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Properties

Label 3150.3.e.i
Level $3150$
Weight $3$
Character orbit 3150.e
Analytic conductor $85.831$
Analytic rank $0$
Dimension $8$
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] = [3150,3,Mod(701,3150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3150.701");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3150.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: \(85.8312832735\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.98344960000.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 6x^{6} + 39x^{4} - 190x^{2} + 225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 630)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - 2 q^{4} - \beta_1 q^{7} + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - 2 q^{4} - \beta_1 q^{7} + 2 \beta_{3} q^{8} + (\beta_{4} - \beta_{3}) q^{11} + ( - \beta_{5} + \beta_{2} + 4) q^{13} + \beta_{4} q^{14} + 4 q^{16} + ( - \beta_{6} + 2 \beta_{4} - 2 \beta_{3}) q^{17} + (\beta_{2} + 4 \beta_1 + 10) q^{19} + (2 \beta_1 - 2) q^{22} + ( - \beta_{7} - 2 \beta_{6} + \cdots - 2 \beta_{3}) q^{23}+ \cdots - 7 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 32 q^{13} + 32 q^{16} + 80 q^{19} - 16 q^{22} - 112 q^{31} - 32 q^{34} + 32 q^{37} - 128 q^{43} - 32 q^{46} + 56 q^{49} - 64 q^{52} - 16 q^{58} - 80 q^{61} - 64 q^{64} - 128 q^{67} - 160 q^{76} - 128 q^{82} + 32 q^{88} - 64 q^{94} + 224 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 6x^{6} + 39x^{4} - 190x^{2} + 225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -4\nu^{7} - 19\nu^{5} - 151\nu^{3} + 1155\nu ) / 450 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - \nu^{4} + 11\nu^{2} + 270 ) / 30 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{7} + 52\nu^{5} + 358\nu^{3} - 615\nu ) / 450 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 7\nu^{4} + 55\nu^{2} - 108 ) / 18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - \nu^{6} + 18\nu^{5} - \nu^{4} + 72\nu^{3} + 11\nu^{2} - 705\nu + 270 ) / 90 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{7} - 8\nu^{6} + 17\nu^{5} - 68\nu^{4} + 113\nu^{3} - 392\nu^{2} - 195\nu + 810 ) / 45 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} - 17\nu^{5} - 113\nu^{3} + 195\nu ) / 15 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} + 12\beta_{3} + 6\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 3\beta_{6} + 12\beta_{4} + 4\beta_{2} - 18 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - 9\beta_{5} - 3\beta_{3} + 3\beta_{2} - 24\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -11\beta_{7} - 33\beta_{6} - 96\beta_{4} + 16\beta_{2} - 126 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -65\beta_{7} + 90\beta_{5} - 582\beta_{3} - 30\beta_{2} + 294\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 11\beta_{7} + 33\beta_{6} + 114\beta_{4} - 166\beta_{2} + 1584 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 673\beta_{7} + 252\beta_{5} + 6456\beta_{3} - 84\beta_{2} + 798\beta_1 ) / 12 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
1.32288 + 2.53225i
1.32288 + 0.296180i
−1.32288 + 0.296180i
−1.32288 + 2.53225i
1.32288 − 2.53225i
1.32288 − 0.296180i
−1.32288 − 0.296180i
−1.32288 − 2.53225i
− 1.41421i 0 −2.00000 0 0 −2.64575 2.82843i 0 0
701.2 − 1.41421i 0 −2.00000 0 0 −2.64575 2.82843i 0 0
701.3 − 1.41421i 0 −2.00000 0 0 2.64575 2.82843i 0 0
701.4 − 1.41421i 0 −2.00000 0 0 2.64575 2.82843i 0 0
701.5 1.41421i 0 −2.00000 0 0 −2.64575 − 2.82843i 0 0
701.6 1.41421i 0 −2.00000 0 0 −2.64575 − 2.82843i 0 0
701.7 1.41421i 0 −2.00000 0 0 2.64575 − 2.82843i 0 0
701.8 1.41421i 0 −2.00000 0 0 2.64575 − 2.82843i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 701.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.3.e.i 8
3.b odd 2 1 inner 3150.3.e.i 8
5.b even 2 1 630.3.e.a ✓ 8
5.c odd 4 2 3150.3.c.d 16
15.d odd 2 1 630.3.e.a ✓ 8
15.e even 4 2 3150.3.c.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.3.e.a ✓ 8 5.b even 2 1
630.3.e.a ✓ 8 15.d odd 2 1
3150.3.c.d 16 5.c odd 4 2
3150.3.c.d 16 15.e even 4 2
3150.3.e.i 8 1.a even 1 1 trivial
3150.3.e.i 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(3150, [\chi])\):

\( T_{11}^{4} + 32T_{11}^{2} + 144 \) Copy content Toggle raw display
\( T_{13}^{4} - 16T_{13}^{3} - 124T_{13}^{2} + 1504T_{13} - 2364 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 32 T^{2} + 144)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 16 T^{3} + \cdots - 2364)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 896 T^{6} + \cdots + 26873856 \) Copy content Toggle raw display
$19$ \( (T^{4} - 40 T^{3} + \cdots - 29916)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 75666805776 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 19925016336 \) Copy content Toggle raw display
$31$ \( (T^{4} + 56 T^{3} + \cdots - 111676)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 16 T^{3} + \cdots - 7664)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 1274875842816 \) Copy content Toggle raw display
$43$ \( (T^{4} + 64 T^{3} + \cdots + 87056)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 3582449565696 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 58107995136 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 21234991124736 \) Copy content Toggle raw display
$61$ \( (T^{4} + 40 T^{3} + \cdots - 42096)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 64 T^{3} + \cdots + 8709264)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 842087039037696 \) Copy content Toggle raw display
$73$ \( (T^{4} - 11100 T^{2} + \cdots - 6997500)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 7376 T^{2} + 7795264)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 2702420361216 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 70331973325056 \) Copy content Toggle raw display
$97$ \( (T^{4} - 112 T^{3} + \cdots + 18658756)^{2} \) Copy content Toggle raw display
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