# Properties

 Label 350.3.k.a Level $350$ Weight $3$ Character orbit 350.k Analytic conductor $9.537$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 350.k (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.53680925261$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + 2 \beta_{2} q^{4} + ( -2 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{6} + ( -3 - 4 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} ) q^{7} -2 \beta_{3} q^{8} + 6 \beta_{1} q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + 2 \beta_{2} q^{4} + ( -2 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{6} + ( -3 - 4 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} ) q^{7} -2 \beta_{3} q^{8} + 6 \beta_{1} q^{9} + ( -3 \beta_{1} - 9 \beta_{2} - 3 \beta_{3} ) q^{11} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{12} + ( -6 + 4 \beta_{1} - 12 \beta_{2} + 2 \beta_{3} ) q^{13} + ( -10 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{14} + ( -4 - 4 \beta_{2} ) q^{16} + ( 10 + 2 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{17} -12 \beta_{2} q^{18} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{19} + ( 8 - 8 \beta_{1} - 11 \beta_{2} - 10 \beta_{3} ) q^{21} + ( -6 + 9 \beta_{3} ) q^{22} + ( -15 + 9 \beta_{1} - 15 \beta_{2} ) q^{23} + ( 8 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{24} + ( 4 + 6 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} ) q^{26} + ( 3 - 6 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{27} + ( 4 + 10 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{28} + ( 12 - 6 \beta_{3} ) q^{29} + ( -14 + 15 \beta_{1} - 7 \beta_{2} - 15 \beta_{3} ) q^{31} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{32} + ( 15 - 12 \beta_{1} - 15 \beta_{2} - 24 \beta_{3} ) q^{33} + ( -4 - 10 \beta_{1} - 8 \beta_{2} - 5 \beta_{3} ) q^{34} + 12 \beta_{3} q^{36} + ( 31 + 24 \beta_{1} + 31 \beta_{2} ) q^{37} + ( 4 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{38} + ( -12 \beta_{1} - 6 \beta_{2} - 12 \beta_{3} ) q^{39} + ( -2 + 20 \beta_{1} - 4 \beta_{2} + 10 \beta_{3} ) q^{41} + ( -20 - 8 \beta_{1} - 4 \beta_{2} + 11 \beta_{3} ) q^{42} + ( 2 + 6 \beta_{3} ) q^{43} + ( 18 + 6 \beta_{1} + 18 \beta_{2} ) q^{44} + ( 15 \beta_{1} - 18 \beta_{2} + 15 \beta_{3} ) q^{46} + ( -29 - \beta_{1} + 29 \beta_{2} - 2 \beta_{3} ) q^{47} + ( -4 - 8 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{48} + ( -25 + 4 \beta_{1} - 40 \beta_{2} + 26 \beta_{3} ) q^{49} + ( 27 + 21 \beta_{1} + 27 \beta_{2} ) q^{51} + ( 24 - 4 \beta_{1} + 12 \beta_{2} + 4 \beta_{3} ) q^{52} + ( 12 \beta_{1} - 39 \beta_{2} + 12 \beta_{3} ) q^{53} + ( -6 - 3 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{54} + ( 4 - 4 \beta_{1} - 16 \beta_{2} + 2 \beta_{3} ) q^{56} -3 q^{57} + ( -12 - 12 \beta_{1} - 12 \beta_{2} ) q^{58} + ( -26 - 25 \beta_{1} - 13 \beta_{2} + 25 \beta_{3} ) q^{59} + ( -7 - 32 \beta_{1} + 7 \beta_{2} - 64 \beta_{3} ) q^{61} + ( -30 + 14 \beta_{1} - 60 \beta_{2} + 7 \beta_{3} ) q^{62} + ( 60 - 18 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{63} + 8 q^{64} + ( -48 - 15 \beta_{1} - 24 \beta_{2} + 15 \beta_{3} ) q^{66} + ( -45 \beta_{1} - 29 \beta_{2} - 45 \beta_{3} ) q^{67} + ( -10 + 4 \beta_{1} + 10 \beta_{2} + 8 \beta_{3} ) q^{68} + ( 3 - 12 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{69} + ( -6 + 30 \beta_{3} ) q^{71} + ( 24 + 24 \beta_{2} ) q^{72} + ( -106 - 16 \beta_{1} - 53 \beta_{2} + 16 \beta_{3} ) q^{73} + ( -31 \beta_{1} - 48 \beta_{2} - 31 \beta_{3} ) q^{74} + ( 2 - 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{76} + ( -42 - 42 \beta_{1} - 21 \beta_{2} ) q^{77} + ( -24 + 6 \beta_{3} ) q^{78} + ( 55 + 15 \beta_{1} + 55 \beta_{2} ) q^{79} + ( -54 \beta_{1} - 9 \beta_{2} - 54 \beta_{3} ) q^{81} + ( 20 + 2 \beta_{1} - 20 \beta_{2} + 4 \beta_{3} ) q^{82} + ( 68 - 8 \beta_{1} + 136 \beta_{2} - 4 \beta_{3} ) q^{83} + ( 22 + 20 \beta_{1} + 38 \beta_{2} + 4 \beta_{3} ) q^{84} + ( 12 - 2 \beta_{1} + 12 \beta_{2} ) q^{86} + ( 48 + 18 \beta_{1} + 24 \beta_{2} - 18 \beta_{3} ) q^{87} + ( -18 \beta_{1} - 12 \beta_{2} - 18 \beta_{3} ) q^{88} + ( -63 + 24 \beta_{1} + 63 \beta_{2} + 48 \beta_{3} ) q^{89} + ( 30 - 44 \beta_{1} + 48 \beta_{2} + 8 \beta_{3} ) q^{91} + ( 30 + 18 \beta_{3} ) q^{92} + ( 69 + 24 \beta_{1} + 69 \beta_{2} ) q^{93} + ( -4 + 29 \beta_{1} - 2 \beta_{2} - 29 \beta_{3} ) q^{94} + ( -8 + 4 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{96} + ( -22 + 52 \beta_{1} - 44 \beta_{2} + 26 \beta_{3} ) q^{97} + ( 52 + 25 \beta_{1} + 44 \beta_{2} + 40 \beta_{3} ) q^{98} + ( 36 - 54 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{3} - 4 q^{4} - 8 q^{7} + O(q^{10})$$ $$4 q + 6 q^{3} - 4 q^{4} - 8 q^{7} + 18 q^{11} - 12 q^{12} - 36 q^{14} - 8 q^{16} + 30 q^{17} + 24 q^{18} + 6 q^{19} + 54 q^{21} - 24 q^{22} - 30 q^{23} + 24 q^{24} + 24 q^{26} + 20 q^{28} + 48 q^{29} - 42 q^{31} + 90 q^{33} + 62 q^{37} + 12 q^{38} + 12 q^{39} - 72 q^{42} + 8 q^{43} + 36 q^{44} + 36 q^{46} - 174 q^{47} - 20 q^{49} + 54 q^{51} + 72 q^{52} + 78 q^{53} - 36 q^{54} + 48 q^{56} - 12 q^{57} - 24 q^{58} - 78 q^{59} - 42 q^{61} + 216 q^{63} + 32 q^{64} - 144 q^{66} + 58 q^{67} - 60 q^{68} - 24 q^{71} + 48 q^{72} - 318 q^{73} + 96 q^{74} - 126 q^{77} - 96 q^{78} + 110 q^{79} + 18 q^{81} + 120 q^{82} + 12 q^{84} + 24 q^{86} + 144 q^{87} + 24 q^{88} - 378 q^{89} + 24 q^{91} + 120 q^{92} + 138 q^{93} - 12 q^{94} - 48 q^{96} + 120 q^{98} + 144 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1 + \beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 0.707107 + 1.22474i −0.707107 − 1.22474i 0.707107 − 1.22474i −0.707107 + 1.22474i
−0.707107 1.22474i 3.62132 + 2.09077i −1.00000 + 1.73205i 0 5.91359i 2.24264 6.63103i 2.82843 4.24264 + 7.34847i 0
101.2 0.707107 + 1.22474i −0.621320 0.358719i −1.00000 + 1.73205i 0 1.01461i −6.24264 + 3.16693i −2.82843 −4.24264 7.34847i 0
201.1 −0.707107 + 1.22474i 3.62132 2.09077i −1.00000 1.73205i 0 5.91359i 2.24264 + 6.63103i 2.82843 4.24264 7.34847i 0
201.2 0.707107 1.22474i −0.621320 + 0.358719i −1.00000 1.73205i 0 1.01461i −6.24264 3.16693i −2.82843 −4.24264 + 7.34847i 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.3.k.a 4
5.b even 2 1 14.3.d.a 4
5.c odd 4 2 350.3.i.a 8
7.d odd 6 1 inner 350.3.k.a 4
15.d odd 2 1 126.3.n.c 4
20.d odd 2 1 112.3.s.b 4
35.c odd 2 1 98.3.d.a 4
35.i odd 6 1 14.3.d.a 4
35.i odd 6 1 98.3.b.b 4
35.j even 6 1 98.3.b.b 4
35.j even 6 1 98.3.d.a 4
35.k even 12 2 350.3.i.a 8
40.e odd 2 1 448.3.s.c 4
40.f even 2 1 448.3.s.d 4
60.h even 2 1 1008.3.cg.l 4
105.g even 2 1 882.3.n.b 4
105.o odd 6 1 882.3.c.f 4
105.o odd 6 1 882.3.n.b 4
105.p even 6 1 126.3.n.c 4
105.p even 6 1 882.3.c.f 4
140.c even 2 1 784.3.s.c 4
140.p odd 6 1 784.3.c.e 4
140.p odd 6 1 784.3.s.c 4
140.s even 6 1 112.3.s.b 4
140.s even 6 1 784.3.c.e 4
280.ba even 6 1 448.3.s.c 4
280.bk odd 6 1 448.3.s.d 4
420.be odd 6 1 1008.3.cg.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.3.d.a 4 5.b even 2 1
14.3.d.a 4 35.i odd 6 1
98.3.b.b 4 35.i odd 6 1
98.3.b.b 4 35.j even 6 1
98.3.d.a 4 35.c odd 2 1
98.3.d.a 4 35.j even 6 1
112.3.s.b 4 20.d odd 2 1
112.3.s.b 4 140.s even 6 1
126.3.n.c 4 15.d odd 2 1
126.3.n.c 4 105.p even 6 1
350.3.i.a 8 5.c odd 4 2
350.3.i.a 8 35.k even 12 2
350.3.k.a 4 1.a even 1 1 trivial
350.3.k.a 4 7.d odd 6 1 inner
448.3.s.c 4 40.e odd 2 1
448.3.s.c 4 280.ba even 6 1
448.3.s.d 4 40.f even 2 1
448.3.s.d 4 280.bk odd 6 1
784.3.c.e 4 140.p odd 6 1
784.3.c.e 4 140.s even 6 1
784.3.s.c 4 140.c even 2 1
784.3.s.c 4 140.p odd 6 1
882.3.c.f 4 105.o odd 6 1
882.3.c.f 4 105.p even 6 1
882.3.n.b 4 105.g even 2 1
882.3.n.b 4 105.o odd 6 1
1008.3.cg.l 4 60.h even 2 1
1008.3.cg.l 4 420.be odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 6 T_{3}^{3} + 9 T_{3}^{2} + 18 T_{3} + 9$$ acting on $$S_{3}^{\mathrm{new}}(350, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T^{2} + T^{4}$$
$3$ $$9 + 18 T + 9 T^{2} - 6 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$2401 + 392 T + 42 T^{2} + 8 T^{3} + T^{4}$$
$11$ $$3969 - 1134 T + 261 T^{2} - 18 T^{3} + T^{4}$$
$13$ $$7056 + 264 T^{2} + T^{4}$$
$17$ $$2601 - 1530 T + 351 T^{2} - 30 T^{3} + T^{4}$$
$19$ $$9 + 18 T + 9 T^{2} - 6 T^{3} + T^{4}$$
$23$ $$3969 + 1890 T + 837 T^{2} + 30 T^{3} + T^{4}$$
$29$ $$( 72 - 24 T + T^{2} )^{2}$$
$31$ $$1447209 - 50526 T - 615 T^{2} + 42 T^{3} + T^{4}$$
$37$ $$36481 + 11842 T + 4035 T^{2} - 62 T^{3} + T^{4}$$
$41$ $$345744 + 1224 T^{2} + T^{4}$$
$43$ $$( -68 - 4 T + T^{2} )^{2}$$
$47$ $$6335289 + 437958 T + 12609 T^{2} + 174 T^{3} + T^{4}$$
$53$ $$1520289 - 96174 T + 4851 T^{2} - 78 T^{3} + T^{4}$$
$59$ $$10517049 - 252954 T - 1215 T^{2} + 78 T^{3} + T^{4}$$
$61$ $$35964009 - 251874 T - 5409 T^{2} + 42 T^{3} + T^{4}$$
$67$ $$10297681 + 186122 T + 6573 T^{2} - 58 T^{3} + T^{4}$$
$71$ $$( -1764 + 12 T + T^{2} )^{2}$$
$73$ $$47485881 + 2191338 T + 40599 T^{2} + 318 T^{3} + T^{4}$$
$79$ $$6630625 - 283250 T + 9525 T^{2} - 110 T^{3} + T^{4}$$
$83$ $$189778176 + 27936 T^{2} + T^{4}$$
$89$ $$71419401 + 3194478 T + 56079 T^{2} + 378 T^{3} + T^{4}$$
$97$ $$6780816 + 11016 T^{2} + T^{4}$$