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Properties

Label 36.7.f.a
Level $36$
Weight $7$
Character orbit 36.f
Analytic conductor $8.282$
Analytic rank $0$
Dimension $68$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [36,7,Mod(7,36)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(36, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.7");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 36.f (of order \(6\), 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: \(8.28194701031\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q - q^{2} - q^{4} - 2 q^{5} - 321 q^{6} + 266 q^{8} + 456 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - q^{2} - q^{4} - 2 q^{5} - 321 q^{6} + 266 q^{8} + 456 q^{9} + 124 q^{10} + 5484 q^{12} - 2 q^{13} - 2604 q^{14} - q^{16} + 6832 q^{17} + 14496 q^{18} + 19972 q^{20} - 9486 q^{21} - 129 q^{22} - 27375 q^{24} - 81252 q^{25} + 74248 q^{26} - 8196 q^{28} + 8566 q^{29} - 20964 q^{30} - 6121 q^{32} - 110322 q^{33} - 20063 q^{34} - 18669 q^{36} - 8 q^{37} + 87507 q^{38} + 27124 q^{40} + 26098 q^{41} - 70638 q^{42} + 251130 q^{44} - 251658 q^{45} + 22296 q^{46} - 314445 q^{48} + 369752 q^{49} + 144063 q^{50} - 62954 q^{52} + 301096 q^{53} - 269943 q^{54} + 40014 q^{56} + 33828 q^{57} + 221392 q^{58} - 89664 q^{60} - 2 q^{61} + 626268 q^{62} - 804478 q^{64} + 283826 q^{65} + 11250 q^{66} + 317707 q^{68} + 740586 q^{69} + 148962 q^{70} - 800925 q^{72} + 356032 q^{73} - 380396 q^{74} - 423891 q^{76} - 854358 q^{77} - 477882 q^{78} - 1230608 q^{80} + 1927536 q^{81} - 965126 q^{82} + 430830 q^{84} - 31252 q^{85} - 554511 q^{86} + 536715 q^{88} - 4867208 q^{89} + 1120620 q^{90} + 778074 q^{92} + 1796442 q^{93} - 2055684 q^{94} + 1215924 q^{96} + 34378 q^{97} + 1937030 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −7.99624 − 0.245338i −26.1985 − 6.52978i 63.8796 + 3.92356i 47.8901 + 82.9481i 207.887 + 58.6412i 146.582 + 84.6293i −509.834 − 47.0458i 643.724 + 342.141i −362.590 − 675.022i
7.2 −7.99537 + 0.272017i −11.5565 + 24.4018i 63.8520 − 4.34975i −96.5692 − 167.263i 85.7612 − 198.245i 171.301 + 98.9006i −509.338 + 52.1467i −461.892 − 564.000i 817.606 + 1311.06i
7.3 −7.96538 + 0.743435i 20.0036 − 18.1344i 62.8946 − 11.8435i 65.0469 + 112.665i −145.854 + 159.319i −77.6346 − 44.8224i −492.175 + 141.096i 71.2872 − 725.506i −601.882 − 849.058i
7.4 −7.33634 − 3.19031i 16.6321 + 21.2691i 43.6438 + 46.8104i 49.0054 + 84.8798i −54.1636 − 209.099i −128.314 − 74.0821i −170.846 − 482.655i −175.748 + 707.498i −88.7272 − 779.049i
7.5 −6.81178 + 4.19519i 23.9803 + 12.4075i 28.8008 − 57.1534i −41.9280 − 72.6214i −215.400 + 16.0844i −391.468 − 226.014i 43.5851 + 510.141i 421.106 + 595.072i 590.265 + 318.785i
7.6 −6.69027 − 4.38637i 25.6380 − 8.46722i 25.5194 + 58.6921i −91.6050 − 158.665i −208.665 − 55.8098i 357.304 + 206.290i 86.7133 − 504.604i 585.612 − 434.165i −83.0997 + 1463.32i
7.7 −6.68379 + 4.39624i −3.84106 − 26.7254i 25.3462 − 58.7671i −60.0263 − 103.969i 143.164 + 161.741i 148.400 + 85.6789i 88.9451 + 504.215i −699.493 + 205.307i 858.273 + 431.015i
7.8 −6.61738 − 4.49559i −13.6355 − 23.3039i 23.5793 + 59.4980i −52.0663 − 90.1815i −14.5340 + 215.510i −501.579 − 289.587i 111.446 − 499.724i −357.148 + 635.520i −60.8768 + 830.834i
7.9 −6.01631 + 5.27296i 0.0605155 + 26.9999i 8.39188 − 63.4474i 102.873 + 178.181i −142.734 − 162.121i 414.387 + 239.246i 284.067 + 425.969i −728.993 + 3.26783i −1558.46 − 529.549i
7.10 −5.06273 + 6.19425i −26.2829 + 6.18155i −12.7375 − 62.7197i 1.13240 + 1.96137i 94.7729 − 194.098i −359.854 − 207.762i 452.988 + 238.633i 652.577 − 324.937i −17.8823 − 2.91553i
7.11 −4.76201 − 6.42832i −18.2042 + 19.9401i −18.6466 + 61.2234i 21.9287 + 37.9815i 214.870 + 22.0680i −90.8361 − 52.4442i 482.359 − 171.680i −66.2113 − 725.987i 139.733 − 321.833i
7.12 −4.30040 − 6.74586i −2.09895 − 26.9183i −27.0132 + 58.0197i 83.4571 + 144.552i −172.561 + 129.919i 488.440 + 282.001i 507.560 − 67.2806i −720.189 + 113.000i 616.228 − 1184.62i
7.13 −2.83302 + 7.48158i 26.2829 − 6.18155i −47.9480 − 42.3909i 1.13240 + 1.96137i −28.2120 + 214.150i 359.854 + 207.762i 452.988 − 238.633i 652.577 − 324.937i −17.8823 + 2.91553i
7.14 −1.55836 + 7.84675i −0.0605155 − 26.9999i −59.1430 − 24.4561i 102.873 + 178.181i 211.956 + 41.6008i −414.387 − 239.246i 284.067 − 425.969i −728.993 + 3.26783i −1558.46 + 529.549i
7.15 −1.27332 − 7.89802i 22.8258 − 14.4216i −60.7573 + 20.1134i 6.47117 + 11.2084i −142.967 − 161.915i −528.387 − 305.064i 236.219 + 454.252i 313.034 − 658.370i 80.2842 − 65.3812i
7.16 −1.17712 − 7.91293i 18.7658 + 19.4125i −61.2288 + 18.6289i −7.24503 − 12.5488i 131.520 − 171.343i 236.226 + 136.385i 219.482 + 462.570i −24.6908 + 728.582i −90.7691 + 72.1007i
7.17 −0.691572 − 7.97005i −26.2979 − 6.11704i −63.0435 + 11.0237i −105.472 − 182.682i −30.5662 + 213.826i 232.923 + 134.478i 131.459 + 494.836i 654.164 + 321.731i −1383.05 + 966.953i
7.18 −0.465355 + 7.98645i 3.84106 + 26.7254i −63.5669 − 7.43308i −60.0263 − 103.969i −215.229 + 18.2396i −148.400 − 85.6789i 88.9451 − 504.215i −699.493 + 205.307i 858.273 − 431.015i
7.19 −0.227250 + 7.99677i −23.9803 − 12.4075i −63.8967 − 3.63453i −41.9280 − 72.6214i 104.670 − 188.945i 391.468 + 226.014i 43.5851 − 510.141i 421.106 + 595.072i 590.265 − 318.785i
7.20 2.23249 − 7.68219i −26.6628 − 4.25395i −54.0319 − 34.3009i 107.818 + 186.746i −92.2042 + 195.331i −217.897 − 125.803i −384.132 + 338.507i 692.808 + 226.845i 1675.32 − 411.367i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.7.f.a ✓ 68
3.b odd 2 1 108.7.f.a 68
4.b odd 2 1 inner 36.7.f.a ✓ 68
9.c even 3 1 inner 36.7.f.a ✓ 68
9.d odd 6 1 108.7.f.a 68
12.b even 2 1 108.7.f.a 68
36.f odd 6 1 inner 36.7.f.a ✓ 68
36.h even 6 1 108.7.f.a 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.7.f.a ✓ 68 1.a even 1 1 trivial
36.7.f.a ✓ 68 4.b odd 2 1 inner
36.7.f.a ✓ 68 9.c even 3 1 inner
36.7.f.a ✓ 68 36.f odd 6 1 inner
108.7.f.a 68 3.b odd 2 1
108.7.f.a 68 9.d odd 6 1
108.7.f.a 68 12.b even 2 1
108.7.f.a 68 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(36, [\chi])\).