# Properties

 Label 25.24.a.a Level $25$ Weight $24$ Character orbit 25.a Self dual yes Analytic conductor $83.801$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$24$$ Character orbit: $$[\chi]$$ $$=$$ 25.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$83.8010093363$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{144169})$$ Defining polynomial: $$x^{2} - x - 36042$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}\cdot 3$$ 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 $$\beta = 12\sqrt{144169}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -540 - \beta ) q^{2} + ( -169740 + 48 \beta ) q^{3} + ( 12663328 + 1080 \beta ) q^{4} + ( -904836528 + 143820 \beta ) q^{6} + ( 679592200 + 985824 \beta ) q^{7} + ( -24729511680 - 4857920 \beta ) q^{8} + ( -17499697083 - 16295040 \beta ) q^{9} +O(q^{10})$$ $$q + ( -540 - \beta ) q^{2} + ( -169740 + 48 \beta ) q^{3} + ( 12663328 + 1080 \beta ) q^{4} + ( -904836528 + 143820 \beta ) q^{6} + ( 679592200 + 985824 \beta ) q^{7} + ( -24729511680 - 4857920 \beta ) q^{8} + ( -17499697083 - 16295040 \beta ) q^{9} + ( 428400984132 + 38671600 \beta ) q^{11} + ( -1073257476480 + 424520544 \beta ) q^{12} + ( -2188054661030 - 1268350272 \beta ) q^{13} + ( -20833017264864 - 1211937160 \beta ) q^{14} + ( 7978293200896 + 18293091840 \beta ) q^{16} + ( -127014073798770 + 23522231424 \beta ) q^{17} + ( 347740341958260 + 26299018683 \beta ) q^{18} + ( 2130300489980 + 137218594320 \beta ) q^{19} + ( 867015818861472 - 134713340160 \beta ) q^{21} + ( -1034171941088880 - 449283648132 \beta ) q^{22} + ( 4072356539504280 + 106334043808 \beta ) q^{23} + ( -643311157570560 - 362433219840 \beta ) q^{24} + ( 27512927329367592 + 2872963807910 \beta ) q^{26} + ( 2712317491358280 - 2592937954080 \beta ) q^{27} + ( 30709219409854720 + 13217772238272 \beta ) q^{28} + ( 10409216800811670 + 6804021206080 \beta ) q^{29} + ( 68857008588500192 - 14814525283200 \beta ) q^{31} + ( -176632831890800640 + 22894623780864 \beta ) q^{32} + ( -34180683383000880 + 13999129854336 \beta ) q^{33} + ( -419741827980662664 + 114312068829810 \beta ) q^{34} + ( -586958150038787424 - 225249109142760 \beta ) q^{36} + ( 448860632204483890 - 173410338010176 \beta ) q^{37} + ( -2849854485795480720 - 76228341422780 \beta ) q^{38} + ( -892505736832514616 + 110263151439840 \beta ) q^{39} + ( -1147217738584157478 - 559210547795200 \beta ) q^{41} + ( 2328505663218698880 - 794270615175072 \beta ) q^{42} + ( 875380384309927900 + 240142500532368 \beta ) q^{43} + ( 6292044420016519296 + 952384217947360 \beta ) q^{44} + ( -4406603009025110688 - 4129776923160600 \beta ) q^{46} + ( -7879872108828390480 - 3634099566813376 \beta ) q^{47} + ( 16874759699788308480 - 2722111335278592 \beta ) q^{48} + ( -6730990852188100407 + 1339916601945600 \beta ) q^{49} + ( 44999181422539146072 - 10089339104250720 \beta ) q^{51} + ( -56145941891956011200 - 18424634547137616 \beta ) q^{52} + ( 70143626700823398210 - 1360721746009152 \beta ) q^{53} + ( 52365611708519899680 - 1312130996155080 \beta ) q^{54} + ( -116228356027144058880 - 27680350662648320 \beta ) q^{56} + ( 136376200724313587760 - 23189229776357760 \beta ) q^{57} + ( -146874743461784344680 - 14083388252094870 \beta ) q^{58} + ( 140436494985670385940 + 31045701436426160 \beta ) q^{59} + ( -90226446258251111818 - 82865402983800000 \beta ) q^{61} + ( 270371737921937051520 - 60857164935572192 \beta ) q^{62} + ( -345387556966967507160 - 28325603459839392 \beta ) q^{63} + ( -446845127234676457472 + 10816158495375360 \beta ) q^{64} + ( -272169070456825941696 + 26621153261659440 \beta ) q^{66} + ( -877116581715778812620 + 131620903771013424 \beta ) q^{67} + ( -1081025095071472165440 + 160694532111347472 \beta ) q^{68} + ( -585280336086202111776 + 177423973300235520 \beta ) q^{69} + ( 1527516755097071664312 + 167496041649300000 \beta ) q^{71} + ( 2076147176051519274240 + 487980510459514560 \beta ) q^{72} + ( 4031704126938803074630 + 117939335115835008 \beta ) q^{73} + ( 3357672141574403878536 - 355219049678988850 \beta ) q^{74} + ( 3103577147256540295040 + 1739944792102275360 \beta ) q^{76} + ( 1082592382178724832800 + 448608889502464768 \beta ) q^{77} + ( -1807146974420404293600 + 832963635055001016 \beta ) q^{78} + ( 3122458407279819990320 - 1561010165657737920 \beta ) q^{79} + ( -1396764290464916987799 + 2104383392623854720 \beta ) q^{81} + ( 12228896445807856225320 + 1449191434393565478 \beta ) q^{82} + ( -3437997041209249488060 - 2448003171672996112 \beta ) q^{83} + ( 7958875953595201238016 - 769542128051262720 \beta ) q^{84} + ( -5458144406459499621648 - 1005057334597406620 \beta ) q^{86} + ( 5013320326918837192440 - 655272153081059040 \beta ) q^{87} + ( -14494257334099636853760 - 3037467492718813440 \beta ) q^{88} + ( 3197546543086535002410 - 304549203196268160 \beta ) q^{89} + ( -27445089081352280073008 - 3018997749874317120 \beta ) q^{91} + ( 53953719508595878490880 + 5744687936971695424 \beta ) q^{92} + ( -26450405720678926039680 + 5819753933818377216 \beta ) q^{93} + ( 79700259003267465913536 + 9842285874907613520 \beta ) q^{94} + ( 52796060834792197128192 - 12364509371322286080 \beta ) q^{96} + ( 15573644423127015250270 - 21536924763862843776 \beta ) q^{97} + ( -24182383808187335501820 + 6007435887137476407 \beta ) q^{98} + ( -20579122566156067990956 - 7657552458185248080 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 1080q^{2} - 339480q^{3} + 25326656q^{4} - 1809673056q^{6} + 1359184400q^{7} - 49459023360q^{8} - 34999394166q^{9} + O(q^{10})$$ $$2q - 1080q^{2} - 339480q^{3} + 25326656q^{4} - 1809673056q^{6} + 1359184400q^{7} - 49459023360q^{8} - 34999394166q^{9} + 856801968264q^{11} - 2146514952960q^{12} - 4376109322060q^{13} - 41666034529728q^{14} + 15956586401792q^{16} - 254028147597540q^{17} + 695480683916520q^{18} + 4260600979960q^{19} + 1734031637722944q^{21} - 2068343882177760q^{22} + 8144713079008560q^{23} - 1286622315141120q^{24} + 55025854658735184q^{26} + 5424634982716560q^{27} + 61418438819709440q^{28} + 20818433601623340q^{29} + 137714017177000384q^{31} - 353265663781601280q^{32} - 68361366766001760q^{33} - 839483655961325328q^{34} - 1173916300077574848q^{36} + 897721264408967780q^{37} - 5699708971590961440q^{38} - 1785011473665029232q^{39} - 2294435477168314956q^{41} + 4657011326437397760q^{42} + 1750760768619855800q^{43} + 12584088840033038592q^{44} - 8813206018050221376q^{46} - 15759744217656780960q^{47} + 33749519399576616960q^{48} - 13461981704376200814q^{49} + 89998362845078292144q^{51} - 112291883783912022400q^{52} + 140287253401646796420q^{53} + 104731223417039799360q^{54} - 232456712054288117760q^{56} + 272752401448627175520q^{57} - 293749486923568689360q^{58} + 280872989971340771880q^{59} - 180452892516502223636q^{61} + 540743475843874103040q^{62} - 690775113933935014320q^{63} - 893690254469352914944q^{64} - 544338140913651883392q^{66} - 1754233163431557625240q^{67} - 2162050190142944330880q^{68} - 1170560672172404223552q^{69} + 3055033510194143328624q^{71} + 4152294352103038548480q^{72} + 8063408253877606149260q^{73} + 6715344283148807757072q^{74} + 6207154294513080590080q^{76} + 2165184764357449665600q^{77} - 3614293948840808587200q^{78} + 6244916814559639980640q^{79} - 2793528580929833975598q^{81} + 24457792891615712450640q^{82} - 6875994082418498976120q^{83} + 15917751907190402476032q^{84} - 10916288812918999243296q^{86} + 10026640653837674384880q^{87} - 28988514668199273707520q^{88} + 6395093086173070004820q^{89} - 54890178162704560146016q^{91} + 107907439017191756981760q^{92} - 52900811441357852079360q^{93} + 159400518006534931827072q^{94} + 105592121669584394256384q^{96} + 31147288846254030500540q^{97} - 48364767616374671003640q^{98} - 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
−5096.35 48964.9 1.75842e7 0 −2.49542e8 5.17135e9 −4.68639e10 −9.17456e10 0
1.2 4016.35 −388445. 7.74247e6 0 −1.56013e9 −3.81217e9 −2.59512e9 5.67462e10 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
$$5$$ $$1$$

## 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 25.24.a.a 2
5.b even 2 1 1.24.a.a 2
5.c odd 4 2 25.24.b.a 4
15.d odd 2 1 9.24.a.b 2
20.d odd 2 1 16.24.a.b 2
35.c odd 2 1 49.24.a.b 2
40.e odd 2 1 64.24.a.g 2
40.f even 2 1 64.24.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.24.a.a 2 5.b even 2 1
9.24.a.b 2 15.d odd 2 1
16.24.a.b 2 20.d odd 2 1
25.24.a.a 2 1.a even 1 1 trivial
25.24.b.a 4 5.c odd 4 2
49.24.a.b 2 35.c odd 2 1
64.24.a.d 2 40.f even 2 1
64.24.a.g 2 40.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 1080 T_{2} - 20468736$$ acting on $$S_{24}^{\mathrm{new}}(\Gamma_0(25))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-20468736 + 1080 T + T^{2}$$
$3$ $$-19020146544 + 339480 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-19714065371291135936 - 1359184400 T + T^{2}$$
$11$ $$15\!\cdots\!24$$$$- 856801968264 T + T^{2}$$
$13$ $$-$$$$28\!\cdots\!24$$$$+ 4376109322060 T + T^{2}$$
$17$ $$46\!\cdots\!64$$$$+ 254028147597540 T + T^{2}$$
$19$ $$-$$$$39\!\cdots\!00$$$$- 4260600979960 T + T^{2}$$
$23$ $$16\!\cdots\!96$$$$- 8144713079008560 T + T^{2}$$
$29$ $$-$$$$85\!\cdots\!00$$$$- 20818433601623340 T + T^{2}$$
$31$ $$18\!\cdots\!64$$$$- 137714017177000384 T + T^{2}$$
$37$ $$-$$$$42\!\cdots\!36$$$$- 897721264408967780 T + T^{2}$$
$41$ $$-$$$$51\!\cdots\!16$$$$+ 2294435477168314956 T + T^{2}$$
$43$ $$-$$$$43\!\cdots\!64$$$$- 1750760768619855800 T + T^{2}$$
$47$ $$-$$$$21\!\cdots\!36$$$$+ 15759744217656780960 T + T^{2}$$
$53$ $$48\!\cdots\!56$$$$-$$$$14\!\cdots\!20$$$$T + T^{2}$$
$59$ $$-$$$$28\!\cdots\!00$$$$-$$$$28\!\cdots\!80$$$$T + T^{2}$$
$61$ $$-$$$$13\!\cdots\!76$$$$+$$$$18\!\cdots\!36$$$$T + T^{2}$$
$67$ $$40\!\cdots\!64$$$$+$$$$17\!\cdots\!40$$$$T + T^{2}$$
$71$ $$17\!\cdots\!44$$$$-$$$$30\!\cdots\!24$$$$T + T^{2}$$
$73$ $$15\!\cdots\!96$$$$-$$$$80\!\cdots\!60$$$$T + T^{2}$$
$79$ $$-$$$$40\!\cdots\!00$$$$-$$$$62\!\cdots\!40$$$$T + T^{2}$$
$83$ $$-$$$$11\!\cdots\!84$$$$+$$$$68\!\cdots\!20$$$$T + T^{2}$$
$89$ $$82\!\cdots\!00$$$$-$$$$63\!\cdots\!20$$$$T + T^{2}$$
$97$ $$-$$$$93\!\cdots\!36$$$$-$$$$31\!\cdots\!40$$$$T + T^{2}$$