# Properties

 Label 3850.2.c.q Level $3850$ Weight $2$ Character orbit 3850.c Analytic conductor $30.742$ 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] = [3850,2,Mod(1849,3850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3850, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3850.1849");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3850.c (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: $$30.7424047782$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 154) 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} - \beta_1) q^{3} - q^{4} + (\beta_{3} - 1) q^{6} - \beta_1 q^{7} + \beta_1 q^{8} + (2 \beta_{3} - 3) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b2 - b1) * q^3 - q^4 + (b3 - 1) * q^6 - b1 * q^7 + b1 * q^8 + (2*b3 - 3) * q^9 $$q - \beta_1 q^{2} + (\beta_{2} - \beta_1) q^{3} - q^{4} + (\beta_{3} - 1) q^{6} - \beta_1 q^{7} + \beta_1 q^{8} + (2 \beta_{3} - 3) q^{9} + q^{11} + ( - \beta_{2} + \beta_1) q^{12} + ( - \beta_{2} - \beta_1) q^{13} - q^{14} + q^{16} + ( - 2 \beta_{2} + 2 \beta_1) q^{17} + ( - 2 \beta_{2} + 3 \beta_1) q^{18} + (\beta_{3} + 5) q^{19} + (\beta_{3} - 1) q^{21} - \beta_1 q^{22} + 4 \beta_1 q^{23} + ( - \beta_{3} + 1) q^{24} + ( - \beta_{3} - 1) q^{26} + ( - 2 \beta_{2} + 10 \beta_1) q^{27} + \beta_1 q^{28} - 2 \beta_{3} q^{29} + 2 q^{31} - \beta_1 q^{32} + (\beta_{2} - \beta_1) q^{33} + ( - 2 \beta_{3} + 2) q^{34} + ( - 2 \beta_{3} + 3) q^{36} + ( - 4 \beta_{2} + 2 \beta_1) q^{37} + ( - \beta_{2} - 5 \beta_1) q^{38} + 4 q^{39} + ( - 2 \beta_{3} + 2) q^{41} + ( - \beta_{2} + \beta_1) q^{42} + ( - 2 \beta_{2} - 6 \beta_1) q^{43} - q^{44} + 4 q^{46} + 2 \beta_1 q^{47} + (\beta_{2} - \beta_1) q^{48} - q^{49} + ( - 4 \beta_{3} + 12) q^{51} + (\beta_{2} + \beta_1) q^{52} + (2 \beta_{2} + 4 \beta_1) q^{53} + ( - 2 \beta_{3} + 10) q^{54} + q^{56} + 4 \beta_{2} q^{57} + 2 \beta_{2} q^{58} + (\beta_{3} - 5) q^{59} + (\beta_{3} - 3) q^{61} - 2 \beta_1 q^{62} + ( - 2 \beta_{2} + 3 \beta_1) q^{63} - q^{64} + (\beta_{3} - 1) q^{66} + ( - 6 \beta_{2} + 2 \beta_1) q^{67} + (2 \beta_{2} - 2 \beta_1) q^{68} + ( - 4 \beta_{3} + 4) q^{69} + (2 \beta_{3} + 2) q^{71} + (2 \beta_{2} - 3 \beta_1) q^{72} + (4 \beta_{2} + 4 \beta_1) q^{73} + ( - 4 \beta_{3} + 2) q^{74} + ( - \beta_{3} - 5) q^{76} - \beta_1 q^{77} - 4 \beta_1 q^{78} + ( - 6 \beta_{3} + 11) q^{81} + (2 \beta_{2} - 2 \beta_1) q^{82} + ( - 5 \beta_{2} - \beta_1) q^{83} + ( - \beta_{3} + 1) q^{84} + ( - 2 \beta_{3} - 6) q^{86} + (2 \beta_{2} - 10 \beta_1) q^{87} + \beta_1 q^{88} - 10 q^{89} + ( - \beta_{3} - 1) q^{91} - 4 \beta_1 q^{92} + (2 \beta_{2} - 2 \beta_1) q^{93} + 2 q^{94} + (\beta_{3} - 1) q^{96} + ( - 2 \beta_{2} - 8 \beta_1) q^{97} + \beta_1 q^{98} + (2 \beta_{3} - 3) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b2 - b1) * q^3 - q^4 + (b3 - 1) * q^6 - b1 * q^7 + b1 * q^8 + (2*b3 - 3) * q^9 + q^11 + (-b2 + b1) * q^12 + (-b2 - b1) * q^13 - q^14 + q^16 + (-2*b2 + 2*b1) * q^17 + (-2*b2 + 3*b1) * q^18 + (b3 + 5) * q^19 + (b3 - 1) * q^21 - b1 * q^22 + 4*b1 * q^23 + (-b3 + 1) * q^24 + (-b3 - 1) * q^26 + (-2*b2 + 10*b1) * q^27 + b1 * q^28 - 2*b3 * q^29 + 2 * q^31 - b1 * q^32 + (b2 - b1) * q^33 + (-2*b3 + 2) * q^34 + (-2*b3 + 3) * q^36 + (-4*b2 + 2*b1) * q^37 + (-b2 - 5*b1) * q^38 + 4 * q^39 + (-2*b3 + 2) * q^41 + (-b2 + b1) * q^42 + (-2*b2 - 6*b1) * q^43 - q^44 + 4 * q^46 + 2*b1 * q^47 + (b2 - b1) * q^48 - q^49 + (-4*b3 + 12) * q^51 + (b2 + b1) * q^52 + (2*b2 + 4*b1) * q^53 + (-2*b3 + 10) * q^54 + q^56 + 4*b2 * q^57 + 2*b2 * q^58 + (b3 - 5) * q^59 + (b3 - 3) * q^61 - 2*b1 * q^62 + (-2*b2 + 3*b1) * q^63 - q^64 + (b3 - 1) * q^66 + (-6*b2 + 2*b1) * q^67 + (2*b2 - 2*b1) * q^68 + (-4*b3 + 4) * q^69 + (2*b3 + 2) * q^71 + (2*b2 - 3*b1) * q^72 + (4*b2 + 4*b1) * q^73 + (-4*b3 + 2) * q^74 + (-b3 - 5) * q^76 - b1 * q^77 - 4*b1 * q^78 + (-6*b3 + 11) * q^81 + (2*b2 - 2*b1) * q^82 + (-5*b2 - b1) * q^83 + (-b3 + 1) * q^84 + (-2*b3 - 6) * q^86 + (2*b2 - 10*b1) * q^87 + b1 * q^88 - 10 * q^89 + (-b3 - 1) * q^91 - 4*b1 * q^92 + (2*b2 - 2*b1) * q^93 + 2 * q^94 + (b3 - 1) * q^96 + (-2*b2 - 8*b1) * q^97 + b1 * q^98 + (2*b3 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 4 q^{6} - 12 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 4 * q^6 - 12 * q^9 $$4 q - 4 q^{4} - 4 q^{6} - 12 q^{9} + 4 q^{11} - 4 q^{14} + 4 q^{16} + 20 q^{19} - 4 q^{21} + 4 q^{24} - 4 q^{26} + 8 q^{31} + 8 q^{34} + 12 q^{36} + 16 q^{39} + 8 q^{41} - 4 q^{44} + 16 q^{46} - 4 q^{49} + 48 q^{51} + 40 q^{54} + 4 q^{56} - 20 q^{59} - 12 q^{61} - 4 q^{64} - 4 q^{66} + 16 q^{69} + 8 q^{71} + 8 q^{74} - 20 q^{76} + 44 q^{81} + 4 q^{84} - 24 q^{86} - 40 q^{89} - 4 q^{91} + 8 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 - 4 * q^6 - 12 * q^9 + 4 * q^11 - 4 * q^14 + 4 * q^16 + 20 * q^19 - 4 * q^21 + 4 * q^24 - 4 * q^26 + 8 * q^31 + 8 * q^34 + 12 * q^36 + 16 * q^39 + 8 * q^41 - 4 * q^44 + 16 * q^46 - 4 * q^49 + 48 * q^51 + 40 * q^54 + 4 * q^56 - 20 * q^59 - 12 * q^61 - 4 * q^64 - 4 * q^66 + 16 * q^69 + 8 * q^71 + 8 * q^74 - 20 * q^76 + 44 * q^81 + 4 * q^84 - 24 * q^86 - 40 * q^89 - 4 * q^91 + 8 * q^94 - 4 * q^96 - 12 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v $$\beta_{2}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 3$$ 2*v^2 + 3
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 3 ) / 2$$ (b3 - 3) / 2 $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 2\beta_1$$ -b2 + 2*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1751$$ $$2201$$ $$2927$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1849.1
 − 1.61803i 0.618034i − 0.618034i 1.61803i
1.00000i 3.23607i −1.00000 0 −3.23607 1.00000i 1.00000i −7.47214 0
1849.2 1.00000i 1.23607i −1.00000 0 1.23607 1.00000i 1.00000i 1.47214 0
1849.3 1.00000i 1.23607i −1.00000 0 1.23607 1.00000i 1.00000i 1.47214 0
1849.4 1.00000i 3.23607i −1.00000 0 −3.23607 1.00000i 1.00000i −7.47214 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 3850.2.c.q 4
5.b even 2 1 inner 3850.2.c.q 4
5.c odd 4 1 154.2.a.d 2
5.c odd 4 1 3850.2.a.bj 2
15.e even 4 1 1386.2.a.m 2
20.e even 4 1 1232.2.a.p 2
35.f even 4 1 1078.2.a.w 2
35.k even 12 2 1078.2.e.n 4
35.l odd 12 2 1078.2.e.q 4
40.i odd 4 1 4928.2.a.bt 2
40.k even 4 1 4928.2.a.bk 2
55.e even 4 1 1694.2.a.l 2
105.k odd 4 1 9702.2.a.cu 2
140.j odd 4 1 8624.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.d 2 5.c odd 4 1
1078.2.a.w 2 35.f even 4 1
1078.2.e.n 4 35.k even 12 2
1078.2.e.q 4 35.l odd 12 2
1232.2.a.p 2 20.e even 4 1
1386.2.a.m 2 15.e even 4 1
1694.2.a.l 2 55.e even 4 1
3850.2.a.bj 2 5.c odd 4 1
3850.2.c.q 4 1.a even 1 1 trivial
3850.2.c.q 4 5.b even 2 1 inner
4928.2.a.bk 2 40.k even 4 1
4928.2.a.bt 2 40.i odd 4 1
8624.2.a.bf 2 140.j odd 4 1
9702.2.a.cu 2 105.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3850, [\chi])$$:

 $$T_{3}^{4} + 12T_{3}^{2} + 16$$ T3^4 + 12*T3^2 + 16 $$T_{13}^{4} + 12T_{13}^{2} + 16$$ T13^4 + 12*T13^2 + 16 $$T_{17}^{4} + 48T_{17}^{2} + 256$$ T17^4 + 48*T17^2 + 256 $$T_{19}^{2} - 10T_{19} + 20$$ T19^2 - 10*T19 + 20 $$T_{37}^{4} + 168T_{37}^{2} + 5776$$ T37^4 + 168*T37^2 + 5776

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4} + 12T^{2} + 16$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 1)^{2}$$
$11$ $$(T - 1)^{4}$$
$13$ $$T^{4} + 12T^{2} + 16$$
$17$ $$T^{4} + 48T^{2} + 256$$
$19$ $$(T^{2} - 10 T + 20)^{2}$$
$23$ $$(T^{2} + 16)^{2}$$
$29$ $$(T^{2} - 20)^{2}$$
$31$ $$(T - 2)^{4}$$
$37$ $$T^{4} + 168T^{2} + 5776$$
$41$ $$(T^{2} - 4 T - 16)^{2}$$
$43$ $$T^{4} + 112T^{2} + 256$$
$47$ $$(T^{2} + 4)^{2}$$
$53$ $$T^{4} + 72T^{2} + 16$$
$59$ $$(T^{2} + 10 T + 20)^{2}$$
$61$ $$(T^{2} + 6 T + 4)^{2}$$
$67$ $$T^{4} + 368 T^{2} + 30976$$
$71$ $$(T^{2} - 4 T - 16)^{2}$$
$73$ $$T^{4} + 192T^{2} + 4096$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 252 T^{2} + 15376$$
$89$ $$(T + 10)^{4}$$
$97$ $$T^{4} + 168T^{2} + 1936$$