# Properties

 Label 450.2.h.g Level $450$ Weight $2$ Character orbit 450.h Analytic conductor $3.593$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 450.h (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.59326809096$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - x^{11} + 6 x^{10} - 26 x^{9} + 61 x^{8} - 120 x^{7} + 465 x^{6} - 600 x^{5} + 1525 x^{4} - 3250 x^{3} + 3750 x^{2} - 3125 x + 15625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{2} + ( -1 + \beta_{4} - \beta_{6} + \beta_{7} ) q^{4} -\beta_{1} q^{5} + ( 1 + \beta_{3} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{7} + \beta_{4} q^{8} +O(q^{10})$$ $$q + \beta_{7} q^{2} + ( -1 + \beta_{4} - \beta_{6} + \beta_{7} ) q^{4} -\beta_{1} q^{5} + ( 1 + \beta_{3} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{7} + \beta_{4} q^{8} -\beta_{8} q^{10} + ( -\beta_{1} + \beta_{2} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{11} + ( \beta_{2} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{13} + ( -\beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{14} + \beta_{6} q^{16} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{17} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{19} + \beta_{2} q^{20} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{22} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{23} + ( -2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{25} + ( 1 - \beta_{1} - \beta_{3} + \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{26} + ( -1 - \beta_{6} - \beta_{8} - \beta_{11} ) q^{28} + ( -\beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{10} ) q^{29} + ( -\beta_{2} - \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} ) q^{31} - q^{32} + ( -\beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{34} + ( 1 - \beta_{3} + 2 \beta_{6} + 3 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{35} + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{37} + ( -\beta_{1} + \beta_{3} - \beta_{6} + \beta_{9} + \beta_{11} ) q^{38} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{8} ) q^{40} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{10} ) q^{41} + ( 3 + \beta_{1} - 2 \beta_{2} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{43} + ( \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{10} - \beta_{11} ) q^{44} + ( -3 - \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{11} ) q^{46} + ( 2 + \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{47} + ( 3 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} ) q^{49} + ( -1 - \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{10} + \beta_{11} ) q^{50} + ( \beta_{1} - \beta_{2} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{52} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{53} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{10} + 3 \beta_{11} ) q^{55} + ( 1 + \beta_{2} - \beta_{7} - \beta_{9} ) q^{56} + ( -2 - \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{9} - \beta_{11} ) q^{58} + ( \beta_{1} - \beta_{2} + \beta_{5} - 6 \beta_{6} - \beta_{8} - \beta_{9} ) q^{59} + ( -\beta_{1} + 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{61} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} ) q^{62} -\beta_{7} q^{64} + ( -2 + \beta_{1} + 2 \beta_{3} + 5 \beta_{4} - 4 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{65} + ( 1 + 2 \beta_{1} - 7 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{10} ) q^{67} + ( -1 + \beta_{1} + \beta_{2} - \beta_{5} + \beta_{11} ) q^{68} + ( -5 + \beta_{1} - \beta_{2} + 3 \beta_{4} - \beta_{5} - 3 \beta_{6} + 4 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{70} + ( -5 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - 5 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{71} + ( -\beta_{1} - \beta_{2} + \beta_{3} - 5 \beta_{4} - 2 \beta_{5} + 5 \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{73} + ( 1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{74} + ( 1 - \beta_{1} + \beta_{5} - \beta_{8} + \beta_{9} ) q^{76} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 4 \beta_{10} + 3 \beta_{11} ) q^{77} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{5} - 2 \beta_{6} - \beta_{8} - 3 \beta_{9} - \beta_{10} ) q^{79} -\beta_{3} q^{80} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{82} + ( 3 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 3 \beta_{7} - \beta_{9} + 2 \beta_{11} ) q^{83} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 6 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + \beta_{10} + \beta_{11} ) q^{85} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{86} + ( \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{11} ) q^{88} + ( \beta_{1} - \beta_{2} - \beta_{3} - 8 \beta_{4} + 8 \beta_{6} - 7 \beta_{7} - \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{89} + ( -3 + 4 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 4 \beta_{8} - \beta_{10} ) q^{91} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{92} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{94} + ( 4 + 2 \beta_{1} - \beta_{2} - 8 \beta_{4} + \beta_{5} + 6 \beta_{6} - 7 \beta_{7} - \beta_{8} - 4 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} ) q^{95} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + \beta_{9} + 3 \beta_{10} ) q^{97} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 3q^{2} - 3q^{4} - q^{5} - 2q^{7} + 3q^{8} + O(q^{10})$$ $$12q + 3q^{2} - 3q^{4} - q^{5} - 2q^{7} + 3q^{8} + q^{10} - q^{11} + 4q^{13} - 8q^{14} - 3q^{16} + 8q^{17} - 8q^{19} - q^{20} - 4q^{22} - 11q^{25} + 16q^{26} - 7q^{28} + 6q^{29} - 3q^{31} - 12q^{32} + 2q^{34} + 18q^{35} - 8q^{37} - 2q^{38} + q^{40} - 20q^{41} + 32q^{43} + 4q^{44} - 10q^{46} + 34q^{49} - 9q^{50} + 4q^{52} - 2q^{53} + 44q^{55} + 7q^{56} - 6q^{58} + 19q^{59} - 26q^{61} - 2q^{62} - 3q^{64} - 16q^{65} - 16q^{67} - 12q^{68} - 23q^{70} - 48q^{71} - 30q^{73} + 8q^{74} + 12q^{76} + 39q^{77} - 18q^{79} + 4q^{80} - 40q^{82} + 29q^{83} - 4q^{85} - 12q^{86} + q^{88} - 62q^{89} - 26q^{91} + 10q^{92} - 6q^{95} + 23q^{97} + 6q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} + 6 x^{10} - 26 x^{9} + 61 x^{8} - 120 x^{7} + 465 x^{6} - 600 x^{5} + 1525 x^{4} - 3250 x^{3} + 3750 x^{2} - 3125 x + 15625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{11} + 27 \nu^{10} - 17 \nu^{9} - 8 \nu^{8} - 467 \nu^{7} - 162 \nu^{6} - 195 \nu^{5} + 1350 \nu^{4} + 6075 \nu^{3} - 12700 \nu^{2} - 21250 \nu - 99375$$$$)/4750$$ $$\beta_{3}$$ $$=$$ $$($$$$6 \nu^{11} + 124 \nu^{10} + 31 \nu^{9} + 199 \nu^{8} - 1339 \nu^{7} + 35 \nu^{6} - 885 \nu^{5} + 11425 \nu^{4} + 25525 \nu^{3} - 1625 \nu^{2} - 68125 \nu - 228125$$$$)/23750$$ $$\beta_{4}$$ $$=$$ $$($$$$31 \nu^{11} + 134 \nu^{10} - 204 \nu^{9} + 2659 \nu^{8} - 2374 \nu^{7} + 21445 \nu^{6} - 37110 \nu^{5} + 103125 \nu^{4} - 164600 \nu^{3} + 440875 \nu^{2} - 171875 \nu + 1503125$$$$)/118750$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{11} - \nu^{10} + 6 \nu^{9} - 26 \nu^{8} + 61 \nu^{7} - 120 \nu^{6} + 465 \nu^{5} - 600 \nu^{4} + 1525 \nu^{3} - 3250 \nu^{2} + 3750 \nu - 3125$$$$)/3125$$ $$\beta_{6}$$ $$=$$ $$($$$$73 \nu^{11} - 43 \nu^{10} + 1058 \nu^{9} - 1743 \nu^{8} + 5448 \nu^{7} - 15455 \nu^{6} + 34120 \nu^{5} - 48225 \nu^{4} + 168450 \nu^{3} - 109625 \nu^{2} + 265625 \nu - 568750$$$$)/118750$$ $$\beta_{7}$$ $$=$$ $$($$$$-117 \nu^{11} - 43 \nu^{10} - 367 \nu^{9} + 157 \nu^{8} - 1677 \nu^{7} - 6145 \nu^{6} + 1345 \nu^{5} - 51075 \nu^{4} + 56825 \nu^{3} - 185625 \nu^{2} + 158750 \nu - 925000$$$$)/118750$$ $$\beta_{8}$$ $$=$$ $$($$$$-32 \nu^{11} + 67 \nu^{10} - 577 \nu^{9} + 1092 \nu^{8} - 4037 \nu^{7} + 11150 \nu^{6} - 24255 \nu^{5} + 47050 \nu^{4} - 113175 \nu^{3} + 119500 \nu^{2} - 258125 \nu + 365625$$$$)/23750$$ $$\beta_{9}$$ $$=$$ $$($$$$182 \nu^{11} + 183 \nu^{10} + 877 \nu^{9} + 558 \nu^{8} + 2387 \nu^{7} + 5400 \nu^{6} + 7355 \nu^{5} + 61400 \nu^{4} + 36425 \nu^{3} + 250750 \nu^{2} + 134375 \nu + 759375$$$$)/118750$$ $$\beta_{10}$$ $$=$$ $$($$$$-296 \nu^{11} + 881 \nu^{10} - 1561 \nu^{9} + 9531 \nu^{8} - 18841 \nu^{7} + 43905 \nu^{6} - 106915 \nu^{5} + 170875 \nu^{4} - 196025 \nu^{3} + 677875 \nu^{2} - 181875 \nu + 131250$$$$)/118750$$ $$\beta_{11}$$ $$=$$ $$($$$$-481 \nu^{11} + 636 \nu^{10} - 2216 \nu^{9} + 11486 \nu^{8} - 16046 \nu^{7} + 45850 \nu^{6} - 116440 \nu^{5} + 103050 \nu^{4} - 217900 \nu^{3} + 740250 \nu^{2} + 400625 \nu + 643750$$$$)/118750$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{4}$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} - 2 \beta_{10} + \beta_{8} + 4 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} + \beta_{2} - 2 \beta_{1} + 4$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{11} - \beta_{9} - \beta_{8} + 13 \beta_{7} - 5 \beta_{6} + 2 \beta_{5} + 9 \beta_{4} + 7 \beta_{3} - 4 \beta_{2} + 7 \beta_{1} - 13$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{10} + 17 \beta_{9} + 7 \beta_{8} - 4 \beta_{6} - \beta_{5} - 32 \beta_{4} + 8 \beta_{3} - 12 \beta_{2} - 7 \beta_{1} - 8$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{11} + \beta_{10} - 3 \beta_{9} + 13 \beta_{8} - 18 \beta_{7} + 48 \beta_{6} + 45 \beta_{5} - 5 \beta_{4} - 24 \beta_{3} + 13 \beta_{2} - 36 \beta_{1} + 14$$ $$\nu^{7}$$ $$=$$ $$-\beta_{11} - 14 \beta_{10} - 71 \beta_{9} - 36 \beta_{8} - 4 \beta_{7} - 52 \beta_{6} + 36 \beta_{5} + 112 \beta_{4} + 30 \beta_{3} + 5 \beta_{2} + 20 \beta_{1} - 240$$ $$\nu^{8}$$ $$=$$ $$-86 \beta_{11} + 82 \beta_{10} + 85 \beta_{9} - 91 \beta_{8} + 156 \beta_{7} - 339 \beta_{6} - 102 \beta_{5} - 4 \beta_{4} + 55 \beta_{3} - 71 \beta_{2} - 163 \beta_{1} - 184$$ $$\nu^{9}$$ $$=$$ $$-222 \beta_{11} + 400 \beta_{10} - 124 \beta_{9} - 74 \beta_{8} - 188 \beta_{7} + 80 \beta_{6} + 123 \beta_{5} - 84 \beta_{4} - 272 \beta_{3} + 24 \beta_{2} - 172 \beta_{1} + 688$$ $$\nu^{10}$$ $$=$$ $$250 \beta_{11} + 48 \beta_{10} - 692 \beta_{9} - 252 \beta_{8} - 2250 \beta_{7} - 346 \beta_{6} - 24 \beta_{5} - 668 \beta_{4} - 28 \beta_{3} + 182 \beta_{2} + 852 \beta_{1} - 317$$ $$\nu^{11}$$ $$=$$ $$292 \beta_{11} - 376 \beta_{10} + 978 \beta_{9} - 548 \beta_{8} - 532 \beta_{7} - 1448 \beta_{6} - 1720 \beta_{5} - 20 \beta_{4} - 1196 \beta_{3} + 1102 \beta_{2} - 1139 \beta_{1} + 36$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{4} - \beta_{6} + \beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1
 1.97131 + 1.05544i 0.220100 − 2.22521i −1.38239 + 1.75756i 1.95382 − 1.08748i −1.01542 + 1.99221i −1.24741 − 1.85579i 1.95382 + 1.08748i −1.01542 − 1.99221i −1.24741 + 1.85579i 1.97131 − 1.05544i 0.220100 + 2.22521i −1.38239 − 1.75756i
0.809017 + 0.587785i 0 0.309017 + 0.951057i −1.97131 1.05544i 0 −5.04842 −0.309017 + 0.951057i 0 −0.974449 2.01257i
91.2 0.809017 + 0.587785i 0 0.309017 + 0.951057i −0.220100 + 2.22521i 0 1.64173 −0.309017 + 0.951057i 0 −1.48601 + 1.67086i
91.3 0.809017 + 0.587785i 0 0.309017 + 0.951057i 1.38239 1.75756i 0 −0.447412 −0.309017 + 0.951057i 0 2.15144 0.609344i
181.1 −0.309017 0.951057i 0 −0.809017 + 0.587785i −1.95382 + 1.08748i 0 0.757055 0.809017 + 0.587785i 0 1.63801 + 1.52214i
181.2 −0.309017 0.951057i 0 −0.809017 + 0.587785i 1.01542 1.99221i 0 4.77988 0.809017 + 0.587785i 0 −2.20849 0.350097i
181.3 −0.309017 0.951057i 0 −0.809017 + 0.587785i 1.24741 + 1.85579i 0 −2.68284 0.809017 + 0.587785i 0 1.37949 1.75983i
271.1 −0.309017 + 0.951057i 0 −0.809017 0.587785i −1.95382 1.08748i 0 0.757055 0.809017 0.587785i 0 1.63801 1.52214i
271.2 −0.309017 + 0.951057i 0 −0.809017 0.587785i 1.01542 + 1.99221i 0 4.77988 0.809017 0.587785i 0 −2.20849 + 0.350097i
271.3 −0.309017 + 0.951057i 0 −0.809017 0.587785i 1.24741 1.85579i 0 −2.68284 0.809017 0.587785i 0 1.37949 + 1.75983i
361.1 0.809017 0.587785i 0 0.309017 0.951057i −1.97131 + 1.05544i 0 −5.04842 −0.309017 0.951057i 0 −0.974449 + 2.01257i
361.2 0.809017 0.587785i 0 0.309017 0.951057i −0.220100 2.22521i 0 1.64173 −0.309017 0.951057i 0 −1.48601 1.67086i
361.3 0.809017 0.587785i 0 0.309017 0.951057i 1.38239 + 1.75756i 0 −0.447412 −0.309017 0.951057i 0 2.15144 + 0.609344i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.h.g yes 12
3.b odd 2 1 450.2.h.f 12
25.d even 5 1 inner 450.2.h.g yes 12
75.j odd 10 1 450.2.h.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.h.f 12 3.b odd 2 1
450.2.h.f 12 75.j odd 10 1
450.2.h.g yes 12 1.a even 1 1 trivial
450.2.h.g yes 12 25.d even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(450, [\chi])$$:

 $$T_{7}^{6} + T_{7}^{5} - 29 T_{7}^{4} - 18 T_{7}^{3} + 124 T_{7}^{2} - 24 T_{7} - 36$$ $$T_{11}^{12} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{3}$$
$3$ $$T^{12}$$
$5$ $$15625 + 3125 T + 3750 T^{2} + 3250 T^{3} + 1525 T^{4} + 600 T^{5} + 465 T^{6} + 120 T^{7} + 61 T^{8} + 26 T^{9} + 6 T^{10} + T^{11} + T^{12}$$
$7$ $$( -36 - 24 T + 124 T^{2} - 18 T^{3} - 29 T^{4} + T^{5} + T^{6} )^{2}$$
$11$ $$16 - 144 T + 832 T^{2} - 2888 T^{3} + 6808 T^{4} - 8528 T^{5} + 8798 T^{6} + 1222 T^{7} + 693 T^{8} - 28 T^{9} - 8 T^{10} + T^{11} + T^{12}$$
$13$ $$126736 + 87576 T - 61988 T^{2} - 117738 T^{3} + 193653 T^{4} - 65918 T^{5} + 24383 T^{6} - 6508 T^{7} + 1768 T^{8} - 248 T^{9} + 62 T^{10} - 4 T^{11} + T^{12}$$
$17$ $$467856 - 952128 T + 2998296 T^{2} - 1661970 T^{3} + 521545 T^{4} - 71858 T^{5} + 24489 T^{6} - 828 T^{7} + 1530 T^{8} - 80 T^{9} + 36 T^{10} - 8 T^{11} + T^{12}$$
$19$ $$92416 + 350208 T + 454656 T^{2} - 331040 T^{3} + 138640 T^{4} + 31888 T^{5} + 74184 T^{6} + 20368 T^{7} + 5440 T^{8} + 760 T^{9} + 96 T^{10} + 8 T^{11} + T^{12}$$
$23$ $$2560000 + 7040000 T + 8224000 T^{2} + 3448000 T^{3} + 848400 T^{4} + 125600 T^{5} + 49400 T^{6} - 1000 T^{7} + 2260 T^{8} - 180 T^{9} + 70 T^{10} + T^{12}$$
$29$ $$1936 + 16984 T + 56732 T^{2} - 32522 T^{3} + 329733 T^{4} - 94922 T^{5} + 43133 T^{6} - 3672 T^{7} + 608 T^{8} + 28 T^{9} + 22 T^{10} - 6 T^{11} + T^{12}$$
$31$ $$26896 + 254528 T + 950856 T^{2} + 392560 T^{3} + 411640 T^{4} - 70072 T^{5} + 6744 T^{6} + 5418 T^{7} + 2465 T^{8} + 260 T^{9} + 76 T^{10} + 3 T^{11} + T^{12}$$
$37$ $$68492176 + 167191752 T + 166698808 T^{2} + 31337456 T^{3} + 15774833 T^{4} + 686394 T^{5} + 535662 T^{6} + 15116 T^{7} + 4288 T^{8} + 4 T^{9} + 63 T^{10} + 8 T^{11} + T^{12}$$
$41$ $$24010000 - 6860000 T + 8379000 T^{2} - 1930250 T^{3} + 1230525 T^{4} - 155800 T^{5} + 47325 T^{6} + 20600 T^{7} + 4910 T^{8} + 690 T^{9} + 180 T^{10} + 20 T^{11} + T^{12}$$
$43$ $$( 2704 - 14976 T - 836 T^{2} + 1108 T^{3} - 34 T^{4} - 16 T^{5} + T^{6} )^{2}$$
$47$ $$6724000000 - 5658000000 T + 1934800000 T^{2} - 21400000 T^{3} + 60850000 T^{4} + 5530000 T^{5} + 1660000 T^{6} + 44000 T^{7} + 13000 T^{8} - 800 T^{9} + 130 T^{10} + T^{12}$$
$53$ $$31798321 + 32723117 T + 81127466 T^{2} + 14420280 T^{3} + 18074625 T^{4} - 4827888 T^{5} + 716519 T^{6} - 65683 T^{7} + 18805 T^{8} - 525 T^{9} - 59 T^{10} + 2 T^{11} + T^{12}$$
$59$ $$70291456 - 110802944 T + 176090112 T^{2} - 110226688 T^{3} + 38413568 T^{4} - 8411008 T^{5} + 1467688 T^{6} - 226988 T^{7} + 30113 T^{8} - 2918 T^{9} + 252 T^{10} - 19 T^{11} + T^{12}$$
$61$ $$32290652416 + 12809449664 T + 11284636144 T^{2} + 386896060 T^{3} - 12890695 T^{4} + 11808664 T^{5} + 4945991 T^{6} + 667276 T^{7} + 77420 T^{8} + 6760 T^{9} + 514 T^{10} + 26 T^{11} + T^{12}$$
$67$ $$3550253056 - 2589282304 T + 1816095232 T^{2} - 449541568 T^{3} + 36495568 T^{4} + 13317232 T^{5} + 2367568 T^{6} + 222272 T^{7} + 47648 T^{8} + 3312 T^{9} + 242 T^{10} + 16 T^{11} + T^{12}$$
$71$ $$4180398336 + 454660992 T + 1260566208 T^{2} + 555182496 T^{3} + 172968688 T^{4} + 50765264 T^{5} + 12658792 T^{6} + 2238576 T^{7} + 271928 T^{8} + 22584 T^{9} + 1288 T^{10} + 48 T^{11} + T^{12}$$
$73$ $$3906250000 - 546875000 T + 1873437500 T^{2} + 689281250 T^{3} + 109500625 T^{4} + 3165000 T^{5} + 680125 T^{6} + 176000 T^{7} + 51150 T^{8} + 6550 T^{9} + 580 T^{10} + 30 T^{11} + T^{12}$$
$79$ $$15083769856 - 11479857152 T + 4013958656 T^{2} - 698673920 T^{3} + 96988480 T^{4} - 14065632 T^{5} + 2079384 T^{6} + 16468 T^{7} + 32045 T^{8} + 2145 T^{9} + 276 T^{10} + 18 T^{11} + T^{12}$$
$83$ $$27081616 + 53476304 T + 47994744 T^{2} + 10649320 T^{3} + 11199140 T^{4} - 2892316 T^{5} + 2072206 T^{6} - 507474 T^{7} + 68765 T^{8} - 5070 T^{9} + 454 T^{10} - 29 T^{11} + T^{12}$$
$89$ $$1387115536 + 519255848 T + 1398308528 T^{2} + 1465315524 T^{3} + 771047873 T^{4} + 209598446 T^{5} + 39954152 T^{6} + 5374514 T^{7} + 531018 T^{8} + 38136 T^{9} + 1933 T^{10} + 62 T^{11} + T^{12}$$
$97$ $$786185521 - 898033092 T + 864566673 T^{2} - 459325441 T^{3} + 188116133 T^{4} - 45797619 T^{5} + 6081717 T^{6} - 82746 T^{7} + 5813 T^{8} - 844 T^{9} + 288 T^{10} - 23 T^{11} + T^{12}$$