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Properties

Label 29.7.f.a
Level $29$
Weight $7$
Character orbit 29.f
Analytic conductor $6.672$
Analytic rank $0$
Dimension $168$
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(2,29)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(29, base_ring=CyclotomicField(28))
 
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("29.2");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 29.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: \(6.67156842498\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 168 q - 14 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} - 368 q^{8} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 168 q - 14 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} - 368 q^{8} - 14 q^{9} + 3700 q^{10} - 1948 q^{11} + 2692 q^{12} - 14 q^{13} - 2662 q^{14} - 16294 q^{15} + 36618 q^{16} + 5222 q^{17} - 5272 q^{18} - 6732 q^{19} + 4726 q^{20} - 84226 q^{21} - 70574 q^{22} + 15350 q^{23} + 59866 q^{24} + 161886 q^{25} + 119340 q^{26} + 81522 q^{27} - 54928 q^{29} - 373896 q^{30} - 156396 q^{31} - 391380 q^{32} - 124334 q^{33} + 27706 q^{34} + 153650 q^{35} + 860794 q^{36} + 423138 q^{37} + 243026 q^{38} - 228790 q^{39} - 372632 q^{40} - 265730 q^{41} - 14 q^{42} - 80604 q^{43} + 653254 q^{44} + 452214 q^{45} - 1387336 q^{46} - 163204 q^{47} - 1078760 q^{48} - 1090538 q^{49} + 149502 q^{50} + 589666 q^{51} + 1361098 q^{52} + 1189890 q^{53} + 2228700 q^{54} + 2372554 q^{55} + 1520404 q^{56} - 1983510 q^{58} + 424472 q^{59} - 6186916 q^{60} - 1567734 q^{61} - 2918944 q^{62} - 2776046 q^{63} - 1891904 q^{64} + 577546 q^{65} + 3590680 q^{66} + 1866466 q^{67} + 3285080 q^{68} + 4224526 q^{69} + 6718894 q^{70} + 1997226 q^{71} - 5842904 q^{72} - 4911462 q^{73} - 5280024 q^{74} - 4270212 q^{75} - 2096902 q^{76} - 262194 q^{77} + 312162 q^{78} + 869460 q^{79} + 10555762 q^{80} + 4125970 q^{81} + 2603762 q^{82} - 898226 q^{83} + 6457100 q^{84} + 4856818 q^{85} - 2465740 q^{87} + 2506064 q^{88} - 8155850 q^{89} - 6343068 q^{90} - 6874574 q^{91} - 13547534 q^{92} - 3967054 q^{93} - 2281378 q^{94} - 2258662 q^{95} + 16911216 q^{96} + 8483976 q^{97} + 3401840 q^{98} - 7403288 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −14.2951 − 1.61068i −32.2642 − 20.2729i 139.361 + 31.8083i 173.520 − 138.378i 428.568 + 341.772i −91.2723 − 399.890i −1071.95 − 375.090i 313.685 + 651.373i −2703.38 + 1698.64i
2.2 −13.6100 − 1.53347i 38.0830 + 23.9291i 120.484 + 27.4997i 11.0707 − 8.82860i −481.613 − 384.074i −16.9210 − 74.1359i −770.253 − 269.523i 561.409 + 1165.78i −164.210 + 103.180i
2.3 −13.4129 − 1.51128i −9.93455 − 6.24229i 115.228 + 26.3000i −151.038 + 120.449i 123.818 + 98.7413i 83.3519 + 365.188i −690.412 − 241.586i −256.572 − 532.778i 2207.89 − 1387.31i
2.4 −9.78132 − 1.10209i 4.86871 + 3.05921i 32.0641 + 7.31843i 67.6441 − 53.9443i −44.2508 − 35.2888i 14.2110 + 62.2623i 289.050 + 101.143i −301.956 − 627.018i −721.099 + 453.097i
2.5 −6.05546 − 0.682286i −30.0097 − 18.8564i −26.1923 − 5.97823i −99.2293 + 79.1327i 168.857 + 134.659i −78.8822 − 345.606i 522.644 + 182.881i 228.721 + 474.943i 654.870 − 411.482i
2.6 −5.51694 − 0.621610i 13.9286 + 8.75193i −32.3452 − 7.38258i 38.7258 − 30.8828i −71.4030 − 56.9420i 8.89145 + 38.9560i 509.236 + 178.189i −198.891 − 413.002i −232.845 + 146.306i
2.7 −2.09772 − 0.236357i 33.1089 + 20.8037i −58.0508 − 13.2497i −157.772 + 125.819i −64.5362 − 51.4659i −105.507 − 462.257i 246.165 + 86.1369i 347.104 + 720.769i 360.700 − 226.643i
2.8 −1.17129 − 0.131973i −34.2542 − 21.5233i −61.0409 − 13.9322i 86.9922 − 69.3739i 37.2812 + 29.7308i 114.959 + 503.669i 140.862 + 49.2898i 393.794 + 817.722i −111.049 + 69.7767i
2.9 3.80574 + 0.428804i 13.3587 + 8.39385i −48.0956 − 10.9775i −108.958 + 86.8915i 47.2405 + 37.6731i 151.820 + 665.167i −409.686 − 143.355i −208.302 − 432.544i −451.927 + 283.964i
2.10 4.02402 + 0.453398i 41.8468 + 26.2941i −46.4083 − 10.5924i 191.158 − 152.443i 156.471 + 124.781i 25.4276 + 111.406i −426.568 − 149.263i 743.475 + 1543.84i 838.340 − 526.764i
2.11 5.23233 + 0.589542i −8.25911 − 5.18955i −35.3657 − 8.07198i 44.1976 − 35.2464i −40.1550 − 32.0225i −75.6796 − 331.574i −498.363 − 174.385i −275.020 − 571.084i 252.036 − 158.365i
2.12 10.4381 + 1.17609i −33.2132 − 20.8692i 45.1758 + 10.3111i −152.668 + 121.748i −322.139 − 256.897i 9.22875 + 40.4338i −175.117 − 61.2762i 351.289 + 729.459i −1736.75 + 1091.27i
2.13 12.6082 + 1.42061i 22.9059 + 14.3928i 94.5537 + 21.5813i −31.1152 + 24.8136i 268.357 + 214.007i −13.1373 − 57.5582i 395.032 + 138.228i 1.22953 + 2.55314i −427.558 + 268.653i
2.14 13.8221 + 1.55738i −21.5996 − 13.5719i 126.230 + 28.8113i 139.632 − 111.353i −277.416 − 221.232i 44.3920 + 194.494i 859.646 + 300.803i −33.9549 − 70.5082i 2103.43 − 1321.67i
3.1 −11.7444 − 7.37947i −17.3573 + 6.07359i 55.7048 + 115.672i 78.4838 − 17.9134i 248.671 + 56.7574i 162.844 + 78.4214i 99.9912 − 887.446i −305.567 + 243.681i −1053.93 − 368.787i
3.2 −10.5151 − 6.60709i 23.9441 − 8.37839i 39.1456 + 81.2866i −108.523 + 24.7696i −307.131 − 70.1007i −161.756 − 77.8977i 36.4592 − 323.584i −66.8348 + 53.2989i 1304.78 + 456.564i
3.3 −7.55873 − 4.74946i −43.7281 + 15.3011i 6.80842 + 14.1378i −185.513 + 42.3421i 403.201 + 92.0279i −234.310 − 112.838i −48.2845 + 428.537i 1108.06 − 883.652i 1603.34 + 561.034i
3.4 −6.17960 − 3.88290i 37.2349 − 13.0290i −4.65801 − 9.67246i 119.027 − 27.1672i −280.687 − 64.0650i 254.881 + 122.744i −61.0698 + 542.009i 646.724 − 515.745i −841.030 − 294.289i
3.5 −4.77604 − 3.00098i −9.72984 + 3.40462i −13.9639 − 28.9964i 155.648 − 35.5257i 56.6873 + 12.9385i −405.979 − 195.509i −60.7445 + 539.122i −486.877 + 388.271i −849.994 − 297.426i
3.6 −2.56297 − 1.61042i −18.3792 + 6.43116i −23.7932 − 49.4071i −13.8056 + 3.15103i 57.4622 + 13.1154i 414.777 + 199.746i −40.2750 + 357.451i −273.520 + 218.125i 40.4577 + 14.1568i
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.14
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 29.7.f.a ✓ 168
29.f odd 28 1 inner 29.7.f.a ✓ 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.7.f.a ✓ 168 1.a even 1 1 trivial
29.7.f.a ✓ 168 29.f odd 28 1 inner

Hecke kernels

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