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

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,3,Mod(101,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.101");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_{2}) q^{3} + (3 \beta_{3} - \beta_1) q^{7} + (\beta_{3} - 2 \beta_{2} - 8 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_{2}) q^{3} + (3 \beta_{3} - \beta_1) q^{7} + (\beta_{3} - 2 \beta_{2} - 8 \beta_1 - 1) q^{9} + ( - 2 \beta_{3} + 4 \beta_{2} + 7 \beta_1 - 10) q^{11} + (3 \beta_{3} - 3 \beta_{2} + \beta_1 - 4) q^{13} + ( - 2 \beta_{3} + \beta_{2} + 16 \beta_1 - 7) q^{17} + ( - 3 \beta_{2} - 1) q^{19} + ( - 3 \beta_{3} + 2 \beta_{2} + \beta_1 + 26) q^{21} + (\beta_{3} + \beta_{2} - 17 \beta_1 - 16) q^{23} + ( - 6 \beta_{2} + 24 \beta_1 - 15) q^{27} + (7 \beta_{3} - 14 \beta_{2} + 7 \beta_1 - 28) q^{29} + (9 \beta_{3} - 9 \beta_{2} - \beta_1 - 8) q^{31} + (12 \beta_{3} - 9 \beta_{2} - 39 \beta_1 + 21) q^{33} + ( - 12 \beta_{2} + 10) q^{37} + (\beta_{3} + 3 \beta_{2} + 23 \beta_1 + 4) q^{39} + ( - 8 \beta_{3} - 8 \beta_{2} + \beta_1 - 7) q^{41} + 23 \beta_1 q^{43} + (3 \beta_{3} - 6 \beta_{2} - 15 \beta_1 + 24) q^{47} + (3 \beta_{3} - 3 \beta_{2} + 24 \beta_1 - 27) q^{49} + (9 \beta_{3} + 6 \beta_{2} - 24 \beta_1 + 6) q^{51} + ( - 16 \beta_{3} + 8 \beta_{2} - 16 \beta_1 + 16) q^{53} + (\beta_{3} + 2 \beta_{2} + 24 \beta_1 - 24) q^{57} + (8 \beta_{3} + 8 \beta_{2} + 17 \beta_1 + 25) q^{59} + ( - 9 \beta_{3} + 25 \beta_1) q^{61} + ( - 23 \beta_{3} + 22 \beta_{2} - 17 \beta_1 - 10) q^{63} + (18 \beta_{3} - 18 \beta_{2} + 49 \beta_1 - 67) q^{67} + (15 \beta_{3} - 33 \beta_{2} + 9 \beta_1) q^{69} + ( - 4 \beta_{3} + 2 \beta_{2} + 32 \beta_1 - 14) q^{71} + (9 \beta_{2} - 17) q^{73} + ( - 17 \beta_{3} - 17 \beta_{2} + 55 \beta_1 + 38) q^{77} + ( - 15 \beta_{3} + 49 \beta_1) q^{79} + (15 \beta_{3} + 15 \beta_{2} + 24 \beta_1 - 24) q^{81} + (3 \beta_{3} - 6 \beta_{2} - 15 \beta_1 + 24) q^{83} + (21 \beta_{3} + 105 \beta_1 - 42) q^{87} + ( - 16 \beta_{3} + 8 \beta_{2} + 128 \beta_1 - 56) q^{89} + ( - 9 \beta_{2} - 80) q^{91} + ( - \beta_{3} + 9 \beta_{2} + 73 \beta_1 + 8) q^{93} + (6 \beta_{3} + 95 \beta_1) q^{97} + ( - 33 \beta_{3} + 3 \beta_{2} + 111 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{3} + q^{7} - 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{3} + q^{7} - 15 q^{9} - 36 q^{11} - 5 q^{13} + 2 q^{19} + 99 q^{21} - 99 q^{23} - 63 q^{29} - 7 q^{31} + 36 q^{33} + 64 q^{37} + 57 q^{39} - 18 q^{41} + 46 q^{43} + 81 q^{47} - 51 q^{49} - 27 q^{51} - 51 q^{57} + 126 q^{59} + 41 q^{61} - 141 q^{63} - 116 q^{67} + 99 q^{69} - 86 q^{73} + 279 q^{77} + 83 q^{79} - 63 q^{81} + 81 q^{83} + 63 q^{87} - 302 q^{91} + 159 q^{93} + 196 q^{97} + 171 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3 \) Copy content Toggle raw display

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 \( T_{7}^{4} - T_{7}^{3} + 75T_{7}^{2} + 74T_{7} + 5476 \) acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3 T^{3} + 12 T^{2} + 27 T + 81 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - T^{3} + 75 T^{2} + 74 T + 5476 \) Copy content Toggle raw display
$11$ \( T^{4} + 36 T^{3} + 441 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{4} + 5 T^{3} + 93 T^{2} + \cdots + 4624 \) Copy content Toggle raw display
$17$ \( T^{4} + 387 T^{2} + 20736 \) Copy content Toggle raw display
$19$ \( (T^{2} - T - 74)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 99 T^{3} + 4059 T^{2} + \cdots + 627264 \) Copy content Toggle raw display
$29$ \( T^{4} + 63 T^{3} + 441 T^{2} + \cdots + 777924 \) Copy content Toggle raw display
$31$ \( T^{4} + 7 T^{3} + 705 T^{2} + \cdots + 430336 \) Copy content Toggle raw display
$37$ \( (T^{2} - 32 T - 932)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 18 T^{3} - 1449 T^{2} + \cdots + 2424249 \) Copy content Toggle raw display
$43$ \( (T^{2} - 23 T + 529)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 81 T^{3} + 2511 T^{2} + \cdots + 104976 \) Copy content Toggle raw display
$53$ \( T^{4} + 4032 T^{2} + \cdots + 1327104 \) Copy content Toggle raw display
$59$ \( T^{4} - 126 T^{3} + 5031 T^{2} + \cdots + 68121 \) Copy content Toggle raw display
$61$ \( T^{4} - 41 T^{3} + 1929 T^{2} + \cdots + 61504 \) Copy content Toggle raw display
$67$ \( T^{4} + 116 T^{3} + 12765 T^{2} + \cdots + 477481 \) Copy content Toggle raw display
$71$ \( T^{4} + 1548 T^{2} + 331776 \) Copy content Toggle raw display
$73$ \( (T^{2} + 43 T - 206)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 83 T^{3} + 7023 T^{2} + \cdots + 17956 \) Copy content Toggle raw display
$83$ \( T^{4} - 81 T^{3} + 2511 T^{2} + \cdots + 104976 \) Copy content Toggle raw display
$89$ \( T^{4} + 24768 T^{2} + \cdots + 84934656 \) Copy content Toggle raw display
$97$ \( T^{4} - 196 T^{3} + \cdots + 86620249 \) Copy content Toggle raw display
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