Properties

Label 450.2.h.d
Level $450$
Weight $2$
Character orbit 450.h
Analytic conductor $3.593$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.1064390625.3
Defining polynomial: \(x^{8} - 2 x^{7} - 3 x^{6} - 5 x^{5} + 36 x^{4} - 35 x^{3} + 23 x^{2} - 171 x + 361\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} + ( -1 - \beta_{2} + \beta_{3} - \beta_{5} ) q^{4} + ( 1 + \beta_{2} + \beta_{4} ) q^{5} + ( -1 - \beta_{6} - \beta_{7} ) q^{7} + \beta_{5} q^{8} +O(q^{10})\) \( q -\beta_{3} q^{2} + ( -1 - \beta_{2} + \beta_{3} - \beta_{5} ) q^{4} + ( 1 + \beta_{2} + \beta_{4} ) q^{5} + ( -1 - \beta_{6} - \beta_{7} ) q^{7} + \beta_{5} q^{8} + ( 1 - \beta_{1} - \beta_{3} ) q^{10} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{11} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{13} + ( \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{14} + \beta_{2} q^{16} + ( -\beta_{1} - \beta_{2} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{17} + ( 2 - 2 \beta_{3} + 2 \beta_{5} ) q^{19} + ( -1 - \beta_{2} - \beta_{5} - \beta_{7} ) q^{20} + ( 1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{22} + ( -4 \beta_{2} + 2 \beta_{3} - 4 \beta_{5} ) q^{23} + ( 2 - 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{25} + ( -2 + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{26} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{28} + ( 5 + 5 \beta_{2} - 3 \beta_{3} - \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{29} + ( 1 - 2 \beta_{3} - \beta_{4} - \beta_{7} ) q^{31} + q^{32} + ( -1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{6} + \beta_{7} ) q^{34} + ( -\beta_{1} - \beta_{2} + 3 \beta_{3} + 4 \beta_{5} - \beta_{6} - \beta_{7} ) q^{35} + ( 5 + 2 \beta_{1} + 7 \beta_{2} + 6 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{37} + ( -2 - 2 \beta_{5} ) q^{38} + ( -1 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{40} + ( -3 + 2 \beta_{1} - 6 \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{41} + ( -4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{43} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{44} + ( -2 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{46} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{47} + ( 4 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{49} + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{50} + ( \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{52} + ( 4 + 3 \beta_{1} + 4 \beta_{2} - 7 \beta_{3} - \beta_{4} + 8 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{53} + ( -7 - 7 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} + 2 \beta_{6} ) q^{55} + ( 1 + \beta_{4} + \beta_{7} ) q^{56} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{58} + ( 2 + 2 \beta_{1} - 5 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} + \beta_{7} ) q^{59} + ( -\beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{61} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{62} -\beta_{3} q^{64} + ( -7 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} - \beta_{6} ) q^{65} + ( -4 + 2 \beta_{3} - 2 \beta_{4} - 8 \beta_{5} - 2 \beta_{7} ) q^{67} + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{68} + ( 2 + \beta_{1} + 8 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} - \beta_{6} ) q^{70} + ( 2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} ) q^{71} + ( 2 - 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} + \beta_{6} + \beta_{7} ) q^{73} + ( 7 - \beta_{1} + 5 \beta_{2} - 6 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{74} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{76} + ( -1 + \beta_{1} + 10 \beta_{2} + 3 \beta_{3} + 9 \beta_{5} - \beta_{7} ) q^{77} + ( -5 - 5 \beta_{2} + 6 \beta_{3} - \beta_{4} - 5 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{79} + ( \beta_{3} + \beta_{6} ) q^{80} + ( -6 - 3 \beta_{2} + 3 \beta_{3} + \beta_{6} + \beta_{7} ) q^{82} + ( -3 - 2 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} - 8 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{83} + ( -5 + 9 \beta_{3} - 10 \beta_{5} - \beta_{6} ) q^{85} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{7} ) q^{86} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{88} + ( 1 + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{89} + ( 2 + 4 \beta_{1} + 12 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{91} + ( -4 + 4 \beta_{3} - 2 \beta_{5} ) q^{92} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{94} + ( 2 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{95} + ( 3 + \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{97} + ( 1 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{2} - 2q^{4} + 4q^{5} - 2q^{7} - 2q^{8} + O(q^{10}) \) \( 8q - 2q^{2} - 2q^{4} + 4q^{5} - 2q^{7} - 2q^{8} + 4q^{10} + 5q^{11} + 6q^{13} - 2q^{14} - 2q^{16} + 2q^{17} + 8q^{19} - q^{20} + 20q^{23} + 14q^{25} - 14q^{26} + 3q^{28} + 18q^{29} + 9q^{31} + 8q^{32} - 3q^{34} + 4q^{35} + 21q^{37} - 12q^{38} - 6q^{40} - 2q^{41} - 32q^{43} - 20q^{46} + 10q^{47} + 22q^{49} - 26q^{50} + 6q^{52} - 7q^{53} - 40q^{55} + 3q^{56} + 18q^{58} + 25q^{59} + 10q^{61} - 6q^{62} - 2q^{64} - 37q^{65} - 2q^{67} + 2q^{68} - 11q^{70} - 24q^{73} + 26q^{74} + 8q^{76} - 35q^{77} - 6q^{79} - q^{80} - 42q^{82} - 11q^{83} + q^{85} - 2q^{86} + 5q^{88} - 9q^{89} - 4q^{91} - 20q^{92} + 10q^{94} + 4q^{95} + q^{97} + 7q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 3 x^{6} - 5 x^{5} + 36 x^{4} - 35 x^{3} + 23 x^{2} - 171 x + 361\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -27571 \nu^{7} + 156 \nu^{6} - 50800 \nu^{5} + 197116 \nu^{4} - 261151 \nu^{3} + 1281772 \nu^{2} - 1105295 \nu + 4276691 \)\()/4238558\)
\(\beta_{3}\)\(=\)\((\)\( -45697 \nu^{7} - 18958 \nu^{6} + 87368 \nu^{5} + 389890 \nu^{4} - 351515 \nu^{3} + 1275844 \nu^{2} - 687257 \nu + 5880747 \)\()/4238558\)
\(\beta_{4}\)\(=\)\((\)\( 2894 \nu^{7} + 7027 \nu^{6} - 3119 \nu^{5} - 38495 \nu^{4} - 16673 \nu^{3} + 24798 \nu^{2} + 23050 \nu - 523849 \)\()/223082\)
\(\beta_{5}\)\(=\)\((\)\( 60697 \nu^{7} + 29865 \nu^{6} - 107003 \nu^{5} - 736723 \nu^{4} + 989612 \nu^{3} + 38090 \nu^{2} + 939005 \nu - 13281190 \)\()/4238558\)
\(\beta_{6}\)\(=\)\((\)\( -72436 \nu^{7} - 26109 \nu^{6} + 188067 \nu^{5} + 265983 \nu^{4} - 1082509 \nu^{3} + 501044 \nu^{2} + 3422970 \nu + 7026371 \)\()/4238558\)
\(\beta_{7}\)\(=\)\((\)\( 5808 \nu^{7} + 2617 \nu^{6} - 8495 \nu^{5} - 68083 \nu^{4} + 17029 \nu^{3} - 19146 \nu^{2} + 101760 \nu - 868243 \)\()/223082\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - \beta_{6} + 4 \beta_{3} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(-3 \beta_{7} + 5 \beta_{5} - \beta_{2} + 5\)
\(\nu^{4}\)\(=\)\(-8 \beta_{6} - 7 \beta_{5} + \beta_{4} + 7 \beta_{3} - 4 \beta_{2} + 8 \beta_{1} - 12\)
\(\nu^{5}\)\(=\)\(2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} + 39 \beta_{3} - 39 \beta_{2} - 4 \beta_{1} + 12\)
\(\nu^{6}\)\(=\)\(-39 \beta_{7} + 16 \beta_{5} + 39 \beta_{4} - 26 \beta_{3} + 6 \beta_{1} + 26\)
\(\nu^{7}\)\(=\)\(71 \beta_{7} - 100 \beta_{6} - 101 \beta_{5} + 164 \beta_{3} - 101 \beta_{2} + 71 \beta_{1}\)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2} + \beta_{3} - \beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
−1.86886 1.45788i
2.36886 0.0809628i
−0.815575 + 1.64827i
1.31557 1.28500i
−0.815575 1.64827i
1.31557 + 1.28500i
−1.86886 + 1.45788i
2.36886 + 0.0809628i
−0.809017 0.587785i 0 0.309017 + 0.951057i −2.17787 + 0.506822i 0 −2.31003 0.309017 0.951057i 0 2.05984 + 0.870094i
91.2 −0.809017 0.587785i 0 0.309017 + 0.951057i 2.05984 0.870094i 0 2.92807 0.309017 0.951057i 0 −2.17787 0.506822i
181.1 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −0.00655751 2.23606i 0 2.63925 −0.809017 0.587785i 0 2.12459 0.697217i
181.2 0.309017 + 0.951057i 0 −0.809017 + 0.587785i 2.12459 + 0.697217i 0 −4.25729 −0.809017 0.587785i 0 −0.00655751 + 2.23606i
271.1 0.309017 0.951057i 0 −0.809017 0.587785i −0.00655751 + 2.23606i 0 2.63925 −0.809017 + 0.587785i 0 2.12459 + 0.697217i
271.2 0.309017 0.951057i 0 −0.809017 0.587785i 2.12459 0.697217i 0 −4.25729 −0.809017 + 0.587785i 0 −0.00655751 2.23606i
361.1 −0.809017 + 0.587785i 0 0.309017 0.951057i −2.17787 0.506822i 0 −2.31003 0.309017 + 0.951057i 0 2.05984 0.870094i
361.2 −0.809017 + 0.587785i 0 0.309017 0.951057i 2.05984 + 0.870094i 0 2.92807 0.309017 + 0.951057i 0 −2.17787 + 0.506822i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.h.d 8
3.b odd 2 1 150.2.g.c 8
15.d odd 2 1 750.2.g.d 8
15.e even 4 2 750.2.h.e 16
25.d even 5 1 inner 450.2.h.d 8
75.h odd 10 1 750.2.g.d 8
75.h odd 10 1 3750.2.a.q 4
75.j odd 10 1 150.2.g.c 8
75.j odd 10 1 3750.2.a.l 4
75.l even 20 2 750.2.h.e 16
75.l even 20 2 3750.2.c.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.g.c 8 3.b odd 2 1
150.2.g.c 8 75.j odd 10 1
450.2.h.d 8 1.a even 1 1 trivial
450.2.h.d 8 25.d even 5 1 inner
750.2.g.d 8 15.d odd 2 1
750.2.g.d 8 75.h odd 10 1
750.2.h.e 16 15.e even 4 2
750.2.h.e 16 75.l even 20 2
3750.2.a.l 4 75.j odd 10 1
3750.2.a.q 4 75.h odd 10 1
3750.2.c.h 8 75.l even 20 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{4} + T_{7}^{3} - 19 T_{7}^{2} - 4 T_{7} + 76 \)
\(T_{11}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( 625 - 500 T + 25 T^{2} + 80 T^{3} - 39 T^{4} + 16 T^{5} + T^{6} - 4 T^{7} + T^{8} \)
$7$ \( ( 76 - 4 T - 19 T^{2} + T^{3} + T^{4} )^{2} \)
$11$ \( 400 - 3600 T + 12800 T^{2} - 6050 T^{3} + 1785 T^{4} - 320 T^{5} + 60 T^{6} - 5 T^{7} + T^{8} \)
$13$ \( 256 + 1216 T + 2112 T^{2} - 872 T^{3} + 705 T^{4} - 143 T^{5} + 32 T^{6} - 6 T^{7} + T^{8} \)
$17$ \( 5776 + 3648 T - 1072 T^{2} - 2054 T^{3} + 2055 T^{4} - 29 T^{5} + 78 T^{6} - 2 T^{7} + T^{8} \)
$19$ \( ( 16 - 24 T + 16 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$23$ \( ( 400 - 200 T + 60 T^{2} - 10 T^{3} + T^{4} )^{2} \)
$29$ \( 1860496 - 1148488 T + 374148 T^{2} - 78946 T^{3} + 13755 T^{4} - 1891 T^{5} + 218 T^{6} - 18 T^{7} + T^{8} \)
$31$ \( 1296 + 1944 T + 1512 T^{2} + 432 T^{3} + 225 T^{4} - 72 T^{5} + 42 T^{6} - 9 T^{7} + T^{8} \)
$37$ \( 11182336 + 1217216 T + 26592 T^{2} - 20272 T^{3} + 16305 T^{4} - 2758 T^{5} + 332 T^{6} - 21 T^{7} + T^{8} \)
$41$ \( 1936 + 18832 T + 69448 T^{2} - 8626 T^{3} + 2775 T^{4} - 611 T^{5} + 118 T^{6} + 2 T^{7} + T^{8} \)
$43$ \( ( -1424 - 784 T - 4 T^{2} + 16 T^{3} + T^{4} )^{2} \)
$47$ \( 774400 + 457600 T + 92800 T^{2} - 18400 T^{3} + 4560 T^{4} - 680 T^{5} + 160 T^{6} - 10 T^{7} + T^{8} \)
$53$ \( 85359121 + 16935087 T + 2102048 T^{2} + 83329 T^{3} + 7455 T^{4} - 191 T^{5} + 168 T^{6} + 7 T^{7} + T^{8} \)
$59$ \( 2310400 - 1368000 T + 822800 T^{2} - 197500 T^{3} + 28785 T^{4} - 3100 T^{5} + 330 T^{6} - 25 T^{7} + T^{8} \)
$61$ \( 400 - 2200 T + 4800 T^{2} - 1400 T^{3} + 785 T^{4} - 215 T^{5} + 60 T^{6} - 10 T^{7} + T^{8} \)
$67$ \( 6885376 + 3064832 T + 795648 T^{2} + 87424 T^{3} + 6800 T^{4} - 1376 T^{5} + 168 T^{6} + 2 T^{7} + T^{8} \)
$71$ \( 102400 + 179200 T + 729600 T^{2} + 92800 T^{3} + 14160 T^{4} + 40 T^{5} - 40 T^{6} + T^{8} \)
$73$ \( 1296 - 27864 T + 228852 T^{2} + 378 T^{3} + 16155 T^{4} + 3087 T^{5} + 372 T^{6} + 24 T^{7} + T^{8} \)
$79$ \( 331776 + 82944 T + 110592 T^{2} + 70632 T^{3} + 21825 T^{4} + 2943 T^{5} + 192 T^{6} + 6 T^{7} + T^{8} \)
$83$ \( 1008016 + 487944 T + 589772 T^{2} + 3722 T^{3} - 3195 T^{4} + 98 T^{5} + 162 T^{6} + 11 T^{7} + T^{8} \)
$89$ \( 1296 - 1944 T + 8532 T^{2} + 378 T^{3} + 1305 T^{4} - 288 T^{5} + 12 T^{6} + 9 T^{7} + T^{8} \)
$97$ \( 130321 + 72561 T + 36902 T^{2} + 18503 T^{3} + 15405 T^{4} - 2863 T^{5} + 222 T^{6} - T^{7} + T^{8} \)
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