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Properties

Label 49.4.e.a
Level $49$
Weight $4$
Character orbit 49.e
Analytic conductor $2.891$
Analytic rank $0$
Dimension $78$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,4,Mod(8,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([12]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.8");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 49.e (of order \(7\), degree \(6\), 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: \(2.89109359028\)
Analytic rank: \(0\)
Dimension: \(78\)
Relative dimension: \(13\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9} + 9 q^{10} - 103 q^{11} + 364 q^{12} - 35 q^{13} + 161 q^{14} - 245 q^{15} - 205 q^{16} - 285 q^{17} + 16 q^{18} + 628 q^{19}+ \cdots + 10768 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −3.44244 − 4.31668i 3.65176 − 1.75859i −5.00318 + 21.9203i −12.6967 + 6.11441i −20.1622 − 9.70961i −15.1671 + 10.6282i 72.0505 − 34.6977i −6.59154 + 8.26553i 70.1015 + 33.7591i
8.2 −2.77071 − 3.47436i −6.15987 + 2.96643i −2.61419 + 11.4535i 1.98451 − 0.955692i 27.3737 + 13.1825i 18.5202 − 0.0393741i 15.0065 − 7.22673i 12.3100 − 15.4362i −8.81894 − 4.24698i
8.3 −2.45388 − 3.07706i 2.79026 − 1.34372i −1.66664 + 7.30203i 16.4856 − 7.93907i −10.9817 − 5.28849i −6.70370 − 17.2644i −1.80909 + 0.871210i −10.8543 + 13.6108i −64.8827 − 31.2459i
8.4 −1.61233 − 2.02180i −3.25484 + 1.56745i 0.292112 − 1.27982i −3.53374 + 1.70176i 8.41693 + 4.05338i −10.4442 + 15.2944i −21.6976 + 10.4490i −8.69714 + 10.9059i 9.13817 + 4.40071i
8.5 −1.46127 − 1.83238i 7.38411 − 3.55600i 0.557879 − 2.44423i −0.458224 + 0.220669i −17.3061 − 8.33419i 12.0280 + 14.0828i −22.1867 + 10.6846i 25.0457 − 31.4064i 1.07394 + 0.517181i
8.6 −0.802750 − 1.00662i −1.70862 + 0.822828i 1.41130 − 6.18330i −15.9369 + 7.67483i 2.19987 + 1.05940i 0.116992 − 18.5199i −16.6372 + 8.01205i −14.5919 + 18.2976i 20.5190 + 9.88143i
8.7 0.153998 + 0.193108i −8.98937 + 4.32905i 1.76659 − 7.73995i 14.2902 − 6.88179i −2.22032 − 1.06925i −13.6848 − 12.4791i 3.54697 − 1.70813i 45.2339 − 56.7215i 3.52959 + 1.69976i
8.8 0.521845 + 0.654373i 5.78023 − 2.78361i 1.62429 − 7.11646i −1.28950 + 0.620991i 4.83790 + 2.32981i −17.3557 − 6.46381i 11.5371 − 5.55599i 8.82835 − 11.0704i −1.07928 − 0.519753i
8.9 0.777062 + 0.974405i −0.189548 + 0.0912813i 1.43453 − 6.28508i 7.66912 − 3.69325i −0.236235 − 0.113765i 18.3768 + 2.30053i 16.2220 − 7.81211i −16.8066 + 21.0748i 9.55811 + 4.60294i
8.10 1.88794 + 2.36740i −5.83533 + 2.81015i −0.260101 + 1.13958i −12.6471 + 6.09051i −17.6695 − 8.50917i −3.94107 + 18.0961i 18.6363 − 8.97477i 9.31994 − 11.6868i −38.2955 − 18.4422i
8.11 2.59695 + 3.25648i 2.96773 − 1.42918i −2.08030 + 9.11440i 9.52410 − 4.58657i 12.3612 + 5.95282i −16.3066 + 8.78031i −5.06162 + 2.43755i −10.0694 + 12.6266i 39.6697 + 19.1039i
8.12 2.64790 + 3.32037i 6.62695 − 3.19137i −2.23327 + 9.78459i −17.1881 + 8.27737i 28.1441 + 13.5535i 16.3388 − 8.72030i −7.79127 + 3.75208i 16.8974 − 21.1887i −72.9964 − 35.1532i
8.13 3.18020 + 3.98784i −4.68696 + 2.25712i −4.00905 + 17.5648i 4.76744 − 2.29588i −23.9065 − 11.5128i 7.38357 − 16.9848i −46.0311 + 22.1674i 0.0387732 − 0.0486201i 24.3170 + 11.7104i
15.1 −1.16701 + 5.11301i −1.24065 + 1.55573i −17.5732 − 8.46282i 0.849213 − 1.06488i −6.50660 − 8.15902i −18.2375 + 3.22369i 37.6195 − 47.1733i 5.12699 + 22.4628i 4.45370 + 5.58476i
15.2 −0.892344 + 3.90961i 5.02886 − 6.30599i −7.28105 − 3.50637i 8.76329 − 10.9888i 20.1665 + 25.2880i 3.97541 − 18.0886i 0.203423 − 0.255085i −8.46803 − 37.1008i 35.1422 + 44.0669i
15.3 −0.846844 + 3.71026i 3.04811 − 3.82222i −5.84116 − 2.81296i −12.1071 + 15.1818i 11.6002 + 14.5461i 13.1709 + 13.0203i −3.59906 + 4.51308i 0.689741 + 3.02195i −46.0758 − 57.7772i
15.4 −0.764068 + 3.34760i −5.99643 + 7.51929i −3.41489 − 1.64452i 0.968529 − 1.21450i −20.5899 − 25.8189i 18.5012 + 0.839908i −9.01256 + 11.3014i −14.5744 − 63.8547i 3.32563 + 4.17021i
15.5 −0.424669 + 1.86059i 0.113640 − 0.142500i 3.92628 + 1.89080i 11.1399 − 13.9690i 0.216876 + 0.271954i 1.18252 + 18.4825i −14.7045 + 18.4389i 6.00067 + 26.2907i 21.2598 + 26.6590i
15.6 −0.379589 + 1.66309i −2.24247 + 2.81197i 4.58597 + 2.20849i −4.66393 + 5.84838i −3.82533 − 4.79682i −14.0409 − 12.0769i −13.9224 + 17.4581i 3.12958 + 13.7116i −7.95600 − 9.97651i
15.7 0.0613960 − 0.268993i 2.25069 − 2.82228i 7.13916 + 3.43804i −2.36861 + 2.97014i −0.620990 − 0.778697i 9.47466 − 15.9132i 2.73935 − 3.43503i 3.10843 + 13.6189i 0.653525 + 0.819495i
See all 78 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.13
Significant digits:
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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 49.4.e.a ✓ 78
49.e even 7 1 inner 49.4.e.a ✓ 78
49.e even 7 1 2401.4.a.d 39
49.f odd 14 1 2401.4.a.c 39
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.e.a ✓ 78 1.a even 1 1 trivial
49.4.e.a ✓ 78 49.e even 7 1 inner
2401.4.a.c 39 49.f odd 14 1
2401.4.a.d 39 49.e even 7 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(49, [\chi])\).