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Properties

Label 920.2.n.a
Level $920$
Weight $2$
Character orbit 920.n
Analytic conductor $7.346$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [920,2,Mod(91,920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(920, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("920.91");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 920.n (of order \(2\), degree \(1\), 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: \(7.34623698596\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{2} + 4 q^{4} - 48 q^{5} - 5 q^{6} + q^{8} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{2} + 4 q^{4} - 48 q^{5} - 5 q^{6} + q^{8} + 48 q^{9} + 2 q^{10} - 3 q^{12} + 12 q^{16} - 7 q^{18} - 4 q^{20} + 8 q^{21} + 4 q^{23} - 2 q^{24} + 48 q^{25} - 7 q^{26} - 12 q^{27} + 5 q^{30} + 8 q^{32} - 12 q^{34} + q^{36} + 8 q^{37} - 10 q^{38} - q^{40} - 10 q^{42} - 12 q^{44} - 48 q^{45} + 12 q^{46} + 33 q^{48} + 48 q^{49} - 2 q^{50} + 7 q^{52} + 32 q^{53} - 17 q^{54} + 12 q^{56} - 15 q^{58} + 12 q^{59} + 3 q^{60} - 8 q^{61} - 15 q^{62} + 7 q^{64} + 88 q^{66} + 10 q^{68} - 20 q^{69} - 3 q^{72} + 22 q^{76} - 19 q^{78} + 32 q^{79} - 12 q^{80} + 56 q^{81} - 7 q^{82} - 8 q^{84} - 18 q^{86} + 12 q^{88} + 7 q^{90} + 14 q^{92} - 25 q^{94} - 65 q^{96} + 10 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1 −1.41243 − 0.0710478i −0.913053 1.98990 + 0.200700i −1.00000 1.28962 + 0.0648705i −2.64750 −2.79634 − 0.424852i −2.16633 1.41243 + 0.0710478i
91.2 −1.41243 + 0.0710478i −0.913053 1.98990 − 0.200700i −1.00000 1.28962 − 0.0648705i −2.64750 −2.79634 + 0.424852i −2.16633 1.41243 − 0.0710478i
91.3 −1.37442 − 0.333129i 2.95751 1.77805 + 0.915718i −1.00000 −4.06485 − 0.985232i −2.10807 −2.13873 − 1.85090i 5.74684 1.37442 + 0.333129i
91.4 −1.37442 + 0.333129i 2.95751 1.77805 − 0.915718i −1.00000 −4.06485 + 0.985232i −2.10807 −2.13873 + 1.85090i 5.74684 1.37442 − 0.333129i
91.5 −1.36018 − 0.387197i −2.78803 1.70016 + 1.05331i −1.00000 3.79221 + 1.07952i 2.03967 −1.90467 − 2.09099i 4.77309 1.36018 + 0.387197i
91.6 −1.36018 + 0.387197i −2.78803 1.70016 − 1.05331i −1.00000 3.79221 − 1.07952i 2.03967 −1.90467 + 2.09099i 4.77309 1.36018 − 0.387197i
91.7 −1.35787 − 0.395203i 1.71241 1.68763 + 1.07327i −1.00000 −2.32524 − 0.676751i 4.89221 −1.86742 − 2.12432i −0.0676366 1.35787 + 0.395203i
91.8 −1.35787 + 0.395203i 1.71241 1.68763 − 1.07327i −1.00000 −2.32524 + 0.676751i 4.89221 −1.86742 + 2.12432i −0.0676366 1.35787 − 0.395203i
91.9 −1.27153 − 0.619041i 0.681538 1.23358 + 1.57426i −1.00000 −0.866595 − 0.421900i 0.0317674 −0.593996 − 2.76535i −2.53551 1.27153 + 0.619041i
91.10 −1.27153 + 0.619041i 0.681538 1.23358 − 1.57426i −1.00000 −0.866595 + 0.421900i 0.0317674 −0.593996 + 2.76535i −2.53551 1.27153 − 0.619041i
91.11 −1.05783 − 0.938612i 0.725937 0.238016 + 1.98579i −1.00000 −0.767919 − 0.681373i −0.223213 1.61210 − 2.32403i −2.47302 1.05783 + 0.938612i
91.12 −1.05783 + 0.938612i 0.725937 0.238016 − 1.98579i −1.00000 −0.767919 + 0.681373i −0.223213 1.61210 + 2.32403i −2.47302 1.05783 − 0.938612i
91.13 −0.956208 − 1.04195i −3.06233 −0.171332 + 1.99265i −1.00000 2.92822 + 3.19080i −4.11381 2.24007 − 1.72687i 6.37786 0.956208 + 1.04195i
91.14 −0.956208 + 1.04195i −3.06233 −0.171332 − 1.99265i −1.00000 2.92822 − 3.19080i −4.11381 2.24007 + 1.72687i 6.37786 0.956208 − 1.04195i
91.15 −0.788676 − 1.17388i 2.23889 −0.755980 + 1.85162i −1.00000 −1.76576 − 2.62819i −3.60854 2.76980 − 0.572900i 2.01264 0.788676 + 1.17388i
91.16 −0.788676 + 1.17388i 2.23889 −0.755980 − 1.85162i −1.00000 −1.76576 + 2.62819i −3.60854 2.76980 + 0.572900i 2.01264 0.788676 − 1.17388i
91.17 −0.773319 − 1.18405i −0.922910 −0.803955 + 1.83130i −1.00000 0.713704 + 1.09277i 3.55561 2.79007 − 0.464255i −2.14824 0.773319 + 1.18405i
91.18 −0.773319 + 1.18405i −0.922910 −0.803955 − 1.83130i −1.00000 0.713704 − 1.09277i 3.55561 2.79007 + 0.464255i −2.14824 0.773319 − 1.18405i
91.19 −0.765311 − 1.18924i −1.60868 −0.828598 + 1.82028i −1.00000 1.23114 + 1.91312i 0.119617 2.79889 − 0.407678i −0.412135 0.765311 + 1.18924i
91.20 −0.765311 + 1.18924i −1.60868 −0.828598 − 1.82028i −1.00000 1.23114 − 1.91312i 0.119617 2.79889 + 0.407678i −0.412135 0.765311 − 1.18924i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
184.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 920.2.n.a ✓ 48
4.b odd 2 1 3680.2.n.a 48
8.b even 2 1 3680.2.n.b 48
8.d odd 2 1 920.2.n.b yes 48
23.b odd 2 1 920.2.n.b yes 48
92.b even 2 1 3680.2.n.b 48
184.e odd 2 1 3680.2.n.a 48
184.h even 2 1 inner 920.2.n.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.n.a ✓ 48 1.a even 1 1 trivial
920.2.n.a ✓ 48 184.h even 2 1 inner
920.2.n.b yes 48 8.d odd 2 1
920.2.n.b yes 48 23.b odd 2 1
3680.2.n.a 48 4.b odd 2 1
3680.2.n.a 48 184.e odd 2 1
3680.2.n.b 48 8.b even 2 1
3680.2.n.b 48 92.b even 2 1