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
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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{5} - \beta_{7} ) q^{3} + ( -3 \beta_{1} - 3 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} ) q^{7} + ( -\beta_{2} - 7 \beta_{3} - \beta_{6} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{5} - \beta_{7} ) q^{3} + ( -3 \beta_{1} - 3 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} ) q^{7} + ( -\beta_{2} - 7 \beta_{3} - \beta_{6} ) q^{9} + ( -12 - 2 \beta_{2} - 5 \beta_{3} - 2 \beta_{6} ) q^{11} + ( -\beta_{1} + 3 \beta_{7} ) q^{13} + ( 9 \beta_{1} + \beta_{4} - 7 \beta_{5} + 2 \beta_{7} ) q^{17} + ( -2 - 3 \beta_{6} ) q^{19} + ( 27 - 3 \beta_{2} + 2 \beta_{3} + \beta_{6} ) q^{21} + ( -15 \beta_{1} + 2 \beta_{4} - 32 \beta_{5} + \beta_{7} ) q^{23} + ( 15 \beta_{1} + 6 \beta_{4} - 9 \beta_{5} ) q^{27} + ( 21 - 7 \beta_{2} + 14 \beta_{3} - 7 \beta_{6} ) q^{29} + ( -8 + 9 \beta_{2} - 8 \beta_{3} ) q^{31} + ( 33 \beta_{1} + 3 \beta_{4} - 6 \beta_{5} + 12 \beta_{7} ) q^{33} + ( -10 \beta_{1} + 12 \beta_{4} - 10 \beta_{5} ) q^{37} + ( -\beta_{2} + 24 \beta_{3} + 4 \beta_{6} ) q^{39} + ( 9 - 8 \beta_{2} + 7 \beta_{3} + 16 \beta_{6} ) q^{41} + 23 \beta_{5} q^{43} + ( -27 \beta_{1} + 3 \beta_{4} - 12 \beta_{5} - 3 \beta_{7} ) q^{47} + ( 27 - 3 \beta_{2} + 27 \beta_{3} ) q^{49} + ( -9 + 9 \beta_{2} + 15 \beta_{3} - 15 \beta_{6} ) q^{51} + ( -8 \beta_{4} - 16 \beta_{5} - 16 \beta_{7} ) q^{53} + ( 23 \beta_{1} - 3 \beta_{4} - \beta_{5} - \beta_{7} ) q^{57} + ( -9 - 8 \beta_{2} + 25 \beta_{3} + 16 \beta_{6} ) q^{59} + ( 9 - 9 \beta_{2} - 16 \beta_{3} + 9 \beta_{6} ) q^{61} + ( -33 \beta_{1} - \beta_{4} - 50 \beta_{5} - 23 \beta_{7} ) q^{63} + ( 49 \beta_{1} - 18 \beta_{7} ) q^{67} + ( -18 - 15 \beta_{2} + 24 \beta_{3} - 18 \beta_{6} ) q^{69} + ( -12 - 4 \beta_{2} - 28 \beta_{3} + 2 \beta_{6} ) q^{71} + ( -17 \beta_{1} + 9 \beta_{4} - 17 \beta_{5} ) q^{73} + ( -21 \beta_{1} + 34 \beta_{4} - 76 \beta_{5} + 17 \beta_{7} ) q^{77} + ( -15 + 15 \beta_{2} + 34 \beta_{3} - 15 \beta_{6} ) q^{79} + ( -54 + 15 \beta_{2} - 39 \beta_{3} - 30 \beta_{6} ) q^{81} + ( 27 \beta_{1} - 3 \beta_{4} + 12 \beta_{5} + 3 \beta_{7} ) q^{83} + ( 21 \beta_{1} - 21 \beta_{4} - 84 \beta_{5} - 21 \beta_{7} ) q^{87} + ( 48 + 16 \beta_{2} + 112 \beta_{3} - 8 \beta_{6} ) q^{89} + ( -71 + 9 \beta_{6} ) q^{91} + ( 7 \beta_{1} + 8 \beta_{4} + 80 \beta_{5} - \beta_{7} ) q^{93} + ( -6 \beta_{1} - 6 \beta_{4} - 101 \beta_{5} - 6 \beta_{7} ) q^{97} + ( -27 + 33 \beta_{2} + 78 \beta_{3} - 30 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 30q^{9} + O(q^{10}) \) \( 8q + 30q^{9} - 72q^{11} - 4q^{19} + 198q^{21} + 126q^{29} - 14q^{31} - 114q^{39} - 36q^{41} + 102q^{49} - 54q^{51} - 252q^{59} + 82q^{61} - 198q^{69} - 166q^{79} - 126q^{81} - 604q^{91} - 342q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 32 \nu^{5} + 16 \nu^{3} + 45 \nu \)\()/432\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - 16 \nu^{4} - 32 \nu^{2} - 51 \)\()/48\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{6} - 16 \nu^{4} - 80 \nu^{2} - 225 \)\()/144\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 8 \nu^{5} + 40 \nu^{3} + 165 \nu \)\()/72\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 13 \nu \)\()/48\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} - 5 \nu^{4} - 7 \nu^{2} - 27 \)\()/9\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{5} + 10 \nu^{3} - 3 \nu \)\()/18\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} + 2 \beta_{5} + \beta_{4}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{6} - 7 \beta_{3} - \beta_{2} - 6\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{7} - 11 \beta_{5} + 2 \beta_{4} - 9 \beta_{1}\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-5 \beta_{6} + 13 \beta_{3} - 5 \beta_{2}\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{7} - \beta_{5} - 2 \beta_{4} + 45 \beta_{1}\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-16 \beta_{6} - 16 \beta_{3} + 32 \beta_{2} - 39\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(13 \beta_{7} + 118 \beta_{5} - 13 \beta_{4}\)\()/3\)

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 \( T_{7}^{8} - 149 T_{7}^{6} + 16725 T_{7}^{4} - 815924 T_{7}^{2} + 29986576 \) acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 15 T^{2} + 144 T^{4} - 1215 T^{6} + 6561 T^{8} \)
$5$ 1
$7$ \( 1 + 47 T^{2} - 3071 T^{4} + 22466 T^{6} + 16809790 T^{8} + 53940866 T^{10} - 17703703871 T^{12} + 650540498447 T^{14} + 33232930569601 T^{16} \)
$11$ \( ( 1 + 36 T + 683 T^{2} + 9036 T^{3} + 100632 T^{4} + 1093356 T^{5} + 9999803 T^{6} + 63776196 T^{7} + 214358881 T^{8} )^{2} \)
$13$ \( 1 + 515 T^{2} + 143653 T^{4} + 33191750 T^{6} + 6388929238 T^{8} + 947989571750 T^{10} + 117182165263813 T^{12} + 11998513838077715 T^{14} + 665416609183179841 T^{16} \)
$17$ \( ( 1 + 769 T^{2} + 298176 T^{4} + 64227649 T^{6} + 6975757441 T^{8} )^{2} \)
$19$ \( ( 1 + T + 648 T^{2} + 361 T^{3} + 130321 T^{4} )^{4} \)
$23$ \( 1 - 433 T^{2} - 338207 T^{4} + 14715938 T^{6} + 145578189886 T^{8} + 4118122805858 T^{10} - 26485323398931167 T^{12} - 9489032379064798993 T^{14} + \)\(61\!\cdots\!61\)\( T^{16} \)
$29$ \( ( 1 - 63 T + 2123 T^{2} - 50400 T^{3} + 1045362 T^{4} - 42386400 T^{5} + 1501557563 T^{6} - 37473869223 T^{7} + 500246412961 T^{8} )^{2} \)
$31$ \( ( 1 + 7 T - 1217 T^{2} - 4592 T^{3} + 632146 T^{4} - 4412912 T^{5} - 1123925057 T^{6} + 6212525767 T^{7} + 852891037441 T^{8} )^{2} \)
$37$ \( ( 1 - 2588 T^{2} + 4206246 T^{4} - 4850328668 T^{6} + 3512479453921 T^{8} )^{2} \)
$41$ \( ( 1 + 18 T + 1913 T^{2} + 32490 T^{3} + 613812 T^{4} + 54615690 T^{5} + 5405680793 T^{6} + 85501876338 T^{7} + 7984925229121 T^{8} )^{2} \)
$43$ \( ( 1 + 3169 T^{2} + 6623760 T^{4} + 10834180369 T^{6} + 11688200277601 T^{8} )^{2} \)
$47$ \( 1 - 7297 T^{2} + 30662449 T^{4} - 93579632206 T^{6} + 226390414696510 T^{8} - 456638753262606286 T^{10} + \)\(73\!\cdots\!89\)\( T^{12} - \)\(84\!\cdots\!77\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \)
$53$ \( ( 1 + 7204 T^{2} + 26018214 T^{4} + 56843025124 T^{6} + 62259690411361 T^{8} )^{2} \)
$59$ \( ( 1 + 126 T + 11993 T^{2} + 844326 T^{3} + 51207492 T^{4} + 2939098806 T^{5} + 145323510473 T^{6} + 5314747238766 T^{7} + 146830437604321 T^{8} )^{2} \)
$61$ \( ( 1 - 41 T - 5513 T^{2} + 10168 T^{3} + 31652794 T^{4} + 37835128 T^{5} - 76332121433 T^{6} - 2112335348801 T^{7} + 191707312997281 T^{8} )^{2} \)
$67$ \( 1 + 5882 T^{2} + 21614089 T^{4} - 160686869974 T^{6} - 934674659122220 T^{8} - 3238020559957340854 T^{10} + \)\(87\!\cdots\!49\)\( T^{12} + \)\(48\!\cdots\!02\)\( T^{14} + \)\(16\!\cdots\!81\)\( T^{16} \)
$71$ \( ( 1 - 18616 T^{2} + 137194926 T^{4} - 473063853496 T^{6} + 645753531245761 T^{8} )^{2} \)
$73$ \( ( 1 - 19055 T^{2} + 146334144 T^{4} - 541128482255 T^{6} + 806460091894081 T^{8} )^{2} \)
$79$ \( ( 1 + 83 T - 5459 T^{2} - 11122 T^{3} + 70528774 T^{4} - 69412402 T^{5} - 212628492179 T^{6} + 20176258808243 T^{7} + 1517108809906561 T^{8} )^{2} \)
$83$ \( 1 - 26017 T^{2} + 413233729 T^{4} - 4389950344606 T^{6} + 35135533688955790 T^{8} - \)\(20\!\cdots\!26\)\( T^{10} + \)\(93\!\cdots\!89\)\( T^{12} - \)\(27\!\cdots\!37\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} \)
$89$ \( ( 1 - 6916 T^{2} + 69013446 T^{4} - 433925338756 T^{6} + 3936588805702081 T^{8} )^{2} \)
$97$ \( 1 + 17834 T^{2} + 72889657 T^{4} + 1214554912058 T^{6} + 24110047553298004 T^{8} + \)\(10\!\cdots\!98\)\( T^{10} + \)\(57\!\cdots\!77\)\( T^{12} + \)\(12\!\cdots\!94\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} \)
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