# Properties

 Label 175.8.b.c Level $175$ Weight $8$ Character orbit 175.b Analytic conductor $54.667$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,8,Mod(99,175)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("175.99");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 175.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$54.6673794597$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 9$$ x^4 - 5*x^2 + 9 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 8 \beta_1) q^{2} + (6 \beta_{3} + 15 \beta_1) q^{3} + ( - 16 \beta_{2} + 20) q^{4} + ( - 63 \beta_{2} - 384) q^{6} - 343 \beta_1 q^{7} + (20 \beta_{3} + 480 \beta_1) q^{8} + ( - 180 \beta_{2} + 378) q^{9}+O(q^{10})$$ q + (b3 + 8*b1) * q^2 + (6*b3 + 15*b1) * q^3 + (-16*b2 + 20) * q^4 + (-63*b2 - 384) * q^6 - 343*b1 * q^7 + (20*b3 + 480*b1) * q^8 + (-180*b2 + 378) * q^9 $$q + (\beta_{3} + 8 \beta_1) q^{2} + (6 \beta_{3} + 15 \beta_1) q^{3} + ( - 16 \beta_{2} + 20) q^{4} + ( - 63 \beta_{2} - 384) q^{6} - 343 \beta_1 q^{7} + (20 \beta_{3} + 480 \beta_1) q^{8} + ( - 180 \beta_{2} + 378) q^{9} + ( - 380 \beta_{2} - 3953) q^{11} + ( - 120 \beta_{3} - 3924 \beta_1) q^{12} + (418 \beta_{3} + 8909 \beta_1) q^{13} + (343 \beta_{2} + 2744) q^{14} + ( - 2688 \beta_{2} - 2160) q^{16} + (2210 \beta_{3} - 1199 \beta_1) q^{17} + ( - 1062 \beta_{3} - 4896 \beta_1) q^{18} + ( - 5542 \beta_{2} + 1806) q^{19} + (2058 \beta_{2} + 5145) q^{21} + ( - 6993 \beta_{3} - 48344 \beta_1) q^{22} + ( - 10690 \beta_{3} - 6922 \beta_1) q^{23} + ( - 3180 \beta_{2} - 12480) q^{24} + ( - 12253 \beta_{2} - 89664) q^{26} + (12690 \beta_{3} - 9045 \beta_1) q^{27} + (5488 \beta_{3} - 6860 \beta_1) q^{28} + (23772 \beta_{2} + 63449) q^{29} + ( - 22554 \beta_{2} + 126384) q^{31} + ( - 21104 \beta_{3} - 74112 \beta_1) q^{32} + ( - 29418 \beta_{3} - 159615 \beta_1) q^{33} + ( - 16481 \beta_{2} - 87648) q^{34} + ( - 9648 \beta_{2} + 134280) q^{36} + ( - 43638 \beta_{3} - 132930 \beta_1) q^{37} + ( - 42530 \beta_{3} - 229400 \beta_1) q^{38} + ( - 59724 \beta_{2} - 243987) q^{39} + (37354 \beta_{2} - 55960) q^{41} + (21609 \beta_{3} + 131712 \beta_1) q^{42} + (24578 \beta_{3} - 473786 \beta_1) q^{43} + (55648 \beta_{2} + 188460) q^{44} + (92442 \beta_{2} + 525736) q^{46} + ( - 56742 \beta_{3} + 135637 \beta_1) q^{47} + ( - 53280 \beta_{3} - 742032 \beta_1) q^{48} - 117649 q^{49} + ( - 25956 \beta_{2} - 565455) q^{51} + ( - 134184 \beta_{3} - 116092 \beta_1) q^{52} + (65224 \beta_{3} + 633896 \beta_1) q^{53} + ( - 92475 \beta_{2} - 486000) q^{54} + (6860 \beta_{2} + 164640) q^{56} + ( - 72294 \beta_{3} - 1435998 \beta_1) q^{57} + (253625 \beta_{3} + 1553560 \beta_1) q^{58} + ( - 170640 \beta_{2} + 680060) q^{59} + (11334 \beta_{2} - 906840) q^{61} + ( - 54048 \beta_{3} + 18696 \beta_1) q^{62} + (61740 \beta_{3} - 129654 \beta_1) q^{63} + ( - 101120 \beta_{2} + 1244992) q^{64} + (394959 \beta_{2} + 2571312) q^{66} + (506344 \beta_{3} - 1094656 \beta_1) q^{67} + (63384 \beta_{3} - 1579820 \beta_1) q^{68} + (201882 \beta_{2} + 2925990) q^{69} + ( - 222048 \beta_{2} - 747464) q^{71} + ( - 78840 \beta_{3} + 23040 \beta_1) q^{72} + ( - 212396 \beta_{3} - 3584894 \beta_1) q^{73} + (482034 \beta_{2} + 2983512) q^{74} + ( - 139736 \beta_{2} + 3937688) q^{76} + (130340 \beta_{3} + 1355879 \beta_1) q^{77} + ( - 721779 \beta_{3} - 4579752 \beta_1) q^{78} + ( - 187504 \beta_{2} + 3971487) q^{79} + ( - 529740 \beta_{2} - 2387799) q^{81} + (242872 \beta_{3} + 1195896 \beta_1) q^{82} + (939444 \beta_{3} + 152356 \beta_1) q^{83} + ( - 41160 \beta_{2} - 1345932) q^{84} + (277162 \beta_{2} + 2708856) q^{86} + (737274 \beta_{3} + 7227543 \beta_1) q^{87} + ( - 261460 \beta_{3} - 2231840 \beta_1) q^{88} + ( - 247538 \beta_{2} + 8971764) q^{89} + (143374 \beta_{2} + 3055787) q^{91} + ( - 103048 \beta_{3} + 7387320 \beta_1) q^{92} + (419994 \beta_{3} - 4058496 \beta_1) q^{93} + (318299 \beta_{2} + 1411552) q^{94} + (761232 \beta_{2} + 6683136) q^{96} + ( - 680782 \beta_{3} + 2129037 \beta_1) q^{97} + ( - 117649 \beta_{3} - 941192 \beta_1) q^{98} + (567900 \beta_{2} + 1515366) q^{99}+O(q^{100})$$ q + (b3 + 8*b1) * q^2 + (6*b3 + 15*b1) * q^3 + (-16*b2 + 20) * q^4 + (-63*b2 - 384) * q^6 - 343*b1 * q^7 + (20*b3 + 480*b1) * q^8 + (-180*b2 + 378) * q^9 + (-380*b2 - 3953) * q^11 + (-120*b3 - 3924*b1) * q^12 + (418*b3 + 8909*b1) * q^13 + (343*b2 + 2744) * q^14 + (-2688*b2 - 2160) * q^16 + (2210*b3 - 1199*b1) * q^17 + (-1062*b3 - 4896*b1) * q^18 + (-5542*b2 + 1806) * q^19 + (2058*b2 + 5145) * q^21 + (-6993*b3 - 48344*b1) * q^22 + (-10690*b3 - 6922*b1) * q^23 + (-3180*b2 - 12480) * q^24 + (-12253*b2 - 89664) * q^26 + (12690*b3 - 9045*b1) * q^27 + (5488*b3 - 6860*b1) * q^28 + (23772*b2 + 63449) * q^29 + (-22554*b2 + 126384) * q^31 + (-21104*b3 - 74112*b1) * q^32 + (-29418*b3 - 159615*b1) * q^33 + (-16481*b2 - 87648) * q^34 + (-9648*b2 + 134280) * q^36 + (-43638*b3 - 132930*b1) * q^37 + (-42530*b3 - 229400*b1) * q^38 + (-59724*b2 - 243987) * q^39 + (37354*b2 - 55960) * q^41 + (21609*b3 + 131712*b1) * q^42 + (24578*b3 - 473786*b1) * q^43 + (55648*b2 + 188460) * q^44 + (92442*b2 + 525736) * q^46 + (-56742*b3 + 135637*b1) * q^47 + (-53280*b3 - 742032*b1) * q^48 - 117649 * q^49 + (-25956*b2 - 565455) * q^51 + (-134184*b3 - 116092*b1) * q^52 + (65224*b3 + 633896*b1) * q^53 + (-92475*b2 - 486000) * q^54 + (6860*b2 + 164640) * q^56 + (-72294*b3 - 1435998*b1) * q^57 + (253625*b3 + 1553560*b1) * q^58 + (-170640*b2 + 680060) * q^59 + (11334*b2 - 906840) * q^61 + (-54048*b3 + 18696*b1) * q^62 + (61740*b3 - 129654*b1) * q^63 + (-101120*b2 + 1244992) * q^64 + (394959*b2 + 2571312) * q^66 + (506344*b3 - 1094656*b1) * q^67 + (63384*b3 - 1579820*b1) * q^68 + (201882*b2 + 2925990) * q^69 + (-222048*b2 - 747464) * q^71 + (-78840*b3 + 23040*b1) * q^72 + (-212396*b3 - 3584894*b1) * q^73 + (482034*b2 + 2983512) * q^74 + (-139736*b2 + 3937688) * q^76 + (130340*b3 + 1355879*b1) * q^77 + (-721779*b3 - 4579752*b1) * q^78 + (-187504*b2 + 3971487) * q^79 + (-529740*b2 - 2387799) * q^81 + (242872*b3 + 1195896*b1) * q^82 + (939444*b3 + 152356*b1) * q^83 + (-41160*b2 - 1345932) * q^84 + (277162*b2 + 2708856) * q^86 + (737274*b3 + 7227543*b1) * q^87 + (-261460*b3 - 2231840*b1) * q^88 + (-247538*b2 + 8971764) * q^89 + (143374*b2 + 3055787) * q^91 + (-103048*b3 + 7387320*b1) * q^92 + (419994*b3 - 4058496*b1) * q^93 + (318299*b2 + 1411552) * q^94 + (761232*b2 + 6683136) * q^96 + (-680782*b3 + 2129037*b1) * q^97 + (-117649*b3 - 941192*b1) * q^98 + (567900*b2 + 1515366) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 80 q^{4} - 1536 q^{6} + 1512 q^{9}+O(q^{10})$$ 4 * q + 80 * q^4 - 1536 * q^6 + 1512 * q^9 $$4 q + 80 q^{4} - 1536 q^{6} + 1512 q^{9} - 15812 q^{11} + 10976 q^{14} - 8640 q^{16} + 7224 q^{19} + 20580 q^{21} - 49920 q^{24} - 358656 q^{26} + 253796 q^{29} + 505536 q^{31} - 350592 q^{34} + 537120 q^{36} - 975948 q^{39} - 223840 q^{41} + 753840 q^{44} + 2102944 q^{46} - 470596 q^{49} - 2261820 q^{51} - 1944000 q^{54} + 658560 q^{56} + 2720240 q^{59} - 3627360 q^{61} + 4979968 q^{64} + 10285248 q^{66} + 11703960 q^{69} - 2989856 q^{71} + 11934048 q^{74} + 15750752 q^{76} + 15885948 q^{79} - 9551196 q^{81} - 5383728 q^{84} + 10835424 q^{86} + 35887056 q^{89} + 12223148 q^{91} + 5646208 q^{94} + 26732544 q^{96} + 6061464 q^{99}+O(q^{100})$$ 4 * q + 80 * q^4 - 1536 * q^6 + 1512 * q^9 - 15812 * q^11 + 10976 * q^14 - 8640 * q^16 + 7224 * q^19 + 20580 * q^21 - 49920 * q^24 - 358656 * q^26 + 253796 * q^29 + 505536 * q^31 - 350592 * q^34 + 537120 * q^36 - 975948 * q^39 - 223840 * q^41 + 753840 * q^44 + 2102944 * q^46 - 470596 * q^49 - 2261820 * q^51 - 1944000 * q^54 + 658560 * q^56 + 2720240 * q^59 - 3627360 * q^61 + 4979968 * q^64 + 10285248 * q^66 + 11703960 * q^69 - 2989856 * q^71 + 11934048 * q^74 + 15750752 * q^76 + 15885948 * q^79 - 9551196 * q^81 - 5383728 * q^84 + 10835424 * q^86 + 35887056 * q^89 + 12223148 * q^91 + 5646208 * q^94 + 26732544 * q^96 + 6061464 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - 2\nu ) / 3$$ (v^3 - 2*v) / 3 $$\beta_{2}$$ $$=$$ $$( -2\nu^{3} + 16\nu ) / 3$$ (-2*v^3 + 16*v) / 3 $$\beta_{3}$$ $$=$$ $$4\nu^{2} - 10$$ 4*v^2 - 10
 $$\nu$$ $$=$$ $$( \beta_{2} + 2\beta_1 ) / 4$$ (b2 + 2*b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 10 ) / 4$$ (b3 + 10) / 4 $$\nu^{3}$$ $$=$$ $$( \beta_{2} + 8\beta_1 ) / 2$$ (b2 + 8*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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 1.65831 − 0.500000i −1.65831 − 0.500000i −1.65831 + 0.500000i 1.65831 + 0.500000i
14.6332i 54.7995i −86.1320 0 −801.895 343.000i 612.665i −815.985 0
99.2 1.36675i 24.7995i 126.132 0 33.8947 343.000i 347.335i 1571.98 0
99.3 1.36675i 24.7995i 126.132 0 33.8947 343.000i 347.335i 1571.98 0
99.4 14.6332i 54.7995i −86.1320 0 −801.895 343.000i 612.665i −815.985 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 175.8.b.c 4
5.b even 2 1 inner 175.8.b.c 4
5.c odd 4 1 35.8.a.a 2
5.c odd 4 1 175.8.a.b 2
15.e even 4 1 315.8.a.c 2
20.e even 4 1 560.8.a.i 2
35.f even 4 1 245.8.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.8.a.a 2 5.c odd 4 1
175.8.a.b 2 5.c odd 4 1
175.8.b.c 4 1.a even 1 1 trivial
175.8.b.c 4 5.b even 2 1 inner
245.8.a.b 2 35.f even 4 1
315.8.a.c 2 15.e even 4 1
560.8.a.i 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 216T_{2}^{2} + 400$$ acting on $$S_{8}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 216T^{2} + 400$$
$3$ $$T^{4} + 3618 T^{2} + 1846881$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 117649)^{2}$$
$11$ $$(T^{2} + 7906 T + 9272609)^{2}$$
$13$ $$T^{4} + \cdots + 51\!\cdots\!25$$
$17$ $$T^{4} + \cdots + 45\!\cdots\!01$$
$19$ $$(T^{2} - 3612 T - 1348143980)^{2}$$
$23$ $$T^{4} + \cdots + 24\!\cdots\!56$$
$29$ $$(T^{2} - 126898 T - 20838975695)^{2}$$
$31$ $$(T^{2} - 252768 T - 6409132848)^{2}$$
$37$ $$T^{4} + \cdots + 43\!\cdots\!96$$
$41$ $$(T^{2} + 111920 T - 58262616304)^{2}$$
$43$ $$T^{4} + \cdots + 39\!\cdots\!00$$
$47$ $$T^{4} + \cdots + 15\!\cdots\!09$$
$53$ $$T^{4} + \cdots + 46\!\cdots\!84$$
$59$ $$(T^{2} - 1360120 T - 818710818800)^{2}$$
$61$ $$(T^{2} + 1813680 T + 816706565136)^{2}$$
$67$ $$T^{4} + \cdots + 10\!\cdots\!04$$
$71$ $$(T^{2} + \cdots - 1610731398080)^{2}$$
$73$ $$T^{4} + \cdots + 11\!\cdots\!24$$
$79$ $$(T^{2} + \cdots + 14225767990465)^{2}$$
$83$ $$T^{4} + \cdots + 15\!\cdots\!04$$
$89$ $$(T^{2} + \cdots + 77796446568160)^{2}$$
$97$ $$T^{4} + \cdots + 25\!\cdots\!69$$