↑

Properties

Label 345.2.n.b
Level $345$
Weight $2$
Character orbit 345.n
Analytic conductor $2.755$
Analytic rank $0$
Dimension $400$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [345,2,Mod(14,345)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(345, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 11, 21]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("345.14");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 345.n (of order \(22\), degree \(10\), 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: \(2.75483886973\)
Analytic rank: \(0\)
Dimension: \(400\)
Relative dimension: \(40\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 400 q - 76 q^{4} - 2 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 400 q - 76 q^{4} - 2 q^{6} + 2 q^{9} - 22 q^{10} - 22 q^{15} - 76 q^{16} - 44 q^{19} - 22 q^{21} - 28 q^{24} - 22 q^{25} - 55 q^{30} - 108 q^{31} + 176 q^{34} + 46 q^{36} - 22 q^{39} + 88 q^{40} + 20 q^{46} - 232 q^{49} - 22 q^{51} - 206 q^{54} + 40 q^{55} - 231 q^{60} - 88 q^{61} + 140 q^{64} - 176 q^{66} + 72 q^{69} + 92 q^{70} + 114 q^{75} - 308 q^{76} - 218 q^{81} + 44 q^{84} - 8 q^{85} + 220 q^{90} - 108 q^{94} - 24 q^{96} - 22 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −2.26077 + 0.663823i −1.63502 − 0.571585i 2.98793 − 1.92023i 1.04208 − 1.97840i 4.07584 + 0.206861i 0.859641 − 1.88235i −2.39435 + 2.76323i 2.34658 + 1.86911i −1.04260 + 5.16447i
14.2 −2.26077 + 0.663823i −0.333080 − 1.69972i 2.98793 − 1.92023i 1.55725 + 1.60467i 1.88133 + 3.62158i −0.859641 + 1.88235i −2.39435 + 2.76323i −2.77812 + 1.13229i −4.58581 − 2.59406i
14.3 −2.24678 + 0.659714i −1.19868 + 1.25026i 2.93029 − 1.88318i 0.369841 + 2.20527i 1.86836 − 3.59985i 0.848919 − 1.85887i −2.27447 + 2.62487i −0.126312 − 2.99734i −2.28580 − 4.71077i
14.4 −2.24678 + 0.659714i 1.40813 − 1.00855i 2.93029 − 1.88318i −0.266437 − 2.22014i −2.49839 + 3.19496i −0.848919 + 1.85887i −2.27447 + 2.62487i 0.965643 − 2.84034i 2.06328 + 4.81239i
14.5 −2.14604 + 0.630135i 0.100087 + 1.72916i 2.52592 − 1.62331i −2.15544 − 0.595052i −1.30439 − 3.64777i 0.865298 − 1.89474i −1.46844 + 1.69467i −2.97997 + 0.346134i 5.00062 − 0.0812096i
14.6 −2.14604 + 0.630135i 1.69731 + 0.345153i 2.52592 − 1.62331i −1.90048 + 1.17821i −3.86000 + 0.328822i −0.865298 + 1.89474i −1.46844 + 1.69467i 2.76174 + 1.17167i 3.33609 − 3.72604i
14.7 −1.77868 + 0.522268i −1.33405 + 1.10468i 1.20843 − 0.776613i 2.18584 − 0.471277i 1.79591 − 2.66160i −1.72036 + 3.76707i 0.684108 − 0.789502i 0.559374 − 2.94739i −3.64178 + 1.97984i
14.8 −1.77868 + 0.522268i 1.28329 − 1.16326i 1.20843 − 0.776613i 2.23007 − 0.163636i −1.67503 + 2.73928i 1.72036 − 3.76707i 0.684108 − 0.789502i 0.293660 − 2.98559i −3.88112 + 1.45575i
14.9 −1.43891 + 0.422502i 0.645956 + 1.60709i 0.209449 − 0.134605i 0.00662602 − 2.23606i −1.60847 − 2.03954i −0.939998 + 2.05831i 1.71963 − 1.98455i −2.16548 + 2.07622i 0.935206 + 3.22029i
14.10 −1.43891 + 0.422502i 1.49880 + 0.868094i 0.209449 − 0.134605i 0.636328 + 2.14362i −2.52342 − 0.615862i 0.939998 − 2.05831i 1.71963 − 1.98455i 1.49283 + 2.60221i −1.82130 − 2.81562i
14.11 −1.31019 + 0.384706i −1.59541 + 0.674298i −0.113912 + 0.0732066i −1.81506 − 1.30597i 1.83088 − 1.49722i −0.0260367 + 0.0570124i 1.90951 − 2.20369i 2.09064 − 2.15156i 2.88049 + 1.01280i
14.12 −1.31019 + 0.384706i 0.894485 − 1.48320i −0.113912 + 0.0732066i −1.37361 + 1.76443i −0.601346 + 2.28739i 0.0260367 − 0.0570124i 1.90951 − 2.20369i −1.39979 − 2.65341i 1.12090 − 2.84017i
14.13 −1.28956 + 0.378649i −1.50146 − 0.863495i −0.162917 + 0.104700i −1.54156 + 1.61975i 2.26318 + 0.545002i −1.67472 + 3.66712i 1.93071 − 2.22816i 1.50875 + 2.59300i 1.37462 − 2.67247i
14.14 −1.28956 + 0.378649i −0.641026 − 1.60906i −0.162917 + 0.104700i −1.93545 − 1.11983i 1.43591 + 1.83226i 1.67472 − 3.66712i 1.93071 − 2.22816i −2.17817 + 2.06290i 2.91991 + 0.711227i
14.15 −0.502689 + 0.147603i −0.554969 + 1.64073i −1.45160 + 0.932885i 1.91371 − 1.15659i 0.0368000 − 0.906695i 2.02283 − 4.42938i 1.27818 − 1.47510i −2.38402 − 1.82111i −0.791285 + 0.863877i
14.16 −0.502689 + 0.147603i 1.70301 − 0.315820i −1.45160 + 0.932885i 2.16204 + 0.570590i −0.809471 + 0.410129i −2.02283 + 4.42938i 1.27818 − 1.47510i 2.80052 − 1.07569i −1.17106 + 0.0322945i
14.17 −0.491934 + 0.144445i −0.748262 + 1.56208i −1.46137 + 0.939167i −1.43003 + 1.71902i 0.142461 − 0.876524i −0.613357 + 1.34306i 1.25474 − 1.44804i −1.88021 − 2.33770i 0.455177 − 1.05220i
14.18 −0.491934 + 0.144445i 1.65267 − 0.518338i −1.46137 + 0.939167i −1.85641 − 1.24650i −0.738134 + 0.493708i 0.613357 − 1.34306i 1.25474 − 1.44804i 2.46265 − 1.71329i 1.09328 + 0.345046i
14.19 −0.305806 + 0.0897928i 0.443370 + 1.67434i −1.59705 + 1.02636i 0.888529 + 2.05195i −0.285929 − 0.472213i −0.740592 + 1.62167i 0.813658 − 0.939012i −2.60685 + 1.48471i −0.455968 − 0.547717i
14.20 −0.305806 + 0.0897928i 1.59420 + 0.677141i −1.59705 + 1.02636i 0.274435 − 2.21916i −0.548319 − 0.0639260i 0.740592 − 1.62167i 0.813658 − 0.939012i 2.08296 + 2.15900i 0.115341 + 0.703276i
See next 80 embeddings (of 400 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
23.d odd 22 1 inner
69.g even 22 1 inner
115.i odd 22 1 inner
345.n even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 345.2.n.b ✓ 400
3.b odd 2 1 inner 345.2.n.b ✓ 400
5.b even 2 1 inner 345.2.n.b ✓ 400
15.d odd 2 1 inner 345.2.n.b ✓ 400
23.d odd 22 1 inner 345.2.n.b ✓ 400
69.g even 22 1 inner 345.2.n.b ✓ 400
115.i odd 22 1 inner 345.2.n.b ✓ 400
345.n even 22 1 inner 345.2.n.b ✓ 400
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.n.b ✓ 400 1.a even 1 1 trivial
345.2.n.b ✓ 400 3.b odd 2 1 inner
345.2.n.b ✓ 400 5.b even 2 1 inner
345.2.n.b ✓ 400 15.d odd 2 1 inner
345.2.n.b ✓ 400 23.d odd 22 1 inner
345.2.n.b ✓ 400 69.g even 22 1 inner
345.2.n.b ✓ 400 115.i odd 22 1 inner
345.2.n.b ✓ 400 345.n even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{200} + 39 T_{2}^{198} + 834 T_{2}^{196} + 12792 T_{2}^{194} + 157651 T_{2}^{192} + 1661220 T_{2}^{190} + \cdots + 279841 \) acting on \(S_{2}^{\mathrm{new}}(345, [\chi])\). Copy content Toggle raw display