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Properties

Label 289.6.a.j
Level $289$
Weight $6$
Character orbit 289.a
Self dual yes
Analytic conductor $46.351$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [289,6,Mod(1,289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("289.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 289.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(46.3509239260\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 16 q^{2} + 448 q^{4} + 768 q^{8} + 2268 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 16 q^{2} + 448 q^{4} + 768 q^{8} + 2268 q^{9} + 304 q^{13} + 4392 q^{15} + 7176 q^{16} + 1896 q^{18} + 9288 q^{19} + 12032 q^{21} + 17692 q^{25} + 29600 q^{26} - 17784 q^{30} + 19032 q^{32} + 38800 q^{33} + 26056 q^{35} + 77816 q^{36} + 36384 q^{38} + 123904 q^{42} + 46520 q^{43} + 71808 q^{47} + 38748 q^{49} + 241632 q^{50} + 6008 q^{52} + 61360 q^{53} + 46680 q^{55} + 256920 q^{59} + 330504 q^{60} - 72496 q^{64} + 10736 q^{66} + 250608 q^{67} - 107696 q^{69} + 273320 q^{70} + 463640 q^{72} + 974048 q^{76} + 482672 q^{77} + 242060 q^{81} + 458584 q^{83} + 1605472 q^{84} + 718272 q^{86} + 1009400 q^{87} + 501088 q^{89} + 903248 q^{93} + 1315264 q^{94} - 156256 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −10.2095 −6.19225 72.2346 37.5538 63.2200 −133.893 −410.777 −204.656 −383.407
1.2 −10.2095 6.19225 72.2346 −37.5538 −63.2200 133.893 −410.777 −204.656 383.407
1.3 −8.95189 −29.0894 48.1363 −65.2984 260.405 −113.506 −144.450 603.192 584.544
1.4 −8.95189 29.0894 48.1363 65.2984 −260.405 113.506 −144.450 603.192 −584.544
1.5 −7.88986 −10.4947 30.2500 −55.9505 82.8016 53.9692 13.8075 −132.862 441.442
1.6 −7.88986 10.4947 30.2500 55.9505 −82.8016 −53.9692 13.8075 −132.862 −441.442
1.7 −5.39360 −17.9903 −2.90907 30.2603 97.0323 −67.7275 188.286 80.6493 −163.212
1.8 −5.39360 17.9903 −2.90907 −30.2603 −97.0323 67.7275 188.286 80.6493 163.212
1.9 −3.49244 −27.6565 −19.8029 −33.0367 96.5885 164.237 180.918 521.880 115.379
1.10 −3.49244 27.6565 −19.8029 33.0367 −96.5885 −164.237 180.918 521.880 −115.379
1.11 −2.31253 −4.07382 −26.6522 −18.8968 9.42082 243.456 135.635 −226.404 43.6993
1.12 −2.31253 4.07382 −26.6522 18.8968 −9.42082 −243.456 135.635 −226.404 −43.6993
1.13 0.319709 −21.7836 −31.8978 82.7344 −6.96442 203.501 −20.4287 231.526 26.4509
1.14 0.319709 21.7836 −31.8978 −82.7344 6.96442 −203.501 −20.4287 231.526 −26.4509
1.15 0.624860 −2.22952 −31.6095 76.0381 −1.39314 −69.7195 −39.7471 −238.029 47.5132
1.16 0.624860 2.22952 −31.6095 −76.0381 1.39314 69.7195 −39.7471 −238.029 −47.5132
1.17 3.75940 −17.0295 −17.8669 36.2064 −64.0207 −106.518 −187.470 47.0037 136.114
1.18 3.75940 17.0295 −17.8669 −36.2064 64.0207 106.518 −187.470 47.0037 −136.114
1.19 5.21577 −1.40483 −4.79575 −86.6347 −7.32727 −100.074 −191.918 −241.026 −451.867
1.20 5.21577 1.40483 −4.79575 86.6347 7.32727 100.074 −191.918 −241.026 451.867
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.6.a.j 28
17.b even 2 1 inner 289.6.a.j 28
17.e odd 16 2 17.6.d.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.6.d.a ✓ 28 17.e odd 16 2
289.6.a.j 28 1.a even 1 1 trivial
289.6.a.j 28 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(289))\):

\( T_{2}^{14} - 8 T_{2}^{13} - 304 T_{2}^{12} + 2304 T_{2}^{11} + 34815 T_{2}^{10} - 244788 T_{2}^{9} + \cdots + 802044928 \) Copy content Toggle raw display
\( T_{3}^{28} - 4536 T_{3}^{26} + 8987104 T_{3}^{24} - 10224129792 T_{3}^{22} + 7383690320856 T_{3}^{20} + \cdots + 21\!\cdots\!32 \) Copy content Toggle raw display