# Properties

 Label 91.2.h.a Level $91$ Weight $2$ Character orbit 91.h Analytic conductor $0.727$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 91.h (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.726638658394$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{2} + 3 \zeta_{6} q^{3} - q^{4} -3 \zeta_{6} q^{5} + 3 \zeta_{6} q^{6} + ( 3 - 2 \zeta_{6} ) q^{7} -3 q^{8} + ( -6 + 6 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + q^{2} + 3 \zeta_{6} q^{3} - q^{4} -3 \zeta_{6} q^{5} + 3 \zeta_{6} q^{6} + ( 3 - 2 \zeta_{6} ) q^{7} -3 q^{8} + ( -6 + 6 \zeta_{6} ) q^{9} -3 \zeta_{6} q^{10} + 3 \zeta_{6} q^{11} -3 \zeta_{6} q^{12} + ( 1 - 4 \zeta_{6} ) q^{13} + ( 3 - 2 \zeta_{6} ) q^{14} + ( 9 - 9 \zeta_{6} ) q^{15} - q^{16} -2 q^{17} + ( -6 + 6 \zeta_{6} ) q^{18} + ( 1 - \zeta_{6} ) q^{19} + 3 \zeta_{6} q^{20} + ( 6 + 3 \zeta_{6} ) q^{21} + 3 \zeta_{6} q^{22} -9 \zeta_{6} q^{24} + ( -4 + 4 \zeta_{6} ) q^{25} + ( 1 - 4 \zeta_{6} ) q^{26} -9 q^{27} + ( -3 + 2 \zeta_{6} ) q^{28} + ( -7 + 7 \zeta_{6} ) q^{29} + ( 9 - 9 \zeta_{6} ) q^{30} + ( -3 + 3 \zeta_{6} ) q^{31} + 5 q^{32} + ( -9 + 9 \zeta_{6} ) q^{33} -2 q^{34} + ( -6 - 3 \zeta_{6} ) q^{35} + ( 6 - 6 \zeta_{6} ) q^{36} + 2 q^{37} + ( 1 - \zeta_{6} ) q^{38} + ( 12 - 9 \zeta_{6} ) q^{39} + 9 \zeta_{6} q^{40} + ( -3 + 3 \zeta_{6} ) q^{41} + ( 6 + 3 \zeta_{6} ) q^{42} + 7 \zeta_{6} q^{43} -3 \zeta_{6} q^{44} + 18 q^{45} -\zeta_{6} q^{47} -3 \zeta_{6} q^{48} + ( 5 - 8 \zeta_{6} ) q^{49} + ( -4 + 4 \zeta_{6} ) q^{50} -6 \zeta_{6} q^{51} + ( -1 + 4 \zeta_{6} ) q^{52} + ( -3 + 3 \zeta_{6} ) q^{53} -9 q^{54} + ( 9 - 9 \zeta_{6} ) q^{55} + ( -9 + 6 \zeta_{6} ) q^{56} + 3 q^{57} + ( -7 + 7 \zeta_{6} ) q^{58} -4 q^{59} + ( -9 + 9 \zeta_{6} ) q^{60} + ( 13 - 13 \zeta_{6} ) q^{61} + ( -3 + 3 \zeta_{6} ) q^{62} + ( -6 + 18 \zeta_{6} ) q^{63} + 7 q^{64} + ( -12 + 9 \zeta_{6} ) q^{65} + ( -9 + 9 \zeta_{6} ) q^{66} + 3 \zeta_{6} q^{67} + 2 q^{68} + ( -6 - 3 \zeta_{6} ) q^{70} -13 \zeta_{6} q^{71} + ( 18 - 18 \zeta_{6} ) q^{72} + ( 13 - 13 \zeta_{6} ) q^{73} + 2 q^{74} -12 q^{75} + ( -1 + \zeta_{6} ) q^{76} + ( 6 + 3 \zeta_{6} ) q^{77} + ( 12 - 9 \zeta_{6} ) q^{78} + 3 \zeta_{6} q^{79} + 3 \zeta_{6} q^{80} -9 \zeta_{6} q^{81} + ( -3 + 3 \zeta_{6} ) q^{82} + ( -6 - 3 \zeta_{6} ) q^{84} + 6 \zeta_{6} q^{85} + 7 \zeta_{6} q^{86} -21 q^{87} -9 \zeta_{6} q^{88} + 6 q^{89} + 18 q^{90} + ( -5 - 6 \zeta_{6} ) q^{91} -9 q^{93} -\zeta_{6} q^{94} -3 q^{95} + 15 \zeta_{6} q^{96} + 5 \zeta_{6} q^{97} + ( 5 - 8 \zeta_{6} ) q^{98} -18 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 3q^{3} - 2q^{4} - 3q^{5} + 3q^{6} + 4q^{7} - 6q^{8} - 6q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 3q^{3} - 2q^{4} - 3q^{5} + 3q^{6} + 4q^{7} - 6q^{8} - 6q^{9} - 3q^{10} + 3q^{11} - 3q^{12} - 2q^{13} + 4q^{14} + 9q^{15} - 2q^{16} - 4q^{17} - 6q^{18} + q^{19} + 3q^{20} + 15q^{21} + 3q^{22} - 9q^{24} - 4q^{25} - 2q^{26} - 18q^{27} - 4q^{28} - 7q^{29} + 9q^{30} - 3q^{31} + 10q^{32} - 9q^{33} - 4q^{34} - 15q^{35} + 6q^{36} + 4q^{37} + q^{38} + 15q^{39} + 9q^{40} - 3q^{41} + 15q^{42} + 7q^{43} - 3q^{44} + 36q^{45} - q^{47} - 3q^{48} + 2q^{49} - 4q^{50} - 6q^{51} + 2q^{52} - 3q^{53} - 18q^{54} + 9q^{55} - 12q^{56} + 6q^{57} - 7q^{58} - 8q^{59} - 9q^{60} + 13q^{61} - 3q^{62} + 6q^{63} + 14q^{64} - 15q^{65} - 9q^{66} + 3q^{67} + 4q^{68} - 15q^{70} - 13q^{71} + 18q^{72} + 13q^{73} + 4q^{74} - 24q^{75} - q^{76} + 15q^{77} + 15q^{78} + 3q^{79} + 3q^{80} - 9q^{81} - 3q^{82} - 15q^{84} + 6q^{85} + 7q^{86} - 42q^{87} - 9q^{88} + 12q^{89} + 36q^{90} - 16q^{91} - 18q^{93} - q^{94} - 6q^{95} + 15q^{96} + 5q^{97} + 2q^{98} - 36q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 1.50000 2.59808i −1.00000 −1.50000 + 2.59808i 1.50000 2.59808i 2.00000 + 1.73205i −3.00000 −3.00000 5.19615i −1.50000 + 2.59808i
74.1 1.00000 1.50000 + 2.59808i −1.00000 −1.50000 2.59808i 1.50000 + 2.59808i 2.00000 1.73205i −3.00000 −3.00000 + 5.19615i −1.50000 2.59808i
 $$n$$: 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.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.h.a yes 2
3.b odd 2 1 819.2.s.a 2
7.b odd 2 1 637.2.h.a 2
7.c even 3 1 91.2.g.a 2
7.c even 3 1 637.2.f.b 2
7.d odd 6 1 637.2.f.a 2
7.d odd 6 1 637.2.g.a 2
13.c even 3 1 91.2.g.a 2
13.c even 3 1 1183.2.e.a 2
13.e even 6 1 1183.2.e.c 2
21.h odd 6 1 819.2.n.c 2
39.i odd 6 1 819.2.n.c 2
91.g even 3 1 637.2.f.b 2
91.g even 3 1 1183.2.e.a 2
91.h even 3 1 inner 91.2.h.a yes 2
91.h even 3 1 8281.2.a.i 1
91.k even 6 1 8281.2.a.c 1
91.l odd 6 1 8281.2.a.g 1
91.m odd 6 1 637.2.f.a 2
91.n odd 6 1 637.2.g.a 2
91.u even 6 1 1183.2.e.c 2
91.v odd 6 1 637.2.h.a 2
91.v odd 6 1 8281.2.a.j 1
273.s odd 6 1 819.2.s.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.a 2 7.c even 3 1
91.2.g.a 2 13.c even 3 1
91.2.h.a yes 2 1.a even 1 1 trivial
91.2.h.a yes 2 91.h even 3 1 inner
637.2.f.a 2 7.d odd 6 1
637.2.f.a 2 91.m odd 6 1
637.2.f.b 2 7.c even 3 1
637.2.f.b 2 91.g even 3 1
637.2.g.a 2 7.d odd 6 1
637.2.g.a 2 91.n odd 6 1
637.2.h.a 2 7.b odd 2 1
637.2.h.a 2 91.v odd 6 1
819.2.n.c 2 21.h odd 6 1
819.2.n.c 2 39.i odd 6 1
819.2.s.a 2 3.b odd 2 1
819.2.s.a 2 273.s odd 6 1
1183.2.e.a 2 13.c even 3 1
1183.2.e.a 2 91.g even 3 1
1183.2.e.c 2 13.e even 6 1
1183.2.e.c 2 91.u even 6 1
8281.2.a.c 1 91.k even 6 1
8281.2.a.g 1 91.l odd 6 1
8281.2.a.i 1 91.h even 3 1
8281.2.a.j 1 91.v odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(91, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$9 - 3 T + T^{2}$$
$5$ $$9 + 3 T + T^{2}$$
$7$ $$7 - 4 T + T^{2}$$
$11$ $$9 - 3 T + T^{2}$$
$13$ $$13 + 2 T + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$49 + 7 T + T^{2}$$
$31$ $$9 + 3 T + T^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$9 + 3 T + T^{2}$$
$43$ $$49 - 7 T + T^{2}$$
$47$ $$1 + T + T^{2}$$
$53$ $$9 + 3 T + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$169 - 13 T + T^{2}$$
$67$ $$9 - 3 T + T^{2}$$
$71$ $$169 + 13 T + T^{2}$$
$73$ $$169 - 13 T + T^{2}$$
$79$ $$9 - 3 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$25 - 5 T + T^{2}$$