Properties

Label 2850.2.d.x
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: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.866025 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
1.00000i 1.00000i −1.00000 0 1.00000 2.73205i 1.00000i −1.00000 0
799.2 1.00000i 1.00000i −1.00000 0 1.00000 0.732051i 1.00000i −1.00000 0
799.3 1.00000i 1.00000i −1.00000 0 1.00000 0.732051i 1.00000i −1.00000 0
799.4 1.00000i 1.00000i −1.00000 0 1.00000 2.73205i 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.x 4
5.b even 2 1 inner 2850.2.d.x 4
5.c odd 4 1 2850.2.a.bd 2
5.c odd 4 1 2850.2.a.bi yes 2
15.e even 4 1 8550.2.a.bs 2
15.e even 4 1 8550.2.a.by 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2850.2.a.bd 2 5.c odd 4 1
2850.2.a.bi yes 2 5.c odd 4 1
2850.2.d.x 4 1.a even 1 1 trivial
2850.2.d.x 4 5.b even 2 1 inner
8550.2.a.bs 2 15.e even 4 1
8550.2.a.by 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 \(S_{2}^{\mathrm{new}}(2850, [\chi])\):

\( T_{7}^{4} + 8 T_{7}^{2} + 4 \)
\( T_{11}^{2} - 4 T_{11} + 1 \)
\( T_{13}^{4} + 8 T_{13}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 4 + 8 T^{2} + T^{4} \)
$11$ \( ( 1 - 4 T + T^{2} )^{2} \)
$13$ \( 4 + 8 T^{2} + T^{4} \)
$17$ \( 676 + 56 T^{2} + T^{4} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( 2209 + 98 T^{2} + T^{4} \)
$29$ \( ( -3 + T^{2} )^{2} \)
$31$ \( ( -11 - 2 T + T^{2} )^{2} \)
$37$ \( ( 4 + T^{2} )^{2} \)
$41$ \( ( -32 - 8 T + T^{2} )^{2} \)
$43$ \( 324 + 72 T^{2} + T^{4} \)
$47$ \( ( 12 + T^{2} )^{2} \)
$53$ \( ( 3 + T^{2} )^{2} \)
$59$ \( ( -18 - 6 T + T^{2} )^{2} \)
$61$ \( ( -71 - 4 T + T^{2} )^{2} \)
$67$ \( 1 + 14 T^{2} + T^{4} \)
$71$ \( ( 22 - 14 T + T^{2} )^{2} \)
$73$ \( 121 + 26 T^{2} + T^{4} \)
$79$ \( ( 69 - 18 T + T^{2} )^{2} \)
$83$ \( 1 + 14 T^{2} + T^{4} \)
$89$ \( ( -183 - 6 T + T^{2} )^{2} \)
$97$ \( 19044 + 312 T^{2} + T^{4} \)
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