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Properties

Label 2100.2.bi.k
Level $2100$
Weight $2$
Character orbit 2100.bi
Analytic conductor $16.769$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(101,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 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("2100.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.bi (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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: 10.0.29471584693248.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 13x^{6} - 36x^{5} + 39x^{4} - 36x^{3} + 54x^{2} - 162x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + ( - \beta_{9} + \beta_{7} - \beta_{5} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{4} + \beta_{3} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + ( - \beta_{9} + \beta_{7} - \beta_{5} + \cdots + 1) q^{7}+ \cdots + (\beta_{9} - \beta_{8} + 2 \beta_{7} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{3} + 5 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{3} + 5 q^{7} - 3 q^{9} - 6 q^{11} - 6 q^{17} + 3 q^{19} + 10 q^{21} - 24 q^{23} - 8 q^{27} + 15 q^{31} + 20 q^{33} + q^{37} + 15 q^{39} - 8 q^{41} + 26 q^{43} - 14 q^{47} - 13 q^{49} - 44 q^{51} + 24 q^{53} - 18 q^{57} + 42 q^{61} + q^{63} - 7 q^{67} - 14 q^{69} + 3 q^{73} + 26 q^{77} + q^{79} + 41 q^{81} + 8 q^{83} + 26 q^{87} + 28 q^{89} - 11 q^{91} + 47 q^{93} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 13x^{6} - 36x^{5} + 39x^{4} - 36x^{3} + 54x^{2} - 162x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 2\nu^{5} + 4\nu^{4} - 4\nu^{3} + 18\nu^{2} - 21\nu + 18 ) / 18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - \nu^{8} - 9\nu^{7} + 7\nu^{6} - 18\nu^{5} + 22\nu^{4} - 51\nu^{3} + 21\nu^{2} - 171\nu + 135 ) / 216 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{9} - 4\nu^{8} + 7\nu^{7} + 28\nu^{6} + 32\nu^{5} - 30\nu^{4} + 123\nu^{3} - 216\nu^{2} - 243\nu - 486 ) / 648 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{9} + 2\nu^{8} - \nu^{7} + 4\nu^{6} + 16\nu^{5} - 58\nu^{4} + 9\nu^{3} + 6\nu^{2} + 9\nu - 486 ) / 216 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{9} - \nu^{7} + 6\nu^{6} - 20\nu^{5} + 22\nu^{4} - 15\nu^{3} + 48\nu^{2} - 63\nu + 216 ) / 72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{9} - 7\nu^{8} + 16\nu^{7} - 23\nu^{6} + 50\nu^{5} - 108\nu^{4} + 114\nu^{3} - 153\nu^{2} - 729 ) / 648 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} - 2\nu^{8} + 2\nu^{7} - 4\nu^{6} + 13\nu^{5} - 36\nu^{4} + 39\nu^{3} - 36\nu^{2} + 54\nu - 162 ) / 81 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13 \nu^{9} + 5 \nu^{8} + 25 \nu^{7} - 35 \nu^{6} - 94 \nu^{5} + 186 \nu^{4} + 231 \nu^{3} + \cdots + 1701 ) / 648 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - 2\beta_{8} + 3\beta_{7} - 2\beta_{6} - 2\beta_{5} + \beta_{4} + 2\beta_{2} + 2\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{9} - \beta_{8} - 2\beta_{7} - 4\beta_{6} - 4\beta_{5} + 2\beta_{4} - 4\beta_{3} + 2\beta_{2} - 2\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{9} - 8\beta_{7} + 4\beta_{6} + 12\beta_{4} - 4\beta_{3} - 2\beta_{2} + 4\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -8\beta_{9} - 10\beta_{8} + 18\beta_{7} + 12\beta_{6} - 4\beta_{5} + 6\beta_{4} - 18\beta_{3} - 8\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16 \beta_{9} - 16 \beta_{8} - 7 \beta_{7} - 10 \beta_{6} + 38 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + \cdots + 30 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 16\beta_{9} - 29\beta_{8} + 88\beta_{7} - 9\beta_{6} + 25\beta_{5} + 26\beta_{3} + 18\beta_{2} + 46\beta _1 + 48 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1 + \beta_{7}\)

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
−1.08831 + 1.34743i
1.15038 + 1.29484i
−1.31611 − 1.12599i
1.72689 + 0.133595i
0.527154 − 1.64988i
−1.08831 − 1.34743i
1.15038 − 1.29484i
−1.31611 + 1.12599i
1.72689 − 0.133595i
0.527154 + 1.64988i
0 −1.71107 + 0.268793i 0 0 0 −2.57325 + 0.615143i 0 2.85550 − 0.919845i 0
101.2 0 −0.546177 − 1.64368i 0 0 0 1.08214 − 2.41433i 0 −2.40338 + 1.79548i 0
101.3 0 0.317079 + 1.70278i 0 0 0 1.73439 + 1.99797i 0 −2.79892 + 1.07983i 0
101.4 0 0.747749 − 1.56233i 0 0 0 0.456468 + 2.60608i 0 −1.88174 − 2.33646i 0
101.5 0 1.69242 + 0.368412i 0 0 0 1.80025 − 1.93884i 0 2.72854 + 1.24701i 0
1601.1 0 −1.71107 − 0.268793i 0 0 0 −2.57325 − 0.615143i 0 2.85550 + 0.919845i 0
1601.2 0 −0.546177 + 1.64368i 0 0 0 1.08214 + 2.41433i 0 −2.40338 − 1.79548i 0
1601.3 0 0.317079 − 1.70278i 0 0 0 1.73439 − 1.99797i 0 −2.79892 − 1.07983i 0
1601.4 0 0.747749 + 1.56233i 0 0 0 0.456468 − 2.60608i 0 −1.88174 + 2.33646i 0
1601.5 0 1.69242 − 0.368412i 0 0 0 1.80025 + 1.93884i 0 2.72854 − 1.24701i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.bi.k 10
3.b odd 2 1 2100.2.bi.j 10
5.b even 2 1 420.2.bh.a ✓ 10
5.c odd 4 2 2100.2.bo.h 20
7.d odd 6 1 2100.2.bi.j 10
15.d odd 2 1 420.2.bh.b yes 10
15.e even 4 2 2100.2.bo.g 20
21.g even 6 1 inner 2100.2.bi.k 10
35.i odd 6 1 420.2.bh.b yes 10
35.i odd 6 1 2940.2.d.a 10
35.j even 6 1 2940.2.d.b 10
35.k even 12 2 2100.2.bo.g 20
105.o odd 6 1 2940.2.d.a 10
105.p even 6 1 420.2.bh.a ✓ 10
105.p even 6 1 2940.2.d.b 10
105.w odd 12 2 2100.2.bo.h 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bh.a ✓ 10 5.b even 2 1
420.2.bh.a ✓ 10 105.p even 6 1
420.2.bh.b yes 10 15.d odd 2 1
420.2.bh.b yes 10 35.i odd 6 1
2100.2.bi.j 10 3.b odd 2 1
2100.2.bi.j 10 7.d odd 6 1
2100.2.bi.k 10 1.a even 1 1 trivial
2100.2.bi.k 10 21.g even 6 1 inner
2100.2.bo.g 20 15.e even 4 2
2100.2.bo.g 20 35.k even 12 2
2100.2.bo.h 20 5.c odd 4 2
2100.2.bo.h 20 105.w odd 12 2
2940.2.d.a 10 35.i odd 6 1
2940.2.d.a 10 105.o odd 6 1
2940.2.d.b 10 35.j even 6 1
2940.2.d.b 10 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2100, [\chi])\):

\( T_{11}^{10} + 6 T_{11}^{9} - 4 T_{11}^{8} - 96 T_{11}^{7} + 84 T_{11}^{6} + 2280 T_{11}^{5} + \cdots + 1728 \) Copy content Toggle raw display
\( T_{13}^{10} + 81T_{13}^{8} + 2183T_{13}^{6} + 22251T_{13}^{4} + 65704T_{13}^{2} + 3888 \) Copy content Toggle raw display
\( T_{19}^{10} - 3 T_{19}^{9} - 12 T_{19}^{8} + 45 T_{19}^{7} + 152 T_{19}^{6} - 519 T_{19}^{5} + \cdots + 192 \) Copy content Toggle raw display
\( T_{37}^{10} - T_{37}^{9} + 68 T_{37}^{8} + 65 T_{37}^{7} + 3634 T_{37}^{6} + 1533 T_{37}^{5} + \cdots + 12544 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 5 T^{9} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( T^{10} + 6 T^{9} + \cdots + 1728 \) Copy content Toggle raw display
$13$ \( T^{10} + 81 T^{8} + \cdots + 3888 \) Copy content Toggle raw display
$17$ \( T^{10} + 6 T^{9} + \cdots + 69696 \) Copy content Toggle raw display
$19$ \( T^{10} - 3 T^{9} + \cdots + 192 \) Copy content Toggle raw display
$23$ \( T^{10} + 24 T^{9} + \cdots + 2462508 \) Copy content Toggle raw display
$29$ \( T^{10} + 142 T^{8} + \cdots + 8748 \) Copy content Toggle raw display
$31$ \( T^{10} - 15 T^{9} + \cdots + 1978032 \) Copy content Toggle raw display
$37$ \( T^{10} - T^{9} + \cdots + 12544 \) Copy content Toggle raw display
$41$ \( (T^{5} + 4 T^{4} + \cdots + 1338)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 13 T^{4} + \cdots - 1559)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 295289856 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 113246208 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 125081856 \) Copy content Toggle raw display
$61$ \( T^{10} - 42 T^{9} + \cdots + 1338672 \) Copy content Toggle raw display
$67$ \( T^{10} + 7 T^{9} + \cdots + 45684081 \) Copy content Toggle raw display
$71$ \( T^{10} + 288 T^{8} + \cdots + 88259328 \) Copy content Toggle raw display
$73$ \( T^{10} - 3 T^{9} + \cdots + 1572528 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 138485824 \) Copy content Toggle raw display
$83$ \( (T^{5} - 4 T^{4} + \cdots + 28794)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} - 28 T^{9} + \cdots + 272484 \) Copy content Toggle raw display
$97$ \( T^{10} + 212 T^{8} + \cdots + 1051392 \) Copy content Toggle raw display
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