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Properties

Label 121.3.h.a
Level $121$
Weight $3$
Character orbit 121.h
Analytic conductor $3.297$
Analytic rank $0$
Dimension $840$
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] = [121,3,Mod(2,121)]
 
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([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.2");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 121.h (of order \(110\), 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: \(3.29701119876\)
Analytic rank: \(0\)
Dimension: \(840\)
Relative dimension: \(21\) over \(\Q(\zeta_{110})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{110}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 840 q - 39 q^{2} - 34 q^{3} - 75 q^{4} - 43 q^{5} - 15 q^{6} - 54 q^{7} - 59 q^{8} - 594 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 840 q - 39 q^{2} - 34 q^{3} - 75 q^{4} - 43 q^{5} - 15 q^{6} - 54 q^{7} - 59 q^{8} - 594 q^{9} - 132 q^{10} + 10 q^{11} - 79 q^{12} - 79 q^{13} + 6 q^{14} + 156 q^{15} + 5 q^{16} - 44 q^{17} + 69 q^{18} - 69 q^{19} - 84 q^{20} + 55 q^{21} - 97 q^{22} - 180 q^{23} + 138 q^{24} + 44 q^{25} - 14 q^{26} + 173 q^{27} + 16 q^{28} - 4 q^{29} + 135 q^{30} - 11 q^{31} - 44 q^{32} - 10 q^{33} - 62 q^{34} - 124 q^{35} + 115 q^{36} - 228 q^{37} + 398 q^{38} + 5 q^{39} + 5 q^{40} + 36 q^{41} + 39 q^{42} - 44 q^{43} - 211 q^{44} + 330 q^{45} - 74 q^{46} - 44 q^{47} + 125 q^{48} - 5 q^{49} - 1143 q^{50} - 300 q^{51} + 132 q^{52} - 774 q^{53} + 649 q^{54} + 384 q^{55} + 483 q^{56} + 780 q^{57} + 723 q^{58} - 100 q^{59} - 97 q^{60} - 54 q^{61} - 849 q^{62} - 101 q^{63} - 287 q^{64} + 187 q^{65} + 141 q^{66} + 5 q^{67} + 216 q^{68} + 112 q^{69} - 628 q^{70} - 611 q^{71} + 854 q^{72} - 630 q^{73} + 226 q^{74} - 14 q^{75} - 1265 q^{76} - 636 q^{77} + 433 q^{78} - 70 q^{79} - 1539 q^{80} - 868 q^{81} - 531 q^{82} - 269 q^{83} - 35 q^{84} - 370 q^{85} - 185 q^{86} + 55 q^{87} - 1287 q^{88} - 1315 q^{89} - 970 q^{90} - 747 q^{91} - 28 q^{92} + 735 q^{93} - 175 q^{94} + 320 q^{95} + 507 q^{96} + 401 q^{97} + 396 q^{98} + 1740 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −3.85069 − 0.110005i 2.73880 − 1.98985i 10.8222 + 0.618838i 3.86893 − 6.41590i −10.7652 + 7.36103i 10.3479 + 2.09675i −26.2525 − 2.25483i 0.760351 − 2.34012i −15.6038 + 24.2801i
2.2 −3.47550 − 0.0992871i −4.31902 + 3.13795i 8.07579 + 0.461790i 2.99922 − 4.97365i 15.3223 − 10.4771i −9.37524 − 1.89967i −14.1649 − 1.21663i 6.02603 − 18.5462i −10.9176 + 16.9881i
2.3 −3.38317 − 0.0966493i −2.37893 + 1.72839i 7.44300 + 0.425606i −4.61936 + 7.66034i 8.21536 − 5.61752i 7.49551 + 1.51879i −11.6513 − 1.00073i −0.109195 + 0.336067i 16.3684 − 25.4698i
2.4 −3.20047 − 0.0914300i 2.46052 − 1.78767i 6.24117 + 0.356883i −2.99780 + 4.97130i −8.03825 + 5.49641i −7.59954 − 1.53987i −7.18192 − 0.616855i 0.0772215 − 0.237663i 10.0489 − 15.6364i
2.5 −3.00938 − 0.0859711i −0.0156601 + 0.0113778i 5.05550 + 0.289084i 0.517797 − 0.858670i 0.0481054 − 0.0328937i −3.09405 − 0.626936i −3.19080 − 0.274058i −2.78104 + 8.55915i −1.63207 + 2.53955i
2.6 −2.18890 − 0.0625320i −1.30459 + 0.947839i 0.793918 + 0.0453979i 2.33368 − 3.86998i 2.91489 − 1.99315i 1.42055 + 0.287841i 6.99209 + 0.600551i −1.97760 + 6.08643i −5.35020 + 8.32508i
2.7 −1.66680 − 0.0476165i 4.20922 − 3.05818i −1.21753 − 0.0696211i −1.38177 + 2.29140i −7.16153 + 4.89693i 11.5839 + 2.34720i 8.67150 + 0.744796i 5.58393 − 17.1856i 2.41223 − 3.75351i
2.8 −1.42546 − 0.0407220i 3.89979 − 2.83336i −1.96321 − 0.112260i 3.30573 − 5.48195i −5.67437 + 3.88003i −11.6344 − 2.35743i 8.47713 + 0.728101i 4.39926 − 13.5395i −4.93542 + 7.67966i
2.9 −1.21524 − 0.0347167i 0.594871 − 0.432199i −2.51787 − 0.143977i −0.691538 + 1.14679i −0.737917 + 0.504575i 4.58308 + 0.928653i 7.89994 + 0.678526i −2.61408 + 8.04530i 0.880200 − 1.36962i
2.10 −1.06069 − 0.0303015i −3.99954 + 2.90584i −2.86933 − 0.164074i −2.07765 + 3.44539i 4.33032 − 2.96100i 0.996879 + 0.201994i 7.26742 + 0.624199i 4.77129 − 14.6845i 2.30814 − 3.59153i
2.11 −0.0773271 − 0.00220906i 1.65883 − 1.20521i −3.98750 − 0.228014i −4.24110 + 7.03308i −0.130935 + 0.0895312i −4.66202 − 0.944648i 0.616138 + 0.0529201i −1.48196 + 4.56101i 0.343489 − 0.534479i
2.12 0.00552359 0.000157796i −3.20029 + 2.32514i −3.99345 − 0.228354i 4.19894 − 6.96317i −0.0180440 + 0.0123381i 7.85131 + 1.59088i −0.0440444 − 0.00378298i 2.05438 − 6.32274i 0.0242920 − 0.0377991i
2.13 0.516740 + 0.0147621i −0.714499 + 0.519114i −3.72667 − 0.213099i 1.16689 − 1.93508i −0.376873 + 0.257699i −12.2257 − 2.47724i −3.98279 − 0.342083i −2.54012 + 7.81770i 0.631547 − 0.982706i
2.14 0.990541 + 0.0282975i 2.30127 − 1.67197i −3.01311 − 0.172296i 3.40305 − 5.64332i 2.32681 − 1.59103i 3.12212 + 0.632623i −6.92897 − 0.595130i −0.280801 + 0.864216i 3.53055 − 5.49364i
2.15 1.72705 + 0.0493378i −3.03901 + 2.20797i −1.01321 − 0.0579376i −0.589093 + 0.976900i −5.35745 + 3.66333i −4.45396 − 0.902490i −8.63267 − 0.741461i 1.57929 − 4.86057i −1.06559 + 1.65809i
2.16 1.76028 + 0.0502871i −1.16832 + 0.848837i −0.897425 − 0.0513166i −4.08088 + 6.76738i −2.09926 + 1.43544i 9.01268 + 1.82621i −8.59528 − 0.738250i −2.13670 + 6.57608i −7.52379 + 11.7073i
2.17 2.10635 + 0.0601735i 3.61347 − 2.62534i 0.439597 + 0.0251371i −0.303526 + 0.503342i 7.76920 − 5.31244i 2.43403 + 0.493197i −7.47347 − 0.641897i 3.38362 − 10.4137i −0.669619 + 1.04195i
2.18 3.06336 + 0.0875131i −1.27011 + 0.922791i 5.38302 + 0.307812i 1.21128 − 2.00869i −3.97156 + 2.71569i 11.1869 + 2.26676i 4.24971 + 0.365008i −2.01951 + 6.21541i 3.88638 − 6.04732i
2.19 3.28200 + 0.0937592i −0.344209 + 0.250083i 6.76926 + 0.387080i 3.84115 − 6.36983i −1.15314 + 0.788498i −6.20703 − 1.25771i 9.09523 + 0.781190i −2.72521 + 8.38735i 13.2039 − 20.5457i
2.20 3.32312 + 0.0949339i 2.34088 − 1.70075i 7.04063 + 0.402598i −2.28781 + 3.79391i 7.94049 − 5.42957i −6.09636 − 1.23528i 10.1095 + 0.868308i −0.193978 + 0.597003i −7.96283 + 12.3904i
See next 80 embeddings (of 840 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.21
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.h odd 110 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.3.h.a ✓ 840
121.h odd 110 1 inner 121.3.h.a ✓ 840
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.3.h.a ✓ 840 1.a even 1 1 trivial
121.3.h.a ✓ 840 121.h odd 110 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(121, [\chi])\).