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Properties

Label 336.2.s.c
Level $336$
Weight $2$
Character orbit 336.s
Analytic conductor $2.683$
Analytic rank $0$
Dimension $40$
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] = [336,2,Mod(155,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.s (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: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{3} - 2 q^{6} - 40 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{3} - 2 q^{6} - 40 q^{7} - 8 q^{10} - 2 q^{12} + 24 q^{13} + 36 q^{16} + 12 q^{18} + 16 q^{19} - 4 q^{21} - 8 q^{22} + 6 q^{24} - 32 q^{27} - 32 q^{30} + 24 q^{33} + 12 q^{34} - 4 q^{36} - 8 q^{37} - 64 q^{39} - 60 q^{40} + 2 q^{42} + 24 q^{43} - 28 q^{45} + 20 q^{46} - 26 q^{48} + 40 q^{49} - 32 q^{51} + 84 q^{52} - 14 q^{54} + 16 q^{55} + 12 q^{58} - 24 q^{60} - 48 q^{61} - 12 q^{64} - 36 q^{66} + 40 q^{67} + 4 q^{69} + 8 q^{70} + 8 q^{72} + 40 q^{75} - 44 q^{76} + 24 q^{78} + 56 q^{81} - 84 q^{82} + 2 q^{84} - 48 q^{85} + 32 q^{87} + 52 q^{88} - 76 q^{90} - 24 q^{91} + 56 q^{93} - 62 q^{96} + 16 q^{97} + 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1 −1.41409 + 0.0184003i 0.827739 − 1.52146i 1.99932 − 0.0520394i −1.23935 + 1.23935i −1.14250 + 2.16672i −1.00000 −2.82627 + 0.110377i −1.62970 − 2.51875i 1.72975 − 1.77536i
155.2 −1.38145 − 0.302633i 1.42461 + 0.985133i 1.81683 + 0.836146i −0.766553 + 0.766553i −1.66990 − 1.79205i −1.00000 −2.25682 − 1.70493i 1.05903 + 2.80686i 1.29094 − 0.826974i
155.3 −1.37293 + 0.339195i −1.63649 + 0.567369i 1.76989 − 0.931385i 0.132854 − 0.132854i 2.05434 − 1.33405i −1.00000 −2.11403 + 1.87907i 2.35619 − 1.85698i −0.137336 + 0.227462i
155.4 −1.25684 − 0.648355i −0.629429 + 1.61364i 1.15927 + 1.62975i 3.08691 − 3.08691i 1.83730 − 1.61998i −1.00000 −0.400356 − 2.79995i −2.20764 − 2.03134i −5.88116 + 1.87833i
155.5 −1.14682 + 0.827528i −0.0404130 + 1.73158i 0.630395 − 1.89805i −0.0573500 + 0.0573500i −1.38658 − 2.01925i −1.00000 0.847742 + 2.69839i −2.99673 − 0.139957i 0.0183114 − 0.113229i
155.6 −0.806993 + 1.16136i −0.0806676 − 1.73017i −0.697526 − 1.87442i −1.66193 + 1.66193i 2.07445 + 1.30255i −1.00000 2.73978 + 0.702564i −2.98699 + 0.279137i −0.588940 − 3.27128i
155.7 −0.724332 − 1.21464i 1.72790 − 0.119805i −0.950686 + 1.75960i 0.793669 − 0.793669i −1.39709 − 2.01200i −1.00000 2.82589 − 0.119796i 2.97129 − 0.414021i −1.53890 − 0.389140i
155.8 −0.244056 + 1.39300i −1.68331 + 0.408017i −1.88087 − 0.679939i −2.26080 + 2.26080i −0.157543 − 2.44442i −1.00000 1.40619 − 2.45410i 2.66704 − 1.37363i −2.59752 − 3.70105i
155.9 −0.230305 − 1.39533i −1.57773 − 0.714681i −1.89392 + 0.642706i −1.31983 + 1.31983i −0.633860 + 2.36606i −1.00000 1.33297 + 2.49463i 1.97846 + 2.25515i 2.14556 + 1.53764i
155.10 −0.153777 + 1.40583i 1.18388 + 1.26429i −1.95271 − 0.432367i 2.84277 − 2.84277i −1.95942 + 1.46992i −1.00000 0.908115 − 2.67868i −0.196841 + 2.99354i 3.55929 + 4.43360i
155.11 0.153777 − 1.40583i 1.26429 + 1.18388i −1.95271 − 0.432367i −2.84277 + 2.84277i 1.85875 − 1.59532i −1.00000 −0.908115 + 2.67868i 0.196841 + 2.99354i 3.55929 + 4.43360i
155.12 0.230305 + 1.39533i −0.714681 − 1.57773i −1.89392 + 0.642706i 1.31983 − 1.31983i 2.03687 − 1.36058i −1.00000 −1.33297 − 2.49463i −1.97846 + 2.25515i 2.14556 + 1.53764i
155.13 0.244056 − 1.39300i 0.408017 − 1.68331i −1.88087 − 0.679939i 2.26080 − 2.26080i −2.24526 − 0.979187i −1.00000 −1.40619 + 2.45410i −2.66704 − 1.37363i −2.59752 − 3.70105i
155.14 0.724332 + 1.21464i −0.119805 + 1.72790i −0.950686 + 1.75960i −0.793669 + 0.793669i −2.18555 + 1.10606i −1.00000 −2.82589 + 0.119796i −2.97129 − 0.414021i −1.53890 − 0.389140i
155.15 0.806993 − 1.16136i −1.73017 − 0.0806676i −0.697526 − 1.87442i 1.66193 − 1.66193i −1.48992 + 1.94426i −1.00000 −2.73978 − 0.702564i 2.98699 + 0.279137i −0.588940 − 3.27128i
155.16 1.14682 − 0.827528i 1.73158 − 0.0404130i 0.630395 − 1.89805i 0.0573500 − 0.0573500i 1.95237 − 1.47928i −1.00000 −0.847742 − 2.69839i 2.99673 − 0.139957i 0.0183114 − 0.113229i
155.17 1.25684 + 0.648355i 1.61364 − 0.629429i 1.15927 + 1.62975i −3.08691 + 3.08691i 2.43617 + 0.255120i −1.00000 0.400356 + 2.79995i 2.20764 − 2.03134i −5.88116 + 1.87833i
155.18 1.37293 − 0.339195i 0.567369 − 1.63649i 1.76989 − 0.931385i −0.132854 + 0.132854i 0.223871 − 2.43924i −1.00000 2.11403 − 1.87907i −2.35619 − 1.85698i −0.137336 + 0.227462i
155.19 1.38145 + 0.302633i 0.985133 + 1.42461i 1.81683 + 0.836146i 0.766553 − 0.766553i 0.929782 + 2.26617i −1.00000 2.25682 + 1.70493i −1.05903 + 2.80686i 1.29094 − 0.826974i
155.20 1.41409 − 0.0184003i −1.52146 + 0.827739i 1.99932 − 0.0520394i 1.23935 − 1.23935i −2.13626 + 1.19850i −1.00000 2.82627 − 0.110377i 1.62970 − 2.51875i 1.72975 − 1.77536i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.s.c ✓ 40
3.b odd 2 1 inner 336.2.s.c ✓ 40
4.b odd 2 1 1344.2.s.c 40
12.b even 2 1 1344.2.s.c 40
16.e even 4 1 1344.2.s.c 40
16.f odd 4 1 inner 336.2.s.c ✓ 40
48.i odd 4 1 1344.2.s.c 40
48.k even 4 1 inner 336.2.s.c ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.s.c ✓ 40 1.a even 1 1 trivial
336.2.s.c ✓ 40 3.b odd 2 1 inner
336.2.s.c ✓ 40 16.f odd 4 1 inner
336.2.s.c ✓ 40 48.k even 4 1 inner
1344.2.s.c 40 4.b odd 2 1
1344.2.s.c 40 12.b even 2 1
1344.2.s.c 40 16.e even 4 1
1344.2.s.c 40 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{40} + 784 T_{5}^{36} + 201200 T_{5}^{32} + 19415584 T_{5}^{28} + 699116768 T_{5}^{24} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display