↑

Properties

Label 1840.2.e.g
Level $1840$
Weight $2$
Character orbit 1840.e
Analytic conductor $14.692$
Analytic rank $0$
Dimension $14$
CM no
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{11} + 39 x^{10} - 10 x^{9} + 2 x^{8} - 26 x^{7} + 297 x^{6} - 116 x^{5} + 24 x^{4} + \cdots + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 920)
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.

\(f(q)\) \(=\) \( q - \beta_{8} q^{3} - \beta_{5} q^{5} + ( - \beta_{13} + \beta_{7}) q^{7} + ( - \beta_{12} - \beta_{10} + \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{3} - \beta_{5} q^{5} + ( - \beta_{13} + \beta_{7}) q^{7} + ( - \beta_{12} - \beta_{10} + \beta_{5} + \cdots + 1) q^{9}+ \cdots + ( - 4 \beta_{12} - 4 \beta_{10} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 2 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 2 q^{5} - 4 q^{9} + 14 q^{11} + 6 q^{15} - 14 q^{19} - 12 q^{21} - 14 q^{25} + 22 q^{29} + 20 q^{31} + 2 q^{35} - 48 q^{39} - 32 q^{41} - 26 q^{45} + 34 q^{49} + 14 q^{51} + 38 q^{55} - 22 q^{59} + 10 q^{61} - 38 q^{65} + 6 q^{69} + 28 q^{71} + 24 q^{75} - 64 q^{79} - 10 q^{81} - 50 q^{85} + 48 q^{89} + 14 q^{91} + 30 q^{95} - 122 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{11} + 39 x^{10} - 10 x^{9} + 2 x^{8} - 26 x^{7} + 297 x^{6} - 116 x^{5} + 24 x^{4} + \cdots + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 224809861 \nu^{13} + 40698207136 \nu^{12} + 13894530424 \nu^{11} + 2084956408 \nu^{10} + \cdots + 1689305356572 ) / 732246302876 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9599816455 \nu^{13} - 39917569898 \nu^{12} - 9824626316 \nu^{11} - 19980270148 \nu^{10} + \cdots - 2008645915756 ) / 732246302876 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14185734445 \nu^{13} + 33377065138 \nu^{12} + 22691432254 \nu^{11} + 49350875674 \nu^{10} + \cdots + 2294924313380 ) / 732246302876 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19958784949 \nu^{13} + 4912313158 \nu^{12} + 390318619 \nu^{11} - 37882617844 \nu^{10} + \cdots - 327723885618 ) / 366123151438 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 40039329007 \nu^{13} - 48607321956 \nu^{12} - 16903078854 \nu^{11} + 91091509614 \nu^{10} + \cdots - 1596402425868 ) / 732246302876 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 51158604007 \nu^{13} + 95455044155 \nu^{12} + 57732941236 \nu^{11} - 86278427068 \nu^{10} + \cdots + 982823726512 ) / 732246302876 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 61621117035 \nu^{13} - 58787127771 \nu^{12} - 866985826 \nu^{11} + 134629395734 \nu^{10} + \cdots - 1737381486820 ) / 732246302876 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 66772094683 \nu^{13} - 15046471791 \nu^{12} - 4521994539 \nu^{11} + 135188822801 \nu^{10} + \cdots + 1230815216570 ) / 366123151438 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 134719081377 \nu^{13} + 166717918 \nu^{12} - 18595581648 \nu^{11} - 277803595150 \nu^{10} + \cdots - 2143125728988 ) / 732246302876 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 159011401463 \nu^{13} - 74011296796 \nu^{12} - 17491051676 \nu^{11} + 317230204266 \nu^{10} + \cdots + 2488082785940 ) / 732246302876 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 161362783524 \nu^{13} + 45302004725 \nu^{12} - 17266584944 \nu^{11} - 359941008954 \nu^{10} + \cdots - 2076944162144 ) / 732246302876 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 189780871574 \nu^{13} + 115971282613 \nu^{12} + 47907054048 \nu^{11} - 369867084278 \nu^{10} + \cdots - 88794449364 ) / 732246302876 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 225258249983 \nu^{13} - 97924060431 \nu^{12} - 6621708034 \nu^{11} + 462876963284 \nu^{10} + \cdots + 1060356158372 ) / 732246302876 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} - \beta_{8} - \beta_{7} + \beta_{4} - \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} - \beta_{9} - \beta_{5} + 6\beta_{4} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5 \beta_{13} - \beta_{10} + \beta_{9} - 3 \beta_{8} - 5 \beta_{7} + \beta_{5} + 3 \beta_{4} - \beta_{3} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{12} + 5\beta_{10} + 7\beta_{9} + 2\beta_{6} - 5\beta_{5} + 5\beta_{3} - 26 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 26 \beta_{13} + 2 \beta_{12} + 9 \beta_{10} - 9 \beta_{9} + 12 \beta_{8} + 24 \beta_{7} + 9 \beta_{5} + \cdots + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{13} + 9 \beta_{11} - 13 \beta_{10} + 13 \beta_{9} - \beta_{8} - \beta_{7} + \cdots + 21 \beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 139 \beta_{13} - 20 \beta_{12} - 2 \beta_{11} + 44 \beta_{10} - 40 \beta_{9} + 53 \beta_{8} + \cdots - 90 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 124 \beta_{12} - 117 \beta_{10} - 241 \beta_{9} - 120 \beta_{6} + 145 \beta_{5} - 145 \beta_{3} + \cdots + 630 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 757 \beta_{13} - 148 \beta_{12} + 28 \beta_{11} - 361 \beta_{10} + 419 \beta_{9} - 251 \beta_{8} + \cdots - 564 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1072 \beta_{13} - 722 \beta_{11} + 843 \beta_{10} - 843 \beta_{9} + 178 \beta_{8} + \cdots - 1359 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4180 \beta_{13} + 984 \beta_{12} + 262 \beta_{11} - 1653 \beta_{10} + 1097 \beta_{9} - 1256 \beta_{8} + \cdots + 3484 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2343 \beta_{12} + 1467 \beta_{10} + 3810 \beta_{9} + 2081 \beta_{6} - 2499 \beta_{5} + 2499 \beta_{3} + \cdots - 9188 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 23331 \beta_{13} + 6226 \beta_{12} - 2064 \beta_{11} + 11782 \beta_{10} - 16240 \beta_{9} + \cdots + 21310 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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
−0.503254 − 0.503254i
0.285770 − 0.285770i
−1.27121 − 1.27121i
−1.71470 − 1.71470i
0.416087 − 0.416087i
1.55369 + 1.55369i
1.23362 + 1.23362i
1.23362 − 1.23362i
1.55369 − 1.55369i
0.416087 + 0.416087i
−1.71470 + 1.71470i
−1.27121 + 1.27121i
0.285770 + 0.285770i
−0.503254 + 0.503254i
0 − 2.98707i 0 0.274289 − 2.21918i 0 − 0.980560i 0 −5.92257 0
369.2 0 − 2.49931i 0 2.11714 + 0.719533i 0 − 2.92777i 0 −3.24657 0
369.3 0 − 1.78665i 0 1.46089 + 1.69287i 0 1.75578i 0 −0.192116 0
369.4 0 − 1.58319i 0 −2.09277 + 0.787606i 0 2.84620i 0 0.493499 0
369.5 0 − 1.40334i 0 0.466981 + 2.18676i 0 − 1.57117i 0 1.03063 0
369.6 0 − 0.356372i 0 0.376266 + 2.20418i 0 − 2.46376i 0 2.87300 0
369.7 0 − 0.189375i 0 −1.60280 − 1.55918i 0 − 1.65661i 0 2.96414 0
369.8 0 0.189375i 0 −1.60280 + 1.55918i 0 1.65661i 0 2.96414 0
369.9 0 0.356372i 0 0.376266 − 2.20418i 0 2.46376i 0 2.87300 0
369.10 0 1.40334i 0 0.466981 − 2.18676i 0 1.57117i 0 1.03063 0
369.11 0 1.58319i 0 −2.09277 − 0.787606i 0 − 2.84620i 0 0.493499 0
369.12 0 1.78665i 0 1.46089 − 1.69287i 0 − 1.75578i 0 −0.192116 0
369.13 0 2.49931i 0 2.11714 − 0.719533i 0 2.92777i 0 −3.24657 0
369.14 0 2.98707i 0 0.274289 + 2.21918i 0 0.980560i 0 −5.92257 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 369.14
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 1840.2.e.g 14
4.b odd 2 1 920.2.e.b ✓ 14
5.b even 2 1 inner 1840.2.e.g 14
5.c odd 4 1 9200.2.a.cz 7
5.c odd 4 1 9200.2.a.dc 7
20.d odd 2 1 920.2.e.b ✓ 14
20.e even 4 1 4600.2.a.bh 7
20.e even 4 1 4600.2.a.bi 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.e.b ✓ 14 4.b odd 2 1
920.2.e.b ✓ 14 20.d odd 2 1
1840.2.e.g 14 1.a even 1 1 trivial
1840.2.e.g 14 5.b even 2 1 inner
4600.2.a.bh 7 20.e even 4 1
4600.2.a.bi 7 20.e even 4 1
9200.2.a.cz 7 5.c odd 4 1
9200.2.a.dc 7 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 \(S_{2}^{\mathrm{new}}(1840, [\chi])\):

\( T_{3}^{14} + 23T_{3}^{12} + 195T_{3}^{10} + 766T_{3}^{8} + 1431T_{3}^{6} + 1095T_{3}^{4} + 149T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{14} + 32T_{7}^{12} + 412T_{7}^{10} + 2745T_{7}^{8} + 10156T_{7}^{6} + 20760T_{7}^{4} + 21488T_{7}^{2} + 8464 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 23 T^{12} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{14} - 2 T^{13} + \cdots + 78125 \) Copy content Toggle raw display
$7$ \( T^{14} + 32 T^{12} + \cdots + 8464 \) Copy content Toggle raw display
$11$ \( (T^{7} - 7 T^{6} + \cdots - 128)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + 75 T^{12} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( T^{14} + 116 T^{12} + \cdots + 2704 \) Copy content Toggle raw display
$19$ \( (T^{7} + 7 T^{6} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$29$ \( (T^{7} - 11 T^{6} + \cdots - 9244)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} - 10 T^{6} + \cdots + 225251)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 1834580224 \) Copy content Toggle raw display
$41$ \( (T^{7} + 16 T^{6} + \cdots + 110153)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + 96 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$47$ \( T^{14} + 392 T^{12} + \cdots + 33085504 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 24445947904 \) Copy content Toggle raw display
$59$ \( (T^{7} + 11 T^{6} + \cdots + 486592)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 5 T^{6} + \cdots + 2669336)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 214213460224 \) Copy content Toggle raw display
$71$ \( (T^{7} - 14 T^{6} + \cdots - 632317)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + 160 T^{12} + \cdots + 256 \) Copy content Toggle raw display
$79$ \( (T^{7} + 32 T^{6} + \cdots + 473312)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 143616256 \) Copy content Toggle raw display
$89$ \( (T^{7} - 24 T^{6} + \cdots - 311456)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 16312357788736 \) Copy content Toggle raw display
show more
show less