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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [300,8,Mod(49,300)] 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("300.49"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 300.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-1458,0,6192] 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: \(93.7155076452\)
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 60)
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.

\(f(q)\) \(=\) \( q + 27 i q^{3} + 1028 i q^{7} - 729 q^{9} + 3096 q^{11} + 13030 i q^{13} + 1878 i q^{17} + 31180 q^{19} - 27756 q^{21} + 33288 i q^{23} - 19683 i q^{27} + 213054 q^{29} - 172696 q^{31} + 83592 i q^{33} + \cdots - 2256984 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1458 q^{9} + 6192 q^{11} + 62360 q^{19} - 55512 q^{21} + 426108 q^{29} - 345392 q^{31} - 703620 q^{39} + 1065300 q^{41} - 466482 q^{49} - 101412 q^{51} - 3016272 q^{59} - 605156 q^{61} - 1797552 q^{69}+ \cdots - 4513968 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
0 − 27.0000i 0 0 0 − 1028.00i 0 −729.000 0
49.2 0 27.0000i 0 0 0 1028.00i 0 −729.000 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 300.8.d.d 2
5.b even 2 1 inner 300.8.d.d 2
5.c odd 4 1 60.8.a.a ✓ 1
5.c odd 4 1 300.8.a.e 1
15.e even 4 1 180.8.a.e 1
20.e even 4 1 240.8.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.8.a.a ✓ 1 5.c odd 4 1
180.8.a.e 1 15.e even 4 1
240.8.a.i 1 20.e even 4 1
300.8.a.e 1 5.c odd 4 1
300.8.d.d 2 1.a even 1 1 trivial
300.8.d.d 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(300, [\chi])\):

\( T_{7}^{2} + 1056784 \) Copy content Toggle raw display
\( T_{11} - 3096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 729 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1056784 \) Copy content Toggle raw display
$11$ \( (T - 3096)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 169780900 \) Copy content Toggle raw display
$17$ \( T^{2} + 3526884 \) Copy content Toggle raw display
$19$ \( (T - 31180)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1108090944 \) Copy content Toggle raw display
$29$ \( (T - 213054)^{2} \) Copy content Toggle raw display
$31$ \( (T + 172696)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 752624356 \) Copy content Toggle raw display
$41$ \( (T - 532650)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 831576200464 \) Copy content Toggle raw display
$47$ \( T^{2} + 536773091904 \) Copy content Toggle raw display
$53$ \( T^{2} + 167341537476 \) Copy content Toggle raw display
$59$ \( (T + 1508136)^{2} \) Copy content Toggle raw display
$61$ \( (T + 302578)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1573348766224 \) Copy content Toggle raw display
$71$ \( (T - 4781280)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 252419827396 \) Copy content Toggle raw display
$79$ \( (T - 1991368)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 65598142135824 \) Copy content Toggle raw display
$89$ \( (T + 7487970)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 294897297785476 \) Copy content Toggle raw display
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