Properties

Label 363.2.e.m
Level $363$
Weight $2$
Character orbit 363.e
Analytic conductor $2.899$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.e (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.89856959337\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.324000000.3
Defining polynomial: \( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} - \beta_{6} q^{3} + \beta_{4} q^{4} + (3 \beta_{6} + 3 \beta_{4} + 3 \beta_{2} + 3) q^{5} - \beta_{3} q^{6} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{7} - \beta_1 q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} - \beta_{6} q^{3} + \beta_{4} q^{4} + (3 \beta_{6} + 3 \beta_{4} + 3 \beta_{2} + 3) q^{5} - \beta_{3} q^{6} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{7} - \beta_1 q^{8} + \beta_{2} q^{9} - 3 \beta_{5} q^{10} - q^{12} - \beta_{7} q^{13} - 6 \beta_{6} q^{14} + 3 \beta_{4} q^{15} + (5 \beta_{6} + 5 \beta_{4} + 5 \beta_{2} + 5) q^{16} + \beta_{3} q^{17} + ( - \beta_{7} - \beta_{5} - \beta_{3} - \beta_1) q^{18} + 4 \beta_1 q^{19} - 3 \beta_{2} q^{20} + 2 \beta_{5} q^{21} - 6 q^{23} + \beta_{7} q^{24} + 4 \beta_{6} q^{25} - 3 \beta_{4} q^{26} + (\beta_{6} + \beta_{4} + \beta_{2} + 1) q^{27} - 2 \beta_{3} q^{28} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{29} + 3 \beta_1 q^{30} + 4 \beta_{2} q^{31} - 3 \beta_{5} q^{32} + 3 q^{34} + 6 \beta_{7} q^{35} + \beta_{6} q^{36} - 11 \beta_{4} q^{37} + ( - 12 \beta_{6} - 12 \beta_{4} - 12 \beta_{2} - 12) q^{38} + \beta_{3} q^{39} + ( - 3 \beta_{7} - 3 \beta_{5} - 3 \beta_{3} - 3 \beta_1) q^{40} - \beta_1 q^{41} + 6 \beta_{2} q^{42} + 2 \beta_{5} q^{43} - 3 q^{45} - 6 \beta_{7} q^{46} + 5 \beta_{4} q^{48} + ( - 5 \beta_{6} - 5 \beta_{4} - 5 \beta_{2} - 5) q^{49} + 4 \beta_{3} q^{50} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{51} - \beta_1 q^{52} - 9 \beta_{2} q^{53} - \beta_{5} q^{54} + 6 q^{56} - 4 \beta_{7} q^{57} - 3 \beta_{6} q^{58} - 6 \beta_{4} q^{59} + ( - 3 \beta_{6} - 3 \beta_{4} - 3 \beta_{2} - 3) q^{60} + ( - 4 \beta_{7} - 4 \beta_{5} - 4 \beta_{3} - 4 \beta_1) q^{62} - 2 \beta_1 q^{63} + \beta_{2} q^{64} + 3 \beta_{5} q^{65} - 2 q^{67} + \beta_{7} q^{68} + 6 \beta_{6} q^{69} + 18 \beta_{4} q^{70} + (6 \beta_{6} + 6 \beta_{4} + 6 \beta_{2} + 6) q^{71} - \beta_{3} q^{72} + ( - 4 \beta_{7} - 4 \beta_{5} - 4 \beta_{3} - 4 \beta_1) q^{73} - 11 \beta_1 q^{74} - 4 \beta_{2} q^{75} + 4 \beta_{5} q^{76} + 3 q^{78} + 15 \beta_{6} q^{80} + \beta_{4} q^{81} + (3 \beta_{6} + 3 \beta_{4} + 3 \beta_{2} + 3) q^{82} + ( - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{84} - 3 \beta_1 q^{85} + 6 \beta_{2} q^{86} + \beta_{5} q^{87} + 9 q^{89} - 3 \beta_{7} q^{90} + 6 \beta_{6} q^{91} - 6 \beta_{4} q^{92} + (4 \beta_{6} + 4 \beta_{4} + 4 \beta_{2} + 4) q^{93} + (12 \beta_{7} + 12 \beta_{5} + 12 \beta_{3} + 12 \beta_1) q^{95} + 3 \beta_1 q^{96} - 7 \beta_{2} q^{97} + 5 \beta_{5} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 2 q^{4} + 6 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 2 q^{4} + 6 q^{5} - 2 q^{9} - 8 q^{12} + 12 q^{14} - 6 q^{15} + 10 q^{16} + 6 q^{20} - 48 q^{23} - 8 q^{25} + 6 q^{26} + 2 q^{27} - 8 q^{31} + 24 q^{34} - 2 q^{36} + 22 q^{37} - 24 q^{38} - 12 q^{42} - 24 q^{45} - 10 q^{48} - 10 q^{49} + 18 q^{53} + 48 q^{56} + 6 q^{58} + 12 q^{59} - 6 q^{60} - 2 q^{64} - 16 q^{67} - 12 q^{69} - 36 q^{70} + 12 q^{71} + 8 q^{75} + 24 q^{78} - 30 q^{80} - 2 q^{81} + 6 q^{82} - 12 q^{86} + 72 q^{89} - 12 q^{91} + 12 q^{92} + 8 q^{93} + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 27\beta_{7} \) Copy content Toggle raw display

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(1\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1
0.535233 + 1.64728i
−0.535233 1.64728i
1.40126 1.01807i
−1.40126 + 1.01807i
1.40126 + 1.01807i
−1.40126 1.01807i
0.535233 1.64728i
−0.535233 + 1.64728i
−1.40126 + 1.01807i −0.309017 0.951057i 0.309017 0.951057i 2.42705 + 1.76336i 1.40126 + 1.01807i −1.07047 + 3.29456i −0.535233 1.64728i −0.809017 + 0.587785i −5.19615
124.2 1.40126 1.01807i −0.309017 0.951057i 0.309017 0.951057i 2.42705 + 1.76336i −1.40126 1.01807i 1.07047 3.29456i 0.535233 + 1.64728i −0.809017 + 0.587785i 5.19615
130.1 −0.535233 + 1.64728i 0.809017 0.587785i −0.809017 0.587785i −0.927051 2.85317i 0.535233 + 1.64728i −2.80252 2.03615i −1.40126 + 1.01807i 0.309017 0.951057i 5.19615
130.2 0.535233 1.64728i 0.809017 0.587785i −0.809017 0.587785i −0.927051 2.85317i −0.535233 1.64728i 2.80252 + 2.03615i 1.40126 1.01807i 0.309017 0.951057i −5.19615
148.1 −0.535233 1.64728i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.927051 + 2.85317i 0.535233 1.64728i −2.80252 + 2.03615i −1.40126 1.01807i 0.309017 + 0.951057i 5.19615
148.2 0.535233 + 1.64728i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.927051 + 2.85317i −0.535233 + 1.64728i 2.80252 2.03615i 1.40126 + 1.01807i 0.309017 + 0.951057i −5.19615
202.1 −1.40126 1.01807i −0.309017 + 0.951057i 0.309017 + 0.951057i 2.42705 1.76336i 1.40126 1.01807i −1.07047 3.29456i −0.535233 + 1.64728i −0.809017 0.587785i −5.19615
202.2 1.40126 + 1.01807i −0.309017 + 0.951057i 0.309017 + 0.951057i 2.42705 1.76336i −1.40126 + 1.01807i 1.07047 + 3.29456i 0.535233 1.64728i −0.809017 0.587785i 5.19615
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 202.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
11.c even 5 3 inner
11.d 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.e.m 8
11.b odd 2 1 inner 363.2.e.m 8
11.c even 5 1 363.2.a.f 2
11.c even 5 3 inner 363.2.e.m 8
11.d odd 10 1 363.2.a.f 2
11.d odd 10 3 inner 363.2.e.m 8
33.f even 10 1 1089.2.a.o 2
33.h odd 10 1 1089.2.a.o 2
44.g even 10 1 5808.2.a.ca 2
44.h odd 10 1 5808.2.a.ca 2
55.h odd 10 1 9075.2.a.bo 2
55.j even 10 1 9075.2.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.f 2 11.c even 5 1
363.2.a.f 2 11.d odd 10 1
363.2.e.m 8 1.a even 1 1 trivial
363.2.e.m 8 11.b odd 2 1 inner
363.2.e.m 8 11.c even 5 3 inner
363.2.e.m 8 11.d odd 10 3 inner
1089.2.a.o 2 33.f even 10 1
1089.2.a.o 2 33.h odd 10 1
5808.2.a.ca 2 44.g even 10 1
5808.2.a.ca 2 44.h odd 10 1
9075.2.a.bo 2 55.h odd 10 1
9075.2.a.bo 2 55.j even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 3T_{2}^{6} + 9T_{2}^{4} + 27T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 12 T^{6} + 144 T^{4} + \cdots + 20736 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{8} + 48 T^{6} + 2304 T^{4} + \cdots + 5308416 \) Copy content Toggle raw display
$23$ \( (T + 6)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 11 T^{3} + 121 T^{2} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} - 9 T^{3} + 81 T^{2} - 729 T + 6561)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 6 T^{3} + 36 T^{2} - 216 T + 1296)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T + 2)^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} - 6 T^{3} + 36 T^{2} - 216 T + 1296)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 48 T^{6} + 2304 T^{4} + \cdots + 5308416 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T - 9)^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} - 7 T^{3} + 49 T^{2} - 343 T + 2401)^{2} \) Copy content Toggle raw display
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