# Properties

 Label 300.2.m.b Level $300$ Weight $2$ Character orbit 300.m Analytic conductor $2.396$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.m (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.26265625.1 Defining polynomial: $$x^{8} - 3 x^{7} + 2 x^{6} + x^{4} + 8 x^{2} - 24 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$5$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + ( -1 + \beta_{6} - \beta_{7} ) q^{5} + ( -\beta_{2} + 2 \beta_{3} + \beta_{6} + \beta_{7} ) q^{7} + ( -1 + \beta_{1} + \beta_{3} + \beta_{6} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + ( -1 + \beta_{6} - \beta_{7} ) q^{5} + ( -\beta_{2} + 2 \beta_{3} + \beta_{6} + \beta_{7} ) q^{7} + ( -1 + \beta_{1} + \beta_{3} + \beta_{6} ) q^{9} + ( 1 - \beta_{2} - \beta_{3} - \beta_{5} ) q^{11} + ( -2 + 3 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{13} + ( \beta_{1} + \beta_{4} + \beta_{6} ) q^{15} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{17} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{19} + ( \beta_{1} + 2 \beta_{3} + \beta_{6} + \beta_{7} ) q^{21} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{23} + ( -1 + 4 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{6} ) q^{25} + \beta_{1} q^{27} + ( -\beta_{3} + \beta_{7} ) q^{29} + ( 2 - \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{31} + ( 1 + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{33} + ( 1 - 2 \beta_{1} + \beta_{2} - 6 \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{35} + ( -2 + 2 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{6} ) q^{37} + ( \beta_{1} + \beta_{2} + \beta_{5} ) q^{39} + ( 3 - \beta_{1} - \beta_{3} - 4 \beta_{4} - 3 \beta_{6} ) q^{41} + ( 2 \beta_{2} - 3 \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{43} + ( 1 - \beta_{1} + \beta_{5} - \beta_{6} ) q^{45} + ( -6 \beta_{1} - 2 \beta_{2} - 7 \beta_{3} - 2 \beta_{4} - 6 \beta_{6} - \beta_{7} ) q^{47} + ( -3 - 2 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{49} + ( -2 - \beta_{2} + 2 \beta_{3} + \beta_{6} + \beta_{7} ) q^{51} + ( -3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{53} + ( -2 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 5 \beta_{6} - 2 \beta_{7} ) q^{55} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{57} + ( -3 - 5 \beta_{1} - 5 \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{59} + ( 8 + 3 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} - 2 \beta_{5} ) q^{61} + ( \beta_{1} + \beta_{3} - \beta_{4} ) q^{63} + ( 4 - \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{65} + ( 1 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 10 \beta_{6} + 2 \beta_{7} ) q^{67} + ( -1 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{69} + ( -3 \beta_{1} - 7 \beta_{3} - 3 \beta_{6} - \beta_{7} ) q^{71} + ( -3 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{5} ) q^{73} + ( -1 + 2 \beta_{1} - 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{75} + ( 3 + \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{77} + ( -4 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 4 \beta_{6} + 4 \beta_{7} ) q^{79} -\beta_{6} q^{81} + ( 4 + 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{83} + ( -5 + 3 \beta_{1} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{85} + ( 2 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{87} + ( -6 + 7 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} + \beta_{5} ) q^{89} + ( 5 - \beta_{1} - \beta_{3} - \beta_{4} - 5 \beta_{6} ) q^{91} + ( -4 + 3 \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{93} + ( -\beta_{1} + \beta_{2} - 6 \beta_{3} - 5 \beta_{6} - 2 \beta_{7} ) q^{95} + ( -2 \beta_{1} + 4 \beta_{2} - \beta_{3} + 4 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{97} + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{3} - 5q^{5} + 8q^{7} - 2q^{9} + O(q^{10})$$ $$8q + 2q^{3} - 5q^{5} + 8q^{7} - 2q^{9} + 8q^{11} + 5q^{15} + 3q^{17} + 5q^{19} + 7q^{21} - 7q^{23} + 5q^{25} + 2q^{27} - 3q^{29} - 3q^{31} + 7q^{33} - 10q^{35} - q^{37} + 10q^{41} - 12q^{43} + 5q^{45} - 33q^{47} - 8q^{49} - 8q^{51} - 19q^{53} - 15q^{55} + 10q^{57} - 38q^{59} + 46q^{61} + 3q^{63} + 25q^{65} - 8q^{67} + 2q^{69} - 25q^{71} - 26q^{73} - 5q^{75} + 23q^{77} - 16q^{79} - 2q^{81} + 8q^{83} - 30q^{85} + 3q^{87} - 30q^{89} + 25q^{91} - 22q^{93} - 25q^{95} - 14q^{97} - 2q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 2 x^{6} + x^{4} + 8 x^{2} - 24 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} + \nu^{3} + 2 \nu^{2} + 4 \nu - 8$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} + \nu^{3} - 2 \nu^{2} + 12 \nu - 12$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-7 \nu^{7} + 9 \nu^{6} + 2 \nu^{5} + 4 \nu^{4} + \nu^{3} - 4 \nu^{2} - 60 \nu + 64$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$11 \nu^{7} - 15 \nu^{6} - 4 \nu^{5} - 8 \nu^{4} + 3 \nu^{3} + 2 \nu^{2} + 92 \nu - 96$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$11 \nu^{7} - 13 \nu^{6} - 6 \nu^{5} - 8 \nu^{4} + 3 \nu^{3} + 4 \nu^{2} + 96 \nu - 88$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$13 \nu^{7} - 21 \nu^{6} - 4 \nu^{5} - 4 \nu^{4} + 5 \nu^{3} + 10 \nu^{2} + 120 \nu - 144$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$19 \nu^{7} - 31 \nu^{6} - 4 \nu^{5} - 8 \nu^{4} + 11 \nu^{3} + 18 \nu^{2} + 180 \nu - 224$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} - 4 \beta_{1} + 4$$$$)/5$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 2 \beta_{1} + 3$$$$)/5$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{4} + 8 \beta_{3} + 2 \beta_{2} + 7 \beta_{1} + 3$$$$)/5$$ $$\nu^{4}$$ $$=$$ $$-\beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{2} + 2 \beta_{1} + 1$$ $$\nu^{5}$$ $$=$$ $$($$$$6 \beta_{7} - 11 \beta_{6} - 9 \beta_{5} + 3 \beta_{4} - 11 \beta_{3} + 6 \beta_{2} + 6 \beta_{1} + 19$$$$)/5$$ $$\nu^{6}$$ $$=$$ $$($$$$2 \beta_{7} - 7 \beta_{6} + 7 \beta_{5} - 9 \beta_{4} - 7 \beta_{3} + 7 \beta_{2} + 12 \beta_{1} - 12$$$$)/5$$ $$\nu^{7}$$ $$=$$ $$($$$$-8 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 6 \beta_{4} - 7 \beta_{3} + 7 \beta_{2} + 57 \beta_{1} + 3$$$$)/5$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1
 1.40799 − 0.132563i −1.21700 + 0.720348i 1.33631 + 0.462894i −0.0272949 − 1.41395i 1.33631 − 0.462894i −0.0272949 + 1.41395i 1.40799 + 0.132563i −1.21700 − 0.720348i
0 −0.309017 + 0.951057i 0 −1.96485 + 1.06740i 0 −1.74037 0 −0.809017 0.587785i 0
61.2 0 −0.309017 + 0.951057i 0 −0.962197 2.01846i 0 1.50430 0 −0.809017 0.587785i 0
121.1 0 0.809017 + 0.587785i 0 −1.57146 1.59076i 0 4.32440 0 0.309017 + 0.951057i 0
121.2 0 0.809017 + 0.587785i 0 1.99851 + 1.00297i 0 −0.0883282 0 0.309017 + 0.951057i 0
181.1 0 0.809017 0.587785i 0 −1.57146 + 1.59076i 0 4.32440 0 0.309017 0.951057i 0
181.2 0 0.809017 0.587785i 0 1.99851 1.00297i 0 −0.0883282 0 0.309017 0.951057i 0
241.1 0 −0.309017 0.951057i 0 −1.96485 1.06740i 0 −1.74037 0 −0.809017 + 0.587785i 0
241.2 0 −0.309017 0.951057i 0 −0.962197 + 2.01846i 0 1.50430 0 −0.809017 + 0.587785i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 241.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.m.b 8
3.b odd 2 1 900.2.n.b 8
5.b even 2 1 1500.2.m.a 8
5.c odd 4 2 1500.2.o.b 16
25.d even 5 1 inner 300.2.m.b 8
25.d even 5 1 7500.2.a.e 4
25.e even 10 1 1500.2.m.a 8
25.e even 10 1 7500.2.a.f 4
25.f odd 20 2 1500.2.o.b 16
25.f odd 20 2 7500.2.d.c 8
75.j odd 10 1 900.2.n.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.m.b 8 1.a even 1 1 trivial
300.2.m.b 8 25.d even 5 1 inner
900.2.n.b 8 3.b odd 2 1
900.2.n.b 8 75.j odd 10 1
1500.2.m.a 8 5.b even 2 1
1500.2.m.a 8 25.e even 10 1
1500.2.o.b 16 5.c odd 4 2
1500.2.o.b 16 25.f odd 20 2
7500.2.a.e 4 25.d even 5 1
7500.2.a.f 4 25.e even 10 1
7500.2.d.c 8 25.f odd 20 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 4 T_{7}^{3} - 4 T_{7}^{2} + 11 T_{7} + 1$$ acting on $$S_{2}^{\mathrm{new}}(300, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$5$ $$625 + 625 T + 250 T^{2} - 25 T^{3} - 45 T^{4} - 5 T^{5} + 10 T^{6} + 5 T^{7} + T^{8}$$
$7$ $$( 1 + 11 T - 4 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$11$ $$361 - 323 T + 1033 T^{2} + 149 T^{3} + 100 T^{4} + 39 T^{5} + 23 T^{6} - 8 T^{7} + T^{8}$$
$13$ $$2025 - 2025 T + 2925 T^{2} - 975 T^{3} + 310 T^{4} - 5 T^{5} - 5 T^{6} + T^{8}$$
$17$ $$81 - 108 T + 333 T^{2} - 21 T^{3} - 20 T^{4} - T^{5} + 13 T^{6} - 3 T^{7} + T^{8}$$
$19$ $$25 - 25 T + 225 T^{2} + 225 T^{3} + 160 T^{4} + 45 T^{5} + 5 T^{6} - 5 T^{7} + T^{8}$$
$23$ $$29241 - 513 T + 12798 T^{2} - 6171 T^{3} + 1705 T^{4} - 61 T^{5} + 18 T^{6} + 7 T^{7} + T^{8}$$
$29$ $$1 + 3 T + 58 T^{2} - 129 T^{3} + 105 T^{4} + 21 T^{5} + 8 T^{6} + 3 T^{7} + T^{8}$$
$31$ $$962361 - 55917 T + 78228 T^{2} - 21699 T^{3} + 3625 T^{4} + 511 T^{5} + 58 T^{6} + 3 T^{7} + T^{8}$$
$37$ $$1042441 - 57176 T + 12657 T^{2} - 343 T^{3} + 1550 T^{4} + 113 T^{5} + 57 T^{6} + T^{7} + T^{8}$$
$41$ $$7317025 + 1704150 T + 228475 T^{2} - 5800 T^{3} + 4085 T^{4} - 520 T^{5} + 155 T^{6} - 10 T^{7} + T^{8}$$
$43$ $$( 131 - 64 T - 19 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$47$ $$16801801 + 4947493 T + 1291508 T^{2} + 275851 T^{3} + 48275 T^{4} + 6151 T^{5} + 568 T^{6} + 33 T^{7} + T^{8}$$
$53$ $$9801 - 9504 T + 5787 T^{2} - 2547 T^{3} + 5080 T^{4} + 1347 T^{5} + 217 T^{6} + 19 T^{7} + T^{8}$$
$59$ $$13697401 + 8874998 T + 2975713 T^{2} + 622126 T^{3} + 94675 T^{4} + 10026 T^{5} + 773 T^{6} + 38 T^{7} + T^{8}$$
$61$ $$62552281 - 11594594 T + 3590217 T^{2} - 782922 T^{3} + 126055 T^{4} - 13818 T^{5} + 1037 T^{6} - 46 T^{7} + T^{8}$$
$67$ $$408321 - 1755972 T + 2888388 T^{2} - 13704 T^{3} + 34630 T^{4} + 2416 T^{5} + 183 T^{6} + 8 T^{7} + T^{8}$$
$71$ $$25 - 650 T + 44875 T^{2} + 10175 T^{3} + 4060 T^{4} + 1395 T^{5} + 275 T^{6} + 25 T^{7} + T^{8}$$
$73$ $$121 - 506 T + 3182 T^{2} - 3618 T^{3} + 1600 T^{4} + 228 T^{5} + 267 T^{6} + 26 T^{7} + T^{8}$$
$79$ $$408321 + 300969 T + 629937 T^{2} - 22443 T^{3} + 2350 T^{4} - 247 T^{5} + 117 T^{6} + 16 T^{7} + T^{8}$$
$83$ $$46908801 - 14876028 T + 9296163 T^{2} - 531546 T^{3} + 43855 T^{4} + 1054 T^{5} - 17 T^{6} - 8 T^{7} + T^{8}$$
$89$ $$97515625 - 11109375 T + 359375 T^{2} + 106875 T^{3} + 57750 T^{4} + 7125 T^{5} + 625 T^{6} + 30 T^{7} + T^{8}$$
$97$ $$64304361 + 13423806 T + 5936247 T^{2} + 188028 T^{3} + 16705 T^{4} - 908 T^{5} + 87 T^{6} + 14 T^{7} + T^{8}$$