Properties

Label 2100.4.k.g
Level $2100$
Weight $4$
Character orbit 2100.k
Analytic conductor $123.904$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,4,Mod(1849,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1849");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2100.k (of order \(2\), degree \(1\), not 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: \(123.904011012\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{3} + 7 i q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{3} + 7 i q^{7} - 9 q^{9} + 4 q^{11} + 54 i q^{13} + 14 i q^{17} - 92 q^{19} - 21 q^{21} - 152 i q^{23} - 27 i q^{27} + 106 q^{29} - 144 q^{31} + 12 i q^{33} - 158 i q^{37} - 162 q^{39} - 390 q^{41} - 508 i q^{43} + 528 i q^{47} - 49 q^{49} - 42 q^{51} + 606 i q^{53} - 276 i q^{57} + 364 q^{59} + 678 q^{61} - 63 i q^{63} - 844 i q^{67} + 456 q^{69} - 8 q^{71} - 422 i q^{73} + 28 i q^{77} - 384 q^{79} + 81 q^{81} - 548 i q^{83} + 318 i q^{87} - 1194 q^{89} - 378 q^{91} - 432 i q^{93} + 1502 i q^{97} - 36 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} + 8 q^{11} - 184 q^{19} - 42 q^{21} + 212 q^{29} - 288 q^{31} - 324 q^{39} - 780 q^{41} - 98 q^{49} - 84 q^{51} + 728 q^{59} + 1356 q^{61} + 912 q^{69} - 16 q^{71} - 768 q^{79} + 162 q^{81} - 2388 q^{89} - 756 q^{91} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
1.00000i
1.00000i
0 3.00000i 0 0 0 7.00000i 0 −9.00000 0
1849.2 0 3.00000i 0 0 0 7.00000i 0 −9.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.4.k.g 2
5.b even 2 1 inner 2100.4.k.g 2
5.c odd 4 1 84.4.a.b 1
5.c odd 4 1 2100.4.a.g 1
15.e even 4 1 252.4.a.a 1
20.e even 4 1 336.4.a.e 1
35.f even 4 1 588.4.a.a 1
35.k even 12 2 588.4.i.h 2
35.l odd 12 2 588.4.i.a 2
40.i odd 4 1 1344.4.a.b 1
40.k even 4 1 1344.4.a.p 1
60.l odd 4 1 1008.4.a.d 1
105.k odd 4 1 1764.4.a.l 1
105.w odd 12 2 1764.4.k.c 2
105.x even 12 2 1764.4.k.n 2
140.j odd 4 1 2352.4.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.b 1 5.c odd 4 1
252.4.a.a 1 15.e even 4 1
336.4.a.e 1 20.e even 4 1
588.4.a.a 1 35.f even 4 1
588.4.i.a 2 35.l odd 12 2
588.4.i.h 2 35.k even 12 2
1008.4.a.d 1 60.l odd 4 1
1344.4.a.b 1 40.i odd 4 1
1344.4.a.p 1 40.k even 4 1
1764.4.a.l 1 105.k odd 4 1
1764.4.k.c 2 105.w odd 12 2
1764.4.k.n 2 105.x even 12 2
2100.4.a.g 1 5.c odd 4 1
2100.4.k.g 2 1.a even 1 1 trivial
2100.4.k.g 2 5.b even 2 1 inner
2352.4.a.v 1 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(2100, [\chi])\):

\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 2916 \) Copy content Toggle raw display
\( T_{17}^{2} + 196 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2916 \) Copy content Toggle raw display
$17$ \( T^{2} + 196 \) Copy content Toggle raw display
$19$ \( (T + 92)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 23104 \) Copy content Toggle raw display
$29$ \( (T - 106)^{2} \) Copy content Toggle raw display
$31$ \( (T + 144)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 24964 \) Copy content Toggle raw display
$41$ \( (T + 390)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 258064 \) Copy content Toggle raw display
$47$ \( T^{2} + 278784 \) Copy content Toggle raw display
$53$ \( T^{2} + 367236 \) Copy content Toggle raw display
$59$ \( (T - 364)^{2} \) Copy content Toggle raw display
$61$ \( (T - 678)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 712336 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 178084 \) Copy content Toggle raw display
$79$ \( (T + 384)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 300304 \) Copy content Toggle raw display
$89$ \( (T + 1194)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2256004 \) Copy content Toggle raw display
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