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Properties

Label 861.2.cd.a
Level $861$
Weight $2$
Character orbit 861.cd
Analytic conductor $6.875$
Analytic rank $0$
Dimension $896$
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] = [861,2,Mod(13,861)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(861, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([0, 20, 31]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("861.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 861.cd (of order \(40\), degree \(16\), 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: \(896\)
Relative dimension: \(56\) over \(\Q(\zeta_{40})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 896 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 896 q - 8 q^{14} + 224 q^{16} + 32 q^{22} - 16 q^{29} + 16 q^{30} - 80 q^{32} - 144 q^{37} + 32 q^{42} - 32 q^{43} - 384 q^{44} - 96 q^{46} + 16 q^{49} + 64 q^{50} - 32 q^{53} + 56 q^{56} - 176 q^{58} + 64 q^{65} + 112 q^{67} - 88 q^{70} - 32 q^{71} + 112 q^{74} - 184 q^{77} - 96 q^{78} - 16 q^{79} - 240 q^{84} - 32 q^{85} - 176 q^{88} - 184 q^{91} - 64 q^{95} - 200 q^{98} + 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −2.41754 − 1.23180i −0.923880 − 0.382683i 3.15161 + 4.33782i −0.395470 + 2.49690i 1.76213 + 2.06319i −1.73235 − 1.99974i −1.42693 − 9.00927i 0.707107 + 0.707107i 4.03174 − 5.54921i
13.2 −2.41754 − 1.23180i 0.923880 + 0.382683i 3.15161 + 4.33782i 0.395470 − 2.49690i −1.76213 − 2.06319i −2.24612 − 1.39819i −1.42693 − 9.00927i 0.707107 + 0.707107i −4.03174 + 5.54921i
13.3 −2.27769 − 1.16054i −0.923880 − 0.382683i 2.66546 + 3.66869i 0.489411 − 3.09002i 1.66019 + 1.94384i 0.300328 − 2.62865i −1.01363 − 6.39984i 0.707107 + 0.707107i −4.70083 + 6.47013i
13.4 −2.27769 − 1.16054i 0.923880 + 0.382683i 2.66546 + 3.66869i −0.489411 + 3.09002i −1.66019 − 1.94384i −2.54931 + 0.707842i −1.01363 − 6.39984i 0.707107 + 0.707107i 4.70083 − 6.47013i
13.5 −2.19788 − 1.11988i −0.923880 − 0.382683i 2.40099 + 3.30469i −0.163554 + 1.03264i 1.60202 + 1.87573i −0.601425 + 2.57649i −0.804496 − 5.07939i 0.707107 + 0.707107i 1.51590 − 2.08646i
13.6 −2.19788 − 1.11988i 0.923880 + 0.382683i 2.40099 + 3.30469i 0.163554 − 1.03264i −1.60202 − 1.87573i 2.45068 − 0.997072i −0.804496 − 5.07939i 0.707107 + 0.707107i −1.51590 + 2.08646i
13.7 −2.00989 − 1.02409i −0.923880 − 0.382683i 1.81534 + 2.49859i −0.406236 + 2.56487i 1.46500 + 1.71529i 2.46826 + 0.952734i −0.384084 − 2.42501i 0.707107 + 0.707107i 3.44315 − 4.73910i
13.8 −2.00989 − 1.02409i 0.923880 + 0.382683i 1.81534 + 2.49859i 0.406236 − 2.56487i −1.46500 − 1.71529i 1.32712 + 2.28883i −0.384084 − 2.42501i 0.707107 + 0.707107i −3.44315 + 4.73910i
13.9 −1.85736 − 0.946372i −0.923880 − 0.382683i 1.37859 + 1.89747i 0.184106 − 1.16240i 1.35382 + 1.58511i 1.44750 − 2.21466i −0.112635 − 0.711150i 0.707107 + 0.707107i −1.44202 + 1.98477i
13.10 −1.85736 − 0.946372i 0.923880 + 0.382683i 1.37859 + 1.89747i −0.184106 + 1.16240i −1.35382 − 1.58511i −1.96096 + 1.77613i −0.112635 − 0.711150i 0.707107 + 0.707107i 1.44202 − 1.98477i
13.11 −1.82331 − 0.929022i −0.923880 − 0.382683i 1.28580 + 1.76975i 0.559035 − 3.52961i 1.32900 + 1.55605i −2.10263 + 1.60591i −0.0600338 − 0.379038i 0.707107 + 0.707107i −4.29838 + 5.91621i
13.12 −1.82331 − 0.929022i 0.923880 + 0.382683i 1.28580 + 1.76975i −0.559035 + 3.52961i −1.32900 − 1.55605i 1.25722 − 2.32796i −0.0600338 − 0.379038i 0.707107 + 0.707107i 4.29838 − 5.91621i
13.13 −1.47180 − 0.749917i −0.923880 − 0.382683i 0.428236 + 0.589416i 0.523250 − 3.30367i 1.07278 + 1.25607i 2.25902 + 1.37725i 0.328546 + 2.07435i 0.707107 + 0.707107i −3.24760 + 4.46993i
13.14 −1.47180 − 0.749917i 0.923880 + 0.382683i 0.428236 + 0.589416i −0.523250 + 3.30367i −1.07278 − 1.25607i 1.71369 + 2.01576i 0.328546 + 2.07435i 0.707107 + 0.707107i 3.24760 − 4.46993i
13.15 −1.43992 − 0.733678i −0.923880 − 0.382683i 0.359526 + 0.494846i −0.535159 + 3.37886i 1.04955 + 1.22886i −2.31028 − 1.28942i 0.350983 + 2.21602i 0.707107 + 0.707107i 3.24958 − 4.47267i
13.16 −1.43992 − 0.733678i 0.923880 + 0.382683i 0.359526 + 0.494846i 0.535159 − 3.37886i −1.04955 − 1.22886i −1.63496 − 2.08012i 0.350983 + 2.21602i 0.707107 + 0.707107i −3.24958 + 4.47267i
13.17 −1.22527 − 0.624304i −0.923880 − 0.382683i −0.0640495 − 0.0881566i −0.111937 + 0.706743i 0.893087 + 1.04567i 0.878078 − 2.49579i 0.453682 + 2.86444i 0.707107 + 0.707107i 0.578376 − 0.796066i
13.18 −1.22527 − 0.624304i 0.923880 + 0.382683i −0.0640495 − 0.0881566i 0.111937 − 0.706743i −0.893087 − 1.04567i −2.32770 + 1.25770i 0.453682 + 2.86444i 0.707107 + 0.707107i −0.578376 + 0.796066i
13.19 −0.855559 − 0.435929i −0.923880 − 0.382683i −0.633624 − 0.872108i −0.374806 + 2.36643i 0.623611 + 0.730154i −0.873143 + 2.49752i 0.462347 + 2.91914i 0.707107 + 0.707107i 1.35227 − 1.86123i
13.20 −0.855559 − 0.435929i 0.923880 + 0.382683i −0.633624 − 0.872108i 0.374806 − 2.36643i −0.623611 − 0.730154i 2.33018 − 1.25309i 0.462347 + 2.91914i 0.707107 + 0.707107i −1.35227 + 1.86123i
See next 80 embeddings (of 896 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
41.h odd 40 1 inner
287.bb even 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 861.2.cd.a ✓ 896
7.b odd 2 1 inner 861.2.cd.a ✓ 896
41.h odd 40 1 inner 861.2.cd.a ✓ 896
287.bb even 40 1 inner 861.2.cd.a ✓ 896
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.cd.a ✓ 896 1.a even 1 1 trivial
861.2.cd.a ✓ 896 7.b odd 2 1 inner
861.2.cd.a ✓ 896 41.h odd 40 1 inner
861.2.cd.a ✓ 896 287.bb even 40 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(861, [\chi])\).