Properties

Label 363.3.h.p
Level $363$
Weight $3$
Character orbit 363.h
Analytic conductor $9.891$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,3,Mod(245,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 8]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.245");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.h (of order \(10\), degree \(4\), not 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: \(9.89103359628\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{3} + 18 q^{4} - 10 q^{6} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{3} + 18 q^{4} - 10 q^{6} + 22 q^{9} - 72 q^{10} + 56 q^{12} - 42 q^{13} - 28 q^{15} - 30 q^{16} + 94 q^{18} + 84 q^{19} + 112 q^{21} + 48 q^{24} + 108 q^{25} + 38 q^{27} + 132 q^{28} - 148 q^{30} + 264 q^{34} - 46 q^{36} + 42 q^{37} - 82 q^{39} - 294 q^{40} + 206 q^{42} + 624 q^{43} - 472 q^{45} - 60 q^{46} + 88 q^{48} - 138 q^{49} + 182 q^{51} + 114 q^{52} - 560 q^{54} - 24 q^{57} + 54 q^{58} + 562 q^{60} + 264 q^{61} - 122 q^{63} - 294 q^{64} + 96 q^{67} - 152 q^{69} - 336 q^{70} - 306 q^{72} + 72 q^{73} - 62 q^{75} + 1632 q^{76} - 776 q^{78} + 250 q^{81} - 798 q^{82} + 328 q^{84} + 330 q^{85} - 1848 q^{87} - 230 q^{90} - 300 q^{91} + 266 q^{93} + 120 q^{94} - 386 q^{96} - 126 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
245.1 −2.05778 2.83229i 2.10061 2.14183i −2.55135 + 7.85225i 4.64306 6.39063i −10.3889 1.54212i −0.560737 + 1.72577i 14.1718 4.60468i −0.174883 8.99830i −27.6545
245.2 −1.69141 2.32802i −1.38470 + 2.66132i −1.32276 + 4.07105i −2.90829 + 4.00292i 8.53770 1.27775i 3.45484 10.6329i 0.767802 0.249474i −5.16520 7.37026i 14.2380
245.3 −0.399998 0.550550i 2.97531 + 0.384091i 1.09296 3.36379i −3.81474 + 5.25053i −0.978656 1.79169i −2.89410 + 8.90713i −4.87795 + 1.58494i 8.70495 + 2.28558i 4.41657
245.4 0.399998 + 0.550550i −2.18131 + 2.05958i 1.09296 3.36379i 3.81474 5.25053i −2.00642 0.377094i −2.89410 + 8.90713i 4.87795 1.58494i 0.516261 8.98518i 4.41657
245.5 1.69141 + 2.32802i 2.68453 + 1.33914i −1.32276 + 4.07105i 2.90829 4.00292i 1.42308 + 8.51468i 3.45484 10.6329i −0.767802 + 0.249474i 5.41340 + 7.18993i 14.2380
245.6 2.05778 + 2.83229i −2.95837 0.498071i −2.55135 + 7.85225i −4.64306 + 6.39063i −4.67699 9.40388i −0.560737 + 1.72577i −14.1718 + 4.60468i 8.50385 + 2.94695i −27.6545
251.1 −3.32956 1.08184i −0.440492 2.96748i 6.67952 + 4.85296i 7.51264 2.44100i −1.74370 + 10.3570i 1.46803 + 1.06658i −8.75863 12.0552i −8.61193 + 2.61430i −27.6545
251.2 −2.73676 0.889226i −0.444035 + 2.96696i 3.46304 + 2.51605i −4.70571 + 1.52898i 3.85351 7.72499i −9.04489 6.57149i −0.474528 0.653131i −8.60567 2.63486i 14.2380
251.3 −0.647210 0.210291i −2.63284 1.43811i −2.86141 2.07894i −6.17237 + 2.00553i 1.40158 + 1.48442i 7.57686 + 5.50491i 3.01474 + 4.14944i 4.86369 + 7.57261i 4.41657
251.4 0.647210 + 0.210291i 0.554129 + 2.94838i −2.86141 2.07894i 6.17237 2.00553i −0.261380 + 2.02475i 7.57686 + 5.50491i −3.01474 4.14944i −8.38588 + 3.26757i 4.41657
251.5 2.73676 + 0.889226i −2.95896 0.494538i 3.46304 + 2.51605i 4.70571 1.52898i −7.65819 3.98461i −9.04489 6.57149i 0.474528 + 0.653131i 8.51086 + 2.92664i 14.2380
251.6 3.32956 + 1.08184i 2.68613 + 1.33594i 6.67952 + 4.85296i −7.51264 + 2.44100i 7.49835 + 7.35404i 1.46803 + 1.06658i 8.75863 + 12.0552i 5.43055 + 7.17698i −27.6545
269.1 −3.32956 + 1.08184i −0.440492 + 2.96748i 6.67952 4.85296i 7.51264 + 2.44100i −1.74370 10.3570i 1.46803 1.06658i −8.75863 + 12.0552i −8.61193 2.61430i −27.6545
269.2 −2.73676 + 0.889226i −0.444035 2.96696i 3.46304 2.51605i −4.70571 1.52898i 3.85351 + 7.72499i −9.04489 + 6.57149i −0.474528 + 0.653131i −8.60567 + 2.63486i 14.2380
269.3 −0.647210 + 0.210291i −2.63284 + 1.43811i −2.86141 + 2.07894i −6.17237 2.00553i 1.40158 1.48442i 7.57686 5.50491i 3.01474 4.14944i 4.86369 7.57261i 4.41657
269.4 0.647210 0.210291i 0.554129 2.94838i −2.86141 + 2.07894i 6.17237 + 2.00553i −0.261380 2.02475i 7.57686 5.50491i −3.01474 + 4.14944i −8.38588 3.26757i 4.41657
269.5 2.73676 0.889226i −2.95896 + 0.494538i 3.46304 2.51605i 4.70571 + 1.52898i −7.65819 + 3.98461i −9.04489 + 6.57149i 0.474528 0.653131i 8.51086 2.92664i 14.2380
269.6 3.32956 1.08184i 2.68613 1.33594i 6.67952 4.85296i −7.51264 2.44100i 7.49835 7.35404i 1.46803 1.06658i 8.75863 12.0552i 5.43055 7.17698i −27.6545
323.1 −2.05778 + 2.83229i 2.10061 + 2.14183i −2.55135 7.85225i 4.64306 + 6.39063i −10.3889 + 1.54212i −0.560737 1.72577i 14.1718 + 4.60468i −0.174883 + 8.99830i −27.6545
323.2 −1.69141 + 2.32802i −1.38470 2.66132i −1.32276 4.07105i −2.90829 4.00292i 8.53770 + 1.27775i 3.45484 + 10.6329i 0.767802 + 0.249474i −5.16520 + 7.37026i 14.2380
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 245.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 3 inner
33.h odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.h.p 24
3.b odd 2 1 inner 363.3.h.p 24
11.b odd 2 1 363.3.h.q 24
11.c even 5 1 363.3.b.k yes 6
11.c even 5 3 inner 363.3.h.p 24
11.d odd 10 1 363.3.b.j 6
11.d odd 10 3 363.3.h.q 24
33.d even 2 1 363.3.h.q 24
33.f even 10 1 363.3.b.j 6
33.f even 10 3 363.3.h.q 24
33.h odd 10 1 363.3.b.k yes 6
33.h odd 10 3 inner 363.3.h.p 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.b.j 6 11.d odd 10 1
363.3.b.j 6 33.f even 10 1
363.3.b.k yes 6 11.c even 5 1
363.3.b.k yes 6 33.h odd 10 1
363.3.h.p 24 1.a even 1 1 trivial
363.3.h.p 24 3.b odd 2 1 inner
363.3.h.p 24 11.c even 5 3 inner
363.3.h.p 24 33.h odd 10 3 inner
363.3.h.q 24 11.b odd 2 1
363.3.h.q 24 11.d odd 10 3
363.3.h.q 24 33.d even 2 1
363.3.h.q 24 33.f even 10 3

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\):

\( T_{2}^{24} - 21 T_{2}^{22} + 330 T_{2}^{20} - 4646 T_{2}^{18} + 61923 T_{2}^{16} - 484686 T_{2}^{14} + 3523315 T_{2}^{12} - 23099850 T_{2}^{10} + 116789127 T_{2}^{8} - 54084122 T_{2}^{6} + \cdots + 4879681 \) Copy content Toggle raw display
\( T_{5}^{24} - 129 T_{5}^{22} + 11454 T_{5}^{20} - 872786 T_{5}^{18} + 61477743 T_{5}^{16} - 3053171538 T_{5}^{14} + 131131745059 T_{5}^{12} - 5034856762854 T_{5}^{10} + \cdots + 17\!\cdots\!01 \) Copy content Toggle raw display
\( T_{7}^{12} + 108 T_{7}^{10} + 190 T_{7}^{9} + 11664 T_{7}^{8} - 61560 T_{7}^{7} + 1295812 T_{7}^{6} - 4432320 T_{7}^{5} + 128251296 T_{7}^{4} - 232486280 T_{7}^{3} + 421070400 T_{7}^{2} + \cdots + 1303210000 \) Copy content Toggle raw display