Properties

Label 450.2.e.n
Level $450$
Weight $2$
Character orbit 450.e
Analytic conductor $3.593$
Analytic rank $0$
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.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{3} + ( -1 + \beta_{2} ) q^{4} + ( 1 - \beta_{1} - \beta_{2} ) q^{6} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{7} - q^{8} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{3} + ( -1 + \beta_{2} ) q^{4} + ( 1 - \beta_{1} - \beta_{2} ) q^{6} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{7} - q^{8} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{9} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + ( 1 - \beta_{3} ) q^{12} + ( -\beta_{1} + 2 \beta_{3} ) q^{13} + ( 2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{14} -\beta_{2} q^{16} + ( -1 - 4 \beta_{1} + 2 \beta_{3} ) q^{17} + ( 1 + 2 \beta_{3} ) q^{18} + ( -3 - 2 \beta_{1} + \beta_{3} ) q^{19} + ( -4 + 3 \beta_{1} - 2 \beta_{3} ) q^{21} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{22} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{23} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{24} + ( -2 \beta_{1} + \beta_{3} ) q^{26} + ( -5 - \beta_{3} ) q^{27} + ( 2 - 2 \beta_{1} + \beta_{3} ) q^{28} + 6 \beta_{2} q^{29} + ( -4 + \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{31} + ( 1 - \beta_{2} ) q^{32} + ( -1 - 3 \beta_{2} - 2 \beta_{3} ) q^{33} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{34} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{36} + 8 q^{37} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{38} + ( 2 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{39} + ( 1 - \beta_{2} ) q^{41} + ( 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{42} + ( \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{43} + ( -1 - 2 \beta_{1} + \beta_{3} ) q^{44} + ( -2 - 4 \beta_{1} + 2 \beta_{3} ) q^{46} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{47} + ( -1 + \beta_{1} + \beta_{2} ) q^{48} + ( -3 + 4 \beta_{1} + 3 \beta_{2} - 8 \beta_{3} ) q^{49} + ( 8 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{51} + ( -\beta_{1} - \beta_{3} ) q^{52} + ( 6 - 2 \beta_{1} + \beta_{3} ) q^{53} + ( \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{54} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{56} + ( 4 + 4 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{57} + ( -6 + 6 \beta_{2} ) q^{58} + ( -1 - 5 \beta_{1} + \beta_{2} + 10 \beta_{3} ) q^{59} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{61} + ( -4 + 2 \beta_{1} - \beta_{3} ) q^{62} + ( -6 + 2 \beta_{1} + 8 \beta_{2} - 5 \beta_{3} ) q^{63} + q^{64} + ( 3 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{66} + ( -7 + \beta_{1} + 7 \beta_{2} - 2 \beta_{3} ) q^{67} + ( 1 + 2 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{68} + ( 6 + 4 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{69} + ( -2 \beta_{1} + \beta_{3} ) q^{71} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{72} + ( 5 + 8 \beta_{1} - 4 \beta_{3} ) q^{73} + 8 \beta_{2} q^{74} + ( 3 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{76} + ( -4 + \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{77} + ( 4 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{78} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{79} + ( 4 \beta_{1} + 7 \beta_{2} - 4 \beta_{3} ) q^{81} + q^{82} + 4 \beta_{2} q^{83} + ( 4 - \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{84} + ( 5 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{86} + ( 6 - 6 \beta_{1} - 6 \beta_{2} ) q^{87} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{88} + ( 8 - 4 \beta_{1} + 2 \beta_{3} ) q^{89} + ( -6 + 4 \beta_{1} - 2 \beta_{3} ) q^{91} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{92} + ( 2 - 2 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{93} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{94} + ( -1 + \beta_{3} ) q^{96} -13 \beta_{2} q^{97} + ( -3 + 8 \beta_{1} - 4 \beta_{3} ) q^{98} + ( -3 + 2 \beta_{1} + 8 \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} - 2q^{3} - 2q^{4} + 2q^{6} - 4q^{7} - 4q^{8} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{2} - 2q^{3} - 2q^{4} + 2q^{6} - 4q^{7} - 4q^{8} + 2q^{9} + 2q^{11} + 4q^{12} + 4q^{14} - 2q^{16} - 4q^{17} + 4q^{18} - 12q^{19} - 16q^{21} - 2q^{22} - 4q^{23} + 2q^{24} - 20q^{27} + 8q^{28} + 12q^{29} - 8q^{31} + 2q^{32} - 10q^{33} - 2q^{34} + 2q^{36} + 32q^{37} - 6q^{38} + 2q^{41} - 8q^{42} - 10q^{43} - 4q^{44} - 8q^{46} + 4q^{47} - 2q^{48} - 6q^{49} + 26q^{51} + 24q^{53} - 10q^{54} + 4q^{56} + 18q^{57} - 12q^{58} - 2q^{59} - 4q^{61} - 16q^{62} - 8q^{63} + 4q^{64} + 4q^{66} - 14q^{67} + 2q^{68} + 8q^{69} - 2q^{72} + 20q^{73} + 16q^{74} + 6q^{76} - 8q^{77} + 12q^{78} + 14q^{81} + 4q^{82} + 8q^{83} + 8q^{84} + 10q^{86} + 12q^{87} - 2q^{88} + 32q^{89} - 24q^{91} - 4q^{92} + 16q^{93} - 4q^{94} - 4q^{96} - 26q^{97} - 12q^{98} + 4q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i
0.500000 0.866025i −1.72474 + 0.158919i −0.500000 0.866025i 0 −0.724745 + 1.57313i 0.224745 0.389270i −1.00000 2.94949 0.548188i 0
151.2 0.500000 0.866025i 0.724745 + 1.57313i −0.500000 0.866025i 0 1.72474 + 0.158919i −2.22474 + 3.85337i −1.00000 −1.94949 + 2.28024i 0
301.1 0.500000 + 0.866025i −1.72474 0.158919i −0.500000 + 0.866025i 0 −0.724745 1.57313i 0.224745 + 0.389270i −1.00000 2.94949 + 0.548188i 0
301.2 0.500000 + 0.866025i 0.724745 1.57313i −0.500000 + 0.866025i 0 1.72474 0.158919i −2.22474 3.85337i −1.00000 −1.94949 2.28024i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.e.n 4
3.b odd 2 1 1350.2.e.j 4
5.b even 2 1 450.2.e.k 4
5.c odd 4 2 90.2.i.b 8
9.c even 3 1 inner 450.2.e.n 4
9.c even 3 1 4050.2.a.bq 2
9.d odd 6 1 1350.2.e.j 4
9.d odd 6 1 4050.2.a.bz 2
15.d odd 2 1 1350.2.e.m 4
15.e even 4 2 270.2.i.b 8
20.e even 4 2 720.2.by.c 8
45.h odd 6 1 1350.2.e.m 4
45.h odd 6 1 4050.2.a.bm 2
45.j even 6 1 450.2.e.k 4
45.j even 6 1 4050.2.a.bs 2
45.k odd 12 2 90.2.i.b 8
45.k odd 12 2 810.2.c.f 4
45.l even 12 2 270.2.i.b 8
45.l even 12 2 810.2.c.e 4
60.l odd 4 2 2160.2.by.d 8
180.v odd 12 2 2160.2.by.d 8
180.x even 12 2 720.2.by.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.i.b 8 5.c odd 4 2
90.2.i.b 8 45.k odd 12 2
270.2.i.b 8 15.e even 4 2
270.2.i.b 8 45.l even 12 2
450.2.e.k 4 5.b even 2 1
450.2.e.k 4 45.j even 6 1
450.2.e.n 4 1.a even 1 1 trivial
450.2.e.n 4 9.c even 3 1 inner
720.2.by.c 8 20.e even 4 2
720.2.by.c 8 180.x even 12 2
810.2.c.e 4 45.l even 12 2
810.2.c.f 4 45.k odd 12 2
1350.2.e.j 4 3.b odd 2 1
1350.2.e.j 4 9.d odd 6 1
1350.2.e.m 4 15.d odd 2 1
1350.2.e.m 4 45.h odd 6 1
2160.2.by.d 8 60.l odd 4 2
2160.2.by.d 8 180.v odd 12 2
4050.2.a.bm 2 45.h odd 6 1
4050.2.a.bq 2 9.c even 3 1
4050.2.a.bs 2 45.j even 6 1
4050.2.a.bz 2 9.d odd 6 1

Hecke kernels

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

\( T_{7}^{4} + 4 T_{7}^{3} + 18 T_{7}^{2} - 8 T_{7} + 4 \)
\( T_{11}^{4} - 2 T_{11}^{3} + 9 T_{11}^{2} + 10 T_{11} + 25 \)
\( T_{17}^{2} + 2 T_{17} - 23 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( 9 + 6 T + T^{2} + 2 T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 4 - 8 T + 18 T^{2} + 4 T^{3} + T^{4} \)
$11$ \( 25 + 10 T + 9 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( 36 + 6 T^{2} + T^{4} \)
$17$ \( ( -23 + 2 T + T^{2} )^{2} \)
$19$ \( ( 3 + 6 T + T^{2} )^{2} \)
$23$ \( 400 - 80 T + 36 T^{2} + 4 T^{3} + T^{4} \)
$29$ \( ( 36 - 6 T + T^{2} )^{2} \)
$31$ \( 100 + 80 T + 54 T^{2} + 8 T^{3} + T^{4} \)
$37$ \( ( -8 + T )^{4} \)
$41$ \( ( 1 - T + T^{2} )^{2} \)
$43$ \( 361 + 190 T + 81 T^{2} + 10 T^{3} + T^{4} \)
$47$ \( 4 + 8 T + 18 T^{2} - 4 T^{3} + T^{4} \)
$53$ \( ( 30 - 12 T + T^{2} )^{2} \)
$59$ \( 22201 - 298 T + 153 T^{2} + 2 T^{3} + T^{4} \)
$61$ \( 4 - 8 T + 18 T^{2} + 4 T^{3} + T^{4} \)
$67$ \( 1849 + 602 T + 153 T^{2} + 14 T^{3} + T^{4} \)
$71$ \( ( -6 + T^{2} )^{2} \)
$73$ \( ( -71 - 10 T + T^{2} )^{2} \)
$79$ \( 2916 + 54 T^{2} + T^{4} \)
$83$ \( ( 16 - 4 T + T^{2} )^{2} \)
$89$ \( ( 40 - 16 T + T^{2} )^{2} \)
$97$ \( ( 169 + 13 T + T^{2} )^{2} \)
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