Properties

Label 75.3.d.c
Level $75$
Weight $3$
Character orbit 75.d
Analytic conductor $2.044$
Analytic rank $0$
Dimension $4$
CM no
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{11})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{2} ) q^{2} + \beta_{1} q^{3} + 7 q^{4} + ( -6 - \beta_{3} ) q^{6} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{8} + ( -3 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{2} ) q^{2} + \beta_{1} q^{3} + 7 q^{4} + ( -6 - \beta_{3} ) q^{6} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{8} + ( -3 + \beta_{3} ) q^{9} + ( 1 + 2 \beta_{3} ) q^{11} + 7 \beta_{1} q^{12} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{13} + 5 q^{16} + ( -\beta_{1} + \beta_{2} ) q^{17} + ( -2 \beta_{1} - 9 \beta_{2} ) q^{18} -7 q^{19} + ( -11 \beta_{1} - 11 \beta_{2} ) q^{22} + ( 6 \beta_{1} - 6 \beta_{2} ) q^{23} + ( -18 - 3 \beta_{3} ) q^{24} + ( -2 - 4 \beta_{3} ) q^{26} + ( -7 \beta_{1} + 9 \beta_{2} ) q^{27} + ( -2 - 4 \beta_{3} ) q^{29} + 42 q^{31} + ( 7 \beta_{1} - 7 \beta_{2} ) q^{32} + ( -7 \beta_{1} + 18 \beta_{2} ) q^{33} + 11 q^{34} + ( -21 + 7 \beta_{3} ) q^{36} + ( 8 \beta_{1} + 8 \beta_{2} ) q^{37} + ( 7 \beta_{1} - 7 \beta_{2} ) q^{38} + ( -24 + 2 \beta_{3} ) q^{39} + ( -1 - 2 \beta_{3} ) q^{41} + ( -10 \beta_{1} - 10 \beta_{2} ) q^{43} + ( 7 + 14 \beta_{3} ) q^{44} -66 q^{46} + ( 14 \beta_{1} - 14 \beta_{2} ) q^{47} + 5 \beta_{1} q^{48} + 49 q^{49} + ( -6 - \beta_{3} ) q^{51} + ( 14 \beta_{1} + 14 \beta_{2} ) q^{52} + ( -14 \beta_{1} + 14 \beta_{2} ) q^{53} + ( 87 - 2 \beta_{3} ) q^{54} -7 \beta_{1} q^{57} + ( 22 \beta_{1} + 22 \beta_{2} ) q^{58} + ( -4 - 8 \beta_{3} ) q^{59} -8 q^{61} + ( -42 \beta_{1} + 42 \beta_{2} ) q^{62} -97 q^{64} + ( 132 - 11 \beta_{3} ) q^{66} + ( -9 \beta_{1} - 9 \beta_{2} ) q^{67} + ( -7 \beta_{1} + 7 \beta_{2} ) q^{68} + ( 36 + 6 \beta_{3} ) q^{69} + ( 2 + 4 \beta_{3} ) q^{71} + ( -6 \beta_{1} - 27 \beta_{2} ) q^{72} + ( -7 \beta_{1} - 7 \beta_{2} ) q^{73} + ( -8 - 16 \beta_{3} ) q^{74} -49 q^{76} + ( 14 \beta_{1} - 36 \beta_{2} ) q^{78} -12 q^{79} + ( -60 - 7 \beta_{3} ) q^{81} + ( 11 \beta_{1} + 11 \beta_{2} ) q^{82} + ( 21 \beta_{1} - 21 \beta_{2} ) q^{83} + ( 10 + 20 \beta_{3} ) q^{86} + ( 14 \beta_{1} - 36 \beta_{2} ) q^{87} + ( -33 \beta_{1} - 33 \beta_{2} ) q^{88} + ( 9 + 18 \beta_{3} ) q^{89} + ( 42 \beta_{1} - 42 \beta_{2} ) q^{92} + 42 \beta_{1} q^{93} -154 q^{94} + ( 42 + 7 \beta_{3} ) q^{96} + ( 14 \beta_{1} + 14 \beta_{2} ) q^{97} + ( -49 \beta_{1} + 49 \beta_{2} ) q^{98} + ( -141 - 7 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 28q^{4} - 22q^{6} - 14q^{9} + O(q^{10}) \) \( 4q + 28q^{4} - 22q^{6} - 14q^{9} + 20q^{16} - 28q^{19} - 66q^{24} + 168q^{31} + 44q^{34} - 98q^{36} - 100q^{39} - 264q^{46} + 196q^{49} - 22q^{51} + 352q^{54} - 32q^{61} - 388q^{64} + 550q^{66} + 132q^{69} - 196q^{76} - 48q^{79} - 226q^{81} - 616q^{94} + 154q^{96} - 550q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{3} + \nu \)\()/3\)
\(\beta_{2}\)\(=\)\( -\nu^{3} + 3 \nu \)
\(\beta_{3}\)\(=\)\( 5 \nu^{2} - 13 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{2} - 3 \beta_{1}\)\()/5\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 13\)\()/5\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{2} - 9 \beta_{1}\)\()/5\)

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
−1.65831 + 0.500000i
−1.65831 0.500000i
1.65831 + 0.500000i
1.65831 0.500000i
−3.31662 1.65831 2.50000i 7.00000 0 −5.50000 + 8.29156i 0 −9.94987 −3.50000 8.29156i 0
74.2 −3.31662 1.65831 + 2.50000i 7.00000 0 −5.50000 8.29156i 0 −9.94987 −3.50000 + 8.29156i 0
74.3 3.31662 −1.65831 2.50000i 7.00000 0 −5.50000 8.29156i 0 9.94987 −3.50000 + 8.29156i 0
74.4 3.31662 −1.65831 + 2.50000i 7.00000 0 −5.50000 + 8.29156i 0 9.94987 −3.50000 8.29156i 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 inner
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.c 4
3.b odd 2 1 inner 75.3.d.c 4
4.b odd 2 1 1200.3.c.d 4
5.b even 2 1 inner 75.3.d.c 4
5.c odd 4 1 75.3.c.c 2
5.c odd 4 1 75.3.c.f yes 2
12.b even 2 1 1200.3.c.d 4
15.d odd 2 1 inner 75.3.d.c 4
15.e even 4 1 75.3.c.c 2
15.e even 4 1 75.3.c.f yes 2
20.d odd 2 1 1200.3.c.d 4
20.e even 4 1 1200.3.l.f 2
20.e even 4 1 1200.3.l.s 2
60.h even 2 1 1200.3.c.d 4
60.l odd 4 1 1200.3.l.f 2
60.l odd 4 1 1200.3.l.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.c.c 2 5.c odd 4 1
75.3.c.c 2 15.e even 4 1
75.3.c.f yes 2 5.c odd 4 1
75.3.c.f yes 2 15.e even 4 1
75.3.d.c 4 1.a even 1 1 trivial
75.3.d.c 4 3.b odd 2 1 inner
75.3.d.c 4 5.b even 2 1 inner
75.3.d.c 4 15.d odd 2 1 inner
1200.3.c.d 4 4.b odd 2 1
1200.3.c.d 4 12.b even 2 1
1200.3.c.d 4 20.d odd 2 1
1200.3.c.d 4 60.h even 2 1
1200.3.l.f 2 20.e even 4 1
1200.3.l.f 2 60.l odd 4 1
1200.3.l.s 2 20.e even 4 1
1200.3.l.s 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 11 \) acting on \(S_{3}^{\mathrm{new}}(75, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -11 + T^{2} )^{2} \)
$3$ \( 81 + 7 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 275 + T^{2} )^{2} \)
$13$ \( ( 100 + T^{2} )^{2} \)
$17$ \( ( -11 + T^{2} )^{2} \)
$19$ \( ( 7 + T )^{4} \)
$23$ \( ( -396 + T^{2} )^{2} \)
$29$ \( ( 1100 + T^{2} )^{2} \)
$31$ \( ( -42 + T )^{4} \)
$37$ \( ( 1600 + T^{2} )^{2} \)
$41$ \( ( 275 + T^{2} )^{2} \)
$43$ \( ( 2500 + T^{2} )^{2} \)
$47$ \( ( -2156 + T^{2} )^{2} \)
$53$ \( ( -2156 + T^{2} )^{2} \)
$59$ \( ( 4400 + T^{2} )^{2} \)
$61$ \( ( 8 + T )^{4} \)
$67$ \( ( 2025 + T^{2} )^{2} \)
$71$ \( ( 1100 + T^{2} )^{2} \)
$73$ \( ( 1225 + T^{2} )^{2} \)
$79$ \( ( 12 + T )^{4} \)
$83$ \( ( -4851 + T^{2} )^{2} \)
$89$ \( ( 22275 + T^{2} )^{2} \)
$97$ \( ( 4900 + T^{2} )^{2} \)
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