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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7406,2,Mod(1,7406)] 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("7406.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7406, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7406.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,20,11,20,-1,11,-20] 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: \(59.1372077370\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 9 x^{19} - 3 x^{18} + 220 x^{17} - 283 x^{16} - 2285 x^{15} + 4163 x^{14} + 13144 x^{13} + \cdots - 5147 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 322)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} - \beta_{12} q^{5} + ( - \beta_1 + 1) q^{6} - q^{7} + q^{8} + (\beta_{11} + \beta_{10} - \beta_1 + 2) q^{9} - \beta_{12} q^{10} + ( - \beta_{16} + \beta_{8} - \beta_{2}) q^{11}+ \cdots + ( - \beta_{17} + \beta_{16} + \beta_{15} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 20 q^{2} + 11 q^{3} + 20 q^{4} - q^{5} + 11 q^{6} - 20 q^{7} + 20 q^{8} + 29 q^{9} - q^{10} - 8 q^{11} + 11 q^{12} + 23 q^{13} - 20 q^{14} + 15 q^{15} + 20 q^{16} - q^{17} + 29 q^{18} + 11 q^{19}+ \cdots + 23 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 9 x^{19} - 3 x^{18} + 220 x^{17} - 283 x^{16} - 2285 x^{15} + 4163 x^{14} + 13144 x^{13} + \cdots - 5147 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 23\!\cdots\!80 \nu^{19} + \cdots + 15\!\cdots\!01 ) / 51\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 36\!\cdots\!26 \nu^{19} + \cdots + 23\!\cdots\!18 ) / 51\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 50\!\cdots\!63 \nu^{19} + \cdots - 32\!\cdots\!42 ) / 51\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 62\!\cdots\!13 \nu^{19} + \cdots - 37\!\cdots\!35 ) / 51\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 63\!\cdots\!86 \nu^{19} + \cdots + 40\!\cdots\!35 ) / 51\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 97\!\cdots\!95 \nu^{19} + \cdots - 60\!\cdots\!21 ) / 51\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\!\cdots\!45 \nu^{19} + \cdots + 68\!\cdots\!16 ) / 51\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!42 \nu^{19} + \cdots + 76\!\cdots\!31 ) / 51\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11\!\cdots\!43 \nu^{19} + \cdots + 72\!\cdots\!40 ) / 51\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11\!\cdots\!43 \nu^{19} + \cdots - 72\!\cdots\!64 ) / 51\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 12\!\cdots\!25 \nu^{19} + \cdots - 82\!\cdots\!26 ) / 51\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!00 \nu^{19} + \cdots - 94\!\cdots\!10 ) / 51\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 18\!\cdots\!15 \nu^{19} + \cdots - 11\!\cdots\!23 ) / 51\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 48\!\cdots\!99 \nu^{19} + \cdots - 30\!\cdots\!97 ) / 51\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 57\!\cdots\!53 \nu^{19} + \cdots - 37\!\cdots\!48 ) / 51\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 59\!\cdots\!81 \nu^{19} + \cdots + 38\!\cdots\!32 ) / 51\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 83\!\cdots\!62 \nu^{19} + \cdots + 52\!\cdots\!60 ) / 51\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 12\!\cdots\!78 \nu^{19} + \cdots + 80\!\cdots\!20 ) / 51\!\cdots\!81 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{10} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{16} + 3 \beta_{14} - \beta_{13} + 2 \beta_{11} + \beta_{10} - 2 \beta_{8} + \beta_{6} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{16} + 11 \beta_{14} - 2 \beta_{13} - \beta_{12} + 11 \beta_{11} + 11 \beta_{10} - 6 \beta_{8} + \cdots + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{19} - \beta_{18} + \beta_{17} - 16 \beta_{16} + 69 \beta_{14} - 15 \beta_{13} - 3 \beta_{12} + \cdots + 53 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 15 \beta_{19} - 2 \beta_{18} + 4 \beta_{17} - 55 \beta_{16} + 2 \beta_{15} + 278 \beta_{14} - 47 \beta_{13} + \cdots + 273 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 96 \beta_{19} - 16 \beta_{18} + 29 \beta_{17} - 228 \beta_{16} + 9 \beta_{15} + 1297 \beta_{14} + \cdots + 725 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 447 \beta_{19} - 43 \beta_{18} + 130 \beta_{17} - 827 \beta_{16} + 67 \beta_{15} + 5307 \beta_{14} + \cdots + 2962 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2160 \beta_{19} - 210 \beta_{18} + 671 \beta_{17} - 3214 \beta_{16} + 314 \beta_{15} + 22613 \beta_{14} + \cdots + 9251 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 9523 \beta_{19} - 631 \beta_{18} + 3008 \beta_{17} - 11957 \beta_{16} + 1675 \beta_{15} + 92110 \beta_{14} + \cdots + 34545 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 41938 \beta_{19} - 2476 \beta_{18} + 13885 \beta_{17} - 45698 \beta_{16} + 7740 \beta_{15} + \cdots + 115695 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 178523 \beta_{19} - 7523 \beta_{18} + 60886 \beta_{17} - 172262 \beta_{16} + 36753 \beta_{15} + \cdots + 417426 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 754750 \beta_{19} - 25551 \beta_{18} + 267235 \beta_{17} - 656668 \beta_{16} + 165326 \beta_{15} + \cdots + 1441533 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 3142989 \beta_{19} - 72365 \beta_{18} + 1146459 \beta_{17} - 2494091 \beta_{16} + 743307 \beta_{15} + \cdots + 5137849 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 13008337 \beta_{19} - 204429 \beta_{18} + 4891961 \beta_{17} - 9521946 \beta_{16} + 3260576 \beta_{15} + \cdots + 17999569 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 53416694 \beta_{19} - 432756 \beta_{18} + 20620613 \beta_{17} - 36342777 \beta_{16} + 14205949 \beta_{15} + \cdots + 63941271 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 218348904 \beta_{19} - 474062 \beta_{18} + 86410824 \beta_{17} - 139104603 \beta_{16} + 61058597 \beta_{15} + \cdots + 225717854 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 888384058 \beta_{19} + 3050386 \beta_{18} + 359373640 \beta_{17} - 532874036 \beta_{16} + \cdots + 802079506 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 3602705458 \beta_{19} + 29144294 \beta_{18} + 1487146048 \beta_{17} - 2045226089 \beta_{16} + \cdots + 2845075213 \) 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
3.96562
3.62310
3.39736
2.81759
2.32577
2.02693
1.43975
0.852523
0.809731
0.795843
0.690504
−0.410773
−0.654517
−1.18897
−1.44387
−1.60222
−1.90175
−2.07084
−2.14393
−2.32784
1.00000 −2.96562 1.00000 −0.263053 −2.96562 −1.00000 1.00000 5.79490 −0.263053
1.2 1.00000 −2.62310 1.00000 −0.677494 −2.62310 −1.00000 1.00000 3.88066 −0.677494
1.3 1.00000 −2.39736 1.00000 −2.04903 −2.39736 −1.00000 1.00000 2.74733 −2.04903
1.4 1.00000 −1.81759 1.00000 −1.74872 −1.81759 −1.00000 1.00000 0.303618 −1.74872
1.5 1.00000 −1.32577 1.00000 2.79230 −1.32577 −1.00000 1.00000 −1.24233 2.79230
1.6 1.00000 −1.02693 1.00000 −1.00042 −1.02693 −1.00000 1.00000 −1.94540 −1.00042
1.7 1.00000 −0.439747 1.00000 −3.80222 −0.439747 −1.00000 1.00000 −2.80662 −3.80222
1.8 1.00000 0.147477 1.00000 3.66054 0.147477 −1.00000 1.00000 −2.97825 3.66054
1.9 1.00000 0.190269 1.00000 0.265180 0.190269 −1.00000 1.00000 −2.96380 0.265180
1.10 1.00000 0.204157 1.00000 −1.86217 0.204157 −1.00000 1.00000 −2.95832 −1.86217
1.11 1.00000 0.309496 1.00000 2.58278 0.309496 −1.00000 1.00000 −2.90421 2.58278
1.12 1.00000 1.41077 1.00000 3.90538 1.41077 −1.00000 1.00000 −1.00972 3.90538
1.13 1.00000 1.65452 1.00000 −3.06858 1.65452 −1.00000 1.00000 −0.262573 −3.06858
1.14 1.00000 2.18897 1.00000 −3.78752 2.18897 −1.00000 1.00000 1.79159 −3.78752
1.15 1.00000 2.44387 1.00000 3.93338 2.44387 −1.00000 1.00000 2.97251 3.93338
1.16 1.00000 2.60222 1.00000 −2.26621 2.60222 −1.00000 1.00000 3.77155 −2.26621
1.17 1.00000 2.90175 1.00000 −1.63209 2.90175 −1.00000 1.00000 5.42018 −1.63209
1.18 1.00000 3.07084 1.00000 −1.04715 3.07084 −1.00000 1.00000 6.43009 −1.04715
1.19 1.00000 3.14393 1.00000 2.26439 3.14393 −1.00000 1.00000 6.88428 2.26439
1.20 1.00000 3.32784 1.00000 2.80070 3.32784 −1.00000 1.00000 8.07452 2.80070
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)
\(23\) \( +1 \)

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 7406.2.a.bu 20
23.b odd 2 1 7406.2.a.bv 20
23.c even 11 2 322.2.i.d ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.i.d ✓ 40 23.c even 11 2
7406.2.a.bu 20 1.a even 1 1 trivial
7406.2.a.bv 20 23.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7406))\):

\( T_{3}^{20} - 11 T_{3}^{19} + 16 T_{3}^{18} + 233 T_{3}^{17} - 878 T_{3}^{16} - 1279 T_{3}^{15} + \cdots - 109 \) Copy content Toggle raw display
\( T_{5}^{20} + T_{5}^{19} - 65 T_{5}^{18} - 87 T_{5}^{17} + 1735 T_{5}^{16} + 2981 T_{5}^{15} + \cdots + 138863 \) Copy content Toggle raw display
\( T_{11}^{20} + 8 T_{11}^{19} - 100 T_{11}^{18} - 896 T_{11}^{17} + 3696 T_{11}^{16} + 39934 T_{11}^{15} + \cdots - 22774784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - 11 T^{19} + \cdots - 109 \) Copy content Toggle raw display
$5$ \( T^{20} + T^{19} + \cdots + 138863 \) Copy content Toggle raw display
$7$ \( (T + 1)^{20} \) Copy content Toggle raw display
$11$ \( T^{20} + 8 T^{19} + \cdots - 22774784 \) Copy content Toggle raw display
$13$ \( T^{20} - 23 T^{19} + \cdots - 1628791 \) Copy content Toggle raw display
$17$ \( T^{20} + T^{19} + \cdots - 11635712 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 72862273319 \) Copy content Toggle raw display
$23$ \( T^{20} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 58812756992 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots - 27662808064 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 26300862934016 \) Copy content Toggle raw display
$41$ \( T^{20} - 5 T^{19} + \cdots - 45212672 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 1370164112384 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 2300831126528 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 55588419277249 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots - 239788679386112 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 45\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 36\!\cdots\!61 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 36\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 136911290980727 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 11\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots - 45666187574272 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots - 10\!\cdots\!96 \) Copy content Toggle raw display
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