Properties

Label 712.2.e.b
Level $712$
Weight $2$
Character orbit 712.e
Analytic conductor $5.685$
Analytic rank $0$
Dimension $10$
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] = [712,2,Mod(177,712)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(712, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("712.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 712 = 2^{3} \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 712.e (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: \(5.68534862392\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 21x^{8} + 143x^{6} + 341x^{4} + 228x^{2} + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{5} + (\beta_{8} - \beta_1) q^{7} + (\beta_{4} + \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{2} q^{5} + (\beta_{8} - \beta_1) q^{7} + (\beta_{4} + \beta_{2} - 1) q^{9} + ( - \beta_{3} + \beta_{2} - 1) q^{11} + \beta_{7} q^{13} + (\beta_{8} + \beta_{7} - \beta_{6}) q^{15} + (\beta_{4} - \beta_{3} + 1) q^{17} + (\beta_{6} + \beta_1) q^{19} + ( - \beta_{4} + 2) q^{21} + ( - \beta_{8} + \beta_{5}) q^{23} + (\beta_{9} - \beta_{2} + 2) q^{25} + (\beta_{6} + \beta_{5} - \beta_1) q^{27} + (2 \beta_{8} - \beta_{7}) q^{29} + (\beta_{7} + \beta_1) q^{31} + ( - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_1) q^{33} + ( - 2 \beta_{8} + \beta_{6} + 2 \beta_1) q^{35} + ( - \beta_{8} + \beta_{6} - \beta_{5} + \beta_1) q^{37} + ( - \beta_{9} - 2 \beta_{4} + 2 \beta_{2} - 1) q^{39} + ( - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_1) q^{41} + (\beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{5}) q^{43} + ( - \beta_{9} - \beta_{4} + 2 \beta_{2} - 5) q^{45} + (\beta_{9} + 1) q^{47} + ( - \beta_{3} - \beta_{2} + 2) q^{49} + (\beta_{8} + \beta_{7} + 2 \beta_{5} - 2 \beta_1) q^{51} + (\beta_{4} + 2 \beta_{3} + \beta_{2} - 4) q^{53} + ( - \beta_{9} + 2 \beta_{2} - 5) q^{55} + ( - \beta_{2} - 2) q^{57} + (2 \beta_{8} + \beta_{6} - \beta_{5}) q^{59} + ( - \beta_{7} + 2 \beta_{5} - 2 \beta_1) q^{61} + (2 \beta_{8} - \beta_{7} - \beta_{5} + 2 \beta_1) q^{63} + (3 \beta_{8} + \beta_{7} - 3 \beta_{6} - \beta_{5} + \beta_1) q^{65} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} + 3) q^{67} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{69} + ( - 2 \beta_{2} + 4) q^{71} + ( - \beta_{9} - \beta_{3} - 2) q^{73} + (4 \beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_1) q^{75} + ( - 2 \beta_{8} - 2 \beta_{6}) q^{77} + (2 \beta_{4} - 2 \beta_{2}) q^{79} + ( - \beta_{4} + 2 \beta_{3} + 1) q^{81} + ( - 2 \beta_{8} + \beta_{6} + \beta_{5} - 2 \beta_1) q^{83} + ( - \beta_{4} - \beta_{2} + 4) q^{85} + (\beta_{9} + 2 \beta_{4} - 3) q^{87} + ( - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1) q^{89} + (\beta_{9} + 2 \beta_{4} - \beta_{3} + \beta_{2}) q^{91} + ( - \beta_{9} - \beta_{4} + 3 \beta_{2} - 5) q^{93} + (3 \beta_{8} - \beta_{7} - 2 \beta_{6} - 4 \beta_1) q^{95} + (\beta_{9} + \beta_{4} + \beta_{2} + 3) q^{97} + (\beta_{9} - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{5} - 12 q^{9} - 12 q^{11} + 2 q^{17} + 24 q^{21} + 20 q^{25} - 44 q^{45} + 12 q^{47} + 14 q^{49} - 34 q^{53} - 48 q^{55} - 22 q^{57} + 36 q^{67} + 14 q^{69} + 36 q^{71} - 26 q^{73} - 12 q^{79} + 22 q^{81} + 42 q^{85} - 36 q^{87} - 2 q^{89} - 8 q^{91} - 42 q^{93} + 30 q^{97} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 21x^{8} + 143x^{6} + 341x^{4} + 228x^{2} + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{8} - 23\nu^{6} - 160\nu^{4} - 313\nu^{2} - 42 ) / 29 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} + 23\nu^{6} + 189\nu^{4} + 603\nu^{2} + 390 ) / 58 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} + 23\nu^{6} + 160\nu^{4} + 342\nu^{2} + 158 ) / 29 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{9} - 23\nu^{7} - 189\nu^{5} - 603\nu^{3} - 390\nu ) / 58 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} + 23\nu^{7} + 189\nu^{5} + 661\nu^{3} + 796\nu ) / 58 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -13\nu^{9} - 241\nu^{7} - 1355\nu^{5} - 2097\nu^{3} + 324\nu ) / 232 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 25\nu^{9} + 517\nu^{7} + 3391\nu^{5} + 7245\nu^{3} + 3196\nu ) / 232 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -8\nu^{8} - 155\nu^{6} - 932\nu^{4} - 1721\nu^{2} - 481 ) / 29 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -10\beta_{4} + 2\beta_{3} - 9\beta_{2} + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{8} - \beta_{7} - 9\beta_{6} - 12\beta_{5} + 58\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{9} + 93\beta_{4} - 24\beta_{3} + 73\beta_{2} - 223 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 19\beta_{8} + 23\beta_{7} + 72\beta_{6} + 116\beta_{5} - 502\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -23\beta_{9} - 852\beta_{4} + 232\beta_{3} - 581\beta_{2} + 1859 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -248\beta_{8} - 340\beta_{7} - 558\beta_{6} - 1061\beta_{5} + 4415\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/712\mathbb{Z}\right)^\times\).

\(n\) \(357\) \(535\) \(537\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
3.02773i
2.79338i
1.75364i
0.874269i
0.436258i
0.436258i
0.874269i
1.75364i
2.79338i
3.02773i
0 3.02773i 0 −1.30883 0 2.79945i 0 −6.16713 0
177.2 0 2.79338i 0 4.18843 0 0.577984i 0 −4.80296 0
177.3 0 1.75364i 0 0.452336 0 0.355209i 0 −0.0752446 0
177.4 0 0.874269i 0 −3.92047 0 3.07093i 0 2.23565 0
177.5 0 0.436258i 0 −0.411457 0 3.20503i 0 2.80968 0
177.6 0 0.436258i 0 −0.411457 0 3.20503i 0 2.80968 0
177.7 0 0.874269i 0 −3.92047 0 3.07093i 0 2.23565 0
177.8 0 1.75364i 0 0.452336 0 0.355209i 0 −0.0752446 0
177.9 0 2.79338i 0 4.18843 0 0.577984i 0 −4.80296 0
177.10 0 3.02773i 0 −1.30883 0 2.79945i 0 −6.16713 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 712.2.e.b 10
4.b odd 2 1 1424.2.e.g 10
89.b even 2 1 inner 712.2.e.b 10
356.d odd 2 1 1424.2.e.g 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
712.2.e.b 10 1.a even 1 1 trivial
712.2.e.b 10 89.b even 2 1 inner
1424.2.e.g 10 4.b odd 2 1
1424.2.e.g 10 356.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 21T_{3}^{8} + 143T_{3}^{6} + 341T_{3}^{4} + 228T_{3}^{2} + 32 \) acting on \(S_{2}^{\mathrm{new}}(712, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 21 T^{8} + 143 T^{6} + \cdots + 32 \) Copy content Toggle raw display
$5$ \( (T^{5} + T^{4} - 17 T^{3} - 21 T^{2} + 4 T + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + 28 T^{8} + 264 T^{6} + \cdots + 32 \) Copy content Toggle raw display
$11$ \( (T^{5} + 6 T^{4} - 16 T^{3} - 92 T^{2} + \cdots + 224)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 98 T^{8} + 3248 T^{6} + \cdots + 100352 \) Copy content Toggle raw display
$17$ \( (T^{5} - T^{4} - 49 T^{3} + 57 T^{2} + 54 T + 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 79 T^{8} + 1601 T^{6} + \cdots + 800 \) Copy content Toggle raw display
$23$ \( T^{10} + 101 T^{8} + 3451 T^{6} + \cdots + 75272 \) Copy content Toggle raw display
$29$ \( T^{10} + 182 T^{8} + 11432 T^{6} + \cdots + 5120000 \) Copy content Toggle raw display
$31$ \( T^{10} + 119 T^{8} + 3693 T^{6} + \cdots + 5000 \) Copy content Toggle raw display
$37$ \( T^{10} + 222 T^{8} + \cdots + 55083008 \) Copy content Toggle raw display
$41$ \( T^{10} + 180 T^{8} + 10888 T^{6} + \cdots + 6422528 \) Copy content Toggle raw display
$43$ \( T^{10} + 303 T^{8} + \cdots + 71808128 \) Copy content Toggle raw display
$47$ \( (T^{5} - 6 T^{4} - 136 T^{3} + 116 T^{2} + \cdots + 9856)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + 17 T^{4} - 29 T^{3} - 1555 T^{2} + \cdots + 1556)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 258 T^{8} + 10580 T^{6} + \cdots + 61952 \) Copy content Toggle raw display
$61$ \( T^{10} + 438 T^{8} + \cdots + 2596610048 \) Copy content Toggle raw display
$67$ \( (T^{5} - 18 T^{4} - 36 T^{3} + 1300 T^{2} + \cdots - 6304)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} - 18 T^{4} + 60 T^{3} + 200 T^{2} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 13 T^{4} - 103 T^{3} - 1129 T^{2} + \cdots + 224)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + 6 T^{4} - 212 T^{3} - 168 T^{2} + \cdots - 28672)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 414 T^{8} + \cdots + 15147008 \) Copy content Toggle raw display
$89$ \( T^{10} + 2 T^{9} + \cdots + 5584059449 \) Copy content Toggle raw display
$97$ \( (T^{5} - 15 T^{4} - 99 T^{3} + 523 T^{2} + \cdots - 820)^{2} \) Copy content Toggle raw display
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