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Properties

Label 880.2.w.a
Level $880$
Weight $2$
Character orbit 880.w
Analytic conductor $7.027$
Analytic rank $0$
Dimension $72$
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] = [880,2,Mod(221,880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(880, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.221");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.w (of order \(4\), degree \(2\), 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: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 12 q^{6} - 4 q^{10} + 12 q^{12} - 8 q^{14} + 8 q^{15} + 4 q^{16} + 56 q^{17} + 20 q^{18} + 8 q^{19} - 8 q^{20} + 4 q^{22} - 24 q^{24} - 24 q^{27} + 20 q^{28} - 20 q^{30} + 20 q^{32} - 8 q^{33} + 4 q^{34} - 60 q^{36} + 12 q^{38} - 40 q^{42} + 8 q^{44} - 20 q^{46} + 40 q^{47} + 20 q^{48} - 24 q^{49} + 32 q^{51} - 84 q^{52} + 72 q^{54} + 8 q^{56} - 8 q^{58} + 32 q^{60} - 16 q^{61} - 96 q^{62} + 36 q^{64} - 40 q^{65} + 20 q^{66} - 76 q^{68} - 16 q^{69} + 24 q^{70} + 20 q^{72} + 56 q^{74} + 56 q^{76} - 132 q^{78} - 16 q^{79} + 56 q^{81} + 36 q^{82} - 40 q^{83} - 152 q^{84} + 16 q^{85} + 56 q^{86} - 36 q^{90} + 88 q^{91} + 80 q^{92} + 48 q^{93} - 76 q^{94} - 32 q^{96} - 64 q^{97} - 60 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1 −1.41184 + 0.0819620i 1.49219 + 1.49219i 1.98656 − 0.231434i 0.707107 − 0.707107i −2.22904 − 1.98443i − 3.93083i −2.78574 + 0.489570i 1.45329i −0.940363 + 1.05628i
221.2 −1.40337 + 0.174823i −0.407797 − 0.407797i 1.93887 − 0.490681i −0.707107 + 0.707107i 0.643580 + 0.500996i 4.32700i −2.63517 + 1.02756i − 2.66740i 0.868712 − 1.11595i
221.3 −1.34722 − 0.430099i 0.0345890 + 0.0345890i 1.63003 + 1.15888i 0.707107 − 0.707107i −0.0317225 − 0.0614759i 2.72072i −1.69758 − 2.26235i − 2.99761i −1.25676 + 0.648506i
221.4 −1.33646 + 0.462464i 1.23249 + 1.23249i 1.57225 − 1.23613i −0.707107 + 0.707107i −2.21716 − 1.07719i − 3.38355i −1.52959 + 2.37915i 0.0380753i 0.618009 − 1.27203i
221.5 −1.31567 + 0.518676i −1.53713 − 1.53713i 1.46195 − 1.36481i −0.707107 + 0.707107i 2.81962 + 1.22508i − 0.0719012i −1.21555 + 2.55391i 1.72555i 0.563557 − 1.29707i
221.6 −1.31160 − 0.528875i −2.14472 − 2.14472i 1.44058 + 1.38734i −0.707107 + 0.707107i 1.67872 + 3.94731i − 0.843062i −1.15573 − 2.58153i 6.19967i 1.30141 − 0.553469i
221.7 −1.27571 + 0.610390i 0.191372 + 0.191372i 1.25485 − 1.55736i 0.707107 − 0.707107i −0.360945 − 0.127322i 1.72361i −0.650220 + 2.75267i − 2.92675i −0.470449 + 1.33367i
221.8 −0.941762 + 1.05503i 2.08238 + 2.08238i −0.226170 − 1.98717i 0.707107 − 0.707107i −4.15807 + 0.235864i − 0.495497i 2.30952 + 1.63283i 5.67259i 0.0800917 + 1.41194i
221.9 −0.936814 − 1.05942i −1.75248 − 1.75248i −0.244760 + 1.98497i 0.707107 − 0.707107i −0.214873 + 3.49837i − 2.53766i 2.33222 − 1.60024i 3.14237i −1.41155 − 0.0866990i
221.10 −0.924017 + 1.07060i −1.65472 − 1.65472i −0.292385 − 1.97851i 0.707107 − 0.707107i 3.30054 − 0.242560i 2.59081i 2.38837 + 1.51515i 2.47619i 0.103653 + 1.41041i
221.11 −0.893505 + 1.09620i 0.578882 + 0.578882i −0.403298 − 1.95892i −0.707107 + 0.707107i −1.15180 + 0.117335i 3.01207i 2.50771 + 1.30821i − 2.32979i −0.143325 − 1.40693i
221.12 −0.855954 − 1.12576i 2.33793 + 2.33793i −0.534684 + 1.92720i 0.707107 − 0.707107i 0.630793 − 4.63311i 3.63387i 2.62724 − 1.04767i 7.93182i −1.40129 − 0.190783i
221.13 −0.820740 − 1.15169i −0.928026 − 0.928026i −0.652771 + 1.89047i −0.707107 + 0.707107i −0.307128 + 1.83047i 0.454189i 2.71299 − 0.799799i − 1.27753i 1.39472 + 0.234015i
221.14 −0.583091 − 1.28841i 1.92615 + 1.92615i −1.32001 + 1.50252i −0.707107 + 0.707107i 1.35855 − 3.60479i − 2.42359i 2.70555 + 0.824609i 4.42007i 1.32335 + 0.498737i
221.15 −0.399487 + 1.35662i −0.776690 − 0.776690i −1.68082 − 1.08390i 0.707107 − 0.707107i 1.36395 − 0.743394i − 2.52910i 2.14191 − 1.84723i − 1.79350i 0.676793 + 1.24175i
221.16 −0.274668 + 1.38728i −1.78994 − 1.78994i −1.84912 − 0.762085i −0.707107 + 0.707107i 2.97480 − 1.99152i 3.39249i 1.56512 − 2.35593i 3.40779i −0.786739 − 1.17518i
221.17 −0.0993049 − 1.41072i 1.33671 + 1.33671i −1.98028 + 0.280183i 0.707107 − 0.707107i 1.75298 − 2.01847i − 1.42871i 0.591912 + 2.76580i 0.573579i −1.06775 − 0.927312i
221.18 −0.0549372 − 1.41315i −1.66048 − 1.66048i −1.99396 + 0.155268i 0.707107 − 0.707107i −2.25527 + 2.43772i 1.68953i 0.328960 + 2.80923i 2.51436i −1.03809 − 0.960399i
221.19 0.107309 − 1.41014i −0.986180 − 0.986180i −1.97697 − 0.302640i −0.707107 + 0.707107i −1.49647 + 1.28482i 2.23618i −0.638910 + 2.75532i − 1.05490i 0.921238 + 1.07300i
221.20 0.163047 + 1.40478i 0.276106 + 0.276106i −1.94683 + 0.458091i −0.707107 + 0.707107i −0.342851 + 0.432887i − 0.125783i −0.960944 − 2.66019i − 2.84753i −1.10862 − 0.878040i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 221.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.w.a ✓ 72
16.e even 4 1 inner 880.2.w.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.w.a ✓ 72 1.a even 1 1 trivial
880.2.w.a ✓ 72 16.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{72} + 8 T_{3}^{69} + 400 T_{3}^{68} + 72 T_{3}^{67} + 32 T_{3}^{66} + 2536 T_{3}^{65} + \cdots + 1327104 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\). Copy content Toggle raw display