Properties

Label 900.2.s.b
Level $900$
Weight $2$
Character orbit 900.s
Analytic conductor $7.187$
Analytic rank $0$
Dimension $4$
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.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.18653618192\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + \zeta_{12} q^{7} + 3 q^{9} +O(q^{10})\) \( q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + \zeta_{12} q^{7} + 3 q^{9} + ( -3 + 3 \zeta_{12}^{2} ) q^{11} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{13} + 6 \zeta_{12}^{3} q^{17} + 4 q^{19} + ( 1 + \zeta_{12}^{2} ) q^{21} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{23} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( 3 - 3 \zeta_{12}^{2} ) q^{29} -5 \zeta_{12}^{2} q^{31} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{33} + 2 \zeta_{12}^{3} q^{37} + ( 2 - \zeta_{12}^{2} ) q^{39} -3 \zeta_{12}^{2} q^{41} -\zeta_{12} q^{43} + 9 \zeta_{12} q^{47} -6 \zeta_{12}^{2} q^{49} + ( -6 + 12 \zeta_{12}^{2} ) q^{51} + 6 \zeta_{12}^{3} q^{53} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{57} -3 \zeta_{12}^{2} q^{59} + ( 13 - 13 \zeta_{12}^{2} ) q^{61} + 3 \zeta_{12} q^{63} + ( -7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{67} + ( 6 - 3 \zeta_{12}^{2} ) q^{69} -12 q^{71} + 10 \zeta_{12}^{3} q^{73} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{77} + ( 11 - 11 \zeta_{12}^{2} ) q^{79} + 9 q^{81} -9 \zeta_{12} q^{83} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{87} -6 q^{89} + q^{91} + ( -5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{93} -11 \zeta_{12} q^{97} + ( -9 + 9 \zeta_{12}^{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} - 6q^{11} + 16q^{19} + 6q^{21} + 6q^{29} - 10q^{31} + 6q^{39} - 6q^{41} - 12q^{49} - 6q^{59} + 26q^{61} + 18q^{69} - 48q^{71} + 22q^{79} + 36q^{81} - 24q^{89} + 4q^{91} - 18q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 −1.73205 0 0 0 −0.866025 + 0.500000i 0 3.00000 0
49.2 0 1.73205 0 0 0 0.866025 0.500000i 0 3.00000 0
349.1 0 −1.73205 0 0 0 −0.866025 0.500000i 0 3.00000 0
349.2 0 1.73205 0 0 0 0.866025 + 0.500000i 0 3.00000 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
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.s.b 4
3.b odd 2 1 2700.2.s.b 4
5.b even 2 1 inner 900.2.s.b 4
5.c odd 4 1 36.2.e.a 2
5.c odd 4 1 900.2.i.b 2
9.c even 3 1 inner 900.2.s.b 4
9.c even 3 1 8100.2.d.h 2
9.d odd 6 1 2700.2.s.b 4
9.d odd 6 1 8100.2.d.c 2
15.d odd 2 1 2700.2.s.b 4
15.e even 4 1 108.2.e.a 2
15.e even 4 1 2700.2.i.b 2
20.e even 4 1 144.2.i.a 2
35.f even 4 1 1764.2.j.b 2
35.k even 12 1 1764.2.i.c 2
35.k even 12 1 1764.2.l.a 2
35.l odd 12 1 1764.2.i.a 2
35.l odd 12 1 1764.2.l.c 2
40.i odd 4 1 576.2.i.f 2
40.k even 4 1 576.2.i.e 2
45.h odd 6 1 2700.2.s.b 4
45.h odd 6 1 8100.2.d.c 2
45.j even 6 1 inner 900.2.s.b 4
45.j even 6 1 8100.2.d.h 2
45.k odd 12 1 36.2.e.a 2
45.k odd 12 1 324.2.a.c 1
45.k odd 12 1 900.2.i.b 2
45.k odd 12 1 8100.2.a.j 1
45.l even 12 1 108.2.e.a 2
45.l even 12 1 324.2.a.a 1
45.l even 12 1 2700.2.i.b 2
45.l even 12 1 8100.2.a.g 1
60.l odd 4 1 432.2.i.c 2
105.k odd 4 1 5292.2.j.a 2
105.w odd 12 1 5292.2.i.a 2
105.w odd 12 1 5292.2.l.c 2
105.x even 12 1 5292.2.i.c 2
105.x even 12 1 5292.2.l.a 2
120.q odd 4 1 1728.2.i.c 2
120.w even 4 1 1728.2.i.d 2
180.v odd 12 1 432.2.i.c 2
180.v odd 12 1 1296.2.a.b 1
180.x even 12 1 144.2.i.a 2
180.x even 12 1 1296.2.a.k 1
315.bs even 12 1 1764.2.l.a 2
315.bt odd 12 1 1764.2.l.c 2
315.bu odd 12 1 5292.2.l.c 2
315.bv even 12 1 5292.2.l.a 2
315.bw odd 12 1 5292.2.i.a 2
315.bx even 12 1 5292.2.i.c 2
315.cb even 12 1 1764.2.j.b 2
315.cf odd 12 1 5292.2.j.a 2
315.cg even 12 1 1764.2.i.c 2
315.ch odd 12 1 1764.2.i.a 2
360.bo even 12 1 576.2.i.e 2
360.bo even 12 1 5184.2.a.f 1
360.br even 12 1 1728.2.i.d 2
360.br even 12 1 5184.2.a.ba 1
360.bt odd 12 1 1728.2.i.c 2
360.bt odd 12 1 5184.2.a.bb 1
360.bu odd 12 1 576.2.i.f 2
360.bu odd 12 1 5184.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 5.c odd 4 1
36.2.e.a 2 45.k odd 12 1
108.2.e.a 2 15.e even 4 1
108.2.e.a 2 45.l even 12 1
144.2.i.a 2 20.e even 4 1
144.2.i.a 2 180.x even 12 1
324.2.a.a 1 45.l even 12 1
324.2.a.c 1 45.k odd 12 1
432.2.i.c 2 60.l odd 4 1
432.2.i.c 2 180.v odd 12 1
576.2.i.e 2 40.k even 4 1
576.2.i.e 2 360.bo even 12 1
576.2.i.f 2 40.i odd 4 1
576.2.i.f 2 360.bu odd 12 1
900.2.i.b 2 5.c odd 4 1
900.2.i.b 2 45.k odd 12 1
900.2.s.b 4 1.a even 1 1 trivial
900.2.s.b 4 5.b even 2 1 inner
900.2.s.b 4 9.c even 3 1 inner
900.2.s.b 4 45.j even 6 1 inner
1296.2.a.b 1 180.v odd 12 1
1296.2.a.k 1 180.x even 12 1
1728.2.i.c 2 120.q odd 4 1
1728.2.i.c 2 360.bt odd 12 1
1728.2.i.d 2 120.w even 4 1
1728.2.i.d 2 360.br even 12 1
1764.2.i.a 2 35.l odd 12 1
1764.2.i.a 2 315.ch odd 12 1
1764.2.i.c 2 35.k even 12 1
1764.2.i.c 2 315.cg even 12 1
1764.2.j.b 2 35.f even 4 1
1764.2.j.b 2 315.cb even 12 1
1764.2.l.a 2 35.k even 12 1
1764.2.l.a 2 315.bs even 12 1
1764.2.l.c 2 35.l odd 12 1
1764.2.l.c 2 315.bt odd 12 1
2700.2.i.b 2 15.e even 4 1
2700.2.i.b 2 45.l even 12 1
2700.2.s.b 4 3.b odd 2 1
2700.2.s.b 4 9.d odd 6 1
2700.2.s.b 4 15.d odd 2 1
2700.2.s.b 4 45.h odd 6 1
5184.2.a.e 1 360.bu odd 12 1
5184.2.a.f 1 360.bo even 12 1
5184.2.a.ba 1 360.br even 12 1
5184.2.a.bb 1 360.bt odd 12 1
5292.2.i.a 2 105.w odd 12 1
5292.2.i.a 2 315.bw odd 12 1
5292.2.i.c 2 105.x even 12 1
5292.2.i.c 2 315.bx even 12 1
5292.2.j.a 2 105.k odd 4 1
5292.2.j.a 2 315.cf odd 12 1
5292.2.l.a 2 105.x even 12 1
5292.2.l.a 2 315.bv even 12 1
5292.2.l.c 2 105.w odd 12 1
5292.2.l.c 2 315.bu odd 12 1
8100.2.a.g 1 45.l even 12 1
8100.2.a.j 1 45.k odd 12 1
8100.2.d.c 2 9.d odd 6 1
8100.2.d.c 2 45.h odd 6 1
8100.2.d.h 2 9.c even 3 1
8100.2.d.h 2 45.j even 6 1

Hecke kernels

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

\( T_{7}^{4} - T_{7}^{2} + 1 \)
\( T_{11}^{2} + 3 T_{11} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 1 - T^{2} + T^{4} \)
$11$ \( ( 9 + 3 T + T^{2} )^{2} \)
$13$ \( 1 - T^{2} + T^{4} \)
$17$ \( ( 36 + T^{2} )^{2} \)
$19$ \( ( -4 + T )^{4} \)
$23$ \( 81 - 9 T^{2} + T^{4} \)
$29$ \( ( 9 - 3 T + T^{2} )^{2} \)
$31$ \( ( 25 + 5 T + T^{2} )^{2} \)
$37$ \( ( 4 + T^{2} )^{2} \)
$41$ \( ( 9 + 3 T + T^{2} )^{2} \)
$43$ \( 1 - T^{2} + T^{4} \)
$47$ \( 6561 - 81 T^{2} + T^{4} \)
$53$ \( ( 36 + T^{2} )^{2} \)
$59$ \( ( 9 + 3 T + T^{2} )^{2} \)
$61$ \( ( 169 - 13 T + T^{2} )^{2} \)
$67$ \( 2401 - 49 T^{2} + T^{4} \)
$71$ \( ( 12 + T )^{4} \)
$73$ \( ( 100 + T^{2} )^{2} \)
$79$ \( ( 121 - 11 T + T^{2} )^{2} \)
$83$ \( 6561 - 81 T^{2} + T^{4} \)
$89$ \( ( 6 + T )^{4} \)
$97$ \( 14641 - 121 T^{2} + T^{4} \)
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