Properties

Label 900.3.b.b
Level $900$
Weight $3$
Character orbit 900.b
Analytic conductor $24.523$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Defining polynomial: \(x^{8} + 7 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \beta_{1} - \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( 2 \beta_{1} - \beta_{3} ) q^{7} + ( \beta_{2} + \beta_{5} ) q^{11} + ( \beta_{1} + \beta_{3} ) q^{13} -2 \beta_{4} q^{17} + ( -8 - \beta_{6} ) q^{19} + ( -\beta_{4} + \beta_{7} ) q^{23} + ( -8 \beta_{2} - \beta_{5} ) q^{29} + ( 2 + \beta_{6} ) q^{31} + ( 17 \beta_{1} + 3 \beta_{3} ) q^{37} + ( -3 \beta_{2} - 4 \beta_{5} ) q^{41} + ( 10 \beta_{1} - 2 \beta_{3} ) q^{43} + ( -7 \beta_{4} - 2 \beta_{7} ) q^{47} + ( -57 + 4 \beta_{6} ) q^{49} + ( -5 \beta_{4} - 4 \beta_{7} ) q^{53} + ( -17 \beta_{2} + 3 \beta_{5} ) q^{59} + ( -10 - 3 \beta_{6} ) q^{61} + 38 \beta_{1} q^{67} + ( 12 \beta_{2} + 2 \beta_{5} ) q^{71} + ( 19 \beta_{1} - 6 \beta_{3} ) q^{73} + ( -17 \beta_{4} + 7 \beta_{7} ) q^{77} + ( -50 - 3 \beta_{6} ) q^{79} + ( -\beta_{4} + 4 \beta_{7} ) q^{83} + ( -21 \beta_{2} + 2 \beta_{5} ) q^{89} + ( 82 - \beta_{6} ) q^{91} + ( 53 \beta_{1} - 2 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 64q^{19} + 16q^{31} - 456q^{49} - 80q^{61} - 400q^{79} + 656q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{6} + 16 \nu^{2} \)\()/3\)
\(\beta_{2}\)\(=\)\( 2 \nu^{7} + \nu^{5} + 13 \nu^{3} + 5 \nu \)
\(\beta_{3}\)\(=\)\( 4 \nu^{7} + \nu^{5} + 29 \nu^{3} + 11 \nu \)
\(\beta_{4}\)\(=\)\( -4 \nu^{7} + 2 \nu^{5} - 26 \nu^{3} + 10 \nu \)
\(\beta_{5}\)\(=\)\( -6 \nu^{6} - 36 \nu^{2} \)
\(\beta_{6}\)\(=\)\( -8 \nu^{7} + 2 \nu^{5} - 58 \nu^{3} + 22 \nu \)
\(\beta_{7}\)\(=\)\( 4 \nu^{4} + 14 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2}\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + 9 \beta_{1}\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{6} + 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2}\)\()/12\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{7} - 14\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{6} + 11 \beta_{4} - 10 \beta_{3} + 22 \beta_{2}\)\()/24\)
\(\nu^{6}\)\(=\)\((\)\(-4 \beta_{5} - 27 \beta_{1}\)\()/6\)
\(\nu^{7}\)\(=\)\((\)\(13 \beta_{6} - 29 \beta_{4} - 26 \beta_{3} + 58 \beta_{2}\)\()/24\)

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−1.14412 + 1.14412i
−0.437016 + 0.437016i
0.437016 + 0.437016i
1.14412 + 1.14412i
1.14412 1.14412i
0.437016 0.437016i
−0.437016 0.437016i
−1.14412 1.14412i
0 0 0 0 0 13.4868i 0 0 0
449.2 0 0 0 0 0 13.4868i 0 0 0
449.3 0 0 0 0 0 5.48683i 0 0 0
449.4 0 0 0 0 0 5.48683i 0 0 0
449.5 0 0 0 0 0 5.48683i 0 0 0
449.6 0 0 0 0 0 5.48683i 0 0 0
449.7 0 0 0 0 0 13.4868i 0 0 0
449.8 0 0 0 0 0 13.4868i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.b.b 8
3.b odd 2 1 inner 900.3.b.b 8
4.b odd 2 1 3600.3.c.k 8
5.b even 2 1 inner 900.3.b.b 8
5.c odd 4 1 180.3.g.a 4
5.c odd 4 1 900.3.g.d 4
12.b even 2 1 3600.3.c.k 8
15.d odd 2 1 inner 900.3.b.b 8
15.e even 4 1 180.3.g.a 4
15.e even 4 1 900.3.g.d 4
20.d odd 2 1 3600.3.c.k 8
20.e even 4 1 720.3.l.c 4
20.e even 4 1 3600.3.l.n 4
40.i odd 4 1 2880.3.l.b 4
40.k even 4 1 2880.3.l.f 4
45.k odd 12 2 1620.3.o.f 8
45.l even 12 2 1620.3.o.f 8
60.h even 2 1 3600.3.c.k 8
60.l odd 4 1 720.3.l.c 4
60.l odd 4 1 3600.3.l.n 4
120.q odd 4 1 2880.3.l.f 4
120.w even 4 1 2880.3.l.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.g.a 4 5.c odd 4 1
180.3.g.a 4 15.e even 4 1
720.3.l.c 4 20.e even 4 1
720.3.l.c 4 60.l odd 4 1
900.3.b.b 8 1.a even 1 1 trivial
900.3.b.b 8 3.b odd 2 1 inner
900.3.b.b 8 5.b even 2 1 inner
900.3.b.b 8 15.d odd 2 1 inner
900.3.g.d 4 5.c odd 4 1
900.3.g.d 4 15.e even 4 1
1620.3.o.f 8 45.k odd 12 2
1620.3.o.f 8 45.l even 12 2
2880.3.l.b 4 40.i odd 4 1
2880.3.l.b 4 120.w even 4 1
2880.3.l.f 4 40.k even 4 1
2880.3.l.f 4 120.q odd 4 1
3600.3.c.k 8 4.b odd 2 1
3600.3.c.k 8 12.b even 2 1
3600.3.c.k 8 20.d odd 2 1
3600.3.c.k 8 60.h even 2 1
3600.3.l.n 4 20.e even 4 1
3600.3.l.n 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 212 T_{7}^{2} + 5476 \) acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 5476 + 212 T^{2} + T^{4} )^{2} \)
$11$ \( ( 26244 + 396 T^{2} + T^{4} )^{2} \)
$13$ \( ( 7396 + 188 T^{2} + T^{4} )^{2} \)
$17$ \( ( -288 + T^{2} )^{4} \)
$19$ \( ( -296 + 16 T + T^{2} )^{4} \)
$23$ \( ( 11664 - 504 T^{2} + T^{4} )^{2} \)
$29$ \( ( 944784 + 2664 T^{2} + T^{4} )^{2} \)
$31$ \( ( -356 - 4 T + T^{2} )^{4} \)
$37$ \( ( 119716 + 3932 T^{2} + T^{4} )^{2} \)
$41$ \( ( 7387524 + 6084 T^{2} + T^{4} )^{2} \)
$43$ \( ( 1600 + 1520 T^{2} + T^{4} )^{2} \)
$47$ \( ( 7884864 - 8496 T^{2} + T^{4} )^{2} \)
$53$ \( ( 1166400 - 9360 T^{2} + T^{4} )^{2} \)
$59$ \( ( 12830724 + 13644 T^{2} + T^{4} )^{2} \)
$61$ \( ( -3140 + 20 T + T^{2} )^{4} \)
$67$ \( ( 5776 + T^{2} )^{4} \)
$71$ \( ( 3504384 + 6624 T^{2} + T^{4} )^{2} \)
$73$ \( ( 3225616 + 9368 T^{2} + T^{4} )^{2} \)
$79$ \( ( -740 + 100 T + T^{2} )^{4} \)
$83$ \( ( 7884864 - 5904 T^{2} + T^{4} )^{2} \)
$89$ \( ( 52099524 + 17316 T^{2} + T^{4} )^{2} \)
$97$ \( ( 118287376 + 23192 T^{2} + T^{4} )^{2} \)
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