Properties

 Label 900.3.f.g Level $900$ Weight $3$ Character orbit 900.f Analytic conductor $24.523$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

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: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - x^{12} + 25 x^{8} - 16 x^{4} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{20}\cdot 5^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} -\beta_{11} q^{7} + ( \beta_{1} - \beta_{4} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} -\beta_{11} q^{7} + ( \beta_{1} - \beta_{4} ) q^{8} + \beta_{6} q^{11} -\beta_{13} q^{13} + ( -\beta_{6} + \beta_{9} ) q^{14} + ( -1 - 2 \beta_{3} - \beta_{5} ) q^{16} + ( 4 \beta_{1} - \beta_{4} + \beta_{10} + \beta_{12} ) q^{17} + ( -1 + 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{19} + ( \beta_{7} - 2 \beta_{11} + \beta_{13} ) q^{22} + ( -3 \beta_{1} + 3 \beta_{4} + 2 \beta_{10} + 3 \beta_{12} ) q^{23} + ( -2 \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{26} + ( \beta_{7} + 2 \beta_{11} - 2 \beta_{13} + \beta_{14} ) q^{28} + ( \beta_{6} + 3 \beta_{9} + \beta_{15} ) q^{29} + ( -2 + 4 \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{31} + ( 2 \beta_{1} + 2 \beta_{4} + 4 \beta_{12} ) q^{32} + ( 11 - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{5} ) q^{34} + 2 \beta_{13} q^{37} + ( \beta_{1} - \beta_{4} - \beta_{10} + 2 \beta_{12} ) q^{38} + ( \beta_{6} - 4 \beta_{8} + \beta_{9} - \beta_{15} ) q^{41} + ( \beta_{7} + \beta_{11} - \beta_{13} + 4 \beta_{14} ) q^{43} + 4 \beta_{9} q^{44} + ( 15 + 2 \beta_{2} + 6 \beta_{3} - 3 \beta_{5} ) q^{46} + ( -10 \beta_{1} + 2 \beta_{4} + 4 \beta_{10} + 2 \beta_{12} ) q^{47} + ( 14 - 8 \beta_{2} - 4 \beta_{5} ) q^{49} + ( -6 \beta_{7} + 4 \beta_{11} + \beta_{13} - \beta_{14} ) q^{52} + ( 22 \beta_{1} - 3 \beta_{4} + 5 \beta_{10} + 3 \beta_{12} ) q^{53} + ( -\beta_{6} + 3 \beta_{8} - 6 \beta_{9} + \beta_{15} ) q^{56} + ( -\beta_{7} - 6 \beta_{11} - 5 \beta_{13} + 4 \beta_{14} ) q^{58} -6 \beta_{6} q^{59} + ( 12 - 14 \beta_{2} - 7 \beta_{5} ) q^{61} + ( \beta_{1} - 3 \beta_{4} - 3 \beta_{10} + 2 \beta_{12} ) q^{62} + ( -18 - 4 \beta_{2} + 4 \beta_{3} - 6 \beta_{5} ) q^{64} + ( -\beta_{7} + 7 \beta_{11} + \beta_{13} - 4 \beta_{14} ) q^{67} + ( -12 \beta_{1} + 4 \beta_{4} - 4 \beta_{10} + 4 \beta_{12} ) q^{68} + ( 3 \beta_{6} + 4 \beta_{8} + 6 \beta_{9} - 2 \beta_{15} ) q^{71} + ( 4 \beta_{7} - 10 \beta_{13} ) q^{73} + ( 4 \beta_{6} - 2 \beta_{8} + 4 \beta_{9} ) q^{74} + ( -28 - \beta_{2} - 2 \beta_{3} - 5 \beta_{5} ) q^{76} + ( 34 \beta_{1} - \beta_{4} + 7 \beta_{10} + \beta_{12} ) q^{77} + ( -6 + 12 \beta_{2} + 8 \beta_{3} - 6 \beta_{5} ) q^{79} + ( 11 \beta_{7} + 2 \beta_{11} - \beta_{13} - 4 \beta_{14} ) q^{82} + ( -22 \beta_{1} - 10 \beta_{4} + 4 \beta_{10} - 10 \beta_{12} ) q^{83} + ( 3 \beta_{6} + 2 \beta_{8} - 3 \beta_{9} + 4 \beta_{15} ) q^{86} + ( 8 \beta_{7} - 4 \beta_{13} + 4 \beta_{14} ) q^{88} + ( -4 \beta_{6} + 8 \beta_{8} - 8 \beta_{9} ) q^{89} + ( -5 + 10 \beta_{2} + 7 \beta_{3} - 5 \beta_{5} ) q^{91} + ( -24 \beta_{1} - 8 \beta_{4} - 12 \beta_{10} - 12 \beta_{12} ) q^{92} + ( 42 + 12 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} ) q^{94} -3 \beta_{7} q^{97} + ( -18 \beta_{1} + 8 \beta_{4} - 8 \beta_{10} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 8q^{4} + O(q^{10})$$ $$16q - 8q^{4} - 16q^{16} + 160q^{34} + 256q^{46} + 160q^{49} + 80q^{61} - 320q^{64} - 456q^{76} + 768q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{12} + 25 x^{8} - 16 x^{4} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3 \nu^{15} + 19 \nu^{11} - 27 \nu^{7} + 192 \nu^{3} - 448 \nu$$$$)/448$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{12} - 3 \nu^{8} - 9 \nu^{4} - 34$$$$)/14$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{12} - 19 \nu^{8} + 139 \nu^{4} - 248$$$$)/56$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{15} + 32 \nu^{13} + 19 \nu^{11} + 96 \nu^{9} - 27 \nu^{7} + 288 \nu^{5} + 1088 \nu^{3} + 1088 \nu$$$$)/448$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{12} + 19 \nu^{8} - 27 \nu^{4} + 220$$$$)/28$$ $$\beta_{6}$$ $$=$$ $$($$$$13 \nu^{15} - 16 \nu^{13} + 67 \nu^{11} - 48 \nu^{9} + 117 \nu^{7} + 304 \nu^{5} + 1408 \nu^{3} - 320 \nu$$$$)/448$$ $$\beta_{7}$$ $$=$$ $$($$$$-15 \nu^{14} + 95 \nu^{10} - 135 \nu^{6} + 960 \nu^{2}$$$$)/448$$ $$\beta_{8}$$ $$=$$ $$($$$$-15 \nu^{15} + 95 \nu^{11} - 135 \nu^{7} + 960 \nu^{3} + 2240 \nu$$$$)/448$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{15} - 5 \nu^{13} - 3 \nu^{11} + 13 \nu^{9} + 19 \nu^{7} - 101 \nu^{5} - 20 \nu^{3} + 208 \nu$$$$)/56$$ $$\beta_{10}$$ $$=$$ $$($$$$-11 \nu^{15} - 24 \nu^{13} - 5 \nu^{11} + 152 \nu^{9} - 323 \nu^{7} - 216 \nu^{5} + 32 \nu^{3} + 1984 \nu$$$$)/448$$ $$\beta_{11}$$ $$=$$ $$($$$$3 \nu^{14} + 13 \nu^{10} - 5 \nu^{6} + 224 \nu^{2}$$$$)/64$$ $$\beta_{12}$$ $$=$$ $$($$$$19 \nu^{15} - 16 \nu^{13} + 29 \nu^{11} - 48 \nu^{9} + 171 \nu^{7} - 592 \nu^{5} + 128 \nu^{3} - 320 \nu$$$$)/448$$ $$\beta_{13}$$ $$=$$ $$($$$$25 \nu^{14} - 9 \nu^{10} + 673 \nu^{6} + 1536 \nu^{2}$$$$)/448$$ $$\beta_{14}$$ $$=$$ $$($$$$-25 \nu^{14} + 9 \nu^{10} - 673 \nu^{6} + 2944 \nu^{2}$$$$)/448$$ $$\beta_{15}$$ $$=$$ $$($$$$-19 \nu^{15} - 16 \nu^{13} - 29 \nu^{11} - 272 \nu^{9} - 395 \nu^{7} - 368 \nu^{5} + 768 \nu^{3} - 3904 \nu$$$$)/224$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{8} - 5 \beta_{1}$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{14} + \beta_{13}$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{15} + 2 \beta_{9} - \beta_{8} + 3 \beta_{6} + 5 \beta_{4} - 5 \beta_{1}$$$$)/20$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{5} + 2 \beta_{3} + 1$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{15} - 10 \beta_{12} - 2 \beta_{9} - \beta_{8} + 7 \beta_{6} - 5 \beta_{4} - 5 \beta_{1}$$$$)/20$$ $$\nu^{6}$$ $$=$$ $$($$$$-4 \beta_{14} + 11 \beta_{13} - 10 \beta_{11} + 11 \beta_{7}$$$$)/20$$ $$\nu^{7}$$ $$=$$ $$($$$$-\beta_{15} - 5 \beta_{12} - 10 \beta_{10} + 8 \beta_{9} + 2 \beta_{6} + 5 \beta_{1}$$$$)/10$$ $$\nu^{8}$$ $$=$$ $$($$$$2 \beta_{5} - 3 \beta_{2} - 23$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-7 \beta_{15} + 20 \beta_{10} + 6 \beta_{9} - 29 \beta_{8} - \beta_{6} + 15 \beta_{4} + 125 \beta_{1}$$$$)/20$$ $$\nu^{10}$$ $$=$$ $$($$$$-30 \beta_{14} - 15 \beta_{13} + 30 \beta_{11} + 67 \beta_{7}$$$$)/20$$ $$\nu^{11}$$ $$=$$ $$($$$$-15 \beta_{15} + 30 \beta_{12} - 30 \beta_{9} + 41 \beta_{8} - 15 \beta_{6} - 45 \beta_{4} + 235 \beta_{1}$$$$)/20$$ $$\nu^{12}$$ $$=$$ $$($$$$-21 \beta_{5} - 18 \beta_{3} - 38 \beta_{2} - 7$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$\beta_{15} + 45 \beta_{12} - 30 \beta_{10} - 28 \beta_{9} + 14 \beta_{8} - 72 \beta_{6} + 70 \beta_{4} + 5 \beta_{1}$$$$)/10$$ $$\nu^{14}$$ $$=$$ $$($$$$-13 \beta_{14} - 33 \beta_{13} + 140 \beta_{11} - 136 \beta_{7}$$$$)/10$$ $$\nu^{15}$$ $$=$$ $$($$$$-13 \beta_{15} + 280 \beta_{12} + 180 \beta_{10} - 206 \beta_{9} - 103 \beta_{8} + 61 \beta_{6} + 35 \beta_{4} - 415 \beta_{1}$$$$)/20$$

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
 −0.404496 − 1.35513i −1.35513 + 0.404496i −0.404496 + 1.35513i −1.35513 − 0.404496i −1.26588 − 0.630504i 0.630504 − 1.26588i −1.26588 + 0.630504i 0.630504 + 1.26588i 1.26588 − 0.630504i −0.630504 − 1.26588i 1.26588 + 0.630504i −0.630504 + 1.26588i 0.404496 − 1.35513i 1.35513 + 0.404496i 0.404496 + 1.35513i 1.35513 − 0.404496i
−1.75963 0.950636i 0 2.19258 + 3.34553i 0 0 −3.98982 −0.677747 7.97124i 0 0
199.2 −1.75963 0.950636i 0 2.19258 + 3.34553i 0 0 3.98982 −0.677747 7.97124i 0 0
199.3 −1.75963 + 0.950636i 0 2.19258 3.34553i 0 0 −3.98982 −0.677747 + 7.97124i 0 0
199.4 −1.75963 + 0.950636i 0 2.19258 3.34553i 0 0 3.98982 −0.677747 + 7.97124i 0 0
199.5 −0.635381 1.89639i 0 −3.19258 + 2.40986i 0 0 −10.1035 6.59853 + 4.52320i 0 0
199.6 −0.635381 1.89639i 0 −3.19258 + 2.40986i 0 0 10.1035 6.59853 + 4.52320i 0 0
199.7 −0.635381 + 1.89639i 0 −3.19258 2.40986i 0 0 −10.1035 6.59853 4.52320i 0 0
199.8 −0.635381 + 1.89639i 0 −3.19258 2.40986i 0 0 10.1035 6.59853 4.52320i 0 0
199.9 0.635381 1.89639i 0 −3.19258 2.40986i 0 0 −10.1035 −6.59853 + 4.52320i 0 0
199.10 0.635381 1.89639i 0 −3.19258 2.40986i 0 0 10.1035 −6.59853 + 4.52320i 0 0
199.11 0.635381 + 1.89639i 0 −3.19258 + 2.40986i 0 0 −10.1035 −6.59853 4.52320i 0 0
199.12 0.635381 + 1.89639i 0 −3.19258 + 2.40986i 0 0 10.1035 −6.59853 4.52320i 0 0
199.13 1.75963 0.950636i 0 2.19258 3.34553i 0 0 −3.98982 0.677747 7.97124i 0 0
199.14 1.75963 0.950636i 0 2.19258 3.34553i 0 0 3.98982 0.677747 7.97124i 0 0
199.15 1.75963 + 0.950636i 0 2.19258 + 3.34553i 0 0 −3.98982 0.677747 + 7.97124i 0 0
199.16 1.75963 + 0.950636i 0 2.19258 + 3.34553i 0 0 3.98982 0.677747 + 7.97124i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 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.g 16
3.b odd 2 1 inner 900.3.f.g 16
4.b odd 2 1 inner 900.3.f.g 16
5.b even 2 1 inner 900.3.f.g 16
5.c odd 4 1 900.3.c.p 8
5.c odd 4 1 900.3.c.q yes 8
12.b even 2 1 inner 900.3.f.g 16
15.d odd 2 1 inner 900.3.f.g 16
15.e even 4 1 900.3.c.p 8
15.e even 4 1 900.3.c.q yes 8
20.d odd 2 1 inner 900.3.f.g 16
20.e even 4 1 900.3.c.p 8
20.e even 4 1 900.3.c.q yes 8
60.h even 2 1 inner 900.3.f.g 16
60.l odd 4 1 900.3.c.p 8
60.l odd 4 1 900.3.c.q yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.3.c.p 8 5.c odd 4 1
900.3.c.p 8 15.e even 4 1
900.3.c.p 8 20.e even 4 1
900.3.c.p 8 60.l odd 4 1
900.3.c.q yes 8 5.c odd 4 1
900.3.c.q yes 8 15.e even 4 1
900.3.c.q yes 8 20.e even 4 1
900.3.c.q yes 8 60.l odd 4 1
900.3.f.g 16 1.a even 1 1 trivial
900.3.f.g 16 3.b odd 2 1 inner
900.3.f.g 16 4.b odd 2 1 inner
900.3.f.g 16 5.b even 2 1 inner
900.3.f.g 16 12.b even 2 1 inner
900.3.f.g 16 15.d odd 2 1 inner
900.3.f.g 16 20.d odd 2 1 inner
900.3.f.g 16 60.h even 2 1 inner

Hecke kernels

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

 $$T_{7}^{4} - 118 T_{7}^{2} + 1625$$ $$T_{29}^{4} - 3048 T_{29}^{2} + 1019200$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 256 + 32 T^{2} + 4 T^{4} + 2 T^{6} + T^{8} )^{2}$$
$3$ $$T^{16}$$
$5$ $$T^{16}$$
$7$ $$( 1625 - 118 T^{2} + T^{4} )^{4}$$
$11$ $$( 8000 + 184 T^{2} + T^{4} )^{4}$$
$13$ $$( 13225 + 234 T^{2} + T^{4} )^{4}$$
$17$ $$( 40768 + 424 T^{2} + T^{4} )^{4}$$
$19$ $$( 65 + 302 T^{2} + T^{4} )^{4}$$
$23$ $$( 15680 - 1784 T^{2} + T^{4} )^{4}$$
$29$ $$( 1019200 - 3048 T^{2} + T^{4} )^{4}$$
$31$ $$( 79625 + 1030 T^{2} + T^{4} )^{4}$$
$37$ $$( 211600 + 936 T^{2} + T^{4} )^{4}$$
$41$ $$( 3515200 - 3752 T^{2} + T^{4} )^{4}$$
$43$ $$( 13456625 - 7358 T^{2} + T^{4} )^{4}$$
$47$ $$( 2708480 - 3296 T^{2} + T^{4} )^{4}$$
$53$ $$( 2896192 + 6888 T^{2} + T^{4} )^{4}$$
$59$ $$( 10368000 + 6624 T^{2} + T^{4} )^{4}$$
$61$ $$( -5659 - 10 T + T^{2} )^{8}$$
$67$ $$( 4564625 - 9502 T^{2} + T^{4} )^{4}$$
$71$ $$( 66248000 + 21304 T^{2} + T^{4} )^{4}$$
$73$ $$( 114490000 + 25000 T^{2} + T^{4} )^{4}$$
$79$ $$( 44994560 + 21568 T^{2} + T^{4} )^{4}$$
$83$ $$( 146232320 - 27104 T^{2} + T^{4} )^{4}$$
$89$ $$( 5324800 - 16512 T^{2} + T^{4} )^{4}$$
$97$ $$( 225 + T^{2} )^{8}$$