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Properties

Label 444.2.p.c
Level $444$
Weight $2$
Character orbit 444.p
Analytic conductor $3.545$
Analytic rank $0$
Dimension $136$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [444,2,Mod(11,444)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("444.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [136,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(136\)
Relative dimension: \(68\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 136 q + 6 q^{4} - 14 q^{9} - 32 q^{10} - 2 q^{12} + 12 q^{13} + 22 q^{16} - 24 q^{18} - 26 q^{21} - 42 q^{22} + 42 q^{24} - 48 q^{25} - 2 q^{28} - 6 q^{30} - 8 q^{33} - 10 q^{34} + 68 q^{36} - 48 q^{37}+ \cdots - 126 q^{96}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.41412 + 0.0160851i 0.366217 − 1.69289i 1.99948 − 0.0454927i −1.53888 + 2.66542i −0.490645 + 2.39985i 1.04351 + 0.602468i −2.82678 + 0.0964942i −2.73177 − 1.23993i 2.13329 − 3.79399i
11.2 −1.41229 + 0.0737273i 1.73191 − 0.0220440i 1.98913 − 0.208249i −1.34547 + 2.33042i −2.44434 + 0.158822i −1.64930 − 0.952223i −2.79387 + 0.440761i 2.99903 − 0.0763566i 1.72838 − 3.39043i
11.3 −1.41035 + 0.104521i 0.750389 − 1.56106i 1.97815 − 0.294822i 1.53437 − 2.65760i −0.895144 + 2.28007i 1.92352 + 1.11055i −2.75906 + 0.622560i −1.87383 − 2.34281i −1.88621 + 3.90851i
11.4 −1.40142 − 0.189831i −0.299887 + 1.70589i 1.92793 + 0.532065i 0.690983 − 1.19682i 0.744097 − 2.33374i −0.418282 − 0.241495i −2.60083 − 1.11162i −2.82014 − 1.02315i −1.19555 + 1.54607i
11.5 −1.38927 + 0.264447i −1.67070 − 0.456919i 1.86014 − 0.734776i 0.836466 − 1.44880i 2.44188 + 0.192972i 3.85015 + 2.22289i −2.38992 + 1.51271i 2.58245 + 1.52674i −0.778945 + 2.23397i
11.6 −1.36097 + 0.384391i −1.39748 + 1.02326i 1.70449 − 1.04629i −0.209441 + 0.362763i 1.50859 − 1.92980i −1.98418 − 1.14557i −1.91757 + 2.07916i 0.905882 − 2.85996i 0.145600 − 0.574217i
11.7 −1.33469 − 0.467563i 1.60135 − 0.660066i 1.56277 + 1.24810i 0.999553 − 1.73128i −2.44592 + 0.132249i −2.64279 − 1.52581i −1.50224 − 2.39651i 2.12863 − 2.11399i −2.14357 + 1.84336i
11.8 −1.31819 − 0.512217i −0.966035 − 1.43763i 1.47527 + 1.35040i −0.463518 + 0.802836i 0.537042 + 2.38989i −0.468102 − 0.270259i −1.25299 − 2.53575i −1.13355 + 2.77760i 1.02223 − 0.820872i
11.9 −1.31764 + 0.513626i 1.34758 + 1.08813i 1.47238 − 1.35355i 1.97449 − 3.41992i −2.33452 − 0.741616i −0.447080 − 0.258122i −1.24485 + 2.53975i 0.631947 + 2.93269i −0.845118 + 5.52038i
11.10 −1.31256 − 0.526494i −1.37948 + 1.04739i 1.44561 + 1.38211i −1.61676 + 2.80030i 2.36209 − 0.648470i 3.79678 + 2.19207i −1.16977 − 2.57520i 0.805944 − 2.88972i 3.59643 − 2.82434i
11.11 −1.29208 − 0.574917i 1.27163 + 1.17599i 1.33894 + 1.48568i 0.0154343 − 0.0267329i −0.966951 − 2.25056i 0.155352 + 0.0896922i −0.875878 − 2.68939i 0.234089 + 2.99085i −0.0353115 + 0.0256677i
11.12 −1.29203 + 0.575038i −0.779927 − 1.54652i 1.33866 − 1.48593i 0.465225 − 0.805793i 1.89699 + 1.54965i −3.88319 − 2.24196i −0.875124 + 2.68964i −1.78343 + 2.41234i −0.137721 + 1.30863i
11.13 −1.25807 − 0.645947i −1.73202 − 0.0109929i 1.16550 + 1.62530i 1.95723 − 3.39002i 2.17190 + 1.13262i −2.51626 − 1.45276i −0.416434 − 2.79760i 2.99976 + 0.0380797i −4.65212 + 3.00063i
11.14 −1.14401 + 0.831409i 1.72929 − 0.0978213i 0.617519 − 1.90228i −0.465225 + 0.805793i −1.89699 + 1.54965i 3.88319 + 2.24196i 0.875124 + 2.68964i 2.98086 − 0.338322i −0.137721 − 1.30863i
11.15 −1.10364 + 0.884301i −1.61614 − 0.622974i 0.436024 − 1.95189i −1.97449 + 3.41992i 2.33452 − 0.741616i 0.447080 + 0.258122i 1.24485 + 2.53975i 2.22381 + 2.01362i −0.845118 − 5.52038i
11.16 −1.08739 − 0.904197i −0.0917777 + 1.72962i 0.364854 + 1.96644i −1.51444 + 2.62309i 1.66371 − 1.79779i −3.67125 − 2.11960i 1.38131 − 2.46820i −2.98315 − 0.317480i 4.01859 − 1.48298i
11.17 −1.05758 − 0.938891i 1.28930 − 1.15659i 0.236969 + 1.98591i 0.248075 − 0.429678i −2.44946 + 0.0126836i 2.93237 + 1.69301i 1.61394 − 2.32276i 0.324585 − 2.98239i −0.665780 + 0.221505i
11.18 −1.01338 + 0.986440i −0.187430 + 1.72188i 0.0538719 − 1.99927i 0.209441 − 0.362763i −1.50859 − 1.92980i 1.98418 + 1.14557i 1.91757 + 2.07916i −2.92974 − 0.645464i 0.145600 + 0.574217i
11.19 −0.923652 + 1.07092i 1.23105 + 1.21841i −0.293733 − 1.97831i −0.836466 + 1.44880i −2.44188 + 0.192972i −3.85015 − 2.22289i 2.38992 + 1.51271i 0.0309745 + 2.99984i −0.778945 − 2.23397i
11.20 −0.882739 − 1.10489i −1.70978 + 0.276855i −0.441545 + 1.95065i −0.0114396 + 0.0198140i 1.81518 + 1.64472i 1.40553 + 0.811486i 2.54501 − 1.23406i 2.84670 − 0.946723i 0.0319904 − 0.00485111i
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 11.68
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
37.e even 6 1 inner
111.h odd 6 1 inner
148.j odd 6 1 inner
444.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 444.2.p.c ✓ 136
3.b odd 2 1 inner 444.2.p.c ✓ 136
4.b odd 2 1 inner 444.2.p.c ✓ 136
12.b even 2 1 inner 444.2.p.c ✓ 136
37.e even 6 1 inner 444.2.p.c ✓ 136
111.h odd 6 1 inner 444.2.p.c ✓ 136
148.j odd 6 1 inner 444.2.p.c ✓ 136
444.p even 6 1 inner 444.2.p.c ✓ 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
444.2.p.c ✓ 136 1.a even 1 1 trivial
444.2.p.c ✓ 136 3.b odd 2 1 inner
444.2.p.c ✓ 136 4.b odd 2 1 inner
444.2.p.c ✓ 136 12.b even 2 1 inner
444.2.p.c ✓ 136 37.e even 6 1 inner
444.2.p.c ✓ 136 111.h odd 6 1 inner
444.2.p.c ✓ 136 148.j odd 6 1 inner
444.2.p.c ✓ 136 444.p even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(444, [\chi])\):

\( T_{5}^{68} + 97 T_{5}^{66} + 5215 T_{5}^{64} + 193422 T_{5}^{62} + 5464516 T_{5}^{60} + 123374900 T_{5}^{58} + \cdots + 2304 \) Copy content Toggle raw display
\( T_{7}^{68} - 131 T_{7}^{66} + 9524 T_{7}^{64} - 476801 T_{7}^{62} + 18145221 T_{7}^{60} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display