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Properties

Label 37.5.g.a
Level $37$
Weight $5$
Character orbit 37.g
Analytic conductor $3.825$
Analytic rank $0$
Dimension $44$
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] = [37,5,Mod(8,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.8");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 37.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: \(3.82468863410\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(11\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 10 q^{2} - 6 q^{3} - 24 q^{4} + 98 q^{5} - 36 q^{6} - 2 q^{7} + 240 q^{8} + 306 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 10 q^{2} - 6 q^{3} - 24 q^{4} + 98 q^{5} - 36 q^{6} - 2 q^{7} + 240 q^{8} + 306 q^{9} + 184 q^{10} - 646 q^{12} - 398 q^{13} + 136 q^{14} - 82 q^{15} + 564 q^{16} - 130 q^{17} - 714 q^{18} - 796 q^{19} + 230 q^{20} + 732 q^{21} + 1036 q^{22} - 2752 q^{23} + 3186 q^{24} - 1770 q^{25} + 2680 q^{26} - 5574 q^{28} - 382 q^{29} - 7182 q^{30} + 3968 q^{31} - 1312 q^{32} - 524 q^{33} + 1710 q^{34} + 3658 q^{35} + 2910 q^{37} - 6896 q^{38} + 2548 q^{39} + 9978 q^{40} + 8220 q^{41} + 5448 q^{42} + 2920 q^{43} + 6792 q^{44} - 9610 q^{45} + 430 q^{46} - 1136 q^{47} - 4726 q^{49} + 6236 q^{50} + 9132 q^{51} - 27704 q^{52} - 4508 q^{53} - 2068 q^{54} + 8176 q^{55} - 16350 q^{56} + 3460 q^{57} + 1284 q^{58} + 4376 q^{59} + 8400 q^{60} + 27020 q^{61} - 20004 q^{62} - 16340 q^{63} - 14802 q^{65} + 3112 q^{66} - 29532 q^{67} + 19028 q^{68} - 17564 q^{69} - 13978 q^{70} - 5978 q^{71} - 7348 q^{72} + 2060 q^{74} - 36080 q^{75} + 27404 q^{76} + 13536 q^{77} + 45660 q^{78} - 19124 q^{79} + 44624 q^{80} + 33574 q^{81} - 6472 q^{82} + 20242 q^{83} - 14200 q^{84} + 18898 q^{86} + 46794 q^{87} - 14108 q^{88} - 52270 q^{89} - 27308 q^{90} + 50672 q^{91} + 53162 q^{92} - 23552 q^{93} + 79644 q^{94} + 13980 q^{95} - 72114 q^{96} + 6742 q^{97} + 77094 q^{98} - 72216 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −7.08279 − 1.89783i −0.547763 − 0.316251i 32.7077 + 18.8838i 12.4417 − 3.33374i 3.27950 + 3.27950i −4.56428 + 7.90557i −112.864 − 112.864i −40.3000 − 69.8016i −94.4487
8.2 −5.18181 − 1.38846i 12.4977 + 7.21555i 11.0669 + 6.38949i −15.5622 + 4.16988i −54.7422 − 54.7422i −10.2546 + 17.7615i 12.2184 + 12.2184i 63.6284 + 110.208i 86.4302
8.3 −4.85521 − 1.30095i −11.3098 − 6.52971i 8.02419 + 4.63277i −23.1849 + 6.21238i 46.4166 + 46.4166i 29.0699 − 50.3505i 23.9360 + 23.9360i 44.7743 + 77.5514i 120.650
8.4 −3.02030 − 0.809288i −8.80383 − 5.08290i −5.38912 − 3.11141i 28.4771 − 7.63041i 22.4767 + 22.4767i −48.9523 + 84.7878i 49.1350 + 49.1350i 11.1717 + 19.3499i −92.1847
8.5 −2.93969 − 0.787689i 4.37688 + 2.52700i −5.83506 − 3.36887i 10.5460 − 2.82580i −10.8762 − 10.8762i 28.0383 − 48.5638i 48.9318 + 48.9318i −27.7286 − 48.0273i −33.2279
8.6 −0.112513 − 0.0301478i 1.29148 + 0.745634i −13.8447 − 7.99322i −28.7245 + 7.69670i −0.122829 − 0.122829i −12.4901 + 21.6335i 2.63458 + 2.63458i −39.3881 − 68.2221i 3.46392
8.7 1.59940 + 0.428557i 12.4348 + 7.17921i −11.4820 − 6.62914i 32.1252 − 8.60791i 16.8114 + 16.8114i −20.2565 + 35.0852i −34.2567 − 34.2567i 62.5821 + 108.395i 55.0698
8.8 2.17412 + 0.582554i −5.52639 − 3.19066i −9.46898 − 5.46692i 33.8656 − 9.07426i −10.1563 − 10.1563i 37.3892 − 64.7600i −42.8670 − 42.8670i −20.1393 − 34.8824i 78.9141
8.9 3.54426 + 0.949682i −11.3758 − 6.56784i −2.19651 − 1.26815i −20.0253 + 5.36575i −34.0816 − 34.0816i −6.99556 + 12.1167i −48.0939 − 48.0939i 45.7729 + 79.2810i −76.0705
8.10 5.54224 + 1.48504i 7.08739 + 4.09191i 14.6547 + 8.46091i −12.9006 + 3.45671i 33.2034 + 33.2034i 8.16147 − 14.1361i 3.74002 + 3.74002i −7.01261 − 12.1462i −76.6317
8.11 6.96627 + 1.86661i −5.95473 − 3.43796i 31.1883 + 18.0066i 24.7624 − 6.63507i −35.0649 − 35.0649i −21.2963 + 36.8862i 102.061 + 102.061i −16.8608 − 29.2038i 184.887
14.1 −7.08279 + 1.89783i −0.547763 + 0.316251i 32.7077 − 18.8838i 12.4417 + 3.33374i 3.27950 − 3.27950i −4.56428 − 7.90557i −112.864 + 112.864i −40.3000 + 69.8016i −94.4487
14.2 −5.18181 + 1.38846i 12.4977 − 7.21555i 11.0669 − 6.38949i −15.5622 − 4.16988i −54.7422 + 54.7422i −10.2546 − 17.7615i 12.2184 − 12.2184i 63.6284 − 110.208i 86.4302
14.3 −4.85521 + 1.30095i −11.3098 + 6.52971i 8.02419 − 4.63277i −23.1849 − 6.21238i 46.4166 − 46.4166i 29.0699 + 50.3505i 23.9360 − 23.9360i 44.7743 − 77.5514i 120.650
14.4 −3.02030 + 0.809288i −8.80383 + 5.08290i −5.38912 + 3.11141i 28.4771 + 7.63041i 22.4767 − 22.4767i −48.9523 − 84.7878i 49.1350 − 49.1350i 11.1717 − 19.3499i −92.1847
14.5 −2.93969 + 0.787689i 4.37688 − 2.52700i −5.83506 + 3.36887i 10.5460 + 2.82580i −10.8762 + 10.8762i 28.0383 + 48.5638i 48.9318 − 48.9318i −27.7286 + 48.0273i −33.2279
14.6 −0.112513 + 0.0301478i 1.29148 − 0.745634i −13.8447 + 7.99322i −28.7245 − 7.69670i −0.122829 + 0.122829i −12.4901 − 21.6335i 2.63458 − 2.63458i −39.3881 + 68.2221i 3.46392
14.7 1.59940 − 0.428557i 12.4348 − 7.17921i −11.4820 + 6.62914i 32.1252 + 8.60791i 16.8114 − 16.8114i −20.2565 − 35.0852i −34.2567 + 34.2567i 62.5821 − 108.395i 55.0698
14.8 2.17412 − 0.582554i −5.52639 + 3.19066i −9.46898 + 5.46692i 33.8656 + 9.07426i −10.1563 + 10.1563i 37.3892 + 64.7600i −42.8670 + 42.8670i −20.1393 + 34.8824i 78.9141
14.9 3.54426 − 0.949682i −11.3758 + 6.56784i −2.19651 + 1.26815i −20.0253 − 5.36575i −34.0816 + 34.0816i −6.99556 − 12.1167i −48.0939 + 48.0939i 45.7729 − 79.2810i −76.0705
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.11
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 37.5.g.a ✓ 44
37.g odd 12 1 inner 37.5.g.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.5.g.a ✓ 44 1.a even 1 1 trivial
37.5.g.a ✓ 44 37.g odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(37, [\chi])\).