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Properties

Label 29.7.c.a
Level $29$
Weight $7$
Character orbit 29.c
Analytic conductor $6.672$
Analytic rank $0$
Dimension $28$
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] = [29,7,Mod(12,29)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(29, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.12");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 29.c (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: \(6.67156842498\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{3} - 4 q^{7} + 354 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{3} - 4 q^{7} + 354 q^{8} - 3714 q^{10} + 1934 q^{11} - 2706 q^{12} + 2648 q^{14} + 16280 q^{15} - 36632 q^{16} - 5236 q^{17} + 5258 q^{18} + 6718 q^{19} + 24828 q^{20} + 7380 q^{21} - 21524 q^{23} + 90648 q^{24} - 73196 q^{25} - 5394 q^{26} + 5824 q^{27} - 12510 q^{29} - 1780 q^{30} + 1150 q^{31} + 104646 q^{32} - 32904 q^{36} - 256832 q^{37} + 127752 q^{39} - 256374 q^{40} + 265716 q^{41} + 80590 q^{43} + 188482 q^{44} - 78008 q^{45} + 405572 q^{46} - 445250 q^{47} - 1016886 q^{48} + 532764 q^{49} - 80230 q^{50} + 619608 q^{52} + 143428 q^{53} + 373816 q^{54} - 533304 q^{55} - 619392 q^{56} + 326848 q^{58} - 957340 q^{59} + 1790622 q^{60} - 115640 q^{61} - 643584 q^{65} - 1821654 q^{66} + 1457028 q^{68} - 1049340 q^{69} + 3024084 q^{70} + 1480560 q^{72} - 627512 q^{73} - 3652060 q^{74} + 586518 q^{75} - 1343752 q^{76} - 115820 q^{77} + 1820304 q^{78} + 235966 q^{79} + 2710496 q^{81} + 796544 q^{82} + 3881892 q^{83} + 219332 q^{84} - 3064608 q^{85} + 900526 q^{87} - 9844332 q^{88} + 3821436 q^{89} - 4192912 q^{90} - 648556 q^{94} - 620200 q^{95} - 620288 q^{97} + 4735296 q^{98} + 4021770 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1 −11.1386 − 11.1386i 25.3142 + 25.3142i 184.138i − 151.054i − 563.931i −332.137 1338.17 − 1338.17i 552.618i −1682.54 + 1682.54i
12.2 −9.26981 − 9.26981i −17.8548 − 17.8548i 107.859i 107.938i 331.022i −104.064 406.562 − 406.562i − 91.4089i 1000.56 − 1000.56i
12.3 −6.90454 − 6.90454i 11.2332 + 11.2332i 31.3454i 10.9400i − 155.120i 617.401 −225.465 + 225.465i − 476.632i 75.5356 − 75.5356i
12.4 −6.25438 − 6.25438i −11.2134 − 11.2134i 14.2346i − 192.776i 140.266i −54.6116 −311.252 + 311.252i − 477.517i −1205.69 + 1205.69i
12.5 −5.36218 − 5.36218i 27.0129 + 27.0129i − 6.49405i 130.759i − 289.696i −330.754 −378.002 + 378.002i 730.397i 701.154 − 701.154i
12.6 −1.83652 − 1.83652i −32.7814 − 32.7814i − 57.2544i − 4.82891i 120.407i 52.1989 −222.686 + 222.686i 1420.24i −8.86838 + 8.86838i
12.7 −0.624225 − 0.624225i 1.83796 + 1.83796i − 63.2207i 41.8066i − 2.29460i −304.112 −79.4144 + 79.4144i − 722.244i 26.0967 − 26.0967i
12.8 0.260765 + 0.260765i 29.0542 + 29.0542i − 63.8640i − 194.251i 15.1526i 282.863 33.3424 − 33.3424i 959.292i 50.6540 − 50.6540i
12.9 2.37438 + 2.37438i −6.54119 − 6.54119i − 52.7246i 220.621i − 31.0626i 573.636 277.149 − 277.149i − 643.426i −523.840 + 523.840i
12.10 4.72749 + 4.72749i −5.84368 − 5.84368i − 19.3016i − 122.935i − 55.2519i −317.611 393.808 − 393.808i − 660.703i 581.174 − 581.174i
12.11 6.98171 + 6.98171i 24.6261 + 24.6261i 33.4886i 68.1677i 343.865i −10.7616 213.022 − 213.022i 483.892i −475.927 + 475.927i
12.12 7.44805 + 7.44805i −22.6252 − 22.6252i 46.9470i − 114.512i − 337.027i 512.680 127.012 − 127.012i 294.796i 852.893 − 852.893i
12.13 8.70689 + 8.70689i −29.1706 − 29.1706i 87.6199i 207.022i − 507.971i −652.893 −205.655 + 205.655i 972.849i −1802.52 + 1802.52i
12.14 10.8910 + 10.8910i 5.95173 + 5.95173i 173.227i − 50.8966i 129.640i 66.1657 −1189.59 + 1189.59i − 658.154i 554.314 − 554.314i
17.1 −11.1386 + 11.1386i 25.3142 − 25.3142i − 184.138i 151.054i 563.931i −332.137 1338.17 + 1338.17i − 552.618i −1682.54 − 1682.54i
17.2 −9.26981 + 9.26981i −17.8548 + 17.8548i − 107.859i − 107.938i − 331.022i −104.064 406.562 + 406.562i 91.4089i 1000.56 + 1000.56i
17.3 −6.90454 + 6.90454i 11.2332 − 11.2332i − 31.3454i − 10.9400i 155.120i 617.401 −225.465 − 225.465i 476.632i 75.5356 + 75.5356i
17.4 −6.25438 + 6.25438i −11.2134 + 11.2134i − 14.2346i 192.776i − 140.266i −54.6116 −311.252 − 311.252i 477.517i −1205.69 − 1205.69i
17.5 −5.36218 + 5.36218i 27.0129 − 27.0129i 6.49405i − 130.759i 289.696i −330.754 −378.002 − 378.002i − 730.397i 701.154 + 701.154i
17.6 −1.83652 + 1.83652i −32.7814 + 32.7814i 57.2544i 4.82891i − 120.407i 52.1989 −222.686 − 222.686i − 1420.24i −8.86838 − 8.86838i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.7.c.a ✓ 28
29.c odd 4 1 inner 29.7.c.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.7.c.a ✓ 28 1.a even 1 1 trivial
29.7.c.a ✓ 28 29.c odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(29, [\chi])\).