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Properties

Label 49.2.e.b
Level $49$
Weight $2$
Character orbit 49.e
Analytic conductor $0.391$
Analytic rank $0$
Dimension $12$
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] = [49,2,Mod(8,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 49.e (of order \(7\), degree \(6\), 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: \(0.391266969904\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\Q(\zeta_{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{11} - \beta_{9} + \cdots + \beta_{2}) q^{2}+ \cdots + (2 \beta_{10} - 2 \beta_{8} + \beta_{7} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{11} - \beta_{9} + \cdots + \beta_{2}) q^{2}+ \cdots + (\beta_{11} - 10 \beta_{10} + 6 \beta_{9} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} + 2 q^{4} - 7 q^{5} - 7 q^{6} - 7 q^{7} + 6 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} + 2 q^{4} - 7 q^{5} - 7 q^{6} - 7 q^{7} + 6 q^{8} - 8 q^{9} + 14 q^{10} - 8 q^{11} - 14 q^{12} + 7 q^{13} - 28 q^{14} - 7 q^{15} + 16 q^{16} + 20 q^{18} - 14 q^{19} + 7 q^{20} + 21 q^{21} + 13 q^{22} - 2 q^{23} + 3 q^{25} - 7 q^{26} + 21 q^{27} + 14 q^{28} - 11 q^{29} + 42 q^{30} - 14 q^{31} - 24 q^{32} + 35 q^{33} - 42 q^{34} + 21 q^{35} - 13 q^{36} - 30 q^{37} + 21 q^{38} + 21 q^{41} - 35 q^{42} + 17 q^{43} - 6 q^{44} - 49 q^{45} - 16 q^{46} - 21 q^{47} + 7 q^{49} - 46 q^{50} + 7 q^{51} - 7 q^{52} + 6 q^{53} + 42 q^{54} - 28 q^{55} - 14 q^{56} + 7 q^{57} - 32 q^{58} + 14 q^{59} - 28 q^{60} - 7 q^{61} + 56 q^{62} + 14 q^{63} + 14 q^{64} + 14 q^{65} - 28 q^{66} + 48 q^{67} + 56 q^{68} - 7 q^{69} + 21 q^{70} - 39 q^{71} - 4 q^{72} + 42 q^{73} + 61 q^{74} + 7 q^{75} - 28 q^{76} + 21 q^{77} - 16 q^{79} + 42 q^{80} - 25 q^{81} + 28 q^{82} - 7 q^{83} + 42 q^{84} + 28 q^{85} + 17 q^{86} + 7 q^{87} - 11 q^{88} - 14 q^{89} - 14 q^{90} - 21 q^{91} + 16 q^{92} - 70 q^{93} - 49 q^{94} - 7 q^{95} - 70 q^{96} - 28 q^{97} - 28 q^{98} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{21}^{2} + \zeta_{21} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{21}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{21}^{5} + \zeta_{21} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{21}^{6} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{21}^{8} + \zeta_{21} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{21}^{9} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \zeta_{21}^{11} + \zeta_{21} \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \zeta_{21}^{8} + \zeta_{21}^{7} \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -\zeta_{21}^{10} - \zeta_{21}^{8} \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( -\zeta_{21}^{11} + \zeta_{21}^{9} - \zeta_{21}^{8} + \zeta_{21}^{6} - \zeta_{21}^{4} + \zeta_{21}^{3} - \zeta_{21} + 1 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( -\zeta_{21}^{11} + \zeta_{21}^{10} - \zeta_{21}^{8} + \zeta_{21}^{7} - \zeta_{21}^{5} + \zeta_{21}^{3} - \zeta_{21}^{2} + \zeta_{21} + 1 \) Copy content Toggle raw display

Display \(\zeta_{21}^j\) in terms of \(\beta_i\)

\(\zeta_{21}\)\(=\) \( ( \beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} + 3\beta_{5} + \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 7 \) Copy content Toggle raw display
\(\zeta_{21}^{2}\)\(=\) \( ( -\beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} - 3\beta_{5} - \beta_{3} + \beta_{2} + 6\beta _1 + 1 ) / 7 \) Copy content Toggle raw display
\(\zeta_{21}^{3}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{21}^{4}\)\(=\) \( ( \beta_{11} - 7 \beta_{10} + \beta_{9} - \beta_{8} - 6 \beta_{7} + 7 \beta_{6} - 4 \beta_{5} + 7 \beta_{4} + \cdots + 6 ) / 7 \) Copy content Toggle raw display
\(\zeta_{21}^{5}\)\(=\) \( ( -\beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} - 3\beta_{5} + 6\beta_{3} + \beta_{2} - \beta _1 + 1 ) / 7 \) Copy content Toggle raw display
\(\zeta_{21}^{6}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\zeta_{21}^{7}\)\(=\) \( ( \beta_{11} + \beta_{9} + 6\beta_{8} + \beta_{7} - 4\beta_{5} + \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 7 \) Copy content Toggle raw display
\(\zeta_{21}^{8}\)\(=\) \( ( -\beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} + 4\beta_{5} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 7 \) Copy content Toggle raw display
\(\zeta_{21}^{9}\)\(=\) \( \beta_{6} \) Copy content Toggle raw display
\(\zeta_{21}^{10}\)\(=\) \( ( \beta_{11} - 6\beta_{9} - \beta_{8} + \beta_{7} - 4\beta_{5} + \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 7 \) Copy content Toggle raw display
\(\zeta_{21}^{11}\)\(=\) \( ( -\beta_{11} - \beta_{9} + \beta_{8} + 6\beta_{7} - 3\beta_{5} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 7 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{21}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
−0.988831 + 0.149042i
0.365341 − 0.930874i
−0.733052 − 0.680173i
0.955573 − 0.294755i
0.0747301 − 0.997204i
0.826239 + 0.563320i
0.0747301 + 0.997204i
0.826239 − 0.563320i
−0.733052 + 0.680173i
0.955573 + 0.294755i
−0.988831 − 0.149042i
0.365341 + 0.930874i
−1.19158 − 1.49419i 1.55929 − 0.750915i −0.367711 + 1.61105i −1.91647 + 0.922924i −2.98003 − 1.43511i 1.91879 − 1.82161i −0.598393 + 0.288171i −0.00295512 + 0.00370560i 3.66265 + 1.76384i
8.2 0.914101 + 1.14625i −0.880843 + 0.424191i −0.0332580 + 0.145713i −0.830509 + 0.399952i −1.29141 − 0.621909i −0.938402 − 2.47374i 2.44440 − 1.17716i −1.27452 + 1.59820i −1.21761 − 0.586371i
15.1 0.0332580 − 0.145713i −1.81507 + 2.27603i 1.78181 + 0.858075i 1.36967 − 1.71752i 0.271281 + 0.340175i −2.64558 + 0.0302261i 0.370666 − 0.464800i −1.21825 − 5.33750i −0.204712 − 0.256700i
15.2 0.367711 − 1.61105i 0.290611 − 0.364415i −0.658322 − 0.317031i −2.42463 + 3.04039i −0.480228 − 0.602187i −1.62586 − 2.08724i 1.30778 − 1.63991i 0.619220 + 2.71298i 4.00665 + 5.02418i
22.1 −1.78181 + 0.858075i 0.590232 − 2.58597i 1.19158 − 1.49419i 0.359497 − 1.57506i 1.16728 + 5.11418i 1.95991 + 1.77729i 0.0391023 − 0.171318i −3.63598 − 1.75100i 0.710963 + 3.11493i
22.2 0.658322 − 0.317031i 0.255779 − 1.12064i −0.914101 + 1.14625i −0.0575591 + 0.252183i −0.186893 − 0.818832i −2.16885 − 1.51528i −0.563561 + 2.46912i 1.51249 + 0.728379i 0.0420574 + 0.184265i
29.1 −1.78181 − 0.858075i 0.590232 + 2.58597i 1.19158 + 1.49419i 0.359497 + 1.57506i 1.16728 − 5.11418i 1.95991 − 1.77729i 0.0391023 + 0.171318i −3.63598 + 1.75100i 0.710963 − 3.11493i
29.2 0.658322 + 0.317031i 0.255779 + 1.12064i −0.914101 − 1.14625i −0.0575591 − 0.252183i −0.186893 + 0.818832i −2.16885 + 1.51528i −0.563561 − 2.46912i 1.51249 − 0.728379i 0.0420574 − 0.184265i
36.1 0.0332580 + 0.145713i −1.81507 − 2.27603i 1.78181 − 0.858075i 1.36967 + 1.71752i 0.271281 − 0.340175i −2.64558 − 0.0302261i 0.370666 + 0.464800i −1.21825 + 5.33750i −0.204712 + 0.256700i
36.2 0.367711 + 1.61105i 0.290611 + 0.364415i −0.658322 + 0.317031i −2.42463 − 3.04039i −0.480228 + 0.602187i −1.62586 + 2.08724i 1.30778 + 1.63991i 0.619220 − 2.71298i 4.00665 − 5.02418i
43.1 −1.19158 + 1.49419i 1.55929 + 0.750915i −0.367711 − 1.61105i −1.91647 − 0.922924i −2.98003 + 1.43511i 1.91879 + 1.82161i −0.598393 − 0.288171i −0.00295512 − 0.00370560i 3.66265 − 1.76384i
43.2 0.914101 − 1.14625i −0.880843 − 0.424191i −0.0332580 − 0.145713i −0.830509 − 0.399952i −1.29141 + 0.621909i −0.938402 + 2.47374i 2.44440 + 1.17716i −1.27452 − 1.59820i −1.21761 + 0.586371i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.2.e.b ✓ 12
3.b odd 2 1 441.2.u.b 12
4.b odd 2 1 784.2.u.b 12
7.b odd 2 1 343.2.e.b 12
7.c even 3 1 343.2.g.b 12
7.c even 3 1 343.2.g.d 12
7.d odd 6 1 343.2.g.a 12
7.d odd 6 1 343.2.g.c 12
49.e even 7 1 inner 49.2.e.b ✓ 12
49.e even 7 1 2401.2.a.c 6
49.f odd 14 1 343.2.e.b 12
49.f odd 14 1 2401.2.a.d 6
49.g even 21 1 343.2.g.b 12
49.g even 21 1 343.2.g.d 12
49.h odd 42 1 343.2.g.a 12
49.h odd 42 1 343.2.g.c 12
147.l odd 14 1 441.2.u.b 12
196.k odd 14 1 784.2.u.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.e.b ✓ 12 1.a even 1 1 trivial
49.2.e.b ✓ 12 49.e even 7 1 inner
343.2.e.b 12 7.b odd 2 1
343.2.e.b 12 49.f odd 14 1
343.2.g.a 12 7.d odd 6 1
343.2.g.a 12 49.h odd 42 1
343.2.g.b 12 7.c even 3 1
343.2.g.b 12 49.g even 21 1
343.2.g.c 12 7.d odd 6 1
343.2.g.c 12 49.h odd 42 1
343.2.g.d 12 7.c even 3 1
343.2.g.d 12 49.g even 21 1
441.2.u.b 12 3.b odd 2 1
441.2.u.b 12 147.l odd 14 1
784.2.u.b 12 4.b odd 2 1
784.2.u.b 12 196.k odd 14 1
2401.2.a.c 6 49.e even 7 1
2401.2.a.d 6 49.f odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 2 T_{2}^{11} + 3 T_{2}^{10} + 4 T_{2}^{9} + 12 T_{2}^{8} + 6 T_{2}^{7} + 7 T_{2}^{6} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(49, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} + 7 T^{10} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{12} + 7 T^{11} + \cdots + 49 \) Copy content Toggle raw display
$7$ \( T^{12} + 7 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} + 8 T^{11} + \cdots + 1849 \) Copy content Toggle raw display
$13$ \( T^{12} - 7 T^{11} + \cdots + 49 \) Copy content Toggle raw display
$17$ \( T^{12} + 35 T^{10} + \cdots + 3087049 \) Copy content Toggle raw display
$19$ \( (T^{6} + 7 T^{5} + \cdots + 2107)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{12} + 11 T^{11} + \cdots + 1681 \) Copy content Toggle raw display
$31$ \( (T^{6} + 7 T^{5} + \cdots - 8183)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 295118041 \) Copy content Toggle raw display
$41$ \( T^{12} - 21 T^{11} + \cdots + 10413529 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 200307409 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 3021810841 \) Copy content Toggle raw display
$53$ \( T^{12} - 6 T^{11} + \cdots + 2985984 \) Copy content Toggle raw display
$59$ \( T^{12} - 14 T^{11} + \cdots + 54125449 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 261242569 \) Copy content Toggle raw display
$67$ \( (T^{6} - 24 T^{5} + \cdots - 293)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 312925003609 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 460917961 \) Copy content Toggle raw display
$79$ \( (T^{6} + 8 T^{5} + \cdots - 21629)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 118178641 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 89755965649 \) Copy content Toggle raw display
$97$ \( (T^{6} + 14 T^{5} + \cdots + 18571)^{2} \) Copy content Toggle raw display
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