Properties

Label 900.3.c.s
Level $900$
Weight $3$
Character orbit 900.c
Analytic conductor $24.523$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.77720518656.9
Defining polynomial: \(x^{8} - 121 x^{4} + 14641\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12} \)
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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{2} ) q^{2} + ( 2 - \beta_{5} ) q^{4} -\beta_{3} q^{7} -4 \beta_{2} q^{8} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{2} ) q^{2} + ( 2 - \beta_{5} ) q^{4} -\beta_{3} q^{7} -4 \beta_{2} q^{8} -\beta_{7} q^{11} -\beta_{4} q^{13} + ( \beta_{6} + \beta_{7} ) q^{14} + ( -8 - 4 \beta_{5} ) q^{16} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{17} -6 \beta_{5} q^{19} + ( 3 \beta_{3} - \beta_{4} ) q^{22} -14 \beta_{2} q^{23} + ( -3 \beta_{6} + \beta_{7} ) q^{26} + ( -2 \beta_{3} + 2 \beta_{4} ) q^{28} + \beta_{6} q^{29} -10 \beta_{5} q^{31} -16 \beta_{1} q^{32} + ( 18 - 3 \beta_{5} ) q^{34} + 3 \beta_{4} q^{37} + ( -12 \beta_{1} - 12 \beta_{2} ) q^{38} + 2 \beta_{6} q^{41} + 4 \beta_{3} q^{43} + ( -6 \beta_{6} - 2 \beta_{7} ) q^{44} + ( -28 - 14 \beta_{5} ) q^{46} + 2 \beta_{2} q^{47} -39 q^{49} + ( -6 \beta_{3} - 2 \beta_{4} ) q^{52} + ( 18 \beta_{1} - 9 \beta_{2} ) q^{53} + 8 \beta_{6} q^{56} + ( \beta_{3} + \beta_{4} ) q^{58} -\beta_{7} q^{59} -58 q^{61} + ( -20 \beta_{1} - 20 \beta_{2} ) q^{62} -64 q^{64} + 2 \beta_{3} q^{67} + ( 12 \beta_{1} - 24 \beta_{2} ) q^{68} + 6 \beta_{7} q^{71} + ( 9 \beta_{6} - 3 \beta_{7} ) q^{74} + ( -72 - 12 \beta_{5} ) q^{76} + ( 88 \beta_{1} - 44 \beta_{2} ) q^{77} + 2 \beta_{5} q^{79} + ( 2 \beta_{3} + 2 \beta_{4} ) q^{82} + 16 \beta_{2} q^{83} + ( -4 \beta_{6} - 4 \beta_{7} ) q^{86} -8 \beta_{4} q^{88} -8 \beta_{6} q^{89} -44 \beta_{5} q^{91} -56 \beta_{1} q^{92} + ( 4 + 2 \beta_{5} ) q^{94} -10 \beta_{4} q^{97} + ( -39 \beta_{1} + 39 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 16q^{4} + O(q^{10}) \) \( 8q + 16q^{4} - 64q^{16} + 144q^{34} - 224q^{46} - 312q^{49} - 464q^{61} - 512q^{64} - 576q^{76} + 32q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 121 x^{4} + 14641\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{2} \)\(/11\)
\(\beta_{2}\)\(=\)\( 2 \nu^{6} \)\(/1331\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{5} - 22 \nu^{3} + 242 \nu \)\()/121\)
\(\beta_{4}\)\(=\)\((\)\( -4 \nu^{7} + 22 \nu^{5} + 242 \nu^{3} + 2662 \nu \)\()/1331\)
\(\beta_{5}\)\(=\)\((\)\( 4 \nu^{4} - 242 \)\()/121\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{5} + 22 \nu^{3} + 242 \nu \)\()/121\)
\(\beta_{7}\)\(=\)\((\)\( 4 \nu^{7} + 22 \nu^{5} - 242 \nu^{3} + 2662 \nu \)\()/1331\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3}\)\()/8\)
\(\nu^{2}\)\(=\)\(11 \beta_{1}\)\(/2\)
\(\nu^{3}\)\(=\)\((\)\(11 \beta_{6} - 11 \beta_{3}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(121 \beta_{5} + 242\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(121 \beta_{7} - 121 \beta_{6} + 121 \beta_{4} - 121 \beta_{3}\)\()/8\)
\(\nu^{6}\)\(=\)\(1331 \beta_{2}\)\(/2\)
\(\nu^{7}\)\(=\)\((\)\(1331 \beta_{7} + 1331 \beta_{6} - 1331 \beta_{4} - 1331 \beta_{3}\)\()/8\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0.858406 + 3.20361i
−0.858406 3.20361i
−0.858406 + 3.20361i
0.858406 3.20361i
−3.20361 0.858406i
3.20361 + 0.858406i
3.20361 0.858406i
−3.20361 + 0.858406i
−1.73205 1.00000i 0 2.00000 + 3.46410i 0 0 9.38083i 8.00000i 0 0
451.2 −1.73205 1.00000i 0 2.00000 + 3.46410i 0 0 9.38083i 8.00000i 0 0
451.3 −1.73205 + 1.00000i 0 2.00000 3.46410i 0 0 9.38083i 8.00000i 0 0
451.4 −1.73205 + 1.00000i 0 2.00000 3.46410i 0 0 9.38083i 8.00000i 0 0
451.5 1.73205 1.00000i 0 2.00000 3.46410i 0 0 9.38083i 8.00000i 0 0
451.6 1.73205 1.00000i 0 2.00000 3.46410i 0 0 9.38083i 8.00000i 0 0
451.7 1.73205 + 1.00000i 0 2.00000 + 3.46410i 0 0 9.38083i 8.00000i 0 0
451.8 1.73205 + 1.00000i 0 2.00000 + 3.46410i 0 0 9.38083i 8.00000i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 451.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.c.s 8
3.b odd 2 1 inner 900.3.c.s 8
4.b odd 2 1 inner 900.3.c.s 8
5.b even 2 1 inner 900.3.c.s 8
5.c odd 4 1 180.3.f.d 4
5.c odd 4 1 180.3.f.g yes 4
12.b even 2 1 inner 900.3.c.s 8
15.d odd 2 1 inner 900.3.c.s 8
15.e even 4 1 180.3.f.d 4
15.e even 4 1 180.3.f.g yes 4
20.d odd 2 1 inner 900.3.c.s 8
20.e even 4 1 180.3.f.d 4
20.e even 4 1 180.3.f.g yes 4
60.h even 2 1 inner 900.3.c.s 8
60.l odd 4 1 180.3.f.d 4
60.l odd 4 1 180.3.f.g yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.f.d 4 5.c odd 4 1
180.3.f.d 4 15.e even 4 1
180.3.f.d 4 20.e even 4 1
180.3.f.d 4 60.l odd 4 1
180.3.f.g yes 4 5.c odd 4 1
180.3.f.g yes 4 15.e even 4 1
180.3.f.g yes 4 20.e even 4 1
180.3.f.g yes 4 60.l odd 4 1
900.3.c.s 8 1.a even 1 1 trivial
900.3.c.s 8 3.b odd 2 1 inner
900.3.c.s 8 4.b odd 2 1 inner
900.3.c.s 8 5.b even 2 1 inner
900.3.c.s 8 12.b even 2 1 inner
900.3.c.s 8 15.d odd 2 1 inner
900.3.c.s 8 20.d odd 2 1 inner
900.3.c.s 8 60.h 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 \(S_{3}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{2} + 88 \)
\( T_{13}^{2} - 264 \)
\( T_{17}^{2} - 108 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 - 4 T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 88 + T^{2} )^{4} \)
$11$ \( ( 264 + T^{2} )^{4} \)
$13$ \( ( -264 + T^{2} )^{4} \)
$17$ \( ( -108 + T^{2} )^{4} \)
$19$ \( ( 432 + T^{2} )^{4} \)
$23$ \( ( 784 + T^{2} )^{4} \)
$29$ \( ( -88 + T^{2} )^{4} \)
$31$ \( ( 1200 + T^{2} )^{4} \)
$37$ \( ( -2376 + T^{2} )^{4} \)
$41$ \( ( -352 + T^{2} )^{4} \)
$43$ \( ( 1408 + T^{2} )^{4} \)
$47$ \( ( 16 + T^{2} )^{4} \)
$53$ \( ( -972 + T^{2} )^{4} \)
$59$ \( ( 264 + T^{2} )^{4} \)
$61$ \( ( 58 + T )^{8} \)
$67$ \( ( 352 + T^{2} )^{4} \)
$71$ \( ( 9504 + T^{2} )^{4} \)
$73$ \( T^{8} \)
$79$ \( ( 48 + T^{2} )^{4} \)
$83$ \( ( 1024 + T^{2} )^{4} \)
$89$ \( ( -5632 + T^{2} )^{4} \)
$97$ \( ( -26400 + T^{2} )^{4} \)
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