Properties

Label 712.2.e.c
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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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: \( x^{10} + 19x^{8} + 113x^{6} + 217x^{4} + 136x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{2} q^{5} + \beta_{8} q^{7} + ( - \beta_{5} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{2} q^{5} + \beta_{8} q^{7} + ( - \beta_{5} - 1) q^{9} + (\beta_{4} + 1) q^{11} - \beta_{6} q^{13} + \beta_{9} q^{15} + (\beta_{3} + 1) q^{17} + (\beta_{9} + \beta_{8} + \beta_{6}) q^{19} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2) q^{21} + (\beta_{7} - \beta_{6} + \beta_1) q^{23} + ( - \beta_{5} + \beta_{3} + \beta_{2}) q^{25} + ( - \beta_{9} - \beta_{7}) q^{27} + (\beta_{8} - \beta_{7}) q^{29} + (2 \beta_{8} - 2 \beta_{7} + \beta_{6} - \beta_1) q^{31} + (\beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_1) q^{33} + ( - \beta_{9} + \beta_{7} + \beta_1) q^{35} + (\beta_{9} - \beta_{6} - \beta_1) q^{37} + ( - \beta_{4} - 1) q^{39} + (\beta_{9} - 2 \beta_{7} - \beta_1) q^{41} + ( - \beta_{8} - \beta_{6} + \beta_1) q^{43} + ( - \beta_{5} - \beta_{4} - \beta_{2} - 1) q^{45} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 1) q^{47} + ( - \beta_{5} - 2 \beta_{4} - \beta_{3} - 2) q^{49} + ( - 2 \beta_{8} + \beta_{7} - \beta_1) q^{51} + (\beta_{4} + \beta_{3} + 2) q^{53} + (\beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 3) q^{55} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 2) q^{57} + ( - \beta_{9} - 2 \beta_{8} + 3 \beta_{7} + \beta_1) q^{59} + (2 \beta_{9} + 3 \beta_{8} - \beta_{7} + 2 \beta_{6} - 2 \beta_1) q^{61} + ( - 3 \beta_{8} - \beta_{7} - 2 \beta_{6}) q^{63} + ( - \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + \beta_1) q^{65} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 3) q^{67} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{69} + ( - \beta_{5} + \beta_{4} - \beta_{3} - 2) q^{71} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 2) q^{73} + ( - 2 \beta_{9} - 2 \beta_{8} + 2 \beta_1) q^{75} + ( - 2 \beta_{9} + 3 \beta_{8} - \beta_{7} + \beta_{6} + 2 \beta_1) q^{77} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{79} + ( - \beta_{4} - \beta_{3} - 3 \beta_{2} - 1) q^{81} + (\beta_{9} - \beta_{8} + 3 \beta_{6} - \beta_1) q^{83} + (\beta_{4} - \beta_{3} - 2 \beta_{2} + 2) q^{85} + (\beta_{5} + \beta_{3} + 3) q^{87} + (\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_1 - 2) q^{89} + (\beta_{5} + \beta_{4} + \beta_{3}) q^{91} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + 3) q^{93} + ( - 2 \beta_{9} - \beta_{7} + 2 \beta_{6} - 3 \beta_1) q^{95} + (2 \beta_{5} + \beta_{3} - 2 \beta_{2} + 1) q^{97} + (\beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{9} + 8 q^{11} + 8 q^{17} + 12 q^{21} - 2 q^{25} - 8 q^{39} - 8 q^{45} - 4 q^{47} - 14 q^{49} + 16 q^{53} + 24 q^{55} + 12 q^{57} - 24 q^{67} + 16 q^{69} - 20 q^{71} - 20 q^{73} + 12 q^{79} - 6 q^{81} + 20 q^{85} + 28 q^{87} - 14 q^{89} - 4 q^{91} + 24 q^{93} + 8 q^{97} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 19x^{8} + 113x^{6} + 217x^{4} + 136x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{9} - 53\nu^{7} - 247\nu^{5} - 87\nu^{3} + 284\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{8} + 53\nu^{6} + 279\nu^{4} + 407\nu^{2} + 164 ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{8} - 53\nu^{6} - 247\nu^{4} - 119\nu^{2} + 60 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} + 15\nu^{6} + 61\nu^{4} + 37\nu^{2} - 28 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{8} - 99\nu^{6} - 561\nu^{4} - 721\nu^{2} - 124 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{9} - 53\nu^{7} - 279\nu^{5} - 407\nu^{3} - 196\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{9} + 53\nu^{7} + 279\nu^{5} + 375\nu^{3} - 28\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{9} + 53\nu^{7} + 279\nu^{5} + 391\nu^{3} + 116\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13\nu^{9} + 251\nu^{7} + 1497\nu^{5} + 2745\nu^{3} + 1372\nu ) / 64 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{8} - \beta_{7} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{4} - \beta_{3} + 2\beta_{2} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{8} + 5\beta_{7} - 9\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -9\beta_{5} + 9\beta_{4} + 11\beta_{3} - 16\beta_{2} + 58 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 55\beta_{8} - 35\beta_{7} + 73\beta_{6} + 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 71\beta_{5} - 77\beta_{4} - 95\beta_{3} + 126\beta_{2} - 462 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6\beta_{9} - 445\beta_{8} + 267\beta_{7} - 583\beta_{6} - 54\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -553\beta_{5} + 659\beta_{4} + 791\beta_{3} - 988\beta_{2} + 3744 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -106\beta_{9} + 3631\beta_{8} - 2075\beta_{7} + 4645\beta_{6} + 582\beta_1 ) / 2 \) 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
0.967164i
2.88577i
0.389125i
1.31374i
2.80350i
2.80350i
1.31374i
0.389125i
2.88577i
0.967164i
0 2.87642i 0 0.427234 0 2.82321i 0 −5.27380 0
177.2 0 2.68448i 0 1.78729 0 5.05891i 0 −4.20643 0
177.3 0 1.77351i 0 −3.39335 0 1.53242i 0 −0.145352 0
177.4 0 1.14423i 0 −1.46162 0 2.16654i 0 1.69074 0
177.5 0 0.255270i 0 2.64044 0 1.18098i 0 2.93484 0
177.6 0 0.255270i 0 2.64044 0 1.18098i 0 2.93484 0
177.7 0 1.14423i 0 −1.46162 0 2.16654i 0 1.69074 0
177.8 0 1.77351i 0 −3.39335 0 1.53242i 0 −0.145352 0
177.9 0 2.68448i 0 1.78729 0 5.05891i 0 −4.20643 0
177.10 0 2.87642i 0 0.427234 0 2.82321i 0 −5.27380 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.c 10
4.b odd 2 1 1424.2.e.h 10
89.b even 2 1 inner 712.2.e.c 10
356.d odd 2 1 1424.2.e.h 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
712.2.e.c 10 1.a even 1 1 trivial
712.2.e.c 10 89.b even 2 1 inner
1424.2.e.h 10 4.b odd 2 1
1424.2.e.h 10 356.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 20T_{3}^{8} + 134T_{3}^{6} + 338T_{3}^{4} + 267T_{3}^{2} + 16 \) 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
$3$ \( T^{10} + 20 T^{8} + 134 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{5} - 12 T^{3} + 6 T^{2} + 23 T - 10)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + 42 T^{8} + 508 T^{6} + \cdots + 3136 \) Copy content Toggle raw display
$11$ \( (T^{5} - 4 T^{4} - 24 T^{3} + 36 T^{2} + \cdots + 128)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 40 T^{8} + 504 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( (T^{5} - 4 T^{4} - 26 T^{3} + 160 T^{2} + \cdots + 122)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 86 T^{8} + 2702 T^{6} + \cdots + 583696 \) Copy content Toggle raw display
$23$ \( T^{10} + 66 T^{8} + 910 T^{6} + \cdots + 11236 \) Copy content Toggle raw display
$29$ \( T^{10} + 60 T^{8} + 1096 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$31$ \( T^{10} + 188 T^{8} + 11306 T^{6} + \cdots + 1012036 \) Copy content Toggle raw display
$37$ \( T^{10} + 164 T^{8} + 8096 T^{6} + \cdots + 4194304 \) Copy content Toggle raw display
$41$ \( T^{10} + 268 T^{8} + \cdots + 239878144 \) Copy content Toggle raw display
$43$ \( T^{10} + 78 T^{8} + 1946 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( (T^{5} + 2 T^{4} - 96 T^{3} - 300 T^{2} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} - 8 T^{4} - 24 T^{3} + 4 T^{2} + 9 T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 394 T^{8} + \cdots + 96510976 \) Copy content Toggle raw display
$61$ \( T^{10} + 380 T^{8} + \cdots + 120472576 \) Copy content Toggle raw display
$67$ \( (T^{5} + 12 T^{4} - 60 T^{3} - 564 T^{2} + \cdots - 1280)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + 10 T^{4} - 76 T^{3} - 1096 T^{2} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 10 T^{4} - 66 T^{3} - 800 T^{2} + \cdots + 5218)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 6 T^{4} - 188 T^{3} + 616 T^{2} + \cdots + 1280)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 454 T^{8} + \cdots + 64513024 \) Copy content Toggle raw display
$89$ \( T^{10} + 14 T^{9} + \cdots + 5584059449 \) Copy content Toggle raw display
$97$ \( (T^{5} - 4 T^{4} - 262 T^{3} + 784 T^{2} + \cdots + 9950)^{2} \) Copy content Toggle raw display
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