Properties

Label 522.2.q.a
Level $522$
Weight $2$
Character orbit 522.q
Analytic conductor $4.168$
Analytic rank $0$
Dimension $180$
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(7,522)] 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("522.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([28, 18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.q (of order \(21\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [180,-15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(180\)
Relative dimension: \(15\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 180 q - 15 q^{2} - 2 q^{3} + 15 q^{4} + 2 q^{5} + 10 q^{6} - 5 q^{7} + 30 q^{8} - 6 q^{9} + 4 q^{10} - 4 q^{11} - 2 q^{12} - 2 q^{14} + 21 q^{15} + 15 q^{16} + 12 q^{17} - 6 q^{18} + 6 q^{19} + 2 q^{20}+ \cdots - 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.955573 − 0.294755i −1.69011 + 0.378868i 0.826239 + 0.563320i 2.03286 − 1.88621i 1.72669 + 0.136132i 2.98569 − 2.03561i −0.623490 − 0.781831i 2.71292 − 1.28065i −2.49851 + 1.20322i
7.2 −0.955573 − 0.294755i −1.54334 + 0.786193i 0.826239 + 0.563320i 0.570156 − 0.529028i 1.70651 − 0.296356i −1.61161 + 1.09878i −0.623490 − 0.781831i 1.76380 − 2.42673i −0.700759 + 0.337468i
7.3 −0.955573 − 0.294755i −1.52228 − 0.826230i 0.826239 + 0.563320i −1.62002 + 1.50316i 1.21112 + 1.23822i −2.58981 + 1.76570i −0.623490 − 0.781831i 1.63469 + 2.51551i 1.99111 − 0.958870i
7.4 −0.955573 − 0.294755i −1.07096 + 1.36127i 0.826239 + 0.563320i −3.02857 + 2.81010i 1.42462 − 0.985119i −2.33072 + 1.58906i −0.623490 − 0.781831i −0.706095 − 2.91572i 3.72231 − 1.79257i
7.5 −0.955573 − 0.294755i −0.937448 − 1.45643i 0.826239 + 0.563320i 0.998281 − 0.926269i 0.466509 + 1.66804i 2.87310 − 1.95885i −0.623490 − 0.781831i −1.24238 + 2.73066i −1.22695 + 0.590869i
7.6 −0.955573 − 0.294755i −0.915999 − 1.47002i 0.826239 + 0.563320i 0.691554 − 0.641669i 0.442009 + 1.67470i −1.01774 + 0.693882i −0.623490 − 0.781831i −1.32189 + 2.69307i −0.849966 + 0.409322i
7.7 −0.955573 − 0.294755i −0.786116 + 1.54338i 0.826239 + 0.563320i −1.85398 + 1.72024i 1.20611 − 1.24310i 3.45434 − 2.35513i −0.623490 − 0.781831i −1.76404 − 2.42655i 2.27866 − 1.09735i
7.8 −0.955573 − 0.294755i 0.168341 + 1.72385i 0.826239 + 0.563320i 0.961466 − 0.892110i 0.347252 − 1.69688i 0.196766 − 0.134153i −0.623490 − 0.781831i −2.94332 + 0.580389i −1.18170 + 0.569079i
7.9 −0.955573 − 0.294755i 0.250715 − 1.71381i 0.826239 + 0.563320i −2.54388 + 2.36037i −0.744731 + 1.56377i 2.09756 − 1.43010i −0.623490 − 0.781831i −2.87428 − 0.859356i 3.12659 − 1.50569i
7.10 −0.955573 − 0.294755i 0.736511 − 1.56766i 0.826239 + 0.563320i 1.89815 − 1.76122i −1.16587 + 1.28092i 0.889695 − 0.606584i −0.623490 − 0.781831i −1.91510 − 2.30920i −2.33294 + 1.12349i
7.11 −0.955573 − 0.294755i 0.761343 + 1.55575i 0.826239 + 0.563320i −0.116336 + 0.107944i −0.268954 − 1.71104i −2.32358 + 1.58419i −0.623490 − 0.781831i −1.84071 + 2.36892i 0.142985 − 0.0688580i
7.12 −0.955573 − 0.294755i 1.54344 + 0.785994i 0.826239 + 0.563320i −1.11925 + 1.03851i −1.24320 − 1.20601i 2.36189 − 1.61031i −0.623490 − 0.781831i 1.76443 + 2.42627i 1.37563 − 0.662467i
7.13 −0.955573 − 0.294755i 1.57954 − 0.710668i 0.826239 + 0.563320i 0.982675 − 0.911790i −1.71884 + 0.213517i 0.110047 − 0.0750290i −0.623490 − 0.781831i 1.98990 − 2.24506i −1.20777 + 0.581633i
7.14 −0.955573 − 0.294755i 1.72851 + 0.110635i 0.826239 + 0.563320i 2.96091 − 2.74732i −1.61911 − 0.615208i −3.30653 + 2.25435i −0.623490 − 0.781831i 2.97552 + 0.382468i −3.63915 + 1.75252i
7.15 −0.955573 − 0.294755i 1.73110 − 0.0573274i 0.826239 + 0.563320i −2.28011 + 2.11564i −1.67109 − 0.455471i −3.27795 + 2.23487i −0.623490 − 0.781831i 2.99343 − 0.198479i 2.80241 − 1.34957i
25.1 0.733052 + 0.680173i −1.72317 − 0.175143i 0.0747301 + 0.997204i −1.30131 + 0.401402i −1.14405 − 1.30044i 0.259861 − 3.46760i −0.623490 + 0.781831i 2.93865 + 0.603605i −1.22695 − 0.590869i
25.2 0.733052 + 0.680173i −1.72042 − 0.200382i 0.0747301 + 0.997204i −0.901478 + 0.278069i −1.12486 − 1.31707i −0.0920504 + 1.22833i −0.623490 + 0.781831i 2.91969 + 0.689484i −0.849966 − 0.409322i
25.3 0.733052 + 0.680173i −1.59510 + 0.675023i 0.0747301 + 0.997204i 2.11179 − 0.651400i −1.62842 − 0.590117i −0.234238 + 3.12569i −0.623490 + 0.781831i 2.08869 − 2.15346i 1.99111 + 0.958870i
25.4 0.733052 + 0.680173i −1.18359 − 1.26456i 0.0747301 + 0.997204i 3.31608 − 1.02288i −0.00751499 − 1.73203i 0.189717 − 2.53159i −0.623490 + 0.781831i −0.198222 + 2.99344i 3.12659 + 1.50569i
25.5 0.733052 + 0.680173i −0.766437 − 1.55325i 0.0747301 + 0.997204i −2.47433 + 0.763231i 0.494638 − 1.65992i 0.0804695 − 1.07379i −0.623490 + 0.781831i −1.82515 + 2.38093i −2.33294 − 1.12349i
See next 80 embeddings (of 180 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 7.15
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
29.d even 7 1 inner
261.q even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 522.2.q.a ✓ 180
9.c even 3 1 inner 522.2.q.a ✓ 180
29.d even 7 1 inner 522.2.q.a ✓ 180
261.q even 21 1 inner 522.2.q.a ✓ 180
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
522.2.q.a ✓ 180 1.a even 1 1 trivial
522.2.q.a ✓ 180 9.c even 3 1 inner
522.2.q.a ✓ 180 29.d even 7 1 inner
522.2.q.a ✓ 180 261.q even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{180} - 2 T_{5}^{179} - 43 T_{5}^{178} + 46 T_{5}^{177} + 720 T_{5}^{176} + 198 T_{5}^{175} + \cdots + 37\!\cdots\!21 \) acting on \(S_{2}^{\mathrm{new}}(522, [\chi])\). Copy content Toggle raw display