Properties

Label 1156.4.a.i
Level $1156$
Weight $4$
Character orbit 1156.a
Self dual yes
Analytic conductor $68.206$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1156,4,Mod(1,1156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1156, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1156.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1156.a (trivial)

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: yes
Analytic conductor: \(68.2062079666\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.889407488.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 14x^{4} + 49x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 68)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1) q^{3} + (3 \beta_{5} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{5} + 7 \beta_{2} - 2 \beta_1) q^{7} + (2 \beta_{4} + 2 \beta_{3} - 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1) q^{3} + (3 \beta_{5} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{5} + 7 \beta_{2} - 2 \beta_1) q^{7} + (2 \beta_{4} + 2 \beta_{3} - 7) q^{9} + ( - 4 \beta_{5} - 11 \beta_{2} + 9 \beta_1) q^{11} + (2 \beta_{4} - 2 \beta_{3} - 16) q^{13} + ( - \beta_{4} - 6 \beta_{3} + 7) q^{15} + ( - 15 \beta_{4} + 10 \beta_{3} + 25) q^{19} + (5 \beta_{4} - 3 \beta_{3} + 25) q^{21} + ( - 13 \beta_{5} + \beta_{2} + 8 \beta_1) q^{23} + ( - 19 \beta_{4} - 5 \beta_{3} + 72) q^{25} + ( - 12 \beta_{5} - 16 \beta_{2} + 20 \beta_1) q^{27} + ( - 41 \beta_{5} - 55 \beta_{2} - \beta_1) q^{29} + (27 \beta_{5} - 9 \beta_{2} - 10 \beta_1) q^{31} + ( - 14 \beta_{4} + 10 \beta_{3} - 128) q^{33} + (33 \beta_{4} - 22 \beta_{3} - 59) q^{35} + (3 \beta_{5} + 31 \beta_{2} + 67 \beta_1) q^{37} + (4 \beta_{5} + 46 \beta_{2} + 10 \beta_1) q^{39} + ( - 24 \beta_{5} - 7 \beta_{2} + 76 \beta_1) q^{41} + (36 \beta_{4} + 25 \beta_{3} - 86) q^{43} + ( - 55 \beta_{5} + 91 \beta_{2} + 37 \beta_1) q^{45} + (20 \beta_{4} - 52 \beta_{3} - 228) q^{47} + ( - 9 \beta_{4} - 19 \beta_{3} - 120) q^{49} + ( - 5 \beta_{4} + 3 \beta_{3} + 177) q^{53} + ( - 19 \beta_{4} + 70 \beta_{3} - 299) q^{55} + ( - 10 \beta_{5} - 150 \beta_{2} + 30 \beta_1) q^{57} + (38 \beta_{4} + 47 \beta_{3} - 120) q^{59} + ( - 43 \beta_{5} - 139 \beta_{2} + 5 \beta_1) q^{61} + (29 \beta_{5} - 179 \beta_{2} + 10 \beta_1) q^{63} + ( - 66 \beta_{5} + 114 \beta_{2} - 10 \beta_1) q^{65} + ( - 13 \beta_{4} - 63 \beta_{3} + 121) q^{67} + ( - 3 \beta_{4} + 17 \beta_{3} - 107) q^{69} + ( - 7 \beta_{5} + 239 \beta_{2} + 28 \beta_1) q^{71} + ( - 22 \beta_{5} + 465 \beta_{2} - 38 \beta_1) q^{73} + (58 \beta_{5} + 123 \beta_{2} + 33 \beta_1) q^{75} + ( - 65 \beta_{4} + 87 \beta_{3} - 373) q^{77} + (107 \beta_{5} + 585 \beta_{2} - 28 \beta_1) q^{79} + ( - 82 \beta_{4} - 34 \beta_{3} - 103) q^{81} + (50 \beta_{4} - 43 \beta_{3} - 24) q^{83} + (43 \beta_{4} + 138 \beta_{3} + 251) q^{87} + (63 \beta_{4} - 51 \beta_{3} - 561) q^{89} + (70 \beta_{5} - 66 \beta_{2} - 12 \beta_1) q^{91} + ( - 7 \beta_{4} - 35 \beta_{3} + 117) q^{93} + (230 \beta_{5} - 910 \beta_{2} + 100 \beta_1) q^{95} + (134 \beta_{5} - 151 \beta_{2} - 286 \beta_1) q^{97} + (96 \beta_{5} + 295 \beta_{2} - 65 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 38 q^{9} - 92 q^{13} + 40 q^{15} + 120 q^{19} + 160 q^{21} + 394 q^{25} - 796 q^{33} - 288 q^{35} - 444 q^{43} - 1328 q^{47} - 738 q^{49} + 1052 q^{53} - 1832 q^{55} - 644 q^{59} + 700 q^{67} - 648 q^{69} - 2368 q^{77} - 782 q^{81} - 44 q^{83} + 1592 q^{87} - 3240 q^{89} + 688 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 14x^{4} + 49x^{2} - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 7\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{4} - 14\nu^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\nu^{2} - 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 12\nu^{3} + 35\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{2} + 7\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{4} + \beta_{3} + 63 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{5} + 24\beta_{2} + 49\beta_1 ) / 2 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.74151
2.53828
−0.203230
0.203230
−2.53828
−2.74151
0 −6.89722 0 6.60506 0 −4.62448 0 20.5717 0
1.2 0 −3.66234 0 −12.6122 0 −18.0121 0 −13.5873 0
1.3 0 −1.00775 0 −19.2173 0 17.7251 0 −25.9844 0
1.4 0 1.00775 0 19.2173 0 −17.7251 0 −25.9844 0
1.5 0 3.66234 0 12.6122 0 18.0121 0 −13.5873 0
1.6 0 6.89722 0 −6.60506 0 4.62448 0 20.5717 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.4.a.i 6
17.b even 2 1 inner 1156.4.a.i 6
17.c even 4 2 1156.4.b.f 6
17.d even 8 2 68.4.e.b 6
51.g odd 8 2 612.4.k.b 6
68.g odd 8 2 272.4.o.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.4.e.b 6 17.d even 8 2
272.4.o.c 6 68.g odd 8 2
612.4.k.b 6 51.g odd 8 2
1156.4.a.i 6 1.a even 1 1 trivial
1156.4.a.i 6 17.b even 2 1 inner
1156.4.b.f 6 17.c even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 62T_{3}^{4} + 700T_{3}^{2} - 648 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1156))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 62 T^{4} + 700 T^{2} + \cdots - 648 \) Copy content Toggle raw display
$5$ \( T^{6} - 572 T^{4} + 81796 T^{2} + \cdots - 2562848 \) Copy content Toggle raw display
$7$ \( T^{6} - 660 T^{4} + 115588 T^{2} + \cdots - 2179872 \) Copy content Toggle raw display
$11$ \( T^{6} - 5278 T^{4} + \cdots - 143888648 \) Copy content Toggle raw display
$13$ \( (T^{3} + 46 T^{2} + 316 T - 3896)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( (T^{3} - 60 T^{2} - 15500 T + 194000)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 12300 T^{4} + \cdots - 32772608 \) Copy content Toggle raw display
$29$ \( T^{6} - 121116 T^{4} + \cdots - 56691570507552 \) Copy content Toggle raw display
$31$ \( T^{6} - 48692 T^{4} + \cdots - 276333674528 \) Copy content Toggle raw display
$37$ \( T^{6} - 262196 T^{4} + \cdots - 294101549568 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 332174590800392 \) Copy content Toggle raw display
$43$ \( (T^{3} + 222 T^{2} - 124844 T - 17200936)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} + 664 T^{2} - 5632 T - 42870784)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 526 T^{2} + 90448 T - 5087584)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} + 322 T^{2} - 226348 T - 71289688)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 105473055616128 \) Copy content Toggle raw display
$67$ \( (T^{3} - 350 T^{2} - 212128 T + 63674784)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} - 391852 T^{4} + \cdots - 10\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{6} - 1471142 T^{4} + \cdots - 61\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{6} - 2530604 T^{4} + \cdots - 12\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( (T^{3} + 22 T^{2} - 215116 T - 11907208)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 1620 T^{2} + 546948 T - 31482432)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 5063638 T^{4} + \cdots - 41\!\cdots\!68 \) Copy content Toggle raw display
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