# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - 9 \beta_{2} + 9) q^{7} - 7 \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + b2 * q^4 + (-9*b2 + 9) * q^7 - 7*b3 * q^8 $$q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - 9 \beta_{2} + 9) q^{7} - 7 \beta_{3} q^{8} + ( - 9 \beta_{3} - 9 \beta_1) q^{11} + ( - 63 \beta_{2} - 63) q^{13} + ( - 9 \beta_{3} + 9 \beta_1) q^{14} + 71 q^{16} + 14 \beta_1 q^{17} + 70 \beta_{2} q^{19} + ( - 81 \beta_{2} + 81) q^{22} - 34 \beta_{3} q^{23} + ( - 63 \beta_{3} - 63 \beta_1) q^{26} + (9 \beta_{2} + 9) q^{28} + ( - 54 \beta_{3} + 54 \beta_1) q^{29} + 196 q^{31} + 15 \beta_1 q^{32} + 126 \beta_{2} q^{34} + (207 \beta_{2} - 207) q^{37} + 70 \beta_{3} q^{38} + ( - 63 \beta_{3} - 63 \beta_1) q^{41} + (144 \beta_{2} + 144) q^{43} + ( - 9 \beta_{3} + 9 \beta_1) q^{44} + 306 q^{46} - 168 \beta_1 q^{47} + 181 \beta_{2} q^{49} + ( - 63 \beta_{2} + 63) q^{52} + 106 \beta_{3} q^{53} + ( - 63 \beta_{3} - 63 \beta_1) q^{56} + (486 \beta_{2} + 486) q^{58} + ( - 63 \beta_{3} + 63 \beta_1) q^{59} - 322 q^{61} + 196 \beta_1 q^{62} - 433 \beta_{2} q^{64} + (378 \beta_{2} - 378) q^{67} + 14 \beta_{3} q^{68} + (198 \beta_{3} + 198 \beta_1) q^{71} + ( - 378 \beta_{2} - 378) q^{73} + (207 \beta_{3} - 207 \beta_1) q^{74} - 70 q^{76} - 162 \beta_1 q^{77} + 488 \beta_{2} q^{79} + ( - 567 \beta_{2} + 567) q^{82} - 364 \beta_{3} q^{83} + (144 \beta_{3} + 144 \beta_1) q^{86} + ( - 567 \beta_{2} - 567) q^{88} + ( - 63 \beta_{3} + 63 \beta_1) q^{89} - 1134 q^{91} + 34 \beta_1 q^{92} - 1512 \beta_{2} q^{94} + ( - 252 \beta_{2} + 252) q^{97} + 181 \beta_{3} q^{98}+O(q^{100})$$ q + b1 * q^2 + b2 * q^4 + (-9*b2 + 9) * q^7 - 7*b3 * q^8 + (-9*b3 - 9*b1) * q^11 + (-63*b2 - 63) * q^13 + (-9*b3 + 9*b1) * q^14 + 71 * q^16 + 14*b1 * q^17 + 70*b2 * q^19 + (-81*b2 + 81) * q^22 - 34*b3 * q^23 + (-63*b3 - 63*b1) * q^26 + (9*b2 + 9) * q^28 + (-54*b3 + 54*b1) * q^29 + 196 * q^31 + 15*b1 * q^32 + 126*b2 * q^34 + (207*b2 - 207) * q^37 + 70*b3 * q^38 + (-63*b3 - 63*b1) * q^41 + (144*b2 + 144) * q^43 + (-9*b3 + 9*b1) * q^44 + 306 * q^46 - 168*b1 * q^47 + 181*b2 * q^49 + (-63*b2 + 63) * q^52 + 106*b3 * q^53 + (-63*b3 - 63*b1) * q^56 + (486*b2 + 486) * q^58 + (-63*b3 + 63*b1) * q^59 - 322 * q^61 + 196*b1 * q^62 - 433*b2 * q^64 + (378*b2 - 378) * q^67 + 14*b3 * q^68 + (198*b3 + 198*b1) * q^71 + (-378*b2 - 378) * q^73 + (207*b3 - 207*b1) * q^74 - 70 * q^76 - 162*b1 * q^77 + 488*b2 * q^79 + (-567*b2 + 567) * q^82 - 364*b3 * q^83 + (144*b3 + 144*b1) * q^86 + (-567*b2 - 567) * q^88 + (-63*b3 + 63*b1) * q^89 - 1134 * q^91 + 34*b1 * q^92 - 1512*b2 * q^94 + (-252*b2 + 252) * q^97 + 181*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 36 q^{7}+O(q^{10})$$ 4 * q + 36 * q^7 $$4 q + 36 q^{7} - 252 q^{13} + 284 q^{16} + 324 q^{22} + 36 q^{28} + 784 q^{31} - 828 q^{37} + 576 q^{43} + 1224 q^{46} + 252 q^{52} + 1944 q^{58} - 1288 q^{61} - 1512 q^{67} - 1512 q^{73} - 280 q^{76} + 2268 q^{82} - 2268 q^{88} - 4536 q^{91} + 1008 q^{97}+O(q^{100})$$ 4 * q + 36 * q^7 - 252 * q^13 + 284 * q^16 + 324 * q^22 + 36 * q^28 + 784 * q^31 - 828 * q^37 + 576 * q^43 + 1224 * q^46 + 252 * q^52 + 1944 * q^58 - 1288 * q^61 - 1512 * q^67 - 1512 * q^73 - 280 * q^76 + 2268 * q^82 - 2268 * q^88 - 4536 * q^91 + 1008 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$3\zeta_{8}$$ 3*v $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$3\zeta_{8}^{3}$$ 3*v^3
 $$\zeta_{8}$$ $$=$$ $$( \beta_1 ) / 3$$ (b1) / 3 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{8}^{3}$$ $$=$$ $$( \beta_{3} ) / 3$$ (b3) / 3

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
−2.12132 + 2.12132i 0 1.00000i 0 0 9.00000 + 9.00000i −14.8492 14.8492i 0 0
107.2 2.12132 2.12132i 0 1.00000i 0 0 9.00000 + 9.00000i 14.8492 + 14.8492i 0 0
143.1 −2.12132 2.12132i 0 1.00000i 0 0 9.00000 9.00000i −14.8492 + 14.8492i 0 0
143.2 2.12132 + 2.12132i 0 1.00000i 0 0 9.00000 9.00000i 14.8492 14.8492i 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
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.f.b yes 4
3.b odd 2 1 inner 225.4.f.b yes 4
5.b even 2 1 225.4.f.a 4
5.c odd 4 1 225.4.f.a 4
5.c odd 4 1 inner 225.4.f.b yes 4
15.d odd 2 1 225.4.f.a 4
15.e even 4 1 225.4.f.a 4
15.e even 4 1 inner 225.4.f.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.4.f.a 4 5.b even 2 1
225.4.f.a 4 5.c odd 4 1
225.4.f.a 4 15.d odd 2 1
225.4.f.a 4 15.e even 4 1
225.4.f.b yes 4 1.a even 1 1 trivial
225.4.f.b yes 4 3.b odd 2 1 inner
225.4.f.b yes 4 5.c odd 4 1 inner
225.4.f.b yes 4 15.e even 4 1 inner

## Hecke kernels

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

 $$T_{2}^{4} + 81$$ T2^4 + 81 $$T_{7}^{2} - 18T_{7} + 162$$ T7^2 - 18*T7 + 162

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 81$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 18 T + 162)^{2}$$
$11$ $$(T^{2} + 1458)^{2}$$
$13$ $$(T^{2} + 126 T + 7938)^{2}$$
$17$ $$T^{4} + 3111696$$
$19$ $$(T^{2} + 4900)^{2}$$
$23$ $$T^{4} + 108243216$$
$29$ $$(T^{2} - 52488)^{2}$$
$31$ $$(T - 196)^{4}$$
$37$ $$(T^{2} + 414 T + 85698)^{2}$$
$41$ $$(T^{2} + 71442)^{2}$$
$43$ $$(T^{2} - 288 T + 41472)^{2}$$
$47$ $$T^{4} + 64524128256$$
$53$ $$T^{4} + 10226063376$$
$59$ $$(T^{2} - 71442)^{2}$$
$61$ $$(T + 322)^{4}$$
$67$ $$(T^{2} + 756 T + 285768)^{2}$$
$71$ $$(T^{2} + 705672)^{2}$$
$73$ $$(T^{2} + 756 T + 285768)^{2}$$
$79$ $$(T^{2} + 238144)^{2}$$
$83$ $$T^{4} + 1421970391296$$
$89$ $$(T^{2} - 71442)^{2}$$
$97$ $$(T^{2} - 504 T + 127008)^{2}$$