Properties

Label 1875.2.a.f
Level $1875$
Weight $2$
Character orbit 1875.a
Self dual yes
Analytic conductor $14.972$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of \(\nu = \zeta_{15} + \zeta_{15}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) 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
1.33826
−0.209057
−1.95630
1.82709
−1.82709 1.00000 1.33826 0 −1.82709 1.44512 1.20906 1.00000 0
1.2 −1.33826 1.00000 −0.209057 0 −1.33826 −1.27977 2.95630 1.00000 0
1.3 0.209057 1.00000 −1.95630 0 0.209057 −0.591023 −0.827091 1.00000 0
1.4 1.95630 1.00000 1.82709 0 1.95630 −4.57433 −0.338261 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.f 4
3.b odd 2 1 5625.2.a.m 4
5.b even 2 1 1875.2.a.g yes 4
5.c odd 4 2 1875.2.b.d 8
15.d odd 2 1 5625.2.a.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.f 4 1.a even 1 1 trivial
1875.2.a.g yes 4 5.b even 2 1
1875.2.b.d 8 5.c odd 4 2
5625.2.a.j 4 15.d odd 2 1
5625.2.a.m 4 3.b odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} - 4 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 5 T^{3} - 10 T - 5 \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + 6 T^{2} - 9 T - 9 \) Copy content Toggle raw display
$13$ \( T^{4} + 7 T^{3} - 6 T^{2} - 92 T - 89 \) Copy content Toggle raw display
$17$ \( T^{4} - 7 T^{3} - 16 T^{2} + 202 T - 359 \) Copy content Toggle raw display
$19$ \( T^{4} + 9 T^{3} + 6 T^{2} - 36 T - 9 \) Copy content Toggle raw display
$23$ \( T^{4} + 10 T^{3} + 20 T^{2} - 10 T - 5 \) Copy content Toggle raw display
$29$ \( T^{4} + 28 T^{3} + 284 T^{2} + \cdots + 1801 \) Copy content Toggle raw display
$31$ \( T^{4} + 10 T^{3} - 125 T - 125 \) Copy content Toggle raw display
$37$ \( T^{4} - 10 T^{3} - 90 T^{2} + \cdots - 1475 \) Copy content Toggle raw display
$41$ \( T^{4} - 70 T^{2} - 135 T + 145 \) Copy content Toggle raw display
$43$ \( T^{4} + T^{3} - 79 T^{2} + 341 T - 419 \) Copy content Toggle raw display
$47$ \( T^{4} - 23 T^{3} + 89 T^{2} + \cdots - 6089 \) Copy content Toggle raw display
$53$ \( T^{4} - 145 T^{2} + 270 T + 2995 \) Copy content Toggle raw display
$59$ \( T^{4} - 4 T^{3} - 154 T^{2} + \cdots + 1531 \) Copy content Toggle raw display
$61$ \( T^{4} + 43 T^{3} + 669 T^{2} + \cdots + 10261 \) Copy content Toggle raw display
$67$ \( T^{4} + 8 T^{3} - 61 T^{2} - 458 T + 151 \) Copy content Toggle raw display
$71$ \( T^{4} + 27 T^{3} + 134 T^{2} + \cdots + 271 \) Copy content Toggle raw display
$73$ \( T^{4} + 15 T^{3} - 60 T^{2} + \cdots + 2745 \) Copy content Toggle raw display
$79$ \( T^{4} - 10 T^{3} - 105 T^{2} + \cdots - 3155 \) Copy content Toggle raw display
$83$ \( T^{4} + 3 T^{3} - 246 T^{2} + \cdots + 13491 \) Copy content Toggle raw display
$89$ \( T^{4} + 9 T^{3} - 4 T^{2} - 96 T + 61 \) Copy content Toggle raw display
$97$ \( T^{4} + 13 T^{3} - 216 T^{2} + \cdots + 14701 \) Copy content Toggle raw display
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