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Properties

Label 1110.2.x.c
Level $1110$
Weight $2$
Character orbit 1110.x
Analytic conductor $8.863$
Analytic rank $0$
Dimension $16$
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] = [1110,2,Mod(751,1110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1110, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1110.751");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.x (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: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{13} + 398 x^{12} - 136 x^{11} + 32 x^{10} - 824 x^{9} + 17825 x^{8} - 11480 x^{7} + \cdots + 20736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{10} - \beta_{6}) q^{2} + \beta_{9} q^{3} - \beta_{9} q^{4} - \beta_{10} q^{5} + \beta_{6} q^{6} + \beta_{2} q^{7} - \beta_{6} q^{8} + ( - \beta_{9} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} - \beta_{6}) q^{2} + \beta_{9} q^{3} - \beta_{9} q^{4} - \beta_{10} q^{5} + \beta_{6} q^{6} + \beta_{2} q^{7} - \beta_{6} q^{8} + ( - \beta_{9} - 1) q^{9} - q^{10} + (\beta_{13} - \beta_{10} + \beta_{6} - 1) q^{11} + (\beta_{9} + 1) q^{12} + ( - \beta_{9} - \beta_{4} - \beta_{3} + \cdots - 2) q^{13}+ \cdots + ( - \beta_{13} + \beta_{10} + \beta_{9} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{3} + 8 q^{4} - 2 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{3} + 8 q^{4} - 2 q^{7} - 8 q^{9} - 16 q^{10} - 12 q^{11} + 8 q^{12} - 12 q^{13} - 8 q^{16} - 6 q^{17} + 6 q^{19} - 2 q^{21} - 6 q^{22} + 8 q^{25} + 16 q^{27} + 2 q^{28} + 8 q^{30} + 6 q^{33} - 4 q^{34} - 6 q^{35} - 16 q^{36} + 18 q^{37} + 12 q^{38} + 12 q^{39} - 8 q^{40} + 6 q^{42} - 6 q^{44} - 4 q^{46} - 60 q^{47} + 16 q^{48} - 4 q^{49} - 12 q^{52} + 12 q^{53} + 6 q^{55} - 6 q^{56} - 6 q^{57} - 12 q^{58} + 12 q^{59} - 4 q^{62} + 4 q^{63} - 16 q^{64} + 22 q^{67} - 6 q^{69} + 2 q^{70} + 2 q^{71} + 24 q^{73} - 8 q^{74} - 16 q^{75} + 6 q^{76} - 58 q^{77} - 36 q^{79} - 8 q^{81} - 8 q^{83} - 4 q^{84} + 8 q^{85} - 2 q^{86} - 42 q^{89} + 8 q^{90} + 6 q^{92} - 6 q^{93} + 6 q^{94} - 6 q^{95} - 12 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{13} + 398 x^{12} - 136 x^{11} + 32 x^{10} - 824 x^{9} + 17825 x^{8} - 11480 x^{7} + \cdots + 20736 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 15\!\cdots\!79 \nu^{15} + \cdots + 66\!\cdots\!56 ) / 10\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 127293331 \nu^{15} - 1300364410 \nu^{14} - 327477924 \nu^{13} + 632205952 \nu^{12} + \cdots + 35657093478528 ) / 61574442553344 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 53\!\cdots\!89 \nu^{15} + \cdots + 85\!\cdots\!60 ) / 13\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!89 \nu^{15} + \cdots + 24\!\cdots\!00 ) / 12\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 64\!\cdots\!55 \nu^{15} + \cdots + 43\!\cdots\!96 ) / 73\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 70\!\cdots\!37 \nu^{15} + \cdots - 20\!\cdots\!00 ) / 45\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 86\!\cdots\!83 \nu^{15} + \cdots + 94\!\cdots\!72 ) / 45\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 30952338089 \nu^{15} - 2291279958 \nu^{14} + 23406559380 \nu^{13} - 241724102080 \nu^{12} + \cdots - 342088009340544 ) / 11\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 82\!\cdots\!45 \nu^{15} + \cdots + 99\!\cdots\!20 ) / 27\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 20\!\cdots\!59 \nu^{15} + \cdots + 11\!\cdots\!16 ) / 61\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 37\!\cdots\!65 \nu^{15} + \cdots - 32\!\cdots\!56 ) / 81\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 32\!\cdots\!43 \nu^{15} + \cdots - 64\!\cdots\!60 ) / 40\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11\!\cdots\!47 \nu^{15} + \cdots - 25\!\cdots\!76 ) / 13\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 26\!\cdots\!81 \nu^{15} + \cdots + 61\!\cdots\!40 ) / 22\!\cdots\!04 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{8} - 6\beta_{6} - \beta_{4} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4 \beta_{15} + 2 \beta_{14} + 2 \beta_{13} - 8 \beta_{10} - 8 \beta_{9} - 2 \beta_{8} + 12 \beta_{7} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4 \beta_{14} + 8 \beta_{12} + 17 \beta_{11} + 4 \beta_{10} - 4 \beta_{9} + 17 \beta_{8} - 24 \beta_{7} + \cdots - 76 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 34 \beta_{14} + 34 \beta_{13} - 48 \beta_{11} - 196 \beta_{10} + 196 \beta_{9} - 4 \beta_{8} + \cdots + 144 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16 \beta_{15} + 8 \beta_{14} - 96 \beta_{13} - 287 \beta_{11} + 96 \beta_{10} - 176 \beta_{9} + \cdots - 92 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1916 \beta_{15} - 574 \beta_{14} - 574 \beta_{13} + 32 \beta_{12} + 160 \beta_{11} + 3992 \beta_{10} + \cdots + 1604 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 656 \beta_{15} - 1916 \beta_{14} + 320 \beta_{13} - 3832 \beta_{12} - 5675 \beta_{11} - 5020 \beta_{10} + \cdots + 21996 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 11350 \beta_{14} - 11350 \beta_{13} + 1312 \beta_{12} + 18328 \beta_{11} + 78396 \beta_{10} + \cdots - 60312 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 18912 \beta_{15} - 9336 \beta_{14} + 36656 \beta_{13} + 89629 \beta_{11} - 36656 \beta_{10} + \cdots + 66460 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 693636 \beta_{15} + 179258 \beta_{14} + 179258 \beta_{13} - 37824 \beta_{12} - 119680 \beta_{11} + \cdots - 476932 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 475488 \beta_{15} + 693636 \beta_{14} - 239360 \beta_{13} + 1387272 \beta_{12} + 2093369 \beta_{11} + \cdots - 7636684 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 4186738 \beta_{14} + 4186738 \beta_{13} - 950976 \beta_{12} - 6551456 \beta_{11} - 28675396 \beta_{10} + \cdots + 23997888 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 11145904 \beta_{15} + 5712648 \beta_{14} - 13102912 \beta_{13} - 29381815 \beta_{11} + 13102912 \beta_{10} + \cdots - 34279916 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 247839420 \beta_{15} - 58763630 \beta_{14} - 58763630 \beta_{13} + 22291808 \beta_{12} + \cdots + 128849140 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-\beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
0.607149 + 0.607149i
−3.13757 − 3.13757i
1.71606 + 1.71606i
−0.917697 − 0.917697i
1.14718 − 1.14718i
−1.91988 + 1.91988i
2.96426 − 2.96426i
−0.459512 + 0.459512i
0.607149 − 0.607149i
−3.13757 + 3.13757i
1.71606 − 1.71606i
−0.917697 + 0.917697i
1.14718 + 1.14718i
−1.91988 − 1.91988i
2.96426 + 2.96426i
−0.459512 − 0.459512i
−0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 − 0.866025i 0.866025 + 0.500000i − 1.00000i −1.98397 + 3.43634i 1.00000i −0.500000 − 0.866025i −1.00000
751.2 −0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 − 0.866025i 0.866025 + 0.500000i − 1.00000i −0.907693 + 1.57217i 1.00000i −0.500000 − 0.866025i −1.00000
751.3 −0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 − 0.866025i 0.866025 + 0.500000i − 1.00000i 0.166951 − 0.289168i 1.00000i −0.500000 − 0.866025i −1.00000
751.4 −0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 − 0.866025i 0.866025 + 0.500000i − 1.00000i 1.35869 − 2.35332i 1.00000i −0.500000 − 0.866025i −1.00000
751.5 0.866025 − 0.500000i −0.500000 + 0.866025i 0.500000 − 0.866025i −0.866025 − 0.500000i 1.00000i −1.41697 + 2.45427i − 1.00000i −0.500000 − 0.866025i −1.00000
751.6 0.866025 − 0.500000i −0.500000 + 0.866025i 0.500000 − 0.866025i −0.866025 − 0.500000i 1.00000i −0.861658 + 1.49244i − 1.00000i −0.500000 − 0.866025i −1.00000
751.7 0.866025 − 0.500000i −0.500000 + 0.866025i 0.500000 − 0.866025i −0.866025 − 0.500000i 1.00000i 0.293092 − 0.507650i − 1.00000i −0.500000 − 0.866025i −1.00000
751.8 0.866025 − 0.500000i −0.500000 + 0.866025i 0.500000 − 0.866025i −0.866025 − 0.500000i 1.00000i 2.35156 − 4.07303i − 1.00000i −0.500000 − 0.866025i −1.00000
841.1 −0.866025 − 0.500000i −0.500000 − 0.866025i 0.500000 + 0.866025i 0.866025 − 0.500000i 1.00000i −1.98397 − 3.43634i − 1.00000i −0.500000 + 0.866025i −1.00000
841.2 −0.866025 − 0.500000i −0.500000 − 0.866025i 0.500000 + 0.866025i 0.866025 − 0.500000i 1.00000i −0.907693 − 1.57217i − 1.00000i −0.500000 + 0.866025i −1.00000
841.3 −0.866025 − 0.500000i −0.500000 − 0.866025i 0.500000 + 0.866025i 0.866025 − 0.500000i 1.00000i 0.166951 + 0.289168i − 1.00000i −0.500000 + 0.866025i −1.00000
841.4 −0.866025 − 0.500000i −0.500000 − 0.866025i 0.500000 + 0.866025i 0.866025 − 0.500000i 1.00000i 1.35869 + 2.35332i − 1.00000i −0.500000 + 0.866025i −1.00000
841.5 0.866025 + 0.500000i −0.500000 − 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i − 1.00000i −1.41697 − 2.45427i 1.00000i −0.500000 + 0.866025i −1.00000
841.6 0.866025 + 0.500000i −0.500000 − 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i − 1.00000i −0.861658 − 1.49244i 1.00000i −0.500000 + 0.866025i −1.00000
841.7 0.866025 + 0.500000i −0.500000 − 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i − 1.00000i 0.293092 + 0.507650i 1.00000i −0.500000 + 0.866025i −1.00000
841.8 0.866025 + 0.500000i −0.500000 − 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i − 1.00000i 2.35156 + 4.07303i 1.00000i −0.500000 + 0.866025i −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 751.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.x.c ✓ 16
37.e even 6 1 inner 1110.2.x.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.x.c ✓ 16 1.a even 1 1 trivial
1110.2.x.c ✓ 16 37.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 2 T_{7}^{15} + 32 T_{7}^{14} + 80 T_{7}^{13} + 767 T_{7}^{12} + 1726 T_{7}^{11} + \cdots + 7744 \) acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} + 2 T^{15} + \cdots + 7744 \) Copy content Toggle raw display
$11$ \( (T^{8} + 6 T^{7} + \cdots + 144)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 12 T^{15} + \cdots + 62001 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 224280576 \) Copy content Toggle raw display
$19$ \( T^{16} - 6 T^{15} + \cdots + 20736 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 732893184 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 1021953024 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 210923421696 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 3512479453921 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 1018387759104 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 20350734336 \) Copy content Toggle raw display
$47$ \( (T^{8} + 30 T^{7} + \cdots - 243648)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 21403873050624 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 32303399897664 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 1018387759104 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 4188342343936 \) Copy content Toggle raw display
$71$ \( T^{16} - 2 T^{15} + \cdots + 56070144 \) Copy content Toggle raw display
$73$ \( (T^{8} - 12 T^{7} + \cdots + 22479232)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 19917869147136 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 215266305024 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 214352087629824 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 80703056646144 \) Copy content Toggle raw display
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