Properties

Label 13.12.b.a
Level $13$
Weight $12$
Character orbit 13.b
Analytic conductor $9.988$
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] = [13,12,Mod(12,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.12");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 13.b (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: \(9.98846134727\)
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: \( x^{12} + 18433 x^{10} + 121088056 x^{8} + 340607607312 x^{6} + 380893885719552 x^{4} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{4}\cdot 13^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - 41) q^{3} + (\beta_{3} - \beta_{2} - 1024) q^{4} + ( - \beta_{5} + 9 \beta_1) q^{5} + (\beta_{6} + \beta_{5} - 101 \beta_1) q^{6} + (\beta_{9} - \beta_{5} + 146 \beta_1) q^{7} + (\beta_{10} + \beta_{6} + \cdots - 1198 \beta_1) q^{8}+ \cdots + (\beta_{4} - 7 \beta_{3} + \cdots + 54564) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - 41) q^{3} + (\beta_{3} - \beta_{2} - 1024) q^{4} + ( - \beta_{5} + 9 \beta_1) q^{5} + (\beta_{6} + \beta_{5} - 101 \beta_1) q^{6} + (\beta_{9} - \beta_{5} + 146 \beta_1) q^{7} + (\beta_{10} + \beta_{6} + \cdots - 1198 \beta_1) q^{8}+ \cdots + (84701 \beta_{11} + \cdots + 1257184535 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 488 q^{3} - 12290 q^{4} + 654644 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 488 q^{3} - 12290 q^{4} + 654644 q^{9} - 333446 q^{10} + 2740298 q^{12} + 3208868 q^{13} - 5367450 q^{14} + 19025698 q^{16} + 12198768 q^{17} - 111171128 q^{22} + 5810592 q^{23} + 6102388 q^{25} - 64543986 q^{26} - 52613336 q^{27} - 244463112 q^{29} + 426504126 q^{30} - 562027560 q^{35} - 1357546052 q^{36} + 3171817788 q^{38} - 2199109744 q^{39} + 4092185498 q^{40} + 1280452314 q^{42} + 2294519976 q^{43} - 14206061378 q^{48} - 3573617796 q^{49} + 7713246552 q^{51} - 5597650396 q^{52} - 4602062760 q^{53} - 6178744976 q^{55} + 20017912662 q^{56} - 13775649944 q^{61} + 239765256 q^{62} - 3560815378 q^{64} - 7598401512 q^{65} + 37979507040 q^{66} + 40844682210 q^{68} - 25419983328 q^{69} + 19351803414 q^{74} + 68016370832 q^{75} - 80478036048 q^{77} + 89375282178 q^{78} + 18046097296 q^{79} - 132677486692 q^{81} - 255687836096 q^{82} + 94507900752 q^{87} + 239343029120 q^{88} - 190413561204 q^{90} + 104793638664 q^{91} - 135236877012 q^{92} - 78363161402 q^{94} + 145093149648 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 18433 x^{10} + 121088056 x^{8} + 340607607312 x^{6} + 380893885719552 x^{4} + \cdots + 14\!\cdots\!56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1291753 \nu^{10} - 21684945577 \nu^{8} - 126271453345720 \nu^{6} + \cdots - 35\!\cdots\!32 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1291753 \nu^{10} - 21684945577 \nu^{8} - 126271453345720 \nu^{6} + \cdots + 15\!\cdots\!68 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3983497 \nu^{10} - 95154992713 \nu^{8} - 746413169769400 \nu^{6} + \cdots + 76\!\cdots\!72 ) / 17\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 30769879 \nu^{11} - 554264573911 \nu^{9} + \cdots - 26\!\cdots\!76 \nu ) / 93\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6069211 \nu^{11} + 106412048731 \nu^{9} + 654153644047720 \nu^{7} + \cdots + 38\!\cdots\!40 \nu ) / 11\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27274907 \nu^{10} - 446842425563 \nu^{8} + \cdots - 40\!\cdots\!08 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23528917 \nu^{10} + 310068528853 \nu^{8} + \cdots - 16\!\cdots\!52 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1464330623 \nu^{11} - 27874125571007 \nu^{9} + \cdots - 18\!\cdots\!12 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7725581 \nu^{11} + 140300856779 \nu^{9} + 897869353591190 \nu^{7} + \cdots + 93\!\cdots\!64 \nu ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6766909607 \nu^{11} + 126801206210663 \nu^{9} + \cdots + 14\!\cdots\!08 \nu ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} - 3072 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{6} + 7\beta_{5} - 5294\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{8} - 11\beta_{7} + 10\beta_{4} - 6934\beta_{3} + 11156\beta_{2} + 16266815 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 80\beta_{11} - 7917\beta_{10} + 2272\beta_{9} - 16893\beta_{6} - 76251\beta_{5} + 31935826\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 30455\beta_{8} + 136351\beta_{7} - 91618\beta_{4} + 46615034\beta_{3} - 109305728\beta_{2} - 98155809667 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 732944 \beta_{11} + 56425589 \beta_{10} - 27642208 \beta_{9} + 173727941 \beta_{6} + \cdots - 205577282642 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 258936911 \beta_{8} - 1261004855 \beta_{7} + 676435890 \beta_{4} - 320122643770 \beta_{3} + \cdots + 632005344997307 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5411487120 \beta_{11} - 401173363373 \beta_{10} + 245488785504 \beta_{9} - 1518491628413 \beta_{6} + \cdots + 13\!\cdots\!10 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2075349569655 \beta_{8} + 10427221687327 \beta_{7} - 4751487722210 \beta_{4} + \cdots - 42\!\cdots\!19 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 38011901777680 \beta_{11} + \cdots - 95\!\cdots\!30 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1
85.7436i
69.4876i
65.0925i
37.0312i
25.1535i
3.38741i
3.38741i
25.1535i
37.0312i
65.0925i
69.4876i
85.7436i
85.7436i −584.699 −5303.96 6975.15i 50134.2i 41147.2i 279178.i 164726. −598075.
12.2 69.4876i 612.411 −2780.53 159.313i 42555.0i 48715.3i 50901.7i 197900. −11070.3
12.3 65.0925i 15.7424 −2189.03 6676.77i 1024.71i 81596.2i 9179.94i −176899. 434608.
12.4 37.0312i −143.287 676.691 5266.14i 5306.08i 16188.3i 100899.i −156616. −195012.
12.5 25.1535i −638.702 1415.30 9287.93i 16065.6i 25239.3i 87114.1i 230793. 233624.
12.6 3.38741i 494.534 2036.53 9091.88i 1675.19i 45027.4i 13836.0i 67417.3 −30797.9
12.7 3.38741i 494.534 2036.53 9091.88i 1675.19i 45027.4i 13836.0i 67417.3 −30797.9
12.8 25.1535i −638.702 1415.30 9287.93i 16065.6i 25239.3i 87114.1i 230793. 233624.
12.9 37.0312i −143.287 676.691 5266.14i 5306.08i 16188.3i 100899.i −156616. −195012.
12.10 65.0925i 15.7424 −2189.03 6676.77i 1024.71i 81596.2i 9179.94i −176899. 434608.
12.11 69.4876i 612.411 −2780.53 159.313i 42555.0i 48715.3i 50901.7i 197900. −11070.3
12.12 85.7436i −584.699 −5303.96 6975.15i 50134.2i 41147.2i 279178.i 164726. −598075.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.12.b.a 12
3.b odd 2 1 117.12.b.b 12
4.b odd 2 1 208.12.f.b 12
13.b even 2 1 inner 13.12.b.a 12
13.d odd 4 2 169.12.a.e 12
39.d odd 2 1 117.12.b.b 12
52.b odd 2 1 208.12.f.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.12.b.a 12 1.a even 1 1 trivial
13.12.b.a 12 13.b even 2 1 inner
117.12.b.b 12 3.b odd 2 1
117.12.b.b 12 39.d odd 2 1
169.12.a.e 12 13.d odd 4 2
208.12.f.b 12 4.b odd 2 1
208.12.f.b 12 52.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$3$ \( (T^{6} + \cdots - 255121008509808)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 33\!\cdots\!09 \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 20\!\cdots\!24)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 11\!\cdots\!40)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 67\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 14\!\cdots\!68)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 44\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 65\!\cdots\!28)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 68\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 79\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 34\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 36\!\cdots\!44 \) Copy content Toggle raw display
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