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Properties

Label 37.6.f.a
Level $37$
Weight $6$
Character orbit 37.f
Analytic conductor $5.934$
Analytic rank $0$
Dimension $84$
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] = [37,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.7");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(5.93420133308\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 9 q^{2} - 3 q^{3} - 39 q^{4} - 30 q^{5} - 12 q^{6} - 213 q^{7} - 846 q^{8} + 87 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - 9 q^{2} - 3 q^{3} - 39 q^{4} - 30 q^{5} - 12 q^{6} - 213 q^{7} - 846 q^{8} + 87 q^{9} + 597 q^{10} + 1449 q^{11} + 1629 q^{12} - 651 q^{13} + 1317 q^{14} - 1611 q^{15} + 2169 q^{16} + 2736 q^{17} + 3012 q^{18} - 1950 q^{19} + 2328 q^{20} - 6018 q^{21} - 8025 q^{22} - 201 q^{23} - 26265 q^{24} - 1122 q^{25} + 5001 q^{26} + 13053 q^{27} + 13005 q^{28} + 3060 q^{29} - 33660 q^{30} + 43572 q^{31} - 9378 q^{32} + 33126 q^{33} - 18177 q^{34} - 37644 q^{35} - 23340 q^{36} + 7650 q^{37} - 8142 q^{38} + 38220 q^{39} + 14247 q^{40} - 16005 q^{41} - 19485 q^{42} + 97800 q^{43} - 60804 q^{44} + 35646 q^{45} + 35346 q^{46} + 66900 q^{47} - 62859 q^{48} + 65319 q^{49} - 177513 q^{50} - 12273 q^{51} + 18411 q^{52} - 61809 q^{53} - 53457 q^{54} + 84612 q^{55} - 86469 q^{56} - 148401 q^{57} + 276405 q^{58} - 219264 q^{59} + 88500 q^{60} + 174900 q^{61} - 8304 q^{62} + 130329 q^{63} + 188808 q^{64} + 5769 q^{65} + 74199 q^{66} + 121734 q^{67} + 304818 q^{68} - 417528 q^{69} - 79206 q^{70} - 108495 q^{71} - 206763 q^{72} + 103104 q^{73} - 451071 q^{74} - 261474 q^{75} - 339435 q^{76} + 306609 q^{77} - 470235 q^{78} + 176982 q^{79} + 141744 q^{80} + 159411 q^{81} + 156621 q^{82} - 13008 q^{83} + 357384 q^{84} + 220143 q^{85} + 225768 q^{86} - 468033 q^{87} + 402342 q^{88} + 125400 q^{89} - 186342 q^{90} - 501915 q^{91} + 304668 q^{92} - 280458 q^{93} - 770124 q^{94} + 603801 q^{95} + 396777 q^{96} + 252795 q^{97} + 506091 q^{98} - 458664 q^{99}+O(q^{100}) \) 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 −9.22175 + 3.35644i 20.4343 + 7.43749i 49.2615 − 41.3353i −2.52359 − 14.3120i −213.404 −35.0759 − 198.925i −158.520 + 274.565i 176.097 + 147.763i 71.3091 + 123.511i
7.2 −8.80438 + 3.20453i −1.91649 − 0.697546i 42.7347 − 35.8587i 11.7475 + 66.6231i 19.1088 30.6173 + 173.639i −111.431 + 193.005i −182.962 − 153.524i −316.925 − 548.930i
7.3 −8.56478 + 3.11732i −13.7002 − 4.98648i 39.1243 − 32.8292i −8.84524 − 50.1639i 132.884 −5.82994 − 33.0633i −86.9207 + 150.551i −23.3175 − 19.5657i 232.135 + 402.069i
7.4 −4.32081 + 1.57265i 11.3185 + 4.11958i −8.31725 + 6.97900i −17.6529 − 100.115i −55.3835 25.9582 + 147.216i 98.5315 − 170.662i −75.0124 − 62.9429i 233.720 + 404.814i
7.5 −4.30890 + 1.56831i −9.50932 − 3.46111i −8.40641 + 7.05382i 11.3007 + 64.0892i 46.4028 −15.5620 − 88.2564i 98.5268 − 170.653i −107.701 − 90.3718i −149.205 − 258.431i
7.6 −4.11172 + 1.49654i 18.9047 + 6.88073i −9.84685 + 8.26249i 5.37663 + 30.4924i −88.0279 5.54212 + 31.4309i 98.1318 − 169.969i 123.893 + 103.958i −67.7404 − 117.330i
7.7 −2.56700 + 0.934311i −26.1362 − 9.51279i −18.7969 + 15.7724i −4.25415 − 24.1265i 75.9795 9.79160 + 55.5309i 77.2231 − 133.754i 406.458 + 341.059i 33.4621 + 57.9580i
7.8 1.13742 − 0.413986i −0.0929136 − 0.0338178i −23.3911 + 19.6275i −6.50032 − 36.8652i −0.119681 −32.3960 − 183.727i −37.8465 + 65.5521i −186.141 − 156.191i −22.6552 − 39.2400i
7.9 2.89803 − 1.05480i −6.79254 − 2.47228i −17.2274 + 14.4555i −2.44498 − 13.8662i −22.2927 42.6160 + 241.688i −84.0223 + 145.531i −146.122 − 122.611i −21.7116 − 37.6057i
7.10 3.32466 − 1.21008i 17.0676 + 6.21210i −14.9243 + 12.5230i 11.1750 + 63.3765i 64.2612 4.06001 + 23.0255i −91.0731 + 157.743i 66.5640 + 55.8539i 113.844 + 197.183i
7.11 5.55021 − 2.02011i −21.3210 − 7.76022i 2.21057 − 1.85489i 16.7997 + 95.2758i −134.013 −9.90823 − 56.1923i −85.9805 + 148.923i 208.216 + 174.714i 285.710 + 494.863i
7.12 6.07724 − 2.21194i 26.7337 + 9.73027i 7.52680 − 6.31573i −15.6640 − 88.8349i 183.990 −7.48537 − 42.4516i −71.7041 + 124.195i 433.863 + 364.054i −291.691 − 505.223i
7.13 7.92295 − 2.88372i −16.0569 − 5.84422i 29.9439 − 25.1259i −10.7255 − 60.8272i −144.071 −7.15427 − 40.5739i 29.8851 − 51.7625i 37.5190 + 31.4822i −260.386 − 451.001i
7.14 9.13765 − 3.32583i 8.08436 + 2.94247i 47.9221 − 40.2114i 4.50430 + 25.5452i 83.6582 2.62190 + 14.8695i 148.573 − 257.337i −129.450 − 108.622i 126.118 + 218.442i
9.1 −1.73586 − 9.84454i −2.77496 + 15.7376i −63.8316 + 23.2328i 6.07450 + 5.09711i 159.746 96.9618 + 81.3607i 179.577 + 311.036i −11.6264 − 4.23168i 39.6343 − 68.6486i
9.2 −1.65307 − 9.37502i 2.86368 − 16.2407i −55.0883 + 20.0505i 1.85987 + 1.56062i −156.991 −84.6976 − 71.0698i 126.724 + 219.493i −27.2148 − 9.90537i 11.5563 − 20.0162i
9.3 −1.00409 − 5.69446i 2.32868 − 13.2066i −1.34847 + 0.490803i −54.2442 − 45.5163i −77.5426 72.3721 + 60.7274i −88.3680 − 153.058i 59.3537 + 21.6030i −204.725 + 354.593i
9.4 −0.998741 − 5.66414i −3.64412 + 20.6668i −1.01484 + 0.369370i −23.7764 − 19.9508i 120.699 −153.664 − 128.939i −88.9186 − 154.011i −185.492 − 67.5136i −89.2576 + 154.599i
9.5 −0.904128 − 5.12756i 0.115410 − 0.654522i 4.59570 − 1.67270i 77.6592 + 65.1638i −3.46045 20.4602 + 17.1681i −96.0386 − 166.344i 227.930 + 82.9598i 263.918 − 457.119i
9.6 −0.244825 − 1.38847i −1.87852 + 10.6536i 28.2022 − 10.2648i −30.4360 − 25.5388i 15.2522 132.908 + 111.523i −43.7152 − 75.7170i 118.374 + 43.0847i −28.0085 + 48.5121i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.14
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 37.6.f.a ✓ 84
37.f even 9 1 inner 37.6.f.a ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.6.f.a ✓ 84 1.a even 1 1 trivial
37.6.f.a ✓ 84 37.f even 9 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(37, [\chi])\).