Properties

Label 900.3.c.q
Level $900$
Weight $3$
Character orbit 900.c
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12239922073600.1
Defining polynomial: \(x^{8} - 2 x^{6} + 4 x^{4} - 32 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{4} q^{7} -\beta_{3} q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{4} q^{7} -\beta_{3} q^{8} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{11} + ( 2 \beta_{2} + \beta_{7} ) q^{13} + ( \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{14} + ( -1 + 2 \beta_{4} + \beta_{7} ) q^{16} + ( 2 \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{17} + ( -1 + 2 \beta_{2} - \beta_{4} - \beta_{7} ) q^{19} + ( -5 + 2 \beta_{2} - 2 \beta_{4} + \beta_{7} ) q^{22} + ( 3 \beta_{1} + 3 \beta_{3} + 3 \beta_{5} + 2 \beta_{6} ) q^{23} + ( -\beta_{1} - 2 \beta_{3} + 2 \beta_{6} ) q^{26} + ( 5 + \beta_{2} - 2 \beta_{4} + 3 \beta_{7} ) q^{28} + ( 4 \beta_{1} - 3 \beta_{3} + 3 \beta_{5} + \beta_{6} ) q^{29} + ( 2 - 4 \beta_{2} + \beta_{4} + 2 \beta_{7} ) q^{31} + ( -2 \beta_{3} - 4 \beta_{5} ) q^{32} + ( -9 - 2 \beta_{2} - 2 \beta_{4} - 3 \beta_{7} ) q^{34} + ( 4 \beta_{2} + 2 \beta_{7} ) q^{37} + ( 2 \beta_{1} - \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{38} + ( 18 \beta_{1} - \beta_{3} + \beta_{5} - 3 \beta_{6} ) q^{41} + ( 5 - 10 \beta_{2} - \beta_{4} + 5 \beta_{7} ) q^{43} + ( 4 \beta_{1} + 4 \beta_{5} + 4 \beta_{6} ) q^{44} + ( 17 - 2 \beta_{2} - 6 \beta_{4} + 3 \beta_{7} ) q^{46} + ( -10 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - 4 \beta_{6} ) q^{47} + ( -6 - 8 \beta_{2} - 4 \beta_{7} ) q^{49} + ( 30 - \beta_{2} + 4 \beta_{4} + 2 \beta_{7} ) q^{52} + ( -16 \beta_{1} - 3 \beta_{3} + 3 \beta_{5} + 5 \beta_{6} ) q^{53} + ( -8 \beta_{1} + \beta_{3} + 4 \beta_{5} + 8 \beta_{6} ) q^{56} + ( -5 - 2 \beta_{2} + 6 \beta_{4} + 9 \beta_{7} ) q^{58} + ( -6 \beta_{1} + 6 \beta_{3} + 6 \beta_{5} ) q^{59} + ( -2 + 14 \beta_{2} + 7 \beta_{7} ) q^{61} + ( -4 \beta_{1} + 3 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} ) q^{62} + ( 22 - 4 \beta_{2} + 4 \beta_{4} - 6 \beta_{7} ) q^{64} + ( 5 - 10 \beta_{2} + 7 \beta_{4} + 5 \beta_{7} ) q^{67} + ( 12 \beta_{1} + 4 \beta_{3} + 4 \beta_{5} - 4 \beta_{6} ) q^{68} + ( -29 \beta_{1} - \beta_{3} - \beta_{5} - 10 \beta_{6} ) q^{71} + ( 20 + 20 \beta_{2} + 10 \beta_{7} ) q^{73} + ( -2 \beta_{1} - 4 \beta_{3} + 4 \beta_{6} ) q^{74} + ( -29 + \beta_{2} + 2 \beta_{4} + 5 \beta_{7} ) q^{76} + ( 32 \beta_{1} + \beta_{3} - \beta_{5} - 7 \beta_{6} ) q^{77} + ( -6 + 12 \beta_{2} + 8 \beta_{4} - 6 \beta_{7} ) q^{79} + ( -55 - 14 \beta_{2} + 2 \beta_{4} + 3 \beta_{7} ) q^{82} + ( 22 \beta_{1} - 10 \beta_{3} - 10 \beta_{5} + 4 \beta_{6} ) q^{83} + ( -10 \beta_{1} + 11 \beta_{3} + 2 \beta_{5} + 11 \beta_{6} ) q^{86} + ( 40 - 4 \beta_{2} + 8 \beta_{7} ) q^{88} + ( 28 \beta_{1} + 4 \beta_{3} - 4 \beta_{5} - 8 \beta_{6} ) q^{89} + ( 5 - 10 \beta_{2} - 7 \beta_{4} + 5 \beta_{7} ) q^{91} + ( -20 \beta_{1} + 8 \beta_{3} + 12 \beta_{5} + 12 \beta_{6} ) q^{92} + ( -54 + 12 \beta_{2} + 4 \beta_{4} - 2 \beta_{7} ) q^{94} + 15 q^{97} + ( 10 \beta_{1} + 8 \beta_{3} - 8 \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{4} + O(q^{10}) \) \( 8q + 4q^{4} + 8q^{13} - 8q^{16} - 32q^{22} + 44q^{28} - 80q^{34} + 16q^{37} + 128q^{46} - 80q^{49} + 236q^{52} - 48q^{58} + 40q^{61} + 160q^{64} + 240q^{73} - 228q^{76} - 496q^{82} + 304q^{88} - 384q^{94} + 120q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{6} + 4 x^{4} - 32 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\( \nu^{3} \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + 6 \nu^{4} + 4 \nu^{2} - 16 \)\()/16\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{3} \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{5} + 4 \nu^{3} - 16 \nu \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{6} + 2 \nu^{4} - 4 \nu^{2} + 24 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{3}\)
\(\nu^{4}\)\(=\)\(\beta_{7} + 2 \beta_{4} - 1\)
\(\nu^{5}\)\(=\)\(4 \beta_{5} + 2 \beta_{3}\)
\(\nu^{6}\)\(=\)\(-6 \beta_{7} + 4 \beta_{4} - 4 \beta_{2} + 22\)
\(\nu^{7}\)\(=\)\(16 \beta_{6} + 8 \beta_{5} + 16 \beta_{1}\)

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
1.89639 + 0.635381i
1.89639 0.635381i
0.950636 + 1.75963i
0.950636 1.75963i
−0.950636 + 1.75963i
−0.950636 1.75963i
−1.89639 + 0.635381i
−1.89639 0.635381i
−1.89639 0.635381i 0 3.19258 + 2.40986i 0 0 10.1035i −4.52320 6.59853i 0 0
451.2 −1.89639 + 0.635381i 0 3.19258 2.40986i 0 0 10.1035i −4.52320 + 6.59853i 0 0
451.3 −0.950636 1.75963i 0 −2.19258 + 3.34553i 0 0 3.98982i 7.97124 + 0.677747i 0 0
451.4 −0.950636 + 1.75963i 0 −2.19258 3.34553i 0 0 3.98982i 7.97124 0.677747i 0 0
451.5 0.950636 1.75963i 0 −2.19258 3.34553i 0 0 3.98982i −7.97124 + 0.677747i 0 0
451.6 0.950636 + 1.75963i 0 −2.19258 + 3.34553i 0 0 3.98982i −7.97124 0.677747i 0 0
451.7 1.89639 0.635381i 0 3.19258 2.40986i 0 0 10.1035i 4.52320 6.59853i 0 0
451.8 1.89639 + 0.635381i 0 3.19258 + 2.40986i 0 0 10.1035i 4.52320 + 6.59853i 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
12.b 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.q yes 8
3.b odd 2 1 inner 900.3.c.q yes 8
4.b odd 2 1 inner 900.3.c.q yes 8
5.b even 2 1 900.3.c.p 8
5.c odd 4 2 900.3.f.g 16
12.b even 2 1 inner 900.3.c.q yes 8
15.d odd 2 1 900.3.c.p 8
15.e even 4 2 900.3.f.g 16
20.d odd 2 1 900.3.c.p 8
20.e even 4 2 900.3.f.g 16
60.h even 2 1 900.3.c.p 8
60.l odd 4 2 900.3.f.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.3.c.p 8 5.b even 2 1
900.3.c.p 8 15.d odd 2 1
900.3.c.p 8 20.d odd 2 1
900.3.c.p 8 60.h even 2 1
900.3.c.q yes 8 1.a even 1 1 trivial
900.3.c.q yes 8 3.b odd 2 1 inner
900.3.c.q yes 8 4.b odd 2 1 inner
900.3.c.q yes 8 12.b even 2 1 inner
900.3.f.g 16 5.c odd 4 2
900.3.f.g 16 15.e even 4 2
900.3.f.g 16 20.e even 4 2
900.3.f.g 16 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{4} + 118 T_{7}^{2} + 1625 \)
\( T_{13}^{2} - 2 T_{13} - 115 \)
\( T_{17}^{4} - 424 T_{17}^{2} + 40768 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 - 32 T^{2} + 4 T^{4} - 2 T^{6} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 1625 + 118 T^{2} + T^{4} )^{2} \)
$11$ \( ( 8000 + 184 T^{2} + T^{4} )^{2} \)
$13$ \( ( -115 - 2 T + T^{2} )^{4} \)
$17$ \( ( 40768 - 424 T^{2} + T^{4} )^{2} \)
$19$ \( ( 65 + 302 T^{2} + T^{4} )^{2} \)
$23$ \( ( 15680 + 1784 T^{2} + T^{4} )^{2} \)
$29$ \( ( 1019200 - 3048 T^{2} + T^{4} )^{2} \)
$31$ \( ( 79625 + 1030 T^{2} + T^{4} )^{2} \)
$37$ \( ( -460 - 4 T + T^{2} )^{4} \)
$41$ \( ( 3515200 - 3752 T^{2} + T^{4} )^{2} \)
$43$ \( ( 13456625 + 7358 T^{2} + T^{4} )^{2} \)
$47$ \( ( 2708480 + 3296 T^{2} + T^{4} )^{2} \)
$53$ \( ( 2896192 - 6888 T^{2} + T^{4} )^{2} \)
$59$ \( ( 10368000 + 6624 T^{2} + T^{4} )^{2} \)
$61$ \( ( -5659 - 10 T + T^{2} )^{4} \)
$67$ \( ( 4564625 + 9502 T^{2} + T^{4} )^{2} \)
$71$ \( ( 66248000 + 21304 T^{2} + T^{4} )^{2} \)
$73$ \( ( -10700 - 60 T + T^{2} )^{4} \)
$79$ \( ( 44994560 + 21568 T^{2} + T^{4} )^{2} \)
$83$ \( ( 146232320 + 27104 T^{2} + T^{4} )^{2} \)
$89$ \( ( 5324800 - 16512 T^{2} + T^{4} )^{2} \)
$97$ \( ( -15 + T )^{8} \)
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