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Properties

Label 13.10.c.a
Level $13$
Weight $10$
Character orbit 13.c
Analytic conductor $6.695$
Analytic rank $0$
Dimension $18$
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] = [13,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.3");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(6.69546587013\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 3 x^{17} + 3193 x^{16} + 896 x^{15} + 6827472 x^{14} + 3327136 x^{13} + 8054385232 x^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{3}\cdot 13^{4} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 2 \beta_{2}) q^{2} + ( - \beta_{8} - \beta_{6} - 18 \beta_{2}) q^{3} + (\beta_{12} - 200 \beta_{2} + \cdots - 200) q^{4}+ \cdots + (\beta_{17} - \beta_{16} - \beta_{15} + \cdots - 3745) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 2 \beta_{2}) q^{2} + ( - \beta_{8} - \beta_{6} - 18 \beta_{2}) q^{3} + (\beta_{12} - 200 \beta_{2} + \cdots - 200) q^{4}+ \cdots + (3409 \beta_{13} - 19573 \beta_{11} + \cdots + 125070211) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 15 q^{2} + 161 q^{3} - 1793 q^{4} - 2280 q^{5} + 2118 q^{6} - 1939 q^{7} - 14478 q^{8} - 33654 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 15 q^{2} + 161 q^{3} - 1793 q^{4} - 2280 q^{5} + 2118 q^{6} - 1939 q^{7} - 14478 q^{8} - 33654 q^{9} + 46923 q^{10} - 5433 q^{11} + 8712 q^{12} - 212524 q^{13} - 9900 q^{14} - 347428 q^{15} + 400127 q^{16} + 248589 q^{17} - 38738 q^{18} - 311001 q^{19} + 927069 q^{20} + 1553030 q^{21} + 1857242 q^{22} + 591609 q^{23} - 4492800 q^{24} + 7008998 q^{25} - 3801525 q^{26} - 11603482 q^{27} + 2697168 q^{28} + 11014155 q^{29} - 6597836 q^{30} - 23148076 q^{31} - 11868417 q^{32} + 14131427 q^{33} + 21859862 q^{34} + 21112794 q^{35} + 10792871 q^{36} - 29215749 q^{37} + 14572188 q^{38} - 71569875 q^{39} + 27322222 q^{40} + 3328377 q^{41} + 39828306 q^{42} + 6074381 q^{43} + 31824624 q^{44} + 32857342 q^{45} + 36693338 q^{46} - 45575052 q^{47} - 30270064 q^{48} + 10293266 q^{49} - 49601730 q^{50} - 136587494 q^{51} - 35278230 q^{52} - 29480016 q^{53} + 152965386 q^{54} - 18710998 q^{55} - 7665444 q^{56} + 523363230 q^{57} - 163479359 q^{58} - 32715855 q^{59} - 188638416 q^{60} - 220502845 q^{61} - 59980476 q^{62} + 166572574 q^{63} - 924604030 q^{64} + 128091756 q^{65} - 48128076 q^{66} + 112659045 q^{67} - 238942419 q^{68} + 86003951 q^{69} + 2040150992 q^{70} - 236450709 q^{71} + 995206683 q^{72} - 211881220 q^{73} - 455580507 q^{74} + 968954813 q^{75} - 365789708 q^{76} - 2399230890 q^{77} - 441111970 q^{78} - 817519096 q^{79} + 580424625 q^{80} + 176914851 q^{81} + 941792217 q^{82} + 2225691456 q^{83} - 1819004068 q^{84} + 1812284636 q^{85} + 291320076 q^{86} + 69564799 q^{87} + 3178375740 q^{88} - 1154379039 q^{89} - 10225809510 q^{90} - 1658338903 q^{91} - 4545506592 q^{92} + 3136878060 q^{93} + 2755131560 q^{94} + 1779441012 q^{95} + 5906965568 q^{96} - 3616470111 q^{97} + 8263323501 q^{98} + 2262149268 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 3193 x^{16} + 896 x^{15} + 6827472 x^{14} + 3327136 x^{13} + 8054385232 x^{12} + \cdots + 15\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 82\!\cdots\!57 \nu^{17} + \cdots - 39\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23\!\cdots\!45 \nu^{17} + \cdots - 32\!\cdots\!00 ) / 96\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 72\!\cdots\!59 \nu^{17} + \cdots + 34\!\cdots\!60 ) / 48\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 51\!\cdots\!85 \nu^{17} + \cdots + 58\!\cdots\!00 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\!\cdots\!55 \nu^{17} + \cdots + 19\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 25\!\cdots\!08 \nu^{17} + \cdots - 13\!\cdots\!00 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 52\!\cdots\!23 \nu^{17} + \cdots + 19\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17\!\cdots\!12 \nu^{17} + \cdots - 87\!\cdots\!60 ) / 32\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21\!\cdots\!75 \nu^{17} + \cdots - 64\!\cdots\!00 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 28\!\cdots\!41 \nu^{17} + \cdots + 19\!\cdots\!00 ) / 39\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 14\!\cdots\!89 \nu^{17} + \cdots - 33\!\cdots\!00 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 82\!\cdots\!45 \nu^{17} + \cdots - 18\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 39\!\cdots\!89 \nu^{17} + \cdots - 48\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 12\!\cdots\!01 \nu^{17} + \cdots + 31\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 37\!\cdots\!87 \nu^{17} + \cdots + 45\!\cdots\!00 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 80\!\cdots\!11 \nu^{17} + \cdots - 25\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} + \beta_{4} - 2\beta_{3} + 708\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - 21\beta_{6} + \beta_{5} + 9\beta_{4} - 1111\beta_{3} - 1111\beta _1 - 1373 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 37 \beta_{15} - 19 \beta_{14} + 19 \beta_{13} + 1507 \beta_{12} + 327 \beta_{8} - 44 \beta_{7} + \cdots - 780567 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 112 \beta_{17} - 48 \beta_{16} + 577 \beta_{15} - 1929 \beta_{14} + 24337 \beta_{12} + \cdots - 4825349 \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 47251 \beta_{13} - 3280 \beta_{11} - 2096 \beta_{10} - 103676 \beta_{9} + 1042167 \beta_{6} + \cdots + 986992551 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 300208 \beta_{17} + 103888 \beta_{16} + 958607 \beta_{15} + 3287801 \beta_{14} - 3287801 \beta_{13} + \cdots + 12016457973 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 10086288 \beta_{17} + 7583344 \beta_{16} + 182981037 \beta_{15} + 94197483 \beta_{14} + \cdots + 1364356391391 \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5458219265 \beta_{13} + 599708080 \beta_{11} + 184936272 \beta_{10} + 3614297012 \beta_{9} + \cdots - 25223627132381 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 22031182608 \beta_{17} - 17420740080 \beta_{16} - 320061824597 \beta_{15} - 174064161539 \beta_{14} + \cdots - 20\!\cdots\!99 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1085597587760 \beta_{17} - 329856769488 \beta_{16} - 9169115795199 \beta_{15} + \cdots - 48\!\cdots\!17 \beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 310266373627163 \beta_{13} - 42239068431248 \beta_{11} - 33382015222128 \beta_{10} + \cdots + 30\!\cdots\!63 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 18\!\cdots\!36 \beta_{17} + 607062520370256 \beta_{16} + \cdots + 89\!\cdots\!33 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 76\!\cdots\!92 \beta_{17} + \cdots + 49\!\cdots\!55 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 24\!\cdots\!41 \beta_{13} + \cdots - 15\!\cdots\!69 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 13\!\cdots\!00 \beta_{17} + \cdots - 79\!\cdots\!27 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 55\!\cdots\!60 \beta_{17} + \cdots - 27\!\cdots\!57 \beta_{2} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
20.5607 − 35.6122i
14.5861 − 25.2639i
12.1287 − 21.0076i
7.02454 − 12.1669i
−2.19214 + 3.79690i
−3.59372 + 6.22451i
−13.4651 + 23.3223i
−16.2753 + 28.1897i
−17.2738 + 29.9191i
20.5607 + 35.6122i
14.5861 + 25.2639i
12.1287 + 21.0076i
7.02454 + 12.1669i
−2.19214 − 3.79690i
−3.59372 − 6.22451i
−13.4651 − 23.3223i
−16.2753 − 28.1897i
−17.2738 − 29.9191i
−19.5607 − 33.8802i −36.0748 − 62.4834i −509.244 + 882.037i −1007.31 −1411.30 + 2444.44i 190.054 − 329.184i 19814.6 7238.71 − 12537.8i 19703.7 + 34127.8i
3.2 −13.5861 − 23.5319i 40.2079 + 69.6421i −113.166 + 196.010i 1952.16 1092.54 − 1892.33i 540.093 − 935.468i −7762.23 6608.15 − 11445.7i −26522.3 − 45938.0i
3.3 −11.1287 − 19.2755i 128.264 + 222.159i 8.30313 − 14.3814i −2488.59 2854.82 − 4944.69i −83.9480 + 145.402i −11765.4 −23061.6 + 39943.8i 27694.8 + 47968.9i
3.4 −6.02454 − 10.4348i −96.9066 − 167.847i 183.410 − 317.675i −26.3134 −1167.64 + 2022.40i −1348.09 + 2334.97i −10589.0 −8940.28 + 15485.0i 158.526 + 274.576i
3.5 3.19214 + 5.52895i 2.91217 + 5.04403i 235.621 − 408.107i −1063.81 −18.5921 + 32.2025i 3237.60 − 5607.69i 6277.28 9824.54 − 17016.6i −3395.82 − 5881.73i
3.6 4.59372 + 7.95656i 74.2673 + 128.635i 213.795 − 370.305i 1229.19 −682.327 + 1181.82i −4166.53 + 7216.65i 8632.44 −1189.76 + 2060.73i 5646.55 + 9780.12i
3.7 14.4651 + 25.0544i −95.3768 − 165.198i −162.481 + 281.425i 1752.49 2759.28 − 4779.21i 2202.24 − 3814.40i 5411.06 −8351.99 + 14466.1i 25350.1 + 43907.6i
3.8 17.2753 + 29.9218i −28.8997 − 50.0557i −340.875 + 590.413i −2017.22 998.503 − 1729.46i −5878.83 + 10182.4i −5865.01 8171.12 − 14152.8i −34848.1 − 60358.7i
3.9 18.2738 + 31.6511i 92.1071 + 159.534i −411.862 + 713.365i 529.393 −3366.29 + 5830.58i 4337.92 − 7513.49i −11392.7 −7125.92 + 12342.5i 9674.01 + 16755.9i
9.1 −19.5607 + 33.8802i −36.0748 + 62.4834i −509.244 − 882.037i −1007.31 −1411.30 − 2444.44i 190.054 + 329.184i 19814.6 7238.71 + 12537.8i 19703.7 − 34127.8i
9.2 −13.5861 + 23.5319i 40.2079 − 69.6421i −113.166 − 196.010i 1952.16 1092.54 + 1892.33i 540.093 + 935.468i −7762.23 6608.15 + 11445.7i −26522.3 + 45938.0i
9.3 −11.1287 + 19.2755i 128.264 − 222.159i 8.30313 + 14.3814i −2488.59 2854.82 + 4944.69i −83.9480 − 145.402i −11765.4 −23061.6 − 39943.8i 27694.8 − 47968.9i
9.4 −6.02454 + 10.4348i −96.9066 + 167.847i 183.410 + 317.675i −26.3134 −1167.64 − 2022.40i −1348.09 − 2334.97i −10589.0 −8940.28 − 15485.0i 158.526 − 274.576i
9.5 3.19214 − 5.52895i 2.91217 − 5.04403i 235.621 + 408.107i −1063.81 −18.5921 − 32.2025i 3237.60 + 5607.69i 6277.28 9824.54 + 17016.6i −3395.82 + 5881.73i
9.6 4.59372 − 7.95656i 74.2673 − 128.635i 213.795 + 370.305i 1229.19 −682.327 − 1181.82i −4166.53 − 7216.65i 8632.44 −1189.76 − 2060.73i 5646.55 − 9780.12i
9.7 14.4651 − 25.0544i −95.3768 + 165.198i −162.481 − 281.425i 1752.49 2759.28 + 4779.21i 2202.24 + 3814.40i 5411.06 −8351.99 − 14466.1i 25350.1 − 43907.6i
9.8 17.2753 − 29.9218i −28.8997 + 50.0557i −340.875 − 590.413i −2017.22 998.503 + 1729.46i −5878.83 − 10182.4i −5865.01 8171.12 + 14152.8i −34848.1 + 60358.7i
9.9 18.2738 − 31.6511i 92.1071 − 159.534i −411.862 − 713.365i 529.393 −3366.29 − 5830.58i 4337.92 + 7513.49i −11392.7 −7125.92 − 12342.5i 9674.01 − 16755.9i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.9
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 13.10.c.a ✓ 18
13.c even 3 1 inner 13.10.c.a ✓ 18
13.c even 3 1 169.10.a.c 9
13.e even 6 1 169.10.a.d 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.10.c.a ✓ 18 1.a even 1 1 trivial
13.10.c.a ✓ 18 13.c even 3 1 inner
169.10.a.c 9 13.c even 3 1
169.10.a.d 9 13.e even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{9} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 28\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 16\!\cdots\!13 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 20\!\cdots\!49 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 19\!\cdots\!21 \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots - 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 94\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 42\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( (T^{9} + \cdots + 79\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{9} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 42\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 58\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{9} + \cdots - 95\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 54\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
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