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Properties

Label 4320.2.d.h
Level $4320$
Weight $2$
Character orbit 4320.d
Analytic conductor $34.495$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4320,2,Mod(3889,4320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4320.3889");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4320.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: \(34.4953736732\)
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: \( x^{16} - 3 x^{15} - 3 x^{14} + 36 x^{13} - 78 x^{12} - 96 x^{11} + 1194 x^{10} + 1456 x^{9} + \cdots + 45658 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 1080)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{5} - \beta_{12} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{5} - \beta_{12} q^{7} - \beta_{6} q^{11} - \beta_{9} q^{13} + (\beta_{13} - \beta_{10} + \beta_{7}) q^{17} + (\beta_{15} + \beta_{7}) q^{19} + ( - \beta_{10} + \beta_{7}) q^{23} + ( - \beta_{3} - \beta_{2} - 2) q^{25} + (\beta_{4} - \beta_{2}) q^{29} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \cdots + 1) q^{31}+ \cdots + (\beta_{12} + 2 \beta_{6} + \cdots + \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{5} - 22 q^{25} - 2 q^{35} - 44 q^{49} - 96 q^{53} - 34 q^{55} + 12 q^{77} - 4 q^{79} + 64 q^{83}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} - 3 x^{14} + 36 x^{13} - 78 x^{12} - 96 x^{11} + 1194 x^{10} + 1456 x^{9} + \cdots + 45658 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 46\!\cdots\!13 \nu^{15} + \cdots - 13\!\cdots\!44 ) / 12\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12\!\cdots\!79 \nu^{15} + \cdots + 19\!\cdots\!54 ) / 23\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!55 \nu^{15} + \cdots + 32\!\cdots\!32 ) / 12\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 68\!\cdots\!85 \nu^{15} + \cdots + 75\!\cdots\!34 ) / 71\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 70\!\cdots\!47 \nu^{15} + \cdots - 13\!\cdots\!70 ) / 71\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 44\!\cdots\!60 \nu^{15} + \cdots - 33\!\cdots\!58 ) / 35\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 37\!\cdots\!97 \nu^{15} + \cdots + 51\!\cdots\!68 ) / 29\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15\!\cdots\!93 \nu^{15} + \cdots - 60\!\cdots\!94 ) / 10\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 69\!\cdots\!31 \nu^{15} + \cdots - 19\!\cdots\!92 ) / 46\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 86\!\cdots\!78 \nu^{15} + \cdots - 64\!\cdots\!14 ) / 48\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!00 \nu^{15} + \cdots + 47\!\cdots\!76 ) / 46\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 26\!\cdots\!78 \nu^{15} + \cdots + 69\!\cdots\!74 ) / 11\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 33\!\cdots\!29 \nu^{15} + \cdots + 48\!\cdots\!98 ) / 14\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 81\!\cdots\!19 \nu^{15} + \cdots - 38\!\cdots\!26 ) / 29\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 44\!\cdots\!43 \nu^{15} + \cdots + 17\!\cdots\!12 ) / 14\!\cdots\!14 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{14} - \beta_{13} - \beta_{6} + \beta_{4} - \beta_{2} - \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 4 \beta_{13} + 2 \beta_{12} + \beta_{11} - \beta_{10} - 3 \beta_{9} + \beta_{8} + 2 \beta_{5} + \cdots + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{15} + \beta_{14} + \beta_{13} + 7 \beta_{12} + 2 \beta_{11} - 6 \beta_{10} - 5 \beta_{9} + \cdots - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 5 \beta_{15} + 2 \beta_{14} + 10 \beta_{13} + 52 \beta_{12} + 5 \beta_{11} - 67 \beta_{10} + \cdots + 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{15} + \beta_{14} - 2 \beta_{13} + 183 \beta_{12} - 23 \beta_{11} - 245 \beta_{10} + \cdots + 46 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 63 \beta_{15} - 93 \beta_{14} - 136 \beta_{13} + 315 \beta_{12} - 88 \beta_{11} - 365 \beta_{10} + \cdots - 268 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1069 \beta_{15} - 1499 \beta_{14} - 990 \beta_{13} + 2307 \beta_{12} - 585 \beta_{11} - 1733 \beta_{10} + \cdots - 5954 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4796 \beta_{15} - 7814 \beta_{14} - 1412 \beta_{13} + 5352 \beta_{12} - 1821 \beta_{11} + \cdots - 32240 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 14276 \beta_{15} - 31975 \beta_{14} + 3214 \beta_{13} - 431 \beta_{12} - 7520 \beta_{11} + \cdots - 129470 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 33577 \beta_{15} - 112910 \beta_{14} + 28382 \beta_{13} - 88920 \beta_{12} - 36151 \beta_{11} + \cdots - 442790 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 66420 \beta_{15} - 350377 \beta_{14} + 131804 \beta_{13} - 596749 \beta_{12} - 140275 \beta_{11} + \cdots - 1349732 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 18892 \beta_{15} - 441224 \beta_{14} + 295627 \beta_{13} - 1487792 \beta_{12} - 184820 \beta_{11} + \cdots - 1706677 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 979469 \beta_{15} - 1076635 \beta_{14} + 2691456 \beta_{13} - 13102785 \beta_{12} - 346745 \beta_{11} + \cdots - 4310888 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 8972462 \beta_{15} + 6331182 \beta_{14} + 11361702 \beta_{13} - 52379620 \beta_{12} + 2817819 \beta_{11} + \cdots + 24189130 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 52306466 \beta_{15} + 65971029 \beta_{14} + 41411014 \beta_{13} - 184022069 \beta_{12} + \cdots + 258516418 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times\).

\(n\) \(2081\) \(2431\) \(3457\) \(3781\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3889.1
1.15135 + 1.37875i
−0.315931 − 0.438405i
1.15135 − 1.37875i
−0.315931 + 0.438405i
1.37578 + 2.65312i
−1.59571 + 0.665253i
1.37578 − 2.65312i
−1.59571 − 0.665253i
−2.05868 − 0.306986i
3.74844 − 1.37690i
−2.05868 + 0.306986i
3.74844 + 1.37690i
0.774725 − 2.71943i
−1.57997 − 0.888702i
0.774725 + 2.71943i
−1.57997 + 0.888702i
0 0 0 −1.27498 − 1.83696i 0 1.23231i 0 0 0
3889.2 0 0 0 −1.27498 − 1.83696i 0 1.23231i 0 0 0
3889.3 0 0 0 −1.27498 + 1.83696i 0 − 1.23231i 0 0 0
3889.4 0 0 0 −1.27498 + 1.83696i 0 − 1.23231i 0 0 0
3889.5 0 0 0 −0.463409 − 2.18752i 0 − 3.14971i 0 0 0
3889.6 0 0 0 −0.463409 − 2.18752i 0 − 3.14971i 0 0 0
3889.7 0 0 0 −0.463409 + 2.18752i 0 3.14971i 0 0 0
3889.8 0 0 0 −0.463409 + 2.18752i 0 3.14971i 0 0 0
3889.9 0 0 0 1.33104 − 1.79676i 0 4.27541i 0 0 0
3889.10 0 0 0 1.33104 − 1.79676i 0 4.27541i 0 0 0
3889.11 0 0 0 1.33104 + 1.79676i 0 − 4.27541i 0 0 0
3889.12 0 0 0 1.33104 + 1.79676i 0 − 4.27541i 0 0 0
3889.13 0 0 0 1.90735 − 1.16705i 0 − 3.04658i 0 0 0
3889.14 0 0 0 1.90735 − 1.16705i 0 − 3.04658i 0 0 0
3889.15 0 0 0 1.90735 + 1.16705i 0 3.04658i 0 0 0
3889.16 0 0 0 1.90735 + 1.16705i 0 3.04658i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3889.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
24.h odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4320.2.d.h 16
3.b odd 2 1 4320.2.d.g 16
4.b odd 2 1 1080.2.d.h yes 16
5.b even 2 1 4320.2.d.g 16
8.b even 2 1 4320.2.d.g 16
8.d odd 2 1 1080.2.d.g ✓ 16
12.b even 2 1 1080.2.d.g ✓ 16
15.d odd 2 1 inner 4320.2.d.h 16
20.d odd 2 1 1080.2.d.g ✓ 16
24.f even 2 1 1080.2.d.h yes 16
24.h odd 2 1 inner 4320.2.d.h 16
40.e odd 2 1 1080.2.d.h yes 16
40.f even 2 1 inner 4320.2.d.h 16
60.h even 2 1 1080.2.d.h yes 16
120.i odd 2 1 4320.2.d.g 16
120.m even 2 1 1080.2.d.g ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.d.g ✓ 16 8.d odd 2 1
1080.2.d.g ✓ 16 12.b even 2 1
1080.2.d.g ✓ 16 20.d odd 2 1
1080.2.d.g ✓ 16 120.m even 2 1
1080.2.d.h yes 16 4.b odd 2 1
1080.2.d.h yes 16 24.f even 2 1
1080.2.d.h yes 16 40.e odd 2 1
1080.2.d.h yes 16 60.h even 2 1
4320.2.d.g 16 3.b odd 2 1
4320.2.d.g 16 5.b even 2 1
4320.2.d.g 16 8.b even 2 1
4320.2.d.g 16 120.i odd 2 1
4320.2.d.h 16 1.a even 1 1 trivial
4320.2.d.h 16 15.d odd 2 1 inner
4320.2.d.h 16 24.h odd 2 1 inner
4320.2.d.h 16 40.f 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 \(S_{2}^{\mathrm{new}}(4320, [\chi])\):

\( T_{7}^{8} + 39T_{7}^{6} + 500T_{7}^{4} + 2356T_{7}^{2} + 2556 \) Copy content Toggle raw display
\( T_{11}^{8} + 43T_{11}^{6} + 556T_{11}^{4} + 2032T_{11}^{2} + 284 \) Copy content Toggle raw display
\( T_{13}^{8} - 88T_{13}^{6} + 2492T_{13}^{4} - 27228T_{13}^{2} + 101104 \) Copy content Toggle raw display
\( T_{53}^{4} + 24T_{53}^{3} + 206T_{53}^{2} + 754T_{53} + 999 \) 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
$5$ \( (T^{8} - 3 T^{7} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 39 T^{6} + \cdots + 2556)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 43 T^{6} + \cdots + 284)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 88 T^{6} + \cdots + 101104)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 77 T^{6} + \cdots + 356)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 81 T^{6} + \cdots + 28836)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 73 T^{6} + \cdots + 356)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 80 T^{6} + \cdots + 72704)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 44 T^{2} + \cdots + 257)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} - 296 T^{6} + \cdots + 1617664)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 264 T^{6} + \cdots + 909936)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 216 T^{6} + \cdots + 1617664)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 140 T^{6} + \cdots + 22784)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 24 T^{3} + \cdots + 999)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + 208 T^{6} + \cdots + 3954416)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 389 T^{6} + \cdots + 44612496)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 468 T^{6} + \cdots + 6470656)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 452 T^{6} + \cdots + 73704816)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 135 T^{6} + \cdots + 207036)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + T^{3} - 165 T^{2} + \cdots + 3364)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 16 T^{3} + \cdots + 213)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} - 464 T^{6} + \cdots + 32757696)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 399 T^{6} + \cdots + 39301056)^{2} \) Copy content Toggle raw display
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