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Properties

Label 150.12.c.g
Level $150$
Weight $12$
Character orbit 150.c
Analytic conductor $115.251$
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] = [150,12,Mod(49,150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 12, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("150.49"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 150.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2048,0,15552,0,0,-118098,0,-518256] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(115.251477084\)
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: no (minimal twist has level 30)
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.

\(f(q)\) \(=\) \( q + 32 i q^{2} - 243 i q^{3} - 1024 q^{4} + 7776 q^{6} + 22876 i q^{7} - 32768 i q^{8} - 59049 q^{9} - 259128 q^{11} + 248832 i q^{12} - 2323462 i q^{13} - 732032 q^{14} + 1048576 q^{16} + 1918266 i q^{17} + \cdots + 15301249272 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2048 q^{4} + 15552 q^{6} - 118098 q^{9} - 518256 q^{11} - 1464064 q^{14} + 2097152 q^{16} + 3227480 q^{19} + 11117736 q^{21} - 15925248 q^{24} + 148701568 q^{26} - 305180580 q^{29} + 113620144 q^{31}+ \cdots + 30602498544 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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
− 32.0000i 243.000i −1024.00 0 7776.00 − 22876.0i 32768.0i −59049.0 0
49.2 32.0000i − 243.000i −1024.00 0 7776.00 22876.0i − 32768.0i −59049.0 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
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 150.12.c.g 2
5.b even 2 1 inner 150.12.c.g 2
5.c odd 4 1 30.12.a.a ✓ 1
5.c odd 4 1 150.12.a.h 1
15.e even 4 1 90.12.a.j 1
20.e even 4 1 240.12.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.12.a.a ✓ 1 5.c odd 4 1
90.12.a.j 1 15.e even 4 1
150.12.a.h 1 5.c odd 4 1
150.12.c.g 2 1.a even 1 1 trivial
150.12.c.g 2 5.b even 2 1 inner
240.12.a.d 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 523311376 \) acting on \(S_{12}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1024 \) Copy content Toggle raw display
$3$ \( T^{2} + 59049 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 523311376 \) Copy content Toggle raw display
$11$ \( (T + 259128)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 5398475665444 \) Copy content Toggle raw display
$17$ \( T^{2} + 3679744446756 \) Copy content Toggle raw display
$19$ \( (T - 1613740)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 10\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T + 152590290)^{2} \) Copy content Toggle raw display
$31$ \( (T - 56810072)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 32\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T - 734063082)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 33\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{2} + 22\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{2} + 65\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( (T + 6317920440)^{2} \) Copy content Toggle raw display
$61$ \( (T + 786138178)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 45\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T - 11719480032)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 53\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( (T + 2698969640)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 10\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T + 74556330210)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 17\!\cdots\!56 \) Copy content Toggle raw display
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