# Properties

 Label 1521.2.b.c Level $1521$ Weight $2$ Character orbit 1521.b Analytic conductor $12.145$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1521.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.1452461474$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{4} - i q^{5} + 2 i q^{7} + 3 i q^{8} +O(q^{10})$$ q + i * q^2 + q^4 - i * q^5 + 2*i * q^7 + 3*i * q^8 $$q + i q^{2} + q^{4} - i q^{5} + 2 i q^{7} + 3 i q^{8} + q^{10} + 2 i q^{11} - 2 q^{14} - q^{16} - 7 q^{17} + 6 i q^{19} - i q^{20} - 2 q^{22} - 6 q^{23} + 4 q^{25} + 2 i q^{28} + q^{29} - 4 i q^{31} + 5 i q^{32} - 7 i q^{34} + 2 q^{35} + i q^{37} - 6 q^{38} + 3 q^{40} + 9 i q^{41} - 6 q^{43} + 2 i q^{44} - 6 i q^{46} - 6 i q^{47} + 3 q^{49} + 4 i q^{50} + 9 q^{53} + 2 q^{55} - 6 q^{56} + i q^{58} + q^{61} + 4 q^{62} - 7 q^{64} + 2 i q^{67} - 7 q^{68} + 2 i q^{70} + 6 i q^{71} + 11 i q^{73} - q^{74} + 6 i q^{76} - 4 q^{77} - 4 q^{79} + i q^{80} - 9 q^{82} - 14 i q^{83} + 7 i q^{85} - 6 i q^{86} - 6 q^{88} + 14 i q^{89} - 6 q^{92} + 6 q^{94} + 6 q^{95} + 2 i q^{97} + 3 i q^{98} +O(q^{100})$$ q + i * q^2 + q^4 - i * q^5 + 2*i * q^7 + 3*i * q^8 + q^10 + 2*i * q^11 - 2 * q^14 - q^16 - 7 * q^17 + 6*i * q^19 - i * q^20 - 2 * q^22 - 6 * q^23 + 4 * q^25 + 2*i * q^28 + q^29 - 4*i * q^31 + 5*i * q^32 - 7*i * q^34 + 2 * q^35 + i * q^37 - 6 * q^38 + 3 * q^40 + 9*i * q^41 - 6 * q^43 + 2*i * q^44 - 6*i * q^46 - 6*i * q^47 + 3 * q^49 + 4*i * q^50 + 9 * q^53 + 2 * q^55 - 6 * q^56 + i * q^58 + q^61 + 4 * q^62 - 7 * q^64 + 2*i * q^67 - 7 * q^68 + 2*i * q^70 + 6*i * q^71 + 11*i * q^73 - q^74 + 6*i * q^76 - 4 * q^77 - 4 * q^79 + i * q^80 - 9 * q^82 - 14*i * q^83 + 7*i * q^85 - 6*i * q^86 - 6 * q^88 + 14*i * q^89 - 6 * q^92 + 6 * q^94 + 6 * q^95 + 2*i * q^97 + 3*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4}+O(q^{10})$$ 2 * q + 2 * q^4 $$2 q + 2 q^{4} + 2 q^{10} - 4 q^{14} - 2 q^{16} - 14 q^{17} - 4 q^{22} - 12 q^{23} + 8 q^{25} + 2 q^{29} + 4 q^{35} - 12 q^{38} + 6 q^{40} - 12 q^{43} + 6 q^{49} + 18 q^{53} + 4 q^{55} - 12 q^{56} + 2 q^{61} + 8 q^{62} - 14 q^{64} - 14 q^{68} - 2 q^{74} - 8 q^{77} - 8 q^{79} - 18 q^{82} - 12 q^{88} - 12 q^{92} + 12 q^{94} + 12 q^{95}+O(q^{100})$$ 2 * q + 2 * q^4 + 2 * q^10 - 4 * q^14 - 2 * q^16 - 14 * q^17 - 4 * q^22 - 12 * q^23 + 8 * q^25 + 2 * q^29 + 4 * q^35 - 12 * q^38 + 6 * q^40 - 12 * q^43 + 6 * q^49 + 18 * q^53 + 4 * q^55 - 12 * q^56 + 2 * q^61 + 8 * q^62 - 14 * q^64 - 14 * q^68 - 2 * q^74 - 8 * q^77 - 8 * q^79 - 18 * q^82 - 12 * q^88 - 12 * q^92 + 12 * q^94 + 12 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1351.1
 − 1.00000i 1.00000i
1.00000i 0 1.00000 1.00000i 0 2.00000i 3.00000i 0 1.00000
1351.2 1.00000i 0 1.00000 1.00000i 0 2.00000i 3.00000i 0 1.00000
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.2.b.c 2
3.b odd 2 1 507.2.b.b 2
13.b even 2 1 inner 1521.2.b.c 2
13.d odd 4 1 1521.2.a.a 1
13.d odd 4 1 1521.2.a.d 1
13.f odd 12 2 117.2.g.b 2
39.d odd 2 1 507.2.b.b 2
39.f even 4 1 507.2.a.b 1
39.f even 4 1 507.2.a.c 1
39.h odd 6 2 507.2.j.d 4
39.i odd 6 2 507.2.j.d 4
39.k even 12 2 39.2.e.a 2
39.k even 12 2 507.2.e.c 2
52.l even 12 2 1872.2.t.j 2
156.l odd 4 1 8112.2.a.w 1
156.l odd 4 1 8112.2.a.bc 1
156.v odd 12 2 624.2.q.c 2
195.bc odd 12 2 975.2.bb.d 4
195.bh even 12 2 975.2.i.f 2
195.bn odd 12 2 975.2.bb.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 39.k even 12 2
117.2.g.b 2 13.f odd 12 2
507.2.a.b 1 39.f even 4 1
507.2.a.c 1 39.f even 4 1
507.2.b.b 2 3.b odd 2 1
507.2.b.b 2 39.d odd 2 1
507.2.e.c 2 39.k even 12 2
507.2.j.d 4 39.h odd 6 2
507.2.j.d 4 39.i odd 6 2
624.2.q.c 2 156.v odd 12 2
975.2.i.f 2 195.bh even 12 2
975.2.bb.d 4 195.bc odd 12 2
975.2.bb.d 4 195.bn odd 12 2
1521.2.a.a 1 13.d odd 4 1
1521.2.a.d 1 13.d odd 4 1
1521.2.b.c 2 1.a even 1 1 trivial
1521.2.b.c 2 13.b even 2 1 inner
1872.2.t.j 2 52.l even 12 2
8112.2.a.w 1 156.l odd 4 1
8112.2.a.bc 1 156.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1521, [\chi])$$:

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{5}^{2} + 1$$ T5^2 + 1 $$T_{7}^{2} + 4$$ T7^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2} + 4$$
$13$ $$T^{2}$$
$17$ $$(T + 7)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T + 6)^{2}$$
$29$ $$(T - 1)^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2} + 1$$
$41$ $$T^{2} + 81$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} + 36$$
$53$ $$(T - 9)^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 1)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$T^{2} + 36$$
$73$ $$T^{2} + 121$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 196$$
$89$ $$T^{2} + 196$$
$97$ $$T^{2} + 4$$