# Properties

 Label 225.10.b.e Level 225 Weight 10 Character orbit 225.b Analytic conductor 115.883 Analytic rank 0 Dimension 2 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} + 496 q^{4} -7680 i q^{7} + 4032 i q^{8} +O(q^{10})$$ $$q + 4 i q^{2} + 496 q^{4} -7680 i q^{7} + 4032 i q^{8} + 86404 q^{11} + 149978 i q^{13} + 30720 q^{14} + 237824 q^{16} + 207622 i q^{17} -716284 q^{19} + 345616 i q^{22} + 1369920 i q^{23} -599912 q^{26} -3809280 i q^{28} -3194402 q^{29} -2349000 q^{31} + 3015680 i q^{32} -830488 q^{34} + 18735710 i q^{37} -2865136 i q^{38} + 29282630 q^{41} + 1516724 i q^{43} + 42856384 q^{44} -5479680 q^{46} -615752 i q^{47} -18628793 q^{49} + 74389088 i q^{52} + 4747430 i q^{53} + 30965760 q^{56} -12777608 i q^{58} + 60616076 q^{59} -126745682 q^{61} -9396000 i q^{62} + 109703168 q^{64} -111182652 i q^{67} + 102980512 i q^{68} + 175551608 q^{71} + 61233350 i q^{73} -74942840 q^{74} -355276864 q^{76} -663582720 i q^{77} -234431160 q^{79} + 117130520 i q^{82} + 118910388 i q^{83} -6066896 q^{86} + 348380928 i q^{88} -316534326 q^{89} + 1151831040 q^{91} + 679480320 i q^{92} + 2463008 q^{94} + 242912258 i q^{97} -74515172 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 992q^{4} + O(q^{10})$$ $$2q + 992q^{4} + 172808q^{11} + 61440q^{14} + 475648q^{16} - 1432568q^{19} - 1199824q^{26} - 6388804q^{29} - 4698000q^{31} - 1660976q^{34} + 58565260q^{41} + 85712768q^{44} - 10959360q^{46} - 37257586q^{49} + 61931520q^{56} + 121232152q^{59} - 253491364q^{61} + 219406336q^{64} + 351103216q^{71} - 149885680q^{74} - 710553728q^{76} - 468862320q^{79} - 12133792q^{86} - 633068652q^{89} + 2303662080q^{91} + 4926016q^{94} + O(q^{100})$$

## 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
 − 1.00000i 1.00000i
4.00000i 0 496.000 0 0 7680.00i 4032.00i 0 0
199.2 4.00000i 0 496.000 0 0 7680.00i 4032.00i 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.e 2
3.b odd 2 1 75.10.b.d 2
5.b even 2 1 inner 225.10.b.e 2
5.c odd 4 1 45.10.a.b 1
5.c odd 4 1 225.10.a.c 1
15.d odd 2 1 75.10.b.d 2
15.e even 4 1 15.10.a.a 1
15.e even 4 1 75.10.a.c 1
60.l odd 4 1 240.10.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.a 1 15.e even 4 1
45.10.a.b 1 5.c odd 4 1
75.10.a.c 1 15.e even 4 1
75.10.b.d 2 3.b odd 2 1
75.10.b.d 2 15.d odd 2 1
225.10.a.c 1 5.c odd 4 1
225.10.b.e 2 1.a even 1 1 trivial
225.10.b.e 2 5.b even 2 1 inner
240.10.a.c 1 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 $$S_{10}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}^{2} + 16$$ $$T_{11} - 86404$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 1008 T^{2} + 262144 T^{4}$$
$3$ 1
$5$ 1
$7$ $$1 - 21724814 T^{2} + 1628413597910449 T^{4}$$
$11$ $$( 1 - 86404 T + 2357947691 T^{2} )^{2}$$
$13$ $$1 + 1284401738 T^{2} +$$$$11\!\cdots\!29$$$$T^{4}$$
$17$ $$1 - 194068858110 T^{2} +$$$$14\!\cdots\!09$$$$T^{4}$$
$19$ $$( 1 + 716284 T + 322687697779 T^{2} )^{2}$$
$23$ $$1 - 1725624516526 T^{2} +$$$$32\!\cdots\!69$$$$T^{4}$$
$29$ $$( 1 + 3194402 T + 14507145975869 T^{2} )^{2}$$
$31$ $$( 1 + 2349000 T + 26439622160671 T^{2} )^{2}$$
$37$ $$1 + 91103349613946 T^{2} +$$$$16\!\cdots\!29$$$$T^{4}$$
$41$ $$( 1 - 29282630 T + 327381934393961 T^{2} )^{2}$$
$43$ $$1 - 1002884772181510 T^{2} +$$$$25\!\cdots\!49$$$$T^{4}$$
$47$ $$1 - 2237881795680030 T^{2} +$$$$12\!\cdots\!89$$$$T^{4}$$
$53$ $$1 - 6576989091999366 T^{2} +$$$$10\!\cdots\!89$$$$T^{4}$$
$59$ $$( 1 - 60616076 T + 8662995818654939 T^{2} )^{2}$$
$61$ $$( 1 + 126745682 T + 11694146092834141 T^{2} )^{2}$$
$67$ $$1 - 42051486686836790 T^{2} +$$$$74\!\cdots\!09$$$$T^{4}$$
$71$ $$( 1 - 175551608 T + 45848500718449031 T^{2} )^{2}$$
$73$ $$1 - 113993650264313326 T^{2} +$$$$34\!\cdots\!69$$$$T^{4}$$
$79$ $$( 1 + 234431160 T + 119851595982618319 T^{2} )^{2}$$
$83$ $$1 - 359740830160770262 T^{2} +$$$$34\!\cdots\!09$$$$T^{4}$$
$89$ $$( 1 + 316534326 T + 350356403707485209 T^{2} )^{2}$$
$97$ $$1 - 1461455752222471870 T^{2} +$$$$57\!\cdots\!89$$$$T^{4}$$