Properties

Label 2575.1.c.b
Level $2575$
Weight $1$
Character orbit 2575.c
Analytic conductor $1.285$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -103
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2575 = 5^{2} \cdot 103 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2575.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.28509240753\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 103)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.10609.1
Artin image: $C_4\times D_5$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{3} ) q^{2} + ( -1 - \beta_{2} ) q^{4} -\beta_{1} q^{7} -\beta_{3} q^{8} - q^{9} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{3} ) q^{2} + ( -1 - \beta_{2} ) q^{4} -\beta_{1} q^{7} -\beta_{3} q^{8} - q^{9} + ( -\beta_{1} - \beta_{3} ) q^{13} + q^{14} -\beta_{1} q^{17} + ( -\beta_{1} - \beta_{3} ) q^{18} + ( 1 + \beta_{2} ) q^{19} + ( -\beta_{1} - \beta_{3} ) q^{23} + ( 2 + \beta_{2} ) q^{26} + \beta_{3} q^{28} -\beta_{2} q^{29} -\beta_{3} q^{32} + q^{34} + ( 1 + \beta_{2} ) q^{36} + ( \beta_{1} + 2 \beta_{3} ) q^{38} + ( -1 - \beta_{2} ) q^{41} + ( 2 + \beta_{2} ) q^{46} + \beta_{2} q^{49} + ( \beta_{1} + 2 \beta_{3} ) q^{52} -\beta_{2} q^{56} -\beta_{3} q^{58} -\beta_{2} q^{59} + \beta_{2} q^{61} + \beta_{1} q^{63} + ( 1 + \beta_{2} ) q^{64} + \beta_{3} q^{68} + \beta_{3} q^{72} + ( -2 - \beta_{2} ) q^{76} + ( 1 + \beta_{2} ) q^{79} + q^{81} + ( -\beta_{1} - 2 \beta_{3} ) q^{82} + \beta_{1} q^{83} - q^{91} + ( \beta_{1} + 2 \beta_{3} ) q^{92} -\beta_{1} q^{97} + \beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} - 4 q^{9} + O(q^{10}) \) \( 4 q - 2 q^{4} - 4 q^{9} + 4 q^{14} + 2 q^{19} + 6 q^{26} + 2 q^{29} + 4 q^{34} + 2 q^{36} - 2 q^{41} + 6 q^{46} - 2 q^{49} + 2 q^{56} + 2 q^{59} - 2 q^{61} + 2 q^{64} - 6 q^{76} + 2 q^{79} + 4 q^{81} - 4 q^{91} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 2 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2575\mathbb{Z}\right)^\times\).

\(n\) \(726\) \(1752\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2574.1
0.618034i
1.61803i
1.61803i
0.618034i
1.61803i 0 −1.61803 0 0 0.618034i 1.00000i −1.00000 0
2574.2 0.618034i 0 0.618034 0 0 1.61803i 1.00000i −1.00000 0
2574.3 0.618034i 0 0.618034 0 0 1.61803i 1.00000i −1.00000 0
2574.4 1.61803i 0 −1.61803 0 0 0.618034i 1.00000i −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
103.b odd 2 1 CM by \(\Q(\sqrt{-103}) \)
5.b even 2 1 inner
515.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2575.1.c.b 4
5.b even 2 1 inner 2575.1.c.b 4
5.c odd 4 1 103.1.b.a 2
5.c odd 4 1 2575.1.d.d 2
15.e even 4 1 927.1.d.b 2
20.e even 4 1 1648.1.c.a 2
103.b odd 2 1 CM 2575.1.c.b 4
515.c odd 2 1 inner 2575.1.c.b 4
515.f even 4 1 103.1.b.a 2
515.f even 4 1 2575.1.d.d 2
1545.m odd 4 1 927.1.d.b 2
2060.m odd 4 1 1648.1.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.1.b.a 2 5.c odd 4 1
103.1.b.a 2 515.f even 4 1
927.1.d.b 2 15.e even 4 1
927.1.d.b 2 1545.m odd 4 1
1648.1.c.a 2 20.e even 4 1
1648.1.c.a 2 2060.m odd 4 1
2575.1.c.b 4 1.a even 1 1 trivial
2575.1.c.b 4 5.b even 2 1 inner
2575.1.c.b 4 103.b odd 2 1 CM
2575.1.c.b 4 515.c odd 2 1 inner
2575.1.d.d 2 5.c odd 4 1
2575.1.d.d 2 515.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2575, [\chi])\):

\( T_{2}^{4} + 3 T_{2}^{2} + 1 \)
\( T_{7}^{4} + 3 T_{7}^{2} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( 1 + 3 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( 1 + 3 T^{2} + T^{4} \)
$17$ \( 1 + 3 T^{2} + T^{4} \)
$19$ \( ( -1 - T + T^{2} )^{2} \)
$23$ \( 1 + 3 T^{2} + T^{4} \)
$29$ \( ( -1 - T + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( -1 + T + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( ( -1 - T + T^{2} )^{2} \)
$61$ \( ( -1 + T + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( -1 - T + T^{2} )^{2} \)
$83$ \( 1 + 3 T^{2} + T^{4} \)
$89$ \( T^{4} \)
$97$ \( 1 + 3 T^{2} + T^{4} \)
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