Properties

 Label 3.22.a.a Level $3$ Weight $22$ Character orbit 3.a Self dual yes Analytic conductor $8.384$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 3.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$8.38432032861$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - 2844 q^{2} - 59049 q^{3} + 5991184 q^{4} + 3109950 q^{5} + 167935356 q^{6} + 363303920 q^{7} - 11074627008 q^{8} + 3486784401 q^{9}+O(q^{10})$$ q - 2844 * q^2 - 59049 * q^3 + 5991184 * q^4 + 3109950 * q^5 + 167935356 * q^6 + 363303920 * q^7 - 11074627008 * q^8 + 3486784401 * q^9 $$q - 2844 q^{2} - 59049 q^{3} + 5991184 q^{4} + 3109950 q^{5} + 167935356 q^{6} + 363303920 q^{7} - 11074627008 q^{8} + 3486784401 q^{9} - 8844697800 q^{10} + 14581833156 q^{11} - 353773424016 q^{12} + 113350790702 q^{13} - 1033236348480 q^{14} - 183639437550 q^{15} + 18931815702784 q^{16} - 8589389597982 q^{17} - 9916414836444 q^{18} - 29202939273796 q^{19} + 18632282680800 q^{20} - 21452733172080 q^{21} - 41470733495664 q^{22} - 155899214954280 q^{23} + 653945650195392 q^{24} - 467165369200625 q^{25} - 322369648756488 q^{26} - 205891132094649 q^{27} + 21\!\cdots\!80 q^{28}+ \cdots + 50\!\cdots\!56 q^{99}+O(q^{100})$$ q - 2844 * q^2 - 59049 * q^3 + 5991184 * q^4 + 3109950 * q^5 + 167935356 * q^6 + 363303920 * q^7 - 11074627008 * q^8 + 3486784401 * q^9 - 8844697800 * q^10 + 14581833156 * q^11 - 353773424016 * q^12 + 113350790702 * q^13 - 1033236348480 * q^14 - 183639437550 * q^15 + 18931815702784 * q^16 - 8589389597982 * q^17 - 9916414836444 * q^18 - 29202939273796 * q^19 + 18632282680800 * q^20 - 21452733172080 * q^21 - 41470733495664 * q^22 - 155899214954280 * q^23 + 653945650195392 * q^24 - 467165369200625 * q^25 - 322369648756488 * q^26 - 205891132094649 * q^27 + 2176620632641280 * q^28 + 2400788707090758 * q^29 + 522270560392200 * q^30 + 2239820676947000 * q^31 - 30616907679636480 * q^32 - 861042666028644 * q^33 + 24428224016660808 * q^34 + 1129857026004000 * q^35 + 20889966914720784 * q^36 - 30785069383298890 * q^37 + 83053159294675824 * q^38 - 6693250840162398 * q^39 - 34441536263529600 * q^40 - 103207571041281030 * q^41 + 61011573141395520 * q^42 - 165557270617488124 * q^43 + 87362445494896704 * q^44 + 10843725147889950 * q^45 + 443377367329972320 * q^46 - 66587216226477408 * q^47 - 1117904785433692416 * q^48 - 426556125795917607 * q^49 + 1328618310006577500 * q^50 + 507194866371239118 * q^51 + 679105443641171168 * q^52 + 435422766592881630 * q^53 + 585554379677181756 * q^54 + 45348772023502200 * q^55 - 4023455404544271360 * q^56 + 1724404361178380004 * q^57 - 6827843082966115752 * q^58 + 5534365798259081316 * q^59 - 1100217660018559200 * q^60 - 7176205164722961202 * q^61 - 6370050005237268000 * q^62 + 1266762441078151920 * q^63 + 47371590276161277952 * q^64 + 352515291543684900 * q^65 + 2448805342185463536 * q^66 - 15755449453068299812 * q^67 - 51460613529196190688 * q^68 + 9205692743835279720 * q^69 - 3213313381955376000 * q^70 + 26457854874259376232 * q^71 - 38614836698387702208 * q^72 + 13471249335464801450 * q^73 + 87552737326102043160 * q^74 + 27585647885927705625 * q^75 - 174960182530138214464 * q^76 + 5297637146360771520 * q^77 + 19035605389421859912 * q^78 - 16886125085525986840 * q^79 + 58877000244873100800 * q^80 + 12157665459056928801 * q^81 + 293522332041403249320 * q^82 - 170687980457962587972 * q^83 - 128527271736834942720 * q^84 - 26712572180244120900 * q^85 + 470844877636136224656 * q^86 - 141764172365002169142 * q^87 - 161488363295587477248 * q^88 - 312592486939587043686 * q^89 - 30839554320599017800 * q^90 + 41180786597136151840 * q^91 - 934020882246643067520 * q^92 - 132259171153043403000 * q^93 + 189374042948101748352 * q^94 - 90819680994541870200 * q^95 + 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.

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 3.10995e6 1.67935e8 3.63304e8 −1.10746e10 3.48678e9 −8.84470e9
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$

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 3.22.a.a 1
3.b odd 2 1 9.22.a.d 1
4.b odd 2 1 48.22.a.e 1
5.b even 2 1 75.22.a.c 1
5.c odd 4 2 75.22.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.22.a.a 1 1.a even 1 1 trivial
9.22.a.d 1 3.b odd 2 1
48.22.a.e 1 4.b odd 2 1
75.22.a.c 1 5.b even 2 1
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 $$S_{22}^{\mathrm{new}}(\Gamma_0(3))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2844$$
$3$ $$T + 59049$$
$5$ $$T - 3109950$$
$7$ $$T - 363303920$$
$11$ $$T - 14581833156$$
$13$ $$T - 113350790702$$
$17$ $$T + 8589389597982$$
$19$ $$T + 29202939273796$$
$23$ $$T + 155899214954280$$
$29$ $$T - 2400788707090758$$
$31$ $$T - 2239820676947000$$
$37$ $$T + 30\!\cdots\!90$$
$41$ $$T + 10\!\cdots\!30$$
$43$ $$T + 16\!\cdots\!24$$
$47$ $$T + 66\!\cdots\!08$$
$53$ $$T - 43\!\cdots\!30$$
$59$ $$T - 55\!\cdots\!16$$
$61$ $$T + 71\!\cdots\!02$$
$67$ $$T + 15\!\cdots\!12$$
$71$ $$T - 26\!\cdots\!32$$
$73$ $$T - 13\!\cdots\!50$$
$79$ $$T + 16\!\cdots\!40$$
$83$ $$T + 17\!\cdots\!72$$
$89$ $$T + 31\!\cdots\!86$$
$97$ $$T - 94\!\cdots\!18$$