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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 $2$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 106 q^{3} - 1536 q^{4} + 104 q^{7} + 17554 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 106 q^{3} - 1536 q^{4} + 104 q^{7} + 17554 q^{9} + 1136 q^{10} + 366 q^{11} + 6784 q^{12} + 98304 q^{16} - 239820 q^{21} - 675570 q^{25} + 97008 q^{26} + 338780 q^{27} - 6656 q^{28} + 350400 q^{30} - 763792 q^{33} + 713632 q^{34} - 1123456 q^{36} + 41652 q^{37} - 30816 q^{38} - 72704 q^{40} + 1729722 q^{41} - 23424 q^{44} - 488496 q^{46} + 114756 q^{47} - 434176 q^{48} + 4003056 q^{49} - 1115964 q^{53} - 2075632 q^{58} + 3248208 q^{62} - 2350900 q^{63} - 6291456 q^{64} - 5246556 q^{65} - 2717994 q^{67} - 5649440 q^{70} + 10643280 q^{71} - 10450370 q^{73} - 1064064 q^{74} - 17737980 q^{75} + 11665236 q^{77} + 8431856 q^{78} + 47300176 q^{81} + 555912 q^{83} + 15348480 q^{84} - 4853096 q^{85} - 12070464 q^{86} + 32064160 q^{90} - 58516956 q^{95} - 53279900 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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:

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 \(S_{8}^{\mathrm{new}}(74, [\chi])\).