↑

Properties

Label 407.2.bg.a
Level $407$
Weight $2$
Character orbit 407.bg
Analytic conductor $3.250$
Analytic rank $0$
Dimension $864$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [407,2,Mod(3,407)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(407, base_ring=CyclotomicField(90))
 
chi = DirichletCharacter(H, H._module([72, 65]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("407.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 407 = 11 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 407.bg (of order \(90\), degree \(24\), 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: \(3.24991136227\)
Analytic rank: \(0\)
Dimension: \(864\)
Relative dimension: \(36\) over \(\Q(\zeta_{90})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{90}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 864 q - 12 q^{2} - 15 q^{3} - 24 q^{4} - 18 q^{5} - 18 q^{7} - 45 q^{8} - 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 864 q - 12 q^{2} - 15 q^{3} - 24 q^{4} - 18 q^{5} - 18 q^{7} - 45 q^{8} - 21 q^{9} - 72 q^{10} - 12 q^{11} - 24 q^{12} - 18 q^{13} + 27 q^{14} - 9 q^{15} - 54 q^{16} - 18 q^{17} - 9 q^{18} - 30 q^{19} - 66 q^{21} - 15 q^{22} - 72 q^{23} - 48 q^{24} + 54 q^{25} - 9 q^{26} - 21 q^{27} - 51 q^{28} - 27 q^{29} + 45 q^{30} - 60 q^{32} - 39 q^{33} - 138 q^{34} + 18 q^{35} + 120 q^{36} + 78 q^{37} - 120 q^{38} - 93 q^{39} + 48 q^{40} - 9 q^{41} - 96 q^{42} + 180 q^{44} - 180 q^{45} - 36 q^{46} - 9 q^{47} - 45 q^{48} - 18 q^{49} + 6 q^{50} - 27 q^{51} + 15 q^{52} + 3 q^{53} + 60 q^{54} - 162 q^{55} - 162 q^{56} - 24 q^{57} - 144 q^{58} - 27 q^{59} + 135 q^{60} + 6 q^{61} - 18 q^{62} + 123 q^{63} - 33 q^{64} - 66 q^{65} + 36 q^{66} - 36 q^{67} - 249 q^{69} + 111 q^{70} + 93 q^{71} + 24 q^{72} + 12 q^{73} - 156 q^{74} + 228 q^{75} + 72 q^{76} + 132 q^{77} - 390 q^{78} + 72 q^{79} - 12 q^{81} + 243 q^{82} - 45 q^{83} + 111 q^{84} - 69 q^{85} + 84 q^{86} + 6 q^{87} - 99 q^{88} + 54 q^{89} - 93 q^{90} + 84 q^{91} - 96 q^{92} - 75 q^{93} + 180 q^{94} - 150 q^{95} + 114 q^{96} + 9 q^{97} - 270 q^{98} - 270 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −0.188081 + 2.68968i 2.08686 + 1.30402i −5.21846 − 0.733407i −0.858893 + 0.579331i −3.89988 + 5.36773i −0.388925 + 0.497802i 1.83296 − 8.62339i 1.33942 + 2.74622i −1.39667 − 2.41911i
3.2 −0.183712 + 2.62721i −0.0273352 − 0.0170809i −4.88792 − 0.686953i 2.62411 − 1.76998i 0.0498969 − 0.0686772i 1.11991 − 1.43342i 1.60762 − 7.56324i −1.31466 − 2.69545i 4.16803 + 7.21924i
3.3 −0.180360 + 2.57927i −2.23026 − 1.39362i −4.63957 − 0.652050i −0.988956 + 0.667059i 3.99677 − 5.50109i −3.16389 + 4.04959i 1.44347 − 6.79098i 1.71676 + 3.51989i −1.54216 − 2.67110i
3.4 −0.178010 + 2.54567i −1.29518 − 0.809321i −4.46819 − 0.627964i −2.31629 + 1.56236i 2.29082 − 3.15304i 2.51298 − 3.21646i 1.33284 − 6.27051i −0.292612 − 0.599943i −3.56492 − 6.17463i
3.5 −0.149921 + 2.14397i 0.0650918 + 0.0406739i −2.59360 − 0.364506i −1.64674 + 1.11074i −0.0969622 + 0.133457i −0.317445 + 0.406310i 0.276636 − 1.30147i −1.31253 − 2.69109i −2.13452 − 3.69709i
3.6 −0.146343 + 2.09281i 1.34850 + 0.842638i −2.37788 − 0.334190i 1.91304 − 1.29036i −1.96082 + 2.69884i −3.06069 + 3.91750i 0.175018 − 0.823397i −0.206692 − 0.423781i 2.42052 + 4.19246i
3.7 −0.138423 + 1.97955i −2.35151 − 1.46939i −1.91891 − 0.269686i 0.608038 − 0.410127i 3.23423 − 4.45153i −0.0882188 + 0.112915i −0.0256743 + 0.120788i 2.05540 + 4.21419i 0.727699 + 1.26041i
3.8 −0.125590 + 1.79602i 1.53970 + 0.962111i −1.22939 − 0.172779i 1.66411 − 1.12246i −1.92135 + 2.64450i 2.71530 − 3.47542i −0.283937 + 1.33582i 0.129904 + 0.266343i 1.80697 + 3.12976i
3.9 −0.118628 + 1.69645i 2.59585 + 1.62207i −0.883348 − 0.124147i −1.48335 + 1.00053i −3.05971 + 4.21132i −0.626199 + 0.801499i −0.391749 + 1.84304i 2.79223 + 5.72492i −1.52139 − 2.63512i
3.10 −0.101164 + 1.44671i −1.02244 − 0.638890i −0.102192 − 0.0143622i 0.772145 − 0.520818i 1.02772 − 1.41454i 0.377664 − 0.483388i −0.571928 + 2.69071i −0.677915 − 1.38993i 0.675359 + 1.16976i
3.11 −0.0866898 + 1.23972i −2.43639 − 1.52243i 0.451143 + 0.0634039i −1.69069 + 1.14038i 2.09860 − 2.88847i 2.13849 − 2.73715i −0.634477 + 2.98498i 2.30311 + 4.72208i −1.26719 − 2.19484i
3.12 −0.0846847 + 1.21105i 1.61464 + 1.00894i 0.521071 + 0.0732318i −2.46762 + 1.66443i −1.35861 + 1.86997i 1.37358 − 1.75810i −0.637626 + 2.99979i 0.273994 + 0.561771i −1.80674 − 3.12936i
3.13 −0.0805366 + 1.15173i −0.172091 − 0.107535i 0.660546 + 0.0928337i −2.74975 + 1.85473i 0.137710 − 0.189542i −1.27325 + 1.62968i −0.640202 + 3.01191i −1.29706 − 2.65937i −1.91469 − 3.31634i
3.14 −0.0769167 + 1.09996i −2.09887 − 1.31152i 0.776539 + 0.109135i 3.40121 − 2.29414i 1.60406 − 2.20780i −0.263420 + 0.337162i −0.638280 + 3.00287i 1.37006 + 2.80903i 2.26186 + 3.91765i
3.15 −0.0608940 + 0.870825i 1.70695 + 1.06662i 1.22591 + 0.172290i 2.45257 − 1.65428i −1.03278 + 1.42151i −0.959521 + 1.22813i −0.587679 + 2.76481i 0.460888 + 0.944960i 1.29124 + 2.23650i
3.16 −0.0460721 + 0.658862i −0.695105 − 0.434350i 1.54856 + 0.217636i −0.0859943 + 0.0580039i 0.318202 − 0.437967i −2.41705 + 3.09369i −0.489377 + 2.30234i −1.02060 − 2.09254i −0.0342546 − 0.0593307i
3.17 −0.0182550 + 0.261058i 0.618736 + 0.386629i 1.91272 + 0.268815i 0.372995 − 0.251589i −0.112228 + 0.154468i 1.10825 − 1.41849i −0.213912 + 1.00638i −1.08176 − 2.21794i 0.0588702 + 0.101966i
3.18 −0.000404257 0.00578114i −1.04690 − 0.654178i 1.98050 + 0.278342i 1.98320 − 1.33768i 0.00420512 − 0.00578785i 2.83455 − 3.62805i −0.00481957 + 0.0226743i −0.647054 − 1.32666i 0.00693162 + 0.0120059i
3.19 0.00972099 − 0.139017i −1.54024 − 0.962451i 1.96131 + 0.275643i −2.62345 + 1.76954i −0.148769 + 0.204763i 0.575810 − 0.737004i 0.115332 − 0.542596i 0.130925 + 0.268436i 0.220493 + 0.381905i
3.20 0.0173868 − 0.248644i −2.84731 − 1.77920i 1.91901 + 0.269700i −0.619205 + 0.417659i −0.491892 + 0.677031i −1.42550 + 1.82456i 0.204069 − 0.960069i 3.62652 + 7.43547i 0.0930822 + 0.161223i
See next 80 embeddings (of 864 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
37.h even 18 1 inner
407.bg even 90 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 407.2.bg.a ✓ 864
11.c even 5 1 inner 407.2.bg.a ✓ 864
37.h even 18 1 inner 407.2.bg.a ✓ 864
407.bg even 90 1 inner 407.2.bg.a ✓ 864
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
407.2.bg.a ✓ 864 1.a even 1 1 trivial
407.2.bg.a ✓ 864 11.c even 5 1 inner
407.2.bg.a ✓ 864 37.h even 18 1 inner
407.2.bg.a ✓ 864 407.bg even 90 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(407, [\chi])\).