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Properties

Label 2160.4.h.h
Level $2160$
Weight $4$
Character orbit 2160.h
Analytic conductor $127.444$
Analytic rank $0$
Dimension $20$
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] = [2160,4,Mod(431,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2160.431");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2160.h (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: \(127.444125612\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 75 x^{18} + 3994 x^{16} - 96395 x^{14} + 1652310 x^{12} - 15841415 x^{10} + 108962369 x^{8} + \cdots + 285610000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{34}\cdot 3^{18}\cdot 5^{6} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{5} + ( - \beta_{4} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{5} + ( - \beta_{4} + \beta_1) q^{7} - \beta_{3} q^{11} + ( - \beta_{6} - 9) q^{13} - \beta_{15} q^{17} + (\beta_{11} - \beta_{4} + 2 \beta_1) q^{19} + (\beta_{17} - \beta_{13} + 3 \beta_{5}) q^{23} - 25 q^{25} + ( - \beta_{19} - \beta_{15} + \cdots + 4 \beta_{9}) q^{29}+ \cdots + ( - 4 \beta_{10} + \beta_{7} + \cdots - 244) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 172 q^{13} - 500 q^{25} - 500 q^{37} - 2200 q^{49} - 2060 q^{61} - 4852 q^{73} + 120 q^{85} - 4996 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 75 x^{18} + 3994 x^{16} - 96395 x^{14} + 1652310 x^{12} - 15841415 x^{10} + 108962369 x^{8} + \cdots + 285610000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 32\!\cdots\!07 \nu^{18} + \cdots - 10\!\cdots\!50 ) / 70\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 77\!\cdots\!24 \nu^{18} + \cdots - 36\!\cdots\!00 ) / 95\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 76\!\cdots\!15 \nu^{19} + \cdots - 50\!\cdots\!00 \nu ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 97\!\cdots\!86 \nu^{18} + \cdots - 30\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 99696748197 \nu^{19} - 7848593826260 \nu^{17} + 426091741080403 \nu^{15} + \cdots - 21\!\cdots\!00 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 67\!\cdots\!84 \nu^{18} + \cdots - 54\!\cdots\!80 ) / 76\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 50\!\cdots\!92 \nu^{18} + \cdots - 62\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 44\!\cdots\!26 \nu^{18} + \cdots - 19\!\cdots\!00 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 253829489107053 \nu^{19} + \cdots + 26\!\cdots\!00 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 97\!\cdots\!04 \nu^{18} + \cdots - 81\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17\!\cdots\!94 \nu^{18} + \cdots - 56\!\cdots\!00 ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 17\!\cdots\!14 \nu^{18} + \cdots + 58\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 52\!\cdots\!97 \nu^{19} + \cdots + 50\!\cdots\!00 \nu ) / 84\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 33\!\cdots\!33 \nu^{19} + \cdots - 35\!\cdots\!00 \nu ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 91\!\cdots\!13 \nu^{19} + \cdots - 92\!\cdots\!00 \nu ) / 84\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 17\!\cdots\!71 \nu^{19} + \cdots + 26\!\cdots\!00 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 17\!\cdots\!29 \nu^{19} + \cdots + 22\!\cdots\!00 \nu ) / 84\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 20\!\cdots\!26 \nu^{19} + \cdots + 21\!\cdots\!00 \nu ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 35\!\cdots\!93 \nu^{19} + \cdots - 36\!\cdots\!00 \nu ) / 40\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{18} + 3\beta_{16} - 2\beta_{15} - 4\beta_{14} - 3\beta_{13} + 3\beta_{9} + 12\beta_{3} ) / 180 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 3 \beta_{12} - 3 \beta_{11} + \beta_{10} - 3 \beta_{8} + \beta_{7} - 2 \beta_{6} - 48 \beta_{4} + \cdots + 677 ) / 90 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{19} - 37\beta_{18} - 124\beta_{15} - 93\beta_{14} + 721\beta_{9} ) / 90 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 258 \beta_{12} - 168 \beta_{11} - 61 \beta_{10} - 183 \beta_{8} - 56 \beta_{7} + 197 \beta_{6} + \cdots - 21297 ) / 90 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 175 \beta_{19} - 1801 \beta_{18} + 1815 \beta_{17} - 7743 \beta_{16} - 5592 \beta_{15} + \cdots - 9357 \beta_{3} ) / 180 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -583\beta_{10} - 580\beta_{7} + 2051\beta_{6} - 5482\beta_{2} - 168875 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 5295 \beta_{19} + 84349 \beta_{18} + 73575 \beta_{17} - 342507 \beta_{16} + 247568 \beta_{15} + \cdots - 371973 \beta_{3} ) / 180 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 203934 \beta_{12} + 137664 \beta_{11} - 44063 \beta_{10} + 132189 \beta_{8} - 45888 \beta_{7} + \cdots - 12029871 ) / 30 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -178475\beta_{19} + 3837393\beta_{18} + 10980896\beta_{15} + 5280107\beta_{14} - 89194729\beta_{9} ) / 90 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 27643422 \beta_{12} + 18856392 \beta_{11} + 5915199 \beta_{10} + 17745597 \beta_{8} + 6285464 \beta_{7} + \cdots + 1582404203 ) / 90 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2280325 \beta_{19} + 57428503 \beta_{18} - 45502285 \beta_{17} + 224628769 \beta_{16} + \cdots + 230984871 \beta_{3} ) / 60 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 52737101\beta_{10} + 56501712\beta_{7} - 194673805\beta_{6} + 462073848\beta_{2} + 14004277445 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 284506275 \beta_{19} - 7692943001 \beta_{18} - 6027112875 \beta_{17} + 29959460703 \beta_{16} + \cdots + 30631428537 \beta_{3} ) / 180 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 55161822678 \beta_{12} - 37858289928 \beta_{11} + 11740172051 \beta_{10} - 35220516153 \beta_{8} + \cdots + 3109009491127 ) / 90 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 12309380855 \beta_{19} - 342762931917 \beta_{18} - 965196344384 \beta_{15} + \cdots + 7968233475061 \beta_{9} ) / 90 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 818805337966 \beta_{12} - 562342069536 \beta_{11} - 174164042247 \beta_{10} - 522492126741 \beta_{8} + \cdots - 46073551529719 ) / 30 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 541624976875 \beta_{19} - 15259001627361 \beta_{18} + 11881960191315 \beta_{17} + \cdots - 60429627783777 \beta_{3} ) / 180 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 4649951571419 \beta_{10} - 5007086361096 \beta_{7} + 17216116758763 \beta_{6} - 40593069085038 \beta_{2} - 12\!\cdots\!67 ) / 9 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 23994222744895 \beta_{19} + 679070637327829 \beta_{18} + 528432719013975 \beta_{17} + \cdots - 26\!\cdots\!33 \beta_{3} ) / 180 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1
−5.77670 + 3.33518i
1.83481 + 1.05933i
−0.669247 − 0.386390i
3.58811 − 2.07159i
2.48813 − 1.43652i
−2.48813 − 1.43652i
−3.58811 − 2.07159i
0.669247 − 0.386390i
−1.83481 + 1.05933i
5.77670 + 3.33518i
5.77670 − 3.33518i
−1.83481 − 1.05933i
0.669247 + 0.386390i
−3.58811 + 2.07159i
−2.48813 + 1.43652i
2.48813 + 1.43652i
3.58811 + 2.07159i
−0.669247 + 0.386390i
1.83481 − 1.05933i
−5.77670 − 3.33518i
0 0 0 − 5.00000i 0 − 29.7098i 0 0 0
431.2 0 0 0 − 5.00000i 0 − 23.8133i 0 0 0
431.3 0 0 0 − 5.00000i 0 − 20.5374i 0 0 0
431.4 0 0 0 − 5.00000i 0 − 19.8361i 0 0 0
431.5 0 0 0 − 5.00000i 0 0 0.000865371i 0 0 0
431.6 0 0 0 − 5.00000i 0 0 0.000865371i 0 0 0
431.7 0 0 0 − 5.00000i 0 19.8361i 0 0 0
431.8 0 0 0 − 5.00000i 0 20.5374i 0 0 0
431.9 0 0 0 − 5.00000i 0 23.8133i 0 0 0
431.10 0 0 0 − 5.00000i 0 29.7098i 0 0 0
431.11 0 0 0 5.00000i 0 − 29.7098i 0 0 0
431.12 0 0 0 5.00000i 0 − 23.8133i 0 0 0
431.13 0 0 0 5.00000i 0 − 20.5374i 0 0 0
431.14 0 0 0 5.00000i 0 − 19.8361i 0 0 0
431.15 0 0 0 5.00000i 0 0 0.000865371i 0 0 0
431.16 0 0 0 5.00000i 0 0 0.000865371i 0 0 0
431.17 0 0 0 5.00000i 0 19.8361i 0 0 0
431.18 0 0 0 5.00000i 0 20.5374i 0 0 0
431.19 0 0 0 5.00000i 0 23.8133i 0 0 0
431.20 0 0 0 5.00000i 0 29.7098i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.20
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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.4.h.h ✓ 20
3.b odd 2 1 inner 2160.4.h.h ✓ 20
4.b odd 2 1 inner 2160.4.h.h ✓ 20
12.b even 2 1 inner 2160.4.h.h ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2160.4.h.h ✓ 20 1.a even 1 1 trivial
2160.4.h.h ✓ 20 3.b odd 2 1 inner
2160.4.h.h ✓ 20 4.b odd 2 1 inner
2160.4.h.h ✓ 20 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} + 2265T_{7}^{8} + 1848411T_{7}^{6} + 648666603T_{7}^{4} + 83069413920T_{7}^{2} + 62208 \) acting on \(S_{4}^{\mathrm{new}}(2160, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{10} \) Copy content Toggle raw display
$7$ \( (T^{10} + 2265 T^{8} + \cdots + 62208)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{5} + 43 T^{4} + \cdots + 12186839)^{4} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 63\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 21\!\cdots\!72)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 3498744406252)^{4} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 7402114924900)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 75\!\cdots\!92)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 91\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 15798471127648)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 39\!\cdots\!27)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 65\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 20\!\cdots\!00)^{4} \) Copy content Toggle raw display
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