# Properties

 Label 49.6.c.f Level $49$ Weight $6$ Character orbit 49.c Analytic conductor $7.859$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,6,Mod(18,49)]

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([4]))

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

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

chi := DirichletCharacter("49.18");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 49.c (of order $$3$$, 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: $$7.85880717084$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{37})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 10x^{2} + 9x + 81$$ x^4 - x^3 + 10*x^2 + 9*x + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 7) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - \beta_1) q^{2} + (\beta_{3} + \beta_{2} + 4 \beta_1 - 4) q^{3} + (2 \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 6) q^{4} + ( - 10 \beta_{2} - 19 \beta_1) q^{5} + ( - 5 \beta_{3} + 41) q^{6} + (24 \beta_{3} + 48) q^{8} + ( - 8 \beta_{2} + 190 \beta_1) q^{9}+O(q^{10})$$ q + (-b2 - b1) * q^2 + (b3 + b2 + 4*b1 - 4) * q^3 + (2*b3 + 2*b2 + 6*b1 - 6) * q^4 + (-10*b2 - 19*b1) * q^5 + (-5*b3 + 41) * q^6 + (24*b3 + 48) * q^8 + (-8*b2 + 190*b1) * q^9 $$q + ( - \beta_{2} - \beta_1) q^{2} + (\beta_{3} + \beta_{2} + 4 \beta_1 - 4) q^{3} + (2 \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 6) q^{4} + ( - 10 \beta_{2} - 19 \beta_1) q^{5} + ( - 5 \beta_{3} + 41) q^{6} + (24 \beta_{3} + 48) q^{8} + ( - 8 \beta_{2} + 190 \beta_1) q^{9} + (29 \beta_{3} + 29 \beta_{2} + \cdots - 389) q^{10}+ \cdots + (2674 \beta_{3} - 33472) q^{99}+O(q^{100})$$ q + (-b2 - b1) * q^2 + (b3 + b2 + 4*b1 - 4) * q^3 + (2*b3 + 2*b2 + 6*b1 - 6) * q^4 + (-10*b2 - 19*b1) * q^5 + (-5*b3 + 41) * q^6 + (24*b3 + 48) * q^8 + (-8*b2 + 190*b1) * q^9 + (29*b3 + 29*b2 + 389*b1 - 389) * q^10 + (23*b3 + 23*b2 + 212*b1 - 212) * q^11 + (-14*b2 - 98*b1) * q^12 + (28*b3 + 462) * q^13 + (-59*b3 + 446) * q^15 + (40*b2 + 1032*b1) * q^16 + (-132*b3 - 132*b2 + 1173*b1 - 1173) * q^17 + (-182*b3 - 182*b2 + 106*b1 - 106) * q^18 + (277*b2 - 180*b1) * q^19 + (-98*b3 + 854) * q^20 + (-235*b3 + 1063) * q^22 + (69*b2 + 6*b1) * q^23 + (-48*b3 - 48*b2 - 696*b1 + 696) * q^24 + (380*b3 + 380*b2 + 936*b1 - 936) * q^25 + (-434*b2 + 574*b1) * q^26 + (401*b3 - 1436) * q^27 + (700*b3 - 3526) * q^29 + (-505*b2 - 2629*b1) * q^30 + (715*b3 + 715*b2 - 1774*b1 + 1774) * q^31 + (-304*b3 - 304*b2 - 4048*b1 + 4048) * q^32 + (-304*b2 - 1699*b1) * q^33 + (-1041*b3 - 3711) * q^34 + (332*b3 - 548) * q^36 + (790*b2 - 5545*b1) * q^37 + (-97*b3 - 97*b2 - 10069*b1 + 10069) * q^38 + (350*b3 + 350*b2 + 812*b1 - 812) * q^39 + (-24*b2 + 7968*b1) * q^40 + (868*b3 - 1750) * q^41 + (-1344*b3 - 6340) * q^43 + (-562*b2 - 2974*b1) * q^44 + (-1748*b3 - 1748*b2 - 650*b1 + 650) * q^45 + (-75*b3 - 75*b2 - 2559*b1 + 2559) * q^46 + (1635*b2 - 11478*b1) * q^47 + (1192*b3 - 5608) * q^48 + (-1316*b3 + 14996) * q^50 + (-645*b2 + 192*b1) * q^51 + (756*b3 + 756*b2 + 700*b1 - 700) * q^52 + (-1818*b3 - 1818*b2 + 1521*b1 - 1521) * q^53 + (1837*b2 + 16273*b1) * q^54 + (-2557*b3 + 12538) * q^55 + (928*b3 - 9529) * q^57 + (4226*b2 + 29426*b1) * q^58 + (-531*b3 - 531*b2 + 32904*b1 - 32904) * q^59 + (1246*b3 + 1246*b2 + 7042*b1 - 7042) * q^60 + (-4154*b2 - 21243*b1) * q^61 + (1059*b3 + 24681) * q^62 + (3072*b3 + 17728) * q^64 + (-4088*b2 + 1582*b1) * q^65 + (2003*b3 + 2003*b2 + 12947*b1 - 12947) * q^66 + (919*b3 + 919*b2 + 21156*b1 - 21156) * q^67 + (-1554*b2 + 2730*b1) * q^68 + (282*b3 - 2577) * q^69 + (2184*b3 - 1104) * q^71 + (-4944*b2 + 16224*b1) * q^72 + (7372*b3 + 7372*b2 + 25253*b1 - 25253) * q^73 + (4755*b3 + 4755*b2 - 23685*b1 + 23685) * q^74 + (-2456*b2 - 17804*b1) * q^75 + (1302*b3 - 19418) * q^76 + (-1162*b3 + 13762) * q^78 + (5193*b2 - 4502*b1) * q^79 + (-11080*b3 - 11080*b2 - 34408*b1 + 34408) * q^80 + (-4984*b3 - 4984*b2 + 25589*b1 - 25589) * q^81 + (2618*b2 + 33866*b1) * q^82 + (-4536*b3 + 52164) * q^83 + (-9222*b3 - 26553) * q^85 + (4996*b2 - 43388*b1) * q^86 + (-6326*b3 - 6326*b2 - 40004*b1 + 40004) * q^87 + (-3984*b3 - 3984*b2 - 10248*b1 + 10248) * q^88 + (9356*b2 - 13333*b1) * q^89 + (2398*b3 - 65326) * q^90 + (426*b3 - 5142) * q^92 + (-1086*b2 - 19359*b1) * q^93 + (9843*b3 + 9843*b2 - 49017*b1 + 49017) * q^94 + (-3463*b3 - 3463*b2 - 99070*b1 + 99070) * q^95 + (5264*b2 + 27440*b1) * q^96 + (-196*b3 - 104566) * q^97 + (2674*b3 - 33472) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 8 q^{3} - 12 q^{4} - 38 q^{5} + 164 q^{6} + 192 q^{8} + 380 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 - 8 * q^3 - 12 * q^4 - 38 * q^5 + 164 * q^6 + 192 * q^8 + 380 * q^9 $$4 q - 2 q^{2} - 8 q^{3} - 12 q^{4} - 38 q^{5} + 164 q^{6} + 192 q^{8} + 380 q^{9} - 778 q^{10} - 424 q^{11} - 196 q^{12} + 1848 q^{13} + 1784 q^{15} + 2064 q^{16} - 2346 q^{17} - 212 q^{18} - 360 q^{19} + 3416 q^{20} + 4252 q^{22} + 12 q^{23} + 1392 q^{24} - 1872 q^{25} + 1148 q^{26} - 5744 q^{27} - 14104 q^{29} - 5258 q^{30} + 3548 q^{31} + 8096 q^{32} - 3398 q^{33} - 14844 q^{34} - 2192 q^{36} - 11090 q^{37} + 20138 q^{38} - 1624 q^{39} + 15936 q^{40} - 7000 q^{41} - 25360 q^{43} - 5948 q^{44} + 1300 q^{45} + 5118 q^{46} - 22956 q^{47} - 22432 q^{48} + 59984 q^{50} + 384 q^{51} - 1400 q^{52} - 3042 q^{53} + 32546 q^{54} + 50152 q^{55} - 38116 q^{57} + 58852 q^{58} - 65808 q^{59} - 14084 q^{60} - 42486 q^{61} + 98724 q^{62} + 70912 q^{64} + 3164 q^{65} - 25894 q^{66} - 42312 q^{67} + 5460 q^{68} - 10308 q^{69} - 4416 q^{71} + 32448 q^{72} - 50506 q^{73} + 47370 q^{74} - 35608 q^{75} - 77672 q^{76} + 55048 q^{78} - 9004 q^{79} + 68816 q^{80} - 51178 q^{81} + 67732 q^{82} + 208656 q^{83} - 106212 q^{85} - 86776 q^{86} + 80008 q^{87} + 20496 q^{88} - 26666 q^{89} - 261304 q^{90} - 20568 q^{92} - 38718 q^{93} + 98034 q^{94} + 198140 q^{95} + 54880 q^{96} - 418264 q^{97} - 133888 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 - 8 * q^3 - 12 * q^4 - 38 * q^5 + 164 * q^6 + 192 * q^8 + 380 * q^9 - 778 * q^10 - 424 * q^11 - 196 * q^12 + 1848 * q^13 + 1784 * q^15 + 2064 * q^16 - 2346 * q^17 - 212 * q^18 - 360 * q^19 + 3416 * q^20 + 4252 * q^22 + 12 * q^23 + 1392 * q^24 - 1872 * q^25 + 1148 * q^26 - 5744 * q^27 - 14104 * q^29 - 5258 * q^30 + 3548 * q^31 + 8096 * q^32 - 3398 * q^33 - 14844 * q^34 - 2192 * q^36 - 11090 * q^37 + 20138 * q^38 - 1624 * q^39 + 15936 * q^40 - 7000 * q^41 - 25360 * q^43 - 5948 * q^44 + 1300 * q^45 + 5118 * q^46 - 22956 * q^47 - 22432 * q^48 + 59984 * q^50 + 384 * q^51 - 1400 * q^52 - 3042 * q^53 + 32546 * q^54 + 50152 * q^55 - 38116 * q^57 + 58852 * q^58 - 65808 * q^59 - 14084 * q^60 - 42486 * q^61 + 98724 * q^62 + 70912 * q^64 + 3164 * q^65 - 25894 * q^66 - 42312 * q^67 + 5460 * q^68 - 10308 * q^69 - 4416 * q^71 + 32448 * q^72 - 50506 * q^73 + 47370 * q^74 - 35608 * q^75 - 77672 * q^76 + 55048 * q^78 - 9004 * q^79 + 68816 * q^80 - 51178 * q^81 + 67732 * q^82 + 208656 * q^83 - 106212 * q^85 - 86776 * q^86 + 80008 * q^87 + 20496 * q^88 - 26666 * q^89 - 261304 * q^90 - 20568 * q^92 - 38718 * q^93 + 98034 * q^94 + 198140 * q^95 + 54880 * q^96 - 418264 * q^97 - 133888 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 10x^{2} + 9x + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 10\nu^{2} - 10\nu + 81 ) / 90$$ (-v^3 + 10*v^2 - 10*v + 81) / 90 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 10\nu^{2} + 190\nu - 81 ) / 90$$ (v^3 - 10*v^2 + 190*v - 81) / 90 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 14 ) / 5$$ (v^3 + 14) / 5
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 19\beta _1 - 19 ) / 2$$ (b3 + b2 + 19*b1 - 19) / 2 $$\nu^{3}$$ $$=$$ $$5\beta_{3} - 14$$ 5*b3 - 14

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
18.1
 1.77069 + 3.06693i −1.27069 − 2.20090i 1.77069 − 3.06693i −1.27069 + 2.20090i
−3.54138 6.13385i −5.04138 + 8.73193i −9.08276 + 15.7318i −39.9138 69.1328i 71.4138 0 −97.9863 70.6689 + 122.402i −282.700 + 489.651i
18.2 2.54138 + 4.40180i 1.04138 1.80373i 3.08276 5.33950i 20.9138 + 36.2238i 10.5862 0 193.986 119.331 + 206.687i −106.300 + 184.117i
30.1 −3.54138 + 6.13385i −5.04138 8.73193i −9.08276 15.7318i −39.9138 + 69.1328i 71.4138 0 −97.9863 70.6689 122.402i −282.700 489.651i
30.2 2.54138 4.40180i 1.04138 + 1.80373i 3.08276 + 5.33950i 20.9138 36.2238i 10.5862 0 193.986 119.331 206.687i −106.300 184.117i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.6.c.f 4
7.b odd 2 1 7.6.c.a 4
7.c even 3 1 49.6.a.e 2
7.c even 3 1 inner 49.6.c.f 4
7.d odd 6 1 7.6.c.a 4
7.d odd 6 1 49.6.a.d 2
21.c even 2 1 63.6.e.d 4
21.g even 6 1 63.6.e.d 4
21.g even 6 1 441.6.a.n 2
21.h odd 6 1 441.6.a.m 2
28.d even 2 1 112.6.i.c 4
28.f even 6 1 112.6.i.c 4
28.f even 6 1 784.6.a.ba 2
28.g odd 6 1 784.6.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.c.a 4 7.b odd 2 1
7.6.c.a 4 7.d odd 6 1
49.6.a.d 2 7.d odd 6 1
49.6.a.e 2 7.c even 3 1
49.6.c.f 4 1.a even 1 1 trivial
49.6.c.f 4 7.c even 3 1 inner
63.6.e.d 4 21.c even 2 1
63.6.e.d 4 21.g even 6 1
112.6.i.c 4 28.d even 2 1
112.6.i.c 4 28.f even 6 1
441.6.a.m 2 21.h odd 6 1
441.6.a.n 2 21.g even 6 1
784.6.a.t 2 28.g odd 6 1
784.6.a.ba 2 28.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(49, [\chi])$$:

 $$T_{2}^{4} + 2T_{2}^{3} + 40T_{2}^{2} - 72T_{2} + 1296$$ T2^4 + 2*T2^3 + 40*T2^2 - 72*T2 + 1296 $$T_{3}^{4} + 8T_{3}^{3} + 85T_{3}^{2} - 168T_{3} + 441$$ T3^4 + 8*T3^3 + 85*T3^2 - 168*T3 + 441

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + \cdots + 1296$$
$3$ $$T^{4} + 8 T^{3} + \cdots + 441$$
$5$ $$T^{4} + 38 T^{3} + \cdots + 11148921$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 424 T^{3} + \cdots + 643687641$$
$13$ $$(T^{2} - 924 T + 184436)^{2}$$
$17$ $$T^{4} + \cdots + 534713400081$$
$19$ $$T^{4} + \cdots + 7876852004329$$
$23$ $$T^{4} + \cdots + 31018606641$$
$29$ $$(T^{2} + 7052 T - 5697324)^{2}$$
$31$ $$T^{4} + \cdots + 248637676526001$$
$37$ $$T^{4} + \cdots + 58604000855625$$
$41$ $$(T^{2} + 3500 T - 24814188)^{2}$$
$43$ $$(T^{2} + 12680 T - 26638832)^{2}$$
$47$ $$T^{4} + \cdots + 10\!\cdots\!81$$
$53$ $$T^{4} + \cdots + 14\!\cdots\!09$$
$59$ $$T^{4} + \cdots + 11\!\cdots\!81$$
$61$ $$T^{4} + \cdots + 35\!\cdots\!49$$
$67$ $$T^{4} + \cdots + 17\!\cdots\!41$$
$71$ $$(T^{2} + 2208 T - 175265856)^{2}$$
$73$ $$T^{4} + \cdots + 18\!\cdots\!01$$
$79$ $$T^{4} + \cdots + 95\!\cdots\!81$$
$83$ $$(T^{2} - 104328 T + 1959796944)^{2}$$
$89$ $$T^{4} + \cdots + 93\!\cdots\!49$$
$97$ $$(T^{2} + 209132 T + 10932626964)^{2}$$