Properties

Label 2100.3.bd.b
Level $2100$
Weight $3$
Character orbit 2100.bd
Analytic conductor $57.221$
Analytic rank $0$
Dimension $2$
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] = [2100,3,Mod(901,2100)]
 
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, 0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.901");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2100.bd (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: \(57.2208555157\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} - 1) q^{3} + ( - 3 \zeta_{6} - 5) q^{7} + 3 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} - 1) q^{3} + ( - 3 \zeta_{6} - 5) q^{7} + 3 \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{11} + ( - 8 \zeta_{6} + 4) q^{13} + (10 \zeta_{6} + 10) q^{17} + ( - 6 \zeta_{6} + 12) q^{19} + (11 \zeta_{6} + 2) q^{21} + 36 \zeta_{6} q^{23} + ( - 6 \zeta_{6} + 3) q^{27} - 51 q^{29} + ( - 7 \zeta_{6} - 7) q^{31} + (3 \zeta_{6} - 6) q^{33} + 22 \zeta_{6} q^{37} + (12 \zeta_{6} - 12) q^{39} + (28 \zeta_{6} - 14) q^{41} - 10 q^{43} + (52 \zeta_{6} - 104) q^{47} + (39 \zeta_{6} + 16) q^{49} - 30 \zeta_{6} q^{51} + ( - 51 \zeta_{6} + 51) q^{53} - 18 q^{57} + (43 \zeta_{6} + 43) q^{59} + ( - 40 \zeta_{6} + 80) q^{61} + ( - 24 \zeta_{6} + 9) q^{63} + ( - 68 \zeta_{6} + 68) q^{67} + ( - 72 \zeta_{6} + 36) q^{69} + (12 \zeta_{6} + 12) q^{73} + (15 \zeta_{6} - 24) q^{77} - 125 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} + ( - 178 \zeta_{6} + 89) q^{83} + (51 \zeta_{6} + 51) q^{87} + ( - 42 \zeta_{6} + 84) q^{89} + (52 \zeta_{6} - 44) q^{91} + 21 \zeta_{6} q^{93} + (170 \zeta_{6} - 85) q^{97} + 9 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 13 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} - 13 q^{7} + 3 q^{9} + 3 q^{11} + 30 q^{17} + 18 q^{19} + 15 q^{21} + 36 q^{23} - 102 q^{29} - 21 q^{31} - 9 q^{33} + 22 q^{37} - 12 q^{39} - 20 q^{43} - 156 q^{47} + 71 q^{49} - 30 q^{51} + 51 q^{53} - 36 q^{57} + 129 q^{59} + 120 q^{61} - 6 q^{63} + 68 q^{67} + 36 q^{73} - 33 q^{77} - 125 q^{79} - 9 q^{81} + 153 q^{87} + 126 q^{89} - 36 q^{91} + 21 q^{93} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.50000 + 0.866025i 0 0 0 −6.50000 + 2.59808i 0 1.50000 2.59808i 0
1501.1 0 −1.50000 0.866025i 0 0 0 −6.50000 2.59808i 0 1.50000 + 2.59808i 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
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.3.bd.b 2
5.b even 2 1 84.3.m.a 2
5.c odd 4 2 2100.3.be.c 4
7.d odd 6 1 inner 2100.3.bd.b 2
15.d odd 2 1 252.3.z.b 2
20.d odd 2 1 336.3.bh.b 2
35.c odd 2 1 588.3.m.a 2
35.i odd 6 1 84.3.m.a 2
35.i odd 6 1 588.3.d.a 2
35.j even 6 1 588.3.d.a 2
35.j even 6 1 588.3.m.a 2
35.k even 12 2 2100.3.be.c 4
60.h even 2 1 1008.3.cg.b 2
105.g even 2 1 1764.3.z.e 2
105.o odd 6 1 1764.3.d.c 2
105.o odd 6 1 1764.3.z.e 2
105.p even 6 1 252.3.z.b 2
105.p even 6 1 1764.3.d.c 2
140.p odd 6 1 2352.3.f.c 2
140.s even 6 1 336.3.bh.b 2
140.s even 6 1 2352.3.f.c 2
420.be odd 6 1 1008.3.cg.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.3.m.a 2 5.b even 2 1
84.3.m.a 2 35.i odd 6 1
252.3.z.b 2 15.d odd 2 1
252.3.z.b 2 105.p even 6 1
336.3.bh.b 2 20.d odd 2 1
336.3.bh.b 2 140.s even 6 1
588.3.d.a 2 35.i odd 6 1
588.3.d.a 2 35.j even 6 1
588.3.m.a 2 35.c odd 2 1
588.3.m.a 2 35.j even 6 1
1008.3.cg.b 2 60.h even 2 1
1008.3.cg.b 2 420.be odd 6 1
1764.3.d.c 2 105.o odd 6 1
1764.3.d.c 2 105.p even 6 1
1764.3.z.e 2 105.g even 2 1
1764.3.z.e 2 105.o odd 6 1
2100.3.bd.b 2 1.a even 1 1 trivial
2100.3.bd.b 2 7.d odd 6 1 inner
2100.3.be.c 4 5.c odd 4 2
2100.3.be.c 4 35.k even 12 2
2352.3.f.c 2 140.p odd 6 1
2352.3.f.c 2 140.s even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2100, [\chi])\):

\( T_{11}^{2} - 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{13}^{2} + 48 \) Copy content Toggle raw display
\( T_{17}^{2} - 30T_{17} + 300 \) Copy content Toggle raw display
\( T_{23}^{2} - 36T_{23} + 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 13T + 49 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} + 48 \) Copy content Toggle raw display
$17$ \( T^{2} - 30T + 300 \) Copy content Toggle raw display
$19$ \( T^{2} - 18T + 108 \) Copy content Toggle raw display
$23$ \( T^{2} - 36T + 1296 \) Copy content Toggle raw display
$29$ \( (T + 51)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 21T + 147 \) Copy content Toggle raw display
$37$ \( T^{2} - 22T + 484 \) Copy content Toggle raw display
$41$ \( T^{2} + 588 \) Copy content Toggle raw display
$43$ \( (T + 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 156T + 8112 \) Copy content Toggle raw display
$53$ \( T^{2} - 51T + 2601 \) Copy content Toggle raw display
$59$ \( T^{2} - 129T + 5547 \) Copy content Toggle raw display
$61$ \( T^{2} - 120T + 4800 \) Copy content Toggle raw display
$67$ \( T^{2} - 68T + 4624 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 36T + 432 \) Copy content Toggle raw display
$79$ \( T^{2} + 125T + 15625 \) Copy content Toggle raw display
$83$ \( T^{2} + 23763 \) Copy content Toggle raw display
$89$ \( T^{2} - 126T + 5292 \) Copy content Toggle raw display
$97$ \( T^{2} + 21675 \) Copy content Toggle raw display
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