Properties

Label 546.2.c.c
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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + i q^{2} + q^{3} - q^{4} - i q^{5} + i q^{6} - i q^{7} - i q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + q^{3} - q^{4} - i q^{5} + i q^{6} - i q^{7} - i q^{8} + q^{9} + q^{10} + 3 i q^{11} - q^{12} + ( - 2 i + 3) q^{13} + q^{14} - i q^{15} + q^{16} + 7 q^{17} + i q^{18} + 3 i q^{19} + i q^{20} - i q^{21} - 3 q^{22} + q^{23} - i q^{24} + 4 q^{25} + (3 i + 2) q^{26} + q^{27} + i q^{28} - q^{29} + q^{30} - 8 i q^{31} + i q^{32} + 3 i q^{33} + 7 i q^{34} - q^{35} - q^{36} - i q^{37} - 3 q^{38} + ( - 2 i + 3) q^{39} - q^{40} + 4 i q^{41} + q^{42} - 5 q^{43} - 3 i q^{44} - i q^{45} + i q^{46} + q^{48} - q^{49} + 4 i q^{50} + 7 q^{51} + (2 i - 3) q^{52} - 6 q^{53} + i q^{54} + 3 q^{55} - q^{56} + 3 i q^{57} - i q^{58} + 10 i q^{59} + i q^{60} - 13 q^{61} + 8 q^{62} - i q^{63} - q^{64} + ( - 3 i - 2) q^{65} - 3 q^{66} - 8 i q^{67} - 7 q^{68} + q^{69} - i q^{70} + 6 i q^{71} - i q^{72} - 13 i q^{73} + q^{74} + 4 q^{75} - 3 i q^{76} + 3 q^{77} + (3 i + 2) q^{78} - 12 q^{79} - i q^{80} + q^{81} - 4 q^{82} + 2 i q^{83} + i q^{84} - 7 i q^{85} - 5 i q^{86} - q^{87} + 3 q^{88} + 12 i q^{89} + q^{90} + ( - 3 i - 2) q^{91} - q^{92} - 8 i q^{93} + 3 q^{95} + i q^{96} + 6 i q^{97} - i q^{98} + 3 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} - 2 q^{12} + 6 q^{13} + 2 q^{14} + 2 q^{16} + 14 q^{17} - 6 q^{22} + 2 q^{23} + 8 q^{25} + 4 q^{26} + 2 q^{27} - 2 q^{29} + 2 q^{30} - 2 q^{35} - 2 q^{36} - 6 q^{38} + 6 q^{39} - 2 q^{40} + 2 q^{42} - 10 q^{43} + 2 q^{48} - 2 q^{49} + 14 q^{51} - 6 q^{52} - 12 q^{53} + 6 q^{55} - 2 q^{56} - 26 q^{61} + 16 q^{62} - 2 q^{64} - 4 q^{65} - 6 q^{66} - 14 q^{68} + 2 q^{69} + 2 q^{74} + 8 q^{75} + 6 q^{77} + 4 q^{78} - 24 q^{79} + 2 q^{81} - 8 q^{82} - 2 q^{87} + 6 q^{88} + 2 q^{90} - 4 q^{91} - 2 q^{92} + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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 1.00000i 1.00000i 1.00000i 1.00000i 1.00000 1.00000
337.2 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 1.00000i 1.00000i 1.00000 1.00000
\(n\): 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.c 2
3.b odd 2 1 1638.2.c.e 2
4.b odd 2 1 4368.2.h.h 2
7.b odd 2 1 3822.2.c.b 2
13.b even 2 1 inner 546.2.c.c 2
13.d odd 4 1 7098.2.a.o 1
13.d odd 4 1 7098.2.a.y 1
39.d odd 2 1 1638.2.c.e 2
52.b odd 2 1 4368.2.h.h 2
91.b odd 2 1 3822.2.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.c 2 1.a even 1 1 trivial
546.2.c.c 2 13.b even 2 1 inner
1638.2.c.e 2 3.b odd 2 1
1638.2.c.e 2 39.d odd 2 1
3822.2.c.b 2 7.b odd 2 1
3822.2.c.b 2 91.b odd 2 1
4368.2.h.h 2 4.b odd 2 1
4368.2.h.h 2 52.b odd 2 1
7098.2.a.o 1 13.d odd 4 1
7098.2.a.y 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 \(S_{2}^{\mathrm{new}}(546, [\chi])\):

\( T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 13 \) Copy content Toggle raw display
$17$ \( (T - 7)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 9 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 64 \) Copy content Toggle raw display
$37$ \( T^{2} + 1 \) Copy content Toggle raw display
$41$ \( T^{2} + 16 \) Copy content Toggle raw display
$43$ \( (T + 5)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 100 \) Copy content Toggle raw display
$61$ \( (T + 13)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( T^{2} + 36 \) Copy content Toggle raw display
$73$ \( T^{2} + 169 \) Copy content Toggle raw display
$79$ \( (T + 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4 \) Copy content Toggle raw display
$89$ \( T^{2} + 144 \) Copy content Toggle raw display
$97$ \( T^{2} + 36 \) Copy content Toggle raw display
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