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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 $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [91,2,Mod(5,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.bb (of order \(12\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 5.8
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(91, [\chi])\).