Properties

Label 363.2.f.c
Level 363
Weight 2
Character orbit 363.f
Analytic conductor 2.899
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: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 363.f (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.89856959337\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.228765625.1
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} -2 \beta_{7} q^{4} + ( -\beta_{4} - 2 \beta_{5} ) q^{5} + ( 2 + \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} -2 \beta_{7} q^{4} + ( -\beta_{4} - 2 \beta_{5} ) q^{5} + ( 2 + \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{9} + ( -2 + 2 \beta_{6} ) q^{12} + ( -\beta_{2} + 5 \beta_{7} ) q^{15} -4 \beta_{4} q^{16} + ( -2 + 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{20} + ( -1 + 2 \beta_{6} ) q^{23} + 6 \beta_{3} q^{25} + ( 3 \beta_{4} - 2 \beta_{5} ) q^{27} + ( -5 + 5 \beta_{3} + 5 \beta_{4} - 5 \beta_{7} ) q^{31} + ( 2 \beta_{1} - 6 \beta_{3} ) q^{36} -7 \beta_{7} q^{37} + ( 8 - 5 \beta_{6} ) q^{45} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{47} + ( 4 \beta_{2} + 4 \beta_{7} ) q^{48} -7 \beta_{4} q^{49} + ( 4 - 8 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} - 8 \beta_{6} + 4 \beta_{7} ) q^{53} + ( -2 \beta_{2} - \beta_{7} ) q^{59} + ( 12 \beta_{4} + 2 \beta_{5} ) q^{60} + ( 8 - 8 \beta_{3} - 8 \beta_{4} + 8 \beta_{7} ) q^{64} -13 q^{67} + ( \beta_{1} - 6 \beta_{3} ) q^{69} + ( 5 \beta_{4} + 10 \beta_{5} ) q^{71} + ( 6 - 6 \beta_{1} + 6 \beta_{2} + 6 \beta_{5} - 6 \beta_{6} + 6 \beta_{7} ) q^{75} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{80} + ( -5 \beta_{2} + \beta_{7} ) q^{81} + ( 5 - 10 \beta_{6} ) q^{89} + ( 4 \beta_{2} + 2 \beta_{7} ) q^{92} + 5 \beta_{5} q^{93} + ( -17 + 17 \beta_{3} + 17 \beta_{4} - 17 \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - q^{3} + 4q^{4} + 5q^{9} + O(q^{10}) \) \( 8q - q^{3} + 4q^{4} + 5q^{9} - 8q^{12} - 11q^{15} - 8q^{16} + 12q^{25} + 8q^{27} - 10q^{31} - 10q^{36} + 14q^{37} + 44q^{45} - 4q^{48} - 14q^{49} + 22q^{60} + 16q^{64} - 104q^{67} - 11q^{69} + 6q^{75} - 7q^{81} - 5q^{93} - 34q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 2 x^{6} + 5 x^{5} + x^{4} + 15 x^{3} - 18 x^{2} - 27 x + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} + 2 \nu^{5} - 5 \nu^{4} - \nu^{3} - 15 \nu^{2} + 18 \nu + 27 \)\()/27\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 16 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{7} - 4 \nu^{6} + 10 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} - 36 \nu^{2} - 54 \nu + 162 \)\()/27\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} - 2 \nu^{6} + 5 \nu^{5} + \nu^{4} + 2 \nu^{3} - 18 \nu^{2} - 27 \nu + 81 \)\()/9\)
\(\beta_{6}\)\(=\)\( \nu^{5} + 16 \)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + 5 \nu^{6} + 10 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} - 75 \nu^{2} + 90 \nu + 135 \)\()/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{7} + \beta_{6} - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - \beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(2 \beta_{5} - 3 \beta_{4}\)
\(\nu^{4}\)\(=\)\(\beta_{7} - 5 \beta_{2}\)
\(\nu^{5}\)\(=\)\(\beta_{6} - 16\)
\(\nu^{6}\)\(=\)\(3 \beta_{3} - 16 \beta_{1}\)
\(\nu^{7}\)\(=\)\(-35 \beta_{7} - 13 \beta_{6} + 13 \beta_{5} + 48 \beta_{4} + 48 \beta_{3} + 13 \beta_{2} - 13 \beta_{1} - 35\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
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 −1.42264 0.987975i −0.618034 1.90211i −1.94946 2.68321i 0 0 0 1.04781 + 2.81107i 0
161.2 0 1.73166 + 0.0369185i −0.618034 1.90211i 1.94946 + 2.68321i 0 0 0 2.99727 + 0.127860i 0
215.1 0 −1.37924 + 1.04771i 1.61803 1.17557i 3.15430 + 1.02489i 0 0 0 0.804606 2.89009i 0
215.2 0 0.570223 1.63550i 1.61803 1.17557i −3.15430 1.02489i 0 0 0 −2.34969 1.86519i 0
233.1 0 −1.37924 1.04771i 1.61803 + 1.17557i 3.15430 1.02489i 0 0 0 0.804606 + 2.89009i 0
233.2 0 0.570223 + 1.63550i 1.61803 + 1.17557i −3.15430 + 1.02489i 0 0 0 −2.34969 + 1.86519i 0
239.1 0 −1.42264 + 0.987975i −0.618034 + 1.90211i −1.94946 + 2.68321i 0 0 0 1.04781 2.81107i 0
239.2 0 1.73166 0.0369185i −0.618034 + 1.90211i 1.94946 2.68321i 0 0 0 2.99727 0.127860i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.2
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 363.2.f.c 8
3.b odd 2 1 inner 363.2.f.c 8
11.b odd 2 1 CM 363.2.f.c 8
11.c even 5 1 33.2.d.a 2
11.c even 5 3 inner 363.2.f.c 8
11.d odd 10 1 33.2.d.a 2
11.d odd 10 3 inner 363.2.f.c 8
33.d even 2 1 inner 363.2.f.c 8
33.f even 10 1 33.2.d.a 2
33.f even 10 3 inner 363.2.f.c 8
33.h odd 10 1 33.2.d.a 2
33.h odd 10 3 inner 363.2.f.c 8
44.g even 10 1 528.2.b.a 2
44.h odd 10 1 528.2.b.a 2
55.h odd 10 1 825.2.f.a 2
55.j even 10 1 825.2.f.a 2
55.k odd 20 2 825.2.d.a 4
55.l even 20 2 825.2.d.a 4
88.k even 10 1 2112.2.b.f 2
88.l odd 10 1 2112.2.b.f 2
88.o even 10 1 2112.2.b.e 2
88.p odd 10 1 2112.2.b.e 2
99.m even 15 2 891.2.g.a 4
99.n odd 30 2 891.2.g.a 4
99.o odd 30 2 891.2.g.a 4
99.p even 30 2 891.2.g.a 4
132.n odd 10 1 528.2.b.a 2
132.o even 10 1 528.2.b.a 2
165.o odd 10 1 825.2.f.a 2
165.r even 10 1 825.2.f.a 2
165.u odd 20 2 825.2.d.a 4
165.v even 20 2 825.2.d.a 4
264.r odd 10 1 2112.2.b.f 2
264.t odd 10 1 2112.2.b.e 2
264.u even 10 1 2112.2.b.e 2
264.w even 10 1 2112.2.b.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.d.a 2 11.c even 5 1
33.2.d.a 2 11.d odd 10 1
33.2.d.a 2 33.f even 10 1
33.2.d.a 2 33.h odd 10 1
363.2.f.c 8 1.a even 1 1 trivial
363.2.f.c 8 3.b odd 2 1 inner
363.2.f.c 8 11.b odd 2 1 CM
363.2.f.c 8 11.c even 5 3 inner
363.2.f.c 8 11.d odd 10 3 inner
363.2.f.c 8 33.d even 2 1 inner
363.2.f.c 8 33.f even 10 3 inner
363.2.f.c 8 33.h odd 10 3 inner
528.2.b.a 2 44.g even 10 1
528.2.b.a 2 44.h odd 10 1
528.2.b.a 2 132.n odd 10 1
528.2.b.a 2 132.o even 10 1
825.2.d.a 4 55.k odd 20 2
825.2.d.a 4 55.l even 20 2
825.2.d.a 4 165.u odd 20 2
825.2.d.a 4 165.v even 20 2
825.2.f.a 2 55.h odd 10 1
825.2.f.a 2 55.j even 10 1
825.2.f.a 2 165.o odd 10 1
825.2.f.a 2 165.r even 10 1
891.2.g.a 4 99.m even 15 2
891.2.g.a 4 99.n odd 30 2
891.2.g.a 4 99.o odd 30 2
891.2.g.a 4 99.p even 30 2
2112.2.b.e 2 88.o even 10 1
2112.2.b.e 2 88.p odd 10 1
2112.2.b.e 2 264.t odd 10 1
2112.2.b.e 2 264.u even 10 1
2112.2.b.f 2 88.k even 10 1
2112.2.b.f 2 88.l odd 10 1
2112.2.b.f 2 264.r odd 10 1
2112.2.b.f 2 264.w even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\):

\( T_{2} \)
\( T_{5}^{8} - 11 T_{5}^{6} + 121 T_{5}^{4} - 1331 T_{5}^{2} + 14641 \)
\( T_{7} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} + 4 T^{4} - 8 T^{6} + 16 T^{8} )^{2} \)
$3$ \( 1 + T - 2 T^{2} - 5 T^{3} + T^{4} - 15 T^{5} - 18 T^{6} + 27 T^{7} + 81 T^{8} \)
$5$ \( ( 1 - 3 T + 4 T^{2} + 3 T^{3} - 29 T^{4} + 15 T^{5} + 100 T^{6} - 375 T^{7} + 625 T^{8} )( 1 + 3 T + 4 T^{2} - 3 T^{3} - 29 T^{4} - 15 T^{5} + 100 T^{6} + 375 T^{7} + 625 T^{8} ) \)
$7$ \( ( 1 + 7 T^{2} + 49 T^{4} + 343 T^{6} + 2401 T^{8} )^{2} \)
$11$ \( \)
$13$ \( ( 1 + 13 T^{2} + 169 T^{4} + 2197 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 17 T^{2} + 289 T^{4} - 4913 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 + 19 T^{2} + 361 T^{4} + 6859 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 9 T + 23 T^{2} )^{4}( 1 + 9 T + 23 T^{2} )^{4} \)
$29$ \( ( 1 - 29 T^{2} + 841 T^{4} - 24389 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 + 5 T - 6 T^{2} - 185 T^{3} - 739 T^{4} - 5735 T^{5} - 5766 T^{6} + 148955 T^{7} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 - 7 T + 12 T^{2} + 175 T^{3} - 1669 T^{4} + 6475 T^{5} + 16428 T^{6} - 354571 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 - 41 T^{2} + 1681 T^{4} - 68921 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 43 T^{2} )^{8} \)
$47$ \( ( 1 - 12 T + 97 T^{2} - 600 T^{3} + 2641 T^{4} - 28200 T^{5} + 214273 T^{6} - 1245876 T^{7} + 4879681 T^{8} )( 1 + 12 T + 97 T^{2} + 600 T^{3} + 2641 T^{4} + 28200 T^{5} + 214273 T^{6} + 1245876 T^{7} + 4879681 T^{8} ) \)
$53$ \( ( 1 - 6 T - 17 T^{2} + 420 T^{3} - 1619 T^{4} + 22260 T^{5} - 47753 T^{6} - 893262 T^{7} + 7890481 T^{8} )( 1 + 6 T - 17 T^{2} - 420 T^{3} - 1619 T^{4} - 22260 T^{5} - 47753 T^{6} + 893262 T^{7} + 7890481 T^{8} ) \)
$59$ \( ( 1 - 15 T + 166 T^{2} - 1605 T^{3} + 14281 T^{4} - 94695 T^{5} + 577846 T^{6} - 3080685 T^{7} + 12117361 T^{8} )( 1 + 15 T + 166 T^{2} + 1605 T^{3} + 14281 T^{4} + 94695 T^{5} + 577846 T^{6} + 3080685 T^{7} + 12117361 T^{8} ) \)
$61$ \( ( 1 + 61 T^{2} + 3721 T^{4} + 226981 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 + 13 T + 67 T^{2} )^{8} \)
$71$ \( ( 1 - 3 T - 62 T^{2} + 399 T^{3} + 3205 T^{4} + 28329 T^{5} - 312542 T^{6} - 1073733 T^{7} + 25411681 T^{8} )( 1 + 3 T - 62 T^{2} - 399 T^{3} + 3205 T^{4} - 28329 T^{5} - 312542 T^{6} + 1073733 T^{7} + 25411681 T^{8} ) \)
$73$ \( ( 1 + 73 T^{2} + 5329 T^{4} + 389017 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 + 79 T^{2} + 6241 T^{4} + 493039 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 - 83 T^{2} + 6889 T^{4} - 571787 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 9 T + 89 T^{2} )^{4}( 1 + 9 T + 89 T^{2} )^{4} \)
$97$ \( ( 1 + 17 T + 192 T^{2} + 1615 T^{3} + 8831 T^{4} + 156655 T^{5} + 1806528 T^{6} + 15515441 T^{7} + 88529281 T^{8} )^{2} \)
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