Properties

Label 8.6.b.a
Level 8
Weight 6
Character orbit 8.b
Analytic conductor 1.283
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 8.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.2830705585\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.218489.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 - \beta_{1} ) q^{2} \) \( + ( -\beta_{1} + \beta_{3} ) q^{3} \) \( + ( 5 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} \) \( + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{5} \) \( + ( -32 - 2 \beta_{2} - 6 \beta_{3} ) q^{6} \) \( + ( 36 + 20 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{7} \) \( + ( -58 - 2 \beta_{1} + 6 \beta_{2} + 10 \beta_{3} ) q^{8} \) \( + ( -65 - 40 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 - \beta_{1} ) q^{2} \) \( + ( -\beta_{1} + \beta_{3} ) q^{3} \) \( + ( 5 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} \) \( + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{5} \) \( + ( -32 - 2 \beta_{2} - 6 \beta_{3} ) q^{6} \) \( + ( 36 + 20 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{7} \) \( + ( -58 - 2 \beta_{1} + 6 \beta_{2} + 10 \beta_{3} ) q^{8} \) \( + ( -65 - 40 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{9} \) \( + ( 152 - 8 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} ) q^{10} \) \( + ( -16 - 37 \beta_{1} + 16 \beta_{2} + 5 \beta_{3} ) q^{11} \) \( + ( 418 + 26 \beta_{1} + 10 \beta_{2} + 22 \beta_{3} ) q^{12} \) \( + ( 6 + 44 \beta_{1} - 6 \beta_{2} - 32 \beta_{3} ) q^{13} \) \( + ( -600 - 24 \beta_{1} - 16 \beta_{2} + 16 \beta_{3} ) q^{14} \) \( + ( -92 + 20 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{15} \) \( + ( -804 + 76 \beta_{1} - 4 \beta_{2} - 28 \beta_{3} ) q^{16} \) \( + ( 74 + 40 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{17} \) \( + ( 1193 + 41 \beta_{1} + 32 \beta_{2} - 32 \beta_{3} ) q^{18} \) \( + ( 48 + 103 \beta_{1} - 48 \beta_{2} - 7 \beta_{3} ) q^{19} \) \( + ( 1044 - 188 \beta_{1} - 28 \beta_{2} - 36 \beta_{3} ) q^{20} \) \( + ( -24 - 208 \beta_{1} + 24 \beta_{2} + 160 \beta_{3} ) q^{21} \) \( + ( -1376 + 64 \beta_{1} + 86 \beta_{2} + 2 \beta_{3} ) q^{22} \) \( + ( 428 - 260 \beta_{1} - 52 \beta_{2} - 52 \beta_{3} ) q^{23} \) \( + ( -2164 - 388 \beta_{1} - 52 \beta_{2} - 44 \beta_{3} ) q^{24} \) \( + ( 629 + 400 \beta_{1} + 80 \beta_{2} + 80 \beta_{3} ) q^{25} \) \( + ( 1480 - 24 \beta_{1} + 28 \beta_{2} + 180 \beta_{3} ) q^{26} \) \( + ( 48 + 166 \beta_{1} - 48 \beta_{2} - 70 \beta_{3} ) q^{27} \) \( + ( 1480 + 552 \beta_{1} - 88 \beta_{2} - 168 \beta_{3} ) q^{28} \) \( + ( -82 + 92 \beta_{1} + 82 \beta_{2} - 256 \beta_{3} ) q^{29} \) \( + ( -472 + 104 \beta_{1} - 16 \beta_{2} + 16 \beta_{3} ) q^{30} \) \( + ( -3376 - 240 \beta_{1} - 48 \beta_{2} - 48 \beta_{3} ) q^{31} \) \( + ( 1848 + 792 \beta_{1} - 8 \beta_{2} + 200 \beta_{3} ) q^{32} \) \( + ( -804 - 360 \beta_{1} - 72 \beta_{2} - 72 \beta_{3} ) q^{33} \) \( + ( -1202 - 50 \beta_{1} - 32 \beta_{2} + 32 \beta_{3} ) q^{34} \) \( + ( -224 - 432 \beta_{1} + 224 \beta_{2} - 16 \beta_{3} ) q^{35} \) \( + ( -2925 - 1097 \beta_{1} + 183 \beta_{2} + 329 \beta_{3} ) q^{36} \) \( + ( 114 + 68 \beta_{1} - 114 \beta_{2} + 160 \beta_{3} ) q^{37} \) \( + ( 3872 - 192 \beta_{1} - 274 \beta_{2} - 54 \beta_{3} ) q^{38} \) \( + ( 9580 + 1340 \beta_{1} + 268 \beta_{2} + 268 \beta_{3} ) q^{39} \) \( + ( 3384 - 1128 \beta_{1} + 184 \beta_{2} - 120 \beta_{3} ) q^{40} \) \( + ( -1958 - 1360 \beta_{1} - 272 \beta_{2} - 272 \beta_{3} ) q^{41} \) \( + ( -6944 + 96 \beta_{1} - 176 \beta_{2} - 912 \beta_{3} ) q^{42} \) \( + ( 96 - 229 \beta_{1} - 96 \beta_{2} + 421 \beta_{3} ) q^{43} \) \( + ( -6262 + 1634 \beta_{1} + 274 \beta_{2} + 398 \beta_{3} ) q^{44} \) \( + ( 462 + 892 \beta_{1} - 462 \beta_{2} + 32 \beta_{3} ) q^{45} \) \( + ( 6904 - 584 \beta_{1} + 208 \beta_{2} - 208 \beta_{3} ) q^{46} \) \( + ( -13032 + 1080 \beta_{1} + 216 \beta_{2} + 216 \beta_{3} ) q^{47} \) \( + ( 9720 + 2008 \beta_{1} + 312 \beta_{2} - 376 \beta_{3} ) q^{48} \) \( + ( 3033 + 960 \beta_{1} + 192 \beta_{2} + 192 \beta_{3} ) q^{49} \) \( + ( -11909 - 389 \beta_{1} - 320 \beta_{2} + 320 \beta_{3} ) q^{50} \) \( + ( -48 - 418 \beta_{1} + 48 \beta_{2} + 322 \beta_{3} ) q^{51} \) \( + ( -10244 - 1396 \beta_{1} - 404 \beta_{2} - 812 \beta_{3} ) q^{52} \) \( + ( -262 + 404 \beta_{1} + 262 \beta_{2} - 928 \beta_{3} ) q^{53} \) \( + ( 5888 - 192 \beta_{1} - 148 \beta_{2} + 324 \beta_{3} ) q^{54} \) \( + ( 19508 - 3100 \beta_{1} - 620 \beta_{2} - 620 \beta_{3} ) q^{55} \) \( + ( 9840 - 1744 \beta_{1} - 400 \beta_{2} + 1040 \beta_{3} ) q^{56} \) \( + ( -52 + 760 \beta_{1} + 152 \beta_{2} + 152 \beta_{3} ) q^{57} \) \( + ( 1960 + 328 \beta_{1} + 1004 \beta_{2} + 1700 \beta_{3} ) q^{58} \) \( + ( -256 + 915 \beta_{1} + 256 \beta_{2} - 1427 \beta_{3} ) q^{59} \) \( + ( 840 + 424 \beta_{1} - 216 \beta_{2} - 40 \beta_{3} ) q^{60} \) \( + ( -1314 - 4644 \beta_{1} + 1314 \beta_{2} + 2016 \beta_{3} ) q^{61} \) \( + ( 10144 + 3232 \beta_{1} + 192 \beta_{2} - 192 \beta_{3} ) q^{62} \) \( + ( -39428 - 1780 \beta_{1} - 356 \beta_{2} - 356 \beta_{3} ) q^{63} \) \( + ( -11536 - 1872 \beta_{1} - 1424 \beta_{2} - 240 \beta_{3} ) q^{64} \) \( + ( -4544 + 560 \beta_{1} + 112 \beta_{2} + 112 \beta_{3} ) q^{65} \) \( + ( 10956 + 588 \beta_{1} + 288 \beta_{2} - 288 \beta_{3} ) q^{66} \) \( + ( 1584 + 4527 \beta_{1} - 1584 \beta_{2} - 1359 \beta_{3} ) q^{67} \) \( + ( 2970 + 1106 \beta_{1} - 174 \beta_{2} - 338 \beta_{3} ) q^{68} \) \( + ( 312 + 1808 \beta_{1} - 312 \beta_{2} - 1184 \beta_{3} ) q^{69} \) \( + ( -16512 + 896 \beta_{1} + 1376 \beta_{2} + 544 \beta_{3} ) q^{70} \) \( + ( 52356 + 1140 \beta_{1} + 228 \beta_{2} + 228 \beta_{3} ) q^{71} \) \( + ( -20086 + 3474 \beta_{1} + 842 \beta_{2} - 2010 \beta_{3} ) q^{72} \) \( + ( 13618 + 6040 \beta_{1} + 1208 \beta_{2} + 1208 \beta_{3} ) q^{73} \) \( + ( 3544 - 456 \beta_{1} - 1004 \beta_{2} - 1188 \beta_{3} ) q^{74} \) \( + ( -480 - 4069 \beta_{1} + 480 \beta_{2} + 3109 \beta_{3} ) q^{75} \) \( + ( 22130 - 4694 \beta_{1} - 742 \beta_{2} - 1018 \beta_{3} ) q^{76} \) \( + ( 1672 + 2416 \beta_{1} - 1672 \beta_{2} + 928 \beta_{3} ) q^{77} \) \( + ( -47368 - 8776 \beta_{1} - 1072 \beta_{2} + 1072 \beta_{3} ) q^{78} \) \( + ( -63000 - 1720 \beta_{1} - 344 \beta_{2} - 344 \beta_{3} ) q^{79} \) \( + ( -10192 - 2832 \beta_{1} + 2224 \beta_{2} + 208 \beta_{3} ) q^{80} \) \( + ( 3557 - 6440 \beta_{1} - 1288 \beta_{2} - 1288 \beta_{3} ) q^{81} \) \( + ( 40310 + 1142 \beta_{1} + 1088 \beta_{2} - 1088 \beta_{3} ) q^{82} \) \( + ( -1408 - 5385 \beta_{1} + 1408 \beta_{2} + 2569 \beta_{3} ) q^{83} \) \( + ( 54352 + 6416 \beta_{1} + 1936 \beta_{2} + 3952 \beta_{3} ) q^{84} \) \( + ( -444 - 856 \beta_{1} + 444 \beta_{2} - 32 \beta_{3} ) q^{85} \) \( + ( -6176 - 384 \beta_{1} - 1418 \beta_{2} - 2718 \beta_{3} ) q^{86} \) \( + ( 82620 + 9420 \beta_{1} + 1884 \beta_{2} + 1884 \beta_{3} ) q^{87} \) \( + ( -37892 + 7084 \beta_{1} - 1732 \beta_{2} + 740 \beta_{3} ) q^{88} \) \( + ( -28670 - 12520 \beta_{1} - 2504 \beta_{2} - 2504 \beta_{3} ) q^{89} \) \( + ( 34088 - 1848 \beta_{1} - 2836 \beta_{2} - 1116 \beta_{3} ) q^{90} \) \( + ( 96 + 5360 \beta_{1} - 96 \beta_{2} - 5168 \beta_{3} ) q^{91} \) \( + ( -14760 - 6280 \beta_{1} + 2040 \beta_{2} + 1288 \beta_{3} ) q^{92} \) \( + ( 288 + 5440 \beta_{1} - 288 \beta_{2} - 4864 \beta_{3} ) q^{93} \) \( + ( -17424 + 13680 \beta_{1} - 864 \beta_{2} + 864 \beta_{3} ) q^{94} \) \( + ( -59260 + 9460 \beta_{1} + 1892 \beta_{2} + 1892 \beta_{3} ) q^{95} \) \( + ( -30736 - 8784 \beta_{1} + 368 \beta_{2} + 5136 \beta_{3} ) q^{96} \) \( + ( -21606 + 5480 \beta_{1} + 1096 \beta_{2} + 1096 \beta_{3} ) q^{97} \) \( + ( -30105 - 2457 \beta_{1} - 768 \beta_{2} + 768 \beta_{3} ) q^{98} \) \( + ( -3456 - 5091 \beta_{1} + 3456 \beta_{2} - 1821 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 20q^{4} \) \(\mathstrut -\mathstrut 116q^{6} \) \(\mathstrut +\mathstrut 96q^{7} \) \(\mathstrut -\mathstrut 248q^{8} \) \(\mathstrut -\mathstrut 164q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 20q^{4} \) \(\mathstrut -\mathstrut 116q^{6} \) \(\mathstrut +\mathstrut 96q^{7} \) \(\mathstrut -\mathstrut 248q^{8} \) \(\mathstrut -\mathstrut 164q^{9} \) \(\mathstrut +\mathstrut 632q^{10} \) \(\mathstrut +\mathstrut 1576q^{12} \) \(\mathstrut -\mathstrut 2384q^{14} \) \(\mathstrut -\mathstrut 416q^{15} \) \(\mathstrut -\mathstrut 3312q^{16} \) \(\mathstrut +\mathstrut 200q^{17} \) \(\mathstrut +\mathstrut 4754q^{18} \) \(\mathstrut +\mathstrut 4624q^{20} \) \(\mathstrut -\mathstrut 5636q^{22} \) \(\mathstrut +\mathstrut 2336q^{23} \) \(\mathstrut -\mathstrut 7792q^{24} \) \(\mathstrut +\mathstrut 1556q^{25} \) \(\mathstrut +\mathstrut 5608q^{26} \) \(\mathstrut +\mathstrut 5152q^{28} \) \(\mathstrut -\mathstrut 2128q^{30} \) \(\mathstrut -\mathstrut 12928q^{31} \) \(\mathstrut +\mathstrut 5408q^{32} \) \(\mathstrut -\mathstrut 2352q^{33} \) \(\mathstrut -\mathstrut 4772q^{34} \) \(\mathstrut -\mathstrut 10164q^{36} \) \(\mathstrut +\mathstrut 15980q^{38} \) \(\mathstrut +\mathstrut 35104q^{39} \) \(\mathstrut +\mathstrut 16032q^{40} \) \(\mathstrut -\mathstrut 4568q^{41} \) \(\mathstrut -\mathstrut 26144q^{42} \) \(\mathstrut -\mathstrut 29112q^{44} \) \(\mathstrut +\mathstrut 29200q^{46} \) \(\mathstrut -\mathstrut 54720q^{47} \) \(\mathstrut +\mathstrut 35616q^{48} \) \(\mathstrut +\mathstrut 9828q^{49} \) \(\mathstrut -\mathstrut 47498q^{50} \) \(\mathstrut -\mathstrut 36560q^{52} \) \(\mathstrut +\mathstrut 23288q^{54} \) \(\mathstrut +\mathstrut 85472q^{55} \) \(\mathstrut +\mathstrut 40768q^{56} \) \(\mathstrut -\mathstrut 2032q^{57} \) \(\mathstrut +\mathstrut 3784q^{58} \) \(\mathstrut +\mathstrut 2592q^{60} \) \(\mathstrut +\mathstrut 34496q^{62} \) \(\mathstrut -\mathstrut 153440q^{63} \) \(\mathstrut -\mathstrut 41920q^{64} \) \(\mathstrut -\mathstrut 19520q^{65} \) \(\mathstrut +\mathstrut 43224q^{66} \) \(\mathstrut +\mathstrut 10344q^{68} \) \(\mathstrut -\mathstrut 68928q^{70} \) \(\mathstrut +\mathstrut 206688q^{71} \) \(\mathstrut -\mathstrut 83272q^{72} \) \(\mathstrut +\mathstrut 39976q^{73} \) \(\mathstrut +\mathstrut 17464q^{74} \) \(\mathstrut +\mathstrut 99944q^{76} \) \(\mathstrut -\mathstrut 174064q^{78} \) \(\mathstrut -\mathstrut 247872q^{79} \) \(\mathstrut -\mathstrut 35520q^{80} \) \(\mathstrut +\mathstrut 29684q^{81} \) \(\mathstrut +\mathstrut 161132q^{82} \) \(\mathstrut +\mathstrut 196672q^{84} \) \(\mathstrut -\mathstrut 18500q^{86} \) \(\mathstrut +\mathstrut 307872q^{87} \) \(\mathstrut -\mathstrut 167216q^{88} \) \(\mathstrut -\mathstrut 84632q^{89} \) \(\mathstrut +\mathstrut 142280q^{90} \) \(\mathstrut -\mathstrut 49056q^{92} \) \(\mathstrut -\mathstrut 98784q^{94} \) \(\mathstrut -\mathstrut 259744q^{95} \) \(\mathstrut -\mathstrut 115648q^{96} \) \(\mathstrut -\mathstrut 99576q^{97} \) \(\mathstrut -\mathstrut 117042q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut -\mathstrut \) \(2\) \(x^{2}\mathstrut -\mathstrut \) \(8\) \(x\mathstrut +\mathstrut \) \(64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 2 \nu + 4 \)\()/4\)
\(\beta_{2}\)\(=\)\( -2 \nu^{2} + 6 \nu + 1 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 7 \nu^{2} + 6 \nu - 20 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(13\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(27\) \(\beta_{1}\mathstrut +\mathstrut \) \(51\)\()/8\)

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
2.38600 1.51888i
2.38600 + 1.51888i
−1.88600 2.10784i
−1.88600 + 2.10784i
−4.77200 3.03776i 23.6095i 13.5440 + 28.9924i 1.38521i −71.7200 + 112.665i 160.704 23.4400 179.495i −314.408 4.20793 6.61022i
5.2 −4.77200 + 3.03776i 23.6095i 13.5440 28.9924i 1.38521i −71.7200 112.665i 160.704 23.4400 + 179.495i −314.408 4.20793 + 6.61022i
5.3 3.77200 4.21569i 3.25452i −3.54400 31.8031i 73.9600i 13.7200 + 12.2760i −112.704 −147.440 105.021i 232.408 311.792 + 278.977i
5.4 3.77200 + 4.21569i 3.25452i −3.54400 + 31.8031i 73.9600i 13.7200 12.2760i −112.704 −147.440 + 105.021i 232.408 311.792 278.977i
\(n\): 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 \(S_{6}^{\mathrm{new}}(8, [\chi])\).