# Properties

 Label 225.10.a.k Level 225 Weight 10 Character orbit 225.a Self dual yes Analytic conductor 115.883 Analytic rank 0 Dimension 2 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: $$2$$ Coefficient field: $$\Q(\sqrt{241})$$ Defining polynomial: $$x^{2} - x - 60$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(-1 + 3\sqrt{241})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 15 - \beta ) q^{2} + ( 255 - 31 \beta ) q^{4} + ( -6944 + 224 \beta ) q^{7} + ( 12947 - 239 \beta ) q^{8} +O(q^{10})$$ $$q + ( 15 - \beta ) q^{2} + ( 255 - 31 \beta ) q^{4} + ( -6944 + 224 \beta ) q^{7} + ( 12947 - 239 \beta ) q^{8} + ( 9572 - 2368 \beta ) q^{11} + ( -14814 - 5344 \beta ) q^{13} + ( -225568 + 10528 \beta ) q^{14} + ( 193183 - 899 \beta ) q^{16} + ( -82238 - 7520 \beta ) q^{17} + ( -45084 + 5728 \beta ) q^{19} + ( 1427036 - 47460 \beta ) q^{22} + ( -380768 - 26272 \beta ) q^{23} + ( 2674238 - 70690 \beta ) q^{26} + ( -5534368 + 279328 \beta ) q^{28} + ( 1254818 - 168576 \beta ) q^{29} + ( 5467584 + 152736 \beta ) q^{31} + ( -3243861 - 85199 \beta ) q^{32} + ( 2842270 - 38082 \beta ) q^{34} + ( -11083414 - 198496 \beta ) q^{37} + ( -3780836 + 136732 \beta ) q^{38} + ( -13276234 - 492096 \beta ) q^{41} + ( 3244412 - 702336 \beta ) q^{43} + ( 42227996 - 973980 \beta ) q^{44} + ( 8527904 - 39584 \beta ) q^{46} + ( -16775384 - 1970528 \beta ) q^{47} + ( 35060921 - 3161088 \beta ) q^{49} + ( 86012318 - 1069150 \beta ) q^{52} + ( 1654422 + 177728 \beta ) q^{53} + ( -118920480 + 4613280 \beta ) q^{56} + ( 110190462 - 3952034 \beta ) q^{58} + ( 15573156 - 4348352 \beta ) q^{59} + ( 170273566 - 950208 \beta ) q^{61} + ( -769152 - 3023808 \beta ) q^{62} + ( -101389753 + 2340965 \beta ) q^{64} + ( 144611188 + 1026560 \beta ) q^{67} + ( 105380350 + 398658 \beta ) q^{68} + ( -107415672 - 4545280 \beta ) q^{71} + ( 116915638 + 1168192 \beta ) q^{73} + ( -58666378 + 7907478 \beta ) q^{74} + ( -107738276 + 3035812 \beta ) q^{76} + ( -353962112 + 19117952 \beta ) q^{77} + ( -21902080 - 19049120 \beta ) q^{79} + ( 67572522 + 5402698 \beta ) q^{82} + ( -189671220 - 7260288 \beta ) q^{83} + ( 429332292 - 14481788 \beta ) q^{86} + ( 430674668 - 33512156 \beta ) q^{88} + ( 218344134 + 9049152 \beta ) q^{89} + ( -545935936 + 34987456 \beta ) q^{91} + ( 344326304 + 4290016 \beta ) q^{92} + ( 816395416 - 14753064 \beta ) q^{94} + ( -892554882 - 13450880 \beta ) q^{97} + ( 2239223511 - 85638329 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 31q^{2} + 541q^{4} - 14112q^{7} + 26133q^{8} + O(q^{10})$$ $$2q + 31q^{2} + 541q^{4} - 14112q^{7} + 26133q^{8} + 21512q^{11} - 24284q^{13} - 461664q^{14} + 387265q^{16} - 156956q^{17} - 95896q^{19} + 2901532q^{22} - 735264q^{23} + 5419166q^{26} - 11348064q^{28} + 2678212q^{29} + 10782432q^{31} - 6402523q^{32} + 5722622q^{34} - 21968332q^{37} - 7698404q^{38} - 26060372q^{41} + 7191160q^{43} + 85429972q^{44} + 17095392q^{46} - 31580240q^{47} + 73282930q^{49} + 173093786q^{52} + 3131116q^{53} - 242454240q^{56} + 224332958q^{58} + 35494664q^{59} + 341497340q^{61} + 1485504q^{62} - 205120471q^{64} + 288195816q^{67} + 210362042q^{68} - 210286064q^{71} + 232663084q^{73} - 125240234q^{74} - 218512364q^{76} - 727042176q^{77} - 24755040q^{79} + 129742346q^{82} - 372082152q^{83} + 873146372q^{86} + 894861492q^{88} + 427639116q^{89} - 1126859328q^{91} + 684362592q^{92} + 1647543896q^{94} - 1771658884q^{97} + 4564085351q^{98} + O(q^{100})$$

## 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
 8.26209 −7.26209
−7.78626 0 −451.374 0 0 −1839.88 7501.08 0 0
1.2 38.7863 0 992.374 0 0 −12272.1 18631.9 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.k 2
3.b odd 2 1 75.10.a.f 2
5.b even 2 1 45.10.a.d 2
5.c odd 4 2 225.10.b.i 4
15.d odd 2 1 15.10.a.d 2
15.e even 4 2 75.10.b.f 4
60.h even 2 1 240.10.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.d 2 15.d odd 2 1
45.10.a.d 2 5.b even 2 1
75.10.a.f 2 3.b odd 2 1
75.10.b.f 4 15.e even 4 2
225.10.a.k 2 1.a even 1 1 trivial
225.10.b.i 4 5.c odd 4 2
240.10.a.r 2 60.h even 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 31 T_{2} - 302$$ $$T_{7}^{2} + 14112 T_{7} + 22579200$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 31 T + 722 T^{2} - 15872 T^{3} + 262144 T^{4}$$
$3$ 1
$5$ 1
$7$ $$1 + 14112 T + 103286414 T^{2} + 569470101984 T^{3} + 1628413597910449 T^{4}$$
$11$ $$1 - 21512 T + 1790961254 T^{2} - 50724170728792 T^{3} + 5559917313492231481 T^{4}$$
$13$ $$1 + 24284 T + 5870669214 T^{2} + 257519662773932 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$
$17$ $$1 + 156956 T + 212670095078 T^{2} + 18613078743463132 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$
$19$ $$1 + 95896 T + 629883192438 T^{2} + 30944459466214984 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$
$23$ $$1 + 735264 T + 3363187908526 T^{2} + 1324322710477931232 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$
$29$ $$1 - 2678212 T + 15397908029438 T^{2} - 38853212438324066228 T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$31$ $$1 - 10782432 T + 69294691361342 T^{2} -$$$$28\!\cdots\!72$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$
$37$ $$1 + 21968332 T + 359210373327534 T^{2} +$$$$28\!\cdots\!64$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$
$41$ $$1 + 26060372 T + 693239183881142 T^{2} +$$$$85\!\cdots\!92$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$
$43$ $$1 - 7191160 T + 750634586008230 T^{2} -$$$$36\!\cdots\!80$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$
$47$ $$1 + 31580240 T + 382042606129310 T^{2} +$$$$35\!\cdots\!80$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$
$53$ $$1 - 3131116 T + 6584849973489806 T^{2} -$$$$10\!\cdots\!28$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$
$59$ $$1 - 35494664 T + 7388006896329158 T^{2} -$$$$30\!\cdots\!96$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$
$61$ $$1 - 341497340 T + 52053805546777278 T^{2} -$$$$39\!\cdots\!40$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$
$67$ $$1 - 288195816 T + 74605839041196758 T^{2} -$$$$78\!\cdots\!52$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$
$71$ $$1 + 210286064 T + 91549406631588686 T^{2} +$$$$96\!\cdots\!84$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$73$ $$1 - 232663084 T + 130536207391012086 T^{2} -$$$$13\!\cdots\!92$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$
$79$ $$1 + 24755040 T + 43090694479668638 T^{2} +$$$$29\!\cdots\!60$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$
$83$ $$1 + 372082152 T + 379908828789982198 T^{2} +$$$$69\!\cdots\!56$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$
$89$ $$1 - 427639116 T + 702028302670151638 T^{2} -$$$$14\!\cdots\!44$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$
$97$ $$1 + 1771658884 T + 2207048700436243398 T^{2} +$$$$13\!\cdots\!28$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$