# Properties

 Label 600.2.k.f Level $600$ Weight $2$ Character orbit 600.k Analytic conductor $4.791$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 600.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.79102412128$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.180227832610816.1 Defining polynomial: $$x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64$$ x^12 + x^10 - 8*x^6 + 16*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 120) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{3} q^{3} + \beta_{2} q^{4} - \beta_{5} q^{6} + (\beta_{10} + \beta_1) q^{7} + (\beta_{10} + \beta_{7}) q^{8} - q^{9}+O(q^{10})$$ q + b1 * q^2 + b3 * q^3 + b2 * q^4 - b5 * q^6 + (b10 + b1) * q^7 + (b10 + b7) * q^8 - q^9 $$q + \beta_1 q^{2} + \beta_{3} q^{3} + \beta_{2} q^{4} - \beta_{5} q^{6} + (\beta_{10} + \beta_1) q^{7} + (\beta_{10} + \beta_{7}) q^{8} - q^{9} + ( - \beta_{8} + \beta_{5} + \beta_{2}) q^{11} + \beta_{9} q^{12} + ( - \beta_{9} - \beta_{6} - 2 \beta_{3} + \beta_1) q^{13} + ( - \beta_{11} + \beta_{4} + \beta_{2} + 2) q^{14} + (\beta_{8} + \beta_{4}) q^{16} + (\beta_{9} - \beta_{6} - \beta_1) q^{17} - \beta_1 q^{18} + ( - \beta_{11} - \beta_{8} + 2 \beta_{4}) q^{19} + ( - \beta_{5} + \beta_{4}) q^{21} + (\beta_{10} - \beta_{9} + 2 \beta_{7} - \beta_{6} - 2 \beta_{3}) q^{22} + (2 \beta_{9} - \beta_{7} - \beta_{6}) q^{23} + ( - \beta_{8} + \beta_{4}) q^{24} + ( - \beta_{11} + 2 \beta_{5} - \beta_{4} + \beta_{2} - 2) q^{26} - \beta_{3} q^{27} + (2 \beta_{6} + 2 \beta_1) q^{28} + (\beta_{11} - \beta_{8} - \beta_{5} + \beta_{4} + 2 \beta_{2}) q^{29} + ( - \beta_{11} - \beta_{8} - 2 \beta_{5} - 2) q^{31} + ( - \beta_{10} - \beta_{7} + 2 \beta_{6} + 4 \beta_{3}) q^{32} + (\beta_{9} - \beta_{7} + \beta_1) q^{33} + ( - \beta_{11} - 2 \beta_{8} + \beta_{4} - \beta_{2} - 2) q^{34} - \beta_{2} q^{36} + ( - \beta_{9} - \beta_{6} + 2 \beta_{3} + \beta_1) q^{37} + ( - 2 \beta_{10} + 2 \beta_{6}) q^{38} + ( - \beta_{11} - \beta_{5} + \beta_{2} + 2) q^{39} + (2 \beta_{11} + 2 \beta_{8} + 4 \beta_{5} - 2) q^{41} + ( - \beta_{10} + \beta_{9} + \beta_{6} + 2 \beta_{3}) q^{42} + ( - 2 \beta_{9} - 2 \beta_{7} - 2 \beta_1) q^{43} + (2 \beta_{8} + 2 \beta_{5} - 4) q^{44} + ( - 2 \beta_{11} - 4 \beta_{8} + 2 \beta_{4}) q^{46} + ( - 2 \beta_{10} - \beta_{7} + \beta_{6}) q^{47} + ( - \beta_{10} + \beta_{7}) q^{48} + (2 \beta_{8} + 2 \beta_{5} + 2 \beta_{2} + 1) q^{49} + ( - \beta_{11} + \beta_{5} - \beta_{2}) q^{51} + (2 \beta_{10} - 2 \beta_{9} - 4 \beta_{3} - 2 \beta_1) q^{52} + ( - \beta_{9} - \beta_{7} + 4 \beta_{3} - \beta_1) q^{53} + \beta_{5} q^{54} + (2 \beta_{11} + 2 \beta_{8} + 2 \beta_{2} + 4) q^{56} + ( - 2 \beta_{10} - \beta_{7} + \beta_{6}) q^{57} + (\beta_{10} + \beta_{9} + 4 \beta_{7} - \beta_{6} + 2 \beta_{3}) q^{58} + ( - \beta_{8} + 3 \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{59} + (2 \beta_{11} + 2 \beta_{8} - 4 \beta_{5}) q^{61} + (2 \beta_{9} - 4 \beta_{3} - 2 \beta_1) q^{62} + ( - \beta_{10} - \beta_1) q^{63} + (2 \beta_{11} + \beta_{8} - 4 \beta_{5} - \beta_{4} + 4) q^{64} + ( - \beta_{11} - 2 \beta_{8} + \beta_{4} + \beta_{2} + 2) q^{66} + ( - 2 \beta_{9} - 2 \beta_{7} - 2 \beta_1) q^{67} + ( - 2 \beta_{10} - 4 \beta_{3} - 2 \beta_1) q^{68} + ( - \beta_{11} + \beta_{8} - 2 \beta_{2}) q^{69} + ( - 2 \beta_{11} - 2 \beta_{8} - 4 \beta_{5} + 4) q^{71} + ( - \beta_{10} - \beta_{7}) q^{72} + (2 \beta_{10} + 2 \beta_{9} + 2 \beta_{7} - 4 \beta_{6} - 4 \beta_1) q^{73} + ( - \beta_{11} - 2 \beta_{5} - \beta_{4} + \beta_{2} - 2) q^{74} + (4 \beta_{11} + 2 \beta_{8} - 2 \beta_{4}) q^{76} - 4 \beta_{3} q^{77} + (\beta_{10} + \beta_{9} + \beta_{6} - 2 \beta_{3} + 2 \beta_1) q^{78} + ( - \beta_{11} - \beta_{8} - 2 \beta_{5} - 2) q^{79} + q^{81} + ( - 4 \beta_{9} + 8 \beta_{3} - 2 \beta_1) q^{82} + ( - 2 \beta_{9} - 2 \beta_{6} - 4 \beta_{3} + 2 \beta_1) q^{83} + (2 \beta_{11} - 2 \beta_{5}) q^{84} + ( - 2 \beta_{11} - 2 \beta_{4} - 2 \beta_{2} + 4) q^{86} + ( - \beta_{10} + 2 \beta_{9} - \beta_{7} - \beta_{6} - \beta_1) q^{87} + ( - 2 \beta_{9} - 2 \beta_{7} + 2 \beta_{6} + 4 \beta_{3} - 4 \beta_1) q^{88} + (2 \beta_{11} - 2 \beta_{8} - 4 \beta_{2} + 2) q^{89} + ( - 4 \beta_{11} - 2 \beta_{8} + 6 \beta_{5} - 2 \beta_{2}) q^{91} + ( - 2 \beta_{10} + 2 \beta_{7} - 8 \beta_{3}) q^{92} + ( - \beta_{7} + \beta_{6} - 2 \beta_{3} - 2 \beta_1) q^{93} + (2 \beta_{11} - 2 \beta_{4}) q^{94} + (2 \beta_{11} + \beta_{8} - \beta_{4} - 4) q^{96} + (4 \beta_{10} - 2 \beta_{9} + 4 \beta_{7} - 2 \beta_{6} - 2 \beta_1) q^{97} + (2 \beta_{10} - 2 \beta_{9} + 2 \beta_{6} + 4 \beta_{3} + \beta_1) q^{98} + (\beta_{8} - \beta_{5} - \beta_{2}) q^{99}+O(q^{100})$$ q + b1 * q^2 + b3 * q^3 + b2 * q^4 - b5 * q^6 + (b10 + b1) * q^7 + (b10 + b7) * q^8 - q^9 + (-b8 + b5 + b2) * q^11 + b9 * q^12 + (-b9 - b6 - 2*b3 + b1) * q^13 + (-b11 + b4 + b2 + 2) * q^14 + (b8 + b4) * q^16 + (b9 - b6 - b1) * q^17 - b1 * q^18 + (-b11 - b8 + 2*b4) * q^19 + (-b5 + b4) * q^21 + (b10 - b9 + 2*b7 - b6 - 2*b3) * q^22 + (2*b9 - b7 - b6) * q^23 + (-b8 + b4) * q^24 + (-b11 + 2*b5 - b4 + b2 - 2) * q^26 - b3 * q^27 + (2*b6 + 2*b1) * q^28 + (b11 - b8 - b5 + b4 + 2*b2) * q^29 + (-b11 - b8 - 2*b5 - 2) * q^31 + (-b10 - b7 + 2*b6 + 4*b3) * q^32 + (b9 - b7 + b1) * q^33 + (-b11 - 2*b8 + b4 - b2 - 2) * q^34 - b2 * q^36 + (-b9 - b6 + 2*b3 + b1) * q^37 + (-2*b10 + 2*b6) * q^38 + (-b11 - b5 + b2 + 2) * q^39 + (2*b11 + 2*b8 + 4*b5 - 2) * q^41 + (-b10 + b9 + b6 + 2*b3) * q^42 + (-2*b9 - 2*b7 - 2*b1) * q^43 + (2*b8 + 2*b5 - 4) * q^44 + (-2*b11 - 4*b8 + 2*b4) * q^46 + (-2*b10 - b7 + b6) * q^47 + (-b10 + b7) * q^48 + (2*b8 + 2*b5 + 2*b2 + 1) * q^49 + (-b11 + b5 - b2) * q^51 + (2*b10 - 2*b9 - 4*b3 - 2*b1) * q^52 + (-b9 - b7 + 4*b3 - b1) * q^53 + b5 * q^54 + (2*b11 + 2*b8 + 2*b2 + 4) * q^56 + (-2*b10 - b7 + b6) * q^57 + (b10 + b9 + 4*b7 - b6 + 2*b3) * q^58 + (-b8 + 3*b5 - 2*b4 + b2) * q^59 + (2*b11 + 2*b8 - 4*b5) * q^61 + (2*b9 - 4*b3 - 2*b1) * q^62 + (-b10 - b1) * q^63 + (2*b11 + b8 - 4*b5 - b4 + 4) * q^64 + (-b11 - 2*b8 + b4 + b2 + 2) * q^66 + (-2*b9 - 2*b7 - 2*b1) * q^67 + (-2*b10 - 4*b3 - 2*b1) * q^68 + (-b11 + b8 - 2*b2) * q^69 + (-2*b11 - 2*b8 - 4*b5 + 4) * q^71 + (-b10 - b7) * q^72 + (2*b10 + 2*b9 + 2*b7 - 4*b6 - 4*b1) * q^73 + (-b11 - 2*b5 - b4 + b2 - 2) * q^74 + (4*b11 + 2*b8 - 2*b4) * q^76 - 4*b3 * q^77 + (b10 + b9 + b6 - 2*b3 + 2*b1) * q^78 + (-b11 - b8 - 2*b5 - 2) * q^79 + q^81 + (-4*b9 + 8*b3 - 2*b1) * q^82 + (-2*b9 - 2*b6 - 4*b3 + 2*b1) * q^83 + (2*b11 - 2*b5) * q^84 + (-2*b11 - 2*b4 - 2*b2 + 4) * q^86 + (-b10 + 2*b9 - b7 - b6 - b1) * q^87 + (-2*b9 - 2*b7 + 2*b6 + 4*b3 - 4*b1) * q^88 + (2*b11 - 2*b8 - 4*b2 + 2) * q^89 + (-4*b11 - 2*b8 + 6*b5 - 2*b2) * q^91 + (-2*b10 + 2*b7 - 8*b3) * q^92 + (-b7 + b6 - 2*b3 - 2*b1) * q^93 + (2*b11 - 2*b4) * q^94 + (2*b11 + b8 - b4 - 4) * q^96 + (4*b10 - 2*b9 + 4*b7 - 2*b6 - 2*b1) * q^97 + (2*b10 - 2*b9 + 2*b6 + 4*b3 + b1) * q^98 + (b8 - b5 - b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 2 q^{4} - 2 q^{6} - 12 q^{9}+O(q^{10})$$ 12 * q - 2 * q^4 - 2 * q^6 - 12 * q^9 $$12 q - 2 q^{4} - 2 q^{6} - 12 q^{9} + 20 q^{14} + 2 q^{16} + 2 q^{24} - 28 q^{26} - 32 q^{31} - 24 q^{34} + 2 q^{36} + 16 q^{39} - 8 q^{41} - 44 q^{44} - 4 q^{46} + 12 q^{49} + 2 q^{54} + 52 q^{56} + 46 q^{64} + 20 q^{66} + 32 q^{71} - 36 q^{74} + 12 q^{76} - 32 q^{79} + 12 q^{81} + 4 q^{84} + 40 q^{86} + 40 q^{89} + 4 q^{94} - 42 q^{96}+O(q^{100})$$ 12 * q - 2 * q^4 - 2 * q^6 - 12 * q^9 + 20 * q^14 + 2 * q^16 + 2 * q^24 - 28 * q^26 - 32 * q^31 - 24 * q^34 + 2 * q^36 + 16 * q^39 - 8 * q^41 - 44 * q^44 - 4 * q^46 + 12 * q^49 + 2 * q^54 + 52 * q^56 + 46 * q^64 + 20 * q^66 + 32 * q^71 - 36 * q^74 + 12 * q^76 - 32 * q^79 + 12 * q^81 + 4 * q^84 + 40 * q^86 + 40 * q^89 + 4 * q^94 - 42 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$( \nu^{11} - \nu^{9} + 2\nu^{7} + 4\nu^{5} + 8\nu^{3} ) / 64$$ (v^11 - v^9 + 2*v^7 + 4*v^5 + 8*v^3) / 64 $$\beta_{4}$$ $$=$$ $$( \nu^{10} + 3\nu^{8} - 2\nu^{6} + 12\nu^{4} - 8\nu^{2} + 32 ) / 32$$ (v^10 + 3*v^8 - 2*v^6 + 12*v^4 - 8*v^2 + 32) / 32 $$\beta_{5}$$ $$=$$ $$( \nu^{10} - \nu^{8} - 6\nu^{6} - 4\nu^{4} + 8\nu^{2} + 32 ) / 32$$ (v^10 - v^8 - 6*v^6 - 4*v^4 + 8*v^2 + 32) / 32 $$\beta_{6}$$ $$=$$ $$( -\nu^{11} + \nu^{9} - 2\nu^{7} + 12\nu^{5} + 8\nu^{3} ) / 32$$ (-v^11 + v^9 - 2*v^7 + 12*v^5 + 8*v^3) / 32 $$\beta_{7}$$ $$=$$ $$( \nu^{11} + 3\nu^{9} - 2\nu^{7} - 4\nu^{5} + 8\nu^{3} + 32\nu ) / 32$$ (v^11 + 3*v^9 - 2*v^7 - 4*v^5 + 8*v^3 + 32*v) / 32 $$\beta_{8}$$ $$=$$ $$( -\nu^{10} - 3\nu^{8} + 2\nu^{6} + 20\nu^{4} + 8\nu^{2} - 32 ) / 32$$ (-v^10 - 3*v^8 + 2*v^6 + 20*v^4 + 8*v^2 - 32) / 32 $$\beta_{9}$$ $$=$$ $$( -\nu^{11} + \nu^{9} + 6\nu^{7} + 4\nu^{5} - 8\nu^{3} - 32\nu ) / 32$$ (-v^11 + v^9 + 6*v^7 + 4*v^5 - 8*v^3 - 32*v) / 32 $$\beta_{10}$$ $$=$$ $$( -\nu^{11} - 3\nu^{9} + 2\nu^{7} + 4\nu^{5} + 24\nu^{3} - 32\nu ) / 32$$ (-v^11 - 3*v^9 + 2*v^7 + 4*v^5 + 24*v^3 - 32*v) / 32 $$\beta_{11}$$ $$=$$ $$( 3\nu^{10} + \nu^{8} + 2\nu^{6} - 12\nu^{4} + 8\nu^{2} + 32 ) / 32$$ (3*v^10 + v^8 + 2*v^6 - 12*v^4 + 8*v^2 + 32) / 32
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\nu^{3}$$ $$=$$ $$\beta_{10} + \beta_{7}$$ b10 + b7 $$\nu^{4}$$ $$=$$ $$\beta_{8} + \beta_{4}$$ b8 + b4 $$\nu^{5}$$ $$=$$ $$-\beta_{10} - \beta_{7} + 2\beta_{6} + 4\beta_{3}$$ -b10 - b7 + 2*b6 + 4*b3 $$\nu^{6}$$ $$=$$ $$2\beta_{11} + \beta_{8} - 4\beta_{5} - \beta_{4} + 4$$ 2*b11 + b8 - 4*b5 - b4 + 4 $$\nu^{7}$$ $$=$$ $$\beta_{10} + 4\beta_{9} + \beta_{7} - 2\beta_{6} + 4\beta_{3} + 4\beta_1$$ b10 + 4*b9 + b7 - 2*b6 + 4*b3 + 4*b1 $$\nu^{8}$$ $$=$$ $$-2\beta_{11} - 5\beta_{8} - 4\beta_{5} + 5\beta_{4} + 4\beta_{2} - 4$$ -2*b11 - 5*b8 - 4*b5 + 5*b4 + 4*b2 - 4 $$\nu^{9}$$ $$=$$ $$-\beta_{10} + 4\beta_{9} + 7\beta_{7} + 2\beta_{6} - 4\beta_{3} - 4\beta_1$$ -b10 + 4*b9 + 7*b7 + 2*b6 - 4*b3 - 4*b1 $$\nu^{10}$$ $$=$$ $$10\beta_{11} + 5\beta_{8} + 4\beta_{5} + 3\beta_{4} - 4\beta_{2} - 12$$ 10*b11 + 5*b8 + 4*b5 + 3*b4 - 4*b2 - 12 $$\nu^{11}$$ $$=$$ $$-7\beta_{10} - 4\beta_{9} + \beta_{7} - 2\beta_{6} + 36\beta_{3} - 12\beta_1$$ -7*b10 - 4*b9 + b7 - 2*b6 + 36*b3 - 12*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$$.

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
301.1
 −1.37729 − 0.321037i −1.37729 + 0.321037i −0.806504 − 1.16170i −0.806504 + 1.16170i −0.450129 − 1.34067i −0.450129 + 1.34067i 0.450129 − 1.34067i 0.450129 + 1.34067i 0.806504 − 1.16170i 0.806504 + 1.16170i 1.37729 − 0.321037i 1.37729 + 0.321037i
−1.37729 0.321037i 1.00000i 1.79387 + 0.884323i 0 −0.321037 + 1.37729i −4.05705 −2.18678 1.79387i −1.00000 0
301.2 −1.37729 + 0.321037i 1.00000i 1.79387 0.884323i 0 −0.321037 1.37729i −4.05705 −2.18678 + 1.79387i −1.00000 0
301.3 −0.806504 1.16170i 1.00000i −0.699104 + 1.87383i 0 1.16170 0.806504i −0.746175 2.74067 0.699104i −1.00000 0
301.4 −0.806504 + 1.16170i 1.00000i −0.699104 1.87383i 0 1.16170 + 0.806504i −0.746175 2.74067 + 0.699104i −1.00000 0
301.5 −0.450129 1.34067i 1.00000i −1.59477 + 1.20695i 0 −1.34067 + 0.450129i 2.64265 2.33596 + 1.59477i −1.00000 0
301.6 −0.450129 + 1.34067i 1.00000i −1.59477 1.20695i 0 −1.34067 0.450129i 2.64265 2.33596 1.59477i −1.00000 0
301.7 0.450129 1.34067i 1.00000i −1.59477 1.20695i 0 −1.34067 0.450129i −2.64265 −2.33596 + 1.59477i −1.00000 0
301.8 0.450129 + 1.34067i 1.00000i −1.59477 + 1.20695i 0 −1.34067 + 0.450129i −2.64265 −2.33596 1.59477i −1.00000 0
301.9 0.806504 1.16170i 1.00000i −0.699104 1.87383i 0 1.16170 + 0.806504i 0.746175 −2.74067 0.699104i −1.00000 0
301.10 0.806504 + 1.16170i 1.00000i −0.699104 + 1.87383i 0 1.16170 0.806504i 0.746175 −2.74067 + 0.699104i −1.00000 0
301.11 1.37729 0.321037i 1.00000i 1.79387 0.884323i 0 −0.321037 1.37729i 4.05705 2.18678 1.79387i −1.00000 0
301.12 1.37729 + 0.321037i 1.00000i 1.79387 + 0.884323i 0 −0.321037 + 1.37729i 4.05705 2.18678 + 1.79387i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 301.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.k.f 12
3.b odd 2 1 1800.2.k.u 12
4.b odd 2 1 2400.2.k.f 12
5.b even 2 1 inner 600.2.k.f 12
5.c odd 4 1 120.2.d.a 6
5.c odd 4 1 120.2.d.b yes 6
8.b even 2 1 inner 600.2.k.f 12
8.d odd 2 1 2400.2.k.f 12
12.b even 2 1 7200.2.k.u 12
15.d odd 2 1 1800.2.k.u 12
15.e even 4 1 360.2.d.e 6
15.e even 4 1 360.2.d.f 6
20.d odd 2 1 2400.2.k.f 12
20.e even 4 1 480.2.d.a 6
20.e even 4 1 480.2.d.b 6
24.f even 2 1 7200.2.k.u 12
24.h odd 2 1 1800.2.k.u 12
40.e odd 2 1 2400.2.k.f 12
40.f even 2 1 inner 600.2.k.f 12
40.i odd 4 1 120.2.d.a 6
40.i odd 4 1 120.2.d.b yes 6
40.k even 4 1 480.2.d.a 6
40.k even 4 1 480.2.d.b 6
60.h even 2 1 7200.2.k.u 12
60.l odd 4 1 1440.2.d.e 6
60.l odd 4 1 1440.2.d.f 6
80.i odd 4 2 3840.2.f.l 12
80.j even 4 2 3840.2.f.m 12
80.s even 4 2 3840.2.f.m 12
80.t odd 4 2 3840.2.f.l 12
120.i odd 2 1 1800.2.k.u 12
120.m even 2 1 7200.2.k.u 12
120.q odd 4 1 1440.2.d.e 6
120.q odd 4 1 1440.2.d.f 6
120.w even 4 1 360.2.d.e 6
120.w even 4 1 360.2.d.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.d.a 6 5.c odd 4 1
120.2.d.a 6 40.i odd 4 1
120.2.d.b yes 6 5.c odd 4 1
120.2.d.b yes 6 40.i odd 4 1
360.2.d.e 6 15.e even 4 1
360.2.d.e 6 120.w even 4 1
360.2.d.f 6 15.e even 4 1
360.2.d.f 6 120.w even 4 1
480.2.d.a 6 20.e even 4 1
480.2.d.a 6 40.k even 4 1
480.2.d.b 6 20.e even 4 1
480.2.d.b 6 40.k even 4 1
600.2.k.f 12 1.a even 1 1 trivial
600.2.k.f 12 5.b even 2 1 inner
600.2.k.f 12 8.b even 2 1 inner
600.2.k.f 12 40.f even 2 1 inner
1440.2.d.e 6 60.l odd 4 1
1440.2.d.e 6 120.q odd 4 1
1440.2.d.f 6 60.l odd 4 1
1440.2.d.f 6 120.q odd 4 1
1800.2.k.u 12 3.b odd 2 1
1800.2.k.u 12 15.d odd 2 1
1800.2.k.u 12 24.h odd 2 1
1800.2.k.u 12 120.i odd 2 1
2400.2.k.f 12 4.b odd 2 1
2400.2.k.f 12 8.d odd 2 1
2400.2.k.f 12 20.d odd 2 1
2400.2.k.f 12 40.e odd 2 1
3840.2.f.l 12 80.i odd 4 2
3840.2.f.l 12 80.t odd 4 2
3840.2.f.m 12 80.j even 4 2
3840.2.f.m 12 80.s even 4 2
7200.2.k.u 12 12.b even 2 1
7200.2.k.u 12 24.f even 2 1
7200.2.k.u 12 60.h even 2 1
7200.2.k.u 12 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{6} - 24T_{7}^{4} + 128T_{7}^{2} - 64$$ acting on $$S_{2}^{\mathrm{new}}(600, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + T^{10} - 8 T^{6} + 16 T^{2} + \cdots + 64$$
$3$ $$(T^{2} + 1)^{6}$$
$5$ $$T^{12}$$
$7$ $$(T^{6} - 24 T^{4} + 128 T^{2} - 64)^{2}$$
$11$ $$(T^{6} + 32 T^{4} + 96 T^{2} + 64)^{2}$$
$13$ $$(T^{6} + 48 T^{4} + 704 T^{2} + 3136)^{2}$$
$17$ $$(T^{6} - 36 T^{4} + 368 T^{2} - 1024)^{2}$$
$19$ $$(T^{6} + 60 T^{4} + 512 T^{2} + 1024)^{2}$$
$23$ $$(T^{6} - 92 T^{4} + 2304 T^{2} + \cdots - 16384)^{2}$$
$29$ $$(T^{6} + 108 T^{4} + 3120 T^{2} + \cdots + 12544)^{2}$$
$31$ $$(T^{3} + 8 T^{2} - 4 T - 64)^{4}$$
$37$ $$(T^{6} + 64 T^{4} + 128 T^{2} + 64)^{2}$$
$41$ $$(T^{3} + 2 T^{2} - 100 T + 56)^{4}$$
$43$ $$(T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 4096)^{2}$$
$47$ $$(T^{6} - 60 T^{4} + 512 T^{2} - 1024)^{2}$$
$53$ $$(T^{6} + 80 T^{4} + 1216 T^{2} + 64)^{2}$$
$59$ $$(T^{6} + 176 T^{4} + 9888 T^{2} + \cdots + 179776)^{2}$$
$61$ $$(T^{6} + 176 T^{4} + 7168 T^{2} + \cdots + 65536)^{2}$$
$67$ $$(T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 4096)^{2}$$
$71$ $$(T^{3} - 8 T^{2} - 80 T + 128)^{4}$$
$73$ $$(T^{6} - 384 T^{4} + 34560 T^{2} + \cdots - 16384)^{2}$$
$79$ $$(T^{3} + 8 T^{2} - 4 T - 64)^{4}$$
$83$ $$(T^{6} + 192 T^{4} + 11264 T^{2} + \cdots + 200704)^{2}$$
$89$ $$(T^{3} - 10 T^{2} - 164 T + 1384)^{4}$$
$97$ $$(T^{6} - 336 T^{4} + 28416 T^{2} + \cdots - 262144)^{2}$$