Properties

Label 450.2.h.a
Level $450$
Weight $2$
Character orbit 450.h
Analytic conductor $3.593$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{3} q^{4} + ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{5} -3 q^{7} + \zeta_{10}^{2} q^{8} +O(q^{10})\) \( q + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{3} q^{4} + ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{5} -3 q^{7} + \zeta_{10}^{2} q^{8} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{10} + ( 1 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{11} -\zeta_{10} q^{13} + ( 3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{14} -\zeta_{10} q^{16} + ( -3 \zeta_{10} - 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{17} + ( -3 \zeta_{10} + 4 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{19} + ( 2 - \zeta_{10} + 2 \zeta_{10}^{2} ) q^{20} + ( 2 - 2 \zeta_{10} + \zeta_{10}^{3} ) q^{22} + ( -5 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{23} -5 \zeta_{10} q^{25} + q^{26} + 3 \zeta_{10}^{3} q^{28} + ( -4 + 4 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{29} + ( 2 \zeta_{10} + \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{31} + q^{32} + ( 3 + 3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{34} + ( 6 - 6 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{35} + ( -7 + 4 \zeta_{10} - 7 \zeta_{10}^{2} ) q^{37} + ( 3 - 4 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{38} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{40} + ( -4 - \zeta_{10} - 4 \zeta_{10}^{2} ) q^{41} + ( -3 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{43} + ( 2 \zeta_{10} - 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{44} + ( 2 - 2 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{46} + ( -8 + 8 \zeta_{10} + \zeta_{10}^{3} ) q^{47} + 2 q^{49} + 5 q^{50} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{52} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{53} + ( 4 \zeta_{10} - 7 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{55} -3 \zeta_{10}^{2} q^{56} + ( -4 \zeta_{10} + 7 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{58} + ( 4 - 2 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{59} + ( 4 - 7 \zeta_{10} + 7 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{61} + ( -2 - \zeta_{10} - 2 \zeta_{10}^{2} ) q^{62} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{64} + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{65} + ( 2 \zeta_{10} - 9 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{67} + ( -6 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{68} + ( 6 \zeta_{10} - 3 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{70} -3 \zeta_{10}^{3} q^{71} + ( 2 + 4 \zeta_{10} - 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{73} + ( 3 + 7 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{74} + ( 1 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{76} + ( -3 + 9 \zeta_{10} - 9 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{77} + ( 2 - 2 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{79} + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{80} + ( 5 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{82} + ( 4 \zeta_{10} + 7 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{83} + ( 12 + 9 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{85} + ( 3 - 5 \zeta_{10} + 5 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{86} + ( -2 + 3 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{88} + ( -2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{89} + 3 \zeta_{10} q^{91} + ( 2 \zeta_{10} + 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{92} + ( -8 \zeta_{10} + 7 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{94} + ( 5 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{95} + ( -9 + 9 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{97} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{2} - q^{4} - 5q^{5} - 12q^{7} - q^{8} + O(q^{10}) \) \( 4q - q^{2} - q^{4} - 5q^{5} - 12q^{7} - q^{8} - 5q^{10} - 3q^{11} - q^{13} + 3q^{14} - q^{16} - 3q^{17} - 10q^{19} + 5q^{20} + 7q^{22} - 9q^{23} - 5q^{25} + 4q^{26} + 3q^{28} - 15q^{29} + 3q^{31} + 4q^{32} + 12q^{34} + 15q^{35} - 17q^{37} + 5q^{38} - 13q^{41} - 16q^{43} + 7q^{44} + q^{46} - 23q^{47} + 8q^{49} + 20q^{50} - q^{52} + 16q^{53} + 15q^{55} + 3q^{56} - 15q^{58} + 10q^{59} - 2q^{61} - 7q^{62} - q^{64} + 5q^{65} + 13q^{67} - 18q^{68} + 15q^{70} - 3q^{71} + 14q^{73} - 2q^{74} + 10q^{76} + 9q^{77} + 10q^{79} + 5q^{80} + 12q^{82} + q^{83} + 30q^{85} - q^{86} - 3q^{88} - 10q^{89} + 3q^{91} + q^{92} - 23q^{94} + 30q^{95} - 22q^{97} - 2q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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
0.809017 0.587785i
−0.309017 + 0.951057i
−0.309017 0.951057i
0.809017 + 0.587785i
−0.809017 0.587785i 0 0.309017 + 0.951057i −0.690983 2.12663i 0 −3.00000 0.309017 0.951057i 0 −0.690983 + 2.12663i
181.1 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −1.80902 + 1.31433i 0 −3.00000 −0.809017 0.587785i 0 −1.80902 1.31433i
271.1 0.309017 0.951057i 0 −0.809017 0.587785i −1.80902 1.31433i 0 −3.00000 −0.809017 + 0.587785i 0 −1.80902 + 1.31433i
361.1 −0.809017 + 0.587785i 0 0.309017 0.951057i −0.690983 + 2.12663i 0 −3.00000 0.309017 + 0.951057i 0 −0.690983 2.12663i
\(n\): e.g. 2-40 or 990-1000
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.a 4
3.b odd 2 1 50.2.d.a 4
12.b even 2 1 400.2.u.c 4
15.d odd 2 1 250.2.d.a 4
15.e even 4 2 250.2.e.b 8
25.d even 5 1 inner 450.2.h.a 4
75.h odd 10 1 250.2.d.a 4
75.h odd 10 1 1250.2.a.d 2
75.j odd 10 1 50.2.d.a 4
75.j odd 10 1 1250.2.a.a 2
75.l even 20 2 250.2.e.b 8
75.l even 20 2 1250.2.b.b 4
300.n even 10 1 400.2.u.c 4
300.n even 10 1 10000.2.a.n 2
300.r even 10 1 10000.2.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.a 4 3.b odd 2 1
50.2.d.a 4 75.j odd 10 1
250.2.d.a 4 15.d odd 2 1
250.2.d.a 4 75.h odd 10 1
250.2.e.b 8 15.e even 4 2
250.2.e.b 8 75.l even 20 2
400.2.u.c 4 12.b even 2 1
400.2.u.c 4 300.n even 10 1
450.2.h.a 4 1.a even 1 1 trivial
450.2.h.a 4 25.d even 5 1 inner
1250.2.a.a 2 75.j odd 10 1
1250.2.a.d 2 75.h odd 10 1
1250.2.b.b 4 75.l even 20 2
10000.2.a.a 2 300.r even 10 1
10000.2.a.n 2 300.n even 10 1

Hecke kernels

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

\( T_{7} + 3 \)
\( T_{11}^{4} + 3 T_{11}^{3} + 19 T_{11}^{2} + 7 T_{11} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 25 + 25 T + 15 T^{2} + 5 T^{3} + T^{4} \)
$7$ \( ( 3 + T )^{4} \)
$11$ \( 1 + 7 T + 19 T^{2} + 3 T^{3} + T^{4} \)
$13$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$17$ \( 81 - 108 T + 54 T^{2} + 3 T^{3} + T^{4} \)
$19$ \( 25 + 25 T + 40 T^{2} + 10 T^{3} + T^{4} \)
$23$ \( 121 - 11 T + 31 T^{2} + 9 T^{3} + T^{4} \)
$29$ \( 25 - 25 T + 85 T^{2} + 15 T^{3} + T^{4} \)
$31$ \( 1 - 7 T + 19 T^{2} - 3 T^{3} + T^{4} \)
$37$ \( 3721 + 1098 T + 184 T^{2} + 17 T^{3} + T^{4} \)
$41$ \( 121 + 77 T + 69 T^{2} + 13 T^{3} + T^{4} \)
$43$ \( ( 11 + 8 T + T^{2} )^{2} \)
$47$ \( 5041 + 1207 T + 249 T^{2} + 23 T^{3} + T^{4} \)
$53$ \( 256 + 64 T + 96 T^{2} - 16 T^{3} + T^{4} \)
$59$ \( 400 - 200 T + 60 T^{2} - 10 T^{3} + T^{4} \)
$61$ \( 361 - 247 T + 64 T^{2} + 2 T^{3} + T^{4} \)
$67$ \( 3481 - 177 T + 79 T^{2} - 13 T^{3} + T^{4} \)
$71$ \( 81 + 27 T + 9 T^{2} + 3 T^{3} + T^{4} \)
$73$ \( 1936 - 704 T + 136 T^{2} - 14 T^{3} + T^{4} \)
$79$ \( 400 + 40 T^{2} - 10 T^{3} + T^{4} \)
$83$ \( 3721 + 1159 T + 141 T^{2} - T^{3} + T^{4} \)
$89$ \( 400 + 200 T + 60 T^{2} + 10 T^{3} + T^{4} \)
$97$ \( 10201 + 2323 T + 304 T^{2} + 22 T^{3} + T^{4} \)
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