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Properties

Label 8619.2.a.bo
Level $8619$
Weight $2$
Character orbit 8619.a
Self dual yes
Analytic conductor $68.823$
Analytic rank $0$
Dimension $20$
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: \(0\)
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: \( x^{20} - 34 x^{18} + 485 x^{16} - 3770 x^{14} + 17340 x^{12} - 47849 x^{10} + 76299 x^{8} - 63137 x^{6} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + \beta_{15} q^{5} + \beta_1 q^{6} + \beta_{18} q^{7} + (\beta_{3} + \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + \beta_{15} q^{5} + \beta_1 q^{6} + \beta_{18} q^{7} + (\beta_{3} + \beta_1) q^{8} + q^{9} + (\beta_{13} + \beta_{9} - \beta_{2} - 1) q^{10} + (\beta_{18} + \beta_{6}) q^{11} + (\beta_{2} + 1) q^{12} + ( - \beta_{12} + \beta_{4}) q^{14} + \beta_{15} q^{15} + (\beta_{11} - \beta_{9} + \beta_{8} + \cdots + 2) q^{16}+ \cdots + (\beta_{18} + \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 20 q^{3} + 28 q^{4} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 20 q^{3} + 28 q^{4} + 20 q^{9} - 18 q^{10} + 28 q^{12} - 2 q^{14} + 44 q^{16} + 20 q^{17} - 2 q^{22} + 16 q^{23} + 44 q^{25} + 20 q^{27} + 46 q^{29} - 18 q^{30} + 10 q^{35} + 28 q^{36} + 70 q^{38} - 44 q^{40} - 2 q^{42} + 12 q^{43} + 44 q^{48} + 32 q^{49} + 20 q^{51} + 46 q^{53} - 2 q^{55} - 18 q^{56} + 30 q^{61} + 64 q^{62} + 40 q^{64} - 2 q^{66} + 28 q^{68} + 16 q^{69} - 18 q^{74} + 44 q^{75} + 148 q^{77} - 6 q^{79} + 20 q^{81} - 54 q^{82} + 46 q^{87} - 18 q^{88} - 18 q^{90} + 64 q^{92} + 42 q^{94} + 28 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 34 x^{18} + 485 x^{16} - 3770 x^{14} + 17340 x^{12} - 47849 x^{10} + 76299 x^{8} - 63137 x^{6} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21 \nu^{18} - 806 \nu^{16} + 11897 \nu^{14} - 85814 \nu^{12} + 318068 \nu^{10} - 611965 \nu^{8} + \cdots + 21668 ) / 152800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4057 \nu^{19} + 212102 \nu^{17} - 4166549 \nu^{15} + 41732238 \nu^{13} - 236007156 \nu^{11} + \cdots + 70809644 \nu ) / 1222400 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5417 \nu^{19} - 184262 \nu^{17} + 2630469 \nu^{15} - 20469678 \nu^{13} + 94274036 \nu^{11} + \cdots - 9697964 \nu ) / 611200 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5417 \nu^{18} - 184262 \nu^{16} + 2630469 \nu^{14} - 20469678 \nu^{12} + 94274036 \nu^{10} + \cdots - 4808364 ) / 305600 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16561 \nu^{18} + 561046 \nu^{16} - 7976077 \nu^{14} + 61831774 \nu^{12} - 283944788 \nu^{10} + \cdots + 9710412 ) / 611200 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5773 \nu^{18} - 181918 \nu^{16} + 2370841 \nu^{14} - 16560422 \nu^{12} + 67199684 \nu^{10} + \cdots - 1789596 ) / 122240 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 32311 \nu^{18} - 936346 \nu^{16} + 10863227 \nu^{14} - 64155474 \nu^{12} + 201462988 \nu^{10} + \cdots - 488212 ) / 611200 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1081 \nu^{18} - 34441 \nu^{16} + 455292 \nu^{14} - 3240454 \nu^{12} + 13481098 \nu^{10} + \cdots - 187052 ) / 19100 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18583 \nu^{18} + 551338 \nu^{16} - 6633131 \nu^{14} + 41586322 \nu^{12} - 145398764 \nu^{10} + \cdots + 551636 ) / 305600 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 45151 \nu^{18} - 1298186 \nu^{16} + 14863107 \nu^{14} - 85628034 \nu^{12} + 254489708 \nu^{10} + \cdots + 7172108 ) / 611200 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 46437 \nu^{19} - 1656782 \nu^{17} + 24850609 \nu^{15} - 203271158 \nu^{13} + 982864996 \nu^{11} + \cdots - 60377404 \nu ) / 1222400 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1081 \nu^{19} - 34441 \nu^{17} + 455292 \nu^{15} - 3240454 \nu^{13} + 13481098 \nu^{11} + \cdots - 91552 \nu ) / 19100 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 49231 \nu^{19} + 1673066 \nu^{17} - 23854067 \nu^{15} + 185336354 \nu^{13} - 852133548 \nu^{11} + \cdots + 42054452 \nu ) / 611200 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 105957 \nu^{19} + 3853902 \nu^{17} - 58832049 \nu^{15} + 488578038 \nu^{13} + \cdots + 97629244 \nu ) / 1222400 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 5083 \nu^{19} - 169842 \nu^{17} + 2376783 \nu^{15} - 18097802 \nu^{13} + 81457948 \nu^{11} + \cdots - 2852548 \nu ) / 48896 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 27067 \nu^{19} - 922802 \nu^{17} + 13196719 \nu^{15} - 102781258 \nu^{13} + 473118876 \nu^{11} + \cdots - 17896004 \nu ) / 122240 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{9} + \beta_{8} + \beta_{7} + 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} - \beta_{18} + \beta_{17} - \beta_{15} + \beta_{14} - \beta_{6} + \beta_{5} + 10\beta_{3} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{13} + 10\beta_{11} - \beta_{10} - 11\beta_{9} + 11\beta_{8} + 13\beta_{7} + \beta_{4} + 48\beta_{2} + 98 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14 \beta_{19} - 11 \beta_{18} + 14 \beta_{17} + 3 \beta_{16} - 15 \beta_{15} + 13 \beta_{14} + \cdots + 183 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 15 \beta_{13} - \beta_{12} + 81 \beta_{11} - 16 \beta_{10} - 96 \beta_{9} + 95 \beta_{8} + 124 \beta_{7} + \cdots + 645 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 140 \beta_{19} - 89 \beta_{18} + 140 \beta_{17} + 50 \beta_{16} - 157 \beta_{15} + 124 \beta_{14} + \cdots + 1218 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 155 \beta_{13} - 20 \beta_{12} + 618 \beta_{11} - 174 \beta_{10} - 775 \beta_{9} + 760 \beta_{8} + \cdots + 4428 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1225 \beta_{19} - 640 \beta_{18} + 1226 \beta_{17} + 569 \beta_{16} - 1416 \beta_{15} + \cdots + 8394 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1375 \beta_{13} - 271 \beta_{12} + 4623 \beta_{11} - 1625 \beta_{10} - 6039 \beta_{9} + 5909 \beta_{8} + \cdots + 31198 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 10014 \beta_{19} - 4320 \beta_{18} + 10035 \beta_{17} + 5543 \beta_{16} - 11810 \beta_{15} + \cdots + 59209 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 11249 \beta_{13} - 3117 \beta_{12} + 34397 \beta_{11} - 14073 \beta_{10} - 46207 \beta_{9} + \cdots + 223464 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 78717 \beta_{19} - 27881 \beta_{18} + 79010 \beta_{17} + 49817 \beta_{16} - 94077 \beta_{15} + \cdots + 424265 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 87600 \beta_{13} - 32687 \beta_{12} + 255981 \beta_{11} - 116852 \beta_{10} - 350058 \beta_{9} + \cdots + 1617931 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 603925 \beta_{19} - 172556 \beta_{18} + 607360 \beta_{17} + 427052 \beta_{16} - 728136 \beta_{15} + \cdots + 3072979 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 660113 \beta_{13} - 322332 \beta_{12} + 1909295 \beta_{11} - 945930 \beta_{10} - 2637431 \beta_{9} + \cdots + 11798746 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 4560796 \beta_{19} - 1015820 \beta_{18} + 4597311 \beta_{17} + 3552726 \beta_{16} + \cdots + 22424328 \beta_1 \) 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
−2.74622
−2.67916
−2.33145
−2.07632
−2.06483
−1.65105
−1.36632
−0.780731
−0.218610
−0.141266
0.141266
0.218610
0.780731
1.36632
1.65105
2.06483
2.07632
2.33145
2.67916
2.74622
−2.74622 1.00000 5.54175 0.255852 −2.74622 −1.34124 −9.72645 1.00000 −0.702626
1.2 −2.67916 1.00000 5.17791 2.95023 −2.67916 3.63383 −8.51413 1.00000 −7.90415
1.3 −2.33145 1.00000 3.43566 −2.34172 −2.33145 −0.681904 −3.34718 1.00000 5.45961
1.4 −2.07632 1.00000 2.31110 4.33839 −2.07632 1.06180 −0.645943 1.00000 −9.00788
1.5 −2.06483 1.00000 2.26351 −2.75305 −2.06483 −2.93154 −0.544098 1.00000 5.68458
1.6 −1.65105 1.00000 0.725965 3.78613 −1.65105 −3.75159 2.10349 1.00000 −6.25108
1.7 −1.36632 1.00000 −0.133182 −2.43054 −1.36632 5.04774 2.91460 1.00000 3.32089
1.8 −0.780731 1.00000 −1.39046 0.503146 −0.780731 −0.270374 2.64704 1.00000 −0.392822
1.9 −0.218610 1.00000 −1.95221 −3.23768 −0.218610 −2.82908 0.863992 1.00000 0.707788
1.10 −0.141266 1.00000 −1.98004 −0.606668 −0.141266 3.63030 0.562247 1.00000 0.0857018
1.11 0.141266 1.00000 −1.98004 0.606668 0.141266 −3.63030 −0.562247 1.00000 0.0857018
1.12 0.218610 1.00000 −1.95221 3.23768 0.218610 2.82908 −0.863992 1.00000 0.707788
1.13 0.780731 1.00000 −1.39046 −0.503146 0.780731 0.270374 −2.64704 1.00000 −0.392822
1.14 1.36632 1.00000 −0.133182 2.43054 1.36632 −5.04774 −2.91460 1.00000 3.32089
1.15 1.65105 1.00000 0.725965 −3.78613 1.65105 3.75159 −2.10349 1.00000 −6.25108
1.16 2.06483 1.00000 2.26351 2.75305 2.06483 2.93154 0.544098 1.00000 5.68458
1.17 2.07632 1.00000 2.31110 −4.33839 2.07632 −1.06180 0.645943 1.00000 −9.00788
1.18 2.33145 1.00000 3.43566 2.34172 2.33145 0.681904 3.34718 1.00000 5.45961
1.19 2.67916 1.00000 5.17791 −2.95023 2.67916 −3.63383 8.51413 1.00000 −7.90415
1.20 2.74622 1.00000 5.54175 −0.255852 2.74622 1.34124 9.72645 1.00000 −0.702626
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
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.bo 20
13.b even 2 1 inner 8619.2.a.bo 20
13.f odd 12 2 663.2.z.e ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.z.e ✓ 20 13.f odd 12 2
8619.2.a.bo 20 1.a even 1 1 trivial
8619.2.a.bo 20 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):

\( T_{2}^{20} - 34 T_{2}^{18} + 485 T_{2}^{16} - 3770 T_{2}^{14} + 17340 T_{2}^{12} - 47849 T_{2}^{10} + \cdots + 16 \) Copy content Toggle raw display
\( T_{5}^{20} - 72 T_{5}^{18} + 2158 T_{5}^{16} - 35037 T_{5}^{14} + 334799 T_{5}^{12} - 1904016 T_{5}^{10} + \cdots + 36864 \) Copy content Toggle raw display
\( T_{7}^{20} - 86 T_{7}^{18} + 3029 T_{7}^{16} - 56834 T_{7}^{14} + 615150 T_{7}^{12} - 3866116 T_{7}^{10} + \cdots + 295936 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 34 T^{18} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T - 1)^{20} \) Copy content Toggle raw display
$5$ \( T^{20} - 72 T^{18} + \cdots + 36864 \) Copy content Toggle raw display
$7$ \( T^{20} - 86 T^{18} + \cdots + 295936 \) Copy content Toggle raw display
$11$ \( T^{20} - 92 T^{18} + \cdots + 883600 \) Copy content Toggle raw display
$13$ \( T^{20} \) Copy content Toggle raw display
$17$ \( (T - 1)^{20} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 21218110241344 \) Copy content Toggle raw display
$23$ \( (T^{10} - 8 T^{9} + \cdots + 270096)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} - 23 T^{9} + \cdots - 1120710)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 10962927616 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 15099494400 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 79105228268544 \) Copy content Toggle raw display
$43$ \( (T^{10} - 6 T^{9} + \cdots - 7963923)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 383533056 \) Copy content Toggle raw display
$53$ \( (T^{10} - 23 T^{9} + \cdots - 110087168)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{10} - 15 T^{9} + \cdots - 7982720)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 555830793732096 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 47\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{10} + 3 T^{9} + \cdots + 61216)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 68451927450624 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
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