↑

Properties

Label 722.2.g.a
Level $722$
Weight $2$
Character orbit 722.g
Analytic conductor $5.765$
Analytic rank $0$
Dimension $288$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [722,2,Mod(39,722)] 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("722.39"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(722, base_ring=CyclotomicField(38)) chi = DirichletCharacter(H, H._module([14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.g (of order \(19\), degree \(18\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [288] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.76519902594\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(16\) over \(\Q(\zeta_{19})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{19}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288 q + 16 q^{2} - q^{3} - 16 q^{4} + q^{5} + q^{6} + 2 q^{7} + 16 q^{8} - 13 q^{9} - q^{10} - 31 q^{11} - q^{12} - 5 q^{13} + 17 q^{14} - 38 q^{15} - 16 q^{16} + 36 q^{17} + 13 q^{18} - q^{19} + q^{20}+ \cdots - 102 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1 0.401695 + 0.915773i −2.42916 − 1.89069i −0.677282 + 0.735724i −0.170086 − 2.05264i 0.755662 − 2.98404i 2.45888 + 2.67106i −0.945817 − 0.324699i 1.58965 + 6.27738i 1.81143 − 0.980295i
39.2 0.401695 + 0.915773i −2.36003 − 1.83688i −0.677282 + 0.735724i 0.0307996 + 0.371696i 0.734157 − 2.89912i 0.295155 + 0.320624i −0.945817 − 0.324699i 1.45914 + 5.76200i −0.328017 + 0.177514i
39.3 0.401695 + 0.915773i −1.79286 − 1.39544i −0.677282 + 0.735724i −0.231736 − 2.79664i 0.557722 − 2.20240i −3.08191 − 3.34785i −0.945817 − 0.324699i 0.530642 + 2.09546i 2.46800 − 1.33562i
39.4 0.401695 + 0.915773i −1.53953 − 1.19827i −0.677282 + 0.735724i 0.267690 + 3.23054i 0.478917 − 1.89120i −1.50703 − 1.63707i −0.945817 − 0.324699i 0.197863 + 0.781344i −2.85091 + 1.54284i
39.5 0.401695 + 0.915773i −1.10234 − 0.857988i −0.677282 + 0.735724i 0.261371 + 3.15428i 0.342917 − 1.35415i 1.63746 + 1.77875i −0.945817 − 0.324699i −0.257439 − 1.01660i −2.78361 + 1.50642i
39.6 0.401695 + 0.915773i −0.564838 − 0.439631i −0.677282 + 0.735724i −0.0388883 − 0.469312i 0.175710 − 0.693862i 3.35287 + 3.64218i −0.945817 − 0.324699i −0.610690 − 2.41156i 0.414162 − 0.224133i
39.7 0.401695 + 0.915773i −0.0347973 − 0.0270838i −0.677282 + 0.735724i 0.171151 + 2.06549i 0.0108247 − 0.0427459i −3.14433 − 3.41565i −0.945817 − 0.324699i −0.735979 − 2.90632i −1.82277 + 0.986433i
39.8 0.401695 + 0.915773i 0.116693 + 0.0908261i −0.677282 + 0.735724i −0.0556816 − 0.671977i −0.0363009 + 0.143349i 0.526720 + 0.572171i −0.945817 − 0.324699i −0.731088 − 2.88700i 0.593012 − 0.320922i
39.9 0.401695 + 0.915773i 0.132731 + 0.103309i −0.677282 + 0.735724i −0.0887255 − 1.07076i −0.0412899 + 0.163050i 0.0991610 + 0.107718i −0.945817 − 0.324699i −0.729512 − 2.88078i 0.944930 − 0.511371i
39.10 0.401695 + 0.915773i 0.206818 + 0.160973i −0.677282 + 0.735724i −0.303516 − 3.66289i −0.0643369 + 0.254061i 0.423727 + 0.460290i −0.945817 − 0.324699i −0.719595 − 2.84162i 3.23246 − 1.74932i
39.11 0.401695 + 0.915773i 0.674456 + 0.524950i −0.677282 + 0.735724i −0.123500 − 1.49042i −0.209809 + 0.828518i −1.20546 − 1.30948i −0.945817 − 0.324699i −0.557138 − 2.20009i 1.31528 − 0.711794i
39.12 0.401695 + 0.915773i 1.20139 + 0.935080i −0.677282 + 0.735724i 0.337952 + 4.07847i −0.373728 + 1.47582i 0.102112 + 0.110924i −0.945817 − 0.324699i −0.167491 − 0.661406i −3.59920 + 1.94779i
39.13 0.401695 + 0.915773i 1.51719 + 1.18087i −0.677282 + 0.735724i 0.156667 + 1.89069i −0.471965 + 1.86375i 3.27119 + 3.55346i −0.945817 − 0.324699i 0.170935 + 0.675007i −1.66852 + 0.902955i
39.14 0.401695 + 0.915773i 2.19032 + 1.70480i −0.677282 + 0.735724i 0.0981252 + 1.18419i −0.681365 + 2.69065i −1.85473 − 2.01478i −0.945817 − 0.324699i 1.15473 + 4.55991i −1.04504 + 0.565546i
39.15 0.401695 + 0.915773i 2.24445 + 1.74693i −0.677282 + 0.735724i −0.330232 − 3.98530i −0.698204 + 2.75714i 2.23039 + 2.42284i −0.945817 − 0.324699i 1.24936 + 4.93360i 3.51698 − 1.90329i
39.16 0.401695 + 0.915773i 2.32865 + 1.81246i −0.677282 + 0.735724i 0.101188 + 1.22115i −0.724396 + 2.86057i 0.210956 + 0.229159i −0.945817 − 0.324699i 1.40114 + 5.53298i −1.07765 + 0.583196i
77.1 0.677282 − 0.735724i −0.694407 − 2.74215i −0.0825793 − 0.996584i 1.13262 − 0.189001i −2.48778 − 1.34632i 0.120897 − 1.45901i −0.789141 − 0.614213i −4.39878 + 2.38050i 0.628052 − 0.961305i
77.2 0.677282 − 0.735724i −0.648406 − 2.56050i −0.0825793 − 0.996584i −3.66119 + 0.610944i −2.32297 − 1.25713i 0.391578 − 4.72565i −0.789141 − 0.614213i −3.49730 + 1.89264i −2.03017 + 3.10741i
77.3 0.677282 − 0.735724i −0.603954 − 2.38496i −0.0825793 − 0.996584i 4.23858 − 0.707294i −2.16372 − 1.17095i −0.215733 + 2.60350i −0.789141 − 0.614213i −2.68486 + 1.45297i 2.35034 − 3.59746i
77.4 0.677282 − 0.735724i −0.473089 − 1.86819i −0.0825793 − 0.996584i −3.12184 + 0.520944i −1.69489 − 0.917226i −0.381486 + 4.60385i −0.789141 − 0.614213i −0.627892 + 0.339798i −1.73110 + 2.64964i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
361.g even 19 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 722.2.g.a ✓ 288
361.g even 19 1 inner 722.2.g.a ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
722.2.g.a ✓ 288 1.a even 1 1 trivial
722.2.g.a ✓ 288 361.g even 19 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{288} + T_{3}^{287} + 31 T_{3}^{286} + 51 T_{3}^{285} + 643 T_{3}^{284} + 1007 T_{3}^{283} + \cdots + 60\!\cdots\!49 \) acting on \(S_{2}^{\mathrm{new}}(722, [\chi])\). Copy content Toggle raw display