Properties

Label 50.3.c.b
Level $50$
Weight $3$
Character orbit 50.c
Analytic conductor $1.362$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [50,3,Mod(7,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 50.c (of order \(4\), degree \(2\), 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: \(1.36240132180\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (i + 1) q^{2} + (3 i - 3) q^{3} + 2 i q^{4} - 6 q^{6} + (3 i + 3) q^{7} + (2 i - 2) q^{8} - 9 i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (i + 1) q^{2} + (3 i - 3) q^{3} + 2 i q^{4} - 6 q^{6} + (3 i + 3) q^{7} + (2 i - 2) q^{8} - 9 i q^{9} + 12 q^{11} + ( - 6 i - 6) q^{12} + ( - 12 i + 12) q^{13} + 6 i q^{14} - 4 q^{16} + ( - 12 i - 12) q^{17} + ( - 9 i + 9) q^{18} + 20 i q^{19} - 18 q^{21} + (12 i + 12) q^{22} + (3 i - 3) q^{23} - 12 i q^{24} + 24 q^{26} + (6 i - 6) q^{28} - 30 i q^{29} - 8 q^{31} + ( - 4 i - 4) q^{32} + (36 i - 36) q^{33} - 24 i q^{34} + 18 q^{36} + (48 i + 48) q^{37} + (20 i - 20) q^{38} + 72 i q^{39} - 48 q^{41} + ( - 18 i - 18) q^{42} + ( - 27 i + 27) q^{43} + 24 i q^{44} - 6 q^{46} + ( - 27 i - 27) q^{47} + ( - 12 i + 12) q^{48} - 31 i q^{49} + 72 q^{51} + (24 i + 24) q^{52} + ( - 12 i + 12) q^{53} - 12 q^{56} + ( - 60 i - 60) q^{57} + ( - 30 i + 30) q^{58} - 60 i q^{59} + 32 q^{61} + ( - 8 i - 8) q^{62} + ( - 27 i + 27) q^{63} - 8 i q^{64} - 72 q^{66} + (3 i + 3) q^{67} + ( - 24 i + 24) q^{68} - 18 i q^{69} - 48 q^{71} + (18 i + 18) q^{72} + ( - 12 i + 12) q^{73} + 96 i q^{74} - 40 q^{76} + (36 i + 36) q^{77} + (72 i - 72) q^{78} + 40 i q^{79} + 81 q^{81} + ( - 48 i - 48) q^{82} + (93 i - 93) q^{83} - 36 i q^{84} + 54 q^{86} + (90 i + 90) q^{87} + (24 i - 24) q^{88} + 30 i q^{89} + 72 q^{91} + ( - 6 i - 6) q^{92} + ( - 24 i + 24) q^{93} - 54 i q^{94} + 24 q^{96} + ( - 12 i - 12) q^{97} + ( - 31 i + 31) q^{98} - 108 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 6 q^{3} - 12 q^{6} + 6 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 6 q^{3} - 12 q^{6} + 6 q^{7} - 4 q^{8} + 24 q^{11} - 12 q^{12} + 24 q^{13} - 8 q^{16} - 24 q^{17} + 18 q^{18} - 36 q^{21} + 24 q^{22} - 6 q^{23} + 48 q^{26} - 12 q^{28} - 16 q^{31} - 8 q^{32} - 72 q^{33} + 36 q^{36} + 96 q^{37} - 40 q^{38} - 96 q^{41} - 36 q^{42} + 54 q^{43} - 12 q^{46} - 54 q^{47} + 24 q^{48} + 144 q^{51} + 48 q^{52} + 24 q^{53} - 24 q^{56} - 120 q^{57} + 60 q^{58} + 64 q^{61} - 16 q^{62} + 54 q^{63} - 144 q^{66} + 6 q^{67} + 48 q^{68} - 96 q^{71} + 36 q^{72} + 24 q^{73} - 80 q^{76} + 72 q^{77} - 144 q^{78} + 162 q^{81} - 96 q^{82} - 186 q^{83} + 108 q^{86} + 180 q^{87} - 48 q^{88} + 144 q^{91} - 12 q^{92} + 48 q^{93} + 48 q^{96} - 24 q^{97} + 62 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
1.00000i
1.00000i
1.00000 + 1.00000i −3.00000 + 3.00000i 2.00000i 0 −6.00000 3.00000 + 3.00000i −2.00000 + 2.00000i 9.00000i 0
43.1 1.00000 1.00000i −3.00000 3.00000i 2.00000i 0 −6.00000 3.00000 3.00000i −2.00000 2.00000i 9.00000i 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
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.3.c.b yes 2
3.b odd 2 1 450.3.g.c 2
4.b odd 2 1 400.3.p.g 2
5.b even 2 1 50.3.c.a 2
5.c odd 4 1 50.3.c.a 2
5.c odd 4 1 inner 50.3.c.b yes 2
15.d odd 2 1 450.3.g.e 2
15.e even 4 1 450.3.g.c 2
15.e even 4 1 450.3.g.e 2
20.d odd 2 1 400.3.p.a 2
20.e even 4 1 400.3.p.a 2
20.e even 4 1 400.3.p.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.3.c.a 2 5.b even 2 1
50.3.c.a 2 5.c odd 4 1
50.3.c.b yes 2 1.a even 1 1 trivial
50.3.c.b yes 2 5.c odd 4 1 inner
400.3.p.a 2 20.d odd 2 1
400.3.p.a 2 20.e even 4 1
400.3.p.g 2 4.b odd 2 1
400.3.p.g 2 20.e even 4 1
450.3.g.c 2 3.b odd 2 1
450.3.g.c 2 15.e even 4 1
450.3.g.e 2 15.d odd 2 1
450.3.g.e 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 6T_{3} + 18 \) acting on \(S_{3}^{\mathrm{new}}(50, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$11$ \( (T - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 24T + 288 \) Copy content Toggle raw display
$17$ \( T^{2} + 24T + 288 \) Copy content Toggle raw display
$19$ \( T^{2} + 400 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$29$ \( T^{2} + 900 \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 96T + 4608 \) Copy content Toggle raw display
$41$ \( (T + 48)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 54T + 1458 \) Copy content Toggle raw display
$47$ \( T^{2} + 54T + 1458 \) Copy content Toggle raw display
$53$ \( T^{2} - 24T + 288 \) Copy content Toggle raw display
$59$ \( T^{2} + 3600 \) Copy content Toggle raw display
$61$ \( (T - 32)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$71$ \( (T + 48)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 24T + 288 \) Copy content Toggle raw display
$79$ \( T^{2} + 1600 \) Copy content Toggle raw display
$83$ \( T^{2} + 186T + 17298 \) Copy content Toggle raw display
$89$ \( T^{2} + 900 \) Copy content Toggle raw display
$97$ \( T^{2} + 24T + 288 \) Copy content Toggle raw display
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