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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [581,2,Mod(1,581)] 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("581.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(581, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 581 = 7 \cdot 83 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 581.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.63930835744\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 25 x^{12} + 23 x^{11} + 243 x^{10} - 197 x^{9} - 1158 x^{8} + 771 x^{7} + 2788 x^{6} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 2) q^{4} + \beta_{6} q^{5} + (\beta_{11} - \beta_{9} + \beta_{8} + \cdots + 1) q^{6} - q^{7} + ( - \beta_{8} + \beta_{7} + 2 \beta_1) q^{8} + ( - \beta_{12} - \beta_{10} - \beta_{6} + \cdots + 1) q^{9}+ \cdots + ( - 3 \beta_{13} + \beta_{12} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + q^{2} + 3 q^{3} + 23 q^{4} - 5 q^{5} + 6 q^{6} - 14 q^{7} + 3 q^{8} + 23 q^{9} + 14 q^{10} + 15 q^{11} + 6 q^{12} - 14 q^{13} - q^{14} + q^{15} + 37 q^{16} + 5 q^{17} + 16 q^{18} + 31 q^{19} - 5 q^{20}+ \cdots + 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 25 x^{12} + 23 x^{11} + 243 x^{10} - 197 x^{9} - 1158 x^{8} + 771 x^{7} + 2788 x^{6} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1795 \nu^{13} + 2955 \nu^{12} + 45246 \nu^{11} - 68440 \nu^{10} - 441123 \nu^{9} + 597047 \nu^{8} + \cdots - 6920 ) / 20092 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 16835 \nu^{13} + 18410 \nu^{12} + 416897 \nu^{11} - 429466 \nu^{10} - 3994335 \nu^{9} + \cdots - 1659711 ) / 120552 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7019 \nu^{13} + 9806 \nu^{12} + 174119 \nu^{11} - 229732 \nu^{10} - 1669494 \nu^{9} + \cdots - 139776 ) / 30138 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17126 \nu^{13} + 22331 \nu^{12} + 418448 \nu^{11} - 523084 \nu^{10} - 3941502 \nu^{9} + \cdots - 409854 ) / 60276 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 69433 \nu^{13} - 89776 \nu^{12} - 1716649 \nu^{11} + 2099354 \nu^{10} + 16413657 \nu^{9} + \cdots + 2715303 ) / 120552 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 69433 \nu^{13} - 89776 \nu^{12} - 1716649 \nu^{11} + 2099354 \nu^{10} + 16413657 \nu^{9} + \cdots + 2715303 ) / 120552 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 72157 \nu^{13} - 98206 \nu^{12} - 1779637 \nu^{11} + 2285318 \nu^{10} + 16959945 \nu^{9} + \cdots + 1844163 ) / 120552 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12359 \nu^{13} - 15113 \nu^{12} - 303196 \nu^{11} + 353360 \nu^{10} + 2869969 \nu^{9} + \cdots + 432472 ) / 20092 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6350 \nu^{13} - 8145 \nu^{12} - 155795 \nu^{11} + 191296 \nu^{10} + 1474610 \nu^{9} - 1692073 \nu^{8} + \cdots + 53317 ) / 10046 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 27783 \nu^{13} + 34810 \nu^{12} + 682797 \nu^{11} - 811410 \nu^{10} - 6475983 \nu^{9} + \cdots - 788095 ) / 40184 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 33417 \nu^{13} - 42266 \nu^{12} - 821571 \nu^{11} + 989230 \nu^{10} + 7799921 \nu^{9} + \cdots + 1040761 ) / 40184 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{7} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - \beta_{11} + 2\beta_{10} - \beta_{5} + \beta_{4} + 8\beta_{2} - \beta _1 + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - \beta_{12} - 2 \beta_{10} - 10 \beta_{8} + 9 \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{13} + 10 \beta_{12} - 12 \beta_{11} + 22 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} - 9 \beta_{5} + \cdots + 146 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 16 \beta_{13} - 12 \beta_{12} + \beta_{11} - 27 \beta_{10} + 2 \beta_{9} - 85 \beta_{8} + 68 \beta_{7} + \cdots - 29 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 19 \beta_{13} + 83 \beta_{12} - 112 \beta_{11} + 192 \beta_{10} - 32 \beta_{9} + 29 \beta_{8} + \cdots + 978 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 176 \beta_{13} - 108 \beta_{12} + 21 \beta_{11} - 273 \beta_{10} + 38 \beta_{9} - 684 \beta_{8} + \cdots - 314 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 233 \beta_{13} + 660 \beta_{12} - 953 \beta_{11} + 1556 \beta_{10} - 362 \beta_{9} + 310 \beta_{8} + \cdots + 6773 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1661 \beta_{13} - 882 \beta_{12} + 286 \beta_{11} - 2481 \beta_{10} + 486 \beta_{9} - 5371 \beta_{8} + \cdots - 3031 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2359 \beta_{13} + 5182 \beta_{12} - 7752 \beta_{11} + 12244 \beta_{10} - 3570 \beta_{9} + 2981 \beta_{8} + \cdots + 47984 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 14469 \beta_{13} - 6940 \beta_{12} + 3226 \beta_{11} - 21396 \beta_{10} + 5226 \beta_{9} - 41662 \beta_{8} + \cdots - 27555 \) 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
−2.79846
−2.52484
−2.14362
−1.71409
−1.42981
−0.296960
−0.284373
−0.0538470
1.01492
1.61383
1.98415
2.39342
2.52665
2.71302
−2.79846 0.207581 5.83135 −4.16061 −0.580905 −1.00000 −10.7219 −2.95691 11.6433
1.2 −2.52484 0.899174 4.37482 2.69792 −2.27027 −1.00000 −5.99603 −2.19149 −6.81182
1.3 −2.14362 −3.01206 2.59511 −0.696178 6.45671 −1.00000 −1.27568 6.07250 1.49234
1.4 −1.71409 3.30031 0.938095 −3.90497 −5.65702 −1.00000 1.82020 7.89205 6.69346
1.5 −1.42981 −1.44693 0.0443697 2.81382 2.06885 −1.00000 2.79619 −0.906379 −4.02324
1.6 −0.296960 −1.70150 −1.91181 −2.93497 0.505278 −1.00000 1.16165 −0.104897 0.871570
1.7 −0.284373 0.0995355 −1.91913 −0.993095 −0.0283052 −1.00000 1.11449 −2.99009 0.282409
1.8 −0.0538470 2.54352 −1.99710 1.17661 −0.136961 −1.00000 0.215232 3.46947 −0.0633569
1.9 1.01492 −2.59943 −0.969945 −2.94800 −2.63821 −1.00000 −3.01425 3.75706 −2.99198
1.10 1.61383 2.65783 0.604453 3.10495 4.28929 −1.00000 −2.25218 4.06405 5.01086
1.11 1.98415 1.69256 1.93686 1.43078 3.35829 −1.00000 −0.125277 −0.135254 2.83889
1.12 2.39342 2.74525 3.72847 −2.90819 6.57055 −1.00000 4.13695 4.53642 −6.96052
1.13 2.52665 −2.87330 4.38397 1.50956 −7.25982 −1.00000 6.02345 5.25584 3.81413
1.14 2.71302 0.487471 5.36050 0.812366 1.32252 −1.00000 9.11713 −2.76237 2.20397
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( +1 \)
\(83\) \( -1 \)

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 581.2.a.d ✓ 14
3.b odd 2 1 5229.2.a.q 14
4.b odd 2 1 9296.2.a.t 14
7.b odd 2 1 4067.2.a.h 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
581.2.a.d ✓ 14 1.a even 1 1 trivial
4067.2.a.h 14 7.b odd 2 1
5229.2.a.q 14 3.b odd 2 1
9296.2.a.t 14 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - T_{2}^{13} - 25 T_{2}^{12} + 23 T_{2}^{11} + 243 T_{2}^{10} - 197 T_{2}^{9} - 1158 T_{2}^{8} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(581))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - T^{13} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{14} - 3 T^{13} + \cdots - 52 \) Copy content Toggle raw display
$5$ \( T^{14} + 5 T^{13} + \cdots - 13754 \) Copy content Toggle raw display
$7$ \( (T + 1)^{14} \) Copy content Toggle raw display
$11$ \( T^{14} - 15 T^{13} + \cdots - 747520 \) Copy content Toggle raw display
$13$ \( T^{14} + 14 T^{13} + \cdots + 124694 \) Copy content Toggle raw display
$17$ \( T^{14} - 5 T^{13} + \cdots + 58802560 \) Copy content Toggle raw display
$19$ \( T^{14} - 31 T^{13} + \cdots - 626432 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 527650816 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 834971562 \) Copy content Toggle raw display
$31$ \( T^{14} - 5 T^{13} + \cdots - 66848176 \) Copy content Toggle raw display
$37$ \( T^{14} - 7 T^{13} + \cdots + 6444878 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 6971776640 \) Copy content Toggle raw display
$43$ \( T^{14} - 35 T^{13} + \cdots + 84606516 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 116608000 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 21210362496 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 138703484 \) Copy content Toggle raw display
$61$ \( T^{14} - 11 T^{13} + \cdots + 99720064 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 106886444 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 1661608681632 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 2952153216 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 16335929272 \) Copy content Toggle raw display
$83$ \( (T - 1)^{14} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 5831393666 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 87712315862 \) Copy content Toggle raw display
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