Properties

Label 100.9.f.b
Level $100$
Weight $9$
Character orbit 100.f
Analytic conductor $40.738$
Analytic rank $0$
Dimension $8$
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] = [100,9,Mod(57,100)]
 
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([0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.57");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 100.f (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: \(40.7378610061\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4 x^{7} + 8 x^{6} + 22254 x^{5} + 4820745 x^{4} + 50131374 x^{3} + 307615702 x^{2} + \cdots + 2405464244 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 9 \beta_1 + 9) q^{3} + (\beta_{7} - \beta_{4} - \beta_{3} + \cdots + 254) q^{7}+ \cdots + ( - 4 \beta_{7} - 4 \beta_{6} + \cdots + 2648 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 9 \beta_1 + 9) q^{3} + (\beta_{7} - \beta_{4} - \beta_{3} + \cdots + 254) q^{7}+ \cdots + (16577 \beta_{7} + \cdots - 34340848 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 70 q^{3} + 2030 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 70 q^{3} + 2030 q^{7} - 420 q^{11} - 33180 q^{13} - 43620 q^{17} + 108668 q^{21} + 663270 q^{23} - 1576040 q^{27} - 3178492 q^{31} + 944020 q^{33} - 5344080 q^{37} - 10185252 q^{41} + 10342710 q^{43} - 19232250 q^{47} - 47126684 q^{51} + 24320640 q^{53} - 88218320 q^{57} - 82515684 q^{61} + 77441350 q^{63} - 100675930 q^{67} - 99290076 q^{71} + 93528520 q^{73} - 134199660 q^{77} - 161920268 q^{81} + 10450350 q^{83} - 164801600 q^{87} - 130681068 q^{91} + 50183620 q^{93} + 179570760 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4 x^{7} + 8 x^{6} + 22254 x^{5} + 4820745 x^{4} + 50131374 x^{3} + 307615702 x^{2} + \cdots + 2405464244 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 17543298991401 \nu^{7} - 16868436332617 \nu^{6} - 231983879936421 \nu^{5} + \cdots - 17\!\cdots\!12 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 83\!\cdots\!98 \nu^{7} + \cdots + 44\!\cdots\!76 ) / 75\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13\!\cdots\!93 \nu^{7} + \cdots + 17\!\cdots\!16 ) / 75\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2504810371918 \nu^{7} + 24975279644456 \nu^{6} - 81235255354622 \nu^{5} + \cdots + 40\!\cdots\!41 ) / 95\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23\!\cdots\!61 \nu^{7} + \cdots - 22\!\cdots\!32 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 35\!\cdots\!87 \nu^{7} + \cdots + 11\!\cdots\!44 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 47\!\cdots\!21 \nu^{7} + \cdots + 80\!\cdots\!52 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + 4\beta_{2} - 39\beta _1 + 41 ) / 80 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 21\beta_{7} + 21\beta_{6} - 11\beta_{5} - 20\beta_{4} + 12\beta_{3} + 16\beta_{2} + 44747\beta _1 + 1 ) / 40 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 165 \beta_{7} + 63 \beta_{6} + 988 \beta_{5} - 1186 \beta_{4} + 8158 \beta_{3} + 42 \beta_{2} + \cdots - 335999 ) / 40 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9453 \beta_{7} - 9321 \beta_{6} + 804 \beta_{5} - 7210 \beta_{4} + 14870 \beta_{3} - 8342 \beta_{2} + \cdots - 19644975 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 234885 \beta_{7} - 929985 \beta_{6} - 2102712 \beta_{5} - 2281412 \beta_{4} + 290290 \beta_{3} + \cdots - 1744509937 ) / 40 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 103276119 \beta_{7} - 108859329 \beta_{6} - 2298436 \beta_{5} + 90808300 \beta_{4} + \cdots - 9000406409 ) / 40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3322505895 \beta_{7} - 744118557 \beta_{6} - 4443569742 \beta_{5} + 7750380034 \beta_{4} + \cdots + 3839292487721 ) / 40 \) Copy content Toggle raw display

Character values

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

\(n\) \(51\) \(77\)
\(\chi(n)\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
1.89682 0.896816i
−29.1758 + 30.1758i
36.4975 35.4975i
−7.21849 + 8.21849i
1.89682 + 0.896816i
−29.1758 30.1758i
36.4975 + 35.4975i
−7.21849 8.21849i
0 −80.1275 + 80.1275i 0 0 0 1045.24 + 1045.24i 0 6279.84i 0
57.2 0 −18.1203 + 18.1203i 0 0 0 190.873 + 190.873i 0 5904.31i 0
57.3 0 29.4437 29.4437i 0 0 0 −1846.83 1846.83i 0 4827.14i 0
57.4 0 103.804 103.804i 0 0 0 1625.71 + 1625.71i 0 14989.6i 0
93.1 0 −80.1275 80.1275i 0 0 0 1045.24 1045.24i 0 6279.84i 0
93.2 0 −18.1203 18.1203i 0 0 0 190.873 190.873i 0 5904.31i 0
93.3 0 29.4437 + 29.4437i 0 0 0 −1846.83 + 1846.83i 0 4827.14i 0
93.4 0 103.804 + 103.804i 0 0 0 1625.71 1625.71i 0 14989.6i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.9.f.b 8
5.b even 2 1 20.9.f.a 8
5.c odd 4 1 20.9.f.a 8
5.c odd 4 1 inner 100.9.f.b 8
15.d odd 2 1 180.9.l.a 8
15.e even 4 1 180.9.l.a 8
20.d odd 2 1 80.9.p.d 8
20.e even 4 1 80.9.p.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.9.f.a 8 5.b even 2 1
20.9.f.a 8 5.c odd 4 1
80.9.p.d 8 20.d odd 2 1
80.9.p.d 8 20.e even 4 1
100.9.f.b 8 1.a even 1 1 trivial
100.9.f.b 8 5.c odd 4 1 inner
180.9.l.a 8 15.d odd 2 1
180.9.l.a 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 70 T_{3}^{7} + 2450 T_{3}^{6} + 774360 T_{3}^{5} + 259170228 T_{3}^{4} + \cdots + 315086072440896 \) acting on \(S_{9}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 315086072440896 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 57\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots - 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 40\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 27\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 88\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 32\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 29\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 83\!\cdots\!04)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 41\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 44\!\cdots\!36 \) Copy content Toggle raw display
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