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Properties

Label 354.2.g.b
Level $354$
Weight $2$
Character orbit 354.g
Analytic conductor $2.827$
Analytic rank $0$
Dimension $280$
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] = [354,2,Mod(11,354)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(354, base_ring=CyclotomicField(58))
 
chi = DirichletCharacter(H, H._module([29, 25]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("354.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 354.g (of order \(58\), degree \(28\), 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: \(2.82670423155\)
Analytic rank: \(0\)
Dimension: \(280\)
Relative dimension: \(10\) over \(\Q(\zeta_{58})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{58}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 280 q + 10 q^{2} + q^{3} - 10 q^{4} - q^{6} + 2 q^{7} + 10 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 280 q + 10 q^{2} + q^{3} - 10 q^{4} - q^{6} + 2 q^{7} + 10 q^{8} - 3 q^{9} - 4 q^{11} + q^{12} + 27 q^{14} + 7 q^{15} - 10 q^{16} + 3 q^{18} + 6 q^{19} + 3 q^{21} + 4 q^{22} + 8 q^{23} - q^{24} - 4 q^{25} + 97 q^{27} + 2 q^{28} - 7 q^{30} + 10 q^{32} - 3 q^{36} - 6 q^{38} + 4 q^{39} - 3 q^{42} - 4 q^{44} - 18 q^{45} - 8 q^{46} - 232 q^{47} + q^{48} - 8 q^{49} + 4 q^{50} - 118 q^{51} + 58 q^{53} - 126 q^{54} - 2 q^{56} + 32 q^{57} - 20 q^{59} - 80 q^{60} - 329 q^{63} - 10 q^{64} - 16 q^{65} - 87 q^{66} - 130 q^{69} + 3 q^{72} - 25 q^{75} + 6 q^{76} - 106 q^{77} - 4 q^{78} - 6 q^{79} - 7 q^{81} + 104 q^{83} + 3 q^{84} + 4 q^{85} + 15 q^{87} + 4 q^{88} + 16 q^{89} - 11 q^{90} + 8 q^{92} - 122 q^{93} - q^{96} + 8 q^{98} - 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −0.976621 + 0.214970i −1.72285 + 0.178276i 0.907575 − 0.419889i 0.135298 + 0.0375654i 1.64425 − 0.544470i −1.37447 + 1.30197i −0.796093 + 0.605174i 2.93644 − 0.614285i −0.140211 − 0.00760200i
11.2 −0.976621 + 0.214970i −1.07546 − 1.35772i 0.907575 − 0.419889i −1.11366 − 0.309205i 1.34218 + 1.09478i −1.48196 + 1.40379i −0.796093 + 0.605174i −0.686781 + 2.92033i 1.15409 + 0.0625729i
11.3 −0.976621 + 0.214970i −1.05934 + 1.37033i 0.907575 − 0.419889i −3.87317 − 1.07538i 0.739995 − 1.56602i −0.391370 + 0.370725i −0.796093 + 0.605174i −0.755592 − 2.90329i 4.01379 + 0.217621i
11.4 −0.976621 + 0.214970i −0.840743 − 1.51432i 0.907575 − 0.419889i 4.02621 + 1.11787i 1.14662 + 1.29818i 1.69268 − 1.60339i −0.796093 + 0.605174i −1.58630 + 2.54630i −4.17238 − 0.226220i
11.5 −0.976621 + 0.214970i −0.836909 + 1.51644i 0.907575 − 0.419889i 0.897171 + 0.249098i 0.491353 − 1.66089i 3.72971 − 3.53297i −0.796093 + 0.605174i −1.59917 − 2.53824i −0.929745 − 0.0504093i
11.6 −0.976621 + 0.214970i −0.233636 + 1.71622i 0.907575 − 0.419889i 2.85285 + 0.792090i −0.140763 − 1.72632i −3.45977 + 3.27726i −0.796093 + 0.605174i −2.89083 − 0.801943i −2.95643 − 0.160293i
11.7 −0.976621 + 0.214970i 0.745416 − 1.56344i 0.907575 − 0.419889i −1.05087 − 0.291772i −0.391895 + 1.68713i 1.62439 − 1.53870i −0.796093 + 0.605174i −1.88871 − 2.33083i 1.08902 + 0.0590451i
11.8 −0.976621 + 0.214970i 0.947783 + 1.44973i 0.907575 − 0.419889i −1.88995 − 0.524743i −1.23727 − 1.21209i 0.743378 − 0.704165i −0.796093 + 0.605174i −1.20342 + 2.74805i 1.95857 + 0.106191i
11.9 −0.976621 + 0.214970i 1.46208 + 0.928615i 0.907575 − 0.419889i 2.23908 + 0.621678i −1.62752 − 0.592600i 1.22747 − 1.16272i −0.796093 + 0.605174i 1.27535 + 2.71542i −2.32038 − 0.125807i
11.10 −0.976621 + 0.214970i 1.73081 + 0.0656435i 0.907575 − 0.419889i −3.54823 − 0.985161i −1.70445 + 0.307963i −3.12490 + 2.96006i −0.796093 + 0.605174i 2.99138 + 0.227232i 3.67705 + 0.199364i
23.1 0.725995 − 0.687699i −1.58067 − 0.708165i 0.0541389 − 0.998533i −1.48491 − 0.243438i −1.63456 + 0.572898i −2.01420 + 3.79918i −0.647386 − 0.762162i 1.99700 + 2.23874i −1.24545 + 0.844434i
23.2 0.725995 − 0.687699i −1.55051 + 0.771956i 0.0541389 − 0.998533i −0.271078 − 0.0444409i −0.594791 + 1.62672i −0.468623 + 0.883917i −0.647386 − 0.762162i 1.80817 − 2.39385i −0.227363 + 0.154156i
23.3 0.725995 − 0.687699i −1.30996 − 1.13314i 0.0541389 − 0.998533i 1.84131 + 0.301868i −1.73028 + 0.0782045i 1.58049 − 2.98111i −0.647386 − 0.762162i 0.431990 + 2.96873i 1.54438 − 1.04712i
23.4 0.725995 − 0.687699i −0.244608 + 1.71469i 0.0541389 − 0.998533i −3.62342 − 0.594030i 1.00161 + 1.41308i −0.837325 + 1.57936i −0.647386 − 0.762162i −2.88033 − 0.838855i −3.03910 + 2.06056i
23.5 0.725995 − 0.687699i 0.0950413 + 1.72944i 0.0541389 − 0.998533i 1.86219 + 0.305290i 1.25834 + 1.19021i 1.84960 − 3.48871i −0.647386 − 0.762162i −2.98193 + 0.328737i 1.56189 − 1.05899i
23.6 0.725995 − 0.687699i 0.399507 − 1.68535i 0.0541389 − 0.998533i −2.51062 − 0.411595i −0.868972 − 1.49829i 0.276889 − 0.522268i −0.647386 − 0.762162i −2.68079 − 1.34661i −2.10575 + 1.42773i
23.7 0.725995 − 0.687699i 0.667477 − 1.59827i 0.0541389 − 0.998533i 3.03432 + 0.497451i −0.614546 − 1.61936i −0.0547647 + 0.103297i −0.647386 − 0.762162i −2.10895 − 2.13362i 2.54500 − 1.72555i
23.8 0.725995 − 0.687699i 1.08874 + 1.34709i 0.0541389 − 0.998533i 1.70784 + 0.279986i 1.71681 + 0.229256i −2.22532 + 4.19739i −0.647386 − 0.762162i −0.629299 + 2.93325i 1.43243 − 0.971213i
23.9 0.725995 − 0.687699i 1.69343 + 0.363726i 0.0541389 − 0.998533i −3.46404 − 0.567901i 1.47956 − 0.900507i 1.50148 − 2.83210i −0.647386 − 0.762162i 2.73541 + 1.23189i −2.90542 + 1.96993i
23.10 0.725995 − 0.687699i 1.73205 + 0.00243249i 0.0541389 − 0.998533i 1.16466 + 0.190936i 1.25913 − 1.18936i 0.141142 − 0.266222i −0.647386 − 0.762162i 2.99999 + 0.00842639i 0.976845 − 0.662318i
See next 80 embeddings (of 280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
177.f even 58 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 354.2.g.b yes 280
3.b odd 2 1 354.2.g.a ✓ 280
59.d odd 58 1 354.2.g.a ✓ 280
177.f even 58 1 inner 354.2.g.b yes 280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
354.2.g.a ✓ 280 3.b odd 2 1
354.2.g.a ✓ 280 59.d odd 58 1
354.2.g.b yes 280 1.a even 1 1 trivial
354.2.g.b yes 280 177.f even 58 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{280} - 23 T_{5}^{278} + 145 T_{5}^{277} + 542 T_{5}^{276} - 2581 T_{5}^{275} + \cdots + 17\!\cdots\!96 \) acting on \(S_{2}^{\mathrm{new}}(354, [\chi])\). Copy content Toggle raw display