Properties

Label 363.3.h.e
Level $363$
Weight $3$
Character orbit 363.h
Analytic conductor $9.891$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.228765625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2x^{6} + 5x^{5} + x^{4} + 15x^{3} - 18x^{2} - 27x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + 3 \beta_{4} q^{3} - 7 \beta_{2} q^{4} + (2 \beta_{7} + 2 \beta_{6} + \cdots + 2 \beta_1) q^{5}+ \cdots - 9 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + 3 \beta_{4} q^{3} - 7 \beta_{2} q^{4} + (2 \beta_{7} + 2 \beta_{6} + \cdots + 2 \beta_1) q^{5}+ \cdots + 15 \beta_{6} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{3} + 14 q^{4} + 16 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{3} + 14 q^{4} + 16 q^{7} - 18 q^{9} + 176 q^{10} - 168 q^{12} - 8 q^{13} - 10 q^{16} + 12 q^{19} - 192 q^{21} + 38 q^{25} - 54 q^{27} - 112 q^{28} - 132 q^{30} + 52 q^{31} - 352 q^{34} + 126 q^{36} - 60 q^{37} - 24 q^{39} + 132 q^{40} + 336 q^{43} + 44 q^{46} - 30 q^{48} - 30 q^{49} + 56 q^{52} + 36 q^{57} - 264 q^{58} - 24 q^{61} + 144 q^{63} - 194 q^{64} + 16 q^{67} + 352 q^{70} + 148 q^{73} + 114 q^{75} + 336 q^{76} + 80 q^{79} - 162 q^{81} + 88 q^{82} - 336 q^{84} - 176 q^{85} - 396 q^{90} + 64 q^{91} + 156 q^{93} + 572 q^{94} - 124 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 2x^{6} + 5x^{5} + x^{4} + 15x^{3} - 18x^{2} - 27x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} - 10\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - 16\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{7} - 4\nu^{6} + 10\nu^{5} + 2\nu^{4} - 5\nu^{3} - 36\nu^{2} - 54\nu + 162 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} + 5\nu^{6} + 10\nu^{5} + 2\nu^{4} - 5\nu^{3} - 75\nu^{2} + 90\nu + 135 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{7} - 7\nu^{6} - 14\nu^{5} + 8\nu^{4} + 7\nu^{3} + 105\nu^{2} - 126\nu - 189 ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\nu^{5} + 31 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{7} + 16\nu^{6} - 40\nu^{5} - 8\nu^{4} - 7\nu^{3} + 144\nu^{2} + 216\nu - 648 ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + 5\beta_{4} - 5\beta_{3} + 5\beta_{2} + \beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{5} + 7\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{6} - 31 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5\beta_{2} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{7} - 13\beta_{6} - 13\beta_{5} - 83\beta_{4} + 83\beta_{3} - 83\beta_{2} - 13\beta _1 - 83 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
245.1
1.42264 + 0.987975i
−1.73166 0.0369185i
−0.570223 + 1.63550i
1.37924 1.04771i
−0.570223 1.63550i
1.37924 + 1.04771i
1.42264 0.987975i
−1.73166 + 0.0369185i
−1.94946 2.68321i 0.927051 + 2.85317i −2.16312 + 6.65740i −3.89893 + 5.36641i 5.84839 8.04962i −2.47214 + 7.60845i 9.46289 3.07468i −7.28115 + 5.29007i 22.0000
245.2 1.94946 + 2.68321i 0.927051 + 2.85317i −2.16312 + 6.65740i 3.89893 5.36641i −5.84839 + 8.04962i −2.47214 + 7.60845i −9.46289 + 3.07468i −7.28115 + 5.29007i 22.0000
251.1 −3.15430 1.02489i −2.42705 + 1.76336i 5.66312 + 4.11450i −6.30860 + 2.04979i 9.46289 3.07468i 6.47214 + 4.70228i −5.84839 8.04962i 2.78115 8.55951i 22.0000
251.2 3.15430 + 1.02489i −2.42705 + 1.76336i 5.66312 + 4.11450i 6.30860 2.04979i −9.46289 + 3.07468i 6.47214 + 4.70228i 5.84839 + 8.04962i 2.78115 8.55951i 22.0000
269.1 −3.15430 + 1.02489i −2.42705 1.76336i 5.66312 4.11450i −6.30860 2.04979i 9.46289 + 3.07468i 6.47214 4.70228i −5.84839 + 8.04962i 2.78115 + 8.55951i 22.0000
269.2 3.15430 1.02489i −2.42705 1.76336i 5.66312 4.11450i 6.30860 + 2.04979i −9.46289 3.07468i 6.47214 4.70228i 5.84839 8.04962i 2.78115 + 8.55951i 22.0000
323.1 −1.94946 + 2.68321i 0.927051 2.85317i −2.16312 6.65740i −3.89893 5.36641i 5.84839 + 8.04962i −2.47214 7.60845i 9.46289 + 3.07468i −7.28115 5.29007i 22.0000
323.2 1.94946 2.68321i 0.927051 2.85317i −2.16312 6.65740i 3.89893 + 5.36641i −5.84839 8.04962i −2.47214 7.60845i −9.46289 3.07468i −7.28115 5.29007i 22.0000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 245.2
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.e 8
3.b odd 2 1 inner 363.3.h.e 8
11.b odd 2 1 363.3.h.d 8
11.c even 5 1 33.3.b.a 2
11.c even 5 3 inner 363.3.h.e 8
11.d odd 10 1 363.3.b.d 2
11.d odd 10 3 363.3.h.d 8
33.d even 2 1 363.3.h.d 8
33.f even 10 1 363.3.b.d 2
33.f even 10 3 363.3.h.d 8
33.h odd 10 1 33.3.b.a 2
33.h odd 10 3 inner 363.3.h.e 8
44.h odd 10 1 528.3.i.a 2
132.o even 10 1 528.3.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.b.a 2 11.c even 5 1
33.3.b.a 2 33.h odd 10 1
363.3.b.d 2 11.d odd 10 1
363.3.b.d 2 33.f even 10 1
363.3.h.d 8 11.b odd 2 1
363.3.h.d 8 11.d odd 10 3
363.3.h.d 8 33.d even 2 1
363.3.h.d 8 33.f even 10 3
363.3.h.e 8 1.a even 1 1 trivial
363.3.h.e 8 3.b odd 2 1 inner
363.3.h.e 8 11.c even 5 3 inner
363.3.h.e 8 33.h odd 10 3 inner
528.3.i.a 2 44.h odd 10 1
528.3.i.a 2 132.o even 10 1

Hecke kernels

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

\( T_{2}^{8} - 11T_{2}^{6} + 121T_{2}^{4} - 1331T_{2}^{2} + 14641 \) Copy content Toggle raw display
\( T_{5}^{8} - 44T_{5}^{6} + 1936T_{5}^{4} - 85184T_{5}^{2} + 3748096 \) Copy content Toggle raw display
\( T_{7}^{4} - 8T_{7}^{3} + 64T_{7}^{2} - 512T_{7} + 4096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 11 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{3} + 9 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 44 T^{6} + \cdots + 3748096 \) Copy content Toggle raw display
$7$ \( (T^{4} - 8 T^{3} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 176 T^{6} + \cdots + 959512576 \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 44)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 6295362011136 \) Copy content Toggle raw display
$31$ \( (T^{4} - 26 T^{3} + \cdots + 456976)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 30 T^{3} + \cdots + 810000)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 176 T^{6} + \cdots + 959512576 \) Copy content Toggle raw display
$43$ \( (T - 42)^{8} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 161343242793216 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 374809600000000 \) Copy content Toggle raw display
$61$ \( (T^{4} + 12 T^{3} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$67$ \( (T - 2)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 161343242793216 \) Copy content Toggle raw display
$73$ \( (T^{4} - 74 T^{3} + \cdots + 29986576)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 40 T^{3} + \cdots + 2560000)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 6295362011136 \) Copy content Toggle raw display
$89$ \( (T^{2} + 14256)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 62 T^{3} + \cdots + 14776336)^{2} \) Copy content Toggle raw display
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