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Properties

Label 207.4.i.b
Level $207$
Weight $4$
Character orbit 207.i
Analytic conductor $12.213$
Analytic rank $0$
Dimension $60$
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] = [207,4,Mod(55,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 10]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.55");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 207.i (of order \(11\), degree \(10\), 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: \(60\)
Relative dimension: \(6\) over \(\Q(\zeta_{11})\)
Twist minimal: no (minimal twist has level 69)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 4 q^{2} - 28 q^{4} + 6 q^{5} - 4 q^{7} + 52 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 4 q^{2} - 28 q^{4} + 6 q^{5} - 4 q^{7} + 52 q^{8} - 78 q^{10} - 10 q^{11} + 50 q^{13} + 224 q^{14} + 260 q^{16} + 662 q^{17} - 4 q^{19} + 735 q^{20} + 622 q^{22} + 438 q^{23} - 754 q^{25} + 40 q^{26} + 672 q^{28} - 1302 q^{29} + 1528 q^{31} - 1588 q^{32} + 29 q^{34} - 950 q^{35} + 316 q^{37} - 3122 q^{38} - 1939 q^{40} + 1500 q^{41} - 1316 q^{43} + 2901 q^{44} - 1980 q^{46} + 1440 q^{47} - 2310 q^{49} - 195 q^{50} + 6189 q^{52} + 148 q^{53} - 606 q^{55} + 432 q^{56} - 2623 q^{58} - 5264 q^{59} + 1482 q^{61} + 2299 q^{62} - 6780 q^{64} + 1446 q^{65} + 388 q^{67} - 5604 q^{68} + 2984 q^{70} + 3316 q^{71} + 2072 q^{73} + 6556 q^{74} + 9841 q^{76} - 9338 q^{77} + 268 q^{79} - 7980 q^{80} + 7742 q^{82} + 3494 q^{83} - 3842 q^{85} + 4792 q^{86} - 7960 q^{88} + 2754 q^{89} - 5436 q^{91} + 17609 q^{92} - 10961 q^{94} + 2396 q^{95} - 5654 q^{97} - 14411 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1 −4.63208 + 1.36010i 0 12.8763 − 8.27507i 2.63408 + 18.3204i 0 −12.3010 + 26.9353i −23.0975 + 26.6560i 0 −37.1190 − 81.2792i
55.2 −4.03831 + 1.18576i 0 8.17192 − 5.25178i −1.29203 − 8.98629i 0 8.66238 − 18.9680i −4.72404 + 5.45184i 0 15.8732 + 34.7574i
55.3 −0.883320 + 0.259366i 0 −6.01705 + 3.86692i −0.869451 − 6.04717i 0 −5.00489 + 10.9592i 9.13500 − 10.5424i 0 2.33643 + 5.11607i
55.4 1.69462 − 0.497587i 0 −4.10587 + 2.63868i 2.65827 + 18.4886i 0 3.62131 − 7.92957i −14.8977 + 17.1928i 0 13.7045 + 30.0086i
55.5 2.93095 − 0.860604i 0 1.11979 − 0.719646i −0.205936 − 1.43232i 0 −7.28337 + 15.9483i −13.3404 + 15.3957i 0 −1.83625 − 4.02082i
55.6 4.57045 − 1.34201i 0 12.3580 − 7.94203i −0.519731 − 3.61481i 0 12.0690 − 26.4275i 20.8687 − 24.0837i 0 −7.22650 − 15.8238i
64.1 −4.63208 − 1.36010i 0 12.8763 + 8.27507i 2.63408 − 18.3204i 0 −12.3010 − 26.9353i −23.0975 − 26.6560i 0 −37.1190 + 81.2792i
64.2 −4.03831 − 1.18576i 0 8.17192 + 5.25178i −1.29203 + 8.98629i 0 8.66238 + 18.9680i −4.72404 − 5.45184i 0 15.8732 − 34.7574i
64.3 −0.883320 − 0.259366i 0 −6.01705 − 3.86692i −0.869451 + 6.04717i 0 −5.00489 − 10.9592i 9.13500 + 10.5424i 0 2.33643 − 5.11607i
64.4 1.69462 + 0.497587i 0 −4.10587 − 2.63868i 2.65827 − 18.4886i 0 3.62131 + 7.92957i −14.8977 − 17.1928i 0 13.7045 − 30.0086i
64.5 2.93095 + 0.860604i 0 1.11979 + 0.719646i −0.205936 + 1.43232i 0 −7.28337 − 15.9483i −13.3404 − 15.3957i 0 −1.83625 + 4.02082i
64.6 4.57045 + 1.34201i 0 12.3580 + 7.94203i −0.519731 + 3.61481i 0 12.0690 + 26.4275i 20.8687 + 24.0837i 0 −7.22650 + 15.8238i
73.1 −2.23641 − 4.89704i 0 −13.7407 + 15.8576i −9.26837 − 5.95642i 0 −1.89824 − 13.2025i 67.0611 + 19.6909i 0 −8.44102 + 58.7086i
73.2 −1.45666 − 3.18964i 0 −2.81307 + 3.24646i 1.51957 + 0.976567i 0 4.44617 + 30.9238i −12.4631 − 3.65950i 0 0.901405 − 6.26941i
73.3 −0.675509 − 1.47916i 0 3.50729 − 4.04763i −3.16001 − 2.03082i 0 −2.13282 − 14.8341i −20.8382 − 6.11864i 0 −0.869282 + 6.04599i
73.4 0.468580 + 1.02605i 0 4.40568 − 5.08442i 16.0845 + 10.3369i 0 1.00849 + 7.01421i 15.9396 + 4.68029i 0 −3.06925 + 21.3471i
73.5 0.936192 + 2.04997i 0 1.91295 − 2.20766i −17.1881 − 11.0461i 0 3.15549 + 21.9469i 23.6153 + 6.93407i 0 6.55289 − 45.5764i
73.6 1.62255 + 3.55289i 0 −4.75146 + 5.48347i −2.37866 − 1.52867i 0 −5.05798 − 35.1790i 2.78946 + 0.819058i 0 1.57171 − 10.9315i
82.1 −2.95596 − 3.41136i 0 −1.76116 + 12.2491i 6.65677 − 14.5763i 0 1.88458 + 0.553362i 16.6135 − 10.6769i 0 −69.4021 + 20.3783i
82.2 −1.56668 − 1.80804i 0 0.323978 − 2.25332i −6.20398 + 13.5848i 0 −24.5496 − 7.20842i −20.6825 + 13.2918i 0 34.2816 − 10.0660i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.4.i.b 60
3.b odd 2 1 69.4.e.b ✓ 60
23.c even 11 1 inner 207.4.i.b 60
69.g even 22 1 1587.4.a.v 30
69.h odd 22 1 69.4.e.b ✓ 60
69.h odd 22 1 1587.4.a.w 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.4.e.b ✓ 60 3.b odd 2 1
69.4.e.b ✓ 60 69.h odd 22 1
207.4.i.b 60 1.a even 1 1 trivial
207.4.i.b 60 23.c even 11 1 inner
1587.4.a.v 30 69.g even 22 1
1587.4.a.w 30 69.h odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} + 4 T_{2}^{59} + 46 T_{2}^{58} + 108 T_{2}^{57} + 783 T_{2}^{56} + 2648 T_{2}^{55} + \cdots + 86\!\cdots\!56 \) acting on \(S_{4}^{\mathrm{new}}(207, [\chi])\). Copy content Toggle raw display