Properties

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

Related objects

Downloads

Learn more

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: 8.0.1698758656.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 210)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_{3} q^{3} - \beta_{5} q^{4} + \beta_{5} q^{6} + ( - \beta_{2} + \beta_1 - 1) q^{7} - \beta_1 q^{8} - \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + \beta_{3} q^{3} - \beta_{5} q^{4} + \beta_{5} q^{6} + ( - \beta_{2} + \beta_1 - 1) q^{7} - \beta_1 q^{8} - \beta_{5} q^{9} + ( - \beta_{7} + \beta_{6} - \beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{11} + \beta_1 q^{12} + (2 \beta_{5} + 2 \beta_{3} - 2) q^{13} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{14} - q^{16} + (\beta_{6} + 2 \beta_{5} + \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{17} - \beta_1 q^{18} + (\beta_{7} - \beta_{6}) q^{19} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{21} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{22} + (\beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 2) q^{23} + q^{24} + (2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{26} + \beta_1 q^{27} + (\beta_{7} + \beta_{5} - \beta_{3} + \beta_1) q^{28} + (\beta_{7} + \beta_{6} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1) q^{29} + (\beta_{7} + \beta_{6} + 4 \beta_{5}) q^{31} + \beta_{3} q^{32} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{33} + ( - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{34} - q^{36} + ( - \beta_{7} - 4 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_1 + 4) q^{37} + (\beta_{6} - \beta_{3} - \beta_{2}) q^{38} + ( - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{39} + (\beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} + 3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{41} + ( - \beta_{7} - \beta_{5} + \beta_{3} - \beta_1) q^{42} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{43} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{2}) q^{44} + ( - \beta_{7} + \beta_{6} - \beta_{4} + 3 \beta_{3} + \beta_{2} - 3 \beta_1) q^{46} + (\beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{47} - \beta_{3} q^{48} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} + 2 \beta_{2} + \beta_1 - 1) q^{49} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{51} + (2 \beta_{5} + 2 \beta_1 + 2) q^{52} + (\beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{53} + q^{54} + (\beta_{6} - \beta_{5} + \beta_1 + 1) q^{56} + ( - \beta_{6} + \beta_{3} + \beta_{2}) q^{57} + ( - 2 \beta_{7} + \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{58} + (\beta_{7} - \beta_{6} - 2 \beta_{4} + 5 \beta_{3} + 2 \beta_{2} - 5 \beta_1 - 2) q^{59} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2}) q^{61} + (\beta_{6} + \beta_{3} + \beta_{2} + 4 \beta_1) q^{62} + (\beta_{7} + \beta_{5} - \beta_{3} + \beta_1) q^{63} + \beta_{5} q^{64} + (\beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{2}) q^{66} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{67} + ( - \beta_{7} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_1 + 2) q^{68} + (\beta_{7} - \beta_{6} + \beta_{4} - 3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{69} + (\beta_{7} - \beta_{6} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{71} + \beta_{3} q^{72} + ( - 2 \beta_{7} - \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{73} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - 4 \beta_{3} - 4 \beta_1) q^{74} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{76} + (3 \beta_{7} + 2 \beta_{6} - 6 \beta_{5} + 4 \beta_{3} - \beta_{2} - \beta_1 - 6) q^{77} + ( - 2 \beta_{5} - 2 \beta_1 - 2) q^{78} + (\beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} + 7 \beta_{3} + \beta_{2} + 7 \beta_1) q^{79} - q^{81} + ( - \beta_{7} + \beta_{6} + 4 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 4) q^{82} + (\beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} + 5 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{83} + ( - \beta_{6} + \beta_{5} - \beta_1 - 1) q^{84} + (\beta_{7} - \beta_{6} - 3 \beta_{4} + \beta_{3} + 3 \beta_{2} - \beta_1) q^{86} + (2 \beta_{7} - \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{87} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{88} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{3} + 2 \beta_1 + 10) q^{89} + ( - 2 \beta_{7} - 4 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} - 6 \beta_1) q^{91} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 2) q^{92} + ( - \beta_{6} - \beta_{3} - \beta_{2} - 4 \beta_1) q^{93} + ( - \beta_{7} + \beta_{6} + \beta_{4} - 3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{94} - \beta_{5} q^{96} + (\beta_{7} - 2 \beta_{6} - 5 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 5) q^{97} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{98} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} + 8 q^{11} - 16 q^{13} + 8 q^{14} - 8 q^{16} + 12 q^{17} + 8 q^{19} - 8 q^{21} - 8 q^{22} - 16 q^{23} + 8 q^{24} + 4 q^{28} + 8 q^{33} - 16 q^{34} - 8 q^{36} + 28 q^{37} - 4 q^{38} - 4 q^{42} - 8 q^{46} + 24 q^{47} - 4 q^{49} + 16 q^{51} + 16 q^{52} + 8 q^{53} + 8 q^{54} + 4 q^{56} + 4 q^{57} + 12 q^{58} - 8 q^{59} - 4 q^{62} + 4 q^{63} + 12 q^{68} + 8 q^{69} + 8 q^{71} - 28 q^{73} - 44 q^{77} - 16 q^{78} - 8 q^{81} + 24 q^{82} - 16 q^{83} - 4 q^{84} + 8 q^{86} - 12 q^{87} + 8 q^{88} + 64 q^{89} - 8 q^{91} - 16 q^{92} + 4 q^{93} - 8 q^{94} - 28 q^{97} + 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 2\nu^{6} + 18\nu^{5} + 28\nu^{4} + 89\nu^{3} + 74\nu^{2} + 104\nu - 16 ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 18\nu^{5} + 8\nu^{4} + 105\nu^{3} + 72\nu^{2} + 248\nu + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 18\nu^{5} - 28\nu^{4} + 89\nu^{3} - 74\nu^{2} + 104\nu + 16 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 18\nu^{5} - 8\nu^{4} + 105\nu^{3} - 72\nu^{2} + 248\nu - 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - 46\nu^{5} - 179\nu^{3} - 168\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 6\nu^{6} + 10\nu^{5} - 92\nu^{4} - 15\nu^{3} - 358\nu^{2} - 120\nu - 336 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 6\nu^{6} + 10\nu^{5} + 92\nu^{4} - 15\nu^{3} + 358\nu^{2} - 120\nu + 336 ) / 64 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{4} + 3\beta_{3} - \beta_{2} - 3\beta _1 - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{7} - 9\beta_{6} - 18\beta_{5} - 5\beta_{4} - 13\beta_{3} - 5\beta_{2} - 13\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -9\beta_{7} + 9\beta_{6} - 17\beta_{4} - 27\beta_{3} + 17\beta_{2} + 27\beta _1 + 74 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 81\beta_{7} + 81\beta_{6} + 178\beta_{5} + 37\beta_{4} + 149\beta_{3} + 37\beta_{2} + 149\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 89\beta_{7} - 89\beta_{6} + 201\beta_{4} + 235\beta_{3} - 201\beta_{2} - 235\beta _1 - 650 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -761\beta_{7} - 761\beta_{6} - 1810\beta_{5} - 325\beta_{4} - 1565\beta_{3} - 325\beta_{2} - 1565\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-\beta_{5}\) \(-1\) \(1\)

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
0.692297i
1.69230i
3.16053i
2.16053i
0.692297i
1.69230i
3.16053i
2.16053i
−0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000i −2.19663 1.47472i 0.707107 0.707107i 1.00000i 0
307.2 −0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000i −0.510472 + 2.59604i 0.707107 0.707107i 1.00000i 0
307.3 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 0 1.00000i −2.52773 0.781409i −0.707107 + 0.707107i 1.00000i 0
307.4 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 0 1.00000i 1.23483 2.33991i −0.707107 + 0.707107i 1.00000i 0
643.1 −0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000i −2.19663 + 1.47472i 0.707107 + 0.707107i 1.00000i 0
643.2 −0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000i −0.510472 2.59604i 0.707107 + 0.707107i 1.00000i 0
643.3 0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000i −2.52773 + 0.781409i −0.707107 0.707107i 1.00000i 0
643.4 0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000i 1.23483 + 2.33991i −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
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.m.a 8
5.b even 2 1 210.2.m.b yes 8
5.c odd 4 1 210.2.m.a 8
5.c odd 4 1 1050.2.m.b 8
7.b odd 2 1 1050.2.m.b 8
15.d odd 2 1 630.2.p.c 8
15.e even 4 1 630.2.p.b 8
20.d odd 2 1 1680.2.cz.a 8
20.e even 4 1 1680.2.cz.b 8
35.c odd 2 1 210.2.m.a 8
35.f even 4 1 210.2.m.b yes 8
35.f even 4 1 inner 1050.2.m.a 8
105.g even 2 1 630.2.p.b 8
105.k odd 4 1 630.2.p.c 8
140.c even 2 1 1680.2.cz.b 8
140.j odd 4 1 1680.2.cz.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.m.a 8 5.c odd 4 1
210.2.m.a 8 35.c odd 2 1
210.2.m.b yes 8 5.b even 2 1
210.2.m.b yes 8 35.f even 4 1
630.2.p.b 8 15.e even 4 1
630.2.p.b 8 105.g even 2 1
630.2.p.c 8 15.d odd 2 1
630.2.p.c 8 105.k odd 4 1
1050.2.m.a 8 1.a even 1 1 trivial
1050.2.m.a 8 35.f even 4 1 inner
1050.2.m.b 8 5.c odd 4 1
1050.2.m.b 8 7.b odd 2 1
1680.2.cz.a 8 20.d odd 2 1
1680.2.cz.a 8 140.j odd 4 1
1680.2.cz.b 8 20.e even 4 1
1680.2.cz.b 8 140.c even 2 1

Hecke kernels

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

\( T_{11}^{4} - 4T_{11}^{3} - 32T_{11}^{2} + 136T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{4} + 8T_{13}^{3} + 32T_{13}^{2} + 32T_{13} + 16 \) Copy content Toggle raw display
\( T_{19}^{4} - 4T_{19}^{3} - 14T_{19}^{2} + 16T_{19} + 32 \) 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
$7$ \( T^{8} + 8 T^{7} + 34 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} - 32 T^{2} + 136 T - 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 8 T^{3} + 32 T^{2} + 32 T + 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 12 T^{7} + 72 T^{6} + \cdots + 40000 \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} - 14 T^{2} + 16 T + 32)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$29$ \( T^{8} + 228 T^{6} + 16324 T^{4} + \cdots + 2560000 \) Copy content Toggle raw display
$31$ \( T^{8} + 100 T^{6} + 2884 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$37$ \( T^{8} - 28 T^{7} + 392 T^{6} + \cdots + 61504 \) Copy content Toggle raw display
$41$ \( T^{8} + 152 T^{6} + 3792 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$43$ \( T^{8} + 32 T^{5} + 16976 T^{4} + \cdots + 59474944 \) Copy content Toggle raw display
$47$ \( T^{8} - 24 T^{7} + 288 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$53$ \( T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$59$ \( (T^{4} + 4 T^{3} - 166 T^{2} - 592 T - 496)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 224 T^{6} + 15328 T^{4} + \cdots + 1364224 \) Copy content Toggle raw display
$67$ \( T^{8} + 64 T^{5} + 1040 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$71$ \( (T^{4} - 4 T^{3} - 126 T^{2} + 1072 T - 2336)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 28 T^{7} + 392 T^{6} + \cdots + 24364096 \) Copy content Toggle raw display
$79$ \( T^{8} + 360 T^{6} + 35536 T^{4} + \cdots + 9634816 \) Copy content Toggle raw display
$83$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( (T^{4} - 32 T^{3} + 272 T^{2} + 224 T - 7312)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 28 T^{7} + 392 T^{6} + \cdots + 64 \) Copy content Toggle raw display
show more
show less