Properties

Label 1875.2.a.o
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: 8.8.13366265625.1
Defining polynomial: \( x^{8} - x^{7} - 12x^{6} + 10x^{5} + 41x^{4} - 20x^{3} - 48x^{2} + 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{5} + \beta_{4} + 1) q^{4} + \beta_1 q^{6} + ( - \beta_{4} + \beta_{2} + 1) q^{7} + (\beta_{6} + 3 \beta_{2}) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{5} + \beta_{4} + 1) q^{4} + \beta_1 q^{6} + ( - \beta_{4} + \beta_{2} + 1) q^{7} + (\beta_{6} + 3 \beta_{2}) q^{8} + q^{9} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 3) q^{11} + (\beta_{5} + \beta_{4} + 1) q^{12} + ( - \beta_{5} - \beta_{3} - \beta_1 + 2) q^{13} + ( - 2 \beta_{7} + 2 \beta_{5} + \beta_{4} - 2 \beta_{2} + 1) q^{14} + (2 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1 + 1) q^{16} + (2 \beta_{7} + \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_1 + 1) q^{17} + \beta_1 q^{18} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} + 3) q^{19} + ( - \beta_{4} + \beta_{2} + 1) q^{21} + ( - \beta_{6} + \beta_{4} - 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{22} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} - 3 \beta_{2} + 3 \beta_1 - 1) q^{23} + (\beta_{6} + 3 \beta_{2}) q^{24} + (2 \beta_{7} - 2 \beta_{5} - \beta_{4} + 2 \beta_1 - 3) q^{26} + q^{27} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{2} + 2 \beta_1 - 4) q^{28} + ( - 2 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_1 - 1) q^{29} + ( - \beta_{7} + \beta_{6} - 3 \beta_{5} - 2 \beta_{3} + \beta_1 + 2) q^{31} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{4} + 2 \beta_1 - 3) q^{32} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 3) q^{33} + ( - 4 \beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 3 \beta_{2} - 6) q^{34} + (\beta_{5} + \beta_{4} + 1) q^{36} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} - 1) q^{37} + (2 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 1) q^{38} + ( - \beta_{5} - \beta_{3} - \beta_1 + 2) q^{39} + (2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - 1) q^{41} + ( - 2 \beta_{7} + 2 \beta_{5} + \beta_{4} - 2 \beta_{2} + 1) q^{42} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 2) q^{43} + ( - 2 \beta_{7} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 4) q^{44} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{3} - 4 \beta_{2} - \beta_1 + 6) q^{46} + (\beta_{6} - \beta_{4} - 4 \beta_{3} + \beta_{2} - 3) q^{47} + (2 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1 + 1) q^{48} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 3) q^{49} + (2 \beta_{7} + \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_1 + 1) q^{51} + (2 \beta_{7} - 2 \beta_{6} + \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 2) q^{52} + ( - 4 \beta_{7} - \beta_{6} + 2 \beta_{4} - \beta_{2} - \beta_1 - 2) q^{53} + \beta_1 q^{54} + (2 \beta_{7} + \beta_{6} + 5 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 5 \beta_{2} - 6 \beta_1 + 6) q^{56} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} + 3) q^{57} + (4 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 4 \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{58} + ( - 4 \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{59} + (\beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{61} + (6 \beta_{7} - \beta_{6} + \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{62} + ( - \beta_{4} + \beta_{2} + 1) q^{63} + ( - 2 \beta_{7} + 3 \beta_{5} - 2 \beta_{2} - 4 \beta_1 + 4) q^{64} + ( - \beta_{6} + \beta_{4} - 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{66} + ( - 3 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - 3 \beta_{2} + \beta_1 + 2) q^{67} + ( - 4 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 6 \beta_{2} + \cdots - 5) q^{68}+ \cdots + (2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 8 q^{3} + 9 q^{4} + q^{6} + 12 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + 8 q^{3} + 9 q^{4} + q^{6} + 12 q^{7} + 3 q^{8} + 8 q^{9} + 12 q^{11} + 9 q^{12} + 14 q^{13} + 16 q^{14} + 15 q^{16} - q^{17} + q^{18} + 16 q^{19} + 12 q^{21} + 18 q^{22} - 4 q^{23} + 3 q^{24} - 34 q^{26} + 8 q^{27} - 21 q^{28} + 2 q^{29} + 13 q^{31} - 18 q^{32} + 12 q^{33} - 37 q^{34} + 9 q^{36} - 8 q^{37} - 24 q^{38} + 14 q^{39} - 12 q^{41} + 16 q^{42} + 20 q^{43} + 47 q^{44} + 33 q^{46} - 15 q^{47} + 15 q^{48} + 30 q^{49} - q^{51} - q^{52} - 4 q^{53} + q^{54} + 60 q^{56} + 16 q^{57} + 2 q^{58} + 14 q^{59} + 10 q^{61} + 4 q^{62} + 12 q^{63} + 41 q^{64} + 18 q^{66} + 19 q^{67} - 33 q^{68} - 4 q^{69} + 21 q^{71} + 3 q^{72} - 19 q^{73} - 9 q^{74} - q^{76} - 11 q^{77} - 34 q^{78} + 10 q^{79} + 8 q^{81} + 24 q^{82} - 27 q^{83} - 21 q^{84} + 42 q^{86} + 2 q^{87} + 53 q^{88} - 9 q^{89} - 12 q^{91} - 63 q^{92} + 13 q^{93} + 14 q^{94} - 18 q^{96} + 24 q^{97} - 24 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 12x^{6} + 10x^{5} + 41x^{4} - 20x^{3} - 48x^{2} + 8x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - \nu^{6} - 12\nu^{5} + 10\nu^{4} + 41\nu^{3} - 20\nu^{2} - 40\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} + 10\nu^{5} - 34\nu^{4} - 21\nu^{3} + 94\nu^{2} + 8\nu - 56 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 3\nu^{6} - 10\nu^{5} + 30\nu^{4} + 21\nu^{3} - 62\nu^{2} - 12\nu + 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} + 10\nu^{5} - 30\nu^{4} - 21\nu^{3} + 66\nu^{2} + 12\nu - 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 3\nu^{6} + 36\nu^{5} - 30\nu^{4} - 115\nu^{3} + 60\nu^{2} + 88\nu - 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} + 3\nu^{6} + 21\nu^{5} - 30\nu^{4} - 52\nu^{3} + 61\nu^{2} + 30\nu - 28 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + 3\beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{5} + 7\beta_{4} - 2\beta_{3} - \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{7} + 10\beta_{6} - \beta_{4} + 24\beta_{2} + 22\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{7} + 59\beta_{5} + 46\beta_{4} - 20\beta_{3} - 2\beta_{2} - 14\beta _1 + 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -26\beta_{7} + 79\beta_{6} - \beta_{5} - 16\beta_{4} + 171\beta_{2} + 136\beta _1 - 44 \) 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
−2.69767
−1.52260
−0.895394
−0.770071
0.741379
1.31354
2.23365
2.59716
−2.69767 1.00000 5.27745 0 −2.69767 −3.56649 −8.84149 1.00000 0
1.2 −1.52260 1.00000 0.318310 0 −1.52260 −0.990985 2.56054 1.00000 0
1.3 −0.895394 1.00000 −1.19827 0 −0.895394 5.08992 2.86371 1.00000 0
1.4 −0.770071 1.00000 −1.40699 0 −0.770071 3.98808 2.62363 1.00000 0
1.5 0.741379 1.00000 −1.45036 0 0.741379 1.03586 −2.55802 1.00000 0
1.6 1.31354 1.00000 −0.274605 0 1.31354 4.19091 −2.98779 1.00000 0
1.7 2.23365 1.00000 2.98921 0 2.23365 −1.03143 2.20956 1.00000 0
1.8 2.59716 1.00000 4.74525 0 2.59716 3.28414 7.12986 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.o yes 8
3.b odd 2 1 5625.2.a.u 8
5.b even 2 1 1875.2.a.n 8
5.c odd 4 2 1875.2.b.g 16
15.d odd 2 1 5625.2.a.bc 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.n 8 5.b even 2 1
1875.2.a.o yes 8 1.a even 1 1 trivial
1875.2.b.g 16 5.c odd 4 2
5625.2.a.u 8 3.b odd 2 1
5625.2.a.bc 8 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{7} - 12T_{2}^{6} + 10T_{2}^{5} + 41T_{2}^{4} - 20T_{2}^{3} - 48T_{2}^{2} + 8T_{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} - 12 T^{6} + 10 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 12 T^{7} + 29 T^{6} + \cdots - 1055 \) Copy content Toggle raw display
$11$ \( T^{8} - 12 T^{7} + 2 T^{6} + 455 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{8} - 14 T^{7} + 43 T^{6} + \cdots + 8731 \) Copy content Toggle raw display
$17$ \( T^{8} + T^{7} - 88 T^{6} - 122 T^{5} + \cdots + 8656 \) Copy content Toggle raw display
$19$ \( T^{8} - 16 T^{7} + 51 T^{6} + \cdots - 14975 \) Copy content Toggle raw display
$23$ \( T^{8} + 4 T^{7} - 104 T^{6} + \cdots + 28720 \) Copy content Toggle raw display
$29$ \( T^{8} - 2 T^{7} - 126 T^{6} + \cdots + 57520 \) Copy content Toggle raw display
$31$ \( T^{8} - 13 T^{7} - 96 T^{6} + \cdots + 801025 \) Copy content Toggle raw display
$37$ \( T^{8} + 8 T^{7} - 31 T^{6} - 313 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$41$ \( T^{8} + 12 T^{7} - 6 T^{6} + \cdots - 48080 \) Copy content Toggle raw display
$43$ \( T^{8} - 20 T^{7} + 6 T^{6} + \cdots - 74369 \) Copy content Toggle raw display
$47$ \( T^{8} + 15 T^{7} - 71 T^{6} + \cdots - 255824 \) Copy content Toggle raw display
$53$ \( T^{8} + 4 T^{7} - 159 T^{6} + \cdots + 28720 \) Copy content Toggle raw display
$59$ \( T^{8} - 14 T^{7} - 164 T^{6} + \cdots + 11920 \) Copy content Toggle raw display
$61$ \( T^{8} - 10 T^{7} - 186 T^{6} + \cdots - 1093919 \) Copy content Toggle raw display
$67$ \( T^{8} - 19 T^{7} - 183 T^{6} + \cdots + 20204221 \) Copy content Toggle raw display
$71$ \( T^{8} - 21 T^{7} - 18 T^{6} + \cdots + 67696 \) Copy content Toggle raw display
$73$ \( T^{8} + 19 T^{7} + 21 T^{6} + \cdots - 379655 \) Copy content Toggle raw display
$79$ \( T^{8} - 10 T^{7} - 320 T^{6} + \cdots + 6951025 \) Copy content Toggle raw display
$83$ \( T^{8} + 27 T^{7} + 132 T^{6} + \cdots - 113744 \) Copy content Toggle raw display
$89$ \( T^{8} + 9 T^{7} - 294 T^{6} + \cdots - 12105680 \) Copy content Toggle raw display
$97$ \( T^{8} - 24 T^{7} + 87 T^{6} + \cdots - 401939 \) Copy content Toggle raw display
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