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Properties

Label 121.4.g.a
Level $121$
Weight $4$
Character orbit 121.g
Analytic conductor $7.139$
Analytic rank $0$
Dimension $1280$
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(4,121)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(121, base_ring=CyclotomicField(110))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 121.g (of order \(55\), degree \(40\), 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: \(1280\)
Relative dimension: \(32\) over \(\Q(\zeta_{55})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{55}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1280 q - 37 q^{2} - 32 q^{3} + 73 q^{4} - 33 q^{5} + 73 q^{6} - 9 q^{7} - 91 q^{8} - 2748 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1280 q - 37 q^{2} - 32 q^{3} + 73 q^{4} - 33 q^{5} + 73 q^{6} - 9 q^{7} - 91 q^{8} - 2748 q^{9} + 48 q^{10} + 43 q^{11} + 323 q^{12} - 287 q^{13} - 120 q^{14} + 632 q^{15} + 785 q^{16} - 13 q^{17} + 443 q^{18} - 192 q^{19} - 302 q^{20} - 81 q^{21} - 1483 q^{22} + 790 q^{23} - 1324 q^{24} + 835 q^{25} + 168 q^{26} + 583 q^{27} - 2 q^{28} + 155 q^{29} + 367 q^{30} - 351 q^{31} - 368 q^{32} + 485 q^{33} - 18 q^{34} - 281 q^{35} + 1823 q^{36} - 1097 q^{37} - 362 q^{38} - 112 q^{39} - 3205 q^{40} - 13 q^{41} - 111 q^{42} + 606 q^{43} + 2539 q^{44} + 1888 q^{45} - 1248 q^{46} - 499 q^{47} - 701 q^{48} - 2081 q^{49} + 3931 q^{50} - 1976 q^{51} + 2724 q^{52} + 1993 q^{53} - 2345 q^{54} - 1893 q^{55} - 167 q^{56} + 3462 q^{57} + 1949 q^{58} - 918 q^{59} - 843 q^{60} + 481 q^{61} + 745 q^{62} + 7284 q^{63} - 5887 q^{64} - 2362 q^{65} + 1329 q^{66} - 264 q^{67} - 754 q^{68} + 565 q^{69} + 5580 q^{70} - 27 q^{71} + 4966 q^{72} + 4223 q^{73} + 1432 q^{74} + 789 q^{75} + 6941 q^{76} - 3441 q^{77} - 7413 q^{78} + 5525 q^{79} + 2135 q^{80} - 18615 q^{81} + 3827 q^{82} - 96 q^{83} + 337 q^{84} - 6563 q^{85} - 963 q^{86} - 2057 q^{87} - 1771 q^{88} + 4114 q^{89} + 15586 q^{90} - 3505 q^{91} - 6694 q^{92} + 9192 q^{93} - 14111 q^{94} - 6295 q^{95} - 8919 q^{96} + 1042 q^{97} - 3048 q^{98} + 13103 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −5.43344 − 0.310695i 0.873694 − 2.68895i 21.4778 + 2.46435i −1.64335 − 3.11450i −5.58261 + 14.3388i 25.7497 + 10.8819i −73.0320 − 12.6387i 15.3763 + 11.1716i 7.96139 + 17.4330i
4.2 −4.96162 − 0.283716i −0.441007 + 1.35728i 16.5893 + 1.90345i −1.29295 − 2.45042i 2.57319 − 6.60918i −18.6947 − 7.90044i −42.5943 − 7.37123i 20.1957 + 14.6731i 5.71992 + 12.5249i
4.3 −4.90410 − 0.280426i −3.02997 + 9.32529i 16.0237 + 1.83855i −1.45061 − 2.74922i 17.4743 − 44.8825i 1.17065 + 0.494721i −39.3447 − 6.80888i −55.9369 − 40.6405i 6.34300 + 13.8892i
4.4 −4.62870 − 0.264679i 2.71005 − 8.34068i 13.4070 + 1.53831i −8.55852 − 16.2202i −14.7516 + 37.8892i −22.6374 − 9.56664i −25.1029 − 4.34423i −40.3791 − 29.3371i 35.3217 + 77.3437i
4.5 −4.44434 − 0.254137i −1.09754 + 3.37787i 11.7397 + 1.34701i 8.19282 + 15.5271i 5.73627 − 14.7335i 1.62617 + 0.687227i −16.7418 − 2.89728i 11.6380 + 8.45553i −32.4657 − 71.0899i
4.6 −4.38498 − 0.250742i 2.61878 − 8.05977i 11.2173 + 1.28707i 9.37767 + 17.7727i −13.5042 + 34.6853i −15.3011 − 6.46629i −14.2425 − 2.46476i −36.2585 − 26.3433i −36.6645 − 80.2841i
4.7 −3.63025 − 0.207585i −1.34595 + 4.14240i 5.18774 + 0.595238i −8.43183 − 15.9801i 5.74602 − 14.7585i 11.1935 + 4.73039i 9.95414 + 1.72263i 6.49556 + 4.71930i 27.2924 + 59.7620i
4.8 −3.50414 − 0.200374i 2.12963 − 6.55432i 4.29102 + 0.492349i −0.534207 − 1.01243i −8.77584 + 22.5406i 28.8627 + 12.1975i 12.7300 + 2.20301i −16.5803 − 12.0463i 1.66907 + 3.65476i
4.9 −3.07035 − 0.175569i 0.824583 − 2.53780i 1.44837 + 0.166184i 1.29762 + 2.45926i −2.97732 + 7.64718i −0.429079 − 0.181330i 19.8248 + 3.43081i 16.0829 + 11.6849i −3.55238 − 7.77862i
4.10 −2.47271 − 0.141395i −1.86923 + 5.75291i −1.85357 − 0.212677i 5.25998 + 9.96877i 5.43550 − 13.9610i 20.5753 + 8.69518i 24.0770 + 4.16669i −7.75847 − 5.63686i −11.5969 − 25.3936i
4.11 −2.45967 − 0.140649i 0.290500 − 0.894067i −1.91764 − 0.220029i −4.41327 − 8.36407i −0.840285 + 2.15826i −21.5269 − 9.09733i 24.1067 + 4.17183i 21.1285 + 15.3507i 9.67880 + 21.1936i
4.12 −1.85939 − 0.106324i −2.64809 + 8.14999i −4.50182 − 0.516536i −2.36375 − 4.47981i 5.79038 − 14.8725i −23.5797 − 9.96488i 22.9969 + 3.97978i −37.5664 − 27.2936i 3.91883 + 8.58103i
4.13 −1.21959 − 0.0697386i 1.08180 − 3.32944i −6.46532 − 0.741827i 5.37679 + 10.1901i −1.55154 + 3.98510i −12.0041 − 5.07296i 17.4628 + 3.02206i 11.9286 + 8.66662i −5.84682 − 12.8027i
4.14 −0.641211 − 0.0366657i 1.82234 − 5.60858i −7.53805 − 0.864910i −9.80678 − 18.5859i −1.37414 + 3.52946i 10.6511 + 4.50120i 9.86457 + 1.70713i −6.29178 − 4.57125i 5.60674 + 12.2771i
4.15 −0.575357 − 0.0329001i 2.68812 − 8.27318i −7.61790 − 0.874073i 0.0851125 + 0.161306i −1.81882 + 4.67159i −4.04277 − 1.70849i 8.89711 + 1.53971i −39.3761 − 28.6084i −0.0436631 − 0.0956088i
4.16 −0.457884 − 0.0261828i −1.82617 + 5.62036i −7.73888 − 0.887954i −4.58861 − 8.69638i 0.983329 − 2.52566i 8.14222 + 3.44093i 7.13558 + 1.23486i −6.41011 − 4.65722i 1.87336 + 4.10208i
4.17 −0.0563200 − 0.00322050i −1.78138 + 5.48253i −7.94469 − 0.911569i 7.94342 + 15.0544i 0.117984 − 0.303039i −23.2081 − 9.80784i 0.889196 + 0.153881i −5.04131 − 3.66272i −0.398891 − 0.873449i
4.18 0.532391 + 0.0304432i −0.448204 + 1.37943i −7.66534 − 0.879516i −3.29729 − 6.24906i −0.280614 + 0.720752i 17.8801 + 7.55618i −8.25779 − 1.42907i 20.1415 + 14.6337i −1.56521 − 3.42733i
4.19 0.874820 + 0.0500240i 1.74018 − 5.35574i −7.18505 − 0.824407i 9.63551 + 18.2613i 1.79026 − 4.59825i 26.7923 + 11.3225i −13.1517 − 2.27599i −3.81222 − 2.76974i 7.51583 + 16.4574i
4.20 1.28494 + 0.0734758i −0.222015 + 0.683293i −6.30217 − 0.723107i 2.87383 + 5.44652i −0.335483 + 0.861681i −1.37445 − 0.580847i −18.1904 − 3.14797i 21.4259 + 15.5668i 3.29253 + 7.20963i
See next 80 embeddings (of 1280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.g even 55 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.4.g.a ✓ 1280
121.g even 55 1 inner 121.4.g.a ✓ 1280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.4.g.a ✓ 1280 1.a even 1 1 trivial
121.4.g.a ✓ 1280 121.g even 55 1 inner

Hecke kernels

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