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Properties

Label 950.3.d.a
Level $950$
Weight $3$
Character orbit 950.d
Analytic conductor $25.886$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [950,3,Mod(949,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.949");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 950.d (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: \(25.8856251142\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 38)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - 2 \beta_{2} q^{3} + 2 q^{4} - 4 q^{6} + \beta_1 q^{7} + 2 \beta_{2} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - 2 \beta_{2} q^{3} + 2 q^{4} - 4 q^{6} + \beta_1 q^{7} + 2 \beta_{2} q^{8} - q^{9} + 5 q^{11} - 4 \beta_{2} q^{12} + 12 \beta_{2} q^{13} + \beta_{3} q^{14} + 4 q^{16} - 5 \beta_1 q^{17} - \beta_{2} q^{18} - 19 q^{19} - 2 \beta_{3} q^{21} + 5 \beta_{2} q^{22} + 2 \beta_1 q^{23} - 8 q^{24} + 24 q^{26} + 20 \beta_{2} q^{27} + 2 \beta_1 q^{28} + 6 \beta_{3} q^{29} + 6 \beta_{3} q^{31} + 4 \beta_{2} q^{32} - 10 \beta_{2} q^{33} - 5 \beta_{3} q^{34} - 2 q^{36} - 18 \beta_{2} q^{37} - 19 \beta_{2} q^{38} - 48 q^{39} + 6 \beta_{3} q^{41} - 4 \beta_1 q^{42} - \beta_1 q^{43} + 10 q^{44} + 2 \beta_{3} q^{46} + \beta_1 q^{47} - 8 \beta_{2} q^{48} + 24 q^{49} + 10 \beta_{3} q^{51} + 24 \beta_{2} q^{52} + 18 \beta_{2} q^{53} + 40 q^{54} + 2 \beta_{3} q^{56} + 38 \beta_{2} q^{57} + 12 \beta_1 q^{58} + 12 \beta_{3} q^{59} + 95 q^{61} + 12 \beta_1 q^{62} - \beta_1 q^{63} + 8 q^{64} - 20 q^{66} + 78 \beta_{2} q^{67} - 10 \beta_1 q^{68} - 4 \beta_{3} q^{69} - 2 \beta_{2} q^{72} + 5 \beta_1 q^{73} - 36 q^{74} - 38 q^{76} + 5 \beta_1 q^{77} - 48 \beta_{2} q^{78} + 6 \beta_{3} q^{79} - 71 q^{81} + 12 \beta_1 q^{82} + 26 \beta_1 q^{83} - 4 \beta_{3} q^{84} - \beta_{3} q^{86} - 24 \beta_1 q^{87} + 10 \beta_{2} q^{88} - 18 \beta_{3} q^{89} + 12 \beta_{3} q^{91} + 4 \beta_1 q^{92} - 24 \beta_1 q^{93} + \beta_{3} q^{94} - 16 q^{96} + 12 \beta_{2} q^{97} + 24 \beta_{2} q^{98} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} - 16 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} - 16 q^{6} - 4 q^{9} + 20 q^{11} + 16 q^{16} - 76 q^{19} - 32 q^{24} + 96 q^{26} - 8 q^{36} - 192 q^{39} + 40 q^{44} + 96 q^{49} + 160 q^{54} + 380 q^{61} + 32 q^{64} - 80 q^{66} - 144 q^{74} - 152 q^{76} - 284 q^{81} - 64 q^{96} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 5\zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 5\zeta_{8}^{3} + 5\zeta_{8} \) Copy content Toggle raw display

Display \(\zeta_{8}^j\) in terms of \(\beta_i\)

\(\zeta_{8}\)\(=\) \( ( \beta_{3} + 5\beta_{2} ) / 10 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_1 ) / 5 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( \beta_{3} - 5\beta_{2} ) / 10 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{8}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1
−0.707107 + 0.707107i
−0.707107 − 0.707107i
0.707107 − 0.707107i
0.707107 + 0.707107i
−1.41421 2.82843 2.00000 0 −4.00000 − 5.00000i −2.82843 −1.00000 0
949.2 −1.41421 2.82843 2.00000 0 −4.00000 5.00000i −2.82843 −1.00000 0
949.3 1.41421 −2.82843 2.00000 0 −4.00000 − 5.00000i 2.82843 −1.00000 0
949.4 1.41421 −2.82843 2.00000 0 −4.00000 5.00000i 2.82843 −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
19.b odd 2 1 inner
95.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.3.d.a 4
5.b even 2 1 inner 950.3.d.a 4
5.c odd 4 1 38.3.b.a ✓ 2
5.c odd 4 1 950.3.c.a 2
15.e even 4 1 342.3.d.a 2
19.b odd 2 1 inner 950.3.d.a 4
20.e even 4 1 304.3.e.c 2
40.i odd 4 1 1216.3.e.j 2
40.k even 4 1 1216.3.e.i 2
60.l odd 4 1 2736.3.o.h 2
95.d odd 2 1 inner 950.3.d.a 4
95.g even 4 1 38.3.b.a ✓ 2
95.g even 4 1 950.3.c.a 2
285.j odd 4 1 342.3.d.a 2
380.j odd 4 1 304.3.e.c 2
760.t even 4 1 1216.3.e.j 2
760.y odd 4 1 1216.3.e.i 2
1140.w even 4 1 2736.3.o.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.b.a ✓ 2 5.c odd 4 1
38.3.b.a ✓ 2 95.g even 4 1
304.3.e.c 2 20.e even 4 1
304.3.e.c 2 380.j odd 4 1
342.3.d.a 2 15.e even 4 1
342.3.d.a 2 285.j odd 4 1
950.3.c.a 2 5.c odd 4 1
950.3.c.a 2 95.g even 4 1
950.3.d.a 4 1.a even 1 1 trivial
950.3.d.a 4 5.b even 2 1 inner
950.3.d.a 4 19.b odd 2 1 inner
950.3.d.a 4 95.d odd 2 1 inner
1216.3.e.i 2 40.k even 4 1
1216.3.e.i 2 760.y odd 4 1
1216.3.e.j 2 40.i odd 4 1
1216.3.e.j 2 760.t even 4 1
2736.3.o.h 2 60.l odd 4 1
2736.3.o.h 2 1140.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 8 \) acting on \(S_{3}^{\mathrm{new}}(950, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$11$ \( (T - 5)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 288)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 625)^{2} \) Copy content Toggle raw display
$19$ \( (T + 19)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1800)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 1800)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 648)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 1800)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 648)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 7200)^{2} \) Copy content Toggle raw display
$61$ \( (T - 95)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 12168)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 625)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 1800)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 16900)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 16200)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 288)^{2} \) Copy content Toggle raw display
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