LMFDB - Embedded newform 729.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,2,Mod(1,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("729.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	729.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:	$5.82109430735$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.1397493.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 $ x^6 - 3x^5 - 3x^4 + 10x^3 + 3x^2 - 6*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.05432$ of defining polynomial
Character	$\chi$	$=$	729.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.40162 q^{2} +3.76778 q^{4} +0.0930834 q^{5} -0.579861 q^{7} -4.24555 q^{8} +O(q^{10})$	\(q-2.40162 q^{2} +3.76778 q^{4} +0.0930834 q^{5} -0.579861 q^{7} -4.24555 q^{8} -0.223551 q^{10} -3.09308 q^{11} +4.20173 q^{13} +1.39261 q^{14} +2.66063 q^{16} -1.99099 q^{17} -3.84542 q^{19} +0.350718 q^{20} +7.42841 q^{22} -4.45282 q^{23} -4.99134 q^{25} -10.0910 q^{26} -2.18479 q^{28} +6.39951 q^{29} +1.65750 q^{31} +2.10127 q^{32} +4.78159 q^{34} -0.0539755 q^{35} +4.03009 q^{37} +9.23525 q^{38} -0.395190 q^{40} -1.09616 q^{41} -6.90112 q^{43} -11.6541 q^{44} +10.6940 q^{46} +3.59319 q^{47} -6.66376 q^{49} +11.9873 q^{50} +15.8312 q^{52} +5.40034 q^{53} -0.287915 q^{55} +2.46183 q^{56} -15.3692 q^{58} -10.2847 q^{59} -13.1963 q^{61} -3.98069 q^{62} -10.3677 q^{64} +0.391112 q^{65} -8.83729 q^{67} -7.50161 q^{68} +0.129629 q^{70} +1.14495 q^{71} +0.195472 q^{73} -9.67876 q^{74} -14.4887 q^{76} +1.79356 q^{77} -7.20799 q^{79} +0.247661 q^{80} +2.63255 q^{82} -14.9004 q^{83} -0.185328 q^{85} +16.5739 q^{86} +13.1318 q^{88} -1.55313 q^{89} -2.43642 q^{91} -16.7772 q^{92} -8.62949 q^{94} -0.357945 q^{95} +5.29553 q^{97} +16.0038 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8}+O(q^{10}) $ 6 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 - 6 * q^8	$ 6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8} + 3 q^{10} - 12 q^{11} - 6 q^{14} - 3 q^{16} - 9 q^{17} + 3 q^{19} - 6 q^{20} + 6 q^{22} - 15 q^{23} - 6 q^{25} - 15 q^{26} - 6 q^{28} - 12 q^{29} - 12 q^{35} + 3 q^{37} + 3 q^{38} + 6 q^{40} - 15 q^{41} - 3 q^{44} + 3 q^{46} - 21 q^{47} - 12 q^{49} - 3 q^{50} + 12 q^{52} - 9 q^{53} - 6 q^{55} + 6 q^{56} - 12 q^{58} - 24 q^{59} - 9 q^{61} + 12 q^{62} - 12 q^{64} + 6 q^{65} - 9 q^{67} + 9 q^{68} + 15 q^{70} - 27 q^{71} - 6 q^{73} + 12 q^{74} + 6 q^{76} + 12 q^{77} + 21 q^{80} - 6 q^{82} - 12 q^{83} + 21 q^{86} + 12 q^{88} - 9 q^{89} - 6 q^{91} - 6 q^{92} + 6 q^{94} - 12 q^{95} + 45 q^{98}+O(q^{100}) $ 6 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 - 6 * q^8 + 3 * q^10 - 12 * q^11 - 6 * q^14 - 3 * q^16 - 9 * q^17 + 3 * q^19 - 6 * q^20 + 6 * q^22 - 15 * q^23 - 6 * q^25 - 15 * q^26 - 6 * q^28 - 12 * q^29 - 12 * q^35 + 3 * q^37 + 3 * q^38 + 6 * q^40 - 15 * q^41 - 3 * q^44 + 3 * q^46 - 21 * q^47 - 12 * q^49 - 3 * q^50 + 12 * q^52 - 9 * q^53 - 6 * q^55 + 6 * q^56 - 12 * q^58 - 24 * q^59 - 9 * q^61 + 12 * q^62 - 12 * q^64 + 6 * q^65 - 9 * q^67 + 9 * q^68 + 15 * q^70 - 27 * q^71 - 6 * q^73 + 12 * q^74 + 6 * q^76 + 12 * q^77 + 21 * q^80 - 6 * q^82 - 12 * q^83 + 21 * q^86 + 12 * q^88 - 9 * q^89 - 6 * q^91 - 6 * q^92 + 6 * q^94 - 12 * q^95 + 45 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.40162	−1.69820	−0.849101	−	0.528230i	$-0.822856\pi$
$2$	−2.40162	−1.69820	−0.849101	+	0.528230i	$0.822856\pi$
$3$	0	0
$3$	0	0
$4$	3.76778	1.88389
$5$	0.0930834	0.0416282	0.0208141	−	0.999783i	$-0.493374\pi$
$5$	0.0930834	0.0416282	0.0208141	+	0.999783i	$0.493374\pi$
$6$	0	0
$7$	−0.579861	−0.219167	−0.109583	−	0.993978i	$-0.534952\pi$
$7$	−0.579861	−0.219167	−0.109583	+	0.993978i	$0.534952\pi$
$8$	−4.24555	−1.50103
$9$	0	0
$10$	−0.223551	−0.0706931
$11$	−3.09308	−0.932600	−0.466300	−	0.884627i	$-0.654413\pi$
$11$	−3.09308	−0.932600	−0.466300	+	0.884627i	$0.654413\pi$
$12$	0	0
$13$	4.20173	1.16535	0.582676	−	0.812705i	$-0.302006\pi$
$13$	4.20173	1.16535	0.582676	+	0.812705i	$0.302006\pi$
$14$	1.39261	0.372190
$15$	0	0
$16$	2.66063	0.665158
$17$	−1.99099	−0.482885	−0.241443	−	0.970415i	$-0.577621\pi$
$17$	−1.99099	−0.482885	−0.241443	+	0.970415i	$0.577621\pi$
$18$	0	0
$19$	−3.84542	−0.882201	−0.441100	−	0.897458i	$-0.645412\pi$
$19$	−3.84542	−0.882201	−0.441100	+	0.897458i	$0.645412\pi$
$20$	0.350718	0.0784230
$21$	0	0
$22$	7.42841	1.58374
$23$	−4.45282	−0.928476	−0.464238	−	0.885710i	$-0.653672\pi$
$23$	−4.45282	−0.928476	−0.464238	+	0.885710i	$0.653672\pi$
$24$	0	0
$25$	−4.99134	−0.998267
$26$	−10.0910	−1.97900
$27$	0	0
$28$	−2.18479	−0.412887
$29$	6.39951	1.18836	0.594179	−	0.804332i	$-0.297477\pi$
$29$	6.39951	1.18836	0.594179	+	0.804332i	$0.297477\pi$
$30$	0	0
$31$	1.65750	0.297696	0.148848	−	0.988860i	$-0.452444\pi$
$31$	1.65750	0.297696	0.148848	+	0.988860i	$0.452444\pi$
$32$	2.10127	0.371456
$33$	0	0
$34$	4.78159	0.820037
$35$	−0.0539755	−0.00912352
$36$	0	0
$37$	4.03009	0.662543	0.331272	−	0.943535i	$-0.392522\pi$
$37$	4.03009	0.662543	0.331272	+	0.943535i	$0.392522\pi$
$38$	9.23525	1.49816
$39$	0	0
$40$	−0.395190	−0.0624850
$41$	−1.09616	−0.171191	−0.0855954	−	0.996330i	$-0.527279\pi$
$41$	−1.09616	−0.171191	−0.0855954	+	0.996330i	$0.527279\pi$
$42$	0	0
$43$	−6.90112	−1.05241	−0.526206	−	0.850357i	$-0.676386\pi$
$43$	−6.90112	−1.05241	−0.526206	+	0.850357i	$0.676386\pi$
$44$	−11.6541	−1.75692
$45$	0	0
$46$	10.6940	1.57674
$47$	3.59319	0.524121	0.262061	−	0.965051i	$-0.415598\pi$
$47$	3.59319	0.524121	0.262061	+	0.965051i	$0.415598\pi$
$48$	0	0
$49$	−6.66376	−0.951966
$50$	11.9873	1.69526
$51$	0	0
$52$	15.8312	2.19540
$53$	5.40034	0.741793	0.370897	−	0.928674i	$-0.379050\pi$
$53$	5.40034	0.741793	0.370897	+	0.928674i	$0.379050\pi$
$54$	0	0
$55$	−0.287915	−0.0388224
$56$	2.46183	0.328976
$57$	0	0
$58$	−15.3692	−2.01807
$59$	−10.2847	−1.33895	−0.669474	−	0.742835i	$-0.733480\pi$
$59$	−10.2847	−1.33895	−0.669474	+	0.742835i	$0.733480\pi$
$60$	0	0
$61$	−13.1963	−1.68962	−0.844808	−	0.535070i	$-0.820285\pi$
$61$	−13.1963	−1.68962	−0.844808	+	0.535070i	$0.820285\pi$
$62$	−3.98069	−0.505548
$63$	0	0
$64$	−10.3677	−1.29596
$65$	0.391112	0.0485114
$66$	0	0
$67$	−8.83729	−1.07965	−0.539824	−	0.841778i	$-0.681509\pi$
$67$	−8.83729	−1.07965	−0.539824	+	0.841778i	$0.681509\pi$
$68$	−7.50161	−0.909703
$69$	0	0
$70$	0.129629	0.0154936
$71$	1.14495	0.135880	0.0679401	−	0.997689i	$-0.478357\pi$
$71$	1.14495	0.135880	0.0679401	+	0.997689i	$0.478357\pi$
$72$	0	0
$73$	0.195472	0.0228783	0.0114391	−	0.999935i	$-0.496359\pi$
$73$	0.195472	0.0228783	0.0114391	+	0.999935i	$0.496359\pi$
$74$	−9.67876	−1.12513
$75$	0	0
$76$	−14.4887	−1.66197
$77$	1.79356	0.204395
$78$	0	0
$79$	−7.20799	−0.810963	−0.405481	−	0.914103i	$-0.632896\pi$
$79$	−7.20799	−0.810963	−0.405481	+	0.914103i	$0.632896\pi$
$80$	0.247661	0.0276893
$81$	0	0
$82$	2.63255	0.290717
$83$	−14.9004	−1.63554	−0.817768	−	0.575548i	$-0.804789\pi$
$83$	−14.9004	−1.63554	−0.817768	+	0.575548i	$0.804789\pi$
$84$	0	0
$85$	−0.185328	−0.0201016
$86$	16.5739	1.78721
$87$	0	0
$88$	13.1318	1.39986
$89$	−1.55313	−0.164631	−0.0823155	−	0.996606i	$-0.526232\pi$
$89$	−1.55313	−0.164631	−0.0823155	+	0.996606i	$0.526232\pi$
$90$	0	0
$91$	−2.43642	−0.255406
$92$	−16.7772	−1.74915
$93$	0	0
$94$	−8.62949	−0.890064
$95$	−0.357945	−0.0367244
$96$	0	0
$97$	5.29553	0.537680	0.268840	−	0.963185i	$-0.413360\pi$
$97$	5.29553	0.537680	0.268840	+	0.963185i	$0.413360\pi$
$98$	16.0038	1.61663
$99$	0	0
$100$	−18.8063	−1.88063
$101$	−7.26898	−0.723291	−0.361645	−	0.932316i	$-0.617785\pi$
$101$	−7.26898	−0.723291	−0.361645	+	0.932316i	$0.617785\pi$
$102$	0	0
$103$	6.40137	0.630746	0.315373	−	0.948968i	$-0.397870\pi$
$103$	6.40137	0.630746	0.315373	+	0.948968i	$0.397870\pi$
$104$	−17.8387	−1.74922
$105$	0	0
$106$	−12.9696	−1.25972
$107$	−5.54365	−0.535925	−0.267963	−	0.963429i	$-0.586350\pi$
$107$	−5.54365	−0.535925	−0.267963	+	0.963429i	$0.586350\pi$
$108$	0	0
$109$	−6.23137	−0.596857	−0.298428	−	0.954432i	$-0.596462\pi$
$109$	−6.23137	−0.596857	−0.298428	+	0.954432i	$0.596462\pi$
$110$	0.691462	0.0659283
$111$	0	0
$112$	−1.54280	−0.145781
$113$	−11.8440	−1.11419	−0.557094	−	0.830449i	$-0.688084\pi$
$113$	−11.8440	−1.11419	−0.557094	+	0.830449i	$0.688084\pi$
$114$	0	0
$115$	−0.414483	−0.0386508
$116$	24.1120	2.23874
$117$	0	0
$118$	24.6998	2.27381
$119$	1.15450	0.105832
$120$	0	0
$121$	−1.43283	−0.130258
$122$	31.6926	2.86931
$123$	0	0
$124$	6.24510	0.560827
$125$	−0.930028	−0.0831842
$126$	0	0
$127$	11.5294	1.02307	0.511533	−	0.859263i	$-0.329078\pi$
$127$	11.5294	1.02307	0.511533	+	0.859263i	$0.329078\pi$
$128$	20.6968	1.82935
$129$	0	0
$130$	−0.939302	−0.0823822
$131$	−9.01177	−0.787362	−0.393681	−	0.919247i	$-0.628799\pi$
$131$	−9.01177	−0.787362	−0.393681	+	0.919247i	$0.628799\pi$
$132$	0	0
$133$	2.22981	0.193349
$134$	21.2238	1.83346
$135$	0	0
$136$	8.45283	0.724824
$137$	11.5202	0.984240	0.492120	−	0.870527i	$-0.336222\pi$
$137$	11.5202	0.984240	0.492120	+	0.870527i	$0.336222\pi$
$138$	0	0
$139$	1.70383	0.144517	0.0722587	−	0.997386i	$-0.476979\pi$
$139$	1.70383	0.144517	0.0722587	+	0.997386i	$0.476979\pi$
$140$	−0.203368	−0.0171877
$141$	0	0
$142$	−2.74973	−0.230752
$143$	−12.9963	−1.08681
$144$	0	0
$145$	0.595688	0.0494692
$146$	−0.469450	−0.0388520
$147$	0	0
$148$	15.1845	1.24816
$149$	−21.6528	−1.77386	−0.886932	−	0.461900i	$-0.847168\pi$
$149$	−21.6528	−1.77386	−0.886932	+	0.461900i	$0.847168\pi$
$150$	0	0
$151$	−4.74152	−0.385860	−0.192930	−	0.981213i	$-0.561799\pi$
$151$	−4.74152	−0.385860	−0.192930	+	0.981213i	$0.561799\pi$
$152$	16.3259	1.32421
$153$	0	0
$154$	−4.30745	−0.347104
$155$	0.154286	0.0123925
$156$	0	0
$157$	−0.209206	−0.0166964	−0.00834822	−	0.999965i	$-0.502657\pi$
$157$	−0.209206	−0.0166964	−0.00834822	+	0.999965i	$0.502657\pi$
$158$	17.3109	1.37718
$159$	0	0
$160$	0.195593	0.0154630
$161$	2.58202	0.203491
$162$	0	0
$163$	5.62384	0.440493	0.220247	−	0.975444i	$-0.429314\pi$
$163$	5.62384	0.440493	0.220247	+	0.975444i	$0.429314\pi$
$164$	−4.13008	−0.322505
$165$	0	0
$166$	35.7852	2.77747
$167$	16.6805	1.29078	0.645388	−	0.763855i	$-0.276696\pi$
$167$	16.6805	1.29078	0.645388	+	0.763855i	$0.276696\pi$
$168$	0	0
$169$	4.65456	0.358043
$170$	0.445087	0.0341366
$171$	0	0
$172$	−26.0019	−1.98263
$173$	19.0021	1.44471	0.722353	−	0.691524i	$-0.243061\pi$
$173$	19.0021	1.44471	0.722353	+	0.691524i	$0.243061\pi$
$174$	0	0
$175$	2.89428	0.218787
$176$	−8.22955	−0.620326
$177$	0	0
$178$	3.73002	0.279577
$179$	16.2352	1.21348	0.606739	−	0.794901i	$-0.292477\pi$
$179$	16.2352	1.21348	0.606739	+	0.794901i	$0.292477\pi$
$180$	0	0
$181$	−2.99158	−0.222362	−0.111181	−	0.993800i	$-0.535463\pi$
$181$	−2.99158	−0.222362	−0.111181	+	0.993800i	$0.535463\pi$
$182$	5.85136	0.433732
$183$	0	0
$184$	18.9046	1.39367
$185$	0.375135	0.0275805
$186$	0	0
$187$	6.15829	0.450338
$188$	13.5384	0.987388
$189$	0	0
$190$	0.859649	0.0623655
$191$	−2.25382	−0.163080	−0.0815402	−	0.996670i	$-0.525984\pi$
$191$	−2.25382	−0.163080	−0.0815402	+	0.996670i	$0.525984\pi$
$192$	0	0
$193$	−0.880273	−0.0633634	−0.0316817	−	0.999498i	$-0.510086\pi$
$193$	−0.880273	−0.0633634	−0.0316817	+	0.999498i	$0.510086\pi$
$194$	−12.7179	−0.913090
$195$	0	0
$196$	−25.1076	−1.79340
$197$	20.2766	1.44464	0.722322	−	0.691557i	$-0.243075\pi$
$197$	20.2766	1.44464	0.722322	+	0.691557i	$0.243075\pi$
$198$	0	0
$199$	−19.0094	−1.34754	−0.673772	−	0.738939i	$-0.735327\pi$
$199$	−19.0094	−1.34754	−0.673772	+	0.738939i	$0.735327\pi$
$200$	21.1910	1.49843
$201$	0	0
$202$	17.4573	1.22829
$203$	−3.71083	−0.260449
$204$	0	0
$205$	−0.102034	−0.00712636
$206$	−15.3737	−1.07113
$207$	0	0
$208$	11.1793	0.775142
$209$	11.8942	0.822740
$210$	0	0
$211$	16.1841	1.11416	0.557079	−	0.830460i	$-0.311922\pi$
$211$	16.1841	1.11416	0.557079	+	0.830460i	$0.311922\pi$
$212$	20.3473	1.39746
$213$	0	0
$214$	13.3138	0.910110
$215$	−0.642380	−0.0438100
$216$	0	0
$217$	−0.961120	−0.0652451
$218$	14.9654	1.01358
$219$	0	0
$220$	−1.08480	−0.0731373
$221$	−8.36559	−0.562731
$222$	0	0
$223$	21.4573	1.43689	0.718443	−	0.695585i	$-0.244855\pi$
$223$	21.4573	1.43689	0.718443	+	0.695585i	$0.244855\pi$
$224$	−1.21845	−0.0814108
$225$	0	0
$226$	28.4448	1.89212
$227$	−19.1142	−1.26866	−0.634329	−	0.773064i	$-0.718723\pi$
$227$	−19.1142	−1.26866	−0.634329	+	0.773064i	$0.718723\pi$
$228$	0	0
$229$	22.4702	1.48487	0.742435	−	0.669918i	$-0.233671\pi$
$229$	22.4702	1.48487	0.742435	+	0.669918i	$0.233671\pi$
$230$	0.995432	0.0656368
$231$	0	0
$232$	−27.1694	−1.78376
$233$	−17.6815	−1.15835	−0.579176	−	0.815203i	$-0.696626\pi$
$233$	−17.6815	−1.15835	−0.579176	+	0.815203i	$0.696626\pi$
$234$	0	0
$235$	0.334467	0.0218182
$236$	−38.7504	−2.52243
$237$	0	0
$238$	−2.77266	−0.179725
$239$	−15.4203	−0.997457	−0.498729	−	0.866758i	$-0.666200\pi$
$239$	−15.4203	−0.997457	−0.498729	+	0.866758i	$0.666200\pi$
$240$	0	0
$241$	13.1514	0.847159	0.423580	−	0.905859i	$-0.360773\pi$
$241$	13.1514	0.847159	0.423580	+	0.905859i	$0.360773\pi$
$242$	3.44113	0.221204
$243$	0	0
$244$	−49.7209	−3.18305
$245$	−0.620286	−0.0396286
$246$	0	0
$247$	−16.1574	−1.02807
$248$	−7.03700	−0.446850
$249$	0	0
$250$	2.23357	0.141264
$251$	17.4166	1.09933	0.549663	−	0.835386i	$-0.314756\pi$
$251$	17.4166	1.09933	0.549663	+	0.835386i	$0.314756\pi$
$252$	0	0
$253$	13.7729	0.865897
$254$	−27.6892	−1.73737
$255$	0	0
$256$	−28.9704	−1.81065
$257$	11.1876	0.697862	0.348931	−	0.937148i	$-0.386545\pi$
$257$	11.1876	0.697862	0.348931	+	0.937148i	$0.386545\pi$
$258$	0	0
$259$	−2.33690	−0.145208
$260$	1.47362	0.0913903
$261$	0	0
$262$	21.6429	1.33710
$263$	−20.7109	−1.27709	−0.638544	−	0.769585i	$-0.720463\pi$
$263$	−20.7109	−1.27709	−0.638544	+	0.769585i	$0.720463\pi$
$264$	0	0
$265$	0.502682	0.0308795
$266$	−5.35516	−0.328346
$267$	0	0
$268$	−33.2970	−2.03394
$269$	−28.2449	−1.72212	−0.861060	−	0.508504i	$-0.830199\pi$
$269$	−28.2449	−1.72212	−0.861060	+	0.508504i	$0.830199\pi$
$270$	0	0
$271$	17.2626	1.04863	0.524316	−	0.851524i	$-0.324321\pi$
$271$	17.2626	1.04863	0.524316	+	0.851524i	$0.324321\pi$
$272$	−5.29728	−0.321195
$273$	0	0
$274$	−27.6672	−1.67144
$275$	15.4386	0.930984
$276$	0	0
$277$	5.16898	0.310574	0.155287	−	0.987869i	$-0.450370\pi$
$277$	5.16898	0.310574	0.155287	+	0.987869i	$0.450370\pi$
$278$	−4.09197	−0.245420
$279$	0	0
$280$	0.229155	0.0136947
$281$	−3.29644	−0.196649	−0.0983246	−	0.995154i	$-0.531348\pi$
$281$	−3.29644	−0.196649	−0.0983246	+	0.995154i	$0.531348\pi$
$282$	0	0
$283$	−9.13072	−0.542765	−0.271382	−	0.962472i	$-0.587481\pi$
$283$	−9.13072	−0.542765	−0.271382	+	0.962472i	$0.587481\pi$
$284$	4.31391	0.255984
$285$	0	0
$286$	31.2122	1.84562
$287$	0.635618	0.0375194
$288$	0	0
$289$	−13.0360	−0.766822
$290$	−1.43062	−0.0840087
$291$	0	0
$292$	0.736497	0.0431002
$293$	−2.82643	−0.165122	−0.0825610	−	0.996586i	$-0.526310\pi$
$293$	−2.82643	−0.165122	−0.0825610	+	0.996586i	$0.526310\pi$
$294$	0	0
$295$	−0.957331	−0.0557380
$296$	−17.1100	−0.994496
$297$	0	0
$298$	52.0017	3.01238
$299$	−18.7095	−1.08200
$300$	0	0
$301$	4.00169	0.230654
$302$	11.3873	0.655268
$303$	0	0
$304$	−10.2312	−0.586802
$305$	−1.22836	−0.0703356
$306$	0	0
$307$	6.29446	0.359244	0.179622	−	0.983736i	$-0.442513\pi$
$307$	6.29446	0.359244	0.179622	+	0.983736i	$0.442513\pi$
$308$	6.75775	0.385058
$309$	0	0
$310$	−0.370536	−0.0210450
$311$	−7.37289	−0.418078	−0.209039	−	0.977907i	$-0.567034\pi$
$311$	−7.37289	−0.418078	−0.209039	+	0.977907i	$0.567034\pi$
$312$	0	0
$313$	−4.27075	−0.241397	−0.120699	−	0.992689i	$-0.538513\pi$
$313$	−4.27075	−0.241397	−0.120699	+	0.992689i	$0.538513\pi$
$314$	0.502433	0.0283539
$315$	0	0
$316$	−27.1582	−1.52777
$317$	16.1476	0.906938	0.453469	−	0.891272i	$-0.350186\pi$
$317$	16.1476	0.906938	0.453469	+	0.891272i	$0.350186\pi$
$318$	0	0
$319$	−19.7942	−1.10826
$320$	−0.965063	−0.0539486
$321$	0	0
$322$	−6.20102	−0.345570
$323$	7.65618	0.426001
$324$	0	0
$325$	−20.9723	−1.16333
$326$	−13.5063	−0.748047
$327$	0	0
$328$	4.65378	0.256962
$329$	−2.08355	−0.114870
$330$	0	0
$331$	19.2282	1.05688	0.528440	−	0.848971i	$-0.322777\pi$
$331$	19.2282	1.05688	0.528440	+	0.848971i	$0.322777\pi$
$332$	−56.1417	−3.08117
$333$	0	0
$334$	−40.0602	−2.19200
$335$	−0.822605	−0.0449437
$336$	0	0
$337$	−29.4628	−1.60494	−0.802469	−	0.596693i	$-0.796481\pi$
$337$	−29.4628	−1.60494	−0.802469	+	0.596693i	$0.796481\pi$
$338$	−11.1785	−0.608030
$339$	0	0
$340$	−0.698275	−0.0378693
$341$	−5.12679	−0.277631
$342$	0	0
$343$	7.92309	0.427806
$344$	29.2990	1.57970
$345$	0	0
$346$	−45.6360	−2.45340
$347$	11.3970	0.611825	0.305912	−	0.952060i	$-0.401039\pi$
$347$	11.3970	0.611825	0.305912	+	0.952060i	$0.401039\pi$
$348$	0	0
$349$	28.1616	1.50746	0.753728	−	0.657186i	$-0.228253\pi$
$349$	28.1616	1.50746	0.753728	+	0.657186i	$0.228253\pi$
$350$	−6.95097	−0.371545
$351$	0	0
$352$	−6.49941	−0.346419
$353$	28.6541	1.52510	0.762552	−	0.646927i	$-0.223946\pi$
$353$	28.6541	1.52510	0.762552	+	0.646927i	$0.223946\pi$
$354$	0	0
$355$	0.106576	0.00565644
$356$	−5.85184	−0.310147
$357$	0	0
$358$	−38.9909	−2.06073
$359$	−31.0322	−1.63782	−0.818909	−	0.573923i	$-0.805421\pi$
$359$	−31.0322	−1.63782	−0.818909	+	0.573923i	$0.805421\pi$
$360$	0	0
$361$	−4.21272	−0.221722
$362$	7.18464	0.377616
$363$	0	0
$364$	−9.17992	−0.481158
$365$	0.0181952	0.000952381 0
$366$	0	0
$367$	24.1242	1.25927	0.629637	−	0.776889i	$-0.283204\pi$
$367$	24.1242	1.25927	0.629637	+	0.776889i	$0.283204\pi$
$368$	−11.8473	−0.617583
$369$	0	0
$370$	−0.900932	−0.0468372
$371$	−3.13145	−0.162577
$372$	0	0
$373$	−12.5856	−0.651658	−0.325829	−	0.945429i	$-0.605643\pi$
$373$	−12.5856	−0.651658	−0.325829	+	0.945429i	$0.605643\pi$
$374$	−14.7899	−0.764766
$375$	0	0
$376$	−15.2551	−0.786721
$377$	26.8890	1.38486
$378$	0	0
$379$	−7.70522	−0.395790	−0.197895	−	0.980223i	$-0.563411\pi$
$379$	−7.70522	−0.395790	−0.197895	+	0.980223i	$0.563411\pi$
$380$	−1.34866	−0.0691848
$381$	0	0
$382$	5.41281	0.276944
$383$	−17.8616	−0.912687	−0.456343	−	0.889804i	$-0.650841\pi$
$383$	−17.8616	−0.912687	−0.456343	+	0.889804i	$0.650841\pi$
$384$	0	0
$385$	0.166951	0.00850859
$386$	2.11408	0.107604
$387$	0	0
$388$	19.9524	1.01293
$389$	27.3885	1.38865	0.694325	−	0.719662i	$-0.255703\pi$
$389$	27.3885	1.38865	0.694325	+	0.719662i	$0.255703\pi$
$390$	0	0
$391$	8.86549	0.448347
$392$	28.2913	1.42893
$393$	0	0
$394$	−48.6966	−2.45330
$395$	−0.670945	−0.0337589
$396$	0	0
$397$	4.21599	0.211594	0.105797	−	0.994388i	$-0.466261\pi$
$397$	4.21599	0.211594	0.105797	+	0.994388i	$0.466261\pi$
$398$	45.6535	2.28840
$399$	0	0
$400$	−13.2801	−0.664005
$401$	15.1725	0.757678	0.378839	−	0.925462i	$-0.376323\pi$
$401$	15.1725	0.757678	0.378839	+	0.925462i	$0.376323\pi$
$402$	0	0
$403$	6.96437	0.346920
$404$	−27.3880	−1.36260
$405$	0	0
$406$	8.91200	0.442295
$407$	−12.4654	−0.617888
$408$	0	0
$409$	4.70961	0.232875	0.116438	−	0.993198i	$-0.462852\pi$
$409$	4.70961	0.232875	0.116438	+	0.993198i	$0.462852\pi$
$410$	0.245047	0.0121020
$411$	0	0
$412$	24.1190	1.18826
$413$	5.96367	0.293453
$414$	0	0
$415$	−1.38698	−0.0680844
$416$	8.82898	0.432876
$417$	0	0
$418$	−28.5654	−1.39718
$419$	19.7911	0.966860	0.483430	−	0.875383i	$-0.339391\pi$
$419$	19.7911	0.966860	0.483430	+	0.875383i	$0.339391\pi$
$420$	0	0
$421$	28.1857	1.37369	0.686844	−	0.726805i	$-0.258996\pi$
$421$	28.1857	1.37369	0.686844	+	0.726805i	$0.258996\pi$
$422$	−38.8680	−1.89206
$423$	0	0
$424$	−22.9274	−1.11345
$425$	9.93768	0.482048
$426$	0	0
$427$	7.65204	0.370308
$428$	−20.8873	−1.00963
$429$	0	0
$430$	1.54275	0.0743982
$431$	−5.19681	−0.250321	−0.125161	−	0.992136i	$-0.539945\pi$
$431$	−5.19681	−0.250321	−0.125161	+	0.992136i	$0.539945\pi$
$432$	0	0
$433$	25.3285	1.21721	0.608605	−	0.793473i	$-0.291730\pi$
$433$	25.3285	1.21721	0.608605	+	0.793473i	$0.291730\pi$
$434$	2.30825	0.110799
$435$	0	0
$436$	−23.4785	−1.12441
$437$	17.1230	0.819102
$438$	0	0
$439$	−15.6612	−0.747470	−0.373735	−	0.927536i	$-0.621923\pi$
$439$	−15.6612	−0.747470	−0.373735	+	0.927536i	$0.621923\pi$
$440$	1.22236	0.0582735
$441$	0	0
$442$	20.0910	0.955631
$443$	−18.2538	−0.867266	−0.433633	−	0.901089i	$-0.642769\pi$
$443$	−18.2538	−0.867266	−0.433633	+	0.901089i	$0.642769\pi$
$444$	0	0
$445$	−0.144570	−0.00685329
$446$	−51.5323	−2.44013
$447$	0	0
$448$	6.01184	0.284033
$449$	28.7216	1.35546	0.677729	−	0.735312i	$-0.262964\pi$
$449$	28.7216	1.35546	0.677729	+	0.735312i	$0.262964\pi$
$450$	0	0
$451$	3.39050	0.159652
$452$	−44.6256	−2.09901
$453$	0	0
$454$	45.9052	2.15444
$455$	−0.226791	−0.0106321
$456$	0	0
$457$	−35.3879	−1.65538	−0.827688	−	0.561188i	$-0.810344\pi$
$457$	−35.3879	−1.65538	−0.827688	+	0.561188i	$0.810344\pi$
$458$	−53.9648	−2.52161
$459$	0	0
$460$	−1.56168	−0.0728139
$461$	2.27673	0.106038	0.0530189	−	0.998594i	$-0.483116\pi$
$461$	2.27673	0.106038	0.0530189	+	0.998594i	$0.483116\pi$
$462$	0	0
$463$	18.3745	0.853937	0.426968	−	0.904267i	$-0.359582\pi$
$463$	18.3745	0.853937	0.426968	+	0.904267i	$0.359582\pi$
$464$	17.0267	0.790446
$465$	0	0
$466$	42.4642	1.96712
$467$	4.65870	0.215579	0.107789	−	0.994174i	$-0.465623\pi$
$467$	4.65870	0.215579	0.107789	+	0.994174i	$0.465623\pi$
$468$	0	0
$469$	5.12440	0.236623
$470$	−0.803263	−0.0370517
$471$	0	0
$472$	43.6640	2.00980
$473$	21.3457	0.981478
$474$	0	0
$475$	19.1938	0.880672
$476$	4.34989	0.199377
$477$	0	0
$478$	37.0338	1.69388
$479$	−14.1759	−0.647713	−0.323857	−	0.946106i	$-0.604980\pi$
$479$	−14.1759	−0.647713	−0.323857	+	0.946106i	$0.604980\pi$
$480$	0	0
$481$	16.9334	0.772096
$482$	−31.5848	−1.43865
$483$	0	0
$484$	−5.39861	−0.245392
$485$	0.492926	0.0223826
$486$	0	0
$487$	−21.4338	−0.971258	−0.485629	−	0.874165i	$-0.661409\pi$
$487$	−21.4338	−0.971258	−0.485629	+	0.874165i	$0.661409\pi$
$488$	56.0256	2.53616
$489$	0	0
$490$	1.48969	0.0672974
$491$	−14.0879	−0.635776	−0.317888	−	0.948128i	$-0.602974\pi$
$491$	−14.0879	−0.635776	−0.317888	+	0.948128i	$0.602974\pi$
$492$	0	0
$493$	−12.7413	−0.573841
$494$	38.8041	1.74588
$495$	0	0
$496$	4.40999	0.198015
$497$	−0.663910	−0.0297804
$498$	0	0
$499$	14.9093	0.667433	0.333717	−	0.942673i	$-0.391697\pi$
$499$	14.9093	0.667433	0.333717	+	0.942673i	$0.391697\pi$
$500$	−3.50414	−0.156710
$501$	0	0
$502$	−41.8281	−1.86688
$503$	−15.8631	−0.707299	−0.353650	−	0.935378i	$-0.615059\pi$
$503$	−15.8631	−0.707299	−0.353650	+	0.935378i	$0.615059\pi$
$504$	0	0
$505$	−0.676622	−0.0301093
$506$	−33.0774	−1.47047
$507$	0	0
$508$	43.4402	1.92735
$509$	33.9267	1.50377	0.751887	−	0.659292i	$-0.229144\pi$
$509$	33.9267	1.50377	0.751887	+	0.659292i	$0.229144\pi$
$510$	0	0
$511$	−0.113347	−0.00501416
$512$	28.1824	1.24550
$513$	0	0
$514$	−26.8683	−1.18511
$515$	0.595862	0.0262568
$516$	0	0
$517$	−11.1140	−0.488795
$518$	5.61234	0.246592
$519$	0	0
$520$	−1.66048	−0.0728170
$521$	−42.7798	−1.87422	−0.937108	−	0.349039i	$-0.886508\pi$
$521$	−42.7798	−1.87422	−0.937108	+	0.349039i	$0.886508\pi$
$522$	0	0
$523$	−2.77785	−0.121467	−0.0607335	−	0.998154i	$-0.519344\pi$
$523$	−2.77785	−0.121467	−0.0607335	+	0.998154i	$0.519344\pi$
$524$	−33.9544	−1.48331
$525$	0	0
$526$	49.7397	2.16875
$527$	−3.30006	−0.143753
$528$	0	0
$529$	−3.17243	−0.137932
$530$	−1.20725	−0.0524396
$531$	0	0
$532$	8.40145	0.364249
$533$	−4.60575	−0.199497
$534$	0	0
$535$	−0.516022	−0.0223096
$536$	37.5192	1.62058
$537$	0	0
$538$	67.8335	2.92451
$539$	20.6116	0.887803
$540$	0	0
$541$	−3.59390	−0.154514	−0.0772570	−	0.997011i	$-0.524616\pi$
$541$	−3.59390	−0.154514	−0.0772570	+	0.997011i	$0.524616\pi$
$542$	−41.4583	−1.78079
$543$	0	0
$544$	−4.18360	−0.179370
$545$	−0.580037	−0.0248461
$546$	0	0
$547$	−39.5858	−1.69257	−0.846284	−	0.532732i	$-0.821165\pi$
$547$	−39.5858	−1.69257	−0.846284	+	0.532732i	$0.821165\pi$
$548$	43.4057	1.85420
$549$	0	0
$550$	−37.0777	−1.58100
$551$	−24.6088	−1.04837
$552$	0	0
$553$	4.17964	0.177736
$554$	−12.4139	−0.527417
$555$	0	0
$556$	6.41968	0.272255
$557$	−11.4346	−0.484501	−0.242250	−	0.970214i	$-0.577886\pi$
$557$	−11.4346	−0.484501	−0.242250	+	0.970214i	$0.577886\pi$
$558$	0	0
$559$	−28.9967	−1.22643
$560$	−0.143609	−0.00606858
$561$	0	0
$562$	7.91680	0.333950
$563$	14.5108	0.611557	0.305778	−	0.952103i	$-0.401083\pi$
$563$	14.5108	0.611557	0.305778	+	0.952103i	$0.401083\pi$
$564$	0	0
$565$	−1.10248	−0.0463816
$566$	21.9285	0.921725
$567$	0	0
$568$	−4.86093	−0.203960
$569$	−1.29918	−0.0544646	−0.0272323	−	0.999629i	$-0.508669\pi$
$569$	−1.29918	−0.0544646	−0.0272323	+	0.999629i	$0.508669\pi$
$570$	0	0
$571$	16.0573	0.671978	0.335989	−	0.941866i	$-0.390929\pi$
$571$	16.0573	0.671978	0.335989	+	0.941866i	$0.390929\pi$
$572$	−48.9673	−2.04743
$573$	0	0
$574$	−1.52651	−0.0637155
$575$	22.2255	0.926867
$576$	0	0
$577$	−8.46033	−0.352208	−0.176104	−	0.984372i	$-0.556350\pi$
$577$	−8.46033	−0.352208	−0.176104	+	0.984372i	$0.556350\pi$
$578$	31.3075	1.30222
$579$	0	0
$580$	2.24442	0.0931947
$581$	8.64019	0.358456
$582$	0	0
$583$	−16.7037	−0.691796
$584$	−0.829886	−0.0343409
$585$	0	0
$586$	6.78802	0.280411
$587$	−18.3852	−0.758838	−0.379419	−	0.925225i	$-0.623876\pi$
$587$	−18.3852	−0.758838	−0.379419	+	0.925225i	$0.623876\pi$
$588$	0	0
$589$	−6.37379	−0.262627
$590$	2.29915	0.0946544
$591$	0	0
$592$	10.7226	0.440696
$593$	13.5128	0.554905	0.277452	−	0.960739i	$-0.410510\pi$
$593$	13.5128	0.554905	0.277452	+	0.960739i	$0.410510\pi$
$594$	0	0
$595$	0.107464	0.00440561
$596$	−81.5830	−3.34177
$597$	0	0
$598$	44.9332	1.83746
$599$	11.0277	0.450580	0.225290	−	0.974292i	$-0.427667\pi$
$599$	11.0277	0.450580	0.225290	+	0.974292i	$0.427667\pi$
$600$	0	0
$601$	25.7979	1.05232	0.526160	−	0.850386i	$-0.323631\pi$
$601$	25.7979	1.05232	0.526160	+	0.850386i	$0.323631\pi$
$602$	−9.61055	−0.391697
$603$	0	0
$604$	−17.8650	−0.726918
$605$	−0.133373	−0.00542239
$606$	0	0
$607$	−14.7899	−0.600302	−0.300151	−	0.953892i	$-0.597037\pi$
$607$	−14.7899	−0.600302	−0.300151	+	0.953892i	$0.597037\pi$
$608$	−8.08027	−0.327698
$609$	0	0
$610$	2.95005	0.119444
$611$	15.0976	0.610785
$612$	0	0
$613$	36.2739	1.46509	0.732545	−	0.680719i	$-0.238332\pi$
$613$	36.2739	1.46509	0.732545	+	0.680719i	$0.238332\pi$
$614$	−15.1169	−0.610069
$615$	0	0
$616$	−7.61464	−0.306803
$617$	−40.3735	−1.62538	−0.812689	−	0.582698i	$-0.801997\pi$
$617$	−40.3735	−1.62538	−0.812689	+	0.582698i	$0.801997\pi$
$618$	0	0
$619$	−6.89330	−0.277065	−0.138533	−	0.990358i	$-0.544239\pi$
$619$	−6.89330	−0.277065	−0.138533	+	0.990358i	$0.544239\pi$
$620$	0.581315	0.0233462
$621$	0	0
$622$	17.7069	0.709981
$623$	0.900598	0.0360817
$624$	0	0
$625$	24.8701	0.994804
$626$	10.2567	0.409941
$627$	0	0
$628$	−0.788242	−0.0314543
$629$	−8.02386	−0.319932
$630$	0	0
$631$	29.8191	1.18708	0.593539	−	0.804805i	$-0.297730\pi$
$631$	29.8191	1.18708	0.593539	+	0.804805i	$0.297730\pi$
$632$	30.6019	1.21728
$633$	0	0
$634$	−38.7804	−1.54016
$635$	1.07319	0.0425884
$636$	0	0
$637$	−27.9993	−1.10937
$638$	47.5382	1.88206
$639$	0	0
$640$	1.92653	0.0761527
$641$	42.8807	1.69369	0.846844	−	0.531842i	$-0.178500\pi$
$641$	42.8807	1.69369	0.846844	+	0.531842i	$0.178500\pi$
$642$	0	0
$643$	−27.4133	−1.08107	−0.540537	−	0.841320i	$-0.681779\pi$
$643$	−27.4133	−1.08107	−0.540537	+	0.841320i	$0.681779\pi$
$644$	9.72848	0.383356
$645$	0	0
$646$	−18.3873	−0.723437
$647$	−16.1623	−0.635407	−0.317703	−	0.948190i	$-0.602912\pi$
$647$	−16.1623	−0.635407	−0.317703	+	0.948190i	$0.602912\pi$
$648$	0	0
$649$	31.8113	1.24870
$650$	50.3674	1.97557
$651$	0	0
$652$	21.1894	0.829842
$653$	32.2463	1.26189	0.630947	−	0.775826i	$-0.282666\pi$
$653$	32.2463	1.26189	0.630947	+	0.775826i	$0.282666\pi$
$654$	0	0
$655$	−0.838847	−0.0327765
$656$	−2.91647	−0.113869
$657$	0	0
$658$	5.00391	0.195073
$659$	27.8186	1.08366	0.541829	−	0.840489i	$-0.317732\pi$
$659$	27.8186	1.08366	0.541829	+	0.840489i	$0.317732\pi$
$660$	0	0
$661$	30.7886	1.19754	0.598769	−	0.800922i	$-0.295657\pi$
$661$	30.7886	1.19754	0.598769	+	0.800922i	$0.295657\pi$
$662$	−46.1789	−1.79480
$663$	0	0
$664$	63.2606	2.45499
$665$	0.207559	0.00804877
$666$	0	0
$667$	−28.4958	−1.10336
$668$	62.8485	2.43168
$669$	0	0
$670$	1.97559	0.0763236
$671$	40.8173	1.57574
$672$	0	0
$673$	−26.1306	−1.00726	−0.503631	−	0.863919i	$-0.668003\pi$
$673$	−26.1306	−1.00726	−0.503631	+	0.863919i	$0.668003\pi$
$674$	70.7584	2.72551
$675$	0	0
$676$	17.5374	0.674515
$677$	18.1945	0.699271	0.349636	−	0.936886i	$-0.386305\pi$
$677$	18.1945	0.699271	0.349636	+	0.936886i	$0.386305\pi$
$678$	0	0
$679$	−3.07068	−0.117842
$680$	0.786818	0.0301731
$681$	0	0
$682$	12.3126	0.471474
$683$	23.4971	0.899092	0.449546	−	0.893257i	$-0.351586\pi$
$683$	23.4971	0.899092	0.449546	+	0.893257i	$0.351586\pi$
$684$	0	0
$685$	1.07234	0.0409721
$686$	−19.0283	−0.726502
$687$	0	0
$688$	−18.3613	−0.700019
$689$	22.6908	0.864449
$690$	0	0
$691$	−44.5379	−1.69430	−0.847151	−	0.531352i	$-0.821684\pi$
$691$	−44.5379	−1.69430	−0.847151	+	0.531352i	$0.821684\pi$
$692$	71.5960	2.72167
$693$	0	0
$694$	−27.3714	−1.03900
$695$	0.158599	0.00601599
$696$	0	0
$697$	2.18243	0.0826655
$698$	−67.6335	−2.55997
$699$	0	0
$700$	10.9050	0.412171
$701$	−25.2567	−0.953934	−0.476967	−	0.878921i	$-0.658264\pi$
$701$	−25.2567	−0.953934	−0.476967	+	0.878921i	$0.658264\pi$
$702$	0	0
$703$	−15.4974	−0.584496
$704$	32.0682	1.20862
$705$	0	0
$706$	−68.8163	−2.58994
$707$	4.21500	0.158521
$708$	0	0
$709$	−15.6840	−0.589026	−0.294513	−	0.955648i	$-0.595157\pi$
$709$	−15.6840	−0.589026	−0.294513	+	0.955648i	$0.595157\pi$
$710$	−0.255954	−0.00960579
$711$	0	0
$712$	6.59387	0.247116
$713$	−7.38054	−0.276403
$714$	0	0
$715$	−1.20974	−0.0452417
$716$	61.1708	2.28606
$717$	0	0
$718$	74.5277	2.78135
$719$	−53.1607	−1.98256	−0.991280	−	0.131771i	$-0.957934\pi$
$719$	−53.1607	−1.98256	−0.991280	+	0.131771i	$0.957934\pi$
$720$	0	0
$721$	−3.71191	−0.138239
$722$	10.1174	0.376529
$723$	0	0
$724$	−11.2716	−0.418907
$725$	−31.9421	−1.18630
$726$	0	0
$727$	0.469172	0.0174006	0.00870032	−	0.999962i	$-0.497231\pi$
$727$	0.469172	0.0174006	0.00870032	+	0.999962i	$0.497231\pi$
$728$	10.3440	0.383372
$729$	0	0
$730$	−0.0436980	−0.00161734
$731$	13.7400	0.508194
$732$	0	0
$733$	−46.4063	−1.71406	−0.857028	−	0.515271i	$-0.827691\pi$
$733$	−46.4063	−1.71406	−0.857028	+	0.515271i	$0.827691\pi$
$734$	−57.9372	−2.13850
$735$	0	0
$736$	−9.35657	−0.344888
$737$	27.3345	1.00688
$738$	0	0
$739$	25.8093	0.949411	0.474706	−	0.880145i	$-0.342555\pi$
$739$	25.8093	0.949411	0.474706	+	0.880145i	$0.342555\pi$
$740$	1.41343	0.0519586
$741$	0	0
$742$	7.52055	0.276088
$743$	34.7096	1.27337	0.636687	−	0.771123i	$-0.280305\pi$
$743$	34.7096	1.27337	0.636687	+	0.771123i	$0.280305\pi$
$744$	0	0
$745$	−2.01551	−0.0738427
$746$	30.2259	1.10665
$747$	0	0
$748$	23.2031	0.848389
$749$	3.21455	0.117457
$750$	0	0
$751$	23.9858	0.875256	0.437628	−	0.899156i	$-0.355819\pi$
$751$	23.9858	0.875256	0.437628	+	0.899156i	$0.355819\pi$
$752$	9.56016	0.348623
$753$	0	0
$754$	−64.5773	−2.35177
$755$	−0.441357	−0.0160626
$756$	0	0
$757$	−8.78780	−0.319398	−0.159699	−	0.987166i	$-0.551052\pi$
$757$	−8.78780	−0.319398	−0.159699	+	0.987166i	$0.551052\pi$
$758$	18.5050	0.672132
$759$	0	0
$760$	1.51967	0.0551243
$761$	13.7539	0.498578	0.249289	−	0.968429i	$-0.419803\pi$
$761$	13.7539	0.498578	0.249289	+	0.968429i	$0.419803\pi$
$762$	0	0
$763$	3.61333	0.130811
$764$	−8.49190	−0.307226
$765$	0	0
$766$	42.8969	1.54993
$767$	−43.2134	−1.56034
$768$	0	0
$769$	−31.3579	−1.13080	−0.565398	−	0.824818i	$-0.691277\pi$
$769$	−31.3579	−1.13080	−0.565398	+	0.824818i	$0.691277\pi$
$770$	−0.400952	−0.0144493
$771$	0	0
$772$	−3.31668	−0.119370
$773$	28.1214	1.01146	0.505729	−	0.862693i	$-0.331224\pi$
$773$	28.1214	1.01146	0.505729	+	0.862693i	$0.331224\pi$
$774$	0	0
$775$	−8.27314	−0.297180
$776$	−22.4824	−0.807073
$777$	0	0
$778$	−65.7767	−2.35821
$779$	4.21518	0.151025
$780$	0	0
$781$	−3.54142	−0.126722
$782$	−21.2916	−0.761385
$783$	0	0
$784$	−17.7298	−0.633207
$785$	−0.0194736	−0.000695042 0
$786$	0	0
$787$	−36.2002	−1.29040	−0.645198	−	0.764015i	$-0.723225\pi$
$787$	−36.2002	−1.29040	−0.645198	+	0.764015i	$0.723225\pi$
$788$	76.3977	2.72155
$789$	0	0
$790$	1.61136	0.0573294
$791$	6.86787	0.244193
$792$	0	0
$793$	−55.4474	−1.96900
$794$	−10.1252	−0.359330
$795$	0	0
$796$	−71.6235	−2.53863
$797$	−29.6001	−1.04849	−0.524245	−	0.851568i	$-0.675652\pi$
$797$	−29.6001	−1.04849	−0.524245	+	0.851568i	$0.675652\pi$
$798$	0	0
$799$	−7.15400	−0.253090
$800$	−10.4881	−0.370812
$801$	0	0
$802$	−36.4386	−1.28669
$803$	−0.604612	−0.0213363
$804$	0	0
$805$	0.240343	0.00847097
$806$	−16.7258	−0.589141
$807$	0	0
$808$	30.8608	1.08568
$809$	−5.75943	−0.202491	−0.101245	−	0.994861i	$-0.532283\pi$
$809$	−5.75943	−0.202491	−0.101245	+	0.994861i	$0.532283\pi$
$810$	0	0
$811$	12.4896	0.438569	0.219284	−	0.975661i	$-0.429628\pi$
$811$	12.4896	0.438569	0.219284	+	0.975661i	$0.429628\pi$
$812$	−13.9816	−0.490658
$813$	0	0
$814$	29.9372	1.04930
$815$	0.523487	0.0183369
$816$	0	0
$817$	26.5377	0.928438
$818$	−11.3107	−0.395470
$819$	0	0
$820$	−0.384442	−0.0134253
$821$	43.0668	1.50304	0.751520	−	0.659710i	$-0.229321\pi$
$821$	43.0668	1.50304	0.751520	+	0.659710i	$0.229321\pi$
$822$	0	0
$823$	10.2841	0.358481	0.179240	−	0.983805i	$-0.442636\pi$
$823$	10.2841	0.358481	0.179240	+	0.983805i	$0.442636\pi$
$824$	−27.1773	−0.946767
$825$	0	0
$826$	−14.3225	−0.498343
$827$	6.09463	0.211931	0.105965	−	0.994370i	$-0.466207\pi$
$827$	6.09463	0.211931	0.105965	+	0.994370i	$0.466207\pi$
$828$	0	0
$829$	−33.6979	−1.17038	−0.585188	−	0.810898i	$-0.698979\pi$
$829$	−33.6979	−1.17038	−0.585188	+	0.810898i	$0.698979\pi$
$830$	3.33101	0.115621
$831$	0	0
$832$	−43.5624	−1.51025
$833$	13.2675	0.459690
$834$	0	0
$835$	1.55268	0.0537326
$836$	44.8148	1.54995
$837$	0	0
$838$	−47.5308	−1.64192
$839$	−42.2262	−1.45781	−0.728905	−	0.684615i	$-0.759971\pi$
$839$	−42.2262	−1.45781	−0.728905	+	0.684615i	$0.759971\pi$
$840$	0	0
$841$	11.9537	0.412197
$842$	−67.6914	−2.33280
$843$	0	0
$844$	60.9781	2.09895
$845$	0.433262	0.0149047
$846$	0	0
$847$	0.830846	0.0285482
$848$	14.3683	0.493409
$849$	0	0
$850$	−23.8665	−0.818616
$851$	−17.9453	−0.615156
$852$	0	0
$853$	−35.7296	−1.22336	−0.611679	−	0.791106i	$-0.709506\pi$
$853$	−35.7296	−1.22336	−0.611679	+	0.791106i	$0.709506\pi$
$854$	−18.3773	−0.628858
$855$	0	0
$856$	23.5358	0.804439
$857$	7.90865	0.270154	0.135077	−	0.990835i	$-0.456872\pi$
$857$	7.90865	0.270154	0.135077	+	0.990835i	$0.456872\pi$
$858$	0	0
$859$	−44.7191	−1.52580	−0.762899	−	0.646518i	$-0.776225\pi$
$859$	−44.7191	−1.52580	−0.762899	+	0.646518i	$0.776225\pi$
$860$	−2.42035	−0.0825332
$861$	0	0
$862$	12.4808	0.425096
$863$	−22.9170	−0.780103	−0.390052	−	0.920793i	$-0.627543\pi$
$863$	−22.9170	−0.780103	−0.390052	+	0.920793i	$0.627543\pi$
$864$	0	0
$865$	1.76878	0.0601405
$866$	−60.8294	−2.06707
$867$	0	0
$868$	−3.62129	−0.122915
$869$	22.2949	0.756304
$870$	0	0
$871$	−37.1319	−1.25817
$872$	26.4556	0.895899
$873$	0	0
$874$	−41.1229	−1.39100
$875$	0.539287	0.0182312
$876$	0	0
$877$	−22.1376	−0.747535	−0.373767	−	0.927522i	$-0.621934\pi$
$877$	−22.1376	−0.747535	−0.373767	+	0.927522i	$0.621934\pi$
$878$	37.6124	1.26936
$879$	0	0
$880$	−0.766035	−0.0258230
$881$	9.86404	0.332328	0.166164	−	0.986098i	$-0.446862\pi$
$881$	9.86404	0.332328	0.166164	+	0.986098i	$0.446862\pi$
$882$	0	0
$883$	47.5731	1.60096	0.800481	−	0.599358i	$-0.204578\pi$
$883$	47.5731	1.60096	0.800481	+	0.599358i	$0.204578\pi$
$884$	−31.5197	−1.06012
$885$	0	0
$886$	43.8388	1.47279
$887$	13.3573	0.448494	0.224247	−	0.974532i	$-0.428008\pi$
$887$	13.3573	0.448494	0.224247	+	0.974532i	$0.428008\pi$
$888$	0	0
$889$	−6.68544	−0.224222
$890$	0.347203	0.0116383
$891$	0	0
$892$	80.8465	2.70694
$893$	−13.8174	−0.462380
$894$	0	0
$895$	1.51123	0.0505149
$896$	−12.0013	−0.400934
$897$	0	0
$898$	−68.9785	−2.30184
$899$	10.6072	0.353769
$900$	0	0
$901$	−10.7520	−0.358201
$902$	−8.14270	−0.271122
$903$	0	0
$904$	50.2842	1.67243
$905$	−0.278467	−0.00925654
$906$	0	0
$907$	36.9606	1.22726	0.613629	−	0.789595i	$-0.289709\pi$
$907$	36.9606	1.22726	0.613629	+	0.789595i	$0.289709\pi$
$908$	−72.0184	−2.39001
$909$	0	0
$910$	0.544665	0.0180555
$911$	−48.4111	−1.60393	−0.801966	−	0.597370i	$-0.796212\pi$
$911$	−48.4111	−1.60393	−0.801966	+	0.597370i	$0.796212\pi$
$912$	0	0
$913$	46.0883	1.52530
$914$	84.9884	2.81117
$915$	0	0
$916$	84.6627	2.79733
$917$	5.22558	0.172564
$918$	0	0
$919$	8.93459	0.294725	0.147363	−	0.989083i	$-0.452922\pi$
$919$	8.93459	0.294725	0.147363	+	0.989083i	$0.452922\pi$
$920$	1.75971	0.0580159
$921$	0	0
$922$	−5.46784	−0.180074
$923$	4.81076	0.158348
$924$	0	0
$925$	−20.1156	−0.661395
$926$	−44.1287	−1.45016
$927$	0	0
$928$	13.4471	0.441423
$929$	−6.24415	−0.204864	−0.102432	−	0.994740i	$-0.532662\pi$
$929$	−6.24415	−0.204864	−0.102432	+	0.994740i	$0.532662\pi$
$930$	0	0
$931$	25.6250	0.839825
$932$	−66.6200	−2.18221
$933$	0	0
$934$	−11.1884	−0.366097
$935$	0.573234	0.0187468
$936$	0	0
$937$	45.8424	1.49760	0.748802	−	0.662794i	$-0.230629\pi$
$937$	45.8424	1.49760	0.748802	+	0.662794i	$0.230629\pi$
$938$	−12.3069	−0.401834
$939$	0	0
$940$	1.26020	0.0411032
$941$	−4.35263	−0.141892	−0.0709459	−	0.997480i	$-0.522602\pi$
$941$	−4.35263	−0.141892	−0.0709459	+	0.997480i	$0.522602\pi$
$942$	0	0
$943$	4.88098	0.158947
$944$	−27.3637	−0.890612
$945$	0	0
$946$	−51.2644	−1.66675
$947$	2.01674	0.0655351	0.0327675	−	0.999463i	$-0.489568\pi$
$947$	2.01674	0.0655351	0.0327675	+	0.999463i	$0.489568\pi$
$948$	0	0
$949$	0.821322	0.0266612
$950$	−46.0962	−1.49556
$951$	0	0
$952$	−4.90147	−0.158857
$953$	−35.7287	−1.15737	−0.578684	−	0.815552i	$-0.696433\pi$
$953$	−35.7287	−1.15737	−0.578684	+	0.815552i	$0.696433\pi$
$954$	0	0
$955$	−0.209793	−0.00678874
$956$	−58.1004	−1.87910
$957$	0	0
$958$	34.0451	1.09995
$959$	−6.68014	−0.215713
$960$	0	0
$961$	−28.2527	−0.911377
$962$	−40.6676	−1.31117
$963$	0	0
$964$	49.5518	1.59596
$965$	−0.0819388	−0.00263770
$966$	0	0
$967$	−0.693317	−0.0222956	−0.0111478	−	0.999938i	$-0.503549\pi$
$967$	−0.693317	−0.0222956	−0.0111478	+	0.999938i	$0.503549\pi$
$968$	6.08317	0.195520
$969$	0	0
$970$	−1.18382	−0.0380103
$971$	47.4942	1.52416	0.762081	−	0.647482i	$-0.224178\pi$
$971$	47.4942	1.52416	0.762081	+	0.647482i	$0.224178\pi$
$972$	0	0
$973$	−0.987988	−0.0316734
$974$	51.4759	1.64939
$975$	0	0
$976$	−35.1105	−1.12386
$977$	−12.6033	−0.403216	−0.201608	−	0.979466i	$-0.564617\pi$
$977$	−12.6033	−0.403216	−0.201608	+	0.979466i	$0.564617\pi$
$978$	0	0
$979$	4.80395	0.153535
$980$	−2.33710	−0.0746560
$981$	0	0
$982$	33.8337	1.07968
$983$	11.5450	0.368229	0.184115	−	0.982905i	$-0.441058\pi$
$983$	11.5450	0.368229	0.184115	+	0.982905i	$0.441058\pi$
$984$	0	0
$985$	1.88741	0.0601379
$986$	30.5999	0.974498
$987$	0	0
$988$	−60.8778	−1.93678
$989$	30.7294	0.977139
$990$	0	0
$991$	18.6935	0.593819	0.296910	−	0.954906i	$-0.404044\pi$
$991$	18.6935	0.593819	0.296910	+	0.954906i	$0.404044\pi$
$992$	3.48286	0.110581
$993$	0	0
$994$	1.59446	0.0505732
$995$	−1.76946	−0.0560958
$996$	0	0
$997$	3.35880	0.106374	0.0531872	−	0.998585i	$-0.483062\pi$
$997$	3.35880	0.106374	0.0531872	+	0.998585i	$0.483062\pi$
$998$	−35.8066	−1.13344
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.a.a.1.1		6
3.2	odd	2		729.2.a.d.1.6		6
9.2	odd	6		729.2.c.b.244.1		12
9.4	even	3		729.2.c.e.487.6		12
9.5	odd	6		729.2.c.b.487.1		12
9.7	even	3		729.2.c.e.244.6		12
27.2	odd	18		243.2.e.a.190.1		12
27.4	even	9		243.2.e.c.136.1		12
27.5	odd	18		81.2.e.a.73.2		12
27.7	even	9		243.2.e.c.109.1		12
27.11	odd	18		81.2.e.a.10.2		12
27.13	even	9		243.2.e.d.55.2		12
27.14	odd	18		243.2.e.a.55.1		12
27.16	even	9		27.2.e.a.13.1	✓	12
27.20	odd	18		243.2.e.b.109.2		12
27.22	even	9		27.2.e.a.25.1	yes	12
27.23	odd	18		243.2.e.b.136.2		12
27.25	even	9		243.2.e.d.190.2		12
108.43	odd	18		432.2.u.c.337.1		12
108.103	odd	18		432.2.u.c.241.1		12
135.22	odd	36		675.2.u.b.349.1		24
135.43	odd	36		675.2.u.b.499.1		24
135.49	even	18		675.2.l.c.376.2		12
135.97	odd	36		675.2.u.b.499.4		24
135.103	odd	36		675.2.u.b.349.4		24
135.124	even	18		675.2.l.c.526.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.e.a.13.1	✓	12	27.16	even	9
27.2.e.a.25.1	yes	12	27.22	even	9
81.2.e.a.10.2		12	27.11	odd	18
81.2.e.a.73.2		12	27.5	odd	18
243.2.e.a.55.1		12	27.14	odd	18
243.2.e.a.190.1		12	27.2	odd	18
243.2.e.b.109.2		12	27.20	odd	18
243.2.e.b.136.2		12	27.23	odd	18
243.2.e.c.109.1		12	27.7	even	9
243.2.e.c.136.1		12	27.4	even	9
243.2.e.d.55.2		12	27.13	even	9
243.2.e.d.190.2		12	27.25	even	9
432.2.u.c.241.1		12	108.103	odd	18
432.2.u.c.337.1		12	108.43	odd	18
675.2.l.c.376.2		12	135.49	even	18
675.2.l.c.526.2		12	135.124	even	18
675.2.u.b.349.1		24	135.22	odd	36
675.2.u.b.349.4		24	135.103	odd	36
675.2.u.b.499.1		24	135.43	odd	36
675.2.u.b.499.4		24	135.97	odd	36
729.2.a.a.1.1		6	1.1	even	1	trivial
729.2.a.d.1.6		6	3.2	odd	2
729.2.c.b.244.1		12	9.2	odd	6
729.2.c.b.487.1		12	9.5	odd	6
729.2.c.e.244.6		12	9.7	even	3
729.2.c.e.487.6		12	9.4	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	729.a (trivial)

\(f(q)\)	\(=\)	\(q-2.40162 q^{2} +3.76778 q^{4} +0.0930834 q^{5} -0.579861 q^{7} -4.24555 q^{8} +O(q^{10})\)	\(q-2.40162 q^{2} +3.76778 q^{4} +0.0930834 q^{5} -0.579861 q^{7} -4.24555 q^{8} -0.223551 q^{10} -3.09308 q^{11} +4.20173 q^{13} +1.39261 q^{14} +2.66063 q^{16} -1.99099 q^{17} -3.84542 q^{19} +0.350718 q^{20} +7.42841 q^{22} -4.45282 q^{23} -4.99134 q^{25} -10.0910 q^{26} -2.18479 q^{28} +6.39951 q^{29} +1.65750 q^{31} +2.10127 q^{32} +4.78159 q^{34} -0.0539755 q^{35} +4.03009 q^{37} +9.23525 q^{38} -0.395190 q^{40} -1.09616 q^{41} -6.90112 q^{43} -11.6541 q^{44} +10.6940 q^{46} +3.59319 q^{47} -6.66376 q^{49} +11.9873 q^{50} +15.8312 q^{52} +5.40034 q^{53} -0.287915 q^{55} +2.46183 q^{56} -15.3692 q^{58} -10.2847 q^{59} -13.1963 q^{61} -3.98069 q^{62} -10.3677 q^{64} +0.391112 q^{65} -8.83729 q^{67} -7.50161 q^{68} +0.129629 q^{70} +1.14495 q^{71} +0.195472 q^{73} -9.67876 q^{74} -14.4887 q^{76} +1.79356 q^{77} -7.20799 q^{79} +0.247661 q^{80} +2.63255 q^{82} -14.9004 q^{83} -0.185328 q^{85} +16.5739 q^{86} +13.1318 q^{88} -1.55313 q^{89} -2.43642 q^{91} -16.7772 q^{92} -8.62949 q^{94} -0.357945 q^{95} +5.29553 q^{97} +16.0038 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8}+O(q^{10}) \) 6 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 - 6 * q^8	\( 6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8} + 3 q^{10} - 12 q^{11} - 6 q^{14} - 3 q^{16} - 9 q^{17} + 3 q^{19} - 6 q^{20} + 6 q^{22} - 15 q^{23} - 6 q^{25} - 15 q^{26} - 6 q^{28} - 12 q^{29} - 12 q^{35} + 3 q^{37} + 3 q^{38} + 6 q^{40} - 15 q^{41} - 3 q^{44} + 3 q^{46} - 21 q^{47} - 12 q^{49} - 3 q^{50} + 12 q^{52} - 9 q^{53} - 6 q^{55} + 6 q^{56} - 12 q^{58} - 24 q^{59} - 9 q^{61} + 12 q^{62} - 12 q^{64} + 6 q^{65} - 9 q^{67} + 9 q^{68} + 15 q^{70} - 27 q^{71} - 6 q^{73} + 12 q^{74} + 6 q^{76} + 12 q^{77} + 21 q^{80} - 6 q^{82} - 12 q^{83} + 21 q^{86} + 12 q^{88} - 9 q^{89} - 6 q^{91} - 6 q^{92} + 6 q^{94} - 12 q^{95} + 45 q^{98}+O(q^{100}) \) 6 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 - 6 * q^8 + 3 * q^10 - 12 * q^11 - 6 * q^14 - 3 * q^16 - 9 * q^17 + 3 * q^19 - 6 * q^20 + 6 * q^22 - 15 * q^23 - 6 * q^25 - 15 * q^26 - 6 * q^28 - 12 * q^29 - 12 * q^35 + 3 * q^37 + 3 * q^38 + 6 * q^40 - 15 * q^41 - 3 * q^44 + 3 * q^46 - 21 * q^47 - 12 * q^49 - 3 * q^50 + 12 * q^52 - 9 * q^53 - 6 * q^55 + 6 * q^56 - 12 * q^58 - 24 * q^59 - 9 * q^61 + 12 * q^62 - 12 * q^64 + 6 * q^65 - 9 * q^67 + 9 * q^68 + 15 * q^70 - 27 * q^71 - 6 * q^73 + 12 * q^74 + 6 * q^76 + 12 * q^77 + 21 * q^80 - 6 * q^82 - 12 * q^83 + 21 * q^86 + 12 * q^88 - 9 * q^89 - 6 * q^91 - 6 * q^92 + 6 * q^94 - 12 * q^95 + 45 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more