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Properties

Label 1110.2.i.p
Level $1110$
Weight $2$
Character orbit 1110.i
Analytic conductor $8.863$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1110,2,Mod(121,1110)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1110.121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1110, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,-5,-5,-5,-5,10,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 15x^{8} - 6x^{7} + 123x^{6} - 62x^{5} + 458x^{4} + 100x^{3} + 844x^{2} - 312x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - 1) q^{2} + \beta_{6} q^{3} + \beta_{6} q^{4} + \beta_{6} q^{5} + q^{6} - \beta_{3} q^{7} + q^{8} + ( - \beta_{6} - 1) q^{9} + q^{10} + ( - \beta_{5} - \beta_{4}) q^{11} + ( - \beta_{6} - 1) q^{12}+ \cdots + ( - \beta_{9} + \beta_{5} + \cdots + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 5 q^{2} - 5 q^{3} - 5 q^{4} - 5 q^{5} + 10 q^{6} + 10 q^{8} - 5 q^{9} + 10 q^{10} - 2 q^{11} - 5 q^{12} + q^{13} - 5 q^{15} - 5 q^{16} - 5 q^{18} - 2 q^{19} - 5 q^{20} + q^{22} - 2 q^{23} - 5 q^{24}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 15x^{8} - 6x^{7} + 123x^{6} - 62x^{5} + 458x^{4} + 100x^{3} + 844x^{2} - 312x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 52206 \nu^{9} - 298647 \nu^{8} + 1382625 \nu^{7} - 3088861 \nu^{6} + 9390789 \nu^{5} + \cdots - 57589572 ) / 131893214 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1026853 \nu^{9} + 4535866 \nu^{8} - 8787045 \nu^{7} + 105828882 \nu^{6} - 91335981 \nu^{5} + \cdots + 1346033064 ) / 1582718568 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 223413 \nu^{9} - 1789641 \nu^{8} + 8285375 \nu^{7} - 21741437 \nu^{6} + 56274267 \nu^{5} + \cdots - 204447812 ) / 263786428 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 246441 \nu^{9} + 898182 \nu^{8} - 4158250 \nu^{7} + 6058312 \nu^{6} - 28242834 \nu^{5} + \cdots + 709537978 ) / 131893214 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4799131 \nu^{9} - 8971790 \nu^{8} + 68403201 \nu^{7} - 12203286 \nu^{6} + 553226781 \nu^{5} + \cdots - 1250447352 ) / 1582718568 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1710145 \nu^{9} - 7020351 \nu^{8} + 32501625 \nu^{7} - 63953489 \nu^{6} + 220751037 \nu^{5} + \cdots - 2060561952 ) / 263786428 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3473146 \nu^{9} - 3300175 \nu^{8} + 47030658 \nu^{7} + 24599493 \nu^{6} + 448160006 \nu^{5} + \cdots + 320551920 ) / 263786428 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 24622127 \nu^{9} + 48442714 \nu^{8} - 358607505 \nu^{7} + 98082762 \nu^{6} + \cdots + 6943311624 ) / 1582718568 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 5\beta_{6} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 2\beta_{4} + 9\beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -11\beta_{9} - 2\beta_{8} - 39\beta_{6} + 11\beta_{5} - 4\beta_{4} - 4\beta_{3} - 17\beta _1 - 39 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -19\beta_{9} - 4\beta_{8} - 4\beta_{7} - 53\beta_{6} - 30\beta_{3} - 93\beta_{2} - 93\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -30\beta_{7} - 119\beta_{5} + 76\beta_{4} - 233\beta_{2} + 363 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 279\beta_{9} + 76\beta_{8} + 745\beta_{6} - 279\beta_{5} + 374\beta_{4} + 374\beta_{3} + 1029\beta _1 + 745 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1327\beta_{9} + 374\beta_{8} + 374\beta_{7} + 3763\beta_{6} + 1084\beta_{3} + 2985\beta_{2} + 2985\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1084\beta_{7} + 3695\beta_{5} - 4486\beta_{4} + 11813\beta_{2} - 9729 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(\beta_{6}\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
1.10642 − 1.91637i
0.201664 − 0.349293i
−1.29048 + 2.23517i
1.73363 − 3.00274i
−0.751235 + 1.30118i
1.10642 + 1.91637i
0.201664 + 0.349293i
−1.29048 − 2.23517i
1.73363 + 3.00274i
−0.751235 − 1.30118i
−0.500000 + 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i 1.00000 −2.44097 − 4.22788i 1.00000 −0.500000 + 0.866025i 1.00000
121.2 −0.500000 + 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i 1.00000 −0.330923 − 0.573175i 1.00000 −0.500000 + 0.866025i 1.00000
121.3 −0.500000 + 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i 1.00000 −0.301577 − 0.522347i 1.00000 −0.500000 + 0.866025i 1.00000
121.4 −0.500000 + 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i 1.00000 0.980808 + 1.69881i 1.00000 −0.500000 + 0.866025i 1.00000
121.5 −0.500000 + 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i 1.00000 2.09266 + 3.62459i 1.00000 −0.500000 + 0.866025i 1.00000
211.1 −0.500000 − 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −2.44097 + 4.22788i 1.00000 −0.500000 − 0.866025i 1.00000
211.2 −0.500000 − 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −0.330923 + 0.573175i 1.00000 −0.500000 − 0.866025i 1.00000
211.3 −0.500000 − 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −0.301577 + 0.522347i 1.00000 −0.500000 − 0.866025i 1.00000
211.4 −0.500000 − 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 0.980808 − 1.69881i 1.00000 −0.500000 − 0.866025i 1.00000
211.5 −0.500000 − 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 2.09266 − 3.62459i 1.00000 −0.500000 − 0.866025i 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.i.p ✓ 10
37.c even 3 1 inner 1110.2.i.p ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.i.p ✓ 10 1.a even 1 1 trivial
1110.2.i.p ✓ 10 37.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{10} + 23T_{7}^{8} - 24T_{7}^{7} + 487T_{7}^{6} - 260T_{7}^{5} + 1110T_{7}^{4} + 1240T_{7}^{3} + 1572T_{7}^{2} + 672T_{7} + 256 \) Copy content Toggle raw display
\( T_{11}^{5} + T_{11}^{4} - 41T_{11}^{3} - T_{11}^{2} + 404T_{11} - 492 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{10} + 23 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{5} + T^{4} - 41 T^{3} + \cdots - 492)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} - T^{9} + \cdots + 9409 \) Copy content Toggle raw display
$17$ \( T^{10} + 53 T^{8} + \cdots + 5184 \) Copy content Toggle raw display
$19$ \( T^{10} + 2 T^{9} + \cdots + 30976 \) Copy content Toggle raw display
$23$ \( (T^{5} + T^{4} - 88 T^{3} + \cdots + 6144)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} - 9 T^{4} + \cdots + 192)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 7 T^{4} + \cdots + 4064)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 25 T^{8} + \cdots + 69343957 \) Copy content Toggle raw display
$41$ \( T^{10} + 10 T^{9} + \cdots + 2359296 \) Copy content Toggle raw display
$43$ \( (T^{5} - 3 T^{4} + \cdots - 24064)^{2} \) Copy content Toggle raw display
$47$ \( (T^{5} - 15 T^{4} + \cdots - 6792)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + 2 T^{9} + \cdots + 331776 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 337971456 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 43992545536 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 3063401104 \) Copy content Toggle raw display
$71$ \( T^{10} - 3 T^{9} + \cdots + 82944 \) Copy content Toggle raw display
$73$ \( (T^{5} + 9 T^{4} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + 15 T^{9} + \cdots + 262144 \) Copy content Toggle raw display
$83$ \( T^{10} + 5 T^{9} + \cdots + 56430144 \) Copy content Toggle raw display
$89$ \( T^{10} + 25 T^{9} + \cdots + 5308416 \) Copy content Toggle raw display
$97$ \( (T^{5} - 3 T^{4} + \cdots + 128)^{2} \) Copy content Toggle raw display
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