# Properties

 Label 35.5.g.a Level $35$ Weight $5$ Character orbit 35.g Analytic conductor $3.618$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 35.g (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.61794870793$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(i)$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 20 q^{3} - 48 q^{5} + 72 q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24 q + 20 q^{3} - 48 q^{5} + 72 q^{6} - 112 q^{10} + 156 q^{11} - 80 q^{12} - 560 q^{13} + 896 q^{15} - 1480 q^{16} + 1320 q^{17} + 340 q^{18} + 180 q^{20} + 196 q^{21} - 2020 q^{22} + 1920 q^{23} - 676 q^{25} + 2208 q^{26} - 340 q^{27} - 5356 q^{30} - 2112 q^{31} - 1200 q^{32} - 6140 q^{33} + 3904 q^{36} + 3980 q^{37} + 9120 q^{38} + 14716 q^{40} + 6384 q^{41} + 4900 q^{42} - 12220 q^{43} - 10528 q^{45} - 8080 q^{46} - 11820 q^{47} - 4040 q^{48} + 10728 q^{50} - 5900 q^{51} + 3600 q^{52} + 24240 q^{53} + 4636 q^{55} - 10584 q^{56} + 6460 q^{57} + 6100 q^{58} - 30088 q^{60} + 440 q^{61} - 16680 q^{62} + 7840 q^{63} - 14652 q^{65} + 4832 q^{66} - 5940 q^{67} - 47040 q^{68} - 6272 q^{70} + 8928 q^{71} + 46720 q^{72} - 2500 q^{73} + 60708 q^{75} + 47816 q^{76} + 5880 q^{77} - 17940 q^{78} + 16140 q^{80} - 11360 q^{81} - 32120 q^{82} + 15120 q^{83} + 18816 q^{85} - 41208 q^{86} - 25460 q^{87} + 52920 q^{88} - 55680 q^{90} - 11172 q^{91} + 19800 q^{92} + 1460 q^{93} - 35508 q^{95} + 20568 q^{96} - 33840 q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1 −5.44757 + 5.44757i 1.36639 + 1.36639i 43.3520i 7.20796 23.9384i −14.8870 −13.0958 + 13.0958i 149.002 + 149.002i 77.2660i 91.1401 + 169.672i
8.2 −4.43769 + 4.43769i −10.8098 10.8098i 23.3862i 8.86080 + 23.3770i 95.9409 13.0958 13.0958i 32.7776 + 32.7776i 152.703i −143.062 64.4186i
8.3 −3.63546 + 3.63546i 2.99367 + 2.99367i 10.4331i −2.80777 + 24.8418i −21.7667 −13.0958 + 13.0958i −20.2382 20.2382i 63.0758i −80.1039 100.519i
8.4 −2.97419 + 2.97419i −1.22954 1.22954i 1.69164i −22.1755 11.5432i 7.31376 13.0958 13.0958i −42.5558 42.5558i 77.9765i 100.286 31.6225i
8.5 −2.88957 + 2.88957i 11.3498 + 11.3498i 0.699277i 24.2963 5.88989i −65.5923 13.0958 13.0958i −44.2126 44.2126i 176.637i −53.1866 + 87.2252i
8.6 −0.0151985 + 0.0151985i 8.40408 + 8.40408i 15.9995i −24.2998 5.87527i −0.255459 −13.0958 + 13.0958i −0.486345 0.486345i 60.2572i 0.458616 0.280025i
8.7 0.896879 0.896879i −9.60523 9.60523i 14.3912i −20.2649 + 14.6401i −17.2295 −13.0958 + 13.0958i 27.2572 + 27.2572i 103.521i −5.04483 + 31.3056i
8.8 1.48373 1.48373i 2.63880 + 2.63880i 11.5971i 8.51221 + 23.5062i 7.83054 13.0958 13.0958i 40.9466 + 40.9466i 67.0734i 47.5067 + 22.2471i
8.9 3.18640 3.18640i 5.77081 + 5.77081i 4.30624i 16.7353 18.5723i 36.7762 −13.0958 + 13.0958i 37.2610 + 37.2610i 14.3955i −5.85321 112.504i
8.10 3.68989 3.68989i −5.41571 5.41571i 11.2306i −14.8082 20.1424i −39.9667 13.0958 13.0958i 17.5987 + 17.5987i 22.3403i −128.964 19.6826i
8.11 5.01495 5.01495i −5.80055 5.80055i 34.2994i 17.0417 + 18.2915i −58.1789 −13.0958 + 13.0958i −91.7707 91.7707i 13.7072i 177.194 + 6.26750i
8.12 5.12784 5.12784i 10.3372 + 10.3372i 36.5894i −22.2980 + 11.3048i 106.015 13.0958 13.0958i −105.579 105.579i 132.716i −56.3718 + 172.310i
22.1 −5.44757 5.44757i 1.36639 1.36639i 43.3520i 7.20796 + 23.9384i −14.8870 −13.0958 13.0958i 149.002 149.002i 77.2660i 91.1401 169.672i
22.2 −4.43769 4.43769i −10.8098 + 10.8098i 23.3862i 8.86080 23.3770i 95.9409 13.0958 + 13.0958i 32.7776 32.7776i 152.703i −143.062 + 64.4186i
22.3 −3.63546 3.63546i 2.99367 2.99367i 10.4331i −2.80777 24.8418i −21.7667 −13.0958 13.0958i −20.2382 + 20.2382i 63.0758i −80.1039 + 100.519i
22.4 −2.97419 2.97419i −1.22954 + 1.22954i 1.69164i −22.1755 + 11.5432i 7.31376 13.0958 + 13.0958i −42.5558 + 42.5558i 77.9765i 100.286 + 31.6225i
22.5 −2.88957 2.88957i 11.3498 11.3498i 0.699277i 24.2963 + 5.88989i −65.5923 13.0958 + 13.0958i −44.2126 + 44.2126i 176.637i −53.1866 87.2252i
22.6 −0.0151985 0.0151985i 8.40408 8.40408i 15.9995i −24.2998 + 5.87527i −0.255459 −13.0958 13.0958i −0.486345 + 0.486345i 60.2572i 0.458616 + 0.280025i
22.7 0.896879 + 0.896879i −9.60523 + 9.60523i 14.3912i −20.2649 14.6401i −17.2295 −13.0958 13.0958i 27.2572 27.2572i 103.521i −5.04483 31.3056i
22.8 1.48373 + 1.48373i 2.63880 2.63880i 11.5971i 8.51221 23.5062i 7.83054 13.0958 + 13.0958i 40.9466 40.9466i 67.0734i 47.5067 22.2471i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 22.12 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 35.5.g.a 24
5.b even 2 1 175.5.g.c 24
5.c odd 4 1 inner 35.5.g.a 24
5.c odd 4 1 175.5.g.c 24

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(35, [\chi])$$.