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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [28,5,Mod(15,28)] 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("28.15"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(28, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 28.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.89435896635\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 10 x^{10} - 29 x^{9} + 174 x^{8} - 96 x^{7} + 88 x^{6} - 3030 x^{5} - 399 x^{4} + \cdots + 117656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{20}\cdot 7^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{2} - 3) q^{4} + (\beta_{9} + 2) q^{5} + (\beta_{10} + \beta_{5} - \beta_{3} + \cdots + 2) q^{6} + (\beta_{5} + \beta_1) q^{7} + ( - \beta_{11} - \beta_{9} + \beta_{7} + \cdots + 15) q^{8}+ \cdots + (81 \beta_{11} + 93 \beta_{10} + \cdots + 80) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 31 q^{4} + 24 q^{5} + 30 q^{6} + 171 q^{8} - 436 q^{9} + 52 q^{10} + 390 q^{12} + 120 q^{13} + 147 q^{14} - 911 q^{16} - 648 q^{17} + 631 q^{18} + 912 q^{20} - 628 q^{22} - 1174 q^{24}+ \cdots - 1029 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 10 x^{10} - 29 x^{9} + 174 x^{8} - 96 x^{7} + 88 x^{6} - 3030 x^{5} - 399 x^{4} + \cdots + 117656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 57 \nu^{11} + 501 \nu^{10} - 1981 \nu^{9} + 5616 \nu^{8} - 12878 \nu^{7} + 12386 \nu^{6} + \cdots - 3433704 ) / 753664 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 129 \nu^{11} + 557 \nu^{10} + 1435 \nu^{9} - 8832 \nu^{8} + 4818 \nu^{7} - 30622 \nu^{6} + \cdots - 8263336 ) / 753664 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2 \nu^{11} - 97 \nu^{10} + 710 \nu^{9} - 1895 \nu^{8} + 4413 \nu^{7} - 20893 \nu^{6} + \cdots - 2590424 ) / 47104 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 173 \nu^{11} + 961 \nu^{10} - 481 \nu^{9} - 1128 \nu^{8} - 16382 \nu^{7} - 27614 \nu^{6} + \cdots - 12261704 ) / 376832 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 321 \nu^{11} + 2845 \nu^{10} - 8421 \nu^{9} + 9392 \nu^{8} - 21758 \nu^{7} + 18610 \nu^{6} + \cdots - 9035816 ) / 753664 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 493 \nu^{11} - 3081 \nu^{10} + 6913 \nu^{9} - 30688 \nu^{8} + 147638 \nu^{7} - 107354 \nu^{6} + \cdots + 34466952 ) / 753664 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 589 \nu^{11} - 4137 \nu^{10} + 5729 \nu^{9} + 11616 \nu^{8} - 4170 \nu^{7} + 91814 \nu^{6} + \cdots + 15901064 ) / 753664 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 353 \nu^{11} - 1757 \nu^{10} + 261 \nu^{9} - 976 \nu^{8} + 51614 \nu^{7} + 9774 \nu^{6} + \cdots + 44050984 ) / 376832 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 281 \nu^{11} - 2941 \nu^{10} + 12413 \nu^{9} - 35160 \nu^{8} + 111782 \nu^{7} - 271802 \nu^{6} + \cdots - 10514904 ) / 376832 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 705 \nu^{11} - 6653 \nu^{10} + 24037 \nu^{9} - 55312 \nu^{8} + 184222 \nu^{7} - 448594 \nu^{6} + \cdots - 19023064 ) / 753664 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1663 \nu^{11} - 10435 \nu^{10} + 13403 \nu^{9} + 1936 \nu^{8} + 147298 \nu^{7} - 8622 \nu^{6} + \cdots + 64023256 ) / 753664 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - 3\beta_{10} + 2\beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + 3\beta_{4} - 2\beta_{2} + 30 ) / 56 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6 \beta_{11} - 4 \beta_{10} + 8 \beta_{9} - 2 \beta_{8} - 3 \beta_{7} - 4 \beta_{6} + 2 \beta_{5} + \cdots + 74 ) / 56 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 14 \beta_{11} + 4 \beta_{9} - 4 \beta_{8} + 17 \beta_{7} + 10 \beta_{6} + 24 \beta_{5} + 62 \beta_{4} + \cdots + 554 ) / 56 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 35 \beta_{11} - 49 \beta_{10} + 10 \beta_{9} + 25 \beta_{8} + 102 \beta_{7} + 67 \beta_{6} + 37 \beta_{5} + \cdots + 230 ) / 56 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 90 \beta_{11} - 60 \beta_{10} - 28 \beta_{9} + 174 \beta_{8} + 285 \beta_{7} + 228 \beta_{6} + \cdots - 4646 ) / 56 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 78 \beta_{11} + 172 \beta_{10} - 584 \beta_{9} + 620 \beta_{8} + 621 \beta_{7} + 930 \beta_{6} + \cdots - 26342 ) / 56 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 13 \beta_{11} + 71 \beta_{10} - 386 \beta_{9} + 357 \beta_{8} + 12 \beta_{7} + 419 \beta_{6} + \cdots - 14014 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 794 \beta_{11} + 1048 \beta_{10} - 10776 \beta_{9} + 7930 \beta_{8} - 10575 \beta_{7} + 5808 \beta_{6} + \cdots - 356550 ) / 56 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 12530 \beta_{11} + 1764 \beta_{10} - 38796 \beta_{9} + 16816 \beta_{8} - 81443 \beta_{7} + \cdots - 1175766 ) / 56 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 92295 \beta_{11} - 7077 \beta_{10} - 132310 \beta_{9} + 18245 \beta_{8} - 397366 \beta_{7} + \cdots - 3200938 ) / 56 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 491150 \beta_{11} - 59464 \beta_{10} - 439852 \beta_{9} - 50510 \beta_{8} - 1496519 \beta_{7} + \cdots - 8555110 ) / 56 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
−0.212454 − 2.55538i
−0.212454 + 2.55538i
3.63313 + 1.03277i
3.63313 − 1.03277i
−1.31008 − 1.63203i
−1.31008 + 1.63203i
3.16763 − 1.81697i
3.16763 + 1.81697i
−1.59179 + 1.13253i
−1.59179 − 1.13253i
−0.686439 + 2.27067i
−0.686439 − 2.27067i
−2.77422 − 2.88161i − 16.9405i −0.607410 + 15.9885i −4.68319 −48.8160 + 46.9966i 18.5203i 47.7577 − 42.6052i −205.980 12.9922 + 13.4951i
15.2 −2.77422 + 2.88161i 16.9405i −0.607410 − 15.9885i −4.68319 −48.8160 − 46.9966i − 18.5203i 47.7577 + 42.6052i −205.980 12.9922 − 13.4951i
15.3 −2.68279 − 2.96693i 5.27316i −1.60531 + 15.9193i 36.6030 15.6451 − 14.1468i − 18.5203i 51.5380 − 37.9452i 53.1938 −98.1982 − 108.599i
15.4 −2.68279 + 2.96693i − 5.27316i −1.60531 − 15.9193i 36.6030 15.6451 + 14.1468i 18.5203i 51.5380 + 37.9452i 53.1938 −98.1982 + 108.599i
15.5 −1.00393 − 3.87197i 10.5366i −13.9843 + 7.77433i −46.3749 40.7974 − 10.5780i 18.5203i 44.1411 + 46.3418i −30.0201 46.5569 + 179.562i
15.6 −1.00393 + 3.87197i − 10.5366i −13.9843 − 7.77433i −46.3749 40.7974 + 10.5780i − 18.5203i 44.1411 − 46.3418i −30.0201 46.5569 − 179.562i
15.7 1.31981 − 3.77599i − 5.99821i −12.5162 − 9.96716i −0.440227 −22.6492 − 7.91648i − 18.5203i −54.1549 + 34.1064i 45.0215 −0.581015 + 1.66229i
15.8 1.31981 + 3.77599i 5.99821i −12.5162 + 9.96716i −0.440227 −22.6492 + 7.91648i 18.5203i −54.1549 − 34.1064i 45.0215 −0.581015 − 1.66229i
15.9 2.79410 − 2.86235i 13.4717i −0.386047 − 15.9953i 36.3145 38.5606 + 37.6412i 18.5203i −46.8629 − 43.5875i −100.486 101.466 − 103.945i
15.10 2.79410 + 2.86235i − 13.4717i −0.386047 + 15.9953i 36.3145 38.5606 − 37.6412i − 18.5203i −46.8629 + 43.5875i −100.486 101.466 + 103.945i
15.11 3.84703 − 1.09562i − 7.79286i 13.5992 − 8.42973i −9.41925 −8.53798 − 29.9793i 18.5203i 43.0809 − 47.3290i 20.2714 −36.2361 + 10.3199i
15.12 3.84703 + 1.09562i 7.79286i 13.5992 + 8.42973i −9.41925 −8.53798 + 29.9793i − 18.5203i 43.0809 + 47.3290i 20.2714 −36.2361 − 10.3199i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 15.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.5.c.a ✓ 12
3.b odd 2 1 252.5.g.a 12
4.b odd 2 1 inner 28.5.c.a ✓ 12
7.b odd 2 1 196.5.c.f 12
8.b even 2 1 448.5.d.e 12
8.d odd 2 1 448.5.d.e 12
12.b even 2 1 252.5.g.a 12
28.d even 2 1 196.5.c.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.5.c.a ✓ 12 1.a even 1 1 trivial
28.5.c.a ✓ 12 4.b odd 2 1 inner
196.5.c.f 12 7.b odd 2 1
196.5.c.f 12 28.d even 2 1
252.5.g.a 12 3.b odd 2 1
252.5.g.a 12 12.b even 2 1
448.5.d.e 12 8.b even 2 1
448.5.d.e 12 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(28, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 3 T^{11} + \cdots + 16777216 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 351298031616 \) Copy content Toggle raw display
$5$ \( (T^{6} - 12 T^{5} + \cdots + 1197056)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 343)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( (T^{6} - 60 T^{5} + \cdots + 73212423424)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 497696355050944)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 43\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 67\!\cdots\!92)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 23\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 58\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 29\!\cdots\!76)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 25\!\cdots\!84)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 88\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 36\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 57\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 87\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 75\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 23\!\cdots\!12)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 98\!\cdots\!36)^{2} \) Copy content Toggle raw display
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