Properties

Label 900.2.k.n
Level $900$
Weight $2$
Character orbit 900.k
Analytic conductor $7.187$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,2,Mod(307,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.k (of order \(4\), 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: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.426337261060096.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - \beta_{2} q^{4} + (\beta_{7} + \beta_{4}) q^{7} + ( - \beta_{11} + \beta_{6} - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - \beta_{2} q^{4} + (\beta_{7} + \beta_{4}) q^{7} + ( - \beta_{11} + \beta_{6} - 1) q^{8} + ( - \beta_{11} - \beta_{10} - \beta_{7}) q^{11} + (\beta_{8} - \beta_{6} - \beta_{5} + \cdots + 1) q^{13}+ \cdots + (2 \beta_{11} + 2 \beta_{10} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{8} + 4 q^{13} + 12 q^{16} - 20 q^{17} - 12 q^{22} - 16 q^{26} + 4 q^{28} + 20 q^{32} - 4 q^{37} + 16 q^{38} - 16 q^{41} - 40 q^{46} + 8 q^{52} + 4 q^{53} + 64 q^{56} + 20 q^{58} - 32 q^{61} - 56 q^{62} - 16 q^{68} - 44 q^{73} + 8 q^{76} + 48 q^{77} - 16 q^{82} - 64 q^{86} - 60 q^{88} + 56 q^{92} + 20 q^{97} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{11} + \nu^{10} - 12\nu^{9} - \nu^{7} + 9\nu^{6} + 16\nu^{5} - 4\nu^{4} - 28\nu^{3} - 12\nu^{2} + 144 ) / 80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} - 8 \nu^{9} - 4 \nu^{8} + 11 \nu^{7} + 12 \nu^{6} + 16 \nu^{5} - 48 \nu^{4} - 60 \nu^{3} + \cdots + 192 ) / 80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{11} + 12 \nu^{10} + 4 \nu^{9} + 4 \nu^{8} - 13 \nu^{7} - 24 \nu^{6} - 4 \nu^{5} + 40 \nu^{4} + \cdots - 64 ) / 160 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{11} + \nu^{10} - 4\nu^{8} - 5\nu^{7} + \nu^{6} + 12\nu^{5} + 8\nu^{4} + 12\nu^{3} - 36\nu^{2} - 16\nu - 64 ) / 80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{11} - \nu^{10} + 4\nu^{8} + 5\nu^{7} - \nu^{6} - 12\nu^{5} - 8\nu^{4} + 68\nu^{3} + 36\nu^{2} + 16\nu - 16 ) / 80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9 \nu^{11} + 4 \nu^{10} + 12 \nu^{9} + 20 \nu^{8} + 11 \nu^{7} - 64 \nu^{6} + 4 \nu^{5} + 24 \nu^{4} + \cdots - 64 ) / 160 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9 \nu^{10} - 4 \nu^{9} + 4 \nu^{8} + 28 \nu^{7} + 27 \nu^{6} - 40 \nu^{5} - 36 \nu^{4} - 48 \nu^{3} + \cdots - 48 ) / 80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9 \nu^{11} + 12 \nu^{10} + 4 \nu^{9} - 36 \nu^{8} - 43 \nu^{7} + 16 \nu^{6} + 76 \nu^{5} + 120 \nu^{4} + \cdots - 64 ) / 160 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5 \nu^{11} - 7 \nu^{10} + 8 \nu^{9} + 12 \nu^{8} + 19 \nu^{7} - 39 \nu^{6} - 60 \nu^{5} - 8 \nu^{4} + \cdots - 144 ) / 80 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{8} - \beta_{4} + \beta_{3} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} + 2\beta_{8} + \beta_{6} - 2\beta_{5} + 2\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{11} + \beta_{9} + 2\beta_{7} - 2\beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{10} + 4\beta_{9} + 4\beta_{8} + \beta_{7} + 5\beta_{6} + 2\beta_{5} + 2\beta_{4} - 2\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -4\beta_{11} - 5\beta_{10} + \beta_{9} + 3\beta_{8} - 4\beta_{7} - 5\beta_{4} + \beta_{3} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -3\beta_{11} + 2\beta_{8} - 5\beta_{6} - 2\beta_{5} - 8\beta_{3} + 2\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -2\beta_{11} + 3\beta_{9} + 2\beta_{7} + 2\beta_{6} + 16\beta_{5} + 5\beta_{4} - 3\beta_{3} + 7\beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -6\beta_{10} + 4\beta_{9} - 9\beta_{7} + 27\beta_{6} + 6\beta_{5} - 2\beta_{4} + 6\beta_{2} + 27 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−0.0912546 1.41127i
−0.760198 1.19252i
1.41127 + 0.0912546i
−1.35818 + 0.394157i
1.19252 + 0.760198i
−0.394157 + 1.35818i
−0.0912546 + 1.41127i
−0.760198 + 1.19252i
1.41127 0.0912546i
−1.35818 0.394157i
1.19252 0.760198i
−0.394157 1.35818i
−1.41127 + 0.0912546i 0 1.98335 0.257569i 0 0 −1.86678 + 1.86678i −2.77552 + 0.544488i 0 0
307.2 −1.19252 + 0.760198i 0 0.844199 1.81310i 0 0 0.611393 0.611393i 0.371591 + 2.80391i 0 0
307.3 0.0912546 1.41127i 0 −1.98335 0.257569i 0 0 1.86678 1.86678i −0.544488 + 2.77552i 0 0
307.4 0.394157 + 1.35818i 0 −1.68928 + 1.07067i 0 0 −2.47817 + 2.47817i −2.12000 1.87233i 0 0
307.5 0.760198 1.19252i 0 −0.844199 1.81310i 0 0 −0.611393 + 0.611393i −2.80391 0.371591i 0 0
307.6 1.35818 + 0.394157i 0 1.68928 + 1.07067i 0 0 2.47817 2.47817i 1.87233 + 2.12000i 0 0
343.1 −1.41127 0.0912546i 0 1.98335 + 0.257569i 0 0 −1.86678 1.86678i −2.77552 0.544488i 0 0
343.2 −1.19252 0.760198i 0 0.844199 + 1.81310i 0 0 0.611393 + 0.611393i 0.371591 2.80391i 0 0
343.3 0.0912546 + 1.41127i 0 −1.98335 + 0.257569i 0 0 1.86678 + 1.86678i −0.544488 2.77552i 0 0
343.4 0.394157 1.35818i 0 −1.68928 1.07067i 0 0 −2.47817 2.47817i −2.12000 + 1.87233i 0 0
343.5 0.760198 + 1.19252i 0 −0.844199 + 1.81310i 0 0 −0.611393 0.611393i −2.80391 + 0.371591i 0 0
343.6 1.35818 0.394157i 0 1.68928 1.07067i 0 0 2.47817 + 2.47817i 1.87233 2.12000i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.k.n 12
3.b odd 2 1 300.2.j.d 12
4.b odd 2 1 inner 900.2.k.n 12
5.b even 2 1 180.2.k.e 12
5.c odd 4 1 180.2.k.e 12
5.c odd 4 1 inner 900.2.k.n 12
12.b even 2 1 300.2.j.d 12
15.d odd 2 1 60.2.j.a 12
15.e even 4 1 60.2.j.a 12
15.e even 4 1 300.2.j.d 12
20.d odd 2 1 180.2.k.e 12
20.e even 4 1 180.2.k.e 12
20.e even 4 1 inner 900.2.k.n 12
60.h even 2 1 60.2.j.a 12
60.l odd 4 1 60.2.j.a 12
60.l odd 4 1 300.2.j.d 12
120.i odd 2 1 960.2.w.g 12
120.m even 2 1 960.2.w.g 12
120.q odd 4 1 960.2.w.g 12
120.w even 4 1 960.2.w.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.j.a 12 15.d odd 2 1
60.2.j.a 12 15.e even 4 1
60.2.j.a 12 60.h even 2 1
60.2.j.a 12 60.l odd 4 1
180.2.k.e 12 5.b even 2 1
180.2.k.e 12 5.c odd 4 1
180.2.k.e 12 20.d odd 2 1
180.2.k.e 12 20.e even 4 1
300.2.j.d 12 3.b odd 2 1
300.2.j.d 12 12.b even 2 1
300.2.j.d 12 15.e even 4 1
300.2.j.d 12 60.l odd 4 1
900.2.k.n 12 1.a even 1 1 trivial
900.2.k.n 12 4.b odd 2 1 inner
900.2.k.n 12 5.c odd 4 1 inner
900.2.k.n 12 20.e even 4 1 inner
960.2.w.g 12 120.i odd 2 1
960.2.w.g 12 120.m even 2 1
960.2.w.g 12 120.q odd 4 1
960.2.w.g 12 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{12} + 200T_{7}^{8} + 7440T_{7}^{4} + 4096 \) Copy content Toggle raw display
\( T_{11}^{6} + 36T_{11}^{4} + 260T_{11}^{2} + 128 \) Copy content Toggle raw display
\( T_{13}^{6} - 2T_{13}^{5} + 2T_{13}^{4} + 32T_{13}^{3} + 144T_{13}^{2} + 96T_{13} + 32 \) Copy content Toggle raw display
\( T_{17}^{6} + 10T_{17}^{5} + 50T_{17}^{4} + 80T_{17}^{3} + 16T_{17}^{2} - 160T_{17} + 800 \) Copy content Toggle raw display
\( T_{19}^{6} - 40T_{19}^{4} + 400T_{19}^{2} - 512 \) Copy content Toggle raw display
\( T_{23}^{12} + 4640T_{23}^{8} + 37120T_{23}^{4} + 65536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 4 T^{9} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 200 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$11$ \( (T^{6} + 36 T^{4} + \cdots + 128)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 2 T^{5} + 2 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 10 T^{5} + \cdots + 800)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 40 T^{4} + \cdots - 512)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 4640 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$29$ \( (T^{6} + 20 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 112 T^{4} + \cdots + 32768)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 2 T^{5} + 2 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 4 T^{2} - 20 T - 64)^{4} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 15352201216 \) Copy content Toggle raw display
$47$ \( T^{12} + 4896 T^{8} + \cdots + 40960000 \) Copy content Toggle raw display
$53$ \( (T^{6} - 2 T^{5} + \cdots + 128)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 100 T^{4} + \cdots - 512)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 8 T^{2} + \cdots + 176)^{4} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 15352201216 \) Copy content Toggle raw display
$71$ \( (T^{6} + 256 T^{4} + \cdots + 204800)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 22 T^{5} + \cdots + 55112)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 304 T^{4} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 16672 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$89$ \( (T^{6} + 72 T^{4} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 10 T^{5} + \cdots + 35912)^{2} \) Copy content Toggle raw display
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