Properties

Label 361.3.d.a
Level $361$
Weight $3$
Character orbit 361.d
Analytic conductor $9.837$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 361.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.83653754341\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4 \zeta_{6} q^{4} + ( 9 - 9 \zeta_{6} ) q^{5} -5 q^{7} -9 \zeta_{6} q^{9} +O(q^{10})\) \( q -4 \zeta_{6} q^{4} + ( 9 - 9 \zeta_{6} ) q^{5} -5 q^{7} -9 \zeta_{6} q^{9} + 3 q^{11} + ( -16 + 16 \zeta_{6} ) q^{16} + ( -15 + 15 \zeta_{6} ) q^{17} -36 q^{20} + 30 \zeta_{6} q^{23} -56 \zeta_{6} q^{25} + 20 \zeta_{6} q^{28} + ( -45 + 45 \zeta_{6} ) q^{35} + ( -36 + 36 \zeta_{6} ) q^{36} + ( 85 - 85 \zeta_{6} ) q^{43} -12 \zeta_{6} q^{44} -81 q^{45} -75 \zeta_{6} q^{47} -24 q^{49} + ( 27 - 27 \zeta_{6} ) q^{55} -103 \zeta_{6} q^{61} + 45 \zeta_{6} q^{63} + 64 q^{64} + 60 q^{68} + ( 25 - 25 \zeta_{6} ) q^{73} -15 q^{77} + 144 \zeta_{6} q^{80} + ( -81 + 81 \zeta_{6} ) q^{81} + 90 q^{83} + 135 \zeta_{6} q^{85} + ( 120 - 120 \zeta_{6} ) q^{92} -27 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} + 9 q^{5} - 10 q^{7} - 9 q^{9} + O(q^{10}) \) \( 2 q - 4 q^{4} + 9 q^{5} - 10 q^{7} - 9 q^{9} + 6 q^{11} - 16 q^{16} - 15 q^{17} - 72 q^{20} + 30 q^{23} - 56 q^{25} + 20 q^{28} - 45 q^{35} - 36 q^{36} + 85 q^{43} - 12 q^{44} - 162 q^{45} - 75 q^{47} - 48 q^{49} + 27 q^{55} - 103 q^{61} + 45 q^{63} + 128 q^{64} + 120 q^{68} + 25 q^{73} - 30 q^{77} + 144 q^{80} - 81 q^{81} + 180 q^{83} + 135 q^{85} + 120 q^{92} - 27 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/361\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 −2.00000 + 3.46410i 4.50000 + 7.79423i 0 −5.00000 0 −4.50000 + 7.79423i 0
293.1 0 0 −2.00000 3.46410i 4.50000 7.79423i 0 −5.00000 0 −4.50000 7.79423i 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
19.c even 3 1 inner
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.3.d.a 2
19.b odd 2 1 CM 361.3.d.a 2
19.c even 3 1 19.3.b.a 1
19.c even 3 1 inner 361.3.d.a 2
19.d odd 6 1 19.3.b.a 1
19.d odd 6 1 inner 361.3.d.a 2
19.e even 9 6 361.3.f.a 6
19.f odd 18 6 361.3.f.a 6
57.f even 6 1 171.3.c.a 1
57.h odd 6 1 171.3.c.a 1
76.f even 6 1 304.3.e.a 1
76.g odd 6 1 304.3.e.a 1
95.h odd 6 1 475.3.c.a 1
95.i even 6 1 475.3.c.a 1
95.l even 12 2 475.3.d.a 2
95.m odd 12 2 475.3.d.a 2
152.k odd 6 1 1216.3.e.b 1
152.l odd 6 1 1216.3.e.a 1
152.o even 6 1 1216.3.e.b 1
152.p even 6 1 1216.3.e.a 1
228.m even 6 1 2736.3.o.a 1
228.n odd 6 1 2736.3.o.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.a 1 19.c even 3 1
19.3.b.a 1 19.d odd 6 1
171.3.c.a 1 57.f even 6 1
171.3.c.a 1 57.h odd 6 1
304.3.e.a 1 76.f even 6 1
304.3.e.a 1 76.g odd 6 1
361.3.d.a 2 1.a even 1 1 trivial
361.3.d.a 2 19.b odd 2 1 CM
361.3.d.a 2 19.c even 3 1 inner
361.3.d.a 2 19.d odd 6 1 inner
361.3.f.a 6 19.e even 9 6
361.3.f.a 6 19.f odd 18 6
475.3.c.a 1 95.h odd 6 1
475.3.c.a 1 95.i even 6 1
475.3.d.a 2 95.l even 12 2
475.3.d.a 2 95.m odd 12 2
1216.3.e.a 1 152.l odd 6 1
1216.3.e.a 1 152.p even 6 1
1216.3.e.b 1 152.k odd 6 1
1216.3.e.b 1 152.o even 6 1
2736.3.o.a 1 228.m even 6 1
2736.3.o.a 1 228.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(361, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 81 - 9 T + T^{2} \)
$7$ \( ( 5 + T )^{2} \)
$11$ \( ( -3 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( 225 + 15 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( 900 - 30 T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( 7225 - 85 T + T^{2} \)
$47$ \( 5625 + 75 T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 10609 + 103 T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 625 - 25 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( ( -90 + T )^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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