Properties

Label 448.2.f.a
Level 448
Weight 2
Character orbit 448.f
Analytic conductor 3.577
Analytic rank 1
Dimension 2
CM No
Inner twists 2

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{3} -2 \beta q^{5} + ( -2 - \beta ) q^{7} + q^{9} +O(q^{10})\) \( q -2 q^{3} -2 \beta q^{5} + ( -2 - \beta ) q^{7} + q^{9} + 2 \beta q^{11} + 2 \beta q^{13} + 4 \beta q^{15} -2 q^{19} + ( 4 + 2 \beta ) q^{21} + 2 \beta q^{23} -7 q^{25} + 4 q^{27} -6 q^{29} -8 q^{31} -4 \beta q^{33} + ( -6 + 4 \beta ) q^{35} + 2 q^{37} -4 \beta q^{39} + 4 \beta q^{41} -6 \beta q^{43} -2 \beta q^{45} + ( 1 + 4 \beta ) q^{49} -6 q^{53} + 12 q^{55} + 4 q^{57} -6 q^{59} + 2 \beta q^{61} + ( -2 - \beta ) q^{63} + 12 q^{65} + 2 \beta q^{67} -4 \beta q^{69} -2 \beta q^{71} -4 \beta q^{73} + 14 q^{75} + ( 6 - 4 \beta ) q^{77} -2 \beta q^{79} -11 q^{81} + 6 q^{83} + 12 q^{87} + 4 \beta q^{89} + ( 6 - 4 \beta ) q^{91} + 16 q^{93} + 4 \beta q^{95} -8 \beta q^{97} + 2 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} - 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 4q^{3} - 4q^{7} + 2q^{9} - 4q^{19} + 8q^{21} - 14q^{25} + 8q^{27} - 12q^{29} - 16q^{31} - 12q^{35} + 4q^{37} + 2q^{49} - 12q^{53} + 24q^{55} + 8q^{57} - 12q^{59} - 4q^{63} + 24q^{65} + 28q^{75} + 12q^{77} - 22q^{81} + 12q^{83} + 24q^{87} + 12q^{91} + 32q^{93} + O(q^{100}) \)

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 + 0.866025i
0.500000 0.866025i
0 −2.00000 0 3.46410i 0 −2.00000 1.73205i 0 1.00000 0
447.2 0 −2.00000 0 3.46410i 0 −2.00000 + 1.73205i 0 1.00000 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
28.d Even 1 yes

Hecke kernels

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