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Properties

Label 75.18.b.a
Level $75$
Weight $18$
Character orbit 75.b
Analytic conductor $137.417$
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] = [75,18,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, 18); 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, 18, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,178912] 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: \(137.416565508\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
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.

\(f(q)\) \(=\) \( q + 204 i q^{2} + 6561 i q^{3} + 89456 q^{4} - 1338444 q^{6} - 20846560 i q^{7} + 44987712 i q^{8} - 43046721 q^{9} + 817372356 q^{11} + 586920816 i q^{12} - 299589758 i q^{13} + 4252698240 q^{14} + 2547683584 q^{16} + \cdots - 35\!\cdots\!76 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 178912 q^{4} - 2676888 q^{6} - 86093442 q^{9} + 1634744712 q^{11} + 8505396480 q^{14} + 5095367168 q^{16} - 157497303928 q^{19} + 273548560320 q^{21} - 590328756864 q^{24} + 122232621264 q^{26} + 327587570484 q^{29}+ \cdots - 70\!\cdots\!52 q^{99}+O(q^{100}) \) 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
− 204.000i − 6561.00i 89456.0 0 −1.33844e6 2.08466e7i − 4.49877e7i −4.30467e7 0
49.2 204.000i 6561.00i 89456.0 0 −1.33844e6 − 2.08466e7i 4.49877e7i −4.30467e7 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 75.18.b.a 2
5.b even 2 1 inner 75.18.b.a 2
5.c odd 4 1 3.18.a.a ✓ 1
5.c odd 4 1 75.18.a.a 1
15.e even 4 1 9.18.a.a 1
20.e even 4 1 48.18.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.18.a.a ✓ 1 5.c odd 4 1
9.18.a.a 1 15.e even 4 1
48.18.a.e 1 20.e even 4 1
75.18.a.a 1 5.c odd 4 1
75.18.b.a 2 1.a even 1 1 trivial
75.18.b.a 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 41616 \) acting on \(S_{18}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 41616 \) Copy content Toggle raw display
$3$ \( T^{2} + 43046721 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 434579063833600 \) Copy content Toggle raw display
$11$ \( (T - 817372356)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 89\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{2} + 20\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( (T + 78748651964)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 49\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T - 163793785242)^{2} \) Copy content Toggle raw display
$31$ \( (T - 1049860831400)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 39\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T - 14660035932090)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 13\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{2} + 31\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{2} + 23\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T - 262797291296124)^{2} \) Copy content Toggle raw display
$61$ \( (T + 13\!\cdots\!62)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 19\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T + 40\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 85\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T + 14\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 69\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T - 38\!\cdots\!26)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 64\!\cdots\!44 \) Copy content Toggle raw display
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