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Properties

Label 2268.4.a.i
Level $2268$
Weight $4$
Character orbit 2268.a
Self dual yes
Analytic conductor $133.816$
Analytic rank $0$
Dimension $9$
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] = [2268,4,Mod(1,2268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2268, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2268.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: \(133.816331893\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2x^{8} - 180x^{7} + 566x^{6} + 8543x^{5} - 33702x^{4} - 75401x^{3} + 385978x^{2} - 453885x + 167250 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{9} \)
Twist minimal: no (minimal twist has level 252)
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 a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{5} + 7 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{5} + 7 q^{7} + ( - \beta_{6} + \beta_{2} + 10) q^{11} + ( - \beta_{8} - \beta_{6} + \beta_{4} + \cdots + 2) q^{13}+ \cdots + ( - 2 \beta_{8} - 7 \beta_{7} + \cdots - 105) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 6 q^{5} + 63 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 6 q^{5} + 63 q^{7} + 93 q^{11} + 18 q^{13} - 27 q^{17} - 45 q^{19} + 246 q^{23} + 315 q^{25} + 318 q^{29} + 18 q^{31} + 42 q^{35} - 36 q^{37} + 57 q^{41} + 171 q^{43} + 1056 q^{47} + 441 q^{49} + 756 q^{53} - 900 q^{55} + 411 q^{59} + 198 q^{61} + 1326 q^{65} + 441 q^{67} + 1758 q^{71} + 27 q^{73} + 651 q^{77} - 72 q^{79} + 558 q^{83} + 1008 q^{85} + 1392 q^{89} + 126 q^{91} + 156 q^{95} - 909 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 180x^{7} + 566x^{6} + 8543x^{5} - 33702x^{4} - 75401x^{3} + 385978x^{2} - 453885x + 167250 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 247259 \nu^{8} + 486963 \nu^{7} + 44293815 \nu^{6} - 141103699 \nu^{5} - 2096626752 \nu^{4} + \cdots + 79965863670 ) / 737514720 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 247259 \nu^{8} - 486963 \nu^{7} - 44293815 \nu^{6} + 141103699 \nu^{5} + 2096626752 \nu^{4} + \cdots - 81440893110 ) / 737514720 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1202071 \nu^{8} + 975807 \nu^{7} + 217679715 \nu^{6} - 418093151 \nu^{5} + \cdots + 158584947630 ) / 368757360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5015341 \nu^{8} - 5005557 \nu^{7} - 908663985 \nu^{6} + 1923167861 \nu^{5} + 44922639168 \nu^{4} + \cdots - 784651957530 ) / 737514720 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1214705 \nu^{8} - 991737 \nu^{7} - 219792645 \nu^{6} + 427024825 \nu^{5} + 10877016672 \nu^{4} + \cdots - 173158790754 ) / 147502944 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7040267 \nu^{8} - 6398019 \nu^{7} - 1273983975 \nu^{6} + 2591893027 \nu^{5} + \cdots - 1100877332790 ) / 737514720 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 254613 \nu^{8} + 278581 \nu^{7} + 46109105 \nu^{6} - 102299093 \nu^{5} - 2271576824 \nu^{4} + \cdots + 46850616570 ) / 20486520 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3196219 \nu^{8} - 2461683 \nu^{7} - 578222295 \nu^{6} + 1100019539 \nu^{5} + 28641136992 \nu^{4} + \cdots - 437348283990 ) / 245838240 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 2 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{8} + 6\beta_{6} + 3\beta_{5} - 3\beta_{4} + 6\beta_{3} - 13\beta_{2} - 4\beta _1 + 367 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 15\beta_{8} + 2\beta_{7} + 8\beta_{6} - 42\beta_{5} + 4\beta_{4} - 21\beta_{3} + 86\beta_{2} + 75\beta _1 - 601 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 405 \beta_{8} + 80 \beta_{7} + 611 \beta_{6} + 540 \beta_{5} - 152 \beta_{4} + 828 \beta_{3} + \cdots + 29152 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3144 \beta_{8} - 132 \beta_{7} - 291 \beta_{6} - 6978 \beta_{5} + 603 \beta_{4} - 3912 \beta_{3} + \cdots - 105151 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 51018 \beta_{8} + 13825 \beta_{7} + 64984 \beta_{6} + 76020 \beta_{5} - 9592 \beta_{4} + \cdots + 2765518 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 469449 \beta_{8} - 56528 \beta_{7} - 157364 \beta_{6} - 942084 \beta_{5} + 57761 \beta_{4} + \cdots - 14952241 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 6288204 \beta_{8} + 1831605 \beta_{7} + 7061097 \beta_{6} + 9943059 \beta_{5} - 754695 \beta_{4} + \cdots + 288638347 ) / 9 \) Copy content Toggle raw display

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

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
5.33582
1.41731
−4.09985
1.13459
9.19269
−11.0282
0.869544
−7.76634
6.94444
0 0 0 −17.5326 0 7.00000 0 0 0
1.2 0 0 0 −15.3368 0 7.00000 0 0 0
1.3 0 0 0 −7.71803 0 7.00000 0 0 0
1.4 0 0 0 −3.21782 0 7.00000 0 0 0
1.5 0 0 0 −0.368846 0 7.00000 0 0 0
1.6 0 0 0 0.666291 0 7.00000 0 0 0
1.7 0 0 0 14.2689 0 7.00000 0 0 0
1.8 0 0 0 16.5244 0 7.00000 0 0 0
1.9 0 0 0 18.7145 0 7.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

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 2268.4.a.i 9
3.b odd 2 1 2268.4.a.h 9
9.c even 3 2 252.4.j.b ✓ 18
9.d odd 6 2 756.4.j.b 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.j.b ✓ 18 9.c even 3 2
756.4.j.b 18 9.d odd 6 2
2268.4.a.h 9 3.b odd 2 1
2268.4.a.i 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{9} - 6 T_{5}^{8} - 702 T_{5}^{7} + 2790 T_{5}^{6} + 157221 T_{5}^{5} - 209574 T_{5}^{4} + \cdots + 7241886 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2268))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} \) Copy content Toggle raw display
$5$ \( T^{9} - 6 T^{8} + \cdots + 7241886 \) Copy content Toggle raw display
$7$ \( (T - 7)^{9} \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots - 8821607523660 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots - 42807996736496 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots - 109582832291940 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots - 20\!\cdots\!75 \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots + 57\!\cdots\!72 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots - 45\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots - 12\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots - 44\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots + 41\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots - 44\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots - 62\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots - 22\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots - 86\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots - 22\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots + 35\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots - 11\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots - 25\!\cdots\!30 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots + 28\!\cdots\!80 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots + 28\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots + 90\!\cdots\!40 \) Copy content Toggle raw display
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