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Properties

Label 686.2.g.f
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} + 14 q^{3} + 2 q^{4} - 7 q^{6} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} + 14 q^{3} + 2 q^{4} - 7 q^{6} - 4 q^{8} - 2 q^{9} + 10 q^{11} - 14 q^{13} + 9 q^{15} + 2 q^{16} - 7 q^{17} - 9 q^{18} - 14 q^{19} - 7 q^{20} + q^{22} + 34 q^{23} - 8 q^{25} + 14 q^{26} - 7 q^{27} + 13 q^{29} - 8 q^{30} - 28 q^{31} + 2 q^{32} - 14 q^{33} - 7 q^{34} - 17 q^{36} - 22 q^{37} - 28 q^{38} - 70 q^{39} - 7 q^{40} + 28 q^{41} + 6 q^{43} + 3 q^{44} - 14 q^{45} - 29 q^{46} - 21 q^{47} + 14 q^{48} + 16 q^{50} - 52 q^{51} + 14 q^{52} + 108 q^{53} - 42 q^{54} - 14 q^{55} + 23 q^{57} + 18 q^{58} - 35 q^{59} - 15 q^{60} + 70 q^{61} - 4 q^{64} - 28 q^{65} + 24 q^{67} - 14 q^{68} + 7 q^{69} + 6 q^{71} - 2 q^{72} + 28 q^{73} + 6 q^{74} + 70 q^{75} + 6 q^{79} - 14 q^{80} + 18 q^{81} - 14 q^{82} - 77 q^{83} - 33 q^{85} + 88 q^{86} - 21 q^{87} + 3 q^{88} + 21 q^{89} - 21 q^{90} - 5 q^{92} + 88 q^{93} + 70 q^{94} - 103 q^{95} + 98 q^{97} - 106 q^{99}+O(q^{100}) \) 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 −0.408049 − 1.03969i 0.826239 − 0.563320i −1.56654 − 0.236117i −0.696376 − 0.873227i 0 0.623490 − 0.781831i 1.28470 − 1.19203i −1.56654 + 0.236117i
67.2 0.955573 − 0.294755i 1.04034 + 2.65074i 0.826239 − 0.563320i −2.44872 − 0.369085i 1.77544 + 2.22633i 0 0.623490 − 0.781831i −3.74496 + 3.47481i −2.44872 + 0.369085i
79.1 −0.988831 + 0.149042i 0.799298 + 0.544952i 0.955573 − 0.294755i −0.277683 + 3.70542i −0.871591 − 0.419736i 0 −0.900969 + 0.433884i −0.754119 − 1.92146i −0.277683 − 3.70542i
79.2 −0.988831 + 0.149042i 1.39228 + 0.949242i 0.955573 − 0.294755i 0.169790 − 2.26569i −1.51821 − 0.731131i 0 −0.900969 + 0.433884i −0.0586340 − 0.149397i 0.169790 + 2.26569i
165.1 −0.988831 − 0.149042i 0.799298 − 0.544952i 0.955573 + 0.294755i −0.277683 − 3.70542i −0.871591 + 0.419736i 0 −0.900969 − 0.433884i −0.754119 + 1.92146i −0.277683 + 3.70542i
165.2 −0.988831 − 0.149042i 1.39228 − 0.949242i 0.955573 + 0.294755i 0.169790 + 2.26569i −1.51821 + 0.731131i 0 −0.900969 − 0.433884i −0.0586340 + 0.149397i 0.169790 − 2.26569i
177.1 −0.733052 − 0.680173i −1.15657 − 0.174325i 0.0747301 + 0.997204i 0.0303424 + 0.0773110i 0.729256 + 0.914459i 0 0.623490 − 0.781831i −1.55945 − 0.481026i 0.0303424 − 0.0773110i
177.2 −0.733052 − 0.680173i 2.12331 + 0.320038i 0.0747301 + 0.997204i 0.768430 + 1.95793i −1.33882 − 1.67883i 0 0.623490 − 0.781831i 1.53932 + 0.474818i 0.768430 − 1.95793i
263.1 0.826239 + 0.563320i −1.49641 − 1.38847i 0.365341 + 0.930874i −1.24896 + 0.385253i −0.454243 − 1.99017i 0 −0.222521 + 0.974928i 0.0872176 + 1.16384i −1.24896 − 0.385253i
263.2 0.826239 + 0.563320i 1.83809 + 1.70550i 0.365341 + 0.930874i 2.84937 − 0.878914i 0.557960 + 2.44458i 0 −0.222521 + 0.974928i 0.245661 + 3.27812i 2.84937 + 0.878914i
275.1 0.0747301 − 0.997204i −0.0761418 + 0.0234866i −0.988831 − 0.149042i 0.327570 + 0.303940i 0.0177309 + 0.0776840i 0 −0.222521 + 0.974928i −2.47347 + 1.68638i 0.327570 − 0.303940i
275.2 0.0747301 − 0.997204i 2.85795 − 0.881562i −0.988831 − 0.149042i −2.43886 − 2.26293i −0.665522 − 2.91584i 0 −0.222521 + 0.974928i 4.91203 − 3.34897i −2.43886 + 2.26293i
373.1 0.826239 − 0.563320i −1.49641 + 1.38847i 0.365341 − 0.930874i −1.24896 − 0.385253i −0.454243 + 1.99017i 0 −0.222521 − 0.974928i 0.0872176 − 1.16384i −1.24896 + 0.385253i
373.2 0.826239 − 0.563320i 1.83809 − 1.70550i 0.365341 − 0.930874i 2.84937 + 0.878914i 0.557960 − 2.44458i 0 −0.222521 − 0.974928i 0.245661 − 3.27812i 2.84937 − 0.878914i
459.1 0.0747301 + 0.997204i −0.0761418 − 0.0234866i −0.988831 + 0.149042i 0.327570 − 0.303940i 0.0177309 − 0.0776840i 0 −0.222521 − 0.974928i −2.47347 − 1.68638i 0.327570 + 0.303940i
459.2 0.0747301 + 0.997204i 2.85795 + 0.881562i −0.988831 + 0.149042i −2.43886 + 2.26293i −0.665522 + 2.91584i 0 −0.222521 − 0.974928i 4.91203 + 3.34897i −2.43886 − 2.26293i
471.1 0.955573 + 0.294755i −0.408049 + 1.03969i 0.826239 + 0.563320i −1.56654 + 0.236117i −0.696376 + 0.873227i 0 0.623490 + 0.781831i 1.28470 + 1.19203i −1.56654 − 0.236117i
471.2 0.955573 + 0.294755i 1.04034 − 2.65074i 0.826239 + 0.563320i −2.44872 + 0.369085i 1.77544 − 2.22633i 0 0.623490 + 0.781831i −3.74496 − 3.47481i −2.44872 − 0.369085i
557.1 0.365341 − 0.930874i −0.0845640 + 1.12843i −0.733052 − 0.680173i 0.853754 + 0.582080i 1.01953 + 0.490980i 0 −0.900969 + 0.433884i 1.70029 + 0.256278i 0.853754 − 0.582080i
557.2 0.365341 − 0.930874i 0.170463 − 2.27467i −0.733052 − 0.680173i 2.98150 + 2.03275i −2.05516 − 0.989712i 0 −0.900969 + 0.433884i −2.17859 − 0.328370i 2.98150 − 2.03275i
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.f 24
7.b odd 2 1 686.2.g.d 24
7.c even 3 1 98.2.g.b ✓ 24
7.c even 3 1 686.2.e.c 24
7.d odd 6 1 686.2.e.d 24
7.d odd 6 1 686.2.g.e 24
21.h odd 6 1 882.2.z.b 24
28.g odd 6 1 784.2.bg.b 24
49.e even 7 1 98.2.g.b ✓ 24
49.f odd 14 1 686.2.g.e 24
49.g even 21 1 686.2.e.c 24
49.g even 21 1 inner 686.2.g.f 24
49.g even 21 1 4802.2.a.o 12
49.h odd 42 1 686.2.e.d 24
49.h odd 42 1 686.2.g.d 24
49.h odd 42 1 4802.2.a.l 12
147.l odd 14 1 882.2.z.b 24
196.k odd 14 1 784.2.bg.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.g.b ✓ 24 7.c even 3 1
98.2.g.b ✓ 24 49.e even 7 1
686.2.e.c 24 7.c even 3 1
686.2.e.c 24 49.g even 21 1
686.2.e.d 24 7.d odd 6 1
686.2.e.d 24 49.h odd 42 1
686.2.g.d 24 7.b odd 2 1
686.2.g.d 24 49.h odd 42 1
686.2.g.e 24 7.d odd 6 1
686.2.g.e 24 49.f odd 14 1
686.2.g.f 24 1.a even 1 1 trivial
686.2.g.f 24 49.g even 21 1 inner
784.2.bg.b 24 28.g odd 6 1
784.2.bg.b 24 196.k odd 14 1
882.2.z.b 24 21.h odd 6 1
882.2.z.b 24 147.l odd 14 1
4802.2.a.l 12 49.h odd 42 1
4802.2.a.o 12 49.g even 21 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 14 T_{3}^{23} + 96 T_{3}^{22} - 427 T_{3}^{21} + 1365 T_{3}^{20} - 3318 T_{3}^{19} + \cdots + 1681 \) acting on \(S_{2}^{\mathrm{new}}(686, [\chi])\). Copy content Toggle raw display