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 $1$

Related objects

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

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

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

$p$ $F_p(T)$
$2$ $$4 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$158 + T$$
$11$ $$-148 + T$$
$13$ $$684 + T$$
$17$ $$-2048 + T$$
$19$ $$-2220 + T$$
$23$ $$1246 + T$$
$29$ $$-270 + T$$
$31$ $$2048 + T$$
$37$ $$-4372 + T$$
$41$ $$-2398 + T$$
$43$ $$2294 + T$$
$47$ $$10682 + T$$
$53$ $$-2964 + T$$
$59$ $$-39740 + T$$
$61$ $$42298 + T$$
$67$ $$32098 + T$$
$71$ $$-4248 + T$$
$73$ $$30104 + T$$
$79$ $$-35280 + T$$
$83$ $$27826 + T$$
$89$ $$-85210 + T$$
$97$ $$-97232 + T$$