# Properties

 Label 350.2.c.d Level $350$ Weight $2$ Character orbit 350.c Analytic conductor $2.795$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,2,Mod(99,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("350.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 350.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: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - 2 i q^{3} - q^{4} + 2 q^{6} - i q^{7} - i q^{8} - q^{9} +O(q^{10})$$ q + i * q^2 - 2*i * q^3 - q^4 + 2 * q^6 - i * q^7 - i * q^8 - q^9 $$q + i q^{2} - 2 i q^{3} - q^{4} + 2 q^{6} - i q^{7} - i q^{8} - q^{9} + 2 i q^{12} - 4 i q^{13} + q^{14} + q^{16} - 6 i q^{17} - i q^{18} - 2 q^{19} - 2 q^{21} - 2 q^{24} + 4 q^{26} - 4 i q^{27} + i q^{28} + 6 q^{29} - 4 q^{31} + i q^{32} + 6 q^{34} + q^{36} - 2 i q^{37} - 2 i q^{38} - 8 q^{39} + 6 q^{41} - 2 i q^{42} + 8 i q^{43} + 12 i q^{47} - 2 i q^{48} - q^{49} - 12 q^{51} + 4 i q^{52} + 6 i q^{53} + 4 q^{54} - q^{56} + 4 i q^{57} + 6 i q^{58} + 6 q^{59} + 8 q^{61} - 4 i q^{62} + i q^{63} - q^{64} + 4 i q^{67} + 6 i q^{68} + i q^{72} + 2 i q^{73} + 2 q^{74} + 2 q^{76} - 8 i q^{78} - 8 q^{79} - 11 q^{81} + 6 i q^{82} - 6 i q^{83} + 2 q^{84} - 8 q^{86} - 12 i q^{87} + 6 q^{89} - 4 q^{91} + 8 i q^{93} - 12 q^{94} + 2 q^{96} + 10 i q^{97} - i q^{98} +O(q^{100})$$ q + i * q^2 - 2*i * q^3 - q^4 + 2 * q^6 - i * q^7 - i * q^8 - q^9 + 2*i * q^12 - 4*i * q^13 + q^14 + q^16 - 6*i * q^17 - i * q^18 - 2 * q^19 - 2 * q^21 - 2 * q^24 + 4 * q^26 - 4*i * q^27 + i * q^28 + 6 * q^29 - 4 * q^31 + i * q^32 + 6 * q^34 + q^36 - 2*i * q^37 - 2*i * q^38 - 8 * q^39 + 6 * q^41 - 2*i * q^42 + 8*i * q^43 + 12*i * q^47 - 2*i * q^48 - q^49 - 12 * q^51 + 4*i * q^52 + 6*i * q^53 + 4 * q^54 - q^56 + 4*i * q^57 + 6*i * q^58 + 6 * q^59 + 8 * q^61 - 4*i * q^62 + i * q^63 - q^64 + 4*i * q^67 + 6*i * q^68 + i * q^72 + 2*i * q^73 + 2 * q^74 + 2 * q^76 - 8*i * q^78 - 8 * q^79 - 11 * q^81 + 6*i * q^82 - 6*i * q^83 + 2 * q^84 - 8 * q^86 - 12*i * q^87 + 6 * q^89 - 4 * q^91 + 8*i * q^93 - 12 * q^94 + 2 * q^96 + 10*i * q^97 - i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} + 2 q^{14} + 2 q^{16} - 4 q^{19} - 4 q^{21} - 4 q^{24} + 8 q^{26} + 12 q^{29} - 8 q^{31} + 12 q^{34} + 2 q^{36} - 16 q^{39} + 12 q^{41} - 2 q^{49} - 24 q^{51} + 8 q^{54} - 2 q^{56} + 12 q^{59} + 16 q^{61} - 2 q^{64} + 4 q^{74} + 4 q^{76} - 16 q^{79} - 22 q^{81} + 4 q^{84} - 16 q^{86} + 12 q^{89} - 8 q^{91} - 24 q^{94} + 4 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 + 2 * q^14 + 2 * q^16 - 4 * q^19 - 4 * q^21 - 4 * q^24 + 8 * q^26 + 12 * q^29 - 8 * q^31 + 12 * q^34 + 2 * q^36 - 16 * q^39 + 12 * q^41 - 2 * q^49 - 24 * q^51 + 8 * q^54 - 2 * q^56 + 12 * q^59 + 16 * q^61 - 2 * q^64 + 4 * q^74 + 4 * q^76 - 16 * q^79 - 22 * q^81 + 4 * q^84 - 16 * q^86 + 12 * q^89 - 8 * q^91 - 24 * q^94 + 4 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 − 1.00000i 1.00000i
1.00000i 2.00000i −1.00000 0 2.00000 1.00000i 1.00000i −1.00000 0
99.2 1.00000i 2.00000i −1.00000 0 2.00000 1.00000i 1.00000i −1.00000 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 350.2.c.d 2
3.b odd 2 1 3150.2.g.j 2
4.b odd 2 1 2800.2.g.h 2
5.b even 2 1 inner 350.2.c.d 2
5.c odd 4 1 14.2.a.a 1
5.c odd 4 1 350.2.a.f 1
7.b odd 2 1 2450.2.c.c 2
15.d odd 2 1 3150.2.g.j 2
15.e even 4 1 126.2.a.b 1
15.e even 4 1 3150.2.a.i 1
20.d odd 2 1 2800.2.g.h 2
20.e even 4 1 112.2.a.c 1
20.e even 4 1 2800.2.a.g 1
35.c odd 2 1 2450.2.c.c 2
35.f even 4 1 98.2.a.a 1
35.f even 4 1 2450.2.a.t 1
35.k even 12 2 98.2.c.a 2
35.l odd 12 2 98.2.c.b 2
40.i odd 4 1 448.2.a.g 1
40.k even 4 1 448.2.a.a 1
45.k odd 12 2 1134.2.f.l 2
45.l even 12 2 1134.2.f.f 2
55.e even 4 1 1694.2.a.e 1
60.l odd 4 1 1008.2.a.h 1
65.f even 4 1 2366.2.d.b 2
65.h odd 4 1 2366.2.a.j 1
65.k even 4 1 2366.2.d.b 2
80.i odd 4 1 1792.2.b.c 2
80.j even 4 1 1792.2.b.g 2
80.s even 4 1 1792.2.b.g 2
80.t odd 4 1 1792.2.b.c 2
85.g odd 4 1 4046.2.a.f 1
95.g even 4 1 5054.2.a.c 1
105.k odd 4 1 882.2.a.i 1
105.w odd 12 2 882.2.g.d 2
105.x even 12 2 882.2.g.c 2
115.e even 4 1 7406.2.a.a 1
120.q odd 4 1 4032.2.a.r 1
120.w even 4 1 4032.2.a.w 1
140.j odd 4 1 784.2.a.b 1
140.w even 12 2 784.2.i.c 2
140.x odd 12 2 784.2.i.i 2
280.s even 4 1 3136.2.a.e 1
280.y odd 4 1 3136.2.a.z 1
420.w even 4 1 7056.2.a.bd 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 5.c odd 4 1
98.2.a.a 1 35.f even 4 1
98.2.c.a 2 35.k even 12 2
98.2.c.b 2 35.l odd 12 2
112.2.a.c 1 20.e even 4 1
126.2.a.b 1 15.e even 4 1
350.2.a.f 1 5.c odd 4 1
350.2.c.d 2 1.a even 1 1 trivial
350.2.c.d 2 5.b even 2 1 inner
448.2.a.a 1 40.k even 4 1
448.2.a.g 1 40.i odd 4 1
784.2.a.b 1 140.j odd 4 1
784.2.i.c 2 140.w even 12 2
784.2.i.i 2 140.x odd 12 2
882.2.a.i 1 105.k odd 4 1
882.2.g.c 2 105.x even 12 2
882.2.g.d 2 105.w odd 12 2
1008.2.a.h 1 60.l odd 4 1
1134.2.f.f 2 45.l even 12 2
1134.2.f.l 2 45.k odd 12 2
1694.2.a.e 1 55.e even 4 1
1792.2.b.c 2 80.i odd 4 1
1792.2.b.c 2 80.t odd 4 1
1792.2.b.g 2 80.j even 4 1
1792.2.b.g 2 80.s even 4 1
2366.2.a.j 1 65.h odd 4 1
2366.2.d.b 2 65.f even 4 1
2366.2.d.b 2 65.k even 4 1
2450.2.a.t 1 35.f even 4 1
2450.2.c.c 2 7.b odd 2 1
2450.2.c.c 2 35.c odd 2 1
2800.2.a.g 1 20.e even 4 1
2800.2.g.h 2 4.b odd 2 1
2800.2.g.h 2 20.d odd 2 1
3136.2.a.e 1 280.s even 4 1
3136.2.a.z 1 280.y odd 4 1
3150.2.a.i 1 15.e even 4 1
3150.2.g.j 2 3.b odd 2 1
3150.2.g.j 2 15.d odd 2 1
4032.2.a.r 1 120.q odd 4 1
4032.2.a.w 1 120.w even 4 1
4046.2.a.f 1 85.g odd 4 1
5054.2.a.c 1 95.g even 4 1
7056.2.a.bd 1 420.w even 4 1
7406.2.a.a 1 115.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(350, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 36$$
$19$ $$(T + 2)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2} + 144$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 6)^{2}$$
$61$ $$(T - 8)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 100$$