Properties

Label 59.11.d.a
Level $59$
Weight $11$
Character orbit 59.d
Analytic conductor $37.486$
Analytic rank $0$
Dimension $1372$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [59,11,Mod(2,59)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(59, base_ring=CyclotomicField(58))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.2");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 59.d (of order \(58\), degree \(28\), minimal)

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: no
Analytic conductor: \(37.4860779077\)
Analytic rank: \(0\)
Dimension: \(1372\)
Relative dimension: \(49\) over \(\Q(\zeta_{58})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{58}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 1372 q - 29 q^{2} - 45 q^{3} + 24549 q^{4} + 3279 q^{5} - 29 q^{6} + 18365 q^{7} - 29 q^{8} - 863866 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1372 q - 29 q^{2} - 45 q^{3} + 24549 q^{4} + 3279 q^{5} - 29 q^{6} + 18365 q^{7} - 29 q^{8} - 863866 q^{9} - 29 q^{10} - 29 q^{11} + 2125191 q^{12} - 29 q^{13} - 29 q^{14} + 525016 q^{15} - 11811419 q^{16} + 2465450 q^{17} - 29 q^{18} - 5161449 q^{19} - 5213773 q^{20} - 3094428 q^{21} - 5288067 q^{22} - 29 q^{23} - 29 q^{24} - 76972146 q^{25} + 18186289 q^{26} + 25282800 q^{27} - 84694487 q^{28} - 56251821 q^{29} - 29 q^{30} - 29 q^{31} - 29 q^{32} - 29 q^{33} - 29 q^{34} + 2533300 q^{35} + 472097749 q^{36} - 29 q^{37} - 29 q^{38} - 29 q^{39} - 29 q^{40} + 1761773 q^{41} - 29 q^{42} - 29 q^{43} - 29 q^{44} + 1704734639 q^{45} - 2985150953 q^{46} + 185032412 q^{47} + 5416429647 q^{48} - 1076336002 q^{49} - 1661476381 q^{50} - 4880264737 q^{51} - 6136559645 q^{52} + 1227310777 q^{53} + 4952615331 q^{54} + 5386244182 q^{55} + 16772894691 q^{56} + 5787309683 q^{57} - 1305938370 q^{59} - 36201517322 q^{60} - 5977396799 q^{61} - 7162306297 q^{62} + 2400099760 q^{63} + 32087866793 q^{64} + 13558367332 q^{65} + 18748519507 q^{66} + 5108918893 q^{67} - 12125472443 q^{68} - 23762763009 q^{69} - 33094877981 q^{70} - 4509247620 q^{71} + 19164343267 q^{72} + 14105547046 q^{73} + 26041025965 q^{74} - 28606023430 q^{75} + 3328782843 q^{76} - 29 q^{77} - 1687450309 q^{78} + 275593977 q^{79} - 5068347253 q^{80} - 30228222398 q^{81} - 29 q^{82} - 29 q^{83} - 19956142245 q^{84} + 19041531887 q^{85} - 4710428163 q^{86} + 14209983096 q^{87} + 28706521209 q^{88} - 29 q^{89} - 29 q^{90} - 29 q^{91} - 29 q^{92} - 29 q^{93} - 31545474805 q^{94} + 17636908389 q^{95} - 29 q^{96} - 29 q^{97} + 81047814844 q^{98} - 185876449895 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −63.5072 3.44326i 228.394 105.666i 3003.31 + 326.630i −2502.91 843.328i −14868.5 + 5924.15i −13636.9 20113.0i −125338. 20548.2i 2770.88 3262.14i 156049. + 62175.6i
2.2 −58.7326 3.18439i −356.429 + 164.902i 2421.38 + 263.341i 953.655 + 321.324i 21459.1 8550.10i −7498.83 11059.9i −81938.2 13433.1i 61621.8 72546.8i −54987.4 21909.0i
2.3 −58.5728 3.17572i −210.180 + 97.2396i 2402.69 + 261.308i −3651.31 1230.27i 12619.6 5028.12i 7707.96 + 11368.4i −80627.0 13218.1i −3507.47 + 4129.31i 209961. + 83656.0i
2.4 −57.0594 3.09367i −43.1529 + 19.9647i 2228.21 + 242.332i 4330.52 + 1459.12i 2524.04 1005.67i 11516.2 + 16985.1i −68646.8 11254.1i −36763.9 + 43281.8i −242583. 96653.9i
2.5 −55.0395 2.98415i 298.048 137.892i 2002.44 + 217.779i 2237.07 + 753.757i −16815.9 + 6700.07i 4424.83 + 6526.14i −53863.9 8830.55i 31590.9 37191.7i −120878. 48162.2i
2.6 −52.5638 2.84992i −130.502 + 60.3767i 1736.83 + 188.892i 3331.84 + 1122.63i 7031.74 2801.70i −8390.68 12375.3i −37561.7 6157.93i −24842.1 + 29246.4i −171934. 68504.9i
2.7 −47.9598 2.60030i −24.3936 + 11.2857i 1275.38 + 138.706i −3095.79 1043.09i 1199.26 477.829i −221.486 326.667i −12271.3 2011.77i −37759.8 + 44454.3i 145761. + 58076.6i
2.8 −47.8443 2.59404i 236.528 109.429i 1264.35 + 137.507i −3457.65 1165.02i −11600.4 + 4622.01i 13550.7 + 19985.9i −11717.2 1920.94i 5743.02 6761.21i 162407. + 64708.7i
2.9 −44.3605 2.40516i 33.8748 15.6722i 944.075 + 102.674i −347.715 117.159i −1540.40 + 613.752i −12957.4 19110.7i 3259.91 + 534.435i −37325.6 + 43943.1i 15143.1 + 6033.55i
2.10 −41.5655 2.25361i 364.487 168.629i 704.611 + 76.6310i 1994.16 + 671.909i −15530.1 + 6187.75i −6225.84 9182.43i 12949.2 + 2122.92i 66187.0 77921.4i −81373.7 32422.3i
2.11 −39.7014 2.15255i −365.544 + 169.118i 553.573 + 60.2047i −2.80624 0.945534i 14876.6 5927.39i 10190.2 + 15029.5i 18329.6 + 3004.98i 66793.5 78635.4i 109.377 + 43.5796i
2.12 −36.1667 1.96090i −322.067 + 149.004i 286.188 + 31.1249i −5356.12 1804.69i 11940.3 4757.45i −16562.6 24428.1i 26311.1 + 4313.48i 43297.6 50973.8i 190174. + 75772.4i
2.13 −33.0583 1.79237i −330.750 + 153.021i 71.6382 + 7.79112i 4318.18 + 1454.96i 11208.3 4465.79i −5648.75 8331.28i 31100.5 + 5098.67i 47752.5 56218.6i −140144. 55838.3i
2.14 −32.5255 1.76348i 427.138 197.615i 36.8041 + 4.00268i −5102.57 1719.25i −14241.4 + 5674.29i −10663.2 15727.1i 31725.7 + 5201.16i 105168. 123813.i 162932. + 64918.0i
2.15 −31.8894 1.72899i −104.626 + 48.4054i −4.05577 0.441091i −182.378 61.4504i 3420.16 1362.72i 6593.18 + 9724.21i 32400.4 + 5311.78i −29623.9 + 34875.9i 5709.68 + 2274.94i
2.16 −29.8836 1.62024i 201.453 93.2019i −127.594 13.8767i 4428.02 + 1491.98i −6171.13 + 2458.80i −6659.33 9821.78i 34032.5 + 5579.35i −6330.97 + 7453.40i −129908. 51760.0i
2.17 −23.8525 1.29324i 123.988 57.3631i −450.729 49.0198i −2276.31 766.978i −3031.61 + 1207.91i −3127.24 4612.33i 34826.2 + 5709.46i −26144.9 + 30780.2i 53303.7 + 21238.1i
2.18 −22.0553 1.19581i −8.21651 + 3.80136i −532.989 57.9661i 4105.81 + 1383.41i 185.764 74.0150i 15487.3 + 22842.0i 34005.8 + 5574.97i −38174.5 + 44942.4i −88900.7 35421.3i
2.19 −18.9233 1.02599i 316.805 146.569i −660.958 71.8835i 463.180 + 156.064i −6145.38 + 2448.54i 14010.4 + 20663.8i 31584.1 + 5177.94i 40655.1 47862.9i −8604.78 3428.46i
2.20 −16.0810 0.871886i 113.519 52.5194i −760.159 82.6723i −3672.04 1237.25i −1871.28 + 745.588i −1377.94 2032.31i 28425.9 + 4660.19i −28099.3 + 33081.0i 57971.3 + 23097.9i
See next 80 embeddings (of 1372 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.49
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.d odd 58 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 59.11.d.a 1372
59.d odd 58 1 inner 59.11.d.a 1372
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.11.d.a 1372 1.a even 1 1 trivial
59.11.d.a 1372 59.d odd 58 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(59, [\chi])\).