# Properties

 Label 252.4.b.f Level $252$ Weight $4$ Character orbit 252.b Analytic conductor $14.868$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 252.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.8684813214$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - x^{11} - 2 x^{10} - 6 x^{9} + 56 x^{7} - 448 x^{6} + 448 x^{5} - 3072 x^{3} - 8192 x^{2} - 32768 x + 262144$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{13}$$ Twist minimal: no (minimal twist has level 84) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + \beta_{3} q^{4} + ( \beta_{2} - \beta_{8} ) q^{5} + ( 1 - \beta_{2} - \beta_{6} ) q^{7} + ( -2 - \beta_{4} ) q^{8} +O(q^{10})$$ $$q -\beta_{2} q^{2} + \beta_{3} q^{4} + ( \beta_{2} - \beta_{8} ) q^{5} + ( 1 - \beta_{2} - \beta_{6} ) q^{7} + ( -2 - \beta_{4} ) q^{8} + ( 5 - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{8} + \beta_{11} ) q^{10} + ( \beta_{1} - 2 \beta_{2} - \beta_{10} - \beta_{11} ) q^{11} + ( -1 + \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{13} + ( -6 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} ) q^{14} + ( 4 - \beta_{1} + \beta_{2} + \beta_{5} - 2 \beta_{6} + 3 \beta_{8} - \beta_{11} ) q^{16} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{17} + ( -7 + \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{19} + ( -14 + \beta_{1} - 5 \beta_{2} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{8} - 3 \beta_{11} ) q^{20} + ( -17 - 4 \beta_{2} + 5 \beta_{3} + \beta_{5} + 4 \beta_{6} - \beta_{7} + 5 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{22} + ( -3 - 3 \beta_{1} + 8 \beta_{2} + 7 \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{23} + ( -18 - 5 \beta_{1} - 19 \beta_{2} + 7 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{25} + ( -28 + 2 \beta_{1} + 4 \beta_{3} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{26} + ( 28 + 3 \beta_{1} + 7 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} + 4 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} - 4 \beta_{10} - \beta_{11} ) q^{28} + ( -21 + \beta_{1} + \beta_{2} + 11 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{8} + \beta_{9} + 2 \beta_{10} - 4 \beta_{11} ) q^{29} + ( -37 - 3 \beta_{1} - 3 \beta_{2} + 13 \beta_{3} - 5 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + 3 \beta_{8} - \beta_{9} + 4 \beta_{10} - 2 \beta_{11} ) q^{31} + ( 14 - 9 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} - \beta_{5} + 2 \beta_{6} - 6 \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{32} + ( -15 + 4 \beta_{1} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} - 3 \beta_{8} - 6 \beta_{10} + \beta_{11} ) q^{34} + ( -5 - 2 \beta_{1} + 33 \beta_{2} - 13 \beta_{3} - 5 \beta_{4} - 3 \beta_{6} - 4 \beta_{7} - 5 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{35} + ( -25 + 3 \beta_{1} + 19 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{37} + ( -20 + 8 \beta_{1} + 10 \beta_{2} - 8 \beta_{3} + \beta_{4} - \beta_{5} + 5 \beta_{6} + 9 \beta_{7} - 6 \beta_{8} + 2 \beta_{9} - 7 \beta_{10} + 2 \beta_{11} ) q^{38} + ( -26 - 6 \beta_{1} + 12 \beta_{2} + 8 \beta_{3} + 8 \beta_{6} + 6 \beta_{7} + 4 \beta_{8} + 6 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{40} + ( 10 + 9 \beta_{1} - 26 \beta_{2} - 20 \beta_{3} - 3 \beta_{6} + 7 \beta_{7} + 4 \beta_{8} + 5 \beta_{10} - 5 \beta_{11} ) q^{41} + ( -2 + 4 \beta_{1} + 8 \beta_{2} + 4 \beta_{4} + 5 \beta_{6} + 3 \beta_{7} - 12 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} - 4 \beta_{11} ) q^{43} + ( -20 + 7 \beta_{1} + 19 \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 10 \beta_{7} - 19 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{44} + ( 75 + 14 \beta_{1} - 11 \beta_{3} - 4 \beta_{4} + \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - 5 \beta_{8} - 3 \beta_{10} - 3 \beta_{11} ) q^{46} + ( -18 - 12 \beta_{1} - 36 \beta_{2} + 2 \beta_{3} - 10 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{47} + ( -27 + 4 \beta_{1} + 34 \beta_{2} - 20 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 5 \beta_{7} - 4 \beta_{8} + 4 \beta_{9} + 2 \beta_{10} - 10 \beta_{11} ) q^{49} + ( 148 + 25 \beta_{2} + 8 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + \beta_{6} + 13 \beta_{7} - 14 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} ) q^{50} + ( -68 - 6 \beta_{1} + 30 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{7} + 2 \beta_{8} - 8 \beta_{9} + 4 \beta_{10} - 6 \beta_{11} ) q^{52} + ( -9 + 7 \beta_{1} + 35 \beta_{2} + 11 \beta_{3} - 3 \beta_{4} - 6 \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} + 4 \beta_{10} - 2 \beta_{11} ) q^{53} + ( 37 - 15 \beta_{1} - 63 \beta_{2} - 35 \beta_{3} - 5 \beta_{4} - 2 \beta_{5} + 10 \beta_{6} - 13 \beta_{7} - 9 \beta_{8} - \beta_{9} - 8 \beta_{10} + 10 \beta_{11} ) q^{55} + ( -4 - 38 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 8 \beta_{6} - 18 \beta_{7} + 14 \beta_{8} + 2 \beta_{9} - 10 \beta_{10} + 8 \beta_{11} ) q^{56} + ( -46 - 18 \beta_{1} + 30 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} - 12 \beta_{7} - 24 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} + 10 \beta_{11} ) q^{58} + ( -90 - 8 \beta_{1} - 48 \beta_{2} - 6 \beta_{3} + 14 \beta_{4} - 4 \beta_{5} + 16 \beta_{6} - 6 \beta_{7} - 4 \beta_{8} + 6 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{59} + ( -1 + 21 \beta_{1} - 57 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} + 9 \beta_{6} + 8 \beta_{7} - 23 \beta_{8} + 3 \beta_{9} + 4 \beta_{10} + 8 \beta_{11} ) q^{61} + ( 46 - 14 \beta_{1} + 40 \beta_{2} - 6 \beta_{3} - 8 \beta_{4} + 6 \beta_{5} - 14 \beta_{6} + 20 \beta_{7} + 8 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} - 10 \beta_{11} ) q^{62} + ( 238 + 3 \beta_{1} - 5 \beta_{2} - 12 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 10 \beta_{6} + 10 \beta_{7} - 23 \beta_{8} + 6 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{64} + ( -18 - 12 \beta_{1} - 36 \beta_{2} + 2 \beta_{3} - 10 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{65} + ( -2 - 22 \beta_{1} + 72 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} - \beta_{6} - 3 \beta_{7} + 16 \beta_{8} - 8 \beta_{9} - 6 \beta_{11} ) q^{67} + ( -138 + 13 \beta_{1} - 5 \beta_{2} + 12 \beta_{3} - 7 \beta_{4} - \beta_{5} - 2 \beta_{6} - 12 \beta_{7} + 25 \beta_{8} + 12 \beta_{9} + 4 \beta_{10} + \beta_{11} ) q^{68} + ( 91 + 8 \beta_{1} - 14 \beta_{2} - 23 \beta_{3} + 8 \beta_{4} + \beta_{5} - 8 \beta_{6} + 7 \beta_{7} + 41 \beta_{8} + 6 \beta_{9} + 7 \beta_{10} - 7 \beta_{11} ) q^{70} + ( -3 + 17 \beta_{1} - 22 \beta_{2} - \beta_{3} + 7 \beta_{4} + 10 \beta_{6} + 7 \beta_{7} - 32 \beta_{8} - 7 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{71} + ( 22 + 4 \beta_{1} - 42 \beta_{2} - 42 \beta_{3} - 2 \beta_{4} - 20 \beta_{6} + 2 \beta_{7} + 6 \beta_{8} + 2 \beta_{9} + 22 \beta_{10} + 2 \beta_{11} ) q^{73} + ( -120 - 8 \beta_{1} + 24 \beta_{2} - 20 \beta_{3} - 5 \beta_{4} + \beta_{5} - 9 \beta_{6} + 15 \beta_{7} + 18 \beta_{8} + 2 \beta_{9} - \beta_{10} - 10 \beta_{11} ) q^{74} + ( -86 + 28 \beta_{1} + 6 \beta_{2} + 6 \beta_{4} + 2 \beta_{5} + 24 \beta_{6} - 14 \beta_{7} + 14 \beta_{8} - 6 \beta_{9} - 22 \beta_{10} + 8 \beta_{11} ) q^{76} + ( -83 + 15 \beta_{1} - 120 \beta_{2} + 5 \beta_{3} + 15 \beta_{4} - 6 \beta_{5} + 19 \beta_{6} - 14 \beta_{7} - 36 \beta_{8} - \beta_{9} - 14 \beta_{10} ) q^{77} + ( -26 + 6 \beta_{1} - 66 \beta_{2} + 34 \beta_{3} + 18 \beta_{4} + 9 \beta_{6} - 17 \beta_{7} + 8 \beta_{8} - 18 \beta_{9} - 4 \beta_{10} + 4 \beta_{11} ) q^{79} + ( 252 + 14 \beta_{1} + 34 \beta_{2} - 24 \beta_{3} - 2 \beta_{4} + 6 \beta_{5} + 8 \beta_{6} - 20 \beta_{7} - 34 \beta_{8} - 8 \beta_{9} - 20 \beta_{10} + 6 \beta_{11} ) q^{80} + ( -215 - 40 \beta_{1} + 31 \beta_{3} + 21 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - 4 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{82} + ( 42 + 20 \beta_{1} + 96 \beta_{2} + 70 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 10 \beta_{6} + 12 \beta_{7} + 16 \beta_{8} + 2 \beta_{9} + 18 \beta_{10} - 18 \beta_{11} ) q^{83} + ( -83 + 15 \beta_{1} + 35 \beta_{2} + 33 \beta_{3} + 31 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 23 \beta_{7} + 5 \beta_{8} + 11 \beta_{9} + 6 \beta_{10} - 16 \beta_{11} ) q^{85} + ( 14 - 12 \beta_{2} + 6 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} + 13 \beta_{6} + 3 \beta_{7} + 32 \beta_{8} - 6 \beta_{9} + 3 \beta_{10} + 12 \beta_{11} ) q^{86} + ( -178 + 4 \beta_{1} + 10 \beta_{2} - 16 \beta_{3} - 16 \beta_{4} + 2 \beta_{5} - 24 \beta_{6} - 6 \beta_{7} + 22 \beta_{8} + 6 \beta_{9} + 18 \beta_{10} + 20 \beta_{11} ) q^{88} + ( 10 - 43 \beta_{1} + 232 \beta_{2} - 28 \beta_{3} + 8 \beta_{4} + 5 \beta_{6} + 15 \beta_{7} - 30 \beta_{8} - 8 \beta_{9} + 9 \beta_{10} - 9 \beta_{11} ) q^{89} + ( 147 - 23 \beta_{1} - 105 \beta_{2} + 25 \beta_{3} - 9 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} + 14 \beta_{7} + 9 \beta_{8} - 5 \beta_{9} - 14 \beta_{11} ) q^{91} + ( -352 - 3 \beta_{1} - 91 \beta_{2} + 6 \beta_{3} + 7 \beta_{4} + 9 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} + 27 \beta_{8} + 6 \beta_{9} + 2 \beta_{10} + 5 \beta_{11} ) q^{92} + ( 352 + 12 \beta_{2} + 32 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 18 \beta_{6} + 30 \beta_{7} + 36 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} - 20 \beta_{11} ) q^{94} + ( 26 - 4 \beta_{1} - 30 \beta_{2} - 58 \beta_{3} + 6 \beta_{4} - 24 \beta_{6} + 2 \beta_{7} + 46 \beta_{8} - 6 \beta_{9} + 18 \beta_{10} - 14 \beta_{11} ) q^{95} + ( -26 - 36 \beta_{1} + 114 \beta_{2} + 58 \beta_{3} - 6 \beta_{4} - 26 \beta_{7} - 34 \beta_{8} + 6 \beta_{9} - 10 \beta_{10} + 22 \beta_{11} ) q^{97} + ( 12 - 30 \beta_{1} + 37 \beta_{2} - 32 \beta_{3} + 11 \beta_{4} + \beta_{5} + 17 \beta_{6} + \beta_{7} - 12 \beta_{8} + 12 \beta_{9} - 15 \beta_{10} + 26 \beta_{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - q^{2} + 5q^{4} + 10q^{7} - 25q^{8} + O(q^{10})$$ $$12q - q^{2} + 5q^{4} + 10q^{7} - 25q^{8} + 56q^{10} - 69q^{14} + 41q^{16} - 84q^{19} - 172q^{20} - 182q^{22} - 216q^{25} - 300q^{26} + 309q^{28} - 200q^{29} - 384q^{31} + 159q^{32} - 164q^{34} - 84q^{35} - 244q^{37} - 268q^{38} - 316q^{40} - 190q^{44} + 894q^{46} - 280q^{47} - 424q^{49} + 1771q^{50} - 796q^{52} + 16q^{53} + 212q^{55} + 7q^{56} - 570q^{58} - 1168q^{59} + 384q^{62} + 2705q^{64} - 280q^{65} - 1552q^{68} + 968q^{70} - 1622q^{74} - 788q^{76} - 968q^{77} + 3060q^{80} - 2540q^{82} + 968q^{83} - 852q^{85} + 258q^{86} - 2186q^{88} + 1648q^{91} - 4298q^{92} + 4256q^{94} - 97q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} - 2 x^{10} - 6 x^{9} + 56 x^{7} - 448 x^{6} + 448 x^{5} - 3072 x^{3} - 8192 x^{2} - 32768 x + 262144$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$4 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{11} + \nu^{10} + 2 \nu^{9} + 6 \nu^{8} - 56 \nu^{6} + 448 \nu^{5} - 448 \nu^{4} + 3072 \nu^{2} + 8192 \nu + 32768$$$$)/32768$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{11} - 7 \nu^{10} + 10 \nu^{9} + 22 \nu^{8} + 48 \nu^{7} - 56 \nu^{6} + 3136 \nu^{4} - 3584 \nu^{3} + 3072 \nu^{2} + 32768 \nu + 98304$$$$)/32768$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{11} - 5 \nu^{10} - 50 \nu^{9} + 98 \nu^{8} + 176 \nu^{7} + 216 \nu^{6} + 896 \nu^{5} - 1344 \nu^{4} + 25088 \nu^{3} - 19456 \nu^{2} + 49152 \nu + 294912$$$$)/32768$$ $$\beta_{5}$$ $$=$$ $$($$$$11 \nu^{11} + 69 \nu^{10} + 26 \nu^{9} - 610 \nu^{8} + 1568 \nu^{7} + 360 \nu^{6} + 3136 \nu^{5} + 33600 \nu^{4} - 15360 \nu^{3} + 138240 \nu^{2} - 925696 \nu - 1310720$$$$)/98304$$ $$\beta_{6}$$ $$=$$ $$($$$$29 \nu^{11} + 91 \nu^{10} + 142 \nu^{9} + 34 \nu^{8} - 80 \nu^{7} + 2264 \nu^{6} - 12928 \nu^{5} - 6464 \nu^{4} - 17920 \nu^{3} - 158720 \nu^{2} - 770048 \nu - 3440640$$$$)/98304$$ $$\beta_{7}$$ $$=$$ $$($$$$31 \nu^{11} + 41 \nu^{10} + 122 \nu^{9} + 182 \nu^{8} - 688 \nu^{7} + 3784 \nu^{6} - 6272 \nu^{5} + 10304 \nu^{4} + 5632 \nu^{3} - 136192 \nu^{2} - 507904 \nu - 3244032$$$$)/98304$$ $$\beta_{8}$$ $$=$$ $$($$$$45 \nu^{11} + 91 \nu^{10} + 158 \nu^{9} + 98 \nu^{8} + 464 \nu^{7} + 5336 \nu^{6} - 10496 \nu^{5} - 17216 \nu^{4} - 33280 \nu^{3} - 171008 \nu^{2} - 901120 \nu - 4685824$$$$)/98304$$ $$\beta_{9}$$ $$=$$ $$($$$$65 \nu^{11} + 111 \nu^{10} + 206 \nu^{9} + 26 \nu^{8} + 1760 \nu^{7} + 1080 \nu^{6} - 12608 \nu^{5} - 19008 \nu^{4} + 52224 \nu^{3} - 224256 \nu^{2} - 1712128 \nu - 5046272$$$$)/98304$$ $$\beta_{10}$$ $$=$$ $$($$$$37 \nu^{11} + 79 \nu^{10} + 66 \nu^{9} + 186 \nu^{8} + 456 \nu^{7} + 1304 \nu^{6} - 9312 \nu^{5} - 5696 \nu^{4} - 18688 \nu^{3} - 109568 \nu^{2} - 823296 \nu - 3031040$$$$)/49152$$ $$\beta_{11}$$ $$=$$ $$($$$$59 \nu^{11} + 101 \nu^{10} + 138 \nu^{9} + 318 \nu^{8} + 384 \nu^{7} + 4648 \nu^{6} - 10560 \nu^{5} - 6592 \nu^{4} - 23552 \nu^{3} - 375808 \nu^{2} - 1130496 \nu - 5128192$$$$)/49152$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{2} + 2$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{8} + 3 \beta_{7} + \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 10$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{11} + \beta_{10} - 2 \beta_{9} - 7 \beta_{8} + 5 \beta_{7} + 3 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 10 \beta_{3} - 5 \beta_{2} + 4 \beta_{1} + 10$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$9 \beta_{11} - 3 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} + 9 \beta_{7} - 33 \beta_{6} + 4 \beta_{5} - 10 \beta_{4} - 26 \beta_{3} + 123 \beta_{2} - 46$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-11 \beta_{11} - 23 \beta_{10} - 22 \beta_{9} + 89 \beta_{8} + 21 \beta_{7} - 5 \beta_{6} + 12 \beta_{5} + 38 \beta_{4} + 14 \beta_{3} - 57 \beta_{2} - 20 \beta_{1} + 898$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$5 \beta_{11} - 15 \beta_{10} + 34 \beta_{9} + 113 \beta_{8} - 179 \beta_{7} + 3 \beta_{6} + 44 \beta_{5} + 94 \beta_{4} + 342 \beta_{3} - 25 \beta_{2} + 132 \beta_{1} - 334$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-99 \beta_{11} + 425 \beta_{10} - 110 \beta_{9} + 9 \beta_{8} + 197 \beta_{7} - 197 \beta_{6} - 164 \beta_{5} + 254 \beta_{4} + 838 \beta_{3} + 2271 \beta_{2} - 212 \beta_{1} + 1010$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$-211 \beta_{11} - 695 \beta_{10} + 834 \beta_{9} + 265 \beta_{8} + 581 \beta_{7} + 507 \beta_{6} - 228 \beta_{5} - 594 \beta_{4} + 2198 \beta_{3} + 6255 \beta_{2} + 604 \beta_{1} + 16946$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$13 \beta_{11} + 1129 \beta_{10} - 1086 \beta_{9} - 1559 \beta_{8} + 293 \beta_{7} + 2875 \beta_{6} + 1276 \beta_{5} + 1038 \beta_{4} - 3626 \beta_{3} + 17455 \beta_{2} + 4940 \beta_{1} + 52306$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$125 \beta_{11} + 4857 \beta_{10} + 2946 \beta_{9} - 1095 \beta_{8} + 6325 \beta_{7} - 16213 \beta_{6} - 836 \beta_{5} - 6130 \beta_{4} - 11114 \beta_{3} - 23873 \beta_{2} + 12396 \beta_{1} + 154098$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 2.82801 − 0.0488466i 2.82801 + 0.0488466i 2.16644 − 1.81839i 2.16644 + 1.81839i 0.965027 − 2.65871i 0.965027 + 2.65871i −0.951271 − 2.66366i −0.951271 + 2.66366i −1.72458 − 2.24184i −1.72458 + 2.24184i −2.78362 − 0.501431i −2.78362 + 0.501431i
−2.82801 0.0488466i 0 7.99523 + 0.276277i 16.6517i 0 15.0420 + 10.8045i −22.5971 1.17185i 0 0.813380 47.0912i
55.2 −2.82801 + 0.0488466i 0 7.99523 0.276277i 16.6517i 0 15.0420 10.8045i −22.5971 + 1.17185i 0 0.813380 + 47.0912i
55.3 −2.16644 1.81839i 0 1.38692 + 7.87886i 4.47531i 0 −14.9825 + 10.8869i 11.3222 19.5910i 0 −8.13786 + 9.69549i
55.4 −2.16644 + 1.81839i 0 1.38692 7.87886i 4.47531i 0 −14.9825 10.8869i 11.3222 + 19.5910i 0 −8.13786 9.69549i
55.5 −0.965027 2.65871i 0 −6.13744 + 5.13145i 19.4608i 0 1.95109 18.4172i 19.5658 + 11.3657i 0 51.7405 18.7802i
55.6 −0.965027 + 2.65871i 0 −6.13744 5.13145i 19.4608i 0 1.95109 + 18.4172i 19.5658 11.3657i 0 51.7405 + 18.7802i
55.7 0.951271 2.66366i 0 −6.19017 5.06772i 10.8462i 0 −15.1344 10.6748i −19.3872 + 11.6677i 0 −28.8905 10.3176i
55.8 0.951271 + 2.66366i 0 −6.19017 + 5.06772i 10.8462i 0 −15.1344 + 10.6748i −19.3872 11.6677i 0 −28.8905 + 10.3176i
55.9 1.72458 2.24184i 0 −2.05167 7.73244i 6.58775i 0 15.1925 + 10.5918i −20.8731 8.73568i 0 14.7687 + 11.3611i
55.10 1.72458 + 2.24184i 0 −2.05167 + 7.73244i 6.58775i 0 15.1925 10.5918i −20.8731 + 8.73568i 0 14.7687 11.3611i
55.11 2.78362 0.501431i 0 7.49713 2.79159i 4.57514i 0 2.93118 18.2868i 19.4694 11.5300i 0 −2.29411 12.7355i
55.12 2.78362 + 0.501431i 0 7.49713 + 2.79159i 4.57514i 0 2.93118 + 18.2868i 19.4694 + 11.5300i 0 −2.29411 + 12.7355i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 55.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.b.f 12
3.b odd 2 1 84.4.b.a 12
4.b odd 2 1 252.4.b.e 12
7.b odd 2 1 252.4.b.e 12
12.b even 2 1 84.4.b.b yes 12
21.c even 2 1 84.4.b.b yes 12
24.f even 2 1 1344.4.b.g 12
24.h odd 2 1 1344.4.b.h 12
28.d even 2 1 inner 252.4.b.f 12
84.h odd 2 1 84.4.b.a 12
168.e odd 2 1 1344.4.b.h 12
168.i even 2 1 1344.4.b.g 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.b.a 12 3.b odd 2 1
84.4.b.a 12 84.h odd 2 1
84.4.b.b yes 12 12.b even 2 1
84.4.b.b yes 12 21.c even 2 1
252.4.b.e 12 4.b odd 2 1
252.4.b.e 12 7.b odd 2 1
252.4.b.f 12 1.a even 1 1 trivial
252.4.b.f 12 28.d even 2 1 inner
1344.4.b.g 12 24.f even 2 1
1344.4.b.g 12 168.i even 2 1
1344.4.b.h 12 24.h odd 2 1
1344.4.b.h 12 168.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(252, [\chi])$$:

 $$T_{5}^{12} + 858 T_{5}^{10} + 249644 T_{5}^{8} + 29440120 T_{5}^{6} + 1456436096 T_{5}^{4} + 30453516288 T_{5}^{2} + 224760987648$$ $$T_{11}^{12} + 10414 T_{11}^{10} + 39015876 T_{11}^{8} + 64285126792 T_{11}^{6} +$$$$45\!\cdots\!56$$$$T_{11}^{4} +$$$$12\!\cdots\!68$$$$T_{11}^{2} +$$$$58\!\cdots\!00$$ $$T_{19}^{6} + 42 T_{19}^{5} - 27084 T_{19}^{4} - 1835880 T_{19}^{3} + 99824480 T_{19}^{2} + 8618904192 T_{19} + 111463534080$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$262144 + 32768 T - 8192 T^{2} + 3072 T^{3} - 448 T^{5} - 448 T^{6} - 56 T^{7} + 6 T^{9} - 2 T^{10} + T^{11} + T^{12}$$
$3$ $$T^{12}$$
$5$ $$224760987648 + 30453516288 T^{2} + 1456436096 T^{4} + 29440120 T^{6} + 249644 T^{8} + 858 T^{10} + T^{12}$$
$7$ $$1628413597910449 - 47475615099430 T + 3626417246662 T^{2} + 33977737094 T^{3} + 10113931583 T^{4} - 460310116 T^{5} + 96999028 T^{6} - 1342012 T^{7} + 85967 T^{8} + 842 T^{9} + 262 T^{10} - 10 T^{11} + T^{12}$$
$11$ $$584217371305280000 + 12525766859191168 T^{2} + 45887475687456 T^{4} + 64285126792 T^{6} + 39015876 T^{8} + 10414 T^{10} + T^{12}$$
$13$ $$62601055759761408 + 2779995917647872 T^{2} + 19645357400064 T^{4} + 39003959808 T^{6} + 30771776 T^{8} + 9976 T^{10} + T^{12}$$
$17$ $$59\!\cdots\!12$$$$+ 9486366618786324480 T^{2} + 8294186258955776 T^{4} + 2811705354936 T^{6} + 454587148 T^{8} + 34826 T^{10} + T^{12}$$
$19$ $$( 111463534080 + 8618904192 T + 99824480 T^{2} - 1835880 T^{3} - 27084 T^{4} + 42 T^{5} + T^{6} )^{2}$$
$23$ $$17\!\cdots\!12$$$$+$$$$16\!\cdots\!32$$$$T^{2} + 61948036290738720 T^{4} + 11431676066248 T^{6} + 1104167940 T^{8} + 53182 T^{10} + T^{12}$$
$29$ $$( -3698977199040 - 95119425984 T + 2688214256 T^{2} - 2852896 T^{3} - 98436 T^{4} + 100 T^{5} + T^{6} )^{2}$$
$31$ $$( 34547086393344 + 622925807616 T + 1743943680 T^{2} - 21663744 T^{3} - 99232 T^{4} + 192 T^{5} + T^{6} )^{2}$$
$37$ $$( 11485380096 - 1692626304 T + 29975392 T^{2} + 3382616 T^{3} - 73420 T^{4} + 122 T^{5} + T^{6} )^{2}$$
$41$ $$18\!\cdots\!00$$$$+$$$$23\!\cdots\!32$$$$T^{2} + 8929551139378138112 T^{4} + 958649263640824 T^{6} + 35274441548 T^{8} + 373146 T^{10} + T^{12}$$
$43$ $$47\!\cdots\!12$$$$+$$$$10\!\cdots\!64$$$$T^{2} + 24122851474400416256 T^{4} + 1464388464157760 T^{6} + 34667163920 T^{8} + 328092 T^{10} + T^{12}$$
$47$ $$( 118618190512128 - 3408906829824 T + 24842032128 T^{2} + 536640 T^{3} - 302608 T^{4} + 140 T^{5} + T^{6} )^{2}$$
$53$ $$( -24896048217408 + 584664527232 T + 5115119536 T^{2} - 37315648 T^{3} - 272924 T^{4} - 8 T^{5} + T^{6} )^{2}$$
$59$ $$( -1915391252459520 + 29557612136448 T + 64356818688 T^{2} - 311343360 T^{3} - 572048 T^{4} + 584 T^{5} + T^{6} )^{2}$$
$61$ $$19\!\cdots\!00$$$$+$$$$41\!\cdots\!28$$$$T^{2} +$$$$13\!\cdots\!56$$$$T^{4} + 165531877369638912 T^{6} + 808904152832 T^{8} + 1590928 T^{10} + T^{12}$$
$67$ $$64\!\cdots\!68$$$$+$$$$50\!\cdots\!44$$$$T^{2} +$$$$13\!\cdots\!28$$$$T^{4} + 154614504282475968 T^{6} + 710502378800 T^{8} + 1407700 T^{10} + T^{12}$$
$71$ $$62\!\cdots\!00$$$$+$$$$22\!\cdots\!72$$$$T^{2} +$$$$30\!\cdots\!24$$$$T^{4} + 208738353017380424 T^{6} + 753608341380 T^{8} + 1375838 T^{10} + T^{12}$$
$73$ $$22\!\cdots\!32$$$$+$$$$48\!\cdots\!36$$$$T^{2} +$$$$49\!\cdots\!40$$$$T^{4} + 1948491776553389568 T^{6} + 3675179437760 T^{8} + 3228968 T^{10} + T^{12}$$
$79$ $$96\!\cdots\!88$$$$+$$$$18\!\cdots\!12$$$$T^{2} +$$$$11\!\cdots\!08$$$$T^{4} + 3351704066729440704 T^{6} + 4957994738864 T^{8} + 3584020 T^{10} + T^{12}$$
$83$ $$( -178662844658221056 - 77003384168448 T + 1331374132224 T^{2} + 508306752 T^{3} - 2160016 T^{4} - 484 T^{5} + T^{6} )^{2}$$
$89$ $$18\!\cdots\!72$$$$+$$$$12\!\cdots\!92$$$$T^{2} +$$$$73\!\cdots\!16$$$$T^{4} + 14989058094622396408 T^{6} + 14072024042060 T^{8} + 6134874 T^{10} + T^{12}$$
$97$ $$29\!\cdots\!88$$$$+$$$$30\!\cdots\!32$$$$T^{2} +$$$$31\!\cdots\!56$$$$T^{4} + 9820780041931402752 T^{6} + 11706611210432 T^{8} + 5772328 T^{10} + T^{12}$$