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Properties

Label 1014.2.i.h
Level $1014$
Weight $2$
Character orbit 1014.i
Analytic conductor $8.097$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1014,2,Mod(361,1014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1014, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("1014.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.i (of order \(6\), degree \(2\), 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: \(8.09683076496\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + ( - \beta_{7} + 1) q^{3} + \beta_{7} q^{4} + (2 \beta_{11} + 2 \beta_{10} + \cdots + 2 \beta_{2}) q^{5}+ \cdots - \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + ( - \beta_{7} + 1) q^{3} + \beta_{7} q^{4} + (2 \beta_{11} + 2 \beta_{10} + \cdots + 2 \beta_{2}) q^{5}+ \cdots + (\beta_{11} - 2 \beta_{10} + \cdots + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} + 6 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{3} + 6 q^{4} - 6 q^{9} + 2 q^{10} + 12 q^{12} + 36 q^{14} - 6 q^{16} - 16 q^{17} - 10 q^{22} - 72 q^{25} - 12 q^{27} - 22 q^{29} - 2 q^{30} + 8 q^{35} + 6 q^{36} + 16 q^{38} + 4 q^{40} + 18 q^{42} + 24 q^{43} + 6 q^{48} + 40 q^{49} - 32 q^{51} + 20 q^{53} - 36 q^{55} + 18 q^{56} - 44 q^{61} + 10 q^{62} - 12 q^{64} - 20 q^{66} + 16 q^{68} + 16 q^{74} - 36 q^{75} + 24 q^{77} + 124 q^{79} - 6 q^{81} + 4 q^{82} + 22 q^{87} + 10 q^{88} - 4 q^{90} - 8 q^{94} + 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -25\nu^{11} + 95\nu^{9} - 361\nu^{7} + 155\nu^{5} - 30\nu^{3} - 1563\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25\nu^{10} - 95\nu^{8} + 361\nu^{6} - 155\nu^{4} + 30\nu^{2} + 1004 ) / 559 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{10} - 20\nu^{8} + 76\nu^{6} - 139\nu^{4} + 124\nu^{2} - 24 ) / 43 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45\nu^{10} - 171\nu^{8} + 538\nu^{6} - 279\nu^{4} + 54\nu^{2} + 242 ) / 559 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 70\nu^{11} - 266\nu^{9} + 899\nu^{7} - 434\nu^{5} + 84\nu^{3} + 1246\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 114\nu^{10} - 545\nu^{8} + 2071\nu^{6} - 2831\nu^{4} + 3379\nu^{2} - 95 ) / 559 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 114\nu^{11} - 545\nu^{9} + 2071\nu^{7} - 2831\nu^{5} + 3379\nu^{3} - 95\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -128\nu^{10} + 710\nu^{8} - 2698\nu^{6} + 4483\nu^{4} - 4402\nu^{2} + 852 ) / 559 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -242\nu^{11} + 1255\nu^{9} - 4769\nu^{7} + 7314\nu^{5} - 7781\nu^{3} + 1506\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -317\nu^{11} + 1540\nu^{9} - 5852\nu^{7} + 8338\nu^{5} - 9548\nu^{3} + 1848\nu ) / 559 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 3\beta_{8} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} + 2\beta_{7} + 4\beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{11} - \beta_{10} + 9\beta_{8} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{5} + 9\beta_{3} - 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -5\beta_{6} - 14\beta_{2} - 28\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -28\beta_{9} - 14\beta_{7} - 19\beta_{5} - 47\beta_{4} + 28\beta_{3} - 28 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -47\beta_{11} + 19\beta_{10} - 89\beta_{8} - 19\beta_{6} - 47\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -89\beta_{9} - 42\beta_{7} - 155\beta_{4} + 42 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -155\beta_{11} + 66\beta_{10} - 286\beta_{8} + 286\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.385418 + 0.222521i
1.07992 − 0.623490i
−1.56052 + 0.900969i
1.56052 − 0.900969i
−1.07992 + 0.623490i
0.385418 − 0.222521i
−1.56052 − 0.900969i
1.07992 + 0.623490i
−0.385418 − 0.222521i
0.385418 + 0.222521i
−1.07992 − 0.623490i
1.56052 + 0.900969i
−0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 − 0.866025i − 3.15883i −0.866025 − 0.500000i −4.06341 − 2.34601i 1.00000i −0.500000 + 0.866025i 1.57942 + 2.73563i
361.2 −0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 − 0.866025i − 2.13706i −0.866025 − 0.500000i 0.0423637 + 0.0244587i 1.00000i −0.500000 + 0.866025i 1.06853 + 1.85075i
361.3 −0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 − 0.866025i 4.29590i −0.866025 − 0.500000i −3.77318 − 2.17845i 1.00000i −0.500000 + 0.866025i −2.14795 − 3.72036i
361.4 0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 − 0.866025i − 4.29590i 0.866025 + 0.500000i 3.77318 + 2.17845i − 1.00000i −0.500000 + 0.866025i −2.14795 − 3.72036i
361.5 0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 − 0.866025i 2.13706i 0.866025 + 0.500000i −0.0423637 − 0.0244587i − 1.00000i −0.500000 + 0.866025i 1.06853 + 1.85075i
361.6 0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 − 0.866025i 3.15883i 0.866025 + 0.500000i 4.06341 + 2.34601i − 1.00000i −0.500000 + 0.866025i 1.57942 + 2.73563i
823.1 −0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 + 0.866025i − 4.29590i −0.866025 + 0.500000i −3.77318 + 2.17845i − 1.00000i −0.500000 − 0.866025i −2.14795 + 3.72036i
823.2 −0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 + 0.866025i 2.13706i −0.866025 + 0.500000i 0.0423637 − 0.0244587i − 1.00000i −0.500000 − 0.866025i 1.06853 − 1.85075i
823.3 −0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 + 0.866025i 3.15883i −0.866025 + 0.500000i −4.06341 + 2.34601i − 1.00000i −0.500000 − 0.866025i 1.57942 − 2.73563i
823.4 0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 + 0.866025i − 3.15883i 0.866025 − 0.500000i 4.06341 − 2.34601i 1.00000i −0.500000 − 0.866025i 1.57942 − 2.73563i
823.5 0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 + 0.866025i − 2.13706i 0.866025 − 0.500000i −0.0423637 + 0.0244587i 1.00000i −0.500000 − 0.866025i 1.06853 − 1.85075i
823.6 0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 + 0.866025i 4.29590i 0.866025 − 0.500000i 3.77318 − 2.17845i 1.00000i −0.500000 − 0.866025i −2.14795 + 3.72036i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.2.i.h 12
13.b even 2 1 inner 1014.2.i.h 12
13.c even 3 1 1014.2.b.f 6
13.c even 3 1 inner 1014.2.i.h 12
13.d odd 4 1 1014.2.e.l 6
13.d odd 4 1 1014.2.e.n 6
13.e even 6 1 1014.2.b.f 6
13.e even 6 1 inner 1014.2.i.h 12
13.f odd 12 1 1014.2.a.l ✓ 3
13.f odd 12 1 1014.2.a.n yes 3
13.f odd 12 1 1014.2.e.l 6
13.f odd 12 1 1014.2.e.n 6
39.h odd 6 1 3042.2.b.o 6
39.i odd 6 1 3042.2.b.o 6
39.k even 12 1 3042.2.a.ba 3
39.k even 12 1 3042.2.a.bh 3
52.l even 12 1 8112.2.a.cj 3
52.l even 12 1 8112.2.a.cm 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1014.2.a.l ✓ 3 13.f odd 12 1
1014.2.a.n yes 3 13.f odd 12 1
1014.2.b.f 6 13.c even 3 1
1014.2.b.f 6 13.e even 6 1
1014.2.e.l 6 13.d odd 4 1
1014.2.e.l 6 13.f odd 12 1
1014.2.e.n 6 13.d odd 4 1
1014.2.e.n 6 13.f odd 12 1
1014.2.i.h 12 1.a even 1 1 trivial
1014.2.i.h 12 13.b even 2 1 inner
1014.2.i.h 12 13.c even 3 1 inner
1014.2.i.h 12 13.e even 6 1 inner
3042.2.a.ba 3 39.k even 12 1
3042.2.a.bh 3 39.k even 12 1
3042.2.b.o 6 39.h odd 6 1
3042.2.b.o 6 39.i odd 6 1
8112.2.a.cj 3 52.l even 12 1
8112.2.a.cm 3 52.l even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1014, [\chi])\):

\( T_{5}^{6} + 33T_{5}^{4} + 314T_{5}^{2} + 841 \) Copy content Toggle raw display
\( T_{7}^{12} - 41T_{7}^{10} + 1263T_{7}^{8} - 17136T_{7}^{6} + 174683T_{7}^{4} - 418T_{7}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} + 33 T^{4} + \cdots + 841)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 41 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} - 41 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} + 8 T^{5} + 52 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 80 T^{10} + \cdots + 16777216 \) Copy content Toggle raw display
$23$ \( (T^{6} + 28 T^{4} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 11 T^{5} + \cdots + 841)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 125 T^{4} + \cdots + 15625)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 152 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 2897022976 \) Copy content Toggle raw display
$43$ \( (T^{6} - 12 T^{5} + \cdots + 10816)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 80 T^{4} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 5 T^{2} - 36 T - 43)^{4} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 777796321 \) Copy content Toggle raw display
$61$ \( (T^{6} + 22 T^{5} + \cdots + 107584)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 116985856 \) Copy content Toggle raw display
$71$ \( T^{12} - 164 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$73$ \( (T^{6} + 229 T^{4} + \cdots + 169)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 31 T^{2} + \cdots - 533)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 425 T^{4} + \cdots + 2455489)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} - 308 T^{10} + \cdots + 9834496 \) Copy content Toggle raw display
$97$ \( T^{12} - 349 T^{10} + \cdots + 88529281 \) Copy content Toggle raw display
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