# Properties

 Label 175.10.a.c Level $175$ Weight $10$ Character orbit 175.a Self dual yes Analytic conductor $90.131$ 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] = [175,10,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, 10, names="a")

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

chi := DirichletCharacter("175.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$10$$ 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: $$90.1312713287$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 12) q^{2} + (54 \beta + 87) q^{3} + (24 \beta - 360) q^{4} + (735 \beta + 1476) q^{6} - 2401 q^{7} + ( - 584 \beta - 10272) q^{8} + (9396 \beta + 11214) q^{9}+O(q^{10})$$ q + (b + 12) * q^2 + (54*b + 87) * q^3 + (24*b - 360) * q^4 + (735*b + 1476) * q^6 - 2401 * q^7 + (-584*b - 10272) * q^8 + (9396*b + 11214) * q^9 $$q + (\beta + 12) q^{2} + (54 \beta + 87) q^{3} + (24 \beta - 360) q^{4} + (735 \beta + 1476) q^{6} - 2401 q^{7} + ( - 584 \beta - 10272) q^{8} + (9396 \beta + 11214) q^{9} + ( - 9124 \beta + 9283) q^{11} + ( - 17352 \beta - 20952) q^{12} + ( - 18334 \beta + 25545) q^{13} + ( - 2401 \beta - 28812) q^{14} + ( - 29568 \beta + 56384) q^{16} + ( - 15186 \beta + 186955) q^{17} + (123966 \beta + 209736) q^{18} + ( - 31138 \beta - 71638) q^{19} + ( - 129654 \beta - 208887) q^{21} + ( - 100205 \beta + 38404) q^{22} + ( - 821442 \beta + 249454) q^{23} + ( - 605496 \beta - 1145952) q^{24} + ( - 194463 \beta + 159868) q^{26} + (360126 \beta + 3322269) q^{27} + ( - 57624 \beta + 864360) q^{28} + (296772 \beta - 5788777) q^{29} + (2191070 \beta - 1976880) q^{31} + (576 \beta + 5699328) q^{32} + ( - 292506 \beta - 3133947) q^{33} + (4723 \beta + 2121972) q^{34} + ( - 3113424 \beta - 2233008) q^{36} + (3997070 \beta + 1602706) q^{37} + ( - 445294 \beta - 1108760) q^{38} + ( - 215628 \beta - 5697873) q^{39} + ( - 10352502 \beta + 529496) q^{41} + ( - 1764735 \beta - 3543876) q^{42} + ( - 9725678 \beta - 7974090) q^{43} + (3507432 \beta - 5093688) q^{44} + ( - 9607850 \beta - 3578088) q^{46} + ( - 9479722 \beta - 32750645) q^{47} + (472320 \beta - 7867968) q^{48} + 5764801 q^{49} + (8774388 \beta + 9704733) q^{51} + (7213320 \beta - 12716328) q^{52} + ( - 5256968 \beta + 12557344) q^{53} + (7643781 \beta + 42748236) q^{54} + (1402184 \beta + 24663072) q^{56} + ( - 6577458 \beta - 19684122) q^{57} + ( - 2227513 \beta - 67091148) q^{58} + ( - 39092800 \beta - 58079604) q^{59} + ( - 15794106 \beta - 22344272) q^{61} + (24315960 \beta - 6194000) q^{62} + ( - 22559796 \beta - 26924814) q^{63} + (20845056 \beta + 39527936) q^{64} + ( - 6644019 \beta - 39947412) q^{66} + (76121736 \beta - 59046248) q^{67} + (9953880 \beta - 70219512) q^{68} + ( - 57994938 \beta - 333160446) q^{69} + (76576000 \beta - 147082912) q^{71} + ( - 103064688 \beta - 159088320) q^{72} + (17548324 \beta + 28709666) q^{73} + (49567546 \beta + 51209032) q^{74} + (9490368 \beta + 19811184) q^{76} + (21906724 \beta - 22288483) q^{77} + ( - 8285409 \beta - 70099500) q^{78} + ( - 28471816 \beta - 346426427) q^{79} + (25792020 \beta + 223886673) q^{81} + ( - 123700528 \beta - 76466064) q^{82} + (92242900 \beta + 270339964) q^{83} + (41662152 \beta + 50305752) q^{84} + ( - 124682226 \beta - 173494504) q^{86} + ( - 286774794 \beta - 375418095) q^{87} + (88300456 \beta - 52727648) q^{88} + (97352322 \beta - 389521852) q^{89} + (44019934 \beta - 61333545) q^{91} + (301706016 \beta - 247520304) q^{92} + (83871570 \beta + 774553680) q^{93} + ( - 146507309 \beta - 468845516) q^{94} + (307813824 \beta + 496090368) q^{96} + (45681470 \beta + 1336519703) q^{97} + (5764801 \beta + 69177612) q^{98} + ( - 15093468 \beta - 581733270) q^{99}+O(q^{100})$$ q + (b + 12) * q^2 + (54*b + 87) * q^3 + (24*b - 360) * q^4 + (735*b + 1476) * q^6 - 2401 * q^7 + (-584*b - 10272) * q^8 + (9396*b + 11214) * q^9 + (-9124*b + 9283) * q^11 + (-17352*b - 20952) * q^12 + (-18334*b + 25545) * q^13 + (-2401*b - 28812) * q^14 + (-29568*b + 56384) * q^16 + (-15186*b + 186955) * q^17 + (123966*b + 209736) * q^18 + (-31138*b - 71638) * q^19 + (-129654*b - 208887) * q^21 + (-100205*b + 38404) * q^22 + (-821442*b + 249454) * q^23 + (-605496*b - 1145952) * q^24 + (-194463*b + 159868) * q^26 + (360126*b + 3322269) * q^27 + (-57624*b + 864360) * q^28 + (296772*b - 5788777) * q^29 + (2191070*b - 1976880) * q^31 + (576*b + 5699328) * q^32 + (-292506*b - 3133947) * q^33 + (4723*b + 2121972) * q^34 + (-3113424*b - 2233008) * q^36 + (3997070*b + 1602706) * q^37 + (-445294*b - 1108760) * q^38 + (-215628*b - 5697873) * q^39 + (-10352502*b + 529496) * q^41 + (-1764735*b - 3543876) * q^42 + (-9725678*b - 7974090) * q^43 + (3507432*b - 5093688) * q^44 + (-9607850*b - 3578088) * q^46 + (-9479722*b - 32750645) * q^47 + (472320*b - 7867968) * q^48 + 5764801 * q^49 + (8774388*b + 9704733) * q^51 + (7213320*b - 12716328) * q^52 + (-5256968*b + 12557344) * q^53 + (7643781*b + 42748236) * q^54 + (1402184*b + 24663072) * q^56 + (-6577458*b - 19684122) * q^57 + (-2227513*b - 67091148) * q^58 + (-39092800*b - 58079604) * q^59 + (-15794106*b - 22344272) * q^61 + (24315960*b - 6194000) * q^62 + (-22559796*b - 26924814) * q^63 + (20845056*b + 39527936) * q^64 + (-6644019*b - 39947412) * q^66 + (76121736*b - 59046248) * q^67 + (9953880*b - 70219512) * q^68 + (-57994938*b - 333160446) * q^69 + (76576000*b - 147082912) * q^71 + (-103064688*b - 159088320) * q^72 + (17548324*b + 28709666) * q^73 + (49567546*b + 51209032) * q^74 + (9490368*b + 19811184) * q^76 + (21906724*b - 22288483) * q^77 + (-8285409*b - 70099500) * q^78 + (-28471816*b - 346426427) * q^79 + (25792020*b + 223886673) * q^81 + (-123700528*b - 76466064) * q^82 + (92242900*b + 270339964) * q^83 + (41662152*b + 50305752) * q^84 + (-124682226*b - 173494504) * q^86 + (-286774794*b - 375418095) * q^87 + (88300456*b - 52727648) * q^88 + (97352322*b - 389521852) * q^89 + (44019934*b - 61333545) * q^91 + (301706016*b - 247520304) * q^92 + (83871570*b + 774553680) * q^93 + (-146507309*b - 468845516) * q^94 + (307813824*b + 496090368) * q^96 + (45681470*b + 1336519703) * q^97 + (5764801*b + 69177612) * q^98 + (-15093468*b - 581733270) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 24 q^{2} + 174 q^{3} - 720 q^{4} + 2952 q^{6} - 4802 q^{7} - 20544 q^{8} + 22428 q^{9}+O(q^{10})$$ 2 * q + 24 * q^2 + 174 * q^3 - 720 * q^4 + 2952 * q^6 - 4802 * q^7 - 20544 * q^8 + 22428 * q^9 $$2 q + 24 q^{2} + 174 q^{3} - 720 q^{4} + 2952 q^{6} - 4802 q^{7} - 20544 q^{8} + 22428 q^{9} + 18566 q^{11} - 41904 q^{12} + 51090 q^{13} - 57624 q^{14} + 112768 q^{16} + 373910 q^{17} + 419472 q^{18} - 143276 q^{19} - 417774 q^{21} + 76808 q^{22} + 498908 q^{23} - 2291904 q^{24} + 319736 q^{26} + 6644538 q^{27} + 1728720 q^{28} - 11577554 q^{29} - 3953760 q^{31} + 11398656 q^{32} - 6267894 q^{33} + 4243944 q^{34} - 4466016 q^{36} + 3205412 q^{37} - 2217520 q^{38} - 11395746 q^{39} + 1058992 q^{41} - 7087752 q^{42} - 15948180 q^{43} - 10187376 q^{44} - 7156176 q^{46} - 65501290 q^{47} - 15735936 q^{48} + 11529602 q^{49} + 19409466 q^{51} - 25432656 q^{52} + 25114688 q^{53} + 85496472 q^{54} + 49326144 q^{56} - 39368244 q^{57} - 134182296 q^{58} - 116159208 q^{59} - 44688544 q^{61} - 12388000 q^{62} - 53849628 q^{63} + 79055872 q^{64} - 79894824 q^{66} - 118092496 q^{67} - 140439024 q^{68} - 666320892 q^{69} - 294165824 q^{71} - 318176640 q^{72} + 57419332 q^{73} + 102418064 q^{74} + 39622368 q^{76} - 44576966 q^{77} - 140199000 q^{78} - 692852854 q^{79} + 447773346 q^{81} - 152932128 q^{82} + 540679928 q^{83} + 100611504 q^{84} - 346989008 q^{86} - 750836190 q^{87} - 105455296 q^{88} - 779043704 q^{89} - 122667090 q^{91} - 495040608 q^{92} + 1549107360 q^{93} - 937691032 q^{94} + 992180736 q^{96} + 2673039406 q^{97} + 138355224 q^{98} - 1163466540 q^{99}+O(q^{100})$$ 2 * q + 24 * q^2 + 174 * q^3 - 720 * q^4 + 2952 * q^6 - 4802 * q^7 - 20544 * q^8 + 22428 * q^9 + 18566 * q^11 - 41904 * q^12 + 51090 * q^13 - 57624 * q^14 + 112768 * q^16 + 373910 * q^17 + 419472 * q^18 - 143276 * q^19 - 417774 * q^21 + 76808 * q^22 + 498908 * q^23 - 2291904 * q^24 + 319736 * q^26 + 6644538 * q^27 + 1728720 * q^28 - 11577554 * q^29 - 3953760 * q^31 + 11398656 * q^32 - 6267894 * q^33 + 4243944 * q^34 - 4466016 * q^36 + 3205412 * q^37 - 2217520 * q^38 - 11395746 * q^39 + 1058992 * q^41 - 7087752 * q^42 - 15948180 * q^43 - 10187376 * q^44 - 7156176 * q^46 - 65501290 * q^47 - 15735936 * q^48 + 11529602 * q^49 + 19409466 * q^51 - 25432656 * q^52 + 25114688 * q^53 + 85496472 * q^54 + 49326144 * q^56 - 39368244 * q^57 - 134182296 * q^58 - 116159208 * q^59 - 44688544 * q^61 - 12388000 * q^62 - 53849628 * q^63 + 79055872 * q^64 - 79894824 * q^66 - 118092496 * q^67 - 140439024 * q^68 - 666320892 * q^69 - 294165824 * q^71 - 318176640 * q^72 + 57419332 * q^73 + 102418064 * q^74 + 39622368 * q^76 - 44576966 * q^77 - 140199000 * q^78 - 692852854 * q^79 + 447773346 * q^81 - 152932128 * q^82 + 540679928 * q^83 + 100611504 * q^84 - 346989008 * q^86 - 750836190 * q^87 - 105455296 * q^88 - 779043704 * q^89 - 122667090 * q^91 - 495040608 * q^92 + 1549107360 * q^93 - 937691032 * q^94 + 992180736 * q^96 + 2673039406 * q^97 + 138355224 * q^98 - 1163466540 * 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
 −1.41421 1.41421
9.17157 −65.7351 −427.882 0 −602.894 −2401.00 −8620.20 −15361.9 0
1.2 14.8284 239.735 −292.118 0 3554.89 −2401.00 −11923.8 37789.9 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 175.10.a.c 2
5.b even 2 1 35.10.a.b 2
5.c odd 4 2 175.10.b.c 4
15.d odd 2 1 315.10.a.b 2
35.c odd 2 1 245.10.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.10.a.b 2 5.b even 2 1
175.10.a.c 2 1.a even 1 1 trivial
175.10.b.c 4 5.c odd 4 2
245.10.a.c 2 35.c odd 2 1
315.10.a.b 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 24T_{2} + 136$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(175))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 24T + 136$$
$3$ $$T^{2} - 174T - 15759$$
$5$ $$T^{2}$$
$7$ $$(T + 2401)^{2}$$
$11$ $$T^{2} - 18566 T - 579804919$$
$13$ $$T^{2} + \cdots - 2036537423$$
$17$ $$T^{2} + \cdots + 33107255257$$
$19$ $$T^{2} + \cdots - 2624597308$$
$23$ $$T^{2} + \cdots - 5335908376796$$
$29$ $$T^{2} + \cdots + 32805350195857$$
$31$ $$T^{2} + \cdots - 34498247424800$$
$37$ $$T^{2} + \cdots - 125243882156764$$
$41$ $$T^{2} + \cdots - 857114015266016$$
$43$ $$T^{2} + \cdots - 693124389149372$$
$47$ $$T^{2} + \cdots + 353683714337753$$
$53$ $$T^{2} + \cdots - 63398812089856$$
$59$ $$T^{2} + \cdots - 88\!\cdots\!84$$
$61$ $$T^{2} + \cdots - 14\!\cdots\!04$$
$67$ $$T^{2} + \cdots - 42\!\cdots\!64$$
$71$ $$T^{2} + \cdots - 25\!\cdots\!56$$
$73$ $$T^{2} + \cdots - 16\!\cdots\!52$$
$79$ $$T^{2} + \cdots + 11\!\cdots\!81$$
$83$ $$T^{2} + \cdots + 50\!\cdots\!96$$
$89$ $$T^{2} + \cdots + 75\!\cdots\!32$$
$97$ $$T^{2} + \cdots + 17\!\cdots\!09$$