# Properties

 Label 225.4.e.b Level $225$ Weight $4$ Character orbit 225.e 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.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 9) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{3} ) q^{2} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( -4 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{4} + ( 12 - 15 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{6} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{7} + ( -16 + \beta_{2} ) q^{8} + ( -3 + 21 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{3} ) q^{2} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( -4 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{4} + ( 12 - 15 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{6} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{7} + ( -16 + \beta_{2} ) q^{8} + ( -3 + 21 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{9} + ( -37 \beta_{1} + 8 \beta_{3} ) q^{11} + ( 52 - 22 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{12} + ( -13 - 2 \beta_{1} - 15 \beta_{2} + 15 \beta_{3} ) q^{13} + ( -34 + 26 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{14} + ( -\beta_{1} + 9 \beta_{3} ) q^{16} + ( -54 - 9 \beta_{2} ) q^{17} + ( -72 - 27 \beta_{2} + 18 \beta_{3} ) q^{18} + ( -52 - 27 \beta_{2} ) q^{19} + ( 34 - 46 \beta_{1} + 8 \beta_{2} - 7 \beta_{3} ) q^{21} + ( -6 + 27 \beta_{1} + 21 \beta_{2} - 21 \beta_{3} ) q^{22} + ( 7 - 26 \beta_{1} - 19 \beta_{2} + 19 \beta_{3} ) q^{23} + ( -24 + 9 \beta_{1} + 18 \beta_{2} + 15 \beta_{3} ) q^{24} + ( -146 - 28 \beta_{2} ) q^{26} + ( 117 + 18 \beta_{1} + 18 \beta_{2} - 36 \beta_{3} ) q^{27} + ( -92 - 18 \beta_{2} ) q^{28} + ( 26 \beta_{1} - \beta_{3} ) q^{29} + ( -23 + 20 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{31} + ( -216 + 207 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} ) q^{32} + ( 6 - 165 \beta_{1} - 21 \beta_{2} + 66 \beta_{3} ) q^{33} + ( -117 \beta_{1} - 63 \beta_{3} ) q^{34} + ( -12 - 141 \beta_{1} - 60 \beta_{2} - 15 \beta_{3} ) q^{36} + ( -2 - 54 \beta_{2} ) q^{37} + ( -241 \beta_{1} - 79 \beta_{3} ) q^{38} + ( 253 - 124 \beta_{1} + 11 \beta_{2} + 17 \beta_{3} ) q^{39} + ( -115 + 17 \beta_{1} - 98 \beta_{2} + 98 \beta_{3} ) q^{41} + ( 162 - 12 \beta_{1} + 60 \beta_{2} - 18 \beta_{3} ) q^{42} + ( 47 \beta_{1} - 6 \beta_{3} ) q^{43} + ( -76 + 79 \beta_{2} ) q^{44} + ( -138 - 12 \beta_{2} ) q^{46} + ( 154 \beta_{1} + 91 \beta_{3} ) q^{47} + ( 88 - 145 \beta_{1} + 17 \beta_{2} - 7 \beta_{3} ) q^{48} + ( 246 - 267 \beta_{1} - 21 \beta_{2} + 21 \beta_{3} ) q^{49} + ( 18 + 117 \beta_{1} + 36 \beta_{2} + 63 \beta_{3} ) q^{51} + ( -358 \beta_{1} - 54 \beta_{3} ) q^{52} + ( 54 + 162 \beta_{2} ) q^{53} + ( 324 - 27 \beta_{1} + 54 \beta_{2} + 81 \beta_{3} ) q^{54} + ( -10 \beta_{1} - 46 \beta_{3} ) q^{56} + ( 164 + 241 \beta_{1} - 2 \beta_{2} + 79 \beta_{3} ) q^{57} + ( -42 + 18 \beta_{1} - 24 \beta_{2} + 24 \beta_{3} ) q^{58} + ( -331 + 467 \beta_{1} + 136 \beta_{2} - 136 \beta_{3} ) q^{59} + ( -272 \beta_{1} + 105 \beta_{3} ) q^{61} + ( -70 - 26 \beta_{2} ) q^{62} + ( -192 - 24 \beta_{1} - 78 \beta_{2} + 57 \beta_{3} ) q^{63} + ( -440 - 153 \beta_{2} ) q^{64} + ( -330 + 222 \beta_{1} + 33 \beta_{2} - 48 \beta_{3} ) q^{66} + ( 527 - 461 \beta_{1} + 66 \beta_{2} - 66 \beta_{3} ) q^{67} + ( 432 - 261 \beta_{1} + 171 \beta_{2} - 171 \beta_{3} ) q^{68} + ( 297 - 204 \beta_{1} - 33 \beta_{2} + 45 \beta_{3} ) q^{69} + ( 756 + 144 \beta_{2} ) q^{71} + ( -333 \beta_{1} + 99 \beta_{2} - 27 \beta_{3} ) q^{72} + ( 106 - 243 \beta_{2} ) q^{73} + ( -380 \beta_{1} - 56 \beta_{3} ) q^{74} + ( 856 - 673 \beta_{1} + 183 \beta_{2} - 183 \beta_{3} ) q^{76} + ( -47 + 118 \beta_{1} + 71 \beta_{2} - 71 \beta_{3} ) q^{77} + ( 78 + 342 \beta_{1} + 90 \beta_{2} + 174 \beta_{3} ) q^{78} + ( 556 \beta_{1} - 309 \beta_{3} ) q^{79} + ( -351 + 351 \beta_{1} - 135 \beta_{2} - 135 \beta_{3} ) q^{81} + ( -1014 - 213 \beta_{2} ) q^{82} + ( 460 \beta_{1} - 107 \beta_{3} ) q^{83} + ( 52 + 218 \beta_{1} + 56 \beta_{2} + 110 \beta_{3} ) q^{84} + ( -34 - \beta_{1} - 35 \beta_{2} + 35 \beta_{3} ) q^{86} + ( 42 + 42 \beta_{1} + 24 \beta_{2} - 51 \beta_{3} ) q^{87} + ( 693 \beta_{1} - 165 \beta_{3} ) q^{88} + ( -162 + 72 \beta_{2} ) q^{89} + ( -425 - 69 \beta_{2} ) q^{91} + ( -430 \beta_{1} + 2 \beta_{3} ) q^{92} + ( 71 + 16 \beta_{1} + 43 \beta_{2} - 17 \beta_{3} ) q^{93} + ( -1218 + 882 \beta_{1} - 336 \beta_{2} + 336 \beta_{3} ) q^{94} + ( 360 + 342 \beta_{1} + 423 \beta_{2} - 198 \beta_{3} ) q^{96} + ( 317 \beta_{1} + 102 \beta_{3} ) q^{97} + ( 324 + 225 \beta_{2} ) q^{98} + ( 504 - 1080 \beta_{1} - 81 \beta_{2} + 279 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 3q^{2} + 3q^{3} - 5q^{4} + 9q^{6} + 7q^{7} - 66q^{8} + 45q^{9} + O(q^{10})$$ $$4q + 3q^{2} + 3q^{3} - 5q^{4} + 9q^{6} + 7q^{7} - 66q^{8} + 45q^{9} - 66q^{11} + 156q^{12} - 11q^{13} - 60q^{14} + 7q^{16} - 198q^{17} - 216q^{18} - 154q^{19} + 21q^{21} - 33q^{22} + 33q^{23} - 99q^{24} - 528q^{26} + 432q^{27} - 332q^{28} + 51q^{29} - 43q^{31} - 423q^{32} - 198q^{33} - 297q^{34} - 225q^{36} + 100q^{37} - 561q^{38} + 759q^{39} - 132q^{41} + 486q^{42} + 88q^{43} - 462q^{44} - 528q^{46} + 399q^{47} + 21q^{48} + 513q^{49} + 297q^{51} - 770q^{52} - 108q^{53} + 1215q^{54} - 66q^{56} + 1221q^{57} - 60q^{58} - 798q^{59} - 439q^{61} - 228q^{62} - 603q^{63} - 1454q^{64} - 990q^{66} + 988q^{67} + 693q^{68} + 891q^{69} + 2736q^{71} - 891q^{72} + 910q^{73} - 816q^{74} + 1529q^{76} - 165q^{77} + 990q^{78} + 803q^{79} - 567q^{81} - 3630q^{82} + 813q^{83} + 642q^{84} - 33q^{86} + 153q^{87} + 1221q^{88} - 792q^{89} - 1562q^{91} - 858q^{92} + 213q^{93} - 2100q^{94} + 1080q^{96} + 736q^{97} + 846q^{98} + 297q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 5 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{3} + \nu^{2} + 2 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 2 \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11$$$$)/3$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 −1.18614 − 1.26217i 1.68614 + 0.396143i −1.18614 + 1.26217i 1.68614 − 0.396143i
−0.686141 1.18843i 5.05842 + 1.18843i 3.05842 5.29734i 0 −2.05842 6.82701i −2.55842 4.43132i −19.3723 24.1753 + 12.0232i 0
76.2 2.18614 + 3.78651i −3.55842 3.78651i −5.55842 + 9.62747i 0 6.55842 21.7518i 6.05842 + 10.4935i −13.6277 −1.67527 + 26.9480i 0
151.1 −0.686141 + 1.18843i 5.05842 1.18843i 3.05842 + 5.29734i 0 −2.05842 + 6.82701i −2.55842 + 4.43132i −19.3723 24.1753 12.0232i 0
151.2 2.18614 3.78651i −3.55842 + 3.78651i −5.55842 9.62747i 0 6.55842 + 21.7518i 6.05842 10.4935i −13.6277 −1.67527 26.9480i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.e.b 4
5.b even 2 1 9.4.c.a 4
5.c odd 4 2 225.4.k.b 8
9.c even 3 1 inner 225.4.e.b 4
9.c even 3 1 2025.4.a.g 2
9.d odd 6 1 2025.4.a.n 2
15.d odd 2 1 27.4.c.a 4
20.d odd 2 1 144.4.i.c 4
45.h odd 6 1 27.4.c.a 4
45.h odd 6 1 81.4.a.a 2
45.j even 6 1 9.4.c.a 4
45.j even 6 1 81.4.a.d 2
45.k odd 12 2 225.4.k.b 8
60.h even 2 1 432.4.i.c 4
180.n even 6 1 432.4.i.c 4
180.n even 6 1 1296.4.a.i 2
180.p odd 6 1 144.4.i.c 4
180.p odd 6 1 1296.4.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.c.a 4 5.b even 2 1
9.4.c.a 4 45.j even 6 1
27.4.c.a 4 15.d odd 2 1
27.4.c.a 4 45.h odd 6 1
81.4.a.a 2 45.h odd 6 1
81.4.a.d 2 45.j even 6 1
144.4.i.c 4 20.d odd 2 1
144.4.i.c 4 180.p odd 6 1
225.4.e.b 4 1.a even 1 1 trivial
225.4.e.b 4 9.c even 3 1 inner
225.4.k.b 8 5.c odd 4 2
225.4.k.b 8 45.k odd 12 2
432.4.i.c 4 60.h even 2 1
432.4.i.c 4 180.n even 6 1
1296.4.a.i 2 180.n even 6 1
1296.4.a.u 2 180.p odd 6 1
2025.4.a.g 2 9.c even 3 1
2025.4.a.n 2 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 3 T_{2}^{3} + 15 T_{2}^{2} + 18 T_{2} + 36$$ acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4}$$
$3$ $$729 - 81 T - 18 T^{2} - 3 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$3844 + 434 T + 111 T^{2} - 7 T^{3} + T^{4}$$
$11$ $$314721 + 37026 T + 3795 T^{2} + 66 T^{3} + T^{4}$$
$13$ $$3334276 - 20086 T + 1947 T^{2} + 11 T^{3} + T^{4}$$
$17$ $$( 1782 + 99 T + T^{2} )^{2}$$
$19$ $$( -4532 + 77 T + T^{2} )^{2}$$
$23$ $$7322436 + 89298 T + 3795 T^{2} - 33 T^{3} + T^{4}$$
$29$ $$412164 - 32742 T + 1959 T^{2} - 51 T^{3} + T^{4}$$
$31$ $$150544 + 16684 T + 1461 T^{2} + 43 T^{3} + T^{4}$$
$37$ $$( -23432 - 50 T + T^{2} )^{2}$$
$41$ $$5606565129 - 9883764 T + 92301 T^{2} + 132 T^{3} + T^{4}$$
$43$ $$2686321 - 144232 T + 6105 T^{2} - 88 T^{3} + T^{4}$$
$47$ $$813276324 + 11378682 T + 187719 T^{2} - 399 T^{3} + T^{4}$$
$53$ $$( -215784 + 54 T + T^{2} )^{2}$$
$59$ $$43678881 + 5273982 T + 630195 T^{2} + 798 T^{3} + T^{4}$$
$61$ $$1829786176 - 18778664 T + 235497 T^{2} + 439 T^{3} + T^{4}$$
$67$ $$43305193801 - 205601812 T + 768045 T^{2} - 988 T^{3} + T^{4}$$
$71$ $$( 296784 - 1368 T + T^{2} )^{2}$$
$73$ $$( -435398 - 455 T + T^{2} )^{2}$$
$79$ $$392522298256 + 503092348 T + 1271325 T^{2} - 803 T^{3} + T^{4}$$
$83$ $$5010940944 - 57550644 T + 590181 T^{2} - 813 T^{3} + T^{4}$$
$89$ $$( -3564 + 396 T + T^{2} )^{2}$$
$97$ $$2459267281 - 36498976 T + 492105 T^{2} - 736 T^{3} + T^{4}$$