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Properties

Label 480.2.bs.b
Level $480$
Weight $2$
Character orbit 480.bs
Analytic conductor $3.833$
Analytic rank $0$
Dimension $352$
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] = [480,2,Mod(59,480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(480, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 1, 4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("480.59");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 480.bs (of order \(8\), 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: \(3.83281929702\)
Analytic rank: \(0\)
Dimension: \(352\)
Relative dimension: \(88\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 352 q - 16 q^{4} - 8 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 352 q - 16 q^{4} - 8 q^{6} - 8 q^{9} - 48 q^{10} - 8 q^{15} - 16 q^{16} - 16 q^{19} - 8 q^{21} + 64 q^{24} - 8 q^{25} + 28 q^{30} + 16 q^{34} - 88 q^{36} - 8 q^{39} - 40 q^{40} - 4 q^{45} - 16 q^{46} + 112 q^{51} + 64 q^{54} - 40 q^{55} - 48 q^{60} - 64 q^{61} - 64 q^{64} - 56 q^{66} + 40 q^{69} - 56 q^{70} - 4 q^{75} - 64 q^{76} - 320 q^{79} - 104 q^{84} + 32 q^{85} - 64 q^{90} - 16 q^{91} + 48 q^{94} - 40 q^{96} - 72 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 −1.41406 + 0.0210692i −0.480423 − 1.66409i 1.99911 − 0.0595860i 1.78194 + 1.35081i 0.714406 + 2.34299i −1.71470 − 1.71470i −2.82560 + 0.126378i −2.53839 + 1.59893i −2.54823 − 1.87257i
59.2 −1.41406 + 0.0210692i 0.836978 + 1.51640i 1.99911 − 0.0595860i 2.21519 + 0.304860i −1.21548 − 2.12664i 1.71470 + 1.71470i −2.82560 + 0.126378i −1.59893 + 2.53839i −3.13883 − 0.384417i
59.3 −1.41080 + 0.0982671i −1.34091 − 1.09635i 1.98069 − 0.277270i 0.399802 − 2.20004i 1.99948 + 1.41495i 1.70685 + 1.70685i −2.76710 + 0.585807i 0.596054 + 2.94019i −0.347848 + 3.14309i
59.4 −1.41080 + 0.0982671i −0.172930 + 1.72340i 1.98069 − 0.277270i −1.27296 + 1.83836i 0.0746158 − 2.44835i −1.70685 − 1.70685i −2.76710 + 0.585807i −2.94019 − 0.596054i 1.61523 − 2.71864i
59.5 −1.40184 + 0.186669i 1.11815 − 1.32277i 1.93031 − 0.523359i 1.42813 − 1.72059i −1.32055 + 2.06304i 0.992713 + 0.992713i −2.60829 + 1.09399i −0.499461 − 2.95813i −1.68083 + 2.67858i
59.6 −1.40184 + 0.186669i 1.72600 + 0.144688i 1.93031 − 0.523359i −0.206800 + 2.22648i −2.44658 + 0.119361i −0.992713 − 0.992713i −2.60829 + 1.09399i 2.95813 + 0.499461i −0.125715 − 3.15978i
59.7 −1.33838 − 0.456881i −0.500017 − 1.65831i 1.58252 + 1.22296i −2.16457 + 0.560936i −0.0884361 + 2.44789i 1.85933 + 1.85933i −1.55927 − 2.35981i −2.49997 + 1.65836i 3.15329 + 0.238203i
59.8 −1.33838 − 0.456881i 0.819035 + 1.52617i 1.58252 + 1.22296i −1.13394 − 1.92722i −0.398904 − 2.41679i −1.85933 − 1.85933i −1.55927 − 2.35981i −1.65836 + 2.49997i 0.637129 + 3.09743i
59.9 −1.28621 − 0.587932i −1.73202 − 0.0108927i 1.30867 + 1.51241i 0.0960729 + 2.23400i 2.22133 + 1.03232i −0.104388 − 0.104388i −0.794033 − 2.71468i 2.99976 + 0.0377325i 1.18987 − 2.92988i
59.10 −1.28621 − 0.587932i −1.21702 + 1.23242i 1.30867 + 1.51241i 1.64761 − 1.51174i 2.28992 − 0.869630i 0.104388 + 0.104388i −0.794033 − 2.71468i −0.0377325 − 2.99976i −3.00798 + 0.975737i
59.11 −1.27573 + 0.610347i −1.72686 + 0.133944i 1.25495 − 1.55727i 1.79178 + 1.33773i 2.12125 − 1.22486i 3.45190 + 3.45190i −0.650498 + 2.75261i 2.96412 − 0.462605i −3.10230 − 0.612973i
59.12 −1.27573 + 0.610347i −1.31579 + 1.12636i 1.25495 − 1.55727i 2.21290 + 0.321059i 0.991113 − 2.24002i −3.45190 − 3.45190i −0.650498 + 2.75261i 0.462605 − 2.96412i −3.01901 + 0.941054i
59.13 −1.21194 + 0.728843i −1.35408 − 1.08003i 0.937575 − 1.76662i −1.81257 + 1.30943i 2.42823 + 0.322013i −0.339133 − 0.339133i 0.151311 + 2.82438i 0.667072 + 2.92490i 1.24235 − 2.90802i
59.14 −1.21194 + 0.728843i −0.193784 + 1.72118i 0.937575 − 1.76662i −0.355776 − 2.20758i −1.01961 − 2.22719i 0.339133 + 0.339133i 0.151311 + 2.82438i −2.92490 − 0.667072i 2.04016 + 2.41614i
59.15 −1.20071 + 0.747184i 0.521825 − 1.65157i 0.883431 − 1.79431i −1.76297 − 1.37548i 0.607468 + 2.37297i −2.15964 − 2.15964i 0.279932 + 2.81454i −2.45540 − 1.72367i 3.14456 + 0.334300i
59.16 −1.20071 + 0.747184i 1.53683 + 0.798854i 0.883431 − 1.79431i −2.21922 − 0.273993i −2.44218 + 0.189097i 2.15964 + 2.15964i 0.279932 + 2.81454i 1.72367 + 2.45540i 2.86937 − 1.32918i
59.17 −1.15076 − 0.822045i 0.804086 − 1.53409i 0.648485 + 1.89195i 1.54506 + 1.61641i −2.18640 + 1.10438i 1.65295 + 1.65295i 0.809018 − 2.71026i −1.70689 − 2.46709i −0.449223 − 3.13021i
59.18 −1.15076 − 0.822045i 1.65334 + 0.516194i 0.648485 + 1.89195i 2.23550 − 0.0504553i −1.47826 − 1.95314i −1.65295 − 1.65295i 0.809018 − 2.71026i 2.46709 + 1.70689i −2.61399 − 1.77962i
59.19 −1.07175 − 0.922690i −0.499713 − 1.65840i 0.297287 + 1.97778i 0.337416 − 2.21046i −0.994622 + 2.23847i −3.13377 − 3.13377i 1.50626 − 2.39399i −2.50057 + 1.65745i −2.40120 + 2.05773i
59.20 −1.07175 − 0.922690i 0.819315 + 1.52602i 0.297287 + 1.97778i −1.32444 + 1.80162i 0.529940 − 2.39148i 3.13377 + 3.13377i 1.50626 − 2.39399i −1.65745 + 2.50057i 3.08181 − 0.708834i
See next 80 embeddings (of 352 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.88
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
32.h odd 8 1 inner
96.o even 8 1 inner
160.y odd 8 1 inner
480.bs even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 480.2.bs.b ✓ 352
3.b odd 2 1 inner 480.2.bs.b ✓ 352
5.b even 2 1 inner 480.2.bs.b ✓ 352
15.d odd 2 1 inner 480.2.bs.b ✓ 352
32.h odd 8 1 inner 480.2.bs.b ✓ 352
96.o even 8 1 inner 480.2.bs.b ✓ 352
160.y odd 8 1 inner 480.2.bs.b ✓ 352
480.bs even 8 1 inner 480.2.bs.b ✓ 352
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.bs.b ✓ 352 1.a even 1 1 trivial
480.2.bs.b ✓ 352 3.b odd 2 1 inner
480.2.bs.b ✓ 352 5.b even 2 1 inner
480.2.bs.b ✓ 352 15.d odd 2 1 inner
480.2.bs.b ✓ 352 32.h odd 8 1 inner
480.2.bs.b ✓ 352 96.o even 8 1 inner
480.2.bs.b ✓ 352 160.y odd 8 1 inner
480.2.bs.b ✓ 352 480.bs even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{176} + 5176 T_{7}^{172} + 12459248 T_{7}^{168} + 18564993728 T_{7}^{164} + 19230427117776 T_{7}^{160} + \cdots + 42\!\cdots\!96 \) acting on \(S_{2}^{\mathrm{new}}(480, [\chi])\). Copy content Toggle raw display