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Properties

Label 28.11.h.a
Level $28$
Weight $11$
Character orbit 28.h
Analytic conductor $17.790$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [28,11,Mod(5,28)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(28, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("28.5");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 28.h (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.7900030749\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 79898 x^{12} + 4721335 x^{11} + 4633670218 x^{10} + 292539163887 x^{9} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{13}\cdot 7^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 12 \beta_1 - 23) q^{3} + ( - \beta_{6} + \beta_{5} + 158 \beta_1 + 160) q^{5} + (\beta_{7} - \beta_{6} + \beta_{5} + \cdots - 1579) q^{7}+ \cdots + ( - \beta_{9} + \beta_{7} + \cdots + 21741) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 12 \beta_1 - 23) q^{3} + ( - \beta_{6} + \beta_{5} + 158 \beta_1 + 160) q^{5} + (\beta_{7} - \beta_{6} + \beta_{5} + \cdots - 1579) q^{7}+ \cdots + ( - 12399 \beta_{13} + \cdots + 2082505311) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 243 q^{3} + 3333 q^{5} - 8810 q^{7} + 152334 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 243 q^{3} + 3333 q^{5} - 8810 q^{7} + 152334 q^{9} + 43575 q^{11} - 295974 q^{15} - 4046547 q^{17} + 10506141 q^{19} - 6849963 q^{21} + 8356047 q^{23} + 16247798 q^{25} + 9638748 q^{29} + 26706309 q^{31} - 68091003 q^{33} + 57757869 q^{35} - 56337841 q^{37} - 47717712 q^{39} + 102867212 q^{43} + 277176762 q^{45} - 494767563 q^{47} - 228310402 q^{49} + 584803989 q^{51} - 280243257 q^{53} - 2426437062 q^{57} - 540103875 q^{59} + 3020130381 q^{61} + 44180910 q^{63} - 156676464 q^{65} - 1385714225 q^{67} + 5251025724 q^{71} + 5335510437 q^{73} - 1896380046 q^{75} - 832139025 q^{77} - 5240397281 q^{79} - 4471861527 q^{81} - 8239454118 q^{85} + 21846639978 q^{87} + 9313203429 q^{89} - 17599015344 q^{91} - 1809384507 q^{93} - 19347043443 q^{95} + 29066774940 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} + 79898 x^{12} + 4721335 x^{11} + 4633670218 x^{10} + 292539163887 x^{9} + \cdots + 11\!\cdots\!96 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 21\!\cdots\!03 \nu^{13} + \cdots + 16\!\cdots\!96 ) / 54\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 35\!\cdots\!53 \nu^{13} + \cdots + 15\!\cdots\!68 ) / 77\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!14 \nu^{13} + \cdots - 11\!\cdots\!52 ) / 11\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 50\!\cdots\!17 \nu^{13} + \cdots + 22\!\cdots\!80 ) / 25\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17\!\cdots\!71 \nu^{13} + \cdots - 83\!\cdots\!92 ) / 58\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15\!\cdots\!39 \nu^{13} + \cdots + 71\!\cdots\!96 ) / 29\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 93\!\cdots\!05 \nu^{13} + \cdots + 41\!\cdots\!48 ) / 25\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 38\!\cdots\!65 \nu^{13} + \cdots - 73\!\cdots\!88 ) / 14\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 73\!\cdots\!85 \nu^{13} + \cdots + 27\!\cdots\!52 ) / 23\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14\!\cdots\!46 \nu^{13} + \cdots - 55\!\cdots\!12 ) / 38\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15\!\cdots\!72 \nu^{13} + \cdots + 52\!\cdots\!48 ) / 38\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 33\!\cdots\!05 \nu^{13} + \cdots - 30\!\cdots\!12 ) / 77\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 16\!\cdots\!33 \nu^{13} + \cdots + 60\!\cdots\!36 ) / 25\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} - 4\beta_{3} + 4\beta_{2} - 7\beta _1 + 3 ) / 42 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7 \beta_{13} - 12 \beta_{12} + 17 \beta_{11} - 28 \beta_{10} - 28 \beta_{9} + 7 \beta_{8} + \cdots - 1478 ) / 294 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3955 \beta_{13} - 1366 \beta_{12} - 3267 \beta_{11} - 9996 \beta_{10} - 19348 \beta_{7} + \cdots - 606540462 ) / 588 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 627124 \beta_{12} - 1318455 \beta_{11} + 1057742 \beta_{9} - 414498 \beta_{8} + 9000439 \beta_{7} + \cdots - 242985586049 ) / 294 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 180840499 \beta_{13} + 146323099 \beta_{12} + 71848065 \beta_{11} + 575756944 \beta_{10} + \cdots + 145228617601 ) / 588 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2847157639 \beta_{13} + 7826264393 \beta_{12} + 4626441496 \beta_{11} + 6520581136 \beta_{10} + \cdots + 14\!\cdots\!54 ) / 42 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 426583942909 \beta_{12} + 540286102934 \beta_{11} - 3961506360108 \beta_{9} - 1003446478739 \beta_{8} + \cdots + 25\!\cdots\!73 ) / 84 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 900942834381072 \beta_{13} + \cdots - 29\!\cdots\!29 ) / 294 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 25\!\cdots\!99 \beta_{13} + \cdots - 85\!\cdots\!18 ) / 588 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 28\!\cdots\!97 \beta_{12} + \cdots - 17\!\cdots\!14 ) / 294 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 87\!\cdots\!13 \beta_{13} + \cdots + 13\!\cdots\!55 ) / 588 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 25\!\cdots\!22 \beta_{13} + \cdots + 10\!\cdots\!80 ) / 42 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 45\!\cdots\!39 \beta_{12} + \cdots + 26\!\cdots\!71 ) / 84 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
107.160 − 185.606i
−64.1547 + 111.119i
31.0272 − 53.7407i
−24.9573 + 43.2273i
101.380 − 175.596i
−99.9066 + 173.043i
−50.0487 + 86.6870i
107.160 + 185.606i
−64.1547 − 111.119i
31.0272 + 53.7407i
−24.9573 − 43.2273i
101.380 + 175.596i
−99.9066 − 173.043i
−50.0487 − 86.6870i
0 −377.314 + 217.843i 0 3295.95 + 1902.92i 0 14685.1 + 8174.48i 0 65386.2 − 113252.i 0
5.2 0 −292.260 + 168.736i 0 −3559.03 − 2054.81i 0 −16353.0 + 3880.07i 0 27419.4 − 47491.7i 0
5.3 0 −105.874 + 61.1265i 0 1184.15 + 683.667i 0 −10934.6 − 12763.6i 0 −22051.6 + 38194.5i 0
5.4 0 −40.9298 + 23.6308i 0 −907.426 − 523.903i 0 16561.9 − 2859.78i 0 −28407.7 + 49203.5i 0
5.5 0 166.589 − 96.1802i 0 5228.83 + 3018.87i 0 −7044.15 + 15259.6i 0 −11023.2 + 19092.8i 0
5.6 0 173.882 − 100.391i 0 −3196.05 − 1845.24i 0 −2101.89 + 16675.0i 0 −9367.86 + 16225.6i 0
5.7 0 354.407 − 204.617i 0 −379.919 − 219.346i 0 781.634 − 16788.8i 0 54211.8 − 93897.5i 0
17.1 0 −377.314 − 217.843i 0 3295.95 − 1902.92i 0 14685.1 − 8174.48i 0 65386.2 + 113252.i 0
17.2 0 −292.260 − 168.736i 0 −3559.03 + 2054.81i 0 −16353.0 − 3880.07i 0 27419.4 + 47491.7i 0
17.3 0 −105.874 − 61.1265i 0 1184.15 − 683.667i 0 −10934.6 + 12763.6i 0 −22051.6 − 38194.5i 0
17.4 0 −40.9298 − 23.6308i 0 −907.426 + 523.903i 0 16561.9 + 2859.78i 0 −28407.7 − 49203.5i 0
17.5 0 166.589 + 96.1802i 0 5228.83 − 3018.87i 0 −7044.15 − 15259.6i 0 −11023.2 − 19092.8i 0
17.6 0 173.882 + 100.391i 0 −3196.05 + 1845.24i 0 −2101.89 − 16675.0i 0 −9367.86 − 16225.6i 0
17.7 0 354.407 + 204.617i 0 −379.919 + 219.346i 0 781.634 + 16788.8i 0 54211.8 + 93897.5i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.11.h.a ✓ 14
3.b odd 2 1 252.11.z.c 14
4.b odd 2 1 112.11.s.c 14
7.b odd 2 1 196.11.h.b 14
7.c even 3 1 196.11.b.a 14
7.c even 3 1 196.11.h.b 14
7.d odd 6 1 inner 28.11.h.a ✓ 14
7.d odd 6 1 196.11.b.a 14
21.g even 6 1 252.11.z.c 14
28.f even 6 1 112.11.s.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.11.h.a ✓ 14 1.a even 1 1 trivial
28.11.h.a ✓ 14 7.d odd 6 1 inner
112.11.s.c 14 4.b odd 2 1
112.11.s.c 14 28.f even 6 1
196.11.b.a 14 7.c even 3 1
196.11.b.a 14 7.d odd 6 1
196.11.h.b 14 7.b odd 2 1
196.11.h.b 14 7.c even 3 1
252.11.z.c 14 3.b odd 2 1
252.11.z.c 14 21.g even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(28, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 18\!\cdots\!47 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 47\!\cdots\!75 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 14\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 57\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 16\!\cdots\!28 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 82\!\cdots\!63 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 68\!\cdots\!47 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 48\!\cdots\!41 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 60\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 48\!\cdots\!43 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 22\!\cdots\!01 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 90\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 16\!\cdots\!40)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 16\!\cdots\!75 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 25\!\cdots\!69 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 17\!\cdots\!03 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 55\!\cdots\!23 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 28\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots + 31\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 15\!\cdots\!75 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 13\!\cdots\!49 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 18\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 14\!\cdots\!63 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 53\!\cdots\!72 \) Copy content Toggle raw display
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