Properties

Label 2850.2.d.v
Level $2850$
Weight $2$
Character orbit 2850.d
Analytic conductor $22.757$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{10})\)
Defining polynomial: \(x^{4} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{1} q^{3} - q^{4} - q^{6} + \beta_{2} q^{7} -\beta_{1} q^{8} - q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{1} q^{3} - q^{4} - q^{6} + \beta_{2} q^{7} -\beta_{1} q^{8} - q^{9} + ( 1 + \beta_{3} ) q^{11} -\beta_{1} q^{12} + ( -2 \beta_{1} - \beta_{2} ) q^{13} -\beta_{3} q^{14} + q^{16} + ( 4 \beta_{1} - \beta_{2} ) q^{17} -\beta_{1} q^{18} + q^{19} -\beta_{3} q^{21} + ( \beta_{1} + \beta_{2} ) q^{22} + ( -\beta_{1} + 2 \beta_{2} ) q^{23} + q^{24} + ( 2 + \beta_{3} ) q^{26} -\beta_{1} q^{27} -\beta_{2} q^{28} + ( 7 - \beta_{3} ) q^{29} + ( -1 + 2 \beta_{3} ) q^{31} + \beta_{1} q^{32} + ( \beta_{1} + \beta_{2} ) q^{33} + ( -4 + \beta_{3} ) q^{34} + q^{36} + 10 \beta_{1} q^{37} + \beta_{1} q^{38} + ( 2 + \beta_{3} ) q^{39} -2 \beta_{3} q^{41} -\beta_{2} q^{42} + ( 6 \beta_{1} + \beta_{2} ) q^{43} + ( -1 - \beta_{3} ) q^{44} + ( 1 - 2 \beta_{3} ) q^{46} + 6 \beta_{1} q^{47} + \beta_{1} q^{48} -3 q^{49} + ( -4 + \beta_{3} ) q^{51} + ( 2 \beta_{1} + \beta_{2} ) q^{52} + ( -5 \beta_{1} + \beta_{2} ) q^{53} + q^{54} + \beta_{3} q^{56} + \beta_{1} q^{57} + ( 7 \beta_{1} - \beta_{2} ) q^{58} + ( -2 + 3 \beta_{3} ) q^{59} + ( -1 - \beta_{3} ) q^{61} + ( -\beta_{1} + 2 \beta_{2} ) q^{62} -\beta_{2} q^{63} - q^{64} + ( -1 - \beta_{3} ) q^{66} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{67} + ( -4 \beta_{1} + \beta_{2} ) q^{68} + ( 1 - 2 \beta_{3} ) q^{69} + ( -2 + \beta_{3} ) q^{71} + \beta_{1} q^{72} + \beta_{1} q^{73} -10 q^{74} - q^{76} + ( 10 \beta_{1} + \beta_{2} ) q^{77} + ( 2 \beta_{1} + \beta_{2} ) q^{78} + ( -1 + 2 \beta_{3} ) q^{79} + q^{81} -2 \beta_{2} q^{82} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{83} + \beta_{3} q^{84} + ( -6 - \beta_{3} ) q^{86} + ( 7 \beta_{1} - \beta_{2} ) q^{87} + ( -\beta_{1} - \beta_{2} ) q^{88} + ( 1 + 2 \beta_{3} ) q^{89} + ( 10 + 2 \beta_{3} ) q^{91} + ( \beta_{1} - 2 \beta_{2} ) q^{92} + ( -\beta_{1} + 2 \beta_{2} ) q^{93} -6 q^{94} - q^{96} + ( 10 \beta_{1} - 3 \beta_{2} ) q^{97} -3 \beta_{1} q^{98} + ( -1 - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} - 4q^{6} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} - 4q^{6} - 4q^{9} + 4q^{11} + 4q^{16} + 4q^{19} + 4q^{24} + 8q^{26} + 28q^{29} - 4q^{31} - 16q^{34} + 4q^{36} + 8q^{39} - 4q^{44} + 4q^{46} - 12q^{49} - 16q^{51} + 4q^{54} - 8q^{59} - 4q^{61} - 4q^{64} - 4q^{66} + 4q^{69} - 8q^{71} - 40q^{74} - 4q^{76} - 4q^{79} + 4q^{81} - 24q^{86} + 4q^{89} + 40q^{91} - 24q^{94} - 4q^{96} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 5 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 5 \nu \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\(5 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{3} + 5 \beta_{2}\)\()/2\)

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
1.58114 1.58114i
−1.58114 + 1.58114i
−1.58114 1.58114i
1.58114 + 1.58114i
1.00000i 1.00000i −1.00000 0 −1.00000 3.16228i 1.00000i −1.00000 0
799.2 1.00000i 1.00000i −1.00000 0 −1.00000 3.16228i 1.00000i −1.00000 0
799.3 1.00000i 1.00000i −1.00000 0 −1.00000 3.16228i 1.00000i −1.00000 0
799.4 1.00000i 1.00000i −1.00000 0 −1.00000 3.16228i 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.v 4
5.b even 2 1 inner 2850.2.d.v 4
5.c odd 4 1 2850.2.a.bf 2
5.c odd 4 1 2850.2.a.bg yes 2
15.e even 4 1 8550.2.a.bq 2
15.e even 4 1 8550.2.a.ca 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2850.2.a.bf 2 5.c odd 4 1
2850.2.a.bg yes 2 5.c odd 4 1
2850.2.d.v 4 1.a even 1 1 trivial
2850.2.d.v 4 5.b even 2 1 inner
8550.2.a.bq 2 15.e even 4 1
8550.2.a.ca 2 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 \(S_{2}^{\mathrm{new}}(2850, [\chi])\):

\( T_{7}^{2} + 10 \)
\( T_{11}^{2} - 2 T_{11} - 9 \)
\( T_{13}^{4} + 28 T_{13}^{2} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 10 + T^{2} )^{2} \)
$11$ \( ( -9 - 2 T + T^{2} )^{2} \)
$13$ \( 36 + 28 T^{2} + T^{4} \)
$17$ \( 36 + 52 T^{2} + T^{4} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( 1521 + 82 T^{2} + T^{4} \)
$29$ \( ( 39 - 14 T + T^{2} )^{2} \)
$31$ \( ( -39 + 2 T + T^{2} )^{2} \)
$37$ \( ( 100 + T^{2} )^{2} \)
$41$ \( ( -40 + T^{2} )^{2} \)
$43$ \( 676 + 92 T^{2} + T^{4} \)
$47$ \( ( 36 + T^{2} )^{2} \)
$53$ \( 225 + 70 T^{2} + T^{4} \)
$59$ \( ( -86 + 4 T + T^{2} )^{2} \)
$61$ \( ( -9 + 2 T + T^{2} )^{2} \)
$67$ \( 6561 + 198 T^{2} + T^{4} \)
$71$ \( ( -6 + 4 T + T^{2} )^{2} \)
$73$ \( ( 1 + T^{2} )^{2} \)
$79$ \( ( -39 + 2 T + T^{2} )^{2} \)
$83$ \( 6561 + 198 T^{2} + T^{4} \)
$89$ \( ( -39 - 2 T + T^{2} )^{2} \)
$97$ \( 100 + 380 T^{2} + T^{4} \)
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