# Properties

 Label 350.4.e.e Level $350$ Weight $4$ Character orbit 350.e Analytic conductor $20.651$ 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] = [350,4,Mod(51,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("350.51");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 350.e (of order $$3$$, degree $$2$$, 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: $$20.6506685020$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + (5 \zeta_{6} - 5) q^{3} + (4 \zeta_{6} - 4) q^{4} - 10 q^{6} + ( - 14 \zeta_{6} + 21) q^{7} - 8 q^{8} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + 2*z * q^2 + (5*z - 5) * q^3 + (4*z - 4) * q^4 - 10 * q^6 + (-14*z + 21) * q^7 - 8 * q^8 + 2*z * q^9 $$q + 2 \zeta_{6} q^{2} + (5 \zeta_{6} - 5) q^{3} + (4 \zeta_{6} - 4) q^{4} - 10 q^{6} + ( - 14 \zeta_{6} + 21) q^{7} - 8 q^{8} + 2 \zeta_{6} q^{9} + ( - 57 \zeta_{6} + 57) q^{11} - 20 \zeta_{6} q^{12} + 70 q^{13} + (14 \zeta_{6} + 28) q^{14} - 16 \zeta_{6} q^{16} + ( - 51 \zeta_{6} + 51) q^{17} + (4 \zeta_{6} - 4) q^{18} - 5 \zeta_{6} q^{19} + (105 \zeta_{6} - 35) q^{21} + 114 q^{22} + 69 \zeta_{6} q^{23} + ( - 40 \zeta_{6} + 40) q^{24} + 140 \zeta_{6} q^{26} - 145 q^{27} + (84 \zeta_{6} - 28) q^{28} + 114 q^{29} + (23 \zeta_{6} - 23) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} + 285 \zeta_{6} q^{33} + 102 q^{34} - 8 q^{36} - 253 \zeta_{6} q^{37} + ( - 10 \zeta_{6} + 10) q^{38} + (350 \zeta_{6} - 350) q^{39} - 42 q^{41} + (140 \zeta_{6} - 210) q^{42} + 124 q^{43} + 228 \zeta_{6} q^{44} + (138 \zeta_{6} - 138) q^{46} + 201 \zeta_{6} q^{47} + 80 q^{48} + ( - 392 \zeta_{6} + 245) q^{49} + 255 \zeta_{6} q^{51} + (280 \zeta_{6} - 280) q^{52} + (393 \zeta_{6} - 393) q^{53} - 290 \zeta_{6} q^{54} + (112 \zeta_{6} - 168) q^{56} + 25 q^{57} + 228 \zeta_{6} q^{58} + (219 \zeta_{6} - 219) q^{59} + 709 \zeta_{6} q^{61} - 46 q^{62} + (14 \zeta_{6} + 28) q^{63} + 64 q^{64} + (570 \zeta_{6} - 570) q^{66} + ( - 419 \zeta_{6} + 419) q^{67} + 204 \zeta_{6} q^{68} - 345 q^{69} - 96 q^{71} - 16 \zeta_{6} q^{72} + (313 \zeta_{6} - 313) q^{73} + ( - 506 \zeta_{6} + 506) q^{74} + 20 q^{76} + ( - 1197 \zeta_{6} + 399) q^{77} - 700 q^{78} - 461 \zeta_{6} q^{79} + ( - 671 \zeta_{6} + 671) q^{81} - 84 \zeta_{6} q^{82} + 588 q^{83} + ( - 140 \zeta_{6} - 280) q^{84} + 248 \zeta_{6} q^{86} + (570 \zeta_{6} - 570) q^{87} + (456 \zeta_{6} - 456) q^{88} + 1017 \zeta_{6} q^{89} + ( - 980 \zeta_{6} + 1470) q^{91} - 276 q^{92} - 115 \zeta_{6} q^{93} + (402 \zeta_{6} - 402) q^{94} + 160 \zeta_{6} q^{96} + 1834 q^{97} + ( - 294 \zeta_{6} + 784) q^{98} + 114 q^{99} +O(q^{100})$$ q + 2*z * q^2 + (5*z - 5) * q^3 + (4*z - 4) * q^4 - 10 * q^6 + (-14*z + 21) * q^7 - 8 * q^8 + 2*z * q^9 + (-57*z + 57) * q^11 - 20*z * q^12 + 70 * q^13 + (14*z + 28) * q^14 - 16*z * q^16 + (-51*z + 51) * q^17 + (4*z - 4) * q^18 - 5*z * q^19 + (105*z - 35) * q^21 + 114 * q^22 + 69*z * q^23 + (-40*z + 40) * q^24 + 140*z * q^26 - 145 * q^27 + (84*z - 28) * q^28 + 114 * q^29 + (23*z - 23) * q^31 + (-32*z + 32) * q^32 + 285*z * q^33 + 102 * q^34 - 8 * q^36 - 253*z * q^37 + (-10*z + 10) * q^38 + (350*z - 350) * q^39 - 42 * q^41 + (140*z - 210) * q^42 + 124 * q^43 + 228*z * q^44 + (138*z - 138) * q^46 + 201*z * q^47 + 80 * q^48 + (-392*z + 245) * q^49 + 255*z * q^51 + (280*z - 280) * q^52 + (393*z - 393) * q^53 - 290*z * q^54 + (112*z - 168) * q^56 + 25 * q^57 + 228*z * q^58 + (219*z - 219) * q^59 + 709*z * q^61 - 46 * q^62 + (14*z + 28) * q^63 + 64 * q^64 + (570*z - 570) * q^66 + (-419*z + 419) * q^67 + 204*z * q^68 - 345 * q^69 - 96 * q^71 - 16*z * q^72 + (313*z - 313) * q^73 + (-506*z + 506) * q^74 + 20 * q^76 + (-1197*z + 399) * q^77 - 700 * q^78 - 461*z * q^79 + (-671*z + 671) * q^81 - 84*z * q^82 + 588 * q^83 + (-140*z - 280) * q^84 + 248*z * q^86 + (570*z - 570) * q^87 + (456*z - 456) * q^88 + 1017*z * q^89 + (-980*z + 1470) * q^91 - 276 * q^92 - 115*z * q^93 + (402*z - 402) * q^94 + 160*z * q^96 + 1834 * q^97 + (-294*z + 784) * q^98 + 114 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 5 q^{3} - 4 q^{4} - 20 q^{6} + 28 q^{7} - 16 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 5 * q^3 - 4 * q^4 - 20 * q^6 + 28 * q^7 - 16 * q^8 + 2 * q^9 $$2 q + 2 q^{2} - 5 q^{3} - 4 q^{4} - 20 q^{6} + 28 q^{7} - 16 q^{8} + 2 q^{9} + 57 q^{11} - 20 q^{12} + 140 q^{13} + 70 q^{14} - 16 q^{16} + 51 q^{17} - 4 q^{18} - 5 q^{19} + 35 q^{21} + 228 q^{22} + 69 q^{23} + 40 q^{24} + 140 q^{26} - 290 q^{27} + 28 q^{28} + 228 q^{29} - 23 q^{31} + 32 q^{32} + 285 q^{33} + 204 q^{34} - 16 q^{36} - 253 q^{37} + 10 q^{38} - 350 q^{39} - 84 q^{41} - 280 q^{42} + 248 q^{43} + 228 q^{44} - 138 q^{46} + 201 q^{47} + 160 q^{48} + 98 q^{49} + 255 q^{51} - 280 q^{52} - 393 q^{53} - 290 q^{54} - 224 q^{56} + 50 q^{57} + 228 q^{58} - 219 q^{59} + 709 q^{61} - 92 q^{62} + 70 q^{63} + 128 q^{64} - 570 q^{66} + 419 q^{67} + 204 q^{68} - 690 q^{69} - 192 q^{71} - 16 q^{72} - 313 q^{73} + 506 q^{74} + 40 q^{76} - 399 q^{77} - 1400 q^{78} - 461 q^{79} + 671 q^{81} - 84 q^{82} + 1176 q^{83} - 700 q^{84} + 248 q^{86} - 570 q^{87} - 456 q^{88} + 1017 q^{89} + 1960 q^{91} - 552 q^{92} - 115 q^{93} - 402 q^{94} + 160 q^{96} + 3668 q^{97} + 1274 q^{98} + 228 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 5 * q^3 - 4 * q^4 - 20 * q^6 + 28 * q^7 - 16 * q^8 + 2 * q^9 + 57 * q^11 - 20 * q^12 + 140 * q^13 + 70 * q^14 - 16 * q^16 + 51 * q^17 - 4 * q^18 - 5 * q^19 + 35 * q^21 + 228 * q^22 + 69 * q^23 + 40 * q^24 + 140 * q^26 - 290 * q^27 + 28 * q^28 + 228 * q^29 - 23 * q^31 + 32 * q^32 + 285 * q^33 + 204 * q^34 - 16 * q^36 - 253 * q^37 + 10 * q^38 - 350 * q^39 - 84 * q^41 - 280 * q^42 + 248 * q^43 + 228 * q^44 - 138 * q^46 + 201 * q^47 + 160 * q^48 + 98 * q^49 + 255 * q^51 - 280 * q^52 - 393 * q^53 - 290 * q^54 - 224 * q^56 + 50 * q^57 + 228 * q^58 - 219 * q^59 + 709 * q^61 - 92 * q^62 + 70 * q^63 + 128 * q^64 - 570 * q^66 + 419 * q^67 + 204 * q^68 - 690 * q^69 - 192 * q^71 - 16 * q^72 - 313 * q^73 + 506 * q^74 + 40 * q^76 - 399 * q^77 - 1400 * q^78 - 461 * q^79 + 671 * q^81 - 84 * q^82 + 1176 * q^83 - 700 * q^84 + 248 * q^86 - 570 * q^87 - 456 * q^88 + 1017 * q^89 + 1960 * q^91 - 552 * q^92 - 115 * q^93 - 402 * q^94 + 160 * q^96 + 3668 * q^97 + 1274 * q^98 + 228 * q^99

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
51.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 1.73205i −2.50000 4.33013i −2.00000 3.46410i 0 −10.0000 14.0000 + 12.1244i −8.00000 1.00000 1.73205i 0
151.1 1.00000 + 1.73205i −2.50000 + 4.33013i −2.00000 + 3.46410i 0 −10.0000 14.0000 12.1244i −8.00000 1.00000 + 1.73205i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.4.e.e 2
5.b even 2 1 14.4.c.a 2
5.c odd 4 2 350.4.j.b 4
7.c even 3 1 inner 350.4.e.e 2
7.c even 3 1 2450.4.a.q 1
7.d odd 6 1 2450.4.a.d 1
15.d odd 2 1 126.4.g.d 2
20.d odd 2 1 112.4.i.a 2
35.c odd 2 1 98.4.c.a 2
35.i odd 6 1 98.4.a.f 1
35.i odd 6 1 98.4.c.a 2
35.j even 6 1 14.4.c.a 2
35.j even 6 1 98.4.a.d 1
35.l odd 12 2 350.4.j.b 4
40.e odd 2 1 448.4.i.e 2
40.f even 2 1 448.4.i.b 2
105.g even 2 1 882.4.g.u 2
105.o odd 6 1 126.4.g.d 2
105.o odd 6 1 882.4.a.f 1
105.p even 6 1 882.4.a.c 1
105.p even 6 1 882.4.g.u 2
140.p odd 6 1 112.4.i.a 2
140.p odd 6 1 784.4.a.p 1
140.s even 6 1 784.4.a.c 1
280.bf even 6 1 448.4.i.b 2
280.bi odd 6 1 448.4.i.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 5.b even 2 1
14.4.c.a 2 35.j even 6 1
98.4.a.d 1 35.j even 6 1
98.4.a.f 1 35.i odd 6 1
98.4.c.a 2 35.c odd 2 1
98.4.c.a 2 35.i odd 6 1
112.4.i.a 2 20.d odd 2 1
112.4.i.a 2 140.p odd 6 1
126.4.g.d 2 15.d odd 2 1
126.4.g.d 2 105.o odd 6 1
350.4.e.e 2 1.a even 1 1 trivial
350.4.e.e 2 7.c even 3 1 inner
350.4.j.b 4 5.c odd 4 2
350.4.j.b 4 35.l odd 12 2
448.4.i.b 2 40.f even 2 1
448.4.i.b 2 280.bf even 6 1
448.4.i.e 2 40.e odd 2 1
448.4.i.e 2 280.bi odd 6 1
784.4.a.c 1 140.s even 6 1
784.4.a.p 1 140.p odd 6 1
882.4.a.c 1 105.p even 6 1
882.4.a.f 1 105.o odd 6 1
882.4.g.u 2 105.g even 2 1
882.4.g.u 2 105.p even 6 1
2450.4.a.d 1 7.d odd 6 1
2450.4.a.q 1 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(350, [\chi])$$:

 $$T_{3}^{2} + 5T_{3} + 25$$ T3^2 + 5*T3 + 25 $$T_{11}^{2} - 57T_{11} + 3249$$ T11^2 - 57*T11 + 3249

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2} + 5T + 25$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 28T + 343$$
$11$ $$T^{2} - 57T + 3249$$
$13$ $$(T - 70)^{2}$$
$17$ $$T^{2} - 51T + 2601$$
$19$ $$T^{2} + 5T + 25$$
$23$ $$T^{2} - 69T + 4761$$
$29$ $$(T - 114)^{2}$$
$31$ $$T^{2} + 23T + 529$$
$37$ $$T^{2} + 253T + 64009$$
$41$ $$(T + 42)^{2}$$
$43$ $$(T - 124)^{2}$$
$47$ $$T^{2} - 201T + 40401$$
$53$ $$T^{2} + 393T + 154449$$
$59$ $$T^{2} + 219T + 47961$$
$61$ $$T^{2} - 709T + 502681$$
$67$ $$T^{2} - 419T + 175561$$
$71$ $$(T + 96)^{2}$$
$73$ $$T^{2} + 313T + 97969$$
$79$ $$T^{2} + 461T + 212521$$
$83$ $$(T - 588)^{2}$$
$89$ $$T^{2} - 1017 T + 1034289$$
$97$ $$(T - 1834)^{2}$$