# Properties

 Label 475.3.c.i Level $475$ Weight $3$ Character orbit 475.c Analytic conductor $12.943$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 475.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.9428125571$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ Defining polynomial: $$x^{14} + 42 x^{12} + 677 x^{10} + 5313 x^{8} + 21125 x^{6} + 40138 x^{4} + 30565 x^{2} + 3675$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{4} q^{3} + ( -2 + \beta_{2} ) q^{4} -\beta_{5} q^{6} + ( 1 + \beta_{9} ) q^{7} + ( -2 \beta_{1} + \beta_{3} ) q^{8} + ( -3 - \beta_{6} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{4} q^{3} + ( -2 + \beta_{2} ) q^{4} -\beta_{5} q^{6} + ( 1 + \beta_{9} ) q^{7} + ( -2 \beta_{1} + \beta_{3} ) q^{8} + ( -3 - \beta_{6} ) q^{9} -\beta_{11} q^{11} + ( 2 \beta_{4} + \beta_{7} ) q^{12} + \beta_{12} q^{13} + ( 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{10} + \beta_{13} ) q^{14} + ( 2 - 2 \beta_{2} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{16} + ( -2 - \beta_{6} - \beta_{8} ) q^{17} + ( -1 - \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{18} + ( 3 - \beta_{2} + 2 \beta_{4} + \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{19} + ( 1 + 2 \beta_{1} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{21} + ( -\beta_{1} + 2 \beta_{4} + 2 \beta_{7} + \beta_{10} - \beta_{12} + \beta_{13} ) q^{22} + ( -2 - \beta_{2} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{23} + ( -4 + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{11} ) q^{24} + ( 2 - 2 \beta_{2} - \beta_{6} + \beta_{8} - \beta_{11} ) q^{26} + ( -1 - \beta_{1} + 4 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} - 2 \beta_{13} ) q^{27} + ( -6 + 4 \beta_{2} + \beta_{6} - 2 \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{28} + ( -5 \beta_{1} + \beta_{3} + 3 \beta_{4} + 3 \beta_{7} - \beta_{12} ) q^{29} + ( 1 + 7 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{31} + ( -1 + 4 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} + 3 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{32} + ( -2 \beta_{1} + \beta_{3} + \beta_{10} - \beta_{12} + \beta_{13} ) q^{33} + ( -1 + \beta_{1} + \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{34} + ( 3 + 4 \beta_{2} - \beta_{5} + 3 \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{36} + ( 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{7} + \beta_{10} + \beta_{12} + \beta_{13} ) q^{37} + ( -2 + 5 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{9} + 2 \beta_{11} - \beta_{12} + \beta_{13} ) q^{38} + ( -3 - \beta_{5} - \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{39} + ( 1 + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{41} + ( -21 + 3 \beta_{2} - 5 \beta_{6} - \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{42} + ( -7 + 3 \beta_{2} + 5 \beta_{5} - \beta_{8} - 3 \beta_{11} ) q^{43} + ( -4 + \beta_{2} + 4 \beta_{5} - \beta_{6} - 2 \beta_{8} - 3 \beta_{9} ) q^{44} + ( -1 + 5 \beta_{1} - 5 \beta_{3} - 6 \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{46} + ( -12 + 8 \beta_{2} + \beta_{6} + 3 \beta_{9} + 2 \beta_{11} ) q^{47} + ( -1 - 4 \beta_{1} - \beta_{3} + 11 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{48} + ( 14 - 2 \beta_{2} - \beta_{8} + 5 \beta_{9} + 2 \beta_{11} ) q^{49} + ( -1 - 6 \beta_{1} + 8 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{12} - 3 \beta_{13} ) q^{51} + ( -1 + 10 \beta_{1} - 5 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + 3 \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{13} ) q^{52} + ( 1 + 11 \beta_{1} - \beta_{3} - 4 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} ) q^{53} + ( 5 + 4 \beta_{2} + 4 \beta_{5} + 4 \beta_{6} + \beta_{8} + 6 \beta_{9} ) q^{54} + ( 1 - 15 \beta_{1} + 7 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} + 3 \beta_{12} ) q^{56} + ( 13 - 4 \beta_{1} - 3 \beta_{2} - 6 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{12} + \beta_{13} ) q^{57} + ( 14 - 4 \beta_{2} + 5 \beta_{5} - 3 \beta_{6} + 4 \beta_{9} + 4 \beta_{11} ) q^{58} + ( 7 \beta_{1} + 2 \beta_{4} + 3 \beta_{7} + \beta_{10} - \beta_{12} + \beta_{13} ) q^{59} + ( 17 - 5 \beta_{2} + 4 \beta_{6} + 2 \beta_{8} + 2 \beta_{11} ) q^{61} + ( -39 + 11 \beta_{2} + 5 \beta_{5} - 4 \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{62} + ( -13 - 5 \beta_{2} - 10 \beta_{5} - 4 \beta_{6} + \beta_{8} ) q^{63} + ( -11 + 4 \beta_{2} - \beta_{5} + 4 \beta_{6} - \beta_{8} - 8 \beta_{9} + 3 \beta_{11} ) q^{64} + ( 8 - 6 \beta_{2} - \beta_{5} - \beta_{8} - 4 \beta_{9} + 2 \beta_{11} ) q^{66} + ( -1 - 2 \beta_{1} + 5 \beta_{3} - 7 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - 3 \beta_{13} ) q^{67} + ( -7 - 2 \beta_{2} - \beta_{8} + 5 \beta_{9} - 3 \beta_{11} ) q^{68} + ( 8 \beta_{1} - \beta_{3} + 15 \beta_{4} + 4 \beta_{7} - \beta_{10} + 2 \beta_{12} - \beta_{13} ) q^{69} + ( 11 \beta_{1} - 7 \beta_{3} - 10 \beta_{4} - 2 \beta_{7} - 2 \beta_{10} + 4 \beta_{12} - 2 \beta_{13} ) q^{71} + ( -1 - 25 \beta_{1} + 7 \beta_{3} + 13 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - 3 \beta_{13} ) q^{72} + ( 18 + 4 \beta_{2} - 10 \beta_{5} + 4 \beta_{6} + \beta_{8} + 6 \beta_{9} + \beta_{11} ) q^{73} + ( -10 - 10 \beta_{2} + 4 \beta_{5} - 5 \beta_{6} + 2 \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{74} + ( -26 - 9 \beta_{1} + 7 \beta_{2} + 5 \beta_{3} - 7 \beta_{4} + 5 \beta_{5} - \beta_{6} - \beta_{8} - 4 \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} - 2 \beta_{13} ) q^{76} + ( 18 - 8 \beta_{2} + 5 \beta_{5} + \beta_{8} - 4 \beta_{9} + 3 \beta_{11} ) q^{77} + ( -1 + 2 \beta_{1} + \beta_{3} + 12 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{78} + ( -1 + 17 \beta_{1} - 9 \beta_{4} + \beta_{5} - \beta_{6} - 5 \beta_{7} - \beta_{8} - \beta_{9} + 3 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{79} + ( 29 - 9 \beta_{2} + 9 \beta_{5} + 5 \beta_{6} + \beta_{9} - 4 \beta_{11} ) q^{81} + ( -7 - 5 \beta_{2} - 5 \beta_{5} - 4 \beta_{6} + \beta_{8} + 6 \beta_{9} ) q^{82} + ( 22 - 4 \beta_{2} + 5 \beta_{5} + 3 \beta_{8} - 4 \beta_{9} - 6 \beta_{11} ) q^{83} + ( -1 - 11 \beta_{1} - \beta_{3} - 10 \beta_{4} + \beta_{5} - \beta_{6} - 5 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - 3 \beta_{13} ) q^{84} + ( -21 \beta_{1} + 5 \beta_{3} - 21 \beta_{4} + \beta_{7} + 2 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} ) q^{86} + ( 26 - 14 \beta_{2} + 6 \beta_{6} - \beta_{8} - 7 \beta_{9} - \beta_{11} ) q^{87} + ( -1 - 11 \beta_{1} + 7 \beta_{3} - 7 \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} - 2 \beta_{12} - 2 \beta_{13} ) q^{88} + ( -1 - 11 \beta_{1} - 5 \beta_{3} + 13 \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{8} - \beta_{9} + 4 \beta_{10} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} ) q^{89} + ( -6 \beta_{1} + 7 \beta_{3} + 10 \beta_{4} - 3 \beta_{7} + 2 \beta_{10} + \beta_{12} + 2 \beta_{13} ) q^{91} + ( -13 + 18 \beta_{2} - 5 \beta_{5} + 9 \beta_{6} - 2 \beta_{8} - 9 \beta_{9} + 2 \beta_{11} ) q^{92} + ( 31 - 2 \beta_{2} - 10 \beta_{5} - \beta_{6} - 2 \beta_{8} - 5 \beta_{9} - 3 \beta_{11} ) q^{93} + ( 1 - 41 \beta_{1} + 6 \beta_{3} - 7 \beta_{4} - \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + 2 \beta_{12} + 2 \beta_{13} ) q^{94} + ( 7 + 6 \beta_{2} + 9 \beta_{5} - \beta_{6} + 5 \beta_{9} + 5 \beta_{11} ) q^{96} + ( 1 - 11 \beta_{1} - 2 \beta_{3} - 16 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - 3 \beta_{12} + \beta_{13} ) q^{97} + ( 30 \beta_{1} - 5 \beta_{3} - 6 \beta_{4} - 4 \beta_{7} + 2 \beta_{10} + 3 \beta_{12} + 2 \beta_{13} ) q^{98} + ( -14 + 5 \beta_{2} - \beta_{5} - 5 \beta_{6} + \beta_{8} + \beta_{9} - 4 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14q - 28q^{4} - 4q^{6} + 20q^{7} - 36q^{9} + O(q^{10})$$ $$14q - 28q^{4} - 4q^{6} + 20q^{7} - 36q^{9} - 4q^{11} + 36q^{16} - 22q^{17} + 39q^{19} - 12q^{23} - 44q^{24} + 30q^{26} - 98q^{28} + 4q^{36} - 37q^{38} - 32q^{39} - 250q^{42} - 90q^{43} - 52q^{44} - 148q^{47} + 234q^{49} + 98q^{54} + 195q^{57} + 274q^{58} + 222q^{61} - 518q^{62} - 198q^{63} - 218q^{64} + 92q^{66} - 80q^{68} + 228q^{73} - 92q^{74} - 351q^{76} + 260q^{77} + 402q^{81} - 58q^{82} + 280q^{83} + 282q^{87} - 302q^{92} + 358q^{93} + 190q^{96} - 180q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} + 42 x^{12} + 677 x^{10} + 5313 x^{8} + 21125 x^{6} + 40138 x^{4} + 30565 x^{2} + 3675$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 6$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 10 \nu$$ $$\beta_{4}$$ $$=$$ $$($$$$-33 \nu^{13} - 1316 \nu^{11} - 19576 \nu^{9} - 134589 \nu^{7} - 419015 \nu^{5} - 459879 \nu^{3} - 47160 \nu$$$$)/1400$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{12} + 79 \nu^{10} + 1164 \nu^{8} + 7946 \nu^{6} + 24705 \nu^{4} + 27471 \nu^{2} + 3465$$$$)/40$$ $$\beta_{6}$$ $$=$$ $$($$$$-9 \nu^{12} - 353 \nu^{10} - 5148 \nu^{8} - 34637 \nu^{6} - 105680 \nu^{4} - 115562 \nu^{2} - 15075$$$$)/160$$ $$\beta_{7}$$ $$=$$ $$($$$$128 \nu^{13} + 5131 \nu^{11} + 76716 \nu^{9} + 529424 \nu^{7} + 1649415 \nu^{5} + 1797789 \nu^{3} + 161685 \nu$$$$)/1400$$ $$\beta_{8}$$ $$=$$ $$($$$$21 \nu^{12} + 842 \nu^{10} + 12612 \nu^{8} + 87493 \nu^{6} + 276205 \nu^{4} + 313973 \nu^{2} + 45470$$$$)/200$$ $$\beta_{9}$$ $$=$$ $$($$$$-89 \nu^{12} - 3553 \nu^{10} - 52908 \nu^{8} - 364237 \nu^{6} - 1138320 \nu^{4} - 1273082 \nu^{2} - 167155$$$$)/800$$ $$\beta_{10}$$ $$=$$ $$($$$$36 \nu^{13} - 791 \nu^{12} + 1372 \nu^{11} - 31507 \nu^{10} + 19192 \nu^{9} - 467852 \nu^{8} + 120988 \nu^{7} - 3209003 \nu^{6} + 326080 \nu^{5} - 9970380 \nu^{4} + 247968 \nu^{3} - 10999058 \nu^{2} - 44580 \nu - 1319745$$$$)/5600$$ $$\beta_{11}$$ $$=$$ $$($$$$-17 \nu^{12} - 684 \nu^{10} - 10274 \nu^{8} - 71311 \nu^{6} - 224085 \nu^{4} - 249771 \nu^{2} - 30990$$$$)/100$$ $$\beta_{12}$$ $$=$$ $$($$$$-947 \nu^{13} - 37919 \nu^{11} - 566884 \nu^{9} - 3922751 \nu^{7} - 12349860 \nu^{5} - 14029386 \nu^{3} - 2133165 \nu$$$$)/5600$$ $$\beta_{13}$$ $$=$$ $$($$$$-113 \nu^{13} + 113 \nu^{12} - 4501 \nu^{11} + 4501 \nu^{10} - 66836 \nu^{9} + 66836 \nu^{8} - 458429 \nu^{7} + 458429 \nu^{6} - 1424340 \nu^{5} + 1424340 \nu^{4} - 1570494 \nu^{3} + 1571294 \nu^{2} - 180535 \nu + 188535$$$$)/800$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 6$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 10 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - 14 \beta_{2} + 58$$ $$\nu^{5}$$ $$=$$ $$\beta_{13} - \beta_{12} - \beta_{11} + 3 \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + 2 \beta_{4} - 18 \beta_{3} + 116 \beta_{1} - 1$$ $$\nu^{6}$$ $$=$$ $$3 \beta_{11} - 28 \beta_{9} - 21 \beta_{8} + 24 \beta_{6} + 19 \beta_{5} + 188 \beta_{2} - 659$$ $$\nu^{7}$$ $$=$$ $$-28 \beta_{13} + 24 \beta_{12} + 24 \beta_{11} - 76 \beta_{10} + 24 \beta_{9} + 24 \beta_{8} - \beta_{7} + 24 \beta_{6} - 24 \beta_{5} - 29 \beta_{4} + 282 \beta_{3} - 1463 \beta_{1} + 24$$ $$\nu^{8}$$ $$=$$ $$-77 \beta_{11} + 541 \beta_{9} + 358 \beta_{8} - 425 \beta_{6} - 288 \beta_{5} - 2584 \beta_{2} + 8194$$ $$\nu^{9}$$ $$=$$ $$551 \beta_{13} - 435 \beta_{12} - 425 \beta_{11} + 1401 \beta_{10} - 425 \beta_{9} - 425 \beta_{8} + 17 \beta_{7} - 425 \beta_{6} + 425 \beta_{5} + 267 \beta_{4} - 4266 \beta_{3} + 19486 \beta_{1} - 425$$ $$\nu^{10}$$ $$=$$ $$1428 \beta_{11} - 9129 \beta_{9} - 5677 \beta_{8} + 6809 \beta_{6} + 4125 \beta_{5} + 36335 \beta_{2} - 108047$$ $$\nu^{11}$$ $$=$$ $$-9425 \beta_{13} + 7105 \beta_{12} + 6809 \beta_{11} - 23043 \beta_{10} + 6809 \beta_{9} + 6809 \beta_{8} - 172 \beta_{7} + 6809 \beta_{6} - 6809 \beta_{5} - 1446 \beta_{4} + 63627 \beta_{3} - 269029 \beta_{1} + 6809$$ $$\nu^{12}$$ $$=$$ $$-23511 \beta_{11} + 144625 \beta_{9} + 86966 \beta_{8} - 104605 \beta_{6} - 58436 \beta_{5} - 519069 \beta_{2} + 1481391$$ $$\nu^{13}$$ $$=$$ $$150497 \beta_{13} - 110477 \beta_{12} - 104605 \beta_{11} + 359707 \beta_{10} - 104605 \beta_{9} - 104605 \beta_{8} + 853 \beta_{7} - 104605 \beta_{6} + 104605 \beta_{5} - 7885 \beta_{4} - 942231 \beta_{3} + 3801025 \beta_{1} - 104605$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 − 3.82982i − 3.01757i − 2.92865i − 2.34822i − 1.41833i − 1.40632i − 0.382406i 0.382406i 1.40632i 1.41833i 2.34822i 2.92865i 3.01757i 3.82982i
3.82982i 0.102062i −10.6675 0 −0.390878 9.96826 25.5353i 8.98958 0
151.2 3.01757i 4.83172i −5.10575 0 14.5801 −6.15539 3.33669i −14.3455 0
151.3 2.92865i 2.02259i −4.57698 0 −5.92346 −8.32815 1.68976i 4.90912 0
151.4 2.34822i 5.73703i −1.51415 0 −13.4718 9.56678 5.83734i −23.9135 0
151.5 1.41833i 1.48141i 1.98835 0 −2.10112 −2.50491 8.49344i 6.80543 0
151.6 1.40632i 2.91656i 2.02225 0 4.10162 11.7208 8.46924i 0.493705 0
151.7 0.382406i 3.15259i 3.85377 0 1.20557 −4.26742 3.00333i −0.938823 0
151.8 0.382406i 3.15259i 3.85377 0 1.20557 −4.26742 3.00333i −0.938823 0
151.9 1.40632i 2.91656i 2.02225 0 4.10162 11.7208 8.46924i 0.493705 0
151.10 1.41833i 1.48141i 1.98835 0 −2.10112 −2.50491 8.49344i 6.80543 0
151.11 2.34822i 5.73703i −1.51415 0 −13.4718 9.56678 5.83734i −23.9135 0
151.12 2.92865i 2.02259i −4.57698 0 −5.92346 −8.32815 1.68976i 4.90912 0
151.13 3.01757i 4.83172i −5.10575 0 14.5801 −6.15539 3.33669i −14.3455 0
151.14 3.82982i 0.102062i −10.6675 0 −0.390878 9.96826 25.5353i 8.98958 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 151.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.3.c.i yes 14
5.b even 2 1 475.3.c.h 14
5.c odd 4 2 475.3.d.d 28
19.b odd 2 1 inner 475.3.c.i yes 14
95.d odd 2 1 475.3.c.h 14
95.g even 4 2 475.3.d.d 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.3.c.h 14 5.b even 2 1
475.3.c.h 14 95.d odd 2 1
475.3.c.i yes 14 1.a even 1 1 trivial
475.3.c.i yes 14 19.b odd 2 1 inner
475.3.d.d 28 5.c odd 4 2
475.3.d.d 28 95.g even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(475, [\chi])$$:

 $$T_{2}^{14} + 42 T_{2}^{12} + 677 T_{2}^{10} + 5313 T_{2}^{8} + 21125 T_{2}^{6} + 40138 T_{2}^{4} + 30565 T_{2}^{2} + 3675$$ $$T_{7}^{7} - 10 T_{7}^{6} - 180 T_{7}^{5} + 1276 T_{7}^{4} + 13005 T_{7}^{3} - 33200 T_{7}^{2} - 383375 T_{7} - 612500$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3675 + 30565 T^{2} + 40138 T^{4} + 21125 T^{6} + 5313 T^{8} + 677 T^{10} + 42 T^{12} + T^{14}$$
$3$ $$6075 + 589225 T^{2} + 580343 T^{4} + 201241 T^{6} + 31508 T^{8} + 2370 T^{10} + 81 T^{12} + T^{14}$$
$5$ $$T^{14}$$
$7$ $$( -612500 - 383375 T - 33200 T^{2} + 13005 T^{3} + 1276 T^{4} - 180 T^{5} - 10 T^{6} + T^{7} )^{2}$$
$11$ $$( 1993920 - 730880 T - 40012 T^{2} + 31953 T^{3} - 355 T^{4} - 391 T^{5} + 2 T^{6} + T^{7} )^{2}$$
$13$ $$51013467967500 + 8022441270445 T^{2} + 454975074013 T^{4} + 11193406931 T^{6} + 116728583 T^{8} + 566395 T^{10} + 1251 T^{12} + T^{14}$$
$17$ $$( 16723125 - 11910375 T - 1170825 T^{2} + 339235 T^{3} - 3548 T^{4} - 1068 T^{5} + 11 T^{6} + T^{7} )^{2}$$
$19$ $$799006685782884121 - 86319281843580279 T + 7780323081149469 T^{2} - 377340803644938 T^{3} + 17649121119507 T^{4} - 878329265577 T^{5} + 56875081783 T^{6} - 3506787212 T^{7} + 157548703 T^{8} - 6739737 T^{9} + 375147 T^{10} - 22218 T^{11} + 1269 T^{12} - 39 T^{13} + T^{14}$$
$23$ $$( -45858750 - 19889125 T + 7705550 T^{2} + 798485 T^{3} - 14224 T^{4} - 1958 T^{5} + 6 T^{6} + T^{7} )^{2}$$
$29$ $$25\!\cdots\!00$$$$+ 2911573957722578125 T^{2} + 12403046323171875 T^{4} + 26542784725875 T^{6} + 31347324845 T^{8} + 20685013 T^{10} + 7133 T^{12} + T^{14}$$
$31$ $$198532687500000000 + 31530364180000000 T^{2} + 970988417350000 T^{4} + 9095831192500 T^{6} + 21987838745 T^{8} + 20700708 T^{10} + 7793 T^{12} + T^{14}$$
$37$ $$39\!\cdots\!00$$$$+ 27710270948876114785 T^{2} + 76417385532268737 T^{4} + 109435659524180 T^{6} + 88936261785 T^{8} + 41210678 T^{10} + 10082 T^{12} + T^{14}$$
$41$ $$12\!\cdots\!00$$$$+ 3635086716163750000 T^{2} + 24868069810375000 T^{4} + 65450728792875 T^{6} + 76211818945 T^{8} + 42676637 T^{10} + 11026 T^{12} + T^{14}$$
$43$ $$( 188721350000 + 25764413000 T + 1110280300 T^{2} + 8775900 T^{3} - 462521 T^{4} - 7645 T^{5} + 45 T^{6} + T^{7} )^{2}$$
$47$ $$( -340225818750 - 4250971125 T + 711726075 T^{2} + 9432540 T^{3} - 442599 T^{4} - 6016 T^{5} + 74 T^{6} + T^{7} )^{2}$$
$53$ $$4771299641111892300 + 1534668894297346585 T^{2} + 120838901803909808 T^{4} + 253422356459496 T^{6} + 204163724843 T^{8} + 78199750 T^{10} + 14281 T^{12} + T^{14}$$
$59$ $$42\!\cdots\!00$$$$+ 7290281695345000000 T^{2} + 34214581580600000 T^{4} + 68410525682500 T^{6} + 68926911845 T^{8} + 36517617 T^{10} + 9654 T^{12} + T^{14}$$
$61$ $$( -121256990560 + 24119558200 T - 815279564 T^{2} - 12789620 T^{3} + 826085 T^{4} - 5294 T^{5} - 111 T^{6} + T^{7} )^{2}$$
$67$ $$7112916874021662675 + 1938954421989617950 T^{2} + 139851331124517072 T^{4} + 1492645887216978 T^{6} + 1214090407480 T^{8} + 307887052 T^{10} + 30124 T^{12} + T^{14}$$
$71$ $$51\!\cdots\!00$$$$+$$$$43\!\cdots\!00$$$$T^{2} + 85741428414502250000 T^{4} + 46182241560276000 T^{6} + 10063270045505 T^{8} + 1046070047 T^{10} + 52055 T^{12} + T^{14}$$
$73$ $$( 694377415625 + 102224812750 T - 8623630200 T^{2} + 62573570 T^{3} + 2550216 T^{4} - 21948 T^{5} - 114 T^{6} + T^{7} )^{2}$$
$79$ $$86\!\cdots\!00$$$$+$$$$10\!\cdots\!00$$$$T^{2} +$$$$54\!\cdots\!00$$$$T^{4} + 137872446445685000 T^{6} + 19362520051545 T^{8} + 1514604143 T^{10} + 61380 T^{12} + T^{14}$$
$83$ $$( 5812813755000 + 83086802500 T - 9766533900 T^{2} - 22768825 T^{3} + 3409113 T^{4} - 19459 T^{5} - 140 T^{6} + T^{7} )^{2}$$
$89$ $$56\!\cdots\!00$$$$+$$$$20\!\cdots\!00$$$$T^{2} + 2267088473068750000 T^{4} + 7879266019818875 T^{6} + 9048071543380 T^{8} + 1293141573 T^{10} + 63367 T^{12} + T^{14}$$
$97$ $$75\!\cdots\!00$$$$+$$$$19\!\cdots\!00$$$$T^{2} + 1397831956952018400 T^{4} + 3349851468242180 T^{6} + 2047317761489 T^{8} + 467216640 T^{10} + 41457 T^{12} + T^{14}$$