# Properties

 Label 74.8.f.a Level $74$ Weight $8$ Character orbit 74.f Analytic conductor $23.116$ Analytic rank $0$ Dimension $60$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [74,8,Mod(7,74)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([16]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("74.7");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 74.f (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$23.1164918858$$ Analytic rank: $$0$$ Dimension: $$60$$ Relative dimension: $$10$$ over $$\Q(\zeta_{9})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$60 q - 39 q^{3} - 624 q^{5} - 918 q^{7} - 15360 q^{8} + 3237 q^{9}+O(q^{10})$$ 60 * q - 39 * q^3 - 624 * q^5 - 918 * q^7 - 15360 * q^8 + 3237 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$60 q - 39 q^{3} - 624 q^{5} - 918 q^{7} - 15360 q^{8} + 3237 q^{9} + 1296 q^{10} + 3447 q^{11} - 2496 q^{12} + 10704 q^{13} + 6744 q^{14} + 19074 q^{15} + 47661 q^{17} - 51792 q^{18} + 20649 q^{19} - 39936 q^{20} + 229161 q^{21} - 18168 q^{22} - 81789 q^{23} - 19968 q^{24} + 224064 q^{25} + 236400 q^{26} - 91689 q^{27} - 91584 q^{28} - 7260 q^{29} + 152592 q^{30} + 439386 q^{31} + 691269 q^{33} - 455064 q^{34} - 829797 q^{35} + 1679616 q^{36} - 441285 q^{37} - 1560816 q^{38} + 314424 q^{39} + 638976 q^{40} + 765417 q^{41} + 1833288 q^{42} - 2860764 q^{43} - 145344 q^{44} - 2301396 q^{45} - 2162472 q^{46} - 120843 q^{47} + 1471290 q^{49} - 2529792 q^{50} + 524880 q^{51} - 902976 q^{52} - 43176 q^{53} - 1934136 q^{54} + 3920940 q^{55} + 1202688 q^{56} - 1371726 q^{57} - 431928 q^{58} + 9107202 q^{59} + 3102912 q^{60} - 2125710 q^{61} + 3174168 q^{62} + 6557916 q^{63} - 7864320 q^{64} + 12416169 q^{65} + 4598712 q^{66} - 6182031 q^{67} - 2520192 q^{68} + 3014298 q^{69} + 5524224 q^{70} - 5518950 q^{71} - 3314688 q^{72} + 6786894 q^{73} + 8713032 q^{74} - 25030830 q^{75} + 1321536 q^{76} + 1185165 q^{77} + 4913208 q^{78} - 26251380 q^{79} - 1327104 q^{80} - 135708 q^{81} + 7311792 q^{82} + 12381114 q^{83} + 3556224 q^{84} - 10853433 q^{85} - 481152 q^{86} - 18206049 q^{87} + 1764864 q^{88} + 1520715 q^{89} - 51806760 q^{90} - 1509615 q^{91} + 13681536 q^{92} - 34211421 q^{93} - 42072000 q^{94} + 52148349 q^{95} + 2555904 q^{96} + 28088142 q^{97} - 22487064 q^{98} - 39791397 q^{99}+O(q^{100})$$ 60 * q - 39 * q^3 - 624 * q^5 - 918 * q^7 - 15360 * q^8 + 3237 * q^9 + 1296 * q^10 + 3447 * q^11 - 2496 * q^12 + 10704 * q^13 + 6744 * q^14 + 19074 * q^15 + 47661 * q^17 - 51792 * q^18 + 20649 * q^19 - 39936 * q^20 + 229161 * q^21 - 18168 * q^22 - 81789 * q^23 - 19968 * q^24 + 224064 * q^25 + 236400 * q^26 - 91689 * q^27 - 91584 * q^28 - 7260 * q^29 + 152592 * q^30 + 439386 * q^31 + 691269 * q^33 - 455064 * q^34 - 829797 * q^35 + 1679616 * q^36 - 441285 * q^37 - 1560816 * q^38 + 314424 * q^39 + 638976 * q^40 + 765417 * q^41 + 1833288 * q^42 - 2860764 * q^43 - 145344 * q^44 - 2301396 * q^45 - 2162472 * q^46 - 120843 * q^47 + 1471290 * q^49 - 2529792 * q^50 + 524880 * q^51 - 902976 * q^52 - 43176 * q^53 - 1934136 * q^54 + 3920940 * q^55 + 1202688 * q^56 - 1371726 * q^57 - 431928 * q^58 + 9107202 * q^59 + 3102912 * q^60 - 2125710 * q^61 + 3174168 * q^62 + 6557916 * q^63 - 7864320 * q^64 + 12416169 * q^65 + 4598712 * q^66 - 6182031 * q^67 - 2520192 * q^68 + 3014298 * q^69 + 5524224 * q^70 - 5518950 * q^71 - 3314688 * q^72 + 6786894 * q^73 + 8713032 * q^74 - 25030830 * q^75 + 1321536 * q^76 + 1185165 * q^77 + 4913208 * q^78 - 26251380 * q^79 - 1327104 * q^80 - 135708 * q^81 + 7311792 * q^82 + 12381114 * q^83 + 3556224 * q^84 - 10853433 * q^85 - 481152 * q^86 - 18206049 * q^87 + 1764864 * q^88 + 1520715 * q^89 - 51806760 * q^90 - 1509615 * q^91 + 13681536 * q^92 - 34211421 * q^93 - 42072000 * q^94 + 52148349 * q^95 + 2555904 * q^96 + 28088142 * q^97 - 22487064 * q^98 - 39791397 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −7.51754 + 2.73616i −85.7137 31.1972i 49.0268 41.1384i −35.4158 200.853i 729.717 −94.2819 534.699i −256.000 + 443.405i 4698.24 + 3942.29i 815.807 + 1413.02i
7.2 −7.51754 + 2.73616i −53.4664 19.4602i 49.0268 41.1384i 64.3355 + 364.865i 455.182 2.60139 + 14.7532i −256.000 + 443.405i 804.620 + 675.157i −1481.97 2566.86i
7.3 −7.51754 + 2.73616i −44.3166 16.1299i 49.0268 41.1384i 7.39799 + 41.9561i 377.286 8.85914 + 50.2427i −256.000 + 443.405i 28.4488 + 23.8714i −170.413 295.165i
7.4 −7.51754 + 2.73616i −23.1121 8.41210i 49.0268 41.1384i −44.0952 250.077i 196.763 212.376 + 1204.44i −256.000 + 443.405i −1211.94 1016.93i 1015.74 + 1759.31i
7.5 −7.51754 + 2.73616i −9.31802 3.39148i 49.0268 41.1384i −43.8374 248.614i 79.3283 −219.309 1243.76i −256.000 + 443.405i −1600.02 1342.57i 1009.80 + 1749.02i
7.6 −7.51754 + 2.73616i 1.79993 + 0.655119i 49.0268 41.1384i −75.0119 425.413i −15.3235 −127.235 721.583i −256.000 + 443.405i −1672.53 1403.42i 1727.90 + 2992.82i
7.7 −7.51754 + 2.73616i 28.9193 + 10.5258i 49.0268 41.1384i 44.4856 + 252.290i −246.202 242.753 + 1376.72i −256.000 + 443.405i −949.803 796.980i −1024.73 1774.88i
7.8 −7.51754 + 2.73616i 44.1810 + 16.0806i 49.0268 41.1384i 32.7777 + 185.892i −376.132 −124.661 706.990i −256.000 + 443.405i 18.0385 + 15.1361i −755.037 1307.76i
7.9 −7.51754 + 2.73616i 64.6665 + 23.5367i 49.0268 41.1384i −53.9336 305.872i −550.533 86.7776 + 492.140i −256.000 + 443.405i 1952.44 + 1638.30i 1242.36 + 2151.84i
7.10 −7.51754 + 2.73616i 72.1175 + 26.2486i 49.0268 41.1384i 26.0390 + 147.674i −613.967 −170.305 965.849i −256.000 + 443.405i 2836.60 + 2380.19i −599.810 1038.90i
9.1 1.38919 + 7.87846i −12.9960 + 73.7039i −60.1403 + 21.8893i 200.010 + 167.828i −598.727 −245.531 206.025i −256.000 443.405i −3208.26 1167.71i −1044.38 + 1808.92i
9.2 1.38919 + 7.87846i −11.0721 + 62.7932i −60.1403 + 21.8893i −319.335 267.954i −510.095 181.169 + 152.019i −256.000 443.405i −1765.29 642.513i 1667.45 2888.11i
9.3 1.38919 + 7.87846i −4.91633 + 27.8819i −60.1403 + 21.8893i −125.695 105.471i −226.496 −1138.83 955.591i −256.000 443.405i 1301.88 + 473.845i 656.333 1136.80i
9.4 1.38919 + 7.87846i −3.34551 + 18.9733i −60.1403 + 21.8893i 421.522 + 353.699i −154.128 −302.680 253.979i −256.000 443.405i 1706.31 + 621.047i −2201.03 + 3812.30i
9.5 1.38919 + 7.87846i −3.25253 + 18.4460i −60.1403 + 21.8893i −67.4246 56.5760i −149.845 665.439 + 558.369i −256.000 443.405i 1725.43 + 628.006i 352.066 609.797i
9.6 1.38919 + 7.87846i −0.914074 + 5.18397i −60.1403 + 21.8893i 145.180 + 121.821i −42.1115 1069.79 + 897.660i −256.000 443.405i 2029.07 + 738.521i −758.078 + 1313.03i
9.7 1.38919 + 7.87846i 5.22442 29.6291i −60.1403 + 21.8893i 90.0964 + 75.5999i 240.690 −323.673 271.594i −256.000 443.405i 1204.52 + 438.408i −470.450 + 814.843i
9.8 1.38919 + 7.87846i 8.76063 49.6840i −60.1403 + 21.8893i −371.412 311.652i 403.604 −422.250 354.310i −256.000 443.405i −336.643 122.528i 1939.38 3359.10i
9.9 1.38919 + 7.87846i 12.4741 70.7441i −60.1403 + 21.8893i −84.3548 70.7821i 574.683 1224.58 + 1027.55i −256.000 443.405i −2794.01 1016.94i 440.469 762.915i
9.10 1.38919 + 7.87846i 13.4960 76.5397i −60.1403 + 21.8893i 125.384 + 105.210i 621.764 −744.639 624.826i −256.000 443.405i −3621.08 1317.97i −654.710 + 1133.99i
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.10 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.f even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.8.f.a 60
37.f even 9 1 inner 74.8.f.a 60

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.8.f.a 60 1.a even 1 1 trivial
74.8.f.a 60 37.f even 9 1 inner