Properties

Label 273.2.cd.a
Level $273$
Weight $2$
Character orbit 273.cd
Analytic conductor $2.180$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.cd (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{4} + ( - 2 \zeta_{12}^{3} + 3 \zeta_{12}) q^{7} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{4} + ( - 2 \zeta_{12}^{3} + 3 \zeta_{12}) q^{7} + (3 \zeta_{12}^{2} - 3) q^{9} + ( - 2 \zeta_{12}^{2} - 2) q^{12} + ( - 4 \zeta_{12}^{3} + \zeta_{12}) q^{13} + ( - 4 \zeta_{12}^{2} + 4) q^{16} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 5 \zeta_{12} - 5) q^{19} + ( - 4 \zeta_{12}^{2} - 1) q^{21} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{25} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} + ( - 4 \zeta_{12}^{2} + 6) q^{28} + (\zeta_{12}^{3} + \zeta_{12}^{2} - 6 \zeta_{12} + 5) q^{31} + 6 \zeta_{12}^{3} q^{36} + (7 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{37} + (2 \zeta_{12}^{2} - 7) q^{39} + (14 \zeta_{12}^{2} - 7) q^{43} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{48} + ( - 3 \zeta_{12}^{2} + 8) q^{49} + ( - 8 \zeta_{12}^{2} + 2) q^{52} + (\zeta_{12}^{3} + 7 \zeta_{12}^{2} + 7 \zeta_{12} + 1) q^{57} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{61} + (9 \zeta_{12}^{3} - 3 \zeta_{12}) q^{63} - 8 \zeta_{12}^{3} q^{64} + (9 \zeta_{12}^{3} + 9 \zeta_{12}^{2} - 2 \zeta_{12} - 7) q^{67} + (8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + \zeta_{12} + 9) q^{73} + ( - 5 \zeta_{12}^{2} - 5) q^{75} + (10 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12} - 10) q^{76} + ( - 14 \zeta_{12}^{3} + 7 \zeta_{12}) q^{79} - 9 \zeta_{12}^{2} q^{81} + (2 \zeta_{12}^{3} - 10 \zeta_{12}) q^{84} + ( - 11 \zeta_{12}^{2} + 6) q^{91} + ( - 7 \zeta_{12}^{3} + 11 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{93} + (3 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 8 \zeta_{12} + 3) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{9} - 12 q^{12} + 8 q^{16} - 16 q^{19} - 12 q^{21} + 16 q^{28} + 22 q^{31} + 22 q^{37} - 24 q^{39} + 26 q^{49} - 8 q^{52} + 18 q^{57} - 10 q^{67} + 20 q^{73} - 30 q^{75} - 28 q^{76} - 18 q^{81} + 2 q^{91} + 6 q^{93} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{3}\) \(-1 + \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0.866025 + 1.50000i −1.73205 + 1.00000i 0 0 −2.59808 + 0.500000i 0 −1.50000 + 2.59808i 0
86.1 0 −0.866025 1.50000i 1.73205 1.00000i 0 0 2.59808 0.500000i 0 −1.50000 + 2.59808i 0
200.1 0 −0.866025 + 1.50000i 1.73205 + 1.00000i 0 0 2.59808 + 0.500000i 0 −1.50000 2.59808i 0
242.1 0 0.866025 1.50000i −1.73205 1.00000i 0 0 −2.59808 0.500000i 0 −1.50000 2.59808i 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 \(\Q(\sqrt{-3}) \)
91.z odd 12 1 inner
273.cd even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.cd.a 4
3.b odd 2 1 CM 273.2.cd.a 4
7.c even 3 1 273.2.cd.b yes 4
13.d odd 4 1 273.2.cd.b yes 4
21.h odd 6 1 273.2.cd.b yes 4
39.f even 4 1 273.2.cd.b yes 4
91.z odd 12 1 inner 273.2.cd.a 4
273.cd even 12 1 inner 273.2.cd.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.cd.a 4 1.a even 1 1 trivial
273.2.cd.a 4 3.b odd 2 1 CM
273.2.cd.a 4 91.z odd 12 1 inner
273.2.cd.a 4 273.cd even 12 1 inner
273.2.cd.b yes 4 7.c even 3 1
273.2.cd.b yes 4 13.d odd 4 1
273.2.cd.b yes 4 21.h odd 6 1
273.2.cd.b yes 4 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{19}^{4} + 16T_{19}^{3} + 65T_{19}^{2} - 22T_{19} + 121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 13T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 23T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 16 T^{3} + 65 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 22 T^{3} + 137 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$37$ \( T^{4} - 22 T^{3} + 221 T^{2} + \cdots + 5329 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 147)^{2} \) 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
$61$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$67$ \( T^{4} + 10 T^{3} + 281 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 20 T^{3} + 389 T^{2} + \cdots + 9409 \) Copy content Toggle raw display
$79$ \( T^{4} + 147 T^{2} + 21609 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 28 T^{3} + 392 T^{2} - 56 T + 4 \) Copy content Toggle raw display
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