Properties

Label 363.2.f.a
Level $363$
Weight $2$
Character orbit 363.f
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.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: 8.0.64000000.1
Defining polynomial: \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + (\beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{3} + \beta_{2} q^{4} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{5} + (\beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{6} + \beta_{7} q^{7} + ( - 3 \beta_{6} + 3 \beta_{4} - 3 \beta_{2} + 3) q^{8} + (\beta_{6} + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + (\beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{3} + \beta_{2} q^{4} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{5} + (\beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{6} + \beta_{7} q^{7} + ( - 3 \beta_{6} + 3 \beta_{4} - 3 \beta_{2} + 3) q^{8} + (\beta_{6} + 2 \beta_1) q^{9} + 2 \beta_{5} q^{10} + ( - \beta_{5} - 1) q^{12} + 3 \beta_1 q^{13} + \beta_{3} q^{14} + ( - 2 \beta_{7} + 4 \beta_{2}) q^{15} - \beta_{4} q^{16} + 6 \beta_{4} q^{17} + ( - 2 \beta_{7} + \beta_{2}) q^{18} + 3 \beta_{3} q^{19} - 2 \beta_1 q^{20} + ( - \beta_{5} + 2) q^{21} + (3 \beta_{6} - 3 \beta_1) q^{24} + ( - 3 \beta_{6} + 3 \beta_{4} - 3 \beta_{2} + 3) q^{25} - 3 \beta_{7} q^{26} + (\beta_{7} - \beta_{5} - 5 \beta_{4} + \beta_{3} - \beta_1) q^{27} + (\beta_{7} - \beta_{5} + \beta_{3} - \beta_1) q^{28} + 2 \beta_{2} q^{29} + ( - 4 \beta_{6} + 4 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 4) q^{30} + 2 \beta_{6} q^{31} + 5 q^{32} + 6 q^{34} - 4 \beta_{6} q^{35} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + \beta_{2} - 1) q^{36} - 8 \beta_{2} q^{37} + ( - 3 \beta_{7} + 3 \beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{38} + (3 \beta_{7} - 3 \beta_{5} - 6 \beta_{4} + 3 \beta_{3} - 3 \beta_1) q^{39} + 6 \beta_{7} q^{40} + (6 \beta_{6} - 6 \beta_{4} + 6 \beta_{2} - 6) q^{41} + ( - 2 \beta_{6} - \beta_1) q^{42} - 3 \beta_{5} q^{43} + ( - 2 \beta_{5} - 8) q^{45} - 2 \beta_{3} q^{47} + (\beta_{7} + \beta_{2}) q^{48} + 5 \beta_{4} q^{49} - 3 \beta_{4} q^{50} + ( - 6 \beta_{7} - 6 \beta_{2}) q^{51} + 3 \beta_{3} q^{52} + 4 \beta_1 q^{53} + (\beta_{5} - 5) q^{54} + 3 \beta_{5} q^{56} + ( - 6 \beta_{6} - 3 \beta_1) q^{57} + ( - 2 \beta_{6} + 2 \beta_{4} - 2 \beta_{2} + 2) q^{58} + 8 \beta_{7} q^{59} + ( - 2 \beta_{7} + 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{60} + ( - 7 \beta_{7} + 7 \beta_{5} - 7 \beta_{3} + 7 \beta_1) q^{61} + 2 \beta_{2} q^{62} + (4 \beta_{6} - 4 \beta_{4} - \beta_{3} + 4 \beta_{2} - 4) q^{63} - 7 \beta_{6} q^{64} - 12 q^{65} + 2 q^{67} + 6 \beta_{6} q^{68} - 4 \beta_{2} q^{70} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{71} + ( - 6 \beta_{7} + 6 \beta_{5} + 3 \beta_{4} - 6 \beta_{3} + 6 \beta_1) q^{72} + \beta_{7} q^{73} + (8 \beta_{6} - 8 \beta_{4} + 8 \beta_{2} - 8) q^{74} + (3 \beta_{6} - 3 \beta_1) q^{75} + 3 \beta_{5} q^{76} + (3 \beta_{5} - 6) q^{78} + 3 \beta_1 q^{79} + 2 \beta_{3} q^{80} + (4 \beta_{7} + 7 \beta_{2}) q^{81} + 6 \beta_{4} q^{82} - 16 \beta_{4} q^{83} + ( - \beta_{7} + 2 \beta_{2}) q^{84} - 12 \beta_{3} q^{85} - 3 \beta_1 q^{86} + ( - 2 \beta_{5} - 2) q^{87} + (8 \beta_{6} - 2 \beta_1) q^{90} + (6 \beta_{6} - 6 \beta_{4} + 6 \beta_{2} - 6) q^{91} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{93} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{94} - 12 \beta_{2} q^{95} + (5 \beta_{6} - 5 \beta_{4} - 5 \beta_{3} + 5 \beta_{2} - 5) q^{96} + 2 \beta_{6} q^{97} + 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{8} + 2 q^{9} - 8 q^{12} + 8 q^{15} + 2 q^{16} - 12 q^{17} + 2 q^{18} + 16 q^{21} + 6 q^{24} + 6 q^{25} + 10 q^{27} + 4 q^{29} + 8 q^{30} + 4 q^{31} + 40 q^{32} + 48 q^{34} - 8 q^{35} - 2 q^{36} - 16 q^{37} + 12 q^{39} - 12 q^{41} - 4 q^{42} - 64 q^{45} + 2 q^{48} - 10 q^{49} + 6 q^{50} - 12 q^{51} - 40 q^{54} - 12 q^{57} + 4 q^{58} - 8 q^{60} + 4 q^{62} - 8 q^{63} - 14 q^{64} - 96 q^{65} + 16 q^{67} + 12 q^{68} - 8 q^{70} - 6 q^{72} - 16 q^{74} + 6 q^{75} - 48 q^{78} + 14 q^{81} - 12 q^{82} + 32 q^{83} + 4 q^{84} - 16 q^{87} + 16 q^{90} - 12 q^{91} + 4 q^{93} - 24 q^{95} - 10 q^{96} + 4 q^{97} + 40 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−0.831254 + 1.14412i
0.831254 1.14412i
1.34500 0.437016i
−1.34500 + 0.437016i
1.34500 + 0.437016i
−1.34500 0.437016i
−0.831254 1.14412i
0.831254 + 1.14412i
−0.809017 0.587785i −1.03598 1.38807i −0.309017 0.951057i 1.66251 + 2.28825i 0.0222369 + 1.73191i −1.34500 + 0.437016i −0.927051 + 2.85317i −0.853491 + 2.87603i 2.82843i
161.2 −0.809017 0.587785i 1.65401 0.514040i −0.309017 0.951057i −1.66251 2.28825i −1.64027 0.556338i 1.34500 0.437016i −0.927051 + 2.85317i 2.47152 1.70046i 2.82843i
215.1 0.309017 + 0.951057i −1.64027 + 0.556338i 0.809017 0.587785i −2.68999 0.874032i −1.03598 1.38807i −0.831254 1.14412i 2.42705 + 1.76336i 2.38098 1.82509i 2.82843i
215.2 0.309017 + 0.951057i 0.0222369 1.73191i 0.809017 0.587785i 2.68999 + 0.874032i 1.65401 0.514040i 0.831254 + 1.14412i 2.42705 + 1.76336i −2.99901 0.0770245i 2.82843i
233.1 0.309017 0.951057i −1.64027 0.556338i 0.809017 + 0.587785i −2.68999 + 0.874032i −1.03598 + 1.38807i −0.831254 + 1.14412i 2.42705 1.76336i 2.38098 + 1.82509i 2.82843i
233.2 0.309017 0.951057i 0.0222369 + 1.73191i 0.809017 + 0.587785i 2.68999 0.874032i 1.65401 + 0.514040i 0.831254 1.14412i 2.42705 1.76336i −2.99901 + 0.0770245i 2.82843i
239.1 −0.809017 + 0.587785i −1.03598 + 1.38807i −0.309017 + 0.951057i 1.66251 2.28825i 0.0222369 1.73191i −1.34500 0.437016i −0.927051 2.85317i −0.853491 2.87603i 2.82843i
239.2 −0.809017 + 0.587785i 1.65401 + 0.514040i −0.309017 + 0.951057i −1.66251 + 2.28825i −1.64027 + 0.556338i 1.34500 + 0.437016i −0.927051 2.85317i 2.47152 + 1.70046i 2.82843i
\(n\): 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.c even 5 3 inner
33.d even 2 1 inner
33.f even 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.a 8
3.b odd 2 1 363.2.f.f 8
11.b odd 2 1 363.2.f.f 8
11.c even 5 1 363.2.d.b yes 2
11.c even 5 3 inner 363.2.f.a 8
11.d odd 10 1 363.2.d.a 2
11.d odd 10 3 363.2.f.f 8
33.d even 2 1 inner 363.2.f.a 8
33.f even 10 1 363.2.d.b yes 2
33.f even 10 3 inner 363.2.f.a 8
33.h odd 10 1 363.2.d.a 2
33.h odd 10 3 363.2.f.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.d.a 2 11.d odd 10 1
363.2.d.a 2 33.h odd 10 1
363.2.d.b yes 2 11.c even 5 1
363.2.d.b yes 2 33.f even 10 1
363.2.f.a 8 1.a even 1 1 trivial
363.2.f.a 8 11.c even 5 3 inner
363.2.f.a 8 33.d even 2 1 inner
363.2.f.a 8 33.f even 10 3 inner
363.2.f.f 8 3.b odd 2 1
363.2.f.f 8 11.b odd 2 1
363.2.f.f 8 11.d odd 10 3
363.2.f.f 8 33.h odd 10 3

Hecke kernels

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

\( T_{2}^{4} + T_{2}^{3} + T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{8} - 8T_{5}^{6} + 64T_{5}^{4} - 512T_{5}^{2} + 4096 \) Copy content Toggle raw display
\( T_{7}^{8} - 2T_{7}^{6} + 4T_{7}^{4} - 8T_{7}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + T^{6} - 4 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} - 8 T^{6} + 64 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 18 T^{6} + 324 T^{4} + \cdots + 104976 \) Copy content Toggle raw display
$17$ \( (T^{4} + 6 T^{3} + 36 T^{2} + 216 T + 1296)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 18 T^{6} + 324 T^{4} + \cdots + 104976 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 8 T^{3} + 64 T^{2} + 512 T + 4096)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 6 T^{3} + 36 T^{2} + 216 T + 1296)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} - 8 T^{6} + 64 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{8} - 32 T^{6} + 1024 T^{4} + \cdots + 1048576 \) Copy content Toggle raw display
$59$ \( T^{8} - 128 T^{6} + \cdots + 268435456 \) Copy content Toggle raw display
$61$ \( T^{8} - 98 T^{6} + 9604 T^{4} + \cdots + 92236816 \) Copy content Toggle raw display
$67$ \( (T - 2)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} - 8 T^{6} + 64 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$73$ \( T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$79$ \( T^{8} - 18 T^{6} + 324 T^{4} + \cdots + 104976 \) Copy content Toggle raw display
$83$ \( (T^{4} - 16 T^{3} + 256 T^{2} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16)^{2} \) Copy content Toggle raw display
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