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Properties

Label 1610.2.c.f
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: \( x^{12} + 27x^{10} + 273x^{8} + 1276x^{6} + 2714x^{4} + 2104x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + q^{5} - \beta_1 q^{6} + (\beta_{10} - 1) q^{7} - q^{8} + (\beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} + q^{5} - \beta_1 q^{6} + (\beta_{10} - 1) q^{7} - q^{8} + (\beta_{2} - 2) q^{9} - q^{10} + \beta_{6} q^{11} + \beta_1 q^{12} + (\beta_{11} + \beta_{10} + \cdots - \beta_{6}) q^{13}+ \cdots + ( - \beta_{11} - 3 \beta_{10} + \cdots + 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{4} + 12 q^{5} - 9 q^{7} - 12 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{4} + 12 q^{5} - 9 q^{7} - 12 q^{8} - 18 q^{9} - 12 q^{10} + 9 q^{14} + 12 q^{16} - 6 q^{17} + 18 q^{18} - 10 q^{19} + 12 q^{20} - 16 q^{21} + 6 q^{23} + 12 q^{25} - 9 q^{28} + 12 q^{29} - 12 q^{32} - 2 q^{33} + 6 q^{34} - 9 q^{35} - 18 q^{36} + 10 q^{38} - 4 q^{39} - 12 q^{40} + 16 q^{42} - 18 q^{45} - 6 q^{46} + 23 q^{49} - 12 q^{50} + 9 q^{56} - 12 q^{58} + 14 q^{61} + 38 q^{63} + 12 q^{64} + 2 q^{66} - 6 q^{68} - 14 q^{69} + 9 q^{70} + 18 q^{71} + 18 q^{72} - 10 q^{76} - 10 q^{77} + 4 q^{78} + 12 q^{80} - 12 q^{81} - 16 q^{84} - 6 q^{85} - 56 q^{89} + 18 q^{90} - 27 q^{91} + 6 q^{92} + 24 q^{93} - 10 q^{95} - 10 q^{97} - 23 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 27x^{10} + 273x^{8} + 1276x^{6} + 2714x^{4} + 2104x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{11} - 279\nu^{9} - 4917\nu^{7} - 37404\nu^{5} - 122434\nu^{3} - 124904\nu ) / 7472 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5 \nu^{11} + 104 \nu^{10} - 279 \nu^{9} + 1320 \nu^{8} - 4917 \nu^{7} - 840 \nu^{6} + \cdots - 88928 ) / 14944 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 19 \nu^{11} + 72 \nu^{10} - 313 \nu^{9} + 1776 \nu^{8} - 1499 \nu^{7} + 15512 \nu^{6} + \cdots - 15584 ) / 7472 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -121\nu^{11} - 3763\nu^{9} - 42777\nu^{7} - 214764\nu^{5} - 449322\nu^{3} - 277464\nu ) / 14944 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 139\nu^{11} + 3273\nu^{9} + 26107\nu^{7} + 78932\nu^{5} + 74142\nu^{3} + 45672\nu ) / 14944 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 65 \nu^{11} + 182 \nu^{10} - 1759 \nu^{9} + 4178 \nu^{8} - 17221 \nu^{7} + 34022 \nu^{6} + \cdots + 57328 ) / 7472 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 163 \nu^{11} - 172 \nu^{10} + 3865 \nu^{9} - 3620 \nu^{8} + 32523 \nu^{7} - 24188 \nu^{6} + \cdots + 28096 ) / 14944 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 163 \nu^{11} + 172 \nu^{10} + 3865 \nu^{9} + 3620 \nu^{8} + 32523 \nu^{7} + 24188 \nu^{6} + \cdots - 28096 ) / 14944 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 65 \nu^{11} - 182 \nu^{10} - 1759 \nu^{9} - 4178 \nu^{8} - 17221 \nu^{7} - 34022 \nu^{6} + \cdots - 57328 ) / 7472 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{11} - \beta_{10} - 2\beta_{9} - \beta_{8} - 3\beta_{5} - \beta_{4} + 2\beta_{3} - 11\beta_{2} + 2\beta _1 + 37 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -13\beta_{11} - \beta_{10} - \beta_{9} - 13\beta_{8} - 10\beta_{7} + 14\beta_{6} - 3\beta_{3} + 43\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 29 \beta_{11} + 11 \beta_{10} + 35 \beta_{9} + 17 \beta_{8} + 46 \beta_{5} + 18 \beta_{4} - 32 \beta_{3} + \cdots - 300 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 142\beta_{11} + 20\beta_{10} + 20\beta_{9} + 142\beta_{8} + 86\beta_{7} - 156\beta_{6} + 43\beta_{3} - 337\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 321 \beta_{11} - 93 \beta_{10} - 433 \beta_{9} - 205 \beta_{8} - 526 \beta_{5} - 240 \beta_{4} + \cdots + 2544 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1459 \beta_{11} - 279 \beta_{10} - 279 \beta_{9} - 1459 \beta_{8} - 724 \beta_{7} + 1590 \beta_{6} + \cdots + 2788 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3234 \beta_{11} + 732 \beta_{10} + 4704 \beta_{9} + 2202 \beta_{8} + 5436 \beta_{5} + 2798 \beta_{4} + \cdots - 22162 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 14533 \beta_{11} + 3381 \beta_{10} + 3381 \beta_{9} + 14533 \beta_{8} + 6148 \beta_{7} + \cdots - 23899 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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
− 3.05564i
− 2.70656i
− 2.27305i
− 1.92363i
− 1.14997i
− 0.384753i
0.384753i
1.14997i
1.92363i
2.27305i
2.70656i
3.05564i
−1.00000 − 3.05564i 1.00000 1.00000 3.05564i −2.38027 − 1.15513i −1.00000 −6.33695 −1.00000
321.2 −1.00000 − 2.70656i 1.00000 1.00000 2.70656i −1.11898 − 2.39747i −1.00000 −4.32549 −1.00000
321.3 −1.00000 − 2.27305i 1.00000 1.00000 2.27305i −2.32808 + 1.25699i −1.00000 −2.16675 −1.00000
321.4 −1.00000 − 1.92363i 1.00000 1.00000 1.92363i 2.17042 − 1.51303i −1.00000 −0.700337 −1.00000
321.5 −1.00000 − 1.14997i 1.00000 1.00000 1.14997i 1.74018 + 1.99293i −1.00000 1.67756 −1.00000
321.6 −1.00000 − 0.384753i 1.00000 1.00000 0.384753i −2.58327 − 0.571602i −1.00000 2.85197 −1.00000
321.7 −1.00000 0.384753i 1.00000 1.00000 − 0.384753i −2.58327 + 0.571602i −1.00000 2.85197 −1.00000
321.8 −1.00000 1.14997i 1.00000 1.00000 − 1.14997i 1.74018 − 1.99293i −1.00000 1.67756 −1.00000
321.9 −1.00000 1.92363i 1.00000 1.00000 − 1.92363i 2.17042 + 1.51303i −1.00000 −0.700337 −1.00000
321.10 −1.00000 2.27305i 1.00000 1.00000 − 2.27305i −2.32808 − 1.25699i −1.00000 −2.16675 −1.00000
321.11 −1.00000 2.70656i 1.00000 1.00000 − 2.70656i −1.11898 + 2.39747i −1.00000 −4.32549 −1.00000
321.12 −1.00000 3.05564i 1.00000 1.00000 − 3.05564i −2.38027 + 1.15513i −1.00000 −6.33695 −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.f yes 12
7.b odd 2 1 1610.2.c.e ✓ 12
23.b odd 2 1 1610.2.c.e ✓ 12
161.c even 2 1 inner 1610.2.c.f yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1610.2.c.e ✓ 12 7.b odd 2 1
1610.2.c.e ✓ 12 23.b odd 2 1
1610.2.c.f yes 12 1.a even 1 1 trivial
1610.2.c.f yes 12 161.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 \(S_{2}^{\mathrm{new}}(1610, [\chi])\):

\( T_{3}^{12} + 27T_{3}^{10} + 273T_{3}^{8} + 1276T_{3}^{6} + 2714T_{3}^{4} + 2104T_{3}^{2} + 256 \) Copy content Toggle raw display
\( T_{17}^{6} + 3T_{17}^{5} - 53T_{17}^{4} - 262T_{17}^{3} + 48T_{17}^{2} + 1840T_{17} + 2304 \) Copy content Toggle raw display
\( T_{19}^{6} + 5T_{19}^{5} - 89T_{19}^{4} - 334T_{19}^{3} + 1858T_{19}^{2} + 3492T_{19} + 704 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 27 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( (T - 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 9 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} + 67 T^{10} + \cdots + 7744 \) Copy content Toggle raw display
$13$ \( T^{12} + 107 T^{10} + \cdots + 46656 \) Copy content Toggle raw display
$17$ \( (T^{6} + 3 T^{5} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 5 T^{5} + \cdots + 704)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 148035889 \) Copy content Toggle raw display
$29$ \( (T^{6} - 6 T^{5} + \cdots - 12288)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 191102976 \) Copy content Toggle raw display
$37$ \( T^{12} + 114 T^{10} + \cdots + 147456 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 220463104 \) Copy content Toggle raw display
$43$ \( T^{12} + 158 T^{10} + \cdots + 11943936 \) Copy content Toggle raw display
$47$ \( T^{12} + 320 T^{10} + \cdots + 67108864 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 91407056896 \) Copy content Toggle raw display
$59$ \( T^{12} + 334 T^{10} + \cdots + 40551424 \) Copy content Toggle raw display
$61$ \( (T^{6} - 7 T^{5} + \cdots + 7424)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 89942409216 \) Copy content Toggle raw display
$71$ \( (T^{6} - 9 T^{5} + \cdots + 192)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 116 T^{10} + \cdots + 1327104 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 217149696 \) Copy content Toggle raw display
$83$ \( (T^{6} - 224 T^{4} + \cdots + 66048)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 28 T^{5} + \cdots + 322944)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 5 T^{5} + \cdots + 145504)^{2} \) Copy content Toggle raw display
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