# Properties

 Label 74.4.c.a Level $74$ Weight $4$ Character orbit 74.c Analytic conductor $4.366$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 74.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.36614134042$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} + 48 x^{6} + 157 x^{5} + 1944 x^{4} + 3005 x^{3} + 12895 x^{2} - 11550 x + 44100$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \beta_{2} ) q^{2} + ( -\beta_{1} - \beta_{2} ) q^{3} -4 \beta_{2} q^{4} + ( -2 \beta_{2} + \beta_{7} ) q^{5} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{6} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{7} -8 q^{8} + ( 3 - 3 \beta_{2} + 5 \beta_{3} + \beta_{6} ) q^{9} +O(q^{10})$$ $$q + ( 2 - 2 \beta_{2} ) q^{2} + ( -\beta_{1} - \beta_{2} ) q^{3} -4 \beta_{2} q^{4} + ( -2 \beta_{2} + \beta_{7} ) q^{5} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{6} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{7} -8 q^{8} + ( 3 - 3 \beta_{2} + 5 \beta_{3} + \beta_{6} ) q^{9} + ( -6 + 2 \beta_{2} - 2 \beta_{5} + 2 \beta_{7} ) q^{10} + ( 10 - 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{11} + ( -4 + 4 \beta_{2} + 4 \beta_{3} ) q^{12} + ( -3 \beta_{1} - 14 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{13} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{14} + ( -8 + 8 \beta_{2} + 5 \beta_{3} + 3 \beta_{5} + \beta_{6} ) q^{15} + ( -16 + 16 \beta_{2} ) q^{16} + ( -1 + \beta_{2} - 8 \beta_{3} - 4 \beta_{5} + 2 \beta_{6} ) q^{17} + ( 10 \beta_{1} - 6 \beta_{2} + 2 \beta_{4} + 2 \beta_{6} ) q^{18} + ( -5 \beta_{1} - 18 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} - 3 \beta_{7} ) q^{19} + ( -12 + 12 \beta_{2} - 4 \beta_{5} ) q^{20} + ( 16 - 16 \beta_{2} - 11 \beta_{3} + 4 \beta_{5} - 3 \beta_{6} ) q^{21} + ( 20 - 20 \beta_{2} + 8 \beta_{3} + 4 \beta_{5} - 4 \beta_{6} ) q^{22} + ( -11 + 6 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} - 7 \beta_{4} - 6 \beta_{5} + 6 \beta_{7} ) q^{23} + ( 8 \beta_{1} + 8 \beta_{2} ) q^{24} + ( 5 - 5 \beta_{2} - 18 \beta_{3} - 5 \beta_{5} + 3 \beta_{6} ) q^{25} + ( -30 - 6 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{26} + ( 83 - \beta_{1} - \beta_{2} + \beta_{3} + 8 \beta_{4} + \beta_{5} - \beta_{7} ) q^{27} + ( -4 \beta_{3} - 4 \beta_{5} - 4 \beta_{6} ) q^{28} + ( 41 + 19 \beta_{1} - 11 \beta_{2} - 19 \beta_{3} + \beta_{4} + 11 \beta_{5} - 11 \beta_{7} ) q^{29} + ( 10 \beta_{1} + 22 \beta_{2} + 2 \beta_{4} + 2 \beta_{6} + 6 \beta_{7} ) q^{30} + ( 2 - 32 \beta_{1} - 16 \beta_{2} + 32 \beta_{3} - 6 \beta_{4} + 16 \beta_{5} - 16 \beta_{7} ) q^{31} + 32 \beta_{2} q^{32} + ( -16 \beta_{1} + 72 \beta_{2} - 4 \beta_{4} - 4 \beta_{6} - 4 \beta_{7} ) q^{33} + ( -16 \beta_{1} - 6 \beta_{2} + 4 \beta_{4} + 4 \beta_{6} - 8 \beta_{7} ) q^{34} + ( -97 + 97 \beta_{2} - 29 \beta_{3} + 2 \beta_{6} ) q^{35} + ( -12 + 20 \beta_{1} - 20 \beta_{3} + 4 \beta_{4} ) q^{36} + ( 104 + 32 \beta_{1} - 76 \beta_{2} - 13 \beta_{3} - 3 \beta_{4} + 14 \beta_{5} + 4 \beta_{6} - 6 \beta_{7} ) q^{37} + ( -30 - 10 \beta_{1} - 6 \beta_{2} + 10 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} - 6 \beta_{7} ) q^{38} + ( -85 + 85 \beta_{2} + 47 \beta_{3} + \beta_{5} + 10 \beta_{6} ) q^{39} + ( 16 \beta_{2} - 8 \beta_{7} ) q^{40} + ( -14 \beta_{1} + 23 \beta_{2} - 15 \beta_{4} - 15 \beta_{6} - 7 \beta_{7} ) q^{41} + ( -22 \beta_{1} - 24 \beta_{2} - 6 \beta_{4} - 6 \beta_{6} + 8 \beta_{7} ) q^{42} + ( 54 + 20 \beta_{1} + 15 \beta_{2} - 20 \beta_{3} + 2 \beta_{4} - 15 \beta_{5} + 15 \beta_{7} ) q^{43} + ( 16 \beta_{1} - 32 \beta_{2} - 8 \beta_{4} - 8 \beta_{6} + 8 \beta_{7} ) q^{44} + ( 25 + 31 \beta_{1} + 17 \beta_{2} - 31 \beta_{3} + 5 \beta_{4} - 17 \beta_{5} + 17 \beta_{7} ) q^{45} + ( -22 + 22 \beta_{2} - 12 \beta_{3} - 12 \beta_{5} + 14 \beta_{6} ) q^{46} + ( 38 + 20 \beta_{1} - 16 \beta_{2} - 20 \beta_{3} + 22 \beta_{4} + 16 \beta_{5} - 16 \beta_{7} ) q^{47} + ( 16 + 16 \beta_{1} - 16 \beta_{3} ) q^{48} + ( 27 - 27 \beta_{2} + 5 \beta_{3} + 8 \beta_{5} - 3 \beta_{6} ) q^{49} + ( -36 \beta_{1} - 20 \beta_{2} + 6 \beta_{4} + 6 \beta_{6} - 10 \beta_{7} ) q^{50} + ( -167 - 5 \beta_{1} + 10 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} + 10 \beta_{7} ) q^{51} + ( -60 + 60 \beta_{2} + 12 \beta_{3} - 4 \beta_{5} + 8 \beta_{6} ) q^{52} + ( 24 - 24 \beta_{2} - 19 \beta_{3} - 22 \beta_{5} - 21 \beta_{6} ) q^{53} + ( 166 - 166 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 16 \beta_{6} ) q^{54} + ( 44 \beta_{1} - 208 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} + 42 \beta_{7} ) q^{55} + ( -8 \beta_{1} - 8 \beta_{2} - 8 \beta_{4} - 8 \beta_{6} - 8 \beta_{7} ) q^{56} + ( -111 + 111 \beta_{2} + 47 \beta_{3} - 11 \beta_{5} + 8 \beta_{6} ) q^{57} + ( 82 - 82 \beta_{2} - 38 \beta_{3} + 22 \beta_{5} - 2 \beta_{6} ) q^{58} + ( 77 - 77 \beta_{2} + 103 \beta_{3} - 29 \beta_{5} + 8 \beta_{6} ) q^{59} + ( 32 + 20 \beta_{1} + 12 \beta_{2} - 20 \beta_{3} + 4 \beta_{4} - 12 \beta_{5} + 12 \beta_{7} ) q^{60} + ( -86 \beta_{1} - 44 \beta_{2} - \beta_{4} - \beta_{6} - 28 \beta_{7} ) q^{61} + ( 4 - 4 \beta_{2} + 64 \beta_{3} + 32 \beta_{5} + 12 \beta_{6} ) q^{62} + ( -283 - 68 \beta_{1} + 18 \beta_{2} + 68 \beta_{3} + 3 \beta_{4} - 18 \beta_{5} + 18 \beta_{7} ) q^{63} + 64 q^{64} + ( -189 + 189 \beta_{2} + 9 \beta_{3} - 17 \beta_{5} + 6 \beta_{6} ) q^{65} + ( 152 - 32 \beta_{1} - 8 \beta_{2} + 32 \beta_{3} - 8 \beta_{4} + 8 \beta_{5} - 8 \beta_{7} ) q^{66} + ( -49 \beta_{1} + 96 \beta_{2} + 16 \beta_{4} + 16 \beta_{6} + 45 \beta_{7} ) q^{67} + ( 4 - 32 \beta_{1} - 16 \beta_{2} + 32 \beta_{3} + 8 \beta_{4} + 16 \beta_{5} - 16 \beta_{7} ) q^{68} + ( 62 \beta_{1} - 100 \beta_{2} + 21 \beta_{4} + 21 \beta_{6} + 11 \beta_{7} ) q^{69} + ( -58 \beta_{1} + 194 \beta_{2} + 4 \beta_{4} + 4 \beta_{6} ) q^{70} + ( 101 \beta_{1} - 249 \beta_{2} + 25 \beta_{4} + 25 \beta_{6} + 31 \beta_{7} ) q^{71} + ( -24 + 24 \beta_{2} - 40 \beta_{3} - 8 \beta_{6} ) q^{72} + ( -1 - 26 \beta_{1} - 53 \beta_{2} + 26 \beta_{3} - 3 \beta_{4} + 53 \beta_{5} - 53 \beta_{7} ) q^{73} + ( 68 + 38 \beta_{1} - 192 \beta_{2} - 64 \beta_{3} + 8 \beta_{4} + 12 \beta_{5} + 14 \beta_{6} + 16 \beta_{7} ) q^{74} + ( -400 - 40 \beta_{1} + 12 \beta_{2} + 40 \beta_{3} - 4 \beta_{4} - 12 \beta_{5} + 12 \beta_{7} ) q^{75} + ( -60 + 60 \beta_{2} + 20 \beta_{3} + 12 \beta_{5} + 8 \beta_{6} ) q^{76} + ( 64 \beta_{1} + 192 \beta_{2} - 28 \beta_{4} - 28 \beta_{6} + 44 \beta_{7} ) q^{77} + ( 94 \beta_{1} + 172 \beta_{2} + 20 \beta_{4} + 20 \beta_{6} + 2 \beta_{7} ) q^{78} + ( -69 \beta_{1} + 78 \beta_{2} + 12 \beta_{4} + 12 \beta_{6} + 35 \beta_{7} ) q^{79} + ( 48 - 16 \beta_{2} + 16 \beta_{5} - 16 \beta_{7} ) q^{80} + ( -18 \beta_{1} - 125 \beta_{2} + 3 \beta_{4} + 3 \beta_{6} + 5 \beta_{7} ) q^{81} + ( 60 - 28 \beta_{1} - 14 \beta_{2} + 28 \beta_{3} - 30 \beta_{4} + 14 \beta_{5} - 14 \beta_{7} ) q^{82} + ( 287 - 287 \beta_{2} - 17 \beta_{3} - 26 \beta_{5} - 46 \beta_{6} ) q^{83} + ( -64 - 44 \beta_{1} + 16 \beta_{2} + 44 \beta_{3} - 12 \beta_{4} - 16 \beta_{5} + 16 \beta_{7} ) q^{84} + ( 425 - 100 \beta_{1} - 53 \beta_{2} + 100 \beta_{3} + 4 \beta_{4} + 53 \beta_{5} - 53 \beta_{7} ) q^{85} + ( 108 - 108 \beta_{2} - 40 \beta_{3} - 30 \beta_{5} - 4 \beta_{6} ) q^{86} + ( -148 \beta_{1} - 563 \beta_{2} - 33 \beta_{4} - 33 \beta_{6} - 32 \beta_{7} ) q^{87} + ( -80 + 32 \beta_{1} + 16 \beta_{2} - 32 \beta_{3} - 16 \beta_{4} - 16 \beta_{5} + 16 \beta_{7} ) q^{88} + ( -566 + 566 \beta_{2} - 10 \beta_{3} - 59 \beta_{5} + 7 \beta_{6} ) q^{89} + ( 50 - 50 \beta_{2} - 62 \beta_{3} - 34 \beta_{5} - 10 \beta_{6} ) q^{90} + ( 357 - 357 \beta_{2} - 113 \beta_{3} - 17 \beta_{5} + 4 \beta_{6} ) q^{91} + ( -24 \beta_{1} + 20 \beta_{2} + 28 \beta_{4} + 28 \beta_{6} - 24 \beta_{7} ) q^{92} + ( 148 \beta_{1} + 588 \beta_{2} + 34 \beta_{4} + 34 \beta_{6} - 54 \beta_{7} ) q^{93} + ( 76 - 76 \beta_{2} - 40 \beta_{3} + 32 \beta_{5} - 44 \beta_{6} ) q^{94} + ( 245 - 245 \beta_{2} + 91 \beta_{3} - \beta_{5} - 4 \beta_{6} ) q^{95} + ( 32 - 32 \beta_{2} - 32 \beta_{3} ) q^{96} + ( 651 + 103 \beta_{1} - 38 \beta_{2} - 103 \beta_{3} - 13 \beta_{4} + 38 \beta_{5} - 38 \beta_{7} ) q^{97} + ( 10 \beta_{1} - 38 \beta_{2} - 6 \beta_{4} - 6 \beta_{6} + 16 \beta_{7} ) q^{98} + ( 6 - 6 \beta_{2} + 124 \beta_{3} + 38 \beta_{5} - 30 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{2} - 5 q^{3} - 16 q^{4} - 10 q^{5} - 20 q^{6} + 3 q^{7} - 64 q^{8} + 7 q^{9} + O(q^{10})$$ $$8 q + 8 q^{2} - 5 q^{3} - 16 q^{4} - 10 q^{5} - 20 q^{6} + 3 q^{7} - 64 q^{8} + 7 q^{9} - 40 q^{10} + 64 q^{11} - 20 q^{12} - 61 q^{13} + 12 q^{14} - 43 q^{15} - 64 q^{16} + 12 q^{17} - 14 q^{18} - 71 q^{19} - 40 q^{20} + 67 q^{21} + 64 q^{22} - 52 q^{23} + 40 q^{24} + 48 q^{25} - 244 q^{26} + 658 q^{27} + 12 q^{28} + 322 q^{29} + 86 q^{30} - 112 q^{31} + 128 q^{32} + 280 q^{33} - 24 q^{34} - 359 q^{35} - 56 q^{36} + 557 q^{37} - 284 q^{38} - 389 q^{39} + 80 q^{40} + 92 q^{41} - 134 q^{42} + 532 q^{43} - 128 q^{44} + 330 q^{45} - 52 q^{46} + 280 q^{47} + 160 q^{48} + 87 q^{49} - 96 q^{50} - 1306 q^{51} - 244 q^{52} + 159 q^{53} + 658 q^{54} - 872 q^{55} - 24 q^{56} - 469 q^{57} + 322 q^{58} + 263 q^{59} + 344 q^{60} - 206 q^{61} - 112 q^{62} - 2328 q^{63} + 512 q^{64} - 731 q^{65} + 1120 q^{66} + 245 q^{67} - 96 q^{68} - 360 q^{69} + 718 q^{70} - 957 q^{71} - 56 q^{72} - 272 q^{73} - 178 q^{74} - 3232 q^{75} - 284 q^{76} + 744 q^{77} + 778 q^{78} + 173 q^{79} + 320 q^{80} - 528 q^{81} + 368 q^{82} + 1217 q^{83} - 536 q^{84} + 2988 q^{85} + 532 q^{86} - 2336 q^{87} - 512 q^{88} - 2136 q^{89} + 330 q^{90} + 1575 q^{91} + 104 q^{92} + 2608 q^{93} + 280 q^{94} + 891 q^{95} + 160 q^{96} + 5262 q^{97} - 174 q^{98} - 176 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 48 x^{6} + 157 x^{5} + 1944 x^{4} + 3005 x^{3} + 12895 x^{2} - 11550 x + 44100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$1161547 \nu^{7} + 2985827 \nu^{6} + 46810062 \nu^{5} + 367759321 \nu^{4} + 2694757026 \nu^{3} + 11047848851 \nu^{2} + 15096834055 \nu + 39927249180$$$$)/ 50470839930$$ $$\beta_{3}$$ $$=$$ $$($$$$98747 \nu^{7} - 212957 \nu^{6} + 4414201 \nu^{5} + 10397849 \nu^{4} + 179938098 \nu^{3} + 2825845 \nu^{2} + 1270074215 \nu - 1219624350$$$$)/ 1201686665$$ $$\beta_{4}$$ $$=$$ $$($$$$410451 \nu^{7} - 313216 \nu^{6} + 18348033 \nu^{5} + 43219617 \nu^{4} + 833723184 \nu^{3} + 11745885 \nu^{2} + 284259150 \nu - 26942923645$$$$)/ 1201686665$$ $$\beta_{5}$$ $$=$$ $$($$$$5600523 \nu^{7} - 60008240 \nu^{6} + 470238386 \nu^{5} - 1552858600 \nu^{4} + 5158399873 \nu^{3} - 36053963593 \nu^{2} + 70661757290 \nu - 102366278700$$$$)/ 8411806655$$ $$\beta_{6}$$ $$=$$ $$($$$$-39157703 \nu^{7} - 41841439 \nu^{6} - 1632820752 \nu^{5} - 9768593357 \nu^{4} - 84651611946 \nu^{3} - 203985740113 \nu^{2} - 507256534355 \nu + 396175255350$$$$)/ 50470839930$$ $$\beta_{7}$$ $$=$$ $$($$$$-7343719 \nu^{7} - 8871239 \nu^{6} - 139078134 \nu^{5} - 2644841047 \nu^{4} - 8006436282 \nu^{3} - 32824442807 \nu^{2} + 50114967155 \nu - 118628497260$$$$)/ 7210119990$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + 3 \beta_{3} + 23 \beta_{2} - 23$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{5} - 5 \beta_{4} + 41 \beta_{3} + \beta_{2} - 41 \beta_{1} - 67$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} - 52 \beta_{6} - 52 \beta_{4} - 937 \beta_{2} - 237 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-342 \beta_{6} + 48 \beta_{5} - 2119 \beta_{3} - 5352 \beta_{2} + 5352$$ $$\nu^{6}$$ $$=$$ $$-150 \beta_{7} + 150 \beta_{5} + 2851 \beta_{4} - 14883 \beta_{3} - 150 \beta_{2} + 14883 \beta_{1} + 48293$$ $$\nu^{7}$$ $$=$$ $$-2251 \beta_{7} + 20735 \beta_{6} + 20735 \beta_{4} + 335106 \beta_{2} + 118901 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/74\mathbb{Z}\right)^\times$$.

 $$n$$ $$39$$ $$\chi(n)$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 3.81550 + 6.60864i 0.810477 + 1.40379i −1.95521 − 3.38653i −2.17076 − 3.75987i 3.81550 − 6.60864i 0.810477 − 1.40379i −1.95521 + 3.38653i −2.17076 + 3.75987i
1.00000 1.73205i −4.31550 7.47467i −2.00000 3.46410i −2.90148 5.02551i −17.2620 8.58362 + 14.8673i −8.00000 −23.7471 + 41.1312i −11.6059
47.2 1.00000 1.73205i −1.31048 2.26981i −2.00000 3.46410i 0.0116171 + 0.0201213i −5.24191 −10.2956 17.8325i −8.00000 10.0653 17.4336i 0.0464682
47.3 1.00000 1.73205i 1.45521 + 2.52050i −2.00000 3.46410i −8.20891 14.2183i 5.82085 −6.65276 11.5229i −8.00000 9.26471 16.0469i −32.8357
47.4 1.00000 1.73205i 1.67076 + 2.89385i −2.00000 3.46410i 6.09877 + 10.5634i 6.68306 9.86474 + 17.0862i −8.00000 7.91709 13.7128i 24.3951
63.1 1.00000 + 1.73205i −4.31550 + 7.47467i −2.00000 + 3.46410i −2.90148 + 5.02551i −17.2620 8.58362 14.8673i −8.00000 −23.7471 41.1312i −11.6059
63.2 1.00000 + 1.73205i −1.31048 + 2.26981i −2.00000 + 3.46410i 0.0116171 0.0201213i −5.24191 −10.2956 + 17.8325i −8.00000 10.0653 + 17.4336i 0.0464682
63.3 1.00000 + 1.73205i 1.45521 2.52050i −2.00000 + 3.46410i −8.20891 + 14.2183i 5.82085 −6.65276 + 11.5229i −8.00000 9.26471 + 16.0469i −32.8357
63.4 1.00000 + 1.73205i 1.67076 2.89385i −2.00000 + 3.46410i 6.09877 10.5634i 6.68306 9.86474 17.0862i −8.00000 7.91709 + 13.7128i 24.3951
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 63.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.4.c.a 8
3.b odd 2 1 666.4.f.a 8
37.c even 3 1 inner 74.4.c.a 8
111.i odd 6 1 666.4.f.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.4.c.a 8 1.a even 1 1 trivial
74.4.c.a 8 37.c even 3 1 inner
666.4.f.a 8 3.b odd 2 1
666.4.f.a 8 111.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(74, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 - 2 T + T^{2} )^{4}$$
$3$ $$48400 - 7040 T + 9384 T^{2} - 984 T^{3} + 1384 T^{4} - 126 T^{5} + 63 T^{6} + 5 T^{7} + T^{8}$$
$5$ $$729 - 31266 T + 1345716 T^{2} + 203268 T^{3} + 42529 T^{4} + 556 T^{5} + 276 T^{6} + 10 T^{7} + T^{8}$$
$7$ $$8611097616 + 127316112 T + 61086232 T^{2} - 318560 T^{3} + 318364 T^{4} - 830 T^{5} + 647 T^{6} - 3 T^{7} + T^{8}$$
$11$ $$( -460800 + 77440 T - 2880 T^{2} - 32 T^{3} + T^{4} )^{2}$$
$13$ $$60535681600 + 14952342880 T + 3408321664 T^{2} + 100390856 T^{3} + 5294096 T^{4} + 50906 T^{5} + 4879 T^{6} + 61 T^{7} + T^{8}$$
$17$ $$1531116688689 - 277059952764 T + 40926188178 T^{2} - 1696020528 T^{3} + 59307643 T^{4} - 358512 T^{5} + 7586 T^{6} - 12 T^{7} + T^{8}$$
$19$ $$6992428262400 - 348204057600 T + 23516753920 T^{2} - 67888960 T^{3} + 12161856 T^{4} + 97504 T^{5} + 7377 T^{6} + 71 T^{7} + T^{8}$$
$23$ $$( -15115776 - 2143124 T - 31000 T^{2} + 26 T^{3} + T^{4} )^{2}$$
$29$ $$( -72087162 + 5980165 T - 40343 T^{2} - 161 T^{3} + T^{4} )^{2}$$
$31$ $$( 712674048 - 8437024 T - 99216 T^{2} + 56 T^{3} + T^{4} )^{2}$$
$37$ $$6582952005840035281 - 72388689065857889 T + 183503318498089 T^{2} + 1449704765042 T^{3} - 12160873546 T^{4} + 28620314 T^{5} + 71521 T^{6} - 557 T^{7} + T^{8}$$
$41$ $$2417419909609430625 - 824886470179500 T + 170803202873050 T^{2} + 344270642160 T^{3} + 10424771171 T^{4} + 11151088 T^{5} + 118138 T^{6} - 92 T^{7} + T^{8}$$
$43$ $$( -712153104 + 12592876 T - 32132 T^{2} - 266 T^{3} + T^{4} )^{2}$$
$47$ $$( 6878004480 + 44250720 T - 300064 T^{2} - 140 T^{3} + T^{4} )^{2}$$
$53$ $$32\!\cdots\!00$$$$+ 452288284015476000 T + 5664614130522400 T^{2} - 1237351403000 T^{3} + 63684370720 T^{4} - 5733230 T^{5} + 304151 T^{6} - 159 T^{7} + T^{8}$$
$59$ $$86030240301901440000 - 1901707229543616000 T + 34786562021395200 T^{2} - 165160003346880 T^{3} + 674321904496 T^{4} - 204461968 T^{5} + 850913 T^{6} - 263 T^{7} + T^{8}$$
$61$ $$48172011272443276729 + 859446811740938778 T + 12363488609419852 T^{2} + 55849334396284 T^{3} + 215571709073 T^{4} + 159504404 T^{5} + 470364 T^{6} + 206 T^{7} + T^{8}$$
$67$ $$58\!\cdots\!24$$$$+ 5693550403411551232 T + 39343641090497024 T^{2} + 149580016132416 T^{3} + 432935571584 T^{4} + 639851628 T^{5} + 743189 T^{6} - 245 T^{7} + T^{8}$$
$71$ $$13\!\cdots\!04$$$$+ 5583104997783101424 T + 219113121634868808 T^{2} + 259689552567888 T^{3} + 715695255988 T^{4} + 470196402 T^{5} + 1415831 T^{6} + 957 T^{7} + T^{8}$$
$73$ $$( -6461408784 - 125038736 T - 581912 T^{2} + 136 T^{3} + T^{4} )^{2}$$
$79$ $$15\!\cdots\!00$$$$+ 2147964517323901440 T + 23254542709526016 T^{2} + 101015996208960 T^{3} + 343674114832 T^{4} + 454099192 T^{5} + 631329 T^{6} - 173 T^{7} + T^{8}$$
$83$ $$16\!\cdots\!16$$$$- 29950889041745651232 T + 531830261605137912 T^{2} - 460374556264440 T^{3} + 1182001307584 T^{4} - 891685270 T^{5} + 1968827 T^{6} - 1217 T^{7} + T^{8}$$
$89$ $$21\!\cdots\!25$$$$+ 5886020719006243200 T + 60008385052774134 T^{2} + 77590761399696 T^{3} + 1214308772191 T^{4} + 2276658096 T^{5} + 3615614 T^{6} + 2136 T^{7} + T^{8}$$
$97$ $$( -492264212614 + 524168339 T + 1604321 T^{2} - 2631 T^{3} + T^{4} )^{2}$$
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