Properties

Label 13.16.a.b
Level $13$
Weight $16$
Character orbit 13.a
Self dual yes
Analytic conductor $18.550$
Analytic rank $0$
Dimension $8$
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] = [13,16,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, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 16 \)
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: \(18.5501556630\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 190031 x^{6} + 1830023 x^{5} + 9448447947 x^{4} - 41019276251 x^{3} + \cdots - 35\!\cdots\!40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 5) q^{2} + ( - \beta_{4} - 6 \beta_1 + 966) q^{3} + (\beta_{2} - 23 \beta_1 + 14767) q^{4} + (\beta_{7} - 17 \beta_{4} + \cdots + 20194) q^{5}+ \cdots + ( - 58 \beta_{7} + 29 \beta_{6} + \cdots + 3088559) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 5) q^{2} + ( - \beta_{4} - 6 \beta_1 + 966) q^{3} + (\beta_{2} - 23 \beta_1 + 14767) q^{4} + (\beta_{7} - 17 \beta_{4} + \cdots + 20194) q^{5}+ \cdots + ( - 5116358386 \beta_{7} + \cdots + 270265408243034) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 39 q^{2} + 7721 q^{3} + 118109 q^{4} + 161733 q^{5} + 2155075 q^{6} + 719199 q^{7} + 8065941 q^{8} + 24692237 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 39 q^{2} + 7721 q^{3} + 118109 q^{4} + 161733 q^{5} + 2155075 q^{6} + 719199 q^{7} + 8065941 q^{8} + 24692237 q^{9} - 79525415 q^{10} + 75027420 q^{11} + 203700857 q^{12} - 501988136 q^{13} + 137285979 q^{14} + 1904607939 q^{15} + 5788167297 q^{16} - 2050915245 q^{17} + 5626249486 q^{18} + 3819057528 q^{19} + 17061442671 q^{20} + 46526595141 q^{21} + 54455336650 q^{22} + 29179323852 q^{23} + 105653721225 q^{24} + 165394806369 q^{25} - 2447192163 q^{26} + 141960763091 q^{27} + 229876282649 q^{28} + 236517959592 q^{29} + 220146114145 q^{30} - 204400904474 q^{31} + 996087331317 q^{32} + 557188054982 q^{33} - 746046220881 q^{34} - 228902911809 q^{35} - 2002180547338 q^{36} - 800886338239 q^{37} - 3083531545758 q^{38} - 484481299757 q^{39} - 4098360527945 q^{40} - 6020078140362 q^{41} - 3128314899541 q^{42} - 2808163767733 q^{43} - 2596146850986 q^{44} - 4921143724224 q^{45} - 4368623463100 q^{46} + 8087628148275 q^{47} - 1194722987691 q^{48} + 2353427994041 q^{49} + 10225408954800 q^{50} + 12865180477813 q^{51} - 7411164594353 q^{52} + 1513535180046 q^{53} - 14374330331543 q^{54} - 7350012609498 q^{55} + 70065508286025 q^{56} + 59477847568230 q^{57} + 82767966893846 q^{58} + 38211220444956 q^{59} + 23149399806403 q^{60} + 45116665409314 q^{61} + 98509076321100 q^{62} + 46413072448548 q^{63} + 16886020732297 q^{64} - 10148505899961 q^{65} - 59696600518478 q^{66} - 28551003552448 q^{67} - 253681283707911 q^{68} - 61911152094332 q^{69} - 431971404492095 q^{70} + 69454351575765 q^{71} - 368242553467386 q^{72} + 84081875465484 q^{73} - 45174753054915 q^{74} - 17697366815988 q^{75} - 583626955980770 q^{76} + 273186043799682 q^{77} - 135227760273775 q^{78} + 133483959013464 q^{79} - 12\!\cdots\!13 q^{80}+ \cdots + 21\!\cdots\!70 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 190031 x^{6} + 1830023 x^{5} + 9448447947 x^{4} - 41019276251 x^{3} + \cdots - 35\!\cdots\!40 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 13\nu - 47510 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 196412982119 \nu^{7} + \cdots - 52\!\cdots\!80 ) / 82\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 254837805504061 \nu^{7} + \cdots + 32\!\cdots\!24 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 118943434869293 \nu^{7} + \cdots + 11\!\cdots\!68 ) / 16\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 41\!\cdots\!87 \nu^{7} + \cdots - 10\!\cdots\!52 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 89\!\cdots\!29 \nu^{7} + \cdots + 14\!\cdots\!20 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 13\beta _1 + 47510 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -10\beta_{7} + 6\beta_{6} - 20\beta_{5} + 372\beta_{4} - 85\beta_{3} - 61\beta_{2} + 90710\beta _1 - 626402 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4032 \beta_{7} - 2176 \beta_{6} + 4856 \beta_{5} + 124176 \beta_{4} + 139 \beta_{3} + \cdots + 4303394100 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 1616904 \beta_{7} + 43576 \beta_{6} - 4541808 \beta_{5} + 40268912 \beta_{4} - 11935786 \beta_{3} + \cdots - 176198220388 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 736331124 \beta_{7} - 433684436 \beta_{6} + 846482568 \beta_{5} + 21944506280 \beta_{4} + \cdots + 432458845229842 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 206718470490 \beta_{7} - 44146729418 \beta_{6} - 684372873876 \beta_{5} + 3904368416948 \beta_{4} + \cdots - 30\!\cdots\!38 \) 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
321.281
262.549
132.292
−17.7136
−30.7395
−66.2217
−252.803
−347.643
−316.281 −3494.95 67265.4 68758.4 1.10538e6 −2.07774e6 −1.09108e7 −2.13426e6 −2.17469e7
1.2 −257.549 2729.54 33563.3 309705. −702989. 3.79395e6 −204823. −6.89851e6 −7.97641e7
1.3 −127.292 1455.66 −16564.8 −180021. −185293. 148892. 6.27966e6 −1.22300e7 2.29152e7
1.4 22.7136 −2447.20 −32252.1 226424. −55584.8 −3.54342e6 −1.47684e6 −8.36010e6 5.14290e6
1.5 35.7395 7402.21 −31490.7 −84298.2 264551. 1.89011e6 −2.29657e6 4.04438e7 −3.01277e6
1.6 71.2217 −5204.44 −27695.5 −315056. −370669. −872980. −4.30631e6 1.27373e7 −2.24388e7
1.7 257.803 4930.69 33694.3 302173. 1.27115e6 −680487. 238799. 9.96284e6 7.79011e7
1.8 352.643 2349.49 91589.2 −165953. 828530. 2.06088e6 2.07429e7 −8.82882e6 −5.85220e7
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 13.16.a.b 8
3.b odd 2 1 117.16.a.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.16.a.b 8 1.a even 1 1 trivial
117.16.a.e 8 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 39 T_{2}^{7} - 189366 T_{2}^{6} + 3864432 T_{2}^{5} + 9422976272 T_{2}^{4} + \cdots - 54\!\cdots\!00 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(13))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots - 54\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots - 15\!\cdots\!92 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 96\!\cdots\!32 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots - 17\!\cdots\!48 \) Copy content Toggle raw display
$13$ \( (T + 62748517)^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots - 21\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 10\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 18\!\cdots\!08 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 13\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots - 19\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots - 24\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 11\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 17\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 12\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots - 82\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 71\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots - 17\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots - 81\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 62\!\cdots\!60 \) Copy content Toggle raw display
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