# Properties

 Label 74.8.b.a Level $74$ Weight $8$ Character orbit 74.b 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(73,74)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("74.73");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 74.b (of order $$2$$, degree $$1$$, 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: $$24$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 106 q^{3} - 1536 q^{4} + 104 q^{7} + 17554 q^{9}+O(q^{10})$$ 24 * q - 106 * q^3 - 1536 * q^4 + 104 * q^7 + 17554 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 106 q^{3} - 1536 q^{4} + 104 q^{7} + 17554 q^{9} + 1136 q^{10} + 366 q^{11} + 6784 q^{12} + 98304 q^{16} - 239820 q^{21} - 675570 q^{25} + 97008 q^{26} + 338780 q^{27} - 6656 q^{28} + 350400 q^{30} - 763792 q^{33} + 713632 q^{34} - 1123456 q^{36} + 41652 q^{37} - 30816 q^{38} - 72704 q^{40} + 1729722 q^{41} - 23424 q^{44} - 488496 q^{46} + 114756 q^{47} - 434176 q^{48} + 4003056 q^{49} - 1115964 q^{53} - 2075632 q^{58} + 3248208 q^{62} - 2350900 q^{63} - 6291456 q^{64} - 5246556 q^{65} - 2717994 q^{67} - 5649440 q^{70} + 10643280 q^{71} - 10450370 q^{73} - 1064064 q^{74} - 17737980 q^{75} + 11665236 q^{77} + 8431856 q^{78} + 47300176 q^{81} + 555912 q^{83} + 15348480 q^{84} - 4853096 q^{85} - 12070464 q^{86} + 32064160 q^{90} - 58516956 q^{95} - 53279900 q^{99}+O(q^{100})$$ 24 * q - 106 * q^3 - 1536 * q^4 + 104 * q^7 + 17554 * q^9 + 1136 * q^10 + 366 * q^11 + 6784 * q^12 + 98304 * q^16 - 239820 * q^21 - 675570 * q^25 + 97008 * q^26 + 338780 * q^27 - 6656 * q^28 + 350400 * q^30 - 763792 * q^33 + 713632 * q^34 - 1123456 * q^36 + 41652 * q^37 - 30816 * q^38 - 72704 * q^40 + 1729722 * q^41 - 23424 * q^44 - 488496 * q^46 + 114756 * q^47 - 434176 * q^48 + 4003056 * q^49 - 1115964 * q^53 - 2075632 * q^58 + 3248208 * q^62 - 2350900 * q^63 - 6291456 * q^64 - 5246556 * q^65 - 2717994 * q^67 - 5649440 * q^70 + 10643280 * q^71 - 10450370 * q^73 - 1064064 * q^74 - 17737980 * q^75 + 11665236 * q^77 + 8431856 * q^78 + 47300176 * q^81 + 555912 * q^83 + 15348480 * q^84 - 4853096 * q^85 - 12070464 * q^86 + 32064160 * q^90 - 58516956 * q^95 - 53279900 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1 8.00000i −91.9686 −64.0000 130.944i 735.749i 433.376 512.000i 6271.23 −1047.55
73.2 8.00000i −61.7342 −64.0000 552.363i 493.873i 606.238 512.000i 1624.11 4418.91
73.3 8.00000i −56.1340 −64.0000 124.677i 449.072i −681.195 512.000i 964.026 −997.419
73.4 8.00000i −35.5669 −64.0000 301.873i 284.535i 1525.43 512.000i −921.994 −2414.98
73.5 8.00000i −31.2675 −64.0000 179.532i 250.140i −1585.88 512.000i −1209.34 1436.25
73.6 8.00000i −28.7630 −64.0000 436.339i 230.104i −218.580 512.000i −1359.69 −3490.71
73.7 8.00000i −0.654487 −64.0000 54.2779i 5.23590i 1545.01 512.000i −2186.57 434.223
73.8 8.00000i 14.3858 −64.0000 303.371i 115.086i 76.6627 512.000i −1980.05 2426.97
73.9 8.00000i 34.9288 −64.0000 125.706i 279.431i −166.083 512.000i −966.977 1005.65
73.10 8.00000i 39.3436 −64.0000 464.624i 314.749i −1250.90 512.000i −639.083 −3716.99
73.11 8.00000i 77.9352 −64.0000 194.832i 623.482i 874.611 512.000i 3886.90 −1558.66
73.12 8.00000i 86.4953 −64.0000 509.039i 691.963i −1106.69 512.000i 5294.44 4072.31
73.13 8.00000i −91.9686 −64.0000 130.944i 735.749i 433.376 512.000i 6271.23 −1047.55
73.14 8.00000i −61.7342 −64.0000 552.363i 493.873i 606.238 512.000i 1624.11 4418.91
73.15 8.00000i −56.1340 −64.0000 124.677i 449.072i −681.195 512.000i 964.026 −997.419
73.16 8.00000i −35.5669 −64.0000 301.873i 284.535i 1525.43 512.000i −921.994 −2414.98
73.17 8.00000i −31.2675 −64.0000 179.532i 250.140i −1585.88 512.000i −1209.34 1436.25
73.18 8.00000i −28.7630 −64.0000 436.339i 230.104i −218.580 512.000i −1359.69 −3490.71
73.19 8.00000i −0.654487 −64.0000 54.2779i 5.23590i 1545.01 512.000i −2186.57 434.223
73.20 8.00000i 14.3858 −64.0000 303.371i 115.086i 76.6627 512.000i −1980.05 2426.97
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 73.24 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.8.b.a 24
37.b even 2 1 inner 74.8.b.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.8.b.a 24 1.a even 1 1 trivial
74.8.b.a 24 37.b even 2 1 inner

## Hecke kernels

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