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Properties

Label 240.3.be.a
Level $240$
Weight $3$
Character orbit 240.be
Analytic conductor $6.540$
Analytic rank $0$
Dimension $96$
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] = [240,3,Mod(133,240)]
 
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, 1, 0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.133");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 240.be (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: \(6.53952634465\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 4 q^{4} + 12 q^{8} + 288 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 4 q^{4} + 12 q^{8} + 288 q^{9} + 24 q^{12} - 28 q^{16} + 32 q^{19} + 84 q^{20} + 4 q^{22} + 124 q^{28} + 120 q^{30} + 140 q^{32} - 148 q^{34} - 96 q^{35} + 12 q^{36} - 104 q^{38} - 284 q^{40} - 60 q^{42} - 48 q^{44} - 28 q^{46} - 144 q^{48} - 416 q^{50} + 96 q^{51} - 352 q^{52} + 336 q^{56} - 452 q^{58} + 128 q^{59} + 32 q^{61} - 8 q^{62} - 44 q^{64} - 72 q^{66} - 352 q^{68} - 96 q^{69} - 676 q^{70} + 36 q^{72} + 96 q^{73} + 32 q^{74} + 192 q^{75} - 148 q^{76} - 216 q^{78} + 404 q^{80} + 864 q^{81} + 328 q^{82} - 320 q^{83} + 216 q^{84} - 48 q^{86} + 140 q^{88} - 384 q^{91} + 408 q^{92} - 340 q^{94} - 768 q^{95} + 768 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
133.1 −1.99999 + 0.00577463i 1.73205 3.99993 − 0.0230984i −4.84784 − 1.22411i −3.46409 + 0.0100020i 1.49053 + 1.49053i −7.99970 + 0.0692948i 3.00000 9.70271 + 2.42022i
133.2 −1.96797 − 0.356513i −1.73205 3.74580 + 1.40321i 3.61750 − 3.45162i 3.40862 + 0.617498i −5.73063 − 5.73063i −6.87135 − 4.09690i 3.00000 −8.34967 + 5.50299i
133.3 −1.96665 + 0.363737i 1.73205 3.73539 − 1.43068i 4.37932 + 2.41279i −3.40633 + 0.630011i 2.06325 + 2.06325i −6.82580 + 4.17235i 3.00000 −9.49019 − 3.15219i
133.4 −1.95539 − 0.420062i −1.73205 3.64710 + 1.64277i −4.78583 + 1.44770i 3.38683 + 0.727568i −3.97229 − 3.97229i −6.44143 − 4.74426i 3.00000 9.96628 − 0.820477i
133.5 −1.88123 − 0.678948i −1.73205 3.07806 + 2.55452i 3.32189 + 3.73699i 3.25839 + 1.17597i 7.38646 + 7.38646i −4.05615 − 6.89548i 3.00000 −3.71201 − 9.28552i
133.6 −1.83332 − 0.799347i 1.73205 2.72209 + 2.93091i 4.62199 + 1.90716i −3.17539 − 1.38451i −7.59279 − 7.59279i −2.64763 − 7.54918i 3.00000 −6.94908 − 7.19099i
133.7 −1.80323 + 0.865089i −1.73205 2.50324 − 3.11990i 4.81867 − 1.33432i 3.12328 − 1.49838i 1.99079 + 1.99079i −1.81492 + 7.79141i 3.00000 −7.53485 + 6.57466i
133.8 −1.73269 − 0.998890i 1.73205 2.00444 + 3.46154i 2.61912 − 4.25913i −3.00111 − 1.73013i 6.38794 + 6.38794i −0.0153792 − 7.99999i 3.00000 −8.79253 + 4.76354i
133.9 −1.64771 + 1.13360i 1.73205 1.42989 − 3.73569i −4.67987 + 1.76034i −2.85392 + 1.96346i −7.28155 − 7.28155i 1.87875 + 7.77627i 3.00000 5.71554 − 8.20564i
133.10 −1.64544 + 1.13690i −1.73205 1.41492 − 3.74139i −3.09795 + 3.92462i 2.84998 − 1.96917i 2.31501 + 2.31501i 1.92544 + 7.76484i 3.00000 0.635577 − 9.97978i
133.11 −1.51353 + 1.30738i 1.73205 0.581525 − 3.95750i −0.202899 + 4.99588i −2.62150 + 2.26445i 7.08950 + 7.08950i 4.29380 + 6.75006i 3.00000 −6.22441 − 7.82666i
133.12 −1.48332 − 1.34155i 1.73205 0.400485 + 3.97990i −2.57336 + 4.28694i −2.56919 − 2.32363i −4.39445 − 4.39445i 4.74519 − 6.44074i 3.00000 9.56826 − 2.90661i
133.13 −1.45762 − 1.36943i −1.73205 0.249325 + 3.99222i −0.972196 − 4.90457i 2.52468 + 2.37192i 2.94220 + 2.94220i 5.10365 − 6.16058i 3.00000 −5.29937 + 8.48037i
133.14 −1.40769 + 1.42071i 1.73205 −0.0368163 − 3.99983i 1.00692 − 4.89756i −2.43819 + 2.46074i −0.348793 − 0.348793i 5.73441 + 5.57822i 3.00000 5.54057 + 8.32479i
133.15 −1.31157 + 1.50990i −1.73205 −0.559569 − 3.96067i −1.07824 − 4.88236i 2.27171 − 2.61522i −2.93846 − 2.93846i 6.71411 + 4.34980i 3.00000 8.78603 + 4.77552i
133.16 −0.995163 − 1.73483i −1.73205 −2.01930 + 3.45289i −0.969482 + 4.90511i 1.72367 + 3.00482i −1.99465 − 1.99465i 7.99972 + 0.0669702i 3.00000 9.47435 − 3.19949i
133.17 −0.936155 − 1.76737i 1.73205 −2.24723 + 3.30907i −4.87785 − 1.09845i −1.62147 − 3.06118i 4.66967 + 4.66967i 7.95213 + 0.873890i 3.00000 2.62505 + 9.64931i
133.18 −0.934839 + 1.76807i −1.73205 −2.25215 − 3.30572i 3.95993 + 3.05270i 1.61919 − 3.06239i −6.71996 − 6.71996i 7.95016 − 0.891646i 3.00000 −9.09928 + 4.14765i
133.19 −0.587204 + 1.91186i 1.73205 −3.31038 − 2.24530i −4.52664 − 2.12357i −1.01707 + 3.31143i 6.85443 + 6.85443i 6.23656 − 5.01053i 3.00000 6.71801 − 7.40731i
133.20 −0.545019 − 1.92431i 1.73205 −3.40591 + 2.09757i 0.962977 − 4.90639i −0.944001 − 3.33300i −6.59004 − 6.59004i 5.89265 + 5.41079i 3.00000 −9.96624 + 0.821015i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 133.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.t odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.3.be.a yes 96
4.b odd 2 1 960.3.be.a 96
5.c odd 4 1 240.3.ba.a ✓ 96
16.e even 4 1 240.3.ba.a ✓ 96
16.f odd 4 1 960.3.ba.a 96
20.e even 4 1 960.3.ba.a 96
80.j even 4 1 960.3.be.a 96
80.t odd 4 1 inner 240.3.be.a yes 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.3.ba.a ✓ 96 5.c odd 4 1
240.3.ba.a ✓ 96 16.e even 4 1
240.3.be.a yes 96 1.a even 1 1 trivial
240.3.be.a yes 96 80.t odd 4 1 inner
960.3.ba.a 96 16.f odd 4 1
960.3.ba.a 96 20.e even 4 1
960.3.be.a 96 4.b odd 2 1
960.3.be.a 96 80.j even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(240, [\chi])\).