Properties

Label 816.2.s.b
Level $816$
Weight $2$
Character orbit 816.s
Analytic conductor $6.516$
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] = [816,2,Mod(157,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("816.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.s (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: \(6.51579280494\)
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: 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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (i + 1) q^{2} + q^{3} + 2 i q^{4} + (i + 1) q^{6} + (3 i + 3) q^{7} + (2 i - 2) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (i + 1) q^{2} + q^{3} + 2 i q^{4} + (i + 1) q^{6} + (3 i + 3) q^{7} + (2 i - 2) q^{8} + q^{9} - 4 q^{11} + 2 i q^{12} + ( - 3 i + 3) q^{13} + 6 i q^{14} - 4 q^{16} + (4 i - 1) q^{17} + (i + 1) q^{18} + ( - 3 i - 3) q^{19} + (3 i + 3) q^{21} + ( - 4 i - 4) q^{22} + (5 i + 5) q^{23} + (2 i - 2) q^{24} + 5 q^{25} + 6 q^{26} + q^{27} + (6 i - 6) q^{28} - 2 q^{29} + ( - 5 i - 5) q^{31} + ( - 4 i - 4) q^{32} - 4 q^{33} + (3 i - 5) q^{34} + 2 i q^{36} - 8 i q^{37} - 6 i q^{38} + ( - 3 i + 3) q^{39} + (3 i + 3) q^{41} + 6 i q^{42} + ( - 3 i + 3) q^{43} - 8 i q^{44} + 10 i q^{46} - 4 q^{48} + 11 i q^{49} + (5 i + 5) q^{50} + (4 i - 1) q^{51} + (6 i + 6) q^{52} + ( - i + 1) q^{53} + (i + 1) q^{54} - 12 q^{56} + ( - 3 i - 3) q^{57} + ( - 2 i - 2) q^{58} + ( - i + 1) q^{59} + 8 i q^{61} - 10 i q^{62} + (3 i + 3) q^{63} - 8 i q^{64} + ( - 4 i - 4) q^{66} + (i - 1) q^{67} + ( - 2 i - 8) q^{68} + (5 i + 5) q^{69} + (5 i - 5) q^{71} + (2 i - 2) q^{72} + ( - 9 i + 9) q^{73} + ( - 8 i + 8) q^{74} + 5 q^{75} + ( - 6 i + 6) q^{76} + ( - 12 i - 12) q^{77} + 6 q^{78} + (3 i - 3) q^{79} + q^{81} + 6 i q^{82} + (7 i + 7) q^{83} + (6 i - 6) q^{84} + 6 q^{86} - 2 q^{87} + ( - 8 i + 8) q^{88} - 4 i q^{89} + 18 q^{91} + (10 i - 10) q^{92} + ( - 5 i - 5) q^{93} + ( - 4 i - 4) q^{96} + ( - 3 i - 3) q^{97} + (11 i - 11) q^{98} - 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{6} + 6 q^{7} - 4 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{6} + 6 q^{7} - 4 q^{8} + 2 q^{9} - 8 q^{11} + 6 q^{13} - 8 q^{16} - 2 q^{17} + 2 q^{18} - 6 q^{19} + 6 q^{21} - 8 q^{22} + 10 q^{23} - 4 q^{24} + 10 q^{25} + 12 q^{26} + 2 q^{27} - 12 q^{28} - 4 q^{29} - 10 q^{31} - 8 q^{32} - 8 q^{33} - 10 q^{34} + 6 q^{39} + 6 q^{41} + 6 q^{43} - 8 q^{48} + 10 q^{50} - 2 q^{51} + 12 q^{52} + 2 q^{53} + 2 q^{54} - 24 q^{56} - 6 q^{57} - 4 q^{58} + 2 q^{59} + 6 q^{63} - 8 q^{66} - 2 q^{67} - 16 q^{68} + 10 q^{69} - 10 q^{71} - 4 q^{72} + 18 q^{73} + 16 q^{74} + 10 q^{75} + 12 q^{76} - 24 q^{77} + 12 q^{78} - 6 q^{79} + 2 q^{81} + 14 q^{83} - 12 q^{84} + 12 q^{86} - 4 q^{87} + 16 q^{88} + 36 q^{91} - 20 q^{92} - 10 q^{93} - 8 q^{96} - 6 q^{97} - 22 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/816\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(511\) \(545\) \(613\)
\(\chi(n)\) \(i\) \(1\) \(1\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
1.00000i
1.00000i
1.00000 1.00000i 1.00000 2.00000i 0 1.00000 1.00000i 3.00000 3.00000i −2.00000 2.00000i 1.00000 0
421.1 1.00000 + 1.00000i 1.00000 2.00000i 0 1.00000 + 1.00000i 3.00000 + 3.00000i −2.00000 + 2.00000i 1.00000 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
272.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 816.2.s.b 2
16.e even 4 1 816.2.bl.a yes 2
17.c even 4 1 816.2.bl.a yes 2
272.s even 4 1 inner 816.2.s.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
816.2.s.b 2 1.a even 1 1 trivial
816.2.s.b 2 272.s even 4 1 inner
816.2.bl.a yes 2 16.e even 4 1
816.2.bl.a yes 2 17.c even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(816, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} - 6T_{7} + 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 17 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$23$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$37$ \( T^{2} + 64 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$43$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$59$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$61$ \( T^{2} + 64 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$71$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$73$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$79$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$83$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$89$ \( T^{2} + 16 \) Copy content Toggle raw display
$97$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
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