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Properties

Label 1100.6.a.i
Level $1100$
Weight $6$
Character orbit 1100.a
Self dual yes
Analytic conductor $176.422$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1100,6,Mod(1,1100)]
 
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, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1100.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1100.a (trivial)

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: yes
Analytic conductor: \(176.422201794\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 1155x^{6} + 1122x^{5} + 365994x^{4} - 476334x^{3} - 28296355x^{2} + 54177943x + 132029745 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} + (\beta_{5} - 2 \beta_1 - 22) q^{7} + ( - \beta_{5} + \beta_{4} + 2 \beta_1 + 47) q^{9} - 121 q^{11} + ( - \beta_{5} - \beta_{2} + 6 \beta_1 + 134) q^{13} + (\beta_{7} + \beta_{6} + 2 \beta_{5} + \cdots - 2) q^{17}+ \cdots + (121 \beta_{5} - 121 \beta_{4} + \cdots - 5687) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 9 q^{3} - 175 q^{7} + 377 q^{9} - 968 q^{11} + 1072 q^{13} - 37 q^{17} - 1356 q^{19} + 5462 q^{21} - 1327 q^{23} - 2670 q^{27} - 11785 q^{29} + 3532 q^{31} + 1089 q^{33} + 8176 q^{37} - 16831 q^{39}+ \cdots - 45617 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 1155x^{6} + 1122x^{5} + 365994x^{4} - 476334x^{3} - 28296355x^{2} + 54177943x + 132029745 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 29993519 \nu^{7} + 972671008 \nu^{6} + 55968546197 \nu^{5} - 1193739185525 \nu^{4} + \cdots - 21\!\cdots\!05 ) / 25734183149700 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 37037373 \nu^{7} - 921476764 \nu^{6} + 45229096399 \nu^{5} + 773761442225 \nu^{4} + \cdots - 40\!\cdots\!35 ) / 25734183149700 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 21022391 \nu^{7} - 265098288 \nu^{6} + 23951024733 \nu^{5} + 274101152025 \nu^{4} + \cdots - 17\!\cdots\!95 ) / 12867091574850 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21022391 \nu^{7} - 265098288 \nu^{6} + 23951024733 \nu^{5} + 274101152025 \nu^{4} + \cdots + 20\!\cdots\!55 ) / 12867091574850 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27803583 \nu^{7} + 99527944 \nu^{6} - 33424966629 \nu^{5} - 121794895375 \nu^{4} + \cdots - 482867449744815 ) / 4289030524950 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 75377227 \nu^{7} - 27532536 \nu^{6} + 83784812576 \nu^{5} + 28968909500 \nu^{4} + \cdots - 12\!\cdots\!15 ) / 2144515262475 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + 289 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{6} - 11\beta_{5} - 3\beta_{4} + 4\beta_{3} - 2\beta_{2} + 516\beta _1 - 47 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{7} - 73\beta_{6} - 719\beta_{5} + 680\beta_{4} - 60\beta_{3} - 36\beta_{2} + 329\beta _1 + 150562 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 187 \beta_{7} - 3587 \beta_{6} - 9365 \beta_{5} - 3106 \beta_{4} + 3588 \beta_{3} - 1260 \beta_{2} + \cdots + 58422 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 6368 \beta_{7} - 74971 \beta_{6} - 521509 \beta_{5} + 443087 \beta_{4} - 65339 \beta_{3} + \cdots + 89500577 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 237057 \beta_{7} - 3019971 \beta_{6} - 6969672 \beta_{5} - 2362314 \beta_{4} + 2698620 \beta_{3} + \cdots + 95589645 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
26.2245
18.9133
9.67677
3.57844
−1.42686
−10.7235
−19.5686
−25.6740
0 −27.2245 0 0 0 −247.835 0 498.172 0
1.2 0 −19.9133 0 0 0 63.9161 0 153.541 0
1.3 0 −10.6768 0 0 0 −123.211 0 −129.007 0
1.4 0 −4.57844 0 0 0 209.239 0 −222.038 0
1.5 0 0.426864 0 0 0 59.7938 0 −242.818 0
1.6 0 9.72346 0 0 0 −220.259 0 −148.454 0
1.7 0 18.5686 0 0 0 50.8505 0 101.795 0
1.8 0 24.6740 0 0 0 32.5053 0 365.809 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(11\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 9 T_{3}^{7} - 1120 T_{3}^{6} - 7975 T_{3}^{5} + 343164 T_{3}^{4} + 1906081 T_{3}^{3} + \cdots + 50395500 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1100))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 9 T^{7} + \cdots + 50395500 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots - 88\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( (T + 121)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 48\!\cdots\!89 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots - 71\!\cdots\!03 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots - 45\!\cdots\!53 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 37\!\cdots\!75 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 36\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 61\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 45\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 74\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 72\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 11\!\cdots\!73 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 22\!\cdots\!93 \) Copy content Toggle raw display
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