Properties

Label 1850.2.a.w
Level $1850$
Weight $2$
Character orbit 1850.a
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + q^{2} + ( 1 + \beta ) q^{3} + q^{4} + ( 1 + \beta ) q^{6} -\beta q^{7} + q^{8} + ( 4 + 2 \beta ) q^{9} +O(q^{10})\) \( q + q^{2} + ( 1 + \beta ) q^{3} + q^{4} + ( 1 + \beta ) q^{6} -\beta q^{7} + q^{8} + ( 4 + 2 \beta ) q^{9} + ( 1 - \beta ) q^{11} + ( 1 + \beta ) q^{12} + ( 2 + \beta ) q^{13} -\beta q^{14} + q^{16} + ( 1 - \beta ) q^{17} + ( 4 + 2 \beta ) q^{18} -5 q^{19} + ( -6 - \beta ) q^{21} + ( 1 - \beta ) q^{22} + 2 q^{23} + ( 1 + \beta ) q^{24} + ( 2 + \beta ) q^{26} + ( 13 + 3 \beta ) q^{27} -\beta q^{28} + ( 4 + 2 \beta ) q^{29} + ( 2 - \beta ) q^{31} + q^{32} -5 q^{33} + ( 1 - \beta ) q^{34} + ( 4 + 2 \beta ) q^{36} - q^{37} -5 q^{38} + ( 8 + 3 \beta ) q^{39} + q^{41} + ( -6 - \beta ) q^{42} + ( -6 - 2 \beta ) q^{43} + ( 1 - \beta ) q^{44} + 2 q^{46} -4 \beta q^{47} + ( 1 + \beta ) q^{48} - q^{49} -5 q^{51} + ( 2 + \beta ) q^{52} + 6 q^{53} + ( 13 + 3 \beta ) q^{54} -\beta q^{56} + ( -5 - 5 \beta ) q^{57} + ( 4 + 2 \beta ) q^{58} -2 q^{59} + ( -4 + \beta ) q^{61} + ( 2 - \beta ) q^{62} + ( -12 - 4 \beta ) q^{63} + q^{64} -5 q^{66} + ( 7 + \beta ) q^{67} + ( 1 - \beta ) q^{68} + ( 2 + 2 \beta ) q^{69} + ( -10 - \beta ) q^{71} + ( 4 + 2 \beta ) q^{72} + ( 3 - 4 \beta ) q^{73} - q^{74} -5 q^{76} + ( 6 - \beta ) q^{77} + ( 8 + 3 \beta ) q^{78} + ( -2 - 4 \beta ) q^{79} + ( 19 + 10 \beta ) q^{81} + q^{82} + ( 1 - \beta ) q^{83} + ( -6 - \beta ) q^{84} + ( -6 - 2 \beta ) q^{86} + ( 16 + 6 \beta ) q^{87} + ( 1 - \beta ) q^{88} + ( 7 - 3 \beta ) q^{89} + ( -6 - 2 \beta ) q^{91} + 2 q^{92} + ( -4 + \beta ) q^{93} -4 \beta q^{94} + ( 1 + \beta ) q^{96} + 14 q^{97} - q^{98} + ( -8 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 2q^{8} + 8q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 2q^{8} + 8q^{9} + 2q^{11} + 2q^{12} + 4q^{13} + 2q^{16} + 2q^{17} + 8q^{18} - 10q^{19} - 12q^{21} + 2q^{22} + 4q^{23} + 2q^{24} + 4q^{26} + 26q^{27} + 8q^{29} + 4q^{31} + 2q^{32} - 10q^{33} + 2q^{34} + 8q^{36} - 2q^{37} - 10q^{38} + 16q^{39} + 2q^{41} - 12q^{42} - 12q^{43} + 2q^{44} + 4q^{46} + 2q^{48} - 2q^{49} - 10q^{51} + 4q^{52} + 12q^{53} + 26q^{54} - 10q^{57} + 8q^{58} - 4q^{59} - 8q^{61} + 4q^{62} - 24q^{63} + 2q^{64} - 10q^{66} + 14q^{67} + 2q^{68} + 4q^{69} - 20q^{71} + 8q^{72} + 6q^{73} - 2q^{74} - 10q^{76} + 12q^{77} + 16q^{78} - 4q^{79} + 38q^{81} + 2q^{82} + 2q^{83} - 12q^{84} - 12q^{86} + 32q^{87} + 2q^{88} + 14q^{89} - 12q^{91} + 4q^{92} - 8q^{93} + 2q^{96} + 28q^{97} - 2q^{98} - 16q^{99} + O(q^{100}) \)

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
1.00000 −1.44949 1.00000 0 −1.44949 2.44949 1.00000 −0.898979 0
1.2 1.00000 3.44949 1.00000 0 3.44949 −2.44949 1.00000 8.89898 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(37\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1850.2.a.w yes 2
5.b even 2 1 1850.2.a.r 2
5.c odd 4 2 1850.2.b.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1850.2.a.r 2 5.b even 2 1
1850.2.a.w yes 2 1.a even 1 1 trivial
1850.2.b.k 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1850))\):

\( T_{3}^{2} - 2 T_{3} - 5 \)
\( T_{7}^{2} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( -5 - 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -6 + T^{2} \)
$11$ \( -5 - 2 T + T^{2} \)
$13$ \( -2 - 4 T + T^{2} \)
$17$ \( -5 - 2 T + T^{2} \)
$19$ \( ( 5 + T )^{2} \)
$23$ \( ( -2 + T )^{2} \)
$29$ \( -8 - 8 T + T^{2} \)
$31$ \( -2 - 4 T + T^{2} \)
$37$ \( ( 1 + T )^{2} \)
$41$ \( ( -1 + T )^{2} \)
$43$ \( 12 + 12 T + T^{2} \)
$47$ \( -96 + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( ( 2 + T )^{2} \)
$61$ \( 10 + 8 T + T^{2} \)
$67$ \( 43 - 14 T + T^{2} \)
$71$ \( 94 + 20 T + T^{2} \)
$73$ \( -87 - 6 T + T^{2} \)
$79$ \( -92 + 4 T + T^{2} \)
$83$ \( -5 - 2 T + T^{2} \)
$89$ \( -5 - 14 T + T^{2} \)
$97$ \( ( -14 + T )^{2} \)
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