Properties

Label 2320.2.d.j
Level $2320$
Weight $2$
Character orbit 2320.d
Analytic conductor $18.525$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2320,2,Mod(929,2320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2320, 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("2320.929");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2320.d (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: \(18.5252932689\)
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: \( x^{16} + 33x^{14} + 432x^{12} + 2881x^{10} + 10441x^{8} + 20304x^{6} + 19480x^{4} + 7344x^{2} + 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 1160)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{8} q^{5} - \beta_{13} q^{7} + (\beta_{10} + \beta_{9} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{8} q^{5} - \beta_{13} q^{7} + (\beta_{10} + \beta_{9} - 1) q^{9} + (\beta_{9} - \beta_{2} - 1) q^{11} + (\beta_{13} + \beta_{11} + \cdots + \beta_{5}) q^{13}+ \cdots + ( - \beta_{15} - \beta_{14} + 2 \beta_{10} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{5} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{5} - 18 q^{9} - 8 q^{11} + 10 q^{15} + 16 q^{19} - 12 q^{21} + 10 q^{25} - 16 q^{29} - 34 q^{31} - 10 q^{35} + 26 q^{39} - 36 q^{41} - 4 q^{45} + 18 q^{49} - 20 q^{55} - 2 q^{59} - 34 q^{61} - 36 q^{65} + 50 q^{69} + 12 q^{71} - 24 q^{75} + 78 q^{79} + 2 q^{85} + 72 q^{89} + 28 q^{91} - 16 q^{95} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 33x^{14} + 432x^{12} + 2881x^{10} + 10441x^{8} + 20304x^{6} + 19480x^{4} + 7344x^{2} + 400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\nu^{14} + 361\nu^{12} + 4638\nu^{10} + 29535\nu^{8} + 96409\nu^{6} + 146846\nu^{4} + 70748\nu^{2} - 6504 ) / 2432 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 44 \nu^{15} - 5 \nu^{14} + 1292 \nu^{13} + 95 \nu^{12} + 15208 \nu^{11} + 4110 \nu^{10} + \cdots + 20920 ) / 12160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11 \nu^{15} - 361 \nu^{13} - 4638 \nu^{11} - 29535 \nu^{9} - 96409 \nu^{7} - 146846 \nu^{5} + \cdots + 21096 \nu ) / 2432 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 33 \nu^{15} + 85 \nu^{14} - 969 \nu^{13} + 2375 \nu^{12} - 11026 \nu^{11} + 25130 \nu^{10} + \cdots + 15240 ) / 12160 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17 \nu^{15} - 741 \nu^{13} - 11904 \nu^{11} - 90637 \nu^{9} - 347677 \nu^{7} - 653328 \nu^{5} + \cdots - 119328 \nu ) / 6080 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 23 \nu^{14} - 513 \nu^{12} - 3514 \nu^{10} - 4603 \nu^{8} + 34591 \nu^{6} + 121702 \nu^{4} + \cdots + 15672 ) / 2432 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 33 \nu^{15} + 85 \nu^{14} + 969 \nu^{13} + 2375 \nu^{12} + 11026 \nu^{11} + 25130 \nu^{10} + \cdots + 15240 ) / 12160 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 35 \nu^{14} + 969 \nu^{12} + 10218 \nu^{10} + 52935 \nu^{8} + 144489 \nu^{6} + 201866 \nu^{4} + \cdots + 13768 ) / 2432 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 35 \nu^{14} - 969 \nu^{12} - 10218 \nu^{10} - 52935 \nu^{8} - 144489 \nu^{6} - 201866 \nu^{4} + \cdots - 4040 ) / 2432 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 23 \nu^{15} + 684 \nu^{13} + 7941 \nu^{11} + 46403 \nu^{9} + 145168 \nu^{7} + 233997 \nu^{5} + \cdots + 33652 \nu ) / 3040 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 83 \nu^{15} + 2869 \nu^{13} + 38326 \nu^{11} + 249303 \nu^{9} + 820453 \nu^{7} + 1290102 \nu^{5} + \cdots + 107512 \nu ) / 12160 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 28 \nu^{15} - 779 \nu^{13} - 8201 \nu^{11} - 41588 \nu^{9} - 105943 \nu^{7} - 124097 \nu^{5} + \cdots + 16908 \nu ) / 3040 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 49 \nu^{14} - 1235 \nu^{12} - 11250 \nu^{10} - 46445 \nu^{8} - 89987 \nu^{6} - 79282 \nu^{4} + \cdots - 16296 ) / 2432 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 29 \nu^{14} - 817 \nu^{12} - 8766 \nu^{10} - 45793 \nu^{8} - 122817 \nu^{6} - 161758 \nu^{4} + \cdots - 10296 ) / 1216 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{12} + 2\beta_{11} - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - 10\beta_{10} - 9\beta_{9} + 2\beta_{8} + \beta_{7} + 2\beta_{5} - 2\beta_{2} + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{15} + 2 \beta_{14} - 14 \beta_{13} - 16 \beta_{12} - 23 \beta_{11} + 14 \beta_{8} - 17 \beta_{6} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 21 \beta_{15} - 4 \beta_{14} + 99 \beta_{10} + 74 \beta_{9} - 35 \beta_{8} - 13 \beta_{7} + \cdots - 202 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 35 \beta_{15} - 35 \beta_{14} + 167 \beta_{13} + 197 \beta_{12} + 233 \beta_{11} - 151 \beta_{8} + \cdots + 35 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 294 \beta_{15} + 86 \beta_{14} - 995 \beta_{10} - 605 \beta_{9} + 450 \beta_{8} + 138 \beta_{7} + \cdots + 1724 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 450 \beta_{15} + 450 \beta_{14} - 1883 \beta_{13} - 2219 \beta_{12} - 2340 \beta_{11} + 1535 \beta_{8} + \cdots - 450 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3549 \beta_{15} - 1272 \beta_{14} + 10102 \beta_{10} + 5065 \beta_{9} - 5198 \beta_{8} - 1389 \beta_{7} + \cdots - 15622 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5198 \beta_{15} - 5198 \beta_{14} + 20586 \beta_{13} + 24036 \beta_{12} + 23683 \beta_{11} - 15450 \beta_{8} + \cdots + 5198 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 40005 \beta_{15} + 16108 \beta_{14} - 103203 \beta_{10} - 43930 \beta_{9} + 57203 \beta_{8} + \cdots + 147530 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 57203 \beta_{15} + 57203 \beta_{14} - 220711 \beta_{13} - 255285 \beta_{12} - 241481 \beta_{11} + \cdots - 57203 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 435190 \beta_{15} - 188166 \beta_{14} + 1058523 \beta_{10} + 395685 \beta_{9} - 613826 \beta_{8} + \cdots - 1434156 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 613826 \beta_{15} - 613826 \beta_{14} + 2336403 \beta_{13} + 2682339 \beta_{12} + 2474756 \beta_{11} + \cdots + 613826 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(581\) \(1857\) \(2031\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
929.1
3.21506i
2.60186i
2.59166i
2.04061i
1.61606i
1.25178i
0.877146i
0.254777i
0.254777i
0.877146i
1.25178i
1.61606i
2.04061i
2.59166i
2.60186i
3.21506i
0 3.21506i 0 −2.23110 0.148957i 0 0.627864i 0 −7.33659 0
929.2 0 2.60186i 0 0.149706 + 2.23105i 0 3.57384i 0 −3.76968 0
929.3 0 2.59166i 0 2.07683 + 0.828709i 0 2.97644i 0 −3.71672 0
929.4 0 2.04061i 0 1.92659 1.13502i 0 3.45115i 0 −1.16410 0
929.5 0 1.61606i 0 1.41675 1.72998i 0 1.06577i 0 0.388359 0
929.6 0 1.25178i 0 −2.17253 + 0.529275i 0 0.133263i 0 1.43304 0
929.7 0 0.877146i 0 −1.53115 + 1.62959i 0 3.05899i 0 2.23061 0
929.8 0 0.254777i 0 −0.635094 + 2.14398i 0 1.59775i 0 2.93509 0
929.9 0 0.254777i 0 −0.635094 2.14398i 0 1.59775i 0 2.93509 0
929.10 0 0.877146i 0 −1.53115 1.62959i 0 3.05899i 0 2.23061 0
929.11 0 1.25178i 0 −2.17253 0.529275i 0 0.133263i 0 1.43304 0
929.12 0 1.61606i 0 1.41675 + 1.72998i 0 1.06577i 0 0.388359 0
929.13 0 2.04061i 0 1.92659 + 1.13502i 0 3.45115i 0 −1.16410 0
929.14 0 2.59166i 0 2.07683 0.828709i 0 2.97644i 0 −3.71672 0
929.15 0 2.60186i 0 0.149706 2.23105i 0 3.57384i 0 −3.76968 0
929.16 0 3.21506i 0 −2.23110 + 0.148957i 0 0.627864i 0 −7.33659 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 929.16
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 2320.2.d.j 16
4.b odd 2 1 1160.2.d.c 16
5.b even 2 1 inner 2320.2.d.j 16
20.d odd 2 1 1160.2.d.c 16
20.e even 4 1 5800.2.a.be 8
20.e even 4 1 5800.2.a.bf 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.2.d.c 16 4.b odd 2 1
1160.2.d.c 16 20.d odd 2 1
2320.2.d.j 16 1.a even 1 1 trivial
2320.2.d.j 16 5.b even 2 1 inner
5800.2.a.be 8 20.e even 4 1
5800.2.a.bf 8 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2320, [\chi])\):

\( T_{3}^{16} + 33T_{3}^{14} + 432T_{3}^{12} + 2881T_{3}^{10} + 10441T_{3}^{8} + 20304T_{3}^{6} + 19480T_{3}^{4} + 7344T_{3}^{2} + 400 \) Copy content Toggle raw display
\( T_{7}^{16} + 47T_{7}^{14} + 865T_{7}^{12} + 7816T_{7}^{10} + 35448T_{7}^{8} + 73872T_{7}^{6} + 61712T_{7}^{4} + 15488T_{7}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 33 T^{14} + \cdots + 400 \) Copy content Toggle raw display
$5$ \( T^{16} + 2 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 47 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{8} + 4 T^{7} - 39 T^{6} + \cdots - 80)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 101 T^{14} + \cdots + 6400 \) Copy content Toggle raw display
$17$ \( T^{16} + 91 T^{14} + \cdots + 50176 \) Copy content Toggle raw display
$19$ \( (T^{8} - 8 T^{7} + \cdots - 5312)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 131 T^{14} + \cdots + 25887744 \) Copy content Toggle raw display
$29$ \( (T + 1)^{16} \) Copy content Toggle raw display
$31$ \( (T^{8} + 17 T^{7} + \cdots + 84)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 280 T^{14} + \cdots + 26214400 \) Copy content Toggle raw display
$41$ \( (T^{8} + 18 T^{7} + \cdots - 434176)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 2879442459664 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 26073206784 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 60390113536 \) Copy content Toggle raw display
$59$ \( (T^{8} + T^{7} + \cdots + 288128)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 17 T^{7} + \cdots - 124928)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 715418238976 \) Copy content Toggle raw display
$71$ \( (T^{8} - 6 T^{7} + \cdots + 240384)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 747159699456 \) Copy content Toggle raw display
$79$ \( (T^{8} - 39 T^{7} + \cdots - 97924)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 8947646464 \) Copy content Toggle raw display
$89$ \( (T^{8} - 36 T^{7} + \cdots - 2757376)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 10983656505600 \) Copy content Toggle raw display
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