# Properties

 Label 35.5.d.a Level $35$ Weight $5$ Character orbit 35.d Analytic conductor $3.618$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 35.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.61794870793$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 6 x^{11} - 109 x^{10} + 570 x^{9} + 5814 x^{8} - 22512 x^{7} - 151120 x^{6} + 300288 x^{5} + 2661184 x^{4} - 1881600 x^{3} - 12922560 x^{2} + 25948800 x + 205833600$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}\cdot 5^{5}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} -\beta_{3} q^{3} + ( 11 - \beta_{1} + \beta_{2} ) q^{4} + \beta_{5} q^{5} + ( \beta_{3} - \beta_{5} - \beta_{8} ) q^{6} + ( -4 - \beta_{10} ) q^{7} + ( -20 + 8 \beta_{1} - 2 \beta_{2} - \beta_{9} ) q^{8} + ( -37 + 2 \beta_{1} - \beta_{11} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} -\beta_{3} q^{3} + ( 11 - \beta_{1} + \beta_{2} ) q^{4} + \beta_{5} q^{5} + ( \beta_{3} - \beta_{5} - \beta_{8} ) q^{6} + ( -4 - \beta_{10} ) q^{7} + ( -20 + 8 \beta_{1} - 2 \beta_{2} - \beta_{9} ) q^{8} + ( -37 + 2 \beta_{1} - \beta_{11} ) q^{9} + ( \beta_{3} - \beta_{4} - \beta_{5} ) q^{10} + ( 5 + 9 \beta_{1} - 5 \beta_{2} - 2 \beta_{7} + \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{11} + ( -9 \beta_{3} + \beta_{4} + 8 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{12} + ( 6 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} - \beta_{10} ) q^{13} + ( 5 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 8 \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{11} ) q^{14} + ( 1 + 5 \beta_{1} - 3 \beta_{2} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{15} + ( 70 - 38 \beta_{1} + 14 \beta_{2} + 4 \beta_{7} + 3 \beta_{9} + 4 \beta_{10} ) q^{16} + ( 6 \beta_{3} - 5 \beta_{4} - 4 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{10} ) q^{17} + ( 82 - 36 \beta_{1} + 16 \beta_{2} + 7 \beta_{7} - \beta_{9} + 7 \beta_{10} + 2 \beta_{11} ) q^{18} + ( -12 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} - 4 \beta_{7} + 4 \beta_{10} ) q^{19} + ( -9 \beta_{3} + 10 \beta_{5} - \beta_{6} + 2 \beta_{8} + \beta_{10} ) q^{20} + ( -63 + 16 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 9 \beta_{5} - \beta_{6} + 3 \beta_{7} - 4 \beta_{8} - 5 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{21} + ( 201 - 27 \beta_{1} - \beta_{7} + 3 \beta_{9} - \beta_{10} + 6 \beta_{11} ) q^{22} + ( -82 + 30 \beta_{1} - 12 \beta_{2} + \beta_{7} + 2 \beta_{9} + \beta_{10} - 4 \beta_{11} ) q^{23} + ( 32 \beta_{3} - 8 \beta_{4} - 46 \beta_{5} + 2 \beta_{6} - 8 \beta_{7} - 2 \beta_{8} + 6 \beta_{10} ) q^{24} -125 q^{25} + ( 15 \beta_{3} + 4 \beta_{4} + 23 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 13 \beta_{8} - 4 \beta_{10} ) q^{26} + ( 17 \beta_{3} - 6 \beta_{4} + 40 \beta_{5} + 3 \beta_{7} - 2 \beta_{8} - 3 \beta_{10} ) q^{27} + ( 118 - 10 \beta_{1} - 14 \beta_{2} - 27 \beta_{3} + 7 \beta_{4} - \beta_{6} + 3 \beta_{7} + 11 \beta_{8} + 3 \beta_{9} - 4 \beta_{10} ) q^{28} + ( -142 - 66 \beta_{1} + 12 \beta_{2} - 8 \beta_{7} + 2 \beta_{9} - 8 \beta_{10} - 7 \beta_{11} ) q^{29} + ( 155 - 35 \beta_{1} + 10 \beta_{2} - 5 \beta_{7} + 5 \beta_{9} - 5 \beta_{10} ) q^{30} + ( -48 \beta_{3} - 12 \beta_{4} - 42 \beta_{5} + 4 \beta_{7} + 6 \beta_{8} - 4 \beta_{10} ) q^{31} + ( -792 + 100 \beta_{1} - 64 \beta_{2} - 4 \beta_{7} - \beta_{9} - 4 \beta_{10} - 8 \beta_{11} ) q^{32} + ( 22 \beta_{3} + 11 \beta_{4} - 14 \beta_{5} + \beta_{6} + 8 \beta_{7} - 31 \beta_{8} - 9 \beta_{10} ) q^{33} + ( -15 \beta_{3} + 2 \beta_{4} + 93 \beta_{5} - 2 \beta_{6} - 12 \beta_{7} + \beta_{8} + 14 \beta_{10} ) q^{34} + ( -33 + 45 \beta_{1} - \beta_{2} + 9 \beta_{3} + \beta_{4} - 4 \beta_{5} - 7 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{35} + ( -390 + 164 \beta_{1} - 32 \beta_{2} + 4 \beta_{7} - 13 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} ) q^{36} + ( 448 + 50 \beta_{1} + 28 \beta_{2} + 6 \beta_{7} - 18 \beta_{9} + 6 \beta_{10} + 6 \beta_{11} ) q^{37} + ( 36 \beta_{3} - 16 \beta_{4} - 98 \beta_{5} - 2 \beta_{6} + 16 \beta_{7} - 10 \beta_{8} - 14 \beta_{10} ) q^{38} + ( 717 + 53 \beta_{1} + 55 \beta_{2} - 20 \beta_{7} - \beta_{9} - 20 \beta_{10} + 7 \beta_{11} ) q^{39} + ( 44 \beta_{3} + \beta_{4} - 14 \beta_{5} - 5 \beta_{7} - 15 \beta_{8} + 5 \beta_{10} ) q^{40} + ( -6 \beta_{3} - 6 \beta_{4} + 108 \beta_{5} + 6 \beta_{7} - 24 \beta_{8} - 6 \beta_{10} ) q^{41} + ( 585 - 181 \beta_{1} + 100 \beta_{2} - 74 \beta_{3} + 12 \beta_{4} - 76 \beta_{5} + 7 \beta_{7} - 10 \beta_{8} + 11 \beta_{9} + 13 \beta_{10} - 8 \beta_{11} ) q^{42} + ( 304 + 64 \beta_{1} + 50 \beta_{2} + 25 \beta_{7} + 12 \beta_{9} + 25 \beta_{10} - 6 \beta_{11} ) q^{43} + ( -999 + 87 \beta_{1} - 95 \beta_{2} - 24 \beta_{7} - 13 \beta_{9} - 24 \beta_{10} + 6 \beta_{11} ) q^{44} + ( -56 \beta_{3} + 5 \beta_{4} - 29 \beta_{5} + \beta_{6} - 10 \beta_{7} - 7 \beta_{8} + 9 \beta_{10} ) q^{45} + ( 780 - 230 \beta_{1} + 62 \beta_{2} + 22 \beta_{7} + 6 \beta_{9} + 22 \beta_{10} + 6 \beta_{11} ) q^{46} + ( -111 \beta_{3} + 10 \beta_{4} + 32 \beta_{5} + 8 \beta_{6} - 4 \beta_{7} + 22 \beta_{8} - 4 \beta_{10} ) q^{47} + ( -74 \beta_{3} + 280 \beta_{5} - 6 \beta_{6} + 24 \beta_{7} + 52 \beta_{8} - 18 \beta_{10} ) q^{48} + ( -842 + 186 \beta_{1} - 48 \beta_{2} - 72 \beta_{3} - 8 \beta_{4} - 87 \beta_{5} - 7 \beta_{6} - 7 \beta_{7} + 2 \beta_{8} + 9 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{49} + ( 125 - 125 \beta_{1} ) q^{50} + ( 579 + 19 \beta_{1} - 43 \beta_{2} - 16 \beta_{7} + 13 \beta_{9} - 16 \beta_{10} + 5 \beta_{11} ) q^{51} + ( 261 \beta_{3} + 15 \beta_{4} - 216 \beta_{5} + 9 \beta_{6} - 15 \beta_{7} + 3 \beta_{8} + 6 \beta_{10} ) q^{52} + ( 1058 - 152 \beta_{1} - 8 \beta_{2} - 12 \beta_{7} - 4 \beta_{9} - 12 \beta_{10} ) q^{53} + ( -47 \beta_{3} - 34 \beta_{4} + 85 \beta_{5} - 2 \beta_{6} - 10 \beta_{7} + 17 \beta_{8} + 12 \beta_{10} ) q^{54} + ( 7 \beta_{3} + 14 \beta_{5} + 8 \beta_{6} - 15 \beta_{7} + 14 \beta_{8} + 7 \beta_{10} ) q^{55} + ( -530 - 60 \beta_{1} - 28 \beta_{2} + 300 \beta_{3} - 14 \beta_{4} - 196 \beta_{5} + 8 \beta_{6} + 4 \beta_{7} - 32 \beta_{8} + 11 \beta_{9} - 8 \beta_{10} - 14 \beta_{11} ) q^{56} + ( -1134 + 94 \beta_{1} - 28 \beta_{2} + 14 \beta_{7} + 10 \beta_{9} + 14 \beta_{10} - 22 \beta_{11} ) q^{57} + ( -1635 + 133 \beta_{1} - 52 \beta_{2} + 25 \beta_{7} - 21 \beta_{9} + 25 \beta_{10} + 30 \beta_{11} ) q^{58} + ( -6 \beta_{3} + 34 \beta_{4} + 20 \beta_{5} - 16 \beta_{6} + 8 \beta_{7} + 22 \beta_{8} + 8 \beta_{10} ) q^{59} + ( -1161 + 305 \beta_{1} - 117 \beta_{2} - 14 \beta_{7} + \beta_{9} - 14 \beta_{10} - 6 \beta_{11} ) q^{60} + ( 174 \beta_{3} - 6 \beta_{4} + 132 \beta_{5} - 18 \beta_{6} + 8 \beta_{7} + 48 \beta_{8} + 10 \beta_{10} ) q^{61} + ( 84 \beta_{3} + 46 \beta_{4} + 182 \beta_{5} + 2 \beta_{6} - 22 \beta_{7} - 44 \beta_{8} + 20 \beta_{10} ) q^{62} + ( -208 - 328 \beta_{1} - 18 \beta_{2} + 21 \beta_{3} - 38 \beta_{4} - 212 \beta_{5} + 8 \beta_{6} + 25 \beta_{7} + 30 \beta_{8} - 18 \beta_{9} + 24 \beta_{10} + 2 \beta_{11} ) q^{63} + ( 2180 - 968 \beta_{1} + 56 \beta_{2} - 12 \beta_{7} + 9 \beta_{9} - 12 \beta_{10} + 24 \beta_{11} ) q^{64} + ( 36 - 140 \beta_{1} + 92 \beta_{2} + 14 \beta_{7} - 16 \beta_{9} + 14 \beta_{10} + \beta_{11} ) q^{65} + ( -619 \beta_{3} + 62 \beta_{4} - 171 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{66} + ( -2062 - 62 \beta_{1} - 202 \beta_{2} + 41 \beta_{7} - 6 \beta_{9} + 41 \beta_{10} + 30 \beta_{11} ) q^{67} + ( -33 \beta_{3} - 69 \beta_{4} + 96 \beta_{5} - 9 \beta_{6} + 9 \beta_{7} - 21 \beta_{8} ) q^{68} + ( 274 \beta_{3} - 32 \beta_{4} + 120 \beta_{5} + 14 \beta_{6} + 10 \beta_{7} - 34 \beta_{8} - 24 \beta_{10} ) q^{69} + ( 1230 - 5 \beta_{1} + 110 \beta_{2} + 94 \beta_{3} - 5 \beta_{4} - \beta_{5} + \beta_{6} + 25 \beta_{7} + 13 \beta_{8} + 14 \beta_{10} + 15 \beta_{11} ) q^{70} + ( -442 + 22 \beta_{1} + 146 \beta_{2} + 40 \beta_{9} + 6 \beta_{11} ) q^{71} + ( 3548 - 380 \beta_{1} + 244 \beta_{2} - 38 \beta_{7} + 59 \beta_{9} - 38 \beta_{10} - 36 \beta_{11} ) q^{72} + ( 60 \beta_{3} + 86 \beta_{4} - 78 \beta_{5} + 16 \beta_{6} - 38 \beta_{7} - 2 \beta_{8} + 22 \beta_{10} ) q^{73} + ( 1270 + 590 \beta_{1} + 322 \beta_{2} + 42 \beta_{7} - 4 \beta_{9} + 42 \beta_{10} - 24 \beta_{11} ) q^{74} + 125 \beta_{3} q^{75} + ( -198 \beta_{3} + 40 \beta_{4} + 228 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} + 8 \beta_{8} - 6 \beta_{10} ) q^{76} + ( 2162 + 190 \beta_{1} - 4 \beta_{2} - 210 \beta_{3} - 73 \beta_{4} - 254 \beta_{5} + \beta_{6} - 38 \beta_{7} - 19 \beta_{8} - 32 \beta_{9} - 59 \beta_{10} + 30 \beta_{11} ) q^{77} + ( 825 + 1657 \beta_{1} - 160 \beta_{2} - 85 \beta_{7} - 47 \beta_{9} - 85 \beta_{10} + 26 \beta_{11} ) q^{78} + ( 11 - 413 \beta_{1} - 127 \beta_{2} + 44 \beta_{7} + 51 \beta_{9} + 44 \beta_{10} - 15 \beta_{11} ) q^{79} + ( -282 \beta_{3} - 5 \beta_{4} + 44 \beta_{5} - 8 \beta_{6} + 35 \beta_{7} + 21 \beta_{8} - 27 \beta_{10} ) q^{80} + ( -1009 + 36 \beta_{1} - 76 \beta_{2} - 52 \beta_{7} - 50 \beta_{9} - 52 \beta_{10} - 8 \beta_{11} ) q^{81} + ( -444 \beta_{3} - 90 \beta_{4} + 78 \beta_{5} - 18 \beta_{6} - 12 \beta_{8} + 18 \beta_{10} ) q^{82} + ( 24 \beta_{3} + 26 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} - 53 \beta_{7} - 70 \beta_{8} + 49 \beta_{10} ) q^{83} + ( -4479 + 1605 \beta_{1} - 285 \beta_{2} - 198 \beta_{3} + 12 \beta_{4} - 34 \beta_{5} + 12 \beta_{6} + 6 \beta_{7} - 16 \beta_{8} - 39 \beta_{9} - 18 \beta_{10} - 36 \beta_{11} ) q^{84} + ( 264 - 400 \beta_{1} - 92 \beta_{2} - 14 \beta_{7} + 6 \beta_{9} - 14 \beta_{10} - 21 \beta_{11} ) q^{85} + ( 1078 + 756 \beta_{1} - 42 \beta_{2} + 44 \beta_{7} - 68 \beta_{9} + 44 \beta_{10} - 38 \beta_{11} ) q^{86} + ( 529 \beta_{3} + 46 \beta_{4} + 348 \beta_{5} - 28 \beta_{6} + 61 \beta_{7} - 10 \beta_{8} - 33 \beta_{10} ) q^{87} + ( 300 - 1624 \beta_{1} + 262 \beta_{2} - 22 \beta_{7} + 66 \beta_{9} - 22 \beta_{10} - 60 \beta_{11} ) q^{88} + ( -318 \beta_{3} + 56 \beta_{4} - 182 \beta_{5} + 16 \beta_{6} + 40 \beta_{7} - 46 \beta_{8} - 56 \beta_{10} ) q^{89} + ( -154 \beta_{3} - \beta_{4} + 54 \beta_{5} - 20 \beta_{6} + 50 \beta_{7} - 35 \beta_{8} - 30 \beta_{10} ) q^{90} + ( -195 - 771 \beta_{1} + 27 \beta_{2} + 330 \beta_{3} - 34 \beta_{4} + 66 \beta_{5} - 20 \beta_{6} - 66 \beta_{7} + 64 \beta_{8} + 9 \beta_{9} - 16 \beta_{10} + 39 \beta_{11} ) q^{91} + ( -5508 + 884 \beta_{1} - 216 \beta_{2} - 38 \beta_{7} - 94 \beta_{9} - 38 \beta_{10} + 8 \beta_{11} ) q^{92} + ( -6030 + 338 \beta_{1} + 100 \beta_{2} - 50 \beta_{7} + 62 \beta_{9} - 50 \beta_{10} - 86 \beta_{11} ) q^{93} + ( 735 \beta_{3} + 2 \beta_{4} - 459 \beta_{5} + 40 \beta_{6} + 18 \beta_{7} - 65 \beta_{8} - 58 \beta_{10} ) q^{94} + ( -480 + 420 \beta_{1} - 10 \beta_{2} + 30 \beta_{7} + 10 \beta_{9} + 30 \beta_{10} + 30 \beta_{11} ) q^{95} + ( 1152 \beta_{3} - 86 \beta_{4} - 308 \beta_{5} + 56 \beta_{6} - 38 \beta_{7} - 126 \beta_{8} - 18 \beta_{10} ) q^{96} + ( 102 \beta_{3} - 121 \beta_{4} + 228 \beta_{5} - 17 \beta_{6} + 82 \beta_{7} + 169 \beta_{8} - 65 \beta_{10} ) q^{97} + ( 5483 - 1325 \beta_{1} - 12 \beta_{2} - 66 \beta_{3} + 40 \beta_{4} + 218 \beta_{5} - 22 \beta_{6} - 88 \beta_{7} - 104 \beta_{8} + 42 \beta_{9} - 28 \beta_{10} + 6 \beta_{11} ) q^{98} + ( 3004 - 2248 \beta_{1} - 40 \beta_{2} + 128 \beta_{7} - 98 \beta_{9} + 128 \beta_{10} + 24 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 6q^{2} + 122q^{4} - 50q^{7} - 186q^{8} - 434q^{9} + O(q^{10})$$ $$12q - 6q^{2} + 122q^{4} - 50q^{7} - 186q^{8} - 434q^{9} + 126q^{11} + 78q^{14} + 50q^{15} + 578q^{16} + 734q^{18} - 642q^{21} + 2264q^{22} - 756q^{23} - 1500q^{25} + 1414q^{28} - 2190q^{29} + 1600q^{30} - 8682q^{32} - 150q^{35} - 3582q^{36} + 5564q^{37} + 8634q^{39} + 5580q^{42} + 3944q^{43} - 11196q^{44} + 7844q^{46} - 8796q^{49} + 750q^{50} + 7206q^{51} + 11760q^{53} - 6606q^{56} - 12900q^{57} - 18496q^{58} - 11700q^{60} - 4310q^{63} + 20146q^{64} - 750q^{65} - 24096q^{67} + 14400q^{70} - 5664q^{71} + 39214q^{72} + 17604q^{74} + 26904q^{77} + 20100q^{78} - 1590q^{79} - 11912q^{81} - 43128q^{84} + 1050q^{85} + 17604q^{86} - 7268q^{88} - 7182q^{91} - 60252q^{92} - 70980q^{93} - 3000q^{95} + 57714q^{98} + 23084q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} - 109 x^{10} + 570 x^{9} + 5814 x^{8} - 22512 x^{7} - 151120 x^{6} + 300288 x^{5} + 2661184 x^{4} - 1881600 x^{3} - 12922560 x^{2} + 25948800 x + 205833600$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-545049904483 \nu^{11} + 3342865644154 \nu^{10} + 54129391887267 \nu^{9} - 292966990118874 \nu^{8} - 2685540664693566 \nu^{7} + 10734745856155716 \nu^{6} + 59841372905197048 \nu^{5} - 112993150153721416 \nu^{4} - 1064249473716996336 \nu^{3} + 949562036062870080 \nu^{2} + 13050276066196902720 \nu - 9321052937904187200$$$$)/ 16073392652512594560$$ $$\beta_{2}$$ $$=$$ $$($$$$230060779665 \nu^{11} - 4634987015638 \nu^{10} - 6235493671465 \nu^{9} + 419908650020022 \nu^{8} - 128431507479198 \nu^{7} - 18595961058901180 \nu^{6} + 13838739407380776 \nu^{5} + 295150124247701496 \nu^{4} + 304080581764103632 \nu^{3} - 1669760914511110720 \nu^{2} - 1135200006369445440 \nu - 88191650417067395520$$$$)/ 5357797550837531520$$ $$\beta_{3}$$ $$=$$ $$($$$$22050672377839 \nu^{11} - 167967216649588 \nu^{10} - 2061629906092179 \nu^{9} + 17828803070304708 \nu^{8} + 86712335037662466 \nu^{7} - 857407951286656872 \nu^{6} - 1386209748632066536 \nu^{5} + 20038103145499233832 \nu^{4} + 20149670014559280720 \nu^{3} - 270854947061320903200 \nu^{2} - 15568457737535265600 \nu + 1787287281635649931200$$$$)/$$$$16\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$8559661080213 \nu^{11} - 144006533906196 \nu^{10} - 392019044756793 \nu^{9} + 13999259022944836 \nu^{8} + 3313818453127822 \nu^{7} - 661245461433450824 \nu^{6} + 382576676517139288 \nu^{5} + 13116511236744603544 \nu^{4} + 5449825601314048240 \nu^{3} - 258060927020299106400 \nu^{2} + 75182814479892964800 \nu + 1126139439825700910400$$$$)/ 53577975508375315200$$ $$\beta_{5}$$ $$=$$ $$($$$$545049904483 \nu^{11} - 3342865644154 \nu^{10} - 54129391887267 \nu^{9} + 292966990118874 \nu^{8} + 2685540664693566 \nu^{7} - 10734745856155716 \nu^{6} - 59841372905197048 \nu^{5} + 112993150153721416 \nu^{4} + 1064249473716996336 \nu^{3} - 949562036062870080 \nu^{2} + 3023116586315691840 \nu + 9321052937904187200$$$$)/ 3214678530502518912$$ $$\beta_{6}$$ $$=$$ $$($$$$2017661537865 \nu^{11} + 30477299744326 \nu^{10} - 339396147765681 \nu^{9} - 4475089774504424 \nu^{8} + 24039237736282390 \nu^{7} + 281351007954963774 \nu^{6} - 815623273198113572 \nu^{5} - 8321735731889443660 \nu^{4} + 11294659105258592368 \nu^{3} + 122037173061585996704 \nu^{2} + 45010973563805110560 \nu - 588570433309672877280$$$$)/ 5357797550837531520$$ $$\beta_{7}$$ $$=$$ $$($$$$10105263298801 \nu^{11} - 7668759064702 \nu^{10} - 1621170695527401 \nu^{9} + 2125916147182962 \nu^{8} + 99104192219776794 \nu^{7} - 127729547061434688 \nu^{6} - 2688063369388328704 \nu^{5} + 3599454843712237888 \nu^{4} + 23478780958709691600 \nu^{3} - 26947796107315842240 \nu^{2} - 31358126519055216000 \nu + 1044654763415655081600$$$$)/ 22961989503589420800$$ $$\beta_{8}$$ $$=$$ $$($$$$-85744361226611 \nu^{11} + 759213715715702 \nu^{10} + 8913921473063211 \nu^{9} - 74715923565584382 \nu^{8} - 469528785793603734 \nu^{7} + 3053611860951235908 \nu^{6} + 12208714035409441544 \nu^{5} - 45544534085049750968 \nu^{4} - 182366786148860809200 \nu^{3} + 379094630341136867520 \nu^{2} - 301620430312290129600 \nu - 3948399140635071259200$$$$)/$$$$16\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$-775009347434 \nu^{11} + 4971706132333 \nu^{10} + 90922162125716 \nu^{9} - 504319655267781 \nu^{8} - 4879645875118962 \nu^{7} + 20195056117009538 \nu^{6} + 136755371484503672 \nu^{5} - 282657801885896480 \nu^{4} - 2771860747306433552 \nu^{3} + 1703397576605990480 \nu^{2} + 34472584683474968640 \nu - 17102586219208062720$$$$)/ 1339449387709382880$$ $$\beta_{10}$$ $$=$$ $$($$$$-19651585016711 \nu^{11} + 155617378754012 \nu^{10} + 2042857147697691 \nu^{9} - 16801182373982112 \nu^{8} - 99508448358846234 \nu^{7} + 769120965371841708 \nu^{6} + 2178004651049365904 \nu^{5} - 13661591936068140368 \nu^{4} - 31793345304297703440 \nu^{3} + 84987832529738771040 \nu^{2} + 40340044140732460800 \nu - 532459746809048505600$$$$)/ 22961989503589420800$$ $$\beta_{11}$$ $$=$$ $$($$$$-15067388428199 \nu^{11} + 153255880811244 \nu^{10} + 1022164060953311 \nu^{9} - 13579439578314540 \nu^{8} - 22460785975807446 \nu^{7} + 503780427285835768 \nu^{6} - 660391719372901120 \nu^{5} - 5263931835779932472 \nu^{4} + 12720328880638085104 \nu^{3} + 48273375146889553120 \nu^{2} - 225527679645652826880 \nu - 54450741744547014720$$$$)/ 10715595101675063040$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + 5 \beta_{1}$$$$)/5$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} + 5 \beta_{1} + 105$$$$)/5$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{10} - 5 \beta_{9} + 6 \beta_{8} - 3 \beta_{6} + 70 \beta_{5} - 6 \beta_{4} - 21 \beta_{3} + 5 \beta_{2} + 125 \beta_{1} + 135$$$$)/5$$ $$\nu^{4}$$ $$=$$ $$($$$$52 \beta_{10} - 5 \beta_{9} - 36 \beta_{8} - 12 \beta_{6} + 120 \beta_{5} - 116 \beta_{4} + 188 \beta_{3} + 150 \beta_{2} + 210 \beta_{1} + 1530$$$$)/5$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{11} + 53 \beta_{10} - 10 \beta_{9} + 49 \beta_{8} + 31 \beta_{7} - 52 \beta_{6} + 760 \beta_{5} - 113 \beta_{4} - 390 \beta_{3} + 58 \beta_{2} - 18 \beta_{1} + 594$$ $$\nu^{6}$$ $$=$$ $$($$$$-120 \beta_{11} + 2540 \beta_{10} - 205 \beta_{9} - 3060 \beta_{8} - 1080 \beta_{7} - 1020 \beta_{6} + 9840 \beta_{5} - 5876 \beta_{4} + 3596 \beta_{3} - 4460 \beta_{2} - 2420 \beta_{1} - 102240$$$$)/5$$ $$\nu^{7}$$ $$=$$ $$($$$$880 \beta_{11} + 10007 \beta_{10} + 10840 \beta_{9} + 49 \beta_{8} + 3595 \beta_{7} - 13762 \beta_{6} + 165600 \beta_{5} - 30149 \beta_{4} - 109284 \beta_{3} - 9460 \beta_{2} - 349100 \beta_{1} - 288660$$$$)/5$$ $$\nu^{8}$$ $$=$$ $$($$$$3480 \beta_{11} + 5364 \beta_{10} + 26265 \beta_{9} - 147552 \beta_{8} - 132300 \beta_{7} - 57984 \beta_{6} + 503040 \beta_{5} - 224672 \beta_{4} - 178624 \beta_{3} - 894940 \beta_{2} - 1166740 \beta_{1} - 13655640$$$$)/5$$ $$\nu^{9}$$ $$=$$ $$($$$$258240 \beta_{11} - 224427 \beta_{10} + 1164820 \beta_{9} - 480129 \beta_{8} - 593715 \beta_{7} - 437058 \beta_{6} + 4215520 \beta_{5} - 993987 \beta_{4} - 3525600 \beta_{3} - 3104200 \beta_{2} - 31724440 \beta_{1} - 43435800$$$$)/5$$ $$\nu^{10}$$ $$=$$ $$268872 \beta_{11} - 1718332 \beta_{10} + 860303 \beta_{9} - 552840 \beta_{8} - 2006412 \beta_{7} - 336480 \beta_{6} + 2738880 \beta_{5} - 605752 \beta_{4} - 3309968 \beta_{3} - 14108460 \beta_{2} - 23955204 \beta_{1} - 191284272$$ $$\nu^{11}$$ $$=$$ $$($$$$20929760 \beta_{11} - 64688969 \beta_{10} + 75805660 \beta_{9} - 23187043 \beta_{8} - 71344585 \beta_{7} + 4962034 \beta_{6} - 110296000 \beta_{5} + 6540743 \beta_{4} + 45082488 \beta_{3} - 301429600 \beta_{2} - 1902160800 \beta_{1} - 3588162480$$$$)/5$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/35\mathbb{Z}\right)^\times$$.

 $$n$$ $$22$$ $$31$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1
 −6.63119 − 2.23607i −6.63119 + 2.23607i −3.62973 + 2.23607i −3.62973 − 2.23607i −2.23331 + 2.23607i −2.23331 − 2.23607i 2.17355 − 2.23607i 2.17355 + 2.23607i 5.81769 + 2.23607i 5.81769 − 2.23607i 7.50299 − 2.23607i 7.50299 + 2.23607i
−7.63119 12.6955i 42.2350 11.1803i 96.8817i 2.03056 48.9579i −200.204 −80.1757 85.3193i
6.2 −7.63119 12.6955i 42.2350 11.1803i 96.8817i 2.03056 + 48.9579i −200.204 −80.1757 85.3193i
6.3 −4.62973 0.296012i 5.43443 11.1803i 1.37046i 15.2405 46.5696i 48.9158 80.9124 51.7620i
6.4 −4.62973 0.296012i 5.43443 11.1803i 1.37046i 15.2405 + 46.5696i 48.9158 80.9124 51.7620i
6.5 −3.23331 14.6970i −5.54568 11.1803i 47.5200i −26.6804 + 41.0994i 69.6640 −135.002 36.1495i
6.6 −3.23331 14.6970i −5.54568 11.1803i 47.5200i −26.6804 41.0994i 69.6640 −135.002 36.1495i
6.7 1.17355 9.10378i −14.6228 11.1803i 10.6837i −40.5002 27.5814i −35.9373 −1.87877 13.1207i
6.8 1.17355 9.10378i −14.6228 11.1803i 10.6837i −40.5002 + 27.5814i −35.9373 −1.87877 13.1207i
6.9 4.81769 14.5704i 7.21016 11.1803i 70.1957i 44.8989 19.6236i −42.3467 −131.296 53.8634i
6.10 4.81769 14.5704i 7.21016 11.1803i 70.1957i 44.8989 + 19.6236i −42.3467 −131.296 53.8634i
6.11 6.50299 5.52807i 26.2889 11.1803i 35.9490i −19.9894 + 44.7373i 66.9085 50.4404 72.7056i
6.12 6.50299 5.52807i 26.2889 11.1803i 35.9490i −19.9894 44.7373i 66.9085 50.4404 72.7056i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 6.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.5.d.a 12
3.b odd 2 1 315.5.h.a 12
4.b odd 2 1 560.5.f.b 12
5.b even 2 1 175.5.d.i 12
5.c odd 4 2 175.5.c.d 24
7.b odd 2 1 inner 35.5.d.a 12
21.c even 2 1 315.5.h.a 12
28.d even 2 1 560.5.f.b 12
35.c odd 2 1 175.5.d.i 12
35.f even 4 2 175.5.c.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.d.a 12 1.a even 1 1 trivial
35.5.d.a 12 7.b odd 2 1 inner
175.5.c.d 24 5.c odd 4 2
175.5.c.d 24 35.f even 4 2
175.5.d.i 12 5.b even 2 1
175.5.d.i 12 35.c odd 2 1
315.5.h.a 12 3.b odd 2 1
315.5.h.a 12 21.c even 2 1
560.5.f.b 12 4.b odd 2 1
560.5.f.b 12 28.d even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(35, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -4200 + 2340 T + 1348 T^{2} - 168 T^{3} - 74 T^{4} + 3 T^{5} + T^{6} )^{2}$$
$3$ $$1640250000 + 18818284500 T^{2} + 1131317100 T^{4} + 21932745 T^{6} + 184351 T^{8} + 703 T^{10} + T^{12}$$
$5$ $$( 125 + T^{2} )^{6}$$
$7$ $$19\!\cdots\!01$$$$+ 3989613314880600050 T + 187699591857106448 T^{2} + 1646421112558950 T^{3} + 58829788440199 T^{4} - 323605339400 T^{5} + 9646987504 T^{6} - 134779400 T^{7} + 10204999 T^{8} + 118950 T^{9} + 5648 T^{10} + 50 T^{11} + T^{12}$$
$11$ $$( -5924759078688 - 19207386432 T + 1079345926 T^{2} + 2403231 T^{3} - 60503 T^{4} - 63 T^{5} + T^{6} )^{2}$$
$13$ $$78\!\cdots\!00$$$$+$$$$15\!\cdots\!00$$$$T^{2} + 7190551872225647100 T^{4} + 551484734188905 T^{6} + 16293804111 T^{8} + 210483 T^{10} + T^{12}$$
$17$ $$20\!\cdots\!00$$$$+$$$$14\!\cdots\!00$$$$T^{2} + 51205934664366582600 T^{4} + 3210151142453445 T^{6} + 63256681251 T^{8} + 446103 T^{10} + T^{12}$$
$19$ $$56\!\cdots\!00$$$$+$$$$32\!\cdots\!00$$$$T^{2} +$$$$25\!\cdots\!00$$$$T^{4} + 7757289179712000 T^{6} + 110690373600 T^{8} + 669120 T^{10} + T^{12}$$
$23$ $$( -449297249971200 + 9085132726560 T + 22096307248 T^{2} - 176255028 T^{3} - 429014 T^{4} + 378 T^{5} + T^{6} )^{2}$$
$29$ $$( 58379986018274952 + 371039296445532 T + 307211233666 T^{2} - 1412379663 T^{3} - 1450655 T^{4} + 1095 T^{5} + T^{6} )^{2}$$
$31$ $$13\!\cdots\!00$$$$+$$$$85\!\cdots\!00$$$$T^{2} +$$$$19\!\cdots\!00$$$$T^{4} + 1854124388830152000 T^{6} + 6618871839600 T^{8} + 4865820 T^{10} + T^{12}$$
$37$ $$( -450366239878419200 - 1267641527871040 T + 2616811213568 T^{2} + 6477845192 T^{3} - 2235124 T^{4} - 2782 T^{5} + T^{6} )^{2}$$
$41$ $$24\!\cdots\!00$$$$+$$$$51\!\cdots\!00$$$$T^{2} +$$$$23\!\cdots\!00$$$$T^{4} +$$$$34\!\cdots\!00$$$$T^{6} + 132022965117600 T^{8} + 19494180 T^{10} + T^{12}$$
$43$ $$( -19871876562948020000 - 14593647943201600 T + 20066821791728 T^{2} + 10788313232 T^{3} - 7327894 T^{4} - 1972 T^{5} + T^{6} )^{2}$$
$47$ $$34\!\cdots\!00$$$$+$$$$28\!\cdots\!00$$$$T^{2} +$$$$37\!\cdots\!00$$$$T^{4} +$$$$16\!\cdots\!45$$$$T^{6} + 314135199997011 T^{8} + 28611243 T^{10} + T^{12}$$
$53$ $$( 582567229534334400 + 2116084946336640 T - 2671170415184 T^{2} - 6414224640 T^{3} + 11303092 T^{4} - 5880 T^{5} + T^{6} )^{2}$$
$59$ $$18\!\cdots\!00$$$$+$$$$44\!\cdots\!00$$$$T^{2} +$$$$10\!\cdots\!00$$$$T^{4} +$$$$81\!\cdots\!00$$$$T^{6} + 1798968500085600 T^{8} + 80120880 T^{10} + T^{12}$$
$61$ $$44\!\cdots\!00$$$$+$$$$21\!\cdots\!00$$$$T^{2} +$$$$82\!\cdots\!00$$$$T^{4} +$$$$10\!\cdots\!00$$$$T^{6} + 5486574012879600 T^{8} + 123491580 T^{10} + T^{12}$$
$67$ $$($$$$28\!\cdots\!00$$$$+ 9936404795821161120 T - 1432610735052432 T^{2} - 788028613288 T^{3} - 30949014 T^{4} + 12048 T^{5} + T^{6} )^{2}$$
$71$ $$($$$$14\!\cdots\!52$$$$+ 674098350997010688 T + 550077005381776 T^{2} - 132760237824 T^{3} - 53126408 T^{4} + 2832 T^{5} + T^{6} )^{2}$$
$73$ $$20\!\cdots\!00$$$$+$$$$17\!\cdots\!00$$$$T^{2} +$$$$15\!\cdots\!00$$$$T^{4} +$$$$45\!\cdots\!80$$$$T^{6} + 5137098503292576 T^{8} + 139132068 T^{10} + T^{12}$$
$79$ $$( -$$$$39\!\cdots\!48$$$$+ 889432476145723512 T + 882448015936566 T^{2} - 226479725803 T^{3} - 91977615 T^{4} + 795 T^{5} + T^{6} )^{2}$$
$83$ $$67\!\cdots\!00$$$$+$$$$60\!\cdots\!00$$$$T^{2} +$$$$16\!\cdots\!00$$$$T^{4} +$$$$22\!\cdots\!20$$$$T^{6} + 10220139492563556 T^{8} + 177197148 T^{10} + T^{12}$$
$89$ $$93\!\cdots\!00$$$$+$$$$14\!\cdots\!00$$$$T^{2} +$$$$32\!\cdots\!00$$$$T^{4} +$$$$20\!\cdots\!00$$$$T^{6} + 44673190948929600 T^{8} + 372631680 T^{10} + T^{12}$$
$97$ $$99\!\cdots\!00$$$$+$$$$22\!\cdots\!00$$$$T^{2} +$$$$10\!\cdots\!00$$$$T^{4} +$$$$13\!\cdots\!45$$$$T^{6} + 180520981634967651 T^{8} + 753417303 T^{10} + T^{12}$$