Properties

Label 75.9.d.a
Level $75$
Weight $9$
Character orbit 75.d
Analytic conductor $30.553$
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: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 75.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.5533957546\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + 81 i q^{3} - 256 q^{4} + 4273 i q^{7} - 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 81 i q^{3} - 256 q^{4} + 4273 i q^{7} - 6561 q^{9} - 20736 i q^{12} + 56447 i q^{13} + 65536 q^{16} - 157967 q^{19} - 346113 q^{21} - 531441 i q^{27} - 1093888 i q^{28} + 1225967 q^{31} + 1679616 q^{36} - 503522 i q^{37} - 4572207 q^{39} - 6837073 i q^{43} + 5308416 i q^{48} - 12493728 q^{49} - 14450432 i q^{52} - 12795327 i q^{57} - 307393 q^{61} - 28035153 i q^{63} - 16777216 q^{64} + 31874833 i q^{67} + 16169282 i q^{73} + 40439552 q^{76} + 18887038 q^{79} + 43046721 q^{81} + 88604928 q^{84} - 241198031 q^{91} + 99303327 i q^{93} + 82132513 i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 512 q^{4} - 13122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 512 q^{4} - 13122 q^{9} + 131072 q^{16} - 315934 q^{19} - 692226 q^{21} + 2451934 q^{31} + 3359232 q^{36} - 9144414 q^{39} - 24987456 q^{49} - 614786 q^{61} - 33554432 q^{64} + 80879104 q^{76} + 37774076 q^{79} + 86093442 q^{81} + 177209856 q^{84} - 482396062 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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 81.0000i −256.000 0 0 4273.00i 0 −6561.00 0
74.2 0 81.0000i −256.000 0 0 4273.00i 0 −6561.00 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 75.9.d.a 2
3.b odd 2 1 CM 75.9.d.a 2
5.b even 2 1 inner 75.9.d.a 2
5.c odd 4 1 75.9.c.a 1
5.c odd 4 1 75.9.c.b yes 1
15.d odd 2 1 inner 75.9.d.a 2
15.e even 4 1 75.9.c.a 1
15.e even 4 1 75.9.c.b yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.9.c.a 1 5.c odd 4 1
75.9.c.a 1 15.e even 4 1
75.9.c.b yes 1 5.c odd 4 1
75.9.c.b yes 1 15.e even 4 1
75.9.d.a 2 1.a even 1 1 trivial
75.9.d.a 2 3.b odd 2 1 CM
75.9.d.a 2 5.b even 2 1 inner
75.9.d.a 2 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{9}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 6561 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 18258529 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3186263809 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 157967)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 1225967)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 253534404484 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 46745567207329 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 307393)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 10\!\cdots\!89 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 261445680395524 \) Copy content Toggle raw display
$79$ \( (T - 18887038)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 67\!\cdots\!69 \) Copy content Toggle raw display
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