# 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 $$2$$ 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{7}^{\mathrm{new}}(8, [\chi])$$.