Properties

Label 363.2.d.e
Level $363$
Weight $2$
Character orbit 363.d
Analytic conductor $2.899$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,2,Mod(362,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.362");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.d (of order \(2\), degree \(1\), 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: \(2.89856959337\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3588489216.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 10x^{6} - 8x^{5} + 8x^{4} + 4x^{3} + 16x^{2} + 32x + 22 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{7} - 1) q^{3} + ( - \beta_{7} - \beta_{4} + 2) q^{4} - \beta_{5} q^{5} + (3 \beta_{6} + \beta_{2} - \beta_1) q^{6} - 2 \beta_{6} q^{7} + (\beta_{6} - \beta_{3} + 2 \beta_{2}) q^{8} + (\beta_{7} - \beta_{5} + \beta_{4} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{7} - 1) q^{3} + ( - \beta_{7} - \beta_{4} + 2) q^{4} - \beta_{5} q^{5} + (3 \beta_{6} + \beta_{2} - \beta_1) q^{6} - 2 \beta_{6} q^{7} + (\beta_{6} - \beta_{3} + 2 \beta_{2}) q^{8} + (\beta_{7} - \beta_{5} + \beta_{4} - 1) q^{9} + (\beta_{6} - 3 \beta_1) q^{10} + ( - 2 \beta_{7} - \beta_{5} + \beta_{4} - 1) q^{12} + ( - 2 \beta_{6} + \beta_1) q^{13} + (2 \beta_{7} - 2 \beta_{4} + 2) q^{14} + (\beta_{7} + \beta_{5} + 2 \beta_{4}) q^{15} + ( - 2 \beta_{7} - 2 \beta_{4} - 1) q^{16} + ( - \beta_{6} + \beta_{3} - 2 \beta_{2}) q^{17} + (2 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{18} + (\beta_{6} - 2 \beta_1) q^{19} + ( - \beta_{7} - \beta_{5} + \beta_{4} - 1) q^{20} + ( - 2 \beta_{3} + 2 \beta_1) q^{21} + (3 \beta_{6} - \beta_{3} - \beta_{2} - 4 \beta_1) q^{24} + ( - 2 \beta_{7} - 2 \beta_{4}) q^{25} + (2 \beta_{7} + \beta_{5} - 2 \beta_{4} + 2) q^{26} + (2 \beta_{7} + 2 \beta_{5} + \beta_{4}) q^{27} + ( - 6 \beta_{6} + 4 \beta_1) q^{28} + ( - \beta_{6} - 3 \beta_{3} - 2 \beta_{2}) q^{29} + (\beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{30} + ( - 3 \beta_{7} - 3 \beta_{4} - 5) q^{31} + \beta_{3} q^{32} + (4 \beta_{7} + 4 \beta_{4} - 3) q^{34} + (2 \beta_{6} + 4 \beta_{2}) q^{35} + (2 \beta_{7} - \beta_{5} + 4 \beta_{4} - 6) q^{36} + (2 \beta_{7} + 2 \beta_{4} - 7) q^{37} + ( - \beta_{7} - 2 \beta_{5} + \beta_{4} - 1) q^{38} + ( - 2 \beta_{3} + \beta_{2} + \beta_1) q^{39} + (4 \beta_{6} + \beta_1) q^{40} - 3 \beta_{3} q^{41} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} + 8) q^{42} - 3 \beta_{6} q^{43} + ( - \beta_{7} - 3 \beta_{4} - 4) q^{45} + ( - \beta_{7} + \beta_{4} - 1) q^{47} + (\beta_{7} - 2 \beta_{5} + 2 \beta_{4} + 3) q^{48} - q^{49} + (2 \beta_{6} - 2 \beta_{3} + 4 \beta_{2}) q^{50} + ( - 3 \beta_{6} + \beta_{3} + \beta_{2} + 4 \beta_1) q^{51} + ( - 7 \beta_{6} + 5 \beta_1) q^{52} + (\beta_{7} + 3 \beta_{5} - \beta_{4} + 1) q^{53} + ( - 6 \beta_{6} + 2 \beta_{3} - 3 \beta_{2} + 7 \beta_1) q^{54} + (2 \beta_{7} + 4 \beta_{5} - 2 \beta_{4} + 2) q^{56} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{57} + 13 q^{58} + ( - 3 \beta_{7} - 2 \beta_{5} + 3 \beta_{4} - 3) q^{59} + (2 \beta_{7} + 3 \beta_{4} - 4) q^{60} + (\beta_{6} - 6 \beta_1) q^{61} + (3 \beta_{6} + 2 \beta_{3} + 6 \beta_{2}) q^{62} + (6 \beta_{6} + 4 \beta_{2} - 4 \beta_1) q^{63} + (5 \beta_{7} + 5 \beta_{4} - 2) q^{64} + (\beta_{6} - \beta_{3} + 2 \beta_{2}) q^{65} + 2 q^{67} + ( - 2 \beta_{6} + 5 \beta_{3} - 4 \beta_{2}) q^{68} + ( - 6 \beta_{7} - 6 \beta_{4} - 2) q^{70} + (\beta_{7} + 4 \beta_{5} - \beta_{4} + 1) q^{71} + (3 \beta_{6} + 4 \beta_{3} - 2 \beta_{2} + \beta_1) q^{72} + (5 \beta_{6} - 2 \beta_1) q^{73} + ( - 2 \beta_{6} + 9 \beta_{3} - 4 \beta_{2}) q^{74} + ( - 2 \beta_{5} + 2 \beta_{4} + 2) q^{75} + (5 \beta_{6} - 4 \beta_1) q^{76} + ( - 3 \beta_{7} + \beta_{5} - 4 \beta_{4} + 8) q^{78} + ( - 3 \beta_{6} + 6 \beta_1) q^{79} + ( - 2 \beta_{7} + 3 \beta_{5} + 2 \beta_{4} - 2) q^{80} + ( - 2 \beta_{7} - 6 \beta_{4} + 1) q^{81} + ( - 3 \beta_{7} - 3 \beta_{4} + 12) q^{82} + ( - 3 \beta_{6} + 4 \beta_{3} - 6 \beta_{2}) q^{83} + ( - 6 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{84} + ( - 4 \beta_{6} - \beta_1) q^{85} + (3 \beta_{7} - 3 \beta_{4} + 3) q^{86} + (9 \beta_{6} + \beta_{3} + 5 \beta_{2}) q^{87} + ( - \beta_{7} - 5 \beta_{5} + \beta_{4} - 1) q^{89} + ( - 3 \beta_{6} + 3 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{90} + (2 \beta_{7} + 2 \beta_{4} - 6) q^{91} + (5 \beta_{7} - 3 \beta_{5} + 3 \beta_{4} + 8) q^{93} + (5 \beta_{6} - 2 \beta_1) q^{94} + (\beta_{6} + 2 \beta_{3} + 2 \beta_{2}) q^{95} + ( - 3 \beta_{6} - \beta_{2} + \beta_1) q^{96} + (\beta_{7} + \beta_{4} + 4) q^{97} + \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 16 q^{4} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 16 q^{4} - 8 q^{9} + 4 q^{12} + 4 q^{15} - 8 q^{16} - 4 q^{27} - 40 q^{31} - 24 q^{34} - 40 q^{36} - 56 q^{37} + 64 q^{42} - 40 q^{45} + 28 q^{48} - 8 q^{49} + 104 q^{58} - 28 q^{60} - 16 q^{64} + 16 q^{67} - 16 q^{70} + 24 q^{75} + 60 q^{78} - 8 q^{81} + 96 q^{82} - 48 q^{91} + 56 q^{93} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 10x^{6} - 8x^{5} + 8x^{4} + 4x^{3} + 16x^{2} + 32x + 22 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} - 4\nu^{4} + 7\nu^{3} - \nu^{2} - 2\nu - 1 ) / 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + \nu^{4} + 2\nu^{3} - 11\nu^{2} + 2\nu - 2 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 7\nu^{5} - \nu^{4} + 7\nu^{3} + 11\nu^{2} + 59 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 10\nu^{5} - 13\nu^{4} + 28\nu^{3} - 19\nu^{2} + 48\nu + 29 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 4\nu^{5} + 10\nu^{4} - 10\nu^{3} + 10\nu^{2} + 8\nu + 3 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 6\nu^{6} - 16\nu^{5} + 19\nu^{4} - 16\nu^{3} + 16\nu^{2} - 36\nu - 26 ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 10\nu^{5} + 10\nu^{4} - 10\nu^{3} + \nu^{2} - 12\nu - 18 ) / 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} - \beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{5} - \beta_{4} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 2\beta_{6} + \beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 3\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{4} + 6\beta_{3} - 2\beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{7} + 4\beta_{6} - 9\beta_{5} + 10\beta_{4} + 9\beta_{3} - 7\beta_{2} - 17\beta _1 - 17 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 30\beta_{6} - 21\beta_{5} + 30\beta_{4} - 18\beta_{2} - 42\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -39\beta_{7} + 85\beta_{6} - 19\beta_{5} + 39\beta_{4} - 44\beta_{3} - 12\beta_{2} - 52\beta _1 + 56 \) Copy content Toggle raw display

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
362.1
−0.697085 0.346269i
−0.697085 + 0.346269i
−0.252986 1.19584i
−0.252986 + 1.19584i
1.25299 1.71348i
1.25299 + 1.71348i
1.69709 + 1.58558i
1.69709 1.58558i
−2.39417 0.366025 1.69293i 3.73205 1.23931i −0.876327 + 4.05317i 2.82843i −4.14682 −2.73205 1.23931i 2.96713i
362.2 −2.39417 0.366025 + 1.69293i 3.73205 1.23931i −0.876327 4.05317i 2.82843i −4.14682 −2.73205 + 1.23931i 2.96713i
362.3 −1.50597 −1.36603 1.06488i 0.267949 2.90931i 2.05719 + 1.60368i 2.82843i 2.60842 0.732051 + 2.90931i 4.38134i
362.4 −1.50597 −1.36603 + 1.06488i 0.267949 2.90931i 2.05719 1.60368i 2.82843i 2.60842 0.732051 2.90931i 4.38134i
362.5 1.50597 −1.36603 1.06488i 0.267949 2.90931i −2.05719 1.60368i 2.82843i −2.60842 0.732051 + 2.90931i 4.38134i
362.6 1.50597 −1.36603 + 1.06488i 0.267949 2.90931i −2.05719 + 1.60368i 2.82843i −2.60842 0.732051 2.90931i 4.38134i
362.7 2.39417 0.366025 1.69293i 3.73205 1.23931i 0.876327 4.05317i 2.82843i 4.14682 −2.73205 1.23931i 2.96713i
362.8 2.39417 0.366025 + 1.69293i 3.73205 1.23931i 0.876327 + 4.05317i 2.82843i 4.14682 −2.73205 + 1.23931i 2.96713i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 362.8
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
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.d.e 8
3.b odd 2 1 inner 363.2.d.e 8
11.b odd 2 1 inner 363.2.d.e 8
11.c even 5 4 363.2.f.i 32
11.d odd 10 4 363.2.f.i 32
33.d even 2 1 inner 363.2.d.e 8
33.f even 10 4 363.2.f.i 32
33.h odd 10 4 363.2.f.i 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.d.e 8 1.a even 1 1 trivial
363.2.d.e 8 3.b odd 2 1 inner
363.2.d.e 8 11.b odd 2 1 inner
363.2.d.e 8 33.d even 2 1 inner
363.2.f.i 32 11.c even 5 4
363.2.f.i 32 11.d odd 10 4
363.2.f.i 32 33.f even 10 4
363.2.f.i 32 33.h odd 10 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 8T_{2}^{2} + 13 \) acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 8 T^{2} + 13)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 2 T^{3} + 4 T^{2} + 6 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 10 T^{2} + 13)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 12 T^{2} + 9)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 24 T^{2} + 117)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} - 104 T^{2} + 2197)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T - 2)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 14 T + 37)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 72 T^{2} + 1053)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 16 T^{2} + 52)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 94 T^{2} + 2197)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 160 T^{2} + 52)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 124 T^{2} + 2116)^{2} \) Copy content Toggle raw display
$67$ \( (T - 2)^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 160 T^{2} + 6292)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 76 T^{2} + 676)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 54)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 260 T^{2} + 8788)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 246 T^{2} + 14157)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 8 T + 13)^{4} \) Copy content Toggle raw display
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