Properties

Label 75.9.c.d
Level $75$
Weight $9$
Character orbit 75.c
Analytic conductor $30.553$
Analytic rank $0$
Dimension $2$
CM discriminant -15
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 75.c (of order \(2\), degree \(1\), 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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
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.

\(f(q)\) \(=\) \( q + 17 i q^{2} - 81 i q^{3} - 33 q^{4} + 1377 q^{6} + 3791 i q^{8} - 6561 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 17 i q^{2} - 81 i q^{3} - 33 q^{4} + 1377 q^{6} + 3791 i q^{8} - 6561 q^{9} + 2673 i q^{12} - 72895 q^{16} - 21118 i q^{17} - 111537 i q^{18} + 203998 q^{19} + 550078 i q^{23} + 307071 q^{24} + 531441 i q^{27} + 1831682 q^{31} - 268719 i q^{32} + 359006 q^{34} + 216513 q^{36} + 3467966 i q^{38} - 9351326 q^{46} + 8065922 i q^{47} + 5904495 i q^{48} - 5764801 q^{49} - 1710558 q^{51} + 12619678 i q^{53} - 9034497 q^{54} - 16523838 i q^{57} + 14324642 q^{61} + 31138594 i q^{62} - 14092897 q^{64} + 696894 i q^{68} + 44556318 q^{69} - 24872751 i q^{72} - 6731934 q^{76} + 69617278 q^{79} + 43046721 q^{81} - 3847202 i q^{83} - 18152574 i q^{92} - 148366242 i q^{93} - 137120674 q^{94} - 21766239 q^{96} - 98001617 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 66 q^{4} + 2754 q^{6} - 13122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 66 q^{4} + 2754 q^{6} - 13122 q^{9} - 145790 q^{16} + 407996 q^{19} + 614142 q^{24} + 3663364 q^{31} + 718012 q^{34} + 433026 q^{36} - 18702652 q^{46} - 11529602 q^{49} - 3421116 q^{51} - 18068994 q^{54} + 28649284 q^{61} - 28185794 q^{64} + 89112636 q^{69} - 13463868 q^{76} + 139234556 q^{79} + 86093442 q^{81} - 274241348 q^{94} - 43532478 q^{96}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
1.00000i
1.00000i
17.0000i 81.0000i −33.0000 0 1377.00 0 3791.00i −6561.00 0
26.2 17.0000i 81.0000i −33.0000 0 1377.00 0 3791.00i −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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
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 75.9.c.d 2
3.b odd 2 1 inner 75.9.c.d 2
5.b even 2 1 inner 75.9.c.d 2
5.c odd 4 1 15.9.d.a 1
5.c odd 4 1 15.9.d.b yes 1
15.d odd 2 1 CM 75.9.c.d 2
15.e even 4 1 15.9.d.a 1
15.e even 4 1 15.9.d.b yes 1
20.e even 4 1 240.9.c.a 1
20.e even 4 1 240.9.c.b 1
60.l odd 4 1 240.9.c.a 1
60.l odd 4 1 240.9.c.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.9.d.a 1 5.c odd 4 1
15.9.d.a 1 15.e even 4 1
15.9.d.b yes 1 5.c odd 4 1
15.9.d.b yes 1 15.e even 4 1
75.9.c.d 2 1.a even 1 1 trivial
75.9.c.d 2 3.b odd 2 1 inner
75.9.c.d 2 5.b even 2 1 inner
75.9.c.d 2 15.d odd 2 1 CM
240.9.c.a 1 20.e even 4 1
240.9.c.a 1 60.l odd 4 1
240.9.c.b 1 20.e even 4 1
240.9.c.b 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{2} + 289 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 289 \) Copy content Toggle raw display
$3$ \( T^{2} + 6561 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 445969924 \) Copy content Toggle raw display
$19$ \( (T - 203998)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 302585806084 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 1831682)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 65059097710084 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 159256272823684 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 14324642)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T - 69617278)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 14800963228804 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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