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Properties

Label 207.4.g.b
Level $207$
Weight $4$
Character orbit 207.g
Analytic conductor $12.213$
Analytic rank $0$
Dimension $128$
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] = [207,4,Mod(68,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.68");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 207.g (of order \(6\), 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: \(12.2133953712\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(64\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 6 q^{2} + 222 q^{4} - 102 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 6 q^{2} + 222 q^{4} - 102 q^{6} - 438 q^{12} - 2 q^{13} - 626 q^{16} - 90 q^{18} - 285 q^{23} - 936 q^{24} - 2158 q^{25} - 558 q^{27} + 102 q^{29} + 34 q^{31} + 3546 q^{32} + 1686 q^{36} + 876 q^{39} - 474 q^{41} - 560 q^{46} + 1926 q^{47} - 3360 q^{48} + 4502 q^{49} + 6198 q^{50} - 654 q^{52} + 462 q^{54} - 300 q^{55} + 1904 q^{58} - 1536 q^{59} + 3560 q^{64} - 3243 q^{69} + 1560 q^{70} - 2418 q^{72} - 8 q^{73} - 2112 q^{75} - 1350 q^{77} - 1332 q^{78} - 3576 q^{81} - 8992 q^{82} - 252 q^{85} - 1842 q^{87} - 3174 q^{92} + 12216 q^{93} - 4990 q^{94} - 5916 q^{95} + 7200 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1 −4.65528 + 2.68773i 4.70824 + 2.19830i 10.4478 − 18.0961i −8.85893 + 15.3441i −27.8266 + 2.42077i −18.2897 + 10.5595i 69.3195i 17.3350 + 20.7002i − 95.2417i
68.2 −4.65528 + 2.68773i 4.70824 + 2.19830i 10.4478 − 18.0961i 8.85893 − 15.3441i −27.8266 + 2.42077i 18.2897 − 10.5595i 69.3195i 17.3350 + 20.7002i 95.2417i
68.3 −4.56386 + 2.63495i −2.52832 − 4.53956i 9.88588 − 17.1228i −7.85208 + 13.6002i 23.5004 + 14.0559i 8.61694 − 4.97499i 62.0359i −14.2152 + 22.9549i − 82.7593i
68.4 −4.56386 + 2.63495i −2.52832 − 4.53956i 9.88588 − 17.1228i 7.85208 − 13.6002i 23.5004 + 14.0559i −8.61694 + 4.97499i 62.0359i −14.2152 + 22.9549i 82.7593i
68.5 −4.40683 + 2.54428i −1.43496 + 4.99408i 8.94675 − 15.4962i −3.35381 + 5.80896i −6.38273 − 25.6590i 26.8642 − 15.5100i 50.3438i −22.8818 − 14.3327i − 34.1321i
68.6 −4.40683 + 2.54428i −1.43496 + 4.99408i 8.94675 − 15.4962i 3.35381 − 5.80896i −6.38273 − 25.6590i −26.8642 + 15.5100i 50.3438i −22.8818 − 14.3327i 34.1321i
68.7 −4.20375 + 2.42704i 3.41512 − 3.91624i 7.78102 − 13.4771i −2.12951 + 3.68841i −4.85147 + 24.7515i 16.5719 − 9.56776i 36.7067i −3.67387 − 26.7489i − 20.6736i
68.8 −4.20375 + 2.42704i 3.41512 − 3.91624i 7.78102 − 13.4771i 2.12951 − 3.68841i −4.85147 + 24.7515i −16.5719 + 9.56776i 36.7067i −3.67387 − 26.7489i 20.6736i
68.9 −4.02772 + 2.32540i −5.03267 + 1.29315i 6.81501 − 11.8039i −6.07728 + 10.5262i 17.2631 − 16.9114i −5.73948 + 3.31369i 26.1842i 23.6555 − 13.0160i − 56.5286i
68.10 −4.02772 + 2.32540i −5.03267 + 1.29315i 6.81501 − 11.8039i 6.07728 − 10.5262i 17.2631 − 16.9114i 5.73948 − 3.31369i 26.1842i 23.6555 − 13.0160i 56.5286i
68.11 −3.40753 + 1.96734i −4.50471 − 2.58989i 3.74084 − 6.47932i −4.02617 + 6.97353i 20.4451 − 0.0371606i −24.7497 + 14.2892i − 2.03943i 13.5849 + 23.3335i − 31.6834i
68.12 −3.40753 + 1.96734i −4.50471 − 2.58989i 3.74084 − 6.47932i 4.02617 − 6.97353i 20.4451 − 0.0371606i 24.7497 − 14.2892i − 2.03943i 13.5849 + 23.3335i 31.6834i
68.13 −3.19780 + 1.84625i 2.16462 + 4.72381i 2.81727 − 4.87965i −4.49709 + 7.78919i −15.6434 − 11.1094i −4.93935 + 2.85174i − 8.73448i −17.6288 + 20.4506i − 33.2110i
68.14 −3.19780 + 1.84625i 2.16462 + 4.72381i 2.81727 − 4.87965i 4.49709 − 7.78919i −15.6434 − 11.1094i 4.93935 − 2.85174i − 8.73448i −17.6288 + 20.4506i 33.2110i
68.15 −2.83976 + 1.63953i 5.15509 − 0.651968i 1.37614 − 2.38354i −7.99746 + 13.8520i −13.5703 + 10.3034i 27.5171 − 15.8870i − 17.2076i 26.1499 − 6.72191i − 52.4484i
68.16 −2.83976 + 1.63953i 5.15509 − 0.651968i 1.37614 − 2.38354i 7.99746 − 13.8520i −13.5703 + 10.3034i −27.5171 + 15.8870i − 17.2076i 26.1499 − 6.72191i 52.4484i
68.17 −2.55334 + 1.47417i −3.17370 + 4.11432i 0.346380 − 0.599948i −9.01797 + 15.6196i 2.03834 − 15.1839i 0.00864380 − 0.00499050i − 21.5443i −6.85520 − 26.1152i − 53.1762i
68.18 −2.55334 + 1.47417i −3.17370 + 4.11432i 0.346380 − 0.599948i 9.01797 − 15.6196i 2.03834 − 15.1839i −0.00864380 + 0.00499050i − 21.5443i −6.85520 − 26.1152i 53.1762i
68.19 −2.49162 + 1.43854i 2.66332 − 4.46169i 0.138779 − 0.240373i −10.8034 + 18.7120i −0.217670 + 14.9481i −25.2827 + 14.5970i − 22.2180i −12.8134 − 23.7659i − 62.1642i
68.20 −2.49162 + 1.43854i 2.66332 − 4.46169i 0.138779 − 0.240373i 10.8034 − 18.7120i −0.217670 + 14.9481i 25.2827 − 14.5970i − 22.2180i −12.8134 − 23.7659i 62.1642i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
23.b odd 2 1 inner
207.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.4.g.b ✓ 128
9.d odd 6 1 inner 207.4.g.b ✓ 128
23.b odd 2 1 inner 207.4.g.b ✓ 128
207.g even 6 1 inner 207.4.g.b ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.4.g.b ✓ 128 1.a even 1 1 trivial
207.4.g.b ✓ 128 9.d odd 6 1 inner
207.4.g.b ✓ 128 23.b odd 2 1 inner
207.4.g.b ✓ 128 207.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} + 3 T_{2}^{63} - 179 T_{2}^{62} - 546 T_{2}^{61} + 17782 T_{2}^{60} + 53022 T_{2}^{59} + \cdots + 55\!\cdots\!24 \) acting on \(S_{4}^{\mathrm{new}}(207, [\chi])\). Copy content Toggle raw display