↑

Properties

Label 690.2.n.b
Level $690$
Weight $2$
Character orbit 690.n
Analytic conductor $5.510$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [690,2,Mod(89,690)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(690, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 11, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("690.89");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.n (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: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 24 q^{2} - 2 q^{3} - 24 q^{4} + 2 q^{6} + 24 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 24 q^{2} - 2 q^{3} - 24 q^{4} + 2 q^{6} + 24 q^{8} - 6 q^{9} + 9 q^{12} + 11 q^{15} - 24 q^{16} + 6 q^{18} + 4 q^{23} + 2 q^{24} - 12 q^{25} - 2 q^{27} + 22 q^{30} + 28 q^{31} + 24 q^{32} + 36 q^{35} - 6 q^{36} - 4 q^{46} - 104 q^{47} + 9 q^{48} + 70 q^{49} - 54 q^{50} - 9 q^{54} - 26 q^{55} + 44 q^{57} - 11 q^{60} + 44 q^{61} - 28 q^{62} + 121 q^{63} - 24 q^{64} - 44 q^{65} + 44 q^{66} - 102 q^{69} - 36 q^{70} - 16 q^{72} - 102 q^{75} - 8 q^{77} - 44 q^{79} + 74 q^{81} - 11 q^{84} + 22 q^{85} + 93 q^{87} + 4 q^{92} - 172 q^{93} + 16 q^{94} - 26 q^{95} + 2 q^{96} - 4 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1 0.142315 + 0.989821i −1.73163 − 0.0381696i −0.959493 + 0.281733i −2.23565 + 0.0433117i −0.208656 − 1.71944i 1.95813 − 1.25842i −0.415415 − 0.909632i 2.99709 + 0.132191i −0.361037 − 2.20673i
89.2 0.142315 + 0.989821i −1.67640 − 0.435517i −0.959493 + 0.281733i 0.584056 + 2.15844i 0.192507 − 1.72132i −1.28454 + 0.825522i −0.415415 − 0.909632i 2.62065 + 1.46020i −2.05335 + 0.885290i
89.3 0.142315 + 0.989821i −1.67505 + 0.440696i −0.959493 + 0.281733i 1.25961 + 1.84754i −0.674594 − 1.59528i 3.27194 − 2.10275i −0.415415 − 0.909632i 2.61157 − 1.47637i −1.64947 + 1.50972i
89.4 0.142315 + 0.989821i −1.56929 − 0.733032i −0.959493 + 0.281733i 1.76486 − 1.37305i 0.502238 − 1.65764i −3.04110 + 1.95439i −0.415415 − 0.909632i 1.92533 + 2.30068i 1.61024 + 1.55149i
89.5 0.142315 + 0.989821i −1.53419 + 0.803900i −0.959493 + 0.281733i −2.02447 − 0.949486i −1.01406 − 1.40417i −4.08214 + 2.62343i −0.415415 − 0.909632i 1.70749 − 2.46667i 0.651709 − 2.13899i
89.6 0.142315 + 0.989821i −1.27489 + 1.17245i −0.959493 + 0.281733i −0.182307 − 2.22862i −1.34195 − 1.09506i 1.96759 − 1.26449i −0.415415 − 0.909632i 0.250714 − 2.98951i 2.17999 − 0.497618i
89.7 0.142315 + 0.989821i −1.11013 + 1.32951i −0.959493 + 0.281733i 1.79300 − 1.33609i −1.47397 − 0.909626i −0.483302 + 0.310599i −0.415415 − 0.909632i −0.535201 − 2.95187i 1.57766 + 1.58461i
89.8 0.142315 + 0.989821i −0.804725 − 1.53376i −0.959493 + 0.281733i −2.06421 − 0.859668i 1.40362 − 1.01481i 0.955977 − 0.614369i −0.415415 − 0.909632i −1.70483 + 2.46851i 0.557150 − 2.16554i
89.9 0.142315 + 0.989821i −0.748393 − 1.56202i −0.959493 + 0.281733i −1.26417 + 1.84441i 1.43961 − 0.963074i −1.66558 + 1.07040i −0.415415 − 0.909632i −1.87982 + 2.33801i −2.00555 − 0.988819i
89.10 0.142315 + 0.989821i −0.730817 − 1.57032i −0.959493 + 0.281733i 2.23496 − 0.0702463i 1.45033 − 0.946859i 3.21136 − 2.06381i −0.415415 − 0.909632i −1.93181 + 2.29523i 0.387600 + 2.20222i
89.11 0.142315 + 0.989821i −0.277794 + 1.70963i −0.959493 + 0.281733i −1.06732 + 1.96490i −1.73176 − 0.0316608i 0.483302 − 0.310599i −0.415415 − 0.909632i −2.84566 − 0.949849i −2.09680 − 0.776821i
89.12 0.142315 + 0.989821i −0.0575173 − 1.73110i −0.959493 + 0.281733i 2.02703 + 0.944001i 1.70529 − 0.303292i −1.08447 + 0.696948i −0.415415 − 0.909632i −2.99338 + 0.199136i −0.645916 + 2.14075i
89.13 0.142315 + 0.989821i −0.0512012 + 1.73129i −0.959493 + 0.281733i −2.23188 + 0.136714i −1.72096 + 0.195709i −1.96759 + 1.26449i −0.415415 − 0.909632i −2.99476 − 0.177289i −0.452953 − 2.18971i
89.14 0.142315 + 0.989821i 0.397135 + 1.68591i −0.959493 + 0.281733i −1.22793 − 1.86874i −1.61223 + 0.633022i 4.08214 − 2.62343i −0.415415 − 0.909632i −2.68457 + 1.33906i 1.67496 − 1.48138i
89.15 0.142315 + 0.989821i 0.479764 − 1.66428i −0.959493 + 0.281733i 0.0269160 − 2.23591i 1.71562 + 0.238029i −2.74772 + 1.76585i −0.415415 − 0.909632i −2.53965 − 1.59692i 2.21698 − 0.291561i
89.16 0.142315 + 0.989821i 0.763868 + 1.55451i −0.959493 + 0.281733i 2.00799 + 0.983855i −1.42998 + 0.977323i −3.27194 + 2.10275i −0.415415 − 0.909632i −1.83301 + 2.37488i −0.688074 + 2.12757i
89.17 0.142315 + 0.989821i 0.943600 − 1.45245i −0.959493 + 0.281733i −2.20932 + 0.344845i 1.57196 + 0.727290i 2.74772 − 1.76585i −0.415415 − 0.909632i −1.21924 − 2.74107i −0.655753 − 2.13775i
89.18 0.142315 + 0.989821i 1.16282 + 1.28368i −0.959493 + 0.281733i −0.275295 − 2.21906i −1.10513 + 1.33367i −1.95813 + 1.25842i −0.415415 − 0.909632i −0.295684 + 2.98539i 2.15729 − 0.588298i
89.19 0.142315 + 0.989821i 1.34594 − 1.09016i −0.959493 + 0.281733i 1.22287 + 1.87206i 1.27061 + 1.17710i 1.08447 − 0.696948i −0.415415 − 0.909632i 0.623112 − 2.93458i −1.67897 + 1.47684i
89.20 0.142315 + 0.989821i 1.42695 + 0.981738i −0.959493 + 0.281733i 2.21959 + 0.270933i −0.768669 + 1.55214i 1.28454 − 0.825522i −0.415415 − 0.909632i 1.07238 + 2.80178i 0.0477058 + 2.23556i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
23.d odd 22 1 inner
345.n even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.n.b yes 240
3.b odd 2 1 690.2.n.a ✓ 240
5.b even 2 1 690.2.n.a ✓ 240
15.d odd 2 1 inner 690.2.n.b yes 240
23.d odd 22 1 inner 690.2.n.b yes 240
69.g even 22 1 690.2.n.a ✓ 240
115.i odd 22 1 690.2.n.a ✓ 240
345.n even 22 1 inner 690.2.n.b yes 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.n.a ✓ 240 3.b odd 2 1
690.2.n.a ✓ 240 5.b even 2 1
690.2.n.a ✓ 240 69.g even 22 1
690.2.n.a ✓ 240 115.i odd 22 1
690.2.n.b yes 240 1.a even 1 1 trivial
690.2.n.b yes 240 15.d odd 2 1 inner
690.2.n.b yes 240 23.d odd 22 1 inner
690.2.n.b yes 240 345.n even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{120} - 83 T_{17}^{118} - 176 T_{17}^{117} + 6090 T_{17}^{116} + 20438 T_{17}^{115} + \cdots + 23\!\cdots\!84 \) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\). Copy content Toggle raw display