Properties

Label 1764.2.e.g
Level 1764
Weight 2
Character orbit 1764.e
Analytic conductor 14.086
Analytic rank 0
Dimension 12
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1764.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.653473922154496.1
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + ( 1 + \beta_{10} ) q^{4} -\beta_{4} q^{5} + ( \beta_{6} - \beta_{9} ) q^{8} +O(q^{10})\) \( q + \beta_{6} q^{2} + ( 1 + \beta_{10} ) q^{4} -\beta_{4} q^{5} + ( \beta_{6} - \beta_{9} ) q^{8} + ( 1 - \beta_{2} - \beta_{5} + \beta_{8} - \beta_{10} ) q^{10} + ( \beta_{3} + \beta_{6} + \beta_{7} ) q^{11} + ( \beta_{2} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{13} + ( -1 + \beta_{8} + \beta_{10} ) q^{16} + ( 2 \beta_{1} - \beta_{4} ) q^{17} + ( \beta_{2} - \beta_{8} + 2 \beta_{10} ) q^{19} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{20} + ( 2 + \beta_{8} + \beta_{10} + \beta_{11} ) q^{22} + ( -2 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{23} + ( -1 - \beta_{2} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{25} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{7} ) q^{26} + \beta_{1} q^{29} + ( -1 + 3 \beta_{2} + \beta_{5} - \beta_{8} - \beta_{11} ) q^{31} + ( -\beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{32} + ( 1 + \beta_{2} - \beta_{5} + \beta_{8} - \beta_{10} ) q^{34} + ( \beta_{2} - 2 \beta_{5} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{37} + ( 2 \beta_{4} - 2 \beta_{9} ) q^{38} + ( -2 \beta_{2} - 2 \beta_{5} ) q^{40} + ( -2 \beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{6} - 2 \beta_{9} ) q^{41} + ( -1 + 2 \beta_{2} + \beta_{5} - 2 \beta_{10} - \beta_{11} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{44} + ( -4 - 2 \beta_{5} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{46} + ( -2 \beta_{1} + 2 \beta_{4} + 4 \beta_{7} ) q^{47} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{50} + ( 2 - 2 \beta_{2} + 2 \beta_{8} + 2 \beta_{11} ) q^{52} + ( -\beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{9} ) q^{53} + ( 2 - 3 \beta_{2} - 2 \beta_{5} + 3 \beta_{8} ) q^{55} + \beta_{2} q^{58} + ( 2 \beta_{1} - 2 \beta_{4} + 4 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} ) q^{59} + ( -5 - \beta_{2} - \beta_{5} - \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{61} + ( 4 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} ) q^{62} + ( -1 - \beta_{2} - 2 \beta_{5} + 3 \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{64} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{9} ) q^{65} + ( 3 \beta_{2} - \beta_{8} + 3 \beta_{10} - \beta_{11} ) q^{67} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{68} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{4} + 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{9} ) q^{71} + ( 1 - 2 \beta_{2} + 3 \beta_{5} - 2 \beta_{8} + \beta_{10} - 2 \beta_{11} ) q^{73} + ( -6 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{74} + ( -6 + 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{10} ) q^{76} + ( 2 - 3 \beta_{2} - 2 \beta_{5} + \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{79} + ( -4 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} ) q^{80} + ( -3 - \beta_{2} - \beta_{5} + \beta_{8} + 3 \beta_{10} ) q^{82} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{83} + ( -4 - \beta_{2} + 2 \beta_{5} - \beta_{8} + \beta_{10} - \beta_{11} ) q^{85} + ( 4 \beta_{1} + 2 \beta_{7} + 2 \beta_{9} ) q^{86} + ( 4 - 3 \beta_{2} + 2 \beta_{8} + \beta_{10} + \beta_{11} ) q^{88} + ( -2 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} + 2 \beta_{9} ) q^{89} + ( -6 \beta_{1} - 2 \beta_{3} + 5 \beta_{4} - 6 \beta_{6} - \beta_{7} ) q^{92} + ( 2 - 2 \beta_{5} + 2 \beta_{8} + 2 \beta_{10} + 4 \beta_{11} ) q^{94} + ( 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} ) q^{95} + ( -1 - 3 \beta_{5} - 3 \beta_{10} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 8q^{4} + O(q^{10}) \) \( 12q + 8q^{4} + 8q^{10} - 20q^{16} + 20q^{22} - 12q^{25} + 16q^{34} + 8q^{37} - 8q^{40} - 36q^{46} + 16q^{52} + 4q^{58} - 56q^{61} - 16q^{64} - 72q^{76} - 56q^{82} - 56q^{85} + 28q^{88} + 24q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{10} + 13 x^{8} - 28 x^{6} + 52 x^{4} - 64 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{11} + 4 \nu^{9} - 5 \nu^{7} + 12 \nu^{5} - 12 \nu^{3} + 16 \nu \)\()/32\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{10} + 4 \nu^{8} - 5 \nu^{6} + 12 \nu^{4} - 12 \nu^{2} + 16 \)\()/16\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{11} + 4 \nu^{9} - 13 \nu^{7} + 28 \nu^{5} - 20 \nu^{3} + 32 \nu \)\()/32\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{9} - 2 \nu^{7} + 9 \nu^{5} - 10 \nu^{3} + 16 \nu \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{10} + 4 \nu^{8} - 13 \nu^{6} + 28 \nu^{4} - 20 \nu^{2} + 32 \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{11} + 4 \nu^{9} - 13 \nu^{7} + 28 \nu^{5} - 52 \nu^{3} + 64 \nu \)\()/32\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{11} - 4 \nu^{9} + 11 \nu^{7} - 28 \nu^{5} + 36 \nu^{3} - 48 \nu \)\()/32\)
\(\beta_{8}\)\(=\)\((\)\( -3 \nu^{10} + 8 \nu^{8} - 23 \nu^{6} + 32 \nu^{4} - 44 \nu^{2} + 16 \)\()/16\)
\(\beta_{9}\)\(=\)\((\)\( 3 \nu^{11} - 8 \nu^{9} + 23 \nu^{7} - 32 \nu^{5} + 44 \nu^{3} + 16 \nu \)\()/32\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{10} + 4 \nu^{8} - 13 \nu^{6} + 28 \nu^{4} - 52 \nu^{2} + 48 \)\()/16\)
\(\beta_{11}\)\(=\)\((\)\( \nu^{10} - 8 \nu^{8} + 21 \nu^{6} - 32 \nu^{4} + 60 \nu^{2} - 48 \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{9} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{10} + \beta_{5} + 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{9} - \beta_{6} - \beta_{4} + 3 \beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{11} + \beta_{10} - \beta_{8} + 2 \beta_{5} + \beta_{2} - 4\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{9} + 2 \beta_{7} - 3 \beta_{6} + 5 \beta_{4} + \beta_{3} - 3 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(2 \beta_{11} + 3 \beta_{10} - 2 \beta_{8} - \beta_{5} + 6 \beta_{2} - 5\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-\beta_{9} + 4 \beta_{7} - 3 \beta_{6} + 9 \beta_{4} - 7 \beta_{3} + 3 \beta_{1}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-5 \beta_{11} - 5 \beta_{10} - 3 \beta_{8} - 2 \beta_{5} + 11 \beta_{2} - 4\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(\beta_{9} - 10 \beta_{7} - 5 \beta_{6} - 5 \beta_{4} - 9 \beta_{3} + 27 \beta_{1}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-18 \beta_{11} - 11 \beta_{10} - 14 \beta_{8} + 9 \beta_{5} - 6 \beta_{2} - 19\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(\beta_{9} - 36 \beta_{7} - 13 \beta_{6} - 9 \beta_{4} - 9 \beta_{3} - 3 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1079.1
−1.35489 + 0.405301i
−1.35489 0.405301i
−1.16947 + 0.795191i
−1.16947 0.795191i
−0.892524 + 1.09700i
−0.892524 1.09700i
0.892524 + 1.09700i
0.892524 1.09700i
1.16947 + 0.795191i
1.16947 0.795191i
1.35489 + 0.405301i
1.35489 0.405301i
−1.35489 0.405301i 0 1.67146 + 1.09828i 3.31339i 0 0 −1.81951 2.16549i 0 −1.34292 + 4.48929i
1079.2 −1.35489 + 0.405301i 0 1.67146 1.09828i 3.31339i 0 0 −1.81951 + 2.16549i 0 −1.34292 4.48929i
1079.3 −1.16947 0.795191i 0 0.735342 + 1.85991i 0.665647i 0 0 0.619022 2.75986i 0 0.529317 0.778457i
1079.4 −1.16947 + 0.795191i 0 0.735342 1.85991i 0.665647i 0 0 0.619022 + 2.75986i 0 0.529317 + 0.778457i
1079.5 −0.892524 1.09700i 0 −0.406803 + 1.95819i 2.56483i 0 0 2.51121 1.30147i 0 2.81361 2.28917i
1079.6 −0.892524 + 1.09700i 0 −0.406803 1.95819i 2.56483i 0 0 2.51121 + 1.30147i 0 2.81361 + 2.28917i
1079.7 0.892524 1.09700i 0 −0.406803 1.95819i 2.56483i 0 0 −2.51121 1.30147i 0 2.81361 + 2.28917i
1079.8 0.892524 + 1.09700i 0 −0.406803 + 1.95819i 2.56483i 0 0 −2.51121 + 1.30147i 0 2.81361 2.28917i
1079.9 1.16947 0.795191i 0 0.735342 1.85991i 0.665647i 0 0 −0.619022 2.75986i 0 0.529317 + 0.778457i
1079.10 1.16947 + 0.795191i 0 0.735342 + 1.85991i 0.665647i 0 0 −0.619022 + 2.75986i 0 0.529317 0.778457i
1079.11 1.35489 0.405301i 0 1.67146 1.09828i 3.31339i 0 0 1.81951 2.16549i 0 −1.34292 4.48929i
1079.12 1.35489 + 0.405301i 0 1.67146 + 1.09828i 3.31339i 0 0 1.81951 + 2.16549i 0 −1.34292 + 4.48929i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1079.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.e.g 12
3.b odd 2 1 inner 1764.2.e.g 12
4.b odd 2 1 inner 1764.2.e.g 12
7.b odd 2 1 252.2.e.a 12
12.b even 2 1 inner 1764.2.e.g 12
21.c even 2 1 252.2.e.a 12
28.d even 2 1 252.2.e.a 12
56.e even 2 1 4032.2.h.h 12
56.h odd 2 1 4032.2.h.h 12
84.h odd 2 1 252.2.e.a 12
168.e odd 2 1 4032.2.h.h 12
168.i even 2 1 4032.2.h.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.e.a 12 7.b odd 2 1
252.2.e.a 12 21.c even 2 1
252.2.e.a 12 28.d even 2 1
252.2.e.a 12 84.h odd 2 1
1764.2.e.g 12 1.a even 1 1 trivial
1764.2.e.g 12 3.b odd 2 1 inner
1764.2.e.g 12 4.b odd 2 1 inner
1764.2.e.g 12 12.b even 2 1 inner
4032.2.h.h 12 56.e even 2 1
4032.2.h.h 12 56.h odd 2 1
4032.2.h.h 12 168.e odd 2 1
4032.2.h.h 12 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{6} + 18 T_{5}^{4} + 80 T_{5}^{2} + 32 \)
\( T_{13}^{3} - 28 T_{13} + 16 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T^{2} + 13 T^{4} - 28 T^{6} + 52 T^{8} - 64 T^{10} + 64 T^{12} \)
$3$ \( \)
$5$ \( ( 1 - 12 T^{2} + 95 T^{4} - 568 T^{6} + 2375 T^{8} - 7500 T^{10} + 15625 T^{12} )^{2} \)
$7$ \( \)
$11$ \( ( 1 + 38 T^{2} + 715 T^{4} + 9068 T^{6} + 86515 T^{8} + 556358 T^{10} + 1771561 T^{12} )^{2} \)
$13$ \( ( 1 + 11 T^{2} + 16 T^{3} + 143 T^{4} + 2197 T^{6} )^{4} \)
$17$ \( ( 1 - 68 T^{2} + 2167 T^{4} - 44168 T^{6} + 626263 T^{8} - 5679428 T^{10} + 24137569 T^{12} )^{2} \)
$19$ \( ( 1 - 50 T^{2} + 1639 T^{4} - 35804 T^{6} + 591679 T^{8} - 6516050 T^{10} + 47045881 T^{12} )^{2} \)
$23$ \( ( 1 - 2 T^{2} + 979 T^{4} + 3772 T^{6} + 517891 T^{8} - 559682 T^{10} + 148035889 T^{12} )^{2} \)
$29$ \( ( 1 - 56 T^{2} + 841 T^{4} )^{6} \)
$31$ \( ( 1 - 58 T^{2} + 3727 T^{4} - 113644 T^{6} + 3581647 T^{8} - 53564218 T^{10} + 887503681 T^{12} )^{2} \)
$37$ \( ( 1 - 2 T + 47 T^{2} - 276 T^{3} + 1739 T^{4} - 2738 T^{5} + 50653 T^{6} )^{4} \)
$41$ \( ( 1 - 132 T^{2} + 10823 T^{4} - 527752 T^{6} + 18193463 T^{8} - 373000452 T^{10} + 4750104241 T^{12} )^{2} \)
$43$ \( ( 1 - 98 T^{2} + 4567 T^{4} - 172988 T^{6} + 8444383 T^{8} - 335042498 T^{10} + 6321363049 T^{12} )^{2} \)
$47$ \( ( 1 + 10 T^{2} + 4591 T^{4} + 70732 T^{6} + 10141519 T^{8} + 48796810 T^{10} + 10779215329 T^{12} )^{2} \)
$53$ \( ( 1 - 184 T^{2} + 16171 T^{4} - 975856 T^{6} + 45424339 T^{8} - 1451848504 T^{10} + 22164361129 T^{12} )^{2} \)
$59$ \( ( 1 + 146 T^{2} + 7799 T^{4} + 306396 T^{6} + 27148319 T^{8} + 1769134706 T^{10} + 42180533641 T^{12} )^{2} \)
$61$ \( ( 1 + 14 T + 187 T^{2} + 1700 T^{3} + 11407 T^{4} + 52094 T^{5} + 226981 T^{6} )^{4} \)
$67$ \( ( 1 - 186 T^{2} + 13175 T^{4} - 680684 T^{6} + 59142575 T^{8} - 3748108506 T^{10} + 90458382169 T^{12} )^{2} \)
$71$ \( ( 1 + 158 T^{2} + 5683 T^{4} - 107140 T^{6} + 28648003 T^{8} + 4015045598 T^{10} + 128100283921 T^{12} )^{2} \)
$73$ \( ( 1 + 47 T^{2} + 352 T^{3} + 3431 T^{4} + 389017 T^{6} )^{4} \)
$79$ \( ( 1 - 162 T^{2} + 19359 T^{4} - 2006332 T^{6} + 120819519 T^{8} - 6309913122 T^{10} + 243087455521 T^{12} )^{2} \)
$83$ \( ( 1 + 370 T^{2} + 65191 T^{4} + 6834652 T^{6} + 449100799 T^{8} + 17559578770 T^{10} + 326940373369 T^{12} )^{2} \)
$89$ \( ( 1 - 308 T^{2} + 49895 T^{4} - 5241384 T^{6} + 395218295 T^{8} - 19324610228 T^{10} + 496981290961 T^{12} )^{2} \)
$97$ \( ( 1 + 135 T^{2} - 128 T^{3} + 13095 T^{4} + 912673 T^{6} )^{4} \)
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