↑

Properties

Label 13.14.c.a
Level $13$
Weight $14$
Character orbit 13.c
Analytic conductor $13.940$
Analytic rank $0$
Dimension $28$
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] = [13,14,Mod(3,13)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(13, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 14, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("13.3"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 13.c (of order \(3\), 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: \(13.9400207637\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 65 q^{2} + 728 q^{3} - 53249 q^{4} + 41860 q^{5} + 143910 q^{6} - 173992 q^{7} + 2616978 q^{8} - 5231838 q^{9} + 2769243 q^{10} + 10986144 q^{11} - 45205464 q^{12} + 25107706 q^{13} - 23731756 q^{14}+ \cdots + 33239809172544 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −82.8702 − 143.535i 451.483 + 781.992i −9638.95 + 16695.1i −67212.0 74829.1 − 129608.i −137812. + 238698.i 1.83738e6 389487. − 674611.i 5.56987e6 + 9.64731e6i
3.2 −82.3546 − 142.642i −255.684 − 442.858i −9468.55 + 16400.0i 48546.3 −42113.5 + 72942.8i 216096. − 374290.i 1.76982e6 666413. − 1.15426e6i −3.99801e6 − 6.92476e6i
3.3 −59.2686 − 102.656i 1072.90 + 1858.33i −2929.54 + 5074.11i 45885.5 127179. − 220281.i −177681. + 307752.i −276538. −1.50509e6 + 2.60689e6i −2.71957e6 − 4.71043e6i
3.4 −56.7104 − 98.2253i −906.506 − 1570.12i −2336.14 + 4046.32i −10940.5 −102817. + 178084.i −89772.2 + 155490.i −399209. −846346. + 1.46591e6i 620440. + 1.07463e6i
3.5 −45.4515 − 78.7243i 108.764 + 188.384i −35.6760 + 61.7927i −2568.18 9886.95 − 17124.7i 77358.1 − 133988.i −738191. 773502. − 1.33975e6i 116728. + 202178.i
3.6 −16.9887 − 29.4253i 797.844 + 1381.91i 3518.77 − 6094.69i −28205.3 27108.7 − 46953.6i 177172. − 306871.i −517460. −475948. + 824367.i 479171. + 829948.i
3.7 −3.81407 − 6.60616i −187.264 − 324.351i 4066.91 − 7044.09i 42407.7 −1428.48 + 2474.20i −178632. + 309400.i −124536. 727026. − 1.25925e6i −161746. − 280152.i
3.8 8.80659 + 15.2535i −449.698 − 778.901i 3940.89 − 6825.82i −51640.9 7920.62 − 13718.9i −75519.2 + 130803.i 283110. 392704. − 680184.i −454780. − 787702.i
3.9 11.2286 + 19.4486i −1142.04 − 1978.08i 3843.84 − 6657.72i 17999.0 25647.2 − 44422.2i 281521. − 487609.i 356614. −1.81136e6 + 3.13737e6i 202104. + 350055.i
3.10 37.0467 + 64.1667i 958.983 + 1661.01i 1351.09 − 2340.15i −22075.6 −71054.3 + 123070.i −250997. + 434740.i 807186. −1.04213e6 + 1.80503e6i −817827. − 1.41652e6i
3.11 40.7956 + 70.6601i 482.143 + 835.097i 767.437 − 1329.24i 41512.4 −39338.7 + 68136.6i 158394. − 274346.i 793627. 332237. − 575452.i 1.69352e6 + 2.93327e6i
3.12 60.7949 + 105.300i −355.726 − 616.135i −3296.05 + 5708.93i −32981.4 43252.7 − 74915.8i 47112.8 − 81601.8i 194532. 544080. − 942374.i −2.00510e6 − 3.47294e6i
3.13 71.5224 + 123.880i −865.569 − 1499.21i −6134.91 + 10626.0i 41380.4 123815. − 214454.i −187839. + 325347.i −583312. −701257. + 1.21461e6i 2.95963e6 + 5.12623e6i
3.14 84.7632 + 146.814i 654.369 + 1133.40i −10273.6 + 17794.4i −1177.50 −110933. + 192141.i 53602.7 − 92842.5i −2.09453e6 −59235.2 + 102598.i −99808.8 − 172874.i
9.1 −82.8702 + 143.535i 451.483 − 781.992i −9638.95 − 16695.1i −67212.0 74829.1 + 129608.i −137812. − 238698.i 1.83738e6 389487. + 674611.i 5.56987e6 − 9.64731e6i
9.2 −82.3546 + 142.642i −255.684 + 442.858i −9468.55 − 16400.0i 48546.3 −42113.5 − 72942.8i 216096. + 374290.i 1.76982e6 666413. + 1.15426e6i −3.99801e6 + 6.92476e6i
9.3 −59.2686 + 102.656i 1072.90 − 1858.33i −2929.54 − 5074.11i 45885.5 127179. + 220281.i −177681. − 307752.i −276538. −1.50509e6 − 2.60689e6i −2.71957e6 + 4.71043e6i
9.4 −56.7104 + 98.2253i −906.506 + 1570.12i −2336.14 − 4046.32i −10940.5 −102817. − 178084.i −89772.2 − 155490.i −399209. −846346. − 1.46591e6i 620440. − 1.07463e6i
9.5 −45.4515 + 78.7243i 108.764 − 188.384i −35.6760 − 61.7927i −2568.18 9886.95 + 17124.7i 77358.1 + 133988.i −738191. 773502. + 1.33975e6i 116728. − 202178.i
9.6 −16.9887 + 29.4253i 797.844 − 1381.91i 3518.77 + 6094.69i −28205.3 27108.7 + 46953.6i 177172. + 306871.i −517460. −475948. − 824367.i 479171. − 829948.i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.14.c.a ✓ 28
13.c even 3 1 inner 13.14.c.a ✓ 28
13.c even 3 1 169.14.a.e 14
13.e even 6 1 169.14.a.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.14.c.a ✓ 28 1.a even 1 1 trivial
13.14.c.a ✓ 28 13.c even 3 1 inner
169.14.a.c 14 13.e even 6 1
169.14.a.e 14 13.c even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{14}^{\mathrm{new}}(13, [\chi])\).