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Properties

Label 245.4.l.d
Level $245$
Weight $4$
Character orbit 245.l
Analytic conductor $14.455$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,4,Mod(68,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 10]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.68");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.l (of order \(12\), degree \(4\), not 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: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(36\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 32 q^{11} + 1152 q^{16} + 64 q^{18} + 1152 q^{22} - 768 q^{23} - 288 q^{25} + 2512 q^{30} - 1840 q^{32} - 9280 q^{36} - 864 q^{37} + 1216 q^{43} + 3552 q^{46} - 4960 q^{50} - 1056 q^{51} + 1384 q^{53} + 15744 q^{57} - 5296 q^{58} - 9104 q^{60} - 736 q^{65} - 1856 q^{67} - 13632 q^{71} + 8528 q^{72} - 10816 q^{78} + 12616 q^{81} + 11200 q^{85} - 8672 q^{86} + 10080 q^{88} + 21248 q^{92} - 416 q^{93} - 5888 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1 −1.42006 − 5.29972i −5.60149 − 1.50091i −19.1423 + 11.0518i −8.43267 + 7.34098i 31.8177i 0 54.7173 + 54.7173i 5.74123 + 3.31470i 50.8800 + 34.2662i
68.2 −1.42006 − 5.29972i 5.60149 + 1.50091i −19.1423 + 11.0518i 8.43267 − 7.34098i − 31.8177i 0 54.7173 + 54.7173i 5.74123 + 3.31470i −50.8800 − 34.2662i
68.3 −1.25845 − 4.69658i −2.23302 − 0.598336i −13.5460 + 7.82078i 3.35008 − 10.6666i 11.2405i 0 26.2728 + 26.2728i −18.7543 − 10.8278i −54.3126 − 2.31057i
68.4 −1.25845 − 4.69658i 2.23302 + 0.598336i −13.5460 + 7.82078i −3.35008 + 10.6666i − 11.2405i 0 26.2728 + 26.2728i −18.7543 − 10.8278i 54.3126 + 2.31057i
68.5 −1.12600 − 4.20228i −9.85536 − 2.64074i −9.46305 + 5.46350i −7.22392 − 8.53317i 44.3884i 0 9.00424 + 9.00424i 66.7720 + 38.5508i −27.7246 + 39.9652i
68.6 −1.12600 − 4.20228i 9.85536 + 2.64074i −9.46305 + 5.46350i 7.22392 + 8.53317i − 44.3884i 0 9.00424 + 9.00424i 66.7720 + 38.5508i 27.7246 − 39.9652i
68.7 −1.02487 − 3.82488i −5.39699 − 1.44612i −6.65111 + 3.84002i 7.44338 − 8.34242i 22.1249i 0 −0.895902 − 0.895902i 3.65358 + 2.10939i −39.5373 − 19.9201i
68.8 −1.02487 − 3.82488i 5.39699 + 1.44612i −6.65111 + 3.84002i −7.44338 + 8.34242i − 22.1249i 0 −0.895902 − 0.895902i 3.65358 + 2.10939i 39.5373 + 19.9201i
68.9 −0.830128 − 3.09808i −1.23485 − 0.330876i −1.98077 + 1.14360i 3.90566 + 10.4760i 4.10032i 0 −12.9563 − 12.9563i −21.9673 − 12.6828i 29.2132 − 20.7964i
68.10 −0.830128 − 3.09808i 1.23485 + 0.330876i −1.98077 + 1.14360i −3.90566 − 10.4760i − 4.10032i 0 −12.9563 − 12.9563i −21.9673 − 12.6828i −29.2132 + 20.7964i
68.11 −0.460426 − 1.71833i −8.48452 − 2.27342i 4.18752 − 2.41767i 11.1530 + 0.781406i 15.6260i 0 −16.1457 − 16.1457i 43.4359 + 25.0777i −3.79242 − 19.5244i
68.12 −0.460426 − 1.71833i 8.48452 + 2.27342i 4.18752 − 2.41767i −11.1530 − 0.781406i − 15.6260i 0 −16.1457 − 16.1457i 43.4359 + 25.0777i 3.79242 + 19.5244i
68.13 −0.429990 − 1.60475i −2.04496 − 0.547944i 4.53789 − 2.61995i 10.0001 + 4.99985i 3.51724i 0 −15.5536 − 15.5536i −19.5011 − 11.2590i 3.72355 − 18.1975i
68.14 −0.429990 − 1.60475i 2.04496 + 0.547944i 4.53789 − 2.61995i −10.0001 − 4.99985i − 3.51724i 0 −15.5536 − 15.5536i −19.5011 − 11.2590i −3.72355 + 18.1975i
68.15 −0.275639 − 1.02870i −7.28620 − 1.95233i 5.94596 − 3.43290i −10.0893 − 4.81725i 8.03343i 0 −11.1948 − 11.1948i 25.8944 + 14.9501i −2.17449 + 11.7067i
68.16 −0.275639 − 1.02870i 7.28620 + 1.95233i 5.94596 − 3.43290i 10.0893 + 4.81725i − 8.03343i 0 −11.1948 − 11.1948i 25.8944 + 14.9501i 2.17449 − 11.7067i
68.17 −0.200282 − 0.747462i −0.595328 − 0.159518i 6.40962 − 3.70059i 3.31282 − 10.6783i 0.476933i 0 −8.42723 − 8.42723i −23.0537 − 13.3101i −8.64509 − 0.337542i
68.18 −0.200282 − 0.747462i 0.595328 + 0.159518i 6.40962 − 3.70059i −3.31282 + 10.6783i − 0.476933i 0 −8.42723 − 8.42723i −23.0537 − 13.3101i 8.64509 + 0.337542i
68.19 −0.150840 − 0.562941i −9.17966 − 2.45968i 6.63405 − 3.83017i −5.84314 + 9.53193i 5.53862i 0 −6.45365 − 6.45365i 54.8334 + 31.6581i 6.24729 + 1.85155i
68.20 −0.150840 − 0.562941i 9.17966 + 2.45968i 6.63405 − 3.83017i 5.84314 − 9.53193i − 5.53862i 0 −6.45365 − 6.45365i 54.8334 + 31.6581i −6.24729 − 1.85155i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
35.f even 4 1 inner
35.k even 12 1 inner
35.l odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.4.l.d 144
5.c odd 4 1 inner 245.4.l.d 144
7.b odd 2 1 inner 245.4.l.d 144
7.c even 3 1 245.4.f.b ✓ 72
7.c even 3 1 inner 245.4.l.d 144
7.d odd 6 1 245.4.f.b ✓ 72
7.d odd 6 1 inner 245.4.l.d 144
35.f even 4 1 inner 245.4.l.d 144
35.k even 12 1 245.4.f.b ✓ 72
35.k even 12 1 inner 245.4.l.d 144
35.l odd 12 1 245.4.f.b ✓ 72
35.l odd 12 1 inner 245.4.l.d 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.f.b ✓ 72 7.c even 3 1
245.4.f.b ✓ 72 7.d odd 6 1
245.4.f.b ✓ 72 35.k even 12 1
245.4.f.b ✓ 72 35.l odd 12 1
245.4.l.d 144 1.a even 1 1 trivial
245.4.l.d 144 5.c odd 4 1 inner
245.4.l.d 144 7.b odd 2 1 inner
245.4.l.d 144 7.c even 3 1 inner
245.4.l.d 144 7.d odd 6 1 inner
245.4.l.d 144 35.f even 4 1 inner
245.4.l.d 144 35.k even 12 1 inner
245.4.l.d 144 35.l odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} - 2160 T_{2}^{68} + 184 T_{2}^{67} - 6344 T_{2}^{65} + 2949766 T_{2}^{64} + \cdots + 28\!\cdots\!96 \) acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\). Copy content Toggle raw display