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Properties

Label 1840.2.e.h
Level $1840$
Weight $2$
Character orbit 1840.e
Analytic conductor $14.692$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 2 x^{14} + 6 x^{13} + 100 x^{12} - 196 x^{11} + 210 x^{10} + 702 x^{9} + 1572 x^{8} + \cdots + 1024 \) 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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} - \beta_{10} q^{5} + (\beta_{13} - \beta_{2}) q^{7} + ( - \beta_{12} - \beta_{10} + \beta_{9} + \cdots - 1) q^{9} + ( - \beta_{12} - 1) q^{11} + (\beta_{15} - \beta_{14} - \beta_{13} + \cdots - 1) q^{13}+ \cdots + (3 \beta_{14} - \beta_{12} + \beta_{11} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{5} - 22 q^{9} - 14 q^{11} - 6 q^{15} + 22 q^{19} + 12 q^{25} - 44 q^{29} - 18 q^{31} - 20 q^{35} + 14 q^{41} + 14 q^{45} - 78 q^{49} + 38 q^{51} - 30 q^{55} + 64 q^{59} + 34 q^{61} + 6 q^{65}+ \cdots + 122 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} + 6 x^{13} + 100 x^{12} - 196 x^{11} + 210 x^{10} + 702 x^{9} + 1572 x^{8} + \cdots + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 56\!\cdots\!89 \nu^{15} + \cdots + 95\!\cdots\!24 ) / 20\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 66\!\cdots\!57 \nu^{15} + \cdots - 43\!\cdots\!08 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 66\!\cdots\!57 \nu^{15} + \cdots - 43\!\cdots\!08 ) / 31\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 36\!\cdots\!99 \nu^{15} + \cdots - 93\!\cdots\!76 ) / 50\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 31\!\cdots\!61 \nu^{15} + \cdots - 14\!\cdots\!44 ) / 20\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 32\!\cdots\!63 \nu^{15} + \cdots - 12\!\cdots\!80 ) / 40\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 84\!\cdots\!19 \nu^{15} + \cdots - 10\!\cdots\!44 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20\!\cdots\!05 \nu^{15} + \cdots - 23\!\cdots\!28 ) / 20\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21\!\cdots\!49 \nu^{15} + \cdots + 44\!\cdots\!56 ) / 20\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21\!\cdots\!67 \nu^{15} + \cdots - 20\!\cdots\!00 ) / 20\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12\!\cdots\!41 \nu^{15} + \cdots - 12\!\cdots\!84 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 25\!\cdots\!67 \nu^{15} + \cdots - 31\!\cdots\!68 ) / 20\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 26\!\cdots\!23 \nu^{15} + \cdots - 44\!\cdots\!52 ) / 20\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 18\!\cdots\!37 \nu^{15} + \cdots - 15\!\cdots\!52 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 32\!\cdots\!01 \nu^{15} + \cdots - 49\!\cdots\!24 ) / 20\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} - 2\beta_{3} + 8\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - 2 \beta_{14} - 2 \beta_{13} + \beta_{12} - 6 \beta_{11} + 6 \beta_{10} + 2 \beta_{9} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{12} - 7\beta_{11} - 5\beta_{10} - 4\beta_{9} + 7\beta_{8} - 5\beta_{7} - 18\beta_{6} + 16\beta _1 - 54 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 11 \beta_{15} + 22 \beta_{13} + 11 \beta_{12} - 39 \beta_{11} - 39 \beta_{10} + 22 \beta_{9} + \cdots + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 13 \beta_{15} - 25 \beta_{13} - 13 \beta_{11} - 18 \beta_{10} + 65 \beta_{8} + 18 \beta_{7} + \cdots - 207 \beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 106 \beta_{15} + 180 \beta_{14} + 206 \beta_{13} - 106 \beta_{12} + 363 \beta_{11} - 363 \beta_{10} + \cdots - 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 16 \beta_{14} - 250 \beta_{12} + 113 \beta_{11} + 153 \beta_{10} + 520 \beta_{9} - 113 \beta_{8} + \cdots + 3352 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 957 \beta_{15} - 1868 \beta_{13} - 957 \beta_{12} + 1948 \beta_{11} + 1948 \beta_{10} - 1868 \beta_{9} + \cdots - 2514 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2136 \beta_{15} + 366 \beta_{14} + 5172 \beta_{13} + 1063 \beta_{11} - 497 \beta_{10} - 10989 \beta_{8} + \cdots + 366 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 8319 \beta_{15} - 13624 \beta_{14} - 16858 \beta_{13} + 8319 \beta_{12} - 25513 \beta_{11} + \cdots + 14772 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2763 \beta_{14} + 8523 \beta_{12} + 7616 \beta_{11} - 4366 \beta_{10} - 25265 \beta_{9} - 7616 \beta_{8} + \cdots - 119772 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 70750 \beta_{15} + 152444 \beta_{13} + 70750 \beta_{12} - 116569 \beta_{11} - 116569 \beta_{10} + \cdots + 298240 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 129196 \beta_{15} - 69708 \beta_{14} - 488396 \beta_{13} - 80835 \beta_{11} + 214549 \beta_{10} + \cdots - 69708 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 594527 \beta_{15} + 1021262 \beta_{14} + 1383542 \beta_{13} - 594527 \beta_{12} + 1890076 \beta_{11} + \cdots - 1986426 ) / 2 \) 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
−0.945903 + 0.945903i
1.65386 − 1.65386i
1.81400 + 1.81400i
−1.13508 − 1.13508i
−2.15699 + 2.15699i
0.303680 + 0.303680i
1.95202 − 1.95202i
−0.485591 + 0.485591i
−0.485591 − 0.485591i
1.95202 + 1.95202i
0.303680 − 0.303680i
−2.15699 − 2.15699i
−1.13508 + 1.13508i
1.81400 − 1.81400i
1.65386 + 1.65386i
−0.945903 − 0.945903i
0 − 3.20935i 0 2.02615 − 0.945903i 0 4.54713i 0 −7.29996 0
369.2 0 − 2.89996i 0 −1.50491 + 1.65386i 0 0.580879i 0 −5.40975 0
369.3 0 − 2.54092i 0 −1.30744 − 1.81400i 0 0.780573i 0 −3.45626 0
369.4 0 − 2.51561i 0 −1.92655 + 1.13508i 0 − 4.64022i 0 −3.32827 0
369.5 0 − 1.69755i 0 0.589417 − 2.15699i 0 − 4.22860i 0 0.118308 0
369.6 0 − 0.540724i 0 2.21535 − 0.303680i 0 − 1.15693i 0 2.70762 0
369.7 0 − 0.493532i 0 1.09069 + 1.95202i 0 4.54439i 0 2.75643 0
369.8 0 − 0.296848i 0 −2.18271 − 0.485591i 0 − 3.46037i 0 2.91188 0
369.9 0 0.296848i 0 −2.18271 + 0.485591i 0 3.46037i 0 2.91188 0
369.10 0 0.493532i 0 1.09069 − 1.95202i 0 − 4.54439i 0 2.75643 0
369.11 0 0.540724i 0 2.21535 + 0.303680i 0 1.15693i 0 2.70762 0
369.12 0 1.69755i 0 0.589417 + 2.15699i 0 4.22860i 0 0.118308 0
369.13 0 2.51561i 0 −1.92655 − 1.13508i 0 4.64022i 0 −3.32827 0
369.14 0 2.54092i 0 −1.30744 + 1.81400i 0 − 0.780573i 0 −3.45626 0
369.15 0 2.89996i 0 −1.50491 − 1.65386i 0 − 0.580879i 0 −5.40975 0
369.16 0 3.20935i 0 2.02615 + 0.945903i 0 − 4.54713i 0 −7.29996 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 369.16
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.h 16
4.b odd 2 1 920.2.e.c ✓ 16
5.b even 2 1 inner 1840.2.e.h 16
5.c odd 4 1 9200.2.a.dd 8
5.c odd 4 1 9200.2.a.de 8
20.d odd 2 1 920.2.e.c ✓ 16
20.e even 4 1 4600.2.a.bj 8
20.e even 4 1 4600.2.a.bk 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.e.c ✓ 16 4.b odd 2 1
920.2.e.c ✓ 16 20.d odd 2 1
1840.2.e.h 16 1.a even 1 1 trivial
1840.2.e.h 16 5.b even 2 1 inner
4600.2.a.bj 8 20.e even 4 1
4600.2.a.bk 8 20.e even 4 1
9200.2.a.dd 8 5.c odd 4 1
9200.2.a.de 8 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 \(S_{2}^{\mathrm{new}}(1840, [\chi])\):

\( T_{3}^{16} + 35T_{3}^{14} + 479T_{3}^{12} + 3218T_{3}^{10} + 10815T_{3}^{8} + 16123T_{3}^{6} + 7441T_{3}^{4} + 1264T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{16} + 95 T_{7}^{14} + 3621 T_{7}^{12} + 69892 T_{7}^{10} + 703096 T_{7}^{8} + 3332320 T_{7}^{6} + \cdots + 541696 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 35 T^{14} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{16} + 2 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 95 T^{14} + \cdots + 541696 \) Copy content Toggle raw display
$11$ \( (T^{8} + 7 T^{7} + \cdots + 440)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 196560400 \) Copy content Toggle raw display
$17$ \( T^{16} + 147 T^{14} + \cdots + 30976 \) Copy content Toggle raw display
$19$ \( (T^{8} - 11 T^{7} + \cdots + 11192)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$29$ \( (T^{8} + 22 T^{7} + \cdots - 400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 9 T^{7} + \cdots - 287276)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 259081216 \) Copy content Toggle raw display
$41$ \( (T^{8} - 7 T^{7} + \cdots + 1197584)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 59754824704 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 5405190400 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 41160294400 \) Copy content Toggle raw display
$59$ \( (T^{8} - 32 T^{7} + \cdots + 29696)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 17 T^{7} + \cdots + 200)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 463227249664 \) Copy content Toggle raw display
$71$ \( (T^{8} + 15 T^{7} + \cdots - 8900000)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 311006982400 \) Copy content Toggle raw display
$79$ \( (T^{8} + 2 T^{7} + \cdots - 5248)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 17501839590400 \) Copy content Toggle raw display
$89$ \( (T^{8} + 46 T^{7} + \cdots + 12804160)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 255 T^{14} + \cdots + 8761600 \) Copy content Toggle raw display
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