Properties

Label 1008.2.r.i
Level 1008
Weight 2
Character orbit 1008.r
Analytic conductor 8.049
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.r (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{18} + 2 \zeta_{18}^{4} ) q^{3} + ( 1 - \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{5} + \zeta_{18}^{3} q^{7} -3 \zeta_{18}^{2} q^{9} +O(q^{10})\) \( q + ( -\zeta_{18} + 2 \zeta_{18}^{4} ) q^{3} + ( 1 - \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{5} + \zeta_{18}^{3} q^{7} -3 \zeta_{18}^{2} q^{9} + ( \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{11} + ( 1 - 2 \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{13} + ( -2 + \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} + \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{15} + ( 2 - \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{17} + ( 3 - \zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{19} + ( -2 \zeta_{18} + \zeta_{18}^{4} ) q^{21} + ( 2 - 3 \zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 5 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{23} + ( -2 \zeta_{18} + 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{25} + ( 6 - 3 \zeta_{18}^{3} ) q^{27} + ( -4 \zeta_{18} + 3 \zeta_{18}^{2} + \zeta_{18}^{3} + 3 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{29} + ( -3 + \zeta_{18} - 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{31} + ( -3 - 3 \zeta_{18}^{2} + 3 \zeta_{18}^{5} ) q^{33} + ( 1 - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{35} + ( -5 + 2 \zeta_{18} + 2 \zeta_{18}^{2} - 3 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{37} + ( -4 + \zeta_{18} + 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{39} + ( -2 + 3 \zeta_{18} - 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} ) q^{41} + ( -3 \zeta_{18} + \zeta_{18}^{2} - 5 \zeta_{18}^{3} + \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{43} + ( -3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{45} + ( \zeta_{18} + 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{47} + ( -1 + \zeta_{18}^{3} ) q^{49} + ( -2 \zeta_{18} + 3 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 4 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{51} + ( -2 - \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{53} + ( \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{55} + ( 4 - 3 \zeta_{18} - 5 \zeta_{18}^{2} + \zeta_{18}^{3} + 6 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{57} + ( 1 - 7 \zeta_{18} - \zeta_{18}^{3} + 7 \zeta_{18}^{4} + 7 \zeta_{18}^{5} ) q^{59} + ( -4 \zeta_{18} + 3 \zeta_{18}^{2} + 4 \zeta_{18}^{3} + 3 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{61} -3 \zeta_{18}^{5} q^{63} + ( -3 \zeta_{18} + 5 \zeta_{18}^{2} - 5 \zeta_{18}^{3} + 5 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{65} + ( -4 + 6 \zeta_{18} - \zeta_{18}^{2} + 4 \zeta_{18}^{3} - 5 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{67} + ( -1 + 2 \zeta_{18} - 7 \zeta_{18}^{2} - 7 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{69} + ( -7 - 3 \zeta_{18} - 3 \zeta_{18}^{2} - 4 \zeta_{18}^{4} + 7 \zeta_{18}^{5} ) q^{71} + ( 1 + 7 \zeta_{18} + 7 \zeta_{18}^{2} - 5 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{73} + ( -4 - 4 \zeta_{18} - 4 \zeta_{18}^{2} + 5 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{75} + ( -\zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{77} + ( -7 \zeta_{18} + 2 \zeta_{18}^{2} - 5 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 7 \zeta_{18}^{5} ) q^{79} + 9 \zeta_{18}^{4} q^{81} + ( 9 \zeta_{18} - 4 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 4 \zeta_{18}^{4} + 9 \zeta_{18}^{5} ) q^{83} + ( 5 - 6 \zeta_{18} + 4 \zeta_{18}^{2} - 5 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{85} + ( -2 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + 7 \zeta_{18}^{3} + \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{87} + ( -6 + 2 \zeta_{18} + 2 \zeta_{18}^{2} + 3 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{89} + ( 1 - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{91} + ( 4 - 3 \zeta_{18} - 5 \zeta_{18}^{2} - 5 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{93} + ( -2 + \zeta_{18} + \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{95} + ( 2 \zeta_{18} + 9 \zeta_{18}^{2} - 5 \zeta_{18}^{3} + 9 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{97} + ( 3 + 3 \zeta_{18} - 6 \zeta_{18}^{3} - 6 \zeta_{18}^{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 3q^{5} + 3q^{7} + O(q^{10}) \) \( 6q + 3q^{5} + 3q^{7} + 3q^{13} - 9q^{15} + 12q^{17} + 18q^{19} + 6q^{23} + 6q^{25} + 27q^{27} + 3q^{29} - 9q^{31} - 18q^{33} + 6q^{35} - 30q^{37} - 18q^{39} - 6q^{41} - 15q^{43} + 9q^{45} + 9q^{47} - 3q^{49} - 9q^{51} - 12q^{53} + 27q^{57} + 3q^{59} + 12q^{61} - 15q^{65} - 12q^{67} - 27q^{69} - 42q^{71} + 6q^{73} - 9q^{75} - 15q^{79} - 6q^{83} + 15q^{85} + 9q^{87} - 36q^{89} + 6q^{91} + 9q^{93} - 6q^{95} - 15q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 + \zeta_{18}\)

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
−0.766044 + 0.642788i
0.939693 + 0.342020i
−0.173648 0.984808i
−0.766044 0.642788i
0.939693 0.342020i
−0.173648 + 0.984808i
0 −1.11334 1.32683i 0 1.43969 2.49362i 0 0.500000 + 0.866025i 0 −0.520945 + 2.95442i 0
337.2 0 −0.592396 + 1.62760i 0 0.326352 0.565258i 0 0.500000 + 0.866025i 0 −2.29813 1.92836i 0
337.3 0 1.70574 0.300767i 0 −0.266044 + 0.460802i 0 0.500000 + 0.866025i 0 2.81908 1.02606i 0
673.1 0 −1.11334 + 1.32683i 0 1.43969 + 2.49362i 0 0.500000 0.866025i 0 −0.520945 2.95442i 0
673.2 0 −0.592396 1.62760i 0 0.326352 + 0.565258i 0 0.500000 0.866025i 0 −2.29813 + 1.92836i 0
673.3 0 1.70574 + 0.300767i 0 −0.266044 0.460802i 0 0.500000 0.866025i 0 2.81908 + 1.02606i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 673.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.r.i 6
3.b odd 2 1 3024.2.r.h 6
4.b odd 2 1 504.2.r.c 6
9.c even 3 1 inner 1008.2.r.i 6
9.c even 3 1 9072.2.a.br 3
9.d odd 6 1 3024.2.r.h 6
9.d odd 6 1 9072.2.a.cc 3
12.b even 2 1 1512.2.r.c 6
36.f odd 6 1 504.2.r.c 6
36.f odd 6 1 4536.2.a.s 3
36.h even 6 1 1512.2.r.c 6
36.h even 6 1 4536.2.a.v 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.c 6 4.b odd 2 1
504.2.r.c 6 36.f odd 6 1
1008.2.r.i 6 1.a even 1 1 trivial
1008.2.r.i 6 9.c even 3 1 inner
1512.2.r.c 6 12.b even 2 1
1512.2.r.c 6 36.h even 6 1
3024.2.r.h 6 3.b odd 2 1
3024.2.r.h 6 9.d odd 6 1
4536.2.a.s 3 36.f odd 6 1
4536.2.a.v 3 36.h even 6 1
9072.2.a.br 3 9.c even 3 1
9072.2.a.cc 3 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{6} - 3 T_{5}^{5} + 9 T_{5}^{4} - 2 T_{5}^{3} + 3 T_{5}^{2} + 1 \)
\( T_{11}^{6} + 9 T_{11}^{4} + 18 T_{11}^{3} + 81 T_{11}^{2} + 81 T_{11} + 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 9 T^{3} + 27 T^{6} \)
$5$ \( 1 - 3 T - 6 T^{2} + 13 T^{3} + 63 T^{4} - 60 T^{5} - 259 T^{6} - 300 T^{7} + 1575 T^{8} + 1625 T^{9} - 3750 T^{10} - 9375 T^{11} + 15625 T^{12} \)
$7$ \( ( 1 - T + T^{2} )^{3} \)
$11$ \( 1 - 24 T^{2} + 18 T^{3} + 312 T^{4} - 216 T^{5} - 3593 T^{6} - 2376 T^{7} + 37752 T^{8} + 23958 T^{9} - 351384 T^{10} + 1771561 T^{12} \)
$13$ \( 1 - 3 T - 21 T^{2} + 60 T^{3} + 285 T^{4} - 417 T^{5} - 3202 T^{6} - 5421 T^{7} + 48165 T^{8} + 131820 T^{9} - 599781 T^{10} - 1113879 T^{11} + 4826809 T^{12} \)
$17$ \( ( 1 - 6 T + 54 T^{2} - 203 T^{3} + 918 T^{4} - 1734 T^{5} + 4913 T^{6} )^{2} \)
$19$ \( ( 1 - 9 T + 63 T^{2} - 323 T^{3} + 1197 T^{4} - 3249 T^{5} + 6859 T^{6} )^{2} \)
$23$ \( 1 - 6 T + 12 T^{2} - 130 T^{3} - 18 T^{4} + 3282 T^{5} - 7909 T^{6} + 75486 T^{7} - 9522 T^{8} - 1581710 T^{9} + 3358092 T^{10} - 38618058 T^{11} + 148035889 T^{12} \)
$29$ \( 1 - 3 T - 42 T^{2} + 157 T^{3} + 657 T^{4} - 1554 T^{5} - 10195 T^{6} - 45066 T^{7} + 552537 T^{8} + 3829073 T^{9} - 29705802 T^{10} - 61533447 T^{11} + 594823321 T^{12} \)
$31$ \( 1 + 9 T - 18 T^{2} - 79 T^{3} + 2367 T^{4} + 1422 T^{5} - 86865 T^{6} + 44082 T^{7} + 2274687 T^{8} - 2353489 T^{9} - 16623378 T^{10} + 257662359 T^{11} + 887503681 T^{12} \)
$37$ \( ( 1 + 15 T + 165 T^{2} + 1167 T^{3} + 6105 T^{4} + 20535 T^{5} + 50653 T^{6} )^{2} \)
$41$ \( 1 + 6 T - 72 T^{2} - 298 T^{3} + 4398 T^{4} + 8034 T^{5} - 177169 T^{6} + 329394 T^{7} + 7393038 T^{8} - 20538458 T^{9} - 203454792 T^{10} + 695137206 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 + 15 T + 42 T^{2} + 199 T^{3} + 6525 T^{4} + 24624 T^{5} - 76509 T^{6} + 1058832 T^{7} + 12064725 T^{8} + 15821893 T^{9} + 143589642 T^{10} + 2205126645 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 - 9 T - 66 T^{2} + 371 T^{3} + 7077 T^{4} - 20028 T^{5} - 269497 T^{6} - 941316 T^{7} + 15633093 T^{8} + 38518333 T^{9} - 322058946 T^{10} - 2064105063 T^{11} + 10779215329 T^{12} \)
$53$ \( ( 1 + 6 T + 162 T^{2} + 617 T^{3} + 8586 T^{4} + 16854 T^{5} + 148877 T^{6} )^{2} \)
$59$ \( 1 - 3 T - 24 T^{2} - 369 T^{3} - 453 T^{4} + 5694 T^{5} + 333403 T^{6} + 335946 T^{7} - 1576893 T^{8} - 75784851 T^{9} - 290816664 T^{10} - 2144772897 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 - 12 T - 48 T^{2} + 478 T^{3} + 8460 T^{4} - 30636 T^{5} - 395685 T^{6} - 1868796 T^{7} + 31479660 T^{8} + 108496918 T^{9} - 664600368 T^{10} - 10135155612 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 + 12 T - 12 T^{2} - 690 T^{3} - 1488 T^{4} + 2856 T^{5} - 34441 T^{6} + 191352 T^{7} - 6679632 T^{8} - 207526470 T^{9} - 241813452 T^{10} + 16201501284 T^{11} + 90458382169 T^{12} \)
$71$ \( ( 1 + 21 T + 249 T^{2} + 2115 T^{3} + 17679 T^{4} + 105861 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( ( 1 - 3 T + 105 T^{2} - 475 T^{3} + 7665 T^{4} - 15987 T^{5} + 389017 T^{6} )^{2} \)
$79$ \( 1 + 15 T + 30 T^{2} + 59 T^{3} + 1125 T^{4} - 89100 T^{5} - 1368633 T^{6} - 7038900 T^{7} + 7021125 T^{8} + 29089301 T^{9} + 1168502430 T^{10} + 46155845985 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 + 6 T - 42 T^{2} - 2610 T^{3} - 9624 T^{4} + 83688 T^{5} + 2759119 T^{6} + 6946104 T^{7} - 66299736 T^{8} - 1492364070 T^{9} - 1993249482 T^{10} + 23634243858 T^{11} + 326940373369 T^{12} \)
$89$ \( ( 1 + 18 T + 318 T^{2} + 3241 T^{3} + 28302 T^{4} + 142578 T^{5} + 704969 T^{6} )^{2} \)
$97$ \( 1 + 15 T + 168 T^{2} + 1477 T^{3} + 2385 T^{4} - 42462 T^{5} - 510111 T^{6} - 4118814 T^{7} + 22440465 T^{8} + 1348018021 T^{9} + 14872919208 T^{10} + 128810103855 T^{11} + 832972004929 T^{12} \)
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