# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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