Properties

 Label 75.22.a.c Level $75$ Weight $22$ Character orbit 75.a Self dual yes Analytic conductor $209.608$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + 2844 q^{2} + 59049 q^{3} + 5991184 q^{4} + 167935356 q^{6} - 363303920 q^{7} + 11074627008 q^{8} + 3486784401 q^{9}+O(q^{10})$$ q + 2844 * q^2 + 59049 * q^3 + 5991184 * q^4 + 167935356 * q^6 - 363303920 * q^7 + 11074627008 * q^8 + 3486784401 * q^9 $$q + 2844 q^{2} + 59049 q^{3} + 5991184 q^{4} + 167935356 q^{6} - 363303920 q^{7} + 11074627008 q^{8} + 3486784401 q^{9} + 14581833156 q^{11} + 353773424016 q^{12} - 113350790702 q^{13} - 1033236348480 q^{14} + 18931815702784 q^{16} + 8589389597982 q^{17} + 9916414836444 q^{18} - 29202939273796 q^{19} - 21452733172080 q^{21} + 41470733495664 q^{22} + 155899214954280 q^{23} + 653945650195392 q^{24} - 322369648756488 q^{26} + 205891132094649 q^{27} - 21\!\cdots\!80 q^{28}+ \cdots + 50\!\cdots\!56 q^{99}+O(q^{100})$$ q + 2844 * q^2 + 59049 * q^3 + 5991184 * q^4 + 167935356 * q^6 - 363303920 * q^7 + 11074627008 * q^8 + 3486784401 * q^9 + 14581833156 * q^11 + 353773424016 * q^12 - 113350790702 * q^13 - 1033236348480 * q^14 + 18931815702784 * q^16 + 8589389597982 * q^17 + 9916414836444 * q^18 - 29202939273796 * q^19 - 21452733172080 * q^21 + 41470733495664 * q^22 + 155899214954280 * q^23 + 653945650195392 * q^24 - 322369648756488 * q^26 + 205891132094649 * q^27 - 2176620632641280 * q^28 + 2400788707090758 * q^29 + 2239820676947000 * q^31 + 30616907679636480 * q^32 + 861042666028644 * q^33 + 24428224016660808 * q^34 + 20889966914720784 * q^36 + 30785069383298890 * q^37 - 83053159294675824 * q^38 - 6693250840162398 * q^39 - 103207571041281030 * q^41 - 61011573141395520 * q^42 + 165557270617488124 * q^43 + 87362445494896704 * q^44 + 443377367329972320 * q^46 + 66587216226477408 * q^47 + 1117904785433692416 * q^48 - 426556125795917607 * q^49 + 507194866371239118 * q^51 - 679105443641171168 * q^52 - 435422766592881630 * q^53 + 585554379677181756 * q^54 - 4023455404544271360 * q^56 - 1724404361178380004 * q^57 + 6827843082966115752 * q^58 + 5534365798259081316 * q^59 - 7176205164722961202 * q^61 + 6370050005237268000 * q^62 - 1266762441078151920 * q^63 + 47371590276161277952 * q^64 + 2448805342185463536 * q^66 + 15755449453068299812 * q^67 + 51460613529196190688 * q^68 + 9205692743835279720 * q^69 + 26457854874259376232 * q^71 + 38614836698387702208 * q^72 - 13471249335464801450 * q^73 + 87552737326102043160 * q^74 - 174960182530138214464 * q^76 - 5297637146360771520 * q^77 - 19035605389421859912 * q^78 - 16886125085525986840 * q^79 + 12157665459056928801 * q^81 - 293522332041403249320 * q^82 + 170687980457962587972 * q^83 - 128527271736834942720 * q^84 + 470844877636136224656 * q^86 + 141764172365002169142 * q^87 + 161488363295587477248 * q^88 - 312592486939587043686 * q^89 + 41180786597136151840 * q^91 + 934020882246643067520 * q^92 + 132259171153043403000 * q^93 + 189374042948101748352 * q^94 + 1807897781574854507520 * q^96 - 949014545286007636418 * q^97 - 1213125621763589674308 * q^98 + 50843708386325399556 * 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
 0
2844.00 59049.0 5.99118e6 0 1.67935e8 −3.63304e8 1.10746e10 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.c 1
5.b even 2 1 3.22.a.a 1
5.c odd 4 2 75.22.b.a 2
15.d odd 2 1 9.22.a.d 1
20.d odd 2 1 48.22.a.e 1

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 2844$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(75))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2844$$
$3$ $$T - 59049$$
$5$ $$T$$
$7$ $$T + 363303920$$
$11$ $$T - 14581833156$$
$13$ $$T + 113350790702$$
$17$ $$T - 8589389597982$$
$19$ $$T + 29202939273796$$
$23$ $$T - 155899214954280$$
$29$ $$T - 2400788707090758$$
$31$ $$T - 2239820676947000$$
$37$ $$T - 30\!\cdots\!90$$
$41$ $$T + 10\!\cdots\!30$$
$43$ $$T - 16\!\cdots\!24$$
$47$ $$T - 66\!\cdots\!08$$
$53$ $$T + 43\!\cdots\!30$$
$59$ $$T - 55\!\cdots\!16$$
$61$ $$T + 71\!\cdots\!02$$
$67$ $$T - 15\!\cdots\!12$$
$71$ $$T - 26\!\cdots\!32$$
$73$ $$T + 13\!\cdots\!50$$
$79$ $$T + 16\!\cdots\!40$$
$83$ $$T - 17\!\cdots\!72$$
$89$ $$T + 31\!\cdots\!86$$
$97$ $$T + 94\!\cdots\!18$$