# Properties

 Label 225.4.b.h Level $225$ Weight $4$ Character orbit 225.b Analytic conductor $13.275$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{19})$$ Defining polynomial: $$x^{4} - 9 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 75) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{3} ) q^{2} + ( -12 + 2 \beta_{2} ) q^{4} + ( -13 \beta_{1} - 4 \beta_{3} ) q^{7} + ( 42 \beta_{1} - 6 \beta_{3} ) q^{8} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{3} ) q^{2} + ( -12 + 2 \beta_{2} ) q^{4} + ( -13 \beta_{1} - 4 \beta_{3} ) q^{7} + ( 42 \beta_{1} - 6 \beta_{3} ) q^{8} + ( -14 - 4 \beta_{2} ) q^{11} + ( 9 \beta_{1} - 16 \beta_{3} ) q^{13} + ( 63 + 9 \beta_{2} ) q^{14} + ( 60 - 32 \beta_{2} ) q^{16} + ( -34 \beta_{1} - 20 \beta_{3} ) q^{17} + ( -3 + 4 \beta_{2} ) q^{19} + ( -62 \beta_{1} - 10 \beta_{3} ) q^{22} + ( -66 \beta_{1} + 12 \beta_{3} ) q^{23} + ( 313 - 25 \beta_{2} ) q^{26} + ( 4 \beta_{1} + 22 \beta_{3} ) q^{28} + ( 46 - 28 \beta_{2} ) q^{29} + ( 61 + 28 \beta_{2} ) q^{31} + ( -332 \beta_{1} + 44 \beta_{3} ) q^{32} + ( 346 + 14 \beta_{2} ) q^{34} + ( 142 \beta_{1} - 24 \beta_{3} ) q^{37} + ( 79 \beta_{1} - 7 \beta_{3} ) q^{38} + ( -196 + 52 \beta_{2} ) q^{41} + ( 345 \beta_{1} - 4 \beta_{3} ) q^{43} + ( 16 + 20 \beta_{2} ) q^{44} + ( -294 + 78 \beta_{2} ) q^{46} + ( -310 \beta_{1} - 32 \beta_{3} ) q^{47} + ( -130 - 104 \beta_{2} ) q^{49} + ( -716 \beta_{1} + 210 \beta_{3} ) q^{52} + ( 424 \beta_{1} - 28 \beta_{3} ) q^{53} + ( 90 + 90 \beta_{2} ) q^{56} + ( -578 \beta_{1} + 74 \beta_{3} ) q^{58} + ( 62 + 64 \beta_{2} ) q^{59} + ( 375 + 56 \beta_{2} ) q^{61} + ( 471 \beta_{1} + 33 \beta_{3} ) q^{62} + ( -688 + 120 \beta_{2} ) q^{64} + ( 179 \beta_{1} - 100 \beta_{3} ) q^{67} + ( -352 \beta_{1} + 172 \beta_{3} ) q^{68} + ( -412 - 20 \beta_{2} ) q^{71} + ( -54 \beta_{1} + 8 \beta_{3} ) q^{73} + ( 598 - 166 \beta_{2} ) q^{74} + ( 188 - 54 \beta_{2} ) q^{76} + ( 486 \beta_{1} + 108 \beta_{3} ) q^{77} + ( 440 - 160 \beta_{2} ) q^{79} + ( 1184 \beta_{1} - 248 \beta_{3} ) q^{82} + ( -78 \beta_{1} - 192 \beta_{3} ) q^{83} + ( 421 - 349 \beta_{2} ) q^{86} + ( -132 \beta_{1} - 84 \beta_{3} ) q^{88} + ( -432 - 144 \beta_{2} ) q^{89} + ( -1099 - 172 \beta_{2} ) q^{91} + ( 1248 \beta_{1} - 276 \beta_{3} ) q^{92} + ( 298 + 278 \beta_{2} ) q^{94} -521 \beta_{1} q^{97} + ( -1846 \beta_{1} - 26 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 48 q^{4} + O(q^{10})$$ $$4 q - 48 q^{4} - 56 q^{11} + 252 q^{14} + 240 q^{16} - 12 q^{19} + 1252 q^{26} + 184 q^{29} + 244 q^{31} + 1384 q^{34} - 784 q^{41} + 64 q^{44} - 1176 q^{46} - 520 q^{49} + 360 q^{56} + 248 q^{59} + 1500 q^{61} - 2752 q^{64} - 1648 q^{71} + 2392 q^{74} + 752 q^{76} + 1760 q^{79} + 1684 q^{86} - 1728 q^{89} - 4396 q^{91} + 1192 q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 4 \nu$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 14 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} - 9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 9$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2} + 7 \beta_{1}$$

## 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
 −2.17945 + 0.500000i 2.17945 − 0.500000i 2.17945 + 0.500000i −2.17945 − 0.500000i
5.35890i 0 −20.7178 0 0 4.43560i 68.1534i 0 0
199.2 3.35890i 0 −3.28220 0 0 30.4356i 15.8466i 0 0
199.3 3.35890i 0 −3.28220 0 0 30.4356i 15.8466i 0 0
199.4 5.35890i 0 −20.7178 0 0 4.43560i 68.1534i 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.h 4
3.b odd 2 1 75.4.b.c 4
5.b even 2 1 inner 225.4.b.h 4
5.c odd 4 1 225.4.a.j 2
5.c odd 4 1 225.4.a.n 2
12.b even 2 1 1200.4.f.v 4
15.d odd 2 1 75.4.b.c 4
15.e even 4 1 75.4.a.d 2
15.e even 4 1 75.4.a.e yes 2
60.h even 2 1 1200.4.f.v 4
60.l odd 4 1 1200.4.a.bl 2
60.l odd 4 1 1200.4.a.bu 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.a.d 2 15.e even 4 1
75.4.a.e yes 2 15.e even 4 1
75.4.b.c 4 3.b odd 2 1
75.4.b.c 4 15.d odd 2 1
225.4.a.j 2 5.c odd 4 1
225.4.a.n 2 5.c odd 4 1
225.4.b.h 4 1.a even 1 1 trivial
225.4.b.h 4 5.b even 2 1 inner
1200.4.a.bl 2 60.l odd 4 1
1200.4.a.bu 2 60.l odd 4 1
1200.4.f.v 4 12.b even 2 1
1200.4.f.v 4 60.h even 2 1

## Hecke kernels

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

 $$T_{2}^{4} + 40 T_{2}^{2} + 324$$ $$T_{7}^{4} + 946 T_{7}^{2} + 18225$$ $$T_{11}^{2} + 28 T_{11} - 108$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$324 + 40 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$18225 + 946 T^{2} + T^{4}$$
$11$ $$( -108 + 28 T + T^{2} )^{2}$$
$13$ $$22877089 + 9890 T^{2} + T^{4}$$
$17$ $$41525136 + 17512 T^{2} + T^{4}$$
$19$ $$( -295 + 6 T + T^{2} )^{2}$$
$23$ $$2624400 + 14184 T^{2} + T^{4}$$
$29$ $$( -12780 - 92 T + T^{2} )^{2}$$
$31$ $$( -11175 - 122 T + T^{2} )^{2}$$
$37$ $$85008400 + 62216 T^{2} + T^{4}$$
$41$ $$( -12960 + 392 T + T^{2} )^{2}$$
$43$ $$14094675841 + 238658 T^{2} + T^{4}$$
$47$ $$5874302736 + 231112 T^{2} + T^{4}$$
$53$ $$27185414400 + 389344 T^{2} + T^{4}$$
$59$ $$( -73980 - 124 T + T^{2} )^{2}$$
$61$ $$( 81041 - 750 T + T^{2} )^{2}$$
$67$ $$24951045681 + 444082 T^{2} + T^{4}$$
$71$ $$( 162144 + 824 T + T^{2} )^{2}$$
$73$ $$2890000 + 8264 T^{2} + T^{4}$$
$79$ $$( -292800 - 880 T + T^{2} )^{2}$$
$83$ $$482096926224 + 1413000 T^{2} + T^{4}$$
$89$ $$( -207360 + 864 T + T^{2} )^{2}$$
$97$ $$( 271441 + T^{2} )^{2}$$