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Properties

Label 686.2.e.e
Level $686$
Weight $2$
Character orbit 686.e
Analytic conductor $5.478$
Analytic rank $0$
Dimension $24$
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(99,686)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(686, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("686.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 686 = 2 \cdot 7^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 686.e (of order \(7\), degree \(6\), 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: \(4\) over \(\Q(\zeta_{7})\)
Twist minimal: no (minimal twist has level 98)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} - 7 q^{3} - 4 q^{4} + 7 q^{6} + 4 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} - 7 q^{3} - 4 q^{4} + 7 q^{6} + 4 q^{8} + 3 q^{9} - 7 q^{11} - 14 q^{13} - 7 q^{15} - 4 q^{16} - 14 q^{17} - 10 q^{18} + 7 q^{20} - 7 q^{22} - 21 q^{23} - 8 q^{25} + 7 q^{26} + 35 q^{27} - 11 q^{29} + 56 q^{31} + 4 q^{32} + 28 q^{33} - 7 q^{34} + 3 q^{36} + 48 q^{37} + 7 q^{38} + 38 q^{39} + 7 q^{40} - 28 q^{41} + 10 q^{43} - 7 q^{45} - 21 q^{46} - 35 q^{47} + 14 q^{48} + 8 q^{50} - 36 q^{51} + 7 q^{52} - 10 q^{53} + 63 q^{54} - 56 q^{55} - 33 q^{57} - 24 q^{58} - 14 q^{59} - 7 q^{60} + 28 q^{61} - 28 q^{62} - 4 q^{64} - 21 q^{66} + 72 q^{67} - 28 q^{68} + 35 q^{69} + 4 q^{72} + 49 q^{73} + q^{74} - 56 q^{75} - 24 q^{78} + 52 q^{79} - 5 q^{81} + 28 q^{82} + 7 q^{83} + 49 q^{85} + 32 q^{86} + 28 q^{87} - 7 q^{89} + 21 q^{90} + 21 q^{92} - 60 q^{93} + 35 q^{94} + 7 q^{95} + 126 q^{97} - 14 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1 0.222521 − 0.974928i −1.79108 + 2.24595i −0.900969 − 0.433884i 0.584442 − 0.732867i 1.79108 + 2.24595i 0 −0.623490 + 0.781831i −1.16874 − 5.12059i −0.584442 − 0.732867i
99.2 0.222521 − 0.974928i 0.335594 − 0.420822i −0.900969 − 0.433884i −0.784813 + 0.984124i −0.335594 − 0.420822i 0 −0.623490 + 0.781831i 0.603095 + 2.64233i 0.784813 + 0.984124i
99.3 0.222521 − 0.974928i 0.743468 − 0.932280i −0.900969 − 0.433884i 1.88321 − 2.36147i −0.743468 − 0.932280i 0 −0.623490 + 0.781831i 0.351162 + 1.53854i −1.88321 − 2.36147i
99.4 0.222521 − 0.974928i 1.18152 − 1.48158i −0.900969 − 0.433884i −2.52885 + 3.17107i −1.18152 − 1.48158i 0 −0.623490 + 0.781831i −0.131528 − 0.576262i 2.52885 + 3.17107i
197.1 0.900969 + 0.433884i −0.643125 − 2.81771i 0.623490 + 0.781831i −0.138970 − 0.608868i 0.643125 − 2.81771i 0 0.222521 + 0.974928i −4.82299 + 2.32263i 0.138970 − 0.608868i
197.2 0.900969 + 0.433884i −0.252266 − 1.10525i 0.623490 + 0.781831i −0.851567 − 3.73096i 0.252266 − 1.10525i 0 0.222521 + 0.974928i 1.54497 − 0.744017i 0.851567 − 3.73096i
197.3 0.900969 + 0.433884i −0.00466617 − 0.0204438i 0.623490 + 0.781831i 0.268304 + 1.17552i 0.00466617 − 0.0204438i 0 0.222521 + 0.974928i 2.70251 − 1.30146i −0.268304 + 1.17552i
197.4 0.900969 + 0.433884i 0.355984 + 1.55967i 0.623490 + 0.781831i 0.0437854 + 0.191836i −0.355984 + 1.55967i 0 0.222521 + 0.974928i 0.397067 − 0.191218i −0.0437854 + 0.191836i
295.1 −0.623490 + 0.781831i −3.01568 − 1.45228i −0.222521 − 0.974928i −1.19709 − 0.576486i 3.01568 − 1.45228i 0 0.900969 + 0.433884i 5.11476 + 6.41370i 1.19709 − 0.576486i
295.2 −0.623490 + 0.781831i −1.60279 − 0.771862i −0.222521 − 0.974928i 2.03762 + 0.981264i 1.60279 − 0.771862i 0 0.900969 + 0.433884i 0.102691 + 0.128770i −2.03762 + 0.981264i
295.3 −0.623490 + 0.781831i 0.567161 + 0.273130i −0.222521 − 0.974928i −0.974055 − 0.469080i −0.567161 + 0.273130i 0 0.900969 + 0.433884i −1.62340 − 2.03568i 0.974055 − 0.469080i
295.4 −0.623490 + 0.781831i 0.625881 + 0.301408i −0.222521 − 0.974928i 1.65798 + 0.798443i −0.625881 + 0.301408i 0 0.900969 + 0.433884i −1.56959 − 1.96820i −1.65798 + 0.798443i
393.1 −0.623490 − 0.781831i −3.01568 + 1.45228i −0.222521 + 0.974928i −1.19709 + 0.576486i 3.01568 + 1.45228i 0 0.900969 − 0.433884i 5.11476 − 6.41370i 1.19709 + 0.576486i
393.2 −0.623490 − 0.781831i −1.60279 + 0.771862i −0.222521 + 0.974928i 2.03762 − 0.981264i 1.60279 + 0.771862i 0 0.900969 − 0.433884i 0.102691 − 0.128770i −2.03762 − 0.981264i
393.3 −0.623490 − 0.781831i 0.567161 − 0.273130i −0.222521 + 0.974928i −0.974055 + 0.469080i −0.567161 − 0.273130i 0 0.900969 − 0.433884i −1.62340 + 2.03568i 0.974055 + 0.469080i
393.4 −0.623490 − 0.781831i 0.625881 − 0.301408i −0.222521 + 0.974928i 1.65798 − 0.798443i −0.625881 − 0.301408i 0 0.900969 − 0.433884i −1.56959 + 1.96820i −1.65798 − 0.798443i
491.1 0.900969 − 0.433884i −0.643125 + 2.81771i 0.623490 − 0.781831i −0.138970 + 0.608868i 0.643125 + 2.81771i 0 0.222521 − 0.974928i −4.82299 − 2.32263i 0.138970 + 0.608868i
491.2 0.900969 − 0.433884i −0.252266 + 1.10525i 0.623490 − 0.781831i −0.851567 + 3.73096i 0.252266 + 1.10525i 0 0.222521 − 0.974928i 1.54497 + 0.744017i 0.851567 + 3.73096i
491.3 0.900969 − 0.433884i −0.00466617 + 0.0204438i 0.623490 − 0.781831i 0.268304 − 1.17552i 0.00466617 + 0.0204438i 0 0.222521 − 0.974928i 2.70251 + 1.30146i −0.268304 − 1.17552i
491.4 0.900969 − 0.433884i 0.355984 − 1.55967i 0.623490 − 0.781831i 0.0437854 − 0.191836i −0.355984 − 1.55967i 0 0.222521 − 0.974928i 0.397067 + 0.191218i −0.0437854 − 0.191836i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 7 T_{3}^{23} + 29 T_{3}^{22} + 77 T_{3}^{21} + 167 T_{3}^{20} + 273 T_{3}^{19} + 697 T_{3}^{18} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(686, [\chi])\). Copy content Toggle raw display