# Properties

 Label 900.4.d.c Level $900$ Weight $4$ Character orbit 900.d Analytic conductor $53.102$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 900.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$53.1017190052$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 12) 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.

 $$f(q)$$ $$=$$ $$q + 8 i q^{7} +O(q^{10})$$ $$q + 8 i q^{7} -36 q^{11} + 10 i q^{13} -18 i q^{17} + 100 q^{19} + 72 i q^{23} -234 q^{29} -16 q^{31} -226 i q^{37} -90 q^{41} -452 i q^{43} -432 i q^{47} + 279 q^{49} + 414 i q^{53} -684 q^{59} + 422 q^{61} + 332 i q^{67} + 360 q^{71} -26 i q^{73} -288 i q^{77} -512 q^{79} -1188 i q^{83} -630 q^{89} -80 q^{91} -1054 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 72q^{11} + 200q^{19} - 468q^{29} - 32q^{31} - 180q^{41} + 558q^{49} - 1368q^{59} + 844q^{61} + 720q^{71} - 1024q^{79} - 1260q^{89} - 160q^{91} + O(q^{100})$$

## 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 8.00000i 0 0 0
649.2 0 0 0 0 0 8.00000i 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 900.4.d.c 2
3.b odd 2 1 300.4.d.e 2
5.b even 2 1 inner 900.4.d.c 2
5.c odd 4 1 36.4.a.a 1
5.c odd 4 1 900.4.a.g 1
12.b even 2 1 1200.4.f.d 2
15.d odd 2 1 300.4.d.e 2
15.e even 4 1 12.4.a.a 1
15.e even 4 1 300.4.a.b 1
20.e even 4 1 144.4.a.g 1
35.f even 4 1 1764.4.a.b 1
35.k even 12 2 1764.4.k.o 2
35.l odd 12 2 1764.4.k.b 2
40.i odd 4 1 576.4.a.b 1
40.k even 4 1 576.4.a.a 1
45.k odd 12 2 324.4.e.a 2
45.l even 12 2 324.4.e.h 2
60.h even 2 1 1200.4.f.d 2
60.l odd 4 1 48.4.a.a 1
60.l odd 4 1 1200.4.a.be 1
105.k odd 4 1 588.4.a.c 1
105.w odd 12 2 588.4.i.e 2
105.x even 12 2 588.4.i.d 2
120.q odd 4 1 192.4.a.l 1
120.w even 4 1 192.4.a.f 1
165.l odd 4 1 1452.4.a.d 1
195.j odd 4 1 2028.4.b.c 2
195.s even 4 1 2028.4.a.c 1
195.u odd 4 1 2028.4.b.c 2
240.z odd 4 1 768.4.d.j 2
240.bb even 4 1 768.4.d.g 2
240.bd odd 4 1 768.4.d.j 2
240.bf even 4 1 768.4.d.g 2
420.w even 4 1 2352.4.a.bk 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 15.e even 4 1
36.4.a.a 1 5.c odd 4 1
48.4.a.a 1 60.l odd 4 1
144.4.a.g 1 20.e even 4 1
192.4.a.f 1 120.w even 4 1
192.4.a.l 1 120.q odd 4 1
300.4.a.b 1 15.e even 4 1
300.4.d.e 2 3.b odd 2 1
300.4.d.e 2 15.d odd 2 1
324.4.e.a 2 45.k odd 12 2
324.4.e.h 2 45.l even 12 2
576.4.a.a 1 40.k even 4 1
576.4.a.b 1 40.i odd 4 1
588.4.a.c 1 105.k odd 4 1
588.4.i.d 2 105.x even 12 2
588.4.i.e 2 105.w odd 12 2
768.4.d.g 2 240.bb even 4 1
768.4.d.g 2 240.bf even 4 1
768.4.d.j 2 240.z odd 4 1
768.4.d.j 2 240.bd odd 4 1
900.4.a.g 1 5.c odd 4 1
900.4.d.c 2 1.a even 1 1 trivial
900.4.d.c 2 5.b even 2 1 inner
1200.4.a.be 1 60.l odd 4 1
1200.4.f.d 2 12.b even 2 1
1200.4.f.d 2 60.h even 2 1
1452.4.a.d 1 165.l odd 4 1
1764.4.a.b 1 35.f even 4 1
1764.4.k.b 2 35.l odd 12 2
1764.4.k.o 2 35.k even 12 2
2028.4.a.c 1 195.s even 4 1
2028.4.b.c 2 195.j odd 4 1
2028.4.b.c 2 195.u odd 4 1
2352.4.a.bk 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 $$S_{4}^{\mathrm{new}}(900, [\chi])$$:

 $$T_{7}^{2} + 64$$ $$T_{11} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$64 + T^{2}$$
$11$ $$( 36 + T )^{2}$$
$13$ $$100 + T^{2}$$
$17$ $$324 + T^{2}$$
$19$ $$( -100 + T )^{2}$$
$23$ $$5184 + T^{2}$$
$29$ $$( 234 + T )^{2}$$
$31$ $$( 16 + T )^{2}$$
$37$ $$51076 + T^{2}$$
$41$ $$( 90 + T )^{2}$$
$43$ $$204304 + T^{2}$$
$47$ $$186624 + T^{2}$$
$53$ $$171396 + T^{2}$$
$59$ $$( 684 + T )^{2}$$
$61$ $$( -422 + T )^{2}$$
$67$ $$110224 + T^{2}$$
$71$ $$( -360 + T )^{2}$$
$73$ $$676 + T^{2}$$
$79$ $$( 512 + T )^{2}$$
$83$ $$1411344 + T^{2}$$
$89$ $$( 630 + T )^{2}$$
$97$ $$1110916 + T^{2}$$