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Properties

Label 17.10.d.a
Level $17$
Weight $10$
Character orbit 17.d
Analytic conductor $8.756$
Analytic rank $0$
Dimension $52$
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] = [17,10,Mod(2,17)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(17, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([7]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.2");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 17.d (of order \(8\), degree \(4\), 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: \(8.75560921479\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(13\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9} - 117636 q^{10} - 115768 q^{11} - 145528 q^{12} + 463660 q^{14} + 288244 q^{15} - 3407880 q^{16} - 271932 q^{17} + 2193336 q^{18} - 406700 q^{19} + 493564 q^{20} + 800964 q^{22} - 4172180 q^{23} + 15078440 q^{24} + 11946576 q^{25} - 14209012 q^{26} - 7973656 q^{27} + 13540804 q^{28} + 7833752 q^{29} + 5936252 q^{31} - 26730116 q^{32} - 1635208 q^{33} - 53493988 q^{34} + 46992776 q^{35} - 12761764 q^{36} + 74606796 q^{37} + 74096776 q^{39} - 69317672 q^{40} - 71362288 q^{41} - 308117664 q^{42} - 3765712 q^{43} + 163659820 q^{44} + 14167408 q^{45} + 72135428 q^{46} + 164621412 q^{48} - 284548508 q^{49} + 226223888 q^{50} - 406733132 q^{51} + 103553016 q^{52} - 244724728 q^{53} + 971226260 q^{54} + 470691652 q^{56} - 440646580 q^{57} - 308561768 q^{58} - 481734984 q^{59} + 379626176 q^{60} + 507966228 q^{61} + 358867236 q^{62} + 1070586456 q^{63} + 366263684 q^{65} - 2244715148 q^{66} - 311910088 q^{67} - 1839196668 q^{68} + 533688136 q^{69} + 34867672 q^{70} + 186006812 q^{71} + 487115192 q^{73} - 214149412 q^{74} + 533015920 q^{75} + 1104236912 q^{76} - 237869092 q^{77} - 2081523384 q^{78} - 136338052 q^{79} - 922534152 q^{80} + 1951892384 q^{82} + 1843663888 q^{83} - 2630025952 q^{84} - 395061976 q^{85} + 1538547296 q^{86} - 369395636 q^{87} - 3388839640 q^{88} + 874363684 q^{90} + 2087197720 q^{91} + 1061427500 q^{92} + 576415420 q^{93} + 6899365112 q^{94} + 5031766876 q^{95} + 490811364 q^{96} - 3419996368 q^{97} - 5630536112 q^{99}+O(q^{100}) \) 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 −29.1798 − 29.1798i −162.034 + 67.1166i 1190.92i −267.860 − 646.671i 6686.56 + 2769.66i −3386.34 + 8175.34i 19810.7 − 19810.7i 7832.33 − 7832.33i −11053.6 + 26685.8i
2.2 −24.8024 − 24.8024i 109.545 − 45.3751i 718.321i −625.855 − 1510.95i −3842.40 − 1591.57i 4127.23 − 9964.00i 5117.27 − 5117.27i −3976.74 + 3976.74i −21952.5 + 52997.9i
2.3 −22.5808 − 22.5808i 147.205 − 60.9745i 507.789i 534.784 + 1291.08i −4700.88 − 1947.17i −3817.95 + 9217.34i −95.0916 + 95.0916i 4033.58 − 4033.58i 17077.8 − 41229.6i
2.4 −19.0841 − 19.0841i −103.159 + 42.7299i 216.409i 813.738 + 1964.54i 2784.17 + 1153.24i 3243.32 − 7830.08i −5641.10 + 5641.10i −5102.02 + 5102.02i 21962.0 − 53021.0i
2.5 −10.5264 − 10.5264i −129.909 + 53.8103i − 290.392i −588.452 − 1420.65i 1933.90 + 801.047i −818.389 + 1975.77i −8446.26 + 8446.26i 62.9403 − 62.9403i −8760.00 + 21148.5i
2.6 −2.79790 − 2.79790i 168.723 − 69.8875i − 496.344i −252.482 − 609.546i −667.609 − 276.533i −532.387 + 1285.30i −2821.24 + 2821.24i 9665.32 − 9665.32i −999.029 + 2411.87i
2.7 −2.23497 − 2.23497i −45.0140 + 18.6454i − 502.010i 265.226 + 640.313i 142.277 + 58.9331i −537.194 + 1296.90i −2266.28 + 2266.28i −12239.4 + 12239.4i 838.309 − 2023.86i
2.8 11.3845 + 11.3845i −244.376 + 101.224i − 252.785i 230.356 + 556.130i −3934.48 − 1629.72i 406.478 − 981.326i 8706.72 − 8706.72i 35555.2 − 35555.2i −3708.77 + 8953.77i
2.9 13.0593 + 13.0593i 116.759 − 48.3631i − 170.907i 663.047 + 1600.74i 2156.38 + 893.204i 2945.54 − 7111.17i 8918.32 − 8918.32i −2624.32 + 2624.32i −12245.6 + 29563.5i
2.10 17.3427 + 17.3427i −21.9931 + 9.10984i 89.5378i −1000.76 − 2416.05i −539.408 − 223.430i 1420.57 − 3429.57i 7326.63 − 7326.63i −13517.3 + 13517.3i 24544.9 − 59256.7i
2.11 21.3276 + 21.3276i −21.6241 + 8.95698i 397.734i 288.795 + 697.212i −652.221 − 270.159i −4324.98 + 10441.4i 2437.03 − 2437.03i −13530.6 + 13530.6i −8710.57 + 21029.2i
2.12 27.1947 + 27.1947i 246.648 − 102.165i 967.098i −306.747 − 740.553i 9485.84 + 3929.16i −1300.23 + 3139.03i −12376.2 + 12376.2i 36479.5 − 36479.5i 11797.2 − 28481.0i
2.13 31.2113 + 31.2113i −114.097 + 47.2607i 1436.29i 205.870 + 497.013i −5036.20 − 2086.06i 3921.77 − 9467.99i −28848.3 + 28848.3i −3133.33 + 3133.33i −9086.97 + 21937.9i
8.1 −29.9491 + 29.9491i 97.1714 − 234.592i − 1281.90i −584.865 − 242.259i 4115.64 + 9936.04i −4412.09 + 1827.55i 23057.9 + 23057.9i −31673.4 − 31673.4i 24771.7 − 10260.8i
8.2 −25.4395 + 25.4395i −16.0088 + 38.6487i − 782.338i 2220.20 + 919.637i −575.948 − 1390.46i 8017.68 − 3321.03i 6877.27 + 6877.27i 12680.5 + 12680.5i −79875.9 + 33085.7i
8.3 −25.0414 + 25.0414i −61.4403 + 148.330i − 742.144i −1194.49 − 494.773i −2175.84 − 5252.94i −4192.59 + 1736.63i 5763.13 + 5763.13i −4308.89 − 4308.89i 42301.4 − 17521.8i
8.4 −14.5474 + 14.5474i 32.3594 − 78.1225i 88.7454i 31.3399 + 12.9814i 665.735 + 1607.23i −4774.06 + 1977.48i −8739.29 − 8739.29i 8861.98 + 8861.98i −644.761 + 267.069i
8.5 −12.2939 + 12.2939i 41.5316 − 100.266i 209.719i −793.507 − 328.681i 722.077 + 1743.25i 7184.85 − 2976.06i −8872.76 − 8872.76i 5589.57 + 5589.57i 13796.1 − 5714.52i
8.6 −4.85709 + 4.85709i −79.5278 + 191.997i 464.817i 1735.92 + 719.040i −546.273 − 1318.82i −3022.19 + 1251.83i −4744.49 − 4744.49i −16620.2 − 16620.2i −11923.9 + 4939.06i
8.7 −0.763172 + 0.763172i −66.0099 + 159.362i 510.835i −1926.04 − 797.791i −71.2437 − 171.997i 7148.81 − 2961.13i −780.599 − 780.599i −7120.96 − 7120.96i 2078.75 − 861.046i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.10.d.a ✓ 52
17.d even 8 1 inner 17.10.d.a ✓ 52
17.e odd 16 2 289.10.a.i 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.10.d.a ✓ 52 1.a even 1 1 trivial
17.10.d.a ✓ 52 17.d even 8 1 inner
289.10.a.i 52 17.e odd 16 2

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(17, [\chi])\).