Properties

Label 75.3.d.a
Level $75$
Weight $3$
Character orbit 75.d
Analytic conductor $2.044$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.04360198270\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{3} -4 q^{4} + 11 i q^{7} -9 q^{9} +O(q^{10})\) \( q + 3 i q^{3} -4 q^{4} + 11 i q^{7} -9 q^{9} -12 i q^{12} + i q^{13} + 16 q^{16} + 37 q^{19} -33 q^{21} -27 i q^{27} -44 i q^{28} -13 q^{31} + 36 q^{36} + 26 i q^{37} -3 q^{39} + 61 i q^{43} + 48 i q^{48} -72 q^{49} -4 i q^{52} + 111 i q^{57} + 47 q^{61} -99 i q^{63} -64 q^{64} -109 i q^{67} + 46 i q^{73} -148 q^{76} + 142 q^{79} + 81 q^{81} + 132 q^{84} -11 q^{91} -39 i q^{93} -169 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{4} - 18q^{9} + O(q^{10}) \) \( 2q - 8q^{4} - 18q^{9} + 32q^{16} + 74q^{19} - 66q^{21} - 26q^{31} + 72q^{36} - 6q^{39} - 144q^{49} + 94q^{61} - 128q^{64} - 296q^{76} + 284q^{79} + 162q^{81} + 264q^{84} - 22q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
1.00000i
1.00000i
0 3.00000i −4.00000 0 0 11.0000i 0 −9.00000 0
74.2 0 3.00000i −4.00000 0 0 11.0000i 0 −9.00000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.3.d.a 2
3.b odd 2 1 CM 75.3.d.a 2
4.b odd 2 1 1200.3.c.a 2
5.b even 2 1 inner 75.3.d.a 2
5.c odd 4 1 75.3.c.a 1
5.c odd 4 1 75.3.c.b yes 1
12.b even 2 1 1200.3.c.a 2
15.d odd 2 1 inner 75.3.d.a 2
15.e even 4 1 75.3.c.a 1
15.e even 4 1 75.3.c.b yes 1
20.d odd 2 1 1200.3.c.a 2
20.e even 4 1 1200.3.l.c 1
20.e even 4 1 1200.3.l.d 1
60.h even 2 1 1200.3.c.a 2
60.l odd 4 1 1200.3.l.c 1
60.l odd 4 1 1200.3.l.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.c.a 1 5.c odd 4 1
75.3.c.a 1 15.e even 4 1
75.3.c.b yes 1 5.c odd 4 1
75.3.c.b yes 1 15.e even 4 1
75.3.d.a 2 1.a even 1 1 trivial
75.3.d.a 2 3.b odd 2 1 CM
75.3.d.a 2 5.b even 2 1 inner
75.3.d.a 2 15.d odd 2 1 inner
1200.3.c.a 2 4.b odd 2 1
1200.3.c.a 2 12.b even 2 1
1200.3.c.a 2 20.d odd 2 1
1200.3.c.a 2 60.h even 2 1
1200.3.l.c 1 20.e even 4 1
1200.3.l.c 1 60.l odd 4 1
1200.3.l.d 1 20.e even 4 1
1200.3.l.d 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(75, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 121 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( -37 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 13 + T )^{2} \)
$37$ \( 676 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 3721 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -47 + T )^{2} \)
$67$ \( 11881 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 2116 + T^{2} \)
$79$ \( ( -142 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 28561 + T^{2} \)
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