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Properties

Label 966.2.t.a
Level $966$
Weight $2$
Character orbit 966.t
Analytic conductor $7.714$
Analytic rank $0$
Dimension $640$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [966,2,Mod(41,966)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 11, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("966.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.t (of order \(22\), degree \(10\), 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: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(640\)
Relative dimension: \(64\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 640 q + 64 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 640 q + 64 q^{4} + 8 q^{9} + 8 q^{15} - 64 q^{16} + 20 q^{18} + 54 q^{21} - 8 q^{22} - 56 q^{25} - 44 q^{30} - 8 q^{36} + 20 q^{37} - 16 q^{39} - 4 q^{42} - 28 q^{43} + 24 q^{46} - 68 q^{49} - 40 q^{51} + 32 q^{57} - 80 q^{58} - 8 q^{60} + 86 q^{63} + 64 q^{64} + 24 q^{67} + 72 q^{70} - 64 q^{72} - 8 q^{78} - 144 q^{79} + 20 q^{81} + 34 q^{84} + 32 q^{85} + 8 q^{88} - 40 q^{91} + 16 q^{93} + 68 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −0.281733 + 0.959493i −1.72699 + 0.132270i −0.841254 − 0.540641i −0.371345 + 2.58276i 0.359638 − 1.69430i −2.27409 − 1.35223i 0.755750 − 0.654861i 2.96501 − 0.456859i −2.37352 − 1.08395i
41.2 −0.281733 + 0.959493i −1.72562 + 0.149127i −0.841254 − 0.540641i 0.250877 − 1.74489i 0.343077 − 1.69773i −0.458506 − 2.60572i 0.755750 − 0.654861i 2.95552 − 0.514671i 1.60353 + 0.732307i
41.3 −0.281733 + 0.959493i −1.72464 + 0.159999i −0.841254 − 0.540641i 0.116658 − 0.811375i 0.332371 − 1.69986i −1.16903 + 2.37347i 0.755750 − 0.654861i 2.94880 − 0.551882i 0.745642 + 0.340523i
41.4 −0.281733 + 0.959493i −1.70014 − 0.330969i −0.841254 − 0.540641i 0.212651 − 1.47902i 0.796546 − 1.53802i 1.89318 + 1.84820i 0.755750 − 0.654861i 2.78092 + 1.12539i 1.35920 + 0.620726i
41.5 −0.281733 + 0.959493i −1.53070 − 0.810522i −0.841254 − 0.540641i 0.429144 − 2.98476i 1.20894 − 1.24035i 0.998575 − 2.45007i 0.755750 − 0.654861i 1.68611 + 2.48134i 2.74295 + 1.25266i
41.6 −0.281733 + 0.959493i −1.26614 − 1.18191i −0.841254 − 0.540641i −0.345722 + 2.40455i 1.49074 − 0.881867i 1.74088 − 1.99232i 0.755750 − 0.654861i 0.206198 + 2.99291i −2.20975 − 1.00916i
41.7 −0.281733 + 0.959493i −1.25848 + 1.19006i −0.841254 − 0.540641i −0.602743 + 4.19217i −0.787299 − 1.54278i −1.84332 + 1.89794i 0.755750 − 0.654861i 0.167522 − 2.99532i −3.85255 − 1.75940i
41.8 −0.281733 + 0.959493i −1.25685 − 1.19177i −0.841254 − 0.540641i −0.205721 + 1.43082i 1.49759 − 0.870178i −2.56843 + 0.634953i 0.755750 − 0.654861i 0.159350 + 2.99576i −1.31490 − 0.600496i
41.9 −0.281733 + 0.959493i −1.13682 + 1.30677i −0.841254 − 0.540641i 0.133168 − 0.926206i −0.933559 − 1.45893i 1.08253 + 2.41415i 0.755750 − 0.654861i −0.415299 − 2.97112i 0.851170 + 0.388716i
41.10 −0.281733 + 0.959493i −0.927012 + 1.46310i −0.841254 − 0.540641i −0.216850 + 1.50823i −1.14266 − 1.30166i −0.318183 − 2.62655i 0.755750 − 0.654861i −1.28130 − 2.71261i −1.38604 − 0.632983i
41.11 −0.281733 + 0.959493i −0.781340 + 1.54580i −0.841254 − 0.540641i −0.0180311 + 0.125409i −1.26306 − 1.18519i 2.64141 − 0.151432i 0.755750 − 0.654861i −1.77902 − 2.41559i −0.115249 − 0.0526324i
41.12 −0.281733 + 0.959493i −0.678660 − 1.59356i −0.841254 − 0.540641i 0.563772 − 3.92112i 1.72021 − 0.202213i −2.63555 + 0.232107i 0.755750 − 0.654861i −2.07884 + 2.16297i 3.60345 + 1.64564i
41.13 −0.281733 + 0.959493i −0.543127 − 1.64469i −0.841254 − 0.540641i 0.342096 − 2.37933i 1.73109 − 0.0577632i 1.62492 + 2.08797i 0.755750 − 0.654861i −2.41003 + 1.78655i 2.18657 + 0.998573i
41.14 −0.281733 + 0.959493i −0.459895 − 1.66988i −0.841254 − 0.540641i −0.0361830 + 0.251658i 1.73180 + 0.0291935i 2.47549 − 0.933780i 0.755750 − 0.654861i −2.57699 + 1.53594i −0.231270 − 0.105618i
41.15 −0.281733 + 0.959493i −0.134057 + 1.72686i −0.841254 − 0.540641i 0.595091 − 4.13895i −1.61914 − 0.615138i 0.717684 + 2.54655i 0.755750 − 0.654861i −2.96406 − 0.462995i 3.80364 + 1.73706i
41.16 −0.281733 + 0.959493i −0.0227709 − 1.73190i −0.841254 − 0.540641i −0.321156 + 2.23369i 1.66816 + 0.466084i −1.72364 − 2.00725i 0.755750 − 0.654861i −2.99896 + 0.0788740i −2.05273 − 0.937451i
41.17 −0.281733 + 0.959493i 0.0227709 + 1.73190i −0.841254 − 0.540641i 0.321156 − 2.23369i −1.66816 − 0.466084i −0.388235 − 2.61711i 0.755750 − 0.654861i −2.99896 + 0.0788740i 2.05273 + 0.937451i
41.18 −0.281733 + 0.959493i 0.134057 − 1.72686i −0.841254 − 0.540641i −0.595091 + 4.13895i 1.61914 + 0.615138i 1.45457 + 2.21003i 0.755750 − 0.654861i −2.96406 − 0.462995i −3.80364 − 1.73706i
41.19 −0.281733 + 0.959493i 0.459895 + 1.66988i −0.841254 − 0.540641i 0.0361830 − 0.251658i −1.73180 − 0.0291935i −2.32681 + 1.25936i 0.755750 − 0.654861i −2.57699 + 1.53594i 0.231270 + 0.105618i
41.20 −0.281733 + 0.959493i 0.543127 + 1.64469i −0.841254 − 0.540641i −0.342096 + 2.37933i −1.73109 + 0.0577632i 0.513888 + 2.59537i 0.755750 − 0.654861i −2.41003 + 1.78655i −2.18657 − 0.998573i
See next 80 embeddings (of 640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner
23.c even 11 1 inner
69.h odd 22 1 inner
161.l odd 22 1 inner
483.v even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.t.a ✓ 640
3.b odd 2 1 inner 966.2.t.a ✓ 640
7.b odd 2 1 inner 966.2.t.a ✓ 640
21.c even 2 1 inner 966.2.t.a ✓ 640
23.c even 11 1 inner 966.2.t.a ✓ 640
69.h odd 22 1 inner 966.2.t.a ✓ 640
161.l odd 22 1 inner 966.2.t.a ✓ 640
483.v even 22 1 inner 966.2.t.a ✓ 640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.t.a ✓ 640 1.a even 1 1 trivial
966.2.t.a ✓ 640 3.b odd 2 1 inner
966.2.t.a ✓ 640 7.b odd 2 1 inner
966.2.t.a ✓ 640 21.c even 2 1 inner
966.2.t.a ✓ 640 23.c even 11 1 inner
966.2.t.a ✓ 640 69.h odd 22 1 inner
966.2.t.a ✓ 640 161.l odd 22 1 inner
966.2.t.a ✓ 640 483.v even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(966, [\chi])\).