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Properties

Label 5002.2.a.j
Level $5002$
Weight $2$
Character orbit 5002.a
Self dual yes
Analytic conductor $39.941$
Analytic rank $1$
Dimension $17$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5002,2,Mod(1,5002)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5002.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5002, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5002 = 2 \cdot 41 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5002.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [17,17,-7,17,-11,-7,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(39.9411710910\)
Analytic rank: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 7 x^{16} - 6 x^{15} + 134 x^{14} - 129 x^{13} - 931 x^{12} + 1519 x^{11} + 3007 x^{10} + \cdots + 724 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} + (\beta_{2} - 1) q^{5} - \beta_1 q^{6} + \beta_{12} q^{7} + q^{8} + (\beta_{15} + \beta_{14} + 1) q^{9} + (\beta_{2} - 1) q^{10} + ( - \beta_{12} + \beta_{8} + \cdots + \beta_1) q^{11}+ \cdots + ( - 2 \beta_{15} + \beta_{14} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 17 q^{2} - 7 q^{3} + 17 q^{4} - 11 q^{5} - 7 q^{6} - 6 q^{7} + 17 q^{8} + 10 q^{9} - 11 q^{10} + 2 q^{11} - 7 q^{12} - 22 q^{13} - 6 q^{14} + 17 q^{16} - 23 q^{17} + 10 q^{18} + 14 q^{19} - 11 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 7 x^{16} - 6 x^{15} + 134 x^{14} - 129 x^{13} - 931 x^{12} + 1519 x^{11} + 3007 x^{10} + \cdots + 724 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 339435068 \nu^{16} + 1339313335 \nu^{15} + 7800971917 \nu^{14} - 31001024178 \nu^{13} + \cdots - 186266265208 ) / 4182661178 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 720722443 \nu^{16} + 2602826721 \nu^{15} + 21669145928 \nu^{14} - 86083828408 \nu^{13} + \cdots + 2111680303532 ) / 8365322356 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1215242559 \nu^{16} + 7378667193 \nu^{15} + 14671610204 \nu^{14} - 152785419076 \nu^{13} + \cdots + 1190025278564 ) / 8365322356 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 878607640 \nu^{16} - 7090935295 \nu^{15} + 1929815969 \nu^{14} + 117737906104 \nu^{13} + \cdots + 350442485076 ) / 4182661178 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1789600597 \nu^{16} + 11952045995 \nu^{15} + 12753103400 \nu^{14} - 223024353808 \nu^{13} + \cdots + 848570712904 ) / 8365322356 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2894011807 \nu^{16} + 17436809307 \nu^{15} + 34348690154 \nu^{14} - 354297352736 \nu^{13} + \cdots + 2664206421236 ) / 8365322356 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3054005023 \nu^{16} - 20271995625 \nu^{15} - 24271655980 \nu^{14} + 389311514240 \nu^{13} + \cdots - 2047075391080 ) / 8365322356 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3098814887 \nu^{16} + 16891264875 \nu^{15} + 50246952958 \nu^{14} - 376710089280 \nu^{13} + \cdots + 4033430304428 ) / 8365322356 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4377266069 \nu^{16} + 26663507099 \nu^{15} + 51635006684 \nu^{14} - 547760055136 \nu^{13} + \cdots + 5105070146248 ) / 8365322356 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5651351497 \nu^{16} + 34787037075 \nu^{15} + 62046304960 \nu^{14} - 696813257028 \nu^{13} + \cdots + 4029117434780 ) / 8365322356 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1426675862 \nu^{16} + 9074074614 \nu^{15} + 14487681185 \nu^{14} - 182692511547 \nu^{13} + \cdots + 1611351126562 ) / 2091330589 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 7003344477 \nu^{16} - 42127538237 \nu^{15} - 85001992082 \nu^{14} + 866105624876 \nu^{13} + \cdots - 7022425918896 ) / 8365322356 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3606272340 \nu^{16} + 22993835963 \nu^{15} + 36039355315 \nu^{14} - 460822519280 \nu^{13} + \cdots + 3634094011344 ) / 4182661178 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 3606272340 \nu^{16} - 22993835963 \nu^{15} - 36039355315 \nu^{14} + 460822519280 \nu^{13} + \cdots - 3650824656056 ) / 4182661178 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 3669400345 \nu^{16} - 22365495238 \nu^{15} - 41640109467 \nu^{14} + 449905821966 \nu^{13} + \cdots - 2743266171400 ) / 4182661178 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} + \beta_{14} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{13} + 2\beta_{12} - 2\beta_{11} + \beta_{9} - \beta_{5} + 3\beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{16} + 10 \beta_{15} + 11 \beta_{14} + 3 \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{16} + 18 \beta_{15} + 16 \beta_{14} + 14 \beta_{13} + 24 \beta_{12} - 22 \beta_{11} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 17 \beta_{16} + 105 \beta_{15} + 117 \beta_{14} + 50 \beta_{13} + 18 \beta_{12} - 14 \beta_{11} + \cdots + 211 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 68 \beta_{16} + 258 \beta_{15} + 223 \beta_{14} + 176 \beta_{13} + 243 \beta_{12} - 201 \beta_{11} + \cdots + 269 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 224 \beta_{16} + 1181 \beta_{15} + 1270 \beta_{14} + 646 \beta_{13} + 249 \beta_{12} - 148 \beta_{11} + \cdots + 1917 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 869 \beta_{16} + 3377 \beta_{15} + 2902 \beta_{14} + 2131 \beta_{13} + 2374 \beta_{12} - 1742 \beta_{11} + \cdots + 3266 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2690 \beta_{16} + 13703 \beta_{15} + 14040 \beta_{14} + 7751 \beta_{13} + 3099 \beta_{12} + \cdots + 18550 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 10083 \beta_{16} + 42231 \beta_{15} + 36245 \beta_{14} + 25295 \beta_{13} + 23098 \beta_{12} + \cdots + 38089 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 30916 \beta_{16} + 160547 \beta_{15} + 157233 \beta_{14} + 90594 \beta_{13} + 36558 \beta_{12} + \cdots + 186544 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 112510 \beta_{16} + 514526 \beta_{15} + 441060 \beta_{14} + 296858 \beta_{13} + 226444 \beta_{12} + \cdots + 435718 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 347653 \beta_{16} + 1882813 \beta_{15} + 1776316 \beta_{14} + 1048336 \beta_{13} + 418778 \beta_{12} + \cdots + 1925413 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1236461 \beta_{16} + 6167015 \beta_{15} + 5275759 \beta_{14} + 3459982 \beta_{13} + 2248544 \beta_{12} + \cdots + 4935861 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 3870268 \beta_{16} + 22035560 \beta_{15} + 20185380 \beta_{14} + 12083340 \beta_{13} + \cdots + 20261272 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
3.38685
3.06503
2.49287
2.38964
2.04981
1.13373
1.08319
0.840126
0.729579
0.651655
−0.870047
−0.985487
−1.12805
−1.20688
−1.62020
−2.15553
−2.85629
1.00000 −3.38685 1.00000 −2.21842 −3.38685 3.03429 1.00000 8.47073 −2.21842
1.2 1.00000 −3.06503 1.00000 0.601580 −3.06503 1.71159 1.00000 6.39441 0.601580
1.3 1.00000 −2.49287 1.00000 −1.93381 −2.49287 −2.47369 1.00000 3.21438 −1.93381
1.4 1.00000 −2.38964 1.00000 0.968966 −2.38964 −3.69400 1.00000 2.71038 0.968966
1.5 1.00000 −2.04981 1.00000 −0.149239 −2.04981 2.69117 1.00000 1.20173 −0.149239
1.6 1.00000 −1.13373 1.00000 2.68789 −1.13373 0.0312694 1.00000 −1.71465 2.68789
1.7 1.00000 −1.08319 1.00000 −4.24775 −1.08319 1.52714 1.00000 −1.82669 −4.24775
1.8 1.00000 −0.840126 1.00000 −3.27319 −0.840126 −0.245011 1.00000 −2.29419 −3.27319
1.9 1.00000 −0.729579 1.00000 0.663184 −0.729579 −4.09241 1.00000 −2.46771 0.663184
1.10 1.00000 −0.651655 1.00000 −0.228546 −0.651655 4.67547 1.00000 −2.57535 −0.228546
1.11 1.00000 0.870047 1.00000 2.14898 0.870047 −2.79906 1.00000 −2.24302 2.14898
1.12 1.00000 0.985487 1.00000 1.21451 0.985487 −1.34729 1.00000 −2.02881 1.21451
1.13 1.00000 1.12805 1.00000 −0.309612 1.12805 −0.311512 1.00000 −1.72749 −0.309612
1.14 1.00000 1.20688 1.00000 −2.36438 1.20688 −1.30363 1.00000 −1.54345 −2.36438
1.15 1.00000 1.62020 1.00000 1.74970 1.62020 −1.06391 1.00000 −0.374953 1.74970
1.16 1.00000 2.15553 1.00000 −4.10207 2.15553 1.87431 1.00000 1.64630 −4.10207
1.17 1.00000 2.85629 1.00000 −2.20778 2.85629 −4.21473 1.00000 5.15839 −2.20778
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(41\) \( +1 \)
\(61\) \( -1 \)

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 5002.2.a.j ✓ 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5002.2.a.j ✓ 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5002))\):

\( T_{3}^{17} + 7 T_{3}^{16} - 6 T_{3}^{15} - 134 T_{3}^{14} - 129 T_{3}^{13} + 931 T_{3}^{12} + 1519 T_{3}^{11} + \cdots - 724 \) Copy content Toggle raw display
\( T_{7}^{17} + 6 T_{7}^{16} - 39 T_{7}^{15} - 282 T_{7}^{14} + 414 T_{7}^{13} + 4681 T_{7}^{12} + \cdots - 368 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{17} \) Copy content Toggle raw display
$3$ \( T^{17} + 7 T^{16} + \cdots - 724 \) Copy content Toggle raw display
$5$ \( T^{17} + 11 T^{16} + \cdots - 64 \) Copy content Toggle raw display
$7$ \( T^{17} + 6 T^{16} + \cdots - 368 \) Copy content Toggle raw display
$11$ \( T^{17} - 2 T^{16} + \cdots - 223604 \) Copy content Toggle raw display
$13$ \( T^{17} + 22 T^{16} + \cdots - 9859472 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots - 469874048 \) Copy content Toggle raw display
$19$ \( T^{17} - 14 T^{16} + \cdots + 20491604 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots + 510460480 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots + 202639360 \) Copy content Toggle raw display
$31$ \( T^{17} + 17 T^{16} + \cdots + 58240064 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots - 15198075904 \) Copy content Toggle raw display
$41$ \( (T + 1)^{17} \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots - 5390435776 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots + 8111150980 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots + 180958025728 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots - 618415248640 \) Copy content Toggle raw display
$61$ \( (T - 1)^{17} \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots + 41289041782772 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 6589086751972 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots - 57165750023872 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 363236223700 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots - 1462892126528 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots + 2383687925168 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots - 17533820116544 \) Copy content Toggle raw display
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