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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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 29.c (of order \(4\), degree \(2\), minimal)

Newform invariants

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.

\(\operatorname{Tr}(f)(q) = \) \( 28q - 2q^{3} - 4q^{7} + 354q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28q - 2q^{3} - 4q^{7} + 354q^{8} - 3714q^{10} + 1934q^{11} - 2706q^{12} + 2648q^{14} + 16280q^{15} - 36632q^{16} - 5236q^{17} + 5258q^{18} + 6718q^{19} + 24828q^{20} + 7380q^{21} - 21524q^{23} + 90648q^{24} - 73196q^{25} - 5394q^{26} + 5824q^{27} - 12510q^{29} - 1780q^{30} + 1150q^{31} + 104646q^{32} - 32904q^{36} - 256832q^{37} + 127752q^{39} - 256374q^{40} + 265716q^{41} + 80590q^{43} + 188482q^{44} - 78008q^{45} + 405572q^{46} - 445250q^{47} - 1016886q^{48} + 532764q^{49} - 80230q^{50} + 619608q^{52} + 143428q^{53} + 373816q^{54} - 533304q^{55} - 619392q^{56} + 326848q^{58} - 957340q^{59} + 1790622q^{60} - 115640q^{61} - 643584q^{65} - 1821654q^{66} + 1457028q^{68} - 1049340q^{69} + 3024084q^{70} + 1480560q^{72} - 627512q^{73} - 3652060q^{74} + 586518q^{75} - 1343752q^{76} - 115820q^{77} + 1820304q^{78} + 235966q^{79} + 2710496q^{81} + 796544q^{82} + 3881892q^{83} + 219332q^{84} - 3064608q^{85} + 900526q^{87} - 9844332q^{88} + 3821436q^{89} - 4192912q^{90} - 648556q^{94} - 620200q^{95} - 620288q^{97} + 4735296q^{98} + 4021770q^{99} + O(q^{100}) \)

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.

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 17.14
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 \(S_{7}^{\mathrm{new}}(29, [\chi])\).