↑

Properties

Label 287.3.bd.a
Level $287$
Weight $3$
Character orbit 287.bd
Analytic conductor $7.820$
Analytic rank $0$
Dimension $864$
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] = [287,3,Mod(5,287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(287, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([50, 33]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("287.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 287 = 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 287.bd (of order \(60\), degree \(16\), 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: \(7.82018358714\)
Analytic rank: \(0\)
Dimension: \(864\)
Relative dimension: \(54\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 864 q - 10 q^{2} - 24 q^{3} - 214 q^{4} - 30 q^{5} - 16 q^{7} - 40 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 864 q - 10 q^{2} - 24 q^{3} - 214 q^{4} - 30 q^{5} - 16 q^{7} - 40 q^{8} - 18 q^{10} - 186 q^{14} - 56 q^{15} + 362 q^{16} - 78 q^{17} - 54 q^{18} + 48 q^{19} - 20 q^{21} + 40 q^{22} - 6 q^{23} - 138 q^{24} + 454 q^{25} - 66 q^{26} + 74 q^{28} - 640 q^{29} - 22 q^{30} + 54 q^{31} - 180 q^{33} - 142 q^{35} - 360 q^{36} - 156 q^{37} - 6 q^{38} - 10 q^{39} - 300 q^{40} - 200 q^{42} + 320 q^{43} + 112 q^{44} - 210 q^{45} + 490 q^{46} + 252 q^{47} + 160 q^{49} + 168 q^{51} + 276 q^{52} + 234 q^{53} - 1164 q^{54} - 110 q^{56} - 656 q^{57} + 106 q^{58} + 378 q^{59} - 486 q^{60} - 30 q^{61} - 480 q^{63} + 720 q^{64} + 42 q^{65} + 2442 q^{66} + 284 q^{67} - 2058 q^{68} + 642 q^{70} + 524 q^{71} + 82 q^{72} - 10 q^{74} - 1512 q^{75} - 640 q^{77} + 1488 q^{78} - 18 q^{79} - 30 q^{80} + 2608 q^{81} + 672 q^{82} - 1420 q^{84} - 44 q^{85} + 202 q^{86} - 30 q^{87} - 742 q^{88} + 1314 q^{89} + 492 q^{92} - 768 q^{93} - 3666 q^{94} - 288 q^{95} + 6492 q^{96} - 690 q^{98} - 1700 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −3.86332 + 0.406051i 0.385359 + 0.103257i 10.8478 − 2.30576i 5.36436 + 5.95772i −1.53069 − 0.242438i 0.114923 − 6.99906i −26.1942 + 8.51102i −7.65639 − 4.42042i −23.1434 − 20.8384i
5.2 −3.78322 + 0.397633i −2.36963 − 0.634941i 10.2421 − 2.17702i −4.89261 − 5.43380i 9.21731 + 1.45988i −4.50728 + 5.35578i −23.4109 + 7.60666i −2.58223 − 1.49085i 20.6705 + 18.6118i
5.3 −3.67223 + 0.385966i 3.60494 + 0.965940i 9.42368 − 2.00307i −3.30438 − 3.66989i −13.6110 − 2.15577i 6.98845 − 0.401949i −19.7858 + 6.42881i 4.26831 + 2.46431i 13.5509 + 12.2013i
5.4 −3.52452 + 0.370441i −5.59949 − 1.50038i 8.37239 − 1.77961i 5.79187 + 6.43253i 20.2913 + 3.21382i −3.57911 + 6.01581i −15.3675 + 4.99319i 21.3089 + 12.3027i −22.7964 − 20.5260i
5.5 −3.51461 + 0.369401i 3.58086 + 0.959489i 8.30344 − 1.76495i −1.51949 − 1.68757i −12.9398 − 2.04946i −6.56906 − 2.41814i −15.0874 + 4.90219i 4.10772 + 2.37160i 5.96381 + 5.36983i
5.6 −3.25516 + 0.342131i −2.98484 − 0.799785i 6.56644 − 1.39574i −1.83918 − 2.04262i 9.98976 + 1.58222i −2.67170 − 6.47009i −8.44572 + 2.74418i 0.475370 + 0.274455i 6.68568 + 6.01981i
5.7 −3.25016 + 0.341606i −3.96280 − 1.06183i 6.53427 − 1.38890i −1.12109 − 1.24510i 13.2425 + 2.09740i 6.77770 − 1.75009i −8.33052 + 2.70675i 6.78206 + 3.91562i 4.06906 + 3.66380i
5.8 −3.15003 + 0.331082i 5.01583 + 1.34399i 5.90050 − 1.25419i 3.23952 + 3.59786i −16.2450 − 2.57296i 4.02106 + 5.72984i −6.12207 + 1.98918i 15.5580 + 8.98244i −11.3958 − 10.2608i
5.9 −3.11963 + 0.327886i 1.13607 + 0.304409i 5.71200 − 1.21412i 1.95582 + 2.17215i −3.64393 − 0.577142i −4.33634 + 5.49510i −5.48808 + 1.78318i −6.59624 − 3.80834i −6.81364 − 6.13503i
5.10 −2.89312 + 0.304080i −0.510002 − 0.136655i 4.36511 − 0.927833i 3.45934 + 3.84199i 1.51705 + 0.240278i 6.80668 + 1.63372i −1.27995 + 0.415880i −7.55280 − 4.36061i −11.1766 − 10.0634i
5.11 −2.46927 + 0.259531i 0.688721 + 0.184542i 2.11735 − 0.450057i −5.13570 − 5.70377i −1.74853 − 0.276940i 2.59590 + 6.50087i 4.33389 − 1.40817i −7.35395 − 4.24580i 14.1617 + 12.7513i
5.12 −2.38273 + 0.250435i 1.40604 + 0.376749i 1.70209 − 0.361791i −4.10188 − 4.55560i −3.44457 − 0.545567i 3.31786 − 6.16375i 5.14935 − 1.67313i −5.95921 − 3.44055i 10.9146 + 9.82752i
5.13 −2.24269 + 0.235717i 4.18226 + 1.12063i 1.06153 − 0.225634i 5.21001 + 5.78631i −9.64368 − 1.52741i −2.17446 − 6.65370i 6.25122 − 2.03114i 8.44125 + 4.87356i −13.0484 − 11.7488i
5.14 −2.22184 + 0.233525i −2.69284 − 0.721545i 0.969471 − 0.206067i 2.30180 + 2.55641i 6.15158 + 0.974314i −6.38644 − 2.86590i 6.39307 − 2.07723i −1.06346 − 0.613988i −5.71124 − 5.14242i
5.15 −2.19003 + 0.230182i 5.08333 + 1.36208i 0.830673 − 0.176565i −3.40707 − 3.78394i −11.4462 − 1.81290i −6.83271 − 1.52119i 6.59871 − 2.14405i 16.1908 + 9.34777i 8.33260 + 7.50271i
5.16 −1.90739 + 0.200474i 1.60329 + 0.429599i −0.314656 + 0.0668823i 1.19878 + 1.33139i −3.14421 − 0.497994i 4.18213 − 5.61335i 7.88286 − 2.56130i −5.40825 − 3.12246i −2.55345 − 2.29914i
5.17 −1.84676 + 0.194102i −4.62713 − 1.23984i −0.539743 + 0.114726i −4.15770 − 4.61759i 8.78585 + 1.39154i 5.25002 + 4.63004i 8.03870 − 2.61193i 12.0789 + 6.97376i 8.57456 + 7.72057i
5.18 −1.80705 + 0.189928i −2.98183 − 0.798979i −0.683242 + 0.145228i 0.710415 + 0.788996i 5.54006 + 0.877459i −2.34483 + 6.59559i 8.11935 − 2.63814i 0.458720 + 0.264842i −1.43361 − 1.29082i
5.19 −1.32950 + 0.139736i −1.23979 − 0.332200i −2.16455 + 0.460089i −5.09510 − 5.65868i 1.69472 + 0.268417i −6.54965 − 2.47023i 7.89905 − 2.56656i −6.36751 − 3.67628i 7.56465 + 6.81124i
5.20 −1.22238 + 0.128477i −2.41703 − 0.647641i −2.43489 + 0.517552i 6.07573 + 6.74778i 3.03773 + 0.481129i 6.66096 + 2.15212i 7.58568 − 2.46474i −2.37164 − 1.36927i −8.29377 − 7.46774i
See next 80 embeddings (of 864 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
41.g even 20 1 inner
287.bd odd 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.3.bd.a ✓ 864
7.d odd 6 1 inner 287.3.bd.a ✓ 864
41.g even 20 1 inner 287.3.bd.a ✓ 864
287.bd odd 60 1 inner 287.3.bd.a ✓ 864
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.3.bd.a ✓ 864 1.a even 1 1 trivial
287.3.bd.a ✓ 864 7.d odd 6 1 inner
287.3.bd.a ✓ 864 41.g even 20 1 inner
287.3.bd.a ✓ 864 287.bd odd 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(287, [\chi])\).