# Properties

 Label 450.2.h.e Level $450$ Weight $2$ Character orbit 450.h Analytic conductor $3.593$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 450.h (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.59326809096$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.58140625.2 Defining polynomial: $$x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$5$$ Twist minimal: no (minimal twist has level 50) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} -\beta_{3} q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} ) q^{7} + \beta_{2} q^{8} +O(q^{10})$$ $$q + \beta_{6} q^{2} -\beta_{3} q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} ) q^{7} + \beta_{2} q^{8} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{10} + ( \beta_{1} - \beta_{5} - \beta_{6} ) q^{11} + ( -3 - \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{13} + ( -\beta_{1} + \beta_{5} + \beta_{6} ) q^{14} + ( -1 + \beta_{2} + \beta_{3} + \beta_{6} ) q^{16} + ( 1 + 2 \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{17} + ( 3 - \beta_{2} - \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{19} + ( -1 + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{20} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{7} ) q^{22} + ( 2 - \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{23} + ( 3 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{25} + ( -3 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{26} + ( -\beta_{1} - \beta_{3} + \beta_{5} - \beta_{7} ) q^{28} + ( 2 \beta_{2} + 5 \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{29} + ( -2 + 5 \beta_{2} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{31} - q^{32} + ( -2 - \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + 3 \beta_{6} ) q^{34} + ( 4 - \beta_{1} - 4 \beta_{2} - 5 \beta_{3} + \beta_{5} - \beta_{7} ) q^{35} + ( -3 + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{7} ) q^{37} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{38} + ( -1 - \beta_{4} ) q^{40} + ( 3 + \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{41} + ( 3 + \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} ) q^{43} + ( -\beta_{2} - \beta_{5} + \beta_{7} ) q^{44} + ( \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{46} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{7} ) q^{47} + ( 2 - \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + \beta_{4} + \beta_{5} ) q^{49} + ( -1 - \beta_{4} + 4 \beta_{6} - \beta_{7} ) q^{50} + ( -2 + 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{52} + ( -4 \beta_{2} - 3 \beta_{5} - 4 \beta_{6} ) q^{53} + ( -4 - \beta_{2} + 4 \beta_{3} - \beta_{5} + \beta_{7} ) q^{55} + ( \beta_{2} + \beta_{5} - \beta_{7} ) q^{56} + ( -2 - 3 \beta_{2} - \beta_{4} + 2 \beta_{6} ) q^{58} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 4 \beta_{7} ) q^{59} + ( 1 + 4 \beta_{1} - \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{61} + ( -5 + \beta_{1} + 5 \beta_{2} + 3 \beta_{3} - \beta_{4} + 3 \beta_{6} + 2 \beta_{7} ) q^{62} -\beta_{6} q^{64} + ( -2 + \beta_{1} + 6 \beta_{3} - 2 \beta_{4} + 3 \beta_{6} - 2 \beta_{7} ) q^{65} + ( -7 \beta_{2} - 3 \beta_{5} + 3 \beta_{7} ) q^{67} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{68} + ( 4 + \beta_{2} - 4 \beta_{3} + \beta_{5} - \beta_{7} ) q^{70} + ( \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{71} + ( 2 - \beta_{1} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} - 8 \beta_{6} - 3 \beta_{7} ) q^{73} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{74} + ( 1 - 3 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{76} + ( -4 + \beta_{1} + 4 \beta_{3} - \beta_{5} - 5 \beta_{6} ) q^{77} + ( -2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{79} + ( -\beta_{6} - \beta_{7} ) q^{80} + ( 3 + 2 \beta_{1} - \beta_{4} - \beta_{5} ) q^{82} + ( -2 - 5 \beta_{2} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{83} + ( 2 - \beta_{1} - 5 \beta_{2} - \beta_{3} - 3 \beta_{4} - 4 \beta_{6} + \beta_{7} ) q^{85} + ( 4 + \beta_{1} - 4 \beta_{3} - \beta_{5} - \beta_{6} ) q^{86} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{88} + ( 4 - 4 \beta_{1} - 4 \beta_{3} + \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{89} + ( 1 + 3 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{6} - 2 \beta_{7} ) q^{91} + ( -2 - \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{92} + ( -\beta_{2} - \beta_{5} + \beta_{7} ) q^{94} + ( 5 - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} ) q^{95} + ( -5 \beta_{2} - 8 \beta_{3} - 5 \beta_{6} ) q^{97} + ( 4 - \beta_{1} - 4 \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{2} - 2q^{4} + 4q^{7} + 2q^{8} + O(q^{10})$$ $$8q + 2q^{2} - 2q^{4} + 4q^{7} + 2q^{8} - q^{11} - 13q^{13} + q^{14} - 2q^{16} + 11q^{17} + 20q^{19} - 5q^{20} + q^{22} + 3q^{23} + 30q^{25} - 22q^{26} - q^{28} + 15q^{29} - 9q^{31} - 8q^{32} - q^{34} + 15q^{35} - 6q^{37} + 15q^{38} - 5q^{40} + 9q^{41} + 12q^{43} - q^{44} + 7q^{46} + q^{47} - 4q^{49} + 5q^{50} - 13q^{52} - 7q^{53} - 25q^{55} + q^{56} - 15q^{58} - 10q^{59} + 6q^{61} - 21q^{62} - 2q^{64} + 10q^{65} - 11q^{67} - 24q^{68} + 25q^{70} + 9q^{71} - 8q^{73} - 24q^{74} - 10q^{76} - 33q^{77} - 10q^{79} + 26q^{82} - 27q^{83} + 5q^{85} + 23q^{86} + q^{88} + 15q^{89} + q^{91} - 7q^{92} - q^{94} + 30q^{95} - 36q^{97} + 19q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-64 \nu^{7} + 206 \nu^{6} - 318 \nu^{5} + 399 \nu^{4} - 732 \nu^{3} - 126 \nu^{2} + 2175 \nu + 235$$$$)/1355$$ $$\beta_{2}$$ $$=$$ $$($$$$406 \nu^{7} - 714 \nu^{6} + 747 \nu^{5} - 1896 \nu^{4} + 2103 \nu^{3} + 4949 \nu^{2} + 1065 \nu - 7800$$$$)/1355$$ $$\beta_{3}$$ $$=$$ $$($$$$-420 \nu^{7} + 776 \nu^{6} - 698 \nu^{5} + 1924 \nu^{4} - 2297 \nu^{3} - 5129 \nu^{2} - 1055 \nu + 10265$$$$)/1355$$ $$\beta_{4}$$ $$=$$ $$($$$$504 \nu^{7} - 877 \nu^{6} + 946 \nu^{5} - 2363 \nu^{4} + 2919 \nu^{3} + 5125 \nu^{2} + 3705 \nu - 11505$$$$)/1355$$ $$\beta_{5}$$ $$=$$ $$($$$$-613 \nu^{7} + 1321 \nu^{6} - 1513 \nu^{5} + 3394 \nu^{4} - 4352 \nu^{3} - 6178 \nu^{2} - 530 \nu + 12985$$$$)/1355$$ $$\beta_{6}$$ $$=$$ $$($$$$-714 \nu^{7} + 1265 \nu^{6} - 1295 \nu^{5} + 3325 \nu^{4} - 3390 \nu^{3} - 8367 \nu^{2} - 2200 \nu + 14605$$$$)/1355$$ $$\beta_{7}$$ $$=$$ $$($$$$-1020 \nu^{7} + 1962 \nu^{6} - 2121 \nu^{5} + 5563 \nu^{4} - 6314 \nu^{3} - 10443 \nu^{2} - 3530 \nu + 21445$$$$)/1355$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 3 \beta_{1} + 3$$$$)/5$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{7} + 3 \beta_{6} - 4 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 11 \beta_{2} + 6 \beta_{1} - 4$$$$)/5$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + 16 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - 6 \beta_{3} + 17 \beta_{2} - 3 \beta_{1} + 2$$$$)/5$$ $$\nu^{4}$$ $$=$$ $$($$$$13 \beta_{7} - 3 \beta_{6} - 6 \beta_{5} + 13 \beta_{4} - 7 \beta_{3} - 6 \beta_{2} - 6 \beta_{1} + 14$$$$)/5$$ $$\nu^{5}$$ $$=$$ $$($$$$16 \beta_{7} - 6 \beta_{6} - 22 \beta_{5} + 6 \beta_{4} + 51 \beta_{3} + 43 \beta_{2} + 8 \beta_{1} - 67$$$$)/5$$ $$\nu^{6}$$ $$=$$ $$($$$$-23 \beta_{7} + 63 \beta_{6} + 46 \beta_{5} + 12 \beta_{4} + 52 \beta_{3} + 156 \beta_{2} - 34 \beta_{1} - 144$$$$)/5$$ $$\nu^{7}$$ $$=$$ $$($$$$-31 \beta_{7} - 9 \beta_{6} + 137 \beta_{5} + 84 \beta_{4} - 31 \beta_{3} - 33 \beta_{2} - 168 \beta_{1} + 62$$$$)/5$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1
 1.66637 + 0.917186i −0.357358 − 1.86824i 1.17421 + 0.0566033i −0.983224 − 0.644389i 1.17421 − 0.0566033i −0.983224 + 0.644389i 1.66637 − 0.917186i −0.357358 + 1.86824i
0.809017 + 0.587785i 0 0.309017 + 0.951057i −2.02373 + 0.951057i 0 −2.77447 −0.309017 + 0.951057i 0 −2.19625 0.420099i
91.2 0.809017 + 0.587785i 0 0.309017 + 0.951057i 2.02373 + 0.951057i 0 3.77447 −0.309017 + 0.951057i 0 1.07822 + 1.95894i
181.1 −0.309017 0.951057i 0 −0.809017 + 0.587785i −2.15743 + 0.587785i 0 1.83337 0.809017 + 0.587785i 0 1.22570 + 1.87020i
181.2 −0.309017 0.951057i 0 −0.809017 + 0.587785i 2.15743 + 0.587785i 0 −0.833366 0.809017 + 0.587785i 0 −0.107666 2.23347i
271.1 −0.309017 + 0.951057i 0 −0.809017 0.587785i −2.15743 0.587785i 0 1.83337 0.809017 0.587785i 0 1.22570 1.87020i
271.2 −0.309017 + 0.951057i 0 −0.809017 0.587785i 2.15743 0.587785i 0 −0.833366 0.809017 0.587785i 0 −0.107666 + 2.23347i
361.1 0.809017 0.587785i 0 0.309017 0.951057i −2.02373 0.951057i 0 −2.77447 −0.309017 0.951057i 0 −2.19625 + 0.420099i
361.2 0.809017 0.587785i 0 0.309017 0.951057i 2.02373 0.951057i 0 3.77447 −0.309017 0.951057i 0 1.07822 1.95894i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.h.e 8
3.b odd 2 1 50.2.d.b 8
12.b even 2 1 400.2.u.d 8
15.d odd 2 1 250.2.d.d 8
15.e even 4 2 250.2.e.c 16
25.d even 5 1 inner 450.2.h.e 8
75.h odd 10 1 250.2.d.d 8
75.h odd 10 1 1250.2.a.f 4
75.j odd 10 1 50.2.d.b 8
75.j odd 10 1 1250.2.a.l 4
75.l even 20 2 250.2.e.c 16
75.l even 20 2 1250.2.b.e 8
300.n even 10 1 400.2.u.d 8
300.n even 10 1 10000.2.a.t 4
300.r even 10 1 10000.2.a.x 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.b 8 3.b odd 2 1
50.2.d.b 8 75.j odd 10 1
250.2.d.d 8 15.d odd 2 1
250.2.d.d 8 75.h odd 10 1
250.2.e.c 16 15.e even 4 2
250.2.e.c 16 75.l even 20 2
400.2.u.d 8 12.b even 2 1
400.2.u.d 8 300.n even 10 1
450.2.h.e 8 1.a even 1 1 trivial
450.2.h.e 8 25.d even 5 1 inner
1250.2.a.f 4 75.h odd 10 1
1250.2.a.l 4 75.j odd 10 1
1250.2.b.e 8 75.l even 20 2
10000.2.a.t 4 300.n even 10 1
10000.2.a.x 4 300.r even 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(450, [\chi])$$:

 $$T_{7}^{4} - 2 T_{7}^{3} - 11 T_{7}^{2} + 12 T_{7} + 16$$ $$T_{11}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$625 - 375 T^{2} + 105 T^{4} - 15 T^{6} + T^{8}$$
$7$ $$( 16 + 12 T - 11 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$11$ $$256 + 64 T + 352 T^{2} - 128 T^{3} + 105 T^{4} - 7 T^{5} - 3 T^{6} + T^{7} + T^{8}$$
$13$ $$39601 + 36218 T + 21988 T^{2} + 8426 T^{3} + 2325 T^{4} + 476 T^{5} + 93 T^{6} + 13 T^{7} + T^{8}$$
$17$ $$11881 - 14279 T + 13302 T^{2} - 6657 T^{3} + 2025 T^{4} - 383 T^{5} + 72 T^{6} - 11 T^{7} + T^{8}$$
$19$ $$6400 - 11200 T + 12000 T^{2} - 8000 T^{3} + 3585 T^{4} - 965 T^{5} + 190 T^{6} - 20 T^{7} + T^{8}$$
$23$ $$256 + 192 T - 192 T^{2} - 296 T^{3} + 705 T^{4} + 49 T^{5} + 63 T^{6} - 3 T^{7} + T^{8}$$
$29$ $$25 + 300 T + 10150 T^{2} - 5700 T^{3} + 1735 T^{4} - 390 T^{5} + 105 T^{6} - 15 T^{7} + T^{8}$$
$31$ $$1597696 + 257856 T + 56432 T^{2} + 228 T^{3} + 805 T^{4} + 237 T^{5} + 87 T^{6} + 9 T^{7} + T^{8}$$
$37$ $$5041 - 4686 T + 3162 T^{2} - 1878 T^{3} + 2380 T^{4} + 348 T^{5} + 47 T^{6} + 6 T^{7} + T^{8}$$
$41$ $$7921 + 11214 T + 9162 T^{2} + 4232 T^{3} + 1425 T^{4} + 128 T^{5} + 27 T^{6} - 9 T^{7} + T^{8}$$
$43$ $$( 176 + 64 T - 39 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$47$ $$256 + 256 T + 32 T^{2} - 132 T^{3} + 125 T^{4} - 33 T^{5} + 17 T^{6} - T^{7} + T^{8}$$
$53$ $$65536 - 73728 T + 232448 T^{2} + 12304 T^{3} - 1695 T^{4} - 206 T^{5} + 108 T^{6} + 7 T^{7} + T^{8}$$
$59$ $$102400 + 281600 T + 320000 T^{2} + 105600 T^{3} + 17360 T^{4} + 1320 T^{5} + 260 T^{6} + 10 T^{7} + T^{8}$$
$61$ $$2920681 - 984384 T + 570837 T^{2} - 56572 T^{3} + 9180 T^{4} + 332 T^{5} - 18 T^{6} - 6 T^{7} + T^{8}$$
$67$ $$891136 + 1204544 T + 730272 T^{2} + 193072 T^{3} + 40505 T^{4} + 3883 T^{5} + 237 T^{6} + 11 T^{7} + T^{8}$$
$71$ $$4096 - 19456 T + 32512 T^{2} + 21432 T^{3} + 5225 T^{4} - 297 T^{5} + 217 T^{6} - 9 T^{7} + T^{8}$$
$73$ $$1175056 + 988608 T + 471808 T^{2} + 141726 T^{3} + 27875 T^{4} + 3231 T^{5} + 208 T^{6} + 8 T^{7} + T^{8}$$
$79$ $$102400 + 25600 T + 25600 T^{2} - 3200 T^{3} - 240 T^{4} + 320 T^{5} + 120 T^{6} + 10 T^{7} + T^{8}$$
$83$ $$13424896 + 6111552 T + 1867728 T^{2} + 352404 T^{3} + 46405 T^{4} + 4359 T^{5} + 403 T^{6} + 27 T^{7} + T^{8}$$
$89$ $$9610000 - 2015000 T + 1025500 T^{2} - 30750 T^{3} + 8525 T^{4} + 750 T^{5} + 70 T^{6} - 15 T^{7} + T^{8}$$
$97$ $$( 1 + 7 T + 124 T^{2} + 18 T^{3} + T^{4} )^{2}$$