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Properties

Label 686.2.g.h
Level $686$
Weight $2$
Character orbit 686.g
Analytic conductor $5.478$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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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: \(36\)
Relative dimension: \(3\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 3 q^{2} + 5 q^{3} + 3 q^{4} + 10 q^{6} + 6 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 3 q^{2} + 5 q^{3} + 3 q^{4} + 10 q^{6} + 6 q^{8} + 10 q^{9} - 7 q^{11} - 2 q^{12} - 20 q^{13} + 14 q^{15} + 3 q^{16} - q^{17} + 32 q^{18} + 44 q^{19} + 14 q^{20} + 14 q^{22} - 21 q^{23} + 2 q^{24} - q^{25} - 3 q^{26} + 20 q^{27} + 22 q^{29} + 24 q^{31} - 3 q^{32} + 14 q^{33} - 2 q^{34} - 20 q^{36} + 13 q^{37} + 19 q^{38} + 3 q^{39} - 7 q^{40} + 16 q^{41} - 48 q^{43} - 98 q^{45} - 21 q^{46} - 40 q^{47} + 4 q^{48} - 100 q^{50} - 36 q^{51} - 18 q^{52} - 10 q^{53} - 4 q^{54} + 98 q^{55} + 38 q^{57} - 24 q^{58} - 13 q^{59} - 7 q^{60} - 27 q^{61} - 22 q^{62} - 6 q^{64} + 21 q^{66} + 86 q^{67} + 34 q^{68} - 182 q^{69} - 3 q^{72} - 5 q^{73} - 13 q^{74} + 3 q^{75} - 4 q^{76} + 76 q^{78} + 66 q^{79} + 2 q^{81} + 8 q^{82} + 110 q^{83} - 98 q^{85} + 18 q^{86} + 110 q^{87} - 62 q^{89} + 42 q^{90} - 42 q^{92} - 46 q^{93} - 23 q^{94} + 7 q^{95} + 2 q^{96} - 64 q^{97} + 28 q^{99}+O(q^{100}) \) 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 −1.12318 − 2.86182i 0.826239 − 0.563320i 2.99159 + 0.450910i 1.91682 + 2.40361i 0 −0.623490 + 0.781831i −4.72932 + 4.38817i −2.99159 + 0.450910i
67.2 −0.955573 + 0.294755i −0.137179 − 0.349528i 0.826239 − 0.563320i −2.11767 − 0.319187i 0.234110 + 0.293565i 0 −0.623490 + 0.781831i 2.09580 − 1.94462i 2.11767 − 0.319187i
67.3 −0.955573 + 0.294755i 0.804788 + 2.05057i 0.826239 − 0.563320i 0.467821 + 0.0705127i −1.37345 − 1.72225i 0 −0.623490 + 0.781831i −1.35798 + 1.26002i −0.467821 + 0.0705127i
79.1 0.988831 − 0.149042i −1.77209 − 1.20819i 0.955573 − 0.294755i −0.100752 + 1.34445i −1.93237 − 0.930578i 0 0.900969 − 0.433884i 0.584550 + 1.48941i 0.100752 + 1.34445i
79.2 0.988831 − 0.149042i 1.27760 + 0.871050i 0.955573 − 0.294755i 0.270182 − 3.60533i 1.39315 + 0.670905i 0 0.900969 − 0.433884i −0.222500 − 0.566921i −0.270182 − 3.60533i
79.3 0.988831 − 0.149042i 1.98332 + 1.35221i 0.955573 − 0.294755i −0.295874 + 3.94817i 2.16271 + 1.04150i 0 0.900969 − 0.433884i 1.00908 + 2.57110i 0.295874 + 3.94817i
165.1 0.988831 + 0.149042i −1.77209 + 1.20819i 0.955573 + 0.294755i −0.100752 − 1.34445i −1.93237 + 0.930578i 0 0.900969 + 0.433884i 0.584550 − 1.48941i 0.100752 − 1.34445i
165.2 0.988831 + 0.149042i 1.27760 − 0.871050i 0.955573 + 0.294755i 0.270182 + 3.60533i 1.39315 − 0.670905i 0 0.900969 + 0.433884i −0.222500 + 0.566921i −0.270182 + 3.60533i
165.3 0.988831 + 0.149042i 1.98332 − 1.35221i 0.955573 + 0.294755i −0.295874 − 3.94817i 2.16271 − 1.04150i 0 0.900969 + 0.433884i 1.00908 − 2.57110i 0.295874 − 3.94817i
177.1 0.733052 + 0.680173i −2.17824 − 0.328316i 0.0747301 + 0.997204i −0.172845 − 0.440402i −1.37345 − 1.72225i 0 −0.623490 + 0.781831i 1.77021 + 0.546036i 0.172845 − 0.440402i
177.2 0.733052 + 0.680173i 0.371290 + 0.0559629i 0.0747301 + 0.997204i 0.782411 + 1.99355i 0.234110 + 0.293565i 0 −0.623490 + 0.781831i −2.73199 − 0.842709i −0.782411 + 1.99355i
177.3 0.733052 + 0.680173i 3.04000 + 0.458206i 0.0747301 + 0.997204i −1.10530 − 2.81625i 1.91682 + 2.40361i 0 −0.623490 + 0.781831i 6.16492 + 1.90163i 1.10530 − 2.81625i
263.1 −0.826239 − 0.563320i −1.91561 − 1.77743i 0.365341 + 0.930874i 2.16774 − 0.668658i 0.581491 + 2.54768i 0 0.222521 − 0.974928i 0.286126 + 3.81809i −2.16774 − 0.668658i
263.2 −0.826239 − 0.563320i −0.539257 − 0.500357i 0.365341 + 0.930874i −2.65055 + 0.817585i 0.163694 + 0.717189i 0 0.222521 − 0.974928i −0.183750 − 2.45197i 2.65055 + 0.817585i
263.3 −0.826239 − 0.563320i 2.12863 + 1.97508i 0.365341 + 0.930874i 3.39627 − 1.04761i −0.646154 − 2.83098i 0 0.222521 − 0.974928i 0.405933 + 5.41679i −3.39627 − 1.04761i
275.1 −0.0747301 + 0.997204i −2.77478 + 0.855906i −0.988831 − 0.149042i −2.60539 − 2.41745i −0.646154 − 2.83098i 0 0.222521 − 0.974928i 4.48812 − 3.05995i 2.60539 − 2.41745i
275.2 −0.0747301 + 0.997204i 0.702951 − 0.216832i −0.988831 − 0.149042i 2.03332 + 1.88665i 0.163694 + 0.717189i 0 0.222521 − 0.974928i −2.03159 + 1.38512i −2.03332 + 1.88665i
275.3 −0.0747301 + 0.997204i 2.49710 − 0.770253i −0.988831 − 0.149042i −1.66294 − 1.54299i 0.581491 + 2.54768i 0 0.222521 − 0.974928i 3.16350 − 2.15684i 1.66294 − 1.54299i
373.1 −0.826239 + 0.563320i −1.91561 + 1.77743i 0.365341 − 0.930874i 2.16774 + 0.668658i 0.581491 − 2.54768i 0 0.222521 + 0.974928i 0.286126 − 3.81809i −2.16774 + 0.668658i
373.2 −0.826239 + 0.563320i −0.539257 + 0.500357i 0.365341 − 0.930874i −2.65055 − 0.817585i 0.163694 − 0.717189i 0 0.222521 + 0.974928i −0.183750 + 2.45197i 2.65055 − 0.817585i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
49.e even 7 1 inner
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.h 36
7.b odd 2 1 686.2.g.g 36
7.c even 3 1 98.2.e.b ✓ 18
7.c even 3 1 inner 686.2.g.h 36
7.d odd 6 1 686.2.e.b 18
7.d odd 6 1 686.2.g.g 36
21.h odd 6 1 882.2.u.g 18
28.g odd 6 1 784.2.u.d 18
49.e even 7 1 inner 686.2.g.h 36
49.f odd 14 1 686.2.g.g 36
49.g even 21 1 98.2.e.b ✓ 18
49.g even 21 1 inner 686.2.g.h 36
49.g even 21 1 4802.2.a.d 9
49.h odd 42 1 686.2.e.b 18
49.h odd 42 1 686.2.g.g 36
49.h odd 42 1 4802.2.a.c 9
147.n odd 42 1 882.2.u.g 18
196.o odd 42 1 784.2.u.d 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.e.b ✓ 18 7.c even 3 1
98.2.e.b ✓ 18 49.g even 21 1
686.2.e.b 18 7.d odd 6 1
686.2.e.b 18 49.h odd 42 1
686.2.g.g 36 7.b odd 2 1
686.2.g.g 36 7.d odd 6 1
686.2.g.g 36 49.f odd 14 1
686.2.g.g 36 49.h odd 42 1
686.2.g.h 36 1.a even 1 1 trivial
686.2.g.h 36 7.c even 3 1 inner
686.2.g.h 36 49.e even 7 1 inner
686.2.g.h 36 49.g even 21 1 inner
784.2.u.d 18 28.g odd 6 1
784.2.u.d 18 196.o odd 42 1
882.2.u.g 18 21.h odd 6 1
882.2.u.g 18 147.n odd 42 1
4802.2.a.c 9 49.h odd 42 1
4802.2.a.d 9 49.g even 21 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} - 5 T_{3}^{35} + 3 T_{3}^{34} + 12 T_{3}^{33} + 5 T_{3}^{32} + 67 T_{3}^{31} + \cdots + 163047361 \) acting on \(S_{2}^{\mathrm{new}}(686, [\chi])\). Copy content Toggle raw display