Properties

Label 522.2.l.a
Level $522$
Weight $2$
Character orbit 522.l
Analytic conductor $4.168$
Analytic rank $0$
Dimension $120$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(41,522)] 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("522.41"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([10, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.l (of order \(12\), degree \(4\), 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: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 24 q^{11} + 12 q^{14} + 20 q^{15} + 60 q^{16} - 28 q^{21} + 8 q^{24} - 60 q^{25} - 24 q^{27} + 12 q^{29} + 12 q^{30} - 16 q^{36} + 16 q^{39} + 12 q^{41} + 104 q^{45} - 24 q^{46} + 24 q^{47} - 60 q^{49}+ \cdots - 24 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −0.965926 − 0.258819i −1.73136 − 0.0490247i 0.866025 + 0.500000i −0.370302 + 0.641382i 1.65967 + 0.495462i 1.44011 + 2.49434i −0.707107 − 0.707107i 2.99519 + 0.169758i 0.523686 − 0.523686i
41.2 −0.965926 − 0.258819i −1.67730 + 0.432031i 0.866025 + 0.500000i −1.49897 + 2.59630i 1.73197 + 0.0168079i −1.76950 − 3.06486i −0.707107 − 0.707107i 2.62670 − 1.44930i 2.11987 − 2.11987i
41.3 −0.965926 − 0.258819i −1.29832 + 1.14646i 0.866025 + 0.500000i 1.72918 − 2.99503i 1.55080 − 0.771369i 1.19537 + 2.07045i −0.707107 − 0.707107i 0.371247 − 2.97694i −2.44544 + 2.44544i
41.4 −0.965926 − 0.258819i −1.04491 − 1.38137i 0.866025 + 0.500000i 1.76675 − 3.06009i 0.651779 + 1.60474i −1.73078 − 2.99780i −0.707107 − 0.707107i −0.816341 + 2.88680i −2.49856 + 2.49856i
41.5 −0.965926 − 0.258819i −0.954417 − 1.44537i 0.866025 + 0.500000i −0.407668 + 0.706102i 0.547808 + 1.64314i 0.0200261 + 0.0346863i −0.707107 − 0.707107i −1.17818 + 2.75897i 0.576530 − 0.576530i
41.6 −0.965926 − 0.258819i −0.891250 + 1.48515i 0.866025 + 0.500000i −1.76344 + 3.05437i 1.24527 − 1.20387i 1.91196 + 3.31162i −0.707107 − 0.707107i −1.41135 − 2.64728i 2.49389 − 2.49389i
41.7 −0.965926 − 0.258819i −0.161095 + 1.72454i 0.866025 + 0.500000i 0.564136 − 0.977112i 0.601950 − 1.62409i −0.572023 − 0.990774i −0.707107 − 0.707107i −2.94810 − 0.555630i −0.797809 + 0.797809i
41.8 −0.965926 − 0.258819i 0.161762 − 1.72448i 0.866025 + 0.500000i 0.404195 − 0.700086i −0.602579 + 1.62385i −0.392833 − 0.680406i −0.707107 − 0.707107i −2.94767 − 0.557911i −0.571617 + 0.571617i
41.9 −0.965926 − 0.258819i 0.369850 + 1.69210i 0.866025 + 0.500000i 0.168220 − 0.291366i 0.0807009 − 1.73017i −1.55615 − 2.69534i −0.707107 − 0.707107i −2.72642 + 1.25165i −0.237900 + 0.237900i
41.10 −0.965926 − 0.258819i 0.423376 − 1.67951i 0.866025 + 0.500000i −1.27726 + 2.21228i −0.843638 + 1.51270i 0.410048 + 0.710224i −0.707107 − 0.707107i −2.64151 − 1.42213i 1.80632 − 1.80632i
41.11 −0.965926 − 0.258819i 1.11378 + 1.32646i 0.866025 + 0.500000i −1.95385 + 3.38416i −0.732514 − 1.56953i −0.723305 − 1.25280i −0.707107 − 0.707107i −0.518994 + 2.95477i 2.76316 − 2.76316i
41.12 −0.965926 − 0.258819i 1.28558 + 1.16072i 0.866025 + 0.500000i 2.13651 − 3.70054i −0.941358 − 1.45391i 1.39109 + 2.40944i −0.707107 − 0.707107i 0.305438 + 2.98441i −3.02148 + 3.02148i
41.13 −0.965926 − 0.258819i 1.59168 − 0.683049i 0.866025 + 0.500000i −0.100602 + 0.174248i −1.71423 + 0.247818i 1.87111 + 3.24085i −0.707107 − 0.707107i 2.06689 − 2.17439i 0.142273 − 0.142273i
41.14 −0.965926 − 0.258819i 1.65473 − 0.511719i 0.866025 + 0.500000i 0.220762 − 0.382371i −1.73079 + 0.0660057i −2.23614 − 3.87310i −0.707107 − 0.707107i 2.47629 − 1.69352i −0.312205 + 0.312205i
41.15 −0.965926 − 0.258819i 1.67552 + 0.438892i 0.866025 + 0.500000i 0.382347 − 0.662244i −1.50484 − 0.857594i −0.483735 − 0.837854i −0.707107 − 0.707107i 2.61475 + 1.47075i −0.540720 + 0.540720i
41.16 0.965926 + 0.258819i −1.70281 − 0.316926i 0.866025 + 0.500000i −0.508834 + 0.881326i −1.56276 − 0.746847i −1.07961 − 1.86995i 0.707107 + 0.707107i 2.79912 + 1.07933i −0.719600 + 0.719600i
41.17 0.965926 + 0.258819i −1.63468 + 0.572547i 0.866025 + 0.500000i 1.02346 − 1.77269i −1.72717 + 0.129950i −1.33363 − 2.30991i 0.707107 + 0.707107i 2.34438 − 1.87187i 1.44739 − 1.44739i
41.18 0.965926 + 0.258819i −1.38233 + 1.04363i 0.866025 + 0.500000i 0.437116 − 0.757106i −1.60534 + 0.650291i 1.19816 + 2.07528i 0.707107 + 0.707107i 0.821689 − 2.88528i 0.618175 − 0.618175i
41.19 0.965926 + 0.258819i −1.15784 + 1.28818i 0.866025 + 0.500000i −1.36966 + 2.37232i −1.45179 + 0.944618i 1.89864 + 3.28854i 0.707107 + 0.707107i −0.318826 − 2.98301i −1.93699 + 1.93699i
41.20 0.965926 + 0.258819i −1.14563 − 1.29905i 0.866025 + 0.500000i 1.77935 − 3.08193i −0.770377 − 1.55130i 0.535608 + 0.927701i 0.707107 + 0.707107i −0.375054 + 2.97646i 2.51639 − 2.51639i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 41.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
29.c odd 4 1 inner
261.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 522.2.l.a ✓ 120
9.d odd 6 1 inner 522.2.l.a ✓ 120
29.c odd 4 1 inner 522.2.l.a ✓ 120
261.l even 12 1 inner 522.2.l.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
522.2.l.a ✓ 120 1.a even 1 1 trivial
522.2.l.a ✓ 120 9.d odd 6 1 inner
522.2.l.a ✓ 120 29.c odd 4 1 inner
522.2.l.a ✓ 120 261.l even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(522, [\chi])\).