# Properties

 Label 175.5.g.c Level $175$ Weight $5$ Character orbit 175.g Analytic conductor $18.090$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 175.g (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.0897435397$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(i)$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 20 q^{3} + 72 q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24 q - 20 q^{3} + 72 q^{6} + 156 q^{11} + 80 q^{12} + 560 q^{13} - 1480 q^{16} - 1320 q^{17} - 340 q^{18} + 196 q^{21} + 2020 q^{22} - 1920 q^{23} + 2208 q^{26} + 340 q^{27} - 2112 q^{31} + 1200 q^{32} + 6140 q^{33} + 3904 q^{36} - 3980 q^{37} - 9120 q^{38} + 6384 q^{41} - 4900 q^{42} + 12220 q^{43} - 8080 q^{46} + 11820 q^{47} + 4040 q^{48} - 5900 q^{51} - 3600 q^{52} - 24240 q^{53} - 10584 q^{56} - 6460 q^{57} - 6100 q^{58} + 440 q^{61} + 16680 q^{62} - 7840 q^{63} + 4832 q^{66} + 5940 q^{67} + 47040 q^{68} + 8928 q^{71} - 46720 q^{72} + 2500 q^{73} + 47816 q^{76} - 5880 q^{77} + 17940 q^{78} - 11360 q^{81} + 32120 q^{82} - 15120 q^{83} - 41208 q^{86} + 25460 q^{87} - 52920 q^{88} - 11172 q^{91} - 19800 q^{92} - 1460 q^{93} + 20568 q^{96} + 33840 q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1 −5.12784 + 5.12784i −10.3372 10.3372i 36.5894i 0 106.015 −13.0958 + 13.0958i 105.579 + 105.579i 132.716i 0
43.2 −5.01495 + 5.01495i 5.80055 + 5.80055i 34.2994i 0 −58.1789 13.0958 13.0958i 91.7707 + 91.7707i 13.7072i 0
43.3 −3.68989 + 3.68989i 5.41571 + 5.41571i 11.2306i 0 −39.9667 −13.0958 + 13.0958i −17.5987 17.5987i 22.3403i 0
43.4 −3.18640 + 3.18640i −5.77081 5.77081i 4.30624i 0 36.7762 13.0958 13.0958i −37.2610 37.2610i 14.3955i 0
43.5 −1.48373 + 1.48373i −2.63880 2.63880i 11.5971i 0 7.83054 −13.0958 + 13.0958i −40.9466 40.9466i 67.0734i 0
43.6 −0.896879 + 0.896879i 9.60523 + 9.60523i 14.3912i 0 −17.2295 13.0958 13.0958i −27.2572 27.2572i 103.521i 0
43.7 0.0151985 0.0151985i −8.40408 8.40408i 15.9995i 0 −0.255459 13.0958 13.0958i 0.486345 + 0.486345i 60.2572i 0
43.8 2.88957 2.88957i −11.3498 11.3498i 0.699277i 0 −65.5923 −13.0958 + 13.0958i 44.2126 + 44.2126i 176.637i 0
43.9 2.97419 2.97419i 1.22954 + 1.22954i 1.69164i 0 7.31376 −13.0958 + 13.0958i 42.5558 + 42.5558i 77.9765i 0
43.10 3.63546 3.63546i −2.99367 2.99367i 10.4331i 0 −21.7667 13.0958 13.0958i 20.2382 + 20.2382i 63.0758i 0
43.11 4.43769 4.43769i 10.8098 + 10.8098i 23.3862i 0 95.9409 −13.0958 + 13.0958i −32.7776 32.7776i 152.703i 0
43.12 5.44757 5.44757i −1.36639 1.36639i 43.3520i 0 −14.8870 13.0958 13.0958i −149.002 149.002i 77.2660i 0
57.1 −5.12784 5.12784i −10.3372 + 10.3372i 36.5894i 0 106.015 −13.0958 13.0958i 105.579 105.579i 132.716i 0
57.2 −5.01495 5.01495i 5.80055 5.80055i 34.2994i 0 −58.1789 13.0958 + 13.0958i 91.7707 91.7707i 13.7072i 0
57.3 −3.68989 3.68989i 5.41571 5.41571i 11.2306i 0 −39.9667 −13.0958 13.0958i −17.5987 + 17.5987i 22.3403i 0
57.4 −3.18640 3.18640i −5.77081 + 5.77081i 4.30624i 0 36.7762 13.0958 + 13.0958i −37.2610 + 37.2610i 14.3955i 0
57.5 −1.48373 1.48373i −2.63880 + 2.63880i 11.5971i 0 7.83054 −13.0958 13.0958i −40.9466 + 40.9466i 67.0734i 0
57.6 −0.896879 0.896879i 9.60523 9.60523i 14.3912i 0 −17.2295 13.0958 + 13.0958i −27.2572 + 27.2572i 103.521i 0
57.7 0.0151985 + 0.0151985i −8.40408 + 8.40408i 15.9995i 0 −0.255459 13.0958 + 13.0958i 0.486345 0.486345i 60.2572i 0
57.8 2.88957 + 2.88957i −11.3498 + 11.3498i 0.699277i 0 −65.5923 −13.0958 13.0958i 44.2126 44.2126i 176.637i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 57.12 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.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.5.g.c 24
5.b even 2 1 35.5.g.a 24
5.c odd 4 1 35.5.g.a 24
5.c odd 4 1 inner 175.5.g.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.g.a 24 5.b even 2 1
35.5.g.a 24 5.c odd 4 1
175.5.g.c 24 1.a even 1 1 trivial
175.5.g.c 24 5.c odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$26\!\cdots\!32$$$$T_{2}^{8} - 38953056000 T_{2}^{7} + 652411699200 T_{2}^{6} +$$$$35\!\cdots\!20$$$$T_{2}^{5} +$$$$18\!\cdots\!84$$$$T_{2}^{4} +$$$$25\!\cdots\!60$$$$T_{2}^{3} +$$$$18\!\cdots\!00$$$$T_{2}^{2} -$$$$56\!\cdots\!00$$$$T_{2} + 87379360000$$">$$T_{2}^{24} + \cdots$$ acting on $$S_{5}^{\mathrm{new}}(175, [\chi])$$.