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Properties

Label 690.3.k.b
Level $690$
Weight $3$
Character orbit 690.k
Analytic conductor $18.801$
Analytic rank $0$
Dimension $48$
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] = [690,3,Mod(277,690)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(690, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("690.277");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 690.k (of order \(4\), degree \(2\), 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: \(18.8011382409\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{2} - 8 q^{5} - 8 q^{7} + 96 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{2} - 8 q^{5} - 8 q^{7} + 96 q^{8} + 8 q^{10} - 32 q^{11} - 24 q^{13} + 24 q^{15} - 192 q^{16} + 72 q^{17} - 144 q^{18} + 32 q^{22} + 24 q^{25} + 48 q^{26} + 16 q^{28} - 24 q^{30} + 24 q^{31} + 192 q^{32} - 24 q^{33} + 288 q^{36} - 128 q^{37} - 16 q^{38} - 16 q^{40} - 40 q^{41} + 48 q^{43} - 136 q^{47} - 80 q^{50} - 48 q^{52} + 144 q^{53} - 144 q^{55} - 32 q^{56} + 96 q^{57} + 8 q^{58} + 128 q^{61} - 24 q^{62} - 24 q^{63} + 184 q^{65} + 48 q^{66} - 144 q^{68} + 40 q^{70} - 40 q^{71} - 288 q^{72} + 40 q^{73} - 72 q^{75} + 32 q^{76} - 104 q^{77} + 96 q^{78} + 32 q^{80} - 432 q^{81} + 40 q^{82} - 88 q^{85} - 96 q^{86} + 120 q^{87} - 64 q^{88} + 24 q^{90} + 144 q^{91} - 96 q^{93} + 312 q^{95} + 480 q^{97} + 584 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1 −1.00000 − 1.00000i −1.22474 + 1.22474i 2.00000i 4.77937 − 1.46889i 2.44949 8.57539 + 8.57539i 2.00000 − 2.00000i − 3.00000i −6.24826 − 3.31048i
277.2 −1.00000 − 1.00000i −1.22474 + 1.22474i 2.00000i −4.35673 + 2.45335i 2.44949 −3.40648 − 3.40648i 2.00000 − 2.00000i − 3.00000i 6.81008 + 1.90338i
277.3 −1.00000 − 1.00000i −1.22474 + 1.22474i 2.00000i −2.46815 + 4.34836i 2.44949 −1.62328 − 1.62328i 2.00000 − 2.00000i − 3.00000i 6.81651 − 1.88022i
277.4 −1.00000 − 1.00000i −1.22474 + 1.22474i 2.00000i 3.38334 + 3.68144i 2.44949 −2.06971 − 2.06971i 2.00000 − 2.00000i − 3.00000i 0.298094 − 7.06478i
277.5 −1.00000 − 1.00000i −1.22474 + 1.22474i 2.00000i 0.0428394 − 4.99982i 2.44949 0.599576 + 0.599576i 2.00000 − 2.00000i − 3.00000i −5.04266 + 4.95698i
277.6 −1.00000 − 1.00000i −1.22474 + 1.22474i 2.00000i −3.65385 − 3.41312i 2.44949 0.283627 + 0.283627i 2.00000 − 2.00000i − 3.00000i 0.240725 + 7.06697i
277.7 −1.00000 − 1.00000i −1.22474 + 1.22474i 2.00000i 4.29808 − 2.55470i 2.44949 4.10164 + 4.10164i 2.00000 − 2.00000i − 3.00000i −6.85278 − 1.74338i
277.8 −1.00000 − 1.00000i −1.22474 + 1.22474i 2.00000i −2.55939 + 4.29529i 2.44949 5.47277 + 5.47277i 2.00000 − 2.00000i − 3.00000i 6.85468 − 1.73590i
277.9 −1.00000 − 1.00000i −1.22474 + 1.22474i 2.00000i 4.00430 + 2.99426i 2.44949 −4.65996 − 4.65996i 2.00000 − 2.00000i − 3.00000i −1.01004 − 6.99856i
277.10 −1.00000 − 1.00000i −1.22474 + 1.22474i 2.00000i −4.46596 − 2.24837i 2.44949 −8.35401 − 8.35401i 2.00000 − 2.00000i − 3.00000i 2.21759 + 6.71434i
277.11 −1.00000 − 1.00000i −1.22474 + 1.22474i 2.00000i −4.94131 − 0.763816i 2.44949 8.62845 + 8.62845i 2.00000 − 2.00000i − 3.00000i 4.17750 + 5.70513i
277.12 −1.00000 − 1.00000i −1.22474 + 1.22474i 2.00000i 1.48797 − 4.77346i 2.44949 −9.54800 − 9.54800i 2.00000 − 2.00000i − 3.00000i −6.26143 + 3.28550i
277.13 −1.00000 − 1.00000i 1.22474 − 1.22474i 2.00000i −4.51369 + 2.15095i −2.44949 −8.14300 − 8.14300i 2.00000 − 2.00000i − 3.00000i 6.66464 + 2.36274i
277.14 −1.00000 − 1.00000i 1.22474 − 1.22474i 2.00000i −1.86672 + 4.63846i −2.44949 2.71586 + 2.71586i 2.00000 − 2.00000i − 3.00000i 6.50519 − 2.77174i
277.15 −1.00000 − 1.00000i 1.22474 − 1.22474i 2.00000i −4.84935 + 1.21811i −2.44949 −0.117427 − 0.117427i 2.00000 − 2.00000i − 3.00000i 6.06746 + 3.63124i
277.16 −1.00000 − 1.00000i 1.22474 − 1.22474i 2.00000i 3.93589 − 3.08362i −2.44949 −1.42240 − 1.42240i 2.00000 − 2.00000i − 3.00000i −7.01952 − 0.852269i
277.17 −1.00000 − 1.00000i 1.22474 − 1.22474i 2.00000i 2.08074 − 4.54648i −2.44949 −1.45838 − 1.45838i 2.00000 − 2.00000i − 3.00000i −6.62723 + 2.46574i
277.18 −1.00000 − 1.00000i 1.22474 − 1.22474i 2.00000i −2.00232 − 4.58156i −2.44949 −5.25476 − 5.25476i 2.00000 − 2.00000i − 3.00000i −2.57924 + 6.58388i
277.19 −1.00000 − 1.00000i 1.22474 − 1.22474i 2.00000i 0.815565 − 4.93304i −2.44949 5.39766 + 5.39766i 2.00000 − 2.00000i − 3.00000i −5.74860 + 4.11747i
277.20 −1.00000 − 1.00000i 1.22474 − 1.22474i 2.00000i −4.82966 − 1.29396i −2.44949 5.84802 + 5.84802i 2.00000 − 2.00000i − 3.00000i 3.53570 + 6.12363i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.3.k.b ✓ 48
5.c odd 4 1 inner 690.3.k.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.3.k.b ✓ 48 1.a even 1 1 trivial
690.3.k.b ✓ 48 5.c odd 4 1 inner