Properties

Label 49.4.c.a
Level $49$
Weight $4$
Character orbit 49.c
Analytic conductor $2.891$
Analytic rank $0$
Dimension $2$
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] = [49,4,Mod(18,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.18");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), 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: \(2.89109359028\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \zeta_{6} q^{2} + ( - 7 \zeta_{6} + 7) q^{3} + ( - 4 \zeta_{6} + 4) q^{4} + 7 \zeta_{6} q^{5} - 14 q^{6} - 24 q^{8} - 22 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \zeta_{6} q^{2} + ( - 7 \zeta_{6} + 7) q^{3} + ( - 4 \zeta_{6} + 4) q^{4} + 7 \zeta_{6} q^{5} - 14 q^{6} - 24 q^{8} - 22 \zeta_{6} q^{9} + ( - 14 \zeta_{6} + 14) q^{10} + ( - 5 \zeta_{6} + 5) q^{11} - 28 \zeta_{6} q^{12} + 14 q^{13} + 49 q^{15} + 16 \zeta_{6} q^{16} + (21 \zeta_{6} - 21) q^{17} + (44 \zeta_{6} - 44) q^{18} + 49 \zeta_{6} q^{19} + 28 q^{20} - 10 q^{22} + 159 \zeta_{6} q^{23} + (168 \zeta_{6} - 168) q^{24} + ( - 76 \zeta_{6} + 76) q^{25} - 28 \zeta_{6} q^{26} + 35 q^{27} + 58 q^{29} - 98 \zeta_{6} q^{30} + ( - 147 \zeta_{6} + 147) q^{31} + (160 \zeta_{6} - 160) q^{32} - 35 \zeta_{6} q^{33} + 42 q^{34} - 88 q^{36} - 219 \zeta_{6} q^{37} + ( - 98 \zeta_{6} + 98) q^{38} + ( - 98 \zeta_{6} + 98) q^{39} - 168 \zeta_{6} q^{40} - 350 q^{41} - 124 q^{43} - 20 \zeta_{6} q^{44} + ( - 154 \zeta_{6} + 154) q^{45} + ( - 318 \zeta_{6} + 318) q^{46} + 525 \zeta_{6} q^{47} + 112 q^{48} - 152 q^{50} + 147 \zeta_{6} q^{51} + ( - 56 \zeta_{6} + 56) q^{52} + (303 \zeta_{6} - 303) q^{53} - 70 \zeta_{6} q^{54} + 35 q^{55} + 343 q^{57} - 116 \zeta_{6} q^{58} + (105 \zeta_{6} - 105) q^{59} + ( - 196 \zeta_{6} + 196) q^{60} - 413 \zeta_{6} q^{61} - 294 q^{62} + 448 q^{64} + 98 \zeta_{6} q^{65} + (70 \zeta_{6} - 70) q^{66} + (415 \zeta_{6} - 415) q^{67} + 84 \zeta_{6} q^{68} + 1113 q^{69} - 432 q^{71} + 528 \zeta_{6} q^{72} + (1113 \zeta_{6} - 1113) q^{73} + (438 \zeta_{6} - 438) q^{74} - 532 \zeta_{6} q^{75} + 196 q^{76} - 196 q^{78} + 103 \zeta_{6} q^{79} + (112 \zeta_{6} - 112) q^{80} + ( - 839 \zeta_{6} + 839) q^{81} + 700 \zeta_{6} q^{82} - 1092 q^{83} - 147 q^{85} + 248 \zeta_{6} q^{86} + ( - 406 \zeta_{6} + 406) q^{87} + (120 \zeta_{6} - 120) q^{88} - 329 \zeta_{6} q^{89} - 308 q^{90} + 636 q^{92} - 1029 \zeta_{6} q^{93} + ( - 1050 \zeta_{6} + 1050) q^{94} + (343 \zeta_{6} - 343) q^{95} + 1120 \zeta_{6} q^{96} + 882 q^{97} - 110 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 7 q^{3} + 4 q^{4} + 7 q^{5} - 28 q^{6} - 48 q^{8} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 7 q^{3} + 4 q^{4} + 7 q^{5} - 28 q^{6} - 48 q^{8} - 22 q^{9} + 14 q^{10} + 5 q^{11} - 28 q^{12} + 28 q^{13} + 98 q^{15} + 16 q^{16} - 21 q^{17} - 44 q^{18} + 49 q^{19} + 56 q^{20} - 20 q^{22} + 159 q^{23} - 168 q^{24} + 76 q^{25} - 28 q^{26} + 70 q^{27} + 116 q^{29} - 98 q^{30} + 147 q^{31} - 160 q^{32} - 35 q^{33} + 84 q^{34} - 176 q^{36} - 219 q^{37} + 98 q^{38} + 98 q^{39} - 168 q^{40} - 700 q^{41} - 248 q^{43} - 20 q^{44} + 154 q^{45} + 318 q^{46} + 525 q^{47} + 224 q^{48} - 304 q^{50} + 147 q^{51} + 56 q^{52} - 303 q^{53} - 70 q^{54} + 70 q^{55} + 686 q^{57} - 116 q^{58} - 105 q^{59} + 196 q^{60} - 413 q^{61} - 588 q^{62} + 896 q^{64} + 98 q^{65} - 70 q^{66} - 415 q^{67} + 84 q^{68} + 2226 q^{69} - 864 q^{71} + 528 q^{72} - 1113 q^{73} - 438 q^{74} - 532 q^{75} + 392 q^{76} - 392 q^{78} + 103 q^{79} - 112 q^{80} + 839 q^{81} + 700 q^{82} - 2184 q^{83} - 294 q^{85} + 248 q^{86} + 406 q^{87} - 120 q^{88} - 329 q^{89} - 616 q^{90} + 1272 q^{92} - 1029 q^{93} + 1050 q^{94} - 343 q^{95} + 1120 q^{96} + 1764 q^{97} - 220 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 1.73205i 3.50000 6.06218i 2.00000 3.46410i 3.50000 + 6.06218i −14.0000 0 −24.0000 −11.0000 19.0526i 7.00000 12.1244i
30.1 −1.00000 + 1.73205i 3.50000 + 6.06218i 2.00000 + 3.46410i 3.50000 6.06218i −14.0000 0 −24.0000 −11.0000 + 19.0526i 7.00000 + 12.1244i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.4.c.a 2
3.b odd 2 1 441.4.e.k 2
7.b odd 2 1 7.4.c.a 2
7.c even 3 1 49.4.a.c 1
7.c even 3 1 inner 49.4.c.a 2
7.d odd 6 1 7.4.c.a 2
7.d odd 6 1 49.4.a.d 1
21.c even 2 1 63.4.e.b 2
21.g even 6 1 63.4.e.b 2
21.g even 6 1 441.4.a.d 1
21.h odd 6 1 441.4.a.e 1
21.h odd 6 1 441.4.e.k 2
28.d even 2 1 112.4.i.c 2
28.f even 6 1 112.4.i.c 2
28.f even 6 1 784.4.a.b 1
28.g odd 6 1 784.4.a.r 1
35.c odd 2 1 175.4.e.a 2
35.f even 4 2 175.4.k.a 4
35.i odd 6 1 175.4.e.a 2
35.i odd 6 1 1225.4.a.c 1
35.j even 6 1 1225.4.a.d 1
35.k even 12 2 175.4.k.a 4
56.e even 2 1 448.4.i.a 2
56.h odd 2 1 448.4.i.f 2
56.j odd 6 1 448.4.i.f 2
56.m even 6 1 448.4.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 7.b odd 2 1
7.4.c.a 2 7.d odd 6 1
49.4.a.c 1 7.c even 3 1
49.4.a.d 1 7.d odd 6 1
49.4.c.a 2 1.a even 1 1 trivial
49.4.c.a 2 7.c even 3 1 inner
63.4.e.b 2 21.c even 2 1
63.4.e.b 2 21.g even 6 1
112.4.i.c 2 28.d even 2 1
112.4.i.c 2 28.f even 6 1
175.4.e.a 2 35.c odd 2 1
175.4.e.a 2 35.i odd 6 1
175.4.k.a 4 35.f even 4 2
175.4.k.a 4 35.k even 12 2
441.4.a.d 1 21.g even 6 1
441.4.a.e 1 21.h odd 6 1
441.4.e.k 2 3.b odd 2 1
441.4.e.k 2 21.h odd 6 1
448.4.i.a 2 56.e even 2 1
448.4.i.a 2 56.m even 6 1
448.4.i.f 2 56.h odd 2 1
448.4.i.f 2 56.j odd 6 1
784.4.a.b 1 28.f even 6 1
784.4.a.r 1 28.g odd 6 1
1225.4.a.c 1 35.i odd 6 1
1225.4.a.d 1 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(49, [\chi])\):

\( T_{2}^{2} + 2T_{2} + 4 \) Copy content Toggle raw display
\( T_{3}^{2} - 7T_{3} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$5$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$13$ \( (T - 14)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 21T + 441 \) Copy content Toggle raw display
$19$ \( T^{2} - 49T + 2401 \) Copy content Toggle raw display
$23$ \( T^{2} - 159T + 25281 \) Copy content Toggle raw display
$29$ \( (T - 58)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 147T + 21609 \) Copy content Toggle raw display
$37$ \( T^{2} + 219T + 47961 \) Copy content Toggle raw display
$41$ \( (T + 350)^{2} \) Copy content Toggle raw display
$43$ \( (T + 124)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 525T + 275625 \) Copy content Toggle raw display
$53$ \( T^{2} + 303T + 91809 \) Copy content Toggle raw display
$59$ \( T^{2} + 105T + 11025 \) Copy content Toggle raw display
$61$ \( T^{2} + 413T + 170569 \) Copy content Toggle raw display
$67$ \( T^{2} + 415T + 172225 \) Copy content Toggle raw display
$71$ \( (T + 432)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1113 T + 1238769 \) Copy content Toggle raw display
$79$ \( T^{2} - 103T + 10609 \) Copy content Toggle raw display
$83$ \( (T + 1092)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 329T + 108241 \) Copy content Toggle raw display
$97$ \( (T - 882)^{2} \) Copy content Toggle raw display
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