# Properties

 Label 4.20.a.a Level $4$ Weight $20$ Character orbit 4.a Self dual yes Analytic conductor $9.153$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$4 = 2^{2}$$ Weight: $$k$$ $$=$$ $$20$$ Character orbit: $$[\chi]$$ $$=$$ 4.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.15266786226$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 36q^{3} - 196290q^{5} - 35905576q^{7} - 1162260171q^{9} + O(q^{10})$$ $$q - 36q^{3} - 196290q^{5} - 35905576q^{7} - 1162260171q^{9} - 12016099980q^{11} - 45529656874q^{13} + 7066440q^{15} + 496563248178q^{17} + 1410273986444q^{19} + 1292600736q^{21} - 7039745388792q^{23} - 19034956564025q^{25} + 83682778968q^{27} + 38996890912134q^{29} + 173641323230816q^{31} + 432579599280q^{33} + 7047905513040q^{35} - 1108106825662306q^{37} + 1639067647464q^{39} - 1444509198124614q^{41} + 4646075748354260q^{43} + 228140048965590q^{45} + 8950457686524048q^{47} - 10109684797481367q^{49} - 17876276934408q^{51} - 32948524384463538q^{53} + 2358640265074200q^{55} - 50769863511984q^{57} + 36999205673523588q^{59} + 82929105285760742q^{61} + 41731620901613496q^{63} + 8937016347797460q^{65} - 186668590860047716q^{67} + 253430833996512q^{69} - 596514630027659112q^{71} + 310786775495585306q^{73} + 685258436304900q^{75} + 431444991055488480q^{77} + 700397513485701872q^{79} + 1350847198802088009q^{81} - 1357882121724855732q^{83} - 97470399984859620q^{85} - 1403888072836824q^{87} - 5991411253779123894q^{89} + 1634768555143329424q^{91} - 6251087636309376q^{93} - 276822680799092760q^{95} + 4531118407744664354q^{97} + 13965834417507896580q^{99} + O(q^{100})$$

## 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
 0
0 −36.0000 0 −196290. 0 −3.59056e7 0 −1.16226e9 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 4.20.a.a 1
3.b odd 2 1 36.20.a.b 1
4.b odd 2 1 16.20.a.b 1
5.b even 2 1 100.20.a.a 1
5.c odd 4 2 100.20.c.a 2
8.b even 2 1 64.20.a.f 1
8.d odd 2 1 64.20.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.20.a.a 1 1.a even 1 1 trivial
16.20.a.b 1 4.b odd 2 1
36.20.a.b 1 3.b odd 2 1
64.20.a.d 1 8.d odd 2 1
64.20.a.f 1 8.b even 2 1
100.20.a.a 1 5.b even 2 1
100.20.c.a 2 5.c odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{20}^{\mathrm{new}}(\Gamma_0(4))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$36 + T$$
$5$ $$196290 + T$$
$7$ $$35905576 + T$$
$11$ $$12016099980 + T$$
$13$ $$45529656874 + T$$
$17$ $$-496563248178 + T$$
$19$ $$-1410273986444 + T$$
$23$ $$7039745388792 + T$$
$29$ $$-38996890912134 + T$$
$31$ $$-173641323230816 + T$$
$37$ $$1108106825662306 + T$$
$41$ $$1444509198124614 + T$$
$43$ $$-4646075748354260 + T$$
$47$ $$-8950457686524048 + T$$
$53$ $$32948524384463538 + T$$
$59$ $$-36999205673523588 + T$$
$61$ $$-82929105285760742 + T$$
$67$ $$186668590860047716 + T$$
$71$ $$596514630027659112 + T$$
$73$ $$-310786775495585306 + T$$
$79$ $$-700397513485701872 + T$$
$83$ $$1357882121724855732 + T$$
$89$ $$5991411253779123894 + T$$
$97$ $$-4531118407744664354 + T$$
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