Properties

Label 13.6.e.a
Level $13$
Weight $6$
Character orbit 13.e
Analytic conductor $2.085$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [13,6,Mod(4,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.4");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 13.e (of order \(6\), degree \(2\), 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: \(2.08498965757\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 335 x^{7} + 4643 x^{6} - 24276 x^{5} + 118427 x^{4} - 639551 x^{3} + 3066180 x^{2} + \cdots + 4777981 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 13^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{6} - \beta_{5} - 2 \beta_{2} - 2) q^{3} + ( - \beta_{8} - \beta_{4} + \cdots + \beta_1) q^{4}+ \cdots + (2 \beta_{9} - 3 \beta_{8} + \cdots + 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{6} - \beta_{5} - 2 \beta_{2} - 2) q^{3} + ( - \beta_{8} - \beta_{4} + \cdots + \beta_1) q^{4}+ \cdots + (532 \beta_{9} + 384 \beta_{8} + \cdots + 37932) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{2} - 10 q^{3} + 63 q^{4} + 168 q^{6} - 276 q^{7} - 215 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{2} - 10 q^{3} + 63 q^{4} + 168 q^{6} - 276 q^{7} - 215 q^{9} + 115 q^{10} - 240 q^{11} - 164 q^{12} - 2015 q^{13} + 3444 q^{14} - 330 q^{15} - 2137 q^{16} + 1851 q^{17} + 4626 q^{19} + 2625 q^{20} - 962 q^{22} - 1374 q^{23} - 6018 q^{24} - 8784 q^{25} - 4017 q^{26} + 8888 q^{27} - 28668 q^{28} - 4071 q^{29} + 20358 q^{30} + 56637 q^{32} + 14142 q^{33} - 17160 q^{35} - 35869 q^{36} + 13491 q^{37} - 26184 q^{38} - 58526 q^{39} + 63074 q^{40} - 23997 q^{41} - 28626 q^{42} + 38120 q^{43} + 130107 q^{45} + 17160 q^{46} - 45442 q^{48} - 15485 q^{49} - 80376 q^{50} - 82860 q^{51} - 128934 q^{52} + 114270 q^{53} - 62070 q^{54} - 68120 q^{55} + 146832 q^{56} + 254625 q^{58} + 18060 q^{59} + 12071 q^{61} - 51450 q^{62} - 40704 q^{63} - 113750 q^{64} - 84747 q^{65} + 166932 q^{66} - 141618 q^{67} + 32763 q^{68} + 104046 q^{69} - 13854 q^{71} + 141825 q^{72} - 87255 q^{74} - 33716 q^{75} - 3822 q^{76} - 45396 q^{77} - 71370 q^{78} + 392 q^{79} - 151365 q^{80} - 79241 q^{81} - 153167 q^{82} + 157380 q^{84} + 76287 q^{85} - 105234 q^{87} - 62608 q^{88} + 234522 q^{89} + 309066 q^{90} + 175968 q^{91} - 207444 q^{92} + 347328 q^{93} + 116866 q^{94} + 42390 q^{95} - 376278 q^{97} - 488517 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 335 x^{7} + 4643 x^{6} - 24276 x^{5} + 118427 x^{4} - 639551 x^{3} + 3066180 x^{2} + \cdots + 4777981 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 8243 \nu^{9} - 1286066 \nu^{8} - 6851006 \nu^{7} - 43031789 \nu^{6} + 179250708 \nu^{5} + \cdots - 1249481914135 ) / 26816081760 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 45921251 \nu^{9} + 16741190 \nu^{8} - 20496950 \nu^{7} - 15284358695 \nu^{6} + \cdots - 138531141390421 ) / 18181303433280 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 35188871 \nu^{9} - 15341618 \nu^{8} - 258160960 \nu^{7} + 10449106397 \nu^{6} + \cdots + 140653480542835 ) / 9090651716640 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1432027 \nu^{9} - 65303 \nu^{8} - 10349269 \nu^{7} - 525927595 \nu^{6} + 6190585932 \nu^{5} + \cdots - 3587048104367 ) / 349640450640 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 121093 \nu^{9} - 597086 \nu^{8} - 330266 \nu^{7} + 47804981 \nu^{6} - 286351572 \nu^{5} + \cdots - 262658326385 ) / 26816081760 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 144710997 \nu^{9} - 110541838 \nu^{8} - 550998642 \nu^{7} + 46493805901 \nu^{6} + \cdots + 463067308196519 ) / 6060434477760 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 412974379 \nu^{9} - 45971365 \nu^{8} - 132156395 \nu^{7} + 134603374420 \nu^{6} + \cdots + 15\!\cdots\!84 ) / 4545325858320 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 987328907 \nu^{9} - 930389966 \nu^{8} - 1659533002 \nu^{7} + 327087926663 \nu^{6} + \cdots + 24\!\cdots\!13 ) / 9090651716640 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 549493069 \nu^{9} - 716872705 \nu^{8} - 1658239475 \nu^{7} + 183208831690 \nu^{6} + \cdots + 10\!\cdots\!74 ) / 4545325858320 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} + 2\beta_{7} - 13\beta_{6} + 24\beta_{4} + 4\beta_{3} + 7\beta_{2} - \beta_1 ) / 52 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 22 \beta_{9} + 19 \beta_{8} - 25 \beta_{7} + 65 \beta_{6} + 13 \beta_{5} + 148 \beta_{4} + \cdots - 443 ) / 52 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 84 \beta_{9} + 51 \beta_{8} + 43 \beta_{7} - 325 \beta_{6} - 403 \beta_{5} - 428 \beta_{4} + \cdots + 5067 ) / 26 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 77 \beta_{9} - \beta_{8} + 70 \beta_{7} + 73 \beta_{6} + 464 \beta_{5} + 632 \beta_{4} + 28 \beta_{3} + \cdots - 7472 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1252 \beta_{9} - 949 \beta_{8} - 15115 \beta_{7} + 43329 \beta_{6} - 27157 \beta_{5} + 5884 \beta_{4} + \cdots + 337179 ) / 52 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 29487 \beta_{9} + 4392 \beta_{8} + 37821 \beta_{7} - 147888 \beta_{6} - 42471 \beta_{5} - 169500 \beta_{4} + \cdots + 685330 ) / 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1454879 \beta_{9} - 212507 \beta_{8} - 697402 \beta_{7} + 4891991 \beta_{6} + 5155020 \beta_{5} + \cdots - 67276164 ) / 52 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 741998 \beta_{9} + 29635 \beta_{8} - 276229 \beta_{7} - 1123735 \beta_{6} - 4011983 \beta_{5} + \cdots + 51711145 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3006318 \beta_{9} + 3653655 \beta_{8} + 53355385 \beta_{7} - 117183313 \beta_{6} + 133531931 \beta_{5} + \cdots - 1754983455 ) / 26 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
4.57535 2.73312i
1.70270 0.292389i
3.17523 + 4.03889i
−6.79550 6.35338i
−2.15779 + 4.47398i
4.57535 + 2.73312i
1.70270 + 0.292389i
3.17523 4.03889i
−6.79550 + 6.35338i
−2.15779 4.47398i
−9.22999 5.32893i −1.94044 + 3.36094i 40.7951 + 70.6592i 21.5719i 35.8205 20.6810i −174.074 + 100.502i 528.526i 113.969 + 197.401i 114.955 199.108i
4.2 −2.80727 1.62078i 4.15714 7.20038i −10.7462 18.6129i 53.0312i −23.3404 + 13.4756i 40.6857 23.4899i 173.398i 86.9363 + 150.578i −85.9518 + 148.873i
4.3 −1.26507 0.730391i −13.6498 + 23.6421i −14.9331 25.8648i 79.0452i 34.5360 19.9394i 53.4173 30.8405i 90.3729i −251.133 434.975i 57.7339 99.9981i
4.4 4.69105 + 2.70838i 11.2615 19.5056i −1.32934 2.30248i 88.6284i 105.657 61.0011i −115.380 + 66.6149i 187.738i −132.145 228.881i −240.040 + 415.761i
4.5 7.11127 + 4.10570i −4.82846 + 8.36314i 17.7135 + 30.6806i 51.3439i −68.6730 + 39.6484i 57.3518 33.1121i 28.1399i 74.8719 + 129.682i 210.802 365.120i
10.1 −9.22999 + 5.32893i −1.94044 3.36094i 40.7951 70.6592i 21.5719i 35.8205 + 20.6810i −174.074 100.502i 528.526i 113.969 197.401i 114.955 + 199.108i
10.2 −2.80727 + 1.62078i 4.15714 + 7.20038i −10.7462 + 18.6129i 53.0312i −23.3404 13.4756i 40.6857 + 23.4899i 173.398i 86.9363 150.578i −85.9518 148.873i
10.3 −1.26507 + 0.730391i −13.6498 23.6421i −14.9331 + 25.8648i 79.0452i 34.5360 + 19.9394i 53.4173 + 30.8405i 90.3729i −251.133 + 434.975i 57.7339 + 99.9981i
10.4 4.69105 2.70838i 11.2615 + 19.5056i −1.32934 + 2.30248i 88.6284i 105.657 + 61.0011i −115.380 66.6149i 187.738i −132.145 + 228.881i −240.040 415.761i
10.5 7.11127 4.10570i −4.82846 8.36314i 17.7135 30.6806i 51.3439i −68.6730 39.6484i 57.3518 + 33.1121i 28.1399i 74.8719 129.682i 210.802 + 365.120i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.6.e.a 10
3.b odd 2 1 117.6.q.c 10
4.b odd 2 1 208.6.w.b 10
13.c even 3 1 169.6.b.d 10
13.e even 6 1 inner 13.6.e.a 10
13.e even 6 1 169.6.b.d 10
13.f odd 12 2 169.6.a.f 10
39.h odd 6 1 117.6.q.c 10
52.i odd 6 1 208.6.w.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.6.e.a 10 1.a even 1 1 trivial
13.6.e.a 10 13.e even 6 1 inner
117.6.q.c 10 3.b odd 2 1
117.6.q.c 10 39.h odd 6 1
169.6.a.f 10 13.f odd 12 2
169.6.b.d 10 13.c even 3 1
169.6.b.d 10 13.e even 6 1
208.6.w.b 10 4.b odd 2 1
208.6.w.b 10 52.i odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 3 T^{9} + \cdots + 5038848 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 36707494464 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 70\!\cdots\!93 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 13\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 15\!\cdots\!12 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 71\!\cdots\!49 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 11\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 66\!\cdots\!75 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 10\!\cdots\!27 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 17\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots + 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 35\!\cdots\!92 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 49\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 98\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 78\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 66\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 78\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 74\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 10\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 69\!\cdots\!12 \) Copy content Toggle raw display
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