Properties

Label 1950.2.y.c
Level $1950$
Weight $2$
Character orbit 1950.y
Analytic conductor $15.571$
Analytic rank $0$
Dimension $4$
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] = [1950,2,Mod(49,1950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1950.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.y (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: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) 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 390)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{12}^{2} q^{2} + (\zeta_{12}^{3} - \zeta_{12}) q^{3} + (\zeta_{12}^{2} - 1) q^{4} + \zeta_{12} q^{6} + ( - 2 \zeta_{12}^{2} + 2) q^{7} + q^{8} + ( - \zeta_{12}^{2} + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12}^{2} q^{2} + (\zeta_{12}^{3} - \zeta_{12}) q^{3} + (\zeta_{12}^{2} - 1) q^{4} + \zeta_{12} q^{6} + ( - 2 \zeta_{12}^{2} + 2) q^{7} + q^{8} + ( - \zeta_{12}^{2} + 1) q^{9} + (3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 3 \zeta_{12} + 4) q^{11} - \zeta_{12}^{3} q^{12} + (3 \zeta_{12}^{3} - 4 \zeta_{12}) q^{13} - 2 q^{14} - \zeta_{12}^{2} q^{16} - 4 \zeta_{12} q^{17} - q^{18} + ( - 2 \zeta_{12}^{2} + 4 \zeta_{12} - 2) q^{19} + 2 \zeta_{12}^{3} q^{21} + ( - 2 \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{22} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{23} + (\zeta_{12}^{3} - \zeta_{12}) q^{24} + (\zeta_{12}^{3} + 3 \zeta_{12}) q^{26} + \zeta_{12}^{3} q^{27} + 2 \zeta_{12}^{2} q^{28} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12}) q^{29} + (2 \zeta_{12}^{2} - 1) q^{31} + (\zeta_{12}^{2} - 1) q^{32} + (4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 2 \zeta_{12} + 3) q^{33} + 4 \zeta_{12}^{3} q^{34} + \zeta_{12}^{2} q^{36} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 3 \zeta_{12}) q^{37} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{38} + ( - 3 \zeta_{12}^{2} + 4) q^{39} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{41} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{42} + (4 \zeta_{12}^{2} - 5 \zeta_{12} + 4) q^{43} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{44} + (\zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{46} + (2 \zeta_{12}^{3} - 4 \zeta_{12} - 7) q^{47} + \zeta_{12} q^{48} + 3 \zeta_{12}^{2} q^{49} + 4 q^{51} + ( - 4 \zeta_{12}^{3} + \zeta_{12}) q^{52} + ( - 6 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{53} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{54} + ( - 2 \zeta_{12}^{2} + 2) q^{56} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 4) q^{57} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + \zeta_{12} + 2) q^{58} + ( - 2 \zeta_{12}^{2} + 5 \zeta_{12} - 2) q^{59} + (12 \zeta_{12}^{3} - 6 \zeta_{12}) q^{61} + ( - \zeta_{12}^{2} + 2) q^{62} - 2 \zeta_{12}^{2} q^{63} + q^{64} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 3) q^{66} + (2 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 2 \zeta_{12}) q^{67} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{68} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{69} + ( - 6 \zeta_{12}^{2} + 2 \zeta_{12} - 6) q^{71} + ( - \zeta_{12}^{2} + 1) q^{72} - 2 q^{73} + (6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 3 \zeta_{12} + 4) q^{74} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{76} + (6 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{77} + ( - \zeta_{12}^{2} - 3) q^{78} + (4 \zeta_{12}^{3} - 8 \zeta_{12} + 7) q^{79} - \zeta_{12}^{2} q^{81} + 2 \zeta_{12} q^{82} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} - 2) q^{83} - 2 \zeta_{12} q^{84} + (5 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{86} + ( - \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{87} + (3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 3 \zeta_{12} + 4) q^{88} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{89} + (8 \zeta_{12}^{3} - 2 \zeta_{12}) q^{91} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{92} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{93} + (2 \zeta_{12}^{3} + 7 \zeta_{12}^{2} + 2 \zeta_{12}) q^{94} - \zeta_{12}^{3} q^{96} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2 \zeta_{12} - 4) q^{97} + ( - 3 \zeta_{12}^{2} + 3) q^{98} + (3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} + 4 q^{7} + 4 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{4} + 4 q^{7} + 4 q^{8} + 2 q^{9} + 12 q^{11} - 8 q^{14} - 2 q^{16} - 4 q^{18} - 12 q^{19} - 12 q^{22} - 6 q^{23} + 4 q^{28} + 4 q^{29} - 2 q^{32} + 6 q^{33} + 2 q^{36} + 8 q^{37} + 10 q^{39} + 24 q^{43} + 6 q^{46} - 28 q^{47} + 6 q^{49} + 16 q^{51} + 4 q^{56} - 16 q^{57} + 4 q^{58} - 12 q^{59} + 6 q^{62} - 4 q^{63} + 4 q^{64} - 12 q^{66} - 16 q^{67} - 4 q^{69} - 36 q^{71} + 2 q^{72} - 8 q^{73} + 8 q^{74} + 12 q^{76} - 14 q^{78} + 28 q^{79} - 2 q^{81} - 8 q^{83} - 6 q^{87} + 12 q^{88} - 12 q^{89} + 14 q^{94} - 8 q^{97} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(1 - \zeta_{12}^{2}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.500000 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0 0.866025 + 0.500000i 1.00000 1.73205i 1.00000 0.500000 0.866025i 0
49.2 −0.500000 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 0 −0.866025 0.500000i 1.00000 1.73205i 1.00000 0.500000 0.866025i 0
199.1 −0.500000 + 0.866025i −0.866025 0.500000i −0.500000 0.866025i 0 0.866025 0.500000i 1.00000 + 1.73205i 1.00000 0.500000 + 0.866025i 0
199.2 −0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 0 −0.866025 + 0.500000i 1.00000 + 1.73205i 1.00000 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.y.c 4
5.b even 2 1 1950.2.y.f 4
5.c odd 4 1 390.2.bb.b 4
5.c odd 4 1 1950.2.bc.b 4
13.e even 6 1 1950.2.y.f 4
15.e even 4 1 1170.2.bs.e 4
65.l even 6 1 inner 1950.2.y.c 4
65.o even 12 1 5070.2.a.bg 2
65.q odd 12 1 5070.2.b.o 4
65.r odd 12 1 390.2.bb.b 4
65.r odd 12 1 1950.2.bc.b 4
65.r odd 12 1 5070.2.b.o 4
65.t even 12 1 5070.2.a.y 2
195.bf even 12 1 1170.2.bs.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.b 4 5.c odd 4 1
390.2.bb.b 4 65.r odd 12 1
1170.2.bs.e 4 15.e even 4 1
1170.2.bs.e 4 195.bf even 12 1
1950.2.y.c 4 1.a even 1 1 trivial
1950.2.y.c 4 65.l even 6 1 inner
1950.2.y.f 4 5.b even 2 1
1950.2.y.f 4 13.e even 6 1
1950.2.bc.b 4 5.c odd 4 1
1950.2.bc.b 4 65.r odd 12 1
5070.2.a.y 2 65.t even 12 1
5070.2.a.bg 2 65.o even 12 1
5070.2.b.o 4 65.q odd 12 1
5070.2.b.o 4 65.r odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 2T_{7} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 12 T^{3} + 51 T^{2} - 36 T + 9 \) Copy content Toggle raw display
$13$ \( T^{4} - 22T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$19$ \( T^{4} + 12 T^{3} + 44 T^{2} - 48 T + 16 \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + 11 T^{2} - 6 T + 1 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + 15 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$31$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 8 T^{3} + 75 T^{2} + 88 T + 121 \) Copy content Toggle raw display
$41$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$43$ \( T^{4} - 24 T^{3} + 215 T^{2} + \cdots + 529 \) Copy content Toggle raw display
$47$ \( (T^{2} + 14 T + 37)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$59$ \( T^{4} + 12 T^{3} + 35 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$61$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$67$ \( T^{4} + 16 T^{3} + 204 T^{2} + \cdots + 2704 \) Copy content Toggle raw display
$71$ \( T^{4} + 36 T^{3} + 536 T^{2} + \cdots + 10816 \) Copy content Toggle raw display
$73$ \( (T + 2)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T + 1)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 4 T - 44)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 12 T^{3} + 44 T^{2} - 48 T + 16 \) Copy content Toggle raw display
$97$ \( T^{4} + 8 T^{3} + 60 T^{2} + 32 T + 16 \) Copy content Toggle raw display
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