Properties

 Label 300.5.b.a Level 300 Weight 5 Character orbit 300.b Analytic conductor 31.011 Analytic rank 0 Dimension 2 CM discriminant -3 Inner twists 4

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$31.0109889252$$ 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: $\mathrm{U}(1)[D_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + 9 i q^{3} + 94 i q^{7} -81 q^{9} +O(q^{10})$$ $$q + 9 i q^{3} + 94 i q^{7} -81 q^{9} + 146 i q^{13} + 46 q^{19} -846 q^{21} -729 i q^{27} + 194 q^{31} + 2062 i q^{37} -1314 q^{39} -3214 i q^{43} -6435 q^{49} + 414 i q^{57} -1966 q^{61} -7614 i q^{63} -5906 i q^{67} -8542 i q^{73} -7682 q^{79} + 6561 q^{81} -13724 q^{91} + 1746 i q^{93} + 18814 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 162q^{9} + O(q^{10})$$ $$2q - 162q^{9} + 92q^{19} - 1692q^{21} + 388q^{31} - 2628q^{39} - 12870q^{49} - 3932q^{61} - 15364q^{79} + 13122q^{81} - 27448q^{91} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 − 1.00000i 1.00000i
0 9.00000i 0 0 0 94.0000i 0 −81.0000 0
149.2 0 9.00000i 0 0 0 94.0000i 0 −81.0000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.5.b.a 2
3.b odd 2 1 CM 300.5.b.a 2
5.b even 2 1 inner 300.5.b.a 2
5.c odd 4 1 12.5.c.a 1
5.c odd 4 1 300.5.g.b 1
15.d odd 2 1 inner 300.5.b.a 2
15.e even 4 1 12.5.c.a 1
15.e even 4 1 300.5.g.b 1
20.e even 4 1 48.5.e.a 1
35.f even 4 1 588.5.c.a 1
40.i odd 4 1 192.5.e.a 1
40.k even 4 1 192.5.e.b 1
45.k odd 12 2 324.5.g.b 2
45.l even 12 2 324.5.g.b 2
60.l odd 4 1 48.5.e.a 1
105.k odd 4 1 588.5.c.a 1
120.q odd 4 1 192.5.e.b 1
120.w even 4 1 192.5.e.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.5.c.a 1 5.c odd 4 1
12.5.c.a 1 15.e even 4 1
48.5.e.a 1 20.e even 4 1
48.5.e.a 1 60.l odd 4 1
192.5.e.a 1 40.i odd 4 1
192.5.e.a 1 120.w even 4 1
192.5.e.b 1 40.k even 4 1
192.5.e.b 1 120.q odd 4 1
300.5.b.a 2 1.a even 1 1 trivial
300.5.b.a 2 3.b odd 2 1 CM
300.5.b.a 2 5.b even 2 1 inner
300.5.b.a 2 15.d odd 2 1 inner
300.5.g.b 1 5.c odd 4 1
300.5.g.b 1 15.e even 4 1
324.5.g.b 2 45.k odd 12 2
324.5.g.b 2 45.l even 12 2
588.5.c.a 1 35.f even 4 1
588.5.c.a 1 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 8836$$ acting on $$S_{5}^{\mathrm{new}}(300, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 81 T^{2}$$
$5$ 1
$7$ $$1 + 4034 T^{2} + 5764801 T^{4}$$
$11$ $$( 1 - 121 T )^{2}( 1 + 121 T )^{2}$$
$13$ $$1 - 35806 T^{2} + 815730721 T^{4}$$
$17$ $$( 1 + 83521 T^{2} )^{2}$$
$19$ $$( 1 - 46 T + 130321 T^{2} )^{2}$$
$23$ $$( 1 + 279841 T^{2} )^{2}$$
$29$ $$( 1 - 841 T )^{2}( 1 + 841 T )^{2}$$
$31$ $$( 1 - 194 T + 923521 T^{2} )^{2}$$
$37$ $$1 + 503522 T^{2} + 3512479453921 T^{4}$$
$41$ $$( 1 - 1681 T )^{2}( 1 + 1681 T )^{2}$$
$43$ $$1 + 3492194 T^{2} + 11688200277601 T^{4}$$
$47$ $$( 1 + 4879681 T^{2} )^{2}$$
$53$ $$( 1 + 7890481 T^{2} )^{2}$$
$59$ $$( 1 - 3481 T )^{2}( 1 + 3481 T )^{2}$$
$61$ $$( 1 + 1966 T + 13845841 T^{2} )^{2}$$
$67$ $$1 - 5421406 T^{2} + 406067677556641 T^{4}$$
$71$ $$( 1 - 5041 T )^{2}( 1 + 5041 T )^{2}$$
$73$ $$1 + 16169282 T^{2} + 806460091894081 T^{4}$$
$79$ $$( 1 + 7682 T + 38950081 T^{2} )^{2}$$
$83$ $$( 1 + 47458321 T^{2} )^{2}$$
$89$ $$( 1 - 7921 T )^{2}( 1 + 7921 T )^{2}$$
$97$ $$1 + 176908034 T^{2} + 7837433594376961 T^{4}$$