# Properties

 Label 64.24.a.d Level $64$ Weight $24$ Character orbit 64.a Self dual yes Analytic conductor $214.531$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$24$$ Character orbit: $$[\chi]$$ $$=$$ 64.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$214.530583901$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{144169})$$ Defining polynomial: $$x^{2} - x - 36042$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{7}\cdot 3$$ Twist minimal: no (minimal twist has level 1) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 192\sqrt{144169}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -169740 - 3 \beta ) q^{3} + ( -36534510 - 940 \beta ) q^{5} + ( -679592200 + 61614 \beta ) q^{7} + ( -17499697083 + 1018440 \beta ) q^{9} +O(q^{10})$$ $$q + ( -169740 - 3 \beta ) q^{3} + ( -36534510 - 940 \beta ) q^{5} + ( -679592200 + 61614 \beta ) q^{7} + ( -17499697083 + 1018440 \beta ) q^{9} + ( -428400984132 + 2416975 \beta ) q^{11} + ( -2188054661030 + 79271892 \beta ) q^{13} + ( 21188669492520 + 269159130 \beta ) q^{15} + ( 127014073798770 + 1470139464 \beta ) q^{17} + ( -2130300489980 + 8576162145 \beta ) q^{19} + ( -867015818861472 - 8419583760 \beta ) q^{21} + ( -4072356539504280 + 6645877738 \beta ) q^{23} + ( -5890137314400425 + 68684878800 \beta ) q^{25} + ( 2712317491358280 + 162058622130 \beta ) q^{27} + ( -10409216800811670 + 425251325380 \beta ) q^{29} + ( 68857008588500192 + 925907830200 \beta ) q^{31} + ( 34180683383000880 + 874945615896 \beta ) q^{33} + ( -282980635625212560 - 1612220631140 \beta ) q^{35} + ( 448860632204483890 + 10838146125636 \beta ) q^{37} + ( -892505736832514616 - 6891446964990 \beta ) q^{39} + ( -1147217738584157478 + 34950659237200 \beta ) q^{41} + ( 875380384309927900 - 15008906283273 \beta ) q^{43} + ( -4448546345147103270 - 20758491106380 \beta ) q^{45} + ( 7879872108828390480 - 227131222925836 \beta ) q^{47} + ( -6730990852188100407 - 83744787621600 \beta ) q^{49} + ( -44999181422539146072 - 630583694015670 \beta ) q^{51} + ( 70143626700823398210 + 85045109125572 \beta ) q^{53} + ( 3576775477530091320 + 314393927776830 \beta ) q^{55} + ( -136376200724313587760 - 1449326861022360 \beta ) q^{57} + ( -140436494985670385940 + 1940356339776635 \beta ) q^{59} + ( 90226446258251111818 - 5179087686487500 \beta ) q^{61} + ( 345387556966967507160 - 1770350216239962 \beta ) q^{63} + ( -316084417404720190380 - 839388349624720 \beta ) q^{65} + ( -877116581715778812620 - 8226306485688339 \beta ) q^{67} + ( 585280336086202111776 + 11088998331264720 \beta ) q^{69} + ( 1527516755097071664312 - 10468502603081250 \beta ) q^{71} + ( -4031704126938803074630 + 7371208444739688 \beta ) q^{73} + ( -95315544675260442900 + 6011840615689275 \beta ) q^{75} + ( 1082592382178724832800 - 28038055593904048 \beta ) q^{77} + ( 3122458407279819990320 + 97563135353608620 \beta ) q^{79} + ( -1396764290464916987799 - 131523962038990920 \beta ) q^{81} + ( -3437997041209249488060 + 153000198229562257 \beta ) q^{83} + ( -11984871543935157451260 - 173104054319746440 \beta ) q^{85} + ( -5013320326918837192440 - 40954509567566190 \beta ) q^{87} + ( 3197546543086535002410 + 19034325199766760 \beta ) q^{89} + ( 27445089081352280073008 - 188687359367144820 \beta ) q^{91} + ( -26450405720678926039680 - 363734620863648576 \beta ) q^{93} + ( -42766680533350429251000 - 311323399187542750 \beta ) q^{95} + ( -15573644423127015250270 - 1346057797741427736 \beta ) q^{97} + ( 20579122566156067990956 - 478597028636578005 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 339480q^{3} - 73069020q^{5} - 1359184400q^{7} - 34999394166q^{9} + O(q^{10})$$ $$2q - 339480q^{3} - 73069020q^{5} - 1359184400q^{7} - 34999394166q^{9} - 856801968264q^{11} - 4376109322060q^{13} + 42377338985040q^{15} + 254028147597540q^{17} - 4260600979960q^{19} - 1734031637722944q^{21} - 8144713079008560q^{23} - 11780274628800850q^{25} + 5424634982716560q^{27} - 20818433601623340q^{29} + 137714017177000384q^{31} + 68361366766001760q^{33} - 565961271250425120q^{35} + 897721264408967780q^{37} - 1785011473665029232q^{39} - 2294435477168314956q^{41} + 1750760768619855800q^{43} - 8897092690294206540q^{45} + 15759744217656780960q^{47} - 13461981704376200814q^{49} - 89998362845078292144q^{51} + 140287253401646796420q^{53} + 7153550955060182640q^{55} - 272752401448627175520q^{57} - 280872989971340771880q^{59} + 180452892516502223636q^{61} + 690775113933935014320q^{63} - 632168834809440380760q^{65} - 1754233163431557625240q^{67} + 1170560672172404223552q^{69} + 3055033510194143328624q^{71} - 8063408253877606149260q^{73} - 190631089350520885800q^{75} + 2165184764357449665600q^{77} + 6244916814559639980640q^{79} - 2793528580929833975598q^{81} - 6875994082418498976120q^{83} - 23969743087870314902520q^{85} - 10026640653837674384880q^{87} + 6395093086173070004820q^{89} + 54890178162704560146016q^{91} - 52900811441357852079360q^{93} - 85533361066700858502000q^{95} - 31147288846254030500540q^{97} + 41158245132312135981912q^{99} + O(q^{100})$$

## 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
 190.348 −189.348
0 −388445. 0 −1.05062e8 0 3.81217e9 0 5.67462e10 0
1.2 0 48964.9 0 3.19930e7 0 −5.17135e9 0 −9.17456e10 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 64.24.a.d 2
4.b odd 2 1 64.24.a.g 2
8.b even 2 1 1.24.a.a 2
8.d odd 2 1 16.24.a.b 2
24.h odd 2 1 9.24.a.b 2
40.f even 2 1 25.24.a.a 2
40.i odd 4 2 25.24.b.a 4
56.h odd 2 1 49.24.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.24.a.a 2 8.b even 2 1
9.24.a.b 2 24.h odd 2 1
16.24.a.b 2 8.d odd 2 1
25.24.a.a 2 40.f even 2 1
25.24.b.a 4 40.i odd 4 2
49.24.a.b 2 56.h odd 2 1
64.24.a.d 2 1.a even 1 1 trivial
64.24.a.g 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 339480 T_{3} - 19020146544$$ acting on $$S_{24}^{\mathrm{new}}(\Gamma_0(64))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-19020146544 + 339480 T + T^{2}$$
$5$ $$-3361250798797500 + 73069020 T + T^{2}$$
$7$ $$-19714065371291135936 + 1359184400 T + T^{2}$$
$11$ $$15\!\cdots\!24$$$$+ 856801968264 T + T^{2}$$
$13$ $$-$$$$28\!\cdots\!24$$$$+ 4376109322060 T + T^{2}$$
$17$ $$46\!\cdots\!64$$$$- 254028147597540 T + T^{2}$$
$19$ $$-$$$$39\!\cdots\!00$$$$+ 4260600979960 T + T^{2}$$
$23$ $$16\!\cdots\!96$$$$+ 8144713079008560 T + T^{2}$$
$29$ $$-$$$$85\!\cdots\!00$$$$+ 20818433601623340 T + T^{2}$$
$31$ $$18\!\cdots\!64$$$$- 137714017177000384 T + T^{2}$$
$37$ $$-$$$$42\!\cdots\!36$$$$- 897721264408967780 T + T^{2}$$
$41$ $$-$$$$51\!\cdots\!16$$$$+ 2294435477168314956 T + T^{2}$$
$43$ $$-$$$$43\!\cdots\!64$$$$- 1750760768619855800 T + T^{2}$$
$47$ $$-$$$$21\!\cdots\!36$$$$- 15759744217656780960 T + T^{2}$$
$53$ $$48\!\cdots\!56$$$$-$$$$14\!\cdots\!20$$$$T + T^{2}$$
$59$ $$-$$$$28\!\cdots\!00$$$$+$$$$28\!\cdots\!80$$$$T + T^{2}$$
$61$ $$-$$$$13\!\cdots\!76$$$$-$$$$18\!\cdots\!36$$$$T + T^{2}$$
$67$ $$40\!\cdots\!64$$$$+$$$$17\!\cdots\!40$$$$T + T^{2}$$
$71$ $$17\!\cdots\!44$$$$-$$$$30\!\cdots\!24$$$$T + T^{2}$$
$73$ $$15\!\cdots\!96$$$$+$$$$80\!\cdots\!60$$$$T + T^{2}$$
$79$ $$-$$$$40\!\cdots\!00$$$$-$$$$62\!\cdots\!40$$$$T + T^{2}$$
$83$ $$-$$$$11\!\cdots\!84$$$$+$$$$68\!\cdots\!20$$$$T + T^{2}$$
$89$ $$82\!\cdots\!00$$$$-$$$$63\!\cdots\!20$$$$T + T^{2}$$
$97$ $$-$$$$93\!\cdots\!36$$$$+$$$$31\!\cdots\!40$$$$T + T^{2}$$