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Properties

Label 229.4.a.a
Level $229$
Weight $4$
Character orbit 229.a
Self dual yes
Analytic conductor $13.511$
Analytic rank $1$
Dimension $26$
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] = [229,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("229.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 229 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(13.5114373913\)
Analytic rank: \(1\)
Dimension: \(26\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 15 q^{2} - 17 q^{3} + 89 q^{4} - 30 q^{5} - 53 q^{6} - 55 q^{7} - 177 q^{8} + 113 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q - 15 q^{2} - 17 q^{3} + 89 q^{4} - 30 q^{5} - 53 q^{6} - 55 q^{7} - 177 q^{8} + 113 q^{9} - 87 q^{10} - 335 q^{11} - 160 q^{12} - 91 q^{13} - 183 q^{14} - 220 q^{15} + 177 q^{16} - 155 q^{17} - 249 q^{18} - 439 q^{19} - 368 q^{20} - 352 q^{21} - 163 q^{22} - 345 q^{23} - 541 q^{24} + 238 q^{25} - 330 q^{26} - 626 q^{27} - 302 q^{28} - 979 q^{29} - 698 q^{30} - 591 q^{31} - 1606 q^{32} - 714 q^{33} - 493 q^{34} - 1972 q^{35} - 700 q^{36} - 330 q^{37} - 532 q^{38} - 1688 q^{39} - 359 q^{40} - 1974 q^{41} - 147 q^{42} - 839 q^{43} - 2819 q^{44} - 980 q^{45} - 458 q^{46} - 1533 q^{47} - 1502 q^{48} + 829 q^{49} - 2633 q^{50} - 3092 q^{51} - 911 q^{52} - 1398 q^{53} - 1595 q^{54} - 1150 q^{55} - 2831 q^{56} + 226 q^{57} - 378 q^{58} - 3156 q^{59} + 2130 q^{60} + 306 q^{61} + 2298 q^{62} - 1925 q^{63} + 3691 q^{64} - 1106 q^{65} + 5904 q^{66} - 220 q^{67} + 1166 q^{68} + 210 q^{69} + 5844 q^{70} - 2026 q^{71} + 7114 q^{72} + 986 q^{73} + 2182 q^{74} + 1893 q^{75} + 1519 q^{76} + 1262 q^{77} + 7863 q^{78} - 1135 q^{79} + 1423 q^{80} + 3682 q^{81} + 11974 q^{82} - 2963 q^{83} + 5264 q^{84} - 876 q^{85} - 429 q^{86} + 4860 q^{87} + 6302 q^{88} - 772 q^{89} + 8446 q^{90} + 126 q^{91} + 4288 q^{92} - 1710 q^{93} + 5241 q^{94} - 1668 q^{95} + 9020 q^{96} - 577 q^{97} + 951 q^{98} - 4579 q^{99}+O(q^{100}) \) 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 −5.60252 2.29426 23.3883 12.9525 −12.8537 18.7618 −86.2133 −21.7364 −72.5665
1.2 −5.45621 −4.16117 21.7702 −18.8307 22.7042 16.7615 −75.1331 −9.68465 102.744
1.3 −4.98125 −5.52627 16.8129 3.70943 27.5277 −14.3100 −43.8991 3.53963 −18.4776
1.4 −4.43586 7.36007 11.6769 −19.5996 −32.6483 16.1006 −16.3101 27.1707 86.9410
1.5 −4.32547 1.83273 10.7097 1.67964 −7.92743 3.65581 −11.7207 −23.6411 −7.26522
1.6 −4.32354 −0.441480 10.6930 22.1876 1.90875 −27.7219 −11.6431 −26.8051 −95.9290
1.7 −3.57467 9.70322 4.77826 4.98066 −34.6858 −29.5708 11.5167 67.1524 −17.8042
1.8 −3.10898 −7.49790 1.66577 −17.5762 23.3108 4.85554 19.6930 29.2185 54.6442
1.9 −2.50500 6.14833 −1.72495 −8.91895 −15.4016 −2.83880 24.3610 10.8020 22.3420
1.10 −2.43661 −9.36300 −2.06293 4.83140 22.8140 −29.6750 24.5194 60.6659 −11.7722
1.11 −2.20735 −3.26143 −3.12760 1.74411 7.19913 −1.10828 24.5625 −16.3630 −3.84986
1.12 −1.77364 3.10622 −4.85422 10.1847 −5.50931 −5.33353 22.7987 −17.3514 −18.0639
1.13 −1.04196 −0.182446 −6.91433 −5.14549 0.190100 33.6983 15.5401 −26.9667 5.36137
1.14 −0.923243 −9.01195 −7.14762 −4.95759 8.32022 27.2042 13.9849 54.2152 4.57706
1.15 0.220221 3.78029 −7.95150 5.93638 0.832500 −8.16759 −3.51286 −12.7094 1.30731
1.16 0.552141 −8.16009 −7.69514 15.5465 −4.50552 4.11260 −8.66594 39.5870 8.58387
1.17 0.996531 7.94578 −7.00693 −11.6386 7.91822 −12.4299 −14.9549 36.1355 −11.5982
1.18 1.68354 2.87700 −5.16568 −0.123929 4.84356 8.29372 −22.1650 −18.7229 −0.208640
1.19 2.57871 −0.140237 −1.35027 15.7771 −0.361629 −32.4086 −24.1116 −26.9803 40.6844
1.20 2.79240 −4.46995 −0.202515 −1.13530 −12.4819 31.0150 −22.9047 −7.01956 −3.17021
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.26
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 229.4.a.a ✓ 26
3.b odd 2 1 2061.4.a.b 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
229.4.a.a ✓ 26 1.a even 1 1 trivial
2061.4.a.b 26 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{26} + 15 T_{2}^{25} - 36 T_{2}^{24} - 1526 T_{2}^{23} - 2896 T_{2}^{22} + 64445 T_{2}^{21} + \cdots + 12897574912 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(229))\). Copy content Toggle raw display