Properties

Label 252.9.d.b
Level $252$
Weight $9$
Character orbit 252.d
Analytic conductor $102.659$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 252.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(102.659409735\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Defining polynomial: \(x^{6} + 2160 x^{4} + 976392 x^{2} + 85162752\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5}\cdot 7^{3} \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} + ( 361 - \beta_{3} ) q^{7} +O(q^{10})\) \( q -\beta_{1} q^{5} + ( 361 - \beta_{3} ) q^{7} + ( -4082 - \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{11} + ( -29 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} - \beta_{4} ) q^{13} + ( -100 \beta_{1} + 45 \beta_{2} + 18 \beta_{3} + 3 \beta_{4} ) q^{17} + ( -64 \beta_{1} + 50 \beta_{2} - 30 \beta_{3} - 5 \beta_{4} ) q^{19} + ( 1934 - 29 \beta_{1} + 6 \beta_{2} - 58 \beta_{3} + 6 \beta_{4} + 5 \beta_{5} ) q^{23} + ( -113119 + 29 \beta_{1} - 11 \beta_{2} + 58 \beta_{3} - 11 \beta_{4} + 15 \beta_{5} ) q^{25} + ( -210722 + 98 \beta_{1} - 30 \beta_{2} + 196 \beta_{3} - 30 \beta_{4} + 22 \beta_{5} ) q^{29} + ( 996 \beta_{1} - 1515 \beta_{2} + 90 \beta_{3} + 15 \beta_{4} ) q^{31} + ( 219136 - 189 \beta_{1} - 1596 \beta_{2} - 2 \beta_{3} - 21 \beta_{4} - 35 \beta_{5} ) q^{35} + ( 530722 - 342 \beta_{1} + 78 \beta_{2} - 684 \beta_{3} + 78 \beta_{4} + 30 \beta_{5} ) q^{37} + ( -2078 \beta_{1} + 1290 \beta_{2} - 540 \beta_{3} - 90 \beta_{4} ) q^{41} + ( -1297230 + 175 \beta_{1} - 70 \beta_{2} + 350 \beta_{3} - 70 \beta_{4} + 105 \beta_{5} ) q^{43} + ( -364 \beta_{1} - 4809 \beta_{2} + 486 \beta_{3} + 81 \beta_{4} ) q^{47} + ( 453185 - 4172 \beta_{1} + 4389 \beta_{2} - 542 \beta_{3} + 119 \beta_{4} - 210 \beta_{5} ) q^{49} + ( 1390110 - 420 \beta_{1} + 180 \beta_{2} - 840 \beta_{3} + 180 \beta_{4} - 300 \beta_{5} ) q^{53} + ( 12948 \beta_{1} - 1655 \beta_{2} - 2250 \beta_{3} - 375 \beta_{4} ) q^{55} + ( -1784 \beta_{1} + 5130 \beta_{2} + 2610 \beta_{3} + 435 \beta_{4} ) q^{59} + ( 10019 \beta_{1} + 3300 \beta_{2} - 120 \beta_{3} - 20 \beta_{4} ) q^{61} + ( -14015872 + 1897 \beta_{1} - 483 \beta_{2} + 3794 \beta_{3} - 483 \beta_{4} + 35 \beta_{5} ) q^{65} + ( 2763250 + 2395 \beta_{1} - 490 \beta_{2} + 4790 \beta_{3} - 490 \beta_{4} - 435 \beta_{5} ) q^{67} + ( 10348142 + 316 \beta_{1} - 219 \beta_{2} + 632 \beta_{3} - 219 \beta_{4} + 560 \beta_{5} ) q^{71} + ( -3346 \beta_{1} + 33803 \beta_{2} + 5118 \beta_{3} + 853 \beta_{4} ) q^{73} + ( 10231742 - 43463 \beta_{1} + 28371 \beta_{2} + 4418 \beta_{3} - 2205 \beta_{4} + 245 \beta_{5} ) q^{77} + ( 31019090 + 1050 \beta_{1} - 471 \beta_{2} + 2100 \beta_{3} - 471 \beta_{4} + 834 \beta_{5} ) q^{79} + ( 43240 \beta_{1} - 29163 \beta_{2} - 4428 \beta_{3} - 738 \beta_{4} ) q^{83} + ( -43868480 - 4480 \beta_{1} + 580 \beta_{2} - 8960 \beta_{3} + 580 \beta_{4} + 2160 \beta_{5} ) q^{85} + ( -25554 \beta_{1} + 21285 \beta_{2} - 14670 \beta_{3} - 2445 \beta_{4} ) q^{89} + ( -29850176 - 39977 \beta_{1} - 23401 \beta_{2} - 4076 \beta_{3} + 224 \beta_{4} - 2583 \beta_{5} ) q^{91} + ( -14318816 - 469 \beta_{1} + 621 \beta_{2} - 938 \beta_{3} + 621 \beta_{4} - 2015 \beta_{5} ) q^{95} + ( -28328 \beta_{1} - 22843 \beta_{2} - 3822 \beta_{3} - 637 \beta_{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2166 q^{7} + O(q^{10}) \) \( 6 q + 2166 q^{7} - 24492 q^{11} + 11604 q^{23} - 678714 q^{25} - 1264332 q^{29} + 1314816 q^{35} + 3184332 q^{37} - 7783380 q^{43} + 2719110 q^{49} + 8340660 q^{53} - 84095232 q^{65} + 16579500 q^{67} + 62088852 q^{71} + 61390452 q^{77} + 186114540 q^{79} - 263210880 q^{85} - 179101056 q^{91} - 85912896 q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 2160 x^{4} + 976392 x^{2} + 85162752\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{5} - 828 \nu^{3} + 5640984 \nu \)\()/183816\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 828 \nu^{3} - 494136 \nu \)\()/20424\)
\(\beta_{3}\)\(=\)\((\)\( 109 \nu^{5} - 1332 \nu^{4} + 212796 \nu^{3} - 2021976 \nu^{2} + 63046152 \nu - 251332416 \)\()/183816\)
\(\beta_{4}\)\(=\)\((\)\( 859 \nu^{5} + 7992 \nu^{4} + 1691604 \nu^{3} + 12131856 \nu^{2} + 531380376 \nu + 1507994496 \)\()/183816\)
\(\beta_{5}\)\(=\)\((\)\( 217 \nu^{5} + 15984 \nu^{4} + 424764 \nu^{3} + 31983984 \nu^{2} + 131733288 \nu + 8574584832 \)\()/183816\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 9 \beta_{1}\)\()/252\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} - \beta_{4} + 6 \beta_{3} - \beta_{2} + 3 \beta_{1} - 30240\)\()/42\)
\(\nu^{3}\)\(=\)\((\)\(9 \beta_{4} + 54 \beta_{3} - 1835 \beta_{2} - 2898 \beta_{1}\)\()/84\)
\(\nu^{4}\)\(=\)\((\)\(-253 \beta_{5} + 322 \beta_{4} - 2070 \beta_{3} + 322 \beta_{2} - 1035 \beta_{1} + 6329904\)\()/7\)
\(\nu^{5}\)\(=\)\((\)\(-621 \beta_{4} - 3726 \beta_{3} + 283309 \beta_{2} + 323496 \beta_{1}\)\()/7\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
39.6746i
21.7042i
10.7169i
10.7169i
21.7042i
39.6746i
0 0 0 964.066i 0 2367.93 + 397.129i 0 0 0
181.2 0 0 0 685.916i 0 −1845.45 + 1535.94i 0 0 0
181.3 0 0 0 333.657i 0 560.525 2334.65i 0 0 0
181.4 0 0 0 333.657i 0 560.525 + 2334.65i 0 0 0
181.5 0 0 0 685.916i 0 −1845.45 1535.94i 0 0 0
181.6 0 0 0 964.066i 0 2367.93 397.129i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.9.d.b 6
3.b odd 2 1 28.9.b.a 6
7.b odd 2 1 inner 252.9.d.b 6
12.b even 2 1 112.9.c.d 6
21.c even 2 1 28.9.b.a 6
21.g even 6 2 196.9.h.b 12
21.h odd 6 2 196.9.h.b 12
84.h odd 2 1 112.9.c.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.9.b.a 6 3.b odd 2 1
28.9.b.a 6 21.c even 2 1
112.9.c.d 6 12.b even 2 1
112.9.c.d 6 84.h odd 2 1
196.9.h.b 12 21.g even 6 2
196.9.h.b 12 21.h odd 6 2
252.9.d.b 6 1.a even 1 1 trivial
252.9.d.b 6 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 1511232 T_{5}^{4} + 593123565600 T_{5}^{2} + \)\(48\!\cdots\!00\)\( \) acting on \(S_{9}^{\mathrm{new}}(252, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( T^{6} \)
$5$ \( 48680661813120000 + 593123565600 T^{2} + 1511232 T^{4} + T^{6} \)
$7$ \( \)\(19\!\cdots\!01\)\( - 71982527613755766 T + 5685379336623 T^{2} - 5377596532 T^{3} + 986223 T^{4} - 2166 T^{5} + T^{6} \)
$11$ \( ( -7013477844792 - 586924020 T + 12246 T^{2} + T^{3} )^{2} \)
$13$ \( \)\(11\!\cdots\!68\)\( + 1550774551274383392 T^{2} + 2691986880 T^{4} + T^{6} \)
$17$ \( \)\(15\!\cdots\!00\)\( + \)\(13\!\cdots\!16\)\( T^{2} + 29101834752 T^{4} + T^{6} \)
$19$ \( \)\(20\!\cdots\!00\)\( + 21924088485169773600 T^{2} + 43801462272 T^{4} + T^{6} \)
$23$ \( ( 6005643700768200 - 62186581620 T - 5802 T^{2} + T^{3} )^{2} \)
$29$ \( ( 106390583279134344 - 638738692596 T + 632166 T^{2} + T^{3} )^{2} \)
$31$ \( \)\(44\!\cdots\!00\)\( + \)\(83\!\cdots\!00\)\( T^{2} + 5071197100032 T^{4} + T^{6} \)
$37$ \( ( 9891859374136705400 - 6206563348980 T - 1592166 T^{2} + T^{3} )^{2} \)
$41$ \( \)\(61\!\cdots\!00\)\( + \)\(87\!\cdots\!00\)\( T^{2} + 17909852667648 T^{4} + T^{6} \)
$43$ \( ( -7677283367599693000 - 2651175088500 T + 3891690 T^{2} + T^{3} )^{2} \)
$47$ \( \)\(22\!\cdots\!12\)\( + \)\(99\!\cdots\!80\)\( T^{2} + 68017743250944 T^{4} + T^{6} \)
$53$ \( ( \)\(11\!\cdots\!00\)\( - 53350797654900 T - 4170330 T^{2} + T^{3} )^{2} \)
$59$ \( \)\(36\!\cdots\!00\)\( + \)\(30\!\cdots\!00\)\( T^{2} + 338441040125952 T^{4} + T^{6} \)
$61$ \( \)\(13\!\cdots\!00\)\( + \)\(53\!\cdots\!00\)\( T^{2} + 225118443807552 T^{4} + T^{6} \)
$67$ \( ( -\)\(22\!\cdots\!00\)\( - 412665478211700 T - 8289750 T^{2} + T^{3} )^{2} \)
$71$ \( ( 22965958767033039240 + 139808421637164 T - 31044426 T^{2} + T^{3} )^{2} \)
$73$ \( \)\(83\!\cdots\!68\)\( + \)\(36\!\cdots\!80\)\( T^{2} + 3553313679653376 T^{4} + T^{6} \)
$79$ \( ( -\)\(17\!\cdots\!68\)\( + 2442387979487532 T - 93057270 T^{2} + T^{3} )^{2} \)
$83$ \( \)\(50\!\cdots\!32\)\( + \)\(13\!\cdots\!52\)\( T^{2} + 4066673581933440 T^{4} + T^{6} \)
$89$ \( \)\(15\!\cdots\!00\)\( + \)\(24\!\cdots\!00\)\( T^{2} + 10030682450092032 T^{4} + T^{6} \)
$97$ \( \)\(12\!\cdots\!28\)\( + \)\(20\!\cdots\!32\)\( T^{2} + 3750067818854400 T^{4} + T^{6} \)
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