# Properties

 Label 900.3.c.u Level $900$ Weight $3$ Character orbit 900.c Analytic conductor $24.523$ 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: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 900.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.85100625.1 Defining polynomial: $$x^{8} - x^{7} - 2 x^{6} + x^{5} + 3 x^{4} + 2 x^{3} - 8 x^{2} - 8 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 60) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{5} ) q^{2} + ( 1 - \beta_{4} ) q^{4} + ( -1 - \beta_{3} + \beta_{5} + \beta_{6} ) q^{7} + ( -3 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{5} ) q^{2} + ( 1 - \beta_{4} ) q^{4} + ( -1 - \beta_{3} + \beta_{5} + \beta_{6} ) q^{7} + ( -3 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{8} + ( 1 - \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + ( -1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{13} + ( 2 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{6} - \beta_{7} ) q^{14} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{16} + ( 1 - 4 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{17} + ( 4 - 5 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{19} + ( -10 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 4 \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{22} + ( -8 + 5 \beta_{1} + \beta_{2} - 8 \beta_{3} + \beta_{4} + 6 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{23} + ( 4 - \beta_{1} - 6 \beta_{2} + \beta_{3} + 4 \beta_{5} - 3 \beta_{7} ) q^{26} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{28} + ( -5 + 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 9 \beta_{5} + 2 \beta_{6} ) q^{29} + ( 3 \beta_{1} - 5 \beta_{2} + 8 \beta_{3} - 5 \beta_{4} - \beta_{6} - 3 \beta_{7} ) q^{31} + ( -7 + 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 7 \beta_{5} + \beta_{7} ) q^{32} + ( -4 - 3 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 3 \beta_{7} ) q^{34} + ( 17 + \beta_{1} - 8 \beta_{2} + 4 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{37} + ( -4 - 2 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} - 4 \beta_{4} - 4 \beta_{6} - 2 \beta_{7} ) q^{38} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{41} + ( -3 \beta_{1} + 11 \beta_{3} + 7 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{43} + ( -28 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 8 \beta_{5} + 4 \beta_{6} ) q^{44} + ( 16 + 10 \beta_{1} - 8 \beta_{2} - 10 \beta_{3} + 8 \beta_{4} - 4 \beta_{6} - 2 \beta_{7} ) q^{46} + ( -2 + 5 \beta_{1} - 7 \beta_{2} + 6 \beta_{3} - 7 \beta_{4} + \beta_{6} - 3 \beta_{7} ) q^{47} + ( 5 - 4 \beta_{3} + 8 \beta_{4} - 24 \beta_{5} - 4 \beta_{7} ) q^{49} + ( 26 - 2 \beta_{1} - 14 \beta_{2} + 14 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} ) q^{52} + ( 43 + 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{53} + ( -16 + 6 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} - 10 \beta_{6} - 2 \beta_{7} ) q^{56} + ( 16 + \beta_{1} + 10 \beta_{2} - \beta_{3} - 8 \beta_{4} + 12 \beta_{5} + 7 \beta_{7} ) q^{58} + ( 1 + 4 \beta_{1} - 5 \beta_{2} - 5 \beta_{4} - 6 \beta_{5} + \beta_{6} + \beta_{7} ) q^{59} + ( -28 - 2 \beta_{1} - 8 \beta_{2} - 4 \beta_{4} + 18 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{61} + ( -4 - 2 \beta_{1} - 4 \beta_{2} + 14 \beta_{3} - 4 \beta_{4} + 12 \beta_{6} + 6 \beta_{7} ) q^{62} + ( -19 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - 6 \beta_{4} + 8 \beta_{5} + \beta_{6} + 8 \beta_{7} ) q^{64} + ( -2 + 11 \beta_{1} - 10 \beta_{2} + \beta_{3} - 10 \beta_{4} - 7 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{67} + ( -18 - 10 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} + 8 \beta_{5} + 4 \beta_{7} ) q^{68} + ( -6 + 4 \beta_{1} + 2 \beta_{2} - 16 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{71} + ( 32 - 2 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} - 4 \beta_{4} - 14 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} ) q^{73} + ( 8 - 7 \beta_{1} - 10 \beta_{2} + 7 \beta_{3} + 8 \beta_{4} - 24 \beta_{5} + 11 \beta_{7} ) q^{74} + ( -8 - 12 \beta_{1} + 16 \beta_{2} + 8 \beta_{5} + 12 \beta_{6} ) q^{76} + ( -20 + 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} + 16 \beta_{4} - 22 \beta_{5} + 2 \beta_{6} ) q^{77} + ( -24 + 19 \beta_{1} + \beta_{2} - 10 \beta_{3} + \beta_{4} + 22 \beta_{5} - 13 \beta_{6} - 17 \beta_{7} ) q^{79} + ( -2 + 8 \beta_{2} - 6 \beta_{5} + 8 \beta_{7} ) q^{82} + ( 6 - 15 \beta_{1} + 14 \beta_{2} - 13 \beta_{3} + 14 \beta_{4} - \beta_{5} + 5 \beta_{6} + 10 \beta_{7} ) q^{83} + ( 36 - 16 \beta_{1} + 8 \beta_{2} + 20 \beta_{3} + 8 \beta_{4} + 16 \beta_{6} + 8 \beta_{7} ) q^{86} + ( -28 + 14 \beta_{1} - 10 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} + 18 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} ) q^{88} + ( 12 - 2 \beta_{1} - 24 \beta_{2} - 4 \beta_{3} + 20 \beta_{4} - 22 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{89} + ( 18 - 8 \beta_{1} + 10 \beta_{3} - 18 \beta_{5} - 2 \beta_{6} + 8 \beta_{7} ) q^{91} + ( 36 + 4 \beta_{1} - 24 \beta_{2} - 16 \beta_{3} + 12 \beta_{4} - 24 \beta_{5} - 4 \beta_{6} + 8 \beta_{7} ) q^{92} + ( -12 + 8 \beta_{1} - 12 \beta_{2} + 16 \beta_{3} - 8 \beta_{4} + 8 \beta_{6} + 4 \beta_{7} ) q^{94} + ( -54 + 2 \beta_{1} + 20 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 18 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{97} + ( 73 - 4 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} - 16 \beta_{4} + 23 \beta_{5} + 4 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{2} + 10q^{4} - 20q^{8} + O(q^{10})$$ $$8q + 4q^{2} + 10q^{4} - 20q^{8} - 16q^{13} + 20q^{14} + 34q^{16} - 68q^{22} + 36q^{26} - 28q^{28} - 64q^{29} - 76q^{32} - 92q^{34} + 112q^{37} - 40q^{38} + 16q^{41} - 172q^{44} + 152q^{46} - 56q^{49} + 128q^{52} + 352q^{53} - 116q^{56} + 204q^{58} - 176q^{61} - 56q^{62} - 110q^{64} - 184q^{68} + 240q^{73} - 132q^{74} - 24q^{76} - 288q^{77} - 40q^{82} + 200q^{86} - 140q^{88} - 80q^{89} + 144q^{92} - 96q^{94} - 432q^{97} + 660q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 2 x^{6} + x^{5} + 3 x^{4} + 2 x^{3} - 8 x^{2} - 8 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{2} + 2 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 3 \nu^{6} + 4 \nu^{5} - 9 \nu^{4} - 17 \nu^{3} + 8 \nu^{2} + 24 \nu + 8$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 3 \nu^{6} + 4 \nu^{5} + 7 \nu^{4} - 17 \nu^{3} - 8 \nu^{2} + 24 \nu + 8$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 3 \nu^{6} + 4 \nu^{5} - \nu^{4} - \nu^{3} + 8 \nu$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{7} + \nu^{6} - 4 \nu^{5} - 3 \nu^{4} + 5 \nu^{3} + 24 \nu^{2} - 24 \nu - 40$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$\nu^{7} - \nu^{5} - \nu^{4} + 2 \nu^{3} + 3 \nu^{2} - 9$$ $$\beta_{7}$$ $$=$$ $$($$$$-27 \nu^{7} - 15 \nu^{6} + 44 \nu^{5} + 29 \nu^{4} - 43 \nu^{3} - 88 \nu^{2} + 40 \nu + 296$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - 3 \beta_{5} + \beta_{3} + \beta_{1} + 2$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{6} + 3 \beta_{5} - \beta_{3} + 3 \beta_{1} + 2$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{6} - 3 \beta_{5} + 4 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + \beta_{1} + 6$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{6} + 3 \beta_{5} + 7 \beta_{3} - 8 \beta_{2} + 3 \beta_{1} + 2$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$4 \beta_{7} + 5 \beta_{6} + 9 \beta_{5} + 8 \beta_{4} + \beta_{3} - 3 \beta_{1} - 10$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-4 \beta_{7} - 5 \beta_{6} - \beta_{5} + 12 \beta_{4} + 3 \beta_{3} - 4 \beta_{2} - \beta_{1} + 26$$$$)/8$$ $$\nu^{7}$$ $$=$$ $$($$$$4 \beta_{7} + 13 \beta_{6} + 9 \beta_{5} + 17 \beta_{3} - 11 \beta_{1} + 46$$$$)/8$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 −1.34966 + 0.422403i −1.34966 − 0.422403i 1.40906 + 0.120653i 1.40906 − 0.120653i −0.600040 − 1.28061i −0.600040 + 1.28061i 1.04064 − 0.957636i 1.04064 + 0.957636i
−1.99281 0.169449i 0 3.94257 + 0.675358i 0 0 12.3959i −7.74236 2.01392i 0 0
451.2 −1.99281 + 0.169449i 0 3.94257 0.675358i 0 0 12.3959i −7.74236 + 2.01392i 0 0
451.3 0.438172 1.95141i 0 −3.61601 1.71011i 0 0 6.33166i −4.92155 + 6.30701i 0 0
451.4 0.438172 + 1.95141i 0 −3.61601 + 1.71011i 0 0 6.33166i −4.92155 6.30701i 0 0
451.5 1.67986 1.08539i 0 1.64388 3.64660i 0 0 0.596540i −1.19648 7.91002i 0 0
451.6 1.67986 + 1.08539i 0 1.64388 + 3.64660i 0 0 0.596540i −1.19648 + 7.91002i 0 0
451.7 1.87477 0.696577i 0 3.02956 2.61185i 0 0 5.46770i 3.86039 7.00695i 0 0
451.8 1.87477 + 0.696577i 0 3.02956 + 2.61185i 0 0 5.46770i 3.86039 + 7.00695i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 451.8 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.c.u 8
3.b odd 2 1 300.3.c.d 8
4.b odd 2 1 inner 900.3.c.u 8
5.b even 2 1 180.3.c.b 8
5.c odd 4 2 900.3.f.f 16
12.b even 2 1 300.3.c.d 8
15.d odd 2 1 60.3.c.a 8
15.e even 4 2 300.3.f.b 16
20.d odd 2 1 180.3.c.b 8
20.e even 4 2 900.3.f.f 16
40.e odd 2 1 2880.3.e.j 8
40.f even 2 1 2880.3.e.j 8
60.h even 2 1 60.3.c.a 8
60.l odd 4 2 300.3.f.b 16
120.i odd 2 1 960.3.e.c 8
120.m even 2 1 960.3.e.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.c.a 8 15.d odd 2 1
60.3.c.a 8 60.h even 2 1
180.3.c.b 8 5.b even 2 1
180.3.c.b 8 20.d odd 2 1
300.3.c.d 8 3.b odd 2 1
300.3.c.d 8 12.b even 2 1
300.3.f.b 16 15.e even 4 2
300.3.f.b 16 60.l odd 4 2
900.3.c.u 8 1.a even 1 1 trivial
900.3.c.u 8 4.b odd 2 1 inner
900.3.f.f 16 5.c odd 4 2
900.3.f.f 16 20.e even 4 2
960.3.e.c 8 120.i odd 2 1
960.3.e.c 8 120.m even 2 1
2880.3.e.j 8 40.e odd 2 1
2880.3.e.j 8 40.f even 2 1

## Hecke kernels

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

 $$T_{7}^{8} + 224 T_{7}^{6} + 12032 T_{7}^{4} + 188416 T_{7}^{2} + 65536$$ $$T_{13}^{4} + 8 T_{13}^{3} - 472 T_{13}^{2} - 5792 T_{13} - 12464$$ $$T_{17}^{4} - 424 T_{17}^{2} + 3840 T_{17} - 8816$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 - 256 T + 48 T^{2} + 64 T^{3} - 52 T^{4} + 16 T^{5} + 3 T^{6} - 4 T^{7} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$65536 + 188416 T^{2} + 12032 T^{4} + 224 T^{6} + T^{8}$$
$11$ $$( 10496 + 208 T^{2} + T^{4} )^{2}$$
$13$ $$( -12464 - 5792 T - 472 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$17$ $$( -8816 + 3840 T - 424 T^{2} + T^{4} )^{2}$$
$19$ $$6544162816 + 173686784 T^{2} + 925952 T^{4} + 1696 T^{6} + T^{8}$$
$23$ $$101419319296 + 1884176384 T^{2} + 4397312 T^{4} + 3616 T^{6} + T^{8}$$
$29$ $$( 1334416 - 34688 T - 2152 T^{2} + 32 T^{3} + T^{4} )^{2}$$
$31$ $$59895709696 + 2731491328 T^{2} + 7432448 T^{4} + 5408 T^{6} + T^{8}$$
$37$ $$( -244784 + 55136 T - 1528 T^{2} - 56 T^{3} + T^{4} )^{2}$$
$41$ $$( 87184 + 7264 T - 1800 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$43$ $$33624411406336 + 62108155904 T^{2} + 40259072 T^{4} + 10816 T^{6} + T^{8}$$
$47$ $$1056981385216 + 9701752832 T^{2} + 15726848 T^{4} + 8032 T^{6} + T^{8}$$
$53$ $$( -478064 - 161344 T + 9752 T^{2} - 176 T^{3} + T^{4} )^{2}$$
$59$ $$173909016576 + 2174459904 T^{2} + 6273792 T^{4} + 4896 T^{6} + T^{8}$$
$61$ $$( -2142704 - 273568 T - 2536 T^{2} + 88 T^{3} + T^{4} )^{2}$$
$67$ $$281086590976 + 15044755456 T^{2} + 57554432 T^{4} + 16064 T^{6} + T^{8}$$
$71$ $$16079971680256 + 101402017792 T^{2} + 64237568 T^{4} + 13952 T^{6} + T^{8}$$
$73$ $$( 4962064 + 325920 T - 1576 T^{2} - 120 T^{3} + T^{4} )^{2}$$
$79$ $$3198642669223936 + 2420601929728 T^{2} + 550899968 T^{4} + 41888 T^{6} + T^{8}$$
$83$ $$4284940379815936 + 2381453017088 T^{2} + 464465408 T^{4} + 36928 T^{6} + T^{8}$$
$89$ $$( 70652944 - 757600 T - 20584 T^{2} + 40 T^{3} + T^{4} )^{2}$$
$97$ $$( -59281776 - 1154592 T + 5880 T^{2} + 216 T^{3} + T^{4} )^{2}$$