Properties

Label 900.3.f.d
Level $900$
Weight $3$
Character orbit 900.f
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.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.18084870400.3
Defining polynomial: \(x^{8} - 12 x^{6} + 40 x^{4} + 17 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 100)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( -1 - \beta_{1} ) q^{4} + ( -2 \beta_{4} + 2 \beta_{7} ) q^{7} + ( -2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{8} +O(q^{10})\) \( q + \beta_{4} q^{2} + ( -1 - \beta_{1} ) q^{4} + ( -2 \beta_{4} + 2 \beta_{7} ) q^{7} + ( -2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{8} + ( \beta_{1} + \beta_{6} ) q^{11} + ( -2 \beta_{3} + 4 \beta_{4} + 2 \beta_{7} ) q^{13} + ( -6 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{6} ) q^{14} + ( -9 + 3 \beta_{1} - 2 \beta_{6} ) q^{16} + ( 2 \beta_{3} - 4 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{17} + ( 2 - \beta_{1} - 2 \beta_{2} + 5 \beta_{6} ) q^{19} + ( 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - 4 \beta_{7} ) q^{22} + ( -2 \beta_{3} + 2 \beta_{4} - 4 \beta_{7} ) q^{23} + ( -20 - 4 \beta_{1} ) q^{26} + ( -8 \beta_{4} - 6 \beta_{5} - 2 \beta_{7} ) q^{28} + ( 10 - 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( 2 \beta_{1} + 2 \beta_{6} ) q^{31} + ( 6 \beta_{3} - 6 \beta_{4} - \beta_{5} + 3 \beta_{7} ) q^{32} + ( 19 + 3 \beta_{1} + \beta_{2} + \beta_{6} ) q^{34} + ( -2 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} + 2 \beta_{7} ) q^{37} + ( -6 \beta_{3} - 3 \beta_{4} - 6 \beta_{5} - 8 \beta_{7} ) q^{38} -17 q^{41} + ( -10 \beta_{3} - 8 \beta_{4} - 2 \beta_{7} ) q^{43} + ( 21 - 3 \beta_{1} + 4 \beta_{2} ) q^{44} + ( 6 - 2 \beta_{1} + 6 \beta_{2} - 6 \beta_{6} ) q^{46} + ( -14 \beta_{3} - 20 \beta_{4} + 6 \beta_{7} ) q^{47} + ( 19 - 8 \beta_{1} + 8 \beta_{2} ) q^{49} + ( -8 \beta_{3} - 24 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} ) q^{52} + ( -8 \beta_{3} + 16 \beta_{4} - 6 \beta_{5} + 8 \beta_{7} ) q^{53} + ( 10 + 2 \beta_{1} + 8 \beta_{2} + 4 \beta_{6} ) q^{56} + ( 12 \beta_{4} + 4 \beta_{5} - 4 \beta_{7} ) q^{58} + ( -6 + 18 \beta_{1} + 6 \beta_{2} ) q^{59} + ( -28 + 10 \beta_{1} - 10 \beta_{2} ) q^{61} + ( 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 8 \beta_{7} ) q^{62} + ( 17 + 5 \beta_{1} - 8 \beta_{2} + 10 \beta_{6} ) q^{64} + ( \beta_{3} - 9 \beta_{4} + 10 \beta_{7} ) q^{67} + ( 8 \beta_{3} + 24 \beta_{4} - 3 \beta_{5} - 9 \beta_{7} ) q^{68} + ( -10 + 22 \beta_{1} + 10 \beta_{2} - 8 \beta_{6} ) q^{71} + ( -2 \beta_{3} + 4 \beta_{4} - 11 \beta_{5} + 2 \beta_{7} ) q^{73} + ( -26 - 10 \beta_{1} + 6 \beta_{2} + 6 \beta_{6} ) q^{74} + ( 5 - 3 \beta_{1} + 20 \beta_{2} - 8 \beta_{6} ) q^{76} + ( -10 \beta_{3} + 20 \beta_{4} + 14 \beta_{5} + 10 \beta_{7} ) q^{77} + ( -2 - 4 \beta_{1} + 2 \beta_{2} - 10 \beta_{6} ) q^{79} -17 \beta_{4} q^{82} + ( -11 \beta_{3} - 5 \beta_{4} - 6 \beta_{7} ) q^{83} + ( -24 + 8 \beta_{1} + 12 \beta_{2} - 12 \beta_{6} ) q^{86} + ( 2 \beta_{3} + 26 \beta_{4} + 7 \beta_{5} - 9 \beta_{7} ) q^{88} + ( -55 - 2 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 4 + 4 \beta_{1} - 4 \beta_{2} + 16 \beta_{6} ) q^{91} + ( 8 \beta_{3} + 16 \beta_{4} + 14 \beta_{5} + 2 \beta_{7} ) q^{92} + ( -60 + 20 \beta_{1} + 8 \beta_{2} - 8 \beta_{6} ) q^{94} + ( -12 \beta_{3} + 24 \beta_{4} + 18 \beta_{5} + 12 \beta_{7} ) q^{97} + ( 27 \beta_{4} + 16 \beta_{5} - 16 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 10q^{4} + O(q^{10}) \) \( 8q - 10q^{4} - 52q^{14} - 62q^{16} - 168q^{26} + 80q^{29} + 158q^{34} - 136q^{41} + 170q^{44} + 68q^{46} + 152q^{49} + 92q^{56} - 224q^{61} + 110q^{64} - 228q^{74} + 90q^{76} - 128q^{86} - 440q^{89} - 408q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 12 x^{6} + 40 x^{4} + 17 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 19 \nu^{6} - 244 \nu^{4} + 896 \nu^{2} + 104 \)\()/55\)
\(\beta_{2}\)\(=\)\((\)\( 23 \nu^{6} - 278 \nu^{4} + 882 \nu^{2} + 453 \)\()/55\)
\(\beta_{3}\)\(=\)\((\)\( -14 \nu^{7} + 174 \nu^{5} - 611 \nu^{3} - 204 \nu \)\()/55\)
\(\beta_{4}\)\(=\)\((\)\( 37 \nu^{7} - 452 \nu^{5} + 1548 \nu^{3} + 437 \nu \)\()/110\)
\(\beta_{5}\)\(=\)\((\)\( 9 \nu^{7} - 104 \nu^{5} + 326 \nu^{3} + 249 \nu \)\()/22\)
\(\beta_{6}\)\(=\)\((\)\( 41 \nu^{6} - 486 \nu^{4} + 1644 \nu^{2} + 401 \)\()/55\)
\(\beta_{7}\)\(=\)\((\)\( -28 \nu^{7} + 348 \nu^{5} - 1277 \nu^{3} + 32 \nu \)\()/55\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - 5 \beta_{4} - 5 \beta_{3}\)\()/10\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{6} - 7 \beta_{2} + 2 \beta_{1} + 32\)\()/10\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{7} + 4 \beta_{5} - 20 \beta_{4} - 10 \beta_{3}\)\()/5\)
\(\nu^{4}\)\(=\)\((\)\(7 \beta_{6} - 10 \beta_{2} - 3 \beta_{1} + 37\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-85 \beta_{7} + 81 \beta_{5} - 265 \beta_{4} - 50 \beta_{3}\)\()/10\)
\(\nu^{6}\)\(=\)\((\)\(154 \beta_{6} - 156 \beta_{2} - 129 \beta_{1} + 406\)\()/5\)
\(\nu^{7}\)\(=\)\((\)\(-620 \beta_{7} + 643 \beta_{5} - 1475 \beta_{4} + 285 \beta_{3}\)\()/10\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
0.220086 0.500000i
0.220086 + 0.500000i
2.53999 + 0.500000i
2.53999 0.500000i
−2.53999 + 0.500000i
−2.53999 0.500000i
−0.220086 0.500000i
−0.220086 + 0.500000i
−1.47492 1.35078i 0 0.350781 + 3.98459i 0 0 10.9190 4.86493 6.35078i 0 0
199.2 −1.47492 + 1.35078i 0 0.350781 3.98459i 0 0 10.9190 4.86493 + 6.35078i 0 0
199.3 −0.758030 1.85078i 0 −2.85078 + 2.80590i 0 0 −4.09573 7.35408 + 3.14922i 0 0
199.4 −0.758030 + 1.85078i 0 −2.85078 2.80590i 0 0 −4.09573 7.35408 3.14922i 0 0
199.5 0.758030 1.85078i 0 −2.85078 2.80590i 0 0 4.09573 −7.35408 + 3.14922i 0 0
199.6 0.758030 + 1.85078i 0 −2.85078 + 2.80590i 0 0 4.09573 −7.35408 3.14922i 0 0
199.7 1.47492 1.35078i 0 0.350781 3.98459i 0 0 −10.9190 −4.86493 6.35078i 0 0
199.8 1.47492 + 1.35078i 0 0.350781 + 3.98459i 0 0 −10.9190 −4.86493 + 6.35078i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.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.f.d 8
3.b odd 2 1 100.3.d.a 8
4.b odd 2 1 inner 900.3.f.d 8
5.b even 2 1 inner 900.3.f.d 8
5.c odd 4 1 900.3.c.l 4
5.c odd 4 1 900.3.c.m 4
12.b even 2 1 100.3.d.a 8
15.d odd 2 1 100.3.d.a 8
15.e even 4 1 100.3.b.d 4
15.e even 4 1 100.3.b.e yes 4
20.d odd 2 1 inner 900.3.f.d 8
20.e even 4 1 900.3.c.l 4
20.e even 4 1 900.3.c.m 4
24.f even 2 1 1600.3.h.o 8
24.h odd 2 1 1600.3.h.o 8
60.h even 2 1 100.3.d.a 8
60.l odd 4 1 100.3.b.d 4
60.l odd 4 1 100.3.b.e yes 4
120.i odd 2 1 1600.3.h.o 8
120.m even 2 1 1600.3.h.o 8
120.q odd 4 1 1600.3.b.o 4
120.q odd 4 1 1600.3.b.p 4
120.w even 4 1 1600.3.b.o 4
120.w even 4 1 1600.3.b.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.3.b.d 4 15.e even 4 1
100.3.b.d 4 60.l odd 4 1
100.3.b.e yes 4 15.e even 4 1
100.3.b.e yes 4 60.l odd 4 1
100.3.d.a 8 3.b odd 2 1
100.3.d.a 8 12.b even 2 1
100.3.d.a 8 15.d odd 2 1
100.3.d.a 8 60.h even 2 1
900.3.c.l 4 5.c odd 4 1
900.3.c.l 4 20.e even 4 1
900.3.c.m 4 5.c odd 4 1
900.3.c.m 4 20.e even 4 1
900.3.f.d 8 1.a even 1 1 trivial
900.3.f.d 8 4.b odd 2 1 inner
900.3.f.d 8 5.b even 2 1 inner
900.3.f.d 8 20.d odd 2 1 inner
1600.3.b.o 4 120.q odd 4 1
1600.3.b.o 4 120.w even 4 1
1600.3.b.p 4 120.q odd 4 1
1600.3.b.p 4 120.w even 4 1
1600.3.h.o 8 24.f even 2 1
1600.3.h.o 8 24.h odd 2 1
1600.3.h.o 8 120.i odd 2 1
1600.3.h.o 8 120.m even 2 1

Hecke kernels

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

\( T_{7}^{4} - 136 T_{7}^{2} + 2000 \)
\( T_{29}^{2} - 20 T_{29} - 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 + 80 T^{2} + 28 T^{4} + 5 T^{6} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 2000 - 136 T^{2} + T^{4} )^{2} \)
$11$ \( ( 125 + 130 T^{2} + T^{4} )^{2} \)
$13$ \( ( 25600 + 336 T^{2} + T^{4} )^{2} \)
$17$ \( ( 24025 + 346 T^{2} + T^{4} )^{2} \)
$19$ \( ( 465125 + 1370 T^{2} + T^{4} )^{2} \)
$23$ \( ( 147920 - 776 T^{2} + T^{4} )^{2} \)
$29$ \( ( -64 - 20 T + T^{2} )^{4} \)
$31$ \( ( 2000 + 520 T^{2} + T^{4} )^{2} \)
$37$ \( ( 739600 + 2376 T^{2} + T^{4} )^{2} \)
$41$ \( ( 17 + T )^{8} \)
$43$ \( ( 128000 - 3056 T^{2} + T^{4} )^{2} \)
$47$ \( ( 5242880 - 4976 T^{2} + T^{4} )^{2} \)
$53$ \( ( 1392400 + 8136 T^{2} + T^{4} )^{2} \)
$59$ \( ( 41472000 + 13680 T^{2} + T^{4} )^{2} \)
$61$ \( ( -3316 + 56 T + T^{2} )^{4} \)
$67$ \( ( 1915805 - 3586 T^{2} + T^{4} )^{2} \)
$71$ \( ( 67712000 + 19120 T^{2} + T^{4} )^{2} \)
$73$ \( ( 9517225 + 6826 T^{2} + T^{4} )^{2} \)
$79$ \( ( 6962000 + 7720 T^{2} + T^{4} )^{2} \)
$83$ \( ( 3621005 - 5426 T^{2} + T^{4} )^{2} \)
$89$ \( ( 2861 + 110 T + T^{2} )^{4} \)
$97$ \( ( 32400 + 23976 T^{2} + T^{4} )^{2} \)
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