# Properties

 Label 76.3.g.a Level $76$ Weight $3$ Character orbit 76.g Analytic conductor $2.071$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 76.g (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.07085000914$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-10})$$ Defining polynomial: $$x^{4} - 10 x^{2} + 100$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -2 \beta_{2} q^{2} + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( -4 + 4 \beta_{2} ) q^{4} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{5} + ( 2 - 4 \beta_{2} - 2 \beta_{3} ) q^{6} + ( -4 + 8 \beta_{2} + \beta_{3} ) q^{7} + 8 q^{8} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q -2 \beta_{2} q^{2} + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( -4 + 4 \beta_{2} ) q^{4} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{5} + ( 2 - 4 \beta_{2} - 2 \beta_{3} ) q^{6} + ( -4 + 8 \beta_{2} + \beta_{3} ) q^{7} + 8 q^{8} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{9} + ( 4 - 4 \beta_{1} + 2 \beta_{3} ) q^{10} + ( 1 - 2 \beta_{2} - 5 \beta_{3} ) q^{11} + ( -8 - 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{12} -6 \beta_{2} q^{13} + ( 16 + 2 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{14} + ( 16 + \beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{15} -16 \beta_{2} q^{16} + ( -16 + 2 \beta_{1} + 16 \beta_{2} - 4 \beta_{3} ) q^{17} + ( 8 + 4 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{18} + ( -6 + 3 \beta_{1} - 9 \beta_{2} - 5 \beta_{3} ) q^{19} + ( 4 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{20} + ( -22 - 5 \beta_{1} + 22 \beta_{2} + 10 \beta_{3} ) q^{21} + ( -4 - 10 \beta_{1} + 2 \beta_{2} + 10 \beta_{3} ) q^{22} + ( -4 - 9 \beta_{1} + 2 \beta_{2} + 9 \beta_{3} ) q^{23} + ( 8 + 8 \beta_{1} + 8 \beta_{2} ) q^{24} + ( 4 \beta_{1} - 9 \beta_{2} + 4 \beta_{3} ) q^{25} + ( -12 + 12 \beta_{2} ) q^{26} + ( -15 + 30 \beta_{2} + \beta_{3} ) q^{27} + ( -16 - 4 \beta_{1} - 16 \beta_{2} ) q^{28} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{29} + ( -16 - 2 \beta_{1} - 16 \beta_{2} ) q^{30} + ( 10 - 20 \beta_{2} + 7 \beta_{3} ) q^{31} + ( -32 + 32 \beta_{2} ) q^{32} + ( 53 + 6 \beta_{1} - 53 \beta_{2} - 12 \beta_{3} ) q^{33} + ( 32 - 8 \beta_{1} + 4 \beta_{3} ) q^{34} + ( 2 + 10 \beta_{1} + 2 \beta_{2} ) q^{35} + ( -16 - 16 \beta_{1} + 8 \beta_{3} ) q^{36} + ( -24 - 10 \beta_{1} + 5 \beta_{3} ) q^{37} + ( -18 - 10 \beta_{1} + 30 \beta_{2} + 4 \beta_{3} ) q^{38} + ( 6 - 12 \beta_{2} - 6 \beta_{3} ) q^{39} + ( -16 + 8 \beta_{1} + 16 \beta_{2} - 16 \beta_{3} ) q^{40} + ( 13 + 4 \beta_{1} - 13 \beta_{2} - 8 \beta_{3} ) q^{41} + ( 44 + 20 \beta_{1} - 10 \beta_{3} ) q^{42} + ( 12 - 10 \beta_{1} + 12 \beta_{2} ) q^{43} + ( 4 + 20 \beta_{1} + 4 \beta_{2} ) q^{44} + 52 q^{45} + ( 4 + 18 \beta_{1} + 4 \beta_{2} ) q^{46} + ( 7 \beta_{1} - 7 \beta_{3} ) q^{47} + ( 16 - 32 \beta_{2} - 16 \beta_{3} ) q^{48} + ( -9 - 16 \beta_{1} + 8 \beta_{3} ) q^{49} + ( -18 + 8 \beta_{1} + 18 \beta_{2} - 16 \beta_{3} ) q^{50} + ( 8 - 10 \beta_{1} - 4 \beta_{2} + 10 \beta_{3} ) q^{51} + 24 q^{52} + ( -4 \beta_{1} + 62 \beta_{2} - 4 \beta_{3} ) q^{53} + ( 60 + 2 \beta_{1} - 30 \beta_{2} - 2 \beta_{3} ) q^{54} + ( -48 + 7 \beta_{1} - 48 \beta_{2} ) q^{55} + ( -32 + 64 \beta_{2} + 8 \beta_{3} ) q^{56} + ( 53 + 2 \beta_{1} - 44 \beta_{2} - 16 \beta_{3} ) q^{57} + ( -4 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{58} + ( 13 - \beta_{1} + 13 \beta_{2} ) q^{59} + ( -32 + 64 \beta_{2} + 4 \beta_{3} ) q^{60} + ( \beta_{1} - 22 \beta_{2} + \beta_{3} ) q^{61} + ( -40 + 14 \beta_{1} + 20 \beta_{2} - 14 \beta_{3} ) q^{62} + ( -72 - 28 \beta_{1} + 36 \beta_{2} + 28 \beta_{3} ) q^{63} + 64 q^{64} + ( 12 - 12 \beta_{1} + 6 \beta_{3} ) q^{65} + ( -106 - 24 \beta_{1} + 12 \beta_{3} ) q^{66} + ( 34 + 21 \beta_{1} - 17 \beta_{2} - 21 \beta_{3} ) q^{67} + ( 8 \beta_{1} - 64 \beta_{2} + 8 \beta_{3} ) q^{68} + ( -96 - 22 \beta_{1} + 11 \beta_{3} ) q^{69} + ( 4 - 8 \beta_{2} - 20 \beta_{3} ) q^{70} + ( -34 + 14 \beta_{1} - 34 \beta_{2} ) q^{71} + ( 16 \beta_{1} + 32 \beta_{2} + 16 \beta_{3} ) q^{72} + ( -13 - 4 \beta_{1} + 13 \beta_{2} + 8 \beta_{3} ) q^{73} + ( 10 \beta_{1} + 48 \beta_{2} + 10 \beta_{3} ) q^{74} + ( -31 + 62 \beta_{2} + 3 \beta_{3} ) q^{75} + ( 60 + 8 \beta_{1} - 24 \beta_{2} + 12 \beta_{3} ) q^{76} + ( 62 + 42 \beta_{1} - 21 \beta_{3} ) q^{77} + ( -24 - 12 \beta_{1} + 12 \beta_{2} + 12 \beta_{3} ) q^{78} + ( 60 - 8 \beta_{1} + 60 \beta_{2} ) q^{79} + ( 32 - 32 \beta_{1} + 16 \beta_{3} ) q^{80} + ( -19 + 2 \beta_{1} + 19 \beta_{2} - 4 \beta_{3} ) q^{81} + ( -26 - 16 \beta_{1} + 8 \beta_{3} ) q^{82} + ( 19 - 38 \beta_{2} - 19 \beta_{3} ) q^{83} + ( -20 \beta_{1} - 88 \beta_{2} - 20 \beta_{3} ) q^{84} + ( 20 \beta_{1} - 92 \beta_{2} + 20 \beta_{3} ) q^{85} + ( 24 - 48 \beta_{2} + 20 \beta_{3} ) q^{86} + ( 12 - 24 \beta_{2} - 5 \beta_{3} ) q^{87} + ( 8 - 16 \beta_{2} - 40 \beta_{3} ) q^{88} + ( 22 \beta_{1} - 8 \beta_{2} + 22 \beta_{3} ) q^{89} -104 \beta_{2} q^{90} + ( 48 + 6 \beta_{1} - 24 \beta_{2} - 6 \beta_{3} ) q^{91} + ( 8 - 16 \beta_{2} - 36 \beta_{3} ) q^{92} + ( -40 + 3 \beta_{1} + 40 \beta_{2} - 6 \beta_{3} ) q^{93} -14 \beta_{1} q^{94} + ( 40 - 20 \beta_{1} - 92 \beta_{2} + 27 \beta_{3} ) q^{95} + ( -64 - 32 \beta_{1} + 32 \beta_{2} + 32 \beta_{3} ) q^{96} + ( -117 - 2 \beta_{1} + 117 \beta_{2} + 4 \beta_{3} ) q^{97} + ( 16 \beta_{1} + 18 \beta_{2} + 16 \beta_{3} ) q^{98} + ( 208 + 26 \beta_{1} - 104 \beta_{2} - 26 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + 6q^{3} - 8q^{4} - 4q^{5} + 32q^{8} + 8q^{9} + O(q^{10})$$ $$4q - 4q^{2} + 6q^{3} - 8q^{4} - 4q^{5} + 32q^{8} + 8q^{9} + 16q^{10} - 24q^{12} - 12q^{13} + 48q^{14} + 48q^{15} - 32q^{16} - 32q^{17} + 16q^{18} - 42q^{19} - 16q^{20} - 44q^{21} - 12q^{22} - 12q^{23} + 48q^{24} - 18q^{25} - 24q^{26} - 96q^{28} - 4q^{29} - 96q^{30} - 64q^{32} + 106q^{33} + 128q^{34} + 12q^{35} - 64q^{36} - 96q^{37} - 12q^{38} - 32q^{40} + 26q^{41} + 176q^{42} + 72q^{43} + 24q^{44} + 208q^{45} + 24q^{46} - 36q^{49} - 36q^{50} + 24q^{51} + 96q^{52} + 124q^{53} + 180q^{54} - 288q^{55} + 124q^{57} - 8q^{58} + 78q^{59} - 44q^{61} - 120q^{62} - 216q^{63} + 256q^{64} + 48q^{65} - 424q^{66} + 102q^{67} - 128q^{68} - 384q^{69} - 204q^{71} + 64q^{72} - 26q^{73} + 96q^{74} + 192q^{76} + 248q^{77} - 72q^{78} + 360q^{79} + 128q^{80} - 38q^{81} - 104q^{82} - 176q^{84} - 184q^{85} - 16q^{89} - 208q^{90} + 144q^{91} - 80q^{93} - 24q^{95} - 192q^{96} - 234q^{97} + 36q^{98} + 624q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 10 x^{2} + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/10$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$10 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$10 \beta_{3}$$

## Character values

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

 $$n$$ $$21$$ $$39$$ $$\chi(n)$$ $$-\beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −2.73861 + 1.58114i 2.73861 − 1.58114i −2.73861 − 1.58114i 2.73861 + 1.58114i
−1.00000 + 1.73205i −1.23861 + 0.715113i −2.00000 3.46410i −3.73861 6.47547i 2.86045i 3.76593i 8.00000 −3.47723 + 6.02273i 14.9545
7.2 −1.00000 + 1.73205i 4.23861 2.44716i −2.00000 3.46410i 1.73861 + 3.01137i 9.78866i 10.0905i 8.00000 7.47723 12.9509i −6.95445
11.1 −1.00000 1.73205i −1.23861 0.715113i −2.00000 + 3.46410i −3.73861 + 6.47547i 2.86045i 3.76593i 8.00000 −3.47723 6.02273i 14.9545
11.2 −1.00000 1.73205i 4.23861 + 2.44716i −2.00000 + 3.46410i 1.73861 3.01137i 9.78866i 10.0905i 8.00000 7.47723 + 12.9509i −6.95445
 $$n$$: 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
76.g odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.3.g.a 4
4.b odd 2 1 76.3.g.b yes 4
19.c even 3 1 76.3.g.b yes 4
76.g odd 6 1 inner 76.3.g.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.g.a 4 1.a even 1 1 trivial
76.3.g.a 4 76.g odd 6 1 inner
76.3.g.b yes 4 4.b odd 2 1
76.3.g.b yes 4 19.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 6 T_{3}^{3} + 5 T_{3}^{2} + 42 T_{3} + 49$$ acting on $$S_{3}^{\mathrm{new}}(76, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + 2 T + T^{2} )^{2}$$
$3$ $$49 + 42 T + 5 T^{2} - 6 T^{3} + T^{4}$$
$5$ $$676 - 104 T + 42 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$1444 + 116 T^{2} + T^{4}$$
$11$ $$61009 + 506 T^{2} + T^{4}$$
$13$ $$( 36 + 6 T + T^{2} )^{2}$$
$17$ $$18496 + 4352 T + 888 T^{2} + 32 T^{3} + T^{4}$$
$19$ $$130321 + 15162 T + 893 T^{2} + 42 T^{3} + T^{4}$$
$23$ $$636804 - 9576 T - 750 T^{2} + 12 T^{3} + T^{4}$$
$29$ $$676 - 104 T + 42 T^{2} + 4 T^{3} + T^{4}$$
$31$ $$36100 + 1580 T^{2} + T^{4}$$
$37$ $$( -174 + 48 T + T^{2} )^{2}$$
$41$ $$96721 + 8086 T + 987 T^{2} - 26 T^{3} + T^{4}$$
$43$ $$322624 + 40896 T + 1160 T^{2} - 72 T^{3} + T^{4}$$
$47$ $$240100 - 490 T^{2} + T^{4}$$
$53$ $$11316496 - 417136 T + 12012 T^{2} - 124 T^{3} + T^{4}$$
$59$ $$247009 - 38766 T + 2525 T^{2} - 78 T^{3} + T^{4}$$
$61$ $$206116 + 19976 T + 1482 T^{2} + 44 T^{3} + T^{4}$$
$67$ $$12552849 + 361386 T - 75 T^{2} - 102 T^{3} + T^{4}$$
$71$ $$2274064 + 307632 T + 15380 T^{2} + 204 T^{3} + T^{4}$$
$73$ $$96721 - 8086 T + 987 T^{2} + 26 T^{3} + T^{4}$$
$79$ $$103225600 - 3657600 T + 53360 T^{2} - 360 T^{3} + T^{4}$$
$83$ $$6385729 + 9386 T^{2} + T^{4}$$
$89$ $$208975936 - 231296 T + 14712 T^{2} + 16 T^{3} + T^{4}$$
$97$ $$184117761 + 3175146 T + 41187 T^{2} + 234 T^{3} + T^{4}$$