# Properties

 Label 64.20.a.h Level $64$ Weight $20$ Character orbit 64.a Self dual yes Analytic conductor $146.443$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$20$$ Character orbit: $$[\chi]$$ $$=$$ 64.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$146.442685796$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 50652 q^{3} + 2377410 q^{5} + 16917544 q^{7} + 1403363637 q^{9}+O(q^{10})$$ q + 50652 * q^3 + 2377410 * q^5 + 16917544 * q^7 + 1403363637 * q^9 $$q + 50652 q^{3} + 2377410 q^{5} + 16917544 q^{7} + 1403363637 q^{9} - 16212108 q^{11} - 50421615062 q^{13} + 120420571320 q^{15} + 225070099506 q^{17} - 1710278572660 q^{19} + 856907438688 q^{21} - 14036534788872 q^{23} - 13421408020025 q^{25} + 12212307114840 q^{27} - 1137835269510 q^{29} + 104626880141728 q^{31} - 821175694416 q^{33} + 40219938281040 q^{35} + 169392327370594 q^{37} - 25\!\cdots\!24 q^{39}+ \cdots - 22\!\cdots\!96 q^{99}+O(q^{100})$$ q + 50652 * q^3 + 2377410 * q^5 + 16917544 * q^7 + 1403363637 * q^9 - 16212108 * q^11 - 50421615062 * q^13 + 120420571320 * q^15 + 225070099506 * q^17 - 1710278572660 * q^19 + 856907438688 * q^21 - 14036534788872 * q^23 - 13421408020025 * q^25 + 12212307114840 * q^27 - 1137835269510 * q^29 + 104626880141728 * q^31 - 821175694416 * q^33 + 40219938281040 * q^35 + 169392327370594 * q^37 - 2553955646120424 * q^39 - 3309984750560838 * q^41 + 1127913532193492 * q^43 + 3336370744240170 * q^45 - 3498693987674256 * q^47 - 11112691890381207 * q^49 + 11400250680177912 * q^51 - 29956294112980302 * q^53 - 38542827680280 * q^55 - 86629030262374320 * q^57 + 58391397642732420 * q^59 - 23373685132672742 * q^61 + 23741466076947528 * q^63 - 119872851864549420 * q^65 - 205102524257382244 * q^67 - 710978560125944544 * q^69 + 177902341950417768 * q^71 + 299853775038660122 * q^73 - 679821159030306300 * q^75 - 274269050422752 * q^77 + 92227090144007440 * q^79 - 1012497699493199799 * q^81 + 1208542823470585932 * q^83 + 535083905266559460 * q^85 - 57633632071220520 * q^87 + 4371201192290304330 * q^89 - 853009891362447728 * q^91 + 5299560732938806656 * q^93 - 4066033381427610600 * q^95 - 635013222218448094 * q^97 - 22751482846316796 * q^99

## 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
 0
0 50652.0 0 2.37741e6 0 1.69175e7 0 1.40336e9 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$

## 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 64.20.a.h 1
4.b odd 2 1 64.20.a.b 1
8.b even 2 1 16.20.a.a 1
8.d odd 2 1 1.20.a.a 1
24.f even 2 1 9.20.a.a 1
40.e odd 2 1 25.20.a.a 1
40.k even 4 2 25.20.b.a 2
56.e even 2 1 49.20.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.20.a.a 1 8.d odd 2 1
9.20.a.a 1 24.f even 2 1
16.20.a.a 1 8.b even 2 1
25.20.a.a 1 40.e odd 2 1
25.20.b.a 2 40.k even 4 2
49.20.a.b 1 56.e even 2 1
64.20.a.b 1 4.b odd 2 1
64.20.a.h 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 50652$$ acting on $$S_{20}^{\mathrm{new}}(\Gamma_0(64))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 50652$$
$5$ $$T - 2377410$$
$7$ $$T - 16917544$$
$11$ $$T + 16212108$$
$13$ $$T + 50421615062$$
$17$ $$T - 225070099506$$
$19$ $$T + 1710278572660$$
$23$ $$T + 14036534788872$$
$29$ $$T + 1137835269510$$
$31$ $$T - 104626880141728$$
$37$ $$T - 169392327370594$$
$41$ $$T + 3309984750560838$$
$43$ $$T - 1127913532193492$$
$47$ $$T + 3498693987674256$$
$53$ $$T + 29\!\cdots\!02$$
$59$ $$T - 58\!\cdots\!20$$
$61$ $$T + 23\!\cdots\!42$$
$67$ $$T + 20\!\cdots\!44$$
$71$ $$T - 17\!\cdots\!68$$
$73$ $$T - 29\!\cdots\!22$$
$79$ $$T - 92\!\cdots\!40$$
$83$ $$T - 12\!\cdots\!32$$
$89$ $$T - 43\!\cdots\!30$$
$97$ $$T + 63\!\cdots\!94$$