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Properties

Label 861.2.z.b
Level $861$
Weight $2$
Character orbit 861.z
Analytic conductor $6.875$
Analytic rank $0$
Dimension $88$
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(64,861)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(861, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("861.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 861.z (of order \(10\), 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: \(6.87511961403\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(22\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 4 q^{2} - 24 q^{4} - 4 q^{5} + 12 q^{8} - 88 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 4 q^{2} - 24 q^{4} - 4 q^{5} + 12 q^{8} - 88 q^{9} - 20 q^{10} + 10 q^{11} - 20 q^{13} - 10 q^{15} - 12 q^{16} + 10 q^{17} - 4 q^{18} + 10 q^{20} - 22 q^{21} - 2 q^{23} + 30 q^{26} + 20 q^{28} - 152 q^{32} - 6 q^{33} - 30 q^{34} + 24 q^{36} + 22 q^{37} - 4 q^{39} + 124 q^{40} + 22 q^{41} - 16 q^{42} - 14 q^{43} + 4 q^{45} - 88 q^{46} - 30 q^{47} + 40 q^{48} + 22 q^{49} - 76 q^{50} + 6 q^{51} + 90 q^{52} + 70 q^{53} + 22 q^{57} - 80 q^{58} - 64 q^{59} + 50 q^{61} + 4 q^{62} + 32 q^{64} - 80 q^{65} - 4 q^{66} + 10 q^{67} + 20 q^{69} - 40 q^{71} - 12 q^{72} - 132 q^{73} - 54 q^{74} + 150 q^{76} + 6 q^{77} - 66 q^{78} - 22 q^{80} + 88 q^{81} + 56 q^{82} + 52 q^{83} - 24 q^{84} - 116 q^{86} - 12 q^{87} - 150 q^{88} + 20 q^{90} + 4 q^{91} - 28 q^{92} + 10 q^{93} + 120 q^{94} - 110 q^{95} + 10 q^{97} - 4 q^{98} - 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −0.842746 − 2.59371i 1.00000i −4.39905 + 3.19610i 3.29688 − 2.39533i 2.59371 − 0.842746i −0.951057 − 0.309017i 7.58435 + 5.51035i −1.00000 −8.99121 − 6.53249i
64.2 −0.831759 − 2.55989i − 1.00000i −4.24318 + 3.08285i 1.31263 − 0.953683i −2.55989 + 0.831759i 0.951057 + 0.309017i 7.06593 + 5.13370i −1.00000 −3.53312 − 2.56696i
64.3 −0.745267 − 2.29370i 1.00000i −3.08758 + 2.24326i −0.353607 + 0.256911i 2.29370 − 0.745267i −0.951057 − 0.309017i 3.54416 + 2.57498i −1.00000 0.852806 + 0.619600i
64.4 −0.705702 − 2.17193i 1.00000i −2.60122 + 1.88989i −3.06975 + 2.23030i 2.17193 − 0.705702i −0.951057 − 0.309017i 2.24529 + 1.63130i −1.00000 7.01037 + 5.09334i
64.5 −0.647076 − 1.99149i − 1.00000i −1.92931 + 1.40172i 2.06079 − 1.49725i −1.99149 + 0.647076i 0.951057 + 0.309017i 0.651802 + 0.473562i −1.00000 −4.31526 − 3.13522i
64.6 −0.553494 − 1.70348i − 1.00000i −0.977453 + 0.710161i −2.15394 + 1.56493i −1.70348 + 0.553494i 0.951057 + 0.309017i −1.14737 − 0.833616i −1.00000 3.85803 + 2.80302i
64.7 −0.504751 − 1.55346i − 1.00000i −0.540442 + 0.392654i 2.88829 − 2.09846i −1.55346 + 0.504751i 0.951057 + 0.309017i −1.76015 − 1.27882i −1.00000 −4.71775 − 3.42765i
64.8 −0.401152 − 1.23462i 1.00000i 0.254676 − 0.185033i −0.503312 + 0.365677i 1.23462 − 0.401152i −0.951057 − 0.309017i −2.43107 − 1.76627i −1.00000 0.653376 + 0.474705i
64.9 −0.351418 − 1.08155i 1.00000i 0.571773 − 0.415417i 0.261160 − 0.189744i 1.08155 − 0.351418i −0.951057 − 0.309017i −2.49027 − 1.80929i −1.00000 −0.296994 − 0.215779i
64.10 −0.134254 − 0.413190i − 1.00000i 1.46533 − 1.06463i −3.52511 + 2.56114i −0.413190 + 0.134254i 0.951057 + 0.309017i −1.33958 − 0.973262i −1.00000 1.53150 + 1.11270i
64.11 −0.0882669 − 0.271658i − 1.00000i 1.55203 − 1.12761i 1.78828 − 1.29927i −0.271658 + 0.0882669i 0.951057 + 0.309017i −0.905489 − 0.657876i −1.00000 −0.510802 − 0.371119i
64.12 −0.0484350 − 0.149068i 1.00000i 1.59816 − 1.16113i 2.76695 − 2.01030i 0.149068 − 0.0484350i −0.951057 − 0.309017i −0.504103 − 0.366252i −1.00000 −0.433688 − 0.315093i
64.13 0.0569231 + 0.175191i 1.00000i 1.59058 − 1.15563i −1.79849 + 1.30668i −0.175191 + 0.0569231i −0.951057 − 0.309017i 0.591050 + 0.429423i −1.00000 −0.331295 − 0.240700i
64.14 0.0713902 + 0.219717i − 1.00000i 1.57486 − 1.14420i −0.0705613 + 0.0512658i 0.219717 − 0.0713902i 0.951057 + 0.309017i 0.737633 + 0.535922i −1.00000 −0.0163013 − 0.0118436i
64.15 0.320300 + 0.985782i − 1.00000i 0.748861 − 0.544079i −2.32459 + 1.68891i 0.985782 − 0.320300i 0.951057 + 0.309017i 2.45331 + 1.78244i −1.00000 −2.40947 − 1.75058i
64.16 0.433022 + 1.33270i 1.00000i 0.0294423 − 0.0213911i −0.818016 + 0.594323i −1.33270 + 0.433022i −0.951057 − 0.309017i 2.30859 + 1.67729i −1.00000 −1.14628 − 0.832818i
64.17 0.462794 + 1.42433i − 1.00000i −0.196516 + 0.142777i 3.11576 − 2.26373i 1.42433 − 0.462794i 0.951057 + 0.309017i 2.12891 + 1.54675i −1.00000 4.66627 + 3.39024i
64.18 0.503660 + 1.55011i 1.00000i −0.531120 + 0.385881i −3.04198 + 2.21013i −1.55011 + 0.503660i −0.951057 − 0.309017i 1.77154 + 1.28710i −1.00000 −4.95805 − 3.60224i
64.19 0.509733 + 1.56880i − 1.00000i −0.583263 + 0.423765i −1.68931 + 1.22736i 1.56880 − 0.509733i 0.951057 + 0.309017i 1.70689 + 1.24013i −1.00000 −2.78657 − 2.02456i
64.20 0.653856 + 2.01236i 1.00000i −2.00404 + 1.45602i 3.29898 − 2.39685i −2.01236 + 0.653856i −0.951057 − 0.309017i −0.816748 − 0.593402i −1.00000 6.98037 + 5.07154i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 861.2.z.b ✓ 88
41.f even 10 1 inner 861.2.z.b ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.z.b ✓ 88 1.a even 1 1 trivial
861.2.z.b ✓ 88 41.f even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{88} - 4 T_{2}^{87} + 42 T_{2}^{86} - 156 T_{2}^{85} + 981 T_{2}^{84} - 3236 T_{2}^{83} + \cdots + 121903681 \) acting on \(S_{2}^{\mathrm{new}}(861, [\chi])\). Copy content Toggle raw display