# Properties

 Label 350.4.j.b 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{2} + 5 \zeta_{12} q^{3} + ( - 4 \zeta_{12}^{2} + 4) q^{4} - 10 q^{6} + (7 \zeta_{12}^{3} + 14 \zeta_{12}) q^{7} + 8 \zeta_{12}^{3} q^{8} - 2 \zeta_{12}^{2} q^{9} +O(q^{10})$$ q + (2*z^3 - 2*z) * q^2 + 5*z * q^3 + (-4*z^2 + 4) * q^4 - 10 * q^6 + (7*z^3 + 14*z) * q^7 + 8*z^3 * q^8 - 2*z^2 * q^9 $$q + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{2} + 5 \zeta_{12} q^{3} + ( - 4 \zeta_{12}^{2} + 4) q^{4} - 10 q^{6} + (7 \zeta_{12}^{3} + 14 \zeta_{12}) q^{7} + 8 \zeta_{12}^{3} q^{8} - 2 \zeta_{12}^{2} q^{9} + ( - 57 \zeta_{12}^{2} + 57) q^{11} + ( - 20 \zeta_{12}^{3} + 20 \zeta_{12}) q^{12} - 70 \zeta_{12}^{3} q^{13} + ( - 14 \zeta_{12}^{2} - 28) q^{14} - 16 \zeta_{12}^{2} q^{16} + 51 \zeta_{12} q^{17} + 4 \zeta_{12} q^{18} + 5 \zeta_{12}^{2} q^{19} + (105 \zeta_{12}^{2} - 35) q^{21} + 114 \zeta_{12}^{3} q^{22} + ( - 69 \zeta_{12}^{3} + 69 \zeta_{12}) q^{23} + (40 \zeta_{12}^{2} - 40) q^{24} + 140 \zeta_{12}^{2} q^{26} - 145 \zeta_{12}^{3} q^{27} + ( - 56 \zeta_{12}^{3} + 84 \zeta_{12}) q^{28} - 114 q^{29} + (23 \zeta_{12}^{2} - 23) q^{31} + 32 \zeta_{12} q^{32} + ( - 285 \zeta_{12}^{3} + 285 \zeta_{12}) q^{33} - 102 q^{34} - 8 q^{36} + ( - 253 \zeta_{12}^{3} + 253 \zeta_{12}) q^{37} - 10 \zeta_{12} q^{38} + ( - 350 \zeta_{12}^{2} + 350) q^{39} - 42 q^{41} + ( - 70 \zeta_{12}^{3} - 140 \zeta_{12}) q^{42} - 124 \zeta_{12}^{3} q^{43} - 228 \zeta_{12}^{2} q^{44} + (138 \zeta_{12}^{2} - 138) q^{46} + (201 \zeta_{12}^{3} - 201 \zeta_{12}) q^{47} - 80 \zeta_{12}^{3} q^{48} + (392 \zeta_{12}^{2} - 245) q^{49} + 255 \zeta_{12}^{2} q^{51} - 280 \zeta_{12} q^{52} + 393 \zeta_{12} q^{53} + 290 \zeta_{12}^{2} q^{54} + (112 \zeta_{12}^{2} - 168) q^{56} + 25 \zeta_{12}^{3} q^{57} + ( - 228 \zeta_{12}^{3} + 228 \zeta_{12}) q^{58} + ( - 219 \zeta_{12}^{2} + 219) q^{59} + 709 \zeta_{12}^{2} q^{61} - 46 \zeta_{12}^{3} q^{62} + ( - 42 \zeta_{12}^{3} + 14 \zeta_{12}) q^{63} - 64 q^{64} + (570 \zeta_{12}^{2} - 570) q^{66} + 419 \zeta_{12} q^{67} + ( - 204 \zeta_{12}^{3} + 204 \zeta_{12}) q^{68} + 345 q^{69} - 96 q^{71} + ( - 16 \zeta_{12}^{3} + 16 \zeta_{12}) q^{72} + 313 \zeta_{12} q^{73} + (506 \zeta_{12}^{2} - 506) q^{74} + 20 q^{76} + ( - 798 \zeta_{12}^{3} + 1197 \zeta_{12}) q^{77} + 700 \zeta_{12}^{3} q^{78} + 461 \zeta_{12}^{2} q^{79} + ( - 671 \zeta_{12}^{2} + 671) q^{81} + ( - 84 \zeta_{12}^{3} + 84 \zeta_{12}) q^{82} - 588 \zeta_{12}^{3} q^{83} + (140 \zeta_{12}^{2} + 280) q^{84} + 248 \zeta_{12}^{2} q^{86} - 570 \zeta_{12} q^{87} + 456 \zeta_{12} q^{88} - 1017 \zeta_{12}^{2} q^{89} + ( - 980 \zeta_{12}^{2} + 1470) q^{91} - 276 \zeta_{12}^{3} q^{92} + (115 \zeta_{12}^{3} - 115 \zeta_{12}) q^{93} + ( - 402 \zeta_{12}^{2} + 402) q^{94} + 160 \zeta_{12}^{2} q^{96} + 1834 \zeta_{12}^{3} q^{97} + ( - 490 \zeta_{12}^{3} - 294 \zeta_{12}) q^{98} - 114 q^{99} +O(q^{100})$$ q + (2*z^3 - 2*z) * q^2 + 5*z * q^3 + (-4*z^2 + 4) * q^4 - 10 * q^6 + (7*z^3 + 14*z) * q^7 + 8*z^3 * q^8 - 2*z^2 * q^9 + (-57*z^2 + 57) * q^11 + (-20*z^3 + 20*z) * q^12 - 70*z^3 * q^13 + (-14*z^2 - 28) * q^14 - 16*z^2 * q^16 + 51*z * q^17 + 4*z * q^18 + 5*z^2 * q^19 + (105*z^2 - 35) * q^21 + 114*z^3 * q^22 + (-69*z^3 + 69*z) * q^23 + (40*z^2 - 40) * q^24 + 140*z^2 * q^26 - 145*z^3 * q^27 + (-56*z^3 + 84*z) * q^28 - 114 * q^29 + (23*z^2 - 23) * q^31 + 32*z * q^32 + (-285*z^3 + 285*z) * q^33 - 102 * q^34 - 8 * q^36 + (-253*z^3 + 253*z) * q^37 - 10*z * q^38 + (-350*z^2 + 350) * q^39 - 42 * q^41 + (-70*z^3 - 140*z) * q^42 - 124*z^3 * q^43 - 228*z^2 * q^44 + (138*z^2 - 138) * q^46 + (201*z^3 - 201*z) * q^47 - 80*z^3 * q^48 + (392*z^2 - 245) * q^49 + 255*z^2 * q^51 - 280*z * q^52 + 393*z * q^53 + 290*z^2 * q^54 + (112*z^2 - 168) * q^56 + 25*z^3 * q^57 + (-228*z^3 + 228*z) * q^58 + (-219*z^2 + 219) * q^59 + 709*z^2 * q^61 - 46*z^3 * q^62 + (-42*z^3 + 14*z) * q^63 - 64 * q^64 + (570*z^2 - 570) * q^66 + 419*z * q^67 + (-204*z^3 + 204*z) * q^68 + 345 * q^69 - 96 * q^71 + (-16*z^3 + 16*z) * q^72 + 313*z * q^73 + (506*z^2 - 506) * q^74 + 20 * q^76 + (-798*z^3 + 1197*z) * q^77 + 700*z^3 * q^78 + 461*z^2 * q^79 + (-671*z^2 + 671) * q^81 + (-84*z^3 + 84*z) * q^82 - 588*z^3 * q^83 + (140*z^2 + 280) * q^84 + 248*z^2 * q^86 - 570*z * q^87 + 456*z * q^88 - 1017*z^2 * q^89 + (-980*z^2 + 1470) * q^91 - 276*z^3 * q^92 + (115*z^3 - 115*z) * q^93 + (-402*z^2 + 402) * q^94 + 160*z^2 * q^96 + 1834*z^3 * q^97 + (-490*z^3 - 294*z) * q^98 - 114 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4} - 40 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q + 8 * q^4 - 40 * q^6 - 4 * q^9 $$4 q + 8 q^{4} - 40 q^{6} - 4 q^{9} + 114 q^{11} - 140 q^{14} - 32 q^{16} + 10 q^{19} + 70 q^{21} - 80 q^{24} + 280 q^{26} - 456 q^{29} - 46 q^{31} - 408 q^{34} - 32 q^{36} + 700 q^{39} - 168 q^{41} - 456 q^{44} - 276 q^{46} - 196 q^{49} + 510 q^{51} + 580 q^{54} - 448 q^{56} + 438 q^{59} + 1418 q^{61} - 256 q^{64} - 1140 q^{66} + 1380 q^{69} - 384 q^{71} - 1012 q^{74} + 80 q^{76} + 922 q^{79} + 1342 q^{81} + 1400 q^{84} + 496 q^{86} - 2034 q^{89} + 3920 q^{91} + 804 q^{94} + 320 q^{96} - 456 q^{99}+O(q^{100})$$ 4 * q + 8 * q^4 - 40 * q^6 - 4 * q^9 + 114 * q^11 - 140 * q^14 - 32 * q^16 + 10 * q^19 + 70 * q^21 - 80 * q^24 + 280 * q^26 - 456 * q^29 - 46 * q^31 - 408 * q^34 - 32 * q^36 + 700 * q^39 - 168 * q^41 - 456 * q^44 - 276 * q^46 - 196 * q^49 + 510 * q^51 + 580 * q^54 - 448 * q^56 + 438 * q^59 + 1418 * q^61 - 256 * q^64 - 1140 * q^66 + 1380 * q^69 - 384 * q^71 - 1012 * q^74 + 80 * q^76 + 922 * q^79 + 1342 * q^81 + 1400 * q^84 + 496 * q^86 - 2034 * q^89 + 3920 * q^91 + 804 * q^94 + 320 * q^96 - 456 * 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 4.33013 2.50000i 2.00000 + 3.46410i 0 −10.0000 12.1244 14.0000i 8.00000i −1.00000 + 1.73205i 0
149.2 1.73205 + 1.00000i −4.33013 + 2.50000i 2.00000 + 3.46410i 0 −10.0000 −12.1244 + 14.0000i 8.00000i −1.00000 + 1.73205i 0
249.1 −1.73205 + 1.00000i 4.33013 + 2.50000i 2.00000 3.46410i 0 −10.0000 12.1244 + 14.0000i 8.00000i −1.00000 1.73205i 0
249.2 1.73205 1.00000i −4.33013 2.50000i 2.00000 3.46410i 0 −10.0000 −12.1244 14.0000i 8.00000i −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
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.b 4
5.b even 2 1 inner 350.4.j.b 4
5.c odd 4 1 14.4.c.a 2
5.c odd 4 1 350.4.e.e 2
7.c even 3 1 inner 350.4.j.b 4
15.e even 4 1 126.4.g.d 2
20.e even 4 1 112.4.i.a 2
35.f even 4 1 98.4.c.a 2
35.j even 6 1 inner 350.4.j.b 4
35.k even 12 1 98.4.a.f 1
35.k even 12 1 98.4.c.a 2
35.k even 12 1 2450.4.a.d 1
35.l odd 12 1 14.4.c.a 2
35.l odd 12 1 98.4.a.d 1
35.l odd 12 1 350.4.e.e 2
35.l odd 12 1 2450.4.a.q 1
40.i odd 4 1 448.4.i.b 2
40.k even 4 1 448.4.i.e 2
105.k odd 4 1 882.4.g.u 2
105.w odd 12 1 882.4.a.c 1
105.w odd 12 1 882.4.g.u 2
105.x even 12 1 126.4.g.d 2
105.x even 12 1 882.4.a.f 1
140.w even 12 1 112.4.i.a 2
140.w even 12 1 784.4.a.p 1
140.x odd 12 1 784.4.a.c 1
280.br even 12 1 448.4.i.e 2
280.bt odd 12 1 448.4.i.b 2

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

 $$T_{3}^{4} - 25T_{3}^{2} + 625$$ T3^4 - 25*T3^2 + 625 $$T_{11}^{2} - 57T_{11} + 3249$$ T11^2 - 57*T11 + 3249

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 4T^{2} + 16$$
$3$ $$T^{4} - 25T^{2} + 625$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 98 T^{2} + 117649$$
$11$ $$(T^{2} - 57 T + 3249)^{2}$$
$13$ $$(T^{2} + 4900)^{2}$$
$17$ $$T^{4} - 2601 T^{2} + \cdots + 6765201$$
$19$ $$(T^{2} - 5 T + 25)^{2}$$
$23$ $$T^{4} - 4761 T^{2} + \cdots + 22667121$$
$29$ $$(T + 114)^{4}$$
$31$ $$(T^{2} + 23 T + 529)^{2}$$
$37$ $$T^{4} - 64009 T^{2} + \cdots + 4097152081$$
$41$ $$(T + 42)^{4}$$
$43$ $$(T^{2} + 15376)^{2}$$
$47$ $$T^{4} - 40401 T^{2} + \cdots + 1632240801$$
$53$ $$T^{4} - 154449 T^{2} + \cdots + 23854493601$$
$59$ $$(T^{2} - 219 T + 47961)^{2}$$
$61$ $$(T^{2} - 709 T + 502681)^{2}$$
$67$ $$T^{4} - 175561 T^{2} + \cdots + 30821664721$$
$71$ $$(T + 96)^{4}$$
$73$ $$T^{4} - 97969 T^{2} + \cdots + 9597924961$$
$79$ $$(T^{2} - 461 T + 212521)^{2}$$
$83$ $$(T^{2} + 345744)^{2}$$
$89$ $$(T^{2} + 1017 T + 1034289)^{2}$$
$97$ $$(T^{2} + 3363556)^{2}$$