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Properties

Label 555.2.bz.a
Level $555$
Weight $2$
Character orbit 555.bz
Analytic conductor $4.432$
Analytic rank $0$
Dimension $600$
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] = [555,2,Mod(56,555)]
 
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([18, 0, 35]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("555.56");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.bz (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: \(600\)
Relative dimension: \(50\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 600 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 600 q + 96 q^{16} - 36 q^{21} - 60 q^{24} + 48 q^{28} - 24 q^{31} - 120 q^{34} + 12 q^{37} - 48 q^{43} - 24 q^{45} + 36 q^{49} - 120 q^{51} + 120 q^{54} - 72 q^{57} + 24 q^{61} - 72 q^{63} - 120 q^{66} - 132 q^{69} - 492 q^{72} - 432 q^{78} + 144 q^{79} - 96 q^{81} + 72 q^{82} - 84 q^{84} - 144 q^{87} - 108 q^{91} + 72 q^{93} + 144 q^{94} + 324 q^{96} + 48 q^{97} + 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
56.1 −0.241379 − 2.75898i −1.33993 + 1.09753i −5.58408 + 0.984625i 0.906308 − 0.422618i 3.35150 + 3.43193i −4.20461 − 1.53035i 2.63083 + 9.81841i 0.590848 − 2.94124i −1.38476 − 2.39847i
56.2 −0.225760 − 2.58045i 0.936532 − 1.45702i −4.63813 + 0.817827i −0.906308 + 0.422618i −3.97120 − 2.08773i −2.36933 − 0.862367i 1.81662 + 6.77973i −1.24582 − 2.72909i 1.29515 + 2.24327i
56.3 −0.215069 − 2.45825i 1.72197 + 0.186575i −4.02715 + 0.710095i 0.906308 − 0.422618i 0.0883047 − 4.27317i −0.657989 − 0.239488i 1.33436 + 4.97991i 2.93038 + 0.642553i −1.23382 − 2.13704i
56.4 −0.215044 − 2.45797i −0.774298 + 1.54934i −4.02573 + 0.709846i −0.906308 + 0.422618i 3.97474 + 1.57002i 2.16488 + 0.787953i 1.33329 + 4.97590i −1.80092 − 2.39931i 1.23368 + 2.13679i
56.5 −0.214142 − 2.44766i 0.610654 + 1.62083i −3.97555 + 0.700998i 0.906308 − 0.422618i 3.83648 − 1.84176i 4.07889 + 1.48459i 1.29530 + 4.83411i −2.25420 + 1.97954i −1.22850 − 2.12783i
56.6 −0.212587 − 2.42988i −1.66667 − 0.471381i −3.88952 + 0.685827i −0.906308 + 0.422618i −0.791087 + 4.15003i 1.32423 + 0.481982i 1.23074 + 4.59318i 2.55560 + 1.57128i 1.21958 + 2.11238i
56.7 −0.212538 − 2.42931i 1.17635 + 1.27130i −3.88678 + 0.685345i −0.906308 + 0.422618i 2.83836 − 3.12793i −3.16958 − 1.15363i 1.22870 + 4.58555i −0.232392 + 2.99099i 1.21930 + 2.11188i
56.8 −0.181379 − 2.07317i 1.32693 − 1.11322i −2.29553 + 0.404764i −0.906308 + 0.422618i −2.54857 − 2.54904i 3.76470 + 1.37024i 0.178256 + 0.665259i 0.521493 − 2.95433i 1.04055 + 1.80228i
56.9 −0.171565 − 1.96100i −0.232941 − 1.71632i −1.84647 + 0.325582i 0.906308 − 0.422618i −3.32573 + 0.751257i −3.59075 − 1.30693i −0.0637105 − 0.237771i −2.89148 + 0.799601i −0.984245 − 1.70476i
56.10 −0.170430 − 1.94802i −1.45708 − 0.936442i −1.79613 + 0.316707i 0.906308 − 0.422618i −1.57588 + 2.99802i −1.02208 − 0.372006i −0.0891560 − 0.332735i 1.24615 + 2.72894i −0.977732 − 1.69348i
56.11 −0.158360 − 1.81006i 0.413495 − 1.68197i −1.28162 + 0.225985i 0.906308 − 0.422618i −3.10995 − 0.482094i 4.69419 + 1.70855i −0.328531 − 1.22610i −2.65804 − 1.39097i −0.908487 − 1.57355i
56.12 −0.140869 − 1.61014i 0.114687 + 1.72825i −0.603084 + 0.106340i 0.906308 − 0.422618i 2.76656 − 0.428119i −2.24187 − 0.815973i −0.580475 − 2.16636i −2.97369 + 0.396416i −0.808144 − 1.39975i
56.13 −0.139891 − 1.59896i 1.19609 + 1.25275i −0.567493 + 0.100064i −0.906308 + 0.422618i 1.83577 − 2.08775i 1.81006 + 0.658808i −0.591459 − 2.20735i −0.138747 + 2.99679i 0.802535 + 1.39003i
56.14 −0.112071 − 1.28098i 1.33068 − 1.10873i 0.341266 − 0.0601744i 0.906308 − 0.422618i −1.56939 − 1.58032i −1.02122 − 0.371692i −0.780945 − 2.91453i 0.541433 − 2.95074i −0.642937 − 1.11360i
56.15 −0.106673 − 1.21928i −0.167574 + 1.72393i 0.494358 − 0.0871687i −0.906308 + 0.422618i 2.11982 + 0.0204235i −2.50877 − 0.913119i −0.792573 − 2.95792i −2.94384 − 0.577772i 0.611967 + 1.05996i
56.16 −0.101340 − 1.15832i 0.453198 − 1.67171i 0.638185 − 0.112529i −0.906308 + 0.422618i −1.98230 − 0.355536i −2.09121 − 0.761139i −0.796898 − 2.97406i −2.58922 − 1.51523i 0.581371 + 1.00696i
56.17 −0.0935222 − 1.06896i −1.57695 + 0.716390i 0.835679 − 0.147353i −0.906308 + 0.422618i 0.913275 + 1.61871i 3.55658 + 1.29449i −0.791119 − 2.95250i 1.97357 − 2.25943i 0.536523 + 0.929286i
56.18 −0.0902000 − 1.03099i −1.16355 − 1.28303i 0.914810 − 0.161306i −0.906308 + 0.422618i −1.21783 + 1.31534i 1.76762 + 0.643361i −0.784539 − 2.92794i −0.292308 + 2.98573i 0.517464 + 0.896275i
56.19 −0.0817416 − 0.934311i −1.39804 − 1.02249i 1.10336 − 0.194552i 0.906308 − 0.422618i −0.841050 + 1.38978i 3.02052 + 1.09938i −0.757445 − 2.82682i 0.909012 + 2.85897i −0.468940 − 0.812228i
56.20 −0.0715293 − 0.817584i 1.47911 + 0.901233i 1.30629 − 0.230334i 0.906308 − 0.422618i 0.631033 − 1.27377i 1.54704 + 0.563075i −0.706584 − 2.63701i 1.37556 + 2.66605i −0.410354 − 0.710753i
See next 80 embeddings (of 600 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 56.50
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
37.i odd 36 1 inner
111.q 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.bz.a ✓ 600
3.b odd 2 1 inner 555.2.bz.a ✓ 600
37.i odd 36 1 inner 555.2.bz.a ✓ 600
111.q even 36 1 inner 555.2.bz.a ✓ 600
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.bz.a ✓ 600 1.a even 1 1 trivial
555.2.bz.a ✓ 600 3.b odd 2 1 inner
555.2.bz.a ✓ 600 37.i odd 36 1 inner
555.2.bz.a ✓ 600 111.q even 36 1 inner

Hecke kernels

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