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Properties

Label 109.2.k.a
Level $109$
Weight $2$
Character orbit 109.k
Analytic conductor $0.870$
Analytic rank $0$
Dimension $162$
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] = [109,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("109.12");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 109 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(0.870369382032\)
Analytic rank: \(0\)
Dimension: \(162\)
Relative dimension: \(9\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 162 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 27 q^{8} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 162 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 27 q^{8} - 36 q^{9} - 18 q^{10} + 18 q^{11} - 18 q^{12} - 27 q^{13} - 18 q^{15} - 36 q^{16} - 18 q^{17} + 45 q^{18} - 18 q^{19} + 45 q^{20} + 18 q^{21} - 72 q^{22} - 9 q^{23} + 63 q^{24} - 18 q^{25} - 18 q^{26} + 36 q^{27} - 18 q^{28} - 54 q^{29} + 36 q^{30} - 36 q^{32} + 180 q^{34} - 27 q^{35} - 90 q^{36} - 45 q^{37} - 18 q^{38} + 27 q^{39} - 27 q^{40} + 27 q^{41} + 36 q^{42} - 18 q^{43} + 54 q^{44} - 63 q^{45} + 27 q^{46} + 126 q^{47} - 135 q^{48} - 36 q^{49} - 108 q^{50} + 54 q^{52} - 27 q^{53} + 234 q^{54} - 18 q^{55} - 81 q^{56} + 90 q^{57} + 36 q^{58} + 90 q^{60} - 90 q^{61} - 54 q^{62} - 99 q^{63} + 135 q^{64} + 45 q^{65} - 72 q^{66} + 135 q^{67} - 27 q^{68} + 9 q^{69} + 90 q^{70} + 54 q^{71} + 45 q^{72} - 18 q^{73} + 90 q^{74} - 63 q^{75} - 27 q^{77} - 135 q^{78} + 72 q^{79} + 180 q^{80} + 108 q^{81} - 90 q^{82} - 9 q^{83} + 252 q^{84} - 72 q^{85} + 162 q^{86} + 198 q^{87} - 18 q^{88} - 63 q^{89} - 90 q^{90} - 9 q^{91} - 108 q^{92} + 9 q^{93} - 18 q^{94} - 72 q^{95} - 117 q^{96} + 54 q^{97} + 18 q^{98} - 72 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1 −0.716786 + 1.96935i 0.289190 − 0.0685393i −1.83248 − 1.53763i −1.06907 + 0.703137i −0.0723093 + 0.618645i 0.245824 + 4.22064i 0.711704 − 0.410903i −2.60196 + 1.30676i −0.618432 − 2.60937i
12.2 −0.575546 + 1.58130i 3.13212 − 0.742326i −0.637163 − 0.534643i −1.97964 + 1.30203i −0.628838 + 5.38006i −0.197391 − 3.38908i −1.70252 + 0.982949i 6.57822 − 3.30371i −0.919523 − 3.87977i
12.3 −0.506788 + 1.39239i 0.394673 − 0.0935391i −0.149821 − 0.125715i 3.59972 − 2.36757i −0.0697725 + 0.596942i −0.0765931 − 1.31505i −2.31549 + 1.33685i −2.53388 + 1.27256i 1.47229 + 6.21206i
12.4 −0.239476 + 0.657956i −1.75991 + 0.417106i 1.15653 + 0.970446i −2.37732 + 1.56359i 0.147019 − 1.25783i −0.0864827 − 1.48485i −2.12822 + 1.22873i 0.242400 − 0.121738i −0.459460 − 1.93861i
12.5 0.164739 − 0.452617i −2.80755 + 0.665402i 1.35437 + 1.13645i 2.27664 − 1.49737i −0.161341 + 1.38036i 0.280853 + 4.82206i 1.57176 − 0.907456i 4.75869 − 2.38991i −0.302683 − 1.27712i
12.6 0.172188 − 0.473083i 0.913964 − 0.216613i 1.33793 + 1.12266i −0.965778 + 0.635202i 0.0548976 − 0.469679i 0.0138284 + 0.237425i 1.63348 − 0.943088i −1.89249 + 0.950444i 0.134208 + 0.566267i
12.7 0.487037 − 1.33812i −0.220603 + 0.0522840i −0.0212778 − 0.0178542i 0.738458 − 0.485692i −0.0374796 + 0.320659i −0.140449 − 2.41141i 2.43219 − 1.40422i −2.63497 + 1.32333i −0.290259 − 1.22470i
12.8 0.779517 − 2.14170i 2.16045 − 0.512036i −2.44716 − 2.05341i −3.03122 + 1.99367i 0.587477 − 5.02619i 0.0940275 + 1.61439i −2.35780 + 1.36128i 1.72446 − 0.866058i 1.90696 + 8.04608i
12.9 0.856462 − 2.35311i −3.02292 + 0.716445i −3.27152 − 2.74513i −1.53889 + 1.01214i −0.903142 + 7.72687i −0.176866 − 3.03667i −4.92424 + 2.84301i 5.94384 − 2.98511i 1.06368 + 4.48803i
20.1 −1.73354 − 2.06595i 0.0958680 + 1.64599i −0.915699 + 5.19319i 1.67264 − 2.24675i 3.23434 − 3.05144i −1.72076 − 3.98917i 7.64508 − 4.41389i 0.279621 − 0.0326830i −7.54126 + 0.439229i
20.2 −1.22633 − 1.46148i −0.153600 − 2.63722i −0.284747 + 1.61488i 2.16989 − 2.91467i −3.66587 + 3.45857i 1.61423 + 3.74221i −0.595147 + 0.343608i −3.95160 + 0.461876i −6.92072 + 0.403086i
20.3 −1.06782 − 1.27258i 0.122056 + 2.09562i −0.131920 + 0.748156i −1.54515 + 2.07549i 2.53651 − 2.39308i 1.08463 + 2.51445i −1.78439 + 1.03022i −1.39703 + 0.163289i 4.29116 − 0.249932i
20.4 −0.361218 − 0.430483i 0.0381135 + 0.654383i 0.292459 − 1.65862i 0.758472 − 1.01881i 0.267933 − 0.252782i −0.206367 − 0.478414i −1.79298 + 1.03518i 2.55295 − 0.298397i −0.712552 + 0.0415014i
20.5 −0.0680874 − 0.0811434i −0.165173 − 2.83591i 0.345348 − 1.95857i −1.92274 + 2.58269i −0.218869 + 0.206492i −0.334469 − 0.775386i −0.365906 + 0.211256i −5.03539 + 0.588553i 0.340483 − 0.0198309i
20.6 0.598140 + 0.712835i −0.0421582 − 0.723828i 0.196933 − 1.11687i 0.0743607 − 0.0998837i 0.490754 − 0.463002i 0.928483 + 2.15247i 2.52568 − 1.45820i 2.45756 − 0.287248i 0.115679 − 0.00673752i
20.7 0.609675 + 0.726582i 0.197532 + 3.39150i 0.191078 − 1.08366i 0.638110 − 0.857130i −2.34377 + 2.21123i −0.491593 − 1.13964i 2.54669 − 1.47033i −8.48351 + 0.991580i 1.01181 − 0.0589315i
20.8 1.32885 + 1.58366i 0.0182805 + 0.313865i −0.394844 + 2.23927i −1.59085 + 2.13689i −0.472763 + 0.446028i −1.74610 − 4.04792i −0.490223 + 0.283030i 2.88154 − 0.336804i −5.49810 + 0.320228i
20.9 1.67094 + 1.99135i −0.175825 − 3.01880i −0.826132 + 4.68523i 0.433774 − 0.582660i 5.71769 − 5.39437i 0.688516 + 1.59616i −6.20783 + 3.58409i −6.10254 + 0.713285i 1.88509 − 0.109794i
28.1 −1.71458 − 2.04335i 2.24596 + 1.12796i −0.888221 + 5.03735i −0.724996 − 1.68073i −1.54604 − 6.52327i 3.52586 + 0.412114i 7.19593 − 4.15457i 1.98056 + 2.66035i −2.19126 + 4.36316i
28.2 −1.68536 − 2.00853i −1.75077 − 0.879270i −0.846470 + 4.80057i 1.17661 + 2.72769i 1.18463 + 4.99836i −3.76592 − 0.440173i 6.52735 − 3.76857i 0.500606 + 0.672430i 3.49565 − 6.96040i
See next 80 embeddings (of 162 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.9
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 109.2.k.a ✓ 162
3.b odd 2 1 981.2.ce.d 162
109.k even 54 1 inner 109.2.k.a ✓ 162
327.t odd 54 1 981.2.ce.d 162
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
109.2.k.a ✓ 162 1.a even 1 1 trivial
109.2.k.a ✓ 162 109.k even 54 1 inner
981.2.ce.d 162 3.b odd 2 1
981.2.ce.d 162 327.t odd 54 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(109, [\chi])\).