# Properties

 Label 900.3.c.t 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: 8.0.4069419264.1 Defining polynomial: $$x^{8} - 7 x^{6} + 50 x^{4} - 84 x^{3} + 55 x^{2} - 12 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 300) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( -1 + \beta_{2} - \beta_{3} + \beta_{5} ) q^{4} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{7} + ( 3 - \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{7} ) q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( -1 + \beta_{2} - \beta_{3} + \beta_{5} ) q^{4} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{7} + ( 3 - \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{7} ) q^{8} + ( 1 + \beta_{1} - 2 \beta_{3} + \beta_{4} + 3 \beta_{6} + 2 \beta_{7} ) q^{11} + ( -2 + \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{13} + ( -2 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{7} ) q^{14} + ( 3 + \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{16} + ( -2 + 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{17} + ( -1 - \beta_{1} + 6 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{19} + ( 1 + \beta_{1} + \beta_{2} - 5 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{22} + ( 1 + \beta_{1} + 5 \beta_{4} - 4 \beta_{5} + 5 \beta_{6} ) q^{23} + ( 10 - \beta_{2} - 4 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{26} + ( -14 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 7 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{28} + ( 6 - 2 \beta_{1} + 5 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} ) q^{29} + ( -4 - 4 \beta_{1} + 11 \beta_{2} + \beta_{3} - 7 \beta_{4} + 3 \beta_{5} - 8 \beta_{6} - \beta_{7} ) q^{31} + ( 12 - 2 \beta_{1} + 8 \beta_{2} - 6 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} ) q^{32} + ( 15 - 3 \beta_{1} + 7 \beta_{2} - 5 \beta_{3} - 15 \beta_{4} + 5 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{34} + ( 24 - 2 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + 8 \beta_{5} + 8 \beta_{6} - 2 \beta_{7} ) q^{37} + ( -20 + \beta_{1} + 5 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} + 6 \beta_{6} + \beta_{7} ) q^{38} + ( 4 - 2 \beta_{1} - 13 \beta_{2} + 2 \beta_{4} - 9 \beta_{5} - 7 \beta_{6} ) q^{41} + ( -3 - 3 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} + 8 \beta_{4} - 4 \beta_{5} - 5 \beta_{6} - 6 \beta_{7} ) q^{43} + ( -4 + 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - 22 \beta_{4} - 4 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} ) q^{44} + ( -11 + 5 \beta_{1} + \beta_{2} - \beta_{3} - 15 \beta_{4} + \beta_{5} - \beta_{6} + 5 \beta_{7} ) q^{46} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 12 \beta_{4} - 10 \beta_{5} + 10 \beta_{6} - 2 \beta_{7} ) q^{47} + ( 2 + 6 \beta_{2} - 12 \beta_{3} - 2 \beta_{5} - 6 \beta_{6} - 4 \beta_{7} ) q^{49} + ( -7 - 2 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} + 14 \beta_{4} - 9 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{52} + ( -38 + 13 \beta_{2} - 12 \beta_{3} + 5 \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{53} + ( 23 - 13 \beta_{2} + 7 \beta_{3} - 14 \beta_{4} + 4 \beta_{5} - \beta_{7} ) q^{56} + ( 1 + 7 \beta_{1} - 3 \beta_{2} + \beta_{3} + 19 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{58} + ( -4 - 4 \beta_{1} + 16 \beta_{2} - 2 \beta_{4} - 4 \beta_{6} ) q^{59} + ( 24 - 7 \beta_{1} - 17 \beta_{2} + 3 \beta_{3} + 7 \beta_{4} - \beta_{5} + 7 \beta_{6} + \beta_{7} ) q^{61} + ( -28 - 7 \beta_{1} + 7 \beta_{2} - 5 \beta_{3} + 21 \beta_{4} + 9 \beta_{5} + 2 \beta_{6} - 7 \beta_{7} ) q^{62} + ( 2 - 2 \beta_{1} + 14 \beta_{2} - 6 \beta_{3} + 30 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} ) q^{64} + ( -1 - \beta_{1} - 12 \beta_{2} + 2 \beta_{3} - 14 \beta_{4} + 12 \beta_{5} - 15 \beta_{6} - 2 \beta_{7} ) q^{67} + ( 10 + 14 \beta_{2} + 2 \beta_{3} + 16 \beta_{4} + 2 \beta_{5} + 12 \beta_{6} - 8 \beta_{7} ) q^{68} + ( -5 - 5 \beta_{1} + 18 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} ) q^{71} + ( -20 - 10 \beta_{1} - 4 \beta_{2} - 18 \beta_{3} + 10 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} - 6 \beta_{7} ) q^{73} + ( 10 + 10 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} + 34 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} + 2 \beta_{7} ) q^{74} + ( 24 + 5 \beta_{1} - 24 \beta_{2} - 8 \beta_{3} + 13 \beta_{4} + 9 \beta_{6} + \beta_{7} ) q^{76} + ( -46 - 2 \beta_{1} + 7 \beta_{2} - 18 \beta_{3} + 2 \beta_{4} - \beta_{5} - 5 \beta_{6} - 6 \beta_{7} ) q^{77} + ( 2 + 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} - 10 \beta_{5} + 14 \beta_{6} + 2 \beta_{7} ) q^{79} + ( -45 - 7 \beta_{1} + 5 \beta_{2} + 15 \beta_{3} - 35 \beta_{4} - 15 \beta_{5} - 7 \beta_{6} - 7 \beta_{7} ) q^{82} + ( -4 - 4 \beta_{1} + 4 \beta_{2} + 16 \beta_{3} - 18 \beta_{4} - 20 \beta_{5} - 16 \beta_{7} ) q^{83} + ( -34 + \beta_{1} + \beta_{2} + 11 \beta_{3} - 3 \beta_{4} + 13 \beta_{5} - 16 \beta_{6} + \beta_{7} ) q^{86} + ( 30 + 8 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 28 \beta_{4} + 4 \beta_{5} + 24 \beta_{6} + 2 \beta_{7} ) q^{88} + ( -16 + 16 \beta_{2} - 12 \beta_{3} + 8 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{89} + ( -3 - 3 \beta_{1} + 38 \beta_{2} - 12 \beta_{3} + 48 \beta_{4} - 2 \beta_{5} + 11 \beta_{6} + 12 \beta_{7} ) q^{91} + ( 48 - 6 \beta_{1} - 16 \beta_{3} - 14 \beta_{4} + 14 \beta_{6} - 2 \beta_{7} ) q^{92} + ( 2 + 12 \beta_{1} + 4 \beta_{3} - 36 \beta_{4} + 4 \beta_{5} + 18 \beta_{6} + 12 \beta_{7} ) q^{94} + ( -35 + 8 \beta_{1} + 28 \beta_{2} - 8 \beta_{4} + 12 \beta_{5} + 4 \beta_{6} ) q^{97} + ( 2 - 2 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} - 42 \beta_{4} - 2 \beta_{5} - 10 \beta_{6} - 18 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{2} - 8q^{4} + 20q^{8} + O(q^{10})$$ $$8q + 2q^{2} - 8q^{4} + 20q^{8} - 8q^{13} - 22q^{14} + 40q^{16} - 4q^{22} + 66q^{26} - 104q^{28} + 32q^{29} + 112q^{32} + 124q^{34} + 176q^{37} - 170q^{38} + 16q^{41} - 40q^{44} - 76q^{46} + 16q^{49} - 56q^{52} - 304q^{53} + 172q^{56} + 12q^{58} + 136q^{61} - 238q^{62} + 16q^{64} + 88q^{68} - 240q^{73} + 108q^{74} + 120q^{76} - 384q^{77} - 320q^{82} - 214q^{86} + 200q^{88} - 128q^{89} + 312q^{92} + 12q^{94} - 216q^{97} + 60q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 7 x^{6} + 50 x^{4} - 84 x^{3} + 55 x^{2} - 12 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-71 \nu^{7} + 137 \nu^{6} + 958 \nu^{5} - 746 \nu^{4} - 7188 \nu^{3} + 9940 \nu^{2} + 9875 \nu - 9803$$$$)/1030$$ $$\beta_{2}$$ $$=$$ $$($$$$457 \nu^{7} - 229 \nu^{6} - 3526 \nu^{5} + 1262 \nu^{4} + 24796 \nu^{3} - 47500 \nu^{2} + 30415 \nu - 5259$$$$)/1030$$ $$\beta_{3}$$ $$=$$ $$($$$$743 \nu^{7} - 41 \nu^{6} - 5354 \nu^{5} + 118 \nu^{4} + 38344 \nu^{3} - 62820 \nu^{2} + 36305 \nu - 6231$$$$)/1030$$ $$\beta_{4}$$ $$=$$ $$($$$$486 \nu^{7} + 143 \nu^{6} - 3308 \nu^{5} - 914 \nu^{4} + 23728 \nu^{3} - 34050 \nu^{2} + 19070 \nu - 2372$$$$)/515$$ $$\beta_{5}$$ $$=$$ $$($$$$-638 \nu^{7} + 56 \nu^{6} + 4474 \nu^{5} - 538 \nu^{4} - 32124 \nu^{3} + 57390 \nu^{2} - 37780 \nu + 4981$$$$)/515$$ $$\beta_{6}$$ $$=$$ $$($$$$1811 \nu^{7} + 553 \nu^{6} - 12598 \nu^{5} - 4154 \nu^{4} + 89368 \nu^{3} - 123760 \nu^{2} + 59265 \nu - 5467$$$$)/1030$$ $$\beta_{7}$$ $$=$$ $$($$$$-1277 \nu^{7} - 256 \nu^{6} + 8976 \nu^{5} + 2018 \nu^{4} - 63856 \nu^{3} + 93290 \nu^{2} - 49845 \nu + 5849$$$$)/515$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} - \beta_{6} - \beta_{3}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{5} + 3 \beta_{4} + 2 \beta_{2} + 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{7} - 3 \beta_{6} - \beta_{5} - 6 \beta_{4} + 4 \beta_{3} - 7 \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$11 \beta_{7} - 3 \beta_{6} + 7 \beta_{5} + 51 \beta_{4} - 33 \beta_{3} + 37 \beta_{2} - 7 \beta_{1} - 51$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$8 \beta_{6} - 7 \beta_{5} - 81 \beta_{4} + 138 \beta_{3} - 101 \beta_{2} + 15 \beta_{1} + 199$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-55 \beta_{7} - 51 \beta_{6} - 25 \beta_{5} + 29 \beta_{4} - 165 \beta_{3} + 27 \beta_{2} - 29 \beta_{1} - 407$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$623 \beta_{7} + 411 \beta_{6} + 218 \beta_{5} + 538 \beta_{4} + 623 \beta_{3} + 330 \beta_{2} + 106 \beta_{1} + 1938$$$$)/4$$

## 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
 0.151747 − 0.0876113i 0.151747 + 0.0876113i 0.845613 + 0.488215i 0.845613 − 0.488215i −2.65095 + 1.53053i −2.65095 − 1.53053i 1.65359 − 0.954702i 1.65359 + 0.954702i
−1.33290 1.49110i 0 −0.446749 + 3.97497i 0 0 6.56834i 6.52255 4.63210i 0 0
451.2 −1.33290 + 1.49110i 0 −0.446749 3.97497i 0 0 6.56834i 6.52255 + 4.63210i 0 0
451.3 −0.177680 1.99209i 0 −3.93686 + 0.707911i 0 0 1.19501i 2.10973 + 7.71680i 0 0
451.4 −0.177680 + 1.99209i 0 −3.93686 0.707911i 0 0 1.19501i 2.10973 7.71680i 0 0
451.5 0.534079 1.92737i 0 −3.42952 2.05874i 0 0 11.9716i −5.79958 + 5.51043i 0 0
451.6 0.534079 + 1.92737i 0 −3.42952 + 2.05874i 0 0 11.9716i −5.79958 5.51043i 0 0
451.7 1.97650 0.305673i 0 3.81313 1.20833i 0 0 0.329898i 7.16731 3.55383i 0 0
451.8 1.97650 + 0.305673i 0 3.81313 + 1.20833i 0 0 0.329898i 7.16731 + 3.55383i 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.t 8
3.b odd 2 1 300.3.c.e 8
4.b odd 2 1 inner 900.3.c.t 8
5.b even 2 1 900.3.c.n 8
5.c odd 4 2 900.3.f.h 16
12.b even 2 1 300.3.c.e 8
15.d odd 2 1 300.3.c.g yes 8
15.e even 4 2 300.3.f.c 16
20.d odd 2 1 900.3.c.n 8
20.e even 4 2 900.3.f.h 16
60.h even 2 1 300.3.c.g yes 8
60.l odd 4 2 300.3.f.c 16

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

## Hecke kernels

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

 $$T_{7}^{8} + 188 T_{7}^{6} + 6470 T_{7}^{4} + 9532 T_{7}^{2} + 961$$ $$T_{13}^{4} + 4 T_{13}^{3} - 418 T_{13}^{2} - 3916 T_{13} - 1559$$ $$T_{17}^{4} - 832 T_{17}^{2} + 1920 T_{17} + 132400$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 - 128 T + 96 T^{2} - 64 T^{3} + 20 T^{4} - 16 T^{5} + 6 T^{6} - 2 T^{7} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$961 + 9532 T^{2} + 6470 T^{4} + 188 T^{6} + T^{8}$$
$11$ $$30824704 + 6673408 T^{2} + 135008 T^{4} + 704 T^{6} + T^{8}$$
$13$ $$( -1559 - 3916 T - 418 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$17$ $$( 132400 + 1920 T - 832 T^{2} + T^{4} )^{2}$$
$19$ $$1099651921 + 32129228 T^{2} + 312950 T^{4} + 1132 T^{6} + T^{8}$$
$23$ $$1611540736 + 86660096 T^{2} + 934496 T^{4} + 1984 T^{6} + T^{8}$$
$29$ $$( 93616 + 29824 T - 1600 T^{2} - 16 T^{3} + T^{4} )^{2}$$
$31$ $$819736484449 + 5243206492 T^{2} + 9582374 T^{4} + 5660 T^{6} + T^{8}$$
$37$ $$( -4548464 + 221728 T - 712 T^{2} - 88 T^{3} + T^{4} )^{2}$$
$41$ $$( 3504448 - 37664 T - 4968 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$43$ $$974581609681 + 6860196428 T^{2} + 13866230 T^{4} + 6892 T^{6} + T^{8}$$
$47$ $$13610196640000 + 54677446400 T^{2} + 43512416 T^{4} + 12016 T^{6} + T^{8}$$
$53$ $$( -4946624 - 135392 T + 4568 T^{2} + 152 T^{3} + T^{4} )^{2}$$
$59$ $$195562066176 + 4566129408 T^{2} + 10470240 T^{4} + 6192 T^{6} + T^{8}$$
$61$ $$( 9745129 + 324236 T - 8098 T^{2} - 68 T^{3} + T^{4} )^{2}$$
$67$ $$23066205847441 + 258205504012 T^{2} + 134011190 T^{4} + 21548 T^{6} + T^{8}$$
$71$ $$11235904000000 + 28259891200 T^{2} + 24817856 T^{4} + 8816 T^{6} + T^{8}$$
$73$ $$( -43602032 - 1495200 T - 8584 T^{2} + 120 T^{3} + T^{4} )^{2}$$
$79$ $$2278988775424 + 24693882880 T^{2} + 45364736 T^{4} + 14528 T^{6} + T^{8}$$
$83$ $$120362665464064 + 3496365751040 T^{2} + 748225376 T^{4} + 48688 T^{6} + T^{8}$$
$89$ $$( -1507328 - 237568 T - 3328 T^{2} + 64 T^{3} + T^{4} )^{2}$$
$97$ $$( 15618033 - 971892 T - 10986 T^{2} + 108 T^{3} + T^{4} )^{2}$$