# Properties

 Label 350.4.e.b Level $350$ Weight $4$ Character orbit 350.e Analytic conductor $20.651$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$20.6506685020$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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

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.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (4 \zeta_{6} - 4) q^{4} + 2 q^{6} + (18 \zeta_{6} + 1) q^{7} + 8 q^{8} + 26 \zeta_{6} q^{9} +O(q^{10})$$ q - 2*z * q^2 + (z - 1) * q^3 + (4*z - 4) * q^4 + 2 * q^6 + (18*z + 1) * q^7 + 8 * q^8 + 26*z * q^9 $$q - 2 \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (4 \zeta_{6} - 4) q^{4} + 2 q^{6} + (18 \zeta_{6} + 1) q^{7} + 8 q^{8} + 26 \zeta_{6} q^{9} + (35 \zeta_{6} - 35) q^{11} - 4 \zeta_{6} q^{12} - 66 q^{13} + ( - 38 \zeta_{6} + 36) q^{14} - 16 \zeta_{6} q^{16} + ( - 59 \zeta_{6} + 59) q^{17} + ( - 52 \zeta_{6} + 52) q^{18} - 137 \zeta_{6} q^{19} + (\zeta_{6} - 19) q^{21} + 70 q^{22} - 7 \zeta_{6} q^{23} + (8 \zeta_{6} - 8) q^{24} + 132 \zeta_{6} q^{26} - 53 q^{27} + (4 \zeta_{6} - 76) q^{28} + 106 q^{29} + (75 \zeta_{6} - 75) q^{31} + (32 \zeta_{6} - 32) q^{32} - 35 \zeta_{6} q^{33} - 118 q^{34} - 104 q^{36} + 11 \zeta_{6} q^{37} + (274 \zeta_{6} - 274) q^{38} + ( - 66 \zeta_{6} + 66) q^{39} - 498 q^{41} + (36 \zeta_{6} + 2) q^{42} - 260 q^{43} - 140 \zeta_{6} q^{44} + (14 \zeta_{6} - 14) q^{46} - 171 \zeta_{6} q^{47} + 16 q^{48} + (360 \zeta_{6} - 323) q^{49} + 59 \zeta_{6} q^{51} + ( - 264 \zeta_{6} + 264) q^{52} + (417 \zeta_{6} - 417) q^{53} + 106 \zeta_{6} q^{54} + (144 \zeta_{6} + 8) q^{56} + 137 q^{57} - 212 \zeta_{6} q^{58} + ( - 17 \zeta_{6} + 17) q^{59} - 51 \zeta_{6} q^{61} + 150 q^{62} + (494 \zeta_{6} - 468) q^{63} + 64 q^{64} + (70 \zeta_{6} - 70) q^{66} + ( - 439 \zeta_{6} + 439) q^{67} + 236 \zeta_{6} q^{68} + 7 q^{69} - 784 q^{71} + 208 \zeta_{6} q^{72} + ( - 295 \zeta_{6} + 295) q^{73} + ( - 22 \zeta_{6} + 22) q^{74} + 548 q^{76} + (35 \zeta_{6} - 665) q^{77} - 132 q^{78} + 495 \zeta_{6} q^{79} + (649 \zeta_{6} - 649) q^{81} + 996 \zeta_{6} q^{82} - 932 q^{83} + ( - 76 \zeta_{6} + 72) q^{84} + 520 \zeta_{6} q^{86} + (106 \zeta_{6} - 106) q^{87} + (280 \zeta_{6} - 280) q^{88} + 873 \zeta_{6} q^{89} + ( - 1188 \zeta_{6} - 66) q^{91} + 28 q^{92} - 75 \zeta_{6} q^{93} + (342 \zeta_{6} - 342) q^{94} - 32 \zeta_{6} q^{96} + 290 q^{97} + ( - 74 \zeta_{6} + 720) q^{98} - 910 q^{99} +O(q^{100})$$ q - 2*z * q^2 + (z - 1) * q^3 + (4*z - 4) * q^4 + 2 * q^6 + (18*z + 1) * q^7 + 8 * q^8 + 26*z * q^9 + (35*z - 35) * q^11 - 4*z * q^12 - 66 * q^13 + (-38*z + 36) * q^14 - 16*z * q^16 + (-59*z + 59) * q^17 + (-52*z + 52) * q^18 - 137*z * q^19 + (z - 19) * q^21 + 70 * q^22 - 7*z * q^23 + (8*z - 8) * q^24 + 132*z * q^26 - 53 * q^27 + (4*z - 76) * q^28 + 106 * q^29 + (75*z - 75) * q^31 + (32*z - 32) * q^32 - 35*z * q^33 - 118 * q^34 - 104 * q^36 + 11*z * q^37 + (274*z - 274) * q^38 + (-66*z + 66) * q^39 - 498 * q^41 + (36*z + 2) * q^42 - 260 * q^43 - 140*z * q^44 + (14*z - 14) * q^46 - 171*z * q^47 + 16 * q^48 + (360*z - 323) * q^49 + 59*z * q^51 + (-264*z + 264) * q^52 + (417*z - 417) * q^53 + 106*z * q^54 + (144*z + 8) * q^56 + 137 * q^57 - 212*z * q^58 + (-17*z + 17) * q^59 - 51*z * q^61 + 150 * q^62 + (494*z - 468) * q^63 + 64 * q^64 + (70*z - 70) * q^66 + (-439*z + 439) * q^67 + 236*z * q^68 + 7 * q^69 - 784 * q^71 + 208*z * q^72 + (-295*z + 295) * q^73 + (-22*z + 22) * q^74 + 548 * q^76 + (35*z - 665) * q^77 - 132 * q^78 + 495*z * q^79 + (649*z - 649) * q^81 + 996*z * q^82 - 932 * q^83 + (-76*z + 72) * q^84 + 520*z * q^86 + (106*z - 106) * q^87 + (280*z - 280) * q^88 + 873*z * q^89 + (-1188*z - 66) * q^91 + 28 * q^92 - 75*z * q^93 + (342*z - 342) * q^94 - 32*z * q^96 + 290 * q^97 + (-74*z + 720) * q^98 - 910 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - q^{3} - 4 q^{4} + 4 q^{6} + 20 q^{7} + 16 q^{8} + 26 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - q^3 - 4 * q^4 + 4 * q^6 + 20 * q^7 + 16 * q^8 + 26 * q^9 $$2 q - 2 q^{2} - q^{3} - 4 q^{4} + 4 q^{6} + 20 q^{7} + 16 q^{8} + 26 q^{9} - 35 q^{11} - 4 q^{12} - 132 q^{13} + 34 q^{14} - 16 q^{16} + 59 q^{17} + 52 q^{18} - 137 q^{19} - 37 q^{21} + 140 q^{22} - 7 q^{23} - 8 q^{24} + 132 q^{26} - 106 q^{27} - 148 q^{28} + 212 q^{29} - 75 q^{31} - 32 q^{32} - 35 q^{33} - 236 q^{34} - 208 q^{36} + 11 q^{37} - 274 q^{38} + 66 q^{39} - 996 q^{41} + 40 q^{42} - 520 q^{43} - 140 q^{44} - 14 q^{46} - 171 q^{47} + 32 q^{48} - 286 q^{49} + 59 q^{51} + 264 q^{52} - 417 q^{53} + 106 q^{54} + 160 q^{56} + 274 q^{57} - 212 q^{58} + 17 q^{59} - 51 q^{61} + 300 q^{62} - 442 q^{63} + 128 q^{64} - 70 q^{66} + 439 q^{67} + 236 q^{68} + 14 q^{69} - 1568 q^{71} + 208 q^{72} + 295 q^{73} + 22 q^{74} + 1096 q^{76} - 1295 q^{77} - 264 q^{78} + 495 q^{79} - 649 q^{81} + 996 q^{82} - 1864 q^{83} + 68 q^{84} + 520 q^{86} - 106 q^{87} - 280 q^{88} + 873 q^{89} - 1320 q^{91} + 56 q^{92} - 75 q^{93} - 342 q^{94} - 32 q^{96} + 580 q^{97} + 1366 q^{98} - 1820 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - q^3 - 4 * q^4 + 4 * q^6 + 20 * q^7 + 16 * q^8 + 26 * q^9 - 35 * q^11 - 4 * q^12 - 132 * q^13 + 34 * q^14 - 16 * q^16 + 59 * q^17 + 52 * q^18 - 137 * q^19 - 37 * q^21 + 140 * q^22 - 7 * q^23 - 8 * q^24 + 132 * q^26 - 106 * q^27 - 148 * q^28 + 212 * q^29 - 75 * q^31 - 32 * q^32 - 35 * q^33 - 236 * q^34 - 208 * q^36 + 11 * q^37 - 274 * q^38 + 66 * q^39 - 996 * q^41 + 40 * q^42 - 520 * q^43 - 140 * q^44 - 14 * q^46 - 171 * q^47 + 32 * q^48 - 286 * q^49 + 59 * q^51 + 264 * q^52 - 417 * q^53 + 106 * q^54 + 160 * q^56 + 274 * q^57 - 212 * q^58 + 17 * q^59 - 51 * q^61 + 300 * q^62 - 442 * q^63 + 128 * q^64 - 70 * q^66 + 439 * q^67 + 236 * q^68 + 14 * q^69 - 1568 * q^71 + 208 * q^72 + 295 * q^73 + 22 * q^74 + 1096 * q^76 - 1295 * q^77 - 264 * q^78 + 495 * q^79 - 649 * q^81 + 996 * q^82 - 1864 * q^83 + 68 * q^84 + 520 * q^86 - 106 * q^87 - 280 * q^88 + 873 * q^89 - 1320 * q^91 + 56 * q^92 - 75 * q^93 - 342 * q^94 - 32 * q^96 + 580 * q^97 + 1366 * q^98 - 1820 * 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.

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 −0.500000 0.866025i −2.00000 3.46410i 0 2.00000 10.0000 15.5885i 8.00000 13.0000 22.5167i 0
151.1 −1.00000 1.73205i −0.500000 + 0.866025i −2.00000 + 3.46410i 0 2.00000 10.0000 + 15.5885i 8.00000 13.0000 + 22.5167i 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.b 2
5.b even 2 1 14.4.c.b 2
5.c odd 4 2 350.4.j.d 4
7.c even 3 1 inner 350.4.e.b 2
7.c even 3 1 2450.4.a.bh 1
7.d odd 6 1 2450.4.a.bf 1
15.d odd 2 1 126.4.g.c 2
20.d odd 2 1 112.4.i.b 2
35.c odd 2 1 98.4.c.e 2
35.i odd 6 1 98.4.a.c 1
35.i odd 6 1 98.4.c.e 2
35.j even 6 1 14.4.c.b 2
35.j even 6 1 98.4.a.b 1
35.l odd 12 2 350.4.j.d 4
40.e odd 2 1 448.4.i.d 2
40.f even 2 1 448.4.i.c 2
105.g even 2 1 882.4.g.d 2
105.o odd 6 1 126.4.g.c 2
105.o odd 6 1 882.4.a.k 1
105.p even 6 1 882.4.a.p 1
105.p even 6 1 882.4.g.d 2
140.p odd 6 1 112.4.i.b 2
140.p odd 6 1 784.4.a.l 1
140.s even 6 1 784.4.a.j 1
280.bf even 6 1 448.4.i.c 2
280.bi odd 6 1 448.4.i.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b 2 5.b even 2 1
14.4.c.b 2 35.j even 6 1
98.4.a.b 1 35.j even 6 1
98.4.a.c 1 35.i odd 6 1
98.4.c.e 2 35.c odd 2 1
98.4.c.e 2 35.i odd 6 1
112.4.i.b 2 20.d odd 2 1
112.4.i.b 2 140.p odd 6 1
126.4.g.c 2 15.d odd 2 1
126.4.g.c 2 105.o odd 6 1
350.4.e.b 2 1.a even 1 1 trivial
350.4.e.b 2 7.c even 3 1 inner
350.4.j.d 4 5.c odd 4 2
350.4.j.d 4 35.l odd 12 2
448.4.i.c 2 40.f even 2 1
448.4.i.c 2 280.bf even 6 1
448.4.i.d 2 40.e odd 2 1
448.4.i.d 2 280.bi odd 6 1
784.4.a.j 1 140.s even 6 1
784.4.a.l 1 140.p odd 6 1
882.4.a.k 1 105.o odd 6 1
882.4.a.p 1 105.p even 6 1
882.4.g.d 2 105.g even 2 1
882.4.g.d 2 105.p even 6 1
2450.4.a.bf 1 7.d odd 6 1
2450.4.a.bh 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 $$S_{4}^{\mathrm{new}}(350, [\chi])$$:

 $$T_{3}^{2} + T_{3} + 1$$ T3^2 + T3 + 1 $$T_{11}^{2} + 35T_{11} + 1225$$ T11^2 + 35*T11 + 1225

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 20T + 343$$
$11$ $$T^{2} + 35T + 1225$$
$13$ $$(T + 66)^{2}$$
$17$ $$T^{2} - 59T + 3481$$
$19$ $$T^{2} + 137T + 18769$$
$23$ $$T^{2} + 7T + 49$$
$29$ $$(T - 106)^{2}$$
$31$ $$T^{2} + 75T + 5625$$
$37$ $$T^{2} - 11T + 121$$
$41$ $$(T + 498)^{2}$$
$43$ $$(T + 260)^{2}$$
$47$ $$T^{2} + 171T + 29241$$
$53$ $$T^{2} + 417T + 173889$$
$59$ $$T^{2} - 17T + 289$$
$61$ $$T^{2} + 51T + 2601$$
$67$ $$T^{2} - 439T + 192721$$
$71$ $$(T + 784)^{2}$$
$73$ $$T^{2} - 295T + 87025$$
$79$ $$T^{2} - 495T + 245025$$
$83$ $$(T + 932)^{2}$$
$89$ $$T^{2} - 873T + 762129$$
$97$ $$(T - 290)^{2}$$