# Properties

 Label 900.3.f.d Level $900$ Weight $3$ Character orbit 900.f Analytic conductor $24.523$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 900.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.18084870400.3 Defining polynomial: $$x^{8} - 12 x^{6} + 40 x^{4} + 17 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{4}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 100) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + ( -1 - \beta_{1} ) q^{4} + ( -2 \beta_{4} + 2 \beta_{7} ) q^{7} + ( -2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{8} +O(q^{10})$$ $$q + \beta_{4} q^{2} + ( -1 - \beta_{1} ) q^{4} + ( -2 \beta_{4} + 2 \beta_{7} ) q^{7} + ( -2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{8} + ( \beta_{1} + \beta_{6} ) q^{11} + ( -2 \beta_{3} + 4 \beta_{4} + 2 \beta_{7} ) q^{13} + ( -6 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{6} ) q^{14} + ( -9 + 3 \beta_{1} - 2 \beta_{6} ) q^{16} + ( 2 \beta_{3} - 4 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{17} + ( 2 - \beta_{1} - 2 \beta_{2} + 5 \beta_{6} ) q^{19} + ( 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - 4 \beta_{7} ) q^{22} + ( -2 \beta_{3} + 2 \beta_{4} - 4 \beta_{7} ) q^{23} + ( -20 - 4 \beta_{1} ) q^{26} + ( -8 \beta_{4} - 6 \beta_{5} - 2 \beta_{7} ) q^{28} + ( 10 - 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( 2 \beta_{1} + 2 \beta_{6} ) q^{31} + ( 6 \beta_{3} - 6 \beta_{4} - \beta_{5} + 3 \beta_{7} ) q^{32} + ( 19 + 3 \beta_{1} + \beta_{2} + \beta_{6} ) q^{34} + ( -2 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} + 2 \beta_{7} ) q^{37} + ( -6 \beta_{3} - 3 \beta_{4} - 6 \beta_{5} - 8 \beta_{7} ) q^{38} -17 q^{41} + ( -10 \beta_{3} - 8 \beta_{4} - 2 \beta_{7} ) q^{43} + ( 21 - 3 \beta_{1} + 4 \beta_{2} ) q^{44} + ( 6 - 2 \beta_{1} + 6 \beta_{2} - 6 \beta_{6} ) q^{46} + ( -14 \beta_{3} - 20 \beta_{4} + 6 \beta_{7} ) q^{47} + ( 19 - 8 \beta_{1} + 8 \beta_{2} ) q^{49} + ( -8 \beta_{3} - 24 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} ) q^{52} + ( -8 \beta_{3} + 16 \beta_{4} - 6 \beta_{5} + 8 \beta_{7} ) q^{53} + ( 10 + 2 \beta_{1} + 8 \beta_{2} + 4 \beta_{6} ) q^{56} + ( 12 \beta_{4} + 4 \beta_{5} - 4 \beta_{7} ) q^{58} + ( -6 + 18 \beta_{1} + 6 \beta_{2} ) q^{59} + ( -28 + 10 \beta_{1} - 10 \beta_{2} ) q^{61} + ( 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 8 \beta_{7} ) q^{62} + ( 17 + 5 \beta_{1} - 8 \beta_{2} + 10 \beta_{6} ) q^{64} + ( \beta_{3} - 9 \beta_{4} + 10 \beta_{7} ) q^{67} + ( 8 \beta_{3} + 24 \beta_{4} - 3 \beta_{5} - 9 \beta_{7} ) q^{68} + ( -10 + 22 \beta_{1} + 10 \beta_{2} - 8 \beta_{6} ) q^{71} + ( -2 \beta_{3} + 4 \beta_{4} - 11 \beta_{5} + 2 \beta_{7} ) q^{73} + ( -26 - 10 \beta_{1} + 6 \beta_{2} + 6 \beta_{6} ) q^{74} + ( 5 - 3 \beta_{1} + 20 \beta_{2} - 8 \beta_{6} ) q^{76} + ( -10 \beta_{3} + 20 \beta_{4} + 14 \beta_{5} + 10 \beta_{7} ) q^{77} + ( -2 - 4 \beta_{1} + 2 \beta_{2} - 10 \beta_{6} ) q^{79} -17 \beta_{4} q^{82} + ( -11 \beta_{3} - 5 \beta_{4} - 6 \beta_{7} ) q^{83} + ( -24 + 8 \beta_{1} + 12 \beta_{2} - 12 \beta_{6} ) q^{86} + ( 2 \beta_{3} + 26 \beta_{4} + 7 \beta_{5} - 9 \beta_{7} ) q^{88} + ( -55 - 2 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 4 + 4 \beta_{1} - 4 \beta_{2} + 16 \beta_{6} ) q^{91} + ( 8 \beta_{3} + 16 \beta_{4} + 14 \beta_{5} + 2 \beta_{7} ) q^{92} + ( -60 + 20 \beta_{1} + 8 \beta_{2} - 8 \beta_{6} ) q^{94} + ( -12 \beta_{3} + 24 \beta_{4} + 18 \beta_{5} + 12 \beta_{7} ) q^{97} + ( 27 \beta_{4} + 16 \beta_{5} - 16 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 10q^{4} + O(q^{10})$$ $$8q - 10q^{4} - 52q^{14} - 62q^{16} - 168q^{26} + 80q^{29} + 158q^{34} - 136q^{41} + 170q^{44} + 68q^{46} + 152q^{49} + 92q^{56} - 224q^{61} + 110q^{64} - 228q^{74} + 90q^{76} - 128q^{86} - 440q^{89} - 408q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 12 x^{6} + 40 x^{4} + 17 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$19 \nu^{6} - 244 \nu^{4} + 896 \nu^{2} + 104$$$$)/55$$ $$\beta_{2}$$ $$=$$ $$($$$$23 \nu^{6} - 278 \nu^{4} + 882 \nu^{2} + 453$$$$)/55$$ $$\beta_{3}$$ $$=$$ $$($$$$-14 \nu^{7} + 174 \nu^{5} - 611 \nu^{3} - 204 \nu$$$$)/55$$ $$\beta_{4}$$ $$=$$ $$($$$$37 \nu^{7} - 452 \nu^{5} + 1548 \nu^{3} + 437 \nu$$$$)/110$$ $$\beta_{5}$$ $$=$$ $$($$$$9 \nu^{7} - 104 \nu^{5} + 326 \nu^{3} + 249 \nu$$$$)/22$$ $$\beta_{6}$$ $$=$$ $$($$$$41 \nu^{6} - 486 \nu^{4} + 1644 \nu^{2} + 401$$$$)/55$$ $$\beta_{7}$$ $$=$$ $$($$$$-28 \nu^{7} + 348 \nu^{5} - 1277 \nu^{3} + 32 \nu$$$$)/55$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - 5 \beta_{4} - 5 \beta_{3}$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{6} - 7 \beta_{2} + 2 \beta_{1} + 32$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{7} + 4 \beta_{5} - 20 \beta_{4} - 10 \beta_{3}$$$$)/5$$ $$\nu^{4}$$ $$=$$ $$($$$$7 \beta_{6} - 10 \beta_{2} - 3 \beta_{1} + 37$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-85 \beta_{7} + 81 \beta_{5} - 265 \beta_{4} - 50 \beta_{3}$$$$)/10$$ $$\nu^{6}$$ $$=$$ $$($$$$154 \beta_{6} - 156 \beta_{2} - 129 \beta_{1} + 406$$$$)/5$$ $$\nu^{7}$$ $$=$$ $$($$$$-620 \beta_{7} + 643 \beta_{5} - 1475 \beta_{4} + 285 \beta_{3}$$$$)/10$$

## Character values

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

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 0.220086 − 0.500000i 0.220086 + 0.500000i 2.53999 + 0.500000i 2.53999 − 0.500000i −2.53999 + 0.500000i −2.53999 − 0.500000i −0.220086 − 0.500000i −0.220086 + 0.500000i
−1.47492 1.35078i 0 0.350781 + 3.98459i 0 0 10.9190 4.86493 6.35078i 0 0
199.2 −1.47492 + 1.35078i 0 0.350781 3.98459i 0 0 10.9190 4.86493 + 6.35078i 0 0
199.3 −0.758030 1.85078i 0 −2.85078 + 2.80590i 0 0 −4.09573 7.35408 + 3.14922i 0 0
199.4 −0.758030 + 1.85078i 0 −2.85078 2.80590i 0 0 −4.09573 7.35408 3.14922i 0 0
199.5 0.758030 1.85078i 0 −2.85078 2.80590i 0 0 4.09573 −7.35408 + 3.14922i 0 0
199.6 0.758030 + 1.85078i 0 −2.85078 + 2.80590i 0 0 4.09573 −7.35408 3.14922i 0 0
199.7 1.47492 1.35078i 0 0.350781 3.98459i 0 0 −10.9190 −4.86493 6.35078i 0 0
199.8 1.47492 + 1.35078i 0 0.350781 + 3.98459i 0 0 −10.9190 −4.86493 + 6.35078i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.f.d 8
3.b odd 2 1 100.3.d.a 8
4.b odd 2 1 inner 900.3.f.d 8
5.b even 2 1 inner 900.3.f.d 8
5.c odd 4 1 900.3.c.l 4
5.c odd 4 1 900.3.c.m 4
12.b even 2 1 100.3.d.a 8
15.d odd 2 1 100.3.d.a 8
15.e even 4 1 100.3.b.d 4
15.e even 4 1 100.3.b.e yes 4
20.d odd 2 1 inner 900.3.f.d 8
20.e even 4 1 900.3.c.l 4
20.e even 4 1 900.3.c.m 4
24.f even 2 1 1600.3.h.o 8
24.h odd 2 1 1600.3.h.o 8
60.h even 2 1 100.3.d.a 8
60.l odd 4 1 100.3.b.d 4
60.l odd 4 1 100.3.b.e yes 4
120.i odd 2 1 1600.3.h.o 8
120.m even 2 1 1600.3.h.o 8
120.q odd 4 1 1600.3.b.o 4
120.q odd 4 1 1600.3.b.p 4
120.w even 4 1 1600.3.b.o 4
120.w even 4 1 1600.3.b.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.3.b.d 4 15.e even 4 1
100.3.b.d 4 60.l odd 4 1
100.3.b.e yes 4 15.e even 4 1
100.3.b.e yes 4 60.l odd 4 1
100.3.d.a 8 3.b odd 2 1
100.3.d.a 8 12.b even 2 1
100.3.d.a 8 15.d odd 2 1
100.3.d.a 8 60.h even 2 1
900.3.c.l 4 5.c odd 4 1
900.3.c.l 4 20.e even 4 1
900.3.c.m 4 5.c odd 4 1
900.3.c.m 4 20.e even 4 1
900.3.f.d 8 1.a even 1 1 trivial
900.3.f.d 8 4.b odd 2 1 inner
900.3.f.d 8 5.b even 2 1 inner
900.3.f.d 8 20.d odd 2 1 inner
1600.3.b.o 4 120.q odd 4 1
1600.3.b.o 4 120.w even 4 1
1600.3.b.p 4 120.q odd 4 1
1600.3.b.p 4 120.w even 4 1
1600.3.h.o 8 24.f even 2 1
1600.3.h.o 8 24.h odd 2 1
1600.3.h.o 8 120.i odd 2 1
1600.3.h.o 8 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$:

 $$T_{7}^{4} - 136 T_{7}^{2} + 2000$$ $$T_{29}^{2} - 20 T_{29} - 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 + 80 T^{2} + 28 T^{4} + 5 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 2000 - 136 T^{2} + T^{4} )^{2}$$
$11$ $$( 125 + 130 T^{2} + T^{4} )^{2}$$
$13$ $$( 25600 + 336 T^{2} + T^{4} )^{2}$$
$17$ $$( 24025 + 346 T^{2} + T^{4} )^{2}$$
$19$ $$( 465125 + 1370 T^{2} + T^{4} )^{2}$$
$23$ $$( 147920 - 776 T^{2} + T^{4} )^{2}$$
$29$ $$( -64 - 20 T + T^{2} )^{4}$$
$31$ $$( 2000 + 520 T^{2} + T^{4} )^{2}$$
$37$ $$( 739600 + 2376 T^{2} + T^{4} )^{2}$$
$41$ $$( 17 + T )^{8}$$
$43$ $$( 128000 - 3056 T^{2} + T^{4} )^{2}$$
$47$ $$( 5242880 - 4976 T^{2} + T^{4} )^{2}$$
$53$ $$( 1392400 + 8136 T^{2} + T^{4} )^{2}$$
$59$ $$( 41472000 + 13680 T^{2} + T^{4} )^{2}$$
$61$ $$( -3316 + 56 T + T^{2} )^{4}$$
$67$ $$( 1915805 - 3586 T^{2} + T^{4} )^{2}$$
$71$ $$( 67712000 + 19120 T^{2} + T^{4} )^{2}$$
$73$ $$( 9517225 + 6826 T^{2} + T^{4} )^{2}$$
$79$ $$( 6962000 + 7720 T^{2} + T^{4} )^{2}$$
$83$ $$( 3621005 - 5426 T^{2} + T^{4} )^{2}$$
$89$ $$( 2861 + 110 T + T^{2} )^{4}$$
$97$ $$( 32400 + 23976 T^{2} + T^{4} )^{2}$$