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Properties

Label 109.16.a.a
Level $109$
Weight $16$
Character orbit 109.a
Self dual yes
Analytic conductor $155.536$
Analytic rank $1$
Dimension $66$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [109,16,Mod(1,109)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(109, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("109.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 109 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 109.a (trivial)

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: yes
Analytic conductor: \(155.535920559\)
Analytic rank: \(1\)
Dimension: \(66\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 384 q^{2} - 9775 q^{3} + 1032192 q^{4} - 550760 q^{5} - 1290263 q^{6} - 3953576 q^{7} - 15117261 q^{8} + 286423921 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q - 384 q^{2} - 9775 q^{3} + 1032192 q^{4} - 550760 q^{5} - 1290263 q^{6} - 3953576 q^{7} - 15117261 q^{8} + 286423921 q^{9} + 6483836 q^{10} - 418640282 q^{11} - 433698156 q^{12} - 251211569 q^{13} - 1592332821 q^{14} - 1067853378 q^{15} + 15344152328 q^{16} - 6669915144 q^{17} - 1130846704 q^{18} - 11923665296 q^{19} - 19811264155 q^{20} - 16836632576 q^{21} - 25066407453 q^{22} - 66240319423 q^{23} - 105321578171 q^{24} + 400217086854 q^{25} - 110366500241 q^{26} - 360361891108 q^{27} + 168304047263 q^{28} - 365318101624 q^{29} - 1696019730217 q^{30} - 581021125356 q^{31} - 945136776447 q^{32} - 80720646410 q^{33} + 509364050231 q^{34} - 1107221269012 q^{35} + 7914914022741 q^{36} + 885376534021 q^{37} - 245887726685 q^{38} - 14886955906 q^{39} + 4617883372968 q^{40} - 5158051651090 q^{41} - 922037781169 q^{42} - 3364734631553 q^{43} - 16009768816826 q^{44} - 13607122605862 q^{45} - 27613809885909 q^{46} - 15436064264767 q^{47} - 58305445438958 q^{48} + 22356385252652 q^{49} - 59201596698313 q^{50} - 31998753555286 q^{51} - 77785825258670 q^{52} - 35536412905909 q^{53} - 72434695159938 q^{54} - 48394117279296 q^{55} - 127390800143296 q^{56} - 41066353710544 q^{57} - 53471659238834 q^{58} - 182936201870706 q^{59} - 34432643772594 q^{60} - 69231508155360 q^{61} - 76728243090752 q^{62} - 989010429598 q^{63} + 206028006415075 q^{64} - 61149791847076 q^{65} + 209354337452312 q^{66} + 49820429470096 q^{67} - 183448493081939 q^{68} + 117351537759646 q^{69} + 488482013120139 q^{70} - 239459552004052 q^{71} + 601977723367037 q^{72} + 110709721135383 q^{73} + 234158962981804 q^{74} - 302768537362063 q^{75} + 418562925992027 q^{76} - 80154467168214 q^{77} + 13\!\cdots\!85 q^{78}+ \cdots - 35\!\cdots\!14 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −354.305 −4011.44 92763.9 70040.4 1.42127e6 −140406. −2.12568e7 1.74274e6 −2.48157e7
1.2 −353.821 2082.60 92421.0 261489. −736868. 2.67149e6 −2.11065e7 −1.00117e7 −9.25202e7
1.3 −334.651 −4439.85 79223.0 −140269. 1.48580e6 −1.94032e6 −1.55462e7 5.36340e6 4.69410e7
1.4 −328.934 310.070 75429.7 −330028. −101993. 36257.6 −1.40329e7 −1.42528e7 1.08557e8
1.5 −319.827 −7573.25 69521.5 −341591. 2.42213e6 2.66650e6 −1.17548e7 4.30051e7 1.09250e8
1.6 −308.551 1567.44 62435.5 165943. −483635. −358816. −9.15394e6 −1.18920e7 −5.12019e7
1.7 −308.549 5842.98 62434.3 −223085. −1.80284e6 2.67205e6 −9.15352e6 1.97915e7 6.88326e7
1.8 −306.868 −1874.14 61399.8 280092. 575114. −3.21333e6 −8.78616e6 −1.08365e7 −8.59512e7
1.9 −306.714 7378.59 61305.3 110226. −2.26312e6 1.02042e6 −8.75280e6 4.00947e7 −3.38080e7
1.10 −275.069 −171.632 42894.9 82311.8 47210.7 1.70364e6 −2.78560e6 −1.43194e7 −2.26414e7
1.11 −265.935 2413.49 37953.4 74697.7 −641830. 2.17525e6 −1.37897e6 −8.52399e6 −1.98647e7
1.12 −255.014 1099.27 32263.9 −131801. −280329. −1.42499e6 128553. −1.31405e7 3.36109e7
1.13 −253.710 6095.20 31600.7 −272913. −1.54641e6 −2.47471e6 296158. 2.28025e7 6.92408e7
1.14 −237.001 −4945.76 23401.7 −176275. 1.17215e6 −1.35556e6 2.21984e6 1.01116e7 4.17773e7
1.15 −224.871 −6095.40 17799.1 81001.9 1.37068e6 284079. 3.36608e6 2.28050e7 −1.82150e7
1.16 −218.339 −1557.07 14904.1 −205275. 339969. 3.17522e6 3.90040e6 −1.19244e7 4.48195e7
1.17 −206.395 5105.43 9831.04 −163344. −1.05374e6 −76922.7 4.73408e6 1.17165e7 3.37134e7
1.18 −188.481 5175.73 2757.27 256292. −975530. −3.65865e6 5.65647e6 1.24393e7 −4.83064e7
1.19 −185.015 −2820.54 1462.51 201560. 521841. −1.93307e6 5.79198e6 −6.39348e6 −3.72915e7
1.20 −166.316 4098.35 −5107.11 321246. −681620. 463286. 6.29922e6 2.44759e6 −5.34282e7
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.66
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(109\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 109.16.a.a ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
109.16.a.a ✓ 66 1.a even 1 1 trivial