Properties

 Label 72.20.a.a Level $72$ Weight $20$ Character orbit 72.a Self dual yes Analytic conductor $164.748$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$20$$ Character orbit: $$[\chi]$$ $$=$$ 72.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$164.748021521$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{1453})$$ Defining polynomial: $$x^{2} - x - 363$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{7}\cdot 3\cdot 5$$ Twist minimal: no (minimal twist has level 8) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 960\sqrt{1453}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -613310 - 44 \beta ) q^{5} + ( 44255256 + 3190 \beta ) q^{7} +O(q^{10})$$ $$q + ( -613310 - 44 \beta ) q^{5} + ( 44255256 + 3190 \beta ) q^{7} + ( 3581893804 + 223467 \beta ) q^{11} + ( -5063461802 + 71660 \beta ) q^{13} + ( 36022539470 - 17338504 \beta ) q^{17} + ( -1560240236116 - 26137813 \beta ) q^{19} + ( 7379603545144 - 184291330 \beta ) q^{23} + ( -16104868999225 + 53971280 \beta ) q^{25} + ( 15124769622522 + 893092484 \beta ) q^{29} + ( -61694781388960 - 3864337064 \beta ) q^{31} + ( -215096133585360 - 3903690164 \beta ) q^{35} + ( 1007696585087262 - 3072429508 \beta ) q^{37} + ( -1270392479752122 + 47315480368 \beta ) q^{41} + ( -2816827546694732 + 16432445197 \beta ) q^{43} + ( 10974169793565168 - 13905445988 \beta ) q^{47} + ( 4186293331532393 + 282348533280 \beta ) q^{49} + ( 4709062533452338 + 810595524548 \beta ) q^{53} + ( -15363426861001640 - 294657873146 \beta ) q^{55} + ( -49271224795203812 - 1148874921753 \beta ) q^{59} + ( 5146072688919910 - 3405518362868 \beta ) q^{61} + ( -1116716180007380 + 178842524688 \beta ) q^{65} + ( 37876814001992252 - 6468769570097 \beta ) q^{67} + ( -8703526283356888 + 4032598137882 \beta ) q^{71} + ( -428754127529916134 + 10723507910184 \beta ) q^{73} + ( 1113097256235937824 + 21315830527312 \beta ) q^{77} + ( -113145960722427536 - 8826033368644 \beta ) q^{79} + ( -383757850730492524 - 1875501918907 \beta ) q^{83} + ( 999487011407779100 + 9048886151560 \beta ) q^{85} + ( -3046272947217587274 - 7105196196264 \beta ) q^{89} + ( 82023827196188688 - 12981111503420 \beta ) q^{91} + ( 2496943855328169560 + 84681152480134 \beta ) q^{95} + ( 774074124761173538 - 186089076998104 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 1226620q^{5} + 88510512q^{7} + O(q^{10})$$ $$2q - 1226620q^{5} + 88510512q^{7} + 7163787608q^{11} - 10126923604q^{13} + 72045078940q^{17} - 3120480472232q^{19} + 14759207090288q^{23} - 32209737998450q^{25} + 30249539245044q^{29} - 123389562777920q^{31} - 430192267170720q^{35} + 2015393170174524q^{37} - 2540784959504244q^{41} - 5633655093389464q^{43} + 21948339587130336q^{47} + 8372586663064786q^{49} + 9418125066904676q^{53} - 30726853722003280q^{55} - 98542449590407624q^{59} + 10292145377839820q^{61} - 2233432360014760q^{65} + 75753628003984504q^{67} - 17407052566713776q^{71} - 857508255059832268q^{73} + 2226194512471875648q^{77} - 226291921444855072q^{79} - 767515701460985048q^{83} + 1998974022815558200q^{85} - 6092545894435174548q^{89} + 164047654392377376q^{91} + 4993887710656339120q^{95} + 1548148249522347076q^{97} + O(q^{100})$$

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
 19.5591 −18.5591
0 0 0 −2.22342e6 0 1.60989e8 0 0 0
1.2 0 0 0 996804. 0 −7.24780e7 0 0 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
$$2$$ $$1$$
$$3$$ $$-1$$

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 72.20.a.a 2
3.b odd 2 1 8.20.a.a 2
12.b even 2 1 16.20.a.e 2
24.f even 2 1 64.20.a.j 2
24.h odd 2 1 64.20.a.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.20.a.a 2 3.b odd 2 1
16.20.a.e 2 12.b even 2 1
64.20.a.j 2 24.f even 2 1
64.20.a.k 2 24.h odd 2 1
72.20.a.a 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 1226620 T_{5} -$$$$22\!\cdots\!00$$ acting on $$S_{20}^{\mathrm{new}}(\Gamma_0(72))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-2216319016700 + 1226620 T + T^{2}$$
$7$ $$-11668133149654464 - 88510512 T + T^{2}$$
$11$ $$-54040584096044956784 - 7163787608 T + T^{2}$$
$13$ $$18762236610718207204 + 10126923604 T + T^{2}$$
$17$ $$-$$$$40\!\cdots\!00$$$$- 72045078940 T + T^{2}$$
$19$ $$15\!\cdots\!56$$$$+ 3120480472232 T + T^{2}$$
$23$ $$89\!\cdots\!36$$$$- 14759207090288 T + T^{2}$$
$29$ $$-$$$$83\!\cdots\!16$$$$- 30249539245044 T + T^{2}$$
$31$ $$-$$$$16\!\cdots\!00$$$$+ 123389562777920 T + T^{2}$$
$37$ $$10\!\cdots\!44$$$$- 2015393170174524 T + T^{2}$$
$41$ $$-$$$$13\!\cdots\!16$$$$+ 2540784959504244 T + T^{2}$$
$43$ $$75\!\cdots\!24$$$$+ 5633655093389464 T + T^{2}$$
$47$ $$12\!\cdots\!24$$$$- 21948339587130336 T + T^{2}$$
$53$ $$-$$$$85\!\cdots\!56$$$$- 9418125066904676 T + T^{2}$$
$59$ $$66\!\cdots\!44$$$$+ 98542449590407624 T + T^{2}$$
$61$ $$-$$$$15\!\cdots\!00$$$$- 10292145377839820 T + T^{2}$$
$67$ $$-$$$$54\!\cdots\!96$$$$- 75753628003984504 T + T^{2}$$
$71$ $$-$$$$21\!\cdots\!56$$$$+ 17407052566713776 T + T^{2}$$
$73$ $$29\!\cdots\!56$$$$+ 857508255059832268 T + T^{2}$$
$79$ $$-$$$$91\!\cdots\!04$$$$+ 226291921444855072 T + T^{2}$$
$83$ $$14\!\cdots\!76$$$$+ 767515701460985048 T + T^{2}$$
$89$ $$92\!\cdots\!76$$$$+ 6092545894435174548 T + T^{2}$$
$97$ $$-$$$$45\!\cdots\!56$$$$- 1548148249522347076 T + T^{2}$$