Properties

Label 363.3.b.h
Level $363$
Weight $3$
Character orbit 363.b
Analytic conductor $9.891$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 33)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{2} + ( -1 + \beta_{2} + \beta_{3} ) q^{3} + ( 1 + \beta_{1} + \beta_{3} ) q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{6} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{7} + ( -3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{8} + ( -3 - 5 \beta_{1} - 3 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{2} + ( -1 + \beta_{2} + \beta_{3} ) q^{3} + ( 1 + \beta_{1} + \beta_{3} ) q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{6} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{7} + ( -3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{8} + ( -3 - 5 \beta_{1} - 3 \beta_{2} ) q^{9} + 2 q^{10} + ( 3 - 5 \beta_{1} + 3 \beta_{3} ) q^{12} + ( 4 + 4 \beta_{1} + 4 \beta_{3} ) q^{13} + ( 4 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{14} + ( 4 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{15} + ( -3 + 5 \beta_{1} + 5 \beta_{3} ) q^{16} + ( 2 \beta_{1} + 16 \beta_{2} - 2 \beta_{3} ) q^{17} + ( -8 + \beta_{1} - 7 \beta_{2} + 5 \beta_{3} ) q^{18} + ( 6 - 6 \beta_{1} - 6 \beta_{3} ) q^{19} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{20} + ( 8 - 10 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{21} + ( -\beta_{1} + 16 \beta_{2} + \beta_{3} ) q^{23} + ( -8 - \beta_{1} + 5 \beta_{2} - 7 \beta_{3} ) q^{24} + ( 21 - \beta_{1} - \beta_{3} ) q^{25} + ( 4 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} ) q^{26} + ( -5 + 16 \beta_{1} + 8 \beta_{2} - 16 \beta_{3} ) q^{27} + 16 q^{28} + ( 24 \beta_{1} + 4 \beta_{2} - 24 \beta_{3} ) q^{29} + ( -2 + 2 \beta_{2} + 2 \beta_{3} ) q^{30} + ( 14 + 5 \beta_{1} + 5 \beta_{3} ) q^{31} + ( \beta_{1} + 19 \beta_{2} - \beta_{3} ) q^{32} + ( 20 - 16 \beta_{1} - 16 \beta_{3} ) q^{34} + ( 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{35} + ( -23 + \beta_{1} + 8 \beta_{2} - 7 \beta_{3} ) q^{36} + ( -26 - 7 \beta_{1} - 7 \beta_{3} ) q^{37} + ( -18 \beta_{1} - 30 \beta_{2} + 18 \beta_{3} ) q^{38} + ( 12 - 20 \beta_{1} + 12 \beta_{3} ) q^{39} + ( 10 + 2 \beta_{1} + 2 \beta_{3} ) q^{40} + ( -4 \beta_{1} + 12 \beta_{2} + 4 \beta_{3} ) q^{41} + ( -16 - 12 \beta_{1} - 20 \beta_{2} + 16 \beta_{3} ) q^{42} + ( -26 - 4 \beta_{1} - 4 \beta_{3} ) q^{43} + ( 4 - 6 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{45} + ( 14 - 16 \beta_{1} - 16 \beta_{3} ) q^{46} + ( -22 \beta_{1} - 14 \beta_{2} + 22 \beta_{3} ) q^{47} + ( 23 - 25 \beta_{1} - 8 \beta_{2} + 7 \beta_{3} ) q^{48} + ( -17 - 4 \beta_{1} - 4 \beta_{3} ) q^{49} + ( -23 \beta_{1} - 25 \beta_{2} + 23 \beta_{3} ) q^{50} + ( -56 - 30 \beta_{1} - 34 \beta_{2} + 20 \beta_{3} ) q^{51} + ( 36 + 4 \beta_{1} + 4 \beta_{3} ) q^{52} + ( 30 \beta_{1} + 38 \beta_{2} - 30 \beta_{3} ) q^{53} + ( 40 - 3 \beta_{1} + 5 \beta_{2} - 13 \beta_{3} ) q^{54} + 16 \beta_{2} q^{56} + ( -30 + 30 \beta_{1} + 12 \beta_{2} - 6 \beta_{3} ) q^{57} + ( 52 - 4 \beta_{1} - 4 \beta_{3} ) q^{58} + ( 23 \beta_{1} - 2 \beta_{2} - 23 \beta_{3} ) q^{59} + ( 16 + 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{60} + ( -4 - 20 \beta_{1} - 20 \beta_{3} ) q^{61} + ( -4 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} ) q^{62} + ( -40 + 12 \beta_{1} + 22 \beta_{2} - 14 \beta_{3} ) q^{63} + ( 9 + \beta_{1} + \beta_{3} ) q^{64} + ( 8 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{65} + ( -6 + 17 \beta_{1} + 17 \beta_{3} ) q^{67} + ( -44 \beta_{1} - 20 \beta_{2} + 44 \beta_{3} ) q^{68} + ( -68 - 33 \beta_{1} - 31 \beta_{2} + 14 \beta_{3} ) q^{69} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{70} + ( 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{71} + ( -16 + 13 \beta_{1} - 17 \beta_{2} - 5 \beta_{3} ) q^{72} + ( 74 + 6 \beta_{1} + 6 \beta_{3} ) q^{73} + ( 12 \beta_{1} - 2 \beta_{2} - 12 \beta_{3} ) q^{74} + ( -25 + 5 \beta_{1} + 22 \beta_{2} + 19 \beta_{3} ) q^{75} + ( -42 + 6 \beta_{1} + 6 \beta_{3} ) q^{76} + ( -32 - 20 \beta_{1} - 28 \beta_{2} + 20 \beta_{3} ) q^{78} + ( 32 + 26 \beta_{1} + 26 \beta_{3} ) q^{79} + ( 2 \beta_{1} - 10 \beta_{2} - 2 \beta_{3} ) q^{80} + ( 37 - 37 \beta_{2} + 35 \beta_{3} ) q^{81} + ( 4 - 12 \beta_{1} - 12 \beta_{3} ) q^{82} + ( 30 \beta_{1} + 24 \beta_{2} - 30 \beta_{3} ) q^{83} + ( -16 + 16 \beta_{2} + 16 \beta_{3} ) q^{84} + ( 24 + 14 \beta_{1} + 14 \beta_{3} ) q^{85} + ( 18 \beta_{1} + 10 \beta_{2} - 18 \beta_{3} ) q^{86} + ( 80 + 16 \beta_{1} - 32 \beta_{2} + 52 \beta_{3} ) q^{87} + ( 39 \beta_{1} - 26 \beta_{2} - 39 \beta_{3} ) q^{89} + ( -6 - 10 \beta_{1} - 6 \beta_{2} ) q^{90} + 64 q^{91} + ( -50 \beta_{1} - 14 \beta_{2} + 50 \beta_{3} ) q^{92} + ( 6 - 25 \beta_{1} + 9 \beta_{2} + 24 \beta_{3} ) q^{93} + ( -58 + 14 \beta_{1} + 14 \beta_{3} ) q^{94} + 12 \beta_{2} q^{95} + ( -72 - 37 \beta_{1} - 39 \beta_{2} + 21 \beta_{3} ) q^{96} + ( 38 - 33 \beta_{1} - 33 \beta_{3} ) q^{97} + ( 9 \beta_{1} + \beta_{2} - 9 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 5 q^{3} + 2 q^{4} + 2 q^{6} - 4 q^{7} - 7 q^{9} + O(q^{10}) \) \( 4 q - 5 q^{3} + 2 q^{4} + 2 q^{6} - 4 q^{7} - 7 q^{9} + 8 q^{10} + 14 q^{12} + 8 q^{13} + 13 q^{15} - 22 q^{16} - 38 q^{18} + 36 q^{19} + 38 q^{21} - 24 q^{24} + 86 q^{25} - 20 q^{27} + 64 q^{28} - 10 q^{30} + 46 q^{31} + 112 q^{34} - 86 q^{36} - 90 q^{37} + 56 q^{39} + 36 q^{40} - 68 q^{42} - 96 q^{43} + 17 q^{45} + 88 q^{46} + 110 q^{48} - 60 q^{49} - 214 q^{51} + 136 q^{52} + 176 q^{54} - 144 q^{57} + 216 q^{58} + 56 q^{60} + 24 q^{61} - 158 q^{63} + 34 q^{64} - 58 q^{67} - 253 q^{69} - 8 q^{70} - 72 q^{72} + 284 q^{73} - 124 q^{75} - 180 q^{76} - 128 q^{78} + 76 q^{79} + 113 q^{81} + 40 q^{82} - 80 q^{84} + 68 q^{85} + 252 q^{87} - 14 q^{90} + 256 q^{91} + 25 q^{93} - 260 q^{94} - 272 q^{96} + 218 q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} + 4 \nu - 9 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} - 2 \nu^{2} + 2 \nu + 6 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu + 3 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} - 3 \beta_{2} + 3\)\()/2\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 4\)

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
−1.18614 + 1.26217i
1.68614 + 0.396143i
1.68614 0.396143i
−1.18614 1.26217i
2.52434i −2.68614 + 1.33591i −2.37228 0.792287i 3.37228 + 6.78073i −6.74456 4.10891i 5.43070 7.17687i 2.00000
122.2 0.792287i 0.186141 2.99422i 3.37228 2.52434i −2.37228 0.147477i 4.74456 5.84096i −8.93070 1.11469i 2.00000
122.3 0.792287i 0.186141 + 2.99422i 3.37228 2.52434i −2.37228 + 0.147477i 4.74456 5.84096i −8.93070 + 1.11469i 2.00000
122.4 2.52434i −2.68614 1.33591i −2.37228 0.792287i 3.37228 6.78073i −6.74456 4.10891i 5.43070 + 7.17687i 2.00000
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.b.h 4
3.b odd 2 1 inner 363.3.b.h 4
11.b odd 2 1 33.3.b.b 4
11.c even 5 4 363.3.h.l 16
11.d odd 10 4 363.3.h.m 16
33.d even 2 1 33.3.b.b 4
33.f even 10 4 363.3.h.m 16
33.h odd 10 4 363.3.h.l 16
44.c even 2 1 528.3.i.d 4
132.d odd 2 1 528.3.i.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.b.b 4 11.b odd 2 1
33.3.b.b 4 33.d even 2 1
363.3.b.h 4 1.a even 1 1 trivial
363.3.b.h 4 3.b odd 2 1 inner
363.3.h.l 16 11.c even 5 4
363.3.h.l 16 33.h odd 10 4
363.3.h.m 16 11.d odd 10 4
363.3.h.m 16 33.f even 10 4
528.3.i.d 4 44.c even 2 1
528.3.i.d 4 132.d odd 2 1

Hecke kernels

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

\( T_{2}^{4} + 7 T_{2}^{2} + 4 \)
\( T_{5}^{4} + 7 T_{5}^{2} + 4 \)
\( T_{7}^{2} + 2 T_{7} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 7 T^{2} + T^{4} \)
$3$ \( 81 + 45 T + 16 T^{2} + 5 T^{3} + T^{4} \)
$5$ \( 4 + 7 T^{2} + T^{4} \)
$7$ \( ( -32 + 2 T + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( ( -128 - 4 T + T^{2} )^{2} \)
$17$ \( 440896 + 1372 T^{2} + T^{4} \)
$19$ \( ( -216 - 18 T + T^{2} )^{2} \)
$23$ \( 662596 + 1639 T^{2} + T^{4} \)
$29$ \( 1937664 + 3552 T^{2} + T^{4} \)
$31$ \( ( -74 - 23 T + T^{2} )^{2} \)
$37$ \( ( 102 + 45 T + T^{2} )^{2} \)
$41$ \( 295936 + 1264 T^{2} + T^{4} \)
$43$ \( ( 444 + 48 T + T^{2} )^{2} \)
$47$ \( 1700416 + 2716 T^{2} + T^{4} \)
$53$ \( 788544 + 8124 T^{2} + T^{4} \)
$59$ \( 824464 + 4003 T^{2} + T^{4} \)
$61$ \( ( -3264 - 12 T + T^{2} )^{2} \)
$67$ \( ( -2174 + 29 T + T^{2} )^{2} \)
$71$ \( 4356 + 231 T^{2} + T^{4} \)
$73$ \( ( 4744 - 142 T + T^{2} )^{2} \)
$79$ \( ( -5216 - 38 T + T^{2} )^{2} \)
$83$ \( 4981824 + 5436 T^{2} + T^{4} \)
$89$ \( 4112784 + 20787 T^{2} + T^{4} \)
$97$ \( ( -6014 - 109 T + T^{2} )^{2} \)
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