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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Newspace parameters

Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.h (of order \(10\), degree \(4\), not minimal)

Newform invariants

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
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

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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + 3 \beta_{4} q^{3} - 7 \beta_{2} q^{4} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_1) q^{5} + ( - 3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 3 \beta_1) q^{6} - 8 \beta_{2} q^{7} - 3 \beta_{5} q^{8} - 9 \beta_{3} q^{9}+O(q^{10}) \) 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} + 2 \beta_{5} + 2 \beta_1) q^{5} + ( - 3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 3 \beta_1) q^{6} - 8 \beta_{2} q^{7} - 3 \beta_{5} q^{8} - 9 \beta_{3} q^{9} + 22 q^{10} - 21 q^{12} - 4 \beta_{3} q^{13} - 8 \beta_{5} q^{14} - 6 \beta_1 q^{15} + ( - 5 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 5) q^{16} + ( - 4 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} - 4 \beta_1) q^{17} + 9 \beta_1 q^{18} - 6 \beta_{4} q^{19} + 14 \beta_{7} q^{20} - 24 q^{21} + 2 \beta_{6} q^{23} - 9 \beta_{7} q^{24} - 19 \beta_{4} q^{25} + 4 \beta_1 q^{26} + ( - 27 \beta_{4} + 27 \beta_{3} - 27 \beta_{2} - 27) q^{27} + ( - 56 \beta_{4} + 56 \beta_{3} - 56 \beta_{2} - 56) q^{28} - 12 \beta_1 q^{29} + 66 \beta_{4} q^{30} + 26 \beta_{3} q^{31} - 7 \beta_{6} q^{32} - 44 q^{34} + 16 \beta_{7} q^{35} - 63 \beta_{4} q^{36} + 30 \beta_{2} q^{37} + (6 \beta_{7} + 6 \beta_{6} + 6 \beta_{5} + 6 \beta_1) q^{38} + ( - 12 \beta_{4} + 12 \beta_{3} - 12 \beta_{2} - 12) q^{39} - 66 \beta_{2} q^{40} + 4 \beta_{5} q^{41} - 24 \beta_{7} q^{42} + 42 q^{43} - 18 \beta_{6} q^{45} + 22 \beta_{3} q^{46} + 26 \beta_{5} q^{47} + 15 \beta_{2} q^{48} + ( - 15 \beta_{4} + 15 \beta_{3} - 15 \beta_{2} - 15) q^{49} + (19 \beta_{7} + 19 \beta_{6} + 19 \beta_{5} + 19 \beta_1) q^{50} + 12 \beta_1 q^{51} - 28 \beta_{4} q^{52} - 18 \beta_{7} q^{53} + 27 \beta_{6} q^{54} + 24 \beta_{6} q^{56} + 18 \beta_{3} q^{57} + 132 \beta_{4} q^{58} + 20 \beta_1 q^{59} + ( - 42 \beta_{7} - 42 \beta_{6} - 42 \beta_{5} - 42 \beta_1) q^{60} + ( - 12 \beta_{4} + 12 \beta_{3} - 12 \beta_{2} - 12) q^{61} - 26 \beta_1 q^{62} - 72 \beta_{4} q^{63} - 97 \beta_{3} q^{64} - 8 \beta_{6} q^{65} + 2 q^{67} - 28 \beta_{7} q^{68} + 6 \beta_{5} q^{69} - 176 \beta_{2} q^{70} + (18 \beta_{7} + 18 \beta_{6} + 18 \beta_{5} + 18 \beta_1) q^{71} + (27 \beta_{7} + 27 \beta_{6} + 27 \beta_{5} + 27 \beta_1) q^{72} - 74 \beta_{2} q^{73} + 30 \beta_{5} q^{74} + 57 \beta_{3} q^{75} + 42 q^{76} + 12 \beta_{6} q^{78} + 40 \beta_{3} q^{79} - 10 \beta_{5} q^{80} + 81 \beta_{2} q^{81} + (44 \beta_{4} - 44 \beta_{3} + 44 \beta_{2} + 44) q^{82} + (12 \beta_{7} + 12 \beta_{6} + 12 \beta_{5} + 12 \beta_1) q^{83} + 168 \beta_{2} q^{84} + 88 \beta_{4} q^{85} + 42 \beta_{7} q^{86} - 36 \beta_{6} q^{87} - 36 \beta_{6} q^{89} - 198 \beta_{3} q^{90} - 32 \beta_{4} q^{91} - 14 \beta_1 q^{92} + (78 \beta_{4} - 78 \beta_{3} + 78 \beta_{2} + 78) q^{93} + (286 \beta_{4} - 286 \beta_{3} + 286 \beta_{2} + 286) q^{94} + 12 \beta_1 q^{95} - 21 \beta_{5} q^{96} - 62 \beta_{3} q^{97} + 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.

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 323.2
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

$p$ $F_p(T)$
$2$ \( T^{8} - 11 T^{6} + 121 T^{4} + \cdots + 14641 \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 44 T^{6} + 1936 T^{4} + \cdots + 3748096 \) Copy content Toggle raw display
$7$ \( (T^{4} - 8 T^{3} + 64 T^{2} - 512 T + 4096)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 176 T^{6} + \cdots + 959512576 \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} + 36 T^{2} - 216 T + 1296)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 44)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} - 1584 T^{6} + \cdots + 6295362011136 \) Copy content Toggle raw display
$31$ \( (T^{4} - 26 T^{3} + 676 T^{2} + \cdots + 456976)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 30 T^{3} + 900 T^{2} + \cdots + 810000)^{2} \) 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
$47$ \( T^{8} - 7436 T^{6} + \cdots + 30\!\cdots\!16 \) 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
$61$ \( (T^{4} + 12 T^{3} + 144 T^{2} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$67$ \( (T - 2)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 161343242793216 \) Copy content Toggle raw display
$73$ \( (T^{4} - 74 T^{3} + 5476 T^{2} + \cdots + 29986576)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 40 T^{3} + 1600 T^{2} + \cdots + 2560000)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 1584 T^{6} + \cdots + 6295362011136 \) Copy content Toggle raw display
$89$ \( (T^{2} + 14256)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 62 T^{3} + 3844 T^{2} + \cdots + 14776336)^{2} \) Copy content Toggle raw display
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