Properties

Label 432.6.s.a
Level $432$
Weight $6$
Character orbit 432.s
Analytic conductor $69.286$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,6,Mod(143,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.143");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 432.s (of order \(6\), degree \(2\), 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: \(69.2858101592\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 8 x^{19} - 12929 x^{18} + 122470 x^{17} + 67551337 x^{16} - 634332392 x^{15} - 188739189566 x^{14} + 1465408138900 x^{13} + \cdots + 12\!\cdots\!83 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{39} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 3 \beta_1 - 6) q^{5} + (\beta_{4} - 3 \beta_1 - 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 3 \beta_1 - 6) q^{5} + (\beta_{4} - 3 \beta_1 - 3) q^{7} + (\beta_{11} - \beta_{5} + 57 \beta_1 - 57) q^{11} + (\beta_{6} - \beta_{4} - 18 \beta_1) q^{13} + (\beta_{17} + \beta_{14} + \beta_{12} - \beta_{4} - \beta_{3} + \beta_{2} - 154 \beta_1 + 77) q^{17} + (\beta_{17} - \beta_{16} + \beta_{14} + \beta_{11} - 2 \beta_{5} + 4 \beta_{3} - 4 \beta_{2} + 5 \beta_1 - 3) q^{19} + ( - \beta_{18} + \beta_{16} - 3 \beta_{12} - 2 \beta_{6} + 7 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + \cdots + 1) q^{23}+ \cdots + ( - 29 \beta_{19} + 29 \beta_{18} + 52 \beta_{17} - 6 \beta_{16} - 3 \beta_{15} + \cdots + 6777) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 87 q^{5} - 87 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 87 q^{5} - 87 q^{7} - 570 q^{11} - 181 q^{13} - 99 q^{23} + 7903 q^{25} - 13191 q^{29} - 6651 q^{31} - 10830 q^{35} - 4856 q^{37} - 846 q^{41} + 15315 q^{47} + 17891 q^{49} - 13308 q^{59} - 24773 q^{61} + 48255 q^{65} + 6402 q^{67} + 43188 q^{71} - 11614 q^{73} + 79317 q^{77} + 171897 q^{79} + 43347 q^{83} - 8850 q^{85} + 210684 q^{95} + 66332 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 8 x^{19} - 12929 x^{18} + 122470 x^{17} + 67551337 x^{16} - 634332392 x^{15} - 188739189566 x^{14} + 1465408138900 x^{13} + \cdots + 12\!\cdots\!83 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 44\!\cdots\!28 \nu^{19} + \cdots - 21\!\cdots\!06 ) / 35\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\!\cdots\!75 \nu^{19} + \cdots + 55\!\cdots\!42 ) / 35\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 65\!\cdots\!61 \nu^{19} + \cdots - 32\!\cdots\!96 ) / 18\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 88\!\cdots\!75 \nu^{19} + \cdots - 54\!\cdots\!34 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 61\!\cdots\!07 \nu^{19} + \cdots - 27\!\cdots\!13 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 72\!\cdots\!79 \nu^{19} + \cdots + 29\!\cdots\!70 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 30\!\cdots\!99 \nu^{19} + \cdots - 17\!\cdots\!62 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10\!\cdots\!75 \nu^{19} + \cdots - 52\!\cdots\!24 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 39\!\cdots\!98 \nu^{19} + \cdots + 20\!\cdots\!87 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13\!\cdots\!57 \nu^{19} + \cdots + 63\!\cdots\!99 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15\!\cdots\!46 \nu^{19} + \cdots + 78\!\cdots\!47 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 57\!\cdots\!71 \nu^{19} + \cdots - 28\!\cdots\!83 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 24\!\cdots\!14 \nu^{19} + \cdots + 12\!\cdots\!93 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 61\!\cdots\!03 \nu^{19} + \cdots - 24\!\cdots\!70 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 54\!\cdots\!25 \nu^{19} + \cdots - 29\!\cdots\!54 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 90\!\cdots\!36 \nu^{19} + \cdots - 43\!\cdots\!99 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 11\!\cdots\!03 \nu^{19} + \cdots - 56\!\cdots\!19 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 54\!\cdots\!71 \nu^{19} + \cdots - 25\!\cdots\!90 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 35\!\cdots\!23 \nu^{19} + \cdots + 18\!\cdots\!02 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 3\beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{19} + \beta_{17} + \beta_{16} + \beta_{15} - \beta_{14} + \beta_{13} - 6 \beta_{11} - 2 \beta_{7} + 2 \beta_{4} + \beta_{3} - 5 \beta_{2} - 4 \beta _1 + 3892 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 66 \beta_{19} - 9 \beta_{18} - 32 \beta_{17} - 120 \beta_{16} - 192 \beta_{15} + 59 \beta_{14} + 486 \beta_{13} + 667 \beta_{12} + 312 \beta_{11} - 180 \beta_{10} - 378 \beta_{9} - 189 \beta_{8} - 162 \beta_{7} + 27 \beta_{6} + \cdots - 6141 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13303 \beta_{19} + 282 \beta_{18} + 12571 \beta_{17} + 13551 \beta_{16} + 15119 \beta_{15} - 13117 \beta_{14} + 4811 \beta_{13} + 6102 \beta_{12} - 66642 \beta_{11} + 5708 \beta_{10} + 6808 \beta_{9} + \cdots + 28538271 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 957138 \beta_{19} - 201795 \beta_{18} - 530928 \beta_{17} - 1550448 \beta_{16} - 2588208 \beta_{15} + 1109313 \beta_{14} + 6472642 \beta_{13} + 9763905 \beta_{12} + \cdots - 804961056 ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 51816000 \beta_{19} + 2118186 \beta_{18} + 46881004 \beta_{17} + 52050900 \beta_{16} + 61421052 \beta_{15} - 51805822 \beta_{14} + 2014216 \beta_{13} + 28596634 \beta_{12} + \cdots + 92700356541 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1430034554 \beta_{19} - 370832028 \beta_{18} - 821967866 \beta_{17} - 2069067126 \beta_{16} - 3550759810 \beta_{15} + 1860204170 \beta_{14} + \cdots - 1682784809307 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 591688955775 \beta_{19} + 38859795210 \beta_{18} + 514717890435 \beta_{17} + 585189374895 \beta_{16} + 732961294239 \beta_{15} - 611482960485 \beta_{14} + \cdots + 996811241729064 ) / 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 6268663795062 \beta_{19} - 1840034079747 \beta_{18} - 3601133649500 \beta_{17} - 8152723165572 \beta_{16} - 14354394662508 \beta_{15} + \cdots - 82\!\cdots\!09 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 748911693167989 \beta_{19} + 72178390297218 \beta_{18} + 625605888480241 \beta_{17} + 726853677735081 \beta_{16} + 971012853963329 \beta_{15} + \cdots + 12\!\cdots\!91 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 80\!\cdots\!88 \beta_{19} + \cdots - 11\!\cdots\!16 ) / 27 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 28\!\cdots\!32 \beta_{19} + \cdots + 46\!\cdots\!81 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 11\!\cdots\!68 \beta_{19} + \cdots - 16\!\cdots\!87 ) / 9 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 32\!\cdots\!01 \beta_{19} + \cdots + 52\!\cdots\!68 ) / 27 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 47\!\cdots\!06 \beta_{19} + \cdots - 68\!\cdots\!49 ) / 9 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 41\!\cdots\!15 \beta_{19} + \cdots + 66\!\cdots\!87 ) / 9 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 58\!\cdots\!02 \beta_{19} + \cdots - 85\!\cdots\!60 ) / 27 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 16\!\cdots\!92 \beta_{19} + \cdots + 25\!\cdots\!09 ) / 9 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 78\!\cdots\!94 \beta_{19} + \cdots - 11\!\cdots\!71 ) / 9 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
56.6364 0.866025i
34.4444 0.866025i
33.8860 0.866025i
30.2198 0.866025i
8.83439 0.866025i
−17.3389 0.866025i
−26.0091 0.866025i
−26.2113 0.866025i
−26.8047 0.866025i
−63.6571 0.866025i
56.6364 + 0.866025i
34.4444 + 0.866025i
33.8860 + 0.866025i
30.2198 + 0.866025i
8.83439 + 0.866025i
−17.3389 + 0.866025i
−26.0091 + 0.866025i
−26.2113 + 0.866025i
−26.8047 + 0.866025i
−63.6571 + 0.866025i
0 0 0 −88.7047 + 51.2137i 0 158.148 + 91.3070i 0 0 0
143.2 0 0 0 −55.4166 + 31.9948i 0 −165.007 95.2670i 0 0 0
143.3 0 0 0 −54.5790 + 31.5112i 0 −155.514 89.7863i 0 0 0
143.4 0 0 0 −49.0797 + 28.3362i 0 75.4185 + 43.5429i 0 0 0
143.5 0 0 0 −17.0016 + 9.81587i 0 59.3512 + 34.2664i 0 0 0
143.6 0 0 0 22.2583 12.8508i 0 −101.723 58.7296i 0 0 0
143.7 0 0 0 35.2636 20.3595i 0 165.669 + 95.6488i 0 0 0
143.8 0 0 0 35.5670 20.5346i 0 −82.3079 47.5205i 0 0 0
143.9 0 0 0 36.4570 21.0484i 0 69.2990 + 40.0098i 0 0 0
143.10 0 0 0 91.7356 52.9636i 0 −66.8335 38.5864i 0 0 0
287.1 0 0 0 −88.7047 51.2137i 0 158.148 91.3070i 0 0 0
287.2 0 0 0 −55.4166 31.9948i 0 −165.007 + 95.2670i 0 0 0
287.3 0 0 0 −54.5790 31.5112i 0 −155.514 + 89.7863i 0 0 0
287.4 0 0 0 −49.0797 28.3362i 0 75.4185 43.5429i 0 0 0
287.5 0 0 0 −17.0016 9.81587i 0 59.3512 34.2664i 0 0 0
287.6 0 0 0 22.2583 + 12.8508i 0 −101.723 + 58.7296i 0 0 0
287.7 0 0 0 35.2636 + 20.3595i 0 165.669 95.6488i 0 0 0
287.8 0 0 0 35.5670 + 20.5346i 0 −82.3079 + 47.5205i 0 0 0
287.9 0 0 0 36.4570 + 21.0484i 0 69.2990 40.0098i 0 0 0
287.10 0 0 0 91.7356 + 52.9636i 0 −66.8335 + 38.5864i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.6.s.a 20
3.b odd 2 1 144.6.s.a 20
4.b odd 2 1 432.6.s.b 20
9.c even 3 1 144.6.s.c yes 20
9.d odd 6 1 432.6.s.b 20
12.b even 2 1 144.6.s.c yes 20
36.f odd 6 1 144.6.s.a 20
36.h even 6 1 inner 432.6.s.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.6.s.a 20 3.b odd 2 1
144.6.s.a 20 36.f odd 6 1
144.6.s.c yes 20 9.c even 3 1
144.6.s.c yes 20 12.b even 2 1
432.6.s.a 20 1.a even 1 1 trivial
432.6.s.a 20 36.h even 6 1 inner
432.6.s.b 20 4.b odd 2 1
432.6.s.b 20 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(432, [\chi])\):

\( T_{5}^{20} + 87 T_{5}^{19} - 15792 T_{5}^{18} - 1593405 T_{5}^{17} + 191955933 T_{5}^{16} + 20754847728 T_{5}^{15} - 799872700815 T_{5}^{14} - 117087697277847 T_{5}^{13} + \cdots + 77\!\cdots\!76 \) Copy content Toggle raw display
\( T_{7}^{20} + 87 T_{7}^{19} - 89196 T_{7}^{18} - 7979553 T_{7}^{17} + 5282435493 T_{7}^{16} + 479254352076 T_{7}^{15} - 177384228292107 T_{7}^{14} + \cdots + 24\!\cdots\!16 \) 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
$5$ \( T^{20} + 87 T^{19} + \cdots + 77\!\cdots\!76 \) Copy content Toggle raw display
$7$ \( T^{20} + 87 T^{19} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{20} + 570 T^{19} + \cdots + 78\!\cdots\!89 \) Copy content Toggle raw display
$13$ \( T^{20} + 181 T^{19} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} + 14795337 T^{18} + \cdots + 71\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{20} + 23722653 T^{18} + \cdots + 36\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{20} + 99 T^{19} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{20} + 13191 T^{19} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{20} + 6651 T^{19} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{10} + 2428 T^{9} + \cdots - 38\!\cdots\!16)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + 846 T^{19} + \cdots + 63\!\cdots\!29 \) Copy content Toggle raw display
$43$ \( T^{20} - 918444987 T^{18} + \cdots + 63\!\cdots\!81 \) Copy content Toggle raw display
$47$ \( T^{20} - 15315 T^{19} + \cdots + 26\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{20} + 4592491344 T^{18} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + 13308 T^{19} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{20} + 24773 T^{19} + \cdots + 61\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{20} - 6402 T^{19} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{10} - 21594 T^{9} + \cdots - 26\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 5807 T^{9} + \cdots - 71\!\cdots\!56)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} - 171897 T^{19} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} - 43347 T^{19} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{20} + 73102018656 T^{18} + \cdots + 55\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{20} - 66332 T^{19} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
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