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Properties

Label 1100.2.n.d
Level $1100$
Weight $2$
Character orbit 1100.n
Analytic conductor $8.784$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1100,2,Mod(201,1100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1100, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1100.201");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1100.n (of order \(5\), degree \(4\), 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: \(8.78354422234\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 13 x^{14} - 15 x^{13} + 59 x^{12} + 4 x^{11} + 369 x^{10} + 618 x^{9} + 1481 x^{8} + \cdots + 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{12} - \beta_{7} + \cdots - \beta_1) q^{3}+ \cdots + (\beta_{13} + \beta_{11} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{12} - \beta_{7} + \cdots - \beta_1) q^{3}+ \cdots + (\beta_{15} + \beta_{14} + 3 \beta_{12} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{3} - 5 q^{11} + q^{13} + 8 q^{17} + 13 q^{19} - 6 q^{21} + 16 q^{23} + 37 q^{27} - 7 q^{29} + 2 q^{31} - 14 q^{33} + 8 q^{37} - 17 q^{39} - 15 q^{41} + 18 q^{47} - 24 q^{49} + 13 q^{51} + 6 q^{53} + 31 q^{57} + 6 q^{59} + 24 q^{61} - 5 q^{63} - 18 q^{67} - 53 q^{69} + 36 q^{71} + 9 q^{73} - 45 q^{77} - 45 q^{79} + 17 q^{81} + 14 q^{83} - 18 q^{87} + 18 q^{89} + 38 q^{91} + 29 q^{93} - 49 q^{97} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 13 x^{14} - 15 x^{13} + 59 x^{12} + 4 x^{11} + 369 x^{10} + 618 x^{9} + 1481 x^{8} + \cdots + 400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 21\!\cdots\!73 \nu^{15} + \cdots - 32\!\cdots\!00 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13\!\cdots\!59 \nu^{15} + \cdots + 78\!\cdots\!40 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 23\!\cdots\!37 \nu^{15} + \cdots + 42\!\cdots\!00 ) / 65\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 52\!\cdots\!93 \nu^{15} + \cdots - 95\!\cdots\!00 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 39\!\cdots\!34 \nu^{15} + \cdots + 92\!\cdots\!80 ) / 65\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24\!\cdots\!32 \nu^{15} + \cdots + 12\!\cdots\!00 ) / 32\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10\!\cdots\!23 \nu^{15} + \cdots + 39\!\cdots\!60 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 42\!\cdots\!89 \nu^{15} + \cdots - 30\!\cdots\!60 ) / 32\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 20\!\cdots\!33 \nu^{15} + \cdots + 85\!\cdots\!40 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 25\!\cdots\!88 \nu^{15} + \cdots + 32\!\cdots\!45 ) / 16\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 28\!\cdots\!68 \nu^{15} + \cdots - 54\!\cdots\!50 ) / 16\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 24\!\cdots\!39 \nu^{15} + \cdots - 65\!\cdots\!40 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 51\!\cdots\!23 \nu^{15} + \cdots + 94\!\cdots\!20 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 52\!\cdots\!85 \nu^{15} + \cdots + 14\!\cdots\!10 ) / 10\!\cdots\!10 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} - 3\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{13} + 2 \beta_{11} - \beta_{10} + 3 \beta_{8} + \beta_{7} - \beta_{6} - 7 \beta_{4} + 4 \beta_{3} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} + 9 \beta_{13} + \beta_{12} + 9 \beta_{11} + \beta_{9} + 20 \beta_{8} + 13 \beta_{7} + \cdots - 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8 \beta_{14} - 13 \beta_{12} + 12 \beta_{11} + 8 \beta_{10} + 8 \beta_{9} + 38 \beta_{8} + 48 \beta_{7} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 14 \beta_{15} - 80 \beta_{13} - 135 \beta_{12} + 3 \beta_{11} + 14 \beta_{10} + 12 \beta_{9} + \cdots + 49 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 64 \beta_{15} + 64 \beta_{14} - 215 \beta_{13} - 425 \beta_{12} + 5 \beta_{10} + 5 \beta_{9} + \cdots - 574 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 151 \beta_{14} - 61 \beta_{13} - 61 \beta_{11} - 29 \beta_{10} - 1648 \beta_{8} - 1328 \beta_{7} + \cdots - 350 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 548 \beta_{15} + 90 \beta_{14} + 746 \beta_{13} + 3903 \beta_{12} - 1310 \beta_{11} - 90 \beta_{9} + \cdots + 1418 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1508 \beta_{15} - 288 \beta_{14} + 12866 \beta_{12} - 7586 \beta_{11} + 288 \beta_{10} - 1220 \beta_{9} + \cdots + 14981 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1169 \beta_{15} - 6167 \beta_{13} + 17308 \beta_{12} - 19571 \beta_{11} - 1169 \beta_{10} - 6078 \beta_{9} + \cdots + 43545 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2427 \beta_{15} + 2427 \beta_{14} + 11141 \beta_{13} + 1268 \beta_{12} - 14662 \beta_{10} + \cdots + 125405 \beta_1 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 13568 \beta_{14} + 186262 \beta_{13} + 186262 \beta_{11} - 45036 \beta_{10} + 349318 \beta_{8} + \cdots - 409892 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 18248 \beta_{15} - 141226 \beta_{14} + 577695 \beta_{13} - 57440 \beta_{12} + 709277 \beta_{11} + \cdots - 1365291 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 149830 \beta_{15} - 418221 \beta_{14} - 1880027 \beta_{12} + 1386133 \beta_{11} + 418221 \beta_{10} + \cdots - 1724643 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(-\beta_{8}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
2.54109 + 1.84621i
0.618641 + 0.449469i
−0.604261 − 0.439021i
−1.24645 − 0.905599i
2.54109 − 1.84621i
0.618641 − 0.449469i
−0.604261 + 0.439021i
−1.24645 + 0.905599i
0.927142 − 2.85345i
0.286660 − 0.882247i
−0.396204 + 1.21939i
−0.626614 + 1.92852i
0.927142 + 2.85345i
0.286660 + 0.882247i
−0.396204 − 1.21939i
−0.626614 − 1.92852i
0 −0.970609 − 2.98723i 0 0 0 −0.679254 + 2.09053i 0 −5.55439 + 4.03550i 0
201.2 0 −0.236300 − 0.727256i 0 0 0 0.650161 − 2.00099i 0 1.95399 − 1.41965i 0
201.3 0 0.230807 + 0.710352i 0 0 0 −1.21535 + 3.74048i 0 1.97572 − 1.43545i 0
201.4 0 0.476102 + 1.46529i 0 0 0 1.24445 − 3.83001i 0 0.506649 − 0.368102i 0
301.1 0 −0.970609 + 2.98723i 0 0 0 −0.679254 − 2.09053i 0 −5.55439 − 4.03550i 0
301.2 0 −0.236300 + 0.727256i 0 0 0 0.650161 + 2.00099i 0 1.95399 + 1.41965i 0
301.3 0 0.230807 − 0.710352i 0 0 0 −1.21535 − 3.74048i 0 1.97572 + 1.43545i 0
301.4 0 0.476102 − 1.46529i 0 0 0 1.24445 + 3.83001i 0 0.506649 + 0.368102i 0
401.1 0 −2.42729 − 1.76353i 0 0 0 1.36631 − 0.992679i 0 1.85465 + 5.70802i 0
401.2 0 −0.750484 − 0.545259i 0 0 0 0.472607 − 0.343369i 0 −0.661131 − 2.03475i 0
401.3 0 1.03728 + 0.753625i 0 0 0 1.45849 − 1.05965i 0 −0.419060 − 1.28974i 0
401.4 0 1.64050 + 1.19189i 0 0 0 −3.29740 + 2.39570i 0 0.343577 + 1.05742i 0
801.1 0 −2.42729 + 1.76353i 0 0 0 1.36631 + 0.992679i 0 1.85465 − 5.70802i 0
801.2 0 −0.750484 + 0.545259i 0 0 0 0.472607 + 0.343369i 0 −0.661131 + 2.03475i 0
801.3 0 1.03728 − 0.753625i 0 0 0 1.45849 + 1.05965i 0 −0.419060 + 1.28974i 0
801.4 0 1.64050 − 1.19189i 0 0 0 −3.29740 − 2.39570i 0 0.343577 − 1.05742i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 201.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1100.2.n.d ✓ 16
5.b even 2 1 1100.2.n.e yes 16
5.c odd 4 2 1100.2.cb.d 32
11.c even 5 1 inner 1100.2.n.d ✓ 16
55.j even 10 1 1100.2.n.e yes 16
55.k odd 20 2 1100.2.cb.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1100.2.n.d ✓ 16 1.a even 1 1 trivial
1100.2.n.d ✓ 16 11.c even 5 1 inner
1100.2.n.e yes 16 5.b even 2 1
1100.2.n.e yes 16 55.j even 10 1
1100.2.cb.d 32 5.c odd 4 2
1100.2.cb.d 32 55.k odd 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 2 T_{3}^{15} + 8 T_{3}^{14} - 5 T_{3}^{13} + 19 T_{3}^{12} - 16 T_{3}^{11} + 389 T_{3}^{10} + \cdots + 400 \) acting on \(S_{2}^{\mathrm{new}}(1100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 2 T^{15} + \cdots + 400 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 26 T^{14} + \cdots + 281961 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{16} - T^{15} + \cdots + 121 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 126990361 \) Copy content Toggle raw display
$19$ \( T^{16} - 13 T^{15} + \cdots + 61716736 \) Copy content Toggle raw display
$23$ \( (T^{8} - 8 T^{7} + \cdots + 15831)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 7743120025 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 4381778025 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 122696778961 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 3401222400 \) Copy content Toggle raw display
$43$ \( (T^{8} - 134 T^{6} + \cdots - 71125)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 4287363654025 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 4303360000 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 1870130025 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 8406166636921 \) Copy content Toggle raw display
$67$ \( (T^{8} + 9 T^{7} + \cdots + 1920301)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 5027034831025 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 57739557809025 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 610938937718041 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 1162689914961 \) Copy content Toggle raw display
$89$ \( (T^{8} - 9 T^{7} + \cdots - 570901)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 10719703296 \) Copy content Toggle raw display
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