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Properties

Label 29.4.d.a
Level $29$
Weight $4$
Character orbit 29.d
Analytic conductor $1.711$
Analytic rank $0$
Dimension $42$
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] = [29,4,Mod(7,29)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(29, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([6]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.7");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 29.d (of order \(7\), 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: \(1.71105539017\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(7\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 5 q^{2} - 5 q^{3} - 23 q^{4} - 7 q^{5} - 11 q^{6} - 31 q^{7} + 38 q^{8} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q - 5 q^{2} - 5 q^{3} - 23 q^{4} - 7 q^{5} - 11 q^{6} - 31 q^{7} + 38 q^{8} - 72 q^{9} + 31 q^{10} - 63 q^{11} + 136 q^{12} + 173 q^{13} + 129 q^{14} - 229 q^{15} - 451 q^{16} - 140 q^{17} + 846 q^{18} - q^{19} - 395 q^{20} - 961 q^{21} - 396 q^{22} + 57 q^{23} - 83 q^{24} + 1028 q^{25} + 141 q^{26} - 35 q^{27} + 356 q^{28} + 472 q^{29} + 2174 q^{30} + 175 q^{31} + 1356 q^{32} - 173 q^{33} - 1469 q^{34} + 477 q^{35} - 712 q^{36} - 417 q^{37} - 2834 q^{38} - 2273 q^{39} + 507 q^{40} + 1104 q^{41} + 2866 q^{42} - 573 q^{43} - 1644 q^{44} - 81 q^{45} - 5984 q^{46} - 223 q^{47} - 2442 q^{48} + 1400 q^{49} - 1234 q^{50} + 92 q^{51} + 2394 q^{52} - 791 q^{53} + 4228 q^{54} + 1505 q^{55} + 4167 q^{56} + 1718 q^{57} + 3635 q^{58} + 6172 q^{59} - 848 q^{60} + 2591 q^{61} + 1333 q^{62} - 1797 q^{63} + 2426 q^{64} - 3483 q^{65} + 679 q^{66} - 2611 q^{67} - 2796 q^{68} + 3665 q^{69} - 13558 q^{70} - 2169 q^{71} - 12175 q^{72} - 1309 q^{73} + 44 q^{74} - 7468 q^{75} + 1679 q^{76} - 1697 q^{77} + 6185 q^{78} + 933 q^{79} + 4604 q^{80} + 2562 q^{81} - 7016 q^{82} + 2553 q^{83} + 5579 q^{84} + 5294 q^{85} + 6824 q^{86} - 3909 q^{87} + 16298 q^{88} - 2543 q^{89} + 16038 q^{90} + 2781 q^{91} - 290 q^{92} + 6515 q^{93} - 3230 q^{94} + 1737 q^{95} - 15242 q^{96} - 5527 q^{97} - 786 q^{98} - 17404 q^{99}+O(q^{100}) \) 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 −0.980555 − 4.29609i −3.05331 − 1.47040i −10.2872 + 4.95403i −0.668979 − 2.93099i −3.32302 + 14.5591i 3.27538 + 1.57734i 9.39047 + 11.7753i −9.67359 − 12.1303i −11.9358 + 5.74798i
7.2 −0.764111 − 3.34779i 8.23892 + 3.96766i −3.41608 + 1.64510i 1.31115 + 5.74451i 6.98743 − 30.6139i −25.5210 − 12.2903i −9.01023 − 11.2985i 35.3034 + 44.2690i 18.2296 − 8.77890i
7.3 −0.160227 − 0.701999i 1.15898 + 0.558133i 6.74062 − 3.24611i −0.752447 − 3.29669i 0.206110 − 0.903028i 19.5988 + 9.43826i −6.95036 − 8.71548i −15.8025 − 19.8157i −2.19371 + 1.05643i
7.4 0.122599 + 0.537140i −8.97814 − 4.32364i 6.93426 − 3.33936i −2.29214 − 10.0425i 1.22169 − 5.35259i −15.9306 − 7.67179i 5.39195 + 6.76129i 45.0788 + 56.5271i 5.11323 − 2.46240i
7.5 0.484814 + 2.12411i 0.796605 + 0.383625i 2.93096 − 1.41147i 3.15414 + 13.8192i −0.428656 + 1.87806i −15.6444 − 7.53397i 15.2864 + 19.1686i −16.3468 − 20.4983i −27.8243 + 13.3995i
7.6 0.902233 + 3.95294i 5.51206 + 2.65447i −7.60398 + 3.66188i −3.78236 − 16.5716i −5.51979 + 24.1838i −10.1766 − 4.90078i −1.11176 − 1.39411i 6.50235 + 8.15369i 62.0941 − 29.9029i
7.7 1.14223 + 5.00442i −5.29861 − 2.55167i −16.5318 + 7.96129i 1.81632 + 7.95783i 6.71744 − 29.4310i 32.5673 + 15.6836i −33.1211 − 41.5326i 4.72996 + 5.93118i −37.7497 + 18.1793i
16.1 −4.50314 − 2.16860i 4.54313 − 5.69691i 10.5875 + 13.2763i −16.6453 − 8.01597i −32.8126 + 15.8017i −4.27206 + 5.35699i −9.98861 − 43.7630i −5.80663 − 25.4405i 57.5728 + 72.1941i
16.2 −3.77959 − 1.82016i −2.29509 + 2.87796i 5.98444 + 7.50425i 15.1796 + 7.31010i 13.9128 − 6.70007i −0.697313 + 0.874402i −1.49198 − 6.53678i 2.99289 + 13.1127i −44.0671 − 55.2584i
16.3 −1.46785 − 0.706881i −3.52482 + 4.41999i −3.33301 − 4.17946i −12.5304 − 6.03430i 8.29832 − 3.99626i −8.89280 + 11.1512i 4.83822 + 21.1976i −1.10385 − 4.83627i 14.1272 + 17.7149i
16.4 −0.689266 − 0.331933i 3.06461 − 3.84290i −4.62301 − 5.79707i 2.78967 + 1.34343i −3.38792 + 1.63153i 13.4697 − 16.8905i 2.62412 + 11.4970i 0.632029 + 2.76910i −1.47690 − 1.85197i
16.5 2.61147 + 1.25762i 2.77920 − 3.48501i 0.250272 + 0.313832i 2.95793 + 1.42446i 11.6406 − 5.60583i −21.6928 + 27.2019i −4.90095 − 21.4725i 1.58675 + 6.95200i 5.93312 + 7.43990i
16.6 2.64247 + 1.27255i −4.95388 + 6.21197i 0.375370 + 0.470699i 13.7847 + 6.63836i −20.9955 + 10.1109i 6.54400 − 8.20591i −4.82818 − 21.1536i −8.03955 − 35.2236i 27.9781 + 35.0834i
16.7 4.24086 + 2.04229i −0.390627 + 0.489830i 8.82604 + 11.0675i −17.7732 − 8.55914i −2.65697 + 1.27953i 16.7517 − 21.0059i 6.44769 + 28.2492i 5.92072 + 25.9404i −57.8936 − 72.5963i
20.1 −4.50314 + 2.16860i 4.54313 + 5.69691i 10.5875 − 13.2763i −16.6453 + 8.01597i −32.8126 − 15.8017i −4.27206 − 5.35699i −9.98861 + 43.7630i −5.80663 + 25.4405i 57.5728 − 72.1941i
20.2 −3.77959 + 1.82016i −2.29509 − 2.87796i 5.98444 − 7.50425i 15.1796 − 7.31010i 13.9128 + 6.70007i −0.697313 − 0.874402i −1.49198 + 6.53678i 2.99289 − 13.1127i −44.0671 + 55.2584i
20.3 −1.46785 + 0.706881i −3.52482 − 4.41999i −3.33301 + 4.17946i −12.5304 + 6.03430i 8.29832 + 3.99626i −8.89280 − 11.1512i 4.83822 − 21.1976i −1.10385 + 4.83627i 14.1272 − 17.7149i
20.4 −0.689266 + 0.331933i 3.06461 + 3.84290i −4.62301 + 5.79707i 2.78967 − 1.34343i −3.38792 − 1.63153i 13.4697 + 16.8905i 2.62412 − 11.4970i 0.632029 − 2.76910i −1.47690 + 1.85197i
20.5 2.61147 − 1.25762i 2.77920 + 3.48501i 0.250272 − 0.313832i 2.95793 − 1.42446i 11.6406 + 5.60583i −21.6928 − 27.2019i −4.90095 + 21.4725i 1.58675 − 6.95200i 5.93312 − 7.43990i
20.6 2.64247 − 1.27255i −4.95388 − 6.21197i 0.375370 − 0.470699i 13.7847 − 6.63836i −20.9955 − 10.1109i 6.54400 + 8.20591i −4.82818 + 21.1536i −8.03955 + 35.2236i 27.9781 − 35.0834i
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.4.d.a ✓ 42
29.d even 7 1 inner 29.4.d.a ✓ 42
29.d even 7 1 841.4.a.h 21
29.e even 14 1 841.4.a.i 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.4.d.a ✓ 42 1.a even 1 1 trivial
29.4.d.a ✓ 42 29.d even 7 1 inner
841.4.a.h 21 29.d even 7 1
841.4.a.i 21 29.e even 14 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(29, [\chi])\).