# Properties

 Label 600.2.b.h Level 600 Weight 2 Character orbit 600.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.79102412128$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.537291533250985984.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{6} q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -\beta_{2} - \beta_{10} ) q^{6} + ( -1 - \beta_{3} - \beta_{10} - \beta_{11} ) q^{7} + ( -\beta_{1} - \beta_{7} + \beta_{8} ) q^{8} + ( \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{6} q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -\beta_{2} - \beta_{10} ) q^{6} + ( -1 - \beta_{3} - \beta_{10} - \beta_{11} ) q^{7} + ( -\beta_{1} - \beta_{7} + \beta_{8} ) q^{8} + ( \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{9} + ( \beta_{7} - \beta_{8} + \beta_{9} ) q^{11} + ( -1 - \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{11} ) q^{12} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{10} + \beta_{11} ) q^{13} + ( -\beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{14} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{16} + ( -\beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{17} + ( \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} ) q^{18} + ( -1 - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{19} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{21} + ( \beta_{2} - \beta_{3} + \beta_{4} ) q^{22} + ( -\beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{23} + ( -\beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{10} ) q^{24} + ( \beta_{1} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{26} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{27} + ( -3 - \beta_{3} + 3 \beta_{4} - 2 \beta_{10} - 2 \beta_{11} ) q^{28} + ( -3 \beta_{1} + \beta_{7} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{29} + ( -1 - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{10} - \beta_{11} ) q^{31} + ( -2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{32} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{8} ) q^{33} + ( -2 - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{10} - 2 \beta_{11} ) q^{34} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{36} + ( 2 \beta_{2} - 2 \beta_{4} ) q^{37} + ( \beta_{1} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} ) q^{38} + ( 1 - 2 \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{39} + ( 3 \beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{41} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{42} + ( -3 + 3 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{43} + ( 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{44} + ( -\beta_{2} - \beta_{3} - 3 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + \beta_{10} + \beta_{11} ) q^{46} + ( -2 \beta_{1} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} ) q^{47} + ( -4 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{48} + ( -4 - 3 \beta_{5} + 3 \beta_{6} ) q^{49} + ( -3 - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{51} + ( 1 + 3 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{10} + \beta_{11} ) q^{52} + ( -\beta_{1} + \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{53} + ( -4 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{10} - 2 \beta_{11} ) q^{54} + ( -\beta_{1} + 4 \beta_{5} + 4 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{56} + ( 4 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{57} + ( 6 + \beta_{2} - \beta_{3} - 3 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + \beta_{10} + \beta_{11} ) q^{58} + ( 2 \beta_{1} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} ) q^{59} + ( 3 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{10} + 3 \beta_{11} ) q^{61} + ( -\beta_{1} + 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} ) q^{62} + ( -1 - 4 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + \beta_{10} - 3 \beta_{11} ) q^{63} + ( 2 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{10} + \beta_{11} ) q^{64} + ( 2 - \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} + 2 \beta_{11} ) q^{66} + ( -5 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{67} + ( -2 \beta_{1} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{68} + ( 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{69} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{71} + ( -2 - 2 \beta_{1} + \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{72} + ( -4 \beta_{2} - 4 \beta_{4} + \beta_{5} - \beta_{6} ) q^{73} + ( -2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} ) q^{74} + ( -5 - \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{10} - \beta_{11} ) q^{76} + ( 3 \beta_{1} - \beta_{7} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{77} + ( 4 + \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{78} + ( 4 + 4 \beta_{2} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{10} + 4 \beta_{11} ) q^{79} + ( -2 - 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{81} + ( 6 - 3 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{10} + 2 \beta_{11} ) q^{82} + ( -2 \beta_{1} - 3 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} + 6 \beta_{8} - 4 \beta_{9} ) q^{83} + ( -1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + \beta_{10} + 3 \beta_{11} ) q^{84} + ( 3 \beta_{1} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 2 \beta_{10} + 2 \beta_{11} ) q^{86} + ( -2 + 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{87} + ( 2 + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{10} - \beta_{11} ) q^{88} + ( 2 \beta_{1} + 3 \beta_{5} + 3 \beta_{6} - 2 \beta_{8} ) q^{89} + ( -1 + 5 \beta_{2} + 5 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{91} + ( 2 \beta_{1} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{92} + ( 3 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + 5 \beta_{11} ) q^{93} + ( 4 + 2 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} ) q^{94} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{11} ) q^{96} + ( -3 - 4 \beta_{2} - 4 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{97} + ( 4 \beta_{1} - 3 \beta_{10} + 3 \beta_{11} ) q^{98} + ( 5 + 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 2q^{3} + 10q^{4} + 7q^{6} - 2q^{9} + O(q^{10})$$ $$12q + 2q^{3} + 10q^{4} + 7q^{6} - 2q^{9} - 3q^{12} - 6q^{16} - 5q^{18} - 4q^{19} - 2q^{22} + 5q^{24} + 8q^{27} - 20q^{28} + 18q^{33} - 2q^{34} + 19q^{36} - 14q^{42} - 40q^{43} - 16q^{46} - 27q^{48} - 36q^{49} - 30q^{51} - 4q^{52} - 30q^{54} + 42q^{57} + 52q^{58} + 10q^{64} + 7q^{66} - 60q^{67} - 39q^{72} + 12q^{73} - 38q^{76} + 54q^{78} - 10q^{81} + 58q^{82} - 34q^{84} + 34q^{88} - 24q^{91} + 28q^{94} - 31q^{96} - 32q^{97} + 58q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 5 x^{10} + 14 x^{8} - 30 x^{6} + 56 x^{4} - 80 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 1$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{10} - 5 \nu^{8} + 14 \nu^{6} - 14 \nu^{4} + 40 \nu^{2} - 48$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{10} + 5 \nu^{8} - 14 \nu^{6} + 30 \nu^{4} - 56 \nu^{2} + 64$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{11} - 6 \nu^{10} - \nu^{9} + 22 \nu^{8} + 10 \nu^{7} - 44 \nu^{6} - 22 \nu^{5} + 100 \nu^{4} + 32 \nu^{3} - 160 \nu^{2} - 16 \nu + 160$$$$)/64$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{11} + 6 \nu^{10} - \nu^{9} - 22 \nu^{8} + 10 \nu^{7} + 44 \nu^{6} - 22 \nu^{5} - 100 \nu^{4} + 32 \nu^{3} + 160 \nu^{2} - 16 \nu - 160$$$$)/64$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{11} + 9 \nu^{9} - 18 \nu^{7} + 38 \nu^{5} - 48 \nu^{3} + 80 \nu$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{11} + 9 \nu^{9} - 18 \nu^{7} + 38 \nu^{5} - 80 \nu^{3} + 112 \nu$$$$)/32$$ $$\beta_{9}$$ $$=$$ $$($$$$3 \nu^{11} - 11 \nu^{9} + 22 \nu^{7} - 34 \nu^{5} + 48 \nu^{3} - 48 \nu$$$$)/32$$ $$\beta_{10}$$ $$=$$ $$($$$$3 \nu^{11} + 2 \nu^{10} - 11 \nu^{9} - 2 \nu^{8} + 22 \nu^{7} + 4 \nu^{6} - 50 \nu^{5} - 12 \nu^{4} + 80 \nu^{3} - 80 \nu$$$$)/32$$ $$\beta_{11}$$ $$=$$ $$($$$$-3 \nu^{11} + 2 \nu^{10} + 11 \nu^{9} - 2 \nu^{8} - 22 \nu^{7} + 4 \nu^{6} + 50 \nu^{5} - 12 \nu^{4} - 80 \nu^{3} + 80 \nu$$$$)/32$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$-\beta_{8} + \beta_{7} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$\beta_{11} - \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7}$$ $$\nu^{6}$$ $$=$$ $$\beta_{11} + \beta_{10} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2$$ $$\nu^{7}$$ $$=$$ $$3 \beta_{11} - 3 \beta_{10} + 4 \beta_{9} - 2 \beta_{8} + 4 \beta_{6} + 4 \beta_{5}$$ $$\nu^{8}$$ $$=$$ $$5 \beta_{11} + 5 \beta_{10} - 6 \beta_{6} + 6 \beta_{5} - 8 \beta_{4} + 2 \beta_{2} + 4$$ $$\nu^{9}$$ $$=$$ $$\beta_{11} - \beta_{10} + 2 \beta_{7} + 8 \beta_{6} + 8 \beta_{5} - 6 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$11 \beta_{11} + 11 \beta_{10} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 12 \beta_{2}$$ $$\nu^{11}$$ $$=$$ $$-7 \beta_{11} + 7 \beta_{10} + 4 \beta_{9} + 8 \beta_{8} + 14 \beta_{7} - 22 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 1.39298 + 0.244153i 1.39298 − 0.244153i 1.26128 + 0.639662i 1.26128 − 0.639662i 0.847808 + 1.13191i 0.847808 − 1.13191i −0.847808 + 1.13191i −0.847808 − 1.13191i −1.26128 + 0.639662i −1.26128 − 0.639662i −1.39298 + 0.244153i −1.39298 − 0.244153i
−1.39298 0.244153i −1.31310 + 1.12950i 1.88078 + 0.680200i 0 2.10489 1.25277i 4.34495i −2.45381 1.40670i 0.448458 2.96629i 0
251.2 −1.39298 + 0.244153i −1.31310 1.12950i 1.88078 0.680200i 0 2.10489 + 1.25277i 4.34495i −2.45381 + 1.40670i 0.448458 + 2.96629i 0
251.3 −1.26128 0.639662i 1.57067 0.730070i 1.18166 + 1.61359i 0 −2.44805 0.0838735i 1.25539i −0.458259 2.79106i 1.93400 2.29339i 0
251.4 −1.26128 + 0.639662i 1.57067 + 0.730070i 1.18166 1.61359i 0 −2.44805 + 0.0838735i 1.25539i −0.458259 + 2.79106i 1.93400 + 2.29339i 0
251.5 −0.847808 1.13191i 0.242431 + 1.71500i −0.562443 + 1.91929i 0 1.73569 1.72840i 3.08957i 2.64930 0.990551i −2.88245 + 0.831539i 0
251.6 −0.847808 + 1.13191i 0.242431 1.71500i −0.562443 1.91929i 0 1.73569 + 1.72840i 3.08957i 2.64930 + 0.990551i −2.88245 0.831539i 0
251.7 0.847808 1.13191i 0.242431 + 1.71500i −0.562443 1.91929i 0 2.14676 + 1.17958i 3.08957i −2.64930 0.990551i −2.88245 + 0.831539i 0
251.8 0.847808 + 1.13191i 0.242431 1.71500i −0.562443 + 1.91929i 0 2.14676 1.17958i 3.08957i −2.64930 + 0.990551i −2.88245 0.831539i 0
251.9 1.26128 0.639662i 1.57067 0.730070i 1.18166 1.61359i 0 1.51406 1.92552i 1.25539i 0.458259 2.79106i 1.93400 2.29339i 0
251.10 1.26128 + 0.639662i 1.57067 + 0.730070i 1.18166 + 1.61359i 0 1.51406 + 1.92552i 1.25539i 0.458259 + 2.79106i 1.93400 + 2.29339i 0
251.11 1.39298 0.244153i −1.31310 + 1.12950i 1.88078 0.680200i 0 −1.55335 + 1.89397i 4.34495i 2.45381 1.40670i 0.448458 2.96629i 0
251.12 1.39298 + 0.244153i −1.31310 1.12950i 1.88078 + 0.680200i 0 −1.55335 1.89397i 4.34495i 2.45381 + 1.40670i 0.448458 + 2.96629i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.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.b.h yes 12
3.b odd 2 1 inner 600.2.b.h yes 12
4.b odd 2 1 2400.2.b.g 12
5.b even 2 1 600.2.b.g 12
5.c odd 4 2 600.2.m.e 24
8.b even 2 1 2400.2.b.g 12
8.d odd 2 1 inner 600.2.b.h yes 12
12.b even 2 1 2400.2.b.g 12
15.d odd 2 1 600.2.b.g 12
15.e even 4 2 600.2.m.e 24
20.d odd 2 1 2400.2.b.h 12
20.e even 4 2 2400.2.m.e 24
24.f even 2 1 inner 600.2.b.h yes 12
24.h odd 2 1 2400.2.b.g 12
40.e odd 2 1 600.2.b.g 12
40.f even 2 1 2400.2.b.h 12
40.i odd 4 2 2400.2.m.e 24
40.k even 4 2 600.2.m.e 24
60.h even 2 1 2400.2.b.h 12
60.l odd 4 2 2400.2.m.e 24
120.i odd 2 1 2400.2.b.h 12
120.m even 2 1 600.2.b.g 12
120.q odd 4 2 600.2.m.e 24
120.w even 4 2 2400.2.m.e 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.b.g 12 5.b even 2 1
600.2.b.g 12 15.d odd 2 1
600.2.b.g 12 40.e odd 2 1
600.2.b.g 12 120.m even 2 1
600.2.b.h yes 12 1.a even 1 1 trivial
600.2.b.h yes 12 3.b odd 2 1 inner
600.2.b.h yes 12 8.d odd 2 1 inner
600.2.b.h yes 12 24.f even 2 1 inner
600.2.m.e 24 5.c odd 4 2
600.2.m.e 24 15.e even 4 2
600.2.m.e 24 40.k even 4 2
600.2.m.e 24 120.q odd 4 2
2400.2.b.g 12 4.b odd 2 1
2400.2.b.g 12 8.b even 2 1
2400.2.b.g 12 12.b even 2 1
2400.2.b.g 12 24.h odd 2 1
2400.2.b.h 12 20.d odd 2 1
2400.2.b.h 12 40.f even 2 1
2400.2.b.h 12 60.h even 2 1
2400.2.b.h 12 120.i odd 2 1
2400.2.m.e 24 20.e even 4 2
2400.2.m.e 24 40.i odd 4 2
2400.2.m.e 24 60.l odd 4 2
2400.2.m.e 24 120.w even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(600, [\chi])$$:

 $$T_{7}^{6} + 30 T_{7}^{4} + 225 T_{7}^{2} + 284$$ $$T_{11}^{6} + 19 T_{11}^{4} + 112 T_{11}^{2} + 200$$ $$T_{23}^{6} - 104 T_{23}^{4} + 2136 T_{23}^{2} - 9088$$ $$T_{43}^{3} + 10 T_{43}^{2} - 29 T_{43} - 148$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 5 T^{2} + 14 T^{4} - 30 T^{6} + 56 T^{8} - 80 T^{10} + 64 T^{12}$$
$3$ $$( 1 - T + T^{2} - 2 T^{3} + 3 T^{4} - 9 T^{5} + 27 T^{6} )^{2}$$
$5$ 1
$7$ $$( 1 - 12 T^{2} + 120 T^{4} - 906 T^{6} + 5880 T^{8} - 28812 T^{10} + 117649 T^{12} )^{2}$$
$11$ $$( 1 - 47 T^{2} + 1091 T^{4} - 15090 T^{6} + 132011 T^{8} - 688127 T^{10} + 1771561 T^{12} )^{2}$$
$13$ $$( 1 - 20 T^{2} + 360 T^{4} - 6710 T^{6} + 60840 T^{8} - 571220 T^{10} + 4826809 T^{12} )^{2}$$
$17$ $$( 1 - 67 T^{2} + 2131 T^{4} - 43546 T^{6} + 615859 T^{8} - 5595907 T^{10} + 24137569 T^{12} )^{2}$$
$19$ $$( 1 + T + 38 T^{2} + 63 T^{3} + 722 T^{4} + 361 T^{5} + 6859 T^{6} )^{4}$$
$23$ $$( 1 + 34 T^{2} + 503 T^{4} + 2412 T^{6} + 266087 T^{8} + 9514594 T^{10} + 148035889 T^{12} )^{2}$$
$29$ $$( 1 + 34 T^{2} + 2127 T^{4} + 58156 T^{6} + 1788807 T^{8} + 24047554 T^{10} + 594823321 T^{12} )^{2}$$
$31$ $$( 1 - 88 T^{2} + 5232 T^{4} - 186430 T^{6} + 5027952 T^{8} - 81269848 T^{10} + 887503681 T^{12} )^{2}$$
$37$ $$( 1 - 114 T^{2} + 7751 T^{4} - 344572 T^{6} + 10611119 T^{8} - 213654354 T^{10} + 2565726409 T^{12} )^{2}$$
$41$ $$( 1 - 91 T^{2} + 6403 T^{4} - 299146 T^{6} + 10763443 T^{8} - 257144251 T^{10} + 4750104241 T^{12} )^{2}$$
$43$ $$( 1 + 10 T + 100 T^{2} + 712 T^{3} + 4300 T^{4} + 18490 T^{5} + 79507 T^{6} )^{4}$$
$47$ $$( 1 + 70 T^{2} + 4799 T^{4} + 313140 T^{6} + 10600991 T^{8} + 341577670 T^{10} + 10779215329 T^{12} )^{2}$$
$53$ $$( 1 + 38 T^{2} + 6351 T^{4} + 175844 T^{6} + 17839959 T^{8} + 299838278 T^{10} + 22164361129 T^{12} )^{2}$$
$59$ $$( 1 - 214 T^{2} + 21991 T^{4} - 1515316 T^{6} + 76550671 T^{8} - 2593115254 T^{10} + 42180533641 T^{12} )^{2}$$
$61$ $$( 1 - 212 T^{2} + 24168 T^{4} - 1817654 T^{6} + 89929128 T^{8} - 2935318292 T^{10} + 51520374361 T^{12} )^{2}$$
$67$ $$( 1 + 15 T + 266 T^{2} + 2093 T^{3} + 17822 T^{4} + 67335 T^{5} + 300763 T^{6} )^{4}$$
$71$ $$( 1 + 190 T^{2} + 23367 T^{4} + 1843444 T^{6} + 117793047 T^{8} + 4828219390 T^{10} + 128100283921 T^{12} )^{2}$$
$73$ $$( 1 - 3 T + 131 T^{2} - 682 T^{3} + 9563 T^{4} - 15987 T^{5} + 389017 T^{6} )^{4}$$
$79$ $$( 1 - 126 T^{2} + 21023 T^{4} - 1571716 T^{6} + 131204543 T^{8} - 4907710206 T^{10} + 243087455521 T^{12} )^{2}$$
$83$ $$( 1 - 23 T + 209 T^{2} - 1486 T^{3} + 17347 T^{4} - 158447 T^{5} + 571787 T^{6} )^{2}( 1 + 23 T + 209 T^{2} + 1486 T^{3} + 17347 T^{4} + 158447 T^{5} + 571787 T^{6} )^{2}$$
$89$ $$( 1 - 347 T^{2} + 57331 T^{4} - 6080690 T^{6} + 454118851 T^{8} - 21771557627 T^{10} + 496981290961 T^{12} )^{2}$$
$97$ $$( 1 + 8 T + 192 T^{2} + 1310 T^{3} + 18624 T^{4} + 75272 T^{5} + 912673 T^{6} )^{4}$$