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Properties

Label 1875.2.a.p
Level $1875$
Weight $2$
Character orbit 1875.a
Self dual yes
Analytic conductor $14.972$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.8.5444000000.1
Defining polynomial: \( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + \beta_1) q^{4} - \beta_1 q^{6} + (\beta_{7} - \beta_{5} + \beta_{3} + 1) q^{7} + (\beta_{3} + \beta_{2} + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + \beta_1) q^{4} - \beta_1 q^{6} + (\beta_{7} - \beta_{5} + \beta_{3} + 1) q^{7} + (\beta_{3} + \beta_{2} + 1) q^{8} + q^{9} + (\beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_1) q^{11} + ( - \beta_{2} - \beta_1) q^{12} + (2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{13} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_1 - 1) q^{14} + (\beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{16} + ( - \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{17} + \beta_1 q^{18} + (2 \beta_{7} + \beta_{6} - \beta_{5} - 1) q^{19} + ( - \beta_{7} + \beta_{5} - \beta_{3} - 1) q^{21} + ( - \beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} + 1) q^{22} + (2 \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_1 + 2) q^{23} + ( - \beta_{3} - \beta_{2} - 1) q^{24} + (2 \beta_{6} + \beta_{5} + 4 \beta_{4} + 3 \beta_1 + 1) q^{26} - q^{27} + ( - 2 \beta_{7} - \beta_{6} - 3 \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{28} + (2 \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2}) q^{29} + ( - \beta_{6} + \beta_{5} + 2 \beta_1 - 4) q^{31} + (\beta_{5} + 3 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{32} + ( - \beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} - \beta_1) q^{33} + ( - 2 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} - 5 \beta_{4} - 2 \beta_{2} - 4) q^{34} + (\beta_{2} + \beta_1) q^{36} + ( - \beta_{7} - \beta_{5} - 3 \beta_{3} + \beta_{2} + 5) q^{37} + (\beta_{7} + 2 \beta_{6} - \beta_{5} - 3 \beta_{4} - 2 \beta_{2} - 2 \beta_1 - 2) q^{38} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 2) q^{39} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{41} + (\beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1 + 1) q^{42} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{43} + ( - 2 \beta_{7} + \beta_{5} - 2 \beta_{4} + 2 \beta_{2} + 2 \beta_1) q^{44} + (2 \beta_{7} + 5 \beta_{6} + \beta_{5} + 3 \beta_{4} - \beta_{2} + 3 \beta_1) q^{46} + (3 \beta_{7} + 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{2} + 3 \beta_1 + 2) q^{47} + ( - \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{48} + ( - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} - 3 \beta_{2} - \beta_1) q^{49} + (\beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{51} + (2 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 6 \beta_{4} + 2 \beta_{3} + \beta_{2} + 4 \beta_1 + 2) q^{52} + (\beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} + 5) q^{53} - \beta_1 q^{54} + ( - \beta_{7} - 3 \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{56} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} + 1) q^{57} + (2 \beta_{7} + \beta_{6} - 3 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{58} + (2 \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1 + 2) q^{59} + ( - 2 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + \cdots - 1) q^{61}+ \cdots + (\beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 8 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 8 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9} + 2 q^{11} - 4 q^{12} + 16 q^{13} + 6 q^{14} + 16 q^{17} + 4 q^{18} - 14 q^{19} - 8 q^{21} + 12 q^{22} + 14 q^{23} - 12 q^{24} + 6 q^{26} - 8 q^{27} + 16 q^{28} + 2 q^{29} - 22 q^{31} - 2 q^{32} - 2 q^{33} - 12 q^{34} + 4 q^{36} + 28 q^{37} - 16 q^{38} - 16 q^{39} + 8 q^{41} - 6 q^{42} + 20 q^{43} + 22 q^{44} - 2 q^{46} + 10 q^{47} - 16 q^{51} + 16 q^{52} + 44 q^{53} - 4 q^{54} + 30 q^{56} + 14 q^{57} + 8 q^{58} + 14 q^{59} - 20 q^{61} + 16 q^{62} + 8 q^{63} + 6 q^{64} - 12 q^{66} + 16 q^{67} - 2 q^{68} - 14 q^{69} + 16 q^{71} + 12 q^{72} + 24 q^{73} + 26 q^{74} - 16 q^{76} + 46 q^{77} - 6 q^{78} - 30 q^{79} + 8 q^{81} + 16 q^{82} + 12 q^{83} - 16 q^{84} + 32 q^{86} - 2 q^{87} + 32 q^{88} + 16 q^{89} - 12 q^{91} - 2 q^{92} + 22 q^{93} + 14 q^{94} + 2 q^{96} + 16 q^{97} + 4 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 4\nu + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 3\nu^{4} - \nu^{3} + 7\nu^{2} - 3\nu - 1 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - 3\nu^{5} - 3\nu^{4} + 11\nu^{3} + 3\nu^{2} - 9\nu - 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - 4\nu^{6} - \nu^{5} + 16\nu^{4} - 5\nu^{3} - 16\nu^{2} + 5\nu + 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 2\beta_{3} + 5\beta_{2} + 7\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 3\beta_{4} + 7\beta_{3} + 9\beta_{2} + 21\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} + 3\beta_{5} + 12\beta_{4} + 16\beta_{3} + 28\beta_{2} + 46\beta _1 + 33 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} + 4\beta_{6} + 13\beta_{5} + 35\beta_{4} + 44\beta_{3} + 62\beta_{2} + 124\beta _1 + 64 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−1.53767
−1.35083
−0.536547
−0.0898194
1.08982
1.53655
2.35083
2.53767
−1.53767 −1.00000 0.364440 0 1.53767 1.68601 2.51496 1.00000 0
1.2 −1.35083 −1.00000 −0.175259 0 1.35083 1.59580 2.93840 1.00000 0
1.3 −0.536547 −1.00000 −1.71212 0 0.536547 −2.57318 1.99173 1.00000 0
1.4 −0.0898194 −1.00000 −1.99193 0 0.0898194 4.36070 0.358553 1.00000 0
1.5 1.08982 −1.00000 −0.812294 0 −1.08982 −3.08724 −3.06489 1.00000 0
1.6 1.53655 −1.00000 0.360976 0 −1.53655 1.49550 −2.51844 1.00000 0
1.7 2.35083 −1.00000 3.52640 0 −2.35083 3.48189 3.58831 1.00000 0
1.8 2.53767 −1.00000 4.43979 0 −2.53767 1.04054 6.19138 1.00000 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
\(3\) \(1\)
\(5\) \(-1\)

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.p 8
3.b odd 2 1 5625.2.a.t 8
5.b even 2 1 1875.2.a.m 8
5.c odd 4 2 1875.2.b.h 16
15.d odd 2 1 5625.2.a.bd 8
25.d even 5 2 375.2.g.d 16
25.e even 10 2 375.2.g.e 16
25.f odd 20 2 75.2.i.a ✓ 16
25.f odd 20 2 375.2.i.c 16
75.l even 20 2 225.2.m.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.i.a ✓ 16 25.f odd 20 2
225.2.m.b 16 75.l even 20 2
375.2.g.d 16 25.d even 5 2
375.2.g.e 16 25.e even 10 2
375.2.i.c 16 25.f odd 20 2
1875.2.a.m 8 5.b even 2 1
1875.2.a.p 8 1.a even 1 1 trivial
1875.2.b.h 16 5.c odd 4 2
5625.2.a.t 8 3.b odd 2 1
5625.2.a.bd 8 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 4T_{2}^{7} - 2T_{2}^{6} + 20T_{2}^{5} - 4T_{2}^{4} - 30T_{2}^{3} + 7T_{2}^{2} + 12T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 4 T^{7} - 2 T^{6} + 20 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 8 T^{7} + 4 T^{6} + 108 T^{5} + \cdots + 505 \) Copy content Toggle raw display
$11$ \( T^{8} - 2 T^{7} - 43 T^{6} + \cdots + 5281 \) Copy content Toggle raw display
$13$ \( T^{8} - 16 T^{7} + 73 T^{6} + \cdots + 281 \) Copy content Toggle raw display
$17$ \( T^{8} - 16 T^{7} + 42 T^{6} + \cdots - 7339 \) Copy content Toggle raw display
$19$ \( T^{8} + 14 T^{7} + 11 T^{6} + \cdots + 2525 \) Copy content Toggle raw display
$23$ \( T^{8} - 14 T^{7} + 11 T^{6} + \cdots - 2095 \) Copy content Toggle raw display
$29$ \( T^{8} - 2 T^{7} - 106 T^{6} + \cdots - 395 \) Copy content Toggle raw display
$31$ \( T^{8} + 22 T^{7} + 169 T^{6} + \cdots + 125 \) Copy content Toggle raw display
$37$ \( T^{8} - 28 T^{7} + 204 T^{6} + \cdots + 93025 \) Copy content Toggle raw display
$41$ \( T^{8} - 8 T^{7} - 56 T^{6} + \cdots + 4705 \) Copy content Toggle raw display
$43$ \( T^{8} - 20 T^{7} + 6 T^{6} + \cdots + 22961 \) Copy content Toggle raw display
$47$ \( T^{8} - 10 T^{7} - 186 T^{6} + \cdots - 6057019 \) Copy content Toggle raw display
$53$ \( T^{8} - 44 T^{7} + 746 T^{6} + \cdots - 200995 \) Copy content Toggle raw display
$59$ \( T^{8} - 14 T^{7} - 29 T^{6} + \cdots - 3595 \) Copy content Toggle raw display
$61$ \( T^{8} + 20 T^{7} - 136 T^{6} + \cdots + 16604261 \) Copy content Toggle raw display
$67$ \( T^{8} - 16 T^{7} - 138 T^{6} + \cdots - 3739 \) Copy content Toggle raw display
$71$ \( T^{8} - 16 T^{7} - 78 T^{6} + \cdots - 159779 \) Copy content Toggle raw display
$73$ \( T^{8} - 24 T^{7} - 34 T^{6} + \cdots - 870295 \) Copy content Toggle raw display
$79$ \( T^{8} + 30 T^{7} + 145 T^{6} + \cdots - 1984975 \) Copy content Toggle raw display
$83$ \( T^{8} - 12 T^{7} - 188 T^{6} + \cdots + 48541 \) Copy content Toggle raw display
$89$ \( T^{8} - 16 T^{7} - 39 T^{6} + 1454 T^{5} + \cdots + 5 \) Copy content Toggle raw display
$97$ \( T^{8} - 16 T^{7} - 108 T^{6} + \cdots - 14719 \) Copy content Toggle raw display
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