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: \( x^{2} - 6 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1850))\):

\( T_{3}^{2} - 2T_{3} - 5 \) Copy content Toggle raw display
\( T_{7}^{2} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

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