Properties

Label 8.7.d.b
Level 8
Weight 7
Character orbit 8.d
Analytic conductor 1.840
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 8.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.84043266896\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.3803625.2
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -13 + 2 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -12 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} \) \( + ( -2 + 6 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{5} \) \( + ( 104 - 12 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{6} \) \( + ( 12 - 20 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{7} \) \( + ( 72 - 22 \beta_{1} - 30 \beta_{2} + 2 \beta_{3} ) q^{8} \) \( + ( -141 - 48 \beta_{1} - 24 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -13 + 2 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -12 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} \) \( + ( -2 + 6 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{5} \) \( + ( 104 - 12 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{6} \) \( + ( 12 - 20 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{7} \) \( + ( 72 - 22 \beta_{1} - 30 \beta_{2} + 2 \beta_{3} ) q^{8} \) \( + ( -141 - 48 \beta_{1} - 24 \beta_{2} ) q^{9} \) \( + ( -496 + 28 \beta_{1} + 68 \beta_{2} + 4 \beta_{3} ) q^{10} \) \( + ( 187 + 114 \beta_{1} + 57 \beta_{2} ) q^{11} \) \( + ( 312 + 70 \beta_{1} - 74 \beta_{2} - 10 \beta_{3} ) q^{12} \) \( + ( -110 + 202 \beta_{1} - 92 \beta_{2} + 18 \beta_{3} ) q^{13} \) \( + ( 1440 + 24 \beta_{1} + 104 \beta_{2} - 24 \beta_{3} ) q^{14} \) \( + ( 124 - 292 \beta_{1} + 168 \beta_{2} + 44 \beta_{3} ) q^{15} \) \( + ( -3664 + 60 \beta_{1} - 84 \beta_{2} - 20 \beta_{3} ) q^{16} \) \( + ( 1242 - 400 \beta_{1} - 200 \beta_{2} ) q^{17} \) \( + ( -2496 - 165 \beta_{1} - 48 \beta_{2} - 48 \beta_{3} ) q^{18} \) \( + ( -429 + 130 \beta_{1} + 65 \beta_{2} ) q^{19} \) \( + ( 8192 - 576 \beta_{1} - 96 \beta_{2} + 32 \beta_{3} ) q^{20} \) \( + ( 88 - 136 \beta_{1} + 48 \beta_{2} - 40 \beta_{3} ) q^{21} \) \( + ( 5928 + 244 \beta_{1} + 114 \beta_{2} + 114 \beta_{3} ) q^{22} \) \( + ( -252 + 676 \beta_{1} - 424 \beta_{2} - 172 \beta_{3} ) q^{23} \) \( + ( -9872 + 556 \beta_{1} + 380 \beta_{2} + 60 \beta_{3} ) q^{24} \) \( + ( -6855 + 1760 \beta_{1} + 880 \beta_{2} ) q^{25} \) \( + ( -14992 + 4 \beta_{1} - 356 \beta_{2} + 220 \beta_{3} ) q^{26} \) \( + ( 1254 - 1212 \beta_{1} - 606 \beta_{2} ) q^{27} \) \( + ( 14080 + 1600 \beta_{1} + 768 \beta_{2} ) q^{28} \) \( + ( 922 - 1742 \beta_{1} + 820 \beta_{2} - 102 \beta_{3} ) q^{29} \) \( + ( 23072 - 776 \beta_{1} - 1656 \beta_{2} - 248 \beta_{3} ) q^{30} \) \( + ( -784 + 1392 \beta_{1} - 608 \beta_{2} + 176 \beta_{3} ) q^{31} \) \( + ( -10592 - 3320 \beta_{1} + 680 \beta_{2} + 40 \beta_{3} ) q^{32} \) \( + ( 21452 - 880 \beta_{1} - 440 \beta_{2} ) q^{33} \) \( + ( -20800 + 1042 \beta_{1} - 400 \beta_{2} - 400 \beta_{3} ) q^{34} \) \( + ( -11680 - 1600 \beta_{1} - 800 \beta_{2} ) q^{35} \) \( + ( -2052 - 2133 \beta_{1} + 1323 \beta_{2} - 213 \beta_{3} ) q^{36} \) \( + ( -370 + 214 \beta_{1} + 156 \beta_{2} + 526 \beta_{3} ) q^{37} \) \( + ( 6760 - 364 \beta_{1} + 130 \beta_{2} + 130 \beta_{3} ) q^{38} \) \( + ( 164 - 956 \beta_{1} + 792 \beta_{2} + 628 \beta_{3} ) q^{39} \) \( + ( -6784 + 7392 \beta_{1} - 1568 \beta_{2} - 544 \beta_{3} ) q^{40} \) \( + ( -30590 + 2208 \beta_{1} + 1104 \beta_{2} ) q^{41} \) \( + ( 9536 + 304 \beta_{1} + 1104 \beta_{2} - 176 \beta_{3} ) q^{42} \) \( + ( 47243 + 4242 \beta_{1} + 2121 \beta_{2} ) q^{43} \) \( + ( 6648 + 4918 \beta_{1} - 3290 \beta_{2} + 358 \beta_{3} ) q^{44} \) \( + ( -2070 + 4290 \beta_{1} - 2220 \beta_{2} - 150 \beta_{3} ) q^{45} \) \( + ( -54816 + 2568 \beta_{1} + 6008 \beta_{2} + 504 \beta_{3} ) q^{46} \) \( + ( 2728 - 3608 \beta_{1} + 880 \beta_{2} - 1848 \beta_{3} ) q^{47} \) \( + ( 39328 - 10296 \beta_{1} - 1304 \beta_{2} + 616 \beta_{3} ) q^{48} \) \( + ( 6225 - 11392 \beta_{1} - 5696 \beta_{2} ) q^{49} \) \( + ( 91520 - 5975 \beta_{1} + 1760 \beta_{2} + 1760 \beta_{3} ) q^{50} \) \( + ( -99946 + 6884 \beta_{1} + 3442 \beta_{2} ) q^{51} \) \( + ( -55296 - 16832 \beta_{1} - 6816 \beta_{2} + 224 \beta_{3} ) q^{52} \) \( + ( -1322 + 4862 \beta_{1} - 3540 \beta_{2} - 2218 \beta_{3} ) q^{53} \) \( + ( -63024 + 648 \beta_{1} - 1212 \beta_{2} - 1212 \beta_{3} ) q^{54} \) \( + ( 5212 - 11076 \beta_{1} + 5864 \beta_{2} + 652 \beta_{3} ) q^{55} \) \( + ( 79104 + 14912 \beta_{1} + 1600 \beta_{2} + 1600 \beta_{3} ) q^{56} \) \( + ( 32812 - 2288 \beta_{1} - 1144 \beta_{2} ) q^{57} \) \( + ( 130352 - 620 \beta_{1} + 1420 \beta_{2} - 1844 \beta_{3} ) q^{58} \) \( + ( 135835 - 654 \beta_{1} - 327 \beta_{2} ) q^{59} \) \( + ( -191744 + 26432 \beta_{1} + 6912 \beta_{2} - 1024 \beta_{3} ) q^{60} \) \( + ( -2902 + 3458 \beta_{1} - 556 \beta_{2} + 2346 \beta_{3} ) q^{61} \) \( + ( -102272 - 544 \beta_{1} - 4064 \beta_{2} + 1568 \beta_{3} ) q^{62} \) \( + ( -7548 + 12324 \beta_{1} - 4776 \beta_{2} + 2772 \beta_{3} ) q^{63} \) \( + ( 125120 - 14992 \beta_{1} - 4560 \beta_{2} - 3280 \beta_{3} ) q^{64} \) \( + ( -63920 + 25120 \beta_{1} + 12560 \beta_{2} ) q^{65} \) \( + ( -45760 + 21012 \beta_{1} - 880 \beta_{2} - 880 \beta_{3} ) q^{66} \) \( + ( -191581 - 11934 \beta_{1} - 5967 \beta_{2} ) q^{67} \) \( + ( -46104 - 15358 \beta_{1} + 13442 \beta_{2} + 642 \beta_{3} ) q^{68} \) \( + ( 11464 - 27352 \beta_{1} + 15888 \beta_{2} + 4424 \beta_{3} ) q^{69} \) \( + ( -83200 - 12480 \beta_{1} - 1600 \beta_{2} - 1600 \beta_{3} ) q^{70} \) \( + ( -14356 + 31180 \beta_{1} - 16824 \beta_{2} - 2468 \beta_{3} ) q^{71} \) \( + ( 204312 - 3378 \beta_{1} + 4470 \beta_{2} - 2346 \beta_{3} ) q^{72} \) \( + ( 119514 - 17072 \beta_{1} - 8536 \beta_{2} ) q^{73} \) \( + ( -5744 - 5572 \beta_{1} - 16092 \beta_{2} + 740 \beta_{3} ) q^{74} \) \( + ( 457835 - 33070 \beta_{1} - 16535 \beta_{2} ) q^{75} \) \( + ( 15288 + 4966 \beta_{1} - 4394 \beta_{2} - 234 \beta_{3} ) q^{76} \) \( + ( 16152 - 26312 \beta_{1} + 10160 \beta_{2} - 5992 \beta_{3} ) q^{77} \) \( + ( 85216 - 7864 \beta_{1} - 20424 \beta_{2} - 328 \beta_{3} ) q^{78} \) \( + ( 9880 - 20904 \beta_{1} + 11024 \beta_{2} + 1144 \beta_{3} ) q^{79} \) \( + ( -265472 + 7616 \beta_{1} + 24256 \beta_{2} + 6848 \beta_{3} ) q^{80} \) \( + ( -167427 + 50832 \beta_{1} + 25416 \beta_{2} ) q^{81} \) \( + ( 114816 - 29486 \beta_{1} + 2208 \beta_{2} + 2208 \beta_{3} ) q^{82} \) \( + ( -869341 + 6178 \beta_{1} + 3089 \beta_{2} ) q^{83} \) \( + ( 145408 + 10496 \beta_{1} + 5760 \beta_{2} + 128 \beta_{3} ) q^{84} \) \( + ( -22084 + 50252 \beta_{1} - 28168 \beta_{2} - 6084 \beta_{3} ) q^{85} \) \( + ( 220584 + 49364 \beta_{1} + 4242 \beta_{2} + 4242 \beta_{3} ) q^{86} \) \( + ( -3916 + 13780 \beta_{1} - 9864 \beta_{2} - 5948 \beta_{3} ) q^{87} \) \( + ( -495888 + 11276 \beta_{1} - 6180 \beta_{2} + 5276 \beta_{3} ) q^{88} \) \( + ( 202234 - 23856 \beta_{1} - 11928 \beta_{2} ) q^{89} \) \( + ( -329040 + 5940 \beta_{1} + 8940 \beta_{2} + 4140 \beta_{3} ) q^{90} \) \( + ( 809632 + 79936 \beta_{1} + 39968 \beta_{2} ) q^{91} \) \( + ( 716032 - 63296 \beta_{1} - 13056 \beta_{2} + 3072 \beta_{3} ) q^{92} \) \( + ( -1312 - 1184 \beta_{1} + 2496 \beta_{2} + 3808 \beta_{3} ) q^{93} \) \( + ( 237248 + 16720 \beta_{1} + 53680 \beta_{2} - 5456 \beta_{3} ) q^{94} \) \( + ( 7228 - 16484 \beta_{1} + 9256 \beta_{2} + 2028 \beta_{3} ) q^{95} \) \( + ( -70464 + 24176 \beta_{1} - 29392 \beta_{2} - 9680 \beta_{3} ) q^{96} \) \( + ( -188998 - 85392 \beta_{1} - 42696 \beta_{2} ) q^{97} \) \( + ( -592384 + 529 \beta_{1} - 11392 \beta_{2} - 11392 \beta_{3} ) q^{98} \) \( + ( -599559 - 30522 \beta_{1} - 15261 \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 48q^{3} \) \(\mathstrut -\mathstrut 44q^{4} \) \(\mathstrut +\mathstrut 396q^{6} \) \(\mathstrut +\mathstrut 248q^{8} \) \(\mathstrut -\mathstrut 660q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 48q^{3} \) \(\mathstrut -\mathstrut 44q^{4} \) \(\mathstrut +\mathstrut 396q^{6} \) \(\mathstrut +\mathstrut 248q^{8} \) \(\mathstrut -\mathstrut 660q^{9} \) \(\mathstrut -\mathstrut 1920q^{10} \) \(\mathstrut +\mathstrut 976q^{11} \) \(\mathstrut +\mathstrut 1368q^{12} \) \(\mathstrut +\mathstrut 5760q^{14} \) \(\mathstrut -\mathstrut 14576q^{16} \) \(\mathstrut +\mathstrut 4168q^{17} \) \(\mathstrut -\mathstrut 10410q^{18} \) \(\mathstrut -\mathstrut 1456q^{19} \) \(\mathstrut +\mathstrut 31680q^{20} \) \(\mathstrut +\mathstrut 24428q^{22} \) \(\mathstrut -\mathstrut 38256q^{24} \) \(\mathstrut -\mathstrut 23900q^{25} \) \(\mathstrut -\mathstrut 59520q^{26} \) \(\mathstrut +\mathstrut 2592q^{27} \) \(\mathstrut +\mathstrut 59520q^{28} \) \(\mathstrut +\mathstrut 90240q^{30} \) \(\mathstrut -\mathstrut 48928q^{32} \) \(\mathstrut +\mathstrut 84048q^{33} \) \(\mathstrut -\mathstrut 81916q^{34} \) \(\mathstrut -\mathstrut 49920q^{35} \) \(\mathstrut -\mathstrut 12900q^{36} \) \(\mathstrut +\mathstrut 26572q^{38} \) \(\mathstrut -\mathstrut 13440q^{40} \) \(\mathstrut -\mathstrut 117944q^{41} \) \(\mathstrut +\mathstrut 38400q^{42} \) \(\mathstrut +\mathstrut 197456q^{43} \) \(\mathstrut +\mathstrut 37144q^{44} \) \(\mathstrut -\mathstrut 213120q^{46} \) \(\mathstrut +\mathstrut 137952q^{48} \) \(\mathstrut +\mathstrut 2116q^{49} \) \(\mathstrut +\mathstrut 357650q^{50} \) \(\mathstrut -\mathstrut 386016q^{51} \) \(\mathstrut -\mathstrut 254400q^{52} \) \(\mathstrut -\mathstrut 253224q^{54} \) \(\mathstrut +\mathstrut 349440q^{56} \) \(\mathstrut +\mathstrut 126672q^{57} \) \(\mathstrut +\mathstrut 516480q^{58} \) \(\mathstrut +\mathstrut 542032q^{59} \) \(\mathstrut -\mathstrut 716160q^{60} \) \(\mathstrut -\mathstrut 407040q^{62} \) \(\mathstrut +\mathstrut 463936q^{64} \) \(\mathstrut -\mathstrut 205440q^{65} \) \(\mathstrut -\mathstrut 142776q^{66} \) \(\mathstrut -\mathstrut 790192q^{67} \) \(\mathstrut -\mathstrut 213848q^{68} \) \(\mathstrut -\mathstrut 360960q^{70} \) \(\mathstrut +\mathstrut 805800q^{72} \) \(\mathstrut +\mathstrut 443912q^{73} \) \(\mathstrut -\mathstrut 32640q^{74} \) \(\mathstrut +\mathstrut 1765200q^{75} \) \(\mathstrut +\mathstrut 70616q^{76} \) \(\mathstrut +\mathstrut 324480q^{78} \) \(\mathstrut -\mathstrut 1032960q^{80} \) \(\mathstrut -\mathstrut 568044q^{81} \) \(\mathstrut +\mathstrut 404708q^{82} \) \(\mathstrut -\mathstrut 3465008q^{83} \) \(\mathstrut +\mathstrut 602880q^{84} \) \(\mathstrut +\mathstrut 989548q^{86} \) \(\mathstrut -\mathstrut 1950448q^{88} \) \(\mathstrut +\mathstrut 761224q^{89} \) \(\mathstrut -\mathstrut 1296000q^{90} \) \(\mathstrut +\mathstrut 3398400q^{91} \) \(\mathstrut +\mathstrut 2743680q^{92} \) \(\mathstrut +\mathstrut 971520q^{94} \) \(\mathstrut -\mathstrut 252864q^{96} \) \(\mathstrut -\mathstrut 926776q^{97} \) \(\mathstrut -\mathstrut 2391262q^{98} \) \(\mathstrut -\mathstrut 2459280q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut +\mathstrut \) \(6\) \(x^{2}\mathstrut -\mathstrut \) \(16\) \(x\mathstrut +\mathstrut \) \(256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} - 6 \nu + 12 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 15 \nu^{2} - 2 \nu + 36 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(12\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(15\) \(\beta_{2}\mathstrut -\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(36\)\()/4\)

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
−2.31174 3.26433i
−2.31174 + 3.26433i
2.81174 2.84502i
2.81174 + 2.84502i
−4.62348 6.52867i −32.4939 −21.2470 + 60.3702i 199.084i 150.235 + 212.142i 19.6656i 492.372 140.406i 326.854 −1299.76 + 920.462i
3.2 −4.62348 + 6.52867i −32.4939 −21.2470 60.3702i 199.084i 150.235 212.142i 19.6656i 492.372 + 140.406i 326.854 −1299.76 920.462i
3.3 5.62348 5.69004i 8.49390 −0.753049 63.9956i 59.7107i 47.7652 48.3306i 483.584i −368.372 355.593i −656.854 339.756 + 335.782i
3.4 5.62348 + 5.69004i 8.49390 −0.753049 + 63.9956i 59.7107i 47.7652 + 48.3306i 483.584i −368.372 + 355.593i −656.854 339.756 335.782i
\(n\): 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 \(T_{3}^{2} \) \(\mathstrut +\mathstrut 24 T_{3} \) \(\mathstrut -\mathstrut 276 \) acting on \(S_{7}^{\mathrm{new}}(8, [\chi])\).