# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 350.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$20.6506685020$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 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: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{2} + \zeta_{12} q^{3} + ( - 4 \zeta_{12}^{2} + 4) q^{4} + 2 q^{6} + (19 \zeta_{12}^{3} - 18 \zeta_{12}) q^{7} - 8 \zeta_{12}^{3} q^{8} - 26 \zeta_{12}^{2} q^{9} +O(q^{10})$$ q + (-2*z^3 + 2*z) * q^2 + z * q^3 + (-4*z^2 + 4) * q^4 + 2 * q^6 + (19*z^3 - 18*z) * q^7 - 8*z^3 * q^8 - 26*z^2 * q^9 $$q + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{2} + \zeta_{12} q^{3} + ( - 4 \zeta_{12}^{2} + 4) q^{4} + 2 q^{6} + (19 \zeta_{12}^{3} - 18 \zeta_{12}) q^{7} - 8 \zeta_{12}^{3} q^{8} - 26 \zeta_{12}^{2} q^{9} + (35 \zeta_{12}^{2} - 35) q^{11} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{12} + 66 \zeta_{12}^{3} q^{13} + (38 \zeta_{12}^{2} - 36) q^{14} - 16 \zeta_{12}^{2} q^{16} + 59 \zeta_{12} q^{17} - 52 \zeta_{12} q^{18} + 137 \zeta_{12}^{2} q^{19} + (\zeta_{12}^{2} - 19) q^{21} + 70 \zeta_{12}^{3} q^{22} + (7 \zeta_{12}^{3} - 7 \zeta_{12}) q^{23} + ( - 8 \zeta_{12}^{2} + 8) q^{24} + 132 \zeta_{12}^{2} q^{26} - 53 \zeta_{12}^{3} q^{27} + (72 \zeta_{12}^{3} + 4 \zeta_{12}) q^{28} - 106 q^{29} + (75 \zeta_{12}^{2} - 75) q^{31} - 32 \zeta_{12} q^{32} + (35 \zeta_{12}^{3} - 35 \zeta_{12}) q^{33} + 118 q^{34} - 104 q^{36} + (11 \zeta_{12}^{3} - 11 \zeta_{12}) q^{37} + 274 \zeta_{12} q^{38} + (66 \zeta_{12}^{2} - 66) q^{39} - 498 q^{41} + (38 \zeta_{12}^{3} - 36 \zeta_{12}) q^{42} + 260 \zeta_{12}^{3} q^{43} + 140 \zeta_{12}^{2} q^{44} + (14 \zeta_{12}^{2} - 14) q^{46} + ( - 171 \zeta_{12}^{3} + 171 \zeta_{12}) q^{47} - 16 \zeta_{12}^{3} q^{48} + ( - 360 \zeta_{12}^{2} + 323) q^{49} + 59 \zeta_{12}^{2} q^{51} + 264 \zeta_{12} q^{52} + 417 \zeta_{12} q^{53} - 106 \zeta_{12}^{2} q^{54} + (144 \zeta_{12}^{2} + 8) q^{56} + 137 \zeta_{12}^{3} q^{57} + (212 \zeta_{12}^{3} - 212 \zeta_{12}) q^{58} + (17 \zeta_{12}^{2} - 17) q^{59} - 51 \zeta_{12}^{2} q^{61} + 150 \zeta_{12}^{3} q^{62} + ( - 26 \zeta_{12}^{3} + 494 \zeta_{12}) q^{63} - 64 q^{64} + (70 \zeta_{12}^{2} - 70) q^{66} + 439 \zeta_{12} q^{67} + ( - 236 \zeta_{12}^{3} + 236 \zeta_{12}) q^{68} - 7 q^{69} - 784 q^{71} + (208 \zeta_{12}^{3} - 208 \zeta_{12}) q^{72} - 295 \zeta_{12} q^{73} + (22 \zeta_{12}^{2} - 22) q^{74} + 548 q^{76} + ( - 630 \zeta_{12}^{3} - 35 \zeta_{12}) q^{77} + 132 \zeta_{12}^{3} q^{78} - 495 \zeta_{12}^{2} q^{79} + (649 \zeta_{12}^{2} - 649) q^{81} + (996 \zeta_{12}^{3} - 996 \zeta_{12}) q^{82} + 932 \zeta_{12}^{3} q^{83} + (76 \zeta_{12}^{2} - 72) q^{84} + 520 \zeta_{12}^{2} q^{86} - 106 \zeta_{12} q^{87} + 280 \zeta_{12} q^{88} - 873 \zeta_{12}^{2} q^{89} + ( - 1188 \zeta_{12}^{2} - 66) q^{91} + 28 \zeta_{12}^{3} q^{92} + (75 \zeta_{12}^{3} - 75 \zeta_{12}) q^{93} + ( - 342 \zeta_{12}^{2} + 342) q^{94} - 32 \zeta_{12}^{2} q^{96} + 290 \zeta_{12}^{3} q^{97} + ( - 646 \zeta_{12}^{3} - 74 \zeta_{12}) q^{98} + 910 q^{99} +O(q^{100})$$ q + (-2*z^3 + 2*z) * q^2 + z * q^3 + (-4*z^2 + 4) * q^4 + 2 * q^6 + (19*z^3 - 18*z) * q^7 - 8*z^3 * q^8 - 26*z^2 * q^9 + (35*z^2 - 35) * q^11 + (-4*z^3 + 4*z) * q^12 + 66*z^3 * q^13 + (38*z^2 - 36) * q^14 - 16*z^2 * q^16 + 59*z * q^17 - 52*z * q^18 + 137*z^2 * q^19 + (z^2 - 19) * q^21 + 70*z^3 * q^22 + (7*z^3 - 7*z) * q^23 + (-8*z^2 + 8) * q^24 + 132*z^2 * q^26 - 53*z^3 * q^27 + (72*z^3 + 4*z) * q^28 - 106 * q^29 + (75*z^2 - 75) * q^31 - 32*z * q^32 + (35*z^3 - 35*z) * q^33 + 118 * q^34 - 104 * q^36 + (11*z^3 - 11*z) * q^37 + 274*z * q^38 + (66*z^2 - 66) * q^39 - 498 * q^41 + (38*z^3 - 36*z) * q^42 + 260*z^3 * q^43 + 140*z^2 * q^44 + (14*z^2 - 14) * q^46 + (-171*z^3 + 171*z) * q^47 - 16*z^3 * q^48 + (-360*z^2 + 323) * q^49 + 59*z^2 * q^51 + 264*z * q^52 + 417*z * q^53 - 106*z^2 * q^54 + (144*z^2 + 8) * q^56 + 137*z^3 * q^57 + (212*z^3 - 212*z) * q^58 + (17*z^2 - 17) * q^59 - 51*z^2 * q^61 + 150*z^3 * q^62 + (-26*z^3 + 494*z) * q^63 - 64 * q^64 + (70*z^2 - 70) * q^66 + 439*z * q^67 + (-236*z^3 + 236*z) * q^68 - 7 * q^69 - 784 * q^71 + (208*z^3 - 208*z) * q^72 - 295*z * q^73 + (22*z^2 - 22) * q^74 + 548 * q^76 + (-630*z^3 - 35*z) * q^77 + 132*z^3 * q^78 - 495*z^2 * q^79 + (649*z^2 - 649) * q^81 + (996*z^3 - 996*z) * q^82 + 932*z^3 * q^83 + (76*z^2 - 72) * q^84 + 520*z^2 * q^86 - 106*z * q^87 + 280*z * q^88 - 873*z^2 * q^89 + (-1188*z^2 - 66) * q^91 + 28*z^3 * q^92 + (75*z^3 - 75*z) * q^93 + (-342*z^2 + 342) * q^94 - 32*z^2 * q^96 + 290*z^3 * q^97 + (-646*z^3 - 74*z) * q^98 + 910 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4} + 8 q^{6} - 52 q^{9}+O(q^{10})$$ 4 * q + 8 * q^4 + 8 * q^6 - 52 * q^9 $$4 q + 8 q^{4} + 8 q^{6} - 52 q^{9} - 70 q^{11} - 68 q^{14} - 32 q^{16} + 274 q^{19} - 74 q^{21} + 16 q^{24} + 264 q^{26} - 424 q^{29} - 150 q^{31} + 472 q^{34} - 416 q^{36} - 132 q^{39} - 1992 q^{41} + 280 q^{44} - 28 q^{46} + 572 q^{49} + 118 q^{51} - 212 q^{54} + 320 q^{56} - 34 q^{59} - 102 q^{61} - 256 q^{64} - 140 q^{66} - 28 q^{69} - 3136 q^{71} - 44 q^{74} + 2192 q^{76} - 990 q^{79} - 1298 q^{81} - 136 q^{84} + 1040 q^{86} - 1746 q^{89} - 2640 q^{91} + 684 q^{94} - 64 q^{96} + 3640 q^{99}+O(q^{100})$$ 4 * q + 8 * q^4 + 8 * q^6 - 52 * q^9 - 70 * q^11 - 68 * q^14 - 32 * q^16 + 274 * q^19 - 74 * q^21 + 16 * q^24 + 264 * q^26 - 424 * q^29 - 150 * q^31 + 472 * q^34 - 416 * q^36 - 132 * q^39 - 1992 * q^41 + 280 * q^44 - 28 * q^46 + 572 * q^49 + 118 * q^51 - 212 * q^54 + 320 * q^56 - 34 * q^59 - 102 * q^61 - 256 * q^64 - 140 * q^66 - 28 * q^69 - 3136 * q^71 - 44 * q^74 + 2192 * q^76 - 990 * q^79 - 1298 * q^81 - 136 * q^84 + 1040 * q^86 - 1746 * q^89 - 2640 * q^91 + 684 * q^94 - 64 * q^96 + 3640 * q^99

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−1.73205 1.00000i −0.866025 + 0.500000i 2.00000 + 3.46410i 0 2.00000 15.5885 + 10.0000i 8.00000i −13.0000 + 22.5167i 0
149.2 1.73205 + 1.00000i 0.866025 0.500000i 2.00000 + 3.46410i 0 2.00000 −15.5885 10.0000i 8.00000i −13.0000 + 22.5167i 0
249.1 −1.73205 + 1.00000i −0.866025 0.500000i 2.00000 3.46410i 0 2.00000 15.5885 10.0000i 8.00000i −13.0000 22.5167i 0
249.2 1.73205 1.00000i 0.866025 + 0.500000i 2.00000 3.46410i 0 2.00000 −15.5885 + 10.0000i 8.00000i −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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{3}^{4} - T_{3}^{2} + 1$$ T3^4 - T3^2 + 1 $$T_{11}^{2} + 35T_{11} + 1225$$ T11^2 + 35*T11 + 1225

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 4T^{2} + 16$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 286 T^{2} + 117649$$
$11$ $$(T^{2} + 35 T + 1225)^{2}$$
$13$ $$(T^{2} + 4356)^{2}$$
$17$ $$T^{4} - 3481 T^{2} + \cdots + 12117361$$
$19$ $$(T^{2} - 137 T + 18769)^{2}$$
$23$ $$T^{4} - 49T^{2} + 2401$$
$29$ $$(T + 106)^{4}$$
$31$ $$(T^{2} + 75 T + 5625)^{2}$$
$37$ $$T^{4} - 121 T^{2} + 14641$$
$41$ $$(T + 498)^{4}$$
$43$ $$(T^{2} + 67600)^{2}$$
$47$ $$T^{4} - 29241 T^{2} + \cdots + 855036081$$
$53$ $$T^{4} - 173889 T^{2} + \cdots + 30237384321$$
$59$ $$(T^{2} + 17 T + 289)^{2}$$
$61$ $$(T^{2} + 51 T + 2601)^{2}$$
$67$ $$T^{4} - 192721 T^{2} + \cdots + 37141383841$$
$71$ $$(T + 784)^{4}$$
$73$ $$T^{4} - 87025 T^{2} + \cdots + 7573350625$$
$79$ $$(T^{2} + 495 T + 245025)^{2}$$
$83$ $$(T^{2} + 868624)^{2}$$
$89$ $$(T^{2} + 873 T + 762129)^{2}$$
$97$ $$(T^{2} + 84100)^{2}$$