↑

Properties

Label 536.2.w.a
Level $536$
Weight $2$
Character orbit 536.w
Analytic conductor $4.280$
Analytic rank $0$
Dimension $660$
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] = [536,2,Mod(149,536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(536, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 11, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("536.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 536 = 2^{3} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 536.w (of order \(22\), degree \(10\), 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: \(4.27998154834\)
Analytic rank: \(0\)
Dimension: \(660\)
Relative dimension: \(66\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 660 q - 11 q^{2} - 7 q^{4} - 13 q^{6} - 22 q^{7} - 38 q^{8} + 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 660 q - 11 q^{2} - 7 q^{4} - 13 q^{6} - 22 q^{7} - 38 q^{8} + 44 q^{9} - 40 q^{10} - 15 q^{12} - 9 q^{14} - 30 q^{15} + q^{16} - 18 q^{17} - q^{18} - 33 q^{20} + 36 q^{22} - 10 q^{23} - 11 q^{24} + 40 q^{25} + 13 q^{26} + 34 q^{28} - 44 q^{30} - 6 q^{31} - 76 q^{32} - 22 q^{33} - q^{34} + 42 q^{36} - 30 q^{38} - 30 q^{39} - 48 q^{40} - 26 q^{41} - 39 q^{42} - 45 q^{44} + 26 q^{46} - 42 q^{47} + 27 q^{48} - 88 q^{49} - 155 q^{50} - 5 q^{52} + 94 q^{54} - 38 q^{55} + 19 q^{56} - 110 q^{57} + 9 q^{58} - 29 q^{60} - 33 q^{62} + 18 q^{63} - 70 q^{64} - 38 q^{65} + 12 q^{66} + 194 q^{68} + 7 q^{70} - 26 q^{71} + 101 q^{72} - 90 q^{73} + 15 q^{74} + 19 q^{76} + 25 q^{78} - 38 q^{79} + 21 q^{80} - 88 q^{81} - 137 q^{82} - 5 q^{84} - 13 q^{86} + 12 q^{87} - 33 q^{88} - 2 q^{89} + 205 q^{90} - 19 q^{92} + 5 q^{94} - 22 q^{95} - 156 q^{96} - 56 q^{97} + 35 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1 −1.41294 + 0.0599870i −0.816300 − 2.78006i 1.99280 − 0.169516i 0.172214 − 0.149224i 1.32015 + 3.87910i 1.64747 − 1.05876i −2.80554 + 0.359058i −4.53864 + 2.91681i −0.234376 + 0.221175i
149.2 −1.40204 + 0.185184i 0.950899 + 3.23846i 1.93141 − 0.519268i 2.86416 − 2.48181i −1.93290 − 4.36435i 1.83469 − 1.17909i −2.61175 + 1.08570i −7.05968 + 4.53698i −3.55607 + 4.00998i
149.3 −1.39982 − 0.201260i 0.0233625 + 0.0795654i 1.91899 + 0.563456i −0.793401 + 0.687486i −0.0166899 − 0.116079i −2.43442 + 1.56451i −2.57284 − 1.17495i 2.51798 − 1.61820i 1.24898 − 0.802676i
149.4 −1.39763 + 0.215958i −0.0330199 − 0.112456i 1.90672 − 0.603658i 2.18103 − 1.88988i 0.0704352 + 0.150040i −2.14197 + 1.37656i −2.53453 + 1.25546i 2.51220 − 1.61450i −2.64014 + 3.11236i
149.5 −1.39540 + 0.229911i −0.548280 − 1.86727i 1.89428 − 0.641637i −0.319827 + 0.277131i 1.19438 + 2.47953i 0.485119 − 0.311767i −2.49576 + 1.33086i −0.662322 + 0.425649i 0.382571 − 0.460241i
149.6 −1.38185 − 0.300829i 0.743111 + 2.53080i 1.81900 + 0.831400i −2.43091 + 2.10639i −0.265527 − 3.72073i 1.30431 − 0.838230i −2.26348 − 1.69608i −3.32899 + 2.13941i 3.99281 − 2.17943i
149.7 −1.33661 − 0.462026i −0.0808773 − 0.275443i 1.57306 + 1.23510i 1.68117 − 1.45675i −0.0191601 + 0.405528i 4.18318 − 2.68836i −1.53193 − 2.37764i 2.45443 − 1.57737i −2.92013 + 1.17036i
149.8 −1.31659 − 0.516315i −0.234387 − 0.798249i 1.46684 + 1.35955i −2.41201 + 2.09002i −0.103555 + 1.17199i −1.03298 + 0.663858i −1.22927 − 2.54733i 1.94150 − 1.24772i 4.25474 − 1.50635i
149.9 −1.30420 + 0.546854i 0.726763 + 2.47513i 1.40190 − 1.42642i −1.12389 + 0.973856i −2.30138 − 2.83064i −4.08154 + 2.62305i −1.04832 + 2.62698i −3.07431 + 1.97574i 0.933225 − 1.88471i
149.10 −1.27564 + 0.610533i −0.123037 − 0.419024i 1.25450 − 1.55764i −2.91607 + 2.52679i 0.412778 + 0.459405i 0.671935 − 0.431826i −0.649294 + 2.75289i 2.36332 − 1.51881i 2.17716 − 5.00363i
149.11 −1.23459 + 0.689775i 0.470175 + 1.60127i 1.04842 − 1.70318i −0.336427 + 0.291516i −1.68499 − 1.65259i 2.37011 − 1.52318i −0.119562 + 2.82590i 0.180764 − 0.116170i 0.214269 − 0.591962i
149.12 −1.21815 − 0.718400i 0.597007 + 2.03322i 0.967802 + 1.75025i 1.82307 − 1.57970i 0.733419 − 2.90567i −3.27580 + 2.10523i 0.0784452 − 2.82734i −1.25380 + 0.805771i −3.35564 + 0.614623i
149.13 −1.20879 − 0.734046i 0.550621 + 1.87524i 0.922352 + 1.77462i 0.594680 − 0.515293i 0.710930 − 2.67096i 0.553491 − 0.355707i 0.187721 − 2.82219i −0.689594 + 0.443175i −1.09709 + 0.186359i
149.14 −1.19889 − 0.750112i −0.812209 − 2.76613i 0.874663 + 1.79860i 0.00141913 − 0.00122968i −1.10116 + 3.92553i −1.97374 + 1.26845i 0.300529 − 2.81242i −4.46803 + 2.87143i −0.00262377 0.000409744i
149.15 −1.14031 + 0.836479i −0.470175 − 1.60127i 0.600605 − 1.90769i 0.336427 − 0.291516i 1.87557 + 1.43265i 2.37011 − 1.52318i 0.910866 + 2.67775i 0.180764 − 0.116170i −0.139784 + 0.613833i
149.16 −1.08528 + 0.906735i 0.123037 + 0.419024i 0.355662 − 1.96812i 2.91607 − 2.52679i −0.513473 − 0.343197i 0.671935 − 0.431826i 1.39857 + 2.45845i 2.36332 − 1.51881i −0.873623 + 5.38638i
149.17 −1.05117 − 0.946072i −0.577214 − 1.96581i 0.209897 + 1.98896i −2.93780 + 2.54562i −1.25305 + 2.61248i 3.00319 − 1.93003i 1.66106 − 2.28930i −1.00747 + 0.647460i 5.49646 + 0.103504i
149.18 −1.03922 + 0.959175i −0.726763 − 2.47513i 0.159967 − 1.99359i 1.12389 − 0.973856i 3.12935 + 1.87511i −4.08154 + 2.62305i 1.74596 + 2.22522i −3.07431 + 1.97574i −0.233873 + 2.09006i
149.19 −0.877363 − 1.10916i 0.403984 + 1.37584i −0.460469 + 1.94627i −1.11741 + 0.968244i 1.17159 − 1.65520i 1.58137 − 1.01628i 2.56272 − 1.19685i 0.794019 − 0.510285i 2.05431 + 0.389888i
149.20 −0.865089 − 1.11876i −0.253957 − 0.864897i −0.503243 + 1.93565i 2.47947 − 2.14847i −0.747916 + 1.03233i −0.696043 + 0.447320i 2.60088 − 1.11150i 1.84021 − 1.18263i −4.54859 − 0.915310i
See next 80 embeddings (of 660 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.66
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
67.e even 11 1 inner
536.w even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 536.2.w.a ✓ 660
8.b even 2 1 inner 536.2.w.a ✓ 660
67.e even 11 1 inner 536.2.w.a ✓ 660
536.w even 22 1 inner 536.2.w.a ✓ 660
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
536.2.w.a ✓ 660 1.a even 1 1 trivial
536.2.w.a ✓ 660 8.b even 2 1 inner
536.2.w.a ✓ 660 67.e even 11 1 inner
536.2.w.a ✓ 660 536.w even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(536, [\chi])\).