# Properties

 Label 74.8.c.b Level $74$ Weight $8$ Character orbit 74.c Analytic conductor $23.116$ Analytic rank $0$ Dimension $24$ 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: $$24$$ Relative dimension: $$12$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 96 q^{2} + 40 q^{3} - 768 q^{4} - 136 q^{5} + 640 q^{6} + 536 q^{7} - 12288 q^{8} - 12014 q^{9}+O(q^{10})$$ 24 * q + 96 * q^2 + 40 * q^3 - 768 * q^4 - 136 * q^5 + 640 * q^6 + 536 * q^7 - 12288 * q^8 - 12014 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 96 q^{2} + 40 q^{3} - 768 q^{4} - 136 q^{5} + 640 q^{6} + 536 q^{7} - 12288 q^{8} - 12014 q^{9} - 2176 q^{10} - 3448 q^{11} + 2560 q^{12} + 186 q^{13} + 8576 q^{14} + 10628 q^{15} - 49152 q^{16} - 1212 q^{17} + 96112 q^{18} + 15980 q^{19} - 8704 q^{20} + 6994 q^{21} - 13792 q^{22} - 62464 q^{23} - 20480 q^{24} - 48368 q^{25} + 2976 q^{26} - 323648 q^{27} + 34304 q^{28} + 200084 q^{29} - 85024 q^{30} + 609672 q^{31} + 393216 q^{32} + 131560 q^{33} + 9696 q^{34} + 43132 q^{35} + 1537792 q^{36} - 557790 q^{37} + 255680 q^{38} - 694460 q^{39} + 69632 q^{40} + 377724 q^{41} - 55952 q^{42} - 304456 q^{43} + 110336 q^{44} + 3686244 q^{45} - 249856 q^{46} - 2156696 q^{47} - 327680 q^{48} - 2929078 q^{49} + 386944 q^{50} + 968912 q^{51} + 11904 q^{52} + 2071746 q^{53} - 1294592 q^{54} - 2492580 q^{55} - 274432 q^{56} - 1669714 q^{57} + 800336 q^{58} + 479512 q^{59} - 1360384 q^{60} + 6713104 q^{61} + 2438688 q^{62} - 7785864 q^{63} + 6291456 q^{64} + 6788906 q^{65} + 2104960 q^{66} + 1823780 q^{67} + 155136 q^{68} - 3604080 q^{69} - 345056 q^{70} - 3510960 q^{71} + 6151168 q^{72} - 27590128 q^{73} + 5276496 q^{74} - 16709392 q^{75} + 1022720 q^{76} - 65144 q^{77} + 5555680 q^{78} + 1130192 q^{79} + 1114112 q^{80} - 10442988 q^{81} + 6043584 q^{82} - 3888560 q^{83} - 895232 q^{84} + 8161728 q^{85} - 1217824 q^{86} - 17683256 q^{87} + 1765376 q^{88} - 14427556 q^{89} + 14744976 q^{90} + 3386000 q^{91} + 1998848 q^{92} - 20025152 q^{93} - 8626784 q^{94} + 21486568 q^{95} - 1310720 q^{96} - 11913436 q^{97} + 23432624 q^{98} + 16388620 q^{99}+O(q^{100})$$ 24 * q + 96 * q^2 + 40 * q^3 - 768 * q^4 - 136 * q^5 + 640 * q^6 + 536 * q^7 - 12288 * q^8 - 12014 * q^9 - 2176 * q^10 - 3448 * q^11 + 2560 * q^12 + 186 * q^13 + 8576 * q^14 + 10628 * q^15 - 49152 * q^16 - 1212 * q^17 + 96112 * q^18 + 15980 * q^19 - 8704 * q^20 + 6994 * q^21 - 13792 * q^22 - 62464 * q^23 - 20480 * q^24 - 48368 * q^25 + 2976 * q^26 - 323648 * q^27 + 34304 * q^28 + 200084 * q^29 - 85024 * q^30 + 609672 * q^31 + 393216 * q^32 + 131560 * q^33 + 9696 * q^34 + 43132 * q^35 + 1537792 * q^36 - 557790 * q^37 + 255680 * q^38 - 694460 * q^39 + 69632 * q^40 + 377724 * q^41 - 55952 * q^42 - 304456 * q^43 + 110336 * q^44 + 3686244 * q^45 - 249856 * q^46 - 2156696 * q^47 - 327680 * q^48 - 2929078 * q^49 + 386944 * q^50 + 968912 * q^51 + 11904 * q^52 + 2071746 * q^53 - 1294592 * q^54 - 2492580 * q^55 - 274432 * q^56 - 1669714 * q^57 + 800336 * q^58 + 479512 * q^59 - 1360384 * q^60 + 6713104 * q^61 + 2438688 * q^62 - 7785864 * q^63 + 6291456 * q^64 + 6788906 * q^65 + 2104960 * q^66 + 1823780 * q^67 + 155136 * q^68 - 3604080 * q^69 - 345056 * q^70 - 3510960 * q^71 + 6151168 * q^72 - 27590128 * q^73 + 5276496 * q^74 - 16709392 * q^75 + 1022720 * q^76 - 65144 * q^77 + 5555680 * q^78 + 1130192 * q^79 + 1114112 * q^80 - 10442988 * q^81 + 6043584 * q^82 - 3888560 * q^83 - 895232 * q^84 + 8161728 * q^85 - 1217824 * q^86 - 17683256 * q^87 + 1765376 * q^88 - 14427556 * q^89 + 14744976 * q^90 + 3386000 * q^91 + 1998848 * q^92 - 20025152 * q^93 - 8626784 * q^94 + 21486568 * q^95 - 1310720 * q^96 - 11913436 * q^97 + 23432624 * q^98 + 16388620 * 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 −42.3772 73.3994i −32.0000 55.4256i 46.6582 + 80.8144i −678.035 −404.125 699.965i −512.000 −2498.15 + 4326.92i 746.531
47.2 4.00000 6.92820i −35.3899 61.2970i −32.0000 55.4256i −144.990 251.131i −566.238 577.803 + 1000.78i −512.000 −1411.38 + 2444.59i −2319.84
47.3 4.00000 6.92820i −27.0658 46.8793i −32.0000 55.4256i 61.0458 + 105.734i −433.053 612.092 + 1060.17i −512.000 −371.614 + 643.655i 976.733
47.4 4.00000 6.92820i −17.1122 29.6391i −32.0000 55.4256i 219.368 + 379.957i −273.795 −84.9006 147.052i −512.000 507.848 879.619i 3509.89
47.5 4.00000 6.92820i −15.9241 27.5813i −32.0000 55.4256i −188.831 327.065i −254.785 −701.348 1214.77i −512.000 586.348 1015.58i −3021.29
47.6 4.00000 6.92820i 1.38293 + 2.39530i −32.0000 55.4256i −125.036 216.569i 22.1268 133.402 + 231.058i −512.000 1089.68 1887.37i −2000.58
47.7 4.00000 6.92820i 7.34337 + 12.7191i −32.0000 55.4256i 179.527 + 310.950i 117.494 −211.144 365.712i −512.000 985.650 1707.20i 2872.43
47.8 4.00000 6.92820i 8.02885 + 13.9064i −32.0000 55.4256i −60.7750 105.265i 128.462 462.740 + 801.489i −512.000 964.575 1670.69i −972.400
47.9 4.00000 6.92820i 27.4375 + 47.5231i −32.0000 55.4256i 8.21796 + 14.2339i 439.000 −392.558 679.931i −512.000 −412.133 + 713.835i 131.487
47.10 4.00000 6.92820i 34.1855 + 59.2110i −32.0000 55.4256i 13.0513 + 22.6056i 546.968 −837.864 1451.22i −512.000 −1243.79 + 2154.31i 208.821
47.11 4.00000 6.92820i 35.4964 + 61.4816i −32.0000 55.4256i 163.203 + 282.676i 567.942 755.643 + 1308.81i −512.000 −1426.49 + 2470.75i 2611.25
47.12 4.00000 6.92820i 43.9945 + 76.2007i −32.0000 55.4256i −239.439 414.721i 703.912 358.261 + 620.527i −512.000 −2777.54 + 4810.83i −3831.03
63.1 4.00000 + 6.92820i −42.3772 + 73.3994i −32.0000 + 55.4256i 46.6582 80.8144i −678.035 −404.125 + 699.965i −512.000 −2498.15 4326.92i 746.531
63.2 4.00000 + 6.92820i −35.3899 + 61.2970i −32.0000 + 55.4256i −144.990 + 251.131i −566.238 577.803 1000.78i −512.000 −1411.38 2444.59i −2319.84
63.3 4.00000 + 6.92820i −27.0658 + 46.8793i −32.0000 + 55.4256i 61.0458 105.734i −433.053 612.092 1060.17i −512.000 −371.614 643.655i 976.733
63.4 4.00000 + 6.92820i −17.1122 + 29.6391i −32.0000 + 55.4256i 219.368 379.957i −273.795 −84.9006 + 147.052i −512.000 507.848 + 879.619i 3509.89
63.5 4.00000 + 6.92820i −15.9241 + 27.5813i −32.0000 + 55.4256i −188.831 + 327.065i −254.785 −701.348 + 1214.77i −512.000 586.348 + 1015.58i −3021.29
63.6 4.00000 + 6.92820i 1.38293 2.39530i −32.0000 + 55.4256i −125.036 + 216.569i 22.1268 133.402 231.058i −512.000 1089.68 + 1887.37i −2000.58
63.7 4.00000 + 6.92820i 7.34337 12.7191i −32.0000 + 55.4256i 179.527 310.950i 117.494 −211.144 + 365.712i −512.000 985.650 + 1707.20i 2872.43
63.8 4.00000 + 6.92820i 8.02885 13.9064i −32.0000 + 55.4256i −60.7750 + 105.265i 128.462 462.740 801.489i −512.000 964.575 + 1670.69i −972.400
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 47.12 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.b 24
37.c even 3 1 inner 74.8.c.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.8.c.b 24 1.a even 1 1 trivial
74.8.c.b 24 37.c even 3 1 inner