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Properties

Label 196.3.g.j
Level $196$
Weight $3$
Character orbit 196.g
Analytic conductor $5.341$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [196,3,Mod(67,196)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(196, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.67");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 196.g (of order \(6\), degree \(2\), not 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: \(5.34061318146\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.1728283481971641.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 4 x^{10} - 6 x^{9} + 6 x^{8} - 8 x^{7} + 9 x^{6} - 16 x^{5} + 24 x^{4} - 48 x^{3} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 28)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{6} q^{3} - \beta_{4} q^{4} + ( - \beta_{5} - \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{11} + \beta_{9} - \beta_{8} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{10} + \beta_{9} + \beta_{8} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{6} q^{3} - \beta_{4} q^{4} + ( - \beta_{5} - \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{11} + \beta_{9} - \beta_{8} + \cdots - 1) q^{6}+ \cdots + ( - 2 \beta_{11} + 13 \beta_{9} + \cdots + 4 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{2} - q^{4} - 4 q^{5} - 12 q^{6} - 26 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + q^{2} - q^{4} - 4 q^{5} - 12 q^{6} - 26 q^{8} + 10 q^{9} - 28 q^{10} + 6 q^{12} - 24 q^{13} - 17 q^{16} - 4 q^{17} - 43 q^{18} + 64 q^{20} + 104 q^{22} + 122 q^{24} + 30 q^{25} - 56 q^{26} - 72 q^{29} + 64 q^{30} + 101 q^{32} + 80 q^{33} - 116 q^{34} - 262 q^{36} - 28 q^{37} - 190 q^{38} + 40 q^{40} + 40 q^{41} - 164 q^{44} + 12 q^{45} - 120 q^{46} + 196 q^{48} + 322 q^{50} + 292 q^{52} - 92 q^{53} - 44 q^{54} + 320 q^{57} + 166 q^{58} + 176 q^{60} - 164 q^{61} - 296 q^{62} - 430 q^{64} + 136 q^{65} - 408 q^{66} + 62 q^{68} + 96 q^{69} - 151 q^{72} - 132 q^{73} - 250 q^{74} + 156 q^{76} + 496 q^{78} + 312 q^{80} + 218 q^{81} - 86 q^{82} - 464 q^{85} + 164 q^{86} + 100 q^{88} + 348 q^{89} - 104 q^{90} - 208 q^{92} - 288 q^{93} - 276 q^{94} + 170 q^{96} - 504 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 4 x^{10} - 6 x^{9} + 6 x^{8} - 8 x^{7} + 9 x^{6} - 16 x^{5} + 24 x^{4} - 48 x^{3} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{11} - 4 \nu^{10} + 40 \nu^{9} - 26 \nu^{8} + 110 \nu^{7} - 12 \nu^{6} + 55 \nu^{5} + \cdots - 896 ) / 544 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{11} - 54 \nu^{10} - 4 \nu^{9} + 6 \nu^{8} + 210 \nu^{7} + 8 \nu^{6} + 3 \nu^{5} + \cdots - 128 ) / 1088 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2 \nu^{11} + 9 \nu^{10} - 22 \nu^{9} + 16 \nu^{8} - 18 \nu^{7} + 78 \nu^{6} - 26 \nu^{5} + \cdots + 384 ) / 272 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3 \nu^{11} - 22 \nu^{10} + 16 \nu^{9} - 58 \nu^{8} + 10 \nu^{7} - 32 \nu^{6} + 107 \nu^{5} - 120 \nu^{4} + \cdots - 32 ) / 272 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 25 \nu^{11} - 2 \nu^{10} + 20 \nu^{9} - 30 \nu^{8} + 38 \nu^{7} - 40 \nu^{6} - 15 \nu^{5} + \cdots - 448 ) / 1088 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7 \nu^{11} - 6 \nu^{10} - 8 \nu^{9} + 46 \nu^{8} - 22 \nu^{7} + 16 \nu^{6} + 23 \nu^{5} - 8 \nu^{4} + \cdots + 288 ) / 272 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7 \nu^{11} + 11 \nu^{10} + 26 \nu^{9} - 22 \nu^{8} + 12 \nu^{7} - 18 \nu^{6} + 23 \nu^{5} - 127 \nu^{4} + \cdots - 256 ) / 272 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6 \nu^{11} - 7 \nu^{10} + 2 \nu^{9} - 20 \nu^{8} + 14 \nu^{7} - 38 \nu^{6} - 10 \nu^{5} + \cdots - 208 ) / 136 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13 \nu^{11} - 50 \nu^{10} + 92 \nu^{9} - 70 \nu^{8} + 134 \nu^{7} - 48 \nu^{6} + 101 \nu^{5} + \cdots - 1408 ) / 272 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29 \nu^{11} - 122 \nu^{10} + 132 \nu^{9} - 198 \nu^{8} + 142 \nu^{7} - 264 \nu^{6} + 37 \nu^{5} + \cdots - 1216 ) / 544 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 8 \nu^{11} + 19 \nu^{10} - 20 \nu^{9} + 30 \nu^{8} - 38 \nu^{7} + 40 \nu^{6} - 70 \nu^{5} + \cdots + 244 ) / 68 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{5} - \beta_{2} + \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{6} + 3\beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{10} + 2\beta_{9} + 2\beta_{8} - \beta_{7} + 5\beta_{5} - \beta_{2} - 3\beta _1 + 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{11} - 4\beta_{10} + \beta_{6} - 9\beta_{5} + 2\beta_{4} - \beta_{3} - 3\beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{11} + 2\beta_{9} - \beta_{8} + 2\beta_{4} + 6\beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -2\beta_{10} - 2\beta_{9} + 2\beta_{8} - 3\beta_{7} + 19\beta_{5} + 9\beta_{2} + 15\beta _1 + 19 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( \beta_{11} + 17\beta_{6} - 13\beta_{5} - 6\beta_{4} - 9\beta_{3} + \beta_{2} + 9\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8\beta_{11} + 8\beta_{9} - 3\beta_{8} + 5\beta_{7} + 5\beta_{6} + 8\beta_{4} - 11\beta_{3} + 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -10\beta_{10} - 6\beta_{9} + 10\beta_{8} + 11\beta_{7} + 41\beta_{5} - 21\beta_{2} + 33\beta _1 + 41 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( \beta_{11} - 4\beta_{10} + 21\beta_{6} + 163\beta_{5} + 10\beta_{4} + 11\beta_{3} + \beta_{2} - 11\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).

\(n\) \(99\) \(101\)
\(\chi(n)\) \(-1\) \(-1 - \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.0455406 − 1.41348i
0.341867 + 1.37227i
−1.24688 − 0.667301i
1.40501 + 0.161055i
−0.563028 + 1.29730i
1.01749 + 0.982201i
0.0455406 + 1.41348i
0.341867 − 1.37227i
−1.24688 + 0.667301i
1.40501 − 0.161055i
−0.563028 − 1.29730i
1.01749 − 0.982201i
−1.84709 − 0.766985i 2.58484 + 1.49236i 2.82347 + 2.83338i −3.40268 − 5.89361i −3.62981 − 4.73905i 0 −3.04204 − 7.39905i −0.0457311 − 0.0792086i 1.76474 + 13.4958i
67.2 −1.64441 + 1.13838i −1.35124 − 0.780139i 1.40817 − 3.74394i 1.71871 + 2.97689i 3.11009 − 0.255361i 0 1.94643 + 7.75960i −3.28276 − 5.68592i −6.21511 − 2.93868i
67.3 0.259316 − 1.98312i −2.58484 − 1.49236i −3.86551 − 1.02851i −3.40268 − 5.89361i −3.62981 + 4.73905i 0 −3.04204 + 7.39905i −0.0457311 − 0.0792086i −12.5701 + 5.21960i
67.4 0.489450 + 1.93919i 3.93608 + 2.27250i −3.52088 + 1.89827i 0.683969 + 1.18467i −2.48028 + 8.74507i 0 −5.40439 − 5.89853i 5.82850 + 10.0953i −1.96252 + 1.90618i
67.5 1.43466 + 1.39347i −3.93608 − 2.27250i 0.116490 + 3.99830i 0.683969 + 1.18467i −2.48028 − 8.74507i 0 −5.40439 + 5.89853i 5.82850 + 10.0953i −0.669537 + 2.65269i
67.6 1.80807 − 0.854909i 1.35124 + 0.780139i 2.53826 − 3.09148i 1.71871 + 2.97689i 3.11009 + 0.255361i 0 1.94643 − 7.75960i −3.28276 − 5.68592i 5.65253 + 3.91310i
79.1 −1.84709 + 0.766985i 2.58484 − 1.49236i 2.82347 − 2.83338i −3.40268 + 5.89361i −3.62981 + 4.73905i 0 −3.04204 + 7.39905i −0.0457311 + 0.0792086i 1.76474 − 13.4958i
79.2 −1.64441 − 1.13838i −1.35124 + 0.780139i 1.40817 + 3.74394i 1.71871 − 2.97689i 3.11009 + 0.255361i 0 1.94643 − 7.75960i −3.28276 + 5.68592i −6.21511 + 2.93868i
79.3 0.259316 + 1.98312i −2.58484 + 1.49236i −3.86551 + 1.02851i −3.40268 + 5.89361i −3.62981 − 4.73905i 0 −3.04204 − 7.39905i −0.0457311 + 0.0792086i −12.5701 − 5.21960i
79.4 0.489450 − 1.93919i 3.93608 − 2.27250i −3.52088 − 1.89827i 0.683969 − 1.18467i −2.48028 − 8.74507i 0 −5.40439 + 5.89853i 5.82850 − 10.0953i −1.96252 − 1.90618i
79.5 1.43466 − 1.39347i −3.93608 + 2.27250i 0.116490 − 3.99830i 0.683969 − 1.18467i −2.48028 + 8.74507i 0 −5.40439 − 5.89853i 5.82850 − 10.0953i −0.669537 − 2.65269i
79.6 1.80807 + 0.854909i 1.35124 − 0.780139i 2.53826 + 3.09148i 1.71871 − 2.97689i 3.11009 − 0.255361i 0 1.94643 + 7.75960i −3.28276 + 5.68592i 5.65253 − 3.91310i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.3.g.j 12
4.b odd 2 1 inner 196.3.g.j 12
7.b odd 2 1 196.3.g.k 12
7.c even 3 1 196.3.c.g 6
7.c even 3 1 inner 196.3.g.j 12
7.d odd 6 1 28.3.c.a ✓ 6
7.d odd 6 1 196.3.g.k 12
21.g even 6 1 252.3.g.a 6
28.d even 2 1 196.3.g.k 12
28.f even 6 1 28.3.c.a ✓ 6
28.f even 6 1 196.3.g.k 12
28.g odd 6 1 196.3.c.g 6
28.g odd 6 1 inner 196.3.g.j 12
56.j odd 6 1 448.3.d.d 6
56.m even 6 1 448.3.d.d 6
84.j odd 6 1 252.3.g.a 6
112.v even 12 2 1792.3.g.g 12
112.x odd 12 2 1792.3.g.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.3.c.a ✓ 6 7.d odd 6 1
28.3.c.a ✓ 6 28.f even 6 1
196.3.c.g 6 7.c even 3 1
196.3.c.g 6 28.g odd 6 1
196.3.g.j 12 1.a even 1 1 trivial
196.3.g.j 12 4.b odd 2 1 inner
196.3.g.j 12 7.c even 3 1 inner
196.3.g.j 12 28.g odd 6 1 inner
196.3.g.k 12 7.b odd 2 1
196.3.g.k 12 7.d odd 6 1
196.3.g.k 12 28.d even 2 1
196.3.g.k 12 28.f even 6 1
252.3.g.a 6 21.g even 6 1
252.3.g.a 6 84.j odd 6 1
448.3.d.d 6 56.j odd 6 1
448.3.d.d 6 56.m even 6 1
1792.3.g.g 12 112.v even 12 2
1792.3.g.g 12 112.x odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(196, [\chi])\):

\( T_{3}^{12} - 32T_{3}^{10} + 768T_{3}^{8} - 7296T_{3}^{6} + 51200T_{3}^{4} - 114688T_{3}^{2} + 200704 \) Copy content Toggle raw display
\( T_{5}^{6} + 2T_{5}^{5} + 32T_{5}^{4} - 120T_{5}^{3} + 720T_{5}^{2} - 896T_{5} + 1024 \) Copy content Toggle raw display
\( T_{11}^{12} - 512 T_{11}^{10} + 182272 T_{11}^{8} - 33955840 T_{11}^{6} + 4603248640 T_{11}^{4} + \cdots + 12036125753344 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$3$ \( T^{12} - 32 T^{10} + \cdots + 200704 \) Copy content Toggle raw display
$5$ \( (T^{6} + 2 T^{5} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 12036125753344 \) Copy content Toggle raw display
$13$ \( (T^{3} + 6 T^{2} + \cdots - 1712)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 2 T^{5} + \cdots + 238144)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 66\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 13153337344 \) Copy content Toggle raw display
$29$ \( (T^{3} + 18 T^{2} + \cdots + 4952)^{4} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 1973822685184 \) Copy content Toggle raw display
$37$ \( (T^{6} + 14 T^{5} + \cdots + 18731584)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 10 T^{2} + \cdots + 15368)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 4096 T^{4} + \cdots + 1477439488)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 21\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{6} + 46 T^{5} + \cdots + 72863296)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{6} + 82 T^{5} + \cdots + 57032704)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 84\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T^{6} + 4928 T^{4} + \cdots + 1438646272)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 66 T^{5} + \cdots + 13601344)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 33\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( (T^{6} + 25600 T^{4} + \cdots + 95209408)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 174 T^{5} + \cdots + 385501908544)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 126 T^{2} + \cdots - 196184)^{4} \) Copy content Toggle raw display
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