Properties

Label 13.6.c.a
Level $13$
Weight $6$
Character orbit 13.c
Analytic conductor $2.085$
Analytic rank $0$
Dimension $8$
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(3,13)]
 
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([2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.3");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 13.c (of order \(3\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 70x^{6} - 133x^{5} + 4766x^{4} - 6777x^{3} + 16825x^{2} + 9696x + 9216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1 - 1) q^{2} + ( - \beta_{5} - 2 \beta_{3} + 2) q^{3} + ( - \beta_{7} + \beta_{4} + \cdots + \beta_1) q^{4}+ \cdots + (7 \beta_{7} + 7 \beta_{6} + \cdots + 6 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_1 - 1) q^{2} + ( - \beta_{5} - 2 \beta_{3} + 2) q^{3} + ( - \beta_{7} + \beta_{4} + \cdots + \beta_1) q^{4}+ \cdots + ( - 4978 \beta_{6} + 1708 \beta_{4} + \cdots - 18108) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5 q^{2} + 8 q^{3} - 17 q^{4} - 20 q^{5} + 6 q^{6} + 68 q^{7} + 18 q^{8} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 5 q^{2} + 8 q^{3} - 17 q^{4} - 20 q^{5} + 6 q^{6} + 68 q^{7} + 18 q^{8} - 30 q^{9} + 303 q^{10} - 480 q^{11} - 24 q^{12} - 234 q^{13} - 1324 q^{14} + 992 q^{15} + 351 q^{16} + 60 q^{17} + 1558 q^{18} + 520 q^{19} + 4429 q^{20} - 1804 q^{21} - 3926 q^{22} + 6644 q^{23} - 3984 q^{24} - 13572 q^{25} + 39 q^{26} - 10240 q^{27} + 9040 q^{28} - 1236 q^{29} + 5524 q^{30} + 24704 q^{31} - 7073 q^{32} + 19454 q^{33} - 3122 q^{34} - 10536 q^{35} + 27623 q^{36} - 11996 q^{37} - 76036 q^{38} + 780 q^{39} - 33298 q^{40} + 24428 q^{41} + 8994 q^{42} + 16304 q^{43} + 75440 q^{44} - 33938 q^{45} + 73066 q^{46} - 1536 q^{47} - 20272 q^{48} + 46378 q^{49} - 20690 q^{50} - 149984 q^{51} - 63622 q^{52} - 49940 q^{53} + 42570 q^{54} + 21712 q^{55} + 29404 q^{56} + 148092 q^{57} - 62107 q^{58} + 52688 q^{59} + 15984 q^{60} + 6380 q^{61} + 59476 q^{62} - 60512 q^{63} - 168510 q^{64} - 8294 q^{65} - 24012 q^{66} + 75256 q^{67} + 35661 q^{68} + 2906 q^{69} + 140272 q^{70} - 31300 q^{71} + 112683 q^{72} - 77116 q^{73} - 64135 q^{74} + 9968 q^{75} + 8196 q^{76} - 135356 q^{77} - 110578 q^{78} + 12032 q^{79} - 50095 q^{80} - 9660 q^{81} - 21795 q^{82} + 76752 q^{83} + 7388 q^{84} - 112154 q^{85} + 100140 q^{86} + 54268 q^{87} - 27492 q^{88} - 154 q^{89} + 318450 q^{90} + 185120 q^{91} + 16160 q^{92} + 37488 q^{93} - 93656 q^{94} + 28352 q^{95} - 375616 q^{96} + 137182 q^{97} - 53863 q^{98} - 146928 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 70x^{6} - 133x^{5} + 4766x^{4} - 6777x^{3} + 16825x^{2} + 9696x + 9216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 319677 \nu^{7} + 174777 \nu^{6} + 21464689 \nu^{5} - 9316963 \nu^{4} + 1509169756 \nu^{3} + \cdots + 3235111584 ) / 5480995715 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 33699079 \nu^{7} - 64388071 \nu^{6} + 2342156938 \nu^{5} - 6542587651 \nu^{4} + \cdots + 318334817568 ) / 526175588640 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 814131 \nu^{7} - 737924 \nu^{6} + 54664767 \nu^{5} - 23727789 \nu^{4} + 3798492578 \nu^{3} + \cdots + 192123818377 ) / 5480995715 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 163476431 \nu^{7} + 338880779 \nu^{6} - 12327003362 \nu^{5} + 33519877559 \nu^{4} + \cdots + 1093886512128 ) / 263087794320 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 319677 \nu^{7} + 174777 \nu^{6} + 21464689 \nu^{5} - 9316963 \nu^{4} + 1424846745 \nu^{3} + \cdots + 6608032024 ) / 337292044 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 87076937 \nu^{7} + 166612553 \nu^{6} - 6060637334 \nu^{5} + 17508240653 \nu^{4} + \cdots + 537815172096 ) / 40475045280 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{4} - 35\beta_{3} - \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{6} + 65\beta_{2} + 40 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 69\beta_{7} - 4\beta_{5} + 2279\beta_{3} + 105\beta _1 - 2279 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -32\beta_{7} + 280\beta_{6} - 280\beta_{5} + 32\beta_{4} - 4016\beta_{3} - 4385\beta_{2} - 4385\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 152\beta_{6} - 4633\beta_{4} + 9329\beta_{2} + 153915 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4544\beta_{7} + 18684\beta_{5} + 349528\beta_{3} + 297601\beta _1 - 349528 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
3.95652 + 6.85289i
1.09142 + 1.89039i
−0.329369 0.570484i
−4.21857 7.30677i
3.95652 6.85289i
1.09142 1.89039i
−0.329369 + 0.570484i
−4.21857 + 7.30677i
−4.45652 7.71892i −1.64205 2.84411i −23.7211 + 41.0862i −58.6393 −14.6357 + 25.3497i 61.9973 107.382i 137.637 116.107 201.104i 261.327 + 452.632i
3.2 −1.59142 2.75641i 12.4354 + 21.5388i 10.9348 18.9396i 44.5576 39.5799 68.5544i −27.1222 + 46.9770i −171.458 −187.780 + 325.244i −70.9097 122.819i
3.3 −0.170631 0.295541i −9.31652 16.1367i 15.9418 27.6120i 13.9092 −3.17937 + 5.50683i −18.2258 + 31.5679i −21.8010 −52.0950 + 90.2311i −2.37335 4.11076i
3.4 3.71857 + 6.44074i 2.52313 + 4.37020i −11.6555 + 20.1878i −9.82751 −18.7649 + 32.5017i 17.3506 30.0522i 64.6219 108.768 188.391i −36.5442 63.2965i
9.1 −4.45652 + 7.71892i −1.64205 + 2.84411i −23.7211 41.0862i −58.6393 −14.6357 25.3497i 61.9973 + 107.382i 137.637 116.107 + 201.104i 261.327 452.632i
9.2 −1.59142 + 2.75641i 12.4354 21.5388i 10.9348 + 18.9396i 44.5576 39.5799 + 68.5544i −27.1222 46.9770i −171.458 −187.780 325.244i −70.9097 + 122.819i
9.3 −0.170631 + 0.295541i −9.31652 + 16.1367i 15.9418 + 27.6120i 13.9092 −3.17937 5.50683i −18.2258 31.5679i −21.8010 −52.0950 90.2311i −2.37335 + 4.11076i
9.4 3.71857 6.44074i 2.52313 4.37020i −11.6555 20.1878i −9.82751 −18.7649 32.5017i 17.3506 + 30.0522i 64.6219 108.768 + 188.391i −36.5442 + 63.2965i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.6.c.a 8
3.b odd 2 1 117.6.g.b 8
4.b odd 2 1 208.6.i.b 8
13.c even 3 1 inner 13.6.c.a 8
13.c even 3 1 169.6.a.d 4
13.e even 6 1 169.6.a.c 4
13.f odd 12 2 169.6.b.c 8
39.i odd 6 1 117.6.g.b 8
52.j odd 6 1 208.6.i.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.6.c.a 8 1.a even 1 1 trivial
13.6.c.a 8 13.c even 3 1 inner
117.6.g.b 8 3.b odd 2 1
117.6.g.b 8 39.i odd 6 1
169.6.a.c 4 13.e even 6 1
169.6.a.d 4 13.c even 3 1
169.6.b.c 8 13.f odd 12 2
208.6.i.b 8 4.b odd 2 1
208.6.i.b 8 52.j odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5 T^{7} + \cdots + 5184 \) Copy content Toggle raw display
$3$ \( T^{8} - 8 T^{7} + \cdots + 58982400 \) Copy content Toggle raw display
$5$ \( (T^{4} + 10 T^{3} + \cdots + 357156)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 72383069214976 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 40\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 14\!\cdots\!09 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 18\!\cdots\!61 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 4693845913600)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 28\!\cdots\!09 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 96\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 63\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 24\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 38\!\cdots\!36)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 13\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 69\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 47\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 34\!\cdots\!48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots - 30\!\cdots\!40)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 91\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
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