Properties

Label 363.2.e.g
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} - \zeta_{10}^{2} q^{3} + \zeta_{10}^{3} q^{4} + 2 \zeta_{10} q^{5} - \zeta_{10} q^{6} + 4 \zeta_{10}^{3} q^{7} + 3 \zeta_{10}^{2} q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} - \zeta_{10}^{2} q^{3} + \zeta_{10}^{3} q^{4} + 2 \zeta_{10} q^{5} - \zeta_{10} q^{6} + 4 \zeta_{10}^{3} q^{7} + 3 \zeta_{10}^{2} q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} + 2 q^{10} + q^{12} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{13} + 4 \zeta_{10}^{2} q^{14} - 2 \zeta_{10}^{3} q^{15} + \zeta_{10} q^{16} - 2 \zeta_{10} q^{17} + \zeta_{10}^{3} q^{18} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{20} + 4 q^{21} + 8 q^{23} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 3) q^{24} - \zeta_{10}^{2} q^{25} + 2 \zeta_{10}^{3} q^{26} + \zeta_{10} q^{27} - 4 \zeta_{10} q^{28} - 6 \zeta_{10}^{3} q^{29} - 2 \zeta_{10}^{2} q^{30} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{31} - 5 q^{32} - 2 q^{34} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 8 \zeta_{10} - 8) q^{35} - \zeta_{10}^{2} q^{36} - 6 \zeta_{10}^{3} q^{37} + 2 \zeta_{10} q^{39} + 6 \zeta_{10}^{3} q^{40} + 2 \zeta_{10}^{2} q^{41} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{42} - 2 q^{45} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{46} + 8 \zeta_{10}^{2} q^{47} - \zeta_{10}^{3} q^{48} - 9 \zeta_{10} q^{49} - \zeta_{10} q^{50} + 2 \zeta_{10}^{3} q^{51} - 2 \zeta_{10}^{2} q^{52} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 6) q^{53} + q^{54} - 12 q^{56} - 6 \zeta_{10}^{2} q^{58} + 4 \zeta_{10}^{3} q^{59} + 2 \zeta_{10} q^{60} + 6 \zeta_{10} q^{61} - 8 \zeta_{10}^{3} q^{62} - 4 \zeta_{10}^{2} q^{63} + (7 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 7 \zeta_{10} - 7) q^{64} - 4 q^{65} - 4 q^{67} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{68} - 8 \zeta_{10}^{2} q^{69} + 8 \zeta_{10}^{3} q^{70} - 3 \zeta_{10} q^{72} - 14 \zeta_{10}^{3} q^{73} - 6 \zeta_{10}^{2} q^{74} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{75} + 2 q^{78} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{79} + 2 \zeta_{10}^{2} q^{80} - \zeta_{10}^{3} q^{81} + 2 \zeta_{10} q^{82} + 12 \zeta_{10} q^{83} + 4 \zeta_{10}^{3} q^{84} - 4 \zeta_{10}^{2} q^{85} - 6 q^{87} - 6 q^{89} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{90} - 8 \zeta_{10}^{2} q^{91} + 8 \zeta_{10}^{3} q^{92} - 8 \zeta_{10} q^{93} + 8 \zeta_{10} q^{94} + 5 \zeta_{10}^{2} q^{96} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{97} - 9 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + q^{3} + q^{4} + 2 q^{5} - q^{6} + 4 q^{7} - 3 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + q^{3} + q^{4} + 2 q^{5} - q^{6} + 4 q^{7} - 3 q^{8} - q^{9} + 8 q^{10} + 4 q^{12} - 2 q^{13} - 4 q^{14} - 2 q^{15} + q^{16} - 2 q^{17} + q^{18} - 2 q^{20} + 16 q^{21} + 32 q^{23} + 3 q^{24} + q^{25} + 2 q^{26} + q^{27} - 4 q^{28} - 6 q^{29} + 2 q^{30} + 8 q^{31} - 20 q^{32} - 8 q^{34} - 8 q^{35} + q^{36} - 6 q^{37} + 2 q^{39} + 6 q^{40} - 2 q^{41} + 4 q^{42} - 8 q^{45} + 8 q^{46} - 8 q^{47} - q^{48} - 9 q^{49} - q^{50} + 2 q^{51} + 2 q^{52} - 6 q^{53} + 4 q^{54} - 48 q^{56} + 6 q^{58} + 4 q^{59} + 2 q^{60} + 6 q^{61} - 8 q^{62} + 4 q^{63} - 7 q^{64} - 16 q^{65} - 16 q^{67} + 2 q^{68} + 8 q^{69} + 8 q^{70} - 3 q^{72} - 14 q^{73} + 6 q^{74} - q^{75} + 8 q^{78} - 4 q^{79} - 2 q^{80} - q^{81} + 2 q^{82} + 12 q^{83} + 4 q^{84} + 4 q^{85} - 24 q^{87} - 24 q^{89} - 2 q^{90} + 8 q^{91} + 8 q^{92} - 8 q^{93} + 8 q^{94} - 5 q^{96} - 2 q^{97} - 36 q^{98}+O(q^{100}) \) 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
0.809017 0.587785i −0.309017 0.951057i −0.309017 + 0.951057i 1.61803 + 1.17557i −0.809017 0.587785i −1.23607 + 3.80423i 0.927051 + 2.85317i −0.809017 + 0.587785i 2.00000
130.1 −0.309017 + 0.951057i 0.809017 0.587785i 0.809017 + 0.587785i −0.618034 1.90211i 0.309017 + 0.951057i 3.23607 + 2.35114i −2.42705 + 1.76336i 0.309017 0.951057i 2.00000
148.1 −0.309017 0.951057i 0.809017 + 0.587785i 0.809017 0.587785i −0.618034 + 1.90211i 0.309017 0.951057i 3.23607 2.35114i −2.42705 1.76336i 0.309017 + 0.951057i 2.00000
202.1 0.809017 + 0.587785i −0.309017 + 0.951057i −0.309017 0.951057i 1.61803 1.17557i −0.809017 + 0.587785i −1.23607 3.80423i 0.927051 2.85317i −0.809017 0.587785i 2.00000
\(n\): 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.g 4
11.b odd 2 1 363.2.e.e 4
11.c even 5 1 363.2.a.b 1
11.c even 5 3 inner 363.2.e.g 4
11.d odd 10 1 33.2.a.a 1
11.d odd 10 3 363.2.e.e 4
33.f even 10 1 99.2.a.b 1
33.h odd 10 1 1089.2.a.j 1
44.g even 10 1 528.2.a.g 1
44.h odd 10 1 5808.2.a.t 1
55.h odd 10 1 825.2.a.a 1
55.j even 10 1 9075.2.a.q 1
55.l even 20 2 825.2.c.a 2
77.l even 10 1 1617.2.a.j 1
88.k even 10 1 2112.2.a.j 1
88.p odd 10 1 2112.2.a.bb 1
99.o odd 30 2 891.2.e.e 2
99.p even 30 2 891.2.e.g 2
132.n odd 10 1 1584.2.a.o 1
143.l odd 10 1 5577.2.a.a 1
165.r even 10 1 2475.2.a.g 1
165.u odd 20 2 2475.2.c.d 2
187.l odd 10 1 9537.2.a.m 1
231.r odd 10 1 4851.2.a.b 1
264.r odd 10 1 6336.2.a.n 1
264.u even 10 1 6336.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 11.d odd 10 1
99.2.a.b 1 33.f even 10 1
363.2.a.b 1 11.c even 5 1
363.2.e.e 4 11.b odd 2 1
363.2.e.e 4 11.d odd 10 3
363.2.e.g 4 1.a even 1 1 trivial
363.2.e.g 4 11.c even 5 3 inner
528.2.a.g 1 44.g even 10 1
825.2.a.a 1 55.h odd 10 1
825.2.c.a 2 55.l even 20 2
891.2.e.e 2 99.o odd 30 2
891.2.e.g 2 99.p even 30 2
1089.2.a.j 1 33.h odd 10 1
1584.2.a.o 1 132.n odd 10 1
1617.2.a.j 1 77.l even 10 1
2112.2.a.j 1 88.k even 10 1
2112.2.a.bb 1 88.p odd 10 1
2475.2.a.g 1 165.r even 10 1
2475.2.c.d 2 165.u odd 20 2
4851.2.a.b 1 231.r odd 10 1
5577.2.a.a 1 143.l odd 10 1
5808.2.a.t 1 44.h odd 10 1
6336.2.a.n 1 264.r odd 10 1
6336.2.a.x 1 264.u even 10 1
9075.2.a.q 1 55.j even 10 1
9537.2.a.m 1 187.l odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{3} + T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 \) 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
$17$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T - 8)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( T^{4} - 8 T^{3} + 64 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$37$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$59$ \( T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 \) Copy content Toggle raw display
$61$ \( T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$67$ \( (T + 4)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 14 T^{3} + 196 T^{2} + \cdots + 38416 \) Copy content Toggle raw display
$79$ \( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \) Copy content Toggle raw display
$83$ \( T^{4} - 12 T^{3} + 144 T^{2} + \cdots + 20736 \) Copy content Toggle raw display
$89$ \( (T + 6)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
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