# Properties

 Label 25.10.a.d Level $25$ Weight $10$ Character orbit 25.a Self dual yes Analytic conductor $12.876$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 25.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.8758959041$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - x^{2} - 652 x + 4000$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2\cdot 5$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( 11 + \beta_{1} ) q^{2} + ( 30 + 2 \beta_{1} - \beta_{2} ) q^{3} + ( 114 + 9 \beta_{1} - \beta_{2} ) q^{4} + ( 1149 + 109 \beta_{1} - 26 \beta_{2} ) q^{6} + ( 1744 + 132 \beta_{1} + 26 \beta_{2} ) q^{7} + ( -24 - 333 \beta_{1} - 33 \beta_{2} ) q^{8} + ( 19422 + 1288 \beta_{1} - 32 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 11 + \beta_{1} ) q^{2} + ( 30 + 2 \beta_{1} - \beta_{2} ) q^{3} + ( 114 + 9 \beta_{1} - \beta_{2} ) q^{4} + ( 1149 + 109 \beta_{1} - 26 \beta_{2} ) q^{6} + ( 1744 + 132 \beta_{1} + 26 \beta_{2} ) q^{7} + ( -24 - 333 \beta_{1} - 33 \beta_{2} ) q^{8} + ( 19422 + 1288 \beta_{1} - 32 \beta_{2} ) q^{9} + ( -18298 + 370 \beta_{1} + 195 \beta_{2} ) q^{11} + ( 47358 + 2065 \beta_{1} - 221 \beta_{2} ) q^{12} + ( 71808 - 2592 \beta_{1} + 460 \beta_{2} ) q^{13} + ( 90810 - 678 \beta_{1} + 492 \beta_{2} ) q^{14} + ( -233100 - 1227 \beta_{1} + 53 \beta_{2} ) q^{16} + ( 111605 - 7016 \beta_{1} + 168 \beta_{2} ) q^{17} + ( 857970 + 19502 \beta_{1} - 2056 \beta_{2} ) q^{18} + ( 273798 - 20934 \beta_{1} - 2549 \beta_{2} ) q^{19} + ( -790314 - 12432 \beta_{1} - 4452 \beta_{2} ) q^{21} + ( 22817 - 35223 \beta_{1} + 4310 \beta_{2} ) q^{22} + ( 1174476 - 17556 \beta_{1} + 3426 \beta_{2} ) q^{23} + ( 933264 + 5763 \beta_{1} + 5943 \beta_{2} ) q^{24} + ( -431212 + 38812 \beta_{1} + 13632 \beta_{2} ) q^{26} + ( 2217186 + 145262 \beta_{1} - 18163 \beta_{2} ) q^{27} + ( -142436 - 16254 \beta_{1} - 826 \beta_{2} ) q^{28} + ( 727252 + 119984 \beta_{1} - 6276 \beta_{2} ) q^{29} + ( 1425052 - 56940 \beta_{1} - 1090 \beta_{2} ) q^{31} + ( -3161324 - 64549 \beta_{1} + 19395 \beta_{2} ) q^{32} + ( -7376475 - 209936 \beta_{1} + 7288 \beta_{2} ) q^{33} + ( -2283337 + 111693 \beta_{1} + 11048 \beta_{2} ) q^{34} + ( 8949420 + 330158 \beta_{1} - 52462 \beta_{2} ) q^{36} + ( -3447538 - 105792 \beta_{1} + 37572 \beta_{2} ) q^{37} + ( -8046751 + 527233 \beta_{1} - 40242 \beta_{2} ) q^{38} + ( -16789428 - 566728 \beta_{1} - 45808 \beta_{2} ) q^{39} + ( 1962517 - 961840 \beta_{1} + 38760 \beta_{2} ) q^{41} + ( -15821946 - 395934 \beta_{1} - 94416 \beta_{2} ) q^{42} + ( -8142408 + 200976 \beta_{1} - 2732 \beta_{2} ) q^{43} + ( -7344842 - 453907 \beta_{1} + 38823 \beta_{2} ) q^{44} + ( 4707822 + 925230 \beta_{1} + 99780 \beta_{2} ) q^{46} + ( 22283108 + 651880 \beta_{1} + 9384 \beta_{2} ) q^{47} + ( -9935964 - 628811 \beta_{1} + 250021 \beta_{2} ) q^{48} + ( -2224403 + 264480 \beta_{1} + 219280 \beta_{2} ) q^{49} + ( -8541210 - 564254 \beta_{1} - 11069 \beta_{2} ) q^{51} + ( -19305256 - 313188 \beta_{1} + 52836 \beta_{2} ) q^{52} + ( 44311790 - 1064528 \beta_{1} - 314856 \beta_{2} ) q^{53} + ( 94277223 + 3434191 \beta_{1} - 581174 \beta_{2} ) q^{54} + ( -56427552 + 305766 \beta_{1} - 255474 \beta_{2} ) q^{56} + ( 84278277 + 1412984 \beta_{1} + 111584 \beta_{2} ) q^{57} + ( 67392976 + 1008192 \beta_{1} - 270608 \beta_{2} ) q^{58} + ( 1775144 - 399752 \beta_{1} + 345528 \beta_{2} ) q^{59} + ( 41835342 + 4664400 \beta_{1} - 199100 \beta_{2} ) q^{61} + ( -13287318 + 1629402 \beta_{1} + 30780 \beta_{2} ) q^{62} + ( 94577904 - 567960 \beta_{1} + 589692 \beta_{2} ) q^{63} + ( 55679836 - 4013787 \beta_{1} + 502893 \beta_{2} ) q^{64} + ( -185766897 - 7561507 \beta_{1} + 384848 \beta_{2} ) q^{66} + ( 29717410 - 7831506 \beta_{1} - 322747 \beta_{2} ) q^{67} + ( -23743334 + 168485 \beta_{1} + 67443 \beta_{2} ) q^{68} + ( -104422626 - 2789424 \beta_{1} - 1007064 \beta_{2} ) q^{69} + ( 98809132 - 4103800 \beta_{1} + 1123200 \beta_{2} ) q^{71} + ( -184127472 + 2658426 \beta_{1} - 536574 \beta_{2} ) q^{72} + ( 60459531 + 11107704 \beta_{1} - 56864 \beta_{2} ) q^{73} + ( -84171626 - 6354430 \beta_{1} + 1007520 \beta_{2} ) q^{74} + ( 29867606 + 4957077 \beta_{1} - 187953 \beta_{2} ) q^{76} + ( 186837678 - 837696 \beta_{1} + 701052 \beta_{2} ) q^{77} + ( -479630676 - 11853908 \beta_{1} - 532664 \beta_{2} ) q^{78} + ( -103098848 + 15961884 \beta_{1} - 728626 \beta_{2} ) q^{79} + ( 467368353 + 10864264 \beta_{1} - 3257696 \beta_{2} ) q^{81} + ( -456738353 + 669117 \beta_{1} + 1892080 \beta_{2} ) q^{82} + ( -243751566 - 8354490 \beta_{1} + 165897 \beta_{2} ) q^{83} + ( 12619236 - 828366 \beta_{1} + 409374 \beta_{2} ) q^{84} + ( 11404580 - 8317604 \beta_{1} - 266544 \beta_{2} ) q^{86} + ( 349578948 + 18508016 \beta_{1} - 2351284 \beta_{2} ) q^{87} + ( -314283408 + 8374839 \beta_{1} - 821061 \beta_{2} ) q^{88} + ( -367574409 + 9709272 \beta_{1} - 1136808 \beta_{2} ) q^{89} + ( 393086120 + 20580336 \beta_{1} + 3929296 \beta_{2} ) q^{91} + ( -63246540 + 3564294 \beta_{1} - 284622 \beta_{2} ) q^{92} + ( 35975730 - 954816 \beta_{1} - 540432 \beta_{2} ) q^{93} + ( 576105932 + 20200476 \beta_{1} - 426664 \beta_{2} ) q^{94} + ( -856922316 - 32380741 \beta_{1} + 3586499 \beta_{2} ) q^{96} + ( -110724742 - 33078720 \beta_{1} - 62616 \beta_{2} ) q^{97} + ( 150976447 - 20953603 \beta_{1} + 4998240 \beta_{2} ) q^{98} + ( -299572596 - 47981644 \beta_{1} + 6483266 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 33q^{2} + 89q^{3} + 341q^{4} + 3421q^{6} + 5258q^{7} - 105q^{8} + 58234q^{9} + O(q^{10})$$ $$3q + 33q^{2} + 89q^{3} + 341q^{4} + 3421q^{6} + 5258q^{7} - 105q^{8} + 58234q^{9} - 54699q^{11} + 141853q^{12} + 215884q^{13} + 272922q^{14} - 699247q^{16} + 334983q^{17} + 2571854q^{18} + 818845q^{19} - 2375394q^{21} + 72761q^{22} + 3526854q^{23} + 2805735q^{24} - 1280004q^{26} + 6633395q^{27} - 428134q^{28} + 2175480q^{29} + 4274066q^{31} - 9464577q^{32} - 22122137q^{33} - 6838963q^{34} + 26795798q^{36} - 10305042q^{37} - 24180495q^{38} - 50414092q^{39} + 5926311q^{41} - 47560254q^{42} - 24429956q^{43} - 21995703q^{44} + 14223246q^{46} + 66858708q^{47} - 29557871q^{48} - 6453929q^{49} - 25634699q^{51} - 57862932q^{52} + 132620514q^{53} + 282250495q^{54} - 169538130q^{56} + 252946415q^{57} + 201908320q^{58} + 5670960q^{59} + 125306926q^{61} - 39831174q^{62} + 284323404q^{63} + 167542401q^{64} - 556915843q^{66} + 88829483q^{67} - 71162559q^{68} - 314274942q^{69} + 297550596q^{71} - 552918990q^{72} + 181321729q^{73} - 251507358q^{74} + 89414865q^{76} + 561214086q^{77} - 1439424692q^{78} - 310025170q^{79} + 1398847363q^{81} - 1368322979q^{82} - 731088801q^{83} + 38267082q^{84} + 33947196q^{86} + 1046385560q^{87} - 943671285q^{88} - 1103860035q^{89} + 1183187656q^{91} - 190024242q^{92} + 107386758q^{93} + 1727891132q^{94} - 2567180449q^{96} - 332236842q^{97} + 457927581q^{98} - 892234522q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 652 x + 4000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{2} + 3 \nu - 436$$$$)/12$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} + 33 \nu - 445$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 4 \beta_{1} + 3$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$-3 \beta_{2} + 132 \beta_{1} + 4351$$$$)/10$$

## 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
 6.48955 22.2334 −27.7229
−20.2014 30.5073 −103.903 0 −616.291 −4010.25 12442.1 −18752.3 0
1.2 21.4187 −210.171 −53.2406 0 −4501.59 9905.49 −12106.7 24489.0 0
1.3 31.7828 268.664 498.143 0 8538.88 −637.237 −440.406 52497.3 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 25.10.a.d yes 3
3.b odd 2 1 225.10.a.m 3
4.b odd 2 1 400.10.a.u 3
5.b even 2 1 25.10.a.c 3
5.c odd 4 2 25.10.b.c 6
15.d odd 2 1 225.10.a.p 3
15.e even 4 2 225.10.b.m 6
20.d odd 2 1 400.10.a.y 3
20.e even 4 2 400.10.c.q 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.10.a.c 3 5.b even 2 1
25.10.a.d yes 3 1.a even 1 1 trivial
25.10.b.c 6 5.c odd 4 2
225.10.a.m 3 3.b odd 2 1
225.10.a.p 3 15.d odd 2 1
225.10.b.m 6 15.e even 4 2
400.10.a.u 3 4.b odd 2 1
400.10.a.y 3 20.d odd 2 1
400.10.c.q 6 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 33 T_{2}^{2} - 394 T_{2} + 13752$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(25))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 33 T + 1142 T^{2} - 20040 T^{3} + 584704 T^{4} - 8650752 T^{5} + 134217728 T^{6}$$
$3$ $$1 - 89 T + 4368 T^{2} - 1780965 T^{3} + 85975344 T^{4} - 34480423521 T^{5} + 7625597484987 T^{6}$$
$5$ 1
$7$ $$1 - 5258 T + 77580657 T^{2} - 449671828900 T^{3} + 3130659343379799 T^{4} - 8562198697813140842 T^{5} +$$$$65\!\cdots\!43$$$$T^{6}$$
$11$ $$1 + 54699 T + 5816187440 T^{2} + 182671409832855 T^{3} + 13714265744571201040 T^{4} +$$$$30\!\cdots\!19$$$$T^{5} +$$$$13\!\cdots\!71$$$$T^{6}$$
$13$ $$1 - 215884 T + 31225156343 T^{2} - 3094983572287480 T^{3} +$$$$33\!\cdots\!39$$$$T^{4} -$$$$24\!\cdots\!36$$$$T^{5} +$$$$11\!\cdots\!17$$$$T^{6}$$
$17$ $$1 - 334983 T + 355030662062 T^{2} - 77675997792847395 T^{3} +$$$$42\!\cdots\!14$$$$T^{4} -$$$$47\!\cdots\!47$$$$T^{5} +$$$$16\!\cdots\!73$$$$T^{6}$$
$19$ $$1 - 818845 T + 468173656712 T^{2} - 302339390932836385 T^{3} +$$$$15\!\cdots\!48$$$$T^{4} -$$$$85\!\cdots\!45$$$$T^{5} +$$$$33\!\cdots\!39$$$$T^{6}$$
$23$ $$1 - 3526854 T + 8700596724993 T^{2} - 13086552252942250620 T^{3} +$$$$15\!\cdots\!59$$$$T^{4} -$$$$11\!\cdots\!26$$$$T^{5} +$$$$58\!\cdots\!47$$$$T^{6}$$
$29$ $$1 - 2175480 T + 32457706855607 T^{2} - 59155485309271560240 T^{3} +$$$$47\!\cdots\!83$$$$T^{4} -$$$$45\!\cdots\!80$$$$T^{5} +$$$$30\!\cdots\!09$$$$T^{6}$$
$31$ $$1 - 4274066 T + 82851493809465 T^{2} -$$$$22\!\cdots\!20$$$$T^{3} +$$$$21\!\cdots\!15$$$$T^{4} -$$$$29\!\cdots\!06$$$$T^{5} +$$$$18\!\cdots\!11$$$$T^{6}$$
$37$ $$1 + 10305042 T + 340842196290747 T^{2} +$$$$25\!\cdots\!40$$$$T^{3} +$$$$44\!\cdots\!19$$$$T^{4} +$$$$17\!\cdots\!18$$$$T^{5} +$$$$21\!\cdots\!33$$$$T^{6}$$
$41$ $$1 - 5926311 T + 232043727124790 T^{2} -$$$$23\!\cdots\!95$$$$T^{3} +$$$$75\!\cdots\!90$$$$T^{4} -$$$$63\!\cdots\!31$$$$T^{5} +$$$$35\!\cdots\!81$$$$T^{6}$$
$43$ $$1 + 24429956 T + 1676050268074193 T^{2} +$$$$24\!\cdots\!00$$$$T^{3} +$$$$84\!\cdots\!99$$$$T^{4} +$$$$61\!\cdots\!44$$$$T^{5} +$$$$12\!\cdots\!07$$$$T^{6}$$
$47$ $$1 - 66858708 T + 4517266130065277 T^{2} -$$$$15\!\cdots\!80$$$$T^{3} +$$$$50\!\cdots\!59$$$$T^{4} -$$$$83\!\cdots\!12$$$$T^{5} +$$$$14\!\cdots\!63$$$$T^{6}$$
$53$ $$1 - 132620514 T + 9209248487675243 T^{2} -$$$$47\!\cdots\!40$$$$T^{3} +$$$$30\!\cdots\!19$$$$T^{4} -$$$$14\!\cdots\!46$$$$T^{5} +$$$$35\!\cdots\!37$$$$T^{6}$$
$59$ $$1 - 5670960 T + 19337695182532817 T^{2} -$$$$14\!\cdots\!80$$$$T^{3} +$$$$16\!\cdots\!63$$$$T^{4} -$$$$42\!\cdots\!60$$$$T^{5} +$$$$65\!\cdots\!19$$$$T^{6}$$
$61$ $$1 - 125306926 T + 22192130203358915 T^{2} -$$$$23\!\cdots\!20$$$$T^{3} +$$$$25\!\cdots\!15$$$$T^{4} -$$$$17\!\cdots\!06$$$$T^{5} +$$$$15\!\cdots\!21$$$$T^{6}$$
$67$ $$1 - 88829483 T + 30562384919894712 T^{2} -$$$$76\!\cdots\!95$$$$T^{3} +$$$$83\!\cdots\!64$$$$T^{4} -$$$$65\!\cdots\!47$$$$T^{5} +$$$$20\!\cdots\!23$$$$T^{6}$$
$71$ $$1 - 297550596 T + 87036096018332165 T^{2} -$$$$15\!\cdots\!20$$$$T^{3} +$$$$39\!\cdots\!15$$$$T^{4} -$$$$62\!\cdots\!56$$$$T^{5} +$$$$96\!\cdots\!91$$$$T^{6}$$
$73$ $$1 - 181321729 T + 94365252526618118 T^{2} -$$$$75\!\cdots\!45$$$$T^{3} +$$$$55\!\cdots\!34$$$$T^{4} -$$$$62\!\cdots\!01$$$$T^{5} +$$$$20\!\cdots\!97$$$$T^{6}$$
$79$ $$1 + 310025170 T + 176092553119892457 T^{2} +$$$$48\!\cdots\!60$$$$T^{3} +$$$$21\!\cdots\!83$$$$T^{4} +$$$$44\!\cdots\!70$$$$T^{5} +$$$$17\!\cdots\!59$$$$T^{6}$$
$83$ $$1 + 731088801 T + 685433348047337568 T^{2} +$$$$27\!\cdots\!65$$$$T^{3} +$$$$12\!\cdots\!04$$$$T^{4} +$$$$25\!\cdots\!09$$$$T^{5} +$$$$65\!\cdots\!27$$$$T^{6}$$
$89$ $$1 + 1103860035 T + 1320664213408319502 T^{2} +$$$$75\!\cdots\!55$$$$T^{3} +$$$$46\!\cdots\!18$$$$T^{4} +$$$$13\!\cdots\!35$$$$T^{5} +$$$$43\!\cdots\!29$$$$T^{6}$$
$97$ $$1 + 332236842 T + 1487745026246658927 T^{2} +$$$$16\!\cdots\!20$$$$T^{3} +$$$$11\!\cdots\!59$$$$T^{4} +$$$$19\!\cdots\!38$$$$T^{5} +$$$$43\!\cdots\!13$$$$T^{6}$$