Properties

Label 2850.2.d.w
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: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{1} q^{3} - q^{4} + q^{6} + ( 2 \beta_{1} - \beta_{2} ) q^{7} -\beta_{1} q^{8} - q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{1} q^{3} - q^{4} + q^{6} + ( 2 \beta_{1} - \beta_{2} ) q^{7} -\beta_{1} q^{8} - q^{9} + ( 1 - \beta_{3} ) q^{11} + \beta_{1} q^{12} -\beta_{2} q^{13} + ( -2 + \beta_{3} ) q^{14} + q^{16} + ( 2 \beta_{1} - \beta_{2} ) q^{17} -\beta_{1} q^{18} - q^{19} + ( 2 - \beta_{3} ) q^{21} + ( \beta_{1} - \beta_{2} ) q^{22} + \beta_{1} q^{23} - q^{24} + \beta_{3} q^{26} + \beta_{1} q^{27} + ( -2 \beta_{1} + \beta_{2} ) q^{28} + ( -3 - 3 \beta_{3} ) q^{29} -3 q^{31} + \beta_{1} q^{32} + ( -\beta_{1} + \beta_{2} ) q^{33} + ( -2 + \beta_{3} ) q^{34} + q^{36} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{37} -\beta_{1} q^{38} -\beta_{3} q^{39} + ( 4 + 2 \beta_{3} ) q^{41} + ( 2 \beta_{1} - \beta_{2} ) q^{42} -\beta_{2} q^{43} + ( -1 + \beta_{3} ) q^{44} - q^{46} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{47} -\beta_{1} q^{48} + ( -3 + 4 \beta_{3} ) q^{49} + ( 2 - \beta_{3} ) q^{51} + \beta_{2} q^{52} + ( 5 \beta_{1} - \beta_{2} ) q^{53} - q^{54} + ( 2 - \beta_{3} ) q^{56} + \beta_{1} q^{57} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{58} + ( -4 + \beta_{3} ) q^{59} + ( -7 + \beta_{3} ) q^{61} -3 \beta_{1} q^{62} + ( -2 \beta_{1} + \beta_{2} ) q^{63} - q^{64} + ( 1 - \beta_{3} ) q^{66} + ( 3 \beta_{1} - 5 \beta_{2} ) q^{67} + ( -2 \beta_{1} + \beta_{2} ) q^{68} + q^{69} + ( -4 - \beta_{3} ) q^{71} + \beta_{1} q^{72} + \beta_{1} q^{73} + ( -2 - 4 \beta_{3} ) q^{74} + q^{76} + ( 8 \beta_{1} - 3 \beta_{2} ) q^{77} -\beta_{2} q^{78} + 5 q^{79} + q^{81} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{82} + ( -\beta_{1} + 3 \beta_{2} ) q^{83} + ( -2 + \beta_{3} ) q^{84} + \beta_{3} q^{86} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{87} + ( -\beta_{1} + \beta_{2} ) q^{88} + ( -7 - 2 \beta_{3} ) q^{89} + ( -6 + 2 \beta_{3} ) q^{91} -\beta_{1} q^{92} + 3 \beta_{1} q^{93} + ( -2 - 4 \beta_{3} ) q^{94} + q^{96} + ( 4 \beta_{1} + \beta_{2} ) q^{97} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{98} + ( -1 + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} + 4q^{11} - 8q^{14} + 4q^{16} - 4q^{19} + 8q^{21} - 4q^{24} - 12q^{29} - 12q^{31} - 8q^{34} + 4q^{36} + 16q^{41} - 4q^{44} - 4q^{46} - 12q^{49} + 8q^{51} - 4q^{54} + 8q^{56} - 16q^{59} - 28q^{61} - 4q^{64} + 4q^{66} + 4q^{69} - 16q^{71} - 8q^{74} + 4q^{76} + 20q^{79} + 4q^{81} - 8q^{84} - 28q^{89} - 24q^{91} - 8q^{94} + 4q^{96} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

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

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
−1.22474 + 1.22474i
1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 1.22474i
1.00000i 1.00000i −1.00000 0 1.00000 4.44949i 1.00000i −1.00000 0
799.2 1.00000i 1.00000i −1.00000 0 1.00000 0.449490i 1.00000i −1.00000 0
799.3 1.00000i 1.00000i −1.00000 0 1.00000 0.449490i 1.00000i −1.00000 0
799.4 1.00000i 1.00000i −1.00000 0 1.00000 4.44949i 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.w 4
5.b even 2 1 inner 2850.2.d.w 4
5.c odd 4 1 2850.2.a.bc 2
5.c odd 4 1 2850.2.a.bj yes 2
15.e even 4 1 8550.2.a.bu 2
15.e even 4 1 8550.2.a.bv 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2850.2.a.bc 2 5.c odd 4 1
2850.2.a.bj yes 2 5.c odd 4 1
2850.2.d.w 4 1.a even 1 1 trivial
2850.2.d.w 4 5.b even 2 1 inner
8550.2.a.bu 2 15.e even 4 1
8550.2.a.bv 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 \(S_{2}^{\mathrm{new}}(2850, [\chi])\):

\( T_{7}^{4} + 20 T_{7}^{2} + 4 \)
\( T_{11}^{2} - 2 T_{11} - 5 \)
\( T_{13}^{2} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 4 + 20 T^{2} + T^{4} \)
$11$ \( ( -5 - 2 T + T^{2} )^{2} \)
$13$ \( ( 6 + T^{2} )^{2} \)
$17$ \( 4 + 20 T^{2} + T^{4} \)
$19$ \( ( 1 + T )^{4} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( -45 + 6 T + T^{2} )^{2} \)
$31$ \( ( 3 + T )^{4} \)
$37$ \( 8464 + 200 T^{2} + T^{4} \)
$41$ \( ( -8 - 8 T + T^{2} )^{2} \)
$43$ \( ( 6 + T^{2} )^{2} \)
$47$ \( 8464 + 200 T^{2} + T^{4} \)
$53$ \( 361 + 62 T^{2} + T^{4} \)
$59$ \( ( 10 + 8 T + T^{2} )^{2} \)
$61$ \( ( 43 + 14 T + T^{2} )^{2} \)
$67$ \( 19881 + 318 T^{2} + T^{4} \)
$71$ \( ( 10 + 8 T + T^{2} )^{2} \)
$73$ \( ( 1 + T^{2} )^{2} \)
$79$ \( ( -5 + T )^{4} \)
$83$ \( 2809 + 110 T^{2} + T^{4} \)
$89$ \( ( 25 + 14 T + T^{2} )^{2} \)
$97$ \( 100 + 44 T^{2} + T^{4} \)
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