↑

Properties

Label 109.6.c.a
Level $109$
Weight $6$
Character orbit 109.c
Analytic conductor $17.482$
Analytic rank $0$
Dimension $90$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [109,6,Mod(45,109)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(109, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("109.45");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 109 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 109.c (of order \(3\), degree \(2\), 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: \(90\)
Relative dimension: \(45\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 90 q + 2 q^{2} + 17 q^{3} + 1434 q^{4} + 33 q^{5} + 97 q^{6} + 50 q^{7} + 78 q^{8} - 3404 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 90 q + 2 q^{2} + 17 q^{3} + 1434 q^{4} + 33 q^{5} + 97 q^{6} + 50 q^{7} + 78 q^{8} - 3404 q^{9} + 580 q^{10} - 504 q^{11} + 588 q^{12} + 353 q^{13} + 35 q^{14} + 1935 q^{15} + 26154 q^{16} + 1102 q^{17} - 2659 q^{18} + 1134 q^{19} + 762 q^{20} + 604 q^{21} + 1315 q^{22} + 1336 q^{23} + 280 q^{24} - 31448 q^{25} + 2267 q^{26} - 25312 q^{27} - 7181 q^{28} + 7925 q^{29} + 23218 q^{30} + 5970 q^{31} + 12638 q^{32} + 7962 q^{33} + 3944 q^{34} + 29560 q^{35} - 29224 q^{36} - 41152 q^{37} - 35324 q^{38} + 11602 q^{39} + 7770 q^{40} - 63294 q^{41} + 8429 q^{42} - 33168 q^{43} - 34745 q^{44} - 80226 q^{45} - 60708 q^{46} - 10375 q^{47} + 123653 q^{48} - 112131 q^{49} + 4308 q^{50} - 64597 q^{51} - 3293 q^{52} + 61071 q^{53} - 114566 q^{54} - 42756 q^{55} - 177571 q^{56} + 65144 q^{57} + 55384 q^{58} + 72417 q^{59} + 225282 q^{60} - 94983 q^{61} + 21761 q^{62} - 72808 q^{63} + 850206 q^{64} - 4504 q^{65} + 89318 q^{66} - 2464 q^{67} + 48368 q^{68} + 96033 q^{69} - 100126 q^{70} - 8760 q^{71} - 65397 q^{72} + 51263 q^{73} - 13732 q^{74} + 132680 q^{75} - 60434 q^{76} - 120236 q^{77} + 337819 q^{78} - 72063 q^{79} - 6907 q^{80} - 391069 q^{81} - 132890 q^{82} + 164367 q^{83} + 181290 q^{84} + 46281 q^{85} + 933094 q^{86} - 125065 q^{87} + 379597 q^{88} + 8428 q^{89} - 555240 q^{90} + 249995 q^{91} + 47430 q^{92} - 475862 q^{93} - 133857 q^{94} + 26797 q^{95} - 223681 q^{96} + 104413 q^{97} - 330555 q^{98} + 67775 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 −11.2661 9.17526 − 15.8920i 94.9251 −37.8697 + 65.5922i −103.369 + 179.041i 53.5759 − 92.7963i −708.921 −46.8709 − 81.1827i 426.644 − 738.968i
45.2 −10.7035 4.28017 − 7.41347i 82.5656 45.9608 − 79.6065i −45.8129 + 79.3503i −76.4362 + 132.391i −541.231 84.8603 + 146.982i −491.943 + 852.071i
45.3 −10.2175 −8.21711 + 14.2324i 72.3978 12.4890 − 21.6316i 83.9585 − 145.420i 123.290 − 213.545i −412.765 −13.5417 − 23.4549i −127.607 + 221.021i
45.4 −10.1354 −5.42954 + 9.40423i 70.7264 −13.6284 + 23.6051i 55.0305 − 95.3157i −63.6205 + 110.194i −392.508 62.5403 + 108.323i 138.130 − 239.247i
45.5 −9.25472 −11.2885 + 19.5523i 53.6498 −37.7475 + 65.3805i 104.472 − 180.951i −30.8289 + 53.3973i −200.363 −133.361 − 230.988i 349.342 − 605.078i
45.6 −8.67693 −1.11102 + 1.92434i 43.2891 8.28920 − 14.3573i 9.64022 − 16.6973i −15.6589 + 27.1220i −97.9545 119.031 + 206.168i −71.9247 + 124.577i
45.7 −8.54185 14.6892 − 25.4424i 40.9633 25.6481 − 44.4238i −125.473 + 217.325i 21.2491 − 36.8045i −76.5628 −310.045 − 537.013i −219.082 + 379.461i
45.8 −8.24785 7.63656 − 13.2269i 36.0271 1.13961 − 1.97386i −62.9853 + 109.094i 38.7612 − 67.1364i −33.2149 4.86579 + 8.42779i −9.39932 + 16.2801i
45.9 −8.01328 −12.6916 + 21.9824i 32.2126 43.8141 − 75.8883i 101.701 − 176.151i −21.7586 + 37.6870i −1.70352 −200.651 − 347.538i −351.095 + 608.114i
45.10 −7.65850 8.03920 − 13.9243i 26.6526 −36.3558 + 62.9701i −61.5683 + 106.639i −93.3856 + 161.749i 40.9527 −7.75763 − 13.4366i 278.431 − 482.256i
45.11 −6.40125 0.144049 − 0.249501i 8.97597 −46.4474 + 80.4493i −0.922096 + 1.59712i 101.783 − 176.293i 147.383 121.458 + 210.372i 297.321 − 514.976i
45.12 −5.51776 0.718560 − 1.24458i −1.55429 40.7904 − 70.6510i −3.96484 + 6.86731i 69.9821 − 121.213i 185.145 120.467 + 208.656i −225.072 + 389.836i
45.13 −5.23078 −14.0442 + 24.3252i −4.63892 −24.1970 + 41.9104i 73.4620 − 127.240i 25.2963 − 43.8145i 191.650 −272.977 − 472.811i 126.569 − 219.224i
45.14 −5.10893 −8.65385 + 14.9889i −5.89879 22.1691 − 38.3981i 44.2120 − 76.5774i −72.1250 + 124.924i 193.622 −28.2783 − 48.9795i −113.261 + 196.173i
45.15 −4.63114 −4.16764 + 7.21857i −10.5525 −24.7683 + 42.9000i 19.3009 − 33.4302i 45.5178 − 78.8392i 197.067 86.7615 + 150.275i 114.706 − 198.676i
45.16 −4.07750 −3.79393 + 6.57128i −15.3740 −9.17624 + 15.8937i 15.4698 − 26.7944i −77.4355 + 134.122i 193.167 92.7121 + 160.582i 37.4161 − 64.8066i
45.17 −3.83643 7.41773 − 12.8479i −17.2818 42.6222 − 73.8239i −28.4576 + 49.2900i −117.631 + 203.744i 189.066 11.4545 + 19.8397i −163.517 + 283.220i
45.18 −3.15518 11.2877 − 19.5508i −22.0448 5.90296 − 10.2242i −35.6147 + 61.6864i −42.3603 + 73.3701i 170.521 −133.323 − 230.922i −18.6249 + 32.2593i
45.19 −2.80528 14.0226 − 24.2878i −24.1304 −43.1340 + 74.7104i −39.3372 + 68.1341i 12.1145 − 20.9830i 157.462 −271.764 − 470.709i 121.003 − 209.584i
45.20 −2.11993 8.64470 − 14.9731i −27.5059 9.49072 − 16.4384i −18.3261 + 31.7418i 72.2258 − 125.099i 126.148 −27.9616 − 48.4310i −20.1196 + 34.8483i
See all 90 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.45
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
109.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 109.6.c.a ✓ 90
109.c even 3 1 inner 109.6.c.a ✓ 90
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
109.6.c.a ✓ 90 1.a even 1 1 trivial
109.6.c.a ✓ 90 109.c even 3 1 inner

Hecke kernels

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