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Properties

Label 686.2.g.e
Level $686$
Weight $2$
Character orbit 686.g
Analytic conductor $5.478$
Analytic rank $0$
Dimension $24$
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] = [686,2,Mod(67,686)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(686, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("686.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 686 = 2 \cdot 7^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 686.g (of order \(21\), degree \(12\), not 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: \(5.47773757866\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{21})\)
Twist minimal: no (minimal twist has level 98)
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} + 7 q^{3} + 2 q^{4} + 7 q^{6} - 4 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} + 7 q^{3} + 2 q^{4} + 7 q^{6} - 4 q^{8} + 19 q^{9} - 11 q^{11} + 14 q^{13} + 9 q^{15} + 2 q^{16} + 7 q^{17} - 9 q^{18} + 14 q^{19} + 7 q^{20} + q^{22} - 29 q^{23} - 8 q^{25} + 7 q^{26} + 7 q^{27} + 13 q^{29} - 8 q^{30} + 28 q^{31} + 2 q^{32} + 14 q^{33} + 7 q^{34} - 17 q^{36} + 20 q^{37} - 35 q^{38} + 56 q^{39} - 14 q^{40} - 28 q^{41} + 6 q^{43} + 3 q^{44} - 7 q^{45} + 34 q^{46} - 42 q^{47} - 14 q^{48} + 16 q^{50} + 32 q^{51} + 7 q^{52} - 60 q^{53} - 21 q^{54} + 14 q^{55} + 23 q^{57} + 18 q^{58} - 49 q^{59} + 6 q^{60} + 14 q^{61} - 4 q^{64} - 28 q^{65} - 21 q^{66} + 24 q^{67} + 14 q^{68} - 7 q^{69} + 6 q^{71} - 2 q^{72} + 35 q^{73} - 15 q^{74} + 56 q^{75} + 6 q^{79} + 14 q^{80} - 45 q^{81} + 14 q^{82} + 77 q^{83} - 33 q^{85} - 38 q^{86} - 63 q^{87} + 3 q^{88} + 21 q^{90} - 5 q^{92} - 38 q^{93} + 35 q^{94} + 86 q^{95} - 98 q^{97} - 106 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 0.955573 − 0.294755i −0.427316 − 1.08878i 0.826239 − 0.563320i 0.0821245 + 0.0123783i −0.729256 − 0.914459i 0 0.623490 − 0.781831i 1.19630 − 1.11001i 0.0821245 − 0.0123783i
67.2 0.955573 − 0.294755i 0.784496 + 1.99886i 0.826239 − 0.563320i 2.07983 + 0.313484i 1.33882 + 1.67883i 0 0.623490 − 0.781831i −1.18087 + 1.09568i 2.07983 − 0.313484i
79.1 −0.988831 + 0.149042i −1.88469 − 1.28496i 0.955573 − 0.294755i −0.269665 + 3.59844i 2.05516 + 0.989712i 0 −0.900969 + 0.433884i 0.804920 + 2.05090i −0.269665 − 3.59844i
79.2 −0.988831 + 0.149042i 0.934966 + 0.637449i 0.955573 − 0.294755i −0.0772188 + 1.03041i −1.01953 − 0.490980i 0 −0.900969 + 0.433884i −0.628203 − 1.60064i −0.0772188 − 1.03041i
165.1 −0.988831 − 0.149042i −1.88469 + 1.28496i 0.955573 + 0.294755i −0.269665 − 3.59844i 2.05516 − 0.989712i 0 −0.900969 − 0.433884i 0.804920 − 2.05090i −0.269665 + 3.59844i
165.2 −0.988831 − 0.149042i 0.934966 − 0.637449i 0.955573 + 0.294755i −0.0772188 − 1.03041i −1.01953 + 0.490980i 0 −0.900969 − 0.433884i −0.628203 + 1.60064i −0.0772188 + 1.03041i
177.1 −0.733052 − 0.680173i −1.10442 − 0.166465i 0.0747301 + 0.997204i −0.578784 − 1.47472i 0.696376 + 0.873227i 0 0.623490 − 0.781831i −1.67467 − 0.516569i −0.578784 + 1.47472i
177.2 −0.733052 − 0.680173i 2.81578 + 0.424410i 0.0747301 + 0.997204i −0.904721 − 2.30519i −1.77544 − 2.22633i 0 0.623490 − 0.781831i 4.88176 + 1.50582i −0.904721 + 2.30519i
263.1 0.826239 + 0.563320i −0.0584109 − 0.0541974i 0.365341 + 0.930874i 0.427005 − 0.131714i −0.0177309 − 0.0776840i 0 −0.222521 + 0.974928i −0.223716 − 2.98528i 0.427005 + 0.131714i
263.2 0.826239 + 0.563320i 2.19243 + 2.03428i 0.365341 + 0.930874i −3.17919 + 0.980650i 0.665522 + 2.91584i 0 −0.222521 + 0.974928i 0.444274 + 5.92843i −3.17919 − 0.980650i
275.1 0.0747301 − 0.997204i −1.95066 + 0.601698i −0.988831 − 0.149042i −0.958118 − 0.889004i 0.454243 + 1.99017i 0 −0.222521 + 0.974928i 0.964304 − 0.657452i −0.958118 + 0.889004i
275.2 0.0747301 − 0.997204i 2.39605 − 0.739084i −0.988831 − 0.149042i 2.18585 + 2.02817i −0.557960 − 2.44458i 0 −0.222521 + 0.974928i 2.71611 − 1.85181i 2.18585 − 2.02817i
373.1 0.826239 − 0.563320i −0.0584109 + 0.0541974i 0.365341 − 0.930874i 0.427005 + 0.131714i −0.0177309 + 0.0776840i 0 −0.222521 − 0.974928i −0.223716 + 2.98528i 0.427005 − 0.131714i
373.2 0.826239 − 0.563320i 2.19243 − 2.03428i 0.365341 − 0.930874i −3.17919 − 0.980650i 0.665522 − 2.91584i 0 −0.222521 − 0.974928i 0.444274 − 5.92843i −3.17919 + 0.980650i
459.1 0.0747301 + 0.997204i −1.95066 − 0.601698i −0.988831 + 0.149042i −0.958118 + 0.889004i 0.454243 − 1.99017i 0 −0.222521 − 0.974928i 0.964304 + 0.657452i −0.958118 − 0.889004i
459.2 0.0747301 + 0.997204i 2.39605 + 0.739084i −0.988831 + 0.149042i 2.18585 − 2.02817i −0.557960 + 2.44458i 0 −0.222521 − 0.974928i 2.71611 + 1.85181i 2.18585 + 2.02817i
471.1 0.955573 + 0.294755i −0.427316 + 1.08878i 0.826239 + 0.563320i 0.0821245 − 0.0123783i −0.729256 + 0.914459i 0 0.623490 + 0.781831i 1.19630 + 1.11001i 0.0821245 + 0.0123783i
471.2 0.955573 + 0.294755i 0.784496 − 1.99886i 0.826239 + 0.563320i 2.07983 − 0.313484i 1.33882 − 1.67883i 0 0.623490 + 0.781831i −1.18087 − 1.09568i 2.07983 + 0.313484i
557.1 0.365341 − 0.930874i −0.125927 + 1.68037i −0.733052 − 0.680173i −1.87725 − 1.27989i 1.51821 + 0.731131i 0 −0.900969 + 0.433884i 0.158698 + 0.0239199i −1.87725 + 1.27989i
557.2 0.365341 − 0.930874i −0.0722934 + 0.964688i −0.733052 − 0.680173i 3.07015 + 2.09319i 0.871591 + 0.419736i 0 −0.900969 + 0.433884i 2.04110 + 0.307646i 3.07015 − 2.09319i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 686.2.g.e 24
7.b odd 2 1 98.2.g.b ✓ 24
7.c even 3 1 686.2.e.d 24
7.c even 3 1 686.2.g.d 24
7.d odd 6 1 686.2.e.c 24
7.d odd 6 1 686.2.g.f 24
21.c even 2 1 882.2.z.b 24
28.d even 2 1 784.2.bg.b 24
49.e even 7 1 686.2.g.d 24
49.f odd 14 1 686.2.g.f 24
49.g even 21 1 686.2.e.d 24
49.g even 21 1 inner 686.2.g.e 24
49.g even 21 1 4802.2.a.l 12
49.h odd 42 1 98.2.g.b ✓ 24
49.h odd 42 1 686.2.e.c 24
49.h odd 42 1 4802.2.a.o 12
147.o even 42 1 882.2.z.b 24
196.p even 42 1 784.2.bg.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.g.b ✓ 24 7.b odd 2 1
98.2.g.b ✓ 24 49.h odd 42 1
686.2.e.c 24 7.d odd 6 1
686.2.e.c 24 49.h odd 42 1
686.2.e.d 24 7.c even 3 1
686.2.e.d 24 49.g even 21 1
686.2.g.d 24 7.c even 3 1
686.2.g.d 24 49.e even 7 1
686.2.g.e 24 1.a even 1 1 trivial
686.2.g.e 24 49.g even 21 1 inner
686.2.g.f 24 7.d odd 6 1
686.2.g.f 24 49.f odd 14 1
784.2.bg.b 24 28.d even 2 1
784.2.bg.b 24 196.p even 42 1
882.2.z.b 24 21.c even 2 1
882.2.z.b 24 147.o even 42 1
4802.2.a.l 12 49.g even 21 1
4802.2.a.o 12 49.h odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 7 T_{3}^{23} + 12 T_{3}^{22} + 28 T_{3}^{21} - 105 T_{3}^{20} - 63 T_{3}^{19} + 617 T_{3}^{18} + \cdots + 1681 \) acting on \(S_{2}^{\mathrm{new}}(686, [\chi])\). Copy content Toggle raw display