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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Newspace parameters

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

Self dual: no
Analytic conductor: \(59.8279367458\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
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

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 i q^{2} -3 q^{3} -4 q^{4} + 6 i q^{5} + 6 i q^{6} + 16 i q^{7} + 8 i q^{8} + 9 q^{9} +O(q^{10})\) \( q -2 i q^{2} -3 q^{3} -4 q^{4} + 6 i q^{5} + 6 i q^{6} + 16 i q^{7} + 8 i q^{8} + 9 q^{9} + 12 q^{10} -12 i q^{11} + 12 q^{12} + 32 q^{14} -18 i q^{15} + 16 q^{16} + 126 q^{17} -18 i q^{18} + 20 i q^{19} -24 i q^{20} -48 i q^{21} -24 q^{22} -168 q^{23} -24 i q^{24} + 89 q^{25} -27 q^{27} -64 i q^{28} + 30 q^{29} -36 q^{30} -88 i q^{31} -32 i q^{32} + 36 i q^{33} -252 i q^{34} -96 q^{35} -36 q^{36} -254 i q^{37} + 40 q^{38} -48 q^{40} + 42 i q^{41} -96 q^{42} + 52 q^{43} + 48 i q^{44} + 54 i q^{45} + 336 i q^{46} + 96 i q^{47} -48 q^{48} + 87 q^{49} -178 i q^{50} -378 q^{51} + 198 q^{53} + 54 i q^{54} + 72 q^{55} -128 q^{56} -60 i q^{57} -60 i q^{58} + 660 i q^{59} + 72 i q^{60} -538 q^{61} -176 q^{62} + 144 i q^{63} -64 q^{64} + 72 q^{66} + 884 i q^{67} -504 q^{68} + 504 q^{69} + 192 i q^{70} + 792 i q^{71} + 72 i q^{72} -218 i q^{73} -508 q^{74} -267 q^{75} -80 i q^{76} + 192 q^{77} -520 q^{79} + 96 i q^{80} + 81 q^{81} + 84 q^{82} -492 i q^{83} + 192 i q^{84} + 756 i q^{85} -104 i q^{86} -90 q^{87} + 96 q^{88} -810 i q^{89} + 108 q^{90} + 672 q^{92} + 264 i q^{93} + 192 q^{94} -120 q^{95} + 96 i q^{96} + 1154 i q^{97} -174 i q^{98} -108 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} - 8q^{4} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} - 8q^{4} + 18q^{9} + 24q^{10} + 24q^{12} + 64q^{14} + 32q^{16} + 252q^{17} - 48q^{22} - 336q^{23} + 178q^{25} - 54q^{27} + 60q^{29} - 72q^{30} - 192q^{35} - 72q^{36} + 80q^{38} - 96q^{40} - 192q^{42} + 104q^{43} - 96q^{48} + 174q^{49} - 756q^{51} + 396q^{53} + 144q^{55} - 256q^{56} - 1076q^{61} - 352q^{62} - 128q^{64} + 144q^{66} - 1008q^{68} + 1008q^{69} - 1016q^{74} - 534q^{75} + 384q^{77} - 1040q^{79} + 162q^{81} + 168q^{82} - 180q^{87} + 192q^{88} + 216q^{90} + 1344q^{92} + 384q^{94} - 240q^{95} + O(q^{100}) \)

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.

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 \(S_{4}^{\mathrm{new}}(1014, [\chi])\):

\( T_{5}^{2} + 36 \)
\( T_{7}^{2} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T^{2} \)
$3$ \( ( 1 + 3 T )^{2} \)
$5$ \( 1 - 214 T^{2} + 15625 T^{4} \)
$7$ \( 1 - 430 T^{2} + 117649 T^{4} \)
$11$ \( 1 - 2518 T^{2} + 1771561 T^{4} \)
$13$ 1
$17$ \( ( 1 - 126 T + 4913 T^{2} )^{2} \)
$19$ \( 1 - 13318 T^{2} + 47045881 T^{4} \)
$23$ \( ( 1 + 168 T + 12167 T^{2} )^{2} \)
$29$ \( ( 1 - 30 T + 24389 T^{2} )^{2} \)
$31$ \( 1 - 51838 T^{2} + 887503681 T^{4} \)
$37$ \( 1 - 36790 T^{2} + 2565726409 T^{4} \)
$41$ \( 1 - 136078 T^{2} + 4750104241 T^{4} \)
$43$ \( ( 1 - 52 T + 79507 T^{2} )^{2} \)
$47$ \( 1 - 198430 T^{2} + 10779215329 T^{4} \)
$53$ \( ( 1 - 198 T + 148877 T^{2} )^{2} \)
$59$ \( 1 + 24842 T^{2} + 42180533641 T^{4} \)
$61$ \( ( 1 + 538 T + 226981 T^{2} )^{2} \)
$67$ \( 1 + 179930 T^{2} + 90458382169 T^{4} \)
$71$ \( 1 - 88558 T^{2} + 128100283921 T^{4} \)
$73$ \( 1 - 730510 T^{2} + 151334226289 T^{4} \)
$79$ \( ( 1 + 520 T + 493039 T^{2} )^{2} \)
$83$ \( 1 - 901510 T^{2} + 326940373369 T^{4} \)
$89$ \( 1 - 753838 T^{2} + 496981290961 T^{4} \)
$97$ \( 1 - 493630 T^{2} + 832972004929 T^{4} \)
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