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Properties

Label 109.12.a.b
Level $109$
Weight $12$
Character orbit 109.a
Self dual yes
Analytic conductor $83.749$
Analytic rank $0$
Dimension $51$
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] = [109,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("109.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 109 \)
Weight: \( k \) \(=\) \( 12 \)
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: \(83.7494066810\)
Analytic rank: \(0\)
Dimension: \(51\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 51 q + 64 q^{2} + 1448 q^{3} + 55296 q^{4} + 28510 q^{5} + 68137 q^{6} + 63799 q^{7} + 81699 q^{8} + 3538177 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 51 q + 64 q^{2} + 1448 q^{3} + 55296 q^{4} + 28510 q^{5} + 68137 q^{6} + 63799 q^{7} + 81699 q^{8} + 3538177 q^{9} + 189116 q^{10} + 2937587 q^{11} + 3924228 q^{12} + 1083565 q^{13} + 5512571 q^{14} + 6468192 q^{15} + 62792200 q^{16} + 14934145 q^{17} + 3321536 q^{18} + 38253131 q^{19} + 70844565 q^{20} + 13493341 q^{21} + 71086867 q^{22} + 31608233 q^{23} + 210715333 q^{24} + 594654619 q^{25} + 222067823 q^{26} + 463838507 q^{27} - 451467133 q^{28} + 294771230 q^{29} + 622876581 q^{30} + 597891001 q^{31} + 1751544211 q^{32} + 856624649 q^{33} + 3600129255 q^{34} + 2154457409 q^{35} + 5810056573 q^{36} + 1049549832 q^{37} + 797284259 q^{38} + 521195904 q^{39} - 1639712948 q^{40} + 959223906 q^{41} - 8573239951 q^{42} + 754083105 q^{43} + 4862133104 q^{44} + 1613232373 q^{45} - 10789507953 q^{46} + 1366944934 q^{47} - 8631968454 q^{48} + 14466656874 q^{49} - 14225400645 q^{50} + 7455316295 q^{51} - 31856411112 q^{52} + 1337552875 q^{53} + 1512110336 q^{54} - 3843522901 q^{55} + 4938777460 q^{56} - 15445555801 q^{57} - 3073486374 q^{58} + 47301482303 q^{59} - 722888958 q^{60} + 13822490170 q^{61} - 1932669698 q^{62} + 5003864768 q^{63} + 78979318895 q^{64} + 21357350656 q^{65} + 45645294008 q^{66} + 20274193709 q^{67} + 45486098723 q^{68} + 31231196738 q^{69} + 83046588809 q^{70} + 93099175414 q^{71} + 102611031811 q^{72} + 57191515984 q^{73} + 165148181624 q^{74} + 124627738655 q^{75} + 159668076691 q^{76} + 110905139448 q^{77} + 184682336925 q^{78} + 76752256986 q^{79} + 311124958373 q^{80} + 388510173363 q^{81} + 254612632946 q^{82} + 245477229195 q^{83} + 321999423280 q^{84} + 170071396859 q^{85} + 429895146204 q^{86} + 186846129658 q^{87} + 556554239670 q^{88} + 273866131305 q^{89} + 917160276496 q^{90} + 326700387184 q^{91} + 272594054018 q^{92} + 142079256652 q^{93} + 663896555109 q^{94} + 204613598519 q^{95} + 1583265529266 q^{96} + 412100927960 q^{97} + 546587755328 q^{98} + 841687453133 q^{99}+O(q^{100}) \) 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 −88.1977 −483.057 5730.83 12763.6 42604.5 −48634.4 −324817. 56197.5 −1.12572e6
1.2 −82.9986 240.275 4840.77 5064.51 −19942.5 75530.4 −231796. −119415. −420347.
1.3 −81.8642 −759.718 4653.76 −6777.51 62193.7 −52407.6 −213318. 400024. 554836.
1.4 −78.1518 −282.297 4059.70 7326.97 22062.0 31719.9 −157218. −97455.3 −572616.
1.5 −78.1071 −249.237 4052.71 −606.166 19467.2 5750.12 −156582. −115028. 47345.9
1.6 −73.7581 783.646 3392.25 −8206.76 −57800.2 17505.1 −99149.4 436954. 605314.
1.7 −73.2300 687.591 3314.63 −6750.06 −50352.3 −59744.2 −92755.6 295634. 494307.
1.8 −72.6946 354.317 3236.50 −9380.62 −25757.0 25663.6 −86397.8 −51606.3 681920.
1.9 −64.9796 204.726 2174.35 11793.6 −13303.0 10527.1 −8210.08 −135234. −766343.
1.10 −64.2070 794.399 2074.54 12560.5 −51006.0 20386.1 −1704.12 453923. −806472.
1.11 −60.7522 −479.737 1642.83 −12250.4 29145.1 −12479.4 24615.2 53000.5 744238.
1.12 −59.8303 −16.1271 1531.66 −4624.94 964.888 −24948.1 30892.6 −176887. 276712.
1.13 −54.0946 −423.863 878.230 12046.7 22928.7 −62864.9 63278.3 2513.02 −651659.
1.14 −43.8325 633.553 −126.708 1372.39 −27770.3 −10933.4 95323.0 224243. −60155.5
1.15 −42.6617 20.8206 −227.978 6960.16 −888.241 −56107.5 97097.1 −176714. −296933.
1.16 −38.8804 −162.174 −536.313 −3760.14 6305.39 60111.5 100479. −150847. 146196.
1.17 −38.2922 −665.223 −581.709 −7760.69 25472.8 −6327.92 100697. 265374. 297174.
1.18 −37.6066 533.772 −633.744 −7366.65 −20073.4 −69549.8 100851. 107766. 277035.
1.19 −22.7804 −510.674 −1529.05 3753.78 11633.4 −10029.2 81486.8 83640.4 −85512.7
1.20 −18.1326 140.513 −1719.21 −7824.83 −2547.86 9888.60 68309.4 −157403. 141885.
See all 51 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.51
Significant digits:
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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 109.12.a.b ✓ 51
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
109.12.a.b ✓ 51 1.a even 1 1 trivial