# Properties

 Label 76.4.a.b Level $76$ Weight $4$ Character orbit 76.a Self dual yes Analytic conductor $4.484$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 76.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.48414516044$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.35529.1 Defining polynomial: $$x^{3} - x^{2} - 52 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{2} ) q^{3} + ( 3 + \beta_{2} ) q^{5} + ( 15 - \beta_{1} - 2 \beta_{2} ) q^{7} + ( 19 + 3 \beta_{1} - 3 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{2} ) q^{3} + ( 3 + \beta_{2} ) q^{5} + ( 15 - \beta_{1} - 2 \beta_{2} ) q^{7} + ( 19 + 3 \beta_{1} - 3 \beta_{2} ) q^{9} + ( 29 - 8 \beta_{1} - 3 \beta_{2} ) q^{11} + ( -2 - 5 \beta_{1} + 7 \beta_{2} ) q^{13} + ( 30 - 2 \beta_{2} ) q^{15} + ( 23 + 13 \beta_{1} - 6 \beta_{2} ) q^{17} + 19 q^{19} + ( -76 + 15 \beta_{1} + 23 \beta_{2} ) q^{21} + ( 38 - 11 \beta_{1} - 9 \beta_{2} ) q^{23} + ( -68 - 6 \beta_{1} + 3 \beta_{2} ) q^{25} + ( -42 + 19 \beta_{1} + 13 \beta_{2} ) q^{27} + ( -24 - 21 \beta_{1} - 23 \beta_{2} ) q^{29} + ( -44 + 16 \beta_{1} - 12 \beta_{2} ) q^{31} + ( -218 - 10 \beta_{1} + 28 \beta_{2} ) q^{33} + ( -33 + 6 \beta_{1} + 17 \beta_{2} ) q^{35} + ( -150 - 16 \beta_{1} - 36 \beta_{2} ) q^{37} + ( 130 - 53 \beta_{1} - 47 \beta_{2} ) q^{39} + ( -58 - 14 \beta_{1} - 26 \beta_{2} ) q^{41} + ( -11 + 22 \beta_{1} + 27 \beta_{2} ) q^{43} + ( -141 + 36 \beta_{1} + 13 \beta_{2} ) q^{45} + ( 59 - 14 \beta_{1} + 41 \beta_{2} ) q^{47} + ( 36 - 39 \beta_{1} - 76 \beta_{2} ) q^{49} + ( 28 + 119 \beta_{1} + 79 \beta_{2} ) q^{51} + ( 46 + 59 \beta_{1} + 31 \beta_{2} ) q^{53} + ( 87 - 30 \beta_{1} + 45 \beta_{2} ) q^{55} + ( 19 \beta_{1} + 19 \beta_{2} ) q^{57} + ( 452 + 25 \beta_{1} + 37 \beta_{2} ) q^{59} + ( -109 - 78 \beta_{1} - 13 \beta_{2} ) q^{61} + ( 525 - 28 \beta_{1} - 107 \beta_{2} ) q^{63} + ( 420 - 72 \beta_{1} + 8 \beta_{2} ) q^{65} + ( 322 - 23 \beta_{1} - 61 \beta_{2} ) q^{67} + ( -446 - \beta_{1} + 61 \beta_{2} ) q^{69} + ( 334 + 50 \beta_{1} - 88 \beta_{2} ) q^{71} + ( -249 + 167 \beta_{1} + 84 \beta_{2} ) q^{73} + ( -6 - 113 \beta_{1} - 95 \beta_{2} ) q^{75} + ( 653 - 104 \beta_{1} - 127 \beta_{2} ) q^{77} + ( -444 - 70 \beta_{1} - 10 \beta_{2} ) q^{79} + ( 181 - 48 \beta_{1} + 12 \beta_{2} ) q^{81} + ( 592 + 26 \beta_{1} + 54 \beta_{2} ) q^{83} + ( -453 + 114 \beta_{1} - 3 \beta_{2} ) q^{85} + ( -1026 - 81 \beta_{1} + 49 \beta_{2} ) q^{87} + ( 632 + 70 \beta_{1} + 68 \beta_{2} ) q^{89} + ( -586 + 35 \beta_{1} + 165 \beta_{2} ) q^{91} + ( -104 + 88 \beta_{1} + 48 \beta_{2} ) q^{93} + ( 57 + 19 \beta_{2} ) q^{95} + ( -16 + 96 \beta_{1} + 54 \beta_{2} ) q^{97} + ( -103 - 146 \beta_{1} - 297 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{3} + 9q^{5} + 44q^{7} + 60q^{9} + O(q^{10})$$ $$3q + q^{3} + 9q^{5} + 44q^{7} + 60q^{9} + 79q^{11} - 11q^{13} + 90q^{15} + 82q^{17} + 57q^{19} - 213q^{21} + 103q^{23} - 210q^{25} - 107q^{27} - 93q^{29} - 116q^{31} - 664q^{33} - 93q^{35} - 466q^{37} + 337q^{39} - 188q^{41} - 11q^{43} - 387q^{45} + 163q^{47} + 69q^{49} + 203q^{51} + 197q^{53} + 231q^{55} + 19q^{57} + 1381q^{59} - 405q^{61} + 1547q^{63} + 1188q^{65} + 943q^{67} - 1339q^{69} + 1052q^{71} - 580q^{73} - 131q^{75} + 1855q^{77} - 1402q^{79} + 495q^{81} + 1802q^{83} - 1245q^{85} - 3159q^{87} + 1966q^{89} - 1723q^{91} - 224q^{93} + 171q^{95} + 48q^{97} - 455q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 52 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - 3 \nu - 34$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$4 \beta_{2} + 3 \beta_{1} + 34$$

## 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
 0.0768183 −6.76918 7.69236
0 −8.47932 0 −5.55614 0 32.0355 0 44.8989 0
1.2 0 1.26314 0 11.0323 0 5.70454 0 −25.4045 0
1.3 0 8.21618 0 3.52382 0 6.26000 0 40.5056 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 76.4.a.b 3
3.b odd 2 1 684.4.a.i 3
4.b odd 2 1 304.4.a.h 3
5.b even 2 1 1900.4.a.c 3
5.c odd 4 2 1900.4.c.c 6
8.b even 2 1 1216.4.a.t 3
8.d odd 2 1 1216.4.a.v 3
19.b odd 2 1 1444.4.a.e 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.a.b 3 1.a even 1 1 trivial
304.4.a.h 3 4.b odd 2 1
684.4.a.i 3 3.b odd 2 1
1216.4.a.t 3 8.b even 2 1
1216.4.a.v 3 8.d odd 2 1
1444.4.a.e 3 19.b odd 2 1
1900.4.a.c 3 5.b even 2 1
1900.4.c.c 6 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} - T_{3}^{2} - 70 T_{3} + 88$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(76))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$88 - 70 T - T^{2} + T^{3}$$
$5$ $$216 - 42 T - 9 T^{2} + T^{3}$$
$7$ $$-1144 + 419 T - 44 T^{2} + T^{3}$$
$11$ $$108888 - 666 T - 79 T^{2} + T^{3}$$
$13$ $$-201816 - 6434 T + 11 T^{2} + T^{3}$$
$17$ $$1022082 - 13065 T - 82 T^{2} + T^{3}$$
$19$ $$( -19 + T )^{3}$$
$23$ $$235392 - 3336 T - 103 T^{2} + T^{3}$$
$29$ $$-2252268 - 32064 T + 93 T^{2} + T^{3}$$
$31$ $$1084416 - 28640 T + 116 T^{2} + T^{3}$$
$37$ $$-15144328 - 1060 T + 466 T^{2} + T^{3}$$
$41$ $$-5041152 - 26556 T + 188 T^{2} + T^{3}$$
$43$ $$2359248 - 45296 T + 11 T^{2} + T^{3}$$
$47$ $$-3850752 - 146664 T - 163 T^{2} + T^{3}$$
$53$ $$-11566104 - 142266 T - 197 T^{2} + T^{3}$$
$59$ $$-52865928 + 555726 T - 1381 T^{2} + T^{3}$$
$61$ $$-847124 - 223668 T + 405 T^{2} + T^{3}$$
$67$ $$1169904 + 83536 T - 943 T^{2} + T^{3}$$
$71$ $$521792928 - 520668 T - 1052 T^{2} + T^{3}$$
$73$ $$-726527962 - 1118827 T + 580 T^{2} + T^{3}$$
$79$ $$18139904 + 427568 T + 1402 T^{2} + T^{3}$$
$83$ $$-91984992 + 917424 T - 1802 T^{2} + T^{3}$$
$89$ $$-47198784 + 955656 T - 1966 T^{2} + T^{3}$$
$97$ $$-82010336 - 418356 T - 48 T^{2} + T^{3}$$