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Properties

Label 561.2.bn.a
Level $561$
Weight $2$
Character orbit 561.bn
Analytic conductor $4.480$
Analytic rank $0$
Dimension $1152$
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] = [561,2,Mod(7,561)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(561, base_ring=CyclotomicField(80))
 
chi = DirichletCharacter(H, H._module([0, 56, 55]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("561.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 561 = 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 561.bn (of order \(80\), degree \(32\), 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: \(4.47960755339\)
Analytic rank: \(0\)
Dimension: \(1152\)
Relative dimension: \(36\) over \(\Q(\zeta_{80})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{80}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1152 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1152 q + 16 q^{11} - 80 q^{13} + 64 q^{14} + 32 q^{22} - 32 q^{23} + 32 q^{25} + 64 q^{26} - 64 q^{31} + 32 q^{37} - 240 q^{38} - 96 q^{42} + 64 q^{49} - 320 q^{52} - 288 q^{55} - 224 q^{58} + 128 q^{59} - 48 q^{69} + 288 q^{70} + 64 q^{75} - 448 q^{77} - 704 q^{80} + 128 q^{86} - 160 q^{88} + 64 q^{91} - 320 q^{92} - 480 q^{94} + 64 q^{97} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.40824 − 2.29805i 0.619094 + 0.785317i −2.38988 + 4.69041i −1.07787 − 0.996372i 0.932859 − 2.52863i 0.127922 + 1.08081i 8.77052 − 0.690255i −0.233445 + 0.972370i −0.771805 + 3.88013i
7.2 −1.30512 − 2.12976i −0.619094 − 0.785317i −1.92457 + 3.77718i −0.191935 − 0.177423i −0.864546 + 2.34346i −0.182226 − 1.53962i 5.57599 − 0.438840i −0.233445 + 0.972370i −0.127370 + 0.640334i
7.3 −1.29073 − 2.10628i −0.619094 − 0.785317i −1.86245 + 3.65527i 1.41249 + 1.30569i −0.855015 + 2.31762i −0.0724431 − 0.612069i 5.17758 − 0.407485i −0.233445 + 0.972370i 0.927009 − 4.66039i
7.4 −1.23541 − 2.01600i 0.619094 + 0.785317i −1.63005 + 3.19915i 0.481288 + 0.444898i 0.818367 − 2.21828i −0.312633 − 2.64143i 3.74900 − 0.295053i −0.233445 + 0.972370i 0.302328 − 1.51991i
7.5 −1.18983 − 1.94163i 0.619094 + 0.785317i −1.44625 + 2.83842i 2.88280 + 2.66483i 0.788177 − 2.13645i 0.190631 + 1.61063i 2.69160 − 0.211833i −0.233445 + 0.972370i 1.74407 − 8.76804i
7.6 −1.16838 − 1.90662i −0.619094 − 0.785317i −1.36210 + 2.67327i −2.18468 − 2.01950i −0.773963 + 2.09792i 0.331965 + 2.80476i 2.22987 − 0.175495i −0.233445 + 0.972370i −1.29788 + 6.52487i
7.7 −1.08981 − 1.77841i 0.619094 + 0.785317i −1.06707 + 2.09424i −2.52253 − 2.33180i 0.721919 − 1.95685i −0.200914 − 1.69752i 0.728635 − 0.0573448i −0.233445 + 0.972370i −1.39782 + 7.02730i
7.8 −0.856832 − 1.39822i −0.619094 − 0.785317i −0.312883 + 0.614068i 1.08487 + 1.00284i −0.567588 + 1.53852i 0.497789 + 4.20579i −2.14294 + 0.168653i −0.233445 + 0.972370i 0.472647 − 2.37616i
7.9 −0.853183 − 1.39227i −0.619094 − 0.785317i −0.302507 + 0.593704i 0.196234 + 0.181397i −0.565171 + 1.53196i −0.372594 − 3.14803i −2.17102 + 0.170863i −0.233445 + 0.972370i 0.0851295 − 0.427975i
7.10 −0.778413 − 1.27025i 0.619094 + 0.785317i −0.0996389 + 0.195552i 0.996357 + 0.921023i 0.515642 − 1.39771i 0.129192 + 1.09154i −2.64443 + 0.208121i −0.233445 + 0.972370i 0.394357 − 1.98256i
7.11 −0.750854 − 1.22528i −0.619094 − 0.785317i −0.0295542 + 0.0580034i 2.72717 + 2.52097i −0.497386 + 1.34822i 0.104374 + 0.881855i −2.77197 + 0.218159i −0.233445 + 0.972370i 1.04119 − 5.23443i
7.12 −0.665730 − 1.08637i −0.619094 − 0.785317i 0.170973 − 0.335553i −1.99960 − 1.84841i −0.440997 + 1.19538i −0.563667 − 4.76240i −3.01876 + 0.237581i −0.233445 + 0.972370i −0.676870 + 3.40285i
7.13 −0.656342 − 1.07105i 0.619094 + 0.785317i 0.191613 − 0.376061i −1.69198 − 1.56405i 0.434778 − 1.17852i 0.550776 + 4.65349i −3.03312 + 0.238712i −0.233445 + 0.972370i −0.564662 + 2.83875i
7.14 −0.463709 − 0.756705i 0.619094 + 0.785317i 0.550405 − 1.08023i 0.594566 + 0.549611i 0.307173 − 0.832630i −0.378342 − 3.19659i −2.84214 + 0.223681i −0.233445 + 0.972370i 0.140188 − 0.704771i
7.15 −0.386274 − 0.630342i 0.619094 + 0.785317i 0.659858 − 1.29504i −1.06464 − 0.984146i 0.255878 − 0.693588i 0.143351 + 1.21117i −2.54521 + 0.200313i −0.233445 + 0.972370i −0.209104 + 1.05124i
7.16 −0.287642 − 0.469389i −0.619094 − 0.785317i 0.770393 − 1.51198i −1.23221 − 1.13904i −0.190542 + 0.516485i 0.398169 + 3.36411i −2.02893 + 0.159681i −0.233445 + 0.972370i −0.180219 + 0.906020i
7.17 −0.0539594 − 0.0880538i −0.619094 − 0.785317i 0.903139 − 1.77251i −3.15349 − 2.91505i −0.0357442 + 0.0968888i −0.0440249 − 0.371965i −0.410716 + 0.0323241i −0.233445 + 0.972370i −0.0865212 + 0.434971i
7.18 −0.0304380 − 0.0496702i 0.619094 + 0.785317i 0.906440 − 1.77899i 2.33137 + 2.15510i 0.0201629 − 0.0546540i 0.0454756 + 0.384221i −0.232103 + 0.0182669i −0.233445 + 0.972370i 0.0360820 − 0.181397i
7.19 0.118739 + 0.193764i 0.619094 + 0.785317i 0.884535 − 1.73600i −2.16206 − 1.99859i −0.0786558 + 0.213206i −0.230451 − 1.94707i 0.894507 − 0.0703992i −0.233445 + 0.972370i 0.130534 − 0.656239i
7.20 0.212801 + 0.347259i −0.619094 − 0.785317i 0.832676 − 1.63422i 0.418818 + 0.387151i 0.140965 − 0.382102i 0.422642 + 3.57089i 1.55673 − 0.122517i −0.233445 + 0.972370i −0.0453171 + 0.227824i
See next 80 embeddings (of 1152 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
17.e odd 16 1 inner
187.t even 80 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 561.2.bn.a ✓ 1152
11.d odd 10 1 inner 561.2.bn.a ✓ 1152
17.e odd 16 1 inner 561.2.bn.a ✓ 1152
187.t even 80 1 inner 561.2.bn.a ✓ 1152
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
561.2.bn.a ✓ 1152 1.a even 1 1 trivial
561.2.bn.a ✓ 1152 11.d odd 10 1 inner
561.2.bn.a ✓ 1152 17.e odd 16 1 inner
561.2.bn.a ✓ 1152 187.t even 80 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(561, [\chi])\).