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Properties

Label 169.10.a.h
Level $169$
Weight $10$
Character orbit 169.a
Self dual yes
Analytic conductor $87.041$
Analytic rank $0$
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: \(0\)
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 −43.3737 −225.369 1369.27 1637.42 9775.06 −1956.78 −37183.1 31108.1 −71021.0
1.2 −36.8678 240.621 847.232 1970.86 −8871.14 9117.25 −12359.3 38215.3 −72661.1
1.3 −36.1368 103.192 793.865 926.370 −3729.02 4023.64 −10185.7 −9034.43 −33476.0
1.4 −35.7320 −157.443 764.774 −2363.03 5625.74 −3406.89 −9032.12 5105.19 84435.6
1.5 −35.3614 −118.041 738.428 −111.377 4174.09 884.005 −8006.80 −5749.36 3938.43
1.6 −30.3833 82.6182 411.145 −433.016 −2510.21 409.354 3064.31 −12857.2 13156.5
1.7 −22.4110 83.2634 −9.74613 −970.434 −1866.02 −11431.9 11692.9 −12750.2 21748.4
1.8 −14.8244 −139.228 −292.237 1387.80 2063.98 −2985.58 11922.3 −298.471 −20573.4
1.9 −12.3426 −225.156 −359.660 −2236.39 2779.01 11037.4 10758.6 31012.0 27602.8
1.10 −8.72715 168.263 −435.837 −932.577 −1468.46 −1292.90 8271.92 8629.37 8138.74
1.11 −6.13624 −213.333 −474.347 517.983 1309.07 2961.86 6052.46 25828.1 −3178.47
1.12 −4.82532 −93.8913 −488.716 1259.17 453.056 −5921.56 4828.78 −10867.4 −6075.91
1.13 −4.74730 109.518 −489.463 −1538.18 −519.914 10378.9 4754.24 −7688.85 7302.18
1.14 −2.76873 273.564 −504.334 1902.60 −757.424 4322.02 2813.96 55154.1 −5267.78
1.15 4.33003 54.1364 −493.251 2633.10 234.412 −3591.16 −4352.76 −16752.3 11401.4
1.16 11.1269 −140.546 −388.192 −992.121 −1563.84 10567.0 −10016.3 70.0974 −11039.2
1.17 13.4402 117.761 −331.361 −2291.59 1582.73 −5059.42 −11334.9 −5815.41 −30799.5
1.18 22.4400 166.502 −8.44752 2749.81 3736.30 4995.16 −11678.8 8039.89 61705.7
1.19 24.5615 −5.41917 91.2697 −546.263 −133.103 −9850.09 −10333.8 −19653.6 −13417.1
1.20 25.3541 −120.880 130.831 −1004.42 −3064.81 −3164.79 −9664.20 −5071.03 −25466.3
See all 27 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.27
Significant digits:
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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.h yes 27
13.b even 2 1 169.10.a.g ✓ 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.10.a.g ✓ 27 13.b even 2 1
169.10.a.h yes 27 1.a even 1 1 trivial

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