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Properties

Label 357.2.bc.a
Level $357$
Weight $2$
Character orbit 357.bc
Analytic conductor $2.851$
Analytic rank $0$
Dimension $192$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [357,2,Mod(97,357)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(357, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 8, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("357.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 357 = 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 357.bc (of order \(16\), degree \(8\), 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: \(2.85065935216\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(24\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q - 32 q^{11} - 32 q^{14} + 64 q^{22} - 32 q^{25} + 48 q^{28} - 128 q^{30} + 64 q^{35} - 64 q^{37} + 48 q^{42} - 128 q^{44} - 32 q^{46} - 32 q^{49} - 80 q^{56} + 32 q^{58} - 176 q^{70} - 64 q^{71} + 128 q^{74} - 96 q^{77} - 128 q^{78} - 128 q^{85} + 96 q^{88} + 64 q^{91} - 320 q^{92} + 64 q^{95} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 −1.01249 − 2.44436i −0.555570 − 0.831470i −3.53554 + 3.53554i −0.364436 + 1.83214i −1.46990 + 2.19986i −2.38385 − 1.14771i 7.33311 + 3.03747i −0.382683 + 0.923880i 4.84740 − 0.964209i
97.2 −1.01249 − 2.44436i 0.555570 + 0.831470i −3.53554 + 3.53554i 0.364436 − 1.83214i 1.46990 − 2.19986i −1.76318 + 1.97261i 7.33311 + 3.03747i −0.382683 + 0.923880i −4.84740 + 0.964209i
97.3 −0.760462 − 1.83592i −0.555570 − 0.831470i −1.37808 + 1.37808i 0.0595354 − 0.299305i −1.10402 + 1.65228i −0.769735 − 2.53131i −0.0938205 − 0.0388617i −0.382683 + 0.923880i −0.594773 + 0.118308i
97.4 −0.760462 − 1.83592i 0.555570 + 0.831470i −1.37808 + 1.37808i −0.0595354 + 0.299305i 1.10402 − 1.65228i 0.257547 + 2.63319i −0.0938205 − 0.0388617i −0.382683 + 0.923880i 0.594773 − 0.118308i
97.5 −0.678477 − 1.63799i −0.555570 − 0.831470i −0.808463 + 0.808463i −0.327699 + 1.64746i −0.984996 + 1.47415i 0.690278 + 2.55412i −1.40320 − 0.581225i −0.382683 + 0.923880i 2.92085 − 0.580993i
97.6 −0.678477 − 1.63799i 0.555570 + 0.831470i −0.808463 + 0.808463i 0.327699 − 1.64746i 0.984996 − 1.47415i −0.339685 − 2.62385i −1.40320 − 0.581225i −0.382683 + 0.923880i −2.92085 + 0.580993i
97.7 −0.622526 − 1.50291i −0.555570 − 0.831470i −0.456986 + 0.456986i 0.801357 − 4.02869i −0.903767 + 1.35258i −1.97462 + 1.76094i −2.03453 − 0.842728i −0.382683 + 0.923880i −6.55362 + 1.30360i
97.8 −0.622526 − 1.50291i 0.555570 + 0.831470i −0.456986 + 0.456986i −0.801357 + 4.02869i 0.903767 − 1.35258i −2.49819 − 0.871240i −2.03453 − 0.842728i −0.382683 + 0.923880i 6.55362 − 1.30360i
97.9 −0.430152 − 1.03848i −0.555570 − 0.831470i 0.520805 − 0.520805i −0.749020 + 3.76558i −0.624484 + 0.934607i 2.47667 − 0.930639i −2.84183 − 1.17712i −0.382683 + 0.923880i 4.23267 − 0.841930i
97.10 −0.430152 − 1.03848i 0.555570 + 0.831470i 0.520805 − 0.520805i 0.749020 − 3.76558i 0.624484 − 0.934607i 2.64429 − 0.0879835i −2.84183 − 1.17712i −0.382683 + 0.923880i −4.23267 + 0.841930i
97.11 −0.0508407 − 0.122740i −0.555570 − 0.831470i 1.40173 − 1.40173i 0.498382 − 2.50553i −0.0738092 + 0.110463i −0.385474 − 2.61752i −0.488795 − 0.202465i −0.382683 + 0.923880i −0.332868 + 0.0662115i
97.12 −0.0508407 − 0.122740i 0.555570 + 0.831470i 1.40173 − 1.40173i −0.498382 + 2.50553i 0.0738092 − 0.110463i 0.645550 + 2.56579i −0.488795 − 0.202465i −0.382683 + 0.923880i 0.332868 − 0.0662115i
97.13 0.0929101 + 0.224305i −0.555570 − 0.831470i 1.37253 − 1.37253i 0.254972 − 1.28183i 0.134885 − 0.201869i 2.38924 + 1.13645i 0.883998 + 0.366164i −0.382683 + 0.923880i 0.311210 − 0.0619035i
97.14 0.0929101 + 0.224305i 0.555570 + 0.831470i 1.37253 − 1.37253i −0.254972 + 1.28183i −0.134885 + 0.201869i 1.77247 − 1.96427i 0.883998 + 0.366164i −0.382683 + 0.923880i −0.311210 + 0.0619035i
97.15 0.347777 + 0.839608i −0.555570 − 0.831470i 0.830220 − 0.830220i −0.621082 + 3.12239i 0.504894 − 0.755628i −1.35265 + 2.27384i 2.66501 + 1.10388i −0.382683 + 0.923880i −2.83758 + 0.564430i
97.16 0.347777 + 0.839608i 0.555570 + 0.831470i 0.830220 − 0.830220i 0.621082 − 3.12239i −0.504894 + 0.755628i −2.11985 − 1.58311i 2.66501 + 1.10388i −0.382683 + 0.923880i 2.83758 − 0.564430i
97.17 0.572289 + 1.38163i −0.555570 − 0.831470i −0.167165 + 0.167165i 0.137019 − 0.688842i 0.830834 − 1.24343i −1.72895 − 2.00268i 2.43663 + 1.00928i −0.382683 + 0.923880i 1.03014 − 0.204907i
97.18 0.572289 + 1.38163i 0.555570 + 0.831470i −0.167165 + 0.167165i −0.137019 + 0.688842i −0.830834 + 1.24343i −0.830945 + 2.51188i 2.43663 + 1.00928i −0.382683 + 0.923880i −1.03014 + 0.204907i
97.19 0.732224 + 1.76775i −0.555570 − 0.831470i −1.17456 + 1.17456i −0.479616 + 2.41119i 1.06302 − 1.59093i 2.43677 − 1.03060i 0.599131 + 0.248168i −0.382683 + 0.923880i −4.61357 + 0.917695i
97.20 0.732224 + 1.76775i 0.555570 + 0.831470i −1.17456 + 1.17456i 0.479616 − 2.41119i −1.06302 + 1.59093i 2.64568 + 0.0196375i 0.599131 + 0.248168i −0.382683 + 0.923880i 4.61357 − 0.917695i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
17.e odd 16 1 inner
119.p even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 357.2.bc.a ✓ 192
7.b odd 2 1 inner 357.2.bc.a ✓ 192
17.e odd 16 1 inner 357.2.bc.a ✓ 192
119.p even 16 1 inner 357.2.bc.a ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
357.2.bc.a ✓ 192 1.a even 1 1 trivial
357.2.bc.a ✓ 192 7.b odd 2 1 inner
357.2.bc.a ✓ 192 17.e odd 16 1 inner
357.2.bc.a ✓ 192 119.p even 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(357, [\chi])\).