LMFDB - Embedded newform 1216.3.d.e.191.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1216,3,Mod(191,1216)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1216.191"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 30 x^{14} + 116 x^{13} + 707 x^{12} - 2372 x^{11} - 7342 x^{10} + 12048 x^{9} + \cdots + 19859428 $ x^16 - 4x^15 - 30x^14 + 116x^13 + 707x^12 - 2372x^11 - 7342x^10 + 12048x^9 + 66793x^8 + 65416x^7 - 419708x^6 - 1201304x^5 + 619800x^4 + 4221760x^3 + 6110408x^2 + 10041872*x + 19859428
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{26} $
Twist minimal:	no (minimal twist has level 608)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.16
Root			$3.77456 - 2.45390i$ of defining polynomial
Character	$\chi$	$=$	1216.191
Dual form			1216.3.d.e.191.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.90780i q^{3} -6.54911 q^{5} +5.90118i q^{7} -15.0865 q^{9} -14.3924i q^{11} -22.2816 q^{13} -32.1417i q^{15} +28.3103 q^{17} -4.35890i q^{19} -28.9618 q^{21} +17.8696i q^{23} +17.8909 q^{25} -29.8711i q^{27} -32.6546 q^{29} -35.9728i q^{31} +70.6348 q^{33} -38.6475i q^{35} +52.6183 q^{37} -109.354i q^{39} -72.2594 q^{41} -58.5219i q^{43} +98.8029 q^{45} +20.2502i q^{47} +14.1761 q^{49} +138.941i q^{51} +67.1154 q^{53} +94.2572i q^{55} +21.3926 q^{57} +64.9303i q^{59} -4.97537 q^{61} -89.0278i q^{63} +145.925 q^{65} +64.2022i q^{67} -87.7003 q^{69} -7.10457i q^{71} +35.7765 q^{73} +87.8047i q^{75} +84.9319 q^{77} -58.6802i q^{79} +10.8230 q^{81} +12.7198i q^{83} -185.407 q^{85} -160.262i q^{87} +20.0413 q^{89} -131.488i q^{91} +176.547 q^{93} +28.5469i q^{95} -50.7104 q^{97} +217.130i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{5} - 40 q^{9} - 40 q^{13} - 32 q^{17} + 96 q^{21} + 88 q^{25} - 144 q^{29} + 88 q^{33} + 56 q^{37} - 104 q^{41} - 40 q^{45} - 144 q^{49} + 320 q^{53} - 8 q^{61} + 336 q^{65} - 392 q^{69} + 72 q^{77}+ \cdots - 240 q^{97}+O(q^{100}) $ 16 * q + 8 * q^5 - 40 * q^9 - 40 * q^13 - 32 * q^17 + 96 * q^21 + 88 * q^25 - 144 * q^29 + 88 * q^33 + 56 * q^37 - 104 * q^41 - 40 * q^45 - 144 * q^49 + 320 * q^53 - 8 * q^61 + 336 * q^65 - 392 * q^69 + 72 * q^77 - 272 * q^81 - 552 * q^85 - 136 * q^89 + 752 * q^93 - 240 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	4.90780i	1.63593i	0.575267	+	0.817966i	$0.304898\pi$
$3$	4.90780i	1.63593i	−0.575267	+	0.817966i	$0.695102\pi$
$4$	0	0
$5$	−6.54911	−1.30982	−0.654911	−	0.755706i	$-0.727294\pi$
$5$	−6.54911	−1.30982	−0.654911	+	0.755706i	$0.727294\pi$
$6$	0	0
$7$	5.90118i	0.843025i	0.906823	+	0.421513i	$0.138501\pi$
$7$	5.90118i	0.843025i	−0.906823	+	0.421513i	$0.861499\pi$
$8$	0	0
$9$	−15.0865	−1.67627
$10$	0	0
$11$	− 14.3924i	− 1.30840i	−0.756323	−	0.654199i	$-0.773006\pi$
$11$	− 14.3924i	− 1.30840i	0.756323	−	0.654199i	$-0.226994\pi$
$12$	0	0
$13$	−22.2816	−1.71397	−0.856985	−	0.515341i	$-0.827665\pi$
$13$	−22.2816	−1.71397	−0.856985	+	0.515341i	$0.827665\pi$
$14$	0	0
$15$	− 32.1417i	− 2.14278i
$16$	0	0
$17$	28.3103	1.66531	0.832655	−	0.553792i	$-0.186820\pi$
$17$	28.3103	1.66531	0.832655	+	0.553792i	$0.186820\pi$
$18$	0	0
$19$	− 4.35890i	− 0.229416i
$19$	− 4.35890i	− 0.229416i
$20$	0	0
$21$	−28.9618	−1.37913
$22$	0	0
$23$	17.8696i	0.776939i	0.921462	+	0.388469i	$0.126996\pi$
$23$	17.8696i	0.776939i	−0.921462	+	0.388469i	$0.873004\pi$
$24$	0	0
$25$	17.8909	0.715635
$26$	0	0
$27$	− 29.8711i	− 1.10634i
$28$	0	0
$29$	−32.6546	−1.12602	−0.563010	−	0.826450i	$-0.690357\pi$
$29$	−32.6546	−1.12602	−0.563010	+	0.826450i	$0.690357\pi$
$30$	0	0
$31$	− 35.9728i	− 1.16041i	−0.814469	−	0.580206i	$-0.802972\pi$
$31$	− 35.9728i	− 1.16041i	0.814469	−	0.580206i	$-0.197028\pi$
$32$	0	0
$33$	70.6348	2.14045
$34$	0	0
$35$	− 38.6475i	− 1.10421i
$36$	0	0
$37$	52.6183	1.42212	0.711058	−	0.703133i	$-0.248216\pi$
$37$	52.6183	1.42212	0.711058	+	0.703133i	$0.248216\pi$
$38$	0	0
$39$	− 109.354i	− 2.80394i
$40$	0	0
$41$	−72.2594	−1.76243	−0.881213	−	0.472720i	$-0.843272\pi$
$41$	−72.2594	−1.76243	−0.881213	+	0.472720i	$0.843272\pi$
$42$	0	0
$43$	− 58.5219i	− 1.36097i	−0.732760	−	0.680487i	$-0.761768\pi$
$43$	− 58.5219i	− 1.36097i	0.732760	−	0.680487i	$-0.238232\pi$
$44$	0	0
$45$	98.8029	2.19562
$46$	0	0
$47$	20.2502i	0.430855i	0.976520	+	0.215427i	$0.0691145\pi$
$47$	20.2502i	0.430855i	−0.976520	+	0.215427i	$0.930886\pi$
$48$	0	0
$49$	14.1761	0.289308
$50$	0	0
$51$	138.941i	2.72433i
$52$	0	0
$53$	67.1154	1.26633	0.633164	−	0.774017i	$-0.281756\pi$
$53$	67.1154	1.26633	0.633164	+	0.774017i	$0.281756\pi$
$54$	0	0
$55$	94.2572i	1.71377i
$56$	0	0
$57$	21.3926	0.375308
$58$	0	0
$59$	64.9303i	1.10051i	0.834995	+	0.550257i	$0.185470\pi$
$59$	64.9303i	1.10051i	−0.834995	+	0.550257i	$0.814530\pi$
$60$	0	0
$61$	−4.97537	−0.0815635	−0.0407818	−	0.999168i	$-0.512985\pi$
$61$	−4.97537	−0.0815635	−0.0407818	+	0.999168i	$0.512985\pi$
$62$	0	0
$63$	− 89.0278i	− 1.41314i
$64$	0	0
$65$	145.925	2.24500
$66$	0	0
$67$	64.2022i	0.958242i	0.877749	+	0.479121i	$0.159044\pi$
$67$	64.2022i	0.958242i	−0.877749	+	0.479121i	$0.840956\pi$
$68$	0	0
$69$	−87.7003	−1.27102
$70$	0	0
$71$	− 7.10457i	− 0.100064i	−0.998748	−	0.0500322i	$-0.984068\pi$
$71$	− 7.10457i	− 0.100064i	0.998748	−	0.0500322i	$-0.0159324\pi$
$72$	0	0
$73$	35.7765	0.490089	0.245045	−	0.969512i	$-0.421197\pi$
$73$	35.7765	0.490089	0.245045	+	0.969512i	$0.421197\pi$
$74$	0	0
$75$	87.8047i	1.17073i
$76$	0	0
$77$	84.9319	1.10301
$78$	0	0
$79$	− 58.6802i	− 0.742787i	−0.928476	−	0.371394i	$-0.878880\pi$
$79$	− 58.6802i	− 0.742787i	0.928476	−	0.371394i	$-0.121120\pi$
$80$	0	0
$81$	10.8230	0.133617
$82$	0	0
$83$	12.7198i	0.153250i	0.997060	+	0.0766251i	$0.0244145\pi$
$83$	12.7198i	0.153250i	−0.997060	+	0.0766251i	$0.975586\pi$
$84$	0	0
$85$	−185.407	−2.18126
$86$	0	0
$87$	− 160.262i	− 1.84209i
$88$	0	0
$89$	20.0413	0.225183	0.112592	−	0.993641i	$-0.464085\pi$
$89$	20.0413	0.225183	0.112592	+	0.993641i	$0.464085\pi$
$90$	0	0
$91$	− 131.488i	− 1.44492i
$92$	0	0
$93$	176.547	1.89836
$94$	0	0
$95$	28.5469i	0.300494i
$96$	0	0
$97$	−50.7104	−0.522787	−0.261394	−	0.965232i	$-0.584182\pi$
$97$	−50.7104	−0.522787	−0.261394	+	0.965232i	$0.584182\pi$
$98$	0	0
$99$	217.130i	2.19323i
$100$	0	0
$101$	−66.1632	−0.655081	−0.327540	−	0.944837i	$-0.606220\pi$
$101$	−66.1632	−0.655081	−0.327540	+	0.944837i	$0.606220\pi$
$102$	0	0
$103$	− 63.4235i	− 0.615762i	−0.951425	−	0.307881i	$-0.900380\pi$
$103$	− 63.4235i	− 0.615762i	0.951425	−	0.307881i	$-0.0996199\pi$
$104$	0	0
$105$	189.674	1.80642
$106$	0	0
$107$	72.2363i	0.675106i	0.941307	+	0.337553i	$0.109599\pi$
$107$	72.2363i	0.675106i	−0.941307	+	0.337553i	$0.890401\pi$
$108$	0	0
$109$	−15.5827	−0.142960	−0.0714802	−	0.997442i	$-0.522772\pi$
$109$	−15.5827	−0.142960	−0.0714802	+	0.997442i	$0.522772\pi$
$110$	0	0
$111$	258.240i	2.32649i
$112$	0	0
$113$	−32.1877	−0.284847	−0.142424	−	0.989806i	$-0.545490\pi$
$113$	−32.1877	−0.284847	−0.142424	+	0.989806i	$0.545490\pi$
$114$	0	0
$115$	− 117.030i	− 1.01765i
$116$	0	0
$117$	336.151	2.87308
$118$	0	0
$119$	167.064i	1.40390i
$120$	0	0
$121$	−86.1402	−0.711903
$122$	0	0
$123$	− 354.635i	− 2.88321i
$124$	0	0
$125$	46.5585	0.372468
$126$	0	0
$127$	− 80.2601i	− 0.631970i	−0.948764	−	0.315985i	$-0.897665\pi$
$127$	− 80.2601i	− 0.631970i	0.948764	−	0.315985i	$-0.102335\pi$
$128$	0	0
$129$	287.213	2.22646
$130$	0	0
$131$	− 163.160i	− 1.24550i	−0.782423	−	0.622748i	$-0.786016\pi$
$131$	− 163.160i	− 1.24550i	0.782423	−	0.622748i	$-0.213984\pi$
$132$	0	0
$133$	25.7226	0.193403
$134$	0	0
$135$	195.629i	1.44910i
$136$	0	0
$137$	60.8760	0.444351	0.222175	−	0.975007i	$-0.428684\pi$
$137$	60.8760	0.444351	0.222175	+	0.975007i	$0.428684\pi$
$138$	0	0
$139$	56.0617i	0.403322i	0.979455	+	0.201661i	$0.0646339\pi$
$139$	56.0617i	0.403322i	−0.979455	+	0.201661i	$0.935366\pi$
$140$	0	0
$141$	−99.3837	−0.704849
$142$	0	0
$143$	320.685i	2.24255i
$144$	0	0
$145$	213.859	1.47489
$146$	0	0
$147$	69.5734i	0.473289i
$148$	0	0
$149$	60.2189	0.404154	0.202077	−	0.979370i	$-0.435231\pi$
$149$	60.2189	0.404154	0.202077	+	0.979370i	$0.435231\pi$
$150$	0	0
$151$	− 19.4247i	− 0.128640i	−0.997929	−	0.0643201i	$-0.979512\pi$
$151$	− 19.4247i	− 0.128640i	0.997929	−	0.0643201i	$-0.0204879\pi$
$152$	0	0
$153$	−427.102	−2.79151
$154$	0	0
$155$	235.590i	1.51993i
$156$	0	0
$157$	161.009	1.02554	0.512768	−	0.858527i	$-0.328620\pi$
$157$	161.009	1.02554	0.512768	+	0.858527i	$0.328620\pi$
$158$	0	0
$159$	329.389i	2.07163i
$160$	0	0
$161$	−105.452	−0.654979
$162$	0	0
$163$	110.317i	0.676792i	0.941004	+	0.338396i	$0.109884\pi$
$163$	110.317i	0.676792i	−0.941004	+	0.338396i	$0.890116\pi$
$164$	0	0
$165$	−462.595	−2.80361
$166$	0	0
$167$	− 245.947i	− 1.47274i	−0.676580	−	0.736369i	$-0.736539\pi$
$167$	− 245.947i	− 1.47274i	0.676580	−	0.736369i	$-0.263461\pi$
$168$	0	0
$169$	327.471	1.93770
$170$	0	0
$171$	65.7603i	0.384563i
$172$	0	0
$173$	22.5590	0.130399	0.0651995	−	0.997872i	$-0.479232\pi$
$173$	22.5590	0.130399	0.0651995	+	0.997872i	$0.479232\pi$
$174$	0	0
$175$	105.577i	0.603298i
$176$	0	0
$177$	−318.665	−1.80037
$178$	0	0
$179$	− 149.865i	− 0.837235i	−0.908163	−	0.418617i	$-0.862515\pi$
$179$	− 149.865i	− 0.837235i	0.908163	−	0.418617i	$-0.137485\pi$
$180$	0	0
$181$	216.352	1.19531	0.597657	−	0.801752i	$-0.296098\pi$
$181$	216.352	1.19531	0.597657	+	0.801752i	$0.296098\pi$
$182$	0	0
$183$	− 24.4181i	− 0.133432i
$184$	0	0
$185$	−344.603	−1.86272
$186$	0	0
$187$	− 407.452i	− 2.17889i
$188$	0	0
$189$	176.274	0.932669
$190$	0	0
$191$	− 349.743i	− 1.83112i	−0.402187	−	0.915558i	$-0.631750\pi$
$191$	− 349.743i	− 1.83112i	0.402187	−	0.915558i	$-0.368250\pi$
$192$	0	0
$193$	263.096	1.36319	0.681595	−	0.731729i	$-0.261286\pi$
$193$	263.096	1.36319	0.681595	+	0.731729i	$0.261286\pi$
$194$	0	0
$195$	716.169i	3.67266i
$196$	0	0
$197$	−173.774	−0.882100	−0.441050	−	0.897482i	$-0.645394\pi$
$197$	−173.774	−0.882100	−0.441050	+	0.897482i	$0.645394\pi$
$198$	0	0
$199$	25.9123i	0.130213i	0.997878	+	0.0651063i	$0.0207387\pi$
$199$	25.9123i	0.130213i	−0.997878	+	0.0651063i	$0.979261\pi$
$200$	0	0
$201$	−315.091	−1.56762
$202$	0	0
$203$	− 192.700i	− 0.949263i
$204$	0	0
$205$	473.235	2.30846
$206$	0	0
$207$	− 269.589i	− 1.30236i
$208$	0	0
$209$	−62.7349	−0.300167
$210$	0	0
$211$	− 395.208i	− 1.87302i	−0.350636	−	0.936512i	$-0.614034\pi$
$211$	− 395.208i	− 1.87302i	0.350636	−	0.936512i	$-0.385966\pi$
$212$	0	0
$213$	34.8678	0.163698
$214$	0	0
$215$	383.266i	1.78263i
$216$	0	0
$217$	212.282	0.978257
$218$	0	0
$219$	175.584i	0.801752i
$220$	0	0
$221$	−630.799	−2.85429
$222$	0	0
$223$	273.035i	1.22437i	0.790713	+	0.612187i	$0.209710\pi$
$223$	273.035i	1.22437i	−0.790713	+	0.612187i	$0.790290\pi$
$224$	0	0
$225$	−269.910	−1.19960
$226$	0	0
$227$	− 193.790i	− 0.853702i	−0.904322	−	0.426851i	$-0.859623\pi$
$227$	− 193.790i	− 0.853702i	0.904322	−	0.426851i	$-0.140377\pi$
$228$	0	0
$229$	89.1608	0.389348	0.194674	−	0.980868i	$-0.437635\pi$
$229$	89.1608	0.389348	0.194674	+	0.980868i	$0.437635\pi$
$230$	0	0
$231$	416.828i	1.80445i
$232$	0	0
$233$	352.701	1.51374	0.756869	−	0.653566i	$-0.226728\pi$
$233$	352.701	1.51374	0.756869	+	0.653566i	$0.226728\pi$
$234$	0	0
$235$	− 132.621i	− 0.564343i
$236$	0	0
$237$	287.990	1.21515
$238$	0	0
$239$	− 335.331i	− 1.40306i	−0.712641	−	0.701529i	$-0.752501\pi$
$239$	− 335.331i	− 1.40306i	0.712641	−	0.701529i	$-0.247499\pi$
$240$	0	0
$241$	−73.6921	−0.305776	−0.152888	−	0.988243i	$-0.548857\pi$
$241$	−73.6921	−0.305776	−0.152888	+	0.988243i	$0.548857\pi$
$242$	0	0
$243$	− 215.723i	− 0.887747i
$244$	0	0
$245$	−92.8409	−0.378943
$246$	0	0
$247$	97.1233i	0.393212i
$248$	0	0
$249$	−62.4260	−0.250707
$250$	0	0
$251$	213.324i	0.849898i	0.905217	+	0.424949i	$0.139708\pi$
$251$	213.324i	0.849898i	−0.905217	+	0.424949i	$0.860292\pi$
$252$	0	0
$253$	257.186	1.01654
$254$	0	0
$255$	− 909.940i	− 3.56839i
$256$	0	0
$257$	−484.932	−1.88689	−0.943447	−	0.331522i	$-0.892438\pi$
$257$	−484.932	−1.88689	−0.943447	+	0.331522i	$0.892438\pi$
$258$	0	0
$259$	310.510i	1.19888i
$260$	0	0
$261$	492.642	1.88752
$262$	0	0
$263$	− 58.3649i	− 0.221920i	−0.993825	−	0.110960i	$-0.964607\pi$
$263$	− 58.3649i	− 0.221920i	0.993825	−	0.110960i	$-0.0353925\pi$
$264$	0	0
$265$	−439.546	−1.65867
$266$	0	0
$267$	98.3586i	0.368384i
$268$	0	0
$269$	8.05769	0.0299542	0.0149771	−	0.999888i	$-0.495232\pi$
$269$	8.05769	0.0299542	0.0149771	+	0.999888i	$0.495232\pi$
$270$	0	0
$271$	− 414.431i	− 1.52927i	−0.644465	−	0.764634i	$-0.722920\pi$
$271$	− 414.431i	− 1.52927i	0.644465	−	0.764634i	$-0.277080\pi$
$272$	0	0
$273$	645.315	2.36379
$274$	0	0
$275$	− 257.492i	− 0.936334i
$276$	0	0
$277$	111.050	0.400901	0.200451	−	0.979704i	$-0.435759\pi$
$277$	111.050	0.400901	0.200451	+	0.979704i	$0.435759\pi$
$278$	0	0
$279$	542.702i	1.94517i
$280$	0	0
$281$	131.151	0.466731	0.233365	−	0.972389i	$-0.425026\pi$
$281$	131.151	0.466731	0.233365	+	0.972389i	$0.425026\pi$
$282$	0	0
$283$	20.5202i	0.0725095i	0.999343	+	0.0362548i	$0.0115428\pi$
$283$	20.5202i	0.0725095i	−0.999343	+	0.0362548i	$0.988457\pi$
$284$	0	0
$285$	−140.102	−0.491587
$286$	0	0
$287$	− 426.416i	− 1.48577i
$288$	0	0
$289$	512.472	1.77326
$290$	0	0
$291$	− 248.876i	− 0.855244i
$292$	0	0
$293$	−503.711	−1.71915	−0.859575	−	0.511010i	$-0.829272\pi$
$293$	−503.711	−1.71915	−0.859575	+	0.511010i	$0.829272\pi$
$294$	0	0
$295$	− 425.236i	− 1.44148i
$296$	0	0
$297$	−429.915	−1.44753
$298$	0	0
$299$	− 398.163i	− 1.33165i
$300$	0	0
$301$	345.348	1.14734
$302$	0	0
$303$	− 324.715i	− 1.07167i
$304$	0	0
$305$	32.5843	0.106834
$306$	0	0
$307$	− 464.831i	− 1.51411i	−0.653353	−	0.757054i	$-0.726638\pi$
$307$	− 464.831i	− 1.51411i	0.653353	−	0.757054i	$-0.273362\pi$
$308$	0	0
$309$	311.269	1.00734
$310$	0	0
$311$	131.725i	0.423553i	0.977318	+	0.211777i	$0.0679249\pi$
$311$	131.725i	0.423553i	−0.977318	+	0.211777i	$0.932075\pi$
$312$	0	0
$313$	−553.953	−1.76982	−0.884908	−	0.465765i	$-0.845779\pi$
$313$	−553.953	−1.76982	−0.884908	+	0.465765i	$0.845779\pi$
$314$	0	0
$315$	583.053i	1.85096i
$316$	0	0
$317$	−226.823	−0.715531	−0.357766	−	0.933811i	$-0.616461\pi$
$317$	−226.823	−0.715531	−0.357766	+	0.933811i	$0.616461\pi$
$318$	0	0
$319$	469.977i	1.47328i
$320$	0	0
$321$	−354.521	−1.10443
$322$	0	0
$323$	− 123.402i	− 0.382048i
$324$	0	0
$325$	−398.638	−1.22658
$326$	0	0
$327$	− 76.4766i	− 0.233874i
$328$	0	0
$329$	−119.500	−0.363222
$330$	0	0
$331$	− 448.730i	− 1.35568i	−0.735209	−	0.677840i	$-0.762916\pi$
$331$	− 448.730i	− 1.35568i	0.735209	−	0.677840i	$-0.237084\pi$
$332$	0	0
$333$	−793.824	−2.38386
$334$	0	0
$335$	− 420.467i	− 1.25513i
$336$	0	0
$337$	−547.144	−1.62357	−0.811786	−	0.583955i	$-0.801505\pi$
$337$	−547.144	−1.62357	−0.811786	+	0.583955i	$0.801505\pi$
$338$	0	0
$339$	− 157.971i	− 0.465991i
$340$	0	0
$341$	−517.734	−1.51828
$342$	0	0
$343$	372.813i	1.08692i
$344$	0	0
$345$	574.359	1.66481
$346$	0	0
$347$	− 113.601i	− 0.327379i	−0.986512	−	0.163690i	$-0.947660\pi$
$347$	− 113.601i	− 0.327379i	0.986512	−	0.163690i	$-0.0523396\pi$
$348$	0	0
$349$	447.002	1.28081	0.640405	−	0.768038i	$-0.278767\pi$
$349$	447.002	1.28081	0.640405	+	0.768038i	$0.278767\pi$
$350$	0	0
$351$	665.576i	1.89623i
$352$	0	0
$353$	206.696	0.585540	0.292770	−	0.956183i	$-0.405423\pi$
$353$	206.696	0.585540	0.292770	+	0.956183i	$0.405423\pi$
$354$	0	0
$355$	46.5286i	0.131067i
$356$	0	0
$357$	−819.916	−2.29668
$358$	0	0
$359$	− 218.425i	− 0.608426i	−0.952604	−	0.304213i	$-0.901607\pi$
$359$	− 218.425i	− 0.608426i	0.952604	−	0.304213i	$-0.0983935\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	− 422.759i	− 1.16462i
$364$	0	0
$365$	−234.304	−0.641930
$366$	0	0
$367$	42.3246i	0.115326i	0.998336	+	0.0576629i	$0.0183649\pi$
$367$	42.3246i	0.115326i	−0.998336	+	0.0576629i	$0.981635\pi$
$368$	0	0
$369$	1090.14	2.95431
$370$	0	0
$371$	396.060i	1.06755i
$372$	0	0
$373$	318.445	0.853741	0.426871	−	0.904313i	$-0.359616\pi$
$373$	318.445	0.853741	0.426871	+	0.904313i	$0.359616\pi$
$374$	0	0
$375$	228.500i	0.609332i
$376$	0	0
$377$	727.597	1.92997
$378$	0	0
$379$	− 609.494i	− 1.60816i	−0.594518	−	0.804082i	$-0.702657\pi$
$379$	− 609.494i	− 1.60816i	0.594518	−	0.804082i	$-0.297343\pi$
$380$	0	0
$381$	393.900	1.03386
$382$	0	0
$383$	− 156.292i	− 0.408074i	−0.978963	−	0.204037i	$-0.934594\pi$
$383$	− 156.292i	− 0.408074i	0.978963	−	0.204037i	$-0.0654063\pi$
$384$	0	0
$385$	−556.229	−1.44475
$386$	0	0
$387$	882.888i	2.28136i
$388$	0	0
$389$	−82.1082	−0.211075	−0.105538	−	0.994415i	$-0.533656\pi$
$389$	−82.1082	−0.211075	−0.105538	+	0.994415i	$0.533656\pi$
$390$	0	0
$391$	505.893i	1.29384i
$392$	0	0
$393$	800.755	2.03755
$394$	0	0
$395$	384.303i	0.972919i
$396$	0	0
$397$	75.1893	0.189394	0.0946969	−	0.995506i	$-0.469812\pi$
$397$	75.1893	0.189394	0.0946969	+	0.995506i	$0.469812\pi$
$398$	0	0
$399$	126.241i	0.316395i
$400$	0	0
$401$	−270.041	−0.673418	−0.336709	−	0.941609i	$-0.609314\pi$
$401$	−270.041	−0.673418	−0.336709	+	0.941609i	$0.609314\pi$
$402$	0	0
$403$	801.532i	1.98891i
$404$	0	0
$405$	−70.8809	−0.175015
$406$	0	0
$407$	− 757.302i	− 1.86069i
$408$	0	0
$409$	−117.254	−0.286685	−0.143342	−	0.989673i	$-0.545785\pi$
$409$	−117.254	−0.286685	−0.143342	+	0.989673i	$0.545785\pi$
$410$	0	0
$411$	298.767i	0.726927i
$412$	0	0
$413$	−383.165	−0.927761
$414$	0	0
$415$	− 83.3032i	− 0.200731i
$416$	0	0
$417$	−275.140	−0.659807
$418$	0	0
$419$	− 196.107i	− 0.468035i	−0.972232	−	0.234018i	$-0.924813\pi$
$419$	− 196.107i	− 0.468035i	0.972232	−	0.234018i	$-0.0751873\pi$
$420$	0	0
$421$	−333.740	−0.792732	−0.396366	−	0.918093i	$-0.629729\pi$
$421$	−333.740	−0.792732	−0.396366	+	0.918093i	$0.629729\pi$
$422$	0	0
$423$	− 305.503i	− 0.722230i
$424$	0	0
$425$	506.495	1.19175
$426$	0	0
$427$	− 29.3606i	− 0.0687601i
$428$	0	0
$429$	−1573.86	−3.66867
$430$	0	0
$431$	− 688.339i	− 1.59707i	−0.601946	−	0.798537i	$-0.705608\pi$
$431$	− 688.339i	− 1.59707i	0.601946	−	0.798537i	$-0.294392\pi$
$432$	0	0
$433$	−63.1972	−0.145952	−0.0729760	−	0.997334i	$-0.523250\pi$
$433$	−63.1972	−0.145952	−0.0729760	+	0.997334i	$0.523250\pi$
$434$	0	0
$435$	1049.57i	2.41281i
$436$	0	0
$437$	77.8917	0.178242
$438$	0	0
$439$	617.043i	1.40556i	0.711405	+	0.702782i	$0.248059\pi$
$439$	617.043i	1.40556i	−0.711405	+	0.702782i	$0.751941\pi$
$440$	0	0
$441$	−213.867	−0.484960
$442$	0	0
$443$	35.8094i	0.0808338i	0.999183	+	0.0404169i	$0.0128686\pi$
$443$	35.8094i	0.0808338i	−0.999183	+	0.0404169i	$0.987131\pi$
$444$	0	0
$445$	−131.253	−0.294950
$446$	0	0
$447$	295.542i	0.661168i
$448$	0	0
$449$	−641.286	−1.42825	−0.714127	−	0.700016i	$-0.753176\pi$
$449$	−641.286	−1.42825	−0.714127	+	0.700016i	$0.753176\pi$
$450$	0	0
$451$	1039.98i	2.30595i
$452$	0	0
$453$	95.3323	0.210447
$454$	0	0
$455$	861.128i	1.89259i
$456$	0	0
$457$	30.6899	0.0671552	0.0335776	−	0.999436i	$-0.489310\pi$
$457$	30.6899	0.0671552	0.0335776	+	0.999436i	$0.489310\pi$
$458$	0	0
$459$	− 845.658i	− 1.84239i
$460$	0	0
$461$	−489.440	−1.06169	−0.530845	−	0.847469i	$-0.678126\pi$
$461$	−489.440	−1.06169	−0.530845	+	0.847469i	$0.678126\pi$
$462$	0	0
$463$	− 174.510i	− 0.376910i	−0.982082	−	0.188455i	$-0.939652\pi$
$463$	− 174.510i	− 0.376910i	0.982082	−	0.188455i	$-0.0603480\pi$
$464$	0	0
$465$	−1156.23	−2.48651
$466$	0	0
$467$	870.782i	1.86463i	0.361649	+	0.932314i	$0.382214\pi$
$467$	870.782i	1.86463i	−0.361649	+	0.932314i	$0.617786\pi$
$468$	0	0
$469$	−378.869	−0.807822
$470$	0	0
$471$	790.200i	1.67771i
$472$	0	0
$473$	−842.268	−1.78069
$474$	0	0
$475$	− 77.9845i	− 0.164178i
$476$	0	0
$477$	−1012.53	−2.12271
$478$	0	0
$479$	120.972i	0.252552i	0.991995	+	0.126276i	$0.0403025\pi$
$479$	120.972i	0.252552i	−0.991995	+	0.126276i	$0.959698\pi$
$480$	0	0
$481$	−1172.42	−2.43747
$482$	0	0
$483$	− 517.535i	− 1.07150i
$484$	0	0
$485$	332.108	0.684758
$486$	0	0
$487$	− 521.190i	− 1.07021i	−0.844787	−	0.535103i	$-0.820273\pi$
$487$	− 521.190i	− 1.07021i	0.844787	−	0.535103i	$-0.179727\pi$
$488$	0	0
$489$	−541.414	−1.10719
$490$	0	0
$491$	511.467i	1.04168i	0.853653	+	0.520842i	$0.174382\pi$
$491$	511.467i	1.04168i	−0.853653	+	0.520842i	$0.825618\pi$
$492$	0	0
$493$	−924.460	−1.87517
$494$	0	0
$495$	− 1422.01i	− 2.87274i
$496$	0	0
$497$	41.9253	0.0843568
$498$	0	0
$499$	− 51.6278i	− 0.103463i	−0.998661	−	0.0517313i	$-0.983526\pi$
$499$	− 51.6278i	− 0.103463i	0.998661	−	0.0517313i	$-0.0164739\pi$
$500$	0	0
$501$	1207.06	2.40930
$502$	0	0
$503$	− 39.1364i	− 0.0778060i	−0.999243	−	0.0389030i	$-0.987614\pi$
$503$	− 39.1364i	− 0.0778060i	0.999243	−	0.0389030i	$-0.0123863\pi$
$504$	0	0
$505$	433.310	0.858040
$506$	0	0
$507$	1607.16i	3.16994i
$508$	0	0
$509$	345.383	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$509$	345.383	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$510$	0	0
$511$	211.124i	0.413158i
$512$	0	0
$513$	−130.205	−0.253811
$514$	0	0
$515$	415.367i	0.806539i
$516$	0	0
$517$	291.448	0.563729
$518$	0	0
$519$	110.715i	0.213324i
$520$	0	0
$521$	−197.584	−0.379240	−0.189620	−	0.981858i	$-0.560726\pi$
$521$	−197.584	−0.379240	−0.189620	+	0.981858i	$0.560726\pi$
$522$	0	0
$523$	− 296.857i	− 0.567604i	−0.958883	−	0.283802i	$-0.908404\pi$
$523$	− 296.857i	− 0.567604i	0.958883	−	0.283802i	$-0.0915959\pi$
$524$	0	0
$525$	−518.151	−0.986955
$526$	0	0
$527$	− 1018.40i	− 1.93245i
$528$	0	0
$529$	209.678	0.396366
$530$	0	0
$531$	− 979.568i	− 1.84476i
$532$	0	0
$533$	1610.06	3.02075
$534$	0	0
$535$	− 473.084i	− 0.884269i
$536$	0	0
$537$	735.507	1.36966
$538$	0	0
$539$	− 204.028i	− 0.378530i
$540$	0	0
$541$	750.030	1.38638	0.693189	−	0.720756i	$-0.256205\pi$
$541$	750.030	1.38638	0.693189	+	0.720756i	$0.256205\pi$
$542$	0	0
$543$	1061.81i	1.95545i
$544$	0	0
$545$	102.053	0.187253
$546$	0	0
$547$	266.667i	0.487508i	0.969837	+	0.243754i	$0.0783789\pi$
$547$	266.667i	0.487508i	−0.969837	+	0.243754i	$0.921621\pi$
$548$	0	0
$549$	75.0607	0.136723
$550$	0	0
$551$	142.338i	0.258327i
$552$	0	0
$553$	346.282	0.626188
$554$	0	0
$555$	− 1691.24i	− 3.04728i
$556$	0	0
$557$	−997.639	−1.79109	−0.895547	−	0.444968i	$-0.853215\pi$
$557$	−997.639	−1.79109	−0.895547	+	0.444968i	$0.853215\pi$
$558$	0	0
$559$	1303.96i	2.33267i
$560$	0	0
$561$	1999.69	3.56451
$562$	0	0
$563$	208.963i	0.371160i	0.982629	+	0.185580i	$0.0594163\pi$
$563$	208.963i	0.371160i	−0.982629	+	0.185580i	$0.940584\pi$
$564$	0	0
$565$	210.801	0.373099
$566$	0	0
$567$	63.8683i	0.112643i
$568$	0	0
$569$	−699.067	−1.22859	−0.614294	−	0.789077i	$-0.710559\pi$
$569$	−699.067	−1.22859	−0.614294	+	0.789077i	$0.710559\pi$
$570$	0	0
$571$	− 378.019i	− 0.662030i	−0.943626	−	0.331015i	$-0.892609\pi$
$571$	− 378.019i	− 0.662030i	0.943626	−	0.331015i	$-0.107391\pi$
$572$	0	0
$573$	1716.47	2.99558
$574$	0	0
$575$	319.702i	0.556004i
$576$	0	0
$577$	−262.611	−0.455131	−0.227565	−	0.973763i	$-0.573077\pi$
$577$	−262.611	−0.455131	−0.227565	+	0.973763i	$0.573077\pi$
$578$	0	0
$579$	1291.22i	2.23009i
$580$	0	0
$581$	−75.0616	−0.129194
$582$	0	0
$583$	− 965.950i	− 1.65686i
$584$	0	0
$585$	−2201.49	−3.76323
$586$	0	0
$587$	− 639.092i	− 1.08874i	−0.838844	−	0.544371i	$-0.816768\pi$
$587$	− 639.092i	− 1.08874i	0.838844	−	0.544371i	$-0.183232\pi$
$588$	0	0
$589$	−156.802	−0.266217
$590$	0	0
$591$	− 852.846i	− 1.44306i
$592$	0	0
$593$	208.076	0.350887	0.175443	−	0.984490i	$-0.443864\pi$
$593$	208.076	0.350887	0.175443	+	0.984490i	$0.443864\pi$
$594$	0	0
$595$	− 1094.12i	− 1.83886i
$596$	0	0
$597$	−127.172	−0.213019
$598$	0	0
$599$	371.464i	0.620140i	0.950714	+	0.310070i	$0.100352\pi$
$599$	371.464i	0.620140i	−0.950714	+	0.310070i	$0.899648\pi$
$600$	0	0
$601$	806.876	1.34256	0.671278	−	0.741206i	$-0.265746\pi$
$601$	806.876	1.34256	0.671278	+	0.741206i	$0.265746\pi$
$602$	0	0
$603$	− 968.584i	− 1.60627i
$604$	0	0
$605$	564.142	0.932466
$606$	0	0
$607$	− 490.659i	− 0.808334i	−0.914685	−	0.404167i	$-0.867561\pi$
$607$	− 490.659i	− 0.808334i	0.914685	−	0.404167i	$-0.132439\pi$
$608$	0	0
$609$	945.734	1.55293
$610$	0	0
$611$	− 451.207i	− 0.738473i
$612$	0	0
$613$	705.454	1.15082	0.575411	−	0.817864i	$-0.304842\pi$
$613$	705.454	1.15082	0.575411	+	0.817864i	$0.304842\pi$
$614$	0	0
$615$	2322.54i	3.77649i
$616$	0	0
$617$	935.186	1.51570	0.757850	−	0.652429i	$-0.226250\pi$
$617$	935.186	1.51570	0.757850	+	0.652429i	$0.226250\pi$
$618$	0	0
$619$	1213.61i	1.96059i	0.197539	+	0.980295i	$0.436705\pi$
$619$	1213.61i	1.96059i	−0.197539	+	0.980295i	$0.563295\pi$
$620$	0	0
$621$	533.784	0.859555
$622$	0	0
$623$	118.267i	0.189835i
$624$	0	0
$625$	−752.189	−1.20350
$626$	0	0
$627$	− 307.890i	− 0.491052i
$628$	0	0
$629$	1489.64	2.36827
$630$	0	0
$631$	− 361.928i	− 0.573578i	−0.957994	−	0.286789i	$-0.907412\pi$
$631$	− 361.928i	− 0.573578i	0.957994	−	0.286789i	$-0.0925879\pi$
$632$	0	0
$633$	1939.60	3.06414
$634$	0	0
$635$	525.633i	0.827768i
$636$	0	0
$637$	−315.867	−0.495866
$638$	0	0
$639$	107.183i	0.167735i
$640$	0	0
$641$	126.497	0.197343	0.0986714	−	0.995120i	$-0.468541\pi$
$641$	126.497	0.197343	0.0986714	+	0.995120i	$0.468541\pi$
$642$	0	0
$643$	− 409.440i	− 0.636765i	−0.947962	−	0.318383i	$-0.896860\pi$
$643$	− 409.440i	− 0.636765i	0.947962	−	0.318383i	$-0.103140\pi$
$644$	0	0
$645$	−1880.99	−2.91627
$646$	0	0
$647$	237.857i	0.367630i	0.982961	+	0.183815i	$0.0588448\pi$
$647$	237.857i	0.367630i	−0.982961	+	0.183815i	$0.941155\pi$
$648$	0	0
$649$	934.501	1.43991
$650$	0	0
$651$	1041.84i	1.60036i
$652$	0	0
$653$	335.904	0.514401	0.257200	−	0.966358i	$-0.417200\pi$
$653$	335.904	0.514401	0.257200	+	0.966358i	$0.417200\pi$
$654$	0	0
$655$	1068.55i	1.63138i
$656$	0	0
$657$	−539.741	−0.821523
$658$	0	0
$659$	25.8130i	0.0391699i	0.999808	+	0.0195850i	$0.00623448\pi$
$659$	25.8130i	0.0391699i	−0.999808	+	0.0195850i	$0.993766\pi$
$660$	0	0
$661$	648.576	0.981204	0.490602	−	0.871384i	$-0.336777\pi$
$661$	648.576	0.981204	0.490602	+	0.871384i	$0.336777\pi$
$662$	0	0
$663$	− 3095.83i	− 4.66943i
$664$	0	0
$665$	−168.460	−0.253324
$666$	0	0
$667$	− 583.524i	− 0.874849i
$668$	0	0
$669$	−1340.00	−2.00299
$670$	0	0
$671$	71.6074i	0.106717i
$672$	0	0
$673$	314.870	0.467860	0.233930	−	0.972253i	$-0.424841\pi$
$673$	314.870	0.467860	0.233930	+	0.972253i	$0.424841\pi$
$674$	0	0
$675$	− 534.419i	− 0.791732i
$676$	0	0
$677$	722.428	1.06710	0.533551	−	0.845768i	$-0.320857\pi$
$677$	722.428	1.06710	0.533551	+	0.845768i	$0.320857\pi$
$678$	0	0
$679$	− 299.251i	− 0.440723i
$680$	0	0
$681$	951.084	1.39660
$682$	0	0
$683$	503.280i	0.736867i	0.929654	+	0.368434i	$0.120106\pi$
$683$	503.280i	0.736867i	−0.929654	+	0.368434i	$0.879894\pi$
$684$	0	0
$685$	−398.684	−0.582021
$686$	0	0
$687$	437.583i	0.636947i
$688$	0	0
$689$	−1495.44	−2.17045
$690$	0	0
$691$	− 197.394i	− 0.285664i	−0.989747	−	0.142832i	$-0.954379\pi$
$691$	− 197.394i	− 0.285664i	0.989747	−	0.142832i	$-0.0456209\pi$
$692$	0	0
$693$	−1281.32	−1.84895
$694$	0	0
$695$	− 367.155i	− 0.528280i
$696$	0	0
$697$	−2045.68	−2.93498
$698$	0	0
$699$	1730.99i	2.47637i
$700$	0	0
$701$	−821.241	−1.17153	−0.585764	−	0.810482i	$-0.699205\pi$
$701$	−821.241	−1.17153	−0.585764	+	0.810482i	$0.699205\pi$
$702$	0	0
$703$	− 229.358i	− 0.326256i
$704$	0	0
$705$	650.875	0.923227
$706$	0	0
$707$	− 390.441i	− 0.552250i
$708$	0	0
$709$	−94.3101	−0.133019	−0.0665093	−	0.997786i	$-0.521186\pi$
$709$	−94.3101	−0.133019	−0.0665093	+	0.997786i	$0.521186\pi$
$710$	0	0
$711$	885.276i	1.24511i
$712$	0	0
$713$	642.819	0.901570
$714$	0	0
$715$	− 2100.20i	− 2.93735i
$716$	0	0
$717$	1645.74	2.29531
$718$	0	0
$719$	585.874i	0.814846i	0.913240	+	0.407423i	$0.133573\pi$
$719$	585.874i	0.814846i	−0.913240	+	0.407423i	$0.866427\pi$
$720$	0	0
$721$	374.273	0.519103
$722$	0	0
$723$	− 361.666i	− 0.500229i
$724$	0	0
$725$	−584.219	−0.805819
$726$	0	0
$727$	596.396i	0.820352i	0.912006	+	0.410176i	$0.134533\pi$
$727$	596.396i	0.820352i	−0.912006	+	0.410176i	$0.865467\pi$
$728$	0	0
$729$	1156.13	1.58591
$730$	0	0
$731$	− 1656.77i	− 2.26644i
$732$	0	0
$733$	106.931	0.145881	0.0729407	−	0.997336i	$-0.476762\pi$
$733$	106.931	0.145881	0.0729407	+	0.997336i	$0.476762\pi$
$734$	0	0
$735$	− 455.644i	− 0.619924i
$736$	0	0
$737$	924.022	1.25376
$738$	0	0
$739$	− 1210.79i	− 1.63842i	−0.573492	−	0.819211i	$-0.694412\pi$
$739$	− 1210.79i	− 1.63842i	0.573492	−	0.819211i	$-0.305588\pi$
$740$	0	0
$741$	−476.661	−0.643268
$742$	0	0
$743$	− 331.817i	− 0.446591i	−0.974751	−	0.223296i	$-0.928318\pi$
$743$	− 331.817i	− 0.446591i	0.974751	−	0.223296i	$-0.0716815\pi$
$744$	0	0
$745$	−394.381	−0.529370
$746$	0	0
$747$	− 191.896i	− 0.256889i
$748$	0	0
$749$	−426.279	−0.569131
$750$	0	0
$751$	623.487i	0.830208i	0.909774	+	0.415104i	$0.136255\pi$
$751$	623.487i	0.830208i	−0.909774	+	0.415104i	$0.863745\pi$
$752$	0	0
$753$	−1046.95	−1.39038
$754$	0	0
$755$	127.214i	0.168496i
$756$	0	0
$757$	1428.64	1.88723	0.943617	−	0.331040i	$-0.107400\pi$
$757$	1428.64	1.88723	0.943617	+	0.331040i	$0.107400\pi$
$758$	0	0
$759$	1262.21i	1.66300i
$760$	0	0
$761$	−1285.24	−1.68888	−0.844440	−	0.535650i	$-0.820067\pi$
$761$	−1285.24	−1.68888	−0.844440	+	0.535650i	$0.820067\pi$
$762$	0	0
$763$	− 91.9562i	− 0.120519i
$764$	0	0
$765$	2797.14	3.65639
$766$	0	0
$767$	− 1446.75i	− 1.88625i
$768$	0	0
$769$	−551.328	−0.716942	−0.358471	−	0.933541i	$-0.616702\pi$
$769$	−551.328	−0.716942	−0.358471	+	0.933541i	$0.616702\pi$
$770$	0	0
$771$	− 2379.95i	− 3.08683i
$772$	0	0
$773$	−447.392	−0.578773	−0.289386	−	0.957212i	$-0.593451\pi$
$773$	−447.392	−0.578773	−0.289386	+	0.957212i	$0.593451\pi$
$774$	0	0
$775$	− 643.585i	− 0.830432i
$776$	0	0
$777$	−1523.92	−1.96129
$778$	0	0
$779$	314.972i	0.404328i
$780$	0	0
$781$	−102.252	−0.130924
$782$	0	0
$783$	975.427i	1.24576i
$784$	0	0
$785$	−1054.47	−1.34327
$786$	0	0
$787$	291.863i	0.370855i	0.982658	+	0.185428i	$0.0593670\pi$
$787$	291.863i	0.370855i	−0.982658	+	0.185428i	$0.940633\pi$
$788$	0	0
$789$	286.443	0.363046
$790$	0	0
$791$	− 189.946i	− 0.240133i
$792$	0	0
$793$	110.859	0.139797
$794$	0	0
$795$	− 2157.20i	− 2.71346i
$796$	0	0
$797$	−224.935	−0.282227	−0.141114	−	0.989993i	$-0.545068\pi$
$797$	−224.935	−0.282227	−0.141114	+	0.989993i	$0.545068\pi$
$798$	0	0
$799$	573.288i	0.717507i
$800$	0	0
$801$	−302.352	−0.377468
$802$	0	0
$803$	− 514.909i	− 0.641231i
$804$	0	0
$805$	690.614	0.857906
$806$	0	0
$807$	39.5455i	0.0490031i
$808$	0	0
$809$	662.505	0.818918	0.409459	−	0.912328i	$-0.365717\pi$
$809$	662.505	0.818918	0.409459	+	0.912328i	$0.365717\pi$
$810$	0	0
$811$	− 364.123i	− 0.448980i	−0.974476	−	0.224490i	$-0.927928\pi$
$811$	− 364.123i	− 0.448980i	0.974476	−	0.224490i	$-0.0720716\pi$
$812$	0	0
$813$	2033.94	2.50178
$814$	0	0
$815$	− 722.479i	− 0.886478i
$816$	0	0
$817$	−255.091	−0.312229
$818$	0	0
$819$	1983.68i	2.42208i
$820$	0	0
$821$	−1237.91	−1.50781	−0.753905	−	0.656983i	$-0.771832\pi$
$821$	−1237.91	−1.50781	−0.753905	+	0.656983i	$0.771832\pi$
$822$	0	0
$823$	− 669.197i	− 0.813119i	−0.913624	−	0.406559i	$-0.866728\pi$
$823$	− 669.197i	− 0.813119i	0.913624	−	0.406559i	$-0.133272\pi$
$824$	0	0
$825$	1263.72	1.53178
$826$	0	0
$827$	− 1128.89i	− 1.36504i	−0.730868	−	0.682519i	$-0.760884\pi$
$827$	− 1128.89i	− 1.36504i	0.730868	−	0.682519i	$-0.239116\pi$
$828$	0	0
$829$	−198.449	−0.239384	−0.119692	−	0.992811i	$-0.538191\pi$
$829$	−198.449	−0.239384	−0.119692	+	0.992811i	$0.538191\pi$
$830$	0	0
$831$	545.009i	0.655847i
$832$	0	0
$833$	401.330	0.481788
$834$	0	0
$835$	1610.74i	1.92903i
$836$	0	0
$837$	−1074.55	−1.28381
$838$	0	0
$839$	− 397.895i	− 0.474249i	−0.971479	−	0.237124i	$-0.923795\pi$
$839$	− 397.895i	− 0.474249i	0.971479	−	0.237124i	$-0.0762049\pi$
$840$	0	0
$841$	225.322	0.267921
$842$	0	0
$843$	643.664i	0.763540i
$844$	0	0
$845$	−2144.64	−2.53804
$846$	0	0
$847$	− 508.329i	− 0.600152i
$848$	0	0
$849$	−100.709	−0.118621
$850$	0	0
$851$	940.268i	1.10490i
$852$	0	0
$853$	−483.040	−0.566284	−0.283142	−	0.959078i	$-0.591377\pi$
$853$	−483.040	−0.566284	−0.283142	+	0.959078i	$0.591377\pi$
$854$	0	0
$855$	− 430.672i	− 0.503710i
$856$	0	0
$857$	−1623.37	−1.89425	−0.947125	−	0.320864i	$-0.896027\pi$
$857$	−1623.37	−1.89425	−0.947125	+	0.320864i	$0.896027\pi$
$858$	0	0
$859$	242.200i	0.281956i	0.990013	+	0.140978i	$0.0450247\pi$
$859$	242.200i	0.281956i	−0.990013	+	0.140978i	$0.954975\pi$
$860$	0	0
$861$	2092.76	2.43062
$862$	0	0
$863$	861.605i	0.998383i	0.866492	+	0.499192i	$0.166370\pi$
$863$	861.605i	0.998383i	−0.866492	+	0.499192i	$0.833630\pi$
$864$	0	0
$865$	−147.742	−0.170800
$866$	0	0
$867$	2515.11i	2.90093i
$868$	0	0
$869$	−844.547	−0.971861
$870$	0	0
$871$	− 1430.53i	− 1.64240i
$872$	0	0
$873$	765.039	0.876334
$874$	0	0
$875$	274.750i	0.314000i
$876$	0	0
$877$	900.709	1.02703	0.513517	−	0.858079i	$-0.328342\pi$
$877$	900.709	1.02703	0.513517	+	0.858079i	$0.328342\pi$
$878$	0	0
$879$	− 2472.11i	− 2.81241i
$880$	0	0
$881$	667.165	0.757282	0.378641	−	0.925544i	$-0.376392\pi$
$881$	667.165	0.757282	0.378641	+	0.925544i	$0.376392\pi$
$882$	0	0
$883$	782.377i	0.886045i	0.896511	+	0.443022i	$0.146094\pi$
$883$	782.377i	0.886045i	−0.896511	+	0.443022i	$0.853906\pi$
$884$	0	0
$885$	2086.97	2.35816
$886$	0	0
$887$	986.381i	1.11204i	0.831168	+	0.556021i	$0.187673\pi$
$887$	986.381i	1.11204i	−0.831168	+	0.556021i	$0.812327\pi$
$888$	0	0
$889$	473.629	0.532766
$890$	0	0
$891$	− 155.768i	− 0.174824i
$892$	0	0
$893$	88.2685	0.0988449
$894$	0	0
$895$	981.483i	1.09663i
$896$	0	0
$897$	1954.10	2.17849
$898$	0	0
$899$	1174.68i	1.30665i
$900$	0	0
$901$	1900.06	2.10883
$902$	0	0
$903$	1694.90i	1.87696i
$904$	0	0
$905$	−1416.91	−1.56565
$906$	0	0
$907$	1433.03i	1.57997i	0.613126	+	0.789985i	$0.289912\pi$
$907$	1433.03i	1.57997i	−0.613126	+	0.789985i	$0.710088\pi$
$908$	0	0
$909$	998.167	1.09809
$910$	0	0
$911$	613.690i	0.673644i	0.941568	+	0.336822i	$0.109352\pi$
$911$	613.690i	0.673644i	−0.941568	+	0.336822i	$0.890648\pi$
$912$	0	0
$913$	183.068	0.200512
$914$	0	0
$915$	159.917i	0.174773i
$916$	0	0
$917$	962.835	1.04998
$918$	0	0
$919$	554.724i	0.603617i	0.953369	+	0.301808i	$0.0975903\pi$
$919$	554.724i	0.603617i	−0.953369	+	0.301808i	$0.902410\pi$
$920$	0	0
$921$	2281.30	2.47698
$922$	0	0
$923$	158.301i	0.171507i
$924$	0	0
$925$	941.388	1.01772
$926$	0	0
$927$	956.835i	1.03218i
$928$	0	0
$929$	1644.41	1.77009	0.885043	−	0.465510i	$-0.154129\pi$
$929$	1644.41	1.77009	0.885043	+	0.465510i	$0.154129\pi$
$930$	0	0
$931$	− 61.7922i	− 0.0663719i
$932$	0	0
$933$	−646.479	−0.692904
$934$	0	0
$935$	2668.45i	2.85395i
$936$	0	0
$937$	92.0136	0.0982002	0.0491001	−	0.998794i	$-0.484365\pi$
$937$	92.0136	0.0982002	0.0491001	+	0.998794i	$0.484365\pi$
$938$	0	0
$939$	− 2718.69i	− 2.89530i
$940$	0	0
$941$	779.771	0.828662	0.414331	−	0.910126i	$-0.364016\pi$
$941$	779.771	0.828662	0.414331	+	0.910126i	$0.364016\pi$
$942$	0	0
$943$	− 1291.25i	− 1.36930i
$944$	0	0
$945$	−1154.44	−1.22163
$946$	0	0
$947$	324.236i	0.342383i	0.985238	+	0.171191i	$0.0547616\pi$
$947$	324.236i	0.342383i	−0.985238	+	0.171191i	$0.945238\pi$
$948$	0	0
$949$	−797.159	−0.839999
$950$	0	0
$951$	− 1113.20i	− 1.17056i
$952$	0	0
$953$	−1230.50	−1.29118	−0.645592	−	0.763682i	$-0.723389\pi$
$953$	−1230.50	−1.29118	−0.645592	+	0.763682i	$0.723389\pi$
$954$	0	0
$955$	2290.51i	2.39844i
$956$	0	0
$957$	−2306.55	−2.41019
$958$	0	0
$959$	359.240i	0.374599i
$960$	0	0
$961$	−333.042	−0.346558
$962$	0	0
$963$	− 1089.79i	− 1.13166i
$964$	0	0
$965$	−1723.04	−1.78554
$966$	0	0
$967$	− 699.842i	− 0.723725i	−0.932232	−	0.361862i	$-0.882141\pi$
$967$	− 699.842i	− 0.723725i	0.932232	−	0.361862i	$-0.117859\pi$
$968$	0	0
$969$	605.630	0.625005
$970$	0	0
$971$	758.268i	0.780915i	0.920621	+	0.390457i	$0.127683\pi$
$971$	758.268i	0.780915i	−0.920621	+	0.390457i	$0.872317\pi$
$972$	0	0
$973$	−330.830	−0.340011
$974$	0	0
$975$	− 1956.43i	− 2.00660i
$976$	0	0
$977$	806.924	0.825920	0.412960	−	0.910749i	$-0.364495\pi$
$977$	806.924	0.825920	0.412960	+	0.910749i	$0.364495\pi$
$978$	0	0
$979$	− 288.442i	− 0.294629i
$980$	0	0
$981$	235.087	0.239641
$982$	0	0
$983$	− 959.544i	− 0.976138i	−0.872805	−	0.488069i	$-0.837701\pi$
$983$	− 959.544i	− 0.976138i	0.872805	−	0.488069i	$-0.162299\pi$
$984$	0	0
$985$	1138.06	1.15539
$986$	0	0
$987$	− 586.481i	− 0.594206i
$988$	0	0
$989$	1045.76	1.05739
$990$	0	0
$991$	282.782i	0.285350i	0.989770	+	0.142675i	$0.0455703\pi$
$991$	282.782i	0.285350i	−0.989770	+	0.142675i	$0.954430\pi$
$992$	0	0
$993$	2202.28	2.21780
$994$	0	0
$995$	− 169.703i	− 0.170555i
$996$	0	0
$997$	−1511.38	−1.51593	−0.757966	−	0.652294i	$-0.773807\pi$
$997$	−1511.38	−1.51593	−0.757966	+	0.652294i	$0.773807\pi$
$998$	0	0
$999$	− 1571.77i	− 1.57334i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.d.e.191.16		16
4.3	odd	2	inner	1216.3.d.e.191.1		16
8.3	odd	2		608.3.d.a.191.16	yes	16
8.5	even	2		608.3.d.a.191.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.3.d.a.191.1	✓	16	8.5	even	2
608.3.d.a.191.16	yes	16	8.3	odd	2
1216.3.d.e.191.1		16	4.3	odd	2	inner
1216.3.d.e.191.16		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.90780i q^{3} -6.54911 q^{5} +5.90118i q^{7} -15.0865 q^{9} -14.3924i q^{11} -22.2816 q^{13} -32.1417i q^{15} +28.3103 q^{17} -4.35890i q^{19} -28.9618 q^{21} +17.8696i q^{23} +17.8909 q^{25} -29.8711i q^{27} -32.6546 q^{29} -35.9728i q^{31} +70.6348 q^{33} -38.6475i q^{35} +52.6183 q^{37} -109.354i q^{39} -72.2594 q^{41} -58.5219i q^{43} +98.8029 q^{45} +20.2502i q^{47} +14.1761 q^{49} +138.941i q^{51} +67.1154 q^{53} +94.2572i q^{55} +21.3926 q^{57} +64.9303i q^{59} -4.97537 q^{61} -89.0278i q^{63} +145.925 q^{65} +64.2022i q^{67} -87.7003 q^{69} -7.10457i q^{71} +35.7765 q^{73} +87.8047i q^{75} +84.9319 q^{77} -58.6802i q^{79} +10.8230 q^{81} +12.7198i q^{83} -185.407 q^{85} -160.262i q^{87} +20.0413 q^{89} -131.488i q^{91} +176.547 q^{93} +28.5469i q^{95} -50.7104 q^{97} +217.130i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{5} - 40 q^{9} - 40 q^{13} - 32 q^{17} + 96 q^{21} + 88 q^{25} - 144 q^{29} + 88 q^{33} + 56 q^{37} - 104 q^{41} - 40 q^{45} - 144 q^{49} + 320 q^{53} - 8 q^{61} + 336 q^{65} - 392 q^{69} + 72 q^{77}+ \cdots - 240 q^{97}+O(q^{100}) \) 16 * q + 8 * q^5 - 40 * q^9 - 40 * q^13 - 32 * q^17 + 96 * q^21 + 88 * q^25 - 144 * q^29 + 88 * q^33 + 56 * q^37 - 104 * q^41 - 40 * q^45 - 144 * q^49 + 320 * q^53 - 8 * q^61 + 336 * q^65 - 392 * q^69 + 72 * q^77 - 272 * q^81 - 552 * q^85 - 136 * q^89 + 752 * q^93 - 240 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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