Properties

Label 8.5.d.b
Level 8
Weight 5
Character orbit 8.d
Analytic conductor 0.827
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 8.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.826959704671\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-15}) \)
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 \(\beta = \sqrt{-15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 - \beta ) q^{2} \) \( + 6 q^{3} \) \( + ( -14 + 2 \beta ) q^{4} \) \( + 8 \beta q^{5} \) \( + ( -6 - 6 \beta ) q^{6} \) \( -16 \beta q^{7} \) \( + ( 44 + 12 \beta ) q^{8} \) \( -45 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 - \beta ) q^{2} \) \( + 6 q^{3} \) \( + ( -14 + 2 \beta ) q^{4} \) \( + 8 \beta q^{5} \) \( + ( -6 - 6 \beta ) q^{6} \) \( -16 \beta q^{7} \) \( + ( 44 + 12 \beta ) q^{8} \) \( -45 q^{9} \) \( + ( 120 - 8 \beta ) q^{10} \) \( -26 q^{11} \) \( + ( -84 + 12 \beta ) q^{12} \) \( -8 \beta q^{13} \) \( + ( -240 + 16 \beta ) q^{14} \) \( + 48 \beta q^{15} \) \( + ( 136 - 56 \beta ) q^{16} \) \( + 226 q^{17} \) \( + ( 45 + 45 \beta ) q^{18} \) \( + 134 q^{19} \) \( + ( -240 - 112 \beta ) q^{20} \) \( -96 \beta q^{21} \) \( + ( 26 + 26 \beta ) q^{22} \) \( + 80 \beta q^{23} \) \( + ( 264 + 72 \beta ) q^{24} \) \( -335 q^{25} \) \( + ( -120 + 8 \beta ) q^{26} \) \( -756 q^{27} \) \( + ( 480 + 224 \beta ) q^{28} \) \( + 88 \beta q^{29} \) \( + ( 720 - 48 \beta ) q^{30} \) \( -320 \beta q^{31} \) \( + ( -976 - 80 \beta ) q^{32} \) \( -156 q^{33} \) \( + ( -226 - 226 \beta ) q^{34} \) \( + 1920 q^{35} \) \( + ( 630 - 90 \beta ) q^{36} \) \( + 456 \beta q^{37} \) \( + ( -134 - 134 \beta ) q^{38} \) \( -48 \beta q^{39} \) \( + ( -1440 + 352 \beta ) q^{40} \) \( + 994 q^{41} \) \( + ( -1440 + 96 \beta ) q^{42} \) \( -1882 q^{43} \) \( + ( 364 - 52 \beta ) q^{44} \) \( -360 \beta q^{45} \) \( + ( 1200 - 80 \beta ) q^{46} \) \( + 544 \beta q^{47} \) \( + ( 816 - 336 \beta ) q^{48} \) \( -1439 q^{49} \) \( + ( 335 + 335 \beta ) q^{50} \) \( + 1356 q^{51} \) \( + ( 240 + 112 \beta ) q^{52} \) \( -984 \beta q^{53} \) \( + ( 756 + 756 \beta ) q^{54} \) \( -208 \beta q^{55} \) \( + ( 2880 - 704 \beta ) q^{56} \) \( + 804 q^{57} \) \( + ( 1320 - 88 \beta ) q^{58} \) \( -5018 q^{59} \) \( + ( -1440 - 672 \beta ) q^{60} \) \( + 536 \beta q^{61} \) \( + ( -4800 + 320 \beta ) q^{62} \) \( + 720 \beta q^{63} \) \( + ( -224 + 1056 \beta ) q^{64} \) \( + 960 q^{65} \) \( + ( 156 + 156 \beta ) q^{66} \) \( + 8006 q^{67} \) \( + ( -3164 + 452 \beta ) q^{68} \) \( + 480 \beta q^{69} \) \( + ( -1920 - 1920 \beta ) q^{70} \) \( -144 \beta q^{71} \) \( + ( -1980 - 540 \beta ) q^{72} \) \( + 386 q^{73} \) \( + ( 6840 - 456 \beta ) q^{74} \) \( -2010 q^{75} \) \( + ( -1876 + 268 \beta ) q^{76} \) \( + 416 \beta q^{77} \) \( + ( -720 + 48 \beta ) q^{78} \) \( -2848 \beta q^{79} \) \( + ( 6720 + 1088 \beta ) q^{80} \) \( -891 q^{81} \) \( + ( -994 - 994 \beta ) q^{82} \) \( -2234 q^{83} \) \( + ( 2880 + 1344 \beta ) q^{84} \) \( + 1808 \beta q^{85} \) \( + ( 1882 + 1882 \beta ) q^{86} \) \( + 528 \beta q^{87} \) \( + ( -1144 - 312 \beta ) q^{88} \) \( -10046 q^{89} \) \( + ( -5400 + 360 \beta ) q^{90} \) \( -1920 q^{91} \) \( + ( -2400 - 1120 \beta ) q^{92} \) \( -1920 \beta q^{93} \) \( + ( 8160 - 544 \beta ) q^{94} \) \( + 1072 \beta q^{95} \) \( + ( -5856 - 480 \beta ) q^{96} \) \( + 8738 q^{97} \) \( + ( 1439 + 1439 \beta ) q^{98} \) \( + 1170 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 12q^{3} \) \(\mathstrut -\mathstrut 28q^{4} \) \(\mathstrut -\mathstrut 12q^{6} \) \(\mathstrut +\mathstrut 88q^{8} \) \(\mathstrut -\mathstrut 90q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 12q^{3} \) \(\mathstrut -\mathstrut 28q^{4} \) \(\mathstrut -\mathstrut 12q^{6} \) \(\mathstrut +\mathstrut 88q^{8} \) \(\mathstrut -\mathstrut 90q^{9} \) \(\mathstrut +\mathstrut 240q^{10} \) \(\mathstrut -\mathstrut 52q^{11} \) \(\mathstrut -\mathstrut 168q^{12} \) \(\mathstrut -\mathstrut 480q^{14} \) \(\mathstrut +\mathstrut 272q^{16} \) \(\mathstrut +\mathstrut 452q^{17} \) \(\mathstrut +\mathstrut 90q^{18} \) \(\mathstrut +\mathstrut 268q^{19} \) \(\mathstrut -\mathstrut 480q^{20} \) \(\mathstrut +\mathstrut 52q^{22} \) \(\mathstrut +\mathstrut 528q^{24} \) \(\mathstrut -\mathstrut 670q^{25} \) \(\mathstrut -\mathstrut 240q^{26} \) \(\mathstrut -\mathstrut 1512q^{27} \) \(\mathstrut +\mathstrut 960q^{28} \) \(\mathstrut +\mathstrut 1440q^{30} \) \(\mathstrut -\mathstrut 1952q^{32} \) \(\mathstrut -\mathstrut 312q^{33} \) \(\mathstrut -\mathstrut 452q^{34} \) \(\mathstrut +\mathstrut 3840q^{35} \) \(\mathstrut +\mathstrut 1260q^{36} \) \(\mathstrut -\mathstrut 268q^{38} \) \(\mathstrut -\mathstrut 2880q^{40} \) \(\mathstrut +\mathstrut 1988q^{41} \) \(\mathstrut -\mathstrut 2880q^{42} \) \(\mathstrut -\mathstrut 3764q^{43} \) \(\mathstrut +\mathstrut 728q^{44} \) \(\mathstrut +\mathstrut 2400q^{46} \) \(\mathstrut +\mathstrut 1632q^{48} \) \(\mathstrut -\mathstrut 2878q^{49} \) \(\mathstrut +\mathstrut 670q^{50} \) \(\mathstrut +\mathstrut 2712q^{51} \) \(\mathstrut +\mathstrut 480q^{52} \) \(\mathstrut +\mathstrut 1512q^{54} \) \(\mathstrut +\mathstrut 5760q^{56} \) \(\mathstrut +\mathstrut 1608q^{57} \) \(\mathstrut +\mathstrut 2640q^{58} \) \(\mathstrut -\mathstrut 10036q^{59} \) \(\mathstrut -\mathstrut 2880q^{60} \) \(\mathstrut -\mathstrut 9600q^{62} \) \(\mathstrut -\mathstrut 448q^{64} \) \(\mathstrut +\mathstrut 1920q^{65} \) \(\mathstrut +\mathstrut 312q^{66} \) \(\mathstrut +\mathstrut 16012q^{67} \) \(\mathstrut -\mathstrut 6328q^{68} \) \(\mathstrut -\mathstrut 3840q^{70} \) \(\mathstrut -\mathstrut 3960q^{72} \) \(\mathstrut +\mathstrut 772q^{73} \) \(\mathstrut +\mathstrut 13680q^{74} \) \(\mathstrut -\mathstrut 4020q^{75} \) \(\mathstrut -\mathstrut 3752q^{76} \) \(\mathstrut -\mathstrut 1440q^{78} \) \(\mathstrut +\mathstrut 13440q^{80} \) \(\mathstrut -\mathstrut 1782q^{81} \) \(\mathstrut -\mathstrut 1988q^{82} \) \(\mathstrut -\mathstrut 4468q^{83} \) \(\mathstrut +\mathstrut 5760q^{84} \) \(\mathstrut +\mathstrut 3764q^{86} \) \(\mathstrut -\mathstrut 2288q^{88} \) \(\mathstrut -\mathstrut 20092q^{89} \) \(\mathstrut -\mathstrut 10800q^{90} \) \(\mathstrut -\mathstrut 3840q^{91} \) \(\mathstrut -\mathstrut 4800q^{92} \) \(\mathstrut +\mathstrut 16320q^{94} \) \(\mathstrut -\mathstrut 11712q^{96} \) \(\mathstrut +\mathstrut 17476q^{97} \) \(\mathstrut +\mathstrut 2878q^{98} \) \(\mathstrut +\mathstrut 2340q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.500000 + 1.93649i
0.500000 1.93649i
−1.00000 3.87298i 6.00000 −14.0000 + 7.74597i 30.9839i −6.00000 23.2379i 61.9677i 44.0000 + 46.4758i −45.0000 120.000 30.9839i
3.2 −1.00000 + 3.87298i 6.00000 −14.0000 7.74597i 30.9839i −6.00000 + 23.2379i 61.9677i 44.0000 46.4758i −45.0000 120.000 + 30.9839i
\(n\): 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.d Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) \(\mathstrut -\mathstrut 6 \) acting on \(S_{5}^{\mathrm{new}}(8, [\chi])\).