# Properties

 Label 175.8.a.b Level $175$ Weight $8$ Character orbit 175.a Self dual yes Analytic conductor $54.667$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,8,Mod(1,175)]

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([0, 0]))

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

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

chi := DirichletCharacter("175.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 175.a (trivial)

## 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: yes Analytic conductor: $$54.6673794597$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 11$$ x^2 - 11 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = 2\sqrt{11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 8) q^{2} + ( - 6 \beta + 15) q^{3} + ( - 16 \beta - 20) q^{4} + (63 \beta - 384) q^{6} + 343 q^{7} + ( - 20 \beta + 480) q^{8} + ( - 180 \beta - 378) q^{9}+O(q^{10})$$ q + (b - 8) * q^2 + (-6*b + 15) * q^3 + (-16*b - 20) * q^4 + (63*b - 384) * q^6 + 343 * q^7 + (-20*b + 480) * q^8 + (-180*b - 378) * q^9 $$q + (\beta - 8) q^{2} + ( - 6 \beta + 15) q^{3} + ( - 16 \beta - 20) q^{4} + (63 \beta - 384) q^{6} + 343 q^{7} + ( - 20 \beta + 480) q^{8} + ( - 180 \beta - 378) q^{9} + (380 \beta - 3953) q^{11} + ( - 120 \beta + 3924) q^{12} + ( - 418 \beta + 8909) q^{13} + (343 \beta - 2744) q^{14} + (2688 \beta - 2160) q^{16} + (2210 \beta + 1199) q^{17} + (1062 \beta - 4896) q^{18} + ( - 5542 \beta - 1806) q^{19} + ( - 2058 \beta + 5145) q^{21} + ( - 6993 \beta + 48344) q^{22} + (10690 \beta - 6922) q^{23} + ( - 3180 \beta + 12480) q^{24} + (12253 \beta - 89664) q^{26} + (12690 \beta + 9045) q^{27} + ( - 5488 \beta - 6860) q^{28} + (23772 \beta - 63449) q^{29} + (22554 \beta + 126384) q^{31} + ( - 21104 \beta + 74112) q^{32} + (29418 \beta - 159615) q^{33} + ( - 16481 \beta + 87648) q^{34} + (9648 \beta + 134280) q^{36} + ( - 43638 \beta + 132930) q^{37} + (42530 \beta - 229400) q^{38} + ( - 59724 \beta + 243987) q^{39} + ( - 37354 \beta - 55960) q^{41} + (21609 \beta - 131712) q^{42} + ( - 24578 \beta - 473786) q^{43} + (55648 \beta - 188460) q^{44} + ( - 92442 \beta + 525736) q^{46} + ( - 56742 \beta - 135637) q^{47} + (53280 \beta - 742032) q^{48} + 117649 q^{49} + (25956 \beta - 565455) q^{51} + ( - 134184 \beta + 116092) q^{52} + ( - 65224 \beta + 633896) q^{53} + ( - 92475 \beta + 486000) q^{54} + ( - 6860 \beta + 164640) q^{56} + ( - 72294 \beta + 1435998) q^{57} + ( - 253625 \beta + 1553560) q^{58} + ( - 170640 \beta - 680060) q^{59} + ( - 11334 \beta - 906840) q^{61} + ( - 54048 \beta - 18696) q^{62} + ( - 61740 \beta - 129654) q^{63} + ( - 101120 \beta - 1244992) q^{64} + ( - 394959 \beta + 2571312) q^{66} + (506344 \beta + 1094656) q^{67} + ( - 63384 \beta - 1579820) q^{68} + (201882 \beta - 2925990) q^{69} + (222048 \beta - 747464) q^{71} + ( - 78840 \beta - 23040) q^{72} + (212396 \beta - 3584894) q^{73} + (482034 \beta - 2983512) q^{74} + (139736 \beta + 3937688) q^{76} + (130340 \beta - 1355879) q^{77} + (721779 \beta - 4579752) q^{78} + ( - 187504 \beta - 3971487) q^{79} + (529740 \beta - 2387799) q^{81} + (242872 \beta - 1195896) q^{82} + ( - 939444 \beta + 152356) q^{83} + ( - 41160 \beta + 1345932) q^{84} + ( - 277162 \beta + 2708856) q^{86} + (737274 \beta - 7227543) q^{87} + (261460 \beta - 2231840) q^{88} + ( - 247538 \beta - 8971764) q^{89} + ( - 143374 \beta + 3055787) q^{91} + ( - 103048 \beta - 7387320) q^{92} + ( - 419994 \beta - 4058496) q^{93} + (318299 \beta - 1411552) q^{94} + ( - 761232 \beta + 6683136) q^{96} + ( - 680782 \beta - 2129037) q^{97} + (117649 \beta - 941192) q^{98} + (567900 \beta - 1515366) q^{99}+O(q^{100})$$ q + (b - 8) * q^2 + (-6*b + 15) * q^3 + (-16*b - 20) * q^4 + (63*b - 384) * q^6 + 343 * q^7 + (-20*b + 480) * q^8 + (-180*b - 378) * q^9 + (380*b - 3953) * q^11 + (-120*b + 3924) * q^12 + (-418*b + 8909) * q^13 + (343*b - 2744) * q^14 + (2688*b - 2160) * q^16 + (2210*b + 1199) * q^17 + (1062*b - 4896) * q^18 + (-5542*b - 1806) * q^19 + (-2058*b + 5145) * q^21 + (-6993*b + 48344) * q^22 + (10690*b - 6922) * q^23 + (-3180*b + 12480) * q^24 + (12253*b - 89664) * q^26 + (12690*b + 9045) * q^27 + (-5488*b - 6860) * q^28 + (23772*b - 63449) * q^29 + (22554*b + 126384) * q^31 + (-21104*b + 74112) * q^32 + (29418*b - 159615) * q^33 + (-16481*b + 87648) * q^34 + (9648*b + 134280) * q^36 + (-43638*b + 132930) * q^37 + (42530*b - 229400) * q^38 + (-59724*b + 243987) * q^39 + (-37354*b - 55960) * q^41 + (21609*b - 131712) * q^42 + (-24578*b - 473786) * q^43 + (55648*b - 188460) * q^44 + (-92442*b + 525736) * q^46 + (-56742*b - 135637) * q^47 + (53280*b - 742032) * q^48 + 117649 * q^49 + (25956*b - 565455) * q^51 + (-134184*b + 116092) * q^52 + (-65224*b + 633896) * q^53 + (-92475*b + 486000) * q^54 + (-6860*b + 164640) * q^56 + (-72294*b + 1435998) * q^57 + (-253625*b + 1553560) * q^58 + (-170640*b - 680060) * q^59 + (-11334*b - 906840) * q^61 + (-54048*b - 18696) * q^62 + (-61740*b - 129654) * q^63 + (-101120*b - 1244992) * q^64 + (-394959*b + 2571312) * q^66 + (506344*b + 1094656) * q^67 + (-63384*b - 1579820) * q^68 + (201882*b - 2925990) * q^69 + (222048*b - 747464) * q^71 + (-78840*b - 23040) * q^72 + (212396*b - 3584894) * q^73 + (482034*b - 2983512) * q^74 + (139736*b + 3937688) * q^76 + (130340*b - 1355879) * q^77 + (721779*b - 4579752) * q^78 + (-187504*b - 3971487) * q^79 + (529740*b - 2387799) * q^81 + (242872*b - 1195896) * q^82 + (-939444*b + 152356) * q^83 + (-41160*b + 1345932) * q^84 + (-277162*b + 2708856) * q^86 + (737274*b - 7227543) * q^87 + (261460*b - 2231840) * q^88 + (-247538*b - 8971764) * q^89 + (-143374*b + 3055787) * q^91 + (-103048*b - 7387320) * q^92 + (-419994*b - 4058496) * q^93 + (318299*b - 1411552) * q^94 + (-761232*b + 6683136) * q^96 + (-680782*b - 2129037) * q^97 + (117649*b - 941192) * q^98 + (567900*b - 1515366) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 16 q^{2} + 30 q^{3} - 40 q^{4} - 768 q^{6} + 686 q^{7} + 960 q^{8} - 756 q^{9}+O(q^{10})$$ 2 * q - 16 * q^2 + 30 * q^3 - 40 * q^4 - 768 * q^6 + 686 * q^7 + 960 * q^8 - 756 * q^9 $$2 q - 16 q^{2} + 30 q^{3} - 40 q^{4} - 768 q^{6} + 686 q^{7} + 960 q^{8} - 756 q^{9} - 7906 q^{11} + 7848 q^{12} + 17818 q^{13} - 5488 q^{14} - 4320 q^{16} + 2398 q^{17} - 9792 q^{18} - 3612 q^{19} + 10290 q^{21} + 96688 q^{22} - 13844 q^{23} + 24960 q^{24} - 179328 q^{26} + 18090 q^{27} - 13720 q^{28} - 126898 q^{29} + 252768 q^{31} + 148224 q^{32} - 319230 q^{33} + 175296 q^{34} + 268560 q^{36} + 265860 q^{37} - 458800 q^{38} + 487974 q^{39} - 111920 q^{41} - 263424 q^{42} - 947572 q^{43} - 376920 q^{44} + 1051472 q^{46} - 271274 q^{47} - 1484064 q^{48} + 235298 q^{49} - 1130910 q^{51} + 232184 q^{52} + 1267792 q^{53} + 972000 q^{54} + 329280 q^{56} + 2871996 q^{57} + 3107120 q^{58} - 1360120 q^{59} - 1813680 q^{61} - 37392 q^{62} - 259308 q^{63} - 2489984 q^{64} + 5142624 q^{66} + 2189312 q^{67} - 3159640 q^{68} - 5851980 q^{69} - 1494928 q^{71} - 46080 q^{72} - 7169788 q^{73} - 5967024 q^{74} + 7875376 q^{76} - 2711758 q^{77} - 9159504 q^{78} - 7942974 q^{79} - 4775598 q^{81} - 2391792 q^{82} + 304712 q^{83} + 2691864 q^{84} + 5417712 q^{86} - 14455086 q^{87} - 4463680 q^{88} - 17943528 q^{89} + 6111574 q^{91} - 14774640 q^{92} - 8116992 q^{93} - 2823104 q^{94} + 13366272 q^{96} - 4258074 q^{97} - 1882384 q^{98} - 3030732 q^{99}+O(q^{100})$$ 2 * q - 16 * q^2 + 30 * q^3 - 40 * q^4 - 768 * q^6 + 686 * q^7 + 960 * q^8 - 756 * q^9 - 7906 * q^11 + 7848 * q^12 + 17818 * q^13 - 5488 * q^14 - 4320 * q^16 + 2398 * q^17 - 9792 * q^18 - 3612 * q^19 + 10290 * q^21 + 96688 * q^22 - 13844 * q^23 + 24960 * q^24 - 179328 * q^26 + 18090 * q^27 - 13720 * q^28 - 126898 * q^29 + 252768 * q^31 + 148224 * q^32 - 319230 * q^33 + 175296 * q^34 + 268560 * q^36 + 265860 * q^37 - 458800 * q^38 + 487974 * q^39 - 111920 * q^41 - 263424 * q^42 - 947572 * q^43 - 376920 * q^44 + 1051472 * q^46 - 271274 * q^47 - 1484064 * q^48 + 235298 * q^49 - 1130910 * q^51 + 232184 * q^52 + 1267792 * q^53 + 972000 * q^54 + 329280 * q^56 + 2871996 * q^57 + 3107120 * q^58 - 1360120 * q^59 - 1813680 * q^61 - 37392 * q^62 - 259308 * q^63 - 2489984 * q^64 + 5142624 * q^66 + 2189312 * q^67 - 3159640 * q^68 - 5851980 * q^69 - 1494928 * q^71 - 46080 * q^72 - 7169788 * q^73 - 5967024 * q^74 + 7875376 * q^76 - 2711758 * q^77 - 9159504 * q^78 - 7942974 * q^79 - 4775598 * q^81 - 2391792 * q^82 + 304712 * q^83 + 2691864 * q^84 + 5417712 * q^86 - 14455086 * q^87 - 4463680 * q^88 - 17943528 * q^89 + 6111574 * q^91 - 14774640 * q^92 - 8116992 * q^93 - 2823104 * q^94 + 13366272 * q^96 - 4258074 * q^97 - 1882384 * q^98 - 3030732 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.31662 3.31662
−14.6332 54.7995 86.1320 0 −801.895 343.000 612.665 815.985 0
1.2 −1.36675 −24.7995 −126.132 0 33.8947 343.000 347.335 −1571.98 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.8.a.b 2
5.b even 2 1 35.8.a.a 2
5.c odd 4 2 175.8.b.c 4
15.d odd 2 1 315.8.a.c 2
20.d odd 2 1 560.8.a.i 2
35.c odd 2 1 245.8.a.b 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 16T_{2} + 20$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(175))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16T + 20$$
$3$ $$T^{2} - 30T - 1359$$
$5$ $$T^{2}$$
$7$ $$(T - 343)^{2}$$
$11$ $$T^{2} + 7906 T + 9272609$$
$13$ $$T^{2} - 17818 T + 71682425$$
$17$ $$T^{2} - 2398 T - 213462799$$
$19$ $$T^{2} + \cdots - 1348143980$$
$23$ $$T^{2} + \cdots - 4980234316$$
$29$ $$T^{2} + \cdots - 20838975695$$
$31$ $$T^{2} + \cdots - 6409132848$$
$37$ $$T^{2} + \cdots - 66117717036$$
$41$ $$T^{2} + \cdots - 58262616304$$
$43$ $$T^{2} + \cdots + 197893738100$$
$47$ $$T^{2} + \cdots - 123267405047$$
$53$ $$T^{2} + \cdots + 214640651072$$
$59$ $$T^{2} + \cdots - 818710818800$$
$61$ $$T^{2} + \cdots + 816706565136$$
$67$ $$T^{2} + \cdots - 10082635080448$$
$71$ $$T^{2} + \cdots - 1610731398080$$
$73$ $$T^{2} + \cdots + 10866534315332$$
$79$ $$T^{2} + \cdots + 14225767990465$$
$83$ $$T^{2} + \cdots - 38809208931248$$
$89$ $$T^{2} + \cdots + 77796446568160$$
$97$ $$T^{2} + \cdots - 15859623239687$$