Properties

Label 5850.2.e.bi
Level $5850$
Weight $2$
Character orbit 5850.e
Analytic conductor $46.712$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5850.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(46.7124851824\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{41})\)
Defining polynomial: \(x^{4} + 21 x^{2} + 100\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1950)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - q^{4} + ( \beta_{1} - \beta_{2} ) q^{7} -\beta_{2} q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} - q^{4} + ( \beta_{1} - \beta_{2} ) q^{7} -\beta_{2} q^{8} + ( -2 + \beta_{3} ) q^{11} -\beta_{2} q^{13} + ( 2 - \beta_{3} ) q^{14} + q^{16} + ( \beta_{1} + 3 \beta_{2} ) q^{17} + ( -1 - \beta_{3} ) q^{19} + ( \beta_{1} - \beta_{2} ) q^{22} + q^{26} + ( -\beta_{1} + \beta_{2} ) q^{28} + ( -1 + 2 \beta_{3} ) q^{29} + ( -2 + 3 \beta_{3} ) q^{31} + \beta_{2} q^{32} + ( -2 - \beta_{3} ) q^{34} + ( -\beta_{1} - 2 \beta_{2} ) q^{37} + ( -\beta_{1} - 2 \beta_{2} ) q^{38} + ( 1 - \beta_{3} ) q^{41} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{43} + ( 2 - \beta_{3} ) q^{44} + 7 \beta_{2} q^{47} + ( -7 + 3 \beta_{3} ) q^{49} + \beta_{2} q^{52} + ( 2 \beta_{1} + 5 \beta_{2} ) q^{53} + ( -2 + \beta_{3} ) q^{56} + ( 2 \beta_{1} + \beta_{2} ) q^{58} -\beta_{3} q^{59} + ( 6 - 3 \beta_{3} ) q^{61} + ( 3 \beta_{1} + \beta_{2} ) q^{62} - q^{64} + ( -2 \beta_{1} - \beta_{2} ) q^{67} + ( -\beta_{1} - 3 \beta_{2} ) q^{68} + ( -1 - \beta_{3} ) q^{71} -12 \beta_{2} q^{73} + ( 1 + \beta_{3} ) q^{74} + ( 1 + \beta_{3} ) q^{76} + ( -3 \beta_{1} + 11 \beta_{2} ) q^{77} + ( 3 - \beta_{3} ) q^{79} -\beta_{1} q^{82} + ( -\beta_{1} + 7 \beta_{2} ) q^{83} + ( 6 - 2 \beta_{3} ) q^{86} + ( -\beta_{1} + \beta_{2} ) q^{88} + ( 6 - 2 \beta_{3} ) q^{89} + ( -2 + \beta_{3} ) q^{91} -7 q^{94} + ( 2 \beta_{1} + 10 \beta_{2} ) q^{97} + ( 3 \beta_{1} - 4 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + O(q^{10}) \) \( 4 q - 4 q^{4} - 6 q^{11} + 6 q^{14} + 4 q^{16} - 6 q^{19} + 4 q^{26} - 2 q^{31} - 10 q^{34} + 2 q^{41} + 6 q^{44} - 22 q^{49} - 6 q^{56} - 2 q^{59} + 18 q^{61} - 4 q^{64} - 6 q^{71} + 6 q^{74} + 6 q^{76} + 10 q^{79} + 20 q^{86} + 20 q^{89} - 6 q^{91} - 28 q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 21 x^{2} + 100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 11 \nu \)\()/10\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 11 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 11\)
\(\nu^{3}\)\(=\)\(10 \beta_{2} - 11 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times\).

\(n\) \(2251\) \(3251\) \(3277\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5149.1
2.70156i
3.70156i
3.70156i
2.70156i
1.00000i 0 −1.00000 0 0 1.70156i 1.00000i 0 0
5149.2 1.00000i 0 −1.00000 0 0 4.70156i 1.00000i 0 0
5149.3 1.00000i 0 −1.00000 0 0 4.70156i 1.00000i 0 0
5149.4 1.00000i 0 −1.00000 0 0 1.70156i 1.00000i 0 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 5850.2.e.bi 4
3.b odd 2 1 1950.2.e.p 4
5.b even 2 1 inner 5850.2.e.bi 4
5.c odd 4 1 5850.2.a.cg 2
5.c odd 4 1 5850.2.a.cj 2
15.d odd 2 1 1950.2.e.p 4
15.e even 4 1 1950.2.a.bc 2
15.e even 4 1 1950.2.a.bg yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.a.bc 2 15.e even 4 1
1950.2.a.bg yes 2 15.e even 4 1
1950.2.e.p 4 3.b odd 2 1
1950.2.e.p 4 15.d odd 2 1
5850.2.a.cg 2 5.c odd 4 1
5850.2.a.cj 2 5.c odd 4 1
5850.2.e.bi 4 1.a even 1 1 trivial
5850.2.e.bi 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5850, [\chi])\):

\( T_{7}^{4} + 25 T_{7}^{2} + 64 \)
\( T_{11}^{2} + 3 T_{11} - 8 \)
\( T_{17}^{4} + 33 T_{17}^{2} + 16 \)
\( T_{19}^{2} + 3 T_{19} - 8 \)
\( T_{29}^{2} - 41 \)
\( T_{31}^{2} + T_{31} - 92 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( 64 + 25 T^{2} + T^{4} \)
$11$ \( ( -8 + 3 T + T^{2} )^{2} \)
$13$ \( ( 1 + T^{2} )^{2} \)
$17$ \( 16 + 33 T^{2} + T^{4} \)
$19$ \( ( -8 + 3 T + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( ( -41 + T^{2} )^{2} \)
$31$ \( ( -92 + T + T^{2} )^{2} \)
$37$ \( 64 + 25 T^{2} + T^{4} \)
$41$ \( ( -10 - T + T^{2} )^{2} \)
$43$ \( 256 + 132 T^{2} + T^{4} \)
$47$ \( ( 49 + T^{2} )^{2} \)
$53$ \( 625 + 114 T^{2} + T^{4} \)
$59$ \( ( -10 + T + T^{2} )^{2} \)
$61$ \( ( -72 - 9 T + T^{2} )^{2} \)
$67$ \( ( 41 + T^{2} )^{2} \)
$71$ \( ( -8 + 3 T + T^{2} )^{2} \)
$73$ \( ( 144 + T^{2} )^{2} \)
$79$ \( ( -4 - 5 T + T^{2} )^{2} \)
$83$ \( 2116 + 133 T^{2} + T^{4} \)
$89$ \( ( -16 - 10 T + T^{2} )^{2} \)
$97$ \( 1600 + 244 T^{2} + T^{4} \)
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