# Properties

 Label 32.6.b.a Level 32 Weight 6 Character orbit 32.b Analytic conductor 5.132 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$32 = 2^{5}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 32.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.13228223402$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.218489.1 Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 8 x + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( -24 - \beta_{3} ) q^{7} + ( -41 - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( -24 - \beta_{3} ) q^{7} + ( -41 - 2 \beta_{3} ) q^{9} + ( 5 \beta_{1} + 8 \beta_{2} ) q^{11} + ( 32 \beta_{1} + 3 \beta_{2} ) q^{13} + ( 104 - \beta_{3} ) q^{15} + ( 50 + 2 \beta_{3} ) q^{17} + ( -7 \beta_{1} - 24 \beta_{2} ) q^{19} + ( -160 \beta_{1} - 12 \beta_{2} ) q^{21} + ( -584 + 13 \beta_{3} ) q^{23} + ( 389 + 20 \beta_{3} ) q^{25} + ( -70 \beta_{1} - 24 \beta_{2} ) q^{27} + ( 256 \beta_{1} - 41 \beta_{2} ) q^{29} + ( 3232 + 12 \beta_{3} ) q^{31} + ( -588 - 18 \beta_{3} ) q^{33} + ( -16 \beta_{1} + 112 \beta_{2} ) q^{35} + ( -160 \beta_{1} + 57 \beta_{2} ) q^{37} + ( -8776 - 67 \beta_{3} ) q^{39} + ( -1142 - 68 \beta_{3} ) q^{41} + ( 421 \beta_{1} - 48 \beta_{2} ) q^{43} + ( -32 \beta_{1} + 231 \beta_{2} ) q^{45} + ( 13680 - 54 \beta_{3} ) q^{47} + ( 2457 + 48 \beta_{3} ) q^{49} + ( 322 \beta_{1} + 24 \beta_{2} ) q^{51} + ( 928 \beta_{1} - 131 \beta_{2} ) q^{53} + ( -21368 + 155 \beta_{3} ) q^{55} + ( -508 + 38 \beta_{3} ) q^{57} + ( -1427 \beta_{1} + 128 \beta_{2} ) q^{59} + ( -2016 \beta_{1} - 657 \beta_{2} ) q^{61} + ( 38360 + 89 \beta_{3} ) q^{63} + ( -4880 + 28 \beta_{3} ) q^{65} + ( -1359 \beta_{1} - 792 \beta_{2} ) q^{67} + ( 1184 \beta_{1} + 156 \beta_{2} ) q^{69} + ( -51672 - 57 \beta_{3} ) q^{71} + ( 9994 + 302 \beta_{3} ) q^{73} + ( 3109 \beta_{1} + 240 \beta_{2} ) q^{75} + ( -928 \beta_{1} + 836 \beta_{2} ) q^{77} + ( 61968 + 86 \beta_{3} ) q^{79} + ( 7421 - 322 \beta_{3} ) q^{81} + ( 2569 \beta_{1} + 704 \beta_{2} ) q^{83} + ( 32 \beta_{1} - 222 \beta_{2} ) q^{85} + ( -76968 - 471 \beta_{3} ) q^{87} + ( -21158 - 626 \beta_{3} ) q^{89} + ( -5168 \beta_{1} - 48 \beta_{2} ) q^{91} + ( 4864 \beta_{1} + 144 \beta_{2} ) q^{93} + ( 64936 - 473 \beta_{3} ) q^{95} + ( -24894 + 274 \beta_{3} ) q^{97} + ( -1821 \beta_{1} + 1728 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 96q^{7} - 164q^{9} + O(q^{10})$$ $$4q - 96q^{7} - 164q^{9} + 416q^{15} + 200q^{17} - 2336q^{23} + 1556q^{25} + 12928q^{31} - 2352q^{33} - 35104q^{39} - 4568q^{41} + 54720q^{47} + 9828q^{49} - 85472q^{55} - 2032q^{57} + 153440q^{63} - 19520q^{65} - 206688q^{71} + 39976q^{73} + 247872q^{79} + 29684q^{81} - 307872q^{87} - 84632q^{89} + 259744q^{95} - 99576q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 8 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 3 \nu^{2} + 2 \nu - 12$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 5 \nu^{2} + 10 \nu - 4$$ $$\beta_{3}$$ $$=$$ $$-4 \nu^{3} + 4 \nu^{2} + 40 \nu + 16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 16$$$$)/64$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 6 \beta_{2} + 20 \beta_{1} + 80$$$$)/64$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{3} + 14 \beta_{2} + 60 \beta_{1} + 496$$$$)/64$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$31$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 2.38600 − 1.51888i −1.88600 + 2.10784i −1.88600 − 2.10784i 2.38600 + 1.51888i
0 23.6095i 0 1.38521i 0 −160.704 0 −314.408 0
17.2 0 3.25452i 0 73.9600i 0 112.704 0 232.408 0
17.3 0 3.25452i 0 73.9600i 0 112.704 0 232.408 0
17.4 0 23.6095i 0 1.38521i 0 −160.704 0 −314.408 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.6.b.a 4
3.b odd 2 1 288.6.d.b 4
4.b odd 2 1 8.6.b.a 4
5.b even 2 1 800.6.d.a 4
5.c odd 4 2 800.6.f.a 8
8.b even 2 1 inner 32.6.b.a 4
8.d odd 2 1 8.6.b.a 4
12.b even 2 1 72.6.d.b 4
16.e even 4 2 256.6.a.n 4
16.f odd 4 2 256.6.a.k 4
20.d odd 2 1 200.6.d.a 4
20.e even 4 2 200.6.f.a 8
24.f even 2 1 72.6.d.b 4
24.h odd 2 1 288.6.d.b 4
40.e odd 2 1 200.6.d.a 4
40.f even 2 1 800.6.d.a 4
40.i odd 4 2 800.6.f.a 8
40.k even 4 2 200.6.f.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.b.a 4 4.b odd 2 1
8.6.b.a 4 8.d odd 2 1
32.6.b.a 4 1.a even 1 1 trivial
32.6.b.a 4 8.b even 2 1 inner
72.6.d.b 4 12.b even 2 1
72.6.d.b 4 24.f even 2 1
200.6.d.a 4 20.d odd 2 1
200.6.d.a 4 40.e odd 2 1
200.6.f.a 8 20.e even 4 2
200.6.f.a 8 40.k even 4 2
256.6.a.k 4 16.f odd 4 2
256.6.a.n 4 16.e even 4 2
288.6.d.b 4 3.b odd 2 1
288.6.d.b 4 24.h odd 2 1
800.6.d.a 4 5.b even 2 1
800.6.d.a 4 40.f even 2 1
800.6.f.a 8 5.c odd 4 2
800.6.f.a 8 40.i odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(32, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 404 T^{2} + 84150 T^{4} - 23855796 T^{6} + 3486784401 T^{8}$$
$5$ $$1 - 7028 T^{2} + 24404246 T^{4} - 68632812500 T^{6} + 95367431640625 T^{8}$$
$7$ $$( 1 + 48 T + 15502 T^{2} + 806736 T^{3} + 282475249 T^{4} )^{2}$$
$11$ $$1 - 296436 T^{2} + 49128544726 T^{4} - 7688786399022036 T^{6} +$$$$67\!\cdots\!01$$$$T^{8}$$
$13$ $$1 - 894228 T^{2} + 396323515894 T^{4} - 123276923449147572 T^{6} +$$$$19\!\cdots\!01$$$$T^{8}$$
$17$ $$( 1 - 100 T + 2767462 T^{2} - 141985700 T^{3} + 2015993900449 T^{4} )^{2}$$
$19$ $$1 - 6794580 T^{2} + 21506967947254 T^{4} - 41658020173929518580 T^{6} +$$$$37\!\cdots\!01$$$$T^{8}$$
$23$ $$( 1 + 1168 T + 10055470 T^{2} + 7517648624 T^{3} + 41426511213649 T^{4} )^{2}$$
$29$ $$1 - 31255380 T^{2} + 976386653995702 T^{4} -$$$$13\!\cdots\!80$$$$T^{6} +$$$$17\!\cdots\!01$$$$T^{8}$$
$31$ $$( 1 - 6464 T + 65013054 T^{2} - 185058832064 T^{3} + 819628286980801 T^{4} )^{2}$$
$37$ $$1 - 241262580 T^{2} + 24149916431784598 T^{4} -$$$$11\!\cdots\!20$$$$T^{6} +$$$$23\!\cdots\!01$$$$T^{8}$$
$41$ $$( 1 + 2284 T + 146603254 T^{2} + 264615563084 T^{3} + 13422659310152401 T^{4} )^{2}$$
$43$ $$1 - 466346868 T^{2} + 96250708269010006 T^{4} -$$$$10\!\cdots\!32$$$$T^{6} +$$$$46\!\cdots\!01$$$$T^{8}$$
$47$ $$( 1 - 27360 T + 591338206 T^{2} - 6274879391520 T^{3} + 52599132235830049 T^{4} )^{2}$$
$53$ $$1 - 1039152180 T^{2} + 595616955270391126 T^{4} -$$$$18\!\cdots\!20$$$$T^{6} +$$$$30\!\cdots\!01$$$$T^{8}$$
$59$ $$1 - 1537424180 T^{2} + 1399694789142612374 T^{4} -$$$$78\!\cdots\!80$$$$T^{6} +$$$$26\!\cdots\!01$$$$T^{8}$$
$61$ $$1 + 741098540 T^{2} + 1478044222094100534 T^{4} +$$$$52\!\cdots\!40$$$$T^{6} +$$$$50\!\cdots\!01$$$$T^{8}$$
$67$ $$1 - 1366835860 T^{2} + 3274116308996825526 T^{4} -$$$$24\!\cdots\!40$$$$T^{6} +$$$$33\!\cdots\!01$$$$T^{8}$$
$71$ $$( 1 + 103344 T + 6217736974 T^{2} + 186456278049744 T^{3} + 3255243551009881201 T^{4} )^{2}$$
$73$ $$( 1 - 19988 T + 2541602870 T^{2} - 41436555000884 T^{3} + 4297625829703557649 T^{4} )^{2}$$
$79$ $$( 1 - 123936 T + 9855929374 T^{2} - 381358061866464 T^{3} + 9468276082626847201 T^{4} )^{2}$$
$83$ $$1 - 10047855188 T^{2} + 55381071937674414326 T^{4} -$$$$15\!\cdots\!12$$$$T^{6} +$$$$24\!\cdots\!01$$$$T^{8}$$
$89$ $$( 1 + 42316 T + 4292401174 T^{2} + 236295059643884 T^{3} + 31181719929966183601 T^{4} )^{2}$$
$97$ $$( 1 + 49788 T + 16391371462 T^{2} + 427546496715516 T^{3} + 73742412689492826049 T^{4} )^{2}$$