# Properties

 Label 175.6.b.b Level $175$ Weight $6$ Character orbit 175.b Analytic conductor $28.067$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 175.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.0671684673$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 8 i q^{2} + i q^{3} - 32 q^{4} - 8 q^{6} - 49 i q^{7} + 242 q^{9} +O(q^{10})$$ q + 8*i * q^2 + i * q^3 - 32 * q^4 - 8 * q^6 - 49*i * q^7 + 242 * q^9 $$q + 8 i q^{2} + i q^{3} - 32 q^{4} - 8 q^{6} - 49 i q^{7} + 242 q^{9} - 453 q^{11} - 32 i q^{12} - 969 i q^{13} + 392 q^{14} - 1024 q^{16} - 1637 i q^{17} + 1936 i q^{18} + 1550 q^{19} + 49 q^{21} - 3624 i q^{22} - 1654 i q^{23} + 7752 q^{26} + 485 i q^{27} + 1568 i q^{28} + 4985 q^{29} + 1192 q^{31} - 8192 i q^{32} - 453 i q^{33} + 13096 q^{34} - 7744 q^{36} + 11018 i q^{37} + 12400 i q^{38} + 969 q^{39} - 1728 q^{41} + 392 i q^{42} - 10814 i q^{43} + 14496 q^{44} + 13232 q^{46} - 26237 i q^{47} - 1024 i q^{48} - 2401 q^{49} + 1637 q^{51} + 31008 i q^{52} + 25936 i q^{53} - 3880 q^{54} + 1550 i q^{57} + 39880 i q^{58} + 4580 q^{59} - 12488 q^{61} + 9536 i q^{62} - 11858 i q^{63} + 32768 q^{64} + 3624 q^{66} + 15848 i q^{67} + 52384 i q^{68} + 1654 q^{69} + 51792 q^{71} + 4846 i q^{73} - 88144 q^{74} - 49600 q^{76} + 22197 i q^{77} + 7752 i q^{78} - 62765 q^{79} + 58321 q^{81} - 13824 i q^{82} - 23644 i q^{83} - 1568 q^{84} + 86512 q^{86} + 4985 i q^{87} + 147300 q^{89} - 47481 q^{91} + 52928 i q^{92} + 1192 i q^{93} + 209896 q^{94} + 8192 q^{96} + 8343 i q^{97} - 19208 i q^{98} - 109626 q^{99} +O(q^{100})$$ q + 8*i * q^2 + i * q^3 - 32 * q^4 - 8 * q^6 - 49*i * q^7 + 242 * q^9 - 453 * q^11 - 32*i * q^12 - 969*i * q^13 + 392 * q^14 - 1024 * q^16 - 1637*i * q^17 + 1936*i * q^18 + 1550 * q^19 + 49 * q^21 - 3624*i * q^22 - 1654*i * q^23 + 7752 * q^26 + 485*i * q^27 + 1568*i * q^28 + 4985 * q^29 + 1192 * q^31 - 8192*i * q^32 - 453*i * q^33 + 13096 * q^34 - 7744 * q^36 + 11018*i * q^37 + 12400*i * q^38 + 969 * q^39 - 1728 * q^41 + 392*i * q^42 - 10814*i * q^43 + 14496 * q^44 + 13232 * q^46 - 26237*i * q^47 - 1024*i * q^48 - 2401 * q^49 + 1637 * q^51 + 31008*i * q^52 + 25936*i * q^53 - 3880 * q^54 + 1550*i * q^57 + 39880*i * q^58 + 4580 * q^59 - 12488 * q^61 + 9536*i * q^62 - 11858*i * q^63 + 32768 * q^64 + 3624 * q^66 + 15848*i * q^67 + 52384*i * q^68 + 1654 * q^69 + 51792 * q^71 + 4846*i * q^73 - 88144 * q^74 - 49600 * q^76 + 22197*i * q^77 + 7752*i * q^78 - 62765 * q^79 + 58321 * q^81 - 13824*i * q^82 - 23644*i * q^83 - 1568 * q^84 + 86512 * q^86 + 4985*i * q^87 + 147300 * q^89 - 47481 * q^91 + 52928*i * q^92 + 1192*i * q^93 + 209896 * q^94 + 8192 * q^96 + 8343*i * q^97 - 19208*i * q^98 - 109626 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 64 q^{4} - 16 q^{6} + 484 q^{9}+O(q^{10})$$ 2 * q - 64 * q^4 - 16 * q^6 + 484 * q^9 $$2 q - 64 q^{4} - 16 q^{6} + 484 q^{9} - 906 q^{11} + 784 q^{14} - 2048 q^{16} + 3100 q^{19} + 98 q^{21} + 15504 q^{26} + 9970 q^{29} + 2384 q^{31} + 26192 q^{34} - 15488 q^{36} + 1938 q^{39} - 3456 q^{41} + 28992 q^{44} + 26464 q^{46} - 4802 q^{49} + 3274 q^{51} - 7760 q^{54} + 9160 q^{59} - 24976 q^{61} + 65536 q^{64} + 7248 q^{66} + 3308 q^{69} + 103584 q^{71} - 176288 q^{74} - 99200 q^{76} - 125530 q^{79} + 116642 q^{81} - 3136 q^{84} + 173024 q^{86} + 294600 q^{89} - 94962 q^{91} + 419792 q^{94} + 16384 q^{96} - 219252 q^{99}+O(q^{100})$$ 2 * q - 64 * q^4 - 16 * q^6 + 484 * q^9 - 906 * q^11 + 784 * q^14 - 2048 * q^16 + 3100 * q^19 + 98 * q^21 + 15504 * q^26 + 9970 * q^29 + 2384 * q^31 + 26192 * q^34 - 15488 * q^36 + 1938 * q^39 - 3456 * q^41 + 28992 * q^44 + 26464 * q^46 - 4802 * q^49 + 3274 * q^51 - 7760 * q^54 + 9160 * q^59 - 24976 * q^61 + 65536 * q^64 + 7248 * q^66 + 3308 * q^69 + 103584 * q^71 - 176288 * q^74 - 99200 * q^76 - 125530 * q^79 + 116642 * q^81 - 3136 * q^84 + 173024 * q^86 + 294600 * q^89 - 94962 * q^91 + 419792 * q^94 + 16384 * q^96 - 219252 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 − 1.00000i 1.00000i
8.00000i 1.00000i −32.0000 0 −8.00000 49.0000i 0 242.000 0
99.2 8.00000i 1.00000i −32.0000 0 −8.00000 49.0000i 0 242.000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.6.b.b 2
5.b even 2 1 inner 175.6.b.b 2
5.c odd 4 1 35.6.a.a 1
5.c odd 4 1 175.6.a.a 1
15.e even 4 1 315.6.a.a 1
20.e even 4 1 560.6.a.c 1
35.f even 4 1 245.6.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.a 1 5.c odd 4 1
175.6.a.a 1 5.c odd 4 1
175.6.b.b 2 1.a even 1 1 trivial
175.6.b.b 2 5.b even 2 1 inner
245.6.a.a 1 35.f even 4 1
315.6.a.a 1 15.e even 4 1
560.6.a.c 1 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 64$$ acting on $$S_{6}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 64$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2401$$
$11$ $$(T + 453)^{2}$$
$13$ $$T^{2} + 938961$$
$17$ $$T^{2} + 2679769$$
$19$ $$(T - 1550)^{2}$$
$23$ $$T^{2} + 2735716$$
$29$ $$(T - 4985)^{2}$$
$31$ $$(T - 1192)^{2}$$
$37$ $$T^{2} + 121396324$$
$41$ $$(T + 1728)^{2}$$
$43$ $$T^{2} + 116942596$$
$47$ $$T^{2} + 688380169$$
$53$ $$T^{2} + 672676096$$
$59$ $$(T - 4580)^{2}$$
$61$ $$(T + 12488)^{2}$$
$67$ $$T^{2} + 251159104$$
$71$ $$(T - 51792)^{2}$$
$73$ $$T^{2} + 23483716$$
$79$ $$(T + 62765)^{2}$$
$83$ $$T^{2} + 559038736$$
$89$ $$(T - 147300)^{2}$$
$97$ $$T^{2} + 69605649$$