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Properties

Label 123.4.g.a
Level $123$
Weight $4$
Character orbit 123.g
Analytic conductor $7.257$
Analytic rank $0$
Dimension $40$
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] = [123,4,Mod(10,123)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(123, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("123.10");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 123 = 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 123.g (of order \(5\), degree \(4\), 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: \(7.25723493071\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{2} - 120 q^{3} - 40 q^{4} - 2 q^{5} + 6 q^{6} - 14 q^{7} - 12 q^{8} + 360 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{2} - 120 q^{3} - 40 q^{4} - 2 q^{5} + 6 q^{6} - 14 q^{7} - 12 q^{8} + 360 q^{9} + 13 q^{10} + 50 q^{11} + 120 q^{12} - 92 q^{13} - 2 q^{14} + 6 q^{15} - 100 q^{16} + 74 q^{17} - 18 q^{18} - 292 q^{19} + 423 q^{20} + 42 q^{21} - 764 q^{22} - 284 q^{23} + 36 q^{24} + 108 q^{25} + 499 q^{26} - 1080 q^{27} - 285 q^{28} - 548 q^{29} - 39 q^{30} + 678 q^{31} + 802 q^{32} - 150 q^{33} + 875 q^{34} - 716 q^{35} - 360 q^{36} - 374 q^{37} - 1442 q^{38} + 276 q^{39} + 1718 q^{40} + 472 q^{41} + 6 q^{42} + 732 q^{43} + 1696 q^{44} - 18 q^{45} + 1280 q^{46} - 476 q^{47} + 300 q^{48} + 200 q^{49} + 1556 q^{50} - 222 q^{51} - 1211 q^{52} - 116 q^{53} + 54 q^{54} - 2556 q^{55} - 2911 q^{56} + 876 q^{57} - 596 q^{58} + 374 q^{59} - 1269 q^{60} - 1218 q^{61} - 209 q^{62} - 126 q^{63} - 550 q^{64} - 5584 q^{65} + 2292 q^{66} - 924 q^{67} + 8250 q^{68} + 852 q^{69} + 5076 q^{70} + 1872 q^{71} - 108 q^{72} + 1928 q^{73} - 3993 q^{74} - 324 q^{75} - 934 q^{76} - 1116 q^{77} - 1497 q^{78} - 2280 q^{79} - 1734 q^{80} + 3240 q^{81} - 3568 q^{82} + 428 q^{83} + 855 q^{84} + 7924 q^{85} + 1171 q^{86} + 1644 q^{87} - 10398 q^{88} - 1048 q^{89} + 117 q^{90} - 4048 q^{91} + 1502 q^{92} - 2034 q^{93} + 5055 q^{94} - 1008 q^{95} - 2406 q^{96} + 164 q^{97} + 5475 q^{98} + 450 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −4.20161 − 3.05265i −3.00000 5.86272 + 18.0436i 3.53757 + 10.8875i 12.6048 + 9.15794i −18.3682 + 13.3453i 17.6089 − 54.1947i 9.00000 18.3723 − 56.5440i
10.2 −4.01879 − 2.91982i −3.00000 5.15318 + 15.8598i −5.25396 − 16.1700i 12.0564 + 8.75946i 20.7309 − 15.0619i 13.3181 − 40.9888i 9.00000 −26.0990 + 80.3246i
10.3 −2.53010 − 1.83822i −3.00000 0.550187 + 1.69330i 0.416616 + 1.28221i 7.59029 + 5.51467i −6.69465 + 4.86394i −6.01064 + 18.4989i 9.00000 1.30291 − 4.00995i
10.4 −1.33228 − 0.967959i −3.00000 −1.63411 − 5.02927i 3.15308 + 9.70419i 3.99684 + 2.90388i 24.3281 − 17.6754i −6.76212 + 20.8117i 9.00000 5.19246 − 15.9808i
10.5 −0.899958 − 0.653858i −3.00000 −2.08974 − 6.43156i −4.69132 − 14.4384i 2.69987 + 1.96157i −6.25286 + 4.54297i −5.07467 + 15.6182i 9.00000 −5.21867 + 16.0614i
10.6 1.07116 + 0.778245i −3.00000 −1.93041 − 5.94120i 4.22883 + 13.0150i −3.21349 − 2.33473i −6.36013 + 4.62090i 5.82910 − 17.9401i 9.00000 −5.59910 + 17.2323i
10.7 1.24393 + 0.903767i −3.00000 −1.74157 − 5.36001i 3.12996 + 9.63302i −3.73179 − 2.71130i 5.96594 − 4.33451i 6.47892 − 19.9401i 9.00000 −4.81256 + 14.8115i
10.8 2.33586 + 1.69710i −3.00000 0.103945 + 0.319910i −2.26383 − 6.96735i −7.00757 − 5.09130i −23.0963 + 16.7804i 6.83762 − 21.0440i 9.00000 6.53630 − 20.1167i
10.9 2.95297 + 2.14546i −3.00000 1.64490 + 5.06248i −3.88449 − 11.9552i −8.85890 − 6.43637i 22.2805 − 16.1877i 3.01947 − 9.29298i 9.00000 14.1787 − 43.6374i
10.10 3.76078 + 2.73237i −3.00000 4.20551 + 12.9432i 0.00951291 + 0.0292777i −11.2823 − 8.19711i −23.8594 + 17.3349i −8.05775 + 24.7992i 9.00000 −0.0442215 + 0.136100i
16.1 −1.57961 + 4.86155i −3.00000 −14.6673 − 10.6564i 1.87323 + 1.36098i 4.73884 − 14.5846i −4.52679 − 13.9320i 41.8917 − 30.4361i 9.00000 −9.57547 + 6.95698i
16.2 −1.10878 + 3.41247i −3.00000 −3.94343 − 2.86507i 6.30623 + 4.58174i 3.32634 − 10.2374i 5.89425 + 18.1406i −9.07322 + 6.59208i 9.00000 −22.6273 + 16.4397i
16.3 −0.989839 + 3.04641i −3.00000 −1.82871 − 1.32863i −15.6632 − 11.3800i 2.96952 − 9.13923i 4.51896 + 13.9079i −14.8738 + 10.8064i 9.00000 50.1722 − 36.4522i
16.4 −0.351798 + 1.08272i −3.00000 5.42361 + 3.94048i 17.0491 + 12.3869i 1.05539 − 3.24817i −5.10719 − 15.7183i −13.5426 + 9.83929i 9.00000 −19.4094 + 14.1018i
16.5 −0.287475 + 0.884758i −3.00000 5.77198 + 4.19359i −4.83521 − 3.51298i 0.862425 − 2.65427i −8.67593 − 26.7018i −11.3906 + 8.27573i 9.00000 4.49814 − 3.26809i
16.6 0.0662656 − 0.203944i −3.00000 6.43493 + 4.67525i −7.70181 − 5.59569i −0.198797 + 0.611833i 4.14937 + 12.7705i 2.76779 − 2.01092i 9.00000 −1.65158 + 1.19994i
16.7 0.791352 − 2.43553i −3.00000 1.16656 + 0.847555i 2.03269 + 1.47684i −2.37406 + 7.30660i −0.802573 − 2.47007i 19.5617 − 14.2124i 9.00000 5.20546 − 3.78198i
16.8 1.22304 − 3.76413i −3.00000 −6.20070 − 4.50507i −4.83142 − 3.51023i −3.66912 + 11.2924i −6.66700 − 20.5189i 1.07432 − 0.780536i 9.00000 −19.1220 + 13.8929i
16.9 1.22413 − 3.76748i −3.00000 −6.22331 − 4.52150i 14.0685 + 10.2214i −3.67239 + 11.3025i 9.23376 + 28.4186i 0.985708 − 0.716159i 9.00000 55.7305 − 40.4905i
16.10 1.63075 − 5.01894i −3.00000 −16.0582 − 11.6670i −7.68008 − 5.57991i −4.89225 + 15.0568i 6.30939 + 19.4183i −50.5879 + 36.7543i 9.00000 −40.5295 + 29.4464i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 123.4.g.a ✓ 40
41.d even 5 1 inner 123.4.g.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
123.4.g.a ✓ 40 1.a even 1 1 trivial
123.4.g.a ✓ 40 41.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 2 T_{2}^{39} + 62 T_{2}^{38} + 136 T_{2}^{37} + 2475 T_{2}^{36} + 3316 T_{2}^{35} + \cdots + 17\!\cdots\!76 \) acting on \(S_{4}^{\mathrm{new}}(123, [\chi])\). Copy content Toggle raw display