Properties

Label 363.2.f.b
Level $363$
Weight $2$
Character orbit 363.f
Analytic conductor $2.899$
Analytic rank $0$
Dimension $8$
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.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: \(\Q(\zeta_{20})\)
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{20}^{7} - \zeta_{20}^{5}) q^{2} + (\zeta_{20}^{6} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20} - 1) q^{3} + ( - \zeta_{20}^{4} - 1) q^{4} + ( - 2 \zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + 2 \zeta_{20}) q^{5} + (\zeta_{20}^{7} + \zeta_{20}^{6} + \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}^{2} + \zeta_{20}) q^{6} + (2 \zeta_{20}^{6} - 2 \zeta_{20}^{4} + \zeta_{20}^{2}) q^{7} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - \zeta_{20}^{3}) q^{8} + ( - \zeta_{20}^{6} - 2 \zeta_{20}^{2} + 2 \zeta_{20} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{20}^{7} - \zeta_{20}^{5}) q^{2} + (\zeta_{20}^{6} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20} - 1) q^{3} + ( - \zeta_{20}^{4} - 1) q^{4} + ( - 2 \zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + 2 \zeta_{20}) q^{5} + (\zeta_{20}^{7} + \zeta_{20}^{6} + \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}^{2} + \zeta_{20}) q^{6} + (2 \zeta_{20}^{6} - 2 \zeta_{20}^{4} + \zeta_{20}^{2}) q^{7} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - \zeta_{20}^{3}) q^{8} + ( - \zeta_{20}^{6} - 2 \zeta_{20}^{2} + 2 \zeta_{20} + 2) q^{9} + ( - 3 \zeta_{20}^{6} - \zeta_{20}^{4} - 2 \zeta_{20}^{2} + 1) q^{10} + ( - \zeta_{20}^{6} + \zeta_{20}^{5} + \zeta_{20}^{4} + 2) q^{12} + ( - \zeta_{20}^{6} + 2 \zeta_{20}^{4} + 2) q^{13} + ( - \zeta_{20}^{5} + 3 \zeta_{20}^{3} - \zeta_{20}) q^{14} + (3 \zeta_{20}^{7} - \zeta_{20}^{4} + 2 \zeta_{20}^{3} - 2 \zeta_{20} - 1) q^{15} + ( - \zeta_{20}^{6} - 3 \zeta_{20}^{4} - \zeta_{20}^{2}) q^{16} + (\zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}^{3} + \zeta_{20}) q^{17} + (2 \zeta_{20}^{7} - 4 \zeta_{20}^{6} - 4 \zeta_{20}^{5} + 2 \zeta_{20}^{4} + \zeta_{20}^{3} - 2 \zeta_{20}^{2} - 3 \zeta_{20} + 2) q^{18} + ( - 2 \zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{2} - 2) q^{19} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{5} - 3 \zeta_{20}) q^{20} + ( - \zeta_{20}^{7} - \zeta_{20}^{6} - \zeta_{20}^{5} + \zeta_{20}^{4} + 3 \zeta_{20}^{3} - 2 \zeta_{20}^{2} - 2 \zeta_{20} + 1) q^{21} + (2 \zeta_{20}^{7} - 5 \zeta_{20}^{5} + 2 \zeta_{20}^{3}) q^{23} + (\zeta_{20}^{7} + 2 \zeta_{20}^{6} - \zeta_{20}^{5} - 4 \zeta_{20}^{4} + 2 \zeta_{20}^{3} + 3 \zeta_{20}^{2} - \zeta_{20} - 1) q^{24} + ( - 3 \zeta_{20}^{4} + 3 \zeta_{20}^{2}) q^{25} + ( - 4 \zeta_{20}^{7} - 3 \zeta_{20}^{3} + 3 \zeta_{20}) q^{26} + (4 \zeta_{20}^{7} - \zeta_{20}^{6} - 3 \zeta_{20}^{5} + 4 \zeta_{20}^{4} + 3 \zeta_{20}^{3} - \zeta_{20}^{2} - 4 \zeta_{20}) q^{27} + ( - \zeta_{20}^{6} + \zeta_{20}^{2}) q^{28} + (2 \zeta_{20}^{3} + 2 \zeta_{20}) q^{29} + (2 \zeta_{20}^{7} + 2 \zeta_{20}^{6} + 3 \zeta_{20}^{4} + \zeta_{20}^{3} + 3 \zeta_{20}^{2} - 2 \zeta_{20} + 2) q^{30} + ( - \zeta_{20}^{6} + 3 \zeta_{20}^{2} - 3) q^{31} + (5 \zeta_{20}^{7} - 2 \zeta_{20}^{5} - \zeta_{20}^{3} - 4 \zeta_{20}) q^{32} - 5 q^{34} + (\zeta_{20}^{7} + \zeta_{20}^{5}) q^{35} + (3 \zeta_{20}^{6} - 2 \zeta_{20}^{5} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{2} - 2 \zeta_{20} - 3) q^{36} - 3 \zeta_{20}^{2} q^{37} + (5 \zeta_{20}^{7} - 5 \zeta_{20}) q^{38} + (3 \zeta_{20}^{6} - \zeta_{20}^{5} - 2 \zeta_{20}^{4} - \zeta_{20}^{3} + \zeta_{20}^{2} - 4) q^{39} + ( - 2 \zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{2} + 1) q^{40} + ( - 2 \zeta_{20}^{7} - \zeta_{20}^{3} + 2 \zeta_{20}) q^{41} + (2 \zeta_{20}^{7} + \zeta_{20}^{6} - 2 \zeta_{20}^{5} - 3 \zeta_{20}^{2} - \zeta_{20} + 3) q^{42} + (2 \zeta_{20}^{6} - 4 \zeta_{20}^{4} + 6 \zeta_{20}^{2} - 3) q^{43} + ( - 3 \zeta_{20}^{7} - 2 \zeta_{20}^{6} - \zeta_{20}^{5} + 2 \zeta_{20}^{4} - 3 \zeta_{20}^{3} + 4) q^{45} + ( - 2 \zeta_{20}^{6} + 4 \zeta_{20}^{4} - 5 \zeta_{20}^{2} - 1) q^{46} + (7 \zeta_{20}^{5} - 4 \zeta_{20}^{3} + 7 \zeta_{20}) q^{47} + (\zeta_{20}^{7} + 4 \zeta_{20}^{4} + 3 \zeta_{20}^{3} + \zeta_{20}^{2} - 3 \zeta_{20} + 4) q^{48} + (4 \zeta_{20}^{6} + 4 \zeta_{20}^{2}) q^{49} + ( - 3 \zeta_{20}^{7} - 3 \zeta_{20}) q^{50} + ( - 2 \zeta_{20}^{7} - 2 \zeta_{20}^{6} + 4 \zeta_{20}^{5} + 4 \zeta_{20}^{4} - \zeta_{20}^{3} - \zeta_{20}^{2} + 3 \zeta_{20} - 2) q^{51} + ( - \zeta_{20}^{6} - 2 \zeta_{20}^{4} - 2 \zeta_{20}^{2} - 1) q^{52} + ( - 3 \zeta_{20}^{7} + 3 \zeta_{20}^{5} + 3 \zeta_{20}) q^{53} + ( - 2 \zeta_{20}^{7} + 5 \zeta_{20}^{6} + 3 \zeta_{20}^{5} + 3 \zeta_{20}^{4} - 4 \zeta_{20}^{3} + 2 \zeta_{20}^{2} + \cdots - 1) q^{54} + \cdots + ( - 8 \zeta_{20}^{7} + 4 \zeta_{20}^{5} + 8 \zeta_{20}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{3} - 6 q^{4} + 10 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{3} - 6 q^{4} + 10 q^{7} + 10 q^{9} + 12 q^{12} + 10 q^{13} - 6 q^{15} + 2 q^{16} - 20 q^{19} + 10 q^{24} + 12 q^{25} - 12 q^{27} + 20 q^{30} - 20 q^{31} - 40 q^{34} - 10 q^{36} - 6 q^{37} - 20 q^{39} + 20 q^{42} + 24 q^{45} - 30 q^{46} + 26 q^{48} + 16 q^{49} - 30 q^{51} - 10 q^{52} + 30 q^{57} + 20 q^{58} + 2 q^{60} + 10 q^{61} + 30 q^{63} - 34 q^{64} - 4 q^{67} - 16 q^{69} + 10 q^{70} + 20 q^{72} + 6 q^{75} - 20 q^{78} - 50 q^{79} - 2 q^{81} - 10 q^{82} - 10 q^{85} - 40 q^{90} - 10 q^{91} + 10 q^{93} + 30 q^{94} - 10 q^{96} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(\zeta_{20}^{2}\)

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
0.587785 0.809017i
−0.587785 + 0.809017i
−0.951057 + 0.309017i
0.951057 0.309017i
−0.951057 0.309017i
0.951057 + 0.309017i
0.587785 + 0.809017i
−0.587785 0.809017i
−0.951057 0.690983i −1.72982 + 0.0877853i −0.190983 0.587785i 0.224514 + 0.309017i 1.70582 + 1.11179i 2.92705 0.951057i −0.951057 + 2.92705i 2.98459 0.303706i 0.449028i
161.2 0.951057 + 0.690983i 1.34786 + 1.08779i −0.190983 0.587785i −0.224514 0.309017i 0.530249 + 1.96589i 2.92705 0.951057i 0.951057 2.92705i 0.633446 + 2.93236i 0.449028i
215.1 −0.587785 1.80902i −0.945746 1.45106i −1.30902 + 0.951057i −2.48990 0.809017i −2.06909 + 2.56378i −0.427051 0.587785i −0.587785 0.427051i −1.21113 + 2.74466i 4.97980i
215.2 0.587785 + 1.80902i −1.67229 0.451057i −1.30902 + 0.951057i 2.48990 + 0.809017i −0.166977 3.29032i −0.427051 0.587785i 0.587785 + 0.427051i 2.59310 + 1.50859i 4.97980i
233.1 −0.587785 + 1.80902i −0.945746 + 1.45106i −1.30902 0.951057i −2.48990 + 0.809017i −2.06909 2.56378i −0.427051 + 0.587785i −0.587785 + 0.427051i −1.21113 2.74466i 4.97980i
233.2 0.587785 1.80902i −1.67229 + 0.451057i −1.30902 0.951057i 2.48990 0.809017i −0.166977 + 3.29032i −0.427051 + 0.587785i 0.587785 0.427051i 2.59310 1.50859i 4.97980i
239.1 −0.951057 + 0.690983i −1.72982 0.0877853i −0.190983 + 0.587785i 0.224514 0.309017i 1.70582 1.11179i 2.92705 + 0.951057i −0.951057 2.92705i 2.98459 + 0.303706i 0.449028i
239.2 0.951057 0.690983i 1.34786 1.08779i −0.190983 + 0.587785i −0.224514 + 0.309017i 0.530249 1.96589i 2.92705 + 0.951057i 0.951057 + 2.92705i 0.633446 2.93236i 0.449028i
\(n\): 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
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.f.b 8
3.b odd 2 1 inner 363.2.f.b 8
11.b odd 2 1 33.2.f.a 8
11.c even 5 1 33.2.f.a 8
11.c even 5 1 363.2.d.f 8
11.c even 5 1 363.2.f.d 8
11.c even 5 1 363.2.f.e 8
11.d odd 10 1 363.2.d.f 8
11.d odd 10 1 inner 363.2.f.b 8
11.d odd 10 1 363.2.f.d 8
11.d odd 10 1 363.2.f.e 8
33.d even 2 1 33.2.f.a 8
33.f even 10 1 363.2.d.f 8
33.f even 10 1 inner 363.2.f.b 8
33.f even 10 1 363.2.f.d 8
33.f even 10 1 363.2.f.e 8
33.h odd 10 1 33.2.f.a 8
33.h odd 10 1 363.2.d.f 8
33.h odd 10 1 363.2.f.d 8
33.h odd 10 1 363.2.f.e 8
44.c even 2 1 528.2.bn.c 8
44.h odd 10 1 528.2.bn.c 8
55.d odd 2 1 825.2.bi.b 8
55.e even 4 1 825.2.bs.a 8
55.e even 4 1 825.2.bs.d 8
55.j even 10 1 825.2.bi.b 8
55.k odd 20 1 825.2.bs.a 8
55.k odd 20 1 825.2.bs.d 8
99.g even 6 2 891.2.u.a 16
99.h odd 6 2 891.2.u.a 16
99.m even 15 2 891.2.u.a 16
99.n odd 30 2 891.2.u.a 16
132.d odd 2 1 528.2.bn.c 8
132.o even 10 1 528.2.bn.c 8
165.d even 2 1 825.2.bi.b 8
165.l odd 4 1 825.2.bs.a 8
165.l odd 4 1 825.2.bs.d 8
165.o odd 10 1 825.2.bi.b 8
165.v even 20 1 825.2.bs.a 8
165.v even 20 1 825.2.bs.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.f.a 8 11.b odd 2 1
33.2.f.a 8 11.c even 5 1
33.2.f.a 8 33.d even 2 1
33.2.f.a 8 33.h odd 10 1
363.2.d.f 8 11.c even 5 1
363.2.d.f 8 11.d odd 10 1
363.2.d.f 8 33.f even 10 1
363.2.d.f 8 33.h odd 10 1
363.2.f.b 8 1.a even 1 1 trivial
363.2.f.b 8 3.b odd 2 1 inner
363.2.f.b 8 11.d odd 10 1 inner
363.2.f.b 8 33.f even 10 1 inner
363.2.f.d 8 11.c even 5 1
363.2.f.d 8 11.d odd 10 1
363.2.f.d 8 33.f even 10 1
363.2.f.d 8 33.h odd 10 1
363.2.f.e 8 11.c even 5 1
363.2.f.e 8 11.d odd 10 1
363.2.f.e 8 33.f even 10 1
363.2.f.e 8 33.h odd 10 1
528.2.bn.c 8 44.c even 2 1
528.2.bn.c 8 44.h odd 10 1
528.2.bn.c 8 132.d odd 2 1
528.2.bn.c 8 132.o even 10 1
825.2.bi.b 8 55.d odd 2 1
825.2.bi.b 8 55.j even 10 1
825.2.bi.b 8 165.d even 2 1
825.2.bi.b 8 165.o odd 10 1
825.2.bs.a 8 55.e even 4 1
825.2.bs.a 8 55.k odd 20 1
825.2.bs.a 8 165.l odd 4 1
825.2.bs.a 8 165.v even 20 1
825.2.bs.d 8 55.e even 4 1
825.2.bs.d 8 55.k odd 20 1
825.2.bs.d 8 165.l odd 4 1
825.2.bs.d 8 165.v even 20 1
891.2.u.a 16 99.g even 6 2
891.2.u.a 16 99.h odd 6 2
891.2.u.a 16 99.m even 15 2
891.2.u.a 16 99.n odd 30 2

Hecke kernels

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

\( T_{2}^{8} + 5T_{2}^{6} + 10T_{2}^{4} + 25 \) Copy content Toggle raw display
\( T_{5}^{8} - 11T_{5}^{6} + 46T_{5}^{4} + 4T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - 5T_{7}^{3} + 5T_{7}^{2} + 5T_{7} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5 T^{6} + 10 T^{4} + 25 \) Copy content Toggle raw display
$3$ \( T^{8} + 6 T^{7} + 13 T^{6} + 10 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} - 11 T^{6} + 46 T^{4} + 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} - 5 T^{3} + 5 T^{2} + 5 T + 5)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} - 5 T^{3} + 5 T^{2} + 5 T + 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 250 T^{4} + 3125 T^{2} + \cdots + 15625 \) Copy content Toggle raw display
$19$ \( (T^{4} + 10 T^{3} + 50 T^{2} + 125 T + 125)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 42 T^{2} + 121)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 160 T^{4} + 1600 T^{2} + \cdots + 6400 \) Copy content Toggle raw display
$31$ \( (T^{4} + 10 T^{3} + 40 T^{2} + 25 T + 25)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 5 T^{6} + 85 T^{4} + 75 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$43$ \( (T^{4} + 50 T^{2} + 125)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 79 T^{6} + 3966 T^{4} + \cdots + 13845841 \) Copy content Toggle raw display
$53$ \( T^{8} - 36 T^{6} + 486 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$59$ \( T^{8} + 4 T^{6} + 46 T^{4} - 11 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( (T^{4} - 5 T^{3} + 125)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + T - 61)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} - 155 T^{6} + 9150 T^{4} + \cdots + 9150625 \) Copy content Toggle raw display
$73$ \( (T^{4} - 2560 T + 20480)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 25 T^{3} + 225 T^{2} + 855 T + 1805)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 315 T^{6} + \cdots + 70644025 \) Copy content Toggle raw display
$89$ \( (T^{4} + 90 T^{2} + 25)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 3 T^{3} + 34 T^{2} - 232 T + 841)^{2} \) Copy content Toggle raw display
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