# Properties

 Label 225.10.a.s Level 225 Weight 10 Character orbit 225.a Self dual yes Analytic conductor 115.883 Analytic rank 1 Dimension 4 CM no Inner twists 2

# Learn more about

## 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.49740556.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}\cdot 3^{2}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 5) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 342 - \beta_{3} ) q^{4} + ( 133 \beta_{1} - 49 \beta_{2} ) q^{7} + ( -684 \beta_{1} + 8 \beta_{2} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 342 - \beta_{3} ) q^{4} + ( 133 \beta_{1} - 49 \beta_{2} ) q^{7} + ( -684 \beta_{1} + 8 \beta_{2} ) q^{8} + ( -27492 - 20 \beta_{3} ) q^{11} + ( 1918 \beta_{1} + 110 \beta_{2} ) q^{13} + ( -106134 + 133 \beta_{3} ) q^{14} + ( 407816 - 172 \beta_{3} ) q^{16} + ( 7448 \beta_{1} + 1632 \beta_{2} ) q^{17} + ( 159220 + 476 \beta_{3} ) q^{19} + ( 10412 \beta_{1} + 160 \beta_{2} ) q^{22} + ( 17005 \beta_{1} + 5199 \beta_{2} ) q^{23} + ( -1654692 + 1918 \beta_{3} ) q^{26} + ( 151620 \beta_{1} + 24024 \beta_{2} ) q^{28} + ( -882930 + 1976 \beta_{3} ) q^{29} + ( -2646928 + 760 \beta_{3} ) q^{31} + ( -204496 \beta_{1} - 2720 \beta_{2} ) q^{32} + ( -6608656 + 7448 \beta_{3} ) q^{34} + ( -335370 \beta_{1} - 51378 \beta_{2} ) q^{37} + ( 247284 \beta_{1} - 3808 \beta_{2} ) q^{38} + ( 4197138 + 24890 \beta_{3} ) q^{41} + ( -719131 \beta_{1} + 132643 \beta_{2} ) q^{43} + ( 5159736 + 20652 \beta_{3} ) q^{44} + ( -15312518 + 17005 \beta_{3} ) q^{46} + ( 1012111 \beta_{1} - 214259 \beta_{2} ) q^{47} + ( 11730257 + 27930 \beta_{3} ) q^{49} + ( 2310648 \beta_{1} - 71664 \beta_{2} ) q^{52} + ( -1349666 \beta_{1} - 259794 \beta_{2} ) q^{53} + ( -78794520 + 83524 \beta_{3} ) q^{56} + ( 2570434 \beta_{1} - 15808 \beta_{2} ) q^{58} + ( -115207260 + 52972 \beta_{3} ) q^{59} + ( 90122642 + 43150 \beta_{3} ) q^{61} + ( 3295968 \beta_{1} - 6080 \beta_{2} ) q^{62} + ( -33748768 - 116432 \beta_{3} ) q^{64} + ( 6669647 \beta_{1} + 1448953 \beta_{2} ) q^{67} + ( 9155872 \beta_{1} - 895168 \beta_{2} ) q^{68} + ( 11902968 - 91200 \beta_{3} ) q^{71} + ( 1847940 \beta_{1} - 90564 \beta_{2} ) q^{73} + ( 294215436 - 335370 \beta_{3} ) q^{74} + ( -292122360 + 3572 \beta_{3} ) q^{76} + ( -1533756 \beta_{1} + 2162748 \beta_{2} ) q^{77} + ( 182010880 - 267976 \beta_{3} ) q^{79} + ( 17058922 \beta_{1} - 199120 \beta_{2} ) q^{82} + ( -12737657 \beta_{1} + 1180353 \beta_{2} ) q^{83} + ( 593976138 - 719131 \beta_{3} ) q^{86} + ( 7146128 \beta_{1} - 247136 \beta_{2} ) q^{88} + ( -395675190 - 185592 \beta_{3} ) q^{89} + ( 118330632 - 357504 \beta_{3} ) q^{91} + ( 21128228 \beta_{1} - 2797928 \beta_{2} ) q^{92} + ( -831775426 + 1012111 \beta_{3} ) q^{94} + ( 14786464 \beta_{1} - 1954216 \beta_{2} ) q^{97} + ( 12121963 \beta_{1} - 223440 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 1368q^{4} + O(q^{10})$$ $$4q + 1368q^{4} - 109968q^{11} - 424536q^{14} + 1631264q^{16} + 636880q^{19} - 6618768q^{26} - 3531720q^{29} - 10587712q^{31} - 26434624q^{34} + 16788552q^{41} + 20638944q^{44} - 61250072q^{46} + 46921028q^{49} - 315178080q^{56} - 460829040q^{59} + 360490568q^{61} - 134995072q^{64} + 47611872q^{71} + 1176861744q^{74} - 1168489440q^{76} + 728043520q^{79} + 2375904552q^{86} - 1582700760q^{89} + 473322528q^{91} - 3327101704q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 45 x^{2} + 304$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + 37 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 7 \nu$$ $$\beta_{3}$$ $$=$$ $$60 \nu^{2} - 1350$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1}$$$$)/30$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 1350$$$$)/60$$ $$\nu^{3}$$ $$=$$ $$($$$$37 \beta_{2} + 14 \beta_{1}$$$$)/30$$

## 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
 2.87724 6.05982 −6.05982 −2.87724
−41.3193 0 1195.29 0 0 5315.22 −28233.0 0 0
1.2 −0.843944 0 −511.288 0 0 −8712.99 863.597 0 0
1.3 0.843944 0 −511.288 0 0 8712.99 −863.597 0 0
1.4 41.3193 0 1195.29 0 0 −5315.22 28233.0 0 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 225.10.a.s 4
3.b odd 2 1 25.10.a.e 4
5.b even 2 1 inner 225.10.a.s 4
5.c odd 4 2 45.10.b.b 4
12.b even 2 1 400.10.a.ba 4
15.d odd 2 1 25.10.a.e 4
15.e even 4 2 5.10.b.a 4
60.h even 2 1 400.10.a.ba 4
60.l odd 4 2 80.10.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.10.b.a 4 15.e even 4 2
25.10.a.e 4 3.b odd 2 1
25.10.a.e 4 15.d odd 2 1
45.10.b.b 4 5.c odd 4 2
80.10.c.c 4 60.l odd 4 2
225.10.a.s 4 1.a even 1 1 trivial
225.10.a.s 4 5.b even 2 1 inner
400.10.a.ba 4 12.b even 2 1
400.10.a.ba 4 60.h even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$

## Hecke kernels

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

 $$T_{2}^{4} - 1708 T_{2}^{2} + 1216$$ $$T_{7}^{4} - 104167728 T_{7}^{2} +$$$$21\!\cdots\!96$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 340 T^{2} - 174912 T^{4} + 89128960 T^{6} + 68719476736 T^{8}$$
$3$ 1
$5$ 1
$7$ $$1 + 57246700 T^{2} + 3508143545353398 T^{4} +$$$$93\!\cdots\!00$$$$T^{6} +$$$$26\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 + 54984 T + 5180465446 T^{2} + 129649395841944 T^{3} + 5559917313492231481 T^{4} )^{2}$$
$13$ $$1 + 35613791860 T^{2} +$$$$53\!\cdots\!58$$$$T^{4} +$$$$40\!\cdots\!40$$$$T^{6} +$$$$12\!\cdots\!41$$$$T^{8}$$
$17$ $$1 + 285780369220 T^{2} +$$$$48\!\cdots\!18$$$$T^{4} +$$$$40\!\cdots\!80$$$$T^{6} +$$$$19\!\cdots\!81$$$$T^{8}$$
$19$ $$( 1 - 318440 T + 505756418358 T^{2} - 102756670480744760 T^{3} +$$$$10\!\cdots\!41$$$$T^{4} )^{2}$$
$23$ $$1 + 5779790962540 T^{2} +$$$$14\!\cdots\!38$$$$T^{4} +$$$$18\!\cdots\!60$$$$T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$29$ $$( 1 + 1765860 T + 26950935551038 T^{2} + 25617588792948032340 T^{3} +$$$$21\!\cdots\!61$$$$T^{4} )^{2}$$
$31$ $$( 1 + 5293856 T + 59464921598526 T^{2} +$$$$13\!\cdots\!76$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4} )^{2}$$
$37$ $$1 + 231603274936660 T^{2} +$$$$44\!\cdots\!58$$$$T^{4} +$$$$39\!\cdots\!40$$$$T^{6} +$$$$28\!\cdots\!41$$$$T^{8}$$
$41$ $$( 1 - 8394276 T + 221313076168966 T^{2} -$$$$27\!\cdots\!36$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4} )^{2}$$
$43$ $$1 + 614109141147100 T^{2} +$$$$57\!\cdots\!98$$$$T^{4} +$$$$15\!\cdots\!00$$$$T^{6} +$$$$63\!\cdots\!01$$$$T^{8}$$
$47$ $$1 + 1368976020813580 T^{2} +$$$$29\!\cdots\!78$$$$T^{4} +$$$$17\!\cdots\!20$$$$T^{6} +$$$$15\!\cdots\!21$$$$T^{8}$$
$53$ $$1 + 7684297973864980 T^{2} +$$$$36\!\cdots\!78$$$$T^{4} +$$$$83\!\cdots\!20$$$$T^{6} +$$$$11\!\cdots\!21$$$$T^{8}$$
$59$ $$( 1 + 230414520 T + 28555631923987078 T^{2} +$$$$19\!\cdots\!80$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4} )^{2}$$
$61$ $$( 1 - 180245284 T + 30154717014478446 T^{2} -$$$$21\!\cdots\!44$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4} )^{2}$$
$67$ $$1 - 41160407446058180 T^{2} +$$$$18\!\cdots\!18$$$$T^{4} -$$$$30\!\cdots\!20$$$$T^{6} +$$$$54\!\cdots\!81$$$$T^{8}$$
$71$ $$( 1 - 23805936 T + 85782754020107086 T^{2} -$$$$10\!\cdots\!16$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4} )^{2}$$
$73$ $$1 + 229489314868712740 T^{2} +$$$$20\!\cdots\!38$$$$T^{4} +$$$$79\!\cdots\!60$$$$T^{6} +$$$$12\!\cdots\!61$$$$T^{8}$$
$79$ $$( 1 - 364021760 T + 220545463862625438 T^{2} -$$$$43\!\cdots\!40$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4} )^{2}$$
$83$ $$1 + 434569632367965820 T^{2} +$$$$10\!\cdots\!18$$$$T^{4} +$$$$15\!\cdots\!80$$$$T^{6} +$$$$12\!\cdots\!81$$$$T^{8}$$
$89$ $$( 1 + 791350380 T + 832192702699668118 T^{2} +$$$$27\!\cdots\!20$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4} )^{2}$$
$97$ $$1 + 2561123777205326980 T^{2} +$$$$27\!\cdots\!78$$$$T^{4} +$$$$14\!\cdots\!20$$$$T^{6} +$$$$33\!\cdots\!21$$$$T^{8}$$
show more
show less