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Properties

Label 121.4.e.a
Level $121$
Weight $4$
Character orbit 121.e
Analytic conductor $7.139$
Analytic rank $0$
Dimension $320$
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] = [121,4,Mod(12,121)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(121, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([18]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.12");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 121.e (of order \(11\), degree \(10\), 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: \(7.13923111069\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(32\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 13 q^{2} - 18 q^{3} - 123 q^{4} - 17 q^{5} - 73 q^{6} - 31 q^{7} + q^{8} + 2618 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q - 13 q^{2} - 18 q^{3} - 123 q^{4} - 17 q^{5} - 73 q^{6} - 31 q^{7} + q^{8} + 2618 q^{9} - 193 q^{10} - 143 q^{11} - 368 q^{12} + 217 q^{13} + 265 q^{14} - 372 q^{15} - 455 q^{16} + 113 q^{17} - 488 q^{18} - 83 q^{19} - 183 q^{20} - 384 q^{21} + 803 q^{22} - 835 q^{23} + 1464 q^{24} - 825 q^{25} - 43 q^{26} - 528 q^{27} + 117 q^{28} - 155 q^{29} - 42 q^{30} + 691 q^{31} + 93 q^{32} - 330 q^{33} - 27 q^{34} + 161 q^{35} - 1918 q^{36} + 907 q^{37} - 73 q^{38} - 708 q^{39} + 3110 q^{40} - 547 q^{41} + 26 q^{42} + 49 q^{43} - 1474 q^{44} - 1933 q^{45} + 303 q^{46} - 141 q^{47} - 774 q^{48} + 1041 q^{49} - 4071 q^{50} + 2331 q^{51} - 1629 q^{52} - 63 q^{53} + 5500 q^{54} + 3443 q^{55} + 122 q^{56} - 2107 q^{57} - 3204 q^{58} - 447 q^{59} - 912 q^{60} - 851 q^{61} - 3390 q^{62} - 7519 q^{63} + 5137 q^{64} + 1397 q^{65} - 2684 q^{66} + 219 q^{67} - 651 q^{68} - 1540 q^{69} - 5775 q^{70} - 493 q^{71} - 5196 q^{72} - 1723 q^{73} - 17 q^{74} - 164 q^{75} - 7186 q^{76} + 5621 q^{77} + 7368 q^{78} - 5035 q^{79} + 1270 q^{80} + 19880 q^{81} - 897 q^{82} - 479 q^{83} + 428 q^{84} + 3673 q^{85} + 3 q^{86} - 1508 q^{87} - 2959 q^{88} - 4159 q^{89} - 15491 q^{90} + 2035 q^{91} + 9129 q^{92} - 6532 q^{93} + 14346 q^{94} + 4605 q^{95} + 8564 q^{96} - 1947 q^{97} + 1123 q^{98} - 14773 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1 −4.37587 + 2.81220i 3.60536 7.91644 − 17.3346i −1.13341 + 7.88302i −15.7766 + 10.1390i 9.94425 − 11.4763i 8.18490 + 56.9273i −14.0014 −17.2090 − 37.6824i
12.2 −4.27677 + 2.74851i 0.395713 7.41313 − 16.2325i 2.55132 − 17.7448i −1.69238 + 1.08762i 6.86910 − 7.92736i 7.12296 + 49.5413i −26.8434 37.8605 + 82.9030i
12.3 −4.24665 + 2.72916i −4.32925 7.26242 − 15.9025i −2.22979 + 15.5086i 18.3848 − 11.8152i −17.6899 + 20.4152i 6.81215 + 47.3796i −8.25763 −32.8561 − 71.9449i
12.4 −4.00704 + 2.57517i 9.58479 6.10154 − 13.3605i 1.21418 − 8.44483i −38.4066 + 24.6824i −14.2772 + 16.4768i 4.53347 + 31.5310i 64.8682 16.8816 + 36.9654i
12.5 −3.83056 + 2.46175i −6.93217 5.28964 − 11.5827i 1.83356 − 12.7527i 26.5541 − 17.0653i −5.27481 + 6.08745i 3.06731 + 21.3336i 21.0549 24.3704 + 53.3637i
12.6 −3.40824 + 2.19034i 3.99579 3.49517 − 7.65336i −0.998129 + 6.94214i −13.6186 + 8.75214i −1.82828 + 2.10995i 0.238522 + 1.65896i −11.0337 −11.8038 − 25.8467i
12.7 −3.38147 + 2.17314i −9.39256 3.38847 − 7.41972i −0.984810 + 6.84950i 31.7606 − 20.4113i 14.6877 − 16.9505i 0.0897200 + 0.624017i 61.2202 −11.5548 − 25.3015i
12.8 −2.68894 + 1.72808i −1.67699 0.920842 − 2.01636i −0.855803 + 5.95224i 4.50935 − 2.89798i 22.9980 − 26.5411i −2.63077 − 18.2974i −24.1877 −7.98474 − 17.4841i
12.9 −2.51632 + 1.61714i −3.50464 0.393415 − 0.861458i 0.713403 − 4.96183i 8.81881 − 5.66751i −11.3410 + 13.0882i −3.00235 − 20.8818i −14.7175 6.22884 + 13.6392i
12.10 −2.15539 + 1.38519i 3.13110 −0.596340 + 1.30580i 1.67614 − 11.6578i −6.74876 + 4.33716i −2.65745 + 3.06687i −3.44046 − 23.9289i −17.1962 12.5355 + 27.4490i
12.11 −2.03789 + 1.30967i 6.35586 −0.885564 + 1.93911i −2.54111 + 17.6738i −12.9525 + 8.32410i −10.0018 + 11.5427i −3.49292 − 24.2938i 13.3970 −17.9684 − 39.3454i
12.12 −1.86881 + 1.20101i 8.25345 −1.27330 + 2.78814i 1.41380 − 9.83316i −15.4241 + 9.91247i 13.2948 − 15.3430i −3.49820 − 24.3305i 41.1194 9.16761 + 20.0743i
12.13 −0.974437 + 0.626233i −4.45981 −2.76596 + 6.05661i −2.66959 + 18.5674i 4.34580 − 2.79288i 0.772606 − 0.891635i −2.41636 − 16.8061i −7.11012 −9.02618 − 19.7646i
12.14 −0.830603 + 0.533796i −7.06801 −2.91836 + 6.39031i 2.53435 − 17.6268i 5.87071 − 3.77288i 8.69424 − 10.0337i −2.11123 − 14.6839i 22.9567 7.30407 + 15.9937i
12.15 −0.552620 + 0.355147i −8.64947 −3.14406 + 6.88453i −0.513245 + 3.56970i 4.77987 − 3.07183i −16.0704 + 18.5462i −1.45545 − 10.1228i 47.8134 −0.984138 − 2.15496i
12.16 −0.237540 + 0.152658i 3.39735 −3.29020 + 7.20453i 1.29652 − 9.01747i −0.807007 + 0.518632i −21.8718 + 25.2413i −0.639750 − 4.44956i −15.4580 1.06861 + 2.33993i
12.17 0.226475 − 0.145547i −1.51217 −3.29321 + 7.21113i −0.368289 + 2.56151i −0.342468 + 0.220091i 8.69122 − 10.0302i 0.610228 + 4.24423i −24.7134 0.289411 + 0.633721i
12.18 0.244761 − 0.157298i 7.46568 −3.28815 + 7.20006i −1.28567 + 8.94205i 1.82731 − 1.17434i 11.4217 − 13.1814i 0.658994 + 4.58340i 28.7363 1.09189 + 2.39090i
12.19 0.631126 − 0.405600i 0.347174 −3.08951 + 6.76509i 2.09930 − 14.6010i 0.219110 − 0.140814i 11.9112 − 13.7462i 1.64819 + 11.4634i −26.8795 −4.59723 − 10.0665i
12.20 1.56857 − 1.00806i 1.69576 −1.87909 + 4.11463i −1.18777 + 8.26114i 2.65993 − 1.70943i −8.93663 + 10.3134i 3.32315 + 23.1130i −24.1244 6.46461 + 14.1555i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.4.e.a ✓ 320
121.e even 11 1 inner 121.4.e.a ✓ 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.4.e.a ✓ 320 1.a even 1 1 trivial
121.4.e.a ✓ 320 121.e even 11 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(121, [\chi])\).