Properties

Label 448.2.f.d
Level 448
Weight 2
Character orbit 448.f
Analytic conductor 3.577
Analytic rank 0
Dimension 8
CM No
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: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} -\beta_{5} q^{5} -\beta_{4} q^{7} + ( 1 + \beta_{7} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} -\beta_{5} q^{5} -\beta_{4} q^{7} + ( 1 + \beta_{7} ) q^{9} + \beta_{1} q^{11} + ( -\beta_{3} - \beta_{5} ) q^{13} + ( \beta_{1} - \beta_{4} - \beta_{6} ) q^{15} + ( \beta_{3} - 2 \beta_{5} ) q^{17} + ( \beta_{2} + \beta_{4} - \beta_{6} ) q^{19} + ( \beta_{3} - \beta_{5} + \beta_{7} ) q^{21} + ( -3 \beta_{1} + 2 \beta_{4} + 2 \beta_{6} ) q^{23} + ( 1 + \beta_{7} ) q^{25} + ( -\beta_{4} + \beta_{6} ) q^{27} + ( 2 + 2 \beta_{7} ) q^{29} + ( 2 \beta_{2} + \beta_{4} - \beta_{6} ) q^{31} -\beta_{3} q^{33} + ( -3 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} ) q^{35} + ( 2 - 2 \beta_{7} ) q^{37} + ( 3 \beta_{1} + \beta_{4} + \beta_{6} ) q^{39} + ( -2 \beta_{3} - 2 \beta_{5} ) q^{41} + ( -\beta_{1} + 2 \beta_{4} + 2 \beta_{6} ) q^{43} + ( \beta_{3} + \beta_{5} ) q^{45} + ( -2 \beta_{2} + \beta_{4} - \beta_{6} ) q^{47} + ( 1 + \beta_{3} - 2 \beta_{7} ) q^{49} + ( -4 \beta_{4} - 4 \beta_{6} ) q^{51} + 2 q^{53} + ( -\beta_{4} + \beta_{6} ) q^{55} + ( -4 - 3 \beta_{7} ) q^{57} + ( 3 \beta_{2} - \beta_{4} + \beta_{6} ) q^{59} + \beta_{5} q^{61} + ( -\beta_{1} - 2 \beta_{2} - \beta_{4} - 2 \beta_{6} ) q^{63} + ( -4 + 3 \beta_{7} ) q^{65} + ( 7 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} ) q^{67} + ( -\beta_{3} + 4 \beta_{5} ) q^{69} + ( 4 \beta_{1} - \beta_{4} - \beta_{6} ) q^{71} + ( \beta_{3} + 4 \beta_{5} ) q^{73} + ( -3 \beta_{2} - \beta_{4} + \beta_{6} ) q^{75} + ( 2 + 2 \beta_{5} + \beta_{7} ) q^{77} + ( -6 \beta_{1} + \beta_{4} + \beta_{6} ) q^{79} + ( -3 - \beta_{7} ) q^{81} + ( -\beta_{2} + 2 \beta_{4} - 2 \beta_{6} ) q^{83} -8 q^{85} + ( -6 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{87} + \beta_{3} q^{89} + ( -5 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} + 4 \beta_{6} ) q^{91} + ( -8 - 4 \beta_{7} ) q^{93} + ( 5 \beta_{1} - \beta_{4} - \beta_{6} ) q^{95} + ( -\beta_{3} + 6 \beta_{5} ) q^{97} + ( -\beta_{1} + 2 \beta_{4} + 2 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{9} + 8q^{25} + 16q^{29} + 16q^{37} + 8q^{49} + 16q^{53} - 32q^{57} - 32q^{65} + 16q^{77} - 24q^{81} - 64q^{85} - 64q^{93} + O(q^{100}) \)

Basis of coefficient ring:

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \zeta_{16}^{4} \)
\(\beta_{2}\)\(=\)\( -\zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}^{3} + \zeta_{16} \)
\(\beta_{3}\)\(=\)\( 2 \zeta_{16}^{7} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{3} + 2 \zeta_{16} \)
\(\beta_{4}\)\(=\)\( -\zeta_{16}^{7} + \zeta_{16}^{6} + \zeta_{16}^{5} + \zeta_{16}^{4} - \zeta_{16}^{3} + \zeta_{16}^{2} + \zeta_{16} \)
\(\beta_{5}\)\(=\)\( \zeta_{16}^{7} - \zeta_{16}^{5} - \zeta_{16}^{3} + \zeta_{16} \)
\(\beta_{6}\)\(=\)\( \zeta_{16}^{7} + \zeta_{16}^{6} - \zeta_{16}^{5} + \zeta_{16}^{4} + \zeta_{16}^{3} + \zeta_{16}^{2} - \zeta_{16} \)
\(\beta_{7}\)\(=\)\( -2 \zeta_{16}^{6} + 2 \zeta_{16}^{2} \)
\(1\)\(=\)\(\beta_0\)
\(\zeta_{16}\)\(=\)\((\)\(-\beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2}\)\()/8\)
\(\zeta_{16}^{2}\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{4} - \beta_{1}\)\()/4\)
\(\zeta_{16}^{3}\)\(=\)\((\)\(\beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2}\)\()/8\)
\(\zeta_{16}^{4}\)\(=\)\(\beta_{1}\)\(/2\)
\(\zeta_{16}^{5}\)\(=\)\((\)\(-\beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2}\)\()/8\)
\(\zeta_{16}^{6}\)\(=\)\((\)\(-\beta_{7} + \beta_{6} + \beta_{4} - \beta_{1}\)\()/4\)
\(\zeta_{16}^{7}\)\(=\)\((\)\(\beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2}\)\()/8\)

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
0.923880 0.382683i
0.923880 + 0.382683i
−0.382683 + 0.923880i
−0.382683 0.923880i
0.382683 + 0.923880i
0.382683 0.923880i
−0.923880 0.382683i
−0.923880 + 0.382683i
0 −2.61313 0 1.08239i 0 −1.08239 + 2.41421i 0 3.82843 0
447.2 0 −2.61313 0 1.08239i 0 −1.08239 2.41421i 0 3.82843 0
447.3 0 −1.08239 0 2.61313i 0 2.61313 + 0.414214i 0 −1.82843 0
447.4 0 −1.08239 0 2.61313i 0 2.61313 0.414214i 0 −1.82843 0
447.5 0 1.08239 0 2.61313i 0 −2.61313 0.414214i 0 −1.82843 0
447.6 0 1.08239 0 2.61313i 0 −2.61313 + 0.414214i 0 −1.82843 0
447.7 0 2.61313 0 1.08239i 0 1.08239 2.41421i 0 3.82843 0
447.8 0 2.61313 0 1.08239i 0 1.08239 + 2.41421i 0 3.82843 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 447.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes
7.b Odd 1 yes
28.d Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{3}^{4} - 8 T_{3}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(448, [\chi])\).