# Properties

 Label 91.2.bb.a Level $91$ Weight $2$ Character orbit 91.bb Analytic conductor $0.727$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 91.bb (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.726638658394$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 2 q^{2} - 12 q^{3} - 6 q^{5} - 6 q^{7} - 16 q^{8} + 8 q^{9}+O(q^{10})$$ 32 * q - 2 * q^2 - 12 * q^3 - 6 * q^5 - 6 * q^7 - 16 * q^8 + 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 2 q^{2} - 12 q^{3} - 6 q^{5} - 6 q^{7} - 16 q^{8} + 8 q^{9} - 10 q^{11} + 28 q^{14} - 44 q^{15} + 12 q^{16} - 4 q^{18} + 12 q^{19} - 26 q^{21} - 8 q^{22} - 12 q^{24} + 24 q^{26} - 6 q^{28} + 16 q^{29} + 24 q^{31} + 4 q^{32} + 48 q^{33} + 28 q^{35} - 8 q^{37} - 6 q^{39} - 132 q^{40} - 16 q^{42} - 42 q^{44} - 24 q^{45} + 12 q^{46} + 30 q^{47} + 88 q^{50} + 36 q^{52} - 12 q^{53} + 78 q^{54} + 40 q^{57} + 26 q^{58} - 54 q^{59} + 16 q^{60} - 48 q^{61} + 24 q^{63} - 8 q^{65} + 12 q^{66} + 16 q^{67} - 48 q^{68} + 50 q^{70} - 36 q^{71} + 22 q^{72} + 66 q^{73} + 12 q^{74} - 176 q^{78} - 32 q^{79} + 138 q^{80} + 16 q^{81} - 58 q^{84} - 84 q^{85} + 42 q^{86} - 24 q^{87} - 60 q^{89} + 48 q^{92} + 6 q^{93} - 72 q^{94} - 42 q^{96} - 86 q^{98} - 24 q^{99}+O(q^{100})$$ 32 * q - 2 * q^2 - 12 * q^3 - 6 * q^5 - 6 * q^7 - 16 * q^8 + 8 * q^9 - 10 * q^11 + 28 * q^14 - 44 * q^15 + 12 * q^16 - 4 * q^18 + 12 * q^19 - 26 * q^21 - 8 * q^22 - 12 * q^24 + 24 * q^26 - 6 * q^28 + 16 * q^29 + 24 * q^31 + 4 * q^32 + 48 * q^33 + 28 * q^35 - 8 * q^37 - 6 * q^39 - 132 * q^40 - 16 * q^42 - 42 * q^44 - 24 * q^45 + 12 * q^46 + 30 * q^47 + 88 * q^50 + 36 * q^52 - 12 * q^53 + 78 * q^54 + 40 * q^57 + 26 * q^58 - 54 * q^59 + 16 * q^60 - 48 * q^61 + 24 * q^63 - 8 * q^65 + 12 * q^66 + 16 * q^67 - 48 * q^68 + 50 * q^70 - 36 * q^71 + 22 * q^72 + 66 * q^73 + 12 * q^74 - 176 * q^78 - 32 * q^79 + 138 * q^80 + 16 * q^81 - 58 * q^84 - 84 * q^85 + 42 * q^86 - 24 * q^87 - 60 * q^89 + 48 * q^92 + 6 * q^93 - 72 * q^94 - 42 * q^96 - 86 * q^98 - 24 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −0.639966 2.38839i −1.77043 + 1.02216i −3.56278 + 2.05697i −1.58199 + 0.423894i 3.57432 + 3.57432i −2.54693 0.716327i 3.69606 + 3.69606i 0.589612 1.02124i 2.02484 + 3.50713i
5.2 −0.411280 1.53492i 0.436133 0.251802i −0.454770 + 0.262561i −0.0769162 + 0.0206096i −0.565867 0.565867i 1.52171 2.16435i −1.65723 1.65723i −1.37319 + 2.37844i 0.0632682 + 0.109584i
5.3 −0.211401 0.788958i −2.60482 + 1.50389i 1.15429 0.666428i 3.03793 0.814012i 1.73717 + 1.73717i 2.28951 + 1.32595i −1.92491 1.92491i 3.02338 5.23665i −1.28444 2.22472i
5.4 −0.186083 0.694471i 1.44134 0.832160i 1.28439 0.741542i −1.87130 + 0.501414i −0.846120 0.846120i −0.783278 + 2.52715i −1.77076 1.77076i −0.115021 + 0.199221i 0.696434 + 1.20626i
5.5 0.200025 + 0.746505i −0.421869 + 0.243566i 1.21479 0.701360i 1.76272 0.472319i −0.266208 0.266208i −2.63734 0.210751i 1.85952 + 1.85952i −1.38135 + 2.39257i 0.705177 + 1.22140i
5.6 0.423548 + 1.58070i −1.71590 + 0.990678i −0.587174 + 0.339005i −2.77274 + 0.742954i −2.29273 2.29273i 1.92960 + 1.81015i 1.52975 + 1.52975i 0.462886 0.801742i −2.34878 4.06820i
5.7 0.493585 + 1.84208i 2.29307 1.32391i −1.41759 + 0.818448i −3.30509 + 0.885596i 3.57057 + 3.57057i −1.12943 2.39257i 0.489646 + 0.489646i 2.00545 3.47355i −3.26268 5.65113i
5.8 0.697597 + 2.60347i −0.657528 + 0.379624i −4.55935 + 2.63234i 2.44137 0.654162i −1.44703 1.44703i 0.722189 2.54528i −6.22207 6.22207i −1.21177 + 2.09885i 3.40618 + 5.89968i
31.1 −2.60347 0.697597i −0.657528 0.379624i 4.55935 + 2.63234i 0.654162 2.44137i 1.44703 + 1.44703i −2.54528 + 0.722189i −6.22207 6.22207i −1.21177 2.09885i −3.40618 + 5.89968i
31.2 −1.84208 0.493585i 2.29307 + 1.32391i 1.41759 + 0.818448i −0.885596 + 3.30509i −3.57057 3.57057i −2.39257 1.12943i 0.489646 + 0.489646i 2.00545 + 3.47355i 3.26268 5.65113i
31.3 −1.58070 0.423548i −1.71590 0.990678i 0.587174 + 0.339005i −0.742954 + 2.77274i 2.29273 + 2.29273i 1.81015 + 1.92960i 1.52975 + 1.52975i 0.462886 + 0.801742i 2.34878 4.06820i
31.4 −0.746505 0.200025i −0.421869 0.243566i −1.21479 0.701360i 0.472319 1.76272i 0.266208 + 0.266208i −0.210751 2.63734i 1.85952 + 1.85952i −1.38135 2.39257i −0.705177 + 1.22140i
31.5 0.694471 + 0.186083i 1.44134 + 0.832160i −1.28439 0.741542i −0.501414 + 1.87130i 0.846120 + 0.846120i 2.52715 0.783278i −1.77076 1.77076i −0.115021 0.199221i −0.696434 + 1.20626i
31.6 0.788958 + 0.211401i −2.60482 1.50389i −1.15429 0.666428i 0.814012 3.03793i −1.73717 1.73717i 1.32595 + 2.28951i −1.92491 1.92491i 3.02338 + 5.23665i 1.28444 2.22472i
31.7 1.53492 + 0.411280i 0.436133 + 0.251802i 0.454770 + 0.262561i −0.0206096 + 0.0769162i 0.565867 + 0.565867i −2.16435 + 1.52171i −1.65723 1.65723i −1.37319 2.37844i −0.0632682 + 0.109584i
31.8 2.38839 + 0.639966i −1.77043 1.02216i 3.56278 + 2.05697i −0.423894 + 1.58199i −3.57432 3.57432i −0.716327 2.54693i 3.69606 + 3.69606i 0.589612 + 1.02124i −2.02484 + 3.50713i
47.1 −2.60347 + 0.697597i −0.657528 + 0.379624i 4.55935 2.63234i 0.654162 + 2.44137i 1.44703 1.44703i −2.54528 0.722189i −6.22207 + 6.22207i −1.21177 + 2.09885i −3.40618 5.89968i
47.2 −1.84208 + 0.493585i 2.29307 1.32391i 1.41759 0.818448i −0.885596 3.30509i −3.57057 + 3.57057i −2.39257 + 1.12943i 0.489646 0.489646i 2.00545 3.47355i 3.26268 + 5.65113i
47.3 −1.58070 + 0.423548i −1.71590 + 0.990678i 0.587174 0.339005i −0.742954 2.77274i 2.29273 2.29273i 1.81015 1.92960i 1.52975 1.52975i 0.462886 0.801742i 2.34878 + 4.06820i
47.4 −0.746505 + 0.200025i −0.421869 + 0.243566i −1.21479 + 0.701360i 0.472319 + 1.76272i 0.266208 0.266208i −0.210751 + 2.63734i 1.85952 1.85952i −1.38135 + 2.39257i −0.705177 1.22140i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 73.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
13.d odd 4 1 inner
91.bb even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.bb.a 32
3.b odd 2 1 819.2.fn.e 32
7.b odd 2 1 637.2.bc.b 32
7.c even 3 1 637.2.i.a 32
7.c even 3 1 637.2.bc.b 32
7.d odd 6 1 inner 91.2.bb.a 32
7.d odd 6 1 637.2.i.a 32
13.d odd 4 1 inner 91.2.bb.a 32
21.g even 6 1 819.2.fn.e 32
39.f even 4 1 819.2.fn.e 32
91.i even 4 1 637.2.bc.b 32
91.z odd 12 1 637.2.i.a 32
91.z odd 12 1 637.2.bc.b 32
91.bb even 12 1 inner 91.2.bb.a 32
91.bb even 12 1 637.2.i.a 32
273.cb odd 12 1 819.2.fn.e 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.bb.a 32 1.a even 1 1 trivial
91.2.bb.a 32 7.d odd 6 1 inner
91.2.bb.a 32 13.d odd 4 1 inner
91.2.bb.a 32 91.bb even 12 1 inner
637.2.i.a 32 7.c even 3 1
637.2.i.a 32 7.d odd 6 1
637.2.i.a 32 91.z odd 12 1
637.2.i.a 32 91.bb even 12 1
637.2.bc.b 32 7.b odd 2 1
637.2.bc.b 32 7.c even 3 1
637.2.bc.b 32 91.i even 4 1
637.2.bc.b 32 91.z odd 12 1
819.2.fn.e 32 3.b odd 2 1
819.2.fn.e 32 21.g even 6 1
819.2.fn.e 32 39.f even 4 1
819.2.fn.e 32 273.cb odd 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(91, [\chi])$$.