Properties

Label 4410.2.b.d
Level $4410$
Weight $2$
Character orbit 4410.b
Analytic conductor $35.214$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(35.2140272914\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
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 \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{16}^{4} q^{2} - q^{4} + q^{5} + \zeta_{16}^{4} q^{8} +O(q^{10})\) \( q -\zeta_{16}^{4} q^{2} - q^{4} + q^{5} + \zeta_{16}^{4} q^{8} -\zeta_{16}^{4} q^{10} + ( \zeta_{16}^{3} - 2 \zeta_{16}^{4} + \zeta_{16}^{5} ) q^{11} + ( -\zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} - 4 \zeta_{16}^{4} - \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{13} + q^{16} + ( \zeta_{16}^{2} - 3 \zeta_{16}^{3} + 3 \zeta_{16}^{5} - \zeta_{16}^{6} ) q^{17} + ( 2 \zeta_{16} - \zeta_{16}^{2} - \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{19} - q^{20} + ( -2 + \zeta_{16} - \zeta_{16}^{7} ) q^{22} + ( \zeta_{16} - \zeta_{16}^{2} - 5 \zeta_{16}^{3} + 2 \zeta_{16}^{4} - 5 \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{23} + q^{25} + ( -4 - \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{26} + ( \zeta_{16} - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{4} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{29} + ( 2 \zeta_{16} - 2 \zeta_{16}^{2} + \zeta_{16}^{3} + 4 \zeta_{16}^{4} + \zeta_{16}^{5} - 2 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{31} -\zeta_{16}^{4} q^{32} + ( 3 \zeta_{16} - \zeta_{16}^{2} - \zeta_{16}^{6} + 3 \zeta_{16}^{7} ) q^{34} + ( -2 + 3 \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{5} - 3 \zeta_{16}^{6} ) q^{37} + ( -\zeta_{16}^{2} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{38} + \zeta_{16}^{4} q^{40} + ( 6 - \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{41} + ( -2 + 2 \zeta_{16} - \zeta_{16}^{2} - 5 \zeta_{16}^{3} + 5 \zeta_{16}^{5} + \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{43} + ( -\zeta_{16}^{3} + 2 \zeta_{16}^{4} - \zeta_{16}^{5} ) q^{44} + ( 2 - 5 \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{6} + 5 \zeta_{16}^{7} ) q^{46} + ( -6 + \zeta_{16} - \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{47} -\zeta_{16}^{4} q^{50} + ( \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} + 4 \zeta_{16}^{4} + \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{52} + ( 5 \zeta_{16} + 2 \zeta_{16}^{2} - \zeta_{16}^{3} + 4 \zeta_{16}^{4} - \zeta_{16}^{5} + 2 \zeta_{16}^{6} + 5 \zeta_{16}^{7} ) q^{53} + ( \zeta_{16}^{3} - 2 \zeta_{16}^{4} + \zeta_{16}^{5} ) q^{55} + ( -2 + 2 \zeta_{16} - 2 \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{5} + 2 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{58} + ( 2 - 2 \zeta_{16} + \zeta_{16}^{2} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} - \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{59} + ( -2 \zeta_{16} - \zeta_{16}^{2} - 6 \zeta_{16}^{4} - \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{61} + ( 4 + \zeta_{16} - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 2 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{62} - q^{64} + ( -\zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} - 4 \zeta_{16}^{4} - \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{65} + ( -5 \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{6} + 5 \zeta_{16}^{7} ) q^{67} + ( -\zeta_{16}^{2} + 3 \zeta_{16}^{3} - 3 \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{68} + ( 3 \zeta_{16}^{2} + 4 \zeta_{16}^{3} + 2 \zeta_{16}^{4} + 4 \zeta_{16}^{5} + 3 \zeta_{16}^{6} ) q^{71} + ( -5 \zeta_{16} - 4 \zeta_{16}^{2} + \zeta_{16}^{3} - 4 \zeta_{16}^{4} + \zeta_{16}^{5} - 4 \zeta_{16}^{6} - 5 \zeta_{16}^{7} ) q^{73} + ( \zeta_{16} - 3 \zeta_{16}^{2} + 2 \zeta_{16}^{4} - 3 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{74} + ( -2 \zeta_{16} + \zeta_{16}^{2} + \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{76} + ( -4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{79} + q^{80} + ( \zeta_{16} + \zeta_{16}^{3} - 6 \zeta_{16}^{4} + \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{82} + ( 2 + 2 \zeta_{16} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{83} + ( \zeta_{16}^{2} - 3 \zeta_{16}^{3} + 3 \zeta_{16}^{5} - \zeta_{16}^{6} ) q^{85} + ( 5 \zeta_{16} + \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{4} - 2 \zeta_{16}^{5} + \zeta_{16}^{6} + 5 \zeta_{16}^{7} ) q^{86} + ( 2 - \zeta_{16} + \zeta_{16}^{7} ) q^{88} + ( -6 - 4 \zeta_{16} + 2 \zeta_{16}^{2} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} - 2 \zeta_{16}^{6} + 4 \zeta_{16}^{7} ) q^{89} + ( -\zeta_{16} + \zeta_{16}^{2} + 5 \zeta_{16}^{3} - 2 \zeta_{16}^{4} + 5 \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{92} + ( -2 \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} + 6 \zeta_{16}^{4} - \zeta_{16}^{5} + \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{94} + ( 2 \zeta_{16} - \zeta_{16}^{2} - \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{95} + ( -4 \zeta_{16}^{3} - 6 \zeta_{16}^{4} - 4 \zeta_{16}^{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + 8q^{5} + O(q^{10}) \) \( 8q - 8q^{4} + 8q^{5} + 8q^{16} - 8q^{20} - 16q^{22} + 8q^{25} - 32q^{26} - 16q^{37} + 48q^{41} - 16q^{43} + 16q^{46} - 48q^{47} - 16q^{58} + 16q^{59} + 32q^{62} - 8q^{64} + 8q^{80} + 16q^{83} + 16q^{88} - 48q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4410\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(2647\) \(3431\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
−0.923880 0.382683i
−0.382683 + 0.923880i
0.382683 0.923880i
0.923880 + 0.382683i
0.923880 0.382683i
0.382683 + 0.923880i
−0.382683 0.923880i
−0.923880 + 0.382683i
1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 1.00000i
881.2 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 1.00000i
881.3 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 1.00000i
881.4 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 1.00000i
881.5 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 1.00000i
881.6 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 1.00000i
881.7 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 1.00000i
881.8 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4410.2.b.d yes 8
3.b odd 2 1 4410.2.b.a 8
7.b odd 2 1 4410.2.b.a 8
21.c even 2 1 inner 4410.2.b.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4410.2.b.a 8 3.b odd 2 1
4410.2.b.a 8 7.b odd 2 1
4410.2.b.d yes 8 1.a even 1 1 trivial
4410.2.b.d yes 8 21.c 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 \(S_{2}^{\mathrm{new}}(4410, [\chi])\):

\( T_{11}^{8} + 24 T_{11}^{6} + 148 T_{11}^{4} + 176 T_{11}^{2} + 4 \)
\( T_{17}^{4} - 40 T_{17}^{2} + 72 T_{17} + 94 \)