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Properties

Label 61.3.h.a
Level $61$
Weight $3$
Character orbit 61.h
Analytic conductor $1.662$
Analytic rank $0$
Dimension $36$
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] = [61,3,Mod(21,61)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.21");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 61.h (of order \(12\), degree \(4\), 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: \(1.66212961260\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(9\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{2} - 6 q^{4} + 6 q^{5} + 8 q^{6} - 18 q^{7} + 12 q^{8} - 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{2} - 6 q^{4} + 6 q^{5} + 8 q^{6} - 18 q^{7} + 12 q^{8} - 80 q^{9} - 50 q^{10} + 6 q^{11} + 16 q^{12} + 34 q^{13} + 84 q^{14} - 16 q^{15} + 6 q^{16} - 28 q^{17} + 68 q^{18} - 6 q^{19} - 96 q^{20} + 36 q^{21} - 2 q^{22} + 64 q^{23} - 198 q^{24} - 32 q^{25} - 46 q^{26} - 144 q^{28} + 170 q^{29} + 262 q^{30} - 134 q^{31} + 16 q^{32} + 22 q^{33} - 96 q^{34} - 110 q^{35} + 240 q^{36} + 26 q^{37} - 36 q^{38} + 174 q^{39} - 298 q^{40} + 236 q^{42} - 192 q^{43} - 106 q^{44} - 492 q^{45} - 18 q^{46} + 100 q^{47} + 696 q^{48} + 378 q^{49} + 26 q^{50} + 56 q^{51} - 52 q^{53} - 352 q^{54} + 384 q^{55} + 92 q^{56} + 384 q^{57} - 52 q^{58} + 40 q^{59} - 48 q^{61} + 100 q^{62} + 264 q^{63} - 600 q^{65} - 354 q^{66} - 170 q^{67} - 626 q^{68} - 198 q^{69} + 684 q^{70} - 118 q^{71} - 568 q^{72} + 256 q^{73} + 54 q^{74} - 90 q^{75} - 146 q^{76} - 350 q^{77} + 462 q^{78} - 336 q^{79} - 360 q^{80} - 60 q^{81} + 384 q^{82} + 108 q^{83} - 774 q^{84} - 288 q^{85} - 852 q^{86} + 196 q^{87} - 242 q^{89} + 294 q^{90} + 764 q^{91} + 244 q^{92} + 96 q^{93} + 534 q^{94} - 288 q^{95} + 734 q^{96} + 180 q^{97} - 676 q^{98} - 910 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −3.28691 + 0.880725i 4.36030i 6.56400 − 3.78973i 6.39151 + 3.69014i −3.84022 − 14.3319i −12.4469 + 3.33514i −8.61283 + 8.61283i −10.0122 −24.2583 − 6.50000i
21.2 −2.98154 + 0.798902i − 0.210245i 4.78725 − 2.76392i −3.53014 − 2.03812i 0.167965 + 0.626856i 7.22427 − 1.93574i −3.33472 + 3.33472i 8.95580 12.1535 + 3.25652i
21.3 −1.21434 + 0.325382i − 1.53723i −2.09535 + 1.20975i 7.19085 + 4.15164i 0.500187 + 1.86672i 4.76901 − 1.27785i 5.70668 − 5.70668i 6.63692 −10.0830 − 2.70173i
21.4 −1.06219 + 0.284612i 2.37315i −2.41686 + 1.39538i −4.62160 − 2.66828i −0.675428 − 2.52073i −6.95024 + 1.86231i 5.28032 − 5.28032i 3.36814 5.66843 + 1.51885i
21.5 −0.724957 + 0.194252i − 4.74578i −2.97627 + 1.71835i −2.99796 − 1.73087i 0.921875 + 3.44048i −6.15656 + 1.64964i 3.94670 − 3.94670i −13.5224 2.50962 + 0.672449i
21.6 0.932749 − 0.249929i 3.54244i −2.65655 + 1.53376i 2.01249 + 1.16191i 0.885360 + 3.30421i 2.71348 − 0.727076i −4.82584 + 4.82584i −3.54888 2.16754 + 0.580791i
21.7 1.93659 − 0.518908i − 3.40445i 0.0170158 − 0.00982409i −0.758060 − 0.437666i −1.76659 − 6.59302i 10.4902 − 2.81085i −5.64288 + 5.64288i −2.59025 −1.69516 − 0.454217i
21.8 2.64708 − 0.709284i − 1.82449i 3.03987 − 1.75507i 2.22717 + 1.28586i −1.29408 − 4.82959i −10.9497 + 2.93397i −0.949258 + 0.949258i 5.67122 6.80756 + 1.82408i
21.9 3.25351 − 0.871776i 3.17835i 6.36126 − 3.67267i −6.14631 − 3.54858i 2.77081 + 10.3408i −0.121755 + 0.0326242i 7.96774 − 7.96774i −1.10194 −23.0907 − 6.18713i
29.1 −0.918927 + 3.42948i − 5.78699i −7.45282 − 4.30289i 4.59835 − 2.65486i 19.8464 + 5.31782i 0.669380 − 2.49816i 11.5631 − 11.5631i −24.4893 4.87924 + 18.2096i
29.2 −0.810867 + 3.02620i 0.928816i −5.03627 − 2.90769i −4.70842 + 2.71841i −2.81078 − 0.753147i 0.0246199 − 0.0918826i 4.02168 − 4.02168i 8.13730 −4.40854 − 16.4529i
29.3 −0.579620 + 2.16317i 5.42090i −0.879251 − 0.507636i 4.55904 − 2.63216i −11.7263 − 3.14206i 2.05540 − 7.67087i −4.72647 + 4.72647i −20.3862 3.05131 + 11.3876i
29.4 −0.371492 + 1.38643i − 0.962184i 1.67993 + 0.969909i 3.93358 − 2.27106i 1.33400 + 0.357443i −1.36559 + 5.09644i −6.02852 + 6.02852i 8.07420 1.68736 + 6.29730i
29.5 0.0143740 − 0.0536445i − 3.60536i 3.46143 + 1.99846i −1.85484 + 1.07089i −0.193408 − 0.0518235i 2.96998 − 11.0841i 0.314043 − 0.314043i −3.99862 0.0307861 + 0.114895i
29.6 0.124134 − 0.463276i 2.75446i 3.26489 + 1.88498i −5.88753 + 3.39917i 1.27607 + 0.341923i −0.189483 + 0.707161i 2.63512 − 2.63512i 1.41297 0.843907 + 3.14951i
29.7 0.554942 − 2.07107i 3.65949i −0.517279 − 0.298651i 3.48998 − 2.01494i 7.57908 + 2.03081i 0.00378603 − 0.0141297i 5.15893 − 5.15893i −4.39189 −2.23635 − 8.34617i
29.8 0.571024 − 2.13109i − 4.02511i −0.751380 − 0.433809i 0.0562301 − 0.0324644i −8.57787 − 2.29843i −2.89526 + 10.8052i 4.88672 − 4.88672i −7.20148 −0.0370760 − 0.138369i
29.9 0.916431 − 3.42017i − 0.116076i −7.39360 − 4.26870i −0.954328 + 0.550982i −0.397000 − 0.106376i 1.15536 − 4.31187i −11.3605 + 11.3605i 8.98653 1.00987 + 3.76890i
32.1 −3.28691 − 0.880725i − 4.36030i 6.56400 + 3.78973i 6.39151 − 3.69014i −3.84022 + 14.3319i −12.4469 − 3.33514i −8.61283 − 8.61283i −10.0122 −24.2583 + 6.50000i
32.2 −2.98154 − 0.798902i 0.210245i 4.78725 + 2.76392i −3.53014 + 2.03812i 0.167965 − 0.626856i 7.22427 + 1.93574i −3.33472 − 3.33472i 8.95580 12.1535 − 3.25652i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.h odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.3.h.a ✓ 36
61.h odd 12 1 inner 61.3.h.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.3.h.a ✓ 36 1.a even 1 1 trivial
61.3.h.a ✓ 36 61.h odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(61, [\chi])\).