Properties

Label 49.3.d.a
Level $49$
Weight $3$
Character orbit 49.d
Analytic conductor $1.335$
Analytic rank $0$
Dimension $2$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,3,Mod(19,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 49.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.33515329537\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + 3 \zeta_{6} q^{2} + (5 \zeta_{6} - 5) q^{4} - 3 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 \zeta_{6} q^{2} + (5 \zeta_{6} - 5) q^{4} - 3 q^{8} - 9 \zeta_{6} q^{9} + ( - 6 \zeta_{6} + 6) q^{11} + 11 \zeta_{6} q^{16} + ( - 27 \zeta_{6} + 27) q^{18} + 18 q^{22} - 18 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} - 54 q^{29} + (45 \zeta_{6} - 45) q^{32} + 45 q^{36} + 38 \zeta_{6} q^{37} + 58 q^{43} + 30 \zeta_{6} q^{44} + ( - 54 \zeta_{6} + 54) q^{46} - 75 q^{50} + ( - 6 \zeta_{6} + 6) q^{53} - 162 \zeta_{6} q^{58} - 91 q^{64} + ( - 118 \zeta_{6} + 118) q^{67} + 114 q^{71} + 27 \zeta_{6} q^{72} + (114 \zeta_{6} - 114) q^{74} + 94 \zeta_{6} q^{79} + (81 \zeta_{6} - 81) q^{81} + 174 \zeta_{6} q^{86} + (18 \zeta_{6} - 18) q^{88} + 90 q^{92} - 54 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} - 5 q^{4} - 6 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} - 5 q^{4} - 6 q^{8} - 9 q^{9} + 6 q^{11} + 11 q^{16} + 27 q^{18} + 36 q^{22} - 18 q^{23} - 25 q^{25} - 108 q^{29} - 45 q^{32} + 90 q^{36} + 38 q^{37} + 116 q^{43} + 30 q^{44} + 54 q^{46} - 150 q^{50} + 6 q^{53} - 162 q^{58} - 182 q^{64} + 118 q^{67} + 228 q^{71} + 27 q^{72} - 114 q^{74} + 94 q^{79} - 81 q^{81} + 174 q^{86} - 18 q^{88} + 180 q^{92} - 108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\zeta_{6}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.500000 0.866025i
0.500000 + 0.866025i
1.50000 2.59808i 0 −2.50000 4.33013i 0 0 0 −3.00000 −4.50000 + 7.79423i 0
31.1 1.50000 + 2.59808i 0 −2.50000 + 4.33013i 0 0 0 −3.00000 −4.50000 7.79423i 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
7.c even 3 1 inner
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 49.3.d.a 2
3.b odd 2 1 441.3.m.a 2
4.b odd 2 1 784.3.s.a 2
7.b odd 2 1 CM 49.3.d.a 2
7.c even 3 1 7.3.b.a 1
7.c even 3 1 inner 49.3.d.a 2
7.d odd 6 1 7.3.b.a 1
7.d odd 6 1 inner 49.3.d.a 2
21.c even 2 1 441.3.m.a 2
21.g even 6 1 63.3.d.a 1
21.g even 6 1 441.3.m.a 2
21.h odd 6 1 63.3.d.a 1
21.h odd 6 1 441.3.m.a 2
28.d even 2 1 784.3.s.a 2
28.f even 6 1 112.3.c.a 1
28.f even 6 1 784.3.s.a 2
28.g odd 6 1 112.3.c.a 1
28.g odd 6 1 784.3.s.a 2
35.i odd 6 1 175.3.d.a 1
35.j even 6 1 175.3.d.a 1
35.k even 12 2 175.3.c.a 2
35.l odd 12 2 175.3.c.a 2
56.j odd 6 1 448.3.c.a 1
56.k odd 6 1 448.3.c.b 1
56.m even 6 1 448.3.c.b 1
56.p even 6 1 448.3.c.a 1
84.j odd 6 1 1008.3.f.a 1
84.n even 6 1 1008.3.f.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.3.b.a 1 7.c even 3 1
7.3.b.a 1 7.d odd 6 1
49.3.d.a 2 1.a even 1 1 trivial
49.3.d.a 2 7.b odd 2 1 CM
49.3.d.a 2 7.c even 3 1 inner
49.3.d.a 2 7.d odd 6 1 inner
63.3.d.a 1 21.g even 6 1
63.3.d.a 1 21.h odd 6 1
112.3.c.a 1 28.f even 6 1
112.3.c.a 1 28.g odd 6 1
175.3.c.a 2 35.k even 12 2
175.3.c.a 2 35.l odd 12 2
175.3.d.a 1 35.i odd 6 1
175.3.d.a 1 35.j even 6 1
441.3.m.a 2 3.b odd 2 1
441.3.m.a 2 21.c even 2 1
441.3.m.a 2 21.g even 6 1
441.3.m.a 2 21.h odd 6 1
448.3.c.a 1 56.j odd 6 1
448.3.c.a 1 56.p even 6 1
448.3.c.b 1 56.k odd 6 1
448.3.c.b 1 56.m even 6 1
784.3.s.a 2 4.b odd 2 1
784.3.s.a 2 28.d even 2 1
784.3.s.a 2 28.f even 6 1
784.3.s.a 2 28.g odd 6 1
1008.3.f.a 1 84.j odd 6 1
1008.3.f.a 1 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3T_{2} + 9 \) acting on \(S_{3}^{\mathrm{new}}(49, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 18T + 324 \) Copy content Toggle raw display
$29$ \( (T + 54)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 38T + 1444 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 58)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 118T + 13924 \) Copy content Toggle raw display
$71$ \( (T - 114)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 94T + 8836 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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