Properties

Label 1014.4.b.g
Level $1014$
Weight $4$
Character orbit 1014.b
Analytic conductor $59.828$
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] = [1014,4,Mod(337,1014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1014.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1014.b (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: \(59.8279367458\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 78)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 3 q^{3} - 4 q^{4} + 2 \beta q^{5} + 3 \beta q^{6} - 2 \beta q^{7} - 4 \beta q^{8} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 3 q^{3} - 4 q^{4} + 2 \beta q^{5} + 3 \beta q^{6} - 2 \beta q^{7} - 4 \beta q^{8} + 9 q^{9} - 8 q^{10} - \beta q^{11} - 12 q^{12} + 8 q^{14} + 6 \beta q^{15} + 16 q^{16} + 6 q^{17} + 9 \beta q^{18} - 18 \beta q^{19} - 8 \beta q^{20} - 6 \beta q^{21} + 4 q^{22} + 20 q^{23} - 12 \beta q^{24} + 109 q^{25} + 27 q^{27} + 8 \beta q^{28} - 14 q^{29} - 24 q^{30} - 76 \beta q^{31} + 16 \beta q^{32} - 3 \beta q^{33} + 6 \beta q^{34} + 16 q^{35} - 36 q^{36} + 129 \beta q^{37} + 72 q^{38} + 32 q^{40} + 42 \beta q^{41} + 24 q^{42} + 188 q^{43} + 4 \beta q^{44} + 18 \beta q^{45} + 20 \beta q^{46} - 127 \beta q^{47} + 48 q^{48} + 327 q^{49} + 109 \beta q^{50} + 18 q^{51} + 366 q^{53} + 27 \beta q^{54} + 8 q^{55} - 32 q^{56} - 54 \beta q^{57} - 14 \beta q^{58} - 275 \beta q^{59} - 24 \beta q^{60} - 14 q^{61} + 304 q^{62} - 18 \beta q^{63} - 64 q^{64} + 12 q^{66} + 224 \beta q^{67} - 24 q^{68} + 60 q^{69} + 16 \beta q^{70} + 463 \beta q^{71} - 36 \beta q^{72} - 127 \beta q^{73} - 516 q^{74} + 327 q^{75} + 72 \beta q^{76} - 8 q^{77} + 1328 q^{79} + 32 \beta q^{80} + 81 q^{81} - 168 q^{82} + 93 \beta q^{83} + 24 \beta q^{84} + 12 \beta q^{85} + 188 \beta q^{86} - 42 q^{87} - 16 q^{88} + 168 \beta q^{89} - 72 q^{90} - 80 q^{92} - 228 \beta q^{93} + 508 q^{94} + 144 q^{95} + 48 \beta q^{96} + 307 \beta q^{97} + 327 \beta q^{98} - 9 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 8 q^{4} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 8 q^{4} + 18 q^{9} - 16 q^{10} - 24 q^{12} + 16 q^{14} + 32 q^{16} + 12 q^{17} + 8 q^{22} + 40 q^{23} + 218 q^{25} + 54 q^{27} - 28 q^{29} - 48 q^{30} + 32 q^{35} - 72 q^{36} + 144 q^{38} + 64 q^{40} + 48 q^{42} + 376 q^{43} + 96 q^{48} + 654 q^{49} + 36 q^{51} + 732 q^{53} + 16 q^{55} - 64 q^{56} - 28 q^{61} + 608 q^{62} - 128 q^{64} + 24 q^{66} - 48 q^{68} + 120 q^{69} - 1032 q^{74} + 654 q^{75} - 16 q^{77} + 2656 q^{79} + 162 q^{81} - 336 q^{82} - 84 q^{87} - 32 q^{88} - 144 q^{90} - 160 q^{92} + 1016 q^{94} + 288 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.00000i
1.00000i
2.00000i 3.00000 −4.00000 4.00000i 6.00000i 4.00000i 8.00000i 9.00000 −8.00000
337.2 2.00000i 3.00000 −4.00000 4.00000i 6.00000i 4.00000i 8.00000i 9.00000 −8.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.4.b.g 2
13.b even 2 1 inner 1014.4.b.g 2
13.d odd 4 1 78.4.a.f 1
13.d odd 4 1 1014.4.a.e 1
39.f even 4 1 234.4.a.c 1
52.f even 4 1 624.4.a.c 1
65.g odd 4 1 1950.4.a.a 1
104.j odd 4 1 2496.4.a.c 1
104.m even 4 1 2496.4.a.l 1
156.l odd 4 1 1872.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.f 1 13.d odd 4 1
234.4.a.c 1 39.f even 4 1
624.4.a.c 1 52.f even 4 1
1014.4.a.e 1 13.d odd 4 1
1014.4.b.g 2 1.a even 1 1 trivial
1014.4.b.g 2 13.b even 2 1 inner
1872.4.a.f 1 156.l odd 4 1
1950.4.a.a 1 65.g odd 4 1
2496.4.a.c 1 104.j odd 4 1
2496.4.a.l 1 104.m even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1014, [\chi])\):

\( T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 16 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1296 \) Copy content Toggle raw display
$23$ \( (T - 20)^{2} \) Copy content Toggle raw display
$29$ \( (T + 14)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 23104 \) Copy content Toggle raw display
$37$ \( T^{2} + 66564 \) Copy content Toggle raw display
$41$ \( T^{2} + 7056 \) Copy content Toggle raw display
$43$ \( (T - 188)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 64516 \) Copy content Toggle raw display
$53$ \( (T - 366)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 302500 \) Copy content Toggle raw display
$61$ \( (T + 14)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 200704 \) Copy content Toggle raw display
$71$ \( T^{2} + 857476 \) Copy content Toggle raw display
$73$ \( T^{2} + 64516 \) Copy content Toggle raw display
$79$ \( (T - 1328)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 34596 \) Copy content Toggle raw display
$89$ \( T^{2} + 112896 \) Copy content Toggle raw display
$97$ \( T^{2} + 376996 \) Copy content Toggle raw display
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