Properties

Label 49.6.c.a
Level $49$
Weight $6$
Character orbit 49.c
Analytic conductor $7.859$
Analytic rank $0$
Dimension $2$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.85880717084\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
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: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q - 11 \zeta_{6} q^{2} + (89 \zeta_{6} - 89) q^{4} + 627 q^{8} + 243 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 11 \zeta_{6} q^{2} + (89 \zeta_{6} - 89) q^{4} + 627 q^{8} + 243 \zeta_{6} q^{9} + ( - 76 \zeta_{6} + 76) q^{11} - 4049 \zeta_{6} q^{16} + ( - 2673 \zeta_{6} + 2673) q^{18} - 836 q^{22} + 4952 \zeta_{6} q^{23} + ( - 3125 \zeta_{6} + 3125) q^{25} + 7282 q^{29} + (24475 \zeta_{6} - 24475) q^{32} - 21627 q^{36} + 8886 \zeta_{6} q^{37} + 11748 q^{43} + 6764 \zeta_{6} q^{44} + ( - 54472 \zeta_{6} + 54472) q^{46} - 34375 q^{50} + (24550 \zeta_{6} - 24550) q^{53} - 80102 \zeta_{6} q^{58} + 139657 q^{64} + (69364 \zeta_{6} - 69364) q^{67} - 2224 q^{71} + 152361 \zeta_{6} q^{72} + ( - 97746 \zeta_{6} + 97746) q^{74} - 80168 \zeta_{6} q^{79} + (59049 \zeta_{6} - 59049) q^{81} - 129228 \zeta_{6} q^{86} + ( - 47652 \zeta_{6} + 47652) q^{88} - 440728 q^{92} + 18468 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 11 q^{2} - 89 q^{4} + 1254 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 11 q^{2} - 89 q^{4} + 1254 q^{8} + 243 q^{9} + 76 q^{11} - 4049 q^{16} + 2673 q^{18} - 1672 q^{22} + 4952 q^{23} + 3125 q^{25} + 14564 q^{29} - 24475 q^{32} - 43254 q^{36} + 8886 q^{37} + 23496 q^{43} + 6764 q^{44} + 54472 q^{46} - 68750 q^{50} - 24550 q^{53} - 80102 q^{58} + 279314 q^{64} - 69364 q^{67} - 4448 q^{71} + 152361 q^{72} + 97746 q^{74} - 80168 q^{79} - 59049 q^{81} - 129228 q^{86} + 47652 q^{88} - 881456 q^{92} + 36936 q^{99}+O(q^{100}) \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
0.500000 + 0.866025i
0.500000 0.866025i
−5.50000 9.52628i 0 −44.5000 + 77.0763i 0 0 0 627.000 121.500 + 210.444i 0
30.1 −5.50000 + 9.52628i 0 −44.5000 77.0763i 0 0 0 627.000 121.500 210.444i 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 49.6.c.a 2
7.b odd 2 1 CM 49.6.c.a 2
7.c even 3 1 49.6.a.b 1
7.c even 3 1 inner 49.6.c.a 2
7.d odd 6 1 49.6.a.b 1
7.d odd 6 1 inner 49.6.c.a 2
21.g even 6 1 441.6.a.a 1
21.h odd 6 1 441.6.a.a 1
28.f even 6 1 784.6.a.g 1
28.g odd 6 1 784.6.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.b 1 7.c even 3 1
49.6.a.b 1 7.d odd 6 1
49.6.c.a 2 1.a even 1 1 trivial
49.6.c.a 2 7.b odd 2 1 CM
49.6.c.a 2 7.c even 3 1 inner
49.6.c.a 2 7.d odd 6 1 inner
441.6.a.a 1 21.g even 6 1
441.6.a.a 1 21.h odd 6 1
784.6.a.g 1 28.f even 6 1
784.6.a.g 1 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(49, [\chi])\):

\( T_{2}^{2} + 11T_{2} + 121 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 11T + 121 \) 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
$11$ \( T^{2} - 76T + 5776 \) 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
$23$ \( T^{2} - 4952 T + 24522304 \) Copy content Toggle raw display
$29$ \( (T - 7282)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 8886 T + 78960996 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 11748)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 24550 T + 602702500 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 69364 T + 4811364496 \) Copy content Toggle raw display
$71$ \( (T + 2224)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 80168 T + 6426908224 \) 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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