Properties

Label 8850.2.a.be
Level 8850
Weight 2
Character orbit 8850.a
Self dual yes
Analytic conductor 70.668
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 8850 = 2 \cdot 3 \cdot 5^{2} \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.6676057888\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 354)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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

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 8850.2.a.be 1
5.b even 2 1 354.2.a.a 1
15.d odd 2 1 1062.2.a.k 1
20.d odd 2 1 2832.2.a.e 1
60.h even 2 1 8496.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
354.2.a.a 1 5.b even 2 1
1062.2.a.k 1 15.d odd 2 1
2832.2.a.e 1 20.d odd 2 1
8496.2.a.j 1 60.h even 2 1
8850.2.a.be 1 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(59\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8850))\):

\( T_{7} - 1 \)
\( T_{11} + 5 \)
\( T_{13} + 1 \)
\( T_{17} + 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( 1 - T \)
$5$ 1
$7$ \( 1 - T + 7 T^{2} \)
$11$ \( 1 + 5 T + 11 T^{2} \)
$13$ \( 1 + T + 13 T^{2} \)
$17$ \( 1 + T + 17 T^{2} \)
$19$ \( 1 + 19 T^{2} \)
$23$ \( 1 - 6 T + 23 T^{2} \)
$29$ \( 1 + 10 T + 29 T^{2} \)
$31$ \( 1 + 8 T + 31 T^{2} \)
$37$ \( 1 + 9 T + 37 T^{2} \)
$41$ \( 1 + 5 T + 41 T^{2} \)
$43$ \( 1 + 3 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 + 4 T + 53 T^{2} \)
$59$ \( 1 + T \)
$61$ \( 1 - 6 T + 61 T^{2} \)
$67$ \( 1 + 4 T + 67 T^{2} \)
$71$ \( 1 - T + 71 T^{2} \)
$73$ \( 1 + 73 T^{2} \)
$79$ \( 1 + 3 T + 79 T^{2} \)
$83$ \( 1 + 7 T + 83 T^{2} \)
$89$ \( 1 - 16 T + 89 T^{2} \)
$97$ \( 1 - 12 T + 97 T^{2} \)
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