# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 35.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.61794870793$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-6})$$ Defining polynomial: $$x^{2} + 6$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} -5 q^{3} + 10 q^{4} + ( -5 + 10 \beta ) q^{5} -5 \beta q^{6} + ( -35 + 14 \beta ) q^{7} + 26 \beta q^{8} -56 q^{9} +O(q^{10})$$ $$q + \beta q^{2} -5 q^{3} + 10 q^{4} + ( -5 + 10 \beta ) q^{5} -5 \beta q^{6} + ( -35 + 14 \beta ) q^{7} + 26 \beta q^{8} -56 q^{9} + ( -60 - 5 \beta ) q^{10} + 89 q^{11} -50 q^{12} + 5 q^{13} + ( -84 - 35 \beta ) q^{14} + ( 25 - 50 \beta ) q^{15} + 4 q^{16} + 485 q^{17} -56 \beta q^{18} + 90 \beta q^{19} + ( -50 + 100 \beta ) q^{20} + ( 175 - 70 \beta ) q^{21} + 89 \beta q^{22} -286 \beta q^{23} -130 \beta q^{24} + ( -575 - 100 \beta ) q^{25} + 5 \beta q^{26} + 685 q^{27} + ( -350 + 140 \beta ) q^{28} + 191 q^{29} + ( 300 + 25 \beta ) q^{30} -430 \beta q^{31} + 420 \beta q^{32} -445 q^{33} + 485 \beta q^{34} + ( -665 - 420 \beta ) q^{35} -560 q^{36} + 666 \beta q^{37} -540 q^{38} -25 q^{39} + ( -1560 - 130 \beta ) q^{40} + 1190 \beta q^{41} + ( 420 + 175 \beta ) q^{42} + 154 \beta q^{43} + 890 q^{44} + ( 280 - 560 \beta ) q^{45} + 1716 q^{46} + 2195 q^{47} -20 q^{48} + ( 49 - 980 \beta ) q^{49} + ( 600 - 575 \beta ) q^{50} -2425 q^{51} + 50 q^{52} -648 \beta q^{53} + 685 \beta q^{54} + ( -445 + 890 \beta ) q^{55} + ( -2184 - 910 \beta ) q^{56} -450 \beta q^{57} + 191 \beta q^{58} -1480 \beta q^{59} + ( 250 - 500 \beta ) q^{60} + 790 \beta q^{61} + 2580 q^{62} + ( 1960 - 784 \beta ) q^{63} -2456 q^{64} + ( -25 + 50 \beta ) q^{65} -445 \beta q^{66} + 836 \beta q^{67} + 4850 q^{68} + 1430 \beta q^{69} + ( 2520 - 665 \beta ) q^{70} + 4454 q^{71} -1456 \beta q^{72} -8650 q^{73} -3996 q^{74} + ( 2875 + 500 \beta ) q^{75} + 900 \beta q^{76} + ( -3115 + 1246 \beta ) q^{77} -25 \beta q^{78} + 5561 q^{79} + ( -20 + 40 \beta ) q^{80} + 1111 q^{81} -7140 q^{82} -1990 q^{83} + ( 1750 - 700 \beta ) q^{84} + ( -2425 + 4850 \beta ) q^{85} -924 q^{86} -955 q^{87} + 2314 \beta q^{88} -330 \beta q^{89} + ( 3360 + 280 \beta ) q^{90} + ( -175 + 70 \beta ) q^{91} -2860 \beta q^{92} + 2150 \beta q^{93} + 2195 \beta q^{94} + ( -5400 - 450 \beta ) q^{95} -2100 \beta q^{96} -9235 q^{97} + ( 5880 + 49 \beta ) q^{98} -4984 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 10 q^{3} + 20 q^{4} - 10 q^{5} - 70 q^{7} - 112 q^{9} + O(q^{10})$$ $$2 q - 10 q^{3} + 20 q^{4} - 10 q^{5} - 70 q^{7} - 112 q^{9} - 120 q^{10} + 178 q^{11} - 100 q^{12} + 10 q^{13} - 168 q^{14} + 50 q^{15} + 8 q^{16} + 970 q^{17} - 100 q^{20} + 350 q^{21} - 1150 q^{25} + 1370 q^{27} - 700 q^{28} + 382 q^{29} + 600 q^{30} - 890 q^{33} - 1330 q^{35} - 1120 q^{36} - 1080 q^{38} - 50 q^{39} - 3120 q^{40} + 840 q^{42} + 1780 q^{44} + 560 q^{45} + 3432 q^{46} + 4390 q^{47} - 40 q^{48} + 98 q^{49} + 1200 q^{50} - 4850 q^{51} + 100 q^{52} - 890 q^{55} - 4368 q^{56} + 500 q^{60} + 5160 q^{62} + 3920 q^{63} - 4912 q^{64} - 50 q^{65} + 9700 q^{68} + 5040 q^{70} + 8908 q^{71} - 17300 q^{73} - 7992 q^{74} + 5750 q^{75} - 6230 q^{77} + 11122 q^{79} - 40 q^{80} + 2222 q^{81} - 14280 q^{82} - 3980 q^{83} + 3500 q^{84} - 4850 q^{85} - 1848 q^{86} - 1910 q^{87} + 6720 q^{90} - 350 q^{91} - 10800 q^{95} - 18470 q^{97} + 11760 q^{98} - 9968 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/35\mathbb{Z}\right)^\times$$.

 $$n$$ $$22$$ $$31$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
34.1
 − 2.44949i 2.44949i
2.44949i −5.00000 10.0000 −5.00000 24.4949i 12.2474i −35.0000 34.2929i 63.6867i −56.0000 −60.0000 + 12.2474i
34.2 2.44949i −5.00000 10.0000 −5.00000 + 24.4949i 12.2474i −35.0000 + 34.2929i 63.6867i −56.0000 −60.0000 12.2474i
 $$n$$: 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
35.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.5.c.c 2
3.b odd 2 1 315.5.e.d 2
4.b odd 2 1 560.5.p.e 2
5.b even 2 1 35.5.c.d yes 2
5.c odd 4 2 175.5.d.e 4
7.b odd 2 1 35.5.c.d yes 2
15.d odd 2 1 315.5.e.c 2
20.d odd 2 1 560.5.p.d 2
21.c even 2 1 315.5.e.c 2
28.d even 2 1 560.5.p.d 2
35.c odd 2 1 inner 35.5.c.c 2
35.f even 4 2 175.5.d.e 4
105.g even 2 1 315.5.e.d 2
140.c even 2 1 560.5.p.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.c 2 1.a even 1 1 trivial
35.5.c.c 2 35.c odd 2 1 inner
35.5.c.d yes 2 5.b even 2 1
35.5.c.d yes 2 7.b odd 2 1
175.5.d.e 4 5.c odd 4 2
175.5.d.e 4 35.f even 4 2
315.5.e.c 2 15.d odd 2 1
315.5.e.c 2 21.c even 2 1
315.5.e.d 2 3.b odd 2 1
315.5.e.d 2 105.g even 2 1
560.5.p.d 2 20.d odd 2 1
560.5.p.d 2 28.d even 2 1
560.5.p.e 2 4.b odd 2 1
560.5.p.e 2 140.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{5}^{\mathrm{new}}(35, [\chi])$$:

 $$T_{2}^{2} + 6$$ $$T_{3} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$6 + T^{2}$$
$3$ $$( 5 + T )^{2}$$
$5$ $$625 + 10 T + T^{2}$$
$7$ $$2401 + 70 T + T^{2}$$
$11$ $$( -89 + T )^{2}$$
$13$ $$( -5 + T )^{2}$$
$17$ $$( -485 + T )^{2}$$
$19$ $$48600 + T^{2}$$
$23$ $$490776 + T^{2}$$
$29$ $$( -191 + T )^{2}$$
$31$ $$1109400 + T^{2}$$
$37$ $$2661336 + T^{2}$$
$41$ $$8496600 + T^{2}$$
$43$ $$142296 + T^{2}$$
$47$ $$( -2195 + T )^{2}$$
$53$ $$2519424 + T^{2}$$
$59$ $$13142400 + T^{2}$$
$61$ $$3744600 + T^{2}$$
$67$ $$4193376 + T^{2}$$
$71$ $$( -4454 + T )^{2}$$
$73$ $$( 8650 + T )^{2}$$
$79$ $$( -5561 + T )^{2}$$
$83$ $$( 1990 + T )^{2}$$
$89$ $$653400 + T^{2}$$
$97$ $$( 9235 + T )^{2}$$