Properties

Label 343.2.g.g
Level $343$
Weight $2$
Character orbit 343.g
Analytic conductor $2.739$
Analytic rank $0$
Dimension $48$
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] = [343,2,Mod(30,343)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(343, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([32]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("343.30");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 343 = 7^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 343.g (of order \(21\), degree \(12\), not 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: \(2.73886878933\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{21})\)
Twist minimal: no (minimal twist has level 49)
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 13 q^{2} + 14 q^{3} - 9 q^{4} + 14 q^{5} - 20 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 13 q^{2} + 14 q^{3} - 9 q^{4} + 14 q^{5} - 20 q^{8} + 6 q^{9} + 14 q^{10} - 3 q^{11} - 21 q^{12} + 14 q^{13} - 12 q^{15} - 3 q^{16} + 7 q^{17} + 2 q^{18} - 21 q^{19} - 14 q^{20} - 20 q^{22} + 15 q^{23} - 28 q^{24} - 4 q^{25} - 7 q^{27} + 12 q^{29} + 11 q^{30} - 35 q^{31} + 45 q^{32} + 14 q^{33} - 70 q^{34} - 12 q^{36} + 15 q^{37} + 28 q^{38} - 7 q^{39} + 42 q^{40} + 42 q^{41} - 30 q^{43} - 50 q^{44} - 7 q^{45} - 78 q^{46} - 21 q^{47} + 84 q^{48} + 40 q^{50} - 52 q^{51} + 70 q^{52} + 11 q^{53} + 77 q^{54} + 7 q^{55} - 12 q^{57} + 16 q^{58} + 28 q^{59} + 56 q^{60} - 7 q^{61} + 28 q^{62} - 32 q^{64} + 14 q^{65} - 154 q^{66} + 11 q^{67} - 77 q^{68} - 70 q^{69} + 19 q^{71} + 170 q^{72} - 7 q^{73} + 34 q^{74} - 112 q^{75} - 119 q^{76} + 28 q^{78} + 15 q^{79} - 70 q^{80} + 64 q^{81} + 14 q^{82} - 26 q^{85} - 33 q^{86} + 112 q^{87} - 77 q^{88} + 14 q^{89} + 182 q^{90} - 38 q^{92} - 80 q^{93} - 14 q^{94} - 61 q^{95} + 70 q^{96} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
30.1 −2.09732 + 1.42993i 1.40598 1.30456i 1.62338 4.13631i 2.38971 + 0.737129i −1.08336 + 4.74653i 0 1.38019 + 6.04701i 0.0507148 0.676742i −6.06605 + 1.87113i
30.2 −0.968018 + 0.659983i −0.630335 + 0.584866i −0.229202 + 0.583997i −3.03447 0.936009i 0.224174 0.982171i 0 −0.684966 3.00103i −0.168936 + 2.25429i 3.55517 1.09662i
30.3 0.174575 0.119023i −1.58372 + 1.46948i −0.714372 + 1.82019i 3.75050 + 1.15688i −0.101576 + 0.445032i 0 0.185965 + 0.814768i 0.124614 1.66286i 0.792437 0.244435i
30.4 1.52543 1.04002i 1.14975 1.06682i 0.514606 1.31120i 1.32986 + 0.410207i 0.644358 2.82312i 0 0.242977 + 1.06455i −0.0403518 + 0.538458i 2.45523 0.757337i
67.1 −2.50546 + 0.772833i 0.539916 + 1.37568i 4.02760 2.74597i −2.04183 0.307757i −2.41591 3.02946i 0 −4.69930 + 5.89274i 0.598158 0.555009i 5.35358 0.806922i
67.2 −1.16438 + 0.359164i −0.671120 1.70999i −0.425690 + 0.290231i 0.478452 + 0.0721150i 1.39561 + 1.75004i 0 1.91089 2.39618i −0.274497 + 0.254696i −0.583002 + 0.0878735i
67.3 0.681025 0.210069i 0.953694 + 2.42997i −1.23281 + 0.840516i −2.65874 0.400741i 1.15995 + 1.45453i 0 −1.55172 + 1.94579i −2.79608 + 2.59438i −1.89485 + 0.285603i
67.4 1.16258 0.358609i −0.190201 0.484624i −0.429482 + 0.292816i 2.56601 + 0.386764i −0.394914 0.495207i 0 −1.91142 + 2.39684i 2.00047 1.85617i 3.12189 0.470550i
79.1 −2.25512 + 0.339905i −0.489949 0.334042i 3.05889 0.943543i 0.260520 3.47640i 1.21844 + 0.586769i 0 −2.46797 + 1.18851i −0.967557 2.46529i 0.594140 + 7.92825i
79.2 −1.23528 + 0.186189i 2.40355 + 1.63871i −0.419885 + 0.129517i −0.186621 + 2.49028i −3.27417 1.57676i 0 2.74561 1.32222i 1.99564 + 5.08481i −0.233134 3.11095i
79.3 −0.251257 + 0.0378709i −1.64786 1.12349i −1.84945 + 0.570480i −0.0348203 + 0.464645i 0.456585 + 0.219880i 0 0.900946 0.433873i 0.357192 + 0.910110i −0.00884768 0.118064i
79.4 1.78609 0.269210i 1.92585 + 1.31302i 1.20650 0.372155i 0.264371 3.52778i 3.79321 + 1.82671i 0 −1.20005 + 0.577915i 0.888840 + 2.26473i −0.477524 6.37211i
116.1 −0.151065 2.01582i −1.33832 0.412818i −2.06305 + 0.310954i 1.66779 1.54748i −0.629993 + 2.76018i 0 0.0388416 + 0.170176i −0.858025 0.584992i −3.37138 3.12819i
116.2 −0.118164 1.57679i 3.07373 + 0.948121i −0.494650 + 0.0745565i −0.291074 + 0.270077i 1.13178 4.95867i 0 −0.527696 2.31199i 6.07018 + 4.13858i 0.460250 + 0.427049i
116.3 0.0658689 + 0.878959i 0.269714 + 0.0831957i 1.20943 0.182293i 1.49036 1.38285i −0.0553598 + 0.242547i 0 0.632162 + 2.76968i −2.41289 1.64508i 1.31364 + 1.21888i
116.4 0.192191 + 2.56461i 0.776689 + 0.239577i −4.56263 + 0.687706i −1.80872 + 1.67825i −0.465149 + 2.03795i 0 −1.49604 6.55456i −1.93287 1.31781i −4.65167 4.31612i
128.1 −2.50546 0.772833i 0.539916 1.37568i 4.02760 + 2.74597i −2.04183 + 0.307757i −2.41591 + 3.02946i 0 −4.69930 5.89274i 0.598158 + 0.555009i 5.35358 + 0.806922i
128.2 −1.16438 0.359164i −0.671120 + 1.70999i −0.425690 0.290231i 0.478452 0.0721150i 1.39561 1.75004i 0 1.91089 + 2.39618i −0.274497 0.254696i −0.583002 0.0878735i
128.3 0.681025 + 0.210069i 0.953694 2.42997i −1.23281 0.840516i −2.65874 + 0.400741i 1.15995 1.45453i 0 −1.55172 1.94579i −2.79608 2.59438i −1.89485 0.285603i
128.4 1.16258 + 0.358609i −0.190201 + 0.484624i −0.429482 0.292816i 2.56601 0.386764i −0.394914 + 0.495207i 0 −1.91142 2.39684i 2.00047 + 1.85617i 3.12189 + 0.470550i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 30.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 343.2.g.g 48
7.b odd 2 1 49.2.g.a 48
7.c even 3 1 343.2.e.c 48
7.c even 3 1 343.2.g.h 48
7.d odd 6 1 343.2.e.d 48
7.d odd 6 1 343.2.g.i 48
21.c even 2 1 441.2.bb.d 48
28.d even 2 1 784.2.bg.c 48
49.e even 7 1 343.2.g.h 48
49.f odd 14 1 343.2.g.i 48
49.g even 21 1 343.2.e.c 48
49.g even 21 1 inner 343.2.g.g 48
49.g even 21 1 2401.2.a.i 24
49.h odd 42 1 49.2.g.a 48
49.h odd 42 1 343.2.e.d 48
49.h odd 42 1 2401.2.a.h 24
147.o even 42 1 441.2.bb.d 48
196.p even 42 1 784.2.bg.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.g.a 48 7.b odd 2 1
49.2.g.a 48 49.h odd 42 1
343.2.e.c 48 7.c even 3 1
343.2.e.c 48 49.g even 21 1
343.2.e.d 48 7.d odd 6 1
343.2.e.d 48 49.h odd 42 1
343.2.g.g 48 1.a even 1 1 trivial
343.2.g.g 48 49.g even 21 1 inner
343.2.g.h 48 7.c even 3 1
343.2.g.h 48 49.e even 7 1
343.2.g.i 48 7.d odd 6 1
343.2.g.i 48 49.f odd 14 1
441.2.bb.d 48 21.c even 2 1
441.2.bb.d 48 147.o even 42 1
784.2.bg.c 48 28.d even 2 1
784.2.bg.c 48 196.p even 42 1
2401.2.a.h 24 49.h odd 42 1
2401.2.a.i 24 49.g even 21 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(343, [\chi])\):

\( T_{2}^{48} + 13 T_{2}^{47} + 85 T_{2}^{46} + 382 T_{2}^{45} + 1339 T_{2}^{44} + 3829 T_{2}^{43} + 9018 T_{2}^{42} + 17506 T_{2}^{41} + 27882 T_{2}^{40} + 36353 T_{2}^{39} + 41812 T_{2}^{38} + 54238 T_{2}^{37} + 79166 T_{2}^{36} + \cdots + 729 \) Copy content Toggle raw display
\( T_{3}^{48} - 14 T_{3}^{47} + 89 T_{3}^{46} - 329 T_{3}^{45} + 679 T_{3}^{44} + 35 T_{3}^{43} - 6061 T_{3}^{42} + 24122 T_{3}^{41} - 49506 T_{3}^{40} + 37289 T_{3}^{39} + 91315 T_{3}^{38} - 314237 T_{3}^{37} + 455848 T_{3}^{36} + \cdots + 4439449 \) Copy content Toggle raw display