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Properties

Label 363.3.g.c
Level $363$
Weight $3$
Character orbit 363.g
Analytic conductor $9.891$
Analytic rank $0$
Dimension $16$
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(40,363)]
 
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([0, 7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.40");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.g (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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: 16.0.6879707136000000000000.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 2 \beta_1) q^{2} + ( - \beta_{14} - \beta_{10} + \cdots + \beta_{2}) q^{3}+ \cdots + ( - 3 \beta_{13} + 3 \beta_{8} + \cdots - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 2 \beta_1) q^{2} + ( - \beta_{14} - \beta_{10} + \cdots + \beta_{2}) q^{3}+ \cdots + ( - 11 \beta_{11} - 37 \beta_{9}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 12 q^{9} - 144 q^{12} + 60 q^{14} + 60 q^{15} - 28 q^{16} - 180 q^{20} - 144 q^{23} - 200 q^{25} + 36 q^{26} + 48 q^{31} + 768 q^{34} + 24 q^{36} + 120 q^{37} - 96 q^{38} + 84 q^{42} + 192 q^{47} - 44 q^{49} + 60 q^{53} + 528 q^{56} - 84 q^{58} - 60 q^{59} - 120 q^{60} - 184 q^{64} + 48 q^{67} - 12 q^{69} - 420 q^{70} + 108 q^{71} + 192 q^{78} - 36 q^{81} + 228 q^{82} + 204 q^{86} - 864 q^{89} + 124 q^{91} - 36 q^{92} + 24 q^{93} + 416 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 362 ) / 209 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 780\nu ) / 209 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{12} + 780\nu^{2} ) / 209 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{13} + 780\nu^{3} ) / 209 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{12} - 1351\nu^{2} ) / 209 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{13} - 2911\nu^{3} ) / 209 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -4\nu^{14} - 2911\nu^{4} ) / 209 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -4\nu^{15} - 2911\nu^{5} ) / 209 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -7\nu^{14} - 5042\nu^{4} ) / 209 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{15} - 723\nu^{5} ) / 19 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 60\nu^{15} - 225\nu^{13} + 840\nu^{11} - 3135\nu^{9} + 11704\nu^{7} - 225\nu^{5} + 60\nu^{3} - 15\nu ) / 209 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 56\nu^{14} - 224\nu^{12} + 840\nu^{10} - 3135\nu^{8} + 11704\nu^{6} - 3136\nu^{4} + 840\nu^{2} - 224 ) / 209 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 104\nu^{14} - 390\nu^{12} + 1456\nu^{10} - 5434\nu^{8} + 20273\nu^{6} - 390\nu^{4} + 104\nu^{2} - 26 ) / 209 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 164\nu^{15} - 615\nu^{13} + 2296\nu^{11} - 8569\nu^{9} + 31977\nu^{7} - 615\nu^{5} + 164\nu^{3} - 41\nu ) / 209 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{10} - 7\beta_{8} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{11} - 11\beta_{9} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -15\beta_{14} + 26\beta_{13} - 26\beta_{8} - 26\beta_{4} + 26 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -15\beta_{15} + 41\beta_{12} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -56\beta_{14} + 97\beta_{13} - 56\beta_{10} + 56\beta_{6} + 56\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -56\beta_{15} + 153\beta_{12} - 56\beta_{11} + 153\beta_{9} + 56\beta_{7} + 209\beta_{5} + 56\beta_{3} - 209\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 209\beta_{2} - 362 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 209\beta_{3} - 780\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -780\beta_{6} - 1351\beta_{4} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -780\beta_{7} - 2911\beta_{5} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -2911\beta_{10} + 5042\beta_{8} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -2911\beta_{11} + 7953\beta_{9} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
1.13551 − 1.56290i
−0.304260 + 0.418778i
0.304260 − 0.418778i
−1.13551 + 1.56290i
1.83730 − 0.596975i
−0.492303 + 0.159959i
0.492303 − 0.159959i
−1.83730 + 0.596975i
1.83730 + 0.596975i
−0.492303 − 0.159959i
0.492303 + 0.159959i
−1.83730 − 0.596975i
1.13551 + 1.56290i
−0.304260 − 0.418778i
0.304260 + 0.418778i
−1.13551 − 1.56290i
−1.96677 + 2.70702i 0.535233 − 1.64728i −2.22373 − 6.84395i 7.00629 − 5.09037i 3.40654 + 4.68870i 7.70959 − 2.50500i 10.1711 + 3.30479i −2.42705 − 1.76336i 28.9778i
40.2 −0.526994 + 0.725345i −0.535233 + 1.64728i 0.987665 + 3.03972i −7.00629 + 5.09037i −0.912780 − 1.25633i 3.05038 − 0.991130i −6.13612 − 1.99374i −2.42705 − 1.76336i − 7.76457i
40.3 0.526994 − 0.725345i −0.535233 + 1.64728i 0.987665 + 3.03972i −7.00629 + 5.09037i 0.912780 + 1.25633i −3.05038 + 0.991130i 6.13612 + 1.99374i −2.42705 − 1.76336i 7.76457i
40.4 1.96677 − 2.70702i 0.535233 − 1.64728i −2.22373 − 6.84395i 7.00629 − 5.09037i −3.40654 − 4.68870i −7.70959 + 2.50500i −10.1711 − 3.30479i −2.42705 − 1.76336i − 28.9778i
94.1 −3.18230 + 1.03399i −1.40126 − 1.01807i 5.82181 − 4.22979i −2.67617 + 8.23639i 5.51190 + 1.79092i −4.76479 − 6.55817i −6.28609 + 8.65206i 0.927051 + 2.85317i − 28.9778i
94.2 −0.852694 + 0.277057i 1.40126 + 1.01807i −2.58574 + 1.87865i 2.67617 − 8.23639i −1.47691 − 0.479877i −1.88524 − 2.59481i 3.79233 − 5.21969i 0.927051 + 2.85317i 7.76457i
94.3 0.852694 − 0.277057i 1.40126 + 1.01807i −2.58574 + 1.87865i 2.67617 − 8.23639i 1.47691 + 0.479877i 1.88524 + 2.59481i −3.79233 + 5.21969i 0.927051 + 2.85317i − 7.76457i
94.4 3.18230 − 1.03399i −1.40126 − 1.01807i 5.82181 − 4.22979i −2.67617 + 8.23639i −5.51190 − 1.79092i 4.76479 + 6.55817i 6.28609 − 8.65206i 0.927051 + 2.85317i 28.9778i
112.1 −3.18230 − 1.03399i −1.40126 + 1.01807i 5.82181 + 4.22979i −2.67617 − 8.23639i 5.51190 − 1.79092i −4.76479 + 6.55817i −6.28609 − 8.65206i 0.927051 − 2.85317i 28.9778i
112.2 −0.852694 − 0.277057i 1.40126 − 1.01807i −2.58574 − 1.87865i 2.67617 + 8.23639i −1.47691 + 0.479877i −1.88524 + 2.59481i 3.79233 + 5.21969i 0.927051 − 2.85317i − 7.76457i
112.3 0.852694 + 0.277057i 1.40126 − 1.01807i −2.58574 − 1.87865i 2.67617 + 8.23639i 1.47691 − 0.479877i 1.88524 − 2.59481i −3.79233 − 5.21969i 0.927051 − 2.85317i 7.76457i
112.4 3.18230 + 1.03399i −1.40126 + 1.01807i 5.82181 + 4.22979i −2.67617 − 8.23639i −5.51190 + 1.79092i 4.76479 − 6.55817i 6.28609 + 8.65206i 0.927051 − 2.85317i − 28.9778i
118.1 −1.96677 − 2.70702i 0.535233 + 1.64728i −2.22373 + 6.84395i 7.00629 + 5.09037i 3.40654 − 4.68870i 7.70959 + 2.50500i 10.1711 − 3.30479i −2.42705 + 1.76336i − 28.9778i
118.2 −0.526994 − 0.725345i −0.535233 − 1.64728i 0.987665 − 3.03972i −7.00629 − 5.09037i −0.912780 + 1.25633i 3.05038 + 0.991130i −6.13612 + 1.99374i −2.42705 + 1.76336i 7.76457i
118.3 0.526994 + 0.725345i −0.535233 − 1.64728i 0.987665 − 3.03972i −7.00629 − 5.09037i 0.912780 − 1.25633i −3.05038 − 0.991130i 6.13612 − 1.99374i −2.42705 + 1.76336i − 7.76457i
118.4 1.96677 + 2.70702i 0.535233 + 1.64728i −2.22373 + 6.84395i 7.00629 + 5.09037i −3.40654 + 4.68870i −7.70959 − 2.50500i −10.1711 + 3.30479i −2.42705 + 1.76336i 28.9778i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
11.c even 5 3 inner
11.d 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.g.c 16
11.b odd 2 1 inner 363.3.g.c 16
11.c even 5 1 363.3.c.b ✓ 4
11.c even 5 3 inner 363.3.g.c 16
11.d odd 10 1 363.3.c.b ✓ 4
11.d odd 10 3 inner 363.3.g.c 16
33.f even 10 1 1089.3.c.d 4
33.h odd 10 1 1089.3.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.c.b ✓ 4 11.c even 5 1
363.3.c.b ✓ 4 11.d odd 10 1
363.3.g.c 16 1.a even 1 1 trivial
363.3.g.c 16 11.b odd 2 1 inner
363.3.g.c 16 11.c even 5 3 inner
363.3.g.c 16 11.d odd 10 3 inner
1089.3.c.d 4 33.f even 10 1
1089.3.c.d 4 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 12T_{2}^{14} + 135T_{2}^{12} - 1512T_{2}^{10} + 16929T_{2}^{8} - 13608T_{2}^{6} + 10935T_{2}^{4} - 8748T_{2}^{2} + 6561 \) acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 12 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$3$ \( (T^{8} + 3 T^{6} + 9 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 75 T^{6} + \cdots + 31640625)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 208827064576 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 13\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 92\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( (T^{2} + 18 T + 78)^{8} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 10\!\cdots\!61 \) Copy content Toggle raw display
$31$ \( (T^{8} - 24 T^{7} + \cdots + 303595776)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 60 T^{7} + \cdots + 580840612641)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 60\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( (T^{4} + 1828 T^{2} + 662596)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 96 T^{7} + \cdots + 526936617216)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 2266617569841)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 16\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( (T^{2} - 6 T - 5034)^{8} \) Copy content Toggle raw display
$71$ \( (T^{8} - 54 T^{7} + \cdots + 541937434896)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{2} + 108 T - 2127)^{8} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 32\!\cdots\!01)^{2} \) Copy content Toggle raw display
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