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Properties

Label 561.2.o.a
Level $561$
Weight $2$
Character orbit 561.o
Analytic conductor $4.480$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [561,2,Mod(100,561)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(561, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("561.100");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 561 = 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 561.o (of order \(8\), degree \(4\), 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: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 8 q^{6} + 16 q^{12} + 16 q^{14} - 32 q^{16} - 24 q^{17} - 16 q^{18} - 16 q^{19} + 48 q^{20} - 8 q^{24} + 24 q^{25} + 32 q^{29} + 40 q^{32} + 56 q^{33} - 88 q^{34} - 96 q^{35} - 8 q^{36} + 8 q^{37} - 48 q^{40} + 72 q^{41} + 16 q^{42} + 16 q^{43} - 16 q^{44} + 8 q^{45} + 88 q^{46} + 80 q^{50} - 8 q^{51} - 96 q^{52} - 8 q^{53} + 8 q^{54} + 16 q^{56} + 48 q^{58} + 48 q^{59} + 8 q^{60} + 32 q^{61} - 56 q^{62} - 56 q^{65} + 72 q^{67} - 56 q^{68} + 16 q^{69} - 128 q^{70} - 32 q^{71} - 40 q^{73} - 64 q^{74} + 32 q^{75} + 40 q^{76} - 32 q^{78} + 24 q^{79} + 176 q^{80} - 40 q^{82} - 40 q^{83} - 32 q^{84} - 88 q^{85} - 48 q^{86} - 40 q^{87} - 16 q^{88} + 32 q^{91} + 136 q^{92} + 48 q^{93} - 8 q^{94} + 8 q^{96} + 40 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
100.1 −1.64827 − 1.64827i −0.382683 − 0.923880i 3.43358i 0.974921 − 0.403825i −0.892037 + 2.15357i 1.00668 + 0.416980i 2.36292 − 2.36292i −0.707107 + 0.707107i −2.27254 − 0.941319i
100.2 −1.53173 − 1.53173i 0.382683 + 0.923880i 2.69239i −0.200170 + 0.0829133i 0.828966 − 2.00130i 0.0418523 + 0.0173358i 1.06056 − 1.06056i −0.707107 + 0.707107i 0.433608 + 0.179606i
100.3 −1.31701 − 1.31701i −0.382683 − 0.923880i 1.46903i 2.48883 − 1.03091i −0.712761 + 1.72076i −3.13080 − 1.29682i −0.699289 + 0.699289i −0.707107 + 0.707107i −4.63553 − 1.92010i
100.4 −1.01573 − 1.01573i 0.382683 + 0.923880i 0.0634092i −3.93823 + 1.63127i 0.549708 − 1.32711i 0.237268 + 0.0982795i −1.96705 + 1.96705i −0.707107 + 0.707107i 5.65710 + 2.34325i
100.5 −0.391626 − 0.391626i −0.382683 − 0.923880i − 1.69326i −2.72343 + 1.12808i −0.211947 + 0.511684i 0.817068 + 0.338441i −1.44638 + 1.44638i −0.707107 + 0.707107i 1.50836 + 0.624781i
100.6 −0.235777 − 0.235777i 0.382683 + 0.923880i − 1.88882i 1.01154 − 0.418993i 0.127602 − 0.308058i 4.41832 + 1.83013i −0.916895 + 0.916895i −0.707107 + 0.707107i −0.337287 − 0.139709i
100.7 0.0134868 + 0.0134868i 0.382683 + 0.923880i − 1.99964i −2.29501 + 0.950624i −0.00729898 + 0.0176213i −1.48405 − 0.614713i 0.0539421 − 0.0539421i −0.707107 + 0.707107i −0.0437731 − 0.0181314i
100.8 0.401548 + 0.401548i 0.382683 + 0.923880i − 1.67752i 2.59487 − 1.07483i −0.217316 + 0.524648i −4.23378 − 1.75369i 1.47670 − 1.47670i −0.707107 + 0.707107i 1.47356 + 0.610369i
100.9 0.607826 + 0.607826i −0.382683 − 0.923880i − 1.26110i −0.790337 + 0.327368i 0.328953 − 0.794163i 1.45167 + 0.601300i 1.98218 − 1.98218i −0.707107 + 0.707107i −0.679370 − 0.281404i
100.10 1.23751 + 1.23751i −0.382683 − 0.923880i 1.06284i 3.24039 − 1.34221i 0.669733 − 1.61688i −2.37695 − 0.984563i 1.15974 − 1.15974i −0.707107 + 0.707107i 5.67099 + 2.34900i
100.11 1.54417 + 1.54417i 0.382683 + 0.923880i 2.76891i 2.75853 − 1.14262i −0.835697 + 2.01755i −1.47958 − 0.612864i −1.18732 + 1.18732i −0.707107 + 0.707107i 6.02403 + 2.49523i
100.12 1.59326 + 1.59326i −0.382683 − 0.923880i 3.07698i −0.0347873 + 0.0144094i 0.862268 − 2.08170i 3.54919 + 1.47012i −1.71591 + 1.71591i −0.707107 + 0.707107i −0.0783833 − 0.0324674i
100.13 1.69705 + 1.69705i 0.382683 + 0.923880i 3.75996i −2.48639 + 1.02990i −0.918437 + 2.21730i 3.42386 + 1.41821i −2.98675 + 2.98675i −0.707107 + 0.707107i −5.96732 − 2.47174i
100.14 1.87372 + 1.87372i −0.382683 − 0.923880i 5.02166i −2.01492 + 0.834609i 1.01405 − 2.44813i −2.24074 − 0.928143i −5.66174 + 5.66174i −0.707107 + 0.707107i −5.33923 − 2.21158i
298.1 −1.95239 + 1.95239i −0.923880 − 0.382683i − 5.62363i −1.16258 + 2.80671i 2.55092 − 1.05662i −0.217114 − 0.524159i 7.07474 + 7.07474i 0.707107 + 0.707107i −3.20998 − 7.74958i
298.2 −1.84281 + 1.84281i 0.923880 + 0.382683i − 4.79186i −0.514971 + 1.24325i −2.40774 + 0.997319i −1.23558 − 2.98296i 5.14486 + 5.14486i 0.707107 + 0.707107i −1.34208 − 3.24006i
298.3 −1.26902 + 1.26902i −0.923880 − 0.382683i − 1.22080i −0.0180261 + 0.0435188i 1.65805 − 0.686786i −0.284422 − 0.686656i −0.988814 − 0.988814i 0.707107 + 0.707107i −0.0323507 − 0.0781014i
298.4 −1.17053 + 1.17053i 0.923880 + 0.382683i − 0.740258i 1.31289 − 3.16961i −1.52936 + 0.633484i 1.60348 + 3.87115i −1.47456 − 1.47456i 0.707107 + 0.707107i 2.17333 + 5.24688i
298.5 −0.956273 + 0.956273i −0.923880 − 0.382683i 0.171086i 1.42133 − 3.43140i 1.24943 − 0.517531i 0.315164 + 0.760873i −2.07615 − 2.07615i 0.707107 + 0.707107i 1.92217 + 4.64054i
298.6 −0.676095 + 0.676095i −0.923880 − 0.382683i 1.08579i −1.06374 + 2.56810i 0.883360 − 0.365900i 1.36197 + 3.28808i −2.08629 − 2.08629i 0.707107 + 0.707107i −1.01709 − 2.45547i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 100.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 561.2.o.a ✓ 56
17.d even 8 1 inner 561.2.o.a ✓ 56
17.e odd 16 1 9537.2.a.ca 28
17.e odd 16 1 9537.2.a.cb 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
561.2.o.a ✓ 56 1.a even 1 1 trivial
561.2.o.a ✓ 56 17.d even 8 1 inner
9537.2.a.ca 28 17.e odd 16 1
9537.2.a.cb 28 17.e odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} + 192 T_{2}^{52} - 8 T_{2}^{51} + 88 T_{2}^{49} + 15198 T_{2}^{48} - 800 T_{2}^{47} + \cdots + 1156 \) acting on \(S_{2}^{\mathrm{new}}(561, [\chi])\). Copy content Toggle raw display