LMFDB - Embedded newform 6004.2.a.h.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, 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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.199808 q^{3} +0.368036 q^{5} +1.63577 q^{7} -2.96008 q^{9} +O(q^{10})$	$q-0.199808 q^{3} +0.368036 q^{5} +1.63577 q^{7} -2.96008 q^{9} -0.834896 q^{11} +0.791812 q^{13} -0.0735364 q^{15} -7.69705 q^{17} -1.00000 q^{19} -0.326840 q^{21} +4.91198 q^{23} -4.86455 q^{25} +1.19087 q^{27} +7.77178 q^{29} -9.19598 q^{31} +0.166819 q^{33} +0.602022 q^{35} +6.82586 q^{37} -0.158210 q^{39} -5.55990 q^{41} +7.32798 q^{43} -1.08941 q^{45} +9.91225 q^{47} -4.32425 q^{49} +1.53793 q^{51} -0.543294 q^{53} -0.307272 q^{55} +0.199808 q^{57} +8.81608 q^{59} +1.62193 q^{61} -4.84201 q^{63} +0.291415 q^{65} +10.9685 q^{67} -0.981452 q^{69} -12.5851 q^{71} +8.46189 q^{73} +0.971975 q^{75} -1.36570 q^{77} -1.00000 q^{79} +8.64229 q^{81} +10.1561 q^{83} -2.83279 q^{85} -1.55286 q^{87} +0.948427 q^{89} +1.29522 q^{91} +1.83743 q^{93} -0.368036 q^{95} +12.0579 q^{97} +2.47136 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * 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
$3$	−0.199808	−0.115359	−0.0576795	−	0.998335i	$-0.518370\pi$
$3$	−0.199808	−0.115359	−0.0576795	+	0.998335i	$0.518370\pi$
$4$	0	0
$5$	0.368036	0.164591	0.0822953	−	0.996608i	$-0.473775\pi$
$5$	0.368036	0.164591	0.0822953	+	0.996608i	$0.473775\pi$
$6$	0	0
$7$	1.63577	0.618264	0.309132	−	0.951019i	$-0.399962\pi$
$7$	1.63577	0.618264	0.309132	+	0.951019i	$0.399962\pi$
$8$	0	0
$9$	−2.96008	−0.986692
$10$	0	0
$11$	−0.834896	−0.251731	−0.125865	−	0.992047i	$-0.540171\pi$
$11$	−0.834896	−0.251731	−0.125865	+	0.992047i	$0.540171\pi$
$12$	0	0
$13$	0.791812	0.219609	0.109805	−	0.993953i	$-0.464978\pi$
$13$	0.791812	0.219609	0.109805	+	0.993953i	$0.464978\pi$
$14$	0	0
$15$	−0.0735364	−0.0189870
$16$	0	0
$17$	−7.69705	−1.86681	−0.933405	−	0.358826i	$-0.883177\pi$
$17$	−7.69705	−1.86681	−0.933405	+	0.358826i	$0.883177\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.326840	−0.0713223
$22$	0	0
$23$	4.91198	1.02422	0.512110	−	0.858920i	$-0.328864\pi$
$23$	4.91198	1.02422	0.512110	+	0.858920i	$0.328864\pi$
$24$	0	0
$25$	−4.86455	−0.972910
$26$	0	0
$27$	1.19087	0.229183
$28$	0	0
$29$	7.77178	1.44318	0.721591	−	0.692319i	$-0.243411\pi$
$29$	7.77178	1.44318	0.721591	+	0.692319i	$0.243411\pi$
$30$	0	0
$31$	−9.19598	−1.65165	−0.825823	−	0.563929i	$-0.809289\pi$
$31$	−9.19598	−1.65165	−0.825823	+	0.563929i	$0.809289\pi$
$32$	0	0
$33$	0.166819	0.0290394
$34$	0	0
$35$	0.602022	0.101760
$36$	0	0
$37$	6.82586	1.12217	0.561083	−	0.827760i	$-0.310385\pi$
$37$	6.82586	1.12217	0.561083	+	0.827760i	$0.310385\pi$
$38$	0	0
$39$	−0.158210	−0.0253339
$40$	0	0
$41$	−5.55990	−0.868311	−0.434156	−	0.900838i	$-0.642953\pi$
$41$	−5.55990	−0.868311	−0.434156	+	0.900838i	$0.642953\pi$
$42$	0	0
$43$	7.32798	1.11751	0.558753	−	0.829334i	$-0.311280\pi$
$43$	7.32798	1.11751	0.558753	+	0.829334i	$0.311280\pi$
$44$	0	0
$45$	−1.08941	−0.162400
$46$	0	0
$47$	9.91225	1.44585	0.722925	−	0.690926i	$-0.242797\pi$
$47$	9.91225	1.44585	0.722925	+	0.690926i	$0.242797\pi$
$48$	0	0
$49$	−4.32425	−0.617750
$50$	0	0
$51$	1.53793	0.215353
$52$	0	0
$53$	−0.543294	−0.0746272	−0.0373136	−	0.999304i	$-0.511880\pi$
$53$	−0.543294	−0.0746272	−0.0373136	+	0.999304i	$0.511880\pi$
$54$	0	0
$55$	−0.307272	−0.0414325
$56$	0	0
$57$	0.199808	0.0264652
$58$	0	0
$59$	8.81608	1.14776	0.573878	−	0.818941i	$-0.305438\pi$
$59$	8.81608	1.14776	0.573878	+	0.818941i	$0.305438\pi$
$60$	0	0
$61$	1.62193	0.207667	0.103834	−	0.994595i	$-0.466889\pi$
$61$	1.62193	0.207667	0.103834	+	0.994595i	$0.466889\pi$
$62$	0	0
$63$	−4.84201	−0.610036
$64$	0	0
$65$	0.291415	0.0361456
$66$	0	0
$67$	10.9685	1.34001	0.670005	−	0.742357i	$-0.266292\pi$
$67$	10.9685	1.34001	0.670005	+	0.742357i	$0.266292\pi$
$68$	0	0
$69$	−0.981452	−0.118153
$70$	0	0
$71$	−12.5851	−1.49358	−0.746791	−	0.665059i	$-0.768407\pi$
$71$	−12.5851	−1.49358	−0.746791	+	0.665059i	$0.768407\pi$
$72$	0	0
$73$	8.46189	0.990390	0.495195	−	0.868782i	$-0.335097\pi$
$73$	8.46189	0.990390	0.495195	+	0.868782i	$0.335097\pi$
$74$	0	0
$75$	0.971975	0.112234
$76$	0	0
$77$	−1.36570	−0.155636
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	8.64229	0.960254
$82$	0	0
$83$	10.1561	1.11478	0.557390	−	0.830251i	$-0.311803\pi$
$83$	10.1561	1.11478	0.557390	+	0.830251i	$0.311803\pi$
$84$	0	0
$85$	−2.83279	−0.307259
$86$	0	0
$87$	−1.55286	−0.166484
$88$	0	0
$89$	0.948427	0.100533	0.0502665	−	0.998736i	$-0.483993\pi$
$89$	0.948427	0.100533	0.0502665	+	0.998736i	$0.483993\pi$
$90$	0	0
$91$	1.29522	0.135776
$92$	0	0
$93$	1.83743	0.190532
$94$	0	0
$95$	−0.368036	−0.0377597
$96$	0	0
$97$	12.0579	1.22429	0.612146	−	0.790745i	$-0.290307\pi$
$97$	12.0579	1.22429	0.612146	+	0.790745i	$0.290307\pi$
$98$	0	0
$99$	2.47136	0.248381
$100$	0	0
$101$	2.78218	0.276837	0.138419	−	0.990374i	$-0.455798\pi$
$101$	2.78218	0.276837	0.138419	+	0.990374i	$0.455798\pi$
$102$	0	0
$103$	8.69808	0.857048	0.428524	−	0.903530i	$-0.359034\pi$
$103$	8.69808	0.857048	0.428524	+	0.903530i	$0.359034\pi$
$104$	0	0
$105$	−0.120289	−0.0117390
$106$	0	0
$107$	13.2094	1.27700	0.638500	−	0.769621i	$-0.279555\pi$
$107$	13.2094	1.27700	0.638500	+	0.769621i	$0.279555\pi$
$108$	0	0
$109$	4.88289	0.467696	0.233848	−	0.972273i	$-0.424868\pi$
$109$	4.88289	0.467696	0.233848	+	0.972273i	$0.424868\pi$
$110$	0	0
$111$	−1.36386	−0.129452
$112$	0	0
$113$	7.21488	0.678719	0.339360	−	0.940657i	$-0.389790\pi$
$113$	7.21488	0.678719	0.339360	+	0.940657i	$0.389790\pi$
$114$	0	0
$115$	1.80778	0.168577
$116$	0	0
$117$	−2.34382	−0.216687
$118$	0	0
$119$	−12.5906	−1.15418
$120$	0	0
$121$	−10.3029	−0.936632
$122$	0	0
$123$	1.11091	0.100168
$124$	0	0
$125$	−3.63051	−0.324722
$126$	0	0
$127$	0.860927	0.0763949	0.0381975	−	0.999270i	$-0.487838\pi$
$127$	0.860927	0.0763949	0.0381975	+	0.999270i	$0.487838\pi$
$128$	0	0
$129$	−1.46419	−0.128915
$130$	0	0
$131$	−4.69880	−0.410536	−0.205268	−	0.978706i	$-0.565807\pi$
$131$	−4.69880	−0.410536	−0.205268	+	0.978706i	$0.565807\pi$
$132$	0	0
$133$	−1.63577	−0.141839
$134$	0	0
$135$	0.438283	0.0377214
$136$	0	0
$137$	−8.87910	−0.758593	−0.379296	−	0.925275i	$-0.623834\pi$
$137$	−8.87910	−0.758593	−0.379296	+	0.925275i	$0.623834\pi$
$138$	0	0
$139$	−17.6293	−1.49529	−0.747647	−	0.664096i	$-0.768817\pi$
$139$	−17.6293	−1.49529	−0.747647	+	0.664096i	$0.768817\pi$
$140$	0	0
$141$	−1.98054	−0.166792
$142$	0	0
$143$	−0.661080	−0.0552823
$144$	0	0
$145$	2.86029	0.237534
$146$	0	0
$147$	0.864019	0.0712631
$148$	0	0
$149$	10.7399	0.879849	0.439925	−	0.898035i	$-0.355005\pi$
$149$	10.7399	0.879849	0.439925	+	0.898035i	$0.355005\pi$
$150$	0	0
$151$	14.9337	1.21528	0.607642	−	0.794211i	$-0.292115\pi$
$151$	14.9337	1.21528	0.607642	+	0.794211i	$0.292115\pi$
$152$	0	0
$153$	22.7839	1.84197
$154$	0	0
$155$	−3.38445	−0.271845
$156$	0	0
$157$	−5.41249	−0.431963	−0.215982	−	0.976397i	$-0.569295\pi$
$157$	−5.41249	−0.431963	−0.215982	+	0.976397i	$0.569295\pi$
$158$	0	0
$159$	0.108554	0.00860893
$160$	0	0
$161$	8.03488	0.633237
$162$	0	0
$163$	22.2241	1.74073	0.870363	−	0.492411i	$-0.163884\pi$
$163$	22.2241	1.74073	0.870363	+	0.492411i	$0.163884\pi$
$164$	0	0
$165$	0.0613952	0.00477961
$166$	0	0
$167$	−16.9222	−1.30948	−0.654738	−	0.755856i	$-0.727221\pi$
$167$	−16.9222	−1.30948	−0.654738	+	0.755856i	$0.727221\pi$
$168$	0	0
$169$	−12.3730	−0.951772
$170$	0	0
$171$	2.96008	0.226363
$172$	0	0
$173$	−5.56858	−0.423371	−0.211686	−	0.977338i	$-0.567895\pi$
$173$	−5.56858	−0.423371	−0.211686	+	0.977338i	$0.567895\pi$
$174$	0	0
$175$	−7.95729	−0.601515
$176$	0	0
$177$	−1.76152	−0.132404
$178$	0	0
$179$	9.67692	0.723287	0.361643	−	0.932317i	$-0.382216\pi$
$179$	9.67692	0.723287	0.361643	+	0.932317i	$0.382216\pi$
$180$	0	0
$181$	3.55244	0.264051	0.132025	−	0.991246i	$-0.457852\pi$
$181$	3.55244	0.264051	0.132025	+	0.991246i	$0.457852\pi$
$182$	0	0
$183$	−0.324075	−0.0239563
$184$	0	0
$185$	2.51216	0.184698
$186$	0	0
$187$	6.42624	0.469933
$188$	0	0
$189$	1.94799	0.141696
$190$	0	0
$191$	−7.10419	−0.514042	−0.257021	−	0.966406i	$-0.582741\pi$
$191$	−7.10419	−0.514042	−0.257021	+	0.966406i	$0.582741\pi$
$192$	0	0
$193$	−14.9768	−1.07806	−0.539028	−	0.842288i	$-0.681208\pi$
$193$	−14.9768	−1.07806	−0.539028	+	0.842288i	$0.681208\pi$
$194$	0	0
$195$	−0.0582270	−0.00416972
$196$	0	0
$197$	27.0625	1.92812	0.964061	−	0.265680i	$-0.0855963\pi$
$197$	27.0625	1.92812	0.964061	+	0.265680i	$0.0855963\pi$
$198$	0	0
$199$	10.3430	0.733198	0.366599	−	0.930379i	$-0.380522\pi$
$199$	10.3430	0.733198	0.366599	+	0.930379i	$0.380522\pi$
$200$	0	0
$201$	−2.19158	−0.154582
$202$	0	0
$203$	12.7129	0.892267
$204$	0	0
$205$	−2.04624	−0.142916
$206$	0	0
$207$	−14.5398	−1.01059
$208$	0	0
$209$	0.834896	0.0577510
$210$	0	0
$211$	25.3068	1.74219	0.871095	−	0.491114i	$-0.163410\pi$
$211$	25.3068	1.74219	0.871095	+	0.491114i	$0.163410\pi$
$212$	0	0
$213$	2.51461	0.172298
$214$	0	0
$215$	2.69696	0.183931
$216$	0	0
$217$	−15.0425	−1.02115
$218$	0	0
$219$	−1.69075	−0.114250
$220$	0	0
$221$	−6.09462	−0.409968
$222$	0	0
$223$	−29.1407	−1.95141	−0.975704	−	0.219093i	$-0.929690\pi$
$223$	−29.1407	−1.95141	−0.975704	+	0.219093i	$0.929690\pi$
$224$	0	0
$225$	14.3994	0.959963
$226$	0	0
$227$	−4.17316	−0.276983	−0.138491	−	0.990364i	$-0.544225\pi$
$227$	−4.17316	−0.276983	−0.138491	+	0.990364i	$0.544225\pi$
$228$	0	0
$229$	−15.7597	−1.04143	−0.520716	−	0.853730i	$-0.674335\pi$
$229$	−15.7597	−1.04143	−0.520716	+	0.853730i	$0.674335\pi$
$230$	0	0
$231$	0.272877	0.0179540
$232$	0	0
$233$	18.4685	1.20991	0.604954	−	0.796260i	$-0.293191\pi$
$233$	18.4685	1.20991	0.604954	+	0.796260i	$0.293191\pi$
$234$	0	0
$235$	3.64806	0.237973
$236$	0	0
$237$	0.199808	0.0129789
$238$	0	0
$239$	−0.288741	−0.0186771	−0.00933855	−	0.999956i	$-0.502973\pi$
$239$	−0.288741	−0.0186771	−0.00933855	+	0.999956i	$0.502973\pi$
$240$	0	0
$241$	−17.4638	−1.12494	−0.562470	−	0.826818i	$-0.690149\pi$
$241$	−17.4638	−1.12494	−0.562470	+	0.826818i	$0.690149\pi$
$242$	0	0
$243$	−5.29941	−0.339957
$244$	0	0
$245$	−1.59148	−0.101676
$246$	0	0
$247$	−0.791812	−0.0503818
$248$	0	0
$249$	−2.02927	−0.128600
$250$	0	0
$251$	−5.32570	−0.336155	−0.168078	−	0.985774i	$-0.553756\pi$
$251$	−5.32570	−0.336155	−0.168078	+	0.985774i	$0.553756\pi$
$252$	0	0
$253$	−4.10099	−0.257827
$254$	0	0
$255$	0.566013	0.0354451
$256$	0	0
$257$	27.4171	1.71023	0.855115	−	0.518438i	$-0.173486\pi$
$257$	27.4171	1.71023	0.855115	+	0.518438i	$0.173486\pi$
$258$	0	0
$259$	11.1656	0.693794
$260$	0	0
$261$	−23.0051	−1.42398
$262$	0	0
$263$	−3.00632	−0.185377	−0.0926887	−	0.995695i	$-0.529546\pi$
$263$	−3.00632	−0.185377	−0.0926887	+	0.995695i	$0.529546\pi$
$264$	0	0
$265$	−0.199952	−0.0122829
$266$	0	0
$267$	−0.189503	−0.0115974
$268$	0	0
$269$	1.27786	0.0779127	0.0389564	−	0.999241i	$-0.487597\pi$
$269$	1.27786	0.0779127	0.0389564	+	0.999241i	$0.487597\pi$
$270$	0	0
$271$	−26.2449	−1.59426	−0.797131	−	0.603807i	$-0.793650\pi$
$271$	−26.2449	−1.59426	−0.797131	+	0.603807i	$0.793650\pi$
$272$	0	0
$273$	−0.258796	−0.0156630
$274$	0	0
$275$	4.06139	0.244911
$276$	0	0
$277$	−8.25986	−0.496287	−0.248144	−	0.968723i	$-0.579820\pi$
$277$	−8.25986	−0.496287	−0.248144	+	0.968723i	$0.579820\pi$
$278$	0	0
$279$	27.2208	1.62967
$280$	0	0
$281$	24.2631	1.44742	0.723709	−	0.690105i	$-0.242436\pi$
$281$	24.2631	1.44742	0.723709	+	0.690105i	$0.242436\pi$
$282$	0	0
$283$	13.0511	0.775808	0.387904	−	0.921700i	$-0.373199\pi$
$283$	13.0511	0.775808	0.387904	+	0.921700i	$0.373199\pi$
$284$	0	0
$285$	0.0735364	0.00435592
$286$	0	0
$287$	−9.09473	−0.536845
$288$	0	0
$289$	42.2446	2.48498
$290$	0	0
$291$	−2.40926	−0.141233
$292$	0	0
$293$	30.1729	1.76272	0.881361	−	0.472444i	$-0.156628\pi$
$293$	30.1729	1.76272	0.881361	+	0.472444i	$0.156628\pi$
$294$	0	0
$295$	3.24463	0.188910
$296$	0	0
$297$	−0.994252	−0.0576924
$298$	0	0
$299$	3.88937	0.224928
$300$	0	0
$301$	11.9869	0.690914
$302$	0	0
$303$	−0.555902	−0.0319357
$304$	0	0
$305$	0.596930	0.0341801
$306$	0	0
$307$	−7.13430	−0.407176	−0.203588	−	0.979057i	$-0.565260\pi$
$307$	−7.13430	−0.407176	−0.203588	+	0.979057i	$0.565260\pi$
$308$	0	0
$309$	−1.73794	−0.0988682
$310$	0	0
$311$	−25.9357	−1.47068	−0.735339	−	0.677699i	$-0.762977\pi$
$311$	−25.9357	−1.47068	−0.735339	+	0.677699i	$0.762977\pi$
$312$	0	0
$313$	26.6883	1.50851	0.754256	−	0.656580i	$-0.227998\pi$
$313$	26.6883	1.50851	0.754256	+	0.656580i	$0.227998\pi$
$314$	0	0
$315$	−1.78203	−0.100406
$316$	0	0
$317$	−13.9922	−0.785882	−0.392941	−	0.919564i	$-0.628542\pi$
$317$	−13.9922	−0.785882	−0.392941	+	0.919564i	$0.628542\pi$
$318$	0	0
$319$	−6.48862	−0.363293
$320$	0	0
$321$	−2.63934	−0.147314
$322$	0	0
$323$	7.69705	0.428275
$324$	0	0
$325$	−3.85181	−0.213660
$326$	0	0
$327$	−0.975640	−0.0539530
$328$	0	0
$329$	16.2142	0.893917
$330$	0	0
$331$	19.7853	1.08750	0.543750	−	0.839247i	$-0.317004\pi$
$331$	19.7853	1.08750	0.543750	+	0.839247i	$0.317004\pi$
$332$	0	0
$333$	−20.2051	−1.10723
$334$	0	0
$335$	4.03678	0.220553
$336$	0	0
$337$	−23.3620	−1.27261	−0.636303	−	0.771439i	$-0.719537\pi$
$337$	−23.3620	−1.27261	−0.636303	+	0.771439i	$0.719537\pi$
$338$	0	0
$339$	−1.44159	−0.0782964
$340$	0	0
$341$	7.67769	0.415770
$342$	0	0
$343$	−18.5239	−1.00020
$344$	0	0
$345$	−0.361209	−0.0194469
$346$	0	0
$347$	31.0337	1.66598	0.832988	−	0.553291i	$-0.186628\pi$
$347$	31.0337	1.66598	0.832988	+	0.553291i	$0.186628\pi$
$348$	0	0
$349$	−23.5381	−1.25996	−0.629982	−	0.776609i	$-0.716938\pi$
$349$	−23.5381	−1.25996	−0.629982	+	0.776609i	$0.716938\pi$
$350$	0	0
$351$	0.942945	0.0503307
$352$	0	0
$353$	−15.7104	−0.836179	−0.418090	−	0.908406i	$-0.637300\pi$
$353$	−15.7104	−0.836179	−0.418090	+	0.908406i	$0.637300\pi$
$354$	0	0
$355$	−4.63178	−0.245830
$356$	0	0
$357$	2.51570	0.133145
$358$	0	0
$359$	−4.07355	−0.214994	−0.107497	−	0.994205i	$-0.534284\pi$
$359$	−4.07355	−0.214994	−0.107497	+	0.994205i	$0.534284\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.05861	0.108049
$364$	0	0
$365$	3.11428	0.163009
$366$	0	0
$367$	30.2156	1.57724	0.788621	−	0.614879i	$-0.210795\pi$
$367$	30.2156	1.57724	0.788621	+	0.614879i	$0.210795\pi$
$368$	0	0
$369$	16.4577	0.856756
$370$	0	0
$371$	−0.888706	−0.0461393
$372$	0	0
$373$	−3.69233	−0.191182	−0.0955908	−	0.995421i	$-0.530474\pi$
$373$	−3.69233	−0.191182	−0.0955908	+	0.995421i	$0.530474\pi$
$374$	0	0
$375$	0.725403	0.0374597
$376$	0	0
$377$	6.15378	0.316936
$378$	0	0
$379$	−23.2410	−1.19381	−0.596905	−	0.802312i	$-0.703603\pi$
$379$	−23.2410	−1.19381	−0.596905	+	0.802312i	$0.703603\pi$
$380$	0	0
$381$	−0.172020	−0.00881285
$382$	0	0
$383$	38.5941	1.97207	0.986034	−	0.166545i	$-0.0532611\pi$
$383$	38.5941	1.97207	0.986034	+	0.166545i	$0.0532611\pi$
$384$	0	0
$385$	−0.502626	−0.0256162
$386$	0	0
$387$	−21.6914	−1.10264
$388$	0	0
$389$	10.4829	0.531503	0.265751	−	0.964042i	$-0.414380\pi$
$389$	10.4829	0.531503	0.265751	+	0.964042i	$0.414380\pi$
$390$	0	0
$391$	−37.8078	−1.91202
$392$	0	0
$393$	0.938856	0.0473590
$394$	0	0
$395$	−0.368036	−0.0185179
$396$	0	0
$397$	36.3419	1.82395	0.911974	−	0.410248i	$-0.134558\pi$
$397$	36.3419	1.82395	0.911974	+	0.410248i	$0.134558\pi$
$398$	0	0
$399$	0.326840	0.0163625
$400$	0	0
$401$	34.5976	1.72772	0.863861	−	0.503731i	$-0.168040\pi$
$401$	34.5976	1.72772	0.863861	+	0.503731i	$0.168040\pi$
$402$	0	0
$403$	−7.28148	−0.362717
$404$	0	0
$405$	3.18067	0.158049
$406$	0	0
$407$	−5.69889	−0.282483
$408$	0	0
$409$	34.0737	1.68484	0.842419	−	0.538823i	$-0.181131\pi$
$409$	34.0737	1.68484	0.842419	+	0.538823i	$0.181131\pi$
$410$	0	0
$411$	1.77411	0.0875106
$412$	0	0
$413$	14.4211	0.709616
$414$	0	0
$415$	3.73782	0.183482
$416$	0	0
$417$	3.52246	0.172496
$418$	0	0
$419$	−17.3630	−0.848238	−0.424119	−	0.905606i	$-0.639416\pi$
$419$	−17.3630	−0.848238	−0.424119	+	0.905606i	$0.639416\pi$
$420$	0	0
$421$	17.2970	0.843004	0.421502	−	0.906827i	$-0.361503\pi$
$421$	17.2970	0.843004	0.421502	+	0.906827i	$0.361503\pi$
$422$	0	0
$423$	−29.3410	−1.42661
$424$	0	0
$425$	37.4427	1.81624
$426$	0	0
$427$	2.65311	0.128393
$428$	0	0
$429$	0.132089	0.00637732
$430$	0	0
$431$	11.9662	0.576392	0.288196	−	0.957571i	$-0.406944\pi$
$431$	11.9662	0.576392	0.288196	+	0.957571i	$0.406944\pi$
$432$	0	0
$433$	−0.306508	−0.0147298	−0.00736491	−	0.999973i	$-0.502344\pi$
$433$	−0.306508	−0.0147298	−0.00736491	+	0.999973i	$0.502344\pi$
$434$	0	0
$435$	−0.571508	−0.0274017
$436$	0	0
$437$	−4.91198	−0.234972
$438$	0	0
$439$	−5.33294	−0.254527	−0.127264	−	0.991869i	$-0.540619\pi$
$439$	−5.33294	−0.254527	−0.127264	+	0.991869i	$0.540619\pi$
$440$	0	0
$441$	12.8001	0.609529
$442$	0	0
$443$	−40.7274	−1.93502	−0.967508	−	0.252839i	$-0.918636\pi$
$443$	−40.7274	−1.93502	−0.967508	+	0.252839i	$0.918636\pi$
$444$	0	0
$445$	0.349055	0.0165468
$446$	0	0
$447$	−2.14592	−0.101499
$448$	0	0
$449$	22.8712	1.07936	0.539679	−	0.841871i	$-0.318546\pi$
$449$	22.8712	1.07936	0.539679	+	0.841871i	$0.318546\pi$
$450$	0	0
$451$	4.64194	0.218580
$452$	0	0
$453$	−2.98386	−0.140194
$454$	0	0
$455$	0.476688	0.0223475
$456$	0	0
$457$	−16.0359	−0.750127	−0.375064	−	0.926999i	$-0.622379\pi$
$457$	−16.0359	−0.750127	−0.375064	+	0.926999i	$0.622379\pi$
$458$	0	0
$459$	−9.16618	−0.427841
$460$	0	0
$461$	−8.13503	−0.378886	−0.189443	−	0.981892i	$-0.560668\pi$
$461$	−8.13503	−0.378886	−0.189443	+	0.981892i	$0.560668\pi$
$462$	0	0
$463$	−5.09937	−0.236988	−0.118494	−	0.992955i	$-0.537807\pi$
$463$	−5.09937	−0.236988	−0.118494	+	0.992955i	$0.537807\pi$
$464$	0	0
$465$	0.676239	0.0313598
$466$	0	0
$467$	6.93225	0.320786	0.160393	−	0.987053i	$-0.448724\pi$
$467$	6.93225	0.320786	0.160393	+	0.987053i	$0.448724\pi$
$468$	0	0
$469$	17.9419	0.828479
$470$	0	0
$471$	1.08146	0.0498309
$472$	0	0
$473$	−6.11810	−0.281311
$474$	0	0
$475$	4.86455	0.223201
$476$	0	0
$477$	1.60819	0.0736341
$478$	0	0
$479$	−6.53767	−0.298714	−0.149357	−	0.988783i	$-0.547720\pi$
$479$	−6.53767	−0.298714	−0.149357	+	0.988783i	$0.547720\pi$
$480$	0	0
$481$	5.40480	0.246438
$482$	0	0
$483$	−1.60543	−0.0730497
$484$	0	0
$485$	4.43773	0.201507
$486$	0	0
$487$	7.22680	0.327478	0.163739	−	0.986504i	$-0.447645\pi$
$487$	7.22680	0.327478	0.163739	+	0.986504i	$0.447645\pi$
$488$	0	0
$489$	−4.44055	−0.200808
$490$	0	0
$491$	−4.46052	−0.201300	−0.100650	−	0.994922i	$-0.532092\pi$
$491$	−4.46052	−0.201300	−0.100650	+	0.994922i	$0.532092\pi$
$492$	0	0
$493$	−59.8198	−2.69415
$494$	0	0
$495$	0.909547	0.0408811
$496$	0	0
$497$	−20.5864	−0.923428
$498$	0	0
$499$	−28.2974	−1.26677	−0.633384	−	0.773838i	$-0.718335\pi$
$499$	−28.2974	−1.26677	−0.633384	+	0.773838i	$0.718335\pi$
$500$	0	0
$501$	3.38118	0.151060
$502$	0	0
$503$	0.190905	0.00851202	0.00425601	−	0.999991i	$-0.498645\pi$
$503$	0.190905	0.00851202	0.00425601	+	0.999991i	$0.498645\pi$
$504$	0	0
$505$	1.02394	0.0455648
$506$	0	0
$507$	2.47223	0.109796
$508$	0	0
$509$	13.8672	0.614653	0.307327	−	0.951604i	$-0.400566\pi$
$509$	13.8672	0.614653	0.307327	+	0.951604i	$0.400566\pi$
$510$	0	0
$511$	13.8417	0.612322
$512$	0	0
$513$	−1.19087	−0.0525782
$514$	0	0
$515$	3.20121	0.141062
$516$	0	0
$517$	−8.27570	−0.363965
$518$	0	0
$519$	1.11265	0.0488397
$520$	0	0
$521$	−0.0288847	−0.00126546	−0.000632730	−	1.00000i	$-0.500201\pi$
$521$	−0.0288847	−0.00126546	−0.000632730		1.00000i	$0.500201\pi$
$522$	0	0
$523$	−28.9209	−1.26462	−0.632311	−	0.774715i	$-0.717893\pi$
$523$	−28.9209	−1.26462	−0.632311	+	0.774715i	$0.717893\pi$
$524$	0	0
$525$	1.58993	0.0693902
$526$	0	0
$527$	70.7819	3.08331
$528$	0	0
$529$	1.12757	0.0490246
$530$	0	0
$531$	−26.0963	−1.13248
$532$	0	0
$533$	−4.40240	−0.190689
$534$	0	0
$535$	4.86153	0.210182
$536$	0	0
$537$	−1.93352	−0.0834377
$538$	0	0
$539$	3.61030	0.155507
$540$	0	0
$541$	19.4812	0.837561	0.418781	−	0.908087i	$-0.362458\pi$
$541$	19.4812	0.837561	0.418781	+	0.908087i	$0.362458\pi$
$542$	0	0
$543$	−0.709805	−0.0304607
$544$	0	0
$545$	1.79708	0.0769784
$546$	0	0
$547$	21.5368	0.920847	0.460423	−	0.887699i	$-0.347698\pi$
$547$	21.5368	0.920847	0.460423	+	0.887699i	$0.347698\pi$
$548$	0	0
$549$	−4.80105	−0.204904
$550$	0	0
$551$	−7.77178	−0.331089
$552$	0	0
$553$	−1.63577	−0.0695601
$554$	0	0
$555$	−0.501949	−0.0213066
$556$	0	0
$557$	34.3623	1.45598	0.727988	−	0.685590i	$-0.240456\pi$
$557$	34.3623	1.45598	0.727988	+	0.685590i	$0.240456\pi$
$558$	0	0
$559$	5.80238	0.245415
$560$	0	0
$561$	−1.28401	−0.0542110
$562$	0	0
$563$	−5.36209	−0.225985	−0.112993	−	0.993596i	$-0.536044\pi$
$563$	−5.36209	−0.225985	−0.112993	+	0.993596i	$0.536044\pi$
$564$	0	0
$565$	2.65533	0.111711
$566$	0	0
$567$	14.1368	0.593690
$568$	0	0
$569$	17.4868	0.733083	0.366541	−	0.930402i	$-0.380542\pi$
$569$	17.4868	0.733083	0.366541	+	0.930402i	$0.380542\pi$
$570$	0	0
$571$	−29.1585	−1.22024	−0.610122	−	0.792307i	$-0.708880\pi$
$571$	−29.1585	−1.22024	−0.610122	+	0.792307i	$0.708880\pi$
$572$	0	0
$573$	1.41947	0.0592994
$574$	0	0
$575$	−23.8946	−0.996473
$576$	0	0
$577$	1.09218	0.0454680	0.0227340	−	0.999742i	$-0.492763\pi$
$577$	1.09218	0.0454680	0.0227340	+	0.999742i	$0.492763\pi$
$578$	0	0
$579$	2.99249	0.124364
$580$	0	0
$581$	16.6131	0.689228
$582$	0	0
$583$	0.453594	0.0187860
$584$	0	0
$585$	−0.862611	−0.0356646
$586$	0	0
$587$	−6.18682	−0.255357	−0.127679	−	0.991816i	$-0.540753\pi$
$587$	−6.18682	−0.255357	−0.127679	+	0.991816i	$0.540753\pi$
$588$	0	0
$589$	9.19598	0.378914
$590$	0	0
$591$	−5.40730	−0.222426
$592$	0	0
$593$	25.1056	1.03096	0.515482	−	0.856900i	$-0.327613\pi$
$593$	25.1056	1.03096	0.515482	+	0.856900i	$0.327613\pi$
$594$	0	0
$595$	−4.63380	−0.189967
$596$	0	0
$597$	−2.06662	−0.0845811
$598$	0	0
$599$	5.53715	0.226242	0.113121	−	0.993581i	$-0.463915\pi$
$599$	5.53715	0.226242	0.113121	+	0.993581i	$0.463915\pi$
$600$	0	0
$601$	10.8556	0.442808	0.221404	−	0.975182i	$-0.428936\pi$
$601$	10.8556	0.442808	0.221404	+	0.975182i	$0.428936\pi$
$602$	0	0
$603$	−32.4675	−1.32218
$604$	0	0
$605$	−3.79185	−0.154161
$606$	0	0
$607$	−2.15324	−0.0873975	−0.0436987	−	0.999045i	$-0.513914\pi$
$607$	−2.15324	−0.0873975	−0.0436987	+	0.999045i	$0.513914\pi$
$608$	0	0
$609$	−2.54013	−0.102931
$610$	0	0
$611$	7.84864	0.317522
$612$	0	0
$613$	−13.8780	−0.560526	−0.280263	−	0.959923i	$-0.590422\pi$
$613$	−13.8780	−0.560526	−0.280263	+	0.959923i	$0.590422\pi$
$614$	0	0
$615$	0.408855	0.0164866
$616$	0	0
$617$	−23.7246	−0.955118	−0.477559	−	0.878600i	$-0.658478\pi$
$617$	−23.7246	−0.955118	−0.477559	+	0.878600i	$0.658478\pi$
$618$	0	0
$619$	−45.1073	−1.81302	−0.906508	−	0.422189i	$-0.861262\pi$
$619$	−45.1073	−1.81302	−0.906508	+	0.422189i	$0.861262\pi$
$620$	0	0
$621$	5.84953	0.234734
$622$	0	0
$623$	1.55141	0.0621559
$624$	0	0
$625$	22.9866	0.919464
$626$	0	0
$627$	−0.166819	−0.00666210
$628$	0	0
$629$	−52.5390	−2.09487
$630$	0	0
$631$	−12.5615	−0.500067	−0.250033	−	0.968237i	$-0.580442\pi$
$631$	−12.5615	−0.500067	−0.250033	+	0.968237i	$0.580442\pi$
$632$	0	0
$633$	−5.05649	−0.200977
$634$	0	0
$635$	0.316852	0.0125739
$636$	0	0
$637$	−3.42399	−0.135664
$638$	0	0
$639$	37.2530	1.47371
$640$	0	0
$641$	−37.6639	−1.48763	−0.743817	−	0.668383i	$-0.766987\pi$
$641$	−37.6639	−1.48763	−0.743817	+	0.668383i	$0.766987\pi$
$642$	0	0
$643$	24.8421	0.979675	0.489838	−	0.871814i	$-0.337056\pi$
$643$	24.8421	0.979675	0.489838	+	0.871814i	$0.337056\pi$
$644$	0	0
$645$	−0.538873	−0.0212181
$646$	0	0
$647$	30.7057	1.20717	0.603583	−	0.797300i	$-0.293739\pi$
$647$	30.7057	1.20717	0.603583	+	0.797300i	$0.293739\pi$
$648$	0	0
$649$	−7.36051	−0.288925
$650$	0	0
$651$	3.00561	0.117799
$652$	0	0
$653$	35.4922	1.38892	0.694459	−	0.719532i	$-0.255644\pi$
$653$	35.4922	1.38892	0.694459	+	0.719532i	$0.255644\pi$
$654$	0	0
$655$	−1.72933	−0.0675703
$656$	0	0
$657$	−25.0479	−0.977210
$658$	0	0
$659$	30.0685	1.17130	0.585651	−	0.810563i	$-0.300839\pi$
$659$	30.0685	1.17130	0.585651	+	0.810563i	$0.300839\pi$
$660$	0	0
$661$	18.7714	0.730122	0.365061	−	0.930984i	$-0.381048\pi$
$661$	18.7714	0.730122	0.365061	+	0.930984i	$0.381048\pi$
$662$	0	0
$663$	1.21775	0.0472936
$664$	0	0
$665$	−0.602022	−0.0233454
$666$	0	0
$667$	38.1748	1.47813
$668$	0	0
$669$	5.82255	0.225113
$670$	0	0
$671$	−1.35415	−0.0522762
$672$	0	0
$673$	−31.8245	−1.22675	−0.613373	−	0.789794i	$-0.710188\pi$
$673$	−31.8245	−1.22675	−0.613373	+	0.789794i	$0.710188\pi$
$674$	0	0
$675$	−5.79305	−0.222974
$676$	0	0
$677$	−18.5185	−0.711723	−0.355862	−	0.934539i	$-0.615813\pi$
$677$	−18.5185	−0.711723	−0.355862	+	0.934539i	$0.615813\pi$
$678$	0	0
$679$	19.7239	0.756935
$680$	0	0
$681$	0.833830	0.0319524
$682$	0	0
$683$	16.1785	0.619053	0.309527	−	0.950891i	$-0.399829\pi$
$683$	16.1785	0.619053	0.309527	+	0.950891i	$0.399829\pi$
$684$	0	0
$685$	−3.26783	−0.124857
$686$	0	0
$687$	3.14891	0.120139
$688$	0	0
$689$	−0.430187	−0.0163888
$690$	0	0
$691$	29.6550	1.12813	0.564065	−	0.825730i	$-0.309237\pi$
$691$	29.6550	1.12813	0.564065	+	0.825730i	$0.309237\pi$
$692$	0	0
$693$	4.04257	0.153565
$694$	0	0
$695$	−6.48820	−0.246111
$696$	0	0
$697$	42.7949	1.62097
$698$	0	0
$699$	−3.69014	−0.139574
$700$	0	0
$701$	−13.4654	−0.508582	−0.254291	−	0.967128i	$-0.581842\pi$
$701$	−13.4654	−0.508582	−0.254291	+	0.967128i	$0.581842\pi$
$702$	0	0
$703$	−6.82586	−0.257442
$704$	0	0
$705$	−0.728911	−0.0274524
$706$	0	0
$707$	4.55102	0.171159
$708$	0	0
$709$	48.1914	1.80986	0.904932	−	0.425556i	$-0.139921\pi$
$709$	48.1914	1.80986	0.904932	+	0.425556i	$0.139921\pi$
$710$	0	0
$711$	2.96008	0.111012
$712$	0	0
$713$	−45.1705	−1.69165
$714$	0	0
$715$	−0.243301	−0.00909895
$716$	0	0
$717$	0.0576927	0.00215457
$718$	0	0
$719$	8.33840	0.310970	0.155485	−	0.987838i	$-0.450306\pi$
$719$	8.33840	0.310970	0.155485	+	0.987838i	$0.450306\pi$
$720$	0	0
$721$	14.2281	0.529881
$722$	0	0
$723$	3.48940	0.129772
$724$	0	0
$725$	−37.8062	−1.40409
$726$	0	0
$727$	30.3168	1.12439	0.562194	−	0.827006i	$-0.309958\pi$
$727$	30.3168	1.12439	0.562194	+	0.827006i	$0.309958\pi$
$728$	0	0
$729$	−24.8680	−0.921037
$730$	0	0
$731$	−56.4039	−2.08617
$732$	0	0
$733$	−11.8013	−0.435889	−0.217945	−	0.975961i	$-0.569935\pi$
$733$	−11.8013	−0.435889	−0.217945	+	0.975961i	$0.569935\pi$
$734$	0	0
$735$	0.317990	0.0117292
$736$	0	0
$737$	−9.15752	−0.337321
$738$	0	0
$739$	41.7684	1.53648	0.768238	−	0.640164i	$-0.221134\pi$
$739$	41.7684	1.53648	0.768238	+	0.640164i	$0.221134\pi$
$740$	0	0
$741$	0.158210	0.00581200
$742$	0	0
$743$	22.4267	0.822756	0.411378	−	0.911465i	$-0.365048\pi$
$743$	22.4267	0.822756	0.411378	+	0.911465i	$0.365048\pi$
$744$	0	0
$745$	3.95268	0.144815
$746$	0	0
$747$	−30.0629	−1.09994
$748$	0	0
$749$	21.6076	0.789523
$750$	0	0
$751$	−33.4452	−1.22043	−0.610216	−	0.792235i	$-0.708917\pi$
$751$	−33.4452	−1.22043	−0.610216	+	0.792235i	$0.708917\pi$
$752$	0	0
$753$	1.06412	0.0387785
$754$	0	0
$755$	5.49612	0.200024
$756$	0	0
$757$	32.5067	1.18148	0.590739	−	0.806863i	$-0.298836\pi$
$757$	32.5067	1.18148	0.590739	+	0.806863i	$0.298836\pi$
$758$	0	0
$759$	0.819410	0.0297427
$760$	0	0
$761$	−8.63621	−0.313062	−0.156531	−	0.987673i	$-0.550031\pi$
$761$	−8.63621	−0.313062	−0.156531	+	0.987673i	$0.550031\pi$
$762$	0	0
$763$	7.98730	0.289160
$764$	0	0
$765$	8.38527	0.303170
$766$	0	0
$767$	6.98068	0.252058
$768$	0	0
$769$	−29.6915	−1.07070	−0.535351	−	0.844630i	$-0.679821\pi$
$769$	−29.6915	−1.07070	−0.535351	+	0.844630i	$0.679821\pi$
$770$	0	0
$771$	−5.47815	−0.197291
$772$	0	0
$773$	−25.7750	−0.927063	−0.463531	−	0.886080i	$-0.653418\pi$
$773$	−25.7750	−0.927063	−0.463531	+	0.886080i	$0.653418\pi$
$774$	0	0
$775$	44.7343	1.60690
$776$	0	0
$777$	−2.23096	−0.0800354
$778$	0	0
$779$	5.55990	0.199204
$780$	0	0
$781$	10.5073	0.375980
$782$	0	0
$783$	9.25517	0.330753
$784$	0	0
$785$	−1.99199	−0.0710971
$786$	0	0
$787$	−8.65415	−0.308487	−0.154244	−	0.988033i	$-0.549294\pi$
$787$	−8.65415	−0.308487	−0.154244	+	0.988033i	$0.549294\pi$
$788$	0	0
$789$	0.600685	0.0213850
$790$	0	0
$791$	11.8019	0.419627
$792$	0	0
$793$	1.28427	0.0456056
$794$	0	0
$795$	0.0399519	0.00141695
$796$	0	0
$797$	26.9202	0.953562	0.476781	−	0.879022i	$-0.341803\pi$
$797$	26.9202	0.953562	0.476781	+	0.879022i	$0.341803\pi$
$798$	0	0
$799$	−76.2951	−2.69913
$800$	0	0
$801$	−2.80742	−0.0991952
$802$	0	0
$803$	−7.06480	−0.249311
$804$	0	0
$805$	2.95712	0.104225
$806$	0	0
$807$	−0.255327	−0.00898794
$808$	0	0
$809$	37.6914	1.32516	0.662580	−	0.748991i	$-0.269461\pi$
$809$	37.6914	1.32516	0.662580	+	0.748991i	$0.269461\pi$
$810$	0	0
$811$	−47.1364	−1.65518	−0.827592	−	0.561331i	$-0.810290\pi$
$811$	−47.1364	−1.65518	−0.827592	+	0.561331i	$0.810290\pi$
$812$	0	0
$813$	5.24393	0.183913
$814$	0	0
$815$	8.17926	0.286507
$816$	0	0
$817$	−7.32798	−0.256374
$818$	0	0
$819$	−3.83396	−0.133969
$820$	0	0
$821$	45.2003	1.57750	0.788750	−	0.614714i	$-0.210728\pi$
$821$	45.2003	1.57750	0.788750	+	0.614714i	$0.210728\pi$
$822$	0	0
$823$	16.6426	0.580126	0.290063	−	0.957008i	$-0.406324\pi$
$823$	16.6426	0.580126	0.290063	+	0.957008i	$0.406324\pi$
$824$	0	0
$825$	−0.811498	−0.0282527
$826$	0	0
$827$	−35.7881	−1.24448	−0.622238	−	0.782828i	$-0.713776\pi$
$827$	−35.7881	−1.24448	−0.622238	+	0.782828i	$0.713776\pi$
$828$	0	0
$829$	−46.7029	−1.62206	−0.811030	−	0.585005i	$-0.801093\pi$
$829$	−46.7029	−1.62206	−0.811030	+	0.585005i	$0.801093\pi$
$830$	0	0
$831$	1.65039	0.0572512
$832$	0	0
$833$	33.2840	1.15322
$834$	0	0
$835$	−6.22796	−0.215527
$836$	0	0
$837$	−10.9512	−0.378529
$838$	0	0
$839$	−41.5492	−1.43444	−0.717219	−	0.696848i	$-0.754585\pi$
$839$	−41.5492	−1.43444	−0.717219	+	0.696848i	$0.754585\pi$
$840$	0	0
$841$	31.4005	1.08278
$842$	0	0
$843$	−4.84797	−0.166973
$844$	0	0
$845$	−4.55372	−0.156653
$846$	0	0
$847$	−16.8533	−0.579085
$848$	0	0
$849$	−2.60771	−0.0894965
$850$	0	0
$851$	33.5285	1.14934
$852$	0	0
$853$	9.16714	0.313877	0.156939	−	0.987608i	$-0.449838\pi$
$853$	9.16714	0.313877	0.156939	+	0.987608i	$0.449838\pi$
$854$	0	0
$855$	1.08941	0.0372572
$856$	0	0
$857$	−0.771816	−0.0263647	−0.0131824	−	0.999913i	$-0.504196\pi$
$857$	−0.771816	−0.0263647	−0.0131824	+	0.999913i	$0.504196\pi$
$858$	0	0
$859$	30.1527	1.02880	0.514399	−	0.857551i	$-0.328015\pi$
$859$	30.1527	1.02880	0.514399	+	0.857551i	$0.328015\pi$
$860$	0	0
$861$	1.81720	0.0619300
$862$	0	0
$863$	−50.1626	−1.70755	−0.853777	−	0.520639i	$-0.825694\pi$
$863$	−50.1626	−1.70755	−0.853777	+	0.520639i	$0.825694\pi$
$864$	0	0
$865$	−2.04944	−0.0696829
$866$	0	0
$867$	−8.44080	−0.286665
$868$	0	0
$869$	0.834896	0.0283219
$870$	0	0
$871$	8.68495	0.294278
$872$	0	0
$873$	−35.6922	−1.20800
$874$	0	0
$875$	−5.93868	−0.200764
$876$	0	0
$877$	−52.6948	−1.77938	−0.889688	−	0.456569i	$-0.849078\pi$
$877$	−52.6948	−1.77938	−0.889688	+	0.456569i	$0.849078\pi$
$878$	0	0
$879$	−6.02879	−0.203346
$880$	0	0
$881$	−12.9221	−0.435358	−0.217679	−	0.976020i	$-0.569849\pi$
$881$	−12.9221	−0.435358	−0.217679	+	0.976020i	$0.569849\pi$
$882$	0	0
$883$	5.56973	0.187436	0.0937182	−	0.995599i	$-0.470125\pi$
$883$	5.56973	0.187436	0.0937182	+	0.995599i	$0.470125\pi$
$884$	0	0
$885$	−0.648303	−0.0217925
$886$	0	0
$887$	44.2487	1.48573	0.742864	−	0.669443i	$-0.233467\pi$
$887$	44.2487	1.48573	0.742864	+	0.669443i	$0.233467\pi$
$888$	0	0
$889$	1.40828	0.0472322
$890$	0	0
$891$	−7.21541	−0.241725
$892$	0	0
$893$	−9.91225	−0.331701
$894$	0	0
$895$	3.56145	0.119046
$896$	0	0
$897$	−0.777125	−0.0259475
$898$	0	0
$899$	−71.4691	−2.38363
$900$	0	0
$901$	4.18176	0.139315
$902$	0	0
$903$	−2.39508	−0.0797032
$904$	0	0
$905$	1.30742	0.0434603
$906$	0	0
$907$	−10.4276	−0.346242	−0.173121	−	0.984901i	$-0.555385\pi$
$907$	−10.4276	−0.346242	−0.173121	+	0.984901i	$0.555385\pi$
$908$	0	0
$909$	−8.23547	−0.273153
$910$	0	0
$911$	−30.7878	−1.02005	−0.510023	−	0.860161i	$-0.670363\pi$
$911$	−30.7878	−1.02005	−0.510023	+	0.860161i	$0.670363\pi$
$912$	0	0
$913$	−8.47931	−0.280624
$914$	0	0
$915$	−0.119271	−0.00394298
$916$	0	0
$917$	−7.68616	−0.253819
$918$	0	0
$919$	−3.34238	−0.110255	−0.0551274	−	0.998479i	$-0.517556\pi$
$919$	−3.34238	−0.110255	−0.0551274	+	0.998479i	$0.517556\pi$
$920$	0	0
$921$	1.42549	0.0469714
$922$	0	0
$923$	−9.96507	−0.328004
$924$	0	0
$925$	−33.2048	−1.09177
$926$	0	0
$927$	−25.7470	−0.845642
$928$	0	0
$929$	−21.4087	−0.702398	−0.351199	−	0.936301i	$-0.614226\pi$
$929$	−21.4087	−0.702398	−0.351199	+	0.936301i	$0.614226\pi$
$930$	0	0
$931$	4.32425	0.141722
$932$	0	0
$933$	5.18215	0.169656
$934$	0	0
$935$	2.36508	0.0773465
$936$	0	0
$937$	−9.75892	−0.318810	−0.159405	−	0.987213i	$-0.550958\pi$
$937$	−9.75892	−0.318810	−0.159405	+	0.987213i	$0.550958\pi$
$938$	0	0
$939$	−5.33253	−0.174021
$940$	0	0
$941$	33.9199	1.10576	0.552878	−	0.833262i	$-0.313529\pi$
$941$	33.9199	1.10576	0.552878	+	0.833262i	$0.313529\pi$
$942$	0	0
$943$	−27.3101	−0.889341
$944$	0	0
$945$	0.716930	0.0233217
$946$	0	0
$947$	−15.8592	−0.515356	−0.257678	−	0.966231i	$-0.582957\pi$
$947$	−15.8592	−0.515356	−0.257678	+	0.966231i	$0.582957\pi$
$948$	0	0
$949$	6.70023	0.217499
$950$	0	0
$951$	2.79576	0.0906586
$952$	0	0
$953$	−2.44446	−0.0791839	−0.0395919	−	0.999216i	$-0.512606\pi$
$953$	−2.44446	−0.0791839	−0.0395919	+	0.999216i	$0.512606\pi$
$954$	0	0
$955$	−2.61460	−0.0846064
$956$	0	0
$957$	1.29648	0.0419092
$958$	0	0
$959$	−14.5242	−0.469010
$960$	0	0
$961$	53.5660	1.72794
$962$	0	0
$963$	−39.1008	−1.26001
$964$	0	0
$965$	−5.51201	−0.177438
$966$	0	0
$967$	35.9304	1.15544	0.577722	−	0.816233i	$-0.303942\pi$
$967$	35.9304	1.15544	0.577722	+	0.816233i	$0.303942\pi$
$968$	0	0
$969$	−1.53793	−0.0494055
$970$	0	0
$971$	−25.7777	−0.827246	−0.413623	−	0.910448i	$-0.635737\pi$
$971$	−25.7777	−0.827246	−0.413623	+	0.910448i	$0.635737\pi$
$972$	0	0
$973$	−28.8375	−0.924486
$974$	0	0
$975$	0.769621	0.0246476
$976$	0	0
$977$	−10.4932	−0.335709	−0.167854	−	0.985812i	$-0.553684\pi$
$977$	−10.4932	−0.335709	−0.167854	+	0.985812i	$0.553684\pi$
$978$	0	0
$979$	−0.791838	−0.0253072
$980$	0	0
$981$	−14.4537	−0.461472
$982$	0	0
$983$	16.0358	0.511462	0.255731	−	0.966748i	$-0.417684\pi$
$983$	16.0358	0.511462	0.255731	+	0.966748i	$0.417684\pi$
$984$	0	0
$985$	9.95996	0.317351
$986$	0	0
$987$	−3.23972	−0.103121
$988$	0	0
$989$	35.9949	1.14457
$990$	0	0
$991$	9.82808	0.312199	0.156100	−	0.987741i	$-0.450108\pi$
$991$	9.82808	0.312199	0.156100	+	0.987741i	$0.450108\pi$
$992$	0	0
$993$	−3.95326	−0.125453
$994$	0	0
$995$	3.80661	0.120678
$996$	0	0
$997$	−15.9579	−0.505393	−0.252696	−	0.967546i	$-0.581317\pi$
$997$	−15.9579	−0.505393	−0.252696	+	0.967546i	$0.581317\pi$
$998$	0	0
$999$	8.12871	0.257181

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.h.1.16	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.h.1.16	✓	31	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q-0.199808 q^{3} +0.368036 q^{5} +1.63577 q^{7} -2.96008 q^{9} +O(q^{10})\)	\(q-0.199808 q^{3} +0.368036 q^{5} +1.63577 q^{7} -2.96008 q^{9} -0.834896 q^{11} +0.791812 q^{13} -0.0735364 q^{15} -7.69705 q^{17} -1.00000 q^{19} -0.326840 q^{21} +4.91198 q^{23} -4.86455 q^{25} +1.19087 q^{27} +7.77178 q^{29} -9.19598 q^{31} +0.166819 q^{33} +0.602022 q^{35} +6.82586 q^{37} -0.158210 q^{39} -5.55990 q^{41} +7.32798 q^{43} -1.08941 q^{45} +9.91225 q^{47} -4.32425 q^{49} +1.53793 q^{51} -0.543294 q^{53} -0.307272 q^{55} +0.199808 q^{57} +8.81608 q^{59} +1.62193 q^{61} -4.84201 q^{63} +0.291415 q^{65} +10.9685 q^{67} -0.981452 q^{69} -12.5851 q^{71} +8.46189 q^{73} +0.971975 q^{75} -1.36570 q^{77} -1.00000 q^{79} +8.64229 q^{81} +10.1561 q^{83} -2.83279 q^{85} -1.55286 q^{87} +0.948427 q^{89} +1.29522 q^{91} +1.83743 q^{93} -0.368036 q^{95} +12.0579 q^{97} +2.47136 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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