Properties

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

Related objects

Downloads

Learn more

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: 6.6.46840000.1
Defining polynomial: \( x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9 \) 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 2) q^{4} - \beta_1 q^{6} + (\beta_{3} - \beta_{2}) q^{7} + (\beta_{5} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 2) q^{4} - \beta_1 q^{6} + (\beta_{3} - \beta_{2}) q^{7} + (\beta_{5} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{8} + q^{9} + (\beta_{5} + \beta_{3}) q^{11} + ( - \beta_{2} - 2) q^{12} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{13} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{14} + (\beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{16} + ( - \beta_{5} - \beta_{3} + 2 \beta_1) q^{17} + \beta_1 q^{18} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{19} + ( - \beta_{3} + \beta_{2}) q^{21} + (2 \beta_{4} + \beta_{2} + 1) q^{22} + (\beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{23} + ( - \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{24} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 7) q^{26} - q^{27} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 - 6) q^{28} + ( - \beta_{2} + 5) q^{29} + (\beta_{5} - 2 \beta_{4} + \beta_{2}) q^{31} + (\beta_{5} + 2 \beta_{4} + 7 \beta_{3} + 2 \beta_{2} + \beta_1 + 8) q^{32} + ( - \beta_{5} - \beta_{3}) q^{33} + ( - 2 \beta_{4} + \beta_{2} + 7) q^{34} + (\beta_{2} + 2) q^{36} + (\beta_{5} + 2 \beta_{4} + \beta_{2} + 2 \beta_1 - 5) q^{37} + ( - \beta_{5} + \beta_{4} - 7 \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{38} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{39} + ( - \beta_{5} - \beta_{3} - \beta_{2} + 2 \beta_1 + 5) q^{41} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{42} + ( - 2 \beta_{4} - \beta_{3} - 2 \beta_1) q^{43} + (\beta_{5} + 7 \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{44} + ( - \beta_{5} + 2 \beta_{4} - 7 \beta_{3} + 2 \beta_1 - 6) q^{46} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2) q^{47} + ( - \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{48} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} + 2 \beta_1) q^{49} + (\beta_{5} + \beta_{3} - 2 \beta_1) q^{51} + ( - 2 \beta_{4} - \beta_{3} + 4 \beta_1 - 6) q^{52} + ( - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 6) q^{53} - \beta_1 q^{54} + ( - \beta_{5} - \beta_{4} - 7 \beta_{3} - \beta_{2} - 5 \beta_1 - 7) q^{56} + (2 \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{57} + ( - \beta_{5} - \beta_{3} - \beta_{2} + 4 \beta_1 - 1) q^{58} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{59} + ( - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 4) q^{61} + ( - \beta_{5} + \beta_{4} - 7 \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{62} + (\beta_{3} - \beta_{2}) q^{63} + (2 \beta_{5} + 4 \beta_{4} + 8 \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 3) q^{64} + ( - 2 \beta_{4} - \beta_{2} - 1) q^{66} + (\beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{67} + (\beta_{5} - 5 \beta_{3} + \beta_{2} + 4 \beta_1 + 1) q^{68} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{69} + (2 \beta_{5} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{71} + (\beta_{5} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{72} + ( - 2 \beta_{5} - \beta_{3} - \beta_{2} + 2 \beta_1 - 6) q^{73} + (3 \beta_{5} + \beta_{4} + 9 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 10) q^{74} + (2 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} + 2 \beta_1 + 6) q^{76} + ( - 6 \beta_{3} - 2 \beta_1) q^{77} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 7) q^{78} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 3) q^{79} + q^{81} + ( - \beta_{5} - 2 \beta_{4} - \beta_{3} + 4 \beta_1 + 6) q^{82} + (\beta_{5} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{83} + (2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 + 6) q^{84} + ( - 2 \beta_{5} - \beta_{4} - 8 \beta_{3} - 2 \beta_{2} - 8) q^{86} + (\beta_{2} - 5) q^{87} + (\beta_{5} + 4 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 8) q^{88} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 4) q^{89} + ( - \beta_{5} + 4 \beta_{4} - \beta_{3} - 2 \beta_{2} + 3) q^{91} + ( - 4 \beta_{4} + 6 \beta_{3} - \beta_{2} - 2 \beta_1 + 5) q^{92} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2}) q^{93} + ( - 4 \beta_{4} + 6 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 4) q^{94} + ( - \beta_{5} - 2 \beta_{4} - 7 \beta_{3} - 2 \beta_{2} - \beta_1 - 8) q^{96} + (3 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 1) q^{97} + (\beta_{5} - \beta_{4} + 7 \beta_{3} - \beta_1 + 6) q^{98} + (\beta_{5} + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - 6 q^{3} + 11 q^{4} - q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} - 6 q^{3} + 11 q^{4} - q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9} - 11 q^{12} - 4 q^{14} + 17 q^{16} + 2 q^{17} + q^{18} - 2 q^{19} + 2 q^{21} + 9 q^{22} - q^{23} - 6 q^{24} + 37 q^{26} - 6 q^{27} - 44 q^{28} + 31 q^{29} - 2 q^{31} + 33 q^{32} + 37 q^{34} + 11 q^{36} - 22 q^{37} + 27 q^{38} + 33 q^{41} + 4 q^{42} - 3 q^{43} - 11 q^{44} - 12 q^{46} - 6 q^{47} - 17 q^{48} + 4 q^{49} - 2 q^{51} - 33 q^{52} + 14 q^{53} - q^{54} - 30 q^{56} + 2 q^{57} - q^{58} - 8 q^{59} + 34 q^{61} + 31 q^{62} - 2 q^{63} + 12 q^{64} - 9 q^{66} + 2 q^{67} + 27 q^{68} + q^{69} - 3 q^{71} + 6 q^{72} - 36 q^{73} + 36 q^{74} + 27 q^{76} + 16 q^{77} - 37 q^{78} + 25 q^{79} + 6 q^{81} + 36 q^{82} - 12 q^{83} + 44 q^{84} - 30 q^{86} - 31 q^{87} + 56 q^{88} + 18 q^{89} + 28 q^{91} + 3 q^{92} + 2 q^{93} - 50 q^{94} - 33 q^{96} + 7 q^{97} + 15 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - \nu^{4} - 8\nu^{3} + 5\nu^{2} + 13\nu - 6 ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + \nu^{4} + 14\nu^{3} - 11\nu^{2} - 43\nu + 24 ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 2\beta_{4} + \beta_{3} + 7\beta_{2} + 2\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{5} + 2\beta_{4} + 15\beta_{3} + 10\beta_{2} + 29\beta _1 + 16 \) 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
−2.38719
−2.02791
−0.364088
0.858825
2.13324
2.78712
−2.38719 −1.00000 3.69868 0 2.38719 −3.31671 −4.05506 1.00000 0
1.2 −2.02791 −1.00000 2.11242 0 2.02791 0.505614 −0.227977 1.00000 0
1.3 −0.364088 −1.00000 −1.86744 0 0.364088 2.24941 1.40809 1.00000 0
1.4 0.858825 −1.00000 −1.26242 0 −0.858825 3.88045 −2.80185 1.00000 0
1.5 2.13324 −1.00000 2.55073 0 −2.13324 −2.16876 1.17484 1.00000 0
1.6 2.78712 −1.00000 5.76803 0 −2.78712 −3.15000 10.5020 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.l yes 6
3.b odd 2 1 5625.2.a.o 6
5.b even 2 1 1875.2.a.i 6
5.c odd 4 2 1875.2.b.e 12
15.d odd 2 1 5625.2.a.r 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.i 6 5.b even 2 1
1875.2.a.l yes 6 1.a even 1 1 trivial
1875.2.b.e 12 5.c odd 4 2
5625.2.a.o 6 3.b odd 2 1
5625.2.a.r 6 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} - 11T_{2}^{4} + 8T_{2}^{3} + 31T_{2}^{2} - 15T_{2} - 9 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} - 11 T^{4} + 8 T^{3} + \cdots - 9 \) Copy content Toggle raw display
$3$ \( (T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 2 T^{5} - 21 T^{4} - 42 T^{3} + \cdots - 100 \) Copy content Toggle raw display
$11$ \( T^{6} - 29 T^{4} - 8 T^{3} + 184 T^{2} + \cdots - 144 \) Copy content Toggle raw display
$13$ \( T^{6} - 56 T^{4} + 74 T^{3} + \cdots + 349 \) Copy content Toggle raw display
$17$ \( T^{6} - 2 T^{5} - 55 T^{4} + 80 T^{3} + \cdots - 576 \) Copy content Toggle raw display
$19$ \( T^{6} + 2 T^{5} - 74 T^{4} + \cdots - 5725 \) Copy content Toggle raw display
$23$ \( T^{6} + T^{5} - 69 T^{4} - 144 T^{3} + \cdots - 720 \) Copy content Toggle raw display
$29$ \( T^{6} - 31 T^{5} + 379 T^{4} + \cdots + 6480 \) Copy content Toggle raw display
$31$ \( T^{6} + 2 T^{5} - 86 T^{4} + \cdots + 3155 \) Copy content Toggle raw display
$37$ \( T^{6} + 22 T^{5} + 59 T^{4} + \cdots - 46100 \) Copy content Toggle raw display
$41$ \( T^{6} - 33 T^{5} + 399 T^{4} + \cdots + 720 \) Copy content Toggle raw display
$43$ \( T^{6} + 3 T^{5} - 76 T^{4} + \cdots - 1289 \) Copy content Toggle raw display
$47$ \( T^{6} + 6 T^{5} - 184 T^{4} + \cdots + 80064 \) Copy content Toggle raw display
$53$ \( T^{6} - 14 T^{5} - 164 T^{4} + \cdots + 14400 \) Copy content Toggle raw display
$59$ \( T^{6} + 8 T^{5} - 69 T^{4} + \cdots - 2880 \) Copy content Toggle raw display
$61$ \( T^{6} - 34 T^{5} + 406 T^{4} + \cdots - 72001 \) Copy content Toggle raw display
$67$ \( T^{6} - 2 T^{5} - 110 T^{4} + 540 T^{3} + \cdots + 59 \) Copy content Toggle raw display
$71$ \( T^{6} + 3 T^{5} - 225 T^{4} + \cdots - 12816 \) Copy content Toggle raw display
$73$ \( T^{6} + 36 T^{5} + 431 T^{4} + \cdots + 20380 \) Copy content Toggle raw display
$79$ \( T^{6} - 25 T^{5} + 150 T^{4} + \cdots + 2725 \) Copy content Toggle raw display
$83$ \( T^{6} + 12 T^{5} - 129 T^{4} + \cdots - 23616 \) Copy content Toggle raw display
$89$ \( T^{6} - 18 T^{5} - 219 T^{4} + \cdots - 42480 \) Copy content Toggle raw display
$97$ \( T^{6} - 7 T^{5} - 310 T^{4} + \cdots - 32291 \) Copy content Toggle raw display
show more
show less