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Properties

Label 1035.2.c.d
Level $1035$
Weight $2$
Character orbit 1035.c
Analytic conductor $8.265$
Analytic rank $0$
Dimension $12$
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] = [1035,2,Mod(206,1035)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1035, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1035.206");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1035.c (of order \(2\), degree \(1\), 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: \(8.26451660920\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 22x^{10} + 177x^{8} + 620x^{6} + 852x^{4} + 280x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + q^{5} - \beta_{9} q^{7} + (\beta_{5} + \beta_{4} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + q^{5} - \beta_{9} q^{7} + (\beta_{5} + \beta_{4} - 2 \beta_1) q^{8} + \beta_1 q^{10} + (\beta_{8} + 2) q^{11} + (\beta_{8} - \beta_{6} + \beta_{3}) q^{13} + ( - 2 \beta_{8} + \beta_{6} - \beta_{2}) q^{14} + (\beta_{6} + \beta_{3} - \beta_{2} + 4) q^{16} + ( - \beta_{7} + \beta_{6} + 1) q^{17} + ( - \beta_{11} - \beta_{9} + \cdots + \beta_1) q^{19}+ \cdots + ( - 2 \beta_{11} + 2 \beta_{10} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 20 q^{4} + 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 20 q^{4} + 12 q^{5} + 20 q^{11} - 4 q^{13} + 4 q^{14} + 44 q^{16} + 12 q^{17} - 20 q^{20} + 16 q^{23} + 12 q^{25} + 8 q^{31} - 40 q^{38} - 56 q^{44} + 24 q^{46} - 20 q^{49} - 32 q^{52} - 60 q^{53} + 20 q^{55} - 48 q^{56} + 44 q^{58} - 12 q^{64} - 4 q^{65} + 8 q^{68} + 4 q^{70} - 20 q^{73} + 28 q^{74} + 44 q^{80} + 28 q^{82} + 28 q^{83} + 12 q^{85} - 80 q^{86} + 32 q^{89} - 4 q^{92} - 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 22x^{10} + 177x^{8} + 620x^{6} + 852x^{4} + 280x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{8} - 15\nu^{6} - 68\nu^{4} - 86\nu^{2} - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} - 15\nu^{7} - 72\nu^{5} - 114\nu^{3} - 24\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} + 15\nu^{7} + 72\nu^{5} + 118\nu^{3} + 48\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{8} + 15\nu^{6} + 72\nu^{4} + 114\nu^{2} + 24 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{10} + 15\nu^{8} + 72\nu^{6} + 118\nu^{4} + 48\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{10} + 16\nu^{8} + 85\nu^{6} + 168\nu^{4} + 102\nu^{2} + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2\nu^{9} + 31\nu^{7} + 155\nu^{5} + 264\nu^{3} + 94\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{11} + 20\nu^{9} + 147\nu^{7} + 478\nu^{5} + 634\nu^{3} + 216\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} + 21\nu^{9} + 161\nu^{7} + 535\nu^{5} + 684\nu^{3} + 178\nu ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{3} - 7\beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} + \beta_{10} - \beta_{9} - 7\beta_{5} - 10\beta_{4} + 38\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{8} + 2\beta_{7} - 9\beta_{6} - 11\beta_{3} + 47\beta_{2} - 154 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11\beta_{11} - 11\beta_{10} + 15\beta_{9} + 41\beta_{5} + 82\beta_{4} - 248\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 30\beta_{8} - 30\beta_{7} + 67\beta_{6} + 93\beta_{3} - 315\beta_{2} + 1014 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -93\beta_{11} + 93\beta_{10} - 153\beta_{9} - 225\beta_{5} - 628\beta_{4} + 1644\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -306\beta_{8} + 310\beta_{7} - 475\beta_{6} - 721\beta_{3} + 2119\beta_{2} - 6762 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 721\beta_{11} - 717\beta_{10} + 1333\beta_{9} + 1185\beta_{5} + 4652\beta_{4} - 11000\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1035\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(622\) \(856\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
206.1
− 2.63209i
− 2.61397i
− 2.33416i
− 1.52686i
− 0.666874i
− 0.122308i
0.122308i
0.666874i
1.52686i
2.33416i
2.61397i
2.63209i
− 2.63209i 0 −4.92787 1.00000 0 2.47509i 7.70641i 0 − 2.63209i
206.2 − 2.61397i 0 −4.83281 1.00000 0 − 2.02066i 7.40488i 0 − 2.61397i
206.3 − 2.33416i 0 −3.44832 1.00000 0 3.14947i 3.38063i 0 − 2.33416i
206.4 − 1.52686i 0 −0.331309 1.00000 0 − 4.86784i − 2.54786i 0 − 1.52686i
206.5 − 0.666874i 0 1.55528 1.00000 0 0.767021i − 2.37092i 0 − 0.666874i
206.6 − 0.122308i 0 1.98504 1.00000 0 2.75454i − 0.487403i 0 − 0.122308i
206.7 0.122308i 0 1.98504 1.00000 0 − 2.75454i 0.487403i 0 0.122308i
206.8 0.666874i 0 1.55528 1.00000 0 − 0.767021i 2.37092i 0 0.666874i
206.9 1.52686i 0 −0.331309 1.00000 0 4.86784i 2.54786i 0 1.52686i
206.10 2.33416i 0 −3.44832 1.00000 0 − 3.14947i − 3.38063i 0 2.33416i
206.11 2.61397i 0 −4.83281 1.00000 0 2.02066i − 7.40488i 0 2.61397i
206.12 2.63209i 0 −4.92787 1.00000 0 − 2.47509i − 7.70641i 0 2.63209i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 206.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
69.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1035.2.c.d yes 12
3.b odd 2 1 1035.2.c.c ✓ 12
23.b odd 2 1 1035.2.c.c ✓ 12
69.c even 2 1 inner 1035.2.c.d yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1035.2.c.c ✓ 12 3.b odd 2 1
1035.2.c.c ✓ 12 23.b odd 2 1
1035.2.c.d yes 12 1.a even 1 1 trivial
1035.2.c.d yes 12 69.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1035, [\chi])\):

\( T_{2}^{12} + 22T_{2}^{10} + 177T_{2}^{8} + 620T_{2}^{6} + 852T_{2}^{4} + 280T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{6} - 10T_{11}^{5} + 18T_{11}^{4} + 56T_{11}^{3} - 122T_{11}^{2} - 96T_{11} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 22 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T - 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 52 T^{10} + \cdots + 26244 \) Copy content Toggle raw display
$11$ \( (T^{6} - 10 T^{5} + \cdots + 144)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} - 66 T^{4} + \cdots - 56)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 6 T^{5} + \cdots - 216)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 100 T^{10} + \cdots + 876096 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 148035889 \) Copy content Toggle raw display
$29$ \( T^{12} + 100 T^{10} + \cdots + 2630884 \) Copy content Toggle raw display
$31$ \( (T^{6} - 4 T^{5} - 62 T^{4} + \cdots - 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 5926844196 \) Copy content Toggle raw display
$41$ \( T^{12} + 292 T^{10} + \cdots + 20232004 \) Copy content Toggle raw display
$43$ \( T^{12} + 244 T^{10} + \cdots + 20736 \) Copy content Toggle raw display
$47$ \( T^{12} + 188 T^{10} + \cdots + 2027776 \) Copy content Toggle raw display
$53$ \( (T^{6} + 30 T^{5} + \cdots - 61236)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 148 T^{10} + \cdots + 16273156 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1087416576 \) Copy content Toggle raw display
$67$ \( T^{12} + 260 T^{10} + \cdots + 24147396 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 7144644676 \) Copy content Toggle raw display
$73$ \( (T^{6} + 10 T^{5} + \cdots + 5752)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 10989748224 \) Copy content Toggle raw display
$83$ \( (T^{6} - 14 T^{5} + \cdots - 8064)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 16 T^{5} + \cdots + 84384)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 94165514496 \) Copy content Toggle raw display
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