Properties

Label 363.2.e.i
Level $363$
Weight $2$
Character orbit 363.e
Analytic conductor $2.899$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) 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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{2} - \zeta_{10}^{2} q^{3} - 2 \zeta_{10}^{3} q^{4} - 4 \zeta_{10} q^{5} - 2 \zeta_{10} q^{6} - \zeta_{10}^{3} q^{7} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{2} - \zeta_{10}^{2} q^{3} - 2 \zeta_{10}^{3} q^{4} - 4 \zeta_{10} q^{5} - 2 \zeta_{10} q^{6} - \zeta_{10}^{3} q^{7} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} - 8 q^{10} - 2 q^{12} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{13} - 2 \zeta_{10}^{2} q^{14} + 4 \zeta_{10}^{3} q^{15} + 4 \zeta_{10} q^{16} - 4 \zeta_{10} q^{17} + 2 \zeta_{10}^{3} q^{18} - 3 \zeta_{10}^{2} q^{19} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 8 \zeta_{10} - 8) q^{20} - q^{21} + 2 q^{23} + 11 \zeta_{10}^{2} q^{25} - 4 \zeta_{10}^{3} q^{26} + \zeta_{10} q^{27} - 2 \zeta_{10} q^{28} - 6 \zeta_{10}^{3} q^{29} + 8 \zeta_{10}^{2} q^{30} + ( - 5 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 5) q^{31} + 8 q^{32} - 8 q^{34} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{35} + 2 \zeta_{10}^{2} q^{36} - 3 \zeta_{10}^{3} q^{37} - 6 \zeta_{10} q^{38} - 2 \zeta_{10} q^{39} - 2 \zeta_{10}^{2} q^{41} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{42} + 12 q^{43} + 4 q^{45} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{46} + 2 \zeta_{10}^{2} q^{47} - 4 \zeta_{10}^{3} q^{48} + 6 \zeta_{10} q^{49} + 22 \zeta_{10} q^{50} + 4 \zeta_{10}^{3} q^{51} - 4 \zeta_{10}^{2} q^{52} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 6) q^{53} + 2 q^{54} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 3) q^{57} - 12 \zeta_{10}^{2} q^{58} + 10 \zeta_{10}^{3} q^{59} + 8 \zeta_{10} q^{60} - 3 \zeta_{10} q^{61} - 10 \zeta_{10}^{3} q^{62} + \zeta_{10}^{2} q^{63} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{64} - 8 q^{65} - q^{67} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 8 \zeta_{10} - 8) q^{68} - 2 \zeta_{10}^{2} q^{69} + 8 \zeta_{10}^{3} q^{70} + 11 \zeta_{10}^{3} q^{73} - 6 \zeta_{10}^{2} q^{74} + ( - 11 \zeta_{10}^{3} + 11 \zeta_{10}^{2} - 11 \zeta_{10} + 11) q^{75} - 6 q^{76} - 4 q^{78} + (11 \zeta_{10}^{3} - 11 \zeta_{10}^{2} + 11 \zeta_{10} - 11) q^{79} - 16 \zeta_{10}^{2} q^{80} - \zeta_{10}^{3} q^{81} - 4 \zeta_{10} q^{82} - 6 \zeta_{10} q^{83} + 2 \zeta_{10}^{3} q^{84} + 16 \zeta_{10}^{2} q^{85} + ( - 24 \zeta_{10}^{3} + 24 \zeta_{10}^{2} - 24 \zeta_{10} + 24) q^{86} - 6 q^{87} + 12 q^{89} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{90} - 2 \zeta_{10}^{2} q^{91} - 4 \zeta_{10}^{3} q^{92} - 5 \zeta_{10} q^{93} + 4 \zeta_{10} q^{94} + 12 \zeta_{10}^{3} q^{95} - 8 \zeta_{10}^{2} q^{96} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} - 5) q^{97} + 12 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + q^{3} - 2 q^{4} - 4 q^{5} - 2 q^{6} - q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + q^{3} - 2 q^{4} - 4 q^{5} - 2 q^{6} - q^{7} - q^{9} - 32 q^{10} - 8 q^{12} + 2 q^{13} + 2 q^{14} + 4 q^{15} + 4 q^{16} - 4 q^{17} + 2 q^{18} + 3 q^{19} - 8 q^{20} - 4 q^{21} + 8 q^{23} - 11 q^{25} - 4 q^{26} + q^{27} - 2 q^{28} - 6 q^{29} - 8 q^{30} + 5 q^{31} + 32 q^{32} - 32 q^{34} - 4 q^{35} - 2 q^{36} - 3 q^{37} - 6 q^{38} - 2 q^{39} + 2 q^{41} - 2 q^{42} + 48 q^{43} + 16 q^{45} + 4 q^{46} - 2 q^{47} - 4 q^{48} + 6 q^{49} + 22 q^{50} + 4 q^{51} + 4 q^{52} - 6 q^{53} + 8 q^{54} - 3 q^{57} + 12 q^{58} + 10 q^{59} + 8 q^{60} - 3 q^{61} - 10 q^{62} - q^{63} + 8 q^{64} - 32 q^{65} - 4 q^{67} - 8 q^{68} + 2 q^{69} + 8 q^{70} + 11 q^{73} + 6 q^{74} + 11 q^{75} - 24 q^{76} - 16 q^{78} - 11 q^{79} + 16 q^{80} - q^{81} - 4 q^{82} - 6 q^{83} + 2 q^{84} - 16 q^{85} + 24 q^{86} - 24 q^{87} + 48 q^{89} + 8 q^{90} + 2 q^{91} - 4 q^{92} - 5 q^{93} + 4 q^{94} + 12 q^{95} + 8 q^{96} - 5 q^{97} + 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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
0.809017 + 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 0.587785i
1.61803 1.17557i −0.309017 0.951057i 0.618034 1.90211i −3.23607 2.35114i −1.61803 1.17557i 0.309017 0.951057i 0 −0.809017 + 0.587785i −8.00000
130.1 −0.618034 + 1.90211i 0.809017 0.587785i −1.61803 1.17557i 1.23607 + 3.80423i 0.618034 + 1.90211i −0.809017 0.587785i 0 0.309017 0.951057i −8.00000
148.1 −0.618034 1.90211i 0.809017 + 0.587785i −1.61803 + 1.17557i 1.23607 3.80423i 0.618034 1.90211i −0.809017 + 0.587785i 0 0.309017 + 0.951057i −8.00000
202.1 1.61803 + 1.17557i −0.309017 + 0.951057i 0.618034 + 1.90211i −3.23607 + 2.35114i −1.61803 + 1.17557i 0.309017 + 0.951057i 0 −0.809017 0.587785i −8.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.e.i 4
11.b odd 2 1 363.2.e.d 4
11.c even 5 1 363.2.a.a 1
11.c even 5 3 inner 363.2.e.i 4
11.d odd 10 1 363.2.a.c yes 1
11.d odd 10 3 363.2.e.d 4
33.f even 10 1 1089.2.a.a 1
33.h odd 10 1 1089.2.a.k 1
44.g even 10 1 5808.2.a.bi 1
44.h odd 10 1 5808.2.a.bh 1
55.h odd 10 1 9075.2.a.b 1
55.j even 10 1 9075.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.a 1 11.c even 5 1
363.2.a.c yes 1 11.d odd 10 1
363.2.e.d 4 11.b odd 2 1
363.2.e.d 4 11.d odd 10 3
363.2.e.i 4 1.a even 1 1 trivial
363.2.e.i 4 11.c even 5 3 inner
1089.2.a.a 1 33.f even 10 1
1089.2.a.k 1 33.h odd 10 1
5808.2.a.bh 1 44.h odd 10 1
5808.2.a.bi 1 44.g even 10 1
9075.2.a.b 1 55.h odd 10 1
9075.2.a.t 1 55.j even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2T_{2}^{3} + 4T_{2}^{2} - 8T_{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \) Copy content Toggle raw display
$19$ \( T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81 \) Copy content Toggle raw display
$23$ \( (T - 2)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( T^{4} - 5 T^{3} + 25 T^{2} - 125 T + 625 \) Copy content Toggle raw display
$37$ \( T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81 \) Copy content Toggle raw display
$41$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$43$ \( (T - 12)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$53$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$59$ \( T^{4} - 10 T^{3} + 100 T^{2} + \cdots + 10000 \) Copy content Toggle raw display
$61$ \( T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81 \) Copy content Toggle raw display
$67$ \( (T + 1)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 11 T^{3} + 121 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
$79$ \( T^{4} + 11 T^{3} + 121 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$89$ \( (T - 12)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625 \) Copy content Toggle raw display
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