↑

Properties

Label 229.12.a.a
Level $229$
Weight $12$
Character orbit 229.a
Self dual yes
Analytic conductor $175.951$
Analytic rank $1$
Dimension $102$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(102\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 102 q - 201 q^{2} - 1721 q^{3} + 100869 q^{4} - 18750 q^{5} - 40727 q^{6} - 101895 q^{7} - 774657 q^{8} + 5620019 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 102 q - 201 q^{2} - 1721 q^{3} + 100869 q^{4} - 18750 q^{5} - 40727 q^{6} - 101895 q^{7} - 774657 q^{8} + 5620019 q^{9} - 836607 q^{10} - 5626361 q^{11} - 3084028 q^{12} - 2599051 q^{13} - 8429901 q^{14} - 10476745 q^{15} + 91144921 q^{16} - 8783951 q^{17} - 42201525 q^{18} - 42930136 q^{19} - 49145288 q^{20} - 83733301 q^{21} - 23246785 q^{22} - 104480106 q^{23} - 47871901 q^{24} + 942218698 q^{25} - 322355424 q^{26} - 357092168 q^{27} - 327130762 q^{28} - 529083811 q^{29} - 506076518 q^{30} - 604755107 q^{31} - 1392293182 q^{32} - 664604700 q^{33} - 417698651 q^{34} - 1918128937 q^{35} + 4184485940 q^{36} - 634892520 q^{37} - 749863354 q^{38} - 3636694382 q^{39} - 2332771439 q^{40} - 4566483216 q^{41} - 1645467399 q^{42} - 2634554154 q^{43} - 12964626839 q^{44} - 3711174005 q^{45} - 2976668364 q^{46} - 8682854778 q^{47} - 13066338062 q^{48} + 24886512259 q^{49} - 15606879863 q^{50} - 17190595805 q^{51} - 9086856651 q^{52} - 11715401991 q^{53} - 8502713303 q^{54} - 14039457220 q^{55} - 31580591087 q^{56} - 17549342450 q^{57} + 36913742792 q^{58} - 34751424252 q^{59} + 53761493850 q^{60} + 5470801749 q^{61} - 10421905920 q^{62} - 34469428433 q^{63} + 37746021923 q^{64} - 67037119511 q^{65} - 163212203400 q^{66} - 78011273630 q^{67} - 131149476790 q^{68} - 73883208378 q^{69} - 269474084056 q^{70} - 129426063793 q^{71} - 356771673134 q^{72} - 65084784580 q^{73} - 173941793150 q^{74} - 207601298232 q^{75} - 192477187773 q^{76} - 104407742308 q^{77} - 90430317789 q^{78} - 172896961791 q^{79} - 220890698897 q^{80} + 305566877650 q^{81} - 19768670730 q^{82} - 116720618231 q^{83} - 192431881744 q^{84} - 120470791301 q^{85} - 63143239455 q^{86} + 14251005171 q^{87} + 218699094910 q^{88} - 127279988242 q^{89} + 511463547946 q^{90} + 83456279688 q^{91} + 245083229476 q^{92} + 70822811877 q^{93} + 456670105827 q^{94} - 76354253763 q^{95} + 1096833229124 q^{96} + 134524240731 q^{97} + 284503650795 q^{98} - 617108531221 q^{99}+O(q^{100}) \) 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 −89.2614 −780.806 5919.60 −10625.1 69695.9 33965.7 −345585. 432512. 948410.
1.2 −88.4569 −241.015 5776.62 10404.9 21319.5 73059.8 −329822. −119059. −920385.
1.3 −85.8000 −521.066 5313.64 8423.19 44707.4 −43440.4 −280192. 94362.3 −722710.
1.4 −84.0557 −213.801 5017.36 −10537.3 17971.2 −40543.5 −249592. −131436. 885718.
1.5 −83.6397 410.313 4947.60 −9759.38 −34318.4 44521.8 −242522. −8790.42 816272.
1.6 −83.2472 807.625 4882.10 −5013.53 −67232.5 −34963.0 −235931. 475111. 417363.
1.7 −82.1362 −494.955 4698.36 −506.543 40653.8 −16512.8 −217691. 67833.7 41605.5
1.8 −80.1839 320.355 4381.46 8021.08 −25687.3 66894.2 −187106. −74519.8 −643161.
1.9 −79.6610 −4.77477 4297.88 1911.80 380.363 23434.7 −179228. −177124. −152296.
1.10 −78.3866 758.390 4096.46 10387.3 −59447.6 36384.3 −160572. 398008. −814228.
1.11 −78.1657 2.39915 4061.87 12005.0 −187.531 −12384.3 −157416. −177141. −938378.
1.12 −75.9208 −491.226 3715.97 9149.98 37294.3 −26722.9 −126633. 64156.3 −694673.
1.13 −73.1132 604.754 3297.54 −13176.9 −44215.5 −55732.9 −91357.9 188581. 963404.
1.14 −72.6157 624.277 3225.03 −2574.00 −45332.3 8381.06 −85471.1 212575. 186912.
1.15 −71.3654 −24.2297 3045.02 −6231.55 1729.16 51297.9 −71152.4 −176560. 444717.
1.16 −68.7235 −765.161 2674.92 −8152.20 52584.5 −38427.2 −43083.9 408325. 560248.
1.17 −67.9033 399.866 2562.85 −2870.22 −27152.2 −57096.6 −34960.1 −17254.4 194898.
1.18 −66.0923 −601.690 2320.20 −10345.5 39767.1 15589.4 −17990.2 184884. 683759.
1.19 −65.9239 57.0821 2297.96 −3662.37 −3763.07 59962.1 −16478.0 −173889. 241438.
1.20 −65.2139 677.919 2204.86 7262.27 −44209.8 −23184.9 −10229.3 282427. −473601.
See next 80 embeddings (of 102 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.102
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.a ✓ 102
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
229.12.a.a ✓ 102 1.a even 1 1 trivial