Properties

Label 1849.2.a.g
Level $1849$
Weight $2$
Character orbit 1849.a
Self dual yes
Analytic conductor $14.764$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7643393337\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} -\beta q^{3} + 4 q^{4} -\beta q^{5} -6 q^{6} + \beta q^{7} + 2 \beta q^{8} + 3 q^{9} +O(q^{10})\) \( q + \beta q^{2} -\beta q^{3} + 4 q^{4} -\beta q^{5} -6 q^{6} + \beta q^{7} + 2 \beta q^{8} + 3 q^{9} -6 q^{10} - q^{11} -4 \beta q^{12} -3 q^{13} + 6 q^{14} + 6 q^{15} + 4 q^{16} -7 q^{17} + 3 \beta q^{18} + 2 \beta q^{19} -4 \beta q^{20} -6 q^{21} -\beta q^{22} + q^{23} -12 q^{24} + q^{25} -3 \beta q^{26} + 4 \beta q^{28} -\beta q^{29} + 6 \beta q^{30} -3 q^{31} + \beta q^{33} -7 \beta q^{34} -6 q^{35} + 12 q^{36} -2 \beta q^{37} + 12 q^{38} + 3 \beta q^{39} -12 q^{40} -5 q^{41} -6 \beta q^{42} -4 q^{44} -3 \beta q^{45} + \beta q^{46} -10 q^{47} -4 \beta q^{48} - q^{49} + \beta q^{50} + 7 \beta q^{51} -12 q^{52} - q^{53} + \beta q^{55} + 12 q^{56} -12 q^{57} -6 q^{58} -10 q^{59} + 24 q^{60} + 3 \beta q^{61} -3 \beta q^{62} + 3 \beta q^{63} -8 q^{64} + 3 \beta q^{65} + 6 q^{66} + 9 q^{67} -28 q^{68} -\beta q^{69} -6 \beta q^{70} + 2 \beta q^{71} + 6 \beta q^{72} -5 \beta q^{73} -12 q^{74} -\beta q^{75} + 8 \beta q^{76} -\beta q^{77} + 18 q^{78} -6 q^{79} -4 \beta q^{80} -9 q^{81} -5 \beta q^{82} + q^{83} -24 q^{84} + 7 \beta q^{85} + 6 q^{87} -2 \beta q^{88} -7 \beta q^{89} -18 q^{90} -3 \beta q^{91} + 4 q^{92} + 3 \beta q^{93} -10 \beta q^{94} -12 q^{95} + 11 q^{97} -\beta q^{98} -3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{4} - 12q^{6} + 6q^{9} + O(q^{10}) \) \( 2q + 8q^{4} - 12q^{6} + 6q^{9} - 12q^{10} - 2q^{11} - 6q^{13} + 12q^{14} + 12q^{15} + 8q^{16} - 14q^{17} - 12q^{21} + 2q^{23} - 24q^{24} + 2q^{25} - 6q^{31} - 12q^{35} + 24q^{36} + 24q^{38} - 24q^{40} - 10q^{41} - 8q^{44} - 20q^{47} - 2q^{49} - 24q^{52} - 2q^{53} + 24q^{56} - 24q^{57} - 12q^{58} - 20q^{59} + 48q^{60} - 16q^{64} + 12q^{66} + 18q^{67} - 56q^{68} - 24q^{74} + 36q^{78} - 12q^{79} - 18q^{81} + 2q^{83} - 48q^{84} + 12q^{87} - 36q^{90} + 8q^{92} - 24q^{95} + 22q^{97} - 6q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.44949
2.44949
−2.44949 2.44949 4.00000 2.44949 −6.00000 −2.44949 −4.89898 3.00000 −6.00000
1.2 2.44949 −2.44949 4.00000 −2.44949 −6.00000 2.44949 4.89898 3.00000 −6.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(43\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1849.2.a.g 2
43.b odd 2 1 inner 1849.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1849.2.a.g 2 1.a even 1 1 trivial
1849.2.a.g 2 43.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 6 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1849))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -6 + T^{2} \)
$3$ \( -6 + T^{2} \)
$5$ \( -6 + T^{2} \)
$7$ \( -6 + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( 3 + T )^{2} \)
$17$ \( ( 7 + T )^{2} \)
$19$ \( -24 + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( -6 + T^{2} \)
$31$ \( ( 3 + T )^{2} \)
$37$ \( -24 + T^{2} \)
$41$ \( ( 5 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( ( 10 + T )^{2} \)
$53$ \( ( 1 + T )^{2} \)
$59$ \( ( 10 + T )^{2} \)
$61$ \( -54 + T^{2} \)
$67$ \( ( -9 + T )^{2} \)
$71$ \( -24 + T^{2} \)
$73$ \( -150 + T^{2} \)
$79$ \( ( 6 + T )^{2} \)
$83$ \( ( -1 + T )^{2} \)
$89$ \( -294 + T^{2} \)
$97$ \( ( -11 + T )^{2} \)
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