# Properties

 Label 74.8.c.a Level $74$ Weight $8$ Character orbit 74.c Analytic conductor $23.116$ Analytic rank $0$ Dimension $22$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [74,8,Mod(47,74)]

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([4]))

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

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

chi := DirichletCharacter("74.47");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 74.c (of order $$3$$, 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: $$22$$ Relative dimension: $$11$$ over $$\Q(\zeta_{3})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$22 q - 88 q^{2} + 40 q^{3} - 704 q^{4} + 475 q^{5} - 640 q^{6} - 588 q^{7} + 11264 q^{8} - 8369 q^{9}+O(q^{10})$$ 22 * q - 88 * q^2 + 40 * q^3 - 704 * q^4 + 475 * q^5 - 640 * q^6 - 588 * q^7 + 11264 * q^8 - 8369 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$22 q - 88 q^{2} + 40 q^{3} - 704 q^{4} + 475 q^{5} - 640 q^{6} - 588 q^{7} + 11264 q^{8} - 8369 q^{9} - 7600 q^{10} - 4904 q^{11} + 2560 q^{12} - 2612 q^{13} + 9408 q^{14} - 3688 q^{15} - 45056 q^{16} + 24651 q^{17} - 66952 q^{18} + 82536 q^{19} + 30400 q^{20} + 50810 q^{21} + 19616 q^{22} + 118832 q^{23} + 20480 q^{24} - 256572 q^{25} + 41792 q^{26} - 363800 q^{27} - 37632 q^{28} - 372170 q^{29} - 29504 q^{30} + 42808 q^{31} - 360448 q^{32} - 170612 q^{33} + 197208 q^{34} + 347316 q^{35} + 1071232 q^{36} - 605757 q^{37} - 1320576 q^{38} + 928324 q^{39} + 243200 q^{40} - 171689 q^{41} + 406480 q^{42} - 794528 q^{43} + 156928 q^{44} - 2199730 q^{45} - 475328 q^{46} - 517424 q^{47} - 327680 q^{48} - 1990061 q^{49} - 2052576 q^{50} + 1046736 q^{51} - 167168 q^{52} - 2880196 q^{53} + 1455200 q^{54} + 120328 q^{55} - 301056 q^{56} - 1881606 q^{57} + 1488680 q^{58} + 2657756 q^{59} + 472064 q^{60} + 1449979 q^{61} - 171232 q^{62} + 7031672 q^{63} + 5767168 q^{64} + 3463904 q^{65} + 2729792 q^{66} + 1415120 q^{67} - 3155328 q^{68} - 7425740 q^{69} + 2778528 q^{70} + 718320 q^{71} - 4284928 q^{72} + 21142228 q^{73} + 2481504 q^{74} - 38086392 q^{75} + 5282304 q^{76} + 3057932 q^{77} + 7426592 q^{78} + 8331416 q^{79} - 3891200 q^{80} + 8804005 q^{81} + 2747024 q^{82} - 1747036 q^{83} - 6503680 q^{84} - 4638522 q^{85} + 3178112 q^{86} + 3200432 q^{87} - 2510848 q^{88} + 16909895 q^{89} + 8798920 q^{90} + 13361368 q^{91} - 3802624 q^{92} + 15700724 q^{93} + 2069696 q^{94} - 32197184 q^{95} + 1310720 q^{96} - 72528842 q^{97} - 15920488 q^{98} + 13715284 q^{99}+O(q^{100})$$ 22 * q - 88 * q^2 + 40 * q^3 - 704 * q^4 + 475 * q^5 - 640 * q^6 - 588 * q^7 + 11264 * q^8 - 8369 * q^9 - 7600 * q^10 - 4904 * q^11 + 2560 * q^12 - 2612 * q^13 + 9408 * q^14 - 3688 * q^15 - 45056 * q^16 + 24651 * q^17 - 66952 * q^18 + 82536 * q^19 + 30400 * q^20 + 50810 * q^21 + 19616 * q^22 + 118832 * q^23 + 20480 * q^24 - 256572 * q^25 + 41792 * q^26 - 363800 * q^27 - 37632 * q^28 - 372170 * q^29 - 29504 * q^30 + 42808 * q^31 - 360448 * q^32 - 170612 * q^33 + 197208 * q^34 + 347316 * q^35 + 1071232 * q^36 - 605757 * q^37 - 1320576 * q^38 + 928324 * q^39 + 243200 * q^40 - 171689 * q^41 + 406480 * q^42 - 794528 * q^43 + 156928 * q^44 - 2199730 * q^45 - 475328 * q^46 - 517424 * q^47 - 327680 * q^48 - 1990061 * q^49 - 2052576 * q^50 + 1046736 * q^51 - 167168 * q^52 - 2880196 * q^53 + 1455200 * q^54 + 120328 * q^55 - 301056 * q^56 - 1881606 * q^57 + 1488680 * q^58 + 2657756 * q^59 + 472064 * q^60 + 1449979 * q^61 - 171232 * q^62 + 7031672 * q^63 + 5767168 * q^64 + 3463904 * q^65 + 2729792 * q^66 + 1415120 * q^67 - 3155328 * q^68 - 7425740 * q^69 + 2778528 * q^70 + 718320 * q^71 - 4284928 * q^72 + 21142228 * q^73 + 2481504 * q^74 - 38086392 * q^75 + 5282304 * q^76 + 3057932 * q^77 + 7426592 * q^78 + 8331416 * q^79 - 3891200 * q^80 + 8804005 * q^81 + 2747024 * q^82 - 1747036 * q^83 - 6503680 * q^84 - 4638522 * q^85 + 3178112 * q^86 + 3200432 * q^87 - 2510848 * q^88 + 16909895 * q^89 + 8798920 * q^90 + 13361368 * q^91 - 3802624 * q^92 + 15700724 * q^93 + 2069696 * q^94 - 32197184 * q^95 + 1310720 * q^96 - 72528842 * q^97 - 15920488 * q^98 + 13715284 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1 −4.00000 + 6.92820i −37.9353 65.7059i −32.0000 55.4256i −64.7023 112.068i 606.965 475.327 + 823.290i 512.000 −1784.68 + 3091.15i 1035.24
47.2 −4.00000 + 6.92820i −33.0633 57.2674i −32.0000 55.4256i 102.843 + 178.129i 529.013 87.2990 + 151.206i 512.000 −1092.87 + 1892.90i −1645.48
47.3 −4.00000 + 6.92820i −27.1156 46.9655i −32.0000 55.4256i 186.038 + 322.227i 433.849 −819.658 1419.69i 512.000 −377.006 + 652.994i −2976.61
47.4 −4.00000 + 6.92820i −13.7514 23.8182i −32.0000 55.4256i −121.418 210.302i 220.023 208.021 + 360.303i 512.000 715.296 1238.93i 1942.69
47.5 −4.00000 + 6.92820i −8.13907 14.0973i −32.0000 55.4256i −200.239 346.825i 130.225 −477.925 827.790i 512.000 961.011 1664.52i 3203.83
47.6 −4.00000 + 6.92820i 0.593488 + 1.02795i −32.0000 55.4256i 95.1239 + 164.759i −9.49581 −57.0072 98.7394i 512.000 1092.80 1892.78i −1521.98
47.7 −4.00000 + 6.92820i 5.98362 + 10.3639i −32.0000 55.4256i 220.554 + 382.010i −95.7379 735.827 + 1274.49i 512.000 1021.89 1769.97i −3528.86
47.8 −4.00000 + 6.92820i 27.1733 + 47.0655i −32.0000 55.4256i 23.6504 + 40.9637i −434.772 472.423 + 818.261i 512.000 −383.273 + 663.848i −378.406
47.9 −4.00000 + 6.92820i 30.5623 + 52.9354i −32.0000 55.4256i −100.205 173.561i −488.996 −620.146 1074.13i 512.000 −774.604 + 1341.65i 1603.29
47.10 −4.00000 + 6.92820i 35.5670 + 61.6039i −32.0000 55.4256i −177.780 307.924i −569.072 291.529 + 504.943i 512.000 −1436.53 + 2488.14i 2844.48
47.11 −4.00000 + 6.92820i 40.1251 + 69.4986i −32.0000 55.4256i 273.637 + 473.952i −642.001 −589.688 1021.37i 512.000 −2126.54 + 3683.28i −4378.18
63.1 −4.00000 6.92820i −37.9353 + 65.7059i −32.0000 + 55.4256i −64.7023 + 112.068i 606.965 475.327 823.290i 512.000 −1784.68 3091.15i 1035.24
63.2 −4.00000 6.92820i −33.0633 + 57.2674i −32.0000 + 55.4256i 102.843 178.129i 529.013 87.2990 151.206i 512.000 −1092.87 1892.90i −1645.48
63.3 −4.00000 6.92820i −27.1156 + 46.9655i −32.0000 + 55.4256i 186.038 322.227i 433.849 −819.658 + 1419.69i 512.000 −377.006 652.994i −2976.61
63.4 −4.00000 6.92820i −13.7514 + 23.8182i −32.0000 + 55.4256i −121.418 + 210.302i 220.023 208.021 360.303i 512.000 715.296 + 1238.93i 1942.69
63.5 −4.00000 6.92820i −8.13907 + 14.0973i −32.0000 + 55.4256i −200.239 + 346.825i 130.225 −477.925 + 827.790i 512.000 961.011 + 1664.52i 3203.83
63.6 −4.00000 6.92820i 0.593488 1.02795i −32.0000 + 55.4256i 95.1239 164.759i −9.49581 −57.0072 + 98.7394i 512.000 1092.80 + 1892.78i −1521.98
63.7 −4.00000 6.92820i 5.98362 10.3639i −32.0000 + 55.4256i 220.554 382.010i −95.7379 735.827 1274.49i 512.000 1021.89 + 1769.97i −3528.86
63.8 −4.00000 6.92820i 27.1733 47.0655i −32.0000 + 55.4256i 23.6504 40.9637i −434.772 472.423 818.261i 512.000 −383.273 663.848i −378.406
63.9 −4.00000 6.92820i 30.5623 52.9354i −32.0000 + 55.4256i −100.205 + 173.561i −488.996 −620.146 + 1074.13i 512.000 −774.604 1341.65i 1603.29
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 47.11 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.8.c.a 22
37.c even 3 1 inner 74.8.c.a 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.8.c.a 22 1.a even 1 1 trivial
74.8.c.a 22 37.c even 3 1 inner