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Properties

Label 1305.2.c.l
Level $1305$
Weight $2$
Character orbit 1305.c
Analytic conductor $10.420$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 20x^{10} + 148x^{8} + 502x^{6} + 792x^{4} + 496x^{2} + 45 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + \beta_{7} q^{5} + \beta_{3} q^{7} + (\beta_{10} + \beta_{9} + \cdots - \beta_{3}) q^{8} + (\beta_{8} - \beta_{6} + \beta_{5} + \cdots - 1) q^{10} + (\beta_{6} - \beta_{2} + 1) q^{11}+ \cdots + (\beta_{10} + 3 \beta_{9} + \cdots + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 16 q^{4} - 10 q^{10} + 12 q^{11} - 16 q^{14} + 16 q^{16} + 20 q^{19} - 14 q^{20} + 8 q^{25} + 56 q^{26} + 12 q^{29} - 16 q^{31} - 4 q^{34} + 16 q^{35} + 16 q^{40} + 32 q^{41} - 68 q^{44} + 20 q^{46}+ \cdots + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 20x^{10} + 148x^{8} + 502x^{6} + 792x^{4} + 496x^{2} + 45 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 17\nu^{9} + 97\nu^{7} + 205\nu^{5} + 105\nu^{3} - 59\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{11} + 37\nu^{9} + 242\nu^{7} + 3\nu^{6} + 668\nu^{5} + 30\nu^{4} + 741\nu^{3} + 66\nu^{2} + 224\nu + 15 ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{11} - 37\nu^{9} - 242\nu^{7} + 3\nu^{6} - 668\nu^{5} + 30\nu^{4} - 741\nu^{3} + 66\nu^{2} - 224\nu + 15 ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{10} + 19\nu^{8} + 127\nu^{6} + 349\nu^{4} + 343\nu^{2} + 43 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2 \nu^{11} + 3 \nu^{10} + 37 \nu^{9} + 54 \nu^{8} + 239 \nu^{7} + 339 \nu^{6} + 626 \nu^{5} + 861 \nu^{4} + \cdots + 45 ) / 12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2\nu^{11} + 37\nu^{9} + 242\nu^{7} + 662\nu^{5} + 693\nu^{3} + 164\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5\nu^{11} + 91\nu^{9} + 575\nu^{7} + 1457\nu^{5} + 1251\nu^{3} + 71\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2 \nu^{11} + 3 \nu^{10} - 37 \nu^{9} + 54 \nu^{8} - 239 \nu^{7} + 339 \nu^{6} - 626 \nu^{5} + \cdots + 45 ) / 12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{10} + 18\nu^{8} + 113\nu^{6} + 289\nu^{4} + 266\nu^{2} + 33 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{9} - \beta_{7} - \beta_{3} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{10} - \beta_{7} - 8\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{10} - 8\beta_{9} - \beta_{8} + 8\beta_{7} - \beta_{5} + \beta_{4} + 8\beta_{3} + 22\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{11} + 10\beta_{10} + 10\beta_{7} + 2\beta_{5} + 2\beta_{4} + 58\beta_{2} - 89 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 56\beta_{10} + 54\beta_{9} + 14\beta_{8} - 56\beta_{7} + 12\beta_{5} - 12\beta_{4} - 54\beta_{3} - 137\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 78\beta_{11} - 80\beta_{10} - 80\beta_{7} + 4\beta_{6} - 28\beta_{5} - 28\beta_{4} - 409\beta_{2} + 567 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 385 \beta_{10} - 353 \beta_{9} - 138 \beta_{8} + 385 \beta_{7} - 108 \beta_{5} + 108 \beta_{4} + \cdots + 894 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -561\beta_{11} + 599\beta_{10} + 599\beta_{7} - 72\beta_{6} + 278\beta_{5} + 278\beta_{4} + 2854\beta_{2} - 3719 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2648 \beta_{10} + 2298 \beta_{9} + 1193 \beta_{8} - 2648 \beta_{7} + 877 \beta_{5} - 877 \beta_{4} + \cdots - 5940 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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
− 2.62500i
− 2.51930i
− 1.78841i
− 1.35513i
− 1.27263i
− 0.328889i
0.328889i
1.27263i
1.35513i
1.78841i
2.51930i
2.62500i
− 2.62500i 0 −4.89062 −1.48188 − 1.67452i 0 − 1.33988i 7.58789i 0 −4.39562 + 3.88994i
784.2 − 2.51930i 0 −4.34685 2.08236 + 0.814730i 0 − 4.08323i 5.91241i 0 2.05255 − 5.24608i
784.3 − 1.78841i 0 −1.19839 0.766993 − 2.10041i 0 4.04635i − 1.43360i 0 −3.75638 − 1.37170i
784.4 − 1.35513i 0 0.163621 1.94590 − 1.10158i 0 − 1.26969i − 2.93199i 0 −1.49278 − 2.63695i
784.5 − 1.27263i 0 0.380419 −1.10723 + 1.94269i 0 − 0.255813i − 3.02939i 0 2.47232 + 1.40909i
784.6 − 0.328889i 0 1.89183 −2.20614 + 0.364635i 0 1.86588i − 1.27998i 0 0.119924 + 0.725574i
784.7 0.328889i 0 1.89183 −2.20614 − 0.364635i 0 − 1.86588i 1.27998i 0 0.119924 − 0.725574i
784.8 1.27263i 0 0.380419 −1.10723 − 1.94269i 0 0.255813i 3.02939i 0 2.47232 − 1.40909i
784.9 1.35513i 0 0.163621 1.94590 + 1.10158i 0 1.26969i 2.93199i 0 −1.49278 + 2.63695i
784.10 1.78841i 0 −1.19839 0.766993 + 2.10041i 0 − 4.04635i 1.43360i 0 −3.75638 + 1.37170i
784.11 2.51930i 0 −4.34685 2.08236 − 0.814730i 0 4.08323i − 5.91241i 0 2.05255 + 5.24608i
784.12 2.62500i 0 −4.89062 −1.48188 + 1.67452i 0 1.33988i − 7.58789i 0 −4.39562 − 3.88994i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 784.12
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.l yes 12
3.b odd 2 1 1305.2.c.k ✓ 12
5.b even 2 1 inner 1305.2.c.l yes 12
5.c odd 4 2 6525.2.a.cf 12
15.d odd 2 1 1305.2.c.k ✓ 12
15.e even 4 2 6525.2.a.ce 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1305.2.c.k ✓ 12 3.b odd 2 1
1305.2.c.k ✓ 12 15.d odd 2 1
1305.2.c.l yes 12 1.a even 1 1 trivial
1305.2.c.l yes 12 5.b even 2 1 inner
6525.2.a.ce 12 15.e even 4 2
6525.2.a.cf 12 5.c odd 4 2

Hecke kernels

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

\( T_{2}^{12} + 20T_{2}^{10} + 148T_{2}^{8} + 502T_{2}^{6} + 792T_{2}^{4} + 496T_{2}^{2} + 45 \) Copy content Toggle raw display
\( T_{7}^{12} + 40T_{7}^{10} + 518T_{7}^{8} + 2412T_{7}^{6} + 4517T_{7}^{4} + 3036T_{7}^{2} + 180 \) Copy content Toggle raw display
\( T_{11}^{6} - 6T_{11}^{5} - 16T_{11}^{4} + 130T_{11}^{3} - 119T_{11}^{2} - 24T_{11} + 30 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 20 T^{10} + \cdots + 45 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} + 40 T^{10} + \cdots + 180 \) Copy content Toggle raw display
$11$ \( (T^{6} - 6 T^{5} - 16 T^{4} + \cdots + 30)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 100 T^{10} + \cdots + 2880 \) Copy content Toggle raw display
$17$ \( T^{12} + 144 T^{10} + \cdots + 8000 \) Copy content Toggle raw display
$19$ \( (T^{6} - 10 T^{5} + \cdots + 9832)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 96 T^{10} + \cdots + 2880 \) Copy content Toggle raw display
$29$ \( (T - 1)^{12} \) Copy content Toggle raw display
$31$ \( (T^{6} + 8 T^{5} + \cdots + 33224)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 16231322880 \) Copy content Toggle raw display
$41$ \( (T^{6} - 16 T^{5} + \cdots - 25920)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 588395520 \) Copy content Toggle raw display
$47$ \( T^{12} + 144 T^{10} + \cdots + 8000 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 550830080 \) Copy content Toggle raw display
$59$ \( (T^{6} + 22 T^{5} + \cdots + 204480)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 26 T^{5} + \cdots + 48)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 24224976180 \) Copy content Toggle raw display
$71$ \( (T^{6} - 10 T^{5} + \cdots - 539040)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 260 T^{10} + \cdots + 33592320 \) Copy content Toggle raw display
$79$ \( (T^{6} - 2 T^{5} + \cdots - 17128)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 314265920 \) Copy content Toggle raw display
$89$ \( (T^{6} + 34 T^{5} + \cdots + 540)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 302330880 \) Copy content Toggle raw display
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