# Properties

 Label 225.10.b.g Level 225 Weight 10 Character orbit 225.b Analytic conductor 115.883 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$115.883063137$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{4729})$$ Defining polynomial: $$x^{4} + 2365 x^{2} + 1397124$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - 5 \beta_{2} ) q^{2} + ( -770 + 19 \beta_{3} ) q^{4} + ( -56 \beta_{1} - 2982 \beta_{2} ) q^{7} + ( 429 \beta_{1} + 12519 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( -\beta_{1} - 5 \beta_{2} ) q^{2} + ( -770 + 19 \beta_{3} ) q^{4} + ( -56 \beta_{1} - 2982 \beta_{2} ) q^{7} + ( 429 \beta_{1} + 12519 \beta_{2} ) q^{8} + ( -16768 - 1952 \beta_{3} ) q^{11} + ( 1384 \beta_{1} - 35573 \beta_{2} ) q^{13} + ( -125832 + 6468 \beta_{3} ) q^{14} + ( 363218 - 19171 \beta_{3} ) q^{16} + ( -2200 \beta_{1} - 96839 \beta_{2} ) q^{17} + ( 201164 + 968 \beta_{3} ) q^{19} + ( -800 \beta_{1} - 1069792 \beta_{2} ) q^{22} + ( -64968 \beta_{1} + 39684 \beta_{2} ) q^{23} + ( 924428 + 58690 \beta_{3} ) q^{26} + ( 155372 \beta_{1} + 2924964 \beta_{2} ) q^{28} + ( -31078 - 12416 \beta_{3} ) q^{29} + ( -2475696 - 75736 \beta_{3} ) q^{31} + ( -316109 \beta_{1} - 6736423 \beta_{2} ) q^{32} + ( -4537180 + 213478 \beta_{3} ) q^{34} + ( -174696 \beta_{1} + 1299733 \beta_{2} ) q^{37} + ( -192452 \beta_{1} - 433732 \beta_{2} ) q^{38} + ( -7340714 + 470096 \beta_{3} ) q^{41} + ( -152384 \beta_{1} - 6975326 \beta_{2} ) q^{43} + ( -30926656 + 1147360 \beta_{3} ) q^{44} + ( -75998496 + 505344 \beta_{3} ) q^{46} + ( -431368 \beta_{1} - 24099452 \beta_{2} ) q^{47} + ( 1077559 + 664832 \beta_{3} ) q^{49} + ( 312390 \beta_{1} + 11850274 \beta_{2} ) q^{52} + ( 929872 \beta_{1} - 15843931 \beta_{2} ) q^{53} + ( 177723000 - 3936660 \beta_{3} ) q^{56} + ( -80666 \beta_{1} - 7182466 \beta_{2} ) q^{58} + ( 93124864 + 1613408 \beta_{3} ) q^{59} + ( 75911686 + 2256688 \beta_{3} ) q^{61} + ( 1794072 \beta_{1} - 32381496 \beta_{2} ) q^{62} + ( -322401682 + 6502275 \beta_{3} ) q^{64} + ( -7444160 \beta_{1} + 6537054 \beta_{2} ) q^{67} + ( 5332082 \beta_{1} + 99269830 \beta_{2} ) q^{68} + ( 110604928 + 7061120 \beta_{3} ) q^{71} + ( 6480208 \beta_{1} + 9900631 \beta_{2} ) q^{73} + ( -180496012 - 1027202 \beta_{3} ) q^{74} + ( -133156936 + 3095148 \beta_{3} ) q^{76} + ( -10593408 \beta_{1} - 14601216 \beta_{2} ) q^{77} + ( 467102400 - 1798040 \beta_{3} ) q^{79} + ( 11571578 \beta_{1} + 314530306 \beta_{2} ) q^{82} + ( 3161088 \beta_{1} + 52550310 \beta_{2} ) q^{83} + ( -319624408 + 15322108 \beta_{3} ) q^{86} + ( 40843296 \beta_{1} + 284989536 \beta_{2} ) q^{88} + ( 107605986 + 9306192 \beta_{3} ) q^{89} + ( -332705016 - 4192496 \beta_{3} ) q^{91} + ( 47282976 \beta_{1} + 698968992 \beta_{2} ) q^{92} + ( -991866016 + 52081216 \beta_{3} ) q^{94} + ( 44039040 \beta_{1} + 107793409 \beta_{2} ) q^{97} + ( 4905929 \beta_{1} + 387527917 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 3042q^{4} + O(q^{10})$$ $$4q - 3042q^{4} - 70976q^{11} - 490392q^{14} + 1414530q^{16} + 806592q^{19} + 3815092q^{26} - 149144q^{29} - 10054256q^{31} - 17721764q^{34} - 28422664q^{41} - 121411904q^{44} - 302983296q^{46} + 5639900q^{49} + 703018680q^{56} + 375726272q^{59} + 308160120q^{61} - 1276602178q^{64} + 456541952q^{71} - 724038452q^{74} - 526437448q^{76} + 1864813520q^{79} - 1247853416q^{86} + 449036328q^{89} - 1339205056q^{91} - 3863301632q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2365 x^{2} + 1397124$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} - 2365 \nu$$$$)/1182$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 1183 \nu$$$$)/591$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 1183$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} - 2 \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 1183$$ $$\nu^{3}$$ $$=$$ $$($$$$2365 \beta_{2} + 2366 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 34.8839i − 33.8839i 33.8839i 34.8839i
43.8839i 0 −1413.79 0 0 7861.50i 39574.2i 0 0
199.2 24.8839i 0 −107.207 0 0 4010.50i 10072.8i 0 0
199.3 24.8839i 0 −107.207 0 0 4010.50i 10072.8i 0 0
199.4 43.8839i 0 −1413.79 0 0 7861.50i 39574.2i 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.b.g 4
3.b odd 2 1 75.10.b.e 4
5.b even 2 1 inner 225.10.b.g 4
5.c odd 4 1 45.10.a.e 2
5.c odd 4 1 225.10.a.j 2
15.d odd 2 1 75.10.b.e 4
15.e even 4 1 15.10.a.c 2
15.e even 4 1 75.10.a.g 2
60.l odd 4 1 240.10.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.c 2 15.e even 4 1
45.10.a.e 2 5.c odd 4 1
75.10.a.g 2 15.e even 4 1
75.10.b.e 4 3.b odd 2 1
75.10.b.e 4 15.d odd 2 1
225.10.a.j 2 5.c odd 4 1
225.10.b.g 4 1.a even 1 1 trivial
225.10.b.g 4 5.b even 2 1 inner
240.10.a.m 2 60.l odd 4 1

## Hecke kernels

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

 $$T_{2}^{4} + 2545 T_{2}^{2} + 1192464$$ $$T_{11}^{2} + 35488 T_{11} - 4189882368$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 497 T^{2} + 159248 T^{4} + 130285568 T^{6} + 68719476736 T^{8}$$
$3$ 1
$5$ 1
$7$ $$1 - 83527164 T^{2} + 4478467599613798 T^{4} -$$$$13\!\cdots\!36$$$$T^{6} +$$$$26\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 + 35488 T + 526013014 T^{2} + 83678847658208 T^{3} + 5559917313492231481 T^{4} )^{2}$$
$13$ $$1 - 27567505292 T^{2} +$$$$36\!\cdots\!18$$$$T^{4} -$$$$31\!\cdots\!68$$$$T^{6} +$$$$12\!\cdots\!41$$$$T^{8}$$
$17$ $$1 - 388734753820 T^{2} +$$$$65\!\cdots\!18$$$$T^{4} -$$$$54\!\cdots\!80$$$$T^{6} +$$$$19\!\cdots\!81$$$$T^{8}$$
$19$ $$( 1 - 403296 T + 684929514838 T^{2} - 130138657763479584 T^{3} +$$$$10\!\cdots\!41$$$$T^{4} )^{2}$$
$23$ $$1 + 2800589695204 T^{2} +$$$$81\!\cdots\!58$$$$T^{4} +$$$$90\!\cdots\!76$$$$T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$29$ $$( 1 + 74572 T + 28833430018078 T^{2} + 1081826889712503068 T^{3} +$$$$21\!\cdots\!61$$$$T^{4} )^{2}$$
$31$ $$( 1 + 5027128 T + 52415931233342 T^{2} +$$$$13\!\cdots\!88$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4} )^{2}$$
$37$ $$1 - 433247572021484 T^{2} +$$$$79\!\cdots\!78$$$$T^{4} -$$$$73\!\cdots\!36$$$$T^{6} +$$$$28\!\cdots\!41$$$$T^{8}$$
$41$ $$( 1 + 14211332 T + 443988635955862 T^{2} +$$$$46\!\cdots\!52$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4} )^{2}$$
$43$ $$1 - 1570463388232460 T^{2} +$$$$11\!\cdots\!98$$$$T^{4} -$$$$39\!\cdots\!40$$$$T^{6} +$$$$63\!\cdots\!01$$$$T^{8}$$
$47$ $$1 + 568239172667780 T^{2} +$$$$55\!\cdots\!78$$$$T^{4} +$$$$71\!\cdots\!20$$$$T^{6} +$$$$15\!\cdots\!21$$$$T^{8}$$
$53$ $$1 - 9086956380170956 T^{2} +$$$$38\!\cdots\!18$$$$T^{4} -$$$$98\!\cdots\!84$$$$T^{6} +$$$$11\!\cdots\!21$$$$T^{8}$$
$59$ $$( 1 - 187863136 T + 23071633420288438 T^{2} -$$$$16\!\cdots\!04$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4} )^{2}$$
$61$ $$( 1 - 154080060 T + 23302683905802238 T^{2} -$$$$18\!\cdots\!60$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4} )^{2}$$
$67$ $$1 + 22768078838174100 T^{2} +$$$$15\!\cdots\!18$$$$T^{4} +$$$$16\!\cdots\!00$$$$T^{6} +$$$$54\!\cdots\!81$$$$T^{8}$$
$71$ $$( 1 - 228270976 T + 45777616900481806 T^{2} -$$$$10\!\cdots\!56$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4} )^{2}$$
$73$ $$1 - 135645128086811996 T^{2} +$$$$11\!\cdots\!58$$$$T^{4} -$$$$47\!\cdots\!24$$$$T^{6} +$$$$12\!\cdots\!61$$$$T^{8}$$
$79$ $$( 1 - 932406760 T + 453226630902929438 T^{2} -$$$$11\!\cdots\!40$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4} )^{2}$$
$83$ $$1 - 702700996120787372 T^{2} +$$$$19\!\cdots\!38$$$$T^{4} -$$$$24\!\cdots\!48$$$$T^{6} +$$$$12\!\cdots\!81$$$$T^{8}$$
$89$ $$( 1 - 224518164 T + 610925899926766678 T^{2} -$$$$78\!\cdots\!76$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4} )^{2}$$
$97$ $$1 + 1619811253917303940 T^{2} +$$$$14\!\cdots\!78$$$$T^{4} +$$$$93\!\cdots\!60$$$$T^{6} +$$$$33\!\cdots\!21$$$$T^{8}$$