Properties

Label 450.2.j.e
Level $450$
Weight $2$
Character orbit 450.j
Analytic conductor $3.593$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i −0.866025 1.50000i 0.500000 0.866025i 0 1.50000 + 0.866025i −1.73205 + 1.00000i 1.00000i −1.50000 + 2.59808i 0
49.2 0.866025 0.500000i 0.866025 + 1.50000i 0.500000 0.866025i 0 1.50000 + 0.866025i 1.73205 1.00000i 1.00000i −1.50000 + 2.59808i 0
349.1 −0.866025 0.500000i −0.866025 + 1.50000i 0.500000 + 0.866025i 0 1.50000 0.866025i −1.73205 1.00000i 1.00000i −1.50000 2.59808i 0
349.2 0.866025 + 0.500000i 0.866025 1.50000i 0.500000 + 0.866025i 0 1.50000 0.866025i 1.73205 + 1.00000i 1.00000i −1.50000 2.59808i 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
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.j.e 4
3.b odd 2 1 1350.2.j.a 4
5.b even 2 1 inner 450.2.j.e 4
5.c odd 4 1 18.2.c.a 2
5.c odd 4 1 450.2.e.i 2
9.c even 3 1 inner 450.2.j.e 4
9.c even 3 1 4050.2.c.c 2
9.d odd 6 1 1350.2.j.a 4
9.d odd 6 1 4050.2.c.r 2
15.d odd 2 1 1350.2.j.a 4
15.e even 4 1 54.2.c.a 2
15.e even 4 1 1350.2.e.c 2
20.e even 4 1 144.2.i.c 2
35.f even 4 1 882.2.f.d 2
35.k even 12 1 882.2.e.g 2
35.k even 12 1 882.2.h.b 2
35.l odd 12 1 882.2.e.i 2
35.l odd 12 1 882.2.h.c 2
40.i odd 4 1 576.2.i.g 2
40.k even 4 1 576.2.i.a 2
45.h odd 6 1 1350.2.j.a 4
45.h odd 6 1 4050.2.c.r 2
45.j even 6 1 inner 450.2.j.e 4
45.j even 6 1 4050.2.c.c 2
45.k odd 12 1 18.2.c.a 2
45.k odd 12 1 162.2.a.c 1
45.k odd 12 1 450.2.e.i 2
45.k odd 12 1 4050.2.a.c 1
45.l even 12 1 54.2.c.a 2
45.l even 12 1 162.2.a.b 1
45.l even 12 1 1350.2.e.c 2
45.l even 12 1 4050.2.a.v 1
60.l odd 4 1 432.2.i.b 2
105.k odd 4 1 2646.2.f.g 2
105.w odd 12 1 2646.2.e.c 2
105.w odd 12 1 2646.2.h.i 2
105.x even 12 1 2646.2.e.b 2
105.x even 12 1 2646.2.h.h 2
120.q odd 4 1 1728.2.i.f 2
120.w even 4 1 1728.2.i.e 2
180.v odd 12 1 432.2.i.b 2
180.v odd 12 1 1296.2.a.f 1
180.x even 12 1 144.2.i.c 2
180.x even 12 1 1296.2.a.g 1
315.bs even 12 1 882.2.h.b 2
315.bt odd 12 1 882.2.h.c 2
315.bu odd 12 1 2646.2.h.i 2
315.bv even 12 1 2646.2.h.h 2
315.bw odd 12 1 2646.2.e.c 2
315.bx even 12 1 2646.2.e.b 2
315.cb even 12 1 882.2.f.d 2
315.cb even 12 1 7938.2.a.x 1
315.cf odd 12 1 2646.2.f.g 2
315.cf odd 12 1 7938.2.a.i 1
315.cg even 12 1 882.2.e.g 2
315.ch odd 12 1 882.2.e.i 2
360.bo even 12 1 576.2.i.a 2
360.bo even 12 1 5184.2.a.o 1
360.br even 12 1 1728.2.i.e 2
360.br even 12 1 5184.2.a.q 1
360.bt odd 12 1 1728.2.i.f 2
360.bt odd 12 1 5184.2.a.p 1
360.bu odd 12 1 576.2.i.g 2
360.bu odd 12 1 5184.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a 2 5.c odd 4 1
18.2.c.a 2 45.k odd 12 1
54.2.c.a 2 15.e even 4 1
54.2.c.a 2 45.l even 12 1
144.2.i.c 2 20.e even 4 1
144.2.i.c 2 180.x even 12 1
162.2.a.b 1 45.l even 12 1
162.2.a.c 1 45.k odd 12 1
432.2.i.b 2 60.l odd 4 1
432.2.i.b 2 180.v odd 12 1
450.2.e.i 2 5.c odd 4 1
450.2.e.i 2 45.k odd 12 1
450.2.j.e 4 1.a even 1 1 trivial
450.2.j.e 4 5.b even 2 1 inner
450.2.j.e 4 9.c even 3 1 inner
450.2.j.e 4 45.j even 6 1 inner
576.2.i.a 2 40.k even 4 1
576.2.i.a 2 360.bo even 12 1
576.2.i.g 2 40.i odd 4 1
576.2.i.g 2 360.bu odd 12 1
882.2.e.g 2 35.k even 12 1
882.2.e.g 2 315.cg even 12 1
882.2.e.i 2 35.l odd 12 1
882.2.e.i 2 315.ch odd 12 1
882.2.f.d 2 35.f even 4 1
882.2.f.d 2 315.cb even 12 1
882.2.h.b 2 35.k even 12 1
882.2.h.b 2 315.bs even 12 1
882.2.h.c 2 35.l odd 12 1
882.2.h.c 2 315.bt odd 12 1
1296.2.a.f 1 180.v odd 12 1
1296.2.a.g 1 180.x even 12 1
1350.2.e.c 2 15.e even 4 1
1350.2.e.c 2 45.l even 12 1
1350.2.j.a 4 3.b odd 2 1
1350.2.j.a 4 9.d odd 6 1
1350.2.j.a 4 15.d odd 2 1
1350.2.j.a 4 45.h odd 6 1
1728.2.i.e 2 120.w even 4 1
1728.2.i.e 2 360.br even 12 1
1728.2.i.f 2 120.q odd 4 1
1728.2.i.f 2 360.bt odd 12 1
2646.2.e.b 2 105.x even 12 1
2646.2.e.b 2 315.bx even 12 1
2646.2.e.c 2 105.w odd 12 1
2646.2.e.c 2 315.bw odd 12 1
2646.2.f.g 2 105.k odd 4 1
2646.2.f.g 2 315.cf odd 12 1
2646.2.h.h 2 105.x even 12 1
2646.2.h.h 2 315.bv even 12 1
2646.2.h.i 2 105.w odd 12 1
2646.2.h.i 2 315.bu odd 12 1
4050.2.a.c 1 45.k odd 12 1
4050.2.a.v 1 45.l even 12 1
4050.2.c.c 2 9.c even 3 1
4050.2.c.c 2 45.j even 6 1
4050.2.c.r 2 9.d odd 6 1
4050.2.c.r 2 45.h odd 6 1
5184.2.a.o 1 360.bo even 12 1
5184.2.a.p 1 360.bt odd 12 1
5184.2.a.q 1 360.br even 12 1
5184.2.a.r 1 360.bu odd 12 1
7938.2.a.i 1 315.cf odd 12 1
7938.2.a.x 1 315.cb even 12 1

Hecke kernels

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

\( T_{7}^{4} - 4 T_{7}^{2} + 16 \)
\( T_{11}^{2} - 3 T_{11} + 9 \)
\( T_{19} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 9 + 3 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 16 - 4 T^{2} + T^{4} \)
$11$ \( ( 9 - 3 T + T^{2} )^{2} \)
$13$ \( 16 - 4 T^{2} + T^{4} \)
$17$ \( ( 9 + T^{2} )^{2} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( 1296 - 36 T^{2} + T^{4} \)
$29$ \( ( 36 - 6 T + T^{2} )^{2} \)
$31$ \( ( 16 - 4 T + T^{2} )^{2} \)
$37$ \( ( 16 + T^{2} )^{2} \)
$41$ \( ( 81 + 9 T + T^{2} )^{2} \)
$43$ \( 1 - T^{2} + T^{4} \)
$47$ \( 1296 - 36 T^{2} + T^{4} \)
$53$ \( ( 144 + T^{2} )^{2} \)
$59$ \( ( 9 - 3 T + T^{2} )^{2} \)
$61$ \( ( 64 + 8 T + T^{2} )^{2} \)
$67$ \( 625 - 25 T^{2} + T^{4} \)
$71$ \( ( 12 + T )^{4} \)
$73$ \( ( 121 + T^{2} )^{2} \)
$79$ \( ( 16 + 4 T + T^{2} )^{2} \)
$83$ \( 20736 - 144 T^{2} + T^{4} \)
$89$ \( ( 6 + T )^{4} \)
$97$ \( 625 - 25 T^{2} + T^{4} \)
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