Properties

Label 900.3.u.b
Level $900$
Weight $3$
Character orbit 900.u
Analytic conductor $24.523$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.u (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12960000.1
Defining polynomial: \(x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + 3 \beta_{1} q^{3} + ( 4 \beta_{1} + \beta_{6} ) q^{7} -9 \beta_{2} q^{9} +O(q^{10})\) \( q + 3 \beta_{1} q^{3} + ( 4 \beta_{1} + \beta_{6} ) q^{7} -9 \beta_{2} q^{9} + ( 1 - \beta_{2} - \beta_{5} - \beta_{7} ) q^{11} + ( 10 \beta_{1} + 10 \beta_{3} - \beta_{4} ) q^{13} + ( -2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{17} + ( 19 + \beta_{5} ) q^{19} + ( -12 \beta_{2} - 3 \beta_{7} ) q^{21} + ( -4 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} - 4 \beta_{6} ) q^{23} -27 \beta_{3} q^{27} + ( 10 - 10 \beta_{2} + 2 \beta_{5} + 2 \beta_{7} ) q^{29} + ( 2 \beta_{2} + 7 \beta_{7} ) q^{31} + ( 3 \beta_{1} - 3 \beta_{3} + 3 \beta_{4} + 6 \beta_{6} ) q^{33} -14 \beta_{3} q^{37} + ( 30 - 3 \beta_{5} ) q^{39} + ( -30 - 15 \beta_{2} ) q^{41} + ( 11 \beta_{1} - 4 \beta_{6} ) q^{43} + ( -34 \beta_{1} - 68 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{47} + ( -27 \beta_{2} - 8 \beta_{7} ) q^{49} + ( -3 + 3 \beta_{2} + 3 \beta_{5} + 3 \beta_{7} ) q^{51} + ( 28 \beta_{1} + 14 \beta_{3} - \beta_{4} + \beta_{6} ) q^{53} + ( 57 \beta_{1} - 3 \beta_{6} ) q^{57} + ( 58 + 29 \beta_{2} - 2 \beta_{5} + \beta_{7} ) q^{59} + ( 22 + 22 \beta_{2} + 3 \beta_{5} - 3 \beta_{7} ) q^{61} + ( -36 \beta_{3} + 9 \beta_{4} + 9 \beta_{6} ) q^{63} + ( -7 \beta_{1} - 7 \beta_{3} - 12 \beta_{4} ) q^{67} + ( 12 + 24 \beta_{2} - 6 \beta_{5} + 12 \beta_{7} ) q^{69} + ( 36 + 72 \beta_{2} + 3 \beta_{5} - 6 \beta_{7} ) q^{71} + ( -37 \beta_{3} + 3 \beta_{4} + 3 \beta_{6} ) q^{73} + ( 64 \beta_{1} - 64 \beta_{3} + 5 \beta_{4} + 10 \beta_{6} ) q^{77} + ( 80 + 80 \beta_{2} - 3 \beta_{5} + 3 \beta_{7} ) q^{79} + ( -81 - 81 \beta_{2} ) q^{81} + ( -52 \beta_{1} - 104 \beta_{3} ) q^{83} + ( 30 \beta_{1} - 30 \beta_{3} - 6 \beta_{4} - 12 \beta_{6} ) q^{87} + ( 24 + 48 \beta_{2} + 4 \beta_{5} - 8 \beta_{7} ) q^{89} + ( 100 - 14 \beta_{5} ) q^{91} + ( 6 \beta_{3} - 21 \beta_{4} - 21 \beta_{6} ) q^{93} + ( \beta_{1} - 13 \beta_{6} ) q^{97} + ( -9 - 18 \beta_{2} + 9 \beta_{5} - 18 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 36q^{9} + O(q^{10}) \) \( 8q + 36q^{9} + 12q^{11} + 152q^{19} + 48q^{21} + 120q^{29} - 8q^{31} + 240q^{39} - 180q^{41} + 108q^{49} - 36q^{51} + 348q^{59} + 88q^{61} + 320q^{79} - 324q^{81} + 800q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 13 \nu \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{6} + 8 \nu^{4} - 24 \nu^{2} + 1 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{7} + 8 \nu^{5} - 20 \nu^{3} + \nu \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{6} + 24 \nu^{4} - 56 \nu^{2} + 39 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{7} + 16 \nu^{5} - 44 \nu^{3} + 31 \nu \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -11 \nu^{6} + 24 \nu^{4} - 56 \nu^{2} - 15 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( 13 \nu^{7} - 32 \nu^{5} + 88 \nu^{3} + 25 \nu \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{5} - 6 \beta_{1}\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{6} + \beta_{4} - 18 \beta_{2}\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} - \beta_{5} + 12 \beta_{3}\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{6} + 2 \beta_{4} - 14 \beta_{2} - 14\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{7} - 10 \beta_{5} + 66 \beta_{3} + 66 \beta_{1}\)\()/12\)
\(\nu^{6}\)\(=\)\((\)\(-2 \beta_{6} + 2 \beta_{4} - 27\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-13 \beta_{7} - 13 \beta_{5} + 174 \beta_{1}\)\()/12\)

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.535233 + 0.309017i
1.40126 0.809017i
0.535233 0.309017i
−1.40126 + 0.809017i
−0.535233 0.309017i
1.40126 + 0.809017i
0.535233 + 0.309017i
−1.40126 0.809017i
0 −2.59808 + 1.50000i 0 0 0 −10.1723 + 5.87298i 0 4.50000 7.79423i 0
149.2 0 −2.59808 + 1.50000i 0 0 0 3.24410 1.87298i 0 4.50000 7.79423i 0
149.3 0 2.59808 1.50000i 0 0 0 −3.24410 + 1.87298i 0 4.50000 7.79423i 0
149.4 0 2.59808 1.50000i 0 0 0 10.1723 5.87298i 0 4.50000 7.79423i 0
749.1 0 −2.59808 1.50000i 0 0 0 −10.1723 5.87298i 0 4.50000 + 7.79423i 0
749.2 0 −2.59808 1.50000i 0 0 0 3.24410 + 1.87298i 0 4.50000 + 7.79423i 0
749.3 0 2.59808 + 1.50000i 0 0 0 −3.24410 1.87298i 0 4.50000 + 7.79423i 0
749.4 0 2.59808 + 1.50000i 0 0 0 10.1723 + 5.87298i 0 4.50000 + 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 749.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.d odd 6 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.u.b 8
3.b odd 2 1 2700.3.u.a 8
5.b even 2 1 inner 900.3.u.b 8
5.c odd 4 1 180.3.o.a 4
5.c odd 4 1 900.3.p.b 4
9.c even 3 1 2700.3.u.a 8
9.d odd 6 1 inner 900.3.u.b 8
15.d odd 2 1 2700.3.u.a 8
15.e even 4 1 540.3.o.a 4
15.e even 4 1 2700.3.p.a 4
20.e even 4 1 720.3.bs.a 4
45.h odd 6 1 inner 900.3.u.b 8
45.j even 6 1 2700.3.u.a 8
45.k odd 12 1 540.3.o.a 4
45.k odd 12 1 1620.3.g.a 4
45.k odd 12 1 2700.3.p.a 4
45.l even 12 1 180.3.o.a 4
45.l even 12 1 900.3.p.b 4
45.l even 12 1 1620.3.g.a 4
60.l odd 4 1 2160.3.bs.a 4
180.v odd 12 1 720.3.bs.a 4
180.x even 12 1 2160.3.bs.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.o.a 4 5.c odd 4 1
180.3.o.a 4 45.l even 12 1
540.3.o.a 4 15.e even 4 1
540.3.o.a 4 45.k odd 12 1
720.3.bs.a 4 20.e even 4 1
720.3.bs.a 4 180.v odd 12 1
900.3.p.b 4 5.c odd 4 1
900.3.p.b 4 45.l even 12 1
900.3.u.b 8 1.a even 1 1 trivial
900.3.u.b 8 5.b even 2 1 inner
900.3.u.b 8 9.d odd 6 1 inner
900.3.u.b 8 45.h odd 6 1 inner
1620.3.g.a 4 45.k odd 12 1
1620.3.g.a 4 45.l even 12 1
2160.3.bs.a 4 60.l odd 4 1
2160.3.bs.a 4 180.x even 12 1
2700.3.p.a 4 15.e even 4 1
2700.3.p.a 4 45.k odd 12 1
2700.3.u.a 8 3.b odd 2 1
2700.3.u.a 8 9.c even 3 1
2700.3.u.a 8 15.d odd 2 1
2700.3.u.a 8 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 152 T_{7}^{6} + 21168 T_{7}^{4} - 294272 T_{7}^{2} + 3748096 \) acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 81 - 9 T^{2} + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( 3748096 - 294272 T^{2} + 21168 T^{4} - 152 T^{6} + T^{8} \)
$11$ \( ( 31329 + 1062 T - 165 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$13$ \( 2560000 - 512000 T^{2} + 100800 T^{4} - 320 T^{6} + T^{8} \)
$17$ \( ( 31329 - 366 T^{2} + T^{4} )^{2} \)
$19$ \( ( 301 - 38 T + T^{2} )^{4} \)
$23$ \( 203928109056 + 693633024 T^{2} + 1907712 T^{4} + 1536 T^{6} + T^{8} \)
$29$ \( ( 176400 + 25200 T + 780 T^{2} - 60 T^{3} + T^{4} )^{2} \)
$31$ \( ( 8620096 - 11744 T + 2952 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$37$ \( ( 196 + T^{2} )^{4} \)
$41$ \( ( 675 + 45 T + T^{2} )^{4} \)
$43$ \( 495504774241 - 1521877202 T^{2} + 3970323 T^{4} - 2162 T^{6} + T^{8} \)
$47$ \( 116876510171136 + 78876647424 T^{2} + 42420672 T^{4} + 7296 T^{6} + T^{8} \)
$53$ \( ( 166464 - 1536 T^{2} + T^{4} )^{2} \)
$59$ \( ( 5489649 - 407682 T + 12435 T^{2} - 174 T^{3} + T^{4} )^{2} \)
$61$ \( ( 3136 + 2464 T + 1992 T^{2} - 44 T^{3} + T^{4} )^{2} \)
$67$ \( 5447219503488961 - 1282588173218 T^{2} + 228189603 T^{4} - 17378 T^{6} + T^{8} \)
$71$ \( ( 5143824 + 11016 T^{2} + T^{4} )^{2} \)
$73$ \( ( 687241 + 3818 T^{2} + T^{4} )^{2} \)
$79$ \( ( 34339600 - 937600 T + 19740 T^{2} - 160 T^{3} + T^{4} )^{2} \)
$83$ \( ( 65804544 + 8112 T^{2} + T^{4} )^{2} \)
$89$ \( ( 1327104 + 9216 T^{2} + T^{4} )^{2} \)
$97$ \( 10567700398061041 - 2084975828522 T^{2} + 308560203 T^{4} - 20282 T^{6} + T^{8} \)
show more
show less