Properties

Label 861.2.z.a
Level $861$
Weight $2$
Character orbit 861.z
Analytic conductor $6.875$
Analytic rank $0$
Dimension $72$
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: \(72\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 12 q^{4} - 12 q^{5} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 12 q^{4} - 12 q^{5} - 72 q^{9} - 12 q^{10} - 10 q^{11} + 20 q^{13} - 10 q^{15} - 32 q^{16} + 10 q^{17} + 50 q^{20} + 18 q^{21} + 20 q^{22} + 2 q^{23} - 4 q^{25} + 30 q^{26} - 20 q^{28} - 8 q^{31} - 6 q^{33} - 30 q^{34} + 12 q^{36} + 6 q^{37} - 12 q^{39} - 52 q^{40} - 10 q^{41} - 46 q^{43} + 12 q^{45} + 48 q^{46} - 30 q^{47} + 40 q^{48} + 18 q^{49} + 36 q^{50} + 6 q^{51} - 110 q^{52} - 70 q^{53} - 26 q^{57} + 80 q^{58} + 32 q^{59} + 10 q^{61} + 60 q^{62} - 4 q^{64} - 80 q^{65} - 4 q^{66} + 10 q^{67} - 20 q^{69} - 40 q^{71} + 28 q^{73} - 38 q^{74} - 50 q^{76} - 6 q^{77} + 26 q^{78} - 30 q^{80} + 72 q^{81} + 24 q^{82} - 28 q^{83} + 12 q^{84} - 92 q^{86} + 44 q^{87} - 10 q^{88} + 12 q^{90} + 68 q^{91} + 36 q^{92} + 10 q^{93} + 40 q^{94} + 30 q^{95} + 10 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −0.752183 2.31498i 1.00000i −3.17533 + 2.30701i −2.87489 + 2.08873i 2.31498 0.752183i 0.951057 + 0.309017i 3.79063 + 2.75405i −1.00000 6.99782 + 5.08421i
64.2 −0.716198 2.20423i 1.00000i −2.72766 + 1.98176i 1.33599 0.970654i 2.20423 0.716198i 0.951057 + 0.309017i 2.57174 + 1.86848i −1.00000 −3.09638 2.24965i
64.3 −0.636263 1.95822i 1.00000i −1.81175 + 1.31631i −2.33402 + 1.69577i −1.95822 + 0.636263i −0.951057 0.309017i 0.398855 + 0.289785i −1.00000 4.80573 + 3.49156i
64.4 −0.592083 1.82224i 1.00000i −1.35198 + 0.982267i 1.10944 0.806055i −1.82224 + 0.592083i −0.951057 0.309017i −0.509774 0.370373i −1.00000 −2.12571 1.54442i
64.5 −0.424654 1.30695i 1.00000i 0.0902446 0.0655665i −1.42837 + 1.03777i −1.30695 + 0.424654i −0.951057 0.309017i −2.34753 1.70558i −1.00000 1.96288 + 1.42612i
64.6 −0.358744 1.10410i 1.00000i 0.527693 0.383392i −1.55355 + 1.12872i 1.10410 0.358744i 0.951057 + 0.309017i −2.49102 1.80983i −1.00000 1.80355 + 1.31035i
64.7 −0.328363 1.01060i 1.00000i 0.704551 0.511886i 1.51935 1.10387i 1.01060 0.328363i 0.951057 + 0.309017i −2.46799 1.79310i −1.00000 −1.61446 1.17298i
64.8 −0.204093 0.628132i 1.00000i 1.26514 0.919176i 2.96102 2.15131i −0.628132 + 0.204093i −0.951057 0.309017i −1.90421 1.38349i −1.00000 −1.95563 1.42085i
64.9 −0.0429603 0.132218i 1.00000i 1.60240 1.16421i −1.31978 + 0.958875i 0.132218 0.0429603i 0.951057 + 0.309017i −0.447712 0.325282i −1.00000 0.183479 + 0.133305i
64.10 0.0790778 + 0.243376i 1.00000i 1.56506 1.13708i −1.80571 + 1.31192i 0.243376 0.0790778i −0.951057 0.309017i 0.814556 + 0.591810i −1.00000 −0.462083 0.335723i
64.11 0.170649 + 0.525203i 1.00000i 1.37132 0.996320i 1.90227 1.38208i 0.525203 0.170649i −0.951057 0.309017i 1.65081 + 1.19939i −1.00000 1.05050 + 0.763230i
64.12 0.304564 + 0.937350i 1.00000i 0.832167 0.604605i 2.63453 1.91410i −0.937350 + 0.304564i 0.951057 + 0.309017i 2.41489 + 1.75452i −1.00000 2.59656 + 1.88651i
64.13 0.348523 + 1.07264i 1.00000i 0.588940 0.427890i 0.357062 0.259421i 1.07264 0.348523i −0.951057 0.309017i 2.48912 + 1.80845i −1.00000 0.402710 + 0.292586i
64.14 0.459506 + 1.41421i 1.00000i −0.170822 + 0.124110i 0.0702561 0.0510440i −1.41421 + 0.459506i 0.951057 + 0.309017i 2.15199 + 1.56351i −1.00000 0.104470 + 0.0759021i
64.15 0.598104 + 1.84077i 1.00000i −1.41269 + 1.02638i −0.0719999 + 0.0523110i 1.84077 0.598104i −0.951057 0.309017i 0.397448 + 0.288763i −1.00000 −0.139356 0.101248i
64.16 0.660739 + 2.03355i 1.00000i −2.08070 + 1.51172i −2.35668 + 1.71223i 2.03355 0.660739i −0.951057 0.309017i −0.989274 0.718750i −1.00000 −5.03904 3.66108i
64.17 0.665430 + 2.04798i 1.00000i −2.13340 + 1.55001i 0.184706 0.134197i −2.04798 + 0.665430i 0.951057 + 0.309017i −1.10978 0.806305i −1.00000 0.397743 + 0.288977i
64.18 0.768948 + 2.36658i 1.00000i −3.39138 + 2.46399i −3.56570 + 2.59063i −2.36658 + 0.768948i 0.951057 + 0.309017i −4.41275 3.20605i −1.00000 −8.87278 6.44645i
127.1 −2.17536 + 1.58049i 1.00000i 1.61621 4.97419i −0.719358 + 2.21396i −1.58049 2.17536i 0.587785 0.809017i 2.68400 + 8.26050i −1.00000 −1.93428 5.95310i
127.2 −1.79621 + 1.30502i 1.00000i 0.905247 2.78606i 0.572446 1.76181i 1.30502 + 1.79621i −0.587785 + 0.809017i 0.637681 + 1.96258i −1.00000 1.27096 + 3.91163i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.18
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.a 72
41.f even 10 1 inner 861.2.z.a 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.z.a 72 1.a even 1 1 trivial
861.2.z.a 72 41.f even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} + 24 T_{2}^{70} + 350 T_{2}^{68} + 4038 T_{2}^{66} + 39947 T_{2}^{64} + 12 T_{2}^{63} + 336414 T_{2}^{62} + 532 T_{2}^{61} + 2462543 T_{2}^{60} + 9832 T_{2}^{59} + 16026179 T_{2}^{58} + 73236 T_{2}^{57} + 93604517 T_{2}^{56} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(861, [\chi])\). Copy content Toggle raw display