Properties

Label 100.6.a.c
Level $100$
Weight $6$
Character orbit 100.a
Self dual yes
Analytic conductor $16.038$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,6,Mod(1,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 100.a (trivial)

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: yes
Analytic conductor: \(16.0383819813\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{409}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 102 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{409}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 10) q^{3} + (6 \beta + 20) q^{7} + (20 \beta + 266) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 10) q^{3} + (6 \beta + 20) q^{7} + (20 \beta + 266) q^{9} + (15 \beta - 30) q^{11} + ( - 24 \beta - 460) q^{13} + (12 \beta - 1455) q^{17} + ( - 45 \beta + 1046) q^{19} + ( - 80 \beta - 2654) q^{21} + (138 \beta - 60) q^{23} + ( - 223 \beta - 8410) q^{27} + ( - 240 \beta + 1776) q^{29} + (150 \beta - 4444) q^{31} + ( - 120 \beta - 5835) q^{33} + (456 \beta - 6070) q^{37} + (700 \beta + 14416) q^{39} + ( - 360 \beta - 6219) q^{41} + ( - 240 \beta - 580) q^{43} + (132 \beta + 600) q^{47} + (240 \beta - 1683) q^{49} + (1335 \beta + 9642) q^{51} + ( - 408 \beta - 13170) q^{53} + ( - 596 \beta + 7945) q^{57} + ( - 1140 \beta - 18348) q^{59} + ( - 1680 \beta + 9602) q^{61} + (1996 \beta + 54400) q^{63} + ( - 639 \beta + 45230) q^{67} + ( - 1320 \beta - 55842) q^{69} + ( - 1980 \beta + 1368) q^{71} + ( - 84 \beta - 6385) q^{73} + (120 \beta + 36210) q^{77} + ( - 390 \beta - 8092) q^{79} + (5780 \beta + 110669) q^{81} + ( - 3147 \beta + 15150) q^{83} + (624 \beta + 80400) q^{87} + (5580 \beta - 23661) q^{89} + ( - 3240 \beta - 68096) q^{91} + (2944 \beta - 16910) q^{93} + (2736 \beta + 1490) q^{97} + (3390 \beta + 114720) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 20 q^{3} + 40 q^{7} + 532 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 20 q^{3} + 40 q^{7} + 532 q^{9} - 60 q^{11} - 920 q^{13} - 2910 q^{17} + 2092 q^{19} - 5308 q^{21} - 120 q^{23} - 16820 q^{27} + 3552 q^{29} - 8888 q^{31} - 11670 q^{33} - 12140 q^{37} + 28832 q^{39} - 12438 q^{41} - 1160 q^{43} + 1200 q^{47} - 3366 q^{49} + 19284 q^{51} - 26340 q^{53} + 15890 q^{57} - 36696 q^{59} + 19204 q^{61} + 108800 q^{63} + 90460 q^{67} - 111684 q^{69} + 2736 q^{71} - 12770 q^{73} + 72420 q^{77} - 16184 q^{79} + 221338 q^{81} + 30300 q^{83} + 160800 q^{87} - 47322 q^{89} - 136192 q^{91} - 33820 q^{93} + 2980 q^{97} + 229440 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
10.6119
−9.61187
0 −30.2237 0 0 0 141.342 0 670.475 0
1.2 0 10.2237 0 0 0 −101.342 0 −138.475 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.6.a.c 2
3.b odd 2 1 900.6.a.s 2
4.b odd 2 1 400.6.a.v 2
5.b even 2 1 100.6.a.e yes 2
5.c odd 4 2 100.6.c.c 4
15.d odd 2 1 900.6.a.m 2
15.e even 4 2 900.6.d.m 4
20.d odd 2 1 400.6.a.p 2
20.e even 4 2 400.6.c.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.6.a.c 2 1.a even 1 1 trivial
100.6.a.e yes 2 5.b even 2 1
100.6.c.c 4 5.c odd 4 2
400.6.a.p 2 20.d odd 2 1
400.6.a.v 2 4.b odd 2 1
400.6.c.m 4 20.e even 4 2
900.6.a.m 2 15.d odd 2 1
900.6.a.s 2 3.b odd 2 1
900.6.d.m 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 20T_{3} - 309 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(100))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 20T - 309 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 40T - 14324 \) Copy content Toggle raw display
$11$ \( T^{2} + 60T - 91125 \) Copy content Toggle raw display
$13$ \( T^{2} + 920T - 23984 \) Copy content Toggle raw display
$17$ \( T^{2} + 2910 T + 2058129 \) Copy content Toggle raw display
$19$ \( T^{2} - 2092 T + 265891 \) Copy content Toggle raw display
$23$ \( T^{2} + 120 T - 7785396 \) Copy content Toggle raw display
$29$ \( T^{2} - 3552 T - 20404224 \) Copy content Toggle raw display
$31$ \( T^{2} + 8888 T + 10546636 \) Copy content Toggle raw display
$37$ \( T^{2} + 12140 T - 48200924 \) Copy content Toggle raw display
$41$ \( T^{2} + 12438 T - 14330439 \) Copy content Toggle raw display
$43$ \( T^{2} + 1160 T - 23222000 \) Copy content Toggle raw display
$47$ \( T^{2} - 1200 T - 6766416 \) Copy content Toggle raw display
$53$ \( T^{2} + 26340 T + 105365124 \) Copy content Toggle raw display
$59$ \( T^{2} + 36696 T - 194887296 \) Copy content Toggle raw display
$61$ \( T^{2} - 19204 T - 1062163196 \) Copy content Toggle raw display
$67$ \( T^{2} - 90460 T + 1878749611 \) Copy content Toggle raw display
$71$ \( T^{2} - 2736 T - 1601572176 \) Copy content Toggle raw display
$73$ \( T^{2} + 12770 T + 37882321 \) Copy content Toggle raw display
$79$ \( T^{2} + 16184 T + 3271564 \) Copy content Toggle raw display
$83$ \( T^{2} - 30300 T - 3821053581 \) Copy content Toggle raw display
$89$ \( T^{2} + 47322 T - 12174944679 \) Copy content Toggle raw display
$97$ \( T^{2} - 2980 T - 3059429564 \) Copy content Toggle raw display
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