Properties

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

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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: \(1\)
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 −48964.9 0 3.19930e7 0 5.17135e9 0 −9.17456e10 0
1.2 0 388445. 0 −1.05062e8 0 −3.81217e9 0 5.67462e10 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.g 2
4.b odd 2 1 64.24.a.d 2
8.b even 2 1 16.24.a.b 2
8.d odd 2 1 1.24.a.a 2
24.f even 2 1 9.24.a.b 2
40.e odd 2 1 25.24.a.a 2
40.k even 4 2 25.24.b.a 4
56.e even 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.d odd 2 1
9.24.a.b 2 24.f even 2 1
16.24.a.b 2 8.b even 2 1
25.24.a.a 2 40.e odd 2 1
25.24.b.a 4 40.k even 4 2
49.24.a.b 2 56.e even 2 1
64.24.a.d 2 4.b odd 2 1
64.24.a.g 2 1.a even 1 1 trivial

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

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