Properties

Label 1014.4.b.d
Level $1014$
Weight $4$
Character orbit 1014.b
Analytic conductor $59.828$
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] = [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 6)
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.

\(f(q)\) \(=\) \( q - \beta q^{2} - 3 q^{3} - 4 q^{4} + 3 \beta q^{5} + 3 \beta q^{6} + 8 \beta q^{7} + 4 \beta q^{8} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - 3 q^{3} - 4 q^{4} + 3 \beta q^{5} + 3 \beta q^{6} + 8 \beta q^{7} + 4 \beta q^{8} + 9 q^{9} + 12 q^{10} - 6 \beta q^{11} + 12 q^{12} + 32 q^{14} - 9 \beta q^{15} + 16 q^{16} + 126 q^{17} - 9 \beta q^{18} + 10 \beta q^{19} - 12 \beta q^{20} - 24 \beta q^{21} - 24 q^{22} - 168 q^{23} - 12 \beta q^{24} + 89 q^{25} - 27 q^{27} - 32 \beta q^{28} + 30 q^{29} - 36 q^{30} - 44 \beta q^{31} - 16 \beta q^{32} + 18 \beta q^{33} - 126 \beta q^{34} - 96 q^{35} - 36 q^{36} - 127 \beta q^{37} + 40 q^{38} - 48 q^{40} + 21 \beta q^{41} - 96 q^{42} + 52 q^{43} + 24 \beta q^{44} + 27 \beta q^{45} + 168 \beta q^{46} + 48 \beta q^{47} - 48 q^{48} + 87 q^{49} - 89 \beta q^{50} - 378 q^{51} + 198 q^{53} + 27 \beta q^{54} + 72 q^{55} - 128 q^{56} - 30 \beta q^{57} - 30 \beta q^{58} + 330 \beta q^{59} + 36 \beta q^{60} - 538 q^{61} - 176 q^{62} + 72 \beta q^{63} - 64 q^{64} + 72 q^{66} + 442 \beta q^{67} - 504 q^{68} + 504 q^{69} + 96 \beta q^{70} + 396 \beta q^{71} + 36 \beta q^{72} - 109 \beta q^{73} - 508 q^{74} - 267 q^{75} - 40 \beta q^{76} + 192 q^{77} - 520 q^{79} + 48 \beta q^{80} + 81 q^{81} + 84 q^{82} - 246 \beta q^{83} + 96 \beta q^{84} + 378 \beta q^{85} - 52 \beta q^{86} - 90 q^{87} + 96 q^{88} - 405 \beta q^{89} + 108 q^{90} + 672 q^{92} + 132 \beta q^{93} + 192 q^{94} - 120 q^{95} + 48 \beta q^{96} + 577 \beta q^{97} - 87 \beta q^{98} - 54 \beta q^{99} +O(q^{100}) \) 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 \( 2 q - 6 q^{3} - 8 q^{4} + 18 q^{9} + 24 q^{10} + 24 q^{12} + 64 q^{14} + 32 q^{16} + 252 q^{17} - 48 q^{22} - 336 q^{23} + 178 q^{25} - 54 q^{27} + 60 q^{29} - 72 q^{30} - 192 q^{35} - 72 q^{36} + 80 q^{38} - 96 q^{40} - 192 q^{42} + 104 q^{43} - 96 q^{48} + 174 q^{49} - 756 q^{51} + 396 q^{53} + 144 q^{55} - 256 q^{56} - 1076 q^{61} - 352 q^{62} - 128 q^{64} + 144 q^{66} - 1008 q^{68} + 1008 q^{69} - 1016 q^{74} - 534 q^{75} + 384 q^{77} - 1040 q^{79} + 162 q^{81} + 168 q^{82} - 180 q^{87} + 192 q^{88} + 216 q^{90} + 1344 q^{92} + 384 q^{94} - 240 q^{95}+O(q^{100}) \) 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 6.00000i 6.00000i 16.0000i 8.00000i 9.00000 12.0000
337.2 2.00000i −3.00000 −4.00000 6.00000i 6.00000i 16.0000i 8.00000i 9.00000 12.0000
\(n\): 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.d 2
13.b even 2 1 inner 1014.4.b.d 2
13.d odd 4 1 6.4.a.a 1
13.d odd 4 1 1014.4.a.g 1
39.f even 4 1 18.4.a.a 1
52.f even 4 1 48.4.a.c 1
65.f even 4 1 150.4.c.d 2
65.g odd 4 1 150.4.a.i 1
65.k even 4 1 150.4.c.d 2
91.i even 4 1 294.4.a.e 1
91.z odd 12 2 294.4.e.h 2
91.bb even 12 2 294.4.e.g 2
104.j odd 4 1 192.4.a.i 1
104.m even 4 1 192.4.a.c 1
117.y odd 12 2 162.4.c.f 2
117.z even 12 2 162.4.c.c 2
143.g even 4 1 726.4.a.f 1
156.l odd 4 1 144.4.a.c 1
195.j odd 4 1 450.4.c.e 2
195.n even 4 1 450.4.a.h 1
195.u odd 4 1 450.4.c.e 2
208.l even 4 1 768.4.d.c 2
208.m odd 4 1 768.4.d.n 2
208.r odd 4 1 768.4.d.n 2
208.s even 4 1 768.4.d.c 2
221.g odd 4 1 1734.4.a.d 1
247.i even 4 1 2166.4.a.i 1
260.l odd 4 1 1200.4.f.j 2
260.s odd 4 1 1200.4.f.j 2
260.u even 4 1 1200.4.a.b 1
273.o odd 4 1 882.4.a.n 1
273.cb odd 12 2 882.4.g.f 2
273.cd even 12 2 882.4.g.i 2
312.w odd 4 1 576.4.a.r 1
312.y even 4 1 576.4.a.q 1
364.p odd 4 1 2352.4.a.e 1
429.l odd 4 1 2178.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 13.d odd 4 1
18.4.a.a 1 39.f even 4 1
48.4.a.c 1 52.f even 4 1
144.4.a.c 1 156.l odd 4 1
150.4.a.i 1 65.g odd 4 1
150.4.c.d 2 65.f even 4 1
150.4.c.d 2 65.k even 4 1
162.4.c.c 2 117.z even 12 2
162.4.c.f 2 117.y odd 12 2
192.4.a.c 1 104.m even 4 1
192.4.a.i 1 104.j odd 4 1
294.4.a.e 1 91.i even 4 1
294.4.e.g 2 91.bb even 12 2
294.4.e.h 2 91.z odd 12 2
450.4.a.h 1 195.n even 4 1
450.4.c.e 2 195.j odd 4 1
450.4.c.e 2 195.u odd 4 1
576.4.a.q 1 312.y even 4 1
576.4.a.r 1 312.w odd 4 1
726.4.a.f 1 143.g even 4 1
768.4.d.c 2 208.l even 4 1
768.4.d.c 2 208.s even 4 1
768.4.d.n 2 208.m odd 4 1
768.4.d.n 2 208.r odd 4 1
882.4.a.n 1 273.o odd 4 1
882.4.g.f 2 273.cb odd 12 2
882.4.g.i 2 273.cd even 12 2
1014.4.a.g 1 13.d odd 4 1
1014.4.b.d 2 1.a even 1 1 trivial
1014.4.b.d 2 13.b even 2 1 inner
1200.4.a.b 1 260.u even 4 1
1200.4.f.j 2 260.l odd 4 1
1200.4.f.j 2 260.s odd 4 1
1734.4.a.d 1 221.g odd 4 1
2166.4.a.i 1 247.i even 4 1
2178.4.a.e 1 429.l odd 4 1
2352.4.a.e 1 364.p odd 4 1

Hecke kernels

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

\( T_{5}^{2} + 36 \) Copy content Toggle raw display
\( T_{7}^{2} + 256 \) 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
$5$ \( T^{2} + 36 \) Copy content Toggle raw display
$7$ \( T^{2} + 256 \) Copy content Toggle raw display
$11$ \( T^{2} + 144 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 126)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 400 \) Copy content Toggle raw display
$23$ \( (T + 168)^{2} \) Copy content Toggle raw display
$29$ \( (T - 30)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 7744 \) Copy content Toggle raw display
$37$ \( T^{2} + 64516 \) Copy content Toggle raw display
$41$ \( T^{2} + 1764 \) Copy content Toggle raw display
$43$ \( (T - 52)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 9216 \) Copy content Toggle raw display
$53$ \( (T - 198)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 435600 \) Copy content Toggle raw display
$61$ \( (T + 538)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 781456 \) Copy content Toggle raw display
$71$ \( T^{2} + 627264 \) Copy content Toggle raw display
$73$ \( T^{2} + 47524 \) Copy content Toggle raw display
$79$ \( (T + 520)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 242064 \) Copy content Toggle raw display
$89$ \( T^{2} + 656100 \) Copy content Toggle raw display
$97$ \( T^{2} + 1331716 \) Copy content Toggle raw display
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