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Properties

Label 229.12.a.b
Level $229$
Weight $12$
Character orbit 229.a
Self dual yes
Analytic conductor $175.951$
Analytic rank $0$
Dimension $107$
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] = [229,12,Mod(1,229)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(229, 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("229.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 229 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 229.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: \(175.950588348\)
Analytic rank: \(0\)
Dimension: \(107\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 107 q + 215 q^{2} + 1195 q^{3} + 116229 q^{4} + 18750 q^{5} + 68137 q^{6} + 99789 q^{7} + 503295 q^{8} + 7096244 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 107 q + 215 q^{2} + 1195 q^{3} + 116229 q^{4} + 18750 q^{5} + 68137 q^{6} + 99789 q^{7} + 503295 q^{8} + 7096244 q^{9} + 163393 q^{10} + 4680903 q^{11} + 2887940 q^{12} + 1856465 q^{13} + 7704819 q^{14} + 13823255 q^{15} + 127845081 q^{16} + 19613189 q^{17} + 31491627 q^{18} + 66018220 q^{19} + 66054712 q^{20} + 63294335 q^{21} + 48904063 q^{22} + 75737498 q^{23} + 286558307 q^{24} + 1186359323 q^{25} + 176662368 q^{26} + 331655368 q^{27} + 189180278 q^{28} + 701585129 q^{29} + 174323482 q^{30} + 196861121 q^{31} + 1661160130 q^{32} + 274644732 q^{33} + 672751525 q^{34} + 2913883563 q^{35} + 8719449140 q^{36} + 751986620 q^{37} + 1310251014 q^{38} + 2137654354 q^{39} + 739228561 q^{40} + 3080026050 q^{41} + 968357241 q^{42} + 3833817338 q^{43} + 8144649833 q^{44} + 2931838495 q^{45} + 6497628532 q^{46} + 5077845642 q^{47} - 835747598 q^{48} + 34773145974 q^{49} + 12830620137 q^{50} + 18692030299 q^{51} + 4600488501 q^{52} + 8357981673 q^{53} + 17210528041 q^{54} + 2065642780 q^{55} + 17985268753 q^{56} + 1704803374 q^{57} - 14454051036 q^{58} + 37511125048 q^{59} - 45562897126 q^{60} + 5247867997 q^{61} + 23479336684 q^{62} + 35756411909 q^{63} + 193907377715 q^{64} + 52745084243 q^{65} + 145705655428 q^{66} + 109696095260 q^{67} + 113297628382 q^{68} + 108574353048 q^{69} + 203428284368 q^{70} + 164182160959 q^{71} + 384360936452 q^{72} + 99223907428 q^{73} + 203465418184 q^{74} + 191539800932 q^{75} + 260405949391 q^{76} + 104852688646 q^{77} + 204902806863 q^{78} + 35845286129 q^{79} + 94467266331 q^{80} + 597373880631 q^{81} - 101434454538 q^{82} + 284535248381 q^{83} - 73372265058 q^{84} - 5818745589 q^{85} + 152224657349 q^{86} - 107566678745 q^{87} - 294979922678 q^{88} + 160140232242 q^{89} - 548871121508 q^{90} - 13791548748 q^{91} + 3796934790 q^{92} - 56884346339 q^{93} - 361242376817 q^{94} + 222650412097 q^{95} - 668711213516 q^{96} - 183910812581 q^{97} - 252245102123 q^{98} + 449685822055 q^{99}+O(q^{100}) \) 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 −89.7626 −178.191 6009.32 −3453.16 15994.9 −4398.55 −355578. −145395. 309965.
1.2 −88.0848 465.774 5710.94 1127.94 −41027.6 −60854.5 −322649. 39798.2 −99354.1
1.3 −86.2267 492.478 5387.04 −1853.64 −42464.7 53266.8 −287914. 65387.5 159834.
1.4 −85.2566 172.794 5220.69 7131.97 −14731.9 −62024.0 −270493. −147289. −608047.
1.5 −83.1368 −833.760 4863.73 7961.86 69316.2 −11512.4 −234091. 518009. −661924.
1.6 −83.1065 581.477 4858.69 9724.92 −48324.5 −11549.4 −233587. 160968. −808204.
1.7 −81.9797 163.432 4672.66 −11668.3 −13398.1 −10773.4 −215169. −150437. 956560.
1.8 −80.7601 −478.705 4474.20 −4754.48 38660.3 64817.2 −195940. 52011.2 383973.
1.9 −77.8064 −38.9575 4005.83 −6590.64 3031.15 −55518.6 −152332. −175629. 512794.
1.10 −77.2257 −547.114 3915.80 −5053.93 42251.3 −83403.2 −144242. 122187. 390293.
1.11 −76.2667 −486.612 3768.62 −518.688 37112.3 52292.8 −131226. 59644.4 39558.6
1.12 −74.9055 46.7252 3562.83 3148.61 −3499.98 13229.0 −113469. −174964. −235849.
1.13 −69.8986 −728.784 2837.81 6611.68 50940.9 77341.6 −55206.4 353979. −462147.
1.14 −69.6578 211.398 2804.20 3614.10 −14725.5 −64863.0 −52675.4 −132458. −251750.
1.15 −66.7307 −287.754 2404.99 −12758.5 19202.1 47394.3 −23822.4 −94344.4 851382.
1.16 −65.6608 449.953 2263.34 6873.79 −29544.3 41398.0 −14139.7 25310.7 −451339.
1.17 −65.6287 824.748 2259.12 −3417.33 −54127.1 85893.2 −13855.6 503062. 224275.
1.18 −64.7646 −650.534 2146.45 2912.33 42131.6 −21417.8 −6376.27 246047. −188616.
1.19 −62.0470 615.089 1801.83 −5340.81 −38164.4 −20468.1 15274.2 201187. 331381.
1.20 −61.6380 −360.202 1751.24 12383.1 22202.1 32273.9 18291.8 −47401.2 −763270.
See next 80 embeddings (of 107 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.107
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(229\) \(1\)

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 229.12.a.b ✓ 107
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
229.12.a.b ✓ 107 1.a even 1 1 trivial