# Properties

 Label 600.2.b.i Level 600 Weight 2 Character orbit 600.b Analytic conductor 4.791 Analytic rank 0 Dimension 16 CM no Inner twists 8

# 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: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{15}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{10} q^{2} + \beta_{5} q^{3} -\beta_{14} q^{4} + ( \beta_{7} + \beta_{12} ) q^{6} + \beta_{15} q^{7} + ( -\beta_{3} - \beta_{5} - \beta_{6} ) q^{8} + ( 1 - \beta_{1} + \beta_{14} ) q^{9} +O(q^{10})$$ $$q -\beta_{10} q^{2} + \beta_{5} q^{3} -\beta_{14} q^{4} + ( \beta_{7} + \beta_{12} ) q^{6} + \beta_{15} q^{7} + ( -\beta_{3} - \beta_{5} - \beta_{6} ) q^{8} + ( 1 - \beta_{1} + \beta_{14} ) q^{9} + ( -\beta_{1} + \beta_{4} - \beta_{7} + \beta_{13} ) q^{11} + ( -\beta_{3} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{12} + ( -\beta_{3} - \beta_{8} + \beta_{11} ) q^{13} + ( -\beta_{1} - \beta_{2} + \beta_{14} ) q^{14} -2 \beta_{12} q^{16} + ( -\beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} ) q^{17} + ( \beta_{3} + \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{15} ) q^{18} + ( 2 + \beta_{13} - \beta_{14} ) q^{19} + ( -\beta_{4} - \beta_{7} + \beta_{12} ) q^{21} + ( \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{11} + \beta_{15} ) q^{22} + \beta_{3} q^{23} + ( 2 + \beta_{1} - \beta_{2} ) q^{24} + ( \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{26} + ( 2 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{27} + ( \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{11} ) q^{28} + ( -\beta_{2} - \beta_{4} - \beta_{7} - \beta_{12} + \beta_{14} ) q^{29} + ( -2 \beta_{12} - \beta_{13} - \beta_{14} ) q^{31} + ( 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{32} + ( \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{33} + ( 2 - \beta_{12} - \beta_{13} + \beta_{14} ) q^{34} + ( -2 - \beta_{1} - \beta_{2} + 2 \beta_{4} + 2 \beta_{12} - \beta_{14} ) q^{36} + ( -\beta_{3} - \beta_{8} + \beta_{11} - 2 \beta_{15} ) q^{37} + ( -\beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{9} - 2 \beta_{10} ) q^{38} + ( -\beta_{2} + \beta_{4} + \beta_{7} + 2 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} ) q^{39} + ( -2 \beta_{4} + 2 \beta_{7} - \beta_{13} + \beta_{14} ) q^{41} + ( -2 \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} ) q^{42} + ( \beta_{5} - \beta_{6} + 2 \beta_{8} + 2 \beta_{11} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{12} ) q^{44} + ( \beta_{12} - \beta_{13} ) q^{46} + \beta_{3} q^{47} + ( 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{15} ) q^{48} + ( -1 + \beta_{13} - \beta_{14} ) q^{49} + ( 2 + \beta_{1} + \beta_{4} - \beta_{7} - \beta_{14} ) q^{51} + ( -2 \beta_{5} + 2 \beta_{6} - 2 \beta_{15} ) q^{52} + ( 3 \beta_{3} - 3 \beta_{9} + 3 \beta_{10} ) q^{53} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{7} + 3 \beta_{13} ) q^{54} + ( 4 \beta_{4} + 2 \beta_{12} - 2 \beta_{14} ) q^{56} + ( 2 \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{57} + ( \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{11} - \beta_{15} ) q^{58} + ( -\beta_{1} + \beta_{4} - \beta_{7} + \beta_{13} ) q^{59} + ( 3 \beta_{13} + 3 \beta_{14} ) q^{61} + ( \beta_{3} - 3 \beta_{5} - 3 \beta_{6} - 2 \beta_{9} ) q^{62} + ( -2 \beta_{3} - \beta_{8} - 3 \beta_{9} + 3 \beta_{10} + \beta_{11} + \beta_{15} ) q^{63} -4 \beta_{13} q^{64} + ( 4 - \beta_{1} + \beta_{2} - 2 \beta_{4} - 2 \beta_{7} - \beta_{12} + 3 \beta_{13} + 3 \beta_{14} ) q^{66} + ( \beta_{5} - \beta_{6} ) q^{67} + ( 2 \beta_{3} - 2 \beta_{9} - 2 \beta_{10} ) q^{68} + ( \beta_{2} - \beta_{14} ) q^{69} + ( 4 \beta_{4} + 4 \beta_{7} + 4 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{71} + ( -\beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{10} - 2 \beta_{11} ) q^{72} + ( 2 \beta_{5} - 2 \beta_{6} ) q^{73} + ( 3 \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{74} + ( 4 - 2 \beta_{12} - 2 \beta_{14} ) q^{76} + ( 4 \beta_{3} - 2 \beta_{9} + 2 \beta_{10} ) q^{77} + ( -3 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} - 4 \beta_{9} - \beta_{11} - \beta_{15} ) q^{78} + ( -4 \beta_{12} + 3 \beta_{13} + 3 \beta_{14} ) q^{79} + ( -3 - 2 \beta_{4} + 2 \beta_{7} - 3 \beta_{13} + 3 \beta_{14} ) q^{81} + ( -\beta_{3} - 3 \beta_{5} + 3 \beta_{6} - 2 \beta_{8} ) q^{82} + ( \beta_{5} + \beta_{6} + 4 \beta_{9} + 4 \beta_{10} ) q^{83} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{7} - 2 \beta_{12} + 6 \beta_{13} ) q^{84} + ( -2 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} - 2 \beta_{7} - 3 \beta_{12} + 3 \beta_{13} + 2 \beta_{14} ) q^{86} + ( -3 \beta_{3} - 2 \beta_{9} + 2 \beta_{10} - \beta_{15} ) q^{87} + ( -2 \beta_{8} - 2 \beta_{11} + 2 \beta_{15} ) q^{88} + ( 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{7} - 2 \beta_{13} ) q^{89} + ( -4 + 4 \beta_{13} - 4 \beta_{14} ) q^{91} + ( -\beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{9} ) q^{92} + ( -2 \beta_{3} - \beta_{8} + 3 \beta_{9} - 3 \beta_{10} + \beta_{11} - 2 \beta_{15} ) q^{93} + ( \beta_{12} - \beta_{13} ) q^{94} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{14} ) q^{96} + ( 2 \beta_{8} + 2 \beta_{11} ) q^{97} + ( -\beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{9} + \beta_{10} ) q^{98} + ( -4 + \beta_{1} + \beta_{4} - \beta_{7} - 3 \beta_{13} + 2 \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 16q^{9} + O(q^{10})$$ $$16q + 16q^{9} + 32q^{19} + 32q^{24} + 32q^{34} - 32q^{36} - 16q^{49} + 32q^{51} - 32q^{54} + 64q^{66} + 64q^{76} - 48q^{81} - 32q^{84} - 64q^{91} + 64q^{96} - 64q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 24 x^{14} + 192 x^{12} + 672 x^{10} + 1092 x^{8} + 880 x^{6} + 352 x^{4} + 64 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3 \nu^{13} + 69 \nu^{11} + 506 \nu^{9} + 1488 \nu^{7} + 1638 \nu^{5} + 594 \nu^{3} + 44 \nu$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{14} - 24 \nu^{12} - 191 \nu^{10} - 651 \nu^{8} - 962 \nu^{6} - 584 \nu^{4} - 142 \nu^{2} - 22$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-7 \nu^{14} - 164 \nu^{12} - 1250 \nu^{10} - 3984 \nu^{8} - 5334 \nu^{6} - 3024 \nu^{4} - 596 \nu^{2} - 16$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$5 \nu^{15} + 4 \nu^{14} + 113 \nu^{13} + 90 \nu^{12} + 799 \nu^{11} + 630 \nu^{10} + 2176 \nu^{9} + 1674 \nu^{8} + 1922 \nu^{7} + 1352 \nu^{6} + 218 \nu^{5} + 20 \nu^{4} - 282 \nu^{3} - 212 \nu^{2} - 56 \nu - 28$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$15 \nu^{14} + 3 \nu^{13} + 352 \nu^{12} + 69 \nu^{11} + 2692 \nu^{10} + 506 \nu^{9} + 8638 \nu^{8} + 1488 \nu^{7} + 11726 \nu^{6} + 1638 \nu^{5} + 6808 \nu^{4} + 594 \nu^{3} + 1496 \nu^{2} + 28 \nu + 76$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$-15 \nu^{14} + 3 \nu^{13} - 352 \nu^{12} + 69 \nu^{11} - 2692 \nu^{10} + 506 \nu^{9} - 8638 \nu^{8} + 1488 \nu^{7} - 11726 \nu^{6} + 1638 \nu^{5} - 6808 \nu^{4} + 594 \nu^{3} - 1496 \nu^{2} + 28 \nu - 76$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{15} - 20 \nu^{14} - 107 \nu^{13} - 474 \nu^{12} - 657 \nu^{11} - 3698 \nu^{10} - 1074 \nu^{9} - 12334 \nu^{8} + 1686 \nu^{7} - 18152 \nu^{6} + 4770 \nu^{5} - 12164 \nu^{4} + 3014 \nu^{3} - 3428 \nu^{2} + 484 \nu - 300$$$$)/16$$ $$\beta_{8}$$ $$=$$ $$($$$$-11 \nu^{15} - 14 \nu^{14} - 260 \nu^{13} - 330 \nu^{12} - 2018 \nu^{11} - 2548 \nu^{10} - 6672 \nu^{9} - 8350 \nu^{8} - 9702 \nu^{7} - 11972 \nu^{6} - 6544 \nu^{5} - 8004 \nu^{4} - 2004 \nu^{3} - 2440 \nu^{2} - 272 \nu - 236$$$$)/16$$ $$\beta_{9}$$ $$=$$ $$($$$$19 \nu^{15} - \nu^{14} + 451 \nu^{13} - 20 \nu^{12} + 3529 \nu^{11} - 100 \nu^{10} + 11832 \nu^{9} + 4 \nu^{8} + 17582 \nu^{7} + 918 \nu^{6} + 11966 \nu^{5} + 1440 \nu^{4} + 3498 \nu^{3} + 664 \nu^{2} + 344 \nu + 72$$$$)/16$$ $$\beta_{10}$$ $$=$$ $$($$$$19 \nu^{15} + \nu^{14} + 451 \nu^{13} + 20 \nu^{12} + 3529 \nu^{11} + 100 \nu^{10} + 11832 \nu^{9} - 4 \nu^{8} + 17582 \nu^{7} - 918 \nu^{6} + 11966 \nu^{5} - 1440 \nu^{4} + 3498 \nu^{3} - 664 \nu^{2} + 344 \nu - 72$$$$)/16$$ $$\beta_{11}$$ $$=$$ $$($$$$11 \nu^{15} - 28 \nu^{14} + 260 \nu^{13} - 658 \nu^{12} + 2018 \nu^{11} - 5048 \nu^{10} + 6672 \nu^{9} - 16318 \nu^{8} + 9702 \nu^{7} - 22640 \nu^{6} + 6544 \nu^{5} - 14052 \nu^{4} + 2004 \nu^{3} - 3632 \nu^{2} + 272 \nu - 268$$$$)/16$$ $$\beta_{12}$$ $$=$$ $$($$$$-11 \nu^{15} - 261 \nu^{13} - 2042 \nu^{11} - 6862 \nu^{9} - 10334 \nu^{7} - 7410 \nu^{5} - 2412 \nu^{3} - 252 \nu$$$$)/8$$ $$\beta_{13}$$ $$=$$ $$($$$$-11 \nu^{15} - 12 \nu^{14} - 258 \nu^{13} - 282 \nu^{12} - 1971 \nu^{11} - 2164 \nu^{10} - 6311 \nu^{9} - 7004 \nu^{8} - 8530 \nu^{7} - 9752 \nu^{6} - 4916 \nu^{5} - 6092 \nu^{4} - 1046 \nu^{3} - 1608 \nu^{2} - 38 \nu - 136$$$$)/8$$ $$\beta_{14}$$ $$=$$ $$($$$$-11 \nu^{15} + 12 \nu^{14} - 258 \nu^{13} + 282 \nu^{12} - 1971 \nu^{11} + 2164 \nu^{10} - 6311 \nu^{9} + 7004 \nu^{8} - 8530 \nu^{7} + 9752 \nu^{6} - 4916 \nu^{5} + 6092 \nu^{4} - 1046 \nu^{3} + 1608 \nu^{2} - 38 \nu + 136$$$$)/8$$ $$\beta_{15}$$ $$=$$ $$($$$$38 \nu^{15} + 896 \nu^{13} + 6921 \nu^{11} + 22674 \nu^{9} + 32332 \nu^{7} + 20960 \nu^{5} + 5842 \nu^{3} + 540 \nu$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{6} - \beta_{5} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{14} - 2 \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{15} - \beta_{14} + \beta_{13} - 3 \beta_{12} + 3 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} - 2 \beta_{7} + 9 \beta_{6} + 9 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 8 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-9 \beta_{14} + 27 \beta_{13} - 18 \beta_{12} - 12 \beta_{11} + 20 \beta_{10} - 20 \beta_{9} - 12 \beta_{8} - 18 \beta_{7} - 8 \beta_{6} + 8 \beta_{5} - 18 \beta_{4} + 12 \beta_{3} + 10 \beta_{2} + 48$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$50 \beta_{15} + 20 \beta_{14} - 9 \beta_{13} + 55 \beta_{12} - 40 \beta_{11} - 31 \beta_{10} - 31 \beta_{9} + 40 \beta_{8} + 29 \beta_{7} - 95 \beta_{6} - 95 \beta_{5} - 29 \beta_{4} + 40 \beta_{3} + 83 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$47 \beta_{14} - 171 \beta_{13} + 124 \beta_{12} + 72 \beta_{11} - 140 \beta_{10} + 140 \beta_{9} + 72 \beta_{8} + 124 \beta_{7} + 37 \beta_{6} - 37 \beta_{5} + 124 \beta_{4} - 67 \beta_{3} - 57 \beta_{2} - 252$$ $$\nu^{7}$$ $$=$$ $$($$$$-672 \beta_{15} - 279 \beta_{14} + 93 \beta_{13} - 756 \beta_{12} + 497 \beta_{11} + 413 \beta_{10} + 413 \beta_{9} - 497 \beta_{8} - 372 \beta_{7} + 1091 \beta_{6} + 1091 \beta_{5} + 372 \beta_{4} - 497 \beta_{3} - 956 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$-540 \beta_{14} + 2120 \beta_{13} - 1580 \beta_{12} - 876 \beta_{11} + 1792 \beta_{10} - 1792 \beta_{9} - 876 \beta_{8} - 1580 \beta_{7} - 396 \beta_{6} + 396 \beta_{5} - 1580 \beta_{4} + 784 \beta_{3} + 680 \beta_{2} + 2910$$ $$\nu^{9}$$ $$=$$ $$4248 \beta_{15} + 1782 \beta_{14} - 534 \beta_{13} + 4812 \beta_{12} - 3054 \beta_{11} - 2610 \beta_{10} - 2610 \beta_{9} + 3054 \beta_{8} + 2316 \beta_{7} - 6517 \beta_{6} - 6517 \beta_{5} - 2316 \beta_{4} + 3054 \beta_{3} + 5723 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$6459 \beta_{14} - 26078 \beta_{13} + 19619 \beta_{12} + 10707 \beta_{11} - 22287 \beta_{10} + 22287 \beta_{9} + 10707 \beta_{8} + 19619 \beta_{7} + 4587 \beta_{6} - 4587 \beta_{5} + 19619 \beta_{4} - 9423 \beta_{3} - 8243 \beta_{2} - 34858$$ $$\nu^{11}$$ $$=$$ $$-52613 \beta_{15} - 22139 \beta_{14} + 6403 \beta_{13} - 59741 \beta_{12} + 37433 \beta_{11} + 32346 \beta_{10} + 32346 \beta_{9} - 37433 \beta_{8} - 28542 \beta_{7} + 79079 \beta_{6} + 79079 \beta_{5} + 28542 \beta_{4} - 37433 \beta_{3} - 69508 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-78431 \beta_{14} + 319865 \beta_{13} - 241434 \beta_{12} - 131044 \beta_{11} + 274428 \beta_{10} - 274428 \beta_{9} - 131044 \beta_{8} - 241434 \beta_{7} - 54976 \beta_{6} + 54976 \beta_{5} - 241434 \beta_{4} + 114548 \beta_{3} + 100538 \beta_{2} + 423384$$ $$\nu^{13}$$ $$=$$ $$646906 \beta_{15} + 272464 \beta_{14} - 77907 \beta_{13} + 735189 \beta_{12} - 458484 \beta_{11} - 397897 \beta_{10} - 397897 \beta_{9} + 458484 \beta_{8} + 350371 \beta_{7} - 965133 \beta_{6} - 965133 \beta_{5} - 350371 \beta_{4} + 458484 \beta_{3} + 848621 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$957558 \beta_{14} - 3919222 \beta_{13} + 2961664 \beta_{12} + 1604464 \beta_{11} - 3367136 \beta_{10} + 3367136 \beta_{9} + 1604464 \beta_{8} + 2961664 \beta_{7} + 667774 \beta_{6} - 667774 \beta_{5} + 2961664 \beta_{4} - 1398722 \beta_{3} - 1229198 \beta_{2} - 5168936$$ $$\nu^{15}$$ $$=$$ $$-7933264 \beta_{15} - 3342241 \beta_{14} + 952247 \beta_{13} - 9018824 \beta_{12} + 5614479 \beta_{11} + 4880807 \beta_{10} + 4880807 \beta_{9} - 5614479 \beta_{8} - 4294488 \beta_{7} + 11804009 \beta_{6} + 11804009 \beta_{5} + 4294488 \beta_{4} - 5614479 \beta_{3} - 10380392 \beta_{1}$$

## 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
 − 0.528036i − 3.49930i 0.528036i 3.49930i − 1.05636i 0.724535i 1.05636i − 0.724535i 0.357857i 2.13875i − 0.357857i − 2.13875i 0.886177i − 2.08509i − 0.886177i 2.08509i
−1.30656 0.541196i −1.13705 + 1.30656i 1.41421 + 1.41421i 0 2.19274 1.09174i 2.27411i −1.08239 2.61313i −0.414214 2.97127i 0
251.2 −1.30656 0.541196i 1.13705 + 1.30656i 1.41421 + 1.41421i 0 −0.778527 2.32248i 2.27411i −1.08239 2.61313i −0.414214 + 2.97127i 0
251.3 −1.30656 + 0.541196i −1.13705 1.30656i 1.41421 1.41421i 0 2.19274 + 1.09174i 2.27411i −1.08239 + 2.61313i −0.414214 + 2.97127i 0
251.4 −1.30656 + 0.541196i 1.13705 1.30656i 1.41421 1.41421i 0 −0.778527 + 2.32248i 2.27411i −1.08239 + 2.61313i −0.414214 2.97127i 0
251.5 −0.541196 1.30656i −1.64533 0.541196i −1.41421 + 1.41421i 0 0.183339 + 2.44262i 3.29066i 2.61313 + 1.08239i 2.41421 + 1.78089i 0
251.6 −0.541196 1.30656i 1.64533 0.541196i −1.41421 + 1.41421i 0 −1.59755 1.85683i 3.29066i 2.61313 + 1.08239i 2.41421 1.78089i 0
251.7 −0.541196 + 1.30656i −1.64533 + 0.541196i −1.41421 1.41421i 0 0.183339 2.44262i 3.29066i 2.61313 1.08239i 2.41421 1.78089i 0
251.8 −0.541196 + 1.30656i 1.64533 + 0.541196i −1.41421 1.41421i 0 −1.59755 + 1.85683i 3.29066i 2.61313 1.08239i 2.41421 + 1.78089i 0
251.9 0.541196 1.30656i −1.64533 0.541196i −1.41421 1.41421i 0 −1.59755 + 1.85683i 3.29066i −2.61313 + 1.08239i 2.41421 + 1.78089i 0
251.10 0.541196 1.30656i 1.64533 0.541196i −1.41421 1.41421i 0 0.183339 2.44262i 3.29066i −2.61313 + 1.08239i 2.41421 1.78089i 0
251.11 0.541196 + 1.30656i −1.64533 + 0.541196i −1.41421 + 1.41421i 0 −1.59755 1.85683i 3.29066i −2.61313 1.08239i 2.41421 1.78089i 0
251.12 0.541196 + 1.30656i 1.64533 + 0.541196i −1.41421 + 1.41421i 0 0.183339 + 2.44262i 3.29066i −2.61313 1.08239i 2.41421 + 1.78089i 0
251.13 1.30656 0.541196i −1.13705 + 1.30656i 1.41421 1.41421i 0 −0.778527 + 2.32248i 2.27411i 1.08239 2.61313i −0.414214 2.97127i 0
251.14 1.30656 0.541196i 1.13705 + 1.30656i 1.41421 1.41421i 0 2.19274 + 1.09174i 2.27411i 1.08239 2.61313i −0.414214 + 2.97127i 0
251.15 1.30656 + 0.541196i −1.13705 1.30656i 1.41421 + 1.41421i 0 −0.778527 2.32248i 2.27411i 1.08239 + 2.61313i −0.414214 + 2.97127i 0
251.16 1.30656 + 0.541196i 1.13705 1.30656i 1.41421 + 1.41421i 0 2.19274 1.09174i 2.27411i 1.08239 + 2.61313i −0.414214 2.97127i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.16 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
5.b even 2 1 inner
8.d odd 2 1 inner
15.d odd 2 1 inner
24.f even 2 1 inner
40.e odd 2 1 inner
120.m 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.i 16
3.b odd 2 1 inner 600.2.b.i 16
4.b odd 2 1 2400.2.b.i 16
5.b even 2 1 inner 600.2.b.i 16
5.c odd 4 2 120.2.m.b 16
8.b even 2 1 2400.2.b.i 16
8.d odd 2 1 inner 600.2.b.i 16
12.b even 2 1 2400.2.b.i 16
15.d odd 2 1 inner 600.2.b.i 16
15.e even 4 2 120.2.m.b 16
20.d odd 2 1 2400.2.b.i 16
20.e even 4 2 480.2.m.b 16
24.f even 2 1 inner 600.2.b.i 16
24.h odd 2 1 2400.2.b.i 16
40.e odd 2 1 inner 600.2.b.i 16
40.f even 2 1 2400.2.b.i 16
40.i odd 4 2 480.2.m.b 16
40.k even 4 2 120.2.m.b 16
60.h even 2 1 2400.2.b.i 16
60.l odd 4 2 480.2.m.b 16
120.i odd 2 1 2400.2.b.i 16
120.m even 2 1 inner 600.2.b.i 16
120.q odd 4 2 120.2.m.b 16
120.w even 4 2 480.2.m.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.m.b 16 5.c odd 4 2
120.2.m.b 16 15.e even 4 2
120.2.m.b 16 40.k even 4 2
120.2.m.b 16 120.q odd 4 2
480.2.m.b 16 20.e even 4 2
480.2.m.b 16 40.i odd 4 2
480.2.m.b 16 60.l odd 4 2
480.2.m.b 16 120.w even 4 2
600.2.b.i 16 1.a even 1 1 trivial
600.2.b.i 16 3.b odd 2 1 inner
600.2.b.i 16 5.b even 2 1 inner
600.2.b.i 16 8.d odd 2 1 inner
600.2.b.i 16 15.d odd 2 1 inner
600.2.b.i 16 24.f even 2 1 inner
600.2.b.i 16 40.e odd 2 1 inner
600.2.b.i 16 120.m even 2 1 inner
2400.2.b.i 16 4.b odd 2 1
2400.2.b.i 16 8.b even 2 1
2400.2.b.i 16 12.b even 2 1
2400.2.b.i 16 20.d odd 2 1
2400.2.b.i 16 24.h odd 2 1
2400.2.b.i 16 40.f even 2 1
2400.2.b.i 16 60.h even 2 1
2400.2.b.i 16 120.i odd 2 1

## Hecke kernels

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

 $$T_{7}^{4} + 16 T_{7}^{2} + 56$$ $$T_{11}^{4} + 24 T_{11}^{2} + 112$$ $$T_{23}^{4} - 8 T_{23}^{2} + 8$$ $$T_{43}^{4} - 112 T_{43}^{2} + 2744$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 16 T^{8} )^{2}$$
$3$ $$( 1 - 4 T^{2} + 14 T^{4} - 36 T^{6} + 81 T^{8} )^{2}$$
$5$ 1
$7$ $$( 1 - 12 T^{2} + 126 T^{4} - 588 T^{6} + 2401 T^{8} )^{4}$$
$11$ $$( 1 - 20 T^{2} + 310 T^{4} - 2420 T^{6} + 14641 T^{8} )^{4}$$
$13$ $$( 1 - 20 T^{2} + 406 T^{4} - 3380 T^{6} + 28561 T^{8} )^{4}$$
$17$ $$( 1 - 52 T^{2} + 1222 T^{4} - 15028 T^{6} + 83521 T^{8} )^{4}$$
$19$ $$( 1 - 4 T + 34 T^{2} - 76 T^{3} + 361 T^{4} )^{8}$$
$23$ $$( 1 + 84 T^{2} + 2814 T^{4} + 44436 T^{6} + 279841 T^{8} )^{4}$$
$29$ $$( 1 + 76 T^{2} + 2838 T^{4} + 63916 T^{6} + 707281 T^{8} )^{4}$$
$31$ $$( 1 - 76 T^{2} + 2854 T^{4} - 73036 T^{6} + 923521 T^{8} )^{4}$$
$37$ $$( 1 - 84 T^{2} + 3702 T^{4} - 114996 T^{6} + 1874161 T^{8} )^{4}$$
$41$ $$( 1 - 84 T^{2} + 3974 T^{4} - 141204 T^{6} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 + 60 T^{2} + 4206 T^{4} + 110940 T^{6} + 3418801 T^{8} )^{4}$$
$47$ $$( 1 + 180 T^{2} + 12510 T^{4} + 397620 T^{6} + 4879681 T^{8} )^{4}$$
$53$ $$( 1 + 68 T^{2} + 4182 T^{4} + 191012 T^{6} + 7890481 T^{8} )^{4}$$
$59$ $$( 1 - 212 T^{2} + 18166 T^{4} - 737972 T^{6} + 12117361 T^{8} )^{4}$$
$61$ $$( 1 - 50 T^{2} + 3721 T^{4} )^{8}$$
$67$ $$( 1 + 252 T^{2} + 24846 T^{4} + 1131228 T^{6} + 20151121 T^{8} )^{4}$$
$71$ $$( 1 + 92 T^{2} + 10150 T^{4} + 463772 T^{6} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 + 228 T^{2} + 23526 T^{4} + 1215012 T^{6} + 28398241 T^{8} )^{4}$$
$79$ $$( 1 - 44 T^{2} - 5466 T^{4} - 274604 T^{6} + 38950081 T^{8} )^{4}$$
$83$ $$( 1 - 196 T^{2} + 22990 T^{4} - 1350244 T^{6} + 47458321 T^{8} )^{4}$$
$89$ $$( 1 - 260 T^{2} + 32230 T^{4} - 2059460 T^{6} + 62742241 T^{8} )^{4}$$
$97$ $$( 1 + 260 T^{2} + 32518 T^{4} + 2446340 T^{6} + 88529281 T^{8} )^{4}$$