# Properties

 Label 175.6.b.d 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: $$\Q(i, \sqrt{65})$$ Defining polynomial: $$x^{4} + 33x^{2} + 256$$ x^4 + 33*x^2 + 256 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (3 \beta_{2} + 3 \beta_1) q^{3} + (\beta_{3} + 15) q^{4} - 48 q^{6} + 49 \beta_{2} q^{7} + (16 \beta_{2} + 47 \beta_1) q^{8} + ( - 9 \beta_{3} + 99) q^{9}+O(q^{10})$$ q + b1 * q^2 + (3*b2 + 3*b1) * q^3 + (b3 + 15) * q^4 - 48 * q^6 + 49*b2 * q^7 + (16*b2 + 47*b1) * q^8 + (-9*b3 + 99) * q^9 $$q + \beta_1 q^{2} + (3 \beta_{2} + 3 \beta_1) q^{3} + (\beta_{3} + 15) q^{4} - 48 q^{6} + 49 \beta_{2} q^{7} + (16 \beta_{2} + 47 \beta_1) q^{8} + ( - 9 \beta_{3} + 99) q^{9} + ( - 97 \beta_{3} - 252) q^{11} + (96 \beta_{2} + 48 \beta_1) q^{12} + ( - 315 \beta_{2} - 53 \beta_1) q^{13} + ( - 49 \beta_{3} + 49) q^{14} + (63 \beta_{3} - 303) q^{16} + (105 \beta_{2} + 251 \beta_1) q^{17} + ( - 144 \beta_{2} + 99 \beta_1) q^{18} + ( - 86 \beta_{3} - 272) q^{19} - 147 \beta_{3} q^{21} + ( - 1552 \beta_{2} - 252 \beta_1) q^{22} + (230 \beta_{2} + 902 \beta_1) q^{23} + ( - 48 \beta_{3} - 2256) q^{24} + (262 \beta_{3} + 586) q^{26} + (567 \beta_{2} + 999 \beta_1) q^{27} + (784 \beta_{2} + 49 \beta_1) q^{28} + ( - 945 \beta_{3} - 2470) q^{29} + ( - 924 \beta_{3} + 264) q^{31} + (1520 \beta_{2} + 1201 \beta_1) q^{32} + ( - 5703 \beta_{2} - 1047 \beta_1) q^{33} + (146 \beta_{3} - 4162) q^{34} + ( - 45 \beta_{3} + 1341) q^{36} + (3822 \beta_{2} - 1260 \beta_1) q^{37} + ( - 1376 \beta_{2} - 272 \beta_1) q^{38} + (945 \beta_{3} + 2544) q^{39} + (3818 \beta_{3} - 1022) q^{41} - 2352 \beta_{2} q^{42} + ( - 14022 \beta_{2} - 922 \beta_1) q^{43} + ( - 1804 \beta_{3} - 5332) q^{44} + (672 \beta_{3} - 15104) q^{46} + (9857 \beta_{2} - 1575 \beta_1) q^{47} + (2304 \beta_{2} - 720 \beta_1) q^{48} - 2401 q^{49} + ( - 315 \beta_{3} - 12048) q^{51} + ( - 5888 \beta_{2} - 1110 \beta_1) q^{52} + ( - 27564 \beta_{2} + 454 \beta_1) q^{53} + (432 \beta_{3} - 16416) q^{54} + ( - 2303 \beta_{3} + 1519) q^{56} + ( - 5202 \beta_{2} - 1074 \beta_1) q^{57} + ( - 15120 \beta_{2} - 2470 \beta_1) q^{58} + (5184 \beta_{3} - 32392) q^{59} + ( - 5706 \beta_{3} - 23070) q^{61} + ( - 14784 \beta_{2} + 264 \beta_1) q^{62} + (4410 \beta_{2} - 441 \beta_1) q^{63} + (1697 \beta_{3} - 28593) q^{64} + (4656 \beta_{3} + 12096) q^{66} + (20388 \beta_{2} - 4568 \beta_1) q^{67} + (5696 \beta_{2} + 3870 \beta_1) q^{68} + ( - 690 \beta_{3} - 43296) q^{69} + ( - 5304 \beta_{3} + 43024) q^{71} + ( - 5328 \beta_{2} + 4509 \beta_1) q^{72} + ( - 4670 \beta_{2} + 4192 \beta_1) q^{73} + ( - 5082 \beta_{3} + 25242) q^{74} + ( - 1648 \beta_{3} - 5456) q^{76} + ( - 17101 \beta_{2} - 4753 \beta_1) q^{77} + (15120 \beta_{2} + 2544 \beta_1) q^{78} + (17635 \beta_{3} + 17080) q^{79} + ( - 3888 \beta_{3} - 23895) q^{81} + (61088 \beta_{2} - 1022 \beta_1) q^{82} + (56876 \beta_{2} + 3924 \beta_1) q^{83} + ( - 2352 \beta_{3} - 2352) q^{84} + (13100 \beta_{3} + 1652) q^{86} + ( - 55605 \beta_{2} - 10245 \beta_1) q^{87} + ( - 78528 \beta_{2} - 13396 \beta_1) q^{88} + ( - 5722 \beta_{3} + 21686) q^{89} + (2597 \beta_{3} + 12838) q^{91} + (18112 \beta_{2} + 13760 \beta_1) q^{92} + ( - 46332 \beta_{2} - 1980 \beta_1) q^{93} + ( - 11432 \beta_{3} + 36632) q^{94} + ( - 4560 \beta_{3} - 57648) q^{96} + (55141 \beta_{2} + 13943 \beta_1) q^{97} - 2401 \beta_1 q^{98} + ( - 6462 \beta_{3} - 10980) q^{99}+O(q^{100})$$ q + b1 * q^2 + (3*b2 + 3*b1) * q^3 + (b3 + 15) * q^4 - 48 * q^6 + 49*b2 * q^7 + (16*b2 + 47*b1) * q^8 + (-9*b3 + 99) * q^9 + (-97*b3 - 252) * q^11 + (96*b2 + 48*b1) * q^12 + (-315*b2 - 53*b1) * q^13 + (-49*b3 + 49) * q^14 + (63*b3 - 303) * q^16 + (105*b2 + 251*b1) * q^17 + (-144*b2 + 99*b1) * q^18 + (-86*b3 - 272) * q^19 - 147*b3 * q^21 + (-1552*b2 - 252*b1) * q^22 + (230*b2 + 902*b1) * q^23 + (-48*b3 - 2256) * q^24 + (262*b3 + 586) * q^26 + (567*b2 + 999*b1) * q^27 + (784*b2 + 49*b1) * q^28 + (-945*b3 - 2470) * q^29 + (-924*b3 + 264) * q^31 + (1520*b2 + 1201*b1) * q^32 + (-5703*b2 - 1047*b1) * q^33 + (146*b3 - 4162) * q^34 + (-45*b3 + 1341) * q^36 + (3822*b2 - 1260*b1) * q^37 + (-1376*b2 - 272*b1) * q^38 + (945*b3 + 2544) * q^39 + (3818*b3 - 1022) * q^41 - 2352*b2 * q^42 + (-14022*b2 - 922*b1) * q^43 + (-1804*b3 - 5332) * q^44 + (672*b3 - 15104) * q^46 + (9857*b2 - 1575*b1) * q^47 + (2304*b2 - 720*b1) * q^48 - 2401 * q^49 + (-315*b3 - 12048) * q^51 + (-5888*b2 - 1110*b1) * q^52 + (-27564*b2 + 454*b1) * q^53 + (432*b3 - 16416) * q^54 + (-2303*b3 + 1519) * q^56 + (-5202*b2 - 1074*b1) * q^57 + (-15120*b2 - 2470*b1) * q^58 + (5184*b3 - 32392) * q^59 + (-5706*b3 - 23070) * q^61 + (-14784*b2 + 264*b1) * q^62 + (4410*b2 - 441*b1) * q^63 + (1697*b3 - 28593) * q^64 + (4656*b3 + 12096) * q^66 + (20388*b2 - 4568*b1) * q^67 + (5696*b2 + 3870*b1) * q^68 + (-690*b3 - 43296) * q^69 + (-5304*b3 + 43024) * q^71 + (-5328*b2 + 4509*b1) * q^72 + (-4670*b2 + 4192*b1) * q^73 + (-5082*b3 + 25242) * q^74 + (-1648*b3 - 5456) * q^76 + (-17101*b2 - 4753*b1) * q^77 + (15120*b2 + 2544*b1) * q^78 + (17635*b3 + 17080) * q^79 + (-3888*b3 - 23895) * q^81 + (61088*b2 - 1022*b1) * q^82 + (56876*b2 + 3924*b1) * q^83 + (-2352*b3 - 2352) * q^84 + (13100*b3 + 1652) * q^86 + (-55605*b2 - 10245*b1) * q^87 + (-78528*b2 - 13396*b1) * q^88 + (-5722*b3 + 21686) * q^89 + (2597*b3 + 12838) * q^91 + (18112*b2 + 13760*b1) * q^92 + (-46332*b2 - 1980*b1) * q^93 + (-11432*b3 + 36632) * q^94 + (-4560*b3 - 57648) * q^96 + (55141*b2 + 13943*b1) * q^97 - 2401*b1 * q^98 + (-6462*b3 - 10980) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 62 q^{4} - 192 q^{6} + 378 q^{9}+O(q^{10})$$ 4 * q + 62 * q^4 - 192 * q^6 + 378 * q^9 $$4 q + 62 q^{4} - 192 q^{6} + 378 q^{9} - 1202 q^{11} + 98 q^{14} - 1086 q^{16} - 1260 q^{19} - 294 q^{21} - 9120 q^{24} + 2868 q^{26} - 11770 q^{29} - 792 q^{31} - 16356 q^{34} + 5274 q^{36} + 12066 q^{39} + 3548 q^{41} - 24936 q^{44} - 59072 q^{46} - 9604 q^{49} - 48822 q^{51} - 64800 q^{54} + 1470 q^{56} - 119200 q^{59} - 103692 q^{61} - 110978 q^{64} + 57696 q^{66} - 174564 q^{69} + 161488 q^{71} + 90804 q^{74} - 25120 q^{76} + 103590 q^{79} - 103356 q^{81} - 14112 q^{84} + 32808 q^{86} + 75300 q^{89} + 56546 q^{91} + 123664 q^{94} - 239712 q^{96} - 56844 q^{99}+O(q^{100})$$ 4 * q + 62 * q^4 - 192 * q^6 + 378 * q^9 - 1202 * q^11 + 98 * q^14 - 1086 * q^16 - 1260 * q^19 - 294 * q^21 - 9120 * q^24 + 2868 * q^26 - 11770 * q^29 - 792 * q^31 - 16356 * q^34 + 5274 * q^36 + 12066 * q^39 + 3548 * q^41 - 24936 * q^44 - 59072 * q^46 - 9604 * q^49 - 48822 * q^51 - 64800 * q^54 + 1470 * q^56 - 119200 * q^59 - 103692 * q^61 - 110978 * q^64 + 57696 * q^66 - 174564 * q^69 + 161488 * q^71 + 90804 * q^74 - 25120 * q^76 + 103590 * q^79 - 103356 * q^81 - 14112 * q^84 + 32808 * q^86 + 75300 * q^89 + 56546 * q^91 + 123664 * q^94 - 239712 * q^96 - 56844 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 33x^{2} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 17\nu ) / 16$$ (v^3 + 17*v) / 16 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 17$$ v^2 + 17
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 17$$ b3 - 17 $$\nu^{3}$$ $$=$$ $$16\beta_{2} - 17\beta_1$$ 16*b2 - 17*b1

## 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
 − 4.53113i − 3.53113i 3.53113i 4.53113i
4.53113i 10.5934i 11.4689 0 −48.0000 49.0000i 196.963i 130.780 0
99.2 3.53113i 13.5934i 19.5311 0 −48.0000 49.0000i 181.963i 58.2198 0
99.3 3.53113i 13.5934i 19.5311 0 −48.0000 49.0000i 181.963i 58.2198 0
99.4 4.53113i 10.5934i 11.4689 0 −48.0000 49.0000i 196.963i 130.780 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.d 4
5.b even 2 1 inner 175.6.b.d 4
5.c odd 4 1 35.6.a.b 2
5.c odd 4 1 175.6.a.d 2
15.e even 4 1 315.6.a.c 2
20.e even 4 1 560.6.a.l 2
35.f even 4 1 245.6.a.c 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 33T_{2}^{2} + 256$$ acting on $$S_{6}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 33T^{2} + 256$$
$3$ $$T^{4} + 297 T^{2} + 20736$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 2401)^{2}$$
$11$ $$(T^{2} + 601 T - 62596)^{2}$$
$13$ $$T^{4} + 257757 T^{2} + \cdots + 1412707396$$
$17$ $$T^{4} + 2048373 T^{2} + \cdots + 1047237035716$$
$19$ $$(T^{2} + 630 T - 20960)^{2}$$
$23$ $$T^{4} + \cdots + 173507485106176$$
$29$ $$(T^{2} + 5885 T - 5853350)^{2}$$
$31$ $$(T^{2} + 396 T - 13834656)^{2}$$
$37$ $$T^{4} + 91237608 T^{2} + \cdots + 35738827414416$$
$41$ $$(T^{2} - 1774 T - 236091496)^{2}$$
$43$ $$T^{4} + 395429172 T^{2} + \cdots + 28\!\cdots\!36$$
$47$ $$T^{4} + 307231073 T^{2} + \cdots + 53\!\cdots\!76$$
$53$ $$T^{4} + 1551378132 T^{2} + \cdots + 59\!\cdots\!16$$
$59$ $$(T^{2} + 59600 T + 451339840)^{2}$$
$61$ $$(T^{2} + 51846 T + 142927344)^{2}$$
$67$ $$T^{4} + 1706204448 T^{2} + \cdots + 30\!\cdots\!36$$
$71$ $$(T^{2} - 80744 T + 1172746624)^{2}$$
$73$ $$T^{4} + 662675592 T^{2} + \cdots + 57\!\cdots\!56$$
$79$ $$(T^{2} - 51795 T - 4382959400)^{2}$$
$83$ $$T^{4} + 6531522512 T^{2} + \cdots + 76\!\cdots\!96$$
$89$ $$(T^{2} - 37650 T - 177665240)^{2}$$
$97$ $$T^{4} + 10958837053 T^{2} + \cdots + 70\!\cdots\!56$$