# Properties

 Label 84.12.a.b Level $84$ Weight $12$ Character orbit 84.a Self dual yes Analytic conductor $64.541$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 84.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.5408271670$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{1000465})$$ Defining polynomial: $$x^{2} - x - 250116$$ x^2 - x - 250116 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{1000465}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 243 q^{3} + ( - 5 \beta - 4455) q^{5} - 16807 q^{7} + 59049 q^{9}+O(q^{10})$$ q - 243 * q^3 + (-5*b - 4455) * q^5 - 16807 * q^7 + 59049 * q^9 $$q - 243 q^{3} + ( - 5 \beta - 4455) q^{5} - 16807 q^{7} + 59049 q^{9} + ( - 115 \beta + 166617) q^{11} + (1458 \beta + 110592) q^{13} + (1215 \beta + 1082565) q^{15} + ( - 2503 \beta + 4120227) q^{17} + (1944 \beta + 4857644) q^{19} + 4084101 q^{21} + (10367 \beta + 8556111) q^{23} + (44550 \beta - 3969475) q^{25} - 14348907 q^{27} + ( - 117800 \beta + 32975262) q^{29} + ( - 34344 \beta + 39729240) q^{31} + (27945 \beta - 40487931) q^{33} + (84035 \beta + 74875185) q^{35} + ( - 483570 \beta + 147033644) q^{37} + ( - 354294 \beta - 26873856) q^{39} + (690591 \beta - 208747935) q^{41} + (1626480 \beta - 149459212) q^{43} + ( - 295245 \beta - 263063295) q^{45} + ( - 2289774 \beta + 241447554) q^{47} + 282475249 q^{49} + (608229 \beta - 1001215161) q^{51} + (3111378 \beta - 2134755648) q^{53} + ( - 320760 \beta - 167011360) q^{55} + ( - 472392 \beta - 1180407492) q^{57} + ( - 4074170 \beta - 2615799582) q^{59} + (8181000 \beta - 924873594) q^{61} - 992436543 q^{63} + ( - 7048350 \beta - 7786077210) q^{65} + ( - 6436746 \beta - 3727394846) q^{67} + ( - 2519181 \beta - 2079134973) q^{69} + (5015153 \beta - 15767760591) q^{71} + ( - 1198638 \beta - 4864591556) q^{73} + ( - 10825650 \beta + 964582425) q^{75} + (1932805 \beta - 2800331919) q^{77} + (13426722 \beta - 25478178222) q^{79} + 3486784401 q^{81} + ( - 86080 \beta - 12122072172) q^{83} + ( - 9450270 \beta - 5834791810) q^{85} + (28625400 \beta - 8012988666) q^{87} + (53744511 \beta + 5742574785) q^{89} + ( - 24504606 \beta - 1858719744) q^{91} + (8345592 \beta - 9654205320) q^{93} + ( - 32948740 \beta - 31365323820) q^{95} + ( - 23050818 \beta - 55400331584) q^{97} + ( - 6790635 \beta + 9838567233) q^{99}+O(q^{100})$$ q - 243 * q^3 + (-5*b - 4455) * q^5 - 16807 * q^7 + 59049 * q^9 + (-115*b + 166617) * q^11 + (1458*b + 110592) * q^13 + (1215*b + 1082565) * q^15 + (-2503*b + 4120227) * q^17 + (1944*b + 4857644) * q^19 + 4084101 * q^21 + (10367*b + 8556111) * q^23 + (44550*b - 3969475) * q^25 - 14348907 * q^27 + (-117800*b + 32975262) * q^29 + (-34344*b + 39729240) * q^31 + (27945*b - 40487931) * q^33 + (84035*b + 74875185) * q^35 + (-483570*b + 147033644) * q^37 + (-354294*b - 26873856) * q^39 + (690591*b - 208747935) * q^41 + (1626480*b - 149459212) * q^43 + (-295245*b - 263063295) * q^45 + (-2289774*b + 241447554) * q^47 + 282475249 * q^49 + (608229*b - 1001215161) * q^51 + (3111378*b - 2134755648) * q^53 + (-320760*b - 167011360) * q^55 + (-472392*b - 1180407492) * q^57 + (-4074170*b - 2615799582) * q^59 + (8181000*b - 924873594) * q^61 - 992436543 * q^63 + (-7048350*b - 7786077210) * q^65 + (-6436746*b - 3727394846) * q^67 + (-2519181*b - 2079134973) * q^69 + (5015153*b - 15767760591) * q^71 + (-1198638*b - 4864591556) * q^73 + (-10825650*b + 964582425) * q^75 + (1932805*b - 2800331919) * q^77 + (13426722*b - 25478178222) * q^79 + 3486784401 * q^81 + (-86080*b - 12122072172) * q^83 + (-9450270*b - 5834791810) * q^85 + (28625400*b - 8012988666) * q^87 + (53744511*b + 5742574785) * q^89 + (-24504606*b - 1858719744) * q^91 + (8345592*b - 9654205320) * q^93 + (-32948740*b - 31365323820) * q^95 + (-23050818*b - 55400331584) * q^97 + (-6790635*b + 9838567233) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 486 q^{3} - 8910 q^{5} - 33614 q^{7} + 118098 q^{9}+O(q^{10})$$ 2 * q - 486 * q^3 - 8910 * q^5 - 33614 * q^7 + 118098 * q^9 $$2 q - 486 q^{3} - 8910 q^{5} - 33614 q^{7} + 118098 q^{9} + 333234 q^{11} + 221184 q^{13} + 2165130 q^{15} + 8240454 q^{17} + 9715288 q^{19} + 8168202 q^{21} + 17112222 q^{23} - 7938950 q^{25} - 28697814 q^{27} + 65950524 q^{29} + 79458480 q^{31} - 80975862 q^{33} + 149750370 q^{35} + 294067288 q^{37} - 53747712 q^{39} - 417495870 q^{41} - 298918424 q^{43} - 526126590 q^{45} + 482895108 q^{47} + 564950498 q^{49} - 2002430322 q^{51} - 4269511296 q^{53} - 334022720 q^{55} - 2360814984 q^{57} - 5231599164 q^{59} - 1849747188 q^{61} - 1984873086 q^{63} - 15572154420 q^{65} - 7454789692 q^{67} - 4158269946 q^{69} - 31535521182 q^{71} - 9729183112 q^{73} + 1929164850 q^{75} - 5600663838 q^{77} - 50956356444 q^{79} + 6973568802 q^{81} - 24244144344 q^{83} - 11669583620 q^{85} - 16025977332 q^{87} + 11485149570 q^{89} - 3717439488 q^{91} - 19308410640 q^{93} - 62730647640 q^{95} - 110800663168 q^{97} + 19677134466 q^{99}+O(q^{100})$$ 2 * q - 486 * q^3 - 8910 * q^5 - 33614 * q^7 + 118098 * q^9 + 333234 * q^11 + 221184 * q^13 + 2165130 * q^15 + 8240454 * q^17 + 9715288 * q^19 + 8168202 * q^21 + 17112222 * q^23 - 7938950 * q^25 - 28697814 * q^27 + 65950524 * q^29 + 79458480 * q^31 - 80975862 * q^33 + 149750370 * q^35 + 294067288 * q^37 - 53747712 * q^39 - 417495870 * q^41 - 298918424 * q^43 - 526126590 * q^45 + 482895108 * q^47 + 564950498 * q^49 - 2002430322 * q^51 - 4269511296 * q^53 - 334022720 * q^55 - 2360814984 * q^57 - 5231599164 * q^59 - 1849747188 * q^61 - 1984873086 * q^63 - 15572154420 * q^65 - 7454789692 * q^67 - 4158269946 * q^69 - 31535521182 * q^71 - 9729183112 * q^73 + 1929164850 * q^75 - 5600663838 * q^77 - 50956356444 * q^79 + 6973568802 * q^81 - 24244144344 * q^83 - 11669583620 * q^85 - 16025977332 * q^87 + 11485149570 * q^89 - 3717439488 * q^91 - 19308410640 * q^93 - 62730647640 * q^95 - 110800663168 * q^97 + 19677134466 * 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
 500.616 −499.616
0 −243.000 0 −9456.16 0 −16807.0 0 59049.0 0
1.2 0 −243.000 0 546.162 0 −16807.0 0 59049.0 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$1$$

## 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 84.12.a.b 2
3.b odd 2 1 252.12.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.12.a.b 2 1.a even 1 1 trivial
252.12.a.d 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 8910T_{5} - 5164600$$ acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(84))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 243)^{2}$$
$5$ $$T^{2} + 8910 T - 5164600$$
$7$ $$(T + 16807)^{2}$$
$11$ $$T^{2} - 333234 T + 14530075064$$
$13$ $$T^{2} - 221184 T - 2114521889796$$
$17$ $$T^{2} - 8240454 T + 10708348302344$$
$19$ $$T^{2} - 9715288 T + 19815811932496$$
$23$ $$T^{2} - 17112222 T - 34317629286064$$
$29$ $$T^{2} - 65950524 T - 12\!\cdots\!56$$
$31$ $$T^{2} + \cdots + 398353702671360$$
$37$ $$T^{2} - 294067288 T - 21\!\cdots\!64$$
$41$ $$T^{2} + 417495870 T - 43\!\cdots\!40$$
$43$ $$T^{2} + 298918424 T - 26\!\cdots\!56$$
$47$ $$T^{2} - 482895108 T - 51\!\cdots\!24$$
$53$ $$T^{2} + 4269511296 T - 51\!\cdots\!56$$
$59$ $$T^{2} + 5231599164 T - 97\!\cdots\!76$$
$61$ $$T^{2} + 1849747188 T - 66\!\cdots\!64$$
$67$ $$T^{2} + 7454789692 T - 27\!\cdots\!24$$
$71$ $$T^{2} + 31535521182 T + 22\!\cdots\!96$$
$73$ $$T^{2} + 9729183112 T + 22\!\cdots\!76$$
$79$ $$T^{2} + 50956356444 T + 46\!\cdots\!24$$
$83$ $$T^{2} + 24244144344 T + 14\!\cdots\!84$$
$89$ $$T^{2} - 11485149570 T - 28\!\cdots\!40$$
$97$ $$T^{2} + 110800663168 T + 25\!\cdots\!96$$