Properties

Label 93.2.c.a
Level $93$
Weight $2$
Character orbit 93.c
Analytic conductor $0.743$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [93,2,Mod(92,93)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(93, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("93.92");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 93.c (of order \(2\), degree \(1\), 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: \(0.742608738798\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.16845963264.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 6x^{6} + 15x^{4} + 8x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + \beta_{6} q^{3} + \beta_{3} q^{4} + \beta_1 q^{5} - \beta_{4} q^{6} - q^{7} - \beta_1 q^{8} + (\beta_{5} - \beta_{3} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + \beta_{6} q^{3} + \beta_{3} q^{4} + \beta_1 q^{5} - \beta_{4} q^{6} - q^{7} - \beta_1 q^{8} + (\beta_{5} - \beta_{3} - \beta_1 - 1) q^{9} + (\beta_{3} + 1) q^{10} + ( - \beta_{6} + \beta_{2}) q^{11} + (\beta_{7} - \beta_{6}) q^{12} + ( - \beta_{7} + \beta_{4}) q^{13} + \beta_{5} q^{14} - \beta_{2} q^{15} + (\beta_{3} - 1) q^{16} + (\beta_{7} - \beta_{6} + \beta_{4} - \beta_{2}) q^{17} + ( - \beta_{5} - 2 \beta_{3} + \beta_1 + 1) q^{18} + ( - 2 \beta_{3} - 1) q^{19} + (\beta_{5} + \beta_1) q^{20} - \beta_{6} q^{21} + ( - \beta_{7} + \beta_{4}) q^{22} + ( - \beta_{7} + 2 \beta_{6} - \beta_{4}) q^{23} + \beta_{2} q^{24} - 2 \beta_{3} q^{25} + ( - \beta_{7} + 3 \beta_{6} - \beta_{4} - \beta_{2}) q^{26} + ( - \beta_{7} + \beta_{4} + \beta_{2}) q^{27} - \beta_{3} q^{28} + (\beta_{7} - 2 \beta_{6} + \beta_{4}) q^{29} + \beta_{7} q^{30} + (\beta_{7} - \beta_{4} + 2 \beta_{3} + 1) q^{31} + (3 \beta_{5} - 3 \beta_1) q^{32} + (3 \beta_1 - 3) q^{33} + (3 \beta_{6} + 2 \beta_{4} + \beta_{2}) q^{34} - \beta_1 q^{35} + ( - 3 \beta_{5} - 3) q^{36} + ( - 3 \beta_{6} - 2 \beta_{4} - \beta_{2}) q^{37} + ( - 3 \beta_{5} + 2 \beta_1) q^{38} + (3 \beta_{5} + 3 \beta_{3} + 3) q^{39} + (2 \beta_{3} + 5) q^{40} + (2 \beta_{5} - 3 \beta_1) q^{41} + \beta_{4} q^{42} + (\beta_{7} - \beta_{4}) q^{43} + ( - \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2}) q^{44} + ( - \beta_{5} + \beta_{3} - 2 \beta_1 + 4) q^{45} + (\beta_{7} - 3 \beta_{6} - 3 \beta_{4} - \beta_{2}) q^{46} + ( - 2 \beta_{5} + 2 \beta_1) q^{47} + (\beta_{7} - 2 \beta_{6}) q^{48} - 6 q^{49} + ( - 4 \beta_{5} + 2 \beta_1) q^{50} + ( - 3 \beta_{5} + 3 \beta_{3} - 3 \beta_1) q^{51} + ( - 3 \beta_{6} - 2 \beta_{4} - \beta_{2}) q^{52} + ( - 2 \beta_{7} + 3 \beta_{6} - \beta_{4} - \beta_{2}) q^{54} + (2 \beta_{7} + 3 \beta_{6} + \beta_{2}) q^{55} + \beta_1 q^{56} + ( - 2 \beta_{7} + \beta_{6}) q^{57} + ( - \beta_{7} + 3 \beta_{6} + 3 \beta_{4} + \beta_{2}) q^{58} + \beta_1 q^{59} + (\beta_{4} - \beta_{2}) q^{60} + (\beta_{7} + 3 \beta_{6} + \beta_{4} + \beta_{2}) q^{61} + (\beta_{7} - 3 \beta_{6} + 3 \beta_{5} + \beta_{4} + \beta_{2} - 2 \beta_1) q^{62} + ( - \beta_{5} + \beta_{3} + \beta_1 + 1) q^{63} + ( - 4 \beta_{3} + 1) q^{64} + ( - \beta_{7} - \beta_{4} + 2 \beta_{2}) q^{65} + (3 \beta_{5} + 3 \beta_{3} + 3) q^{66} + ( - 2 \beta_{3} + 2) q^{67} + ( - \beta_{7} + 4 \beta_{6} - \beta_{4} - 2 \beta_{2}) q^{68} + (3 \beta_{5} - 3 \beta_{3} + 3) q^{69} + ( - \beta_{3} - 1) q^{70} + ( - 4 \beta_{5} + 3 \beta_1) q^{71} + (\beta_{5} - \beta_{3} + 2 \beta_1 - 4) q^{72} + (3 \beta_{6} + 2 \beta_{4} + \beta_{2}) q^{73} + (3 \beta_{7} - 6 \beta_{6} + 3 \beta_{4}) q^{74} + ( - 2 \beta_{7} + 2 \beta_{6}) q^{75} + (\beta_{3} - 6) q^{76} + (\beta_{6} - \beta_{2}) q^{77} + (3 \beta_{5} - 3 \beta_{3} - 3 \beta_1 + 6) q^{78} + (\beta_{7} - 3 \beta_{6} - 3 \beta_{4} - \beta_{2}) q^{79} + \beta_{5} q^{80} + (4 \beta_{5} + 2 \beta_{3} + 2 \beta_1 - 1) q^{81} + ( - 5 \beta_{3} + 1) q^{82} + (2 \beta_{6} - 2 \beta_{2}) q^{83} + ( - \beta_{7} + \beta_{6}) q^{84} + ( - 3 \beta_{7} - 3 \beta_{6} + \beta_{4} - \beta_{2}) q^{85} + (\beta_{7} - 3 \beta_{6} + \beta_{4} + \beta_{2}) q^{86} + ( - 3 \beta_{5} + 3 \beta_{3} - 3) q^{87} + ( - 2 \beta_{7} - 3 \beta_{6} - \beta_{2}) q^{88} + ( - \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2}) q^{89} + ( - 2 \beta_{5} - \beta_{3} - \beta_1 - 4) q^{90} + (\beta_{7} - \beta_{4}) q^{91} + (2 \beta_{7} - 5 \beta_{6} + 2 \beta_{4} + \beta_{2}) q^{92} + (2 \beta_{7} - \beta_{6} - 3 \beta_{5} - 3 \beta_{3} - 3) q^{93} + (4 \beta_{3} - 2) q^{94} + ( - 2 \beta_{5} - 3 \beta_1) q^{95} + (3 \beta_{4} + 3 \beta_{2}) q^{96} + ( - 6 \beta_{3} - 1) q^{97} + 6 \beta_{5} q^{98} + ( - 3 \beta_{6} - 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{4} - 8 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{4} - 8 q^{7} - 4 q^{9} + 4 q^{10} - 12 q^{16} + 16 q^{18} + 8 q^{25} + 4 q^{28} - 24 q^{33} - 24 q^{36} + 12 q^{39} + 32 q^{40} + 28 q^{45} - 48 q^{49} - 12 q^{51} + 4 q^{63} + 24 q^{64} + 12 q^{66} + 24 q^{67} + 36 q^{69} - 4 q^{70} - 28 q^{72} - 52 q^{76} + 60 q^{78} - 16 q^{81} + 28 q^{82} - 36 q^{87} - 28 q^{90} - 12 q^{93} - 32 q^{94} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 6x^{6} + 15x^{4} + 8x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} - 7\nu^{4} + 29\nu^{2} + 6 ) / 33 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} + 7\nu^{5} + 4\nu^{3} - 39\nu ) / 33 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} + 4\nu^{2} - 72 ) / 33 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{7} + 7\nu^{5} + 4\nu^{3} - 138\nu ) / 33 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{6} + 34\nu^{4} - 89\nu^{2} - 15 ) / 33 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} + 34\nu^{5} - 89\nu^{3} - 15\nu ) / 33 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} - 41\nu^{5} + 118\nu^{3} + 21\nu ) / 33 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 3\beta_{6} - \beta_{4} + 4\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + \beta_{3} + 6\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 18\beta_{7} + 24\beta_{6} - \beta_{4} + 4\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{5} - 11\beta_{3} + 23\beta _1 - 25 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 69\beta_{7} + 90\beta_{6} + 14\beta_{4} - 47\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/93\mathbb{Z}\right)^\times\).

\(n\) \(32\) \(34\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
92.1
−0.444099 + 0.707107i
0.444099 0.707107i
1.95007 + 0.707107i
−1.95007 0.707107i
1.95007 0.707107i
−1.95007 + 0.707107i
−0.444099 0.707107i
0.444099 + 0.707107i
2.07431i −1.46676 0.921201i −2.30278 0.628052i −1.91086 + 3.04252i −1.00000 0.628052i 1.30278 + 2.70236i −1.30278
92.2 2.07431i 1.46676 + 0.921201i −2.30278 0.628052i 1.91086 3.04252i −1.00000 0.628052i 1.30278 + 2.70236i −1.30278
92.3 0.835000i −0.590434 + 1.62831i 1.30278 2.75782i 1.35964 + 0.493012i −1.00000 2.75782i −2.30278 1.92282i 2.30278
92.4 0.835000i 0.590434 1.62831i 1.30278 2.75782i −1.35964 0.493012i −1.00000 2.75782i −2.30278 1.92282i 2.30278
92.5 0.835000i −0.590434 1.62831i 1.30278 2.75782i 1.35964 0.493012i −1.00000 2.75782i −2.30278 + 1.92282i 2.30278
92.6 0.835000i 0.590434 + 1.62831i 1.30278 2.75782i −1.35964 + 0.493012i −1.00000 2.75782i −2.30278 + 1.92282i 2.30278
92.7 2.07431i −1.46676 + 0.921201i −2.30278 0.628052i −1.91086 3.04252i −1.00000 0.628052i 1.30278 2.70236i −1.30278
92.8 2.07431i 1.46676 0.921201i −2.30278 0.628052i 1.91086 + 3.04252i −1.00000 0.628052i 1.30278 2.70236i −1.30278
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 92.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.b odd 2 1 inner
93.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.2.c.a 8
3.b odd 2 1 inner 93.2.c.a 8
4.b odd 2 1 1488.2.h.e 8
12.b even 2 1 1488.2.h.e 8
31.b odd 2 1 inner 93.2.c.a 8
93.c even 2 1 inner 93.2.c.a 8
124.d even 2 1 1488.2.h.e 8
372.b odd 2 1 1488.2.h.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.2.c.a 8 1.a even 1 1 trivial
93.2.c.a 8 3.b odd 2 1 inner
93.2.c.a 8 31.b odd 2 1 inner
93.2.c.a 8 93.c even 2 1 inner
1488.2.h.e 8 4.b odd 2 1
1488.2.h.e 8 12.b even 2 1
1488.2.h.e 8 124.d even 2 1
1488.2.h.e 8 372.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(93, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 5 T^{2} + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{6} + 6 T^{4} + 18 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} + 8 T^{2} + 3)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 30 T^{2} + 108)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 66 T^{2} + 972)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 13)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 48 T^{2} + 108)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 48 T^{2} + 108)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 10 T^{2} + 961)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 126 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 80 T^{2} + 1587)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 44 T^{2} + 432)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 8 T^{2} + 3)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 126 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 6 T - 4)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 128 T^{2} + 2523)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 126 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 198 T^{2} + 324)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 120 T^{2} + 1728)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 66 T^{2} + 972)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 4 T - 113)^{4} \) Copy content Toggle raw display
show more
show less