Properties

Label 528.2.t.b
Level $528$
Weight $2$
Character orbit 528.t
Analytic conductor $4.216$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [528,2,Mod(133,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("528.133");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 528.t (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: \(4.21610122672\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 12 q^{8} + 8 q^{10} + 16 q^{14} - 8 q^{15} + 20 q^{16} + 8 q^{19} - 8 q^{20} - 4 q^{22} - 16 q^{24} - 20 q^{28} + 20 q^{30} + 8 q^{31} + 40 q^{32} - 40 q^{33} + 20 q^{34} - 24 q^{35} + 4 q^{36} - 60 q^{38} - 56 q^{40} + 20 q^{42} + 8 q^{44} + 76 q^{46} - 64 q^{47} + 32 q^{48} - 40 q^{49} - 12 q^{50} - 8 q^{51} + 36 q^{52} - 4 q^{54} - 20 q^{56} + 12 q^{58} - 16 q^{59} - 12 q^{60} - 64 q^{61} + 68 q^{62} + 24 q^{63} + 60 q^{64} + 16 q^{65} + 24 q^{67} - 4 q^{68} - 16 q^{69} - 116 q^{70} - 12 q^{76} + 28 q^{78} - 72 q^{79} + 40 q^{80} - 40 q^{81} - 24 q^{82} + 16 q^{85} - 64 q^{86} - 12 q^{90} + 16 q^{91} + 20 q^{92} + 32 q^{93} + 104 q^{94} + 128 q^{95} - 20 q^{96} - 32 q^{97} + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
133.1 −1.40318 + 0.176320i −0.707107 + 0.707107i 1.93782 0.494818i −0.177505 0.177505i 0.867520 1.11687i 0.297421i −2.63186 + 1.03600i 1.00000i 0.280370 + 0.217774i
133.2 −1.39548 0.229436i 0.707107 0.707107i 1.89472 + 0.640345i −1.44191 1.44191i −1.14899 + 0.824517i 3.86773i −2.49712 1.32830i 1.00000i 1.68133 + 2.34298i
133.3 −1.27899 0.603477i 0.707107 0.707107i 1.27163 + 1.54368i 0.360029 + 0.360029i −1.33111 + 0.477660i 3.61419i −0.694825 2.74175i 1.00000i −0.243204 0.677743i
133.4 −1.09834 0.890868i −0.707107 + 0.707107i 0.412708 + 1.95695i −1.78499 1.78499i 1.40658 0.146706i 4.29781i 1.29009 2.51707i 1.00000i 0.370337 + 3.55072i
133.5 −1.07628 + 0.917404i −0.707107 + 0.707107i 0.316740 1.97476i 2.94133 + 2.94133i 0.112340 1.40974i 4.25298i 1.47075 + 2.41597i 1.00000i −5.86407 0.467295i
133.6 −0.899005 + 1.09169i −0.707107 + 0.707107i −0.383579 1.96287i −0.245391 0.245391i −0.136250 1.40763i 2.09093i 2.48769 + 1.34588i 1.00000i 0.488500 0.0472835i
133.7 −0.611904 1.27498i −0.707107 + 0.707107i −1.25115 + 1.56033i 2.47469 + 2.47469i 1.33423 + 0.468865i 0.523941i 2.75497 + 0.640412i 1.00000i 1.64090 4.66945i
133.8 −0.610414 + 1.27569i 0.707107 0.707107i −1.25479 1.55740i −1.71726 1.71726i 0.470424 + 1.33368i 0.688369i 2.75271 0.650069i 1.00000i 3.23893 1.14246i
133.9 −0.532522 1.31012i 0.707107 0.707107i −1.43284 + 1.39534i −1.78100 1.78100i −1.30295 0.549847i 3.86812i 2.59108 + 1.13415i 1.00000i −1.38491 + 3.28176i
133.10 0.112225 + 1.40975i 0.707107 0.707107i −1.97481 + 0.316420i 0.198134 + 0.198134i 1.07620 + 0.917491i 2.99121i −0.667697 2.74849i 1.00000i −0.257084 + 0.301556i
133.11 0.116618 1.40940i 0.707107 0.707107i −1.97280 0.328724i 2.29623 + 2.29623i −0.914133 1.07906i 2.56341i −0.693367 + 2.74212i 1.00000i 3.50408 2.96851i
133.12 0.138138 + 1.40745i −0.707107 + 0.707107i −1.96184 + 0.388846i −1.48034 1.48034i −1.09290 0.897539i 0.896581i −0.818286 2.70747i 1.00000i 1.87902 2.28800i
133.13 0.392209 1.35874i −0.707107 + 0.707107i −1.69234 1.06582i 1.57974 + 1.57974i 0.683440 + 1.23811i 2.84773i −2.11192 + 1.88143i 1.00000i 2.76604 1.52686i
133.14 0.747359 + 1.20061i 0.707107 0.707107i −0.882909 + 1.79457i 2.98823 + 2.98823i 1.37742 + 0.320494i 1.35380i −2.81442 + 0.281161i 1.00000i −1.35440 + 5.82096i
133.15 0.757184 1.19443i 0.707107 0.707107i −0.853344 1.80881i −0.248435 0.248435i −0.309182 1.38000i 1.46803i −2.80665 0.350341i 1.00000i −0.484850 + 0.108628i
133.16 1.06297 + 0.932783i −0.707107 + 0.707107i 0.259832 + 1.98305i −1.50956 1.50956i −1.41121 + 0.0920596i 2.25600i −1.57356 + 2.35030i 1.00000i −0.196532 3.01271i
133.17 1.37459 0.332432i −0.707107 + 0.707107i 1.77898 0.913914i 1.37452 + 1.37452i −0.736915 + 1.20704i 2.32592i 2.14154 1.84764i 1.00000i 2.34633 + 1.43246i
133.18 1.38370 + 0.292167i 0.707107 0.707107i 1.82928 + 0.808547i −2.65097 2.65097i 1.18502 0.771833i 3.16552i 2.29495 + 1.65325i 1.00000i −2.89363 4.44268i
133.19 1.40742 0.138462i 0.707107 0.707107i 1.96166 0.389747i 0.582740 + 0.582740i 0.897288 1.09310i 4.89753i 2.70691 0.820152i 1.00000i 0.900847 + 0.739473i
133.20 1.41369 + 0.0384766i −0.707107 + 0.707107i 1.99704 + 0.108788i −1.75827 1.75827i −1.02684 + 0.972423i 1.52399i 2.81901 + 0.230632i 1.00000i −2.41800 2.55331i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 133.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.2.t.b 40
4.b odd 2 1 2112.2.t.b 40
16.e even 4 1 inner 528.2.t.b 40
16.f odd 4 1 2112.2.t.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
528.2.t.b 40 1.a even 1 1 trivial
528.2.t.b 40 16.e even 4 1 inner
2112.2.t.b 40 4.b odd 2 1
2112.2.t.b 40 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{40} + 48 T_{5}^{37} + 664 T_{5}^{36} + 656 T_{5}^{35} + 1152 T_{5}^{34} + 23200 T_{5}^{33} + 183656 T_{5}^{32} + 334592 T_{5}^{31} + 563840 T_{5}^{30} + 4020032 T_{5}^{29} + 25164320 T_{5}^{28} + \cdots + 16777216 \) acting on \(S_{2}^{\mathrm{new}}(528, [\chi])\). Copy content Toggle raw display