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Properties

Label 343.2.e.c
Level $343$
Weight $2$
Character orbit 343.e
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(50,343)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(343, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("343.50");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 343 = 7^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 343.e (of order \(7\), degree \(6\), 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: \(8\) over \(\Q(\zeta_{7})\)
Twist minimal: no (minimal twist has level 49)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 5 q^{2} - 7 q^{3} - 3 q^{4} - 7 q^{5} - 20 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 5 q^{2} - 7 q^{3} - 3 q^{4} - 7 q^{5} - 20 q^{8} + 9 q^{9} - 7 q^{10} + 6 q^{11} - 42 q^{12} + 14 q^{13} - 12 q^{15} - 15 q^{16} + 7 q^{17} - 4 q^{18} + 42 q^{19} - 14 q^{20} - 20 q^{22} + 12 q^{23} - 49 q^{24} - 13 q^{25} - 21 q^{26} - 7 q^{27} + 12 q^{29} - 22 q^{30} + 70 q^{31} + 15 q^{32} - 7 q^{33} - 70 q^{34} - 12 q^{36} - 9 q^{37} + 7 q^{38} - 28 q^{39} + 63 q^{40} + 42 q^{41} - 30 q^{43} + 37 q^{44} - 28 q^{45} + 9 q^{46} - 21 q^{47} + 84 q^{48} + 40 q^{50} - q^{51} - 77 q^{52} + 20 q^{53} - 7 q^{54} + 7 q^{55} - 12 q^{57} + 31 q^{58} + 7 q^{59} + 35 q^{60} - 7 q^{61} + 28 q^{62} - 32 q^{64} - 49 q^{65} + 119 q^{66} - 22 q^{67} + 154 q^{68} - 70 q^{69} + 19 q^{71} - 46 q^{72} - 28 q^{73} - 47 q^{74} - 7 q^{75} - 119 q^{76} + 28 q^{78} - 30 q^{79} + 140 q^{80} + 61 q^{81} - 112 q^{82} - 26 q^{85} + 24 q^{86} - 77 q^{87} + 28 q^{88} - 28 q^{89} + 182 q^{90} - 38 q^{92} + 34 q^{93} + 28 q^{94} - 67 q^{95} + 28 q^{96} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
50.1 −1.51385 − 1.89831i −0.582756 + 0.280641i −0.866793 + 3.79767i 1.40704 − 0.677593i 1.41495 + 0.681404i 0 4.14619 − 1.99670i −1.60962 + 2.01840i −3.41633 − 1.64522i
50.2 −1.12619 − 1.41219i −2.10003 + 1.01132i −0.280953 + 1.23093i −3.18734 + 1.53494i 3.79321 + 1.82671i 0 −1.20005 + 0.577915i 1.51689 − 1.90212i 5.75717 + 2.77251i
50.3 −0.604489 − 0.758006i 2.62847 − 1.26580i 0.235877 − 1.03344i −0.660703 + 0.318178i −2.54837 − 1.22723i 0 −2.67297 + 1.28723i 3.43611 − 4.30874i 0.640569 + 0.308482i
50.4 0.0583220 + 0.0731334i −0.993167 + 0.478284i 0.443095 − 1.94133i −2.28832 + 1.10200i −0.0929020 − 0.0447392i 0 0.336373 − 0.161989i −1.11284 + 1.39546i −0.214052 − 0.103082i
50.5 0.158426 + 0.198659i 1.79691 − 0.865345i 0.430675 − 1.88691i 0.419805 − 0.202167i 0.456585 + 0.219880i 0 0.900946 − 0.433873i 0.609583 − 0.764392i 0.106670 + 0.0513697i
50.6 0.778887 + 0.976693i −2.62094 + 1.26218i 0.0977773 − 0.428390i 2.24996 − 1.08352i −3.27417 − 1.57676i 0 2.74561 − 1.32222i 3.40576 − 4.27068i 2.81073 + 1.35358i
50.7 1.42193 + 1.78304i 0.534263 − 0.257288i −0.712313 + 3.12085i −3.14091 + 1.51258i 1.21844 + 0.586769i 0 −2.46797 + 1.18851i −1.65123 + 2.07058i −7.16313 − 3.44958i
50.8 1.60445 + 2.01191i −2.08817 + 1.00561i −1.02850 + 4.50616i −0.427867 + 0.206050i −5.37356 − 2.58777i 0 −6.07919 + 2.92758i 1.47874 − 1.85428i −1.10105 − 0.530235i
99.1 −0.525663 + 2.30308i 1.75326 − 2.19852i −3.22592 − 1.55352i 0.969239 − 1.21539i 4.14175 + 5.19359i 0 2.32789 − 2.91908i −1.09201 − 4.78439i 2.28964 + 2.87112i
99.2 −0.311083 + 1.36294i −1.23669 + 1.55076i 0.0410944 + 0.0197900i 1.89847 − 2.38060i −1.72889 − 2.16796i 0 −1.78303 + 2.23585i −0.207895 − 0.910846i 2.65405 + 3.32807i
99.3 −0.270726 + 1.18613i −0.324596 + 0.407031i 0.468327 + 0.225535i −1.61795 + 2.02885i −0.394914 − 0.495207i 0 −1.91142 + 2.39684i 0.607251 + 2.66054i −1.96846 − 2.46837i
99.4 −0.158588 + 0.694820i 1.62757 − 2.04091i 1.34431 + 0.647387i 1.67642 − 2.10217i 1.15995 + 1.45453i 0 −1.55172 + 1.94579i −0.848762 − 3.71867i 1.19477 + 1.49819i
99.5 0.0510692 − 0.223749i 0.140708 − 0.176442i 1.75448 + 0.844914i −1.21368 + 1.52190i −0.0322928 − 0.0404939i 0 0.564834 − 0.708279i 0.656230 + 2.87513i 0.278542 + 0.349281i
99.6 0.271146 − 1.18797i −1.14533 + 1.43620i 0.464192 + 0.223543i −0.301679 + 0.378294i 1.39561 + 1.75004i 0 1.91089 − 2.39618i −0.0833247 − 0.365069i 0.367602 + 0.460958i
99.7 0.459438 − 2.01293i −1.26684 + 1.58857i −2.03887 − 0.981868i 0.141711 − 0.177700i 2.61564 + 3.27991i 0 −0.338532 + 0.424506i −0.251100 − 1.10014i −0.292590 − 0.366897i
99.8 0.583439 − 2.55621i 0.921419 − 1.15542i −4.39188 − 2.11502i 1.28744 − 1.61440i −2.41591 − 3.02946i 0 −4.69930 + 5.89274i 0.181573 + 0.795525i −3.37561 − 4.23288i
148.1 −2.31711 + 1.11586i −0.180865 + 0.792421i 2.87689 − 3.60750i −0.549044 + 2.40552i −0.465149 − 2.03795i 0 −1.49604 + 6.55456i 2.10769 + 1.01501i −1.41203 − 6.18652i
148.2 −1.66340 + 0.801049i 0.349013 − 1.52912i 0.878225 − 1.10126i −0.309680 + 1.35680i 0.644358 + 2.82312i 0 0.242977 − 1.06455i 0.486494 + 0.234283i −0.571740 − 2.50496i
148.3 −0.794135 + 0.382435i −0.0628073 + 0.275177i −0.762586 + 0.956252i 0.452405 − 1.98212i −0.0553598 − 0.242547i 0 0.632162 − 2.76968i 2.63113 + 1.26709i 0.398761 + 1.74708i
148.4 −0.190364 + 0.0916746i −0.480744 + 2.10628i −1.21915 + 1.52876i −0.873366 + 3.82647i −0.101576 − 0.445032i 0 0.185965 − 0.814768i −1.50239 − 0.723511i −0.184532 − 0.808488i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

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

Hecke kernels

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

\( T_{2}^{48} - 5 T_{2}^{47} + 22 T_{2}^{46} - 59 T_{2}^{45} + 178 T_{2}^{44} - 437 T_{2}^{43} + 1350 T_{2}^{42} + \cdots + 729 \) Copy content Toggle raw display
\( T_{3}^{48} + 7 T_{3}^{47} + 32 T_{3}^{46} + 112 T_{3}^{45} + 376 T_{3}^{44} + 1295 T_{3}^{43} + \cdots + 4439449 \) Copy content Toggle raw display