LMFDB - Embedded newform 152.4.a.d.1.3

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			1.3
Root			$-3.22412$ 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)

$f(q)$	$=$	$q+2.49563 q^{3} -15.7941 q^{5} +30.5820 q^{7} -20.7718 q^{9} +39.5545 q^{11} +80.7559 q^{13} -39.4163 q^{15} +74.9991 q^{17} +19.0000 q^{19} +76.3212 q^{21} -54.9828 q^{23} +124.455 q^{25} -119.221 q^{27} -105.853 q^{29} +245.493 q^{31} +98.7132 q^{33} -483.016 q^{35} -167.493 q^{37} +201.536 q^{39} +433.978 q^{41} +356.648 q^{43} +328.073 q^{45} -403.858 q^{47} +592.257 q^{49} +187.170 q^{51} -489.379 q^{53} -624.729 q^{55} +47.4169 q^{57} +8.75307 q^{59} -801.910 q^{61} -635.244 q^{63} -1275.47 q^{65} -917.754 q^{67} -137.216 q^{69} +104.852 q^{71} +813.135 q^{73} +310.593 q^{75} +1209.65 q^{77} -392.850 q^{79} +263.309 q^{81} -122.905 q^{83} -1184.55 q^{85} -264.169 q^{87} +15.2554 q^{89} +2469.67 q^{91} +612.659 q^{93} -300.089 q^{95} -637.620 q^{97} -821.619 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.49563	0.480284	0.240142	−	0.970738i	$-0.422806\pi$
$3$	2.49563	0.480284	0.240142	+	0.970738i	$0.422806\pi$
$4$	0	0
$5$	−15.7941	−1.41267	−0.706336	−	0.707877i	$-0.749653\pi$
$5$	−15.7941	−1.41267	−0.706336	+	0.707877i	$0.749653\pi$
$6$	0	0
$7$	30.5820	1.65127	0.825636	−	0.564203i	$-0.190817\pi$
$7$	30.5820	1.65127	0.825636	+	0.564203i	$0.190817\pi$
$8$	0	0
$9$	−20.7718	−0.769328
$10$	0	0
$11$	39.5545	1.08419	0.542097	−	0.840316i	$-0.317631\pi$
$11$	39.5545	1.08419	0.542097	+	0.840316i	$0.317631\pi$
$12$	0	0
$13$	80.7559	1.72290	0.861448	−	0.507846i	$-0.169558\pi$
$13$	80.7559	1.72290	0.861448	+	0.507846i	$0.169558\pi$
$14$	0	0
$15$	−39.4163	−0.678483
$16$	0	0
$17$	74.9991	1.07000	0.534998	−	0.844853i	$-0.320312\pi$
$17$	74.9991	1.07000	0.534998	+	0.844853i	$0.320312\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	76.3212	0.793079
$22$	0	0
$23$	−54.9828	−0.498465	−0.249233	−	0.968444i	$-0.580178\pi$
$23$	−54.9828	−0.498465	−0.249233	+	0.968444i	$0.580178\pi$
$24$	0	0
$25$	124.455	0.995639
$26$	0	0
$27$	−119.221	−0.849779
$28$	0	0
$29$	−105.853	−0.677806	−0.338903	−	0.940821i	$-0.610056\pi$
$29$	−105.853	−0.677806	−0.338903	+	0.940821i	$0.610056\pi$
$30$	0	0
$31$	245.493	1.42232	0.711159	−	0.703031i	$-0.248171\pi$
$31$	245.493	1.42232	0.711159	+	0.703031i	$0.248171\pi$
$32$	0	0
$33$	98.7132	0.520720
$34$	0	0
$35$	−483.016	−2.33270
$36$	0	0
$37$	−167.493	−0.744208	−0.372104	−	0.928191i	$-0.621363\pi$
$37$	−167.493	−0.744208	−0.372104	+	0.928191i	$0.621363\pi$
$38$	0	0
$39$	201.536	0.827479
$40$	0	0
$41$	433.978	1.65307	0.826536	−	0.562883i	$-0.190308\pi$
$41$	433.978	1.65307	0.826536	+	0.562883i	$0.190308\pi$
$42$	0	0
$43$	356.648	1.26485	0.632423	−	0.774623i	$-0.282061\pi$
$43$	356.648	1.26485	0.632423	+	0.774623i	$0.282061\pi$
$44$	0	0
$45$	328.073	1.08681
$46$	0	0
$47$	−403.858	−1.25338	−0.626688	−	0.779270i	$-0.715590\pi$
$47$	−403.858	−1.25338	−0.626688	+	0.779270i	$0.715590\pi$
$48$	0	0
$49$	592.257	1.72670
$50$	0	0
$51$	187.170	0.513902
$52$	0	0
$53$	−489.379	−1.26833	−0.634163	−	0.773199i	$-0.718655\pi$
$53$	−489.379	−1.26833	−0.634163	+	0.773199i	$0.718655\pi$
$54$	0	0
$55$	−624.729	−1.53161
$56$	0	0
$57$	47.4169	0.110185
$58$	0	0
$59$	8.75307	0.0193144	0.00965722	−	0.999953i	$-0.496926\pi$
$59$	8.75307	0.0193144	0.00965722	+	0.999953i	$0.496926\pi$
$60$	0	0
$61$	−801.910	−1.68318	−0.841590	−	0.540117i	$-0.818380\pi$
$61$	−801.910	−1.68318	−0.841590	+	0.540117i	$0.818380\pi$
$62$	0	0
$63$	−635.244	−1.27037
$64$	0	0
$65$	−1275.47	−2.43388
$66$	0	0
$67$	−917.754	−1.67345	−0.836727	−	0.547619i	$-0.815534\pi$
$67$	−917.754	−1.67345	−0.836727	+	0.547619i	$0.815534\pi$
$68$	0	0
$69$	−137.216	−0.239405
$70$	0	0
$71$	104.852	0.175263	0.0876317	−	0.996153i	$-0.472070\pi$
$71$	104.852	0.175263	0.0876317	+	0.996153i	$0.472070\pi$
$72$	0	0
$73$	813.135	1.30370	0.651851	−	0.758347i	$-0.273993\pi$
$73$	813.135	1.30370	0.651851	+	0.758347i	$0.273993\pi$
$74$	0	0
$75$	310.593	0.478189
$76$	0	0
$77$	1209.65	1.79030
$78$	0	0
$79$	−392.850	−0.559481	−0.279741	−	0.960076i	$-0.590249\pi$
$79$	−392.850	−0.559481	−0.279741	+	0.960076i	$0.590249\pi$
$80$	0	0
$81$	263.309	0.361193
$82$	0	0
$83$	−122.905	−0.162537	−0.0812683	−	0.996692i	$-0.525897\pi$
$83$	−122.905	−0.162537	−0.0812683	+	0.996692i	$0.525897\pi$
$84$	0	0
$85$	−1184.55	−1.51155
$86$	0	0
$87$	−264.169	−0.325539
$88$	0	0
$89$	15.2554	0.0181693	0.00908465	−	0.999959i	$-0.497108\pi$
$89$	15.2554	0.0181693	0.00908465	+	0.999959i	$0.497108\pi$
$90$	0	0
$91$	2469.67	2.84497
$92$	0	0
$93$	612.659	0.683116
$94$	0	0
$95$	−300.089	−0.324089
$96$	0	0
$97$	−637.620	−0.667428	−0.333714	−	0.942674i	$-0.608302\pi$
$97$	−637.620	−0.667428	−0.333714	+	0.942674i	$0.608302\pi$
$98$	0	0
$99$	−821.619	−0.834100
$100$	0	0
$101$	230.764	0.227345	0.113673	−	0.993518i	$-0.463738\pi$
$101$	230.764	0.227345	0.113673	+	0.993518i	$0.463738\pi$
$102$	0	0
$103$	56.5197	0.0540684	0.0270342	−	0.999635i	$-0.491394\pi$
$103$	56.5197	0.0540684	0.0270342	+	0.999635i	$0.491394\pi$
$104$	0	0
$105$	−1205.43	−1.12036
$106$	0	0
$107$	413.201	0.373324	0.186662	−	0.982424i	$-0.440233\pi$
$107$	413.201	0.373324	0.186662	+	0.982424i	$0.440233\pi$
$108$	0	0
$109$	677.546	0.595387	0.297693	−	0.954662i	$-0.403783\pi$
$109$	677.546	0.595387	0.297693	+	0.954662i	$0.403783\pi$
$110$	0	0
$111$	−418.000	−0.357431
$112$	0	0
$113$	−368.845	−0.307062	−0.153531	−	0.988144i	$-0.549064\pi$
$113$	−368.845	−0.307062	−0.153531	+	0.988144i	$0.549064\pi$
$114$	0	0
$115$	868.406	0.704167
$116$	0	0
$117$	−1677.45	−1.32547
$118$	0	0
$119$	2293.62	1.76686
$120$	0	0
$121$	233.557	0.175475
$122$	0	0
$123$	1083.05	0.793944
$124$	0	0
$125$	8.60906	0.00616014
$126$	0	0
$127$	225.445	0.157519	0.0787597	−	0.996894i	$-0.474904\pi$
$127$	225.445	0.157519	0.0787597	+	0.996894i	$0.474904\pi$
$128$	0	0
$129$	890.061	0.607485
$130$	0	0
$131$	−1426.63	−0.951490	−0.475745	−	0.879583i	$-0.657821\pi$
$131$	−1426.63	−0.951490	−0.475745	+	0.879583i	$0.657821\pi$
$132$	0	0
$133$	581.058	0.378828
$134$	0	0
$135$	1882.99	1.20046
$136$	0	0
$137$	2724.50	1.69905	0.849525	−	0.527549i	$-0.176889\pi$
$137$	2724.50	1.69905	0.849525	+	0.527549i	$0.176889\pi$
$138$	0	0
$139$	−1366.68	−0.833961	−0.416980	−	0.908916i	$-0.636912\pi$
$139$	−1366.68	−0.833961	−0.416980	+	0.908916i	$0.636912\pi$
$140$	0	0
$141$	−1007.88	−0.601976
$142$	0	0
$143$	3194.26	1.86795
$144$	0	0
$145$	1671.85	0.957517
$146$	0	0
$147$	1478.05	0.829305
$148$	0	0
$149$	154.205	0.0847848	0.0423924	−	0.999101i	$-0.486502\pi$
$149$	154.205	0.0847848	0.0423924	+	0.999101i	$0.486502\pi$
$150$	0	0
$151$	935.176	0.503997	0.251998	−	0.967728i	$-0.418912\pi$
$151$	935.176	0.503997	0.251998	+	0.967728i	$0.418912\pi$
$152$	0	0
$153$	−1557.87	−0.823178
$154$	0	0
$155$	−3877.35	−2.00927
$156$	0	0
$157$	−3229.12	−1.64148	−0.820739	−	0.571303i	$-0.806438\pi$
$157$	−3229.12	−1.64148	−0.820739	+	0.571303i	$0.806438\pi$
$158$	0	0
$159$	−1221.31	−0.609157
$160$	0	0
$161$	−1681.48	−0.823101
$162$	0	0
$163$	2588.37	1.24379	0.621893	−	0.783102i	$-0.286364\pi$
$163$	2588.37	1.24379	0.621893	+	0.783102i	$0.286364\pi$
$164$	0	0
$165$	−1559.09	−0.735606
$166$	0	0
$167$	−190.389	−0.0882202	−0.0441101	−	0.999027i	$-0.514045\pi$
$167$	−190.389	−0.0882202	−0.0441101	+	0.999027i	$0.514045\pi$
$168$	0	0
$169$	4324.51	1.96837
$170$	0	0
$171$	−394.665	−0.176496
$172$	0	0
$173$	−2375.41	−1.04393	−0.521964	−	0.852968i	$-0.674800\pi$
$173$	−2375.41	−1.04393	−0.521964	+	0.852968i	$0.674800\pi$
$174$	0	0
$175$	3806.08	1.64407
$176$	0	0
$177$	21.8444	0.00927641
$178$	0	0
$179$	−2121.18	−0.885723	−0.442862	−	0.896590i	$-0.646037\pi$
$179$	−2121.18	−0.885723	−0.442862	+	0.896590i	$0.646037\pi$
$180$	0	0
$181$	2023.93	0.831148	0.415574	−	0.909559i	$-0.363581\pi$
$181$	2023.93	0.831148	0.415574	+	0.909559i	$0.363581\pi$
$182$	0	0
$183$	−2001.27	−0.808404
$184$	0	0
$185$	2645.41	1.05132
$186$	0	0
$187$	2966.55	1.16008
$188$	0	0
$189$	−3646.01	−1.40322
$190$	0	0
$191$	−2328.39	−0.882074	−0.441037	−	0.897489i	$-0.645389\pi$
$191$	−2328.39	−0.882074	−0.441037	+	0.897489i	$0.645389\pi$
$192$	0	0
$193$	351.034	0.130922	0.0654612	−	0.997855i	$-0.479148\pi$
$193$	351.034	0.130922	0.0654612	+	0.997855i	$0.479148\pi$
$194$	0	0
$195$	−3183.10	−1.16896
$196$	0	0
$197$	1820.58	0.658430	0.329215	−	0.944255i	$-0.393216\pi$
$197$	1820.58	0.658430	0.329215	+	0.944255i	$0.393216\pi$
$198$	0	0
$199$	94.4932	0.0336605	0.0168303	−	0.999858i	$-0.494643\pi$
$199$	94.4932	0.0336605	0.0168303	+	0.999858i	$0.494643\pi$
$200$	0	0
$201$	−2290.37	−0.803733
$202$	0	0
$203$	−3237.19	−1.11924
$204$	0	0
$205$	−6854.31	−2.33525
$206$	0	0
$207$	1142.09	0.383483
$208$	0	0
$209$	751.535	0.248731
$210$	0	0
$211$	1271.36	0.414805	0.207402	−	0.978256i	$-0.433499\pi$
$211$	1271.36	0.414805	0.207402	+	0.978256i	$0.433499\pi$
$212$	0	0
$213$	261.673	0.0841761
$214$	0	0
$215$	−5632.96	−1.78681
$216$	0	0
$217$	7507.66	2.34863
$218$	0	0
$219$	2029.28	0.626147
$220$	0	0
$221$	6056.61	1.84349
$222$	0	0
$223$	−804.197	−0.241493	−0.120747	−	0.992683i	$-0.538529\pi$
$223$	−804.197	−0.241493	−0.120747	+	0.992683i	$0.538529\pi$
$224$	0	0
$225$	−2585.16	−0.765973
$226$	0	0
$227$	−5727.99	−1.67480	−0.837401	−	0.546588i	$-0.815926\pi$
$227$	−5727.99	−1.67480	−0.837401	+	0.546588i	$0.815926\pi$
$228$	0	0
$229$	−1556.61	−0.449185	−0.224592	−	0.974453i	$-0.572105\pi$
$229$	−1556.61	−0.449185	−0.224592	+	0.974453i	$0.572105\pi$
$230$	0	0
$231$	3018.85	0.859850
$232$	0	0
$233$	−4819.55	−1.35511	−0.677553	−	0.735474i	$-0.736959\pi$
$233$	−4819.55	−1.35511	−0.677553	+	0.735474i	$0.736959\pi$
$234$	0	0
$235$	6378.59	1.77061
$236$	0	0
$237$	−980.406	−0.268710
$238$	0	0
$239$	−5514.00	−1.49235	−0.746174	−	0.665751i	$-0.768111\pi$
$239$	−5514.00	−1.49235	−0.746174	+	0.665751i	$0.768111\pi$
$240$	0	0
$241$	−213.424	−0.0570451	−0.0285225	−	0.999593i	$-0.509080\pi$
$241$	−213.424	−0.0570451	−0.0285225	+	0.999593i	$0.509080\pi$
$242$	0	0
$243$	3876.08	1.02325
$244$	0	0
$245$	−9354.20	−2.43926
$246$	0	0
$247$	1534.36	0.395259
$248$	0	0
$249$	−306.724	−0.0780636
$250$	0	0
$251$	1606.05	0.403876	0.201938	−	0.979398i	$-0.435276\pi$
$251$	1606.05	0.403876	0.201938	+	0.979398i	$0.435276\pi$
$252$	0	0
$253$	−2174.81	−0.540432
$254$	0	0
$255$	−2956.18	−0.725974
$256$	0	0
$257$	2127.48	0.516377	0.258188	−	0.966095i	$-0.416874\pi$
$257$	2127.48	0.516377	0.258188	+	0.966095i	$0.416874\pi$
$258$	0	0
$259$	−5122.27	−1.22889
$260$	0	0
$261$	2198.76	0.521455
$262$	0	0
$263$	−647.537	−0.151821	−0.0759103	−	0.997115i	$-0.524186\pi$
$263$	−647.537	−0.151821	−0.0759103	+	0.997115i	$0.524186\pi$
$264$	0	0
$265$	7729.31	1.79173
$266$	0	0
$267$	38.0717	0.00872642
$268$	0	0
$269$	−2967.65	−0.672642	−0.336321	−	0.941747i	$-0.609183\pi$
$269$	−2967.65	−0.672642	−0.336321	+	0.941747i	$0.609183\pi$
$270$	0	0
$271$	−8060.51	−1.80679	−0.903397	−	0.428806i	$-0.858934\pi$
$271$	−8060.51	−1.80679	−0.903397	+	0.428806i	$0.858934\pi$
$272$	0	0
$273$	6163.38	1.36639
$274$	0	0
$275$	4922.75	1.07947
$276$	0	0
$277$	1297.72	0.281489	0.140745	−	0.990046i	$-0.455050\pi$
$277$	1297.72	0.281489	0.140745	+	0.990046i	$0.455050\pi$
$278$	0	0
$279$	−5099.34	−1.09423
$280$	0	0
$281$	5629.57	1.19513	0.597566	−	0.801819i	$-0.296134\pi$
$281$	5629.57	1.19513	0.597566	+	0.801819i	$0.296134\pi$
$282$	0	0
$283$	−6674.39	−1.40195	−0.700974	−	0.713187i	$-0.747251\pi$
$283$	−6674.39	−1.40195	−0.700974	+	0.713187i	$0.747251\pi$
$284$	0	0
$285$	−748.909	−0.155655
$286$	0	0
$287$	13271.9	2.72967
$288$	0	0
$289$	711.859	0.144893
$290$	0	0
$291$	−1591.26	−0.320555
$292$	0	0
$293$	4239.27	0.845259	0.422630	−	0.906302i	$-0.361107\pi$
$293$	4239.27	0.845259	0.422630	+	0.906302i	$0.361107\pi$
$294$	0	0
$295$	−138.247	−0.0272850
$296$	0	0
$297$	−4715.71	−0.921325
$298$	0	0
$299$	−4440.18	−0.858803
$300$	0	0
$301$	10907.0	2.08860
$302$	0	0
$303$	575.901	0.109190
$304$	0	0
$305$	12665.5	2.37778
$306$	0	0
$307$	−3884.16	−0.722087	−0.361043	−	0.932549i	$-0.617579\pi$
$307$	−3884.16	−0.722087	−0.361043	+	0.932549i	$0.617579\pi$
$308$	0	0
$309$	141.052	0.0259682
$310$	0	0
$311$	−5303.28	−0.966950	−0.483475	−	0.875358i	$-0.660626\pi$
$311$	−5303.28	−0.966950	−0.483475	+	0.875358i	$0.660626\pi$
$312$	0	0
$313$	−6241.98	−1.12721	−0.563607	−	0.826043i	$-0.690587\pi$
$313$	−6241.98	−1.12721	−0.563607	+	0.826043i	$0.690587\pi$
$314$	0	0
$315$	10033.1	1.79461
$316$	0	0
$317$	1307.84	0.231722	0.115861	−	0.993265i	$-0.463037\pi$
$317$	1307.84	0.231722	0.115861	+	0.993265i	$0.463037\pi$
$318$	0	0
$319$	−4186.95	−0.734872
$320$	0	0
$321$	1031.20	0.179301
$322$	0	0
$323$	1424.98	0.245474
$324$	0	0
$325$	10050.5	1.71538
$326$	0	0
$327$	1690.90	0.285954
$328$	0	0
$329$	−12350.8	−2.06966
$330$	0	0
$331$	6829.84	1.13415	0.567073	−	0.823668i	$-0.308076\pi$
$331$	6829.84	1.13415	0.567073	+	0.823668i	$0.308076\pi$
$332$	0	0
$333$	3479.14	0.572540
$334$	0	0
$335$	14495.1	2.36404
$336$	0	0
$337$	−2106.49	−0.340498	−0.170249	−	0.985401i	$-0.554457\pi$
$337$	−2106.49	−0.340498	−0.170249	+	0.985401i	$0.554457\pi$
$338$	0	0
$339$	−920.499	−0.147477
$340$	0	0
$341$	9710.35	1.54207
$342$	0	0
$343$	7622.78	1.19998
$344$	0	0
$345$	2167.22	0.338200
$346$	0	0
$347$	−1126.07	−0.174209	−0.0871046	−	0.996199i	$-0.527761\pi$
$347$	−1126.07	−0.174209	−0.0871046	+	0.996199i	$0.527761\pi$
$348$	0	0
$349$	−3089.62	−0.473878	−0.236939	−	0.971525i	$-0.576144\pi$
$349$	−3089.62	−0.473878	−0.236939	+	0.971525i	$0.576144\pi$
$350$	0	0
$351$	−9627.77	−1.46408
$352$	0	0
$353$	−4124.98	−0.621956	−0.310978	−	0.950417i	$-0.600657\pi$
$353$	−4124.98	−0.621956	−0.310978	+	0.950417i	$0.600657\pi$
$354$	0	0
$355$	−1656.06	−0.247590
$356$	0	0
$357$	5724.02	0.848592
$358$	0	0
$359$	−2914.71	−0.428502	−0.214251	−	0.976779i	$-0.568731\pi$
$359$	−2914.71	−0.428502	−0.214251	+	0.976779i	$0.568731\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	582.870	0.0842775
$364$	0	0
$365$	−12842.8	−1.84170
$366$	0	0
$367$	8029.35	1.14204	0.571020	−	0.820936i	$-0.306548\pi$
$367$	8029.35	1.14204	0.571020	+	0.820936i	$0.306548\pi$
$368$	0	0
$369$	−9014.52	−1.27175
$370$	0	0
$371$	−14966.2	−2.09435
$372$	0	0
$373$	3620.93	0.502640	0.251320	−	0.967904i	$-0.419135\pi$
$373$	3620.93	0.502640	0.251320	+	0.967904i	$0.419135\pi$
$374$	0	0
$375$	21.4850	0.00295862
$376$	0	0
$377$	−8548.23	−1.16779
$378$	0	0
$379$	4972.26	0.673899	0.336949	−	0.941523i	$-0.390605\pi$
$379$	4972.26	0.673899	0.336949	+	0.941523i	$0.390605\pi$
$380$	0	0
$381$	562.625	0.0756540
$382$	0	0
$383$	11226.8	1.49781	0.748905	−	0.662678i	$-0.230580\pi$
$383$	11226.8	1.49781	0.748905	+	0.662678i	$0.230580\pi$
$384$	0	0
$385$	−19105.4	−2.52910
$386$	0	0
$387$	−7408.25	−0.973081
$388$	0	0
$389$	−5501.50	−0.717063	−0.358531	−	0.933518i	$-0.616722\pi$
$389$	−5501.50	−0.717063	−0.358531	+	0.933518i	$0.616722\pi$
$390$	0	0
$391$	−4123.66	−0.533356
$392$	0	0
$393$	−3560.33	−0.456985
$394$	0	0
$395$	6204.72	0.790363
$396$	0	0
$397$	−8595.21	−1.08660	−0.543301	−	0.839538i	$-0.682826\pi$
$397$	−8595.21	−1.08660	−0.543301	+	0.839538i	$0.682826\pi$
$398$	0	0
$399$	1450.10	0.181945
$400$	0	0
$401$	8690.87	1.08230	0.541149	−	0.840927i	$-0.317990\pi$
$401$	8690.87	1.08230	0.541149	+	0.840927i	$0.317990\pi$
$402$	0	0
$403$	19825.0	2.45051
$404$	0	0
$405$	−4158.75	−0.510246
$406$	0	0
$407$	−6625.10	−0.806865
$408$	0	0
$409$	866.231	0.104725	0.0523623	−	0.998628i	$-0.483325\pi$
$409$	866.231	0.104725	0.0523623	+	0.998628i	$0.483325\pi$
$410$	0	0
$411$	6799.34	0.816026
$412$	0	0
$413$	267.686	0.0318934
$414$	0	0
$415$	1941.17	0.229611
$416$	0	0
$417$	−3410.73	−0.400538
$418$	0	0
$419$	3305.35	0.385387	0.192693	−	0.981259i	$-0.438278\pi$
$419$	3305.35	0.385387	0.192693	+	0.981259i	$0.438278\pi$
$420$	0	0
$421$	−9719.31	−1.12515	−0.562577	−	0.826745i	$-0.690190\pi$
$421$	−9719.31	−1.12515	−0.562577	+	0.826745i	$0.690190\pi$
$422$	0	0
$423$	8388.87	0.964257
$424$	0	0
$425$	9334.00	1.06533
$426$	0	0
$427$	−24524.0	−2.77939
$428$	0	0
$429$	7971.67	0.897147
$430$	0	0
$431$	−3282.60	−0.366862	−0.183431	−	0.983033i	$-0.558720\pi$
$431$	−3282.60	−0.366862	−0.183431	+	0.983033i	$0.558720\pi$
$432$	0	0
$433$	14447.4	1.60346	0.801728	−	0.597690i	$-0.203915\pi$
$433$	14447.4	1.60346	0.801728	+	0.597690i	$0.203915\pi$
$434$	0	0
$435$	4172.32	0.459880
$436$	0	0
$437$	−1044.67	−0.114356
$438$	0	0
$439$	13632.1	1.48206	0.741028	−	0.671474i	$-0.234339\pi$
$439$	13632.1	1.48206	0.741028	+	0.671474i	$0.234339\pi$
$440$	0	0
$441$	−12302.3	−1.32840
$442$	0	0
$443$	−4906.36	−0.526203	−0.263102	−	0.964768i	$-0.584745\pi$
$443$	−4906.36	−0.526203	−0.263102	+	0.964768i	$0.584745\pi$
$444$	0	0
$445$	−240.946	−0.0256672
$446$	0	0
$447$	384.837	0.0407207
$448$	0	0
$449$	10710.2	1.12571	0.562856	−	0.826555i	$-0.309703\pi$
$449$	10710.2	1.12571	0.562856	+	0.826555i	$0.309703\pi$
$450$	0	0
$451$	17165.8	1.79225
$452$	0	0
$453$	2333.85	0.242061
$454$	0	0
$455$	−39006.4	−4.01900
$456$	0	0
$457$	−11451.2	−1.17214	−0.586069	−	0.810261i	$-0.699325\pi$
$457$	−11451.2	−1.17214	−0.586069	+	0.810261i	$0.699325\pi$
$458$	0	0
$459$	−8941.44	−0.909261
$460$	0	0
$461$	9681.27	0.978095	0.489047	−	0.872257i	$-0.337345\pi$
$461$	9681.27	0.978095	0.489047	+	0.872257i	$0.337345\pi$
$462$	0	0
$463$	−10252.5	−1.02911	−0.514553	−	0.857459i	$-0.672042\pi$
$463$	−10252.5	−1.02911	−0.514553	+	0.857459i	$0.672042\pi$
$464$	0	0
$465$	−9676.42	−0.965018
$466$	0	0
$467$	3693.37	0.365972	0.182986	−	0.983116i	$-0.441424\pi$
$467$	3693.37	0.365972	0.182986	+	0.983116i	$0.441424\pi$
$468$	0	0
$469$	−28066.7	−2.76333
$470$	0	0
$471$	−8058.69	−0.788375
$472$	0	0
$473$	14107.0	1.37134
$474$	0	0
$475$	2364.64	0.228415
$476$	0	0
$477$	10165.3	0.975759
$478$	0	0
$479$	−1875.00	−0.178854	−0.0894271	−	0.995993i	$-0.528504\pi$
$479$	−1875.00	−0.178854	−0.0894271	+	0.995993i	$0.528504\pi$
$480$	0	0
$481$	−13526.0	−1.28219
$482$	0	0
$483$	−4196.35	−0.395322
$484$	0	0
$485$	10070.7	0.942857
$486$	0	0
$487$	−11284.2	−1.04997	−0.524986	−	0.851111i	$-0.675929\pi$
$487$	−11284.2	−1.04997	−0.524986	+	0.851111i	$0.675929\pi$
$488$	0	0
$489$	6459.61	0.597370
$490$	0	0
$491$	1073.04	0.0986261	0.0493130	−	0.998783i	$-0.484297\pi$
$491$	1073.04	0.0986261	0.0493130	+	0.998783i	$0.484297\pi$
$492$	0	0
$493$	−7938.86	−0.725250
$494$	0	0
$495$	12976.8	1.17831
$496$	0	0
$497$	3206.60	0.289407
$498$	0	0
$499$	−7055.10	−0.632925	−0.316462	−	0.948605i	$-0.602495\pi$
$499$	−7055.10	−0.632925	−0.316462	+	0.948605i	$0.602495\pi$
$500$	0	0
$501$	−475.141	−0.0423707
$502$	0	0
$503$	31.3527	0.00277922	0.00138961	−	0.999999i	$-0.499558\pi$
$503$	31.3527	0.00277922	0.00138961	+	0.999999i	$0.499558\pi$
$504$	0	0
$505$	−3644.72	−0.321164
$506$	0	0
$507$	10792.4	0.945376
$508$	0	0
$509$	−11355.8	−0.988875	−0.494437	−	0.869213i	$-0.664626\pi$
$509$	−11355.8	−0.988875	−0.494437	+	0.869213i	$0.664626\pi$
$510$	0	0
$511$	24867.3	2.15277
$512$	0	0
$513$	−2265.19	−0.194953
$514$	0	0
$515$	−892.680	−0.0763809
$516$	0	0
$517$	−15974.4	−1.35890
$518$	0	0
$519$	−5928.15	−0.501381
$520$	0	0
$521$	−21111.7	−1.77528	−0.887640	−	0.460539i	$-0.847656\pi$
$521$	−21111.7	−1.77528	−0.887640	+	0.460539i	$0.847656\pi$
$522$	0	0
$523$	2439.98	0.204002	0.102001	−	0.994784i	$-0.467476\pi$
$523$	2439.98	0.204002	0.102001	+	0.994784i	$0.467476\pi$
$524$	0	0
$525$	9498.55	0.789620
$526$	0	0
$527$	18411.7	1.52188
$528$	0	0
$529$	−9143.90	−0.751533
$530$	0	0
$531$	−181.817	−0.0148591
$532$	0	0
$533$	35046.3	2.84807
$534$	0	0
$535$	−6526.15	−0.527383
$536$	0	0
$537$	−5293.68	−0.425398
$538$	0	0
$539$	23426.4	1.87207
$540$	0	0
$541$	15539.3	1.23491	0.617456	−	0.786605i	$-0.288163\pi$
$541$	15539.3	1.23491	0.617456	+	0.786605i	$0.288163\pi$
$542$	0	0
$543$	5050.98	0.399187
$544$	0	0
$545$	−10701.3	−0.841085
$546$	0	0
$547$	−10699.1	−0.836306	−0.418153	−	0.908377i	$-0.637322\pi$
$547$	−10699.1	−0.836306	−0.418153	+	0.908377i	$0.637322\pi$
$548$	0	0
$549$	16657.1	1.29492
$550$	0	0
$551$	−2011.20	−0.155499
$552$	0	0
$553$	−12014.1	−0.923856
$554$	0	0
$555$	6601.95	0.504932
$556$	0	0
$557$	12253.5	0.932129	0.466064	−	0.884751i	$-0.345672\pi$
$557$	12253.5	0.932129	0.466064	+	0.884751i	$0.345672\pi$
$558$	0	0
$559$	28801.4	2.17920
$560$	0	0
$561$	7403.40	0.557169
$562$	0	0
$563$	2109.18	0.157889	0.0789445	−	0.996879i	$-0.474845\pi$
$563$	2109.18	0.157889	0.0789445	+	0.996879i	$0.474845\pi$
$564$	0	0
$565$	5825.59	0.433778
$566$	0	0
$567$	8052.52	0.596427
$568$	0	0
$569$	3099.14	0.228335	0.114167	−	0.993462i	$-0.463580\pi$
$569$	3099.14	0.228335	0.114167	+	0.993462i	$0.463580\pi$
$570$	0	0
$571$	6643.00	0.486867	0.243433	−	0.969918i	$-0.421726\pi$
$571$	6643.00	0.486867	0.243433	+	0.969918i	$0.421726\pi$
$572$	0	0
$573$	−5810.79	−0.423646
$574$	0	0
$575$	−6842.87	−0.496291
$576$	0	0
$577$	−6116.00	−0.441269	−0.220635	−	0.975357i	$-0.570813\pi$
$577$	−6116.00	−0.441269	−0.220635	+	0.975357i	$0.570813\pi$
$578$	0	0
$579$	876.051	0.0628799
$580$	0	0
$581$	−3758.67	−0.268392
$582$	0	0
$583$	−19357.1	−1.37511
$584$	0	0
$585$	26493.9	1.87245
$586$	0	0
$587$	−19056.8	−1.33996	−0.669981	−	0.742378i	$-0.733698\pi$
$587$	−19056.8	−1.33996	−0.669981	+	0.742378i	$0.733698\pi$
$588$	0	0
$589$	4664.37	0.326302
$590$	0	0
$591$	4543.48	0.316233
$592$	0	0
$593$	22651.4	1.56860	0.784301	−	0.620380i	$-0.213022\pi$
$593$	22651.4	1.56860	0.784301	+	0.620380i	$0.213022\pi$
$594$	0	0
$595$	−36225.8	−2.49599
$596$	0	0
$597$	235.820	0.0161666
$598$	0	0
$599$	−22822.3	−1.55675	−0.778377	−	0.627797i	$-0.783957\pi$
$599$	−22822.3	−1.55675	−0.778377	+	0.627797i	$0.783957\pi$
$600$	0	0
$601$	22694.1	1.54028	0.770141	−	0.637873i	$-0.220186\pi$
$601$	22694.1	1.54028	0.770141	+	0.637873i	$0.220186\pi$
$602$	0	0
$603$	19063.4	1.28744
$604$	0	0
$605$	−3688.83	−0.247888
$606$	0	0
$607$	14021.1	0.937561	0.468781	−	0.883315i	$-0.344693\pi$
$607$	14021.1	0.937561	0.468781	+	0.883315i	$0.344693\pi$
$608$	0	0
$609$	−8078.81	−0.537553
$610$	0	0
$611$	−32613.9	−2.15944
$612$	0	0
$613$	−4842.03	−0.319034	−0.159517	−	0.987195i	$-0.550994\pi$
$613$	−4842.03	−0.319034	−0.159517	+	0.987195i	$0.550994\pi$
$614$	0	0
$615$	−17105.8	−1.12158
$616$	0	0
$617$	10369.8	0.676616	0.338308	−	0.941035i	$-0.390145\pi$
$617$	10369.8	0.676616	0.338308	+	0.941035i	$0.390145\pi$
$618$	0	0
$619$	−21706.1	−1.40944	−0.704720	−	0.709485i	$-0.748928\pi$
$619$	−21706.1	−1.40944	−0.704720	+	0.709485i	$0.748928\pi$
$620$	0	0
$621$	6555.08	0.423585
$622$	0	0
$623$	466.540	0.0300024
$624$	0	0
$625$	−15692.8	−1.00434
$626$	0	0
$627$	1875.55	0.119461
$628$	0	0
$629$	−12561.8	−0.796300
$630$	0	0
$631$	2159.37	0.136233	0.0681167	−	0.997677i	$-0.478301\pi$
$631$	2159.37	0.136233	0.0681167	+	0.997677i	$0.478301\pi$
$632$	0	0
$633$	3172.83	0.199224
$634$	0	0
$635$	−3560.70	−0.222523
$636$	0	0
$637$	47828.3	2.97492
$638$	0	0
$639$	−2177.98	−0.134835
$640$	0	0
$641$	13128.4	0.808956	0.404478	−	0.914548i	$-0.367453\pi$
$641$	13128.4	0.808956	0.404478	+	0.914548i	$0.367453\pi$
$642$	0	0
$643$	19778.4	1.21304	0.606520	−	0.795068i	$-0.292565\pi$
$643$	19778.4	1.21304	0.606520	+	0.795068i	$0.292565\pi$
$644$	0	0
$645$	−14057.8	−0.858176
$646$	0	0
$647$	25540.6	1.55194	0.775969	−	0.630771i	$-0.217261\pi$
$647$	25540.6	1.55194	0.775969	+	0.630771i	$0.217261\pi$
$648$	0	0
$649$	346.223	0.0209406
$650$	0	0
$651$	18736.3	1.12801
$652$	0	0
$653$	28195.4	1.68969	0.844847	−	0.535008i	$-0.179691\pi$
$653$	28195.4	1.68969	0.844847	+	0.535008i	$0.179691\pi$
$654$	0	0
$655$	22532.4	1.34414
$656$	0	0
$657$	−16890.3	−1.00297
$658$	0	0
$659$	12042.9	0.711876	0.355938	−	0.934510i	$-0.384161\pi$
$659$	12042.9	0.711876	0.355938	+	0.934510i	$0.384161\pi$
$660$	0	0
$661$	20530.6	1.20809	0.604044	−	0.796951i	$-0.293555\pi$
$661$	20530.6	1.20809	0.604044	+	0.796951i	$0.293555\pi$
$662$	0	0
$663$	15115.0	0.885399
$664$	0	0
$665$	−9177.31	−0.535159
$666$	0	0
$667$	5820.08	0.337863
$668$	0	0
$669$	−2006.98	−0.115985
$670$	0	0
$671$	−31719.1	−1.82489
$672$	0	0
$673$	−13865.1	−0.794143	−0.397072	−	0.917788i	$-0.629974\pi$
$673$	−13865.1	−0.794143	−0.397072	+	0.917788i	$0.629974\pi$
$674$	0	0
$675$	−14837.6	−0.846074
$676$	0	0
$677$	33998.6	1.93009	0.965047	−	0.262077i	$-0.0844076\pi$
$677$	33998.6	1.93009	0.965047	+	0.262077i	$0.0844076\pi$
$678$	0	0
$679$	−19499.7	−1.10211
$680$	0	0
$681$	−14294.9	−0.804380
$682$	0	0
$683$	12811.9	0.717767	0.358884	−	0.933382i	$-0.383158\pi$
$683$	12811.9	0.717767	0.358884	+	0.933382i	$0.383158\pi$
$684$	0	0
$685$	−43031.2	−2.40020
$686$	0	0
$687$	−3884.71	−0.215736
$688$	0	0
$689$	−39520.2	−2.18519
$690$	0	0
$691$	−18409.1	−1.01348	−0.506739	−	0.862099i	$-0.669149\pi$
$691$	−18409.1	−1.01348	−0.506739	+	0.862099i	$0.669149\pi$
$692$	0	0
$693$	−25126.7	−1.37733
$694$	0	0
$695$	21585.6	1.17811
$696$	0	0
$697$	32547.9	1.76878
$698$	0	0
$699$	−12027.8	−0.650835
$700$	0	0
$701$	−5403.16	−0.291119	−0.145560	−	0.989349i	$-0.546498\pi$
$701$	−5403.16	−0.291119	−0.145560	+	0.989349i	$0.546498\pi$
$702$	0	0
$703$	−3182.37	−0.170733
$704$	0	0
$705$	15918.6	0.850394
$706$	0	0
$707$	7057.22	0.375409
$708$	0	0
$709$	−10920.3	−0.578448	−0.289224	−	0.957261i	$-0.593397\pi$
$709$	−10920.3	−0.578448	−0.289224	+	0.957261i	$0.593397\pi$
$710$	0	0
$711$	8160.21	0.430425
$712$	0	0
$713$	−13497.9	−0.708976
$714$	0	0
$715$	−50450.5	−2.63880
$716$	0	0
$717$	−13760.9	−0.716750
$718$	0	0
$719$	−13208.7	−0.685121	−0.342560	−	0.939496i	$-0.611294\pi$
$719$	−13208.7	−0.685121	−0.342560	+	0.939496i	$0.611294\pi$
$720$	0	0
$721$	1728.48	0.0892817
$722$	0	0
$723$	−532.627	−0.0273978
$724$	0	0
$725$	−13173.9	−0.674850
$726$	0	0
$727$	−36346.1	−1.85420	−0.927099	−	0.374818i	$-0.877705\pi$
$727$	−36346.1	−1.85420	−0.927099	+	0.374818i	$0.877705\pi$
$728$	0	0
$729$	2563.90	0.130260
$730$	0	0
$731$	26748.3	1.35338
$732$	0	0
$733$	14611.4	0.736267	0.368134	−	0.929773i	$-0.379997\pi$
$733$	14611.4	0.736267	0.368134	+	0.929773i	$0.379997\pi$
$734$	0	0
$735$	−23344.6	−1.17153
$736$	0	0
$737$	−36301.3	−1.81435
$738$	0	0
$739$	−29945.8	−1.49063	−0.745314	−	0.666714i	$-0.767700\pi$
$739$	−29945.8	−1.49063	−0.745314	+	0.666714i	$0.767700\pi$
$740$	0	0
$741$	3829.19	0.189837
$742$	0	0
$743$	30613.4	1.51157	0.755784	−	0.654820i	$-0.227256\pi$
$743$	30613.4	1.51157	0.755784	+	0.654820i	$0.227256\pi$
$744$	0	0
$745$	−2435.53	−0.119773
$746$	0	0
$747$	2552.96	0.125044
$748$	0	0
$749$	12636.5	0.616459
$750$	0	0
$751$	−18280.7	−0.888247	−0.444123	−	0.895966i	$-0.646485\pi$
$751$	−18280.7	−0.888247	−0.444123	+	0.895966i	$0.646485\pi$
$752$	0	0
$753$	4008.10	0.193975
$754$	0	0
$755$	−14770.3	−0.711982
$756$	0	0
$757$	24217.6	1.16275	0.581377	−	0.813635i	$-0.302514\pi$
$757$	24217.6	1.16275	0.581377	+	0.813635i	$0.302514\pi$
$758$	0	0
$759$	−5427.53	−0.259561
$760$	0	0
$761$	12065.7	0.574745	0.287372	−	0.957819i	$-0.407218\pi$
$761$	12065.7	0.574745	0.287372	+	0.957819i	$0.407218\pi$
$762$	0	0
$763$	20720.7	0.983145
$764$	0	0
$765$	24605.2	1.16288
$766$	0	0
$767$	706.861	0.0332768
$768$	0	0
$769$	−26520.6	−1.24364	−0.621819	−	0.783161i	$-0.713606\pi$
$769$	−26520.6	−1.24364	−0.621819	+	0.783161i	$0.713606\pi$
$770$	0	0
$771$	5309.41	0.248007
$772$	0	0
$773$	−29188.7	−1.35814	−0.679072	−	0.734072i	$-0.737617\pi$
$773$	−29188.7	−1.35814	−0.679072	+	0.734072i	$0.737617\pi$
$774$	0	0
$775$	30552.8	1.41612
$776$	0	0
$777$	−12783.3	−0.590215
$778$	0	0
$779$	8245.58	0.379241
$780$	0	0
$781$	4147.39	0.190019
$782$	0	0
$783$	12619.8	0.575985
$784$	0	0
$785$	51001.2	2.31887
$786$	0	0
$787$	−14545.4	−0.658814	−0.329407	−	0.944188i	$-0.606849\pi$
$787$	−14545.4	−0.658814	−0.329407	+	0.944188i	$0.606849\pi$
$788$	0	0
$789$	−1616.01	−0.0729170
$790$	0	0
$791$	−11280.0	−0.507043
$792$	0	0
$793$	−64758.9	−2.89994
$794$	0	0
$795$	19289.5	0.860538
$796$	0	0
$797$	14218.4	0.631922	0.315961	−	0.948772i	$-0.397673\pi$
$797$	14218.4	0.631922	0.315961	+	0.948772i	$0.397673\pi$
$798$	0	0
$799$	−30288.9	−1.34111
$800$	0	0
$801$	−316.882	−0.0139781
$802$	0	0
$803$	32163.1	1.41346
$804$	0	0
$805$	26557.6	1.16277
$806$	0	0
$807$	−7406.15	−0.323059
$808$	0	0
$809$	−6707.25	−0.291489	−0.145744	−	0.989322i	$-0.546558\pi$
$809$	−6707.25	−0.291489	−0.145744	+	0.989322i	$0.546558\pi$
$810$	0	0
$811$	29240.1	1.26604	0.633021	−	0.774135i	$-0.281815\pi$
$811$	29240.1	1.26604	0.633021	+	0.774135i	$0.281815\pi$
$812$	0	0
$813$	−20116.0	−0.867773
$814$	0	0
$815$	−40881.1	−1.75706
$816$	0	0
$817$	6776.32	0.290176
$818$	0	0
$819$	−51299.7	−2.18871
$820$	0	0
$821$	16904.2	0.718586	0.359293	−	0.933225i	$-0.383018\pi$
$821$	16904.2	0.718586	0.359293	+	0.933225i	$0.383018\pi$
$822$	0	0
$823$	25386.1	1.07522	0.537609	−	0.843194i	$-0.319328\pi$
$823$	25386.1	1.07522	0.537609	+	0.843194i	$0.319328\pi$
$824$	0	0
$825$	12285.3	0.518450
$826$	0	0
$827$	12343.1	0.518996	0.259498	−	0.965744i	$-0.416443\pi$
$827$	12343.1	0.518996	0.259498	+	0.965744i	$0.416443\pi$
$828$	0	0
$829$	−73.2483	−0.00306878	−0.00153439	−	0.999999i	$-0.500488\pi$
$829$	−73.2483	−0.00306878	−0.00153439	+	0.999999i	$0.500488\pi$
$830$	0	0
$831$	3238.63	0.135195
$832$	0	0
$833$	44418.7	1.84756
$834$	0	0
$835$	3007.04	0.124626
$836$	0	0
$837$	−29267.9	−1.20866
$838$	0	0
$839$	31028.4	1.27678	0.638391	−	0.769713i	$-0.279600\pi$
$839$	31028.4	1.27678	0.638391	+	0.769713i	$0.279600\pi$
$840$	0	0
$841$	−13184.2	−0.540579
$842$	0	0
$843$	14049.3	0.574003
$844$	0	0
$845$	−68301.9	−2.78066
$846$	0	0
$847$	7142.62	0.289756
$848$	0	0
$849$	−16656.8	−0.673333
$850$	0	0
$851$	9209.23	0.370962
$852$	0	0
$853$	−15219.0	−0.610891	−0.305445	−	0.952210i	$-0.598805\pi$
$853$	−15219.0	−0.610891	−0.305445	+	0.952210i	$0.598805\pi$
$854$	0	0
$855$	6233.40	0.249331
$856$	0	0
$857$	−16775.1	−0.668644	−0.334322	−	0.942459i	$-0.608507\pi$
$857$	−16775.1	−0.668644	−0.334322	+	0.942459i	$0.608507\pi$
$858$	0	0
$859$	40428.7	1.60583	0.802917	−	0.596091i	$-0.203280\pi$
$859$	40428.7	1.60583	0.802917	+	0.596091i	$0.203280\pi$
$860$	0	0
$861$	33121.7	1.31102
$862$	0	0
$863$	28177.2	1.11143	0.555715	−	0.831373i	$-0.312445\pi$
$863$	28177.2	1.11143	0.555715	+	0.831373i	$0.312445\pi$
$864$	0	0
$865$	37517.6	1.47473
$866$	0	0
$867$	1776.54	0.0695897
$868$	0	0
$869$	−15539.0	−0.606586
$870$	0	0
$871$	−74114.0	−2.88319
$872$	0	0
$873$	13244.6	0.513471
$874$	0	0
$875$	263.282	0.0101721
$876$	0	0
$877$	31702.0	1.22064	0.610319	−	0.792156i	$-0.291041\pi$
$877$	31702.0	1.22064	0.610319	+	0.792156i	$0.291041\pi$
$878$	0	0
$879$	10579.6	0.405964
$880$	0	0
$881$	17767.5	0.679457	0.339728	−	0.940524i	$-0.389665\pi$
$881$	17767.5	0.679457	0.339728	+	0.940524i	$0.389665\pi$
$882$	0	0
$883$	−4318.69	−0.164593	−0.0822964	−	0.996608i	$-0.526225\pi$
$883$	−4318.69	−0.164593	−0.0822964	+	0.996608i	$0.526225\pi$
$884$	0	0
$885$	−345.013	−0.0131045
$886$	0	0
$887$	38999.6	1.47630	0.738150	−	0.674637i	$-0.235700\pi$
$887$	38999.6	1.47630	0.738150	+	0.674637i	$0.235700\pi$
$888$	0	0
$889$	6894.54	0.260107
$890$	0	0
$891$	10415.1	0.391602
$892$	0	0
$893$	−7673.30	−0.287544
$894$	0	0
$895$	33502.2	1.25124
$896$	0	0
$897$	−11081.0	−0.412469
$898$	0	0
$899$	−25986.1	−0.964055
$900$	0	0
$901$	−36702.9	−1.35711
$902$	0	0
$903$	27219.8	1.00312
$904$	0	0
$905$	−31966.3	−1.17414
$906$	0	0
$907$	40814.6	1.49419	0.747093	−	0.664719i	$-0.231449\pi$
$907$	40814.6	1.49419	0.747093	+	0.664719i	$0.231449\pi$
$908$	0	0
$909$	−4793.40	−0.174903
$910$	0	0
$911$	−11344.0	−0.412560	−0.206280	−	0.978493i	$-0.566136\pi$
$911$	−11344.0	−0.412560	−0.206280	+	0.978493i	$0.566136\pi$
$912$	0	0
$913$	−4861.43	−0.176221
$914$	0	0
$915$	31608.3	1.14201
$916$	0	0
$917$	−43629.1	−1.57117
$918$	0	0
$919$	−8806.92	−0.316119	−0.158060	−	0.987430i	$-0.550524\pi$
$919$	−8806.92	−0.316119	−0.158060	+	0.987430i	$0.550524\pi$
$920$	0	0
$921$	−9693.41	−0.346807
$922$	0	0
$923$	8467.45	0.301961
$924$	0	0
$925$	−20845.3	−0.740963
$926$	0	0
$927$	−1174.02	−0.0415963
$928$	0	0
$929$	37674.2	1.33052	0.665258	−	0.746614i	$-0.268322\pi$
$929$	37674.2	1.33052	0.665258	+	0.746614i	$0.268322\pi$
$930$	0	0
$931$	11252.9	0.396132
$932$	0	0
$933$	−13235.0	−0.464410
$934$	0	0
$935$	−46854.1	−1.63882
$936$	0	0
$937$	28079.1	0.978979	0.489490	−	0.872009i	$-0.337183\pi$
$937$	28079.1	0.978979	0.489490	+	0.872009i	$0.337183\pi$
$938$	0	0
$939$	−15577.7	−0.541382
$940$	0	0
$941$	−55220.4	−1.91300	−0.956500	−	0.291731i	$-0.905769\pi$
$941$	−55220.4	−1.91300	−0.956500	+	0.291731i	$0.905769\pi$
$942$	0	0
$943$	−23861.3	−0.823999
$944$	0	0
$945$	57585.5	1.98228
$946$	0	0
$947$	13657.5	0.468647	0.234323	−	0.972159i	$-0.424713\pi$
$947$	13657.5	0.468647	0.234323	+	0.972159i	$0.424713\pi$
$948$	0	0
$949$	65665.4	2.24614
$950$	0	0
$951$	3263.89	0.111292
$952$	0	0
$953$	9373.39	0.318609	0.159304	−	0.987230i	$-0.449075\pi$
$953$	9373.39	0.318609	0.159304	+	0.987230i	$0.449075\pi$
$954$	0	0
$955$	36774.9	1.24608
$956$	0	0
$957$	−10449.1	−0.352947
$958$	0	0
$959$	83320.6	2.80559
$960$	0	0
$961$	30475.8	1.02299
$962$	0	0
$963$	−8582.94	−0.287208
$964$	0	0
$965$	−5544.29	−0.184950
$966$	0	0
$967$	−50513.6	−1.67984	−0.839921	−	0.542709i	$-0.817399\pi$
$967$	−50513.6	−1.67984	−0.839921	+	0.542709i	$0.817399\pi$
$968$	0	0
$969$	3556.22	0.117897
$970$	0	0
$971$	21215.2	0.701162	0.350581	−	0.936532i	$-0.385984\pi$
$971$	21215.2	0.701162	0.350581	+	0.936532i	$0.385984\pi$
$972$	0	0
$973$	−41795.9	−1.37710
$974$	0	0
$975$	25082.2	0.823870
$976$	0	0
$977$	−10649.2	−0.348720	−0.174360	−	0.984682i	$-0.555786\pi$
$977$	−10649.2	−0.348720	−0.174360	+	0.984682i	$0.555786\pi$
$978$	0	0
$979$	603.419	0.0196990
$980$	0	0
$981$	−14073.9	−0.458047
$982$	0	0
$983$	−43021.6	−1.39591	−0.697953	−	0.716143i	$-0.745906\pi$
$983$	−43021.6	−1.39591	−0.697953	+	0.716143i	$0.745906\pi$
$984$	0	0
$985$	−28754.4	−0.930145
$986$	0	0
$987$	−30822.9	−0.994026
$988$	0	0
$989$	−19609.5	−0.630482
$990$	0	0
$991$	1894.52	0.0607278	0.0303639	−	0.999539i	$-0.490333\pi$
$991$	1894.52	0.0607278	0.0303639	+	0.999539i	$0.490333\pi$
$992$	0	0
$993$	17044.7	0.544711
$994$	0	0
$995$	−1492.44	−0.0475513
$996$	0	0
$997$	48718.3	1.54757	0.773784	−	0.633450i	$-0.218362\pi$
$997$	48718.3	1.54757	0.773784	+	0.633450i	$0.218362\pi$
$998$	0	0
$999$	19968.6	0.632412

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	152.4.a.d.1.3	✓	5
3.2	odd	2		1368.4.a.m.1.4		5
4.3	odd	2		304.4.a.k.1.3		5
8.3	odd	2		1216.4.a.bb.1.3		5
8.5	even	2		1216.4.a.ba.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.a.d.1.3	✓	5	1.1	even	1	trivial
304.4.a.k.1.3		5	4.3	odd	2
1216.4.a.ba.1.3		5	8.5	even	2
1216.4.a.bb.1.3		5	8.3	odd	2
1368.4.a.m.1.4		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)

\(f(q)\)	\(=\)	\(q+2.49563 q^{3} -15.7941 q^{5} +30.5820 q^{7} -20.7718 q^{9} +39.5545 q^{11} +80.7559 q^{13} -39.4163 q^{15} +74.9991 q^{17} +19.0000 q^{19} +76.3212 q^{21} -54.9828 q^{23} +124.455 q^{25} -119.221 q^{27} -105.853 q^{29} +245.493 q^{31} +98.7132 q^{33} -483.016 q^{35} -167.493 q^{37} +201.536 q^{39} +433.978 q^{41} +356.648 q^{43} +328.073 q^{45} -403.858 q^{47} +592.257 q^{49} +187.170 q^{51} -489.379 q^{53} -624.729 q^{55} +47.4169 q^{57} +8.75307 q^{59} -801.910 q^{61} -635.244 q^{63} -1275.47 q^{65} -917.754 q^{67} -137.216 q^{69} +104.852 q^{71} +813.135 q^{73} +310.593 q^{75} +1209.65 q^{77} -392.850 q^{79} +263.309 q^{81} -122.905 q^{83} -1184.55 q^{85} -264.169 q^{87} +15.2554 q^{89} +2469.67 q^{91} +612.659 q^{93} -300.089 q^{95} -637.620 q^{97} -821.619 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} + 22 q^{7} + 83 q^{9} + 6 q^{11} + 82 q^{13} + 204 q^{15} + 234 q^{17} + 95 q^{19} + 308 q^{21} - 60 q^{23} + 549 q^{25} - 190 q^{27} + 210 q^{29} - 224 q^{31} + 704 q^{33} + 42 q^{35} + 614 q^{37}+ \cdots + 126 q^{99}+O(q^{100}) \) 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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