# Properties

 Label 300.2.j.d Level $300$ Weight $2$ Character orbit 300.j Analytic conductor $2.396$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: 12.0.426337261060096.1 Defining polynomial: $$x^{12} - 4 x^{9} - 3 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 32 x^{3} + 64$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{9} q^{3} + ( \beta_{2} + \beta_{8} + \beta_{9} ) q^{4} -\beta_{11} q^{6} + ( -\beta_{1} - \beta_{5} - \beta_{6} - \beta_{11} ) q^{7} + ( 1 - \beta_{4} + \beta_{5} - \beta_{10} - \beta_{11} ) q^{8} + \beta_{10} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{9} q^{3} + ( \beta_{2} + \beta_{8} + \beta_{9} ) q^{4} -\beta_{11} q^{6} + ( -\beta_{1} - \beta_{5} - \beta_{6} - \beta_{11} ) q^{7} + ( 1 - \beta_{4} + \beta_{5} - \beta_{10} - \beta_{11} ) q^{8} + \beta_{10} q^{9} + ( -\beta_{4} - \beta_{6} + \beta_{7} - \beta_{11} ) q^{11} + ( 1 + \beta_{6} + \beta_{10} ) q^{12} + ( 1 - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{10} - \beta_{11} ) q^{13} + ( -2 \beta_{2} - \beta_{4} + \beta_{6} - 2 \beta_{9} + 2 \beta_{10} ) q^{14} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{16} + ( 1 - \beta_{1} + \beta_{5} - \beta_{6} + \beta_{10} + \beta_{11} ) q^{17} + \beta_{3} q^{18} + ( -\beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{19} + ( -\beta_{1} + \beta_{3} + \beta_{7} + \beta_{11} ) q^{21} + ( -2 \beta_{2} + 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{22} + ( -2 \beta_{3} - \beta_{5} + \beta_{7} - \beta_{8} - 4 \beta_{9} + \beta_{11} ) q^{23} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{8} + \beta_{9} ) q^{24} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - 2 \beta_{9} ) q^{26} -\beta_{2} q^{27} + ( 2 \beta_{3} - 2 \beta_{5} + \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{28} + ( -\beta_{1} - \beta_{3} + \beta_{5} + \beta_{8} ) q^{29} + ( -2 \beta_{1} + 2 \beta_{3} - 2 \beta_{7} - 2 \beta_{11} ) q^{31} + ( -1 + 2 \beta_{1} + 4 \beta_{2} - \beta_{5} + \beta_{6} - \beta_{10} - \beta_{11} ) q^{32} + ( \beta_{3} + \beta_{4} + \beta_{7} - \beta_{8} ) q^{33} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{9} - 2 \beta_{10} ) q^{34} + ( -\beta_{2} - \beta_{7} + \beta_{9} ) q^{36} + ( -1 - \beta_{1} + \beta_{5} - \beta_{6} - \beta_{10} + \beta_{11} ) q^{37} + ( -2 - 2 \beta_{4} - 4 \beta_{9} + 2 \beta_{10} ) q^{38} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{8} + \beta_{9} ) q^{39} + ( -\beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{11} ) q^{41} + ( -2 - 2 \beta_{2} - \beta_{7} - \beta_{8} - 2 \beta_{10} ) q^{42} + ( 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} + 2 \beta_{8} + 4 \beta_{9} ) q^{43} + ( 2 \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{44} + ( -4 + 2 \beta_{2} + 2 \beta_{7} - 2 \beta_{9} + 4 \beta_{11} ) q^{46} + ( 4 \beta_{2} - \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} ) q^{47} + ( -1 + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{48} + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{8} - \beta_{10} ) q^{49} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{7} + \beta_{9} + \beta_{11} ) q^{51} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{5} + 2 \beta_{11} ) q^{52} + ( -1 - \beta_{3} - \beta_{4} - \beta_{7} + \beta_{8} + \beta_{10} ) q^{53} -\beta_{5} q^{54} + ( -6 - 2 \beta_{2} - \beta_{4} - \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{11} ) q^{56} + ( -2 \beta_{1} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{11} ) q^{57} + ( 2 + \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{58} + ( -4 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - 4 \beta_{9} ) q^{59} + ( -4 + 3 \beta_{1} - 3 \beta_{3} - \beta_{4} - \beta_{6} - 4 \beta_{7} - 2 \beta_{11} ) q^{61} + ( 4 - 4 \beta_{2} - 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{10} ) q^{62} + ( -\beta_{3} + \beta_{4} - \beta_{5} + \beta_{11} ) q^{63} + ( -\beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} + \beta_{8} + 3 \beta_{9} + 2 \beta_{10} ) q^{64} + ( -2 + \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{66} + ( -2 \beta_{1} - 4 \beta_{2} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{67} + ( -2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{11} ) q^{68} + ( -\beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - 4 \beta_{10} ) q^{69} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - 4 \beta_{9} + 4 \beta_{11} ) q^{71} + ( 1 - \beta_{5} - \beta_{6} + \beta_{10} - \beta_{11} ) q^{72} + ( -5 - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} + 2 \beta_{8} + 5 \beta_{10} ) q^{73} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{9} - 2 \beta_{10} ) q^{74} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{7} + 2 \beta_{9} + 4 \beta_{11} ) q^{76} + ( -4 + 2 \beta_{1} - \beta_{5} - \beta_{7} - \beta_{8} - 4 \beta_{10} - \beta_{11} ) q^{77} + ( 2 + \beta_{5} - \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{78} + ( 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + 4 \beta_{9} ) q^{79} - q^{81} + ( -2 - 4 \beta_{2} - 2 \beta_{6} - 2 \beta_{10} ) q^{82} + ( -2 \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} + 4 \beta_{9} - \beta_{11} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{10} ) q^{84} + ( 4 - 2 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} - 4 \beta_{11} ) q^{86} + ( \beta_{1} + \beta_{5} + \beta_{6} + \beta_{11} ) q^{87} + ( -4 - 2 \beta_{3} + 2 \beta_{5} + \beta_{7} - \beta_{8} + 2 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} ) q^{88} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} + 2 \beta_{8} ) q^{89} + ( -2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{7} + 4 \beta_{9} - 2 \beta_{11} ) q^{91} + ( -6 - 4 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} - 6 \beta_{10} + 2 \beta_{11} ) q^{92} + ( 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{11} ) q^{93} + ( -2 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{94} + ( 4 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{7} - \beta_{9} - 2 \beta_{11} ) q^{96} + ( 3 + 6 \beta_{1} - 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} - 4 \beta_{11} ) q^{97} + ( -2 - \beta_{3} + 2 \beta_{4} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} ) q^{98} + ( \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 4q^{6} + 12q^{8} + O(q^{10})$$ $$12q - 4q^{6} + 12q^{8} + 8q^{12} + 4q^{13} + 12q^{16} + 20q^{17} - 12q^{22} + 16q^{26} + 4q^{28} - 20q^{32} - 8q^{33} + 4q^{36} - 4q^{37} - 16q^{38} + 16q^{41} - 20q^{42} - 40q^{46} - 16q^{48} + 8q^{52} - 4q^{53} - 64q^{56} + 20q^{58} - 32q^{61} + 56q^{62} - 24q^{66} + 16q^{68} + 12q^{72} - 44q^{73} + 8q^{76} - 48q^{77} + 24q^{78} - 12q^{81} - 16q^{82} + 64q^{86} - 60q^{88} - 56q^{92} + 16q^{93} + 44q^{96} + 20q^{97} - 24q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{9} - 3 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 32 x^{3} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{11} + \nu^{10} + 4 \nu^{9} - 2 \nu^{8} - 3 \nu^{7} + 5 \nu^{6} + 24 \nu^{5} + 2 \nu^{4} - 8 \nu^{3} - 24 \nu^{2} - 48 \nu$$$$)/160$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{11} - 8 \nu^{9} - 4 \nu^{8} + 11 \nu^{7} + 12 \nu^{6} + 16 \nu^{5} - 48 \nu^{4} - 60 \nu^{3} + 16 \nu^{2} + 64 \nu + 192$$$$)/80$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{11} + 2 \nu^{9} + 6 \nu^{8} + 11 \nu^{7} - 8 \nu^{6} - 14 \nu^{5} + 2 \nu^{4} - 20 \nu^{3} + 36 \nu^{2} - 16 \nu - 48$$$$)/80$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{11} + 4 \nu^{10} + 2 \nu^{9} + \nu^{7} + 16 \nu^{6} - 6 \nu^{5} + 4 \nu^{4} + 8 \nu^{3} - 48 \nu^{2} - 64$$$$)/160$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{10} - 4 \nu^{9} - \nu^{8} - 2 \nu^{7} + 7 \nu^{6} + 20 \nu^{5} - \nu^{4} - 18 \nu^{3} - 18 \nu^{2} - 24 \nu + 72$$$$)/40$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{10} + 3 \nu^{9} - 8 \nu^{8} - 6 \nu^{7} - 9 \nu^{6} + 15 \nu^{5} + 52 \nu^{4} + 6 \nu^{3} - 24 \nu^{2} - 72 \nu - 64$$$$)/80$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{11} + 4 \nu^{10} + \nu^{9} + 2 \nu^{8} - 7 \nu^{7} - 20 \nu^{6} + \nu^{5} + 18 \nu^{4} + 18 \nu^{3} + 64 \nu^{2} - 72 \nu$$$$)/80$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{11} - 9 \nu^{10} - 6 \nu^{9} - 2 \nu^{8} + 17 \nu^{7} + 35 \nu^{6} - 26 \nu^{5} - 38 \nu^{4} - 28 \nu^{3} + 56 \nu^{2} + 192 \nu$$$$)/160$$ $$\beta_{10}$$ $$=$$ $$($$$$-3 \nu^{11} - \nu^{10} + 4 \nu^{8} + 5 \nu^{7} - \nu^{6} - 12 \nu^{5} - 8 \nu^{4} - 12 \nu^{3} + 36 \nu^{2} + 16 \nu + 64$$$$)/80$$ $$\beta_{11}$$ $$=$$ $$($$$$9 \nu^{11} + 6 \nu^{10} - 2 \nu^{9} - 20 \nu^{8} - 31 \nu^{7} + 34 \nu^{6} + 46 \nu^{5} + 16 \nu^{4} - 88 \nu^{3} - 192 \nu^{2} + 64$$$$)/160$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} + \beta_{8} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{11} - \beta_{10} + \beta_{5} - \beta_{4} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{9} + \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_{1} + 2$$ $$\nu^{5}$$ $$=$$ $$-\beta_{11} - \beta_{10} + \beta_{6} - \beta_{5} + 4 \beta_{2} + 2 \beta_{1} - 1$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{10} + 3 \beta_{9} + \beta_{8} + \beta_{6} + 4 \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} - \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$-3 \beta_{11} - 5 \beta_{10} - 2 \beta_{8} + 2 \beta_{7} + 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 5$$ $$\nu^{8}$$ $$=$$ $$-\beta_{9} - 5 \beta_{7} + 5 \beta_{6} + 5 \beta_{4} - 5 \beta_{3} + \beta_{2} + 5 \beta_{1} + 2$$ $$\nu^{9}$$ $$=$$ $$\beta_{11} + 5 \beta_{10} - 5 \beta_{6} + \beta_{5} + 20 \beta_{2} + 2 \beta_{1} + 5$$ $$\nu^{10}$$ $$=$$ $$-2 \beta_{10} - 3 \beta_{9} + 7 \beta_{8} - \beta_{6} + 20 \beta_{5} + \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + 5 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$3 \beta_{11} - 27 \beta_{10} + 8 \beta_{9} + 6 \beta_{8} - 6 \beta_{7} - 3 \beta_{5} + 13 \beta_{4} - 2 \beta_{3} + 27$$

## Character values

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

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\beta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −1.35818 − 0.394157i −0.760198 + 1.19252i −0.394157 − 1.35818i −0.0912546 + 1.41127i 1.19252 − 0.760198i 1.41127 − 0.0912546i −1.35818 + 0.394157i −0.760198 − 1.19252i −0.394157 + 1.35818i −0.0912546 − 1.41127i 1.19252 + 0.760198i 1.41127 + 0.0912546i
−1.35818 0.394157i 0.707107 + 0.707107i 1.68928 + 1.07067i 0 −0.681664 1.23909i 2.47817 2.47817i −1.87233 2.12000i 1.00000i 0
7.2 −0.760198 + 1.19252i 0.707107 + 0.707107i −0.844199 1.81310i 0 −1.38078 + 0.305697i −0.611393 + 0.611393i 2.80391 + 0.371591i 1.00000i 0
7.3 −0.394157 1.35818i −0.707107 0.707107i −1.68928 + 1.07067i 0 −0.681664 + 1.23909i −2.47817 + 2.47817i 2.12000 + 1.87233i 1.00000i 0
7.4 −0.0912546 + 1.41127i −0.707107 0.707107i −1.98335 0.257569i 0 1.06244 0.933389i 1.86678 1.86678i 0.544488 2.77552i 1.00000i 0
7.5 1.19252 0.760198i −0.707107 0.707107i 0.844199 1.81310i 0 −1.38078 0.305697i 0.611393 0.611393i −0.371591 2.80391i 1.00000i 0
7.6 1.41127 0.0912546i 0.707107 + 0.707107i 1.98335 0.257569i 0 1.06244 + 0.933389i −1.86678 + 1.86678i 2.77552 0.544488i 1.00000i 0
43.1 −1.35818 + 0.394157i 0.707107 0.707107i 1.68928 1.07067i 0 −0.681664 + 1.23909i 2.47817 + 2.47817i −1.87233 + 2.12000i 1.00000i 0
43.2 −0.760198 1.19252i 0.707107 0.707107i −0.844199 + 1.81310i 0 −1.38078 0.305697i −0.611393 0.611393i 2.80391 0.371591i 1.00000i 0
43.3 −0.394157 + 1.35818i −0.707107 + 0.707107i −1.68928 1.07067i 0 −0.681664 1.23909i −2.47817 2.47817i 2.12000 1.87233i 1.00000i 0
43.4 −0.0912546 1.41127i −0.707107 + 0.707107i −1.98335 + 0.257569i 0 1.06244 + 0.933389i 1.86678 + 1.86678i 0.544488 + 2.77552i 1.00000i 0
43.5 1.19252 + 0.760198i −0.707107 + 0.707107i 0.844199 + 1.81310i 0 −1.38078 + 0.305697i 0.611393 + 0.611393i −0.371591 + 2.80391i 1.00000i 0
43.6 1.41127 + 0.0912546i 0.707107 0.707107i 1.98335 + 0.257569i 0 1.06244 0.933389i −1.86678 1.86678i 2.77552 + 0.544488i 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.6 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.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.j.d 12
3.b odd 2 1 900.2.k.n 12
4.b odd 2 1 inner 300.2.j.d 12
5.b even 2 1 60.2.j.a 12
5.c odd 4 1 60.2.j.a 12
5.c odd 4 1 inner 300.2.j.d 12
12.b even 2 1 900.2.k.n 12
15.d odd 2 1 180.2.k.e 12
15.e even 4 1 180.2.k.e 12
15.e even 4 1 900.2.k.n 12
20.d odd 2 1 60.2.j.a 12
20.e even 4 1 60.2.j.a 12
20.e even 4 1 inner 300.2.j.d 12
40.e odd 2 1 960.2.w.g 12
40.f even 2 1 960.2.w.g 12
40.i odd 4 1 960.2.w.g 12
40.k even 4 1 960.2.w.g 12
60.h even 2 1 180.2.k.e 12
60.l odd 4 1 180.2.k.e 12
60.l odd 4 1 900.2.k.n 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.j.a 12 5.b even 2 1
60.2.j.a 12 5.c odd 4 1
60.2.j.a 12 20.d odd 2 1
60.2.j.a 12 20.e even 4 1
180.2.k.e 12 15.d odd 2 1
180.2.k.e 12 15.e even 4 1
180.2.k.e 12 60.h even 2 1
180.2.k.e 12 60.l odd 4 1
300.2.j.d 12 1.a even 1 1 trivial
300.2.j.d 12 4.b odd 2 1 inner
300.2.j.d 12 5.c odd 4 1 inner
300.2.j.d 12 20.e even 4 1 inner
900.2.k.n 12 3.b odd 2 1
900.2.k.n 12 12.b even 2 1
900.2.k.n 12 15.e even 4 1
900.2.k.n 12 60.l odd 4 1
960.2.w.g 12 40.e odd 2 1
960.2.w.g 12 40.f even 2 1
960.2.w.g 12 40.i odd 4 1
960.2.w.g 12 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(300, [\chi])$$:

 $$T_{7}^{12} + 200 T_{7}^{8} + 7440 T_{7}^{4} + 4096$$ $$T_{19}^{6} - 40 T_{19}^{4} + 400 T_{19}^{2} - 512$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 - 32 T^{3} - 12 T^{4} + 8 T^{5} + 8 T^{6} + 4 T^{7} - 3 T^{8} - 4 T^{9} + T^{12}$$
$3$ $$( 1 + T^{4} )^{3}$$
$5$ $$T^{12}$$
$7$ $$4096 + 7440 T^{4} + 200 T^{8} + T^{12}$$
$11$ $$( 128 + 260 T^{2} + 36 T^{4} + T^{6} )^{2}$$
$13$ $$( 32 + 96 T + 144 T^{2} + 32 T^{3} + 2 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$17$ $$( 800 + 160 T + 16 T^{2} - 80 T^{3} + 50 T^{4} - 10 T^{5} + T^{6} )^{2}$$
$19$ $$( -512 + 400 T^{2} - 40 T^{4} + T^{6} )^{2}$$
$23$ $$65536 + 37120 T^{4} + 4640 T^{8} + T^{12}$$
$29$ $$( 64 + 100 T^{2} + 20 T^{4} + T^{6} )^{2}$$
$31$ $$( 32768 + 3648 T^{2} + 112 T^{4} + T^{6} )^{2}$$
$37$ $$( 32 - 96 T + 144 T^{2} - 32 T^{3} + 2 T^{4} + 2 T^{5} + T^{6} )^{2}$$
$41$ $$( 64 - 20 T - 4 T^{2} + T^{3} )^{4}$$
$43$ $$15352201216 + 24350720 T^{4} + 9600 T^{8} + T^{12}$$
$47$ $$40960000 + 901376 T^{4} + 4896 T^{8} + T^{12}$$
$53$ $$( 128 + 256 T + 256 T^{2} - 16 T^{3} + 2 T^{4} + 2 T^{5} + T^{6} )^{2}$$
$59$ $$( -512 + 1860 T^{2} - 100 T^{4} + T^{6} )^{2}$$
$61$ $$( 176 - 100 T + 8 T^{2} + T^{3} )^{4}$$
$67$ $$15352201216 + 24350720 T^{4} + 9600 T^{8} + T^{12}$$
$71$ $$( 204800 + 15616 T^{2} + 256 T^{4} + T^{6} )^{2}$$
$73$ $$( 55112 - 4648 T + 196 T^{2} + 640 T^{3} + 242 T^{4} + 22 T^{5} + T^{6} )^{2}$$
$79$ $$( -2048 + 12864 T^{2} - 304 T^{4} + T^{6} )^{2}$$
$83$ $$65536 + 10264832 T^{4} + 16672 T^{8} + T^{12}$$
$89$ $$( 1024 + 1040 T^{2} + 72 T^{4} + T^{6} )^{2}$$
$97$ $$( 35912 + 47704 T + 31684 T^{2} + 2048 T^{3} + 50 T^{4} - 10 T^{5} + T^{6} )^{2}$$