Properties

 Label 448.6.a.s Level $448$ Weight $6$ Character orbit 448.a Self dual yes Analytic conductor $71.852$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,6,Mod(1,448)]

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, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("448.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 448.a (trivial)

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: yes Analytic conductor: $$71.8519512762$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{61})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 15$$ x^2 - x - 15 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 224) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 $$\beta = \sqrt{61}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 7) q^{3} + (7 \beta - 17) q^{5} + 49 q^{7} + (14 \beta - 133) q^{9}+O(q^{10})$$ q + (-b - 7) * q^3 + (7*b - 17) * q^5 + 49 * q^7 + (14*b - 133) * q^9 $$q + ( - \beta - 7) q^{3} + (7 \beta - 17) q^{5} + 49 q^{7} + (14 \beta - 133) q^{9} + ( - 46 \beta + 210) q^{11} + (49 \beta + 245) q^{13} + ( - 32 \beta - 308) q^{15} + ( - 14 \beta - 528) q^{17} + ( - 247 \beta + 623) q^{19} + ( - 49 \beta - 343) q^{21} + (16 \beta + 252) q^{23} + ( - 238 \beta + 153) q^{25} + (278 \beta + 1778) q^{27} + (546 \beta + 1952) q^{29} + ( - 642 \beta - 1022) q^{31} + (112 \beta + 1336) q^{33} + (343 \beta - 833) q^{35} + ( - 14 \beta + 3744) q^{37} + ( - 588 \beta - 4704) q^{39} + (1134 \beta + 3916) q^{41} + ( - 1090 \beta + 5166) q^{43} + ( - 1169 \beta + 8239) q^{45} + (850 \beta - 20986) q^{47} + 2401 q^{49} + (626 \beta + 4550) q^{51} + (2688 \beta - 16406) q^{53} + (2252 \beta - 23212) q^{55} + (1106 \beta + 10706) q^{57} + ( - 1639 \beta + 24199) q^{59} + ( - 441 \beta + 359) q^{61} + (686 \beta - 6517) q^{63} + (882 \beta + 16758) q^{65} + (3696 \beta + 6412) q^{67} + ( - 364 \beta - 2740) q^{69} + (396 \beta - 51996) q^{71} + (5404 \beta - 27050) q^{73} + (1513 \beta + 13447) q^{75} + ( - 2254 \beta + 10290) q^{77} + (1108 \beta - 32284) q^{79} + ( - 7126 \beta + 2915) q^{81} + ( - 527 \beta - 23905) q^{83} + ( - 3458 \beta + 2998) q^{85} + ( - 5774 \beta - 46970) q^{87} + ( - 14448 \beta - 8694) q^{89} + (2401 \beta + 12005) q^{91} + (5516 \beta + 46316) q^{93} + (8560 \beta - 116060) q^{95} + (6146 \beta - 48648) q^{97} + (9058 \beta - 67214) q^{99}+O(q^{100})$$ q + (-b - 7) * q^3 + (7*b - 17) * q^5 + 49 * q^7 + (14*b - 133) * q^9 + (-46*b + 210) * q^11 + (49*b + 245) * q^13 + (-32*b - 308) * q^15 + (-14*b - 528) * q^17 + (-247*b + 623) * q^19 + (-49*b - 343) * q^21 + (16*b + 252) * q^23 + (-238*b + 153) * q^25 + (278*b + 1778) * q^27 + (546*b + 1952) * q^29 + (-642*b - 1022) * q^31 + (112*b + 1336) * q^33 + (343*b - 833) * q^35 + (-14*b + 3744) * q^37 + (-588*b - 4704) * q^39 + (1134*b + 3916) * q^41 + (-1090*b + 5166) * q^43 + (-1169*b + 8239) * q^45 + (850*b - 20986) * q^47 + 2401 * q^49 + (626*b + 4550) * q^51 + (2688*b - 16406) * q^53 + (2252*b - 23212) * q^55 + (1106*b + 10706) * q^57 + (-1639*b + 24199) * q^59 + (-441*b + 359) * q^61 + (686*b - 6517) * q^63 + (882*b + 16758) * q^65 + (3696*b + 6412) * q^67 + (-364*b - 2740) * q^69 + (396*b - 51996) * q^71 + (5404*b - 27050) * q^73 + (1513*b + 13447) * q^75 + (-2254*b + 10290) * q^77 + (1108*b - 32284) * q^79 + (-7126*b + 2915) * q^81 + (-527*b - 23905) * q^83 + (-3458*b + 2998) * q^85 + (-5774*b - 46970) * q^87 + (-14448*b - 8694) * q^89 + (2401*b + 12005) * q^91 + (5516*b + 46316) * q^93 + (8560*b - 116060) * q^95 + (6146*b - 48648) * q^97 + (9058*b - 67214) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 14 q^{3} - 34 q^{5} + 98 q^{7} - 266 q^{9}+O(q^{10})$$ 2 * q - 14 * q^3 - 34 * q^5 + 98 * q^7 - 266 * q^9 $$2 q - 14 q^{3} - 34 q^{5} + 98 q^{7} - 266 q^{9} + 420 q^{11} + 490 q^{13} - 616 q^{15} - 1056 q^{17} + 1246 q^{19} - 686 q^{21} + 504 q^{23} + 306 q^{25} + 3556 q^{27} + 3904 q^{29} - 2044 q^{31} + 2672 q^{33} - 1666 q^{35} + 7488 q^{37} - 9408 q^{39} + 7832 q^{41} + 10332 q^{43} + 16478 q^{45} - 41972 q^{47} + 4802 q^{49} + 9100 q^{51} - 32812 q^{53} - 46424 q^{55} + 21412 q^{57} + 48398 q^{59} + 718 q^{61} - 13034 q^{63} + 33516 q^{65} + 12824 q^{67} - 5480 q^{69} - 103992 q^{71} - 54100 q^{73} + 26894 q^{75} + 20580 q^{77} - 64568 q^{79} + 5830 q^{81} - 47810 q^{83} + 5996 q^{85} - 93940 q^{87} - 17388 q^{89} + 24010 q^{91} + 92632 q^{93} - 232120 q^{95} - 97296 q^{97} - 134428 q^{99}+O(q^{100})$$ 2 * q - 14 * q^3 - 34 * q^5 + 98 * q^7 - 266 * q^9 + 420 * q^11 + 490 * q^13 - 616 * q^15 - 1056 * q^17 + 1246 * q^19 - 686 * q^21 + 504 * q^23 + 306 * q^25 + 3556 * q^27 + 3904 * q^29 - 2044 * q^31 + 2672 * q^33 - 1666 * q^35 + 7488 * q^37 - 9408 * q^39 + 7832 * q^41 + 10332 * q^43 + 16478 * q^45 - 41972 * q^47 + 4802 * q^49 + 9100 * q^51 - 32812 * q^53 - 46424 * q^55 + 21412 * q^57 + 48398 * q^59 + 718 * q^61 - 13034 * q^63 + 33516 * q^65 + 12824 * q^67 - 5480 * q^69 - 103992 * q^71 - 54100 * q^73 + 26894 * q^75 + 20580 * q^77 - 64568 * q^79 + 5830 * q^81 - 47810 * q^83 + 5996 * q^85 - 93940 * q^87 - 17388 * q^89 + 24010 * q^91 + 92632 * q^93 - 232120 * q^95 - 97296 * q^97 - 134428 * q^99

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 4.40512 −3.40512
0 −14.8102 0 37.6717 0 49.0000 0 −23.6565 0
1.2 0 0.810250 0 −71.6717 0 49.0000 0 −242.343 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.6.a.s 2
4.b odd 2 1 448.6.a.y 2
8.b even 2 1 224.6.a.d yes 2
8.d odd 2 1 224.6.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.6.a.c 2 8.d odd 2 1
224.6.a.d yes 2 8.b even 2 1
448.6.a.s 2 1.a even 1 1 trivial
448.6.a.y 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(448))$$:

 $$T_{3}^{2} + 14T_{3} - 12$$ T3^2 + 14*T3 - 12 $$T_{5}^{2} + 34T_{5} - 2700$$ T5^2 + 34*T5 - 2700

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 14T - 12$$
$5$ $$T^{2} + 34T - 2700$$
$7$ $$(T - 49)^{2}$$
$11$ $$T^{2} - 420T - 84976$$
$13$ $$T^{2} - 490T - 86436$$
$17$ $$T^{2} + 1056 T + 266828$$
$19$ $$T^{2} - 1246 T - 3333420$$
$23$ $$T^{2} - 504T + 47888$$
$29$ $$T^{2} - 3904 T - 14374772$$
$31$ $$T^{2} + 2044 T - 24097520$$
$37$ $$T^{2} - 7488 T + 14005580$$
$41$ $$T^{2} - 7832 T - 63108260$$
$43$ $$T^{2} - 10332 T - 45786544$$
$47$ $$T^{2} + 41972 T + 396339696$$
$53$ $$T^{2} + 32812 T - 171589148$$
$59$ $$T^{2} - 48398 T + 421726020$$
$61$ $$T^{2} - 718 T - 11734460$$
$67$ $$T^{2} - 12824 T - 792171632$$
$71$ $$T^{2} + \cdots + 2694018240$$
$73$ $$T^{2} + \cdots - 1049693676$$
$79$ $$T^{2} + 64568 T + 967369152$$
$83$ $$T^{2} + 47810 T + 554507556$$
$89$ $$T^{2} + \cdots - 12657841308$$
$97$ $$T^{2} + 97296 T + 62455628$$