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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [150,16,Mod(49,150)] 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("150.49"); S:= CuspForms(chi, 16); N := Newforms(S);
 
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, 16, names="a")
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 16 \)
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,-32768,0,559872,0,0,-9565938,0,220510104] 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: \(214.040257650\)
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 6)
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.

\(f(q)\) \(=\) \( q - 128 i q^{2} + 2187 i q^{3} - 16384 q^{4} + 279936 q^{6} + 2025056 i q^{7} + 2097152 i q^{8} - 4782969 q^{9} + 110255052 q^{11} - 35831808 i q^{12} - 56047862 i q^{13} + 259207168 q^{14} + 268435456 q^{16} + \cdots - 527346495809388 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32768 q^{4} + 559872 q^{6} - 9565938 q^{9} + 220510104 q^{11} + 518414336 q^{14} + 536870912 q^{16} - 4326376360 q^{19} - 8857594944 q^{21} - 9172942848 q^{24} - 14348252672 q^{26} - 129487438140 q^{29}+ \cdots - 10\!\cdots\!76 q^{99}+O(q^{100}) \) 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
− 128.000i 2187.00i −16384.0 0 279936. 2.02506e6i 2.09715e6i −4.78297e6 0
49.2 128.000i − 2187.00i −16384.0 0 279936. − 2.02506e6i − 2.09715e6i −4.78297e6 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 150.16.c.i 2
5.b even 2 1 inner 150.16.c.i 2
5.c odd 4 1 6.16.a.a ✓ 1
5.c odd 4 1 150.16.a.h 1
15.e even 4 1 18.16.a.f 1
20.e even 4 1 48.16.a.c 1
60.l odd 4 1 144.16.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.16.a.a ✓ 1 5.c odd 4 1
18.16.a.f 1 15.e even 4 1
48.16.a.c 1 20.e even 4 1
144.16.a.o 1 60.l odd 4 1
150.16.a.h 1 5.c odd 4 1
150.16.c.i 2 1.a even 1 1 trivial
150.16.c.i 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 4100851803136 \) acting on \(S_{16}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16384 \) Copy content Toggle raw display
$3$ \( T^{2} + 4782969 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4100851803136 \) Copy content Toggle raw display
$11$ \( (T - 110255052)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 31\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{2} + 37\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T + 2163188180)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 38\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T + 64743719070)^{2} \) Copy content Toggle raw display
$31$ \( (T + 20237611048)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 23\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T + 772359114198)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 17\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{2} + 11\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{2} + 88\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( (T + 28930359275340)^{2} \) Copy content Toggle raw display
$61$ \( (T - 42393077399702)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 27\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T + 27194529024648)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 83\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T + 62882111078120)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 49\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T + 554198786115210)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 19\!\cdots\!76 \) Copy content Toggle raw display
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