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Properties

Label 196.4.f.c
Level $196$
Weight $4$
Character orbit 196.f
Analytic conductor $11.564$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [196,4,Mod(19,196)]
 
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, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.19");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 196.f (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: \(11.5643743611\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 78 x^{14} + 280 x^{13} + 2659 x^{12} - 8424 x^{11} - 49830 x^{10} + 138796 x^{9} + \cdots + 19109188 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{26} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_1 + 1) q^{2} + \beta_{3} q^{3} + (\beta_{12} + \beta_{10} + \cdots - 2 \beta_1) q^{4}+ \cdots + (3 \beta_{12} + 3 \beta_{9} + \cdots - 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_1 + 1) q^{2} + \beta_{3} q^{3} + (\beta_{12} + \beta_{10} + \cdots - 2 \beta_1) q^{4}+ \cdots + (15 \beta_{10} + 5 \beta_{9} + \cdots - 19 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} + 16 q^{4} - 64 q^{8} - 104 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{2} + 16 q^{4} - 64 q^{8} - 104 q^{9} - 64 q^{16} - 88 q^{18} + 480 q^{22} + 472 q^{25} - 1184 q^{29} - 256 q^{30} - 1152 q^{32} - 1952 q^{36} - 1392 q^{37} + 1184 q^{44} + 816 q^{46} + 3376 q^{50} + 1168 q^{53} - 384 q^{57} + 560 q^{58} - 2944 q^{60} - 6656 q^{64} - 448 q^{65} + 1184 q^{72} + 496 q^{74} + 15360 q^{78} + 4984 q^{81} + 2048 q^{85} - 240 q^{86} - 3776 q^{88} - 7616 q^{92} + 2304 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 78 x^{14} + 280 x^{13} + 2659 x^{12} - 8424 x^{11} - 49830 x^{10} + 138796 x^{9} + \cdots + 19109188 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 42\!\cdots\!88 \nu^{15} + \cdots - 30\!\cdots\!54 ) / 48\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 25\!\cdots\!02 \nu^{15} + \cdots - 55\!\cdots\!16 ) / 12\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 51\!\cdots\!63 \nu^{15} + \cdots - 41\!\cdots\!84 ) / 24\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 58\!\cdots\!68 \nu^{15} + \cdots + 73\!\cdots\!88 ) / 24\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 73\!\cdots\!16 \nu^{15} + \cdots + 13\!\cdots\!06 ) / 24\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 80\!\cdots\!28 \nu^{15} + \cdots - 97\!\cdots\!34 ) / 24\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19\!\cdots\!16 \nu^{15} + \cdots + 21\!\cdots\!18 ) / 48\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 22\!\cdots\!97 \nu^{15} + \cdots + 82\!\cdots\!92 ) / 47\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 25\!\cdots\!79 \nu^{15} + \cdots - 74\!\cdots\!09 ) / 45\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 30\!\cdots\!52 \nu^{15} + \cdots - 16\!\cdots\!86 ) / 48\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 59\!\cdots\!91 \nu^{15} + \cdots + 13\!\cdots\!76 ) / 47\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 77\!\cdots\!68 \nu^{15} + \cdots - 89\!\cdots\!54 ) / 48\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 30\!\cdots\!74 \nu^{15} + \cdots + 30\!\cdots\!48 ) / 12\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 62\!\cdots\!16 \nu^{15} + \cdots - 24\!\cdots\!52 ) / 24\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 63\!\cdots\!37 \nu^{15} + \cdots - 19\!\cdots\!44 ) / 24\!\cdots\!90 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 2\beta_{4} + 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{12} + 2 \beta_{10} - 3 \beta_{9} - 2 \beta_{8} + 3 \beta_{7} - \beta_{6} - 2 \beta_{4} + \cdots + 40 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{15} - 3 \beta_{13} - 6 \beta_{11} - 6 \beta_{10} + 2 \beta_{9} + 30 \beta_{7} + 4 \beta_{6} + \cdots + 140 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{14} - 12 \beta_{13} - 27 \beta_{12} + 8 \beta_{11} - 63 \beta_{9} - 36 \beta_{8} + \cdots + 462 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 55 \beta_{15} + 20 \beta_{14} - 95 \beta_{13} - 14 \beta_{12} - 140 \beta_{11} - 378 \beta_{10} + \cdots + 3640 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 21 \beta_{15} - 61 \beta_{14} - 387 \beta_{13} - 240 \beta_{12} + 232 \beta_{11} - 829 \beta_{10} + \cdots + 5386 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 588 \beta_{15} + 1017 \beta_{14} - 2338 \beta_{13} - 1734 \beta_{12} - 2170 \beta_{11} - 10248 \beta_{10} + \cdots + 75044 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1228 \beta_{15} - 1064 \beta_{14} - 9212 \beta_{13} + 2307 \beta_{12} + 3752 \beta_{11} - 31149 \beta_{10} + \cdots + 70104 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1269 \beta_{15} + 28767 \beta_{14} - 56241 \beta_{13} - 68184 \beta_{12} - 22926 \beta_{11} + \cdots + 1366748 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 37836 \beta_{15} - 6021 \beta_{14} - 190572 \beta_{13} + 150437 \beta_{12} + 41212 \beta_{11} + \cdots + 1203266 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 112827 \beta_{15} + 637250 \beta_{14} - 1343595 \beta_{13} - 1743474 \beta_{12} - 90662 \beta_{11} + \cdots + 23483680 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 849078 \beta_{15} + 348778 \beta_{14} - 3640242 \beta_{13} + 3844038 \beta_{12} + 245532 \beta_{11} + \cdots + 25508930 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 3111134 \beta_{15} + 12469075 \beta_{14} - 31209126 \beta_{13} - 34309394 \beta_{12} + \cdots + 402777964 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 15289359 \beta_{15} + 17048486 \beta_{14} - 67221869 \beta_{13} + 70258363 \beta_{12} - 1575028 \beta_{11} + \cdots + 566532608 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 48752373 \beta_{15} + 232963361 \beta_{14} - 692337495 \beta_{13} - 546579060 \beta_{12} + \cdots + 7230140580 ) / 8 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
4.52578 − 0.143277i
−3.02211 − 0.143277i
1.10894 − 1.32242i
−3.68996 − 1.32242i
3.52211 − 0.722748i
−4.02578 − 0.722748i
4.18996 + 0.456399i
−0.608943 + 0.456399i
4.52578 + 0.143277i
−3.02211 + 0.143277i
1.10894 + 1.32242i
−3.68996 + 1.32242i
3.52211 + 0.722748i
−4.02578 + 0.722748i
4.18996 − 0.456399i
−0.608943 − 0.456399i
−2.63019 − 1.04025i −3.77394 − 6.53666i 5.83577 + 5.47209i 7.57561 + 4.37378i 3.12644 + 21.1185i 0 −9.65685 − 20.4633i −14.9853 + 25.9553i −15.3755 − 19.3844i
19.2 −2.63019 − 1.04025i 3.77394 + 6.53666i 5.83577 + 5.47209i −7.57561 − 4.37378i −3.12644 − 21.1185i 0 −9.65685 − 20.4633i −14.9853 + 25.9553i 15.3755 + 19.3844i
19.3 −0.0690906 − 2.82758i −2.39945 − 4.15597i −7.99045 + 0.390719i −14.7855 − 8.53640i −11.5856 + 7.07178i 0 1.65685 + 22.5667i 1.98528 − 3.43861i −23.1158 + 42.3969i
19.4 −0.0690906 − 2.82758i 2.39945 + 4.15597i −7.99045 + 0.390719i 14.7855 + 8.53640i 11.5856 − 7.07178i 0 1.65685 + 22.5667i 1.98528 − 3.43861i 23.1158 − 42.3969i
19.5 2.21597 + 1.75769i −3.77394 − 6.53666i 1.82108 + 7.78997i −7.57561 − 4.37378i 3.12644 − 21.1185i 0 −9.65685 + 20.4633i −14.9853 + 25.9553i −9.09962 − 23.0077i
19.6 2.21597 + 1.75769i 3.77394 + 6.53666i 1.82108 + 7.78997i 7.57561 + 4.37378i −3.12644 + 21.1185i 0 −9.65685 + 20.4633i −14.9853 + 25.9553i 9.09962 + 23.0077i
19.7 2.48330 − 1.35396i −2.39945 − 4.15597i 4.33360 − 6.72458i 14.7855 + 8.53640i −11.5856 − 7.07178i 0 1.65685 − 22.5667i 1.98528 − 3.43861i 48.2747 + 1.17957i
19.8 2.48330 − 1.35396i 2.39945 + 4.15597i 4.33360 − 6.72458i −14.7855 − 8.53640i 11.5856 + 7.07178i 0 1.65685 − 22.5667i 1.98528 − 3.43861i −48.2747 − 1.17957i
31.1 −2.63019 + 1.04025i −3.77394 + 6.53666i 5.83577 − 5.47209i 7.57561 − 4.37378i 3.12644 − 21.1185i 0 −9.65685 + 20.4633i −14.9853 − 25.9553i −15.3755 + 19.3844i
31.2 −2.63019 + 1.04025i 3.77394 − 6.53666i 5.83577 − 5.47209i −7.57561 + 4.37378i −3.12644 + 21.1185i 0 −9.65685 + 20.4633i −14.9853 − 25.9553i 15.3755 − 19.3844i
31.3 −0.0690906 + 2.82758i −2.39945 + 4.15597i −7.99045 − 0.390719i −14.7855 + 8.53640i −11.5856 − 7.07178i 0 1.65685 − 22.5667i 1.98528 + 3.43861i −23.1158 − 42.3969i
31.4 −0.0690906 + 2.82758i 2.39945 − 4.15597i −7.99045 − 0.390719i 14.7855 − 8.53640i 11.5856 + 7.07178i 0 1.65685 − 22.5667i 1.98528 + 3.43861i 23.1158 + 42.3969i
31.5 2.21597 − 1.75769i −3.77394 + 6.53666i 1.82108 − 7.78997i −7.57561 + 4.37378i 3.12644 + 21.1185i 0 −9.65685 − 20.4633i −14.9853 − 25.9553i −9.09962 + 23.0077i
31.6 2.21597 − 1.75769i 3.77394 − 6.53666i 1.82108 − 7.78997i 7.57561 − 4.37378i −3.12644 − 21.1185i 0 −9.65685 − 20.4633i −14.9853 − 25.9553i 9.09962 − 23.0077i
31.7 2.48330 + 1.35396i −2.39945 + 4.15597i 4.33360 + 6.72458i 14.7855 − 8.53640i −11.5856 + 7.07178i 0 1.65685 + 22.5667i 1.98528 + 3.43861i 48.2747 − 1.17957i
31.8 2.48330 + 1.35396i 2.39945 − 4.15597i 4.33360 + 6.72458i −14.7855 + 8.53640i 11.5856 − 7.07178i 0 1.65685 + 22.5667i 1.98528 + 3.43861i −48.2747 + 1.17957i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
28.d even 2 1 inner
28.f even 6 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 196.4.f.c 16
4.b odd 2 1 inner 196.4.f.c 16
7.b odd 2 1 inner 196.4.f.c 16
7.c even 3 1 28.4.d.b ✓ 8
7.c even 3 1 inner 196.4.f.c 16
7.d odd 6 1 28.4.d.b ✓ 8
7.d odd 6 1 inner 196.4.f.c 16
21.g even 6 1 252.4.b.d 8
21.h odd 6 1 252.4.b.d 8
28.d even 2 1 inner 196.4.f.c 16
28.f even 6 1 28.4.d.b ✓ 8
28.f even 6 1 inner 196.4.f.c 16
28.g odd 6 1 28.4.d.b ✓ 8
28.g odd 6 1 inner 196.4.f.c 16
56.j odd 6 1 448.4.f.d 8
56.k odd 6 1 448.4.f.d 8
56.m even 6 1 448.4.f.d 8
56.p even 6 1 448.4.f.d 8
84.j odd 6 1 252.4.b.d 8
84.n even 6 1 252.4.b.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.d.b ✓ 8 7.c even 3 1
28.4.d.b ✓ 8 7.d odd 6 1
28.4.d.b ✓ 8 28.f even 6 1
28.4.d.b ✓ 8 28.g odd 6 1
196.4.f.c 16 1.a even 1 1 trivial
196.4.f.c 16 4.b odd 2 1 inner
196.4.f.c 16 7.b odd 2 1 inner
196.4.f.c 16 7.c even 3 1 inner
196.4.f.c 16 7.d odd 6 1 inner
196.4.f.c 16 28.d even 2 1 inner
196.4.f.c 16 28.f even 6 1 inner
196.4.f.c 16 28.g odd 6 1 inner
252.4.b.d 8 21.g even 6 1
252.4.b.d 8 21.h odd 6 1
252.4.b.d 8 84.j odd 6 1
252.4.b.d 8 84.n even 6 1
448.4.f.d 8 56.j odd 6 1
448.4.f.d 8 56.k odd 6 1
448.4.f.d 8 56.m even 6 1
448.4.f.d 8 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(196, [\chi])\):

\( T_{3}^{8} + 80T_{3}^{6} + 5088T_{3}^{4} + 104960T_{3}^{2} + 1721344 \) Copy content Toggle raw display
\( T_{5}^{8} - 368T_{5}^{6} + 113120T_{5}^{4} - 8207872T_{5}^{2} + 497468416 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 4 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + 80 T^{6} + \cdots + 1721344)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 368 T^{6} + \cdots + 497468416)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 1720 T^{6} + \cdots + 73984)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 7792 T^{2} + 11798816)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 32602090110976)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 2128 T^{6} + \cdots + 481702626304)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 17\!\cdots\!04)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 148 T - 4892)^{8} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 348 T^{3} + \cdots + 576192016)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 58304 T^{2} + 342946304)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 34360 T^{2} + 141396752)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 32\!\cdots\!44)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 292 T^{3} + \cdots + 26737482256)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 46\!\cdots\!44)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 41549059572736)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 14\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 275576 T^{2} + 9248152592)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 32\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 11\!\cdots\!24)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1267920 T^{2} + 340971272992)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 43\!\cdots\!36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 1135775350784)^{4} \) Copy content Toggle raw display
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