Properties

Label 900.1.t.a
Level $900$
Weight $1$
Character orbit 900.t
Analytic conductor $0.449$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -20
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 900.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.449158511370\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.1620.1
Artin image: $C_{12}\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{2} -\zeta_{12} q^{3} -\zeta_{12}^{4} q^{4} + q^{6} -\zeta_{12}^{5} q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{12}^{5} q^{2} -\zeta_{12} q^{3} -\zeta_{12}^{4} q^{4} + q^{6} -\zeta_{12}^{5} q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + \zeta_{12}^{5} q^{12} + \zeta_{12}^{4} q^{14} -\zeta_{12}^{2} q^{16} -\zeta_{12} q^{18} - q^{21} + \zeta_{12} q^{23} -\zeta_{12}^{4} q^{24} -\zeta_{12}^{3} q^{27} -\zeta_{12}^{3} q^{28} -\zeta_{12}^{2} q^{29} + \zeta_{12} q^{32} + q^{36} -\zeta_{12}^{4} q^{41} -\zeta_{12}^{5} q^{42} -2 \zeta_{12}^{5} q^{43} - q^{46} -\zeta_{12}^{5} q^{47} + \zeta_{12}^{3} q^{48} + \zeta_{12}^{2} q^{54} + \zeta_{12}^{2} q^{56} + \zeta_{12} q^{58} + \zeta_{12}^{2} q^{61} + \zeta_{12} q^{63} - q^{64} -\zeta_{12} q^{67} -\zeta_{12}^{2} q^{69} + \zeta_{12}^{5} q^{72} + \zeta_{12}^{4} q^{81} + \zeta_{12}^{3} q^{82} + \zeta_{12}^{5} q^{83} + \zeta_{12}^{4} q^{84} + 2 \zeta_{12}^{4} q^{86} + \zeta_{12}^{3} q^{87} + q^{89} -\zeta_{12}^{5} q^{92} + \zeta_{12}^{4} q^{94} -\zeta_{12}^{2} q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 4q^{6} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{4} + 4q^{6} + 2q^{9} - 2q^{14} - 2q^{16} - 4q^{21} + 2q^{24} - 2q^{29} + 4q^{36} + 2q^{41} - 4q^{46} + 2q^{54} + 2q^{56} + 2q^{61} - 4q^{64} - 2q^{69} - 2q^{81} - 2q^{84} - 4q^{86} + 4q^{89} - 2q^{94} - 2q^{96} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(\zeta_{12}^{4}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0 1.00000 0.866025 + 0.500000i 1.00000i 0.500000 0.866025i 0
151.2 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 0 1.00000 −0.866025 0.500000i 1.00000i 0.500000 0.866025i 0
751.1 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i 0 1.00000 0.866025 0.500000i 1.00000i 0.500000 + 0.866025i 0
751.2 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 0 1.00000 −0.866025 + 0.500000i 1.00000i 0.500000 + 0.866025i 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner
45.j even 6 1 inner
180.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.1.t.a 4
3.b odd 2 1 2700.1.t.a 4
4.b odd 2 1 inner 900.1.t.a 4
5.b even 2 1 inner 900.1.t.a 4
5.c odd 4 1 180.1.p.a 2
5.c odd 4 1 180.1.p.b yes 2
9.c even 3 1 inner 900.1.t.a 4
9.d odd 6 1 2700.1.t.a 4
12.b even 2 1 2700.1.t.a 4
15.d odd 2 1 2700.1.t.a 4
15.e even 4 1 540.1.p.a 2
15.e even 4 1 540.1.p.b 2
20.d odd 2 1 CM 900.1.t.a 4
20.e even 4 1 180.1.p.a 2
20.e even 4 1 180.1.p.b yes 2
36.f odd 6 1 inner 900.1.t.a 4
36.h even 6 1 2700.1.t.a 4
40.i odd 4 1 2880.1.bu.a 2
40.i odd 4 1 2880.1.bu.b 2
40.k even 4 1 2880.1.bu.a 2
40.k even 4 1 2880.1.bu.b 2
45.h odd 6 1 2700.1.t.a 4
45.j even 6 1 inner 900.1.t.a 4
45.k odd 12 1 180.1.p.a 2
45.k odd 12 1 180.1.p.b yes 2
45.k odd 12 1 1620.1.f.b 1
45.k odd 12 1 1620.1.f.d 1
45.l even 12 1 540.1.p.a 2
45.l even 12 1 540.1.p.b 2
45.l even 12 1 1620.1.f.a 1
45.l even 12 1 1620.1.f.c 1
60.h even 2 1 2700.1.t.a 4
60.l odd 4 1 540.1.p.a 2
60.l odd 4 1 540.1.p.b 2
180.n even 6 1 2700.1.t.a 4
180.p odd 6 1 inner 900.1.t.a 4
180.v odd 12 1 540.1.p.a 2
180.v odd 12 1 540.1.p.b 2
180.v odd 12 1 1620.1.f.a 1
180.v odd 12 1 1620.1.f.c 1
180.x even 12 1 180.1.p.a 2
180.x even 12 1 180.1.p.b yes 2
180.x even 12 1 1620.1.f.b 1
180.x even 12 1 1620.1.f.d 1
360.bo even 12 1 2880.1.bu.a 2
360.bo even 12 1 2880.1.bu.b 2
360.bu odd 12 1 2880.1.bu.a 2
360.bu odd 12 1 2880.1.bu.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.1.p.a 2 5.c odd 4 1
180.1.p.a 2 20.e even 4 1
180.1.p.a 2 45.k odd 12 1
180.1.p.a 2 180.x even 12 1
180.1.p.b yes 2 5.c odd 4 1
180.1.p.b yes 2 20.e even 4 1
180.1.p.b yes 2 45.k odd 12 1
180.1.p.b yes 2 180.x even 12 1
540.1.p.a 2 15.e even 4 1
540.1.p.a 2 45.l even 12 1
540.1.p.a 2 60.l odd 4 1
540.1.p.a 2 180.v odd 12 1
540.1.p.b 2 15.e even 4 1
540.1.p.b 2 45.l even 12 1
540.1.p.b 2 60.l odd 4 1
540.1.p.b 2 180.v odd 12 1
900.1.t.a 4 1.a even 1 1 trivial
900.1.t.a 4 4.b odd 2 1 inner
900.1.t.a 4 5.b even 2 1 inner
900.1.t.a 4 9.c even 3 1 inner
900.1.t.a 4 20.d odd 2 1 CM
900.1.t.a 4 36.f odd 6 1 inner
900.1.t.a 4 45.j even 6 1 inner
900.1.t.a 4 180.p odd 6 1 inner
1620.1.f.a 1 45.l even 12 1
1620.1.f.a 1 180.v odd 12 1
1620.1.f.b 1 45.k odd 12 1
1620.1.f.b 1 180.x even 12 1
1620.1.f.c 1 45.l even 12 1
1620.1.f.c 1 180.v odd 12 1
1620.1.f.d 1 45.k odd 12 1
1620.1.f.d 1 180.x even 12 1
2700.1.t.a 4 3.b odd 2 1
2700.1.t.a 4 9.d odd 6 1
2700.1.t.a 4 12.b even 2 1
2700.1.t.a 4 15.d odd 2 1
2700.1.t.a 4 36.h even 6 1
2700.1.t.a 4 45.h odd 6 1
2700.1.t.a 4 60.h even 2 1
2700.1.t.a 4 180.n even 6 1
2880.1.bu.a 2 40.i odd 4 1
2880.1.bu.a 2 40.k even 4 1
2880.1.bu.a 2 360.bo even 12 1
2880.1.bu.a 2 360.bu odd 12 1
2880.1.bu.b 2 40.i odd 4 1
2880.1.bu.b 2 40.k even 4 1
2880.1.bu.b 2 360.bo even 12 1
2880.1.bu.b 2 360.bu odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 1 - T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( 1 - T^{2} + T^{4} \)
$29$ \( ( 1 + T + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 1 - T + T^{2} )^{2} \)
$43$ \( 16 - 4 T^{2} + T^{4} \)
$47$ \( 1 - T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 1 - T + T^{2} )^{2} \)
$67$ \( 1 - T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( 1 - T^{2} + T^{4} \)
$89$ \( ( -1 + T )^{4} \)
$97$ \( T^{4} \)
show more
show less