↑

Properties

Label 342.2.x.c
Level $342$
Weight $2$
Character orbit 342.x
Analytic conductor $2.731$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [342,2,Mod(29,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([3, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.x (of order \(18\), 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: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 6 q^{3} - 30 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 6 q^{3} - 30 q^{8} + 6 q^{9} + 3 q^{13} + 3 q^{14} + 27 q^{15} - 27 q^{17} - 18 q^{18} + 3 q^{19} + 18 q^{23} + 3 q^{24} - 6 q^{27} + 3 q^{28} + 57 q^{33} + 27 q^{34} + 9 q^{38} - 30 q^{39} + 9 q^{41} - 3 q^{43} - 9 q^{44} + 27 q^{45} - 30 q^{49} - 132 q^{50} - 66 q^{51} + 6 q^{52} + 27 q^{54} + 102 q^{57} - 54 q^{59} + 6 q^{60} - 24 q^{61} - 3 q^{62} - 30 q^{64} + 36 q^{65} - 63 q^{66} + 51 q^{67} - 18 q^{68} + 3 q^{69} - 18 q^{71} - 3 q^{72} - 66 q^{73} - 6 q^{74} + 3 q^{78} - 51 q^{79} - 30 q^{81} - 36 q^{83} + 60 q^{84} - 3 q^{86} + 54 q^{87} - 27 q^{89} + 36 q^{90} - 69 q^{91} + 69 q^{93} + 18 q^{94} - 27 q^{95} + 81 q^{97} + 60 q^{98} + 51 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −0.939693 − 0.342020i −1.72877 − 0.106574i 0.766044 + 0.642788i −0.945697 + 0.166752i 1.58806 + 0.691421i 1.38681 + 2.40203i −0.500000 − 0.866025i 2.97728 + 0.368485i 0.945697 + 0.166752i
29.2 −0.939693 − 0.342020i −1.68190 − 0.413794i 0.766044 + 0.642788i 2.73856 − 0.482882i 1.43894 + 0.964081i −0.587572 − 1.01770i −0.500000 − 0.866025i 2.65755 + 1.39192i −2.73856 − 0.482882i
29.3 −0.939693 − 0.342020i −1.31833 − 1.12339i 0.766044 + 0.642788i −3.92640 + 0.692330i 0.854602 + 1.50654i −1.66637 − 2.88624i −0.500000 − 0.866025i 0.475984 + 2.96200i 3.92640 + 0.692330i
29.4 −0.939693 − 0.342020i −1.26200 + 1.18632i 0.766044 + 0.642788i −0.400356 + 0.0705935i 1.59164 − 0.683144i 0.301141 + 0.521592i −0.500000 − 0.866025i 0.185301 − 2.99427i 0.400356 + 0.0705935i
29.5 −0.939693 − 0.342020i −0.599055 + 1.62516i 0.766044 + 0.642788i 2.00011 − 0.352673i 1.11876 − 1.32226i −1.97653 − 3.42345i −0.500000 − 0.866025i −2.28227 − 1.94712i −2.00011 − 0.352673i
29.6 −0.939693 − 0.342020i 0.125873 − 1.72747i 0.766044 + 0.642788i −1.99507 + 0.351785i −0.709112 + 1.58024i 2.49195 + 4.31618i −0.500000 − 0.866025i −2.96831 − 0.434885i 1.99507 + 0.351785i
29.7 −0.939693 − 0.342020i 1.10498 + 1.33380i 0.766044 + 0.642788i −3.19630 + 0.563594i −0.582152 − 1.63129i −1.88874 − 3.27140i −0.500000 − 0.866025i −0.558051 + 2.94764i 3.19630 + 0.563594i
29.8 −0.939693 − 0.342020i 1.18444 − 1.26377i 0.766044 + 0.642788i 4.17291 − 0.735797i −1.54524 + 0.782453i −0.956453 − 1.65663i −0.500000 − 0.866025i −0.194220 − 2.99371i −4.17291 − 0.735797i
29.9 −0.939693 − 0.342020i 1.48855 + 0.885558i 0.766044 + 0.642788i 2.61254 − 0.460661i −1.09590 − 1.34127i 1.48426 + 2.57081i −0.500000 − 0.866025i 1.43157 + 2.63640i −2.61254 − 0.460661i
29.10 −0.939693 − 0.342020i 1.68621 − 0.395835i 0.766044 + 0.642788i −1.06030 + 0.186959i −1.71991 − 0.204756i 0.471814 + 0.817206i −0.500000 − 0.866025i 2.68663 − 1.33492i 1.06030 + 0.186959i
41.1 0.173648 − 0.984808i −1.71328 − 0.254334i −0.939693 − 0.342020i −0.343019 + 0.408794i −0.547977 + 1.64308i −1.00033 + 1.73262i −0.500000 + 0.866025i 2.87063 + 0.871489i 0.343019 + 0.408794i
41.2 0.173648 − 0.984808i −1.46934 + 0.917084i −0.939693 − 0.342020i −2.70096 + 3.21888i 0.648004 + 1.60627i 2.35519 − 4.07931i −0.500000 + 0.866025i 1.31791 − 2.69502i 2.70096 + 3.21888i
41.3 0.173648 − 0.984808i −1.28007 − 1.16680i −0.939693 − 0.342020i 0.837833 − 0.998490i −1.37136 + 1.05801i 1.63174 − 2.82626i −0.500000 + 0.866025i 0.277146 + 2.98717i −0.837833 − 0.998490i
41.4 0.173648 − 0.984808i −1.08055 + 1.35367i −0.939693 − 0.342020i 2.37150 − 2.82624i 1.14547 + 1.29919i 0.0861637 − 0.149240i −0.500000 + 0.866025i −0.664827 − 2.92541i −2.37150 − 2.82624i
41.5 0.173648 − 0.984808i −0.366266 + 1.69288i −0.939693 − 0.342020i −0.275077 + 0.327824i 1.60356 + 0.654667i −0.767495 + 1.32934i −0.500000 + 0.866025i −2.73170 − 1.24009i 0.275077 + 0.327824i
41.6 0.173648 − 0.984808i 0.0630297 − 1.73090i −0.939693 − 0.342020i −2.13165 + 2.54041i −1.69366 − 0.362640i −1.49646 + 2.59195i −0.500000 + 0.866025i −2.99205 − 0.218197i 2.13165 + 2.54041i
41.7 0.173648 − 0.984808i 0.619296 − 1.61755i −0.939693 − 0.342020i 1.98845 − 2.36975i −1.48544 − 0.890773i 1.01921 − 1.76532i −0.500000 + 0.866025i −2.23294 − 2.00349i −1.98845 − 2.36975i
41.8 0.173648 − 0.984808i 1.33622 − 1.10205i −0.939693 − 0.342020i −0.394525 + 0.470176i −0.853270 − 1.50729i −0.0395313 + 0.0684702i −0.500000 + 0.866025i 0.570991 − 2.94516i 0.394525 + 0.470176i
41.9 0.173648 − 0.984808i 1.43331 + 0.972427i −0.939693 − 0.342020i −1.90002 + 2.26436i 1.20655 − 1.24268i −0.700879 + 1.21396i −0.500000 + 0.866025i 1.10877 + 2.78758i 1.90002 + 2.26436i
41.10 0.173648 − 0.984808i 1.45763 + 0.935576i −0.939693 − 0.342020i 2.54747 − 3.03596i 1.17448 − 1.27303i −0.913964 + 1.58303i −0.500000 + 0.866025i 1.24940 + 2.72746i −2.54747 − 3.03596i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.x even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.x.c ✓ 60
9.d odd 6 1 342.2.bf.c yes 60
19.f odd 18 1 342.2.bf.c yes 60
171.x even 18 1 inner 342.2.x.c ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.x.c ✓ 60 1.a even 1 1 trivial
342.2.x.c ✓ 60 171.x even 18 1 inner
342.2.bf.c yes 60 9.d odd 6 1
342.2.bf.c yes 60 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{60} - 18 T_{5}^{57} + 99 T_{5}^{56} + 9 T_{5}^{55} - 6207 T_{5}^{54} - 2232 T_{5}^{53} + \cdots + 80\!\cdots\!64 \) acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display