Properties

Label 990.2.n.e
Level $990$
Weight $2$
Character orbit 990.n
Analytic conductor $7.905$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 990.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.90518980011\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 330)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} + ( -2 + 2 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{7} -\zeta_{10}^{2} q^{8} +O(q^{10})\) \( q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} + ( -2 + 2 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{7} -\zeta_{10}^{2} q^{8} - q^{10} + ( -1 - \zeta_{10} - \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{11} + ( -4 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{13} + ( 2 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{14} -\zeta_{10} q^{16} + ( -5 + 2 \zeta_{10} - 5 \zeta_{10}^{2} ) q^{17} + ( 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{19} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{20} + ( -2 + 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{22} + ( -5 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{23} + \zeta_{10}^{2} q^{25} + ( 1 - \zeta_{10} + 4 \zeta_{10}^{3} ) q^{26} + ( 2 + 2 \zeta_{10}^{2} ) q^{28} + ( 1 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{29} + ( 2 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{31} - q^{32} + ( -3 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{34} + ( 2 - 2 \zeta_{10}^{3} ) q^{35} + ( 1 - \zeta_{10} + 7 \zeta_{10}^{3} ) q^{37} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{38} + \zeta_{10}^{3} q^{40} + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{41} + ( 3 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{43} + ( -2 + 4 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{44} + ( -5 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{46} + ( 7 \zeta_{10} - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{47} + ( -4 + 3 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{49} + \zeta_{10} q^{50} + ( -\zeta_{10} + 5 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{52} + ( 6 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{53} + ( 3 - 2 \zeta_{10} + 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{55} + ( 2 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{56} + ( -\zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{58} + ( -5 + 5 \zeta_{10} + 11 \zeta_{10}^{3} ) q^{59} + ( -1 + \zeta_{10} - 2 \zeta_{10}^{3} ) q^{62} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{64} + ( 4 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{65} + ( -2 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{67} + ( -3 - 2 \zeta_{10} + 2 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{68} + ( 2 - 2 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{70} + ( 4 + 4 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{71} + ( 12 - 12 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{73} + ( -\zeta_{10} + 8 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{74} + ( 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{76} + ( -2 \zeta_{10} - 4 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{77} + ( 1 + 4 \zeta_{10} - 4 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{79} + \zeta_{10}^{2} q^{80} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{82} + 4 \zeta_{10} q^{83} + ( 5 \zeta_{10} - 2 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{85} + ( 3 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{86} + ( 2 + \zeta_{10} + 2 \zeta_{10}^{3} ) q^{88} + ( -16 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{89} + ( -8 \zeta_{10} + 2 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{91} + ( -3 + 3 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{92} + ( 7 - 7 \zeta_{10} + 7 \zeta_{10}^{2} ) q^{94} + ( 2 - 2 \zeta_{10} ) q^{95} + ( -10 - 4 \zeta_{10} + 4 \zeta_{10}^{2} + 10 \zeta_{10}^{3} ) q^{97} + ( -1 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{2} - q^{4} - q^{5} - 4q^{7} + q^{8} + O(q^{10}) \) \( 4q + q^{2} - q^{4} - q^{5} - 4q^{7} + q^{8} - 4q^{10} - q^{11} - 2q^{13} + 4q^{14} - q^{16} - 13q^{17} + 6q^{19} - q^{20} - 9q^{22} - 14q^{23} - q^{25} + 7q^{26} + 6q^{28} + q^{29} - 4q^{32} - 2q^{34} + 6q^{35} + 10q^{37} + 4q^{38} + q^{40} - 6q^{41} + 22q^{43} - q^{44} - 11q^{46} + 21q^{47} - 9q^{49} + q^{50} - 7q^{52} + 14q^{53} + 4q^{55} + 4q^{56} - q^{58} - 4q^{59} - 5q^{62} - q^{64} + 18q^{65} + 2q^{67} - 13q^{68} + 4q^{70} + 16q^{71} + 30q^{73} - 10q^{74} - 4q^{76} - 4q^{77} + 11q^{79} - q^{80} - 4q^{82} + 4q^{83} + 12q^{85} - 7q^{86} + 11q^{88} - 60q^{89} - 18q^{91} - 4q^{92} + 14q^{94} + 6q^{95} - 38q^{97} + 4q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/990\mathbb{Z}\right)^\times\).

\(n\) \(397\) \(541\) \(551\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
0.809017 + 0.587785i
−0.309017 + 0.951057i
−0.309017 0.951057i
0.809017 0.587785i
0.809017 0.587785i 0 0.309017 0.951057i −0.809017 0.587785i 0 −1.00000 + 3.07768i −0.309017 0.951057i 0 −1.00000
181.1 −0.309017 0.951057i 0 −0.809017 + 0.587785i 0.309017 0.951057i 0 −1.00000 + 0.726543i 0.809017 + 0.587785i 0 −1.00000
361.1 −0.309017 + 0.951057i 0 −0.809017 0.587785i 0.309017 + 0.951057i 0 −1.00000 0.726543i 0.809017 0.587785i 0 −1.00000
631.1 0.809017 + 0.587785i 0 0.309017 + 0.951057i −0.809017 + 0.587785i 0 −1.00000 3.07768i −0.309017 + 0.951057i 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 990.2.n.e 4
3.b odd 2 1 330.2.m.b 4
11.c even 5 1 inner 990.2.n.e 4
33.f even 10 1 3630.2.a.bb 2
33.h odd 10 1 330.2.m.b 4
33.h odd 10 1 3630.2.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.b 4 3.b odd 2 1
330.2.m.b 4 33.h odd 10 1
990.2.n.e 4 1.a even 1 1 trivial
990.2.n.e 4 11.c even 5 1 inner
3630.2.a.bb 2 33.f even 10 1
3630.2.a.bj 2 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(990, [\chi])\):

\( T_{7}^{4} + 4 T_{7}^{3} + 16 T_{7}^{2} + 24 T_{7} + 16 \)
\( T_{13}^{4} + 2 T_{13}^{3} + 24 T_{13}^{2} + 133 T_{13} + 361 \)
\( T_{17}^{4} + 13 T_{17}^{3} + 94 T_{17}^{2} + 372 T_{17} + 961 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$7$ \( 16 + 24 T + 16 T^{2} + 4 T^{3} + T^{4} \)
$11$ \( 121 + 11 T - 9 T^{2} + T^{3} + T^{4} \)
$13$ \( 361 + 133 T + 24 T^{2} + 2 T^{3} + T^{4} \)
$17$ \( 961 + 372 T + 94 T^{2} + 13 T^{3} + T^{4} \)
$19$ \( 16 - 16 T + 16 T^{2} - 6 T^{3} + T^{4} \)
$23$ \( ( 1 + 7 T + T^{2} )^{2} \)
$29$ \( 1 + 4 T + 6 T^{2} - T^{3} + T^{4} \)
$31$ \( 25 - 25 T + 10 T^{2} + T^{4} \)
$37$ \( 3025 - 275 T + 60 T^{2} - 10 T^{3} + T^{4} \)
$41$ \( 16 + 16 T + 16 T^{2} + 6 T^{3} + T^{4} \)
$43$ \( ( -1 - 11 T + T^{2} )^{2} \)
$47$ \( 2401 - 686 T + 196 T^{2} - 21 T^{3} + T^{4} \)
$53$ \( 16 - 24 T + 76 T^{2} - 14 T^{3} + T^{4} \)
$59$ \( 1681 - 861 T + 166 T^{2} + 4 T^{3} + T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( -31 - T + T^{2} )^{2} \)
$71$ \( 256 + 64 T + 96 T^{2} - 16 T^{3} + T^{4} \)
$73$ \( 32400 - 5400 T + 540 T^{2} - 30 T^{3} + T^{4} \)
$79$ \( 841 - 406 T + 96 T^{2} - 11 T^{3} + T^{4} \)
$83$ \( 256 - 64 T + 16 T^{2} - 4 T^{3} + T^{4} \)
$89$ \( ( 220 + 30 T + T^{2} )^{2} \)
$97$ \( 55696 + 7552 T + 744 T^{2} + 38 T^{3} + T^{4} \)
show more
show less