# Properties

 Label 225.4.k.b Level $225$ Weight $4$ Character orbit 225.k Analytic conductor $13.275$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.k (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.303595776.1 Defining polynomial: $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 9) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{7} ) q^{2} + ( \beta_{4} - \beta_{5} - \beta_{7} ) q^{3} + ( -3 + 3 \beta_{2} - 4 \beta_{3} - 3 \beta_{6} ) q^{4} + ( -9 + 3 \beta_{2} - 18 \beta_{3} - 3 \beta_{6} ) q^{6} + ( 2 \beta_{1} - 3 \beta_{7} ) q^{7} + ( 16 \beta_{1} - \beta_{4} + 16 \beta_{5} ) q^{8} + ( -27 + 3 \beta_{2} - 24 \beta_{3} - 6 \beta_{6} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{7} ) q^{2} + ( \beta_{4} - \beta_{5} - \beta_{7} ) q^{3} + ( -3 + 3 \beta_{2} - 4 \beta_{3} - 3 \beta_{6} ) q^{4} + ( -9 + 3 \beta_{2} - 18 \beta_{3} - 3 \beta_{6} ) q^{6} + ( 2 \beta_{1} - 3 \beta_{7} ) q^{7} + ( 16 \beta_{1} - \beta_{4} + 16 \beta_{5} ) q^{8} + ( -27 + 3 \beta_{2} - 24 \beta_{3} - 6 \beta_{6} ) q^{9} + ( -29 - 8 \beta_{2} - 29 \beta_{3} ) q^{11} + ( 30 \beta_{1} + 5 \beta_{4} + 52 \beta_{5} - 2 \beta_{7} ) q^{12} + ( 15 \beta_{1} + 15 \beta_{4} + 13 \beta_{5} + 15 \beta_{7} ) q^{13} + ( -8 + 8 \beta_{2} - 34 \beta_{3} - 8 \beta_{6} ) q^{14} + ( 8 - 9 \beta_{2} + 8 \beta_{3} ) q^{16} + ( -54 \beta_{1} - 9 \beta_{4} - 54 \beta_{5} ) q^{17} + ( 72 \beta_{1} + 27 \beta_{4} + 72 \beta_{5} + 18 \beta_{7} ) q^{18} + ( 25 - 27 \beta_{6} ) q^{19} + ( -27 + 7 \beta_{2} - 53 \beta_{3} - 8 \beta_{6} ) q^{21} + ( 21 \beta_{1} + 21 \beta_{4} - 6 \beta_{5} + 21 \beta_{7} ) q^{22} + ( 19 \beta_{1} + 19 \beta_{4} - 7 \beta_{5} + 19 \beta_{7} ) q^{23} + ( 18 + 15 \beta_{2} - 24 \beta_{3} + 18 \beta_{6} ) q^{24} + ( -118 + 28 \beta_{6} ) q^{26} + ( 135 \beta_{1} + 18 \beta_{4} + 117 \beta_{5} + 36 \beta_{7} ) q^{27} + ( 92 \beta_{1} + 18 \beta_{4} + 92 \beta_{5} ) q^{28} + ( -25 - \beta_{2} - 25 \beta_{3} ) q^{29} + ( 3 - 3 \beta_{2} + 23 \beta_{3} + 3 \beta_{6} ) q^{31} + ( -9 \beta_{1} - 9 \beta_{4} - 216 \beta_{5} - 9 \beta_{7} ) q^{32} + ( 159 \beta_{1} + 21 \beta_{4} - 6 \beta_{5} + 66 \beta_{7} ) q^{33} + ( 180 - 63 \beta_{2} + 180 \beta_{3} ) q^{34} + ( -108 + 15 \beta_{2} - 156 \beta_{3} + 60 \beta_{6} ) q^{36} + ( -2 \beta_{1} - 54 \beta_{4} - 2 \beta_{5} ) q^{37} + ( 241 \beta_{1} - 79 \beta_{7} ) q^{38} + ( -135 + 17 \beta_{2} + 107 \beta_{3} + 11 \beta_{6} ) q^{39} + ( 98 - 98 \beta_{2} + 115 \beta_{3} + 98 \beta_{6} ) q^{41} + ( 150 \beta_{1} + 60 \beta_{4} + 162 \beta_{5} + 18 \beta_{7} ) q^{42} + ( -47 \beta_{1} - 6 \beta_{7} ) q^{43} + ( 155 + 79 \beta_{6} ) q^{44} + ( -126 + 12 \beta_{6} ) q^{46} + ( 154 \beta_{1} - 91 \beta_{7} ) q^{47} + ( 57 \beta_{1} - 17 \beta_{4} - 88 \beta_{5} - 7 \beta_{7} ) q^{48} + ( -21 + 21 \beta_{2} + 246 \beta_{3} - 21 \beta_{6} ) q^{49} + ( 162 - 63 \beta_{2} + 180 \beta_{3} - 36 \beta_{6} ) q^{51} + ( -358 \beta_{1} + 54 \beta_{7} ) q^{52} + ( -54 \beta_{1} - 162 \beta_{4} - 54 \beta_{5} ) q^{53} + ( -324 + 81 \beta_{2} - 54 \beta_{3} + 54 \beta_{6} ) q^{54} + ( -56 + 46 \beta_{2} - 56 \beta_{3} ) q^{56} + ( 405 \beta_{1} - 2 \beta_{4} + 164 \beta_{5} - 79 \beta_{7} ) q^{57} + ( 24 \beta_{1} + 24 \beta_{4} + 42 \beta_{5} + 24 \beta_{7} ) q^{58} + ( 136 - 136 \beta_{2} - 331 \beta_{3} + 136 \beta_{6} ) q^{59} + ( -167 - 105 \beta_{2} - 167 \beta_{3} ) q^{61} + ( -70 \beta_{1} - 26 \beta_{4} - 70 \beta_{5} ) q^{62} + ( 216 \beta_{1} + 78 \beta_{4} + 192 \beta_{5} + 57 \beta_{7} ) q^{63} + ( 287 - 153 \beta_{6} ) q^{64} + ( -189 + 48 \beta_{2} + 174 \beta_{3} - 33 \beta_{6} ) q^{66} + ( 66 \beta_{1} + 66 \beta_{4} + 527 \beta_{5} + 66 \beta_{7} ) q^{67} + ( -171 \beta_{1} - 171 \beta_{4} - 432 \beta_{5} - 171 \beta_{7} ) q^{68} + ( -171 + 45 \beta_{2} + 159 \beta_{3} - 33 \beta_{6} ) q^{69} + ( 612 - 144 \beta_{6} ) q^{71} + ( -333 \beta_{1} + 99 \beta_{4} + 27 \beta_{7} ) q^{72} + ( -106 \beta_{1} + 243 \beta_{4} - 106 \beta_{5} ) q^{73} + ( 436 - 56 \beta_{2} + 436 \beta_{3} ) q^{74} + ( -183 + 183 \beta_{2} - 856 \beta_{3} - 183 \beta_{6} ) q^{76} + ( 71 \beta_{1} + 71 \beta_{4} - 47 \beta_{5} + 71 \beta_{7} ) q^{77} + ( -420 \beta_{1} - 90 \beta_{4} - 78 \beta_{5} + 174 \beta_{7} ) q^{78} + ( -247 - 309 \beta_{2} - 247 \beta_{3} ) q^{79} + ( 135 \beta_{2} + 216 \beta_{3} + 135 \beta_{6} ) q^{81} + ( -1014 \beta_{1} - 213 \beta_{4} - 1014 \beta_{5} ) q^{82} + ( -460 \beta_{1} - 107 \beta_{7} ) q^{83} + ( -324 + 110 \beta_{2} - 328 \beta_{3} + 56 \beta_{6} ) q^{84} + ( 35 - 35 \beta_{2} + 34 \beta_{3} + 35 \beta_{6} ) q^{86} + ( 84 \beta_{1} + 24 \beta_{4} + 42 \beta_{5} + 51 \beta_{7} ) q^{87} + ( -693 \beta_{1} - 165 \beta_{7} ) q^{88} + ( 234 + 72 \beta_{6} ) q^{89} + ( -356 + 69 \beta_{6} ) q^{91} + ( -430 \beta_{1} - 2 \beta_{7} ) q^{92} + ( -87 \beta_{1} - 43 \beta_{4} - 71 \beta_{5} - 17 \beta_{7} ) q^{93} + ( -336 + 336 \beta_{2} - 1218 \beta_{3} - 336 \beta_{6} ) q^{94} + ( 81 + 198 \beta_{2} + 144 \beta_{3} - 423 \beta_{6} ) q^{96} + ( 317 \beta_{1} - 102 \beta_{7} ) q^{97} + ( -324 \beta_{1} - 225 \beta_{4} - 324 \beta_{5} ) q^{98} + ( 216 + 279 \beta_{2} + 801 \beta_{3} - 81 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 10q^{4} + 18q^{6} - 90q^{9} + O(q^{10})$$ $$8q + 10q^{4} + 18q^{6} - 90q^{9} - 132q^{11} + 120q^{14} + 14q^{16} + 308q^{19} + 42q^{21} + 198q^{24} - 1056q^{26} - 102q^{29} - 86q^{31} + 594q^{34} - 450q^{36} - 1518q^{39} - 264q^{41} + 924q^{44} - 1056q^{46} - 1026q^{49} + 594q^{51} - 2430q^{54} - 132q^{56} + 1596q^{59} - 878q^{61} + 2908q^{64} - 1980q^{66} - 1782q^{69} + 5472q^{71} + 1632q^{74} + 3058q^{76} - 1606q^{79} - 1134q^{81} - 1284q^{84} - 66q^{86} + 1584q^{89} - 3124q^{91} + 4200q^{94} + 2160q^{96} - 594q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 32 \nu^{5} + 16 \nu^{3} + 45 \nu$$$$)/432$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - 16 \nu^{4} - 32 \nu^{2} - 51$$$$)/48$$ $$\beta_{3}$$ $$=$$ $$($$$$-5 \nu^{6} - 16 \nu^{4} - 80 \nu^{2} - 225$$$$)/144$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 8 \nu^{5} + 40 \nu^{3} + 165 \nu$$$$)/72$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 13 \nu$$$$)/48$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} - 5 \nu^{4} - 7 \nu^{2} - 27$$$$)/9$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{5} + 10 \nu^{3} - 3 \nu$$$$)/18$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} + 2 \beta_{5} + \beta_{4}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{6} - 7 \beta_{3} - \beta_{2} - 6$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{7} - 11 \beta_{5} + 2 \beta_{4} - 9 \beta_{1}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-5 \beta_{6} + 13 \beta_{3} - 5 \beta_{2}$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{7} - \beta_{5} - 2 \beta_{4} + 45 \beta_{1}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-16 \beta_{6} - 16 \beta_{3} + 32 \beta_{2} - 39$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$13 \beta_{7} + 118 \beta_{5} - 13 \beta_{4}$$$$)/3$$

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$\beta_{3}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.396143 + 1.68614i −1.26217 + 1.18614i 1.26217 − 1.18614i 0.396143 − 1.68614i −0.396143 − 1.68614i −1.26217 − 1.18614i 1.26217 + 1.18614i 0.396143 + 1.68614i
−3.78651 + 2.18614i −3.78651 + 3.55842i 5.55842 9.62747i 0 6.55842 21.7518i −10.4935 + 6.05842i 13.6277i 1.67527 26.9480i 0
49.2 −1.18843 + 0.686141i −1.18843 + 5.05842i −3.05842 + 5.29734i 0 −2.05842 6.82701i −4.43132 + 2.55842i 19.3723i −24.1753 12.0232i 0
49.3 1.18843 0.686141i 1.18843 5.05842i −3.05842 + 5.29734i 0 −2.05842 6.82701i 4.43132 2.55842i 19.3723i −24.1753 12.0232i 0
49.4 3.78651 2.18614i 3.78651 3.55842i 5.55842 9.62747i 0 6.55842 21.7518i 10.4935 6.05842i 13.6277i 1.67527 26.9480i 0
124.1 −3.78651 2.18614i −3.78651 3.55842i 5.55842 + 9.62747i 0 6.55842 + 21.7518i −10.4935 6.05842i 13.6277i 1.67527 + 26.9480i 0
124.2 −1.18843 0.686141i −1.18843 5.05842i −3.05842 5.29734i 0 −2.05842 + 6.82701i −4.43132 2.55842i 19.3723i −24.1753 + 12.0232i 0
124.3 1.18843 + 0.686141i 1.18843 + 5.05842i −3.05842 5.29734i 0 −2.05842 + 6.82701i 4.43132 + 2.55842i 19.3723i −24.1753 + 12.0232i 0
124.4 3.78651 + 2.18614i 3.78651 + 3.55842i 5.55842 + 9.62747i 0 6.55842 + 21.7518i 10.4935 + 6.05842i 13.6277i 1.67527 + 26.9480i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 124.4 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
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 21 T_{2}^{6} + 405 T_{2}^{4} - 756 T_{2}^{2} + 1296$$ acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1296 - 756 T^{2} + 405 T^{4} - 21 T^{6} + T^{8}$$
$3$ $$531441 + 32805 T^{2} + 1296 T^{4} + 45 T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$14776336 - 665012 T^{2} + 26085 T^{4} - 173 T^{6} + T^{8}$$
$11$ $$( 314721 + 37026 T + 3795 T^{2} + 66 T^{3} + T^{4} )^{2}$$
$13$ $$11117396444176 - 12580223348 T^{2} + 10901253 T^{4} - 3773 T^{6} + T^{8}$$
$17$ $$( 3175524 + 6237 T^{2} + T^{4} )^{2}$$
$19$ $$( -4532 - 77 T + T^{2} )^{4}$$
$23$ $$53618068974096 - 47603156436 T^{2} + 34940565 T^{4} - 6501 T^{6} + T^{8}$$
$29$ $$( 412164 + 32742 T + 1959 T^{2} + 51 T^{3} + T^{4} )^{2}$$
$31$ $$( 150544 + 16684 T + 1461 T^{2} + 43 T^{3} + T^{4} )^{2}$$
$37$ $$( 549058624 + 49364 T^{2} + T^{4} )^{2}$$
$41$ $$( 5606565129 - 9883764 T + 92301 T^{2} + 132 T^{3} + T^{4} )^{2}$$
$43$ $$7216320515041 - 11997109586 T^{2} + 17258835 T^{4} - 4466 T^{6} + T^{8}$$
$47$ $$661418379178952976 - 175860432472788 T^{2} + 45945163845 T^{4} - 216237 T^{6} + T^{8}$$
$53$ $$( 46562734656 + 434484 T^{2} + T^{4} )^{2}$$
$59$ $$( 43678881 - 5273982 T + 630195 T^{2} - 798 T^{3} + T^{4} )^{2}$$
$61$ $$( 1829786176 - 18778664 T + 235497 T^{2} + 439 T^{3} + T^{4} )^{2}$$
$67$ $$18\!\cdots\!01$$$$- 24248570048094746 T^{2} + 270234329115 T^{4} - 559946 T^{6} + T^{8}$$
$71$ $$( 296784 - 1368 T + T^{2} )^{4}$$
$73$ $$( 189571418404 + 1077821 T^{2} + T^{4} )^{2}$$
$79$ $$( 392522298256 - 503092348 T + 1271325 T^{2} + 803 T^{3} + T^{4} )^{2}$$
$83$ $$25109529144255611136 - 2602647649726992 T^{2} + 264758147505 T^{4} - 519393 T^{6} + T^{8}$$
$89$ $$( -3564 - 396 T + T^{2} )^{4}$$
$97$ $$6047995559397132961 - 1088260201584434 T^{2} + 193359372915 T^{4} - 442514 T^{6} + T^{8}$$