Properties

Label 850.2.d.a
Level $850$
Weight $2$
Character orbit 850.d
Analytic conductor $6.787$
Analytic rank $0$
Dimension $2$
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] = [850,2,Mod(849,850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(850, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("850.849");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 850.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: \(6.78728417181\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 170)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/850\mathbb{Z}\right)^\times\).

\(n\) \(477\) \(751\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
849.1
1.00000i
1.00000i
1.00000i −3.00000 −1.00000 0 3.00000i −4.00000 1.00000i 6.00000 0
849.2 1.00000i −3.00000 −1.00000 0 3.00000i −4.00000 1.00000i 6.00000 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
85.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 850.2.d.a 2
5.b even 2 1 850.2.d.h 2
5.c odd 4 1 170.2.b.b 2
5.c odd 4 1 850.2.b.a 2
15.e even 4 1 1530.2.c.b 2
17.b even 2 1 850.2.d.h 2
20.e even 4 1 1360.2.c.a 2
85.c even 2 1 inner 850.2.d.a 2
85.f odd 4 1 2890.2.a.l 1
85.g odd 4 1 170.2.b.b 2
85.g odd 4 1 850.2.b.a 2
85.i odd 4 1 2890.2.a.a 1
255.o even 4 1 1530.2.c.b 2
340.r even 4 1 1360.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.b.b 2 5.c odd 4 1
170.2.b.b 2 85.g odd 4 1
850.2.b.a 2 5.c odd 4 1
850.2.b.a 2 85.g odd 4 1
850.2.d.a 2 1.a even 1 1 trivial
850.2.d.a 2 85.c even 2 1 inner
850.2.d.h 2 5.b even 2 1
850.2.d.h 2 17.b even 2 1
1360.2.c.a 2 20.e even 4 1
1360.2.c.a 2 340.r even 4 1
1530.2.c.b 2 15.e even 4 1
1530.2.c.b 2 255.o even 4 1
2890.2.a.a 1 85.i odd 4 1
2890.2.a.l 1 85.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(850, [\chi])\):

\( T_{3} + 3 \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 17 \) Copy content Toggle raw display
$19$ \( (T + 7)^{2} \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 9 \) Copy content Toggle raw display
$31$ \( T^{2} + 49 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 64 \) Copy content Toggle raw display
$43$ \( T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{2} + 81 \) Copy content Toggle raw display
$53$ \( T^{2} + 121 \) Copy content Toggle raw display
$59$ \( (T - 5)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 1 \) Copy content Toggle raw display
$67$ \( T^{2} + 100 \) Copy content Toggle raw display
$71$ \( T^{2} + 1 \) Copy content Toggle raw display
$73$ \( (T - 9)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T - 1)^{2} \) Copy content Toggle raw display
$97$ \( (T - 1)^{2} \) Copy content Toggle raw display
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