LMFDB - Embedded newform 3380.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, 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("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.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:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 19x^{7} + 16x^{6} + 106x^{5} - 87x^{4} - 153x^{3} + 149x^{2} - 26x + 1 $ x^9 - x^8 - 19x^7 + 16x^6 + 106x^5 - 87x^4 - 153x^3 + 149x^2 - 26*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.40622$ of defining polynomial
Character	$\chi$	$=$	3380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.40622 q^{3} -1.00000 q^{5} -2.33790 q^{7} +8.60234 q^{9} +O(q^{10})$	$q-3.40622 q^{3} -1.00000 q^{5} -2.33790 q^{7} +8.60234 q^{9} +6.16565 q^{11} +3.40622 q^{15} +3.07520 q^{17} +0.0902681 q^{19} +7.96342 q^{21} +4.81954 q^{23} +1.00000 q^{25} -19.0828 q^{27} +5.63201 q^{29} -8.34078 q^{31} -21.0016 q^{33} +2.33790 q^{35} -0.794425 q^{37} +3.95907 q^{41} -8.75108 q^{43} -8.60234 q^{45} +3.67871 q^{47} -1.53420 q^{49} -10.4748 q^{51} +8.08456 q^{53} -6.16565 q^{55} -0.307473 q^{57} -0.379577 q^{59} -4.96367 q^{61} -20.1114 q^{63} -0.376774 q^{67} -16.4164 q^{69} +9.04130 q^{71} -5.48133 q^{73} -3.40622 q^{75} -14.4147 q^{77} -4.79283 q^{79} +39.1932 q^{81} -11.6189 q^{83} -3.07520 q^{85} -19.1839 q^{87} +11.7118 q^{89} +28.4105 q^{93} -0.0902681 q^{95} -2.43658 q^{97} +53.0390 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9	$ 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + 7 q^{11} + q^{15} + 13 q^{17} + 4 q^{19} + 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} - 13 q^{31} - 34 q^{33} + q^{35} - q^{37} + 6 q^{41} + q^{43} - 12 q^{45} + 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} - 38 q^{57} - 15 q^{59} + 21 q^{61} + 17 q^{63} + 7 q^{67} + 15 q^{69} + 7 q^{71} + 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} - 45 q^{83} - 13 q^{85} + 28 q^{87} + 41 q^{89} + 11 q^{93} - 4 q^{95} - 8 q^{97} + 81 q^{99}+O(q^{100}) $ 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9 + 7 * q^11 + q^15 + 13 * q^17 + 4 * q^19 + 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 - 13 * q^31 - 34 * q^33 + q^35 - q^37 + 6 * q^41 + q^43 - 12 * q^45 + 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 - 38 * q^57 - 15 * q^59 + 21 * q^61 + 17 * q^63 + 7 * q^67 + 15 * q^69 + 7 * q^71 + 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 - 45 * q^83 - 13 * q^85 + 28 * q^87 + 41 * q^89 + 11 * q^93 - 4 * q^95 - 8 * q^97 + 81 * q^99

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	0
$2$	0	0
$3$	−3.40622	−1.96658	−0.983291	−	0.182040i	$-0.941730\pi$
$3$	−3.40622	−1.96658	−0.983291	+	0.182040i	$0.941730\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.33790	−0.883645	−0.441822	−	0.897103i	$-0.645668\pi$
$7$	−2.33790	−0.883645	−0.441822	+	0.897103i	$0.645668\pi$
$8$	0	0
$9$	8.60234	2.86745
$10$	0	0
$11$	6.16565	1.85901	0.929507	−	0.368806i	$-0.120233\pi$
$11$	6.16565	1.85901	0.929507	+	0.368806i	$0.120233\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.40622	0.879482
$16$	0	0
$17$	3.07520	0.745844	0.372922	−	0.927863i	$-0.378356\pi$
$17$	3.07520	0.745844	0.372922	+	0.927863i	$0.378356\pi$
$18$	0	0
$19$	0.0902681	0.0207089	0.0103545	−	0.999946i	$-0.496704\pi$
$19$	0.0902681	0.0207089	0.0103545	+	0.999946i	$0.496704\pi$
$20$	0	0
$21$	7.96342	1.73776
$22$	0	0
$23$	4.81954	1.00494	0.502472	−	0.864594i	$-0.332424\pi$
$23$	4.81954	1.00494	0.502472	+	0.864594i	$0.332424\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−19.0828	−3.67249
$28$	0	0
$29$	5.63201	1.04584	0.522919	−	0.852382i	$-0.324843\pi$
$29$	5.63201	1.04584	0.522919	+	0.852382i	$0.324843\pi$
$30$	0	0
$31$	−8.34078	−1.49805	−0.749024	−	0.662543i	$-0.769477\pi$
$31$	−8.34078	−1.49805	−0.749024	+	0.662543i	$0.769477\pi$
$32$	0	0
$33$	−21.0016	−3.65590
$34$	0	0
$35$	2.33790	0.395178
$36$	0	0
$37$	−0.794425	−0.130603	−0.0653014	−	0.997866i	$-0.520801\pi$
$37$	−0.794425	−0.130603	−0.0653014	+	0.997866i	$0.520801\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.95907	0.618302	0.309151	−	0.951013i	$-0.399955\pi$
$41$	3.95907	0.618302	0.309151	+	0.951013i	$0.399955\pi$
$42$	0	0
$43$	−8.75108	−1.33453	−0.667264	−	0.744822i	$-0.732535\pi$
$43$	−8.75108	−1.33453	−0.667264	+	0.744822i	$0.732535\pi$
$44$	0	0
$45$	−8.60234	−1.28236
$46$	0	0
$47$	3.67871	0.536595	0.268298	−	0.963336i	$-0.413539\pi$
$47$	3.67871	0.536595	0.268298	+	0.963336i	$0.413539\pi$
$48$	0	0
$49$	−1.53420	−0.219172
$50$	0	0
$51$	−10.4748	−1.46676
$52$	0	0
$53$	8.08456	1.11050	0.555250	−	0.831683i	$-0.312623\pi$
$53$	8.08456	1.11050	0.555250	+	0.831683i	$0.312623\pi$
$54$	0	0
$55$	−6.16565	−0.831376
$56$	0	0
$57$	−0.307473	−0.0407258
$58$	0	0
$59$	−0.379577	−0.0494167	−0.0247084	−	0.999695i	$-0.507866\pi$
$59$	−0.379577	−0.0494167	−0.0247084	+	0.999695i	$0.507866\pi$
$60$	0	0
$61$	−4.96367	−0.635533	−0.317766	−	0.948169i	$-0.602933\pi$
$61$	−4.96367	−0.635533	−0.317766	+	0.948169i	$0.602933\pi$
$62$	0	0
$63$	−20.1114	−2.53380
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−0.376774	−0.0460302	−0.0230151	−	0.999735i	$-0.507327\pi$
$67$	−0.376774	−0.0460302	−0.0230151	+	0.999735i	$0.507327\pi$
$68$	0	0
$69$	−16.4164	−1.97630
$70$	0	0
$71$	9.04130	1.07300	0.536502	−	0.843899i	$-0.319745\pi$
$71$	9.04130	1.07300	0.536502	+	0.843899i	$0.319745\pi$
$72$	0	0
$73$	−5.48133	−0.641541	−0.320770	−	0.947157i	$-0.603942\pi$
$73$	−5.48133	−0.641541	−0.320770	+	0.947157i	$0.603942\pi$
$74$	0	0
$75$	−3.40622	−0.393316
$76$	0	0
$77$	−14.4147	−1.64271
$78$	0	0
$79$	−4.79283	−0.539235	−0.269617	−	0.962967i	$-0.586897\pi$
$79$	−4.79283	−0.539235	−0.269617	+	0.962967i	$0.586897\pi$
$80$	0	0
$81$	39.1932	4.35480
$82$	0	0
$83$	−11.6189	−1.27534	−0.637670	−	0.770310i	$-0.720102\pi$
$83$	−11.6189	−1.27534	−0.637670	+	0.770310i	$0.720102\pi$
$84$	0	0
$85$	−3.07520	−0.333552
$86$	0	0
$87$	−19.1839	−2.05673
$88$	0	0
$89$	11.7118	1.24145	0.620724	−	0.784029i	$-0.286839\pi$
$89$	11.7118	1.24145	0.620724	+	0.784029i	$0.286839\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	28.4105	2.94603
$94$	0	0
$95$	−0.0902681	−0.00926131
$96$	0	0
$97$	−2.43658	−0.247397	−0.123698	−	0.992320i	$-0.539476\pi$
$97$	−2.43658	−0.247397	−0.123698	+	0.992320i	$0.539476\pi$
$98$	0	0
$99$	53.0390	5.33062
$100$	0	0
$101$	−5.52834	−0.550090	−0.275045	−	0.961431i	$-0.588693\pi$
$101$	−5.52834	−0.550090	−0.275045	+	0.961431i	$0.588693\pi$
$102$	0	0
$103$	8.46999	0.834573	0.417287	−	0.908775i	$-0.362981\pi$
$103$	8.46999	0.834573	0.417287	+	0.908775i	$0.362981\pi$
$104$	0	0
$105$	−7.96342	−0.777150
$106$	0	0
$107$	11.3891	1.10103	0.550514	−	0.834826i	$-0.314432\pi$
$107$	11.3891	1.10103	0.550514	+	0.834826i	$0.314432\pi$
$108$	0	0
$109$	9.04624	0.866473	0.433236	−	0.901280i	$-0.357372\pi$
$109$	9.04624	0.866473	0.433236	+	0.901280i	$0.357372\pi$
$110$	0	0
$111$	2.70599	0.256841
$112$	0	0
$113$	−2.99157	−0.281423	−0.140712	−	0.990051i	$-0.544939\pi$
$113$	−2.99157	−0.281423	−0.140712	+	0.990051i	$0.544939\pi$
$114$	0	0
$115$	−4.81954	−0.449425
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.18951	−0.659061
$120$	0	0
$121$	27.0152	2.45593
$122$	0	0
$123$	−13.4855	−1.21594
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	10.6776	0.947482	0.473741	−	0.880664i	$-0.342903\pi$
$127$	10.6776	0.947482	0.473741	+	0.880664i	$0.342903\pi$
$128$	0	0
$129$	29.8081	2.62446
$130$	0	0
$131$	−20.6072	−1.80046	−0.900231	−	0.435412i	$-0.856603\pi$
$131$	−20.6072	−1.80046	−0.900231	+	0.435412i	$0.856603\pi$
$132$	0	0
$133$	−0.211038	−0.0182993
$134$	0	0
$135$	19.0828	1.64239
$136$	0	0
$137$	0.0850150	0.00726332	0.00363166	−	0.999993i	$-0.498844\pi$
$137$	0.0850150	0.00726332	0.00363166	+	0.999993i	$0.498844\pi$
$138$	0	0
$139$	11.0101	0.933864	0.466932	−	0.884293i	$-0.345359\pi$
$139$	11.0101	0.933864	0.466932	+	0.884293i	$0.345359\pi$
$140$	0	0
$141$	−12.5305	−1.05526
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−5.63201	−0.467713
$146$	0	0
$147$	5.22584	0.431020
$148$	0	0
$149$	−4.41512	−0.361701	−0.180850	−	0.983511i	$-0.557885\pi$
$149$	−4.41512	−0.361701	−0.180850	+	0.983511i	$0.557885\pi$
$150$	0	0
$151$	−15.7555	−1.28216	−0.641081	−	0.767473i	$-0.721514\pi$
$151$	−15.7555	−1.28216	−0.641081	+	0.767473i	$0.721514\pi$
$152$	0	0
$153$	26.4539	2.13867
$154$	0	0
$155$	8.34078	0.669947
$156$	0	0
$157$	3.40120	0.271445	0.135722	−	0.990747i	$-0.456664\pi$
$157$	3.40120	0.271445	0.135722	+	0.990747i	$0.456664\pi$
$158$	0	0
$159$	−27.5378	−2.18389
$160$	0	0
$161$	−11.2676	−0.888013
$162$	0	0
$163$	−14.4332	−1.13050	−0.565249	−	0.824920i	$-0.691220\pi$
$163$	−14.4332	−1.13050	−0.565249	+	0.824920i	$0.691220\pi$
$164$	0	0
$165$	21.0016	1.63497
$166$	0	0
$167$	18.9815	1.46883	0.734417	−	0.678698i	$-0.237456\pi$
$167$	18.9815	1.46883	0.734417	+	0.678698i	$0.237456\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0.776517	0.0593817
$172$	0	0
$173$	11.0970	0.843691	0.421845	−	0.906668i	$-0.361383\pi$
$173$	11.0970	0.843691	0.421845	+	0.906668i	$0.361383\pi$
$174$	0	0
$175$	−2.33790	−0.176729
$176$	0	0
$177$	1.29292	0.0971821
$178$	0	0
$179$	5.27165	0.394021	0.197011	−	0.980401i	$-0.436877\pi$
$179$	5.27165	0.394021	0.197011	+	0.980401i	$0.436877\pi$
$180$	0	0
$181$	16.1744	1.20223	0.601117	−	0.799161i	$-0.294723\pi$
$181$	16.1744	1.20223	0.601117	+	0.799161i	$0.294723\pi$
$182$	0	0
$183$	16.9074	1.24983
$184$	0	0
$185$	0.794425	0.0584073
$186$	0	0
$187$	18.9606	1.38653
$188$	0	0
$189$	44.6138	3.24517
$190$	0	0
$191$	−6.75365	−0.488677	−0.244338	−	0.969690i	$-0.578571\pi$
$191$	−6.75365	−0.488677	−0.244338	+	0.969690i	$0.578571\pi$
$192$	0	0
$193$	−0.481519	−0.0346605	−0.0173302	−	0.999850i	$-0.505517\pi$
$193$	−0.481519	−0.0346605	−0.0173302	+	0.999850i	$0.505517\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.8881	−1.55946	−0.779732	−	0.626113i	$-0.784645\pi$
$197$	−21.8881	−1.55946	−0.779732	+	0.626113i	$0.784645\pi$
$198$	0	0
$199$	23.5567	1.66989	0.834947	−	0.550331i	$-0.185498\pi$
$199$	23.5567	1.66989	0.834947	+	0.550331i	$0.185498\pi$
$200$	0	0
$201$	1.28337	0.0905223
$202$	0	0
$203$	−13.1671	−0.924149
$204$	0	0
$205$	−3.95907	−0.276513
$206$	0	0
$207$	41.4593	2.88162
$208$	0	0
$209$	0.556561	0.0384981
$210$	0	0
$211$	−11.0661	−0.761824	−0.380912	−	0.924611i	$-0.624390\pi$
$211$	−11.0661	−0.761824	−0.380912	+	0.924611i	$0.624390\pi$
$212$	0	0
$213$	−30.7967	−2.11015
$214$	0	0
$215$	8.75108	0.596819
$216$	0	0
$217$	19.4999	1.32374
$218$	0	0
$219$	18.6706	1.26164
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−24.3465	−1.63036	−0.815182	−	0.579205i	$-0.803363\pi$
$223$	−24.3465	−1.63036	−0.815182	+	0.579205i	$0.803363\pi$
$224$	0	0
$225$	8.60234	0.573489
$226$	0	0
$227$	−0.100791	−0.00668971	−0.00334485	−	0.999994i	$-0.501065\pi$
$227$	−0.100791	−0.00668971	−0.00334485	+	0.999994i	$0.501065\pi$
$228$	0	0
$229$	5.60506	0.370393	0.185196	−	0.982702i	$-0.440708\pi$
$229$	5.60506	0.370393	0.185196	+	0.982702i	$0.440708\pi$
$230$	0	0
$231$	49.0996	3.23052
$232$	0	0
$233$	13.2785	0.869902	0.434951	−	0.900454i	$-0.356766\pi$
$233$	13.2785	0.869902	0.434951	+	0.900454i	$0.356766\pi$
$234$	0	0
$235$	−3.67871	−0.239973
$236$	0	0
$237$	16.3254	1.06045
$238$	0	0
$239$	12.9817	0.839715	0.419858	−	0.907590i	$-0.362080\pi$
$239$	12.9817	0.839715	0.419858	+	0.907590i	$0.362080\pi$
$240$	0	0
$241$	−1.49222	−0.0961223	−0.0480612	−	0.998844i	$-0.515304\pi$
$241$	−1.49222	−0.0961223	−0.0480612	+	0.998844i	$0.515304\pi$
$242$	0	0
$243$	−76.2524	−4.89159
$244$	0	0
$245$	1.53420	0.0980167
$246$	0	0
$247$	0	0
$248$	0	0
$249$	39.5765	2.50806
$250$	0	0
$251$	−11.2640	−0.710975	−0.355487	−	0.934681i	$-0.615685\pi$
$251$	−11.2640	−0.710975	−0.355487	+	0.934681i	$0.615685\pi$
$252$	0	0
$253$	29.7156	1.86820
$254$	0	0
$255$	10.4748	0.655957
$256$	0	0
$257$	15.0148	0.936600	0.468300	−	0.883569i	$-0.344867\pi$
$257$	15.0148	0.936600	0.468300	+	0.883569i	$0.344867\pi$
$258$	0	0
$259$	1.85729	0.115406
$260$	0	0
$261$	48.4484	2.99888
$262$	0	0
$263$	11.4305	0.704837	0.352418	−	0.935843i	$-0.385359\pi$
$263$	11.4305	0.704837	0.352418	+	0.935843i	$0.385359\pi$
$264$	0	0
$265$	−8.08456	−0.496631
$266$	0	0
$267$	−39.8930	−2.44141
$268$	0	0
$269$	25.3906	1.54809	0.774047	−	0.633129i	$-0.218230\pi$
$269$	25.3906	1.54809	0.774047	+	0.633129i	$0.218230\pi$
$270$	0	0
$271$	−10.6327	−0.645892	−0.322946	−	0.946417i	$-0.604673\pi$
$271$	−10.6327	−0.645892	−0.322946	+	0.946417i	$0.604673\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.16565	0.371803
$276$	0	0
$277$	25.9901	1.56159	0.780797	−	0.624785i	$-0.214813\pi$
$277$	25.9901	1.56159	0.780797	+	0.624785i	$0.214813\pi$
$278$	0	0
$279$	−71.7502	−4.29557
$280$	0	0
$281$	7.65920	0.456909	0.228455	−	0.973555i	$-0.426633\pi$
$281$	7.65920	0.456909	0.228455	+	0.973555i	$0.426633\pi$
$282$	0	0
$283$	19.5025	1.15930	0.579651	−	0.814865i	$-0.303189\pi$
$283$	19.5025	1.15930	0.579651	+	0.814865i	$0.303189\pi$
$284$	0	0
$285$	0.307473	0.0182131
$286$	0	0
$287$	−9.25592	−0.546360
$288$	0	0
$289$	−7.54317	−0.443716
$290$	0	0
$291$	8.29952	0.486526
$292$	0	0
$293$	12.7879	0.747078	0.373539	−	0.927614i	$-0.378144\pi$
$293$	12.7879	0.747078	0.373539	+	0.927614i	$0.378144\pi$
$294$	0	0
$295$	0.379577	0.0220998
$296$	0	0
$297$	−117.658	−6.82720
$298$	0	0
$299$	0	0
$300$	0	0
$301$	20.4592	1.17925
$302$	0	0
$303$	18.8307	1.08180
$304$	0	0
$305$	4.96367	0.284219
$306$	0	0
$307$	−32.5882	−1.85990	−0.929952	−	0.367680i	$-0.880152\pi$
$307$	−32.5882	−1.85990	−0.929952	+	0.367680i	$0.880152\pi$
$308$	0	0
$309$	−28.8507	−1.64126
$310$	0	0
$311$	4.51961	0.256284	0.128142	−	0.991756i	$-0.459099\pi$
$311$	4.51961	0.256284	0.128142	+	0.991756i	$0.459099\pi$
$312$	0	0
$313$	−2.40976	−0.136208	−0.0681038	−	0.997678i	$-0.521695\pi$
$313$	−2.40976	−0.136208	−0.0681038	+	0.997678i	$0.521695\pi$
$314$	0	0
$315$	20.1114	1.13315
$316$	0	0
$317$	−11.6751	−0.655739	−0.327870	−	0.944723i	$-0.606331\pi$
$317$	−11.6751	−0.655739	−0.327870	+	0.944723i	$0.606331\pi$
$318$	0	0
$319$	34.7250	1.94423
$320$	0	0
$321$	−38.7939	−2.16526
$322$	0	0
$323$	0.277592	0.0154456
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−30.8135	−1.70399
$328$	0	0
$329$	−8.60048	−0.474160
$330$	0	0
$331$	−22.3827	−1.23026	−0.615131	−	0.788425i	$-0.710897\pi$
$331$	−22.3827	−1.23026	−0.615131	+	0.788425i	$0.710897\pi$
$332$	0	0
$333$	−6.83392	−0.374496
$334$	0	0
$335$	0.376774	0.0205854
$336$	0	0
$337$	12.6686	0.690102	0.345051	−	0.938584i	$-0.387862\pi$
$337$	12.6686	0.690102	0.345051	+	0.938584i	$0.387862\pi$
$338$	0	0
$339$	10.1900	0.553442
$340$	0	0
$341$	−51.4263	−2.78489
$342$	0	0
$343$	19.9522	1.07731
$344$	0	0
$345$	16.4164	0.883830
$346$	0	0
$347$	7.56502	0.406111	0.203056	−	0.979167i	$-0.434913\pi$
$347$	7.56502	0.406111	0.203056	+	0.979167i	$0.434913\pi$
$348$	0	0
$349$	18.3617	0.982879	0.491439	−	0.870912i	$-0.336471\pi$
$349$	18.3617	0.982879	0.491439	+	0.870912i	$0.336471\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.27380	0.0677976	0.0338988	−	0.999425i	$-0.489208\pi$
$353$	1.27380	0.0677976	0.0338988	+	0.999425i	$0.489208\pi$
$354$	0	0
$355$	−9.04130	−0.479862
$356$	0	0
$357$	24.4891	1.29610
$358$	0	0
$359$	11.0008	0.580598	0.290299	−	0.956936i	$-0.406245\pi$
$359$	11.0008	0.580598	0.290299	+	0.956936i	$0.406245\pi$
$360$	0	0
$361$	−18.9919	−0.999571
$362$	0	0
$363$	−92.0198	−4.82979
$364$	0	0
$365$	5.48133	0.286906
$366$	0	0
$367$	5.95825	0.311018	0.155509	−	0.987834i	$-0.450298\pi$
$367$	5.95825	0.311018	0.155509	+	0.987834i	$0.450298\pi$
$368$	0	0
$369$	34.0572	1.77295
$370$	0	0
$371$	−18.9009	−0.981288
$372$	0	0
$373$	−21.7186	−1.12455	−0.562274	−	0.826951i	$-0.690073\pi$
$373$	−21.7186	−1.12455	−0.562274	+	0.826951i	$0.690073\pi$
$374$	0	0
$375$	3.40622	0.175896
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−29.5331	−1.51701	−0.758507	−	0.651665i	$-0.774071\pi$
$379$	−29.5331	−1.51701	−0.758507	+	0.651665i	$0.774071\pi$
$380$	0	0
$381$	−36.3702	−1.86330
$382$	0	0
$383$	26.8889	1.37396	0.686980	−	0.726676i	$-0.258936\pi$
$383$	26.8889	1.37396	0.686980	+	0.726676i	$0.258936\pi$
$384$	0	0
$385$	14.4147	0.734641
$386$	0	0
$387$	−75.2798	−3.82669
$388$	0	0
$389$	−3.11775	−0.158076	−0.0790380	−	0.996872i	$-0.525185\pi$
$389$	−3.11775	−0.158076	−0.0790380	+	0.996872i	$0.525185\pi$
$390$	0	0
$391$	14.8210	0.749532
$392$	0	0
$393$	70.1928	3.54076
$394$	0	0
$395$	4.79283	0.241153
$396$	0	0
$397$	21.2851	1.06827	0.534133	−	0.845400i	$-0.320638\pi$
$397$	21.2851	1.06827	0.534133	+	0.845400i	$0.320638\pi$
$398$	0	0
$399$	0.718842	0.0359871
$400$	0	0
$401$	−7.30134	−0.364611	−0.182306	−	0.983242i	$-0.558356\pi$
$401$	−7.30134	−0.364611	−0.182306	+	0.983242i	$0.558356\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−39.1932	−1.94753
$406$	0	0
$407$	−4.89815	−0.242792
$408$	0	0
$409$	4.26732	0.211005	0.105503	−	0.994419i	$-0.466355\pi$
$409$	4.26732	0.211005	0.105503	+	0.994419i	$0.466355\pi$
$410$	0	0
$411$	−0.289580	−0.0142839
$412$	0	0
$413$	0.887415	0.0436668
$414$	0	0
$415$	11.6189	0.570349
$416$	0	0
$417$	−37.5028	−1.83652
$418$	0	0
$419$	−8.14300	−0.397811	−0.198906	−	0.980019i	$-0.563739\pi$
$419$	−8.14300	−0.397811	−0.198906	+	0.980019i	$0.563739\pi$
$420$	0	0
$421$	−7.36448	−0.358923	−0.179461	−	0.983765i	$-0.557436\pi$
$421$	−7.36448	−0.358923	−0.179461	+	0.983765i	$0.557436\pi$
$422$	0	0
$423$	31.6455	1.53866
$424$	0	0
$425$	3.07520	0.149169
$426$	0	0
$427$	11.6046	0.561585
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.1661	0.971369	0.485684	−	0.874134i	$-0.338570\pi$
$431$	20.1661	0.971369	0.485684	+	0.874134i	$0.338570\pi$
$432$	0	0
$433$	11.0657	0.531783	0.265892	−	0.964003i	$-0.414334\pi$
$433$	11.0657	0.531783	0.265892	+	0.964003i	$0.414334\pi$
$434$	0	0
$435$	19.1839	0.919796
$436$	0	0
$437$	0.435051	0.0208113
$438$	0	0
$439$	34.5470	1.64884	0.824419	−	0.565980i	$-0.191502\pi$
$439$	34.5470	1.64884	0.824419	+	0.565980i	$0.191502\pi$
$440$	0	0
$441$	−13.1977	−0.628464
$442$	0	0
$443$	−1.22275	−0.0580947	−0.0290473	−	0.999578i	$-0.509247\pi$
$443$	−1.22275	−0.0580947	−0.0290473	+	0.999578i	$0.509247\pi$
$444$	0	0
$445$	−11.7118	−0.555193
$446$	0	0
$447$	15.0389	0.711315
$448$	0	0
$449$	−18.2491	−0.861228	−0.430614	−	0.902536i	$-0.641703\pi$
$449$	−18.2491	−0.861228	−0.430614	+	0.902536i	$0.641703\pi$
$450$	0	0
$451$	24.4102	1.14943
$452$	0	0
$453$	53.6666	2.52148
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.2180	1.08609	0.543046	−	0.839703i	$-0.317271\pi$
$457$	23.2180	1.08609	0.543046	+	0.839703i	$0.317271\pi$
$458$	0	0
$459$	−58.6833	−2.73910
$460$	0	0
$461$	−29.6014	−1.37868	−0.689338	−	0.724440i	$-0.742099\pi$
$461$	−29.6014	−1.37868	−0.689338	+	0.724440i	$0.742099\pi$
$462$	0	0
$463$	30.4362	1.41449	0.707244	−	0.706969i	$-0.249938\pi$
$463$	30.4362	1.41449	0.707244	+	0.706969i	$0.249938\pi$
$464$	0	0
$465$	−28.4105	−1.31751
$466$	0	0
$467$	15.6000	0.721880	0.360940	−	0.932589i	$-0.382456\pi$
$467$	15.6000	0.721880	0.360940	+	0.932589i	$0.382456\pi$
$468$	0	0
$469$	0.880861	0.0406744
$470$	0	0
$471$	−11.5852	−0.533819
$472$	0	0
$473$	−53.9561	−2.48090
$474$	0	0
$475$	0.0902681	0.00414178
$476$	0	0
$477$	69.5462	3.18430
$478$	0	0
$479$	41.5259	1.89737	0.948684	−	0.316227i	$-0.102416\pi$
$479$	41.5259	1.89737	0.948684	+	0.316227i	$0.102416\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	38.3800	1.74635
$484$	0	0
$485$	2.43658	0.110639
$486$	0	0
$487$	9.84163	0.445967	0.222983	−	0.974822i	$-0.428420\pi$
$487$	9.84163	0.445967	0.222983	+	0.974822i	$0.428420\pi$
$488$	0	0
$489$	49.1628	2.22322
$490$	0	0
$491$	29.1322	1.31472	0.657359	−	0.753577i	$-0.271673\pi$
$491$	29.1322	1.31472	0.657359	+	0.753577i	$0.271673\pi$
$492$	0	0
$493$	17.3195	0.780032
$494$	0	0
$495$	−53.0390	−2.38393
$496$	0	0
$497$	−21.1377	−0.948155
$498$	0	0
$499$	−33.1347	−1.48331	−0.741656	−	0.670781i	$-0.765959\pi$
$499$	−33.1347	−1.48331	−0.741656	+	0.670781i	$0.765959\pi$
$500$	0	0
$501$	−64.6553	−2.88858
$502$	0	0
$503$	24.3323	1.08492	0.542462	−	0.840080i	$-0.317492\pi$
$503$	24.3323	1.08492	0.542462	+	0.840080i	$0.317492\pi$
$504$	0	0
$505$	5.52834	0.246008
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.9484	0.751223	0.375611	−	0.926777i	$-0.377433\pi$
$509$	16.9484	0.751223	0.375611	+	0.926777i	$0.377433\pi$
$510$	0	0
$511$	12.8148	0.566894
$512$	0	0
$513$	−1.72257	−0.0760532
$514$	0	0
$515$	−8.46999	−0.373232
$516$	0	0
$517$	22.6817	0.997538
$518$	0	0
$519$	−37.7989	−1.65919
$520$	0	0
$521$	−16.0211	−0.701896	−0.350948	−	0.936395i	$-0.614141\pi$
$521$	−16.0211	−0.701896	−0.350948	+	0.936395i	$0.614141\pi$
$522$	0	0
$523$	−5.05699	−0.221127	−0.110563	−	0.993869i	$-0.535266\pi$
$523$	−5.05699	−0.221127	−0.110563	+	0.993869i	$0.535266\pi$
$524$	0	0
$525$	7.96342	0.347552
$526$	0	0
$527$	−25.6495	−1.11731
$528$	0	0
$529$	0.227981	0.00991221
$530$	0	0
$531$	−3.26525	−0.141700
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−11.3891	−0.492395
$536$	0	0
$537$	−17.9564	−0.774876
$538$	0	0
$539$	−9.45936	−0.407444
$540$	0	0
$541$	−44.1770	−1.89932	−0.949659	−	0.313286i	$-0.898570\pi$
$541$	−44.1770	−1.89932	−0.949659	+	0.313286i	$0.898570\pi$
$542$	0	0
$543$	−55.0936	−2.36429
$544$	0	0
$545$	−9.04624	−0.387498
$546$	0	0
$547$	16.3624	0.699605	0.349803	−	0.936823i	$-0.386249\pi$
$547$	16.3624	0.699605	0.349803	+	0.936823i	$0.386249\pi$
$548$	0	0
$549$	−42.6992	−1.82236
$550$	0	0
$551$	0.508391	0.0216582
$552$	0	0
$553$	11.2052	0.476492
$554$	0	0
$555$	−2.70599	−0.114863
$556$	0	0
$557$	29.8749	1.26584	0.632919	−	0.774218i	$-0.281857\pi$
$557$	29.8749	1.26584	0.632919	+	0.774218i	$0.281857\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−64.5839	−2.72673
$562$	0	0
$563$	16.9922	0.716135	0.358067	−	0.933696i	$-0.383436\pi$
$563$	16.9922	0.716135	0.358067	+	0.933696i	$0.383436\pi$
$564$	0	0
$565$	2.99157	0.125856
$566$	0	0
$567$	−91.6300	−3.84810
$568$	0	0
$569$	16.8326	0.705660	0.352830	−	0.935687i	$-0.385219\pi$
$569$	16.8326	0.705660	0.352830	+	0.935687i	$0.385219\pi$
$570$	0	0
$571$	1.09953	0.0460138	0.0230069	−	0.999735i	$-0.492676\pi$
$571$	1.09953	0.0460138	0.0230069	+	0.999735i	$0.492676\pi$
$572$	0	0
$573$	23.0044	0.961023
$574$	0	0
$575$	4.81954	0.200989
$576$	0	0
$577$	−21.1153	−0.879042	−0.439521	−	0.898232i	$-0.644852\pi$
$577$	−21.1153	−0.879042	−0.439521	+	0.898232i	$0.644852\pi$
$578$	0	0
$579$	1.64016	0.0681627
$580$	0	0
$581$	27.1639	1.12695
$582$	0	0
$583$	49.8466	2.06443
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.2938	1.33291	0.666453	−	0.745547i	$-0.267812\pi$
$587$	32.2938	1.33291	0.666453	+	0.745547i	$0.267812\pi$
$588$	0	0
$589$	−0.752906	−0.0310229
$590$	0	0
$591$	74.5558	3.06681
$592$	0	0
$593$	−12.5827	−0.516711	−0.258356	−	0.966050i	$-0.583181\pi$
$593$	−12.5827	−0.516711	−0.258356	+	0.966050i	$0.583181\pi$
$594$	0	0
$595$	7.18951	0.294741
$596$	0	0
$597$	−80.2395	−3.28398
$598$	0	0
$599$	9.46110	0.386570	0.193285	−	0.981143i	$-0.438086\pi$
$599$	9.46110	0.386570	0.193285	+	0.981143i	$0.438086\pi$
$600$	0	0
$601$	8.02016	0.327149	0.163574	−	0.986531i	$-0.447698\pi$
$601$	8.02016	0.327149	0.163574	+	0.986531i	$0.447698\pi$
$602$	0	0
$603$	−3.24114	−0.131989
$604$	0	0
$605$	−27.0152	−1.09833
$606$	0	0
$607$	−26.5832	−1.07898	−0.539490	−	0.841992i	$-0.681383\pi$
$607$	−26.5832	−1.07898	−0.539490	+	0.841992i	$0.681383\pi$
$608$	0	0
$609$	44.8500	1.81742
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−44.8547	−1.81166	−0.905831	−	0.423639i	$-0.860753\pi$
$613$	−44.8547	−1.81166	−0.905831	+	0.423639i	$0.860753\pi$
$614$	0	0
$615$	13.4855	0.543786
$616$	0	0
$617$	20.9169	0.842082	0.421041	−	0.907042i	$-0.361665\pi$
$617$	20.9169	0.842082	0.421041	+	0.907042i	$0.361665\pi$
$618$	0	0
$619$	−9.55179	−0.383919	−0.191959	−	0.981403i	$-0.561484\pi$
$619$	−9.55179	−0.383919	−0.191959	+	0.981403i	$0.561484\pi$
$620$	0	0
$621$	−91.9704	−3.69064
$622$	0	0
$623$	−27.3811	−1.09700
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.89577	−0.0757098
$628$	0	0
$629$	−2.44301	−0.0974093
$630$	0	0
$631$	4.38539	0.174580	0.0872899	−	0.996183i	$-0.472179\pi$
$631$	4.38539	0.174580	0.0872899	+	0.996183i	$0.472179\pi$
$632$	0	0
$633$	37.6937	1.49819
$634$	0	0
$635$	−10.6776	−0.423727
$636$	0	0
$637$	0	0
$638$	0	0
$639$	77.7763	3.07678
$640$	0	0
$641$	−19.1325	−0.755690	−0.377845	−	0.925869i	$-0.623335\pi$
$641$	−19.1325	−0.755690	−0.377845	+	0.925869i	$0.623335\pi$
$642$	0	0
$643$	17.7043	0.698190	0.349095	−	0.937087i	$-0.386489\pi$
$643$	17.7043	0.698190	0.349095	+	0.937087i	$0.386489\pi$
$644$	0	0
$645$	−29.8081	−1.17369
$646$	0	0
$647$	−30.0594	−1.18176	−0.590879	−	0.806761i	$-0.701219\pi$
$647$	−30.0594	−1.18176	−0.590879	+	0.806761i	$0.701219\pi$
$648$	0	0
$649$	−2.34034	−0.0918663
$650$	0	0
$651$	−66.4211	−2.60325
$652$	0	0
$653$	−19.3677	−0.757915	−0.378958	−	0.925414i	$-0.623717\pi$
$653$	−19.3677	−0.757915	−0.378958	+	0.925414i	$0.623717\pi$
$654$	0	0
$655$	20.6072	0.805191
$656$	0	0
$657$	−47.1522	−1.83958
$658$	0	0
$659$	8.47410	0.330104	0.165052	−	0.986285i	$-0.447221\pi$
$659$	8.47410	0.330104	0.165052	+	0.986285i	$0.447221\pi$
$660$	0	0
$661$	14.9713	0.582318	0.291159	−	0.956675i	$-0.405959\pi$
$661$	14.9713	0.582318	0.291159	+	0.956675i	$0.405959\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.211038	0.00818371
$666$	0	0
$667$	27.1437	1.05101
$668$	0	0
$669$	82.9296	3.20625
$670$	0	0
$671$	−30.6043	−1.18146
$672$	0	0
$673$	35.3418	1.36233	0.681163	−	0.732132i	$-0.261475\pi$
$673$	35.3418	1.36233	0.681163	+	0.732132i	$0.261475\pi$
$674$	0	0
$675$	−19.0828	−0.734497
$676$	0	0
$677$	−31.1737	−1.19810	−0.599051	−	0.800711i	$-0.704455\pi$
$677$	−31.1737	−1.19810	−0.599051	+	0.800711i	$0.704455\pi$
$678$	0	0
$679$	5.69648	0.218611
$680$	0	0
$681$	0.343315	0.0131559
$682$	0	0
$683$	−48.5452	−1.85753	−0.928766	−	0.370667i	$-0.879129\pi$
$683$	−48.5452	−1.85753	−0.928766	+	0.370667i	$0.879129\pi$
$684$	0	0
$685$	−0.0850150	−0.00324826
$686$	0	0
$687$	−19.0921	−0.728408
$688$	0	0
$689$	0	0
$690$	0	0
$691$	17.9719	0.683684	0.341842	−	0.939757i	$-0.388949\pi$
$691$	17.9719	0.683684	0.341842	+	0.939757i	$0.388949\pi$
$692$	0	0
$693$	−124.000	−4.71037
$694$	0	0
$695$	−11.0101	−0.417637
$696$	0	0
$697$	12.1749	0.461157
$698$	0	0
$699$	−45.2294	−1.71073
$700$	0	0
$701$	13.0567	0.493147	0.246573	−	0.969124i	$-0.420695\pi$
$701$	13.0567	0.493147	0.246573	+	0.969124i	$0.420695\pi$
$702$	0	0
$703$	−0.0717113	−0.00270464
$704$	0	0
$705$	12.5305	0.471926
$706$	0	0
$707$	12.9247	0.486084
$708$	0	0
$709$	14.9427	0.561186	0.280593	−	0.959827i	$-0.409469\pi$
$709$	14.9427	0.561186	0.280593	+	0.959827i	$0.409469\pi$
$710$	0	0
$711$	−41.2295	−1.54623
$712$	0	0
$713$	−40.1987	−1.50545
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−44.2185	−1.65137
$718$	0	0
$719$	10.7997	0.402761	0.201381	−	0.979513i	$-0.435457\pi$
$719$	10.7997	0.402761	0.201381	+	0.979513i	$0.435457\pi$
$720$	0	0
$721$	−19.8020	−0.737466
$722$	0	0
$723$	5.08283	0.189032
$724$	0	0
$725$	5.63201	0.209168
$726$	0	0
$727$	−25.1935	−0.934376	−0.467188	−	0.884158i	$-0.654733\pi$
$727$	−25.1935	−0.934376	−0.467188	+	0.884158i	$0.654733\pi$
$728$	0	0
$729$	142.153	5.26491
$730$	0	0
$731$	−26.9113	−0.995350
$732$	0	0
$733$	32.1714	1.18828	0.594139	−	0.804363i	$-0.297493\pi$
$733$	32.1714	1.18828	0.594139	+	0.804363i	$0.297493\pi$
$734$	0	0
$735$	−5.22584	−0.192758
$736$	0	0
$737$	−2.32306	−0.0855708
$738$	0	0
$739$	−44.1036	−1.62238	−0.811188	−	0.584785i	$-0.801179\pi$
$739$	−44.1036	−1.62238	−0.811188	+	0.584785i	$0.801179\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.4544	1.63087	0.815437	−	0.578846i	$-0.196497\pi$
$743$	44.4544	1.63087	0.815437	+	0.578846i	$0.196497\pi$
$744$	0	0
$745$	4.41512	0.161758
$746$	0	0
$747$	−99.9497	−3.65697
$748$	0	0
$749$	−26.6267	−0.972918
$750$	0	0
$751$	37.6126	1.37250	0.686252	−	0.727364i	$-0.259255\pi$
$751$	37.6126	1.37250	0.686252	+	0.727364i	$0.259255\pi$
$752$	0	0
$753$	38.3675	1.39819
$754$	0	0
$755$	15.7555	0.573400
$756$	0	0
$757$	35.3294	1.28407	0.642034	−	0.766676i	$-0.278091\pi$
$757$	35.3294	1.28407	0.642034	+	0.766676i	$0.278091\pi$
$758$	0	0
$759$	−101.218	−3.67398
$760$	0	0
$761$	−27.3170	−0.990240	−0.495120	−	0.868825i	$-0.664876\pi$
$761$	−27.3170	−0.990240	−0.495120	+	0.868825i	$0.664876\pi$
$762$	0	0
$763$	−21.1492	−0.765654
$764$	0	0
$765$	−26.4539	−0.956442
$766$	0	0
$767$	0	0
$768$	0	0
$769$	23.1958	0.836460	0.418230	−	0.908341i	$-0.362651\pi$
$769$	23.1958	0.836460	0.418230	+	0.908341i	$0.362651\pi$
$770$	0	0
$771$	−51.1439	−1.84190
$772$	0	0
$773$	−6.52005	−0.234510	−0.117255	−	0.993102i	$-0.537409\pi$
$773$	−6.52005	−0.234510	−0.117255	+	0.993102i	$0.537409\pi$
$774$	0	0
$775$	−8.34078	−0.299610
$776$	0	0
$777$	−6.32634	−0.226956
$778$	0	0
$779$	0.357377	0.0128044
$780$	0	0
$781$	55.7455	1.99473
$782$	0	0
$783$	−107.475	−3.84083
$784$	0	0
$785$	−3.40120	−0.121394
$786$	0	0
$787$	40.7311	1.45191	0.725954	−	0.687744i	$-0.241399\pi$
$787$	40.7311	1.45191	0.725954	+	0.687744i	$0.241399\pi$
$788$	0	0
$789$	−38.9349	−1.38612
$790$	0	0
$791$	6.99401	0.248678
$792$	0	0
$793$	0	0
$794$	0	0
$795$	27.5378	0.976665
$796$	0	0
$797$	4.48208	0.158763	0.0793816	−	0.996844i	$-0.474705\pi$
$797$	4.48208	0.158763	0.0793816	+	0.996844i	$0.474705\pi$
$798$	0	0
$799$	11.3128	0.400217
$800$	0	0
$801$	100.749	3.55979
$802$	0	0
$803$	−33.7959	−1.19263
$804$	0	0
$805$	11.2676	0.397132
$806$	0	0
$807$	−86.4860	−3.04445
$808$	0	0
$809$	32.7884	1.15278	0.576390	−	0.817175i	$-0.304461\pi$
$809$	32.7884	1.15278	0.576390	+	0.817175i	$0.304461\pi$
$810$	0	0
$811$	29.5808	1.03872	0.519361	−	0.854555i	$-0.326170\pi$
$811$	29.5808	1.03872	0.519361	+	0.854555i	$0.326170\pi$
$812$	0	0
$813$	36.2174	1.27020
$814$	0	0
$815$	14.4332	0.505574
$816$	0	0
$817$	−0.789943	−0.0276366
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−47.1035	−1.64392	−0.821962	−	0.569543i	$-0.807120\pi$
$821$	−47.1035	−1.64392	−0.821962	+	0.569543i	$0.807120\pi$
$822$	0	0
$823$	−16.4099	−0.572012	−0.286006	−	0.958228i	$-0.592328\pi$
$823$	−16.4099	−0.572012	−0.286006	+	0.958228i	$0.592328\pi$
$824$	0	0
$825$	−21.0016	−0.731180
$826$	0	0
$827$	7.42042	0.258033	0.129017	−	0.991642i	$-0.458818\pi$
$827$	7.42042	0.258033	0.129017	+	0.991642i	$0.458818\pi$
$828$	0	0
$829$	−48.0555	−1.66904	−0.834518	−	0.550980i	$-0.814254\pi$
$829$	−48.0555	−1.66904	−0.834518	+	0.550980i	$0.814254\pi$
$830$	0	0
$831$	−88.5280	−3.07100
$832$	0	0
$833$	−4.71798	−0.163468
$834$	0	0
$835$	−18.9815	−0.656883
$836$	0	0
$837$	159.165	5.50156
$838$	0	0
$839$	−12.5460	−0.433136	−0.216568	−	0.976268i	$-0.569486\pi$
$839$	−12.5460	−0.433136	−0.216568	+	0.976268i	$0.569486\pi$
$840$	0	0
$841$	2.71952	0.0937766
$842$	0	0
$843$	−26.0889	−0.898550
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−63.1590	−2.17017
$848$	0	0
$849$	−66.4298	−2.27986
$850$	0	0
$851$	−3.82877	−0.131248
$852$	0	0
$853$	−8.26494	−0.282986	−0.141493	−	0.989939i	$-0.545190\pi$
$853$	−8.26494	−0.282986	−0.141493	+	0.989939i	$0.545190\pi$
$854$	0	0
$855$	−0.776517	−0.0265563
$856$	0	0
$857$	15.0212	0.513115	0.256558	−	0.966529i	$-0.417412\pi$
$857$	15.0212	0.513115	0.256558	+	0.966529i	$0.417412\pi$
$858$	0	0
$859$	22.3863	0.763810	0.381905	−	0.924202i	$-0.375268\pi$
$859$	22.3863	0.763810	0.381905	+	0.924202i	$0.375268\pi$
$860$	0	0
$861$	31.5277	1.07446
$862$	0	0
$863$	−4.17657	−0.142172	−0.0710860	−	0.997470i	$-0.522646\pi$
$863$	−4.17657	−0.142172	−0.0710860	+	0.997470i	$0.522646\pi$
$864$	0	0
$865$	−11.0970	−0.377310
$866$	0	0
$867$	25.6937	0.872604
$868$	0	0
$869$	−29.5509	−1.00244
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−20.9603	−0.709397
$874$	0	0
$875$	2.33790	0.0790356
$876$	0	0
$877$	39.2315	1.32475	0.662377	−	0.749171i	$-0.269548\pi$
$877$	39.2315	1.32475	0.662377	+	0.749171i	$0.269548\pi$
$878$	0	0
$879$	−43.5585	−1.46919
$880$	0	0
$881$	8.53832	0.287663	0.143832	−	0.989602i	$-0.454058\pi$
$881$	8.53832	0.287663	0.143832	+	0.989602i	$0.454058\pi$
$882$	0	0
$883$	44.7515	1.50601	0.753004	−	0.658016i	$-0.228604\pi$
$883$	44.7515	1.50601	0.753004	+	0.658016i	$0.228604\pi$
$884$	0	0
$885$	−1.29292	−0.0434611
$886$	0	0
$887$	37.1035	1.24581	0.622906	−	0.782296i	$-0.285952\pi$
$887$	37.1035	1.24581	0.622906	+	0.782296i	$0.285952\pi$
$888$	0	0
$889$	−24.9632	−0.837237
$890$	0	0
$891$	241.652	8.09563
$892$	0	0
$893$	0.332070	0.0111123
$894$	0	0
$895$	−5.27165	−0.176212
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−46.9753	−1.56671
$900$	0	0
$901$	24.8616	0.828260
$902$	0	0
$903$	−69.6885	−2.31909
$904$	0	0
$905$	−16.1744	−0.537655
$906$	0	0
$907$	−37.6342	−1.24962	−0.624812	−	0.780776i	$-0.714824\pi$
$907$	−37.6342	−1.24962	−0.624812	+	0.780776i	$0.714824\pi$
$908$	0	0
$909$	−47.5566	−1.57735
$910$	0	0
$911$	39.8183	1.31924	0.659619	−	0.751600i	$-0.270718\pi$
$911$	39.8183	1.31924	0.659619	+	0.751600i	$0.270718\pi$
$912$	0	0
$913$	−71.6380	−2.37087
$914$	0	0
$915$	−16.9074	−0.558940
$916$	0	0
$917$	48.1777	1.59097
$918$	0	0
$919$	−34.1947	−1.12798	−0.563989	−	0.825782i	$-0.690734\pi$
$919$	−34.1947	−1.12798	−0.563989	+	0.825782i	$0.690734\pi$
$920$	0	0
$921$	111.002	3.65766
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−0.794425	−0.0261205
$926$	0	0
$927$	72.8617	2.39309
$928$	0	0
$929$	30.5866	1.00351	0.501757	−	0.865009i	$-0.332687\pi$
$929$	30.5866	1.00351	0.501757	+	0.865009i	$0.332687\pi$
$930$	0	0
$931$	−0.138490	−0.00453881
$932$	0	0
$933$	−15.3948	−0.504003
$934$	0	0
$935$	−18.9606	−0.620077
$936$	0	0
$937$	−39.0385	−1.27533	−0.637666	−	0.770313i	$-0.720100\pi$
$937$	−39.0385	−1.27533	−0.637666	+	0.770313i	$0.720100\pi$
$938$	0	0
$939$	8.20816	0.267863
$940$	0	0
$941$	6.92575	0.225773	0.112886	−	0.993608i	$-0.463990\pi$
$941$	6.92575	0.225773	0.112886	+	0.993608i	$0.463990\pi$
$942$	0	0
$943$	19.0809	0.621359
$944$	0	0
$945$	−44.6138	−1.45129
$946$	0	0
$947$	−15.2145	−0.494405	−0.247202	−	0.968964i	$-0.579511\pi$
$947$	−15.2145	−0.494405	−0.247202	+	0.968964i	$0.579511\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	39.7680	1.28957
$952$	0	0
$953$	−5.06168	−0.163964	−0.0819819	−	0.996634i	$-0.526125\pi$
$953$	−5.06168	−0.163964	−0.0819819	+	0.996634i	$0.526125\pi$
$954$	0	0
$955$	6.75365	0.218543
$956$	0	0
$957$	−118.281	−3.82348
$958$	0	0
$959$	−0.198757	−0.00641820
$960$	0	0
$961$	38.5686	1.24415
$962$	0	0
$963$	97.9732	3.15714
$964$	0	0
$965$	0.481519	0.0155006
$966$	0	0
$967$	−48.3231	−1.55397	−0.776983	−	0.629522i	$-0.783251\pi$
$967$	−48.3231	−1.55397	−0.776983	+	0.629522i	$0.783251\pi$
$968$	0	0
$969$	−0.945539	−0.0303751
$970$	0	0
$971$	45.8463	1.47128	0.735638	−	0.677374i	$-0.236882\pi$
$971$	45.8463	1.47128	0.735638	+	0.677374i	$0.236882\pi$
$972$	0	0
$973$	−25.7405	−0.825204
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−43.2717	−1.38438	−0.692192	−	0.721714i	$-0.743355\pi$
$977$	−43.2717	−1.38438	−0.692192	+	0.721714i	$0.743355\pi$
$978$	0	0
$979$	72.2108	2.30787
$980$	0	0
$981$	77.8188	2.48456
$982$	0	0
$983$	−0.695243	−0.0221748	−0.0110874	−	0.999939i	$-0.503529\pi$
$983$	−0.695243	−0.0221748	−0.0110874	+	0.999939i	$0.503529\pi$
$984$	0	0
$985$	21.8881	0.697414
$986$	0	0
$987$	29.2951	0.932474
$988$	0	0
$989$	−42.1762	−1.34113
$990$	0	0
$991$	21.1973	0.673355	0.336677	−	0.941620i	$-0.390697\pi$
$991$	21.1973	0.673355	0.336677	+	0.941620i	$0.390697\pi$
$992$	0	0
$993$	76.2403	2.41941
$994$	0	0
$995$	−23.5567	−0.746799
$996$	0	0
$997$	−36.7860	−1.16503	−0.582513	−	0.812822i	$-0.697930\pi$
$997$	−36.7860	−1.16503	−0.582513	+	0.812822i	$0.697930\pi$
$998$	0	0
$999$	15.1599	0.479637

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.r.1.1	✓	9
13.5	odd	4		3380.2.f.j.3041.2		18
13.8	odd	4		3380.2.f.j.3041.1		18
13.12	even	2		3380.2.a.s.1.1	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.2.a.r.1.1	✓	9	1.1	even	1	trivial
3380.2.a.s.1.1	yes	9	13.12	even	2
3380.2.f.j.3041.1		18	13.8	odd	4
3380.2.f.j.3041.2		18	13.5	odd	4

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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.a (trivial)

\(f(q)\)	\(=\)	\(q-3.40622 q^{3} -1.00000 q^{5} -2.33790 q^{7} +8.60234 q^{9} +O(q^{10})\)	\(q-3.40622 q^{3} -1.00000 q^{5} -2.33790 q^{7} +8.60234 q^{9} +6.16565 q^{11} +3.40622 q^{15} +3.07520 q^{17} +0.0902681 q^{19} +7.96342 q^{21} +4.81954 q^{23} +1.00000 q^{25} -19.0828 q^{27} +5.63201 q^{29} -8.34078 q^{31} -21.0016 q^{33} +2.33790 q^{35} -0.794425 q^{37} +3.95907 q^{41} -8.75108 q^{43} -8.60234 q^{45} +3.67871 q^{47} -1.53420 q^{49} -10.4748 q^{51} +8.08456 q^{53} -6.16565 q^{55} -0.307473 q^{57} -0.379577 q^{59} -4.96367 q^{61} -20.1114 q^{63} -0.376774 q^{67} -16.4164 q^{69} +9.04130 q^{71} -5.48133 q^{73} -3.40622 q^{75} -14.4147 q^{77} -4.79283 q^{79} +39.1932 q^{81} -11.6189 q^{83} -3.07520 q^{85} -19.1839 q^{87} +11.7118 q^{89} +28.4105 q^{93} -0.0902681 q^{95} -2.43658 q^{97} +53.0390 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9	\( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + 7 q^{11} + q^{15} + 13 q^{17} + 4 q^{19} + 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} - 13 q^{31} - 34 q^{33} + q^{35} - q^{37} + 6 q^{41} + q^{43} - 12 q^{45} + 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} - 38 q^{57} - 15 q^{59} + 21 q^{61} + 17 q^{63} + 7 q^{67} + 15 q^{69} + 7 q^{71} + 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} - 45 q^{83} - 13 q^{85} + 28 q^{87} + 41 q^{89} + 11 q^{93} - 4 q^{95} - 8 q^{97} + 81 q^{99}+O(q^{100}) \) 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9 + 7 * q^11 + q^15 + 13 * q^17 + 4 * q^19 + 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 - 13 * q^31 - 34 * q^33 + q^35 - q^37 + 6 * q^41 + q^43 - 12 * q^45 + 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 - 38 * q^57 - 15 * q^59 + 21 * q^61 + 17 * q^63 + 7 * q^67 + 15 * q^69 + 7 * q^71 + 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 - 45 * q^83 - 13 * q^85 + 28 * q^87 + 41 * q^89 + 11 * q^93 - 4 * q^95 - 8 * q^97 + 81 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more