Properties

Label 13.12.a.a
Level $13$
Weight $12$
Character orbit 13.a
Self dual yes
Analytic conductor $9.988$
Analytic rank $1$
Dimension $5$
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] = [13,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 12 \)
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: \(9.98846134727\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 6448x^{3} - 12116x^{2} + 9682560x + 66650112 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 8) q^{2} + ( - \beta_{4} - 2 \beta_1 - 99) q^{3} + (2 \beta_{4} - 2 \beta_{3} + \cdots + 595) q^{4}+ \cdots + ( - 161 \beta_{4} + 207 \beta_{3} + \cdots + 34245) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 8) q^{2} + ( - \beta_{4} - 2 \beta_1 - 99) q^{3} + (2 \beta_{4} - 2 \beta_{3} + \cdots + 595) q^{4}+ \cdots + (313410796 \beta_{4} + \cdots - 7363777140) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 41 q^{2} - 496 q^{3} + 2993 q^{4} - 2542 q^{5} + 35441 q^{6} - 36296 q^{7} - 200037 q^{8} + 172645 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 41 q^{2} - 496 q^{3} + 2993 q^{4} - 2542 q^{5} + 35441 q^{6} - 36296 q^{7} - 200037 q^{8} + 172645 q^{9} - 1004055 q^{10} - 1718912 q^{11} - 3670067 q^{12} + 1856465 q^{13} - 3264661 q^{14} - 15356636 q^{15} - 9077455 q^{16} - 6022762 q^{17} - 25785896 q^{18} - 6984968 q^{19} + 38011501 q^{20} + 30001972 q^{21} + 35725110 q^{22} - 26355744 q^{23} + 88388907 q^{24} + 159029879 q^{25} - 15223013 q^{26} + 112641236 q^{27} + 56979871 q^{28} - 81933754 q^{29} + 564471025 q^{30} - 13953380 q^{31} + 399845123 q^{32} - 703422928 q^{33} - 86504907 q^{34} - 754497044 q^{35} + 70511186 q^{36} - 417857846 q^{37} - 903900222 q^{38} - 184161328 q^{39} - 1116423525 q^{40} - 2690154174 q^{41} - 1731320681 q^{42} - 2194005968 q^{43} + 2402296286 q^{44} + 953635766 q^{45} + 4518855072 q^{46} - 4632149016 q^{47} + 5192309977 q^{48} + 2068996185 q^{49} + 2031214610 q^{50} + 4379956420 q^{51} + 1111279949 q^{52} + 3964085286 q^{53} + 12032576291 q^{54} + 3173729472 q^{55} + 10863941385 q^{56} - 15850418664 q^{57} + 16463234370 q^{58} - 6213900336 q^{59} - 4723327387 q^{60} - 13653194690 q^{61} + 7267753516 q^{62} - 29514629404 q^{63} - 17414214535 q^{64} - 943826806 q^{65} + 12784796606 q^{66} - 11630839736 q^{67} - 29781833135 q^{68} - 10491265752 q^{69} + 22288071195 q^{70} - 55420684056 q^{71} - 8164244886 q^{72} - 10807393382 q^{73} - 24555721283 q^{74} - 7282468148 q^{75} + 86244951178 q^{76} - 29595239248 q^{77} + 13158995213 q^{78} - 4898325368 q^{79} - 19551386423 q^{80} + 67432983205 q^{81} + 101980594164 q^{82} - 17839206992 q^{83} + 115847226287 q^{84} + 74975179056 q^{85} - 6756156273 q^{86} + 166439707672 q^{87} - 5163165966 q^{88} + 65706244882 q^{89} - 197230060630 q^{90} - 13476450728 q^{91} - 91088158752 q^{92} + 189811093880 q^{93} + 152425406559 q^{94} + 51462465912 q^{95} - 181515906965 q^{96} - 66619160654 q^{97} - 394227632340 q^{98} - 35714076944 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 6448x^{3} - 12116x^{2} + 9682560x + 66650112 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{4} + 133\nu^{3} - 17726\nu^{2} - 457048\nu - 169824 ) / 24096 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 127\nu^{3} - 6256\nu^{2} - 427348\nu + 5904896 ) / 16064 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{4} + 124\nu^{3} + 11527\nu^{2} - 448642\nu - 22129536 ) / 24096 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{4} - 2\beta_{3} + \beta_{2} + 3\beta _1 + 2579 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 54\beta_{4} + 106\beta_{3} - 21\beta_{2} + 3427\beta _1 + 10481 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5654\beta_{4} - 9910\beta_{3} + 8923\beta_{2} + 10887\beta _1 + 8898241 \) 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
63.7031
52.4714
−7.05285
−44.2939
−63.8277
−71.7031 −710.121 3093.33 10598.3 50917.9 −48491.8 −74953.5 327125. −759930.
1.2 −60.4714 −54.5312 1608.79 −233.583 3297.58 81481.1 26559.7 −174173. 14125.1
1.3 −0.947150 680.292 −2047.10 −10968.3 −644.339 −18559.5 3878.68 285650. 10388.7
1.4 36.2939 −248.328 −730.754 8212.56 −9012.78 −51189.3 −100852. −115480. 298066.
1.5 55.8277 −163.312 1068.74 −10150.9 −9117.33 463.524 −54670.1 −150476. −566704.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 13.12.a.a 5
3.b odd 2 1 117.12.a.b 5
4.b odd 2 1 208.12.a.f 5
13.b even 2 1 169.12.a.b 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.12.a.a 5 1.a even 1 1 trivial
117.12.a.b 5 3.b odd 2 1
169.12.a.b 5 13.b even 2 1
208.12.a.f 5 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 41T_{2}^{4} - 5776T_{2}^{3} - 137132T_{2}^{2} + 8660928T_{2} + 8321280 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(13))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 41 T^{4} + \cdots + 8321280 \) Copy content Toggle raw display
$3$ \( T^{5} + \cdots - 1068353853588 \) Copy content Toggle raw display
$5$ \( T^{5} + \cdots + 22\!\cdots\!50 \) Copy content Toggle raw display
$7$ \( T^{5} + \cdots + 17\!\cdots\!68 \) Copy content Toggle raw display
$11$ \( T^{5} + \cdots + 66\!\cdots\!48 \) Copy content Toggle raw display
$13$ \( (T - 371293)^{5} \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots + 16\!\cdots\!82 \) Copy content Toggle raw display
$19$ \( T^{5} + \cdots + 15\!\cdots\!48 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots + 29\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots + 67\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots - 84\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots + 47\!\cdots\!06 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots - 16\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots - 47\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots - 51\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots - 27\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots + 88\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots - 40\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots - 33\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots - 15\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots - 49\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots - 50\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots + 37\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
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