Properties

Label 49.2.c.a
Level $49$
Weight $2$
Character orbit 49.c
Analytic conductor $0.391$
Analytic rank $0$
Dimension $2$
CM discriminant -7
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.391266969904\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} -3 q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} -3 q^{8} + 3 \zeta_{6} q^{9} + ( -4 + 4 \zeta_{6} ) q^{11} + \zeta_{6} q^{16} + ( 3 - 3 \zeta_{6} ) q^{18} + 4 q^{22} -8 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} + 2 q^{29} + ( -5 + 5 \zeta_{6} ) q^{32} + 3 q^{36} + 6 \zeta_{6} q^{37} -12 q^{43} + 4 \zeta_{6} q^{44} + ( -8 + 8 \zeta_{6} ) q^{46} -5 q^{50} + ( 10 - 10 \zeta_{6} ) q^{53} -2 \zeta_{6} q^{58} + 7 q^{64} + ( -4 + 4 \zeta_{6} ) q^{67} + 16 q^{71} -9 \zeta_{6} q^{72} + ( 6 - 6 \zeta_{6} ) q^{74} -8 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + 12 \zeta_{6} q^{86} + ( 12 - 12 \zeta_{6} ) q^{88} -8 q^{92} -12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{4} - 6q^{8} + 3q^{9} + O(q^{10}) \) \( 2q - q^{2} + q^{4} - 6q^{8} + 3q^{9} - 4q^{11} + q^{16} + 3q^{18} + 8q^{22} - 8q^{23} + 5q^{25} + 4q^{29} - 5q^{32} + 6q^{36} + 6q^{37} - 24q^{43} + 4q^{44} - 8q^{46} - 10q^{50} + 10q^{53} - 2q^{58} + 14q^{64} - 4q^{67} + 32q^{71} - 9q^{72} + 6q^{74} - 8q^{79} - 9q^{81} + 12q^{86} + 12q^{88} - 16q^{92} - 24q^{99} + O(q^{100}) \)

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
−0.500000 0.866025i 0 0.500000 0.866025i 0 0 0 −3.00000 1.50000 + 2.59808i 0
30.1 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 0 0 −3.00000 1.50000 2.59808i 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 49.2.c.a 2
3.b odd 2 1 441.2.e.d 2
4.b odd 2 1 784.2.i.f 2
7.b odd 2 1 CM 49.2.c.a 2
7.c even 3 1 49.2.a.a 1
7.c even 3 1 inner 49.2.c.a 2
7.d odd 6 1 49.2.a.a 1
7.d odd 6 1 inner 49.2.c.a 2
21.c even 2 1 441.2.e.d 2
21.g even 6 1 441.2.a.c 1
21.g even 6 1 441.2.e.d 2
21.h odd 6 1 441.2.a.c 1
21.h odd 6 1 441.2.e.d 2
28.d even 2 1 784.2.i.f 2
28.f even 6 1 784.2.a.f 1
28.f even 6 1 784.2.i.f 2
28.g odd 6 1 784.2.a.f 1
28.g odd 6 1 784.2.i.f 2
35.i odd 6 1 1225.2.a.c 1
35.j even 6 1 1225.2.a.c 1
35.k even 12 2 1225.2.b.c 2
35.l odd 12 2 1225.2.b.c 2
56.j odd 6 1 3136.2.a.n 1
56.k odd 6 1 3136.2.a.o 1
56.m even 6 1 3136.2.a.o 1
56.p even 6 1 3136.2.a.n 1
77.h odd 6 1 5929.2.a.c 1
77.i even 6 1 5929.2.a.c 1
84.j odd 6 1 7056.2.a.bg 1
84.n even 6 1 7056.2.a.bg 1
91.r even 6 1 8281.2.a.d 1
91.s odd 6 1 8281.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.a.a 1 7.c even 3 1
49.2.a.a 1 7.d odd 6 1
49.2.c.a 2 1.a even 1 1 trivial
49.2.c.a 2 7.b odd 2 1 CM
49.2.c.a 2 7.c even 3 1 inner
49.2.c.a 2 7.d odd 6 1 inner
441.2.a.c 1 21.g even 6 1
441.2.a.c 1 21.h odd 6 1
441.2.e.d 2 3.b odd 2 1
441.2.e.d 2 21.c even 2 1
441.2.e.d 2 21.g even 6 1
441.2.e.d 2 21.h odd 6 1
784.2.a.f 1 28.f even 6 1
784.2.a.f 1 28.g odd 6 1
784.2.i.f 2 4.b odd 2 1
784.2.i.f 2 28.d even 2 1
784.2.i.f 2 28.f even 6 1
784.2.i.f 2 28.g odd 6 1
1225.2.a.c 1 35.i odd 6 1
1225.2.a.c 1 35.j even 6 1
1225.2.b.c 2 35.k even 12 2
1225.2.b.c 2 35.l odd 12 2
3136.2.a.n 1 56.j odd 6 1
3136.2.a.n 1 56.p even 6 1
3136.2.a.o 1 56.k odd 6 1
3136.2.a.o 1 56.m even 6 1
5929.2.a.c 1 77.h odd 6 1
5929.2.a.c 1 77.i even 6 1
7056.2.a.bg 1 84.j odd 6 1
7056.2.a.bg 1 84.n even 6 1
8281.2.a.d 1 91.r even 6 1
8281.2.a.d 1 91.s odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(49, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( 16 + 4 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( 64 + 8 T + T^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 36 - 6 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 12 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( 100 - 10 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( T^{2} \)
$67$ \( 16 + 4 T + T^{2} \)
$71$ \( ( -16 + T )^{2} \)
$73$ \( T^{2} \)
$79$ \( 64 + 8 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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