Properties

Label 1950.2.e.i
Level $1950$
Weight $2$
Character orbit 1950.e
Analytic conductor $15.571$
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] = [1950,2,Mod(1249,1950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1950.1249");
 
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.e (of order \(2\), degree \(1\), 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: \(2\)
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: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Character values

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

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 1.00000 4.00000i 1.00000i −1.00000 0
1249.2 1.00000i 1.00000i −1.00000 0 1.00000 4.00000i 1.00000i −1.00000 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.e.i 2
3.b odd 2 1 5850.2.e.bb 2
5.b even 2 1 inner 1950.2.e.i 2
5.c odd 4 1 78.2.a.a 1
5.c odd 4 1 1950.2.a.w 1
15.d odd 2 1 5850.2.e.bb 2
15.e even 4 1 234.2.a.c 1
15.e even 4 1 5850.2.a.d 1
20.e even 4 1 624.2.a.h 1
35.f even 4 1 3822.2.a.j 1
40.i odd 4 1 2496.2.a.t 1
40.k even 4 1 2496.2.a.b 1
45.k odd 12 2 2106.2.e.q 2
45.l even 12 2 2106.2.e.j 2
55.e even 4 1 9438.2.a.t 1
60.l odd 4 1 1872.2.a.c 1
65.f even 4 1 1014.2.b.b 2
65.h odd 4 1 1014.2.a.d 1
65.k even 4 1 1014.2.b.b 2
65.o even 12 2 1014.2.i.d 4
65.q odd 12 2 1014.2.e.f 2
65.r odd 12 2 1014.2.e.c 2
65.t even 12 2 1014.2.i.d 4
120.q odd 4 1 7488.2.a.bk 1
120.w even 4 1 7488.2.a.bz 1
195.j odd 4 1 3042.2.b.g 2
195.s even 4 1 3042.2.a.f 1
195.u odd 4 1 3042.2.b.g 2
260.p even 4 1 8112.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 5.c odd 4 1
234.2.a.c 1 15.e even 4 1
624.2.a.h 1 20.e even 4 1
1014.2.a.d 1 65.h odd 4 1
1014.2.b.b 2 65.f even 4 1
1014.2.b.b 2 65.k even 4 1
1014.2.e.c 2 65.r odd 12 2
1014.2.e.f 2 65.q odd 12 2
1014.2.i.d 4 65.o even 12 2
1014.2.i.d 4 65.t even 12 2
1872.2.a.c 1 60.l odd 4 1
1950.2.a.w 1 5.c odd 4 1
1950.2.e.i 2 1.a even 1 1 trivial
1950.2.e.i 2 5.b even 2 1 inner
2106.2.e.j 2 45.l even 12 2
2106.2.e.q 2 45.k odd 12 2
2496.2.a.b 1 40.k even 4 1
2496.2.a.t 1 40.i odd 4 1
3042.2.a.f 1 195.s even 4 1
3042.2.b.g 2 195.j odd 4 1
3042.2.b.g 2 195.u odd 4 1
3822.2.a.j 1 35.f even 4 1
5850.2.a.d 1 15.e even 4 1
5850.2.e.bb 2 3.b odd 2 1
5850.2.e.bb 2 15.d odd 2 1
7488.2.a.bk 1 120.q odd 4 1
7488.2.a.bz 1 120.w even 4 1
8112.2.a.v 1 260.p even 4 1
9438.2.a.t 1 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\):

\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 4 \) Copy content Toggle raw display
\( T_{31} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T + 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 100 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 256 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T + 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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