↑

Properties

Label 12.6.b.a
Level $12$
Weight $6$
Character orbit 12.b
Analytic conductor $1.925$
Analytic rank $0$
Dimension $8$
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] = [12,6,Mod(11,12)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(12, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("12.11");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 12.b (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: \(1.92460583776\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 7x^{6} + 6x^{5} - 11x^{4} - 73x^{3} + 223x^{2} - 768x + 912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{18}\cdot 3^{5} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_1 q^{3} + ( - \beta_{3} + 1) q^{4} + (\beta_{5} + \beta_{2}) q^{5} + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots + 3) q^{6}+ \cdots + ( - \beta_{7} + 2 \beta_{6} + \beta_{5} + \cdots - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_1 q^{3} + ( - \beta_{3} + 1) q^{4} + (\beta_{5} + \beta_{2}) q^{5} + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots + 3) q^{6}+ \cdots + ( - 702 \beta_{7} + 180 \beta_{6} + \cdots - 267 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} + 24 q^{6} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} + 24 q^{6} - 24 q^{9} - 272 q^{10} - 696 q^{12} + 112 q^{13} + 2336 q^{16} + 3696 q^{18} - 336 q^{21} - 8304 q^{22} - 11232 q^{24} - 1368 q^{25} + 19248 q^{28} + 25008 q^{30} + 4224 q^{33} - 36416 q^{34} - 42072 q^{36} + 6448 q^{37} + 53824 q^{40} + 59088 q^{42} - 17664 q^{45} - 69600 q^{46} - 74976 q^{48} - 21544 q^{49} + 84208 q^{52} + 78120 q^{54} + 43344 q^{57} - 62576 q^{58} - 64704 q^{60} + 59632 q^{61} + 38528 q^{64} + 33360 q^{66} - 92928 q^{69} - 30432 q^{70} + 20544 q^{72} - 129200 q^{73} - 86256 q^{76} - 83760 q^{78} + 200520 q^{81} + 154720 q^{82} + 167856 q^{84} + 229376 q^{85} - 150336 q^{88} - 251472 q^{90} - 346128 q^{93} + 386112 q^{94} + 331392 q^{96} - 392048 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 7x^{6} + 6x^{5} - 11x^{4} - 73x^{3} + 223x^{2} - 768x + 912 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 9436 \nu^{7} + 30607 \nu^{6} - 62704 \nu^{5} - 231 \nu^{4} - 423169 \nu^{3} - 610094 \nu^{2} + \cdots + 4238706 ) / 953786 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5694 \nu^{7} - 2252 \nu^{6} - 73822 \nu^{5} - 139528 \nu^{4} - 251008 \nu^{3} - 81692 \nu^{2} + \cdots + 5096288 ) / 476893 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8084 \nu^{7} + 72516 \nu^{6} + 129032 \nu^{5} + 488016 \nu^{4} + 1228252 \nu^{3} + \cdots - 1507299 ) / 476893 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 98436 \nu^{7} - 85191 \nu^{6} + 753588 \nu^{5} + 1046763 \nu^{4} - 436119 \nu^{3} - 10020102 \nu^{2} + \cdots - 78787206 ) / 953786 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 70008 \nu^{7} + 87389 \nu^{6} - 396078 \nu^{5} - 1183333 \nu^{4} + 723461 \nu^{3} + \cdots + 43615544 ) / 476893 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 164992 \nu^{7} - 286197 \nu^{6} - 1331716 \nu^{5} - 2951719 \nu^{4} - 3169901 \nu^{3} + \cdots + 53217788 ) / 953786 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 83652 \nu^{7} - 63236 \nu^{6} - 485532 \nu^{5} - 1647824 \nu^{4} - 1820252 \nu^{3} + \cdots + 26548384 ) / 476893 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{4} + 3\beta_{3} + 18\beta_{2} - 6\beta _1 + 9 ) / 72 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} - 3\beta_{6} - 3\beta_{5} - 5\beta_{4} + 3\beta_{2} - 12\beta _1 - 117 ) / 72 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -21\beta_{7} + 12\beta_{6} + 18\beta_{5} - 22\beta_{4} - 24\beta_{3} - 198\beta_{2} - 54\beta _1 - 684 ) / 144 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{7} + 9\beta_{6} - 7\beta_{5} - 5\beta_{4} + 6\beta_{3} - 33\beta_{2} + 48\beta _1 + 273 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 72\beta_{7} - 12\beta_{6} + 24\beta_{5} + 130\beta_{4} + 69\beta_{3} - 198\beta_{2} + 54\beta _1 + 7299 ) / 72 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 111 \beta_{7} - 402 \beta_{6} + 120 \beta_{5} + 496 \beta_{4} + 846 \beta_{3} + 6744 \beta_{2} + \cdots - 18108 ) / 144 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 21 \beta_{7} - 648 \beta_{6} - 174 \beta_{5} - 1096 \beta_{4} - 1329 \beta_{3} - 180 \beta_{2} + \cdots - 30987 ) / 72 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−2.29661 + 1.35416i
−2.29661 − 1.35416i
0.713157 − 2.93555i
0.713157 + 2.93555i
1.67284 + 0.274906i
1.67284 − 0.274906i
0.410606 − 2.17330i
0.410606 + 2.17330i
−5.41443 − 1.63829i 4.17995 + 15.0176i 26.6320 + 17.7408i 73.1202i 1.97116 − 88.1596i − 51.8209i −115.132 − 139.687i −208.056 + 125.545i 119.792 − 395.904i
11.2 −5.41443 + 1.63829i 4.17995 − 15.0176i 26.6320 − 17.7408i − 73.1202i 1.97116 + 88.1596i 51.8209i −115.132 + 139.687i −208.056 − 125.545i 119.792 + 395.904i
11.3 −1.91937 − 5.32128i −14.9174 − 4.52459i −24.6320 + 20.4270i − 35.2908i 4.55538 + 88.0639i − 190.564i 155.976 + 91.8667i 202.056 + 134.990i −187.792 + 67.7362i
11.4 −1.91937 + 5.32128i −14.9174 + 4.52459i −24.6320 − 20.4270i 35.2908i 4.55538 − 88.0639i 190.564i 155.976 − 91.8667i 202.056 − 134.990i −187.792 − 67.7362i
11.5 1.91937 − 5.32128i 14.9174 + 4.52459i −24.6320 − 20.4270i − 35.2908i 52.7086 − 70.6951i 190.564i −155.976 + 91.8667i 202.056 + 134.990i −187.792 − 67.7362i
11.6 1.91937 + 5.32128i 14.9174 − 4.52459i −24.6320 + 20.4270i 35.2908i 52.7086 + 70.6951i − 190.564i −155.976 − 91.8667i 202.056 − 134.990i −187.792 + 67.7362i
11.7 5.41443 − 1.63829i −4.17995 − 15.0176i 26.6320 − 17.7408i 73.1202i −47.2352 − 74.4637i 51.8209i 115.132 − 139.687i −208.056 + 125.545i 119.792 + 395.904i
11.8 5.41443 + 1.63829i −4.17995 + 15.0176i 26.6320 + 17.7408i − 73.1202i −47.2352 + 74.4637i − 51.8209i 115.132 + 139.687i −208.056 − 125.545i 119.792 − 395.904i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.6.b.a ✓ 8
3.b odd 2 1 inner 12.6.b.a ✓ 8
4.b odd 2 1 inner 12.6.b.a ✓ 8
8.b even 2 1 192.6.c.e 8
8.d odd 2 1 192.6.c.e 8
12.b even 2 1 inner 12.6.b.a ✓ 8
24.f even 2 1 192.6.c.e 8
24.h odd 2 1 192.6.c.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.6.b.a ✓ 8 1.a even 1 1 trivial
12.6.b.a ✓ 8 3.b odd 2 1 inner
12.6.b.a ✓ 8 4.b odd 2 1 inner
12.6.b.a ✓ 8 12.b even 2 1 inner
192.6.c.e 8 8.b even 2 1
192.6.c.e 8 8.d odd 2 1
192.6.c.e 8 24.f even 2 1
192.6.c.e 8 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(12, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 4 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 3486784401 \) Copy content Toggle raw display
$5$ \( (T^{4} + 6592 T^{2} + 6658816)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 39000 T^{2} + 97519248)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 139200 T^{2} + 153453312)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 28 T - 167996)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 3215624175616)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1880280 T^{2} + 181855208592)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 29300325101568)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 182708761705216)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 292116151966608)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 1612 T + 481444)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 27\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 76\!\cdots\!08)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 15\!\cdots\!68)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 517240473144064)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 11\!\cdots\!68)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 14908 T - 227168636)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 56\!\cdots\!12)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 46\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 32300 T - 1848977948)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 28\!\cdots\!28)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 345810854880000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 46\!\cdots\!56)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 98012 T + 1690976836)^{4} \) Copy content Toggle raw display
show more
show less