↑

Properties

Label 2768.2.b.d
Level $2768$
Weight $2$
Character orbit 2768.b
Analytic conductor $22.103$
Analytic rank $0$
Dimension $14$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2768,2,Mod(1729,2768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2768.1729");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2768 = 2^{4} \cdot 173 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2768.b (of order \(2\), degree \(1\), 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: \(22.1025912795\)
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: \( x^{14} + 25x^{12} + 234x^{10} + 1068x^{8} + 2520x^{6} + 2913x^{4} + 1297x^{2} + 43 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 692)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{4} q^{5} - \beta_{3} q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{4} q^{5} - \beta_{3} q^{7} + (\beta_{2} - 1) q^{9} + \beta_{7} q^{11} - \beta_{9} q^{13} + \beta_{8} q^{15} - \beta_{10} q^{17} - \beta_{12} q^{19} + ( - \beta_{11} + \beta_{9} + \cdots + \beta_{2}) q^{21}+ \cdots + ( - \beta_{10} + \beta_{7} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 8 q^{9} - 4 q^{13} + 4 q^{15} + 2 q^{21} + 6 q^{23} - 8 q^{25} + 4 q^{29} - 10 q^{31} + 10 q^{33} + 2 q^{35} - 14 q^{37} + 8 q^{41} + 8 q^{43} - 20 q^{47} - 10 q^{49} - 4 q^{51} - 4 q^{55} - 8 q^{57} - 12 q^{67} + 20 q^{73} + 2 q^{77} - 10 q^{81} + 44 q^{83} - 18 q^{85} + 18 q^{89} + 18 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 25x^{12} + 234x^{10} + 1068x^{8} + 2520x^{6} + 2913x^{4} + 1297x^{2} + 43 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{13} - 24\nu^{11} + 205\nu^{9} + 4224\nu^{7} + 21995\nu^{5} + 43608\nu^{3} + 25430\nu ) / 1083 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 33\nu^{13} + 757\nu^{11} + 6184\nu^{9} + 23081\nu^{7} + 41222\nu^{5} + 35680\nu^{3} + 16132\nu ) / 1083 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{12} + 36\nu^{10} - 127\nu^{8} - 2726\nu^{6} - 10069\nu^{4} - 11984\nu^{2} - 3128 ) / 361 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 33\nu^{13} + 757\nu^{11} + 6184\nu^{9} + 23081\nu^{7} + 41222\nu^{5} + 36763\nu^{3} + 22630\nu ) / 1083 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\nu^{13} + 132\nu^{11} - 586\nu^{9} - 12402\nu^{7} - 52202\nu^{5} - 79560\nu^{3} - 35897\nu ) / 1083 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 68\nu^{12} + 1538\nu^{10} + 12163\nu^{8} + 41938\nu^{6} + 60449\nu^{4} + 26669\nu^{2} + 1419 ) / 1083 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 113\nu^{12} + 2439\nu^{10} + 18200\nu^{8} + 60252\nu^{6} + 89914\nu^{4} + 50874\nu^{2} + 3595 ) / 1083 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 125\nu^{13} + 2944\nu^{11} + 24912\nu^{9} + 96278\nu^{7} + 172017\nu^{5} + 120985\nu^{3} + 15992\nu ) / 1083 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 148\nu^{12} + 3220\nu^{10} + 24179\nu^{8} + 79109\nu^{6} + 109141\nu^{4} + 44029\nu^{2} - 1371 ) / 1083 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 238\nu^{13} + 5383\nu^{11} + 43112\nu^{9} + 156530\nu^{7} + 261931\nu^{5} + 172942\nu^{3} + 25002\nu ) / 1083 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -282\nu^{12} - 6272\nu^{10} - 48710\nu^{8} - 167209\nu^{6} - 250951\nu^{4} - 129062\nu^{2} - 5237 ) / 1083 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{4} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - \beta_{11} + 2\beta_{9} + 3\beta_{8} - 8\beta_{2} + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{10} - \beta_{7} - 13\beta_{6} + 17\beta_{4} - 2\beta_{3} + 47\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{13} + 9\beta_{11} - 26\beta_{9} - 43\beta_{8} + 2\beta_{5} + 74\beta_{2} - 202 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{12} + 17\beta_{10} + 18\beta_{7} + 143\beta_{6} - 204\beta_{4} + 36\beta_{3} - 426\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 193\beta_{13} - 71\beta_{11} + 284\beta_{9} + 488\beta_{8} - 38\beta_{5} - 733\beta_{2} + 1864 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 20\beta_{12} - 213\beta_{10} - 231\beta_{7} - 1505\beta_{6} + 2217\beta_{4} - 455\beta_{3} + 4135\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -2122\beta_{13} + 586\beta_{11} - 2981\beta_{9} - 5187\beta_{8} + 493\beta_{5} + 7425\beta_{2} - 18256 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -273\beta_{12} + 2395\beta_{10} + 2615\beta_{7} + 15593\beta_{6} - 23259\beta_{4} + 5094\beta_{3} - 41367\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 22452\beta_{13} - 5216\beta_{11} + 30882\beta_{9} + 53899\beta_{8} - 5587\beta_{5} - 75745\beta_{2} + 183403 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3214 \beta_{12} - 25666 \beta_{10} - 28039 \beta_{7} - 160526 \beta_{6} + 240656 \beta_{4} + \cdots + 419303 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2768\mathbb{Z}\right)^\times\).

\(n\) \(693\) \(1039\) \(1905\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1729.1
− 3.20019i
− 2.19264i
− 1.98288i
− 1.66595i
− 1.48112i
− 1.00709i
− 0.189658i
0.189658i
1.00709i
1.48112i
1.66595i
1.98288i
2.19264i
3.20019i
0 − 3.20019i 0 2.04593i 0 − 0.0881275i 0 −7.24119 0
1729.2 0 − 2.19264i 0 − 1.04827i 0 2.52149i 0 −1.80767 0
1729.3 0 − 1.98288i 0 − 2.42785i 0 2.57203i 0 −0.931802 0
1729.4 0 − 1.66595i 0 − 2.24682i 0 − 1.96387i 0 0.224594 0
1729.5 0 − 1.48112i 0 1.37407i 0 − 4.42263i 0 0.806272 0
1729.6 0 − 1.00709i 0 3.75168i 0 − 0.315748i 0 1.98577 0
1729.7 0 − 0.189658i 0 2.60949i 0 4.18364i 0 2.96403 0
1729.8 0 0.189658i 0 − 2.60949i 0 − 4.18364i 0 2.96403 0
1729.9 0 1.00709i 0 − 3.75168i 0 0.315748i 0 1.98577 0
1729.10 0 1.48112i 0 − 1.37407i 0 4.42263i 0 0.806272 0
1729.11 0 1.66595i 0 2.24682i 0 1.96387i 0 0.224594 0
1729.12 0 1.98288i 0 2.42785i 0 − 2.57203i 0 −0.931802 0
1729.13 0 2.19264i 0 1.04827i 0 − 2.52149i 0 −1.80767 0
1729.14 0 3.20019i 0 − 2.04593i 0 0.0881275i 0 −7.24119 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1729.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2768.2.b.d 14
4.b odd 2 1 692.2.b.a ✓ 14
12.b even 2 1 6228.2.b.d 14
173.b even 2 1 inner 2768.2.b.d 14
692.d odd 2 1 692.2.b.a ✓ 14
2076.h even 2 1 6228.2.b.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
692.2.b.a ✓ 14 4.b odd 2 1
692.2.b.a ✓ 14 692.d odd 2 1
2768.2.b.d 14 1.a even 1 1 trivial
2768.2.b.d 14 173.b even 2 1 inner
6228.2.b.d 14 12.b even 2 1
6228.2.b.d 14 2076.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 25T_{3}^{12} + 234T_{3}^{10} + 1068T_{3}^{8} + 2520T_{3}^{6} + 2913T_{3}^{4} + 1297T_{3}^{2} + 43 \) acting on \(S_{2}^{\mathrm{new}}(2768, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 25 T^{12} + \cdots + 43 \) Copy content Toggle raw display
$5$ \( T^{14} + 39 T^{12} + \cdots + 24768 \) Copy content Toggle raw display
$7$ \( T^{14} + 54 T^{12} + \cdots + 43 \) Copy content Toggle raw display
$11$ \( T^{14} + 92 T^{12} + \cdots + 139707 \) Copy content Toggle raw display
$13$ \( (T^{7} + 2 T^{6} + \cdots + 683)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 129588928 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 1287069163 \) Copy content Toggle raw display
$23$ \( (T^{7} - 3 T^{6} - 48 T^{5} + \cdots - 56)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} - 2 T^{6} + \cdots - 11637)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + 5 T^{6} + \cdots - 2616)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} + 7 T^{6} + \cdots + 2223)^{2} \) Copy content Toggle raw display
$41$ \( (T^{7} - 4 T^{6} + \cdots - 363)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} - 4 T^{6} - 158 T^{5} + \cdots + 24)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} + 10 T^{6} + \cdots - 18624)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 28587030208 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 161168128 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 9737019072 \) Copy content Toggle raw display
$67$ \( (T^{7} + 6 T^{6} + \cdots - 3974008)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 44007235483 \) Copy content Toggle raw display
$73$ \( (T^{7} - 10 T^{6} + \cdots - 3642551)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 2037154627 \) Copy content Toggle raw display
$83$ \( (T^{7} - 22 T^{6} + \cdots + 535944)^{2} \) Copy content Toggle raw display
$89$ \( (T^{7} - 9 T^{6} + \cdots + 743927)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 141990785728 \) Copy content Toggle raw display
show more
show less