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Properties

Label 95.2.i.a
Level $95$
Weight $2$
Character orbit 95.i
Analytic conductor $0.759$
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] = [95,2,Mod(49,95)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(95, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("95.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 95.i (of order \(6\), 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: \(0.758578819202\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - \beta_{2} - \beta_1 + 1) q^{5} + 2 \beta_{3} q^{7} - 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - \beta_{2} - \beta_1 + 1) q^{5} + 2 \beta_{3} q^{7} - 3 \beta_{2} q^{9} + ( - \beta_{3} - 4 \beta_{2} + \beta_1) q^{10} - q^{11} + (\beta_{3} - \beta_1) q^{13} + (8 \beta_{2} - 8) q^{14} + ( - 4 \beta_{2} + 4) q^{16} - \beta_1 q^{17} - 3 \beta_{3} q^{18} + (3 \beta_{2} + 2) q^{19} + ( - 2 \beta_{3} + 2) q^{20} - \beta_1 q^{22} + ( - 3 \beta_{3} + 3 \beta_1) q^{23} + (2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{25} - 4 q^{26} + (4 \beta_{3} - 4 \beta_1) q^{28} + 9 \beta_{2} q^{29} - 7 q^{31} + ( - 4 \beta_{3} + 4 \beta_1) q^{32} - 4 \beta_{2} q^{34} + ( - 8 \beta_{2} + 2 \beta_1 + 8) q^{35} + ( - 6 \beta_{2} + 6) q^{36} - \beta_{3} q^{37} + (3 \beta_{3} + 2 \beta_1) q^{38} + (2 \beta_{2} - 2) q^{41} - \beta_1 q^{43} - 2 \beta_{2} q^{44} + (3 \beta_{3} - 3) q^{45} + 12 q^{46} + (3 \beta_{3} - 3 \beta_1) q^{47} - 9 q^{49} + (3 \beta_{3} - 8) q^{50} - 2 \beta_1 q^{52} + ( - 2 \beta_{3} + 2 \beta_1) q^{53} + (\beta_{2} + \beta_1 - 1) q^{55} + 9 \beta_{3} q^{58} + ( - 9 \beta_{2} + 9) q^{59} + 7 \beta_{2} q^{61} - 7 \beta_1 q^{62} + ( - 6 \beta_{3} + 6 \beta_1) q^{63} + 8 q^{64} + (\beta_{3} + 4) q^{65} + ( - 5 \beta_{3} + 5 \beta_1) q^{67} - 2 \beta_{3} q^{68} + ( - 8 \beta_{3} + 8 \beta_{2} + 8 \beta_1) q^{70} + (\beta_{2} - 1) q^{71} - 5 \beta_1 q^{73} + ( - 4 \beta_{2} + 4) q^{74} + (10 \beta_{2} - 6) q^{76} - 2 \beta_{3} q^{77} + ( - \beta_{2} + 1) q^{79} + (4 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{80} + (9 \beta_{2} - 9) q^{81} + (2 \beta_{3} - 2 \beta_1) q^{82} - 3 \beta_{3} q^{83} + (\beta_{3} + 4 \beta_{2} - \beta_1) q^{85} - 4 \beta_{2} q^{86} - 11 \beta_{2} q^{89} + (12 \beta_{2} - 3 \beta_1 - 12) q^{90} - 8 \beta_{2} q^{91} + 6 \beta_1 q^{92} - 12 q^{94} + ( - 3 \beta_{3} - 2 \beta_{2} + \cdots + 5) q^{95}+ \cdots + 3 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 2 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 2 q^{5} - 6 q^{9} - 8 q^{10} - 4 q^{11} - 16 q^{14} + 8 q^{16} + 14 q^{19} + 8 q^{20} + 6 q^{25} - 16 q^{26} + 18 q^{29} - 28 q^{31} - 8 q^{34} + 16 q^{35} + 12 q^{36} - 4 q^{41} - 4 q^{44} - 12 q^{45} + 48 q^{46} - 36 q^{49} - 32 q^{50} - 2 q^{55} + 18 q^{59} + 14 q^{61} + 32 q^{64} + 16 q^{65} + 16 q^{70} - 2 q^{71} + 8 q^{74} - 4 q^{76} + 2 q^{79} - 8 q^{80} - 18 q^{81} + 8 q^{85} - 8 q^{86} - 22 q^{89} - 24 q^{90} - 16 q^{91} - 48 q^{94} + 16 q^{95} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\)

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
−1.73205 − 1.00000i 0 1.00000 + 1.73205i 2.23205 + 0.133975i 0 − 4.00000i 0 −1.50000 − 2.59808i −3.73205 − 2.46410i
49.2 1.73205 + 1.00000i 0 1.00000 + 1.73205i −1.23205 − 1.86603i 0 4.00000i 0 −1.50000 − 2.59808i −0.267949 − 4.46410i
64.1 −1.73205 + 1.00000i 0 1.00000 − 1.73205i 2.23205 − 0.133975i 0 4.00000i 0 −1.50000 + 2.59808i −3.73205 + 2.46410i
64.2 1.73205 − 1.00000i 0 1.00000 − 1.73205i −1.23205 + 1.86603i 0 − 4.00000i 0 −1.50000 + 2.59808i −0.267949 + 4.46410i
\(n\): 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.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.2.i.a ✓ 4
3.b odd 2 1 855.2.be.a 4
5.b even 2 1 inner 95.2.i.a ✓ 4
5.c odd 4 1 475.2.e.a 2
5.c odd 4 1 475.2.e.c 2
15.d odd 2 1 855.2.be.a 4
19.c even 3 1 inner 95.2.i.a ✓ 4
19.c even 3 1 1805.2.b.b 2
19.d odd 6 1 1805.2.b.a 2
57.h odd 6 1 855.2.be.a 4
95.h odd 6 1 1805.2.b.a 2
95.i even 6 1 inner 95.2.i.a ✓ 4
95.i even 6 1 1805.2.b.b 2
95.l even 12 1 9025.2.a.a 1
95.l even 12 1 9025.2.a.j 1
95.m odd 12 1 475.2.e.a 2
95.m odd 12 1 475.2.e.c 2
95.m odd 12 1 9025.2.a.b 1
95.m odd 12 1 9025.2.a.i 1
285.n odd 6 1 855.2.be.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.i.a ✓ 4 1.a even 1 1 trivial
95.2.i.a ✓ 4 5.b even 2 1 inner
95.2.i.a ✓ 4 19.c even 3 1 inner
95.2.i.a ✓ 4 95.i even 6 1 inner
475.2.e.a 2 5.c odd 4 1
475.2.e.a 2 95.m odd 12 1
475.2.e.c 2 5.c odd 4 1
475.2.e.c 2 95.m odd 12 1
855.2.be.a 4 3.b odd 2 1
855.2.be.a 4 15.d odd 2 1
855.2.be.a 4 57.h odd 6 1
855.2.be.a 4 285.n odd 6 1
1805.2.b.a 2 19.d odd 6 1
1805.2.b.a 2 95.h odd 6 1
1805.2.b.b 2 19.c even 3 1
1805.2.b.b 2 95.i even 6 1
9025.2.a.a 1 95.l even 12 1
9025.2.a.b 1 95.m odd 12 1
9025.2.a.i 1 95.m odd 12 1
9025.2.a.j 1 95.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 4T_{2}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(95, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} - 7 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$31$ \( (T + 7)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$53$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$59$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$71$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$79$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
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