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Properties

Label 63.7.m.a
Level $63$
Weight $7$
Character orbit 63.m
Analytic conductor $14.493$
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] = [63,7,Mod(10,63)] 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("63.10"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 63.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-12,0,-80,-315] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.4934072681\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 12 \zeta_{6} q^{2} + (80 \zeta_{6} - 80) q^{4} + (105 \zeta_{6} - 210) q^{5} - 343 q^{7} + 192 q^{8} + (1260 \zeta_{6} + 1260) q^{10} + ( - 1479 \zeta_{6} + 1479) q^{11} + (560 \zeta_{6} - 280) q^{13} + \cdots - 1411788 \zeta_{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{2} - 80 q^{4} - 315 q^{5} - 686 q^{7} + 384 q^{8} + 3780 q^{10} + 1479 q^{11} + 4116 q^{14} + 2816 q^{16} + 5229 q^{17} + 11907 q^{19} - 35496 q^{22} - 5913 q^{23} + 17450 q^{25} + 10080 q^{26}+ \cdots - 1411788 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
0.500000 + 0.866025i
0.500000 − 0.866025i
−6.00000 − 10.3923i 0 −40.0000 + 69.2820i −157.500 + 90.9327i 0 −343.000 192.000 0 1890.00 + 1091.19i
19.1 −6.00000 + 10.3923i 0 −40.0000 − 69.2820i −157.500 − 90.9327i 0 −343.000 192.000 0 1890.00 − 1091.19i
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.7.m.a 2
3.b odd 2 1 7.7.d.a ✓ 2
7.c even 3 1 441.7.d.a 2
7.d odd 6 1 inner 63.7.m.a 2
7.d odd 6 1 441.7.d.a 2
12.b even 2 1 112.7.s.a 2
21.c even 2 1 49.7.d.b 2
21.g even 6 1 7.7.d.a ✓ 2
21.g even 6 1 49.7.b.a 2
21.h odd 6 1 49.7.b.a 2
21.h odd 6 1 49.7.d.b 2
84.j odd 6 1 112.7.s.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.7.d.a ✓ 2 3.b odd 2 1
7.7.d.a ✓ 2 21.g even 6 1
49.7.b.a 2 21.g even 6 1
49.7.b.a 2 21.h odd 6 1
49.7.d.b 2 21.c even 2 1
49.7.d.b 2 21.h odd 6 1
63.7.m.a 2 1.a even 1 1 trivial
63.7.m.a 2 7.d odd 6 1 inner
112.7.s.a 2 12.b even 2 1
112.7.s.a 2 84.j odd 6 1
441.7.d.a 2 7.c even 3 1
441.7.d.a 2 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 12T_{2} + 144 \) acting on \(S_{7}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 315T + 33075 \) Copy content Toggle raw display
$7$ \( (T + 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 1479 T + 2187441 \) Copy content Toggle raw display
$13$ \( T^{2} + 235200 \) Copy content Toggle raw display
$17$ \( T^{2} - 5229 T + 9114147 \) Copy content Toggle raw display
$19$ \( T^{2} - 11907 T + 47258883 \) Copy content Toggle raw display
$23$ \( T^{2} + 5913 T + 34963569 \) Copy content Toggle raw display
$29$ \( (T + 3978)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 22197 T + 164235603 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 3791726929 \) Copy content Toggle raw display
$41$ \( T^{2} + 12226636800 \) Copy content Toggle raw display
$43$ \( (T + 17414)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 53109 T + 940188627 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 3661823169 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 46538854803 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 26486008563 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 72241075729 \) Copy content Toggle raw display
$71$ \( (T + 101922)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 100898977347 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 131211297361 \) Copy content Toggle raw display
$83$ \( T^{2} + 46995076800 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 1781061603363 \) Copy content Toggle raw display
$97$ \( T^{2} + 2292467116800 \) Copy content Toggle raw display
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