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Properties

Label 690.2.m.h
Level $690$
Weight $2$
Character orbit 690.m
Analytic conductor $5.510$
Analytic rank $0$
Dimension $30$
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(31,690)]
 
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([0, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("690.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.m (of order \(11\), 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: \(30\)
Relative dimension: \(3\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{2} - 3 q^{3} - 3 q^{4} + 3 q^{5} + 3 q^{6} + 8 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{2} - 3 q^{3} - 3 q^{4} + 3 q^{5} + 3 q^{6} + 8 q^{7} + 3 q^{8} - 3 q^{9} - 3 q^{10} - 18 q^{11} - 3 q^{12} + 13 q^{13} - 8 q^{14} + 3 q^{15} - 3 q^{16} - 6 q^{17} + 3 q^{18} + 4 q^{19} + 3 q^{20} - 3 q^{21} - 4 q^{22} - 23 q^{23} - 30 q^{24} - 3 q^{25} + 9 q^{26} - 3 q^{27} + 8 q^{28} + 18 q^{29} - 3 q^{30} - 8 q^{31} + 3 q^{32} + 4 q^{33} - 5 q^{34} + 3 q^{35} - 3 q^{36} - 32 q^{37} - 15 q^{38} + 13 q^{39} - 3 q^{40} + 35 q^{41} + 3 q^{42} + 48 q^{43} - 18 q^{44} - 30 q^{45} + q^{46} + 8 q^{47} - 3 q^{48} - 11 q^{49} + 3 q^{50} + 27 q^{51} + 2 q^{52} + 26 q^{53} + 3 q^{54} - 4 q^{55} - 8 q^{56} - 29 q^{57} - 7 q^{58} + 55 q^{59} + 3 q^{60} + 21 q^{61} + 8 q^{62} - 3 q^{63} - 3 q^{64} - 2 q^{65} + 7 q^{66} + 4 q^{67} - 28 q^{68} - 45 q^{69} - 14 q^{70} - 41 q^{71} + 3 q^{72} - 39 q^{73} + 32 q^{74} - 3 q^{75} + 4 q^{76} - 33 q^{77} - 2 q^{78} + 18 q^{79} + 3 q^{80} - 3 q^{81} + 31 q^{82} - 85 q^{83} - 3 q^{84} - 5 q^{85} + 40 q^{86} + 18 q^{87} - 15 q^{88} + 43 q^{89} - 3 q^{90} + 38 q^{91} + 10 q^{92} + 36 q^{93} - 19 q^{94} - 15 q^{95} + 3 q^{96} + 43 q^{97} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0.142315 − 0.989821i 0.415415 + 0.909632i −0.959493 − 0.281733i 0.654861 + 0.755750i 0.959493 − 0.281733i −3.23498 − 2.07900i −0.415415 + 0.909632i −0.654861 + 0.755750i 0.841254 − 0.540641i
31.2 0.142315 − 0.989821i 0.415415 + 0.909632i −0.959493 − 0.281733i 0.654861 + 0.755750i 0.959493 − 0.281733i 0.208237 + 0.133826i −0.415415 + 0.909632i −0.654861 + 0.755750i 0.841254 − 0.540641i
31.3 0.142315 − 0.989821i 0.415415 + 0.909632i −0.959493 − 0.281733i 0.654861 + 0.755750i 0.959493 − 0.281733i 2.49910 + 1.60607i −0.415415 + 0.909632i −0.654861 + 0.755750i 0.841254 − 0.540641i
121.1 −0.415415 + 0.909632i −0.959493 − 0.281733i −0.654861 − 0.755750i −0.841254 + 0.540641i 0.654861 − 0.755750i −0.588355 + 4.09210i 0.959493 − 0.281733i 0.841254 + 0.540641i −0.142315 − 0.989821i
121.2 −0.415415 + 0.909632i −0.959493 − 0.281733i −0.654861 − 0.755750i −0.841254 + 0.540641i 0.654861 − 0.755750i −0.0949893 + 0.660665i 0.959493 − 0.281733i 0.841254 + 0.540641i −0.142315 − 0.989821i
121.3 −0.415415 + 0.909632i −0.959493 − 0.281733i −0.654861 − 0.755750i −0.841254 + 0.540641i 0.654861 − 0.755750i 0.626720 − 4.35894i 0.959493 − 0.281733i 0.841254 + 0.540641i −0.142315 − 0.989821i
151.1 0.654861 + 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i −0.415415 + 0.909632i 0.142315 + 0.989821i −4.28731 − 1.25887i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.959493 + 0.281733i
151.2 0.654861 + 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i −0.415415 + 0.909632i 0.142315 + 0.989821i 1.94609 + 0.571424i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.959493 + 0.281733i
151.3 0.654861 + 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i −0.415415 + 0.909632i 0.142315 + 0.989821i 4.34479 + 1.27575i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.959493 + 0.281733i
211.1 −0.415415 − 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i −0.841254 − 0.540641i 0.654861 + 0.755750i −0.588355 − 4.09210i 0.959493 + 0.281733i 0.841254 − 0.540641i −0.142315 + 0.989821i
211.2 −0.415415 − 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i −0.841254 − 0.540641i 0.654861 + 0.755750i −0.0949893 − 0.660665i 0.959493 + 0.281733i 0.841254 − 0.540641i −0.142315 + 0.989821i
211.3 −0.415415 − 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i −0.841254 − 0.540641i 0.654861 + 0.755750i 0.626720 + 4.35894i 0.959493 + 0.281733i 0.841254 − 0.540641i −0.142315 + 0.989821i
271.1 0.959493 + 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i 0.142315 − 0.989821i −0.841254 + 0.540641i −1.89055 − 4.13973i 0.654861 + 0.755750i −0.142315 − 0.989821i 0.415415 − 0.909632i
271.2 0.959493 + 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i 0.142315 − 0.989821i −0.841254 + 0.540641i 0.0700250 + 0.153333i 0.654861 + 0.755750i −0.142315 − 0.989821i 0.415415 − 0.909632i
271.3 0.959493 + 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i 0.142315 − 0.989821i −0.841254 + 0.540641i 1.63201 + 3.57361i 0.654861 + 0.755750i −0.142315 − 0.989821i 0.415415 − 0.909632i
301.1 −0.841254 − 0.540641i −0.142315 − 0.989821i 0.415415 + 0.909632i 0.959493 + 0.281733i −0.415415 + 0.909632i −1.50479 + 1.73662i 0.142315 − 0.989821i −0.959493 + 0.281733i −0.654861 − 0.755750i
301.2 −0.841254 − 0.540641i −0.142315 − 0.989821i 0.415415 + 0.909632i 0.959493 + 0.281733i −0.415415 + 0.909632i 1.32702 − 1.53147i 0.142315 − 0.989821i −0.959493 + 0.281733i −0.654861 − 0.755750i
301.3 −0.841254 − 0.540641i −0.142315 − 0.989821i 0.415415 + 0.909632i 0.959493 + 0.281733i −0.415415 + 0.909632i 2.94698 − 3.40099i 0.142315 − 0.989821i −0.959493 + 0.281733i −0.654861 − 0.755750i
331.1 0.959493 − 0.281733i −0.654861 − 0.755750i 0.841254 − 0.540641i 0.142315 + 0.989821i −0.841254 − 0.540641i −1.89055 + 4.13973i 0.654861 − 0.755750i −0.142315 + 0.989821i 0.415415 + 0.909632i
331.2 0.959493 − 0.281733i −0.654861 − 0.755750i 0.841254 − 0.540641i 0.142315 + 0.989821i −0.841254 − 0.540641i 0.0700250 − 0.153333i 0.654861 − 0.755750i −0.142315 + 0.989821i 0.415415 + 0.909632i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.m.h ✓ 30
23.c even 11 1 inner 690.2.m.h ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.m.h ✓ 30 1.a even 1 1 trivial
690.2.m.h ✓ 30 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{30} - 8 T_{7}^{29} + 48 T_{7}^{28} - 190 T_{7}^{27} + 532 T_{7}^{26} - 523 T_{7}^{25} + \cdots + 7930971136 \) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\). Copy content Toggle raw display