# Properties

 Label 175.6.b.c Level 175 Weight 6 Character orbit 175.b Analytic conductor 28.067 Analytic rank 0 Dimension 4 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{57})$$ Defining polynomial: $$x^{4} + 29 x^{2} + 196$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + 5 \beta_{2} ) q^{2} + ( 6 \beta_{1} + 6 \beta_{2} ) q^{3} + ( 2 - 9 \beta_{3} ) q^{4} + ( -84 - 30 \beta_{3} ) q^{6} + 49 \beta_{2} q^{7} + ( -11 \beta_{1} - \beta_{2} ) q^{8} + ( -261 - 36 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + 5 \beta_{2} ) q^{2} + ( 6 \beta_{1} + 6 \beta_{2} ) q^{3} + ( 2 - 9 \beta_{3} ) q^{4} + ( -84 - 30 \beta_{3} ) q^{6} + 49 \beta_{2} q^{7} + ( -11 \beta_{1} - \beta_{2} ) q^{8} + ( -261 - 36 \beta_{3} ) q^{9} + ( 260 - 124 \beta_{3} ) q^{11} + ( -42 \beta_{1} - 798 \beta_{2} ) q^{12} + ( -126 \beta_{1} + 112 \beta_{2} ) q^{13} + ( -196 - 49 \beta_{3} ) q^{14} + ( 178 - 243 \beta_{3} ) q^{16} + ( -76 \beta_{1} + 862 \beta_{2} ) q^{17} + ( -441 \beta_{1} - 1989 \beta_{2} ) q^{18} + ( 1624 + 18 \beta_{3} ) q^{19} -294 \beta_{3} q^{21} + ( -360 \beta_{1} - 1056 \beta_{2} ) q^{22} + ( -568 \beta_{1} - 1328 \beta_{2} ) q^{23} + ( 924 + 6 \beta_{3} ) q^{24} + ( 812 + 392 \beta_{3} ) q^{26} + ( -324 \beta_{1} - 3348 \beta_{2} ) q^{27} + ( -441 \beta_{1} - 343 \beta_{2} ) q^{28} + ( -3222 - 252 \beta_{3} ) q^{29} + ( -280 + 540 \beta_{3} ) q^{31} + ( -1389 \beta_{1} - 3759 \beta_{2} ) q^{32} + ( 816 \beta_{1} - 9600 \beta_{2} ) q^{33} + ( -2688 - 558 \beta_{3} ) q^{34} + ( 4014 + 2601 \beta_{3} ) q^{36} + ( 540 \beta_{1} + 3386 \beta_{2} ) q^{37} + ( 1714 \beta_{1} + 8462 \beta_{2} ) q^{38} + ( 10584 - 672 \beta_{3} ) q^{39} + ( -2478 - 1092 \beta_{3} ) q^{41} + ( -1470 \beta_{1} - 5586 \beta_{2} ) q^{42} + ( 4788 \beta_{1} + 3904 \beta_{2} ) q^{43} + ( 16144 - 1472 \beta_{3} ) q^{44} + ( 10992 + 3600 \beta_{3} ) q^{46} + ( 3748 \beta_{1} + 7724 \beta_{2} ) q^{47} + ( -390 \beta_{1} - 20802 \beta_{2} ) q^{48} -2401 q^{49} + ( 6384 - 5172 \beta_{3} ) q^{51} + ( -1260 \beta_{1} + 15092 \beta_{2} ) q^{52} + ( 208 \beta_{1} - 4630 \beta_{2} ) q^{53} + ( 16632 + 4644 \beta_{3} ) q^{54} + ( -490 + 539 \beta_{3} ) q^{56} + ( 9852 \beta_{1} + 11364 \beta_{2} ) q^{57} + ( -4482 \beta_{1} - 20898 \beta_{2} ) q^{58} + ( 20944 + 2050 \beta_{3} ) q^{59} + ( -29974 - 4806 \beta_{3} ) q^{61} + ( 2420 \beta_{1} + 8860 \beta_{2} ) q^{62} + ( -1764 \beta_{1} - 14553 \beta_{2} ) q^{63} + ( 34622 + 1539 \beta_{3} ) q^{64} + ( 30240 + 6336 \beta_{3} ) q^{66} + ( -1944 \beta_{1} + 11420 \beta_{2} ) q^{67} + ( -7910 \beta_{1} + 3542 \beta_{2} ) q^{68} + ( 47712 + 7968 \beta_{3} ) q^{69} + ( 50808 - 4200 \beta_{3} ) q^{71} + ( 2907 \beta_{1} + 5841 \beta_{2} ) q^{72} + ( 5256 \beta_{1} - 6098 \beta_{2} ) q^{73} + ( -18944 - 5546 \beta_{3} ) q^{74} + ( 980 - 14742 \beta_{3} ) q^{76} + ( -6076 \beta_{1} + 6664 \beta_{2} ) q^{77} + ( 7224 \beta_{1} + 40152 \beta_{2} ) q^{78} + ( -18176 - 14904 \beta_{3} ) q^{79} + ( -36207 + 11340 \beta_{3} ) q^{81} + ( -7938 \beta_{1} - 33138 \beta_{2} ) q^{82} + ( -15750 \beta_{1} - 66654 \beta_{2} ) q^{83} + ( 37044 + 2058 \beta_{3} ) q^{84} + ( -63496 - 23056 \beta_{3} ) q^{86} + ( -20844 \beta_{1} - 42012 \beta_{2} ) q^{87} + ( -2736 \beta_{1} + 18960 \beta_{2} ) q^{88} + ( -53242 + 22208 \beta_{3} ) q^{89} + ( -11662 + 6174 \beta_{3} ) q^{91} + ( 10816 \beta_{1} + 80864 \beta_{2} ) q^{92} + ( 1560 \beta_{1} + 46920 \beta_{2} ) q^{93} + ( -68376 - 22716 \beta_{3} ) q^{94} + ( 116676 + 22554 \beta_{3} ) q^{96} + ( 8820 \beta_{1} + 14798 \beta_{2} ) q^{97} + ( -2401 \beta_{1} - 12005 \beta_{2} ) q^{98} + ( -5364 + 27468 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 10q^{4} - 396q^{6} - 1116q^{9} + O(q^{10})$$ $$4q - 10q^{4} - 396q^{6} - 1116q^{9} + 792q^{11} - 882q^{14} + 226q^{16} + 6532q^{19} - 588q^{21} + 3708q^{24} + 4032q^{26} - 13392q^{29} - 40q^{31} - 11868q^{34} + 21258q^{36} + 40992q^{39} - 12096q^{41} + 61632q^{44} + 51168q^{46} - 9604q^{49} + 15192q^{51} + 75816q^{54} - 882q^{56} + 87876q^{59} - 129508q^{61} + 141566q^{64} + 133632q^{66} + 206784q^{69} + 194832q^{71} - 86868q^{74} - 25564q^{76} - 102512q^{79} - 122148q^{81} + 152292q^{84} - 300096q^{86} - 168552q^{89} - 34300q^{91} - 318936q^{94} + 511812q^{96} + 33480q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 29 x^{2} + 196$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 15 \nu$$$$)/14$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 15$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 15$$ $$\nu^{3}$$ $$=$$ $$14 \beta_{2} - 15 \beta_{1}$$

## 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
 − 3.27492i 4.27492i − 4.27492i 3.27492i
8.27492i 25.6495i −36.4743 0 −212.248 49.0000i 37.0241i −414.897 0
99.2 0.725083i 19.6495i 31.4743 0 14.2475 49.0000i 46.0241i −143.103 0
99.3 0.725083i 19.6495i 31.4743 0 14.2475 49.0000i 46.0241i −143.103 0
99.4 8.27492i 25.6495i −36.4743 0 −212.248 49.0000i 37.0241i −414.897 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.c 4
5.b even 2 1 inner 175.6.b.c 4
5.c odd 4 1 7.6.a.b 2
5.c odd 4 1 175.6.a.c 2
15.e even 4 1 63.6.a.f 2
20.e even 4 1 112.6.a.h 2
35.f even 4 1 49.6.a.f 2
35.k even 12 2 49.6.c.d 4
35.l odd 12 2 49.6.c.e 4
40.i odd 4 1 448.6.a.w 2
40.k even 4 1 448.6.a.u 2
55.e even 4 1 847.6.a.c 2
60.l odd 4 1 1008.6.a.bq 2
105.k odd 4 1 441.6.a.l 2
140.j odd 4 1 784.6.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.b 2 5.c odd 4 1
49.6.a.f 2 35.f even 4 1
49.6.c.d 4 35.k even 12 2
49.6.c.e 4 35.l odd 12 2
63.6.a.f 2 15.e even 4 1
112.6.a.h 2 20.e even 4 1
175.6.a.c 2 5.c odd 4 1
175.6.b.c 4 1.a even 1 1 trivial
175.6.b.c 4 5.b even 2 1 inner
441.6.a.l 2 105.k odd 4 1
448.6.a.u 2 40.k even 4 1
448.6.a.w 2 40.i odd 4 1
784.6.a.v 2 140.j odd 4 1
847.6.a.c 2 55.e even 4 1
1008.6.a.bq 2 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 69 T_{2}^{2} + 36$$ acting on $$S_{6}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 59 T^{2} + 1764 T^{4} - 60416 T^{6} + 1048576 T^{8}$$
$3$ $$1 + 72 T^{2} + 100926 T^{4} + 4251528 T^{6} + 3486784401 T^{8}$$
$5$ 1
$7$ $$( 1 + 2401 T^{2} )^{2}$$
$11$ $$( 1 - 396 T + 142198 T^{2} - 63776196 T^{3} + 25937424601 T^{4} )^{2}$$
$13$ $$1 - 971456 T^{2} + 483935131182 T^{4} - 133923459057662144 T^{6} +$$$$19\!\cdots\!01$$$$T^{8}$$
$17$ $$1 - 3894812 T^{2} + 7557700009734 T^{4} - 7851917235395570588 T^{6} +$$$$40\!\cdots\!01$$$$T^{8}$$
$19$ $$( 1 - 3266 T + 7614270 T^{2} - 8086939334 T^{3} + 6131066257801 T^{4} )^{2}$$
$23$ $$1 - 14370716 T^{2} + 114438939827814 T^{4} -$$$$59\!\cdots\!84$$$$T^{6} +$$$$17\!\cdots\!01$$$$T^{8}$$
$29$ $$( 1 + 6696 T + 51326470 T^{2} + 137342653704 T^{3} + 420707233300201 T^{4} )^{2}$$
$31$ $$( 1 + 20 T + 53103102 T^{2} + 572583020 T^{3} + 819628286980801 T^{4} )^{2}$$
$37$ $$1 - 249646316 T^{2} + 25036606707861462 T^{4} -$$$$12\!\cdots\!84$$$$T^{6} +$$$$23\!\cdots\!01$$$$T^{8}$$
$41$ $$( 1 + 6048 T + 223864366 T^{2} + 700698303648 T^{3} + 13422659310152401 T^{4} )^{2}$$
$43$ $$1 + 69887332 T^{2} + 41464568025667254 T^{4} +$$$$15\!\cdots\!68$$$$T^{6} +$$$$46\!\cdots\!01$$$$T^{8}$$
$47$ $$1 - 448581164 T^{2} + 128102309424078822 T^{4} -$$$$23\!\cdots\!36$$$$T^{6} +$$$$27\!\cdots\!01$$$$T^{8}$$
$53$ $$1 - 1626727436 T^{2} + 1011280212489797334 T^{4} -$$$$28\!\cdots\!64$$$$T^{6} +$$$$30\!\cdots\!01$$$$T^{8}$$
$59$ $$( 1 - 43938 T + 1852599934 T^{2} - 31412343849462 T^{3} + 511116753300641401 T^{4} )^{2}$$
$61$ $$( 1 + 64754 T + 2408321418 T^{2} + 54690988874954 T^{3} + 713342911662882601 T^{4} )^{2}$$
$67$ $$1 - 4985671724 T^{2} + 9826827410456194614 T^{4} -$$$$90\!\cdots\!76$$$$T^{6} +$$$$33\!\cdots\!01$$$$T^{8}$$
$71$ $$( 1 - 97416 T + 5729557966 T^{2} - 175760806457016 T^{3} + 3255243551009881201 T^{4} )^{2}$$
$73$ $$1 - 7352672444 T^{2} + 21990800559226590630 T^{4} -$$$$31\!\cdots\!56$$$$T^{6} +$$$$18\!\cdots\!01$$$$T^{8}$$
$79$ $$( 1 + 51256 T + 3645565854 T^{2} + 157717602787144 T^{3} + 9468276082626847201 T^{4} )^{2}$$
$83$ $$1 - 1776439640 T^{2} - 17030759318944523202 T^{4} -$$$$27\!\cdots\!60$$$$T^{6} +$$$$24\!\cdots\!01$$$$T^{8}$$
$89$ $$( 1 + 84276 T + 5915697430 T^{2} + 470602194123924 T^{3} + 31181719929966183601 T^{4} )^{2}$$
$97$ $$1 - 31916456540 T^{2} +$$$$40\!\cdots\!98$$$$T^{4} -$$$$23\!\cdots\!60$$$$T^{6} +$$$$54\!\cdots\!01$$$$T^{8}$$