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Properties

Label 2268.2.f.a
Level $2268$
Weight $2$
Character orbit 2268.f
Analytic conductor $18.110$
Analytic rank $0$
Dimension $16$
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] = [2268,2,Mod(1133,2268)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.1133");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.f (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: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 28 x^{14} - 44 x^{13} + 40 x^{12} - 164 x^{11} + 466 x^{10} - 188 x^{9} + \cdots + 1926 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{8} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5} + \beta_{10} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{5} + \beta_{10} q^{7} - \beta_{4} q^{11} + \beta_{13} q^{13} + \beta_{6} q^{17} + \beta_{15} q^{19} + ( - \beta_{12} + \beta_{4}) q^{23} + (\beta_{7} + 1) q^{25} - \beta_{11} q^{29} + \beta_{14} q^{31} + (\beta_{4} + \beta_{3} + \beta_{2}) q^{35} + (\beta_{10} - \beta_{9} + \cdots + 2 \beta_{7}) q^{37}+ \cdots + (\beta_{15} + 2 \beta_{13}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{7} + 16 q^{25} - 8 q^{37} - 8 q^{43} + 16 q^{49} + 8 q^{67} + 56 q^{79} + 24 q^{85} + 36 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 28 x^{14} - 44 x^{13} + 40 x^{12} - 164 x^{11} + 466 x^{10} - 188 x^{9} + \cdots + 1926 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 42\!\cdots\!15 \nu^{15} + \cdots + 25\!\cdots\!65 ) / 59\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 46\!\cdots\!64 \nu^{15} + \cdots + 28\!\cdots\!86 ) / 59\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 56\!\cdots\!99 \nu^{15} + \cdots + 72\!\cdots\!39 ) / 59\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 42\!\cdots\!84 \nu^{15} + \cdots - 25\!\cdots\!80 ) / 30\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 64624928328648 \nu^{15} + 748419060000303 \nu^{14} + \cdots - 78\!\cdots\!62 ) / 43\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 93\!\cdots\!74 \nu^{15} + \cdots + 14\!\cdots\!27 ) / 59\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\!\cdots\!18 \nu^{15} + \cdots - 57\!\cdots\!05 ) / 88\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 30\!\cdots\!31 \nu^{15} + \cdots + 12\!\cdots\!75 ) / 17\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 119023179720 \nu^{15} - 669640924282 \nu^{14} + 1609639658494 \nu^{13} + \cdots - 339217856552421 ) / 52900736658373 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 40\!\cdots\!97 \nu^{15} + \cdots + 15\!\cdots\!69 ) / 17\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!68 \nu^{15} + \cdots - 22\!\cdots\!01 ) / 30\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 42\!\cdots\!75 \nu^{15} + \cdots + 10\!\cdots\!72 ) / 10\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 71\!\cdots\!94 \nu^{15} + \cdots + 42\!\cdots\!07 ) / 17\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 88\!\cdots\!70 \nu^{15} + \cdots - 34\!\cdots\!92 ) / 17\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 12\!\cdots\!28 \nu^{15} + \cdots + 18\!\cdots\!32 ) / 17\!\cdots\!87 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{15} + \beta_{14} - 2 \beta_{13} - \beta_{12} + 2 \beta_{11} + \beta_{10} + \beta_{8} + \cdots + 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{15} + 4 \beta_{14} - 8 \beta_{13} + \beta_{12} - 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \cdots + 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} + 12 \beta_{14} - 12 \beta_{13} + 6 \beta_{12} + 18 \beta_{10} - 18 \beta_{9} + \cdots - 12 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 16 \beta_{15} - 2 \beta_{14} + 22 \beta_{13} + 25 \beta_{12} - 4 \beta_{11} + 40 \beta_{10} + \cdots - 78 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 103 \beta_{15} - 104 \beta_{14} + 304 \beta_{13} + 184 \beta_{12} - 88 \beta_{11} + 94 \beta_{10} + \cdots - 264 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 330 \beta_{15} - 324 \beta_{14} + 996 \beta_{13} + 201 \beta_{12} - 216 \beta_{11} - 282 \beta_{10} + \cdots + 474 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 233 \beta_{15} - 301 \beta_{14} + 812 \beta_{13} - 751 \beta_{12} - 152 \beta_{11} - 2251 \beta_{10} + \cdots + 4122 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 572 \beta_{15} + 604 \beta_{14} - 1802 \beta_{13} - 1355 \beta_{12} + 416 \beta_{11} - 1655 \beta_{10} + \cdots + 3087 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 11637 \beta_{15} + 14031 \beta_{14} - 39312 \beta_{13} - 15741 \beta_{12} + 9432 \beta_{11} + \cdots + 1626 ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 28930 \beta_{15} + 32300 \beta_{14} - 93232 \beta_{13} - 8005 \beta_{12} + 22168 \beta_{11} + \cdots - 137334 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1403 \beta_{15} - 16286 \beta_{14} + 25600 \beta_{13} + 129904 \beta_{12} - 724 \beta_{11} + \cdots - 611088 ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 145968 \beta_{15} - 214164 \beta_{14} + 557196 \beta_{13} + 314103 \beta_{12} - 114216 \beta_{11} + \cdots - 581010 ) / 6 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1198793 \beta_{15} - 1685410 \beta_{14} + 4452800 \beta_{13} + 1256486 \beta_{12} - 950276 \beta_{11} + \cdots + 1741980 ) / 12 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1894630 \beta_{15} - 2581772 \beta_{14} + 6898696 \beta_{13} - 1546769 \beta_{12} - 1494856 \beta_{11} + \cdots + 20472486 ) / 12 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 5074401 \beta_{15} + 7270119 \beta_{14} - 19077204 \beta_{13} - 20386095 \beta_{12} + 4035492 \beta_{11} + \cdots + 69455994 ) / 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1133.1
−1.22392 − 1.59632i
−1.22392 + 1.59632i
1.28086 + 0.424858i
1.28086 − 0.424858i
2.14537 − 2.51197i
2.14537 + 2.51197i
−0.600499 − 0.0912347i
−0.600499 + 0.0912347i
1.60050 + 0.426403i
1.60050 − 0.426403i
−1.14537 − 0.580120i
−1.14537 + 0.580120i
−0.280860 − 1.50699i
−0.280860 + 1.50699i
2.22392 − 1.07868i
2.22392 + 1.07868i
0 0 0 −3.15351 0 −2.44383 − 1.01375i 0 0 0
1133.2 0 0 0 −3.15351 0 −2.44383 + 1.01375i 0 0 0
1133.3 0 0 0 −2.89853 0 2.28052 − 1.34136i 0 0 0
1133.4 0 0 0 −2.89853 0 2.28052 + 1.34136i 0 0 0
1133.5 0 0 0 −2.30880 0 −1.91449 − 1.82612i 0 0 0
1133.6 0 0 0 −2.30880 0 −1.91449 + 1.82612i 0 0 0
1133.7 0 0 0 −0.568621 0 1.07781 − 2.41626i 0 0 0
1133.8 0 0 0 −0.568621 0 1.07781 + 2.41626i 0 0 0
1133.9 0 0 0 0.568621 0 1.07781 − 2.41626i 0 0 0
1133.10 0 0 0 0.568621 0 1.07781 + 2.41626i 0 0 0
1133.11 0 0 0 2.30880 0 −1.91449 − 1.82612i 0 0 0
1133.12 0 0 0 2.30880 0 −1.91449 + 1.82612i 0 0 0
1133.13 0 0 0 2.89853 0 2.28052 − 1.34136i 0 0 0
1133.14 0 0 0 2.89853 0 2.28052 + 1.34136i 0 0 0
1133.15 0 0 0 3.15351 0 −2.44383 − 1.01375i 0 0 0
1133.16 0 0 0 3.15351 0 −2.44383 + 1.01375i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1133.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.f.a ✓ 16
3.b odd 2 1 inner 2268.2.f.a ✓ 16
7.b odd 2 1 inner 2268.2.f.a ✓ 16
9.c even 3 2 2268.2.x.k 32
9.d odd 6 2 2268.2.x.k 32
21.c even 2 1 inner 2268.2.f.a ✓ 16
63.l odd 6 2 2268.2.x.k 32
63.o even 6 2 2268.2.x.k 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2268.2.f.a ✓ 16 1.a even 1 1 trivial
2268.2.f.a ✓ 16 3.b odd 2 1 inner
2268.2.f.a ✓ 16 7.b odd 2 1 inner
2268.2.f.a ✓ 16 21.c even 2 1 inner
2268.2.x.k 32 9.c even 3 2
2268.2.x.k 32 9.d odd 6 2
2268.2.x.k 32 63.l odd 6 2
2268.2.x.k 32 63.o even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 24T_{5}^{6} + 189T_{5}^{4} - 504T_{5}^{2} + 144 \) acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 24 T^{6} + \cdots + 144)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 2 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 36 T^{6} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 42 T^{6} + \cdots + 576)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 96 T^{6} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 96 T^{6} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 108 T^{6} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 72 T^{6} + \cdots + 324)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 144 T^{6} + \cdots + 186624)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} + \cdots + 1417)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 204 T^{6} + \cdots + 97344)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 2 T^{3} + \cdots - 122)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 204 T^{6} + \cdots + 97344)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 144 T^{6} + \cdots + 26244)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 420 T^{6} + \cdots + 11614464)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 342 T^{6} + \cdots + 13927824)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 2 T^{3} - 39 T^{2} + \cdots + 4)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 504 T^{6} + \cdots + 54756)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 594 T^{6} + \cdots + 336208896)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 14 T^{3} + \cdots - 296)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 276 T^{6} + \cdots + 14197824)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 432 T^{6} + \cdots + 24681024)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 312 T^{6} + \cdots + 27373824)^{2} \) Copy content Toggle raw display
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