# Properties

 Label 225.4.b.f Level $225$ Weight $4$ Character orbit 225.b Analytic conductor $13.275$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + 7 q^{4} -6 i q^{7} + 15 i q^{8} +O(q^{10})$$ $$q + i q^{2} + 7 q^{4} -6 i q^{7} + 15 i q^{8} + 43 q^{11} -28 i q^{13} + 6 q^{14} + 41 q^{16} + 91 i q^{17} + 35 q^{19} + 43 i q^{22} -162 i q^{23} + 28 q^{26} -42 i q^{28} + 160 q^{29} + 42 q^{31} + 161 i q^{32} -91 q^{34} + 314 i q^{37} + 35 i q^{38} + 203 q^{41} + 92 i q^{43} + 301 q^{44} + 162 q^{46} + 196 i q^{47} + 307 q^{49} -196 i q^{52} -82 i q^{53} + 90 q^{56} + 160 i q^{58} -280 q^{59} -518 q^{61} + 42 i q^{62} + 167 q^{64} -141 i q^{67} + 637 i q^{68} -412 q^{71} -763 i q^{73} -314 q^{74} + 245 q^{76} -258 i q^{77} -510 q^{79} + 203 i q^{82} -777 i q^{83} -92 q^{86} + 645 i q^{88} -945 q^{89} -168 q^{91} -1134 i q^{92} -196 q^{94} -1246 i q^{97} + 307 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 14 q^{4} + O(q^{10})$$ $$2 q + 14 q^{4} + 86 q^{11} + 12 q^{14} + 82 q^{16} + 70 q^{19} + 56 q^{26} + 320 q^{29} + 84 q^{31} - 182 q^{34} + 406 q^{41} + 602 q^{44} + 324 q^{46} + 614 q^{49} + 180 q^{56} - 560 q^{59} - 1036 q^{61} + 334 q^{64} - 824 q^{71} - 628 q^{74} + 490 q^{76} - 1020 q^{79} - 184 q^{86} - 1890 q^{89} - 336 q^{91} - 392 q^{94} + O(q^{100})$$

## 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
1.00000i 0 7.00000 0 0 6.00000i 15.0000i 0 0
199.2 1.00000i 0 7.00000 0 0 6.00000i 15.0000i 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 225.4.b.f 2
3.b odd 2 1 25.4.b.b 2
5.b even 2 1 inner 225.4.b.f 2
5.c odd 4 1 225.4.a.c 1
5.c odd 4 1 225.4.a.e 1
12.b even 2 1 400.4.c.e 2
15.d odd 2 1 25.4.b.b 2
15.e even 4 1 25.4.a.a 1
15.e even 4 1 25.4.a.b yes 1
60.h even 2 1 400.4.c.e 2
60.l odd 4 1 400.4.a.c 1
60.l odd 4 1 400.4.a.s 1
105.k odd 4 1 1225.4.a.h 1
105.k odd 4 1 1225.4.a.i 1
120.q odd 4 1 1600.4.a.h 1
120.q odd 4 1 1600.4.a.bs 1
120.w even 4 1 1600.4.a.i 1
120.w even 4 1 1600.4.a.bt 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 15.e even 4 1
25.4.a.b yes 1 15.e even 4 1
25.4.b.b 2 3.b odd 2 1
25.4.b.b 2 15.d odd 2 1
225.4.a.c 1 5.c odd 4 1
225.4.a.e 1 5.c odd 4 1
225.4.b.f 2 1.a even 1 1 trivial
225.4.b.f 2 5.b even 2 1 inner
400.4.a.c 1 60.l odd 4 1
400.4.a.s 1 60.l odd 4 1
400.4.c.e 2 12.b even 2 1
400.4.c.e 2 60.h even 2 1
1225.4.a.h 1 105.k odd 4 1
1225.4.a.i 1 105.k odd 4 1
1600.4.a.h 1 120.q odd 4 1
1600.4.a.i 1 120.w even 4 1
1600.4.a.bs 1 120.q odd 4 1
1600.4.a.bt 1 120.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}^{2} + 1$$ $$T_{7}^{2} + 36$$ $$T_{11} - 43$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$36 + T^{2}$$
$11$ $$( -43 + T )^{2}$$
$13$ $$784 + T^{2}$$
$17$ $$8281 + T^{2}$$
$19$ $$( -35 + T )^{2}$$
$23$ $$26244 + T^{2}$$
$29$ $$( -160 + T )^{2}$$
$31$ $$( -42 + T )^{2}$$
$37$ $$98596 + T^{2}$$
$41$ $$( -203 + T )^{2}$$
$43$ $$8464 + T^{2}$$
$47$ $$38416 + T^{2}$$
$53$ $$6724 + T^{2}$$
$59$ $$( 280 + T )^{2}$$
$61$ $$( 518 + T )^{2}$$
$67$ $$19881 + T^{2}$$
$71$ $$( 412 + T )^{2}$$
$73$ $$582169 + T^{2}$$
$79$ $$( 510 + T )^{2}$$
$83$ $$603729 + T^{2}$$
$89$ $$( 945 + T )^{2}$$
$97$ $$1552516 + T^{2}$$