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: \(\Q(\zeta_{10})\)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{2} q^{3} + \zeta_{10}^{3} q^{4} + 2 \zeta_{10} q^{5} -\zeta_{10} q^{6} + 4 \zeta_{10}^{3} q^{7} + 3 \zeta_{10}^{2} q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) 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} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} + 2 q^{10} + q^{12} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) 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 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{20} + 4 q^{21} + 8 q^{23} + ( 3 - 3 \zeta_{10} + 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 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{31} -5 q^{32} -2 q^{34} + ( -8 + 8 \zeta_{10} - 8 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) 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 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{42} -2 q^{45} + ( 8 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) 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 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) 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 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{64} -4 q^{65} -4 q^{67} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) 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} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{75} + 2 q^{78} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) 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 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) 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 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{97} -9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{2} + q^{3} + q^{4} + 2q^{5} - q^{6} + 4q^{7} - 3q^{8} - q^{9} + O(q^{10}) \) \( 4q + q^{2} + q^{3} + q^{4} + 2q^{5} - q^{6} + 4q^{7} - 3q^{8} - q^{9} + 8q^{10} + 4q^{12} - 2q^{13} - 4q^{14} - 2q^{15} + q^{16} - 2q^{17} + q^{18} - 2q^{20} + 16q^{21} + 32q^{23} + 3q^{24} + q^{25} + 2q^{26} + q^{27} - 4q^{28} - 6q^{29} + 2q^{30} + 8q^{31} - 20q^{32} - 8q^{34} - 8q^{35} + q^{36} - 6q^{37} + 2q^{39} + 6q^{40} - 2q^{41} + 4q^{42} - 8q^{45} + 8q^{46} - 8q^{47} - q^{48} - 9q^{49} - q^{50} + 2q^{51} + 2q^{52} - 6q^{53} + 4q^{54} - 48q^{56} + 6q^{58} + 4q^{59} + 2q^{60} + 6q^{61} - 8q^{62} + 4q^{63} - 7q^{64} - 16q^{65} - 16q^{67} + 2q^{68} + 8q^{69} + 8q^{70} - 3q^{72} - 14q^{73} + 6q^{74} - q^{75} + 8q^{78} - 4q^{79} - 2q^{80} - q^{81} + 2q^{82} + 12q^{83} + 4q^{84} + 4q^{85} - 24q^{87} - 24q^{89} - 2q^{90} + 8q^{91} + 8q^{92} - 8q^{93} + 8q^{94} - 5q^{96} - 2q^{97} - 36q^{98} + O(q^{100}) \)

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 \(S_{2}^{\mathrm{new}}(363, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T - T^{2} + 3 T^{3} - T^{4} + 6 T^{5} - 4 T^{6} - 8 T^{7} + 16 T^{8} \)
$3$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$5$ \( 1 - 2 T - T^{2} + 12 T^{3} - 19 T^{4} + 60 T^{5} - 25 T^{6} - 250 T^{7} + 625 T^{8} \)
$7$ \( 1 - 4 T + 9 T^{2} - 8 T^{3} - 31 T^{4} - 56 T^{5} + 441 T^{6} - 1372 T^{7} + 2401 T^{8} \)
$11$ 1
$13$ \( 1 + 2 T - 9 T^{2} - 44 T^{3} + 29 T^{4} - 572 T^{5} - 1521 T^{6} + 4394 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 2 T - 13 T^{2} - 60 T^{3} + 101 T^{4} - 1020 T^{5} - 3757 T^{6} + 9826 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} \)
$23$ \( ( 1 - 8 T + 23 T^{2} )^{4} \)
$29$ \( 1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 3828 T^{5} + 5887 T^{6} + 146334 T^{7} + 707281 T^{8} \)
$31$ \( 1 - 8 T + 33 T^{2} - 16 T^{3} - 895 T^{4} - 496 T^{5} + 31713 T^{6} - 238328 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 6 T - T^{2} - 228 T^{3} - 1331 T^{4} - 8436 T^{5} - 1369 T^{6} + 303918 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 2 T - 37 T^{2} - 156 T^{3} + 1205 T^{4} - 6396 T^{5} - 62197 T^{6} + 137842 T^{7} + 2825761 T^{8} \)
$43$ \( ( 1 + 43 T^{2} )^{4} \)
$47$ \( 1 + 8 T + 17 T^{2} - 240 T^{3} - 2719 T^{4} - 11280 T^{5} + 37553 T^{6} + 830584 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 6 T - 17 T^{2} - 420 T^{3} - 1619 T^{4} - 22260 T^{5} - 47753 T^{6} + 893262 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 4 T - 43 T^{2} + 408 T^{3} + 905 T^{4} + 24072 T^{5} - 149683 T^{6} - 821516 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 6 T - 25 T^{2} + 516 T^{3} - 1571 T^{4} + 31476 T^{5} - 93025 T^{6} - 1361886 T^{7} + 13845841 T^{8} \)
$67$ \( ( 1 + 4 T + 67 T^{2} )^{4} \)
$71$ \( 1 - 71 T^{2} + 5041 T^{4} - 357911 T^{6} + 25411681 T^{8} \)
$73$ \( 1 + 14 T + 123 T^{2} + 700 T^{3} + 821 T^{4} + 51100 T^{5} + 655467 T^{6} + 5446238 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 4 T - 63 T^{2} - 568 T^{3} + 2705 T^{4} - 44872 T^{5} - 393183 T^{6} + 1972156 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 12 T + 61 T^{2} + 264 T^{3} - 8231 T^{4} + 21912 T^{5} + 420229 T^{6} - 6861444 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 + 6 T + 89 T^{2} )^{4} \)
$97$ \( 1 + 2 T - 93 T^{2} - 380 T^{3} + 8261 T^{4} - 36860 T^{5} - 875037 T^{6} + 1825346 T^{7} + 88529281 T^{8} \)
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