Properties

Label 450.4.c.e
Level 450
Weight 4
Character orbit 450.c
Analytic conductor 26.551
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.5508595026\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 6)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -2 i q^{2} -4 q^{4} + 16 i q^{7} + 8 i q^{8} +O(q^{10})\) \( q -2 i q^{2} -4 q^{4} + 16 i q^{7} + 8 i q^{8} -12 q^{11} + 38 i q^{13} + 32 q^{14} + 16 q^{16} -126 i q^{17} -20 q^{19} + 24 i q^{22} -168 i q^{23} + 76 q^{26} -64 i q^{28} + 30 q^{29} -88 q^{31} -32 i q^{32} -252 q^{34} -254 i q^{37} + 40 i q^{38} -42 q^{41} -52 i q^{43} + 48 q^{44} -336 q^{46} -96 i q^{47} + 87 q^{49} -152 i q^{52} -198 i q^{53} -128 q^{56} -60 i q^{58} -660 q^{59} -538 q^{61} + 176 i q^{62} -64 q^{64} -884 i q^{67} + 504 i q^{68} -792 q^{71} + 218 i q^{73} -508 q^{74} + 80 q^{76} -192 i q^{77} + 520 q^{79} + 84 i q^{82} + 492 i q^{83} -104 q^{86} -96 i q^{88} + 810 q^{89} -608 q^{91} + 672 i q^{92} -192 q^{94} -1154 i q^{97} -174 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{4} + O(q^{10}) \) \( 2q - 8q^{4} - 24q^{11} + 64q^{14} + 32q^{16} - 40q^{19} + 152q^{26} + 60q^{29} - 176q^{31} - 504q^{34} - 84q^{41} + 96q^{44} - 672q^{46} + 174q^{49} - 256q^{56} - 1320q^{59} - 1076q^{61} - 128q^{64} - 1584q^{71} - 1016q^{74} + 160q^{76} + 1040q^{79} - 208q^{86} + 1620q^{89} - 1216q^{91} - 384q^{94} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
2.00000i 0 −4.00000 0 0 16.0000i 8.00000i 0 0
199.2 2.00000i 0 −4.00000 0 0 16.0000i 8.00000i 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.4.c.e 2
3.b odd 2 1 150.4.c.d 2
5.b even 2 1 inner 450.4.c.e 2
5.c odd 4 1 18.4.a.a 1
5.c odd 4 1 450.4.a.h 1
12.b even 2 1 1200.4.f.j 2
15.d odd 2 1 150.4.c.d 2
15.e even 4 1 6.4.a.a 1
15.e even 4 1 150.4.a.i 1
20.e even 4 1 144.4.a.c 1
35.f even 4 1 882.4.a.n 1
35.k even 12 2 882.4.g.f 2
35.l odd 12 2 882.4.g.i 2
40.i odd 4 1 576.4.a.q 1
40.k even 4 1 576.4.a.r 1
45.k odd 12 2 162.4.c.c 2
45.l even 12 2 162.4.c.f 2
55.e even 4 1 2178.4.a.e 1
60.h even 2 1 1200.4.f.j 2
60.l odd 4 1 48.4.a.c 1
60.l odd 4 1 1200.4.a.b 1
105.k odd 4 1 294.4.a.e 1
105.w odd 12 2 294.4.e.g 2
105.x even 12 2 294.4.e.h 2
120.q odd 4 1 192.4.a.c 1
120.w even 4 1 192.4.a.i 1
165.l odd 4 1 726.4.a.f 1
195.j odd 4 1 1014.4.b.d 2
195.s even 4 1 1014.4.a.g 1
195.u odd 4 1 1014.4.b.d 2
240.z odd 4 1 768.4.d.c 2
240.bb even 4 1 768.4.d.n 2
240.bd odd 4 1 768.4.d.c 2
240.bf even 4 1 768.4.d.n 2
255.o even 4 1 1734.4.a.d 1
285.j odd 4 1 2166.4.a.i 1
420.w even 4 1 2352.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 15.e even 4 1
18.4.a.a 1 5.c odd 4 1
48.4.a.c 1 60.l odd 4 1
144.4.a.c 1 20.e even 4 1
150.4.a.i 1 15.e even 4 1
150.4.c.d 2 3.b odd 2 1
150.4.c.d 2 15.d odd 2 1
162.4.c.c 2 45.k odd 12 2
162.4.c.f 2 45.l even 12 2
192.4.a.c 1 120.q odd 4 1
192.4.a.i 1 120.w even 4 1
294.4.a.e 1 105.k odd 4 1
294.4.e.g 2 105.w odd 12 2
294.4.e.h 2 105.x even 12 2
450.4.a.h 1 5.c odd 4 1
450.4.c.e 2 1.a even 1 1 trivial
450.4.c.e 2 5.b even 2 1 inner
576.4.a.q 1 40.i odd 4 1
576.4.a.r 1 40.k even 4 1
726.4.a.f 1 165.l odd 4 1
768.4.d.c 2 240.z odd 4 1
768.4.d.c 2 240.bd odd 4 1
768.4.d.n 2 240.bb even 4 1
768.4.d.n 2 240.bf even 4 1
882.4.a.n 1 35.f even 4 1
882.4.g.f 2 35.k even 12 2
882.4.g.i 2 35.l odd 12 2
1014.4.a.g 1 195.s even 4 1
1014.4.b.d 2 195.j odd 4 1
1014.4.b.d 2 195.u odd 4 1
1200.4.a.b 1 60.l odd 4 1
1200.4.f.j 2 12.b even 2 1
1200.4.f.j 2 60.h even 2 1
1734.4.a.d 1 255.o even 4 1
2166.4.a.i 1 285.j odd 4 1
2178.4.a.e 1 55.e even 4 1
2352.4.a.e 1 420.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{2} + 256 \)
\( T_{11} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T^{2} \)
$3$ 1
$5$ 1
$7$ \( 1 - 430 T^{2} + 117649 T^{4} \)
$11$ \( ( 1 + 12 T + 1331 T^{2} )^{2} \)
$13$ \( 1 - 2950 T^{2} + 4826809 T^{4} \)
$17$ \( 1 + 6050 T^{2} + 24137569 T^{4} \)
$19$ \( ( 1 + 20 T + 6859 T^{2} )^{2} \)
$23$ \( 1 + 3890 T^{2} + 148035889 T^{4} \)
$29$ \( ( 1 - 30 T + 24389 T^{2} )^{2} \)
$31$ \( ( 1 + 88 T + 29791 T^{2} )^{2} \)
$37$ \( 1 - 36790 T^{2} + 2565726409 T^{4} \)
$41$ \( ( 1 + 42 T + 68921 T^{2} )^{2} \)
$43$ \( 1 - 156310 T^{2} + 6321363049 T^{4} \)
$47$ \( 1 - 198430 T^{2} + 10779215329 T^{4} \)
$53$ \( 1 - 258550 T^{2} + 22164361129 T^{4} \)
$59$ \( ( 1 + 660 T + 205379 T^{2} )^{2} \)
$61$ \( ( 1 + 538 T + 226981 T^{2} )^{2} \)
$67$ \( 1 + 179930 T^{2} + 90458382169 T^{4} \)
$71$ \( ( 1 + 792 T + 357911 T^{2} )^{2} \)
$73$ \( 1 - 730510 T^{2} + 151334226289 T^{4} \)
$79$ \( ( 1 - 520 T + 493039 T^{2} )^{2} \)
$83$ \( 1 - 901510 T^{2} + 326940373369 T^{4} \)
$89$ \( ( 1 - 810 T + 704969 T^{2} )^{2} \)
$97$ \( 1 - 493630 T^{2} + 832972004929 T^{4} \)
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