Properties

Label 2850.2.d.f
Level $2850$
Weight $2$
Character orbit 2850.d
Analytic conductor $22.757$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.7573645761\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 570)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -i q^{2} -i q^{3} - q^{4} - q^{6} + 4 i q^{7} + i q^{8} - q^{9} +O(q^{10})\) \( q -i q^{2} -i q^{3} - q^{4} - q^{6} + 4 i q^{7} + i q^{8} - q^{9} + i q^{12} -2 i q^{13} + 4 q^{14} + q^{16} -2 i q^{17} + i q^{18} + q^{19} + 4 q^{21} + q^{24} -2 q^{26} + i q^{27} -4 i q^{28} -10 q^{29} -i q^{32} -2 q^{34} + q^{36} + 2 i q^{37} -i q^{38} -2 q^{39} + 2 q^{41} -4 i q^{42} + 4 i q^{43} -i q^{48} -9 q^{49} -2 q^{51} + 2 i q^{52} + 6 i q^{53} + q^{54} -4 q^{56} -i q^{57} + 10 i q^{58} -8 q^{59} + 6 q^{61} -4 i q^{63} - q^{64} + 12 i q^{67} + 2 i q^{68} -i q^{72} + 14 i q^{73} + 2 q^{74} - q^{76} + 2 i q^{78} + q^{81} -2 i q^{82} + 12 i q^{83} -4 q^{84} + 4 q^{86} + 10 i q^{87} -10 q^{89} + 8 q^{91} - q^{96} + 2 i q^{97} + 9 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + 8q^{14} + 2q^{16} + 2q^{19} + 8q^{21} + 2q^{24} - 4q^{26} - 20q^{29} - 4q^{34} + 2q^{36} - 4q^{39} + 4q^{41} - 18q^{49} - 4q^{51} + 2q^{54} - 8q^{56} - 16q^{59} + 12q^{61} - 2q^{64} + 4q^{74} - 2q^{76} + 2q^{81} - 8q^{84} + 8q^{86} - 20q^{89} + 16q^{91} - 2q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times\).

\(n\) \(1027\) \(1351\) \(1901\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 4.00000i 1.00000i −1.00000 0
799.2 1.00000i 1.00000i −1.00000 0 −1.00000 4.00000i 1.00000i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2850.2.d.f 2
5.b even 2 1 inner 2850.2.d.f 2
5.c odd 4 1 570.2.a.d 1
5.c odd 4 1 2850.2.a.p 1
15.e even 4 1 1710.2.a.t 1
15.e even 4 1 8550.2.a.b 1
20.e even 4 1 4560.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.a.d 1 5.c odd 4 1
1710.2.a.t 1 15.e even 4 1
2850.2.a.p 1 5.c odd 4 1
2850.2.d.f 2 1.a even 1 1 trivial
2850.2.d.f 2 5.b even 2 1 inner
4560.2.a.a 1 20.e even 4 1
8550.2.a.b 1 15.e even 4 1

Hecke kernels

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

\( T_{7}^{2} + 16 \)
\( T_{11} \)
\( T_{13}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 10 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( 8 + T )^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 196 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( 4 + T^{2} \)
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