# 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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.84043266896$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.3803625.2 Defining polynomial: $$x^{4} - x^{3} + 6 x^{2} - 16 x + 256$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 48q^{3} - 44q^{4} + 396q^{6} + 248q^{8} - 660q^{9} + O(q^{10})$$ $$4q + 2q^{2} - 48q^{3} - 44q^{4} + 396q^{6} + 248q^{8} - 660q^{9} - 1920q^{10} + 976q^{11} + 1368q^{12} + 5760q^{14} - 14576q^{16} + 4168q^{17} - 10410q^{18} - 1456q^{19} + 31680q^{20} + 24428q^{22} - 38256q^{24} - 23900q^{25} - 59520q^{26} + 2592q^{27} + 59520q^{28} + 90240q^{30} - 48928q^{32} + 84048q^{33} - 81916q^{34} - 49920q^{35} - 12900q^{36} + 26572q^{38} - 13440q^{40} - 117944q^{41} + 38400q^{42} + 197456q^{43} + 37144q^{44} - 213120q^{46} + 137952q^{48} + 2116q^{49} + 357650q^{50} - 386016q^{51} - 254400q^{52} - 253224q^{54} + 349440q^{56} + 126672q^{57} + 516480q^{58} + 542032q^{59} - 716160q^{60} - 407040q^{62} + 463936q^{64} - 205440q^{65} - 142776q^{66} - 790192q^{67} - 213848q^{68} - 360960q^{70} + 805800q^{72} + 443912q^{73} - 32640q^{74} + 1765200q^{75} + 70616q^{76} + 324480q^{78} - 1032960q^{80} - 568044q^{81} + 404708q^{82} - 3465008q^{83} + 602880q^{84} + 989548q^{86} - 1950448q^{88} + 761224q^{89} - 1296000q^{90} + 3398400q^{91} + 2743680q^{92} + 971520q^{94} - 252864q^{96} - 926776q^{97} - 2391262q^{98} - 2459280q^{99} + O(q^{100})$$

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

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.7.d.b 4
3.b odd 2 1 72.7.b.b 4
4.b odd 2 1 32.7.d.b 4
8.b even 2 1 32.7.d.b 4
8.d odd 2 1 inner 8.7.d.b 4
12.b even 2 1 288.7.b.b 4
16.e even 4 2 256.7.c.l 8
16.f odd 4 2 256.7.c.l 8
24.f even 2 1 72.7.b.b 4
24.h odd 2 1 288.7.b.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.7.d.b 4 1.a even 1 1 trivial
8.7.d.b 4 8.d odd 2 1 inner
32.7.d.b 4 4.b odd 2 1
32.7.d.b 4 8.b even 2 1
72.7.b.b 4 3.b odd 2 1
72.7.b.b 4 24.f even 2 1
256.7.c.l 8 16.e even 4 2
256.7.c.l 8 16.f odd 4 2
288.7.b.b 4 12.b even 2 1
288.7.b.b 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 24 T_{3} - 276$$ acting on $$S_{7}^{\mathrm{new}}(8, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 24 T^{2} - 128 T^{3} + 4096 T^{4}$$
$3$ $$( 1 + 24 T + 1182 T^{2} + 17496 T^{3} + 531441 T^{4} )^{2}$$
$5$ $$1 - 19300 T^{2} + 256155750 T^{4} - 4711914062500 T^{6} + 59604644775390625 T^{8}$$
$7$ $$1 - 236356 T^{2} + 28021959366 T^{4} - 3271471277679556 T^{6} +$$$$19\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 - 488 T + 2238078 T^{2} - 864521768 T^{3} + 3138428376721 T^{4} )^{2}$$
$13$ $$1 - 4994596 T^{2} + 30260873415846 T^{4} -$$$$11\!\cdots\!76$$$$T^{6} +$$$$54\!\cdots\!61$$$$T^{8}$$
$17$ $$( 1 - 2084 T + 32560902 T^{2} - 50302693796 T^{3} + 582622237229761 T^{4} )^{2}$$
$19$ $$( 1 + 728 T + 92449758 T^{2} + 34249401368 T^{3} + 2213314919066161 T^{4} )^{2}$$
$23$ $$1 - 212117956 T^{2} + 23026671552237126 T^{4} -$$$$46\!\cdots\!76$$$$T^{6} +$$$$48\!\cdots\!41$$$$T^{8}$$
$29$ $$1 - 1409719204 T^{2} + 1160515330165289766 T^{4} -$$$$49\!\cdots\!64$$$$T^{6} +$$$$12\!\cdots\!81$$$$T^{8}$$
$31$ $$1 - 2758360324 T^{2} + 3362667870952277766 T^{4} -$$$$21\!\cdots\!64$$$$T^{6} +$$$$62\!\cdots\!21$$$$T^{8}$$
$37$ $$1 - 8121202276 T^{2} + 29600495645847907686 T^{4} -$$$$53\!\cdots\!56$$$$T^{6} +$$$$43\!\cdots\!61$$$$T^{8}$$
$41$ $$( 1 + 58972 T + 9857729958 T^{2} + 280123147300252 T^{3} + 22563490300366186081 T^{4} )^{2}$$
$43$ $$( 1 - 98728 T + 13190101374 T^{2} - 624095531101672 T^{3} + 39959630797262576401 T^{4} )^{2}$$
$47$ $$1 - 13578767236 T^{2} +$$$$16\!\cdots\!86$$$$T^{4} -$$$$15\!\cdots\!76$$$$T^{6} +$$$$13\!\cdots\!81$$$$T^{8}$$
$53$ $$1 - 42194545636 T^{2} +$$$$11\!\cdots\!86$$$$T^{4} -$$$$20\!\cdots\!76$$$$T^{6} +$$$$24\!\cdots\!81$$$$T^{8}$$
$59$ $$( 1 - 271016 T + 102678575166 T^{2} - 11431599505249256 T^{3} +$$$$17\!\cdots\!81$$$$T^{4} )^{2}$$
$61$ $$1 - 160868902564 T^{2} +$$$$11\!\cdots\!46$$$$T^{4} -$$$$42\!\cdots\!44$$$$T^{6} +$$$$70\!\cdots\!41$$$$T^{8}$$
$67$ $$( 1 + 395096 T + 204987839262 T^{2} + 35739744961443224 T^{3} +$$$$81\!\cdots\!61$$$$T^{4} )^{2}$$
$71$ $$1 - 164364621124 T^{2} +$$$$21\!\cdots\!06$$$$T^{4} -$$$$26\!\cdots\!84$$$$T^{6} +$$$$26\!\cdots\!81$$$$T^{8}$$
$73$ $$( 1 - 221956 T + 284381984742 T^{2} - 33589539530201284 T^{3} +$$$$22\!\cdots\!21$$$$T^{4} )^{2}$$
$79$ $$1 - 828257339524 T^{2} +$$$$28\!\cdots\!06$$$$T^{4} -$$$$48\!\cdots\!84$$$$T^{6} +$$$$34\!\cdots\!81$$$$T^{8}$$
$83$ $$( 1 + 1732504 T + 1400265667422 T^{2} + 566425504623285976 T^{3} +$$$$10\!\cdots\!61$$$$T^{4} )^{2}$$
$89$ $$( 1 - 380612 T + 970422538278 T^{2} - 189157043115248132 T^{3} +$$$$24\!\cdots\!21$$$$T^{4} )^{2}$$
$97$ $$( 1 + 463388 T + 953987784774 T^{2} + 385989231420039452 T^{3} +$$$$69\!\cdots\!41$$$$T^{4} )^{2}$$