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Properties

Label 242.3.h.a
Level $242$
Weight $3$
Character orbit 242.h
Analytic conductor $6.594$
Analytic rank $0$
Dimension $880$
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] = [242,3,Mod(7,242)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(242, base_ring=CyclotomicField(110))
 
chi = DirichletCharacter(H, H._module([7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("242.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 242.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: \(6.59402239752\)
Analytic rank: \(0\)
Dimension: \(880\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 880 q + 4 q^{3} - 44 q^{4} + 4 q^{5} + 20 q^{6} + 30 q^{7} - 652 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 880 q + 4 q^{3} - 44 q^{4} + 4 q^{5} + 20 q^{6} + 30 q^{7} - 652 q^{9} + 88 q^{10} - 106 q^{11} + 36 q^{12} + 80 q^{13} - 60 q^{14} - 268 q^{15} + 88 q^{16} - 30 q^{17} - 40 q^{18} + 30 q^{19} - 8 q^{20} + 64 q^{22} + 48 q^{23} + 40 q^{24} + 118 q^{25} + 104 q^{26} + 28 q^{27} + 40 q^{28} + 10 q^{29} + 60 q^{30} + 16 q^{31} + 14 q^{33} - 80 q^{34} - 70 q^{35} - 128 q^{36} + 166 q^{37} - 520 q^{38} - 130 q^{39} - 80 q^{40} - 250 q^{41} - 80 q^{42} + 12 q^{44} + 40 q^{45} + 160 q^{46} + 62 q^{47} + 16 q^{48} - 104 q^{49} + 80 q^{50} + 228 q^{51} + 40 q^{52} + 854 q^{53} - 150 q^{55} - 48 q^{56} + 20 q^{57} - 168 q^{58} - 48 q^{59} - 104 q^{60} - 50 q^{61} - 20 q^{62} + 1230 q^{63} - 176 q^{64} - 462 q^{65} - 40 q^{66} + 186 q^{67} - 60 q^{68} - 38 q^{69} + 644 q^{70} + 444 q^{71} + 80 q^{72} + 642 q^{73} + 40 q^{74} - 318 q^{75} + 132 q^{76} - 2 q^{77} - 40 q^{78} + 334 q^{79} - 24 q^{80} - 2126 q^{81} + 528 q^{82} + 150 q^{83} + 120 q^{84} + 462 q^{85} + 56 q^{86} + 192 q^{88} + 1842 q^{89} + 1776 q^{90} + 1748 q^{91} + 304 q^{92} - 1014 q^{93} + 20 q^{94} - 198 q^{95} - 546 q^{97} - 1056 q^{98} - 1342 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.38605 − 0.280849i −1.80308 + 5.54931i 1.84225 + 0.778540i 4.79184 − 6.21417i 4.05767 − 7.18520i 0.175077 + 1.01167i −2.33479 − 1.59649i −20.2626 − 14.7216i −8.38696 + 7.26734i
7.2 −1.38605 − 0.280849i −1.17429 + 3.61408i 1.84225 + 0.778540i −1.61886 + 2.09938i 2.64263 − 4.67948i −2.20643 − 12.7497i −2.33479 − 1.59649i −4.40147 − 3.19785i 2.83343 − 2.45518i
7.3 −1.38605 − 0.280849i −1.05174 + 3.23691i 1.84225 + 0.778540i −1.30450 + 1.69170i 2.36684 − 4.19113i 0.352886 + 2.03913i −2.33479 − 1.59649i −2.09030 − 1.51869i 2.28320 − 1.97841i
7.4 −1.38605 − 0.280849i −0.693610 + 2.13471i 1.84225 + 0.778540i −4.82559 + 6.25793i 1.56091 − 2.76401i 1.83009 + 10.5751i −2.33479 − 1.59649i 3.20525 + 2.32875i 8.44602 − 7.31852i
7.5 −1.38605 − 0.280849i −0.316722 + 0.974769i 1.84225 + 0.778540i 2.06287 − 2.67518i 0.712754 − 1.26212i 0.945190 + 5.46173i −2.33479 − 1.59649i 6.43129 + 4.67261i −3.61056 + 3.12856i
7.6 −1.38605 − 0.280849i 0.354297 − 1.09041i 1.84225 + 0.778540i 2.96757 − 3.84842i −0.797314 + 1.41186i −1.33845 − 7.73417i −2.33479 − 1.59649i 6.21768 + 4.51741i −5.19402 + 4.50064i
7.7 −1.38605 − 0.280849i 0.471549 − 1.45128i 1.84225 + 0.778540i −3.50023 + 4.53918i −1.06118 + 1.87911i −1.43391 − 8.28579i −2.33479 − 1.59649i 5.39730 + 3.92137i 6.12631 − 5.30848i
7.8 −1.38605 − 0.280849i 0.473239 − 1.45648i 1.84225 + 0.778540i 4.42630 − 5.74012i −1.06498 + 1.88584i 1.79505 + 10.3726i −2.33479 − 1.59649i 5.38377 + 3.91154i −7.74716 + 6.71295i
7.9 −1.38605 − 0.280849i 0.913112 − 2.81027i 1.84225 + 0.778540i −3.52436 + 4.57047i −2.05488 + 3.63871i −0.110291 − 0.637308i −2.33479 − 1.59649i 0.217316 + 0.157889i 6.16854 − 5.34507i
7.10 −1.38605 − 0.280849i 1.51513 − 4.66308i 1.84225 + 0.778540i −1.28814 + 1.67049i −3.40966 + 6.03773i 0.634275 + 3.66513i −2.33479 − 1.59649i −12.1676 − 8.84027i 2.25457 − 1.95360i
7.11 −1.38605 − 0.280849i 1.71611 − 5.28166i 1.84225 + 0.778540i 3.64626 − 4.72855i −3.86196 + 6.83865i −1.40046 − 8.09252i −2.33479 − 1.59649i −17.6697 − 12.8378i −6.38189 + 5.52994i
7.12 1.38605 + 0.280849i −1.51474 + 4.66189i 1.84225 + 0.778540i −5.29094 + 6.86141i −3.40879 + 6.03618i −1.19002 − 6.87648i 2.33479 + 1.59649i −12.1576 − 8.83303i −9.26051 + 8.02428i
7.13 1.38605 + 0.280849i −1.38159 + 4.25210i 1.84225 + 0.778540i 4.27277 − 5.54103i −3.10915 + 5.50559i 1.47607 + 8.52937i 2.33479 + 1.59649i −8.89041 − 6.45926i 7.47846 − 6.48012i
7.14 1.38605 + 0.280849i −1.20236 + 3.70047i 1.84225 + 0.778540i −1.52486 + 1.97747i −2.70579 + 4.79134i 0.962456 + 5.56150i 2.33479 + 1.59649i −4.96667 − 3.60850i −2.66889 + 2.31261i
7.15 1.38605 + 0.280849i −0.514181 + 1.58249i 1.84225 + 0.778540i 3.85862 − 5.00394i −1.15712 + 2.04899i −1.57821 − 9.11961i 2.33479 + 1.59649i 5.04127 + 3.66270i 6.75358 − 5.85201i
7.16 1.38605 + 0.280849i −0.214924 + 0.661467i 1.84225 + 0.778540i −0.421854 + 0.547070i −0.483666 + 0.856462i −0.297035 − 1.71640i 2.33479 + 1.59649i 6.88981 + 5.00574i −0.738354 + 0.639787i
7.17 1.38605 + 0.280849i 0.139104 − 0.428119i 1.84225 + 0.778540i −3.65698 + 4.74245i 0.313042 − 0.554326i 0.278170 + 1.60739i 2.33479 + 1.59649i 7.11722 + 5.17096i −6.40066 + 5.54620i
7.18 1.38605 + 0.280849i 0.539863 − 1.66153i 1.84225 + 0.778540i 2.96897 − 3.85022i 1.21491 − 2.15133i 2.14801 + 12.4122i 2.33479 + 1.59649i 4.81193 + 3.49607i 5.19646 − 4.50275i
7.19 1.38605 + 0.280849i 1.03909 − 3.19800i 1.84225 + 0.778540i −5.33195 + 6.91460i 2.33839 − 4.14075i 2.06415 + 11.9276i 2.33479 + 1.59649i −1.86633 − 1.35597i −9.33229 + 8.08648i
7.20 1.38605 + 0.280849i 1.07190 − 3.29898i 1.84225 + 0.778540i 3.37347 − 4.37479i 2.41222 − 4.27149i −0.434813 − 2.51254i 2.33479 + 1.59649i −2.45313 − 1.78231i 5.90444 − 5.11623i
See next 80 embeddings (of 880 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.22
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 242.3.h.a ✓ 880
121.h odd 110 1 inner 242.3.h.a ✓ 880
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
242.3.h.a ✓ 880 1.a even 1 1 trivial
242.3.h.a ✓ 880 121.h odd 110 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(242, [\chi])\).