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Properties

Label 900.3.c.r
Level $900$
Weight $3$
Character orbit 900.c
Analytic conductor $24.523$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,3,Mod(451,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.451");
 
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.c (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: 8.0.6080256576.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 7x^{6} - 12x^{5} + 12x^{4} - 48x^{3} + 112x^{2} - 192x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{7} + 1) q^{4} + (\beta_{4} + \beta_{2}) q^{7} + ( - \beta_{4} + \beta_{3} + \cdots + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{7} + 1) q^{4} + (\beta_{4} + \beta_{2}) q^{7} + ( - \beta_{4} + \beta_{3} + \cdots + \beta_1) q^{8}+ \cdots + (12 \beta_{4} + 12 \beta_{3} + \cdots - 12 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 10 q^{4} + 20 q^{14} - 46 q^{16} - 84 q^{26} - 184 q^{29} - 12 q^{34} + 256 q^{41} + 348 q^{44} + 112 q^{46} + 24 q^{49} + 244 q^{56} + 304 q^{61} + 10 q^{64} - 252 q^{74} - 24 q^{76} + 280 q^{86} + 560 q^{89} - 376 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 7x^{6} - 12x^{5} + 12x^{4} - 48x^{3} + 112x^{2} - 192x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 5\nu^{6} + 17\nu^{5} + 10\nu^{4} - 64\nu^{3} - 88\nu^{2} + 176\nu - 544 ) / 112 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{7} + 15\nu^{6} - 23\nu^{5} - 2\nu^{4} - 60\nu^{3} + 152\nu^{2} - 528\nu + 736 ) / 224 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} + 25\nu^{6} - \nu^{5} + 34\nu^{4} + 12\nu^{3} + 104\nu^{2} - 880\nu + 32 ) / 224 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 3\nu^{6} + 11\nu^{5} + 2\nu^{4} - 4\nu^{3} + 40\nu^{2} + 80\nu - 352 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 36\nu^{6} - 58\nu^{5} + 61\nu^{4} - 116\nu^{3} + 292\nu^{2} - 976\nu + 1744 ) / 112 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 8\nu^{6} + 30\nu^{5} - 33\nu^{4} + 60\nu^{3} - 180\nu^{2} + 528\nu - 848 ) / 56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\nu^{7} - 20\nu^{6} + 26\nu^{5} - 37\nu^{4} + 52\nu^{3} - 324\nu^{2} + 592\nu - 384 ) / 112 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{5} - 2\beta_{3} - 2\beta_{2} + 3 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} - 2\beta _1 - 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} - 3\beta_{2} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} - 2\beta_{6} + 3\beta_{5} + 3\beta_{3} - 25\beta_{2} + 2\beta _1 + 11 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} + 2\beta_{6} + \beta_{5} + 8\beta_{4} + 11\beta_{3} + 7\beta_{2} + 2\beta _1 + 105 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3\beta_{7} + 10\beta_{6} + 9\beta_{5} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 71\beta_{7} - 2\beta_{6} + 43\beta_{5} + 40\beta_{4} - \beta_{3} + 11\beta_{2} - 38\beta _1 - 237 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
−1.51328 − 1.30766i
−1.51328 + 1.30766i
0.670410 + 1.88429i
0.670410 − 1.88429i
1.96705 − 0.361553i
1.96705 + 0.361553i
0.375825 + 1.96437i
0.375825 − 1.96437i
−1.88911 − 0.656712i 0 3.13746 + 2.48120i 0 0 9.55505i −4.29756 − 6.74766i 0 0
451.2 −1.88911 + 0.656712i 0 3.13746 − 2.48120i 0 0 − 9.55505i −4.29756 + 6.74766i 0 0
451.3 −1.29664 − 1.52274i 0 −0.637459 + 3.94888i 0 0 − 0.837253i 6.83966 − 4.14959i 0 0
451.4 −1.29664 + 1.52274i 0 −0.637459 − 3.94888i 0 0 0.837253i 6.83966 + 4.14959i 0 0
451.5 1.29664 − 1.52274i 0 −0.637459 − 3.94888i 0 0 − 0.837253i −6.83966 − 4.14959i 0 0
451.6 1.29664 + 1.52274i 0 −0.637459 + 3.94888i 0 0 0.837253i −6.83966 + 4.14959i 0 0
451.7 1.88911 − 0.656712i 0 3.13746 − 2.48120i 0 0 9.55505i 4.29756 − 6.74766i 0 0
451.8 1.88911 + 0.656712i 0 3.13746 + 2.48120i 0 0 − 9.55505i 4.29756 + 6.74766i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 451.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.c.r 8
3.b odd 2 1 300.3.c.f 8
4.b odd 2 1 inner 900.3.c.r 8
5.b even 2 1 inner 900.3.c.r 8
5.c odd 4 2 180.3.f.h 8
12.b even 2 1 300.3.c.f 8
15.d odd 2 1 300.3.c.f 8
15.e even 4 2 60.3.f.b ✓ 8
20.d odd 2 1 inner 900.3.c.r 8
20.e even 4 2 180.3.f.h 8
60.h even 2 1 300.3.c.f 8
60.l odd 4 2 60.3.f.b ✓ 8
120.q odd 4 2 960.3.j.e 8
120.w even 4 2 960.3.j.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.b ✓ 8 15.e even 4 2
60.3.f.b ✓ 8 60.l odd 4 2
180.3.f.h 8 5.c odd 4 2
180.3.f.h 8 20.e even 4 2
300.3.c.f 8 3.b odd 2 1
300.3.c.f 8 12.b even 2 1
300.3.c.f 8 15.d odd 2 1
300.3.c.f 8 60.h even 2 1
900.3.c.r 8 1.a even 1 1 trivial
900.3.c.r 8 4.b odd 2 1 inner
900.3.c.r 8 5.b even 2 1 inner
900.3.c.r 8 20.d odd 2 1 inner
960.3.j.e 8 120.q odd 4 2
960.3.j.e 8 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{4} + 92T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{13}^{4} - 84T_{13}^{2} + 1536 \) Copy content Toggle raw display
\( T_{17}^{4} - 1044T_{17}^{2} + 221184 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 5 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 92 T^{2} + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 348 T^{2} + 24576)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 84 T^{2} + 1536)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 1044 T^{2} + 221184)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1008 T^{2} + 221184)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 368 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 46 T + 16)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2304 T^{2} + 393216)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 756 T^{2} + 124416)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 64 T - 1028)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 2528 T^{2} + 1364224)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 3296 T^{2} + 614656)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 756 T^{2} + 124416)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 16668 T^{2} + 69033984)^{2} \) Copy content Toggle raw display
$61$ \( (T - 38)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 9392 T^{2} + 7573504)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 17136 T^{2} + 884736)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 4176 T^{2} + 3538944)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 2304 T^{2} + 393216)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 4064 T^{2} + 2027776)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 140 T + 3988)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 45888 T^{2} + 495550464)^{2} \) Copy content Toggle raw display
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