# Properties

 Label 315.5.e.c Level $315$ Weight $5$ Character orbit 315.e Analytic conductor $32.562$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,5,Mod(244,315)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("315.244");

S:= CuspForms(chi, 5);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$32.5615383714$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 6$$ x^2 + 6 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 10 q^{4} + ( - 10 \beta - 5) q^{5} + ( - 14 \beta + 35) q^{7} + 26 \beta q^{8}+O(q^{10})$$ q + b * q^2 + 10 * q^4 + (-10*b - 5) * q^5 + (-14*b + 35) * q^7 + 26*b * q^8 $$q + \beta q^{2} + 10 q^{4} + ( - 10 \beta - 5) q^{5} + ( - 14 \beta + 35) q^{7} + 26 \beta q^{8} + ( - 5 \beta + 60) q^{10} - 89 q^{11} - 5 q^{13} + (35 \beta + 84) q^{14} + 4 q^{16} + 485 q^{17} + 90 \beta q^{19} + ( - 100 \beta - 50) q^{20} - 89 \beta q^{22} - 286 \beta q^{23} + (100 \beta - 575) q^{25} - 5 \beta q^{26} + ( - 140 \beta + 350) q^{28} - 191 q^{29} - 430 \beta q^{31} + 420 \beta q^{32} + 485 \beta q^{34} + ( - 280 \beta - 1015) q^{35} - 666 \beta q^{37} - 540 q^{38} + ( - 130 \beta + 1560) q^{40} - 1190 \beta q^{41} - 154 \beta q^{43} - 890 q^{44} + 1716 q^{46} + 2195 q^{47} + ( - 980 \beta + 49) q^{49} + ( - 575 \beta - 600) q^{50} - 50 q^{52} - 648 \beta q^{53} + (890 \beta + 445) q^{55} + (910 \beta + 2184) q^{56} - 191 \beta q^{58} + 1480 \beta q^{59} + 790 \beta q^{61} + 2580 q^{62} - 2456 q^{64} + (50 \beta + 25) q^{65} - 836 \beta q^{67} + 4850 q^{68} + ( - 1015 \beta + 1680) q^{70} - 4454 q^{71} + 8650 q^{73} + 3996 q^{74} + 900 \beta q^{76} + (1246 \beta - 3115) q^{77} + 5561 q^{79} + ( - 40 \beta - 20) q^{80} + 7140 q^{82} - 1990 q^{83} + ( - 4850 \beta - 2425) q^{85} + 924 q^{86} - 2314 \beta q^{88} + 330 \beta q^{89} + (70 \beta - 175) q^{91} - 2860 \beta q^{92} + 2195 \beta q^{94} + ( - 450 \beta + 5400) q^{95} + 9235 q^{97} + (49 \beta + 5880) q^{98} +O(q^{100})$$ q + b * q^2 + 10 * q^4 + (-10*b - 5) * q^5 + (-14*b + 35) * q^7 + 26*b * q^8 + (-5*b + 60) * q^10 - 89 * q^11 - 5 * q^13 + (35*b + 84) * q^14 + 4 * q^16 + 485 * q^17 + 90*b * q^19 + (-100*b - 50) * q^20 - 89*b * q^22 - 286*b * q^23 + (100*b - 575) * q^25 - 5*b * q^26 + (-140*b + 350) * q^28 - 191 * q^29 - 430*b * q^31 + 420*b * q^32 + 485*b * q^34 + (-280*b - 1015) * q^35 - 666*b * q^37 - 540 * q^38 + (-130*b + 1560) * q^40 - 1190*b * q^41 - 154*b * q^43 - 890 * q^44 + 1716 * q^46 + 2195 * q^47 + (-980*b + 49) * q^49 + (-575*b - 600) * q^50 - 50 * q^52 - 648*b * q^53 + (890*b + 445) * q^55 + (910*b + 2184) * q^56 - 191*b * q^58 + 1480*b * q^59 + 790*b * q^61 + 2580 * q^62 - 2456 * q^64 + (50*b + 25) * q^65 - 836*b * q^67 + 4850 * q^68 + (-1015*b + 1680) * q^70 - 4454 * q^71 + 8650 * q^73 + 3996 * q^74 + 900*b * q^76 + (1246*b - 3115) * q^77 + 5561 * q^79 + (-40*b - 20) * q^80 + 7140 * q^82 - 1990 * q^83 + (-4850*b - 2425) * q^85 + 924 * q^86 - 2314*b * q^88 + 330*b * q^89 + (70*b - 175) * q^91 - 2860*b * q^92 + 2195*b * q^94 + (-450*b + 5400) * q^95 + 9235 * q^97 + (49*b + 5880) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 20 q^{4} - 10 q^{5} + 70 q^{7}+O(q^{10})$$ 2 * q + 20 * q^4 - 10 * q^5 + 70 * q^7 $$2 q + 20 q^{4} - 10 q^{5} + 70 q^{7} + 120 q^{10} - 178 q^{11} - 10 q^{13} + 168 q^{14} + 8 q^{16} + 970 q^{17} - 100 q^{20} - 1150 q^{25} + 700 q^{28} - 382 q^{29} - 2030 q^{35} - 1080 q^{38} + 3120 q^{40} - 1780 q^{44} + 3432 q^{46} + 4390 q^{47} + 98 q^{49} - 1200 q^{50} - 100 q^{52} + 890 q^{55} + 4368 q^{56} + 5160 q^{62} - 4912 q^{64} + 50 q^{65} + 9700 q^{68} + 3360 q^{70} - 8908 q^{71} + 17300 q^{73} + 7992 q^{74} - 6230 q^{77} + 11122 q^{79} - 40 q^{80} + 14280 q^{82} - 3980 q^{83} - 4850 q^{85} + 1848 q^{86} - 350 q^{91} + 10800 q^{95} + 18470 q^{97} + 11760 q^{98}+O(q^{100})$$ 2 * q + 20 * q^4 - 10 * q^5 + 70 * q^7 + 120 * q^10 - 178 * q^11 - 10 * q^13 + 168 * q^14 + 8 * q^16 + 970 * q^17 - 100 * q^20 - 1150 * q^25 + 700 * q^28 - 382 * q^29 - 2030 * q^35 - 1080 * q^38 + 3120 * q^40 - 1780 * q^44 + 3432 * q^46 + 4390 * q^47 + 98 * q^49 - 1200 * q^50 - 100 * q^52 + 890 * q^55 + 4368 * q^56 + 5160 * q^62 - 4912 * q^64 + 50 * q^65 + 9700 * q^68 + 3360 * q^70 - 8908 * q^71 + 17300 * q^73 + 7992 * q^74 - 6230 * q^77 + 11122 * q^79 - 40 * q^80 + 14280 * q^82 - 3980 * q^83 - 4850 * q^85 + 1848 * q^86 - 350 * q^91 + 10800 * q^95 + 18470 * q^97 + 11760 * q^98

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
244.1
 − 2.44949i 2.44949i
2.44949i 0 10.0000 −5.00000 + 24.4949i 0 35.0000 + 34.2929i 63.6867i 0 60.0000 + 12.2474i
244.2 2.44949i 0 10.0000 −5.00000 24.4949i 0 35.0000 34.2929i 63.6867i 0 60.0000 12.2474i
 $$n$$: 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
35.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.5.e.c 2
3.b odd 2 1 35.5.c.d yes 2
5.b even 2 1 315.5.e.d 2
7.b odd 2 1 315.5.e.d 2
12.b even 2 1 560.5.p.d 2
15.d odd 2 1 35.5.c.c 2
15.e even 4 2 175.5.d.e 4
21.c even 2 1 35.5.c.c 2
35.c odd 2 1 inner 315.5.e.c 2
60.h even 2 1 560.5.p.e 2
84.h odd 2 1 560.5.p.e 2
105.g even 2 1 35.5.c.d yes 2
105.k odd 4 2 175.5.d.e 4
420.o odd 2 1 560.5.p.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.c 2 15.d odd 2 1
35.5.c.c 2 21.c even 2 1
35.5.c.d yes 2 3.b odd 2 1
35.5.c.d yes 2 105.g even 2 1
175.5.d.e 4 15.e even 4 2
175.5.d.e 4 105.k odd 4 2
315.5.e.c 2 1.a even 1 1 trivial
315.5.e.c 2 35.c odd 2 1 inner
315.5.e.d 2 5.b even 2 1
315.5.e.d 2 7.b odd 2 1
560.5.p.d 2 12.b even 2 1
560.5.p.d 2 420.o odd 2 1
560.5.p.e 2 60.h even 2 1
560.5.p.e 2 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{5}^{\mathrm{new}}(315, [\chi])$$:

 $$T_{2}^{2} + 6$$ T2^2 + 6 $$T_{13} + 5$$ T13 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 6$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 10T + 625$$
$7$ $$T^{2} - 70T + 2401$$
$11$ $$(T + 89)^{2}$$
$13$ $$(T + 5)^{2}$$
$17$ $$(T - 485)^{2}$$
$19$ $$T^{2} + 48600$$
$23$ $$T^{2} + 490776$$
$29$ $$(T + 191)^{2}$$
$31$ $$T^{2} + 1109400$$
$37$ $$T^{2} + 2661336$$
$41$ $$T^{2} + 8496600$$
$43$ $$T^{2} + 142296$$
$47$ $$(T - 2195)^{2}$$
$53$ $$T^{2} + 2519424$$
$59$ $$T^{2} + 13142400$$
$61$ $$T^{2} + 3744600$$
$67$ $$T^{2} + 4193376$$
$71$ $$(T + 4454)^{2}$$
$73$ $$(T - 8650)^{2}$$
$79$ $$(T - 5561)^{2}$$
$83$ $$(T + 1990)^{2}$$
$89$ $$T^{2} + 653400$$
$97$ $$(T - 9235)^{2}$$