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Properties

Label 363.3.h.r
Level $363$
Weight $3$
Character orbit 363.h
Analytic conductor $9.891$
Analytic rank $0$
Dimension $48$
Inner twists $16$

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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: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{3} + 44 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{3} + 44 q^{4} + 12 q^{9} - 320 q^{12} - 68 q^{15} - 92 q^{16} + 88 q^{25} + 232 q^{27} + 8 q^{31} + 464 q^{34} - 164 q^{36} + 244 q^{37} - 404 q^{42} - 208 q^{45} - 540 q^{48} - 100 q^{49} + 460 q^{58} - 24 q^{60} + 1276 q^{64} - 512 q^{67} - 128 q^{69} + 784 q^{70} - 684 q^{75} + 2112 q^{78} - 348 q^{81} + 380 q^{82} - 120 q^{91} - 196 q^{93} + 156 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
245.1 −2.25308 − 3.10110i −1.78315 + 2.41254i −3.30438 + 10.1698i 0.364947 − 0.502306i 11.4991 + 0.0940783i −0.595614 + 1.83311i 24.4005 − 7.92821i −2.64073 − 8.60387i −2.37996
245.2 −2.25308 − 3.10110i 2.86066 + 0.903678i −3.30438 + 10.1698i −0.364947 + 0.502306i −3.64290 − 10.9073i 0.595614 − 1.83311i 24.4005 − 7.92821i 7.36673 + 5.17023i 2.37996
245.3 −1.27494 − 1.75481i −2.76975 + 1.15261i −0.217804 + 0.670333i −3.13305 + 4.31227i 5.55387 + 3.39087i −3.85076 + 11.8514i −6.79761 + 2.20868i 6.34300 − 6.38485i 11.5617
245.4 −1.27494 − 1.75481i 2.91826 − 0.695539i −0.217804 + 0.670333i 3.13305 − 4.31227i −4.94115 − 4.23421i 3.85076 − 11.8514i −6.79761 + 2.20868i 8.03245 − 4.05953i −11.5617
245.5 −1.11555 − 1.53542i −1.21873 − 2.74130i 0.123002 − 0.378560i −4.85423 + 6.68127i −2.84949 + 4.92930i −1.11421 + 3.42918i −7.93844 + 2.57935i −6.02940 + 6.68179i 15.6737
245.6 −1.11555 − 1.53542i −0.625321 − 2.93411i 0.123002 − 0.378560i 4.85423 − 6.68127i −3.80751 + 4.23326i 1.11421 − 3.42918i −7.93844 + 2.57935i −8.21795 + 3.66951i −15.6737
245.7 1.11555 + 1.53542i −1.21873 − 2.74130i 0.123002 − 0.378560i −4.85423 + 6.68127i 2.84949 − 4.92930i 1.11421 − 3.42918i 7.93844 − 2.57935i −6.02940 + 6.68179i −15.6737
245.8 1.11555 + 1.53542i −0.625321 − 2.93411i 0.123002 − 0.378560i 4.85423 − 6.68127i 3.80751 − 4.23326i −1.11421 + 3.42918i 7.93844 − 2.57935i −8.21795 + 3.66951i 15.6737
245.9 1.27494 + 1.75481i −2.76975 + 1.15261i −0.217804 + 0.670333i −3.13305 + 4.31227i −5.55387 − 3.39087i 3.85076 − 11.8514i 6.79761 − 2.20868i 6.34300 − 6.38485i −11.5617
245.10 1.27494 + 1.75481i 2.91826 − 0.695539i −0.217804 + 0.670333i 3.13305 − 4.31227i 4.94115 + 4.23421i −3.85076 + 11.8514i 6.79761 − 2.20868i 8.03245 − 4.05953i 11.5617
245.11 2.25308 + 3.10110i −1.78315 + 2.41254i −3.30438 + 10.1698i 0.364947 − 0.502306i −11.4991 − 0.0940783i 0.595614 − 1.83311i −24.4005 + 7.92821i −2.64073 − 8.60387i 2.37996
245.12 2.25308 + 3.10110i 2.86066 + 0.903678i −3.30438 + 10.1698i −0.364947 + 0.502306i 3.64290 + 10.9073i −0.595614 + 1.83311i −24.4005 + 7.92821i 7.36673 + 5.17023i −2.37996
251.1 −3.64556 − 1.18452i −2.84549 − 0.950362i 8.65099 + 6.28531i −0.590496 + 0.191864i 9.24770 + 6.83513i −1.55934 − 1.13293i −15.0804 − 20.7563i 7.19362 + 5.40849i 2.37996
251.2 −3.64556 − 1.18452i 0.0245432 + 2.99990i 8.65099 + 6.28531i 0.590496 − 0.191864i 3.46395 − 10.9654i 1.55934 + 1.13293i −15.0804 − 20.7563i −8.99880 + 0.147254i −2.37996
251.3 −2.06290 − 0.670277i −1.95209 − 2.27801i 0.570219 + 0.414289i 5.06938 − 1.64714i 2.50007 + 6.00775i −10.0814 − 7.32457i 4.20115 + 5.78239i −1.37867 + 8.89378i −11.5617
251.4 −2.06290 − 0.670277i 1.56329 + 2.56049i 0.570219 + 0.414289i −5.06938 + 1.64714i −1.50867 − 6.32988i 10.0814 + 7.32457i 4.20115 + 5.78239i −4.11226 + 8.00558i 11.5617
251.5 −1.80499 − 0.586478i 2.23052 − 2.00619i −0.322023 − 0.233963i 7.85430 − 2.55202i −5.20266 + 2.31300i −2.91704 − 2.11935i 4.90622 + 6.75284i 0.950429 − 8.94968i −15.6737
251.6 −1.80499 − 0.586478i 2.59727 − 1.50140i −0.322023 − 0.233963i −7.85430 + 2.55202i −5.56859 + 1.18679i 2.91704 + 2.11935i 4.90622 + 6.75284i 4.49157 − 7.79909i 15.6737
251.7 1.80499 + 0.586478i 2.23052 − 2.00619i −0.322023 − 0.233963i 7.85430 − 2.55202i 5.20266 − 2.31300i 2.91704 + 2.11935i −4.90622 − 6.75284i 0.950429 − 8.94968i 15.6737
251.8 1.80499 + 0.586478i 2.59727 − 1.50140i −0.322023 − 0.233963i −7.85430 + 2.55202i 5.56859 − 1.18679i −2.91704 − 2.11935i −4.90622 − 6.75284i 4.49157 − 7.79909i −15.6737
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 245.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
33.d even 2 1 inner
33.f even 10 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.r 48
3.b odd 2 1 inner 363.3.h.r 48
11.b odd 2 1 inner 363.3.h.r 48
11.c even 5 1 363.3.b.n ✓ 12
11.c even 5 3 inner 363.3.h.r 48
11.d odd 10 1 363.3.b.n ✓ 12
11.d odd 10 3 inner 363.3.h.r 48
33.d even 2 1 inner 363.3.h.r 48
33.f even 10 1 363.3.b.n ✓ 12
33.f even 10 3 inner 363.3.h.r 48
33.h odd 10 1 363.3.b.n ✓ 12
33.h odd 10 3 inner 363.3.h.r 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.b.n ✓ 12 11.c even 5 1
363.3.b.n ✓ 12 11.d odd 10 1
363.3.b.n ✓ 12 33.f even 10 1
363.3.b.n ✓ 12 33.h odd 10 1
363.3.h.r 48 1.a even 1 1 trivial
363.3.h.r 48 3.b odd 2 1 inner
363.3.h.r 48 11.b odd 2 1 inner
363.3.h.r 48 11.c even 5 3 inner
363.3.h.r 48 11.d odd 10 3 inner
363.3.h.r 48 33.d even 2 1 inner
363.3.h.r 48 33.f even 10 3 inner
363.3.h.r 48 33.h odd 10 3 inner

Hecke kernels

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

\( T_{2}^{24} - 23 T_{2}^{22} + 390 T_{2}^{20} - 6022 T_{2}^{18} + 90023 T_{2}^{16} - 642838 T_{2}^{14} + \cdots + 3844124001 \) Copy content Toggle raw display
\( T_{5}^{24} - 97 T_{5}^{22} + 7434 T_{5}^{20} - 530270 T_{5}^{18} + 36826499 T_{5}^{16} + \cdots + 311374044081 \) Copy content Toggle raw display
\( T_{7}^{24} + 172 T_{7}^{22} + 26940 T_{7}^{20} + 4186412 T_{7}^{18} + 650123504 T_{7}^{16} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display