Properties

Label 100.9.f.c
Level $100$
Weight $9$
Character orbit 100.f
Analytic conductor $40.738$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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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: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 8 x^{10} + 1412 x^{9} + 550393 x^{8} - 1456736 x^{7} + 2420672 x^{6} + \cdots + 547748010000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{6}\cdot 5^{12} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + ( - \beta_{11} - 20 \beta_{8} + 7 \beta_{7}) q^{7} + ( - \beta_{10} - \beta_{3} - 2002 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + ( - \beta_{11} - 20 \beta_{8} + 7 \beta_{7}) q^{7} + ( - \beta_{10} - \beta_{3} - 2002 \beta_1) q^{9} + ( - 2 \beta_{5} - \beta_{2} - 35) q^{11} + (\beta_{9} - 46 \beta_{6} + 295 \beta_{4}) q^{13} + (15 \beta_{11} + 183 \beta_{8} - 429 \beta_{7}) q^{17} + (16 \beta_{10} - 23 \beta_{3} + 21333 \beta_1) q^{19} + ( - 85 \beta_{5} + 17 \beta_{2} - 76764) q^{21} + ( - 18 \beta_{9} + \cdots + 3816 \beta_{4}) q^{23}+ \cdots + (5403 \beta_{10} + \cdots - 7567530 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 420 q^{11} - 921168 q^{21} + 3833112 q^{31} + 3587532 q^{41} + 46092564 q^{51} + 31354704 q^{61} - 29589384 q^{71} + 104018868 q^{81} + 433229088 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 8 x^{10} + 1412 x^{9} + 550393 x^{8} - 1456736 x^{7} + 2420672 x^{6} + \cdots + 547748010000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 73\!\cdots\!16 \nu^{11} + \cdots + 92\!\cdots\!00 ) / 70\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14\!\cdots\!63 \nu^{11} + \cdots + 77\!\cdots\!00 ) / 93\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 64\!\cdots\!93 \nu^{11} + \cdots + 11\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 16\!\cdots\!48 \nu^{11} + \cdots + 14\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 27\!\cdots\!53 \nu^{11} + \cdots + 15\!\cdots\!00 ) / 37\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 51\!\cdots\!96 \nu^{11} + \cdots - 43\!\cdots\!00 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 48\!\cdots\!71 \nu^{11} + \cdots + 93\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14\!\cdots\!67 \nu^{11} + \cdots - 28\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 47\!\cdots\!52 \nu^{11} + \cdots - 39\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 89\!\cdots\!13 \nu^{11} + \cdots - 11\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\!\cdots\!49 \nu^{11} + \cdots + 26\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 12\beta_{6} - \beta_{5} + \beta_{3} - 100\beta _1 + 100 ) / 300 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -25\beta_{10} - 132\beta_{8} - 300\beta_{7} + 132\beta_{6} + 300\beta_{4} - 3\beta_{3} - 209000\beta_1 ) / 600 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 225 \beta_{11} - 25 \beta_{10} - 25110 \beta_{8} - 675 \beta_{7} + 1049 \beta_{5} + 1049 \beta_{3} + \cdots - 213400 ) / 600 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 300 \beta_{11} + 300 \beta_{9} - 154824 \beta_{8} - 312900 \beta_{7} - 154824 \beta_{6} + \cdots - 111214600 ) / 600 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 22525 \beta_{10} + 196875 \beta_{9} - 22101186 \beta_{6} + 564393 \beta_{5} - 1118625 \beta_{4} + \cdots - 189045800 ) / 600 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 396450 \beta_{11} + 7330625 \beta_{10} + 396450 \beta_{9} + 139724028 \beta_{8} + \cdots + 60727073000 \beta_1 ) / 600 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 145712175 \beta_{11} + 17239875 \beta_{10} + 16507567290 \beta_{8} + 1214035725 \beta_{7} + \cdots + 144154564600 ) / 600 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 391656600 \beta_{11} - 391656600 \beta_{9} + 111159201936 \beta_{8} + 174966189000 \beta_{7} + \cdots + 33846102675400 ) / 600 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 12335561525 \beta_{10} - 99935411175 \beta_{9} + 11422804595274 \beta_{6} - 173980088617 \beta_{5} + \cdots + 102937668792200 ) / 600 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 335214781350 \beta_{11} - 2317860984225 \beta_{10} - 335214781350 \beta_{9} + \cdots - 19\!\cdots\!00 \beta_1 ) / 600 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 65723608796175 \beta_{11} - 8526937895975 \beta_{10} + \cdots - 71\!\cdots\!00 ) / 600 \) 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
−16.9725 + 16.9725i
17.6536 17.6536i
−0.905864 + 0.905864i
1.54363 1.54363i
15.2041 15.2041i
−14.5230 + 14.5230i
−16.9725 16.9725i
17.6536 + 17.6536i
−0.905864 0.905864i
1.54363 + 1.54363i
15.2041 + 15.2041i
−14.5230 14.5230i
0 −90.6200 + 90.6200i 0 0 0 2970.95 + 2970.95i 0 9862.98i 0
57.2 0 −67.0124 + 67.0124i 0 0 0 −1824.09 1824.09i 0 2420.33i 0
57.3 0 −11.9100 + 11.9100i 0 0 0 −2673.78 2673.78i 0 6277.30i 0
57.4 0 11.9100 11.9100i 0 0 0 2673.78 + 2673.78i 0 6277.30i 0
57.5 0 67.0124 67.0124i 0 0 0 1824.09 + 1824.09i 0 2420.33i 0
57.6 0 90.6200 90.6200i 0 0 0 −2970.95 2970.95i 0 9862.98i 0
93.1 0 −90.6200 90.6200i 0 0 0 2970.95 2970.95i 0 9862.98i 0
93.2 0 −67.0124 67.0124i 0 0 0 −1824.09 + 1824.09i 0 2420.33i 0
93.3 0 −11.9100 11.9100i 0 0 0 −2673.78 + 2673.78i 0 6277.30i 0
93.4 0 11.9100 + 11.9100i 0 0 0 2673.78 2673.78i 0 6277.30i 0
93.5 0 67.0124 + 67.0124i 0 0 0 1824.09 1824.09i 0 2420.33i 0
93.6 0 90.6200 + 90.6200i 0 0 0 −2970.95 + 2970.95i 0 9862.98i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.9.f.c 12
5.b even 2 1 inner 100.9.f.c 12
5.c odd 4 2 inner 100.9.f.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.9.f.c 12 1.a even 1 1 trivial
100.9.f.c 12 5.b even 2 1 inner
100.9.f.c 12 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 350491707T_{3}^{8} + 21787141765832883T_{3}^{4} + 1751237556926466380409 \) acting on \(S_{9}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 17\!\cdots\!09 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 28\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( (T^{3} + 105 T^{2} + \cdots - 823037986125)^{4} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 71\!\cdots\!49 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 97\!\cdots\!69)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 28\!\cdots\!64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots + 42\!\cdots\!24)^{4} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 45\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots + 15\!\cdots\!19)^{4} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 39\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 13\!\cdots\!88)^{4} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 82\!\cdots\!29 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots - 90\!\cdots\!32)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 73\!\cdots\!89 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 16\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 27\!\cdots\!29 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 72\!\cdots\!89)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 98\!\cdots\!44 \) Copy content Toggle raw display
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