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Properties

Label 74.6.e.a
Level $74$
Weight $6$
Character orbit 74.e
Analytic conductor $11.868$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [74,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.11");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(11.8684026662\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 20 q^{3} + 224 q^{4} - 90 q^{5} - 156 q^{7} - 586 q^{9} - 736 q^{10} + 464 q^{11} + 320 q^{12} + 72 q^{13} + 3960 q^{15} - 3584 q^{16} + 1530 q^{17} - 1944 q^{19} - 1440 q^{20} - 192 q^{21} - 3028 q^{25}+ \cdots - 401224 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −3.46410 + 2.00000i −15.4275 + 26.7212i 8.00000 − 13.8564i −17.2671 − 9.96916i − 123.420i 41.1449 − 71.2650i 64.0000i −354.516 − 614.041i 79.7533
11.2 −3.46410 + 2.00000i −6.30602 + 10.9223i 8.00000 − 13.8564i 35.8827 + 20.7169i − 50.4481i −39.6912 + 68.7472i 64.0000i 41.9683 + 72.6912i −165.735
11.3 −3.46410 + 2.00000i −5.84366 + 10.1215i 8.00000 − 13.8564i 5.13154 + 2.96270i − 46.7492i −44.9330 + 77.8262i 64.0000i 53.2034 + 92.1510i −23.7016
11.4 −3.46410 + 2.00000i −1.00232 + 1.73606i 8.00000 − 13.8564i −82.7641 − 47.7839i − 8.01854i 45.7270 − 79.2015i 64.0000i 119.491 + 206.964i 382.271
11.5 −3.46410 + 2.00000i 7.51352 − 13.0138i 8.00000 − 13.8564i −14.6785 − 8.47462i 60.1081i −106.799 + 184.981i 64.0000i 8.59417 + 14.8855i 67.7969
11.6 −3.46410 + 2.00000i 7.85694 − 13.6086i 8.00000 − 13.8564i −3.80965 − 2.19950i 62.8556i 79.6803 − 138.010i 64.0000i −1.96314 − 3.40026i 17.5960
11.7 −3.46410 + 2.00000i 8.20905 − 14.2185i 8.00000 − 13.8564i 94.8422 + 54.7572i 65.6724i −14.1291 + 24.4724i 64.0000i −13.2769 − 22.9963i −438.057
11.8 3.46410 − 2.00000i −13.1695 + 22.8102i 8.00000 − 13.8564i 6.19329 + 3.57570i 105.356i −65.8065 + 113.980i − 64.0000i −225.371 − 390.354i 28.6056
11.9 3.46410 − 2.00000i −10.1358 + 17.5557i 8.00000 − 13.8564i −38.3325 − 22.1313i 81.0863i 82.3063 − 142.559i − 64.0000i −83.9682 − 145.437i −177.050
11.10 3.46410 − 2.00000i −1.15082 + 1.99328i 8.00000 − 13.8564i 49.1611 + 28.3832i 9.20657i −34.8611 + 60.3812i − 64.0000i 118.851 + 205.856i 227.065
11.11 3.46410 − 2.00000i −0.787837 + 1.36457i 8.00000 − 13.8564i −44.3176 − 25.5868i 6.30270i −116.697 + 202.124i − 64.0000i 120.259 + 208.294i −204.694
11.12 3.46410 − 2.00000i −0.156146 + 0.270453i 8.00000 − 13.8564i −53.2403 − 30.7383i 1.24917i 45.8234 − 79.3684i − 64.0000i 121.451 + 210.360i −245.906
11.13 3.46410 − 2.00000i 7.72221 − 13.3753i 8.00000 − 13.8564i 53.6293 + 30.9629i − 61.7777i 57.7242 − 99.9812i − 64.0000i 2.23485 + 3.87087i 247.703
11.14 3.46410 − 2.00000i 12.6779 − 21.9587i 8.00000 − 13.8564i −35.4304 − 20.4557i − 101.423i −7.48959 + 12.9724i − 64.0000i −199.957 − 346.335i −163.646
27.1 −3.46410 − 2.00000i −15.4275 − 26.7212i 8.00000 + 13.8564i −17.2671 + 9.96916i 123.420i 41.1449 + 71.2650i − 64.0000i −354.516 + 614.041i 79.7533
27.2 −3.46410 − 2.00000i −6.30602 − 10.9223i 8.00000 + 13.8564i 35.8827 − 20.7169i 50.4481i −39.6912 − 68.7472i − 64.0000i 41.9683 − 72.6912i −165.735
27.3 −3.46410 − 2.00000i −5.84366 − 10.1215i 8.00000 + 13.8564i 5.13154 − 2.96270i 46.7492i −44.9330 − 77.8262i − 64.0000i 53.2034 − 92.1510i −23.7016
27.4 −3.46410 − 2.00000i −1.00232 − 1.73606i 8.00000 + 13.8564i −82.7641 + 47.7839i 8.01854i 45.7270 + 79.2015i − 64.0000i 119.491 − 206.964i 382.271
27.5 −3.46410 − 2.00000i 7.51352 + 13.0138i 8.00000 + 13.8564i −14.6785 + 8.47462i − 60.1081i −106.799 − 184.981i − 64.0000i 8.59417 − 14.8855i 67.7969
27.6 −3.46410 − 2.00000i 7.85694 + 13.6086i 8.00000 + 13.8564i −3.80965 + 2.19950i − 62.8556i 79.6803 + 138.010i − 64.0000i −1.96314 + 3.40026i 17.5960
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.14
Significant digits:
Format:

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 74.6.e.a ✓ 28
37.e even 6 1 inner 74.6.e.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.6.e.a ✓ 28 1.a even 1 1 trivial
74.6.e.a ✓ 28 37.e even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(74, [\chi])\).