Properties

Label 49.7.d.a
Level $49$
Weight $7$
Character orbit 49.d
Analytic conductor $11.273$
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] = [49,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.19");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
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: \(11.2726500974\)
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.

\(f(q)\) \(=\) \( q - 9 \zeta_{6} q^{2} + (17 \zeta_{6} - 17) q^{4} - 423 q^{8} - 729 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 9 \zeta_{6} q^{2} + (17 \zeta_{6} - 17) q^{4} - 423 q^{8} - 729 \zeta_{6} q^{9} + (1962 \zeta_{6} - 1962) q^{11} + 4895 \zeta_{6} q^{16} + (6561 \zeta_{6} - 6561) q^{18} + 17658 q^{22} + 22734 \zeta_{6} q^{23} + (15625 \zeta_{6} - 15625) q^{25} - 21222 q^{29} + ( - 16983 \zeta_{6} + 16983) q^{32} + 12393 q^{36} - 101194 \zeta_{6} q^{37} - 126614 q^{43} - 33354 \zeta_{6} q^{44} + ( - 204606 \zeta_{6} + 204606) q^{46} + 140625 q^{50} + (50346 \zeta_{6} - 50346) q^{53} + 190998 \zeta_{6} q^{58} + 160433 q^{64} + ( - 53926 \zeta_{6} + 53926) q^{67} - 242478 q^{71} + 308367 \zeta_{6} q^{72} + (910746 \zeta_{6} - 910746) q^{74} - 929378 \zeta_{6} q^{79} + (531441 \zeta_{6} - 531441) q^{81} + 1139526 \zeta_{6} q^{86} + ( - 829926 \zeta_{6} + 829926) q^{88} - 386478 q^{92} + 1430298 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{2} - 17 q^{4} - 846 q^{8} - 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 9 q^{2} - 17 q^{4} - 846 q^{8} - 729 q^{9} - 1962 q^{11} + 4895 q^{16} - 6561 q^{18} + 35316 q^{22} + 22734 q^{23} - 15625 q^{25} - 42444 q^{29} + 16983 q^{32} + 24786 q^{36} - 101194 q^{37} - 253228 q^{43} - 33354 q^{44} + 204606 q^{46} + 281250 q^{50} - 50346 q^{53} + 190998 q^{58} + 320866 q^{64} + 53926 q^{67} - 484956 q^{71} + 308367 q^{72} - 910746 q^{74} - 929378 q^{79} - 531441 q^{81} + 1139526 q^{86} + 829926 q^{88} - 772956 q^{92} + 2860596 q^{99}+O(q^{100}) \) 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
−4.50000 + 7.79423i 0 −8.50000 14.7224i 0 0 0 −423.000 −364.500 + 631.333i 0
31.1 −4.50000 7.79423i 0 −8.50000 + 14.7224i 0 0 0 −423.000 −364.500 631.333i 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 49.7.d.a 2
7.b odd 2 1 CM 49.7.d.a 2
7.c even 3 1 7.7.b.a 1
7.c even 3 1 inner 49.7.d.a 2
7.d odd 6 1 7.7.b.a 1
7.d odd 6 1 inner 49.7.d.a 2
21.g even 6 1 63.7.d.a 1
21.h odd 6 1 63.7.d.a 1
28.f even 6 1 112.7.c.a 1
28.g odd 6 1 112.7.c.a 1
35.i odd 6 1 175.7.d.a 1
35.j even 6 1 175.7.d.a 1
35.k even 12 2 175.7.c.a 2
35.l odd 12 2 175.7.c.a 2
56.j odd 6 1 448.7.c.a 1
56.k odd 6 1 448.7.c.b 1
56.m even 6 1 448.7.c.b 1
56.p even 6 1 448.7.c.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.7.b.a 1 7.c even 3 1
7.7.b.a 1 7.d odd 6 1
49.7.d.a 2 1.a even 1 1 trivial
49.7.d.a 2 7.b odd 2 1 CM
49.7.d.a 2 7.c even 3 1 inner
49.7.d.a 2 7.d odd 6 1 inner
63.7.d.a 1 21.g even 6 1
63.7.d.a 1 21.h odd 6 1
112.7.c.a 1 28.f even 6 1
112.7.c.a 1 28.g odd 6 1
175.7.c.a 2 35.k even 12 2
175.7.c.a 2 35.l odd 12 2
175.7.d.a 1 35.i odd 6 1
175.7.d.a 1 35.j even 6 1
448.7.c.a 1 56.j odd 6 1
448.7.c.a 1 56.p even 6 1
448.7.c.b 1 56.k odd 6 1
448.7.c.b 1 56.m even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 9T_{2} + 81 \) acting on \(S_{7}^{\mathrm{new}}(49, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 9T + 81 \) 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
$11$ \( T^{2} + 1962 T + 3849444 \) 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
$23$ \( T^{2} - 22734 T + 516834756 \) Copy content Toggle raw display
$29$ \( (T + 21222)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 101194 T + 10240225636 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 126614)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 50346 T + 2534719716 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 53926 T + 2908013476 \) Copy content Toggle raw display
$71$ \( (T + 242478)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 929378 T + 863743466884 \) 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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