# Properties

 Label 136.4.k.b Level $136$ Weight $4$ Character orbit 136.k Analytic conductor $8.024$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$136 = 2^{3} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 136.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.02425976078$$ Analytic rank: $$0$$ Dimension: $$14$$ Relative dimension: $$7$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ Defining polynomial: $$x^{14} + 119 x^{12} + 5319 x^{10} + 112122 x^{8} + 1120191 x^{6} + 4382607 x^{4} + 1699337 x^{2} + 2704$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{15}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{3} -\beta_{7} q^{5} + ( 1 - \beta_{5} - \beta_{8} ) q^{7} + ( \beta_{2} - \beta_{3} + 6 \beta_{5} - \beta_{11} ) q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{3} -\beta_{7} q^{5} + ( 1 - \beta_{5} - \beta_{8} ) q^{7} + ( \beta_{2} - \beta_{3} + 6 \beta_{5} - \beta_{11} ) q^{9} + ( -6 - \beta_{2} + 6 \beta_{5} + \beta_{6} - \beta_{12} ) q^{11} + ( -8 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( -5 \beta_{5} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{15} + ( 9 + \beta_{2} - \beta_{3} - \beta_{4} + 4 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{17} + ( -2 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{19} + ( -7 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} + \beta_{12} + \beta_{13} ) q^{21} + ( -14 + \beta_{1} - 4 \beta_{2} + 3 \beta_{4} + 14 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} - 3 \beta_{11} - \beta_{12} ) q^{23} + ( -4 \beta_{2} + 4 \beta_{3} + 30 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} + 2 \beta_{13} ) q^{25} + ( -8 + 6 \beta_{2} + 2 \beta_{4} + 8 \beta_{5} - 5 \beta_{6} - 4 \beta_{8} - 2 \beta_{11} + \beta_{12} ) q^{27} + ( 7 - 2 \beta_{3} - \beta_{4} + 7 \beta_{5} - 5 \beta_{7} - 4 \beta_{9} - \beta_{11} ) q^{29} + ( -32 - \beta_{1} - 10 \beta_{3} + \beta_{4} - 32 \beta_{5} - 3 \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} - 3 \beta_{13} ) q^{31} + ( -4 - \beta_{1} - 5 \beta_{2} - 5 \beta_{3} - 4 \beta_{4} + 5 \beta_{6} - 5 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} - \beta_{12} - \beta_{13} ) q^{33} + ( 28 + 6 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 5 \beta_{6} + 5 \beta_{7} - 5 \beta_{8} + 5 \beta_{9} - \beta_{12} - \beta_{13} ) q^{35} + ( 39 + \beta_{1} - 16 \beta_{3} - 5 \beta_{4} + 39 \beta_{5} + 7 \beta_{7} + 4 \beta_{9} - \beta_{10} - 5 \beta_{11} + 2 \beta_{13} ) q^{37} + ( -20 + 12 \beta_{3} + 2 \beta_{4} - 20 \beta_{5} - 13 \beta_{7} + 12 \beta_{9} + 2 \beta_{11} - \beta_{13} ) q^{39} + ( 50 + \beta_{1} - 16 \beta_{2} - \beta_{4} - 50 \beta_{5} + 18 \beta_{6} - 4 \beta_{8} + \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{41} + ( 19 \beta_{2} - 19 \beta_{3} + 23 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} - \beta_{10} + 5 \beta_{11} - \beta_{12} + \beta_{13} ) q^{43} + ( 5 - 2 \beta_{1} + 22 \beta_{2} + \beta_{4} - 5 \beta_{5} + 5 \beta_{6} + 8 \beta_{8} - 2 \beta_{10} - \beta_{11} + 4 \beta_{12} ) q^{45} + ( -21 + 5 \beta_{1} + 18 \beta_{2} + 18 \beta_{3} + 5 \beta_{4} + 3 \beta_{6} - 3 \beta_{7} - 3 \beta_{12} - 3 \beta_{13} ) q^{47} + ( -9 \beta_{2} + 9 \beta_{3} - 71 \beta_{5} + 11 \beta_{6} + 11 \beta_{7} - 10 \beta_{8} - 10 \beta_{9} - 3 \beta_{10} + 4 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} ) q^{49} + ( -41 + \beta_{1} + 23 \beta_{2} - 28 \beta_{3} + \beta_{4} + 37 \beta_{5} - 17 \beta_{6} - 4 \beta_{7} + 9 \beta_{8} - \beta_{9} - 3 \beta_{10} - 3 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} ) q^{51} + ( -24 \beta_{2} + 24 \beta_{3} - 79 \beta_{5} - 20 \beta_{6} - 20 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} - 3 \beta_{10} + 9 \beta_{11} + \beta_{12} - \beta_{13} ) q^{53} + ( 117 + 5 \beta_{1} - 18 \beta_{2} - 18 \beta_{3} - 3 \beta_{4} - 29 \beta_{6} + 29 \beta_{7} - 5 \beta_{8} + 5 \beta_{9} + \beta_{12} + \beta_{13} ) q^{55} + ( -6 - \beta_{1} - 50 \beta_{2} - 4 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} + 20 \beta_{8} - \beta_{10} + 4 \beta_{11} ) q^{57} + ( -33 \beta_{2} + 33 \beta_{3} - 17 \beta_{5} + 12 \beta_{6} + 12 \beta_{7} - 6 \beta_{8} - 6 \beta_{9} + 3 \beta_{10} + 5 \beta_{11} + 4 \beta_{12} - 4 \beta_{13} ) q^{59} + ( -87 - \beta_{1} - 4 \beta_{2} + \beta_{4} + 87 \beta_{5} + 7 \beta_{6} + 12 \beta_{8} - \beta_{10} - \beta_{11} + 4 \beta_{12} ) q^{61} + ( 33 + 2 \beta_{1} + 58 \beta_{3} + 8 \beta_{4} + 33 \beta_{5} - 11 \beta_{7} + 11 \beta_{9} - 2 \beta_{10} + 8 \beta_{11} + \beta_{13} ) q^{63} + ( -4 + 3 \beta_{1} - 58 \beta_{3} + 6 \beta_{4} - 4 \beta_{5} + 34 \beta_{7} + 8 \beta_{9} - 3 \beta_{10} + 6 \beta_{11} + 4 \beta_{13} ) q^{65} + ( -3 + 3 \beta_{1} + 17 \beta_{2} + 17 \beta_{3} - 3 \beta_{4} + 10 \beta_{6} - 10 \beta_{7} + 13 \beta_{8} - 13 \beta_{9} + 4 \beta_{12} + 4 \beta_{13} ) q^{67} + ( 226 + \beta_{1} + 71 \beta_{2} + 71 \beta_{3} - 4 \beta_{4} - 21 \beta_{6} + 21 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} + 2 \beta_{12} + 2 \beta_{13} ) q^{69} + ( -241 - 2 \beta_{1} - 34 \beta_{3} + 6 \beta_{4} - 241 \beta_{5} + 20 \beta_{7} + 5 \beta_{9} + 2 \beta_{10} + 6 \beta_{11} - 8 \beta_{13} ) q^{71} + ( -90 + \beta_{1} + 48 \beta_{3} - 3 \beta_{4} - 90 \beta_{5} - 20 \beta_{7} + 4 \beta_{9} - \beta_{10} - 3 \beta_{11} - 2 \beta_{13} ) q^{73} + ( 37 - 5 \beta_{1} - 33 \beta_{2} - 7 \beta_{4} - 37 \beta_{5} + 30 \beta_{6} + 6 \beta_{8} - 5 \beta_{10} + 7 \beta_{11} - 2 \beta_{12} ) q^{75} + ( 49 \beta_{2} - 49 \beta_{3} + 70 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} + 8 \beta_{8} + 8 \beta_{9} + 3 \beta_{10} - 4 \beta_{11} + 4 \beta_{12} - 4 \beta_{13} ) q^{77} + ( 89 + 6 \beta_{1} - 36 \beta_{2} - 4 \beta_{4} - 89 \beta_{5} + 33 \beta_{6} + 5 \beta_{8} + 6 \beta_{10} + 4 \beta_{11} - 5 \beta_{12} ) q^{79} + ( 93 - 3 \beta_{1} + 55 \beta_{2} + 55 \beta_{3} - 6 \beta_{4} - 23 \beta_{6} + 23 \beta_{7} - 16 \beta_{8} + 16 \beta_{9} + 7 \beta_{12} + 7 \beta_{13} ) q^{81} + ( -13 \beta_{2} + 13 \beta_{3} - 33 \beta_{5} + 10 \beta_{6} + 10 \beta_{7} + 10 \beta_{8} + 10 \beta_{9} - \beta_{10} - 11 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} ) q^{83} + ( -141 - 2 \beta_{1} + 62 \beta_{2} - 12 \beta_{3} - 5 \beta_{4} + 118 \beta_{5} + \beta_{6} - 32 \beta_{7} - 8 \beta_{8} - 8 \beta_{9} - 3 \beta_{10} + 12 \beta_{11} - 7 \beta_{12} - \beta_{13} ) q^{85} + ( 42 \beta_{2} - 42 \beta_{3} + 19 \beta_{5} - 13 \beta_{6} - 13 \beta_{7} - 9 \beta_{8} - 9 \beta_{9} - 3 \beta_{10} - 13 \beta_{11} + 5 \beta_{12} - 5 \beta_{13} ) q^{87} + ( -40 + 9 \beta_{1} - 67 \beta_{2} - 67 \beta_{3} + 14 \beta_{4} + 11 \beta_{6} - 11 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 12 \beta_{12} - 12 \beta_{13} ) q^{89} + ( 92 - 10 \beta_{1} - 142 \beta_{2} - 92 \beta_{5} + 11 \beta_{6} - 6 \beta_{8} - 10 \beta_{10} + 9 \beta_{12} ) q^{91} + ( -17 \beta_{2} + 17 \beta_{3} + 302 \beta_{5} - 25 \beta_{6} - 25 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} + \beta_{10} - 16 \beta_{11} + 4 \beta_{12} - 4 \beta_{13} ) q^{93} + ( -161 + 3 \beta_{1} - 4 \beta_{2} - 13 \beta_{4} + 161 \beta_{5} - 36 \beta_{6} - 30 \beta_{8} + 3 \beta_{10} + 13 \beta_{11} - 16 \beta_{12} ) q^{95} + ( 99 - 5 \beta_{1} - 50 \beta_{3} - 10 \beta_{4} + 99 \beta_{5} - 20 \beta_{7} - 16 \beta_{9} + 5 \beta_{10} - 10 \beta_{11} - 6 \beta_{13} ) q^{97} + ( 211 + 3 \beta_{1} - 113 \beta_{3} - 19 \beta_{4} + 211 \beta_{5} - 32 \beta_{7} - 20 \beta_{9} - 3 \beta_{10} - 19 \beta_{11} + 12 \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q + 6 q^{3} + 6 q^{5} + 10 q^{7} + O(q^{10})$$ $$14 q + 6 q^{3} + 6 q^{5} + 10 q^{7} - 66 q^{11} - 124 q^{13} + 130 q^{17} - 148 q^{21} - 162 q^{23} - 204 q^{27} + 158 q^{29} - 350 q^{31} + 116 q^{33} + 236 q^{35} + 582 q^{37} - 320 q^{39} + 878 q^{41} - 26 q^{45} - 448 q^{47} - 590 q^{51} + 1460 q^{55} + 292 q^{57} - 1130 q^{61} + 114 q^{63} + 20 q^{65} - 64 q^{67} + 2076 q^{69} - 3274 q^{71} - 1426 q^{73} + 946 q^{75} + 1718 q^{79} + 166 q^{81} - 2018 q^{85} + 476 q^{89} + 2128 q^{91} - 2444 q^{95} + 1926 q^{97} + 3870 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} + 119 x^{12} + 5319 x^{10} + 112122 x^{8} + 1120191 x^{6} + 4382607 x^{4} + 1699337 x^{2} + 2704$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-16553 \nu^{12} - 2958786 \nu^{10} - 553258413 \nu^{8} - 35042935761 \nu^{6} - 776715718098 \nu^{4} - 5240890807749 \nu^{2} - 1462819625392$$$$)/ 53247141288$$ $$\beta_{2}$$ $$=$$ $$($$$$9031 \nu^{12} + 574494 \nu^{10} - 2543421 \nu^{8} - 708095937 \nu^{6} - 12509813490 \nu^{4} - 52342238181 \nu^{2} - 29043895248 \nu - 1244553232$$$$)/ 29043895248$$ $$\beta_{3}$$ $$=$$ $$($$$$9031 \nu^{12} + 574494 \nu^{10} - 2543421 \nu^{8} - 708095937 \nu^{6} - 12509813490 \nu^{4} - 52342238181 \nu^{2} + 29043895248 \nu - 1244553232$$$$)/ 29043895248$$ $$\beta_{4}$$ $$=$$ $$($$$$-9031 \nu^{12} - 574494 \nu^{10} + 2543421 \nu^{8} + 708095937 \nu^{6} + 12509813490 \nu^{4} + 81386133429 \nu^{2} + 480468824824$$$$)/ 14521947624$$ $$\beta_{5}$$ $$=$$ $$($$$$-5983429 \nu^{13} - 712145454 \nu^{11} - 31833327273 \nu^{9} - 670840961865 \nu^{7} - 6693378067758 \nu^{5} - 26060390244033 \nu^{3} - 9487413190220 \nu$$$$)/ 377570638224$$ $$\beta_{6}$$ $$=$$ $$($$$$-16718259 \nu^{13} - 19735339 \nu^{12} - 1997695269 \nu^{11} - 2196992499 \nu^{10} - 89620518609 \nu^{9} - 89819078388 \nu^{8} - 1893984144588 \nu^{7} - 1686473706255 \nu^{6} - 19003094129961 \nu^{5} - 14410751366271 \nu^{4} - 76569890579577 \nu^{3} - 44078757852480 \nu^{2} - 51529530090165 \nu - 2125260748364$$$$)/ 1038319255116$$ $$\beta_{7}$$ $$=$$ $$($$$$-16718259 \nu^{13} + 19735339 \nu^{12} - 1997695269 \nu^{11} + 2196992499 \nu^{10} - 89620518609 \nu^{9} + 89819078388 \nu^{8} - 1893984144588 \nu^{7} + 1686473706255 \nu^{6} - 19003094129961 \nu^{5} + 14410751366271 \nu^{4} - 76569890579577 \nu^{3} + 44078757852480 \nu^{2} - 51529530090165 \nu + 2125260748364$$$$)/ 1038319255116$$ $$\beta_{8}$$ $$=$$ $$($$$$394782247 \nu^{13} + 30952792 \nu^{12} + 46939392582 \nu^{11} + 2586850656 \nu^{10} + 2094008337087 \nu^{9} + 56784667056 \nu^{8} + 43968354527715 \nu^{7} - 163613325720 \nu^{6} + 436113015710646 \nu^{5} - 15301683043296 \nu^{4} + 1681302766451415 \nu^{3} - 100278228438336 \nu^{2} + 553556528507924 \nu + 21859444421360$$$$)/ 4153277020464$$ $$\beta_{9}$$ $$=$$ $$($$$$394782247 \nu^{13} - 30952792 \nu^{12} + 46939392582 \nu^{11} - 2586850656 \nu^{10} + 2094008337087 \nu^{9} - 56784667056 \nu^{8} + 43968354527715 \nu^{7} + 163613325720 \nu^{6} + 436113015710646 \nu^{5} + 15301683043296 \nu^{4} + 1681302766451415 \nu^{3} + 100278228438336 \nu^{2} + 553556528507924 \nu - 21859444421360$$$$)/ 4153277020464$$ $$\beta_{10}$$ $$=$$ $$($$$$-778104491 \nu^{13} - 93193189950 \nu^{11} - 4199517040215 \nu^{9} - 89390253958059 \nu^{7} - 903371091656622 \nu^{5} - 3581518823081679 \nu^{3} - 1424787091162888 \nu$$$$)/ 2076638510232$$ $$\beta_{11}$$ $$=$$ $$($$$$-197687963 \nu^{13} - 23515736826 \nu^{11} - 1050433671063 \nu^{9} - 22119341247183 \nu^{7} - 220556221085274 \nu^{5} - 858631979860383 \nu^{3} - 313807418169676 \nu$$$$)/ 377570638224$$ $$\beta_{12}$$ $$=$$ $$($$$$-143212757 \nu^{13} - 11120005 \nu^{12} - 17048541404 \nu^{11} - 1318184036 \nu^{10} - 762476943733 \nu^{9} - 58709091363 \nu^{8} - 16088764865267 \nu^{7} - 1235688069941 \nu^{6} - 161046078537220 \nu^{5} - 12386311463292 \nu^{4} - 633026078177693 \nu^{3} - 48633301863731 \nu^{2} - 257557705863182 \nu - 10332452117624$$$$)/ 230737612248$$ $$\beta_{13}$$ $$=$$ $$($$$$143212757 \nu^{13} - 11120005 \nu^{12} + 17048541404 \nu^{11} - 1318184036 \nu^{10} + 762476943733 \nu^{9} - 58709091363 \nu^{8} + 16088764865267 \nu^{7} - 1235688069941 \nu^{6} + 161046078537220 \nu^{5} - 12386311463292 \nu^{4} + 633026078177693 \nu^{3} - 48633301863731 \nu^{2} + 257557705863182 \nu - 10332452117624$$$$)/ 230737612248$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} + \beta_{2} - 33$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{13} + \beta_{12} - 4 \beta_{11} - 4 \beta_{9} - 4 \beta_{8} - 5 \beta_{7} - 5 \beta_{6} + 16 \beta_{5} - 60 \beta_{3} + 60 \beta_{2}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-7 \beta_{13} - 7 \beta_{12} - 16 \beta_{9} + 16 \beta_{8} - 23 \beta_{7} + 23 \beta_{6} - 75 \beta_{4} - 136 \beta_{3} - 136 \beta_{2} + 3 \beta_{1} + 1851$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$59 \beta_{13} - 59 \beta_{12} + 267 \beta_{11} + 9 \beta_{10} + 212 \beta_{9} + 212 \beta_{8} + 235 \beta_{7} + 235 \beta_{6} - 2347 \beta_{5} + 2104 \beta_{3} - 2104 \beta_{2}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$400 \beta_{13} + 400 \beta_{12} + 1000 \beta_{9} - 1000 \beta_{8} + 1328 \beta_{7} - 1328 \beta_{6} + 2945 \beta_{4} + 7638 \beta_{3} + 7638 \beta_{2} - 189 \beta_{1} - 61061$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-1389 \beta_{13} + 1389 \beta_{12} - 6950 \beta_{11} - 336 \beta_{10} - 4776 \beta_{9} - 4776 \beta_{8} - 5067 \beta_{7} - 5067 \beta_{6} + 81766 \beta_{5} - 40585 \beta_{3} + 40585 \beta_{2}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-9472 \beta_{13} - 9472 \beta_{12} - 25150 \beta_{9} + 25150 \beta_{8} - 31688 \beta_{7} + 31688 \beta_{6} - 60492 \beta_{4} - 192121 \beta_{3} - 192121 \beta_{2} + 4485 \beta_{1} + 1124836$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$125251 \beta_{13} - 125251 \beta_{12} + 665760 \beta_{11} + 37224 \beta_{10} + 415996 \beta_{9} + 415996 \beta_{8} + 439475 \beta_{7} + 439475 \beta_{6} - 8967788 \beta_{5} + 3320388 \beta_{3} - 3320388 \beta_{2}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$855515 \beta_{13} + 855515 \beta_{12} + 2370092 \beta_{9} - 2370092 \beta_{8} + 2880691 \beta_{7} - 2880691 \beta_{6} + 5129345 \beta_{4} + 18248008 \beta_{3} + 18248008 \beta_{2} - 392517 \beta_{1} - 89151457$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$-5594021 \beta_{13} + 5594021 \beta_{12} - 30731683 \beta_{11} - 1859493 \beta_{10} - 18061028 \beta_{9} - 18061028 \beta_{8} - 19289761 \beta_{7} - 19289761 \beta_{6} + 443658627 \beta_{5} - 140628400 \beta_{3} + 140628400 \beta_{2}$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$-38091144 \beta_{13} - 38091144 \beta_{12} - 108692148 \beta_{9} + 108692148 \beta_{8} - 128834664 \beta_{7} + 128834664 \beta_{6} - 221758407 \beta_{4} - 840525510 \beta_{3} - 840525510 \beta_{2} + 16832295 \beta_{1} + 3699241435$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$124623174 \beta_{13} - 124623174 \beta_{12} + 696412656 \beta_{11} + 44274816 \beta_{10} + 393809340 \beta_{9} + 393809340 \beta_{8} + 426413058 \beta_{7} + 426413058 \beta_{6} - 10436158464 \beta_{5} + 3038627173 \beta_{3} - 3038627173 \beta_{2}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$$.

 $$n$$ $$69$$ $$103$$ $$105$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 5.11455i 4.87012i 0.657265i − 0.0399724i − 3.13314i − 3.80701i − 6.66182i − 5.11455i − 4.87012i − 0.657265i 0.0399724i 3.13314i 3.80701i 6.66182i
0 −5.11455 5.11455i 0 7.20340 + 7.20340i 0 19.2581 19.2581i 0 25.3172i 0
81.2 0 −4.87012 4.87012i 0 −8.46115 8.46115i 0 −8.46698 + 8.46698i 0 20.4362i 0
81.3 0 −0.657265 0.657265i 0 13.8302 + 13.8302i 0 −14.0096 + 14.0096i 0 26.1360i 0
81.4 0 0.0399724 + 0.0399724i 0 −1.97903 1.97903i 0 −4.30175 + 4.30175i 0 26.9968i 0
81.5 0 3.13314 + 3.13314i 0 −13.6954 13.6954i 0 −0.366316 + 0.366316i 0 7.36691i 0
81.6 0 3.80701 + 3.80701i 0 2.78003 + 2.78003i 0 25.7370 25.7370i 0 1.98658i 0
81.7 0 6.66182 + 6.66182i 0 3.32200 + 3.32200i 0 −12.8505 + 12.8505i 0 61.7597i 0
89.1 0 −5.11455 + 5.11455i 0 7.20340 7.20340i 0 19.2581 + 19.2581i 0 25.3172i 0
89.2 0 −4.87012 + 4.87012i 0 −8.46115 + 8.46115i 0 −8.46698 8.46698i 0 20.4362i 0
89.3 0 −0.657265 + 0.657265i 0 13.8302 13.8302i 0 −14.0096 14.0096i 0 26.1360i 0
89.4 0 0.0399724 0.0399724i 0 −1.97903 + 1.97903i 0 −4.30175 4.30175i 0 26.9968i 0
89.5 0 3.13314 3.13314i 0 −13.6954 + 13.6954i 0 −0.366316 0.366316i 0 7.36691i 0
89.6 0 3.80701 3.80701i 0 2.78003 2.78003i 0 25.7370 + 25.7370i 0 1.98658i 0
89.7 0 6.66182 6.66182i 0 3.32200 3.32200i 0 −12.8505 12.8505i 0 61.7597i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 89.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.4.k.b 14
4.b odd 2 1 272.4.o.f 14
17.c even 4 1 inner 136.4.k.b 14
17.d even 8 2 2312.4.a.l 14
68.f odd 4 1 272.4.o.f 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.k.b 14 1.a even 1 1 trivial
136.4.k.b 14 17.c even 4 1 inner
272.4.o.f 14 4.b odd 2 1
272.4.o.f 14 68.f odd 4 1
2312.4.a.l 14 17.d even 8 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{14} - \cdots$$ acting on $$S_{4}^{\mathrm{new}}(136, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{14}$$
$3$ $$346112 - 8246784 T + 98247744 T^{2} + 122689088 T^{3} + 79351072 T^{4} - 33003936 T^{5} + 6988144 T^{6} + 126400 T^{7} + 59408 T^{8} - 26400 T^{9} + 6044 T^{10} + 140 T^{11} + 18 T^{12} - 6 T^{13} + T^{14}$$
$5$ $$5698471938048 - 999277056000 T + 87616000000 T^{2} - 56580550400 T^{3} + 73901762336 T^{4} - 19094038624 T^{5} + 2492938000 T^{6} - 37998208 T^{7} + 2879888 T^{8} - 958768 T^{9} + 160456 T^{10} - 176 T^{11} + 18 T^{12} - 6 T^{13} + T^{14}$$
$7$ $$181426562039808 + 569474568757248 T + 893753595986944 T^{2} + 319524912989184 T^{3} + 61060686928896 T^{4} + 6311609489792 T^{5} + 381298007440 T^{6} + 10963466224 T^{7} + 161097608 T^{8} + 9099688 T^{9} + 1047160 T^{10} + 24676 T^{11} + 50 T^{12} - 10 T^{13} + T^{14}$$
$11$ $$27\!\cdots\!28$$$$- 10182113340648910848 T + 185453283204408384 T^{2} + 255061619196560064 T^{3} + 38665528242357536 T^{4} + 1600623865000032 T^{5} + 32412222656496 T^{6} - 54137359040 T^{7} + 18036119120 T^{8} + 685803552 T^{9} + 12974140 T^{10} - 45404 T^{11} + 2178 T^{12} + 66 T^{13} + T^{14}$$
$13$ $$( -237436285696 - 4295709760 T + 807446880 T^{2} + 14204848 T^{3} - 463008 T^{4} - 7788 T^{5} + 62 T^{6} + T^{7} )^{2}$$
$17$ $$69\!\cdots\!17$$$$-$$$$18\!\cdots\!70$$$$T +$$$$55\!\cdots\!35$$$$T^{2} -$$$$80\!\cdots\!76$$$$T^{3} + 17351411533914787773 T^{4} - 207875325547903998 T^{5} + 4109575288903203 T^{6} - 44186330027592 T^{7} + 836469629331 T^{8} - 8612106942 T^{9} + 146316909 T^{10} - 1383516 T^{11} + 19295 T^{12} - 130 T^{13} + T^{14}$$
$19$ $$19\!\cdots\!04$$$$+$$$$13\!\cdots\!52$$$$T^{2} + 32385667791356915712 T^{4} + 30439545731249408 T^{6} + 8927223280704 T^{8} + 1060588928 T^{10} + 54516 T^{12} + T^{14}$$
$23$ $$88\!\cdots\!12$$$$+$$$$16\!\cdots\!56$$$$T +$$$$16\!\cdots\!64$$$$T^{2} -$$$$25\!\cdots\!56$$$$T^{3} +$$$$13\!\cdots\!36$$$$T^{4} + 25871187437233284608 T^{5} + 248951461561244176 T^{6} - 545646739702288 T^{7} + 7886150783496 T^{8} + 122976893944 T^{9} + 1012187512 T^{10} - 1578492 T^{11} + 13122 T^{12} + 162 T^{13} + T^{14}$$
$29$ $$58\!\cdots\!32$$$$+$$$$65\!\cdots\!96$$$$T +$$$$36\!\cdots\!44$$$$T^{2} +$$$$46\!\cdots\!16$$$$T^{3} + 16954540060633330208 T^{4} + 1843782857098869024 T^{5} + 119097614803188240 T^{6} + 1049532091882240 T^{7} + 3989457735952 T^{8} - 36936914480 T^{9} + 491184520 T^{10} + 2927248 T^{11} + 12482 T^{12} - 158 T^{13} + T^{14}$$
$31$ $$20\!\cdots\!72$$$$-$$$$24\!\cdots\!24$$$$T +$$$$15\!\cdots\!04$$$$T^{2} +$$$$46\!\cdots\!12$$$$T^{3} +$$$$22\!\cdots\!08$$$$T^{4} +$$$$12\!\cdots\!72$$$$T^{5} + 366448951108826896 T^{6} + 3531391830517776 T^{7} + 171101760753800 T^{8} + 998229859016 T^{9} + 2916263416 T^{10} + 827580 T^{11} + 61250 T^{12} + 350 T^{13} + T^{14}$$
$37$ $$19\!\cdots\!12$$$$-$$$$84\!\cdots\!24$$$$T +$$$$18\!\cdots\!24$$$$T^{2} -$$$$22\!\cdots\!36$$$$T^{3} +$$$$15\!\cdots\!12$$$$T^{4} -$$$$55\!\cdots\!00$$$$T^{5} +$$$$13\!\cdots\!80$$$$T^{6} - 517890067352740352 T^{7} + 3635046432120336 T^{8} - 12575682726448 T^{9} + 23482582600 T^{10} - 24209760 T^{11} + 169362 T^{12} - 582 T^{13} + T^{14}$$
$41$ $$20\!\cdots\!28$$$$+$$$$30\!\cdots\!44$$$$T +$$$$22\!\cdots\!56$$$$T^{2} -$$$$30\!\cdots\!00$$$$T^{3} +$$$$16\!\cdots\!64$$$$T^{4} -$$$$22\!\cdots\!56$$$$T^{5} +$$$$14\!\cdots\!84$$$$T^{6} - 1467332986752853504 T^{7} + 7622864235671576 T^{8} - 16651063622952 T^{9} + 25276691916 T^{10} - 74107072 T^{11} + 385442 T^{12} - 878 T^{13} + T^{14}$$
$43$ $$18\!\cdots\!76$$$$+$$$$53\!\cdots\!24$$$$T^{2} +$$$$23\!\cdots\!68$$$$T^{4} +$$$$23\!\cdots\!84$$$$T^{6} + 7573422664465280 T^{8} + 97747014320 T^{10} + 532972 T^{12} + T^{14}$$
$47$ $$( 1675965455400960 - 204778432364544 T + 1023101804544 T^{2} + 19091078144 T^{3} - 23297152 T^{4} - 332208 T^{5} + 224 T^{6} + T^{7} )^{2}$$
$53$ $$84\!\cdots\!84$$$$+$$$$13\!\cdots\!36$$$$T^{2} +$$$$58\!\cdots\!52$$$$T^{4} +$$$$71\!\cdots\!56$$$$T^{6} + 403616910680487424 T^{8} + 1170896726800 T^{10} + 1711272 T^{12} + T^{14}$$
$59$ $$48\!\cdots\!00$$$$+$$$$25\!\cdots\!36$$$$T^{2} +$$$$54\!\cdots\!92$$$$T^{4} +$$$$57\!\cdots\!36$$$$T^{6} + 329405632321597824 T^{8} + 1033488251696 T^{10} + 1627836 T^{12} + T^{14}$$
$61$ $$77\!\cdots\!92$$$$+$$$$50\!\cdots\!16$$$$T +$$$$16\!\cdots\!84$$$$T^{2} +$$$$32\!\cdots\!08$$$$T^{3} +$$$$43\!\cdots\!84$$$$T^{4} +$$$$41\!\cdots\!08$$$$T^{5} +$$$$27\!\cdots\!12$$$$T^{6} + 12607773525708836608 T^{7} + 39148342370993168 T^{8} + 77672606302800 T^{9} + 106613283528 T^{10} + 197208256 T^{11} + 638450 T^{12} + 1130 T^{13} + T^{14}$$
$67$ $$( -11156453550526464 + 684019203379200 T - 13714445875968 T^{2} + 92839926336 T^{3} - 43216768 T^{4} - 929364 T^{5} + 32 T^{6} + T^{7} )^{2}$$
$71$ $$12\!\cdots\!28$$$$-$$$$17\!\cdots\!32$$$$T +$$$$12\!\cdots\!04$$$$T^{2} +$$$$90\!\cdots\!72$$$$T^{3} +$$$$12\!\cdots\!72$$$$T^{4} +$$$$68\!\cdots\!32$$$$T^{5} +$$$$22\!\cdots\!60$$$$T^{6} +$$$$42\!\cdots\!00$$$$T^{7} + 606217536261960072 T^{8} + 1049148833062904 T^{9} + 2742822135992 T^{10} + 4872692012 T^{11} + 5359538 T^{12} + 3274 T^{13} + T^{14}$$
$73$ $$80\!\cdots\!92$$$$-$$$$11\!\cdots\!00$$$$T +$$$$78\!\cdots\!00$$$$T^{2} +$$$$92\!\cdots\!36$$$$T^{3} +$$$$45\!\cdots\!68$$$$T^{4} +$$$$54\!\cdots\!68$$$$T^{5} +$$$$35\!\cdots\!44$$$$T^{6} +$$$$13\!\cdots\!08$$$$T^{7} + 343880258758544344 T^{8} + 509871971527256 T^{9} + 451096802924 T^{10} + 388458720 T^{11} + 1016738 T^{12} + 1426 T^{13} + T^{14}$$
$79$ $$10\!\cdots\!92$$$$+$$$$56\!\cdots\!40$$$$T +$$$$15\!\cdots\!00$$$$T^{2} +$$$$11\!\cdots\!28$$$$T^{3} +$$$$40\!\cdots\!08$$$$T^{4} -$$$$49\!\cdots\!48$$$$T^{5} +$$$$61\!\cdots\!16$$$$T^{6} +$$$$19\!\cdots\!16$$$$T^{7} + 571735266712609352 T^{8} - 573143731907272 T^{9} + 317182310776 T^{10} + 332144356 T^{11} + 1475762 T^{12} - 1718 T^{13} + T^{14}$$
$83$ $$72\!\cdots\!04$$$$+$$$$12\!\cdots\!28$$$$T^{2} +$$$$78\!\cdots\!52$$$$T^{4} +$$$$20\!\cdots\!92$$$$T^{6} + 227846261347735936 T^{8} + 943515416752 T^{10} + 1628796 T^{12} + T^{14}$$
$89$ $$( -3417965880465162240 - 23429016836186112 T + 143083326421248 T^{2} + 1053994048208 T^{3} + 586425480 T^{4} - 2440160 T^{5} - 238 T^{6} + T^{7} )^{2}$$
$97$ $$53\!\cdots\!08$$$$+$$$$69\!\cdots\!28$$$$T +$$$$44\!\cdots\!24$$$$T^{2} -$$$$43\!\cdots\!60$$$$T^{3} +$$$$20\!\cdots\!80$$$$T^{4} -$$$$39\!\cdots\!92$$$$T^{5} +$$$$54\!\cdots\!88$$$$T^{6} -$$$$20\!\cdots\!64$$$$T^{7} + 1161308162674107800 T^{8} - 2035777796458120 T^{9} + 1621177144684 T^{10} + 703219712 T^{11} + 1854738 T^{12} - 1926 T^{13} + T^{14}$$