Properties

Label 1875.2.a.h
Level $1875$
Weight $2$
Character orbit 1875.a
Self dual yes
Analytic conductor $14.972$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.5125.1
Defining polynomial: \( x^{4} - 2x^{3} - 6x^{2} + 7x + 11 \) 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + \beta_1 q^{6} + ( - 2 \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 4) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + \beta_1 q^{6} + ( - 2 \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 4) q^{8} + q^{9} + (\beta_{3} - 3 \beta_{2} - 3) q^{11} + (\beta_{2} + \beta_1 + 2) q^{12} + ( - \beta_{3} + 2 \beta_{2} + 1) q^{13} + ( - 2 \beta_{3} + \beta_{2} + 4) q^{14} + (2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 1) q^{16} + ( - \beta_1 + 1) q^{17} + \beta_1 q^{18} + ( - \beta_{3} + 2 \beta_1) q^{19} + ( - 2 \beta_{2} + \beta_1 - 1) q^{21} + ( - 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 1) q^{22} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{23} + (\beta_{3} + \beta_{2} + \beta_1 + 4) q^{24} + (\beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{26} + q^{27} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{28} + ( - \beta_{2} - \beta_1 - 5) q^{29} + (\beta_{3} - 2 \beta_{2} + 5) q^{31} + (2 \beta_{3} + 7 \beta_{2} + 2 \beta_1 + 6) q^{32} + (\beta_{3} - 3 \beta_{2} - 3) q^{33} + ( - \beta_{2} - 4) q^{34} + (\beta_{2} + \beta_1 + 2) q^{36} + (2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{37} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 + 7) q^{38} + ( - \beta_{3} + 2 \beta_{2} + 1) q^{39} + (4 \beta_{2} - 1) q^{41} + ( - 2 \beta_{3} + \beta_{2} + 4) q^{42} + (2 \beta_{3} - \beta_{2} + 4) q^{43} + ( - \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 8) q^{44} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 3) q^{46} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{47} + (2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 1) q^{48} + ( - 4 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{49} + ( - \beta_1 + 1) q^{51} + 3 q^{52} + ( - 2 \beta_{3} - \beta_1 - 1) q^{53} + \beta_1 q^{54} + (\beta_{3} - 3 \beta_{2} + 2 \beta_1 - 1) q^{56} + ( - \beta_{3} + 2 \beta_1) q^{57} + ( - \beta_{3} - \beta_{2} - 6 \beta_1 - 4) q^{58} + ( - 3 \beta_{3} - 3 \beta_{2} - 6) q^{59} + ( - 2 \beta_{3} + 4 \beta_{2} + 1) q^{61} + ( - \beta_{3} + 3 \beta_{2} + 5 \beta_1 + 1) q^{62} + ( - 2 \beta_{2} + \beta_1 - 1) q^{63} + (5 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 8) q^{64} + ( - 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 1) q^{66} + (\beta_{2} - 2 \beta_1 + 2) q^{67} + ( - \beta_{3} - 2 \beta_1 - 2) q^{68} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{69} + (4 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{71} + (\beta_{3} + \beta_{2} + \beta_1 + 4) q^{72} + ( - 2 \beta_{3} - \beta_{2} - 4 \beta_1 + 5) q^{73} + (\beta_{3} + 5 \beta_{2} - 2) q^{74} + ( - \beta_{2} + 5 \beta_1 + 7) q^{76} + ( - \beta_{3} + 6 \beta_{2} - 5 \beta_1 + 10) q^{77} + (\beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{78} + ( - 3 \beta_{3} - \beta_{2} - 3 \beta_1 + 9) q^{79} + q^{81} + (4 \beta_{3} - \beta_1) q^{82} + (4 \beta_{3} + 5) q^{83} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{84} + (\beta_{3} + 6 \beta_{2} + 4 \beta_1 + 2) q^{86} + ( - \beta_{2} - \beta_1 - 5) q^{87} + ( - 11 \beta_{2} - 4 \beta_1 - 11) q^{88} + ( - \beta_{3} + 4 \beta_{2} - 7) q^{89} + ( - 3 \beta_{2} + 3 \beta_1 - 6) q^{91} + ( - \beta_{3} - 2 \beta_1 - 5) q^{92} + (\beta_{3} - 2 \beta_{2} + 5) q^{93} + ( - \beta_{3} - 7 \beta_{2} - 6) q^{94} + (2 \beta_{3} + 7 \beta_{2} + 2 \beta_1 + 6) q^{96} + (4 \beta_{3} - 2 \beta_{2} + 3) q^{97} + ( - 3 \beta_{3} - 13 \beta_{2} + \beta_1 - 8) q^{98} + (\beta_{3} - 3 \beta_{2} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 4 q^{3} + 8 q^{4} + 2 q^{6} + 2 q^{7} + 15 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 4 q^{3} + 8 q^{4} + 2 q^{6} + 2 q^{7} + 15 q^{8} + 4 q^{9} - 7 q^{11} + 8 q^{12} + q^{13} + 16 q^{14} + 4 q^{16} + 2 q^{17} + 2 q^{18} + 5 q^{19} + 2 q^{21} - 6 q^{22} + q^{23} + 15 q^{24} + 3 q^{26} + 4 q^{27} + 9 q^{28} - 20 q^{29} + 23 q^{31} + 12 q^{32} - 7 q^{33} - 14 q^{34} + 8 q^{36} + 2 q^{37} + 35 q^{38} + q^{39} - 12 q^{41} + 16 q^{42} + 16 q^{43} - 29 q^{44} - 17 q^{46} + 2 q^{47} + 4 q^{48} + 8 q^{49} + 2 q^{51} + 12 q^{52} - 4 q^{53} + 2 q^{54} + 5 q^{56} + 5 q^{57} - 25 q^{58} - 15 q^{59} - 2 q^{61} + 9 q^{62} + 2 q^{63} + 23 q^{64} - 6 q^{66} + 2 q^{67} - 11 q^{68} + q^{69} - 2 q^{71} + 15 q^{72} + 16 q^{73} - 19 q^{74} + 40 q^{76} + 19 q^{77} + 3 q^{78} + 35 q^{79} + 4 q^{81} - 6 q^{82} + 16 q^{83} + 9 q^{84} + 3 q^{86} - 20 q^{87} - 30 q^{88} - 35 q^{89} - 12 q^{91} - 23 q^{92} + 23 q^{93} - 9 q^{94} + 12 q^{96} + 12 q^{97} - q^{98} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 6x^{2} + 7x + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 4 \) 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
−1.70636
−1.12233
2.12233
2.70636
−1.70636 1.00000 0.911672 0 −1.70636 −3.94243 1.85708 1.00000 0
1.2 −1.12233 1.00000 −0.740367 0 −1.12233 1.11373 3.07561 1.00000 0
1.3 2.12233 1.00000 2.50430 0 2.12233 4.35840 1.07029 1.00000 0
1.4 2.70636 1.00000 5.32440 0 2.70636 0.470294 8.99702 1.00000 0
\(n\): e.g. 2-40 or 990-1000
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.h 4
3.b odd 2 1 5625.2.a.i 4
5.b even 2 1 1875.2.a.e 4
5.c odd 4 2 1875.2.b.c 8
15.d odd 2 1 5625.2.a.n 4
25.d even 5 2 75.2.g.b 8
25.e even 10 2 375.2.g.b 8
25.f odd 20 4 375.2.i.b 16
75.j odd 10 2 225.2.h.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.b 8 25.d even 5 2
225.2.h.c 8 75.j odd 10 2
375.2.g.b 8 25.e even 10 2
375.2.i.b 16 25.f odd 20 4
1875.2.a.e 4 5.b even 2 1
1875.2.a.h 4 1.a even 1 1 trivial
1875.2.b.c 8 5.c odd 4 2
5625.2.a.i 4 3.b odd 2 1
5625.2.a.n 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2T_{2}^{3} - 6T_{2}^{2} + 7T_{2} + 11 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} - 6 T^{2} + 7 T + 11 \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} - 16 T^{2} + 27 T - 9 \) Copy content Toggle raw display
$11$ \( T^{4} + 7 T^{3} - 6 T^{2} - 92 T - 109 \) Copy content Toggle raw display
$13$ \( T^{4} - T^{3} - 14 T^{2} + 24 T - 9 \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} - 6 T^{2} + 7 T + 11 \) Copy content Toggle raw display
$19$ \( T^{4} - 5 T^{3} - 35 T^{2} + 75 T + 275 \) Copy content Toggle raw display
$23$ \( T^{4} - T^{3} - 24 T^{2} - 46 T - 19 \) Copy content Toggle raw display
$29$ \( T^{4} + 20 T^{3} + 140 T^{2} + \cdots + 405 \) Copy content Toggle raw display
$31$ \( T^{4} - 23 T^{3} + 184 T^{2} + \cdots + 711 \) Copy content Toggle raw display
$37$ \( T^{4} - 2 T^{3} - 46 T^{2} + 47 T + 1 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T - 11)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 16 T^{3} + 61 T^{2} + 24 T - 99 \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} - 36 T^{2} + 37 T + 311 \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} - 34 T^{2} - 51 T + 261 \) Copy content Toggle raw display
$59$ \( T^{4} + 15 T^{3} - 45 T^{2} + \cdots - 3645 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} - 56 T^{2} + 78 T - 9 \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} - 31 T^{2} + 12 T + 171 \) Copy content Toggle raw display
$71$ \( T^{4} + 2 T^{3} - 216 T^{2} + \cdots + 911 \) Copy content Toggle raw display
$73$ \( T^{4} - 16 T^{3} - 49 T^{2} + \cdots - 4869 \) Copy content Toggle raw display
$79$ \( T^{4} - 35 T^{3} + 320 T^{2} + \cdots - 9845 \) Copy content Toggle raw display
$83$ \( T^{4} - 16 T^{3} - 54 T^{2} + \cdots - 979 \) Copy content Toggle raw display
$89$ \( T^{4} + 35 T^{3} + 420 T^{2} + \cdots + 3305 \) Copy content Toggle raw display
$97$ \( T^{4} - 12 T^{3} - 86 T^{2} + \cdots + 2101 \) Copy content Toggle raw display
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