# Properties

 Label 91.2.u.a Level $91$ Weight $2$ Character orbit 91.u Analytic conductor $0.727$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 91.u (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.726638658394$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + \zeta_{6} ) q^{2} - q^{3} + ( 1 - \zeta_{6} ) q^{4} + ( -1 - \zeta_{6} ) q^{5} + ( 2 - \zeta_{6} ) q^{6} + ( -3 + \zeta_{6} ) q^{7} + ( 1 - 2 \zeta_{6} ) q^{8} -2 q^{9} +O(q^{10})$$ $$q + ( -2 + \zeta_{6} ) q^{2} - q^{3} + ( 1 - \zeta_{6} ) q^{4} + ( -1 - \zeta_{6} ) q^{5} + ( 2 - \zeta_{6} ) q^{6} + ( -3 + \zeta_{6} ) q^{7} + ( 1 - 2 \zeta_{6} ) q^{8} -2 q^{9} + 3 q^{10} + ( 3 - 6 \zeta_{6} ) q^{11} + ( -1 + \zeta_{6} ) q^{12} + ( -3 + 4 \zeta_{6} ) q^{13} + ( 5 - 4 \zeta_{6} ) q^{14} + ( 1 + \zeta_{6} ) q^{15} + 5 \zeta_{6} q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + ( 4 - 2 \zeta_{6} ) q^{18} + ( -1 + 2 \zeta_{6} ) q^{19} + ( -2 + \zeta_{6} ) q^{20} + ( 3 - \zeta_{6} ) q^{21} + 9 \zeta_{6} q^{22} + ( -1 + 2 \zeta_{6} ) q^{24} -2 \zeta_{6} q^{25} + ( 2 - 7 \zeta_{6} ) q^{26} + 5 q^{27} + ( -2 + 3 \zeta_{6} ) q^{28} + ( -3 + 3 \zeta_{6} ) q^{29} -3 q^{30} + ( 2 - \zeta_{6} ) q^{31} + ( -3 - 3 \zeta_{6} ) q^{32} + ( -3 + 6 \zeta_{6} ) q^{33} + ( 6 - 12 \zeta_{6} ) q^{34} + ( 4 + \zeta_{6} ) q^{35} + ( -2 + 2 \zeta_{6} ) q^{36} -3 \zeta_{6} q^{38} + ( 3 - 4 \zeta_{6} ) q^{39} + ( -3 + 3 \zeta_{6} ) q^{40} + ( -3 - 3 \zeta_{6} ) q^{41} + ( -5 + 4 \zeta_{6} ) q^{42} -11 \zeta_{6} q^{43} + ( -3 - 3 \zeta_{6} ) q^{44} + ( 2 + 2 \zeta_{6} ) q^{45} + ( -5 - 5 \zeta_{6} ) q^{47} -5 \zeta_{6} q^{48} + ( 8 - 5 \zeta_{6} ) q^{49} + ( 2 + 2 \zeta_{6} ) q^{50} + ( 6 - 6 \zeta_{6} ) q^{51} + ( 1 + 3 \zeta_{6} ) q^{52} + 9 \zeta_{6} q^{53} + ( -10 + 5 \zeta_{6} ) q^{54} + ( -9 + 9 \zeta_{6} ) q^{55} + ( -1 + 5 \zeta_{6} ) q^{56} + ( 1 - 2 \zeta_{6} ) q^{57} + ( 3 - 6 \zeta_{6} ) q^{58} + ( -2 - 2 \zeta_{6} ) q^{59} + ( 2 - \zeta_{6} ) q^{60} + 7 q^{61} + ( -3 + 3 \zeta_{6} ) q^{62} + ( 6 - 2 \zeta_{6} ) q^{63} - q^{64} + ( 7 - 5 \zeta_{6} ) q^{65} -9 \zeta_{6} q^{66} + ( -5 + 10 \zeta_{6} ) q^{67} + 6 \zeta_{6} q^{68} + ( -9 + 3 \zeta_{6} ) q^{70} + ( 2 - \zeta_{6} ) q^{71} + ( -2 + 4 \zeta_{6} ) q^{72} + ( -10 + 5 \zeta_{6} ) q^{73} + 2 \zeta_{6} q^{75} + ( 1 + \zeta_{6} ) q^{76} + ( -3 + 15 \zeta_{6} ) q^{77} + ( -2 + 7 \zeta_{6} ) q^{78} + ( 5 - 5 \zeta_{6} ) q^{79} + ( 5 - 10 \zeta_{6} ) q^{80} + q^{81} + 9 q^{82} + ( 2 - 4 \zeta_{6} ) q^{83} + ( 2 - 3 \zeta_{6} ) q^{84} + ( 12 - 6 \zeta_{6} ) q^{85} + ( 11 + 11 \zeta_{6} ) q^{86} + ( 3 - 3 \zeta_{6} ) q^{87} -9 q^{88} + ( -8 + 4 \zeta_{6} ) q^{89} -6 q^{90} + ( 5 - 11 \zeta_{6} ) q^{91} + ( -2 + \zeta_{6} ) q^{93} + 15 q^{94} + ( 3 - 3 \zeta_{6} ) q^{95} + ( 3 + 3 \zeta_{6} ) q^{96} + ( -6 + 3 \zeta_{6} ) q^{97} + ( -11 + 13 \zeta_{6} ) q^{98} + ( -6 + 12 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{2} - 2q^{3} + q^{4} - 3q^{5} + 3q^{6} - 5q^{7} - 4q^{9} + O(q^{10})$$ $$2q - 3q^{2} - 2q^{3} + q^{4} - 3q^{5} + 3q^{6} - 5q^{7} - 4q^{9} + 6q^{10} - q^{12} - 2q^{13} + 6q^{14} + 3q^{15} + 5q^{16} - 6q^{17} + 6q^{18} - 3q^{20} + 5q^{21} + 9q^{22} - 2q^{25} - 3q^{26} + 10q^{27} - q^{28} - 3q^{29} - 6q^{30} + 3q^{31} - 9q^{32} + 9q^{35} - 2q^{36} - 3q^{38} + 2q^{39} - 3q^{40} - 9q^{41} - 6q^{42} - 11q^{43} - 9q^{44} + 6q^{45} - 15q^{47} - 5q^{48} + 11q^{49} + 6q^{50} + 6q^{51} + 5q^{52} + 9q^{53} - 15q^{54} - 9q^{55} + 3q^{56} - 6q^{59} + 3q^{60} + 14q^{61} - 3q^{62} + 10q^{63} - 2q^{64} + 9q^{65} - 9q^{66} + 6q^{68} - 15q^{70} + 3q^{71} - 15q^{73} + 2q^{75} + 3q^{76} + 9q^{77} + 3q^{78} + 5q^{79} + 2q^{81} + 18q^{82} + q^{84} + 18q^{85} + 33q^{86} + 3q^{87} - 18q^{88} - 12q^{89} - 12q^{90} - q^{91} - 3q^{93} + 30q^{94} + 3q^{95} + 9q^{96} - 9q^{97} - 9q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/91\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$66$$ $$\chi(n)$$ $$\zeta_{6}$$ $$-1 + \zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
30.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.50000 + 0.866025i −1.00000 0.500000 0.866025i −1.50000 0.866025i 1.50000 0.866025i −2.50000 + 0.866025i 1.73205i −2.00000 3.00000
88.1 −1.50000 0.866025i −1.00000 0.500000 + 0.866025i −1.50000 + 0.866025i 1.50000 + 0.866025i −2.50000 0.866025i 1.73205i −2.00000 3.00000
 $$n$$: 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
91.u even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.u.a yes 2
3.b odd 2 1 819.2.do.c 2
7.b odd 2 1 637.2.u.a 2
7.c even 3 1 91.2.k.a 2
7.c even 3 1 637.2.q.c 2
7.d odd 6 1 637.2.k.b 2
7.d odd 6 1 637.2.q.b 2
13.e even 6 1 91.2.k.a 2
13.f odd 12 2 1183.2.e.e 4
21.h odd 6 1 819.2.bm.a 2
39.h odd 6 1 819.2.bm.a 2
91.k even 6 1 637.2.q.c 2
91.l odd 6 1 637.2.q.b 2
91.p odd 6 1 637.2.u.a 2
91.t odd 6 1 637.2.k.b 2
91.u even 6 1 inner 91.2.u.a yes 2
91.w even 12 2 8281.2.a.w 2
91.x odd 12 2 1183.2.e.e 4
91.bd odd 12 2 8281.2.a.s 2
273.x odd 6 1 819.2.do.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.a 2 7.c even 3 1
91.2.k.a 2 13.e even 6 1
91.2.u.a yes 2 1.a even 1 1 trivial
91.2.u.a yes 2 91.u even 6 1 inner
637.2.k.b 2 7.d odd 6 1
637.2.k.b 2 91.t odd 6 1
637.2.q.b 2 7.d odd 6 1
637.2.q.b 2 91.l odd 6 1
637.2.q.c 2 7.c even 3 1
637.2.q.c 2 91.k even 6 1
637.2.u.a 2 7.b odd 2 1
637.2.u.a 2 91.p odd 6 1
819.2.bm.a 2 21.h odd 6 1
819.2.bm.a 2 39.h odd 6 1
819.2.do.c 2 3.b odd 2 1
819.2.do.c 2 273.x odd 6 1
1183.2.e.e 4 13.f odd 12 2
1183.2.e.e 4 91.x odd 12 2
8281.2.a.s 2 91.bd odd 12 2
8281.2.a.w 2 91.w even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3 T_{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(91, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + 3 T + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$3 + 3 T + T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$27 + T^{2}$$
$13$ $$13 + 2 T + T^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$3 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$9 + 3 T + T^{2}$$
$31$ $$3 - 3 T + T^{2}$$
$37$ $$T^{2}$$
$41$ $$27 + 9 T + T^{2}$$
$43$ $$121 + 11 T + T^{2}$$
$47$ $$75 + 15 T + T^{2}$$
$53$ $$81 - 9 T + T^{2}$$
$59$ $$12 + 6 T + T^{2}$$
$61$ $$( -7 + T )^{2}$$
$67$ $$75 + T^{2}$$
$71$ $$3 - 3 T + T^{2}$$
$73$ $$75 + 15 T + T^{2}$$
$79$ $$25 - 5 T + T^{2}$$
$83$ $$12 + T^{2}$$
$89$ $$48 + 12 T + T^{2}$$
$97$ $$27 + 9 T + T^{2}$$