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Properties

Label 169.10.a.f
Level $169$
Weight $10$
Character orbit 169.a
Self dual yes
Analytic conductor $87.041$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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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: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 7679 x^{18} + 24599364 x^{16} - 42662336000 x^{14} + 43527566862400 x^{12} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{25}\cdot 3^{10}\cdot 13^{12} \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{4} + 16) q^{3} + (\beta_{4} + \beta_{3} + 256) q^{4} + ( - \beta_{12} + \beta_{11} + 6 \beta_1) q^{5} + (\beta_{15} - \beta_{12} + \cdots + 39 \beta_1) q^{6}+ \cdots + (\beta_{5} + 44 \beta_{4} + \cdots + 6464) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{4} + 16) q^{3} + (\beta_{4} + \beta_{3} + 256) q^{4} + ( - \beta_{12} + \beta_{11} + 6 \beta_1) q^{5} + (\beta_{15} - \beta_{12} + \cdots + 39 \beta_1) q^{6}+ \cdots + ( - 4442 \beta_{19} + \cdots - 5095796 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 326 q^{3} + 5118 q^{4} + 129526 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 326 q^{3} + 5118 q^{4} + 129526 q^{9} + 88390 q^{10} + 427652 q^{12} + 473556 q^{14} + 1189618 q^{16} - 99312 q^{17} - 5073532 q^{22} + 6252378 q^{23} + 1529274 q^{25} + 18052718 q^{27} + 5424828 q^{29} + 30045516 q^{30} - 15098160 q^{35} + 54078886 q^{36} + 39021096 q^{38} + 134360674 q^{40} + 121600068 q^{42} + 53242058 q^{43} + 82771076 q^{48} + 14055298 q^{49} + 10208934 q^{51} - 36429786 q^{53} - 131454160 q^{55} + 127272672 q^{56} + 536425792 q^{61} - 890759796 q^{62} - 343337066 q^{64} - 445758060 q^{66} + 336594090 q^{68} + 1083885582 q^{69} + 1083869262 q^{74} - 1218826882 q^{75} + 1470187374 q^{77} + 1556703616 q^{79} + 2139127516 q^{81} + 3719322754 q^{82} + 1288246146 q^{87} - 4109160928 q^{88} + 6489878934 q^{90} + 10148843820 q^{92} + 4894712828 q^{94} + 9251202540 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 7679 x^{18} + 24599364 x^{16} - 42662336000 x^{14} + 43527566862400 x^{12} + \cdots + 25\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\!\cdots\!39 \nu^{18} + \cdots + 78\!\cdots\!92 ) / 76\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14\!\cdots\!19 \nu^{18} + \cdots - 27\!\cdots\!32 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!19 \nu^{18} + \cdots + 74\!\cdots\!32 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 41\!\cdots\!61 \nu^{18} + \cdots - 42\!\cdots\!08 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35\!\cdots\!21 \nu^{18} + \cdots + 41\!\cdots\!48 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 47\!\cdots\!07 \nu^{18} + \cdots - 11\!\cdots\!44 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12\!\cdots\!21 \nu^{18} + \cdots - 41\!\cdots\!80 ) / 64\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 30\!\cdots\!79 \nu^{18} + \cdots - 10\!\cdots\!12 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 79\!\cdots\!37 \nu^{18} + \cdots - 39\!\cdots\!04 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11\!\cdots\!09 \nu^{19} + \cdots + 37\!\cdots\!48 \nu ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 56\!\cdots\!69 \nu^{19} + \cdots - 22\!\cdots\!48 \nu ) / 80\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 18\!\cdots\!91 \nu^{19} + \cdots - 30\!\cdots\!48 \nu ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 19\!\cdots\!47 \nu^{19} + \cdots + 22\!\cdots\!64 \nu ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 59\!\cdots\!99 \nu^{19} + \cdots - 60\!\cdots\!24 \nu ) / 45\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 90\!\cdots\!97 \nu^{19} + \cdots - 10\!\cdots\!04 \nu ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 11\!\cdots\!97 \nu^{19} + \cdots + 23\!\cdots\!84 \nu ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 11\!\cdots\!91 \nu^{19} + \cdots + 15\!\cdots\!64 \nu ) / 80\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 36\!\cdots\!07 \nu^{19} + \cdots - 43\!\cdots\!84 \nu ) / 20\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 768 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} - 3\beta_{12} + 34\beta_{11} + 1273\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4 \beta_{10} - 2 \beta_{9} - 5 \beta_{8} - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} + 1705 \beta_{4} + \cdots + 976993 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 26 \beta_{19} + 18 \beta_{18} - 322 \beta_{17} + 1997 \beta_{16} + 46 \beta_{15} + \cdots + 1838895 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 11572 \beta_{10} - 5042 \beta_{9} - 12555 \beta_{8} - 886 \beta_{7} + 263 \beta_{6} + \cdots + 1410181379 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 106534 \beta_{19} + 27214 \beta_{18} - 1142942 \beta_{17} + 3451391 \beta_{16} + \cdots + 2801848045 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 24905532 \beta_{10} - 10467606 \beta_{9} - 25376833 \beta_{8} + 3427486 \beta_{7} + \cdots + 2147205185833 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 281218322 \beta_{19} + 14718346 \beta_{18} - 2724133882 \beta_{17} + 5786352621 \beta_{16} + \cdots + 4392574977559 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 48024360308 \beta_{10} - 19956364066 \beta_{9} - 47729119123 \beta_{8} + 12648051770 \beta_{7} + \cdots + 33\!\cdots\!51 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 622278000598 \beta_{19} - 39214696770 \beta_{18} - 5581682625806 \beta_{17} + \cdots + 70\!\cdots\!65 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 87836036476316 \beta_{10} - 36203846890630 \beta_{9} - 86631068734185 \beta_{8} + \cdots + 53\!\cdots\!85 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 12\!\cdots\!34 \beta_{19} - 171163539454438 \beta_{18} + \cdots + 11\!\cdots\!27 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 15\!\cdots\!24 \beta_{10} + \cdots + 86\!\cdots\!79 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 24\!\cdots\!98 \beta_{19} + \cdots + 18\!\cdots\!97 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 27\!\cdots\!68 \beta_{10} + \cdots + 14\!\cdots\!33 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 44\!\cdots\!54 \beta_{19} + \cdots + 30\!\cdots\!67 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 46\!\cdots\!68 \beta_{10} + \cdots + 23\!\cdots\!07 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 80\!\cdots\!70 \beta_{19} + \cdots + 50\!\cdots\!93 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−41.3599
−40.5664
−36.6943
−34.2355
−27.6771
−20.2564
−19.7704
−13.9570
−5.95885
−2.58166
2.58166
5.95885
13.9570
19.7704
20.2564
27.6771
34.2355
36.6943
40.5664
41.3599
−41.3599 214.052 1198.64 −1662.90 −8853.18 −124.522 −28399.6 26135.3 68777.5
1.2 −40.5664 −100.154 1133.63 −1429.76 4062.90 6531.30 −25217.2 −9652.08 58000.3
1.3 −36.6943 −130.026 834.471 1417.57 4771.23 −7507.92 −11832.9 −2776.14 −52016.6
1.4 −34.2355 264.006 660.066 −402.267 −9038.36 −9369.20 −5069.12 50016.1 13771.8
1.5 −27.6771 −44.2389 254.024 2467.22 1224.41 −3580.38 7140.04 −17725.9 −68285.5
1.6 −20.2564 −99.3079 −101.676 −1799.37 2011.63 1935.26 12430.9 −9820.93 36448.7
1.7 −19.7704 99.5109 −121.130 1521.53 −1967.37 9243.80 12517.3 −9780.58 −30081.3
1.8 −13.9570 25.6151 −317.202 −792.583 −357.510 900.794 11573.2 −19026.9 11062.1
1.9 −5.95885 −250.658 −476.492 −841.807 1493.64 −1268.09 5890.28 43146.7 5016.20
1.10 −2.58166 184.202 −505.335 −581.690 −475.547 −10922.8 2626.41 14247.4 1501.73
1.11 2.58166 184.202 −505.335 581.690 475.547 10922.8 −2626.41 14247.4 1501.73
1.12 5.95885 −250.658 −476.492 841.807 −1493.64 1268.09 −5890.28 43146.7 5016.20
1.13 13.9570 25.6151 −317.202 792.583 357.510 −900.794 −11573.2 −19026.9 11062.1
1.14 19.7704 99.5109 −121.130 −1521.53 1967.37 −9243.80 −12517.3 −9780.58 −30081.3
1.15 20.2564 −99.3079 −101.676 1799.37 −2011.63 −1935.26 −12430.9 −9820.93 36448.7
1.16 27.6771 −44.2389 254.024 −2467.22 −1224.41 3580.38 −7140.04 −17725.9 −68285.5
1.17 34.2355 264.006 660.066 402.267 9038.36 9369.20 5069.12 50016.1 13771.8
1.18 36.6943 −130.026 834.471 −1417.57 −4771.23 7507.92 11832.9 −2776.14 −52016.6
1.19 40.5664 −100.154 1133.63 1429.76 −4062.90 −6531.30 25217.2 −9652.08 58000.3
1.20 41.3599 214.052 1198.64 1662.90 8853.18 124.522 28399.6 26135.3 68777.5
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.10.a.f 20
13.b even 2 1 inner 169.10.a.f 20
13.f odd 12 2 13.10.e.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.10.e.a ✓ 20 13.f odd 12 2
169.10.a.f 20 1.a even 1 1 trivial
169.10.a.f 20 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 7679 T_{2}^{18} + 24599364 T_{2}^{16} - 42662336000 T_{2}^{14} + 43527566862400 T_{2}^{12} + \cdots + 25\!\cdots\!36 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(169))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$3$ \( (T^{10} + \cdots - 38\!\cdots\!68)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{20} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 92\!\cdots\!29)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots + 43\!\cdots\!52)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 17\!\cdots\!93)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 64\!\cdots\!49 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 60\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 30\!\cdots\!96)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 12\!\cdots\!35)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 52\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 54\!\cdots\!24 \) Copy content Toggle raw display
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