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Properties

Label 280.3.bi.c
Level $280$
Weight $3$
Character orbit 280.bi
Analytic conductor $7.629$
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] = [280,3,Mod(179,280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(280, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("280.179");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 280.bi (of order \(6\), 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: \(7.62944740209\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(88\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 176 q + 14 q^{4} - 24 q^{6} + 284 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 176 q + 14 q^{4} - 24 q^{6} + 284 q^{9} - 24 q^{10} + 32 q^{11} + 2 q^{14} + 50 q^{16} + 48 q^{20} + 16 q^{24} + 98 q^{25} - 90 q^{26} - 32 q^{30} - 256 q^{34} + 154 q^{35} - 68 q^{36} - 84 q^{40} - 328 q^{41} + 174 q^{44} - 26 q^{46} + 240 q^{49} - 96 q^{50} - 76 q^{51} - 116 q^{54} + 228 q^{56} + 244 q^{59} + 90 q^{60} - 268 q^{64} + 8 q^{65} - 304 q^{66} + 98 q^{70} - 98 q^{74} - 38 q^{75} - 612 q^{76} + 112 q^{80} - 168 q^{81} - 20 q^{84} - 16 q^{86} + 20 q^{89} + 800 q^{90} - 280 q^{91} + 226 q^{94} - 408 q^{96} - 88 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1 −1.99950 + 0.0446276i 2.20583 + 1.27354i 3.99602 − 0.178466i −4.70583 − 1.68971i −4.46740 − 2.44800i −0.0496201 + 6.99982i −7.98208 + 0.535176i −1.25621 − 2.17582i 9.48473 + 3.16858i
179.2 −1.99947 − 0.0461599i −0.351343 − 0.202848i 3.99574 + 0.184590i 4.73765 + 1.59834i 0.693135 + 0.421805i −3.56482 − 6.02429i −7.98083 − 0.553525i −4.41771 − 7.65169i −9.39899 − 3.41452i
179.3 −1.99914 + 0.0586333i 3.69058 + 2.13076i 3.99312 − 0.234432i 4.38427 + 2.40379i −7.50293 − 4.04330i 4.99894 + 4.90006i −7.96907 + 0.702793i 4.58028 + 7.93327i −8.90571 − 4.54845i
179.4 −1.99735 − 0.102982i −2.72251 − 1.57184i 3.97879 + 0.411382i −4.39336 + 2.38713i 5.27592 + 3.41988i 1.49836 − 6.83776i −7.90466 − 1.23142i 0.441358 + 0.764454i 9.02090 − 4.31549i
179.5 −1.98991 + 0.200657i −1.00268 − 0.578895i 3.91947 − 0.798578i 0.934527 − 4.91189i 2.11139 + 0.950755i −5.85885 + 3.83065i −7.63915 + 2.37557i −3.82976 − 6.63334i −0.874020 + 9.96173i
179.6 −1.98672 − 0.230065i −3.41033 − 1.96896i 3.89414 + 0.914150i −2.13688 − 4.52037i 6.32240 + 4.69637i 6.60205 + 2.32658i −7.52627 − 2.71207i 3.25359 + 5.63538i 3.20542 + 9.47234i
179.7 −1.97899 + 0.289105i 4.96018 + 2.86376i 3.83284 − 1.14427i 2.00369 − 4.58096i −10.6441 − 4.23335i −4.51160 − 5.35215i −7.25435 + 3.37260i 11.9022 + 20.6153i −2.64091 + 9.64498i
179.8 −1.89161 + 0.649460i 2.89501 + 1.67143i 3.15640 − 2.45705i −4.99093 − 0.300983i −6.56177 − 1.28151i 5.66904 − 4.10633i −4.37494 + 6.69776i 1.08738 + 1.88340i 9.63639 − 2.67207i
179.9 −1.87713 + 0.690197i −2.06839 − 1.19419i 3.04726 − 2.59118i 2.16509 + 4.50693i 4.70687 + 0.814048i 4.51954 + 5.34545i −3.93167 + 6.96720i −1.64784 − 2.85414i −7.17483 − 6.96576i
179.10 −1.85318 − 0.752151i −4.22362 − 2.43851i 2.86854 + 2.78774i 1.52881 + 4.76054i 5.99299 + 7.69578i −6.27352 + 3.10531i −3.21911 − 7.32375i 7.39262 + 12.8044i 0.747478 − 9.97202i
179.11 −1.84854 − 0.763468i 3.95690 + 2.28452i 2.83423 + 2.82261i −2.18422 + 4.49769i −5.57035 − 7.24400i −6.96711 − 0.677745i −3.08424 − 7.38156i 5.93805 + 10.2850i 7.47147 − 6.64659i
179.12 −1.82405 − 0.820259i 1.83471 + 1.05927i 2.65435 + 2.99239i 0.390131 + 4.98476i −2.47773 − 3.43710i 6.56196 − 2.43735i −2.38714 − 7.63554i −2.25590 − 3.90734i 3.37717 − 9.41248i
179.13 −1.79219 − 0.887732i 1.68939 + 0.975368i 2.42386 + 3.18196i −0.436965 − 4.98087i −2.16183 − 3.24776i 2.96691 − 6.34015i −1.51929 − 7.85441i −2.59732 − 4.49868i −3.63856 + 9.31455i
179.14 −1.78147 + 0.909040i 0.872699 + 0.503853i 2.34729 − 3.23886i −2.59516 + 4.27377i −2.01271 − 0.104282i −6.99108 + 0.353338i −1.23738 + 7.90373i −3.99226 − 6.91481i 0.738178 − 9.97272i
179.15 −1.76168 + 0.946824i −4.66176 − 2.69147i 2.20705 − 3.33601i 4.95786 − 0.647782i 10.7609 + 0.327648i 2.76299 − 6.43163i −0.729513 + 7.96667i 9.98801 + 17.2997i −8.12084 + 5.83541i
179.16 −1.74805 − 0.971760i 1.31174 + 0.757334i 2.11136 + 3.39737i 4.64738 − 1.84441i −1.55704 − 2.59856i −2.92749 + 6.35844i −0.389340 − 7.99052i −3.35289 − 5.80738i −9.91618 − 1.29201i
179.17 −1.74227 − 0.982090i −3.22427 − 1.86153i 2.07100 + 3.42213i 4.73238 − 1.61386i 3.78935 + 6.40981i 5.82754 + 3.87812i −0.247390 − 7.99617i 2.43060 + 4.20993i −9.83004 − 1.83585i
179.18 −1.70081 + 1.05225i −4.66176 − 2.69147i 1.78554 − 3.57936i −4.95786 + 0.647782i 10.7609 − 0.327648i −2.76299 + 6.43163i 0.729513 + 7.96667i 9.98801 + 17.2997i 7.75077 − 6.31867i
179.19 −1.67799 + 1.08828i 0.872699 + 0.503853i 1.63129 − 3.65224i 2.59516 − 4.27377i −2.01271 + 0.104282i 6.99108 − 0.353338i 1.23738 + 7.90373i −3.99226 − 6.91481i 0.296412 + 9.99561i
179.20 −1.55679 − 1.25555i −1.88933 − 1.09081i 0.847203 + 3.90925i −4.70848 − 1.68234i 1.57174 + 4.07030i −6.98992 + 0.375549i 3.58933 − 7.14959i −2.12028 − 3.67244i 5.21786 + 8.53076i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 179.88
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
8.d odd 2 1 inner
35.j even 6 1 inner
40.e odd 2 1 inner
56.k odd 6 1 inner
280.bi odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.3.bi.c ✓ 176
5.b even 2 1 inner 280.3.bi.c ✓ 176
7.c even 3 1 inner 280.3.bi.c ✓ 176
8.d odd 2 1 inner 280.3.bi.c ✓ 176
35.j even 6 1 inner 280.3.bi.c ✓ 176
40.e odd 2 1 inner 280.3.bi.c ✓ 176
56.k odd 6 1 inner 280.3.bi.c ✓ 176
280.bi odd 6 1 inner 280.3.bi.c ✓ 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.3.bi.c ✓ 176 1.a even 1 1 trivial
280.3.bi.c ✓ 176 5.b even 2 1 inner
280.3.bi.c ✓ 176 7.c even 3 1 inner
280.3.bi.c ✓ 176 8.d odd 2 1 inner
280.3.bi.c ✓ 176 35.j even 6 1 inner
280.3.bi.c ✓ 176 40.e odd 2 1 inner
280.3.bi.c ✓ 176 56.k odd 6 1 inner
280.3.bi.c ✓ 176 280.bi odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(280, [\chi])\):

\( T_{3}^{88} - 269 T_{3}^{86} + 38942 T_{3}^{84} - 3897649 T_{3}^{82} + 298736174 T_{3}^{80} + \cdots + 38\!\cdots\!24 \) Copy content Toggle raw display
\( T_{13}^{44} - 3527 T_{13}^{42} + 5721799 T_{13}^{40} - 5675878877 T_{13}^{38} + 3859785844348 T_{13}^{36} + \cdots + 58\!\cdots\!36 \) Copy content Toggle raw display