Properties

Label 900.3.u.a
Level $900$
Weight $3$
Character orbit 900.u
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.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: 8.0.303595776.1
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 36)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_{5} - \beta_1) q^{3} + ( - 3 \beta_{7} - 2 \beta_{5} - 3 \beta_{4} - 3 \beta_1) q^{7} + ( - \beta_{6} - 7 \beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - \beta_{5} - \beta_1) q^{3} + ( - 3 \beta_{7} - 2 \beta_{5} - 3 \beta_{4} - 3 \beta_1) q^{7} + ( - \beta_{6} - 7 \beta_{3} - \beta_{2}) q^{9} + ( - 2 \beta_{6} - 5 \beta_{3} - 2 \beta_{2} - 12) q^{11} + (3 \beta_{7} - \beta_1) q^{13} + (2 \beta_{7} - 7 \beta_{5} + \beta_{4} + 9 \beta_1) q^{17} + ( - 3 \beta_{6} - 2) q^{19} + (\beta_{6} + 2 \beta_{3} - 3 \beta_{2} + 27) q^{21} + (\beta_{7} - 32 \beta_{5} + 2 \beta_{4} - 15 \beta_1) q^{23} + ( - 9 \beta_{5} + 6 \beta_{4} + 15 \beta_1) q^{27} + ( - 7 \beta_{6} + 14 \beta_{3} - 7 \beta_{2} + 21) q^{29} + ( - 8 \beta_{3} + 9 \beta_{2} - 8) q^{31} + (12 \beta_{7} - 6 \beta_{5} + 3 \beta_{4} + 33 \beta_1) q^{33} + ( - 10 \beta_{5} + 12 \beta_{4} - 10 \beta_1) q^{37} + (4 \beta_{6} + 24 \beta_{3} - \beta_{2}) q^{39} + (16 \beta_{6} + 7 \beta_{3} - 8 \beta_{2} + 9) q^{41} + 23 \beta_{5} q^{43} + ( - 3 \beta_{7} - 12 \beta_{5} + 3 \beta_{4} - 27 \beta_1) q^{47} + (27 \beta_{3} - 3 \beta_{2} + 27) q^{49} + ( - 15 \beta_{6} + 15 \beta_{3} + 9 \beta_{2} - 9) q^{51} + ( - 16 \beta_{7} - 16 \beta_{5} - 8 \beta_{4}) q^{53} + ( - \beta_{7} - \beta_{5} - 3 \beta_{4} + 23 \beta_1) q^{57} + (16 \beta_{6} + 25 \beta_{3} - 8 \beta_{2} - 9) q^{59} + (9 \beta_{6} - 16 \beta_{3} - 9 \beta_{2} + 9) q^{61} + ( - 23 \beta_{7} - 50 \beta_{5} - \beta_{4} - 33 \beta_1) q^{63} + ( - 18 \beta_{7} + 49 \beta_1) q^{67} + ( - 18 \beta_{6} + 24 \beta_{3} - 15 \beta_{2} - 18) q^{69} + (2 \beta_{6} - 28 \beta_{3} - 4 \beta_{2} - 12) q^{71} + ( - 17 \beta_{5} + 9 \beta_{4} - 17 \beta_1) q^{73} + (17 \beta_{7} - 76 \beta_{5} + 34 \beta_{4} - 21 \beta_1) q^{77} + ( - 15 \beta_{6} + 34 \beta_{3} + 15 \beta_{2} - 15) q^{79} + ( - 30 \beta_{6} - 39 \beta_{3} + 15 \beta_{2} - 54) q^{81} + (3 \beta_{7} + 12 \beta_{5} - 3 \beta_{4} + 27 \beta_1) q^{83} + ( - 21 \beta_{7} - 84 \beta_{5} - 21 \beta_{4} + 21 \beta_1) q^{87} + ( - 8 \beta_{6} + 112 \beta_{3} + 16 \beta_{2} + 48) q^{89} + (9 \beta_{6} - 71) q^{91} + ( - \beta_{7} + 80 \beta_{5} + 8 \beta_{4} + 7 \beta_1) q^{93} + ( - 6 \beta_{7} - 101 \beta_{5} - 6 \beta_{4} - 6 \beta_1) q^{97} + ( - 30 \beta_{6} + 78 \beta_{3} + 33 \beta_{2} - 27) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 30 q^{9} - 72 q^{11} - 4 q^{19} + 198 q^{21} + 126 q^{29} - 14 q^{31} - 114 q^{39} - 36 q^{41} + 102 q^{49} - 54 q^{51} - 252 q^{59} + 82 q^{61} - 198 q^{69} - 166 q^{79} - 126 q^{81} - 604 q^{91} - 342 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 16\nu^{4} - 32\nu^{2} - 51 ) / 48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 8\nu^{5} + 40\nu^{3} + 165\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 13\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 5\nu^{4} - 7\nu^{2} - 27 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 2\nu^{5} + 10\nu^{3} - 3\nu ) / 18 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} + 2\beta_{5} + \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - 7\beta_{3} - \beta_{2} - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{7} - 11\beta_{5} + 2\beta_{4} - 9\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{6} + 13\beta_{3} - 5\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - \beta_{5} - 2\beta_{4} + 45\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -16\beta_{6} - 16\beta_{3} + 32\beta_{2} - 39 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13\beta_{7} + 118\beta_{5} - 13\beta_{4} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1 + \beta_{3}\) \(1\) \(-1\)

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
−0.396143 1.68614i
−1.26217 1.18614i
1.26217 + 1.18614i
0.396143 + 1.68614i
−0.396143 + 1.68614i
−1.26217 + 1.18614i
1.26217 1.18614i
0.396143 1.68614i
0 −2.92048 0.686141i 0 0 0 −7.89542 + 4.55842i 0 8.05842 + 4.00772i 0
149.2 0 −2.05446 2.18614i 0 0 0 −7.02939 + 4.05842i 0 −0.558422 + 8.98266i 0
149.3 0 2.05446 + 2.18614i 0 0 0 7.02939 4.05842i 0 −0.558422 + 8.98266i 0
149.4 0 2.92048 + 0.686141i 0 0 0 7.89542 4.55842i 0 8.05842 + 4.00772i 0
749.1 0 −2.92048 + 0.686141i 0 0 0 −7.89542 4.55842i 0 8.05842 4.00772i 0
749.2 0 −2.05446 + 2.18614i 0 0 0 −7.02939 4.05842i 0 −0.558422 8.98266i 0
749.3 0 2.05446 2.18614i 0 0 0 7.02939 + 4.05842i 0 −0.558422 8.98266i 0
749.4 0 2.92048 0.686141i 0 0 0 7.89542 + 4.55842i 0 8.05842 4.00772i 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.a 8
3.b odd 2 1 2700.3.u.b 8
5.b even 2 1 inner 900.3.u.a 8
5.c odd 4 1 36.3.g.a 4
5.c odd 4 1 900.3.p.a 4
9.c even 3 1 2700.3.u.b 8
9.d odd 6 1 inner 900.3.u.a 8
15.d odd 2 1 2700.3.u.b 8
15.e even 4 1 108.3.g.a 4
15.e even 4 1 2700.3.p.b 4
20.e even 4 1 144.3.q.b 4
40.i odd 4 1 576.3.q.d 4
40.k even 4 1 576.3.q.g 4
45.h odd 6 1 inner 900.3.u.a 8
45.j even 6 1 2700.3.u.b 8
45.k odd 12 1 108.3.g.a 4
45.k odd 12 1 324.3.c.b 4
45.k odd 12 1 2700.3.p.b 4
45.l even 12 1 36.3.g.a 4
45.l even 12 1 324.3.c.b 4
45.l even 12 1 900.3.p.a 4
60.l odd 4 1 432.3.q.b 4
120.q odd 4 1 1728.3.q.h 4
120.w even 4 1 1728.3.q.g 4
180.v odd 12 1 144.3.q.b 4
180.v odd 12 1 1296.3.e.e 4
180.x even 12 1 432.3.q.b 4
180.x even 12 1 1296.3.e.e 4
360.bo even 12 1 1728.3.q.h 4
360.br even 12 1 576.3.q.d 4
360.bt odd 12 1 576.3.q.g 4
360.bu odd 12 1 1728.3.q.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.g.a 4 5.c odd 4 1
36.3.g.a 4 45.l even 12 1
108.3.g.a 4 15.e even 4 1
108.3.g.a 4 45.k odd 12 1
144.3.q.b 4 20.e even 4 1
144.3.q.b 4 180.v odd 12 1
324.3.c.b 4 45.k odd 12 1
324.3.c.b 4 45.l even 12 1
432.3.q.b 4 60.l odd 4 1
432.3.q.b 4 180.x even 12 1
576.3.q.d 4 40.i odd 4 1
576.3.q.d 4 360.br even 12 1
576.3.q.g 4 40.k even 4 1
576.3.q.g 4 360.bt odd 12 1
900.3.p.a 4 5.c odd 4 1
900.3.p.a 4 45.l even 12 1
900.3.u.a 8 1.a even 1 1 trivial
900.3.u.a 8 5.b even 2 1 inner
900.3.u.a 8 9.d odd 6 1 inner
900.3.u.a 8 45.h odd 6 1 inner
1296.3.e.e 4 180.v odd 12 1
1296.3.e.e 4 180.x even 12 1
1728.3.q.g 4 120.w even 4 1
1728.3.q.g 4 360.bu odd 12 1
1728.3.q.h 4 120.q odd 4 1
1728.3.q.h 4 360.bo even 12 1
2700.3.p.b 4 15.e even 4 1
2700.3.p.b 4 45.k odd 12 1
2700.3.u.b 8 3.b odd 2 1
2700.3.u.b 8 9.c even 3 1
2700.3.u.b 8 15.d odd 2 1
2700.3.u.b 8 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 149T_{7}^{6} + 16725T_{7}^{4} - 815924T_{7}^{2} + 29986576 \) acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 15 T^{6} + 144 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 149 T^{6} + \cdots + 29986576 \) Copy content Toggle raw display
$11$ \( (T^{4} + 36 T^{3} + 441 T^{2} + 324 T + 81)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 161 T^{6} + \cdots + 21381376 \) Copy content Toggle raw display
$17$ \( (T^{4} - 387 T^{2} + 20736)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + T - 74)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 1683 T^{6} + \cdots + 393460125696 \) Copy content Toggle raw display
$29$ \( (T^{4} - 63 T^{3} + 441 T^{2} + \cdots + 777924)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 7 T^{3} + 705 T^{2} + \cdots + 430336)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2888 T^{2} + 868624)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 18 T^{3} - 1449 T^{2} + \cdots + 2424249)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 529 T^{2} + 279841)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 1539 T^{6} + \cdots + 11019960576 \) Copy content Toggle raw display
$53$ \( (T^{4} - 4032 T^{2} + 1327104)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 126 T^{3} + 5031 T^{2} + \cdots + 68121)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 41 T^{3} + 1929 T^{2} + \cdots + 61504)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 12074 T^{6} + \cdots + 227988105361 \) Copy content Toggle raw display
$71$ \( (T^{4} + 1548 T^{2} + 331776)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 2261 T^{2} + 42436)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 83 T^{3} + 7023 T^{2} + \cdots + 17956)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 1539 T^{6} + \cdots + 11019960576 \) Copy content Toggle raw display
$89$ \( (T^{4} + 24768 T^{2} + 84934656)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 19802 T^{6} + \cdots + 75\!\cdots\!01 \) Copy content Toggle raw display
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