Properties

Label 16.5.c.a
Level 16
Weight 5
Character orbit 16.c
Analytic conductor 1.654
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 16 = 2^{4} \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 16.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.65391940934\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 8\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} + 18 q^{5} + 2 \beta q^{7} -111 q^{9} +O(q^{10})\) \( q -\beta q^{3} + 18 q^{5} + 2 \beta q^{7} -111 q^{9} + 9 \beta q^{11} + 178 q^{13} -18 \beta q^{15} -126 q^{17} -29 \beta q^{19} + 384 q^{21} + 54 \beta q^{23} -301 q^{25} + 30 \beta q^{27} -1422 q^{29} -24 \beta q^{31} + 1728 q^{33} + 36 \beta q^{35} + 530 q^{37} -178 \beta q^{39} + 162 q^{41} -111 \beta q^{43} -1998 q^{45} + 252 \beta q^{47} + 1633 q^{49} + 126 \beta q^{51} + 594 q^{53} + 162 \beta q^{55} -5568 q^{57} -171 \beta q^{59} + 626 q^{61} -222 \beta q^{63} + 3204 q^{65} + 79 \beta q^{67} + 10368 q^{69} -558 \beta q^{71} -6686 q^{73} + 301 \beta q^{75} -3456 q^{77} + 100 \beta q^{79} -3231 q^{81} -333 \beta q^{83} -2268 q^{85} + 1422 \beta q^{87} + 8226 q^{89} + 356 \beta q^{91} -4608 q^{93} -522 \beta q^{95} -1598 q^{97} -999 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 36q^{5} - 222q^{9} + O(q^{10}) \) \( 2q + 36q^{5} - 222q^{9} + 356q^{13} - 252q^{17} + 768q^{21} - 602q^{25} - 2844q^{29} + 3456q^{33} + 1060q^{37} + 324q^{41} - 3996q^{45} + 3266q^{49} + 1188q^{53} - 11136q^{57} + 1252q^{61} + 6408q^{65} + 20736q^{69} - 13372q^{73} - 6912q^{77} - 6462q^{81} - 4536q^{85} + 16452q^{89} - 9216q^{93} - 3196q^{97} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(15\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
0.500000 + 0.866025i
0.500000 0.866025i
0 13.8564i 0 18.0000 0 27.7128i 0 −111.000 0
15.2 0 13.8564i 0 18.0000 0 27.7128i 0 −111.000 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
4.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{5}^{\mathrm{new}}(16, [\chi])\).