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Properties

Label 861.2.bo.b
Level $861$
Weight $2$
Character orbit 861.bo
Analytic conductor $6.875$
Analytic rank $0$
Dimension $192$
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] = [861,2,Mod(43,861)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(861, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 0, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("861.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 861.bo (of order \(20\), degree \(8\), 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: \(6.87511961403\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(24\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 60 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 60 q^{4} + 16 q^{10} - 20 q^{11} + 8 q^{12} - 4 q^{13} + 20 q^{15} - 36 q^{16} + 8 q^{17} - 4 q^{18} - 4 q^{19} - 8 q^{22} + 16 q^{23} + 48 q^{25} + 24 q^{26} - 32 q^{28} + 12 q^{29} + 8 q^{30} + 40 q^{31} + 28 q^{34} - 20 q^{37} + 8 q^{38} - 64 q^{40} - 4 q^{41} + 16 q^{42} + 20 q^{44} + 16 q^{45} - 40 q^{46} + 44 q^{47} - 16 q^{48} - 12 q^{51} - 180 q^{52} + 96 q^{53} + 76 q^{55} - 28 q^{57} + 4 q^{58} - 32 q^{59} + 36 q^{64} + 96 q^{65} - 24 q^{66} + 52 q^{67} + 200 q^{68} - 12 q^{69} + 8 q^{70} - 96 q^{71} + 12 q^{72} + 40 q^{74} + 32 q^{75} - 228 q^{76} - 68 q^{78} - 200 q^{80} - 192 q^{81} + 52 q^{82} + 192 q^{83} - 24 q^{85} - 288 q^{86} - 40 q^{87} - 76 q^{88} - 12 q^{89} - 184 q^{92} - 36 q^{93} + 112 q^{94} - 92 q^{95} - 20 q^{96} + 72 q^{97} - 4 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −2.65041 + 0.861170i 0.707107 − 0.707107i 4.66501 − 3.38933i 1.66808 + 2.29591i −1.26518 + 2.48306i −0.453990 − 0.891007i −6.16931 + 8.49133i − 1.00000i −6.39825 − 4.64860i
43.2 −2.51599 + 0.817496i −0.707107 + 0.707107i 4.04389 − 2.93806i −2.47111 − 3.40119i 1.20102 − 2.35713i 0.453990 + 0.891007i −4.66261 + 6.41754i − 1.00000i 8.99776 + 6.53726i
43.3 −2.45617 + 0.798059i 0.707107 − 0.707107i 3.77786 − 2.74477i −1.78570 − 2.45781i −1.17246 + 2.30109i −0.453990 − 0.891007i −4.05258 + 5.57790i − 1.00000i 6.34748 + 4.61171i
43.4 −2.29113 + 0.744435i −0.707107 + 0.707107i 3.07708 − 2.23563i 1.52470 + 2.09857i 1.09368 − 2.14647i 0.453990 + 0.891007i −2.55373 + 3.51491i − 1.00000i −5.05553 − 3.67306i
43.5 −1.89351 + 0.615239i −0.707107 + 0.707107i 1.58883 − 1.15435i 0.120879 + 0.166375i 0.903874 − 1.77395i 0.453990 + 0.891007i 0.0422493 − 0.0581512i − 1.00000i −0.331245 − 0.240664i
43.6 −1.82592 + 0.593277i 0.707107 − 0.707107i 1.36397 − 0.990983i 1.31669 + 1.81226i −0.871610 + 1.71063i −0.453990 − 0.891007i 0.354386 − 0.487771i − 1.00000i −3.47934 − 2.52789i
43.7 −1.56543 + 0.508641i 0.707107 − 0.707107i 0.573837 − 0.416917i −0.247429 − 0.340556i −0.747266 + 1.46659i −0.453990 − 0.891007i 1.24874 − 1.71874i − 1.00000i 0.560554 + 0.407266i
43.8 −1.16408 + 0.378234i 0.707107 − 0.707107i −0.406005 + 0.294980i −0.588220 − 0.809616i −0.555680 + 1.09058i −0.453990 − 0.891007i 1.79994 − 2.47740i − 1.00000i 0.990961 + 0.719975i
43.9 −0.954046 + 0.309988i −0.707107 + 0.707107i −0.803923 + 0.584084i 1.99937 + 2.75190i 0.455418 − 0.893807i 0.453990 + 0.891007i 1.76519 − 2.42957i − 1.00000i −2.76055 − 2.00565i
43.10 −0.919553 + 0.298781i −0.707107 + 0.707107i −0.861727 + 0.626081i −2.04859 − 2.81964i 0.438952 − 0.861492i 0.453990 + 0.891007i 1.74197 − 2.39762i − 1.00000i 2.72624 + 1.98073i
43.11 −0.486735 + 0.158150i 0.707107 − 0.707107i −1.40613 + 1.02162i 2.59347 + 3.56960i −0.232345 + 0.456003i −0.453990 − 0.891007i 1.12448 − 1.54772i − 1.00000i −1.82686 − 1.32729i
43.12 −0.230040 + 0.0747444i −0.707107 + 0.707107i −1.57070 + 1.14118i 1.44989 + 1.99561i 0.109810 − 0.215515i 0.453990 + 0.891007i 0.560372 − 0.771286i − 1.00000i −0.482694 − 0.350698i
43.13 −0.190195 + 0.0617981i −0.707107 + 0.707107i −1.58568 + 1.15206i −0.518357 − 0.713458i 0.0907903 − 0.178186i 0.453990 + 0.891007i 0.465487 − 0.640688i − 1.00000i 0.142679 + 0.103663i
43.14 0.397213 − 0.129062i 0.707107 − 0.707107i −1.47691 + 1.07304i 0.300688 + 0.413862i 0.189611 − 0.372133i −0.453990 − 0.891007i −0.939142 + 1.29262i − 1.00000i 0.172851 + 0.125584i
43.15 0.559941 − 0.181936i −0.707107 + 0.707107i −1.33760 + 0.971824i −0.993905 − 1.36799i −0.267290 + 0.524586i 0.453990 + 0.891007i −1.26429 + 1.74015i − 1.00000i −0.805415 − 0.585168i
43.16 0.561903 − 0.182573i 0.707107 − 0.707107i −1.33563 + 0.970393i 0.0138296 + 0.0190348i 0.268227 − 0.526424i −0.453990 − 0.891007i −1.26788 + 1.74508i − 1.00000i 0.0112461 + 0.00817080i
43.17 1.25741 − 0.408557i −0.707107 + 0.707107i −0.203875 + 0.148124i 0.682334 + 0.939153i −0.600229 + 1.17802i 0.453990 + 0.891007i −1.75008 + 2.40878i − 1.00000i 1.24167 + 0.902127i
43.18 1.54260 − 0.501221i 0.707107 − 0.707107i 0.510356 − 0.370795i −2.11596 − 2.91238i 0.736366 − 1.44520i −0.453990 − 0.891007i −1.30533 + 1.79664i − 1.00000i −4.72383 − 3.43206i
43.19 1.73325 − 0.563167i −0.707107 + 0.707107i 1.06897 − 0.776650i −0.628378 − 0.864889i −0.827374 + 1.62381i 0.453990 + 0.891007i −0.727012 + 1.00065i − 1.00000i −1.57621 − 1.14519i
43.20 1.83773 − 0.597113i 0.707107 − 0.707107i 1.40266 − 1.01909i 1.26155 + 1.73638i 0.877245 − 1.72169i −0.453990 − 0.891007i −0.302367 + 0.416172i − 1.00000i 3.35521 + 2.43770i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.g even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 861.2.bo.b ✓ 192
41.g even 20 1 inner 861.2.bo.b ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.bo.b ✓ 192 1.a even 1 1 trivial
861.2.bo.b ✓ 192 41.g even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{192} - 78 T_{2}^{190} + 3237 T_{2}^{188} - 95048 T_{2}^{186} - 60 T_{2}^{185} + \cdots + 15\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(861, [\chi])\). Copy content Toggle raw display