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Properties

Label 1100.6.b.i
Level $1100$
Weight $6$
Character orbit 1100.b
Analytic conductor $176.422$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1100,6,Mod(749,1100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1100.749");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1100.b (of order \(2\), degree \(1\), not 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: \(176.422201794\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 2231 x^{14} + 1950697 x^{12} + 841157794 x^{10} + 184347539486 x^{8} + 19011199757266 x^{6} + \cdots + 36\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{14}\cdot 5^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{9} + \beta_{8}) q^{3} + (\beta_{10} - 2 \beta_{9}) q^{7} + ( - \beta_{2} - \beta_1 - 47) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} + \beta_{8}) q^{3} + (\beta_{10} - 2 \beta_{9}) q^{7} + ( - \beta_{2} - \beta_1 - 47) q^{9} + 121 q^{11} + (\beta_{12} - \beta_{10} + \cdots - 5 \beta_{8}) q^{13}+ \cdots + ( - 121 \beta_{2} - 121 \beta_1 - 5687) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 754 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 754 q^{9} + 1936 q^{11} + 708 q^{19} - 1176 q^{21} - 5486 q^{29} + 7536 q^{31} + 20054 q^{39} + 17512 q^{41} - 54682 q^{49} + 9414 q^{51} - 40800 q^{59} + 21022 q^{61} - 144904 q^{69} + 63724 q^{71} + 143358 q^{79} - 14456 q^{81} + 29922 q^{89} + 265282 q^{91} - 91234 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 2231 x^{14} + 1950697 x^{12} + 841157794 x^{10} + 184347539486 x^{8} + 19011199757266 x^{6} + \cdots + 36\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11678598862180 \nu^{14} + \cdots - 10\!\cdots\!11 ) / 36\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 58392994310900 \nu^{14} + \cdots + 17\!\cdots\!77 ) / 36\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 26\!\cdots\!95 \nu^{14} + \cdots - 41\!\cdots\!09 ) / 14\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 47\!\cdots\!93 \nu^{14} + \cdots + 12\!\cdots\!63 ) / 14\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19\!\cdots\!02 \nu^{14} + \cdots + 80\!\cdots\!85 ) / 28\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 67\!\cdots\!65 \nu^{14} + \cdots - 80\!\cdots\!12 ) / 95\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 41\!\cdots\!13 \nu^{14} + \cdots + 69\!\cdots\!86 ) / 28\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 78\!\cdots\!43 \nu^{15} + \cdots - 17\!\cdots\!73 \nu ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 78\!\cdots\!43 \nu^{15} + \cdots + 12\!\cdots\!33 \nu ) / 41\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 29\!\cdots\!29 \nu^{15} + \cdots - 64\!\cdots\!07 \nu ) / 11\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 87\!\cdots\!76 \nu^{15} + \cdots - 41\!\cdots\!41 \nu ) / 91\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 30\!\cdots\!31 \nu^{15} + \cdots - 25\!\cdots\!59 \nu ) / 27\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12\!\cdots\!71 \nu^{15} + \cdots - 65\!\cdots\!66 \nu ) / 11\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 14\!\cdots\!56 \nu^{15} + \cdots - 62\!\cdots\!21 \nu ) / 55\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 21\!\cdots\!39 \nu^{15} + \cdots - 12\!\cdots\!54 \nu ) / 55\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{9} + 5\beta_{8} ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 5\beta _1 - 1393 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{14} - \beta_{13} + 4\beta_{12} - 15\beta_{11} - 17\beta_{10} - 948\beta_{9} - 2415\beta_{8} ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -49\beta_{7} + 77\beta_{6} + 88\beta_{5} + 181\beta_{4} + 21\beta_{3} - 340\beta_{2} + 2966\beta _1 + 671767 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 135 \beta_{15} - 3205 \beta_{14} - 460 \beta_{13} - 4895 \beta_{12} + 12390 \beta_{11} + \cdots + 1286665 \beta_{8} ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 37835 \beta_{7} - 86599 \beta_{6} - 75755 \beta_{5} - 176645 \beta_{4} - 26670 \beta_{3} + \cdots - 357061603 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 205665 \beta_{15} + 1800585 \beta_{14} + 824598 \beta_{13} + 3978813 \beta_{12} + \cdots - 713544890 \beta_{8} ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 24403371 \beta_{7} + 69791979 \beta_{6} + 53575827 \beta_{5} + 131839674 \beta_{4} + \cdots + 197482475914 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 186236361 \beta_{15} - 1001824847 \beta_{14} - 790011254 \beta_{13} - 2875838770 \beta_{12} + \cdots + 404625580416 \beta_{8} ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 15473594935 \beta_{7} - 49189316903 \beta_{6} - 35702615110 \beta_{5} - 89838206710 \beta_{4} + \cdots - 111689955247228 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 137132352009 \beta_{15} + 566878487458 \beta_{14} + 638666427451 \beta_{13} + \cdots - 232765266333919 \beta_{8} ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 9903842142170 \beta_{7} + 32383511642434 \beta_{6} + 23100578496890 \beta_{5} + 58783704339230 \beta_{4} + \cdots + 64\!\cdots\!53 ) / 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 90993771245166 \beta_{15} - 327663206213142 \beta_{14} - 476369588987724 \beta_{13} + \cdots + 13\!\cdots\!31 \beta_{8} ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 63\!\cdots\!54 \beta_{7} + \cdots - 37\!\cdots\!87 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 56\!\cdots\!82 \beta_{15} + \cdots - 79\!\cdots\!37 \beta_{8} ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
749.1
− 23.6316i
− 22.3000i
− 24.8760i
− 17.8741i
− 13.1518i
− 6.06056i
− 4.75672i
− 2.15294i
2.15294i
4.75672i
6.06056i
13.1518i
17.8741i
24.8760i
22.3000i
23.6316i
0 − 26.6316i 0 0 0 − 210.828i 0 −466.240 0
749.2 0 − 25.3000i 0 0 0 150.442i 0 −397.092 0
749.3 0 − 21.8760i 0 0 0 48.9354i 0 −235.558 0
749.4 0 − 14.8741i 0 0 0 − 166.492i 0 21.7626 0
749.5 0 − 10.1518i 0 0 0 122.703i 0 139.941 0
749.6 0 − 9.06056i 0 0 0 199.386i 0 160.906 0
749.7 0 − 7.75672i 0 0 0 − 86.6415i 0 182.833 0
749.8 0 − 5.15294i 0 0 0 47.7885i 0 216.447 0
749.9 0 5.15294i 0 0 0 − 47.7885i 0 216.447 0
749.10 0 7.75672i 0 0 0 86.6415i 0 182.833 0
749.11 0 9.06056i 0 0 0 − 199.386i 0 160.906 0
749.12 0 10.1518i 0 0 0 − 122.703i 0 139.941 0
749.13 0 14.8741i 0 0 0 166.492i 0 21.7626 0
749.14 0 21.8760i 0 0 0 − 48.9354i 0 −235.558 0
749.15 0 25.3000i 0 0 0 − 150.442i 0 −397.092 0
749.16 0 26.6316i 0 0 0 210.828i 0 −466.240 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 749.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1100.6.b.i 16
5.b even 2 1 inner 1100.6.b.i 16
5.c odd 4 1 1100.6.a.h ✓ 8
5.c odd 4 1 1100.6.a.k yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1100.6.a.h ✓ 8 5.c odd 4 1
1100.6.a.k yes 8 5.c odd 4 1
1100.6.b.i 16 1.a even 1 1 trivial
1100.6.b.i 16 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 2321 T_{3}^{14} + 2087326 T_{3}^{12} + 924020173 T_{3}^{10} + 214678632158 T_{3}^{8} + \cdots + 64\!\cdots\!76 \) acting on \(S_{6}^{\mathrm{new}}(1100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 64\!\cdots\!76 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T - 121)^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 79\!\cdots\!25 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 56\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 11\!\cdots\!89)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 12\!\cdots\!29 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 14\!\cdots\!19)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 62\!\cdots\!25)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 91\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 75\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 30\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 36\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 89\!\cdots\!08)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 64\!\cdots\!09 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 34\!\cdots\!19)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 57\!\cdots\!81 \) Copy content Toggle raw display
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