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Properties

Label 74.5.g.b
Level $74$
Weight $5$
Character orbit 74.g
Analytic conductor $7.649$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [74,5,Mod(23,74)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(74, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.23");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 74.g (of order \(12\), degree \(4\), 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: \(7.64937726820\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 28 q^{2} - 102 q^{5} + 40 q^{6} - 48 q^{7} + 448 q^{8} + 556 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 28 q^{2} - 102 q^{5} + 40 q^{6} - 48 q^{7} + 448 q^{8} + 556 q^{9} - 120 q^{10} + 80 q^{12} - 88 q^{13} - 192 q^{14} + 12 q^{15} + 896 q^{16} - 540 q^{17} - 1112 q^{18} + 976 q^{19} - 816 q^{20} - 618 q^{21} + 752 q^{22} + 1996 q^{23} - 160 q^{24} + 414 q^{25} - 2504 q^{26} + 1584 q^{28} + 852 q^{29} + 804 q^{30} + 300 q^{31} - 1792 q^{32} + 316 q^{33} + 300 q^{34} + 796 q^{35} - 3678 q^{37} - 328 q^{38} + 3630 q^{39} - 2784 q^{40} - 6426 q^{41} - 1348 q^{42} - 5548 q^{43} - 1280 q^{44} + 10164 q^{45} + 3992 q^{46} - 7624 q^{47} - 7388 q^{49} + 1768 q^{50} + 15382 q^{51} - 704 q^{52} + 13538 q^{53} - 1108 q^{54} - 10352 q^{55} + 2400 q^{56} - 12946 q^{57} + 3000 q^{58} + 5540 q^{59} - 3024 q^{60} - 26794 q^{61} + 2880 q^{62} - 61204 q^{63} + 39498 q^{65} + 1264 q^{66} + 23910 q^{67} + 1200 q^{68} + 19510 q^{69} + 524 q^{70} - 15992 q^{71} + 8896 q^{72} - 16780 q^{74} + 56644 q^{75} + 7808 q^{76} + 18960 q^{77} + 11892 q^{78} - 1792 q^{79} - 1920 q^{80} - 62170 q^{81} - 2128 q^{82} - 2516 q^{83} - 896 q^{84} - 11096 q^{86} + 39608 q^{87} - 5120 q^{88} + 25460 q^{89} + 20328 q^{90} - 20162 q^{91} - 20512 q^{92} + 8394 q^{93} - 31552 q^{94} + 7470 q^{95} + 1280 q^{96} - 12272 q^{97} + 14752 q^{98} + 23376 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.732051 − 2.73205i −13.9920 + 8.07829i −6.92820 + 4.00000i −9.30753 + 34.7362i 32.3132 + 32.3132i −30.9246 − 53.5629i 16.0000 + 16.0000i 90.0175 − 155.915i 101.715
23.2 −0.732051 − 2.73205i −9.27942 + 5.35748i −6.92820 + 4.00000i 1.47068 − 5.48866i 21.4299 + 21.4299i 20.7504 + 35.9408i 16.0000 + 16.0000i 16.9051 − 29.2805i −16.0719
23.3 −0.732051 − 2.73205i −7.91112 + 4.56749i −6.92820 + 4.00000i 9.46674 − 35.3304i 18.2700 + 18.2700i −10.9589 − 18.9814i 16.0000 + 16.0000i 1.22390 − 2.11986i −103.454
23.4 −0.732051 − 2.73205i 0.515191 − 0.297445i −6.92820 + 4.00000i −2.76875 + 10.3331i −1.18978 − 1.18978i −5.68857 − 9.85290i 16.0000 + 16.0000i −40.3231 + 69.8416i 30.2575
23.5 −0.732051 − 2.73205i 4.65870 − 2.68970i −6.92820 + 4.00000i 6.50685 − 24.2839i −10.7588 − 10.7588i −14.6009 − 25.2895i 16.0000 + 16.0000i −26.0310 + 45.0870i −71.1082
23.6 −0.732051 − 2.73205i 6.81857 − 3.93670i −6.92820 + 4.00000i −7.04485 + 26.2917i −15.7468 − 15.7468i 41.4796 + 71.8448i 16.0000 + 16.0000i −9.50474 + 16.4627i 76.9875
23.7 −0.732051 − 2.73205i 14.8600 − 8.57940i −6.92820 + 4.00000i −4.77059 + 17.8041i −34.3176 − 34.3176i −40.6359 − 70.3834i 16.0000 + 16.0000i 106.712 − 184.831i 52.1339
29.1 −0.732051 + 2.73205i −13.9920 − 8.07829i −6.92820 − 4.00000i −9.30753 − 34.7362i 32.3132 − 32.3132i −30.9246 + 53.5629i 16.0000 − 16.0000i 90.0175 + 155.915i 101.715
29.2 −0.732051 + 2.73205i −9.27942 − 5.35748i −6.92820 − 4.00000i 1.47068 + 5.48866i 21.4299 − 21.4299i 20.7504 − 35.9408i 16.0000 − 16.0000i 16.9051 + 29.2805i −16.0719
29.3 −0.732051 + 2.73205i −7.91112 − 4.56749i −6.92820 − 4.00000i 9.46674 + 35.3304i 18.2700 − 18.2700i −10.9589 + 18.9814i 16.0000 − 16.0000i 1.22390 + 2.11986i −103.454
29.4 −0.732051 + 2.73205i 0.515191 + 0.297445i −6.92820 − 4.00000i −2.76875 − 10.3331i −1.18978 + 1.18978i −5.68857 + 9.85290i 16.0000 − 16.0000i −40.3231 − 69.8416i 30.2575
29.5 −0.732051 + 2.73205i 4.65870 + 2.68970i −6.92820 − 4.00000i 6.50685 + 24.2839i −10.7588 + 10.7588i −14.6009 + 25.2895i 16.0000 − 16.0000i −26.0310 − 45.0870i −71.1082
29.6 −0.732051 + 2.73205i 6.81857 + 3.93670i −6.92820 − 4.00000i −7.04485 − 26.2917i −15.7468 + 15.7468i 41.4796 − 71.8448i 16.0000 − 16.0000i −9.50474 − 16.4627i 76.9875
29.7 −0.732051 + 2.73205i 14.8600 + 8.57940i −6.92820 − 4.00000i −4.77059 − 17.8041i −34.3176 + 34.3176i −40.6359 + 70.3834i 16.0000 − 16.0000i 106.712 + 184.831i 52.1339
45.1 2.73205 + 0.732051i −15.3929 − 8.88709i 6.92820 + 4.00000i −1.55696 + 0.417185i −35.5484 − 35.5484i −4.44075 + 7.69161i 16.0000 + 16.0000i 117.461 + 203.448i −4.55909
45.2 2.73205 + 0.732051i −6.14320 − 3.54678i 6.92820 + 4.00000i 13.7399 − 3.68160i −14.1871 − 14.1871i 17.6478 − 30.5669i 16.0000 + 16.0000i −15.3407 − 26.5709i 40.2333
45.3 2.73205 + 0.732051i −3.71174 − 2.14297i 6.92820 + 4.00000i −38.9095 + 10.4258i −8.57189 − 8.57189i 32.4545 − 56.2129i 16.0000 + 16.0000i −31.3153 − 54.2398i −113.935
45.4 2.73205 + 0.732051i −0.984681 − 0.568506i 6.92820 + 4.00000i −27.3382 + 7.32524i −2.27402 − 2.27402i −37.5434 + 65.0271i 16.0000 + 16.0000i −39.8536 − 69.0285i −80.0517
45.5 2.73205 + 0.732051i 6.81652 + 3.93552i 6.92820 + 4.00000i 31.3386 − 8.39715i 15.7421 + 15.7421i −18.6320 + 32.2716i 16.0000 + 16.0000i −9.52334 − 16.4949i 91.7657
45.6 2.73205 + 0.732051i 9.05180 + 5.22606i 6.92820 + 4.00000i 8.32031 − 2.22942i 20.9042 + 20.9042i 41.9891 − 72.7273i 16.0000 + 16.0000i 14.1234 + 24.4624i 24.3635
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.g odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.5.g.b ✓ 28
37.g odd 12 1 inner 74.5.g.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.5.g.b ✓ 28 1.a even 1 1 trivial
74.5.g.b ✓ 28 37.g odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} - 845 T_{3}^{26} + 452259 T_{3}^{24} - 5418 T_{3}^{23} - 146796902 T_{3}^{22} + \cdots + 34\!\cdots\!36 \) acting on \(S_{5}^{\mathrm{new}}(74, [\chi])\). Copy content Toggle raw display