# Properties

 Label 25.10.a.c Level $25$ Weight $10$ Character orbit 25.a Self dual yes Analytic conductor $12.876$ Analytic rank $1$ 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: $$1$$ 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
 −27.7229 22.2334 6.48955
−31.7828 −268.664 498.143 0 8538.88 637.237 440.406 52497.3 0
1.2 −21.4187 210.171 −53.2406 0 −4501.59 −9905.49 12106.7 24489.0 0
1.3 20.2014 −30.5073 −103.903 0 −616.291 4010.25 −12442.1 −18752.3 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.c 3
3.b odd 2 1 225.10.a.p 3
4.b odd 2 1 400.10.a.y 3
5.b even 2 1 25.10.a.d yes 3
5.c odd 4 2 25.10.b.c 6
15.d odd 2 1 225.10.a.m 3
15.e even 4 2 225.10.b.m 6
20.d odd 2 1 400.10.a.u 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 1.a even 1 1 trivial
25.10.a.d yes 3 5.b even 2 1
25.10.b.c 6 5.c odd 4 2
225.10.a.m 3 15.d odd 2 1
225.10.a.p 3 3.b odd 2 1
225.10.b.m 6 15.e even 4 2
400.10.a.u 3 20.d odd 2 1
400.10.a.y 3 4.b 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
$7$ $$1 + 5258 T + 77580657 T^{2} + 449671828900 T^{3} + 3130659343379799 T^{4} + 8562198697813140842 T^{5} +$$$$65\!\cdots\!43$$$$T^{6}$$
$11$ $$1 + 54699 T + 5816187440 T^{2} + 182671409832855 T^{3} + 13714265744571201040 T^{4} +$$$$30\!\cdots\!19$$$$T^{5} +$$$$13\!\cdots\!71$$$$T^{6}$$
$13$ $$1 + 215884 T + 31225156343 T^{2} + 3094983572287480 T^{3} +$$$$33\!\cdots\!39$$$$T^{4} +$$$$24\!\cdots\!36$$$$T^{5} +$$$$11\!\cdots\!17$$$$T^{6}$$
$17$ $$1 + 334983 T + 355030662062 T^{2} + 77675997792847395 T^{3} +$$$$42\!\cdots\!14$$$$T^{4} +$$$$47\!\cdots\!47$$$$T^{5} +$$$$16\!\cdots\!73$$$$T^{6}$$
$19$ $$1 - 818845 T + 468173656712 T^{2} - 302339390932836385 T^{3} +$$$$15\!\cdots\!48$$$$T^{4} -$$$$85\!\cdots\!45$$$$T^{5} +$$$$33\!\cdots\!39$$$$T^{6}$$
$23$ $$1 + 3526854 T + 8700596724993 T^{2} + 13086552252942250620 T^{3} +$$$$15\!\cdots\!59$$$$T^{4} +$$$$11\!\cdots\!26$$$$T^{5} +$$$$58\!\cdots\!47$$$$T^{6}$$
$29$ $$1 - 2175480 T + 32457706855607 T^{2} - 59155485309271560240 T^{3} +$$$$47\!\cdots\!83$$$$T^{4} -$$$$45\!\cdots\!80$$$$T^{5} +$$$$30\!\cdots\!09$$$$T^{6}$$
$31$ $$1 - 4274066 T + 82851493809465 T^{2} -$$$$22\!\cdots\!20$$$$T^{3} +$$$$21\!\cdots\!15$$$$T^{4} -$$$$29\!\cdots\!06$$$$T^{5} +$$$$18\!\cdots\!11$$$$T^{6}$$
$37$ $$1 - 10305042 T + 340842196290747 T^{2} -$$$$25\!\cdots\!40$$$$T^{3} +$$$$44\!\cdots\!19$$$$T^{4} -$$$$17\!\cdots\!18$$$$T^{5} +$$$$21\!\cdots\!33$$$$T^{6}$$
$41$ $$1 - 5926311 T + 232043727124790 T^{2} -$$$$23\!\cdots\!95$$$$T^{3} +$$$$75\!\cdots\!90$$$$T^{4} -$$$$63\!\cdots\!31$$$$T^{5} +$$$$35\!\cdots\!81$$$$T^{6}$$
$43$ $$1 - 24429956 T + 1676050268074193 T^{2} -$$$$24\!\cdots\!00$$$$T^{3} +$$$$84\!\cdots\!99$$$$T^{4} -$$$$61\!\cdots\!44$$$$T^{5} +$$$$12\!\cdots\!07$$$$T^{6}$$
$47$ $$1 + 66858708 T + 4517266130065277 T^{2} +$$$$15\!\cdots\!80$$$$T^{3} +$$$$50\!\cdots\!59$$$$T^{4} +$$$$83\!\cdots\!12$$$$T^{5} +$$$$14\!\cdots\!63$$$$T^{6}$$
$53$ $$1 + 132620514 T + 9209248487675243 T^{2} +$$$$47\!\cdots\!40$$$$T^{3} +$$$$30\!\cdots\!19$$$$T^{4} +$$$$14\!\cdots\!46$$$$T^{5} +$$$$35\!\cdots\!37$$$$T^{6}$$
$59$ $$1 - 5670960 T + 19337695182532817 T^{2} -$$$$14\!\cdots\!80$$$$T^{3} +$$$$16\!\cdots\!63$$$$T^{4} -$$$$42\!\cdots\!60$$$$T^{5} +$$$$65\!\cdots\!19$$$$T^{6}$$
$61$ $$1 - 125306926 T + 22192130203358915 T^{2} -$$$$23\!\cdots\!20$$$$T^{3} +$$$$25\!\cdots\!15$$$$T^{4} -$$$$17\!\cdots\!06$$$$T^{5} +$$$$15\!\cdots\!21$$$$T^{6}$$
$67$ $$1 + 88829483 T + 30562384919894712 T^{2} +$$$$76\!\cdots\!95$$$$T^{3} +$$$$83\!\cdots\!64$$$$T^{4} +$$$$65\!\cdots\!47$$$$T^{5} +$$$$20\!\cdots\!23$$$$T^{6}$$
$71$ $$1 - 297550596 T + 87036096018332165 T^{2} -$$$$15\!\cdots\!20$$$$T^{3} +$$$$39\!\cdots\!15$$$$T^{4} -$$$$62\!\cdots\!56$$$$T^{5} +$$$$96\!\cdots\!91$$$$T^{6}$$
$73$ $$1 + 181321729 T + 94365252526618118 T^{2} +$$$$75\!\cdots\!45$$$$T^{3} +$$$$55\!\cdots\!34$$$$T^{4} +$$$$62\!\cdots\!01$$$$T^{5} +$$$$20\!\cdots\!97$$$$T^{6}$$
$79$ $$1 + 310025170 T + 176092553119892457 T^{2} +$$$$48\!\cdots\!60$$$$T^{3} +$$$$21\!\cdots\!83$$$$T^{4} +$$$$44\!\cdots\!70$$$$T^{5} +$$$$17\!\cdots\!59$$$$T^{6}$$
$83$ $$1 - 731088801 T + 685433348047337568 T^{2} -$$$$27\!\cdots\!65$$$$T^{3} +$$$$12\!\cdots\!04$$$$T^{4} -$$$$25\!\cdots\!09$$$$T^{5} +$$$$65\!\cdots\!27$$$$T^{6}$$
$89$ $$1 + 1103860035 T + 1320664213408319502 T^{2} +$$$$75\!\cdots\!55$$$$T^{3} +$$$$46\!\cdots\!18$$$$T^{4} +$$$$13\!\cdots\!35$$$$T^{5} +$$$$43\!\cdots\!29$$$$T^{6}$$
$97$ $$1 - 332236842 T + 1487745026246658927 T^{2} -$$$$16\!\cdots\!20$$$$T^{3} +$$$$11\!\cdots\!59$$$$T^{4} -$$$$19\!\cdots\!38$$$$T^{5} +$$$$43\!\cdots\!13$$$$T^{6}$$