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Properties

Label 343.2.g.i
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 8 q^{2} + 7 q^{3} + 12 q^{4} + 7 q^{5} - 20 q^{8} - 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 8 q^{2} + 7 q^{3} + 12 q^{4} + 7 q^{5} - 20 q^{8} - 15 q^{9} + 7 q^{10} - 3 q^{11} - 63 q^{12} - 14 q^{13} - 12 q^{15} + 18 q^{16} + 14 q^{17} + 2 q^{18} + 21 q^{19} + 14 q^{20} - 20 q^{22} - 27 q^{23} - 77 q^{24} + 17 q^{25} - 21 q^{26} + 7 q^{27} + 12 q^{29} + 11 q^{30} + 35 q^{31} - 60 q^{32} + 7 q^{33} + 70 q^{34} - 12 q^{36} - 6 q^{37} + 35 q^{38} + 35 q^{39} + 105 q^{40} - 42 q^{41} - 30 q^{43} + 13 q^{44} - 35 q^{45} + 69 q^{46} - 42 q^{47} - 84 q^{48} + 40 q^{50} + 53 q^{51} - 7 q^{52} - 31 q^{53} + 70 q^{54} - 7 q^{55} - 12 q^{57} - 47 q^{58} + 35 q^{59} - 91 q^{60} - 14 q^{61} - 28 q^{62} - 32 q^{64} + 35 q^{65} - 35 q^{66} + 11 q^{67} + 77 q^{68} + 70 q^{69} + 19 q^{71} - 124 q^{72} - 35 q^{73} + 13 q^{74} - 119 q^{75} + 119 q^{76} + 28 q^{78} + 15 q^{79} + 70 q^{80} - 125 q^{81} - 98 q^{82} - 26 q^{85} + 9 q^{86} + 35 q^{87} + 49 q^{88} - 14 q^{89} - 182 q^{90} - 38 q^{92} + 46 q^{93} + 14 q^{94} + 128 q^{95} + 98 q^{96} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
30.1 −1.67022 + 1.13874i −1.02667 + 0.952613i 0.762229 − 1.94213i 2.17405 + 0.670606i 0.629993 − 2.76018i 0 0.0388416 + 0.170176i −0.0776051 + 1.03557i −4.39478 + 1.35561i
30.2 −1.30646 + 0.890730i 2.35796 − 2.18787i 0.182757 − 0.465658i −0.379430 − 0.117039i −1.13178 + 4.95867i 0 −0.527696 − 2.31199i 0.549024 − 7.32622i 0.599961 − 0.185063i
30.3 0.728266 − 0.496523i 0.206906 − 0.191981i −0.446846 + 1.13855i 1.94277 + 0.599264i 0.0553598 − 0.242547i 0 0.632162 + 2.76968i −0.218237 + 2.91217i 1.71240 − 0.528205i
30.4 2.12492 − 1.44875i 0.595824 − 0.552844i 1.68574 − 4.29521i −2.35776 − 0.727274i 0.465149 − 2.03795i 0 −1.49604 − 6.55456i −0.174820 + 2.33282i −6.06370 + 1.87040i
67.1 −1.97297 + 0.608580i 0.742320 + 1.89140i 1.86976 − 1.27478i 0.224748 + 0.0338753i −2.61564 − 3.27991i 0 −0.338532 + 0.424506i −0.827200 + 0.767530i −0.464037 + 0.0699423i
67.2 −0.219307 + 0.0676471i −0.0824493 − 0.210077i −1.60896 + 1.09697i −1.92485 − 0.290124i 0.0322928 + 0.0404939i 0 0.564834 − 0.708279i 2.16182 − 2.00588i 0.441758 − 0.0665842i
67.3 1.33589 − 0.412066i 0.724654 + 1.84639i −0.0376858 + 0.0256938i 3.01090 + 0.453820i 1.72889 + 2.16796i 0 −1.78303 + 2.23585i −0.684868 + 0.635465i 4.20922 − 0.634437i
67.4 2.25736 − 0.696303i −1.02734 − 2.61763i 2.95835 − 2.01697i 1.53718 + 0.231692i −4.14175 − 5.19359i 0 2.32789 − 2.91908i −3.59740 + 3.33790i 3.63128 − 0.547328i
79.1 −2.54459 + 0.383535i −1.91497 − 1.30560i 4.41670 − 1.36237i −0.0354891 + 0.473569i 5.37356 + 2.58777i 0 −6.07919 + 2.92758i 0.866482 + 2.20776i −0.0913253 − 1.21865i
79.2 −0.0924964 + 0.0139416i −0.910789 − 0.620966i −1.90278 + 0.586931i −0.189803 + 2.53274i 0.0929020 + 0.0447392i 0 0.336373 − 0.161989i −0.652084 − 1.66148i −0.0177544 − 0.236916i
79.3 0.958697 − 0.144500i 2.41045 + 1.64342i −1.01293 + 0.312446i −0.0548015 + 0.731275i 2.54837 + 1.22723i 0 −2.67297 + 1.28723i 2.01343 + 5.13013i 0.0531314 + 0.708990i
79.4 2.40091 − 0.361879i −0.534420 − 0.364361i 3.72227 − 1.14817i 0.116705 − 1.55733i −1.41495 − 0.681404i 0 4.14619 − 1.99670i −0.943177 − 2.40318i −0.283364 − 3.78123i
116.1 −0.189695 − 2.53130i 1.83277 + 0.565334i −4.39385 + 0.662266i 1.83323 − 1.70099i 1.08336 − 4.74653i 0 1.38019 + 6.04701i 0.560718 + 0.382291i −4.65347 − 4.31779i
116.2 −0.0875534 − 1.16832i −0.821676 − 0.253454i 0.620357 − 0.0935038i −2.32784 + 2.15992i −0.224174 + 0.982171i 0 −0.684966 − 3.00103i −1.86780 − 1.27345i 2.72729 + 2.53055i
116.3 0.0157896 + 0.210698i −2.06446 − 0.636802i 1.93352 − 0.291431i 2.87713 − 2.66959i 0.101576 − 0.445032i 0 0.185965 + 0.814768i 1.37777 + 0.939349i 0.607905 + 0.564054i
116.4 0.137969 + 1.84107i 1.49877 + 0.462308i −1.39283 + 0.209936i 1.02018 − 0.946588i −0.644358 + 2.82312i 0 0.242977 + 1.06455i −0.446142 − 0.304175i 1.88349 + 1.74762i
128.1 −1.97297 − 0.608580i 0.742320 − 1.89140i 1.86976 + 1.27478i 0.224748 − 0.0338753i −2.61564 + 3.27991i 0 −0.338532 − 0.424506i −0.827200 − 0.767530i −0.464037 − 0.0699423i
128.2 −0.219307 − 0.0676471i −0.0824493 + 0.210077i −1.60896 − 1.09697i −1.92485 + 0.290124i 0.0322928 − 0.0404939i 0 0.564834 + 0.708279i 2.16182 + 2.00588i 0.441758 + 0.0665842i
128.3 1.33589 + 0.412066i 0.724654 − 1.84639i −0.0376858 − 0.0256938i 3.01090 − 0.453820i 1.72889 − 2.16796i 0 −1.78303 − 2.23585i −0.684868 − 0.635465i 4.20922 + 0.634437i
128.4 2.25736 + 0.696303i −1.02734 + 2.61763i 2.95835 + 2.01697i 1.53718 − 0.231692i −4.14175 + 5.19359i 0 2.32789 + 2.91908i −3.59740 − 3.33790i 3.63128 + 0.547328i
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.i 48
7.b odd 2 1 343.2.g.h 48
7.c even 3 1 49.2.g.a ✓ 48
7.c even 3 1 343.2.e.d 48
7.d odd 6 1 343.2.e.c 48
7.d odd 6 1 343.2.g.g 48
21.h odd 6 1 441.2.bb.d 48
28.g odd 6 1 784.2.bg.c 48
49.e even 7 1 49.2.g.a ✓ 48
49.f odd 14 1 343.2.g.g 48
49.g even 21 1 343.2.e.d 48
49.g even 21 1 inner 343.2.g.i 48
49.g even 21 1 2401.2.a.h 24
49.h odd 42 1 343.2.e.c 48
49.h odd 42 1 343.2.g.h 48
49.h odd 42 1 2401.2.a.i 24
147.l odd 14 1 441.2.bb.d 48
196.k odd 14 1 784.2.bg.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.g.a ✓ 48 7.c even 3 1
49.2.g.a ✓ 48 49.e even 7 1
343.2.e.c 48 7.d odd 6 1
343.2.e.c 48 49.h odd 42 1
343.2.e.d 48 7.c even 3 1
343.2.e.d 48 49.g even 21 1
343.2.g.g 48 7.d odd 6 1
343.2.g.g 48 49.f odd 14 1
343.2.g.h 48 7.b odd 2 1
343.2.g.h 48 49.h odd 42 1
343.2.g.i 48 1.a even 1 1 trivial
343.2.g.i 48 49.g even 21 1 inner
441.2.bb.d 48 21.h odd 6 1
441.2.bb.d 48 147.l odd 14 1
784.2.bg.c 48 28.g odd 6 1
784.2.bg.c 48 196.k odd 14 1
2401.2.a.h 24 49.g even 21 1
2401.2.a.i 24 49.h odd 42 1

Hecke kernels

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

\( T_{2}^{48} - 8 T_{2}^{47} + 22 T_{2}^{46} + 4 T_{2}^{45} - 194 T_{2}^{44} + 616 T_{2}^{43} - 810 T_{2}^{42} + \cdots + 729 \) Copy content Toggle raw display
\( T_{3}^{48} - 7 T_{3}^{47} + 26 T_{3}^{46} - 70 T_{3}^{45} + 154 T_{3}^{44} - 77 T_{3}^{43} + \cdots + 4439449 \) Copy content Toggle raw display