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Properties

Label 444.2.t.a
Level $444$
Weight $2$
Character orbit 444.t
Analytic conductor $3.545$
Analytic rank $0$
Dimension $144$
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(47,444)] 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("444.47"); 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, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(72\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 2 q^{4} - 2 q^{9} - 16 q^{10} - 12 q^{12} + 6 q^{16} + 12 q^{18} - 14 q^{21} + 2 q^{22} + 24 q^{24} + 52 q^{25} - 34 q^{28} - 4 q^{30} - 8 q^{33} - 18 q^{34} - 12 q^{36} - 24 q^{37} + 26 q^{40}+ \cdots - 40 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −1.41044 − 0.103185i −1.04682 + 1.37992i 1.97871 + 0.291073i −1.21965 + 0.704163i 1.61887 − 1.83828i 1.86102 − 1.07446i −2.76082 − 0.614715i −0.808344 − 2.88904i 1.79290 − 0.867333i
47.2 −1.40954 − 0.114932i 1.30412 − 1.13986i 1.97358 + 0.324000i −1.69009 + 0.975771i −1.96920 + 1.45679i −3.61172 + 2.08523i −2.74460 − 0.683517i 0.401433 − 2.97302i 2.49438 − 1.18114i
47.3 −1.40921 − 0.118891i −0.180480 + 1.72262i 1.97173 + 0.335085i 1.93366 − 1.11640i 0.459138 − 2.40607i −2.01799 + 1.16509i −2.73874 − 0.706625i −2.93485 − 0.621798i −2.85765 + 1.34334i
47.4 −1.39532 − 0.230405i −1.44601 − 0.953443i 1.89383 + 0.642978i 1.12751 − 0.650966i 1.79797 + 1.66352i 3.87109 − 2.23497i −2.49435 − 1.33351i 1.18189 + 2.75738i −1.72321 + 0.648521i
47.5 −1.38040 + 0.307391i −0.523387 − 1.65108i 1.81102 − 0.848648i −1.44965 + 0.836955i 1.23001 + 2.11827i 0.137650 − 0.0794721i −2.23907 + 1.72817i −2.45213 + 1.72831i 1.74383 − 1.60095i
47.6 −1.37232 + 0.341686i −1.72176 + 0.188561i 1.76650 − 0.937801i −2.82081 + 1.62860i 2.29837 − 0.847064i −2.03089 + 1.17254i −2.10377 + 1.89055i 2.92889 − 0.649311i 3.31458 − 3.19878i
47.7 −1.36147 − 0.382637i 0.108550 − 1.72865i 1.70718 + 1.04190i 2.89925 − 1.67388i −0.809232 + 2.31196i −2.17590 + 1.25626i −1.92559 − 2.07173i −2.97643 − 0.375289i −4.58771 + 1.16957i
47.8 −1.35070 − 0.419050i 1.10677 + 1.33231i 1.64879 + 1.13202i −3.05308 + 1.76270i −0.936613 − 2.26335i 0.0392709 − 0.0226731i −1.75266 − 2.21995i −0.550114 + 2.94913i 4.86246 − 1.10148i
47.9 −1.34530 + 0.436074i 1.42115 − 0.990119i 1.61968 − 1.17330i 2.22266 − 1.28325i −1.48011 + 1.95174i 1.77411 − 1.02428i −1.66731 + 2.28475i 1.03933 − 2.81421i −2.43056 + 2.69561i
47.10 −1.28089 + 0.599439i 0.934814 + 1.45812i 1.28135 − 1.53563i −0.00571820 + 0.00330141i −2.07145 − 1.30733i 2.58583 − 1.49293i −0.720743 + 2.73506i −1.25224 + 2.72615i 0.00534538 − 0.00765645i
47.11 −1.25314 − 0.655470i 1.72795 − 0.119081i 1.14072 + 1.64279i 0.728147 − 0.420396i −2.24342 − 0.983396i 2.10591 − 1.21585i −0.352679 − 2.80635i 2.97164 − 0.411531i −1.18803 + 0.0495363i
47.12 −1.23768 + 0.684215i −1.44861 − 0.949496i 1.06370 − 1.69368i 3.11142 − 1.79638i 2.44257 + 0.184014i −2.98520 + 1.72351i −0.157679 + 2.82403i 1.19691 + 2.75089i −2.62183 + 4.35222i
47.13 −1.19898 + 0.749969i −1.48373 + 0.893613i 0.875092 − 1.79839i 2.62563 − 1.51591i 1.10878 − 2.18417i 2.60625 − 1.50472i 0.299523 + 2.81252i 1.40291 − 2.65176i −2.01118 + 3.78668i
47.14 −1.19422 − 0.757516i −1.72795 + 0.119081i 0.852339 + 1.80929i 0.728147 − 0.420396i 2.15377 + 1.16674i −2.10591 + 1.21585i 0.352679 − 2.80635i 2.97164 − 0.411531i −1.18803 − 0.0495363i
47.15 −1.14051 + 0.836207i −0.148398 + 1.72568i 0.601516 − 1.90740i −0.132398 + 0.0764400i −1.27378 − 2.09225i −4.49571 + 2.59560i 0.908949 + 2.67840i −2.95596 − 0.512177i 0.0870812 − 0.197892i
47.16 −1.08314 + 0.909291i −0.352611 − 1.69578i 0.346378 − 1.96978i −0.704512 + 0.406750i 1.92388 + 1.51614i −0.769242 + 0.444122i 1.41593 + 2.44850i −2.75133 + 1.19590i 0.393230 − 1.08117i
47.17 −1.03826 − 0.960218i −1.10677 − 1.33231i 0.155965 + 1.99391i −3.05308 + 1.76270i −0.130194 + 2.44603i −0.0392709 + 0.0226731i 1.75266 − 2.21995i −0.550114 + 2.94913i 4.86246 + 1.10148i
47.18 −1.01211 − 0.987745i −0.108550 + 1.72865i 0.0487194 + 1.99941i 2.89925 − 1.67388i 1.81733 − 1.64235i 2.17590 − 1.25626i 1.92559 − 2.07173i −2.97643 − 0.375289i −4.58771 − 1.16957i
47.19 −0.970392 + 1.02876i 1.69365 + 0.362720i −0.116680 − 1.99659i −2.33849 + 1.35013i −2.01665 + 1.39037i −1.17929 + 0.680865i 2.16723 + 1.81744i 2.73687 + 1.22864i 0.880299 − 3.71589i
47.20 −0.897196 − 1.09318i 1.44601 + 0.953443i −0.390078 + 1.96159i 1.12751 − 0.650966i −0.255072 − 2.43617i −3.87109 + 2.23497i 2.49435 − 1.33351i 1.18189 + 2.75738i −1.72321 − 0.648521i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 47.72
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.c even 3 1 inner
111.i odd 6 1 inner
148.i odd 6 1 inner
444.t 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.t.a ✓ 144
3.b odd 2 1 inner 444.2.t.a ✓ 144
4.b odd 2 1 inner 444.2.t.a ✓ 144
12.b even 2 1 inner 444.2.t.a ✓ 144
37.c even 3 1 inner 444.2.t.a ✓ 144
111.i odd 6 1 inner 444.2.t.a ✓ 144
148.i odd 6 1 inner 444.2.t.a ✓ 144
444.t even 6 1 inner 444.2.t.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
444.2.t.a ✓ 144 1.a even 1 1 trivial
444.2.t.a ✓ 144 3.b odd 2 1 inner
444.2.t.a ✓ 144 4.b odd 2 1 inner
444.2.t.a ✓ 144 12.b even 2 1 inner
444.2.t.a ✓ 144 37.c even 3 1 inner
444.2.t.a ✓ 144 111.i odd 6 1 inner
444.2.t.a ✓ 144 148.i odd 6 1 inner
444.2.t.a ✓ 144 444.t even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(444, [\chi])\).