Properties

Label 2850.2.a.bc
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: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−2.44949
2.44949
−1.00000 −1.00000 1.00000 0 1.00000 −4.44949 −1.00000 1.00000 0
1.2 −1.00000 −1.00000 1.00000 0 1.00000 0.449490 −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.bc 2
3.b odd 2 1 8550.2.a.bv 2
5.b even 2 1 2850.2.a.bj yes 2
5.c odd 4 2 2850.2.d.w 4
15.d odd 2 1 8550.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2850.2.a.bc 2 1.a even 1 1 trivial
2850.2.a.bj yes 2 5.b even 2 1
2850.2.d.w 4 5.c odd 4 2
8550.2.a.bu 2 15.d odd 2 1
8550.2.a.bv 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2850))\):

\( T_{7}^{2} + 4 T_{7} - 2 \)
\( T_{11}^{2} - 2 T_{11} - 5 \)
\( T_{13}^{2} - 6 \)
\( T_{23} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -2 + 4 T + T^{2} \)
$11$ \( -5 - 2 T + T^{2} \)
$13$ \( -6 + T^{2} \)
$17$ \( -2 + 4 T + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( -45 - 6 T + T^{2} \)
$31$ \( ( 3 + T )^{2} \)
$37$ \( -92 + 4 T + T^{2} \)
$41$ \( -8 - 8 T + T^{2} \)
$43$ \( -6 + T^{2} \)
$47$ \( -92 + 4 T + T^{2} \)
$53$ \( 19 - 10 T + T^{2} \)
$59$ \( 10 - 8 T + T^{2} \)
$61$ \( 43 + 14 T + T^{2} \)
$67$ \( -141 + 6 T + T^{2} \)
$71$ \( 10 + 8 T + T^{2} \)
$73$ \( ( -1 + T )^{2} \)
$79$ \( ( 5 + T )^{2} \)
$83$ \( -53 + 2 T + T^{2} \)
$89$ \( 25 - 14 T + T^{2} \)
$97$ \( 10 + 8 T + T^{2} \)
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