# Properties

 Label 448.7.c.d Level $448$ Weight $7$ Character orbit 448.c Analytic conductor $103.064$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,7,Mod(321,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("448.321");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 448.c (of order $$2$$, degree $$1$$, not 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: $$103.064229462$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-510})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 510$$ x^2 + 510 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 7) 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 $$\beta = 2\sqrt{-510}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - \beta q^{5} + ( - 7 \beta + 133) q^{7} - 1311 q^{9} +O(q^{10})$$ q + b * q^3 - b * q^5 + (-7*b + 133) * q^7 - 1311 * q^9 $$q + \beta q^{3} - \beta q^{5} + ( - 7 \beta + 133) q^{7} - 1311 q^{9} - 874 q^{11} + 49 \beta q^{13} + 2040 q^{15} + 132 \beta q^{17} + 69 \beta q^{19} + (133 \beta + 14280) q^{21} + 4738 q^{23} + 13585 q^{25} - 582 \beta q^{27} - 11146 q^{29} + 608 \beta q^{31} - 874 \beta q^{33} + ( - 133 \beta - 14280) q^{35} - 3002 q^{37} - 99960 q^{39} - 1274 \beta q^{41} - 31418 q^{43} + 1311 \beta q^{45} + 1604 \beta q^{47} + ( - 1862 \beta - 82271) q^{49} - 269280 q^{51} + 76406 q^{53} + 874 \beta q^{55} - 140760 q^{57} + 2507 \beta q^{59} - 6091 \beta q^{61} + (9177 \beta - 174363) q^{63} + 99960 q^{65} - 495242 q^{67} + 4738 \beta q^{69} - 184406 q^{71} + 1350 \beta q^{73} + 13585 \beta q^{75} + (6118 \beta - 116242) q^{77} - 534934 q^{79} + 231561 q^{81} - 15827 \beta q^{83} + 269280 q^{85} - 11146 \beta q^{87} + 13938 \beta q^{89} + (6517 \beta + 699720) q^{91} - 1240320 q^{93} + 140760 q^{95} - 18032 \beta q^{97} + 1145814 q^{99} +O(q^{100})$$ q + b * q^3 - b * q^5 + (-7*b + 133) * q^7 - 1311 * q^9 - 874 * q^11 + 49*b * q^13 + 2040 * q^15 + 132*b * q^17 + 69*b * q^19 + (133*b + 14280) * q^21 + 4738 * q^23 + 13585 * q^25 - 582*b * q^27 - 11146 * q^29 + 608*b * q^31 - 874*b * q^33 + (-133*b - 14280) * q^35 - 3002 * q^37 - 99960 * q^39 - 1274*b * q^41 - 31418 * q^43 + 1311*b * q^45 + 1604*b * q^47 + (-1862*b - 82271) * q^49 - 269280 * q^51 + 76406 * q^53 + 874*b * q^55 - 140760 * q^57 + 2507*b * q^59 - 6091*b * q^61 + (9177*b - 174363) * q^63 + 99960 * q^65 - 495242 * q^67 + 4738*b * q^69 - 184406 * q^71 + 1350*b * q^73 + 13585*b * q^75 + (6118*b - 116242) * q^77 - 534934 * q^79 + 231561 * q^81 - 15827*b * q^83 + 269280 * q^85 - 11146*b * q^87 + 13938*b * q^89 + (6517*b + 699720) * q^91 - 1240320 * q^93 + 140760 * q^95 - 18032*b * q^97 + 1145814 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 266 q^{7} - 2622 q^{9}+O(q^{10})$$ 2 * q + 266 * q^7 - 2622 * q^9 $$2 q + 266 q^{7} - 2622 q^{9} - 1748 q^{11} + 4080 q^{15} + 28560 q^{21} + 9476 q^{23} + 27170 q^{25} - 22292 q^{29} - 28560 q^{35} - 6004 q^{37} - 199920 q^{39} - 62836 q^{43} - 164542 q^{49} - 538560 q^{51} + 152812 q^{53} - 281520 q^{57} - 348726 q^{63} + 199920 q^{65} - 990484 q^{67} - 368812 q^{71} - 232484 q^{77} - 1069868 q^{79} + 463122 q^{81} + 538560 q^{85} + 1399440 q^{91} - 2480640 q^{93} + 281520 q^{95} + 2291628 q^{99}+O(q^{100})$$ 2 * q + 266 * q^7 - 2622 * q^9 - 1748 * q^11 + 4080 * q^15 + 28560 * q^21 + 9476 * q^23 + 27170 * q^25 - 22292 * q^29 - 28560 * q^35 - 6004 * q^37 - 199920 * q^39 - 62836 * q^43 - 164542 * q^49 - 538560 * q^51 + 152812 * q^53 - 281520 * q^57 - 348726 * q^63 + 199920 * q^65 - 990484 * q^67 - 368812 * q^71 - 232484 * q^77 - 1069868 * q^79 + 463122 * q^81 + 538560 * q^85 + 1399440 * q^91 - 2480640 * q^93 + 281520 * q^95 + 2291628 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
321.1
 − 22.5832i 22.5832i
0 45.1664i 0 45.1664i 0 133.000 + 316.165i 0 −1311.00 0
321.2 0 45.1664i 0 45.1664i 0 133.000 316.165i 0 −1311.00 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
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.7.c.d 2
4.b odd 2 1 448.7.c.c 2
7.b odd 2 1 inner 448.7.c.d 2
8.b even 2 1 7.7.b.b 2
8.d odd 2 1 112.7.c.b 2
24.h odd 2 1 63.7.d.d 2
28.d even 2 1 448.7.c.c 2
40.f even 2 1 175.7.d.e 2
40.i odd 4 2 175.7.c.c 4
56.e even 2 1 112.7.c.b 2
56.h odd 2 1 7.7.b.b 2
56.j odd 6 2 49.7.d.d 4
56.p even 6 2 49.7.d.d 4
168.i even 2 1 63.7.d.d 2
280.c odd 2 1 175.7.d.e 2
280.s even 4 2 175.7.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.7.b.b 2 8.b even 2 1
7.7.b.b 2 56.h odd 2 1
49.7.d.d 4 56.j odd 6 2
49.7.d.d 4 56.p even 6 2
63.7.d.d 2 24.h odd 2 1
63.7.d.d 2 168.i even 2 1
112.7.c.b 2 8.d odd 2 1
112.7.c.b 2 56.e even 2 1
175.7.c.c 4 40.i odd 4 2
175.7.c.c 4 280.s even 4 2
175.7.d.e 2 40.f even 2 1
175.7.d.e 2 280.c odd 2 1
448.7.c.c 2 4.b odd 2 1
448.7.c.c 2 28.d even 2 1
448.7.c.d 2 1.a even 1 1 trivial
448.7.c.d 2 7.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{7}^{\mathrm{new}}(448, [\chi])$$:

 $$T_{3}^{2} + 2040$$ T3^2 + 2040 $$T_{11} + 874$$ T11 + 874

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2040$$
$5$ $$T^{2} + 2040$$
$7$ $$T^{2} - 266T + 117649$$
$11$ $$(T + 874)^{2}$$
$13$ $$T^{2} + 4898040$$
$17$ $$T^{2} + 35544960$$
$19$ $$T^{2} + 9712440$$
$23$ $$(T - 4738)^{2}$$
$29$ $$(T + 11146)^{2}$$
$31$ $$T^{2} + 754114560$$
$37$ $$(T + 3002)^{2}$$
$41$ $$T^{2} + 3311075040$$
$43$ $$(T + 31418)^{2}$$
$47$ $$T^{2} + 5248544640$$
$53$ $$(T - 76406)^{2}$$
$59$ $$T^{2} + 12821499960$$
$61$ $$T^{2} + 75684573240$$
$67$ $$(T + 495242)^{2}$$
$71$ $$(T + 184406)^{2}$$
$73$ $$T^{2} + 3717900000$$
$79$ $$(T + 534934)^{2}$$
$83$ $$T^{2} + 511007615160$$
$89$ $$T^{2} + 396306401760$$
$97$ $$T^{2} + 663312168960$$