Properties

Label 2760.2.k.c
Level $2760$
Weight $2$
Character orbit 2760.k
Analytic conductor $22.039$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 31x^{10} + 359x^{8} + 1957x^{6} + 5132x^{4} + 5744x^{2} + 1600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} - \beta_{10} q^{5} + \beta_1 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} - \beta_{10} q^{5} + \beta_1 q^{7} - q^{9} + \beta_{2} q^{11} - \beta_{7} q^{13} + \beta_{4} q^{15} + ( - \beta_{10} - \beta_{9} + \beta_{7} - 2 \beta_{6} + \beta_1) q^{17} + (\beta_{10} - \beta_{9} + \beta_{8} + \beta_{4} + \beta_{3} + 2) q^{19} - \beta_{5} q^{21} - \beta_{6} q^{23} + ( - 2 \beta_{9} - 2 \beta_{6} + \beta_{4} + \beta_{3} + 1) q^{25} - \beta_{6} q^{27} + (\beta_{8} - \beta_{5} + \beta_{4} + \beta_{3} + 2) q^{29} + (\beta_{10} - \beta_{9} - \beta_{8} + 3 \beta_{5} - \beta_{2} - 2) q^{31} + \beta_{11} q^{33} + ( - \beta_{11} - \beta_{8} - \beta_{5}) q^{35} + (\beta_{11} - \beta_{10} - \beta_{9} - \beta_{7} + 2 \beta_{6} - \beta_1) q^{37} - \beta_{8} q^{39} + ( - 2 \beta_{8} + 3 \beta_{5} - \beta_{2} - 4) q^{41} + ( - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{4} + \beta_{3}) q^{43} + \beta_{10} q^{45} + (\beta_{11} - \beta_{10} - \beta_{9} + 2 \beta_{6} - 2 \beta_1) q^{47} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 3) q^{49} + (\beta_{8} - \beta_{5} + \beta_{4} + \beta_{3} + 2) q^{51} + ( - \beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} - \beta_1) q^{53} + (\beta_{8} + \beta_{7} - 2 \beta_{5} - \beta_{2}) q^{55} + (\beta_{10} + \beta_{9} - \beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{3}) q^{57} + ( - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{59} + (\beta_{8} + \beta_{4} + \beta_{3} + 2 \beta_{2} - 4) q^{61} - \beta_1 q^{63} + (\beta_{11} - \beta_{7} - 2 \beta_{5} + \beta_{2}) q^{65} + ( - \beta_{11} + 3 \beta_{7} + 4 \beta_{6} - \beta_{4} + \beta_{3} - \beta_1) q^{67} + q^{69} + ( - \beta_{10} + \beta_{9} + \beta_{5} - \beta_{2} + 4) q^{71} + ( - 2 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + 3 \beta_{7} - \beta_{4} + \beta_{3}) q^{73} + (\beta_{10} + \beta_{9} + \beta_{6} + 2 \beta_{3} + 2) q^{75} + (\beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3}) q^{77} + ( - \beta_{10} + \beta_{9} + 2 \beta_{8} - 4) q^{79} + q^{81} + ( - \beta_{7} - 6 \beta_{6} + \beta_1) q^{83} + ( - 2 \beta_{11} - 2 \beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + \cdots - 4) q^{85}+ \cdots - \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{9} - 4 q^{15} + 12 q^{19} - 6 q^{21} + 4 q^{25} + 6 q^{29} - 2 q^{31} - 2 q^{35} + 4 q^{39} - 22 q^{41} + 22 q^{49} + 6 q^{51} - 16 q^{55} + 18 q^{59} - 60 q^{61} - 12 q^{65} + 12 q^{69} + 54 q^{71} + 16 q^{75} - 56 q^{79} + 12 q^{81} - 38 q^{85} + 28 q^{89} - 12 q^{91} - 52 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 31x^{10} + 359x^{8} + 1957x^{6} + 5132x^{4} + 5744x^{2} + 1600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{10} - 29\nu^{8} - 291\nu^{6} - 1175\nu^{4} - 1632\nu^{2} - 400 ) / 80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{11} - 6 \nu^{10} - 27 \nu^{9} - 174 \nu^{8} - 243 \nu^{7} - 1786 \nu^{6} - 793 \nu^{5} - 7730 \nu^{4} - 432 \nu^{3} - 12352 \nu^{2} + 784 \nu - 3360 ) / 320 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 6 \nu^{10} + 27 \nu^{9} - 174 \nu^{8} + 243 \nu^{7} - 1786 \nu^{6} + 793 \nu^{5} - 7730 \nu^{4} + 432 \nu^{3} - 12352 \nu^{2} - 784 \nu - 3360 ) / 320 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{10} - 87\nu^{8} - 893\nu^{6} - 3865\nu^{4} - 6256\nu^{2} - 2000 ) / 80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{11} - 143\nu^{9} - 1447\nu^{7} - 6213\nu^{5} - 10200\nu^{3} - 3696\nu ) / 320 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{11} - 87\nu^{9} - 893\nu^{7} - 3865\nu^{5} - 6176\nu^{3} - 1520\nu ) / 160 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6\nu^{10} + 169\nu^{8} + 1676\nu^{6} + 7025\nu^{4} + 11192\nu^{2} + 3600 ) / 80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2 \nu^{11} - 9 \nu^{10} - 56 \nu^{9} - 251 \nu^{8} - 554 \nu^{7} - 2459 \nu^{6} - 2348 \nu^{5} - 10145 \nu^{4} - 3944 \nu^{3} - 15688 \nu^{2} - 1616 \nu - 4560 ) / 160 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2 \nu^{11} + 9 \nu^{10} - 56 \nu^{9} + 251 \nu^{8} - 554 \nu^{7} + 2459 \nu^{6} - 2348 \nu^{5} + 10145 \nu^{4} - 3944 \nu^{3} + 15688 \nu^{2} - 1616 \nu + 4560 ) / 160 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 9\nu^{11} + 251\nu^{9} + 2459\nu^{7} + 10145\nu^{5} + 15688\nu^{3} + 4560\nu ) / 160 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{9} + 2\beta_{7} - 4\beta_{6} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{10} + 2\beta_{9} + 4\beta_{8} + 13\beta_{5} - 11\beta_{4} - 11\beta_{3} + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{11} - 21\beta_{10} - 21\beta_{9} - 26\beta_{7} + 52\beta_{6} + 4\beta_{4} - 4\beta_{3} + 63\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 34\beta_{10} - 34\beta_{9} - 68\beta_{8} - 157\beta_{5} + 119\beta_{4} + 119\beta_{3} + 12\beta_{2} - 244 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 80\beta_{11} + 305\beta_{10} + 305\beta_{9} + 306\beta_{7} - 604\beta_{6} - 92\beta_{4} + 92\beta_{3} - 639\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -466\beta_{10} + 466\beta_{9} + 916\beta_{8} + 1853\beta_{5} - 1331\beta_{4} - 1331\beta_{3} - 264\beta_{2} + 2396 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1196 \beta_{11} - 3949 \beta_{10} - 3949 \beta_{9} - 3578 \beta_{7} + 6884 \beta_{6} + 1444 \beta_{4} - 1444 \beta_{3} + 6911 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5970 \beta_{10} - 5970 \beta_{9} - 11476 \beta_{8} - 21693 \beta_{5} + 15263 \beta_{4} + 15263 \beta_{3} + 4084 \beta_{2} - 25252 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 16024 \beta_{11} + 48729 \beta_{10} + 48729 \beta_{9} + 42002 \beta_{7} - 78604 \beta_{6} - 19644 \beta_{4} + 19644 \beta_{3} - 77471 \beta_1 \) 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
2.70731i
1.70731i
3.43090i
2.43090i
1.63473i
0.634730i
1.70731i
2.70731i
2.43090i
3.43090i
0.634730i
1.63473i
0 1.00000i 0 −2.21432 0.311108i 0 2.70731i 0 −1.00000 0
2209.2 0 1.00000i 0 −2.21432 0.311108i 0 1.70731i 0 −1.00000 0
2209.3 0 1.00000i 0 0.539189 2.17009i 0 3.43090i 0 −1.00000 0
2209.4 0 1.00000i 0 0.539189 2.17009i 0 2.43090i 0 −1.00000 0
2209.5 0 1.00000i 0 1.67513 + 1.48119i 0 1.63473i 0 −1.00000 0
2209.6 0 1.00000i 0 1.67513 + 1.48119i 0 0.634730i 0 −1.00000 0
2209.7 0 1.00000i 0 −2.21432 + 0.311108i 0 1.70731i 0 −1.00000 0
2209.8 0 1.00000i 0 −2.21432 + 0.311108i 0 2.70731i 0 −1.00000 0
2209.9 0 1.00000i 0 0.539189 + 2.17009i 0 2.43090i 0 −1.00000 0
2209.10 0 1.00000i 0 0.539189 + 2.17009i 0 3.43090i 0 −1.00000 0
2209.11 0 1.00000i 0 1.67513 1.48119i 0 0.634730i 0 −1.00000 0
2209.12 0 1.00000i 0 1.67513 1.48119i 0 1.63473i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2209.12
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.c 12
5.b even 2 1 inner 2760.2.k.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.k.c 12 1.a even 1 1 trivial
2760.2.k.c 12 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 31T_{7}^{10} + 359T_{7}^{8} + 1957T_{7}^{6} + 5132T_{7}^{4} + 5744T_{7}^{2} + 1600 \) acting on \(S_{2}^{\mathrm{new}}(2760, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} - T^{4} + 16 T^{3} - 5 T^{2} + 125)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 31 T^{10} + 359 T^{8} + \cdots + 1600 \) Copy content Toggle raw display
$11$ \( (T^{6} - 32 T^{4} - 46 T^{3} + 120 T^{2} + \cdots - 160)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 48 T^{10} + 844 T^{8} + \cdots + 25600 \) Copy content Toggle raw display
$17$ \( T^{12} + 91 T^{10} + 2567 T^{8} + \cdots + 118336 \) Copy content Toggle raw display
$19$ \( (T^{6} - 6 T^{5} - 42 T^{4} + 266 T^{3} + \cdots + 800)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$29$ \( (T^{6} - 3 T^{5} - 41 T^{4} + 3 T^{3} + \cdots + 344)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + T^{5} - 141 T^{4} + 195 T^{3} + \cdots - 1600)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 191 T^{10} + \cdots + 30913600 \) Copy content Toggle raw display
$41$ \( (T^{6} + 11 T^{5} - 115 T^{4} - 1645 T^{3} + \cdots - 1336)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 188 T^{10} + 10384 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$47$ \( T^{12} + 228 T^{10} + 12472 T^{8} + \cdots + 4194304 \) Copy content Toggle raw display
$53$ \( T^{12} + 431 T^{10} + \cdots + 13601344 \) Copy content Toggle raw display
$59$ \( (T^{6} - 9 T^{5} - 145 T^{4} + 1221 T^{3} + \cdots + 5000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 30 T^{5} + 246 T^{4} + \cdots + 117200)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 623 T^{10} + \cdots + 217581199936 \) Copy content Toggle raw display
$71$ \( (T^{6} - 27 T^{5} + 217 T^{4} - 251 T^{3} + \cdots - 5608)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 608 T^{10} + \cdots + 89718784 \) Copy content Toggle raw display
$79$ \( (T^{6} + 28 T^{5} + 208 T^{4} + \cdots + 71168)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 295 T^{10} + \cdots + 547185664 \) Copy content Toggle raw display
$89$ \( (T^{6} - 14 T^{5} - 204 T^{4} + \cdots - 405760)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 592 T^{10} + \cdots + 876160000 \) Copy content Toggle raw display
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