Properties

Label 13.10.a.a
Level $13$
Weight $10$
Character orbit 13.a
Self dual yes
Analytic conductor $6.695$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [13,10,Mod(1,13)]
 
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([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 13.a (trivial)

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: yes
Analytic conductor: \(6.69546587013\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 1602x^{2} + 1544x + 342272 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 8) q^{2} + ( - \beta_{3} + 2 \beta_1 - 41) q^{3} + (6 \beta_{3} + \beta_{2} + 15 \beta_1 + 352) q^{4} + (\beta_{3} - 9 \beta_{2} + 18 \beta_1 + 113) q^{5} + ( - 12 \beta_{3} + 14 \beta_{2} + \cdots - 1152) q^{6}+ \cdots + (125 \beta_{3} - 33 \beta_{2} + \cdots - 7400) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 8) q^{2} + ( - \beta_{3} + 2 \beta_1 - 41) q^{3} + (6 \beta_{3} + \beta_{2} + 15 \beta_1 + 352) q^{4} + (\beta_{3} - 9 \beta_{2} + 18 \beta_1 + 113) q^{5} + ( - 12 \beta_{3} + 14 \beta_{2} + \cdots - 1152) q^{6}+ \cdots + ( - 3141970 \beta_{3} + \cdots + 536706688) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 33 q^{2} - 163 q^{3} + 1429 q^{4} + 471 q^{5} - 4529 q^{6} - 11241 q^{7} - 45543 q^{8} - 29953 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 33 q^{2} - 163 q^{3} + 1429 q^{4} + 471 q^{5} - 4529 q^{6} - 11241 q^{7} - 45543 q^{8} - 29953 q^{9} - 67831 q^{10} - 40140 q^{11} - 155479 q^{12} - 114244 q^{13} - 277653 q^{14} + 83307 q^{15} + 726609 q^{16} + 78717 q^{17} + 1691026 q^{18} + 209664 q^{19} + 870843 q^{20} + 1138431 q^{21} + 1364090 q^{22} - 4257444 q^{23} + 3561573 q^{24} - 2900157 q^{25} + 942513 q^{26} - 2077801 q^{27} + 4035181 q^{28} - 1647936 q^{29} - 744143 q^{30} - 11366002 q^{31} - 29458959 q^{32} - 14413222 q^{33} + 26257659 q^{34} - 13789797 q^{35} - 11587714 q^{36} + 4636891 q^{37} + 25172466 q^{38} + 4655443 q^{39} + 22536791 q^{40} + 13859538 q^{41} + 75564923 q^{42} - 33368081 q^{43} + 66489222 q^{44} - 17423928 q^{45} + 71369332 q^{46} - 3943005 q^{47} - 620787 q^{48} + 23294923 q^{49} - 4217748 q^{50} - 19664471 q^{51} - 40813669 q^{52} - 171019326 q^{53} - 64946915 q^{54} - 121160538 q^{55} - 281552967 q^{56} - 47829030 q^{57} + 79964734 q^{58} - 63389388 q^{59} + 37708135 q^{60} + 77050190 q^{61} - 95878740 q^{62} - 155695476 q^{63} + 768962465 q^{64} - 13452231 q^{65} - 42396374 q^{66} - 41174072 q^{67} - 717615423 q^{68} + 546642556 q^{69} + 409056389 q^{70} + 252460989 q^{71} + 562579254 q^{72} + 594415068 q^{73} - 957058539 q^{74} + 533318748 q^{75} - 326897170 q^{76} + 561950454 q^{77} + 129352769 q^{78} + 115998984 q^{79} - 509107233 q^{80} + 437803700 q^{81} - 875148240 q^{82} - 79577862 q^{83} + 108899441 q^{84} + 549463469 q^{85} - 589924887 q^{86} - 1087526510 q^{87} - 2327564370 q^{88} - 1152240276 q^{89} + 877550038 q^{90} + 321054201 q^{91} - 4213481460 q^{92} + 1618266556 q^{93} + 1859909503 q^{94} - 1273705170 q^{95} + 3171454029 q^{96} + 1049098084 q^{97} + 420532254 q^{98} + 2132181050 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 1602x^{2} + 1544x + 342272 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{3} + 17\nu^{2} - 3586\nu - 12856 ) / 332 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 105\nu^{2} + 1306\nu - 84248 ) / 664 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 6\beta_{3} + \beta_{2} - \beta _1 + 800 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -34\beta_{3} + 105\beta_{2} + 1201\beta _1 - 248 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
36.6235
16.5360
−15.3567
−36.8028
−44.6235 −51.0278 1479.26 151.187 2277.04 5436.58 −43162.5 −17079.2 −6746.48
1.2 −24.5360 49.9972 90.0171 1814.98 −1226.73 −8707.31 10353.8 −17183.3 −44532.3
1.3 7.35673 42.6243 −457.879 −1236.25 313.575 892.010 −7135.13 −17866.2 −9094.74
1.4 28.8028 −204.594 317.603 −258.914 −5892.88 −8862.28 −5599.18 22175.6 −7457.46
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.10.a.a 4
3.b odd 2 1 117.10.a.c 4
4.b odd 2 1 208.10.a.g 4
5.b even 2 1 325.10.a.a 4
13.b even 2 1 169.10.a.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.10.a.a 4 1.a even 1 1 trivial
117.10.a.c 4 3.b odd 2 1
169.10.a.a 4 13.b even 2 1
208.10.a.g 4 4.b odd 2 1
325.10.a.a 4 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 33T_{2}^{3} - 1194T_{2}^{2} - 24936T_{2} + 232000 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(13))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 33 T^{3} + \cdots + 232000 \) Copy content Toggle raw display
$3$ \( T^{4} + 163 T^{3} + \cdots + 22248576 \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 87830562190 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 374218195104754 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 27\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T + 28561)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 25\!\cdots\!18 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 50\!\cdots\!08 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots - 17\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 37\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 58\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 61\!\cdots\!62 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 15\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 26\!\cdots\!40 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 10\!\cdots\!50 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 27\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 26\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 60\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 45\!\cdots\!54 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 47\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 12\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 34\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 21\!\cdots\!60 \) Copy content Toggle raw display
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