# Properties

 Label 75.22.a.h Level $75$ Weight $22$ Character orbit 75.a Self dual yes Analytic conductor $209.608$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 3512166x^{2} + 363520480x + 2321089280000$$ x^4 - x^3 - 3512166*x^2 + 363520480*x + 2321089280000 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{5}\cdot 3^{4}\cdot 5^{2}\cdot 7^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 224) q^{2} + 59049 q^{3} + (\beta_{2} + 294 \beta_1 - 290854) q^{4} + (59049 \beta_1 + 13226976) q^{6} + (11 \beta_{3} + 908 \beta_{2} + \cdots + 58725955) q^{7}+ \cdots + 3486784401 q^{9}+O(q^{10})$$ q + (b1 + 224) * q^2 + 59049 * q^3 + (b2 + 294*b1 - 290854) * q^4 + (59049*b1 + 13226976) * q^6 + (11*b3 + 908*b2 - 325419*b1 + 58725955) * q^7 + (22*b3 - 39*b2 - 1957002*b1 - 18724672) * q^8 + 3486784401 * q^9 $$q + (\beta_1 + 224) q^{2} + 59049 q^{3} + (\beta_{2} + 294 \beta_1 - 290854) q^{4} + (59049 \beta_1 + 13226976) q^{6} + (11 \beta_{3} + 908 \beta_{2} + \cdots + 58725955) q^{7}+ \cdots + (6506339692266 \beta_{3} + \cdots + 27\!\cdots\!22) q^{99}+O(q^{100})$$ q + (b1 + 224) * q^2 + 59049 * q^3 + (b2 + 294*b1 - 290854) * q^4 + (59049*b1 + 13226976) * q^6 + (11*b3 + 908*b2 - 325419*b1 + 58725955) * q^7 + (22*b3 - 39*b2 - 1957002*b1 - 18724672) * q^8 + 3486784401 * q^9 + (1866*b3 + 83816*b2 + 7011318*b1 + 7871225222) * q^11 + (59049*b2 + 17360406*b1 - 17174637846) * q^12 + (18293*b3 - 346764*b2 + 66729643*b1 + 6737720767) * q^13 + (30272*b3 - 165200*b2 + 407342624*b1 - 558428916192) * q^14 + (19734*b3 - 3116001*b2 - 790822022*b1 - 2830973406372) * q^16 + (192599*b3 - 6564036*b2 + 455906569*b1 + 736408091421) * q^17 + (3486784401*b1 + 781039705824) * q^18 + (586223*b3 - 16202852*b2 - 2258407055*b1 - 6067027920693) * q^19 + (649539*b3 + 53616492*b2 - 19215666531*b1 + 3467708916795) * q^21 + (3590528*b3 + 57571488*b2 + 42546848108*b1 + 14065359981632) * q^22 + (-612447*b3 + 197897380*b2 + 92175566591*b1 - 2610725494423) * q^23 + (1299078*b3 - 2302911*b2 - 115559011098*b1 - 1105673156928) * q^24 + (9493440*b3 + 951477584*b2 - 133021807518*b1 + 118721396665760) * q^26 + 205891132094649 * q^27 + (1631520*b3 - 168831376*b2 + 81243860640*b1 + 467096845396608) * q^28 + (-37260652*b3 - 625646384*b2 - 1898891144724*b1 + 1182705744959522) * q^29 + (26406105*b3 - 1748479612*b2 - 1623434513977*b1 - 273325562694319) * q^31 + (-96218342*b3 + 1158469141*b2 - 63358457474*b1 - 1983315759417316) * q^32 + (110185434*b3 + 4949250984*b2 + 414011316582*b1 + 464787978133878) * q^33 + (35863872*b3 + 10741096304*b2 - 1947935158450*b1 + 966188565944544) * q^34 + (3486784401*b2 + 1025114613894*b1 - 1014145190168454) * q^36 + (-55639375*b3 - 29887727132*b2 - 10933054426897*b1 - 1200633118476989) * q^37 + (192241984*b3 + 27789578480*b2 - 12942784187740*b1 - 5323629204537568) * q^38 + (1080183357*b3 - 20476067436*b2 + 3940318689507*b1 + 397855673570583) * q^39 + (-406366070*b3 + 80957187432*b2 + 20760794179510*b1 + 72427728003826084) * q^41 + (1787531328*b3 - 9754894800*b2 + 24053174604576*b1 - 32974669072221408) * q^42 + (131528276*b3 + 112886973904*b2 - 34884316682196*b1 - 112764228896582544) * q^43 + (714021312*b3 - 1406875444*b2 + 25554346092104*b1 + 61352221988362104) * q^44 + (3780491968*b3 + 520505360*b2 + 85134052179328*b1 + 161265127697573472) * q^46 + (-4433365233*b3 + 69373023772*b2 + 182144480961105*b1 - 203516376578266145) * q^47 + (1165272966*b3 - 183996743049*b2 - 46697249577078*b1 - 167166148672860228) * q^48 + (-17028362736*b3 + 286232554048*b2 - 77388435911696*b1 + 386314611175395945) * q^49 + (11372778351*b3 - 387599761764*b2 + 26920826992881*b1 + 43484161390318629) * q^51 + (-8544834848*b3 + 676580605458*b2 + 358881710079660*b1 - 221250881635780044) * q^52 + (2662901670*b3 + 417497193816*b2 + 688343826113050*b1 - 174491566098976528) * q^53 + (205891132094649*b1 + 46119613589201376) * q^54 + (-65672172896*b3 + 552524529808*b2 - 450956148670624*b1 + 1418431863299712832) * q^56 + (34615881927*b3 - 956762207748*b2 - 133356678190695*b1 - 358251931689000957) * q^57 + (-48640190720*b3 - 3257473097408*b2 + 797301573918702*b1 - 3069664811850813504) * q^58 + (114174962638*b3 + 446448193528*b2 + 627652253829106*b1 + 1655565323647249994) * q^59 + (-46220774006*b3 - 8613861731992*b2 + 782225835745974*b1 + 1847524106133744840) * q^61 + (-13750437184*b3 + 69265130384*b2 - 1107630262439312*b1 - 2911996609261268128) * q^62 + (38354628411*b3 + 3166000236108*b2 - 1134665892989019*b1 + 204764743827827955) * q^63 + (-105959244578*b3 + 2039329111599*b2 + 157286708954778*b1 + 5381386908866118468) * q^64 + (212017087872*b3 + 3399538794912*b2 + 2512348833929292*b1 + 830545441555387968) * q^66 + (49933885584*b3 + 1487324599872*b2 - 11173717991431312*b1 + 6049840901189733572) * q^67 + (-134036675168*b3 + 9749244407006*b2 + 4277994077468916*b1 - 4749962761563922772) * q^68 + (-36164382903*b3 + 11685642391620*b2 + 5442875031631959*b1 - 154160729720183727) * q^69 + (216435933208*b3 + 16701469383520*b2 - 17989364733802776*b1 - 9343082988558115056) * q^71 + (76709256822*b3 - 135984591639*b2 - 6823644046325802*b1 - 65288894243441472) * q^72 + (280208264930*b3 + 8085803378120*b2 - 8400701500746658*b1 + 9315520352840265320) * q^73 + (-709608451904*b3 - 3320243928816*b2 - 14226168723565622*b1 - 19465359921343002848) * q^74 + (-438089513312*b3 + 19867481809076*b2 + 9889931120603320*b1 - 11201335413630501400) * q^76 + (-946736025532*b3 + 9427665355152*b2 + 28284749730452028*b1 + 78804216184382676196) * q^77 + (560578138560*b3 + 56183799857616*b2 - 7854804712130382*b1 + 7010379751716462240) * q^78 + (1889866705991*b3 + 22366462515452*b2 + 24237494373585689*b1 - 27443877015009336657) * q^79 + 12157665459056928801 * q^81 + (1400699481984*b3 - 23286861577056*b2 + 107154635899916362*b1 + 52673558608397945984) * q^82 + (2040266248020*b3 - 29749749737520*b2 + 85676302259279148*b1 + 37234470999090804552) * q^83 + (96339624480*b3 - 9969323921424*b2 + 4797368726931360*b1 + 27581601623824305792) * q^84 + (2606623892224*b3 - 66944520401600*b2 - 68889781183265156*b1 - 86532940717233491968) * q^86 + (-2200204239948*b3 - 36943793328816*b2 - 112127623204807476*b1 + 69837591534114814578) * q^87 + (-6892510287992*b3 - 64686587353684*b2 - 26745968476423544*b1 + 29121944910995290752) * q^88 + (-2577327858138*b3 - 74274018438312*b2 + 27461469126211482*b1 + 139618675996364292048) * q^89 + (-6203586142842*b3 - 208715307631912*b2 + 479874928838142522*b1 + 7469999634971521398) * q^91 + (4834386050912*b3 - 171079133637008*b2 - 26304163480354144*b1 + 191101761098509056192) * q^92 + (1559254094145*b3 - 103245972608988*b2 - 95862184615827873*b1 - 16139601151536842631) * q^93 + (-2623423335104*b3 - 27393044098320*b2 - 161782462689014888*b1 + 274275408981281627296) * q^94 + (-5681596876758*b3 + 68406444306909*b2 - 3741253555382226*b1 - 117112812277833092484) * q^96 + (-6938741707524*b3 + 533330659339120*b2 - 280914628682512252*b1 + 59131594661689334106) * q^97 + (-9641431331840*b3 - 888797614546688*b2 + 500338573434486745*b1 - 49389900335565151264) * q^98 + (6506339692266*b3 + 292248321354216*b2 + 24446954232850518*b1 + 27445265320827362022) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 897 q^{2} + 236196 q^{3} - 1163123 q^{4} + 52966953 q^{6} + 234577504 q^{7} - 76855629 q^{8} + 13947137604 q^{9}+O(q^{10})$$ 4 * q + 897 * q^2 + 236196 * q^3 - 1163123 * q^4 + 52966953 * q^6 + 234577504 * q^7 - 76855629 * q^8 + 13947137604 * q^9 $$4 q + 897 q^{2} + 236196 q^{3} - 1163123 q^{4} + 52966953 q^{6} + 234577504 q^{7} - 76855629 q^{8} + 13947137604 q^{9} + 31491830256 q^{11} - 68681250027 q^{12} + 27017977768 q^{13} - 2233308126672 q^{14} - 11324681311775 q^{16} + 2946095028888 q^{17} + 3127645607697 q^{18} - 24270353300752 q^{19} + 13851567033696 q^{21} + 56303932793676 q^{22} - 10350924920928 q^{23} - 4538248036821 q^{24} + 474751622871378 q^{26} + 823564528378596 q^{27} + 18\!\cdots\!68 q^{28}+ \cdots + 10\!\cdots\!56 q^{99}+O(q^{100})$$ 4 * q + 897 * q^2 + 236196 * q^3 - 1163123 * q^4 + 52966953 * q^6 + 234577504 * q^7 - 76855629 * q^8 + 13947137604 * q^9 + 31491830256 * q^11 - 68681250027 * q^12 + 27017977768 * q^13 - 2233308126672 * q^14 - 11324681311775 * q^16 + 2946095028888 * q^17 + 3127645607697 * q^18 - 24270353300752 * q^19 + 13851567033696 * q^21 + 56303932793676 * q^22 - 10350924920928 * q^23 - 4538248036821 * q^24 + 474751622871378 * q^26 + 823564528378596 * q^27 + 1868468795909968 * q^28 + 4728924677079096 * q^29 - 1094923910405536 * q^31 - 7933327650814221 * q^32 + 1859561084786544 * q^33 + 3862795623387294 * q^34 - 4055559132844323 * q^36 - 4813435696247096 * q^37 - 21307487199674508 * q^38 + 1595384569222632 * q^39 + 289731591445930344 * q^41 - 131874611571854928 * q^42 - 451091912658458000 * q^43 + 245434444420437276 * q^44 + 645145648102459824 * q^46 - 813883435638492480 * q^47 - 668711106779001975 * q^48 + 1545180753004755300 * q^49 + 173963965360807512 * q^51 - 884645329958480822 * q^52 - 697278335404085208 * q^53 + 184684345488900153 * q^54 + 5673275878853478000 * q^56 - 1433140092056104848 * q^57 - 12277858736996428626 * q^58 + 6622888614569598192 * q^59 + 7390887218011683320 * q^61 - 11649094150323079392 * q^62 + 817921181772715104 * q^63 + 21525702776885072473 * q^64 + 3324690927533774124 * q^66 + 24188188449376788688 * q^67 - 18995582935459304346 * q^68 - 611211765655877472 * q^69 - 37390337803999713312 * q^71 - 267979008326243229 * q^72 + 37253672904265201432 * q^73 - 77875663243460100102 * q^74 - 44795472028972724668 * q^76 + 315245139112859776128 * q^77 + 28033608578931999522 * q^78 - 109751291042259570400 * q^79 + 48630661836227715204 * q^81 + 210801413757052759338 * q^82 + 149023592088638482896 * q^83 + 110331213929687700432 * q^84 - 346200583098972939204 * q^86 + 279238273256843539704 * q^87 + 116461091469581805156 * q^88 + 558502237151273959848 * q^89 + 30360075980445717184 * q^91 + 764380916144075558544 * q^92 - 64654161985536495264 * q^93 + 1096939878232058257512 * q^94 - 468455064452928935829 * q^96 + 236244923748673777528 * q^97 - 197058383612642903463 * q^98 + 109805222495560636656 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 3512166x^{2} + 363520480x + 2321089280000$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 154\nu - 1756122$$ v^2 + 154*v - 1756122 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 711\nu^{2} - 2080768\nu - 978048758 ) / 22$$ (v^3 + 711*v^2 - 2080768*v - 978048758) / 22
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 154\beta _1 + 1756122$$ b2 - 154*b1 + 1756122 $$\nu^{3}$$ $$=$$ $$22\beta_{3} - 711\beta_{2} + 2190262\beta _1 - 270553984$$ 22*b3 - 711*b2 + 2190262*b1 - 270553984

## 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
 −1711.97 −849.272 1069.36 1492.88
−1487.97 59049.0 116903. 0 −8.78631e7 1.26833e9 2.94655e9 3.48678e9 0
1.2 −625.272 59049.0 −1.70619e6 0 −3.69217e7 −3.78633e8 2.37812e9 3.48678e9 0
1.3 1293.36 59049.0 −424379. 0 7.63714e7 −1.27960e9 −3.26124e9 3.48678e9 0
1.4 1716.88 59049.0 850541. 0 1.01380e8 6.24477e8 −2.14029e9 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.h 4
5.b even 2 1 15.22.a.e 4
5.c odd 4 2 75.22.b.h 8
15.d odd 2 1 45.22.a.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.22.a.e 4 5.b even 2 1
45.22.a.g 4 15.d odd 2 1
75.22.a.h 4 1.a even 1 1 trivial
75.22.b.h 8 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 897T_{2}^{3} - 3210438T_{2}^{2} + 1891862624T_{2} + 2065963121664$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(75))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + \cdots + 2065963121664$$
$3$ $$(T - 59049)^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + \cdots + 38\!\cdots\!00$$
$11$ $$T^{4} + \cdots + 28\!\cdots\!64$$
$13$ $$T^{4} + \cdots + 26\!\cdots\!84$$
$17$ $$T^{4} + \cdots + 31\!\cdots\!56$$
$19$ $$T^{4} + \cdots + 34\!\cdots\!00$$
$23$ $$T^{4} + \cdots - 97\!\cdots\!00$$
$29$ $$T^{4} + \cdots - 51\!\cdots\!00$$
$31$ $$T^{4} + \cdots + 15\!\cdots\!00$$
$37$ $$T^{4} + \cdots + 50\!\cdots\!00$$
$41$ $$T^{4} + \cdots - 70\!\cdots\!00$$
$43$ $$T^{4} + \cdots - 16\!\cdots\!04$$
$47$ $$T^{4} + \cdots - 21\!\cdots\!44$$
$53$ $$T^{4} + \cdots + 34\!\cdots\!00$$
$59$ $$T^{4} + \cdots + 97\!\cdots\!00$$
$61$ $$T^{4} + \cdots + 27\!\cdots\!76$$
$67$ $$T^{4} + \cdots + 28\!\cdots\!56$$
$71$ $$T^{4} + \cdots - 17\!\cdots\!44$$
$73$ $$T^{4} + \cdots - 99\!\cdots\!00$$
$79$ $$T^{4} + \cdots - 12\!\cdots\!00$$
$83$ $$T^{4} + \cdots - 79\!\cdots\!56$$
$89$ $$T^{4} + \cdots + 85\!\cdots\!00$$
$97$ $$T^{4} + \cdots - 16\!\cdots\!24$$