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Properties

Label 300.9.g.e
Level $300$
Weight $9$
Character orbit 300.g
Analytic conductor $122.214$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,9,Mod(101,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.101");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 300.g (of order \(2\), degree \(1\), 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: \(122.213583018\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 17581 x^{8} - 268094 x^{7} + 129938570 x^{6} - 2805075950 x^{5} + 497042336337 x^{4} + \cdots + 51\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{17}\cdot 5^{7}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 14) q^{3} + ( - \beta_{3} + 2 \beta_1 - 104) q^{7} + (\beta_{4} - \beta_{2} + 17 \beta_1 - 1240) q^{9} + ( - \beta_{7} - 8 \beta_1 - 2) q^{11} + (2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \cdots + 3430) q^{13}+ \cdots + ( - 102 \beta_{9} + 528 \beta_{8} + \cdots + 5247444) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 137 q^{3} - 1048 q^{7} - 12455 q^{9} + 34472 q^{13} - 376030 q^{19} - 142540 q^{21} - 1458782 q^{27} - 410860 q^{31} - 523275 q^{33} - 110344 q^{37} + 3527870 q^{39} - 6252148 q^{43} + 13891530 q^{49}+ \cdots + 52292505 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 17581 x^{8} - 268094 x^{7} + 129938570 x^{6} - 2805075950 x^{5} + 497042336337 x^{4} + \cdots + 51\!\cdots\!56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 31\!\cdots\!99 \nu^{9} + \cdots + 83\!\cdots\!94 ) / 20\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 20\!\cdots\!89 \nu^{9} + \cdots - 99\!\cdots\!34 ) / 69\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 43\!\cdots\!82 \nu^{9} + \cdots - 10\!\cdots\!33 ) / 10\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15\!\cdots\!52 \nu^{9} + \cdots + 17\!\cdots\!06 ) / 20\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 28\!\cdots\!76 \nu^{9} + \cdots - 69\!\cdots\!70 ) / 20\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 58\!\cdots\!66 \nu^{9} + \cdots - 67\!\cdots\!54 ) / 31\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 53\!\cdots\!66 \nu^{9} + \cdots + 36\!\cdots\!24 ) / 20\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 32\!\cdots\!36 \nu^{9} + \cdots + 29\!\cdots\!88 ) / 69\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 43\!\cdots\!51 \nu^{9} + \cdots - 62\!\cdots\!76 ) / 31\!\cdots\!75 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 3\beta_{6} - 5\beta_{5} + 10\beta_{4} - 24\beta_{3} + 105\beta_{2} + 3521\beta _1 + 1645 ) / 5670 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 18 \beta_{9} - 126 \beta_{8} - 252 \beta_{7} - 24 \beta_{6} - 593 \beta_{5} + 574 \beta_{4} + \cdots - 19933400 ) / 5670 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 477 \beta_{9} - 1197 \beta_{8} - 1260 \beta_{7} - 24816 \beta_{6} + 17531 \beta_{5} + \cdots + 140784414 ) / 1890 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 89478 \beta_{9} + 979020 \beta_{8} + 3251934 \beta_{7} - 1069734 \beta_{6} + 4077899 \beta_{5} + \cdots + 56555124980 ) / 5670 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 12096225 \beta_{9} + 20022975 \beta_{8} + 60883200 \beta_{7} + 540939723 \beta_{6} + \cdots - 4892621889835 ) / 5670 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 151267842 \beta_{9} - 1705873806 \beta_{8} - 6682521132 \beta_{7} + 656273328 \beta_{6} + \cdots + 5722741427884 ) / 1890 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 73494705564 \beta_{9} - 25578553329 \beta_{8} - 80276140665 \beta_{7} - 2593989467772 \beta_{6} + \cdots + 38\!\cdots\!50 ) / 5670 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 10937784134394 \beta_{9} + 17339543872560 \beta_{8} + 91365578955378 \beta_{7} + \cdots - 17\!\cdots\!60 ) / 5670 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 77721907903605 \beta_{9} - 229931190642195 \beta_{8} + \cdots - 76\!\cdots\!00 ) / 1890 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
18.8523 + 21.8350i
18.8523 − 21.8350i
−5.98515 + 56.3224i
−5.98515 − 56.3224i
−16.1674 + 79.4621i
−16.1674 − 79.4621i
−23.3576 + 73.0318i
−23.3576 − 73.0318i
27.1579 + 64.9569i
27.1579 − 64.9569i
0 −78.0015 − 21.8350i 0 0 0 1300.30 0 5607.47 + 3406.32i 0
101.2 0 −78.0015 + 21.8350i 0 0 0 1300.30 0 5607.47 − 3406.32i 0
101.3 0 −58.2133 − 56.3224i 0 0 0 −2295.51 0 216.575 + 6557.42i 0
101.4 0 −58.2133 + 56.3224i 0 0 0 −2295.51 0 216.575 − 6557.42i 0
101.5 0 −15.7090 − 79.4621i 0 0 0 2840.81 0 −6067.46 + 2496.53i 0
101.6 0 −15.7090 + 79.4621i 0 0 0 2840.81 0 −6067.46 − 2496.53i 0
101.7 0 35.0336 − 73.0318i 0 0 0 −4179.12 0 −4106.29 − 5117.14i 0
101.8 0 35.0336 + 73.0318i 0 0 0 −4179.12 0 −4106.29 + 5117.14i 0
101.9 0 48.3901 − 64.9569i 0 0 0 1809.51 0 −1877.79 − 6286.54i 0
101.10 0 48.3901 + 64.9569i 0 0 0 1809.51 0 −1877.79 + 6286.54i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.9.g.e ✓ 10
3.b odd 2 1 inner 300.9.g.e ✓ 10
5.b even 2 1 300.9.g.g yes 10
5.c odd 4 2 300.9.b.e 20
15.d odd 2 1 300.9.g.g yes 10
15.e even 4 2 300.9.b.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.9.b.e 20 5.c odd 4 2
300.9.b.e 20 15.e even 4 2
300.9.g.e ✓ 10 1.a even 1 1 trivial
300.9.g.e ✓ 10 3.b odd 2 1 inner
300.9.g.g yes 10 5.b even 2 1
300.9.g.g yes 10 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{5} + 524T_{7}^{4} - 17747597T_{7}^{3} + 8663960872T_{7}^{2} + 64044622169696T_{7} - 64122838801930496 \) acting on \(S_{9}^{\mathrm{new}}(300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( (T^{5} + \cdots - 64\!\cdots\!96)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots + 17\!\cdots\!84)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots + 50\!\cdots\!11)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 82\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 97\!\cdots\!12)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 25\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 66\!\cdots\!84)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots + 45\!\cdots\!29)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 97\!\cdots\!92)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 22\!\cdots\!28)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 29\!\cdots\!64)^{2} \) Copy content Toggle raw display
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