Properties

Label 109.6.k.a
Level $109$
Weight $6$
Character orbit 109.k
Analytic conductor $17.482$
Analytic rank $0$
Dimension $810$
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] = [109,6,Mod(12,109)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(109, base_ring=CyclotomicField(54))
 
chi = DirichletCharacter(H, H._module([29]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("109.12");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 109 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 109.k (of order \(54\), degree \(18\), 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: \(17.4818363596\)
Analytic rank: \(0\)
Dimension: \(810\)
Relative dimension: \(45\) over \(\Q(\zeta_{54})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{54}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 810 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 27 q^{8} + 1692 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 810 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 27 q^{8} + 1692 q^{9} - 18 q^{10} - 3438 q^{11} - 18 q^{12} + 5283 q^{13} + 9684 q^{14} - 18 q^{15} - 306 q^{16} - 18 q^{17} + 17055 q^{18} - 18 q^{19} - 3735 q^{20} - 36630 q^{21} - 22644 q^{22} + 7803 q^{23} - 36765 q^{24} - 18 q^{25} - 18 q^{26} + 22716 q^{27} - 18 q^{28} - 18342 q^{29} - 15354 q^{30} - 216 q^{31} + 16398 q^{32} + 74088 q^{34} + 84213 q^{35} - 63378 q^{36} + 27369 q^{37} - 18 q^{38} - 35073 q^{39} - 42543 q^{40} + 254043 q^{41} - 101592 q^{42} - 18 q^{43} + 102510 q^{44} - 109359 q^{45} + 76167 q^{46} - 233874 q^{47} - 39825 q^{48} - 20880 q^{49} - 114840 q^{50} - 48528 q^{51} - 41130 q^{52} - 81333 q^{53} + 53316 q^{54} - 18 q^{55} - 151281 q^{56} - 51894 q^{57} - 181602 q^{58} - 50256 q^{59} + 177282 q^{60} - 216378 q^{61} + 142686 q^{62} - 357219 q^{63} + 1400823 q^{64} + 144315 q^{65} + 447048 q^{66} + 621927 q^{67} - 27 q^{68} - 187785 q^{69} - 253890 q^{70} - 352458 q^{71} + 153405 q^{72} - 18 q^{73} - 569016 q^{74} + 172737 q^{75} - 344853 q^{77} + 887895 q^{78} - 434808 q^{79} - 2044260 q^{80} + 706212 q^{81} + 358308 q^{82} + 166023 q^{83} - 151218 q^{84} - 79668 q^{85} - 1293750 q^{86} + 788166 q^{87} - 1563282 q^{88} + 707427 q^{89} - 1503450 q^{90} - 32049 q^{91} + 19278 q^{92} + 1395315 q^{93} + 2038914 q^{94} + 571248 q^{95} + 78003 q^{96} + 137286 q^{97} + 1283526 q^{98} + 835416 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1 −3.65593 + 10.0446i 26.4787 6.27556i −63.0143 52.8753i −5.72252 + 3.76376i −33.7688 + 288.910i −0.671442 11.5282i 465.258 268.617i 444.585 223.279i −16.8843 71.2404i
12.2 −3.62689 + 9.96480i −5.02656 + 1.19132i −61.6295 51.7133i −49.7432 + 32.7166i 6.35956 54.4095i −2.37763 40.8224i 444.960 256.898i −193.306 + 97.0817i −145.601 614.341i
12.3 −3.51872 + 9.66760i 3.96132 0.938850i −56.5677 47.4659i 40.2791 26.4920i −4.86234 + 41.6000i 11.4295 + 196.238i 372.817 215.246i −202.342 + 101.620i 114.383 + 482.620i
12.4 −3.38854 + 9.30995i −29.6099 + 7.01769i −50.6794 42.5251i −18.4132 + 12.1106i 35.0003 299.447i 6.36974 + 109.364i 293.073 169.206i 610.348 306.528i −50.3547 212.463i
12.5 −3.38086 + 9.28883i −14.4411 + 3.42259i −50.3388 42.2393i 52.2807 34.3855i 17.0313 145.712i −11.6256 199.603i 288.601 166.624i −20.3226 + 10.2064i 142.648 + 601.879i
12.6 −2.94203 + 8.08315i 14.6409 3.46996i −32.1684 26.9925i 16.9489 11.1474i −15.0258 + 128.554i −7.07504 121.474i 74.4419 42.9791i −14.8367 + 7.45126i 40.2425 + 169.796i
12.7 −2.77911 + 7.63554i 13.2124 3.13139i −26.0646 21.8708i −82.1562 + 54.0350i −12.8088 + 109.586i −11.0768 190.181i 14.2496 8.22703i −52.3916 + 26.3121i −184.265 777.476i
12.8 −2.76985 + 7.61010i −6.96070 + 1.64972i −25.7281 21.5885i −57.0888 + 37.5479i 6.72559 57.5411i 8.48020 + 145.599i 11.1212 6.42083i −171.423 + 86.0918i −127.616 538.454i
12.9 −2.75736 + 7.57578i 11.2738 2.67195i −25.2759 21.2090i 68.8677 45.2950i −10.8439 + 92.7755i −0.319872 5.49199i 6.94979 4.01246i −97.1929 + 48.8121i 153.252 + 646.621i
12.10 −2.40029 + 6.59473i −19.6016 + 4.64566i −13.2157 11.0893i 6.30294 4.14551i 16.4125 140.418i −4.37114 75.0495i −89.6352 + 51.7509i 145.487 73.0662i 12.2097 + 51.5166i
12.11 −2.29356 + 6.30151i −19.1346 + 4.53497i −9.93518 8.33661i 81.3756 53.5216i 15.3091 130.978i 10.1737 + 174.676i −110.520 + 63.8087i 128.413 64.4913i 150.627 + 635.544i
12.12 −2.26029 + 6.21011i 21.4109 5.07447i −8.94307 7.50412i −38.7861 + 25.5100i −16.8819 + 144.433i 9.77571 + 167.842i −116.329 + 67.1626i 215.522 108.239i −70.7519 298.526i
12.13 −2.15628 + 5.92433i −10.7516 + 2.54819i −5.93468 4.97979i −16.2841 + 10.7102i 8.08725 69.1909i 1.48030 + 25.4159i −132.418 + 76.4514i −108.048 + 54.2638i −28.3379 119.567i
12.14 −1.64009 + 4.50610i 28.9642 6.86463i 6.89838 + 5.78843i 79.8742 52.5341i −16.5710 + 141.774i 0.0919795 + 1.57923i −170.288 + 98.3159i 574.647 288.599i 105.723 + 446.082i
12.15 −1.63205 + 4.48401i −25.4903 + 6.04131i 7.07062 + 5.93296i −84.8211 + 55.7877i 14.5121 124.159i −13.6673 234.658i −170.383 + 98.3705i 396.106 198.932i −111.721 471.387i
12.16 −1.56667 + 4.30438i 4.68011 1.10921i 8.44018 + 7.08215i 34.1751 22.4773i −2.55772 + 21.8827i −4.41415 75.7881i −170.649 + 98.5244i −196.480 + 98.6758i 43.2099 + 182.317i
12.17 −1.15065 + 3.16140i 7.70306 1.82566i 15.8430 + 13.2939i −12.9234 + 8.49987i −3.09193 + 26.4531i −5.68504 97.6083i −153.491 + 88.6180i −161.149 + 80.9319i −12.0011 50.6365i
12.18 −0.966472 + 2.65536i −18.2988 + 4.33691i 18.3966 + 15.4365i 33.8630 22.2720i 6.16927 52.7815i −3.68670 63.2983i −137.080 + 79.1429i 98.8862 49.6625i 26.4126 + 111.444i
12.19 −0.718640 + 1.97445i 25.6230 6.07276i 21.1314 + 17.7314i −18.6909 + 12.2932i −6.42335 + 54.9553i −12.4742 214.173i −108.425 + 62.5990i 402.506 202.146i −10.8402 45.7386i
12.20 −0.634883 + 1.74433i 6.41049 1.51931i 21.8738 + 18.3543i 44.4897 29.2614i −1.41973 + 12.1466i 12.1549 + 208.692i −97.3458 + 56.2026i −178.367 + 89.5791i 22.7956 + 96.1821i
See next 80 embeddings (of 810 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.45
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
109.k even 54 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 109.6.k.a 810
109.k even 54 1 inner 109.6.k.a 810
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
109.6.k.a 810 1.a even 1 1 trivial
109.6.k.a 810 109.k even 54 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(109, [\chi])\).