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Properties

Label 361.2.k.a
Level $361$
Weight $2$
Character orbit 361.k
Analytic conductor $2.883$
Analytic rank $0$
Dimension $3348$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [361,2,Mod(4,361)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(361, base_ring=CyclotomicField(342)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("361.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 361.k (of order \(171\), degree \(108\), 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: \(2.88259951297\)
Analytic rank: \(0\)
Dimension: \(3348\)
Relative dimension: \(31\) over \(\Q(\zeta_{171})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{171}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3348 q - 108 q^{2} - 111 q^{3} - 114 q^{4} - 108 q^{5} - 117 q^{6} - 111 q^{7} - 120 q^{8} - 117 q^{9} - 123 q^{10} - 111 q^{11} - 117 q^{12} - 111 q^{13} - 111 q^{14} - 117 q^{15} - 96 q^{16} - 3 q^{17}+ \cdots - 123 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.68057 − 0.0492526i 2.14428 − 1.42921i 5.18437 + 0.190579i 0.225586 + 0.466865i −5.81829 + 3.72547i −1.02509 + 0.416356i −8.53377 − 0.470821i 1.40091 − 3.36025i −0.581703 − 1.26257i
4.2 −2.59231 − 0.0476309i −1.12649 + 0.750827i 4.71913 + 0.173477i 0.724103 + 1.49858i 2.95597 − 1.89272i −2.70404 + 1.09828i −7.04756 − 0.388825i −0.449173 + 1.07740i −1.80572 − 3.91927i
4.3 −2.57446 − 0.0473030i −1.65408 + 1.10248i 4.62696 + 0.170089i −0.912865 − 1.88924i 4.31052 − 2.76004i 3.30104 − 1.34077i −6.76191 − 0.373065i 0.366117 − 0.878175i 2.26077 + 4.90695i
4.4 −2.10682 − 0.0387106i 0.975583 − 0.650245i 2.43855 + 0.0896418i 1.36894 + 2.83311i −2.08055 + 1.33218i 0.0179410 − 0.00728700i −0.926162 − 0.0510978i −0.625468 + 1.50026i −2.77444 − 6.02186i
4.5 −2.03170 − 0.0373303i 1.47869 − 0.985572i 2.12774 + 0.0782165i −1.15639 − 2.39322i −3.04103 + 1.94718i 3.42774 − 1.39223i −0.262102 − 0.0144606i 0.0607472 − 0.145709i 2.26009 + 4.90547i
4.6 −2.00564 − 0.0368515i 0.00350894 − 0.00233877i 2.02258 + 0.0743507i −1.36836 − 2.83191i −0.00712385 + 0.00456143i −3.83080 + 1.55594i −0.0479653 − 0.00264632i −1.15441 + 2.76898i 2.64007 + 5.73021i
4.7 −1.88090 − 0.0345596i −0.161179 + 0.107429i 1.53795 + 0.0565354i 0.926139 + 1.91671i 0.306874 − 0.196493i 2.75639 − 1.11955i 0.865952 + 0.0477759i −1.13997 + 2.73436i −1.67573 − 3.63715i
4.8 −1.74837 − 0.0321244i −2.71203 + 1.80762i 1.05711 + 0.0388596i −0.570558 − 1.18081i 4.79969 − 3.07326i −1.39625 + 0.567108i 1.64505 + 0.0907600i 2.93321 − 7.03564i 0.959612 + 2.08282i
4.9 −1.29904 − 0.0238685i −1.21388 + 0.809073i −0.311709 − 0.0114585i −0.0974005 − 0.201577i 1.59619 − 1.02205i −0.926892 + 0.376471i 2.99923 + 0.165472i −0.335510 + 0.804760i 0.121716 + 0.264182i
4.10 −1.15091 − 0.0211468i 2.46964 − 1.64606i −0.674499 − 0.0247948i 0.997299 + 2.06398i −2.87715 + 1.84225i −1.75516 + 0.712886i 3.07448 + 0.169624i 2.23520 − 5.36139i −1.10416 − 2.39655i
4.11 −0.937912 − 0.0172331i −2.24235 + 1.49457i −1.11927 − 0.0411447i 1.35012 + 2.79416i 2.12888 − 1.36313i 4.41634 − 1.79376i 2.92236 + 0.161231i 1.63998 − 3.93370i −1.21814 − 2.64394i
4.12 −0.933975 − 0.0171608i 1.29845 − 0.865439i −1.12664 − 0.0414155i −0.880440 − 1.82213i −1.22757 + 0.786016i −2.85494 + 1.15958i 2.91697 + 0.160934i −0.217434 + 0.521541i 0.791040 + 1.71694i
4.13 −0.720055 − 0.0132303i −1.41262 + 0.941535i −1.48035 − 0.0544180i 1.90263 + 3.93762i 1.02962 − 0.659268i −4.52008 + 1.83590i 2.50338 + 0.138115i −0.0454172 + 0.108939i −1.31790 − 2.86047i
4.14 −0.495679 − 0.00910759i 2.27739 − 1.51792i −1.75304 − 0.0644421i −0.130251 − 0.269563i −1.14268 + 0.731660i 4.23959 − 1.72197i 1.85838 + 0.102529i 1.72799 − 4.14479i 0.0621075 + 0.134803i
4.15 −0.351568 − 0.00645970i −0.533085 + 0.355312i −1.87509 − 0.0689290i −1.63080 − 3.37505i 0.189711 − 0.121473i 1.84053 − 0.747557i 1.36096 + 0.0750865i −0.996479 + 2.39017i 0.551535 + 1.19709i
4.16 0.0510612 0.000938197i −1.90048 + 1.26671i −1.99604 − 0.0733752i −0.539848 − 1.11725i −0.0982292 + 0.0628965i 0.439274 − 0.178418i −0.203836 − 0.0112460i 0.852866 − 2.04570i −0.0265171 − 0.0575549i
4.17 0.126711 + 0.00232818i −0.167706 + 0.111779i −1.98260 − 0.0728810i −0.267723 − 0.554071i −0.0215104 + 0.0137732i 1.97597 − 0.802569i −0.504126 − 0.0278134i −1.13878 + 2.73150i −0.0326333 − 0.0708300i
4.18 0.328494 + 0.00603573i 0.508770 − 0.339105i −1.89078 − 0.0695056i 0.597036 + 1.23561i 0.169175 − 0.108323i −2.54147 + 1.03226i −1.27679 − 0.0704424i −1.01056 + 2.42394i 0.188665 + 0.409493i
4.19 0.554324 + 0.0101851i 2.33009 − 1.55305i −1.69148 − 0.0621793i −1.51974 − 3.14522i 1.30744 − 0.837161i −1.92218 + 0.780722i −2.04415 − 0.112779i 1.86295 − 4.46850i −0.810397 − 1.75895i
4.20 0.564438 + 0.0103710i 0.721659 − 0.480999i −1.68017 − 0.0617635i 1.58305 + 3.27624i 0.412320 − 0.264010i 2.92391 − 1.18759i −2.07506 − 0.114484i −0.864981 + 2.07476i 0.859559 + 1.86565i
See next 80 embeddings (of 3348 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.31
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
361.k even 171 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.2.k.a ✓ 3348
361.k even 171 1 inner 361.2.k.a ✓ 3348
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
361.2.k.a ✓ 3348 1.a even 1 1 trivial
361.2.k.a ✓ 3348 361.k even 171 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(361, [\chi])\).