Properties

Label 363.2.f.e
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{20}^{7} - \zeta_{20}) q^{2} + (\zeta_{20}^{7} + \zeta_{20}^{4} + \zeta_{20}^{3} - \zeta_{20}^{2} - \zeta_{20} + 1) q^{3} + ( - \zeta_{20}^{4} + \zeta_{20}^{2}) q^{4} + ( - \zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}) q^{5} + ( - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{3} + \zeta_{20}^{2} - \zeta_{20} + 1) q^{6} + ( - \zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{2} - 1) q^{7} + (\zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}) q^{8} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{6} - 2 \zeta_{20}^{5} + \zeta_{20}^{4} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{2} - 2 \zeta_{20}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{20}^{7} - \zeta_{20}) q^{2} + (\zeta_{20}^{7} + \zeta_{20}^{4} + \zeta_{20}^{3} - \zeta_{20}^{2} - \zeta_{20} + 1) q^{3} + ( - \zeta_{20}^{4} + \zeta_{20}^{2}) q^{4} + ( - \zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}) q^{5} + ( - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{3} + \zeta_{20}^{2} - \zeta_{20} + 1) q^{6} + ( - \zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{2} - 1) q^{7} + (\zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}) q^{8} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{6} - 2 \zeta_{20}^{5} + \zeta_{20}^{4} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{2} - 2 \zeta_{20}) q^{9} + (\zeta_{20}^{6} - 3 \zeta_{20}^{4} + 4 \zeta_{20}^{2} - 2) q^{10} + (\zeta_{20}^{6} + \zeta_{20}^{5} - \zeta_{20}^{4} + 1) q^{12} + ( - \zeta_{20}^{4} + 2 \zeta_{20}^{2} - 2) q^{13} + (3 \zeta_{20}^{7} + \zeta_{20}^{3} - \zeta_{20}) q^{14} + ( - 2 \zeta_{20}^{5} - \zeta_{20}^{4} + 3 \zeta_{20}^{3} + \zeta_{20}^{2} - 2 \zeta_{20}) q^{15} + (3 \zeta_{20}^{6} - \zeta_{20}^{2} + 1) q^{16} + ( - 3 \zeta_{20}^{7} + \zeta_{20}^{5} - 4 \zeta_{20}^{3} + 2 \zeta_{20}) q^{17} + (4 \zeta_{20}^{7} + 2 \zeta_{20}^{6} - 3 \zeta_{20}^{5} + 2 \zeta_{20}^{3} - \zeta_{20} + 2) q^{18} + ( - 4 \zeta_{20}^{6} + 3 \zeta_{20}^{4} - 2 \zeta_{20}^{2} + 1) q^{19} + ( - \zeta_{20}^{7} + 3 \zeta_{20}^{5} - 3 \zeta_{20}^{3} + \zeta_{20}) q^{20} + ( - 3 \zeta_{20}^{7} - \zeta_{20}^{6} + 2 \zeta_{20}^{5} - \zeta_{20}^{4} - \zeta_{20}^{3} + 4 \zeta_{20}) q^{21} + ( - 2 \zeta_{20}^{7} - 3 \zeta_{20}^{5} - 2 \zeta_{20}^{3}) q^{23} + ( - 3 \zeta_{20}^{6} - \zeta_{20}^{5} + 2 \zeta_{20}^{4} - \zeta_{20}^{3} - \zeta_{20}^{2} + 4) q^{24} + ( - 3 \zeta_{20}^{4} - 3) q^{25} + (3 \zeta_{20}^{5} - 4 \zeta_{20}^{3} + 3 \zeta_{20}) q^{26} + (\zeta_{20}^{7} - 4 \zeta_{20}^{6} - \zeta_{20}^{5} - \zeta_{20}^{2} + 3 \zeta_{20} + 1) q^{27} + ( - \zeta_{20}^{2} - 1) q^{28} + (2 \zeta_{20}^{5} - 2 \zeta_{20}) q^{29} + ( - \zeta_{20}^{7} + 4 \zeta_{20}^{6} + 2 \zeta_{20}^{5} - 5 \zeta_{20}^{4} - 2 \zeta_{20}^{3} + 2 \zeta_{20}^{2} + \cdots + 1) q^{30}+ \cdots + (4 \zeta_{20}^{5} - 8 \zeta_{20}^{3} + 8 \zeta_{20}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 4 q^{4} + 10 q^{6} - 10 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 4 q^{4} + 10 q^{6} - 10 q^{7} - 10 q^{9} + 12 q^{12} - 10 q^{13} + 4 q^{15} + 12 q^{16} + 20 q^{18} - 10 q^{19} + 20 q^{24} - 18 q^{25} - 2 q^{27} - 10 q^{28} + 30 q^{30} + 10 q^{31} - 40 q^{34} - 6 q^{37} + 10 q^{39} + 10 q^{40} - 10 q^{42} + 24 q^{45} - 20 q^{46} - 14 q^{48} - 24 q^{49} + 10 q^{51} - 20 q^{52} - 10 q^{57} - 8 q^{60} - 20 q^{61} - 10 q^{63} + 26 q^{64} - 4 q^{67} + 34 q^{69} + 20 q^{72} - 80 q^{73} + 6 q^{75} - 20 q^{78} + 10 q^{79} - 2 q^{81} + 20 q^{82} - 10 q^{84} - 30 q^{85} + 30 q^{90} + 10 q^{91} - 70 q^{94} + 30 q^{96} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(1 - \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{6}\)

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
−1.53884 1.11803i −0.0877853 1.72982i 0.500000 + 1.53884i −1.53884 2.11803i −1.79892 + 2.76007i −0.690983 + 0.224514i −0.224514 + 0.690983i −2.98459 + 0.303706i 4.97980i
161.2 1.53884 + 1.11803i 1.08779 1.34786i 0.500000 + 1.53884i 1.53884 + 2.11803i 3.18088 0.857960i −0.690983 + 0.224514i 0.224514 0.690983i −0.633446 2.93236i 4.97980i
215.1 −0.363271 1.11803i −0.451057 + 1.67229i 0.500000 0.363271i −0.363271 0.118034i 2.03353 0.103198i −1.80902 2.48990i −2.48990 1.80902i −2.59310 1.50859i 0.449028i
215.2 0.363271 + 1.11803i 1.45106 0.945746i 0.500000 0.363271i 0.363271 + 0.118034i 1.58450 + 1.27877i −1.80902 2.48990i 2.48990 + 1.80902i 1.21113 2.74466i 0.449028i
233.1 −0.363271 + 1.11803i −0.451057 1.67229i 0.500000 + 0.363271i −0.363271 + 0.118034i 2.03353 + 0.103198i −1.80902 + 2.48990i −2.48990 + 1.80902i −2.59310 + 1.50859i 0.449028i
233.2 0.363271 1.11803i 1.45106 + 0.945746i 0.500000 + 0.363271i 0.363271 0.118034i 1.58450 1.27877i −1.80902 + 2.48990i 2.48990 1.80902i 1.21113 + 2.74466i 0.449028i
239.1 −1.53884 + 1.11803i −0.0877853 + 1.72982i 0.500000 1.53884i −1.53884 + 2.11803i −1.79892 2.76007i −0.690983 0.224514i −0.224514 0.690983i −2.98459 0.303706i 4.97980i
239.2 1.53884 1.11803i 1.08779 + 1.34786i 0.500000 1.53884i 1.53884 2.11803i 3.18088 + 0.857960i −0.690983 0.224514i 0.224514 + 0.690983i −0.633446 + 2.93236i 4.97980i
\(n\): 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.e 8
3.b odd 2 1 inner 363.2.f.e 8
11.b odd 2 1 363.2.f.d 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.b 8
11.c even 5 1 363.2.f.d 8
11.d odd 10 1 33.2.f.a 8
11.d odd 10 1 363.2.d.f 8
11.d odd 10 1 363.2.f.b 8
11.d odd 10 1 inner 363.2.f.e 8
33.d even 2 1 363.2.f.d 8
33.f even 10 1 33.2.f.a 8
33.f even 10 1 363.2.d.f 8
33.f even 10 1 363.2.f.b 8
33.f even 10 1 inner 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.b 8
33.h odd 10 1 363.2.f.d 8
44.g even 10 1 528.2.bn.c 8
44.h odd 10 1 528.2.bn.c 8
55.h odd 10 1 825.2.bi.b 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
55.l even 20 1 825.2.bs.a 8
55.l even 20 1 825.2.bs.d 8
99.m even 15 2 891.2.u.a 16
99.n odd 30 2 891.2.u.a 16
99.o odd 30 2 891.2.u.a 16
99.p even 30 2 891.2.u.a 16
132.n odd 10 1 528.2.bn.c 8
132.o even 10 1 528.2.bn.c 8
165.o odd 10 1 825.2.bi.b 8
165.r even 10 1 825.2.bi.b 8
165.u odd 20 1 825.2.bs.a 8
165.u odd 20 1 825.2.bs.d 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.c even 5 1
33.2.f.a 8 11.d odd 10 1
33.2.f.a 8 33.f even 10 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 11.c even 5 1
363.2.f.b 8 11.d odd 10 1
363.2.f.b 8 33.f even 10 1
363.2.f.b 8 33.h odd 10 1
363.2.f.d 8 11.b odd 2 1
363.2.f.d 8 11.c even 5 1
363.2.f.d 8 33.d even 2 1
363.2.f.d 8 33.h odd 10 1
363.2.f.e 8 1.a even 1 1 trivial
363.2.f.e 8 3.b odd 2 1 inner
363.2.f.e 8 11.d odd 10 1 inner
363.2.f.e 8 33.f even 10 1 inner
528.2.bn.c 8 44.g even 10 1
528.2.bn.c 8 44.h odd 10 1
528.2.bn.c 8 132.n odd 10 1
528.2.bn.c 8 132.o even 10 1
825.2.bi.b 8 55.h odd 10 1
825.2.bi.b 8 55.j even 10 1
825.2.bi.b 8 165.o odd 10 1
825.2.bi.b 8 165.r even 10 1
825.2.bs.a 8 55.k odd 20 1
825.2.bs.a 8 55.l even 20 1
825.2.bs.a 8 165.u odd 20 1
825.2.bs.a 8 165.v even 20 1
825.2.bs.d 8 55.k odd 20 1
825.2.bs.d 8 55.l even 20 1
825.2.bs.d 8 165.u odd 20 1
825.2.bs.d 8 165.v even 20 1
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 99.o odd 30 2
891.2.u.a 16 99.p even 30 2

Hecke kernels

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

\( T_{2}^{8} + 10T_{2}^{4} + 25T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{5}^{8} + 4T_{5}^{6} + 46T_{5}^{4} - 11T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 5T_{7}^{3} + 15T_{7}^{2} + 15T_{7} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 10 T^{4} + 25 T^{2} + 25 \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + 13 T^{6} - 30 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 4 T^{6} + 46 T^{4} - 11 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} + 5 T^{3} + 15 T^{2} + 15 T + 5)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 5 T^{3} + 15 T^{2} + 15 T + 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 25 T^{6} + 250 T^{4} + \cdots + 15625 \) Copy content Toggle raw display
$19$ \( (T^{4} + 5 T^{3} + 125)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 42 T^{2} + 121)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 20 T^{6} + 160 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$31$ \( (T^{4} - 5 T^{3} + 40 T^{2} - 50 T + 25)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 15 T^{6} + 85 T^{4} - 25 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$43$ \( (T^{4} + 50 T^{2} + 125)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 44 T^{6} + 3966 T^{4} + \cdots + 13845841 \) Copy content Toggle raw display
$53$ \( T^{8} + 9 T^{6} + 486 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$59$ \( T^{8} - 11 T^{6} + 46 T^{4} + 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( (T^{4} + 10 T^{3} + 50 T^{2} + 125 T + 125)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + T - 61)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} + 20 T^{6} + 9150 T^{4} + \cdots + 9150625 \) Copy content Toggle raw display
$73$ \( (T^{4} + 40 T^{3} + 640 T^{2} + \cdots + 20480)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 5 T^{3} + 35 T^{2} + 665 T + 1805)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 65 T^{6} + 37285 T^{4} + \cdots + 70644025 \) Copy content Toggle raw display
$89$ \( (T^{4} + 90 T^{2} + 25)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 8 T^{3} + 34 T^{2} - 87 T + 841)^{2} \) Copy content Toggle raw display
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