Properties

Label 2760.2.k.e
Level $2760$
Weight $2$
Character orbit 2760.k
Analytic conductor $22.039$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2760,2,Mod(2209,2760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2760, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2760.2209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2760.k (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: \(22.0387109579\)
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: \( x^{16} - 2 x^{15} + 2 x^{14} + 141 x^{12} - 298 x^{11} + 314 x^{10} + 144 x^{9} + 4788 x^{8} - 10568 x^{7} + 11688 x^{6} + 8928 x^{5} + 8320 x^{4} - 18624 x^{3} + 28800 x^{2} + \cdots + 20736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_{5} q^{5} + \beta_1 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_{5} q^{5} + \beta_1 q^{7} - q^{9} + \beta_{14} q^{11} - \beta_{15} q^{13} + \beta_{4} q^{15} + ( - \beta_{15} + \beta_1) q^{17} + (\beta_{11} + \beta_{7} - \beta_{5}) q^{19} + \beta_{8} q^{21} - \beta_{3} q^{23} + (\beta_{14} - \beta_{12} - \beta_{11} - \beta_{10} - \beta_{8} + \beta_{5} - \beta_{4} + \beta_{2}) q^{25} - \beta_{3} q^{27} + (\beta_{13} - \beta_{12} + \beta_{9} - \beta_{7} - \beta_{6} + 1) q^{29} + ( - \beta_{14} + \beta_{11} + \beta_{8} + \beta_{7} - \beta_{5} + 2) q^{31} - \beta_{10} q^{33} + (\beta_{15} - \beta_{11} + \beta_{10} - \beta_{8} + 2 \beta_{6} + \beta_{3} - \beta_{2} - \beta_1) q^{35} + ( - \beta_{13} - \beta_{12} - \beta_{10} + \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2}) q^{37} + \beta_{11} q^{39} + ( - \beta_{14} + \beta_{8} + \beta_{6} - \beta_{4} - 2) q^{41} + ( - 2 \beta_{15} - \beta_{10} + \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{43} + \beta_{5} q^{45} + ( - \beta_{15} + \beta_{10} + 2 \beta_{6} + 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{47} + ( - 2 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + \beta_{11} - \beta_{9} + 2 \beta_{7} + 3 \beta_{6} + \cdots - 2) q^{49}+ \cdots - \beta_{14} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{5} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{5} - 16 q^{9} - 2 q^{15} - 8 q^{19} + 6 q^{21} + 22 q^{29} + 30 q^{31} + 2 q^{35} - 4 q^{39} - 22 q^{41} + 2 q^{45} - 38 q^{49} + 2 q^{51} + 12 q^{55} - 6 q^{59} - 4 q^{61} + 20 q^{65} + 16 q^{69} - 10 q^{71} - 4 q^{75} - 84 q^{79} + 16 q^{81} + 22 q^{85} + 16 q^{89} + 36 q^{91} + 76 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} + 141 x^{12} - 298 x^{11} + 314 x^{10} + 144 x^{9} + 4788 x^{8} - 10568 x^{7} + 11688 x^{6} + 8928 x^{5} + 8320 x^{4} - 18624 x^{3} + 28800 x^{2} + \cdots + 20736 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 47\!\cdots\!29 \nu^{15} + \cdots + 17\!\cdots\!64 ) / 44\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 22\!\cdots\!20 \nu^{15} + \cdots + 11\!\cdots\!08 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 58\!\cdots\!49 \nu^{15} + \cdots + 10\!\cdots\!00 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 34\!\cdots\!37 \nu^{15} + \cdots + 19\!\cdots\!80 ) / 44\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 919401865937407 \nu^{15} + \cdots + 84\!\cdots\!04 ) / 99\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 21\!\cdots\!91 \nu^{15} + \cdots - 86\!\cdots\!36 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16\!\cdots\!09 \nu^{15} + \cdots + 20\!\cdots\!36 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12\!\cdots\!84 \nu^{15} + \cdots + 23\!\cdots\!96 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 65\!\cdots\!94 \nu^{15} + \cdots - 76\!\cdots\!32 ) / 44\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 22\!\cdots\!51 \nu^{15} + 419250683036573 \nu^{14} + \cdots + 46\!\cdots\!60 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 68\!\cdots\!68 \nu^{15} + \cdots - 19\!\cdots\!76 ) / 44\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 96\!\cdots\!83 \nu^{15} + \cdots + 21\!\cdots\!08 ) / 44\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 48\!\cdots\!51 \nu^{15} + \cdots - 66\!\cdots\!84 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11\!\cdots\!83 \nu^{15} + \cdots - 10\!\cdots\!68 ) / 44\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 12\!\cdots\!07 \nu^{15} + \cdots - 20\!\cdots\!60 ) / 49\!\cdots\!60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{15} + \beta_{13} + \beta_{12} + 2\beta_{10} + \beta_{7} + \beta_{4} - 10\beta_{3} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{15} - 2 \beta_{12} - \beta_{11} - \beta_{9} + \beta_{8} + 6 \beta_{7} + 8 \beta_{6} - 8 \beta_{5} + 6 \beta_{4} + \beta_{3} - \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 18 \beta_{14} - 9 \beta_{13} + 9 \beta_{12} + 18 \beta_{11} + 22 \beta_{8} + 7 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 7 \beta_{4} - 74 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7 \beta_{15} - 4 \beta_{14} + 18 \beta_{13} - 7 \beta_{11} - 4 \beta_{10} - 15 \beta_{9} + 11 \beta_{8} + 70 \beta_{7} - 66 \beta_{6} - 66 \beta_{5} - 70 \beta_{4} - 23 \beta_{3} + 15 \beta_{2} + 11 \beta _1 + 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 146 \beta_{15} - 81 \beta_{13} - 81 \beta_{12} - 146 \beta_{10} - 47 \beta_{7} + 34 \beta_{6} + 34 \beta_{5} - 47 \beta_{4} + 610 \beta_{3} + 206 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 39 \beta_{15} + 60 \beta_{14} + 146 \beta_{12} + 39 \beta_{11} - 60 \beta_{10} + 175 \beta_{9} - 91 \beta_{8} - 412 \beta_{7} - 626 \beta_{6} + 626 \beta_{5} - 412 \beta_{4} - 295 \beta_{3} + 175 \beta_{2} + 91 \beta _1 - 295 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1162 \beta_{14} + 741 \beta_{13} - 741 \beta_{12} - 1178 \beta_{11} - 16 \beta_{9} - 1862 \beta_{8} - 307 \beta_{7} - 418 \beta_{6} - 434 \beta_{5} - 323 \beta_{4} + 5242 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 171 \beta_{15} + 684 \beta_{14} - 1194 \beta_{13} + 171 \beta_{11} + 684 \beta_{10} + 1859 \beta_{9} - 671 \beta_{8} - 5634 \beta_{7} + 4830 \beta_{6} + 4830 \beta_{5} + 5634 \beta_{4} + 3275 \beta_{3} + \cdots - 3275 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 9586 \beta_{15} + 6809 \beta_{13} + 6809 \beta_{12} + 9234 \beta_{10} + 2287 \beta_{7} - 4986 \beta_{6} - 4522 \beta_{5} + 1823 \beta_{4} - 46066 \beta_{3} + 416 \beta_{2} - 16718 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 279 \beta_{15} - 7084 \beta_{14} - 10114 \beta_{12} - 279 \beta_{11} + 7084 \beta_{10} - 18815 \beta_{9} + 4507 \beta_{8} + 32548 \beta_{7} + 50874 \beta_{6} - 50874 \beta_{5} + 32548 \beta_{4} + \cdots + 34423 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 73690 \beta_{14} - 62605 \beta_{13} + 62605 \beta_{12} + 78826 \beta_{11} + 7056 \beta_{9} + 150294 \beta_{8} + 8267 \beta_{7} + 45842 \beta_{6} + 54338 \beta_{5} + 16763 \beta_{4} + \cdots - 410922 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 6045 \beta_{15} - 70028 \beta_{14} + 89242 \beta_{13} + 6045 \beta_{11} - 70028 \beta_{10} - 185051 \beta_{9} + 26295 \beta_{8} + 460674 \beta_{7} - 383102 \beta_{6} - 383102 \beta_{5} + \cdots + 352323 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 654978 \beta_{15} - 575697 \beta_{13} - 575697 \beta_{12} - 591746 \beta_{10} - 128023 \beta_{7} + 574026 \beta_{6} + 447674 \beta_{5} - 1671 \beta_{4} + 3706946 \beta_{3} + \cdots + 1355710 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 110657 \beta_{15} + 673516 \beta_{14} + 817234 \beta_{12} - 110657 \beta_{11} - 673516 \beta_{10} + 1787335 \beta_{9} - 106851 \beta_{8} - 2668676 \beta_{7} - 4182986 \beta_{6} + \cdots - 3551007 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2760\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1657\) \(1841\) \(2071\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2209.1
2.09040 + 2.09040i
1.83968 + 1.83968i
1.27105 + 1.27105i
1.16408 + 1.16408i
−0.477740 0.477740i
−0.793869 0.793869i
−1.90917 1.90917i
−2.18444 2.18444i
2.09040 2.09040i
1.83968 1.83968i
1.27105 1.27105i
1.16408 1.16408i
−0.477740 + 0.477740i
−0.793869 + 0.793869i
−1.90917 + 1.90917i
−2.18444 + 2.18444i
0 1.00000i 0 −2.09040 + 0.793869i 0 4.15840i 0 −1.00000 0
2209.2 0 1.00000i 0 −1.83968 1.27105i 0 0.440442i 0 −1.00000 0
2209.3 0 1.00000i 0 −1.27105 1.83968i 0 2.20929i 0 −1.00000 0
2209.4 0 1.00000i 0 −1.16408 + 1.90917i 0 5.26833i 0 −1.00000 0
2209.5 0 1.00000i 0 0.477740 + 2.18444i 0 0.612117i 0 −1.00000 0
2209.6 0 1.00000i 0 0.793869 2.09040i 0 0.418853i 0 −1.00000 0
2209.7 0 1.00000i 0 1.90917 1.16408i 0 2.97849i 0 −1.00000 0
2209.8 0 1.00000i 0 2.18444 + 0.477740i 0 3.93141i 0 −1.00000 0
2209.9 0 1.00000i 0 −2.09040 0.793869i 0 4.15840i 0 −1.00000 0
2209.10 0 1.00000i 0 −1.83968 + 1.27105i 0 0.440442i 0 −1.00000 0
2209.11 0 1.00000i 0 −1.27105 + 1.83968i 0 2.20929i 0 −1.00000 0
2209.12 0 1.00000i 0 −1.16408 1.90917i 0 5.26833i 0 −1.00000 0
2209.13 0 1.00000i 0 0.477740 2.18444i 0 0.612117i 0 −1.00000 0
2209.14 0 1.00000i 0 0.793869 + 2.09040i 0 0.418853i 0 −1.00000 0
2209.15 0 1.00000i 0 1.90917 + 1.16408i 0 2.97849i 0 −1.00000 0
2209.16 0 1.00000i 0 2.18444 0.477740i 0 3.93141i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2209.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2760.2.k.e 16
5.b even 2 1 inner 2760.2.k.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.k.e 16 1.a even 1 1 trivial
2760.2.k.e 16 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 75 T_{7}^{14} + 2107 T_{7}^{12} + 27753 T_{7}^{10} + 172808 T_{7}^{8} + 439568 T_{7}^{6} + 265728 T_{7}^{4} + 57344 T_{7}^{2} + 4096 \) acting on \(S_{2}^{\mathrm{new}}(2760, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + 2 T^{15} + 2 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 75 T^{14} + 2107 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$11$ \( (T^{8} - 42 T^{6} + 26 T^{5} + 424 T^{4} + \cdots + 1280)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 72 T^{14} + 1884 T^{12} + \cdots + 65536 \) Copy content Toggle raw display
$17$ \( T^{16} + 111 T^{14} + 4707 T^{12} + \cdots + 1327104 \) Copy content Toggle raw display
$19$ \( (T^{8} + 4 T^{7} - 64 T^{6} - 354 T^{5} + \cdots + 2272)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$29$ \( (T^{8} - 11 T^{7} - 61 T^{6} + 1079 T^{5} + \cdots - 97184)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 15 T^{7} + 31 T^{6} + 479 T^{5} + \cdots + 23040)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 219 T^{14} + 14711 T^{12} + \cdots + 4000000 \) Copy content Toggle raw display
$41$ \( (T^{8} + 11 T^{7} - 35 T^{6} - 989 T^{5} + \cdots + 2880)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 444 T^{14} + \cdots + 1459727908864 \) Copy content Toggle raw display
$47$ \( T^{16} + 508 T^{14} + \cdots + 16384000000 \) Copy content Toggle raw display
$53$ \( T^{16} + 343 T^{14} + \cdots + 14285030400 \) Copy content Toggle raw display
$59$ \( (T^{8} + 3 T^{7} - 181 T^{6} - 991 T^{5} + \cdots - 5312)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 2 T^{7} - 112 T^{6} - 18 T^{5} + \cdots - 8416)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + 223 T^{14} + \cdots + 113550336 \) Copy content Toggle raw display
$71$ \( (T^{8} + 5 T^{7} - 383 T^{6} - 2179 T^{5} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 201140469760000 \) Copy content Toggle raw display
$79$ \( (T^{8} + 42 T^{7} + 416 T^{6} + \cdots - 21994496)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 971 T^{14} + \cdots + 9836503142400 \) Copy content Toggle raw display
$89$ \( (T^{8} - 8 T^{7} - 316 T^{6} + \cdots + 4654592)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 1208 T^{14} + \cdots + 142871904256 \) Copy content Toggle raw display
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