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Properties

Label 19.13.d.a
Level $19$
Weight $13$
Character orbit 19.d
Analytic conductor $17.366$
Analytic rank $0$
Dimension $38$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [19,13,Mod(8,19)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(19, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 13, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("19.8"); S:= CuspForms(chi, 13); N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 19.d (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: \(17.3658825282\)
Analytic rank: \(0\)
Dimension: \(38\)
Relative dimension: \(19\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 3 q^{2} + 1377 q^{3} + 37325 q^{4} + 3023 q^{5} - 26867 q^{6} + 202236 q^{7} + 3465518 q^{9} - 4210176 q^{10} + 225284 q^{11} + 212517 q^{13} - 21278490 q^{14} + 46075869 q^{15} - 72759187 q^{16}+ \cdots + 231693405580 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −103.778 − 59.9160i 1112.21 + 642.136i 5131.87 + 8888.65i 9865.83 − 17088.1i −76948.4 − 133279.i 136607. − 739092.i 558956. + 968140.i −2.04771e6 + 1.18224e6i
8.2 −101.162 − 58.4061i −509.434 − 294.122i 4774.55 + 8269.76i −5618.03 + 9730.71i 34357.0 + 59508.1i 37646.3 − 636988.i −92705.3 − 160570.i 1.13667e6 − 656254.i
8.3 −79.9424 − 46.1548i −112.900 − 65.1830i 2212.52 + 3832.20i 6369.31 − 11032.0i 6017.01 + 10421.8i −43502.9 − 30374.1i −257223. − 445523.i −1.01836e6 + 587948.i
8.4 −73.2301 − 42.2794i 681.442 + 393.431i 1527.10 + 2645.02i −8480.07 + 14687.9i −33268.1 − 57622.0i −130670. 88093.1i 43855.2 + 75959.5i 1.24199e6 − 717065.i
8.5 −63.8174 − 36.8450i −1254.11 − 724.058i 667.109 + 1155.47i 6597.60 − 11427.4i 53355.9 + 92415.1i −120010. 203516.i 782800. + 1.35585e6i −842084. + 486177.i
8.6 −45.4503 − 26.2407i 618.493 + 357.087i −670.847 − 1161.94i −4028.61 + 6977.76i −18740.5 − 32459.4i 120053. 285378.i −10698.1 − 18529.6i 366203. − 211428.i
8.7 −39.7413 − 22.9446i −202.195 − 116.737i −995.087 − 1723.54i 9461.86 − 16388.4i 5357.00 + 9278.59i 162465. 279290.i −238465. − 413034.i −752053. + 434198.i
8.8 −38.8416 − 22.4252i −812.168 − 468.905i −1042.22 − 1805.17i −14355.5 + 24864.5i 21030.6 + 36426.1i 137365. 277195.i 174024. + 301418.i 1.11518e6 − 643851.i
8.9 −14.2608 − 8.23346i 844.490 + 487.567i −1912.42 − 3312.41i 13891.8 − 24061.3i −8028.72 − 13906.2i −187143. 130432.i 209722. + 363249.i −396216. + 228756.i
8.10 −5.19053 − 2.99675i −395.326 − 228.241i −2030.04 − 3516.13i −3340.18 + 5785.37i 1367.97 + 2369.39i −150303. 48883.5i −161532. − 279782.i 34674.6 − 20019.4i
8.11 10.6262 + 6.13502i 1081.85 + 624.608i −1972.72 − 3416.86i −3733.97 + 6467.43i 7663.97 + 13274.4i 109319. − 98668.8i 514551. + 891228.i −79355.6 + 45816.0i
8.12 26.7100 + 15.4210i 374.102 + 215.988i −1572.38 − 2723.45i −3570.15 + 6183.68i 6661.52 + 11538.1i 32696.9 − 223320.i −172419. − 298638.i −190717. + 110111.i
8.13 30.7257 + 17.7395i −822.069 − 474.622i −1418.62 − 2457.13i 6636.69 − 11495.1i −16839.1 − 29166.2i 61912.6 − 245984.i 184812. + 320103.i 407834. − 235463.i
8.14 65.7272 + 37.9476i 202.251 + 116.770i 832.046 + 1441.15i −13303.8 + 23042.9i 8862.26 + 15349.9i −78949.2 − 184570.i −238450. − 413008.i −1.74884e6 + 1.00970e6i
8.15 66.5813 + 38.4407i 303.254 + 175.084i 907.380 + 1571.63i 9795.04 − 16965.5i 13460.7 + 23314.6i 53709.9 − 175385.i −204412. − 354052.i 1.30433e6 − 753057.i
8.16 73.8192 + 42.6195i −962.676 − 555.801i 1584.85 + 2745.04i −7644.85 + 13241.3i −47376.0 − 82057.6i 76020.2 − 78956.7i 352109. + 609872.i −1.12867e6 + 651640.i
8.17 87.9394 + 50.7718i 1114.77 + 643.610i 3107.56 + 5382.45i −509.092 + 881.773i 65354.5 + 113197.i −101293. 215183.i 562748. + 974708.i −89538.5 + 51695.1i
8.18 94.4468 + 54.5289i −618.668 − 357.188i 3898.79 + 6752.91i 6399.15 − 11083.7i −38954.2 − 67470.6i −232754. 403687.i −10553.4 − 18279.0i 1.20876e6 − 697877.i
8.19 107.339 + 61.9720i 45.1791 + 26.0842i 5633.06 + 9756.75i −2921.54 + 5060.26i 3232.98 + 5599.69i 217949. 888694.i −264360. − 457884.i −627189. + 362108.i
12.1 −103.778 + 59.9160i 1112.21 − 642.136i 5131.87 − 8888.65i 9865.83 + 17088.1i −76948.4 + 133279.i 136607. 739092.i 558956. − 968140.i −2.04771e6 − 1.18224e6i
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.19
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.13.d.a ✓ 38
19.d odd 6 1 inner 19.13.d.a ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.13.d.a ✓ 38 1.a even 1 1 trivial
19.13.d.a ✓ 38 19.d odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(19, [\chi])\).