# Properties

 Label 91.2.bc.a Level $91$ Weight $2$ Character orbit 91.bc 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.bc (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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 8q^{2} - 12q^{4} - 16q^{8} + 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 8q^{2} - 12q^{4} - 16q^{8} + 8q^{9} - 4q^{11} - 32q^{14} - 8q^{15} + 12q^{16} - 4q^{18} + 16q^{21} + 4q^{22} - 12q^{23} + 24q^{28} + 4q^{29} + 12q^{30} + 4q^{32} - 20q^{35} + 4q^{37} - 36q^{39} - 28q^{42} - 48q^{43} + 24q^{44} + 84q^{46} + 24q^{49} - 44q^{50} + 72q^{53} + 60q^{56} - 92q^{57} - 16q^{58} + 76q^{60} + 48q^{63} + 4q^{65} - 56q^{67} + 56q^{70} + 84q^{71} - 128q^{72} - 24q^{74} + 148q^{78} - 80q^{79} + 28q^{81} - 64q^{84} + 36q^{85} - 48q^{86} - 228q^{88} - 48q^{91} + 24q^{92} + 108q^{93} - 84q^{95} - 32q^{98} - 60q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1 −0.607529 2.26733i −2.39046 1.38013i −3.03963 + 1.75493i 1.12678 + 1.12678i −1.67694 + 6.25841i −0.437995 2.60925i 2.50608 + 2.50608i 2.30952 + 4.00020i 1.87023 3.23933i
6.2 −0.607529 2.26733i 2.39046 + 1.38013i −3.03963 + 1.75493i −1.12678 1.12678i 1.67694 6.25841i 1.68394 2.04067i 2.50608 + 2.50608i 2.30952 + 4.00020i −1.87023 + 3.23933i
6.3 −0.357317 1.33353i −0.928437 0.536034i 0.0814361 0.0470171i −2.02925 2.02925i −0.383068 + 1.42963i 1.39092 + 2.25063i −2.04421 2.04421i −0.925336 1.60273i −1.98097 + 3.43114i
6.4 −0.357317 1.33353i 0.928437 + 0.536034i 0.0814361 0.0470171i 2.02925 + 2.02925i 0.383068 1.42963i −2.32989 + 1.25364i −2.04421 2.04421i −0.925336 1.60273i 1.98097 3.43114i
6.5 0.218036 + 0.813723i −1.24122 0.716618i 1.11745 0.645157i 1.01306 + 1.01306i 0.312498 1.16626i 2.43848 + 1.02656i 1.96000 + 1.96000i −0.472916 0.819114i −0.603466 + 1.04523i
6.6 0.218036 + 0.813723i 1.24122 + 0.716618i 1.11745 0.645157i −1.01306 1.01306i −0.312498 + 1.16626i −2.62506 0.330214i 1.96000 + 1.96000i −0.472916 0.819114i 0.603466 1.04523i
6.7 0.612835 + 2.28713i −1.04118 0.601128i −3.12335 + 1.80327i 2.25527 + 2.25527i 0.736784 2.74971i −2.60590 0.457468i −2.68981 2.68981i −0.777291 1.34631i −3.77599 + 6.54020i
6.8 0.612835 + 2.28713i 1.04118 + 0.601128i −3.12335 + 1.80327i −2.25527 2.25527i −0.736784 + 2.74971i 2.48551 + 0.906771i −2.68981 2.68981i −0.777291 1.34631i 3.77599 6.54020i
20.1 −2.47996 + 0.664504i −2.33506 1.34815i 3.97660 2.29589i −0.281686 + 0.281686i 6.68671 + 1.79170i −0.201802 + 2.63804i −4.70528 + 4.70528i 2.13500 + 3.69793i 0.511390 0.885754i
20.2 −2.47996 + 0.664504i 2.33506 + 1.34815i 3.97660 2.29589i 0.281686 0.281686i −6.68671 1.79170i 1.14426 2.38551i −4.70528 + 4.70528i 2.13500 + 3.69793i −0.511390 + 0.885754i
20.3 −0.892257 + 0.239080i −0.988196 0.570535i −0.993087 + 0.573359i 2.80028 2.80028i 1.01813 + 0.272807i 2.62900 0.297264i 2.05537 2.05537i −0.848979 1.47047i −1.82908 + 3.16806i
20.4 −0.892257 + 0.239080i 0.988196 + 0.570535i −0.993087 + 0.573359i −2.80028 + 2.80028i −1.01813 0.272807i 2.12815 + 1.57194i 2.05537 2.05537i −0.848979 1.47047i 1.82908 3.16806i
20.5 −0.112775 + 0.0302180i −2.25055 1.29935i −1.72025 + 0.993184i −1.24841 + 1.24841i 0.293070 + 0.0785278i −2.18126 1.49736i 0.329103 0.329103i 1.87664 + 3.25044i 0.103066 0.178515i
20.6 −0.112775 + 0.0302180i 2.25055 + 1.29935i −1.72025 + 0.993184i 1.24841 1.24841i −0.293070 0.0785278i −2.63771 + 0.206123i 0.329103 0.329103i 1.87664 + 3.25044i −0.103066 + 0.178515i
20.7 1.61897 0.433802i −0.552306 0.318874i 0.700831 0.404625i 1.42145 1.42145i −1.03250 0.276656i −1.11484 + 2.39940i −1.41124 + 1.41124i −1.29664 2.24584i 1.68466 2.91792i
20.8 1.61897 0.433802i 0.552306 + 0.318874i 0.700831 0.404625i −1.42145 + 1.42145i 1.03250 + 0.276656i 0.234216 2.63536i −1.41124 + 1.41124i −1.29664 2.24584i −1.68466 + 2.91792i
41.1 −2.47996 0.664504i −2.33506 + 1.34815i 3.97660 + 2.29589i −0.281686 0.281686i 6.68671 1.79170i −0.201802 2.63804i −4.70528 4.70528i 2.13500 3.69793i 0.511390 + 0.885754i
41.2 −2.47996 0.664504i 2.33506 1.34815i 3.97660 + 2.29589i 0.281686 + 0.281686i −6.68671 + 1.79170i 1.14426 + 2.38551i −4.70528 4.70528i 2.13500 3.69793i −0.511390 0.885754i
41.3 −0.892257 0.239080i −0.988196 + 0.570535i −0.993087 0.573359i 2.80028 + 2.80028i 1.01813 0.272807i 2.62900 + 0.297264i 2.05537 + 2.05537i −0.848979 + 1.47047i −1.82908 3.16806i
41.4 −0.892257 0.239080i 0.988196 0.570535i −0.993087 0.573359i −2.80028 2.80028i −1.01813 + 0.272807i 2.12815 1.57194i 2.05537 + 2.05537i −0.848979 + 1.47047i 1.82908 + 3.16806i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 76.8 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.b odd 2 1 inner
13.f odd 12 1 inner
91.bc 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.bc.a 32
3.b odd 2 1 819.2.fm.g 32
7.b odd 2 1 inner 91.2.bc.a 32
7.c even 3 1 637.2.x.b 32
7.c even 3 1 637.2.bb.b 32
7.d odd 6 1 637.2.x.b 32
7.d odd 6 1 637.2.bb.b 32
13.f odd 12 1 inner 91.2.bc.a 32
21.c even 2 1 819.2.fm.g 32
39.k even 12 1 819.2.fm.g 32
91.w even 12 1 637.2.bb.b 32
91.x odd 12 1 637.2.x.b 32
91.ba even 12 1 637.2.x.b 32
91.bc even 12 1 inner 91.2.bc.a 32
91.bd odd 12 1 637.2.bb.b 32
273.ca odd 12 1 819.2.fm.g 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.bc.a 32 1.a even 1 1 trivial
91.2.bc.a 32 7.b odd 2 1 inner
91.2.bc.a 32 13.f odd 12 1 inner
91.2.bc.a 32 91.bc even 12 1 inner
637.2.x.b 32 7.c even 3 1
637.2.x.b 32 7.d odd 6 1
637.2.x.b 32 91.x odd 12 1
637.2.x.b 32 91.ba even 12 1
637.2.bb.b 32 7.c even 3 1
637.2.bb.b 32 7.d odd 6 1
637.2.bb.b 32 91.w even 12 1
637.2.bb.b 32 91.bd odd 12 1
819.2.fm.g 32 3.b odd 2 1
819.2.fm.g 32 21.c even 2 1
819.2.fm.g 32 39.k even 12 1
819.2.fm.g 32 273.ca odd 12 1

## Hecke kernels

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