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Properties

Label 690.2.q.b
Level $690$
Weight $2$
Character orbit 690.q
Analytic conductor $5.510$
Analytic rank $0$
Dimension $160$
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] = [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.

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 16 q^{4} + 16 q^{5} - 2 q^{6} - 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + 16 q^{4} + 16 q^{5} - 2 q^{6} - 46 q^{9} + 12 q^{11} + 12 q^{14} - 16 q^{16} - 8 q^{18} - 16 q^{20} + 70 q^{21} - 4 q^{23} + 2 q^{24} - 16 q^{25} + 42 q^{27} + 2 q^{30} - 4 q^{31} - 16 q^{33} + 2 q^{36} - 72 q^{38} + 140 q^{39} - 44 q^{41} + 44 q^{43} - 12 q^{44} + 2 q^{45} + 4 q^{46} + 70 q^{49} + 2 q^{51} + 52 q^{53} - 62 q^{54} + 10 q^{55} + 54 q^{56} - 94 q^{57} - 36 q^{58} - 44 q^{61} + 16 q^{64} - 54 q^{66} - 44 q^{67} - 30 q^{69} - 12 q^{70} - 36 q^{72} - 28 q^{73} + 24 q^{74} + 88 q^{77} - 54 q^{78} - 44 q^{79} + 16 q^{80} - 66 q^{81} - 28 q^{82} - 4 q^{83} - 4 q^{84} - 158 q^{86} + 156 q^{87} - 80 q^{89} + 8 q^{90} + 4 q^{92} + 4 q^{93} + 24 q^{94} - 2 q^{96} + 88 q^{98} - 58 q^{99}+O(q^{100}) \) 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 −1.68349 − 0.407274i −0.415415 + 0.909632i 0.959493 − 0.281733i 0.567541 + 1.63643i 0.0904807 − 0.0784019i 0.989821 − 0.142315i 2.66826 + 1.37128i −0.755750 − 0.654861i
11.2 −0.540641 − 0.841254i −1.23589 + 1.21350i −0.415415 + 0.909632i 0.959493 − 0.281733i 1.68903 + 0.383629i −1.57741 + 1.36683i 0.989821 − 0.142315i 0.0548407 − 2.99950i −0.755750 − 0.654861i
11.3 −0.540641 − 0.841254i −1.01936 − 1.40032i −0.415415 + 0.909632i 0.959493 − 0.281733i −0.626917 + 1.61461i 2.43170 − 2.10708i 0.989821 − 0.142315i −0.921804 + 2.85487i −0.755750 − 0.654861i
11.4 −0.540641 − 0.841254i −0.329943 + 1.70033i −0.415415 + 0.909632i 0.959493 − 0.281733i 1.60879 − 0.641705i 0.461235 − 0.399662i 0.989821 − 0.142315i −2.78228 − 1.12203i −0.755750 − 0.654861i
11.5 −0.540641 − 0.841254i −0.0752308 − 1.73042i −0.415415 + 0.909632i 0.959493 − 0.281733i −1.41505 + 0.998822i −2.94981 + 2.55602i 0.989821 − 0.142315i −2.98868 + 0.260361i −0.755750 − 0.654861i
11.6 −0.540641 − 0.841254i 0.869657 + 1.49790i −0.415415 + 0.909632i 0.959493 − 0.281733i 0.789940 − 1.54143i 3.70222 − 3.20799i 0.989821 − 0.142315i −1.48739 + 2.60531i −0.755750 − 0.654861i
11.7 −0.540641 − 0.841254i 0.989774 − 1.42139i −0.415415 + 0.909632i 0.959493 − 0.281733i −1.73086 − 0.0641895i −0.315364 + 0.273265i 0.989821 − 0.142315i −1.04070 − 2.81371i −0.755750 − 0.654861i
11.8 −0.540641 − 0.841254i 1.72873 − 0.107190i −0.415415 + 0.909632i 0.959493 − 0.281733i −1.02480 − 1.39635i 1.12640 − 0.976034i 0.989821 − 0.142315i 2.97702 − 0.370607i −0.755750 − 0.654861i
11.9 0.540641 + 0.841254i −1.68374 + 0.406235i −0.415415 + 0.909632i 0.959493 − 0.281733i −1.25204 − 1.19682i 0.920755 − 0.797839i −0.989821 + 0.142315i 2.66995 − 1.36799i 0.755750 + 0.654861i
11.10 0.540641 + 0.841254i −1.50413 + 0.858826i −0.415415 + 0.909632i 0.959493 − 0.281733i −1.53569 − 0.801041i −1.03609 + 0.897775i −0.989821 + 0.142315i 1.52484 − 2.58358i 0.755750 + 0.654861i
11.11 0.540641 + 0.841254i −1.16057 − 1.28572i −0.415415 + 0.909632i 0.959493 − 0.281733i 0.454160 − 1.67145i −1.93677 + 1.67822i −0.989821 + 0.142315i −0.306134 + 2.98434i 0.755750 + 0.654861i
11.12 0.540641 + 0.841254i 0.244221 + 1.71475i −0.415415 + 0.909632i 0.959493 − 0.281733i −1.31050 + 1.13251i 3.34933 − 2.90221i −0.989821 + 0.142315i −2.88071 + 0.837554i 0.755750 + 0.654861i
11.13 0.540641 + 0.841254i 0.416404 − 1.68125i −0.415415 + 0.909632i 0.959493 − 0.281733i 1.63948 − 0.558652i 1.57600 − 1.36561i −0.989821 + 0.142315i −2.65322 − 1.40016i 0.755750 + 0.654861i
11.14 0.540641 + 0.841254i 1.36664 + 1.06409i −0.415415 + 0.909632i 0.959493 − 0.281733i −0.156311 + 1.72498i −3.41511 + 2.95921i −0.989821 + 0.142315i 0.735409 + 2.90847i 0.755750 + 0.654861i
11.15 0.540641 + 0.841254i 1.43420 − 0.971113i −0.415415 + 0.909632i 0.959493 − 0.281733i 1.59234 + 0.681505i −3.34609 + 2.89940i −0.989821 + 0.142315i 1.11388 − 2.78555i 0.755750 + 0.654861i
11.16 0.540641 + 0.841254i 1.64273 + 0.549041i −0.415415 + 0.909632i 0.959493 − 0.281733i 0.426243 + 1.67878i 0.918508 − 0.795892i −0.989821 + 0.142315i 2.39711 + 1.80385i 0.755750 + 0.654861i
191.1 −0.755750 − 0.654861i −1.58079 − 0.707884i 0.142315 + 0.989821i −0.415415 − 0.909632i 0.731117 + 1.57018i 0.325415 + 1.10826i 0.540641 − 0.841254i 1.99780 + 2.23803i −0.281733 + 0.959493i
191.2 −0.755750 − 0.654861i −1.30022 + 1.14430i 0.142315 + 0.989821i −0.415415 − 0.909632i 1.73200 − 0.0133460i 0.519422 + 1.76899i 0.540641 − 0.841254i 0.381134 − 2.97569i −0.281733 + 0.959493i
191.3 −0.755750 − 0.654861i −0.982602 + 1.42636i 0.142315 + 0.989821i −0.415415 − 0.909632i 1.67667 − 0.434500i −0.819457 − 2.79081i 0.540641 − 0.841254i −1.06898 − 2.80308i −0.281733 + 0.959493i
191.4 −0.755750 − 0.654861i −0.340196 − 1.69831i 0.142315 + 0.989821i −0.415415 − 0.909632i −0.855055 + 1.50628i 0.680617 + 2.31797i 0.540641 − 0.841254i −2.76853 + 1.15552i −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.b yes 160
3.b odd 2 1 690.2.q.a ✓ 160
23.d odd 22 1 690.2.q.a ✓ 160
69.g even 22 1 inner 690.2.q.b yes 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.q.a ✓ 160 3.b odd 2 1
690.2.q.a ✓ 160 23.d odd 22 1
690.2.q.b yes 160 1.a even 1 1 trivial
690.2.q.b yes 160 69.g even 22 1 inner

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