Properties

Label 450.3.d.f
Level $450$
Weight $3$
Character orbit 450.d
Analytic conductor $12.262$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 450.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.2616118962\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} -2 q^{4} + 4 q^{7} + 2 \beta q^{8} +O(q^{10})\) \( q -\beta q^{2} -2 q^{4} + 4 q^{7} + 2 \beta q^{8} + 12 \beta q^{11} -8 q^{13} -4 \beta q^{14} + 4 q^{16} + 9 \beta q^{17} -16 q^{19} + 24 q^{22} + 12 \beta q^{23} + 8 \beta q^{26} -8 q^{28} + 3 \beta q^{29} + 44 q^{31} -4 \beta q^{32} + 18 q^{34} + 34 q^{37} + 16 \beta q^{38} + 33 \beta q^{41} + 40 q^{43} -24 \beta q^{44} + 24 q^{46} + 60 \beta q^{47} -33 q^{49} + 16 q^{52} -27 \beta q^{53} + 8 \beta q^{56} + 6 q^{58} + 24 \beta q^{59} + 50 q^{61} -44 \beta q^{62} -8 q^{64} -8 q^{67} -18 \beta q^{68} -36 \beta q^{71} + 16 q^{73} -34 \beta q^{74} + 32 q^{76} + 48 \beta q^{77} -76 q^{79} + 66 q^{82} -84 \beta q^{83} -40 \beta q^{86} -48 q^{88} + 9 \beta q^{89} -32 q^{91} -24 \beta q^{92} + 120 q^{94} -176 q^{97} + 33 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} + 8 q^{7} + O(q^{10}) \) \( 2 q - 4 q^{4} + 8 q^{7} - 16 q^{13} + 8 q^{16} - 32 q^{19} + 48 q^{22} - 16 q^{28} + 88 q^{31} + 36 q^{34} + 68 q^{37} + 80 q^{43} + 48 q^{46} - 66 q^{49} + 32 q^{52} + 12 q^{58} + 100 q^{61} - 16 q^{64} - 16 q^{67} + 32 q^{73} + 64 q^{76} - 152 q^{79} + 132 q^{82} - 96 q^{88} - 64 q^{91} + 240 q^{94} - 352 q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
1.41421i
1.41421i
1.41421i 0 −2.00000 0 0 4.00000 2.82843i 0 0
251.2 1.41421i 0 −2.00000 0 0 4.00000 2.82843i 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.3.d.f 2
3.b odd 2 1 inner 450.3.d.f 2
4.b odd 2 1 3600.3.l.d 2
5.b even 2 1 18.3.b.a 2
5.c odd 4 2 450.3.b.b 4
12.b even 2 1 3600.3.l.d 2
15.d odd 2 1 18.3.b.a 2
15.e even 4 2 450.3.b.b 4
20.d odd 2 1 144.3.e.b 2
20.e even 4 2 3600.3.c.b 4
35.c odd 2 1 882.3.b.a 2
35.i odd 6 2 882.3.s.d 4
35.j even 6 2 882.3.s.b 4
40.e odd 2 1 576.3.e.f 2
40.f even 2 1 576.3.e.c 2
45.h odd 6 2 162.3.d.b 4
45.j even 6 2 162.3.d.b 4
55.d odd 2 1 2178.3.c.d 2
60.h even 2 1 144.3.e.b 2
60.l odd 4 2 3600.3.c.b 4
65.d even 2 1 3042.3.c.e 2
65.g odd 4 2 3042.3.d.a 4
80.k odd 4 2 2304.3.h.c 4
80.q even 4 2 2304.3.h.f 4
105.g even 2 1 882.3.b.a 2
105.o odd 6 2 882.3.s.b 4
105.p even 6 2 882.3.s.d 4
120.i odd 2 1 576.3.e.c 2
120.m even 2 1 576.3.e.f 2
165.d even 2 1 2178.3.c.d 2
180.n even 6 2 1296.3.q.f 4
180.p odd 6 2 1296.3.q.f 4
195.e odd 2 1 3042.3.c.e 2
195.n even 4 2 3042.3.d.a 4
240.t even 4 2 2304.3.h.c 4
240.bm odd 4 2 2304.3.h.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 5.b even 2 1
18.3.b.a 2 15.d odd 2 1
144.3.e.b 2 20.d odd 2 1
144.3.e.b 2 60.h even 2 1
162.3.d.b 4 45.h odd 6 2
162.3.d.b 4 45.j even 6 2
450.3.b.b 4 5.c odd 4 2
450.3.b.b 4 15.e even 4 2
450.3.d.f 2 1.a even 1 1 trivial
450.3.d.f 2 3.b odd 2 1 inner
576.3.e.c 2 40.f even 2 1
576.3.e.c 2 120.i odd 2 1
576.3.e.f 2 40.e odd 2 1
576.3.e.f 2 120.m even 2 1
882.3.b.a 2 35.c odd 2 1
882.3.b.a 2 105.g even 2 1
882.3.s.b 4 35.j even 6 2
882.3.s.b 4 105.o odd 6 2
882.3.s.d 4 35.i odd 6 2
882.3.s.d 4 105.p even 6 2
1296.3.q.f 4 180.n even 6 2
1296.3.q.f 4 180.p odd 6 2
2178.3.c.d 2 55.d odd 2 1
2178.3.c.d 2 165.d even 2 1
2304.3.h.c 4 80.k odd 4 2
2304.3.h.c 4 240.t even 4 2
2304.3.h.f 4 80.q even 4 2
2304.3.h.f 4 240.bm odd 4 2
3042.3.c.e 2 65.d even 2 1
3042.3.c.e 2 195.e odd 2 1
3042.3.d.a 4 65.g odd 4 2
3042.3.d.a 4 195.n even 4 2
3600.3.c.b 4 20.e even 4 2
3600.3.c.b 4 60.l odd 4 2
3600.3.l.d 2 4.b odd 2 1
3600.3.l.d 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(450, [\chi])\):

\( T_{7} - 4 \)
\( T_{11}^{2} + 288 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -4 + T )^{2} \)
$11$ \( 288 + T^{2} \)
$13$ \( ( 8 + T )^{2} \)
$17$ \( 162 + T^{2} \)
$19$ \( ( 16 + T )^{2} \)
$23$ \( 288 + T^{2} \)
$29$ \( 18 + T^{2} \)
$31$ \( ( -44 + T )^{2} \)
$37$ \( ( -34 + T )^{2} \)
$41$ \( 2178 + T^{2} \)
$43$ \( ( -40 + T )^{2} \)
$47$ \( 7200 + T^{2} \)
$53$ \( 1458 + T^{2} \)
$59$ \( 1152 + T^{2} \)
$61$ \( ( -50 + T )^{2} \)
$67$ \( ( 8 + T )^{2} \)
$71$ \( 2592 + T^{2} \)
$73$ \( ( -16 + T )^{2} \)
$79$ \( ( 76 + T )^{2} \)
$83$ \( 14112 + T^{2} \)
$89$ \( 162 + T^{2} \)
$97$ \( ( 176 + T )^{2} \)
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