# Properties

 Label 49.7.d.d Level $49$ Weight $7$ Character orbit 49.d Analytic conductor $11.273$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,7,Mod(19,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5]))

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

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

chi := DirichletCharacter("49.19");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 49.d (of order $$6$$, degree $$2$$, 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: $$11.2726500974$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{170})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 170x^{2} + 28900$$ x^4 + 170*x^2 + 28900 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 7) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (8 \beta_1 + 8) q^{2} + \beta_{3} q^{3} + (\beta_{3} - \beta_{2}) q^{5} + 8 \beta_{2} q^{6} + 512 q^{8} + (1311 \beta_1 + 1311) q^{9}+O(q^{10})$$ q + (8*b1 + 8) * q^2 + b3 * q^3 + (b3 - b2) * q^5 + 8*b2 * q^6 + 512 * q^8 + (1311*b1 + 1311) * q^9 $$q + (8 \beta_1 + 8) q^{2} + \beta_{3} q^{3} + (\beta_{3} - \beta_{2}) q^{5} + 8 \beta_{2} q^{6} + 512 q^{8} + (1311 \beta_1 + 1311) q^{9} + 8 \beta_{3} q^{10} + 874 \beta_1 q^{11} - 49 \beta_{2} q^{13} + 2040 q^{15} + (4096 \beta_1 + 4096) q^{16} - 132 \beta_{3} q^{17} + 10488 \beta_1 q^{18} + ( - 69 \beta_{3} + 69 \beta_{2}) q^{19} - 6992 q^{22} + ( - 4738 \beta_1 - 4738) q^{23} + 512 \beta_{3} q^{24} + 13585 \beta_1 q^{25} + (392 \beta_{3} - 392 \beta_{2}) q^{26} + 582 \beta_{2} q^{27} + 11146 q^{29} + (16320 \beta_1 + 16320) q^{30} - 608 \beta_{3} q^{31} + ( - 874 \beta_{3} + 874 \beta_{2}) q^{33} - 1056 \beta_{2} q^{34} + ( - 3002 \beta_1 - 3002) q^{37} - 552 \beta_{3} q^{38} - 99960 \beta_1 q^{39} + (512 \beta_{3} - 512 \beta_{2}) q^{40} - 1274 \beta_{2} q^{41} + 31418 q^{43} + 1311 \beta_{3} q^{45} - 37904 \beta_1 q^{46} + (1604 \beta_{3} - 1604 \beta_{2}) q^{47} + 4096 \beta_{2} q^{48} - 108680 q^{50} + ( - 269280 \beta_1 - 269280) q^{51} - 76406 \beta_1 q^{53} + ( - 4656 \beta_{3} + 4656 \beta_{2}) q^{54} + 874 \beta_{2} q^{55} - 140760 q^{57} + (89168 \beta_1 + 89168) q^{58} + 2507 \beta_{3} q^{59} + (6091 \beta_{3} - 6091 \beta_{2}) q^{61} - 4864 \beta_{2} q^{62} + 262144 q^{64} + ( - 99960 \beta_1 - 99960) q^{65} - 6992 \beta_{3} q^{66} + 495242 \beta_1 q^{67} - 4738 \beta_{2} q^{69} - 184406 q^{71} + (671232 \beta_1 + 671232) q^{72} - 1350 \beta_{3} q^{73} - 24016 \beta_1 q^{74} + ( - 13585 \beta_{3} + 13585 \beta_{2}) q^{75} + 799680 q^{78} + (534934 \beta_1 + 534934) q^{79} + 4096 \beta_{3} q^{80} + 231561 \beta_1 q^{81} + (10192 \beta_{3} - 10192 \beta_{2}) q^{82} + 15827 \beta_{2} q^{83} - 269280 q^{85} + (251344 \beta_1 + 251344) q^{86} + 11146 \beta_{3} q^{87} + 447488 \beta_1 q^{88} + (13938 \beta_{3} - 13938 \beta_{2}) q^{89} + 10488 \beta_{2} q^{90} + ( - 1240320 \beta_1 - 1240320) q^{93} + 12832 \beta_{3} q^{94} + 140760 \beta_1 q^{95} - 18032 \beta_{2} q^{97} - 1145814 q^{99}+O(q^{100})$$ q + (8*b1 + 8) * q^2 + b3 * q^3 + (b3 - b2) * q^5 + 8*b2 * q^6 + 512 * q^8 + (1311*b1 + 1311) * q^9 + 8*b3 * q^10 + 874*b1 * q^11 - 49*b2 * q^13 + 2040 * q^15 + (4096*b1 + 4096) * q^16 - 132*b3 * q^17 + 10488*b1 * q^18 + (-69*b3 + 69*b2) * q^19 - 6992 * q^22 + (-4738*b1 - 4738) * q^23 + 512*b3 * q^24 + 13585*b1 * q^25 + (392*b3 - 392*b2) * q^26 + 582*b2 * q^27 + 11146 * q^29 + (16320*b1 + 16320) * q^30 - 608*b3 * q^31 + (-874*b3 + 874*b2) * q^33 - 1056*b2 * q^34 + (-3002*b1 - 3002) * q^37 - 552*b3 * q^38 - 99960*b1 * q^39 + (512*b3 - 512*b2) * q^40 - 1274*b2 * q^41 + 31418 * q^43 + 1311*b3 * q^45 - 37904*b1 * q^46 + (1604*b3 - 1604*b2) * q^47 + 4096*b2 * q^48 - 108680 * q^50 + (-269280*b1 - 269280) * q^51 - 76406*b1 * q^53 + (-4656*b3 + 4656*b2) * q^54 + 874*b2 * q^55 - 140760 * q^57 + (89168*b1 + 89168) * q^58 + 2507*b3 * q^59 + (6091*b3 - 6091*b2) * q^61 - 4864*b2 * q^62 + 262144 * q^64 + (-99960*b1 - 99960) * q^65 - 6992*b3 * q^66 + 495242*b1 * q^67 - 4738*b2 * q^69 - 184406 * q^71 + (671232*b1 + 671232) * q^72 - 1350*b3 * q^73 - 24016*b1 * q^74 + (-13585*b3 + 13585*b2) * q^75 + 799680 * q^78 + (534934*b1 + 534934) * q^79 + 4096*b3 * q^80 + 231561*b1 * q^81 + (10192*b3 - 10192*b2) * q^82 + 15827*b2 * q^83 - 269280 * q^85 + (251344*b1 + 251344) * q^86 + 11146*b3 * q^87 + 447488*b1 * q^88 + (13938*b3 - 13938*b2) * q^89 + 10488*b2 * q^90 + (-1240320*b1 - 1240320) * q^93 + 12832*b3 * q^94 + 140760*b1 * q^95 - 18032*b2 * q^97 - 1145814 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 16 q^{2} + 2048 q^{8} + 2622 q^{9}+O(q^{10})$$ 4 * q + 16 * q^2 + 2048 * q^8 + 2622 * q^9 $$4 q + 16 q^{2} + 2048 q^{8} + 2622 q^{9} - 1748 q^{11} + 8160 q^{15} + 8192 q^{16} - 20976 q^{18} - 27968 q^{22} - 9476 q^{23} - 27170 q^{25} + 44584 q^{29} + 32640 q^{30} - 6004 q^{37} + 199920 q^{39} + 125672 q^{43} + 75808 q^{46} - 434720 q^{50} - 538560 q^{51} + 152812 q^{53} - 563040 q^{57} + 178336 q^{58} + 1048576 q^{64} - 199920 q^{65} - 990484 q^{67} - 737624 q^{71} + 1342464 q^{72} + 48032 q^{74} + 3198720 q^{78} + 1069868 q^{79} - 463122 q^{81} - 1077120 q^{85} + 502688 q^{86} - 894976 q^{88} - 2480640 q^{93} - 281520 q^{95} - 4583256 q^{99}+O(q^{100})$$ 4 * q + 16 * q^2 + 2048 * q^8 + 2622 * q^9 - 1748 * q^11 + 8160 * q^15 + 8192 * q^16 - 20976 * q^18 - 27968 * q^22 - 9476 * q^23 - 27170 * q^25 + 44584 * q^29 + 32640 * q^30 - 6004 * q^37 + 199920 * q^39 + 125672 * q^43 + 75808 * q^46 - 434720 * q^50 - 538560 * q^51 + 152812 * q^53 - 563040 * q^57 + 178336 * q^58 + 1048576 * q^64 - 199920 * q^65 - 990484 * q^67 - 737624 * q^71 + 1342464 * q^72 + 48032 * q^74 + 3198720 * q^78 + 1069868 * q^79 - 463122 * q^81 - 1077120 * q^85 + 502688 * q^86 - 894976 * q^88 - 2480640 * q^93 - 281520 * q^95 - 4583256 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 170x^{2} + 28900$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 170$$ (v^2) / 170 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 340\nu ) / 85$$ (v^3 + 340*v) / 85 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 170\nu ) / 85$$ (-v^3 + 170*v) / 85
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 6$$ (b3 + b2) / 6 $$\nu^{2}$$ $$=$$ $$170\beta_1$$ 170*b1 $$\nu^{3}$$ $$=$$ $$( -170\beta_{3} + 85\beta_{2} ) / 3$$ (-170*b3 + 85*b2) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/49\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$1 + \beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −6.51920 + 11.2916i 6.51920 − 11.2916i −6.51920 − 11.2916i 6.51920 + 11.2916i
4.00000 6.92820i −39.1152 + 22.5832i 0 −39.1152 22.5832i 361.331i 0 512.000 655.500 1135.36i −312.922 + 180.665i
19.2 4.00000 6.92820i 39.1152 22.5832i 0 39.1152 + 22.5832i 361.331i 0 512.000 655.500 1135.36i 312.922 180.665i
31.1 4.00000 + 6.92820i −39.1152 22.5832i 0 −39.1152 + 22.5832i 361.331i 0 512.000 655.500 + 1135.36i −312.922 180.665i
31.2 4.00000 + 6.92820i 39.1152 + 22.5832i 0 39.1152 22.5832i 361.331i 0 512.000 655.500 + 1135.36i 312.922 + 180.665i
 $$n$$: 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
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.7.d.d 4
7.b odd 2 1 inner 49.7.d.d 4
7.c even 3 1 7.7.b.b 2
7.c even 3 1 inner 49.7.d.d 4
7.d odd 6 1 7.7.b.b 2
7.d odd 6 1 inner 49.7.d.d 4
21.g even 6 1 63.7.d.d 2
21.h odd 6 1 63.7.d.d 2
28.f even 6 1 112.7.c.b 2
28.g odd 6 1 112.7.c.b 2
35.i odd 6 1 175.7.d.e 2
35.j even 6 1 175.7.d.e 2
35.k even 12 2 175.7.c.c 4
35.l odd 12 2 175.7.c.c 4
56.j odd 6 1 448.7.c.d 2
56.k odd 6 1 448.7.c.c 2
56.m even 6 1 448.7.c.c 2
56.p even 6 1 448.7.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.7.b.b 2 7.c even 3 1
7.7.b.b 2 7.d odd 6 1
49.7.d.d 4 1.a even 1 1 trivial
49.7.d.d 4 7.b odd 2 1 inner
49.7.d.d 4 7.c even 3 1 inner
49.7.d.d 4 7.d odd 6 1 inner
63.7.d.d 2 21.g even 6 1
63.7.d.d 2 21.h odd 6 1
112.7.c.b 2 28.f even 6 1
112.7.c.b 2 28.g odd 6 1
175.7.c.c 4 35.k even 12 2
175.7.c.c 4 35.l odd 12 2
175.7.d.e 2 35.i odd 6 1
175.7.d.e 2 35.j even 6 1
448.7.c.c 2 56.k odd 6 1
448.7.c.c 2 56.m even 6 1
448.7.c.d 2 56.j odd 6 1
448.7.c.d 2 56.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 8T_{2} + 64$$ acting on $$S_{7}^{\mathrm{new}}(49, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 8 T + 64)^{2}$$
$3$ $$T^{4} - 2040 T^{2} + \cdots + 4161600$$
$5$ $$T^{4} - 2040 T^{2} + \cdots + 4161600$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 874 T + 763876)^{2}$$
$13$ $$(T^{2} + 4898040)^{2}$$
$17$ $$T^{4} - 35544960 T^{2} + \cdots + 12\!\cdots\!00$$
$19$ $$T^{4} - 9712440 T^{2} + \cdots + 94331490753600$$
$23$ $$(T^{2} + 4738 T + 22448644)^{2}$$
$29$ $$(T - 11146)^{4}$$
$31$ $$T^{4} - 754114560 T^{2} + \cdots + 56\!\cdots\!00$$
$37$ $$(T^{2} + 3002 T + 9012004)^{2}$$
$41$ $$(T^{2} + 3311075040)^{2}$$
$43$ $$(T - 31418)^{4}$$
$47$ $$T^{4} - 5248544640 T^{2} + \cdots + 27\!\cdots\!00$$
$53$ $$(T^{2} - 76406 T + 5837876836)^{2}$$
$59$ $$T^{4} - 12821499960 T^{2} + \cdots + 16\!\cdots\!00$$
$61$ $$T^{4} - 75684573240 T^{2} + \cdots + 57\!\cdots\!00$$
$67$ $$(T^{2} + 495242 T + 245264638564)^{2}$$
$71$ $$(T + 184406)^{4}$$
$73$ $$T^{4} - 3717900000 T^{2} + \cdots + 13\!\cdots\!00$$
$79$ $$(T^{2} - 534934 T + 286154384356)^{2}$$
$83$ $$(T^{2} + 511007615160)^{2}$$
$89$ $$T^{4} - 396306401760 T^{2} + \cdots + 15\!\cdots\!00$$
$97$ $$(T^{2} + 663312168960)^{2}$$