↑

Properties

Label 555.2.ch.a
Level $555$
Weight $2$
Character orbit 555.ch
Analytic conductor $4.432$
Analytic rank $0$
Dimension $456$
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] = [555,2,Mod(13,555)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(555, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([0, 27, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("555.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.ch (of order \(36\), degree \(12\), 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: \(4.43169731218\)
Analytic rank: \(0\)
Dimension: \(456\)
Relative dimension: \(38\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 456 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 456 q - 48 q^{14} - 48 q^{17} + 24 q^{20} + 48 q^{25} - 48 q^{30} - 120 q^{35} - 24 q^{37} - 168 q^{38} + 48 q^{39} - 72 q^{40} - 60 q^{41} - 360 q^{44} + 48 q^{48} + 48 q^{49} + 180 q^{50} - 60 q^{52} + 36 q^{53} - 24 q^{57} + 180 q^{58} - 72 q^{60} - 144 q^{61} - 48 q^{62} + 228 q^{64} + 36 q^{65} - 24 q^{67} - 168 q^{70} - 96 q^{71} + 180 q^{73} + 60 q^{74} - 120 q^{76} - 36 q^{77} - 72 q^{78} - 96 q^{79} + 264 q^{80} - 120 q^{83} - 240 q^{86} + 72 q^{88} - 24 q^{89} + 48 q^{91} - 72 q^{93} + 72 q^{94} + 12 q^{95} + 48 q^{97} - 180 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −2.69596 − 0.475371i 0.819152 + 0.573576i 5.16285 + 1.87913i −0.834960 + 2.07433i −1.93574 − 1.93574i −0.114339 + 1.30690i −8.28400 − 4.78277i 0.342020 + 0.939693i 3.23710 − 5.19540i
13.2 −2.61069 − 0.460335i −0.819152 − 0.573576i 4.72440 + 1.71954i 0.403768 − 2.19931i 1.87451 + 1.87451i −0.0749771 + 0.856992i −6.95078 − 4.01304i 0.342020 + 0.939693i −2.06653 + 5.55585i
13.3 −2.58096 − 0.455092i −0.819152 − 0.573576i 4.57485 + 1.66511i 1.26646 + 1.84285i 1.85317 + 1.85317i 0.0116663 − 0.133347i −6.51040 − 3.75878i 0.342020 + 0.939693i −2.43001 − 5.33266i
13.4 −2.27239 − 0.400684i 0.819152 + 0.573576i 3.12382 + 1.13698i −0.192145 − 2.22780i −1.63161 − 1.63161i −0.244257 + 2.79187i −2.64636 − 1.52787i 0.342020 + 0.939693i −0.456012 + 5.13941i
13.5 −2.05293 − 0.361987i −0.819152 − 0.573576i 2.20409 + 0.802224i −1.66998 + 1.48700i 1.47403 + 1.47403i 0.414298 − 4.73545i −0.623821 − 0.360163i 0.342020 + 0.939693i 3.96663 − 2.44819i
13.6 −1.99274 − 0.351374i 0.819152 + 0.573576i 1.96816 + 0.716351i 0.945027 + 2.02655i −1.43082 − 1.43082i 0.193609 − 2.21296i −0.165550 − 0.0955805i 0.342020 + 0.939693i −1.17111 − 4.37045i
13.7 −1.90592 − 0.336065i 0.819152 + 0.573576i 1.64021 + 0.596987i −2.11501 + 0.725754i −1.36848 − 1.36848i −0.0564030 + 0.644689i 0.426595 + 0.246295i 0.342020 + 0.939693i 4.27495 − 0.672447i
13.8 −1.77221 − 0.312489i 0.819152 + 0.573576i 1.16370 + 0.423551i 0.927585 − 2.03460i −1.27247 − 1.27247i 0.0483125 − 0.552214i 1.18695 + 0.685286i 0.342020 + 0.939693i −2.27966 + 3.31588i
13.9 −1.66842 − 0.294187i −0.819152 − 0.573576i 0.817687 + 0.297614i −1.54602 − 1.61549i 1.19795 + 1.19795i −0.0748633 + 0.855691i 1.65767 + 0.957059i 0.342020 + 0.939693i 2.10416 + 3.15013i
13.10 −1.53571 − 0.270787i −0.819152 − 0.573576i 0.405690 + 0.147659i 1.54113 + 1.62016i 1.10266 + 1.10266i −0.0464819 + 0.531290i 2.11792 + 1.22278i 0.342020 + 0.939693i −1.92802 − 2.90541i
13.11 −1.43597 − 0.253200i −0.819152 − 0.573576i 0.118505 + 0.0431325i −1.60916 − 1.55261i 1.03105 + 1.03105i 0.0492115 − 0.562490i 2.36629 + 1.36618i 0.342020 + 0.939693i 1.91758 + 2.63694i
13.12 −1.33026 − 0.234561i 0.819152 + 0.573576i −0.164807 − 0.0599847i 1.92605 + 1.13592i −0.955148 − 0.955148i −0.442473 + 5.05749i 2.54479 + 1.46924i 0.342020 + 0.939693i −2.29571 − 1.96285i
13.13 −1.09297 − 0.192720i −0.819152 − 0.573576i −0.721949 − 0.262768i 2.15510 − 0.596263i 0.784767 + 0.784767i 0.0107619 − 0.123009i 2.66070 + 1.53616i 0.342020 + 0.939693i −2.47037 + 0.236365i
13.14 −0.951829 − 0.167833i 0.819152 + 0.573576i −1.00158 − 0.364544i −1.58194 − 1.58034i −0.683427 − 0.683427i 0.271210 − 3.09994i 2.56619 + 1.48159i 0.342020 + 0.939693i 1.24050 + 1.76971i
13.15 −0.668255 − 0.117831i 0.819152 + 0.573576i −1.44670 − 0.526557i 2.17361 − 0.524793i −0.479817 − 0.479817i 0.182204 − 2.08260i 2.08003 + 1.20091i 0.342020 + 0.939693i −1.51437 + 0.0945755i
13.16 −0.574523 − 0.101304i −0.819152 − 0.573576i −1.55957 − 0.567638i −1.30061 + 1.81890i 0.412516 + 0.412516i −0.247692 + 2.83113i 1.84896 + 1.06750i 0.342020 + 0.939693i 0.931494 − 0.913243i
13.17 −0.262197 − 0.0462324i 0.819152 + 0.573576i −1.81278 − 0.659796i −1.70097 + 1.45145i −0.188261 − 0.188261i 0.269739 − 3.08313i 0.905944 + 0.523047i 0.342020 + 0.939693i 0.513092 − 0.301926i
13.18 −0.130403 − 0.0229935i −0.819152 − 0.573576i −1.86291 − 0.678043i 0.337396 − 2.21047i 0.0936311 + 0.0936311i 0.390019 − 4.45793i 0.456686 + 0.263668i 0.342020 + 0.939693i −0.0948237 + 0.280493i
13.19 0.0436440 + 0.00769561i 0.819152 + 0.573576i −1.87754 − 0.683369i 0.608488 + 2.15168i 0.0313370 + 0.0313370i −0.0107255 + 0.122593i −0.153444 − 0.0885910i 0.342020 + 0.939693i 0.00999831 + 0.0985907i
13.20 0.129206 + 0.0227826i −0.819152 − 0.573576i −1.86321 − 0.678153i −2.18817 + 0.460358i −0.0927722 − 0.0927722i 0.0862263 − 0.985571i −0.452533 − 0.261270i 0.342020 + 0.939693i −0.293213 + 0.00962918i
See next 80 embeddings (of 456 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.bc even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 555.2.ch.a yes 456
5.c odd 4 1 555.2.bx.a ✓ 456
37.i odd 36 1 555.2.bx.a ✓ 456
185.bc even 36 1 inner 555.2.ch.a yes 456
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.bx.a ✓ 456 5.c odd 4 1
555.2.bx.a ✓ 456 37.i odd 36 1
555.2.ch.a yes 456 1.a even 1 1 trivial
555.2.ch.a yes 456 185.bc even 36 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(555, [\chi])\).