# Properties

 Label 225.12.b.d Level $225$ Weight $12$ Character orbit 225.b Analytic conductor $172.877$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$172.877215626$$ 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 1) 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.

 $$f(q)$$ $$=$$ $$q + 24 i q^{2} + 1472 q^{4} -16744 i q^{7} + 84480 i q^{8} +O(q^{10})$$ $$q + 24 i q^{2} + 1472 q^{4} -16744 i q^{7} + 84480 i q^{8} -534612 q^{11} + 577738 i q^{13} + 401856 q^{14} + 987136 q^{16} + 6905934 i q^{17} -10661420 q^{19} -12830688 i q^{22} + 18643272 i q^{23} -13865712 q^{26} -24647168 i q^{28} + 128406630 q^{29} -52843168 q^{31} + 196706304 i q^{32} -165742416 q^{34} -182213314 i q^{37} -255874080 i q^{38} -308120442 q^{41} + 17125708 i q^{43} -786948864 q^{44} -447438528 q^{46} -2687348496 i q^{47} + 1696965207 q^{49} + 850430336 i q^{52} -1596055698 i q^{53} + 1414533120 q^{56} + 3081759120 i q^{58} -5189203740 q^{59} + 6956478662 q^{61} -1268236032 i q^{62} -2699296768 q^{64} -15481826884 i q^{67} + 10165534848 i q^{68} -9791485272 q^{71} -1463791322 i q^{73} + 4373119536 q^{74} -15693610240 q^{76} + 8951543328 i q^{77} -38116845680 q^{79} -7394890608 i q^{82} -29335099668 i q^{83} -411016992 q^{86} -45164021760 i q^{88} -24992917110 q^{89} + 9673645072 q^{91} + 27442896384 i q^{92} + 64496363904 q^{94} + 75013568546 i q^{97} + 40727164968 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2944q^{4} + O(q^{10})$$ $$2q + 2944q^{4} - 1069224q^{11} + 803712q^{14} + 1974272q^{16} - 21322840q^{19} - 27731424q^{26} + 256813260q^{29} - 105686336q^{31} - 331484832q^{34} - 616240884q^{41} - 1573897728q^{44} - 894877056q^{46} + 3393930414q^{49} + 2829066240q^{56} - 10378407480q^{59} + 13912957324q^{61} - 5398593536q^{64} - 19582970544q^{71} + 8746239072q^{74} - 31387220480q^{76} - 76233691360q^{79} - 822033984q^{86} - 49985834220q^{89} + 19347290144q^{91} + 128992727808q^{94} + O(q^{100})$$

## 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
24.0000i 0 1472.00 0 0 16744.0i 84480.0i 0 0
199.2 24.0000i 0 1472.00 0 0 16744.0i 84480.0i 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 225.12.b.d 2
3.b odd 2 1 25.12.b.b 2
5.b even 2 1 inner 225.12.b.d 2
5.c odd 4 1 9.12.a.b 1
5.c odd 4 1 225.12.a.b 1
15.d odd 2 1 25.12.b.b 2
15.e even 4 1 1.12.a.a 1
15.e even 4 1 25.12.a.b 1
20.e even 4 1 144.12.a.d 1
45.k odd 12 2 81.12.c.b 2
45.l even 12 2 81.12.c.d 2
60.l odd 4 1 16.12.a.a 1
105.k odd 4 1 49.12.a.a 1
105.w odd 12 2 49.12.c.c 2
105.x even 12 2 49.12.c.b 2
120.q odd 4 1 64.12.a.f 1
120.w even 4 1 64.12.a.b 1
165.l odd 4 1 121.12.a.b 1
195.s even 4 1 169.12.a.a 1
240.z odd 4 1 256.12.b.c 2
240.bb even 4 1 256.12.b.e 2
240.bd odd 4 1 256.12.b.c 2
240.bf even 4 1 256.12.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a 1 15.e even 4 1
9.12.a.b 1 5.c odd 4 1
16.12.a.a 1 60.l odd 4 1
25.12.a.b 1 15.e even 4 1
25.12.b.b 2 3.b odd 2 1
25.12.b.b 2 15.d odd 2 1
49.12.a.a 1 105.k odd 4 1
49.12.c.b 2 105.x even 12 2
49.12.c.c 2 105.w odd 12 2
64.12.a.b 1 120.w even 4 1
64.12.a.f 1 120.q odd 4 1
81.12.c.b 2 45.k odd 12 2
81.12.c.d 2 45.l even 12 2
121.12.a.b 1 165.l odd 4 1
144.12.a.d 1 20.e even 4 1
169.12.a.a 1 195.s even 4 1
225.12.a.b 1 5.c odd 4 1
225.12.b.d 2 1.a even 1 1 trivial
225.12.b.d 2 5.b even 2 1 inner
256.12.b.c 2 240.z odd 4 1
256.12.b.c 2 240.bd odd 4 1
256.12.b.e 2 240.bb even 4 1
256.12.b.e 2 240.bf even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{12}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}^{2} + 576$$ $$T_{11} + 534612$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$576 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$280361536 + T^{2}$$
$11$ $$( 534612 + T )^{2}$$
$13$ $$333781196644 + T^{2}$$
$17$ $$47691924412356 + T^{2}$$
$19$ $$( 10661420 + T )^{2}$$
$23$ $$347571590865984 + T^{2}$$
$29$ $$( -128406630 + T )^{2}$$
$31$ $$( 52843168 + T )^{2}$$
$37$ $$33201691798862596 + T^{2}$$
$41$ $$( 308120442 + T )^{2}$$
$43$ $$293289874501264 + T^{2}$$
$47$ $$7221841938953462016 + T^{2}$$
$53$ $$2547393791118267204 + T^{2}$$
$59$ $$( 5189203740 + T )^{2}$$
$61$ $$( -6956478662 + T )^{2}$$
$67$ $$23\!\cdots\!56$$$$+ T^{2}$$
$71$ $$( 9791485272 + T )^{2}$$
$73$ $$2142685034362507684 + T^{2}$$
$79$ $$( 38116845680 + T )^{2}$$
$83$ $$86\!\cdots\!24$$$$+ T^{2}$$
$89$ $$( 24992917110 + T )^{2}$$
$97$ $$56\!\cdots\!16$$$$+ T^{2}$$