# Properties

 Label 950.4.b.i Level $950$ Weight $4$ Character orbit 950.b Analytic conductor $56.052$ 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] = [950,4,Mod(799,950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("950.799");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 950.b (of order $$2$$, degree $$1$$, 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: $$56.0518145055$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{73})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 37x^{2} + 324$$ x^4 + 37*x^2 + 324 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 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.

 $$f(q)$$ $$=$$ $$q - 2 \beta_{2} q^{2} + (5 \beta_{2} + \beta_1) q^{3} - 4 q^{4} + (2 \beta_{3} + 8) q^{6} + (11 \beta_{2} + 4 \beta_1) q^{7} + 8 \beta_{2} q^{8} + ( - 9 \beta_{3} - 7) q^{9}+O(q^{10})$$ q - 2*b2 * q^2 + (5*b2 + b1) * q^3 - 4 * q^4 + (2*b3 + 8) * q^6 + (11*b2 + 4*b1) * q^7 + 8*b2 * q^8 + (-9*b3 - 7) * q^9 $$q - 2 \beta_{2} q^{2} + (5 \beta_{2} + \beta_1) q^{3} - 4 q^{4} + (2 \beta_{3} + 8) q^{6} + (11 \beta_{2} + 4 \beta_1) q^{7} + 8 \beta_{2} q^{8} + ( - 9 \beta_{3} - 7) q^{9} + (\beta_{3} - 9) q^{11} + ( - 20 \beta_{2} - 4 \beta_1) q^{12} + (15 \beta_{2} + 13 \beta_1) q^{13} + (8 \beta_{3} + 14) q^{14} + 16 q^{16} + (41 \beta_{2} + 2 \beta_1) q^{17} + (32 \beta_{2} + 18 \beta_1) q^{18} - 19 q^{19} + ( - 27 \beta_{3} - 100) q^{21} + (16 \beta_{2} - 2 \beta_1) q^{22} + (43 \beta_{2} + 13 \beta_1) q^{23} + ( - 8 \beta_{3} - 32) q^{24} + (26 \beta_{3} + 4) q^{26} + ( - 107 \beta_{2} - 25 \beta_1) q^{27} + ( - 44 \beta_{2} - 16 \beta_1) q^{28} + (21 \beta_{3} - 12) q^{29} + ( - 44 \beta_{3} + 128) q^{31} - 32 \beta_{2} q^{32} + ( - 22 \beta_{2} - 4 \beta_1) q^{33} + (4 \beta_{3} + 78) q^{34} + (36 \beta_{3} + 28) q^{36} + ( - 82 \beta_{2} + 28 \beta_1) q^{37} + 38 \beta_{2} q^{38} + ( - 67 \beta_{3} - 242) q^{39} + (10 \beta_{3} - 30) q^{41} + (254 \beta_{2} + 54 \beta_1) q^{42} + (342 \beta_{2} + 7 \beta_1) q^{43} + ( - 4 \beta_{3} + 36) q^{44} + (26 \beta_{3} + 60) q^{46} + (230 \beta_{2} + 71 \beta_1) q^{47} + (80 \beta_{2} + 16 \beta_1) q^{48} + ( - 72 \beta_{3} + 6) q^{49} + ( - 49 \beta_{3} - 192) q^{51} + ( - 60 \beta_{2} - 52 \beta_1) q^{52} + ( - 601 \beta_{2} + 17 \beta_1) q^{53} + ( - 50 \beta_{3} - 164) q^{54} + ( - 32 \beta_{3} - 56) q^{56} + ( - 95 \beta_{2} - 19 \beta_1) q^{57} + ( - 18 \beta_{2} - 42 \beta_1) q^{58} + ( - 25 \beta_{3} + 156) q^{59} + (111 \beta_{3} + 101) q^{61} + ( - 168 \beta_{2} + 88 \beta_1) q^{62} + ( - 824 \beta_{2} - 127 \beta_1) q^{63} - 64 q^{64} + ( - 8 \beta_{3} - 36) q^{66} + ( - 573 \beta_{2} + 77 \beta_1) q^{67} + ( - 164 \beta_{2} - 8 \beta_1) q^{68} + ( - 95 \beta_{3} - 354) q^{69} + (116 \beta_{3} + 42) q^{71} + ( - 128 \beta_{2} - 72 \beta_1) q^{72} + (97 \beta_{2} - 184 \beta_1) q^{73} + (56 \beta_{3} - 220) q^{74} + 76 q^{76} + ( - 16 \beta_{2} - 25 \beta_1) q^{77} + (618 \beta_{2} + 134 \beta_1) q^{78} + (58 \beta_{3} - 704) q^{79} + ( - 36 \beta_{3} + 589) q^{81} + (40 \beta_{2} - 20 \beta_1) q^{82} + ( - 238 \beta_{2} + 194 \beta_1) q^{83} + (108 \beta_{3} + 400) q^{84} + (14 \beta_{3} + 670) q^{86} + (423 \beta_{2} + 93 \beta_1) q^{87} + ( - 64 \beta_{2} + 8 \beta_1) q^{88} + (188 \beta_{3} + 24) q^{89} + ( - 151 \beta_{3} - 950) q^{91} + ( - 172 \beta_{2} - 52 \beta_1) q^{92} + ( - 372 \beta_{2} - 92 \beta_1) q^{93} + (142 \beta_{3} + 318) q^{94} + (32 \beta_{3} + 128) q^{96} + ( - 698 \beta_{2} - 102 \beta_1) q^{97} + (132 \beta_{2} + 144 \beta_1) q^{98} + (65 \beta_{3} - 99) q^{99}+O(q^{100})$$ q - 2*b2 * q^2 + (5*b2 + b1) * q^3 - 4 * q^4 + (2*b3 + 8) * q^6 + (11*b2 + 4*b1) * q^7 + 8*b2 * q^8 + (-9*b3 - 7) * q^9 + (b3 - 9) * q^11 + (-20*b2 - 4*b1) * q^12 + (15*b2 + 13*b1) * q^13 + (8*b3 + 14) * q^14 + 16 * q^16 + (41*b2 + 2*b1) * q^17 + (32*b2 + 18*b1) * q^18 - 19 * q^19 + (-27*b3 - 100) * q^21 + (16*b2 - 2*b1) * q^22 + (43*b2 + 13*b1) * q^23 + (-8*b3 - 32) * q^24 + (26*b3 + 4) * q^26 + (-107*b2 - 25*b1) * q^27 + (-44*b2 - 16*b1) * q^28 + (21*b3 - 12) * q^29 + (-44*b3 + 128) * q^31 - 32*b2 * q^32 + (-22*b2 - 4*b1) * q^33 + (4*b3 + 78) * q^34 + (36*b3 + 28) * q^36 + (-82*b2 + 28*b1) * q^37 + 38*b2 * q^38 + (-67*b3 - 242) * q^39 + (10*b3 - 30) * q^41 + (254*b2 + 54*b1) * q^42 + (342*b2 + 7*b1) * q^43 + (-4*b3 + 36) * q^44 + (26*b3 + 60) * q^46 + (230*b2 + 71*b1) * q^47 + (80*b2 + 16*b1) * q^48 + (-72*b3 + 6) * q^49 + (-49*b3 - 192) * q^51 + (-60*b2 - 52*b1) * q^52 + (-601*b2 + 17*b1) * q^53 + (-50*b3 - 164) * q^54 + (-32*b3 - 56) * q^56 + (-95*b2 - 19*b1) * q^57 + (-18*b2 - 42*b1) * q^58 + (-25*b3 + 156) * q^59 + (111*b3 + 101) * q^61 + (-168*b2 + 88*b1) * q^62 + (-824*b2 - 127*b1) * q^63 - 64 * q^64 + (-8*b3 - 36) * q^66 + (-573*b2 + 77*b1) * q^67 + (-164*b2 - 8*b1) * q^68 + (-95*b3 - 354) * q^69 + (116*b3 + 42) * q^71 + (-128*b2 - 72*b1) * q^72 + (97*b2 - 184*b1) * q^73 + (56*b3 - 220) * q^74 + 76 * q^76 + (-16*b2 - 25*b1) * q^77 + (618*b2 + 134*b1) * q^78 + (58*b3 - 704) * q^79 + (-36*b3 + 589) * q^81 + (40*b2 - 20*b1) * q^82 + (-238*b2 + 194*b1) * q^83 + (108*b3 + 400) * q^84 + (14*b3 + 670) * q^86 + (423*b2 + 93*b1) * q^87 + (-64*b2 + 8*b1) * q^88 + (188*b3 + 24) * q^89 + (-151*b3 - 950) * q^91 + (-172*b2 - 52*b1) * q^92 + (-372*b2 - 92*b1) * q^93 + (142*b3 + 318) * q^94 + (32*b3 + 128) * q^96 + (-698*b2 - 102*b1) * q^97 + (132*b2 + 144*b1) * q^98 + (65*b3 - 99) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 16 q^{4} + 36 q^{6} - 46 q^{9}+O(q^{10})$$ 4 * q - 16 * q^4 + 36 * q^6 - 46 * q^9 $$4 q - 16 q^{4} + 36 q^{6} - 46 q^{9} - 34 q^{11} + 72 q^{14} + 64 q^{16} - 76 q^{19} - 454 q^{21} - 144 q^{24} + 68 q^{26} - 6 q^{29} + 424 q^{31} + 320 q^{34} + 184 q^{36} - 1102 q^{39} - 100 q^{41} + 136 q^{44} + 292 q^{46} - 120 q^{49} - 866 q^{51} - 756 q^{54} - 288 q^{56} + 574 q^{59} + 626 q^{61} - 256 q^{64} - 160 q^{66} - 1606 q^{69} + 400 q^{71} - 768 q^{74} + 304 q^{76} - 2700 q^{79} + 2284 q^{81} + 1816 q^{84} + 2708 q^{86} + 472 q^{89} - 4102 q^{91} + 1556 q^{94} + 576 q^{96} - 266 q^{99}+O(q^{100})$$ 4 * q - 16 * q^4 + 36 * q^6 - 46 * q^9 - 34 * q^11 + 72 * q^14 + 64 * q^16 - 76 * q^19 - 454 * q^21 - 144 * q^24 + 68 * q^26 - 6 * q^29 + 424 * q^31 + 320 * q^34 + 184 * q^36 - 1102 * q^39 - 100 * q^41 + 136 * q^44 + 292 * q^46 - 120 * q^49 - 866 * q^51 - 756 * q^54 - 288 * q^56 + 574 * q^59 + 626 * q^61 - 256 * q^64 - 160 * q^66 - 1606 * q^69 + 400 * q^71 - 768 * q^74 + 304 * q^76 - 2700 * q^79 + 2284 * q^81 + 1816 * q^84 + 2708 * q^86 + 472 * q^89 - 4102 * q^91 + 1556 * q^94 + 576 * q^96 - 266 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 37x^{2} + 324$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 19\nu ) / 18$$ (v^3 + 19*v) / 18 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 19$$ v^2 + 19
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 19$$ b3 - 19 $$\nu^{3}$$ $$=$$ $$18\beta_{2} - 19\beta_1$$ 18*b2 - 19*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
799.1
 − 4.77200i 3.77200i − 3.77200i 4.77200i
2.00000i 0.227998i −4.00000 0 0.455996 8.08801i 8.00000i 26.9480 0
799.2 2.00000i 8.77200i −4.00000 0 17.5440 26.0880i 8.00000i −49.9480 0
799.3 2.00000i 8.77200i −4.00000 0 17.5440 26.0880i 8.00000i −49.9480 0
799.4 2.00000i 0.227998i −4.00000 0 0.455996 8.08801i 8.00000i 26.9480 0
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.4.b.i 4
5.b even 2 1 inner 950.4.b.i 4
5.c odd 4 1 38.4.a.c 2
5.c odd 4 1 950.4.a.e 2
15.e even 4 1 342.4.a.h 2
20.e even 4 1 304.4.a.c 2
35.f even 4 1 1862.4.a.e 2
40.i odd 4 1 1216.4.a.g 2
40.k even 4 1 1216.4.a.p 2
95.g even 4 1 722.4.a.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.a.c 2 5.c odd 4 1
304.4.a.c 2 20.e even 4 1
342.4.a.h 2 15.e even 4 1
722.4.a.f 2 95.g even 4 1
950.4.a.e 2 5.c odd 4 1
950.4.b.i 4 1.a even 1 1 trivial
950.4.b.i 4 5.b even 2 1 inner
1216.4.a.g 2 40.i odd 4 1
1216.4.a.p 2 40.k even 4 1
1862.4.a.e 2 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(950, [\chi])$$:

 $$T_{3}^{4} + 77T_{3}^{2} + 4$$ T3^4 + 77*T3^2 + 4 $$T_{7}^{4} + 746T_{7}^{2} + 44521$$ T7^4 + 746*T7^2 + 44521

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 4)^{2}$$
$3$ $$T^{4} + 77T^{2} + 4$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 746 T^{2} + 44521$$
$11$ $$(T^{2} + 17 T + 54)^{2}$$
$13$ $$T^{4} + 6313 T^{2} + \cdots + 9072144$$
$17$ $$T^{4} + 3346 T^{2} + \cdots + 2331729$$
$19$ $$(T + 19)^{4}$$
$23$ $$T^{4} + 8833 T^{2} + \cdots + 3069504$$
$29$ $$(T^{2} + 3 T - 8046)^{2}$$
$31$ $$(T^{2} - 212 T - 24096)^{2}$$
$37$ $$T^{4} + 47048 T^{2} + \cdots + 25928464$$
$41$ $$(T^{2} + 50 T - 1200)^{2}$$
$43$ $$T^{4} + 230953 T^{2} + \cdots + 12924961344$$
$47$ $$T^{4} + 259657 T^{2} + \cdots + 2934172224$$
$53$ $$T^{4} + 753529 T^{2} + \cdots + 134114158656$$
$59$ $$(T^{2} - 287 T + 9186)^{2}$$
$61$ $$(T^{2} - 313 T - 200366)^{2}$$
$67$ $$T^{4} + 964273 T^{2} + \cdots + 70611369984$$
$71$ $$(T^{2} - 200 T - 235572)^{2}$$
$73$ $$T^{4} + 1307186 T^{2} + \cdots + 338899786801$$
$79$ $$(T^{2} + 1350 T + 394232)^{2}$$
$83$ $$T^{4} + 1598164 T^{2} + \cdots + 330201935424$$
$89$ $$(T^{2} - 236 T - 631104)^{2}$$
$97$ $$T^{4} + 1216964 T^{2} + \cdots + 52320157696$$