Properties

 Label 900.6.d.a Level $900$ Weight $6$ Character orbit 900.d Analytic conductor $144.345$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 900.d (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$144.345437832$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 4) 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.

 $$f(q)$$ $$=$$ $$q + 88 i q^{7} +O(q^{10})$$ $$q + 88 i q^{7} -540 q^{11} -418 i q^{13} + 594 i q^{17} -836 q^{19} + 4104 i q^{23} -594 q^{29} + 4256 q^{31} + 298 i q^{37} -17226 q^{41} -12100 i q^{43} -1296 i q^{47} + 9063 q^{49} -19494 i q^{53} -7668 q^{59} -34738 q^{61} -21812 i q^{67} + 46872 q^{71} + 67562 i q^{73} -47520 i q^{77} + 76912 q^{79} -67716 i q^{83} + 29754 q^{89} + 36784 q^{91} + 122398 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + O(q^{10})$$ $$2 q - 1080 q^{11} - 1672 q^{19} - 1188 q^{29} + 8512 q^{31} - 34452 q^{41} + 18126 q^{49} - 15336 q^{59} - 69476 q^{61} + 93744 q^{71} + 153824 q^{79} + 59508 q^{89} + 73568 q^{91} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
0 0 0 0 0 88.0000i 0 0 0
649.2 0 0 0 0 0 88.0000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 900.6.d.a 2
3.b odd 2 1 100.6.c.b 2
5.b even 2 1 inner 900.6.d.a 2
5.c odd 4 1 36.6.a.a 1
5.c odd 4 1 900.6.a.h 1
12.b even 2 1 400.6.c.f 2
15.d odd 2 1 100.6.c.b 2
15.e even 4 1 4.6.a.a 1
15.e even 4 1 100.6.a.b 1
20.e even 4 1 144.6.a.c 1
40.i odd 4 1 576.6.a.bc 1
40.k even 4 1 576.6.a.bd 1
45.k odd 12 2 324.6.e.d 2
45.l even 12 2 324.6.e.a 2
60.h even 2 1 400.6.c.f 2
60.l odd 4 1 16.6.a.b 1
60.l odd 4 1 400.6.a.d 1
105.k odd 4 1 196.6.a.e 1
105.w odd 12 2 196.6.e.d 2
105.x even 12 2 196.6.e.g 2
120.q odd 4 1 64.6.a.b 1
120.w even 4 1 64.6.a.f 1
165.l odd 4 1 484.6.a.a 1
195.j odd 4 1 676.6.d.a 2
195.s even 4 1 676.6.a.a 1
195.u odd 4 1 676.6.d.a 2
240.z odd 4 1 256.6.b.c 2
240.bb even 4 1 256.6.b.g 2
240.bd odd 4 1 256.6.b.c 2
240.bf even 4 1 256.6.b.g 2
420.w even 4 1 784.6.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 15.e even 4 1
16.6.a.b 1 60.l odd 4 1
36.6.a.a 1 5.c odd 4 1
64.6.a.b 1 120.q odd 4 1
64.6.a.f 1 120.w even 4 1
100.6.a.b 1 15.e even 4 1
100.6.c.b 2 3.b odd 2 1
100.6.c.b 2 15.d odd 2 1
144.6.a.c 1 20.e even 4 1
196.6.a.e 1 105.k odd 4 1
196.6.e.d 2 105.w odd 12 2
196.6.e.g 2 105.x even 12 2
256.6.b.c 2 240.z odd 4 1
256.6.b.c 2 240.bd odd 4 1
256.6.b.g 2 240.bb even 4 1
256.6.b.g 2 240.bf even 4 1
324.6.e.a 2 45.l even 12 2
324.6.e.d 2 45.k odd 12 2
400.6.a.d 1 60.l odd 4 1
400.6.c.f 2 12.b even 2 1
400.6.c.f 2 60.h even 2 1
484.6.a.a 1 165.l odd 4 1
576.6.a.bc 1 40.i odd 4 1
576.6.a.bd 1 40.k even 4 1
676.6.a.a 1 195.s even 4 1
676.6.d.a 2 195.j odd 4 1
676.6.d.a 2 195.u odd 4 1
784.6.a.d 1 420.w even 4 1
900.6.a.h 1 5.c odd 4 1
900.6.d.a 2 1.a even 1 1 trivial
900.6.d.a 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(900, [\chi])$$:

 $$T_{7}^{2} + 7744$$ $$T_{11} + 540$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$7744 + T^{2}$$
$11$ $$( 540 + T )^{2}$$
$13$ $$174724 + T^{2}$$
$17$ $$352836 + T^{2}$$
$19$ $$( 836 + T )^{2}$$
$23$ $$16842816 + T^{2}$$
$29$ $$( 594 + T )^{2}$$
$31$ $$( -4256 + T )^{2}$$
$37$ $$88804 + T^{2}$$
$41$ $$( 17226 + T )^{2}$$
$43$ $$146410000 + T^{2}$$
$47$ $$1679616 + T^{2}$$
$53$ $$380016036 + T^{2}$$
$59$ $$( 7668 + T )^{2}$$
$61$ $$( 34738 + T )^{2}$$
$67$ $$475763344 + T^{2}$$
$71$ $$( -46872 + T )^{2}$$
$73$ $$4564623844 + T^{2}$$
$79$ $$( -76912 + T )^{2}$$
$83$ $$4585456656 + T^{2}$$
$89$ $$( -29754 + T )^{2}$$
$97$ $$14981270404 + T^{2}$$