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Properties

Label 3150.3.e.j
Level $3150$
Weight $3$
Character orbit 3150.e
Analytic conductor $85.831$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 48x^{10} + 825x^{8} + 6568x^{6} + 25440x^{4} + 43968x^{2} + 23104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - 2 q^{4} - \beta_{4} q^{7} - 2 \beta_{5} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} - 2 q^{4} - \beta_{4} q^{7} - 2 \beta_{5} q^{8} + (\beta_{11} + \beta_{10}) q^{11} + ( - \beta_{9} - \beta_{6} - 4) q^{13} + \beta_{2} q^{14} + 4 q^{16} + ( - \beta_{11} - \beta_{7} - \beta_{2}) q^{17} + ( - 2 \beta_{6} - \beta_{4} - \beta_1 - 7) q^{19} + (\beta_{9} - 2 \beta_{4} + \cdots + \beta_1) q^{22}+ \cdots + 7 \beta_{5} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 24 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 24 q^{4} - 40 q^{13} + 48 q^{16} - 80 q^{19} + 16 q^{31} + 8 q^{34} - 144 q^{37} - 48 q^{43} + 144 q^{46} + 84 q^{49} + 80 q^{52} - 120 q^{58} - 120 q^{61} - 96 q^{64} - 256 q^{67} + 216 q^{73} + 160 q^{76} - 48 q^{79} + 200 q^{82} - 128 q^{94} - 360 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 48x^{10} + 825x^{8} + 6568x^{6} + 25440x^{4} + 43968x^{2} + 23104 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{10} + 52\nu^{8} + 961\nu^{6} + 7532\nu^{4} + 21224\nu^{2} + 8192 ) / 864 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -13\nu^{11} - 548\nu^{9} - 7533\nu^{7} - 42140\nu^{5} - 100744\nu^{3} - 104640\nu ) / 10944 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{10} - 40\nu^{8} - 497\nu^{6} - 2272\nu^{4} - 3256\nu^{2} ) / 96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{10} - 218\nu^{8} - 3173\nu^{6} - 19138\nu^{4} - 45736\nu^{2} - 28768 ) / 432 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -83\nu^{11} - 3680\nu^{9} - 54947\nu^{7} - 342376\nu^{5} - 849160\nu^{3} - 569824\nu ) / 32832 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25\nu^{10} + 1096\nu^{8} + 16057\nu^{6} + 97184\nu^{4} + 230600\nu^{2} + 138176 ) / 864 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 35\nu^{11} + 1566\nu^{9} + 23707\nu^{7} + 150118\nu^{5} + 374208\nu^{3} + 265424\nu ) / 5472 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -119\nu^{11} - 5294\nu^{9} - 79631\nu^{7} - 503926\nu^{5} - 1281184\nu^{3} - 845776\nu ) / 16416 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -19\nu^{10} - 838\nu^{8} - 12379\nu^{6} - 75374\nu^{4} - 176912\nu^{2} - 100544 ) / 432 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -33\nu^{11} - 1432\nu^{9} - 20689\nu^{7} - 123568\nu^{5} - 291864\nu^{3} - 183264\nu ) / 3648 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -205\nu^{11} - 8928\nu^{9} - 129149\nu^{7} - 761240\nu^{5} - 1685304\nu^{3} - 742816\nu ) / 10944 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 3\beta_{5} - \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} - 3\beta_{6} - 5\beta_{4} + \beta_{3} - 4\beta _1 - 48 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - 2\beta_{10} + 5\beta_{8} - 4\beta_{7} - 28\beta_{5} + 7\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 34\beta_{9} + 93\beta_{6} + 125\beta_{4} - 16\beta_{3} + 73\beta _1 + 672 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -75\beta_{11} + 186\beta_{10} - 393\beta_{8} + 194\beta_{7} + 1764\beta_{5} - 545\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -263\beta_{9} - 747\beta_{6} - 997\beta_{4} + 95\beta_{3} - 434\beta _1 - 4024 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1737\beta_{11} - 4662\beta_{10} + 8721\beta_{8} - 3668\beta_{7} - 36492\beta_{5} + 12803\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 17102\beta_{9} + 50379\beta_{6} + 68347\beta_{4} - 5456\beta_{3} + 24767\beta _1 + 240000 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -12911\beta_{11} + 35938\beta_{10} - 62581\beta_{8} + 24786\beta_{7} + 253964\beta_{5} - 95205\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -365939\beta_{9} - 1102911\beta_{6} - 1515073\beta_{4} + 109115\beta_{3} - 496418\beta _1 - 4970712 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 846087 \beta_{11} - 2399754 \beta_{10} + 4018287 \beta_{8} - 1552924 \beta_{7} + \cdots + 6227821 \beta_{2} ) / 6 \) 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
− 1.95713i
− 4.62557i
2.59055i
2.09246i
− 3.28567i
0.942724i
− 2.59055i
4.62557i
1.95713i
− 0.942724i
3.28567i
− 2.09246i
− 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
701.9 1.41421i 0 −2.00000 0 0 −2.64575 − 2.82843i 0 0
701.10 1.41421i 0 −2.00000 0 0 2.64575 − 2.82843i 0 0
701.11 1.41421i 0 −2.00000 0 0 2.64575 − 2.82843i 0 0
701.12 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 701.12
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.j 12
3.b odd 2 1 inner 3150.3.e.j 12
5.b even 2 1 3150.3.e.k 12
5.c odd 4 2 630.3.c.a ✓ 24
15.d odd 2 1 3150.3.e.k 12
15.e even 4 2 630.3.c.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.3.c.a ✓ 24 5.c odd 4 2
630.3.c.a ✓ 24 15.e even 4 2
3150.3.e.j 12 1.a even 1 1 trivial
3150.3.e.j 12 3.b odd 2 1 inner
3150.3.e.k 12 5.b even 2 1
3150.3.e.k 12 15.d odd 2 1

Hecke kernels

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

\( T_{11}^{12} + 928 T_{11}^{10} + 293824 T_{11}^{8} + 37390848 T_{11}^{6} + 1974961152 T_{11}^{4} + \cdots + 155223392256 \) Copy content Toggle raw display
\( T_{13}^{6} + 20T_{13}^{5} - 440T_{13}^{4} - 6832T_{13}^{3} + 39652T_{13}^{2} + 380816T_{13} + 512208 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 155223392256 \) Copy content Toggle raw display
$13$ \( (T^{6} + 20 T^{5} + \cdots + 512208)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 48322295627776 \) Copy content Toggle raw display
$19$ \( (T^{6} + 40 T^{5} + \cdots - 480000)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 2425365504 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{6} - 8 T^{5} + \cdots - 21913344)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 72 T^{5} + \cdots + 32608256)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( (T^{6} + 24 T^{5} + \cdots - 114172672)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 147591786397696 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 63\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{6} + 60 T^{5} + \cdots + 661414464)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 128 T^{5} + \cdots + 9173508096)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{6} - 108 T^{5} + \cdots - 64981716400)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 24 T^{5} + \cdots - 3651922944)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{6} + 180 T^{5} + \cdots - 596188761264)^{2} \) Copy content Toggle raw display
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