Properties

Label 450.6.f.e
Level $450$
Weight $6$
Character orbit 450.f
Analytic conductor $72.173$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(72.1727189158\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 3457 x^{8} + 2937456 x^{4} + 12960000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4}\cdot 5^{8} \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 \beta_{3} + 2 \beta_{4} ) q^{2} + 16 \beta_{2} q^{4} + ( -12 + 12 \beta_{2} - \beta_{6} ) q^{7} + ( 32 \beta_{3} - 32 \beta_{4} ) q^{8} +O(q^{10})\) \( q + ( 2 \beta_{3} + 2 \beta_{4} ) q^{2} + 16 \beta_{2} q^{4} + ( -12 + 12 \beta_{2} - \beta_{6} ) q^{7} + ( 32 \beta_{3} - 32 \beta_{4} ) q^{8} + ( 2 \beta_{1} + 8 \beta_{3} + 2 \beta_{9} - \beta_{11} ) q^{11} + ( 23 + 23 \beta_{2} - 2 \beta_{5} + \beta_{8} ) q^{13} + ( -48 \beta_{4} + 4 \beta_{10} ) q^{14} -256 q^{16} + ( -11 \beta_{1} + 167 \beta_{3} + 167 \beta_{4} - 8 \beta_{10} - 8 \beta_{11} ) q^{17} + ( 448 \beta_{2} - 4 \beta_{5} - 4 \beta_{6} - \beta_{7} - \beta_{8} ) q^{19} + ( -32 + 32 \beta_{2} - 12 \beta_{6} + 8 \beta_{7} ) q^{22} + ( 632 \beta_{3} - 632 \beta_{4} + 2 \beta_{9} - 18 \beta_{10} + 18 \beta_{11} ) q^{23} + ( 4 \beta_{1} + 92 \beta_{3} + 4 \beta_{9} - 4 \beta_{11} ) q^{26} + ( -192 - 192 \beta_{2} + 16 \beta_{5} ) q^{28} + ( 29 \beta_{1} - 499 \beta_{4} - 29 \beta_{9} + 17 \beta_{10} ) q^{29} + ( -4876 + 4 \beta_{5} - 4 \beta_{6} - 13 \beta_{7} + 13 \beta_{8} ) q^{31} + ( -512 \beta_{3} - 512 \beta_{4} ) q^{32} + ( 1336 \beta_{2} - 10 \beta_{5} - 10 \beta_{6} - 22 \beta_{7} - 22 \beta_{8} ) q^{34} + ( -2147 + 2147 \beta_{2} + 49 \beta_{6} + 8 \beta_{7} ) q^{37} + ( 896 \beta_{3} - 896 \beta_{4} - 8 \beta_{9} + 20 \beta_{10} - 20 \beta_{11} ) q^{38} + ( 5 \beta_{1} - 1563 \beta_{3} + 5 \beta_{9} - 2 \beta_{11} ) q^{41} + ( -1340 - 1340 \beta_{2} + 26 \beta_{5} - 28 \beta_{8} ) q^{43} + ( -32 \beta_{1} - 128 \beta_{4} + 32 \beta_{9} + 16 \beta_{10} ) q^{44} + ( -5056 - 68 \beta_{5} + 68 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} ) q^{46} + ( -38 \beta_{1} - 5920 \beta_{3} - 5920 \beta_{4} + 9 \beta_{10} + 9 \beta_{11} ) q^{47} + ( -5081 \beta_{2} + 44 \beta_{5} + 44 \beta_{6} + 50 \beta_{7} + 50 \beta_{8} ) q^{49} + ( -368 + 368 \beta_{2} - 32 \beta_{6} + 16 \beta_{7} ) q^{52} + ( 6373 \beta_{3} - 6373 \beta_{4} + 61 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} ) q^{53} + ( -768 \beta_{3} + 64 \beta_{11} ) q^{56} + ( -1996 - 1996 \beta_{2} - 48 \beta_{5} + 116 \beta_{8} ) q^{58} + ( -66 \beta_{1} - 16252 \beta_{4} + 66 \beta_{9} - 63 \beta_{10} ) q^{59} + ( -12100 + 113 \beta_{5} - 113 \beta_{6} + 72 \beta_{7} - 72 \beta_{8} ) q^{61} + ( 104 \beta_{1} - 9752 \beta_{3} - 9752 \beta_{4} + 68 \beta_{10} + 68 \beta_{11} ) q^{62} -4096 \beta_{2} q^{64} + ( -2796 + 2796 \beta_{2} + 8 \beta_{6} - 218 \beta_{7} ) q^{67} + ( 2672 \beta_{3} - 2672 \beta_{4} - 176 \beta_{9} + 128 \beta_{10} - 128 \beta_{11} ) q^{68} + ( -100 \beta_{1} - 5084 \beta_{3} - 100 \beta_{9} - 28 \beta_{11} ) q^{71} + ( 13249 + 13249 \beta_{2} - 245 \beta_{5} - 143 \beta_{8} ) q^{73} + ( -32 \beta_{1} - 8588 \beta_{4} + 32 \beta_{9} - 228 \beta_{10} ) q^{74} + ( -7168 + 64 \beta_{5} - 64 \beta_{6} - 16 \beta_{7} + 16 \beta_{8} ) q^{76} + ( 332 \beta_{1} - 27496 \beta_{3} - 27496 \beta_{4} + 70 \beta_{10} + 70 \beta_{11} ) q^{77} + ( -41492 \beta_{2} - 26 \beta_{5} - 26 \beta_{6} + 39 \beta_{7} + 39 \beta_{8} ) q^{79} + ( 6252 - 6252 \beta_{2} - 28 \beta_{6} + 20 \beta_{7} ) q^{82} + ( 20156 \beta_{3} - 20156 \beta_{4} + 36 \beta_{9} + 23 \beta_{10} - 23 \beta_{11} ) q^{83} + ( -112 \beta_{1} - 5360 \beta_{3} - 112 \beta_{9} - 8 \beta_{11} ) q^{86} + ( -512 - 512 \beta_{2} + 192 \beta_{5} - 128 \beta_{8} ) q^{88} + ( 115 \beta_{1} - 63817 \beta_{4} - 115 \beta_{9} - 108 \beta_{10} ) q^{89} + ( -38752 - 13 \beta_{5} + 13 \beta_{6} - 108 \beta_{7} + 108 \beta_{8} ) q^{91} + ( -32 \beta_{1} - 10112 \beta_{3} - 10112 \beta_{4} - 288 \beta_{10} - 288 \beta_{11} ) q^{92} + ( -47360 \beta_{2} + 112 \beta_{5} + 112 \beta_{6} - 76 \beta_{7} - 76 \beta_{8} ) q^{94} + ( 52593 - 52593 \beta_{2} - 261 \beta_{6} + 407 \beta_{7} ) q^{97} + ( -10162 \beta_{3} + 10162 \beta_{4} + 400 \beta_{9} - 376 \beta_{10} + 376 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 144q^{7} + O(q^{10}) \) \( 12q - 144q^{7} + 276q^{13} - 3072q^{16} - 384q^{22} - 2304q^{28} - 58512q^{31} - 25764q^{37} - 16080q^{43} - 60672q^{46} - 4416q^{52} - 23952q^{58} - 145200q^{61} - 33552q^{67} + 158988q^{73} - 86016q^{76} + 75024q^{82} - 6144q^{88} - 465024q^{91} + 631116q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 3457 x^{8} + 2937456 x^{4} + 12960000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 5 \nu^{9} - 4603 \nu^{5} + 78117552 \nu \)\()/3381264\)
\(\beta_{2}\)\(=\)\((\)\( 17 \nu^{10} + 59489 \nu^{6} + 53782272 \nu^{2} \)\()/ 112708800 \)
\(\beta_{3}\)\(=\)\((\)\( 523 \nu^{11} - 750 \nu^{9} + 1885411 \nu^{7} + 690450 \nu^{5} + 1667910888 \nu^{3} + 3498055200 \nu \)\()/ 5071896000 \)
\(\beta_{4}\)\(=\)\((\)\( -523 \nu^{11} - 750 \nu^{9} - 1885411 \nu^{7} + 690450 \nu^{5} - 1667910888 \nu^{3} + 3498055200 \nu \)\()/ 5071896000 \)
\(\beta_{5}\)\(=\)\((\)\( -689 \nu^{10} + 1500 \nu^{8} - 2521553 \nu^{6} - 1380900 \nu^{4} - 2206394064 \nu^{2} - 4629225600 \)\()/33812640\)
\(\beta_{6}\)\(=\)\((\)\( -689 \nu^{10} - 1500 \nu^{8} - 2521553 \nu^{6} + 1380900 \nu^{4} - 2206394064 \nu^{2} + 4629225600 \)\()/33812640\)
\(\beta_{7}\)\(=\)\((\)\( -89 \nu^{10} + 1398 \nu^{8} - 256193 \nu^{6} + 2770518 \nu^{4} - 166812984 \nu^{2} + 361173600 \)\()/3381264\)
\(\beta_{8}\)\(=\)\((\)\( -89 \nu^{10} - 1398 \nu^{8} - 256193 \nu^{6} - 2770518 \nu^{4} - 166812984 \nu^{2} - 361173600 \)\()/3381264\)
\(\beta_{9}\)\(=\)\((\)\( 443 \nu^{11} + 1536401 \nu^{7} + 1398173958 \nu^{3} \)\()/ 126797400 \)
\(\beta_{10}\)\(=\)\((\)\( 16159 \nu^{11} - 56400 \nu^{9} + 54778063 \nu^{7} - 150954000 \nu^{5} + 45623651904 \nu^{3} - 96239361600 \nu \)\()/ 1014379200 \)
\(\beta_{11}\)\(=\)\((\)\( -16159 \nu^{11} - 56400 \nu^{9} - 54778063 \nu^{7} - 150954000 \nu^{5} - 45623651904 \nu^{3} - 96239361600 \nu \)\()/ 1014379200 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(5 \beta_{4} + 5 \beta_{3} + \beta_{1}\)\()/30\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{8} + \beta_{7} + 5 \beta_{6} + 5 \beta_{5} + 1700 \beta_{2}\)\()/60\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{11} - 3 \beta_{10} + 43 \beta_{9} + 265 \beta_{4} - 265 \beta_{3}\)\()/30\)
\(\nu^{4}\)\(=\)\((\)\(-25 \beta_{8} + 25 \beta_{7} + 233 \beta_{6} - 233 \beta_{5} - 69140\)\()/60\)
\(\nu^{5}\)\(=\)\((\)\(-75 \beta_{11} - 75 \beta_{10} + 19345 \beta_{4} + 19345 \beta_{3} - 1771 \beta_{1}\)\()/30\)
\(\nu^{6}\)\(=\)\((\)\(-241 \beta_{8} - 241 \beta_{7} - 10385 \beta_{6} - 10385 \beta_{5} - 2890100 \beta_{2}\)\()/60\)
\(\nu^{7}\)\(=\)\((\)\(-723 \beta_{11} + 723 \beta_{10} - 73123 \beta_{9} - 1127065 \beta_{4} + 1127065 \beta_{3}\)\()/30\)
\(\nu^{8}\)\(=\)\((\)\(-23015 \beta_{8} + 23015 \beta_{7} - 461753 \beta_{6} + 461753 \beta_{5} + 121518740\)\()/60\)
\(\nu^{9}\)\(=\)\((\)\(-69045 \beta_{11} - 69045 \beta_{10} - 60308545 \beta_{4} - 60308545 \beta_{3} + 3033691 \beta_{1}\)\()/30\)
\(\nu^{10}\)\(=\)\((\)\(-2320319 \beta_{8} - 2320319 \beta_{7} + 20522465 \beta_{6} + 20522465 \beta_{5} + 5133048500 \beta_{2}\)\()/60\)
\(\nu^{11}\)\(=\)\((\)\(-6960957 \beta_{11} + 6960957 \beta_{10} + 126475603 \beta_{9} + 3072477865 \beta_{4} - 3072477865 \beta_{3}\)\()/30\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−1.02615 + 1.02615i
−4.38999 + 4.38999i
4.70903 4.70903i
1.02615 1.02615i
4.38999 4.38999i
−4.70903 + 4.70903i
−1.02615 1.02615i
−4.38999 4.38999i
4.70903 + 4.70903i
1.02615 + 1.02615i
4.38999 + 4.38999i
−4.70903 4.70903i
−2.82843 + 2.82843i 0 16.0000i 0 0 −148.726 148.726i 45.2548 + 45.2548i 0 0
107.2 −2.82843 + 2.82843i 0 16.0000i 0 0 9.67825 + 9.67825i 45.2548 + 45.2548i 0 0
107.3 −2.82843 + 2.82843i 0 16.0000i 0 0 103.048 + 103.048i 45.2548 + 45.2548i 0 0
107.4 2.82843 2.82843i 0 16.0000i 0 0 −148.726 148.726i −45.2548 45.2548i 0 0
107.5 2.82843 2.82843i 0 16.0000i 0 0 9.67825 + 9.67825i −45.2548 45.2548i 0 0
107.6 2.82843 2.82843i 0 16.0000i 0 0 103.048 + 103.048i −45.2548 45.2548i 0 0
143.1 −2.82843 2.82843i 0 16.0000i 0 0 −148.726 + 148.726i 45.2548 45.2548i 0 0
143.2 −2.82843 2.82843i 0 16.0000i 0 0 9.67825 9.67825i 45.2548 45.2548i 0 0
143.3 −2.82843 2.82843i 0 16.0000i 0 0 103.048 103.048i 45.2548 45.2548i 0 0
143.4 2.82843 + 2.82843i 0 16.0000i 0 0 −148.726 + 148.726i −45.2548 + 45.2548i 0 0
143.5 2.82843 + 2.82843i 0 16.0000i 0 0 9.67825 9.67825i −45.2548 + 45.2548i 0 0
143.6 2.82843 + 2.82843i 0 16.0000i 0 0 103.048 103.048i −45.2548 + 45.2548i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.6.f.e 12
3.b odd 2 1 inner 450.6.f.e 12
5.b even 2 1 90.6.f.c 12
5.c odd 4 1 90.6.f.c 12
5.c odd 4 1 inner 450.6.f.e 12
15.d odd 2 1 90.6.f.c 12
15.e even 4 1 90.6.f.c 12
15.e even 4 1 inner 450.6.f.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.6.f.c 12 5.b even 2 1
90.6.f.c 12 5.c odd 4 1
90.6.f.c 12 15.d odd 2 1
90.6.f.c 12 15.e even 4 1
450.6.f.e 12 1.a even 1 1 trivial
450.6.f.e 12 3.b odd 2 1 inner
450.6.f.e 12 5.c odd 4 1 inner
450.6.f.e 12 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 72 T_{7}^{5} + 2592 T_{7}^{4} - 2863904 T_{7}^{3} + 994519296 T_{7}^{2} - 18710687232 T_{7} + 176009564672 \) acting on \(S_{6}^{\mathrm{new}}(450, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 256 + T^{4} )^{3} \)
$3$ \( T^{12} \)
$5$ \( T^{12} \)
$7$ \( ( 176009564672 - 18710687232 T + 994519296 T^{2} - 2863904 T^{3} + 2592 T^{4} + 72 T^{5} + T^{6} )^{2} \)
$11$ \( ( 11425693455114752 + 197714929152 T^{2} + 873984 T^{4} + T^{6} )^{2} \)
$13$ \( ( 858377073723912 - 6431592483432 T + 24095111076 T^{2} - 20012544 T^{3} + 9522 T^{4} - 138 T^{5} + T^{6} )^{2} \)
$17$ \( \)\(12\!\cdots\!56\)\( + \)\(32\!\cdots\!88\)\( T^{4} + 36394503656208 T^{8} + T^{12} \)
$19$ \( ( 3598103289892864 + 290733646848 T^{2} + 3150912 T^{4} + T^{6} )^{2} \)
$23$ \( \)\(33\!\cdots\!96\)\( + \)\(12\!\cdots\!68\)\( T^{4} + 817412062990848 T^{8} + T^{12} \)
$29$ \( ( -\)\(14\!\cdots\!08\)\( + 2757815228592012 T^{2} - 99286806 T^{4} + T^{6} )^{2} \)
$31$ \( ( -24637853824 + 48854928 T + 14628 T^{2} + T^{3} )^{4} \)
$37$ \( ( \)\(26\!\cdots\!32\)\( + 43305453681533263368 T + 3573402827978916 T^{2} - 45620864064 T^{3} + 82972962 T^{4} + 12882 T^{5} + T^{6} )^{2} \)
$41$ \( ( 42758891860940388872 + 60296985035532 T^{2} + 19723014 T^{4} + T^{6} )^{2} \)
$43$ \( ( \)\(23\!\cdots\!00\)\( + 10102393239552000000 T + 2144430864000000 T^{2} - 154159776000 T^{3} + 32320800 T^{4} + 8040 T^{5} + T^{6} )^{2} \)
$47$ \( \)\(66\!\cdots\!00\)\( + \)\(63\!\cdots\!00\)\( T^{4} + 167318145434880000 T^{8} + T^{12} \)
$53$ \( \)\(19\!\cdots\!16\)\( + \)\(11\!\cdots\!08\)\( T^{4} + 349750677852599568 T^{8} + T^{12} \)
$59$ \( ( -\)\(61\!\cdots\!12\)\( + 687297345346728192 T^{2} - 2091673824 T^{4} + T^{6} )^{2} \)
$61$ \( ( -27672667200000 - 753759600 T + 36300 T^{2} + T^{3} )^{4} \)
$67$ \( ( \)\(44\!\cdots\!68\)\( + \)\(25\!\cdots\!24\)\( T + 7417011616032162816 T^{2} + 49084745335552 T^{3} + 140717088 T^{4} + 16776 T^{5} + T^{6} )^{2} \)
$71$ \( ( \)\(14\!\cdots\!28\)\( + 409117028199877632 T^{2} + 1451005536 T^{4} + T^{6} )^{2} \)
$73$ \( ( \)\(24\!\cdots\!08\)\( + \)\(18\!\cdots\!24\)\( T + 6861886046773797636 T^{2} + 215200411183232 T^{3} + 3159648018 T^{4} - 79494 T^{5} + T^{6} )^{2} \)
$79$ \( ( \)\(40\!\cdots\!44\)\( + 8973277790695596288 T^{2} + 5549266992 T^{4} + T^{6} )^{2} \)
$83$ \( \)\(12\!\cdots\!56\)\( + \)\(28\!\cdots\!88\)\( T^{4} + 10139988275384260608 T^{8} + T^{12} \)
$89$ \( ( -\)\(34\!\cdots\!52\)\( + \)\(18\!\cdots\!52\)\( T^{2} - 28444006134 T^{4} + T^{6} )^{2} \)
$97$ \( ( \)\(20\!\cdots\!92\)\( + \)\(39\!\cdots\!68\)\( T + 38965876000676906436 T^{2} - 2606705752132224 T^{3} + 49788425682 T^{4} - 315558 T^{5} + T^{6} )^{2} \)
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