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Properties

Label 28.12.e.a
Level $28$
Weight $12$
Character orbit 28.e
Analytic conductor $21.514$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [28,12,Mod(9,28)] 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("28.9"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(28, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 28.e (of order \(3\), degree \(2\), 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: \(21.5136090557\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
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: \( x^{14} - 5 x^{13} - 381616 x^{12} - 6529891 x^{11} + 52342937589 x^{10} + 1150852380738 x^{9} + \cdots + 61\!\cdots\!52 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{42}\cdot 3^{5}\cdot 7^{11} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_{2} + 35 \beta_1) q^{3} + ( - \beta_{6} - \beta_{4} - 531 \beta_1 + 531) q^{5} + (\beta_{9} - 5 \beta_{3} - 17 \beta_{2} + \cdots - 6811) q^{7} + ( - \beta_{13} - \beta_{6} + \cdots - 42173) q^{9}+ \cdots + (231010 \beta_{11} + \cdots - 23137191470) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 243 q^{3} + 3719 q^{5} - 83356 q^{7} - 294938 q^{9} - 72905 q^{11} - 2193572 q^{13} + 1372294 q^{15} - 261533 q^{17} + 13374085 q^{19} - 13411307 q^{21} - 11167331 q^{23} - 2989084 q^{25} - 342323586 q^{27}+ \cdots - 323900282804 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 5 x^{13} - 381616 x^{12} - 6529891 x^{11} + 52342937589 x^{10} + 1150852380738 x^{9} + \cdots + 61\!\cdots\!52 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 10\!\cdots\!19 \nu^{13} + \cdots + 63\!\cdots\!20 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10\!\cdots\!19 \nu^{13} + \cdots + 63\!\cdots\!20 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21\!\cdots\!45 \nu^{13} + \cdots + 11\!\cdots\!28 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!73 \nu^{13} + \cdots + 71\!\cdots\!12 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 56\!\cdots\!13 \nu^{13} + \cdots - 66\!\cdots\!78 ) / 43\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 35\!\cdots\!63 \nu^{13} + \cdots - 53\!\cdots\!28 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14\!\cdots\!69 \nu^{13} + \cdots - 30\!\cdots\!64 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 33\!\cdots\!11 \nu^{13} + \cdots + 23\!\cdots\!84 ) / 77\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 46\!\cdots\!37 \nu^{13} + \cdots + 31\!\cdots\!28 ) / 77\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 62\!\cdots\!89 \nu^{13} + \cdots + 27\!\cdots\!16 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15\!\cdots\!83 \nu^{13} + \cdots - 92\!\cdots\!32 ) / 77\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!19 \nu^{13} + \cdots - 76\!\cdots\!36 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 41\!\cdots\!99 \nu^{13} + \cdots + 26\!\cdots\!56 ) / 99\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - 2\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} - 4\beta_{3} + 64\beta_{2} - 4\beta _1 + 218095 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{13} - 68 \beta_{11} - 17 \beta_{10} - 584 \beta_{9} + 692 \beta_{8} - 380 \beta_{7} + \cdots + 15242065 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 820 \beta_{13} + 68 \beta_{12} - 67196 \beta_{11} + 4776 \beta_{10} - 180254 \beta_{9} + \cdots + 46933691909 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2975370 \beta_{13} - 48100 \beta_{12} - 20056625 \beta_{11} - 8952165 \beta_{10} + \cdots + 10596211727448 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1100471892 \beta_{13} + 107716620 \beta_{12} - 50024021638 \beta_{11} + 3489806423 \beta_{10} + \cdots + 24\!\cdots\!86 ) / 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1182997476577 \beta_{13} - 24806663231 \beta_{12} - 6676114045863 \beta_{11} - 2791000032962 \beta_{10} + \cdots + 44\!\cdots\!77 ) / 32 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 274377911659476 \beta_{13} + 22428734955012 \beta_{12} + \cdots + 34\!\cdots\!78 ) / 32 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 21\!\cdots\!03 \beta_{13} + \cdots + 80\!\cdots\!65 ) / 32 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 59\!\cdots\!11 \beta_{13} + \cdots + 49\!\cdots\!83 ) / 32 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 39\!\cdots\!60 \beta_{13} + \cdots + 13\!\cdots\!10 ) / 32 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 11\!\cdots\!17 \beta_{13} + \cdots + 72\!\cdots\!33 ) / 32 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 68\!\cdots\!16 \beta_{13} + \cdots + 22\!\cdots\!08 ) / 32 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(15\) \(17\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−332.826 − 0.866025i
−247.631 − 0.866025i
−121.238 − 0.866025i
33.0232 − 0.866025i
126.688 − 0.866025i
150.857 − 0.866025i
393.628 − 0.866025i
−332.826 + 0.866025i
−247.631 + 0.866025i
−121.238 + 0.866025i
33.0232 + 0.866025i
126.688 + 0.866025i
150.857 + 0.866025i
393.628 + 0.866025i
0 −315.826 − 547.027i 0 −483.543 + 837.520i 0 19347.3 + 40037.6i 0 −110919. + 192117.i 0
9.2 0 −230.631 − 399.464i 0 −1135.19 + 1966.21i 0 2206.92 − 44412.3i 0 −17807.4 + 30843.3i 0
9.3 0 −104.238 − 180.546i 0 5578.64 − 9662.49i 0 −41531.9 − 15888.1i 0 66842.2 − 115774.i 0
9.4 0 50.0232 + 86.6427i 0 −5807.66 + 10059.2i 0 −37034.4 + 24612.6i 0 83568.9 − 144746.i 0
9.5 0 143.688 + 248.874i 0 4302.39 − 7451.96i 0 17422.8 + 40911.8i 0 47281.3 − 81893.6i 0
9.6 0 167.857 + 290.736i 0 −1051.29 + 1820.89i 0 39765.4 − 19900.8i 0 32221.7 − 55809.6i 0
9.7 0 410.628 + 711.228i 0 456.142 − 790.062i 0 −41854.1 − 15018.7i 0 −248657. + 430686.i 0
25.1 0 −315.826 + 547.027i 0 −483.543 − 837.520i 0 19347.3 − 40037.6i 0 −110919. − 192117.i 0
25.2 0 −230.631 + 399.464i 0 −1135.19 − 1966.21i 0 2206.92 + 44412.3i 0 −17807.4 − 30843.3i 0
25.3 0 −104.238 + 180.546i 0 5578.64 + 9662.49i 0 −41531.9 + 15888.1i 0 66842.2 + 115774.i 0
25.4 0 50.0232 − 86.6427i 0 −5807.66 − 10059.2i 0 −37034.4 − 24612.6i 0 83568.9 + 144746.i 0
25.5 0 143.688 − 248.874i 0 4302.39 + 7451.96i 0 17422.8 − 40911.8i 0 47281.3 + 81893.6i 0
25.6 0 167.857 − 290.736i 0 −1051.29 − 1820.89i 0 39765.4 + 19900.8i 0 32221.7 + 55809.6i 0
25.7 0 410.628 − 711.228i 0 456.142 + 790.062i 0 −41854.1 + 15018.7i 0 −248657. − 430686.i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.12.e.a ✓ 14
3.b odd 2 1 252.12.k.c 14
4.b odd 2 1 112.12.i.d 14
7.b odd 2 1 196.12.e.h 14
7.c even 3 1 inner 28.12.e.a ✓ 14
7.c even 3 1 196.12.a.e 7
7.d odd 6 1 196.12.a.f 7
7.d odd 6 1 196.12.e.h 14
21.h odd 6 1 252.12.k.c 14
28.g odd 6 1 112.12.i.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.12.e.a ✓ 14 1.a even 1 1 trivial
28.12.e.a ✓ 14 7.c even 3 1 inner
112.12.i.d 14 4.b odd 2 1
112.12.i.d 14 28.g odd 6 1
196.12.a.e 7 7.c even 3 1
196.12.a.f 7 7.d odd 6 1
196.12.e.h 14 7.b odd 2 1
196.12.e.h 14 7.d odd 6 1
252.12.k.c 14 3.b odd 2 1
252.12.k.c 14 21.h odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(28, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 23\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 11\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 26\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( (T^{7} + \cdots + 11\!\cdots\!28)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 74\!\cdots\!29 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 54\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 87\!\cdots\!49 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 10\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 91\!\cdots\!29 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 13\!\cdots\!09 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 84\!\cdots\!20)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 66\!\cdots\!60)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 16\!\cdots\!69 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 15\!\cdots\!89 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 52\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 25\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 43\!\cdots\!49 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 40\!\cdots\!28)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
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