Properties

 Label 900.2.n.b Level $900$ Weight $2$ Character orbit 900.n Analytic conductor $7.187$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 900.n (of order $$5$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$7.18653618192$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.26265625.1 Defining polynomial: $$x^{8} - 3 x^{7} + 2 x^{6} + x^{4} + 8 x^{2} - 24 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$5$$ Twist minimal: no (minimal twist has level 300) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{3} - \beta_{6} - \beta_{7} ) q^{5} + ( 1 + \beta_{2} + \beta_{6} - \beta_{7} ) q^{7} +O(q^{10})$$ $$q + ( 1 - \beta_{3} - \beta_{6} - \beta_{7} ) q^{5} + ( 1 + \beta_{2} + \beta_{6} - \beta_{7} ) q^{7} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{11} + ( -2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{13} + ( -1 - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{17} + ( 1 + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{19} + ( 3 \beta_{1} - \beta_{2} - 2 \beta_{5} ) q^{23} + ( -2 + 3 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{6} ) q^{25} + ( 2 \beta_{3} + \beta_{7} ) q^{29} + ( -1 - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{31} + ( \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{35} + ( 1 - \beta_{1} - \beta_{3} - 3 \beta_{4} - \beta_{6} ) q^{37} + ( 1 - 3 \beta_{1} - 3 \beta_{3} - 4 \beta_{4} - \beta_{6} ) q^{41} + ( -2 - 2 \beta_{2} + \beta_{3} - \beta_{6} + 2 \beta_{7} ) q^{43} + ( 4 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 4 \beta_{6} - \beta_{7} ) q^{47} + ( -2 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{49} + ( 5 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 5 \beta_{6} - \beta_{7} ) q^{53} + ( -6 + 3 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - \beta_{5} + 7 \beta_{6} + 2 \beta_{7} ) q^{55} + ( 5 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - 5 \beta_{6} - \beta_{7} ) q^{59} + ( 6 + \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{5} ) q^{61} + ( -2 - 6 \beta_{1} - \beta_{2} - 3 \beta_{4} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{65} + ( 3 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 10 \beta_{6} - 2 \beta_{7} ) q^{67} + ( 3 \beta_{1} + 6 \beta_{3} + 3 \beta_{6} - \beta_{7} ) q^{71} + ( -5 - 2 \beta_{2} + 5 \beta_{3} - 2 \beta_{5} ) q^{73} + ( -4 - \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{77} + ( -3 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{6} - 4 \beta_{7} ) q^{79} + ( -2 + 8 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} + 7 \beta_{6} - 2 \beta_{7} ) q^{83} + ( -3 - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{85} + ( 9 - 8 \beta_{1} + 3 \beta_{2} - 9 \beta_{3} + \beta_{5} ) q^{89} + ( 4 + \beta_{4} - 4 \beta_{6} ) q^{91} + ( 1 + \beta_{1} + \beta_{2} + 3 \beta_{3} + 5 \beta_{6} - 2 \beta_{7} ) q^{95} + ( -6 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 6 \beta_{6} + 4 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 5q^{5} + 8q^{7} + O(q^{10})$$ $$8q + 5q^{5} + 8q^{7} - 8q^{11} - 3q^{17} + 5q^{19} + 7q^{23} + 5q^{25} + 3q^{29} - 3q^{31} + 10q^{35} - q^{37} - 10q^{41} - 12q^{43} + 33q^{47} - 8q^{49} + 19q^{53} - 15q^{55} + 38q^{59} + 46q^{61} - 25q^{65} - 8q^{67} + 25q^{71} - 26q^{73} - 23q^{77} - 16q^{79} - 8q^{83} - 30q^{85} + 30q^{89} + 25q^{91} + 25q^{95} - 14q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 2 x^{6} + x^{4} + 8 x^{2} - 24 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} + \nu^{3} + 2 \nu^{2} + 4 \nu - 8$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} + \nu^{3} - 2 \nu^{2} + 12 \nu - 12$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-7 \nu^{7} + 9 \nu^{6} + 2 \nu^{5} + 4 \nu^{4} + \nu^{3} - 4 \nu^{2} - 60 \nu + 64$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$11 \nu^{7} - 15 \nu^{6} - 4 \nu^{5} - 8 \nu^{4} + 3 \nu^{3} + 2 \nu^{2} + 92 \nu - 96$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$11 \nu^{7} - 13 \nu^{6} - 6 \nu^{5} - 8 \nu^{4} + 3 \nu^{3} + 4 \nu^{2} + 96 \nu - 88$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$13 \nu^{7} - 21 \nu^{6} - 4 \nu^{5} - 4 \nu^{4} + 5 \nu^{3} + 10 \nu^{2} + 120 \nu - 144$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$19 \nu^{7} - 31 \nu^{6} - 4 \nu^{5} - 8 \nu^{4} + 11 \nu^{3} + 18 \nu^{2} + 180 \nu - 224$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} - 4 \beta_{1} + 4$$$$)/5$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 2 \beta_{1} + 3$$$$)/5$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{4} + 8 \beta_{3} + 2 \beta_{2} + 7 \beta_{1} + 3$$$$)/5$$ $$\nu^{4}$$ $$=$$ $$-\beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{2} + 2 \beta_{1} + 1$$ $$\nu^{5}$$ $$=$$ $$($$$$6 \beta_{7} - 11 \beta_{6} - 9 \beta_{5} + 3 \beta_{4} - 11 \beta_{3} + 6 \beta_{2} + 6 \beta_{1} + 19$$$$)/5$$ $$\nu^{6}$$ $$=$$ $$($$$$2 \beta_{7} - 7 \beta_{6} + 7 \beta_{5} - 9 \beta_{4} - 7 \beta_{3} + 7 \beta_{2} + 12 \beta_{1} - 12$$$$)/5$$ $$\nu^{7}$$ $$=$$ $$($$$$-8 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 6 \beta_{4} - 7 \beta_{3} + 7 \beta_{2} + 57 \beta_{1} + 3$$$$)/5$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{3}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 1.33631 − 0.462894i −0.0272949 + 1.41395i 1.40799 − 0.132563i −1.21700 + 0.720348i 1.40799 + 0.132563i −1.21700 − 0.720348i 1.33631 + 0.462894i −0.0272949 − 1.41395i
0 0 0 −1.99851 + 1.00297i 0 −0.0883282 0 0 0
181.2 0 0 0 1.57146 1.59076i 0 4.32440 0 0 0
361.1 0 0 0 0.962197 + 2.01846i 0 1.50430 0 0 0
361.2 0 0 0 1.96485 1.06740i 0 −1.74037 0 0 0
541.1 0 0 0 0.962197 2.01846i 0 1.50430 0 0 0
541.2 0 0 0 1.96485 + 1.06740i 0 −1.74037 0 0 0
721.1 0 0 0 −1.99851 1.00297i 0 −0.0883282 0 0 0
721.2 0 0 0 1.57146 + 1.59076i 0 4.32440 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 721.2 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 900.2.n.b 8
3.b odd 2 1 300.2.m.b 8
15.d odd 2 1 1500.2.m.a 8
15.e even 4 2 1500.2.o.b 16
25.d even 5 1 inner 900.2.n.b 8
75.h odd 10 1 1500.2.m.a 8
75.h odd 10 1 7500.2.a.f 4
75.j odd 10 1 300.2.m.b 8
75.j odd 10 1 7500.2.a.e 4
75.l even 20 2 1500.2.o.b 16
75.l even 20 2 7500.2.d.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.m.b 8 3.b odd 2 1
300.2.m.b 8 75.j odd 10 1
900.2.n.b 8 1.a even 1 1 trivial
900.2.n.b 8 25.d even 5 1 inner
1500.2.m.a 8 15.d odd 2 1
1500.2.m.a 8 75.h odd 10 1
1500.2.o.b 16 15.e even 4 2
1500.2.o.b 16 75.l even 20 2
7500.2.a.e 4 75.j odd 10 1
7500.2.a.f 4 75.h odd 10 1
7500.2.d.c 8 75.l even 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 4 T_{7}^{3} - 4 T_{7}^{2} + 11 T_{7} + 1$$ acting on $$S_{2}^{\mathrm{new}}(900, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$625 - 625 T + 250 T^{2} + 25 T^{3} - 45 T^{4} + 5 T^{5} + 10 T^{6} - 5 T^{7} + T^{8}$$
$7$ $$( 1 + 11 T - 4 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$11$ $$361 + 323 T + 1033 T^{2} - 149 T^{3} + 100 T^{4} - 39 T^{5} + 23 T^{6} + 8 T^{7} + T^{8}$$
$13$ $$2025 - 2025 T + 2925 T^{2} - 975 T^{3} + 310 T^{4} - 5 T^{5} - 5 T^{6} + T^{8}$$
$17$ $$81 + 108 T + 333 T^{2} + 21 T^{3} - 20 T^{4} + T^{5} + 13 T^{6} + 3 T^{7} + T^{8}$$
$19$ $$25 - 25 T + 225 T^{2} + 225 T^{3} + 160 T^{4} + 45 T^{5} + 5 T^{6} - 5 T^{7} + T^{8}$$
$23$ $$29241 + 513 T + 12798 T^{2} + 6171 T^{3} + 1705 T^{4} + 61 T^{5} + 18 T^{6} - 7 T^{7} + T^{8}$$
$29$ $$1 - 3 T + 58 T^{2} + 129 T^{3} + 105 T^{4} - 21 T^{5} + 8 T^{6} - 3 T^{7} + T^{8}$$
$31$ $$962361 - 55917 T + 78228 T^{2} - 21699 T^{3} + 3625 T^{4} + 511 T^{5} + 58 T^{6} + 3 T^{7} + T^{8}$$
$37$ $$1042441 - 57176 T + 12657 T^{2} - 343 T^{3} + 1550 T^{4} + 113 T^{5} + 57 T^{6} + T^{7} + T^{8}$$
$41$ $$7317025 - 1704150 T + 228475 T^{2} + 5800 T^{3} + 4085 T^{4} + 520 T^{5} + 155 T^{6} + 10 T^{7} + T^{8}$$
$43$ $$( 131 - 64 T - 19 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$47$ $$16801801 - 4947493 T + 1291508 T^{2} - 275851 T^{3} + 48275 T^{4} - 6151 T^{5} + 568 T^{6} - 33 T^{7} + T^{8}$$
$53$ $$9801 + 9504 T + 5787 T^{2} + 2547 T^{3} + 5080 T^{4} - 1347 T^{5} + 217 T^{6} - 19 T^{7} + T^{8}$$
$59$ $$13697401 - 8874998 T + 2975713 T^{2} - 622126 T^{3} + 94675 T^{4} - 10026 T^{5} + 773 T^{6} - 38 T^{7} + T^{8}$$
$61$ $$62552281 - 11594594 T + 3590217 T^{2} - 782922 T^{3} + 126055 T^{4} - 13818 T^{5} + 1037 T^{6} - 46 T^{7} + T^{8}$$
$67$ $$408321 - 1755972 T + 2888388 T^{2} - 13704 T^{3} + 34630 T^{4} + 2416 T^{5} + 183 T^{6} + 8 T^{7} + T^{8}$$
$71$ $$25 + 650 T + 44875 T^{2} - 10175 T^{3} + 4060 T^{4} - 1395 T^{5} + 275 T^{6} - 25 T^{7} + T^{8}$$
$73$ $$121 - 506 T + 3182 T^{2} - 3618 T^{3} + 1600 T^{4} + 228 T^{5} + 267 T^{6} + 26 T^{7} + T^{8}$$
$79$ $$408321 + 300969 T + 629937 T^{2} - 22443 T^{3} + 2350 T^{4} - 247 T^{5} + 117 T^{6} + 16 T^{7} + T^{8}$$
$83$ $$46908801 + 14876028 T + 9296163 T^{2} + 531546 T^{3} + 43855 T^{4} - 1054 T^{5} - 17 T^{6} + 8 T^{7} + T^{8}$$
$89$ $$97515625 + 11109375 T + 359375 T^{2} - 106875 T^{3} + 57750 T^{4} - 7125 T^{5} + 625 T^{6} - 30 T^{7} + T^{8}$$
$97$ $$64304361 + 13423806 T + 5936247 T^{2} + 188028 T^{3} + 16705 T^{4} - 908 T^{5} + 87 T^{6} + 14 T^{7} + T^{8}$$