↑

Properties

Label 69.4.c.c
Level $69$
Weight $4$
Character orbit 69.c
Analytic conductor $4.071$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [69,4,Mod(68,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.68");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 69.c (of order \(2\), degree \(1\), 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: \(4.07113179040\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 1604 x^{14} + 1093114 x^{12} - 406721600 x^{10} + 88188665665 x^{8} - 10937417469260 x^{6} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{11} q^{2} - \beta_{10} q^{3} + (\beta_1 - 2) q^{4} - \beta_{12} q^{5} + ( - \beta_{14} - \beta_{11} + \beta_{10} + \cdots + 5) q^{6}+ \cdots + (3 \beta_{15} + 3 \beta_{11} + \cdots + 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{11} q^{2} - \beta_{10} q^{3} + (\beta_1 - 2) q^{4} - \beta_{12} q^{5} + ( - \beta_{14} - \beta_{11} + \beta_{10} + \cdots + 5) q^{6}+ \cdots + (3 \beta_{13} + 27 \beta_{12} + \cdots + 24 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{3} - 24 q^{4} + 72 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{3} - 24 q^{4} + 72 q^{6} - 6 q^{9} - 180 q^{12} - 52 q^{13} - 568 q^{16} + 84 q^{18} - 492 q^{24} + 1360 q^{25} + 972 q^{27} + 116 q^{31} + 936 q^{36} - 882 q^{39} + 344 q^{46} - 1284 q^{48} - 3224 q^{49} - 936 q^{52} - 828 q^{54} + 480 q^{55} + 3712 q^{58} - 1256 q^{64} - 1146 q^{69} - 2328 q^{70} + 96 q^{72} + 1148 q^{73} + 606 q^{75} + 4668 q^{78} + 4122 q^{81} + 7432 q^{82} - 3552 q^{85} - 5358 q^{87} + 1326 q^{93} - 1760 q^{94} - 6084 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 1604 x^{14} + 1093114 x^{12} - 406721600 x^{10} + 88188665665 x^{8} - 10937417469260 x^{6} + \cdots + 20\!\cdots\!56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 99\!\cdots\!05 \nu^{14} + \cdots + 71\!\cdots\!86 ) / 14\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 99\!\cdots\!39 \nu^{14} + \cdots - 55\!\cdots\!52 ) / 91\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 82\!\cdots\!38 \nu^{14} + \cdots - 14\!\cdots\!72 ) / 57\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17\!\cdots\!21 \nu^{15} + \cdots - 11\!\cdots\!32 ) / 52\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\!\cdots\!21 \nu^{15} + \cdots - 42\!\cdots\!72 ) / 52\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17\!\cdots\!21 \nu^{15} + \cdots - 42\!\cdots\!72 ) / 52\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11\!\cdots\!45 \nu^{15} + \cdots - 50\!\cdots\!52 \nu ) / 26\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 93\!\cdots\!83 \nu^{15} + \cdots - 62\!\cdots\!60 \nu ) / 87\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13\!\cdots\!09 \nu^{15} + \cdots + 35\!\cdots\!32 ) / 10\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13\!\cdots\!09 \nu^{15} + \cdots + 35\!\cdots\!32 ) / 10\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 38\!\cdots\!57 \nu^{15} + \cdots - 14\!\cdots\!72 \nu ) / 26\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 38\!\cdots\!57 \nu^{15} + \cdots + 41\!\cdots\!76 \nu ) / 26\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 40\!\cdots\!29 \nu^{15} + \cdots - 53\!\cdots\!12 \nu ) / 26\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 17\!\cdots\!25 \nu^{15} + \cdots - 15\!\cdots\!88 ) / 52\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 17\!\cdots\!25 \nu^{15} + \cdots + 15\!\cdots\!88 ) / 52\!\cdots\!08 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{12} + \beta_{11} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -13\beta_{15} + 13\beta_{14} - 6\beta_{10} - 6\beta_{9} + 2\beta_{3} - 7\beta _1 + 202 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 103 \beta_{15} + 103 \beta_{14} + 21 \beta_{13} + 265 \beta_{12} + 809 \beta_{11} + 32 \beta_{10} + \cdots - 15 \beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4980 \beta_{15} + 4980 \beta_{14} - 1376 \beta_{10} - 1376 \beta_{9} - 48 \beta_{6} - 104 \beta_{5} + \cdots + 46998 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 48546 \beta_{15} + 48546 \beta_{14} + 10507 \beta_{13} + 65499 \beta_{12} + 370865 \beta_{11} + \cdots - 9003 \beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1644347 \beta_{15} + 1644347 \beta_{14} - 596904 \beta_{10} - 596904 \beta_{9} - 47396 \beta_{6} + \cdots + 10007228 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15953479 \beta_{15} + 15953479 \beta_{14} + 4429791 \beta_{13} + 14352761 \beta_{12} + \cdots - 3742527 \beta_{5} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 489522072 \beta_{15} + 489522072 \beta_{14} - 284129244 \beta_{10} - 284129244 \beta_{9} + \cdots + 1704944858 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4011202094 \beta_{15} + 4011202094 \beta_{14} + 1590613911 \beta_{13} + 2460684387 \beta_{12} + \cdots - 1281760443 \beta_{5} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 126652748183 \beta_{15} + 126652748183 \beta_{14} - 111145062416 \beta_{10} - 111145062416 \beta_{9} + \cdots + 114284351572 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 665086726423 \beta_{15} + 665086726423 \beta_{14} + 474378405055 \beta_{13} + \cdots - 363257475207 \beta_{5} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 24276641148832 \beta_{15} + 24276641148832 \beta_{14} - 34242795278844 \beta_{10} + \cdots - 85059286867086 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 14305398804118 \beta_{15} - 14305398804118 \beta_{14} + 104782003240167 \beta_{13} + \cdots - 74374347709419 \beta_{5} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 115250700798347 \beta_{15} + 115250700798347 \beta_{14} + \cdots - 63\!\cdots\!32 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 67\!\cdots\!33 \beta_{15} + \cdots - 14\!\cdots\!11 \beta_{5} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
16.9991 + 4.36469i
−16.9991 + 4.36469i
12.7008 + 3.29311i
−12.7008 + 3.29311i
19.2408 + 2.14761i
−19.2408 + 2.14761i
4.41718 + 1.86886i
−4.41718 + 1.86886i
4.41718 − 1.86886i
−4.41718 − 1.86886i
19.2408 − 2.14761i
−19.2408 − 2.14761i
12.7008 − 3.29311i
−12.7008 − 3.29311i
16.9991 − 4.36469i
−16.9991 − 4.36469i
− 4.36469i 0.167520 + 5.19345i −11.0506 −16.9991 22.6678 − 0.731173i 23.9477i 13.3148i −26.9439 + 1.74001i 74.1959i
68.2 − 4.36469i 0.167520 + 5.19345i −11.0506 16.9991 22.6678 − 0.731173i − 23.9477i 13.3148i −26.9439 + 1.74001i − 74.1959i
68.3 − 3.29311i 5.18870 + 0.278177i −2.84459 −12.7008 0.916069 − 17.0870i − 26.0706i − 16.9774i 26.8452 + 2.88676i 41.8250i
68.4 − 3.29311i 5.18870 + 0.278177i −2.84459 12.7008 0.916069 − 17.0870i 26.0706i − 16.9774i 26.8452 + 2.88676i − 41.8250i
68.5 − 2.14761i −2.24586 − 4.68574i 3.38777 −19.2408 −10.0631 + 4.82323i 3.50800i − 24.4565i −16.9122 + 21.0470i 41.3218i
68.6 − 2.14761i −2.24586 − 4.68574i 3.38777 19.2408 −10.0631 + 4.82323i − 3.50800i − 24.4565i −16.9122 + 21.0470i − 41.3218i
68.7 − 1.86886i −4.61036 + 2.39678i 4.50738 −4.41718 4.47924 + 8.61610i − 30.2080i − 23.3745i 15.5109 − 22.1001i 8.25507i
68.8 − 1.86886i −4.61036 + 2.39678i 4.50738 4.41718 4.47924 + 8.61610i 30.2080i − 23.3745i 15.5109 − 22.1001i − 8.25507i
68.9 1.86886i −4.61036 − 2.39678i 4.50738 −4.41718 4.47924 − 8.61610i 30.2080i 23.3745i 15.5109 + 22.1001i − 8.25507i
68.10 1.86886i −4.61036 − 2.39678i 4.50738 4.41718 4.47924 − 8.61610i − 30.2080i 23.3745i 15.5109 + 22.1001i 8.25507i
68.11 2.14761i −2.24586 + 4.68574i 3.38777 −19.2408 −10.0631 − 4.82323i − 3.50800i 24.4565i −16.9122 − 21.0470i − 41.3218i
68.12 2.14761i −2.24586 + 4.68574i 3.38777 19.2408 −10.0631 − 4.82323i 3.50800i 24.4565i −16.9122 − 21.0470i 41.3218i
68.13 3.29311i 5.18870 − 0.278177i −2.84459 −12.7008 0.916069 + 17.0870i 26.0706i 16.9774i 26.8452 − 2.88676i − 41.8250i
68.14 3.29311i 5.18870 − 0.278177i −2.84459 12.7008 0.916069 + 17.0870i − 26.0706i 16.9774i 26.8452 − 2.88676i 41.8250i
68.15 4.36469i 0.167520 − 5.19345i −11.0506 −16.9991 22.6678 + 0.731173i − 23.9477i − 13.3148i −26.9439 − 1.74001i − 74.1959i
68.16 4.36469i 0.167520 − 5.19345i −11.0506 16.9991 22.6678 + 0.731173i 23.9477i − 13.3148i −26.9439 − 1.74001i 74.1959i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.b odd 2 1 inner
69.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.4.c.c ✓ 16
3.b odd 2 1 inner 69.4.c.c ✓ 16
23.b odd 2 1 inner 69.4.c.c ✓ 16
69.c even 2 1 inner 69.4.c.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.4.c.c ✓ 16 1.a even 1 1 trivial
69.4.c.c ✓ 16 3.b odd 2 1 inner
69.4.c.c ✓ 16 23.b odd 2 1 inner
69.4.c.c ✓ 16 69.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 38T_{2}^{6} + 465T_{2}^{4} + 2156T_{2}^{2} + 3328 \) acting on \(S_{4}^{\mathrm{new}}(69, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 38 T^{6} + \cdots + 3328)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + 3 T^{7} + \cdots + 531441)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 840 T^{6} + \cdots + 336705120)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 2178 T^{6} + \cdots + 4377166560)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 6789322039680)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 13 T^{3} + \cdots + 1144912)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 458554999171680)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 31\!\cdots\!40)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 48\!\cdots\!41 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 19\!\cdots\!92)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 29 T^{3} + \cdots + 324525184)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 18\!\cdots\!40)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 77\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 88\!\cdots\!40)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 11\!\cdots\!32)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 53\!\cdots\!40)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 287 T^{3} + \cdots + 2018470216)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 25\!\cdots\!40)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 13\!\cdots\!20)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 70\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
show more
show less