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Properties

Label 432.8.i.c
Level $432$
Weight $8$
Character orbit 432.i
Analytic conductor $134.950$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.145");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(134.950331009\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{21} \)
Twist minimal: no (minimal twist has level 9)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{10} + 30 \beta_{7} + 30) q^{5} + (\beta_{11} - 3 \beta_{10} + \cdots - 2 \beta_1) q^{7} + (9 \beta_{11} - \beta_{10} + \cdots - 2 \beta_1) q^{11} + (7 \beta_{11} - 6 \beta_{10} + \cdots - 308) q^{13}+ \cdots + ( - 9097 \beta_{11} + \cdots - 54440 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 180 q^{5} + 84 q^{7} - 8460 q^{11} - 1848 q^{13} - 30564 q^{17} - 24432 q^{19} - 51588 q^{23} + 4746 q^{25} + 414648 q^{29} - 8196 q^{31} + 2210616 q^{35} + 139344 q^{37} + 1731582 q^{41} - 408372 q^{43}+ \cdots + 9977226 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{10} - 5 \nu^{9} + 631 \nu^{8} - 2494 \nu^{7} + 213005 \nu^{6} - 630307 \nu^{5} + \cdots + 2618829225 ) / 119996640 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} - 5 \nu^{9} + 631 \nu^{8} - 2494 \nu^{7} + 213005 \nu^{6} - 630307 \nu^{5} + \cdots + 4058788905 ) / 119996640 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} - 5 \nu^{9} + 631 \nu^{8} - 2494 \nu^{7} + 213005 \nu^{6} - 630307 \nu^{5} + \cdots + 89736389865 ) / 239993280 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 443 \nu^{10} - 2215 \nu^{9} + 183791 \nu^{8} - 721874 \nu^{7} + 26320567 \nu^{6} + \cdots + 18976459761 ) / 239993280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2515 \nu^{10} - 12575 \nu^{9} + 916771 \nu^{8} - 3591634 \nu^{7} + 119421359 \nu^{6} + \cdots + 502007984397 ) / 239993280 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25 \nu^{10} - 125 \nu^{9} + 8092 \nu^{8} - 31618 \nu^{7} + 902081 \nu^{6} - 2596105 \nu^{5} + \cdots + 1988761716 ) / 1481440 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3916394 \nu^{11} + 21540167 \nu^{10} - 1457064271 \nu^{9} + 6395237967 \nu^{8} + \cdots + 87187467749373 ) / 780837815928960 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 21966890 \nu^{11} + 852873845 \nu^{10} - 8693680945 \nu^{9} + 258696041061 \nu^{8} + \cdots + 70\!\cdots\!55 ) / 86759757325440 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 433021779 \nu^{11} - 335930102 \nu^{10} + 149474174962 \nu^{9} + 44897749129 \nu^{8} + \cdots + 27\!\cdots\!97 ) / 390418907964480 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 674669385 \nu^{11} - 3350347411 \nu^{10} + 236441521415 \nu^{9} - 894769481542 \nu^{8} + \cdots - 27\!\cdots\!12 ) / 390418907964480 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1399146876 \nu^{11} - 7693681027 \nu^{10} + 519257690903 \nu^{9} - 2277954898105 \nu^{8} + \cdots - 18\!\cdots\!03 ) / 780837815928960 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + \beta _1 + 12 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} - \beta_{2} - 714 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 36 \beta_{11} + 16 \beta_{10} + 32 \beta_{9} + 16 \beta_{8} - 1080 \beta_{7} - 8 \beta_{6} + \cdots - 6984 ) / 72 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 18 \beta_{11} + 8 \beta_{10} + 16 \beta_{9} + 8 \beta_{8} - 540 \beta_{7} - 14 \beta_{6} + \cdots + 103221 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 22212 \beta_{11} - 8856 \beta_{10} - 13264 \beta_{9} - 8312 \beta_{8} + 2370600 \beta_{7} + \cdots + 4322268 ) / 216 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 11196 \beta_{11} - 4468 \beta_{10} - 6712 \beta_{9} - 4196 \beta_{8} + 1188000 \beta_{7} + \cdots - 21509748 ) / 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1180224 \beta_{11} + 411656 \beta_{10} + 552848 \beta_{9} + 373384 \beta_{8} - 176405040 \beta_{7} + \cdots - 246596076 ) / 72 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3619170 \beta_{11} + 1266300 \beta_{10} + 1705640 \beta_{9} + 1149580 \beta_{8} - 537534900 \beta_{7} + \cdots + 3443103315 ) / 54 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 171131364 \beta_{11} - 54820952 \beta_{10} - 67594448 \beta_{9} - 46506424 \beta_{8} + \cdots + 38050923240 ) / 72 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 446002812 \beta_{11} - 143415236 \beta_{10} - 177561464 \beta_{9} - 122043412 \beta_{8} + \cdots - 246838018302 ) / 36 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 70631755956 \beta_{11} + 21735158760 \beta_{10} + 24639480272 \beta_{9} + 16831083496 \beta_{8} + \cdots - 16651687740228 ) / 216 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{7}\)

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
0.500000 − 2.70685i
0.500000 + 9.80854i
0.500000 + 1.48508i
0.500000 − 11.4952i
0.500000 + 9.08282i
0.500000 − 6.17443i
0.500000 + 2.70685i
0.500000 − 9.80854i
0.500000 − 1.48508i
0.500000 + 11.4952i
0.500000 − 9.08282i
0.500000 + 6.17443i
0 0 0 −167.952 + 290.901i 0 −442.025 − 765.610i 0 0 0
145.2 0 0 0 −52.7641 + 91.3900i 0 −761.419 − 1318.82i 0 0 0
145.3 0 0 0 −47.9866 + 83.1153i 0 189.000 + 327.358i 0 0 0
145.4 0 0 0 −32.6274 + 56.5123i 0 118.194 + 204.717i 0 0 0
145.5 0 0 0 145.304 − 251.673i 0 555.940 + 962.916i 0 0 0
145.6 0 0 0 246.026 − 426.130i 0 382.311 + 662.182i 0 0 0
289.1 0 0 0 −167.952 − 290.901i 0 −442.025 + 765.610i 0 0 0
289.2 0 0 0 −52.7641 − 91.3900i 0 −761.419 + 1318.82i 0 0 0
289.3 0 0 0 −47.9866 − 83.1153i 0 189.000 − 327.358i 0 0 0
289.4 0 0 0 −32.6274 − 56.5123i 0 118.194 − 204.717i 0 0 0
289.5 0 0 0 145.304 + 251.673i 0 555.940 − 962.916i 0 0 0
289.6 0 0 0 246.026 + 426.130i 0 382.311 − 662.182i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.6
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 432.8.i.c 12
3.b odd 2 1 144.8.i.c 12
4.b odd 2 1 27.8.c.a 12
9.c even 3 1 inner 432.8.i.c 12
9.d odd 6 1 144.8.i.c 12
12.b even 2 1 9.8.c.a ✓ 12
36.f odd 6 1 27.8.c.a 12
36.f odd 6 1 81.8.a.c 6
36.h even 6 1 9.8.c.a ✓ 12
36.h even 6 1 81.8.a.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.8.c.a ✓ 12 12.b even 2 1
9.8.c.a ✓ 12 36.h even 6 1
27.8.c.a 12 4.b odd 2 1
27.8.c.a 12 36.f odd 6 1
81.8.a.c 6 36.f odd 6 1
81.8.a.e 6 36.h even 6 1
144.8.i.c 12 3.b odd 2 1
144.8.i.c 12 9.d odd 6 1
432.8.i.c 12 1.a even 1 1 trivial
432.8.i.c 12 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 180 T_{5}^{11} + 248202 T_{5}^{10} + 26226720 T_{5}^{9} + 37979473779 T_{5}^{8} + \cdots + 10\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(432, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 83\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 50\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 16\!\cdots\!76)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 73\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 53\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 17\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 45\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 27\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 56\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 54\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 74\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 75\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 21\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 81\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 34\!\cdots\!16)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 58\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 83\!\cdots\!69 \) Copy content Toggle raw display
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