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Properties

Label 432.6.i.f
Level $432$
Weight $6$
Character orbit 432.i
Analytic conductor $69.286$
Analytic rank $0$
Dimension $16$
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(145,432)]
 
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([0, 0, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.145");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 432.i (of order \(3\), 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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 943 x^{14} + 354586 x^{12} + 67201893 x^{10} + 6662403684 x^{8} + 324029196504 x^{6} + \cdots + 6387206400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{28} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{10} - 3 \beta_{8}) q^{5} + (\beta_{9} - 12 \beta_{8} - \beta_{2} + 12) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} - 3 \beta_{8}) q^{5} + (\beta_{9} - 12 \beta_{8} - \beta_{2} + 12) q^{7} + (\beta_{12} - \beta_{10} + 75 \beta_{8} + \cdots - 75) q^{11}+ \cdots + (67 \beta_{15} + 42 \beta_{14} + \cdots - 36441) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 25 q^{5} + 93 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 25 q^{5} + 93 q^{7} - 596 q^{11} - 89 q^{13} - 554 q^{17} + 874 q^{19} + 133 q^{23} - 4453 q^{25} + 1089 q^{29} + 3517 q^{31} - 8670 q^{35} + 19884 q^{37} - 11958 q^{41} - 6358 q^{43} + 18651 q^{47} - 22873 q^{49} - 27148 q^{53} - 33562 q^{55} - 5252 q^{59} - 28727 q^{61} - 1951 q^{65} + 76426 q^{67} + 57104 q^{71} + 159070 q^{73} - 11553 q^{77} + 97043 q^{79} - 87151 q^{83} - 132178 q^{85} + 21024 q^{89} - 469698 q^{91} - 30844 q^{95} - 291814 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 943 x^{14} + 354586 x^{12} + 67201893 x^{10} + 6662403684 x^{8} + 324029196504 x^{6} + \cdots + 6387206400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 96\!\cdots\!97 \nu^{14} + \cdots + 68\!\cdots\!76 ) / 92\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17\!\cdots\!38 \nu^{14} + \cdots - 20\!\cdots\!80 ) / 77\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 61\!\cdots\!49 \nu^{14} + \cdots - 49\!\cdots\!60 ) / 46\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 52\!\cdots\!89 \nu^{14} + \cdots + 24\!\cdots\!40 ) / 28\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 41\!\cdots\!63 \nu^{14} + \cdots - 60\!\cdots\!00 ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 64\!\cdots\!33 \nu^{14} + \cdots - 14\!\cdots\!50 ) / 23\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 39\!\cdots\!66 \nu^{14} + \cdots - 49\!\cdots\!40 ) / 77\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 100595975854811 \nu^{15} + \cdots + 26\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13\!\cdots\!77 \nu^{15} + \cdots - 30\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 19\!\cdots\!73 \nu^{15} + \cdots - 50\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 16\!\cdots\!67 \nu^{15} + \cdots - 44\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 15\!\cdots\!17 \nu^{15} + \cdots - 41\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!78 \nu^{15} + \cdots + 36\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13\!\cdots\!81 \nu^{15} + \cdots - 35\!\cdots\!00 ) / 83\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 13\!\cdots\!09 \nu^{15} + \cdots - 36\!\cdots\!00 ) / 68\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{14} + 4\beta_{11} - 4\beta_{9} + 42\beta_{8} - 2\beta_{5} + \beta_{4} + 2\beta_{2} - 21 ) / 162 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -9\beta_{7} + 9\beta_{6} - 2\beta_{5} - 9\beta_{3} - 2\beta_{2} + 27\beta _1 - 12728 ) / 108 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 198 \beta_{15} - 1258 \beta_{14} + 36 \beta_{13} + 198 \beta_{12} - 3056 \beta_{11} - 5724 \beta_{10} + \cdots + 74526 ) / 648 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4203 \beta_{7} - 4860 \beta_{6} + 397 \beta_{5} + 184 \beta_{4} + 3564 \beta_{3} + 2332 \beta_{2} + \cdots + 4859927 ) / 216 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 23526 \beta_{15} + 118696 \beta_{14} - 13374 \beta_{13} - 16722 \beta_{12} + 307748 \beta_{11} + \cdots - 10657188 ) / 324 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 896094 \beta_{7} + 1119303 \beta_{6} + 51276 \beta_{5} - 20581 \beta_{4} - 741033 \beta_{3} + \cdots - 999571336 ) / 216 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 9717930 \beta_{15} - 48817378 \beta_{14} + 9048006 \beta_{13} + 4228074 \beta_{12} - 126821582 \beta_{11} + \cdots + 5810178981 ) / 648 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 188311311 \beta_{7} - 250922655 \beta_{6} - 40980571 \beta_{5} - 4640421 \beta_{4} + \cdots + 210484548989 ) / 216 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1904817186 \beta_{15} + 10355896262 \beta_{14} - 2526761106 \beta_{13} - 238524390 \beta_{12} + \cdots - 1519000697793 ) / 648 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 39599393193 \beta_{7} + 55917296406 \beta_{6} + 14894126605 \beta_{5} + 3316706698 \beta_{4} + \cdots - 44796101680963 ) / 216 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 361520606574 \beta_{15} - 2225449776502 \beta_{14} + 651125985192 \beta_{13} - 99644689422 \beta_{12} + \cdots + 383110015537890 ) / 648 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 8362376487909 \beta_{7} - 12446299801233 \beta_{6} - 4413807960225 \beta_{5} - 1232571906979 \beta_{4} + \cdots + 95\!\cdots\!77 ) / 216 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 66579807458592 \beta_{15} + 481614568454240 \beta_{14} - 160684275023550 \beta_{13} + \cdots - 94\!\cdots\!91 ) / 648 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 887228590411395 \beta_{7} + \cdots - 10\!\cdots\!89 ) / 108 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 59\!\cdots\!00 \beta_{15} + \cdots + 11\!\cdots\!38 ) / 324 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
− 15.0522i
5.93418i
2.68595i
14.4255i
− 14.7582i
13.3608i
0.0140385i
− 8.34214i
15.0522i
− 5.93418i
− 2.68595i
− 14.4255i
14.7582i
− 13.3608i
− 0.0140385i
8.34214i
0 0 0 −46.3431 + 80.2687i 0 85.1399 + 147.467i 0 0 0
145.2 0 0 0 −38.2195 + 66.1981i 0 45.4692 + 78.7549i 0 0 0
145.3 0 0 0 −23.8592 + 41.3253i 0 −82.8535 − 143.507i 0 0 0
145.4 0 0 0 2.41676 − 4.18594i 0 −104.276 − 180.611i 0 0 0
145.5 0 0 0 4.40840 − 7.63556i 0 −42.3054 − 73.2752i 0 0 0
145.6 0 0 0 11.6242 − 20.1337i 0 96.0253 + 166.321i 0 0 0
145.7 0 0 0 35.2589 − 61.0702i 0 19.5485 + 33.8589i 0 0 0
145.8 0 0 0 42.2136 − 73.1160i 0 29.7517 + 51.5315i 0 0 0
289.1 0 0 0 −46.3431 − 80.2687i 0 85.1399 − 147.467i 0 0 0
289.2 0 0 0 −38.2195 − 66.1981i 0 45.4692 − 78.7549i 0 0 0
289.3 0 0 0 −23.8592 − 41.3253i 0 −82.8535 + 143.507i 0 0 0
289.4 0 0 0 2.41676 + 4.18594i 0 −104.276 + 180.611i 0 0 0
289.5 0 0 0 4.40840 + 7.63556i 0 −42.3054 + 73.2752i 0 0 0
289.6 0 0 0 11.6242 + 20.1337i 0 96.0253 − 166.321i 0 0 0
289.7 0 0 0 35.2589 + 61.0702i 0 19.5485 − 33.8589i 0 0 0
289.8 0 0 0 42.2136 + 73.1160i 0 29.7517 − 51.5315i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.6.i.f 16
3.b odd 2 1 144.6.i.f 16
4.b odd 2 1 216.6.i.b 16
9.c even 3 1 inner 432.6.i.f 16
9.d odd 6 1 144.6.i.f 16
12.b even 2 1 72.6.i.b ✓ 16
36.f odd 6 1 216.6.i.b 16
36.f odd 6 1 648.6.a.i 8
36.h even 6 1 72.6.i.b ✓ 16
36.h even 6 1 648.6.a.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.6.i.b ✓ 16 12.b even 2 1
72.6.i.b ✓ 16 36.h even 6 1
144.6.i.f 16 3.b odd 2 1
144.6.i.f 16 9.d odd 6 1
216.6.i.b 16 4.b odd 2 1
216.6.i.b 16 36.f odd 6 1
432.6.i.f 16 1.a even 1 1 trivial
432.6.i.f 16 9.c even 3 1 inner
648.6.a.h 8 36.h even 6 1
648.6.a.i 8 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 25 T_{5}^{15} + 15039 T_{5}^{14} + 113630 T_{5}^{13} + 153864845 T_{5}^{12} + \cdots + 39\!\cdots\!76 \) acting on \(S_{6}^{\mathrm{new}}(432, [\chi])\). 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
$5$ \( T^{16} + \cdots + 39\!\cdots\!76 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 43\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 72\!\cdots\!92)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 70\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 51\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 10\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 60\!\cdots\!49 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 30\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 27\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 16\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 11\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 29\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 44\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 17\!\cdots\!80)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 30\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 21\!\cdots\!69 \) Copy content Toggle raw display
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