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Properties

Label 58.7.f.a
Level $58$
Weight $7$
Character orbit 58.f
Analytic conductor $13.343$
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] = [58,7,Mod(3,58)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(58, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("58.3");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 58.f (of order \(28\), degree \(12\), 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: \(13.3431368500\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(7\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q + 56 q^{2} + 28 q^{3} + 588 q^{5} + 610 q^{7} - 1792 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q + 56 q^{2} + 28 q^{3} + 588 q^{5} + 610 q^{7} - 1792 q^{8} - 2096 q^{10} - 600 q^{11} - 896 q^{12} + 8816 q^{14} + 28160 q^{15} + 14336 q^{16} + 11890 q^{17} - 14648 q^{18} + 34182 q^{19} - 23104 q^{20} + 52044 q^{21} + 35280 q^{22} - 35690 q^{23} - 17920 q^{24} - 47674 q^{25} - 46248 q^{26} + 98704 q^{27} + 78698 q^{29} + 267520 q^{30} + 169710 q^{31} - 57344 q^{32} - 220976 q^{33} + 37464 q^{34} - 153664 q^{35} - 20800 q^{36} - 209618 q^{37} - 14000 q^{38} + 67296 q^{39} + 67328 q^{40} + 38078 q^{41} + 288064 q^{42} + 364376 q^{43} - 18432 q^{44} - 650106 q^{45} + 80208 q^{46} + 166990 q^{47} - 28672 q^{48} - 158370 q^{49} - 653336 q^{50} - 1054830 q^{51} - 332928 q^{52} + 2470 q^{53} - 587696 q^{54} + 183274 q^{55} - 282112 q^{56} + 196360 q^{58} + 722428 q^{59} + 901120 q^{60} + 609098 q^{61} - 404880 q^{62} - 1782004 q^{63} + 357356 q^{65} + 648912 q^{66} - 1472310 q^{67} - 380480 q^{68} - 1943988 q^{69} - 2121232 q^{70} + 113120 q^{71} + 276736 q^{72} + 4246286 q^{73} + 1947088 q^{74} + 3353544 q^{75} + 246144 q^{76} + 2440308 q^{77} - 1050432 q^{78} - 3040534 q^{79} - 7081546 q^{81} + 1852496 q^{82} + 662780 q^{83} + 685696 q^{84} - 2606000 q^{85} - 234268 q^{87} + 147456 q^{88} + 5204902 q^{89} + 3947168 q^{90} + 7752598 q^{91} + 302400 q^{92} + 9294460 q^{93} - 128000 q^{94} + 6073690 q^{95} - 7400384 q^{97} - 1333016 q^{98} - 402816 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 4.78980 + 3.00963i −42.6986 + 14.9409i 13.8843 + 28.8310i −49.1801 + 11.2250i −249.484 − 56.9431i −286.600 − 138.020i −20.2677 + 179.881i 1029.99 − 821.387i −269.346 − 94.2482i
3.2 4.78980 + 3.00963i −26.7774 + 9.36983i 13.8843 + 28.8310i 133.148 − 30.3901i −156.458 − 35.7105i 185.283 + 89.2274i −20.2677 + 179.881i 59.2820 − 47.2758i 729.212 + 255.162i
3.3 4.78980 + 3.00963i −24.6057 + 8.60990i 13.8843 + 28.8310i −148.723 + 33.9450i −143.769 − 32.8143i 467.067 + 224.928i −20.2677 + 179.881i −38.6464 + 30.8194i −814.514 − 285.011i
3.4 4.78980 + 3.00963i 6.65696 − 2.32937i 13.8843 + 28.8310i −67.6185 + 15.4335i 38.8960 + 8.87777i −412.264 − 198.536i −20.2677 + 179.881i −531.066 + 423.511i −370.328 − 129.583i
3.5 4.78980 + 3.00963i 19.1602 − 6.70443i 13.8843 + 28.8310i −127.831 + 29.1767i 111.951 + 25.5521i 338.987 + 163.247i −20.2677 + 179.881i −247.792 + 197.608i −700.097 − 244.975i
3.6 4.78980 + 3.00963i 20.9819 − 7.34189i 13.8843 + 28.8310i 110.202 − 25.1530i 122.595 + 27.9816i 218.054 + 105.009i −20.2677 + 179.881i −183.618 + 146.430i 603.548 + 211.191i
3.7 4.78980 + 3.00963i 43.1242 − 15.0898i 13.8843 + 28.8310i 118.526 − 27.0527i 251.971 + 57.5107i −66.1040 − 31.8340i −20.2677 + 179.881i 1062.04 − 846.949i 649.132 + 227.141i
11.1 5.33941 − 1.86834i −4.58623 − 40.7039i 25.0186 − 19.9517i 77.0202 − 159.934i −100.537 − 208.766i 84.9274 − 106.496i 96.3081 − 153.273i −925.052 + 211.137i 112.431 − 997.854i
11.2 5.33941 − 1.86834i −4.49162 − 39.8642i 25.0186 − 19.9517i −88.9436 + 184.693i −98.4626 − 204.460i −284.158 + 356.323i 96.3081 − 153.273i −858.261 + 195.892i −129.836 + 1152.33i
11.3 5.33941 − 1.86834i −2.77414 − 24.6212i 25.0186 − 19.9517i −15.8285 + 32.8682i −60.8131 − 126.280i 155.831 − 195.406i 96.3081 − 153.273i 112.215 − 25.6122i −23.1058 + 205.070i
11.4 5.33941 − 1.86834i 0.272097 + 2.41493i 25.0186 − 19.9517i −36.9008 + 76.6253i 5.96475 + 12.3859i 107.794 − 135.170i 96.3081 − 153.273i 704.965 − 160.904i −53.8664 + 478.077i
11.5 5.33941 − 1.86834i 0.807171 + 7.16384i 25.0186 − 19.9517i 71.7677 − 149.027i 17.6943 + 36.7426i −132.446 + 166.082i 96.3081 − 153.273i 660.053 − 150.653i 104.764 − 929.803i
11.6 5.33941 − 1.86834i 3.93563 + 34.9297i 25.0186 − 19.9517i −7.59633 + 15.7739i 86.2745 + 179.151i −282.485 + 354.225i 96.3081 − 153.273i −493.871 + 112.723i −11.0888 + 98.4161i
11.7 5.33941 − 1.86834i 4.83984 + 42.9548i 25.0186 − 19.9517i 78.1614 − 162.304i 106.096 + 220.311i 364.935 − 457.614i 96.3081 − 153.273i −1110.97 + 253.571i 114.097 − 1012.64i
15.1 −5.62129 + 0.633367i −39.2248 + 24.6466i 31.1977 − 7.12067i −25.9997 − 20.7341i 204.884 − 163.389i 70.8972 − 310.621i −170.861 + 59.7869i 614.832 − 1276.71i 159.284 + 100.085i
15.2 −5.62129 + 0.633367i −19.1829 + 12.0534i 31.1977 − 7.12067i −36.1433 − 28.8233i 100.198 − 79.9053i −87.2437 + 382.240i −170.861 + 59.7869i −93.6033 + 194.369i 221.428 + 139.132i
15.3 −5.62129 + 0.633367i −6.63798 + 4.17092i 31.1977 − 7.12067i −4.19334 − 3.34407i 34.6722 − 27.6502i 19.1296 − 83.8122i −170.861 + 59.7869i −289.635 + 601.433i 25.6900 + 16.1421i
15.4 −5.62129 + 0.633367i 1.12998 − 0.710011i 31.1977 − 7.12067i 172.425 + 137.504i −5.90222 + 4.70686i 117.365 − 514.209i −170.861 + 59.7869i −315.529 + 655.202i −1056.34 − 663.742i
15.5 −5.62129 + 0.633367i 16.1374 − 10.1398i 31.1977 − 7.12067i −174.686 − 139.307i −84.2906 + 67.2195i −54.0540 + 236.826i −170.861 + 59.7869i −158.702 + 329.547i 1070.19 + 672.446i
15.6 −5.62129 + 0.633367i 23.7639 − 14.9319i 31.1977 − 7.12067i 87.1390 + 69.4910i −124.126 + 98.9876i −98.5121 + 431.610i −170.861 + 59.7869i 25.4624 − 52.8732i −533.846 − 335.438i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.f odd 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 58.7.f.a ✓ 84
29.f odd 28 1 inner 58.7.f.a ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.7.f.a ✓ 84 1.a even 1 1 trivial
58.7.f.a ✓ 84 29.f odd 28 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{84} - 28 T_{3}^{83} + 392 T_{3}^{82} - 70580 T_{3}^{81} - 2052491 T_{3}^{80} + \cdots + 35\!\cdots\!96 \) acting on \(S_{7}^{\mathrm{new}}(58, [\chi])\). Copy content Toggle raw display