Properties

Label 17.14.d.a
Level $17$
Weight $14$
Character orbit 17.d
Analytic conductor $18.229$
Analytic rank $0$
Dimension $76$
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] = [17,14,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, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.2");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 14 \)
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: \(18.2292579218\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(19\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 4 q^{2} - 4 q^{3} - 62720 q^{5} - 309100 q^{6} - 4 q^{7} + 32764 q^{8} - 1963432 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 76 q - 4 q^{2} - 4 q^{3} - 62720 q^{5} - 309100 q^{6} - 4 q^{7} + 32764 q^{8} - 1963432 q^{9} + 132604 q^{10} - 9895604 q^{11} + 12816200 q^{12} + 37393836 q^{14} - 98442536 q^{15} - 1275068424 q^{16} + 211848832 q^{17} + 967464696 q^{18} + 873061160 q^{19} - 917962756 q^{20} + 1581701700 q^{22} - 714117372 q^{23} - 3711140824 q^{24} - 4726381084 q^{25} + 1875457164 q^{26} - 6853971688 q^{27} - 12603753788 q^{28} - 13884159964 q^{29} + 12906743796 q^{31} + 14102669884 q^{32} - 33841465600 q^{33} - 26009026532 q^{34} + 76578836296 q^{35} + 52707857756 q^{36} - 41625753216 q^{37} - 41783782976 q^{39} - 234552924392 q^{40} + 188928190052 q^{41} + 544442698848 q^{42} - 104267338520 q^{43} - 506716756436 q^{44} + 454315145428 q^{45} + 477188224388 q^{46} + 528693922212 q^{48} + 24323318216 q^{49} - 1153170486192 q^{50} - 1025605613156 q^{51} + 625522999288 q^{52} + 139498299540 q^{53} + 225287597780 q^{54} - 20007854460 q^{56} - 1028224570072 q^{57} + 593957237976 q^{58} + 41873422480 q^{59} - 1271476363264 q^{60} - 645012831456 q^{61} - 487267180124 q^{62} + 1171950117492 q^{63} + 3780977575124 q^{65} - 7961154307340 q^{66} + 1909626396112 q^{67} - 3697914901308 q^{68} + 4417827885088 q^{69} - 237304712168 q^{70} + 7229599444948 q^{71} + 1423258534652 q^{73} - 5445425704740 q^{74} - 9298935028700 q^{75} - 18698821281616 q^{76} + 2269796192072 q^{77} + 22223859461640 q^{78} + 3992708433980 q^{79} + 22472184030968 q^{80} + 4755056335264 q^{82} - 33692459244000 q^{83} + 54425187275168 q^{84} - 35929160386916 q^{85} - 37071578257440 q^{86} - 25421422626992 q^{87} + 77676455896104 q^{88} + 54722681228644 q^{90} - 23580241666176 q^{91} - 47667786363348 q^{92} - 47998771243112 q^{93} + 9265665779832 q^{94} + 33460963865016 q^{95} + 61616278224996 q^{96} + 57930719765064 q^{97} + 62210603991868 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −119.518 119.518i 860.523 356.440i 20377.2i −19631.6 47394.9i −145449. 60247.1i −71661.0 + 173005.i 1.45636e6 1.45636e6i −513906. + 513906.i −3.31822e6 + 8.01089e6i
2.2 −112.431 112.431i −344.690 + 142.775i 17089.3i 24752.8 + 59758.6i 54806.0 + 22701.4i −106446. + 256984.i 1.00033e6 1.00033e6i −1.02893e6 + 1.02893e6i 3.93572e6 9.50168e6i
2.3 −105.777 105.777i −2066.93 + 856.152i 14185.4i −10075.4 24324.3i 309194. + 128073.i 38475.1 92887.2i 633964. 633964.i 2.41187e6 2.41187e6i −1.50719e6 + 3.63869e6i
2.4 −85.9808 85.9808i 1709.90 708.262i 6593.38i 10234.5 + 24708.2i −207915. 86121.2i 131539. 317563.i −137450. + 137450.i 1.29475e6 1.29475e6i 1.24446e6 3.00439e6i
2.5 −82.3937 82.3937i 15.7751 6.53428i 5385.46i −3541.82 8550.70i −1838.16 761.390i 149811. 361675.i −231242. + 231242.i −1.12715e6 + 1.12715e6i −412701. + 996348.i
2.6 −57.7630 57.7630i 1786.66 740.060i 1518.87i −6597.89 15928.7i −145951. 60454.9i −202896. + 489834.i −560929. + 560929.i 1.51712e6 1.51712e6i −538977. + 1.30120e6i
2.7 −53.0086 53.0086i −1055.64 + 437.260i 2572.17i −1152.06 2781.33i 79136.6 + 32779.4i −89142.6 + 215209.i −570594. + 570594.i −204179. + 204179.i −86365.1 + 208504.i
2.8 −12.1255 12.1255i −1763.94 + 730.648i 7897.95i 22183.3 + 53555.3i 30248.1 + 12529.2i 220223. 531666.i −195098. + 195098.i 1.45029e6 1.45029e6i 380400. 918366.i
2.9 −10.2774 10.2774i 768.584 318.358i 7980.75i 18240.0 + 44035.3i −11171.0 4627.18i −83946.8 + 202666.i −166215. + 166215.i −637987. + 637987.i 265109. 640031.i
2.10 −6.44357 6.44357i 278.134 115.207i 8108.96i −21890.2 52847.5i −2534.52 1049.83i 63144.2 152443.i −105036. + 105036.i −1.06327e6 + 1.06327e6i −199476. + 481577.i
2.11 19.3378 + 19.3378i 1598.43 662.090i 7444.10i 3162.32 + 7634.51i 43713.4 + 18106.7i 140421. 339006.i 302368. 302368.i 989248. 989248.i −86482.3 + 208787.i
2.12 35.2841 + 35.2841i −1889.98 + 782.855i 5702.06i −14202.2 34287.2i −94308.6 39063.9i −85800.7 + 207141.i 490240. 490240.i 1.83181e6 1.83181e6i 708680. 1.71090e6i
2.13 43.7143 + 43.7143i −624.013 + 258.475i 4370.11i 8305.42 + 20051.1i −38577.4 15979.3i −53765.5 + 129801.i 549145. 549145.i −804773. + 804773.i −513453. + 1.23958e6i
2.14 73.9767 + 73.9767i 1820.07 753.899i 2753.11i −14191.6 34261.6i 190414. + 78872.1i −88983.0 + 214824.i 402351. 402351.i 1.61695e6 1.61695e6i 1.48471e6 3.58441e6i
2.15 78.3701 + 78.3701i 286.744 118.773i 4091.76i 8240.02 + 19893.2i 31780.4 + 13163.9i −71677.4 + 173045.i 321337. 321337.i −1.05924e6 + 1.05924e6i −913259. + 2.20480e6i
2.16 90.1042 + 90.1042i −914.334 + 378.730i 8045.55i −6954.34 16789.3i −116511. 48260.2i 215725. 520806.i 13196.1 13196.1i −434786. + 434786.i 886168. 2.13940e6i
2.17 114.094 + 114.094i −2071.82 + 858.176i 17842.8i 18497.9 + 44657.9i −334294. 138469.i −142277. + 343487.i −1.10109e6 + 1.10109e6i 2.42861e6 2.42861e6i −2.98469e6 + 7.20568e6i
2.18 115.717 + 115.717i 1642.47 680.335i 18588.8i 20808.7 + 50236.6i 268789. + 111336.i 141600. 341853.i −1.20309e6 + 1.20309e6i 1.10751e6 1.10751e6i −3.40531e6 + 8.22115e6i
2.19 119.375 + 119.375i 54.9833 22.7748i 20308.8i −19729.1 47630.4i 9282.36 + 3844.88i −102559. + 247601.i −1.44644e6 + 1.44644e6i −1.12485e6 + 1.12485e6i 3.33071e6 8.04104e6i
8.1 −125.778 + 125.778i −585.692 + 1413.99i 23448.2i −37851.1 15678.5i −104181. 251515.i −47682.6 + 19750.8i 1.91889e6 + 1.91889e6i −528964. 528964.i 6.73284e6 2.78883e6i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.19
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 17.14.d.a 76
17.d even 8 1 inner 17.14.d.a 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.14.d.a 76 1.a even 1 1 trivial
17.14.d.a 76 17.d even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{14}^{\mathrm{new}}(17, [\chi])\).