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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,-1458,0,6312] 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.

\(f(q)\) \(=\) \( q - 27 i q^{3} + 832 i q^{7} - 729 q^{9} + 3156 q^{11} - 7690 i q^{13} - 258 i q^{17} - 45740 q^{19} + 22464 q^{21} + 104832 i q^{23} + 19683 i q^{27} - 38646 q^{29} + 192224 q^{31} - 85212 i q^{33} - 403454 i q^{37} + \cdots - 2300724 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1458 q^{9} + 6312 q^{11} - 91480 q^{19} + 44928 q^{21} - 77292 q^{29} + 384448 q^{31} - 415260 q^{39} + 172020 q^{41} + 262638 q^{49} - 13932 q^{51} - 397992 q^{59} + 2419564 q^{61} + 5660928 q^{69}+ \cdots - 4601448 q^{99}+O(q^{100}) \) 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 0 832.000i 0 −729.000 0
49.2 0 27.0000i 0 0 0 − 832.000i 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.e 2
5.b even 2 1 inner 300.8.d.e 2
5.c odd 4 1 60.8.a.b ✓ 1
5.c odd 4 1 300.8.a.h 1
15.e even 4 1 180.8.a.b 1
20.e even 4 1 240.8.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.8.a.b ✓ 1 5.c odd 4 1
180.8.a.b 1 15.e even 4 1
240.8.a.n 1 20.e even 4 1
300.8.a.h 1 5.c odd 4 1
300.8.d.e 2 1.a even 1 1 trivial
300.8.d.e 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 \(S_{8}^{\mathrm{new}}(300, [\chi])\):

\( T_{7}^{2} + 692224 \) Copy content Toggle raw display
\( T_{11} - 3156 \) 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
$7$ \( T^{2} + 692224 \) Copy content Toggle raw display
$11$ \( (T - 3156)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 59136100 \) Copy content Toggle raw display
$17$ \( T^{2} + 66564 \) Copy content Toggle raw display
$19$ \( (T + 45740)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 10989748224 \) Copy content Toggle raw display
$29$ \( (T + 38646)^{2} \) Copy content Toggle raw display
$31$ \( (T - 192224)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 162775130116 \) Copy content Toggle raw display
$41$ \( (T - 86010)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16217513104 \) Copy content Toggle raw display
$47$ \( T^{2} + 361528017984 \) Copy content Toggle raw display
$53$ \( T^{2} + 2651119907076 \) Copy content Toggle raw display
$59$ \( (T + 198996)^{2} \) Copy content Toggle raw display
$61$ \( (T - 1209782)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 489143574544 \) Copy content Toggle raw display
$71$ \( (T + 4939320)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1626476811556 \) Copy content Toggle raw display
$79$ \( (T + 6559712)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 9661827289104 \) Copy content Toggle raw display
$89$ \( (T + 5542410)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 20370292115716 \) Copy content Toggle raw display
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