# Properties

 Label 25.24.b.a Level $25$ Weight $24$ Character orbit 25.b Analytic conductor $83.801$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$24$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$83.8010093363$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} + \cdots)$$ Defining polynomial: $$x^{4} + 72085 x^{2} + 1299025764$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{8}\cdot 3^{2}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 1) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 54 \beta_{1} + \beta_{2} ) q^{2} + ( -16974 \beta_{1} + 48 \beta_{2} ) q^{3} + ( -12663328 + 108 \beta_{3} ) q^{4} + ( -904836528 - 14382 \beta_{3} ) q^{6} + ( -67959220 \beta_{1} - 985824 \beta_{2} ) q^{7} + ( -2472951168 \beta_{1} - 4857920 \beta_{2} ) q^{8} + ( 17499697083 - 1629504 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 54 \beta_{1} + \beta_{2} ) q^{2} + ( -16974 \beta_{1} + 48 \beta_{2} ) q^{3} + ( -12663328 + 108 \beta_{3} ) q^{4} + ( -904836528 - 14382 \beta_{3} ) q^{6} + ( -67959220 \beta_{1} - 985824 \beta_{2} ) q^{7} + ( -2472951168 \beta_{1} - 4857920 \beta_{2} ) q^{8} + ( 17499697083 - 1629504 \beta_{3} ) q^{9} + ( 428400984132 - 3867160 \beta_{3} ) q^{11} + ( 107325747648 \beta_{1} - 424520544 \beta_{2} ) q^{12} + ( -218805466103 \beta_{1} - 1268350272 \beta_{2} ) q^{13} + ( 20833017264864 - 121193716 \beta_{3} ) q^{14} + ( 7978293200896 - 1829309184 \beta_{3} ) q^{16} + ( 12701407379877 \beta_{1} - 23522231424 \beta_{2} ) q^{17} + ( 34774034195826 \beta_{1} + 26299018683 \beta_{2} ) q^{18} + ( -2130300489980 + 13721859432 \beta_{3} ) q^{19} + ( 867015818861472 + 13471334016 \beta_{3} ) q^{21} + ( 103417194108888 \beta_{1} + 449283648132 \beta_{2} ) q^{22} + ( 407235653950428 \beta_{1} + 106334043808 \beta_{2} ) q^{23} + ( 643311157570560 - 36243321984 \beta_{3} ) q^{24} + ( 27512927329367592 - 287296380791 \beta_{3} ) q^{26} + ( -271231749135828 \beta_{1} + 2592937954080 \beta_{2} ) q^{27} + ( 3070921940985472 \beta_{1} + 13217772238272 \beta_{2} ) q^{28} + ( -10409216800811670 + 680402120608 \beta_{3} ) q^{29} + ( 68857008588500192 + 1481452528320 \beta_{3} ) q^{31} + ( 17663283189080064 \beta_{1} - 22894623780864 \beta_{2} ) q^{32} + ( -3418068338300088 \beta_{1} + 13999129854336 \beta_{2} ) q^{33} + ( 419741827980662664 + 11431206882981 \beta_{3} ) q^{34} + ( -586958150038787424 + 22524910914276 \beta_{3} ) q^{36} + ( -44886063220448389 \beta_{1} + 173410338010176 \beta_{2} ) q^{37} + ( -284985448579548072 \beta_{1} - 76228341422780 \beta_{2} ) q^{38} + ( 892505736832514616 + 11026315143984 \beta_{3} ) q^{39} + ( -1147217738584157478 + 55921054779520 \beta_{3} ) q^{41} + ( -232850566321869888 \beta_{1} + 794270615175072 \beta_{2} ) q^{42} + ( 87538038430992790 \beta_{1} + 240142500532368 \beta_{2} ) q^{43} + ( -6292044420016519296 + 95238421794736 \beta_{3} ) q^{44} + ( -4406603009025110688 + 412977692316060 \beta_{3} ) q^{46} + ( 787987210882839048 \beta_{1} + 3634099566813376 \beta_{2} ) q^{47} + ( 1687475969978830848 \beta_{1} - 2722111335278592 \beta_{2} ) q^{48} + ( 6730990852188100407 + 133991660194560 \beta_{3} ) q^{49} + ( 44999181422539146072 + 1008933910425072 \beta_{3} ) q^{51} + ( 5614594189195601120 \beta_{1} + 18424634547137616 \beta_{2} ) q^{52} + ( 7014362670082339821 \beta_{1} - 1360721746009152 \beta_{2} ) q^{53} + ( -52365611708519899680 - 131213099615508 \beta_{3} ) q^{54} + ( -116228356027144058880 + 2768035066264832 \beta_{3} ) q^{56} + ( -13637620072431358776 \beta_{1} + 23189229776357760 \beta_{2} ) q^{57} + ( -14687474346178434468 \beta_{1} - 14083388252094870 \beta_{2} ) q^{58} + ( -140436494985670385940 + 3104570143642616 \beta_{3} ) q^{59} + ( -90226446258251111818 + 8286540298380000 \beta_{3} ) q^{61} + ( -27037173792193705152 \beta_{1} + 60857164935572192 \beta_{2} ) q^{62} + ( -34538755696696750716 \beta_{1} - 28325603459839392 \beta_{2} ) q^{63} + ( 446845127234676457472 + 1081615849537536 \beta_{3} ) q^{64} + ( -272169070456825941696 - 2662115326165944 \beta_{3} ) q^{66} + ( 87711658171577881262 \beta_{1} - 131620903771013424 \beta_{2} ) q^{67} + ( -108102509507147216544 \beta_{1} + 160694532111347472 \beta_{2} ) q^{68} + ( 585280336086202111776 + 17742397330023552 \beta_{3} ) q^{69} + ( 1527516755097071664312 - 16749604164930000 \beta_{3} ) q^{71} + ( -207614717605151927424 \beta_{1} - 487980510459514560 \beta_{2} ) q^{72} + ( 403170412693880307463 \beta_{1} + 117939335115835008 \beta_{2} ) q^{73} + ( -3357672141574403878536 - 35521904967898885 \beta_{3} ) q^{74} + ( 3103577147256540295040 - 173994479210227536 \beta_{3} ) q^{76} + ( -108259238217872483280 \beta_{1} - 448608889502464768 \beta_{2} ) q^{77} + ( -180714697442040429360 \beta_{1} + 832963635055001016 \beta_{2} ) q^{78} + ( -3122458407279819990320 - 156101016565773792 \beta_{3} ) q^{79} + ( -1396764290464916987799 - 210438339262385472 \beta_{3} ) q^{81} + ( -1222889644580785622532 \beta_{1} - 1449191434393565478 \beta_{2} ) q^{82} + ( -343799704120924948806 \beta_{1} - 2448003171672996112 \beta_{2} ) q^{83} + ( -7958875953595201238016 - 76954212805126272 \beta_{3} ) q^{84} + ( -5458144406459499621648 + 100505733459740662 \beta_{3} ) q^{86} + ( -501332032691883719244 \beta_{1} + 655272153081059040 \beta_{2} ) q^{87} + ( -1449425733409963685376 \beta_{1} - 3037467492718813440 \beta_{2} ) q^{88} + ( -3197546543086535002410 - 30454920319626816 \beta_{3} ) q^{89} + ( -27445089081352280073008 + 301899774987431712 \beta_{3} ) q^{91} + ( -5395371950859587849088 \beta_{1} - 5744687936971695424 \beta_{2} ) q^{92} + ( -2645040572067892603968 \beta_{1} + 5819753933818377216 \beta_{2} ) q^{93} + ( -79700259003267465913536 + 984228587490761352 \beta_{3} ) q^{94} + ( 52796060834792197128192 + 1236450937132228608 \beta_{3} ) q^{96} + ( -1557364442312701525027 \beta_{1} + 21536924763862843776 \beta_{2} ) q^{97} + ( -2418238380818733550182 \beta_{1} + 6007435887137476407 \beta_{2} ) q^{98} + ( 20579122566156067990956 - 765755245818524808 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 50653312q^{4} - 3619346112q^{6} + 69998788332q^{9} + O(q^{10})$$ $$4q - 50653312q^{4} - 3619346112q^{6} + 69998788332q^{9} + 1713603936528q^{11} + 83332069059456q^{14} + 31913172803584q^{16} - 8521201959920q^{19} + 3468063275445888q^{21} + 2573244630282240q^{24} + 110051709317470368q^{26} - 41636867203246680q^{29} + 275428034354000768q^{31} + 1678967311922650656q^{34} - 2347832600155149696q^{36} + 3570022947330058464q^{39} - 4588870954336629912q^{41} - 25168177680066077184q^{44} - 17626412036100442752q^{46} + 26923963408752401628q^{49} + 179996725690156584288q^{51} - 209462446834079598720q^{54} - 464913424108576235520q^{56} - 561745979942681543760q^{59} - 360905785033004447272q^{61} + 1787380508938705829888q^{64} - 1088676281827303766784q^{66} + 2341121344344808447104q^{69} + 6110067020388286657248q^{71} - 13430688566297615514144q^{74} + 12414308589026161180160q^{76} - 12489833629119279961280q^{79} - 5587057161859667951196q^{81} - 31835503814380804952064q^{84} - 21832577625837998486592q^{86} - 12790186172346140009640q^{89} - 109780356325409120292032q^{91} - 318801036013069863654144q^{94} + 211184243339168788512768q^{96} + 82316490264624271963824q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 72085 x^{2} + 1299025764$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-5 \nu^{3} - 180215 \nu$$$$)/18021$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{3} + 216254 \nu$$$$)/6007$$ $$\beta_{3}$$ $$=$$ $$240 \nu^{2} + 8650200$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$5 \beta_{2} + 6 \beta_{1}$$$$)/120$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 8650200$$$$)/240$$ $$\nu^{3}$$ $$=$$ $$($$$$-180215 \beta_{2} - 648762 \beta_{1}$$$$)/120$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/25\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
24.1
 − 190.348i − 189.348i 189.348i 190.348i
5096.35i 48964.9i −1.75842e7 0 −2.49542e8 5.17135e9i 4.68639e10i 9.17456e10 0
24.2 4016.35i 388445.i −7.74247e6 0 −1.56013e9 3.81217e9i 2.59512e9i −5.67462e10 0
24.3 4016.35i 388445.i −7.74247e6 0 −1.56013e9 3.81217e9i 2.59512e9i −5.67462e10 0
24.4 5096.35i 48964.9i −1.75842e7 0 −2.49542e8 5.17135e9i 4.68639e10i 9.17456e10 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 42103872 T_{2}^{2} +$$$$41\!\cdots\!96$$ acting on $$S_{24}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$418969153437696 + 42103872 T^{2} + T^{4}$$
$3$ $$36\!\cdots\!36$$$$+ 153286963488 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$38\!\cdots\!96$$$$+ 41275512975785631872 T^{2} + T^{4}$$
$11$ $$($$$$15\!\cdots\!24$$$$- 856801968264 T + T^{2} )^{2}$$
$13$ $$81\!\cdots\!76$$$$+$$$$76\!\cdots\!48$$$$T^{2} + T^{4}$$
$17$ $$21\!\cdots\!96$$$$+$$$$55\!\cdots\!72$$$$T^{2} + T^{4}$$
$19$ $$( -$$$$39\!\cdots\!00$$$$+ 4260600979960 T + T^{2} )^{2}$$
$23$ $$26\!\cdots\!16$$$$+$$$$33\!\cdots\!08$$$$T^{2} + T^{4}$$
$29$ $$( -$$$$85\!\cdots\!00$$$$+ 20818433601623340 T + T^{2} )^{2}$$
$31$ $$($$$$18\!\cdots\!64$$$$- 137714017177000384 T + T^{2} )^{2}$$
$37$ $$17\!\cdots\!96$$$$+$$$$16\!\cdots\!72$$$$T^{2} + T^{4}$$
$41$ $$( -$$$$51\!\cdots\!16$$$$+ 2294435477168314956 T + T^{2} )^{2}$$
$43$ $$18\!\cdots\!96$$$$+$$$$39\!\cdots\!28$$$$T^{2} + T^{4}$$
$47$ $$44\!\cdots\!96$$$$+$$$$67\!\cdots\!72$$$$T^{2} + T^{4}$$
$53$ $$23\!\cdots\!36$$$$+$$$$99\!\cdots\!88$$$$T^{2} + T^{4}$$
$59$ $$( -$$$$28\!\cdots\!00$$$$+$$$$28\!\cdots\!80$$$$T + T^{2} )^{2}$$
$61$ $$( -$$$$13\!\cdots\!76$$$$+$$$$18\!\cdots\!36$$$$T + T^{2} )^{2}$$
$67$ $$16\!\cdots\!96$$$$+$$$$22\!\cdots\!72$$$$T^{2} + T^{4}$$
$71$ $$($$$$17\!\cdots\!44$$$$-$$$$30\!\cdots\!24$$$$T + T^{2} )^{2}$$
$73$ $$25\!\cdots\!16$$$$+$$$$33\!\cdots\!08$$$$T^{2} + T^{4}$$
$79$ $$( -$$$$40\!\cdots\!00$$$$+$$$$62\!\cdots\!40$$$$T + T^{2} )^{2}$$
$83$ $$12\!\cdots\!56$$$$+$$$$27\!\cdots\!68$$$$T^{2} + T^{4}$$
$89$ $$($$$$82\!\cdots\!00$$$$+$$$$63\!\cdots\!20$$$$T + T^{2} )^{2}$$
$97$ $$88\!\cdots\!96$$$$+$$$$19\!\cdots\!72$$$$T^{2} + T^{4}$$