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Properties

Label 440.2.r.h
Level $440$
Weight $2$
Character orbit 440.r
Analytic conductor $3.513$
Analytic rank $0$
Dimension $56$
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: \(56\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{2} + 4 q^{3} - 4 q^{6} - 26 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{2} + 4 q^{3} - 4 q^{6} - 26 q^{8} - 10 q^{10} + 56 q^{11} + 8 q^{12} + 8 q^{16} + 12 q^{17} + 12 q^{18} - 24 q^{20} + 4 q^{22} - 16 q^{25} + 40 q^{27} + 32 q^{28} + 18 q^{30} + 14 q^{32} + 4 q^{33} - 24 q^{35} + 44 q^{36} - 56 q^{38} + 42 q^{40} - 8 q^{41} + 16 q^{42} - 16 q^{43} - 16 q^{46} - 96 q^{48} - 22 q^{50} - 32 q^{51} - 76 q^{52} - 40 q^{56} + 16 q^{57} - 64 q^{60} + 4 q^{62} - 52 q^{65} - 4 q^{66} + 52 q^{67} + 8 q^{68} + 8 q^{70} + 50 q^{72} - 36 q^{73} + 52 q^{75} + 68 q^{76} - 20 q^{78} + 68 q^{80} - 96 q^{81} - 36 q^{82} - 16 q^{83} + 52 q^{86} - 26 q^{88} - 112 q^{90} - 48 q^{91} - 32 q^{92} - 88 q^{96} - 16 q^{97} + 76 q^{98}+O(q^{100}) \) 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 −1.39709 + 0.219381i −2.26841 + 2.26841i 1.90374 − 0.612991i −1.72799 + 1.41917i 2.67153 − 3.66682i 1.64033 − 1.64033i −2.52523 + 1.27405i − 7.29135i 2.10283 − 2.36180i
67.2 −1.39135 + 0.253268i −1.54165 + 1.54165i 1.87171 − 0.704770i 2.22939 + 0.172624i 1.75452 − 2.53542i −1.78407 + 1.78407i −2.42571 + 1.45463i − 1.75334i −3.14559 + 0.324455i
67.3 −1.37936 + 0.312035i 1.39630 − 1.39630i 1.80527 − 0.860817i −1.69864 + 1.45418i −1.49031 + 2.36169i −0.184718 + 0.184718i −2.22151 + 1.75068i − 0.899303i 1.88928 − 2.53587i
67.4 −1.32280 − 0.500201i 0.320579 − 0.320579i 1.49960 + 1.32333i 0.839266 + 2.07259i −0.584416 + 0.263708i 3.29065 − 3.29065i −1.32173 − 2.50060i 2.79446i −0.0734673 − 3.16142i
67.5 −1.27693 − 0.607813i 1.77760 − 1.77760i 1.26113 + 1.55228i −1.99723 − 1.00552i −3.35033 + 1.18943i −0.419731 + 0.419731i −0.666881 − 2.74869i − 3.31974i 1.93916 + 2.49793i
67.6 −1.23640 + 0.686522i −0.354863 + 0.354863i 1.05738 − 1.69763i 0.0254014 − 2.23592i 0.195132 − 0.682375i 1.50617 − 1.50617i −0.141877 + 2.82487i 2.74814i 1.50360 + 2.78194i
67.7 −1.19214 + 0.760789i 0.612152 − 0.612152i 0.842401 − 1.81394i 0.578627 + 2.15991i −0.264053 + 1.19549i −2.51292 + 2.51292i 0.375762 + 2.80336i 2.25054i −2.33304 − 2.13470i
67.8 −0.769234 + 1.18671i 1.81243 − 1.81243i −0.816559 − 1.82571i 1.82281 − 1.29513i 0.756646 + 3.54501i −0.258603 + 0.258603i 2.79472 + 0.435382i − 3.56981i 0.134774 + 3.15940i
67.9 −0.607813 − 1.27693i 1.77760 − 1.77760i −1.26113 + 1.55228i 1.99723 + 1.00552i −3.35033 − 1.18943i 0.419731 − 0.419731i 2.74869 + 0.666881i − 3.31974i 0.0700454 − 3.16150i
67.10 −0.566038 + 1.29599i 0.112791 − 0.112791i −1.35920 − 1.46716i −1.73275 − 1.41336i 0.0823324 + 0.210020i −0.285323 + 0.285323i 2.67080 − 0.931047i 2.97456i 2.81251 − 1.44561i
67.11 −0.500201 − 1.32280i 0.320579 − 0.320579i −1.49960 + 1.32333i −0.839266 − 2.07259i −0.584416 − 0.263708i −3.29065 + 3.29065i 2.50060 + 1.32173i 2.79446i −2.32182 + 2.14689i
67.12 0.0562174 + 1.41310i −2.08443 + 2.08443i −1.99368 + 0.158881i −0.691474 + 2.12647i −3.06267 − 2.82831i −3.17264 + 3.17264i −0.336594 − 2.80833i − 5.68966i −3.04377 − 0.857575i
67.13 0.219381 − 1.39709i −2.26841 + 2.26841i −1.90374 − 0.612991i 1.72799 − 1.41917i 2.67153 + 3.66682i −1.64033 + 1.64033i −1.27405 + 2.52523i − 7.29135i −1.60362 − 2.72551i
67.14 0.253268 − 1.39135i −1.54165 + 1.54165i −1.87171 − 0.704770i −2.22939 − 0.172624i 1.75452 + 2.53542i 1.78407 − 1.78407i −1.45463 + 2.42571i − 1.75334i −0.804815 + 3.05815i
67.15 0.288646 + 1.38444i −0.394495 + 0.394495i −1.83337 + 0.799229i 2.08550 − 0.806650i −0.660026 − 0.432287i 1.98915 − 1.98915i −1.63568 − 2.30750i 2.68875i 1.71873 + 2.65442i
67.16 0.312035 − 1.37936i 1.39630 − 1.39630i −1.80527 − 0.860817i 1.69864 − 1.45418i −1.49031 − 2.36169i 0.184718 − 0.184718i −1.75068 + 2.22151i − 0.899303i −1.47580 − 2.79679i
67.17 0.587834 + 1.28625i 2.19650 − 2.19650i −1.30890 + 1.51221i 0.364744 + 2.20612i 4.11644 + 1.53408i 2.10143 − 2.10143i −2.71450 − 0.794652i − 6.64924i −2.62322 + 1.76599i
67.18 0.686522 − 1.23640i −0.354863 + 0.354863i −1.05738 − 1.69763i −0.0254014 + 2.23592i 0.195132 + 0.682375i −1.50617 + 1.50617i −2.82487 + 0.141877i 2.74814i 2.74706 + 1.56642i
67.19 0.739128 + 1.20569i 0.926588 − 0.926588i −0.907380 + 1.78232i 2.10254 + 0.761129i 1.80204 + 0.432311i −3.49994 + 3.49994i −2.81960 + 0.223343i 1.28287i 0.636361 + 3.09759i
67.20 0.760789 − 1.19214i 0.612152 − 0.612152i −0.842401 − 1.81394i −0.578627 − 2.15991i −0.264053 − 1.19549i 2.51292 − 2.51292i −2.80336 − 0.375762i 2.25054i −3.01512 − 0.953427i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.28
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.h ✓ 56
5.c odd 4 1 inner 440.2.r.h ✓ 56
8.d odd 2 1 inner 440.2.r.h ✓ 56
40.k even 4 1 inner 440.2.r.h ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.r.h ✓ 56 1.a even 1 1 trivial
440.2.r.h ✓ 56 5.c odd 4 1 inner
440.2.r.h ✓ 56 8.d odd 2 1 inner
440.2.r.h ✓ 56 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 \(S_{2}^{\mathrm{new}}(440, [\chi])\):

\( T_{3}^{28} - 2 T_{3}^{27} + 2 T_{3}^{26} - 4 T_{3}^{25} + 216 T_{3}^{24} - 460 T_{3}^{23} + 496 T_{3}^{22} + \cdots + 1600 \) Copy content Toggle raw display
\( T_{7}^{56} + 1876 T_{7}^{52} + 1368214 T_{7}^{48} + 496266676 T_{7}^{44} + 96780924913 T_{7}^{40} + \cdots + 6553600000000 \) Copy content Toggle raw display
\( T_{23}^{56} + 13344 T_{23}^{52} + 69748358 T_{23}^{48} + 187570250724 T_{23}^{44} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display