LMFDB - Embedded newform 616.4.a.i.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [616,4,Mod(1,616)] 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("616.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(616, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 616 = 2^{3} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	616.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$36.3451765635$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 47x^{4} + 10x^{3} + 612x^{2} + 240x - 1440 $ x^6 - x^5 - 47x^4 + 10x^3 + 612x^2 + 240x - 1440
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.66461$ of defining polynomial
Character	$\chi$	$=$	616.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.94660 q^{3} +4.15482 q^{5} +7.00000 q^{7} -2.53111 q^{9} -11.0000 q^{11} +78.7489 q^{13} -20.5522 q^{15} -132.636 q^{17} +36.2244 q^{19} -34.6262 q^{21} +184.528 q^{23} -107.737 q^{25} +146.079 q^{27} -136.235 q^{29} -164.772 q^{31} +54.4126 q^{33} +29.0837 q^{35} -259.061 q^{37} -389.539 q^{39} +481.046 q^{41} +206.135 q^{43} -10.5163 q^{45} -137.434 q^{47} +49.0000 q^{49} +656.099 q^{51} +514.780 q^{53} -45.7030 q^{55} -179.188 q^{57} +122.650 q^{59} -365.067 q^{61} -17.7178 q^{63} +327.187 q^{65} +426.198 q^{67} -912.785 q^{69} +616.442 q^{71} -63.7880 q^{73} +532.935 q^{75} -77.0000 q^{77} +131.660 q^{79} -654.253 q^{81} +1315.46 q^{83} -551.079 q^{85} +673.901 q^{87} +1148.54 q^{89} +551.242 q^{91} +815.061 q^{93} +150.506 q^{95} -38.2946 q^{97} +27.8423 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} + 42 q^{7} + 60 q^{9} - 66 q^{11} - 6 q^{13} + 126 q^{15} - 14 q^{17} + 80 q^{19} + 254 q^{23} + 220 q^{25} + 90 q^{27} + 132 q^{29} - 52 q^{31} + 14 q^{35} - 518 q^{37} + 332 q^{39} + 486 q^{41}+ \cdots - 660 q^{99}+O(q^{100}) $ 6 * q + 2 * q^5 + 42 * q^7 + 60 * q^9 - 66 * q^11 - 6 * q^13 + 126 * q^15 - 14 * q^17 + 80 * q^19 + 254 * q^23 + 220 * q^25 + 90 * q^27 + 132 * q^29 - 52 * q^31 + 14 * q^35 - 518 * q^37 + 332 * q^39 + 486 * q^41 + 428 * q^43 - 244 * q^45 + 790 * q^47 + 294 * q^49 + 640 * q^51 - 40 * q^53 - 22 * q^55 + 2276 * q^57 + 436 * q^59 + 1034 * q^61 + 420 * q^63 + 800 * q^65 + 562 * q^67 + 530 * q^69 + 2474 * q^71 - 902 * q^73 + 1986 * q^75 - 462 * q^77 + 1636 * q^79 + 570 * q^81 + 3016 * q^83 + 236 * q^85 + 4340 * q^87 + 1750 * q^89 - 42 * q^91 + 150 * q^93 + 3628 * q^95 - 1250 * q^97 - 660 * 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
$3$	−4.94660	−0.951974	−0.475987	−	0.879452i	$-0.657909\pi$
$3$	−4.94660	−0.951974	−0.475987	+	0.879452i	$0.657909\pi$
$4$	0	0
$5$	4.15482	0.371618	0.185809	−	0.982586i	$-0.440509\pi$
$5$	4.15482	0.371618	0.185809	+	0.982586i	$0.440509\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−2.53111	−0.0937450
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	78.7489	1.68008	0.840039	−	0.542527i	$-0.182532\pi$
$13$	78.7489	1.68008	0.840039	+	0.542527i	$0.182532\pi$
$14$	0	0
$15$	−20.5522	−0.353771
$16$	0	0
$17$	−132.636	−1.89229	−0.946147	−	0.323737i	$-0.895061\pi$
$17$	−132.636	−1.89229	−0.946147	+	0.323737i	$0.895061\pi$
$18$	0	0
$19$	36.2244	0.437392	0.218696	−	0.975793i	$-0.429820\pi$
$19$	36.2244	0.437392	0.218696	+	0.975793i	$0.429820\pi$
$20$	0	0
$21$	−34.6262	−0.359812
$22$	0	0
$23$	184.528	1.67290	0.836449	−	0.548045i	$-0.184628\pi$
$23$	184.528	1.67290	0.836449	+	0.548045i	$0.184628\pi$
$24$	0	0
$25$	−107.737	−0.861900
$26$	0	0
$27$	146.079	1.04122
$28$	0	0
$29$	−136.235	−0.872353	−0.436176	−	0.899861i	$-0.643668\pi$
$29$	−136.235	−0.872353	−0.436176	+	0.899861i	$0.643668\pi$
$30$	0	0
$31$	−164.772	−0.954642	−0.477321	−	0.878729i	$-0.658392\pi$
$31$	−164.772	−0.954642	−0.477321	+	0.878729i	$0.658392\pi$
$32$	0	0
$33$	54.4126	0.287031
$34$	0	0
$35$	29.0837	0.140459
$36$	0	0
$37$	−259.061	−1.15106	−0.575532	−	0.817779i	$-0.695205\pi$
$37$	−259.061	−1.15106	−0.575532	+	0.817779i	$0.695205\pi$
$38$	0	0
$39$	−389.539	−1.59939
$40$	0	0
$41$	481.046	1.83236	0.916180	−	0.400768i	$-0.131257\pi$
$41$	481.046	1.83236	0.916180	+	0.400768i	$0.131257\pi$
$42$	0	0
$43$	206.135	0.731055	0.365527	−	0.930801i	$-0.380889\pi$
$43$	206.135	0.731055	0.365527	+	0.930801i	$0.380889\pi$
$44$	0	0
$45$	−10.5163	−0.0348373
$46$	0	0
$47$	−137.434	−0.426527	−0.213263	−	0.976995i	$-0.568409\pi$
$47$	−137.434	−0.426527	−0.213263	+	0.976995i	$0.568409\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	656.099	1.80142
$52$	0	0
$53$	514.780	1.33416	0.667080	−	0.744986i	$-0.267544\pi$
$53$	514.780	1.33416	0.667080	+	0.744986i	$0.267544\pi$
$54$	0	0
$55$	−45.7030	−0.112047
$56$	0	0
$57$	−179.188	−0.416386
$58$	0	0
$59$	122.650	0.270639	0.135319	−	0.990802i	$-0.456794\pi$
$59$	122.650	0.270639	0.135319	+	0.990802i	$0.456794\pi$
$60$	0	0
$61$	−365.067	−0.766262	−0.383131	−	0.923694i	$-0.625154\pi$
$61$	−365.067	−0.766262	−0.383131	+	0.923694i	$0.625154\pi$
$62$	0	0
$63$	−17.7178	−0.0354323
$64$	0	0
$65$	327.187	0.624347
$66$	0	0
$67$	426.198	0.777140	0.388570	−	0.921419i	$-0.372969\pi$
$67$	426.198	0.777140	0.388570	+	0.921419i	$0.372969\pi$
$68$	0	0
$69$	−912.785	−1.59256
$70$	0	0
$71$	616.442	1.03040	0.515198	−	0.857071i	$-0.327718\pi$
$71$	616.442	1.03040	0.515198	+	0.857071i	$0.327718\pi$
$72$	0	0
$73$	−63.7880	−0.102272	−0.0511358	−	0.998692i	$-0.516284\pi$
$73$	−63.7880	−0.102272	−0.0511358	+	0.998692i	$0.516284\pi$
$74$	0	0
$75$	532.935	0.820506
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	131.660	0.187505	0.0937523	−	0.995596i	$-0.470114\pi$
$79$	131.660	0.187505	0.0937523	+	0.995596i	$0.470114\pi$
$80$	0	0
$81$	−654.253	−0.897467
$82$	0	0
$83$	1315.46	1.73964	0.869820	−	0.493370i	$-0.164235\pi$
$83$	1315.46	1.73964	0.869820	+	0.493370i	$0.164235\pi$
$84$	0	0
$85$	−551.079	−0.703211
$86$	0	0
$87$	673.901	0.830457
$88$	0	0
$89$	1148.54	1.36792	0.683962	−	0.729517i	$-0.260255\pi$
$89$	1148.54	1.36792	0.683962	+	0.729517i	$0.260255\pi$
$90$	0	0
$91$	551.242	0.635010
$92$	0	0
$93$	815.061	0.908794
$94$	0	0
$95$	150.506	0.162543
$96$	0	0
$97$	−38.2946	−0.0400848	−0.0200424	−	0.999799i	$-0.506380\pi$
$97$	−38.2946	−0.0400848	−0.0200424	+	0.999799i	$0.506380\pi$
$98$	0	0
$99$	27.8423	0.0282652
$100$	0	0
$101$	1722.54	1.69702	0.848509	−	0.529180i	$-0.177501\pi$
$101$	1722.54	1.69702	0.848509	+	0.529180i	$0.177501\pi$
$102$	0	0
$103$	1603.05	1.53352	0.766761	−	0.641933i	$-0.221867\pi$
$103$	1603.05	1.53352	0.766761	+	0.641933i	$0.221867\pi$
$104$	0	0
$105$	−143.866	−0.133713
$106$	0	0
$107$	608.658	0.549918	0.274959	−	0.961456i	$-0.411336\pi$
$107$	608.658	0.549918	0.274959	+	0.961456i	$0.411336\pi$
$108$	0	0
$109$	86.7416	0.0762233	0.0381116	−	0.999273i	$-0.487866\pi$
$109$	86.7416	0.0762233	0.0381116	+	0.999273i	$0.487866\pi$
$110$	0	0
$111$	1281.47	1.09578
$112$	0	0
$113$	703.471	0.585637	0.292819	−	0.956168i	$-0.405407\pi$
$113$	703.471	0.585637	0.292819	+	0.956168i	$0.405407\pi$
$114$	0	0
$115$	766.679	0.621680
$116$	0	0
$117$	−199.322	−0.157499
$118$	0	0
$119$	−928.453	−0.715220
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−2379.54	−1.74436
$124$	0	0
$125$	−966.982	−0.691916
$126$	0	0
$127$	2300.41	1.60731	0.803654	−	0.595097i	$-0.202886\pi$
$127$	2300.41	1.60731	0.803654	+	0.595097i	$0.202886\pi$
$128$	0	0
$129$	−1019.67	−0.695945
$130$	0	0
$131$	−2128.36	−1.41951	−0.709755	−	0.704449i	$-0.751194\pi$
$131$	−2128.36	−1.41951	−0.709755	+	0.704449i	$0.751194\pi$
$132$	0	0
$133$	253.571	0.165319
$134$	0	0
$135$	606.931	0.386935
$136$	0	0
$137$	1624.74	1.01322	0.506610	−	0.862176i	$-0.330899\pi$
$137$	1624.74	1.01322	0.506610	+	0.862176i	$0.330899\pi$
$138$	0	0
$139$	−2125.78	−1.29717	−0.648583	−	0.761144i	$-0.724638\pi$
$139$	−2125.78	−1.29717	−0.648583	+	0.761144i	$0.724638\pi$
$140$	0	0
$141$	679.830	0.406043
$142$	0	0
$143$	−866.237	−0.506562
$144$	0	0
$145$	−566.032	−0.324182
$146$	0	0
$147$	−242.384	−0.135996
$148$	0	0
$149$	−2827.03	−1.55436	−0.777179	−	0.629279i	$-0.783350\pi$
$149$	−2827.03	−1.55436	−0.777179	+	0.629279i	$0.783350\pi$
$150$	0	0
$151$	1528.99	0.824022	0.412011	−	0.911179i	$-0.364827\pi$
$151$	1528.99	0.824022	0.412011	+	0.911179i	$0.364827\pi$
$152$	0	0
$153$	335.717	0.177393
$154$	0	0
$155$	−684.597	−0.354762
$156$	0	0
$157$	−1261.67	−0.641349	−0.320675	−	0.947189i	$-0.603910\pi$
$157$	−1261.67	−0.641349	−0.320675	+	0.947189i	$0.603910\pi$
$158$	0	0
$159$	−2546.41	−1.27009
$160$	0	0
$161$	1291.69	0.632296
$162$	0	0
$163$	1931.11	0.927953	0.463976	−	0.885848i	$-0.346422\pi$
$163$	1931.11	0.927953	0.463976	+	0.885848i	$0.346422\pi$
$164$	0	0
$165$	226.075	0.106666
$166$	0	0
$167$	1558.03	0.721941	0.360971	−	0.932577i	$-0.382445\pi$
$167$	1558.03	0.721941	0.360971	+	0.932577i	$0.382445\pi$
$168$	0	0
$169$	4004.38	1.82266
$170$	0	0
$171$	−91.6882	−0.0410033
$172$	0	0
$173$	−3483.48	−1.53089	−0.765445	−	0.643501i	$-0.777481\pi$
$173$	−3483.48	−1.53089	−0.765445	+	0.643501i	$0.777481\pi$
$174$	0	0
$175$	−754.162	−0.325768
$176$	0	0
$177$	−606.702	−0.257641
$178$	0	0
$179$	1118.33	0.466970	0.233485	−	0.972360i	$-0.424987\pi$
$179$	1118.33	0.466970	0.233485	+	0.972360i	$0.424987\pi$
$180$	0	0
$181$	1367.11	0.561415	0.280708	−	0.959793i	$-0.409431\pi$
$181$	1367.11	0.561415	0.280708	+	0.959793i	$0.409431\pi$
$182$	0	0
$183$	1805.84	0.729462
$184$	0	0
$185$	−1076.35	−0.427757
$186$	0	0
$187$	1459.00	0.570548
$188$	0	0
$189$	1022.55	0.393543
$190$	0	0
$191$	2887.04	1.09371	0.546856	−	0.837226i	$-0.315824\pi$
$191$	2887.04	1.09371	0.546856	+	0.837226i	$0.315824\pi$
$192$	0	0
$193$	−3261.14	−1.21628	−0.608140	−	0.793830i	$-0.708084\pi$
$193$	−3261.14	−1.21628	−0.608140	+	0.793830i	$0.708084\pi$
$194$	0	0
$195$	−1618.47	−0.594363
$196$	0	0
$197$	−3711.47	−1.34229	−0.671145	−	0.741326i	$-0.734197\pi$
$197$	−3711.47	−1.34229	−0.671145	+	0.741326i	$0.734197\pi$
$198$	0	0
$199$	−4991.72	−1.77816	−0.889080	−	0.457752i	$-0.848655\pi$
$199$	−4991.72	−1.77816	−0.889080	+	0.457752i	$0.848655\pi$
$200$	0	0
$201$	−2108.23	−0.739818
$202$	0	0
$203$	−953.646	−0.329718
$204$	0	0
$205$	1998.66	0.680938
$206$	0	0
$207$	−467.060	−0.156826
$208$	0	0
$209$	−398.469	−0.131879
$210$	0	0
$211$	3119.80	1.01790	0.508948	−	0.860797i	$-0.330035\pi$
$211$	3119.80	1.01790	0.508948	+	0.860797i	$0.330035\pi$
$212$	0	0
$213$	−3049.29	−0.980911
$214$	0	0
$215$	856.455	0.271673
$216$	0	0
$217$	−1153.40	−0.360821
$218$	0	0
$219$	315.534	0.0973599
$220$	0	0
$221$	−10445.0	−3.17920
$222$	0	0
$223$	−3603.52	−1.08211	−0.541053	−	0.840988i	$-0.681974\pi$
$223$	−3603.52	−1.08211	−0.541053	+	0.840988i	$0.681974\pi$
$224$	0	0
$225$	272.696	0.0807988
$226$	0	0
$227$	1364.41	0.398938	0.199469	−	0.979904i	$-0.436078\pi$
$227$	1364.41	0.398938	0.199469	+	0.979904i	$0.436078\pi$
$228$	0	0
$229$	3094.45	0.892955	0.446478	−	0.894795i	$-0.352678\pi$
$229$	3094.45	0.892955	0.446478	+	0.894795i	$0.352678\pi$
$230$	0	0
$231$	380.888	0.108488
$232$	0	0
$233$	6664.22	1.87377	0.936883	−	0.349643i	$-0.113697\pi$
$233$	6664.22	1.87377	0.936883	+	0.349643i	$0.113697\pi$
$234$	0	0
$235$	−571.012	−0.158505
$236$	0	0
$237$	−651.268	−0.178500
$238$	0	0
$239$	−2363.74	−0.639738	−0.319869	−	0.947462i	$-0.603639\pi$
$239$	−2363.74	−0.639738	−0.319869	+	0.947462i	$0.603639\pi$
$240$	0	0
$241$	−2276.63	−0.608510	−0.304255	−	0.952591i	$-0.598407\pi$
$241$	−2276.63	−0.608510	−0.304255	+	0.952591i	$0.598407\pi$
$242$	0	0
$243$	−707.793	−0.186852
$244$	0	0
$245$	203.586	0.0530883
$246$	0	0
$247$	2852.63	0.734853
$248$	0	0
$249$	−6507.04	−1.65609
$250$	0	0
$251$	−2181.46	−0.548577	−0.274288	−	0.961647i	$-0.588442\pi$
$251$	−2181.46	−0.548577	−0.274288	+	0.961647i	$0.588442\pi$
$252$	0	0
$253$	−2029.80	−0.504398
$254$	0	0
$255$	2725.97	0.669439
$256$	0	0
$257$	−6355.96	−1.54270	−0.771350	−	0.636411i	$-0.780418\pi$
$257$	−6355.96	−1.54270	−0.771350	+	0.636411i	$0.780418\pi$
$258$	0	0
$259$	−1813.43	−0.435061
$260$	0	0
$261$	344.827	0.0817787
$262$	0	0
$263$	7573.22	1.77561	0.887804	−	0.460222i	$-0.152230\pi$
$263$	7573.22	1.77561	0.887804	+	0.460222i	$0.152230\pi$
$264$	0	0
$265$	2138.82	0.495798
$266$	0	0
$267$	−5681.39	−1.30223
$268$	0	0
$269$	5598.66	1.26898	0.634492	−	0.772930i	$-0.281210\pi$
$269$	5598.66	1.26898	0.634492	+	0.772930i	$0.281210\pi$
$270$	0	0
$271$	−1988.46	−0.445722	−0.222861	−	0.974850i	$-0.571540\pi$
$271$	−1988.46	−0.445722	−0.222861	+	0.974850i	$0.571540\pi$
$272$	0	0
$273$	−2726.78	−0.604513
$274$	0	0
$275$	1185.11	0.259873
$276$	0	0
$277$	−866.991	−0.188059	−0.0940297	−	0.995569i	$-0.529975\pi$
$277$	−866.991	−0.188059	−0.0940297	+	0.995569i	$0.529975\pi$
$278$	0	0
$279$	417.056	0.0894928
$280$	0	0
$281$	6965.30	1.47870	0.739351	−	0.673321i	$-0.235133\pi$
$281$	6965.30	1.47870	0.739351	+	0.673321i	$0.235133\pi$
$282$	0	0
$283$	−3403.02	−0.714800	−0.357400	−	0.933951i	$-0.616337\pi$
$283$	−3403.02	−0.714800	−0.357400	+	0.933951i	$0.616337\pi$
$284$	0	0
$285$	−744.494	−0.154737
$286$	0	0
$287$	3367.32	0.692567
$288$	0	0
$289$	12679.4	2.58078
$290$	0	0
$291$	189.428	0.0381597
$292$	0	0
$293$	7240.81	1.44373	0.721864	−	0.692034i	$-0.243285\pi$
$293$	7240.81	1.44373	0.721864	+	0.692034i	$0.243285\pi$
$294$	0	0
$295$	509.589	0.100574
$296$	0	0
$297$	−1606.87	−0.313939
$298$	0	0
$299$	14531.3	2.81060
$300$	0	0
$301$	1442.95	0.276313
$302$	0	0
$303$	−8520.71	−1.61552
$304$	0	0
$305$	−1516.79	−0.284757
$306$	0	0
$307$	−6366.22	−1.18352	−0.591758	−	0.806116i	$-0.701566\pi$
$307$	−6366.22	−1.18352	−0.591758	+	0.806116i	$0.701566\pi$
$308$	0	0
$309$	−7929.63	−1.45987
$310$	0	0
$311$	−7207.03	−1.31406	−0.657031	−	0.753864i	$-0.728188\pi$
$311$	−7207.03	−1.31406	−0.657031	+	0.753864i	$0.728188\pi$
$312$	0	0
$313$	−5677.60	−1.02529	−0.512647	−	0.858600i	$-0.671335\pi$
$313$	−5677.60	−1.02529	−0.512647	+	0.858600i	$0.671335\pi$
$314$	0	0
$315$	−73.6143	−0.0131673
$316$	0	0
$317$	−2108.47	−0.373576	−0.186788	−	0.982400i	$-0.559808\pi$
$317$	−2108.47	−0.373576	−0.186788	+	0.982400i	$0.559808\pi$
$318$	0	0
$319$	1498.59	0.263024
$320$	0	0
$321$	−3010.79	−0.523507
$322$	0	0
$323$	−4804.67	−0.827675
$324$	0	0
$325$	−8484.20	−1.44806
$326$	0	0
$327$	−429.076	−0.0725626
$328$	0	0
$329$	−962.036	−0.161212
$330$	0	0
$331$	4998.97	0.830115	0.415057	−	0.909795i	$-0.363761\pi$
$331$	4998.97	0.830115	0.415057	+	0.909795i	$0.363761\pi$
$332$	0	0
$333$	655.713	0.107906
$334$	0	0
$335$	1770.78	0.288800
$336$	0	0
$337$	−6200.00	−1.00218	−0.501091	−	0.865394i	$-0.667068\pi$
$337$	−6200.00	−1.00218	−0.501091	+	0.865394i	$0.667068\pi$
$338$	0	0
$339$	−3479.79	−0.557511
$340$	0	0
$341$	1812.49	0.287835
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−3792.46	−0.591823
$346$	0	0
$347$	−1480.21	−0.228997	−0.114498	−	0.993423i	$-0.536526\pi$
$347$	−1480.21	−0.228997	−0.114498	+	0.993423i	$0.536526\pi$
$348$	0	0
$349$	−2557.76	−0.392303	−0.196152	−	0.980574i	$-0.562844\pi$
$349$	−2557.76	−0.392303	−0.196152	+	0.980574i	$0.562844\pi$
$350$	0	0
$351$	11503.5	1.74933
$352$	0	0
$353$	6895.80	1.03973	0.519867	−	0.854247i	$-0.325981\pi$
$353$	6895.80	1.03973	0.519867	+	0.854247i	$0.325981\pi$
$354$	0	0
$355$	2561.20	0.382914
$356$	0	0
$357$	4592.69	0.680871
$358$	0	0
$359$	−2118.03	−0.311380	−0.155690	−	0.987806i	$-0.549760\pi$
$359$	−2118.03	−0.311380	−0.155690	+	0.987806i	$0.549760\pi$
$360$	0	0
$361$	−5546.79	−0.808688
$362$	0	0
$363$	−598.539	−0.0865431
$364$	0	0
$365$	−265.028	−0.0380060
$366$	0	0
$367$	−413.567	−0.0588229	−0.0294115	−	0.999567i	$-0.509363\pi$
$367$	−413.567	−0.0588229	−0.0294115	+	0.999567i	$0.509363\pi$
$368$	0	0
$369$	−1217.58	−0.171774
$370$	0	0
$371$	3603.46	0.504265
$372$	0	0
$373$	−2591.14	−0.359689	−0.179844	−	0.983695i	$-0.557559\pi$
$373$	−2591.14	−0.359689	−0.179844	+	0.983695i	$0.557559\pi$
$374$	0	0
$375$	4783.28	0.658686
$376$	0	0
$377$	−10728.4	−1.46562
$378$	0	0
$379$	6627.36	0.898218	0.449109	−	0.893477i	$-0.351742\pi$
$379$	6627.36	0.898218	0.449109	+	0.893477i	$0.351742\pi$
$380$	0	0
$381$	−11379.2	−1.53012
$382$	0	0
$383$	9668.47	1.28991	0.644955	−	0.764221i	$-0.276876\pi$
$383$	9668.47	1.28991	0.644955	+	0.764221i	$0.276876\pi$
$384$	0	0
$385$	−319.921	−0.0423498
$386$	0	0
$387$	−521.752	−0.0685327
$388$	0	0
$389$	3385.35	0.441245	0.220622	−	0.975359i	$-0.429191\pi$
$389$	3385.35	0.441245	0.220622	+	0.975359i	$0.429191\pi$
$390$	0	0
$391$	−24475.0	−3.16562
$392$	0	0
$393$	10528.2	1.35134
$394$	0	0
$395$	547.022	0.0696801
$396$	0	0
$397$	10491.3	1.32631	0.663153	−	0.748484i	$-0.269218\pi$
$397$	10491.3	1.32631	0.663153	+	0.748484i	$0.269218\pi$
$398$	0	0
$399$	−1254.32	−0.157379
$400$	0	0
$401$	−4544.60	−0.565951	−0.282975	−	0.959127i	$-0.591322\pi$
$401$	−4544.60	−0.565951	−0.282975	+	0.959127i	$0.591322\pi$
$402$	0	0
$403$	−12975.6	−1.60387
$404$	0	0
$405$	−2718.30	−0.333515
$406$	0	0
$407$	2849.67	0.347059
$408$	0	0
$409$	1203.82	0.145539	0.0727693	−	0.997349i	$-0.476816\pi$
$409$	1203.82	0.145539	0.0727693	+	0.997349i	$0.476816\pi$
$410$	0	0
$411$	−8036.95	−0.964559
$412$	0	0
$413$	858.551	0.102292
$414$	0	0
$415$	5465.48	0.646482
$416$	0	0
$417$	10515.4	1.23487
$418$	0	0
$419$	−1582.56	−0.184518	−0.0922591	−	0.995735i	$-0.529409\pi$
$419$	−1582.56	−0.184518	−0.0922591	+	0.995735i	$0.529409\pi$
$420$	0	0
$421$	−7545.83	−0.873542	−0.436771	−	0.899573i	$-0.643878\pi$
$421$	−7545.83	−0.873542	−0.436771	+	0.899573i	$0.643878\pi$
$422$	0	0
$423$	347.860	0.0399847
$424$	0	0
$425$	14289.9	1.63097
$426$	0	0
$427$	−2555.47	−0.289620
$428$	0	0
$429$	4284.93	0.482234
$430$	0	0
$431$	−4787.62	−0.535062	−0.267531	−	0.963549i	$-0.586208\pi$
$431$	−4787.62	−0.535062	−0.267531	+	0.963549i	$0.586208\pi$
$432$	0	0
$433$	−560.566	−0.0622149	−0.0311075	−	0.999516i	$-0.509903\pi$
$433$	−560.566	−0.0622149	−0.0311075	+	0.999516i	$0.509903\pi$
$434$	0	0
$435$	2799.94	0.308613
$436$	0	0
$437$	6684.41	0.731713
$438$	0	0
$439$	13922.3	1.51362	0.756808	−	0.653637i	$-0.226758\pi$
$439$	13922.3	1.51362	0.756808	+	0.653637i	$0.226758\pi$
$440$	0	0
$441$	−124.025	−0.0133921
$442$	0	0
$443$	−13604.9	−1.45912	−0.729560	−	0.683917i	$-0.760275\pi$
$443$	−13604.9	−1.45912	−0.729560	+	0.683917i	$0.760275\pi$
$444$	0	0
$445$	4771.99	0.508346
$446$	0	0
$447$	13984.2	1.47971
$448$	0	0
$449$	17220.8	1.81002	0.905011	−	0.425389i	$-0.139863\pi$
$449$	17220.8	1.81002	0.905011	+	0.425389i	$0.139863\pi$
$450$	0	0
$451$	−5291.50	−0.552477
$452$	0	0
$453$	−7563.30	−0.784448
$454$	0	0
$455$	2290.31	0.235981
$456$	0	0
$457$	7913.94	0.810062	0.405031	−	0.914303i	$-0.367261\pi$
$457$	7913.94	0.810062	0.405031	+	0.914303i	$0.367261\pi$
$458$	0	0
$459$	−19375.3	−1.97029
$460$	0	0
$461$	−16954.5	−1.71291	−0.856454	−	0.516224i	$-0.827337\pi$
$461$	−16954.5	−1.71291	−0.856454	+	0.516224i	$0.827337\pi$
$462$	0	0
$463$	−9959.12	−0.999653	−0.499827	−	0.866125i	$-0.666603\pi$
$463$	−9959.12	−0.999653	−0.499827	+	0.866125i	$0.666603\pi$
$464$	0	0
$465$	3386.43	0.337725
$466$	0	0
$467$	−8066.96	−0.799345	−0.399673	−	0.916658i	$-0.630876\pi$
$467$	−8066.96	−0.799345	−0.399673	+	0.916658i	$0.630876\pi$
$468$	0	0
$469$	2983.39	0.293731
$470$	0	0
$471$	6240.96	0.610548
$472$	0	0
$473$	−2267.49	−0.220421
$474$	0	0
$475$	−3902.73	−0.376989
$476$	0	0
$477$	−1302.97	−0.125071
$478$	0	0
$479$	13773.7	1.31386	0.656929	−	0.753953i	$-0.271855\pi$
$479$	13773.7	1.31386	0.656929	+	0.753953i	$0.271855\pi$
$480$	0	0
$481$	−20400.8	−1.93388
$482$	0	0
$483$	−6389.49	−0.601930
$484$	0	0
$485$	−159.107	−0.0148963
$486$	0	0
$487$	14147.3	1.31637	0.658187	−	0.752855i	$-0.271324\pi$
$487$	14147.3	1.31637	0.658187	+	0.752855i	$0.271324\pi$
$488$	0	0
$489$	−9552.44	−0.883387
$490$	0	0
$491$	8324.94	0.765172	0.382586	−	0.923920i	$-0.375034\pi$
$491$	8324.94	0.765172	0.382586	+	0.923920i	$0.375034\pi$
$492$	0	0
$493$	18069.7	1.65075
$494$	0	0
$495$	115.680	0.0105039
$496$	0	0
$497$	4315.09	0.389453
$498$	0	0
$499$	7980.12	0.715910	0.357955	−	0.933739i	$-0.383474\pi$
$499$	7980.12	0.715910	0.357955	+	0.933739i	$0.383474\pi$
$500$	0	0
$501$	−7706.97	−0.687269
$502$	0	0
$503$	−15689.1	−1.39074	−0.695370	−	0.718652i	$-0.744759\pi$
$503$	−15689.1	−1.39074	−0.695370	+	0.718652i	$0.744759\pi$
$504$	0	0
$505$	7156.83	0.630643
$506$	0	0
$507$	−19808.1	−1.73513
$508$	0	0
$509$	1889.31	0.164523	0.0822614	−	0.996611i	$-0.473786\pi$
$509$	1889.31	0.164523	0.0822614	+	0.996611i	$0.473786\pi$
$510$	0	0
$511$	−446.516	−0.0386550
$512$	0	0
$513$	5291.62	0.455421
$514$	0	0
$515$	6660.36	0.569885
$516$	0	0
$517$	1511.77	0.128603
$518$	0	0
$519$	17231.4	1.45737
$520$	0	0
$521$	13099.0	1.10149	0.550746	−	0.834673i	$-0.314343\pi$
$521$	13099.0	1.10149	0.550746	+	0.834673i	$0.314343\pi$
$522$	0	0
$523$	−9682.19	−0.809508	−0.404754	−	0.914426i	$-0.632643\pi$
$523$	−9682.19	−0.809508	−0.404754	+	0.914426i	$0.632643\pi$
$524$	0	0
$525$	3730.54	0.310122
$526$	0	0
$527$	21854.7	1.80646
$528$	0	0
$529$	21883.4	1.79859
$530$	0	0
$531$	−310.442	−0.0253710
$532$	0	0
$533$	37881.8	3.07850
$534$	0	0
$535$	2528.86	0.204359
$536$	0	0
$537$	−5531.92	−0.444544
$538$	0	0
$539$	−539.000	−0.0430730
$540$	0	0
$541$	11748.4	0.933644	0.466822	−	0.884351i	$-0.345399\pi$
$541$	11748.4	0.933644	0.466822	+	0.884351i	$0.345399\pi$
$542$	0	0
$543$	−6762.53	−0.534453
$544$	0	0
$545$	360.396	0.0283260
$546$	0	0
$547$	7583.28	0.592756	0.296378	−	0.955071i	$-0.404221\pi$
$547$	7583.28	0.592756	0.296378	+	0.955071i	$0.404221\pi$
$548$	0	0
$549$	924.025	0.0718332
$550$	0	0
$551$	−4935.04	−0.381560
$552$	0	0
$553$	921.617	0.0708701
$554$	0	0
$555$	5324.29	0.407213
$556$	0	0
$557$	−4090.51	−0.311168	−0.155584	−	0.987823i	$-0.549726\pi$
$557$	−4090.51	−0.311168	−0.155584	+	0.987823i	$0.549726\pi$
$558$	0	0
$559$	16232.9	1.22823
$560$	0	0
$561$	−7217.09	−0.543147
$562$	0	0
$563$	1451.52	0.108658	0.0543288	−	0.998523i	$-0.482698\pi$
$563$	1451.52	0.108658	0.0543288	+	0.998523i	$0.482698\pi$
$564$	0	0
$565$	2922.79	0.217633
$566$	0	0
$567$	−4579.77	−0.339211
$568$	0	0
$569$	−7066.86	−0.520665	−0.260332	−	0.965519i	$-0.583832\pi$
$569$	−7066.86	−0.520665	−0.260332	+	0.965519i	$0.583832\pi$
$570$	0	0
$571$	12465.8	0.913623	0.456812	−	0.889563i	$-0.348991\pi$
$571$	12465.8	0.913623	0.456812	+	0.889563i	$0.348991\pi$
$572$	0	0
$573$	−14281.1	−1.04119
$574$	0	0
$575$	−19880.5	−1.44187
$576$	0	0
$577$	2823.31	0.203702	0.101851	−	0.994800i	$-0.467524\pi$
$577$	2823.31	0.203702	0.101851	+	0.994800i	$0.467524\pi$
$578$	0	0
$579$	16131.6	1.15787
$580$	0	0
$581$	9208.19	0.657522
$582$	0	0
$583$	−5662.58	−0.402264
$584$	0	0
$585$	−828.148	−0.0585294
$586$	0	0
$587$	−8955.15	−0.629674	−0.314837	−	0.949146i	$-0.601950\pi$
$587$	−8955.15	−0.629674	−0.314837	+	0.949146i	$0.601950\pi$
$588$	0	0
$589$	−5968.77	−0.417553
$590$	0	0
$591$	18359.2	1.27782
$592$	0	0
$593$	18367.5	1.27194	0.635971	−	0.771713i	$-0.280600\pi$
$593$	18367.5	1.27194	0.635971	+	0.771713i	$0.280600\pi$
$594$	0	0
$595$	−3857.56	−0.265789
$596$	0	0
$597$	24692.1	1.69276
$598$	0	0
$599$	−13651.5	−0.931195	−0.465598	−	0.884997i	$-0.654161\pi$
$599$	−13651.5	−0.931195	−0.465598	+	0.884997i	$0.654161\pi$
$600$	0	0
$601$	184.678	0.0125344	0.00626718	−	0.999980i	$-0.498005\pi$
$601$	184.678	0.0125344	0.00626718	+	0.999980i	$0.498005\pi$
$602$	0	0
$603$	−1078.76	−0.0728530
$604$	0	0
$605$	502.733	0.0337835
$606$	0	0
$607$	14497.8	0.969435	0.484717	−	0.874671i	$-0.338922\pi$
$607$	14497.8	0.969435	0.484717	+	0.874671i	$0.338922\pi$
$608$	0	0
$609$	4717.31	0.313883
$610$	0	0
$611$	−10822.7	−0.716598
$612$	0	0
$613$	−20987.2	−1.38281	−0.691407	−	0.722466i	$-0.743009\pi$
$613$	−20987.2	−1.38281	−0.691407	+	0.722466i	$0.743009\pi$
$614$	0	0
$615$	−9886.57	−0.648236
$616$	0	0
$617$	−22195.3	−1.44822	−0.724108	−	0.689687i	$-0.757748\pi$
$617$	−22195.3	−1.44822	−0.724108	+	0.689687i	$0.757748\pi$
$618$	0	0
$619$	−24885.9	−1.61591	−0.807956	−	0.589243i	$-0.799426\pi$
$619$	−24885.9	−1.61591	−0.807956	+	0.589243i	$0.799426\pi$
$620$	0	0
$621$	26955.5	1.74185
$622$	0	0
$623$	8039.80	0.517027
$624$	0	0
$625$	9449.55	0.604771
$626$	0	0
$627$	1971.07	0.125545
$628$	0	0
$629$	34360.9	2.17815
$630$	0	0
$631$	−14226.2	−0.897521	−0.448761	−	0.893652i	$-0.648134\pi$
$631$	−14226.2	−0.897521	−0.448761	+	0.893652i	$0.648134\pi$
$632$	0	0
$633$	−15432.4	−0.969011
$634$	0	0
$635$	9557.77	0.597305
$636$	0	0
$637$	3858.69	0.240011
$638$	0	0
$639$	−1560.28	−0.0965945
$640$	0	0
$641$	20663.6	1.27326	0.636632	−	0.771168i	$-0.280327\pi$
$641$	20663.6	1.27326	0.636632	+	0.771168i	$0.280327\pi$
$642$	0	0
$643$	−23272.0	−1.42731	−0.713653	−	0.700499i	$-0.752961\pi$
$643$	−23272.0	−1.42731	−0.713653	+	0.700499i	$0.752961\pi$
$644$	0	0
$645$	−4236.54	−0.258626
$646$	0	0
$647$	−5842.03	−0.354983	−0.177491	−	0.984122i	$-0.556798\pi$
$647$	−5842.03	−0.354983	−0.177491	+	0.984122i	$0.556798\pi$
$648$	0	0
$649$	−1349.15	−0.0816007
$650$	0	0
$651$	5705.43	0.343492
$652$	0	0
$653$	−3424.26	−0.205209	−0.102605	−	0.994722i	$-0.532718\pi$
$653$	−3424.26	−0.205209	−0.102605	+	0.994722i	$0.532718\pi$
$654$	0	0
$655$	−8842.96	−0.527516
$656$	0	0
$657$	161.455	0.00958745
$658$	0	0
$659$	10678.8	0.631242	0.315621	−	0.948885i	$-0.397787\pi$
$659$	10678.8	0.631242	0.315621	+	0.948885i	$0.397787\pi$
$660$	0	0
$661$	11619.7	0.683743	0.341872	−	0.939747i	$-0.388939\pi$
$661$	11619.7	0.683743	0.341872	+	0.939747i	$0.388939\pi$
$662$	0	0
$663$	51667.0	3.02652
$664$	0	0
$665$	1053.54	0.0614355
$666$	0	0
$667$	−25139.1	−1.45936
$668$	0	0
$669$	17825.2	1.03014
$670$	0	0
$671$	4015.73	0.231037
$672$	0	0
$673$	3393.16	0.194349	0.0971745	−	0.995267i	$-0.469020\pi$
$673$	3393.16	0.194349	0.0971745	+	0.995267i	$0.469020\pi$
$674$	0	0
$675$	−15738.2	−0.897425
$676$	0	0
$677$	12327.2	0.699815	0.349907	−	0.936784i	$-0.386213\pi$
$677$	12327.2	0.699815	0.349907	+	0.936784i	$0.386213\pi$
$678$	0	0
$679$	−268.062	−0.0151506
$680$	0	0
$681$	−6749.18	−0.379778
$682$	0	0
$683$	−9198.34	−0.515322	−0.257661	−	0.966235i	$-0.582952\pi$
$683$	−9198.34	−0.515322	−0.257661	+	0.966235i	$0.582952\pi$
$684$	0	0
$685$	6750.51	0.376531
$686$	0	0
$687$	−15307.0	−0.850070
$688$	0	0
$689$	40538.3	2.24149
$690$	0	0
$691$	−32.2193	−0.00177378	−0.000886888	−	1.00000i	$-0.500282\pi$
$691$	−32.2193	−0.00177378	−0.000886888		1.00000i	$0.500282\pi$
$692$	0	0
$693$	194.896	0.0106832
$694$	0	0
$695$	−8832.22	−0.482050
$696$	0	0
$697$	−63804.1	−3.46736
$698$	0	0
$699$	−32965.3	−1.78378
$700$	0	0
$701$	22431.6	1.20860	0.604302	−	0.796756i	$-0.293452\pi$
$701$	22431.6	1.20860	0.604302	+	0.796756i	$0.293452\pi$
$702$	0	0
$703$	−9384.34	−0.503467
$704$	0	0
$705$	2824.57	0.150893
$706$	0	0
$707$	12057.8	0.641413
$708$	0	0
$709$	−26769.5	−1.41798	−0.708992	−	0.705217i	$-0.750850\pi$
$709$	−26769.5	−1.41798	−0.708992	+	0.705217i	$0.750850\pi$
$710$	0	0
$711$	−333.245	−0.0175776
$712$	0	0
$713$	−30404.9	−1.59702
$714$	0	0
$715$	−3599.06	−0.188248
$716$	0	0
$717$	11692.5	0.609014
$718$	0	0
$719$	−2983.35	−0.154743	−0.0773714	−	0.997002i	$-0.524653\pi$
$719$	−2983.35	−0.154743	−0.0773714	+	0.997002i	$0.524653\pi$
$720$	0	0
$721$	11221.3	0.579617
$722$	0	0
$723$	11261.6	0.579286
$724$	0	0
$725$	14677.6	0.751880
$726$	0	0
$727$	−17322.3	−0.883697	−0.441848	−	0.897090i	$-0.645677\pi$
$727$	−17322.3	−0.883697	−0.441848	+	0.897090i	$0.645677\pi$
$728$	0	0
$729$	21166.0	1.07534
$730$	0	0
$731$	−27341.0	−1.38337
$732$	0	0
$733$	−28634.3	−1.44288	−0.721440	−	0.692477i	$-0.756519\pi$
$733$	−28634.3	−1.44288	−0.721440	+	0.692477i	$0.756519\pi$
$734$	0	0
$735$	−1007.06	−0.0505387
$736$	0	0
$737$	−4688.18	−0.234317
$738$	0	0
$739$	19999.2	0.995511	0.497755	−	0.867317i	$-0.334158\pi$
$739$	19999.2	0.995511	0.497755	+	0.867317i	$0.334158\pi$
$740$	0	0
$741$	−14110.8	−0.699561
$742$	0	0
$743$	14504.9	0.716194	0.358097	−	0.933684i	$-0.383426\pi$
$743$	14504.9	0.716194	0.358097	+	0.933684i	$0.383426\pi$
$744$	0	0
$745$	−11745.8	−0.577628
$746$	0	0
$747$	−3329.57	−0.163082
$748$	0	0
$749$	4260.61	0.207849
$750$	0	0
$751$	−8529.99	−0.414466	−0.207233	−	0.978292i	$-0.566446\pi$
$751$	−8529.99	−0.414466	−0.207233	+	0.978292i	$0.566446\pi$
$752$	0	0
$753$	10790.8	0.522231
$754$	0	0
$755$	6352.67	0.306222
$756$	0	0
$757$	−24502.3	−1.17642	−0.588210	−	0.808708i	$-0.700167\pi$
$757$	−24502.3	−1.17642	−0.588210	+	0.808708i	$0.700167\pi$
$758$	0	0
$759$	10040.6	0.480174
$760$	0	0
$761$	−10932.2	−0.520752	−0.260376	−	0.965507i	$-0.583846\pi$
$761$	−10932.2	−0.520752	−0.260376	+	0.965507i	$0.583846\pi$
$762$	0	0
$763$	607.191	0.0288097
$764$	0	0
$765$	1394.85	0.0659225
$766$	0	0
$767$	9658.56	0.454694
$768$	0	0
$769$	−16794.5	−0.787551	−0.393775	−	0.919207i	$-0.628831\pi$
$769$	−16794.5	−0.787551	−0.393775	+	0.919207i	$0.628831\pi$
$770$	0	0
$771$	31440.4	1.46861
$772$	0	0
$773$	−1173.11	−0.0545844	−0.0272922	−	0.999627i	$-0.508688\pi$
$773$	−1173.11	−0.0545844	−0.0272922	+	0.999627i	$0.508688\pi$
$774$	0	0
$775$	17752.1	0.822805
$776$	0	0
$777$	8970.31	0.414167
$778$	0	0
$779$	17425.6	0.801460
$780$	0	0
$781$	−6780.86	−0.310676
$782$	0	0
$783$	−19901.0	−0.908308
$784$	0	0
$785$	−5241.99	−0.238337
$786$	0	0
$787$	12596.4	0.570536	0.285268	−	0.958448i	$-0.407917\pi$
$787$	12596.4	0.570536	0.285268	+	0.958448i	$0.407917\pi$
$788$	0	0
$789$	−37461.7	−1.69033
$790$	0	0
$791$	4924.30	0.221350
$792$	0	0
$793$	−28748.6	−1.28738
$794$	0	0
$795$	−10579.9	−0.471987
$796$	0	0
$797$	−17386.3	−0.772715	−0.386357	−	0.922349i	$-0.626267\pi$
$797$	−17386.3	−0.772715	−0.386357	+	0.922349i	$0.626267\pi$
$798$	0	0
$799$	18228.7	0.807114
$800$	0	0
$801$	−2907.09	−0.128236
$802$	0	0
$803$	701.668	0.0308360
$804$	0	0
$805$	5366.75	0.234973
$806$	0	0
$807$	−27694.4	−1.20804
$808$	0	0
$809$	18593.3	0.808042	0.404021	−	0.914750i	$-0.367612\pi$
$809$	18593.3	0.808042	0.404021	+	0.914750i	$0.367612\pi$
$810$	0	0
$811$	35158.6	1.52230	0.761149	−	0.648577i	$-0.224635\pi$
$811$	35158.6	1.52230	0.761149	+	0.648577i	$0.224635\pi$
$812$	0	0
$813$	9836.15	0.424316
$814$	0	0
$815$	8023.42	0.344844
$816$	0	0
$817$	7467.14	0.319758
$818$	0	0
$819$	−1395.26	−0.0595290
$820$	0	0
$821$	23965.9	1.01878	0.509388	−	0.860537i	$-0.329872\pi$
$821$	23965.9	1.01878	0.509388	+	0.860537i	$0.329872\pi$
$822$	0	0
$823$	−3801.13	−0.160995	−0.0804975	−	0.996755i	$-0.525651\pi$
$823$	−3801.13	−0.160995	−0.0804975	+	0.996755i	$0.525651\pi$
$824$	0	0
$825$	−5862.28	−0.247392
$826$	0	0
$827$	−34857.4	−1.46567	−0.732836	−	0.680406i	$-0.761804\pi$
$827$	−34857.4	−1.46567	−0.732836	+	0.680406i	$0.761804\pi$
$828$	0	0
$829$	−10194.9	−0.427121	−0.213560	−	0.976930i	$-0.568506\pi$
$829$	−10194.9	−0.427121	−0.213560	+	0.976930i	$0.568506\pi$
$830$	0	0
$831$	4288.66	0.179028
$832$	0	0
$833$	−6499.17	−0.270328
$834$	0	0
$835$	6473.34	0.268287
$836$	0	0
$837$	−24069.7	−0.993989
$838$	0	0
$839$	−8625.67	−0.354936	−0.177468	−	0.984127i	$-0.556791\pi$
$839$	−8625.67	−0.354936	−0.177468	+	0.984127i	$0.556791\pi$
$840$	0	0
$841$	−5829.00	−0.239001
$842$	0	0
$843$	−34454.6	−1.40769
$844$	0	0
$845$	16637.5	0.677334
$846$	0	0
$847$	847.000	0.0343604
$848$	0	0
$849$	16833.4	0.680471
$850$	0	0
$851$	−47803.9	−1.92561
$852$	0	0
$853$	−5592.35	−0.224476	−0.112238	−	0.993681i	$-0.535802\pi$
$853$	−5592.35	−0.224476	−0.112238	+	0.993681i	$0.535802\pi$
$854$	0	0
$855$	−380.948	−0.0152376
$856$	0	0
$857$	−21762.5	−0.867436	−0.433718	−	0.901049i	$-0.642799\pi$
$857$	−21762.5	−0.867436	−0.433718	+	0.901049i	$0.642799\pi$
$858$	0	0
$859$	−2505.66	−0.0995251	−0.0497626	−	0.998761i	$-0.515846\pi$
$859$	−2505.66	−0.0995251	−0.0497626	+	0.998761i	$0.515846\pi$
$860$	0	0
$861$	−16656.8	−0.659306
$862$	0	0
$863$	−1406.17	−0.0554653	−0.0277327	−	0.999615i	$-0.508829\pi$
$863$	−1406.17	−0.0554653	−0.0277327	+	0.999615i	$0.508829\pi$
$864$	0	0
$865$	−14473.2	−0.568907
$866$	0	0
$867$	−62719.8	−2.45683
$868$	0	0
$869$	−1448.26	−0.0565347
$870$	0	0
$871$	33562.6	1.30566
$872$	0	0
$873$	96.9280	0.00375775
$874$	0	0
$875$	−6768.87	−0.261520
$876$	0	0
$877$	−17598.1	−0.677589	−0.338795	−	0.940860i	$-0.610019\pi$
$877$	−17598.1	−0.677589	−0.338795	+	0.940860i	$0.610019\pi$
$878$	0	0
$879$	−35817.4	−1.37439
$880$	0	0
$881$	−39785.5	−1.52146	−0.760731	−	0.649068i	$-0.775159\pi$
$881$	−39785.5	−1.52146	−0.760731	+	0.649068i	$0.775159\pi$
$882$	0	0
$883$	−13462.3	−0.513073	−0.256537	−	0.966535i	$-0.582581\pi$
$883$	−13462.3	−0.513073	−0.256537	+	0.966535i	$0.582581\pi$
$884$	0	0
$885$	−2520.74	−0.0957442
$886$	0	0
$887$	−7376.62	−0.279237	−0.139618	−	0.990205i	$-0.544588\pi$
$887$	−7376.62	−0.279237	−0.139618	+	0.990205i	$0.544588\pi$
$888$	0	0
$889$	16102.8	0.607505
$890$	0	0
$891$	7196.79	0.270596
$892$	0	0
$893$	−4978.46	−0.186560
$894$	0	0
$895$	4646.45	0.173535
$896$	0	0
$897$	−71880.8	−2.67562
$898$	0	0
$899$	22447.7	0.832784
$900$	0	0
$901$	−68278.4	−2.52462
$902$	0	0
$903$	−7137.69	−0.263043
$904$	0	0
$905$	5680.08	0.208632
$906$	0	0
$907$	−38667.7	−1.41559	−0.707794	−	0.706419i	$-0.750310\pi$
$907$	−38667.7	−1.41559	−0.707794	+	0.706419i	$0.750310\pi$
$908$	0	0
$909$	−4359.94	−0.159087
$910$	0	0
$911$	39881.9	1.45044	0.725218	−	0.688519i	$-0.241739\pi$
$911$	39881.9	1.45044	0.725218	+	0.688519i	$0.241739\pi$
$912$	0	0
$913$	−14470.0	−0.524521
$914$	0	0
$915$	7502.94	0.271081
$916$	0	0
$917$	−14898.5	−0.536524
$918$	0	0
$919$	873.284	0.0313460	0.0156730	−	0.999877i	$-0.495011\pi$
$919$	873.284	0.0313460	0.0156730	+	0.999877i	$0.495011\pi$
$920$	0	0
$921$	31491.2	1.12668
$922$	0	0
$923$	48544.1	1.73115
$924$	0	0
$925$	27910.6	0.992102
$926$	0	0
$927$	−4057.49	−0.143760
$928$	0	0
$929$	−34892.4	−1.23227	−0.616136	−	0.787639i	$-0.711303\pi$
$929$	−34892.4	−1.23227	−0.616136	+	0.787639i	$0.711303\pi$
$930$	0	0
$931$	1775.00	0.0624846
$932$	0	0
$933$	35650.3	1.25095
$934$	0	0
$935$	6061.87	0.212026
$936$	0	0
$937$	18321.5	0.638781	0.319391	−	0.947623i	$-0.396522\pi$
$937$	18321.5	0.638781	0.319391	+	0.947623i	$0.396522\pi$
$938$	0	0
$939$	28084.8	0.976053
$940$	0	0
$941$	−14649.0	−0.507486	−0.253743	−	0.967272i	$-0.581662\pi$
$941$	−14649.0	−0.507486	−0.253743	+	0.967272i	$0.581662\pi$
$942$	0	0
$943$	88766.2	3.06535
$944$	0	0
$945$	4248.51	0.146248
$946$	0	0
$947$	19010.5	0.652333	0.326166	−	0.945312i	$-0.394243\pi$
$947$	19010.5	0.652333	0.326166	+	0.945312i	$0.394243\pi$
$948$	0	0
$949$	−5023.23	−0.171824
$950$	0	0
$951$	10429.8	0.355635
$952$	0	0
$953$	23486.8	0.798334	0.399167	−	0.916878i	$-0.369299\pi$
$953$	23486.8	0.798334	0.399167	+	0.916878i	$0.369299\pi$
$954$	0	0
$955$	11995.1	0.406444
$956$	0	0
$957$	−7412.91	−0.250392
$958$	0	0
$959$	11373.2	0.382961
$960$	0	0
$961$	−2641.26	−0.0886595
$962$	0	0
$963$	−1540.58	−0.0515520
$964$	0	0
$965$	−13549.4	−0.451992
$966$	0	0
$967$	41551.7	1.38181	0.690906	−	0.722945i	$-0.257212\pi$
$967$	41551.7	1.38181	0.690906	+	0.722945i	$0.257212\pi$
$968$	0	0
$969$	23766.8	0.787926
$970$	0	0
$971$	18176.3	0.600727	0.300364	−	0.953825i	$-0.402892\pi$
$971$	18176.3	0.600727	0.300364	+	0.953825i	$0.402892\pi$
$972$	0	0
$973$	−14880.4	−0.490282
$974$	0	0
$975$	41968.0	1.37851
$976$	0	0
$977$	20485.6	0.670821	0.335410	−	0.942072i	$-0.391125\pi$
$977$	20485.6	0.670821	0.335410	+	0.942072i	$0.391125\pi$
$978$	0	0
$979$	−12634.0	−0.412445
$980$	0	0
$981$	−219.553	−0.00714555
$982$	0	0
$983$	−41366.4	−1.34220	−0.671101	−	0.741366i	$-0.734178\pi$
$983$	−41366.4	−1.34220	−0.671101	+	0.741366i	$0.734178\pi$
$984$	0	0
$985$	−15420.5	−0.498819
$986$	0	0
$987$	4758.81	0.153470
$988$	0	0
$989$	38037.7	1.22298
$990$	0	0
$991$	−22512.7	−0.721635	−0.360818	−	0.932636i	$-0.617502\pi$
$991$	−22512.7	−0.721635	−0.360818	+	0.932636i	$0.617502\pi$
$992$	0	0
$993$	−24727.9	−0.790248
$994$	0	0
$995$	−20739.7	−0.660797
$996$	0	0
$997$	37348.8	1.18641	0.593203	−	0.805053i	$-0.297863\pi$
$997$	37348.8	1.18641	0.593203	+	0.805053i	$0.297863\pi$
$998$	0	0
$999$	−37843.3	−1.19851

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	616.4.a.i.1.2	✓	6
4.3	odd	2		1232.4.a.bb.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.4.a.i.1.2	✓	6	1.1	even	1	trivial
1232.4.a.bb.1.5		6	4.3	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:	\( N \)	\(=\)	\( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	616.a (trivial)

\(f(q)\)	\(=\)	\(q-4.94660 q^{3} +4.15482 q^{5} +7.00000 q^{7} -2.53111 q^{9} -11.0000 q^{11} +78.7489 q^{13} -20.5522 q^{15} -132.636 q^{17} +36.2244 q^{19} -34.6262 q^{21} +184.528 q^{23} -107.737 q^{25} +146.079 q^{27} -136.235 q^{29} -164.772 q^{31} +54.4126 q^{33} +29.0837 q^{35} -259.061 q^{37} -389.539 q^{39} +481.046 q^{41} +206.135 q^{43} -10.5163 q^{45} -137.434 q^{47} +49.0000 q^{49} +656.099 q^{51} +514.780 q^{53} -45.7030 q^{55} -179.188 q^{57} +122.650 q^{59} -365.067 q^{61} -17.7178 q^{63} +327.187 q^{65} +426.198 q^{67} -912.785 q^{69} +616.442 q^{71} -63.7880 q^{73} +532.935 q^{75} -77.0000 q^{77} +131.660 q^{79} -654.253 q^{81} +1315.46 q^{83} -551.079 q^{85} +673.901 q^{87} +1148.54 q^{89} +551.242 q^{91} +815.061 q^{93} +150.506 q^{95} -38.2946 q^{97} +27.8423 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} + 42 q^{7} + 60 q^{9} - 66 q^{11} - 6 q^{13} + 126 q^{15} - 14 q^{17} + 80 q^{19} + 254 q^{23} + 220 q^{25} + 90 q^{27} + 132 q^{29} - 52 q^{31} + 14 q^{35} - 518 q^{37} + 332 q^{39} + 486 q^{41}+ \cdots - 660 q^{99}+O(q^{100}) \) 6 * q + 2 * q^5 + 42 * q^7 + 60 * q^9 - 66 * q^11 - 6 * q^13 + 126 * q^15 - 14 * q^17 + 80 * q^19 + 254 * q^23 + 220 * q^25 + 90 * q^27 + 132 * q^29 - 52 * q^31 + 14 * q^35 - 518 * q^37 + 332 * q^39 + 486 * q^41 + 428 * q^43 - 244 * q^45 + 790 * q^47 + 294 * q^49 + 640 * q^51 - 40 * q^53 - 22 * q^55 + 2276 * q^57 + 436 * q^59 + 1034 * q^61 + 420 * q^63 + 800 * q^65 + 562 * q^67 + 530 * q^69 + 2474 * q^71 - 902 * q^73 + 1986 * q^75 - 462 * q^77 + 1636 * q^79 + 570 * q^81 + 3016 * q^83 + 236 * q^85 + 4340 * q^87 + 1750 * q^89 - 42 * q^91 + 150 * q^93 + 3628 * q^95 - 1250 * q^97 - 660 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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