Properties

Label 2850.2.d.k
Level $2850$
Weight $2$
Character orbit 2850.d
Analytic conductor $22.757$
Analytic rank $1$
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: \(1\)
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.

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

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.k 2
5.b even 2 1 inner 2850.2.d.k 2
5.c odd 4 1 570.2.a.m 1
5.c odd 4 1 2850.2.a.a 1
15.e even 4 1 1710.2.a.f 1
15.e even 4 1 8550.2.a.t 1
20.e even 4 1 4560.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.a.m 1 5.c odd 4 1
1710.2.a.f 1 15.e even 4 1
2850.2.a.a 1 5.c odd 4 1
2850.2.d.k 2 1.a even 1 1 trivial
2850.2.d.k 2 5.b even 2 1 inner
4560.2.a.k 1 20.e even 4 1
8550.2.a.t 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 \(S_{2}^{\mathrm{new}}(2850, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 64 + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( 144 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( -12 + T )^{2} \)
$83$ \( 64 + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( 324 + T^{2} \)
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