Properties

Label 1305.2.c.i
Level $1305$
Weight $2$
Character orbit 1305.c
Analytic conductor $10.420$
Analytic rank $0$
Dimension $10$
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] = [1305,2,Mod(784,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.784");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.7025129046016.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 2x^{8} + 2x^{7} + 23x^{6} - 36x^{5} + 28x^{4} - 2x^{3} + x^{2} - 2x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
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.

\(f(q)\) \(=\) \( q + (\beta_{9} + \beta_{2} + \beta_1) q^{2} + ( - \beta_{7} + \beta_{5} + \beta_{2} + \cdots - 1) q^{4}+ \cdots + ( - 2 \beta_{9} - \beta_{8} + \cdots - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} + \beta_{2} + \beta_1) q^{2} + ( - \beta_{7} + \beta_{5} + \beta_{2} + \cdots - 1) q^{4}+ \cdots + ( - 6 \beta_{9} - 2 \beta_{4} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 16 q^{4} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 16 q^{4} - 2 q^{5} + 10 q^{11} - 16 q^{14} + 28 q^{16} - 4 q^{19} + 6 q^{20} - 2 q^{25} - 24 q^{26} - 10 q^{29} + 4 q^{31} + 12 q^{34} + 10 q^{35} - 30 q^{40} + 38 q^{41} - 44 q^{44} + 60 q^{46} - 18 q^{49} + 20 q^{50} + 10 q^{55} + 32 q^{56} - 44 q^{59} - 16 q^{61} - 16 q^{64} + 26 q^{65} - 60 q^{70} + 36 q^{71} + 4 q^{74} + 104 q^{76} - 32 q^{79} - 24 q^{80} + 30 q^{85} + 4 q^{89} - 72 q^{91} - 4 q^{94} - 36 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 2x^{8} + 2x^{7} + 23x^{6} - 36x^{5} + 28x^{4} - 2x^{3} + x^{2} - 2x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8190 \nu^{9} - 16246 \nu^{8} + 17565 \nu^{7} + 16143 \nu^{6} + 188739 \nu^{5} - 288291 \nu^{4} + \cdots - 18260 ) / 71579 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8852 \nu^{9} + 1845 \nu^{8} - 369 \nu^{7} - 21171 \nu^{6} - 255280 \nu^{5} - 76611 \nu^{4} + \cdots + 17988 ) / 71579 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 9130 \nu^{9} + 10070 \nu^{8} - 2014 \nu^{7} - 35825 \nu^{6} - 226133 \nu^{5} + 139941 \nu^{4} + \cdots + 10305 ) / 71579 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21643 \nu^{9} + 61268 \nu^{8} - 55201 \nu^{7} - 40903 \nu^{6} - 440793 \nu^{5} + 1260561 \nu^{4} + \cdots + 160648 ) / 71579 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 22439 \nu^{9} + 45682 \nu^{8} - 37768 \nu^{7} - 46300 \nu^{6} - 513883 \nu^{5} + 847098 \nu^{4} + \cdots + 33598 ) / 71579 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 30637 \nu^{9} + 70404 \nu^{8} - 71344 \nu^{7} - 59260 \nu^{6} - 668826 \nu^{5} + 1329065 \nu^{4} + \cdots + 62451 ) / 71579 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 40942 \nu^{9} - 72754 \nu^{8} + 71814 \nu^{7} + 83898 \nu^{6} + 977491 \nu^{5} - 1247779 \nu^{4} + \cdots - 80707 ) / 71579 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 57741 \nu^{9} + 83913 \nu^{8} - 59730 \nu^{7} - 155264 \nu^{6} - 1410168 \nu^{5} + 1338660 \nu^{4} + \cdots + 90456 ) / 71579 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{6} - 2\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{7} - \beta_{6} + 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 6\beta_{6} + 5\beta_{5} + 6\beta_{3} + 7\beta_{2} - 7\beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{9} + 6\beta_{8} + 6\beta_{7} + \beta_{5} + 2\beta_{4} + 9\beta_{3} - 35\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -26\beta_{9} + 9\beta_{8} + 34\beta_{6} + 47\beta_{4} + 34\beta_{3} - 46\beta_{2} - 46\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -12\beta_{9} + 34\beta_{8} - 34\beta_{7} + 66\beta_{6} - 12\beta_{5} + 23\beta_{4} - 207\beta_{2} + 23 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -66\beta_{7} + 195\beta_{6} - 141\beta_{5} - 195\beta_{3} - 296\beta_{2} + 296\beta _1 + 256 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 101\beta_{9} - 195\beta_{8} - 195\beta_{7} - 101\beta_{5} - 190\beta_{4} - 451\beta_{3} + 1238\beta _1 + 190 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times\).

\(n\) \(146\) \(262\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
784.1
0.664979 0.664979i
1.75525 1.75525i
0.410556 + 0.410556i
−1.47774 1.47774i
−0.353040 0.353040i
−0.353040 + 0.353040i
−1.47774 + 1.47774i
0.410556 0.410556i
1.75525 + 1.75525i
0.664979 + 0.664979i
2.68032i 0 −5.18413 −0.117379 + 2.23299i 0 3.20454i 8.53449i 0 5.98512 + 0.314615i
784.2 2.45441i 0 −4.02413 −1.55285 1.60893i 0 2.54244i 4.96805i 0 −3.94898 + 3.81133i
784.3 1.88448i 0 −1.55125 2.18424 0.478651i 0 3.97545i 0.845662i 0 −0.902006 4.11614i
784.4 1.10415i 0 0.780857 −2.10939 0.741948i 0 2.02596i 3.07048i 0 −0.819220 + 2.32908i
784.5 0.146109i 0 1.97865 0.595378 2.15535i 0 2.71260i 0.581318i 0 −0.314916 0.0869902i
784.6 0.146109i 0 1.97865 0.595378 + 2.15535i 0 2.71260i 0.581318i 0 −0.314916 + 0.0869902i
784.7 1.10415i 0 0.780857 −2.10939 + 0.741948i 0 2.02596i 3.07048i 0 −0.819220 2.32908i
784.8 1.88448i 0 −1.55125 2.18424 + 0.478651i 0 3.97545i 0.845662i 0 −0.902006 + 4.11614i
784.9 2.45441i 0 −4.02413 −1.55285 + 1.60893i 0 2.54244i 4.96805i 0 −3.94898 3.81133i
784.10 2.68032i 0 −5.18413 −0.117379 2.23299i 0 3.20454i 8.53449i 0 5.98512 0.314615i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 784.10
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 1305.2.c.i 10
3.b odd 2 1 435.2.c.d 10
5.b even 2 1 inner 1305.2.c.i 10
5.c odd 4 1 6525.2.a.bn 5
5.c odd 4 1 6525.2.a.br 5
15.d odd 2 1 435.2.c.d 10
15.e even 4 1 2175.2.a.x 5
15.e even 4 1 2175.2.a.y 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.c.d 10 3.b odd 2 1
435.2.c.d 10 15.d odd 2 1
1305.2.c.i 10 1.a even 1 1 trivial
1305.2.c.i 10 5.b even 2 1 inner
2175.2.a.x 5 15.e even 4 1
2175.2.a.y 5 15.e even 4 1
6525.2.a.bn 5 5.c odd 4 1
6525.2.a.br 5 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1305, [\chi])\):

\( T_{2}^{10} + 18T_{2}^{8} + 111T_{2}^{6} + 266T_{2}^{4} + 193T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{10} + 44T_{7}^{8} + 734T_{7}^{6} + 5824T_{7}^{4} + 22017T_{7}^{2} + 31684 \) Copy content Toggle raw display
\( T_{11}^{5} - 5T_{11}^{4} - 16T_{11}^{3} + 88T_{11}^{2} - 5T_{11} - 191 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 18 T^{8} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{9} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 44 T^{8} + \cdots + 31684 \) Copy content Toggle raw display
$11$ \( (T^{5} - 5 T^{4} + \cdots - 191)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 48 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{10} + 60 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T^{5} + 2 T^{4} + \cdots - 352)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 105 T^{8} + \cdots + 126736 \) Copy content Toggle raw display
$29$ \( (T + 1)^{10} \) Copy content Toggle raw display
$31$ \( (T^{5} - 2 T^{4} + \cdots + 2672)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 201 T^{8} + \cdots + 12475024 \) Copy content Toggle raw display
$41$ \( (T^{5} - 19 T^{4} + \cdots - 656)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 248755984 \) Copy content Toggle raw display
$47$ \( T^{10} + 212 T^{8} + \cdots + 8549776 \) Copy content Toggle raw display
$53$ \( T^{10} + 149 T^{8} + \cdots + 8690704 \) Copy content Toggle raw display
$59$ \( (T^{5} + 22 T^{4} + \cdots - 8744)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 8 T^{4} + \cdots + 200)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 400 T^{8} + \cdots + 4129024 \) Copy content Toggle raw display
$71$ \( (T^{5} - 18 T^{4} + \cdots - 176)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 393 T^{8} + \cdots + 1281424 \) Copy content Toggle raw display
$79$ \( (T^{5} + 16 T^{4} + \cdots - 14488)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 1746905616 \) Copy content Toggle raw display
$89$ \( (T^{5} - 2 T^{4} + \cdots - 165482)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 147962896 \) Copy content Toggle raw display
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