q^{78} - 5 q^{79} + 6 q^{81} - 13 q^{82} - 20 q^{83} + 7 q^{84} - 37 q^{86} - 3 q^{87} + 28 q^{88} - 4 q^{89} + 34 q^{91} + 61 q^{92} - 11 q^{93} + 41 q^{94} + 16 q^{96} - 7 q^{97} - 41 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 11x^{4} - x^{3} + 29x^{2} + 3x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{4} - 6\nu^{3} + 5\nu^{2} - 1 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - \nu^{4} - 10\nu^{3} + 5\nu^{2} + 24\nu - 1 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - \nu^{4} - 10\nu^{3} + 9\nu^{2} + 24\nu - 13 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + \nu^{4} - 12\nu^{3} - 7\nu^{2} + 34\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 6\beta_{4} - 9\beta_{3} + \beta_{2} + \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + \beta_{4} - 10\beta_{3} + 11\beta_{2} + 37\beta _1 + 2 \) 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
−2.44028
−2.16056
−0.246759
0.141689
2.01887
2.68704
−2.44028 1.00000 3.95498 0 −2.44028 3.44028 −4.77071 1.00000 0
1.2 −2.16056 1.00000 2.66802 0 −2.16056 3.16056 −1.44329 1.00000 0
1.3 −0.246759 1.00000 −1.93911 0 −0.246759 1.24676 0.972011 1.00000 0
1.4 0.141689 1.00000 −1.97992 0 0.141689 0.858311 −0.563913 1.00000 0
1.5 2.01887 1.00000 2.07584 0 2.01887 −1.01887 0.153106 1.00000 0
1.6 2.68704 1.00000 5.22020 0 2.68704 −1.68704 8.65280 1.00000 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
\(3\) \(-1\)
\(5\) \(1\)

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 1875.2.a.k 6
3.b odd 2 1 5625.2.a.q 6
5.b even 2 1 1875.2.a.j 6
5.c odd 4 2 1875.2.b.f 12
15.d odd 2 1 5625.2.a.p 6
25.d even 5 2 375.2.g.c 12
25.e even 10 2 75.2.g.c 12
25.f odd 20 4 375.2.i.d 24
75.h odd 10 2 225.2.h.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.c 12 25.e even 10 2
225.2.h.d 12 75.h odd 10 2
375.2.g.c 12 25.d even 5 2
375.2.i.d 24 25.f odd 20 4
1875.2.a.j 6 5.b even 2 1
1875.2.a.k 6 1.a even 1 1 trivial
1875.2.b.f 12 5.c odd 4 2
5625.2.a.p 6 15.d odd 2 1
5625.2.a.q 6 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 11T_{2}^{4} - T_{2}^{3} + 29T_{2}^{2} + 3T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 11 T^{4} - T^{3} + 29 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 6 T^{5} + 4 T^{4} + 25 T^{3} + \cdots + 20 \) Copy content Toggle raw display
$11$ \( T^{6} - 3 T^{5} - 24 T^{4} + 66 T^{3} + \cdots + 244 \) Copy content Toggle raw display
$13$ \( T^{6} - 6 T^{5} - 26 T^{4} + 173 T^{3} + \cdots + 101 \) Copy content Toggle raw display
$17$ \( T^{6} - 13 T^{5} + 15 T^{4} + \cdots + 4639 \) Copy content Toggle raw display
$19$ \( T^{6} - 11 T^{5} + 11 T^{4} + \cdots - 380 \) Copy content Toggle raw display
$23$ \( T^{6} - 13 T^{5} + 6 T^{4} + \cdots - 2020 \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{5} - 41 T^{4} + \cdots + 2105 \) Copy content Toggle raw display
$31$ \( T^{6} + 11 T^{5} - 46 T^{4} + \cdots - 2900 \) Copy content Toggle raw display
$37$ \( T^{6} - 21 T^{5} + 139 T^{4} + \cdots - 6025 \) Copy content Toggle raw display
$41$ \( T^{6} + T^{5} - 181 T^{4} + \cdots - 82655 \) Copy content Toggle raw display
$43$ \( T^{6} - 2 T^{5} - 91 T^{4} + \cdots - 6284 \) Copy content Toggle raw display
$47$ \( T^{6} - 14 T^{5} - 4 T^{4} + \cdots + 2284 \) Copy content Toggle raw display
$53$ \( T^{6} - 23 T^{5} + 61 T^{4} + \cdots - 34495 \) Copy content Toggle raw display
$59$ \( T^{6} - 9 T^{5} - 89 T^{4} + \cdots + 3920 \) Copy content Toggle raw display
$61$ \( T^{6} - 11 T^{5} - 139 T^{4} + \cdots + 168269 \) Copy content Toggle raw display
$67$ \( T^{6} - 8 T^{5} - 155 T^{4} + \cdots + 14684 \) Copy content Toggle raw display
$71$ \( T^{6} + 8 T^{5} - 150 T^{4} + \cdots - 196 \) Copy content Toggle raw display
$73$ \( T^{6} - 13 T^{5} - 74 T^{4} + \cdots + 22205 \) Copy content Toggle raw display
$79$ \( T^{6} + 5 T^{5} - 160 T^{4} + \cdots - 8000 \) Copy content Toggle raw display
$83$ \( T^{6} + 20 T^{5} + 26 T^{4} + \cdots + 41036 \) Copy content Toggle raw display
$89$ \( T^{6} + 4 T^{5} - 464 T^{4} + \cdots - 377055 \) Copy content Toggle raw display
$97$ \( T^{6} + 7 T^{5} - 215 T^{4} + \cdots + 2399 \) Copy content Toggle raw display
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