Properties

Label 1050.2.m.f
Level $1050$
Weight $2$
Character orbit 1050.m
Analytic conductor $8.384$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,2,Mod(307,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.m (of order \(4\), degree \(2\), 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: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.1871773696.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 31x^{4} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_1 q^{3} + \beta_{2} q^{4} + \beta_{2} q^{6} + ( - \beta_{7} - \beta_{4}) q^{7} + \beta_{7} q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_1 q^{3} + \beta_{2} q^{4} + \beta_{2} q^{6} + ( - \beta_{7} - \beta_{4}) q^{7} + \beta_{7} q^{8} + \beta_{2} q^{9} + ( - \beta_{5} + \beta_{3}) q^{11} + \beta_{7} q^{12} + (\beta_{6} - \beta_{4} - 3 \beta_1) q^{13} + ( - \beta_{5} - \beta_{2} + 1) q^{14} - q^{16} + ( - \beta_{6} - \beta_{4} + \beta_1) q^{17} + \beta_{7} q^{18} + ( - \beta_{5} + \beta_{3}) q^{19} + ( - \beta_{5} - \beta_{2} + 1) q^{21} + (\beta_{6} - \beta_{4} - \beta_1) q^{22} + (2 \beta_{6} + 2 \beta_{4} - 2 \beta_1) q^{23} - q^{24} + ( - \beta_{5} - \beta_{3} - 4 \beta_{2}) q^{26} + \beta_{7} q^{27} + \beta_{6} q^{28} + (2 \beta_{5} + 2 \beta_{3} + 4 \beta_{2}) q^{31} - \beta_1 q^{32} + (\beta_{6} - \beta_{4} - \beta_1) q^{33} + ( - \beta_{5} + \beta_{3}) q^{34} - q^{36} + 6 \beta_1 q^{37} + (\beta_{6} - \beta_{4} - \beta_1) q^{38} + ( - \beta_{5} - \beta_{3} - 4 \beta_{2}) q^{39} + (\beta_{5} + \beta_{3} + 8 \beta_{2}) q^{41} + \beta_{6} q^{42} + (6 \beta_{7} + \beta_{6} + \beta_{4} - \beta_1) q^{43} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2}) q^{44} + (2 \beta_{5} - 2 \beta_{3}) q^{46} + ( - \beta_{6} - \beta_{4} + \beta_1) q^{47} - \beta_1 q^{48} + (\beta_{5} + \beta_{3} + \beta_{2} + 6) q^{49} + ( - \beta_{5} + \beta_{3}) q^{51} + ( - 2 \beta_{7} + \beta_{6} + \beta_{4} - \beta_1) q^{52} + (6 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} - 2 \beta_1) q^{53} - q^{54} - \beta_{3} q^{56} + (\beta_{6} - \beta_{4} - \beta_1) q^{57} + (\beta_{5} - \beta_{3} - 6) q^{59} + (\beta_{5} + \beta_{3} - 10 \beta_{2}) q^{61} + ( - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_1) q^{62} + \beta_{6} q^{63} - \beta_{2} q^{64} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2}) q^{66} + (\beta_{6} - \beta_{4} - 7 \beta_1) q^{67} + (\beta_{6} - \beta_{4} - \beta_1) q^{68} + (2 \beta_{5} - 2 \beta_{3}) q^{69} + ( - \beta_{5} + \beta_{3} + 6) q^{71} - \beta_1 q^{72} + (2 \beta_{6} - 2 \beta_{4} + 6 \beta_1) q^{73} + 6 \beta_{2} q^{74} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2}) q^{76} + ( - 6 \beta_{7} - \beta_{6} + \beta_{4} + 7 \beta_1) q^{77} + ( - 2 \beta_{7} + \beta_{6} + \beta_{4} - \beta_1) q^{78} + (2 \beta_{5} + 2 \beta_{3} + 8 \beta_{2}) q^{79} - q^{81} + (6 \beta_{7} - \beta_{6} - \beta_{4} + \beta_1) q^{82} - 12 \beta_1 q^{83} - \beta_{3} q^{84} + (\beta_{5} - \beta_{3} - 6) q^{86} + (\beta_{6} + \beta_{4} - \beta_1) q^{88} + (3 \beta_{5} - 3 \beta_{3} - 6) q^{89} + (3 \beta_{5} + \beta_{3} + 10 \beta_{2} + 4) q^{91} + ( - 2 \beta_{6} + 2 \beta_{4} + 2 \beta_1) q^{92} + ( - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_1) q^{93} + ( - \beta_{5} + \beta_{3}) q^{94} - \beta_{2} q^{96} + 8 \beta_{7} q^{97} + ( - \beta_{7} - \beta_{6} - \beta_{4} + 7 \beta_1) q^{98} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{11} + 4 q^{14} - 8 q^{16} - 8 q^{19} + 4 q^{21} - 8 q^{24} - 8 q^{34} - 8 q^{36} + 16 q^{46} + 48 q^{49} - 8 q^{51} - 8 q^{54} + 4 q^{56} - 40 q^{59} + 16 q^{69} + 40 q^{71} - 8 q^{81} + 4 q^{84} - 40 q^{86} - 24 q^{89} + 40 q^{91} - 8 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 31x^{4} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 19\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 40\nu^{2} ) / 63 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{6} + 9\nu^{4} - 97\nu^{2} + 108 ) / 63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 40\nu^{3} - 63\nu ) / 63 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{6} - 9\nu^{4} - 97\nu^{2} - 108 ) / 63 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 3\nu^{5} + 40\nu^{3} + 120\nu ) / 63 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{7} + 97\nu^{3} ) / 189 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} - \beta_{4} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{3} + 8\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} + 2\beta_{6} + 2\beta_{4} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{5} + 7\beta_{3} - 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -19\beta_{6} + 19\beta_{4} + 61\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -20\beta_{5} - 20\beta_{3} - 97\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 240\beta_{7} - 97\beta_{6} - 97\beta_{4} + 97\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(\beta_{2}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−0.921201 0.921201i
1.62831 + 1.62831i
−1.62831 1.62831i
0.921201 + 0.921201i
−0.921201 + 0.921201i
1.62831 1.62831i
−1.62831 + 1.62831i
0.921201 0.921201i
−0.707107 0.707107i −0.707107 0.707107i 1.00000i 0 1.00000i −2.54951 + 0.707107i 0.707107 0.707107i 1.00000i 0
307.2 −0.707107 0.707107i −0.707107 0.707107i 1.00000i 0 1.00000i 2.54951 + 0.707107i 0.707107 0.707107i 1.00000i 0
307.3 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000i −2.54951 0.707107i −0.707107 + 0.707107i 1.00000i 0
307.4 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000i 2.54951 0.707107i −0.707107 + 0.707107i 1.00000i 0
643.1 −0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000i −2.54951 0.707107i 0.707107 + 0.707107i 1.00000i 0
643.2 −0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000i 2.54951 0.707107i 0.707107 + 0.707107i 1.00000i 0
643.3 0.707107 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000i −2.54951 + 0.707107i −0.707107 0.707107i 1.00000i 0
643.4 0.707107 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000i 2.54951 + 0.707107i −0.707107 0.707107i 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
35.f even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.m.f yes 8
5.b even 2 1 inner 1050.2.m.f yes 8
5.c odd 4 2 1050.2.m.e 8
7.b odd 2 1 1050.2.m.e 8
35.c odd 2 1 1050.2.m.e 8
35.f even 4 2 inner 1050.2.m.f yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.m.e 8 5.c odd 4 2
1050.2.m.e 8 7.b odd 2 1
1050.2.m.e 8 35.c odd 2 1
1050.2.m.f yes 8 1.a even 1 1 trivial
1050.2.m.f yes 8 5.b even 2 1 inner
1050.2.m.f yes 8 35.f even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{2} + 2T_{11} - 12 \) Copy content Toggle raw display
\( T_{13}^{8} + 1904T_{13}^{4} + 256 \) Copy content Toggle raw display
\( T_{19}^{2} + 2T_{19} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 12 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T - 12)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 1904T^{4} + 256 \) Copy content Toggle raw display
$17$ \( T^{8} + 496 T^{4} + 20736 \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T - 12)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 7936 T^{4} + \cdots + 5308416 \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} + 112 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 124 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 5488 T^{4} + 20736 \) Copy content Toggle raw display
$47$ \( T^{8} + 496 T^{4} + 20736 \) Copy content Toggle raw display
$53$ \( T^{8} + 15904 T^{4} + \cdots + 1679616 \) Copy content Toggle raw display
$59$ \( (T^{2} + 10 T + 12)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 268 T^{2} + 11664)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 12784 T^{4} + \cdots + 1679616 \) Copy content Toggle raw display
$71$ \( (T^{2} - 10 T + 12)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + 30464 T^{4} + 65536 \) Copy content Toggle raw display
$79$ \( (T^{4} + 176 T^{2} + 256)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 20736)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T - 108)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 4096)^{2} \) Copy content Toggle raw display
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