Properties

Label 363.3.b.c
Level $363$
Weight $3$
Character orbit 363.b
Analytic conductor $9.891$
Analytic rank $0$
Dimension $2$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.89103359628\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Defining polynomial: \(x^{2} - x + 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - \beta ) q^{3} + 4 q^{4} + ( -3 + 6 \beta ) q^{5} + ( 6 - 5 \beta ) q^{9} +O(q^{10})\) \( q + ( 3 - \beta ) q^{3} + 4 q^{4} + ( -3 + 6 \beta ) q^{5} + ( 6 - 5 \beta ) q^{9} + ( 12 - 4 \beta ) q^{12} + ( 9 + 15 \beta ) q^{15} + 16 q^{16} + ( -12 + 24 \beta ) q^{20} + ( -9 + 18 \beta ) q^{23} -74 q^{25} + ( 3 - 16 \beta ) q^{27} + 37 q^{31} + ( 24 - 20 \beta ) q^{36} + 25 q^{37} + ( 72 + 21 \beta ) q^{45} + ( 24 - 48 \beta ) q^{47} + ( 48 - 16 \beta ) q^{48} -49 q^{49} + ( 24 - 48 \beta ) q^{53} + ( -15 + 30 \beta ) q^{59} + ( 36 + 60 \beta ) q^{60} + 64 q^{64} -35 q^{67} + ( 27 + 45 \beta ) q^{69} + ( 15 - 30 \beta ) q^{71} + ( -222 + 74 \beta ) q^{75} + ( -48 + 96 \beta ) q^{80} + ( -39 - 35 \beta ) q^{81} + ( 45 - 90 \beta ) q^{89} + ( -36 + 72 \beta ) q^{92} + ( 111 - 37 \beta ) q^{93} + 95 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{3} + 8 q^{4} + 7 q^{9} + O(q^{10}) \) \( 2 q + 5 q^{3} + 8 q^{4} + 7 q^{9} + 20 q^{12} + 33 q^{15} + 32 q^{16} - 148 q^{25} - 10 q^{27} + 74 q^{31} + 28 q^{36} + 50 q^{37} + 165 q^{45} + 80 q^{48} - 98 q^{49} + 132 q^{60} + 128 q^{64} - 70 q^{67} + 99 q^{69} - 370 q^{75} - 113 q^{81} + 185 q^{93} + 190 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
122.1
0.500000 + 1.65831i
0.500000 1.65831i
0 2.50000 1.65831i 4.00000 9.94987i 0 0 0 3.50000 8.29156i 0
122.2 0 2.50000 + 1.65831i 4.00000 9.94987i 0 0 0 3.50000 + 8.29156i 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
3.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.b.c 2
3.b odd 2 1 inner 363.3.b.c 2
11.b odd 2 1 CM 363.3.b.c 2
11.c even 5 4 363.3.h.f 8
11.d odd 10 4 363.3.h.f 8
33.d even 2 1 inner 363.3.b.c 2
33.f even 10 4 363.3.h.f 8
33.h odd 10 4 363.3.h.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.b.c 2 1.a even 1 1 trivial
363.3.b.c 2 3.b odd 2 1 inner
363.3.b.c 2 11.b odd 2 1 CM
363.3.b.c 2 33.d even 2 1 inner
363.3.h.f 8 11.c even 5 4
363.3.h.f 8 11.d odd 10 4
363.3.h.f 8 33.f even 10 4
363.3.h.f 8 33.h odd 10 4

Hecke kernels

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

\( T_{2} \)
\( T_{5}^{2} + 99 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 - 5 T + T^{2} \)
$5$ \( 99 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( 891 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( -37 + T )^{2} \)
$37$ \( ( -25 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( 6336 + T^{2} \)
$53$ \( 6336 + T^{2} \)
$59$ \( 2475 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( ( 35 + T )^{2} \)
$71$ \( 2475 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( 22275 + T^{2} \)
$97$ \( ( -95 + T )^{2} \)
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