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Properties

Label 17.16.d.a
Level $17$
Weight $16$
Character orbit 17.d
Analytic conductor $24.258$
Analytic rank $0$
Dimension $84$
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] = [17,16,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, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.2");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 16 \)
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: \(24.2578958670\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(21\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 4 q^{2} - 4 q^{3} + 311340 q^{5} + 2536916 q^{6} - 4 q^{7} + 131068 q^{8} - 22872304 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - 4 q^{2} - 4 q^{3} + 311340 q^{5} + 2536916 q^{6} - 4 q^{7} + 131068 q^{8} - 22872304 q^{9} + 50859004 q^{10} + 173485880 q^{11} - 459486808 q^{12} + 191894956 q^{14} + 61419124 q^{15} - 16106127368 q^{16} - 1368740940 q^{17} - 16518942216 q^{18} - 8402365644 q^{19} + 30664687612 q^{20} - 12471915068 q^{22} + 16507799788 q^{23} - 102643171288 q^{24} + 101046442456 q^{25} + 126862227980 q^{26} + 97803284312 q^{27} + 414734173508 q^{28} + 38095836992 q^{29} - 454861482244 q^{31} - 468965196260 q^{32} - 147308439688 q^{33} + 11669079836 q^{34} - 2924467999672 q^{35} - 2934044727460 q^{36} + 394585837036 q^{37} - 2049082749656 q^{39} + 1067763905336 q^{40} - 1250739155368 q^{41} + 2976945531552 q^{42} + 6045876906752 q^{43} + 7067482919212 q^{44} - 7360916195192 q^{45} + 4040103349700 q^{46} - 27764019978300 q^{48} + 5309400161556 q^{49} + 60265792775696 q^{50} + 25692923319316 q^{51} - 41057261584392 q^{52} + 21014630263184 q^{53} + 6006589850708 q^{54} + 99098835581092 q^{56} + 16023832352684 q^{57} - 133237277458056 q^{58} - 162308533044504 q^{59} + 61330541070368 q^{60} + 94725123914692 q^{61} - 6812150187804 q^{62} - 147056237955912 q^{63} + 50336584810604 q^{65} - 12430884533996 q^{66} + 98450049742392 q^{67} + 337258481859492 q^{68} - 375302224065944 q^{69} - 550930595572168 q^{70} - 86064611343844 q^{71} - 279987543361456 q^{73} + 679912619862428 q^{74} + 395700269536000 q^{75} + 490105619704784 q^{76} + 276632087661980 q^{77} + 136457598976488 q^{78} + 98637443768060 q^{79} - 308164031478024 q^{80} - 316613168950528 q^{82} - 649911724895168 q^{83} + 13\!\cdots\!76 q^{84}+ \cdots + 34\!\cdots\!08 q^{99}+O(q^{100}) \) 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 −245.820 − 245.820i −1672.50 + 692.772i 88087.0i −25565.3 − 61720.1i 581431. + 240837.i 1.16718e6 − 2.81783e6i 1.35985e7 − 1.35985e7i −7.82889e6 + 7.82889e6i −8.88757e6 + 2.14565e7i
2.2 −212.195 − 212.195i 5316.82 − 2202.30i 57285.2i 29926.6 + 72249.3i −1.59552e6 − 660884.i −50910.9 + 122910.i 5.20242e6 − 5.20242e6i 1.32722e7 − 1.32722e7i 8.98064e6 − 2.16812e7i
2.3 −200.374 − 200.374i 476.536 − 197.388i 47531.5i 16556.1 + 39969.9i −135037. − 55934.1i −1.20371e6 + 2.90602e6i 2.95821e6 − 2.95821e6i −9.95808e6 + 9.95808e6i 4.69153e6 − 1.13263e7i
2.4 −182.282 − 182.282i −5695.73 + 2359.25i 33685.3i 87740.6 + 211825.i 1.46828e6 + 608180.i −90193.3 + 217746.i 167216. − 167216.i 1.67290e7 − 1.67290e7i 2.26183e7 − 5.46053e7i
2.5 −170.250 − 170.250i −4130.31 + 1710.83i 25201.8i −124080. − 299556.i 994451. + 411915.i −905698. + 2.18655e6i −1.28814e6 + 1.28814e6i 3.98630e6 − 3.98630e6i −2.98746e7 + 7.21238e7i
2.6 −129.673 − 129.673i 4061.21 − 1682.21i 862.087i −105067. − 253655.i −744765. − 308492.i 456097. − 1.10112e6i −4.13733e6 + 4.13733e6i 3.51739e6 − 3.51739e6i −1.92678e7 + 4.65166e7i
2.7 −123.094 − 123.094i −906.241 + 375.377i − 2463.77i 17515.8 + 42286.8i 157759. + 65346.1i 987054. − 2.38296e6i −4.33682e6 + 4.33682e6i −9.46584e6 + 9.46584e6i 3.04917e6 − 7.36134e6i
2.8 −89.2325 − 89.2325i 2770.30 − 1147.50i − 16843.1i 120460. + 290817.i −349595. − 144807.i 148211. − 357812.i −4.42692e6 + 4.42692e6i −3.78838e6 + 3.78838e6i 1.52013e7 − 3.66993e7i
2.9 −40.7523 − 40.7523i −3075.62 + 1273.97i − 29446.5i 8583.59 + 20722.6i 177256. + 73421.8i −428729. + 1.03504e6i −2.53539e6 + 2.53539e6i −2.30974e6 + 2.30974e6i 494694. − 1.19430e6i
2.10 −27.3745 − 27.3745i −6006.82 + 2488.11i − 31269.3i −53706.7 − 129659.i 232545. + 96323.1i 789571. − 1.90619e6i −1.75299e6 + 1.75299e6i 1.97450e7 − 1.97450e7i −2.07917e6 + 5.01956e6i
2.11 −22.5360 − 22.5360i 2542.58 − 1053.17i − 31752.3i −22347.0 − 53950.4i −81033.7 − 33565.3i −1.46128e6 + 3.52785e6i −1.45403e6 + 1.45403e6i −4.79067e6 + 4.79067e6i −712213. + 1.71944e6i
2.12 15.5628 + 15.5628i 6353.60 − 2631.75i − 32283.6i 16618.5 + 40120.5i 139837. + 57922.5i 389638. − 940668.i 1.01239e6 − 1.01239e6i 2.32959e7 − 2.32959e7i −365758. + 883018.i
2.13 50.7927 + 50.7927i −431.133 + 178.581i − 27608.2i −62169.6 − 150091.i −30969.0 − 12827.8i 775764. − 1.87286e6i 3.06667e6 − 3.06667e6i −9.99222e6 + 9.99222e6i 4.46575e6 − 1.07813e7i
2.14 74.8074 + 74.8074i −4152.39 + 1719.98i − 21575.7i 90601.2 + 218731.i −439296. − 181962.i −964925. + 2.32953e6i 4.06531e6 − 4.06531e6i 4.13780e6 − 4.13780e6i −9.58503e6 + 2.31403e7i
2.15 115.076 + 115.076i 833.789 − 345.367i − 6282.82i 89161.7 + 215255.i 135693. + 56205.9i 1.12431e6 − 2.71432e6i 4.49383e6 − 4.49383e6i −9.57028e6 + 9.57028e6i −1.45104e7 + 3.50312e7i
2.16 122.476 + 122.476i 2916.40 − 1208.01i − 2767.50i −62547.2 − 151002.i 505140. + 209236.i −281393. + 679342.i 4.35223e6 − 4.35223e6i −3.10010e6 + 3.10010e6i 1.08336e7 − 2.61546e7i
2.17 160.402 + 160.402i −3356.58 + 1390.34i 18689.4i −110139. − 265899.i −761415. − 315388.i −944161. + 2.27941e6i 2.25823e6 − 2.25823e6i −812613. + 812613.i 2.49841e7 − 6.03171e7i
2.18 175.124 + 175.124i −5939.27 + 2460.13i 28568.5i 6790.58 + 16393.9i −1.47093e6 − 609280.i 856416. − 2.06757e6i 735433. − 735433.i 1.90765e7 − 1.90765e7i −1.68177e6 + 4.06015e6i
2.19 183.168 + 183.168i 4576.47 − 1895.63i 34333.1i 63067.3 + 152258.i 1.18548e6 + 491043.i −951603. + 2.29737e6i −286685. + 286685.i 7.20441e6 − 7.20441e6i −1.63369e7 + 3.94407e7i
2.20 225.107 + 225.107i 5049.07 − 2091.39i 68578.6i −91778.2 − 221572.i 1.60737e6 + 665794.i 1.36178e6 − 3.28763e6i −8.06122e6 + 8.06122e6i 1.09729e7 − 1.09729e7i 2.92176e7 − 7.05375e7i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.21
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 17.16.d.a ✓ 84
17.d even 8 1 inner 17.16.d.a ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.16.d.a ✓ 84 1.a even 1 1 trivial
17.16.d.a ✓ 84 17.d even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(17, [\chi])\).