Properties

Label 450.6.a.c
Level 450
Weight 6
Character orbit 450.a
Self dual yes
Analytic conductor 72.173
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.1727189158\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{2} + 16q^{4} - 158q^{7} - 64q^{8} + O(q^{10}) \) \( q - 4q^{2} + 16q^{4} - 158q^{7} - 64q^{8} + 148q^{11} - 684q^{13} + 632q^{14} + 256q^{16} + 2048q^{17} + 2220q^{19} - 592q^{22} - 1246q^{23} + 2736q^{26} - 2528q^{28} + 270q^{29} - 2048q^{31} - 1024q^{32} - 8192q^{34} + 4372q^{37} - 8880q^{38} + 2398q^{41} - 2294q^{43} + 2368q^{44} + 4984q^{46} - 10682q^{47} + 8157q^{49} - 10944q^{52} + 2964q^{53} + 10112q^{56} - 1080q^{58} + 39740q^{59} - 42298q^{61} + 8192q^{62} + 4096q^{64} - 32098q^{67} + 32768q^{68} + 4248q^{71} - 30104q^{73} - 17488q^{74} + 35520q^{76} - 23384q^{77} + 35280q^{79} - 9592q^{82} - 27826q^{83} + 9176q^{86} - 9472q^{88} + 85210q^{89} + 108072q^{91} - 19936q^{92} + 42728q^{94} + 97232q^{97} - 32628q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−4.00000 0 16.0000 0 0 −158.000 −64.0000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.6.a.c 1
3.b odd 2 1 50.6.a.e 1
5.b even 2 1 450.6.a.w 1
5.c odd 4 2 90.6.c.a 2
12.b even 2 1 400.6.a.k 1
15.d odd 2 1 50.6.a.c 1
15.e even 4 2 10.6.b.a 2
20.e even 4 2 720.6.f.a 2
60.h even 2 1 400.6.a.c 1
60.l odd 4 2 80.6.c.c 2
120.q odd 4 2 320.6.c.a 2
120.w even 4 2 320.6.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.b.a 2 15.e even 4 2
50.6.a.c 1 15.d odd 2 1
50.6.a.e 1 3.b odd 2 1
80.6.c.c 2 60.l odd 4 2
90.6.c.a 2 5.c odd 4 2
320.6.c.a 2 120.q odd 4 2
320.6.c.b 2 120.w even 4 2
400.6.a.c 1 60.h even 2 1
400.6.a.k 1 12.b even 2 1
450.6.a.c 1 1.a even 1 1 trivial
450.6.a.w 1 5.b even 2 1
720.6.f.a 2 20.e even 4 2

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} + 158 \)
\( T_{11} - 148 \)
\( T_{17} - 2048 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T \)
$3$ 1
$5$ 1
$7$ \( 1 + 158 T + 16807 T^{2} \)
$11$ \( 1 - 148 T + 161051 T^{2} \)
$13$ \( 1 + 684 T + 371293 T^{2} \)
$17$ \( 1 - 2048 T + 1419857 T^{2} \)
$19$ \( 1 - 2220 T + 2476099 T^{2} \)
$23$ \( 1 + 1246 T + 6436343 T^{2} \)
$29$ \( 1 - 270 T + 20511149 T^{2} \)
$31$ \( 1 + 2048 T + 28629151 T^{2} \)
$37$ \( 1 - 4372 T + 69343957 T^{2} \)
$41$ \( 1 - 2398 T + 115856201 T^{2} \)
$43$ \( 1 + 2294 T + 147008443 T^{2} \)
$47$ \( 1 + 10682 T + 229345007 T^{2} \)
$53$ \( 1 - 2964 T + 418195493 T^{2} \)
$59$ \( 1 - 39740 T + 714924299 T^{2} \)
$61$ \( 1 + 42298 T + 844596301 T^{2} \)
$67$ \( 1 + 32098 T + 1350125107 T^{2} \)
$71$ \( 1 - 4248 T + 1804229351 T^{2} \)
$73$ \( 1 + 30104 T + 2073071593 T^{2} \)
$79$ \( 1 - 35280 T + 3077056399 T^{2} \)
$83$ \( 1 + 27826 T + 3939040643 T^{2} \)
$89$ \( 1 - 85210 T + 5584059449 T^{2} \)
$97$ \( 1 - 97232 T + 8587340257 T^{2} \)
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