Properties

Label 100.6.e.a
Level $100$
Weight $6$
Character orbit 100.e
Analytic conductor $16.038$
Analytic rank $0$
Dimension $2$
CM discriminant -20
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.7");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(16.0383819813\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (4 i - 4) q^{2} + ( - 19 i - 19) q^{3} - 32 i q^{4} + 152 q^{6} + (183 i - 183) q^{7} + (128 i + 128) q^{8} + 479 i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (4 i - 4) q^{2} + ( - 19 i - 19) q^{3} - 32 i q^{4} + 152 q^{6} + (183 i - 183) q^{7} + (128 i + 128) q^{8} + 479 i q^{9} + (608 i - 608) q^{12} - 1464 i q^{14} - 1024 q^{16} + ( - 1916 i - 1916) q^{18} + 6954 q^{21} + ( - 2419 i - 2419) q^{23} - 4864 i q^{24} + ( - 4484 i + 4484) q^{27} + (5856 i + 5856) q^{28} + 1686 i q^{29} + ( - 4096 i + 4096) q^{32} + 15328 q^{36} - 4548 q^{41} + (27816 i - 27816) q^{42} + (5931 i + 5931) q^{43} + 19352 q^{46} + ( - 16667 i + 16667) q^{47} + (19456 i + 19456) q^{48} - 50171 i q^{49} + 35872 i q^{54} - 46848 q^{56} + ( - 6744 i - 6744) q^{58} - 25448 q^{61} + ( - 87657 i - 87657) q^{63} + 32768 i q^{64} + ( - 50217 i + 50217) q^{67} + 91922 i q^{69} + (61312 i - 61312) q^{72} - 53995 q^{81} + ( - 18192 i + 18192) q^{82} + (81631 i + 81631) q^{83} - 222528 i q^{84} - 47448 q^{86} + ( - 32034 i + 32034) q^{87} + 149286 i q^{89} + (77408 i - 77408) q^{92} + 133336 i q^{94} - 155648 q^{96} + (200684 i + 200684) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} - 38 q^{3} + 304 q^{6} - 366 q^{7} + 256 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} - 38 q^{3} + 304 q^{6} - 366 q^{7} + 256 q^{8} - 1216 q^{12} - 2048 q^{16} - 3832 q^{18} + 13908 q^{21} - 4838 q^{23} + 8968 q^{27} + 11712 q^{28} + 8192 q^{32} + 30656 q^{36} - 9096 q^{41} - 55632 q^{42} + 11862 q^{43} + 38704 q^{46} + 33334 q^{47} + 38912 q^{48} - 93696 q^{56} - 13488 q^{58} - 50896 q^{61} - 175314 q^{63} + 100434 q^{67} - 122624 q^{72} - 107990 q^{81} + 36384 q^{82} + 163262 q^{83} - 94896 q^{86} + 64068 q^{87} - 154816 q^{92} - 311296 q^{96} + 401368 q^{98}+O(q^{100}) \) 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
−4.00000 + 4.00000i −19.0000 19.0000i 32.0000i 0 152.000 −183.000 + 183.000i 128.000 + 128.000i 479.000i 0
43.1 −4.00000 4.00000i −19.0000 + 19.0000i 32.0000i 0 152.000 −183.000 183.000i 128.000 128.000i 479.000i 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
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 100.6.e.a 2
4.b odd 2 1 100.6.e.c yes 2
5.b even 2 1 100.6.e.c yes 2
5.c odd 4 1 inner 100.6.e.a 2
5.c odd 4 1 100.6.e.c yes 2
20.d odd 2 1 CM 100.6.e.a 2
20.e even 4 1 inner 100.6.e.a 2
20.e even 4 1 100.6.e.c yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.6.e.a 2 1.a even 1 1 trivial
100.6.e.a 2 5.c odd 4 1 inner
100.6.e.a 2 20.d odd 2 1 CM
100.6.e.a 2 20.e even 4 1 inner
100.6.e.c yes 2 4.b odd 2 1
100.6.e.c yes 2 5.b even 2 1
100.6.e.c yes 2 5.c odd 4 1
100.6.e.c yes 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 38T_{3} + 722 \) acting on \(S_{6}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 8T + 32 \) Copy content Toggle raw display
$3$ \( T^{2} + 38T + 722 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 366T + 66978 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4838 T + 11703122 \) Copy content Toggle raw display
$29$ \( T^{2} + 2842596 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T + 4548)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 11862 T + 70353522 \) Copy content Toggle raw display
$47$ \( T^{2} - 33334 T + 555577778 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 25448)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 100434 T + 5043494178 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 163262 T + 13327240322 \) Copy content Toggle raw display
$89$ \( T^{2} + 22286309796 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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