Properties

Label 49.4.c.d
Level $49$
Weight $4$
Character orbit 49.c
Analytic conductor $2.891$
Analytic rank $0$
Dimension $2$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.89109359028\)
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.

\(f(q)\) \(=\) \( q + 5 \zeta_{6} q^{2} + (17 \zeta_{6} - 17) q^{4} - 45 q^{8} + 27 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 \zeta_{6} q^{2} + (17 \zeta_{6} - 17) q^{4} - 45 q^{8} + 27 \zeta_{6} q^{9} + ( - 68 \zeta_{6} + 68) q^{11} - 89 \zeta_{6} q^{16} + (135 \zeta_{6} - 135) q^{18} + 340 q^{22} + 40 \zeta_{6} q^{23} + ( - 125 \zeta_{6} + 125) q^{25} - 166 q^{29} + ( - 85 \zeta_{6} + 85) q^{32} - 459 q^{36} - 450 \zeta_{6} q^{37} - 180 q^{43} + 1156 \zeta_{6} q^{44} + (200 \zeta_{6} - 200) q^{46} + 625 q^{50} + (590 \zeta_{6} - 590) q^{53} - 830 \zeta_{6} q^{58} - 287 q^{64} + ( - 740 \zeta_{6} + 740) q^{67} + 688 q^{71} - 1215 \zeta_{6} q^{72} + ( - 2250 \zeta_{6} + 2250) q^{74} + 1384 \zeta_{6} q^{79} + (729 \zeta_{6} - 729) q^{81} - 900 \zeta_{6} q^{86} + (3060 \zeta_{6} - 3060) q^{88} - 680 q^{92} + 1836 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{2} - 17 q^{4} - 90 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{2} - 17 q^{4} - 90 q^{8} + 27 q^{9} + 68 q^{11} - 89 q^{16} - 135 q^{18} + 680 q^{22} + 40 q^{23} + 125 q^{25} - 332 q^{29} + 85 q^{32} - 918 q^{36} - 450 q^{37} - 360 q^{43} + 1156 q^{44} - 200 q^{46} + 1250 q^{50} - 590 q^{53} - 830 q^{58} - 574 q^{64} + 740 q^{67} + 1376 q^{71} - 1215 q^{72} + 2250 q^{74} + 1384 q^{79} - 729 q^{81} - 900 q^{86} - 3060 q^{88} - 1360 q^{92} + 3672 q^{99}+O(q^{100}) \) 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
2.50000 + 4.33013i 0 −8.50000 + 14.7224i 0 0 0 −45.0000 13.5000 + 23.3827i 0
30.1 2.50000 4.33013i 0 −8.50000 14.7224i 0 0 0 −45.0000 13.5000 23.3827i 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 49.4.c.d 2
3.b odd 2 1 441.4.e.a 2
7.b odd 2 1 CM 49.4.c.d 2
7.c even 3 1 49.4.a.a 1
7.c even 3 1 inner 49.4.c.d 2
7.d odd 6 1 49.4.a.a 1
7.d odd 6 1 inner 49.4.c.d 2
21.c even 2 1 441.4.e.a 2
21.g even 6 1 441.4.a.m 1
21.g even 6 1 441.4.e.a 2
21.h odd 6 1 441.4.a.m 1
21.h odd 6 1 441.4.e.a 2
28.f even 6 1 784.4.a.k 1
28.g odd 6 1 784.4.a.k 1
35.i odd 6 1 1225.4.a.l 1
35.j even 6 1 1225.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.a 1 7.c even 3 1
49.4.a.a 1 7.d odd 6 1
49.4.c.d 2 1.a even 1 1 trivial
49.4.c.d 2 7.b odd 2 1 CM
49.4.c.d 2 7.c even 3 1 inner
49.4.c.d 2 7.d odd 6 1 inner
441.4.a.m 1 21.g even 6 1
441.4.a.m 1 21.h odd 6 1
441.4.e.a 2 3.b odd 2 1
441.4.e.a 2 21.c even 2 1
441.4.e.a 2 21.g even 6 1
441.4.e.a 2 21.h odd 6 1
784.4.a.k 1 28.f even 6 1
784.4.a.k 1 28.g odd 6 1
1225.4.a.l 1 35.i odd 6 1
1225.4.a.l 1 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(49, [\chi])\):

\( T_{2}^{2} - 5T_{2} + 25 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 5T + 25 \) 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
$11$ \( T^{2} - 68T + 4624 \) 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
$23$ \( T^{2} - 40T + 1600 \) Copy content Toggle raw display
$29$ \( (T + 166)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 450T + 202500 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 180)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 590T + 348100 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 740T + 547600 \) Copy content Toggle raw display
$71$ \( (T - 688)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 1384 T + 1915456 \) 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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