Properties

Label 2850.2.a.bf
Level $2850$
Weight $2$
Character orbit 2850.a
Self dual yes
Analytic conductor $22.757$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.7573645761\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
Defining polynomial: \(x^{2} - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−3.16228
3.16228
−1.00000 1.00000 1.00000 0 −1.00000 −3.16228 −1.00000 1.00000 0
1.2 −1.00000 1.00000 1.00000 0 −1.00000 3.16228 −1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(19\) \(1\)

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 2850.2.a.bf 2
3.b odd 2 1 8550.2.a.ca 2
5.b even 2 1 2850.2.a.bg yes 2
5.c odd 4 2 2850.2.d.v 4
15.d odd 2 1 8550.2.a.bq 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2850.2.a.bf 2 1.a even 1 1 trivial
2850.2.a.bg yes 2 5.b even 2 1
2850.2.d.v 4 5.c odd 4 2
8550.2.a.bq 2 15.d odd 2 1
8550.2.a.ca 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2850))\):

\( T_{7}^{2} - 10 \)
\( T_{11}^{2} - 2 T_{11} - 9 \)
\( T_{13}^{2} + 4 T_{13} - 6 \)
\( T_{23}^{2} + 2 T_{23} - 39 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -10 + T^{2} \)
$11$ \( -9 - 2 T + T^{2} \)
$13$ \( -6 + 4 T + T^{2} \)
$17$ \( 6 + 8 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( -39 + 2 T + T^{2} \)
$29$ \( 39 + 14 T + T^{2} \)
$31$ \( -39 + 2 T + T^{2} \)
$37$ \( ( 10 + T )^{2} \)
$41$ \( -40 + T^{2} \)
$43$ \( 26 - 12 T + T^{2} \)
$47$ \( ( 6 + T )^{2} \)
$53$ \( 15 + 10 T + T^{2} \)
$59$ \( -86 - 4 T + T^{2} \)
$61$ \( -9 + 2 T + T^{2} \)
$67$ \( -81 + 6 T + T^{2} \)
$71$ \( -6 + 4 T + T^{2} \)
$73$ \( ( -1 + T )^{2} \)
$79$ \( -39 - 2 T + T^{2} \)
$83$ \( -81 - 6 T + T^{2} \)
$89$ \( -39 + 2 T + T^{2} \)
$97$ \( 10 + 20 T + T^{2} \)
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