# Properties

 Label 225.10.a.p Level $225$ Weight $10$ Character orbit 225.a Self dual yes Analytic conductor $115.883$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$115.883063137$$ 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: no (minimal twist has level 25) 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.

 $$f(q)$$ $$=$$ $$q + ( 11 + \beta_{1} ) q^{2} + ( 114 + 9 \beta_{1} - \beta_{2} ) q^{4} + ( -1744 - 132 \beta_{1} - 26 \beta_{2} ) q^{7} + ( -24 - 333 \beta_{1} - 33 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( 11 + \beta_{1} ) q^{2} + ( 114 + 9 \beta_{1} - \beta_{2} ) q^{4} + ( -1744 - 132 \beta_{1} - 26 \beta_{2} ) q^{7} + ( -24 - 333 \beta_{1} - 33 \beta_{2} ) q^{8} + ( 18298 - 370 \beta_{1} - 195 \beta_{2} ) q^{11} + ( -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} + ( 273798 - 20934 \beta_{1} - 2549 \beta_{2} ) q^{19} + ( -22817 + 35223 \beta_{1} - 4310 \beta_{2} ) q^{22} + ( 1174476 - 17556 \beta_{1} + 3426 \beta_{2} ) q^{23} + ( 431212 - 38812 \beta_{1} - 13632 \beta_{2} ) q^{26} + ( 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} + ( -2283337 + 111693 \beta_{1} + 11048 \beta_{2} ) q^{34} + ( 3447538 + 105792 \beta_{1} - 37572 \beta_{2} ) q^{37} + ( -8046751 + 527233 \beta_{1} - 40242 \beta_{2} ) q^{38} + ( -1962517 + 961840 \beta_{1} - 38760 \beta_{2} ) q^{41} + ( 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} + ( -2224403 + 264480 \beta_{1} + 219280 \beta_{2} ) q^{49} + ( 19305256 + 313188 \beta_{1} - 52836 \beta_{2} ) q^{52} + ( 44311790 - 1064528 \beta_{1} - 314856 \beta_{2} ) q^{53} + ( 56427552 - 305766 \beta_{1} + 255474 \beta_{2} ) q^{56} + ( -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} + ( 55679836 - 4013787 \beta_{1} + 502893 \beta_{2} ) q^{64} + ( -29717410 + 7831506 \beta_{1} + 322747 \beta_{2} ) q^{67} + ( -23743334 + 168485 \beta_{1} + 67443 \beta_{2} ) q^{68} + ( -98809132 + 4103800 \beta_{1} - 1123200 \beta_{2} ) q^{71} + ( -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} + ( -103098848 + 15961884 \beta_{1} - 728626 \beta_{2} ) q^{79} + ( 456738353 - 669117 \beta_{1} - 1892080 \beta_{2} ) q^{82} + ( -243751566 - 8354490 \beta_{1} + 165897 \beta_{2} ) q^{83} + ( -11404580 + 8317604 \beta_{1} + 266544 \beta_{2} ) q^{86} + ( 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} + ( 576105932 + 20200476 \beta_{1} - 426664 \beta_{2} ) q^{94} + ( 110724742 + 33078720 \beta_{1} + 62616 \beta_{2} ) q^{97} + ( 150976447 - 20953603 \beta_{1} + 4998240 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 33q^{2} + 341q^{4} - 5258q^{7} - 105q^{8} + O(q^{10})$$ $$3q + 33q^{2} + 341q^{4} - 5258q^{7} - 105q^{8} + 54699q^{11} - 215884q^{13} - 272922q^{14} - 699247q^{16} + 334983q^{17} + 818845q^{19} - 72761q^{22} + 3526854q^{23} + 1280004q^{26} + 428134q^{28} - 2175480q^{29} + 4274066q^{31} - 9464577q^{32} - 6838963q^{34} + 10305042q^{37} - 24180495q^{38} - 5926311q^{41} + 24429956q^{43} + 21995703q^{44} + 14223246q^{46} + 66858708q^{47} - 6453929q^{49} + 57862932q^{52} + 132620514q^{53} + 169538130q^{56} - 201908320q^{58} - 5670960q^{59} + 125306926q^{61} - 39831174q^{62} + 167542401q^{64} - 88829483q^{67} - 71162559q^{68} - 297550596q^{71} - 181321729q^{73} + 251507358q^{74} + 89414865q^{76} + 561214086q^{77} - 310025170q^{79} + 1368322979q^{82} - 731088801q^{83} - 33947196q^{86} + 943671285q^{88} + 1103860035q^{89} + 1183187656q^{91} - 190024242q^{92} + 1727891132q^{94} + 332236842q^{97} + 457927581q^{98} + O(q^{100})$$

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 0 −103.903 0 0 4010.25 12442.1 0 0
1.2 21.4187 0 −53.2406 0 0 −9905.49 −12106.7 0 0
1.3 31.7828 0 498.143 0 0 637.237 −440.406 0 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
$$3$$ $$-1$$
$$5$$ $$1$$

## 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 225.10.a.p 3
3.b odd 2 1 25.10.a.c 3
5.b even 2 1 225.10.a.m 3
5.c odd 4 2 225.10.b.m 6
12.b even 2 1 400.10.a.y 3
15.d odd 2 1 25.10.a.d yes 3
15.e even 4 2 25.10.b.c 6
60.h even 2 1 400.10.a.u 3
60.l odd 4 2 400.10.c.q 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.10.a.c 3 3.b odd 2 1
25.10.a.d yes 3 15.d odd 2 1
25.10.b.c 6 15.e even 4 2
225.10.a.m 3 5.b even 2 1
225.10.a.p 3 1.a even 1 1 trivial
225.10.b.m 6 5.c odd 4 2
400.10.a.u 3 60.h even 2 1
400.10.a.y 3 12.b even 2 1
400.10.c.q 6 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2}^{3} - 33 T_{2}^{2} - 394 T_{2} + 13752$$ $$T_{7}^{3} + 5258 T_{7}^{2} - 43480164 T_{7} + 25313297688$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$13752 - 394 T - 33 T^{2} + T^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3}$$
$7$ $$25313297688 - 43480164 T + 5258 T^{2} + T^{3}$$
$11$ $$75283351667163 - 1257655633 T - 54699 T^{2} + T^{3}$$
$13$ $$-1483699912993984 - 588341776 T + 215884 T^{2} + T^{3}$$
$17$ $$1773847472341707 - 732967429 T - 334983 T^{2} + T^{3}$$
$19$ $$226123024842854125 - 499889436625 T - 818845 T^{2} + T^{3}$$
$23$ $$-381747315559395816 + 3297138740604 T - 3526854 T^{2} + T^{3}$$
$29$ $$-3964526545895424000 - 11063731072000 T + 2175480 T^{2} + T^{3}$$
$31$ $$-817098195664566648 + 3532627327452 T - 4274066 T^{2} + T^{3}$$
$37$ $$17\!\cdots\!28$$$$- 49043023094484 T - 10305042 T^{2} + T^{3}$$
$41$ $$-$$$$15\!\cdots\!47$$$$- 750102076057093 T + 5926311 T^{2} + T^{3}$$
$43$ $$-$$$$33\!\cdots\!84$$$$+ 168272432263664 T - 24429956 T^{2} + T^{3}$$
$47$ $$-$$$$14\!\cdots\!08$$$$+ 1159874710756976 T - 66858708 T^{2} + T^{3}$$
$53$ $$40\!\cdots\!84$$$$- 690042287731156 T - 132620514 T^{2} + T^{3}$$
$59$ $$49\!\cdots\!00$$$$- 6651292273432000 T + 5670960 T^{2} + T^{3}$$
$61$ $$62\!\cdots\!12$$$$- 12890308075143508 T - 125306926 T^{2} + T^{3}$$
$67$ $$27\!\cdots\!93$$$$- 51057218268990129 T + 88829483 T^{2} + T^{3}$$
$71$ $$-$$$$11\!\cdots\!32$$$$- 50509406137014928 T + 297550596 T^{2} + T^{3}$$
$73$ $$-$$$$13\!\cdots\!09$$$$- 82249507598185621 T + 181321729 T^{2} + T^{3}$$
$79$ $$-$$$$26\!\cdots\!00$$$$- 183462234827962500 T + 310025170 T^{2} + T^{3}$$
$83$ $$-$$$$83\!\cdots\!41$$$$+ 124612582244716359 T + 731088801 T^{2} + T^{3}$$
$89$ $$19\!\cdots\!75$$$$+ 269595002285863875 T - 1103860035 T^{2} + T^{3}$$
$97$ $$34\!\cdots\!08$$$$- 792948149717036724 T - 332236842 T^{2} + T^{3}$$