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Properties

Label 75.6.b.c
Level $75$
Weight $6$
Character orbit 75.b
Analytic conductor $12.029$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,6,Mod(49,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.49"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.0287864860\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

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

\(f(q)\) \(=\) \( q + 4 i q^{2} - 9 i q^{3} + 16 q^{4} + 36 q^{6} - 225 i q^{7} + 192 i q^{8} - 81 q^{9} - 434 q^{11} - 144 i q^{12} - 613 i q^{13} + 900 q^{14} - 256 q^{16} - 878 i q^{17} - 324 i q^{18} + 731 q^{19} - 2025 q^{21} + \cdots + 35154 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 32 q^{4} + 72 q^{6} - 162 q^{9} - 868 q^{11} + 1800 q^{14} - 512 q^{16} + 1462 q^{19} - 4050 q^{21} + 3456 q^{24} + 4904 q^{26} + 15164 q^{29} + 4350 q^{31} + 7024 q^{34} - 2592 q^{36} - 11034 q^{39}+ \cdots + 70308 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
− 1.00000i
1.00000i
− 4.00000i 9.00000i 16.0000 0 36.0000 225.000i − 192.000i −81.0000 0
49.2 4.00000i − 9.00000i 16.0000 0 36.0000 − 225.000i 192.000i −81.0000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 75.6.b.c 2
3.b odd 2 1 225.6.b.c 2
5.b even 2 1 inner 75.6.b.c 2
5.c odd 4 1 75.6.a.b ✓ 1
5.c odd 4 1 75.6.a.d yes 1
15.d odd 2 1 225.6.b.c 2
15.e even 4 1 225.6.a.b 1
15.e even 4 1 225.6.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.6.a.b ✓ 1 5.c odd 4 1
75.6.a.d yes 1 5.c odd 4 1
75.6.b.c 2 1.a even 1 1 trivial
75.6.b.c 2 5.b even 2 1 inner
225.6.a.b 1 15.e even 4 1
225.6.a.g 1 15.e even 4 1
225.6.b.c 2 3.b odd 2 1
225.6.b.c 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 16 \) acting on \(S_{6}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 50625 \) Copy content Toggle raw display
$11$ \( (T + 434)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 375769 \) Copy content Toggle raw display
$17$ \( T^{2} + 770884 \) Copy content Toggle raw display
$19$ \( (T - 731)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 8122500 \) Copy content Toggle raw display
$29$ \( (T - 7582)^{2} \) Copy content Toggle raw display
$31$ \( (T - 2175)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 86676100 \) Copy content Toggle raw display
$41$ \( (T + 12040)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1256641 \) Copy content Toggle raw display
$47$ \( T^{2} + 892694884 \) Copy content Toggle raw display
$53$ \( T^{2} + 32947600 \) Copy content Toggle raw display
$59$ \( (T - 5174)^{2} \) Copy content Toggle raw display
$61$ \( (T + 38717)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1005333849 \) Copy content Toggle raw display
$71$ \( (T - 64472)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 391644100 \) Copy content Toggle raw display
$79$ \( (T - 105000)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 11009124 \) Copy content Toggle raw display
$89$ \( (T - 65376)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 7946474449 \) Copy content Toggle raw display
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