Properties

Label 900.3.p.a
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: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{2} - \beta_{3} ) q^{3} + ( -\beta_{1} + 3 \beta_{3} ) q^{7} + ( -1 - 8 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( \beta_{2} - \beta_{3} ) q^{3} + ( -\beta_{1} + 3 \beta_{3} ) q^{7} + ( -1 - 8 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{9} + ( -10 + 7 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{11} + ( -4 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{13} + ( -7 + 16 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{17} + ( -1 - 3 \beta_{2} ) q^{19} + ( 26 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{21} + ( -16 - 17 \beta_{1} + \beta_{2} + \beta_{3} ) q^{23} + ( -15 + 24 \beta_{1} - 6 \beta_{2} ) q^{27} + ( -28 + 7 \beta_{1} - 14 \beta_{2} + 7 \beta_{3} ) q^{29} + ( -8 - \beta_{1} - 9 \beta_{2} + 9 \beta_{3} ) q^{31} + ( 21 - 39 \beta_{1} - 9 \beta_{2} + 12 \beta_{3} ) q^{33} + ( 10 - 12 \beta_{2} ) q^{37} + ( 4 + 23 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{39} + ( -7 + \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{41} + 23 \beta_{1} q^{43} + ( 24 - 15 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{47} + ( -27 + 24 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{49} + ( 6 - 24 \beta_{1} + 6 \beta_{2} + 9 \beta_{3} ) q^{51} + ( 16 - 16 \beta_{1} + 8 \beta_{2} - 16 \beta_{3} ) q^{53} + ( -24 + 24 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{57} + ( 25 + 17 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{59} + ( 25 \beta_{1} - 9 \beta_{3} ) q^{61} + ( -10 - 17 \beta_{1} + 22 \beta_{2} - 23 \beta_{3} ) q^{63} + ( -67 + 49 \beta_{1} - 18 \beta_{2} + 18 \beta_{3} ) q^{67} + ( 9 \beta_{1} - 33 \beta_{2} + 15 \beta_{3} ) q^{69} + ( -14 + 32 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{71} + ( -17 + 9 \beta_{2} ) q^{73} + ( 38 + 55 \beta_{1} - 17 \beta_{2} - 17 \beta_{3} ) q^{77} + ( 49 \beta_{1} - 15 \beta_{3} ) q^{79} + ( -24 + 24 \beta_{1} + 15 \beta_{2} + 15 \beta_{3} ) q^{81} + ( 24 - 15 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{83} + ( -42 + 105 \beta_{1} + 21 \beta_{3} ) q^{87} + ( -56 + 128 \beta_{1} + 8 \beta_{2} - 16 \beta_{3} ) q^{89} + ( -80 - 9 \beta_{2} ) q^{91} + ( 8 + 73 \beta_{1} + 9 \beta_{2} - \beta_{3} ) q^{93} + ( 95 \beta_{1} + 6 \beta_{3} ) q^{97} + ( -3 + 111 \beta_{1} + 3 \beta_{2} - 33 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 3q^{3} + q^{7} - 15q^{9} + O(q^{10}) \) \( 4q - 3q^{3} + q^{7} - 15q^{9} - 36q^{11} - 5q^{13} + 2q^{19} + 99q^{21} - 99q^{23} - 63q^{29} - 7q^{31} + 36q^{33} + 64q^{37} + 57q^{39} - 18q^{41} + 46q^{43} + 81q^{47} - 51q^{49} - 27q^{51} - 51q^{57} + 126q^{59} + 41q^{61} - 141q^{63} - 116q^{67} + 99q^{69} - 86q^{73} + 279q^{77} + 83q^{79} - 63q^{81} + 81q^{83} + 63q^{87} - 302q^{91} + 159q^{93} + 196q^{97} + 171q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} + \nu^{2} + 2 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11\)\()/3\)

Character values

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

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

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
−1.18614 + 1.26217i
1.68614 0.396143i
−1.18614 1.26217i
1.68614 + 0.396143i
0 −2.18614 2.05446i 0 0 0 −4.05842 + 7.02939i 0 0.558422 + 8.98266i 0
101.2 0 0.686141 + 2.92048i 0 0 0 4.55842 7.89542i 0 −8.05842 + 4.00772i 0
401.1 0 −2.18614 + 2.05446i 0 0 0 −4.05842 7.02939i 0 0.558422 8.98266i 0
401.2 0 0.686141 2.92048i 0 0 0 4.55842 + 7.89542i 0 −8.05842 4.00772i 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.a 4
3.b odd 2 1 2700.3.p.b 4
5.b even 2 1 36.3.g.a 4
5.c odd 4 2 900.3.u.a 8
9.c even 3 1 2700.3.p.b 4
9.d odd 6 1 inner 900.3.p.a 4
15.d odd 2 1 108.3.g.a 4
15.e even 4 2 2700.3.u.b 8
20.d odd 2 1 144.3.q.b 4
40.e odd 2 1 576.3.q.g 4
40.f even 2 1 576.3.q.d 4
45.h odd 6 1 36.3.g.a 4
45.h odd 6 1 324.3.c.b 4
45.j even 6 1 108.3.g.a 4
45.j even 6 1 324.3.c.b 4
45.k odd 12 2 2700.3.u.b 8
45.l even 12 2 900.3.u.a 8
60.h even 2 1 432.3.q.b 4
120.i odd 2 1 1728.3.q.g 4
120.m even 2 1 1728.3.q.h 4
180.n even 6 1 144.3.q.b 4
180.n even 6 1 1296.3.e.e 4
180.p odd 6 1 432.3.q.b 4
180.p odd 6 1 1296.3.e.e 4
360.z odd 6 1 1728.3.q.h 4
360.bd even 6 1 576.3.q.g 4
360.bh odd 6 1 576.3.q.d 4
360.bk even 6 1 1728.3.q.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.g.a 4 5.b even 2 1
36.3.g.a 4 45.h odd 6 1
108.3.g.a 4 15.d odd 2 1
108.3.g.a 4 45.j even 6 1
144.3.q.b 4 20.d odd 2 1
144.3.q.b 4 180.n even 6 1
324.3.c.b 4 45.h odd 6 1
324.3.c.b 4 45.j even 6 1
432.3.q.b 4 60.h even 2 1
432.3.q.b 4 180.p odd 6 1
576.3.q.d 4 40.f even 2 1
576.3.q.d 4 360.bh odd 6 1
576.3.q.g 4 40.e odd 2 1
576.3.q.g 4 360.bd even 6 1
900.3.p.a 4 1.a even 1 1 trivial
900.3.p.a 4 9.d odd 6 1 inner
900.3.u.a 8 5.c odd 4 2
900.3.u.a 8 45.l even 12 2
1296.3.e.e 4 180.n even 6 1
1296.3.e.e 4 180.p odd 6 1
1728.3.q.g 4 120.i odd 2 1
1728.3.q.g 4 360.bk even 6 1
1728.3.q.h 4 120.m even 2 1
1728.3.q.h 4 360.z odd 6 1
2700.3.p.b 4 3.b odd 2 1
2700.3.p.b 4 9.c even 3 1
2700.3.u.b 8 15.e even 4 2
2700.3.u.b 8 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - T_{7}^{3} + 75 T_{7}^{2} + 74 T_{7} + 5476 \) acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T + 12 T^{2} + 27 T^{3} + 81 T^{4} \)
$5$ 1
$7$ \( 1 - T - 23 T^{2} + 74 T^{3} - 1874 T^{4} + 3626 T^{5} - 55223 T^{6} - 117649 T^{7} + 5764801 T^{8} \)
$11$ \( 1 + 36 T + 683 T^{2} + 9036 T^{3} + 100632 T^{4} + 1093356 T^{5} + 9999803 T^{6} + 63776196 T^{7} + 214358881 T^{8} \)
$13$ \( 1 + 5 T - 245 T^{2} - 340 T^{3} + 40114 T^{4} - 57460 T^{5} - 6997445 T^{6} + 24134045 T^{7} + 815730721 T^{8} \)
$17$ \( 1 - 769 T^{2} + 298176 T^{4} - 64227649 T^{6} + 6975757441 T^{8} \)
$19$ \( ( 1 - T + 648 T^{2} - 361 T^{3} + 130321 T^{4} )^{2} \)
$23$ \( 1 + 99 T + 5117 T^{2} + 183150 T^{3} + 4870902 T^{4} + 96886350 T^{5} + 1431946397 T^{6} + 14655553011 T^{7} + 78310985281 T^{8} \)
$29$ \( 1 + 63 T + 2123 T^{2} + 50400 T^{3} + 1045362 T^{4} + 42386400 T^{5} + 1501557563 T^{6} + 37473869223 T^{7} + 500246412961 T^{8} \)
$31$ \( 1 + 7 T - 1217 T^{2} - 4592 T^{3} + 632146 T^{4} - 4412912 T^{5} - 1123925057 T^{6} + 6212525767 T^{7} + 852891037441 T^{8} \)
$37$ \( ( 1 - 32 T + 1806 T^{2} - 43808 T^{3} + 1874161 T^{4} )^{2} \)
$41$ \( 1 + 18 T + 1913 T^{2} + 32490 T^{3} + 613812 T^{4} + 54615690 T^{5} + 5405680793 T^{6} + 85501876338 T^{7} + 7984925229121 T^{8} \)
$43$ \( ( 1 - 23 T - 1320 T^{2} - 42527 T^{3} + 3418801 T^{4} )^{2} \)
$47$ \( 1 - 81 T + 6929 T^{2} - 384102 T^{3} + 22437966 T^{4} - 848481318 T^{5} + 33811309649 T^{6} - 873116441649 T^{7} + 23811286661761 T^{8} \)
$53$ \( 1 - 7204 T^{2} + 26018214 T^{4} - 56843025124 T^{6} + 62259690411361 T^{8} \)
$59$ \( 1 - 126 T + 11993 T^{2} - 844326 T^{3} + 51207492 T^{4} - 2939098806 T^{5} + 145323510473 T^{6} - 5314747238766 T^{7} + 146830437604321 T^{8} \)
$61$ \( 1 - 41 T - 5513 T^{2} + 10168 T^{3} + 31652794 T^{4} + 37835128 T^{5} - 76332121433 T^{6} - 2112335348801 T^{7} + 191707312997281 T^{8} \)
$67$ \( 1 + 116 T + 3787 T^{2} + 80156 T^{3} + 12934456 T^{4} + 359820284 T^{5} + 76312295227 T^{6} + 10493172331604 T^{7} + 406067677556641 T^{8} \)
$71$ \( 1 - 18616 T^{2} + 137194926 T^{4} - 473063853496 T^{6} + 645753531245761 T^{8} \)
$73$ \( ( 1 + 43 T + 10452 T^{2} + 229147 T^{3} + 28398241 T^{4} )^{2} \)
$79$ \( 1 - 83 T - 5459 T^{2} + 11122 T^{3} + 70528774 T^{4} + 69412402 T^{5} - 212628492179 T^{6} - 20176258808243 T^{7} + 1517108809906561 T^{8} \)
$83$ \( 1 - 81 T + 16289 T^{2} - 1142262 T^{3} + 166474326 T^{4} - 7869042918 T^{5} + 773048590769 T^{6} - 26482170242889 T^{7} + 2252292232139041 T^{8} \)
$89$ \( 1 - 6916 T^{2} + 69013446 T^{4} - 433925338756 T^{6} + 3936588805702081 T^{8} \)
$97$ \( 1 - 196 T + 10291 T^{2} - 1824172 T^{3} + 341030200 T^{4} - 17163634348 T^{5} + 911054830771 T^{6} - 163262512966084 T^{7} + 7837433594376961 T^{8} \)
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