# Properties

 Label 75.22.a.e Level $75$ Weight $22$ Character orbit 75.a Self dual yes Analytic conductor $209.608$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 6395796x - 2792983104$$ x^3 - x^2 - 6395796*x - 2792983104 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}\cdot 3\cdot 5\cdot 7$$ Twist minimal: no (minimal twist has level 15) 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 268) q^{2} - 59049 q^{3} + (\beta_{2} + 120 \beta_1 + 2238318) q^{4} + ( - 59049 \beta_1 + 15825132) q^{6} + ( - 264 \beta_{2} + 153760 \beta_1 + 525814764) q^{7} + ( - 803 \beta_{2} + 1890196 \beta_1 + 474100006) q^{8} + 3486784401 q^{9}+O(q^{10})$$ q + (b1 - 268) * q^2 - 59049 * q^3 + (b2 + 120*b1 + 2238318) * q^4 + (-59049*b1 + 15825132) * q^6 + (-264*b2 + 153760*b1 + 525814764) * q^7 + (-803*b2 + 1890196*b1 + 474100006) * q^8 + 3486784401 * q^9 $$q + (\beta_1 - 268) q^{2} - 59049 q^{3} + (\beta_{2} + 120 \beta_1 + 2238318) q^{4} + ( - 59049 \beta_1 + 15825132) q^{6} + ( - 264 \beta_{2} + 153760 \beta_1 + 525814764) q^{7} + ( - 803 \beta_{2} + 1890196 \beta_1 + 474100006) q^{8} + 3486784401 q^{9} + ( - 7808 \beta_{2} + \cdots + 28162718864) q^{11}+ \cdots + ( - 27224812603008 \beta_{2} + \cdots + 98\!\cdots\!64) q^{99}+O(q^{100})$$ q + (b1 - 268) * q^2 - 59049 * q^3 + (b2 + 120*b1 + 2238318) * q^4 + (-59049*b1 + 15825132) * q^6 + (-264*b2 + 153760*b1 + 525814764) * q^7 + (-803*b2 + 1890196*b1 + 474100006) * q^8 + 3486784401 * q^9 + (-7808*b2 + 9117600*b1 + 28162718864) * q^11 + (-59049*b2 - 7085880*b1 - 132170439582) * q^12 + (-89144*b2 + 104131392*b1 - 355198062282) * q^13 + (397432*b2 + 136021564*b1 + 514581979072) * q^14 + (534213*b2 - 411245596*b1 + 3237737878550) * q^16 + (729992*b2 - 1409034048*b1 - 4616822158622) * q^17 + (3486784401*b1 - 934458219468) * q^18 + (-9362104*b2 - 14111812896*b1 + 8957650249812) * q^19 + (15588936*b2 - 9079374240*b1 - 31048835999436) * q^21 + (16324384*b2 + 18407461904*b1 + 31324306957056) * q^22 + (10535432*b2 + 18765594464*b1 - 25265117784328) * q^23 + (47416347*b2 - 111614183604*b1 - 27995131254294) * q^24 + (186411304*b2 - 466560067866*b1 + 539146178504552) * q^26 - 205891132094649 * q^27 + (322839956*b2 + 921516311424*b1 - 660556424974312) * q^28 + (732279424*b2 - 993243207040*b1 - 303392922896886) * q^29 + (-479108872*b2 + 477897067264*b1 - 4923704756606944) * q^31 + (779688861*b2 + 23627871620*b1 - 3615221588691866) * q^32 + (461054592*b2 - 538385162400*b1 - 1662980386200336) * q^33 + (-2082816664*b2 - 3920737889006*b1 - 4770098715448104) * q^34 + (3486784401*b2 + 418414128120*b1 + 7804532286877518) * q^36 + (1056894152*b2 - 6894023111104*b1 + 11294499909514134) * q^37 + (-5470590904*b2 - 12456434350716*b1 - 62571186450993728) * q^38 + (5263864056*b2 - 6148854566208*b1 + 20974090379689818) * q^39 + (26907294352*b2 + 47659581755648*b1 + 1451864625405546) * q^41 + (-23467962168*b2 - 8031937332636*b1 - 30385551282222528) * q^42 + (53393843840*b2 - 12561144913856*b1 + 25526910116705860) * q^43 + (19714618288*b2 + 47137183129088*b1 + 11031300130421664) * q^44 + (9041390728*b2 - 47810215256*b1 + 86784011018774416) * q^46 + (68756675576*b2 + 178318406875104*b1 - 59685136446539480) * q^47 + (-31544743437*b2 + 24283641198204*b1 - 191185183990498950) * q^48 + (-319467118208*b2 - 38706400316416*b1 + 324590512270227945) * q^49 + (-43105297608*b2 + 83202051500352*b1 + 272618731644470478) * q^51 + (-451669183570*b2 + 457101167897840*b1 - 1388778829757012412) * q^52 + (-411233665296*b2 - 13520998705664*b1 - 1337869814576843578) * q^53 + (-205891132094649*b1 + 55178823401365932) * q^54 + (-209940281628*b2 - 38640651236208*b1 + 3026987059849185720) * q^56 + (552822879096*b2 + 833288439695904*b1 - 528940289601148788) * q^57 + (-1669137115392*b2 + 557912463748874*b1 - 4153312119996087416) * q^58 + (-1595336337024*b2 + 434925331623008*b1 - 1492246460343526352) * q^59 + (961515250192*b2 - 437849824268416*b1 + 4557750515913238550) * q^61 + (920114556120*b2 - 5553949175822352*b1 + 3356995469362136208) * q^62 + (-920511081864*b2 + 536127969497760*b1 + 1833402716930696364) * q^63 + (-1816350808459*b2 - 1416212555174044*b1 - 5720178213444258378) * q^64 + (-963938550816*b2 - 1086942217969296*b1 - 1849669001507199744) * q^66 + (-536812296576*b2 - 6230202201438656*b1 - 3339354295109713212) * q^67 + (-3529202290918*b2 - 6882319330511216*b1 - 5756688514932986196) * q^68 + (-622106724168*b2 - 1108089587504736*b1 + 1491879940046784072) * q^69 + (8510370755328*b2 + 7945981682866048*b1 + 10513525059014066432) * q^71 + (-2799887874003*b2 + 6590705927632596*b1 + 1653084505434806406) * q^72 + (2078961676560*b2 - 6166519714978048*b1 - 26877162613795978258) * q^73 + (-7869536413400*b2 + 10418949529361222*b1 - 32420288281020321048) * q^74 + (12226676181484*b2 - 47123183236932224*b1 - 55127916343601603352) * q^76 + (-11284940248960*b2 + 1604293120992128*b1 + 35746317998272134848) * q^77 + (-11007401089896*b2 + 27549905447419434*b1 - 31836042694515291048) * q^78 + (-28313946671352*b2 - 10734583048382272*b1 - 33883127166957598320) * q^79 + 12157665459056928801 * q^81 + (22824149068752*b2 + 65752643762046410*b1 + 202822422346777073128) * q^82 + (11083286184960*b2 - 77743160475758592*b1 + 32907085470311429076) * q^83 + (-19063376561844*b2 - 54414616673275776*b1 + 39005196338308149288) * q^84 + (-61843662778176*b2 + 111554603212414532*b1 - 60381737382966789296) * q^86 + (-43240367707776*b2 + 58650018132504960*b1 + 17915048704138221414) * q^87 + (-5294124105104*b2 + 24280827834381760*b1 + 132333856180861508896) * q^88 + (120822099984240*b2 + 65549330041667328*b1 + 16564107621445798242) * q^89 + (49828662348656*b2 - 85731769774759488*b1 + 52303687100299092296) * q^91 + (-30487416146864*b2 + 62803853166586720*b1 + 29525498480760708064) * q^92 + (28290899782728*b2 - 28219343924871936*b1 + 290739842172883436256) * q^93 + (114855995318456*b2 + 126558582888873592*b1 + 776302460198703604848) * q^94 + (-46039847553189*b2 - 1395202191289380*b1 + 213475219590665995434) * q^96 + (227359737881344*b2 - 497730817948697216*b1 - 255822414225813915202) * q^97 + (256161749789568*b2 - 234310755788115223*b1 - 252114880665633188588) * q^98 + (-27224812603008*b2 + 31791105454557600*b1 + 98197328824743640464) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 803 q^{2} - 177147 q^{3} + 6715073 q^{4} + 47416347 q^{6} + 1577598316 q^{7} + 1424191017 q^{8} + 10460353203 q^{9}+O(q^{10})$$ 3 * q - 803 * q^2 - 177147 * q^3 + 6715073 * q^4 + 47416347 * q^6 + 1577598316 * q^7 + 1424191017 * q^8 + 10460353203 * q^9 $$3 q - 803 q^{2} - 177147 q^{3} + 6715073 q^{4} + 47416347 q^{6} + 1577598316 q^{7} + 1424191017 q^{8} + 10460353203 q^{9} + 84497282000 q^{11} - 396518345577 q^{12} - 1065489966310 q^{13} + 1543881561348 q^{14} + 9712801855841 q^{16} - 13851876239906 q^{17} - 2799887874003 q^{18} + 26858848298644 q^{19} - 93155602961484 q^{21} + 93991312008688 q^{22} - 75776598293952 q^{23} - 84097055362833 q^{24} + 16\!\cdots\!86 q^{26}+ \cdots + 29\!\cdots\!00 q^{99}+O(q^{100})$$ 3 * q - 803 * q^2 - 177147 * q^3 + 6715073 * q^4 + 47416347 * q^6 + 1577598316 * q^7 + 1424191017 * q^8 + 10460353203 * q^9 + 84497282000 * q^11 - 396518345577 * q^12 - 1065489966310 * q^13 + 1543881561348 * q^14 + 9712801855841 * q^16 - 13851876239906 * q^17 - 2799887874003 * q^18 + 26858848298644 * q^19 - 93155602961484 * q^21 + 93991312008688 * q^22 - 75776598293952 * q^23 - 84097055362833 * q^24 + 1616971789034486 * q^26 - 617673396283947 * q^27 - 1980748081451468 * q^28 - 911172744177122 * q^29 - 14770635893644696 * q^31 - 10845641917892839 * q^32 - 4989480004818000 * q^33 - 14314214801416654 * q^34 + 23414011787976273 * q^36 + 33876604648537146 * q^37 - 187726010316740996 * q^38 + 62916117020639190 * q^39 + 4403226550677934 * q^41 - 91164662316038052 * q^42 + 76568115811359884 * q^43 + 33141017859775792 * q^44 + 260351976204717264 * q^46 - 178877159689418912 * q^47 - 573531236785555209 * q^48 + 973733149877485627 * q^49 + 817939440090209394 * q^51 - 4165878936433955826 * q^52 - 4013622553495571102 * q^53 + 165330579072003147 * q^54 + 9080922748836602580 * q^56 - 1585988133186629556 * q^57 - 12459376778387397982 * q^58 - 4476302860362619024 * q^59 + 13672812736400197042 * q^61 + 10065431538796030152 * q^62 + 5500745199272668716 * q^63 - 17161949036537140719 * q^64 - 5550092982801017712 * q^66 - 10024292550718281716 * q^67 - 17276944334927178886 * q^68 + 4474532352659571648 * q^69 + 31548512648354310016 * q^71 + 4965847022119925817 * q^72 - 80637656440064589382 * q^73 - 97250438023995188522 * q^74 - 165430884440717923764 * q^76 + 107240569572877645632 * q^77 - 95480567170697363814 * q^78 - 101660087769974505880 * q^79 + 36472996377170786403 * q^81 + 608532996859944197042 * q^82 + 98643502167172343676 * q^83 + 116961193461627733932 * q^84 - 181033595702025175180 * q^86 + 53803839370914876978 * q^87 + 397025854664543013552 * q^88 + 49757751372279077814 * q^89 + 156825279702460168744 * q^91 + 88639329782864857776 * q^92 + 872191278883825654104 * q^93 + 2329033824323004369680 * q^94 + 640424309609654250111 * q^96 - 767965200855128324166 * q^97 - 756579208914437470555 * q^98 + 294623804804498082000 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6395796x - 2792983104$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 656\nu - 4263646$$ v^2 - 656*v - 4263646
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 656\beta _1 + 4263646$$ b2 + 656*b1 + 4263646

## 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
 −2272.56 −451.072 2724.63
−2540.56 −59049.0 4.35728e6 0 1.50017e8 −4.55015e8 −5.74199e9 3.48678e9 0
1.2 −719.072 −59049.0 −1.58009e6 0 4.24605e7 1.45023e9 2.64420e9 3.48678e9 0
1.3 2456.63 −59049.0 3.93788e6 0 −1.45062e8 5.82386e8 4.52198e9 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.e 3
5.b even 2 1 15.22.a.d 3
5.c odd 4 2 75.22.b.e 6
15.d odd 2 1 45.22.a.b 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.22.a.d 3 5.b even 2 1
45.22.a.b 3 15.d odd 2 1
75.22.a.e 3 1.a even 1 1 trivial
75.22.b.e 6 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 803T_{2}^{2} - 6180860T_{2} - 4487879424$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(75))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + \cdots - 4487879424$$
$3$ $$(T + 59049)^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} + \cdots + 38\!\cdots\!80$$
$11$ $$T^{3} + \cdots + 25\!\cdots\!36$$
$13$ $$T^{3} + \cdots + 10\!\cdots\!28$$
$17$ $$T^{3} + \cdots - 16\!\cdots\!92$$
$19$ $$T^{3} + \cdots + 39\!\cdots\!00$$
$23$ $$T^{3} + \cdots - 12\!\cdots\!80$$
$29$ $$T^{3} + \cdots - 19\!\cdots\!00$$
$31$ $$T^{3} + \cdots + 10\!\cdots\!00$$
$37$ $$T^{3} + \cdots + 18\!\cdots\!60$$
$41$ $$T^{3} + \cdots - 86\!\cdots\!00$$
$43$ $$T^{3} + \cdots + 19\!\cdots\!04$$
$47$ $$T^{3} + \cdots - 62\!\cdots\!68$$
$53$ $$T^{3} + \cdots - 96\!\cdots\!40$$
$59$ $$T^{3} + \cdots - 67\!\cdots\!00$$
$61$ $$T^{3} + \cdots - 41\!\cdots\!32$$
$67$ $$T^{3} + \cdots + 29\!\cdots\!48$$
$71$ $$T^{3} + \cdots + 14\!\cdots\!52$$
$73$ $$T^{3} + \cdots + 10\!\cdots\!20$$
$79$ $$T^{3} + \cdots - 61\!\cdots\!00$$
$83$ $$T^{3} + \cdots + 10\!\cdots\!92$$
$89$ $$T^{3} + \cdots + 26\!\cdots\!00$$
$97$ $$T^{3} + \cdots - 16\!\cdots\!52$$