LMFDB - Embedded newform 6039.2.a.m.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.325798 q^{2} -1.89386 q^{4} -3.19527 q^{5} -2.43974 q^{7} -1.26861 q^{8} +O(q^{10})$	\(q+0.325798 q^{2} -1.89386 q^{4} -3.19527 q^{5} -2.43974 q^{7} -1.26861 q^{8} -1.04101 q^{10} +1.00000 q^{11} -5.49298 q^{13} -0.794861 q^{14} +3.37440 q^{16} -2.18684 q^{17} +7.07354 q^{19} +6.05139 q^{20} +0.325798 q^{22} +6.98301 q^{23} +5.20977 q^{25} -1.78960 q^{26} +4.62052 q^{28} +1.30591 q^{29} +1.11334 q^{31} +3.63659 q^{32} -0.712466 q^{34} +7.79564 q^{35} +5.67409 q^{37} +2.30454 q^{38} +4.05355 q^{40} -8.17397 q^{41} -6.15103 q^{43} -1.89386 q^{44} +2.27505 q^{46} +4.94933 q^{47} -1.04767 q^{49} +1.69733 q^{50} +10.4029 q^{52} +1.99344 q^{53} -3.19527 q^{55} +3.09508 q^{56} +0.425461 q^{58} +5.77756 q^{59} +1.00000 q^{61} +0.362723 q^{62} -5.56401 q^{64} +17.5516 q^{65} +10.3143 q^{67} +4.14155 q^{68} +2.53980 q^{70} -3.75706 q^{71} +8.52155 q^{73} +1.84860 q^{74} -13.3963 q^{76} -2.43974 q^{77} -10.5168 q^{79} -10.7821 q^{80} -2.66306 q^{82} -12.9448 q^{83} +6.98754 q^{85} -2.00399 q^{86} -1.26861 q^{88} +5.85329 q^{89} +13.4014 q^{91} -13.2248 q^{92} +1.61248 q^{94} -22.6019 q^{95} +8.17042 q^{97} -0.341329 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) $ 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8	$ 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) $ 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8 - 12 * q^10 + 25 * q^11 - 4 * q^13 - 14 * q^14 + 21 * q^16 - 16 * q^17 - 18 * q^19 - 28 * q^20 - 5 * q^22 - 8 * q^23 + 29 * q^25 - 16 * q^26 + 18 * q^28 - 28 * q^29 - 8 * q^31 - 35 * q^32 + 6 * q^34 - 22 * q^35 + 4 * q^37 + 4 * q^38 - 12 * q^40 - 58 * q^41 - 26 * q^43 + 25 * q^44 + 8 * q^46 - 20 * q^47 + 23 * q^49 - 27 * q^50 - 2 * q^52 - 36 * q^53 - 12 * q^55 - 70 * q^56 + 12 * q^58 - 18 * q^59 + 25 * q^61 - 42 * q^62 + 35 * q^64 - 76 * q^65 - 8 * q^67 - 28 * q^68 + 76 * q^70 - 24 * q^71 + 2 * q^73 - 40 * q^74 - 64 * q^76 - 4 * q^77 - 22 * q^79 - 36 * q^80 + 30 * q^82 - 14 * q^83 - 70 * q^86 - 15 * q^88 - 82 * q^89 - 6 * q^91 - 48 * q^92 - 16 * q^94 - 34 * q^95 + 16 * q^97 - 35 * 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 $

$2$	0.325798	0.230374	0.115187	−	0.993344i	$-0.463253\pi$
$2$	0.325798	0.230374	0.115187	+	0.993344i	$0.463253\pi$
$3$	0	0
$3$	0	0
$4$	−1.89386	−0.946928
$5$	−3.19527	−1.42897	−0.714485	−	0.699651i	$-0.753339\pi$
$5$	−3.19527	−1.42897	−0.714485	+	0.699651i	$0.753339\pi$
$6$	0	0
$7$	−2.43974	−0.922135	−0.461067	−	0.887365i	$-0.652533\pi$
$7$	−2.43974	−0.922135	−0.461067	+	0.887365i	$0.652533\pi$
$8$	−1.26861	−0.448521
$9$	0	0
$10$	−1.04101	−0.329197
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.49298	−1.52348	−0.761740	−	0.647883i	$-0.775655\pi$
$13$	−5.49298	−1.52348	−0.761740	+	0.647883i	$0.775655\pi$
$14$	−0.794861	−0.212436
$15$	0	0
$16$	3.37440	0.843601
$17$	−2.18684	−0.530386	−0.265193	−	0.964195i	$-0.585436\pi$
$17$	−2.18684	−0.530386	−0.265193	+	0.964195i	$0.585436\pi$
$18$	0	0
$19$	7.07354	1.62278	0.811391	−	0.584504i	$-0.198711\pi$
$19$	7.07354	1.62278	0.811391	+	0.584504i	$0.198711\pi$
$20$	6.05139	1.35313
$21$	0	0
$22$	0.325798	0.0694603
$23$	6.98301	1.45606	0.728029	−	0.685546i	$-0.240437\pi$
$23$	6.98301	1.45606	0.728029	+	0.685546i	$0.240437\pi$
$24$	0	0
$25$	5.20977	1.04195
$26$	−1.78960	−0.350970
$27$	0	0
$28$	4.62052	0.873195
$29$	1.30591	0.242501	0.121250	−	0.992622i	$-0.461310\pi$
$29$	1.30591	0.242501	0.121250	+	0.992622i	$0.461310\pi$
$30$	0	0
$31$	1.11334	0.199961	0.0999807	−	0.994989i	$-0.468122\pi$
$31$	1.11334	0.199961	0.0999807	+	0.994989i	$0.468122\pi$
$32$	3.63659	0.642864
$33$	0	0
$34$	−0.712466	−0.122187
$35$	7.79564	1.31770
$36$	0	0
$37$	5.67409	0.932814	0.466407	−	0.884570i	$-0.345548\pi$
$37$	5.67409	0.932814	0.466407	+	0.884570i	$0.345548\pi$
$38$	2.30454	0.373846
$39$	0	0
$40$	4.05355	0.640923
$41$	−8.17397	−1.27656	−0.638280	−	0.769804i	$-0.720354\pi$
$41$	−8.17397	−1.27656	−0.638280	+	0.769804i	$0.720354\pi$
$42$	0	0
$43$	−6.15103	−0.938023	−0.469012	−	0.883192i	$-0.655390\pi$
$43$	−6.15103	−0.938023	−0.469012	+	0.883192i	$0.655390\pi$
$44$	−1.89386	−0.285510
$45$	0	0
$46$	2.27505	0.335438
$47$	4.94933	0.721933	0.360967	−	0.932579i	$-0.382447\pi$
$47$	4.94933	0.721933	0.360967	+	0.932579i	$0.382447\pi$
$48$	0	0
$49$	−1.04767	−0.149667
$50$	1.69733	0.240039
$51$	0	0
$52$	10.4029	1.44263
$53$	1.99344	0.273820	0.136910	−	0.990583i	$-0.456283\pi$
$53$	1.99344	0.273820	0.136910	+	0.990583i	$0.456283\pi$
$54$	0	0
$55$	−3.19527	−0.430851
$56$	3.09508	0.413597
$57$	0	0
$58$	0.425461	0.0558658
$59$	5.77756	0.752175	0.376087	−	0.926584i	$-0.377269\pi$
$59$	5.77756	0.752175	0.376087	+	0.926584i	$0.377269\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	0.362723	0.0460658
$63$	0	0
$64$	−5.56401	−0.695501
$65$	17.5516	2.17701
$66$	0	0
$67$	10.3143	1.26009	0.630046	−	0.776558i	$-0.283036\pi$
$67$	10.3143	1.26009	0.630046	+	0.776558i	$0.283036\pi$
$68$	4.14155	0.502237
$69$	0	0
$70$	2.53980	0.303564
$71$	−3.75706	−0.445881	−0.222941	−	0.974832i	$-0.571566\pi$
$71$	−3.75706	−0.445881	−0.222941	+	0.974832i	$0.571566\pi$
$72$	0	0
$73$	8.52155	0.997372	0.498686	−	0.866783i	$-0.333816\pi$
$73$	8.52155	0.997372	0.498686	+	0.866783i	$0.333816\pi$
$74$	1.84860	0.214896
$75$	0	0
$76$	−13.3963	−1.53666
$77$	−2.43974	−0.278034
$78$	0	0
$79$	−10.5168	−1.18323	−0.591614	−	0.806222i	$-0.701509\pi$
$79$	−10.5168	−1.18323	−0.591614	+	0.806222i	$0.701509\pi$
$80$	−10.7821	−1.20548
$81$	0	0
$82$	−2.66306	−0.294086
$83$	−12.9448	−1.42088	−0.710438	−	0.703760i	$-0.751503\pi$
$83$	−12.9448	−1.42088	−0.710438	+	0.703760i	$0.751503\pi$
$84$	0	0
$85$	6.98754	0.757905
$86$	−2.00399	−0.216096
$87$	0	0
$88$	−1.26861	−0.135234
$89$	5.85329	0.620447	0.310224	−	0.950664i	$-0.399596\pi$
$89$	5.85329	0.620447	0.310224	+	0.950664i	$0.399596\pi$
$90$	0	0
$91$	13.4014	1.40485
$92$	−13.2248	−1.37878
$93$	0	0
$94$	1.61248	0.166314
$95$	−22.6019	−2.31891
$96$	0	0
$97$	8.17042	0.829581	0.414790	−	0.909917i	$-0.363855\pi$
$97$	8.17042	0.829581	0.414790	+	0.909917i	$0.363855\pi$
$98$	−0.341329	−0.0344794
$99$	0	0
$100$	−9.86656	−0.986656
$101$	2.03011	0.202004	0.101002	−	0.994886i	$-0.467795\pi$
$101$	2.03011	0.202004	0.101002	+	0.994886i	$0.467795\pi$
$102$	0	0
$103$	10.2184	1.00685	0.503426	−	0.864038i	$-0.332073\pi$
$103$	10.2184	1.00685	0.503426	+	0.864038i	$0.332073\pi$
$104$	6.96845	0.683313
$105$	0	0
$106$	0.649458	0.0630809
$107$	8.01823	0.775151	0.387576	−	0.921838i	$-0.373313\pi$
$107$	8.01823	0.775151	0.387576	+	0.921838i	$0.373313\pi$
$108$	0	0
$109$	11.2686	1.07933	0.539666	−	0.841879i	$-0.318550\pi$
$109$	11.2686	1.07933	0.539666	+	0.841879i	$0.318550\pi$
$110$	−1.04101	−0.0992567
$111$	0	0
$112$	−8.23266	−0.777913
$113$	−16.4586	−1.54830	−0.774148	−	0.633004i	$-0.781822\pi$
$113$	−16.4586	−1.54830	−0.774148	+	0.633004i	$0.781822\pi$
$114$	0	0
$115$	−22.3126	−2.08066
$116$	−2.47320	−0.229631
$117$	0	0
$118$	1.88232	0.173281
$119$	5.33531	0.489087
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.325798	0.0294963
$123$	0	0
$124$	−2.10850	−0.189349
$125$	−0.670288	−0.0599524
$126$	0	0
$127$	−3.85068	−0.341692	−0.170846	−	0.985298i	$-0.554650\pi$
$127$	−3.85068	−0.341692	−0.170846	+	0.985298i	$0.554650\pi$
$128$	−9.08592	−0.803090
$129$	0	0
$130$	5.71826	0.501525
$131$	−16.1737	−1.41310	−0.706551	−	0.707662i	$-0.749750\pi$
$131$	−16.1737	−1.41310	−0.706551	+	0.707662i	$0.749750\pi$
$132$	0	0
$133$	−17.2576	−1.49642
$134$	3.36037	0.290292
$135$	0	0
$136$	2.77424	0.237889
$137$	2.12637	0.181668	0.0908342	−	0.995866i	$-0.471047\pi$
$137$	2.12637	0.181668	0.0908342	+	0.995866i	$0.471047\pi$
$138$	0	0
$139$	−18.8009	−1.59467	−0.797336	−	0.603535i	$-0.793758\pi$
$139$	−18.8009	−1.59467	−0.797336	+	0.603535i	$0.793758\pi$
$140$	−14.7638	−1.24777
$141$	0	0
$142$	−1.22404	−0.102719
$143$	−5.49298	−0.459346
$144$	0	0
$145$	−4.17273	−0.346526
$146$	2.77630	0.229768
$147$	0	0
$148$	−10.7459	−0.883308
$149$	−11.5221	−0.943925	−0.471962	−	0.881619i	$-0.656454\pi$
$149$	−11.5221	−0.943925	−0.471962	+	0.881619i	$0.656454\pi$
$150$	0	0
$151$	−7.05497	−0.574126	−0.287063	−	0.957912i	$-0.592679\pi$
$151$	−7.05497	−0.574126	−0.287063	+	0.957912i	$0.592679\pi$
$152$	−8.97356	−0.727851
$153$	0	0
$154$	−0.794861	−0.0640517
$155$	−3.55742	−0.285739
$156$	0	0
$157$	−7.95441	−0.634831	−0.317416	−	0.948287i	$-0.602815\pi$
$157$	−7.95441	−0.634831	−0.317416	+	0.948287i	$0.602815\pi$
$158$	−3.42633	−0.272584
$159$	0	0
$160$	−11.6199	−0.918634
$161$	−17.0367	−1.34268
$162$	0	0
$163$	−5.97697	−0.468152	−0.234076	−	0.972218i	$-0.575206\pi$
$163$	−5.97697	−0.468152	−0.234076	+	0.972218i	$0.575206\pi$
$164$	15.4803	1.20881
$165$	0	0
$166$	−4.21738	−0.327333
$167$	−19.7952	−1.53180	−0.765901	−	0.642958i	$-0.777707\pi$
$167$	−19.7952	−1.53180	−0.765901	+	0.642958i	$0.777707\pi$
$168$	0	0
$169$	17.1729	1.32099
$170$	2.27652	0.174601
$171$	0	0
$172$	11.6492	0.888241
$173$	10.8408	0.824213	0.412106	−	0.911136i	$-0.364793\pi$
$173$	10.8408	0.824213	0.412106	+	0.911136i	$0.364793\pi$
$174$	0	0
$175$	−12.7105	−0.960823
$176$	3.37440	0.254355
$177$	0	0
$178$	1.90699	0.142935
$179$	−21.2287	−1.58670	−0.793352	−	0.608763i	$-0.791666\pi$
$179$	−21.2287	−1.58670	−0.793352	+	0.608763i	$0.791666\pi$
$180$	0	0
$181$	21.8262	1.62233	0.811164	−	0.584819i	$-0.198835\pi$
$181$	21.8262	1.62233	0.811164	+	0.584819i	$0.198835\pi$
$182$	4.36616	0.323641
$183$	0	0
$184$	−8.85871	−0.653073
$185$	−18.1303	−1.33296
$186$	0	0
$187$	−2.18684	−0.159917
$188$	−9.37331	−0.683619
$189$	0	0
$190$	−7.36364	−0.534215
$191$	4.95742	0.358706	0.179353	−	0.983785i	$-0.442600\pi$
$191$	4.95742	0.358706	0.179353	+	0.983785i	$0.442600\pi$
$192$	0	0
$193$	−10.5522	−0.759563	−0.379782	−	0.925076i	$-0.624001\pi$
$193$	−10.5522	−0.759563	−0.379782	+	0.925076i	$0.624001\pi$
$194$	2.66190	0.191114
$195$	0	0
$196$	1.98414	0.141724
$197$	−23.7665	−1.69329	−0.846645	−	0.532158i	$-0.821381\pi$
$197$	−23.7665	−1.69329	−0.846645	+	0.532158i	$0.821381\pi$
$198$	0	0
$199$	21.1491	1.49922	0.749612	−	0.661878i	$-0.230240\pi$
$199$	21.1491	1.49922	0.749612	+	0.661878i	$0.230240\pi$
$200$	−6.60917	−0.467339
$201$	0	0
$202$	0.661405	0.0465363
$203$	−3.18607	−0.223618
$204$	0	0
$205$	26.1181	1.82417
$206$	3.32914	0.231952
$207$	0	0
$208$	−18.5355	−1.28521
$209$	7.07354	0.489287
$210$	0	0
$211$	−7.93199	−0.546060	−0.273030	−	0.962005i	$-0.588026\pi$
$211$	−7.93199	−0.546060	−0.273030	+	0.962005i	$0.588026\pi$
$212$	−3.77529	−0.259288
$213$	0	0
$214$	2.61232	0.178575
$215$	19.6542	1.34041
$216$	0	0
$217$	−2.71625	−0.184391
$218$	3.67127	0.248650
$219$	0	0
$220$	6.05139	0.407985
$221$	12.0123	0.808032
$222$	0	0
$223$	17.0803	1.14378	0.571889	−	0.820331i	$-0.306211\pi$
$223$	17.0803	1.14378	0.571889	+	0.820331i	$0.306211\pi$
$224$	−8.87233	−0.592808
$225$	0	0
$226$	−5.36218	−0.356687
$227$	11.7426	0.779385	0.389692	−	0.920945i	$-0.372581\pi$
$227$	11.7426	0.779385	0.389692	+	0.920945i	$0.372581\pi$
$228$	0	0
$229$	29.0130	1.91723	0.958616	−	0.284702i	$-0.0918947\pi$
$229$	29.0130	1.91723	0.958616	+	0.284702i	$0.0918947\pi$
$230$	−7.26940	−0.479330
$231$	0	0
$232$	−1.65668	−0.108767
$233$	11.2515	0.737111	0.368556	−	0.929606i	$-0.379852\pi$
$233$	11.2515	0.737111	0.368556	+	0.929606i	$0.379852\pi$
$234$	0	0
$235$	−15.8145	−1.03162
$236$	−10.9419	−0.712255
$237$	0	0
$238$	1.73823	0.112673
$239$	3.89825	0.252157	0.126078	−	0.992020i	$-0.459761\pi$
$239$	3.89825	0.252157	0.126078	+	0.992020i	$0.459761\pi$
$240$	0	0
$241$	17.1124	1.10231	0.551153	−	0.834404i	$-0.314188\pi$
$241$	17.1124	1.10231	0.551153	+	0.834404i	$0.314188\pi$
$242$	0.325798	0.0209431
$243$	0	0
$244$	−1.89386	−0.121242
$245$	3.34760	0.213870
$246$	0	0
$247$	−38.8548	−2.47227
$248$	−1.41239	−0.0896869
$249$	0	0
$250$	−0.218378	−0.0138114
$251$	−7.79593	−0.492075	−0.246037	−	0.969260i	$-0.579129\pi$
$251$	−7.79593	−0.492075	−0.246037	+	0.969260i	$0.579129\pi$
$252$	0	0
$253$	6.98301	0.439018
$254$	−1.25454	−0.0787169
$255$	0	0
$256$	8.16785	0.510491
$257$	3.52530	0.219902	0.109951	−	0.993937i	$-0.464931\pi$
$257$	3.52530	0.219902	0.109951	+	0.993937i	$0.464931\pi$
$258$	0	0
$259$	−13.8433	−0.860180
$260$	−33.2402	−2.06147
$261$	0	0
$262$	−5.26935	−0.325542
$263$	22.3919	1.38074	0.690371	−	0.723455i	$-0.257447\pi$
$263$	22.3919	1.38074	0.690371	+	0.723455i	$0.257447\pi$
$264$	0	0
$265$	−6.36959	−0.391281
$266$	−5.62248	−0.344737
$267$	0	0
$268$	−19.5338	−1.19322
$269$	−1.75703	−0.107128	−0.0535639	−	0.998564i	$-0.517058\pi$
$269$	−1.75703	−0.107128	−0.0535639	+	0.998564i	$0.517058\pi$
$270$	0	0
$271$	−8.48259	−0.515281	−0.257640	−	0.966241i	$-0.582945\pi$
$271$	−8.48259	−0.515281	−0.257640	+	0.966241i	$0.582945\pi$
$272$	−7.37926	−0.447434
$273$	0	0
$274$	0.692768	0.0418516
$275$	5.20977	0.314161
$276$	0	0
$277$	−16.5287	−0.993112	−0.496556	−	0.868005i	$-0.665402\pi$
$277$	−16.5287	−0.993112	−0.496556	+	0.868005i	$0.665402\pi$
$278$	−6.12529	−0.367371
$279$	0	0
$280$	−9.88961	−0.591017
$281$	−23.2682	−1.38806	−0.694032	−	0.719944i	$-0.744167\pi$
$281$	−23.2682	−1.38806	−0.694032	+	0.719944i	$0.744167\pi$
$282$	0	0
$283$	−5.71648	−0.339809	−0.169905	−	0.985461i	$-0.554346\pi$
$283$	−5.71648	−0.339809	−0.169905	+	0.985461i	$0.554346\pi$
$284$	7.11534	0.422218
$285$	0	0
$286$	−1.78960	−0.105821
$287$	19.9424	1.17716
$288$	0	0
$289$	−12.2177	−0.718691
$290$	−1.35946	−0.0798305
$291$	0	0
$292$	−16.1386	−0.944440
$293$	17.8726	1.04413	0.522063	−	0.852907i	$-0.325162\pi$
$293$	17.8726	1.04413	0.522063	+	0.852907i	$0.325162\pi$
$294$	0	0
$295$	−18.4609	−1.07483
$296$	−7.19820	−0.418387
$297$	0	0
$298$	−3.75386	−0.217455
$299$	−38.3576	−2.21828
$300$	0	0
$301$	15.0069	0.864984
$302$	−2.29849	−0.132263
$303$	0	0
$304$	23.8690	1.36898
$305$	−3.19527	−0.182961
$306$	0	0
$307$	−8.15793	−0.465598	−0.232799	−	0.972525i	$-0.574788\pi$
$307$	−8.15793	−0.465598	−0.232799	+	0.972525i	$0.574788\pi$
$308$	4.62052	0.263278
$309$	0	0
$310$	−1.15900	−0.0658267
$311$	−12.4016	−0.703229	−0.351614	−	0.936145i	$-0.614367\pi$
$311$	−12.4016	−0.703229	−0.351614	+	0.936145i	$0.614367\pi$
$312$	0	0
$313$	21.1502	1.19548	0.597739	−	0.801691i	$-0.296066\pi$
$313$	21.1502	1.19548	0.597739	+	0.801691i	$0.296066\pi$
$314$	−2.59153	−0.146248
$315$	0	0
$316$	19.9172	1.12043
$317$	22.1743	1.24544	0.622718	−	0.782447i	$-0.286029\pi$
$317$	22.1743	1.24544	0.622718	+	0.782447i	$0.286029\pi$
$318$	0	0
$319$	1.30591	0.0731167
$320$	17.7785	0.993851
$321$	0	0
$322$	−5.55053	−0.309319
$323$	−15.4687	−0.860700
$324$	0	0
$325$	−28.6172	−1.58740
$326$	−1.94728	−0.107850
$327$	0	0
$328$	10.3696	0.572564
$329$	−12.0751	−0.665720
$330$	0	0
$331$	9.42859	0.518242	0.259121	−	0.965845i	$-0.416567\pi$
$331$	9.42859	0.518242	0.259121	+	0.965845i	$0.416567\pi$
$332$	24.5156	1.34547
$333$	0	0
$334$	−6.44924	−0.352887
$335$	−32.9570	−1.80063
$336$	0	0
$337$	0.242919	0.0132326	0.00661632	−	0.999978i	$-0.497894\pi$
$337$	0.242919	0.0132326	0.00661632	+	0.999978i	$0.497894\pi$
$338$	5.59488	0.304321
$339$	0	0
$340$	−13.2334	−0.717682
$341$	1.11334	0.0602906
$342$	0	0
$343$	19.6342	1.06015
$344$	7.80325	0.420723
$345$	0	0
$346$	3.53191	0.189877
$347$	−25.6414	−1.37650	−0.688251	−	0.725473i	$-0.741621\pi$
$347$	−25.6414	−1.37650	−0.688251	+	0.725473i	$0.741621\pi$
$348$	0	0
$349$	0.596781	0.0319450	0.0159725	−	0.999872i	$-0.494916\pi$
$349$	0.596781	0.0319450	0.0159725	+	0.999872i	$0.494916\pi$
$350$	−4.14105	−0.221348
$351$	0	0
$352$	3.63659	0.193831
$353$	10.2063	0.543227	0.271614	−	0.962406i	$-0.412443\pi$
$353$	10.2063	0.543227	0.271614	+	0.962406i	$0.412443\pi$
$354$	0	0
$355$	12.0048	0.637151
$356$	−11.0853	−0.587519
$357$	0	0
$358$	−6.91625	−0.365535
$359$	−23.8311	−1.25776	−0.628880	−	0.777503i	$-0.716486\pi$
$359$	−23.8311	−1.25776	−0.628880	+	0.777503i	$0.716486\pi$
$360$	0	0
$361$	31.0350	1.63342
$362$	7.11092	0.373741
$363$	0	0
$364$	−25.3804	−1.33030
$365$	−27.2287	−1.42521
$366$	0	0
$367$	10.7314	0.560175	0.280087	−	0.959974i	$-0.409637\pi$
$367$	10.7314	0.560175	0.280087	+	0.959974i	$0.409637\pi$
$368$	23.5635	1.22833
$369$	0	0
$370$	−5.90680	−0.307080
$371$	−4.86347	−0.252499
$372$	0	0
$373$	13.9065	0.720052	0.360026	−	0.932942i	$-0.382768\pi$
$373$	13.9065	0.720052	0.360026	+	0.932942i	$0.382768\pi$
$374$	−0.712466	−0.0368407
$375$	0	0
$376$	−6.27876	−0.323802
$377$	−7.17332	−0.369445
$378$	0	0
$379$	−22.5155	−1.15655	−0.578273	−	0.815843i	$-0.696273\pi$
$379$	−22.5155	−1.15655	−0.578273	+	0.815843i	$0.696273\pi$
$380$	42.8047	2.19584
$381$	0	0
$382$	1.61512	0.0826365
$383$	−4.48187	−0.229013	−0.114506	−	0.993423i	$-0.536529\pi$
$383$	−4.48187	−0.229013	−0.114506	+	0.993423i	$0.536529\pi$
$384$	0	0
$385$	7.79564	0.397302
$386$	−3.43788	−0.174983
$387$	0	0
$388$	−15.4736	−0.785553
$389$	−16.7099	−0.847228	−0.423614	−	0.905843i	$-0.639239\pi$
$389$	−16.7099	−0.847228	−0.423614	+	0.905843i	$0.639239\pi$
$390$	0	0
$391$	−15.2707	−0.772273
$392$	1.32909	0.0671289
$393$	0	0
$394$	−7.74306	−0.390089
$395$	33.6039	1.69080
$396$	0	0
$397$	30.3142	1.52143	0.760714	−	0.649088i	$-0.224849\pi$
$397$	30.3142	1.52143	0.760714	+	0.649088i	$0.224849\pi$
$398$	6.89034	0.345382
$399$	0	0
$400$	17.5799	0.878994
$401$	−22.1670	−1.10697	−0.553484	−	0.832860i	$-0.686702\pi$
$401$	−22.1670	−1.10697	−0.553484	+	0.832860i	$0.686702\pi$
$402$	0	0
$403$	−6.11555	−0.304637
$404$	−3.84474	−0.191283
$405$	0	0
$406$	−1.03801	−0.0515158
$407$	5.67409	0.281254
$408$	0	0
$409$	−26.6861	−1.31954	−0.659772	−	0.751466i	$-0.729347\pi$
$409$	−26.6861	−1.31954	−0.659772	+	0.751466i	$0.729347\pi$
$410$	8.50921	0.420240
$411$	0	0
$412$	−19.3522	−0.953416
$413$	−14.0957	−0.693606
$414$	0	0
$415$	41.3622	2.03039
$416$	−19.9757	−0.979391
$417$	0	0
$418$	2.30454	0.112719
$419$	−23.0411	−1.12563	−0.562816	−	0.826582i	$-0.690282\pi$
$419$	−23.0411	−1.12563	−0.562816	+	0.826582i	$0.690282\pi$
$420$	0	0
$421$	0.567324	0.0276497	0.0138248	−	0.999904i	$-0.495599\pi$
$421$	0.567324	0.0276497	0.0138248	+	0.999904i	$0.495599\pi$
$422$	−2.58422	−0.125798
$423$	0	0
$424$	−2.52890	−0.122814
$425$	−11.3929	−0.552638
$426$	0	0
$427$	−2.43974	−0.118067
$428$	−15.1854	−0.734013
$429$	0	0
$430$	6.40330	0.308795
$431$	26.0709	1.25579	0.627894	−	0.778299i	$-0.283917\pi$
$431$	26.0709	1.25579	0.627894	+	0.778299i	$0.283917\pi$
$432$	0	0
$433$	7.40004	0.355623	0.177812	−	0.984065i	$-0.443098\pi$
$433$	7.40004	0.355623	0.177812	+	0.984065i	$0.443098\pi$
$434$	−0.884949	−0.0424789
$435$	0	0
$436$	−21.3410	−1.02205
$437$	49.3946	2.36286
$438$	0	0
$439$	18.4556	0.880840	0.440420	−	0.897792i	$-0.354830\pi$
$439$	18.4556	0.880840	0.440420	+	0.897792i	$0.354830\pi$
$440$	4.05355	0.193246
$441$	0	0
$442$	3.91356	0.186149
$443$	−33.8994	−1.61061	−0.805306	−	0.592860i	$-0.797999\pi$
$443$	−33.8994	−1.61061	−0.805306	+	0.592860i	$0.797999\pi$
$444$	0	0
$445$	−18.7029	−0.886601
$446$	5.56471	0.263497
$447$	0	0
$448$	13.5747	0.641346
$449$	−33.6473	−1.58792	−0.793958	−	0.607972i	$-0.791983\pi$
$449$	−33.6473	−1.58792	−0.793958	+	0.607972i	$0.791983\pi$
$450$	0	0
$451$	−8.17397	−0.384897
$452$	31.1702	1.46613
$453$	0	0
$454$	3.82572	0.179550
$455$	−42.8213	−2.00749
$456$	0	0
$457$	−11.4368	−0.534990	−0.267495	−	0.963559i	$-0.586196\pi$
$457$	−11.4368	−0.534990	−0.267495	+	0.963559i	$0.586196\pi$
$458$	9.45236	0.441680
$459$	0	0
$460$	42.2569	1.97024
$461$	−40.7897	−1.89977	−0.949884	−	0.312603i	$-0.898799\pi$
$461$	−40.7897	−1.89977	−0.949884	+	0.312603i	$0.898799\pi$
$462$	0	0
$463$	−27.1559	−1.26204	−0.631022	−	0.775765i	$-0.717364\pi$
$463$	−27.1559	−1.26204	−0.631022	+	0.775765i	$0.717364\pi$
$464$	4.40665	0.204574
$465$	0	0
$466$	3.66572	0.169811
$467$	−26.6896	−1.23505	−0.617524	−	0.786552i	$-0.711864\pi$
$467$	−26.6896	−1.23505	−0.617524	+	0.786552i	$0.711864\pi$
$468$	0	0
$469$	−25.1642	−1.16198
$470$	−5.15231	−0.237658
$471$	0	0
$472$	−7.32947	−0.337366
$473$	−6.15103	−0.282825
$474$	0	0
$475$	36.8515	1.69086
$476$	−10.1043	−0.463130
$477$	0	0
$478$	1.27004	0.0580903
$479$	0.208264	0.00951583	0.00475791	−	0.999989i	$-0.498486\pi$
$479$	0.208264	0.00951583	0.00475791	+	0.999989i	$0.498486\pi$
$480$	0	0
$481$	−31.1677	−1.42112
$482$	5.57517	0.253942
$483$	0	0
$484$	−1.89386	−0.0860844
$485$	−26.1067	−1.18545
$486$	0	0
$487$	2.70487	0.122569	0.0612846	−	0.998120i	$-0.480480\pi$
$487$	2.70487	0.122569	0.0612846	+	0.998120i	$0.480480\pi$
$488$	−1.26861	−0.0574272
$489$	0	0
$490$	1.09064	0.0492701
$491$	6.99640	0.315743	0.157872	−	0.987460i	$-0.449537\pi$
$491$	6.99640	0.315743	0.157872	+	0.987460i	$0.449537\pi$
$492$	0	0
$493$	−2.85580	−0.128619
$494$	−12.6588	−0.569547
$495$	0	0
$496$	3.75685	0.168688
$497$	9.16626	0.411163
$498$	0	0
$499$	38.3930	1.71871	0.859354	−	0.511381i	$-0.170866\pi$
$499$	38.3930	1.71871	0.859354	+	0.511381i	$0.170866\pi$
$500$	1.26943	0.0567706
$501$	0	0
$502$	−2.53990	−0.113361
$503$	22.2395	0.991609	0.495804	−	0.868434i	$-0.334873\pi$
$503$	22.2395	0.991609	0.495804	+	0.868434i	$0.334873\pi$
$504$	0	0
$505$	−6.48676	−0.288657
$506$	2.27505	0.101138
$507$	0	0
$508$	7.29263	0.323558
$509$	4.29211	0.190244	0.0951222	−	0.995466i	$-0.469676\pi$
$509$	4.29211	0.190244	0.0951222	+	0.995466i	$0.469676\pi$
$510$	0	0
$511$	−20.7904	−0.919712
$512$	20.8329	0.920693
$513$	0	0
$514$	1.14854	0.0506597
$515$	−32.6507	−1.43876
$516$	0	0
$517$	4.94933	0.217671
$518$	−4.51011	−0.198163
$519$	0	0
$520$	−22.2661	−0.976433
$521$	−16.6313	−0.728629	−0.364315	−	0.931276i	$-0.618697\pi$
$521$	−16.6313	−0.728629	−0.364315	+	0.931276i	$0.618697\pi$
$522$	0	0
$523$	37.5986	1.64407	0.822036	−	0.569436i	$-0.192838\pi$
$523$	37.5986	1.64407	0.822036	+	0.569436i	$0.192838\pi$
$524$	30.6307	1.33811
$525$	0	0
$526$	7.29522	0.318087
$527$	−2.43469	−0.106057
$528$	0	0
$529$	25.7624	1.12011
$530$	−2.07520	−0.0901408
$531$	0	0
$532$	32.6834	1.41700
$533$	44.8995	1.94481
$534$	0	0
$535$	−25.6204	−1.10767
$536$	−13.0848	−0.565178
$537$	0	0
$538$	−0.572435	−0.0246794
$539$	−1.04767	−0.0451264
$540$	0	0
$541$	−26.0619	−1.12049	−0.560244	−	0.828328i	$-0.689292\pi$
$541$	−26.0619	−1.12049	−0.560244	+	0.828328i	$0.689292\pi$
$542$	−2.76361	−0.118707
$543$	0	0
$544$	−7.95263	−0.340966
$545$	−36.0062	−1.54233
$546$	0	0
$547$	−23.8538	−1.01992	−0.509958	−	0.860199i	$-0.670339\pi$
$547$	−23.8538	−1.01992	−0.509958	+	0.860199i	$0.670339\pi$
$548$	−4.02705	−0.172027
$549$	0	0
$550$	1.69733	0.0723745
$551$	9.23737	0.393525
$552$	0	0
$553$	25.6581	1.09110
$554$	−5.38501	−0.228787
$555$	0	0
$556$	35.6062	1.51004
$557$	−10.3144	−0.437035	−0.218518	−	0.975833i	$-0.570122\pi$
$557$	−10.3144	−0.437035	−0.218518	+	0.975833i	$0.570122\pi$
$558$	0	0
$559$	33.7875	1.42906
$560$	26.3056	1.11161
$561$	0	0
$562$	−7.58072	−0.319773
$563$	28.1196	1.18510	0.592550	−	0.805534i	$-0.298121\pi$
$563$	28.1196	1.18510	0.592550	+	0.805534i	$0.298121\pi$
$564$	0	0
$565$	52.5898	2.21247
$566$	−1.86241	−0.0782831
$567$	0	0
$568$	4.76625	0.199987
$569$	9.90370	0.415185	0.207592	−	0.978215i	$-0.433437\pi$
$569$	9.90370	0.415185	0.207592	+	0.978215i	$0.433437\pi$
$570$	0	0
$571$	−3.66816	−0.153508	−0.0767539	−	0.997050i	$-0.524456\pi$
$571$	−3.66816	−0.153508	−0.0767539	+	0.997050i	$0.524456\pi$
$572$	10.4029	0.434968
$573$	0	0
$574$	6.49718	0.271187
$575$	36.3799	1.51715
$576$	0	0
$577$	−5.20381	−0.216637	−0.108319	−	0.994116i	$-0.534547\pi$
$577$	−5.20381	−0.216637	−0.108319	+	0.994116i	$0.534547\pi$
$578$	−3.98051	−0.165568
$579$	0	0
$580$	7.90254	0.328135
$581$	31.5819	1.31024
$582$	0	0
$583$	1.99344	0.0825599
$584$	−10.8105	−0.447342
$585$	0	0
$586$	5.82284	0.240539
$587$	−44.0007	−1.81610	−0.908052	−	0.418857i	$-0.862431\pi$
$587$	−44.0007	−1.81610	−0.908052	+	0.418857i	$0.862431\pi$
$588$	0	0
$589$	7.87524	0.324494
$590$	−6.01452	−0.247614
$591$	0	0
$592$	19.1467	0.786922
$593$	2.89921	0.119056	0.0595282	−	0.998227i	$-0.481040\pi$
$593$	2.89921	0.119056	0.0595282	+	0.998227i	$0.481040\pi$
$594$	0	0
$595$	−17.0478	−0.698891
$596$	21.8211	0.893829
$597$	0	0
$598$	−12.4968	−0.511032
$599$	30.9885	1.26615	0.633077	−	0.774088i	$-0.281791\pi$
$599$	30.9885	1.26615	0.633077	+	0.774088i	$0.281791\pi$
$600$	0	0
$601$	−10.1733	−0.414976	−0.207488	−	0.978238i	$-0.566529\pi$
$601$	−10.1733	−0.414976	−0.207488	+	0.978238i	$0.566529\pi$
$602$	4.88922	0.199270
$603$	0	0
$604$	13.3611	0.543656
$605$	−3.19527	−0.129906
$606$	0	0
$607$	17.6514	0.716449	0.358225	−	0.933635i	$-0.383382\pi$
$607$	17.6514	0.716449	0.358225	+	0.933635i	$0.383382\pi$
$608$	25.7236	1.04323
$609$	0	0
$610$	−1.04101	−0.0421494
$611$	−27.1866	−1.09985
$612$	0	0
$613$	−33.8743	−1.36817	−0.684086	−	0.729402i	$-0.739799\pi$
$613$	−33.8743	−1.36817	−0.684086	+	0.729402i	$0.739799\pi$
$614$	−2.65783	−0.107261
$615$	0	0
$616$	3.09508	0.124704
$617$	−23.1224	−0.930874	−0.465437	−	0.885081i	$-0.654103\pi$
$617$	−23.1224	−0.930874	−0.465437	+	0.885081i	$0.654103\pi$
$618$	0	0
$619$	−5.54937	−0.223048	−0.111524	−	0.993762i	$-0.535573\pi$
$619$	−5.54937	−0.223048	−0.111524	+	0.993762i	$0.535573\pi$
$620$	6.73724	0.270574
$621$	0	0
$622$	−4.04040	−0.162005
$623$	−14.2805	−0.572136
$624$	0	0
$625$	−23.9071	−0.956285
$626$	6.89067	0.275407
$627$	0	0
$628$	15.0645	0.601139
$629$	−12.4083	−0.494751
$630$	0	0
$631$	−22.1090	−0.880147	−0.440074	−	0.897962i	$-0.645048\pi$
$631$	−22.1090	−0.880147	−0.440074	+	0.897962i	$0.645048\pi$
$632$	13.3416	0.530702
$633$	0	0
$634$	7.22435	0.286915
$635$	12.3040	0.488268
$636$	0	0
$637$	5.75484	0.228015
$638$	0.425461	0.0168442
$639$	0	0
$640$	29.0320	1.14759
$641$	−47.9201	−1.89273	−0.946366	−	0.323097i	$-0.895276\pi$
$641$	−47.9201	−1.89273	−0.946366	+	0.323097i	$0.895276\pi$
$642$	0	0
$643$	6.68620	0.263678	0.131839	−	0.991271i	$-0.457912\pi$
$643$	6.68620	0.263678	0.131839	+	0.991271i	$0.457912\pi$
$644$	32.2651	1.27142
$645$	0	0
$646$	−5.03966	−0.198283
$647$	−41.8915	−1.64693	−0.823463	−	0.567371i	$-0.807961\pi$
$647$	−41.8915	−1.64693	−0.823463	+	0.567371i	$0.807961\pi$
$648$	0	0
$649$	5.77756	0.226789
$650$	−9.32342	−0.365695
$651$	0	0
$652$	11.3195	0.443306
$653$	12.8115	0.501351	0.250676	−	0.968071i	$-0.419347\pi$
$653$	12.8115	0.501351	0.250676	+	0.968071i	$0.419347\pi$
$654$	0	0
$655$	51.6794	2.01928
$656$	−27.5823	−1.07691
$657$	0	0
$658$	−3.93403	−0.153364
$659$	−4.14966	−0.161648	−0.0808239	−	0.996728i	$-0.525755\pi$
$659$	−4.14966	−0.161648	−0.0808239	+	0.996728i	$0.525755\pi$
$660$	0	0
$661$	46.0312	1.79041	0.895203	−	0.445658i	$-0.147030\pi$
$661$	46.0312	1.79041	0.895203	+	0.445658i	$0.147030\pi$
$662$	3.07181	0.119389
$663$	0	0
$664$	16.4219	0.637293
$665$	55.1427	2.13834
$666$	0	0
$667$	9.11915	0.353095
$668$	37.4893	1.45051
$669$	0	0
$670$	−10.7373	−0.414819
$671$	1.00000	0.0386046
$672$	0	0
$673$	48.6214	1.87422	0.937108	−	0.349040i	$-0.113492\pi$
$673$	48.6214	1.87422	0.937108	+	0.349040i	$0.113492\pi$
$674$	0.0791424	0.00304845
$675$	0	0
$676$	−32.5229	−1.25088
$677$	−0.895305	−0.0344094	−0.0172047	−	0.999852i	$-0.505477\pi$
$677$	−0.895305	−0.0344094	−0.0172047	+	0.999852i	$0.505477\pi$
$678$	0	0
$679$	−19.9337	−0.764985
$680$	−8.86446	−0.339936
$681$	0	0
$682$	0.362723	0.0138894
$683$	39.3797	1.50682	0.753411	−	0.657549i	$-0.228407\pi$
$683$	39.3797	1.50682	0.753411	+	0.657549i	$0.228407\pi$
$684$	0	0
$685$	−6.79435	−0.259599
$686$	6.39678	0.244230
$687$	0	0
$688$	−20.7561	−0.791317
$689$	−10.9499	−0.417159
$690$	0	0
$691$	18.6563	0.709719	0.354860	−	0.934920i	$-0.384529\pi$
$691$	18.6563	0.709719	0.354860	+	0.934920i	$0.384529\pi$
$692$	−20.5310	−0.780470
$693$	0	0
$694$	−8.35390	−0.317110
$695$	60.0741	2.27874
$696$	0	0
$697$	17.8751	0.677069
$698$	0.194430	0.00735928
$699$	0	0
$700$	24.0718	0.909830
$701$	−19.8001	−0.747840	−0.373920	−	0.927461i	$-0.621987\pi$
$701$	−19.8001	−0.747840	−0.373920	+	0.927461i	$0.621987\pi$
$702$	0	0
$703$	40.1359	1.51375
$704$	−5.56401	−0.209702
$705$	0	0
$706$	3.32519	0.125145
$707$	−4.95294	−0.186275
$708$	0	0
$709$	−29.3177	−1.10105	−0.550525	−	0.834819i	$-0.685572\pi$
$709$	−29.3177	−1.10105	−0.550525	+	0.834819i	$0.685572\pi$
$710$	3.91115	0.146783
$711$	0	0
$712$	−7.42554	−0.278284
$713$	7.77445	0.291155
$714$	0	0
$715$	17.5516	0.656392
$716$	40.2040	1.50249
$717$	0	0
$718$	−7.76413	−0.289755
$719$	−11.4219	−0.425964	−0.212982	−	0.977056i	$-0.568318\pi$
$719$	−11.4219	−0.425964	−0.212982	+	0.977056i	$0.568318\pi$
$720$	0	0
$721$	−24.9303	−0.928453
$722$	10.1111	0.376297
$723$	0	0
$724$	−41.3356	−1.53623
$725$	6.80347	0.252675
$726$	0	0
$727$	27.6488	1.02544	0.512719	−	0.858557i	$-0.328638\pi$
$727$	27.6488	1.02544	0.512719	+	0.858557i	$0.328638\pi$
$728$	−17.0012	−0.630106
$729$	0	0
$730$	−8.87104	−0.328332
$731$	13.4513	0.497514
$732$	0	0
$733$	−19.5408	−0.721757	−0.360878	−	0.932613i	$-0.617523\pi$
$733$	−19.5408	−0.721757	−0.360878	+	0.932613i	$0.617523\pi$
$734$	3.49627	0.129050
$735$	0	0
$736$	25.3943	0.936048
$737$	10.3143	0.379932
$738$	0	0
$739$	−7.52813	−0.276927	−0.138463	−	0.990368i	$-0.544216\pi$
$739$	−7.52813	−0.276927	−0.138463	+	0.990368i	$0.544216\pi$
$740$	34.3361	1.26222
$741$	0	0
$742$	−1.58451	−0.0581691
$743$	39.6681	1.45528	0.727640	−	0.685959i	$-0.240617\pi$
$743$	39.6681	1.45528	0.727640	+	0.685959i	$0.240617\pi$
$744$	0	0
$745$	36.8162	1.34884
$746$	4.53071	0.165881
$747$	0	0
$748$	4.14155	0.151430
$749$	−19.5624	−0.714794
$750$	0	0
$751$	−31.2179	−1.13916	−0.569579	−	0.821937i	$-0.692894\pi$
$751$	−31.2179	−1.13916	−0.569579	+	0.821937i	$0.692894\pi$
$752$	16.7010	0.609023
$753$	0	0
$754$	−2.33705	−0.0851103
$755$	22.5426	0.820408
$756$	0	0
$757$	16.1303	0.586266	0.293133	−	0.956072i	$-0.405302\pi$
$757$	16.1303	0.586266	0.293133	+	0.956072i	$0.405302\pi$
$758$	−7.33551	−0.266438
$759$	0	0
$760$	28.6730	1.04008
$761$	22.6027	0.819347	0.409673	−	0.912232i	$-0.365643\pi$
$761$	22.6027	0.819347	0.409673	+	0.912232i	$0.365643\pi$
$762$	0	0
$763$	−27.4924	−0.995290
$764$	−9.38864	−0.339669
$765$	0	0
$766$	−1.46018	−0.0527585
$767$	−31.7361	−1.14592
$768$	0	0
$769$	−16.0026	−0.577067	−0.288534	−	0.957470i	$-0.593168\pi$
$769$	−16.0026	−0.577067	−0.288534	+	0.957470i	$0.593168\pi$
$770$	2.53980	0.0915280
$771$	0	0
$772$	19.9843	0.719252
$773$	1.88372	0.0677527	0.0338763	−	0.999426i	$-0.489215\pi$
$773$	1.88372	0.0677527	0.0338763	+	0.999426i	$0.489215\pi$
$774$	0	0
$775$	5.80024	0.208351
$776$	−10.3651	−0.372084
$777$	0	0
$778$	−5.44406	−0.195179
$779$	−57.8189	−2.07158
$780$	0	0
$781$	−3.75706	−0.134438
$782$	−4.97516	−0.177911
$783$	0	0
$784$	−3.53526	−0.126259
$785$	25.4165	0.907154
$786$	0	0
$787$	−39.4207	−1.40520	−0.702599	−	0.711586i	$-0.747977\pi$
$787$	−39.4207	−1.40520	−0.702599	+	0.711586i	$0.747977\pi$
$788$	45.0102	1.60342
$789$	0	0
$790$	10.9481	0.389515
$791$	40.1547	1.42774
$792$	0	0
$793$	−5.49298	−0.195062
$794$	9.87630	0.350497
$795$	0	0
$796$	−40.0534	−1.41966
$797$	3.14778	0.111500	0.0557501	−	0.998445i	$-0.482245\pi$
$797$	3.14778	0.111500	0.0557501	+	0.998445i	$0.482245\pi$
$798$	0	0
$799$	−10.8234	−0.382903
$800$	18.9458	0.669836
$801$	0	0
$802$	−7.22196	−0.255016
$803$	8.52155	0.300719
$804$	0	0
$805$	54.4370	1.91865
$806$	−1.99243	−0.0701804
$807$	0	0
$808$	−2.57542	−0.0906029
$809$	−32.5405	−1.14406	−0.572032	−	0.820231i	$-0.693845\pi$
$809$	−32.5405	−1.14406	−0.572032	+	0.820231i	$0.693845\pi$
$810$	0	0
$811$	21.8424	0.766990	0.383495	−	0.923543i	$-0.374720\pi$
$811$	21.8424	0.766990	0.383495	+	0.923543i	$0.374720\pi$
$812$	6.03396	0.211750
$813$	0	0
$814$	1.84860	0.0647935
$815$	19.0980	0.668975
$816$	0	0
$817$	−43.5096	−1.52221
$818$	−8.69427	−0.303988
$819$	0	0
$820$	−49.4639	−1.72735
$821$	−2.11622	−0.0738564	−0.0369282	−	0.999318i	$-0.511757\pi$
$821$	−2.11622	−0.0738564	−0.0369282	+	0.999318i	$0.511757\pi$
$822$	0	0
$823$	−54.3017	−1.89284	−0.946419	−	0.322942i	$-0.895328\pi$
$823$	−54.3017	−1.89284	−0.946419	+	0.322942i	$0.895328\pi$
$824$	−12.9632	−0.451594
$825$	0	0
$826$	−4.59236	−0.159789
$827$	40.7368	1.41656	0.708278	−	0.705933i	$-0.249472\pi$
$827$	40.7368	1.41656	0.708278	+	0.705933i	$0.249472\pi$
$828$	0	0
$829$	−53.2182	−1.84834	−0.924172	−	0.381976i	$-0.875244\pi$
$829$	−53.2182	−1.84834	−0.924172	+	0.381976i	$0.875244\pi$
$830$	13.4757	0.467748
$831$	0	0
$832$	30.5630	1.05958
$833$	2.29109	0.0793814
$834$	0	0
$835$	63.2512	2.18890
$836$	−13.3963	−0.463319
$837$	0	0
$838$	−7.50674	−0.259316
$839$	18.7515	0.647375	0.323688	−	0.946164i	$-0.395077\pi$
$839$	18.7515	0.647375	0.323688	+	0.946164i	$0.395077\pi$
$840$	0	0
$841$	−27.2946	−0.941193
$842$	0.184833	0.00636976
$843$	0	0
$844$	15.0220	0.517080
$845$	−54.8720	−1.88765
$846$	0	0
$847$	−2.43974	−0.0838304
$848$	6.72667	0.230995
$849$	0	0
$850$	−3.71179	−0.127313
$851$	39.6222	1.35823
$852$	0	0
$853$	−13.8163	−0.473060	−0.236530	−	0.971624i	$-0.576010\pi$
$853$	−13.8163	−0.473060	−0.236530	+	0.971624i	$0.576010\pi$
$854$	−0.794861	−0.0271996
$855$	0	0
$856$	−10.1720	−0.347672
$857$	29.3010	1.00090	0.500452	−	0.865765i	$-0.333167\pi$
$857$	29.3010	1.00090	0.500452	+	0.865765i	$0.333167\pi$
$858$	0	0
$859$	47.9401	1.63569	0.817847	−	0.575436i	$-0.195167\pi$
$859$	47.9401	1.63569	0.817847	+	0.575436i	$0.195167\pi$
$860$	−37.2223	−1.26927
$861$	0	0
$862$	8.49382	0.289301
$863$	−5.09654	−0.173488	−0.0867441	−	0.996231i	$-0.527646\pi$
$863$	−5.09654	−0.173488	−0.0867441	+	0.996231i	$0.527646\pi$
$864$	0	0
$865$	−34.6394	−1.17777
$866$	2.41091	0.0819262
$867$	0	0
$868$	5.14419	0.174605
$869$	−10.5168	−0.356756
$870$	0	0
$871$	−56.6563	−1.91972
$872$	−14.2954	−0.484103
$873$	0	0
$874$	16.0926	0.544342
$875$	1.63533	0.0552842
$876$	0	0
$877$	8.24100	0.278279	0.139139	−	0.990273i	$-0.455566\pi$
$877$	8.24100	0.278279	0.139139	+	0.990273i	$0.455566\pi$
$878$	6.01280	0.202922
$879$	0	0
$880$	−10.7821	−0.363466
$881$	−0.911510	−0.0307096	−0.0153548	−	0.999882i	$-0.504888\pi$
$881$	−0.911510	−0.0307096	−0.0153548	+	0.999882i	$0.504888\pi$
$882$	0	0
$883$	13.8790	0.467066	0.233533	−	0.972349i	$-0.424971\pi$
$883$	13.8790	0.467066	0.233533	+	0.972349i	$0.424971\pi$
$884$	−22.7495	−0.765148
$885$	0	0
$886$	−11.0444	−0.371042
$887$	−39.6412	−1.33102	−0.665510	−	0.746389i	$-0.731786\pi$
$887$	−39.6412	−1.33102	−0.665510	+	0.746389i	$0.731786\pi$
$888$	0	0
$889$	9.39465	0.315086
$890$	−6.09335	−0.204250
$891$	0	0
$892$	−32.3475	−1.08308
$893$	35.0093	1.17154
$894$	0	0
$895$	67.8314	2.26735
$896$	22.1673	0.740557
$897$	0	0
$898$	−10.9622	−0.365814
$899$	1.45391	0.0484908
$900$	0	0
$901$	−4.35933	−0.145230
$902$	−2.66306	−0.0886702
$903$	0	0
$904$	20.8795	0.694443
$905$	−69.7406	−2.31826
$906$	0	0
$907$	−26.4502	−0.878263	−0.439131	−	0.898423i	$-0.644714\pi$
$907$	−26.4502	−0.878263	−0.439131	+	0.898423i	$0.644714\pi$
$908$	−22.2388	−0.738021
$909$	0	0
$910$	−13.9511	−0.462474
$911$	−44.0317	−1.45884	−0.729418	−	0.684068i	$-0.760209\pi$
$911$	−44.0317	−1.45884	−0.729418	+	0.684068i	$0.760209\pi$
$912$	0	0
$913$	−12.9448	−0.428410
$914$	−3.72608	−0.123248
$915$	0	0
$916$	−54.9464	−1.81548
$917$	39.4596	1.30307
$918$	0	0
$919$	−24.0917	−0.794712	−0.397356	−	0.917664i	$-0.630072\pi$
$919$	−24.0917	−0.794712	−0.397356	+	0.917664i	$0.630072\pi$
$920$	28.3060	0.933221
$921$	0	0
$922$	−13.2892	−0.437656
$923$	20.6375	0.679291
$924$	0	0
$925$	29.5607	0.971950
$926$	−8.84734	−0.290742
$927$	0	0
$928$	4.74904	0.155895
$929$	11.5309	0.378316	0.189158	−	0.981947i	$-0.439424\pi$
$929$	11.5309	0.378316	0.189158	+	0.981947i	$0.439424\pi$
$930$	0	0
$931$	−7.41074	−0.242877
$932$	−21.3087	−0.697991
$933$	0	0
$934$	−8.69542	−0.284523
$935$	6.98754	0.228517
$936$	0	0
$937$	9.40768	0.307336	0.153668	−	0.988123i	$-0.450891\pi$
$937$	9.40768	0.307336	0.153668	+	0.988123i	$0.450891\pi$
$938$	−8.19844	−0.267688
$939$	0	0
$940$	29.9503	0.976871
$941$	55.8237	1.81980	0.909900	−	0.414828i	$-0.136158\pi$
$941$	55.8237	1.81980	0.909900	+	0.414828i	$0.136158\pi$
$942$	0	0
$943$	−57.0790	−1.85875
$944$	19.4958	0.634535
$945$	0	0
$946$	−2.00399	−0.0651554
$947$	21.0771	0.684915	0.342457	−	0.939533i	$-0.388741\pi$
$947$	21.0771	0.684915	0.342457	+	0.939533i	$0.388741\pi$
$948$	0	0
$949$	−46.8087	−1.51948
$950$	12.0061	0.389531
$951$	0	0
$952$	−6.76842	−0.219366
$953$	−39.2308	−1.27081	−0.635405	−	0.772179i	$-0.719167\pi$
$953$	−39.2308	−1.27081	−0.635405	+	0.772179i	$0.719167\pi$
$954$	0	0
$955$	−15.8403	−0.512580
$956$	−7.38273	−0.238774
$957$	0	0
$958$	0.0678519	0.00219220
$959$	−5.18780	−0.167523
$960$	0	0
$961$	−29.7605	−0.960015
$962$	−10.1544	−0.327389
$963$	0	0
$964$	−32.4084	−1.04380
$965$	33.7171	1.08539
$966$	0	0
$967$	−16.0438	−0.515933	−0.257966	−	0.966154i	$-0.583052\pi$
$967$	−16.0438	−0.515933	−0.257966	+	0.966154i	$0.583052\pi$
$968$	−1.26861	−0.0407746
$969$	0	0
$970$	−8.50551	−0.273095
$971$	2.24322	0.0719885	0.0359942	−	0.999352i	$-0.488540\pi$
$971$	2.24322	0.0719885	0.0359942	+	0.999352i	$0.488540\pi$
$972$	0	0
$973$	45.8693	1.47050
$974$	0.881239	0.0282367
$975$	0	0
$976$	3.37440	0.108012
$977$	−14.6332	−0.468158	−0.234079	−	0.972218i	$-0.575207\pi$
$977$	−14.6332	−0.468158	−0.234079	+	0.972218i	$0.575207\pi$
$978$	0	0
$979$	5.85329	0.187072
$980$	−6.33987	−0.202520
$981$	0	0
$982$	2.27941	0.0727389
$983$	38.8491	1.23909	0.619547	−	0.784960i	$-0.287316\pi$
$983$	38.8491	1.23909	0.619547	+	0.784960i	$0.287316\pi$
$984$	0	0
$985$	75.9403	2.41966
$986$	−0.930413	−0.0296304
$987$	0	0
$988$	73.5855	2.34107
$989$	−42.9527	−1.36582
$990$	0	0
$991$	−3.67987	−0.116895	−0.0584474	−	0.998290i	$-0.518615\pi$
$991$	−3.67987	−0.116895	−0.0584474	+	0.998290i	$0.518615\pi$
$992$	4.04875	0.128548
$993$	0	0
$994$	2.98634	0.0947211
$995$	−67.5773	−2.14234
$996$	0	0
$997$	−24.7417	−0.783579	−0.391789	−	0.920055i	$-0.628144\pi$
$997$	−24.7417	−0.783579	−0.391789	+	0.920055i	$0.628144\pi$
$998$	12.5084	0.395945
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.m.1.15	✓	25
3.2	odd	2		6039.2.a.p.1.11	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.15	✓	25	1.1	even	1	trivial
6039.2.a.p.1.11	yes	25	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+0.325798 q^{2} -1.89386 q^{4} -3.19527 q^{5} -2.43974 q^{7} -1.26861 q^{8} +O(q^{10})\)	\(q+0.325798 q^{2} -1.89386 q^{4} -3.19527 q^{5} -2.43974 q^{7} -1.26861 q^{8} -1.04101 q^{10} +1.00000 q^{11} -5.49298 q^{13} -0.794861 q^{14} +3.37440 q^{16} -2.18684 q^{17} +7.07354 q^{19} +6.05139 q^{20} +0.325798 q^{22} +6.98301 q^{23} +5.20977 q^{25} -1.78960 q^{26} +4.62052 q^{28} +1.30591 q^{29} +1.11334 q^{31} +3.63659 q^{32} -0.712466 q^{34} +7.79564 q^{35} +5.67409 q^{37} +2.30454 q^{38} +4.05355 q^{40} -8.17397 q^{41} -6.15103 q^{43} -1.89386 q^{44} +2.27505 q^{46} +4.94933 q^{47} -1.04767 q^{49} +1.69733 q^{50} +10.4029 q^{52} +1.99344 q^{53} -3.19527 q^{55} +3.09508 q^{56} +0.425461 q^{58} +5.77756 q^{59} +1.00000 q^{61} +0.362723 q^{62} -5.56401 q^{64} +17.5516 q^{65} +10.3143 q^{67} +4.14155 q^{68} +2.53980 q^{70} -3.75706 q^{71} +8.52155 q^{73} +1.84860 q^{74} -13.3963 q^{76} -2.43974 q^{77} -10.5168 q^{79} -10.7821 q^{80} -2.66306 q^{82} -12.9448 q^{83} +6.98754 q^{85} -2.00399 q^{86} -1.26861 q^{88} +5.85329 q^{89} +13.4014 q^{91} -13.2248 q^{92} +1.61248 q^{94} -22.6019 q^{95} +8.17042 q^{97} -0.341329 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8 - 12 * q^10 + 25 * q^11 - 4 * q^13 - 14 * q^14 + 21 * q^16 - 16 * q^17 - 18 * q^19 - 28 * q^20 - 5 * q^22 - 8 * q^23 + 29 * q^25 - 16 * q^26 + 18 * q^28 - 28 * q^29 - 8 * q^31 - 35 * q^32 + 6 * q^34 - 22 * q^35 + 4 * q^37 + 4 * q^38 - 12 * q^40 - 58 * q^41 - 26 * q^43 + 25 * q^44 + 8 * q^46 - 20 * q^47 + 23 * q^49 - 27 * q^50 - 2 * q^52 - 36 * q^53 - 12 * q^55 - 70 * q^56 + 12 * q^58 - 18 * q^59 + 25 * q^61 - 42 * q^62 + 35 * q^64 - 76 * q^65 - 8 * q^67 - 28 * q^68 + 76 * q^70 - 24 * q^71 + 2 * q^73 - 40 * q^74 - 64 * q^76 - 4 * q^77 - 22 * q^79 - 36 * q^80 + 30 * q^82 - 14 * q^83 - 70 * q^86 - 15 * q^88 - 82 * q^89 - 6 * q^91 - 48 * q^92 - 16 * q^94 - 34 * q^95 + 16 * q^97 - 35 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more