Properties

Label 546.2.c.d
Level $546$
Weight $2$
Character orbit 546.c
Analytic conductor $4.360$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + i q^{2} + q^{3} - q^{4} -3 i q^{5} + i q^{6} + i q^{7} -i q^{8} + q^{9} +O(q^{10})\) \( q + i q^{2} + q^{3} - q^{4} -3 i q^{5} + i q^{6} + i q^{7} -i q^{8} + q^{9} + 3 q^{10} -5 i q^{11} - q^{12} + ( -3 - 2 i ) q^{13} - q^{14} -3 i q^{15} + q^{16} + 3 q^{17} + i q^{18} -i q^{19} + 3 i q^{20} + i q^{21} + 5 q^{22} - q^{23} -i q^{24} -4 q^{25} + ( 2 - 3 i ) q^{26} + q^{27} -i q^{28} + 5 q^{29} + 3 q^{30} + i q^{32} -5 i q^{33} + 3 i q^{34} + 3 q^{35} - q^{36} -7 i q^{37} + q^{38} + ( -3 - 2 i ) q^{39} -3 q^{40} - q^{42} - q^{43} + 5 i q^{44} -3 i q^{45} -i q^{46} + 8 i q^{47} + q^{48} - q^{49} -4 i q^{50} + 3 q^{51} + ( 3 + 2 i ) q^{52} + 14 q^{53} + i q^{54} -15 q^{55} + q^{56} -i q^{57} + 5 i q^{58} + 14 i q^{59} + 3 i q^{60} -3 q^{61} + i q^{63} - q^{64} + ( -6 + 9 i ) q^{65} + 5 q^{66} + 8 i q^{67} -3 q^{68} - q^{69} + 3 i q^{70} -10 i q^{71} -i q^{72} + 11 i q^{73} + 7 q^{74} -4 q^{75} + i q^{76} + 5 q^{77} + ( 2 - 3 i ) q^{78} -3 i q^{80} + q^{81} + 6 i q^{83} -i q^{84} -9 i q^{85} -i q^{86} + 5 q^{87} -5 q^{88} -16 i q^{89} + 3 q^{90} + ( 2 - 3 i ) q^{91} + q^{92} -8 q^{94} -3 q^{95} + i q^{96} -2 i q^{97} -i q^{98} -5 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{4} + 2q^{9} + 6q^{10} - 2q^{12} - 6q^{13} - 2q^{14} + 2q^{16} + 6q^{17} + 10q^{22} - 2q^{23} - 8q^{25} + 4q^{26} + 2q^{27} + 10q^{29} + 6q^{30} + 6q^{35} - 2q^{36} + 2q^{38} - 6q^{39} - 6q^{40} - 2q^{42} - 2q^{43} + 2q^{48} - 2q^{49} + 6q^{51} + 6q^{52} + 28q^{53} - 30q^{55} + 2q^{56} - 6q^{61} - 2q^{64} - 12q^{65} + 10q^{66} - 6q^{68} - 2q^{69} + 14q^{74} - 8q^{75} + 10q^{77} + 4q^{78} + 2q^{81} + 10q^{87} - 10q^{88} + 6q^{90} + 4q^{91} + 2q^{92} - 16q^{94} - 6q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.00000i
1.00000i
1.00000i 1.00000 −1.00000 3.00000i 1.00000i 1.00000i 1.00000i 1.00000 3.00000
337.2 1.00000i 1.00000 −1.00000 3.00000i 1.00000i 1.00000i 1.00000i 1.00000 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.c.d 2
3.b odd 2 1 1638.2.c.g 2
4.b odd 2 1 4368.2.h.b 2
7.b odd 2 1 3822.2.c.a 2
13.b even 2 1 inner 546.2.c.d 2
13.d odd 4 1 7098.2.a.p 1
13.d odd 4 1 7098.2.a.x 1
39.d odd 2 1 1638.2.c.g 2
52.b odd 2 1 4368.2.h.b 2
91.b odd 2 1 3822.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.d 2 1.a even 1 1 trivial
546.2.c.d 2 13.b even 2 1 inner
1638.2.c.g 2 3.b odd 2 1
1638.2.c.g 2 39.d odd 2 1
3822.2.c.a 2 7.b odd 2 1
3822.2.c.a 2 91.b odd 2 1
4368.2.h.b 2 4.b odd 2 1
4368.2.h.b 2 52.b odd 2 1
7098.2.a.p 1 13.d odd 4 1
7098.2.a.x 1 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\):

\( T_{5}^{2} + 9 \)
\( T_{11}^{2} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( 9 + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( 25 + T^{2} \)
$13$ \( 13 + 6 T + T^{2} \)
$17$ \( ( -3 + T )^{2} \)
$19$ \( 1 + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( ( -5 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 49 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( ( -14 + T )^{2} \)
$59$ \( 196 + T^{2} \)
$61$ \( ( 3 + T )^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( 100 + T^{2} \)
$73$ \( 121 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( 256 + T^{2} \)
$97$ \( 4 + T^{2} \)
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