Properties

Label 363.3.h.b
Level $363$
Weight $3$
Character orbit 363.h
Analytic conductor $9.891$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.h (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.89103359628\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + 3 \zeta_{10}^{3} q^{3} + 4 \zeta_{10}^{2} q^{4} - 11 \zeta_{10}^{2} q^{7} - 9 \zeta_{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 \zeta_{10}^{3} q^{3} + 4 \zeta_{10}^{2} q^{4} - 11 \zeta_{10}^{2} q^{7} - 9 \zeta_{10} q^{9} - 12 q^{12} - 22 \zeta_{10} q^{13} + (16 \zeta_{10}^{3} - 16 \zeta_{10}^{2} + 16 \zeta_{10} - 16) q^{16} + 11 \zeta_{10}^{3} q^{19} + 33 q^{21} - 25 \zeta_{10}^{3} q^{25} + ( - 27 \zeta_{10}^{3} + 27 \zeta_{10}^{2} - 27 \zeta_{10} + 27) q^{27} + ( - 44 \zeta_{10}^{3} + 44 \zeta_{10}^{2} - 44 \zeta_{10} + 44) q^{28} - 59 \zeta_{10} q^{31} - 36 \zeta_{10}^{3} q^{36} + 47 \zeta_{10}^{2} q^{37} + ( - 66 \zeta_{10}^{3} + 66 \zeta_{10}^{2} - 66 \zeta_{10} + 66) q^{39} + 22 q^{43} - 48 \zeta_{10}^{2} q^{48} + (72 \zeta_{10}^{3} - 72 \zeta_{10}^{2} + 72 \zeta_{10} - 72) q^{49} - 88 \zeta_{10}^{3} q^{52} - 33 \zeta_{10} q^{57} + (121 \zeta_{10}^{3} - 121 \zeta_{10}^{2} + 121 \zeta_{10} - 121) q^{61} + 99 \zeta_{10}^{3} q^{63} - 64 \zeta_{10} q^{64} - 13 q^{67} - 143 \zeta_{10}^{2} q^{73} + 75 \zeta_{10} q^{75} - 44 q^{76} + 11 \zeta_{10} q^{79} + 81 \zeta_{10}^{2} q^{81} + 132 \zeta_{10}^{2} q^{84} + 242 \zeta_{10}^{3} q^{91} + ( - 177 \zeta_{10}^{3} + 177 \zeta_{10}^{2} - 177 \zeta_{10} + 177) q^{93} + 169 \zeta_{10} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} - 4 q^{4} + 11 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{3} - 4 q^{4} + 11 q^{7} - 9 q^{9} - 48 q^{12} - 22 q^{13} - 16 q^{16} + 11 q^{19} + 132 q^{21} - 25 q^{25} + 27 q^{27} + 44 q^{28} - 59 q^{31} - 36 q^{36} - 47 q^{37} + 66 q^{39} + 88 q^{43} + 48 q^{48} - 72 q^{49} - 88 q^{52} - 33 q^{57} - 121 q^{61} + 99 q^{63} - 64 q^{64} - 52 q^{67} + 143 q^{73} + 75 q^{75} - 176 q^{76} + 11 q^{79} - 81 q^{81} - 132 q^{84} + 242 q^{91} + 177 q^{93} + 169 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(\zeta_{10}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
245.1
0.809017 0.587785i
−0.309017 + 0.951057i
−0.309017 0.951057i
0.809017 + 0.587785i
0 −0.927051 2.85317i 1.23607 3.80423i 0 0 −3.39919 + 10.4616i 0 −7.28115 + 5.29007i 0
251.1 0 2.42705 1.76336i −3.23607 2.35114i 0 0 8.89919 + 6.46564i 0 2.78115 8.55951i 0
269.1 0 2.42705 + 1.76336i −3.23607 + 2.35114i 0 0 8.89919 6.46564i 0 2.78115 + 8.55951i 0
323.1 0 −0.927051 + 2.85317i 1.23607 + 3.80423i 0 0 −3.39919 10.4616i 0 −7.28115 5.29007i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
11.c even 5 3 inner
33.h odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.h.b 4
3.b odd 2 1 CM 363.3.h.b 4
11.b odd 2 1 363.3.h.a 4
11.c even 5 1 363.3.b.a 1
11.c even 5 3 inner 363.3.h.b 4
11.d odd 10 1 363.3.b.b yes 1
11.d odd 10 3 363.3.h.a 4
33.d even 2 1 363.3.h.a 4
33.f even 10 1 363.3.b.b yes 1
33.f even 10 3 363.3.h.a 4
33.h odd 10 1 363.3.b.a 1
33.h odd 10 3 inner 363.3.h.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.b.a 1 11.c even 5 1
363.3.b.a 1 33.h odd 10 1
363.3.b.b yes 1 11.d odd 10 1
363.3.b.b yes 1 33.f even 10 1
363.3.h.a 4 11.b odd 2 1
363.3.h.a 4 11.d odd 10 3
363.3.h.a 4 33.d even 2 1
363.3.h.a 4 33.f even 10 3
363.3.h.b 4 1.a even 1 1 trivial
363.3.h.b 4 3.b odd 2 1 CM
363.3.h.b 4 11.c even 5 3 inner
363.3.h.b 4 33.h odd 10 3 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{4} - 11T_{7}^{3} + 121T_{7}^{2} - 1331T_{7} + 14641 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 11 T^{3} + 121 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 22 T^{3} + 484 T^{2} + \cdots + 234256 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 11 T^{3} + 121 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 59 T^{3} + 3481 T^{2} + \cdots + 12117361 \) Copy content Toggle raw display
$37$ \( T^{4} + 47 T^{3} + 2209 T^{2} + \cdots + 4879681 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T - 22)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 121 T^{3} + \cdots + 214358881 \) Copy content Toggle raw display
$67$ \( (T + 13)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 143 T^{3} + \cdots + 418161601 \) Copy content Toggle raw display
$79$ \( T^{4} - 11 T^{3} + 121 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 169 T^{3} + \cdots + 815730721 \) Copy content Toggle raw display
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