# Properties

 Label 49.6.a.e Level $49$ Weight $6$ Character orbit 49.a Self dual yes Analytic conductor $7.859$ Analytic rank $0$ 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] = [49,6,Mod(1,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("49.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 49.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: $$7.85880717084$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{37})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 9$$ x^2 - x - 9 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 7) 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 $$\beta = \sqrt{37}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + (\beta + 4) q^{3} + (2 \beta + 6) q^{4} + (10 \beta + 19) q^{5} + (5 \beta + 41) q^{6} + ( - 24 \beta + 48) q^{8} + (8 \beta - 190) q^{9}+O(q^{10})$$ q + (b + 1) * q^2 + (b + 4) * q^3 + (2*b + 6) * q^4 + (10*b + 19) * q^5 + (5*b + 41) * q^6 + (-24*b + 48) * q^8 + (8*b - 190) * q^9 $$q + (\beta + 1) q^{2} + (\beta + 4) q^{3} + (2 \beta + 6) q^{4} + (10 \beta + 19) q^{5} + (5 \beta + 41) q^{6} + ( - 24 \beta + 48) q^{8} + (8 \beta - 190) q^{9} + (29 \beta + 389) q^{10} + (23 \beta + 212) q^{11} + (14 \beta + 98) q^{12} + ( - 28 \beta + 462) q^{13} + (59 \beta + 446) q^{15} + ( - 40 \beta - 1032) q^{16} + ( - 132 \beta + 1173) q^{17} + ( - 182 \beta + 106) q^{18} + ( - 277 \beta + 180) q^{19} + (98 \beta + 854) q^{20} + (235 \beta + 1063) q^{22} + ( - 69 \beta - 6) q^{23} + ( - 48 \beta - 696) q^{24} + (380 \beta + 936) q^{25} + (434 \beta - 574) q^{26} + ( - 401 \beta - 1436) q^{27} + ( - 700 \beta - 3526) q^{29} + (505 \beta + 2629) q^{30} + (715 \beta - 1774) q^{31} + ( - 304 \beta - 4048) q^{32} + (304 \beta + 1699) q^{33} + (1041 \beta - 3711) q^{34} + ( - 332 \beta - 548) q^{36} + ( - 790 \beta + 5545) q^{37} + ( - 97 \beta - 10069) q^{38} + (350 \beta + 812) q^{39} + (24 \beta - 7968) q^{40} + ( - 868 \beta - 1750) q^{41} + (1344 \beta - 6340) q^{43} + (562 \beta + 2974) q^{44} + ( - 1748 \beta - 650) q^{45} + ( - 75 \beta - 2559) q^{46} + ( - 1635 \beta + 11478) q^{47} + ( - 1192 \beta - 5608) q^{48} + (1316 \beta + 14996) q^{50} + (645 \beta - 192) q^{51} + (756 \beta + 700) q^{52} + ( - 1818 \beta + 1521) q^{53} + ( - 1837 \beta - 16273) q^{54} + (2557 \beta + 12538) q^{55} + ( - 928 \beta - 9529) q^{57} + ( - 4226 \beta - 29426) q^{58} + ( - 531 \beta + 32904) q^{59} + (1246 \beta + 7042) q^{60} + (4154 \beta + 21243) q^{61} + ( - 1059 \beta + 24681) q^{62} + ( - 3072 \beta + 17728) q^{64} + (4088 \beta - 1582) q^{65} + (2003 \beta + 12947) q^{66} + (919 \beta + 21156) q^{67} + (1554 \beta - 2730) q^{68} + ( - 282 \beta - 2577) q^{69} + ( - 2184 \beta - 1104) q^{71} + (4944 \beta - 16224) q^{72} + (7372 \beta + 25253) q^{73} + (4755 \beta - 23685) q^{74} + (2456 \beta + 17804) q^{75} + ( - 1302 \beta - 19418) q^{76} + (1162 \beta + 13762) q^{78} + ( - 5193 \beta + 4502) q^{79} + ( - 11080 \beta - 34408) q^{80} + ( - 4984 \beta + 25589) q^{81} + ( - 2618 \beta - 33866) q^{82} + (4536 \beta + 52164) q^{83} + (9222 \beta - 26553) q^{85} + ( - 4996 \beta + 43388) q^{86} + ( - 6326 \beta - 40004) q^{87} + ( - 3984 \beta - 10248) q^{88} + ( - 9356 \beta + 13333) q^{89} + ( - 2398 \beta - 65326) q^{90} + ( - 426 \beta - 5142) q^{92} + (1086 \beta + 19359) q^{93} + (9843 \beta - 49017) q^{94} + ( - 3463 \beta - 99070) q^{95} + ( - 5264 \beta - 27440) q^{96} + (196 \beta - 104566) q^{97} + ( - 2674 \beta - 33472) q^{99}+O(q^{100})$$ q + (b + 1) * q^2 + (b + 4) * q^3 + (2*b + 6) * q^4 + (10*b + 19) * q^5 + (5*b + 41) * q^6 + (-24*b + 48) * q^8 + (8*b - 190) * q^9 + (29*b + 389) * q^10 + (23*b + 212) * q^11 + (14*b + 98) * q^12 + (-28*b + 462) * q^13 + (59*b + 446) * q^15 + (-40*b - 1032) * q^16 + (-132*b + 1173) * q^17 + (-182*b + 106) * q^18 + (-277*b + 180) * q^19 + (98*b + 854) * q^20 + (235*b + 1063) * q^22 + (-69*b - 6) * q^23 + (-48*b - 696) * q^24 + (380*b + 936) * q^25 + (434*b - 574) * q^26 + (-401*b - 1436) * q^27 + (-700*b - 3526) * q^29 + (505*b + 2629) * q^30 + (715*b - 1774) * q^31 + (-304*b - 4048) * q^32 + (304*b + 1699) * q^33 + (1041*b - 3711) * q^34 + (-332*b - 548) * q^36 + (-790*b + 5545) * q^37 + (-97*b - 10069) * q^38 + (350*b + 812) * q^39 + (24*b - 7968) * q^40 + (-868*b - 1750) * q^41 + (1344*b - 6340) * q^43 + (562*b + 2974) * q^44 + (-1748*b - 650) * q^45 + (-75*b - 2559) * q^46 + (-1635*b + 11478) * q^47 + (-1192*b - 5608) * q^48 + (1316*b + 14996) * q^50 + (645*b - 192) * q^51 + (756*b + 700) * q^52 + (-1818*b + 1521) * q^53 + (-1837*b - 16273) * q^54 + (2557*b + 12538) * q^55 + (-928*b - 9529) * q^57 + (-4226*b - 29426) * q^58 + (-531*b + 32904) * q^59 + (1246*b + 7042) * q^60 + (4154*b + 21243) * q^61 + (-1059*b + 24681) * q^62 + (-3072*b + 17728) * q^64 + (4088*b - 1582) * q^65 + (2003*b + 12947) * q^66 + (919*b + 21156) * q^67 + (1554*b - 2730) * q^68 + (-282*b - 2577) * q^69 + (-2184*b - 1104) * q^71 + (4944*b - 16224) * q^72 + (7372*b + 25253) * q^73 + (4755*b - 23685) * q^74 + (2456*b + 17804) * q^75 + (-1302*b - 19418) * q^76 + (1162*b + 13762) * q^78 + (-5193*b + 4502) * q^79 + (-11080*b - 34408) * q^80 + (-4984*b + 25589) * q^81 + (-2618*b - 33866) * q^82 + (4536*b + 52164) * q^83 + (9222*b - 26553) * q^85 + (-4996*b + 43388) * q^86 + (-6326*b - 40004) * q^87 + (-3984*b - 10248) * q^88 + (-9356*b + 13333) * q^89 + (-2398*b - 65326) * q^90 + (-426*b - 5142) * q^92 + (1086*b + 19359) * q^93 + (9843*b - 49017) * q^94 + (-3463*b - 99070) * q^95 + (-5264*b - 27440) * q^96 + (196*b - 104566) * q^97 + (-2674*b - 33472) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 8 q^{3} + 12 q^{4} + 38 q^{5} + 82 q^{6} + 96 q^{8} - 380 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 8 * q^3 + 12 * q^4 + 38 * q^5 + 82 * q^6 + 96 * q^8 - 380 * q^9 $$2 q + 2 q^{2} + 8 q^{3} + 12 q^{4} + 38 q^{5} + 82 q^{6} + 96 q^{8} - 380 q^{9} + 778 q^{10} + 424 q^{11} + 196 q^{12} + 924 q^{13} + 892 q^{15} - 2064 q^{16} + 2346 q^{17} + 212 q^{18} + 360 q^{19} + 1708 q^{20} + 2126 q^{22} - 12 q^{23} - 1392 q^{24} + 1872 q^{25} - 1148 q^{26} - 2872 q^{27} - 7052 q^{29} + 5258 q^{30} - 3548 q^{31} - 8096 q^{32} + 3398 q^{33} - 7422 q^{34} - 1096 q^{36} + 11090 q^{37} - 20138 q^{38} + 1624 q^{39} - 15936 q^{40} - 3500 q^{41} - 12680 q^{43} + 5948 q^{44} - 1300 q^{45} - 5118 q^{46} + 22956 q^{47} - 11216 q^{48} + 29992 q^{50} - 384 q^{51} + 1400 q^{52} + 3042 q^{53} - 32546 q^{54} + 25076 q^{55} - 19058 q^{57} - 58852 q^{58} + 65808 q^{59} + 14084 q^{60} + 42486 q^{61} + 49362 q^{62} + 35456 q^{64} - 3164 q^{65} + 25894 q^{66} + 42312 q^{67} - 5460 q^{68} - 5154 q^{69} - 2208 q^{71} - 32448 q^{72} + 50506 q^{73} - 47370 q^{74} + 35608 q^{75} - 38836 q^{76} + 27524 q^{78} + 9004 q^{79} - 68816 q^{80} + 51178 q^{81} - 67732 q^{82} + 104328 q^{83} - 53106 q^{85} + 86776 q^{86} - 80008 q^{87} - 20496 q^{88} + 26666 q^{89} - 130652 q^{90} - 10284 q^{92} + 38718 q^{93} - 98034 q^{94} - 198140 q^{95} - 54880 q^{96} - 209132 q^{97} - 66944 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 8 * q^3 + 12 * q^4 + 38 * q^5 + 82 * q^6 + 96 * q^8 - 380 * q^9 + 778 * q^10 + 424 * q^11 + 196 * q^12 + 924 * q^13 + 892 * q^15 - 2064 * q^16 + 2346 * q^17 + 212 * q^18 + 360 * q^19 + 1708 * q^20 + 2126 * q^22 - 12 * q^23 - 1392 * q^24 + 1872 * q^25 - 1148 * q^26 - 2872 * q^27 - 7052 * q^29 + 5258 * q^30 - 3548 * q^31 - 8096 * q^32 + 3398 * q^33 - 7422 * q^34 - 1096 * q^36 + 11090 * q^37 - 20138 * q^38 + 1624 * q^39 - 15936 * q^40 - 3500 * q^41 - 12680 * q^43 + 5948 * q^44 - 1300 * q^45 - 5118 * q^46 + 22956 * q^47 - 11216 * q^48 + 29992 * q^50 - 384 * q^51 + 1400 * q^52 + 3042 * q^53 - 32546 * q^54 + 25076 * q^55 - 19058 * q^57 - 58852 * q^58 + 65808 * q^59 + 14084 * q^60 + 42486 * q^61 + 49362 * q^62 + 35456 * q^64 - 3164 * q^65 + 25894 * q^66 + 42312 * q^67 - 5460 * q^68 - 5154 * q^69 - 2208 * q^71 - 32448 * q^72 + 50506 * q^73 - 47370 * q^74 + 35608 * q^75 - 38836 * q^76 + 27524 * q^78 + 9004 * q^79 - 68816 * q^80 + 51178 * q^81 - 67732 * q^82 + 104328 * q^83 - 53106 * q^85 + 86776 * q^86 - 80008 * q^87 - 20496 * q^88 + 26666 * q^89 - 130652 * q^90 - 10284 * q^92 + 38718 * q^93 - 98034 * q^94 - 198140 * q^95 - 54880 * q^96 - 209132 * q^97 - 66944 * 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
 −2.54138 3.54138
−5.08276 −2.08276 −6.16553 −41.8276 10.5862 0 193.986 −238.662 212.600
1.2 7.08276 10.0828 18.1655 79.8276 71.4138 0 −97.9863 −141.338 565.400
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 49.6.a.e 2
3.b odd 2 1 441.6.a.m 2
4.b odd 2 1 784.6.a.t 2
7.b odd 2 1 49.6.a.d 2
7.c even 3 2 49.6.c.f 4
7.d odd 6 2 7.6.c.a 4
21.c even 2 1 441.6.a.n 2
21.g even 6 2 63.6.e.d 4
28.d even 2 1 784.6.a.ba 2
28.f even 6 2 112.6.i.c 4

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(49))$$:

 $$T_{2}^{2} - 2T_{2} - 36$$ T2^2 - 2*T2 - 36 $$T_{3}^{2} - 8T_{3} - 21$$ T3^2 - 8*T3 - 21

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 36$$
$3$ $$T^{2} - 8T - 21$$
$5$ $$T^{2} - 38T - 3339$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 424T + 25371$$
$13$ $$T^{2} - 924T + 184436$$
$17$ $$T^{2} - 2346 T + 731241$$
$19$ $$T^{2} - 360 T - 2806573$$
$23$ $$T^{2} + 12T - 176121$$
$29$ $$T^{2} + 7052 T - 5697324$$
$31$ $$T^{2} + 3548 T - 15768249$$
$37$ $$T^{2} - 11090 T + 7655325$$
$41$ $$T^{2} + 3500 T - 24814188$$
$43$ $$T^{2} + 12680 T - 26638832$$
$47$ $$T^{2} - 22956 T + 32835159$$
$53$ $$T^{2} - 3042 T - 119976147$$
$59$ $$T^{2} + \cdots + 1072240659$$
$61$ $$T^{2} - 42486 T - 187196443$$
$67$ $$T^{2} - 42312 T + 416327579$$
$71$ $$T^{2} + 2208 T - 175265856$$
$73$ $$T^{2} + \cdots - 1373102199$$
$79$ $$T^{2} - 9004 T - 977520209$$
$83$ $$T^{2} + \cdots + 1959796944$$
$89$ $$T^{2} + \cdots - 3061016343$$
$97$ $$T^{2} + \cdots + 10932626964$$