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Properties

Label 117.3.bb.a
Level $117$
Weight $3$
Character orbit 117.bb
Analytic conductor $3.188$
Analytic rank $0$
Dimension $104$
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] = [117,3,Mod(7,117)]
 
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([8, 11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 117.bb (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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 2 q^{2} - 2 q^{3} - 6 q^{4} - 2 q^{5} + 10 q^{6} - 6 q^{7} + 18 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 104 q - 2 q^{2} - 2 q^{3} - 6 q^{4} - 2 q^{5} + 10 q^{6} - 6 q^{7} + 18 q^{8} - 2 q^{9} - 12 q^{10} - 14 q^{11} - 54 q^{12} - 2 q^{13} - 4 q^{14} - 44 q^{15} + 162 q^{16} - 12 q^{17} + 34 q^{18} - 36 q^{19} - 82 q^{20} + 106 q^{21} + 2 q^{22} - 66 q^{24} + 52 q^{26} + 4 q^{27} + 56 q^{28} + 74 q^{29} - 438 q^{30} - 34 q^{31} + 22 q^{32} - 68 q^{33} + 6 q^{34} - 106 q^{35} - 6 q^{36} + 14 q^{37} - 216 q^{38} - 176 q^{39} - 36 q^{40} - 26 q^{41} - 220 q^{42} + 496 q^{44} - 50 q^{45} + 118 q^{47} + 130 q^{48} + 92 q^{50} + 76 q^{52} - 208 q^{53} + 904 q^{54} - 4 q^{55} - 86 q^{57} - 76 q^{58} + 430 q^{59} - 358 q^{60} - 4 q^{61} - 282 q^{62} - 326 q^{63} + 406 q^{65} - 412 q^{66} - 72 q^{67} - 1068 q^{68} + 204 q^{69} - 106 q^{70} - 8 q^{71} - 642 q^{72} + 90 q^{73} + 356 q^{74} + 204 q^{75} - 194 q^{76} + 432 q^{77} + 646 q^{78} - 16 q^{79} - 278 q^{80} - 182 q^{81} - 12 q^{82} - 536 q^{83} + 310 q^{84} - 234 q^{85} + 414 q^{86} + 434 q^{87} - 6 q^{88} + 400 q^{89} + 798 q^{90} + 120 q^{91} + 48 q^{92} - 116 q^{93} - 244 q^{94} + 858 q^{95} + 1528 q^{96} + 344 q^{97} + 280 q^{98} - 524 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.985740 + 3.67883i 1.51073 − 2.59185i −9.09802 − 5.25275i −1.72850 + 6.45085i 8.04580 + 8.11262i −6.33066 + 6.33066i 17.5199 − 17.5199i −4.43538 − 7.83118i −22.0278 − 12.7177i
7.2 −0.954072 + 3.56064i −1.50245 + 2.59666i −8.30384 − 4.79422i 1.83667 − 6.85456i −7.81234 − 7.82709i −1.57948 + 1.57948i 14.5667 − 14.5667i −4.48530 − 7.80270i 22.6543 + 13.0795i
7.3 −0.849385 + 3.16995i −2.32988 − 1.88988i −5.86301 − 3.38501i 0.483124 − 1.80304i 7.96979 − 5.78037i 6.39811 − 6.39811i 6.42800 − 6.42800i 1.85670 + 8.80640i 5.30519 + 3.06295i
7.4 −0.825818 + 3.08200i 1.49477 + 2.60109i −5.35262 − 3.09034i −1.67984 + 6.26926i −9.25095 + 2.45885i 5.59737 − 5.59737i 4.91998 − 4.91998i −4.53132 + 7.77606i −17.9346 − 10.3545i
7.5 −0.738153 + 2.75483i 2.39100 − 1.81193i −3.58010 − 2.06697i 1.54818 − 5.77788i 3.22664 + 7.92428i 5.82885 − 5.82885i 0.270108 − 0.270108i 2.43379 − 8.66468i 14.7743 + 8.52992i
7.6 −0.687085 + 2.56424i 2.77842 + 1.13153i −2.63912 − 1.52369i 0.672006 − 2.50796i −4.81053 + 6.34707i −6.84868 + 6.84868i −1.78820 + 1.78820i 6.43927 + 6.28775i 5.96928 + 3.44636i
7.7 −0.579256 + 2.16181i −1.71409 − 2.46209i −0.873789 − 0.504482i 0.365478 − 1.36398i 6.31547 − 2.27936i −7.55072 + 7.55072i −4.73348 + 4.73348i −3.12377 + 8.44050i 2.73696 + 1.58019i
7.8 −0.533716 + 1.99185i −1.28337 + 2.71163i −0.218531 − 0.126169i −0.459354 + 1.71433i −4.71622 − 4.00353i −2.37210 + 2.37210i −5.46461 + 5.46461i −5.70591 − 6.96007i −3.16954 − 1.82993i
7.9 −0.412698 + 1.54021i −2.94208 + 0.586681i 1.26217 + 0.728715i −1.42093 + 5.30297i 0.310577 − 4.77354i 2.67174 − 2.67174i −6.15332 + 6.15332i 8.31161 − 3.45212i −7.58129 − 4.37706i
7.10 −0.326843 + 1.21980i 0.212026 − 2.99250i 2.08303 + 1.20264i −2.20981 + 8.24713i 3.58094 + 1.23671i 4.91549 − 4.91549i −5.71960 + 5.71960i −8.91009 − 1.26898i −9.33755 − 5.39104i
7.11 −0.227045 + 0.847343i −2.93694 + 0.611861i 2.79766 + 1.61523i 2.29731 − 8.57367i 0.148361 − 2.62752i 0.760861 − 0.760861i −4.48504 + 4.48504i 8.25125 − 3.59400i 6.74325 + 3.89322i
7.12 −0.203464 + 0.759340i 2.94965 − 0.547339i 2.92890 + 1.69100i −0.980067 + 3.65766i −0.184532 + 2.35115i −1.50909 + 1.50909i −4.10347 + 4.10347i 8.40084 − 3.22892i −2.57800 − 1.48841i
7.13 −0.150066 + 0.560053i 1.88011 + 2.33777i 3.17296 + 1.83191i 1.19056 − 4.44321i −1.59142 + 0.702140i 8.23071 − 8.23071i −3.14207 + 3.14207i −1.93038 + 8.79054i 2.30977 + 1.33355i
7.14 −0.00774623 + 0.0289093i 1.19499 − 2.75172i 3.46333 + 1.99955i 1.76908 − 6.60228i 0.0702937 + 0.0558619i −3.33116 + 3.33116i −0.169286 + 0.169286i −6.14397 − 6.57659i 0.177164 + 0.102286i
7.15 0.107592 − 0.401541i −0.191564 + 2.99388i 3.31444 + 1.91359i −0.322695 + 1.20431i 1.18155 + 0.399039i −3.52152 + 3.52152i 2.30079 − 2.30079i −8.92661 − 1.14704i 0.448861 + 0.259150i
7.16 0.224300 − 0.837100i −2.71604 − 1.27402i 2.81368 + 1.62448i −1.33823 + 4.99435i −1.67569 + 1.98783i −8.46705 + 8.46705i 4.44216 − 4.44216i 5.75373 + 6.92059i 3.88061 + 2.24047i
7.17 0.254432 − 0.949552i −1.28144 − 2.71254i 2.62719 + 1.51681i 0.323014 − 1.20550i −2.90174 + 0.526641i 5.37476 − 5.37476i 4.88921 − 4.88921i −5.71580 + 6.95195i −1.06250 − 0.613437i
7.18 0.449246 − 1.67661i −2.83824 + 0.971803i 0.854911 + 0.493583i −0.762199 + 2.84456i 0.354267 + 5.19519i 6.90561 − 6.90561i 6.12106 − 6.12106i 7.11120 − 5.51642i 4.42680 + 2.55582i
7.19 0.524404 − 1.95710i 2.63868 + 1.42737i −0.0911449 − 0.0526225i 2.10179 − 7.84399i 4.17724 − 4.41564i −7.76479 + 7.76479i 5.58001 − 5.58001i 4.92522 + 7.53274i −14.2493 − 8.22684i
7.20 0.535255 − 1.99760i 2.51826 − 1.63045i −0.239808 − 0.138453i −0.486029 + 1.81388i −1.90908 − 5.90318i 0.855863 − 0.855863i 5.44445 − 5.44445i 3.68326 − 8.21180i 3.36327 + 1.94178i
See next 80 embeddings (of 104 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.26
Significant digits:
Format:

Inner twists

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

Hecke kernels

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