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Properties

Label 2760.2.k.d
Level $2760$
Weight $2$
Character orbit 2760.k
Analytic conductor $22.039$
Analytic rank $0$
Dimension $14$
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + 2 x^{12} + 112 x^{10} - 228 x^{9} + 232 x^{8} + 40 x^{7} + 1316 x^{6} - 2688 x^{5} + \cdots + 128 \) 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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - \beta_{12} q^{5} + (\beta_{10} + \beta_{2}) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - \beta_{12} q^{5} + (\beta_{10} + \beta_{2}) q^{7} - q^{9} + ( - \beta_{13} + \beta_{12} - \beta_{3} + 1) q^{11} + (\beta_{11} + \beta_{6} + \cdots - \beta_{2}) q^{13}+ \cdots + (\beta_{13} - \beta_{12} + \beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 2 q^{5} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 2 q^{5} - 14 q^{9} + 8 q^{11} - 4 q^{15} + 8 q^{19} - 10 q^{21} - 10 q^{25} + 26 q^{29} - 18 q^{31} - 10 q^{35} + 12 q^{39} - 2 q^{41} - 2 q^{45} + 4 q^{49} - 6 q^{51} - 32 q^{55} - 10 q^{59} - 24 q^{61} - 36 q^{65} - 14 q^{69} - 38 q^{71} + 4 q^{79} + 14 q^{81} - 14 q^{85} - 16 q^{89} - 4 q^{91} - 8 q^{95} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 2 x^{12} + 112 x^{10} - 228 x^{9} + 232 x^{8} + 40 x^{7} + 1316 x^{6} - 2688 x^{5} + \cdots + 128 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 116988090327 \nu^{13} + 68234098889 \nu^{12} - 308655154773 \nu^{11} + \cdots - 318466493335408 ) / 128200415802260 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 194190133762 \nu^{13} - 266049636186 \nu^{12} + 132920384567 \nu^{11} + \cdots - 21707654432624 ) / 102560332641808 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2062338941532 \nu^{13} + 6062498519609 \nu^{12} - 4783436083738 \nu^{11} + \cdots - 25\!\cdots\!28 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2329442214398 \nu^{13} + 2117319031143 \nu^{12} + 850790793356 \nu^{11} + \cdots + 123956557952800 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1514451233960 \nu^{13} + 2380599533434 \nu^{12} - 1145071395649 \nu^{11} + \cdots - 695434444231776 ) / 512801663209040 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1591778632998 \nu^{13} - 4626875105067 \nu^{12} + 6362179169853 \nu^{11} + \cdots + 419216905502336 ) / 512801663209040 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3613642983413 \nu^{13} - 9640223751246 \nu^{12} + 8730556878882 \nu^{11} + \cdots - 12\!\cdots\!08 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4182638689979 \nu^{13} - 7315218283548 \nu^{12} + 10280624951786 \nu^{11} + \cdots - 22\!\cdots\!84 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4836949296793 \nu^{13} + 12194822580816 \nu^{12} - 11576272833422 \nu^{11} + \cdots + 14\!\cdots\!68 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1340208934261 \nu^{13} - 1569597711166 \nu^{12} + 237624600458 \nu^{11} + \cdots - 117033020155680 ) / 256400831604520 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3010317262932 \nu^{13} + 5789174739407 \nu^{12} - 6126710659516 \nu^{11} + \cdots + 544036069727520 ) / 512801663209040 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3043233035284 \nu^{13} + 5204248744284 \nu^{12} - 4603403990867 \nu^{11} + \cdots + 469099688649520 ) / 512801663209040 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3654886191974 \nu^{13} + 6481548159069 \nu^{12} - 6026261968137 \nu^{11} + \cdots + 593381374257200 ) / 512801663209040 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} - \beta_{12} - \beta_{9} - \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{10} - \beta_{6} - \beta_{5} + 2\beta_{4} + 8\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 4 \beta_{13} + 3 \beta_{12} + \beta_{11} - \beta_{10} - 4 \beta_{9} - \beta_{8} - 3 \beta_{7} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - \beta_{12} - \beta_{9} + 2\beta_{8} + \beta_{7} - 8\beta_{6} + 8\beta_{5} - 12\beta_{3} - 10\beta _1 - 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 21 \beta_{13} + 37 \beta_{12} - 12 \beta_{11} + 14 \beta_{10} + 21 \beta_{9} - 12 \beta_{8} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{13} + 12 \beta_{12} + 28 \beta_{11} + 98 \beta_{10} - 14 \beta_{9} - 14 \beta_{7} + \cdots - 268 \beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 356 \beta_{13} - 178 \beta_{12} - 126 \beta_{11} + 152 \beta_{10} + 356 \beta_{9} + 126 \beta_{8} + \cdots - 154 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 160 \beta_{13} + 160 \beta_{12} + 120 \beta_{9} - 312 \beta_{8} - 120 \beta_{7} + 910 \beta_{6} + \cdots + 2560 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1662 \beta_{13} - 3482 \beta_{12} + 1276 \beta_{11} - 1536 \beta_{10} - 1662 \beta_{9} + 1276 \beta_{8} + \cdots - 1608 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1152 \beta_{13} - 1152 \beta_{12} - 3256 \beta_{11} - 9404 \beta_{10} + 1736 \beta_{9} + \cdots + 25056 \beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 34284 \beta_{13} + 16136 \beta_{12} + 12772 \beta_{11} - 15148 \beta_{10} - 34284 \beta_{9} + \cdots + 16572 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 18508 \beta_{13} - 18508 \beta_{12} - 10940 \beta_{9} + 33224 \beta_{8} + 10940 \beta_{7} + \cdots - 247336 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 158940 \beta_{13} + 338452 \beta_{12} - 127336 \beta_{11} + 148080 \beta_{10} + 158940 \beta_{9} + \cdots + 170128 \) Copy content Toggle raw display

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

Character values

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

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

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
−2.23554 + 2.23554i
−1.34837 + 1.34837i
−0.371989 + 0.371989i
0.262528 − 0.262528i
1.17338 − 1.17338i
1.31001 − 1.31001i
2.20998 − 2.20998i
−2.23554 − 2.23554i
−1.34837 − 1.34837i
−0.371989 − 0.371989i
0.262528 + 0.262528i
1.17338 + 1.17338i
1.31001 + 1.31001i
2.20998 + 2.20998i
0 − 1.00000i 0 −2.23554 + 0.0487327i 0 − 0.346855i 0 −1.00000 0
2209.2 0 − 1.00000i 0 −1.34837 − 1.78378i 0 − 0.125051i 0 −1.00000 0
2209.3 0 − 1.00000i 0 −0.371989 + 2.20491i 0 − 4.05381i 0 −1.00000 0
2209.4 0 − 1.00000i 0 0.262528 − 2.22060i 0 − 3.25152i 0 −1.00000 0
2209.5 0 − 1.00000i 0 1.17338 + 1.90346i 0 1.78370i 0 −1.00000 0
2209.6 0 − 1.00000i 0 1.31001 − 1.81215i 0 − 2.34779i 0 −1.00000 0
2209.7 0 − 1.00000i 0 2.20998 − 0.340571i 0 3.34134i 0 −1.00000 0
2209.8 0 1.00000i 0 −2.23554 − 0.0487327i 0 0.346855i 0 −1.00000 0
2209.9 0 1.00000i 0 −1.34837 + 1.78378i 0 0.125051i 0 −1.00000 0
2209.10 0 1.00000i 0 −0.371989 − 2.20491i 0 4.05381i 0 −1.00000 0
2209.11 0 1.00000i 0 0.262528 + 2.22060i 0 3.25152i 0 −1.00000 0
2209.12 0 1.00000i 0 1.17338 − 1.90346i 0 − 1.78370i 0 −1.00000 0
2209.13 0 1.00000i 0 1.31001 + 1.81215i 0 2.34779i 0 −1.00000 0
2209.14 0 1.00000i 0 2.20998 + 0.340571i 0 − 3.34134i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2209.14
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.d ✓ 14
5.b even 2 1 inner 2760.2.k.d ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.k.d ✓ 14 1.a even 1 1 trivial
2760.2.k.d ✓ 14 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{14} + 47T_{7}^{12} + 831T_{7}^{10} + 6853T_{7}^{8} + 26116T_{7}^{6} + 37456T_{7}^{4} + 4672T_{7}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(2760, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} - 2 T^{13} + \cdots + 78125 \) Copy content Toggle raw display
$7$ \( T^{14} + 47 T^{12} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T^{7} - 4 T^{6} + \cdots + 512)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + 120 T^{12} + \cdots + 10240000 \) Copy content Toggle raw display
$17$ \( T^{14} + 123 T^{12} + \cdots + 8620096 \) Copy content Toggle raw display
$19$ \( (T^{7} - 4 T^{6} + \cdots + 6320)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$29$ \( (T^{7} - 13 T^{6} + \cdots + 944)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + 9 T^{6} + \cdots - 896)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + 363 T^{12} + \cdots + 34480384 \) Copy content Toggle raw display
$41$ \( (T^{7} + T^{6} + \cdots + 15056)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 66407228416 \) Copy content Toggle raw display
$47$ \( T^{14} + 228 T^{12} + \cdots + 20070400 \) Copy content Toggle raw display
$53$ \( T^{14} + 303 T^{12} + \cdots + 23116864 \) Copy content Toggle raw display
$59$ \( (T^{7} + 5 T^{6} + \cdots + 53744)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} + 12 T^{6} + \cdots - 248800)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 860872220224 \) Copy content Toggle raw display
$71$ \( (T^{7} + 19 T^{6} + \cdots + 250000)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 104857600 \) Copy content Toggle raw display
$79$ \( (T^{7} - 2 T^{6} + \cdots - 10496)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 8826978304 \) Copy content Toggle raw display
$89$ \( (T^{7} + 8 T^{6} + \cdots - 387008)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 122332057600 \) Copy content Toggle raw display
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