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Properties

Label 123.3.m.a
Level $123$
Weight $3$
Character orbit 123.m
Analytic conductor $3.352$
Analytic rank $0$
Dimension $208$
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] = [123,3,Mod(2,123)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(123, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 13]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("123.2");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 123 = 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 123.m (of order \(20\), degree \(8\), 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: \(3.35150725163\)
Analytic rank: \(0\)
Dimension: \(208\)
Relative dimension: \(26\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 208 q - 12 q^{3} - 108 q^{4} + 46 q^{6} - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 208 q - 12 q^{3} - 108 q^{4} + 46 q^{6} - 16 q^{7} - 4 q^{10} - 36 q^{12} - 52 q^{13} + 16 q^{15} - 140 q^{16} - 30 q^{18} + 60 q^{19} - 10 q^{21} + 12 q^{22} - 46 q^{24} - 64 q^{25} - 42 q^{27} + 4 q^{28} - 112 q^{30} - 8 q^{31} - 170 q^{33} - 156 q^{34} - 130 q^{36} - 88 q^{37} - 50 q^{39} + 392 q^{40} + 408 q^{42} - 20 q^{43} - 100 q^{45} + 480 q^{46} + 902 q^{48} - 20 q^{49} + 48 q^{51} + 932 q^{52} + 308 q^{54} - 120 q^{55} - 258 q^{57} - 260 q^{58} - 100 q^{60} - 300 q^{61} + 218 q^{63} - 880 q^{64} - 1034 q^{66} - 668 q^{67} + 138 q^{69} - 800 q^{70} - 338 q^{72} + 500 q^{75} - 1348 q^{76} - 410 q^{78} + 140 q^{79} - 728 q^{81} - 500 q^{82} + 30 q^{84} + 168 q^{85} - 210 q^{87} + 208 q^{88} + 800 q^{90} - 774 q^{93} + 2936 q^{94} + 544 q^{96} + 540 q^{97} - 526 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −1.19218 − 3.66916i −0.149838 − 2.99626i −8.80537 + 6.39747i −2.02040 + 1.46791i −10.8151 + 4.12186i 1.27384 + 2.50005i 21.4863 + 15.6107i −8.95510 + 0.897905i 7.79469 + 5.66317i
2.2 −1.06603 − 3.28089i 0.739812 + 2.90735i −6.39176 + 4.64389i 2.44404 − 1.77570i 8.75004 − 5.52655i −5.29072 − 10.3836i 10.8863 + 7.90937i −7.90536 + 4.30178i −8.43130 − 6.12569i
2.3 −1.04572 − 3.21839i −2.63037 + 1.44262i −6.02845 + 4.37992i −0.616234 + 0.447720i 7.39353 + 6.95700i 2.11295 + 4.14691i 9.44947 + 6.86544i 4.83772 − 7.58923i 2.08535 + 1.51509i
2.4 −0.964398 − 2.96811i 3.00000 + 0.00428293i −4.64355 + 3.37374i 7.18584 − 5.22082i −2.88048 − 8.90846i 4.55885 + 8.94724i 4.39256 + 3.19138i 8.99996 + 0.0256976i −22.4260 − 16.2934i
2.5 −0.862042 − 2.65309i 1.89680 + 2.32425i −3.05971 + 2.22301i −6.66647 + 4.84348i 4.53134 − 7.03599i 3.34893 + 6.57264i −0.491962 − 0.357432i −1.80430 + 8.81728i 18.5970 + 13.5115i
2.6 −0.837964 − 2.57899i −2.06031 − 2.18062i −2.71293 + 1.97106i 5.97498 − 4.34108i −3.89734 + 7.14080i −4.28267 − 8.40522i −1.41859 − 1.03067i −0.510237 + 8.98552i −16.2024 − 11.7717i
2.7 −0.730205 − 2.24734i 2.24988 − 1.98445i −1.28127 + 0.930895i −2.06027 + 1.49687i −6.10261 − 3.60719i −1.62243 − 3.18419i −4.61918 − 3.35603i 1.12392 − 8.92955i 4.86841 + 3.53710i
2.8 −0.622473 − 1.91578i −2.39094 − 1.81202i −0.0466546 + 0.0338966i −5.11444 + 3.71586i −1.98313 + 5.70843i −0.375448 − 0.736859i −6.42464 − 4.66778i 2.43316 + 8.66486i 10.3023 + 7.48509i
2.9 −0.424004 − 1.30495i −2.96080 + 0.483367i 1.71295 − 1.24453i 3.34558 − 2.43070i 1.88616 + 3.65875i 1.93798 + 3.80350i −6.79058 − 4.93364i 8.53271 − 2.86231i −4.59049 − 3.33518i
2.10 −0.320473 − 0.986314i −1.72602 + 2.45374i 2.36596 − 1.71897i −5.95069 + 4.32343i 2.97330 + 0.916041i −4.10753 − 8.06148i −5.80970 − 4.22100i −3.04170 − 8.47042i 6.17129 + 4.48371i
2.11 −0.289092 − 0.889734i 0.484191 + 2.96067i 2.52802 − 1.83671i 3.01633 − 2.19149i 2.49423 − 1.28671i 3.51723 + 6.90295i −5.39242 − 3.91783i −8.53112 + 2.86706i −2.82184 − 2.05019i
2.12 −0.261342 − 0.804328i 2.73511 + 1.23254i 2.65742 − 1.93073i 1.68623 − 1.22512i 0.276570 − 2.52204i −4.34245 − 8.52255i −4.98425 − 3.62127i 5.96167 + 6.74229i −1.42608 − 1.03611i
2.13 −0.173876 − 0.535135i 0.389245 − 2.97464i 2.97993 − 2.16505i 3.56046 − 2.58683i −1.65951 + 0.308920i 2.85926 + 5.61161i −3.49758 − 2.54114i −8.69698 − 2.31573i −2.00338 − 1.45554i
2.14 0.173876 + 0.535135i 2.97464 − 0.389245i 2.97993 − 2.16505i −3.56046 + 2.58683i 0.725517 + 1.52415i 2.85926 + 5.61161i 3.49758 + 2.54114i 8.69698 − 2.31573i −2.00338 − 1.45554i
2.15 0.261342 + 0.804328i −1.23254 − 2.73511i 2.65742 − 1.93073i −1.68623 + 1.22512i 1.87781 − 1.70617i −4.34245 − 8.52255i 4.98425 + 3.62127i −5.96167 + 6.74229i −1.42608 − 1.03611i
2.16 0.289092 + 0.889734i −2.96067 − 0.484191i 2.52802 − 1.83671i −3.01633 + 2.19149i −0.425105 − 2.77418i 3.51723 + 6.90295i 5.39242 + 3.91783i 8.53112 + 2.86706i −2.82184 − 2.05019i
2.17 0.320473 + 0.986314i −2.45374 + 1.72602i 2.36596 − 1.71897i 5.95069 − 4.32343i −2.48876 − 1.86702i −4.10753 − 8.06148i 5.80970 + 4.22100i 3.04170 − 8.47042i 6.17129 + 4.48371i
2.18 0.424004 + 1.30495i −0.483367 + 2.96080i 1.71295 − 1.24453i −3.34558 + 2.43070i −4.06865 + 0.624623i 1.93798 + 3.80350i 6.79058 + 4.93364i −8.53271 − 2.86231i −4.59049 − 3.33518i
2.19 0.622473 + 1.91578i 1.81202 + 2.39094i −0.0466546 + 0.0338966i 5.11444 − 3.71586i −3.45256 + 4.95972i −0.375448 − 0.736859i 6.42464 + 4.66778i −2.43316 + 8.66486i 10.3023 + 7.48509i
2.20 0.730205 + 2.24734i 1.98445 − 2.24988i −1.28127 + 0.930895i 2.06027 − 1.49687i 6.50530 + 2.81686i −1.62243 − 3.18419i 4.61918 + 3.35603i −1.12392 − 8.92955i 4.86841 + 3.53710i
See next 80 embeddings (of 208 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
41.g even 20 1 inner
123.m odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 123.3.m.a ✓ 208
3.b odd 2 1 inner 123.3.m.a ✓ 208
41.g even 20 1 inner 123.3.m.a ✓ 208
123.m odd 20 1 inner 123.3.m.a ✓ 208
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
123.3.m.a ✓ 208 1.a even 1 1 trivial
123.3.m.a ✓ 208 3.b odd 2 1 inner
123.3.m.a ✓ 208 41.g even 20 1 inner
123.3.m.a ✓ 208 123.m odd 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(123, [\chi])\).