↑

Properties

Label 240.4.bb.a
Level $240$
Weight $4$
Character orbit 240.bb
Analytic conductor $14.160$
Analytic rank $0$
Dimension $280$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,4,Mod(173,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.173");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.bb (of order \(4\), 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: \(14.1604584014\)
Analytic rank: \(0\)
Dimension: \(280\)
Relative dimension: \(140\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 280 q + 12 q^{4} - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 280 q + 12 q^{4} - 4 q^{6} - 36 q^{10} + 52 q^{12} - 4 q^{15} - 140 q^{16} - 224 q^{18} + 24 q^{19} - 4 q^{21} - 252 q^{22} - 108 q^{24} + 172 q^{28} + 632 q^{30} - 16 q^{31} - 4 q^{33} - 924 q^{34} - 260 q^{36} + 216 q^{39} + 1276 q^{40} + 220 q^{42} + 856 q^{43} + 248 q^{45} - 92 q^{46} - 2192 q^{48} - 4 q^{51} - 744 q^{52} - 1104 q^{54} + 108 q^{57} - 332 q^{58} - 1112 q^{60} + 904 q^{61} - 1376 q^{63} - 1596 q^{64} + 1644 q^{66} - 8 q^{67} - 108 q^{69} + 4924 q^{70} + 2272 q^{72} + 1704 q^{75} - 2612 q^{76} - 3652 q^{78} - 8 q^{81} + 1632 q^{82} - 2880 q^{84} - 504 q^{85} - 108 q^{87} - 3620 q^{88} + 964 q^{90} - 8 q^{91} + 420 q^{94} + 1392 q^{96} - 8 q^{97} + 2656 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
173.1 −2.82805 − 0.0459492i 5.17898 + 0.422093i 7.99578 + 0.259893i 10.5532 + 3.69195i −14.6270 − 1.43167i 12.0982 + 12.0982i −22.6005 − 1.10239i 26.6437 + 4.37202i −29.6753 − 10.9259i
173.2 −2.82777 + 0.0610482i −2.62306 + 4.48548i 7.99255 − 0.345260i −10.6390 + 3.43694i 7.14358 − 12.8440i 19.8046 + 19.8046i −22.5800 + 1.46425i −13.2391 − 23.5314i 29.8747 − 10.3684i
173.3 −2.82128 − 0.200891i −3.98393 − 3.33592i 7.91929 + 1.13354i 10.6215 + 3.49058i 10.5696 + 10.2119i −16.0583 − 16.0583i −22.1148 − 4.78897i 4.74333 + 26.5801i −29.2650 − 11.9817i
173.4 −2.80980 − 0.324108i −0.406075 + 5.18026i 7.78991 + 1.82135i 3.06996 + 10.7506i 2.81995 − 14.4239i −0.0561253 − 0.0561253i −21.2977 − 7.64240i −26.6702 − 4.20715i −5.14160 − 31.2020i
173.5 −2.80967 + 0.325160i 0.664819 − 5.15345i 7.78854 − 1.82719i −10.0034 + 4.99318i −0.192229 + 14.6957i −21.0353 − 21.0353i −21.2891 + 7.66633i −26.1160 − 6.85222i 26.4827 − 17.2819i
173.6 −2.80650 − 0.351474i 4.57020 − 2.47249i 7.75293 + 1.97282i −2.41545 + 10.9163i −13.6953 + 5.33276i 1.24307 + 1.24307i −21.0652 − 8.26169i 14.7735 − 22.5996i 10.6158 − 29.7877i
173.7 −2.80305 − 0.378042i 1.43920 − 4.99287i 7.71417 + 2.11934i 5.55621 − 9.70199i −5.92165 + 13.4512i 7.15975 + 7.15975i −20.8220 − 8.85690i −22.8574 − 14.3714i −19.2421 + 25.0947i
173.8 −2.80202 − 0.385578i −4.52370 − 2.55658i 7.70266 + 2.16080i −4.90170 − 10.0485i 11.6898 + 8.90784i 3.96668 + 3.96668i −20.7499 − 9.02457i 13.9278 + 23.1304i 9.86017 + 30.0462i
173.9 −2.78345 + 0.502418i 2.00280 + 4.79466i 7.49515 − 2.79691i 9.48243 − 5.92314i −7.98361 − 12.3394i −17.6187 − 17.6187i −19.4571 + 11.5508i −18.9776 + 19.2055i −23.4179 + 21.2509i
173.10 −2.78065 + 0.517680i −1.99967 − 4.79597i 7.46401 − 2.87897i −0.820040 + 11.1502i 8.04315 + 12.3007i 23.2337 + 23.2337i −19.2644 + 11.8694i −19.0027 + 19.1807i −3.49201 − 31.4294i
173.11 −2.77653 + 0.539320i 3.90906 + 3.42334i 7.41827 − 2.99488i −11.0468 + 1.72278i −12.6999 − 7.39678i −1.53166 − 1.53166i −18.9819 + 12.3162i 3.56149 + 26.7641i 29.7427 − 10.7411i
173.12 −2.74259 + 0.691519i −3.84920 + 3.49051i 7.04360 − 3.79311i −5.38346 − 9.79889i 8.14301 − 12.2348i −9.80185 − 9.80185i −16.6947 + 15.2737i 2.63262 − 26.8713i 21.5408 + 23.1516i
173.13 −2.68978 + 0.874697i 4.80990 − 1.96593i 6.46981 − 4.70548i −0.924600 − 11.1420i −11.2180 + 9.49513i −5.61418 − 5.61418i −13.2865 + 18.3158i 19.2702 − 18.9119i 12.2329 + 29.1609i
173.14 −2.67121 − 0.929856i −5.08400 + 1.07374i 6.27074 + 4.96768i −9.53337 + 5.84079i 14.5789 + 1.85922i −15.4016 − 15.4016i −12.1312 − 19.1006i 24.6942 − 10.9178i 30.8967 − 6.73731i
173.15 −2.66312 − 0.952770i −2.02043 + 4.78726i 6.18446 + 5.07469i 9.84397 − 5.30060i 9.94180 − 10.8241i 7.56633 + 7.56633i −11.6350 − 19.4069i −18.8358 − 19.3446i −31.2660 + 4.73711i
173.16 −2.65923 + 0.963597i −5.14813 + 0.704829i 6.14296 − 5.12485i 10.1381 − 4.71360i 13.0109 − 6.83502i 15.8196 + 15.8196i −11.3972 + 19.5475i 26.0064 − 7.25710i −22.4176 + 22.3036i
173.17 −2.64163 − 1.01085i 5.17507 − 0.467635i 5.95637 + 5.34057i −7.44509 − 8.34090i −14.1433 − 3.99589i −21.4609 − 21.4609i −10.3360 − 20.1288i 26.5626 − 4.84008i 11.2358 + 29.5594i
173.18 −2.64108 − 1.01226i 3.48984 + 3.84981i 5.95064 + 5.34695i −2.31863 − 10.9373i −5.31994 − 13.7003i 16.9657 + 16.9657i −10.3036 − 20.1454i −2.64202 + 26.8704i −4.94772 + 31.2333i
173.19 −2.53383 + 1.25686i −5.14985 − 0.692154i 4.84059 − 6.36936i −2.57413 + 10.8800i 13.9188 − 4.71886i −6.37710 − 6.37710i −4.25980 + 22.2228i 26.0418 + 7.12897i −7.15225 − 30.8033i
173.20 −2.50044 + 1.32205i −3.09812 + 4.17153i 4.50437 − 6.61141i 4.70489 + 10.1422i 2.23169 − 14.5265i −11.7582 − 11.7582i −2.52227 + 22.4864i −7.80327 − 25.8478i −25.1728 − 19.1398i
See next 80 embeddings (of 280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 173.140
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
80.i odd 4 1 inner
240.bb even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.4.bb.a ✓ 280
3.b odd 2 1 inner 240.4.bb.a ✓ 280
5.c odd 4 1 240.4.bf.a yes 280
15.e even 4 1 240.4.bf.a yes 280
16.e even 4 1 240.4.bf.a yes 280
48.i odd 4 1 240.4.bf.a yes 280
80.i odd 4 1 inner 240.4.bb.a ✓ 280
240.bb even 4 1 inner 240.4.bb.a ✓ 280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.4.bb.a ✓ 280 1.a even 1 1 trivial
240.4.bb.a ✓ 280 3.b odd 2 1 inner
240.4.bb.a ✓ 280 80.i odd 4 1 inner
240.4.bb.a ✓ 280 240.bb even 4 1 inner
240.4.bf.a yes 280 5.c odd 4 1
240.4.bf.a yes 280 15.e even 4 1
240.4.bf.a yes 280 16.e even 4 1
240.4.bf.a yes 280 48.i odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(240, [\chi])\).