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Properties

Label 1872.4.n.d
Level $1872$
Weight $4$
Character orbit 1872.n
Analytic conductor $110.452$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1872,4,Mod(1871,1872)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1872, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1872.1871");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1872.n (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: \(110.451575531\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 92 x^{14} + 25462 x^{12} - 1363472 x^{10} + 140075083 x^{8} - 211475216 x^{6} + \cdots + 18293490313921 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + \beta_{10} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + \beta_{10} q^{7} + \beta_{4} q^{11} + (\beta_{13} - \beta_{11} + 11) q^{13} + ( - 5 \beta_{9} + 29 \beta_{8}) q^{17} + (4 \beta_{12} - 3 \beta_{10}) q^{19} + \beta_{15} q^{23} + (11 \beta_1 + 73) q^{25} + ( - 12 \beta_{9} - 7 \beta_{8}) q^{29} - 9 \beta_{12} q^{31} - \beta_{14} q^{35} - 7 \beta_{11} q^{37} + ( - 2 \beta_{5} - 15 \beta_{2}) q^{41} + ( - \beta_{7} - \beta_{6}) q^{43} + ( - 9 \beta_{4} - 22 \beta_{3}) q^{47} + (35 \beta_1 + 167) q^{49} + ( - 17 \beta_{9} - 221 \beta_{8}) q^{53} - \beta_{6} q^{55} + (20 \beta_{4} - \beta_{3}) q^{59} + (36 \beta_1 - 286) q^{61} + ( - 77 \beta_{9} - 132 \beta_{8} + \cdots + 14 \beta_{2}) q^{65}+ \cdots + (10 \beta_{13} - 11 \beta_{11} - 5 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 176 q^{13} + 1168 q^{25} + 2672 q^{49} - 4576 q^{61}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 92 x^{14} + 25462 x^{12} - 1363472 x^{10} + 140075083 x^{8} - 211475216 x^{6} + \cdots + 18293490313921 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 218268618 \nu^{14} - 16457508719 \nu^{12} + 3865782794319 \nu^{10} - 173331556795332 \nu^{8} + \cdots - 69\!\cdots\!19 ) / 62\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 93\!\cdots\!61 \nu^{14} + \cdots - 30\!\cdots\!67 ) / 31\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 62\!\cdots\!19 \nu^{14} + \cdots + 13\!\cdots\!99 ) / 79\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 52\!\cdots\!85 \nu^{14} + \cdots + 11\!\cdots\!41 ) / 23\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29\!\cdots\!21 \nu^{14} + \cdots - 86\!\cdots\!83 ) / 10\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 32\!\cdots\!76 \nu^{14} + \cdots + 26\!\cdots\!26 ) / 26\!\cdots\!79 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 86\!\cdots\!08 \nu^{14} + \cdots - 73\!\cdots\!23 ) / 53\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 489503891234490 \nu^{15} + \cdots - 28\!\cdots\!13 \nu ) / 23\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4451326 \nu^{15} - 287913948 \nu^{13} + 102254719491 \nu^{11} - 2985571984412 \nu^{9} + \cdots + 26\!\cdots\!82 \nu ) / 40\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 65\!\cdots\!81 \nu^{15} + \cdots - 19\!\cdots\!95 \nu ) / 54\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 35\!\cdots\!91 \nu^{15} + \cdots + 84\!\cdots\!14 \nu ) / 17\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!28 \nu^{15} + \cdots - 41\!\cdots\!95 \nu ) / 54\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 16\!\cdots\!77 \nu^{15} + \cdots - 93\!\cdots\!03 ) / 17\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 10\!\cdots\!44 \nu^{15} + \cdots - 15\!\cdots\!15 \nu ) / 53\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 21\!\cdots\!12 \nu^{15} + \cdots - 32\!\cdots\!76 \nu ) / 53\!\cdots\!48 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{12} + \beta_{11} + \beta_{10} + 4\beta_{8} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + 4\beta_{3} - 4\beta_{2} + 5\beta _1 + 46 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} + 3 \beta_{14} - 26 \beta_{13} + 42 \beta_{12} + 136 \beta_{11} - 60 \beta_{10} + \cdots + 13 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 30\beta_{7} + 45\beta_{6} + 104\beta_{5} + 72\beta_{4} - 376\beta_{3} - 1016\beta_{2} - 1806\beta _1 - 21230 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 115 \beta_{15} + 265 \beta_{14} + 934 \beta_{13} + 6571 \beta_{12} - 5157 \beta_{11} + \cdots - 467 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3912 \beta_{7} - 5244 \beta_{6} - 1762 \beta_{5} + 22482 \beta_{4} - 64884 \beta_{3} + 17916 \beta_{2} + \cdots - 539602 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 32676 \beta_{15} - 65772 \beta_{14} + 339794 \beta_{13} - 216943 \beta_{12} - 1598589 \beta_{11} + \cdots - 169897 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 453404 \beta_{7} - 621131 \beta_{6} - 2834384 \beta_{5} - 39312 \beta_{4} + 300352 \beta_{3} + \cdots + 223841038 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4105557 \beta_{15} - 8672931 \beta_{14} - 11541822 \beta_{13} - 79065216 \beta_{12} + \cdots + 5770911 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 98793243 \beta_{7} + 133825167 \beta_{6} - 55957332 \beta_{5} - 854357868 \beta_{4} + 2412578732 \beta_{3} + \cdots + 13210791406 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 378011689 \beta_{15} + 791318473 \beta_{14} - 4447707792 \beta_{13} + 3338464123 \beta_{12} + \cdots + 2223853896 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1416151686 \beta_{7} + 1918615962 \beta_{6} + 11355618296 \beta_{5} - 7727012856 \beta_{4} + \cdots - 831082022405 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 79901410992 \beta_{15} + 167508860688 \beta_{14} + 215145047260 \beta_{13} + \cdots - 107572523630 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1615772944787 \beta_{7} - 2187753270221 \beta_{6} + 2274786827768 \beta_{5} + 12531334853016 \beta_{4} + \cdots - 191050201712198 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 3648176439447 \beta_{15} - 7645624275171 \beta_{14} + 73082592378266 \beta_{13} + \cdots - 36541296189133 \beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1871.1
−8.18568 + 4.89549i
−8.18568 − 4.89549i
8.18568 − 4.89549i
8.18568 + 4.89549i
3.08133 + 1.69828i
3.08133 − 1.69828i
−3.08133 − 1.69828i
−3.08133 + 1.69828i
3.08133 − 4.52671i
3.08133 + 4.52671i
−3.08133 + 4.52671i
−3.08133 − 4.52671i
−8.18568 − 7.72392i
−8.18568 + 7.72392i
8.18568 + 7.72392i
8.18568 − 7.72392i
0 0 0 −17.8465 0 −29.8899 0 0 0
1871.2 0 0 0 −17.8465 0 −29.8899 0 0 0
1871.3 0 0 0 −17.8465 0 29.8899 0 0 0
1871.4 0 0 0 −17.8465 0 29.8899 0 0 0
1871.5 0 0 0 −8.80347 0 −11.2514 0 0 0
1871.6 0 0 0 −8.80347 0 −11.2514 0 0 0
1871.7 0 0 0 −8.80347 0 11.2514 0 0 0
1871.8 0 0 0 −8.80347 0 11.2514 0 0 0
1871.9 0 0 0 8.80347 0 −11.2514 0 0 0
1871.10 0 0 0 8.80347 0 −11.2514 0 0 0
1871.11 0 0 0 8.80347 0 11.2514 0 0 0
1871.12 0 0 0 8.80347 0 11.2514 0 0 0
1871.13 0 0 0 17.8465 0 −29.8899 0 0 0
1871.14 0 0 0 17.8465 0 −29.8899 0 0 0
1871.15 0 0 0 17.8465 0 29.8899 0 0 0
1871.16 0 0 0 17.8465 0 29.8899 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1871.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
13.b even 2 1 inner
39.d odd 2 1 inner
52.b odd 2 1 inner
156.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1872.4.n.d ✓ 16
3.b odd 2 1 inner 1872.4.n.d ✓ 16
4.b odd 2 1 inner 1872.4.n.d ✓ 16
12.b even 2 1 inner 1872.4.n.d ✓ 16
13.b even 2 1 inner 1872.4.n.d ✓ 16
39.d odd 2 1 inner 1872.4.n.d ✓ 16
52.b odd 2 1 inner 1872.4.n.d ✓ 16
156.h even 2 1 inner 1872.4.n.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1872.4.n.d ✓ 16 1.a even 1 1 trivial
1872.4.n.d ✓ 16 3.b odd 2 1 inner
1872.4.n.d ✓ 16 4.b odd 2 1 inner
1872.4.n.d ✓ 16 12.b even 2 1 inner
1872.4.n.d ✓ 16 13.b even 2 1 inner
1872.4.n.d ✓ 16 39.d odd 2 1 inner
1872.4.n.d ✓ 16 52.b odd 2 1 inner
1872.4.n.d ✓ 16 156.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1872, [\chi])\):

\( T_{5}^{4} - 396T_{5}^{2} + 24684 \) Copy content Toggle raw display
\( T_{7}^{4} - 1020T_{7}^{2} + 113100 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} - 396 T^{2} + 24684)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 1020 T^{2} + 113100)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 972 T^{2} + 221676)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 44 T^{3} + \cdots + 4826809)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 6364 T^{2} + 33124)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 13884 T^{2} + 37463244)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 45672 T^{2} + 111670416)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 17476 T^{2} + 72965764)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 45684 T^{2} + 29681964)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 38808 T^{2} + 237065136)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 111276 T^{2} + 2543464044)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 176616 T^{2} + 1005033744)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 284652 T^{2} + 19441116396)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 230044 T^{2} + 6454836964)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 398532 T^{2} + 38494213836)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 572 T - 73724)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 364500 T^{2} + 31173187500)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 1343172 T^{2} + 403406206476)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 650760 T^{2} + 34955012400)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 2251722268224)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 606372 T^{2} + 21836483916)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 785004 T^{2} + 32701387884)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 420552 T^{2} + 44190382896)^{4} \) Copy content Toggle raw display
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