# Properties

 Label 76.7.c.b Level $76$ Weight $7$ Character orbit 76.c Analytic conductor $17.484$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 76.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.4841103551$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 5090 x^{6} + 8905881 x^{4} + 5831691048 x^{2} + 827887219200$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{8}\cdot 3\cdot 5$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{7} q^{5} + ( 46 + \beta_{2} + 2 \beta_{3} - \beta_{7} ) q^{7} + ( -545 + \beta_{2} - 5 \beta_{3} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{7} q^{5} + ( 46 + \beta_{2} + 2 \beta_{3} - \beta_{7} ) q^{7} + ( -545 + \beta_{2} - 5 \beta_{3} + \beta_{7} ) q^{9} + ( 112 + 2 \beta_{2} + 6 \beta_{3} + 9 \beta_{7} ) q^{11} + ( -16 \beta_{1} + \beta_{4} ) q^{13} + ( 5 \beta_{1} - \beta_{4} - \beta_{6} ) q^{15} + ( 206 + \beta_{2} + 27 \beta_{3} - 22 \beta_{7} ) q^{17} + ( 771 - 34 \beta_{1} - 5 \beta_{2} - 23 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 9 \beta_{7} ) q^{19} + ( -7 \beta_{1} - 4 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{21} + ( -2328 + 3 \beta_{2} + 48 \beta_{3} - 50 \beta_{7} ) q^{23} + ( -1475 - 10 \beta_{2} + 80 \beta_{3} - 99 \beta_{7} ) q^{25} + ( -132 \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{27} + ( -145 \beta_{1} - 2 \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{29} + ( -209 \beta_{1} + 7 \beta_{4} - 7 \beta_{6} ) q^{31} + ( 139 \beta_{1} - 21 \beta_{4} + 4 \beta_{5} - 9 \beta_{6} ) q^{33} + ( -12710 + 110 \beta_{2} + 150 \beta_{3} + 109 \beta_{7} ) q^{35} + ( -7 \beta_{1} - 11 \beta_{4} + 2 \beta_{5} - 13 \beta_{6} ) q^{37} + ( 20982 + 111 \beta_{2} + 576 \beta_{3} + 162 \beta_{7} ) q^{39} + ( -513 \beta_{1} + 35 \beta_{4} + 4 \beta_{5} + 9 \beta_{6} ) q^{41} + ( -42068 + 100 \beta_{2} - 360 \beta_{3} + 391 \beta_{7} ) q^{43} + ( -6880 + 20 \beta_{2} + 170 \beta_{3} - 925 \beta_{7} ) q^{45} + ( -71560 + 22 \beta_{2} - 1020 \beta_{3} + 23 \beta_{7} ) q^{47} + ( 55951 - 199 \beta_{2} + 555 \beta_{3} + 842 \beta_{7} ) q^{49} + ( 1023 \beta_{1} - 8 \beta_{4} + 2 \beta_{5} + 22 \beta_{6} ) q^{51} + ( -29 \beta_{1} - 60 \beta_{4} + 4 \beta_{5} - \beta_{6} ) q^{53} + ( 136650 + 130 \beta_{2} + 1220 \beta_{3} - 781 \beta_{7} ) q^{55} + ( 42906 + 549 \beta_{1} - 501 \beta_{2} - 93 \beta_{3} + 29 \beta_{4} - 10 \beta_{5} - 9 \beta_{6} - 1059 \beta_{7} ) q^{57} + ( 964 \beta_{1} - 9 \beta_{4} - 20 \beta_{5} + 11 \beta_{6} ) q^{59} + ( -79408 - 180 \beta_{2} - 1702 \beta_{3} - 77 \beta_{7} ) q^{61} + ( 40172 - 892 \beta_{2} - 2098 \beta_{3} + 1301 \beta_{7} ) q^{63} + ( -2790 \beta_{1} + 62 \beta_{4} - 20 \beta_{5} + 12 \beta_{6} ) q^{65} + ( -1509 \beta_{1} + 86 \beta_{4} - 48 \beta_{5} + 8 \beta_{6} ) q^{67} + ( -1084 \beta_{1} - 7 \beta_{4} + 6 \beta_{5} + 50 \beta_{6} ) q^{69} + ( 4556 \beta_{1} + 74 \beta_{4} - 48 \beta_{5} + 40 \beta_{6} ) q^{71} + ( -106742 + 895 \beta_{2} - 789 \beta_{3} - 290 \beta_{7} ) q^{73} + ( 2410 \beta_{1} + 49 \beta_{4} - 20 \beta_{5} + 99 \beta_{6} ) q^{75} + ( 231402 + 924 \beta_{2} + 3306 \beta_{3} + 2963 \beta_{7} ) q^{77} + ( 7625 \beta_{1} - 111 \beta_{4} - 48 \beta_{5} - 31 \beta_{6} ) q^{79} + ( -228251 - 98 \beta_{2} - 2714 \beta_{3} + 562 \beta_{7} ) q^{81} + ( 54774 + 882 \beta_{2} - 1644 \beta_{3} - 140 \beta_{7} ) q^{83} + ( -229610 + 570 \beta_{2} - 1030 \beta_{3} + 3223 \beta_{7} ) q^{85} + ( 183846 - 2469 \beta_{2} - 2790 \beta_{3} + 3516 \beta_{7} ) q^{87} + ( 12081 \beta_{1} - 41 \beta_{4} - 16 \beta_{5} - 141 \beta_{6} ) q^{89} + ( -16028 \beta_{1} + 349 \beta_{4} - 46 \beta_{5} + 73 \beta_{6} ) q^{91} + ( 271068 + 1674 \beta_{2} + 9354 \beta_{3} - 9330 \beta_{7} ) q^{93} + ( 78420 + 10000 \beta_{1} - 720 \beta_{2} - 690 \beta_{3} - 68 \beta_{4} + 30 \beta_{5} + 122 \beta_{6} - 395 \beta_{7} ) q^{95} + ( -7162 \beta_{1} + 96 \beta_{4} + 60 \beta_{5} - 112 \beta_{6} ) q^{97} + ( -106804 - 1720 \beta_{2} - 2350 \beta_{3} - 7831 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{5} + 362q^{7} - 4348q^{9} + O(q^{10})$$ $$8q + 2q^{5} + 362q^{7} - 4348q^{9} + 902q^{11} + 1550q^{17} + 6232q^{19} - 18820q^{23} - 12158q^{25} - 101762q^{35} + 167028q^{39} - 335042q^{43} - 57230q^{45} - 570394q^{47} + 448182q^{49} + 1089198q^{55} + 341316q^{57} - 632014q^{61} + 328174q^{63} - 852938q^{73} + 1850530q^{77} - 1819456q^{81} + 441200q^{83} - 1828374q^{85} + 1483380q^{87} + 2131176q^{93} + 627950q^{95} - 865394q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5090 x^{6} + 8905881 x^{4} + 5831691048 x^{2} + 827887219200$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{6} - 46859 \nu^{4} - 67416687 \nu^{2} - 4315657896$$$$)/21128796$$ $$\beta_{3}$$ $$=$$ $$($$$$-55 \nu^{6} - 204128 \nu^{4} - 211681665 \nu^{2} - 40102527312$$$$)/42257592$$ $$\beta_{4}$$ $$=$$ $$($$$$55 \nu^{7} + 204128 \nu^{5} + 211681665 \nu^{3} + 41750573400 \nu$$$$)/42257592$$ $$\beta_{5}$$ $$=$$ $$($$$$157 \nu^{7} + 518666 \nu^{5} + 500211621 \nu^{3} + 121944861000 \nu$$$$)/84515184$$ $$\beta_{6}$$ $$=$$ $$($$$$106 \nu^{7} + 361397 \nu^{5} + 334817847 \nu^{3} + 48252931560 \nu$$$$)/21128796$$ $$\beta_{7}$$ $$=$$ $$($$$$-89 \nu^{6} - 308974 \nu^{4} - 293772453 \nu^{2} - 46015049520$$$$)/14085864$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} - 5 \beta_{3} + \beta_{2} - 1274$$ $$\nu^{3}$$ $$=$$ $$-\beta_{6} + 2 \beta_{5} + \beta_{4} - 1590 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-1625 \beta_{7} + 8221 \beta_{3} - 2285 \beta_{2} + 2026546$$ $$\nu^{5}$$ $$=$$ $$1625 \beta_{6} - 4570 \beta_{5} + 259 \beta_{4} + 2626950 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$2182297 \beta_{7} - 12036101 \beta_{3} + 4631833 \beta_{2} - 3347179418$$ $$\nu^{7}$$ $$=$$ $$-2182297 \beta_{6} + 9263666 \beta_{5} - 4041695 \beta_{4} - 4389286830 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$39$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 − 42.5965i − 40.3415i − 38.1507i − 13.8790i 13.8790i 38.1507i 40.3415i 42.5965i
0 42.5965i 0 103.439 0 −619.422 0 −1085.46 0
37.2 0 40.3415i 0 −186.796 0 202.293 0 −898.437 0
37.3 0 38.1507i 0 102.472 0 512.609 0 −726.473 0
37.4 0 13.8790i 0 −18.1155 0 85.5200 0 536.374 0
37.5 0 13.8790i 0 −18.1155 0 85.5200 0 536.374 0
37.6 0 38.1507i 0 102.472 0 512.609 0 −726.473 0
37.7 0 40.3415i 0 −186.796 0 202.293 0 −898.437 0
37.8 0 42.5965i 0 103.439 0 −619.422 0 −1085.46 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 37.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.7.c.b 8
3.b odd 2 1 684.7.h.c 8
4.b odd 2 1 304.7.e.c 8
19.b odd 2 1 inner 76.7.c.b 8
57.d even 2 1 684.7.h.c 8
76.d even 2 1 304.7.e.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.7.c.b 8 1.a even 1 1 trivial
76.7.c.b 8 19.b odd 2 1 inner
304.7.e.c 8 4.b odd 2 1
304.7.e.c 8 76.d even 2 1
684.7.h.c 8 3.b odd 2 1
684.7.h.c 8 57.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 5090 T_{3}^{6} + 8905881 T_{3}^{4} + 5831691048 T_{3}^{2} + 827887219200$$ acting on $$S_{7}^{\mathrm{new}}(76, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$827887219200 + 5831691048 T^{2} + 8905881 T^{4} + 5090 T^{6} + T^{8}$$
$5$ $$( 35868000 + 1475200 T - 28210 T^{2} - T^{3} + T^{4} )^{2}$$
$7$ $$( -5493140806 + 93234493 T - 330963 T^{2} - 181 T^{3} + T^{4} )^{2}$$
$11$ $$( -70045150776 - 1452421460 T - 4134886 T^{2} - 451 T^{3} + T^{4} )^{2}$$
$13$ $$40\!\cdots\!00$$$$+ 21740366264471651808 T^{2} + 90968814465705 T^{4} + 20512698 T^{6} + T^{8}$$
$17$ $$( 56395405380078 - 6203796653 T - 20243875 T^{2} - 775 T^{3} + T^{4} )^{2}$$
$19$ $$48\!\cdots\!21$$$$-$$$$64\!\cdots\!12$$$$T +$$$$30\!\cdots\!08$$$$T^{2} - 33132962622494782952 T^{3} + 9232740094257854 T^{4} - 704269149992 T^{5} + 140039728 T^{6} - 6232 T^{7} + T^{8}$$
$23$ $$( -329689946145300 - 563918448080 T - 54276811 T^{2} + 9410 T^{3} + T^{4} )^{2}$$
$29$ $$14\!\cdots\!00$$$$+$$$$29\!\cdots\!48$$$$T^{2} + 1586649112632837801 T^{4} + 2802614130 T^{6} + T^{8}$$
$31$ $$25\!\cdots\!00$$$$+$$$$90\!\cdots\!92$$$$T^{2} + 11248102188712326096 T^{4} + 5739788496 T^{6} + T^{8}$$
$37$ $$85\!\cdots\!00$$$$+$$$$45\!\cdots\!52$$$$T^{2} + 43719014470366470096 T^{4} + 13281482256 T^{6} + T^{8}$$
$41$ $$29\!\cdots\!00$$$$+$$$$18\!\cdots\!28$$$$T^{2} +$$$$38\!\cdots\!52$$$$T^{4} + 33188295504 T^{6} + T^{8}$$
$43$ $$( -4346662716173125856 - 316371500354144 T + 2774402490 T^{2} + 167521 T^{3} + T^{4} )^{2}$$
$47$ $$( -$$$$15\!\cdots\!12$$$$- 2432258045959076 T + 9897704330 T^{2} + 285197 T^{3} + T^{4} )^{2}$$
$53$ $$14\!\cdots\!00$$$$+$$$$94\!\cdots\!00$$$$T^{2} +$$$$14\!\cdots\!53$$$$T^{4} + 70469336370 T^{6} + T^{8}$$
$59$ $$72\!\cdots\!00$$$$+$$$$24\!\cdots\!72$$$$T^{2} +$$$$77\!\cdots\!13$$$$T^{4} + 54182885322 T^{6} + T^{8}$$
$61$ $$( -7254911790602711680 - 3823129093392224 T - 23927920458 T^{2} + 316007 T^{3} + T^{4} )^{2}$$
$67$ $$22\!\cdots\!00$$$$+$$$$10\!\cdots\!52$$$$T^{2} +$$$$49\!\cdots\!57$$$$T^{4} + 439547155026 T^{6} + T^{8}$$
$71$ $$77\!\cdots\!00$$$$+$$$$43\!\cdots\!72$$$$T^{2} +$$$$74\!\cdots\!28$$$$T^{4} + 474354989544 T^{6} + T^{8}$$
$73$ $$($$$$16\!\cdots\!62$$$$- 17773225199911541 T - 193182556047 T^{2} + 426469 T^{3} + T^{4} )^{2}$$
$79$ $$30\!\cdots\!00$$$$+$$$$30\!\cdots\!92$$$$T^{2} +$$$$30\!\cdots\!52$$$$T^{4} + 1008954259536 T^{6} + T^{8}$$
$83$ $$($$$$46\!\cdots\!88$$$$+ 34337886760600384 T - 287093649748 T^{2} - 220600 T^{3} + T^{4} )^{2}$$
$89$ $$16\!\cdots\!00$$$$+$$$$31\!\cdots\!32$$$$T^{2} +$$$$15\!\cdots\!80$$$$T^{4} + 2584899501672 T^{6} + T^{8}$$
$97$ $$24\!\cdots\!00$$$$+$$$$45\!\cdots\!68$$$$T^{2} +$$$$65\!\cdots\!76$$$$T^{4} + 1538138511624 T^{6} + T^{8}$$
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