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Properties

Label 117.3.w.a
Level $117$
Weight $3$
Character orbit 117.w
Analytic conductor $3.188$
Analytic rank $0$
Dimension $104$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,3,Mod(58,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([4, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.58");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 117.w (of order \(12\), degree \(4\), 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: \(3.18801909302\)
Analytic rank: \(0\)
Dimension: \(104\)
Relative dimension: \(26\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 2 q^{2} - 2 q^{3} - 2 q^{5} - 32 q^{6} + 18 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 104 q - 2 q^{2} - 2 q^{3} - 2 q^{5} - 32 q^{6} + 18 q^{8} - 2 q^{9} - 12 q^{10} + 22 q^{11} + 54 q^{12} - 2 q^{13} - 4 q^{14} + 70 q^{15} - 324 q^{16} - 12 q^{17} - 86 q^{18} - 36 q^{19} + 134 q^{20} - 32 q^{21} - 4 q^{22} - 6 q^{23} + 30 q^{24} + 52 q^{26} + 4 q^{27} + 56 q^{28} - 148 q^{29} - 246 q^{30} + 62 q^{31} + 70 q^{32} + 52 q^{33} - 18 q^{34} - 106 q^{35} + 96 q^{36} + 14 q^{37} + 216 q^{38} + 118 q^{39} - 36 q^{40} - 26 q^{41} + 278 q^{42} - 198 q^{43} + 496 q^{44} - 80 q^{45} - 8 q^{47} + 58 q^{48} - 6 q^{49} - 292 q^{50} + 28 q^{52} - 208 q^{53} + 58 q^{54} - 4 q^{55} + 186 q^{56} - 140 q^{57} + 146 q^{58} - 434 q^{59} + 140 q^{60} + 2 q^{61} - 282 q^{62} - 266 q^{63} - 260 q^{65} - 412 q^{66} + 138 q^{67} + 534 q^{68} - 294 q^{69} + 212 q^{70} - 8 q^{71} - 306 q^{72} + 90 q^{73} - 178 q^{74} - 60 q^{75} - 98 q^{76} - 432 q^{77} + 352 q^{78} - 16 q^{79} - 278 q^{80} - 230 q^{81} - 12 q^{82} - 50 q^{83} + 1066 q^{84} + 204 q^{85} + 330 q^{86} - 154 q^{87} + 400 q^{89} - 798 q^{90} + 120 q^{91} + 48 q^{92} - 128 q^{93} + 122 q^{94} + 958 q^{96} - 232 q^{97} + 280 q^{98} + 940 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
58.1 −2.66555 + 2.66555i 2.99690 − 0.136394i − 10.2103i 0.151696 + 0.566137i −7.62482 + 8.35195i −1.59453 + 0.427252i 16.5540 + 16.5540i 8.96279 − 0.817519i −1.91342 − 1.10471i
58.2 −2.59584 + 2.59584i −1.41809 + 2.64368i − 9.47679i −0.898750 − 3.35418i −3.18143 − 10.5437i 13.2917 − 3.56149i 14.2169 + 14.2169i −4.97805 − 7.49794i 11.0399 + 6.37391i
58.3 −2.46426 + 2.46426i −1.28173 − 2.71241i − 8.14515i 2.29040 + 8.54788i 9.84260 + 3.52555i 1.69578 − 0.454383i 10.2147 + 10.2147i −5.71431 + 6.95317i −26.7083 − 15.4201i
58.4 −2.05515 + 2.05515i −2.55279 + 1.57584i − 4.44728i 1.02938 + 3.84171i 2.00779 − 8.48495i −11.7531 + 3.14923i 0.919224 + 0.919224i 4.03348 − 8.04556i −10.0108 − 5.77976i
58.5 −1.88215 + 1.88215i 1.48796 + 2.60499i − 3.08498i −1.58439 − 5.91301i −7.70355 − 2.10241i −9.40849 + 2.52100i −1.72221 − 1.72221i −4.57193 + 7.75225i 14.1112 + 8.14711i
58.6 −1.82132 + 1.82132i 1.56629 − 2.55866i − 2.63444i −0.750566 − 2.80115i 1.80743 + 7.51287i −0.00333442 0.000893455i −2.48713 − 2.48713i −4.09348 − 8.01520i 6.46883 + 3.73478i
58.7 −1.74157 + 1.74157i −2.81069 − 1.04883i − 2.06613i −0.760473 − 2.83812i 6.72161 − 3.06840i 4.20536 − 1.12682i −3.36797 − 3.36797i 6.79992 + 5.89586i 6.26721 + 3.61837i
58.8 −1.43729 + 1.43729i 2.21903 + 2.01889i − 0.131606i 1.39764 + 5.21605i −6.09112 + 0.287655i 6.72649 − 1.80236i −5.56000 − 5.56000i 0.848165 + 8.95995i −9.50579 − 5.48817i
58.9 −0.781876 + 0.781876i 2.42393 − 1.76765i 2.77734i 1.97793 + 7.38172i −0.513129 + 3.27729i −7.60192 + 2.03693i −5.29904 − 5.29904i 2.75084 − 8.56930i −7.31808 − 4.22509i
58.10 −0.656548 + 0.656548i −1.43494 + 2.63457i 3.13789i 0.963843 + 3.59711i −0.787615 − 2.67182i 2.03338 − 0.544842i −4.68636 − 4.68636i −4.88191 − 7.56088i −2.99448 − 1.72886i
58.11 −0.547586 + 0.547586i −1.26837 − 2.71868i 3.40030i −0.822537 − 3.06975i 2.18325 + 0.794174i −7.72490 + 2.06988i −4.05230 − 4.05230i −5.78249 + 6.89658i 2.13136 + 1.23054i
58.12 −0.513828 + 0.513828i 2.98609 − 0.288554i 3.47196i −2.21291 − 8.25869i −1.38607 + 1.68260i 10.0413 − 2.69055i −3.83930 − 3.83930i 8.83347 − 1.72330i 5.38060 + 3.10649i
58.13 −0.0203305 + 0.0203305i −2.72600 + 1.25256i 3.99917i −2.28573 − 8.53046i 0.0299559 − 0.0808862i −1.33641 + 0.358091i −0.162627 − 0.162627i 5.86220 − 6.82895i 0.219899 + 0.126959i
58.14 0.0727269 − 0.0727269i −0.510508 − 2.95624i 3.98942i 0.969534 + 3.61835i −0.252126 − 0.177871i 10.9873 − 2.94403i 0.581046 + 0.581046i −8.47876 + 3.01838i 0.333663 + 0.192640i
58.15 0.549055 − 0.549055i 2.56691 + 1.55273i 3.39708i 0.0757358 + 0.282650i 2.26191 − 0.556840i −2.35467 + 0.630931i 4.06140 + 4.06140i 4.17805 + 7.97144i 0.196773 + 0.113607i
58.16 0.574886 − 0.574886i 0.355272 + 2.97889i 3.33901i −0.368282 − 1.37445i 1.91676 + 1.50828i −4.95189 + 1.32685i 4.21910 + 4.21910i −8.74756 + 2.11663i −1.00187 − 0.578431i
58.17 0.982617 − 0.982617i −2.77223 − 1.14662i 2.06893i 1.58072 + 5.89931i −3.85073 + 1.59736i −10.8853 + 2.91671i 5.96343 + 5.96343i 6.37054 + 6.35738i 7.35000 + 4.24353i
58.18 1.20073 − 1.20073i −2.90912 + 0.732830i 1.11649i 0.331796 + 1.23828i −2.61313 + 4.37300i 8.36838 − 2.24230i 6.14353 + 6.14353i 7.92592 − 4.26378i 1.88524 + 1.08844i
58.19 1.20903 − 1.20903i 2.63140 − 1.44075i 1.07649i 0.734621 + 2.74164i 1.43953 − 4.92335i 3.07395 − 0.823664i 6.13763 + 6.13763i 4.84850 − 7.58235i 4.20291 + 2.42655i
58.20 1.52354 − 1.52354i 1.61877 − 2.52578i − 0.642329i −1.83169 − 6.83596i −1.38187 − 6.31438i −7.99700 + 2.14279i 5.11554 + 5.11554i −3.75916 − 8.17733i −13.2055 − 7.62419i
See next 80 embeddings (of 104 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.w odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.3.w.a ✓ 104
3.b odd 2 1 351.3.z.a 104
9.c even 3 1 117.3.bb.a yes 104
9.d odd 6 1 351.3.be.a 104
13.f odd 12 1 117.3.bb.a yes 104
39.k even 12 1 351.3.be.a 104
117.w odd 12 1 inner 117.3.w.a ✓ 104
117.x even 12 1 351.3.z.a 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.3.w.a ✓ 104 1.a even 1 1 trivial
117.3.w.a ✓ 104 117.w odd 12 1 inner
117.3.bb.a yes 104 9.c even 3 1
117.3.bb.a yes 104 13.f odd 12 1
351.3.z.a 104 3.b odd 2 1
351.3.z.a 104 117.x even 12 1
351.3.be.a 104 9.d odd 6 1
351.3.be.a 104 39.k even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(117, [\chi])\).