# Properties

 Label 75.22.a.d Level $75$ Weight $22$ Character orbit 75.a Self dual yes Analytic conductor $209.608$ 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] = [75,22,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 75.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: $$209.608008215$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{649})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 162$$ x^2 - x - 162 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2\cdot 3^{2}\cdot 7$$ Twist minimal: no (minimal twist has level 3) 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 $$\beta = 63\sqrt{649}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 333) q^{2} - 59049 q^{3} + (666 \beta + 589618) q^{4} + (59049 \beta + 19663317) q^{6} + (182016 \beta - 339948056) q^{7} + (1285756 \beta - 1213527924) q^{8} + 3486784401 q^{9}+O(q^{10})$$ q + (-b - 333) * q^2 - 59049 * q^3 + (666*b + 589618) * q^4 + (59049*b + 19663317) * q^6 + (182016*b - 339948056) * q^7 + (1285756*b - 1213527924) * q^8 + 3486784401 * q^9 $$q + ( - \beta - 333) q^{2} - 59049 q^{3} + (666 \beta + 589618) q^{4} + (59049 \beta + 19663317) q^{6} + (182016 \beta - 339948056) q^{7} + (1285756 \beta - 1213527924) q^{8} + 3486784401 q^{9} + (31263232 \beta + 109934561484) q^{11} + ( - 39326634 \beta - 34816353282) q^{12} + (475236864 \beta + 24234454978) q^{13} + (279336728 \beta - 355648853448) q^{14} + ( - 611332056 \beta - 4144368220280) q^{16} + ( - 1677304320 \beta + 5666764520718) q^{17} + ( - 3486784401 \beta - 1161099205533) q^{18} + ( - 22384332288 \beta + 5980292505812) q^{19} + ( - 10747862784 \beta + 20073592758744) q^{21} + ( - 120345217740 \beta - 117138574281564) q^{22} + (73613782528 \beta + 73254195031752) q^{23} + ( - 75922606044 \beta + 71657610384276) q^{24} + ( - 182488330690 \beta - 12\!\cdots\!58) q^{26}+ \cdots + (10\!\cdots\!32 \beta + 38\!\cdots\!84) q^{99}+O(q^{100})$$ q + (-b - 333) * q^2 - 59049 * q^3 + (666*b + 589618) * q^4 + (59049*b + 19663317) * q^6 + (182016*b - 339948056) * q^7 + (1285756*b - 1213527924) * q^8 + 3486784401 * q^9 + (31263232*b + 109934561484) * q^11 + (-39326634*b - 34816353282) * q^12 + (475236864*b + 24234454978) * q^13 + (279336728*b - 355648853448) * q^14 + (-611332056*b - 4144368220280) * q^16 + (-1677304320*b + 5666764520718) * q^17 + (-3486784401*b - 1161099205533) * q^18 + (-22384332288*b + 5980292505812) * q^19 + (-10747862784*b + 20073592758744) * q^21 + (-120345217740*b - 117138574281564) * q^22 + (73613782528*b + 73254195031752) * q^23 + (-75922606044*b + 71657610384276) * q^24 + (-182488330690*b - 1232223681984858) * q^26 - 205891132094649 * q^27 + (-119085495408*b + 111815643477328) * q^28 + (1163540203264*b - 899260021837026) * q^29 + (-1732114342656*b + 5584553763472496) * q^31 + (1651516028016*b + 5499745757967024) * q^32 + (-1846062586368*b - 6491525921068716) * q^33 + (-5108222182158*b + 2433503743706826) * q^34 + (2322198411066*b + 2055870844948818) * q^36 + (-15038204442624*b - 6368132429330006) * q^37 + (1473690146092*b + 55667938833910332) * q^38 + (-28062261582336*b - 1431020331995922) * q^39 + (-52192512395776*b + 61486010308234026) * q^41 + (-16494554451672*b + 21000709147250952) * q^42 + (-12139771405824*b - 144227709081135020) * q^43 + (91649782273720*b + 118452619567796184) * q^44 + (-97767584613576*b - 214013990697580584) * q^46 + (6195754260992*b - 418621872870798480) * q^47 + (36098546574744*b + 244720799039313720) * q^48 + (-123751970721792*b - 357642698470735335) * q^49 + (99043142791680*b - 334616778183877182) * q^51 + (296348356293300*b + 829575374121022948) * q^52 + (-283816914561792*b + 21503982006387882) * q^53 + (205891132094649*b + 68561746987518117) * q^54 + (-657971751305120*b + 1015365160025284320) * q^56 + (1321772437274112*b - 353130292175692788) * q^57 + (511801134150114*b - 2697687515052145926) * q^58 + (-2106913298778112*b - 1761911665451928612) * q^59 + (1794014554180608*b - 889511564225506930) * q^61 + (-5007759687368048*b + 2602064021838738768) * q^62 + (634650549532416*b - 1185325578811074456) * q^63 + (-4767644351391840*b + 2605846006731741472) * q^64 + (7106264762329260*b + 6916915672752072636) * q^66 + (3370226524784640*b + 8227034333810805148) * q^67 + (2785096352248428*b + 463749167992163004) * q^68 + (-4346820244495872*b - 4325586962429923848) * q^69 + (-830682487764480*b + 8689613565575210472) * q^71 + (4483153964292156*b - 4231310235581113524) * q^72 + (-26617229816020992*b - 25445573134236994538) * q^73 + (11375854508723798*b + 40857213196837643742) * q^74 + (-9215330426115192*b - 34875056468046395032) * q^76 + (9381974200394752*b - 22714225491908012832) * q^77 + (10775753438913810*b + 72761576197523880042) * q^78 + (3613605017944320*b - 27027892797095295520) * q^79 + 12157665459056928801 * q^81 + (-44105903680440618*b + 113966859589901947998) * q^82 + (-34423999299854848*b - 55554214638833338644) * q^83 + (7031879418346992*b - 6602601931692741072) * q^84 + (148270252959274412*b + 79298433632623292604) * q^86 + (-68705885462535936*b + 53100405029454548274) * q^87 + (103410217008931536*b - 29866259797357770864) * q^88 + (-75474030898664448*b + 113460383982724032762) * q^89 + (-157144789499060736*b + 214577087342592500176) * q^91 + (92191305117741136*b + 169479140905068324624) * q^92 + (102279619819494144*b - 329762315179287416304) * q^93 + (416558686701888144*b + 123441557984417559888) * q^94 + (-97520369938316784*b - 324754487262194800176) * q^96 + (-328931501848427520*b + 64165734626397873118) * q^97 + (398852104721092071*b + 437865368685575165307) * q^98 + (109008149662444032*b + 383318114113186611084) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 666 q^{2} - 118098 q^{3} + 1179236 q^{4} + 39326634 q^{6} - 679896112 q^{7} - 2427055848 q^{8} + 6973568802 q^{9}+O(q^{10})$$ 2 * q - 666 * q^2 - 118098 * q^3 + 1179236 * q^4 + 39326634 * q^6 - 679896112 * q^7 - 2427055848 * q^8 + 6973568802 * q^9 $$2 q - 666 q^{2} - 118098 q^{3} + 1179236 q^{4} + 39326634 q^{6} - 679896112 q^{7} - 2427055848 q^{8} + 6973568802 q^{9} + 219869122968 q^{11} - 69632706564 q^{12} + 48468909956 q^{13} - 711297706896 q^{14} - 8288736440560 q^{16} + 11333529041436 q^{17} - 2322198411066 q^{18} + 11960585011624 q^{19} + 40147185517488 q^{21} - 234277148563128 q^{22} + 146508390063504 q^{23} + 143315220768552 q^{24} - 24\!\cdots\!16 q^{26}+ \cdots + 76\!\cdots\!68 q^{99}+O(q^{100})$$ 2 * q - 666 * q^2 - 118098 * q^3 + 1179236 * q^4 + 39326634 * q^6 - 679896112 * q^7 - 2427055848 * q^8 + 6973568802 * q^9 + 219869122968 * q^11 - 69632706564 * q^12 + 48468909956 * q^13 - 711297706896 * q^14 - 8288736440560 * q^16 + 11333529041436 * q^17 - 2322198411066 * q^18 + 11960585011624 * q^19 + 40147185517488 * q^21 - 234277148563128 * q^22 + 146508390063504 * q^23 + 143315220768552 * q^24 - 2464447363969716 * q^26 - 411782264189298 * q^27 + 223631286954656 * q^28 - 1798520043674052 * q^29 + 11169107526944992 * q^31 + 10999491515934048 * q^32 - 12983051842137432 * q^33 + 4867007487413652 * q^34 + 4111741689897636 * q^36 - 12736264858660012 * q^37 + 111335877667820664 * q^38 - 2862040663991844 * q^39 + 122972020616468052 * q^41 + 42001418294501904 * q^42 - 288455418162270040 * q^43 + 236905239135592368 * q^44 - 428027981395161168 * q^46 - 837243745741596960 * q^47 + 489441598078627440 * q^48 - 715285396941470670 * q^49 - 669233556367754364 * q^51 + 1659150748242045896 * q^52 + 43007964012775764 * q^53 + 137123493975036234 * q^54 + 2030730320050568640 * q^56 - 706260584351385576 * q^57 - 5395375030104291852 * q^58 - 3523823330903857224 * q^59 - 1779023128451013860 * q^61 + 5204128043677477536 * q^62 - 2370651157622148912 * q^63 + 5211692013463482944 * q^64 + 13833831345504145272 * q^66 + 16454068667621610296 * q^67 + 927498335984326008 * q^68 - 8651173924859847696 * q^69 + 17379227131150420944 * q^71 - 8462620471162227048 * q^72 - 50891146268473989076 * q^73 + 81714426393675287484 * q^74 - 69750112936092790064 * q^76 - 45428450983816025664 * q^77 + 145523152395047760084 * q^78 - 54055785594190591040 * q^79 + 24315330918113857602 * q^81 + 227933719179803895996 * q^82 - 111108429277666677288 * q^83 - 13205203863385482144 * q^84 + 158596867265246585208 * q^86 + 106200810058909096548 * q^87 - 59732519594715541728 * q^88 + 226920767965448065524 * q^89 + 429154174685185000352 * q^91 + 338958281810136649248 * q^92 - 659524630358574832608 * q^93 + 246883115968835119776 * q^94 - 649508974524389600352 * q^96 + 128331469252795746236 * q^97 + 875730737371150330614 * q^98 + 766636228226373222168 * 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
 13.2377 −12.2377
−1937.96 −59049.0 1.65852e6 0 1.14434e8 −4.78205e7 8.50053e8 3.48678e9 0
1.2 1271.96 −59049.0 −479282. 0 −7.51077e7 −6.32076e8 −3.27711e9 3.48678e9 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
$$3$$ $$1$$
$$5$$ $$1$$

## 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 75.22.a.d 2
5.b even 2 1 3.22.a.c 2
5.c odd 4 2 75.22.b.d 4
15.d odd 2 1 9.22.a.e 2
20.d odd 2 1 48.22.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.22.a.c 2 5.b even 2 1
9.22.a.e 2 15.d odd 2 1
48.22.a.g 2 20.d odd 2 1
75.22.a.d 2 1.a even 1 1 trivial
75.22.b.d 4 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 666T_{2} - 2464992$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(75))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 666 T - 2464992$$
$3$ $$(T + 59049)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 679896112 T + 30\!\cdots\!00$$
$11$ $$T^{2} - 219869122968 T + 95\!\cdots\!12$$
$13$ $$T^{2} - 48468909956 T - 58\!\cdots\!92$$
$17$ $$T^{2} - 11333529041436 T + 24\!\cdots\!24$$
$19$ $$T^{2} - 11960585011624 T - 12\!\cdots\!20$$
$23$ $$T^{2} - 146508390063504 T - 85\!\cdots\!00$$
$29$ $$T^{2} + \cdots - 26\!\cdots\!00$$
$31$ $$T^{2} + \cdots + 23\!\cdots\!00$$
$37$ $$T^{2} + \cdots - 54\!\cdots\!20$$
$41$ $$T^{2} + \cdots - 32\!\cdots\!80$$
$43$ $$T^{2} + \cdots + 20\!\cdots\!44$$
$47$ $$T^{2} + \cdots + 17\!\cdots\!16$$
$53$ $$T^{2} + \cdots - 20\!\cdots\!60$$
$59$ $$T^{2} + \cdots - 83\!\cdots\!20$$
$61$ $$T^{2} + \cdots - 74\!\cdots\!84$$
$67$ $$T^{2} + \cdots + 38\!\cdots\!04$$
$71$ $$T^{2} + \cdots + 73\!\cdots\!84$$
$73$ $$T^{2} + \cdots - 11\!\cdots\!40$$
$79$ $$T^{2} + \cdots + 69\!\cdots\!00$$
$83$ $$T^{2} + \cdots + 33\!\cdots\!12$$
$89$ $$T^{2} + \cdots - 17\!\cdots\!80$$
$97$ $$T^{2} + \cdots - 27\!\cdots\!76$$