Properties

Label 432.6.a.r
Level 432
Weight 6
Character orbit 432.a
Self dual yes
Analytic conductor 69.286
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 432.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.2858101592\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 108)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 9\sqrt{41}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{5} + ( 16 - 3 \beta ) q^{7} +O(q^{10})\) \( q -\beta q^{5} + ( 16 - 3 \beta ) q^{7} + ( 243 - 8 \beta ) q^{11} + ( 104 - 12 \beta ) q^{13} + ( -972 - 20 \beta ) q^{17} + ( 316 - 6 \beta ) q^{19} + ( 3402 - 4 \beta ) q^{23} + 196 q^{25} + ( -5832 + 38 \beta ) q^{29} + ( -1664 - 81 \beta ) q^{31} + ( 9963 - 16 \beta ) q^{35} + ( -4978 - 60 \beta ) q^{37} + ( -6804 - 58 \beta ) q^{41} + ( -2480 + 162 \beta ) q^{43} + ( 9234 + 264 \beta ) q^{47} + ( 13338 - 96 \beta ) q^{49} + ( 5832 + 341 \beta ) q^{53} + ( 26568 - 243 \beta ) q^{55} + ( 972 - 376 \beta ) q^{59} + ( 4088 + 720 \beta ) q^{61} + ( 39852 - 104 \beta ) q^{65} + ( -45032 + 450 \beta ) q^{67} + ( -22356 + 788 \beta ) q^{71} + ( -60607 - 504 \beta ) q^{73} + ( 83592 - 857 \beta ) q^{77} + ( -14384 + 840 \beta ) q^{79} + ( -7533 - 1752 \beta ) q^{83} + ( 66420 + 972 \beta ) q^{85} + ( 89424 + 358 \beta ) q^{89} + ( 121220 - 504 \beta ) q^{91} + ( 19926 - 316 \beta ) q^{95} + ( 44471 - 1272 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 32q^{7} + O(q^{10}) \) \( 2q + 32q^{7} + 486q^{11} + 208q^{13} - 1944q^{17} + 632q^{19} + 6804q^{23} + 392q^{25} - 11664q^{29} - 3328q^{31} + 19926q^{35} - 9956q^{37} - 13608q^{41} - 4960q^{43} + 18468q^{47} + 26676q^{49} + 11664q^{53} + 53136q^{55} + 1944q^{59} + 8176q^{61} + 79704q^{65} - 90064q^{67} - 44712q^{71} - 121214q^{73} + 167184q^{77} - 28768q^{79} - 15066q^{83} + 132840q^{85} + 178848q^{89} + 242440q^{91} + 39852q^{95} + 88942q^{97} + O(q^{100}) \)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.70156
−2.70156
0 0 0 −57.6281 0 −156.884 0 0 0
1.2 0 0 0 57.6281 0 188.884 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.6.a.r 2
3.b odd 2 1 432.6.a.q 2
4.b odd 2 1 108.6.a.b 2
12.b even 2 1 108.6.a.c yes 2
36.f odd 6 2 324.6.e.g 4
36.h even 6 2 324.6.e.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.a.b 2 4.b odd 2 1
108.6.a.c yes 2 12.b even 2 1
324.6.e.f 4 36.h even 6 2
324.6.e.g 4 36.f odd 6 2
432.6.a.q 2 3.b odd 2 1
432.6.a.r 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(432))\):

\( T_{5}^{2} - 3321 \)
\( T_{7}^{2} - 32 T_{7} - 29633 \)
\( T_{11}^{2} - 486 T_{11} - 153495 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 2929 T^{2} + 9765625 T^{4} \)
$7$ \( 1 - 32 T + 3981 T^{2} - 537824 T^{3} + 282475249 T^{4} \)
$11$ \( 1 - 486 T + 168607 T^{2} - 78270786 T^{3} + 25937424601 T^{4} \)
$13$ \( 1 - 208 T + 275178 T^{2} - 77228944 T^{3} + 137858491849 T^{4} \)
$17$ \( 1 + 1944 T + 2456098 T^{2} + 2760202008 T^{3} + 2015993900449 T^{4} \)
$19$ \( 1 - 632 T + 4932498 T^{2} - 1564894568 T^{3} + 6131066257801 T^{4} \)
$23$ \( 1 - 6804 T + 24393154 T^{2} - 43792877772 T^{3} + 41426511213649 T^{4} \)
$29$ \( 1 + 11664 T + 70238998 T^{2} + 239242041936 T^{3} + 420707233300201 T^{4} \)
$31$ \( 1 + 3328 T + 38238117 T^{2} + 95277814528 T^{3} + 819628286980801 T^{4} \)
$37$ \( 1 + 9956 T + 151512798 T^{2} + 690388435892 T^{3} + 4808584372417849 T^{4} \)
$41$ \( 1 + 13608 T + 266834974 T^{2} + 1576571183208 T^{3} + 13422659310152401 T^{4} \)
$43$ \( 1 + 4960 T + 213010962 T^{2} + 729161877280 T^{3} + 21611482313284249 T^{4} \)
$47$ \( 1 - 18468 T + 312496354 T^{2} - 4235543589276 T^{3} + 52599132235830049 T^{4} \)
$53$ \( 1 - 11664 T + 484234009 T^{2} - 4877832230352 T^{3} + 174887470365513049 T^{4} \)
$59$ \( 1 - 1944 T + 961283686 T^{2} - 1389812837256 T^{3} + 511116753300641401 T^{4} \)
$61$ \( 1 - 8176 T - 15702054 T^{2} - 6905419356976 T^{3} + 713342911662882601 T^{4} \)
$67$ \( 1 + 90064 T + 4055628738 T^{2} + 121597667636848 T^{3} + 1822837804551761449 T^{4} \)
$71$ \( 1 + 44712 T + 2046094414 T^{2} + 80670702741912 T^{3} + 3255243551009881201 T^{4} \)
$73$ \( 1 + 121214 T + 6975764499 T^{2} + 251285300073902 T^{3} + 4297625829703557649 T^{4} \)
$79$ \( 1 + 28768 T + 4017714654 T^{2} + 88520758486432 T^{3} + 9468276082626847201 T^{4} \)
$83$ \( 1 + 15066 T - 2258995409 T^{2} + 59345586327438 T^{3} + 15516041187205853449 T^{4} \)
$89$ \( 1 - 178848 T + 18739138030 T^{2} - 998697864334752 T^{3} + 31181719929966183601 T^{4} \)
$97$ \( 1 - 88942 T + 13779025491 T^{2} - 763775217138094 T^{3} + 73742412689492826049 T^{4} \)
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