# Properties

 Label 900.4.d.a 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: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 100) 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.

 $$f(q)$$ $$=$$ $$q + 26 i q^{7}+O(q^{10})$$ q + 26*i * q^7 $$q + 26 i q^{7} - 45 q^{11} - 44 i q^{13} - 117 i q^{17} + 91 q^{19} - 18 i q^{23} + 144 q^{29} + 26 q^{31} - 214 i q^{37} + 459 q^{41} + 460 i q^{43} + 468 i q^{47} - 333 q^{49} + 558 i q^{53} - 72 q^{59} - 118 q^{61} + 251 i q^{67} - 108 q^{71} - 299 i q^{73} - 1170 i q^{77} + 898 q^{79} + 927 i q^{83} + 351 q^{89} + 1144 q^{91} + 386 i q^{97} +O(q^{100})$$ q + 26*i * q^7 - 45 * q^11 - 44*i * q^13 - 117*i * q^17 + 91 * q^19 - 18*i * q^23 + 144 * q^29 + 26 * q^31 - 214*i * q^37 + 459 * q^41 + 460*i * q^43 + 468*i * q^47 - 333 * q^49 + 558*i * q^53 - 72 * q^59 - 118 * q^61 + 251*i * q^67 - 108 * q^71 - 299*i * q^73 - 1170*i * q^77 + 898 * q^79 + 927*i * q^83 + 351 * q^89 + 1144 * q^91 + 386*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 90 q^{11} + 182 q^{19} + 288 q^{29} + 52 q^{31} + 918 q^{41} - 666 q^{49} - 144 q^{59} - 236 q^{61} - 216 q^{71} + 1796 q^{79} + 702 q^{89} + 2288 q^{91}+O(q^{100})$$ 2 * q - 90 * q^11 + 182 * q^19 + 288 * q^29 + 52 * q^31 + 918 * q^41 - 666 * q^49 - 144 * q^59 - 236 * q^61 - 216 * q^71 + 1796 * q^79 + 702 * q^89 + 2288 * q^91

## 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 26.0000i 0 0 0
649.2 0 0 0 0 0 26.0000i 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 900.4.d.a 2
3.b odd 2 1 100.4.c.b 2
5.b even 2 1 inner 900.4.d.a 2
5.c odd 4 1 900.4.a.c 1
5.c odd 4 1 900.4.a.p 1
12.b even 2 1 400.4.c.l 2
15.d odd 2 1 100.4.c.b 2
15.e even 4 1 100.4.a.b 1
15.e even 4 1 100.4.a.c yes 1
60.h even 2 1 400.4.c.l 2
60.l odd 4 1 400.4.a.i 1
60.l odd 4 1 400.4.a.l 1
120.q odd 4 1 1600.4.a.y 1
120.q odd 4 1 1600.4.a.bd 1
120.w even 4 1 1600.4.a.x 1
120.w even 4 1 1600.4.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.4.a.b 1 15.e even 4 1
100.4.a.c yes 1 15.e even 4 1
100.4.c.b 2 3.b odd 2 1
100.4.c.b 2 15.d odd 2 1
400.4.a.i 1 60.l odd 4 1
400.4.a.l 1 60.l odd 4 1
400.4.c.l 2 12.b even 2 1
400.4.c.l 2 60.h even 2 1
900.4.a.c 1 5.c odd 4 1
900.4.a.p 1 5.c odd 4 1
900.4.d.a 2 1.a even 1 1 trivial
900.4.d.a 2 5.b even 2 1 inner
1600.4.a.x 1 120.w even 4 1
1600.4.a.y 1 120.q odd 4 1
1600.4.a.bc 1 120.w even 4 1
1600.4.a.bd 1 120.q odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(900, [\chi])$$:

 $$T_{7}^{2} + 676$$ T7^2 + 676 $$T_{11} + 45$$ T11 + 45

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 676$$
$11$ $$(T + 45)^{2}$$
$13$ $$T^{2} + 1936$$
$17$ $$T^{2} + 13689$$
$19$ $$(T - 91)^{2}$$
$23$ $$T^{2} + 324$$
$29$ $$(T - 144)^{2}$$
$31$ $$(T - 26)^{2}$$
$37$ $$T^{2} + 45796$$
$41$ $$(T - 459)^{2}$$
$43$ $$T^{2} + 211600$$
$47$ $$T^{2} + 219024$$
$53$ $$T^{2} + 311364$$
$59$ $$(T + 72)^{2}$$
$61$ $$(T + 118)^{2}$$
$67$ $$T^{2} + 63001$$
$71$ $$(T + 108)^{2}$$
$73$ $$T^{2} + 89401$$
$79$ $$(T - 898)^{2}$$
$83$ $$T^{2} + 859329$$
$89$ $$(T - 351)^{2}$$
$97$ $$T^{2} + 148996$$