Properties

Label 49.2.e.a
Level $49$
Weight $2$
Character orbit 49.e
Analytic conductor $0.391$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{14}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{14}^{5} - \zeta_{14}^{4} + \cdots - 1) q^{2}+ \cdots + (\zeta_{14}^{5} - \zeta_{14}^{4} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{14}^{5} - \zeta_{14}^{4} + \cdots - 1) q^{2}+ \cdots + (5 \zeta_{14}^{5} - 6 \zeta_{14}^{4} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{3} - 7 q^{4} + 6 q^{5} - 2 q^{6} + 7 q^{7} - q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} - 3 q^{3} - 7 q^{4} + 6 q^{5} - 2 q^{6} + 7 q^{7} - q^{8} - 6 q^{9} - 3 q^{10} + 2 q^{11} + 7 q^{14} + 4 q^{15} - 3 q^{16} + 4 q^{17} - 4 q^{18} - 8 q^{19} - 8 q^{22} - 10 q^{23} + 11 q^{24} + 11 q^{25} + 28 q^{26} - 9 q^{27} - 14 q^{28} - 16 q^{29} - 2 q^{30} - 10 q^{31} + 21 q^{32} + 6 q^{33} + 5 q^{34} + 14 q^{36} + 4 q^{37} + 4 q^{38} - 7 q^{39} - 15 q^{40} - 14 q^{42} + 12 q^{43} - 7 q^{44} - 6 q^{45} + 19 q^{46} - 15 q^{47} + 12 q^{48} + 7 q^{49} - 30 q^{50} + 12 q^{51} - 26 q^{53} + 8 q^{54} - 19 q^{55} + 4 q^{57} + q^{58} + 11 q^{59} - 7 q^{60} - 8 q^{61} + 5 q^{62} - 7 q^{63} + 13 q^{64} + 35 q^{65} - 3 q^{66} - 12 q^{67} - 28 q^{68} + 12 q^{69} + 14 q^{70} + 5 q^{71} + 8 q^{72} + 4 q^{73} - 2 q^{74} - 16 q^{75} + 28 q^{76} + 35 q^{77} - 7 q^{78} + 60 q^{79} - 10 q^{80} - 23 q^{81} - 14 q^{82} - 14 q^{83} - 14 q^{84} - 17 q^{85} - 20 q^{86} + 36 q^{87} + 16 q^{88} + 13 q^{89} + 10 q^{90} - 70 q^{91} + 28 q^{92} + 5 q^{93} - 31 q^{94} - q^{95} - 28 q^{96} + 21 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-\zeta_{14}\)

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
−0.623490 + 0.781831i
0.222521 + 0.974928i
0.900969 + 0.433884i
0.900969 0.433884i
0.222521 0.974928i
−0.623490 0.781831i
−0.500000 0.626980i −0.500000 + 0.240787i 0.301938 1.32288i 2.52446 1.21572i 0.400969 + 0.193096i −1.14795 + 2.38374i −2.42543 + 1.16802i −1.67845 + 2.10471i −2.02446 0.974928i
15.1 −0.500000 + 2.19064i −0.500000 + 0.626980i −2.74698 1.32288i 0.153989 0.193096i −1.12349 1.40881i 2.06853 1.64960i 1.46950 1.84270i 0.524459 + 2.29780i 0.346011 + 0.433884i
22.1 −0.500000 + 0.240787i −0.500000 + 2.19064i −1.05496 + 1.32288i 0.321552 1.40881i −0.277479 1.21572i 2.57942 0.588735i 0.455927 1.99755i −1.84601 0.888992i 0.178448 + 0.781831i
29.1 −0.500000 0.240787i −0.500000 2.19064i −1.05496 1.32288i 0.321552 + 1.40881i −0.277479 + 1.21572i 2.57942 + 0.588735i 0.455927 + 1.99755i −1.84601 + 0.888992i 0.178448 0.781831i
36.1 −0.500000 2.19064i −0.500000 0.626980i −2.74698 + 1.32288i 0.153989 + 0.193096i −1.12349 + 1.40881i 2.06853 + 1.64960i 1.46950 + 1.84270i 0.524459 2.29780i 0.346011 0.433884i
43.1 −0.500000 + 0.626980i −0.500000 0.240787i 0.301938 + 1.32288i 2.52446 + 1.21572i 0.400969 0.193096i −1.14795 2.38374i −2.42543 1.16802i −1.67845 2.10471i −2.02446 + 0.974928i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.1
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.a 6
3.b odd 2 1 441.2.u.a 6
4.b odd 2 1 784.2.u.a 6
7.b odd 2 1 343.2.e.a 6
7.c even 3 2 343.2.g.f 12
7.d odd 6 2 343.2.g.e 12
49.e even 7 1 inner 49.2.e.a 6
49.e even 7 1 2401.2.a.b 3
49.f odd 14 1 343.2.e.a 6
49.f odd 14 1 2401.2.a.a 3
49.g even 21 2 343.2.g.f 12
49.h odd 42 2 343.2.g.e 12
147.l odd 14 1 441.2.u.a 6
196.k odd 14 1 784.2.u.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.e.a 6 1.a even 1 1 trivial
49.2.e.a 6 49.e even 7 1 inner
343.2.e.a 6 7.b odd 2 1
343.2.e.a 6 49.f odd 14 1
343.2.g.e 12 7.d odd 6 2
343.2.g.e 12 49.h odd 42 2
343.2.g.f 12 7.c even 3 2
343.2.g.f 12 49.g even 21 2
441.2.u.a 6 3.b odd 2 1
441.2.u.a 6 147.l odd 14 1
784.2.u.a 6 4.b odd 2 1
784.2.u.a 6 196.k odd 14 1
2401.2.a.a 3 49.f odd 14 1
2401.2.a.b 3 49.e even 7 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 3T_{2}^{5} + 9T_{2}^{4} + 13T_{2}^{3} + 11T_{2}^{2} + 5T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(49, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} + 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} - 7 T^{5} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{6} - 2 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{6} + 14 T^{4} + \cdots + 41209 \) Copy content Toggle raw display
$17$ \( T^{6} - 4 T^{5} + \cdots + 1681 \) Copy content Toggle raw display
$19$ \( (T^{3} + 4 T^{2} - 11 T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 10 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$29$ \( T^{6} + 16 T^{5} + \cdots + 6889 \) Copy content Toggle raw display
$31$ \( (T^{3} + 5 T^{2} + 6 T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 4 T^{5} + \cdots + 10816 \) Copy content Toggle raw display
$41$ \( T^{6} + 56 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$43$ \( T^{6} - 12 T^{5} + \cdots + 841 \) Copy content Toggle raw display
$47$ \( T^{6} + 15 T^{5} + \cdots + 9409 \) Copy content Toggle raw display
$53$ \( T^{6} + 26 T^{5} + \cdots + 53824 \) Copy content Toggle raw display
$59$ \( T^{6} - 11 T^{5} + \cdots + 57121 \) Copy content Toggle raw display
$61$ \( T^{6} + 8 T^{5} + \cdots + 1849 \) Copy content Toggle raw display
$67$ \( (T^{3} + 6 T^{2} + 5 T - 13)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} - 5 T^{5} + \cdots + 85849 \) Copy content Toggle raw display
$73$ \( T^{6} - 4 T^{5} + \cdots + 5041 \) Copy content Toggle raw display
$79$ \( (T^{3} - 30 T^{2} + \cdots - 937)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 14 T^{5} + \cdots + 625681 \) Copy content Toggle raw display
$89$ \( T^{6} - 13 T^{5} + \cdots + 187489 \) Copy content Toggle raw display
$97$ \( (T^{3} - 7 T - 7)^{2} \) Copy content Toggle raw display
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