Properties

Label 8.4.b.a
Level 8
Weight 4
Character orbit 8.b
Analytic conductor 0.472
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 8.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.472015280046\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( -1 - \beta ) q^{2} \) \( + 2 \beta q^{3} \) \( + ( -6 + 2 \beta ) q^{4} \) \( -4 \beta q^{5} \) \( + ( 14 - 2 \beta ) q^{6} \) \( -8 q^{7} \) \( + ( 20 + 4 \beta ) q^{8} \) \(- q^{9}\) \(+O(q^{10})\) \( q\) \( + ( -1 - \beta ) q^{2} \) \( + 2 \beta q^{3} \) \( + ( -6 + 2 \beta ) q^{4} \) \( -4 \beta q^{5} \) \( + ( 14 - 2 \beta ) q^{6} \) \( -8 q^{7} \) \( + ( 20 + 4 \beta ) q^{8} \) \(- q^{9}\) \( + ( -28 + 4 \beta ) q^{10} \) \( -6 \beta q^{11} \) \( + ( -28 - 12 \beta ) q^{12} \) \( + 20 \beta q^{13} \) \( + ( 8 + 8 \beta ) q^{14} \) \( + 56 q^{15} \) \( + ( 8 - 24 \beta ) q^{16} \) \( -14 q^{17} \) \( + ( 1 + \beta ) q^{18} \) \( -14 \beta q^{19} \) \( + ( 56 + 24 \beta ) q^{20} \) \( -16 \beta q^{21} \) \( + ( -42 + 6 \beta ) q^{22} \) \( -152 q^{23} \) \( + ( -56 + 40 \beta ) q^{24} \) \( + 13 q^{25} \) \( + ( 140 - 20 \beta ) q^{26} \) \( + 52 \beta q^{27} \) \( + ( 48 - 16 \beta ) q^{28} \) \( -60 \beta q^{29} \) \( + ( -56 - 56 \beta ) q^{30} \) \( + 224 q^{31} \) \( + ( -176 + 16 \beta ) q^{32} \) \( + 84 q^{33} \) \( + ( 14 + 14 \beta ) q^{34} \) \( + 32 \beta q^{35} \) \( + ( 6 - 2 \beta ) q^{36} \) \( + 92 \beta q^{37} \) \( + ( -98 + 14 \beta ) q^{38} \) \( -280 q^{39} \) \( + ( 112 - 80 \beta ) q^{40} \) \( -70 q^{41} \) \( + ( -112 + 16 \beta ) q^{42} \) \( -166 \beta q^{43} \) \( + ( 84 + 36 \beta ) q^{44} \) \( + 4 \beta q^{45} \) \( + ( 152 + 152 \beta ) q^{46} \) \( + 336 q^{47} \) \( + ( 336 + 16 \beta ) q^{48} \) \( -279 q^{49} \) \( + ( -13 - 13 \beta ) q^{50} \) \( -28 \beta q^{51} \) \( + ( -280 - 120 \beta ) q^{52} \) \( + 12 \beta q^{53} \) \( + ( 364 - 52 \beta ) q^{54} \) \( -168 q^{55} \) \( + ( -160 - 32 \beta ) q^{56} \) \( + 196 q^{57} \) \( + ( -420 + 60 \beta ) q^{58} \) \( + 202 \beta q^{59} \) \( + ( -336 + 112 \beta ) q^{60} \) \( + 36 \beta q^{61} \) \( + ( -224 - 224 \beta ) q^{62} \) \( + 8 q^{63} \) \( + ( 288 + 160 \beta ) q^{64} \) \( + 560 q^{65} \) \( + ( -84 - 84 \beta ) q^{66} \) \( + 66 \beta q^{67} \) \( + ( 84 - 28 \beta ) q^{68} \) \( -304 \beta q^{69} \) \( + ( 224 - 32 \beta ) q^{70} \) \( -72 q^{71} \) \( + ( -20 - 4 \beta ) q^{72} \) \( -294 q^{73} \) \( + ( 644 - 92 \beta ) q^{74} \) \( + 26 \beta q^{75} \) \( + ( 196 + 84 \beta ) q^{76} \) \( + 48 \beta q^{77} \) \( + ( 280 + 280 \beta ) q^{78} \) \( -464 q^{79} \) \( + ( -672 - 32 \beta ) q^{80} \) \( -755 q^{81} \) \( + ( 70 + 70 \beta ) q^{82} \) \( -206 \beta q^{83} \) \( + ( 224 + 96 \beta ) q^{84} \) \( + 56 \beta q^{85} \) \( + ( -1162 + 166 \beta ) q^{86} \) \( + 840 q^{87} \) \( + ( 168 - 120 \beta ) q^{88} \) \( + 266 q^{89} \) \( + ( 28 - 4 \beta ) q^{90} \) \( -160 \beta q^{91} \) \( + ( 912 - 304 \beta ) q^{92} \) \( + 448 \beta q^{93} \) \( + ( -336 - 336 \beta ) q^{94} \) \( -392 q^{95} \) \( + ( -224 - 352 \beta ) q^{96} \) \( + 994 q^{97} \) \( + ( 279 + 279 \beta ) q^{98} \) \( + 6 \beta q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 12q^{4} \) \(\mathstrut +\mathstrut 28q^{6} \) \(\mathstrut -\mathstrut 16q^{7} \) \(\mathstrut +\mathstrut 40q^{8} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 12q^{4} \) \(\mathstrut +\mathstrut 28q^{6} \) \(\mathstrut -\mathstrut 16q^{7} \) \(\mathstrut +\mathstrut 40q^{8} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut -\mathstrut 56q^{10} \) \(\mathstrut -\mathstrut 56q^{12} \) \(\mathstrut +\mathstrut 16q^{14} \) \(\mathstrut +\mathstrut 112q^{15} \) \(\mathstrut +\mathstrut 16q^{16} \) \(\mathstrut -\mathstrut 28q^{17} \) \(\mathstrut +\mathstrut 2q^{18} \) \(\mathstrut +\mathstrut 112q^{20} \) \(\mathstrut -\mathstrut 84q^{22} \) \(\mathstrut -\mathstrut 304q^{23} \) \(\mathstrut -\mathstrut 112q^{24} \) \(\mathstrut +\mathstrut 26q^{25} \) \(\mathstrut +\mathstrut 280q^{26} \) \(\mathstrut +\mathstrut 96q^{28} \) \(\mathstrut -\mathstrut 112q^{30} \) \(\mathstrut +\mathstrut 448q^{31} \) \(\mathstrut -\mathstrut 352q^{32} \) \(\mathstrut +\mathstrut 168q^{33} \) \(\mathstrut +\mathstrut 28q^{34} \) \(\mathstrut +\mathstrut 12q^{36} \) \(\mathstrut -\mathstrut 196q^{38} \) \(\mathstrut -\mathstrut 560q^{39} \) \(\mathstrut +\mathstrut 224q^{40} \) \(\mathstrut -\mathstrut 140q^{41} \) \(\mathstrut -\mathstrut 224q^{42} \) \(\mathstrut +\mathstrut 168q^{44} \) \(\mathstrut +\mathstrut 304q^{46} \) \(\mathstrut +\mathstrut 672q^{47} \) \(\mathstrut +\mathstrut 672q^{48} \) \(\mathstrut -\mathstrut 558q^{49} \) \(\mathstrut -\mathstrut 26q^{50} \) \(\mathstrut -\mathstrut 560q^{52} \) \(\mathstrut +\mathstrut 728q^{54} \) \(\mathstrut -\mathstrut 336q^{55} \) \(\mathstrut -\mathstrut 320q^{56} \) \(\mathstrut +\mathstrut 392q^{57} \) \(\mathstrut -\mathstrut 840q^{58} \) \(\mathstrut -\mathstrut 672q^{60} \) \(\mathstrut -\mathstrut 448q^{62} \) \(\mathstrut +\mathstrut 16q^{63} \) \(\mathstrut +\mathstrut 576q^{64} \) \(\mathstrut +\mathstrut 1120q^{65} \) \(\mathstrut -\mathstrut 168q^{66} \) \(\mathstrut +\mathstrut 168q^{68} \) \(\mathstrut +\mathstrut 448q^{70} \) \(\mathstrut -\mathstrut 144q^{71} \) \(\mathstrut -\mathstrut 40q^{72} \) \(\mathstrut -\mathstrut 588q^{73} \) \(\mathstrut +\mathstrut 1288q^{74} \) \(\mathstrut +\mathstrut 392q^{76} \) \(\mathstrut +\mathstrut 560q^{78} \) \(\mathstrut -\mathstrut 928q^{79} \) \(\mathstrut -\mathstrut 1344q^{80} \) \(\mathstrut -\mathstrut 1510q^{81} \) \(\mathstrut +\mathstrut 140q^{82} \) \(\mathstrut +\mathstrut 448q^{84} \) \(\mathstrut -\mathstrut 2324q^{86} \) \(\mathstrut +\mathstrut 1680q^{87} \) \(\mathstrut +\mathstrut 336q^{88} \) \(\mathstrut +\mathstrut 532q^{89} \) \(\mathstrut +\mathstrut 56q^{90} \) \(\mathstrut +\mathstrut 1824q^{92} \) \(\mathstrut -\mathstrut 672q^{94} \) \(\mathstrut -\mathstrut 784q^{95} \) \(\mathstrut -\mathstrut 448q^{96} \) \(\mathstrut +\mathstrut 1988q^{97} \) \(\mathstrut +\mathstrut 558q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 + 1.32288i
0.500000 1.32288i
−1.00000 2.64575i 5.29150i −6.00000 + 5.29150i 10.5830i 14.0000 5.29150i −8.00000 20.0000 + 10.5830i −1.00000 −28.0000 + 10.5830i
5.2 −1.00000 + 2.64575i 5.29150i −6.00000 5.29150i 10.5830i 14.0000 + 5.29150i −8.00000 20.0000 10.5830i −1.00000 −28.0000 10.5830i
\(n\): 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
8.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{4}^{\mathrm{new}}(8, [\chi])\).