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Properties

Label 912.2.bq.a
Level $912$
Weight $2$
Character orbit 912.bq
Analytic conductor $7.282$
Analytic rank $0$
Dimension $320$
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] = [912,2,Mod(277,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.277");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bq (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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(80\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 320 q + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q + 24 q^{8} - 8 q^{10} + 16 q^{15} - 4 q^{16} + 8 q^{19} - 8 q^{24} - 40 q^{26} - 40 q^{28} + 96 q^{31} - 60 q^{32} + 28 q^{34} - 48 q^{35} - 4 q^{36} - 16 q^{38} + 20 q^{40} + 56 q^{44} - 104 q^{46} - 320 q^{49} + 88 q^{50} + 8 q^{51} + 16 q^{52} + 4 q^{54} + 56 q^{56} + 56 q^{58} - 48 q^{60} - 32 q^{61} - 24 q^{62} - 72 q^{64} - 48 q^{66} + 48 q^{67} + 32 q^{68} + 64 q^{69} + 48 q^{70} + 40 q^{74} - 56 q^{76} - 24 q^{78} + 48 q^{79} + 16 q^{80} + 160 q^{81} + 80 q^{82} + 80 q^{83} - 16 q^{85} - 64 q^{86} + 24 q^{92} + 128 q^{94} + 48 q^{95} + 40 q^{96} + 4 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1 −1.41392 + 0.0286883i 0.258819 + 0.965926i 1.99835 − 0.0811260i −1.78169 + 0.477402i −0.393661 − 1.35832i − 2.71325i −2.82319 + 0.172035i −0.866025 + 0.500000i 2.50547 − 0.726123i
277.2 −1.41195 − 0.0800517i 0.258819 + 0.965926i 1.98718 + 0.226057i 3.51365 − 0.941479i −0.288115 − 1.38455i − 2.76193i −2.78770 − 0.478258i −0.866025 + 0.500000i −5.03645 + 1.04804i
277.3 −1.40998 − 0.109381i −0.258819 − 0.965926i 1.97607 + 0.308448i −1.46117 + 0.391518i 0.259275 + 1.39024i − 1.10864i −2.75248 − 0.651049i −0.866025 + 0.500000i 2.10304 − 0.392209i
277.4 −1.39762 + 0.215985i −0.258819 − 0.965926i 1.90670 − 0.603732i 1.04576 − 0.280211i 0.570357 + 1.29410i 2.89896i −2.53445 + 1.25561i −0.866025 + 0.500000i −1.40106 + 0.617498i
277.5 −1.39617 − 0.225195i −0.258819 − 0.965926i 1.89857 + 0.628821i 1.96870 − 0.527513i 0.143833 + 1.40688i 1.35820i −2.50912 − 1.30549i −0.866025 + 0.500000i −2.86744 + 0.293154i
277.6 −1.37931 + 0.312237i 0.258819 + 0.965926i 1.80502 − 0.861347i −0.281437 + 0.0754108i −0.658591 − 1.25150i 0.0423372i −2.22074 + 1.75166i −0.866025 + 0.500000i 0.364644 − 0.191890i
277.7 −1.37728 − 0.321097i −0.258819 − 0.965926i 1.79379 + 0.884480i −3.77079 + 1.01038i 0.0463102 + 1.41346i − 4.29536i −2.18655 − 1.79416i −0.866025 + 0.500000i 5.51785 − 0.180786i
277.8 −1.35327 − 0.410667i 0.258819 + 0.965926i 1.66270 + 1.11149i 2.49893 − 0.669585i 0.0464209 − 1.41345i 2.08745i −1.79364 − 2.18697i −0.866025 + 0.500000i −3.65671 − 0.120095i
277.9 −1.33270 + 0.473186i −0.258819 − 0.965926i 1.55219 − 1.26123i 1.19360 − 0.319824i 0.801991 + 1.16482i − 4.90644i −1.47181 + 2.41532i −0.866025 + 0.500000i −1.43938 + 0.991025i
277.10 −1.29399 − 0.570614i 0.258819 + 0.965926i 1.34880 + 1.47673i −2.47161 + 0.662266i 0.216262 − 1.39758i − 0.703016i −0.902685 − 2.68051i −0.866025 + 0.500000i 3.57613 + 0.553371i
277.11 −1.27886 + 0.603757i −0.258819 − 0.965926i 1.27096 − 1.54424i −2.62618 + 0.703682i 0.914177 + 1.07902i − 0.218741i −0.693027 + 2.74221i −0.866025 + 0.500000i 2.93365 − 2.48548i
277.12 −1.25159 + 0.658418i 0.258819 + 0.965926i 1.13297 − 1.64814i −0.343204 + 0.0919613i −0.959919 − 1.03854i 2.80793i −0.332855 + 2.80877i −0.866025 + 0.500000i 0.369003 − 0.341070i
277.13 −1.24354 + 0.673496i −0.258819 − 0.965926i 1.09281 − 1.67504i 2.25016 − 0.602928i 0.972400 + 1.02686i − 1.69431i −0.230817 + 2.81899i −0.866025 + 0.500000i −2.39210 + 2.26524i
277.14 −1.20443 − 0.741179i −0.258819 − 0.965926i 0.901307 + 1.78540i −3.70703 + 0.993296i −0.404194 + 1.35522i 4.51201i 0.237737 − 2.81842i −0.866025 + 0.500000i 5.20107 + 1.55122i
277.15 −1.19971 − 0.748793i 0.258819 + 0.965926i 0.878617 + 1.79667i −0.598330 + 0.160322i 0.412771 − 1.35263i 4.63727i 0.291250 − 2.81339i −0.866025 + 0.500000i 0.837872 + 0.255686i
277.16 −1.19899 + 0.749952i 0.258819 + 0.965926i 0.875144 − 1.79837i 3.01995 − 0.809194i −1.03472 − 0.964031i 0.967369i 0.299402 + 2.81254i −0.866025 + 0.500000i −3.01403 + 3.23503i
277.17 −1.17518 − 0.786736i −0.258819 − 0.965926i 0.762094 + 1.84911i 1.01275 − 0.271366i −0.455770 + 1.33876i − 3.11289i 0.559165 − 2.77260i −0.866025 + 0.500000i −1.40366 − 0.477863i
277.18 −1.05563 − 0.941088i 0.258819 + 0.965926i 0.228709 + 1.98688i 0.515915 − 0.138239i 0.635804 − 1.26323i − 5.04592i 1.62840 − 2.31264i −0.866025 + 0.500000i −0.674710 − 0.339592i
277.19 −1.04792 + 0.949665i 0.258819 + 0.965926i 0.196273 − 1.99035i −0.929831 + 0.249147i −1.18853 − 0.766422i − 1.54064i 1.68448 + 2.27212i −0.866025 + 0.500000i 0.737782 − 1.14411i
277.20 −1.04049 + 0.957803i 0.258819 + 0.965926i 0.165228 − 1.99316i −3.40004 + 0.911037i −1.19446 − 0.757136i 3.29859i 1.73714 + 2.23212i −0.866025 + 0.500000i 2.66510 − 4.20449i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner
19.c even 3 1 inner
304.v even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bq.a ✓ 320
16.e even 4 1 inner 912.2.bq.a ✓ 320
19.c even 3 1 inner 912.2.bq.a ✓ 320
304.v even 12 1 inner 912.2.bq.a ✓ 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.bq.a ✓ 320 1.a even 1 1 trivial
912.2.bq.a ✓ 320 16.e even 4 1 inner
912.2.bq.a ✓ 320 19.c even 3 1 inner
912.2.bq.a ✓ 320 304.v even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(912, [\chi])\).