Properties

Label 3150.2.d.f
Level $3150$
Weight $2$
Character orbit 3150.d
Analytic conductor $25.153$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3150,2,Mod(3149,3150)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3150.3149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.d (of order \(2\), degree \(1\), not 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: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.7442857984.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta_1 q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + \beta_1 q^{7} + q^{8} - \beta_{3} q^{11} + ( - \beta_{7} + 1) q^{13} + \beta_1 q^{14} + q^{16} + (\beta_{6} - \beta_{3}) q^{17} + ( - \beta_{6} - \beta_{5} + \beta_{2} - \beta_1) q^{19} - \beta_{3} q^{22} + (\beta_{7} - \beta_{2} - \beta_1 - 1) q^{23} + ( - \beta_{7} + 1) q^{26} + \beta_1 q^{28} + (\beta_{5} + \beta_{3} - \beta_{2} + \beta_1) q^{29} + ( - 2 \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{31} + q^{32} + (\beta_{6} - \beta_{3}) q^{34} + (\beta_{6} - \beta_{3} - \beta_{2} + \beta_1) q^{37} + ( - \beta_{6} - \beta_{5} + \beta_{2} - \beta_1) q^{38} + ( - \beta_{7} + 1) q^{41} + (\beta_{6} + \beta_{5} - \beta_{3}) q^{43} - \beta_{3} q^{44} + (\beta_{7} - \beta_{2} - \beta_1 - 1) q^{46} + ( - \beta_{6} + 2 \beta_{5} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{47} + (\beta_{6} - \beta_{4} + \beta_{2}) q^{49} + ( - \beta_{7} + 1) q^{52} + (\beta_{7} + \beta_{2} + \beta_1 - 1) q^{53} + \beta_1 q^{56} + (\beta_{5} + \beta_{3} - \beta_{2} + \beta_1) q^{58} + ( - \beta_{7} - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{59} + (\beta_{6} + 2 \beta_{5} + \beta_{2} - \beta_1) q^{61} + ( - 2 \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{62} + q^{64} + (\beta_{6} + 3 \beta_{5} - \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{67} + (\beta_{6} - \beta_{3}) q^{68} + (\beta_{6} + \beta_{5} - \beta_{2} + \beta_1) q^{71} + (2 \beta_{2} + 2 \beta_1 - 6) q^{73} + (\beta_{6} - \beta_{3} - \beta_{2} + \beta_1) q^{74} + ( - \beta_{6} - \beta_{5} + \beta_{2} - \beta_1) q^{76} + (\beta_{7} - \beta_{6} - 3 \beta_{5} - \beta_{4} - \beta_1) q^{77} + ( - \beta_{7} - 3 \beta_{2} - 3 \beta_1 - 1) q^{79} + ( - \beta_{7} + 1) q^{82} + (2 \beta_{5} - 2 \beta_{2} + 2 \beta_1) q^{83} + (\beta_{6} + \beta_{5} - \beta_{3}) q^{86} - \beta_{3} q^{88} + (2 \beta_{7} + \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{89} + (\beta_{7} - \beta_{6} + 4 \beta_{5} - \beta_{4} + \beta_1) q^{91} + (\beta_{7} - \beta_{2} - \beta_1 - 1) q^{92} + ( - \beta_{6} + 2 \beta_{5} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{94} + ( - 3 \beta_{7} + \beta_{2} + \beta_1 + 3) q^{97} + (\beta_{6} - \beta_{4} + \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} + 8 q^{13} + 8 q^{16} - 8 q^{23} + 8 q^{26} + 8 q^{32} + 8 q^{41} - 8 q^{46} - 4 q^{49} + 8 q^{52} - 8 q^{53} + 8 q^{64} - 48 q^{73} - 4 q^{77} - 8 q^{79} + 8 q^{82} - 8 q^{89} - 4 q^{91} - 8 q^{92} + 24 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 26\nu^{5} + 18\nu^{4} + 223\nu^{3} + 234\nu^{2} + 774\nu + 324 ) / 216 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 26\nu^{5} + 18\nu^{4} - 223\nu^{3} + 234\nu^{2} - 774\nu + 324 ) / 216 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 20\nu^{5} + 73\nu^{3} - 54\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 26\nu^{5} + 18\nu^{4} - 223\nu^{3} + 450\nu^{2} - 558\nu + 1836 ) / 216 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} - 112\nu^{5} - 665\nu^{3} - 954\nu ) / 216 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 26\nu^{5} + 187\nu^{3} + 306\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 22\nu^{4} + 123\nu^{2} + 138 ) / 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} - \beta _1 - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{6} - 13\beta_{5} - 13\beta_{3} + 7\beta_{2} - 7\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13\beta_{5} - 26\beta_{4} + 13\beta_{3} + 25\beta_{2} + 25\beta _1 + 146 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 50\beta_{6} + 187\beta_{5} + 163\beta_{3} - 73\beta_{2} + 73\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 24\beta_{7} - 163\beta_{5} + 326\beta_{4} - 163\beta_{3} - 427\beta_{2} - 427\beta _1 - 1766 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -854\beta_{6} - 2737\beta_{5} - 2113\beta_{3} + 895\beta_{2} - 895\beta_1 ) / 2 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3149.1
2.73923i
2.73923i
1.91681i
1.91681i
0.916813i
0.916813i
3.73923i
3.73923i
1.00000 0 1.00000 0 0 −1.93693 1.80230i 1.00000 0 0
3149.2 1.00000 0 1.00000 0 0 −1.93693 + 1.80230i 1.00000 0 0
3149.3 1.00000 0 1.00000 0 0 −1.35539 2.27220i 1.00000 0 0
3149.4 1.00000 0 1.00000 0 0 −1.35539 + 2.27220i 1.00000 0 0
3149.5 1.00000 0 1.00000 0 0 0.648285 2.56510i 1.00000 0 0
3149.6 1.00000 0 1.00000 0 0 0.648285 + 2.56510i 1.00000 0 0
3149.7 1.00000 0 1.00000 0 0 2.64404 0.0951965i 1.00000 0 0
3149.8 1.00000 0 1.00000 0 0 2.64404 + 0.0951965i 1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3149.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.2.d.f 8
3.b odd 2 1 3150.2.d.c 8
5.b even 2 1 3150.2.d.a 8
5.c odd 4 1 630.2.b.b yes 8
5.c odd 4 1 3150.2.b.f 8
7.b odd 2 1 3150.2.d.d 8
15.d odd 2 1 3150.2.d.d 8
15.e even 4 1 630.2.b.a 8
15.e even 4 1 3150.2.b.e 8
20.e even 4 1 5040.2.f.i 8
21.c even 2 1 3150.2.d.a 8
35.c odd 2 1 3150.2.d.c 8
35.f even 4 1 630.2.b.a 8
35.f even 4 1 3150.2.b.e 8
60.l odd 4 1 5040.2.f.f 8
105.g even 2 1 inner 3150.2.d.f 8
105.k odd 4 1 630.2.b.b yes 8
105.k odd 4 1 3150.2.b.f 8
140.j odd 4 1 5040.2.f.f 8
420.w even 4 1 5040.2.f.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.b.a 8 15.e even 4 1
630.2.b.a 8 35.f even 4 1
630.2.b.b yes 8 5.c odd 4 1
630.2.b.b yes 8 105.k odd 4 1
3150.2.b.e 8 15.e even 4 1
3150.2.b.e 8 35.f even 4 1
3150.2.b.f 8 5.c odd 4 1
3150.2.b.f 8 105.k odd 4 1
3150.2.d.a 8 5.b even 2 1
3150.2.d.a 8 21.c even 2 1
3150.2.d.c 8 3.b odd 2 1
3150.2.d.c 8 35.c odd 2 1
3150.2.d.d 8 7.b odd 2 1
3150.2.d.d 8 15.d odd 2 1
3150.2.d.f 8 1.a even 1 1 trivial
3150.2.d.f 8 105.g even 2 1 inner
5040.2.f.f 8 60.l odd 4 1
5040.2.f.f 8 140.j odd 4 1
5040.2.f.i 8 20.e even 4 1
5040.2.f.i 8 420.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3150, [\chi])\):

\( T_{11}^{8} + 52T_{11}^{6} + 820T_{11}^{4} + 4320T_{11}^{2} + 5184 \) Copy content Toggle raw display
\( T_{13}^{4} - 4T_{13}^{3} - 22T_{13}^{2} + 24T_{13} + 72 \) Copy content Toggle raw display
\( T_{23}^{4} + 4T_{23}^{3} - 44T_{23}^{2} - 96T_{23} + 288 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{6} - 24 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 52 T^{6} + 820 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} - 22 T^{2} + 24 T + 72)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 100 T^{6} + 2356 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$19$ \( T^{8} + 108 T^{6} + 3460 T^{4} + \cdots + 82944 \) Copy content Toggle raw display
$23$ \( (T^{4} + 4 T^{3} - 44 T^{2} - 96 T + 288)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 104 T^{6} + 3280 T^{4} + \cdots + 82944 \) Copy content Toggle raw display
$31$ \( T^{8} + 216 T^{6} + 13840 T^{4} + \cdots + 1327104 \) Copy content Toggle raw display
$37$ \( T^{8} + 144 T^{6} + 3640 T^{4} + \cdots + 15376 \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} - 22 T^{2} + 24 T + 72)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 116 T^{6} + 1412 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( T^{8} + 444 T^{6} + \cdots + 144576576 \) Copy content Toggle raw display
$53$ \( (T^{4} + 4 T^{3} - 52 T^{2} + 288)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 194 T^{2} + 312 T + 5112)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 252 T^{6} + 19252 T^{4} + \cdots + 4981824 \) Copy content Toggle raw display
$67$ \( T^{8} + 356 T^{6} + \cdots + 23970816 \) Copy content Toggle raw display
$71$ \( T^{8} + 108 T^{6} + 3460 T^{4} + \cdots + 82944 \) Copy content Toggle raw display
$73$ \( (T^{4} + 24 T^{3} + 112 T^{2} - 576 T - 2448)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} - 268 T^{2} - 416 T + 14368)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 240 T^{6} + 13120 T^{4} + \cdots + 1327104 \) Copy content Toggle raw display
$89$ \( (T^{4} + 4 T^{3} - 238 T^{2} - 2376 T - 5832)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 12 T^{3} - 212 T^{2} + \cdots + 14112)^{2} \) Copy content Toggle raw display
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