Properties

Label 1875.2.b.a
Level $1875$
Weight $2$
Character orbit 1875.b
Analytic conductor $14.972$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.9719503790\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 1) q^{4} - \beta_{2} q^{6} + 2 \beta_{3} q^{7} + (\beta_{3} + 2 \beta_1) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 1) q^{4} - \beta_{2} q^{6} + 2 \beta_{3} q^{7} + (\beta_{3} + 2 \beta_1) q^{8} - q^{9} - 3 q^{11} + (\beta_{3} + \beta_1) q^{12} - \beta_{3} q^{13} - 2 \beta_{2} q^{14} + 3 \beta_{2} q^{16} + (\beta_{3} - 2 \beta_1) q^{17} - \beta_1 q^{18} + ( - 6 \beta_{2} - 3) q^{19} - 2 q^{21} - 3 \beta_1 q^{22} + (7 \beta_{3} + \beta_1) q^{23} + ( - 2 \beta_{2} - 1) q^{24} + \beta_{2} q^{26} - \beta_{3} q^{27} + (2 \beta_{3} + 2 \beta_1) q^{28} + ( - \beta_{2} + 2) q^{29} + ( - 6 \beta_{2} - 1) q^{31} + (5 \beta_{3} + \beta_1) q^{32} - 3 \beta_{3} q^{33} + ( - 3 \beta_{2} + 2) q^{34} + ( - \beta_{2} - 1) q^{36} + 2 \beta_{3} q^{37} + ( - 6 \beta_{3} + 3 \beta_1) q^{38} + q^{39} + ( - \beta_{2} - 11) q^{41} - 2 \beta_1 q^{42} + ( - 9 \beta_{3} - \beta_1) q^{43} + ( - 3 \beta_{2} - 3) q^{44} + ( - 6 \beta_{2} - 1) q^{46} + (8 \beta_{3} + 2 \beta_1) q^{47} + 3 \beta_1 q^{48} + 3 q^{49} + (2 \beta_{2} - 1) q^{51} + ( - \beta_{3} - \beta_1) q^{52} + (8 \beta_{3} - 2 \beta_1) q^{53} + \beta_{2} q^{54} + ( - 4 \beta_{2} - 2) q^{56} + ( - 3 \beta_{3} - 6 \beta_1) q^{57} + ( - \beta_{3} + 3 \beta_1) q^{58} + (8 \beta_{2} + 9) q^{59} + ( - 6 \beta_{2} - 1) q^{61} + ( - 6 \beta_{3} + 5 \beta_1) q^{62} - 2 \beta_{3} q^{63} + (2 \beta_{2} - 1) q^{64} + 3 \beta_{2} q^{66} + ( - 3 \beta_{3} - 10 \beta_1) q^{67} + ( - \beta_{3} + \beta_1) q^{68} + ( - \beta_{2} - 7) q^{69} + ( - 5 \beta_{2} + 2) q^{71} + ( - \beta_{3} - 2 \beta_1) q^{72} + (6 \beta_{3} - 6 \beta_1) q^{73} - 2 \beta_{2} q^{74} + ( - 3 \beta_{2} - 9) q^{76} - 6 \beta_{3} q^{77} + \beta_1 q^{78} + (3 \beta_{2} + 14) q^{79} + q^{81} + ( - \beta_{3} - 10 \beta_1) q^{82} + 9 \beta_{3} q^{83} + ( - 2 \beta_{2} - 2) q^{84} + (8 \beta_{2} + 1) q^{86} + (2 \beta_{3} - \beta_1) q^{87} + ( - 3 \beta_{3} - 6 \beta_1) q^{88} + ( - 10 \beta_{2} - 5) q^{89} + 2 q^{91} + (8 \beta_{3} + 7 \beta_1) q^{92} + ( - \beta_{3} - 6 \beta_1) q^{93} + ( - 6 \beta_{2} - 2) q^{94} + ( - \beta_{2} - 5) q^{96} + (\beta_{3} + 3 \beta_1) q^{97} + 3 \beta_1 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 2 q^{6} - 4 q^{9} - 12 q^{11} + 4 q^{14} - 6 q^{16} - 8 q^{21} - 2 q^{26} + 10 q^{29} + 8 q^{31} + 14 q^{34} - 2 q^{36} + 4 q^{39} - 42 q^{41} - 6 q^{44} + 8 q^{46} + 12 q^{49} - 8 q^{51} - 2 q^{54} + 20 q^{59} + 8 q^{61} - 8 q^{64} - 6 q^{66} - 26 q^{69} + 18 q^{71} + 4 q^{74} - 30 q^{76} + 50 q^{79} + 4 q^{81} - 4 q^{84} - 12 q^{86} + 8 q^{91} + 4 q^{94} - 18 q^{96} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times\).

\(n\) \(626\) \(1252\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
1.61803i
0.618034i
0.618034i
1.61803i
1.61803i 1.00000i −0.618034 0 1.61803 2.00000i 2.23607i −1.00000 0
1249.2 0.618034i 1.00000i 1.61803 0 −0.618034 2.00000i 2.23607i −1.00000 0
1249.3 0.618034i 1.00000i 1.61803 0 −0.618034 2.00000i 2.23607i −1.00000 0
1249.4 1.61803i 1.00000i −0.618034 0 1.61803 2.00000i 2.23607i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1875.2.b.a 4
5.b even 2 1 inner 1875.2.b.a 4
5.c odd 4 1 1875.2.a.b 2
5.c odd 4 1 1875.2.a.c yes 2
15.e even 4 1 5625.2.a.b 2
15.e even 4 1 5625.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.b 2 5.c odd 4 1
1875.2.a.c yes 2 5.c odd 4 1
1875.2.b.a 4 1.a even 1 1 trivial
1875.2.b.a 4 5.b even 2 1 inner
5625.2.a.b 2 15.e even 4 1
5625.2.a.g 2 15.e even 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T + 3)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 18T^{2} + 1 \) Copy content Toggle raw display
$19$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 87T^{2} + 1681 \) Copy content Toggle raw display
$29$ \( (T^{2} - 5 T + 5)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 41)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 21 T + 109)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 147T^{2} + 5041 \) Copy content Toggle raw display
$47$ \( T^{4} + 108T^{2} + 1936 \) Copy content Toggle raw display
$53$ \( T^{4} + 172T^{2} + 5776 \) Copy content Toggle raw display
$59$ \( (T^{2} - 10 T - 55)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 41)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 258 T^{2} + 14641 \) Copy content Toggle raw display
$71$ \( (T^{2} - 9 T - 11)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 252T^{2} + 1296 \) Copy content Toggle raw display
$79$ \( (T^{2} - 25 T + 145)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 125)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 23T^{2} + 121 \) Copy content Toggle raw display
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