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Properties

Label 37.4.f.a
Level $37$
Weight $4$
Character orbit 37.f
Analytic conductor $2.183$
Analytic rank $0$
Dimension $54$
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] = [37,4,Mod(7,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([16]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.7");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 37.f (of order \(9\), degree \(6\), 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: \(2.18307067021\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(9\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 54 q - 9 q^{2} - 12 q^{3} + 9 q^{4} + 6 q^{5} - 12 q^{6} + 96 q^{7} + 18 q^{8} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q - 9 q^{2} - 12 q^{3} + 9 q^{4} + 6 q^{5} - 12 q^{6} + 96 q^{7} + 18 q^{8} + 42 q^{9} - 63 q^{10} - 135 q^{11} - 315 q^{12} + 258 q^{13} - 15 q^{14} + 108 q^{15} - 87 q^{16} - 72 q^{17} - 660 q^{18} + 156 q^{19} + 636 q^{20} - 240 q^{21} + 63 q^{22} - 255 q^{23} + 87 q^{24} + 618 q^{25} - 417 q^{26} - 1095 q^{27} - 483 q^{28} - 63 q^{29} - 540 q^{30} - 708 q^{31} + 2502 q^{32} + 339 q^{33} + 633 q^{34} + 192 q^{35} + 2364 q^{36} + 1812 q^{37} + 210 q^{38} + 834 q^{39} - 465 q^{40} + 1050 q^{41} - 2061 q^{42} + 1056 q^{43} + 684 q^{44} - 1479 q^{45} - 3570 q^{46} - 1635 q^{47} - 1875 q^{48} - 2196 q^{49} + 795 q^{50} + 1047 q^{51} - 1389 q^{52} - 420 q^{53} + 1335 q^{54} - 621 q^{55} + 5403 q^{56} + 2196 q^{57} - 4293 q^{58} - 2130 q^{59} - 6612 q^{60} - 3222 q^{61} - 240 q^{62} - 1143 q^{63} - 2088 q^{64} - 3330 q^{65} + 5007 q^{66} + 522 q^{67} - 3054 q^{68} + 7227 q^{69} + 9978 q^{70} - 1656 q^{71} + 381 q^{72} + 1968 q^{73} + 7515 q^{74} + 6996 q^{75} + 2229 q^{76} + 843 q^{77} + 1329 q^{78} - 2508 q^{79} + 10992 q^{80} + 723 q^{81} + 141 q^{82} + 3048 q^{83} - 4920 q^{84} - 4365 q^{85} - 5496 q^{86} - 663 q^{87} + 6438 q^{88} - 2436 q^{89} - 17880 q^{90} - 7794 q^{91} - 11196 q^{92} + 3564 q^{93} - 708 q^{94} - 5922 q^{95} - 9663 q^{96} - 4359 q^{97} - 1725 q^{98} + 9399 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −4.66153 + 1.69666i 5.32520 + 1.93821i 12.7229 − 10.6758i 0.0986035 + 0.559208i −28.1121 5.58163 + 31.6550i −21.3523 + 36.9832i 3.91790 + 3.28750i −1.40843 − 2.43947i
7.2 −4.64220 + 1.68962i −9.11793 − 3.31866i 12.5669 − 10.5448i 2.20425 + 12.5009i 47.9346 −2.19292 − 12.4366i −20.7605 + 35.9583i 51.4400 + 43.1633i −31.3544 − 54.3074i
7.3 −2.90338 + 1.05674i −0.568534 − 0.206929i 1.18455 − 0.993954i −1.93043 − 10.9480i 1.86934 −2.36329 − 13.4029i 9.97001 − 17.2686i −20.4028 − 17.1200i 17.1740 + 29.7462i
7.4 −0.520529 + 0.189457i −2.44355 − 0.889380i −5.89330 + 4.94507i 2.45216 + 13.9069i 1.44044 2.86977 + 16.2753i 4.34650 − 7.52835i −15.5033 − 13.0088i −3.91118 − 6.77436i
7.5 −0.449881 + 0.163743i 8.42342 + 3.06587i −5.95277 + 4.99497i 0.173844 + 0.985916i −4.29155 −0.934908 − 5.30213i 3.77516 − 6.53877i 40.8712 + 34.2950i −0.239646 − 0.415080i
7.6 1.15642 − 0.420901i −7.56068 − 2.75186i −4.96821 + 4.16883i −2.39934 − 13.6073i −9.90156 −1.98493 − 11.2571i −8.91319 + 15.4381i 28.9080 + 24.2567i −8.50199 − 14.7259i
7.7 3.26662 − 1.18895i 2.72365 + 0.991328i 3.12885 − 2.62542i −1.78901 − 10.1460i 10.0758 3.63308 + 20.6042i −6.80580 + 11.7880i −14.2476 − 11.9552i −17.9071 − 31.0159i
7.8 3.36470 − 1.22465i 2.39058 + 0.870100i 3.69310 − 3.09888i 1.93182 + 10.9559i 9.10917 −6.25019 − 35.4466i −5.69142 + 9.85783i −15.7254 − 13.1952i 19.9172 + 34.4976i
7.9 5.17677 − 1.88419i −7.22905 − 2.63116i 17.1205 − 14.3658i 1.19779 + 6.79301i −42.3808 3.26056 + 18.4916i 39.5248 − 68.4590i 24.6530 + 20.6863i 19.0000 + 32.9090i
9.1 −0.819858 − 4.64964i −0.377417 + 2.14044i −13.4295 + 4.88793i −14.7991 − 12.4179i 10.2617 2.57615 + 2.16164i 14.8519 + 25.7243i 20.9327 + 7.61887i −45.6058 + 78.9915i
9.2 −0.763041 − 4.32742i 1.56086 − 8.85207i −10.6268 + 3.86784i 8.55885 + 7.18172i −39.4977 26.9224 + 22.5906i 7.26976 + 12.5916i −50.5512 − 18.3991i 24.5476 − 42.5177i
9.3 −0.547370 − 3.10429i −1.50530 + 8.53698i −1.81945 + 0.662227i 9.73173 + 8.16589i 27.3252 7.61350 + 6.38848i −9.55705 − 16.5533i −45.2424 − 16.4669i 20.0224 − 34.6799i
9.4 −0.377881 − 2.14307i 0.499079 − 2.83042i 3.06758 − 1.11651i −0.311669 − 0.261521i −6.25437 −17.4581 − 14.6491i −12.2565 − 21.2288i 17.6095 + 6.40934i −0.442684 + 0.766752i
9.5 0.0242210 + 0.137364i −0.232052 + 1.31603i 7.49926 − 2.72951i 2.32000 + 1.94671i −0.186396 10.6043 + 8.89810i 1.11451 + 1.93038i 23.6936 + 8.62377i −0.211215 + 0.365835i
9.6 0.324898 + 1.84259i 1.31718 − 7.47007i 4.22796 − 1.53885i −16.1799 − 13.5766i 14.1922 14.7830 + 12.4044i 11.6932 + 20.2532i −28.6953 − 10.4442i 19.7592 − 34.2239i
9.7 0.516618 + 2.92989i −1.06977 + 6.06697i −0.799793 + 0.291101i −1.62838 − 1.36638i −18.3282 −10.1228 − 8.49404i 10.6343 + 18.4191i −10.2920 − 3.74599i 3.16207 − 5.47687i
9.8 0.570453 + 3.23520i 1.27671 − 7.24060i −2.62358 + 0.954904i 14.7707 + 12.3941i 24.1531 −17.5700 − 14.7430i 8.55451 + 14.8168i −25.4247 − 9.25382i −31.6713 + 54.8564i
9.9 0.930400 + 5.27656i 0.0443828 − 0.251707i −19.4589 + 7.08246i −1.63583 − 1.37262i 1.36944 17.2717 + 14.4927i −34.0438 − 58.9656i 25.3103 + 9.21220i 5.72076 − 9.90865i
12.1 −4.19194 + 3.51746i −1.62324 − 1.36206i 3.81069 − 21.6115i −8.00579 + 2.91387i 11.5955 22.8525 − 8.31764i 38.1545 + 66.0856i −3.90880 − 22.1679i 23.3104 − 40.3748i
12.2 −3.00315 + 2.51994i 7.73107 + 6.48714i 1.27962 − 7.25709i 8.80505 − 3.20478i −39.5648 −6.68269 + 2.43230i −1.23679 − 2.14219i 12.9980 + 73.7153i −18.3671 + 31.8127i
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.4.f.a ✓ 54
37.f even 9 1 inner 37.4.f.a ✓ 54
37.f even 9 1 1369.4.a.h 27
37.h even 18 1 1369.4.a.i 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.4.f.a ✓ 54 1.a even 1 1 trivial
37.4.f.a ✓ 54 37.f even 9 1 inner
1369.4.a.h 27 37.f even 9 1
1369.4.a.i 27 37.h even 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(37, [\chi])\).