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Properties

Label 109.8.a.a
Level $109$
Weight $8$
Character orbit 109.a
Self dual yes
Analytic conductor $34.050$
Analytic rank $1$
Dimension $30$
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] = [109,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("109.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 109 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(34.0499677778\)
Analytic rank: \(1\)
Dimension: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 24 q^{2} - 136 q^{3} + 1728 q^{4} - 1070 q^{5} - 2159 q^{6} - 501 q^{7} - 3309 q^{8} + 18094 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 24 q^{2} - 136 q^{3} + 1728 q^{4} - 1070 q^{5} - 2159 q^{6} - 501 q^{7} - 3309 q^{8} + 18094 q^{9} - 6004 q^{10} - 28187 q^{11} - 17868 q^{12} - 13479 q^{13} - 24165 q^{14} - 20688 q^{15} + 101640 q^{16} - 57723 q^{17} - 29152 q^{18} - 113499 q^{19} - 217915 q^{20} - 12257 q^{21} - 71133 q^{22} - 140323 q^{23} - 567611 q^{24} + 311834 q^{25} - 388361 q^{26} - 432901 q^{27} + 604927 q^{28} - 175126 q^{29} + 998183 q^{30} - 239379 q^{31} - 163791 q^{32} - 265235 q^{33} - 567561 q^{34} - 1752007 q^{35} - 40683 q^{36} - 526134 q^{37} - 1156781 q^{38} - 2402266 q^{39} - 3929432 q^{40} - 2063788 q^{41} - 8561809 q^{42} - 2625775 q^{43} - 5890346 q^{44} - 5764597 q^{45} - 6077517 q^{46} - 4301248 q^{47} - 7747502 q^{48} + 864635 q^{49} - 9075313 q^{50} - 3663853 q^{51} - 11996446 q^{52} - 3280939 q^{53} - 11204466 q^{54} - 5986961 q^{55} - 11667808 q^{56} - 6728047 q^{57} - 8099362 q^{58} - 13898757 q^{59} - 15463314 q^{60} - 2209438 q^{61} - 6460976 q^{62} - 4517284 q^{63} + 2351235 q^{64} - 5418346 q^{65} - 10224712 q^{66} - 5870869 q^{67} - 17533379 q^{68} - 9359828 q^{69} + 991179 q^{70} - 10924606 q^{71} - 10729843 q^{72} - 11393430 q^{73} - 7314764 q^{74} - 3522673 q^{75} - 10897925 q^{76} - 3136830 q^{77} + 22728413 q^{78} - 4869920 q^{79} - 32623771 q^{80} + 17171374 q^{81} - 5658878 q^{82} - 15975941 q^{83} + 16719812 q^{84} + 7113689 q^{85} - 10498616 q^{86} - 2113602 q^{87} + 26335826 q^{88} - 18161151 q^{89} + 25760030 q^{90} - 27586516 q^{91} + 2795570 q^{92} - 1301132 q^{93} + 30185605 q^{94} + 5624421 q^{95} - 7092032 q^{96} + 4907686 q^{97} - 9138506 q^{98} - 68365609 q^{99}+O(q^{100}) \) 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 −21.5077 69.5717 334.582 −485.477 −1496.33 1793.39 −4443.12 2653.22 10441.5
1.2 −21.5055 −33.8744 334.488 189.786 728.488 1114.18 −4440.62 −1039.52 −4081.45
1.3 −19.5917 −68.2799 255.833 298.706 1337.72 −1277.25 −2504.46 2475.15 −5852.14
1.4 −18.0707 −26.5084 198.551 −397.530 479.027 121.323 −1274.91 −1484.30 7183.65
1.5 −16.8795 −14.2285 156.918 458.247 240.170 −254.734 −488.121 −1984.55 −7734.99
1.6 −16.3584 40.3294 139.596 344.091 −659.722 810.127 −189.686 −560.542 −5628.76
1.7 −15.3058 59.5520 106.267 42.1382 −911.491 −119.271 332.636 1359.44 −644.959
1.8 −14.1481 81.7536 72.1685 −283.962 −1156.66 −741.768 789.909 4496.64 4017.51
1.9 −11.9726 −53.3584 15.3429 −167.341 638.839 −1508.45 1348.80 660.121 2003.50
1.10 −10.4734 15.8860 −18.3084 −518.985 −166.380 166.675 1532.34 −1934.64 5435.52
1.11 −8.25915 −1.58984 −59.7865 244.574 13.1307 84.6888 1550.96 −2184.47 −2019.98
1.12 −6.33251 −82.4354 −87.8993 −460.382 522.024 −811.947 1367.19 4608.60 2915.37
1.13 −4.85523 −5.37598 −104.427 55.1628 26.1016 289.383 1128.48 −2158.10 −267.828
1.14 −2.98427 −59.7654 −119.094 −223.151 178.356 −318.715 737.395 1384.91 665.943
1.15 −1.87467 59.8111 −124.486 26.7668 −112.126 −455.965 473.327 1390.37 −50.1789
1.16 −0.396446 −56.9279 −127.843 −248.317 22.5689 1245.17 101.428 1053.79 98.4443
1.17 2.09462 −68.0677 −123.613 438.454 −142.576 −950.367 −527.034 2446.21 918.396
1.18 2.37161 39.8433 −122.375 426.158 94.4925 −109.740 −593.792 −599.514 1010.68
1.19 3.19951 38.0332 −117.763 −189.926 121.688 1395.02 −786.322 −740.478 −607.669
1.20 7.47181 86.8353 −72.1721 9.40940 648.817 −1521.99 −1495.65 5353.37 70.3052
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
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 109.8.a.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
109.8.a.a ✓ 30 1.a even 1 1 trivial