Properties

Label 1875.2.a.c
Level $1875$
Weight $2$
Character orbit 1875.a
Self dual yes
Analytic conductor $14.972$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{3} + (\beta - 1) q^{4} + \beta q^{6} - 2 q^{7} + ( - 2 \beta + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + q^{3} + (\beta - 1) q^{4} + \beta q^{6} - 2 q^{7} + ( - 2 \beta + 1) q^{8} + q^{9} - 3 q^{11} + (\beta - 1) q^{12} - q^{13} - 2 \beta q^{14} - 3 \beta q^{16} + ( - 2 \beta - 1) q^{17} + \beta q^{18} + ( - 6 \beta + 3) q^{19} - 2 q^{21} - 3 \beta q^{22} + ( - \beta + 7) q^{23} + ( - 2 \beta + 1) q^{24} - \beta q^{26} + q^{27} + ( - 2 \beta + 2) q^{28} + ( - \beta - 2) q^{29} + (6 \beta - 1) q^{31} + (\beta - 5) q^{32} - 3 q^{33} + ( - 3 \beta - 2) q^{34} + (\beta - 1) q^{36} - 2 q^{37} + ( - 3 \beta - 6) q^{38} - q^{39} + (\beta - 11) q^{41} - 2 \beta q^{42} + (\beta - 9) q^{43} + ( - 3 \beta + 3) q^{44} + (6 \beta - 1) q^{46} + (2 \beta - 8) q^{47} - 3 \beta q^{48} - 3 q^{49} + ( - 2 \beta - 1) q^{51} + ( - \beta + 1) q^{52} + (2 \beta + 8) q^{53} + \beta q^{54} + (4 \beta - 2) q^{56} + ( - 6 \beta + 3) q^{57} + ( - 3 \beta - 1) q^{58} + (8 \beta - 9) q^{59} + (6 \beta - 1) q^{61} + (5 \beta + 6) q^{62} - 2 q^{63} + (2 \beta + 1) q^{64} - 3 \beta q^{66} + ( - 10 \beta + 3) q^{67} + ( - \beta - 1) q^{68} + ( - \beta + 7) q^{69} + (5 \beta + 2) q^{71} + ( - 2 \beta + 1) q^{72} + (6 \beta + 6) q^{73} - 2 \beta q^{74} + (3 \beta - 9) q^{76} + 6 q^{77} - \beta q^{78} + (3 \beta - 14) q^{79} + q^{81} + ( - 10 \beta + 1) q^{82} + 9 q^{83} + ( - 2 \beta + 2) q^{84} + ( - 8 \beta + 1) q^{86} + ( - \beta - 2) q^{87} + (6 \beta - 3) q^{88} + ( - 10 \beta + 5) q^{89} + 2 q^{91} + (7 \beta - 8) q^{92} + (6 \beta - 1) q^{93} + ( - 6 \beta + 2) q^{94} + (\beta - 5) q^{96} + (3 \beta - 1) q^{97} - 3 \beta q^{98} - 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} - 4 q^{7} + 2 q^{9} - 6 q^{11} - q^{12} - 2 q^{13} - 2 q^{14} - 3 q^{16} - 4 q^{17} + q^{18} - 4 q^{21} - 3 q^{22} + 13 q^{23} - q^{26} + 2 q^{27} + 2 q^{28} - 5 q^{29} + 4 q^{31} - 9 q^{32} - 6 q^{33} - 7 q^{34} - q^{36} - 4 q^{37} - 15 q^{38} - 2 q^{39} - 21 q^{41} - 2 q^{42} - 17 q^{43} + 3 q^{44} + 4 q^{46} - 14 q^{47} - 3 q^{48} - 6 q^{49} - 4 q^{51} + q^{52} + 18 q^{53} + q^{54} - 5 q^{58} - 10 q^{59} + 4 q^{61} + 17 q^{62} - 4 q^{63} + 4 q^{64} - 3 q^{66} - 4 q^{67} - 3 q^{68} + 13 q^{69} + 9 q^{71} + 18 q^{73} - 2 q^{74} - 15 q^{76} + 12 q^{77} - q^{78} - 25 q^{79} + 2 q^{81} - 8 q^{82} + 18 q^{83} + 2 q^{84} - 6 q^{86} - 5 q^{87} + 4 q^{91} - 9 q^{92} + 4 q^{93} - 2 q^{94} - 9 q^{96} + q^{97} - 3 q^{98} - 6 q^{99}+O(q^{100}) \) 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
−0.618034
1.61803
−0.618034 1.00000 −1.61803 0 −0.618034 −2.00000 2.23607 1.00000 0
1.2 1.61803 1.00000 0.618034 0 1.61803 −2.00000 −2.23607 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.c yes 2
3.b odd 2 1 5625.2.a.b 2
5.b even 2 1 1875.2.a.b 2
5.c odd 4 2 1875.2.b.a 4
15.d odd 2 1 5625.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.b 2 5.b even 2 1
1875.2.a.c yes 2 1.a even 1 1 trivial
1875.2.b.a 4 5.c odd 4 2
5625.2.a.b 2 3.b odd 2 1
5625.2.a.g 2 15.d odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$19$ \( T^{2} - 45 \) Copy content Toggle raw display
$23$ \( T^{2} - 13T + 41 \) Copy content Toggle raw display
$29$ \( T^{2} + 5T + 5 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 41 \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 21T + 109 \) Copy content Toggle raw display
$43$ \( T^{2} + 17T + 71 \) Copy content Toggle raw display
$47$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$53$ \( T^{2} - 18T + 76 \) Copy content Toggle raw display
$59$ \( T^{2} + 10T - 55 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 41 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 121 \) Copy content Toggle raw display
$71$ \( T^{2} - 9T - 11 \) Copy content Toggle raw display
$73$ \( T^{2} - 18T + 36 \) Copy content Toggle raw display
$79$ \( T^{2} + 25T + 145 \) Copy content Toggle raw display
$83$ \( (T - 9)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 125 \) Copy content Toggle raw display
$97$ \( T^{2} - T - 11 \) Copy content Toggle raw display
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