LMFDB - Embedded newform 6384.2.a.bu.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6384,2,Mod(1,6384)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6384.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6384, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6384.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,0,0,3,0,3,0,-4,0,2,0,0,0,12,0,3,0,-3,0,8,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$50.9764966504$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 399)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.210756$ of defining polynomial
Character	$\chi$	$=$	6384.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.53407 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.42151 q^{11} -0.421512 q^{13} +2.53407 q^{15} +6.53407 q^{17} +1.00000 q^{19} -1.00000 q^{21} +1.57849 q^{23} +1.42151 q^{25} -1.00000 q^{27} +6.02372 q^{29} -3.48965 q^{31} +2.42151 q^{33} -2.53407 q^{35} -2.84302 q^{37} +0.421512 q^{39} +8.42151 q^{41} -11.4897 q^{43} -2.53407 q^{45} -8.02372 q^{47} +1.00000 q^{49} -6.53407 q^{51} -10.8667 q^{53} +6.13628 q^{55} -1.00000 q^{57} -14.1363 q^{59} +6.84302 q^{61} +1.00000 q^{63} +1.06814 q^{65} +9.91116 q^{67} -1.57849 q^{69} +13.3771 q^{71} +7.06814 q^{73} -1.42151 q^{75} -2.42151 q^{77} -14.1363 q^{79} +1.00000 q^{81} -2.11256 q^{83} -16.5578 q^{85} -6.02372 q^{87} -9.71477 q^{89} -0.421512 q^{91} +3.48965 q^{93} -2.53407 q^{95} -4.97930 q^{97} -2.42151 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} - 4 q^{11} + 2 q^{13} + 12 q^{17} + 3 q^{19} - 3 q^{21} + 8 q^{23} + q^{25} - 3 q^{27} - 8 q^{29} + 8 q^{31} + 4 q^{33} - 2 q^{37} - 2 q^{39} + 22 q^{41} - 16 q^{43} + 2 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 - 4 * q^11 + 2 * q^13 + 12 * q^17 + 3 * q^19 - 3 * q^21 + 8 * q^23 + q^25 - 3 * q^27 - 8 * q^29 + 8 * q^31 + 4 * q^33 - 2 * q^37 - 2 * q^39 + 22 * q^41 - 16 * q^43 + 2 * q^47 + 3 * q^49 - 12 * q^51 - 12 * q^55 - 3 * q^57 - 12 * q^59 + 14 * q^61 + 3 * q^63 - 12 * q^65 + 8 * q^67 - 8 * q^69 + 26 * q^71 + 6 * q^73 - q^75 - 4 * q^77 - 12 * q^79 + 3 * q^81 - 2 * q^83 - 16 * q^85 + 8 * q^87 - 2 * q^89 + 2 * q^91 - 8 * q^93 + 22 * q^97 - 4 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.53407	−1.13327	−0.566635	−	0.823969i	$-0.691755\pi$
$5$	−2.53407	−1.13327	−0.566635	+	0.823969i	$0.691755\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.42151	−0.730113	−0.365057	−	0.930985i	$-0.618950\pi$
$11$	−2.42151	−0.730113	−0.365057	+	0.930985i	$0.618950\pi$
$12$	0	0
$13$	−0.421512	−0.116906	−0.0584532	−	0.998290i	$-0.518617\pi$
$13$	−0.421512	−0.116906	−0.0584532	+	0.998290i	$0.518617\pi$
$14$	0	0
$15$	2.53407	0.654294
$16$	0	0
$17$	6.53407	1.58474	0.792372	−	0.610038i	$-0.208846\pi$
$17$	6.53407	1.58474	0.792372	+	0.610038i	$0.208846\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.57849	0.329138	0.164569	−	0.986366i	$-0.447377\pi$
$23$	1.57849	0.329138	0.164569	+	0.986366i	$0.447377\pi$
$24$	0	0
$25$	1.42151	0.284302
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.02372	1.11858	0.559289	−	0.828973i	$-0.311074\pi$
$29$	6.02372	1.11858	0.559289	+	0.828973i	$0.311074\pi$
$30$	0	0
$31$	−3.48965	−0.626760	−0.313380	−	0.949628i	$-0.601461\pi$
$31$	−3.48965	−0.626760	−0.313380	+	0.949628i	$0.601461\pi$
$32$	0	0
$33$	2.42151	0.421531
$34$	0	0
$35$	−2.53407	−0.428336
$36$	0	0
$37$	−2.84302	−0.467390	−0.233695	−	0.972310i	$-0.575082\pi$
$37$	−2.84302	−0.467390	−0.233695	+	0.972310i	$0.575082\pi$
$38$	0	0
$39$	0.421512	0.0674959
$40$	0	0
$41$	8.42151	1.31522	0.657610	−	0.753359i	$-0.271568\pi$
$41$	8.42151	1.31522	0.657610	+	0.753359i	$0.271568\pi$
$42$	0	0
$43$	−11.4897	−1.75216	−0.876078	−	0.482170i	$-0.839849\pi$
$43$	−11.4897	−1.75216	−0.876078	+	0.482170i	$0.839849\pi$
$44$	0	0
$45$	−2.53407	−0.377757
$46$	0	0
$47$	−8.02372	−1.17038	−0.585190	−	0.810896i	$-0.698980\pi$
$47$	−8.02372	−1.17038	−0.585190	+	0.810896i	$0.698980\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.53407	−0.914953
$52$	0	0
$53$	−10.8667	−1.49266	−0.746331	−	0.665575i	$-0.768186\pi$
$53$	−10.8667	−1.49266	−0.746331	+	0.665575i	$0.768186\pi$
$54$	0	0
$55$	6.13628	0.827416
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−14.1363	−1.84039	−0.920194	−	0.391464i	$-0.871969\pi$
$59$	−14.1363	−1.84039	−0.920194	+	0.391464i	$0.871969\pi$
$60$	0	0
$61$	6.84302	0.876159	0.438080	−	0.898936i	$-0.355659\pi$
$61$	6.84302	0.876159	0.438080	+	0.898936i	$0.355659\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	1.06814	0.132487
$66$	0	0
$67$	9.91116	1.21084	0.605421	−	0.795906i	$-0.293005\pi$
$67$	9.91116	1.21084	0.605421	+	0.795906i	$0.293005\pi$
$68$	0	0
$69$	−1.57849	−0.190028
$70$	0	0
$71$	13.3771	1.58757	0.793784	−	0.608199i	$-0.208108\pi$
$71$	13.3771	1.58757	0.793784	+	0.608199i	$0.208108\pi$
$72$	0	0
$73$	7.06814	0.827263	0.413632	−	0.910444i	$-0.364260\pi$
$73$	7.06814	0.827263	0.413632	+	0.910444i	$0.364260\pi$
$74$	0	0
$75$	−1.42151	−0.164142
$76$	0	0
$77$	−2.42151	−0.275957
$78$	0	0
$79$	−14.1363	−1.59046	−0.795228	−	0.606311i	$-0.792649\pi$
$79$	−14.1363	−1.59046	−0.795228	+	0.606311i	$0.792649\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.11256	−0.231883	−0.115942	−	0.993256i	$-0.536989\pi$
$83$	−2.11256	−0.231883	−0.115942	+	0.993256i	$0.536989\pi$
$84$	0	0
$85$	−16.5578	−1.79594
$86$	0	0
$87$	−6.02372	−0.645811
$88$	0	0
$89$	−9.71477	−1.02976	−0.514882	−	0.857261i	$-0.672164\pi$
$89$	−9.71477	−1.02976	−0.514882	+	0.857261i	$0.672164\pi$
$90$	0	0
$91$	−0.421512	−0.0441864
$92$	0	0
$93$	3.48965	0.361860
$94$	0	0
$95$	−2.53407	−0.259990
$96$	0	0
$97$	−4.97930	−0.505572	−0.252786	−	0.967522i	$-0.581347\pi$
$97$	−4.97930	−0.505572	−0.252786	+	0.967522i	$0.581347\pi$
$98$	0	0
$99$	−2.42151	−0.243371
$100$	0	0
$101$	−1.69105	−0.168265	−0.0841327	−	0.996455i	$-0.526812\pi$
$101$	−1.69105	−0.168265	−0.0841327	+	0.996455i	$0.526812\pi$
$102$	0	0
$103$	−3.15698	−0.311066	−0.155533	−	0.987831i	$-0.549710\pi$
$103$	−3.15698	−0.311066	−0.155533	+	0.987831i	$0.549710\pi$
$104$	0	0
$105$	2.53407	0.247300
$106$	0	0
$107$	16.5341	1.59841	0.799204	−	0.601059i	$-0.205254\pi$
$107$	16.5341	1.59841	0.799204	+	0.601059i	$0.205254\pi$
$108$	0	0
$109$	1.77488	0.170003	0.0850015	−	0.996381i	$-0.472910\pi$
$109$	1.77488	0.170003	0.0850015	+	0.996381i	$0.472910\pi$
$110$	0	0
$111$	2.84302	0.269848
$112$	0	0
$113$	10.0237	0.942952	0.471476	−	0.881879i	$-0.343721\pi$
$113$	10.0237	0.942952	0.471476	+	0.881879i	$0.343721\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	−0.421512	−0.0389688
$118$	0	0
$119$	6.53407	0.598977
$120$	0	0
$121$	−5.13628	−0.466935
$122$	0	0
$123$	−8.42151	−0.759342
$124$	0	0
$125$	9.06814	0.811079
$126$	0	0
$127$	−2.08884	−0.185354	−0.0926771	−	0.995696i	$-0.529542\pi$
$127$	−2.08884	−0.185354	−0.0926771	+	0.995696i	$0.529542\pi$
$128$	0	0
$129$	11.4897	1.01161
$130$	0	0
$131$	13.9349	1.21750	0.608748	−	0.793363i	$-0.291672\pi$
$131$	13.9349	1.21750	0.608748	+	0.793363i	$0.291672\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	2.53407	0.218098
$136$	0	0
$137$	16.9793	1.45064	0.725320	−	0.688412i	$-0.241692\pi$
$137$	16.9793	1.45064	0.725320	+	0.688412i	$0.241692\pi$
$138$	0	0
$139$	1.06814	0.0905985	0.0452992	−	0.998973i	$-0.485576\pi$
$139$	1.06814	0.0905985	0.0452992	+	0.998973i	$0.485576\pi$
$140$	0	0
$141$	8.02372	0.675719
$142$	0	0
$143$	1.02070	0.0853549
$144$	0	0
$145$	−15.2645	−1.26765
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	14.0474	1.15081	0.575406	−	0.817868i	$-0.304844\pi$
$149$	14.0474	1.15081	0.575406	+	0.817868i	$0.304844\pi$
$150$	0	0
$151$	14.7542	1.20068	0.600339	−	0.799745i	$-0.295032\pi$
$151$	14.7542	1.20068	0.600339	+	0.799745i	$0.295032\pi$
$152$	0	0
$153$	6.53407	0.528248
$154$	0	0
$155$	8.84302	0.710289
$156$	0	0
$157$	−16.1363	−1.28782	−0.643908	−	0.765103i	$-0.722688\pi$
$157$	−16.1363	−1.28782	−0.643908	+	0.765103i	$0.722688\pi$
$158$	0	0
$159$	10.8667	0.861789
$160$	0	0
$161$	1.57849	0.124402
$162$	0	0
$163$	6.64663	0.520604	0.260302	−	0.965527i	$-0.416178\pi$
$163$	6.64663	0.520604	0.260302	+	0.965527i	$0.416178\pi$
$164$	0	0
$165$	−6.13628	−0.477709
$166$	0	0
$167$	14.3614	1.11132	0.555659	−	0.831410i	$-0.312466\pi$
$167$	14.3614	1.11132	0.555659	+	0.831410i	$0.312466\pi$
$168$	0	0
$169$	−12.8223	−0.986333
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−5.71477	−0.434486	−0.217243	−	0.976118i	$-0.569706\pi$
$173$	−5.71477	−0.434486	−0.217243	+	0.976118i	$0.569706\pi$
$174$	0	0
$175$	1.42151	0.107456
$176$	0	0
$177$	14.1363	1.06255
$178$	0	0
$179$	9.60221	0.717703	0.358851	−	0.933395i	$-0.383168\pi$
$179$	9.60221	0.717703	0.358851	+	0.933395i	$0.383168\pi$
$180$	0	0
$181$	−0.979304	−0.0727911	−0.0363956	−	0.999337i	$-0.511588\pi$
$181$	−0.979304	−0.0727911	−0.0363956	+	0.999337i	$0.511588\pi$
$182$	0	0
$183$	−6.84302	−0.505851
$184$	0	0
$185$	7.20442	0.529680
$186$	0	0
$187$	−15.8223	−1.15704
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	11.7148	0.847651	0.423825	−	0.905744i	$-0.360687\pi$
$191$	11.7148	0.847651	0.423825	+	0.905744i	$0.360687\pi$
$192$	0	0
$193$	17.8223	1.28288	0.641440	−	0.767174i	$-0.278337\pi$
$193$	17.8223	1.28288	0.641440	+	0.767174i	$0.278337\pi$
$194$	0	0
$195$	−1.06814	−0.0764911
$196$	0	0
$197$	−19.1156	−1.36193	−0.680965	−	0.732316i	$-0.738439\pi$
$197$	−19.1156	−1.36193	−0.680965	+	0.732316i	$0.738439\pi$
$198$	0	0
$199$	−16.0474	−1.13757	−0.568787	−	0.822485i	$-0.692587\pi$
$199$	−16.0474	−1.13757	−0.568787	+	0.822485i	$0.692587\pi$
$200$	0	0
$201$	−9.91116	−0.699080
$202$	0	0
$203$	6.02372	0.422782
$204$	0	0
$205$	−21.3407	−1.49050
$206$	0	0
$207$	1.57849	0.109713
$208$	0	0
$209$	−2.42151	−0.167499
$210$	0	0
$211$	9.29326	0.639774	0.319887	−	0.947456i	$-0.396355\pi$
$211$	9.29326	0.639774	0.319887	+	0.947456i	$0.396355\pi$
$212$	0	0
$213$	−13.3771	−0.916583
$214$	0	0
$215$	29.1156	1.98567
$216$	0	0
$217$	−3.48965	−0.236893
$218$	0	0
$219$	−7.06814	−0.477621
$220$	0	0
$221$	−2.75419	−0.185267
$222$	0	0
$223$	8.33268	0.557997	0.278999	−	0.960292i	$-0.409998\pi$
$223$	8.33268	0.557997	0.278999	+	0.960292i	$0.409998\pi$
$224$	0	0
$225$	1.42151	0.0947675
$226$	0	0
$227$	−16.8905	−1.12106	−0.560530	−	0.828134i	$-0.689403\pi$
$227$	−16.8905	−1.12106	−0.560530	+	0.828134i	$0.689403\pi$
$228$	0	0
$229$	6.04744	0.399626	0.199813	−	0.979834i	$-0.435966\pi$
$229$	6.04744	0.399626	0.199813	+	0.979834i	$0.435966\pi$
$230$	0	0
$231$	2.42151	0.159324
$232$	0	0
$233$	21.2044	1.38915	0.694574	−	0.719421i	$-0.255593\pi$
$233$	21.2044	1.38915	0.694574	+	0.719421i	$0.255593\pi$
$234$	0	0
$235$	20.3327	1.32636
$236$	0	0
$237$	14.1363	0.918250
$238$	0	0
$239$	5.80361	0.375404	0.187702	−	0.982226i	$-0.439896\pi$
$239$	5.80361	0.375404	0.187702	+	0.982226i	$0.439896\pi$
$240$	0	0
$241$	−12.9793	−0.836070	−0.418035	−	0.908431i	$-0.637281\pi$
$241$	−12.9793	−0.836070	−0.418035	+	0.908431i	$0.637281\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.53407	−0.161896
$246$	0	0
$247$	−0.421512	−0.0268202
$248$	0	0
$249$	2.11256	0.133878
$250$	0	0
$251$	−10.3377	−0.652508	−0.326254	−	0.945282i	$-0.605787\pi$
$251$	−10.3377	−0.652508	−0.326254	+	0.945282i	$0.605787\pi$
$252$	0	0
$253$	−3.82233	−0.240308
$254$	0	0
$255$	16.5578	1.03689
$256$	0	0
$257$	−15.1757	−0.946634	−0.473317	−	0.880892i	$-0.656944\pi$
$257$	−15.1757	−0.946634	−0.473317	+	0.880892i	$0.656944\pi$
$258$	0	0
$259$	−2.84302	−0.176657
$260$	0	0
$261$	6.02372	0.372859
$262$	0	0
$263$	8.51035	0.524771	0.262385	−	0.964963i	$-0.415491\pi$
$263$	8.51035	0.524771	0.262385	+	0.964963i	$0.415491\pi$
$264$	0	0
$265$	27.5371	1.69159
$266$	0	0
$267$	9.71477	0.594534
$268$	0	0
$269$	15.1757	0.925279	0.462639	−	0.886547i	$-0.346902\pi$
$269$	15.1757	0.925279	0.462639	+	0.886547i	$0.346902\pi$
$270$	0	0
$271$	−0.843024	−0.0512100	−0.0256050	−	0.999672i	$-0.508151\pi$
$271$	−0.843024	−0.0512100	−0.0256050	+	0.999672i	$0.508151\pi$
$272$	0	0
$273$	0.421512	0.0255111
$274$	0	0
$275$	−3.44221	−0.207573
$276$	0	0
$277$	10.5578	0.634356	0.317178	−	0.948366i	$-0.397265\pi$
$277$	10.5578	0.634356	0.317178	+	0.948366i	$0.397265\pi$
$278$	0	0
$279$	−3.48965	−0.208920
$280$	0	0
$281$	15.0444	0.897475	0.448737	−	0.893664i	$-0.351874\pi$
$281$	15.0444	0.897475	0.448737	+	0.893664i	$0.351874\pi$
$282$	0	0
$283$	−1.24581	−0.0740559	−0.0370279	−	0.999314i	$-0.511789\pi$
$283$	−1.24581	−0.0740559	−0.0370279	+	0.999314i	$0.511789\pi$
$284$	0	0
$285$	2.53407	0.150105
$286$	0	0
$287$	8.42151	0.497106
$288$	0	0
$289$	25.6941	1.51142
$290$	0	0
$291$	4.97930	0.291892
$292$	0	0
$293$	20.4215	1.19304	0.596519	−	0.802599i	$-0.296550\pi$
$293$	20.4215	1.19304	0.596519	+	0.802599i	$0.296550\pi$
$294$	0	0
$295$	35.8223	2.08566
$296$	0	0
$297$	2.42151	0.140510
$298$	0	0
$299$	−0.665351	−0.0384783
$300$	0	0
$301$	−11.4897	−0.662253
$302$	0	0
$303$	1.69105	0.0971481
$304$	0	0
$305$	−17.3407	−0.992926
$306$	0	0
$307$	2.64663	0.151051	0.0755255	−	0.997144i	$-0.475937\pi$
$307$	2.64663	0.151051	0.0755255	+	0.997144i	$0.475937\pi$
$308$	0	0
$309$	3.15698	0.179594
$310$	0	0
$311$	5.09186	0.288733	0.144367	−	0.989524i	$-0.453886\pi$
$311$	5.09186	0.288733	0.144367	+	0.989524i	$0.453886\pi$
$312$	0	0
$313$	19.9586	1.12813	0.564064	−	0.825731i	$-0.309237\pi$
$313$	19.9586	1.12813	0.564064	+	0.825731i	$0.309237\pi$
$314$	0	0
$315$	−2.53407	−0.142779
$316$	0	0
$317$	−6.81930	−0.383010	−0.191505	−	0.981492i	$-0.561337\pi$
$317$	−6.81930	−0.383010	−0.191505	+	0.981492i	$0.561337\pi$
$318$	0	0
$319$	−14.5865	−0.816688
$320$	0	0
$321$	−16.5341	−0.922842
$322$	0	0
$323$	6.53407	0.363565
$324$	0	0
$325$	−0.599184	−0.0332367
$326$	0	0
$327$	−1.77488	−0.0981513
$328$	0	0
$329$	−8.02372	−0.442362
$330$	0	0
$331$	5.91116	0.324907	0.162453	−	0.986716i	$-0.448059\pi$
$331$	5.91116	0.324907	0.162453	+	0.986716i	$0.448059\pi$
$332$	0	0
$333$	−2.84302	−0.155797
$334$	0	0
$335$	−25.1156	−1.37221
$336$	0	0
$337$	4.08884	0.222733	0.111367	−	0.993779i	$-0.464477\pi$
$337$	4.08884	0.222733	0.111367	+	0.993779i	$0.464477\pi$
$338$	0	0
$339$	−10.0237	−0.544414
$340$	0	0
$341$	8.45023	0.457606
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	17.3534	0.931578	0.465789	−	0.884896i	$-0.345771\pi$
$347$	17.3534	0.931578	0.465789	+	0.884896i	$0.345771\pi$
$348$	0	0
$349$	−4.70674	−0.251946	−0.125973	−	0.992034i	$-0.540205\pi$
$349$	−4.70674	−0.251946	−0.125973	+	0.992034i	$0.540205\pi$
$350$	0	0
$351$	0.421512	0.0224986
$352$	0	0
$353$	20.6704	1.10017	0.550086	−	0.835108i	$-0.314595\pi$
$353$	20.6704	1.10017	0.550086	+	0.835108i	$0.314595\pi$
$354$	0	0
$355$	−33.8985	−1.79915
$356$	0	0
$357$	−6.53407	−0.345820
$358$	0	0
$359$	−8.10756	−0.427901	−0.213950	−	0.976845i	$-0.568633\pi$
$359$	−8.10756	−0.427901	−0.213950	+	0.976845i	$0.568633\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.13628	0.269585
$364$	0	0
$365$	−17.9112	−0.937513
$366$	0	0
$367$	−28.8905	−1.50807	−0.754035	−	0.656834i	$-0.771895\pi$
$367$	−28.8905	−1.50807	−0.754035	+	0.656834i	$0.771895\pi$
$368$	0	0
$369$	8.42151	0.438406
$370$	0	0
$371$	−10.8667	−0.564173
$372$	0	0
$373$	11.9586	0.619193	0.309597	−	0.950868i	$-0.399806\pi$
$373$	11.9586	0.619193	0.309597	+	0.950868i	$0.399806\pi$
$374$	0	0
$375$	−9.06814	−0.468277
$376$	0	0
$377$	−2.53907	−0.130769
$378$	0	0
$379$	15.3821	0.790125	0.395063	−	0.918654i	$-0.370723\pi$
$379$	15.3821	0.790125	0.395063	+	0.918654i	$0.370723\pi$
$380$	0	0
$381$	2.08884	0.107014
$382$	0	0
$383$	0.617907	0.0315736	0.0157868	−	0.999875i	$-0.494975\pi$
$383$	0.617907	0.0315736	0.0157868	+	0.999875i	$0.494975\pi$
$384$	0	0
$385$	6.13628	0.312734
$386$	0	0
$387$	−11.4897	−0.584052
$388$	0	0
$389$	−11.0681	−0.561177	−0.280588	−	0.959828i	$-0.590530\pi$
$389$	−11.0681	−0.561177	−0.280588	+	0.959828i	$0.590530\pi$
$390$	0	0
$391$	10.3140	0.521599
$392$	0	0
$393$	−13.9349	−0.702922
$394$	0	0
$395$	35.8223	1.80242
$396$	0	0
$397$	−10.4502	−0.524482	−0.262241	−	0.965002i	$-0.584462\pi$
$397$	−10.4502	−0.524482	−0.262241	+	0.965002i	$0.584462\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	3.26953	0.163273	0.0816364	−	0.996662i	$-0.473985\pi$
$401$	3.26953	0.163273	0.0816364	+	0.996662i	$0.473985\pi$
$402$	0	0
$403$	1.47093	0.0732722
$404$	0	0
$405$	−2.53407	−0.125919
$406$	0	0
$407$	6.88441	0.341248
$408$	0	0
$409$	7.40082	0.365947	0.182973	−	0.983118i	$-0.441428\pi$
$409$	7.40082	0.365947	0.182973	+	0.983118i	$0.441428\pi$
$410$	0	0
$411$	−16.9793	−0.837527
$412$	0	0
$413$	−14.1363	−0.695601
$414$	0	0
$415$	5.35337	0.262787
$416$	0	0
$417$	−1.06814	−0.0523071
$418$	0	0
$419$	31.8461	1.55578	0.777891	−	0.628400i	$-0.216290\pi$
$419$	31.8461	1.55578	0.777891	+	0.628400i	$0.216290\pi$
$420$	0	0
$421$	40.0061	1.94978	0.974888	−	0.222696i	$-0.0714858\pi$
$421$	40.0061	1.94978	0.974888	+	0.222696i	$0.0714858\pi$
$422$	0	0
$423$	−8.02372	−0.390127
$424$	0	0
$425$	9.28826	0.450547
$426$	0	0
$427$	6.84302	0.331157
$428$	0	0
$429$	−1.02070	−0.0492797
$430$	0	0
$431$	−3.91616	−0.188635	−0.0943175	−	0.995542i	$-0.530067\pi$
$431$	−3.91616	−0.188635	−0.0943175	+	0.995542i	$0.530067\pi$
$432$	0	0
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	0	0
$435$	15.2645	0.731878
$436$	0	0
$437$	1.57849	0.0755093
$438$	0	0
$439$	−17.4483	−0.832760	−0.416380	−	0.909191i	$-0.636701\pi$
$439$	−17.4483	−0.832760	−0.416380	+	0.909191i	$0.636701\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	3.03942	0.144407	0.0722036	−	0.997390i	$-0.476997\pi$
$443$	3.03942	0.144407	0.0722036	+	0.997390i	$0.476997\pi$
$444$	0	0
$445$	24.6179	1.16700
$446$	0	0
$447$	−14.0474	−0.664421
$448$	0	0
$449$	35.9349	1.69587	0.847936	−	0.530099i	$-0.177845\pi$
$449$	35.9349	1.69587	0.847936	+	0.530099i	$0.177845\pi$
$450$	0	0
$451$	−20.3928	−0.960259
$452$	0	0
$453$	−14.7542	−0.693212
$454$	0	0
$455$	1.06814	0.0500752
$456$	0	0
$457$	−24.4215	−1.14239	−0.571195	−	0.820814i	$-0.693520\pi$
$457$	−24.4215	−1.14239	−0.571195	+	0.820814i	$0.693520\pi$
$458$	0	0
$459$	−6.53407	−0.304984
$460$	0	0
$461$	10.1313	0.471861	0.235930	−	0.971770i	$-0.424186\pi$
$461$	10.1313	0.471861	0.235930	+	0.971770i	$0.424186\pi$
$462$	0	0
$463$	5.86372	0.272510	0.136255	−	0.990674i	$-0.456493\pi$
$463$	5.86372	0.272510	0.136255	+	0.990674i	$0.456493\pi$
$464$	0	0
$465$	−8.84302	−0.410085
$466$	0	0
$467$	14.9556	0.692062	0.346031	−	0.938223i	$-0.387529\pi$
$467$	14.9556	0.692062	0.346031	+	0.938223i	$0.387529\pi$
$468$	0	0
$469$	9.91116	0.457655
$470$	0	0
$471$	16.1363	0.743521
$472$	0	0
$473$	27.8223	1.27927
$474$	0	0
$475$	1.42151	0.0652234
$476$	0	0
$477$	−10.8667	−0.497554
$478$	0	0
$479$	33.7098	1.54024	0.770119	−	0.637900i	$-0.220197\pi$
$479$	33.7098	1.54024	0.770119	+	0.637900i	$0.220197\pi$
$480$	0	0
$481$	1.19837	0.0546409
$482$	0	0
$483$	−1.57849	−0.0718237
$484$	0	0
$485$	12.6179	0.572950
$486$	0	0
$487$	−21.5084	−0.974637	−0.487319	−	0.873224i	$-0.662025\pi$
$487$	−21.5084	−0.974637	−0.487319	+	0.873224i	$0.662025\pi$
$488$	0	0
$489$	−6.64663	−0.300571
$490$	0	0
$491$	17.3534	0.783147	0.391573	−	0.920147i	$-0.371931\pi$
$491$	17.3534	0.783147	0.391573	+	0.920147i	$0.371931\pi$
$492$	0	0
$493$	39.3594	1.77266
$494$	0	0
$495$	6.13628	0.275805
$496$	0	0
$497$	13.3771	0.600045
$498$	0	0
$499$	32.9379	1.47450	0.737252	−	0.675618i	$-0.236123\pi$
$499$	32.9379	1.47450	0.737252	+	0.675618i	$0.236123\pi$
$500$	0	0
$501$	−14.3614	−0.641620
$502$	0	0
$503$	38.2074	1.70359	0.851793	−	0.523879i	$-0.175515\pi$
$503$	38.2074	1.70359	0.851793	+	0.523879i	$0.175515\pi$
$504$	0	0
$505$	4.28523	0.190690
$506$	0	0
$507$	12.8223	0.569460
$508$	0	0
$509$	−8.82430	−0.391130	−0.195565	−	0.980691i	$-0.562654\pi$
$509$	−8.82430	−0.391130	−0.195565	+	0.980691i	$0.562654\pi$
$510$	0	0
$511$	7.06814	0.312676
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	19.4295	0.854510
$518$	0	0
$519$	5.71477	0.250851
$520$	0	0
$521$	−24.2913	−1.06422	−0.532110	−	0.846675i	$-0.678601\pi$
$521$	−24.2913	−1.06422	−0.532110	+	0.846675i	$0.678601\pi$
$522$	0	0
$523$	−19.0969	−0.835047	−0.417524	−	0.908666i	$-0.637102\pi$
$523$	−19.0969	−0.835047	−0.417524	+	0.908666i	$0.637102\pi$
$524$	0	0
$525$	−1.42151	−0.0620399
$526$	0	0
$527$	−22.8016	−0.993255
$528$	0	0
$529$	−20.5084	−0.891668
$530$	0	0
$531$	−14.1363	−0.613462
$532$	0	0
$533$	−3.54977	−0.153757
$534$	0	0
$535$	−41.8985	−1.81143
$536$	0	0
$537$	−9.60221	−0.414366
$538$	0	0
$539$	−2.42151	−0.104302
$540$	0	0
$541$	−36.2312	−1.55770	−0.778850	−	0.627210i	$-0.784197\pi$
$541$	−36.2312	−1.55770	−0.778850	+	0.627210i	$0.784197\pi$
$542$	0	0
$543$	0.979304	0.0420260
$544$	0	0
$545$	−4.49768	−0.192659
$546$	0	0
$547$	29.9586	1.28094	0.640469	−	0.767984i	$-0.278740\pi$
$547$	29.9586	1.28094	0.640469	+	0.767984i	$0.278740\pi$
$548$	0	0
$549$	6.84302	0.292053
$550$	0	0
$551$	6.02372	0.256619
$552$	0	0
$553$	−14.1363	−0.601136
$554$	0	0
$555$	−7.20442	−0.305811
$556$	0	0
$557$	−14.4402	−0.611852	−0.305926	−	0.952055i	$-0.598966\pi$
$557$	−14.4402	−0.611852	−0.305926	+	0.952055i	$0.598966\pi$
$558$	0	0
$559$	4.84302	0.204838
$560$	0	0
$561$	15.8223	0.668019
$562$	0	0
$563$	17.9112	0.754866	0.377433	−	0.926037i	$-0.376807\pi$
$563$	17.9112	0.754866	0.377433	+	0.926037i	$0.376807\pi$
$564$	0	0
$565$	−25.4008	−1.06862
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−8.50535	−0.356563	−0.178281	−	0.983980i	$-0.557054\pi$
$569$	−8.50535	−0.356563	−0.178281	+	0.983980i	$0.557054\pi$
$570$	0	0
$571$	6.80163	0.284639	0.142320	−	0.989821i	$-0.454544\pi$
$571$	6.80163	0.284639	0.142320	+	0.989821i	$0.454544\pi$
$572$	0	0
$573$	−11.7148	−0.489391
$574$	0	0
$575$	2.24384	0.0935746
$576$	0	0
$577$	−14.2726	−0.594175	−0.297087	−	0.954850i	$-0.596015\pi$
$577$	−14.2726	−0.594175	−0.297087	+	0.954850i	$0.596015\pi$
$578$	0	0
$579$	−17.8223	−0.740671
$580$	0	0
$581$	−2.11256	−0.0876437
$582$	0	0
$583$	26.3140	1.08981
$584$	0	0
$585$	1.06814	0.0441622
$586$	0	0
$587$	−40.2963	−1.66321	−0.831603	−	0.555371i	$-0.812576\pi$
$587$	−40.2963	−1.66321	−0.831603	+	0.555371i	$0.812576\pi$
$588$	0	0
$589$	−3.48965	−0.143789
$590$	0	0
$591$	19.1156	0.786310
$592$	0	0
$593$	26.1313	1.07308	0.536542	−	0.843874i	$-0.319730\pi$
$593$	26.1313	1.07308	0.536542	+	0.843874i	$0.319730\pi$
$594$	0	0
$595$	−16.5578	−0.678803
$596$	0	0
$597$	16.0474	0.656778
$598$	0	0
$599$	−3.06314	−0.125157	−0.0625783	−	0.998040i	$-0.519932\pi$
$599$	−3.06314	−0.125157	−0.0625783	+	0.998040i	$0.519932\pi$
$600$	0	0
$601$	28.1363	1.14770	0.573851	−	0.818959i	$-0.305449\pi$
$601$	28.1363	1.14770	0.573851	+	0.818959i	$0.305449\pi$
$602$	0	0
$603$	9.91116	0.403614
$604$	0	0
$605$	13.0157	0.529163
$606$	0	0
$607$	−28.8430	−1.17070	−0.585351	−	0.810780i	$-0.699043\pi$
$607$	−28.8430	−1.17070	−0.585351	+	0.810780i	$0.699043\pi$
$608$	0	0
$609$	−6.02372	−0.244094
$610$	0	0
$611$	3.38209	0.136825
$612$	0	0
$613$	22.2852	0.900092	0.450046	−	0.893005i	$-0.351408\pi$
$613$	22.2852	0.900092	0.450046	+	0.893005i	$0.351408\pi$
$614$	0	0
$615$	21.3407	0.860540
$616$	0	0
$617$	4.13628	0.166520	0.0832602	−	0.996528i	$-0.473467\pi$
$617$	4.13628	0.166520	0.0832602	+	0.996528i	$0.473467\pi$
$618$	0	0
$619$	9.06814	0.364479	0.182240	−	0.983254i	$-0.441665\pi$
$619$	9.06814	0.364479	0.182240	+	0.983254i	$0.441665\pi$
$620$	0	0
$621$	−1.57849	−0.0633426
$622$	0	0
$623$	−9.71477	−0.389214
$624$	0	0
$625$	−30.0869	−1.20347
$626$	0	0
$627$	2.42151	0.0967059
$628$	0	0
$629$	−18.5765	−0.740694
$630$	0	0
$631$	41.6259	1.65710	0.828551	−	0.559913i	$-0.189166\pi$
$631$	41.6259	1.65710	0.828551	+	0.559913i	$0.189166\pi$
$632$	0	0
$633$	−9.29326	−0.369374
$634$	0	0
$635$	5.29326	0.210057
$636$	0	0
$637$	−0.421512	−0.0167009
$638$	0	0
$639$	13.3771	0.529190
$640$	0	0
$641$	3.09186	0.122121	0.0610606	−	0.998134i	$-0.480552\pi$
$641$	3.09186	0.122121	0.0610606	+	0.998134i	$0.480552\pi$
$642$	0	0
$643$	−2.93186	−0.115621	−0.0578106	−	0.998328i	$-0.518412\pi$
$643$	−2.93186	−0.115621	−0.0578106	+	0.998328i	$0.518412\pi$
$644$	0	0
$645$	−29.1156	−1.14643
$646$	0	0
$647$	5.93489	0.233324	0.116662	−	0.993172i	$-0.462781\pi$
$647$	5.93489	0.233324	0.116662	+	0.993172i	$0.462781\pi$
$648$	0	0
$649$	34.2312	1.34369
$650$	0	0
$651$	3.48965	0.136770
$652$	0	0
$653$	1.82233	0.0713132	0.0356566	−	0.999364i	$-0.488648\pi$
$653$	1.82233	0.0713132	0.0356566	+	0.999364i	$0.488648\pi$
$654$	0	0
$655$	−35.3120	−1.37975
$656$	0	0
$657$	7.06814	0.275754
$658$	0	0
$659$	−10.3978	−0.405040	−0.202520	−	0.979278i	$-0.564913\pi$
$659$	−10.3978	−0.405040	−0.202520	+	0.979278i	$0.564913\pi$
$660$	0	0
$661$	23.2231	0.903276	0.451638	−	0.892201i	$-0.350840\pi$
$661$	23.2231	0.903276	0.451638	+	0.892201i	$0.350840\pi$
$662$	0	0
$663$	2.75419	0.106964
$664$	0	0
$665$	−2.53407	−0.0982670
$666$	0	0
$667$	9.50837	0.368166
$668$	0	0
$669$	−8.33268	−0.322160
$670$	0	0
$671$	−16.5705	−0.639696
$672$	0	0
$673$	13.5498	0.522305	0.261153	−	0.965298i	$-0.415897\pi$
$673$	13.5498	0.522305	0.261153	+	0.965298i	$0.415897\pi$
$674$	0	0
$675$	−1.42151	−0.0547140
$676$	0	0
$677$	−12.6466	−0.486049	−0.243025	−	0.970020i	$-0.578140\pi$
$677$	−12.6466	−0.486049	−0.243025	+	0.970020i	$0.578140\pi$
$678$	0	0
$679$	−4.97930	−0.191088
$680$	0	0
$681$	16.8905	0.647244
$682$	0	0
$683$	13.7799	0.527273	0.263636	−	0.964622i	$-0.415078\pi$
$683$	13.7799	0.527273	0.263636	+	0.964622i	$0.415078\pi$
$684$	0	0
$685$	−43.0267	−1.64397
$686$	0	0
$687$	−6.04744	−0.230724
$688$	0	0
$689$	4.58046	0.174502
$690$	0	0
$691$	−1.11559	−0.0424389	−0.0212194	−	0.999775i	$-0.506755\pi$
$691$	−1.11559	−0.0424389	−0.0212194	+	0.999775i	$0.506755\pi$
$692$	0	0
$693$	−2.42151	−0.0919856
$694$	0	0
$695$	−2.70674	−0.102673
$696$	0	0
$697$	55.0267	2.08429
$698$	0	0
$699$	−21.2044	−0.802025
$700$	0	0
$701$	16.7067	0.631005	0.315502	−	0.948925i	$-0.397827\pi$
$701$	16.7067	0.631005	0.315502	+	0.948925i	$0.397827\pi$
$702$	0	0
$703$	−2.84302	−0.107227
$704$	0	0
$705$	−20.3327	−0.765773
$706$	0	0
$707$	−1.69105	−0.0635984
$708$	0	0
$709$	−25.1443	−0.944314	−0.472157	−	0.881514i	$-0.656525\pi$
$709$	−25.1443	−0.944314	−0.472157	+	0.881514i	$0.656525\pi$
$710$	0	0
$711$	−14.1363	−0.530152
$712$	0	0
$713$	−5.50837	−0.206290
$714$	0	0
$715$	−2.58651	−0.0967302
$716$	0	0
$717$	−5.80361	−0.216740
$718$	0	0
$719$	−32.2488	−1.20268	−0.601339	−	0.798994i	$-0.705366\pi$
$719$	−32.2488	−1.20268	−0.601339	+	0.798994i	$0.705366\pi$
$720$	0	0
$721$	−3.15698	−0.117572
$722$	0	0
$723$	12.9793	0.482706
$724$	0	0
$725$	8.56279	0.318014
$726$	0	0
$727$	30.2312	1.12121	0.560606	−	0.828083i	$-0.310568\pi$
$727$	30.2312	1.12121	0.560606	+	0.828083i	$0.310568\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−75.0742	−2.77672
$732$	0	0
$733$	25.0267	0.924384	0.462192	−	0.886780i	$-0.347063\pi$
$733$	25.0267	0.924384	0.462192	+	0.886780i	$0.347063\pi$
$734$	0	0
$735$	2.53407	0.0934706
$736$	0	0
$737$	−24.0000	−0.884051
$738$	0	0
$739$	−5.53104	−0.203463	−0.101731	−	0.994812i	$-0.532438\pi$
$739$	−5.53104	−0.203463	−0.101731	+	0.994812i	$0.532438\pi$
$740$	0	0
$741$	0.421512	0.0154846
$742$	0	0
$743$	−21.6971	−0.795989	−0.397995	−	0.917388i	$-0.630294\pi$
$743$	−21.6971	−0.795989	−0.397995	+	0.917388i	$0.630294\pi$
$744$	0	0
$745$	−35.5972	−1.30418
$746$	0	0
$747$	−2.11256	−0.0772945
$748$	0	0
$749$	16.5341	0.604142
$750$	0	0
$751$	48.9854	1.78750	0.893751	−	0.448564i	$-0.148065\pi$
$751$	48.9854	1.78750	0.893751	+	0.448564i	$0.148065\pi$
$752$	0	0
$753$	10.3377	0.376726
$754$	0	0
$755$	−37.3881	−1.36069
$756$	0	0
$757$	−4.04139	−0.146887	−0.0734434	−	0.997299i	$-0.523399\pi$
$757$	−4.04139	−0.146887	−0.0734434	+	0.997299i	$0.523399\pi$
$758$	0	0
$759$	3.82233	0.138742
$760$	0	0
$761$	−36.9429	−1.33918	−0.669590	−	0.742731i	$-0.733530\pi$
$761$	−36.9429	−1.33918	−0.669590	+	0.742731i	$0.733530\pi$
$762$	0	0
$763$	1.77488	0.0642551
$764$	0	0
$765$	−16.5578	−0.598648
$766$	0	0
$767$	5.95861	0.215153
$768$	0	0
$769$	24.1363	0.870377	0.435188	−	0.900339i	$-0.356682\pi$
$769$	24.1363	0.870377	0.435188	+	0.900339i	$0.356682\pi$
$770$	0	0
$771$	15.1757	0.546539
$772$	0	0
$773$	−25.5371	−0.918506	−0.459253	−	0.888306i	$-0.651883\pi$
$773$	−25.5371	−0.918506	−0.459253	+	0.888306i	$0.651883\pi$
$774$	0	0
$775$	−4.96058	−0.178189
$776$	0	0
$777$	2.84302	0.101993
$778$	0	0
$779$	8.42151	0.301732
$780$	0	0
$781$	−32.3928	−1.15911
$782$	0	0
$783$	−6.02372	−0.215270
$784$	0	0
$785$	40.8905	1.45944
$786$	0	0
$787$	19.0969	0.680730	0.340365	−	0.940293i	$-0.389449\pi$
$787$	19.0969	0.680730	0.340365	+	0.940293i	$0.389449\pi$
$788$	0	0
$789$	−8.51035	−0.302976
$790$	0	0
$791$	10.0237	0.356403
$792$	0	0
$793$	−2.88441	−0.102429
$794$	0	0
$795$	−27.5371	−0.976640
$796$	0	0
$797$	8.02872	0.284392	0.142196	−	0.989839i	$-0.454584\pi$
$797$	8.02872	0.284392	0.142196	+	0.989839i	$0.454584\pi$
$798$	0	0
$799$	−52.4276	−1.85475
$800$	0	0
$801$	−9.71477	−0.343254
$802$	0	0
$803$	−17.1156	−0.603996
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	−15.1757	−0.534210
$808$	0	0
$809$	−29.0742	−1.02219	−0.511097	−	0.859523i	$-0.670761\pi$
$809$	−29.0742	−1.02219	−0.511097	+	0.859523i	$0.670761\pi$
$810$	0	0
$811$	−4.66535	−0.163823	−0.0819113	−	0.996640i	$-0.526102\pi$
$811$	−4.66535	−0.163823	−0.0819113	+	0.996640i	$0.526102\pi$
$812$	0	0
$813$	0.843024	0.0295661
$814$	0	0
$815$	−16.8430	−0.589985
$816$	0	0
$817$	−11.4897	−0.401972
$818$	0	0
$819$	−0.421512	−0.0147288
$820$	0	0
$821$	−17.5972	−0.614147	−0.307073	−	0.951686i	$-0.599350\pi$
$821$	−17.5972	−0.614147	−0.307073	+	0.951686i	$0.599350\pi$
$822$	0	0
$823$	7.93989	0.276767	0.138384	−	0.990379i	$-0.455809\pi$
$823$	7.93989	0.276767	0.138384	+	0.990379i	$0.455809\pi$
$824$	0	0
$825$	3.44221	0.119842
$826$	0	0
$827$	31.7859	1.10531	0.552653	−	0.833412i	$-0.313616\pi$
$827$	31.7859	1.10531	0.552653	+	0.833412i	$0.313616\pi$
$828$	0	0
$829$	−48.9793	−1.70112	−0.850561	−	0.525877i	$-0.823737\pi$
$829$	−48.9793	−1.70112	−0.850561	+	0.525877i	$0.823737\pi$
$830$	0	0
$831$	−10.5578	−0.366246
$832$	0	0
$833$	6.53407	0.226392
$834$	0	0
$835$	−36.3928	−1.25942
$836$	0	0
$837$	3.48965	0.120620
$838$	0	0
$839$	−36.2726	−1.25227	−0.626134	−	0.779716i	$-0.715364\pi$
$839$	−36.2726	−1.25227	−0.626134	+	0.779716i	$0.715364\pi$
$840$	0	0
$841$	7.28523	0.251215
$842$	0	0
$843$	−15.0444	−0.518157
$844$	0	0
$845$	32.4927	1.11778
$846$	0	0
$847$	−5.13628	−0.176485
$848$	0	0
$849$	1.24581	0.0427562
$850$	0	0
$851$	−4.48768	−0.153836
$852$	0	0
$853$	−34.6654	−1.18692	−0.593460	−	0.804864i	$-0.702238\pi$
$853$	−34.6654	−1.18692	−0.593460	+	0.804864i	$0.702238\pi$
$854$	0	0
$855$	−2.53407	−0.0866634
$856$	0	0
$857$	−4.87175	−0.166416	−0.0832078	−	0.996532i	$-0.526517\pi$
$857$	−4.87175	−0.166416	−0.0832078	+	0.996532i	$0.526517\pi$
$858$	0	0
$859$	37.1156	1.26637	0.633184	−	0.774002i	$-0.281748\pi$
$859$	37.1156	1.26637	0.633184	+	0.774002i	$0.281748\pi$
$860$	0	0
$861$	−8.42151	−0.287004
$862$	0	0
$863$	29.1520	0.992345	0.496172	−	0.868224i	$-0.334738\pi$
$863$	29.1520	0.992345	0.496172	+	0.868224i	$0.334738\pi$
$864$	0	0
$865$	14.4816	0.492390
$866$	0	0
$867$	−25.6941	−0.872616
$868$	0	0
$869$	34.2312	1.16121
$870$	0	0
$871$	−4.17767	−0.141555
$872$	0	0
$873$	−4.97930	−0.168524
$874$	0	0
$875$	9.06814	0.306559
$876$	0	0
$877$	14.4502	0.487950	0.243975	−	0.969782i	$-0.421549\pi$
$877$	14.4502	0.487950	0.243975	+	0.969782i	$0.421549\pi$
$878$	0	0
$879$	−20.4215	−0.688800
$880$	0	0
$881$	10.5341	0.354902	0.177451	−	0.984130i	$-0.443215\pi$
$881$	10.5341	0.354902	0.177451	+	0.984130i	$0.443215\pi$
$882$	0	0
$883$	5.11559	0.172153	0.0860766	−	0.996289i	$-0.472567\pi$
$883$	5.11559	0.172153	0.0860766	+	0.996289i	$0.472567\pi$
$884$	0	0
$885$	−35.8223	−1.20415
$886$	0	0
$887$	0.617907	0.0207473	0.0103736	−	0.999946i	$-0.496698\pi$
$887$	0.617907	0.0207473	0.0103736	+	0.999946i	$0.496698\pi$
$888$	0	0
$889$	−2.08884	−0.0700573
$890$	0	0
$891$	−2.42151	−0.0811237
$892$	0	0
$893$	−8.02372	−0.268504
$894$	0	0
$895$	−24.3327	−0.813352
$896$	0	0
$897$	0.665351	0.0222154
$898$	0	0
$899$	−21.0207	−0.701079
$900$	0	0
$901$	−71.0041	−2.36549
$902$	0	0
$903$	11.4897	0.382352
$904$	0	0
$905$	2.48163	0.0824920
$906$	0	0
$907$	8.27256	0.274686	0.137343	−	0.990524i	$-0.456144\pi$
$907$	8.27256	0.274686	0.137343	+	0.990524i	$0.456144\pi$
$908$	0	0
$909$	−1.69105	−0.0560885
$910$	0	0
$911$	5.77988	0.191496	0.0957480	−	0.995406i	$-0.469476\pi$
$911$	5.77988	0.191496	0.0957480	+	0.995406i	$0.469476\pi$
$912$	0	0
$913$	5.11559	0.169301
$914$	0	0
$915$	17.3407	0.573266
$916$	0	0
$917$	13.9349	0.460170
$918$	0	0
$919$	−4.62791	−0.152661	−0.0763303	−	0.997083i	$-0.524320\pi$
$919$	−4.62791	−0.152661	−0.0763303	+	0.997083i	$0.524320\pi$
$920$	0	0
$921$	−2.64663	−0.0872094
$922$	0	0
$923$	−5.63860	−0.185597
$924$	0	0
$925$	−4.04139	−0.132880
$926$	0	0
$927$	−3.15698	−0.103689
$928$	0	0
$929$	−29.9536	−0.982746	−0.491373	−	0.870949i	$-0.663505\pi$
$929$	−29.9536	−0.982746	−0.491373	+	0.870949i	$0.663505\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−5.09186	−0.166700
$934$	0	0
$935$	40.0949	1.31124
$936$	0	0
$937$	4.58651	0.149835	0.0749174	−	0.997190i	$-0.476131\pi$
$937$	4.58651	0.149835	0.0749174	+	0.997190i	$0.476131\pi$
$938$	0	0
$939$	−19.9586	−0.651325
$940$	0	0
$941$	7.41082	0.241586	0.120793	−	0.992678i	$-0.461456\pi$
$941$	7.41082	0.241586	0.120793	+	0.992678i	$0.461456\pi$
$942$	0	0
$943$	13.2933	0.432888
$944$	0	0
$945$	2.53407	0.0824333
$946$	0	0
$947$	−59.6320	−1.93778	−0.968890	−	0.247493i	$-0.920393\pi$
$947$	−59.6320	−1.93778	−0.968890	+	0.247493i	$0.920393\pi$
$948$	0	0
$949$	−2.97930	−0.0967123
$950$	0	0
$951$	6.81930	0.221131
$952$	0	0
$953$	25.7986	0.835699	0.417849	−	0.908516i	$-0.362784\pi$
$953$	25.7986	0.835699	0.417849	+	0.908516i	$0.362784\pi$
$954$	0	0
$955$	−29.6860	−0.960618
$956$	0	0
$957$	14.5865	0.471515
$958$	0	0
$959$	16.9793	0.548290
$960$	0	0
$961$	−18.8223	−0.607172
$962$	0	0
$963$	16.5341	0.532803
$964$	0	0
$965$	−45.1630	−1.45385
$966$	0	0
$967$	−55.5845	−1.78748	−0.893739	−	0.448587i	$-0.851927\pi$
$967$	−55.5845	−1.78748	−0.893739	+	0.448587i	$0.851927\pi$
$968$	0	0
$969$	−6.53407	−0.209905
$970$	0	0
$971$	48.4877	1.55604	0.778022	−	0.628237i	$-0.216223\pi$
$971$	48.4877	1.55604	0.778022	+	0.628237i	$0.216223\pi$
$972$	0	0
$973$	1.06814	0.0342430
$974$	0	0
$975$	0.599184	0.0191892
$976$	0	0
$977$	53.0505	1.69723	0.848617	−	0.529007i	$-0.177435\pi$
$977$	53.0505	1.69723	0.848617	+	0.529007i	$0.177435\pi$
$978$	0	0
$979$	23.5244	0.751844
$980$	0	0
$981$	1.77488	0.0566677
$982$	0	0
$983$	−12.1777	−0.388407	−0.194204	−	0.980961i	$-0.562212\pi$
$983$	−12.1777	−0.388407	−0.194204	+	0.980961i	$0.562212\pi$
$984$	0	0
$985$	48.4402	1.54343
$986$	0	0
$987$	8.02372	0.255398
$988$	0	0
$989$	−18.1363	−0.576700
$990$	0	0
$991$	2.04139	0.0648469	0.0324235	−	0.999474i	$-0.489677\pi$
$991$	2.04139	0.0648469	0.0324235	+	0.999474i	$0.489677\pi$
$992$	0	0
$993$	−5.91116	−0.187585
$994$	0	0
$995$	40.6654	1.28918
$996$	0	0
$997$	−45.6447	−1.44558	−0.722790	−	0.691067i	$-0.757141\pi$
$997$	−45.6447	−1.44558	−0.722790	+	0.691067i	$0.757141\pi$
$998$	0	0
$999$	2.84302	0.0899493

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.bu.1.1		3
4.3	odd	2		399.2.a.e.1.1	✓	3
12.11	even	2		1197.2.a.m.1.3		3
20.19	odd	2		9975.2.a.x.1.3		3
28.27	even	2		2793.2.a.w.1.1		3
76.75	even	2		7581.2.a.l.1.3		3
84.83	odd	2		8379.2.a.bq.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.e.1.1	✓	3	4.3	odd	2
1197.2.a.m.1.3		3	12.11	even	2
2793.2.a.w.1.1		3	28.27	even	2
6384.2.a.bu.1.1		3	1.1	even	1	trivial
7581.2.a.l.1.3		3	76.75	even	2
8379.2.a.bq.1.3		3	84.83	odd	2
9975.2.a.x.1.3		3	20.19	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6384.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.53407 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.42151 q^{11} -0.421512 q^{13} +2.53407 q^{15} +6.53407 q^{17} +1.00000 q^{19} -1.00000 q^{21} +1.57849 q^{23} +1.42151 q^{25} -1.00000 q^{27} +6.02372 q^{29} -3.48965 q^{31} +2.42151 q^{33} -2.53407 q^{35} -2.84302 q^{37} +0.421512 q^{39} +8.42151 q^{41} -11.4897 q^{43} -2.53407 q^{45} -8.02372 q^{47} +1.00000 q^{49} -6.53407 q^{51} -10.8667 q^{53} +6.13628 q^{55} -1.00000 q^{57} -14.1363 q^{59} +6.84302 q^{61} +1.00000 q^{63} +1.06814 q^{65} +9.91116 q^{67} -1.57849 q^{69} +13.3771 q^{71} +7.06814 q^{73} -1.42151 q^{75} -2.42151 q^{77} -14.1363 q^{79} +1.00000 q^{81} -2.11256 q^{83} -16.5578 q^{85} -6.02372 q^{87} -9.71477 q^{89} -0.421512 q^{91} +3.48965 q^{93} -2.53407 q^{95} -4.97930 q^{97} -2.42151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} - 4 q^{11} + 2 q^{13} + 12 q^{17} + 3 q^{19} - 3 q^{21} + 8 q^{23} + q^{25} - 3 q^{27} - 8 q^{29} + 8 q^{31} + 4 q^{33} - 2 q^{37} - 2 q^{39} + 22 q^{41} - 16 q^{43} + 2 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 - 4 * q^11 + 2 * q^13 + 12 * q^17 + 3 * q^19 - 3 * q^21 + 8 * q^23 + q^25 - 3 * q^27 - 8 * q^29 + 8 * q^31 + 4 * q^33 - 2 * q^37 - 2 * q^39 + 22 * q^41 - 16 * q^43 + 2 * q^47 + 3 * q^49 - 12 * q^51 - 12 * q^55 - 3 * q^57 - 12 * q^59 + 14 * q^61 + 3 * q^63 - 12 * q^65 + 8 * q^67 - 8 * q^69 + 26 * q^71 + 6 * q^73 - q^75 - 4 * q^77 - 12 * q^79 + 3 * q^81 - 2 * q^83 - 16 * q^85 + 8 * q^87 - 2 * q^89 + 2 * q^91 - 8 * q^93 + 22 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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