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Properties

Label 37.5.d.a
Level $37$
Weight $5$
Character orbit 37.d
Analytic conductor $3.825$
Analytic rank $0$
Dimension $22$
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(6,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.6");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 37.d (of order \(4\), 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: \(3.82468863410\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 4 q^{2} + 4 q^{5} + 30 q^{6} - 4 q^{7} + 24 q^{8} - 522 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 4 q^{2} + 4 q^{5} + 30 q^{6} - 4 q^{7} + 24 q^{8} - 522 q^{9} - 196 q^{10} - 320 q^{12} + 392 q^{13} + 434 q^{14} + 874 q^{15} + 360 q^{16} - 596 q^{17} - 1722 q^{18} + 646 q^{19} + 1684 q^{20} - 946 q^{22} + 748 q^{23} + 1428 q^{24} + 188 q^{26} - 2 q^{29} - 2612 q^{31} + 1288 q^{32} - 1048 q^{33} - 6900 q^{34} - 262 q^{35} - 2958 q^{37} + 2312 q^{38} + 6302 q^{39} + 3378 q^{42} - 8170 q^{43} + 7536 q^{44} - 10946 q^{45} + 17744 q^{46} - 2440 q^{47} - 7394 q^{49} - 7112 q^{50} + 11502 q^{51} - 1708 q^{52} + 6392 q^{53} + 7966 q^{54} - 9886 q^{55} + 3096 q^{56} + 2006 q^{57} + 8380 q^{59} - 13404 q^{60} + 838 q^{61} - 2392 q^{63} - 21058 q^{66} - 2960 q^{68} - 25432 q^{69} - 7556 q^{70} + 17780 q^{71} - 12104 q^{72} + 5476 q^{74} + 39284 q^{75} + 16468 q^{76} - 1042 q^{79} - 6404 q^{80} + 45278 q^{81} - 22550 q^{82} + 16292 q^{83} + 60004 q^{84} - 48892 q^{86} + 6852 q^{87} + 30740 q^{88} - 8900 q^{89} - 30604 q^{90} - 4238 q^{91} - 45068 q^{92} - 1336 q^{93} - 6018 q^{94} - 26832 q^{96} - 48346 q^{97} - 11490 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −4.79951 + 4.79951i − 9.08864i − 30.0706i 28.7458 + 28.7458i 43.6210 + 43.6210i −27.0960 67.5321 + 67.5321i −1.60340 −275.932
6.2 −4.49357 + 4.49357i 1.11295i − 24.3844i −20.0629 − 20.0629i −5.00112 − 5.00112i 12.9592 37.6759 + 37.6759i 79.7613 180.308
6.3 −2.96882 + 2.96882i 12.0524i − 1.62777i 12.4327 + 12.4327i −35.7815 − 35.7815i 3.96994 −42.6685 − 42.6685i −64.2611 −73.8209
6.4 −1.66963 + 1.66963i − 12.0727i 10.4247i −10.5974 − 10.5974i 20.1569 + 20.1569i −70.1035 −44.1194 − 44.1194i −64.7500 35.3874
6.5 −1.06504 + 1.06504i − 7.18291i 13.7314i 4.77919 + 4.77919i 7.65009 + 7.65009i 92.9327 −31.6651 − 31.6651i 29.4058 −10.1801
6.6 0.0187538 − 0.0187538i 10.3409i 15.9993i −25.3535 − 25.3535i 0.193932 + 0.193932i −50.4781 0.600108 + 0.600108i −25.9349 −0.950947
6.7 1.54938 − 1.54938i 2.90201i 11.1988i 16.4199 + 16.4199i 4.49631 + 4.49631i −20.3621 42.1413 + 42.1413i 72.5784 50.8813
6.8 3.27015 − 3.27015i − 6.05522i − 5.38772i −31.3848 − 31.3848i −19.8014 − 19.8014i 28.5333 34.7037 + 34.7037i 44.3344 −205.266
6.9 3.32520 − 3.32520i − 17.6884i − 6.11389i 20.8291 + 20.8291i −58.8175 − 58.8175i −11.0392 32.8733 + 32.8733i −231.880 138.522
6.10 3.75580 − 3.75580i 16.1382i − 12.2121i −0.0666033 − 0.0666033i 60.6118 + 60.6118i 61.7579 14.2265 + 14.2265i −179.440 −0.500298
6.11 5.07729 − 5.07729i − 0.458614i − 35.5577i 6.25838 + 6.25838i −2.32852 − 2.32852i −23.0741 −99.3000 − 99.3000i 80.7897 63.5512
31.1 −4.79951 − 4.79951i 9.08864i 30.0706i 28.7458 − 28.7458i 43.6210 − 43.6210i −27.0960 67.5321 − 67.5321i −1.60340 −275.932
31.2 −4.49357 − 4.49357i − 1.11295i 24.3844i −20.0629 + 20.0629i −5.00112 + 5.00112i 12.9592 37.6759 − 37.6759i 79.7613 180.308
31.3 −2.96882 − 2.96882i − 12.0524i 1.62777i 12.4327 − 12.4327i −35.7815 + 35.7815i 3.96994 −42.6685 + 42.6685i −64.2611 −73.8209
31.4 −1.66963 − 1.66963i 12.0727i − 10.4247i −10.5974 + 10.5974i 20.1569 − 20.1569i −70.1035 −44.1194 + 44.1194i −64.7500 35.3874
31.5 −1.06504 − 1.06504i 7.18291i − 13.7314i 4.77919 − 4.77919i 7.65009 − 7.65009i 92.9327 −31.6651 + 31.6651i 29.4058 −10.1801
31.6 0.0187538 + 0.0187538i − 10.3409i − 15.9993i −25.3535 + 25.3535i 0.193932 − 0.193932i −50.4781 0.600108 − 0.600108i −25.9349 −0.950947
31.7 1.54938 + 1.54938i − 2.90201i − 11.1988i 16.4199 − 16.4199i 4.49631 − 4.49631i −20.3621 42.1413 − 42.1413i 72.5784 50.8813
31.8 3.27015 + 3.27015i 6.05522i 5.38772i −31.3848 + 31.3848i −19.8014 + 19.8014i 28.5333 34.7037 − 34.7037i 44.3344 −205.266
31.9 3.32520 + 3.32520i 17.6884i 6.11389i 20.8291 − 20.8291i −58.8175 + 58.8175i −11.0392 32.8733 − 32.8733i −231.880 138.522
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.11
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.5.d.a ✓ 22
37.d odd 4 1 inner 37.5.d.a ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.5.d.a ✓ 22 1.a even 1 1 trivial
37.5.d.a ✓ 22 37.d odd 4 1 inner

Hecke kernels

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