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Properties

Label 1610.2.c.g
Level $1610$
Weight $2$
Character orbit 1610.c
Analytic conductor $12.856$
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] = [1610,2,Mod(321,1610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1610, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1610.321");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1610 = 2 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1610.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: \(12.8559147254\)
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: \( x^{12} + 23x^{10} + 191x^{8} + 698x^{6} + 1036x^{4} + 360x^{2} + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} - q^{5} - \beta_1 q^{6} - \beta_{4} q^{7} + q^{8} + (\beta_{4} + \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_1 q^{3} + q^{4} - q^{5} - \beta_1 q^{6} - \beta_{4} q^{7} + q^{8} + (\beta_{4} + \beta_{3} - 1) q^{9} - q^{10} + ( - \beta_{11} - \beta_{5}) q^{11} - \beta_1 q^{12} + ( - \beta_{9} - \beta_{7} + \beta_{5} + \cdots + 1) q^{13}+ \cdots + ( - \beta_{11} + 3 \beta_{9} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 12 q^{4} - 12 q^{5} - q^{7} + 12 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} + 12 q^{4} - 12 q^{5} - q^{7} + 12 q^{8} - 10 q^{9} - 12 q^{10} - q^{14} + 12 q^{16} + 10 q^{17} - 10 q^{18} + 2 q^{19} - 12 q^{20} + 4 q^{21} - 14 q^{23} + 12 q^{25} - q^{28} + 16 q^{29} + 12 q^{32} - 6 q^{33} + 10 q^{34} + q^{35} - 10 q^{36} + 2 q^{38} - 40 q^{39} - 12 q^{40} + 4 q^{42} + 10 q^{45} - 14 q^{46} - 25 q^{49} + 12 q^{50} - q^{56} + 16 q^{58} + 30 q^{61} - 58 q^{63} + 12 q^{64} - 6 q^{66} + 10 q^{68} - 30 q^{69} + q^{70} + 38 q^{71} - 10 q^{72} + 2 q^{76} - 2 q^{77} - 40 q^{78} - 12 q^{80} - 12 q^{81} - 12 q^{83} + 4 q^{84} - 10 q^{85} - 12 q^{89} + 10 q^{90} - 5 q^{91} - 14 q^{92} - 12 q^{93} - 2 q^{95} - 14 q^{97} - 25 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 23x^{10} + 191x^{8} + 698x^{6} + 1036x^{4} + 360x^{2} + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 17\nu^{8} + 81\nu^{6} + 68\nu^{4} - 8\nu^{3} - 172\nu^{2} - 48\nu - 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 17\nu^{9} + 81\nu^{7} + 60\nu^{5} - 260\nu^{3} + 16\nu^{2} - 192\nu + 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} - 17\nu^{9} - 81\nu^{7} - 60\nu^{5} + 260\nu^{3} + 16\nu^{2} + 192\nu + 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} + 17\nu^{7} + 85\nu^{5} + 124\nu^{3} + 4\nu^{2} + 4\nu + 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{10} - 21\nu^{8} - 153\nu^{6} - 464\nu^{4} + 4\nu^{3} - 508\nu^{2} + 24\nu - 64 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3 \nu^{11} + 4 \nu^{10} + 67 \nu^{9} + 76 \nu^{8} + 531 \nu^{7} + 476 \nu^{6} + 1796 \nu^{5} + \cdots + 224 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} - \nu^{10} + 23 \nu^{9} - 21 \nu^{8} + 191 \nu^{7} - 157 \nu^{6} + 698 \nu^{5} - 520 \nu^{4} + \cdots - 112 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3 \nu^{11} - 4 \nu^{10} + 67 \nu^{9} - 76 \nu^{8} + 531 \nu^{7} - 476 \nu^{6} + 1796 \nu^{5} + \cdots - 192 ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{11} + \nu^{10} - 25 \nu^{9} + 25 \nu^{8} - 225 \nu^{7} + 225 \nu^{6} - 868 \nu^{5} + 868 \nu^{4} + \cdots + 256 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -7\nu^{11} - 159\nu^{9} - 1295\nu^{7} - 4572\nu^{5} - 6228\nu^{3} - 16\nu^{2} - 1248\nu - 64 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{8} + \beta_{5} - \beta_{4} - \beta_{2} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{10} + \beta_{9} + \beta_{7} + 2\beta_{6} - 9\beta_{4} - 11\beta_{3} + 2\beta_{2} + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{11} + 14 \beta_{9} - 11 \beta_{8} + 3 \beta_{7} - 15 \beta_{5} + 16 \beta_{4} + \beta_{3} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 28 \beta_{10} - 13 \beta_{9} - 3 \beta_{8} - 12 \beta_{7} - 26 \beta_{6} - 3 \beta_{5} + 83 \beta_{4} + \cdots - 180 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 32 \beta_{11} - 155 \beta_{9} + 108 \beta_{8} - 47 \beta_{7} + 164 \beta_{5} - 181 \beta_{4} + \cdots + 47 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 306 \beta_{10} + 136 \beta_{9} + 53 \beta_{8} + 117 \beta_{7} + 268 \beta_{6} + 53 \beta_{5} + \cdots + 1425 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 374 \beta_{11} + 1569 \beta_{9} - 1025 \beta_{8} + 544 \beta_{7} - 1629 \beta_{5} + 1837 \beta_{4} + \cdots - 544 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3070 \beta_{10} - 1335 \beta_{9} - 650 \beta_{8} - 1085 \beta_{7} - 2586 \beta_{6} - 650 \beta_{5} + \cdots - 12017 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3886 \beta_{11} - 15218 \beta_{9} + 9597 \beta_{8} - 5621 \beta_{7} + 15569 \beta_{5} - 17804 \beta_{4} + \cdots + 5621 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1610\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(967\) \(1151\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
3.03990i
2.29466i
2.20192i
1.78977i
0.548835i
0.374934i
− 0.374934i
− 0.548835i
− 1.78977i
− 2.20192i
− 2.29466i
− 3.03990i
1.00000 − 3.03990i 1.00000 −1.00000 − 3.03990i 2.62049 − 0.364763i 1.00000 −6.24097 −1.00000
321.2 1.00000 − 2.29466i 1.00000 −1.00000 − 2.29466i 0.632744 + 2.56898i 1.00000 −2.26549 −1.00000
321.3 1.00000 − 2.20192i 1.00000 −1.00000 − 2.20192i 0.424228 − 2.61152i 1.00000 −1.84846 −1.00000
321.4 1.00000 − 1.78977i 1.00000 −1.00000 − 1.78977i −0.398355 + 2.61559i 1.00000 −0.203290 −1.00000
321.5 1.00000 − 0.548835i 1.00000 −1.00000 − 0.548835i −1.84939 − 1.89202i 1.00000 2.69878 −1.00000
321.6 1.00000 − 0.374934i 1.00000 −1.00000 − 0.374934i −1.92971 − 1.81003i 1.00000 2.85942 −1.00000
321.7 1.00000 0.374934i 1.00000 −1.00000 0.374934i −1.92971 + 1.81003i 1.00000 2.85942 −1.00000
321.8 1.00000 0.548835i 1.00000 −1.00000 0.548835i −1.84939 + 1.89202i 1.00000 2.69878 −1.00000
321.9 1.00000 1.78977i 1.00000 −1.00000 1.78977i −0.398355 − 2.61559i 1.00000 −0.203290 −1.00000
321.10 1.00000 2.20192i 1.00000 −1.00000 2.20192i 0.424228 + 2.61152i 1.00000 −1.84846 −1.00000
321.11 1.00000 2.29466i 1.00000 −1.00000 2.29466i 0.632744 − 2.56898i 1.00000 −2.26549 −1.00000
321.12 1.00000 3.03990i 1.00000 −1.00000 3.03990i 2.62049 + 0.364763i 1.00000 −6.24097 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 321.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1610.2.c.g ✓ 12
7.b odd 2 1 1610.2.c.h yes 12
23.b odd 2 1 1610.2.c.h yes 12
161.c even 2 1 inner 1610.2.c.g ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1610.2.c.g ✓ 12 1.a even 1 1 trivial
1610.2.c.g ✓ 12 161.c even 2 1 inner
1610.2.c.h yes 12 7.b odd 2 1
1610.2.c.h yes 12 23.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1610, [\chi])\):

\( T_{3}^{12} + 23T_{3}^{10} + 191T_{3}^{8} + 698T_{3}^{6} + 1036T_{3}^{4} + 360T_{3}^{2} + 32 \) Copy content Toggle raw display
\( T_{17}^{6} - 5T_{17}^{5} - 57T_{17}^{4} + 170T_{17}^{3} + 1016T_{17}^{2} - 432T_{17} - 1984 \) Copy content Toggle raw display
\( T_{19}^{6} - T_{19}^{5} - 39T_{19}^{4} + 12T_{19}^{3} + 312T_{19}^{2} + 384T_{19} + 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 23 T^{10} + \cdots + 32 \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} + 91 T^{10} + \cdots + 618272 \) Copy content Toggle raw display
$13$ \( T^{12} + 79 T^{10} + \cdots + 128 \) Copy content Toggle raw display
$17$ \( (T^{6} - 5 T^{5} + \cdots - 1984)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - T^{5} - 39 T^{4} + \cdots + 128)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 148035889 \) Copy content Toggle raw display
$29$ \( (T^{6} - 8 T^{5} + \cdots - 25856)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 81 T^{10} + \cdots + 512 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 101844992 \) Copy content Toggle raw display
$41$ \( T^{12} + 141 T^{10} + \cdots + 32768 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 153755648 \) Copy content Toggle raw display
$47$ \( T^{12} + 324 T^{10} + \cdots + 33554432 \) Copy content Toggle raw display
$53$ \( T^{12} + 134 T^{10} + \cdots + 2048 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 692515328 \) Copy content Toggle raw display
$61$ \( (T^{6} - 15 T^{5} + \cdots - 51472)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 298 T^{10} + \cdots + 1438208 \) Copy content Toggle raw display
$71$ \( (T^{6} - 19 T^{5} + \cdots + 29632)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 74121740288 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 140582912 \) Copy content Toggle raw display
$83$ \( (T^{6} + 6 T^{5} + \cdots + 401408)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 6 T^{5} + \cdots + 210752)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 7 T^{5} + \cdots - 1168)^{2} \) Copy content Toggle raw display
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