Properties

Label 13.12.a.b
Level $13$
Weight $12$
Character orbit 13.a
Self dual yes
Analytic conductor $9.988$
Analytic rank $0$
Dimension $6$
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: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 13546x^{4} + 130998x^{3} + 49403509x^{2} - 776207317x - 22123683244 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 9) q^{2} + (\beta_{4} - 2 \beta_1 + 80) q^{3} + (3 \beta_{4} - 5 \beta_{3} + \cdots + 2550) q^{4}+ \cdots + ( - 6 \beta_{5} + 283 \beta_{4} + \cdots + 146078) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 9) q^{2} + (\beta_{4} - 2 \beta_1 + 80) q^{3} + (3 \beta_{4} - 5 \beta_{3} + \cdots + 2550) q^{4}+ \cdots + (3379140 \beta_{5} + \cdots - 39305917264) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 55 q^{2} + 476 q^{3} + 15309 q^{4} + 3312 q^{5} - 57797 q^{6} - 4176 q^{7} + 158493 q^{8} + 876218 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 55 q^{2} + 476 q^{3} + 15309 q^{4} + 3312 q^{5} - 57797 q^{6} - 4176 q^{7} + 158493 q^{8} + 876218 q^{9} + 997497 q^{10} + 275060 q^{11} + 3949049 q^{12} - 2227758 q^{13} + 6462587 q^{14} + 5951652 q^{15} + 25038945 q^{16} + 18470848 q^{17} + 1544758 q^{18} + 2382612 q^{19} + 37821799 q^{20} - 67640772 q^{21} - 52649718 q^{22} + 25001944 q^{23} - 243039615 q^{24} - 14063202 q^{25} - 20421115 q^{26} + 77250908 q^{27} - 340836927 q^{28} - 142876028 q^{29} - 838796927 q^{30} - 158397468 q^{31} + 739784589 q^{32} - 115057792 q^{33} - 668802009 q^{34} + 1377003692 q^{35} + 3099344006 q^{36} + 47994456 q^{37} + 2673019714 q^{38} - 176735468 q^{39} + 242886231 q^{40} + 112037548 q^{41} - 5282633557 q^{42} + 1399191924 q^{43} - 1571975050 q^{44} + 7736061780 q^{45} - 2701412412 q^{46} - 3383597640 q^{47} + 1090782789 q^{48} + 7189538970 q^{49} - 12848613144 q^{50} + 8959562860 q^{51} - 5684124537 q^{52} + 546961604 q^{53} - 38372021519 q^{54} - 7803526248 q^{55} - 6807872407 q^{56} + 918537576 q^{57} + 5714690406 q^{58} + 10067834260 q^{59} + 2453022955 q^{60} + 15731821572 q^{61} - 7829475572 q^{62} + 29876175732 q^{63} + 2237284569 q^{64} - 1229722416 q^{65} + 12031833058 q^{66} + 50546073444 q^{67} + 15412804265 q^{68} + 10879166680 q^{69} + 2924449065 q^{70} - 2646136112 q^{71} - 8720745402 q^{72} + 4198695060 q^{73} + 5050454541 q^{74} - 7695720336 q^{75} + 59928748062 q^{76} + 9015828840 q^{77} + 21459621521 q^{78} - 92124930312 q^{79} + 89421404931 q^{80} + 67208776622 q^{81} - 150798850248 q^{82} + 20440296092 q^{83} - 419349915667 q^{84} - 124095891228 q^{85} + 116436457677 q^{86} - 161197597808 q^{87} - 158617438842 q^{88} + 20540234076 q^{89} - 279059693450 q^{90} + 1550519568 q^{91} + 446253814012 q^{92} + 142195723000 q^{93} + 92592053391 q^{94} + 82521342544 q^{95} - 2651637759 q^{96} - 203942467020 q^{97} + 711378915442 q^{98} - 235408311580 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 13546x^{4} + 130998x^{3} + 49403509x^{2} - 776207317x - 22123683244 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -1925\nu^{5} - 86014\nu^{4} + 15916032\nu^{3} + 462431442\nu^{2} - 26906396267\nu - 114096339268 ) / 502193280 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4549 \nu^{5} + 297830 \nu^{4} - 39656088 \nu^{3} - 2035001754 \nu^{2} + 79116342019 \nu + 1739047527524 ) / 1004386560 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5015 \nu^{5} + 381698 \nu^{4} - 44872104 \nu^{3} - 2440298814 \nu^{2} + 100337716769 \nu + 1234012729676 ) / 1004386560 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1745\nu^{5} + 170366\nu^{4} - 12096168\nu^{3} - 1100201058\nu^{2} + 15895857863\nu + 715844708372 ) / 251096640 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{4} - 5\beta_{3} - 2\beta_{2} - 13\beta _1 + 4517 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 58\beta_{5} - 166\beta_{4} + 50\beta_{3} - 52\beta_{2} + 6187\beta _1 - 59786 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 1254\beta_{5} + 26301\beta_{4} - 38115\beta_{3} - 8502\beta_{2} - 160011\beta _1 + 28173671 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 423516\beta_{5} - 1827024\beta_{4} + 915360\beta_{3} - 791376\beta_{2} + 41203985\beta _1 - 727365996 \) 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
−89.6176
−79.8045
−15.2405
36.5093
73.0961
76.0571
−80.6176 827.494 4451.20 7679.45 −66710.6 −6800.32 −193740. 507599. −619099.
1.2 −70.8045 −19.0788 2965.27 −11885.7 1350.86 −62220.3 −64947.1 −176783. 841561.
1.3 −6.24050 −573.204 −2009.06 −3613.82 3577.08 14401.4 25318.0 151416. 22552.1
1.4 45.5093 469.277 23.0963 4406.29 21356.5 63013.3 −92151.9 43074.3 200527.
1.5 82.0961 −696.954 4691.77 6795.70 −57217.2 67467.4 217044. 308598. 557901.
1.6 85.0571 468.466 5186.72 −69.9037 39846.4 −80037.3 266970. 42313.3 −5945.81
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.b 6
3.b odd 2 1 117.12.a.d 6
4.b odd 2 1 208.12.a.h 6
13.b even 2 1 169.12.a.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.12.a.b 6 1.a even 1 1 trivial
117.12.a.d 6 3.b odd 2 1
169.12.a.c 6 13.b even 2 1
208.12.a.h 6 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 55T_{2}^{5} - 12286T_{2}^{4} + 603264T_{2}^{3} + 39388912T_{2}^{2} - 1594524928T_{2} - 11319915520 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(13))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + \cdots - 11319915520 \) Copy content Toggle raw display
$3$ \( T^{6} + \cdots - 13\!\cdots\!12 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots - 69\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots - 20\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots - 18\!\cdots\!92 \) Copy content Toggle raw display
$13$ \( (T + 371293)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 14\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 20\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 21\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 82\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 54\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots - 36\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 89\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 38\!\cdots\!60 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 46\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 24\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 28\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 32\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 12\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 32\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 73\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 25\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 50\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
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