# Properties

 Label 35.3.c.c Level $35$ Weight $3$ Character orbit 35.c Analytic conductor $0.954$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 35.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.953680925261$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{10})$$ Defining polynomial: $$x^{4} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{3} q^{3} -5 q^{4} + ( \beta_{2} - \beta_{3} ) q^{5} + ( -2 \beta_{2} + \beta_{3} ) q^{6} + ( \beta_{1} - 2 \beta_{3} ) q^{7} + \beta_{1} q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{3} q^{3} -5 q^{4} + ( \beta_{2} - \beta_{3} ) q^{5} + ( -2 \beta_{2} + \beta_{3} ) q^{6} + ( \beta_{1} - 2 \beta_{3} ) q^{7} + \beta_{1} q^{8} + q^{9} + ( \beta_{2} + 4 \beta_{3} ) q^{10} + 14 q^{11} -5 \beta_{3} q^{12} -\beta_{3} q^{13} + ( 9 + 4 \beta_{2} - 2 \beta_{3} ) q^{14} + ( -5 + 5 \beta_{1} ) q^{15} -11 q^{16} + 2 \beta_{3} q^{17} -\beta_{1} q^{18} + ( -6 \beta_{2} + 3 \beta_{3} ) q^{19} + ( -5 \beta_{2} + 5 \beta_{3} ) q^{20} + ( -20 + 2 \beta_{2} - \beta_{3} ) q^{21} -14 \beta_{1} q^{22} + 4 \beta_{1} q^{23} + ( 2 \beta_{2} - \beta_{3} ) q^{24} + ( -20 - 5 \beta_{1} ) q^{25} + ( 2 \beta_{2} - \beta_{3} ) q^{26} -8 \beta_{3} q^{27} + ( -5 \beta_{1} + 10 \beta_{3} ) q^{28} + 14 q^{29} + ( 45 + 5 \beta_{1} ) q^{30} + ( 8 \beta_{2} - 4 \beta_{3} ) q^{31} + 15 \beta_{1} q^{32} + 14 \beta_{3} q^{33} + ( -4 \beta_{2} + 2 \beta_{3} ) q^{34} + ( 10 - 10 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{35} -5 q^{36} -6 \beta_{1} q^{37} -27 \beta_{3} q^{38} -10 q^{39} + ( -\beta_{2} - 4 \beta_{3} ) q^{40} + ( -4 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 20 \beta_{1} + 9 \beta_{3} ) q^{42} + 14 \beta_{1} q^{43} -70 q^{44} + ( \beta_{2} - \beta_{3} ) q^{45} + 36 q^{46} + 14 \beta_{3} q^{47} -11 \beta_{3} q^{48} + ( 31 - 8 \beta_{2} + 4 \beta_{3} ) q^{49} + ( -45 + 20 \beta_{1} ) q^{50} + 20 q^{51} + 5 \beta_{3} q^{52} -18 \beta_{1} q^{53} + ( 16 \beta_{2} - 8 \beta_{3} ) q^{54} + ( 14 \beta_{2} - 14 \beta_{3} ) q^{55} + ( -9 - 4 \beta_{2} + 2 \beta_{3} ) q^{56} -30 \beta_{1} q^{57} -14 \beta_{1} q^{58} + ( 2 \beta_{2} - \beta_{3} ) q^{59} + ( 25 - 25 \beta_{1} ) q^{60} + ( -14 \beta_{2} + 7 \beta_{3} ) q^{61} + 36 \beta_{3} q^{62} + ( \beta_{1} - 2 \beta_{3} ) q^{63} + 91 q^{64} + ( 5 - 5 \beta_{1} ) q^{65} + ( -28 \beta_{2} + 14 \beta_{3} ) q^{66} + 34 \beta_{1} q^{67} -10 \beta_{3} q^{68} + ( 8 \beta_{2} - 4 \beta_{3} ) q^{69} + ( -90 - 10 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{70} -16 q^{71} + \beta_{1} q^{72} + 20 \beta_{3} q^{73} -54 q^{74} + ( -10 \beta_{2} - 15 \beta_{3} ) q^{75} + ( 30 \beta_{2} - 15 \beta_{3} ) q^{76} + ( 14 \beta_{1} - 28 \beta_{3} ) q^{77} + 10 \beta_{1} q^{78} -76 q^{79} + ( -11 \beta_{2} + 11 \beta_{3} ) q^{80} -89 q^{81} -18 \beta_{3} q^{82} + 23 \beta_{3} q^{83} + ( 100 - 10 \beta_{2} + 5 \beta_{3} ) q^{84} + ( -10 + 10 \beta_{1} ) q^{85} + 126 q^{86} + 14 \beta_{3} q^{87} + 14 \beta_{1} q^{88} + ( 12 \beta_{2} - 6 \beta_{3} ) q^{89} + ( \beta_{2} + 4 \beta_{3} ) q^{90} + ( 20 - 2 \beta_{2} + \beta_{3} ) q^{91} -20 \beta_{1} q^{92} + 40 \beta_{1} q^{93} + ( -28 \beta_{2} + 14 \beta_{3} ) q^{94} + ( 135 + 15 \beta_{1} ) q^{95} + ( 30 \beta_{2} - 15 \beta_{3} ) q^{96} -22 \beta_{3} q^{97} + ( -31 \beta_{1} - 36 \beta_{3} ) q^{98} + 14 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 20q^{4} + 4q^{9} + O(q^{10})$$ $$4q - 20q^{4} + 4q^{9} + 56q^{11} + 36q^{14} - 20q^{15} - 44q^{16} - 80q^{21} - 80q^{25} + 56q^{29} + 180q^{30} + 40q^{35} - 20q^{36} - 40q^{39} - 280q^{44} + 144q^{46} + 124q^{49} - 180q^{50} + 80q^{51} - 36q^{56} + 100q^{60} + 364q^{64} + 20q^{65} - 360q^{70} - 64q^{71} - 216q^{74} - 304q^{79} - 356q^{81} + 400q^{84} - 40q^{85} + 504q^{86} + 80q^{91} + 540q^{95} + 56q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$3 \nu^{2}$$$$/5$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 10 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 5 \nu$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{1}$$$$/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-10 \beta_{3} + 5 \beta_{2}$$$$)/3$$

## 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
 −1.58114 − 1.58114i 1.58114 + 1.58114i −1.58114 + 1.58114i 1.58114 − 1.58114i
3.00000i −3.16228 −5.00000 1.58114 4.74342i 9.48683i 6.32456 + 3.00000i 3.00000i 1.00000 −14.2302 4.74342i
34.2 3.00000i 3.16228 −5.00000 −1.58114 + 4.74342i 9.48683i −6.32456 + 3.00000i 3.00000i 1.00000 14.2302 + 4.74342i
34.3 3.00000i −3.16228 −5.00000 1.58114 + 4.74342i 9.48683i 6.32456 3.00000i 3.00000i 1.00000 −14.2302 + 4.74342i
34.4 3.00000i 3.16228 −5.00000 −1.58114 4.74342i 9.48683i −6.32456 3.00000i 3.00000i 1.00000 14.2302 4.74342i
 $$n$$: 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
5.b even 2 1 inner
7.b odd 2 1 inner
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 35.3.c.c 4
3.b odd 2 1 315.3.e.c 4
4.b odd 2 1 560.3.p.f 4
5.b even 2 1 inner 35.3.c.c 4
5.c odd 4 1 175.3.d.b 2
5.c odd 4 1 175.3.d.h 2
7.b odd 2 1 inner 35.3.c.c 4
7.c even 3 2 245.3.i.c 8
7.d odd 6 2 245.3.i.c 8
15.d odd 2 1 315.3.e.c 4
20.d odd 2 1 560.3.p.f 4
21.c even 2 1 315.3.e.c 4
28.d even 2 1 560.3.p.f 4
35.c odd 2 1 inner 35.3.c.c 4
35.f even 4 1 175.3.d.b 2
35.f even 4 1 175.3.d.h 2
35.i odd 6 2 245.3.i.c 8
35.j even 6 2 245.3.i.c 8
105.g even 2 1 315.3.e.c 4
140.c even 2 1 560.3.p.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.c.c 4 1.a even 1 1 trivial
35.3.c.c 4 5.b even 2 1 inner
35.3.c.c 4 7.b odd 2 1 inner
35.3.c.c 4 35.c odd 2 1 inner
175.3.d.b 2 5.c odd 4 1
175.3.d.b 2 35.f even 4 1
175.3.d.h 2 5.c odd 4 1
175.3.d.h 2 35.f even 4 1
245.3.i.c 8 7.c even 3 2
245.3.i.c 8 7.d odd 6 2
245.3.i.c 8 35.i odd 6 2
245.3.i.c 8 35.j even 6 2
315.3.e.c 4 3.b odd 2 1
315.3.e.c 4 15.d odd 2 1
315.3.e.c 4 21.c even 2 1
315.3.e.c 4 105.g even 2 1
560.3.p.f 4 4.b odd 2 1
560.3.p.f 4 20.d odd 2 1
560.3.p.f 4 28.d even 2 1
560.3.p.f 4 140.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 $$S_{3}^{\mathrm{new}}(35, [\chi])$$:

 $$T_{2}^{2} + 9$$ $$T_{3}^{2} - 10$$