↑

Properties

Label 1845.4.a.s
Level $1845$
Weight $4$
Character orbit 1845.a
Self dual yes
Analytic conductor $108.859$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1845,4,Mod(1,1845)] 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("1845.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1845, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1845 = 3^{2} \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1845.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,-5,0,91,-70,0,40] 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: \(108.858523961\)
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: \( x^{14} - 5 x^{13} - 89 x^{12} + 433 x^{11} + 3100 x^{10} - 14427 x^{9} - 53983 x^{8} + 233727 x^{7} + \cdots - 2084736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 615)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 7) q^{4} - 5 q^{5} + (\beta_{6} + 3) q^{7} + ( - \beta_{3} - 5 \beta_1 - 4) q^{8} + 5 \beta_1 q^{10} + (\beta_{10} - \beta_{9} + \beta_{6} + \cdots - 3) q^{11} + (\beta_{8} - \beta_{4} + \beta_1 + 6) q^{13}+ \cdots + ( - 2 \beta_{13} - 11 \beta_{12} + \cdots + 757) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 5 q^{2} + 91 q^{4} - 70 q^{5} + 40 q^{7} - 81 q^{8} + 25 q^{10} - 33 q^{11} + 91 q^{13} - 84 q^{14} + 331 q^{16} - 174 q^{17} + 323 q^{19} - 455 q^{20} + 110 q^{22} - 520 q^{23} + 350 q^{25} - 220 q^{26}+ \cdots + 9192 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 5 x^{13} - 89 x^{12} + 433 x^{11} + 3100 x^{10} - 14427 x^{9} - 53983 x^{8} + 233727 x^{7} + \cdots - 2084736 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 21\nu - 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1158660373 \nu^{13} - 15450685482 \nu^{12} - 51087891293 \nu^{11} + 1130531683500 \nu^{10} + \cdots - 519434938283760 ) / 18051785519568 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9818086552 \nu^{13} + 164383165437 \nu^{12} + 202342223366 \nu^{11} + \cdots + 12\!\cdots\!32 ) / 90258927597840 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3489037715 \nu^{13} - 1822236778 \nu^{12} + 317002499801 \nu^{11} + \cdots - 13\!\cdots\!68 ) / 12034523679712 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 143848027787 \nu^{13} - 224181829428 \nu^{12} + 13619338066861 \nu^{11} + \cdots - 92\!\cdots\!88 ) / 180517855195680 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 162889120919 \nu^{13} + 55900337691 \nu^{12} - 14580750021967 \nu^{11} + \cdots + 75\!\cdots\!76 ) / 180517855195680 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 32140534573 \nu^{13} + 3702723093 \nu^{12} + 2846999838569 \nu^{11} + \cdots - 11\!\cdots\!52 ) / 30086309199280 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 66422412883 \nu^{13} + 10871461998 \nu^{12} + 5848090022749 \nu^{11} + \cdots - 29\!\cdots\!72 ) / 60172618398560 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 311987939473 \nu^{13} - 152206758708 \nu^{12} - 26948150596079 \nu^{11} + \cdots + 12\!\cdots\!52 ) / 180517855195680 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 36552017459 \nu^{13} - 19171216785 \nu^{12} - 3155780850049 \nu^{11} + \cdots + 11\!\cdots\!08 ) / 18051785519568 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 36587580245 \nu^{13} - 2971363179 \nu^{12} + 3274821255973 \nu^{11} + \cdots - 16\!\cdots\!96 ) / 18051785519568 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 21\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} + 2\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 29\beta_{2} + 2\beta _1 + 322 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{13} - 2 \beta_{12} + 2 \beta_{10} - \beta_{9} + 2 \beta_{8} - 4 \beta_{6} - 3 \beta_{5} + \cdots + 147 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{13} + \beta_{12} + 45 \beta_{11} + 48 \beta_{10} - 4 \beta_{9} - 4 \beta_{8} + 4 \beta_{7} + \cdots + 7785 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 52 \beta_{13} - 100 \beta_{12} - 5 \beta_{11} + 112 \beta_{10} - 61 \beta_{9} + 86 \beta_{8} + \cdots + 4128 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 134 \beta_{13} + 48 \beta_{12} + 1555 \beta_{11} + 1774 \beta_{10} - 255 \beta_{9} - 238 \beta_{8} + \cdots + 201361 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2094 \beta_{13} - 3668 \beta_{12} - 422 \beta_{11} + 4545 \beta_{10} - 2527 \beta_{9} + 2826 \beta_{8} + \cdots + 99279 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6153 \beta_{13} + 1822 \beta_{12} + 49005 \beta_{11} + 60058 \beta_{10} - 11318 \beta_{9} + \cdots + 5427432 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 76050 \beta_{13} - 120205 \beta_{12} - 22204 \beta_{11} + 163180 \beta_{10} - 90509 \beta_{9} + \cdots + 2034868 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 240728 \beta_{13} + 64946 \beta_{12} + 1483458 \beta_{11} + 1949826 \beta_{10} - 429080 \beta_{9} + \cdots + 150103527 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2604420 \beta_{13} - 3742682 \beta_{12} - 952080 \beta_{11} + 5507092 \beta_{10} - 3028138 \beta_{9} + \cdots + 31152772 \) 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
5.39368
5.05518
5.04778
3.95718
3.43302
2.69239
1.15585
−0.295122
−1.29533
−2.69918
−3.81353
−4.01759
−4.13962
−5.47471
−5.39368 0 21.0917 −5.00000 0 −1.41190 −70.6126 0 26.9684
1.2 −5.05518 0 17.5548 −5.00000 0 12.2491 −48.3013 0 25.2759
1.3 −5.04778 0 17.4800 −5.00000 0 −12.7256 −47.8531 0 25.2389
1.4 −3.95718 0 7.65931 −5.00000 0 35.6205 1.34816 0 19.7859
1.5 −3.43302 0 3.78565 −5.00000 0 5.77356 14.4680 0 17.1651
1.6 −2.69239 0 −0.751048 −5.00000 0 −17.0007 23.5612 0 13.4619
1.7 −1.15585 0 −6.66400 −5.00000 0 8.87787 16.9494 0 5.77927
1.8 0.295122 0 −7.91290 −5.00000 0 22.6261 −4.69624 0 −1.47561
1.9 1.29533 0 −6.32213 −5.00000 0 −23.1041 −18.5518 0 −6.47663
1.10 2.69918 0 −0.714432 −5.00000 0 11.9424 −23.5218 0 −13.4959
1.11 3.81353 0 6.54301 −5.00000 0 23.9474 −5.55628 0 −19.0676
1.12 4.01759 0 8.14100 −5.00000 0 −21.2183 0.566485 0 −20.0879
1.13 4.13962 0 9.13646 −5.00000 0 −35.0560 4.70451 0 −20.6981
1.14 5.47471 0 21.9725 −5.00000 0 29.4797 76.4954 0 −27.3736
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)
\(41\) \( -1 \)

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 1845.4.a.s 14
3.b odd 2 1 615.4.a.k ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
615.4.a.k ✓ 14 3.b odd 2 1
1845.4.a.s 14 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1845))\):

\( T_{2}^{14} + 5 T_{2}^{13} - 89 T_{2}^{12} - 433 T_{2}^{11} + 3100 T_{2}^{10} + 14427 T_{2}^{9} + \cdots - 2084736 \) Copy content Toggle raw display
\( T_{7}^{14} - 40 T_{7}^{13} - 2347 T_{7}^{12} + 105673 T_{7}^{11} + 1643997 T_{7}^{10} + \cdots + 22\!\cdots\!12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 5 T^{13} + \cdots - 2084736 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( (T + 5)^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 22\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots - 36\!\cdots\!60 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots - 16\!\cdots\!92 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 91\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 74\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 21\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T - 41)^{14} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 48\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 19\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 12\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 90\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 66\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 44\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 32\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 90\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 73\!\cdots\!76 \) Copy content Toggle raw display
show more
show less