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Properties

Label 17.9.e.a
Level $17$
Weight $9$
Character orbit 17.e
Analytic conductor $6.925$
Analytic rank $0$
Dimension $88$
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] = [17,9,Mod(3,17)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(17, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.3");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 17.e (of order \(16\), degree \(8\), 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: \(6.92543637104\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(11\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9} + 21496 q^{10} - 27968 q^{11} + 172024 q^{12} + 19984 q^{13} - 141320 q^{14} - 272144 q^{15} + 134896 q^{17} + 770032 q^{18} + 82816 q^{19} - 602120 q^{20} - 1209728 q^{21} - 694280 q^{22} + 176224 q^{23} + 2453128 q^{24} - 1715920 q^{25} - 925352 q^{26} + 2939320 q^{27} + 4516072 q^{28} - 275528 q^{29} - 7843400 q^{30} - 5352712 q^{31} - 4838880 q^{32} + 9334968 q^{34} + 8050928 q^{35} + 16654136 q^{36} + 4647416 q^{37} + 8405200 q^{38} - 19334424 q^{39} - 49015240 q^{40} - 8023472 q^{41} + 19462408 q^{42} + 21418088 q^{43} + 55649944 q^{44} + 12876632 q^{45} - 42199192 q^{46} - 28631408 q^{47} - 56707840 q^{48} - 14300136 q^{49} + 16156760 q^{51} + 58777584 q^{52} + 72829624 q^{53} - 12773160 q^{54} - 47916072 q^{55} - 42610448 q^{56} + 8039072 q^{57} + 43505424 q^{58} - 3667112 q^{59} + 24554384 q^{60} - 170296 q^{61} + 25385392 q^{62} - 94112864 q^{63} + 6935048 q^{64} - 125391128 q^{65} - 220878144 q^{66} + 225348832 q^{68} + 176211296 q^{69} + 87384336 q^{70} + 40919176 q^{71} + 234186544 q^{72} + 167466192 q^{73} + 111150568 q^{74} - 125599592 q^{75} - 61416384 q^{76} - 212184488 q^{77} - 541008288 q^{78} - 176854504 q^{79} - 279231000 q^{80} + 92322504 q^{81} + 91580672 q^{82} + 462895984 q^{83} - 504357528 q^{85} - 617697952 q^{86} + 70542376 q^{87} + 711198008 q^{88} + 426540736 q^{89} + 1253929784 q^{90} + 566661608 q^{91} + 461913448 q^{92} - 38907432 q^{93} - 713174760 q^{94} - 741593960 q^{95} - 1781388112 q^{96} - 482346152 q^{97} - 1202501264 q^{98} + 147226040 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −27.0831 + 11.2182i −27.1070 − 5.39191i 426.625 − 426.625i 583.700 + 873.569i 794.627 − 158.061i 1223.56 − 1831.18i −3896.51 + 9407.00i −5355.86 − 2218.47i −25608.2 − 17110.9i
3.2 −22.4842 + 9.31327i −74.8365 − 14.8859i 237.784 − 237.784i −616.975 − 923.368i 1821.28 − 362.275i −1390.42 + 2080.92i −747.643 + 1804.97i −682.659 − 282.767i 22471.8 + 15015.2i
3.3 −22.2906 + 9.23308i 137.900 + 27.4300i 230.603 − 230.603i −329.917 − 493.756i −3327.13 + 661.808i 1362.42 − 2039.00i −647.436 + 1563.05i 12202.4 + 5054.39i 11913.0 + 7959.98i
3.4 −13.4623 + 5.57628i 21.6403 + 4.30452i −30.8798 + 30.8798i 145.447 + 217.677i −315.332 + 62.7234i −798.155 + 1194.52i 1671.05 − 4034.27i −5611.80 − 2324.48i −3171.89 − 2119.39i
3.5 −6.36107 + 2.63484i −117.068 − 23.2862i −147.499 + 147.499i 111.853 + 167.401i 806.031 − 160.330i 1131.04 − 1692.72i 1224.13 − 2955.32i 7101.04 + 2941.35i −1152.58 − 770.130i
3.6 −0.895127 + 0.370774i 94.5743 + 18.8120i −180.356 + 180.356i 144.314 + 215.981i −91.6310 + 18.2265i −558.699 + 836.152i 189.488 − 457.465i 2528.84 + 1047.48i −209.259 − 139.823i
3.7 7.18313 − 2.97535i 13.8016 + 2.74532i −138.275 + 138.275i −619.572 − 927.255i 107.307 − 21.3447i 1478.79 − 2213.17i −1343.52 + 3243.54i −5878.63 − 2435.01i −7209.38 − 4817.15i
3.8 14.1769 − 5.87227i −81.9629 − 16.3034i −14.5181 + 14.5181i 92.1783 + 137.955i −1257.72 + 250.176i −1650.04 + 2469.45i −1623.87 + 3920.37i 390.549 + 161.771i 2116.91 + 1414.47i
3.9 17.9614 − 7.43985i 63.2751 + 12.5862i 86.2406 − 86.2406i 501.158 + 750.037i 1230.15 − 244.692i 1576.86 − 2359.94i −997.218 + 2407.50i −2216.25 − 918.001i 14581.7 + 9743.15i
3.10 23.7585 − 9.84110i 127.695 + 25.4000i 286.601 − 286.601i −409.123 − 612.296i 3283.80 − 653.188i −2516.58 + 3766.33i 1469.42 − 3547.49i 9599.16 + 3976.10i −15745.8 − 10521.0i
3.11 27.7894 − 11.5107i −68.5000 − 13.6255i 458.733 − 458.733i −99.6540 − 149.143i −2060.41 + 409.841i 1333.93 − 1996.37i 4520.79 − 10914.2i −1554.97 − 644.090i −4486.07 − 2997.49i
5.1 −11.1165 − 26.8376i −47.6889 − 71.3715i −415.663 + 415.663i −37.5525 + 188.789i −1385.31 + 2073.26i −554.124 − 2785.77i 8905.68 + 3688.85i −308.869 + 745.676i 5484.10 − 1090.86i
5.2 −8.96365 − 21.6402i 54.4002 + 81.4157i −206.931 + 206.931i 159.606 − 802.392i 1274.22 − 1907.01i −201.955 − 1015.30i 792.983 + 328.464i −1158.34 + 2796.49i −18794.6 + 3738.47i
5.3 −7.92082 − 19.1225i 14.4343 + 21.6025i −121.913 + 121.913i −137.968 + 693.613i 298.763 − 447.130i 656.146 + 3298.67i −1598.44 − 662.094i 2252.47 − 5437.94i 14356.5 − 2855.68i
5.4 −4.22853 − 10.2086i −68.8606 − 103.057i 94.6848 − 94.6848i 115.657 − 581.449i −760.887 + 1138.75i 185.869 + 934.427i −3980.37 − 1648.72i −3368.21 + 8131.58i −6424.83 + 1277.98i
5.5 −2.38470 − 5.75717i −3.79216 − 5.67536i 153.561 − 153.561i −48.4134 + 243.391i −23.6309 + 35.3661i −289.248 − 1454.15i −2724.11 − 1128.36i 2492.96 − 6018.53i 1516.69 − 301.689i
5.6 0.0196634 + 0.0474717i 81.7173 + 122.299i 181.017 − 181.017i −109.490 + 550.445i −4.19888 + 6.28407i −258.327 − 1298.70i 24.3054 + 10.0676i −5768.43 + 13926.2i −28.2835 + 5.62594i
5.7 3.04523 + 7.35183i 29.4808 + 44.1212i 136.243 − 136.243i 219.386 − 1102.93i −234.595 + 351.097i 626.138 + 3147.81i 3298.60 + 1366.32i 1433.23 − 3460.12i 8776.61 − 1745.78i
5.8 5.12891 + 12.3823i −61.6614 − 92.2828i 54.0040 − 54.0040i −194.848 + 979.566i 826.416 − 1236.82i 756.561 + 3803.49i 4115.54 + 1704.71i −2203.20 + 5319.00i −13128.6 + 2611.45i
5.9 5.24507 + 12.6627i −28.0383 − 41.9623i 48.1857 − 48.1857i 8.01568 − 40.2975i 384.294 − 575.136i −709.313 − 3565.96i 4104.55 + 1700.16i 1536.10 − 3708.47i 552.319 − 109.863i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.e odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.9.e.a ✓ 88
17.e odd 16 1 inner 17.9.e.a ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.9.e.a ✓ 88 1.a even 1 1 trivial
17.9.e.a ✓ 88 17.e odd 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(17, [\chi])\).