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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1445,4,Mod(1,1445)] 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("1445.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1445.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,-4,12,72,-90] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(85.2577599583\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 4 x^{17} - 100 x^{16} + 384 x^{15} + 4081 x^{14} - 15112 x^{13} - 87122 x^{12} + 314604 x^{11} + \cdots + 2267392 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 85)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{7} + 1) q^{3} + (\beta_{2} + 4) q^{4} - 5 q^{5} + ( - \beta_{10} - \beta_{7} - \beta_{6} + \cdots + 2) q^{6} + (\beta_{14} - \beta_{3} - 2 \beta_1 + 4) q^{7} + (\beta_{9} - \beta_{8} + 3 \beta_{3} + \cdots - 2) q^{8}+ \cdots + (11 \beta_{17} - 3 \beta_{16} + \cdots + 120) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 4 q^{2} + 12 q^{3} + 72 q^{4} - 90 q^{5} + 24 q^{6} + 56 q^{7} - 48 q^{8} + 194 q^{9} + 20 q^{10} + 40 q^{11} + 96 q^{12} - 36 q^{13} + 412 q^{14} - 60 q^{15} + 156 q^{16} - 220 q^{18} + 180 q^{19}+ \cdots + 3360 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 4 x^{17} - 100 x^{16} + 384 x^{15} + 4081 x^{14} - 15112 x^{13} - 87122 x^{12} + 314604 x^{11} + \cdots + 2267392 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!29 \nu^{17} + \cdots - 13\!\cdots\!28 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 24\!\cdots\!11 \nu^{17} + \cdots - 11\!\cdots\!32 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 41\!\cdots\!41 \nu^{17} + \cdots - 76\!\cdots\!32 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\!\cdots\!19 \nu^{17} + \cdots - 16\!\cdots\!48 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14\!\cdots\!17 \nu^{17} + \cdots - 54\!\cdots\!48 ) / 92\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!81 \nu^{17} + \cdots + 10\!\cdots\!92 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 25\!\cdots\!79 \nu^{17} + \cdots + 11\!\cdots\!28 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 31\!\cdots\!29 \nu^{17} + \cdots + 29\!\cdots\!88 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 57\!\cdots\!21 \nu^{17} + \cdots - 66\!\cdots\!92 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 88\!\cdots\!75 \nu^{17} + \cdots - 12\!\cdots\!64 ) / 27\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 29\!\cdots\!03 \nu^{17} + \cdots - 79\!\cdots\!56 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 49\!\cdots\!93 \nu^{17} + \cdots - 44\!\cdots\!56 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 54\!\cdots\!17 \nu^{17} + \cdots - 32\!\cdots\!04 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 21\!\cdots\!69 \nu^{17} + \cdots + 78\!\cdots\!08 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 10\!\cdots\!39 \nu^{17} + \cdots + 23\!\cdots\!28 ) / 57\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{8} - 3\beta_{3} + \beta_{2} + 19\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} - \beta_{16} + \beta_{14} + \beta_{13} - 2 \beta_{12} + \beta_{9} + \beta_{8} + 3 \beta_{7} + \cdots + 234 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{17} - \beta_{16} + \beta_{12} + 6 \beta_{10} - 32 \beta_{9} + 34 \beta_{8} + 12 \beta_{7} + \cdots + 141 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 35 \beta_{17} - 33 \beta_{16} + 2 \beta_{15} + 48 \beta_{14} + 34 \beta_{13} - 77 \beta_{12} + \cdots + 5301 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 53 \beta_{17} - 49 \beta_{16} - 2 \beta_{15} + 18 \beta_{14} - 29 \beta_{13} - 2 \beta_{12} + \cdots + 5956 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 990 \beta_{17} - 866 \beta_{16} + 156 \beta_{15} + 1695 \beta_{14} + 868 \beta_{13} - 2432 \beta_{12} + \cdots + 129730 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1916 \beta_{17} - 1790 \beta_{16} - 4 \beta_{15} + 1336 \beta_{14} - 1924 \beta_{13} - 1744 \beta_{12} + \cdots + 212074 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 26144 \beta_{17} - 21380 \beta_{16} + 7428 \beta_{15} + 53626 \beta_{14} + 19156 \beta_{13} + \cdots + 3321472 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 59660 \beta_{17} - 58000 \beta_{16} + 4448 \beta_{15} + 65340 \beta_{14} - 85196 \beta_{13} + \cdots + 6993890 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 668975 \beta_{17} - 522159 \beta_{16} + 286392 \beta_{15} + 1616615 \beta_{14} + 363391 \beta_{13} + \cdots + 87595894 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1726367 \beta_{17} - 1762111 \beta_{16} + 317536 \beta_{15} + 2662712 \beta_{14} - 3188320 \beta_{13} + \cdots + 221701391 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 16815299 \beta_{17} - 12874933 \beta_{16} + 9900862 \beta_{15} + 47630142 \beta_{14} + 5105020 \beta_{13} + \cdots + 2360297415 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 47982229 \beta_{17} - 51539953 \beta_{16} + 15539826 \beta_{15} + 98181058 \beta_{14} + \cdots + 6873769010 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 417717664 \beta_{17} - 323270328 \beta_{16} + 321889104 \beta_{15} + 1388108721 \beta_{14} + \cdots + 64673472144 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1302722518 \beta_{17} - 1473396524 \beta_{16} + 643876608 \beta_{15} + 3405115660 \beta_{14} + \cdots + 210350270338 \) 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
5.47811
5.11518
4.74538
3.47905
3.20788
2.83688
2.05027
1.94047
0.545023
0.127036
−0.377720
−1.57809
−2.79850
−3.39344
−3.53281
−4.34429
−4.50572
−4.99469
−5.47811 7.02735 22.0096 −5.00000 −38.4966 −5.82890 −76.7463 22.3836 27.3905
1.2 −5.11518 2.11968 18.1651 −5.00000 −10.8426 −14.8030 −51.9962 −22.5069 25.5759
1.3 −4.74538 −9.20696 14.5186 −5.00000 43.6905 −17.7728 −30.9332 57.7682 23.7269
1.4 −3.47905 −7.09451 4.10378 −5.00000 24.6822 −12.1453 13.5551 23.3321 17.3952
1.5 −3.20788 −3.66771 2.29051 −5.00000 11.7656 29.0219 18.3154 −13.5479 16.0394
1.6 −2.83688 6.41828 0.0479002 −5.00000 −18.2079 −3.50988 22.5592 14.1944 14.1844
1.7 −2.05027 3.13670 −3.79639 −5.00000 −6.43109 19.1082 24.1858 −17.1611 10.2514
1.8 −1.94047 9.45226 −4.23456 −5.00000 −18.3419 −15.9679 23.7408 62.3452 9.70237
1.9 −0.545023 3.71535 −7.70295 −5.00000 −2.02495 −7.55312 8.55847 −13.1962 2.72512
1.10 −0.127036 −4.67779 −7.98386 −5.00000 0.594250 −0.388425 2.03053 −5.11828 0.635182
1.11 0.377720 −8.15996 −7.85733 −5.00000 −3.08218 26.3048 −5.98964 39.5849 −1.88860
1.12 1.57809 2.98484 −5.50962 −5.00000 4.71036 4.92348 −21.3195 −18.0907 −7.89047
1.13 2.79850 9.63040 −0.168384 −5.00000 26.9507 19.8611 −22.8592 65.7446 −13.9925
1.14 3.39344 1.08368 3.51542 −5.00000 3.67738 −17.6031 −15.2181 −25.8256 −16.9672
1.15 3.53281 −5.08946 4.48076 −5.00000 −17.9801 6.06165 −12.4328 −1.09735 −17.6641
1.16 4.34429 1.66844 10.8729 −5.00000 7.24821 −21.9596 12.4806 −24.2163 −21.7215
1.17 4.50572 −5.73681 12.3016 −5.00000 −25.8485 31.7731 19.3816 5.91098 −22.5286
1.18 4.99469 8.39622 16.9470 −5.00000 41.9366 36.4776 44.6874 43.4966 −24.9735
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(17\) \( +1 \)

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 1445.4.a.t 18
17.b even 2 1 1445.4.a.s 18
17.d even 8 2 85.4.e.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.4.e.a ✓ 36 17.d even 8 2
1445.4.a.s 18 17.b even 2 1
1445.4.a.t 18 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1445))\):

\( T_{2}^{18} + 4 T_{2}^{17} - 100 T_{2}^{16} - 384 T_{2}^{15} + 4081 T_{2}^{14} + 15112 T_{2}^{13} + \cdots + 2267392 \) Copy content Toggle raw display
\( T_{3}^{18} - 12 T_{3}^{17} - 268 T_{3}^{16} + 3488 T_{3}^{15} + 27169 T_{3}^{14} - 407628 T_{3}^{13} + \cdots - 1227026926600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 4 T^{17} + \cdots + 2267392 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots - 1227026926600 \) Copy content Toggle raw display
$5$ \( (T + 5)^{18} \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots - 18\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots - 30\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{18} \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots - 36\!\cdots\!32 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 15\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots - 53\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots - 59\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 10\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 52\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 59\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 14\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 43\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 12\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 42\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 39\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 16\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 74\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 80\!\cdots\!00 \) Copy content Toggle raw display
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