# Properties

 Label 84.2.b.b Level $84$ Weight $2$ Character orbit 84.b Analytic conductor $0.671$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 84.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.670743376979$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.2312.1 Defining polynomial: $$x^{4} - x^{3} - 2 x + 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$43$$ $$73$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 1.28078 + 0.599676i 1.28078 − 0.599676i −0.780776 + 1.17915i −0.780776 − 1.17915i
−1.28078 0.599676i 1.00000 1.28078 + 1.53610i 3.33513i −1.28078 0.599676i 1.56155 2.13578i −0.719224 2.73546i 1.00000 2.00000 4.27156i
55.2 −1.28078 + 0.599676i 1.00000 1.28078 1.53610i 3.33513i −1.28078 + 0.599676i 1.56155 + 2.13578i −0.719224 + 2.73546i 1.00000 2.00000 + 4.27156i
55.3 0.780776 1.17915i 1.00000 −0.780776 1.84130i 1.69614i 0.780776 1.17915i −2.56155 + 0.662153i −2.78078 0.516994i 1.00000 2.00000 + 1.32431i
55.4 0.780776 + 1.17915i 1.00000 −0.780776 + 1.84130i 1.69614i 0.780776 + 1.17915i −2.56155 0.662153i −2.78078 + 0.516994i 1.00000 2.00000 1.32431i
 $$n$$: 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
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.2.b.b yes 4
3.b odd 2 1 252.2.b.d 4
4.b odd 2 1 84.2.b.a 4
7.b odd 2 1 84.2.b.a 4
7.c even 3 2 588.2.o.a 8
7.d odd 6 2 588.2.o.c 8
8.b even 2 1 1344.2.b.e 4
8.d odd 2 1 1344.2.b.f 4
12.b even 2 1 252.2.b.e 4
21.c even 2 1 252.2.b.e 4
24.f even 2 1 4032.2.b.n 4
24.h odd 2 1 4032.2.b.j 4
28.d even 2 1 inner 84.2.b.b yes 4
28.f even 6 2 588.2.o.a 8
28.g odd 6 2 588.2.o.c 8
56.e even 2 1 1344.2.b.e 4
56.h odd 2 1 1344.2.b.f 4
84.h odd 2 1 252.2.b.d 4
168.e odd 2 1 4032.2.b.j 4
168.i even 2 1 4032.2.b.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.b.a 4 4.b odd 2 1
84.2.b.a 4 7.b odd 2 1
84.2.b.b yes 4 1.a even 1 1 trivial
84.2.b.b yes 4 28.d even 2 1 inner
252.2.b.d 4 3.b odd 2 1
252.2.b.d 4 84.h odd 2 1
252.2.b.e 4 12.b even 2 1
252.2.b.e 4 21.c even 2 1
588.2.o.a 8 7.c even 3 2
588.2.o.a 8 28.f even 6 2
588.2.o.c 8 7.d odd 6 2
588.2.o.c 8 28.g odd 6 2
1344.2.b.e 4 8.b even 2 1
1344.2.b.e 4 56.e even 2 1
1344.2.b.f 4 8.d odd 2 1
1344.2.b.f 4 56.h odd 2 1
4032.2.b.j 4 24.h odd 2 1
4032.2.b.j 4 168.e odd 2 1
4032.2.b.n 4 24.f even 2 1
4032.2.b.n 4 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{19}^{2} + 6 T_{19} - 8$$ acting on $$S_{2}^{\mathrm{new}}(84, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{3} + T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$32 + 14 T^{2} + T^{4}$$
$7$ $$49 + 14 T - 2 T^{2} + 2 T^{3} + T^{4}$$
$11$ $$8 + 10 T^{2} + T^{4}$$
$13$ $$128 + 40 T^{2} + T^{4}$$
$17$ $$512 + 46 T^{2} + T^{4}$$
$19$ $$( -8 + 6 T + T^{2} )^{2}$$
$23$ $$8 + 10 T^{2} + T^{4}$$
$29$ $$( 2 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$( -8 + 6 T + T^{2} )^{2}$$
$41$ $$128 + 62 T^{2} + T^{4}$$
$43$ $$5408 + 148 T^{2} + T^{4}$$
$47$ $$( -64 - 4 T + T^{2} )^{2}$$
$53$ $$( -52 - 8 T + T^{2} )^{2}$$
$59$ $$( 4 + T )^{4}$$
$61$ $$2048 + 112 T^{2} + T^{4}$$
$67$ $$512 + 124 T^{2} + T^{4}$$
$71$ $$2312 + 170 T^{2} + T^{4}$$
$73$ $$512 + 56 T^{2} + T^{4}$$
$79$ $$128 + 28 T^{2} + T^{4}$$
$83$ $$( -64 - 4 T + T^{2} )^{2}$$
$89$ $$128 + 62 T^{2} + T^{4}$$
$97$ $$8192 + 184 T^{2} + T^{4}$$