LMFDB - Embedded newform 152.4.a.d.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [152,4,Mod(1,152)] 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("152.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 152 = 2^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	152.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$8.96829032087$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 106x^{3} - 401x^{2} + 356x + 2112 $ x^5 - 106x^3 - 401x^2 + 356*x + 2112
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.15594$ of defining polynomial
Character	$\chi$	$=$	152.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+8.62694 q^{3} +4.10677 q^{5} -11.2105 q^{7} +47.4241 q^{9} +71.9427 q^{11} -33.7445 q^{13} +35.4289 q^{15} +18.4226 q^{17} +19.0000 q^{19} -96.7120 q^{21} -86.8656 q^{23} -108.134 q^{25} +176.198 q^{27} +192.838 q^{29} -203.026 q^{31} +620.645 q^{33} -46.0388 q^{35} +281.026 q^{37} -291.112 q^{39} -139.880 q^{41} -70.7620 q^{43} +194.760 q^{45} -275.829 q^{47} -217.326 q^{49} +158.931 q^{51} +296.114 q^{53} +295.452 q^{55} +163.912 q^{57} -884.504 q^{59} -67.8043 q^{61} -531.646 q^{63} -138.581 q^{65} +737.836 q^{67} -749.384 q^{69} -700.052 q^{71} -91.3241 q^{73} -932.869 q^{75} -806.511 q^{77} -1139.23 q^{79} +239.596 q^{81} +71.6383 q^{83} +75.6575 q^{85} +1663.61 q^{87} -296.333 q^{89} +378.292 q^{91} -1751.50 q^{93} +78.0287 q^{95} -28.5351 q^{97} +3411.82 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{3} + 22 q^{7} + 83 q^{9} + 6 q^{11} + 82 q^{13} + 204 q^{15} + 234 q^{17} + 95 q^{19} + 308 q^{21} - 60 q^{23} + 549 q^{25} - 190 q^{27} + 210 q^{29} - 224 q^{31} + 704 q^{33} + 42 q^{35} + 614 q^{37}+ \cdots + 126 q^{99}+O(q^{100}) $ 5 * q + 2 * q^3 + 22 * q^7 + 83 * q^9 + 6 * q^11 + 82 * q^13 + 204 * q^15 + 234 * q^17 + 95 * q^19 + 308 * q^21 - 60 * q^23 + 549 * q^25 - 190 * q^27 + 210 * q^29 - 224 * q^31 + 704 * q^33 + 42 * q^35 + 614 * q^37 - 132 * q^39 + 194 * q^41 - 38 * q^43 - 516 * q^45 - 1066 * q^47 + 489 * q^49 - 1382 * q^51 - 42 * q^53 - 618 * q^55 + 38 * q^57 - 1090 * q^59 - 276 * q^61 - 1790 * q^63 - 184 * q^65 - 314 * q^67 - 268 * q^69 - 1276 * q^71 - 106 * q^73 - 2066 * q^75 - 1226 * q^77 + 52 * q^79 - 283 * q^81 - 580 * q^83 - 3498 * q^85 + 900 * q^87 + 810 * q^89 + 2522 * q^91 - 1332 * q^93 + 2694 * q^97 + 126 * 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
$3$	8.62694	1.66026	0.830128	−	0.557573i	$-0.188267\pi$
$3$	8.62694	1.66026	0.830128	+	0.557573i	$0.188267\pi$
$4$	0	0
$5$	4.10677	0.367321	0.183660	−	0.982990i	$-0.441205\pi$
$5$	4.10677	0.367321	0.183660	+	0.982990i	$0.441205\pi$
$6$	0	0
$7$	−11.2105	−0.605308	−0.302654	−	0.953100i	$-0.597873\pi$
$7$	−11.2105	−0.605308	−0.302654	+	0.953100i	$0.597873\pi$
$8$	0	0
$9$	47.4241	1.75645
$10$	0	0
$11$	71.9427	1.97196	0.985979	−	0.166869i	$-0.0533656\pi$
$11$	71.9427	1.97196	0.985979	+	0.166869i	$0.0533656\pi$
$12$	0	0
$13$	−33.7445	−0.719927	−0.359964	−	0.932966i	$-0.617211\pi$
$13$	−33.7445	−0.719927	−0.359964	+	0.932966i	$0.617211\pi$
$14$	0	0
$15$	35.4289	0.609847
$16$	0	0
$17$	18.4226	0.262832	0.131416	−	0.991327i	$-0.458048\pi$
$17$	18.4226	0.262832	0.131416	+	0.991327i	$0.458048\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	−96.7120	−1.00497
$22$	0	0
$23$	−86.8656	−0.787510	−0.393755	−	0.919216i	$-0.628824\pi$
$23$	−86.8656	−0.787510	−0.393755	+	0.919216i	$0.628824\pi$
$24$	0	0
$25$	−108.134	−0.865075
$26$	0	0
$27$	176.198	1.25590
$28$	0	0
$29$	192.838	1.23480	0.617400	−	0.786649i	$-0.288186\pi$
$29$	192.838	1.23480	0.617400	+	0.786649i	$0.288186\pi$
$30$	0	0
$31$	−203.026	−1.17628	−0.588139	−	0.808760i	$-0.700139\pi$
$31$	−203.026	−1.17628	−0.588139	+	0.808760i	$0.700139\pi$
$32$	0	0
$33$	620.645	3.27395
$34$	0	0
$35$	−46.0388	−0.222342
$36$	0	0
$37$	281.026	1.24866	0.624330	−	0.781161i	$-0.285372\pi$
$37$	281.026	1.24866	0.624330	+	0.781161i	$0.285372\pi$
$38$	0	0
$39$	−291.112	−1.19526
$40$	0	0
$41$	−139.880	−0.532821	−0.266410	−	0.963860i	$-0.585838\pi$
$41$	−139.880	−0.532821	−0.266410	+	0.963860i	$0.585838\pi$
$42$	0	0
$43$	−70.7620	−0.250956	−0.125478	−	0.992096i	$-0.540046\pi$
$43$	−70.7620	−0.250956	−0.125478	+	0.992096i	$0.540046\pi$
$44$	0	0
$45$	194.760	0.645180
$46$	0	0
$47$	−275.829	−0.856038	−0.428019	−	0.903770i	$-0.640788\pi$
$47$	−275.829	−0.856038	−0.428019	+	0.903770i	$0.640788\pi$
$48$	0	0
$49$	−217.326	−0.633602
$50$	0	0
$51$	158.931	0.436368
$52$	0	0
$53$	296.114	0.767441	0.383721	−	0.923449i	$-0.374643\pi$
$53$	296.114	0.767441	0.383721	+	0.923449i	$0.374643\pi$
$54$	0	0
$55$	295.452	0.724342
$56$	0	0
$57$	163.912	0.380889
$58$	0	0
$59$	−884.504	−1.95174	−0.975870	−	0.218353i	$-0.929932\pi$
$59$	−884.504	−1.95174	−0.975870	+	0.218353i	$0.929932\pi$
$60$	0	0
$61$	−67.8043	−0.142319	−0.0711594	−	0.997465i	$-0.522670\pi$
$61$	−67.8043	−0.142319	−0.0711594	+	0.997465i	$0.522670\pi$
$62$	0	0
$63$	−531.646	−1.06319
$64$	0	0
$65$	−138.581	−0.264444
$66$	0	0
$67$	737.836	1.34539	0.672694	−	0.739921i	$-0.265137\pi$
$67$	737.836	1.34539	0.672694	+	0.739921i	$0.265137\pi$
$68$	0	0
$69$	−749.384	−1.30747
$70$	0	0
$71$	−700.052	−1.17015	−0.585077	−	0.810978i	$-0.698936\pi$
$71$	−700.052	−1.17015	−0.585077	+	0.810978i	$0.698936\pi$
$72$	0	0
$73$	−91.3241	−0.146420	−0.0732101	−	0.997317i	$-0.523324\pi$
$73$	−91.3241	−0.146420	−0.0732101	+	0.997317i	$0.523324\pi$
$74$	0	0
$75$	−932.869	−1.43625
$76$	0	0
$77$	−806.511	−1.19364
$78$	0	0
$79$	−1139.23	−1.62245	−0.811225	−	0.584735i	$-0.801199\pi$
$79$	−1139.23	−1.62245	−0.811225	+	0.584735i	$0.801199\pi$
$80$	0	0
$81$	239.596	0.328663
$82$	0	0
$83$	71.6383	0.0947388	0.0473694	−	0.998877i	$-0.484916\pi$
$83$	71.6383	0.0947388	0.0473694	+	0.998877i	$0.484916\pi$
$84$	0	0
$85$	75.6575	0.0965436
$86$	0	0
$87$	1663.61	2.05008
$88$	0	0
$89$	−296.333	−0.352935	−0.176468	−	0.984306i	$-0.556467\pi$
$89$	−296.333	−0.352935	−0.176468	+	0.984306i	$0.556467\pi$
$90$	0	0
$91$	378.292	0.435778
$92$	0	0
$93$	−1751.50	−1.95292
$94$	0	0
$95$	78.0287	0.0842692
$96$	0	0
$97$	−28.5351	−0.0298691	−0.0149346	−	0.999888i	$-0.504754\pi$
$97$	−28.5351	−0.0298691	−0.0149346	+	0.999888i	$0.504754\pi$
$98$	0	0
$99$	3411.82	3.46364
$100$	0	0
$101$	179.849	0.177185	0.0885924	−	0.996068i	$-0.471763\pi$
$101$	179.849	0.177185	0.0885924	+	0.996068i	$0.471763\pi$
$102$	0	0
$103$	−1928.59	−1.84495	−0.922474	−	0.386060i	$-0.873836\pi$
$103$	−1928.59	−1.84495	−0.922474	+	0.386060i	$0.873836\pi$
$104$	0	0
$105$	−397.174	−0.369145
$106$	0	0
$107$	1773.58	1.60242	0.801208	−	0.598385i	$-0.204191\pi$
$107$	1773.58	1.60242	0.801208	+	0.598385i	$0.204191\pi$
$108$	0	0
$109$	1151.75	1.01209	0.506046	−	0.862506i	$-0.331106\pi$
$109$	1151.75	1.01209	0.506046	+	0.862506i	$0.331106\pi$
$110$	0	0
$111$	2424.40	2.07310
$112$	0	0
$113$	1498.88	1.24782	0.623908	−	0.781498i	$-0.285544\pi$
$113$	1498.88	1.24782	0.623908	+	0.781498i	$0.285544\pi$
$114$	0	0
$115$	−356.737	−0.289269
$116$	0	0
$117$	−1600.31	−1.26452
$118$	0	0
$119$	−206.526	−0.159094
$120$	0	0
$121$	3844.75	2.88862
$122$	0	0
$123$	−1206.74	−0.884618
$124$	0	0
$125$	−957.430	−0.685081
$126$	0	0
$127$	1691.58	1.18192	0.590959	−	0.806701i	$-0.298749\pi$
$127$	1691.58	1.18192	0.590959	+	0.806701i	$0.298749\pi$
$128$	0	0
$129$	−610.460	−0.416651
$130$	0	0
$131$	−2520.38	−1.68096	−0.840482	−	0.541840i	$-0.817728\pi$
$131$	−2520.38	−1.68096	−0.840482	+	0.541840i	$0.817728\pi$
$132$	0	0
$133$	−212.999	−0.138867
$134$	0	0
$135$	723.604	0.461318
$136$	0	0
$137$	1180.48	0.736169	0.368084	−	0.929792i	$-0.380014\pi$
$137$	1180.48	0.736169	0.368084	+	0.929792i	$0.380014\pi$
$138$	0	0
$139$	1200.12	0.732326	0.366163	−	0.930551i	$-0.380671\pi$
$139$	1200.12	0.732326	0.366163	+	0.930551i	$0.380671\pi$
$140$	0	0
$141$	−2379.56	−1.42124
$142$	0	0
$143$	−2427.67	−1.41967
$144$	0	0
$145$	791.944	0.453568
$146$	0	0
$147$	−1874.85	−1.05194
$148$	0	0
$149$	−246.291	−0.135416	−0.0677079	−	0.997705i	$-0.521569\pi$
$149$	−246.291	−0.135416	−0.0677079	+	0.997705i	$0.521569\pi$
$150$	0	0
$151$	949.844	0.511902	0.255951	−	0.966690i	$-0.417611\pi$
$151$	949.844	0.511902	0.255951	+	0.966690i	$0.417611\pi$
$152$	0	0
$153$	873.676	0.461650
$154$	0	0
$155$	−833.783	−0.432071
$156$	0	0
$157$	−436.191	−0.221732	−0.110866	−	0.993835i	$-0.535362\pi$
$157$	−436.191	−0.221732	−0.110866	+	0.993835i	$0.535362\pi$
$158$	0	0
$159$	2554.56	1.27415
$160$	0	0
$161$	973.803	0.476686
$162$	0	0
$163$	323.466	0.155434	0.0777172	−	0.996975i	$-0.475237\pi$
$163$	323.466	0.155434	0.0777172	+	0.996975i	$0.475237\pi$
$164$	0	0
$165$	2548.85	1.20259
$166$	0	0
$167$	3428.12	1.58848	0.794238	−	0.607606i	$-0.207870\pi$
$167$	3428.12	1.58848	0.794238	+	0.607606i	$0.207870\pi$
$168$	0	0
$169$	−1058.31	−0.481705
$170$	0	0
$171$	901.058	0.402957
$172$	0	0
$173$	−3268.84	−1.43656	−0.718280	−	0.695754i	$-0.755071\pi$
$173$	−3268.84	−1.43656	−0.718280	+	0.695754i	$0.755071\pi$
$174$	0	0
$175$	1212.24	0.523637
$176$	0	0
$177$	−7630.57	−3.24039
$178$	0	0
$179$	−3900.63	−1.62875	−0.814377	−	0.580336i	$-0.802921\pi$
$179$	−3900.63	−1.62875	−0.814377	+	0.580336i	$0.802921\pi$
$180$	0	0
$181$	3128.05	1.28456	0.642282	−	0.766469i	$-0.277988\pi$
$181$	3128.05	1.28456	0.642282	+	0.766469i	$0.277988\pi$
$182$	0	0
$183$	−584.943	−0.236286
$184$	0	0
$185$	1154.11	0.458659
$186$	0	0
$187$	1325.37	0.518293
$188$	0	0
$189$	−1975.26	−0.760205
$190$	0	0
$191$	2878.40	1.09044	0.545219	−	0.838294i	$-0.316447\pi$
$191$	2878.40	1.09044	0.545219	+	0.838294i	$0.316447\pi$
$192$	0	0
$193$	−617.191	−0.230188	−0.115094	−	0.993355i	$-0.536717\pi$
$193$	−617.191	−0.230188	−0.115094	+	0.993355i	$0.536717\pi$
$194$	0	0
$195$	−1195.53	−0.439045
$196$	0	0
$197$	2850.18	1.03080	0.515399	−	0.856950i	$-0.327644\pi$
$197$	2850.18	1.03080	0.515399	+	0.856950i	$0.327644\pi$
$198$	0	0
$199$	988.062	0.351969	0.175985	−	0.984393i	$-0.443689\pi$
$199$	988.062	0.351969	0.175985	+	0.984393i	$0.443689\pi$
$200$	0	0
$201$	6365.27	2.23369
$202$	0	0
$203$	−2161.81	−0.747434
$204$	0	0
$205$	−574.457	−0.195716
$206$	0	0
$207$	−4119.52	−1.38322
$208$	0	0
$209$	1366.91	0.452398
$210$	0	0
$211$	2249.02	0.733786	0.366893	−	0.930263i	$-0.380422\pi$
$211$	2249.02	0.733786	0.366893	+	0.930263i	$0.380422\pi$
$212$	0	0
$213$	−6039.31	−1.94275
$214$	0	0
$215$	−290.604	−0.0921814
$216$	0	0
$217$	2276.02	0.712010
$218$	0	0
$219$	−787.848	−0.243095
$220$	0	0
$221$	−621.662	−0.189220
$222$	0	0
$223$	5406.85	1.62363	0.811814	−	0.583916i	$-0.198480\pi$
$223$	5406.85	1.62363	0.811814	+	0.583916i	$0.198480\pi$
$224$	0	0
$225$	−5128.18	−1.51946
$226$	0	0
$227$	−352.967	−0.103204	−0.0516019	−	0.998668i	$-0.516433\pi$
$227$	−352.967	−0.103204	−0.0516019	+	0.998668i	$0.516433\pi$
$228$	0	0
$229$	767.432	0.221456	0.110728	−	0.993851i	$-0.464682\pi$
$229$	767.432	0.221456	0.110728	+	0.993851i	$0.464682\pi$
$230$	0	0
$231$	−6957.72	−1.98175
$232$	0	0
$233$	−755.415	−0.212399	−0.106199	−	0.994345i	$-0.533868\pi$
$233$	−755.415	−0.212399	−0.106199	+	0.994345i	$0.533868\pi$
$234$	0	0
$235$	−1132.77	−0.314441
$236$	0	0
$237$	−9828.08	−2.69368
$238$	0	0
$239$	−5257.18	−1.42284	−0.711420	−	0.702767i	$-0.751947\pi$
$239$	−5257.18	−1.42284	−0.711420	+	0.702767i	$0.751947\pi$
$240$	0	0
$241$	−643.756	−0.172066	−0.0860331	−	0.996292i	$-0.527419\pi$
$241$	−643.756	−0.172066	−0.0860331	+	0.996292i	$0.527419\pi$
$242$	0	0
$243$	−2690.36	−0.710233
$244$	0	0
$245$	−892.507	−0.232735
$246$	0	0
$247$	−641.146	−0.165163
$248$	0	0
$249$	618.019	0.157291
$250$	0	0
$251$	3464.64	0.871258	0.435629	−	0.900126i	$-0.356526\pi$
$251$	3464.64	0.871258	0.435629	+	0.900126i	$0.356526\pi$
$252$	0	0
$253$	−6249.34	−1.55294
$254$	0	0
$255$	652.692	0.160287
$256$	0	0
$257$	3303.56	0.801831	0.400915	−	0.916115i	$-0.368692\pi$
$257$	3303.56	0.801831	0.400915	+	0.916115i	$0.368692\pi$
$258$	0	0
$259$	−3150.43	−0.755824
$260$	0	0
$261$	9145.19	2.16886
$262$	0	0
$263$	612.607	0.143631	0.0718155	−	0.997418i	$-0.477121\pi$
$263$	612.607	0.143631	0.0718155	+	0.997418i	$0.477121\pi$
$264$	0	0
$265$	1216.07	0.281897
$266$	0	0
$267$	−2556.45	−0.585963
$268$	0	0
$269$	4683.09	1.06146	0.530731	−	0.847540i	$-0.321918\pi$
$269$	4683.09	1.06146	0.530731	+	0.847540i	$0.321918\pi$
$270$	0	0
$271$	3816.91	0.855574	0.427787	−	0.903880i	$-0.359293\pi$
$271$	3816.91	0.855574	0.427787	+	0.903880i	$0.359293\pi$
$272$	0	0
$273$	3263.50	0.723502
$274$	0	0
$275$	−7779.48	−1.70589
$276$	0	0
$277$	8588.70	1.86298	0.931489	−	0.363770i	$-0.118511\pi$
$277$	8588.70	1.86298	0.931489	+	0.363770i	$0.118511\pi$
$278$	0	0
$279$	−9628.34	−2.06607
$280$	0	0
$281$	4233.53	0.898759	0.449380	−	0.893341i	$-0.351645\pi$
$281$	4233.53	0.898759	0.449380	+	0.893341i	$0.351645\pi$
$282$	0	0
$283$	7951.72	1.67025	0.835125	−	0.550061i	$-0.185395\pi$
$283$	7951.72	1.67025	0.835125	+	0.550061i	$0.185395\pi$
$284$	0	0
$285$	673.149	0.139908
$286$	0	0
$287$	1568.12	0.322521
$288$	0	0
$289$	−4573.61	−0.930920
$290$	0	0
$291$	−246.171	−0.0495904
$292$	0	0
$293$	8691.51	1.73298	0.866491	−	0.499193i	$-0.166370\pi$
$293$	8691.51	1.73298	0.866491	+	0.499193i	$0.166370\pi$
$294$	0	0
$295$	−3632.46	−0.716915
$296$	0	0
$297$	12676.1	2.47658
$298$	0	0
$299$	2931.24	0.566950
$300$	0	0
$301$	793.275	0.151906
$302$	0	0
$303$	1551.55	0.294172
$304$	0	0
$305$	−278.457	−0.0522767
$306$	0	0
$307$	1816.21	0.337644	0.168822	−	0.985647i	$-0.446004\pi$
$307$	1816.21	0.337644	0.168822	+	0.985647i	$0.446004\pi$
$308$	0	0
$309$	−16637.8	−3.06308
$310$	0	0
$311$	−10503.2	−1.91505	−0.957526	−	0.288347i	$-0.906894\pi$
$311$	−10503.2	−1.91505	−0.957526	+	0.288347i	$0.906894\pi$
$312$	0	0
$313$	4617.82	0.833912	0.416956	−	0.908927i	$-0.363097\pi$
$313$	4617.82	0.833912	0.416956	+	0.908927i	$0.363097\pi$
$314$	0	0
$315$	−2183.35	−0.390533
$316$	0	0
$317$	−999.131	−0.177025	−0.0885123	−	0.996075i	$-0.528211\pi$
$317$	−999.131	−0.177025	−0.0885123	+	0.996075i	$0.528211\pi$
$318$	0	0
$319$	13873.3	2.43497
$320$	0	0
$321$	15300.6	2.66042
$322$	0	0
$323$	350.029	0.0602977
$324$	0	0
$325$	3648.95	0.622791
$326$	0	0
$327$	9936.12	1.68033
$328$	0	0
$329$	3092.17	0.518166
$330$	0	0
$331$	−8285.70	−1.37590	−0.687951	−	0.725757i	$-0.741490\pi$
$331$	−8285.70	−1.37590	−0.687951	+	0.725757i	$0.741490\pi$
$332$	0	0
$333$	13327.4	2.19321
$334$	0	0
$335$	3030.13	0.494189
$336$	0	0
$337$	−6313.43	−1.02052	−0.510259	−	0.860021i	$-0.670450\pi$
$337$	−6313.43	−1.02052	−0.510259	+	0.860021i	$0.670450\pi$
$338$	0	0
$339$	12930.8	2.07169
$340$	0	0
$341$	−14606.3	−2.31957
$342$	0	0
$343$	6281.51	0.988832
$344$	0	0
$345$	−3077.55	−0.480260
$346$	0	0
$347$	1267.35	0.196066	0.0980331	−	0.995183i	$-0.468745\pi$
$347$	1267.35	0.196066	0.0980331	+	0.995183i	$0.468745\pi$
$348$	0	0
$349$	−5488.56	−0.841823	−0.420911	−	0.907102i	$-0.638290\pi$
$349$	−5488.56	−0.841823	−0.420911	+	0.907102i	$0.638290\pi$
$350$	0	0
$351$	−5945.71	−0.904155
$352$	0	0
$353$	−9253.29	−1.39519	−0.697596	−	0.716491i	$-0.745747\pi$
$353$	−9253.29	−1.39519	−0.697596	+	0.716491i	$0.745747\pi$
$354$	0	0
$355$	−2874.96	−0.429822
$356$	0	0
$357$	−1781.69	−0.264137
$358$	0	0
$359$	−12158.7	−1.78750	−0.893748	−	0.448569i	$-0.851934\pi$
$359$	−12158.7	−1.78750	−0.893748	+	0.448569i	$0.851934\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	33168.5	4.79585
$364$	0	0
$365$	−375.047	−0.0537832
$366$	0	0
$367$	−8906.82	−1.26685	−0.633423	−	0.773806i	$-0.718351\pi$
$367$	−8906.82	−1.26685	−0.633423	+	0.773806i	$0.718351\pi$
$368$	0	0
$369$	−6633.70	−0.935872
$370$	0	0
$371$	−3319.57	−0.464538
$372$	0	0
$373$	−465.593	−0.0646314	−0.0323157	−	0.999478i	$-0.510288\pi$
$373$	−465.593	−0.0646314	−0.0323157	+	0.999478i	$0.510288\pi$
$374$	0	0
$375$	−8259.69	−1.13741
$376$	0	0
$377$	−6507.25	−0.888966
$378$	0	0
$379$	−6221.96	−0.843274	−0.421637	−	0.906765i	$-0.638544\pi$
$379$	−6221.96	−0.843274	−0.421637	+	0.906765i	$0.638544\pi$
$380$	0	0
$381$	14593.2	1.96229
$382$	0	0
$383$	−9367.45	−1.24975	−0.624875	−	0.780725i	$-0.714850\pi$
$383$	−9367.45	−1.24975	−0.624875	+	0.780725i	$0.714850\pi$
$384$	0	0
$385$	−3312.16	−0.438450
$386$	0	0
$387$	−3355.83	−0.440791
$388$	0	0
$389$	1205.29	0.157097	0.0785487	−	0.996910i	$-0.474971\pi$
$389$	1205.29	0.157097	0.0785487	+	0.996910i	$0.474971\pi$
$390$	0	0
$391$	−1600.29	−0.206982
$392$	0	0
$393$	−21743.1	−2.79083
$394$	0	0
$395$	−4678.56	−0.595960
$396$	0	0
$397$	−3135.03	−0.396329	−0.198164	−	0.980169i	$-0.563498\pi$
$397$	−3135.03	−0.396329	−0.198164	+	0.980169i	$0.563498\pi$
$398$	0	0
$399$	−1837.53	−0.230555
$400$	0	0
$401$	2780.30	0.346238	0.173119	−	0.984901i	$-0.444615\pi$
$401$	2780.30	0.346238	0.173119	+	0.984901i	$0.444615\pi$
$402$	0	0
$403$	6851.03	0.846834
$404$	0	0
$405$	983.965	0.120725
$406$	0	0
$407$	20217.8	2.46231
$408$	0	0
$409$	−8405.57	−1.01621	−0.508103	−	0.861296i	$-0.669653\pi$
$409$	−8405.57	−1.01621	−0.508103	+	0.861296i	$0.669653\pi$
$410$	0	0
$411$	10183.9	1.22223
$412$	0	0
$413$	9915.70	1.18140
$414$	0	0
$415$	294.202	0.0347996
$416$	0	0
$417$	10353.4	1.21585
$418$	0	0
$419$	9077.31	1.05837	0.529183	−	0.848508i	$-0.322498\pi$
$419$	9077.31	1.05837	0.529183	+	0.848508i	$0.322498\pi$
$420$	0	0
$421$	1790.52	0.207280	0.103640	−	0.994615i	$-0.466951\pi$
$421$	1790.52	0.207280	0.103640	+	0.994615i	$0.466951\pi$
$422$	0	0
$423$	−13080.9	−1.50359
$424$	0	0
$425$	−1992.12	−0.227369
$426$	0	0
$427$	760.117	0.0861467
$428$	0	0
$429$	−20943.4	−2.35701
$430$	0	0
$431$	−2415.56	−0.269962	−0.134981	−	0.990848i	$-0.543097\pi$
$431$	−2415.56	−0.269962	−0.134981	+	0.990848i	$0.543097\pi$
$432$	0	0
$433$	−16288.6	−1.80780	−0.903901	−	0.427741i	$-0.859310\pi$
$433$	−16288.6	−1.80780	−0.903901	+	0.427741i	$0.859310\pi$
$434$	0	0
$435$	6832.05	0.753039
$436$	0	0
$437$	−1650.45	−0.180667
$438$	0	0
$439$	13844.1	1.50511	0.752555	−	0.658529i	$-0.228821\pi$
$439$	13844.1	1.50511	0.752555	+	0.658529i	$0.228821\pi$
$440$	0	0
$441$	−10306.5	−1.11289
$442$	0	0
$443$	−268.946	−0.0288442	−0.0144221	−	0.999896i	$-0.504591\pi$
$443$	−268.946	−0.0288442	−0.0144221	+	0.999896i	$0.504591\pi$
$444$	0	0
$445$	−1216.97	−0.129641
$446$	0	0
$447$	−2124.74	−0.224825
$448$	0	0
$449$	8651.55	0.909336	0.454668	−	0.890661i	$-0.349758\pi$
$449$	8651.55	0.909336	0.454668	+	0.890661i	$0.349758\pi$
$450$	0	0
$451$	−10063.4	−1.05070
$452$	0	0
$453$	8194.25	0.849888
$454$	0	0
$455$	1553.56	0.160070
$456$	0	0
$457$	5394.14	0.552138	0.276069	−	0.961138i	$-0.410968\pi$
$457$	5394.14	0.552138	0.276069	+	0.961138i	$0.410968\pi$
$458$	0	0
$459$	3246.02	0.330090
$460$	0	0
$461$	−8578.19	−0.866651	−0.433325	−	0.901238i	$-0.642660\pi$
$461$	−8578.19	−0.866651	−0.433325	+	0.901238i	$0.642660\pi$
$462$	0	0
$463$	−3657.01	−0.367074	−0.183537	−	0.983013i	$-0.558755\pi$
$463$	−3657.01	−0.367074	−0.183537	+	0.983013i	$0.558755\pi$
$464$	0	0
$465$	−7193.00	−0.717349
$466$	0	0
$467$	−522.490	−0.0517729	−0.0258865	−	0.999665i	$-0.508241\pi$
$467$	−522.490	−0.0517729	−0.0258865	+	0.999665i	$0.508241\pi$
$468$	0	0
$469$	−8271.48	−0.814374
$470$	0	0
$471$	−3763.00	−0.368131
$472$	0	0
$473$	−5090.81	−0.494875
$474$	0	0
$475$	−2054.55	−0.198462
$476$	0	0
$477$	14042.9	1.34797
$478$	0	0
$479$	1053.20	0.100464	0.0502318	−	0.998738i	$-0.484004\pi$
$479$	1053.20	0.100464	0.0502318	+	0.998738i	$0.484004\pi$
$480$	0	0
$481$	−9483.10	−0.898945
$482$	0	0
$483$	8400.94	0.791420
$484$	0	0
$485$	−117.187	−0.0109716
$486$	0	0
$487$	1658.43	0.154314	0.0771568	−	0.997019i	$-0.475416\pi$
$487$	1658.43	0.154314	0.0771568	+	0.997019i	$0.475416\pi$
$488$	0	0
$489$	2790.52	0.258061
$490$	0	0
$491$	−13223.7	−1.21543	−0.607714	−	0.794156i	$-0.707913\pi$
$491$	−13223.7	−1.21543	−0.607714	+	0.794156i	$0.707913\pi$
$492$	0	0
$493$	3552.59	0.324545
$494$	0	0
$495$	14011.6	1.27227
$496$	0	0
$497$	7847.91	0.708304
$498$	0	0
$499$	−17102.5	−1.53429	−0.767145	−	0.641474i	$-0.778323\pi$
$499$	−17102.5	−1.53429	−0.767145	+	0.641474i	$0.778323\pi$
$500$	0	0
$501$	29574.2	2.63728
$502$	0	0
$503$	15576.3	1.38074	0.690369	−	0.723457i	$-0.257448\pi$
$503$	15576.3	1.38074	0.690369	+	0.723457i	$0.257448\pi$
$504$	0	0
$505$	738.600	0.0650837
$506$	0	0
$507$	−9129.94	−0.799753
$508$	0	0
$509$	12121.3	1.05554	0.527769	−	0.849388i	$-0.323029\pi$
$509$	12121.3	1.05554	0.527769	+	0.849388i	$0.323029\pi$
$510$	0	0
$511$	1023.79	0.0886293
$512$	0	0
$513$	3347.76	0.288123
$514$	0	0
$515$	−7920.28	−0.677688
$516$	0	0
$517$	−19843.9	−1.68807
$518$	0	0
$519$	−28200.1	−2.38506
$520$	0	0
$521$	9281.30	0.780463	0.390232	−	0.920717i	$-0.372395\pi$
$521$	9281.30	0.780463	0.390232	+	0.920717i	$0.372395\pi$
$522$	0	0
$523$	67.9208	0.00567872	0.00283936	−	0.999996i	$-0.499096\pi$
$523$	67.9208	0.00567872	0.00283936	+	0.999996i	$0.499096\pi$
$524$	0	0
$525$	10457.9	0.869371
$526$	0	0
$527$	−3740.27	−0.309163
$528$	0	0
$529$	−4621.38	−0.379829
$530$	0	0
$531$	−41946.8	−3.42813
$532$	0	0
$533$	4720.20	0.383592
$534$	0	0
$535$	7283.70	0.588601
$536$	0	0
$537$	−33650.5	−2.70415
$538$	0	0
$539$	−15635.0	−1.24944
$540$	0	0
$541$	−15455.6	−1.22826	−0.614130	−	0.789205i	$-0.710493\pi$
$541$	−15455.6	−1.22826	−0.614130	+	0.789205i	$0.710493\pi$
$542$	0	0
$543$	26985.5	2.13270
$544$	0	0
$545$	4730.00	0.371763
$546$	0	0
$547$	2401.02	0.187679	0.0938393	−	0.995587i	$-0.470086\pi$
$547$	2401.02	0.187679	0.0938393	+	0.995587i	$0.470086\pi$
$548$	0	0
$549$	−3215.56	−0.249976
$550$	0	0
$551$	3663.93	0.283283
$552$	0	0
$553$	12771.3	0.982081
$554$	0	0
$555$	9956.45	0.761491
$556$	0	0
$557$	−10802.4	−0.821743	−0.410871	−	0.911693i	$-0.634775\pi$
$557$	−10802.4	−0.821743	−0.410871	+	0.911693i	$0.634775\pi$
$558$	0	0
$559$	2387.83	0.180670
$560$	0	0
$561$	11433.9	0.860499
$562$	0	0
$563$	4215.18	0.315540	0.157770	−	0.987476i	$-0.449570\pi$
$563$	4215.18	0.315540	0.157770	+	0.987476i	$0.449570\pi$
$564$	0	0
$565$	6155.58	0.458349
$566$	0	0
$567$	−2685.98	−0.198943
$568$	0	0
$569$	22102.0	1.62841	0.814205	−	0.580577i	$-0.197173\pi$
$569$	22102.0	1.62841	0.814205	+	0.580577i	$0.197173\pi$
$570$	0	0
$571$	10430.8	0.764475	0.382238	−	0.924064i	$-0.375154\pi$
$571$	10430.8	0.764475	0.382238	+	0.924064i	$0.375154\pi$
$572$	0	0
$573$	24831.8	1.81040
$574$	0	0
$575$	9393.16	0.681255
$576$	0	0
$577$	−4064.22	−0.293234	−0.146617	−	0.989193i	$-0.546838\pi$
$577$	−4064.22	−0.293234	−0.146617	+	0.989193i	$0.546838\pi$
$578$	0	0
$579$	−5324.47	−0.382172
$580$	0	0
$581$	−803.098	−0.0573462
$582$	0	0
$583$	21303.2	1.51336
$584$	0	0
$585$	−6572.09	−0.464483
$586$	0	0
$587$	−2114.64	−0.148689	−0.0743444	−	0.997233i	$-0.523686\pi$
$587$	−2114.64	−0.148689	−0.0743444	+	0.997233i	$0.523686\pi$
$588$	0	0
$589$	−3857.50	−0.269857
$590$	0	0
$591$	24588.4	1.71139
$592$	0	0
$593$	−9666.66	−0.669414	−0.334707	−	0.942322i	$-0.608637\pi$
$593$	−9666.66	−0.669414	−0.334707	+	0.942322i	$0.608637\pi$
$594$	0	0
$595$	−848.155	−0.0584386
$596$	0	0
$597$	8523.95	0.584359
$598$	0	0
$599$	12610.6	0.860195	0.430097	−	0.902783i	$-0.358479\pi$
$599$	12610.6	0.860195	0.430097	+	0.902783i	$0.358479\pi$
$600$	0	0
$601$	14326.9	0.972389	0.486194	−	0.873851i	$-0.338385\pi$
$601$	14326.9	0.972389	0.486194	+	0.873851i	$0.338385\pi$
$602$	0	0
$603$	34991.2	2.36311
$604$	0	0
$605$	15789.5	1.06105
$606$	0	0
$607$	9058.13	0.605697	0.302848	−	0.953039i	$-0.402062\pi$
$607$	9058.13	0.605697	0.302848	+	0.953039i	$0.402062\pi$
$608$	0	0
$609$	−18649.8	−1.24093
$610$	0	0
$611$	9307.72	0.616285
$612$	0	0
$613$	25609.2	1.68735	0.843674	−	0.536856i	$-0.180388\pi$
$613$	25609.2	1.68735	0.843674	+	0.536856i	$0.180388\pi$
$614$	0	0
$615$	−4955.81	−0.324939
$616$	0	0
$617$	−13536.5	−0.883238	−0.441619	−	0.897203i	$-0.645596\pi$
$617$	−13536.5	−0.883238	−0.441619	+	0.897203i	$0.645596\pi$
$618$	0	0
$619$	−3618.91	−0.234986	−0.117493	−	0.993074i	$-0.537486\pi$
$619$	−3618.91	−0.234986	−0.117493	+	0.993074i	$0.537486\pi$
$620$	0	0
$621$	−15305.5	−0.989032
$622$	0	0
$623$	3322.03	0.213635
$624$	0	0
$625$	9584.85	0.613431
$626$	0	0
$627$	11792.3	0.751097
$628$	0	0
$629$	5177.24	0.328187
$630$	0	0
$631$	−15012.4	−0.947125	−0.473562	−	0.880760i	$-0.657032\pi$
$631$	−15012.4	−0.947125	−0.473562	+	0.880760i	$0.657032\pi$
$632$	0	0
$633$	19402.2	1.21827
$634$	0	0
$635$	6946.95	0.434144
$636$	0	0
$637$	7333.55	0.456147
$638$	0	0
$639$	−33199.4	−2.05532
$640$	0	0
$641$	13969.4	0.860778	0.430389	−	0.902644i	$-0.358376\pi$
$641$	13969.4	0.860778	0.430389	+	0.902644i	$0.358376\pi$
$642$	0	0
$643$	−22659.9	−1.38977	−0.694883	−	0.719122i	$-0.744544\pi$
$643$	−22659.9	−1.38977	−0.694883	+	0.719122i	$0.744544\pi$
$644$	0	0
$645$	−2507.02	−0.153045
$646$	0	0
$647$	−15175.9	−0.922145	−0.461073	−	0.887362i	$-0.652535\pi$
$647$	−15175.9	−0.922145	−0.461073	+	0.887362i	$0.652535\pi$
$648$	0	0
$649$	−63633.6	−3.84875
$650$	0	0
$651$	19635.1	1.18212
$652$	0	0
$653$	17656.6	1.05813	0.529063	−	0.848582i	$-0.322543\pi$
$653$	17656.6	1.05813	0.529063	+	0.848582i	$0.322543\pi$
$654$	0	0
$655$	−10350.6	−0.617453
$656$	0	0
$657$	−4330.96	−0.257180
$658$	0	0
$659$	−22595.1	−1.33563	−0.667815	−	0.744327i	$-0.732770\pi$
$659$	−22595.1	−1.33563	−0.667815	+	0.744327i	$0.732770\pi$
$660$	0	0
$661$	6333.17	0.372665	0.186333	−	0.982487i	$-0.440340\pi$
$661$	6333.17	0.372665	0.186333	+	0.982487i	$0.440340\pi$
$662$	0	0
$663$	−5363.04	−0.314153
$664$	0	0
$665$	−874.738	−0.0510088
$666$	0	0
$667$	−16751.0	−0.972417
$668$	0	0
$669$	46644.5	2.69564
$670$	0	0
$671$	−4878.02	−0.280647
$672$	0	0
$673$	−8536.63	−0.488949	−0.244475	−	0.969656i	$-0.578615\pi$
$673$	−8536.63	−0.488949	−0.244475	+	0.969656i	$0.578615\pi$
$674$	0	0
$675$	−19053.0	−1.08645
$676$	0	0
$677$	−8711.29	−0.494538	−0.247269	−	0.968947i	$-0.579533\pi$
$677$	−8711.29	−0.494538	−0.247269	+	0.968947i	$0.579533\pi$
$678$	0	0
$679$	319.892	0.0180800
$680$	0	0
$681$	−3045.03	−0.171345
$682$	0	0
$683$	14190.3	0.794987	0.397493	−	0.917605i	$-0.369880\pi$
$683$	14190.3	0.794987	0.397493	+	0.917605i	$0.369880\pi$
$684$	0	0
$685$	4847.96	0.270410
$686$	0	0
$687$	6620.59	0.367673
$688$	0	0
$689$	−9992.23	−0.552502
$690$	0	0
$691$	222.886	0.0122706	0.00613530	−	0.999981i	$-0.498047\pi$
$691$	222.886	0.0122706	0.00613530	+	0.999981i	$0.498047\pi$
$692$	0	0
$693$	−38248.1	−2.09657
$694$	0	0
$695$	4928.64	0.268999
$696$	0	0
$697$	−2576.96	−0.140042
$698$	0	0
$699$	−6516.92	−0.352636
$700$	0	0
$701$	17181.5	0.925727	0.462864	−	0.886430i	$-0.346822\pi$
$701$	17181.5	0.925727	0.462864	+	0.886430i	$0.346822\pi$
$702$	0	0
$703$	5339.50	0.286462
$704$	0	0
$705$	−9772.31	−0.522052
$706$	0	0
$707$	−2016.19	−0.107251
$708$	0	0
$709$	12842.1	0.680247	0.340124	−	0.940381i	$-0.389531\pi$
$709$	12842.1	0.680247	0.340124	+	0.940381i	$0.389531\pi$
$710$	0	0
$711$	−54027.0	−2.84975
$712$	0	0
$713$	17636.0	0.926330
$714$	0	0
$715$	−9969.91	−0.521473
$716$	0	0
$717$	−45353.4	−2.36228
$718$	0	0
$719$	25006.6	1.29706	0.648532	−	0.761187i	$-0.275383\pi$
$719$	25006.6	1.29706	0.648532	+	0.761187i	$0.275383\pi$
$720$	0	0
$721$	21620.4	1.11676
$722$	0	0
$723$	−5553.64	−0.285674
$724$	0	0
$725$	−20852.5	−1.06820
$726$	0	0
$727$	−17760.1	−0.906030	−0.453015	−	0.891503i	$-0.649652\pi$
$727$	−17760.1	−0.906030	−0.453015	+	0.891503i	$0.649652\pi$
$728$	0	0
$729$	−29678.7	−1.50783
$730$	0	0
$731$	−1303.62	−0.0659592
$732$	0	0
$733$	1998.35	0.100697	0.0503483	−	0.998732i	$-0.483967\pi$
$733$	1998.35	0.100697	0.0503483	+	0.998732i	$0.483967\pi$
$734$	0	0
$735$	−7699.60	−0.386400
$736$	0	0
$737$	53081.9	2.65305
$738$	0	0
$739$	−24084.1	−1.19885	−0.599423	−	0.800433i	$-0.704603\pi$
$739$	−24084.1	−1.19885	−0.599423	+	0.800433i	$0.704603\pi$
$740$	0	0
$741$	−5531.13	−0.274212
$742$	0	0
$743$	−21478.3	−1.06051	−0.530257	−	0.847837i	$-0.677905\pi$
$743$	−21478.3	−1.06051	−0.530257	+	0.847837i	$0.677905\pi$
$744$	0	0
$745$	−1011.46	−0.0497411
$746$	0	0
$747$	3397.38	0.166404
$748$	0	0
$749$	−19882.7	−0.969956
$750$	0	0
$751$	−14198.0	−0.689868	−0.344934	−	0.938627i	$-0.612099\pi$
$751$	−14198.0	−0.689868	−0.344934	+	0.938627i	$0.612099\pi$
$752$	0	0
$753$	29889.2	1.44651
$754$	0	0
$755$	3900.79	0.188032
$756$	0	0
$757$	36489.7	1.75197	0.875984	−	0.482339i	$-0.160213\pi$
$757$	36489.7	1.75197	0.875984	+	0.482339i	$0.160213\pi$
$758$	0	0
$759$	−53912.7	−2.57827
$760$	0	0
$761$	−785.209	−0.0374032	−0.0187016	−	0.999825i	$-0.505953\pi$
$761$	−785.209	−0.0374032	−0.0187016	+	0.999825i	$0.505953\pi$
$762$	0	0
$763$	−12911.7	−0.612628
$764$	0	0
$765$	3587.99	0.169574
$766$	0	0
$767$	29847.2	1.40511
$768$	0	0
$769$	−535.416	−0.0251074	−0.0125537	−	0.999921i	$-0.503996\pi$
$769$	−535.416	−0.0251074	−0.0125537	+	0.999921i	$0.503996\pi$
$770$	0	0
$771$	28499.6	1.33124
$772$	0	0
$773$	7042.02	0.327664	0.163832	−	0.986488i	$-0.447615\pi$
$773$	7042.02	0.327664	0.163832	+	0.986488i	$0.447615\pi$
$774$	0	0
$775$	21954.1	1.01757
$776$	0	0
$777$	−27178.6	−1.25486
$778$	0	0
$779$	−2657.73	−0.122237
$780$	0	0
$781$	−50363.7	−2.30749
$782$	0	0
$783$	33977.7	1.55078
$784$	0	0
$785$	−1791.34	−0.0814466
$786$	0	0
$787$	12781.6	0.578927	0.289463	−	0.957189i	$-0.406523\pi$
$787$	12781.6	0.578927	0.289463	+	0.957189i	$0.406523\pi$
$788$	0	0
$789$	5284.93	0.238464
$790$	0	0
$791$	−16803.2	−0.755313
$792$	0	0
$793$	2288.02	0.102459
$794$	0	0
$795$	10491.0	0.468022
$796$	0	0
$797$	8827.50	0.392329	0.196164	−	0.980571i	$-0.437151\pi$
$797$	8827.50	0.392329	0.196164	+	0.980571i	$0.437151\pi$
$798$	0	0
$799$	−5081.48	−0.224994
$800$	0	0
$801$	−14053.3	−0.619913
$802$	0	0
$803$	−6570.10	−0.288735
$804$	0	0
$805$	3999.19	0.175097
$806$	0	0
$807$	40400.8	1.76230
$808$	0	0
$809$	30681.6	1.33338	0.666691	−	0.745334i	$-0.267710\pi$
$809$	30681.6	1.33338	0.666691	+	0.745334i	$0.267710\pi$
$810$	0	0
$811$	17311.7	0.749565	0.374783	−	0.927113i	$-0.377717\pi$
$811$	17311.7	0.749565	0.374783	+	0.927113i	$0.377717\pi$
$812$	0	0
$813$	32928.2	1.42047
$814$	0	0
$815$	1328.40	0.0570944
$816$	0	0
$817$	−1344.48	−0.0575733
$818$	0	0
$819$	17940.2	0.765421
$820$	0	0
$821$	−13869.9	−0.589602	−0.294801	−	0.955559i	$-0.595253\pi$
$821$	−13869.9	−0.589602	−0.294801	+	0.955559i	$0.595253\pi$
$822$	0	0
$823$	−30858.5	−1.30700	−0.653499	−	0.756928i	$-0.726699\pi$
$823$	−30858.5	−1.30700	−0.653499	+	0.756928i	$0.726699\pi$
$824$	0	0
$825$	−67113.1	−2.83222
$826$	0	0
$827$	−10118.6	−0.425462	−0.212731	−	0.977111i	$-0.568236\pi$
$827$	−10118.6	−0.425462	−0.212731	+	0.977111i	$0.568236\pi$
$828$	0	0
$829$	19625.2	0.822210	0.411105	−	0.911588i	$-0.365143\pi$
$829$	19625.2	0.822210	0.411105	+	0.911588i	$0.365143\pi$
$830$	0	0
$831$	74094.2	3.09302
$832$	0	0
$833$	−4003.70	−0.166531
$834$	0	0
$835$	14078.5	0.583481
$836$	0	0
$837$	−35772.8	−1.47728
$838$	0	0
$839$	−19083.0	−0.785241	−0.392621	−	0.919701i	$-0.628431\pi$
$839$	−19083.0	−0.785241	−0.392621	+	0.919701i	$0.628431\pi$
$840$	0	0
$841$	12797.7	0.524731
$842$	0	0
$843$	36522.4	1.49217
$844$	0	0
$845$	−4346.22	−0.176940
$846$	0	0
$847$	−43101.4	−1.74850
$848$	0	0
$849$	68599.0	2.77304
$850$	0	0
$851$	−24411.5	−0.983332
$852$	0	0
$853$	−26666.7	−1.07040	−0.535200	−	0.844726i	$-0.679764\pi$
$853$	−26666.7	−1.07040	−0.535200	+	0.844726i	$0.679764\pi$
$854$	0	0
$855$	3700.44	0.148015
$856$	0	0
$857$	12467.5	0.496944	0.248472	−	0.968639i	$-0.420072\pi$
$857$	12467.5	0.496944	0.248472	+	0.968639i	$0.420072\pi$
$858$	0	0
$859$	31848.1	1.26501	0.632505	−	0.774557i	$-0.282027\pi$
$859$	31848.1	1.26501	0.632505	+	0.774557i	$0.282027\pi$
$860$	0	0
$861$	13528.1	0.535467
$862$	0	0
$863$	12043.9	0.475064	0.237532	−	0.971380i	$-0.423662\pi$
$863$	12043.9	0.475064	0.237532	+	0.971380i	$0.423662\pi$
$864$	0	0
$865$	−13424.4	−0.527679
$866$	0	0
$867$	−39456.2	−1.54556
$868$	0	0
$869$	−81959.3	−3.19940
$870$	0	0
$871$	−24897.9	−0.968582
$872$	0	0
$873$	−1353.25	−0.0524636
$874$	0	0
$875$	10733.2	0.414685
$876$	0	0
$877$	−1833.06	−0.0705792	−0.0352896	−	0.999377i	$-0.511235\pi$
$877$	−1833.06	−0.0705792	−0.0352896	+	0.999377i	$0.511235\pi$
$878$	0	0
$879$	74981.2	2.87719
$880$	0	0
$881$	−11322.0	−0.432970	−0.216485	−	0.976286i	$-0.569459\pi$
$881$	−11322.0	−0.432970	−0.216485	+	0.976286i	$0.569459\pi$
$882$	0	0
$883$	41569.1	1.58427	0.792136	−	0.610345i	$-0.208969\pi$
$883$	41569.1	1.58427	0.792136	+	0.610345i	$0.208969\pi$
$884$	0	0
$885$	−31337.0	−1.19026
$886$	0	0
$887$	−1082.37	−0.0409723	−0.0204862	−	0.999790i	$-0.506521\pi$
$887$	−1082.37	−0.0409723	−0.0204862	+	0.999790i	$0.506521\pi$
$888$	0	0
$889$	−18963.4	−0.715425
$890$	0	0
$891$	17237.2	0.648111
$892$	0	0
$893$	−5240.75	−0.196388
$894$	0	0
$895$	−16019.0	−0.598276
$896$	0	0
$897$	25287.6	0.941281
$898$	0	0
$899$	−39151.3	−1.45247
$900$	0	0
$901$	5455.19	0.201708
$902$	0	0
$903$	6843.54	0.252202
$904$	0	0
$905$	12846.2	0.471847
$906$	0	0
$907$	42893.6	1.57030	0.785148	−	0.619308i	$-0.212587\pi$
$907$	42893.6	1.57030	0.785148	+	0.619308i	$0.212587\pi$
$908$	0	0
$909$	8529.19	0.311216
$910$	0	0
$911$	11952.0	0.434672	0.217336	−	0.976097i	$-0.430263\pi$
$911$	11952.0	0.434672	0.217336	+	0.976097i	$0.430263\pi$
$912$	0	0
$913$	5153.85	0.186821
$914$	0	0
$915$	−2402.23	−0.0867927
$916$	0	0
$917$	28254.6	1.01750
$918$	0	0
$919$	15304.8	0.549356	0.274678	−	0.961536i	$-0.411429\pi$
$919$	15304.8	0.549356	0.274678	+	0.961536i	$0.411429\pi$
$920$	0	0
$921$	15668.3	0.560575
$922$	0	0
$923$	23622.9	0.842426
$924$	0	0
$925$	−30388.6	−1.08019
$926$	0	0
$927$	−91461.6	−3.24056
$928$	0	0
$929$	−25079.2	−0.885707	−0.442854	−	0.896594i	$-0.646034\pi$
$929$	−25079.2	−0.885707	−0.442854	+	0.896594i	$0.646034\pi$
$930$	0	0
$931$	−4129.19	−0.145358
$932$	0	0
$933$	−90610.4	−3.17948
$934$	0	0
$935$	5443.00	0.190380
$936$	0	0
$937$	24930.3	0.869196	0.434598	−	0.900625i	$-0.356890\pi$
$937$	24930.3	0.869196	0.434598	+	0.900625i	$0.356890\pi$
$938$	0	0
$939$	39837.6	1.38451
$940$	0	0
$941$	−13315.9	−0.461302	−0.230651	−	0.973037i	$-0.574086\pi$
$941$	−13315.9	−0.461302	−0.230651	+	0.973037i	$0.574086\pi$
$942$	0	0
$943$	12150.8	0.419601
$944$	0	0
$945$	−8111.93	−0.279239
$946$	0	0
$947$	14215.0	0.487776	0.243888	−	0.969803i	$-0.421577\pi$
$947$	14215.0	0.487776	0.243888	+	0.969803i	$0.421577\pi$
$948$	0	0
$949$	3081.69	0.105412
$950$	0	0
$951$	−8619.44	−0.293906
$952$	0	0
$953$	−32263.0	−1.09664	−0.548321	−	0.836268i	$-0.684733\pi$
$953$	−32263.0	−1.09664	−0.548321	+	0.836268i	$0.684733\pi$
$954$	0	0
$955$	11820.9	0.400541
$956$	0	0
$957$	119684.	4.04268
$958$	0	0
$959$	−13233.7	−0.445609
$960$	0	0
$961$	11428.7	0.383628
$962$	0	0
$963$	84110.5	2.81456
$964$	0	0
$965$	−2534.66	−0.0845530
$966$	0	0
$967$	−12881.5	−0.428377	−0.214188	−	0.976792i	$-0.568711\pi$
$967$	−12881.5	−0.428377	−0.214188	+	0.976792i	$0.568711\pi$
$968$	0	0
$969$	3019.68	0.100110
$970$	0	0
$971$	−7956.70	−0.262969	−0.131484	−	0.991318i	$-0.541974\pi$
$971$	−7956.70	−0.262969	−0.131484	+	0.991318i	$0.541974\pi$
$972$	0	0
$973$	−13454.0	−0.443282
$974$	0	0
$975$	31479.2	1.03399
$976$	0	0
$977$	−31419.9	−1.02888	−0.514438	−	0.857528i	$-0.671999\pi$
$977$	−31419.9	−1.02888	−0.514438	+	0.857528i	$0.671999\pi$
$978$	0	0
$979$	−21319.0	−0.695974
$980$	0	0
$981$	54620.9	1.77769
$982$	0	0
$983$	12250.4	0.397484	0.198742	−	0.980052i	$-0.436314\pi$
$983$	12250.4	0.397484	0.198742	+	0.980052i	$0.436314\pi$
$984$	0	0
$985$	11705.1	0.378634
$986$	0	0
$987$	26675.9	0.860289
$988$	0	0
$989$	6146.78	0.197630
$990$	0	0
$991$	25270.1	0.810022	0.405011	−	0.914312i	$-0.367268\pi$
$991$	25270.1	0.810022	0.405011	+	0.914312i	$0.367268\pi$
$992$	0	0
$993$	−71480.3	−2.28435
$994$	0	0
$995$	4057.75	0.129286
$996$	0	0
$997$	43576.2	1.38422	0.692112	−	0.721790i	$-0.256680\pi$
$997$	43576.2	1.38422	0.692112	+	0.721790i	$0.256680\pi$
$998$	0	0
$999$	49516.2	1.56819

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	152.4.a.d.1.5	✓	5
3.2	odd	2		1368.4.a.m.1.3		5
4.3	odd	2		304.4.a.k.1.1		5
8.3	odd	2		1216.4.a.bb.1.5		5
8.5	even	2		1216.4.a.ba.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.a.d.1.5	✓	5	1.1	even	1	trivial
304.4.a.k.1.1		5	4.3	odd	2
1216.4.a.ba.1.1		5	8.5	even	2
1216.4.a.bb.1.5		5	8.3	odd	2
1368.4.a.m.1.3		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 152 = 2^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	152.a (trivial)

\(f(q)\)	\(=\)	\(q+8.62694 q^{3} +4.10677 q^{5} -11.2105 q^{7} +47.4241 q^{9} +71.9427 q^{11} -33.7445 q^{13} +35.4289 q^{15} +18.4226 q^{17} +19.0000 q^{19} -96.7120 q^{21} -86.8656 q^{23} -108.134 q^{25} +176.198 q^{27} +192.838 q^{29} -203.026 q^{31} +620.645 q^{33} -46.0388 q^{35} +281.026 q^{37} -291.112 q^{39} -139.880 q^{41} -70.7620 q^{43} +194.760 q^{45} -275.829 q^{47} -217.326 q^{49} +158.931 q^{51} +296.114 q^{53} +295.452 q^{55} +163.912 q^{57} -884.504 q^{59} -67.8043 q^{61} -531.646 q^{63} -138.581 q^{65} +737.836 q^{67} -749.384 q^{69} -700.052 q^{71} -91.3241 q^{73} -932.869 q^{75} -806.511 q^{77} -1139.23 q^{79} +239.596 q^{81} +71.6383 q^{83} +75.6575 q^{85} +1663.61 q^{87} -296.333 q^{89} +378.292 q^{91} -1751.50 q^{93} +78.0287 q^{95} -28.5351 q^{97} +3411.82 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} + 22 q^{7} + 83 q^{9} + 6 q^{11} + 82 q^{13} + 204 q^{15} + 234 q^{17} + 95 q^{19} + 308 q^{21} - 60 q^{23} + 549 q^{25} - 190 q^{27} + 210 q^{29} - 224 q^{31} + 704 q^{33} + 42 q^{35} + 614 q^{37}+ \cdots + 126 q^{99}+O(q^{100}) \) 5 * q + 2 * q^3 + 22 * q^7 + 83 * q^9 + 6 * q^11 + 82 * q^13 + 204 * q^15 + 234 * q^17 + 95 * q^19 + 308 * q^21 - 60 * q^23 + 549 * q^25 - 190 * q^27 + 210 * q^29 - 224 * q^31 + 704 * q^33 + 42 * q^35 + 614 * q^37 - 132 * q^39 + 194 * q^41 - 38 * q^43 - 516 * q^45 - 1066 * q^47 + 489 * q^49 - 1382 * q^51 - 42 * q^53 - 618 * q^55 + 38 * q^57 - 1090 * q^59 - 276 * q^61 - 1790 * q^63 - 184 * q^65 - 314 * q^67 - 268 * q^69 - 1276 * q^71 - 106 * q^73 - 2066 * q^75 - 1226 * q^77 + 52 * q^79 - 283 * q^81 - 580 * q^83 - 3498 * q^85 + 900 * q^87 + 810 * q^89 + 2522 * q^91 - 1332 * q^93 + 2694 * q^97 + 126 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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