Properties

Label 1100.6.b.b
Level $1100$
Weight $6$
Character orbit 1100.b
Analytic conductor $176.422$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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: \(4\)
Coefficient field: \(\Q(i, \sqrt{1761})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 881x^{2} + 193600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 220)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 5 \beta_{2} - \beta_1) q^{3} + (34 \beta_{2} + 5 \beta_1) q^{7} + (19 \beta_{3} - 297) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 5 \beta_{2} - \beta_1) q^{3} + (34 \beta_{2} + 5 \beta_1) q^{7} + (19 \beta_{3} - 297) q^{9} - 121 q^{11} + ( - 398 \beta_{2} + 18 \beta_1) q^{13} + (56 \beta_{2} - 97 \beta_1) q^{17} + ( - 7 \beta_{3} - 1716) q^{19} + ( - 113 \beta_{3} + 2880) q^{21} + (1157 \beta_{2} + 92 \beta_1) q^{23} + (4450 \beta_{2} + 225 \beta_1) q^{27} + (139 \beta_{3} + 2446) q^{29} + (113 \beta_{3} - 2288) q^{31} + (605 \beta_{2} + 121 \beta_1) q^{33} + (691 \beta_{2} + 341 \beta_1) q^{37} + (634 \beta_{3} - 40) q^{39} + ( - 248 \beta_{3} + 9958) q^{41} + ( - 3800 \beta_{2} - 300 \beta_1) q^{43} + (5941 \beta_{2} + 712 \beta_1) q^{47} + (655 \beta_{3} + 1183) q^{49} + (761 \beta_{3} - 41560) q^{51} + ( - 1911 \beta_{2} - 441 \beta_1) q^{53} + (7040 \beta_{2} + 1653 \beta_1) q^{57} + (562 \beta_{3} - 25700) q^{59} + ( - 581 \beta_{3} - 21798) q^{61} + ( - 30998 \beta_{2} - 2682 \beta_1) q^{63} + (2597 \beta_{2} - 222 \beta_1) q^{67} + ( - 3142 \beta_{3} + 63620) q^{69} + ( - 921 \beta_{3} - 84) q^{71} + ( - 23100 \beta_{2} + 254 \beta_1) q^{73} + ( - 4114 \beta_{2} - 605 \beta_1) q^{77} + ( - 2818 \beta_{3} + 9576) q^{79} + ( - 6308 \beta_{3} + 115829) q^{81} + (22732 \beta_{2} + 2518 \beta_1) q^{83} + (18350 \beta_{2} - 1195 \beta_1) q^{87} + ( - 499 \beta_{3} + 40658) q^{89} + ( - 2846 \beta_{3} + 14528) q^{91} + (36300 \beta_{2} + 3305 \beta_1) q^{93} + ( - 25703 \beta_{2} - 118 \beta_1) q^{97} + ( - 2299 \beta_{3} + 35937) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 1150 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 1150 q^{9} - 484 q^{11} - 6878 q^{19} + 11294 q^{21} + 10062 q^{29} - 8926 q^{31} + 1108 q^{39} + 39336 q^{41} + 6042 q^{49} - 164718 q^{51} - 101676 q^{59} - 88354 q^{61} + 248196 q^{69} - 2178 q^{71} + 32668 q^{79} + 450700 q^{81} + 161634 q^{89} + 52420 q^{91} + 139150 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 881x^{2} + 193600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} - 881\nu ) / 440 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 441\nu ) / 220 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 441 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 441 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 881\beta_{2} + 882\beta_1 ) / 2 \) Copy content Toggle raw display

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
21.4821i
20.4821i
20.4821i
21.4821i
0 30.4821i 0 0 0 170.411i 0 −686.161 0
749.2 0 11.4821i 0 0 0 39.4107i 0 111.161 0
749.3 0 11.4821i 0 0 0 39.4107i 0 111.161 0
749.4 0 30.4821i 0 0 0 170.411i 0 −686.161 0
\(n\): e.g. 2-40 or 990-1000
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.b 4
5.b even 2 1 inner 1100.6.b.b 4
5.c odd 4 1 220.6.a.a 2
5.c odd 4 1 1100.6.a.c 2
20.e even 4 1 880.6.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.6.a.a 2 5.c odd 4 1
880.6.a.h 2 20.e even 4 1
1100.6.a.c 2 5.c odd 4 1
1100.6.b.b 4 1.a even 1 1 trivial
1100.6.b.b 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 1061T_{3}^{2} + 122500 \) acting on \(S_{6}^{\mathrm{new}}(1100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 1061 T^{2} + 122500 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 30593 T^{2} + 45104656 \) Copy content Toggle raw display
$11$ \( (T + 121)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 255412987456 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 16946000368704 \) Copy content Toggle raw display
$19$ \( (T^{2} + 3439 T + 2935108)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 2009442332304 \) Copy content Toggle raw display
$29$ \( (T^{2} - 5031 T - 2178330)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4463 T - 641960)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{2} - 19668 T + 69630420)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 252174400000000 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 52\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{2} + 50838 T + 507075240)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 44177 T + 339290602)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 41547615713536 \) Copy content Toggle raw display
$71$ \( (T^{2} + 1089 T - 373141620)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{2} - 16334 T - 3429379952)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 70\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{2} - 80817 T + 1523224182)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 69\!\cdots\!24 \) Copy content Toggle raw display
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