Properties

Label 450.3.g.b
Level $450$
Weight $3$
Character orbit 450.g
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.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.2616118962\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - i ) q^{2} + 2 i q^{4} + ( -2 - 2 i ) q^{7} + ( 2 - 2 i ) q^{8} +O(q^{10})\) \( q + ( -1 - i ) q^{2} + 2 i q^{4} + ( -2 - 2 i ) q^{7} + ( 2 - 2 i ) q^{8} + 8 q^{11} + ( -3 + 3 i ) q^{13} + 4 i q^{14} -4 q^{16} + ( 7 + 7 i ) q^{17} -20 i q^{19} + ( -8 - 8 i ) q^{22} + ( -2 + 2 i ) q^{23} + 6 q^{26} + ( 4 - 4 i ) q^{28} -40 i q^{29} + 52 q^{31} + ( 4 + 4 i ) q^{32} -14 i q^{34} + ( 3 + 3 i ) q^{37} + ( -20 + 20 i ) q^{38} + 8 q^{41} + ( 42 - 42 i ) q^{43} + 16 i q^{44} + 4 q^{46} + ( -18 - 18 i ) q^{47} -41 i q^{49} + ( -6 - 6 i ) q^{52} + ( 53 - 53 i ) q^{53} -8 q^{56} + ( -40 + 40 i ) q^{58} + 20 i q^{59} -48 q^{61} + ( -52 - 52 i ) q^{62} -8 i q^{64} + ( -62 - 62 i ) q^{67} + ( -14 + 14 i ) q^{68} + 28 q^{71} + ( 47 - 47 i ) q^{73} -6 i q^{74} + 40 q^{76} + ( -16 - 16 i ) q^{77} + ( -8 - 8 i ) q^{82} + ( 18 - 18 i ) q^{83} -84 q^{86} + ( 16 - 16 i ) q^{88} -80 i q^{89} + 12 q^{91} + ( -4 - 4 i ) q^{92} + 36 i q^{94} + ( 63 + 63 i ) q^{97} + ( -41 + 41 i ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 4q^{7} + 4q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 4q^{7} + 4q^{8} + 16q^{11} - 6q^{13} - 8q^{16} + 14q^{17} - 16q^{22} - 4q^{23} + 12q^{26} + 8q^{28} + 104q^{31} + 8q^{32} + 6q^{37} - 40q^{38} + 16q^{41} + 84q^{43} + 8q^{46} - 36q^{47} - 12q^{52} + 106q^{53} - 16q^{56} - 80q^{58} - 96q^{61} - 104q^{62} - 124q^{67} - 28q^{68} + 56q^{71} + 94q^{73} + 80q^{76} - 32q^{77} - 16q^{82} + 36q^{83} - 168q^{86} + 32q^{88} + 24q^{91} - 8q^{92} + 126q^{97} - 82q^{98} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
1.00000i
1.00000i
−1.00000 1.00000i 0 2.00000i 0 0 −2.00000 2.00000i 2.00000 2.00000i 0 0
343.1 −1.00000 + 1.00000i 0 2.00000i 0 0 −2.00000 + 2.00000i 2.00000 + 2.00000i 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
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.3.g.b 2
3.b odd 2 1 50.3.c.c 2
5.b even 2 1 90.3.g.b 2
5.c odd 4 1 90.3.g.b 2
5.c odd 4 1 inner 450.3.g.b 2
12.b even 2 1 400.3.p.b 2
15.d odd 2 1 10.3.c.a 2
15.e even 4 1 10.3.c.a 2
15.e even 4 1 50.3.c.c 2
20.d odd 2 1 720.3.bh.c 2
20.e even 4 1 720.3.bh.c 2
60.h even 2 1 80.3.p.c 2
60.l odd 4 1 80.3.p.c 2
60.l odd 4 1 400.3.p.b 2
105.g even 2 1 490.3.f.b 2
105.k odd 4 1 490.3.f.b 2
120.i odd 2 1 320.3.p.h 2
120.m even 2 1 320.3.p.a 2
120.q odd 4 1 320.3.p.a 2
120.w even 4 1 320.3.p.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.3.c.a 2 15.d odd 2 1
10.3.c.a 2 15.e even 4 1
50.3.c.c 2 3.b odd 2 1
50.3.c.c 2 15.e even 4 1
80.3.p.c 2 60.h even 2 1
80.3.p.c 2 60.l odd 4 1
90.3.g.b 2 5.b even 2 1
90.3.g.b 2 5.c odd 4 1
320.3.p.a 2 120.m even 2 1
320.3.p.a 2 120.q odd 4 1
320.3.p.h 2 120.i odd 2 1
320.3.p.h 2 120.w even 4 1
400.3.p.b 2 12.b even 2 1
400.3.p.b 2 60.l odd 4 1
450.3.g.b 2 1.a even 1 1 trivial
450.3.g.b 2 5.c odd 4 1 inner
490.3.f.b 2 105.g even 2 1
490.3.f.b 2 105.k odd 4 1
720.3.bh.c 2 20.d odd 2 1
720.3.bh.c 2 20.e even 4 1

Hecke kernels

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

\( T_{7}^{2} + 4 T_{7} + 8 \)
\( T_{11} - 8 \)
\( T_{17}^{2} - 14 T_{17} + 98 \)