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Properties

Label 440.2.r.g
Level $440$
Weight $2$
Character orbit 440.r
Analytic conductor $3.513$
Analytic rank $0$
Dimension $48$
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] = [440,2,Mod(67,440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(440, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("440.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.r (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: \(3.51341768894\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{3} + 12 q^{6} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{3} + 12 q^{6} + 6 q^{8} - 10 q^{10} - 48 q^{11} - 8 q^{12} - 24 q^{16} - 4 q^{17} - 60 q^{18} + 24 q^{20} - 48 q^{25} - 8 q^{26} + 4 q^{28} - 58 q^{30} + 10 q^{32} - 12 q^{33} + 16 q^{35} - 12 q^{36} + 6 q^{40} - 72 q^{41} + 80 q^{42} + 16 q^{43} + 48 q^{48} + 46 q^{50} - 8 q^{51} + 16 q^{52} + 128 q^{56} - 32 q^{57} - 80 q^{58} + 96 q^{60} - 24 q^{62} + 44 q^{65} - 12 q^{66} + 52 q^{67} - 92 q^{68} - 48 q^{70} - 6 q^{72} + 12 q^{73} + 108 q^{75} + 20 q^{76} + 116 q^{78} - 12 q^{80} - 104 q^{81} + 60 q^{82} + 16 q^{83} + 28 q^{86} - 6 q^{88} + 40 q^{90} + 64 q^{91} - 8 q^{96} + 80 q^{97} - 64 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 −1.41263 + 0.0669602i 2.41113 − 2.41113i 1.99103 − 0.189180i 1.99828 + 1.00343i −3.24458 + 3.56748i −0.519342 + 0.519342i −2.79992 + 0.400561i − 8.62713i −2.89002 − 1.28367i
67.2 −1.34885 − 0.424980i −0.899469 + 0.899469i 1.63878 + 1.14647i 0.413339 + 2.19753i 1.59550 − 0.830991i −1.85075 + 1.85075i −1.72325 − 2.24286i 1.38191i 0.376376 − 3.13980i
67.3 −1.27802 + 0.605520i 0.467060 − 0.467060i 1.26669 − 1.54774i 2.19718 + 0.415216i −0.314099 + 0.879727i 2.58203 − 2.58203i −0.681673 + 2.74505i 2.56371i −3.05947 + 0.799781i
67.4 −1.21011 − 0.731875i −2.18495 + 2.18495i 0.928717 + 1.77129i −1.20524 − 1.88346i 4.24313 − 1.04491i −2.44905 + 2.44905i 0.172520 − 2.82316i − 6.54800i 0.0800093 + 3.16127i
67.5 −1.09303 + 0.897377i −1.98444 + 1.98444i 0.389427 − 1.96172i 0.475248 − 2.18498i 0.388259 − 3.94984i 0.0645339 − 0.0645339i 1.33475 + 2.49368i − 4.87598i 1.44129 + 2.81473i
67.6 −1.06890 − 0.925992i 1.46879 − 1.46879i 0.285078 + 1.97958i 0.974080 − 2.01275i −2.93007 + 0.209896i 2.70426 − 2.70426i 1.52836 − 2.37994i − 1.31467i −2.90498 + 1.24943i
67.7 −0.925992 − 1.06890i 1.46879 − 1.46879i −0.285078 + 1.97958i −0.974080 + 2.01275i −2.93007 − 0.209896i −2.70426 + 2.70426i 2.37994 − 1.52836i − 1.31467i 3.05341 − 0.822602i
67.8 −0.906769 + 1.08525i 1.76752 − 1.76752i −0.355539 − 1.96814i −2.15099 + 0.610934i 0.315470 + 3.52094i 3.11838 − 3.11838i 2.45832 + 1.39880i − 3.24826i 1.28744 − 2.88834i
67.9 −0.731875 − 1.21011i −2.18495 + 2.18495i −0.928717 + 1.77129i 1.20524 + 1.88346i 4.24313 + 1.04491i 2.44905 − 2.44905i 2.82316 − 0.172520i − 6.54800i 1.39710 − 2.83692i
67.10 −0.424980 − 1.34885i −0.899469 + 0.899469i −1.63878 + 1.14647i −0.413339 − 2.19753i 1.59550 + 0.830991i 1.85075 − 1.85075i 2.24286 + 1.72325i 1.38191i −2.78848 + 1.49144i
67.11 −0.374636 + 1.36369i 0.858294 − 0.858294i −1.71930 − 1.02177i 0.910269 + 2.04240i 0.848898 + 1.49199i −0.0679451 + 0.0679451i 2.03749 − 1.96179i 1.52666i −3.12622 + 0.476165i
67.12 −0.192464 + 1.40106i −0.637664 + 0.637664i −1.92591 − 0.539307i 1.18568 − 1.89583i −0.770675 − 1.01613i −3.02537 + 3.02537i 1.12627 − 2.59452i 2.18677i 2.42796 + 2.02608i
67.13 0.0669602 − 1.41263i 2.41113 − 2.41113i −1.99103 − 0.189180i −1.99828 − 1.00343i −3.24458 − 3.56748i 0.519342 − 0.519342i −0.400561 + 2.79992i − 8.62713i −1.55128 + 2.75564i
67.14 0.241906 + 1.39337i 1.67246 − 1.67246i −1.88296 + 0.674130i 0.00759330 − 2.23606i 2.73493 + 1.92577i 1.04087 − 1.04087i −1.39481 − 2.46059i − 2.59422i 3.11749 − 0.530336i
67.15 0.605520 − 1.27802i 0.467060 − 0.467060i −1.26669 − 1.54774i −2.19718 − 0.415216i −0.314099 − 0.879727i −2.58203 + 2.58203i −2.74505 + 0.681673i 2.56371i −1.86109 + 2.55663i
67.16 0.747494 + 1.20052i −0.751520 + 0.751520i −0.882505 + 1.79477i −0.791440 + 2.09132i −1.46397 − 0.340459i 0.878706 − 0.878706i −2.81432 + 0.282111i 1.87044i −3.10227 + 0.613110i
67.17 0.796560 + 1.16854i 0.812785 − 0.812785i −0.730983 + 1.86163i −2.21213 + 0.326307i 1.59721 + 0.302342i −1.60804 + 1.60804i −2.75767 + 0.628715i 1.67876i −2.14340 − 2.32505i
67.18 0.897377 − 1.09303i −1.98444 + 1.98444i −0.389427 − 1.96172i −0.475248 + 2.18498i 0.388259 + 3.94984i −0.0645339 + 0.0645339i −2.49368 − 1.33475i − 4.87598i 1.96177 + 2.48021i
67.19 1.08525 − 0.906769i 1.76752 − 1.76752i 0.355539 − 1.96814i 2.15099 − 0.610934i 0.315470 − 3.52094i −3.11838 + 3.11838i −1.39880 − 2.45832i − 3.24826i 1.78039 − 2.61347i
67.20 1.16854 + 0.796560i 0.812785 − 0.812785i 0.730983 + 1.86163i 2.21213 − 0.326307i 1.59721 − 0.302342i 1.60804 − 1.60804i −0.628715 + 2.75767i 1.67876i 2.84489 + 1.38079i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.d odd 2 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 440.2.r.g ✓ 48
5.c odd 4 1 inner 440.2.r.g ✓ 48
8.d odd 2 1 inner 440.2.r.g ✓ 48
40.k even 4 1 inner 440.2.r.g ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.r.g ✓ 48 1.a even 1 1 trivial
440.2.r.g ✓ 48 5.c odd 4 1 inner
440.2.r.g ✓ 48 8.d odd 2 1 inner
440.2.r.g ✓ 48 40.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(440, [\chi])\):

\( T_{3}^{24} - 6 T_{3}^{23} + 18 T_{3}^{22} - 24 T_{3}^{21} + 175 T_{3}^{20} - 946 T_{3}^{19} + \cdots + 166464 \) Copy content Toggle raw display
\( T_{7}^{48} + 1330 T_{7}^{44} + 706257 T_{7}^{40} + 191861728 T_{7}^{36} + 28377798272 T_{7}^{32} + \cdots + 16777216 \) Copy content Toggle raw display
\( T_{23}^{48} + 12568 T_{23}^{44} + 50915510 T_{23}^{40} + 79741981748 T_{23}^{36} + 48169502811969 T_{23}^{32} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display