# Properties

 Label 315.2.d.c Level $315$ Weight $2$ Character orbit 315.d Analytic conductor $2.515$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 315.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ 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 105) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{4} + ( - 2 i - 1) q^{5} - i q^{7} + 3 i q^{8} +O(q^{10})$$ q + i * q^2 + q^4 + (-2*i - 1) * q^5 - i * q^7 + 3*i * q^8 $$q + i q^{2} + q^{4} + ( - 2 i - 1) q^{5} - i q^{7} + 3 i q^{8} + ( - i + 2) q^{10} + 6 q^{11} + 2 i q^{13} + q^{14} - q^{16} - 4 i q^{17} + 6 q^{19} + ( - 2 i - 1) q^{20} + 6 i q^{22} + (4 i - 3) q^{25} - 2 q^{26} - i q^{28} - 2 q^{29} - 10 q^{31} + 5 i q^{32} + 4 q^{34} + (i - 2) q^{35} - 4 i q^{37} + 6 i q^{38} + ( - 3 i + 6) q^{40} - 2 q^{41} + 4 i q^{43} + 6 q^{44} - q^{49} + ( - 3 i - 4) q^{50} + 2 i q^{52} + 6 i q^{53} + ( - 12 i - 6) q^{55} + 3 q^{56} - 2 i q^{58} - 8 q^{59} - 2 q^{61} - 10 i q^{62} - 7 q^{64} + ( - 2 i + 4) q^{65} - 16 i q^{67} - 4 i q^{68} + ( - 2 i - 1) q^{70} - 10 q^{71} + 6 i q^{73} + 4 q^{74} + 6 q^{76} - 6 i q^{77} - 4 q^{79} + (2 i + 1) q^{80} - 2 i q^{82} + 8 i q^{83} + (4 i - 8) q^{85} - 4 q^{86} + 18 i q^{88} + 6 q^{89} + 2 q^{91} + ( - 12 i - 6) q^{95} - 2 i q^{97} - i q^{98} +O(q^{100})$$ q + i * q^2 + q^4 + (-2*i - 1) * q^5 - i * q^7 + 3*i * q^8 + (-i + 2) * q^10 + 6 * q^11 + 2*i * q^13 + q^14 - q^16 - 4*i * q^17 + 6 * q^19 + (-2*i - 1) * q^20 + 6*i * q^22 + (4*i - 3) * q^25 - 2 * q^26 - i * q^28 - 2 * q^29 - 10 * q^31 + 5*i * q^32 + 4 * q^34 + (i - 2) * q^35 - 4*i * q^37 + 6*i * q^38 + (-3*i + 6) * q^40 - 2 * q^41 + 4*i * q^43 + 6 * q^44 - q^49 + (-3*i - 4) * q^50 + 2*i * q^52 + 6*i * q^53 + (-12*i - 6) * q^55 + 3 * q^56 - 2*i * q^58 - 8 * q^59 - 2 * q^61 - 10*i * q^62 - 7 * q^64 + (-2*i + 4) * q^65 - 16*i * q^67 - 4*i * q^68 + (-2*i - 1) * q^70 - 10 * q^71 + 6*i * q^73 + 4 * q^74 + 6 * q^76 - 6*i * q^77 - 4 * q^79 + (2*i + 1) * q^80 - 2*i * q^82 + 8*i * q^83 + (4*i - 8) * q^85 - 4 * q^86 + 18*i * q^88 + 6 * q^89 + 2 * q^91 + (-12*i - 6) * q^95 - 2*i * q^97 - i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{5}+O(q^{10})$$ 2 * q + 2 * q^4 - 2 * q^5 $$2 q + 2 q^{4} - 2 q^{5} + 4 q^{10} + 12 q^{11} + 2 q^{14} - 2 q^{16} + 12 q^{19} - 2 q^{20} - 6 q^{25} - 4 q^{26} - 4 q^{29} - 20 q^{31} + 8 q^{34} - 4 q^{35} + 12 q^{40} - 4 q^{41} + 12 q^{44} - 2 q^{49} - 8 q^{50} - 12 q^{55} + 6 q^{56} - 16 q^{59} - 4 q^{61} - 14 q^{64} + 8 q^{65} - 2 q^{70} - 20 q^{71} + 8 q^{74} + 12 q^{76} - 8 q^{79} + 2 q^{80} - 16 q^{85} - 8 q^{86} + 12 q^{89} + 4 q^{91} - 12 q^{95}+O(q^{100})$$ 2 * q + 2 * q^4 - 2 * q^5 + 4 * q^10 + 12 * q^11 + 2 * q^14 - 2 * q^16 + 12 * q^19 - 2 * q^20 - 6 * q^25 - 4 * q^26 - 4 * q^29 - 20 * q^31 + 8 * q^34 - 4 * q^35 + 12 * q^40 - 4 * q^41 + 12 * q^44 - 2 * q^49 - 8 * q^50 - 12 * q^55 + 6 * q^56 - 16 * q^59 - 4 * q^61 - 14 * q^64 + 8 * q^65 - 2 * q^70 - 20 * q^71 + 8 * q^74 + 12 * q^76 - 8 * q^79 + 2 * q^80 - 16 * q^85 - 8 * q^86 + 12 * q^89 + 4 * q^91 - 12 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 − 1.00000i 1.00000i
1.00000i 0 1.00000 −1.00000 + 2.00000i 0 1.00000i 3.00000i 0 2.00000 + 1.00000i
64.2 1.00000i 0 1.00000 −1.00000 2.00000i 0 1.00000i 3.00000i 0 2.00000 1.00000i
 $$n$$: 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 315.2.d.c 2
3.b odd 2 1 105.2.d.a 2
4.b odd 2 1 5040.2.t.e 2
5.b even 2 1 inner 315.2.d.c 2
5.c odd 4 1 1575.2.a.e 1
5.c odd 4 1 1575.2.a.i 1
7.b odd 2 1 2205.2.d.f 2
12.b even 2 1 1680.2.t.f 2
15.d odd 2 1 105.2.d.a 2
15.e even 4 1 525.2.a.b 1
15.e even 4 1 525.2.a.c 1
20.d odd 2 1 5040.2.t.e 2
21.c even 2 1 735.2.d.a 2
21.g even 6 2 735.2.q.b 4
21.h odd 6 2 735.2.q.a 4
35.c odd 2 1 2205.2.d.f 2
60.h even 2 1 1680.2.t.f 2
60.l odd 4 1 8400.2.a.bj 1
60.l odd 4 1 8400.2.a.ch 1
105.g even 2 1 735.2.d.a 2
105.k odd 4 1 3675.2.a.d 1
105.k odd 4 1 3675.2.a.l 1
105.o odd 6 2 735.2.q.a 4
105.p even 6 2 735.2.q.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.a 2 3.b odd 2 1
105.2.d.a 2 15.d odd 2 1
315.2.d.c 2 1.a even 1 1 trivial
315.2.d.c 2 5.b even 2 1 inner
525.2.a.b 1 15.e even 4 1
525.2.a.c 1 15.e even 4 1
735.2.d.a 2 21.c even 2 1
735.2.d.a 2 105.g even 2 1
735.2.q.a 4 21.h odd 6 2
735.2.q.a 4 105.o odd 6 2
735.2.q.b 4 21.g even 6 2
735.2.q.b 4 105.p even 6 2
1575.2.a.e 1 5.c odd 4 1
1575.2.a.i 1 5.c odd 4 1
1680.2.t.f 2 12.b even 2 1
1680.2.t.f 2 60.h even 2 1
2205.2.d.f 2 7.b odd 2 1
2205.2.d.f 2 35.c odd 2 1
3675.2.a.d 1 105.k odd 4 1
3675.2.a.l 1 105.k odd 4 1
5040.2.t.e 2 4.b odd 2 1
5040.2.t.e 2 20.d odd 2 1
8400.2.a.bj 1 60.l odd 4 1
8400.2.a.ch 1 60.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 $$S_{2}^{\mathrm{new}}(315, [\chi])$$:

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{11} - 6$$ T11 - 6 $$T_{29} + 2$$ T29 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 5$$
$7$ $$T^{2} + 1$$
$11$ $$(T - 6)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 16$$
$19$ $$(T - 6)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T + 2)^{2}$$
$31$ $$(T + 10)^{2}$$
$37$ $$T^{2} + 16$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 8)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 256$$
$71$ $$(T + 10)^{2}$$
$73$ $$T^{2} + 36$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 64$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 4$$