# Properties

 Label 49.6.a.d Level $49$ Weight $6$ Character orbit 49.a Self dual yes Analytic conductor $7.859$ 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] = [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: $$1$$ 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.d 2
3.b odd 2 1 441.6.a.n 2
4.b odd 2 1 784.6.a.ba 2
7.b odd 2 1 49.6.a.e 2
7.c even 3 2 7.6.c.a 4
7.d odd 6 2 49.6.c.f 4
21.c even 2 1 441.6.a.m 2
21.h odd 6 2 63.6.e.d 4
28.d even 2 1 784.6.a.t 2
28.g odd 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.c even 3 2
49.6.a.d 2 1.a even 1 1 trivial
49.6.a.e 2 7.b odd 2 1
49.6.c.f 4 7.d odd 6 2
63.6.e.d 4 21.h odd 6 2
112.6.i.c 4 28.g odd 6 2
441.6.a.m 2 21.c even 2 1
441.6.a.n 2 3.b odd 2 1
784.6.a.t 2 28.d even 2 1
784.6.a.ba 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(49))$$:

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

## Hecke characteristic polynomials

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