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Properties

Label 13.8.e.a
Level $13$
Weight $8$
Character orbit 13.e
Analytic conductor $4.061$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [13,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.4");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(4.06100533129\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 1279 x^{12} + 629380 x^{10} + 148562016 x^{8} + 16872573312 x^{6} + 790180980480 x^{4} + \cdots + 4669637050368 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{3}\cdot 13^{4} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{6} - \beta_{4} + 4 \beta_{3} + 4) q^{3} + (\beta_{7} + \beta_{6} - 55 \beta_{3} + \cdots + \beta_1) q^{4}+ \cdots + (2 \beta_{11} - \beta_{10} + \cdots + 17 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{6} - \beta_{4} + 4 \beta_{3} + 4) q^{3} + (\beta_{7} + \beta_{6} - 55 \beta_{3} + \cdots + \beta_1) q^{4}+ \cdots + (3405 \beta_{13} + 3405 \beta_{12} + \cdots + 3596288) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 3 q^{2} + 26 q^{3} + 383 q^{4} - 264 q^{6} - 2772 q^{7} - 3491 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 3 q^{2} + 26 q^{3} + 383 q^{4} - 264 q^{6} - 2772 q^{7} - 3491 q^{9} - 509 q^{10} + 6516 q^{11} + 38380 q^{12} + 5109 q^{13} - 47916 q^{14} - 19266 q^{15} - 633 q^{16} - 38403 q^{17} + 43254 q^{19} + 89409 q^{20} - 125882 q^{22} - 68550 q^{23} + 224454 q^{24} + 39380 q^{25} + 361959 q^{26} - 432400 q^{27} - 234780 q^{28} + 221583 q^{29} - 157890 q^{30} - 878067 q^{32} + 219756 q^{33} + 659616 q^{35} + 635675 q^{36} - 565347 q^{37} + 953640 q^{38} + 30082 q^{39} - 77182 q^{40} + 1716057 q^{41} + 206406 q^{42} - 882692 q^{43} - 4616853 q^{45} - 3267480 q^{46} + 1133630 q^{48} + 2653979 q^{49} + 145128 q^{50} - 38964 q^{51} + 4962178 q^{52} - 1791210 q^{53} + 9597234 q^{54} + 2453536 q^{55} - 7457952 q^{56} - 14978127 q^{58} - 1189032 q^{59} + 3858525 q^{61} + 1904142 q^{62} + 7718520 q^{63} + 14570922 q^{64} + 10424661 q^{65} - 31956396 q^{66} + 4674330 q^{67} + 9569619 q^{68} - 13743840 q^{69} - 25203126 q^{71} - 6851607 q^{72} + 8640105 q^{74} + 10316152 q^{75} + 12934506 q^{76} + 19599336 q^{77} + 50085750 q^{78} - 30688440 q^{79} + 29127963 q^{80} + 13886221 q^{81} - 37359719 q^{82} - 96398508 q^{84} - 22436313 q^{85} + 12769686 q^{87} + 27045536 q^{88} + 2123160 q^{89} + 41193546 q^{90} + 37205532 q^{91} - 47685588 q^{92} + 36638964 q^{93} + 19138810 q^{94} - 43420386 q^{95} - 68556288 q^{97} + 16173003 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 1279 x^{12} + 629380 x^{10} + 148562016 x^{8} + 16872573312 x^{6} + 790180980480 x^{4} + \cdots + 4669637050368 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1054513 \nu^{12} - 1163482171 \nu^{10} - 463409859688 \nu^{8} - 80046208978752 \nu^{6} + \cdots + 44\!\cdots\!68 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 124535171 \nu^{13} - 172984934657 \nu^{11} - 93500560218296 \nu^{9} + \cdots - 10\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 38514673 \nu^{12} + 38644774891 \nu^{10} + 13645656502648 \nu^{8} + \cdots + 46\!\cdots\!72 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 255090683 \nu^{12} - 256426706561 \nu^{10} - 90782111635808 \nu^{8} + \cdots + 37\!\cdots\!88 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13570841413 \nu^{13} - 180749360389 \nu^{12} + 12086872800271 \nu^{11} + \cdots - 21\!\cdots\!96 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 90777752893 \nu^{13} + 736701892504 \nu^{12} - 95124348537631 \nu^{11} + \cdots - 10\!\cdots\!44 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 117312010813 \nu^{13} + 1183029064020 \nu^{12} + 136138707608671 \nu^{11} + \cdots - 70\!\cdots\!20 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 117312010813 \nu^{13} - 1183029064020 \nu^{12} + 136138707608671 \nu^{11} + \cdots + 70\!\cdots\!20 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 288283218065 \nu^{13} + 8453701114932 \nu^{12} + 290032520116955 \nu^{11} + \cdots - 21\!\cdots\!52 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 288283218065 \nu^{13} + 8453701114932 \nu^{12} - 290032520116955 \nu^{11} + \cdots - 21\!\cdots\!52 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 673250292779 \nu^{13} - 12171829886856 \nu^{12} - 729912943754393 \nu^{11} + \cdots + 14\!\cdots\!16 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 673250292779 \nu^{13} + 12171829886856 \nu^{12} - 729912943754393 \nu^{11} + \cdots - 14\!\cdots\!16 ) / 21\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - 2\beta_{2} + \beta _1 - 183 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 6 \beta_{7} - 10 \beta_{6} + \cdots + 173 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{13} + 3 \beta_{12} + 7 \beta_{11} + 7 \beta_{10} + 17 \beta_{9} - 17 \beta_{8} + \cdots + 54159 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 449 \beta_{13} - 449 \beta_{12} + 581 \beta_{11} - 581 \beta_{10} - 125 \beta_{9} - 125 \beta_{8} + \cdots - 129847 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2265 \beta_{13} - 2265 \beta_{12} - 4925 \beta_{11} - 4925 \beta_{10} - 8299 \beta_{9} + 8299 \beta_{8} + \cdots - 17739453 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 179827 \beta_{13} + 179827 \beta_{12} - 267391 \beta_{11} + 267391 \beta_{10} - 51833 \beta_{9} + \cdots + 70177157 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1221675 \beta_{13} + 1221675 \beta_{12} + 2522455 \beta_{11} + 2522455 \beta_{10} + 3425633 \beta_{9} + \cdots + 6155642823 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 70953161 \beta_{13} - 70953161 \beta_{12} + 114440429 \beta_{11} - 114440429 \beta_{10} + \cdots - 33942873535 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 577854801 \beta_{13} - 577854801 \beta_{12} - 1153547669 \beta_{11} - 1153547669 \beta_{10} + \cdots - 2230336988517 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 28059663691 \beta_{13} + 28059663691 \beta_{12} - 47572113271 \beta_{11} + 47572113271 \beta_{10} + \cdots + 15480788538317 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 256767707235 \beta_{13} + 256767707235 \beta_{12} + 501083296495 \beta_{11} + 501083296495 \beta_{10} + \cdots + 835582948516959 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 11162205575345 \beta_{13} - 11162205575345 \beta_{12} + 19529218262645 \beta_{11} + \cdots - 68\!\cdots\!75 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
− 16.7657i
− 16.7213i
− 8.41902i
0.679146i
4.51724i
14.7644i
20.2132i
16.7657i
16.7213i
8.41902i
− 0.679146i
− 4.51724i
− 14.7644i
− 20.2132i
−14.5196 − 8.38287i 40.5114 − 70.1679i 76.5449 + 132.580i − 223.318i −1176.42 + 679.204i −577.559 + 333.454i − 420.649i −2188.85 − 3791.20i −1872.05 + 3242.48i
4.2 −14.4811 − 8.36065i −21.2782 + 36.8549i 75.8010 + 131.291i − 94.5127i 616.262 − 355.799i 1225.76 − 707.691i − 394.655i 187.977 + 325.587i −790.187 + 1368.64i
4.3 −7.29108 − 4.20951i 0.573744 − 0.993754i −28.5601 − 49.4675i 399.024i −8.36643 + 4.83036i −817.728 + 472.116i 1558.53i 1092.84 + 1892.86i 1679.70 − 2909.32i
4.4 0.588158 + 0.339573i −31.2267 + 54.0862i −63.7694 − 110.452i − 439.155i −36.7324 + 21.2075i −1128.14 + 651.329i − 173.548i −856.712 − 1483.87i 149.125 − 258.293i
4.5 3.91205 + 2.25862i 21.9195 − 37.9656i −53.7973 − 93.1796i − 49.0275i 171.500 − 99.0156i 736.478 − 425.206i − 1064.24i 132.573 + 229.623i 110.735 − 191.798i
4.6 12.7863 + 7.38219i −22.6982 + 39.3144i 44.9933 + 77.9308i 248.787i −580.452 + 335.124i 497.017 − 286.953i − 561.243i 63.0878 + 109.271i −1836.59 + 3181.07i
4.7 17.5052 + 10.1066i 25.1984 − 43.6449i 140.288 + 242.985i − 228.046i 882.204 − 509.341i −1321.83 + 763.159i 3084.03i −176.416 − 305.562i 2304.77 − 3991.98i
10.1 −14.5196 + 8.38287i 40.5114 + 70.1679i 76.5449 − 132.580i 223.318i −1176.42 − 679.204i −577.559 − 333.454i 420.649i −2188.85 + 3791.20i −1872.05 − 3242.48i
10.2 −14.4811 + 8.36065i −21.2782 − 36.8549i 75.8010 − 131.291i 94.5127i 616.262 + 355.799i 1225.76 + 707.691i 394.655i 187.977 − 325.587i −790.187 − 1368.64i
10.3 −7.29108 + 4.20951i 0.573744 + 0.993754i −28.5601 + 49.4675i − 399.024i −8.36643 − 4.83036i −817.728 − 472.116i − 1558.53i 1092.84 − 1892.86i 1679.70 + 2909.32i
10.4 0.588158 − 0.339573i −31.2267 − 54.0862i −63.7694 + 110.452i 439.155i −36.7324 − 21.2075i −1128.14 − 651.329i 173.548i −856.712 + 1483.87i 149.125 + 258.293i
10.5 3.91205 − 2.25862i 21.9195 + 37.9656i −53.7973 + 93.1796i 49.0275i 171.500 + 99.0156i 736.478 + 425.206i 1064.24i 132.573 − 229.623i 110.735 + 191.798i
10.6 12.7863 − 7.38219i −22.6982 − 39.3144i 44.9933 − 77.9308i − 248.787i −580.452 − 335.124i 497.017 + 286.953i 561.243i 63.0878 − 109.271i −1836.59 − 3181.07i
10.7 17.5052 − 10.1066i 25.1984 + 43.6449i 140.288 − 242.985i 228.046i 882.204 + 509.341i −1321.83 − 763.159i − 3084.03i −176.416 + 305.562i 2304.77 + 3991.98i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.7
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 13.8.e.a ✓ 14
3.b odd 2 1 117.8.q.b 14
4.b odd 2 1 208.8.w.a 14
13.c even 3 1 169.8.b.d 14
13.e even 6 1 inner 13.8.e.a ✓ 14
13.e even 6 1 169.8.b.d 14
13.f odd 12 2 169.8.a.g 14
39.h odd 6 1 117.8.q.b 14
52.i odd 6 1 208.8.w.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.8.e.a ✓ 14 1.a even 1 1 trivial
13.8.e.a ✓ 14 13.e even 6 1 inner
117.8.q.b 14 3.b odd 2 1
117.8.q.b 14 39.h odd 6 1
169.8.a.g 14 13.f odd 12 2
169.8.b.d 14 13.c even 3 1
169.8.b.d 14 13.e even 6 1
208.8.w.a 14 4.b odd 2 1
208.8.w.a 14 52.i odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 4669637050368 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 61\!\cdots\!44 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 38\!\cdots\!73 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 30\!\cdots\!29 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 27\!\cdots\!72 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 79\!\cdots\!81 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 25\!\cdots\!75 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 15\!\cdots\!83 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 12\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots - 32\!\cdots\!76)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots - 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 20\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 35\!\cdots\!68 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 28\!\cdots\!28 \) Copy content Toggle raw display
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