Properties

Label 448.2.f.b
Level 448
Weight 2
Character orbit 448.f
Analytic conductor 3.577
Analytic rank 0
Dimension 2
CM disc. -7
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 448.f (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{7} -3 q^{9} +O(q^{10})\) \( q -\beta q^{7} -3 q^{9} -2 \beta q^{11} -2 \beta q^{23} + 5 q^{25} + 2 q^{29} -6 q^{37} -2 \beta q^{43} -7 q^{49} + 10 q^{53} + 3 \beta q^{63} + 6 \beta q^{67} -2 \beta q^{71} -14 q^{77} + 6 \beta q^{79} + 9 q^{81} + 6 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{9} + O(q^{10}) \) \( 2q - 6q^{9} + 10q^{25} + 4q^{29} - 12q^{37} - 14q^{49} + 20q^{53} - 28q^{77} + 18q^{81} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
447.1
0.500000 + 1.32288i
0.500000 1.32288i
0 0 0 0 0 2.64575i 0 −3.00000 0
447.2 0 0 0 0 0 2.64575i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.b Odd 1 CM by \(\Q(\sqrt{-7}) \) yes
4.b Odd 1 yes
28.d Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(448, [\chi])\).