Properties

Label 450.3.b.b
Level $450$
Weight $3$
Character orbit 450.b
Analytic conductor $12.262$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 450.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.2616118962\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{2} + 2 q^{4} + 4 \zeta_{8}^{2} q^{7} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} +O(q^{10})\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{2} + 2 q^{4} + 4 \zeta_{8}^{2} q^{7} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} + ( 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{11} + 8 \zeta_{8}^{2} q^{13} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{14} + 4 q^{16} + ( -9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{17} + 16 q^{19} + 24 \zeta_{8}^{2} q^{22} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{23} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{26} + 8 \zeta_{8}^{2} q^{28} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{29} + 44 q^{31} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{32} -18 q^{34} + 34 \zeta_{8}^{2} q^{37} + ( 16 \zeta_{8} - 16 \zeta_{8}^{3} ) q^{38} + ( 33 \zeta_{8} + 33 \zeta_{8}^{3} ) q^{41} -40 \zeta_{8}^{2} q^{43} + ( 24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{44} + 24 q^{46} + ( -60 \zeta_{8} + 60 \zeta_{8}^{3} ) q^{47} + 33 q^{49} + 16 \zeta_{8}^{2} q^{52} + ( -27 \zeta_{8} + 27 \zeta_{8}^{3} ) q^{53} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{56} -6 \zeta_{8}^{2} q^{58} + ( -24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{59} + 50 q^{61} + ( 44 \zeta_{8} - 44 \zeta_{8}^{3} ) q^{62} + 8 q^{64} -8 \zeta_{8}^{2} q^{67} + ( -18 \zeta_{8} + 18 \zeta_{8}^{3} ) q^{68} + ( -36 \zeta_{8} - 36 \zeta_{8}^{3} ) q^{71} -16 \zeta_{8}^{2} q^{73} + ( 34 \zeta_{8} + 34 \zeta_{8}^{3} ) q^{74} + 32 q^{76} + ( -48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{77} + 76 q^{79} + 66 \zeta_{8}^{2} q^{82} + ( -84 \zeta_{8} + 84 \zeta_{8}^{3} ) q^{83} + ( -40 \zeta_{8} - 40 \zeta_{8}^{3} ) q^{86} + 48 \zeta_{8}^{2} q^{88} + ( -9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{89} -32 q^{91} + ( 24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{92} -120 q^{94} -176 \zeta_{8}^{2} q^{97} + ( 33 \zeta_{8} - 33 \zeta_{8}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} + O(q^{10}) \) \( 4q + 8q^{4} + 16q^{16} + 64q^{19} + 176q^{31} - 72q^{34} + 96q^{46} + 132q^{49} + 200q^{61} + 32q^{64} + 128q^{76} + 304q^{79} - 128q^{91} - 480q^{94} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−1.41421 0 2.00000 0 0 4.00000i −2.82843 0 0
449.2 −1.41421 0 2.00000 0 0 4.00000i −2.82843 0 0
449.3 1.41421 0 2.00000 0 0 4.00000i 2.82843 0 0
449.4 1.41421 0 2.00000 0 0 4.00000i 2.82843 0 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.3.b.b 4
3.b odd 2 1 inner 450.3.b.b 4
4.b odd 2 1 3600.3.c.b 4
5.b even 2 1 inner 450.3.b.b 4
5.c odd 4 1 18.3.b.a 2
5.c odd 4 1 450.3.d.f 2
12.b even 2 1 3600.3.c.b 4
15.d odd 2 1 inner 450.3.b.b 4
15.e even 4 1 18.3.b.a 2
15.e even 4 1 450.3.d.f 2
20.d odd 2 1 3600.3.c.b 4
20.e even 4 1 144.3.e.b 2
20.e even 4 1 3600.3.l.d 2
35.f even 4 1 882.3.b.a 2
35.k even 12 2 882.3.s.d 4
35.l odd 12 2 882.3.s.b 4
40.i odd 4 1 576.3.e.c 2
40.k even 4 1 576.3.e.f 2
45.k odd 12 2 162.3.d.b 4
45.l even 12 2 162.3.d.b 4
55.e even 4 1 2178.3.c.d 2
60.h even 2 1 3600.3.c.b 4
60.l odd 4 1 144.3.e.b 2
60.l odd 4 1 3600.3.l.d 2
65.f even 4 1 3042.3.d.a 4
65.h odd 4 1 3042.3.c.e 2
65.k even 4 1 3042.3.d.a 4
80.i odd 4 1 2304.3.h.f 4
80.j even 4 1 2304.3.h.c 4
80.s even 4 1 2304.3.h.c 4
80.t odd 4 1 2304.3.h.f 4
105.k odd 4 1 882.3.b.a 2
105.w odd 12 2 882.3.s.d 4
105.x even 12 2 882.3.s.b 4
120.q odd 4 1 576.3.e.f 2
120.w even 4 1 576.3.e.c 2
165.l odd 4 1 2178.3.c.d 2
180.v odd 12 2 1296.3.q.f 4
180.x even 12 2 1296.3.q.f 4
195.j odd 4 1 3042.3.d.a 4
195.s even 4 1 3042.3.c.e 2
195.u odd 4 1 3042.3.d.a 4
240.z odd 4 1 2304.3.h.c 4
240.bb even 4 1 2304.3.h.f 4
240.bd odd 4 1 2304.3.h.c 4
240.bf even 4 1 2304.3.h.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 5.c odd 4 1
18.3.b.a 2 15.e even 4 1
144.3.e.b 2 20.e even 4 1
144.3.e.b 2 60.l odd 4 1
162.3.d.b 4 45.k odd 12 2
162.3.d.b 4 45.l even 12 2
450.3.b.b 4 1.a even 1 1 trivial
450.3.b.b 4 3.b odd 2 1 inner
450.3.b.b 4 5.b even 2 1 inner
450.3.b.b 4 15.d odd 2 1 inner
450.3.d.f 2 5.c odd 4 1
450.3.d.f 2 15.e even 4 1
576.3.e.c 2 40.i odd 4 1
576.3.e.c 2 120.w even 4 1
576.3.e.f 2 40.k even 4 1
576.3.e.f 2 120.q odd 4 1
882.3.b.a 2 35.f even 4 1
882.3.b.a 2 105.k odd 4 1
882.3.s.b 4 35.l odd 12 2
882.3.s.b 4 105.x even 12 2
882.3.s.d 4 35.k even 12 2
882.3.s.d 4 105.w odd 12 2
1296.3.q.f 4 180.v odd 12 2
1296.3.q.f 4 180.x even 12 2
2178.3.c.d 2 55.e even 4 1
2178.3.c.d 2 165.l odd 4 1
2304.3.h.c 4 80.j even 4 1
2304.3.h.c 4 80.s even 4 1
2304.3.h.c 4 240.z odd 4 1
2304.3.h.c 4 240.bd odd 4 1
2304.3.h.f 4 80.i odd 4 1
2304.3.h.f 4 80.t odd 4 1
2304.3.h.f 4 240.bb even 4 1
2304.3.h.f 4 240.bf even 4 1
3042.3.c.e 2 65.h odd 4 1
3042.3.c.e 2 195.s even 4 1
3042.3.d.a 4 65.f even 4 1
3042.3.d.a 4 65.k even 4 1
3042.3.d.a 4 195.j odd 4 1
3042.3.d.a 4 195.u odd 4 1
3600.3.c.b 4 4.b odd 2 1
3600.3.c.b 4 12.b even 2 1
3600.3.c.b 4 20.d odd 2 1
3600.3.c.b 4 60.h even 2 1
3600.3.l.d 2 20.e even 4 1
3600.3.l.d 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 16 \) acting on \(S_{3}^{\mathrm{new}}(450, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 16 + T^{2} )^{2} \)
$11$ \( ( 288 + T^{2} )^{2} \)
$13$ \( ( 64 + T^{2} )^{2} \)
$17$ \( ( -162 + T^{2} )^{2} \)
$19$ \( ( -16 + T )^{4} \)
$23$ \( ( -288 + T^{2} )^{2} \)
$29$ \( ( 18 + T^{2} )^{2} \)
$31$ \( ( -44 + T )^{4} \)
$37$ \( ( 1156 + T^{2} )^{2} \)
$41$ \( ( 2178 + T^{2} )^{2} \)
$43$ \( ( 1600 + T^{2} )^{2} \)
$47$ \( ( -7200 + T^{2} )^{2} \)
$53$ \( ( -1458 + T^{2} )^{2} \)
$59$ \( ( 1152 + T^{2} )^{2} \)
$61$ \( ( -50 + T )^{4} \)
$67$ \( ( 64 + T^{2} )^{2} \)
$71$ \( ( 2592 + T^{2} )^{2} \)
$73$ \( ( 256 + T^{2} )^{2} \)
$79$ \( ( -76 + T )^{4} \)
$83$ \( ( -14112 + T^{2} )^{2} \)
$89$ \( ( 162 + T^{2} )^{2} \)
$97$ \( ( 30976 + T^{2} )^{2} \)
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