Properties

Label 363.3.b.i
Level $363$
Weight $3$
Character orbit 363.b
Analytic conductor $9.891$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.89103359628\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-7})\)
Defining polynomial: \(x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( -1 + \beta_{3} ) q^{3} -3 q^{4} + \beta_{3} q^{5} + ( -\beta_{1} - \beta_{2} ) q^{6} + \beta_{1} q^{7} + \beta_{2} q^{8} + ( -7 - 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( -1 + \beta_{3} ) q^{3} -3 q^{4} + \beta_{3} q^{5} + ( -\beta_{1} - \beta_{2} ) q^{6} + \beta_{1} q^{7} + \beta_{2} q^{8} + ( -7 - 2 \beta_{3} ) q^{9} -\beta_{1} q^{10} + ( 3 - 3 \beta_{3} ) q^{12} -3 \beta_{1} q^{13} + 7 \beta_{3} q^{14} + ( -8 - \beta_{3} ) q^{15} -19 q^{16} + 8 \beta_{2} q^{17} + ( 2 \beta_{1} - 7 \beta_{2} ) q^{18} -2 \beta_{1} q^{19} -3 \beta_{3} q^{20} + ( -\beta_{1} + 8 \beta_{2} ) q^{21} -11 \beta_{3} q^{23} + ( -\beta_{1} - \beta_{2} ) q^{24} + 17 q^{25} -21 \beta_{3} q^{26} + ( 23 - 5 \beta_{3} ) q^{27} -3 \beta_{1} q^{28} + 8 \beta_{2} q^{29} + ( \beta_{1} - 8 \beta_{2} ) q^{30} + 30 q^{31} -15 \beta_{2} q^{32} -56 q^{34} + 8 \beta_{2} q^{35} + ( 21 + 6 \beta_{3} ) q^{36} -10 q^{37} -14 \beta_{3} q^{38} + ( 3 \beta_{1} - 24 \beta_{2} ) q^{39} -\beta_{1} q^{40} + 16 \beta_{2} q^{41} + ( -56 - 7 \beta_{3} ) q^{42} + 2 \beta_{1} q^{43} + ( 16 - 7 \beta_{3} ) q^{45} + 11 \beta_{1} q^{46} + 13 \beta_{3} q^{47} + ( 19 - 19 \beta_{3} ) q^{48} + 7 q^{49} + 17 \beta_{2} q^{50} + ( -8 \beta_{1} - 8 \beta_{2} ) q^{51} + 9 \beta_{1} q^{52} -15 \beta_{3} q^{53} + ( 5 \beta_{1} + 23 \beta_{2} ) q^{54} + 7 \beta_{3} q^{56} + ( 2 \beta_{1} - 16 \beta_{2} ) q^{57} -56 q^{58} + 12 \beta_{3} q^{59} + ( 24 + 3 \beta_{3} ) q^{60} + 13 \beta_{1} q^{61} + 30 \beta_{2} q^{62} + ( -7 \beta_{1} - 16 \beta_{2} ) q^{63} + 29 q^{64} -24 \beta_{2} q^{65} -42 q^{67} -24 \beta_{2} q^{68} + ( 88 + 11 \beta_{3} ) q^{69} -56 q^{70} + 23 \beta_{3} q^{71} + ( 2 \beta_{1} - 7 \beta_{2} ) q^{72} -10 \beta_{1} q^{73} -10 \beta_{2} q^{74} + ( -17 + 17 \beta_{3} ) q^{75} + 6 \beta_{1} q^{76} + ( 168 + 21 \beta_{3} ) q^{78} -3 \beta_{1} q^{79} -19 \beta_{3} q^{80} + ( 17 + 28 \beta_{3} ) q^{81} -112 q^{82} + 8 \beta_{2} q^{83} + ( 3 \beta_{1} - 24 \beta_{2} ) q^{84} -8 \beta_{1} q^{85} + 14 \beta_{3} q^{86} + ( -8 \beta_{1} - 8 \beta_{2} ) q^{87} -22 \beta_{3} q^{89} + ( 7 \beta_{1} + 16 \beta_{2} ) q^{90} -168 q^{91} + 33 \beta_{3} q^{92} + ( -30 + 30 \beta_{3} ) q^{93} -13 \beta_{1} q^{94} -16 \beta_{2} q^{95} + ( 15 \beta_{1} + 15 \beta_{2} ) q^{96} + 74 q^{97} + 7 \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 12 q^{4} - 28 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{3} - 12 q^{4} - 28 q^{9} + 12 q^{12} - 32 q^{15} - 76 q^{16} + 68 q^{25} + 92 q^{27} + 120 q^{31} - 224 q^{34} + 84 q^{36} - 40 q^{37} - 224 q^{42} + 64 q^{45} + 76 q^{48} + 28 q^{49} - 224 q^{58} + 96 q^{60} + 116 q^{64} - 168 q^{67} + 352 q^{69} - 224 q^{70} - 68 q^{75} + 672 q^{78} + 68 q^{81} - 448 q^{82} - 672 q^{91} - 120 q^{93} + 296 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{2} - 2 \nu + 8 \)
\(\beta_{2}\)\(=\)\( 4 \nu^{3} - 6 \nu^{2} + 32 \nu - 15 \)
\(\beta_{3}\)\(=\)\( 4 \nu^{3} - 6 \nu^{2} + 34 \nu - 16 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{2} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - \beta_{2} + \beta_{1} - 7\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-13 \beta_{3} + 14 \beta_{2} + 3 \beta_{1} - 22\)\()/4\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
122.1
0.500000 0.0913379i
0.500000 + 2.73709i
0.500000 2.73709i
0.500000 + 0.0913379i
2.64575i −1.00000 2.82843i −3.00000 2.82843i −7.48331 + 2.64575i 7.48331 2.64575i −7.00000 + 5.65685i −7.48331
122.2 2.64575i −1.00000 + 2.82843i −3.00000 2.82843i 7.48331 + 2.64575i −7.48331 2.64575i −7.00000 5.65685i 7.48331
122.3 2.64575i −1.00000 2.82843i −3.00000 2.82843i 7.48331 2.64575i −7.48331 2.64575i −7.00000 + 5.65685i 7.48331
122.4 2.64575i −1.00000 + 2.82843i −3.00000 2.82843i −7.48331 2.64575i 7.48331 2.64575i −7.00000 5.65685i −7.48331
\(n\): e.g. 2-40 or 990-1000
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 363.3.b.i 4
3.b odd 2 1 inner 363.3.b.i 4
11.b odd 2 1 inner 363.3.b.i 4
11.c even 5 4 363.3.h.k 16
11.d odd 10 4 363.3.h.k 16
33.d even 2 1 inner 363.3.b.i 4
33.f even 10 4 363.3.h.k 16
33.h odd 10 4 363.3.h.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.b.i 4 1.a even 1 1 trivial
363.3.b.i 4 3.b odd 2 1 inner
363.3.b.i 4 11.b odd 2 1 inner
363.3.b.i 4 33.d even 2 1 inner
363.3.h.k 16 11.c even 5 4
363.3.h.k 16 11.d odd 10 4
363.3.h.k 16 33.f even 10 4
363.3.h.k 16 33.h odd 10 4

Hecke kernels

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

\( T_{2}^{2} + 7 \)
\( T_{5}^{2} + 8 \)
\( T_{7}^{2} - 56 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 7 + T^{2} )^{2} \)
$3$ \( ( 9 + 2 T + T^{2} )^{2} \)
$5$ \( ( 8 + T^{2} )^{2} \)
$7$ \( ( -56 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( ( -504 + T^{2} )^{2} \)
$17$ \( ( 448 + T^{2} )^{2} \)
$19$ \( ( -224 + T^{2} )^{2} \)
$23$ \( ( 968 + T^{2} )^{2} \)
$29$ \( ( 448 + T^{2} )^{2} \)
$31$ \( ( -30 + T )^{4} \)
$37$ \( ( 10 + T )^{4} \)
$41$ \( ( 1792 + T^{2} )^{2} \)
$43$ \( ( -224 + T^{2} )^{2} \)
$47$ \( ( 1352 + T^{2} )^{2} \)
$53$ \( ( 1800 + T^{2} )^{2} \)
$59$ \( ( 1152 + T^{2} )^{2} \)
$61$ \( ( -9464 + T^{2} )^{2} \)
$67$ \( ( 42 + T )^{4} \)
$71$ \( ( 4232 + T^{2} )^{2} \)
$73$ \( ( -5600 + T^{2} )^{2} \)
$79$ \( ( -504 + T^{2} )^{2} \)
$83$ \( ( 448 + T^{2} )^{2} \)
$89$ \( ( 3872 + T^{2} )^{2} \)
$97$ \( ( -74 + T )^{4} \)
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