LMFDB - Embedded newform 6561.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6561,2,Mod(1,6561)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6561, 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("6561.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6561 = 3^{8} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6561.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:	$52.3898487662$
Analytic rank:	$1$
Dimension:	$72$
Twist minimal:	no (minimal twist has level 81)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6561.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.76978 q^{2} +5.67166 q^{4} -1.02650 q^{5} +1.57628 q^{7} -10.1697 q^{8} +O(q^{10})$	\(q-2.76978 q^{2} +5.67166 q^{4} -1.02650 q^{5} +1.57628 q^{7} -10.1697 q^{8} +2.84319 q^{10} -0.0651872 q^{11} -0.867704 q^{13} -4.36594 q^{14} +16.8244 q^{16} -4.70464 q^{17} +1.45171 q^{19} -5.82198 q^{20} +0.180554 q^{22} +4.89447 q^{23} -3.94629 q^{25} +2.40335 q^{26} +8.94012 q^{28} +1.10030 q^{29} -5.63858 q^{31} -26.2605 q^{32} +13.0308 q^{34} -1.61806 q^{35} +4.60005 q^{37} -4.02091 q^{38} +10.4392 q^{40} -2.77441 q^{41} +9.25820 q^{43} -0.369720 q^{44} -13.5566 q^{46} +7.25610 q^{47} -4.51535 q^{49} +10.9303 q^{50} -4.92132 q^{52} -2.03831 q^{53} +0.0669149 q^{55} -16.0302 q^{56} -3.04759 q^{58} -8.73279 q^{59} +2.60883 q^{61} +15.6176 q^{62} +39.0869 q^{64} +0.890702 q^{65} -6.35407 q^{67} -26.6831 q^{68} +4.48166 q^{70} +10.1342 q^{71} +2.42588 q^{73} -12.7411 q^{74} +8.23361 q^{76} -0.102753 q^{77} -8.74862 q^{79} -17.2703 q^{80} +7.68448 q^{82} +8.48714 q^{83} +4.82933 q^{85} -25.6431 q^{86} +0.662933 q^{88} +14.7977 q^{89} -1.36774 q^{91} +27.7598 q^{92} -20.0978 q^{94} -1.49019 q^{95} -6.16431 q^{97} +12.5065 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8}+O(q^{10}) $ 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8	$ 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38} - 72 q^{41} - 90 q^{44} - 90 q^{47} + 18 q^{49} - 45 q^{50} - 45 q^{53} + 9 q^{55} - 108 q^{56} + 18 q^{58} - 108 q^{59} - 72 q^{62} + 9 q^{64} - 72 q^{65} - 108 q^{68} - 126 q^{71} - 90 q^{74} - 72 q^{77} - 144 q^{80} - 18 q^{82} - 108 q^{83} - 90 q^{86} - 108 q^{89} - 72 q^{92} - 144 q^{95} - 81 q^{98}+O(q^{100}) $ 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8 - 36 * q^11 - 36 * q^14 + 45 * q^16 - 36 * q^17 - 54 * q^20 - 54 * q^23 + 36 * q^25 - 45 * q^26 + 9 * q^28 - 54 * q^29 - 63 * q^32 - 72 * q^35 - 54 * q^38 - 72 * q^41 - 90 * q^44 - 90 * q^47 + 18 * q^49 - 45 * q^50 - 45 * q^53 + 9 * q^55 - 108 * q^56 + 18 * q^58 - 108 * q^59 - 72 * q^62 + 9 * q^64 - 72 * q^65 - 108 * q^68 - 126 * q^71 - 90 * q^74 - 72 * q^77 - 144 * q^80 - 18 * q^82 - 108 * q^83 - 90 * q^86 - 108 * q^89 - 72 * q^92 - 144 * q^95 - 81 * 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 $

$2$	−2.76978	−1.95853	−0.979264	−	0.202589i	$-0.935064\pi$
$2$	−2.76978	−1.95853	−0.979264	+	0.202589i	$0.935064\pi$
$3$	0	0
$3$	0	0
$4$	5.67166	2.83583
$5$	−1.02650	−0.459067	−0.229533	−	0.973301i	$-0.573720\pi$
$5$	−1.02650	−0.459067	−0.229533	+	0.973301i	$0.573720\pi$
$6$	0	0
$7$	1.57628	0.595777	0.297889	−	0.954601i	$-0.403718\pi$
$7$	1.57628	0.595777	0.297889	+	0.954601i	$0.403718\pi$
$8$	−10.1697	−3.59552
$9$	0	0
$10$	2.84319	0.899095
$11$	−0.0651872	−0.0196547	−0.00982734	−	0.999952i	$-0.503128\pi$
$11$	−0.0651872	−0.0196547	−0.00982734	+	0.999952i	$0.503128\pi$
$12$	0	0
$13$	−0.867704	−0.240658	−0.120329	−	0.992734i	$-0.538395\pi$
$13$	−0.867704	−0.240658	−0.120329	+	0.992734i	$0.538395\pi$
$14$	−4.36594	−1.16685
$15$	0	0
$16$	16.8244	4.20610
$17$	−4.70464	−1.14104	−0.570521	−	0.821283i	$-0.693259\pi$
$17$	−4.70464	−1.14104	−0.570521	+	0.821283i	$0.693259\pi$
$18$	0	0
$19$	1.45171	0.333045	0.166523	−	0.986038i	$-0.446746\pi$
$19$	1.45171	0.333045	0.166523	+	0.986038i	$0.446746\pi$
$20$	−5.82198	−1.30184
$21$	0	0
$22$	0.180554	0.0384942
$23$	4.89447	1.02057	0.510284	−	0.860006i	$-0.329540\pi$
$23$	4.89447	1.02057	0.510284	+	0.860006i	$0.329540\pi$
$24$	0	0
$25$	−3.94629	−0.789258
$26$	2.40335	0.471335
$27$	0	0
$28$	8.94012	1.68952
$29$	1.10030	0.204321	0.102161	−	0.994768i	$-0.467424\pi$
$29$	1.10030	0.204321	0.102161	+	0.994768i	$0.467424\pi$
$30$	0	0
$31$	−5.63858	−1.01272	−0.506359	−	0.862323i	$-0.669009\pi$
$31$	−5.63858	−1.01272	−0.506359	+	0.862323i	$0.669009\pi$
$32$	−26.2605	−4.64224
$33$	0	0
$34$	13.0308	2.23476
$35$	−1.61806	−0.273502
$36$	0	0
$37$	4.60005	0.756243	0.378122	−	0.925756i	$-0.376570\pi$
$37$	4.60005	0.756243	0.378122	+	0.925756i	$0.376570\pi$
$38$	−4.02091	−0.652278
$39$	0	0
$40$	10.4392	1.65059
$41$	−2.77441	−0.433289	−0.216645	−	0.976251i	$-0.569511\pi$
$41$	−2.77441	−0.433289	−0.216645	+	0.976251i	$0.569511\pi$
$42$	0	0
$43$	9.25820	1.41186	0.705931	−	0.708280i	$-0.250529\pi$
$43$	9.25820	1.41186	0.705931	+	0.708280i	$0.250529\pi$
$44$	−0.369720	−0.0557373
$45$	0	0
$46$	−13.5566	−1.99881
$47$	7.25610	1.05841	0.529205	−	0.848494i	$-0.322490\pi$
$47$	7.25610	1.05841	0.529205	+	0.848494i	$0.322490\pi$
$48$	0	0
$49$	−4.51535	−0.645049
$50$	10.9303	1.54578
$51$	0	0
$52$	−4.92132	−0.682465
$53$	−2.03831	−0.279984	−0.139992	−	0.990153i	$-0.544708\pi$
$53$	−2.03831	−0.279984	−0.139992	+	0.990153i	$0.544708\pi$
$54$	0	0
$55$	0.0669149	0.00902281
$56$	−16.0302	−2.14213
$57$	0	0
$58$	−3.04759	−0.400168
$59$	−8.73279	−1.13691	−0.568456	−	0.822714i	$-0.692459\pi$
$59$	−8.73279	−1.13691	−0.568456	+	0.822714i	$0.692459\pi$
$60$	0	0
$61$	2.60883	0.334027	0.167013	−	0.985955i	$-0.446588\pi$
$61$	2.60883	0.334027	0.167013	+	0.985955i	$0.446588\pi$
$62$	15.6176	1.98344
$63$	0	0
$64$	39.0869	4.88586
$65$	0.890702	0.110478
$66$	0	0
$67$	−6.35407	−0.776273	−0.388137	−	0.921602i	$-0.626881\pi$
$67$	−6.35407	−0.776273	−0.388137	+	0.921602i	$0.626881\pi$
$68$	−26.6831	−3.23580
$69$	0	0
$70$	4.48166	0.535660
$71$	10.1342	1.20270	0.601352	−	0.798984i	$-0.294629\pi$
$71$	10.1342	1.20270	0.601352	+	0.798984i	$0.294629\pi$
$72$	0	0
$73$	2.42588	0.283928	0.141964	−	0.989872i	$-0.454658\pi$
$73$	2.42588	0.283928	0.141964	+	0.989872i	$0.454658\pi$
$74$	−12.7411	−1.48112
$75$	0	0
$76$	8.23361	0.944460
$77$	−0.102753	−0.0117098
$78$	0	0
$79$	−8.74862	−0.984297	−0.492148	−	0.870511i	$-0.663788\pi$
$79$	−8.74862	−0.984297	−0.492148	+	0.870511i	$0.663788\pi$
$80$	−17.2703	−1.93088
$81$	0	0
$82$	7.68448	0.848609
$83$	8.48714	0.931584	0.465792	−	0.884894i	$-0.345769\pi$
$83$	8.48714	0.931584	0.465792	+	0.884894i	$0.345769\pi$
$84$	0	0
$85$	4.82933	0.523815
$86$	−25.6431	−2.76517
$87$	0	0
$88$	0.662933	0.0706689
$89$	14.7977	1.56856	0.784278	−	0.620409i	$-0.213034\pi$
$89$	14.7977	1.56856	0.784278	+	0.620409i	$0.213034\pi$
$90$	0	0
$91$	−1.36774	−0.143378
$92$	27.7598	2.89416
$93$	0	0
$94$	−20.0978	−2.07293
$95$	−1.49019	−0.152890
$96$	0	0
$97$	−6.16431	−0.625891	−0.312946	−	0.949771i	$-0.601316\pi$
$97$	−6.16431	−0.625891	−0.312946	+	0.949771i	$0.601316\pi$
$98$	12.5065	1.26335
$99$	0	0
$100$	−22.3820	−2.23820
$101$	−3.37634	−0.335958	−0.167979	−	0.985791i	$-0.553724\pi$
$101$	−3.37634	−0.335958	−0.167979	+	0.985791i	$0.553724\pi$
$102$	0	0
$103$	18.4329	1.81625	0.908124	−	0.418702i	$-0.137515\pi$
$103$	18.4329	1.81625	0.908124	+	0.418702i	$0.137515\pi$
$104$	8.82427	0.865291
$105$	0	0
$106$	5.64567	0.548356
$107$	−12.0674	−1.16660	−0.583299	−	0.812258i	$-0.698238\pi$
$107$	−12.0674	−1.16660	−0.583299	+	0.812258i	$0.698238\pi$
$108$	0	0
$109$	−7.03651	−0.673975	−0.336988	−	0.941509i	$-0.609408\pi$
$109$	−7.03651	−0.673975	−0.336988	+	0.941509i	$0.609408\pi$
$110$	−0.185339	−0.0176714
$111$	0	0
$112$	26.5200	2.50590
$113$	1.00574	0.0946124	0.0473062	−	0.998880i	$-0.484936\pi$
$113$	1.00574	0.0946124	0.0473062	+	0.998880i	$0.484936\pi$
$114$	0	0
$115$	−5.02420	−0.468509
$116$	6.24054	0.579420
$117$	0	0
$118$	24.1879	2.22667
$119$	−7.41582	−0.679807
$120$	0	0
$121$	−10.9958	−0.999614
$122$	−7.22589	−0.654201
$123$	0	0
$124$	−31.9801	−2.87190
$125$	9.18340	0.821389
$126$	0	0
$127$	12.6910	1.12615	0.563073	−	0.826407i	$-0.309619\pi$
$127$	12.6910	1.12615	0.563073	+	0.826407i	$0.309619\pi$
$128$	−55.7409	−4.92685
$129$	0	0
$130$	−2.46704	−0.216374
$131$	−15.3035	−1.33708	−0.668538	−	0.743678i	$-0.733080\pi$
$131$	−15.3035	−1.33708	−0.668538	+	0.743678i	$0.733080\pi$
$132$	0	0
$133$	2.28830	0.198421
$134$	17.5994	1.52035
$135$	0	0
$136$	47.8447	4.10265
$137$	−9.44344	−0.806808	−0.403404	−	0.915022i	$-0.632173\pi$
$137$	−9.44344	−0.806808	−0.403404	+	0.915022i	$0.632173\pi$
$138$	0	0
$139$	−5.27932	−0.447786	−0.223893	−	0.974614i	$-0.571877\pi$
$139$	−5.27932	−0.447786	−0.223893	+	0.974614i	$0.571877\pi$
$140$	−9.17707	−0.775604
$141$	0	0
$142$	−28.0694	−2.35553
$143$	0.0565632	0.00473005
$144$	0	0
$145$	−1.12947	−0.0937970
$146$	−6.71915	−0.556081
$147$	0	0
$148$	26.0899	2.14458
$149$	13.0595	1.06987	0.534937	−	0.844892i	$-0.320336\pi$
$149$	13.0595	1.06987	0.534937	+	0.844892i	$0.320336\pi$
$150$	0	0
$151$	−15.6294	−1.27190	−0.635951	−	0.771729i	$-0.719392\pi$
$151$	−15.6294	−1.27190	−0.635951	+	0.771729i	$0.719392\pi$
$152$	−14.7634	−1.19747
$153$	0	0
$154$	0.284603	0.0229340
$155$	5.78803	0.464905
$156$	0	0
$157$	−5.94939	−0.474813	−0.237406	−	0.971410i	$-0.576297\pi$
$157$	−5.94939	−0.474813	−0.237406	+	0.971410i	$0.576297\pi$
$158$	24.2317	1.92777
$159$	0	0
$160$	26.9565	2.13110
$161$	7.71505	0.608031
$162$	0	0
$163$	11.0829	0.868081	0.434041	−	0.900893i	$-0.357087\pi$
$163$	11.0829	0.868081	0.434041	+	0.900893i	$0.357087\pi$
$164$	−15.7355	−1.22873
$165$	0	0
$166$	−23.5075	−1.82453
$167$	−5.01481	−0.388057	−0.194029	−	0.980996i	$-0.562155\pi$
$167$	−5.01481	−0.388057	−0.194029	+	0.980996i	$0.562155\pi$
$168$	0	0
$169$	−12.2471	−0.942084
$170$	−13.3762	−1.02591
$171$	0	0
$172$	52.5094	4.00380
$173$	−5.14385	−0.391080	−0.195540	−	0.980696i	$-0.562646\pi$
$173$	−5.14385	−0.391080	−0.195540	+	0.980696i	$0.562646\pi$
$174$	0	0
$175$	−6.22045	−0.470222
$176$	−1.09674	−0.0826696
$177$	0	0
$178$	−40.9864	−3.07206
$179$	−2.37449	−0.177478	−0.0887389	−	0.996055i	$-0.528284\pi$
$179$	−2.37449	−0.177478	−0.0887389	+	0.996055i	$0.528284\pi$
$180$	0	0
$181$	−20.9493	−1.55715	−0.778573	−	0.627554i	$-0.784056\pi$
$181$	−20.9493	−1.55715	−0.778573	+	0.627554i	$0.784056\pi$
$182$	3.78834	0.280811
$183$	0	0
$184$	−49.7752	−3.66948
$185$	−4.72197	−0.347166
$186$	0	0
$187$	0.306682	0.0224268
$188$	41.1541	3.00147
$189$	0	0
$190$	4.12749	0.299439
$191$	9.98001	0.722128	0.361064	−	0.932541i	$-0.382414\pi$
$191$	9.98001	0.722128	0.361064	+	0.932541i	$0.382414\pi$
$192$	0	0
$193$	−25.7797	−1.85567	−0.927833	−	0.372995i	$-0.878331\pi$
$193$	−25.7797	−1.85567	−0.927833	+	0.372995i	$0.878331\pi$
$194$	17.0738	1.22583
$195$	0	0
$196$	−25.6095	−1.82925
$197$	−20.4214	−1.45497	−0.727484	−	0.686125i	$-0.759310\pi$
$197$	−20.4214	−1.45497	−0.727484	+	0.686125i	$0.759310\pi$
$198$	0	0
$199$	6.45863	0.457840	0.228920	−	0.973445i	$-0.426481\pi$
$199$	6.45863	0.457840	0.228920	+	0.973445i	$0.426481\pi$
$200$	40.1325	2.83780
$201$	0	0
$202$	9.35169	0.657983
$203$	1.73438	0.121730
$204$	0	0
$205$	2.84794	0.198909
$206$	−51.0550	−3.55717
$207$	0	0
$208$	−14.5986	−1.01223
$209$	−0.0946329	−0.00654590
$210$	0	0
$211$	6.98586	0.480927	0.240463	−	0.970658i	$-0.422701\pi$
$211$	6.98586	0.480927	0.240463	+	0.970658i	$0.422701\pi$
$212$	−11.5606	−0.793987
$213$	0	0
$214$	33.4239	2.28481
$215$	−9.50358	−0.648139
$216$	0	0
$217$	−8.88797	−0.603355
$218$	19.4896	1.32000
$219$	0	0
$220$	0.379519	0.0255871
$221$	4.08223	0.274601
$222$	0	0
$223$	25.2814	1.69297	0.846483	−	0.532416i	$-0.178716\pi$
$223$	25.2814	1.69297	0.846483	+	0.532416i	$0.178716\pi$
$224$	−41.3939	−2.76574
$225$	0	0
$226$	−2.78569	−0.185301
$227$	25.6191	1.70040	0.850200	−	0.526460i	$-0.176481\pi$
$227$	25.6191	1.70040	0.850200	+	0.526460i	$0.176481\pi$
$228$	0	0
$229$	−21.7624	−1.43810	−0.719050	−	0.694959i	$-0.755423\pi$
$229$	−21.7624	−1.43810	−0.719050	+	0.694959i	$0.755423\pi$
$230$	13.9159	0.917588
$231$	0	0
$232$	−11.1897	−0.734641
$233$	−0.335566	−0.0219837	−0.0109918	−	0.999940i	$-0.503499\pi$
$233$	−0.335566	−0.0219837	−0.0109918	+	0.999940i	$0.503499\pi$
$234$	0	0
$235$	−7.44842	−0.485881
$236$	−49.5294	−3.22409
$237$	0	0
$238$	20.5402	1.33142
$239$	−11.2702	−0.729007	−0.364503	−	0.931202i	$-0.618761\pi$
$239$	−11.2702	−0.729007	−0.364503	+	0.931202i	$0.618761\pi$
$240$	0	0
$241$	5.90282	0.380234	0.190117	−	0.981761i	$-0.439113\pi$
$241$	5.90282	0.380234	0.190117	+	0.981761i	$0.439113\pi$
$242$	30.4558	1.95777
$243$	0	0
$244$	14.7964	0.947244
$245$	4.63502	0.296121
$246$	0	0
$247$	−1.25966	−0.0801499
$248$	57.3425	3.64125
$249$	0	0
$250$	−25.4360	−1.60871
$251$	−6.89779	−0.435385	−0.217692	−	0.976017i	$-0.569853\pi$
$251$	−6.89779	−0.435385	−0.217692	+	0.976017i	$0.569853\pi$
$252$	0	0
$253$	−0.319057	−0.0200589
$254$	−35.1513	−2.20559
$255$	0	0
$256$	76.2161	4.76351
$257$	−22.2297	−1.38665	−0.693326	−	0.720624i	$-0.743855\pi$
$257$	−22.2297	−1.38665	−0.693326	+	0.720624i	$0.743855\pi$
$258$	0	0
$259$	7.25096	0.450553
$260$	5.05176	0.313297
$261$	0	0
$262$	42.3874	2.61870
$263$	6.46654	0.398744	0.199372	−	0.979924i	$-0.436110\pi$
$263$	6.46654	0.398744	0.199372	+	0.979924i	$0.436110\pi$
$264$	0	0
$265$	2.09234	0.128531
$266$	−6.33808	−0.388613
$267$	0	0
$268$	−36.0381	−2.20138
$269$	−21.8425	−1.33176	−0.665881	−	0.746058i	$-0.731944\pi$
$269$	−21.8425	−1.33176	−0.665881	+	0.746058i	$0.731944\pi$
$270$	0	0
$271$	9.08253	0.551725	0.275862	−	0.961197i	$-0.411037\pi$
$271$	9.08253	0.551725	0.275862	+	0.961197i	$0.411037\pi$
$272$	−79.1528	−4.79934
$273$	0	0
$274$	26.1562	1.58016
$275$	0.257247	0.0155126
$276$	0	0
$277$	19.3722	1.16396	0.581982	−	0.813202i	$-0.302277\pi$
$277$	19.3722	1.16396	0.581982	+	0.813202i	$0.302277\pi$
$278$	14.6225	0.877002
$279$	0	0
$280$	16.4551	0.983381
$281$	−21.2821	−1.26958	−0.634792	−	0.772683i	$-0.718914\pi$
$281$	−21.2821	−1.26958	−0.634792	+	0.772683i	$0.718914\pi$
$282$	0	0
$283$	−5.27010	−0.313275	−0.156637	−	0.987656i	$-0.550065\pi$
$283$	−5.27010	−0.313275	−0.156637	+	0.987656i	$0.550065\pi$
$284$	57.4775	3.41066
$285$	0	0
$286$	−0.156667	−0.00926393
$287$	−4.37324	−0.258144
$288$	0	0
$289$	5.13363	0.301978
$290$	3.12837	0.183704
$291$	0	0
$292$	13.7588	0.805171
$293$	−8.96285	−0.523615	−0.261808	−	0.965120i	$-0.584319\pi$
$293$	−8.96285	−0.523615	−0.261808	+	0.965120i	$0.584319\pi$
$294$	0	0
$295$	8.96424	0.521918
$296$	−46.7810	−2.71909
$297$	0	0
$298$	−36.1718	−2.09538
$299$	−4.24695	−0.245608
$300$	0	0
$301$	14.5935	0.841156
$302$	43.2900	2.49106
$303$	0	0
$304$	24.4242	1.40082
$305$	−2.67798	−0.153341
$306$	0	0
$307$	6.94196	0.396198	0.198099	−	0.980182i	$-0.436523\pi$
$307$	6.94196	0.396198	0.198099	+	0.980182i	$0.436523\pi$
$308$	−0.582781	−0.0332070
$309$	0	0
$310$	−16.0315	−0.910530
$311$	15.6636	0.888203	0.444102	−	0.895976i	$-0.353523\pi$
$311$	15.6636	0.888203	0.444102	+	0.895976i	$0.353523\pi$
$312$	0	0
$313$	15.1016	0.853590	0.426795	−	0.904348i	$-0.359643\pi$
$313$	15.1016	0.853590	0.426795	+	0.904348i	$0.359643\pi$
$314$	16.4785	0.929934
$315$	0	0
$316$	−49.6192	−2.79130
$317$	16.9334	0.951073	0.475536	−	0.879696i	$-0.342254\pi$
$317$	16.9334	0.951073	0.475536	+	0.879696i	$0.342254\pi$
$318$	0	0
$319$	−0.0717256	−0.00401586
$320$	−40.1229	−2.24294
$321$	0	0
$322$	−21.3690	−1.19085
$323$	−6.82978	−0.380019
$324$	0	0
$325$	3.42421	0.189941
$326$	−30.6972	−1.70016
$327$	0	0
$328$	28.2148	1.55790
$329$	11.4376	0.630577
$330$	0	0
$331$	−12.3258	−0.677489	−0.338744	−	0.940878i	$-0.610002\pi$
$331$	−12.3258	−0.677489	−0.338744	+	0.940878i	$0.610002\pi$
$332$	48.1362	2.64182
$333$	0	0
$334$	13.8899	0.760021
$335$	6.52248	0.356361
$336$	0	0
$337$	−30.6618	−1.67026	−0.835128	−	0.550056i	$-0.814606\pi$
$337$	−30.6618	−1.67026	−0.835128	+	0.550056i	$0.814606\pi$
$338$	33.9217	1.84510
$339$	0	0
$340$	27.3903	1.48545
$341$	0.367563	0.0199047
$342$	0	0
$343$	−18.1514	−0.980083
$344$	−94.1529	−5.07639
$345$	0	0
$346$	14.2473	0.765940
$347$	19.1510	1.02808	0.514041	−	0.857766i	$-0.328148\pi$
$347$	19.1510	1.02808	0.514041	+	0.857766i	$0.328148\pi$
$348$	0	0
$349$	9.25035	0.495160	0.247580	−	0.968867i	$-0.420365\pi$
$349$	9.25035	0.495160	0.247580	+	0.968867i	$0.420365\pi$
$350$	17.2293	0.920943
$351$	0	0
$352$	1.71185	0.0912418
$353$	9.71544	0.517101	0.258550	−	0.965998i	$-0.416755\pi$
$353$	9.71544	0.517101	0.258550	+	0.965998i	$0.416755\pi$
$354$	0	0
$355$	−10.4028	−0.552121
$356$	83.9277	4.44816
$357$	0	0
$358$	6.57681	0.347595
$359$	21.1030	1.11377	0.556887	−	0.830589i	$-0.311996\pi$
$359$	21.1030	1.11377	0.556887	+	0.830589i	$0.311996\pi$
$360$	0	0
$361$	−16.8925	−0.889081
$362$	58.0247	3.04971
$363$	0	0
$364$	−7.75738	−0.406597
$365$	−2.49018	−0.130342
$366$	0	0
$367$	15.9885	0.834594	0.417297	−	0.908770i	$-0.362978\pi$
$367$	15.9885	0.834594	0.417297	+	0.908770i	$0.362978\pi$
$368$	82.3466	4.29262
$369$	0	0
$370$	13.0788	0.679934
$371$	−3.21295	−0.166808
$372$	0	0
$373$	22.8172	1.18143	0.590716	−	0.806880i	$-0.298846\pi$
$373$	22.8172	1.18143	0.590716	+	0.806880i	$0.298846\pi$
$374$	−0.849441	−0.0439235
$375$	0	0
$376$	−73.7922	−3.80554
$377$	−0.954737	−0.0491715
$378$	0	0
$379$	7.67781	0.394383	0.197191	−	0.980365i	$-0.436818\pi$
$379$	7.67781	0.394383	0.197191	+	0.980365i	$0.436818\pi$
$380$	−8.45184	−0.433570
$381$	0	0
$382$	−27.6424	−1.41431
$383$	5.33193	0.272449	0.136225	−	0.990678i	$-0.456503\pi$
$383$	5.33193	0.272449	0.136225	+	0.990678i	$0.456503\pi$
$384$	0	0
$385$	0.105477	0.00537558
$386$	71.4041	3.63437
$387$	0	0
$388$	−34.9619	−1.77492
$389$	−24.5641	−1.24545	−0.622725	−	0.782441i	$-0.713974\pi$
$389$	−24.5641	−1.24545	−0.622725	+	0.782441i	$0.713974\pi$
$390$	0	0
$391$	−23.0267	−1.16451
$392$	45.9196	2.31929
$393$	0	0
$394$	56.5628	2.84959
$395$	8.98050	0.451858
$396$	0	0
$397$	19.6178	0.984592	0.492296	−	0.870428i	$-0.336158\pi$
$397$	19.6178	0.984592	0.492296	+	0.870428i	$0.336158\pi$
$398$	−17.8890	−0.896693
$399$	0	0
$400$	−66.3940	−3.31970
$401$	29.0700	1.45169	0.725844	−	0.687859i	$-0.241449\pi$
$401$	29.0700	1.45169	0.725844	+	0.687859i	$0.241449\pi$
$402$	0	0
$403$	4.89262	0.243719
$404$	−19.1494	−0.952720
$405$	0	0
$406$	−4.80385	−0.238411
$407$	−0.299864	−0.0148637
$408$	0	0
$409$	−22.1964	−1.09754	−0.548771	−	0.835973i	$-0.684904\pi$
$409$	−22.1964	−1.09754	−0.548771	+	0.835973i	$0.684904\pi$
$410$	−7.88815	−0.389568
$411$	0	0
$412$	104.545	5.15057
$413$	−13.7653	−0.677346
$414$	0	0
$415$	−8.71208	−0.427659
$416$	22.7863	1.11719
$417$	0	0
$418$	0.262112	0.0128203
$419$	−1.50488	−0.0735184	−0.0367592	−	0.999324i	$-0.511703\pi$
$419$	−1.50488	−0.0735184	−0.0367592	+	0.999324i	$0.511703\pi$
$420$	0	0
$421$	15.3117	0.746246	0.373123	−	0.927782i	$-0.378287\pi$
$421$	15.3117	0.746246	0.373123	+	0.927782i	$0.378287\pi$
$422$	−19.3493	−0.941908
$423$	0	0
$424$	20.7290	1.00669
$425$	18.5659	0.900577
$426$	0	0
$427$	4.11225	0.199006
$428$	−68.4421	−3.30827
$429$	0	0
$430$	26.3228	1.26940
$431$	−12.1143	−0.583526	−0.291763	−	0.956491i	$-0.594242\pi$
$431$	−12.1143	−0.583526	−0.291763	+	0.956491i	$0.594242\pi$
$432$	0	0
$433$	−27.1109	−1.30287	−0.651433	−	0.758707i	$-0.725832\pi$
$433$	−27.1109	−1.30287	−0.651433	+	0.758707i	$0.725832\pi$
$434$	24.6177	1.18169
$435$	0	0
$436$	−39.9087	−1.91128
$437$	7.10536	0.339895
$438$	0	0
$439$	−16.0586	−0.766434	−0.383217	−	0.923658i	$-0.625184\pi$
$439$	−16.0586	−0.766434	−0.383217	+	0.923658i	$0.625184\pi$
$440$	−0.680503	−0.0324417
$441$	0	0
$442$	−11.3069	−0.537813
$443$	−11.9088	−0.565803	−0.282901	−	0.959149i	$-0.591297\pi$
$443$	−11.9088	−0.565803	−0.282901	+	0.959149i	$0.591297\pi$
$444$	0	0
$445$	−15.1899	−0.720072
$446$	−70.0237	−3.31572
$447$	0	0
$448$	61.6118	2.91089
$449$	5.47513	0.258387	0.129194	−	0.991619i	$-0.458761\pi$
$449$	5.47513	0.258387	0.129194	+	0.991619i	$0.458761\pi$
$450$	0	0
$451$	0.180856	0.00851616
$452$	5.70424	0.268305
$453$	0	0
$454$	−70.9592	−3.33028
$455$	1.40399	0.0658203
$456$	0	0
$457$	−10.6164	−0.496616	−0.248308	−	0.968681i	$-0.579874\pi$
$457$	−10.6164	−0.496616	−0.248308	+	0.968681i	$0.579874\pi$
$458$	60.2770	2.81656
$459$	0	0
$460$	−28.4955	−1.32861
$461$	−8.85220	−0.412288	−0.206144	−	0.978522i	$-0.566092\pi$
$461$	−8.85220	−0.412288	−0.206144	+	0.978522i	$0.566092\pi$
$462$	0	0
$463$	−33.4362	−1.55391	−0.776956	−	0.629555i	$-0.783237\pi$
$463$	−33.4362	−1.55391	−0.776956	+	0.629555i	$0.783237\pi$
$464$	18.5119	0.859396
$465$	0	0
$466$	0.929444	0.0430557
$467$	−24.9577	−1.15491	−0.577453	−	0.816424i	$-0.695953\pi$
$467$	−24.9577	−1.15491	−0.577453	+	0.816424i	$0.695953\pi$
$468$	0	0
$469$	−10.0158	−0.462486
$470$	20.6305	0.951612
$471$	0	0
$472$	88.8096	4.08779
$473$	−0.603516	−0.0277497
$474$	0	0
$475$	−5.72887	−0.262859
$476$	−42.0600	−1.92782
$477$	0	0
$478$	31.2158	1.42778
$479$	6.91527	0.315967	0.157983	−	0.987442i	$-0.449501\pi$
$479$	6.91527	0.315967	0.157983	+	0.987442i	$0.449501\pi$
$480$	0	0
$481$	−3.99148	−0.181996
$482$	−16.3495	−0.744699
$483$	0	0
$484$	−62.3642	−2.83473
$485$	6.32769	0.287326
$486$	0	0
$487$	−5.49461	−0.248985	−0.124492	−	0.992221i	$-0.539730\pi$
$487$	−5.49461	−0.248985	−0.124492	+	0.992221i	$0.539730\pi$
$488$	−26.5310	−1.20100
$489$	0	0
$490$	−12.8380	−0.579961
$491$	21.6996	0.979290	0.489645	−	0.871922i	$-0.337126\pi$
$491$	21.6996	0.979290	0.489645	+	0.871922i	$0.337126\pi$
$492$	0	0
$493$	−5.17653	−0.233139
$494$	3.48896	0.156976
$495$	0	0
$496$	−94.8658	−4.25960
$497$	15.9743	0.716544
$498$	0	0
$499$	−25.4696	−1.14018	−0.570088	−	0.821584i	$-0.693091\pi$
$499$	−25.4696	−1.14018	−0.570088	+	0.821584i	$0.693091\pi$
$500$	52.0852	2.32932
$501$	0	0
$502$	19.1053	0.852713
$503$	17.1716	0.765645	0.382822	−	0.923822i	$-0.374952\pi$
$503$	17.1716	0.765645	0.382822	+	0.923822i	$0.374952\pi$
$504$	0	0
$505$	3.46582	0.154227
$506$	0.883716	0.0392860
$507$	0	0
$508$	71.9791	3.19356
$509$	−2.55583	−0.113285	−0.0566426	−	0.998395i	$-0.518040\pi$
$509$	−2.55583	−0.113285	−0.0566426	+	0.998395i	$0.518040\pi$
$510$	0	0
$511$	3.82387	0.169158
$512$	−99.6198	−4.40261
$513$	0	0
$514$	61.5713	2.71579
$515$	−18.9214	−0.833779
$516$	0	0
$517$	−0.473005	−0.0208027
$518$	−20.0835	−0.882420
$519$	0	0
$520$	−9.05815	−0.397226
$521$	−20.9658	−0.918527	−0.459263	−	0.888300i	$-0.651887\pi$
$521$	−20.9658	−0.918527	−0.459263	+	0.888300i	$0.651887\pi$
$522$	0	0
$523$	8.55805	0.374217	0.187109	−	0.982339i	$-0.440088\pi$
$523$	8.55805	0.374217	0.187109	+	0.982339i	$0.440088\pi$
$524$	−86.7965	−3.79172
$525$	0	0
$526$	−17.9109	−0.780950
$527$	26.5275	1.15556
$528$	0	0
$529$	0.955871	0.0415596
$530$	−5.79531	−0.251732
$531$	0	0
$532$	12.9785	0.562688
$533$	2.40736	0.104274
$534$	0	0
$535$	12.3872	0.535546
$536$	64.6189	2.79111
$537$	0	0
$538$	60.4989	2.60829
$539$	0.294343	0.0126782
$540$	0	0
$541$	28.6896	1.23346	0.616732	−	0.787173i	$-0.288456\pi$
$541$	28.6896	1.23346	0.616732	+	0.787173i	$0.288456\pi$
$542$	−25.1566	−1.08057
$543$	0	0
$544$	123.546	5.29700
$545$	7.22301	0.309400
$546$	0	0
$547$	−13.0843	−0.559444	−0.279722	−	0.960081i	$-0.590242\pi$
$547$	−13.0843	−0.559444	−0.279722	+	0.960081i	$0.590242\pi$
$548$	−53.5600	−2.28797
$549$	0	0
$550$	−0.712518	−0.0303819
$551$	1.59732	0.0680482
$552$	0	0
$553$	−13.7903	−0.586422
$554$	−53.6567	−2.27966
$555$	0	0
$556$	−29.9425	−1.26985
$557$	−14.7451	−0.624770	−0.312385	−	0.949956i	$-0.601128\pi$
$557$	−14.7451	−0.624770	−0.312385	+	0.949956i	$0.601128\pi$
$558$	0	0
$559$	−8.03338	−0.339776
$560$	−27.2229	−1.15038
$561$	0	0
$562$	58.9467	2.48651
$563$	−37.8745	−1.59622	−0.798111	−	0.602511i	$-0.794167\pi$
$563$	−37.8745	−1.59622	−0.798111	+	0.602511i	$0.794167\pi$
$564$	0	0
$565$	−1.03240	−0.0434334
$566$	14.5970	0.613558
$567$	0	0
$568$	−103.061	−4.32435
$569$	42.6869	1.78953	0.894764	−	0.446539i	$-0.147343\pi$
$569$	42.6869	1.78953	0.894764	+	0.446539i	$0.147343\pi$
$570$	0	0
$571$	−3.39061	−0.141893	−0.0709463	−	0.997480i	$-0.522602\pi$
$571$	−3.39061	−0.141893	−0.0709463	+	0.997480i	$0.522602\pi$
$572$	0.320807	0.0134136
$573$	0	0
$574$	12.1129	0.505582
$575$	−19.3150	−0.805491
$576$	0	0
$577$	−12.7811	−0.532084	−0.266042	−	0.963961i	$-0.585716\pi$
$577$	−12.7811	−0.532084	−0.266042	+	0.963961i	$0.585716\pi$
$578$	−14.2190	−0.591432
$579$	0	0
$580$	−6.40594	−0.265992
$581$	13.3781	0.555017
$582$	0	0
$583$	0.132872	0.00550299
$584$	−24.6704	−1.02087
$585$	0	0
$586$	24.8251	1.02551
$587$	0.575993	0.0237738	0.0118869	−	0.999929i	$-0.496216\pi$
$587$	0.575993	0.0237738	0.0118869	+	0.999929i	$0.496216\pi$
$588$	0	0
$589$	−8.18559	−0.337281
$590$	−24.8289	−1.02219
$591$	0	0
$592$	77.3931	3.18084
$593$	−11.0775	−0.454897	−0.227449	−	0.973790i	$-0.573038\pi$
$593$	−11.0775	−0.454897	−0.227449	+	0.973790i	$0.573038\pi$
$594$	0	0
$595$	7.61237	0.312077
$596$	74.0689	3.03398
$597$	0	0
$598$	11.7631	0.481029
$599$	−20.6148	−0.842298	−0.421149	−	0.906991i	$-0.638373\pi$
$599$	−20.6148	−0.842298	−0.421149	+	0.906991i	$0.638373\pi$
$600$	0	0
$601$	8.61329	0.351343	0.175672	−	0.984449i	$-0.443790\pi$
$601$	8.61329	0.351343	0.175672	+	0.984449i	$0.443790\pi$
$602$	−40.4207	−1.64743
$603$	0	0
$604$	−88.6447	−3.60690
$605$	11.2872	0.458889
$606$	0	0
$607$	−33.3602	−1.35405	−0.677025	−	0.735960i	$-0.736731\pi$
$607$	−33.3602	−1.35405	−0.677025	+	0.735960i	$0.736731\pi$
$608$	−38.1227	−1.54608
$609$	0	0
$610$	7.41740	0.300322
$611$	−6.29615	−0.254715
$612$	0	0
$613$	−21.8223	−0.881393	−0.440696	−	0.897656i	$-0.645268\pi$
$613$	−21.8223	−0.881393	−0.440696	+	0.897656i	$0.645268\pi$
$614$	−19.2277	−0.775966
$615$	0	0
$616$	1.04497	0.0421029
$617$	30.9823	1.24730	0.623650	−	0.781704i	$-0.285649\pi$
$617$	30.9823	1.24730	0.623650	+	0.781704i	$0.285649\pi$
$618$	0	0
$619$	−29.4947	−1.18549	−0.592745	−	0.805390i	$-0.701956\pi$
$619$	−29.4947	−1.18549	−0.592745	+	0.805390i	$0.701956\pi$
$620$	32.8277	1.31839
$621$	0	0
$622$	−43.3848	−1.73957
$623$	23.3253	0.934510
$624$	0	0
$625$	10.3046	0.412186
$626$	−41.8279	−1.67178
$627$	0	0
$628$	−33.7429	−1.34649
$629$	−21.6416	−0.862906
$630$	0	0
$631$	29.5214	1.17523	0.587615	−	0.809141i	$-0.300067\pi$
$631$	29.5214	1.17523	0.587615	+	0.809141i	$0.300067\pi$
$632$	88.9707	3.53906
$633$	0	0
$634$	−46.9016	−1.86270
$635$	−13.0274	−0.516976
$636$	0	0
$637$	3.91798	0.155236
$638$	0.198664	0.00786518
$639$	0	0
$640$	57.2183	2.26175
$641$	−19.8458	−0.783863	−0.391931	−	0.919994i	$-0.628193\pi$
$641$	−19.8458	−0.783863	−0.391931	+	0.919994i	$0.628193\pi$
$642$	0	0
$643$	31.5387	1.24376	0.621882	−	0.783111i	$-0.286368\pi$
$643$	31.5387	1.24376	0.621882	+	0.783111i	$0.286368\pi$
$644$	43.7572	1.72427
$645$	0	0
$646$	18.9169	0.744277
$647$	−12.1273	−0.476772	−0.238386	−	0.971170i	$-0.576618\pi$
$647$	−12.1273	−0.476772	−0.238386	+	0.971170i	$0.576618\pi$
$648$	0	0
$649$	0.569266	0.0223456
$650$	−9.48430	−0.372005
$651$	0	0
$652$	62.8586	2.46173
$653$	−14.3644	−0.562124	−0.281062	−	0.959690i	$-0.590687\pi$
$653$	−14.3644	−0.562124	−0.281062	+	0.959690i	$0.590687\pi$
$654$	0	0
$655$	15.7092	0.613807
$656$	−46.6777	−1.82246
$657$	0	0
$658$	−31.6797	−1.23500
$659$	11.0388	0.430010	0.215005	−	0.976613i	$-0.431023\pi$
$659$	11.0388	0.430010	0.215005	+	0.976613i	$0.431023\pi$
$660$	0	0
$661$	2.91712	0.113463	0.0567314	−	0.998389i	$-0.481932\pi$
$661$	2.91712	0.113463	0.0567314	+	0.998389i	$0.481932\pi$
$662$	34.1398	1.32688
$663$	0	0
$664$	−86.3115	−3.34953
$665$	−2.34895	−0.0910884
$666$	0	0
$667$	5.38540	0.208524
$668$	−28.4423	−1.10046
$669$	0	0
$670$	−18.0658	−0.697943
$671$	−0.170063	−0.00656519
$672$	0	0
$673$	−20.8690	−0.804440	−0.402220	−	0.915543i	$-0.631761\pi$
$673$	−20.8690	−0.804440	−0.402220	+	0.915543i	$0.631761\pi$
$674$	84.9264	3.27124
$675$	0	0
$676$	−69.4613	−2.67159
$677$	−8.55673	−0.328862	−0.164431	−	0.986389i	$-0.552579\pi$
$677$	−8.55673	−0.328862	−0.164431	+	0.986389i	$0.552579\pi$
$678$	0	0
$679$	−9.71668	−0.372892
$680$	−49.1128	−1.88339
$681$	0	0
$682$	−1.01807	−0.0389838
$683$	−1.49428	−0.0571770	−0.0285885	−	0.999591i	$-0.509101\pi$
$683$	−1.49428	−0.0571770	−0.0285885	+	0.999591i	$0.509101\pi$
$684$	0	0
$685$	9.69374	0.370379
$686$	50.2753	1.91952
$687$	0	0
$688$	155.764	5.93844
$689$	1.76865	0.0673803
$690$	0	0
$691$	2.32693	0.0885205	0.0442603	−	0.999020i	$-0.485907\pi$
$691$	2.32693	0.0885205	0.0442603	+	0.999020i	$0.485907\pi$
$692$	−29.1742	−1.10904
$693$	0	0
$694$	−53.0441	−2.01353
$695$	5.41925	0.205564
$696$	0	0
$697$	13.0526	0.494401
$698$	−25.6214	−0.969785
$699$	0	0
$700$	−35.2803	−1.33347
$701$	−39.9569	−1.50915	−0.754575	−	0.656214i	$-0.772157\pi$
$701$	−39.9569	−1.50915	−0.754575	+	0.656214i	$0.772157\pi$
$702$	0	0
$703$	6.67794	0.251863
$704$	−2.54796	−0.0960300
$705$	0	0
$706$	−26.9096	−1.01276
$707$	−5.32205	−0.200156
$708$	0	0
$709$	−20.8458	−0.782879	−0.391440	−	0.920204i	$-0.628023\pi$
$709$	−20.8458	−0.782879	−0.391440	+	0.920204i	$0.628023\pi$
$710$	28.8133	1.08134
$711$	0	0
$712$	−150.488	−5.63978
$713$	−27.5979	−1.03355
$714$	0	0
$715$	−0.0580623	−0.00217141
$716$	−13.4673	−0.503297
$717$	0	0
$718$	−58.4506	−2.18136
$719$	−2.68683	−0.100202	−0.0501009	−	0.998744i	$-0.515954\pi$
$719$	−2.68683	−0.100202	−0.0501009	+	0.998744i	$0.515954\pi$
$720$	0	0
$721$	29.0554	1.08208
$722$	46.7885	1.74129
$723$	0	0
$724$	−118.817	−4.41580
$725$	−4.34211	−0.161262
$726$	0	0
$727$	45.4478	1.68557	0.842783	−	0.538254i	$-0.180916\pi$
$727$	45.4478	1.68557	0.842783	+	0.538254i	$0.180916\pi$
$728$	13.9095	0.515521
$729$	0	0
$730$	6.89724	0.255278
$731$	−43.5565	−1.61099
$732$	0	0
$733$	−26.1099	−0.964391	−0.482195	−	0.876064i	$-0.660160\pi$
$733$	−26.1099	−0.964391	−0.482195	+	0.876064i	$0.660160\pi$
$734$	−44.2846	−1.63457
$735$	0	0
$736$	−128.531	−4.73773
$737$	0.414204	0.0152574
$738$	0	0
$739$	−5.01174	−0.184360	−0.0921799	−	0.995742i	$-0.529384\pi$
$739$	−5.01174	−0.184360	−0.0921799	+	0.995742i	$0.529384\pi$
$740$	−26.7814	−0.984504
$741$	0	0
$742$	8.89915	0.326698
$743$	−25.6085	−0.939484	−0.469742	−	0.882804i	$-0.655653\pi$
$743$	−25.6085	−0.939484	−0.469742	+	0.882804i	$0.655653\pi$
$744$	0	0
$745$	−13.4056	−0.491143
$746$	−63.1986	−2.31387
$747$	0	0
$748$	1.73940	0.0635987
$749$	−19.0215	−0.695032
$750$	0	0
$751$	3.54567	0.129384	0.0646918	−	0.997905i	$-0.479394\pi$
$751$	3.54567	0.129384	0.0646918	+	0.997905i	$0.479394\pi$
$752$	122.080	4.45179
$753$	0	0
$754$	2.64441	0.0963037
$755$	16.0437	0.583888
$756$	0	0
$757$	−19.5557	−0.710765	−0.355382	−	0.934721i	$-0.615649\pi$
$757$	−19.5557	−0.710765	−0.355382	+	0.934721i	$0.615649\pi$
$758$	−21.2658	−0.772409
$759$	0	0
$760$	15.1547	0.549720
$761$	−16.4168	−0.595108	−0.297554	−	0.954705i	$-0.596171\pi$
$761$	−16.4168	−0.595108	−0.297554	+	0.954705i	$0.596171\pi$
$762$	0	0
$763$	−11.0915	−0.401539
$764$	56.6032	2.04783
$765$	0	0
$766$	−14.7683	−0.533599
$767$	7.57747	0.273607
$768$	0	0
$769$	−39.0809	−1.40929	−0.704647	−	0.709558i	$-0.748895\pi$
$769$	−39.0809	−1.40929	−0.704647	+	0.709558i	$0.748895\pi$
$770$	−0.292147	−0.0105282
$771$	0	0
$772$	−146.214	−5.26236
$773$	52.9436	1.90425	0.952124	−	0.305712i	$-0.0988943\pi$
$773$	52.9436	1.90425	0.952124	+	0.305712i	$0.0988943\pi$
$774$	0	0
$775$	22.2515	0.799296
$776$	62.6891	2.25041
$777$	0	0
$778$	68.0371	2.43925
$779$	−4.02763	−0.144305
$780$	0	0
$781$	−0.660617	−0.0236388
$782$	63.7789	2.28073
$783$	0	0
$784$	−75.9680	−2.71314
$785$	6.10707	0.217971
$786$	0	0
$787$	−23.1163	−0.824008	−0.412004	−	0.911182i	$-0.635171\pi$
$787$	−23.1163	−0.824008	−0.412004	+	0.911182i	$0.635171\pi$
$788$	−115.823	−4.12604
$789$	0	0
$790$	−24.8740	−0.884976
$791$	1.58533	0.0563679
$792$	0	0
$793$	−2.26370	−0.0803862
$794$	−54.3370	−1.92835
$795$	0	0
$796$	36.6312	1.29836
$797$	−42.0255	−1.48862	−0.744310	−	0.667834i	$-0.767221\pi$
$797$	−42.0255	−1.48862	−0.744310	+	0.667834i	$0.767221\pi$
$798$	0	0
$799$	−34.1373	−1.20769
$800$	103.632	3.66393
$801$	0	0
$802$	−80.5175	−2.84317
$803$	−0.158136	−0.00558051
$804$	0	0
$805$	−7.91954	−0.279127
$806$	−13.5515	−0.477330
$807$	0	0
$808$	34.3362	1.20794
$809$	−51.4434	−1.80865	−0.904327	−	0.426841i	$-0.859626\pi$
$809$	−51.4434	−1.80865	−0.904327	+	0.426841i	$0.859626\pi$
$810$	0	0
$811$	−54.0236	−1.89702	−0.948512	−	0.316742i	$-0.897411\pi$
$811$	−54.0236	−1.89702	−0.948512	+	0.316742i	$0.897411\pi$
$812$	9.83684	0.345205
$813$	0	0
$814$	0.830557	0.0291110
$815$	−11.3767	−0.398507
$816$	0	0
$817$	13.4402	0.470214
$818$	61.4791	2.14957
$819$	0	0
$820$	16.1525	0.564071
$821$	−1.15670	−0.0403690	−0.0201845	−	0.999796i	$-0.506425\pi$
$821$	−1.15670	−0.0403690	−0.0201845	+	0.999796i	$0.506425\pi$
$822$	0	0
$823$	9.97692	0.347774	0.173887	−	0.984766i	$-0.444367\pi$
$823$	9.97692	0.347774	0.173887	+	0.984766i	$0.444367\pi$
$824$	−187.457	−6.53036
$825$	0	0
$826$	38.1268	1.32660
$827$	−32.4499	−1.12839	−0.564196	−	0.825641i	$-0.690814\pi$
$827$	−32.4499	−1.12839	−0.564196	+	0.825641i	$0.690814\pi$
$828$	0	0
$829$	50.4234	1.75128	0.875638	−	0.482967i	$-0.160441\pi$
$829$	50.4234	1.75128	0.875638	+	0.482967i	$0.160441\pi$
$830$	24.1305	0.837583
$831$	0	0
$832$	−33.9158	−1.17582
$833$	21.2431	0.736029
$834$	0	0
$835$	5.14772	0.178144
$836$	−0.536726	−0.0185631
$837$	0	0
$838$	4.16819	0.143988
$839$	−47.0026	−1.62271	−0.811356	−	0.584553i	$-0.801270\pi$
$839$	−47.0026	−1.62271	−0.811356	+	0.584553i	$0.801270\pi$
$840$	0	0
$841$	−27.7893	−0.958253
$842$	−42.4099	−1.46154
$843$	0	0
$844$	39.6214	1.36383
$845$	12.5717	0.432479
$846$	0	0
$847$	−17.3324	−0.595547
$848$	−34.2934	−1.17764
$849$	0	0
$850$	−51.4233	−1.76380
$851$	22.5148	0.771798
$852$	0	0
$853$	−53.9906	−1.84860	−0.924302	−	0.381663i	$-0.875352\pi$
$853$	−53.9906	−1.84860	−0.924302	+	0.381663i	$0.875352\pi$
$854$	−11.3900	−0.389758
$855$	0	0
$856$	122.721	4.19453
$857$	−27.4106	−0.936329	−0.468164	−	0.883641i	$-0.655084\pi$
$857$	−27.4106	−0.936329	−0.468164	+	0.883641i	$0.655084\pi$
$858$	0	0
$859$	−23.8997	−0.815447	−0.407723	−	0.913105i	$-0.633677\pi$
$859$	−23.8997	−0.815447	−0.407723	+	0.913105i	$0.633677\pi$
$860$	−53.9011	−1.83801
$861$	0	0
$862$	33.5539	1.14285
$863$	−22.4986	−0.765861	−0.382930	−	0.923777i	$-0.625085\pi$
$863$	−22.4986	−0.765861	−0.382930	+	0.923777i	$0.625085\pi$
$864$	0	0
$865$	5.28018	0.179532
$866$	75.0910	2.55170
$867$	0	0
$868$	−50.4096	−1.71101
$869$	0.570298	0.0193460
$870$	0	0
$871$	5.51345	0.186816
$872$	71.5590	2.42329
$873$	0	0
$874$	−19.6803	−0.665695
$875$	14.4756	0.489365
$876$	0	0
$877$	−33.3225	−1.12522	−0.562611	−	0.826722i	$-0.690203\pi$
$877$	−33.3225	−1.12522	−0.562611	+	0.826722i	$0.690203\pi$
$878$	44.4787	1.50108
$879$	0	0
$880$	1.12580	0.0379509
$881$	−28.3617	−0.955530	−0.477765	−	0.878488i	$-0.658553\pi$
$881$	−28.3617	−0.955530	−0.477765	+	0.878488i	$0.658553\pi$
$882$	0	0
$883$	−10.2707	−0.345638	−0.172819	−	0.984954i	$-0.555288\pi$
$883$	−10.2707	−0.345638	−0.172819	+	0.984954i	$0.555288\pi$
$884$	23.1530	0.778721
$885$	0	0
$886$	32.9846	1.10814
$887$	−36.5348	−1.22672	−0.613360	−	0.789804i	$-0.710182\pi$
$887$	−36.5348	−1.22672	−0.613360	+	0.789804i	$0.710182\pi$
$888$	0	0
$889$	20.0046	0.670932
$890$	42.0727	1.41028
$891$	0	0
$892$	143.387	4.80096
$893$	10.5338	0.352499
$894$	0	0
$895$	2.43743	0.0814742
$896$	−87.8632	−2.93531
$897$	0	0
$898$	−15.1649	−0.506059
$899$	−6.20414	−0.206920
$900$	0	0
$901$	9.58953	0.319473
$902$	−0.500930	−0.0166791
$903$	0	0
$904$	−10.2281	−0.340181
$905$	21.5045	0.714834
$906$	0	0
$907$	−18.3570	−0.609533	−0.304766	−	0.952427i	$-0.598578\pi$
$907$	−18.3570	−0.609533	−0.304766	+	0.952427i	$0.598578\pi$
$908$	145.303	4.82205
$909$	0	0
$910$	−3.88875	−0.128911
$911$	5.62526	0.186373	0.0931866	−	0.995649i	$-0.470295\pi$
$911$	5.62526	0.186373	0.0931866	+	0.995649i	$0.470295\pi$
$912$	0	0
$913$	−0.553253	−0.0183100
$914$	29.4052	0.972636
$915$	0	0
$916$	−123.429	−4.07821
$917$	−24.1226	−0.796600
$918$	0	0
$919$	11.1653	0.368309	0.184155	−	0.982897i	$-0.441045\pi$
$919$	11.1653	0.368309	0.184155	+	0.982897i	$0.441045\pi$
$920$	51.0945	1.68454
$921$	0	0
$922$	24.5186	0.807478
$923$	−8.79345	−0.289440
$924$	0	0
$925$	−18.1531	−0.596871
$926$	92.6107	3.04338
$927$	0	0
$928$	−28.8945	−0.948508
$929$	21.9619	0.720546	0.360273	−	0.932847i	$-0.382684\pi$
$929$	21.9619	0.720546	0.360273	+	0.932847i	$0.382684\pi$
$930$	0	0
$931$	−6.55498	−0.214831
$932$	−1.90322	−0.0623420
$933$	0	0
$934$	69.1273	2.26191
$935$	−0.314811	−0.0102954
$936$	0	0
$937$	−41.5334	−1.35684	−0.678418	−	0.734676i	$-0.737334\pi$
$937$	−41.5334	−1.35684	−0.678418	+	0.734676i	$0.737334\pi$
$938$	27.7415	0.905792
$939$	0	0
$940$	−42.2449	−1.37788
$941$	28.6779	0.934873	0.467436	−	0.884027i	$-0.345178\pi$
$941$	28.6779	0.934873	0.467436	+	0.884027i	$0.345178\pi$
$942$	0	0
$943$	−13.5793	−0.442201
$944$	−146.924	−4.78197
$945$	0	0
$946$	1.67160	0.0543485
$947$	33.0613	1.07435	0.537174	−	0.843471i	$-0.319492\pi$
$947$	33.0613	1.07435	0.537174	+	0.843471i	$0.319492\pi$
$948$	0	0
$949$	−2.10495	−0.0683295
$950$	15.8677	0.514816
$951$	0	0
$952$	75.4165	2.44426
$953$	−8.25310	−0.267344	−0.133672	−	0.991026i	$-0.542677\pi$
$953$	−8.25310	−0.267344	−0.133672	+	0.991026i	$0.542677\pi$
$954$	0	0
$955$	−10.2445	−0.331505
$956$	−63.9206	−2.06734
$957$	0	0
$958$	−19.1538	−0.618830
$959$	−14.8855	−0.480678
$960$	0	0
$961$	0.793576	0.0255992
$962$	11.0555	0.356444
$963$	0	0
$964$	33.4788	1.07828
$965$	26.4630	0.851875
$966$	0	0
$967$	25.9097	0.833201	0.416600	−	0.909090i	$-0.363221\pi$
$967$	25.9097	0.833201	0.416600	+	0.909090i	$0.363221\pi$
$968$	111.823	3.59414
$969$	0	0
$970$	−17.5263	−0.562736
$971$	−25.2199	−0.809345	−0.404673	−	0.914462i	$-0.632615\pi$
$971$	−25.2199	−0.809345	−0.404673	+	0.914462i	$0.632615\pi$
$972$	0	0
$973$	−8.32168	−0.266781
$974$	15.2188	0.487643
$975$	0	0
$976$	43.8921	1.40495
$977$	3.79777	0.121501	0.0607506	−	0.998153i	$-0.480651\pi$
$977$	3.79777	0.121501	0.0607506	+	0.998153i	$0.480651\pi$
$978$	0	0
$979$	−0.964622	−0.0308295
$980$	26.2883	0.839748
$981$	0	0
$982$	−60.1031	−1.91797
$983$	−22.3748	−0.713646	−0.356823	−	0.934172i	$-0.616140\pi$
$983$	−22.3748	−0.713646	−0.356823	+	0.934172i	$0.616140\pi$
$984$	0	0
$985$	20.9627	0.667927
$986$	14.3378	0.456609
$987$	0	0
$988$	−7.14434	−0.227292
$989$	45.3140	1.44090
$990$	0	0
$991$	0.580001	0.0184244	0.00921218	−	0.999958i	$-0.497068\pi$
$991$	0.580001	0.0184244	0.00921218	+	0.999958i	$0.497068\pi$
$992$	148.072	4.70129
$993$	0	0
$994$	−44.2451	−1.40337
$995$	−6.62982	−0.210179
$996$	0	0
$997$	0.489673	0.0155081	0.00775406	−	0.999970i	$-0.497532\pi$
$997$	0.489673	0.0155081	0.00775406	+	0.999970i	$0.497532\pi$
$998$	70.5451	2.23307
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.c.1.1		72
3.2	odd	2		6561.2.a.d.1.72		72
81.2	odd	54		729.2.g.b.433.1		144
81.13	even	27		729.2.g.d.541.1		144
81.14	odd	54		243.2.g.a.181.8		144
81.25	even	27		729.2.g.d.190.1		144
81.29	odd	54		243.2.g.a.145.8		144
81.40	even	27		729.2.g.c.298.8		144
81.41	odd	54		729.2.g.b.298.1		144
81.52	even	27		81.2.g.a.31.1	✓	144
81.56	odd	54		729.2.g.a.190.8		144
81.67	even	27		81.2.g.a.34.1	yes	144
81.68	odd	54		729.2.g.a.541.8		144
81.79	even	27		729.2.g.c.433.8		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.31.1	✓	144	81.52	even	27
81.2.g.a.34.1	yes	144	81.67	even	27
243.2.g.a.145.8		144	81.29	odd	54
243.2.g.a.181.8		144	81.14	odd	54
729.2.g.a.190.8		144	81.56	odd	54
729.2.g.a.541.8		144	81.68	odd	54
729.2.g.b.298.1		144	81.41	odd	54
729.2.g.b.433.1		144	81.2	odd	54
729.2.g.c.298.8		144	81.40	even	27
729.2.g.c.433.8		144	81.79	even	27
729.2.g.d.190.1		144	81.25	even	27
729.2.g.d.541.1		144	81.13	even	27
6561.2.a.c.1.1		72	1.1	even	1	trivial
6561.2.a.d.1.72		72	3.2	odd	2

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:	\( N \)	\(=\)	\( 6561 = 3^{8} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6561.a (trivial)

\(f(q)\)	\(=\)	\(q-2.76978 q^{2} +5.67166 q^{4} -1.02650 q^{5} +1.57628 q^{7} -10.1697 q^{8} +O(q^{10})\)	\(q-2.76978 q^{2} +5.67166 q^{4} -1.02650 q^{5} +1.57628 q^{7} -10.1697 q^{8} +2.84319 q^{10} -0.0651872 q^{11} -0.867704 q^{13} -4.36594 q^{14} +16.8244 q^{16} -4.70464 q^{17} +1.45171 q^{19} -5.82198 q^{20} +0.180554 q^{22} +4.89447 q^{23} -3.94629 q^{25} +2.40335 q^{26} +8.94012 q^{28} +1.10030 q^{29} -5.63858 q^{31} -26.2605 q^{32} +13.0308 q^{34} -1.61806 q^{35} +4.60005 q^{37} -4.02091 q^{38} +10.4392 q^{40} -2.77441 q^{41} +9.25820 q^{43} -0.369720 q^{44} -13.5566 q^{46} +7.25610 q^{47} -4.51535 q^{49} +10.9303 q^{50} -4.92132 q^{52} -2.03831 q^{53} +0.0669149 q^{55} -16.0302 q^{56} -3.04759 q^{58} -8.73279 q^{59} +2.60883 q^{61} +15.6176 q^{62} +39.0869 q^{64} +0.890702 q^{65} -6.35407 q^{67} -26.6831 q^{68} +4.48166 q^{70} +10.1342 q^{71} +2.42588 q^{73} -12.7411 q^{74} +8.23361 q^{76} -0.102753 q^{77} -8.74862 q^{79} -17.2703 q^{80} +7.68448 q^{82} +8.48714 q^{83} +4.82933 q^{85} -25.6431 q^{86} +0.662933 q^{88} +14.7977 q^{89} -1.36774 q^{91} +27.7598 q^{92} -20.0978 q^{94} -1.49019 q^{95} -6.16431 q^{97} +12.5065 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8}+O(q^{10}) \) 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38} - 72 q^{41} - 90 q^{44} - 90 q^{47} + 18 q^{49} - 45 q^{50} - 45 q^{53} + 9 q^{55} - 108 q^{56} + 18 q^{58} - 108 q^{59} - 72 q^{62} + 9 q^{64} - 72 q^{65} - 108 q^{68} - 126 q^{71} - 90 q^{74} - 72 q^{77} - 144 q^{80} - 18 q^{82} - 108 q^{83} - 90 q^{86} - 108 q^{89} - 72 q^{92} - 144 q^{95} - 81 q^{98}+O(q^{100}) \) 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8 - 36 * q^11 - 36 * q^14 + 45 * q^16 - 36 * q^17 - 54 * q^20 - 54 * q^23 + 36 * q^25 - 45 * q^26 + 9 * q^28 - 54 * q^29 - 63 * q^32 - 72 * q^35 - 54 * q^38 - 72 * q^41 - 90 * q^44 - 90 * q^47 + 18 * q^49 - 45 * q^50 - 45 * q^53 + 9 * q^55 - 108 * q^56 + 18 * q^58 - 108 * q^59 - 72 * q^62 + 9 * q^64 - 72 * q^65 - 108 * q^68 - 126 * q^71 - 90 * q^74 - 72 * q^77 - 144 * q^80 - 18 * q^82 - 108 * q^83 - 90 * q^86 - 108 * q^89 - 72 * q^92 - 144 * q^95 - 81 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more