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 $4$

Related objects

Downloads

Learn more

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: \( x^{2} - x + 3 \) Copy content Toggle raw display
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.

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

Character values

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

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

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 \(S_{3}^{\mathrm{new}}(363, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5}^{2} + 99 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

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