# Properties

 Label 300.3.c.f Level $300$ Weight $3$ Character orbit 300.c Analytic conductor $8.174$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 300.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.17440793081$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.6080256576.2 Defining polynomial: $$x^{8} - 3 x^{7} + 7 x^{6} - 12 x^{5} + 12 x^{4} - 48 x^{3} + 112 x^{2} - 192 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} -\beta_{2} q^{3} + ( 1 - \beta_{7} ) q^{4} + ( -1 - \beta_{6} ) q^{6} + ( 3 \beta_{2} - \beta_{3} + \beta_{5} ) q^{7} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{8} -3 q^{9} +O(q^{10})$$ $$q + \beta_{5} q^{2} -\beta_{2} q^{3} + ( 1 - \beta_{7} ) q^{4} + ( -1 - \beta_{6} ) q^{6} + ( 3 \beta_{2} - \beta_{3} + \beta_{5} ) q^{7} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{8} -3 q^{9} + ( -1 + \beta_{4} + \beta_{6} - 3 \beta_{7} ) q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{12} + ( \beta_{2} + \beta_{3} + 3 \beta_{5} ) q^{13} + ( -2 + 4 \beta_{6} - 2 \beta_{7} ) q^{14} + ( -5 - 2 \beta_{4} + 2 \beta_{6} - 3 \beta_{7} ) q^{16} + ( -4 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{17} -3 \beta_{5} q^{18} + ( -2 + 2 \beta_{4} - 6 \beta_{6} + 2 \beta_{7} ) q^{19} + ( 9 - \beta_{4} - \beta_{6} - \beta_{7} ) q^{21} + ( -2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} - 2 \beta_{5} ) q^{22} + ( 6 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{23} + ( 3 - 2 \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{24} + ( 10 - 2 \beta_{7} ) q^{26} + 3 \beta_{2} q^{27} + ( 2 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} - 2 \beta_{5} ) q^{28} + ( 23 + 3 \beta_{4} + 3 \beta_{6} + 3 \beta_{7} ) q^{29} + ( -8 \beta_{6} + 8 \beta_{7} ) q^{31} + ( -\beta_{1} + 17 \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{32} + ( -4 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{33} + ( 2 - 8 \beta_{4} - 2 \beta_{7} ) q^{34} + ( -3 + 3 \beta_{7} ) q^{36} + ( 3 \beta_{2} + 3 \beta_{3} + 9 \beta_{5} ) q^{37} + ( -4 \beta_{1} - 20 \beta_{2} + 4 \beta_{3} - 4 \beta_{5} ) q^{38} + ( -1 + \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{39} + ( -32 + 6 \beta_{4} + 6 \beta_{6} + 6 \beta_{7} ) q^{41} + ( -2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} + 10 \beta_{5} ) q^{42} + ( 4 \beta_{2} - 8 \beta_{3} + 8 \beta_{5} ) q^{43} + ( -42 - 4 \beta_{4} + 4 \beta_{6} - 6 \beta_{7} ) q^{44} + ( 16 + 4 \beta_{6} + 4 \beta_{7} ) q^{46} + ( -12 \beta_{2} - 8 \beta_{3} + 8 \beta_{5} ) q^{47} + ( -\beta_{1} + 9 \beta_{2} + 7 \beta_{3} + 5 \beta_{5} ) q^{48} + ( 3 + 6 \beta_{4} + 6 \beta_{6} + 6 \beta_{7} ) q^{49} + ( 3 - 3 \beta_{4} - 3 \beta_{6} + 9 \beta_{7} ) q^{51} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 10 \beta_{5} ) q^{52} + ( -3 \beta_{2} - 3 \beta_{3} - 9 \beta_{5} ) q^{53} + ( 3 + 3 \beta_{6} ) q^{54} + ( -30 + 4 \beta_{4} + 12 \beta_{6} - 2 \beta_{7} ) q^{56} + ( -6 \beta_{2} - 6 \beta_{3} - 18 \beta_{5} ) q^{57} + ( 6 \beta_{1} - 18 \beta_{2} - 6 \beta_{3} + 20 \beta_{5} ) q^{58} + ( -9 + 9 \beta_{4} - 23 \beta_{6} + 5 \beta_{7} ) q^{59} + 38 q^{61} + ( -16 \beta_{2} + 16 \beta_{3} ) q^{62} + ( -9 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} ) q^{63} + ( 7 - 2 \beta_{4} + 18 \beta_{6} + \beta_{7} ) q^{64} + ( 2 - 8 \beta_{4} - 2 \beta_{7} ) q^{66} + ( 18 \beta_{2} + 14 \beta_{3} - 14 \beta_{5} ) q^{67} + ( -2 \beta_{1} + 50 \beta_{2} + 14 \beta_{3} + 10 \beta_{5} ) q^{68} + ( 18 + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{69} + ( -10 + 10 \beta_{4} - 6 \beta_{6} - 14 \beta_{7} ) q^{71} + ( 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} ) q^{72} + ( 8 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{73} + ( 30 - 6 \beta_{7} ) q^{74} + ( -4 - 8 \beta_{4} - 24 \beta_{6} + 4 \beta_{7} ) q^{76} + ( 8 \beta_{1} - 10 \beta_{2} - 10 \beta_{3} - 22 \beta_{5} ) q^{77} + ( -2 \beta_{1} - 10 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{78} + ( -8 \beta_{6} + 8 \beta_{7} ) q^{79} + 9 q^{81} + ( 12 \beta_{1} - 36 \beta_{2} - 12 \beta_{3} - 38 \beta_{5} ) q^{82} + ( -24 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} ) q^{83} + ( 26 - 4 \beta_{4} + 4 \beta_{6} - 10 \beta_{7} ) q^{84} + ( -36 + 12 \beta_{6} - 16 \beta_{7} ) q^{86} + ( -23 \beta_{2} - 9 \beta_{3} + 9 \beta_{5} ) q^{87} + ( -2 \beta_{1} + 34 \beta_{2} - 2 \beta_{3} - 38 \beta_{5} ) q^{88} + ( -70 + 4 \beta_{4} + 4 \beta_{6} + 4 \beta_{7} ) q^{89} + ( 2 - 2 \beta_{4} + 14 \beta_{6} - 10 \beta_{7} ) q^{91} + ( 8 \beta_{1} + 16 \beta_{5} ) q^{92} + ( 8 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} - 16 \beta_{5} ) q^{93} + ( -52 - 4 \beta_{6} - 16 \beta_{7} ) q^{94} + ( 55 - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{96} + ( 8 \beta_{1} + 20 \beta_{2} + 20 \beta_{3} + 68 \beta_{5} ) q^{97} + ( 12 \beta_{1} - 36 \beta_{2} - 12 \beta_{3} - 3 \beta_{5} ) q^{98} + ( 3 - 3 \beta_{4} - 3 \beta_{6} + 9 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 10q^{4} - 6q^{6} - 24q^{9} + O(q^{10})$$ $$8q + 10q^{4} - 6q^{6} - 24q^{9} - 20q^{14} - 46q^{16} + 72q^{21} + 18q^{24} + 84q^{26} + 184q^{29} - 12q^{34} - 30q^{36} - 256q^{41} - 348q^{44} + 112q^{46} + 24q^{49} + 18q^{54} - 244q^{56} + 304q^{61} + 10q^{64} - 12q^{66} + 144q^{69} + 252q^{74} - 24q^{76} + 72q^{81} + 204q^{84} - 280q^{86} - 560q^{89} - 376q^{94} + 426q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 7 x^{6} - 12 x^{5} + 12 x^{4} - 48 x^{3} + 112 x^{2} - 192 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3 \nu^{7} - 13 \nu^{6} + 5 \nu^{5} - 30 \nu^{4} + 108 \nu^{3} + 264 \nu^{2} + 368 \nu - 160$$$$)/224$$ $$\beta_{2}$$ $$=$$ $$($$$$3 \nu^{7} - \nu^{6} - 19 \nu^{5} - 12 \nu^{4} + 4 \nu^{3} - 96 \nu^{2} + 80 \nu + 608$$$$)/224$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - 12 \nu^{6} + 10 \nu^{5} - 11 \nu^{4} - 8 \nu^{3} - 4 \nu^{2} + 400 \nu - 320$$$$)/112$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 9 \nu^{6} - 11 \nu^{5} + 10 \nu^{4} - 22 \nu^{3} + 24 \nu^{2} - 160 \nu + 408$$$$)/56$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{7} - 15 \nu^{6} + 23 \nu^{5} + 2 \nu^{4} + 60 \nu^{3} - 152 \nu^{2} + 528 \nu - 736$$$$)/224$$ $$\beta_{6}$$ $$=$$ $$($$$$-5 \nu^{7} + 18 \nu^{6} - 36 \nu^{5} + 41 \nu^{4} - 72 \nu^{3} + 244 \nu^{2} - 656 \nu + 928$$$$)/112$$ $$\beta_{7}$$ $$=$$ $$($$$$-11 \nu^{7} + 20 \nu^{6} - 26 \nu^{5} + 37 \nu^{4} - 52 \nu^{3} + 324 \nu^{2} - 592 \nu + 384$$$$)/112$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} + 2 \beta_{1} - 3$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{5} - 3 \beta_{3} + \beta_{2} + 2 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{7} + 9 \beta_{6} + 25 \beta_{5} + \beta_{4} - \beta_{3} - 5 \beta_{2} - 2 \beta_{1} + 13$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-3 \beta_{7} - 5 \beta_{6} - 7 \beta_{5} + 3 \beta_{4} - \beta_{3} - 37 \beta_{2} - 2 \beta_{1} + 103$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$13 \beta_{7} - 21 \beta_{6} + 19 \beta_{4} - 9$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-73 \beta_{7} + 49 \beta_{6} - 11 \beta_{5} + 41 \beta_{4} + 3 \beta_{3} - 81 \beta_{2} + 38 \beta_{1} - 235$$$$)/4$$

## Character values

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

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 −1.51328 − 1.30766i −1.51328 + 1.30766i 0.670410 + 1.88429i 0.670410 − 1.88429i 1.96705 − 0.361553i 1.96705 + 0.361553i 0.375825 + 1.96437i 0.375825 − 1.96437i
−1.88911 0.656712i 1.73205i 3.13746 + 2.48120i 0 1.13746 3.27203i 9.55505i −4.29756 6.74766i −3.00000 0
151.2 −1.88911 + 0.656712i 1.73205i 3.13746 2.48120i 0 1.13746 + 3.27203i 9.55505i −4.29756 + 6.74766i −3.00000 0
151.3 −1.29664 1.52274i 1.73205i −0.637459 + 3.94888i 0 −2.63746 + 2.24584i 0.837253i 6.83966 4.14959i −3.00000 0
151.4 −1.29664 + 1.52274i 1.73205i −0.637459 3.94888i 0 −2.63746 2.24584i 0.837253i 6.83966 + 4.14959i −3.00000 0
151.5 1.29664 1.52274i 1.73205i −0.637459 3.94888i 0 −2.63746 2.24584i 0.837253i −6.83966 4.14959i −3.00000 0
151.6 1.29664 + 1.52274i 1.73205i −0.637459 + 3.94888i 0 −2.63746 + 2.24584i 0.837253i −6.83966 + 4.14959i −3.00000 0
151.7 1.88911 0.656712i 1.73205i 3.13746 2.48120i 0 1.13746 + 3.27203i 9.55505i 4.29756 6.74766i −3.00000 0
151.8 1.88911 + 0.656712i 1.73205i 3.13746 + 2.48120i 0 1.13746 3.27203i 9.55505i 4.29756 + 6.74766i −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 151.8 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
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(300, [\chi])$$:

 $$T_{7}^{4} + 92 T_{7}^{2} + 64$$ $$T_{13}^{4} - 84 T_{13}^{2} + 1536$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 - 80 T^{2} + 24 T^{4} - 5 T^{6} + T^{8}$$
$3$ $$( 3 + T^{2} )^{4}$$
$5$ $$T^{8}$$
$7$ $$( 64 + 92 T^{2} + T^{4} )^{2}$$
$11$ $$( 24576 + 348 T^{2} + T^{4} )^{2}$$
$13$ $$( 1536 - 84 T^{2} + T^{4} )^{2}$$
$17$ $$( 221184 - 1044 T^{2} + T^{4} )^{2}$$
$19$ $$( 221184 + 1008 T^{2} + T^{4} )^{2}$$
$23$ $$( 1024 + 368 T^{2} + T^{4} )^{2}$$
$29$ $$( 16 - 46 T + T^{2} )^{4}$$
$31$ $$( 393216 + 2304 T^{2} + T^{4} )^{2}$$
$37$ $$( 124416 - 756 T^{2} + T^{4} )^{2}$$
$41$ $$( -1028 + 64 T + T^{2} )^{4}$$
$43$ $$( 1364224 + 2528 T^{2} + T^{4} )^{2}$$
$47$ $$( 614656 + 3296 T^{2} + T^{4} )^{2}$$
$53$ $$( 124416 - 756 T^{2} + T^{4} )^{2}$$
$59$ $$( 69033984 + 16668 T^{2} + T^{4} )^{2}$$
$61$ $$( -38 + T )^{8}$$
$67$ $$( 7573504 + 9392 T^{2} + T^{4} )^{2}$$
$71$ $$( 884736 + 17136 T^{2} + T^{4} )^{2}$$
$73$ $$( 3538944 - 4176 T^{2} + T^{4} )^{2}$$
$79$ $$( 393216 + 2304 T^{2} + T^{4} )^{2}$$
$83$ $$( 2027776 + 4064 T^{2} + T^{4} )^{2}$$
$89$ $$( 3988 + 140 T + T^{2} )^{4}$$
$97$ $$( 495550464 - 45888 T^{2} + T^{4} )^{2}$$