Properties

Label 17.6.d.a
Level $17$
Weight $6$
Character orbit 17.d
Analytic conductor $2.727$
Analytic rank $0$
Dimension $28$
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] = [17,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.2");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(2.72652493682\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 4 q^{2} - 4 q^{3} + 40 q^{5} - 364 q^{6} - 4 q^{7} + 124 q^{8} - 136 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 4 q^{2} - 4 q^{3} + 40 q^{5} - 364 q^{6} - 4 q^{7} + 124 q^{8} - 136 q^{9} - 676 q^{10} + 1228 q^{11} + 3080 q^{12} + 2124 q^{14} - 4376 q^{15} - 7176 q^{16} - 3128 q^{17} + 1896 q^{18} - 4440 q^{19} + 12284 q^{20} + 14820 q^{22} + 1668 q^{23} - 18136 q^{24} - 14564 q^{25} + 10092 q^{26} + 20792 q^{27} + 5572 q^{28} + 11708 q^{29} - 2732 q^{31} - 29636 q^{32} - 38800 q^{33} - 37284 q^{34} + 26056 q^{35} - 13924 q^{36} - 28056 q^{37} + 29728 q^{39} + 87448 q^{40} + 17372 q^{41} + 97728 q^{42} + 50760 q^{43} + 65836 q^{44} - 63812 q^{45} - 160252 q^{46} + 45732 q^{48} - 33312 q^{49} - 241632 q^{50} + 19612 q^{51} + 6008 q^{52} - 56340 q^{53} - 296380 q^{54} + 202884 q^{56} + 288848 q^{57} + 22616 q^{58} + 57712 q^{59} + 540416 q^{60} + 216792 q^{61} - 59804 q^{62} - 228348 q^{63} - 74956 q^{65} - 346076 q^{66} - 250608 q^{67} - 275388 q^{68} - 107696 q^{69} - 178088 q^{70} - 245324 q^{71} + 398132 q^{73} + 396156 q^{74} + 44740 q^{75} + 79280 q^{76} + 475712 q^{77} + 721800 q^{78} - 41284 q^{79} - 337672 q^{80} + 250704 q^{82} - 657120 q^{83} - 1605472 q^{84} - 495036 q^{85} + 718272 q^{86} + 12448 q^{87} - 404696 q^{88} + 1554724 q^{90} + 406016 q^{91} + 366252 q^{92} - 16112 q^{93} + 734392 q^{94} + 497736 q^{95} - 423708 q^{96} - 308456 q^{97} + 57628 q^{99}+O(q^{100}) \) 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 −6.32994 6.32994i 26.8751 11.1320i 48.1363i −24.9886 60.3278i −240.583 99.6526i −43.4369 + 104.866i 102.142 102.142i 426.521 426.521i −223.695 + 540.048i
2.2 −5.57898 5.57898i −9.69582 + 4.01614i 30.2500i 21.4113 + 51.6915i 76.4987 + 31.6868i −20.6531 + 49.8610i −9.76339 + 9.76339i −93.9474 + 93.9474i 168.932 407.839i
2.3 −1.63520 1.63520i 3.76372 1.55898i 26.6522i −7.23148 17.4583i −8.70370 3.60519i 93.1665 224.924i −95.9083 + 95.9083i −160.092 + 160.092i −16.7230 + 40.3729i
2.4 0.226068 + 0.226068i −20.1254 + 8.33623i 31.8978i −31.6611 76.4366i −6.43429 2.66517i −77.8766 + 188.011i 14.4453 14.4453i 163.714 163.714i 10.1223 24.4375i
2.5 2.65830 + 2.65830i 15.7332 6.51691i 17.8669i 13.8556 + 33.4503i 59.1474 + 24.4997i −40.7626 + 98.4095i 132.561 132.561i 33.2366 33.2366i −52.0887 + 125.753i
2.6 5.25239 + 5.25239i −18.5982 + 7.70364i 23.1753i 32.1442 + 77.6030i −138.148 57.2227i 39.9387 96.4206i 46.3508 46.3508i 114.721 114.721i −238.767 + 576.436i
2.7 7.23579 + 7.23579i 8.11856 3.36282i 72.7132i −35.2492 85.0992i 83.0768 + 34.4115i 7.61170 18.3763i −294.592 + 294.592i −117.224 + 117.224i 360.704 870.815i
8.1 −7.43061 + 7.43061i −9.60499 + 23.1885i 78.4279i 51.4729 + 21.3208i −100.934 243.676i −84.7940 + 35.1228i 344.988 + 344.988i −273.623 273.623i −540.902 + 224.049i
8.2 −5.77048 + 5.77048i 8.99350 21.7122i 34.5969i 33.8199 + 14.0087i 73.3931 + 177.187i 191.069 79.1434i 14.9851 + 14.9851i −218.711 218.711i −275.994 + 114.320i
8.3 −3.68811 + 3.68811i −0.537605 + 1.29789i 4.79575i −80.0401 33.1537i −2.80402 6.76951i −92.4561 + 38.2966i −135.707 135.707i 170.431 + 170.431i 417.471 172.922i
8.4 −0.441843 + 0.441843i −0.853199 + 2.05980i 31.6095i 70.2501 + 29.0985i −0.533130 1.28709i −64.4124 + 26.6805i −28.1054 28.1054i 168.312 + 168.312i −43.8965 + 18.1825i
8.5 2.46953 2.46953i −10.5837 + 25.5512i 19.8029i −30.5219 12.6426i 36.9628 + 89.2361i 151.735 62.8506i 127.929 + 127.929i −369.025 369.025i −106.596 + 44.1535i
8.6 3.81385 3.81385i 6.88457 16.6208i 2.90907i −27.9569 11.5801i −37.1326 89.6461i 62.5721 25.9182i 133.138 + 133.138i −57.0277 57.0277i −150.788 + 62.4585i
8.7 7.21923 7.21923i −2.36967 + 5.72090i 72.2346i 34.6952 + 14.3712i 24.1933 + 58.4077i −123.701 + 51.2388i −290.463 290.463i 144.714 + 144.714i 354.222 146.724i
9.1 −6.32994 + 6.32994i 26.8751 + 11.1320i 48.1363i −24.9886 + 60.3278i −240.583 + 99.6526i −43.4369 104.866i 102.142 + 102.142i 426.521 + 426.521i −223.695 540.048i
9.2 −5.57898 + 5.57898i −9.69582 4.01614i 30.2500i 21.4113 51.6915i 76.4987 31.6868i −20.6531 49.8610i −9.76339 9.76339i −93.9474 93.9474i 168.932 + 407.839i
9.3 −1.63520 + 1.63520i 3.76372 + 1.55898i 26.6522i −7.23148 + 17.4583i −8.70370 + 3.60519i 93.1665 + 224.924i −95.9083 95.9083i −160.092 160.092i −16.7230 40.3729i
9.4 0.226068 0.226068i −20.1254 8.33623i 31.8978i −31.6611 + 76.4366i −6.43429 + 2.66517i −77.8766 188.011i 14.4453 + 14.4453i 163.714 + 163.714i 10.1223 + 24.4375i
9.5 2.65830 2.65830i 15.7332 + 6.51691i 17.8669i 13.8556 33.4503i 59.1474 24.4997i −40.7626 98.4095i 132.561 + 132.561i 33.2366 + 33.2366i −52.0887 125.753i
9.6 5.25239 5.25239i −18.5982 7.70364i 23.1753i 32.1442 77.6030i −138.148 + 57.2227i 39.9387 + 96.4206i 46.3508 + 46.3508i 114.721 + 114.721i −238.767 576.436i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.7
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 17.6.d.a 28
17.d even 8 1 inner 17.6.d.a 28
17.e odd 16 2 289.6.a.j 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.6.d.a 28 1.a even 1 1 trivial
17.6.d.a 28 17.d even 8 1 inner
289.6.a.j 28 17.e odd 16 2

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(17, [\chi])\).