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Properties

Label 2610.2.e.k
Level $2610$
Weight $2$
Character orbit 2610.e
Analytic conductor $20.841$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2610,2,Mod(2089,2610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2610, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2610.2089");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2610 = 2 \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2610.e (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: \(20.8409549276\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + 2 x^{12} + 10 x^{11} + 74 x^{10} - 66 x^{9} + 34 x^{8} + 74 x^{7} + 745 x^{6} + \cdots + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} - q^{4} + \beta_{9} q^{5} + ( - \beta_{13} + \beta_{6}) q^{7} - \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} - q^{4} + \beta_{9} q^{5} + ( - \beta_{13} + \beta_{6}) q^{7} - \beta_{6} q^{8} - \beta_{11} q^{10} + (\beta_{7} - \beta_{5} - \beta_1 + 1) q^{11} + (\beta_{8} - \beta_{6}) q^{13} + (\beta_{7} - 1) q^{14} + q^{16} + (\beta_{13} - \beta_{12} + \cdots + \beta_1) q^{17}+ \cdots + (\beta_{13} + 2 \beta_{12} + \cdots + \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{4} + 4 q^{11} - 16 q^{14} + 14 q^{16} - 8 q^{19} + 20 q^{26} - 14 q^{29} + 8 q^{31} - 14 q^{35} + 24 q^{41} - 4 q^{44} - 4 q^{46} - 18 q^{49} + 8 q^{50} + 4 q^{55} + 16 q^{56} - 8 q^{59} + 28 q^{61} - 14 q^{64} - 24 q^{65} - 8 q^{70} + 32 q^{71} - 36 q^{74} + 8 q^{76} - 20 q^{85} + 20 q^{86} - 24 q^{89} + 8 q^{91} + 20 q^{94} - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 2 x^{12} + 10 x^{11} + 74 x^{10} - 66 x^{9} + 34 x^{8} + 74 x^{7} + 745 x^{6} + \cdots + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 207706388381 \nu^{13} - 2552345284482 \nu^{12} + 4508371212286 \nu^{11} + \cdots + 161533896874552 ) / 451617133655448 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 51223684802 \nu^{13} + 112490535374 \nu^{12} - 142256101053 \nu^{11} + \cdots + 7146295607136 ) / 25089840758636 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 51223684802 \nu^{13} - 112490535374 \nu^{12} + 142256101053 \nu^{11} + 466417196307 \nu^{10} + \cdots - 7146295607136 ) / 25089840758636 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1285696017187 \nu^{13} + 5862485591380 \nu^{12} - 6122384772122 \nu^{11} + \cdots + 367989677590152 ) / 451617133655448 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1976530231665 \nu^{13} - 5417411226592 \nu^{12} + 4412609510800 \nu^{11} + \cdots + 139099710670088 ) / 451617133655448 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 223321737723 \nu^{13} - 395419790644 \nu^{12} + 334152940072 \nu^{11} + \cdots - 16936945866388 ) / 25089840758636 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4622348790511 \nu^{13} - 13596443257438 \nu^{12} + 16873112616932 \nu^{11} + \cdots - 12\!\cdots\!76 ) / 451617133655448 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2521702295932 \nu^{13} - 4434098475971 \nu^{12} + 3614945732802 \nu^{11} + \cdots - 191098339889164 ) / 225808566827724 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3002819473790 \nu^{13} + 5798525408944 \nu^{12} - 4717038481077 \nu^{11} + \cdots + 615958984228148 ) / 225808566827724 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3546227887277 \nu^{13} + 5829261337831 \nu^{12} - 5188747812207 \nu^{11} + \cdots - 115499367053812 ) / 225808566827724 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7240748727229 \nu^{13} + 12074566716358 \nu^{12} - 8943696494810 \nu^{11} + \cdots - 214620921102024 ) / 451617133655448 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 7651185726083 \nu^{13} + 13372547186762 \nu^{12} - 10575193182652 \nu^{11} + \cdots + 12\!\cdots\!04 ) / 451617133655448 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 6182111867818 \nu^{13} + 11651458784663 \nu^{12} - 11768004815508 \nu^{11} + \cdots + 567687957941500 ) / 225808566827724 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} - \beta_{12} - \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - \beta_{8} + 8 \beta_{6} + \cdots + 2 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{13} - 4 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + \cdots - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 6 \beta_{12} + 6 \beta_{11} - 9 \beta_{10} + 9 \beta_{9} - 11 \beta_{7} + 22 \beta_{5} + 11 \beta_{4} + \cdots - 60 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 24 \beta_{13} + 22 \beta_{12} + 52 \beta_{11} - 34 \beta_{10} - 22 \beta_{9} + 28 \beta_{8} + \cdots - 72 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 125 \beta_{13} + 157 \beta_{12} + 157 \beta_{11} - 158 \beta_{10} - 158 \beta_{9} + 129 \beta_{8} + \cdots - 242 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 282 \beta_{13} + 608 \beta_{12} + 234 \beta_{11} - 234 \beta_{10} - 454 \beta_{9} + 334 \beta_{8} + \cdots + 988 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 994 \beta_{12} - 994 \beta_{11} + 861 \beta_{10} - 861 \beta_{9} + 1427 \beta_{7} - 2730 \beta_{5} + \cdots + 5924 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3308 \beta_{13} - 2550 \beta_{12} - 7032 \beta_{11} + 5582 \beta_{10} + 2550 \beta_{9} - 3860 \beta_{8} + \cdots + 12320 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 16389 \beta_{13} - 22041 \beta_{12} - 22041 \beta_{11} + 19818 \beta_{10} + 19818 \beta_{9} + \cdots + 31262 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 38662 \beta_{13} - 81368 \beta_{12} - 28462 \beta_{11} + 28462 \beta_{10} + 66422 \beta_{9} + \cdots - 147644 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 131614 \beta_{12} + 131614 \beta_{11} - 102425 \beta_{10} + 102425 \beta_{9} - 189067 \beta_{7} + \cdots - 745324 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 450480 \beta_{13} + 323038 \beta_{12} + 942668 \beta_{11} - 779466 \beta_{10} - 323038 \beta_{9} + \cdots - 1738744 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2610\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(1567\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2089.1
0.862996 − 0.862996i
−1.67046 + 1.67046i
2.40869 − 2.40869i
1.16010 − 1.16010i
−1.42461 + 1.42461i
0.171452 − 0.171452i
−0.508168 + 0.508168i
0.862996 + 0.862996i
−1.67046 − 1.67046i
2.40869 + 2.40869i
1.16010 + 1.16010i
−1.42461 − 1.42461i
0.171452 + 0.171452i
−0.508168 − 0.508168i
− 1.00000i 0 −1.00000 −2.23163 + 0.140865i 0 0.546791i 1.00000i 0 0.140865 + 2.23163i
2089.2 − 1.00000i 0 −1.00000 −2.04108 − 0.913241i 0 − 0.776224i 1.00000i 0 −0.913241 + 2.04108i
2089.3 − 1.00000i 0 −1.00000 −0.693128 + 2.12593i 0 − 1.50042i 1.00000i 0 2.12593 + 0.693128i
2089.4 − 1.00000i 0 −1.00000 0.183693 − 2.22851i 0 − 1.21131i 1.00000i 0 −2.22851 − 0.183693i
2089.5 − 1.00000i 0 −1.00000 1.27443 − 1.83734i 0 − 4.80878i 1.00000i 0 −1.83734 − 1.27443i
2089.6 − 1.00000i 0 −1.00000 1.57565 + 1.58661i 0 − 4.01247i 1.00000i 0 1.58661 − 1.57565i
2089.7 − 1.00000i 0 −1.00000 1.93205 + 1.12569i 0 3.76241i 1.00000i 0 1.12569 − 1.93205i
2089.8 1.00000i 0 −1.00000 −2.23163 − 0.140865i 0 − 0.546791i − 1.00000i 0 0.140865 − 2.23163i
2089.9 1.00000i 0 −1.00000 −2.04108 + 0.913241i 0 0.776224i − 1.00000i 0 −0.913241 − 2.04108i
2089.10 1.00000i 0 −1.00000 −0.693128 − 2.12593i 0 1.50042i − 1.00000i 0 2.12593 − 0.693128i
2089.11 1.00000i 0 −1.00000 0.183693 + 2.22851i 0 1.21131i − 1.00000i 0 −2.22851 + 0.183693i
2089.12 1.00000i 0 −1.00000 1.27443 + 1.83734i 0 4.80878i − 1.00000i 0 −1.83734 + 1.27443i
2089.13 1.00000i 0 −1.00000 1.57565 − 1.58661i 0 4.01247i − 1.00000i 0 1.58661 + 1.57565i
2089.14 1.00000i 0 −1.00000 1.93205 − 1.12569i 0 − 3.76241i − 1.00000i 0 1.12569 + 1.93205i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2089.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2610.2.e.k yes 14
3.b odd 2 1 2610.2.e.j ✓ 14
5.b even 2 1 inner 2610.2.e.k yes 14
15.d odd 2 1 2610.2.e.j ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2610.2.e.j ✓ 14 3.b odd 2 1
2610.2.e.j ✓ 14 15.d odd 2 1
2610.2.e.k yes 14 1.a even 1 1 trivial
2610.2.e.k yes 14 5.b 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 \(S_{2}^{\mathrm{new}}(2610, [\chi])\):

\( T_{7}^{14} + 58T_{7}^{12} + 1181T_{7}^{10} + 9924T_{7}^{8} + 30884T_{7}^{6} + 39440T_{7}^{4} + 19776T_{7}^{2} + 3136 \) Copy content Toggle raw display
\( T_{11}^{7} - 2T_{11}^{6} - 38T_{11}^{5} + 112T_{11}^{4} + 152T_{11}^{3} - 632T_{11}^{2} + 480T_{11} - 64 \) Copy content Toggle raw display
\( T_{13}^{14} + 84T_{13}^{12} + 2672T_{13}^{10} + 40528T_{13}^{8} + 300160T_{13}^{6} + 982272T_{13}^{4} + 1019904T_{13}^{2} + 16384 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 18 T^{11} + \cdots + 78125 \) Copy content Toggle raw display
$7$ \( T^{14} + 58 T^{12} + \cdots + 3136 \) Copy content Toggle raw display
$11$ \( (T^{7} - 2 T^{6} - 38 T^{5} + \cdots - 64)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + 84 T^{12} + \cdots + 16384 \) Copy content Toggle raw display
$17$ \( T^{14} + 102 T^{12} + \cdots + 102400 \) Copy content Toggle raw display
$19$ \( (T^{7} + 4 T^{6} + \cdots - 2752)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 128 T^{12} + \cdots + 331776 \) Copy content Toggle raw display
$29$ \( (T + 1)^{14} \) Copy content Toggle raw display
$31$ \( (T^{7} - 4 T^{6} + \cdots - 8000)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + 258 T^{12} + \cdots + 3444736 \) Copy content Toggle raw display
$41$ \( (T^{7} - 12 T^{6} + \cdots - 118480)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 1646411776 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 3573170176 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 1073741824 \) Copy content Toggle raw display
$59$ \( (T^{7} + 4 T^{6} + \cdots + 5504)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 14 T^{6} + \cdots + 288000)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 92104966144 \) Copy content Toggle raw display
$71$ \( (T^{7} - 16 T^{6} + \cdots - 89600)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 1822888820736 \) Copy content Toggle raw display
$79$ \( (T^{7} - 306 T^{5} + \cdots + 568064)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 586802176 \) Copy content Toggle raw display
$89$ \( (T^{7} + 12 T^{6} + \cdots - 3003136)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 2120308752384 \) Copy content Toggle raw display
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