Properties

Label 900.3.p.b
Level $900$
Weight $3$
Character orbit 900.p
Analytic conductor $24.523$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta_{1} q^{3} + ( -4 \beta_{1} - \beta_{2} ) q^{7} + ( -9 + 9 \beta_{1} ) q^{9} +O(q^{10})\) \( q + 3 \beta_{1} q^{3} + ( -4 \beta_{1} - \beta_{2} ) q^{7} + ( -9 + 9 \beta_{1} ) q^{9} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{11} + ( -10 + 10 \beta_{1} + \beta_{2} - \beta_{3} ) q^{13} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{17} + ( -19 + \beta_{3} ) q^{19} + ( 12 - 12 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{21} + ( -4 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{23} -27 q^{27} + ( -20 + 10 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{29} + ( -2 + 2 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} ) q^{31} + ( 3 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{33} -14 q^{37} + ( -30 - 3 \beta_{3} ) q^{39} + ( -15 - 15 \beta_{1} ) q^{41} + ( 11 \beta_{1} - 4 \beta_{2} ) q^{43} + ( -68 + 34 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{47} + ( -27 + 27 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{49} + ( -6 + 3 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} ) q^{51} + ( -14 + 28 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{53} + ( -57 \beta_{1} + 3 \beta_{2} ) q^{57} + ( -29 - 29 \beta_{1} - \beta_{2} - \beta_{3} ) q^{59} + ( 22 \beta_{1} - 3 \beta_{2} ) q^{61} + ( 36 + 9 \beta_{3} ) q^{63} + ( -7 + 7 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{67} + ( 12 - 24 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} ) q^{69} + ( -36 + 72 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{71} + ( 37 + 3 \beta_{3} ) q^{73} + ( -64 - 64 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{77} + ( -80 \beta_{1} - 3 \beta_{2} ) q^{79} -81 \beta_{1} q^{81} + ( 104 - 52 \beta_{1} ) q^{83} + ( -30 - 30 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{87} + ( 24 - 48 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{89} + ( 100 + 14 \beta_{3} ) q^{91} + ( -6 - 21 \beta_{3} ) q^{93} + ( -\beta_{1} + 13 \beta_{2} ) q^{97} + ( -9 + 18 \beta_{1} + 18 \beta_{2} - 9 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{3} - 8q^{7} - 18q^{9} + O(q^{10}) \) \( 4q + 6q^{3} - 8q^{7} - 18q^{9} + 6q^{11} - 20q^{13} - 76q^{19} + 24q^{21} - 24q^{23} - 108q^{27} - 60q^{29} - 4q^{31} + 18q^{33} - 56q^{37} - 120q^{39} - 90q^{41} + 22q^{43} - 204q^{47} - 54q^{49} - 18q^{51} - 114q^{57} - 174q^{59} + 44q^{61} + 144q^{63} - 14q^{67} + 148q^{73} - 384q^{77} - 160q^{79} - 162q^{81} + 312q^{83} - 180q^{87} + 400q^{91} - 24q^{93} - 2q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{3} + 10 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{3} + 20 \nu \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/6\)
\(\nu^{2}\)\(=\)\(5 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{3} + 10 \beta_{2}\)\()/6\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
1.93649 1.11803i
−1.93649 + 1.11803i
1.93649 + 1.11803i
−1.93649 1.11803i
0 1.50000 2.59808i 0 0 0 −5.87298 + 10.1723i 0 −4.50000 7.79423i 0
101.2 0 1.50000 2.59808i 0 0 0 1.87298 3.24410i 0 −4.50000 7.79423i 0
401.1 0 1.50000 + 2.59808i 0 0 0 −5.87298 10.1723i 0 −4.50000 + 7.79423i 0
401.2 0 1.50000 + 2.59808i 0 0 0 1.87298 + 3.24410i 0 −4.50000 + 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 8 T_{7}^{3} + 108 T_{7}^{2} - 352 T_{7} + 1936 \) acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 9 - 3 T + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 1936 - 352 T + 108 T^{2} + 8 T^{3} + T^{4} \)
$11$ \( 31329 + 1062 T - 165 T^{2} - 6 T^{3} + T^{4} \)
$13$ \( 1600 + 800 T + 360 T^{2} + 20 T^{3} + T^{4} \)
$17$ \( 31329 + 366 T^{2} + T^{4} \)
$19$ \( ( 301 + 38 T + T^{2} )^{2} \)
$23$ \( 451584 - 16128 T - 480 T^{2} + 24 T^{3} + T^{4} \)
$29$ \( 176400 - 25200 T + 780 T^{2} + 60 T^{3} + T^{4} \)
$31$ \( 8620096 - 11744 T + 2952 T^{2} + 4 T^{3} + T^{4} \)
$37$ \( ( 14 + T )^{4} \)
$41$ \( ( 675 + 45 T + T^{2} )^{2} \)
$43$ \( 703921 + 18458 T + 1323 T^{2} - 22 T^{3} + T^{4} \)
$47$ \( 10810944 + 670752 T + 17160 T^{2} + 204 T^{3} + T^{4} \)
$53$ \( 166464 + 1536 T^{2} + T^{4} \)
$59$ \( 5489649 + 407682 T + 12435 T^{2} + 174 T^{3} + T^{4} \)
$61$ \( 3136 + 2464 T + 1992 T^{2} - 44 T^{3} + T^{4} \)
$67$ \( 73805281 - 120274 T + 8787 T^{2} + 14 T^{3} + T^{4} \)
$71$ \( 5143824 + 11016 T^{2} + T^{4} \)
$73$ \( ( 829 - 74 T + T^{2} )^{2} \)
$79$ \( 34339600 + 937600 T + 19740 T^{2} + 160 T^{3} + T^{4} \)
$83$ \( ( 8112 - 156 T + T^{2} )^{2} \)
$89$ \( 1327104 + 9216 T^{2} + T^{4} \)
$97$ \( 102799321 - 20278 T + 10143 T^{2} + 2 T^{3} + T^{4} \)
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