# Properties

 Label 370.4.a.a Level $370$ Weight $4$ Character orbit 370.a Self dual yes Analytic conductor $21.831$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [370,4,Mod(1,370)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(370, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("370.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 370.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: $$21.8307067021$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{5} - 4 q^{6} + 2 q^{7} + 8 q^{8} - 23 q^{9}+O(q^{10})$$ q + 2 * q^2 - 2 * q^3 + 4 * q^4 + 5 * q^5 - 4 * q^6 + 2 * q^7 + 8 * q^8 - 23 * q^9 $$q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{5} - 4 q^{6} + 2 q^{7} + 8 q^{8} - 23 q^{9} + 10 q^{10} - 72 q^{11} - 8 q^{12} + 2 q^{13} + 4 q^{14} - 10 q^{15} + 16 q^{16} - 66 q^{17} - 46 q^{18} + 38 q^{19} + 20 q^{20} - 4 q^{21} - 144 q^{22} - 36 q^{23} - 16 q^{24} + 25 q^{25} + 4 q^{26} + 100 q^{27} + 8 q^{28} - 90 q^{29} - 20 q^{30} - 70 q^{31} + 32 q^{32} + 144 q^{33} - 132 q^{34} + 10 q^{35} - 92 q^{36} + 37 q^{37} + 76 q^{38} - 4 q^{39} + 40 q^{40} - 438 q^{41} - 8 q^{42} + 272 q^{43} - 288 q^{44} - 115 q^{45} - 72 q^{46} - 198 q^{47} - 32 q^{48} - 339 q^{49} + 50 q^{50} + 132 q^{51} + 8 q^{52} - 354 q^{53} + 200 q^{54} - 360 q^{55} + 16 q^{56} - 76 q^{57} - 180 q^{58} - 498 q^{59} - 40 q^{60} + 542 q^{61} - 140 q^{62} - 46 q^{63} + 64 q^{64} + 10 q^{65} + 288 q^{66} + 2 q^{67} - 264 q^{68} + 72 q^{69} + 20 q^{70} + 408 q^{71} - 184 q^{72} - 358 q^{73} + 74 q^{74} - 50 q^{75} + 152 q^{76} - 144 q^{77} - 8 q^{78} + 722 q^{79} + 80 q^{80} + 421 q^{81} - 876 q^{82} - 174 q^{83} - 16 q^{84} - 330 q^{85} + 544 q^{86} + 180 q^{87} - 576 q^{88} - 102 q^{89} - 230 q^{90} + 4 q^{91} - 144 q^{92} + 140 q^{93} - 396 q^{94} + 190 q^{95} - 64 q^{96} - 574 q^{97} - 678 q^{98} + 1656 q^{99}+O(q^{100})$$ q + 2 * q^2 - 2 * q^3 + 4 * q^4 + 5 * q^5 - 4 * q^6 + 2 * q^7 + 8 * q^8 - 23 * q^9 + 10 * q^10 - 72 * q^11 - 8 * q^12 + 2 * q^13 + 4 * q^14 - 10 * q^15 + 16 * q^16 - 66 * q^17 - 46 * q^18 + 38 * q^19 + 20 * q^20 - 4 * q^21 - 144 * q^22 - 36 * q^23 - 16 * q^24 + 25 * q^25 + 4 * q^26 + 100 * q^27 + 8 * q^28 - 90 * q^29 - 20 * q^30 - 70 * q^31 + 32 * q^32 + 144 * q^33 - 132 * q^34 + 10 * q^35 - 92 * q^36 + 37 * q^37 + 76 * q^38 - 4 * q^39 + 40 * q^40 - 438 * q^41 - 8 * q^42 + 272 * q^43 - 288 * q^44 - 115 * q^45 - 72 * q^46 - 198 * q^47 - 32 * q^48 - 339 * q^49 + 50 * q^50 + 132 * q^51 + 8 * q^52 - 354 * q^53 + 200 * q^54 - 360 * q^55 + 16 * q^56 - 76 * q^57 - 180 * q^58 - 498 * q^59 - 40 * q^60 + 542 * q^61 - 140 * q^62 - 46 * q^63 + 64 * q^64 + 10 * q^65 + 288 * q^66 + 2 * q^67 - 264 * q^68 + 72 * q^69 + 20 * q^70 + 408 * q^71 - 184 * q^72 - 358 * q^73 + 74 * q^74 - 50 * q^75 + 152 * q^76 - 144 * q^77 - 8 * q^78 + 722 * q^79 + 80 * q^80 + 421 * q^81 - 876 * q^82 - 174 * q^83 - 16 * q^84 - 330 * q^85 + 544 * q^86 + 180 * q^87 - 576 * q^88 - 102 * q^89 - 230 * q^90 + 4 * q^91 - 144 * q^92 + 140 * q^93 - 396 * q^94 + 190 * q^95 - 64 * q^96 - 574 * q^97 - 678 * q^98 + 1656 * 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
 0
2.00000 −2.00000 4.00000 5.00000 −4.00000 2.00000 8.00000 −23.0000 10.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$37$$ $$-1$$

## 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 370.4.a.a 1
5.b even 2 1 1850.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.4.a.a 1 1.a even 1 1 trivial
1850.4.a.c 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(370))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T + 2$$
$5$ $$T - 5$$
$7$ $$T - 2$$
$11$ $$T + 72$$
$13$ $$T - 2$$
$17$ $$T + 66$$
$19$ $$T - 38$$
$23$ $$T + 36$$
$29$ $$T + 90$$
$31$ $$T + 70$$
$37$ $$T - 37$$
$41$ $$T + 438$$
$43$ $$T - 272$$
$47$ $$T + 198$$
$53$ $$T + 354$$
$59$ $$T + 498$$
$61$ $$T - 542$$
$67$ $$T - 2$$
$71$ $$T - 408$$
$73$ $$T + 358$$
$79$ $$T - 722$$
$83$ $$T + 174$$
$89$ $$T + 102$$
$97$ $$T + 574$$