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Properties

Label 169.10.a.e
Level $169$
Weight $10$
Character orbit 169.a
Self dual yes
Analytic conductor $87.041$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3841x^{8} + 5134480x^{6} - 2823572208x^{4} + 614223235584x^{2} - 43308450164736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4}\cdot 13^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 256) q^{4} + (\beta_{5} - 6 \beta_1) q^{5} + (\beta_{8} - 6 \beta_1) q^{6} + (\beta_{9} + \beta_{5} + 22 \beta_1) q^{7} + ( - \beta_{9} + \beta_{8} + \cdots + 144 \beta_1) q^{8}+ \cdots + (\beta_{4} - 33 \beta_{3} + \cdots + 6727) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 256) q^{4} + (\beta_{5} - 6 \beta_1) q^{5} + (\beta_{8} - 6 \beta_1) q^{6} + (\beta_{9} + \beta_{5} + 22 \beta_1) q^{7} + ( - \beta_{9} + \beta_{8} + \cdots + 144 \beta_1) q^{8}+ \cdots + ( - 45438 \beta_{9} + \cdots - 7727926 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{3} + 2562 q^{4} + 67304 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{3} + 2562 q^{4} + 67304 q^{9} - 45562 q^{10} - 46298 q^{12} + 169350 q^{14} - 209470 q^{16} + 1189686 q^{17} + 3516760 q^{22} - 2210916 q^{23} + 10109316 q^{25} - 7880006 q^{27} + 7039224 q^{29} + 7573410 q^{30} + 39493506 q^{35} - 29007196 q^{36} + 50219652 q^{38} - 23263606 q^{40} + 400842 q^{42} + 92334370 q^{43} + 102213790 q^{48} + 100714448 q^{49} - 77969322 q^{51} + 29507556 q^{53} + 138590608 q^{55} - 437436342 q^{56} - 238375816 q^{61} + 565735320 q^{62} - 272247742 q^{64} - 495101088 q^{66} + 550420050 q^{68} - 505082484 q^{69} + 1257672774 q^{74} + 1443387976 q^{75} - 530898576 q^{77} - 441679756 q^{79} + 1486784810 q^{81} - 518795296 q^{82} + 4793242032 q^{87} + 3927690208 q^{88} + 2656725444 q^{90} - 4121025444 q^{92} + 6226353478 q^{94} - 1047837276 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3841x^{8} + 5134480x^{6} - 2823572208x^{4} + 614223235584x^{2} - 43308450164736 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 768 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -101\nu^{8} + 335285\nu^{6} - 359803808\nu^{4} + 134078143536\nu^{2} - 13050018918144 ) / 12944968704 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -128\nu^{8} + 466637\nu^{6} - 584699369\nu^{4} + 281289968700\nu^{2} - 36497540643552 ) / 202265136 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 103037 \nu^{9} - 407466221 \nu^{7} + 517098453440 \nu^{5} - 225182669584944 \nu^{3} + 24\!\cdots\!24 \nu ) / 49242660950016 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3871\nu^{8} - 20860879\nu^{6} + 32030013856\nu^{4} - 15796747994256\nu^{2} + 1898872235880192 ) / 6472484352 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18149 \nu^{9} + 73154197 \nu^{7} - 101232420736 \nu^{5} + 55804164878768 \nu^{3} - 10\!\cdots\!36 \nu ) / 2735703386112 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -101\nu^{9} + 335285\nu^{7} - 359803808\nu^{5} + 134078143536\nu^{3} - 12972349105920\nu ) / 12944968704 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 118481 \nu^{9} + 432033281 \nu^{7} - 531812876480 \nu^{5} + 244210927479792 \nu^{3} - 29\!\cdots\!40 \nu ) / 8207110158336 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 768 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{8} + \beta_{7} + 1168\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} - 6\beta_{4} + 410\beta_{3} + 1657\beta_{2} + 896615 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2045\beta_{9} + 2005\beta_{8} + 1653\beta_{7} - 1392\beta_{5} + 1556172\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3085\beta_{6} - 13662\beta_{4} + 871634\beta_{3} + 2442489\beta_{2} + 1194377627 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -3338893\beta_{9} + 3045285\beta_{8} + 2439845\beta_{7} - 3945264\beta_{5} + 2164532636\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -6678717\beta_{6} - 23978622\beta_{4} + 1304765106\beta_{3} + 3532803545\beta_{2} + 1661122838379 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -5126337581\beta_{9} + 4165989317\beta_{8} + 3538276037\beta_{7} - 8138029104\beta_{5} + 3063857781212\beta_1 \) 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
−38.5911
−36.6220
−24.9226
−15.7959
−11.8282
11.8282
15.7959
24.9226
36.6220
38.5911
−38.5911 −58.0612 977.274 2510.84 2240.65 8007.46 −17955.4 −16311.9 −96896.0
1.2 −36.6220 126.985 829.173 −2095.65 −4650.45 −3353.27 −11615.5 −3557.77 76746.8
1.3 −24.9226 −252.690 109.136 −825.595 6297.70 −9476.84 10040.4 44169.4 20576.0
1.4 −15.7959 217.907 −262.489 1801.57 −3442.04 −6896.47 12233.8 27800.4 −28457.4
1.5 −11.8282 −35.1406 −372.094 −443.831 415.650 6276.11 10457.2 −18448.1 5249.72
1.6 11.8282 −35.1406 −372.094 443.831 −415.650 −6276.11 −10457.2 −18448.1 5249.72
1.7 15.7959 217.907 −262.489 −1801.57 3442.04 6896.47 −12233.8 27800.4 −28457.4
1.8 24.9226 −252.690 109.136 825.595 −6297.70 9476.84 −10040.4 44169.4 20576.0
1.9 36.6220 126.985 829.173 2095.65 4650.45 3353.27 11615.5 −3557.77 76746.8
1.10 38.5911 −58.0612 977.274 −2510.84 −2240.65 −8007.46 17955.4 −16311.9 −96896.0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.e 10
13.b even 2 1 inner 169.10.a.e 10
13.d odd 4 2 13.10.b.a ✓ 10
39.f even 4 2 117.10.b.c 10
52.f even 4 2 208.10.f.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.10.b.a ✓ 10 13.d odd 4 2
117.10.b.c 10 39.f even 4 2
169.10.a.e 10 1.a even 1 1 trivial
169.10.a.e 10 13.b even 2 1 inner
208.10.f.b 10 52.f even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 3841T_{2}^{8} + 5134480T_{2}^{6} - 2823572208T_{2}^{4} + 614223235584T_{2}^{2} - 43308450164736 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(169))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots - 43308450164736 \) Copy content Toggle raw display
$3$ \( (T^{5} + T^{4} + \cdots + 14266185264)^{2} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots - 94\!\cdots\!60)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 35\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots + 20\!\cdots\!84)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 22\!\cdots\!60)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 58\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 56\!\cdots\!88)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 61\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots + 58\!\cdots\!16)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 45\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 10\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 51\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 62\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 72\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 26\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 62\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 19\!\cdots\!44 \) Copy content Toggle raw display
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