# Properties

 Label 74.8.f.b Level $74$ Weight $8$ Character orbit 74.f Analytic conductor $23.116$ Analytic rank $0$ Dimension $66$ 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: $$66$$ Relative dimension: $$11$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$66 q - 39 q^{3} + 459 q^{5} - 918 q^{7} + 16896 q^{8} + 3237 q^{9}+O(q^{10})$$ 66 * q - 39 * q^3 + 459 * q^5 - 918 * q^7 + 16896 * q^8 + 3237 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$66 q - 39 q^{3} + 459 q^{5} - 918 q^{7} + 16896 q^{8} + 3237 q^{9} + 4704 q^{10} + 4539 q^{11} - 2496 q^{12} - 10542 q^{13} + 6744 q^{14} - 39894 q^{15} - 78822 q^{17} + 51792 q^{18} + 79461 q^{19} + 29376 q^{20} - 5835 q^{21} + 18168 q^{22} - 199479 q^{23} + 19968 q^{24} + 356301 q^{25} + 186744 q^{26} - 91689 q^{27} - 91584 q^{28} - 38526 q^{29} + 319152 q^{30} + 1917246 q^{31} - 902847 q^{33} - 205776 q^{34} - 1574793 q^{35} + 3079296 q^{36} - 396513 q^{37} - 692880 q^{38} + 2514306 q^{39} + 470016 q^{40} + 2989350 q^{41} + 46680 q^{42} + 6006348 q^{43} - 145344 q^{44} + 1083726 q^{45} - 588744 q^{46} - 894135 q^{47} + 7973742 q^{49} + 2012400 q^{50} - 524880 q^{51} - 2262720 q^{52} + 1457844 q^{53} + 1452312 q^{54} + 2415690 q^{55} - 1202688 q^{56} + 2734338 q^{57} - 3423360 q^{58} + 3099522 q^{59} - 3102912 q^{60} - 10968396 q^{61} - 3067320 q^{62} - 10727970 q^{63} - 8650752 q^{64} - 13497300 q^{65} - 4598712 q^{66} - 1978119 q^{67} + 5026176 q^{68} - 7251108 q^{69} + 435744 q^{70} - 19555002 q^{71} + 3314688 q^{72} - 31332840 q^{73} + 7607472 q^{74} + 44758074 q^{75} + 5085504 q^{76} + 1982007 q^{77} - 22512264 q^{78} + 10522902 q^{79} + 4816896 q^{80} - 7548876 q^{81} - 7150320 q^{82} + 19771926 q^{83} + 3556224 q^{84} - 6684072 q^{85} + 22982496 q^{86} - 67769667 q^{87} - 2323968 q^{88} - 42018963 q^{89} - 25158672 q^{90} + 4847565 q^{91} - 1096704 q^{92} - 14558475 q^{93} + 3175680 q^{94} + 4508415 q^{95} - 2555904 q^{96} - 39286662 q^{97} + 71630904 q^{98} + 66155463 q^{99}+O(q^{100})$$ 66 * q - 39 * q^3 + 459 * q^5 - 918 * q^7 + 16896 * q^8 + 3237 * q^9 + 4704 * q^10 + 4539 * q^11 - 2496 * q^12 - 10542 * q^13 + 6744 * q^14 - 39894 * q^15 - 78822 * q^17 + 51792 * q^18 + 79461 * q^19 + 29376 * q^20 - 5835 * q^21 + 18168 * q^22 - 199479 * q^23 + 19968 * q^24 + 356301 * q^25 + 186744 * q^26 - 91689 * q^27 - 91584 * q^28 - 38526 * q^29 + 319152 * q^30 + 1917246 * q^31 - 902847 * q^33 - 205776 * q^34 - 1574793 * q^35 + 3079296 * q^36 - 396513 * q^37 - 692880 * q^38 + 2514306 * q^39 + 470016 * q^40 + 2989350 * q^41 + 46680 * q^42 + 6006348 * q^43 - 145344 * q^44 + 1083726 * q^45 - 588744 * q^46 - 894135 * q^47 + 7973742 * q^49 + 2012400 * q^50 - 524880 * q^51 - 2262720 * q^52 + 1457844 * q^53 + 1452312 * q^54 + 2415690 * q^55 - 1202688 * q^56 + 2734338 * q^57 - 3423360 * q^58 + 3099522 * q^59 - 3102912 * q^60 - 10968396 * q^61 - 3067320 * q^62 - 10727970 * q^63 - 8650752 * q^64 - 13497300 * q^65 - 4598712 * q^66 - 1978119 * q^67 + 5026176 * q^68 - 7251108 * q^69 + 435744 * q^70 - 19555002 * q^71 + 3314688 * q^72 - 31332840 * q^73 + 7607472 * q^74 + 44758074 * q^75 + 5085504 * q^76 + 1982007 * q^77 - 22512264 * q^78 + 10522902 * q^79 + 4816896 * q^80 - 7548876 * q^81 - 7150320 * q^82 + 19771926 * q^83 + 3556224 * q^84 - 6684072 * q^85 + 22982496 * q^86 - 67769667 * q^87 - 2323968 * q^88 - 42018963 * q^89 - 25158672 * q^90 + 4847565 * q^91 - 1096704 * q^92 - 14558475 * q^93 + 3175680 * q^94 + 4508415 * q^95 - 2555904 * q^96 - 39286662 * q^97 + 71630904 * q^98 + 66155463 * 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 −83.1185 30.2527i 49.0268 41.1384i 77.4784 + 439.402i −707.623 −71.3442 404.613i 256.000 443.405i 4318.12 + 3623.33i 1784.72 + 3091.23i
7.2 7.51754 2.73616i −62.7580 22.8420i 49.0268 41.1384i 1.66323 + 9.43265i −534.285 141.286 + 801.274i 256.000 443.405i 1741.46 + 1461.26i 38.3126 + 66.3594i
7.3 7.51754 2.73616i −56.1799 20.4478i 49.0268 41.1384i −76.5259 434.000i −478.283 −144.408 818.976i 256.000 443.405i 1062.73 + 891.735i −1762.78 3053.23i
7.4 7.51754 2.73616i −36.4042 13.2501i 49.0268 41.1384i −1.10281 6.25436i −309.925 125.532 + 711.929i 256.000 443.405i −525.635 441.061i −25.4034 43.9999i
7.5 7.51754 2.73616i −19.5402 7.11207i 49.0268 41.1384i 69.5532 + 394.456i −166.354 −65.8495 373.451i 256.000 443.405i −1344.10 1127.83i 1602.16 + 2775.03i
7.6 7.51754 2.73616i 6.89307 + 2.50887i 49.0268 41.1384i −27.8528 157.961i 58.6836 117.132 + 664.288i 256.000 443.405i −1634.12 1371.19i −641.591 1111.27i
7.7 7.51754 2.73616i 25.6856 + 9.34879i 49.0268 41.1384i 46.1726 + 261.858i 218.672 −206.971 1173.79i 256.000 443.405i −1102.99 925.518i 1063.59 + 1842.19i
7.8 7.51754 2.73616i 28.9563 + 10.5392i 49.0268 41.1384i −48.7600 276.532i 246.517 −219.425 1244.42i 256.000 443.405i −947.948 795.422i −1123.19 1945.43i
7.9 7.51754 2.73616i 46.7880 + 17.0295i 49.0268 41.1384i 58.4927 + 331.728i 398.326 201.162 + 1140.85i 256.000 443.405i 223.779 + 187.773i 1347.38 + 2333.74i
7.10 7.51754 2.73616i 65.8840 + 23.9798i 49.0268 41.1384i 42.5352 + 241.229i 560.898 44.8916 + 254.593i 256.000 443.405i 2090.33 + 1754.00i 979.802 + 1697.07i
7.11 7.51754 2.73616i 79.5512 + 28.9543i 49.0268 41.1384i −57.6869 327.158i 677.253 −6.84167 38.8010i 256.000 443.405i 3814.71 + 3200.92i −1328.82 2301.59i
9.1 −1.38919 7.87846i −14.0898 + 79.9071i −60.1403 + 21.8893i −46.9471 39.3933i 649.118 −514.898 432.050i 256.000 + 443.405i −4131.51 1503.75i −245.140 + 424.595i
9.2 −1.38919 7.87846i −11.0757 + 62.8133i −60.1403 + 21.8893i 48.5860 + 40.7685i 510.258 745.438 + 625.497i 256.000 + 443.405i −1767.73 643.401i 253.698 439.418i
9.3 −1.38919 7.87846i −8.32711 + 47.2254i −60.1403 + 21.8893i −300.074 251.792i 383.631 −303.488 254.657i 256.000 + 443.405i −105.789 38.5041i −1566.88 + 2713.91i
9.4 −1.38919 7.87846i −6.48463 + 36.7762i −60.1403 + 21.8893i 381.143 + 319.817i 298.748 975.665 + 818.680i 256.000 + 443.405i 744.672 + 271.038i 1990.19 3447.11i
9.5 −1.38919 7.87846i −2.39984 + 13.6102i −60.1403 + 21.8893i −34.7022 29.1186i 110.561 −140.013 117.485i 256.000 + 443.405i 1875.63 + 682.674i −181.202 + 313.852i
9.6 −1.38919 7.87846i −0.307798 + 1.74561i −60.1403 + 21.8893i 323.926 + 271.806i 14.1803 −1225.86 1028.62i 256.000 + 443.405i 2052.16 + 746.923i 1691.42 2929.63i
9.7 −1.38919 7.87846i 2.48533 14.0950i −60.1403 + 21.8893i −356.271 298.947i −114.500 1364.62 + 1145.05i 256.000 + 443.405i 1862.62 + 677.936i −1860.32 + 3222.16i
9.8 −1.38919 7.87846i 6.70779 38.0417i −60.1403 + 21.8893i 124.780 + 104.703i −309.029 25.1771 + 21.1261i 256.000 + 443.405i 652.928 + 237.646i 651.556 1128.53i
9.9 −1.38919 7.87846i 9.76159 55.3608i −60.1403 + 21.8893i −282.513 237.057i −449.718 −1337.20 1122.04i 256.000 + 443.405i −914.416 332.820i −1475.18 + 2555.09i
See all 66 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.11 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.b 66
37.f even 9 1 inner 74.8.f.b 66

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.8.f.b 66 1.a even 1 1 trivial
74.8.f.b 66 37.f even 9 1 inner