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Properties

Label 2610.2.e.h
Level $2610$
Weight $2$
Character orbit 2610.e
Analytic conductor $20.841$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

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: \(10\)
Coefficient field: 10.0.16198757453824.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} + 4x^{8} - 6x^{7} + 37x^{6} - 74x^{5} + 57x^{4} + 48x^{3} - 74x^{2} + 20x + 41 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 870)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{4} - \beta_{5} q^{5} + ( - \beta_{4} - \beta_{3}) q^{7} - \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{4} - \beta_{5} q^{5} + ( - \beta_{4} - \beta_{3}) q^{7} - \beta_1 q^{8} + \beta_{3} q^{10} + (\beta_{9} - \beta_{8} + \beta_{5} + \cdots - 2) q^{11}+ \cdots + ( - \beta_{8} + 4 \beta_{7} + \cdots - 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{4} - 16 q^{11} - 4 q^{14} + 10 q^{16} - 16 q^{19} + 4 q^{25} + 8 q^{26} + 10 q^{29} + 24 q^{31} + 14 q^{35} - 20 q^{41} + 16 q^{44} + 4 q^{46} - 46 q^{49} - 16 q^{50} - 12 q^{55} + 4 q^{56} + 12 q^{59} + 28 q^{61} - 10 q^{64} - 8 q^{65} + 28 q^{70} + 16 q^{76} - 72 q^{79} - 16 q^{85} + 12 q^{86} + 12 q^{89} + 48 q^{91} - 12 q^{94} - 48 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{9} + 4x^{8} - 6x^{7} + 37x^{6} - 74x^{5} + 57x^{4} + 48x^{3} - 74x^{2} + 20x + 41 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 255 \nu^{9} - 3138 \nu^{8} + 13587 \nu^{7} - 7540 \nu^{6} + 13785 \nu^{5} - 138010 \nu^{4} + \cdots + 76686 ) / 69023 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 132280 \nu^{9} - 422567 \nu^{8} + 260128 \nu^{7} - 1051551 \nu^{6} + 4932184 \nu^{5} + \cdots + 2636177 ) / 2829943 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 797036 \nu^{9} - 2276478 \nu^{8} + 1441231 \nu^{7} - 8157605 \nu^{6} + 28822957 \nu^{5} + \cdots + 38664599 ) / 14149715 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 809224 \nu^{9} - 4567472 \nu^{8} + 8046549 \nu^{7} - 9655430 \nu^{6} + 40701913 \nu^{5} + \cdots + 56467086 ) / 14149715 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 958862 \nu^{9} + 3522651 \nu^{8} - 3288332 \nu^{7} + 7461230 \nu^{6} - 35263954 \nu^{5} + \cdots - 24022228 ) / 14149715 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 998106 \nu^{9} - 3915093 \nu^{8} + 2753991 \nu^{7} - 3261120 \nu^{6} + 37685807 \nu^{5} + \cdots - 2411661 ) / 14149715 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1164828 \nu^{9} + 5528959 \nu^{8} - 8605843 \nu^{7} + 12152220 \nu^{6} - 51661731 \nu^{5} + \cdots - 51697597 ) / 14149715 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 305980 \nu^{9} + 1086557 \nu^{8} - 822490 \nu^{7} + 2164092 \nu^{6} - 11547382 \nu^{5} + \cdots - 4992611 ) / 2829943 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1577292 \nu^{9} - 5847576 \nu^{8} + 3480147 \nu^{7} - 7073715 \nu^{6} + 61496649 \nu^{5} + \cdots - 51058202 ) / 14149715 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{6} - 2\beta_{3} + 2\beta_{2} - 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{9} + 4\beta_{8} - 5\beta_{7} - 5\beta_{6} - 2\beta_{5} + 2\beta_{4} - 8\beta_{2} - 14\beta _1 + 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{9} - 3 \beta_{8} - 6 \beta_{7} - 3 \beta_{6} + 5 \beta_{5} + 4 \beta_{4} - 7 \beta_{3} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6 \beta_{9} - 24 \beta_{8} - 29 \beta_{7} - 7 \beta_{6} + 50 \beta_{5} - 2 \beta_{4} + 16 \beta_{3} + \cdots + 22 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 12 \beta_{9} + 51 \beta_{8} - 43 \beta_{7} + 2 \beta_{6} + 10 \beta_{5} + 29 \beta_{4} + 14 \beta_{3} + \cdots + 26 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 102 \beta_{9} + 4 \beta_{8} - 199 \beta_{7} + 83 \beta_{6} + 216 \beta_{5} + 76 \beta_{4} + \cdots - 366 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - \beta_{9} + 55 \beta_{8} - 102 \beta_{7} + 70 \beta_{6} + 113 \beta_{5} - \beta_{4} + 136 \beta_{3} + \cdots - 112 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 220 \beta_{9} + 1884 \beta_{8} - 881 \beta_{7} + 959 \beta_{6} - 76 \beta_{5} + 668 \beta_{4} + \cdots - 2046 ) / 4 \) 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
0.810796 − 1.11953i
0.881335 + 0.797777i
2.38193 + 0.664146i
−1.38447 + 1.77624i
−0.689587 − 0.118639i
0.810796 + 1.11953i
0.881335 − 0.797777i
2.38193 − 0.664146i
−1.38447 − 1.77624i
−0.689587 + 0.118639i
− 1.00000i 0 −1.00000 −2.21754 + 0.287221i 0 − 4.87402i 1.00000i 0 0.287221 + 2.21754i
2089.2 − 1.00000i 0 −1.00000 −1.62237 + 1.53881i 0 1.28709i 1.00000i 0 1.53881 + 1.62237i
2089.3 − 1.00000i 0 −1.00000 0.468629 − 2.18641i 0 − 1.19992i 1.00000i 0 −2.18641 − 0.468629i
2089.4 − 1.00000i 0 −1.00000 1.53065 + 1.63006i 0 − 2.29875i 1.00000i 0 1.63006 − 1.53065i
2089.5 − 1.00000i 0 −1.00000 1.84063 − 1.26968i 0 5.08559i 1.00000i 0 −1.26968 − 1.84063i
2089.6 1.00000i 0 −1.00000 −2.21754 − 0.287221i 0 4.87402i − 1.00000i 0 0.287221 − 2.21754i
2089.7 1.00000i 0 −1.00000 −1.62237 − 1.53881i 0 − 1.28709i − 1.00000i 0 1.53881 − 1.62237i
2089.8 1.00000i 0 −1.00000 0.468629 + 2.18641i 0 1.19992i − 1.00000i 0 −2.18641 + 0.468629i
2089.9 1.00000i 0 −1.00000 1.53065 − 1.63006i 0 2.29875i − 1.00000i 0 1.63006 + 1.53065i
2089.10 1.00000i 0 −1.00000 1.84063 + 1.26968i 0 − 5.08559i − 1.00000i 0 −1.26968 + 1.84063i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2089.10
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.h 10
3.b odd 2 1 870.2.e.g ✓ 10
5.b even 2 1 inner 2610.2.e.h 10
15.d odd 2 1 870.2.e.g ✓ 10
15.e even 4 1 4350.2.a.ca 5
15.e even 4 1 4350.2.a.cb 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
870.2.e.g ✓ 10 3.b odd 2 1
870.2.e.g ✓ 10 15.d odd 2 1
2610.2.e.h 10 1.a even 1 1 trivial
2610.2.e.h 10 5.b even 2 1 inner
4350.2.a.ca 5 15.e even 4 1
4350.2.a.cb 5 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2610, [\chi])\):

\( T_{7}^{10} + 58T_{7}^{8} + 1049T_{7}^{6} + 6092T_{7}^{4} + 12144T_{7}^{2} + 7744 \) Copy content Toggle raw display
\( T_{11}^{5} + 8T_{11}^{4} - 12T_{11}^{3} - 160T_{11}^{2} + 704 \) Copy content Toggle raw display
\( T_{13}^{10} + 96T_{13}^{8} + 3008T_{13}^{6} + 35072T_{13}^{4} + 163840T_{13}^{2} + 262144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 2 T^{8} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 58 T^{8} + \cdots + 7744 \) Copy content Toggle raw display
$11$ \( (T^{5} + 8 T^{4} + \cdots + 704)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 96 T^{8} + \cdots + 262144 \) Copy content Toggle raw display
$17$ \( T^{10} + 54 T^{8} + \cdots + 23104 \) Copy content Toggle raw display
$19$ \( (T^{5} + 8 T^{4} - 15 T^{3} + \cdots - 32)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 76 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$29$ \( (T - 1)^{10} \) Copy content Toggle raw display
$31$ \( (T^{5} - 12 T^{4} + \cdots - 1408)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 174 T^{8} + \cdots + 1000000 \) Copy content Toggle raw display
$41$ \( (T^{5} + 10 T^{4} + \cdots - 56)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 234 T^{8} + \cdots + 51380224 \) Copy content Toggle raw display
$47$ \( T^{10} + 242 T^{8} + \cdots + 7573504 \) Copy content Toggle raw display
$53$ \( T^{10} + 304 T^{8} + \cdots + 4194304 \) Copy content Toggle raw display
$59$ \( (T^{5} - 6 T^{4} + \cdots - 2752)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 14 T^{4} + \cdots - 512)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 1714622464 \) Copy content Toggle raw display
$71$ \( (T^{5} - 192 T^{3} + \cdots - 28672)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 356 T^{8} + \cdots + 30647296 \) Copy content Toggle raw display
$79$ \( (T^{5} + 36 T^{4} + \cdots - 13376)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 1291396096 \) Copy content Toggle raw display
$89$ \( (T^{5} - 6 T^{4} + \cdots - 17152)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 18618056704 \) Copy content Toggle raw display
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