Properties

Label 1050.6.g.o
Level $1050$
Weight $6$
Character orbit 1050.g
Analytic conductor $168.403$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,6,Mod(799,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.799");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1050.g (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: \(168.403010804\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 i q^{2} + 9 i q^{3} - 16 q^{4} + 36 q^{6} + 49 i q^{7} + 64 i q^{8} - 81 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 4 i q^{2} + 9 i q^{3} - 16 q^{4} + 36 q^{6} + 49 i q^{7} + 64 i q^{8} - 81 q^{9} + 216 q^{11} - 144 i q^{12} - 998 i q^{13} + 196 q^{14} + 256 q^{16} + 1302 i q^{17} + 324 i q^{18} - 884 q^{19} - 441 q^{21} - 864 i q^{22} + 2268 i q^{23} - 576 q^{24} - 3992 q^{26} - 729 i q^{27} - 784 i q^{28} + 1482 q^{29} + 8360 q^{31} - 1024 i q^{32} + 1944 i q^{33} + 5208 q^{34} + 1296 q^{36} - 4714 i q^{37} + 3536 i q^{38} + 8982 q^{39} - 9786 q^{41} + 1764 i q^{42} - 19436 i q^{43} - 3456 q^{44} + 9072 q^{46} + 22200 i q^{47} + 2304 i q^{48} - 2401 q^{49} - 11718 q^{51} + 15968 i q^{52} - 26790 i q^{53} - 2916 q^{54} - 3136 q^{56} - 7956 i q^{57} - 5928 i q^{58} - 28092 q^{59} - 38866 q^{61} - 33440 i q^{62} - 3969 i q^{63} - 4096 q^{64} + 7776 q^{66} + 23948 i q^{67} - 20832 i q^{68} - 20412 q^{69} - 20628 q^{71} - 5184 i q^{72} - 290 i q^{73} - 18856 q^{74} + 14144 q^{76} + 10584 i q^{77} - 35928 i q^{78} + 99544 q^{79} + 6561 q^{81} + 39144 i q^{82} - 19308 i q^{83} + 7056 q^{84} - 77744 q^{86} + 13338 i q^{87} + 13824 i q^{88} - 36390 q^{89} + 48902 q^{91} - 36288 i q^{92} + 75240 i q^{93} + 88800 q^{94} + 9216 q^{96} - 79078 i q^{97} + 9604 i q^{98} - 17496 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} + 72 q^{6} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} + 72 q^{6} - 162 q^{9} + 432 q^{11} + 392 q^{14} + 512 q^{16} - 1768 q^{19} - 882 q^{21} - 1152 q^{24} - 7984 q^{26} + 2964 q^{29} + 16720 q^{31} + 10416 q^{34} + 2592 q^{36} + 17964 q^{39} - 19572 q^{41} - 6912 q^{44} + 18144 q^{46} - 4802 q^{49} - 23436 q^{51} - 5832 q^{54} - 6272 q^{56} - 56184 q^{59} - 77732 q^{61} - 8192 q^{64} + 15552 q^{66} - 40824 q^{69} - 41256 q^{71} - 37712 q^{74} + 28288 q^{76} + 199088 q^{79} + 13122 q^{81} + 14112 q^{84} - 155488 q^{86} - 72780 q^{89} + 97804 q^{91} + 177600 q^{94} + 18432 q^{96} - 34992 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
1.00000i
1.00000i
4.00000i 9.00000i −16.0000 0 36.0000 49.0000i 64.0000i −81.0000 0
799.2 4.00000i 9.00000i −16.0000 0 36.0000 49.0000i 64.0000i −81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.6.g.o 2
5.b even 2 1 inner 1050.6.g.o 2
5.c odd 4 1 42.6.a.a 1
5.c odd 4 1 1050.6.a.n 1
15.e even 4 1 126.6.a.k 1
20.e even 4 1 336.6.a.j 1
35.f even 4 1 294.6.a.h 1
35.k even 12 2 294.6.e.h 2
35.l odd 12 2 294.6.e.r 2
60.l odd 4 1 1008.6.a.x 1
105.k odd 4 1 882.6.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.a 1 5.c odd 4 1
126.6.a.k 1 15.e even 4 1
294.6.a.h 1 35.f even 4 1
294.6.e.h 2 35.k even 12 2
294.6.e.r 2 35.l odd 12 2
336.6.a.j 1 20.e even 4 1
882.6.a.o 1 105.k odd 4 1
1008.6.a.x 1 60.l odd 4 1
1050.6.a.n 1 5.c odd 4 1
1050.6.g.o 2 1.a even 1 1 trivial
1050.6.g.o 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} - 216 \) acting on \(S_{6}^{\mathrm{new}}(1050, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T - 216)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 996004 \) Copy content Toggle raw display
$17$ \( T^{2} + 1695204 \) Copy content Toggle raw display
$19$ \( (T + 884)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 5143824 \) Copy content Toggle raw display
$29$ \( (T - 1482)^{2} \) Copy content Toggle raw display
$31$ \( (T - 8360)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 22221796 \) Copy content Toggle raw display
$41$ \( (T + 9786)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 377758096 \) Copy content Toggle raw display
$47$ \( T^{2} + 492840000 \) Copy content Toggle raw display
$53$ \( T^{2} + 717704100 \) Copy content Toggle raw display
$59$ \( (T + 28092)^{2} \) Copy content Toggle raw display
$61$ \( (T + 38866)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 573506704 \) Copy content Toggle raw display
$71$ \( (T + 20628)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 84100 \) Copy content Toggle raw display
$79$ \( (T - 99544)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 372798864 \) Copy content Toggle raw display
$89$ \( (T + 36390)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 6253330084 \) Copy content Toggle raw display
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