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Properties

Label 240.3.bm.b
Level $240$
Weight $3$
Character orbit 240.bm
Analytic conductor $6.540$
Analytic rank $0$
Dimension $176$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,3,Mod(29,240)]
 
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, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.29");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 240.bm (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: \(176\)
Relative dimension: \(88\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 176 q - 8 q^{4} - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 176 q - 8 q^{4} - 4 q^{6} - 24 q^{10} - 124 q^{15} + 32 q^{16} - 64 q^{19} - 40 q^{21} - 152 q^{24} - 128 q^{30} - 16 q^{31} + 288 q^{34} - 392 q^{36} - 136 q^{40} - 52 q^{45} - 344 q^{46} + 1328 q^{49} - 208 q^{51} + 76 q^{54} - 72 q^{60} + 432 q^{61} - 32 q^{64} + 72 q^{66} + 440 q^{69} - 416 q^{70} + 92 q^{75} - 192 q^{76} - 1056 q^{79} + 640 q^{81} - 128 q^{84} - 384 q^{85} + 152 q^{90} - 400 q^{91} + 664 q^{94} + 520 q^{96} - 200 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.99857 − 0.0756676i −2.88555 + 0.820736i 3.98855 + 0.302454i −4.46008 − 2.26002i 5.82907 − 1.42195i 6.37621 −7.94850 − 0.906278i 7.65278 − 4.73655i 8.74276 + 4.85429i
29.2 −1.99857 − 0.0756676i 0.820736 − 2.88555i 3.98855 + 0.302454i 2.26002 + 4.46008i −1.85864 + 5.70486i −6.37621 −7.94850 − 0.906278i −7.65278 − 4.73655i −4.17932 − 9.08478i
29.3 −1.99007 + 0.199051i −2.56290 − 1.55935i 3.92076 − 0.792251i 0.346845 − 4.98796i 5.41073 + 2.59306i −10.7538 −7.64488 + 2.35707i 4.13688 + 7.99289i 0.302613 + 9.99542i
29.4 −1.99007 + 0.199051i −1.55935 − 2.56290i 3.92076 − 0.792251i 4.98796 − 0.346845i 3.61336 + 4.78995i 10.7538 −7.64488 + 2.35707i −4.13688 + 7.99289i −9.85734 + 1.68310i
29.5 −1.96631 − 0.365531i −0.313469 + 2.98358i 3.73277 + 1.43750i −4.35166 + 2.46232i 1.70697 − 5.75207i −3.69702 −6.81435 − 4.19101i −8.80347 − 1.87052i 9.45679 − 3.25103i
29.6 −1.96631 − 0.365531i 2.98358 − 0.313469i 3.73277 + 1.43750i −2.46232 + 4.35166i −5.98123 − 0.474213i 3.69702 −6.81435 − 4.19101i 8.80347 − 1.87052i 6.43236 − 7.65668i
29.7 −1.95938 + 0.401031i 1.30612 + 2.70075i 3.67835 − 1.57154i 3.97085 + 3.03848i −3.64228 − 4.76800i −6.46656 −6.57705 + 4.55438i −5.58808 + 7.05502i −8.99893 − 4.36110i
29.8 −1.95938 + 0.401031i 2.70075 + 1.30612i 3.67835 − 1.57154i −3.03848 − 3.97085i −5.81559 − 1.47611i 6.46656 −6.57705 + 4.55438i 5.58808 + 7.05502i 7.54597 + 6.56189i
29.9 −1.83945 − 0.785125i −2.43426 + 1.75339i 2.76716 + 2.88840i 4.73977 − 1.59203i 5.85433 − 1.31407i −3.64271 −2.82230 − 7.48563i 2.85127 − 8.53641i −9.96852 − 0.792857i
29.10 −1.83945 − 0.785125i 1.75339 − 2.43426i 2.76716 + 2.88840i 1.59203 − 4.73977i −5.13647 + 3.10108i 3.64271 −2.82230 − 7.48563i −2.85127 − 8.53641i −6.64977 + 7.46864i
29.11 −1.82687 + 0.813975i −2.55369 + 1.57438i 2.67489 − 2.97405i 2.35461 + 4.41087i 3.38374 − 4.95483i 4.35081 −2.46586 + 7.61049i 4.04265 − 8.04096i −7.89190 − 6.14149i
29.12 −1.82687 + 0.813975i 1.57438 − 2.55369i 2.67489 − 2.97405i −4.41087 − 2.35461i −0.797547 + 5.94676i −4.35081 −2.46586 + 7.61049i −4.04265 − 8.04096i 9.97468 + 0.711220i
29.13 −1.81797 + 0.833659i −0.863556 + 2.87302i 2.61003 − 3.03113i 2.34891 − 4.41391i −0.825202 − 5.94298i 3.90689 −2.21802 + 7.68638i −7.50854 − 4.96204i −0.590548 + 9.98255i
29.14 −1.81797 + 0.833659i 2.87302 − 0.863556i 2.61003 − 3.03113i 4.41391 − 2.34891i −4.50316 + 3.96504i −3.90689 −2.21802 + 7.68638i 7.50854 − 4.96204i −6.06617 + 7.94994i
29.15 −1.74055 − 0.985139i 1.94076 + 2.28767i 2.05900 + 3.42936i 4.87496 + 1.11122i −1.12430 − 5.89372i 11.5568 −0.205395 − 7.99736i −1.46691 + 8.87965i −7.39038 − 6.73664i
29.16 −1.74055 − 0.985139i 2.28767 + 1.94076i 2.05900 + 3.42936i −1.11122 − 4.87496i −2.06989 − 5.63166i −11.5568 −0.205395 − 7.99736i 1.46691 + 8.87965i −2.86838 + 9.57979i
29.17 −1.71783 − 1.02424i −2.86515 − 0.889334i 1.90187 + 3.51894i −0.191431 + 4.99633i 4.01094 + 4.46232i 4.77266 0.337157 − 7.99289i 7.41817 + 5.09615i 5.44629 − 8.38677i
29.18 −1.71783 − 1.02424i −0.889334 − 2.86515i 1.90187 + 3.51894i −4.99633 + 0.191431i −1.40688 + 5.83273i −4.77266 0.337157 − 7.99289i −7.41817 + 5.09615i 8.77891 + 4.78860i
29.19 −1.41867 + 1.40975i 0.869256 + 2.87131i 0.0252249 − 3.99992i −3.72009 + 3.34080i −5.28100 − 2.84799i 11.3807 5.60309 + 5.71011i −7.48879 + 4.99180i 0.567883 − 9.98386i
29.20 −1.41867 + 1.40975i 2.87131 + 0.869256i 0.0252249 − 3.99992i −3.34080 + 3.72009i −5.29885 + 2.81463i −11.3807 5.60309 + 5.71011i 7.48879 + 4.99180i −0.504911 − 9.98725i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.88
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner
80.q even 4 1 inner
240.bm 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.bm.b ✓ 176
3.b odd 2 1 inner 240.3.bm.b ✓ 176
5.b even 2 1 inner 240.3.bm.b ✓ 176
15.d odd 2 1 inner 240.3.bm.b ✓ 176
16.e even 4 1 inner 240.3.bm.b ✓ 176
48.i odd 4 1 inner 240.3.bm.b ✓ 176
80.q even 4 1 inner 240.3.bm.b ✓ 176
240.bm odd 4 1 inner 240.3.bm.b ✓ 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.3.bm.b ✓ 176 1.a even 1 1 trivial
240.3.bm.b ✓ 176 3.b odd 2 1 inner
240.3.bm.b ✓ 176 5.b even 2 1 inner
240.3.bm.b ✓ 176 15.d odd 2 1 inner
240.3.bm.b ✓ 176 16.e even 4 1 inner
240.3.bm.b ✓ 176 48.i odd 4 1 inner
240.3.bm.b ✓ 176 80.q even 4 1 inner
240.3.bm.b ✓ 176 240.bm odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{44} - 1244 T_{7}^{42} + 710608 T_{7}^{40} - 247364768 T_{7}^{38} + 58742905960 T_{7}^{36} + \cdots + 85\!\cdots\!24 \) acting on \(S_{3}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display