Properties

Label 900.2.k.l
Level $900$
Weight $2$
Character orbit 900.k
Analytic conductor $7.187$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{2} + 2 \zeta_{24}^{4} q^{4} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{7} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} +O(q^{10})\) \( q + ( \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{2} + 2 \zeta_{24}^{4} q^{4} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{7} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} -2 \zeta_{24}^{6} q^{11} + ( -4 \zeta_{24} + \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{13} + ( -3 - 3 \zeta_{24}^{2} - \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{14} + ( -4 + 4 \zeta_{24}^{4} ) q^{16} + 6 \zeta_{24}^{3} q^{17} + ( 2 - 6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{19} + ( 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{22} + ( -2 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{23} + ( -3 - 3 \zeta_{24}^{2} + 5 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{26} + ( -2 \zeta_{24} - 8 \zeta_{24}^{3} + 4 \zeta_{24}^{7} ) q^{28} + ( -2 + 4 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{29} + ( -1 + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{31} + ( -4 \zeta_{24} + 4 \zeta_{24}^{7} ) q^{32} + ( -6 + 6 \zeta_{24}^{2} + 6 \zeta_{24}^{4} ) q^{34} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{37} + ( -6 \zeta_{24} - \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{38} + ( 2 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{41} + ( 2 \zeta_{24} + \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{43} + ( 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{44} + ( 2 - 2 \zeta_{24}^{2} - 6 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{46} + ( 6 \zeta_{24} - 4 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{47} + ( 4 - 8 \zeta_{24}^{4} - 6 \zeta_{24}^{6} ) q^{49} + ( -8 \zeta_{24} + 4 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{52} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{53} + ( 2 - 8 \zeta_{24}^{2} - 8 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{56} + ( -4 \zeta_{24} + 6 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{58} + ( 8 + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{59} + 3 q^{61} + ( -2 \zeta_{24} - \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{62} -8 q^{64} + ( 9 \zeta_{24} - 2 \zeta_{24}^{3} - 9 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{67} + 12 \zeta_{24}^{7} q^{68} + ( 2 - 4 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{71} + ( 4 \zeta_{24} - 4 \zeta_{24}^{5} ) q^{73} + ( 8 \zeta_{24}^{4} + 8 \zeta_{24}^{6} ) q^{74} + ( -6 \zeta_{24}^{2} + 4 \zeta_{24}^{4} - 6 \zeta_{24}^{6} ) q^{76} + ( 4 \zeta_{24} + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{77} + ( 4 + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{79} + ( -4 \zeta_{24} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{82} + ( 4 \zeta_{24} + 6 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{83} + ( 1 - \zeta_{24}^{2} + 3 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{86} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} ) q^{88} + ( 4 - 8 \zeta_{24}^{4} ) q^{89} + ( 7 - 14 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{91} + ( 4 \zeta_{24} - 8 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{92} + ( 2 + 2 \zeta_{24}^{2} - 10 \zeta_{24}^{4} + 8 \zeta_{24}^{6} ) q^{94} + ( \zeta_{24} + 8 \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{97} + ( 8 \zeta_{24} + 2 \zeta_{24}^{3} - 10 \zeta_{24}^{5} - 10 \zeta_{24}^{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{4} + O(q^{10}) \) \( 8q + 8q^{4} - 28q^{14} - 16q^{16} + 16q^{19} - 4q^{26} - 24q^{34} + 16q^{41} - 8q^{46} - 16q^{56} + 64q^{59} + 24q^{61} - 64q^{64} + 32q^{74} + 16q^{76} + 32q^{79} + 20q^{86} - 24q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{24}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−0.965926 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
0.258819 0.965926i
−1.22474 0.707107i 0 1.00000 + 1.73205i 0 0 3.15660 3.15660i 2.82843i 0 0
307.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 0 0 1.74238 1.74238i 2.82843i 0 0
307.3 1.22474 0.707107i 0 1.00000 1.73205i 0 0 −1.74238 + 1.74238i 2.82843i 0 0
307.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 0 0 −3.15660 + 3.15660i 2.82843i 0 0
343.1 −1.22474 0.707107i 0 1.00000 + 1.73205i 0 0 1.74238 + 1.74238i 2.82843i 0 0
343.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 0 0 3.15660 + 3.15660i 2.82843i 0 0
343.3 1.22474 0.707107i 0 1.00000 1.73205i 0 0 −3.15660 3.15660i 2.82843i 0 0
343.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 0 0 −1.74238 1.74238i 2.82843i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 343.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
20.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.k.l 8
3.b odd 2 1 300.2.j.c yes 8
4.b odd 2 1 900.2.k.g 8
5.b even 2 1 inner 900.2.k.l 8
5.c odd 4 2 900.2.k.g 8
12.b even 2 1 300.2.j.a 8
15.d odd 2 1 300.2.j.c yes 8
15.e even 4 2 300.2.j.a 8
20.d odd 2 1 900.2.k.g 8
20.e even 4 2 inner 900.2.k.l 8
60.h even 2 1 300.2.j.a 8
60.l odd 4 2 300.2.j.c yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.j.a 8 12.b even 2 1
300.2.j.a 8 15.e even 4 2
300.2.j.a 8 60.h even 2 1
300.2.j.c yes 8 3.b odd 2 1
300.2.j.c yes 8 15.d odd 2 1
300.2.j.c yes 8 60.l odd 4 2
900.2.k.g 8 4.b odd 2 1
900.2.k.g 8 5.c odd 4 2
900.2.k.g 8 20.d odd 2 1
900.2.k.l 8 1.a even 1 1 trivial
900.2.k.l 8 5.b even 2 1 inner
900.2.k.l 8 20.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{8} + 434 T_{7}^{4} + 14641 \)
\( T_{11}^{2} + 4 \)
\( T_{13}^{8} + 1106 T_{13}^{4} + 28561 \)
\( T_{17}^{4} + 1296 \)
\( T_{19}^{2} - 4 T_{19} - 23 \)
\( T_{23}^{8} + 3104 T_{23}^{4} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( 14641 + 434 T^{4} + T^{8} \)
$11$ \( ( 4 + T^{2} )^{4} \)
$13$ \( 28561 + 1106 T^{4} + T^{8} \)
$17$ \( ( 1296 + T^{4} )^{2} \)
$19$ \( ( -23 - 4 T + T^{2} )^{4} \)
$23$ \( 256 + 3104 T^{4} + T^{8} \)
$29$ \( ( 16 + 56 T^{2} + T^{4} )^{2} \)
$31$ \( ( 1 + 14 T^{2} + T^{4} )^{2} \)
$37$ \( 1048576 + 14336 T^{4} + T^{8} \)
$41$ \( ( -8 - 4 T + T^{2} )^{4} \)
$43$ \( 14641 + 434 T^{4} + T^{8} \)
$47$ \( 20736 + 27936 T^{4} + T^{8} \)
$53$ \( 4096 + 896 T^{4} + T^{8} \)
$59$ \( ( 52 - 16 T + T^{2} )^{4} \)
$61$ \( ( -3 + T )^{8} \)
$67$ \( 22667121 + 25074 T^{4} + T^{8} \)
$71$ \( ( 576 + 96 T^{2} + T^{4} )^{2} \)
$73$ \( ( 256 + T^{4} )^{2} \)
$79$ \( ( 4 - 8 T + T^{2} )^{4} \)
$83$ \( 20736 + 27936 T^{4} + T^{8} \)
$89$ \( ( 48 + T^{2} )^{4} \)
$97$ \( 13845841 + 10514 T^{4} + T^{8} \)
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