# Properties

 Label 350.2.o.a Level 350 Weight 2 Character orbit 350.o Analytic conductor 2.795 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 350.o (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$\zeta_{24}^{4}$$ $$\zeta_{24}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 0.258819 − 0.965926i −0.258819 + 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i 0.965926 − 0.258819i −0.965926 + 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i
−0.965926 0.258819i −0.707107 2.63896i 0.866025 + 0.500000i 0 2.73205i 2.19067 1.48356i −0.707107 0.707107i −3.86603 + 2.23205i 0
143.2 0.965926 + 0.258819i 0.707107 + 2.63896i 0.866025 + 0.500000i 0 2.73205i −2.19067 + 1.48356i 0.707107 + 0.707107i −3.86603 + 2.23205i 0
157.1 −0.258819 + 0.965926i 0.707107 0.189469i −0.866025 0.500000i 0 0.732051i 1.48356 + 2.19067i 0.707107 0.707107i −2.13397 + 1.23205i 0
157.2 0.258819 0.965926i −0.707107 + 0.189469i −0.866025 0.500000i 0 0.732051i −1.48356 2.19067i −0.707107 + 0.707107i −2.13397 + 1.23205i 0
243.1 −0.258819 0.965926i 0.707107 + 0.189469i −0.866025 + 0.500000i 0 0.732051i 1.48356 2.19067i 0.707107 + 0.707107i −2.13397 1.23205i 0
243.2 0.258819 + 0.965926i −0.707107 0.189469i −0.866025 + 0.500000i 0 0.732051i −1.48356 + 2.19067i −0.707107 0.707107i −2.13397 1.23205i 0
257.1 −0.965926 + 0.258819i −0.707107 + 2.63896i 0.866025 0.500000i 0 2.73205i 2.19067 + 1.48356i −0.707107 + 0.707107i −3.86603 2.23205i 0
257.2 0.965926 0.258819i 0.707107 2.63896i 0.866025 0.500000i 0 2.73205i −2.19067 1.48356i 0.707107 0.707107i −3.86603 2.23205i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 257.2 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
35.k even 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.o.a 8
5.b even 2 1 inner 350.2.o.a 8
5.c odd 4 2 350.2.o.b yes 8
7.d odd 6 1 350.2.o.b yes 8
35.i odd 6 1 350.2.o.b yes 8
35.k even 12 2 inner 350.2.o.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.o.a 8 1.a even 1 1 trivial
350.2.o.a 8 5.b even 2 1 inner
350.2.o.a 8 35.k even 12 2 inner
350.2.o.b yes 8 5.c odd 4 2
350.2.o.b yes 8 7.d odd 6 1
350.2.o.b yes 8 35.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 12 T_{3}^{6} + 44 T_{3}^{4} - 48 T_{3}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(350, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8}$$
$3$ $$1 + 12 T^{2} + 74 T^{4} + 312 T^{6} + 1027 T^{8} + 2808 T^{10} + 5994 T^{12} + 8748 T^{14} + 6561 T^{16}$$
$5$ 
$7$ $$1 + 71 T^{4} + 2401 T^{8}$$
$11$ $$( 1 - 6 T + 8 T^{2} - 36 T^{3} + 267 T^{4} - 396 T^{5} + 968 T^{6} - 7986 T^{7} + 14641 T^{8} )^{2}$$
$13$ $$1 - 452 T^{4} + 106470 T^{8} - 12909572 T^{12} + 815730721 T^{16}$$
$17$ $$1 - 72 T^{2} + 2569 T^{4} - 60552 T^{6} + 1123152 T^{8} - 17499528 T^{10} + 214565449 T^{12} - 1737904968 T^{14} + 6975757441 T^{16}$$
$19$ $$( 1 - 19 T^{2} + 361 T^{4} )^{4}$$
$23$ $$1 + 697 T^{4} + 205968 T^{8} + 195049177 T^{12} + 78310985281 T^{16}$$
$29$ $$( 1 - 46 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{4}( 1 + 7 T + 31 T^{2} )^{4}$$
$37$ $$( 1 - 529 T^{4} + 1874161 T^{8} )( 1 + 2591 T^{4} + 1874161 T^{8} )$$
$41$ $$( 1 - 38 T^{2} - 165 T^{4} - 63878 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$1 - 5444 T^{4} + 14231334 T^{8} - 18611952644 T^{12} + 11688200277601 T^{16}$$
$47$ $$1 + 144 T^{2} + 11689 T^{4} + 687888 T^{6} + 33208656 T^{8} + 1519544592 T^{10} + 57038591209 T^{12} + 1552207007376 T^{14} + 23811286661761 T^{16}$$
$53$ $$1 + 108 T^{2} + 5578 T^{4} + 182520 T^{6} + 5887011 T^{8} + 512698680 T^{10} + 44013103018 T^{12} + 2393751001932 T^{14} + 62259690411361 T^{16}$$
$59$ $$( 1 - 18 T + 152 T^{2} - 972 T^{3} + 6987 T^{4} - 57348 T^{5} + 529112 T^{6} - 3696822 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 6 T + 56 T^{2} - 264 T^{3} - 1053 T^{4} - 16104 T^{5} + 208376 T^{6} - 1361886 T^{7} + 13845841 T^{8} )^{2}$$
$67$ $$1 + 324 T^{2} + 52042 T^{4} + 5524200 T^{6} + 427630467 T^{8} + 24798133800 T^{10} + 1048704639082 T^{12} + 29308515822756 T^{14} + 406067677556641 T^{16}$$
$71$ $$( 1 - 6 T + 43 T^{2} - 426 T^{3} + 5041 T^{4} )^{4}$$
$73$ $$1 + 3934 T^{4} - 12921885 T^{8} + 111718680094 T^{12} + 806460091894081 T^{16}$$
$79$ $$( 1 - 36 T + 697 T^{2} - 9540 T^{3} + 98112 T^{4} - 753660 T^{5} + 4349977 T^{6} - 17749404 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 + 3122 T^{4} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 12 T - 43 T^{2} + 108 T^{3} + 14232 T^{4} + 9612 T^{5} - 340603 T^{6} + 8459628 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$1 - 22322 T^{4} + 289141683 T^{8} - 1976150610482 T^{12} + 7837433594376961 T^{16}$$