Properties

Label 1014.4.b.e
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.

\(f(q)\) \(=\) \( q - \beta q^{2} - 3 q^{3} - 4 q^{4} + 10 \beta q^{5} + 3 \beta q^{6} - 16 \beta q^{7} + 4 \beta q^{8} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - 3 q^{3} - 4 q^{4} + 10 \beta q^{5} + 3 \beta q^{6} - 16 \beta q^{7} + 4 \beta q^{8} + 9 q^{9} + 40 q^{10} + 25 \beta q^{11} + 12 q^{12} - 64 q^{14} - 30 \beta q^{15} + 16 q^{16} + 30 q^{17} - 9 \beta q^{18} + 60 \beta q^{19} - 40 \beta q^{20} + 48 \beta q^{21} + 100 q^{22} + 20 q^{23} - 12 \beta q^{24} - 275 q^{25} - 27 q^{27} + 64 \beta q^{28} + 82 q^{29} - 120 q^{30} + 22 \beta q^{31} - 16 \beta q^{32} - 75 \beta q^{33} - 30 \beta q^{34} + 640 q^{35} - 36 q^{36} - 153 \beta q^{37} + 240 q^{38} - 160 q^{40} - 54 \beta q^{41} + 192 q^{42} + 356 q^{43} - 100 \beta q^{44} + 90 \beta q^{45} - 20 \beta q^{46} - 89 \beta q^{47} - 48 q^{48} - 681 q^{49} + 275 \beta q^{50} - 90 q^{51} + 198 q^{53} + 27 \beta q^{54} - 1000 q^{55} + 256 q^{56} - 180 \beta q^{57} - 82 \beta q^{58} + 47 \beta q^{59} + 120 \beta q^{60} - 62 q^{61} + 88 q^{62} - 144 \beta q^{63} - 64 q^{64} - 300 q^{66} + 70 \beta q^{67} - 120 q^{68} - 60 q^{69} - 640 \beta q^{70} + 389 \beta q^{71} + 36 \beta q^{72} + 31 \beta q^{73} - 612 q^{74} + 825 q^{75} - 240 \beta q^{76} + 1600 q^{77} - 1096 q^{79} + 160 \beta q^{80} + 81 q^{81} - 216 q^{82} + 231 \beta q^{83} - 192 \beta q^{84} + 300 \beta q^{85} - 356 \beta q^{86} - 246 q^{87} - 400 q^{88} + 612 \beta q^{89} + 360 q^{90} - 80 q^{92} - 66 \beta q^{93} - 356 q^{94} - 2400 q^{95} + 48 \beta q^{96} - 307 \beta q^{97} + 681 \beta q^{98} + 225 \beta q^{99} +O(q^{100}) \) 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 \( 2 q - 6 q^{3} - 8 q^{4} + 18 q^{9} + 80 q^{10} + 24 q^{12} - 128 q^{14} + 32 q^{16} + 60 q^{17} + 200 q^{22} + 40 q^{23} - 550 q^{25} - 54 q^{27} + 164 q^{29} - 240 q^{30} + 1280 q^{35} - 72 q^{36} + 480 q^{38} - 320 q^{40} + 384 q^{42} + 712 q^{43} - 96 q^{48} - 1362 q^{49} - 180 q^{51} + 396 q^{53} - 2000 q^{55} + 512 q^{56} - 124 q^{61} + 176 q^{62} - 128 q^{64} - 600 q^{66} - 240 q^{68} - 120 q^{69} - 1224 q^{74} + 1650 q^{75} + 3200 q^{77} - 2192 q^{79} + 162 q^{81} - 432 q^{82} - 492 q^{87} - 800 q^{88} + 720 q^{90} - 160 q^{92} - 712 q^{94} - 4800 q^{95}+O(q^{100}) \) 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 20.0000i 6.00000i 32.0000i 8.00000i 9.00000 40.0000
337.2 2.00000i −3.00000 −4.00000 20.0000i 6.00000i 32.0000i 8.00000i 9.00000 40.0000
\(n\): 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.e 2
13.b even 2 1 inner 1014.4.b.e 2
13.d odd 4 1 78.4.a.d 1
13.d odd 4 1 1014.4.a.d 1
39.f even 4 1 234.4.a.f 1
52.f even 4 1 624.4.a.e 1
65.g odd 4 1 1950.4.a.h 1
104.j odd 4 1 2496.4.a.r 1
104.m even 4 1 2496.4.a.i 1
156.l odd 4 1 1872.4.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.d 1 13.d odd 4 1
234.4.a.f 1 39.f even 4 1
624.4.a.e 1 52.f even 4 1
1014.4.a.d 1 13.d odd 4 1
1014.4.b.e 2 1.a even 1 1 trivial
1014.4.b.e 2 13.b even 2 1 inner
1872.4.a.r 1 156.l odd 4 1
1950.4.a.h 1 65.g odd 4 1
2496.4.a.i 1 104.m even 4 1
2496.4.a.r 1 104.j odd 4 1

Hecke kernels

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

\( T_{5}^{2} + 400 \) Copy content Toggle raw display
\( T_{7}^{2} + 1024 \) 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
$5$ \( T^{2} + 400 \) Copy content Toggle raw display
$7$ \( T^{2} + 1024 \) Copy content Toggle raw display
$11$ \( T^{2} + 2500 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 30)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 14400 \) Copy content Toggle raw display
$23$ \( (T - 20)^{2} \) Copy content Toggle raw display
$29$ \( (T - 82)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1936 \) Copy content Toggle raw display
$37$ \( T^{2} + 93636 \) Copy content Toggle raw display
$41$ \( T^{2} + 11664 \) Copy content Toggle raw display
$43$ \( (T - 356)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 31684 \) Copy content Toggle raw display
$53$ \( (T - 198)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 8836 \) Copy content Toggle raw display
$61$ \( (T + 62)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 19600 \) Copy content Toggle raw display
$71$ \( T^{2} + 605284 \) Copy content Toggle raw display
$73$ \( T^{2} + 3844 \) Copy content Toggle raw display
$79$ \( (T + 1096)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 213444 \) Copy content Toggle raw display
$89$ \( T^{2} + 1498176 \) Copy content Toggle raw display
$97$ \( T^{2} + 376996 \) Copy content Toggle raw display
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