Properties

Label 756.2.k.e
Level 756
Weight 2
Character orbit 756.k
Analytic conductor 6.037
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 756.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{5} + ( 1 + \beta_{2} + \beta_{4} ) q^{7} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{5} + ( 1 + \beta_{2} + \beta_{4} ) q^{7} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{11} + ( -\beta_{1} + \beta_{3} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{17} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{19} + ( \beta_{2} - \beta_{3} + 5 \beta_{4} ) q^{23} + ( -5 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} ) q^{25} + ( -2 - \beta_{2} - \beta_{5} ) q^{29} + ( -\beta_{2} - \beta_{3} + \beta_{5} ) q^{31} + ( -6 + \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{35} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{37} + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{41} + ( -4 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{43} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{47} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{49} + ( 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} ) q^{53} + ( 5 - 4 \beta_{1} - \beta_{2} + 4 \beta_{3} - \beta_{5} ) q^{55} + ( 2 - \beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{59} + ( 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{61} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 9 \beta_{4} - 2 \beta_{5} ) q^{65} + ( 5 + \beta_{1} + 5 \beta_{4} + \beta_{5} ) q^{67} + ( 2 + 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{5} ) q^{71} + ( 2 + \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{73} + ( -7 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 7 \beta_{4} + \beta_{5} ) q^{77} + ( -2 \beta_{1} + \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{79} + ( 5 + \beta_{1} - \beta_{3} ) q^{83} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{85} + ( -2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{89} + ( 4 + 3 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{91} + ( -14 - 5 \beta_{2} - 5 \beta_{3} - 14 \beta_{4} + 5 \beta_{5} ) q^{95} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{5} + 2q^{7} + O(q^{10}) \) \( 6q - q^{5} + 2q^{7} - 5q^{11} - 4q^{13} + 4q^{17} - 3q^{19} - 14q^{23} - 10q^{25} - 10q^{29} + 2q^{31} - 26q^{35} + 24q^{41} - 18q^{43} + 9q^{47} - 6q^{53} + 16q^{55} + 5q^{59} - 7q^{61} - 24q^{65} + 16q^{67} + 22q^{71} + q^{73} - 31q^{77} + 8q^{79} + 34q^{83} + 10q^{85} + 3q^{89} + 5q^{91} - 32q^{95} + 28q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 5 \nu^{3} - \nu^{2} + 3 \nu - 6 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 8 \nu^{3} + 8 \nu^{2} - 21 \nu + 12 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + 4 \nu^{4} - 11 \nu^{3} + 20 \nu^{2} - 15 \nu + 9 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 6 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - 4 \nu^{4} + 14 \nu^{3} - 20 \nu^{2} + 30 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{4} + \beta_{3} + \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} - 5\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} - 5 \beta_{1} - 5\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{5} + 11 \beta_{4} - 9 \beta_{3} - 7 \beta_{2} - 7 \beta_{1} + 16\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-17 \beta_{5} - 16 \beta_{4} + 2 \beta_{3} - 8 \beta_{2} + 12 \beta_{1} + 31\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(1\) \(\beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.500000 + 0.224437i
0.500000 + 2.05195i
0.500000 1.41036i
0.500000 0.224437i
0.500000 2.05195i
0.500000 + 1.41036i
0 0 0 −2.14400 3.71351i 0 1.40545 2.24159i 0 0 0
109.2 0 0 0 0.433463 + 0.750780i 0 −2.25729 1.38008i 0 0 0
109.3 0 0 0 1.21053 + 2.09671i 0 1.85185 + 1.88962i 0 0 0
541.1 0 0 0 −2.14400 + 3.71351i 0 1.40545 + 2.24159i 0 0 0
541.2 0 0 0 0.433463 0.750780i 0 −2.25729 + 1.38008i 0 0 0
541.3 0 0 0 1.21053 2.09671i 0 1.85185 1.88962i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 541.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.k.e 6
3.b odd 2 1 756.2.k.f yes 6
7.c even 3 1 inner 756.2.k.e 6
7.c even 3 1 5292.2.a.x 3
7.d odd 6 1 5292.2.a.v 3
9.c even 3 1 2268.2.i.j 6
9.c even 3 1 2268.2.l.k 6
9.d odd 6 1 2268.2.i.k 6
9.d odd 6 1 2268.2.l.j 6
21.g even 6 1 5292.2.a.w 3
21.h odd 6 1 756.2.k.f yes 6
21.h odd 6 1 5292.2.a.u 3
63.g even 3 1 2268.2.i.j 6
63.h even 3 1 2268.2.l.k 6
63.j odd 6 1 2268.2.l.j 6
63.n odd 6 1 2268.2.i.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.k.e 6 1.a even 1 1 trivial
756.2.k.e 6 7.c even 3 1 inner
756.2.k.f yes 6 3.b odd 2 1
756.2.k.f yes 6 21.h odd 6 1
2268.2.i.j 6 9.c even 3 1
2268.2.i.j 6 63.g even 3 1
2268.2.i.k 6 9.d odd 6 1
2268.2.i.k 6 63.n odd 6 1
2268.2.l.j 6 9.d odd 6 1
2268.2.l.j 6 63.j odd 6 1
2268.2.l.k 6 9.c even 3 1
2268.2.l.k 6 63.h even 3 1
5292.2.a.u 3 21.h odd 6 1
5292.2.a.v 3 7.d odd 6 1
5292.2.a.w 3 21.g even 6 1
5292.2.a.x 3 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\):

\( T_{5}^{6} + T_{5}^{5} + 13 T_{5}^{4} - 30 T_{5}^{3} + 135 T_{5}^{2} - 108 T_{5} + 81 \)
\( T_{13}^{3} + 2 T_{13}^{2} - 11 T_{13} - 21 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + T - 2 T^{2} - 35 T^{3} - 25 T^{4} + 52 T^{5} + 541 T^{6} + 260 T^{7} - 625 T^{8} - 4375 T^{9} - 1250 T^{10} + 3125 T^{11} + 15625 T^{12} \)
$7$ \( 1 - 2 T + 2 T^{2} + 19 T^{3} + 14 T^{4} - 98 T^{5} + 343 T^{6} \)
$11$ \( 1 + 5 T + 4 T^{2} + 11 T^{3} - 25 T^{4} - 718 T^{5} - 3005 T^{6} - 7898 T^{7} - 3025 T^{8} + 14641 T^{9} + 58564 T^{10} + 805255 T^{11} + 1771561 T^{12} \)
$13$ \( ( 1 + 2 T + 28 T^{2} + 31 T^{3} + 364 T^{4} + 338 T^{5} + 2197 T^{6} )^{2} \)
$17$ \( 1 - 4 T - 20 T^{2} + 146 T^{3} + 104 T^{4} - 1480 T^{5} + 4195 T^{6} - 25160 T^{7} + 30056 T^{8} + 717298 T^{9} - 1670420 T^{10} - 5679428 T^{11} + 24137569 T^{12} \)
$19$ \( 1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 1026 T^{7} - 55233 T^{8} - 459553 T^{9} - 1563852 T^{10} + 7428297 T^{11} + 47045881 T^{12} \)
$23$ \( 1 + 14 T + 88 T^{2} + 350 T^{3} + 1046 T^{4} + 602 T^{5} - 10061 T^{6} + 13846 T^{7} + 553334 T^{8} + 4258450 T^{9} + 24626008 T^{10} + 90108802 T^{11} + 148035889 T^{12} \)
$29$ \( ( 1 + 5 T + 75 T^{2} + 227 T^{3} + 2175 T^{4} + 4205 T^{5} + 24389 T^{6} )^{2} \)
$31$ \( 1 - 2 T - 78 T^{2} + 42 T^{3} + 3976 T^{4} - 200 T^{5} - 142097 T^{6} - 6200 T^{7} + 3820936 T^{8} + 1251222 T^{9} - 72034638 T^{10} - 57258302 T^{11} + 887503681 T^{12} \)
$37$ \( 1 - 54 T^{2} - 274 T^{3} + 918 T^{4} + 7398 T^{5} + 12183 T^{6} + 273726 T^{7} + 1256742 T^{8} - 13878922 T^{9} - 101204694 T^{10} + 2565726409 T^{12} \)
$41$ \( ( 1 - 12 T + 132 T^{2} - 903 T^{3} + 5412 T^{4} - 20172 T^{5} + 68921 T^{6} )^{2} \)
$43$ \( ( 1 + 9 T + 99 T^{2} + 493 T^{3} + 4257 T^{4} + 16641 T^{5} + 79507 T^{6} )^{2} \)
$47$ \( 1 - 9 T - 6 T^{2} + 423 T^{3} - 1947 T^{4} + 1260 T^{5} + 36583 T^{6} + 59220 T^{7} - 4300923 T^{8} + 43917129 T^{9} - 29278086 T^{10} - 2064105063 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 + 6 T - 24 T^{2} + 222 T^{3} + 6 T^{4} - 17166 T^{5} + 10591 T^{6} - 909798 T^{7} + 16854 T^{8} + 33050694 T^{9} - 189371544 T^{10} + 2509172958 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 - 5 T - 134 T^{2} + 223 T^{3} + 13355 T^{4} - 4526 T^{5} - 908765 T^{6} - 267034 T^{7} + 46488755 T^{8} + 45799517 T^{9} - 1623726374 T^{10} - 3574621495 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 + 7 T - 45 T^{2} + 84 T^{3} + 1093 T^{4} - 27251 T^{5} - 184994 T^{6} - 1662311 T^{7} + 4067053 T^{8} + 19066404 T^{9} - 623062845 T^{10} + 5912174107 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 - 16 T - 4 T^{2} + 234 T^{3} + 15028 T^{4} - 85604 T^{5} - 250049 T^{6} - 5735468 T^{7} + 67460692 T^{8} + 70378542 T^{9} - 80604484 T^{10} - 21602001712 T^{11} + 90458382169 T^{12} \)
$71$ \( ( 1 - 11 T + 141 T^{2} - 1373 T^{3} + 10011 T^{4} - 55451 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( 1 - T - 136 T^{2} + 477 T^{3} + 8461 T^{4} - 27386 T^{5} - 503183 T^{6} - 1999178 T^{7} + 45088669 T^{8} + 185561109 T^{9} - 3862160776 T^{10} - 2073071593 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 - 8 T - 18 T^{2} + 126 T^{3} - 2882 T^{4} + 21658 T^{5} + 208339 T^{6} + 1710982 T^{7} - 17986562 T^{8} + 62122914 T^{9} - 701101458 T^{10} - 24616451192 T^{11} + 243087455521 T^{12} \)
$83$ \( ( 1 - 17 T + 333 T^{2} - 2921 T^{3} + 27639 T^{4} - 117113 T^{5} + 571787 T^{6} )^{2} \)
$89$ \( 1 - 3 T - 33 T^{2} - 1704 T^{3} + 393 T^{4} + 31803 T^{5} + 1804174 T^{6} + 2830467 T^{7} + 3112953 T^{8} - 1201267176 T^{9} - 2070493953 T^{10} - 16752178347 T^{11} + 496981290961 T^{12} \)
$97$ \( ( 1 - 14 T + 251 T^{2} - 2660 T^{3} + 24347 T^{4} - 131726 T^{5} + 912673 T^{6} )^{2} \)
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