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Properties

Label 880.2.w.b
Level $880$
Weight $2$
Character orbit 880.w
Analytic conductor $7.027$
Analytic rank $0$
Dimension $88$
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: \(88\)
Relative dimension: \(44\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 12 q^{6} - 4 q^{10} + 12 q^{12} + 8 q^{15} + 4 q^{16} - 56 q^{17} + 20 q^{18} + 8 q^{19} - 4 q^{22} - 24 q^{24} - 24 q^{27} - 60 q^{28} + 20 q^{30} + 20 q^{32} + 8 q^{33} + 4 q^{34} + 68 q^{36} - 52 q^{38} - 40 q^{42} + 92 q^{46} + 40 q^{47} + 20 q^{48} - 136 q^{49} + 32 q^{51} + 84 q^{52} - 104 q^{54} + 8 q^{56} - 8 q^{58} + 32 q^{59} - 32 q^{60} - 16 q^{61} + 24 q^{62} + 36 q^{64} + 40 q^{65} + 20 q^{66} + 68 q^{68} - 16 q^{69} - 16 q^{70} + 20 q^{72} + 56 q^{74} - 120 q^{76} + 108 q^{78} - 16 q^{79} - 216 q^{81} + 36 q^{82} - 40 q^{83} + 152 q^{84} + 16 q^{85} - 80 q^{86} - 36 q^{90} - 72 q^{91} - 32 q^{92} + 48 q^{93} + 100 q^{94} - 32 q^{96} + 64 q^{97} - 60 q^{98}+O(q^{100}) \) 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 −1.41393 − 0.0284804i 2.10040 + 2.10040i 1.99838 + 0.0805385i 0.707107 − 0.707107i −2.91000 − 3.02964i 1.58783i −2.82327 − 0.170790i 5.82338i −1.01994 + 0.979658i
221.2 −1.40932 − 0.117591i −1.01809 − 1.01809i 1.97234 + 0.331446i −0.707107 + 0.707107i 1.31509 + 1.55452i 0.122450i −2.74068 − 0.699042i − 0.926998i 1.07969 − 0.913388i
221.3 −1.39236 + 0.247668i 1.86750 + 1.86750i 1.87732 − 0.689684i −0.707107 + 0.707107i −3.06275 − 2.13771i 4.57807i −2.44309 + 1.42524i 3.97514i 0.809418 − 1.15967i
221.4 −1.36788 + 0.359029i −1.90728 − 1.90728i 1.74220 − 0.982219i 0.707107 − 0.707107i 3.29370 + 1.92416i 4.40121i −2.03047 + 1.96906i 4.27543i −0.713366 + 1.22111i
221.5 −1.36160 − 0.382176i 1.35509 + 1.35509i 1.70788 + 1.04074i −0.707107 + 0.707107i −1.32720 − 2.36296i − 1.84283i −1.92770 − 2.06978i 0.672526i 1.23303 − 0.692554i
221.6 −1.27608 + 0.609597i −0.814927 − 0.814927i 1.25678 − 1.55579i 0.707107 − 0.707107i 1.53669 + 0.543138i − 2.85838i −0.655353 + 2.75146i − 1.67179i −0.471278 + 1.33338i
221.7 −1.22282 − 0.710423i 0.963314 + 0.963314i 0.990599 + 1.73744i 0.707107 − 0.707107i −0.493604 − 1.86232i 1.58600i 0.0229924 − 2.82833i − 1.14405i −1.36701 + 0.362323i
221.8 −1.22198 − 0.711866i −1.87795 − 1.87795i 0.986493 + 1.73978i 0.707107 − 0.707107i 0.957978 + 3.63168i 1.95491i 0.0330098 − 2.82823i 4.05340i −1.36744 + 0.360708i
221.9 −1.21359 + 0.726086i −0.769481 − 0.769481i 0.945599 − 1.76234i −0.707107 + 0.707107i 1.49254 + 0.375125i − 3.80392i 0.132040 + 2.82534i − 1.81580i 0.344717 − 1.37156i
221.10 −1.18337 + 0.774364i 1.17805 + 1.17805i 0.800720 − 1.83272i −0.707107 + 0.707107i −2.30631 − 0.481827i − 0.298936i 0.471643 + 2.78883i − 0.224393i 0.289209 − 1.38433i
221.11 −0.968126 − 1.03089i 0.131873 + 0.131873i −0.125464 + 1.99606i −0.707107 + 0.707107i 0.00827677 − 0.263616i − 0.197974i 2.17918 − 1.80310i − 2.96522i 1.41352 + 0.0443802i
221.12 −0.862938 + 1.12042i 0.668515 + 0.668515i −0.510676 − 1.93370i 0.707107 − 0.707107i −1.32590 + 0.172130i − 1.48466i 2.60724 + 1.09650i − 2.10617i 0.182066 + 1.40244i
221.13 −0.812857 + 1.15727i −2.28261 − 2.28261i −0.678526 − 1.88138i 0.707107 − 0.707107i 4.49703 − 0.786151i − 3.10941i 2.72880 + 0.744061i 7.42064i 0.243533 + 1.39309i
221.14 −0.809110 − 1.15989i −0.515612 − 0.515612i −0.690682 + 1.87695i 0.707107 − 0.707107i −0.180866 + 1.01524i 0.0196650i 2.73590 − 0.717548i − 2.46829i −1.39229 − 0.248038i
221.15 −0.738289 − 1.20620i 1.27200 + 1.27200i −0.909860 + 1.78105i 0.707107 − 0.707107i 0.595190 − 2.47340i − 3.62311i 2.82006 − 0.217455i 0.235979i −1.37496 − 0.330867i
221.16 −0.672086 + 1.24431i −1.75891 − 1.75891i −1.09660 − 1.67256i −0.707107 + 0.707107i 3.37077 − 1.00649i − 0.320588i 2.81819 − 0.240403i 3.18755i −0.404622 − 1.35509i
221.17 −0.620569 − 1.27078i −2.21073 − 2.21073i −1.22979 + 1.57722i −0.707107 + 0.707107i −1.43745 + 4.18127i 4.41832i 2.76748 + 0.584020i 6.77466i 1.33739 + 0.459772i
221.18 −0.608974 + 1.27638i 0.286475 + 0.286475i −1.25830 − 1.55457i −0.707107 + 0.707107i −0.540107 + 0.191196i 0.877622i 2.75049 − 0.659383i − 2.83586i −0.471929 − 1.33315i
221.19 −0.345697 + 1.37131i 1.23996 + 1.23996i −1.76099 − 0.948115i 0.707107 − 0.707107i −2.12902 + 1.27172i − 1.06237i 1.90893 − 2.08710i 0.0749939i 0.725219 + 1.21411i
221.20 −0.192943 − 1.40099i −0.500430 − 0.500430i −1.92555 + 0.540623i 0.707107 − 0.707107i −0.604543 + 0.797652i 3.49311i 1.12893 + 2.59336i − 2.49914i −1.12708 − 0.854218i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 221.44
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.b ✓ 88
16.e even 4 1 inner 880.2.w.b ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.w.b ✓ 88 1.a even 1 1 trivial
880.2.w.b ✓ 88 16.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{88} + 8 T_{3}^{85} + 720 T_{3}^{84} + 72 T_{3}^{83} + 32 T_{3}^{82} + 5096 T_{3}^{81} + \cdots + 4294967296 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\). Copy content Toggle raw display