# Properties

 Label 900.3.c.n 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + ( -1 + \beta_{7} ) q^{4} + ( -\beta_{5} - \beta_{6} - \beta_{7} ) q^{7} + ( -3 - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{8} +O(q^{10})$$ $$q + \beta_{4} q^{2} + ( -1 + \beta_{7} ) q^{4} + ( -\beta_{5} - \beta_{6} - \beta_{7} ) q^{7} + ( -3 - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{8} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{11} + ( 2 - \beta_{1} + 4 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{13} + ( -2 - \beta_{1} + \beta_{2} + 4 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{7} ) q^{14} + ( 3 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{16} + ( 2 - 2 \beta_{1} + 5 \beta_{2} - \beta_{3} + 6 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{17} + ( -1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 7 \beta_{4} - 4 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{19} + ( -1 - \beta_{1} - 2 \beta_{2} - 8 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} + 3 \beta_{7} ) q^{22} + ( -1 - \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 5 \beta_{4} - 5 \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{23} + ( 10 + 2 \beta_{2} + 5 \beta_{4} + 8 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{26} + ( 14 - \beta_{1} + 4 \beta_{2} + \beta_{4} + 7 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{28} + ( 6 - 2 \beta_{1} + 7 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 5 \beta_{7} ) q^{29} + ( -4 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - 7 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{31} + ( -12 + 2 \beta_{1} + 6 \beta_{3} + 6 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} ) q^{32} + ( 15 - 3 \beta_{1} + \beta_{4} - 15 \beta_{5} + 5 \beta_{7} ) q^{34} + ( -24 + 2 \beta_{1} - 10 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 8 \beta_{7} ) q^{37} + ( 20 - \beta_{1} - 5 \beta_{2} + \beta_{4} + 3 \beta_{5} - 5 \beta_{7} ) q^{38} + ( 4 - 2 \beta_{1} - 5 \beta_{2} + 9 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 9 \beta_{7} ) q^{41} + ( 3 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} - 8 \beta_{5} + 3 \beta_{6} + 4 \beta_{7} ) q^{43} + ( -4 + 2 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} - 10 \beta_{4} - 22 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{44} + ( -11 + 5 \beta_{1} - 6 \beta_{2} - 5 \beta_{4} - 15 \beta_{5} + \beta_{7} ) q^{46} + ( -2 - 2 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} + 10 \beta_{4} + 12 \beta_{5} + 4 \beta_{6} + 10 \beta_{7} ) q^{47} + ( 2 - 6 \beta_{2} + 14 \beta_{3} - 8 \beta_{4} - 4 \beta_{6} - 2 \beta_{7} ) q^{49} + ( 7 + 2 \beta_{1} - 16 \beta_{3} + 14 \beta_{4} - 14 \beta_{5} + 9 \beta_{7} ) q^{52} + ( 38 - \beta_{2} - 7 \beta_{3} + 8 \beta_{4} + 4 \beta_{6} - 5 \beta_{7} ) q^{53} + ( 23 - 11 \beta_{3} + 17 \beta_{4} - 14 \beta_{5} - \beta_{6} + 4 \beta_{7} ) q^{56} + ( -1 - 7 \beta_{1} + 4 \beta_{2} + 5 \beta_{4} - 19 \beta_{5} + 8 \beta_{6} + \beta_{7} ) q^{58} + ( -4 - 4 \beta_{1} - 12 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} ) q^{59} + ( 24 - 7 \beta_{1} + 14 \beta_{2} - 2 \beta_{3} + 23 \beta_{4} + 7 \beta_{5} + 8 \beta_{6} - \beta_{7} ) q^{61} + ( 28 + 7 \beta_{1} - 9 \beta_{2} + 4 \beta_{3} - 9 \beta_{4} - 21 \beta_{5} - 9 \beta_{7} ) q^{62} + ( 2 - 2 \beta_{1} + 8 \beta_{2} - 6 \beta_{4} + 30 \beta_{5} + 6 \beta_{7} ) q^{64} + ( 1 + \beta_{1} + 14 \beta_{2} + 14 \beta_{3} - 25 \beta_{4} + 14 \beta_{5} + \beta_{6} - 12 \beta_{7} ) q^{67} + ( -10 - 12 \beta_{2} + 4 \beta_{3} + 12 \beta_{4} - 16 \beta_{5} + 8 \beta_{6} - 2 \beta_{7} ) q^{68} + ( -5 - 5 \beta_{1} - 15 \beta_{4} + \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{71} + ( 20 + 10 \beta_{1} - 18 \beta_{2} - 14 \beta_{3} - 18 \beta_{4} - 10 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{73} + ( 10 + 10 \beta_{1} - 4 \beta_{2} - 20 \beta_{4} + 34 \beta_{5} - 8 \beta_{6} + 2 \beta_{7} ) q^{74} + ( 24 + 5 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} + 19 \beta_{4} + 13 \beta_{5} - 4 \beta_{6} ) q^{76} + ( 46 + 2 \beta_{1} + 3 \beta_{2} - 19 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{77} + ( 2 + 2 \beta_{1} + 12 \beta_{2} + 12 \beta_{3} - 18 \beta_{4} - 6 \beta_{5} - 10 \beta_{7} ) q^{79} + ( 45 + 7 \beta_{1} + 13 \beta_{4} + 35 \beta_{5} + 15 \beta_{7} ) q^{82} + ( 4 + 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 20 \beta_{4} + 18 \beta_{5} + 12 \beta_{6} + 20 \beta_{7} ) q^{83} + ( -34 + \beta_{1} - 17 \beta_{2} - 24 \beta_{3} + 11 \beta_{4} - 3 \beta_{5} + 13 \beta_{7} ) q^{86} + ( -30 - 8 \beta_{1} - 16 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 28 \beta_{5} + 6 \beta_{6} - 4 \beta_{7} ) q^{88} + ( -16 + 4 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - 4 \beta_{6} + 8 \beta_{7} ) q^{89} + ( -3 - 3 \beta_{1} + 14 \beta_{2} + 14 \beta_{3} - 37 \beta_{4} + 48 \beta_{5} + 15 \beta_{6} - 2 \beta_{7} ) q^{91} + ( -48 + 6 \beta_{1} - 20 \beta_{2} - 16 \beta_{3} - 6 \beta_{4} + 14 \beta_{5} - 4 \beta_{6} ) q^{92} + ( 2 + 12 \beta_{1} + 6 \beta_{2} - 8 \beta_{3} - 8 \beta_{4} - 36 \beta_{5} + 4 \beta_{7} ) q^{94} + ( 35 - 8 \beta_{1} + 4 \beta_{2} + 12 \beta_{3} + 24 \beta_{4} + 8 \beta_{5} + 8 \beta_{6} - 12 \beta_{7} ) q^{97} + ( -2 + 2 \beta_{1} + 8 \beta_{2} + 12 \beta_{4} + 42 \beta_{5} + 16 \beta_{6} + 2 \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 $$x^{8} - 7 x^{6} + 50 x^{4} - 84 x^{3} + 55 x^{2} - 12 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-73 \nu^{7} - 599 \nu^{6} - 16 \nu^{5} + 3897 \nu^{4} + 291 \nu^{3} - 20680 \nu^{2} + 20620 \nu - 5669$$$$)/515$$ $$\beta_{2}$$ $$=$$ $$($$$$-291 \nu^{7} + 32 \nu^{6} + 1968 \nu^{5} - 381 \nu^{4} - 14163 \nu^{3} + 27350 \nu^{2} - 21000 \nu + 3582$$$$)/515$$ $$\beta_{3}$$ $$=$$ $$($$$$347 \nu^{7} - 24 \nu^{6} - 2506 \nu^{5} + 157 \nu^{4} + 17961 \nu^{3} - 30040 \nu^{2} + 17810 \nu - 1399$$$$)/515$$ $$\beta_{4}$$ $$=$$ $$($$$$347 \nu^{7} - 24 \nu^{6} - 2506 \nu^{5} + 157 \nu^{4} + 17961 \nu^{3} - 30040 \nu^{2} + 16780 \nu - 1399$$$$)/515$$ $$\beta_{5}$$ $$=$$ $$($$$$-486 \nu^{7} - 143 \nu^{6} + 3308 \nu^{5} + 914 \nu^{4} - 23728 \nu^{3} + 34050 \nu^{2} - 19070 \nu + 2372$$$$)/515$$ $$\beta_{6}$$ $$=$$ $$($$$$924 \nu^{7} + 132 \nu^{6} - 6302 \nu^{5} - 606 \nu^{4} + 45672 \nu^{3} - 72710 \nu^{2} + 44700 \nu - 5953$$$$)/515$$ $$\beta_{7}$$ $$=$$ $$($$$$1134 \nu^{7} + 162 \nu^{6} - 8062 \nu^{5} - 1446 \nu^{4} + 57082 \nu^{3} - 85630 \nu^{2} + 45870 \nu - 5363$$$$)/515$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{4} + \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} - 2 \beta_{6} - 7 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{1} + 7$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + 4 \beta_{6} + 6 \beta_{5} - 6 \beta_{4} + 3 \beta_{3} + 3 \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-7 \beta_{7} - 11 \beta_{6} - 22 \beta_{5} + 11 \beta_{4} + 7 \beta_{3} - 4 \beta_{2} - 22$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$19 \beta_{7} + 58 \beta_{6} + 55 \beta_{5} - 19 \beta_{4} - 88 \beta_{3} + 36 \beta_{2} + 11 \beta_{1} + 195$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-25 \beta_{7} - 26 \beta_{6} + 29 \beta_{5} - 23 \beta_{4} + 190 \beta_{3} - 22 \beta_{2} - 29 \beta_{1} - 407$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-25 \beta_{7} - 81 \beta_{6} - 403 \beta_{5} + 330 \beta_{4} - 598 \beta_{3} - 25 \beta_{2} + 81 \beta_{1} + 997$$$$)/2$$

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

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