# Properties

 Label 225.4.k.a 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: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 45) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{7} q^{2} + ( 3 \beta_{1} + \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{3} + ( -1 + \beta_{2} - \beta_{3} - \beta_{6} ) q^{4} + ( -9 + \beta_{2} + 7 \beta_{3} + \beta_{6} ) q^{6} + ( -5 \beta_{1} - \beta_{7} ) q^{7} + ( \beta_{1} - 7 \beta_{4} + \beta_{5} ) q^{8} + ( 27 - 6 \beta_{2} + 3 \beta_{3} + 3 \beta_{6} ) q^{9} +O(q^{10})$$ $$q -\beta_{7} q^{2} + ( 3 \beta_{1} + \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{3} + ( -1 + \beta_{2} - \beta_{3} - \beta_{6} ) q^{4} + ( -9 + \beta_{2} + 7 \beta_{3} + \beta_{6} ) q^{6} + ( -5 \beta_{1} - \beta_{7} ) q^{7} + ( \beta_{1} - 7 \beta_{4} + \beta_{5} ) q^{8} + ( 27 - 6 \beta_{2} + 3 \beta_{3} + 3 \beta_{6} ) q^{9} + ( 21 - 5 \beta_{2} + 21 \beta_{3} ) q^{11} + ( -15 \beta_{1} + \beta_{4} - 7 \beta_{5} + 2 \beta_{7} ) q^{12} + ( 8 \beta_{1} + 8 \beta_{4} - 52 \beta_{5} + 8 \beta_{7} ) q^{13} + ( 4 - 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{6} ) q^{14} + ( 63 - 7 \beta_{2} + 63 \beta_{3} ) q^{16} + ( 51 \beta_{1} + 25 \beta_{4} + 51 \beta_{5} ) q^{17} + ( -27 \beta_{1} - 3 \beta_{4} - 51 \beta_{5} - 24 \beta_{7} ) q^{18} + ( -30 - 25 \beta_{6} ) q^{19} + ( -9 + 6 \beta_{2} + 12 \beta_{3} - 9 \beta_{6} ) q^{21} + ( -21 \beta_{1} - 21 \beta_{4} - 61 \beta_{5} - 21 \beta_{7} ) q^{22} + ( 23 \beta_{1} + 23 \beta_{4} - 122 \beta_{5} + 23 \beta_{7} ) q^{23} + ( -63 - 9 \beta_{2} - 111 \beta_{3} - 6 \beta_{6} ) q^{24} + ( -64 - 52 \beta_{6} ) q^{26} + ( 135 \beta_{1} + 18 \beta_{4} + 117 \beta_{5} + 36 \beta_{7} ) q^{27} + ( 4 \beta_{1} - 4 \beta_{4} + 4 \beta_{5} ) q^{28} + ( 182 - 39 \beta_{2} + 182 \beta_{3} ) q^{29} -6 \beta_{3} q^{31} + ( -7 \beta_{1} - 7 \beta_{4} - 127 \beta_{5} - 7 \beta_{7} ) q^{32} + ( 93 \beta_{1} + 37 \beta_{4} + 101 \beta_{5} + 11 \beta_{7} ) q^{33} + ( -251 + 51 \beta_{2} - 251 \beta_{3} ) q^{34} + ( 27 + 21 \beta_{2} + 3 \beta_{3} - 24 \beta_{6} ) q^{36} + ( -324 \beta_{1} - 10 \beta_{4} - 324 \beta_{5} ) q^{37} + ( 200 \beta_{1} + 5 \beta_{7} ) q^{38} + ( 144 + 44 \beta_{2} - 40 \beta_{3} + 68 \beta_{6} ) q^{39} + ( 60 - 60 \beta_{2} + 149 \beta_{3} + 60 \beta_{6} ) q^{41} + ( 60 \beta_{1} - 12 \beta_{4} + 36 \beta_{5} ) q^{42} + ( -92 \beta_{1} - 87 \beta_{7} ) q^{43} + ( 40 - 21 \beta_{6} ) q^{44} + ( -184 - 122 \beta_{6} ) q^{46} + ( 445 \beta_{1} - 11 \beta_{7} ) q^{47} + ( 231 \beta_{1} + 119 \beta_{4} + 175 \beta_{5} + 49 \beta_{7} ) q^{48} + ( 9 - 9 \beta_{2} + 319 \beta_{3} + 9 \beta_{6} ) q^{49} + ( 225 - 77 \beta_{2} + 451 \beta_{3} + 76 \beta_{6} ) q^{51} + ( -64 \beta_{1} - 52 \beta_{7} ) q^{52} + ( -62 \beta_{1} + 100 \beta_{4} - 62 \beta_{5} ) q^{53} + ( -243 + 99 \beta_{2} + 45 \beta_{3} + 18 \beta_{6} ) q^{54} + ( 60 + 36 \beta_{2} + 60 \beta_{3} ) q^{56} + ( -240 \beta_{1} + 20 \beta_{4} + 190 \beta_{5} - 35 \beta_{7} ) q^{57} + ( -182 \beta_{1} - 182 \beta_{4} - 494 \beta_{5} - 182 \beta_{7} ) q^{58} + ( 49 - 49 \beta_{2} + 67 \beta_{3} + 49 \beta_{6} ) q^{59} + ( 26 + 195 \beta_{2} + 26 \beta_{3} ) q^{61} + ( 6 \beta_{1} + 6 \beta_{4} + 6 \beta_{5} ) q^{62} + ( -117 \beta_{1} + 27 \beta_{4} - 36 \beta_{5} - 9 \beta_{7} ) q^{63} + ( -392 - 71 \beta_{6} ) q^{64} + ( -378 + 82 \beta_{2} - 290 \beta_{3} + 19 \beta_{6} ) q^{66} + ( 184 \beta_{1} + 184 \beta_{4} + 395 \beta_{5} + 184 \beta_{7} ) q^{67} + ( 51 \beta_{1} + 51 \beta_{4} + 251 \beta_{5} + 51 \beta_{7} ) q^{68} + ( 414 + 99 \beta_{2} - 60 \beta_{3} + 168 \beta_{6} ) q^{69} + ( 252 + 110 \beta_{6} ) q^{71} + ( -171 \beta_{1} - 171 \beta_{4} + 189 \beta_{5} - 27 \beta_{7} ) q^{72} + ( 303 \beta_{1} - 205 \beta_{4} + 303 \beta_{5} ) q^{73} + ( 404 - 324 \beta_{2} + 404 \beta_{3} ) q^{74} + ( 5 - 5 \beta_{2} - 195 \beta_{3} + 5 \beta_{6} ) q^{76} + ( 4 \beta_{1} + 4 \beta_{4} + 44 \beta_{5} + 4 \beta_{7} ) q^{77} + ( -504 \beta_{1} + 40 \beta_{4} + 392 \beta_{5} - 76 \beta_{7} ) q^{78} + ( -458 + 76 \beta_{2} - 458 \beta_{3} ) q^{79} + ( 486 - 270 \beta_{2} + 135 \beta_{3} + 135 \beta_{6} ) q^{81} + ( -629 \beta_{1} - 149 \beta_{4} - 629 \beta_{5} ) q^{82} + ( 33 \beta_{1} + 453 \beta_{7} ) q^{83} + ( -36 - 12 \beta_{2} - 60 \beta_{3} ) q^{84} + ( 5 - 5 \beta_{2} - 691 \beta_{3} + 5 \beta_{6} ) q^{86} + ( 780 \beta_{1} + 325 \beta_{4} + 806 \beta_{5} + 104 \beta_{7} ) q^{87} + ( -152 \beta_{1} + 107 \beta_{7} ) q^{88} + ( 375 - 315 \beta_{6} ) q^{89} + ( -364 - 92 \beta_{6} ) q^{91} + ( -184 \beta_{1} - 122 \beta_{7} ) q^{92} + ( -6 \beta_{4} + 6 \beta_{5} + 6 \beta_{7} ) q^{93} + ( -456 + 456 \beta_{2} - 544 \beta_{3} - 456 \beta_{6} ) q^{94} + ( -126 + 134 \beta_{2} - 310 \beta_{3} + 113 \beta_{6} ) q^{96} + ( 450 \beta_{1} - 131 \beta_{7} ) q^{97} + ( -391 \beta_{1} - 319 \beta_{4} - 391 \beta_{5} ) q^{98} + ( 432 - 168 \beta_{2} + 624 \beta_{3} + 111 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{4} - 102 q^{6} + 180 q^{9} + O(q^{10})$$ $$8 q + 2 q^{4} - 102 q^{6} + 180 q^{9} + 74 q^{11} + 24 q^{14} + 238 q^{16} - 140 q^{19} - 72 q^{21} - 54 q^{24} - 304 q^{26} + 650 q^{29} + 24 q^{31} - 902 q^{34} + 342 q^{36} + 1128 q^{39} - 476 q^{41} + 404 q^{44} - 984 q^{46} - 1258 q^{49} - 462 q^{51} - 1998 q^{54} + 312 q^{56} - 170 q^{59} + 494 q^{61} - 2852 q^{64} - 1776 q^{66} + 3078 q^{69} + 1576 q^{71} + 968 q^{74} + 790 q^{76} - 1680 q^{79} + 2268 q^{81} - 72 q^{84} + 2774 q^{86} + 4260 q^{89} - 2544 q^{91} + 1264 q^{94} + 48 q^{96} + 180 q^{99} + O(q^{100})$$

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
−2.92048 + 1.68614i 4.97494 + 1.50000i 1.68614 2.92048i 0 −17.0584 + 4.00772i 1.40965 0.813859i 15.6060i 22.5000 + 14.9248i 0
49.2 −2.05446 + 1.18614i 4.97494 1.50000i −1.18614 + 2.05446i 0 −8.44158 + 8.98266i −6.38458 + 3.68614i 24.6060i 22.5000 14.9248i 0
49.3 2.05446 1.18614i −4.97494 + 1.50000i −1.18614 + 2.05446i 0 −8.44158 + 8.98266i 6.38458 3.68614i 24.6060i 22.5000 14.9248i 0
49.4 2.92048 1.68614i −4.97494 1.50000i 1.68614 2.92048i 0 −17.0584 + 4.00772i −1.40965 + 0.813859i 15.6060i 22.5000 + 14.9248i 0
124.1 −2.92048 1.68614i 4.97494 1.50000i 1.68614 + 2.92048i 0 −17.0584 4.00772i 1.40965 + 0.813859i 15.6060i 22.5000 14.9248i 0
124.2 −2.05446 1.18614i 4.97494 + 1.50000i −1.18614 2.05446i 0 −8.44158 8.98266i −6.38458 3.68614i 24.6060i 22.5000 + 14.9248i 0
124.3 2.05446 + 1.18614i −4.97494 1.50000i −1.18614 2.05446i 0 −8.44158 8.98266i 6.38458 + 3.68614i 24.6060i 22.5000 + 14.9248i 0
124.4 2.92048 + 1.68614i −4.97494 + 1.50000i 1.68614 + 2.92048i 0 −17.0584 4.00772i −1.40965 0.813859i 15.6060i 22.5000 14.9248i 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.a 8
5.b even 2 1 inner 225.4.k.a 8
5.c odd 4 1 45.4.e.a 4
5.c odd 4 1 225.4.e.a 4
9.c even 3 1 inner 225.4.k.a 8
15.e even 4 1 135.4.e.a 4
45.j even 6 1 inner 225.4.k.a 8
45.k odd 12 1 45.4.e.a 4
45.k odd 12 1 225.4.e.a 4
45.k odd 12 1 405.4.a.d 2
45.k odd 12 1 2025.4.a.l 2
45.l even 12 1 135.4.e.a 4
45.l even 12 1 405.4.a.e 2
45.l even 12 1 2025.4.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.a 4 5.c odd 4 1
45.4.e.a 4 45.k odd 12 1
135.4.e.a 4 15.e even 4 1
135.4.e.a 4 45.l even 12 1
225.4.e.a 4 5.c odd 4 1
225.4.e.a 4 45.k odd 12 1
225.4.k.a 8 1.a even 1 1 trivial
225.4.k.a 8 5.b even 2 1 inner
225.4.k.a 8 9.c even 3 1 inner
225.4.k.a 8 45.j even 6 1 inner
405.4.a.d 2 45.k odd 12 1
405.4.a.e 2 45.l even 12 1
2025.4.a.j 2 45.l even 12 1
2025.4.a.l 2 45.k odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 17 T_{2}^{6} + 225 T_{2}^{4} - 1088 T_{2}^{2} + 4096$$ acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4096 - 1088 T^{2} + 225 T^{4} - 17 T^{6} + T^{8}$$
$3$ $$( 729 - 45 T^{2} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$20736 - 8208 T^{2} + 3105 T^{4} - 57 T^{6} + T^{8}$$
$11$ $$( 18496 - 5032 T + 1233 T^{2} - 37 T^{3} + T^{4} )^{2}$$
$13$ $$46262633168896 - 49842593792 T^{2} + 46897920 T^{4} - 7328 T^{6} + T^{8}$$
$17$ $$( 13498276 + 13277 T^{2} + T^{4} )^{2}$$
$19$ $$( -4850 + 35 T + T^{2} )^{4}$$
$23$ $$32803644436359696 - 8036738541972 T^{2} + 1787845365 T^{4} - 44373 T^{6} + T^{8}$$
$29$ $$( 192044164 - 4503850 T + 91767 T^{2} - 325 T^{3} + T^{4} )^{2}$$
$31$ $$( 36 - 6 T + T^{2} )^{4}$$
$37$ $$( 10188076096 + 205172 T^{2} + T^{4} )^{2}$$
$41$ $$( 241460521 - 3698282 T + 72183 T^{2} + 238 T^{3} + T^{4} )^{2}$$
$43$ $$13039671009356759296 - 467966603595152 T^{2} + 13183297185 T^{4} - 129593 T^{6} + T^{8}$$
$47$ $$16\!\cdots\!16$$$$- 16635920189170688 T^{2} + 125595352305 T^{4} - 407897 T^{6} + T^{8}$$
$53$ $$( 4893841936 + 190088 T^{2} + T^{4} )^{2}$$
$59$ $$( 324072004 - 1530170 T + 25227 T^{2} + 85 T^{3} + T^{4} )^{2}$$
$61$ $$( 89074790116 + 73718138 T + 359463 T^{2} - 247 T^{3} + T^{4} )^{2}$$
$67$ $$12\!\cdots\!81$$$$- 26095280341430178 T^{2} + 515765811555 T^{4} - 742242 T^{6} + T^{8}$$
$71$ $$( -61016 - 394 T + T^{2} )^{4}$$
$73$ $$( 33224540176 + 1022273 T^{2} + T^{4} )^{2}$$
$79$ $$( 16576047504 + 108148320 T + 576852 T^{2} + 840 T^{3} + T^{4} )^{2}$$
$83$ $$75\!\cdots\!76$$$$- 9485403337370158992 T^{2} + 9236578850865 T^{4} - 3460833 T^{6} + T^{8}$$
$89$ $$( -535050 - 1065 T + T^{2} )^{4}$$
$97$ $$23\!\cdots\!36$$$$- 12558608747445428 T^{2} + 648217239525 T^{4} - 814637 T^{6} + T^{8}$$