# Properties

 Label 300.4.d.e Level $300$ Weight $4$ Character orbit 300.d Analytic conductor $17.701$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 300.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.7005730017$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q -3 i q^{3} + 8 i q^{7} -9 q^{9} +O(q^{10})$$ $$q -3 i q^{3} + 8 i q^{7} -9 q^{9} + 36 q^{11} + 10 i q^{13} + 18 i q^{17} + 100 q^{19} + 24 q^{21} -72 i q^{23} + 27 i q^{27} + 234 q^{29} -16 q^{31} -108 i q^{33} -226 i q^{37} + 30 q^{39} + 90 q^{41} -452 i q^{43} + 432 i q^{47} + 279 q^{49} + 54 q^{51} -414 i q^{53} -300 i q^{57} + 684 q^{59} + 422 q^{61} -72 i q^{63} + 332 i q^{67} -216 q^{69} -360 q^{71} -26 i q^{73} + 288 i q^{77} -512 q^{79} + 81 q^{81} + 1188 i q^{83} -702 i q^{87} + 630 q^{89} -80 q^{91} + 48 i q^{93} -1054 i q^{97} -324 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 18q^{9} + O(q^{10})$$ $$2q - 18q^{9} + 72q^{11} + 200q^{19} + 48q^{21} + 468q^{29} - 32q^{31} + 60q^{39} + 180q^{41} + 558q^{49} + 108q^{51} + 1368q^{59} + 844q^{61} - 432q^{69} - 720q^{71} - 1024q^{79} + 162q^{81} + 1260q^{89} - 160q^{91} - 648q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 1.00000i − 1.00000i
0 3.00000i 0 0 0 8.00000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 8.00000i 0 −9.00000 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 300.4.d.e 2
3.b odd 2 1 900.4.d.c 2
4.b odd 2 1 1200.4.f.d 2
5.b even 2 1 inner 300.4.d.e 2
5.c odd 4 1 12.4.a.a 1
5.c odd 4 1 300.4.a.b 1
15.d odd 2 1 900.4.d.c 2
15.e even 4 1 36.4.a.a 1
15.e even 4 1 900.4.a.g 1
20.d odd 2 1 1200.4.f.d 2
20.e even 4 1 48.4.a.a 1
20.e even 4 1 1200.4.a.be 1
35.f even 4 1 588.4.a.c 1
35.k even 12 2 588.4.i.e 2
35.l odd 12 2 588.4.i.d 2
40.i odd 4 1 192.4.a.f 1
40.k even 4 1 192.4.a.l 1
45.k odd 12 2 324.4.e.h 2
45.l even 12 2 324.4.e.a 2
55.e even 4 1 1452.4.a.d 1
60.l odd 4 1 144.4.a.g 1
65.f even 4 1 2028.4.b.c 2
65.h odd 4 1 2028.4.a.c 1
65.k even 4 1 2028.4.b.c 2
80.i odd 4 1 768.4.d.g 2
80.j even 4 1 768.4.d.j 2
80.s even 4 1 768.4.d.j 2
80.t odd 4 1 768.4.d.g 2
105.k odd 4 1 1764.4.a.b 1
105.w odd 12 2 1764.4.k.o 2
105.x even 12 2 1764.4.k.b 2
120.q odd 4 1 576.4.a.a 1
120.w even 4 1 576.4.a.b 1
140.j odd 4 1 2352.4.a.bk 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 5.c odd 4 1
36.4.a.a 1 15.e even 4 1
48.4.a.a 1 20.e even 4 1
144.4.a.g 1 60.l odd 4 1
192.4.a.f 1 40.i odd 4 1
192.4.a.l 1 40.k even 4 1
300.4.a.b 1 5.c odd 4 1
300.4.d.e 2 1.a even 1 1 trivial
300.4.d.e 2 5.b even 2 1 inner
324.4.e.a 2 45.l even 12 2
324.4.e.h 2 45.k odd 12 2
576.4.a.a 1 120.q odd 4 1
576.4.a.b 1 120.w even 4 1
588.4.a.c 1 35.f even 4 1
588.4.i.d 2 35.l odd 12 2
588.4.i.e 2 35.k even 12 2
768.4.d.g 2 80.i odd 4 1
768.4.d.g 2 80.t odd 4 1
768.4.d.j 2 80.j even 4 1
768.4.d.j 2 80.s even 4 1
900.4.a.g 1 15.e even 4 1
900.4.d.c 2 3.b odd 2 1
900.4.d.c 2 15.d odd 2 1
1200.4.a.be 1 20.e even 4 1
1200.4.f.d 2 4.b odd 2 1
1200.4.f.d 2 20.d odd 2 1
1452.4.a.d 1 55.e even 4 1
1764.4.a.b 1 105.k odd 4 1
1764.4.k.b 2 105.x even 12 2
1764.4.k.o 2 105.w odd 12 2
2028.4.a.c 1 65.h odd 4 1
2028.4.b.c 2 65.f even 4 1
2028.4.b.c 2 65.k even 4 1
2352.4.a.bk 1 140.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 64$$ acting on $$S_{4}^{\mathrm{new}}(300, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$64 + T^{2}$$
$11$ $$( -36 + T )^{2}$$
$13$ $$100 + T^{2}$$
$17$ $$324 + T^{2}$$
$19$ $$( -100 + T )^{2}$$
$23$ $$5184 + T^{2}$$
$29$ $$( -234 + T )^{2}$$
$31$ $$( 16 + T )^{2}$$
$37$ $$51076 + T^{2}$$
$41$ $$( -90 + T )^{2}$$
$43$ $$204304 + T^{2}$$
$47$ $$186624 + T^{2}$$
$53$ $$171396 + T^{2}$$
$59$ $$( -684 + T )^{2}$$
$61$ $$( -422 + T )^{2}$$
$67$ $$110224 + T^{2}$$
$71$ $$( 360 + T )^{2}$$
$73$ $$676 + T^{2}$$
$79$ $$( 512 + T )^{2}$$
$83$ $$1411344 + T^{2}$$
$89$ $$( -630 + T )^{2}$$
$97$ $$1110916 + T^{2}$$