# Properties

 Label 975.1.cj.a Level $975$ Weight $1$ Character orbit 975.cj Analytic conductor $0.487$ Analytic rank $0$ Dimension $16$ Projective image $A_{5}$ CM/RM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 975.cj (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.486588387317$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{30})$$ Coefficient field: $$\Q(\zeta_{60})$$ Defining polynomial: $$x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$A_{5}$$ Projective field: Galois closure of 5.1.594140625.2

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{60}^{11} + \zeta_{60}^{17} ) q^{2} + \zeta_{60}^{23} q^{3} + ( -\zeta_{60}^{4} + \zeta_{60}^{22} - \zeta_{60}^{28} ) q^{4} + \zeta_{60}^{9} q^{5} + ( \zeta_{60}^{4} - \zeta_{60}^{10} ) q^{6} -\zeta_{60}^{10} q^{7} + ( \zeta_{60}^{3} - \zeta_{60}^{9} + \zeta_{60}^{15} - \zeta_{60}^{21} ) q^{8} -\zeta_{60}^{16} q^{9} +O(q^{10})$$ $$q + ( -\zeta_{60}^{11} + \zeta_{60}^{17} ) q^{2} + \zeta_{60}^{23} q^{3} + ( -\zeta_{60}^{4} + \zeta_{60}^{22} - \zeta_{60}^{28} ) q^{4} + \zeta_{60}^{9} q^{5} + ( \zeta_{60}^{4} - \zeta_{60}^{10} ) q^{6} -\zeta_{60}^{10} q^{7} + ( \zeta_{60}^{3} - \zeta_{60}^{9} + \zeta_{60}^{15} - \zeta_{60}^{21} ) q^{8} -\zeta_{60}^{16} q^{9} + ( -\zeta_{60}^{20} + \zeta_{60}^{26} ) q^{10} + ( \zeta_{60}^{11} - \zeta_{60}^{17} ) q^{11} + ( -\zeta_{60}^{15} + \zeta_{60}^{21} - \zeta_{60}^{27} ) q^{12} + \zeta_{60}^{6} q^{13} + ( \zeta_{60}^{21} - \zeta_{60}^{27} ) q^{14} -\zeta_{60}^{2} q^{15} + ( -\zeta_{60}^{2} + \zeta_{60}^{8} - \zeta_{60}^{14} + \zeta_{60}^{20} - \zeta_{60}^{26} ) q^{16} + \zeta_{60}^{7} q^{17} + ( \zeta_{60}^{3} + \zeta_{60}^{27} ) q^{18} -\zeta_{60}^{22} q^{19} + ( -\zeta_{60} + \zeta_{60}^{7} - \zeta_{60}^{13} ) q^{20} + \zeta_{60}^{3} q^{21} + ( \zeta_{60}^{4} - \zeta_{60}^{22} + 2 \zeta_{60}^{28} ) q^{22} + \zeta_{60}^{29} q^{23} + ( \zeta_{60}^{2} - \zeta_{60}^{8} + \zeta_{60}^{14} + \zeta_{60}^{26} ) q^{24} + \zeta_{60}^{18} q^{25} + ( -\zeta_{60}^{17} + \zeta_{60}^{23} ) q^{26} + \zeta_{60}^{9} q^{27} + ( \zeta_{60}^{2} - \zeta_{60}^{8} + \zeta_{60}^{14} ) q^{28} + ( \zeta_{60}^{13} - \zeta_{60}^{19} ) q^{30} + ( \zeta_{60} - \zeta_{60}^{7} + \zeta_{60}^{13} - \zeta_{60}^{19} + 2 \zeta_{60}^{25} ) q^{32} + ( -\zeta_{60}^{4} + \zeta_{60}^{10} ) q^{33} + ( -\zeta_{60}^{18} + \zeta_{60}^{24} ) q^{34} -\zeta_{60}^{19} q^{35} + ( \zeta_{60}^{8} - \zeta_{60}^{14} + \zeta_{60}^{20} ) q^{36} + \zeta_{60}^{26} q^{37} + ( -\zeta_{60}^{3} + \zeta_{60}^{9} ) q^{38} + \zeta_{60}^{29} q^{39} + ( 1 + \zeta_{60}^{12} - \zeta_{60}^{18} + \zeta_{60}^{24} ) q^{40} + \zeta_{60}^{11} q^{41} + ( -\zeta_{60}^{14} + \zeta_{60}^{20} ) q^{42} + ( \zeta_{60}^{4} + \zeta_{60}^{16} ) q^{43} + ( -\zeta_{60}^{3} + 2 \zeta_{60}^{9} - 2 \zeta_{60}^{15} + \zeta_{60}^{21} ) q^{44} -\zeta_{60}^{25} q^{45} + ( \zeta_{60}^{10} - \zeta_{60}^{16} ) q^{46} + ( -\zeta_{60}^{9} + \zeta_{60}^{27} ) q^{47} + ( -\zeta_{60} + \zeta_{60}^{7} - \zeta_{60}^{13} + \zeta_{60}^{19} - \zeta_{60}^{25} ) q^{48} + ( -\zeta_{60}^{5} - \zeta_{60}^{29} ) q^{50} - q^{51} + ( \zeta_{60}^{4} - \zeta_{60}^{10} + \zeta_{60}^{28} ) q^{52} + ( -\zeta_{60}^{15} + \zeta_{60}^{21} ) q^{53} + ( -\zeta_{60}^{20} + \zeta_{60}^{26} ) q^{54} + ( \zeta_{60}^{20} - \zeta_{60}^{26} ) q^{55} + ( -\zeta_{60} - \zeta_{60}^{13} + \zeta_{60}^{19} - \zeta_{60}^{25} ) q^{56} + \zeta_{60}^{15} q^{57} + \zeta_{60} q^{59} + ( -1 + \zeta_{60}^{6} - \zeta_{60}^{24} ) q^{60} + \zeta_{60}^{4} q^{61} + \zeta_{60}^{26} q^{63} + ( -1 + 2 \zeta_{60}^{6} - 2 \zeta_{60}^{12} + \zeta_{60}^{18} - \zeta_{60}^{24} ) q^{64} + \zeta_{60}^{15} q^{65} + ( \zeta_{60}^{15} - 2 \zeta_{60}^{21} + \zeta_{60}^{27} ) q^{66} + ( \zeta_{60}^{8} - \zeta_{60}^{26} ) q^{67} + ( \zeta_{60}^{5} - \zeta_{60}^{11} + \zeta_{60}^{29} ) q^{68} -\zeta_{60}^{22} q^{69} + ( -1 + \zeta_{60}^{6} ) q^{70} + ( \zeta_{60} - \zeta_{60}^{7} - \zeta_{60}^{19} + \zeta_{60}^{25} ) q^{72} -\zeta_{60}^{24} q^{73} + ( \zeta_{60}^{7} - \zeta_{60}^{13} ) q^{74} -\zeta_{60}^{11} q^{75} + ( \zeta_{60}^{14} - \zeta_{60}^{20} + \zeta_{60}^{26} ) q^{76} + ( -\zeta_{60}^{21} + \zeta_{60}^{27} ) q^{77} + ( \zeta_{60}^{10} - \zeta_{60}^{16} ) q^{78} + ( \zeta_{60}^{5} - \zeta_{60}^{11} + \zeta_{60}^{17} - \zeta_{60}^{23} + \zeta_{60}^{29} ) q^{80} -\zeta_{60}^{2} q^{81} + ( -\zeta_{60}^{22} + \zeta_{60}^{28} ) q^{82} -\zeta_{60}^{27} q^{83} + ( \zeta_{60} - \zeta_{60}^{7} + \zeta_{60}^{25} ) q^{84} + \zeta_{60}^{16} q^{85} + ( -\zeta_{60}^{3} - \zeta_{60}^{15} + \zeta_{60}^{21} - \zeta_{60}^{27} ) q^{86} + ( 2 \zeta_{60}^{2} - \zeta_{60}^{8} + \zeta_{60}^{14} - 2 \zeta_{60}^{20} + 2 \zeta_{60}^{26} ) q^{88} -\zeta_{60}^{29} q^{89} + ( -\zeta_{60}^{6} + \zeta_{60}^{12} ) q^{90} -\zeta_{60}^{16} q^{91} + ( \zeta_{60}^{3} - \zeta_{60}^{21} + \zeta_{60}^{27} ) q^{92} + ( \zeta_{60}^{8} - \zeta_{60}^{14} + \zeta_{60}^{20} - \zeta_{60}^{26} ) q^{94} + \zeta_{60} q^{95} + ( 1 - \zeta_{60}^{6} + \zeta_{60}^{12} - 2 \zeta_{60}^{18} + \zeta_{60}^{24} ) q^{96} + ( -\zeta_{60}^{4} + \zeta_{60}^{22} ) q^{97} + ( -\zeta_{60}^{3} - \zeta_{60}^{27} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 6q^{4} - 6q^{6} - 8q^{7} - 2q^{9} + O(q^{10})$$ $$16q - 6q^{4} - 6q^{6} - 8q^{7} - 2q^{9} + 6q^{10} + 4q^{13} + 2q^{15} + 2q^{19} + 8q^{22} - 8q^{24} + 4q^{25} - 6q^{28} + 6q^{33} - 8q^{34} - 4q^{36} - 2q^{37} + 4q^{40} - 6q^{42} + 4q^{43} + 6q^{46} - 16q^{51} - 4q^{52} + 6q^{54} - 6q^{55} - 8q^{60} + 2q^{61} - 2q^{63} + 8q^{64} + 4q^{67} + 2q^{69} - 12q^{70} + 4q^{73} + 4q^{76} + 6q^{78} + 2q^{81} + 4q^{82} + 2q^{85} + 4q^{88} - 8q^{90} - 2q^{91} - 2q^{94} - 4q^{96} - 4q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$-\zeta_{60}^{10}$$ $$-1$$ $$-\zeta_{60}^{18}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
146.1
 −0.207912 + 0.978148i 0.207912 − 0.978148i 0.406737 − 0.913545i −0.406737 + 0.913545i 0.994522 − 0.104528i −0.994522 + 0.104528i 0.994522 + 0.104528i −0.994522 − 0.104528i 0.406737 + 0.913545i −0.406737 − 0.913545i −0.207912 − 0.978148i 0.207912 + 0.978148i −0.743145 + 0.669131i 0.743145 − 0.669131i −0.743145 − 0.669131i 0.743145 + 0.669131i
−0.336408 1.58268i −0.994522 0.104528i −1.47815 + 0.658114i −0.951057 0.309017i 0.169131 + 1.60917i −0.500000 + 0.866025i 0.587785 + 0.809017i 0.978148 + 0.207912i −0.169131 + 1.60917i
146.2 0.336408 + 1.58268i 0.994522 + 0.104528i −1.47815 + 0.658114i 0.951057 + 0.309017i 0.169131 + 1.60917i −0.500000 + 0.866025i −0.587785 0.809017i 0.978148 + 0.207912i −0.169131 + 1.60917i
191.1 −0.251377 0.564602i 0.207912 0.978148i 0.413545 0.459289i −0.587785 + 0.809017i −0.604528 + 0.128496i −0.500000 0.866025i −0.951057 0.309017i −0.913545 0.406737i 0.604528 + 0.128496i
191.2 0.251377 + 0.564602i −0.207912 + 0.978148i 0.413545 0.459289i 0.587785 0.809017i −0.604528 + 0.128496i −0.500000 0.866025i 0.951057 + 0.309017i −0.913545 0.406737i 0.604528 + 0.128496i
341.1 −0.614648 0.0646021i −0.743145 0.669131i −0.604528 0.128496i 0.587785 0.809017i 0.413545 + 0.459289i −0.500000 + 0.866025i 0.951057 + 0.309017i 0.104528 + 0.994522i −0.413545 + 0.459289i
341.2 0.614648 + 0.0646021i 0.743145 + 0.669131i −0.604528 0.128496i −0.587785 + 0.809017i 0.413545 + 0.459289i −0.500000 + 0.866025i −0.951057 0.309017i 0.104528 + 0.994522i −0.413545 + 0.459289i
386.1 −0.614648 + 0.0646021i −0.743145 + 0.669131i −0.604528 + 0.128496i 0.587785 + 0.809017i 0.413545 0.459289i −0.500000 0.866025i 0.951057 0.309017i 0.104528 0.994522i −0.413545 0.459289i
386.2 0.614648 0.0646021i 0.743145 0.669131i −0.604528 + 0.128496i −0.587785 0.809017i 0.413545 0.459289i −0.500000 0.866025i −0.951057 + 0.309017i 0.104528 0.994522i −0.413545 0.459289i
536.1 −0.251377 + 0.564602i 0.207912 + 0.978148i 0.413545 + 0.459289i −0.587785 0.809017i −0.604528 0.128496i −0.500000 + 0.866025i −0.951057 + 0.309017i −0.913545 + 0.406737i 0.604528 0.128496i
536.2 0.251377 0.564602i −0.207912 0.978148i 0.413545 + 0.459289i 0.587785 + 0.809017i −0.604528 0.128496i −0.500000 + 0.866025i 0.951057 0.309017i −0.913545 + 0.406737i 0.604528 0.128496i
581.1 −0.336408 + 1.58268i −0.994522 + 0.104528i −1.47815 0.658114i −0.951057 + 0.309017i 0.169131 1.60917i −0.500000 0.866025i 0.587785 0.809017i 0.978148 0.207912i −0.169131 1.60917i
581.2 0.336408 1.58268i 0.994522 0.104528i −1.47815 0.658114i 0.951057 0.309017i 0.169131 1.60917i −0.500000 0.866025i −0.587785 + 0.809017i 0.978148 0.207912i −0.169131 1.60917i
731.1 −1.20243 1.08268i 0.406737 0.913545i 0.169131 + 1.60917i −0.951057 + 0.309017i −1.47815 + 0.658114i −0.500000 + 0.866025i 0.587785 0.809017i −0.669131 0.743145i 1.47815 + 0.658114i
731.2 1.20243 + 1.08268i −0.406737 + 0.913545i 0.169131 + 1.60917i 0.951057 0.309017i −1.47815 + 0.658114i −0.500000 + 0.866025i −0.587785 + 0.809017i −0.669131 0.743145i 1.47815 + 0.658114i
971.1 −1.20243 + 1.08268i 0.406737 + 0.913545i 0.169131 1.60917i −0.951057 0.309017i −1.47815 0.658114i −0.500000 0.866025i 0.587785 + 0.809017i −0.669131 + 0.743145i 1.47815 0.658114i
971.2 1.20243 1.08268i −0.406737 0.913545i 0.169131 1.60917i 0.951057 + 0.309017i −1.47815 0.658114i −0.500000 0.866025i −0.587785 0.809017i −0.669131 + 0.743145i 1.47815 0.658114i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 971.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.c even 3 1 inner
25.d even 5 1 inner
39.i odd 6 1 inner
75.j odd 10 1 inner
325.y even 15 1 inner
975.cj odd 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.1.cj.a 16
3.b odd 2 1 inner 975.1.cj.a 16
13.c even 3 1 inner 975.1.cj.a 16
25.d even 5 1 inner 975.1.cj.a 16
39.i odd 6 1 inner 975.1.cj.a 16
75.j odd 10 1 inner 975.1.cj.a 16
325.y even 15 1 inner 975.1.cj.a 16
975.cj odd 30 1 inner 975.1.cj.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.1.cj.a 16 1.a even 1 1 trivial
975.1.cj.a 16 3.b odd 2 1 inner
975.1.cj.a 16 13.c even 3 1 inner
975.1.cj.a 16 25.d even 5 1 inner
975.1.cj.a 16 39.i odd 6 1 inner
975.1.cj.a 16 75.j odd 10 1 inner
975.1.cj.a 16 325.y even 15 1 inner
975.1.cj.a 16 975.cj odd 30 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(975, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} - 5 T^{4} - 14 T^{6} + 39 T^{8} + 26 T^{10} + 10 T^{12} + 4 T^{14} + T^{16}$$
$3$ $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
$5$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$7$ $$( 1 + T + T^{2} )^{8}$$
$11$ $$1 - T^{2} - 5 T^{4} - 14 T^{6} + 39 T^{8} + 26 T^{10} + 10 T^{12} + 4 T^{14} + T^{16}$$
$13$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$
$17$ $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
$19$ $$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$
$23$ $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
$29$ $$T^{16}$$
$31$ $$T^{16}$$
$37$ $$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$
$41$ $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
$43$ $$( 1 + T + 2 T^{2} - T^{3} + T^{4} )^{4}$$
$47$ $$( 1 + T^{2} + 6 T^{4} - 4 T^{6} + T^{8} )^{2}$$
$53$ $$( 1 - 4 T^{2} + 6 T^{4} + T^{6} + T^{8} )^{2}$$
$59$ $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
$61$ $$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$
$67$ $$( 1 - 3 T + 5 T^{2} - 8 T^{3} + 9 T^{4} - 2 T^{5} - 2 T^{7} + T^{8} )^{2}$$
$71$ $$T^{16}$$
$73$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$
$79$ $$T^{16}$$
$83$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$89$ $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
$97$ $$( 1 + 3 T + 5 T^{2} + 8 T^{3} + 9 T^{4} + 2 T^{5} + 2 T^{7} + T^{8} )^{2}$$