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Properties

Label 8619.2.a.bn
Level $8619$
Weight $2$
Character orbit 8619.a
Self dual yes
Analytic conductor $68.823$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8619,2,Mod(1,8619)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8619, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8619.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8619.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: \(68.8230615021\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 20x^{14} + 156x^{12} - 602x^{10} + 1212x^{8} - 1259x^{6} + 665x^{4} - 168x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 663)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + \beta_{14} q^{5} + \beta_1 q^{6} + ( - \beta_{15} - \beta_{14} + \beta_{3}) q^{7} + (\beta_{14} + \beta_{13} - \beta_{3}) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + \beta_{14} q^{5} + \beta_1 q^{6} + ( - \beta_{15} - \beta_{14} + \beta_{3}) q^{7} + (\beta_{14} + \beta_{13} - \beta_{3}) q^{8} + q^{9} - \beta_{6} q^{10} + (\beta_{13} + \beta_{12} + \beta_{3}) q^{11} + (\beta_{2} + 1) q^{12} + ( - \beta_{9} + \beta_{8} + \beta_{7} + \cdots - 2) q^{14}+ \cdots + (\beta_{13} + \beta_{12} + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{3} + 8 q^{4} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{3} + 8 q^{4} + 16 q^{9} + 2 q^{10} + 8 q^{12} - 26 q^{14} - 16 q^{17} - 2 q^{22} - 42 q^{23} - 14 q^{25} + 16 q^{27} - 58 q^{29} + 2 q^{30} - 30 q^{35} + 8 q^{36} - 62 q^{38} + 4 q^{40} - 26 q^{42} - 6 q^{43} + 4 q^{49} - 16 q^{51} - 26 q^{53} - 18 q^{55} - 74 q^{56} - 58 q^{61} - 40 q^{62} - 36 q^{64} - 2 q^{66} - 8 q^{68} - 42 q^{69} - 34 q^{74} - 14 q^{75} + 8 q^{77} - 14 q^{79} + 16 q^{81} - 6 q^{82} - 58 q^{87} - 10 q^{88} + 2 q^{90} - 64 q^{92} + 50 q^{94} - 54 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 20x^{14} + 156x^{12} - 602x^{10} + 1212x^{8} - 1259x^{6} + 665x^{4} - 168x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} - 20\nu^{13} + 156\nu^{11} - 602\nu^{9} + 1212\nu^{7} - 1259\nu^{5} + 657\nu^{3} - 128\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{14} + 20\nu^{12} - 156\nu^{10} + 602\nu^{8} - 1212\nu^{6} + 1255\nu^{4} - 633\nu^{2} + 112 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\nu^{14} - 296\nu^{12} + 2260\nu^{10} - 8406\nu^{8} + 15788\nu^{6} - 14213\nu^{4} + 5563\nu^{2} - 724 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -15\nu^{14} + 296\nu^{12} - 2260\nu^{10} + 8406\nu^{8} - 15788\nu^{6} + 14229\nu^{4} - 5659\nu^{2} + 788 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 45\nu^{14} - 872\nu^{12} + 6476\nu^{10} - 23042\nu^{8} + 40116\nu^{6} - 31487\nu^{4} + 10001\nu^{2} - 1020 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19\nu^{14} - 376\nu^{12} + 2884\nu^{10} - 10814\nu^{8} + 20620\nu^{6} - 19089\nu^{4} + 7775\nu^{2} - 1060 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 47\nu^{14} - 920\nu^{12} + 6948\nu^{10} - 25462\nu^{8} + 46876\nu^{6} - 41173\nu^{4} + 15899\nu^{2} - 2132 ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21\nu^{15} - 416\nu^{13} + 3196\nu^{11} - 12018\nu^{9} + 23044\nu^{7} - 21607\nu^{5} + 9089\nu^{3} - 1316\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 71 \nu^{15} + 1384 \nu^{13} - 10372 \nu^{11} + 37446 \nu^{9} - 66844 \nu^{7} + 54909 \nu^{5} + \cdots + 2084 \nu ) / 32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 199 \nu^{15} - 3896 \nu^{13} + 29412 \nu^{11} - 107590 \nu^{9} + 197020 \nu^{7} - 170717 \nu^{5} + \cdots - 8180 \nu ) / 64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 189 \nu^{15} + 3720 \nu^{13} - 28300 \nu^{11} + 104738 \nu^{9} - 195444 \nu^{7} + 174799 \nu^{5} + \cdots + 9180 \nu ) / 64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 197 \nu^{15} - 3880 \nu^{13} + 29548 \nu^{11} - 109554 \nu^{9} + 205140 \nu^{7} - 184871 \nu^{5} + \cdots - 10460 \nu ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 121 \nu^{15} + 2376 \nu^{13} - 18012 \nu^{11} + 66298 \nu^{9} - 122628 \nu^{7} + 108163 \nu^{5} + \cdots + 5612 \nu ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{13} - \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{5} + 6\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} + 6\beta_{14} + 6\beta_{13} + \beta_{11} - \beta_{10} - 8\beta_{3} + 20\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{8} + 9\beta_{6} + 10\beta_{5} - \beta_{4} + 34\beta_{2} + 73 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -10\beta_{15} + 35\beta_{14} + 33\beta_{13} + 11\beta_{11} - 13\beta_{10} - 53\beta_{3} + 107\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{9} - 14\beta_{8} - \beta_{7} + 62\beta_{6} + 77\beta_{5} - 10\beta_{4} + 193\beta_{2} + 397 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -71\beta_{15} + 209\beta_{14} + 178\beta_{13} + \beta_{12} + 88\beta_{11} - 116\beta_{10} - 334\beta_{3} + 590\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 20\beta_{9} - 135\beta_{8} - 18\beta_{7} + 391\beta_{6} + 539\beta_{5} - 70\beta_{4} + 1101\beta_{2} + 2211 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 443 \beta_{15} + 1267 \beta_{14} + 955 \beta_{13} + 20 \beta_{12} + 627 \beta_{11} + \cdots + 3312 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 244\beta_{9} - 1114\beta_{8} - 204\beta_{7} + 2377\beta_{6} + 3599\beta_{5} - 423\beta_{4} + 6309\beta_{2} + 12510 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2596 \beta_{15} + 7735 \beta_{14} + 5127 \beta_{13} + 244 \beta_{12} + 4226 \beta_{11} + \cdots + 18819 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2362 \beta_{9} - 8436 \beta_{8} - 1874 \beta_{7} + 14215 \beta_{6} + 23385 \beta_{5} - 2352 \beta_{4} + \cdots + 71585 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 14693 \beta_{15} + 47343 \beta_{14} + 27617 \beta_{13} + 2362 \beta_{12} + 27611 \beta_{11} + \cdots + 107884 \beta_1 \) 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
−2.45691
−2.29812
−1.96679
−1.64424
−0.978320
−0.729584
−0.598113
−0.513139
0.513139
0.598113
0.729584
0.978320
1.64424
1.96679
2.29812
2.45691
−2.45691 1.00000 4.03639 −0.284332 −2.45691 4.70317 −5.00321 1.00000 0.698577
1.2 −2.29812 1.00000 3.28136 1.20591 −2.29812 −0.486164 −2.94471 1.00000 −2.77133
1.3 −1.96679 1.00000 1.86826 −0.562071 −1.96679 3.08512 0.259096 1.00000 1.10548
1.4 −1.64424 1.00000 0.703537 −3.13210 −1.64424 1.47189 2.13170 1.00000 5.14993
1.5 −0.978320 1.00000 −1.04289 1.22925 −0.978320 −4.65625 2.97692 1.00000 −1.20260
1.6 −0.729584 1.00000 −1.46771 0.945208 −0.729584 −0.355853 2.52998 1.00000 −0.689609
1.7 −0.598113 1.00000 −1.64226 −1.38228 −0.598113 −1.33422 2.17848 1.00000 0.826761
1.8 −0.513139 1.00000 −1.73669 4.12599 −0.513139 −0.609920 1.91744 1.00000 −2.11721
1.9 0.513139 1.00000 −1.73669 −4.12599 0.513139 0.609920 −1.91744 1.00000 −2.11721
1.10 0.598113 1.00000 −1.64226 1.38228 0.598113 1.33422 −2.17848 1.00000 0.826761
1.11 0.729584 1.00000 −1.46771 −0.945208 0.729584 0.355853 −2.52998 1.00000 −0.689609
1.12 0.978320 1.00000 −1.04289 −1.22925 0.978320 4.65625 −2.97692 1.00000 −1.20260
1.13 1.64424 1.00000 0.703537 3.13210 1.64424 −1.47189 −2.13170 1.00000 5.14993
1.14 1.96679 1.00000 1.86826 0.562071 1.96679 −3.08512 −0.259096 1.00000 1.10548
1.15 2.29812 1.00000 3.28136 −1.20591 2.29812 0.486164 2.94471 1.00000 −2.77133
1.16 2.45691 1.00000 4.03639 0.284332 2.45691 −4.70317 5.00321 1.00000 0.698577
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(-1\)
\(17\) \(1\)

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 8619.2.a.bn 16
13.b even 2 1 inner 8619.2.a.bn 16
13.f odd 12 2 663.2.z.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.z.d ✓ 16 13.f odd 12 2
8619.2.a.bn 16 1.a even 1 1 trivial
8619.2.a.bn 16 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):

\( T_{2}^{16} - 20T_{2}^{14} + 156T_{2}^{12} - 602T_{2}^{10} + 1212T_{2}^{8} - 1259T_{2}^{6} + 665T_{2}^{4} - 168T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{5}^{16} - 33T_{5}^{14} + 347T_{5}^{12} - 1436T_{5}^{10} + 2871T_{5}^{8} - 2939T_{5}^{6} + 1470T_{5}^{4} - 299T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{16} - 58T_{7}^{14} + 1153T_{7}^{12} - 9134T_{7}^{10} + 27762T_{7}^{8} - 34768T_{7}^{6} + 16573T_{7}^{4} - 3143T_{7}^{2} + 196 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 20 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T - 1)^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 33 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{16} - 58 T^{14} + \cdots + 196 \) Copy content Toggle raw display
$11$ \( T^{16} - 83 T^{14} + \cdots + 43264 \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( (T + 1)^{16} \) Copy content Toggle raw display
$19$ \( T^{16} - 184 T^{14} + \cdots + 20702500 \) Copy content Toggle raw display
$23$ \( (T^{8} + 21 T^{7} + \cdots - 518)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 29 T^{7} + \cdots - 246074)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 10276687876 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 6305630464 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 420660100 \) Copy content Toggle raw display
$43$ \( (T^{8} + 3 T^{7} + \cdots - 56636)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 528633693184 \) Copy content Toggle raw display
$53$ \( (T^{8} + 13 T^{7} + \cdots + 36088)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 2428161761536 \) Copy content Toggle raw display
$61$ \( (T^{8} + 29 T^{7} + \cdots - 880976)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 2608383272401 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 23854784466496 \) Copy content Toggle raw display
$73$ \( T^{16} - 365 T^{14} + \cdots + 6724 \) Copy content Toggle raw display
$79$ \( (T^{8} + 7 T^{7} + \cdots + 17872504)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 242861056 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 17701770304 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 16072882810000 \) Copy content Toggle raw display
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