Properties

Label 100.4.e.a
Level $100$
Weight $4$
Character orbit 100.e
Analytic conductor $5.900$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,4,Mod(7,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.7");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 100.e (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: \(5.90019100057\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 i - 2) q^{2} + 8 i q^{4} + ( - 16 i + 16) q^{8} - 27 i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 i - 2) q^{2} + 8 i q^{4} + ( - 16 i + 16) q^{8} - 27 i q^{9} + ( - 37 i + 37) q^{13} - 64 q^{16} + ( - 99 i - 99) q^{17} + (54 i - 54) q^{18} - 148 q^{26} - 284 i q^{29} + (128 i + 128) q^{32} + 396 i q^{34} + 216 q^{36} + (91 i + 91) q^{37} + 472 q^{41} + 343 i q^{49} + (296 i + 296) q^{52} + ( - 27 i + 27) q^{53} + (568 i - 568) q^{58} - 468 q^{61} - 512 i q^{64} + ( - 792 i + 792) q^{68} + ( - 432 i - 432) q^{72} + (253 i - 253) q^{73} - 364 i q^{74} - 729 q^{81} + ( - 944 i - 944) q^{82} + 176 i q^{89} + (611 i + 611) q^{97} + ( - 686 i + 686) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 32 q^{8} + 74 q^{13} - 128 q^{16} - 198 q^{17} - 108 q^{18} - 296 q^{26} + 256 q^{32} + 432 q^{36} + 182 q^{37} + 944 q^{41} + 592 q^{52} + 54 q^{53} - 1136 q^{58} - 936 q^{61} + 1584 q^{68} - 864 q^{72} - 506 q^{73} - 1458 q^{81} - 1888 q^{82} + 1222 q^{97} + 1372 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
1.00000i
1.00000i
−2.00000 2.00000i 0 8.00000i 0 0 0 16.0000 16.0000i 27.0000i 0
43.1 −2.00000 + 2.00000i 0 8.00000i 0 0 0 16.0000 + 16.0000i 27.0000i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
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 100.4.e.a 2
4.b odd 2 1 CM 100.4.e.a 2
5.b even 2 1 20.4.e.a 2
5.c odd 4 1 20.4.e.a 2
5.c odd 4 1 inner 100.4.e.a 2
15.d odd 2 1 180.4.k.a 2
15.e even 4 1 180.4.k.a 2
20.d odd 2 1 20.4.e.a 2
20.e even 4 1 20.4.e.a 2
20.e even 4 1 inner 100.4.e.a 2
40.e odd 2 1 320.4.n.c 2
40.f even 2 1 320.4.n.c 2
40.i odd 4 1 320.4.n.c 2
40.k even 4 1 320.4.n.c 2
60.h even 2 1 180.4.k.a 2
60.l odd 4 1 180.4.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.e.a 2 5.b even 2 1
20.4.e.a 2 5.c odd 4 1
20.4.e.a 2 20.d odd 2 1
20.4.e.a 2 20.e even 4 1
100.4.e.a 2 1.a even 1 1 trivial
100.4.e.a 2 4.b odd 2 1 CM
100.4.e.a 2 5.c odd 4 1 inner
100.4.e.a 2 20.e even 4 1 inner
180.4.k.a 2 15.d odd 2 1
180.4.k.a 2 15.e even 4 1
180.4.k.a 2 60.h even 2 1
180.4.k.a 2 60.l odd 4 1
320.4.n.c 2 40.e odd 2 1
320.4.n.c 2 40.f even 2 1
320.4.n.c 2 40.i odd 4 1
320.4.n.c 2 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{4}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 74T + 2738 \) Copy content Toggle raw display
$17$ \( T^{2} + 198T + 19602 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 80656 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 182T + 16562 \) Copy content Toggle raw display
$41$ \( (T - 472)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 54T + 1458 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 468)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 506T + 128018 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 30976 \) Copy content Toggle raw display
$97$ \( T^{2} - 1222 T + 746642 \) Copy content Toggle raw display
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