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Properties

Label 2160.2.br.b
Level $2160$
Weight $2$
Character orbit 2160.br
Analytic conductor $17.248$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2160,2,Mod(719,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2160.719");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2160.br (of order \(6\), degree \(2\), not 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: \(17.2476868366\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - 42x^{12} - 239x^{10} + 1858x^{8} + 26493x^{6} + 128697x^{4} + 265752x^{2} + 197136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 720)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{3} + \beta_{2}) q^{5} + (\beta_{13} + \beta_{12} + \cdots + \beta_{10}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{3} + \beta_{2}) q^{5} + (\beta_{13} + \beta_{12} + \cdots + \beta_{10}) q^{7}+ \cdots + ( - 2 \beta_{5} + \beta_{4} + 3 \beta_{3} + \cdots - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{5} - 26 q^{25} - 48 q^{29} + 48 q^{41} + 8 q^{49} + 16 q^{61} - 66 q^{65}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + x^{14} - 42x^{12} - 239x^{10} + 1858x^{8} + 26493x^{6} + 128697x^{4} + 265752x^{2} + 197136 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11491627 \nu^{14} + 716260255 \nu^{12} - 2813934666 \nu^{10} - 27565397693 \nu^{8} + \cdots + 30537958761228 ) / 6477215372160 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 181882283 \nu^{14} - 390014753 \nu^{12} - 4010431498 \nu^{10} - 33319568125 \nu^{8} + \cdots + 37337857042956 ) / 6477215372160 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 133839417 \nu^{14} + 410194795 \nu^{12} + 3927193646 \nu^{10} + 15161682463 \nu^{8} + \cdots - 5353992729348 ) / 2159071790720 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 201931 \nu^{14} + 191105 \nu^{12} + 8331498 \nu^{10} + 30416669 \nu^{8} - 434479786 \nu^{6} + \cdots - 18484232844 ) / 1948327680 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 343510995 \nu^{14} - 28259449 \nu^{12} - 15114688042 \nu^{10} - 62051282229 \nu^{8} + \cdots + 44140086506412 ) / 3238607686080 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1433018271 \nu^{14} + 546184067 \nu^{12} - 62157581122 \nu^{10} - 308460142377 \nu^{8} + \cdots + 228196665189468 ) / 6477215372160 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3410967413 \nu^{14} + 6441613439 \nu^{12} + 123692622166 \nu^{10} + \cdots - 218628167488884 ) / 12954430744320 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 17156638989 \nu^{15} + 82593949831 \nu^{13} + 656312837350 \nu^{11} + \cdots + 27\!\cdots\!96 \nu ) / 958627875079680 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 22189659027 \nu^{15} - 10481353127 \nu^{13} + 1192723052890 \nu^{11} + \cdots - 41\!\cdots\!92 \nu ) / 958627875079680 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8112651691 \nu^{15} + 5997015519 \nu^{13} - 369449025066 \nu^{11} + \cdots + 14\!\cdots\!32 \nu ) / 319542625026560 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 31997648543 \nu^{15} - 75269151357 \nu^{13} - 1268854319810 \nu^{11} + \cdots + 356015462688348 \nu ) / 958627875079680 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 8001920247 \nu^{15} - 6758117829 \nu^{13} - 331613812498 \nu^{11} + \cdots + 722895601321980 \nu ) / 79885656256640 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 133501246795 \nu^{15} + 161788086657 \nu^{13} + 5283791977066 \nu^{11} + \cdots - 11\!\cdots\!56 \nu ) / 958627875079680 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 163380696627 \nu^{15} - 99856891321 \nu^{13} - 6809825065114 \nu^{11} + \cdots + 16\!\cdots\!76 \nu ) / 958627875079680 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 21511560697 \nu^{15} - 33343322251 \nu^{13} - 822018900878 \nu^{11} + \cdots + 17\!\cdots\!08 \nu ) / 95862787507968 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} - \beta_{13} + \beta_{12} - 2\beta_{11} - \beta_{10} - \beta_{9} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{4} - 3\beta_{3} - \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{15} + \beta_{14} + 16\beta_{13} + 11\beta_{12} - 7\beta_{11} - 8\beta_{10} - 2\beta_{9} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{7} - \beta_{6} + 18\beta_{5} + 12\beta_{4} - 22\beta_{3} - 8\beta_{2} - 2\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 42 \beta_{15} - 70 \beta_{14} + 113 \beta_{13} + 172 \beta_{12} - 47 \beta_{11} + 107 \beta_{10} + \cdots - 30 \beta_{8} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 35\beta_{7} - 43\beta_{6} + 122\beta_{5} + 51\beta_{4} - 201\beta_{3} - 59\beta_{2} + 153\beta _1 + 47 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 426 \beta_{15} - 551 \beta_{14} + 1297 \beta_{13} + 1505 \beta_{12} + 500 \beta_{11} + \cdots + 282 \beta_{8} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 467\beta_{7} - 558\beta_{6} + 1190\beta_{5} - 283\beta_{4} - 1357\beta_{3} + 171\beta_{2} + 1095\beta _1 - 877 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4584 \beta_{15} - 2647 \beta_{14} + 13820 \beta_{13} + 8443 \beta_{12} + 10363 \beta_{11} + \cdots + 3798 \beta_{8} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 956\beta_{7} - 3729\beta_{6} + 8186\beta_{5} + 2086\beta_{4} - 4828\beta_{3} + 3972\beta_{2} + 7798\beta _1 - 15904 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 26400 \beta_{15} - 18986 \beta_{14} + 81415 \beta_{13} + 38384 \beta_{12} + 102911 \beta_{11} + \cdots + 35310 \beta_{8} ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 8665 \beta_{7} - 30325 \beta_{6} + 34100 \beta_{5} - 9793 \beta_{4} + 28127 \beta_{3} + 50905 \beta_{2} + \cdots - 151081 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 68334 \beta_{15} + 1709 \beta_{14} + 341663 \beta_{13} - 100355 \beta_{12} + 1023928 \beta_{11} + \cdots + 338424 \beta_{8} ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 190443 \beta_{7} - 151888 \beta_{6} - 112788 \beta_{5} - 268045 \beta_{4} + 794453 \beta_{3} + \cdots - 1340655 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 505722 \beta_{15} + 1586233 \beta_{14} - 789548 \beta_{13} - 5325457 \beta_{12} + \cdots + 2408400 \beta_{8} ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
719.1
2.90064 − 0.687839i
−2.90064 + 0.687839i
−1.23225 − 2.03500i
1.23225 + 2.03500i
−0.549213 + 2.04544i
0.549213 − 2.04544i
0.127353 + 1.39709i
−0.127353 − 1.39709i
2.90064 + 0.687839i
−2.90064 − 0.687839i
−1.23225 + 2.03500i
1.23225 − 2.03500i
−0.549213 − 2.04544i
0.549213 + 2.04544i
0.127353 − 1.39709i
−0.127353 + 1.39709i
0 0 0 −1.44342 + 1.70779i 0 −1.71352 − 2.96790i 0 0 0
719.2 0 0 0 −1.44342 + 1.70779i 0 1.71352 + 2.96790i 0 0 0
719.3 0 0 0 0.170177 + 2.22958i 0 −0.252704 − 0.437696i 0 0 0
719.4 0 0 0 0.170177 + 2.22958i 0 0.252704 + 0.437696i 0 0 0
719.5 0 0 0 0.757279 − 2.10393i 0 −1.71352 − 2.96790i 0 0 0
719.6 0 0 0 0.757279 − 2.10393i 0 1.71352 + 2.96790i 0 0 0
719.7 0 0 0 2.01596 − 0.967414i 0 −0.252704 − 0.437696i 0 0 0
719.8 0 0 0 2.01596 − 0.967414i 0 0.252704 + 0.437696i 0 0 0
1439.1 0 0 0 −1.44342 − 1.70779i 0 −1.71352 + 2.96790i 0 0 0
1439.2 0 0 0 −1.44342 − 1.70779i 0 1.71352 − 2.96790i 0 0 0
1439.3 0 0 0 0.170177 − 2.22958i 0 −0.252704 + 0.437696i 0 0 0
1439.4 0 0 0 0.170177 − 2.22958i 0 0.252704 − 0.437696i 0 0 0
1439.5 0 0 0 0.757279 + 2.10393i 0 −1.71352 + 2.96790i 0 0 0
1439.6 0 0 0 0.757279 + 2.10393i 0 1.71352 − 2.96790i 0 0 0
1439.7 0 0 0 2.01596 + 0.967414i 0 −0.252704 + 0.437696i 0 0 0
1439.8 0 0 0 2.01596 + 0.967414i 0 0.252704 − 0.437696i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 719.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
9.d odd 6 1 inner
20.d odd 2 1 inner
36.h even 6 1 inner
45.h odd 6 1 inner
180.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.2.br.b 16
3.b odd 2 1 720.2.br.b ✓ 16
4.b odd 2 1 inner 2160.2.br.b 16
5.b even 2 1 inner 2160.2.br.b 16
9.c even 3 1 720.2.br.b ✓ 16
9.d odd 6 1 inner 2160.2.br.b 16
12.b even 2 1 720.2.br.b ✓ 16
15.d odd 2 1 720.2.br.b ✓ 16
20.d odd 2 1 inner 2160.2.br.b 16
36.f odd 6 1 720.2.br.b ✓ 16
36.h even 6 1 inner 2160.2.br.b 16
45.h odd 6 1 inner 2160.2.br.b 16
45.j even 6 1 720.2.br.b ✓ 16
60.h even 2 1 720.2.br.b ✓ 16
180.n even 6 1 inner 2160.2.br.b 16
180.p odd 6 1 720.2.br.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.br.b ✓ 16 3.b odd 2 1
720.2.br.b ✓ 16 9.c even 3 1
720.2.br.b ✓ 16 12.b even 2 1
720.2.br.b ✓ 16 15.d odd 2 1
720.2.br.b ✓ 16 36.f odd 6 1
720.2.br.b ✓ 16 45.j even 6 1
720.2.br.b ✓ 16 60.h even 2 1
720.2.br.b ✓ 16 180.p odd 6 1
2160.2.br.b 16 1.a even 1 1 trivial
2160.2.br.b 16 4.b odd 2 1 inner
2160.2.br.b 16 5.b even 2 1 inner
2160.2.br.b 16 9.d odd 6 1 inner
2160.2.br.b 16 20.d odd 2 1 inner
2160.2.br.b 16 36.h even 6 1 inner
2160.2.br.b 16 45.h odd 6 1 inner
2160.2.br.b 16 180.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2160, [\chi])\):

\( T_{7}^{8} + 12T_{7}^{6} + 141T_{7}^{4} + 36T_{7}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{8} + 33T_{11}^{6} + 891T_{11}^{4} + 6534T_{11}^{2} + 39204 \) 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
$5$ \( (T^{8} - 3 T^{7} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 12 T^{6} + 141 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 33 T^{6} + \cdots + 39204)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 33 T^{6} + \cdots + 4356)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 66 T^{2} + 1056)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 66 T^{2} + 792)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} - 60 T^{6} + \cdots + 751689)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 12 T^{3} + 49 T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 99 T^{6} + \cdots + 3175524)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 66 T^{2} + 264)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 12 T^{3} + 49 T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 27 T^{6} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 12 T^{6} + 141 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( (T^{8} + 231 T^{6} + \cdots + 160579584)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 4 T^{3} + \cdots + 841)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 180 T^{6} + \cdots + 16867449)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 198 T^{2} + 7128)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 132 T^{2} + 1056)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} - 165 T^{6} + \cdots + 10036224)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 276 T^{6} + \cdots + 181360089)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 343 T^{2} + 9604)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} - 99 T^{6} + \cdots + 5645376)^{2} \) Copy content Toggle raw display
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