Properties

Label 363.3.h.f
Level $363$
Weight $3$
Character orbit 363.h
Analytic conductor $9.891$
Analytic rank $0$
Dimension $8$
CM discriminant -11
Inner twists $16$

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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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{3} - \beta_1) q^{3} + 4 \beta_{7} q^{4} + ( - 6 \beta_{5} - 3 \beta_{4}) q^{5} + ( - 6 \beta_{7} + 5 \beta_{6} - 5 \beta_{5} + \beta_{4} + \beta_{3} - 5 \beta_{2} + 5 \beta_1 - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{3} - \beta_1) q^{3} + 4 \beta_{7} q^{4} + ( - 6 \beta_{5} - 3 \beta_{4}) q^{5} + ( - 6 \beta_{7} + 5 \beta_{6} - 5 \beta_{5} + \beta_{4} + \beta_{3} - 5 \beta_{2} + 5 \beta_1 - 6) q^{9} + ( - 4 \beta_{6} + 12) q^{12} + (9 \beta_{7} - 15 \beta_{2}) q^{15} - 16 \beta_{4} q^{16} + (12 \beta_{7} - 24 \beta_{6} + 24 \beta_{5} + 12 \beta_{4} + 12 \beta_{3} + \cdots + 12) q^{20}+ \cdots + ( - 95 \beta_{7} + 95 \beta_{4} + 95 \beta_{3} - 95) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5 q^{3} - 8 q^{4} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 5 q^{3} - 8 q^{4} - 7 q^{9} + 80 q^{12} - 33 q^{15} - 32 q^{16} + 148 q^{25} + 10 q^{27} - 74 q^{31} - 28 q^{36} - 50 q^{37} + 660 q^{45} - 80 q^{48} + 98 q^{49} - 132 q^{60} - 128 q^{64} - 280 q^{67} - 99 q^{69} + 370 q^{75} + 113 q^{81} - 185 q^{93} - 190 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + \nu^{6} + 2\nu^{5} - 5\nu^{4} - \nu^{3} - 15\nu^{2} + 18\nu + 27 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 16\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{7} - 4\nu^{6} + 10\nu^{5} + 2\nu^{4} - 5\nu^{3} - 36\nu^{2} - 54\nu + 162 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} + 5\nu^{5} + \nu^{4} + 2\nu^{3} - 18\nu^{2} - 27\nu + 81 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{5} + 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 5\nu^{6} + 10\nu^{5} + 2\nu^{4} - 5\nu^{3} - 75\nu^{2} + 90\nu + 135 ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{7} + \beta_{6} - \beta_{5} - 3\beta_{4} - 3\beta_{3} - \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} - 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{3} - 16\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -35\beta_{7} - 13\beta_{6} + 13\beta_{5} + 48\beta_{4} + 48\beta_{3} + 13\beta_{2} - 13\beta _1 - 35 \) Copy content Toggle raw display

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(\beta_{7}\)

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
1.37924 + 1.04771i
−0.570223 1.63550i
1.37924 1.04771i
−0.570223 + 1.63550i
1.42264 + 0.987975i
−1.73166 0.0369185i
0 −0.804606 + 2.89009i 1.23607 3.80423i −5.84839 + 8.04962i 0 0 0 −7.70522 4.65077i 0
245.2 0 2.34969 + 1.86519i 1.23607 3.80423i 5.84839 8.04962i 0 0 0 2.04210 + 8.76526i 0
251.1 0 −2.99727 + 0.127860i −3.23607 2.35114i 9.46289 3.07468i 0 0 0 8.96730 0.766464i 0
251.2 0 −1.04781 + 2.81107i −3.23607 2.35114i −9.46289 + 3.07468i 0 0 0 −6.80418 5.89093i 0
269.1 0 −2.99727 0.127860i −3.23607 + 2.35114i 9.46289 + 3.07468i 0 0 0 8.96730 + 0.766464i 0
269.2 0 −1.04781 2.81107i −3.23607 + 2.35114i −9.46289 3.07468i 0 0 0 −6.80418 + 5.89093i 0
323.1 0 −0.804606 2.89009i 1.23607 + 3.80423i −5.84839 8.04962i 0 0 0 −7.70522 + 4.65077i 0
323.2 0 2.34969 1.86519i 1.23607 + 3.80423i 5.84839 + 8.04962i 0 0 0 2.04210 8.76526i 0
\(n\): 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
3.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
33.d even 2 1 inner
33.f even 10 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.f 8
3.b odd 2 1 inner 363.3.h.f 8
11.b odd 2 1 CM 363.3.h.f 8
11.c even 5 1 363.3.b.c 2
11.c even 5 3 inner 363.3.h.f 8
11.d odd 10 1 363.3.b.c 2
11.d odd 10 3 inner 363.3.h.f 8
33.d even 2 1 inner 363.3.h.f 8
33.f even 10 1 363.3.b.c 2
33.f even 10 3 inner 363.3.h.f 8
33.h odd 10 1 363.3.b.c 2
33.h odd 10 3 inner 363.3.h.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.b.c 2 11.c even 5 1
363.3.b.c 2 11.d odd 10 1
363.3.b.c 2 33.f even 10 1
363.3.b.c 2 33.h odd 10 1
363.3.h.f 8 1.a even 1 1 trivial
363.3.h.f 8 3.b odd 2 1 inner
363.3.h.f 8 11.b odd 2 1 CM
363.3.h.f 8 11.c even 5 3 inner
363.3.h.f 8 11.d odd 10 3 inner
363.3.h.f 8 33.d even 2 1 inner
363.3.h.f 8 33.f even 10 3 inner
363.3.h.f 8 33.h odd 10 3 inner

Hecke kernels

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

\( T_{2} \) Copy content Toggle raw display
\( T_{5}^{8} - 99T_{5}^{6} + 9801T_{5}^{4} - 970299T_{5}^{2} + 96059601 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 5 T^{7} + 16 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} - 99 T^{6} + 9801 T^{4} + \cdots + 96059601 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{2} + 891)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} + 37 T^{3} + 1369 T^{2} + \cdots + 1874161)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 25 T^{3} + 625 T^{2} + \cdots + 390625)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} - 6336 T^{6} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{8} - 6336 T^{6} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{8} - 2475 T^{6} + \cdots + 37523281640625 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T + 35)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} - 2475 T^{6} + \cdots + 37523281640625 \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{2} + 22275)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 95 T^{3} + 9025 T^{2} + \cdots + 81450625)^{2} \) Copy content Toggle raw display
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