↑

Properties

Label 690.2.q.a
Level $690$
Weight $2$
Character orbit 690.q
Analytic conductor $5.510$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [690,2,Mod(11,690)]
 
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, 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("690.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.q (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: \(160\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 16 q^{4} - 16 q^{5} - 2 q^{6} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + 16 q^{4} - 16 q^{5} - 2 q^{6} + 42 q^{9} - 12 q^{11} - 12 q^{14} - 16 q^{16} - 8 q^{18} + 16 q^{20} + 62 q^{21} + 4 q^{23} + 2 q^{24} - 16 q^{25} + 42 q^{27} - 2 q^{30} - 4 q^{31} + 16 q^{33} + 2 q^{36} + 72 q^{38} - 124 q^{39} + 44 q^{41} + 44 q^{43} + 12 q^{44} - 2 q^{45} + 4 q^{46} + 70 q^{49} - 2 q^{51} - 52 q^{53} + 92 q^{54} + 10 q^{55} - 54 q^{56} - 38 q^{57} - 36 q^{58} - 44 q^{61} - 220 q^{63} + 16 q^{64} - 34 q^{66} - 44 q^{67} + 22 q^{69} - 12 q^{70} - 36 q^{72} - 28 q^{73} - 24 q^{74} - 88 q^{77} - 54 q^{78} - 44 q^{79} - 16 q^{80} - 66 q^{81} - 28 q^{82} + 4 q^{83} - 18 q^{84} + 158 q^{86} - 64 q^{87} + 80 q^{89} - 8 q^{90} - 4 q^{92} + 4 q^{93} + 24 q^{94} - 2 q^{96} - 88 q^{98} + 190 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −0.540641 − 0.841254i −1.73087 + 0.0639912i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.989611 + 1.42150i 0.918508 − 0.795892i 0.989821 − 0.142315i 2.99181 − 0.221521i 0.755750 + 0.654861i
11.2 −0.540641 − 0.841254i −1.61107 + 0.635963i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.40602 + 1.01149i −3.41511 + 2.95921i 0.989821 − 0.142315i 2.19110 − 2.04916i 0.755750 + 0.654861i
11.3 −0.540641 − 0.841254i −1.10251 − 1.33584i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.527715 + 1.64970i −3.34609 + 2.89940i 0.989821 − 0.142315i −0.568927 + 2.94556i 0.755750 + 0.654861i
11.4 −0.540641 − 0.841254i −0.717428 + 1.57648i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.71409 − 0.248772i 3.34933 − 2.90221i 0.989821 − 0.142315i −1.97059 − 2.26203i 0.755750 + 0.654861i
11.5 −0.540641 − 0.841254i 0.0741266 − 1.73046i −0.415415 + 0.909632i −0.959493 + 0.281733i −1.49583 + 0.873200i 1.57600 − 1.36561i 0.989821 − 0.142315i −2.98901 − 0.256547i 0.755750 + 0.654861i
11.6 −0.540641 − 0.841254i 1.20125 + 1.24780i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.400274 − 1.68516i −1.03609 + 0.897775i 0.989821 − 0.142315i −0.114014 + 2.99783i 0.755750 + 0.654861i
11.7 −0.540641 − 0.841254i 1.47579 − 0.906664i −0.415415 + 0.909632i −0.959493 + 0.281733i −1.56061 − 0.751335i −1.93677 + 1.67822i 0.989821 − 0.142315i 1.35592 − 2.67609i 0.755750 + 0.654861i
11.8 −0.540641 − 0.841254i 1.50109 + 0.864143i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.0845842 − 1.72998i 0.920755 − 0.797839i 0.989821 − 0.142315i 1.50651 + 2.59431i 0.755750 + 0.654861i
11.9 0.540641 + 0.841254i −1.62851 − 0.589888i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.384191 − 1.68890i 1.12640 − 0.976034i −0.989821 + 0.142315i 2.30406 + 1.92127i −0.755750 − 0.654861i
11.10 0.540641 + 0.841254i −1.25644 + 1.19221i −0.415415 + 0.909632i −0.959493 + 0.281733i −1.68223 − 0.412423i 3.70222 − 3.20799i −0.989821 + 0.142315i 0.157263 − 2.99588i −0.755750 − 0.654861i
11.11 0.540641 + 0.841254i −0.549229 − 1.64266i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.08496 − 1.35013i −0.315364 + 0.273265i −0.989821 + 0.142315i −2.39669 + 1.80440i −0.755750 − 0.654861i
11.12 0.540641 + 0.841254i −0.162462 + 1.72441i −0.415415 + 0.909632i −0.959493 + 0.281733i −1.53850 + 0.795618i 0.461235 − 0.399662i −0.989821 + 0.142315i −2.94721 − 0.560302i −0.755750 − 0.654861i
11.13 0.540641 + 0.841254i 0.559698 − 1.63913i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.68152 − 0.415331i −2.94981 + 2.55602i −0.989821 + 0.142315i −2.37348 − 1.83483i −0.755750 − 0.654861i
11.14 0.540641 + 0.841254i 0.843944 + 1.51253i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.816154 + 1.52771i −1.57741 + 1.36683i −0.989821 + 0.142315i −1.57552 + 2.55299i −0.755750 − 0.654861i
11.15 0.540641 + 0.841254i 1.37259 − 1.05641i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.63079 + 0.583554i 2.43170 − 2.10708i −0.989821 + 0.142315i 0.767988 − 2.90003i −0.755750 − 0.654861i
11.16 0.540641 + 0.841254i 1.73004 + 0.0835168i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.865069 + 1.50055i 0.0904807 − 0.0784019i −0.989821 + 0.142315i 2.98605 + 0.288974i −0.755750 − 0.654861i
191.1 −0.755750 − 0.654861i −1.72862 + 0.108947i 0.142315 + 0.989821i 0.415415 + 0.909632i 1.37775 + 1.04967i −0.501114 − 1.70664i 0.540641 − 0.841254i 2.97626 − 0.376654i 0.281733 − 0.959493i
191.2 −0.755750 − 0.654861i −1.23307 − 1.21636i 0.142315 + 0.989821i 0.415415 + 0.909632i 0.135347 + 1.72675i 0.0399458 + 0.136043i 0.540641 − 0.841254i 0.0409354 + 2.99972i 0.281733 − 0.959493i
191.3 −0.755750 − 0.654861i −0.906618 + 1.47582i 0.142315 + 0.989821i 0.415415 + 0.909632i 1.65163 − 0.521642i −0.302591 − 1.03053i 0.540641 − 0.841254i −1.35609 − 2.67601i 0.281733 − 0.959493i
191.4 −0.755750 − 0.654861i 0.359391 − 1.69435i 0.142315 + 0.989821i 0.415415 + 0.909632i −1.38118 + 1.04516i 0.816450 + 2.78057i 0.540641 − 0.841254i −2.74168 − 1.21787i 0.281733 − 0.959493i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
69.g 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.q.a ✓ 160
3.b odd 2 1 690.2.q.b yes 160
23.d odd 22 1 690.2.q.b yes 160
69.g even 22 1 inner 690.2.q.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.q.a ✓ 160 1.a even 1 1 trivial
690.2.q.a ✓ 160 69.g even 22 1 inner
690.2.q.b yes 160 3.b odd 2 1
690.2.q.b yes 160 23.d odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{160} + 12 T_{11}^{159} + 166 T_{11}^{158} + 1020 T_{11}^{157} + 8067 T_{11}^{156} + \cdots + 12\!\cdots\!44 \) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\). Copy content Toggle raw display