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Properties

Label 96.2.n.a
Level $96$
Weight $2$
Character orbit 96.n
Analytic conductor $0.767$
Analytic rank $0$
Dimension $32$
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] = [96,2,Mod(13,96)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(96, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("96.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 96.n (of order \(8\), 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: \(0.766563859404\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{10} - 16 q^{12} - 32 q^{14} - 8 q^{16} - 8 q^{18} - 32 q^{20} - 8 q^{22} - 16 q^{23} - 8 q^{24} + 40 q^{26} + 40 q^{28} - 48 q^{31} + 40 q^{32} + 40 q^{34} - 48 q^{35} + 40 q^{38} + 8 q^{40} - 16 q^{43} + 8 q^{44} - 32 q^{46} + 24 q^{50} + 16 q^{51} - 8 q^{52} - 32 q^{53} + 8 q^{54} + 32 q^{55} + 56 q^{56} - 32 q^{58} + 64 q^{59} + 48 q^{60} - 32 q^{61} + 48 q^{62} + 16 q^{63} + 24 q^{64} + 48 q^{66} + 16 q^{67} - 8 q^{68} - 32 q^{69} - 24 q^{70} + 64 q^{71} - 32 q^{74} + 32 q^{75} - 56 q^{76} - 32 q^{77} + 24 q^{78} - 56 q^{80} - 40 q^{82} - 64 q^{86} - 48 q^{88} - 48 q^{91} - 80 q^{92} - 32 q^{94} - 40 q^{96} - 80 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.29526 − 0.567706i 0.382683 − 0.923880i 1.35542 + 1.47066i 0.825824 − 0.342068i −1.02017 + 0.979416i 1.17750 − 1.17750i −0.920723 − 2.67437i −0.707107 − 0.707107i −1.26385 − 0.0257578i
13.2 −1.26685 − 0.628571i −0.382683 + 0.923880i 1.20980 + 1.59260i −3.09318 + 1.28124i 1.06552 − 0.929869i −1.73503 + 1.73503i −0.531562 − 2.77803i −0.707107 − 0.707107i 4.72394 + 0.321153i
13.3 −0.603367 + 1.27904i 0.382683 − 0.923880i −1.27190 − 1.54346i 3.68816 − 1.52768i 0.950782 + 1.04691i −1.63704 + 1.63704i 2.74158 − 0.695531i −0.707107 − 0.707107i −0.271341 + 5.63906i
13.4 0.333592 − 1.37431i 0.382683 − 0.923880i −1.77743 − 0.916914i −1.20409 + 0.498752i −1.14203 − 0.834122i 2.59422 − 2.59422i −1.85306 + 2.13686i −0.707107 − 0.707107i 0.283762 + 1.82117i
13.5 0.605567 + 1.27800i −0.382683 + 0.923880i −1.26658 + 1.54783i 1.60930 − 0.666593i −1.41246 + 0.0704008i −0.589445 + 0.589445i −2.74513 − 0.681373i −0.707107 − 0.707107i 1.82644 + 1.65302i
13.6 0.884039 − 1.10385i −0.382683 + 0.923880i −0.436951 − 1.95168i 2.14986 − 0.890503i 0.681514 + 1.23917i −1.10001 + 1.10001i −2.54064 − 1.24304i −0.707107 − 0.707107i 0.917585 − 3.16036i
13.7 1.34827 − 0.426820i 0.382683 − 0.923880i 1.63565 − 1.15093i −1.46213 + 0.605634i 0.121630 − 1.40897i −3.54889 + 3.54889i 1.71405 − 2.24989i −0.707107 − 0.707107i −1.71285 + 1.44062i
13.8 1.40823 + 0.129991i −0.382683 + 0.923880i 1.96620 + 0.366115i −2.51374 + 1.04122i −0.659001 + 1.25129i 2.01027 − 2.01027i 2.72127 + 0.771162i −0.707107 − 0.707107i −3.67526 + 1.13951i
37.1 −1.29526 + 0.567706i 0.382683 + 0.923880i 1.35542 − 1.47066i 0.825824 + 0.342068i −1.02017 − 0.979416i 1.17750 + 1.17750i −0.920723 + 2.67437i −0.707107 + 0.707107i −1.26385 + 0.0257578i
37.2 −1.26685 + 0.628571i −0.382683 − 0.923880i 1.20980 − 1.59260i −3.09318 − 1.28124i 1.06552 + 0.929869i −1.73503 − 1.73503i −0.531562 + 2.77803i −0.707107 + 0.707107i 4.72394 − 0.321153i
37.3 −0.603367 − 1.27904i 0.382683 + 0.923880i −1.27190 + 1.54346i 3.68816 + 1.52768i 0.950782 − 1.04691i −1.63704 − 1.63704i 2.74158 + 0.695531i −0.707107 + 0.707107i −0.271341 − 5.63906i
37.4 0.333592 + 1.37431i 0.382683 + 0.923880i −1.77743 + 0.916914i −1.20409 − 0.498752i −1.14203 + 0.834122i 2.59422 + 2.59422i −1.85306 − 2.13686i −0.707107 + 0.707107i 0.283762 − 1.82117i
37.5 0.605567 − 1.27800i −0.382683 − 0.923880i −1.26658 − 1.54783i 1.60930 + 0.666593i −1.41246 − 0.0704008i −0.589445 − 0.589445i −2.74513 + 0.681373i −0.707107 + 0.707107i 1.82644 − 1.65302i
37.6 0.884039 + 1.10385i −0.382683 − 0.923880i −0.436951 + 1.95168i 2.14986 + 0.890503i 0.681514 − 1.23917i −1.10001 − 1.10001i −2.54064 + 1.24304i −0.707107 + 0.707107i 0.917585 + 3.16036i
37.7 1.34827 + 0.426820i 0.382683 + 0.923880i 1.63565 + 1.15093i −1.46213 − 0.605634i 0.121630 + 1.40897i −3.54889 − 3.54889i 1.71405 + 2.24989i −0.707107 + 0.707107i −1.71285 − 1.44062i
37.8 1.40823 − 0.129991i −0.382683 − 0.923880i 1.96620 − 0.366115i −2.51374 − 1.04122i −0.659001 − 1.25129i 2.01027 + 2.01027i 2.72127 − 0.771162i −0.707107 + 0.707107i −3.67526 − 1.13951i
61.1 −1.36002 − 0.387756i −0.923880 − 0.382683i 1.69929 + 1.05471i 0.705805 + 1.70396i 1.10810 + 0.878696i 3.24150 − 3.24150i −1.90209 − 2.09333i 0.707107 + 0.707107i −0.299183 − 2.59110i
61.2 −1.13314 + 0.846160i 0.923880 + 0.382683i 0.568026 − 1.91764i 0.750897 + 1.81283i −1.37070 + 0.348115i 0.638460 − 0.638460i 0.978976 + 2.65360i 0.707107 + 0.707107i −2.38481 − 1.41881i
61.3 −0.947612 + 1.04978i −0.923880 − 0.382683i −0.204063 − 1.98956i −1.48656 − 3.58888i 1.27721 − 0.607232i 1.03821 − 1.03821i 2.28197 + 1.67111i 0.707107 + 0.707107i 5.17620 + 1.84030i
61.4 −0.607577 − 1.27705i 0.923880 + 0.382683i −1.26170 + 1.55181i −1.35803 − 3.27858i −0.0726234 − 1.41235i 2.48546 − 2.48546i 2.74831 + 0.668405i 0.707107 + 0.707107i −3.36179 + 3.72626i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.2.n.a ✓ 32
3.b odd 2 1 288.2.v.d 32
4.b odd 2 1 384.2.n.a 32
8.b even 2 1 768.2.n.a 32
8.d odd 2 1 768.2.n.b 32
12.b even 2 1 1152.2.v.c 32
32.g even 8 1 inner 96.2.n.a ✓ 32
32.g even 8 1 768.2.n.a 32
32.h odd 8 1 384.2.n.a 32
32.h odd 8 1 768.2.n.b 32
96.o even 8 1 1152.2.v.c 32
96.p odd 8 1 288.2.v.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.n.a ✓ 32 1.a even 1 1 trivial
96.2.n.a ✓ 32 32.g even 8 1 inner
288.2.v.d 32 3.b odd 2 1
288.2.v.d 32 96.p odd 8 1
384.2.n.a 32 4.b odd 2 1
384.2.n.a 32 32.h odd 8 1
768.2.n.a 32 8.b even 2 1
768.2.n.a 32 32.g even 8 1
768.2.n.b 32 8.d odd 2 1
768.2.n.b 32 32.h odd 8 1
1152.2.v.c 32 12.b even 2 1
1152.2.v.c 32 96.o even 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(96, [\chi])\).