# Properties

 Label 74.8.e.a Level $74$ Weight $8$ Character orbit 74.e Analytic conductor $23.116$ Analytic rank $0$ Dimension $48$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [74,8,Mod(11,74)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5]))

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

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

chi := DirichletCharacter("74.11");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 74.e (of order $$6$$, degree $$2$$, 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: $$48$$ Relative dimension: $$24$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48 q + 28 q^{3} + 1536 q^{4} + 252 q^{5} + 52 q^{7} - 24028 q^{9}+O(q^{10})$$ 48 * q + 28 * q^3 + 1536 * q^4 + 252 * q^5 + 52 * q^7 - 24028 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$48 q + 28 q^{3} + 1536 q^{4} + 252 q^{5} + 52 q^{7} - 24028 q^{9} + 4864 q^{10} - 8352 q^{11} - 1792 q^{12} - 5136 q^{13} - 16632 q^{15} - 98304 q^{16} + 28980 q^{17} - 66096 q^{19} + 16128 q^{20} - 64344 q^{21} + 675072 q^{25} + 533952 q^{26} - 444416 q^{27} - 3328 q^{28} - 114528 q^{30} + 120916 q^{33} + 120992 q^{34} + 865908 q^{35} - 3075584 q^{36} - 500268 q^{37} + 879360 q^{38} - 669984 q^{39} + 155648 q^{40} + 1024524 q^{41} - 589728 q^{42} - 267264 q^{44} - 729888 q^{46} - 2144712 q^{47} - 229376 q^{48} - 3570900 q^{49} + 2919168 q^{50} - 328704 q^{52} - 2231700 q^{53} + 3784320 q^{54} + 3781260 q^{55} - 1487472 q^{57} + 468064 q^{58} - 10302480 q^{59} + 10141752 q^{61} + 4388256 q^{62} + 9348424 q^{63} - 12582912 q^{64} + 1169736 q^{65} - 1473864 q^{67} + 14887176 q^{69} - 310528 q^{70} + 3017772 q^{71} + 16773224 q^{73} - 2565696 q^{74} - 10623504 q^{75} - 4230144 q^{76} + 15133908 q^{77} - 763328 q^{78} - 2776836 q^{79} - 29469256 q^{81} - 1844556 q^{83} - 8236032 q^{84} - 9520960 q^{85} - 1972416 q^{86} - 51259224 q^{87} + 33361344 q^{89} - 28098496 q^{90} + 91933608 q^{91} + 4193280 q^{92} - 9874476 q^{93} - 28885728 q^{94} - 40830084 q^{95} + 11612160 q^{98} + 50165480 q^{99}+O(q^{100})$$ 48 * q + 28 * q^3 + 1536 * q^4 + 252 * q^5 + 52 * q^7 - 24028 * q^9 + 4864 * q^10 - 8352 * q^11 - 1792 * q^12 - 5136 * q^13 - 16632 * q^15 - 98304 * q^16 + 28980 * q^17 - 66096 * q^19 + 16128 * q^20 - 64344 * q^21 + 675072 * q^25 + 533952 * q^26 - 444416 * q^27 - 3328 * q^28 - 114528 * q^30 + 120916 * q^33 + 120992 * q^34 + 865908 * q^35 - 3075584 * q^36 - 500268 * q^37 + 879360 * q^38 - 669984 * q^39 + 155648 * q^40 + 1024524 * q^41 - 589728 * q^42 - 267264 * q^44 - 729888 * q^46 - 2144712 * q^47 - 229376 * q^48 - 3570900 * q^49 + 2919168 * q^50 - 328704 * q^52 - 2231700 * q^53 + 3784320 * q^54 + 3781260 * q^55 - 1487472 * q^57 + 468064 * q^58 - 10302480 * q^59 + 10141752 * q^61 + 4388256 * q^62 + 9348424 * q^63 - 12582912 * q^64 + 1169736 * q^65 - 1473864 * q^67 + 14887176 * q^69 - 310528 * q^70 + 3017772 * q^71 + 16773224 * q^73 - 2565696 * q^74 - 10623504 * q^75 - 4230144 * q^76 + 15133908 * q^77 - 763328 * q^78 - 2776836 * q^79 - 29469256 * q^81 - 1844556 * q^83 - 8236032 * q^84 - 9520960 * q^85 - 1972416 * q^86 - 51259224 * q^87 + 33361344 * q^89 - 28098496 * q^90 + 91933608 * q^91 + 4193280 * q^92 - 9874476 * q^93 - 28885728 * q^94 - 40830084 * q^95 + 11612160 * q^98 + 50165480 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −6.92820 + 4.00000i −39.6506 + 68.6769i 32.0000 55.4256i 342.164 + 197.548i 634.410i −54.0205 + 93.5662i 512.000i −2050.84 3552.16i −3160.77
11.2 −6.92820 + 4.00000i −34.7601 + 60.2062i 32.0000 55.4256i −366.211 211.432i 556.161i −785.705 + 1360.88i 512.000i −1323.02 2291.55i 3382.91
11.3 −6.92820 + 4.00000i −28.4196 + 49.2242i 32.0000 55.4256i −302.692 174.760i 454.714i 498.049 862.646i 512.000i −521.849 903.869i 2796.15
11.4 −6.92820 + 4.00000i −27.2721 + 47.2366i 32.0000 55.4256i 55.1235 + 31.8256i 436.353i −71.7224 + 124.227i 512.000i −394.030 682.480i −509.209
11.5 −6.92820 + 4.00000i −11.4976 + 19.9144i 32.0000 55.4256i 145.889 + 84.2291i 183.961i 862.198 1493.37i 512.000i 829.111 + 1436.06i −1347.67
11.6 −6.92820 + 4.00000i −3.53023 + 6.11453i 32.0000 55.4256i −112.662 65.0457i 56.4836i −91.9227 + 159.215i 512.000i 1068.57 + 1850.83i 1040.73
11.7 −6.92820 + 4.00000i 3.17200 5.49407i 32.0000 55.4256i 434.558 + 250.892i 50.7521i −92.5229 + 160.254i 512.000i 1073.38 + 1859.14i −4014.27
11.8 −6.92820 + 4.00000i 14.4393 25.0096i 32.0000 55.4256i −258.190 149.066i 231.029i −432.865 + 749.744i 512.000i 676.512 + 1171.75i 2385.05
11.9 −6.92820 + 4.00000i 18.6201 32.2509i 32.0000 55.4256i 100.040 + 57.7582i 297.921i −87.6118 + 151.748i 512.000i 400.086 + 692.969i −924.132
11.10 −6.92820 + 4.00000i 34.1570 59.1616i 32.0000 55.4256i 121.654 + 70.2371i 546.511i 713.486 1235.79i 512.000i −1239.90 2147.56i −1123.79
11.11 −6.92820 + 4.00000i 38.1188 66.0237i 32.0000 55.4256i −423.248 244.362i 609.901i 318.681 551.972i 512.000i −1812.59 3139.49i 3909.80
11.12 −6.92820 + 4.00000i 43.6230 75.5572i 32.0000 55.4256i 194.940 + 112.548i 697.968i −763.043 + 1321.63i 512.000i −2712.43 4698.07i −1800.78
11.13 6.92820 4.00000i −45.3894 + 78.6168i 32.0000 55.4256i −72.4779 41.8451i 726.231i −443.344 + 767.894i 512.000i −3026.90 5242.75i −669.522
11.14 6.92820 4.00000i −38.6069 + 66.8691i 32.0000 55.4256i 420.227 + 242.618i 617.710i 571.245 989.425i 512.000i −1887.48 3269.21i 3881.89
11.15 6.92820 4.00000i −20.8766 + 36.1593i 32.0000 55.4256i −98.5923 56.9223i 334.025i 197.476 342.039i 512.000i 221.837 + 384.233i −910.756
11.16 6.92820 4.00000i −20.6208 + 35.7163i 32.0000 55.4256i −65.0909 37.5802i 329.933i 14.5370 25.1789i 512.000i 243.062 + 420.996i −601.284
11.17 6.92820 4.00000i −12.2622 + 21.2388i 32.0000 55.4256i −470.905 271.877i 196.195i −473.212 + 819.628i 512.000i 792.776 + 1373.13i −4350.04
11.18 6.92820 4.00000i −8.96322 + 15.5248i 32.0000 55.4256i 442.893 + 255.705i 143.412i −521.070 + 902.519i 512.000i 932.821 + 1615.69i 4091.27
11.19 6.92820 4.00000i 5.03657 8.72359i 32.0000 55.4256i 153.813 + 88.8037i 80.5851i 660.537 1144.08i 512.000i 1042.77 + 1806.12i 1420.86
11.20 6.92820 4.00000i 16.6391 28.8198i 32.0000 55.4256i 214.814 + 124.023i 266.226i −443.171 + 767.595i 512.000i 539.780 + 934.927i 1984.37
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.24 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.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.8.e.a 48
37.e even 6 1 inner 74.8.e.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.8.e.a 48 1.a even 1 1 trivial
74.8.e.a 48 37.e even 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(74, [\chi])$$.