Properties

Label 32.2.g.a
Level 32
Weight 2
Character orbit 32.g
Analytic conductor 0.256
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 32 = 2^{5} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 32.g (of order \(8\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.255521286468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{2} \) \( + ( \zeta_{8} + \zeta_{8}^{2} ) q^{3} \) \( -2 q^{4} \) \( + ( -1 - \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{5} \) \( + ( 1 + \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} \) \( + ( 1 + \zeta_{8}^{2} ) q^{7} \) \( + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} \) \( + ( -1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{2} \) \( + ( \zeta_{8} + \zeta_{8}^{2} ) q^{3} \) \( -2 q^{4} \) \( + ( -1 - \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{5} \) \( + ( 1 + \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} \) \( + ( 1 + \zeta_{8}^{2} ) q^{7} \) \( + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} \) \( + ( -1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{9} \) \( + ( 1 - \zeta_{8} + 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{10} \) \( + ( -2 + 2 \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{11} \) \( + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{12} \) \( + ( 1 - \zeta_{8}^{3} ) q^{13} \) \( -2 \zeta_{8}^{3} q^{14} \) \( + ( -3 \zeta_{8} - 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{15} \) \( + 4 q^{16} \) \( + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{17} \) \( + ( -1 + 2 \zeta_{8} - \zeta_{8}^{2} ) q^{18} \) \( + ( -2 - 3 \zeta_{8} + 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{19} \) \( + ( 2 + 2 \zeta_{8} + 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{20} \) \( + ( -1 + \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{21} \) \( + ( 1 + \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{22} \) \( + ( 3 - 3 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{23} \) \( + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{24} \) \( + ( 1 + 5 \zeta_{8} + \zeta_{8}^{2} ) q^{25} \) \( + ( -1 - \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{26} \) \( + ( 3 + 3 \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{27} \) \( + ( -2 - 2 \zeta_{8}^{2} ) q^{28} \) \( + ( -1 + 2 \zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{29} \) \( + ( -6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{30} \) \( -4 q^{31} \) \( + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{32} \) \( + ( 2 - \zeta_{8} + \zeta_{8}^{3} ) q^{33} \) \( -4 q^{34} \) \( + ( 1 - 3 \zeta_{8} - 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{35} \) \( + ( 2 - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{36} \) \( + ( 1 + \zeta_{8} ) q^{37} \) \( + ( -1 + 5 \zeta_{8} + 5 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{38} \) \( + ( 1 + 2 \zeta_{8} + \zeta_{8}^{2} ) q^{39} \) \( + ( -2 + 2 \zeta_{8} - 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{40} \) \( + ( -3 + 3 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{41} \) \( + ( 2 + 2 \zeta_{8} ) q^{42} \) \( + ( 4 - 4 \zeta_{8} - 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{43} \) \( + ( 4 - 4 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{44} \) \( + ( 2 - 3 \zeta_{8} + 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{45} \) \( + ( 4 - 6 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{46} \) \( + ( 4 \zeta_{8} + 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{47} \) \( + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{48} \) \( -5 \zeta_{8}^{2} q^{49} \) \( + ( 5 - 5 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{50} \) \( + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{51} \) \( + ( -2 + 2 \zeta_{8}^{3} ) q^{52} \) \( + ( 1 - \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{53} \) \( + ( 4 - 4 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{54} \) \( + ( -5 + 5 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{55} \) \( + 4 \zeta_{8}^{3} q^{56} \) \( + ( -5 - 4 \zeta_{8} - 5 \zeta_{8}^{2} ) q^{57} \) \( + ( 1 + 3 \zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{58} \) \( + ( -4 - 4 \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{59} \) \( + ( 6 \zeta_{8} + 4 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{60} \) \( + ( 1 + \zeta_{8}^{3} ) q^{61} \) \( + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{62} \) \( + ( -2 + \zeta_{8} - \zeta_{8}^{3} ) q^{63} \) \( -8 q^{64} \) \( + ( -2 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{65} \) \( + ( -2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{66} \) \( + ( -2 + 3 \zeta_{8} + 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{67} \) \( + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{68} \) \( + ( -1 - \zeta_{8} + 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{69} \) \( + ( -2 - 4 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{70} \) \( + ( -3 + 4 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{71} \) \( + ( 2 - 4 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{72} \) \( + ( 7 - 7 \zeta_{8}^{2} ) q^{73} \) \( + ( 1 - \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{74} \) \( + ( -1 + \zeta_{8} + 6 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{75} \) \( + ( 4 + 6 \zeta_{8} - 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{76} \) \( + ( -1 + 3 \zeta_{8} - 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{77} \) \( + ( 2 - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{78} \) \( -6 \zeta_{8}^{2} q^{79} \) \( + ( -4 - 4 \zeta_{8} - 8 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{80} \) \( + ( 5 \zeta_{8} + 3 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{81} \) \( + ( 4 + 6 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{82} \) \( + ( 4 - \zeta_{8} + \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{83} \) \( + ( 2 - 2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{84} \) \( + ( 2 - 2 \zeta_{8} + 6 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{85} \) \( + ( -7 - 7 \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{86} \) \( + ( -1 + \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{87} \) \( + ( -2 - 2 \zeta_{8} + 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{88} \) \( + ( 3 - 8 \zeta_{8} + 3 \zeta_{8}^{2} ) q^{89} \) \( + ( -5 + \zeta_{8} + \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{90} \) \( + ( 1 + \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{91} \) \( + ( -6 + 6 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{92} \) \( + ( -4 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{93} \) \( + ( 8 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{94} \) \( + ( 16 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{95} \) \( + ( 4 + 4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{96} \) \( + ( -10 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{97} \) \( + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{98} \) \( + ( 5 - 2 \zeta_{8} - 2 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut 4q^{10} \) \(\mathstrut -\mathstrut 8q^{11} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut +\mathstrut 16q^{16} \) \(\mathstrut -\mathstrut 4q^{18} \) \(\mathstrut -\mathstrut 8q^{19} \) \(\mathstrut +\mathstrut 8q^{20} \) \(\mathstrut -\mathstrut 4q^{21} \) \(\mathstrut +\mathstrut 4q^{22} \) \(\mathstrut +\mathstrut 12q^{23} \) \(\mathstrut -\mathstrut 8q^{24} \) \(\mathstrut +\mathstrut 4q^{25} \) \(\mathstrut -\mathstrut 4q^{26} \) \(\mathstrut +\mathstrut 12q^{27} \) \(\mathstrut -\mathstrut 8q^{28} \) \(\mathstrut -\mathstrut 4q^{29} \) \(\mathstrut -\mathstrut 24q^{30} \) \(\mathstrut -\mathstrut 16q^{31} \) \(\mathstrut +\mathstrut 8q^{33} \) \(\mathstrut -\mathstrut 16q^{34} \) \(\mathstrut +\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 8q^{36} \) \(\mathstrut +\mathstrut 4q^{37} \) \(\mathstrut -\mathstrut 4q^{38} \) \(\mathstrut +\mathstrut 4q^{39} \) \(\mathstrut -\mathstrut 8q^{40} \) \(\mathstrut -\mathstrut 12q^{41} \) \(\mathstrut +\mathstrut 8q^{42} \) \(\mathstrut +\mathstrut 16q^{43} \) \(\mathstrut +\mathstrut 16q^{44} \) \(\mathstrut +\mathstrut 8q^{45} \) \(\mathstrut +\mathstrut 16q^{46} \) \(\mathstrut +\mathstrut 20q^{50} \) \(\mathstrut +\mathstrut 8q^{51} \) \(\mathstrut -\mathstrut 8q^{52} \) \(\mathstrut +\mathstrut 4q^{53} \) \(\mathstrut +\mathstrut 16q^{54} \) \(\mathstrut -\mathstrut 20q^{55} \) \(\mathstrut -\mathstrut 20q^{57} \) \(\mathstrut +\mathstrut 4q^{58} \) \(\mathstrut -\mathstrut 16q^{59} \) \(\mathstrut +\mathstrut 4q^{61} \) \(\mathstrut -\mathstrut 8q^{63} \) \(\mathstrut -\mathstrut 32q^{64} \) \(\mathstrut -\mathstrut 8q^{65} \) \(\mathstrut -\mathstrut 8q^{67} \) \(\mathstrut -\mathstrut 4q^{69} \) \(\mathstrut -\mathstrut 8q^{70} \) \(\mathstrut -\mathstrut 12q^{71} \) \(\mathstrut +\mathstrut 8q^{72} \) \(\mathstrut +\mathstrut 28q^{73} \) \(\mathstrut +\mathstrut 4q^{74} \) \(\mathstrut -\mathstrut 4q^{75} \) \(\mathstrut +\mathstrut 16q^{76} \) \(\mathstrut -\mathstrut 4q^{77} \) \(\mathstrut +\mathstrut 8q^{78} \) \(\mathstrut -\mathstrut 16q^{80} \) \(\mathstrut +\mathstrut 16q^{82} \) \(\mathstrut +\mathstrut 16q^{83} \) \(\mathstrut +\mathstrut 8q^{84} \) \(\mathstrut +\mathstrut 8q^{85} \) \(\mathstrut -\mathstrut 28q^{86} \) \(\mathstrut -\mathstrut 4q^{87} \) \(\mathstrut -\mathstrut 8q^{88} \) \(\mathstrut +\mathstrut 12q^{89} \) \(\mathstrut -\mathstrut 20q^{90} \) \(\mathstrut +\mathstrut 4q^{91} \) \(\mathstrut -\mathstrut 24q^{92} \) \(\mathstrut +\mathstrut 32q^{94} \) \(\mathstrut +\mathstrut 64q^{95} \) \(\mathstrut +\mathstrut 16q^{96} \) \(\mathstrut -\mathstrut 40q^{97} \) \(\mathstrut +\mathstrut 20q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(\zeta_{8}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
1.41421i 0.707107 + 1.70711i −2.00000 −3.12132 1.29289i 2.41421 1.00000i 1.00000 + 1.00000i 2.82843i −0.292893 + 0.292893i −1.82843 + 4.41421i
13.1 1.41421i 0.707107 1.70711i −2.00000 −3.12132 + 1.29289i 2.41421 + 1.00000i 1.00000 1.00000i 2.82843i −0.292893 0.292893i −1.82843 4.41421i
21.1 1.41421i −0.707107 + 0.292893i −2.00000 1.12132 2.70711i −0.414214 1.00000i 1.00000 + 1.00000i 2.82843i −1.70711 + 1.70711i 3.82843 + 1.58579i
29.1 1.41421i −0.707107 0.292893i −2.00000 1.12132 + 2.70711i −0.414214 + 1.00000i 1.00000 1.00000i 2.82843i −1.70711 1.70711i 3.82843 1.58579i
\(n\): 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
32.g Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{4} \) \(\mathstrut +\mathstrut 2 T_{3}^{2} \) \(\mathstrut +\mathstrut 4 T_{3} \) \(\mathstrut +\mathstrut 2 \) acting on \(S_{2}^{\mathrm{new}}(32, [\chi])\).