# Properties

 Label 76.6.a.b Level $76$ Weight $6$ Character orbit 76.a Self dual yes Analytic conductor $12.189$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 76.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.1891703058$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 140 x^{2} - 84 x + 3103$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 3 + \beta_{3} ) q^{3} + ( -26 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{5} + ( 10 - \beta_{1} + 3 \beta_{2} + 5 \beta_{3} ) q^{7} + ( 218 + \beta_{1} - 12 \beta_{2} - 3 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 3 + \beta_{3} ) q^{3} + ( -26 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{5} + ( 10 - \beta_{1} + 3 \beta_{2} + 5 \beta_{3} ) q^{7} + ( 218 + \beta_{1} - 12 \beta_{2} - 3 \beta_{3} ) q^{9} + ( 176 - 3 \beta_{1} - 10 \beta_{2} - \beta_{3} ) q^{11} + ( 202 - 14 \beta_{1} + 15 \beta_{2} + 10 \beta_{3} ) q^{13} + ( 674 + 37 \beta_{1} - 6 \beta_{2} - 48 \beta_{3} ) q^{15} + ( 75 - 36 \beta_{1} + 48 \beta_{2} + 30 \beta_{3} ) q^{17} -361 q^{19} + ( 1754 + 31 \beta_{1} - 75 \beta_{2} - 56 \beta_{3} ) q^{21} + ( 1450 + 4 \beta_{1} + 95 \beta_{2} - 42 \beta_{3} ) q^{23} + ( 2895 + 107 \beta_{1} - 132 \beta_{2} - 97 \beta_{3} ) q^{25} + ( 1157 - 137 \beta_{1} + 24 \beta_{2} + 101 \beta_{3} ) q^{27} + ( 1382 - 67 \beta_{1} + 19 \beta_{2} + 144 \beta_{3} ) q^{29} + ( -174 - 5 \beta_{1} + 150 \beta_{2} + 82 \beta_{3} ) q^{31} + ( 2800 - 151 \beta_{1} - 90 \beta_{2} + 226 \beta_{3} ) q^{33} + ( 4922 + 199 \beta_{1} - 186 \beta_{2} - 317 \beta_{3} ) q^{35} + ( -5004 + 133 \beta_{1} + 330 \beta_{2} + 358 \beta_{3} ) q^{37} + ( 3778 + 50 \beta_{1} - 411 \beta_{2} - 146 \beta_{3} ) q^{39} + ( -2580 + 13 \beta_{1} - 320 \beta_{2} - 58 \beta_{3} ) q^{41} + ( -3146 - 81 \beta_{1} + 378 \beta_{2} - 15 \beta_{3} ) q^{43} + ( -17464 + 7 \beta_{1} + 960 \beta_{2} + 725 \beta_{3} ) q^{45} + ( -1424 - 299 \beta_{1} + 32 \beta_{2} - 435 \beta_{3} ) q^{47} + ( -5602 + 144 \beta_{1} - 324 \beta_{2} - 516 \beta_{3} ) q^{49} + ( 8169 + 246 \beta_{1} - 1080 \beta_{2} - 921 \beta_{3} ) q^{51} + ( -19254 + 53 \beta_{1} + 251 \beta_{2} - 266 \beta_{3} ) q^{53} + ( -18200 + 9 \beta_{1} + 648 \beta_{2} + 57 \beta_{3} ) q^{55} + ( -1083 - 361 \beta_{3} ) q^{57} + ( 5779 - 661 \beta_{1} - 420 \beta_{2} - 391 \beta_{3} ) q^{59} + ( -8072 - 611 \beta_{1} - 570 \beta_{2} + 97 \beta_{3} ) q^{61} + ( -9968 - 403 \beta_{1} + 462 \beta_{2} + 1847 \beta_{3} ) q^{63} + ( -2178 + 538 \beta_{1} - 734 \beta_{2} - 1318 \beta_{3} ) q^{65} + ( 1667 + 1426 \beta_{1} - 660 \beta_{2} + 823 \beta_{3} ) q^{67} + ( -36886 + 1138 \beta_{1} + 885 \beta_{2} + 990 \beta_{3} ) q^{69} + ( 30112 + 376 \beta_{1} - 130 \beta_{2} - 426 \beta_{3} ) q^{71} + ( -26095 - 1382 \beta_{1} + 96 \beta_{2} + 160 \beta_{3} ) q^{73} + ( -20899 - 611 \beta_{1} + 3336 \beta_{2} + 5817 \beta_{3} ) q^{75} + ( -1990 - 467 \beta_{1} + 12 \beta_{2} + 1021 \beta_{3} ) q^{77} + ( 29016 + 17 \beta_{1} + 1326 \beta_{2} - 532 \beta_{3} ) q^{79} + ( 10953 - 1224 \beta_{1} - 1512 \beta_{2} - 556 \beta_{3} ) q^{81} + ( 23412 + 1344 \beta_{1} + 834 \beta_{2} + 408 \beta_{3} ) q^{83} + ( 17682 + 1179 \beta_{1} - 3186 \beta_{2} - 4797 \beta_{3} ) q^{85} + ( 74818 - 298 \beta_{1} - 3279 \beta_{2} - 438 \beta_{3} ) q^{87} + ( -742 - 953 \beta_{1} + 1494 \beta_{2} - 4478 \beta_{3} ) q^{89} + ( 40577 - 383 \beta_{1} - 444 \beta_{2} - 1655 \beta_{3} ) q^{91} + ( 3082 + 1832 \beta_{1} - 654 \beta_{2} - 1926 \beta_{3} ) q^{93} + ( 9386 - 361 \beta_{1} - 722 \beta_{2} - 1083 \beta_{3} ) q^{95} + ( 3510 - 908 \beta_{1} + 1104 \beta_{2} - 4466 \beta_{3} ) q^{97} + ( 110652 - 1635 \beta_{1} - 4176 \beta_{2} + 595 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 10q^{3} - 110q^{5} + 30q^{7} + 878q^{9} + O(q^{10})$$ $$4q + 10q^{3} - 110q^{5} + 30q^{7} + 878q^{9} + 706q^{11} + 788q^{13} + 2792q^{15} + 240q^{17} - 1444q^{19} + 7128q^{21} + 5884q^{23} + 11774q^{25} + 4426q^{27} + 5240q^{29} - 860q^{31} + 10748q^{33} + 20322q^{35} - 20732q^{37} + 15404q^{39} - 10204q^{41} - 12554q^{43} - 71306q^{45} - 4826q^{47} - 21376q^{49} + 34518q^{51} - 76484q^{53} - 72914q^{55} - 3610q^{57} + 23898q^{59} - 32482q^{61} - 43566q^{63} - 6076q^{65} + 5022q^{67} - 149524q^{69} + 121300q^{71} - 104700q^{73} - 95230q^{75} - 10002q^{77} + 117128q^{79} + 44924q^{81} + 92832q^{83} + 80322q^{85} + 300148q^{87} + 5988q^{89} + 165618q^{91} + 16180q^{93} + 39710q^{95} + 22972q^{97} + 441418q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 140 x^{2} - 84 x + 3103$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$4 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 5 \nu^{2} - 79 \nu + 287$$$$)/12$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{2} - 2 \nu - 71$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/4$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{3} + \beta_{1} + 142$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$40 \beta_{3} + 48 \beta_{2} + 89 \beta_{1} + 272$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 4.80512 −5.81615 11.0547 −10.0437
0 −25.7605 0 −109.245 0 −177.299 0 420.606 0
1.2 0 −9.77006 0 −24.1428 0 64.5622 0 −147.546 0
1.3 0 17.5489 0 87.4671 0 76.9277 0 64.9636 0
1.4 0 27.9817 0 −64.0791 0 65.8093 0 539.976 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$19$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.6.a.b 4
3.b odd 2 1 684.6.a.e 4
4.b odd 2 1 304.6.a.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.6.a.b 4 1.a even 1 1 trivial
304.6.a.k 4 4.b odd 2 1
684.6.a.e 4 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 10 T_{3}^{3} - 875 T_{3}^{2} + 5988 T_{3} + 123588$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(76))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$123588 + 5988 T - 875 T^{2} - 10 T^{3} + T^{4}$$
$5$ $$-14782624 - 809300 T - 6087 T^{2} + 110 T^{3} + T^{4}$$
$7$ $$-57950181 + 2204622 T - 22476 T^{2} - 30 T^{3} + T^{4}$$
$11$ $$842790380 + 17856160 T + 14469 T^{2} - 706 T^{3} + T^{4}$$
$13$ $$15521087536 + 176779672 T - 298941 T^{2} - 788 T^{3} + T^{4}$$
$17$ $$933010219809 + 260798832 T - 4066146 T^{2} - 240 T^{3} + T^{4}$$
$19$ $$( 361 + T )^{4}$$
$23$ $$-69330313326016 + 58928167792 T - 4333677 T^{2} - 5884 T^{3} + T^{4}$$
$29$ $$-37858523058292 + 68597297852 T - 21188385 T^{2} - 5240 T^{3} + T^{4}$$
$31$ $$129248370446848 - 11154181696 T - 23116560 T^{2} + 860 T^{3} + T^{4}$$
$37$ $$-4701007501700240 - 2332526971120 T - 49568352 T^{2} + 20732 T^{3} + T^{4}$$
$41$ $$1168932051626240 - 507593819200 T - 74685948 T^{2} + 10204 T^{3} + T^{4}$$
$43$ $$-2367148348334144 - 1398101372356 T - 108385095 T^{2} + 12554 T^{3} + T^{4}$$
$47$ $$1826425856030336 + 842949181576 T - 286986975 T^{2} + 4826 T^{3} + T^{4}$$
$53$ $$75211347963000464 + 20504517033688 T + 1958495643 T^{2} + 76484 T^{3} + T^{4}$$
$59$ $$-129409346201894748 + 26521861709340 T - 1139036835 T^{2} - 23898 T^{3} + T^{4}$$
$61$ $$-385211274135945332 - 50337215709824 T - 1347049563 T^{2} + 32482 T^{3} + T^{4}$$
$67$ $$508871346363957168 - 557781344136 T - 4416190023 T^{2} - 5022 T^{3} + T^{4}$$
$71$ $$406332945301902704 - 81045731523248 T + 5020185504 T^{2} - 121300 T^{3} + T^{4}$$
$73$ $$1025450251801515009 - 149622116529924 T - 174343674 T^{2} + 104700 T^{3} + T^{4}$$
$79$ $$-3407677020392545664 + 129375307451968 T + 1970108940 T^{2} - 117128 T^{3} + T^{4}$$
$83$ $$-2078981839792132416 + 202856324775840 T - 2488100004 T^{2} - 92832 T^{3} + T^{4}$$
$89$ $$14\!\cdots\!48$$$$+ 117828675020736 T - 24926212848 T^{2} - 5988 T^{3} + T^{4}$$
$97$ $$12\!\cdots\!72$$$$+ 281032633234496 T - 22191139356 T^{2} - 22972 T^{3} + T^{4}$$