# Properties

 Label 175.3.c.d Level $175$ Weight $3$ Character orbit 175.c Analytic conductor $4.768$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 175.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.76840462631$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 35) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{3} q^{3} + 3 q^{4} + \beta_{2} q^{6} - 7 \beta_1 q^{7} + 7 \beta_1 q^{8} + 11 q^{9}+O(q^{10})$$ q + b1 * q^2 + b3 * q^3 + 3 * q^4 + b2 * q^6 - 7*b1 * q^7 + 7*b1 * q^8 + 11 * q^9 $$q + \beta_1 q^{2} + \beta_{3} q^{3} + 3 q^{4} + \beta_{2} q^{6} - 7 \beta_1 q^{7} + 7 \beta_1 q^{8} + 11 q^{9} + 2 q^{11} + 3 \beta_{3} q^{12} - 3 \beta_{3} q^{13} + 7 q^{14} + 5 q^{16} - 6 \beta_{3} q^{17} + 11 \beta_1 q^{18} - 3 \beta_{2} q^{19} - 7 \beta_{2} q^{21} + 2 \beta_1 q^{22} + 26 \beta_1 q^{23} + 7 \beta_{2} q^{24} - 3 \beta_{2} q^{26} + 2 \beta_{3} q^{27} - 21 \beta_1 q^{28} + 22 q^{29} + 12 \beta_{2} q^{31} + 33 \beta_1 q^{32} + 2 \beta_{3} q^{33} - 6 \beta_{2} q^{34} + 33 q^{36} - 14 \beta_1 q^{37} + 3 \beta_{3} q^{38} - 60 q^{39} + 6 \beta_{2} q^{41} + 7 \beta_{3} q^{42} - 34 \beta_1 q^{43} + 6 q^{44} - 26 q^{46} + 6 \beta_{3} q^{47} + 5 \beta_{3} q^{48} - 49 q^{49} - 120 q^{51} - 9 \beta_{3} q^{52} - 34 \beta_1 q^{53} + 2 \beta_{2} q^{54} + 49 q^{56} - 60 \beta_1 q^{57} + 22 \beta_1 q^{58} - 9 \beta_{2} q^{59} - 21 \beta_{2} q^{61} - 12 \beta_{3} q^{62} - 77 \beta_1 q^{63} - 13 q^{64} + 2 \beta_{2} q^{66} - 14 \beta_1 q^{67} - 18 \beta_{3} q^{68} + 26 \beta_{2} q^{69} + 62 q^{71} + 77 \beta_1 q^{72} - 12 \beta_{3} q^{73} + 14 q^{74} - 9 \beta_{2} q^{76} - 14 \beta_1 q^{77} - 60 \beta_1 q^{78} - 38 q^{79} - 59 q^{81} - 6 \beta_{3} q^{82} + 9 \beta_{3} q^{83} - 21 \beta_{2} q^{84} + 34 q^{86} + 22 \beta_{3} q^{87} + 14 \beta_1 q^{88} - 6 \beta_{2} q^{89} + 21 \beta_{2} q^{91} + 78 \beta_1 q^{92} + 240 \beta_1 q^{93} + 6 \beta_{2} q^{94} + 33 \beta_{2} q^{96} + 6 \beta_{3} q^{97} - 49 \beta_1 q^{98} + 22 q^{99}+O(q^{100})$$ q + b1 * q^2 + b3 * q^3 + 3 * q^4 + b2 * q^6 - 7*b1 * q^7 + 7*b1 * q^8 + 11 * q^9 + 2 * q^11 + 3*b3 * q^12 - 3*b3 * q^13 + 7 * q^14 + 5 * q^16 - 6*b3 * q^17 + 11*b1 * q^18 - 3*b2 * q^19 - 7*b2 * q^21 + 2*b1 * q^22 + 26*b1 * q^23 + 7*b2 * q^24 - 3*b2 * q^26 + 2*b3 * q^27 - 21*b1 * q^28 + 22 * q^29 + 12*b2 * q^31 + 33*b1 * q^32 + 2*b3 * q^33 - 6*b2 * q^34 + 33 * q^36 - 14*b1 * q^37 + 3*b3 * q^38 - 60 * q^39 + 6*b2 * q^41 + 7*b3 * q^42 - 34*b1 * q^43 + 6 * q^44 - 26 * q^46 + 6*b3 * q^47 + 5*b3 * q^48 - 49 * q^49 - 120 * q^51 - 9*b3 * q^52 - 34*b1 * q^53 + 2*b2 * q^54 + 49 * q^56 - 60*b1 * q^57 + 22*b1 * q^58 - 9*b2 * q^59 - 21*b2 * q^61 - 12*b3 * q^62 - 77*b1 * q^63 - 13 * q^64 + 2*b2 * q^66 - 14*b1 * q^67 - 18*b3 * q^68 + 26*b2 * q^69 + 62 * q^71 + 77*b1 * q^72 - 12*b3 * q^73 + 14 * q^74 - 9*b2 * q^76 - 14*b1 * q^77 - 60*b1 * q^78 - 38 * q^79 - 59 * q^81 - 6*b3 * q^82 + 9*b3 * q^83 - 21*b2 * q^84 + 34 * q^86 + 22*b3 * q^87 + 14*b1 * q^88 - 6*b2 * q^89 + 21*b2 * q^91 + 78*b1 * q^92 + 240*b1 * q^93 + 6*b2 * q^94 + 33*b2 * q^96 + 6*b3 * q^97 - 49*b1 * q^98 + 22 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{4} + 44 q^{9}+O(q^{10})$$ 4 * q + 12 * q^4 + 44 * q^9 $$4 q + 12 q^{4} + 44 q^{9} + 8 q^{11} + 28 q^{14} + 20 q^{16} + 88 q^{29} + 132 q^{36} - 240 q^{39} + 24 q^{44} - 104 q^{46} - 196 q^{49} - 480 q^{51} + 196 q^{56} - 52 q^{64} + 248 q^{71} + 56 q^{74} - 152 q^{79} - 236 q^{81} + 136 q^{86} + 88 q^{99}+O(q^{100})$$ 4 * q + 12 * q^4 + 44 * q^9 + 8 * q^11 + 28 * q^14 + 20 * q^16 + 88 * q^29 + 132 * q^36 - 240 * q^39 + 24 * q^44 - 104 * q^46 - 196 * q^49 - 480 * q^51 + 196 * q^56 - 52 * q^64 + 248 * q^71 + 56 * q^74 - 152 * q^79 - 236 * q^81 + 136 * q^86 + 88 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v $$\beta_{2}$$ $$=$$ $$2\nu^{3} + 8\nu$$ 2*v^3 + 8*v $$\beta_{3}$$ $$=$$ $$4\nu^{2} + 6$$ 4*v^2 + 6
 $$\nu$$ $$=$$ $$( \beta_{2} - 2\beta_1 ) / 4$$ (b2 - 2*b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 6 ) / 4$$ (b3 - 6) / 4 $$\nu^{3}$$ $$=$$ $$( -\beta_{2} + 4\beta_1 ) / 2$$ (-b2 + 4*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
174.1
 1.61803i − 0.618034i − 1.61803i 0.618034i
1.00000i −4.47214 3.00000 0 4.47214i 7.00000i 7.00000i 11.0000 0
174.2 1.00000i 4.47214 3.00000 0 4.47214i 7.00000i 7.00000i 11.0000 0
174.3 1.00000i −4.47214 3.00000 0 4.47214i 7.00000i 7.00000i 11.0000 0
174.4 1.00000i 4.47214 3.00000 0 4.47214i 7.00000i 7.00000i 11.0000 0
 $$n$$: 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 175.3.c.d 4
5.b even 2 1 inner 175.3.c.d 4
5.c odd 4 1 35.3.d.a 2
5.c odd 4 1 175.3.d.f 2
7.b odd 2 1 inner 175.3.c.d 4
15.e even 4 1 315.3.h.b 2
20.e even 4 1 560.3.f.a 2
35.c odd 2 1 inner 175.3.c.d 4
35.f even 4 1 35.3.d.a 2
35.f even 4 1 175.3.d.f 2
35.k even 12 2 245.3.h.b 4
35.l odd 12 2 245.3.h.b 4
105.k odd 4 1 315.3.h.b 2
140.j odd 4 1 560.3.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.a 2 5.c odd 4 1
35.3.d.a 2 35.f even 4 1
175.3.c.d 4 1.a even 1 1 trivial
175.3.c.d 4 5.b even 2 1 inner
175.3.c.d 4 7.b odd 2 1 inner
175.3.c.d 4 35.c odd 2 1 inner
175.3.d.f 2 5.c odd 4 1
175.3.d.f 2 35.f even 4 1
245.3.h.b 4 35.k even 12 2
245.3.h.b 4 35.l odd 12 2
315.3.h.b 2 15.e even 4 1
315.3.h.b 2 105.k odd 4 1
560.3.f.a 2 20.e even 4 1
560.3.f.a 2 140.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 1$$ acting on $$S_{3}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T^{2} - 20)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 49)^{2}$$
$11$ $$(T - 2)^{4}$$
$13$ $$(T^{2} - 180)^{2}$$
$17$ $$(T^{2} - 720)^{2}$$
$19$ $$(T^{2} + 180)^{2}$$
$23$ $$(T^{2} + 676)^{2}$$
$29$ $$(T - 22)^{4}$$
$31$ $$(T^{2} + 2880)^{2}$$
$37$ $$(T^{2} + 196)^{2}$$
$41$ $$(T^{2} + 720)^{2}$$
$43$ $$(T^{2} + 1156)^{2}$$
$47$ $$(T^{2} - 720)^{2}$$
$53$ $$(T^{2} + 1156)^{2}$$
$59$ $$(T^{2} + 1620)^{2}$$
$61$ $$(T^{2} + 8820)^{2}$$
$67$ $$(T^{2} + 196)^{2}$$
$71$ $$(T - 62)^{4}$$
$73$ $$(T^{2} - 2880)^{2}$$
$79$ $$(T + 38)^{4}$$
$83$ $$(T^{2} - 1620)^{2}$$
$89$ $$(T^{2} + 720)^{2}$$
$97$ $$(T^{2} - 720)^{2}$$