Properties

Label 1840.2.e.f
Level $1840$
Weight $2$
Character orbit 1840.e
Analytic conductor $14.692$
Analytic rank $0$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,2,Mod(369,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1840.369");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.e (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: \(14.6924739719\)
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: \( x^{12} + 24x^{10} + 188x^{8} + 530x^{6} + 508x^{4} + 80x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 460)
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.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{5}) q^{3} - \beta_{8} q^{5} + (\beta_{10} - \beta_{6} - \beta_1) q^{7} + (\beta_{10} + \beta_{7} + \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_{5}) q^{3} - \beta_{8} q^{5} + (\beta_{10} - \beta_{6} - \beta_1) q^{7} + (\beta_{10} + \beta_{7} + \beta_{2} - 2) q^{9} + ( - \beta_{9} + \beta_{8} + \beta_{4} + \cdots - 1) q^{11}+ \cdots + (2 \beta_{9} - 2 \beta_{8} + 4 \beta_{2} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 20 q^{9} - 4 q^{11} - 2 q^{15} + 8 q^{19} + 8 q^{25} - 10 q^{29} - 18 q^{31} + 10 q^{35} - 16 q^{39} - 2 q^{41} + 2 q^{45} - 38 q^{49} + 24 q^{51} - 16 q^{55} - 22 q^{59} - 8 q^{61} + 38 q^{65} - 8 q^{69} + 34 q^{71} - 16 q^{75} + 20 q^{79} + 28 q^{81} + 6 q^{85} + 48 q^{89} + 8 q^{91} - 12 q^{95} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 24x^{10} + 188x^{8} + 530x^{6} + 508x^{4} + 80x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{10} - 23\nu^{8} - 8\nu^{7} - 157\nu^{6} - 144\nu^{5} - 261\nu^{4} - 752\nu^{3} + 121\nu^{2} - 1064\nu + 159 ) / 96 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{10} + 69\nu^{8} + 503\nu^{6} + 1215\nu^{4} + 869\nu^{2} + 83 ) / 48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{10} + 169\nu^{8} + 1331\nu^{6} + 3779\nu^{4} + 3689\nu^{2} + 415 ) / 96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9\nu^{10} + 215\nu^{8} + 1661\nu^{6} + 4493\nu^{4} + 3895\nu^{2} + 497 ) / 96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{11} - 169\nu^{9} - 1339\nu^{7} - 3875\nu^{5} - 3913\nu^{3} - 663\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + 24\nu^{9} + 188\nu^{7} + 529\nu^{5} + 496\nu^{3} + 58\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27 \nu^{11} - \nu^{10} - 645 \nu^{9} - 23 \nu^{8} - 5007 \nu^{7} - 157 \nu^{6} - 13815 \nu^{5} + \cdots + 159 ) / 96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27 \nu^{11} + \nu^{10} + 645 \nu^{9} + 23 \nu^{8} + 5007 \nu^{7} + 173 \nu^{6} + 13815 \nu^{5} + \cdots - 143 ) / 96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 27 \nu^{11} - \nu^{10} + 645 \nu^{9} - 23 \nu^{8} + 5007 \nu^{7} - 173 \nu^{6} + 13815 \nu^{5} + \cdots + 143 ) / 96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27 \nu^{11} - \nu^{10} + 645 \nu^{9} - 23 \nu^{8} + 5007 \nu^{7} - 157 \nu^{6} + 13815 \nu^{5} + \cdots + 159 ) / 96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -37\nu^{11} - 891\nu^{9} - 7017\nu^{7} - 19993\nu^{5} - 19547\nu^{3} - 3245\nu ) / 96 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + 2\beta_{4} - 2\beta_{3} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + 4\beta_{10} - 4\beta_{9} - 4\beta_{8} - 3\beta_{7} + 2\beta_{6} + 3\beta_{5} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -9\beta_{10} - 7\beta_{9} + 7\beta_{8} - 9\beta_{7} - 22\beta_{4} + 22\beta_{3} + 2\beta_{2} + 66 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 22\beta_{11} - 73\beta_{10} + 75\beta_{9} + 75\beta_{8} + 51\beta_{7} - 56\beta_{6} - 72\beta_{5} + 22\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 46\beta_{10} + 28\beta_{9} - 28\beta_{8} + 46\beta_{7} + 110\beta_{4} - 110\beta_{3} - 12\beta_{2} - 309 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 208 \beta_{11} + 707 \beta_{10} - 731 \beta_{9} - 731 \beta_{8} - 475 \beta_{7} + 632 \beta_{6} + \cdots - 232 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 955 \beta_{10} - 483 \beta_{9} + 483 \beta_{8} - 955 \beta_{7} - 2146 \beta_{4} + 2170 \beta_{3} + \cdots + 5972 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 941 \beta_{11} - 3488 \beta_{10} + 3592 \beta_{9} + 3592 \beta_{8} + 2271 \beta_{7} - 3376 \beta_{6} + \cdots + 1217 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 9895 \beta_{10} + 4265 \beta_{9} - 4265 \beta_{8} + 9895 \beta_{7} + 20802 \beta_{4} - 21354 \beta_{3} + \cdots - 58206 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 16710 \beta_{11} + 69215 \beta_{10} - 70753 \beta_{9} - 70753 \beta_{8} - 43769 \beta_{7} + \cdots - 25446 \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
369.1
3.16223i
1.65047i
1.26443i
0.420790i
3.08006i
0.116918i
0.116918i
3.08006i
0.420790i
1.26443i
1.65047i
3.16223i
0 3.21923i 0 1.59013 1.57210i 0 2.43185i 0 −7.36343 0
369.2 0 2.80150i 0 −2.17393 0.523461i 0 4.50896i 0 −4.84843 0
369.3 0 2.40050i 0 −0.817027 + 2.08146i 0 4.41307i 0 −2.76241 0
369.4 0 1.73961i 0 1.77747 + 1.35668i 0 3.32224i 0 −0.0262434 0
369.5 0 0.873449i 0 −1.89824 1.18182i 0 0.992530i 0 2.23709 0
369.6 0 0.486391i 0 1.52160 1.63852i 0 1.80495i 0 2.76342 0
369.7 0 0.486391i 0 1.52160 + 1.63852i 0 1.80495i 0 2.76342 0
369.8 0 0.873449i 0 −1.89824 + 1.18182i 0 0.992530i 0 2.23709 0
369.9 0 1.73961i 0 1.77747 1.35668i 0 3.32224i 0 −0.0262434 0
369.10 0 2.40050i 0 −0.817027 2.08146i 0 4.41307i 0 −2.76241 0
369.11 0 2.80150i 0 −2.17393 + 0.523461i 0 4.50896i 0 −4.84843 0
369.12 0 3.21923i 0 1.59013 + 1.57210i 0 2.43185i 0 −7.36343 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 369.12
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 1840.2.e.f 12
4.b odd 2 1 460.2.c.a 12
5.b even 2 1 inner 1840.2.e.f 12
5.c odd 4 1 9200.2.a.cx 6
5.c odd 4 1 9200.2.a.cy 6
12.b even 2 1 4140.2.f.b 12
20.d odd 2 1 460.2.c.a 12
20.e even 4 1 2300.2.a.n 6
20.e even 4 1 2300.2.a.o 6
60.h even 2 1 4140.2.f.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.c.a 12 4.b odd 2 1
460.2.c.a 12 20.d odd 2 1
1840.2.e.f 12 1.a even 1 1 trivial
1840.2.e.f 12 5.b even 2 1 inner
2300.2.a.n 6 20.e even 4 1
2300.2.a.o 6 20.e even 4 1
4140.2.f.b 12 12.b even 2 1
4140.2.f.b 12 60.h even 2 1
9200.2.a.cx 6 5.c odd 4 1
9200.2.a.cy 6 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1840, [\chi])\):

\( T_{3}^{12} + 28T_{3}^{10} + 286T_{3}^{8} + 1296T_{3}^{6} + 2497T_{3}^{4} + 1604T_{3}^{2} + 256 \) Copy content Toggle raw display
\( T_{7}^{12} + 61T_{7}^{10} + 1380T_{7}^{8} + 14312T_{7}^{6} + 68992T_{7}^{4} + 139536T_{7}^{2} + 82944 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 28 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} + 61 T^{10} + \cdots + 82944 \) Copy content Toggle raw display
$11$ \( (T^{6} + 2 T^{5} + \cdots - 256)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 108 T^{10} + \cdots + 1401856 \) Copy content Toggle raw display
$17$ \( T^{12} + 57 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{6} - 4 T^{5} + \cdots + 256)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$29$ \( (T^{6} + 5 T^{5} + \cdots + 11862)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 9 T^{5} + \cdots + 916)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 3996262656 \) Copy content Toggle raw display
$41$ \( (T^{6} + T^{5} - 64 T^{4} + \cdots - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 8399355904 \) Copy content Toggle raw display
$47$ \( T^{12} + 228 T^{10} + \cdots + 215296 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 149426176 \) Copy content Toggle raw display
$59$ \( (T^{6} + 11 T^{5} + \cdots - 360576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 4 T^{5} + \cdots - 171088)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 341 T^{10} + \cdots + 55115776 \) Copy content Toggle raw display
$71$ \( (T^{6} - 17 T^{5} + \cdots - 16108)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 376 T^{10} + \cdots + 30294016 \) Copy content Toggle raw display
$79$ \( (T^{6} - 10 T^{5} + \cdots - 128)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 401124622336 \) Copy content Toggle raw display
$89$ \( (T^{6} - 24 T^{5} + \cdots + 180432)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 2091049984 \) Copy content Toggle raw display
show more
show less