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Properties

Label 169.10.a.g
Level $169$
Weight $10$
Character orbit 169.a
Self dual yes
Analytic conductor $87.041$
Analytic rank $1$
Dimension $27$
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] = [169,10,Mod(1,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 169.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: \(87.0410563117\)
Analytic rank: \(1\)
Dimension: \(27\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 27 q - 65 q^{2} + q^{3} + 7169 q^{4} - 3238 q^{5} - 8490 q^{6} - 17378 q^{7} - 54204 q^{8} + 191118 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 27 q - 65 q^{2} + q^{3} + 7169 q^{4} - 3238 q^{5} - 8490 q^{6} - 17378 q^{7} - 54204 q^{8} + 191118 q^{9} + 11697 q^{10} - 164171 q^{11} - 181941 q^{12} - 77651 q^{14} - 614110 q^{15} + 3012565 q^{16} + 105667 q^{17} - 584849 q^{18} - 2140031 q^{19} - 3722382 q^{20} - 2953252 q^{21} + 3789362 q^{22} + 3586420 q^{23} - 6334338 q^{24} + 10278737 q^{25} - 4332980 q^{27} - 5293616 q^{28} - 4203616 q^{29} - 34788079 q^{30} - 18634294 q^{31} - 38778128 q^{32} - 15170356 q^{33} - 10546914 q^{34} + 16260796 q^{35} + 112071987 q^{36} - 7214770 q^{37} - 60128169 q^{38} - 46270051 q^{40} - 67544533 q^{41} - 31249012 q^{42} - 55585357 q^{43} - 246583161 q^{44} - 172236942 q^{45} - 168715272 q^{46} - 175569972 q^{47} - 48573948 q^{48} + 144680057 q^{49} - 3455750 q^{50} + 122191382 q^{51} + 105504478 q^{53} + 359394409 q^{54} - 2471426 q^{55} + 260765426 q^{56} + 307372146 q^{57} + 777873375 q^{58} - 267562255 q^{59} + 745460915 q^{60} - 371334848 q^{61} + 52175323 q^{62} - 431551952 q^{63} + 642079732 q^{64} + 973549503 q^{66} - 213848603 q^{67} - 278629116 q^{68} - 201174310 q^{69} + 122562066 q^{70} - 970709202 q^{71} - 773619646 q^{72} - 271733895 q^{73} - 1068711482 q^{74} + 1236853937 q^{75} - 2585634844 q^{76} - 244285066 q^{77} + 1052001020 q^{79} - 3293699123 q^{80} - 456846001 q^{81} - 3832425 q^{82} - 1491498691 q^{83} - 5153327787 q^{84} - 2713007958 q^{85} - 3421126005 q^{86} - 1622618562 q^{87} + 6574070508 q^{88} - 4616501291 q^{89} - 2822436128 q^{90} + 3684890562 q^{92} - 3324407374 q^{93} - 5582614303 q^{94} - 2892286998 q^{95} - 12902380548 q^{96} - 3944038157 q^{97} - 7822213576 q^{98} - 5866875443 q^{99}+O(q^{100}) \) 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 −43.2607 160.179 1359.49 2049.68 −6929.44 −8578.81 −36662.9 5974.20 −88670.5
1.2 −43.1482 195.708 1349.77 −457.758 −8444.48 6911.97 −36148.3 18618.8 19751.4
1.3 −43.0325 −29.4845 1339.79 −1666.12 1268.79 −10498.9 −35622.0 −18813.7 71697.1
1.4 −40.6641 −254.108 1141.57 −1761.73 10333.1 1279.48 −25601.0 44887.8 71639.4
1.5 −34.8323 −234.206 701.286 469.408 8157.91 9293.31 −6593.27 35169.3 −16350.5
1.6 −27.6084 218.573 250.226 409.297 −6034.47 2909.47 7227.17 28091.3 −11300.0
1.7 −25.8386 −15.7946 155.636 −715.059 408.111 −8658.96 9207.97 −19433.5 18476.2
1.8 −25.3541 −120.880 130.831 1004.42 3064.81 3164.79 9664.20 −5071.03 −25466.3
1.9 −24.5615 −5.41917 91.2697 546.263 133.103 9850.09 10333.8 −19653.6 −13417.1
1.10 −22.4400 166.502 −8.44752 −2749.81 −3736.30 −4995.16 11678.8 8039.89 61705.7
1.11 −13.4402 117.761 −331.361 2291.59 −1582.73 5059.42 11334.9 −5815.41 −30799.5
1.12 −11.1269 −140.546 −388.192 992.121 1563.84 −10567.0 10016.3 70.0974 −11039.2
1.13 −4.33003 54.1364 −493.251 −2633.10 −234.412 3591.16 4352.76 −16752.3 11401.4
1.14 2.76873 273.564 −504.334 −1902.60 757.424 −4322.02 −2813.96 55154.1 −5267.78
1.15 4.74730 109.518 −489.463 1538.18 519.914 −10378.9 −4754.24 −7688.85 7302.18
1.16 4.82532 −93.8913 −488.716 −1259.17 −453.056 5921.56 −4828.78 −10867.4 −6075.91
1.17 6.13624 −213.333 −474.347 −517.983 −1309.07 −2961.86 −6052.46 25828.1 −3178.47
1.18 8.72715 168.263 −435.837 932.577 1468.46 1292.90 −8271.92 8629.37 8138.74
1.19 12.3426 −225.156 −359.660 2236.39 −2779.01 −11037.4 −10758.6 31012.0 27602.8
1.20 14.8244 −139.228 −292.237 −1387.80 −2063.98 2985.58 −11922.3 −298.471 −20573.4
See all 27 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.27
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \( +1 \)

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 169.10.a.g ✓ 27
13.b even 2 1 169.10.a.h yes 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.10.a.g ✓ 27 1.a even 1 1 trivial
169.10.a.h yes 27 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{27} + 65 T_{2}^{26} - 8384 T_{2}^{25} - 596247 T_{2}^{24} + 29229748 T_{2}^{23} + \cdots + 22\!\cdots\!48 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(169))\). Copy content Toggle raw display