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Properties

Label 13.5.f.a
Level $13$
Weight $5$
Character orbit 13.f
Analytic conductor $1.344$
Analytic rank $0$
Dimension $16$
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] = [13,5,Mod(2,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.2");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 13.f (of order \(12\), degree \(4\), 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: \(1.34380952009\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 152 x^{14} + 9190 x^{12} + 285720 x^{10} + 4862025 x^{8} + 43573680 x^{6} + 169417008 x^{4} + \cdots + 3779136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2}\cdot 13^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{12} q^{2} + ( - \beta_{14} + \beta_{12} + \beta_{9} + \cdots - 1) q^{3}+ \cdots + (3 \beta_{14} - 2 \beta_{13} + \cdots - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{12} q^{2} + ( - \beta_{14} + \beta_{12} + \beta_{9} + \cdots - 1) q^{3}+ \cdots + (14 \beta_{15} - 254 \beta_{14} + \cdots + 608) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} - 2 q^{3} - 6 q^{4} + 8 q^{5} - 38 q^{6} + 56 q^{7} + 90 q^{8} - 164 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{2} - 2 q^{3} - 6 q^{4} + 8 q^{5} - 38 q^{6} + 56 q^{7} + 90 q^{8} - 164 q^{9} - 486 q^{10} - 100 q^{11} + 294 q^{13} + 808 q^{14} + 346 q^{15} + 230 q^{16} + 984 q^{17} + 2434 q^{18} - 1498 q^{19} - 3962 q^{20} - 1076 q^{21} - 1524 q^{22} - 1014 q^{23} - 2142 q^{24} + 614 q^{26} + 3352 q^{27} + 5764 q^{28} + 814 q^{29} + 9162 q^{30} + 4060 q^{31} - 4996 q^{32} - 5636 q^{33} - 2502 q^{34} - 4892 q^{35} - 15750 q^{36} - 1790 q^{37} + 6982 q^{39} + 18816 q^{40} + 4280 q^{41} - 1204 q^{42} - 1368 q^{43} + 10736 q^{44} - 6806 q^{45} - 15246 q^{46} + 1484 q^{47} - 3002 q^{48} - 11820 q^{49} - 13574 q^{50} + 1432 q^{52} + 7204 q^{53} + 13240 q^{54} + 6936 q^{55} + 8124 q^{56} + 12736 q^{57} + 3030 q^{58} - 2380 q^{59} - 6472 q^{60} - 162 q^{61} + 19614 q^{62} + 12004 q^{63} - 5248 q^{65} - 23872 q^{66} - 14854 q^{67} - 6444 q^{68} + 2412 q^{69} - 34524 q^{70} - 8050 q^{71} + 15420 q^{72} - 15448 q^{73} - 2882 q^{74} + 8280 q^{75} + 10622 q^{76} - 11672 q^{78} - 17064 q^{79} + 2564 q^{80} + 2128 q^{81} - 5346 q^{82} + 12788 q^{83} + 25948 q^{84} + 35382 q^{85} + 67260 q^{86} + 29342 q^{87} + 40836 q^{88} + 20492 q^{89} + 8996 q^{91} - 49884 q^{92} - 78920 q^{93} - 30606 q^{94} - 98574 q^{95} - 94664 q^{96} + 50944 q^{97} + 61484 q^{98} - 21632 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 152 x^{14} + 9190 x^{12} + 285720 x^{10} + 4862025 x^{8} + 43573680 x^{6} + 169417008 x^{4} + \cdots + 3779136 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 403105 \nu^{15} - 60681659 \nu^{13} - 3635180779 \nu^{11} - 112263516645 \nu^{9} + \cdots + 5545371476736 ) / 11090742953472 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 450212 \nu^{15} - 12268161 \nu^{14} - 40141192 \nu^{13} - 1670382423 \nu^{12} + \cdots - 35916509215296 ) / 33272228860416 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 450212 \nu^{15} - 12268161 \nu^{14} + 40141192 \nu^{13} - 1670382423 \nu^{12} + \cdots - 35916509215296 ) / 33272228860416 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 53905 \nu^{15} - 3031335 \nu^{14} - 1946495 \nu^{13} - 399803697 \nu^{12} + \cdots + 22901572977984 ) / 3696914317824 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 53905 \nu^{15} + 3031335 \nu^{14} - 1946495 \nu^{13} + 399803697 \nu^{12} + \cdots - 22901572977984 ) / 3696914317824 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 958933 \nu^{14} + 127064675 \nu^{12} + 6269679835 \nu^{10} + 142169221341 \nu^{8} + \cdots - 17722730943936 ) / 616152386304 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1611011 \nu^{15} + 20386710 \nu^{14} + 235643173 \nu^{13} + 2770864650 \nu^{12} + \cdots - 127810286301312 ) / 11090742953472 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 64316 \nu^{15} + 10690839 \nu^{14} - 5734456 \nu^{13} + 1454826609 \nu^{12} + \cdots - 46660774261824 ) / 4753175551488 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8483506 \nu^{15} - 22455 \nu^{14} + 1257390830 \nu^{13} - 34398657 \nu^{12} + \cdots - 121083795836352 ) / 33272228860416 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1611011 \nu^{15} - 20386710 \nu^{14} + 235643173 \nu^{13} - 2770864650 \nu^{12} + \cdots + 116719543347840 ) / 11090742953472 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3060949 \nu^{15} + 30779505 \nu^{14} + 538883915 \nu^{13} + 4066826247 \nu^{12} + \cdots + 262228572037440 ) / 11090742953472 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1668943 \nu^{15} + 3478479 \nu^{14} + 244791977 \nu^{13} + 476788953 \nu^{12} + \cdots - 4839006028608 ) / 3696914317824 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3060949 \nu^{15} - 30779505 \nu^{14} + 538883915 \nu^{13} - 4066826247 \nu^{12} + \cdots - 262228572037440 ) / 11090742953472 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1668943 \nu^{15} - 3478479 \nu^{14} + 244791977 \nu^{13} - 476788953 \nu^{12} + \cdots + 4839006028608 ) / 3696914317824 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 7320853 \nu^{15} + 2876799 \nu^{14} + 1089833795 \nu^{13} + 381194025 \nu^{12} + \cdots - 53168192831808 ) / 3696914317824 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 6 \beta_{15} + 8 \beta_{14} + \beta_{13} + 12 \beta_{12} + \beta_{11} + 8 \beta_{10} + 4 \beta_{9} + \cdots - 4 ) / 26 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7 \beta_{14} - 4 \beta_{13} - 7 \beta_{12} + 4 \beta_{11} - 7 \beta_{10} + 2 \beta_{8} + 7 \beta_{7} + \cdots - 242 ) / 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 92 \beta_{15} - 114 \beta_{14} + 15 \beta_{13} - 184 \beta_{12} + 15 \beta_{11} - 140 \beta_{10} + \cdots - 242 ) / 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 407 \beta_{14} + 175 \beta_{13} + 407 \beta_{12} - 175 \beta_{11} + 381 \beta_{10} - 380 \beta_{8} + \cdots + 7318 ) / 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3476 \beta_{15} + 3430 \beta_{14} - 1479 \beta_{13} + 7212 \beta_{12} - 1479 \beta_{11} + \cdots + 17616 ) / 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 17913 \beta_{14} - 8533 \beta_{13} - 17913 \beta_{12} + 8533 \beta_{11} - 18797 \beta_{10} + \cdots - 271710 ) / 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11468 \beta_{15} - 8888 \beta_{14} + 7017 \beta_{13} - 24998 \beta_{12} + 7017 \beta_{11} - 19004 \beta_{10} + \cdots - 73312 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 772903 \beta_{14} + 430867 \beta_{13} + 772903 \beta_{12} - 430867 \beta_{11} + 918997 \beta_{10} + \cdots + 11600462 ) / 13 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 6901292 \beta_{15} + 4408926 \beta_{14} - 5008851 \beta_{13} + 15652900 \beta_{12} - 5008851 \beta_{11} + \cdots + 48370256 ) / 13 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 34597169 \beta_{14} - 21896869 \beta_{13} - 34597169 \beta_{12} + 21896869 \beta_{11} + \cdots - 536617966 ) / 13 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 332946236 \beta_{15} - 187007200 \beta_{14} + 263030493 \beta_{13} - 775259574 \beta_{12} + \cdots - 2420792496 ) / 13 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 1614876399 \beta_{14} + 1111784059 \beta_{13} + 1614876399 \beta_{12} - 1111784059 \beta_{11} + \cdots + 25885701678 ) / 13 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 16408659212 \beta_{15} + 8537243750 \beta_{14} - 13536977019 \beta_{13} + 38821589996 \beta_{12} + \cdots + 121041181744 ) / 13 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 5983739437 \beta_{14} - 4333135633 \beta_{13} - 5983739437 \beta_{12} + 4333135633 \beta_{11} + \cdots - 98119308806 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 817505384348 \beta_{15} - 407544050424 \beta_{14} + 689836505013 \beta_{13} - 1952234100238 \beta_{12} + \cdots - 6064117949264 ) / 13 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
− 4.89748i
5.33868i
0.816521i
− 3.98977i
4.34562i
− 7.09996i
3.68702i
− 0.200628i
4.89748i
− 5.33868i
− 0.816521i
3.98977i
− 4.34562i
7.09996i
− 3.68702i
0.200628i
−6.34141 − 1.69918i 7.28678 + 12.6211i 23.4699 + 13.5504i −20.2793 + 20.2793i −24.7631 − 92.4170i 21.3318 − 5.71584i −51.5321 − 51.5321i −65.6944 + 113.786i 163.058 − 94.1415i
2.2 −3.41998 − 0.916381i −5.16500 − 8.94604i −2.99988 − 1.73198i 8.65321 − 8.65321i 9.46622 + 35.3284i −2.94871 + 0.790103i 48.7300 + 48.7300i −12.8545 + 22.2646i −37.5235 + 21.6642i
2.3 2.38644 + 0.639444i 4.25709 + 7.37349i −8.57021 − 4.94801i 16.9748 − 16.9748i 5.44434 + 20.3185i −50.2192 + 13.4562i −45.2402 − 45.2402i 4.25442 − 7.36887i 51.3637 − 29.6548i
2.4 5.50893 + 1.47611i −4.28079 − 7.41455i 14.3130 + 8.26362i −29.3294 + 29.3294i −12.6379 − 47.1652i 45.8361 − 12.2817i 2.12627 + 2.12627i 3.84960 − 6.66770i −204.867 + 118.280i
6.1 −1.88469 − 7.03375i −0.939567 + 1.62738i −32.0652 + 18.5128i 26.6717 − 26.6717i 13.2174 + 3.54158i 7.85376 − 29.3106i 108.263 + 108.263i 38.7344 + 67.0900i −237.870 − 137.334i
6.2 −0.387376 − 1.44571i 3.29308 − 5.70378i 11.9164 − 6.87993i −10.5663 + 10.5663i −9.52166 − 2.55132i −14.3010 + 53.3722i −31.4958 − 31.4958i 18.8113 + 32.5821i 19.3688 + 11.1826i
6.3 0.540374 + 2.01670i −8.21171 + 14.2231i 10.0813 − 5.82045i 17.2508 − 17.2508i −33.1212 − 8.87479i −2.95016 + 11.0101i 40.8071 + 40.8071i −94.3643 − 163.444i 44.1117 + 25.4679i
6.4 1.59772 + 5.96275i 2.76012 − 4.78067i −19.1453 + 11.0536i −5.37551 + 5.37551i 32.9158 + 8.81977i 23.3974 − 87.3205i −26.6579 − 26.6579i 25.2635 + 43.7576i −40.6414 − 23.4643i
7.1 −6.34141 + 1.69918i 7.28678 − 12.6211i 23.4699 − 13.5504i −20.2793 − 20.2793i −24.7631 + 92.4170i 21.3318 + 5.71584i −51.5321 + 51.5321i −65.6944 − 113.786i 163.058 + 94.1415i
7.2 −3.41998 + 0.916381i −5.16500 + 8.94604i −2.99988 + 1.73198i 8.65321 + 8.65321i 9.46622 − 35.3284i −2.94871 − 0.790103i 48.7300 − 48.7300i −12.8545 − 22.2646i −37.5235 − 21.6642i
7.3 2.38644 − 0.639444i 4.25709 − 7.37349i −8.57021 + 4.94801i 16.9748 + 16.9748i 5.44434 − 20.3185i −50.2192 − 13.4562i −45.2402 + 45.2402i 4.25442 + 7.36887i 51.3637 + 29.6548i
7.4 5.50893 − 1.47611i −4.28079 + 7.41455i 14.3130 − 8.26362i −29.3294 − 29.3294i −12.6379 + 47.1652i 45.8361 + 12.2817i 2.12627 − 2.12627i 3.84960 + 6.66770i −204.867 − 118.280i
11.1 −1.88469 + 7.03375i −0.939567 − 1.62738i −32.0652 − 18.5128i 26.6717 + 26.6717i 13.2174 − 3.54158i 7.85376 + 29.3106i 108.263 − 108.263i 38.7344 − 67.0900i −237.870 + 137.334i
11.2 −0.387376 + 1.44571i 3.29308 + 5.70378i 11.9164 + 6.87993i −10.5663 − 10.5663i −9.52166 + 2.55132i −14.3010 − 53.3722i −31.4958 + 31.4958i 18.8113 − 32.5821i 19.3688 − 11.1826i
11.3 0.540374 − 2.01670i −8.21171 − 14.2231i 10.0813 + 5.82045i 17.2508 + 17.2508i −33.1212 + 8.87479i −2.95016 − 11.0101i 40.8071 − 40.8071i −94.3643 + 163.444i 44.1117 − 25.4679i
11.4 1.59772 − 5.96275i 2.76012 + 4.78067i −19.1453 − 11.0536i −5.37551 − 5.37551i 32.9158 − 8.81977i 23.3974 + 87.3205i −26.6579 + 26.6579i 25.2635 − 43.7576i −40.6414 + 23.4643i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.5.f.a ✓ 16
3.b odd 2 1 117.5.bd.c 16
13.c even 3 1 169.5.d.d 16
13.e even 6 1 169.5.d.c 16
13.f odd 12 1 inner 13.5.f.a ✓ 16
13.f odd 12 1 169.5.d.c 16
13.f odd 12 1 169.5.d.d 16
39.k even 12 1 117.5.bd.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.5.f.a ✓ 16 1.a even 1 1 trivial
13.5.f.a ✓ 16 13.f odd 12 1 inner
117.5.bd.c 16 3.b odd 2 1
117.5.bd.c 16 39.k even 12 1
169.5.d.c 16 13.e even 6 1
169.5.d.c 16 13.f odd 12 1
169.5.d.d 16 13.c even 3 1
169.5.d.d 16 13.f odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 2116736064 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 151617258383616 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 82\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 44\!\cdots\!81 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 91\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 45\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 38\!\cdots\!41 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 94\!\cdots\!29 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 87\!\cdots\!21 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 47\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 20\!\cdots\!72)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 63\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 57\!\cdots\!89 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 77\!\cdots\!28)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 37\!\cdots\!24 \) Copy content Toggle raw display
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