LMFDB - Embedded newform 588.6.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,6,Mod(1,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("588.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	588.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$94.3056860500$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 699x^{2} - 686x + 59664 $ x^4 - 2x^3 - 699x^2 - 686*x + 59664
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2}\cdot 7 $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$26.2941$ of defining polynomial
Character	$\chi$	$=$	588.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} +31.9616 q^{5} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} +31.9616 q^{5} +81.0000 q^{9} +260.884 q^{11} -769.735 q^{13} +287.654 q^{15} +1553.32 q^{17} -750.049 q^{19} +754.854 q^{23} -2103.46 q^{25} +729.000 q^{27} +6008.93 q^{29} +6420.04 q^{31} +2347.95 q^{33} -4775.73 q^{37} -6927.62 q^{39} +5423.27 q^{41} -11896.4 q^{43} +2588.89 q^{45} +17428.0 q^{47} +13979.9 q^{51} +37650.6 q^{53} +8338.26 q^{55} -6750.44 q^{57} -22078.1 q^{59} +8173.38 q^{61} -24602.0 q^{65} +13001.7 q^{67} +6793.69 q^{69} -12349.6 q^{71} -43600.3 q^{73} -18931.1 q^{75} +76749.5 q^{79} +6561.00 q^{81} -21893.6 q^{83} +49646.5 q^{85} +54080.4 q^{87} +136967. q^{89} +57780.4 q^{93} -23972.8 q^{95} +93050.1 q^{97} +21131.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 36 q^{3} + 324 q^{9}+O(q^{10}) $ 4 * q + 36 * q^3 + 324 * q^9	$ 4 q + 36 q^{3} + 324 q^{9} + 462 q^{11} + 602 q^{13} + 228 q^{17} + 358 q^{19} + 2148 q^{23} + 5454 q^{25} + 2916 q^{27} - 5532 q^{29} + 830 q^{31} + 4158 q^{33} + 3914 q^{37} + 5418 q^{39} + 8316 q^{41} - 14518 q^{43} + 41700 q^{47} + 2052 q^{51} - 22164 q^{53} - 3892 q^{55} + 3222 q^{57} + 32886 q^{59} + 83732 q^{61} + 93192 q^{65} + 80034 q^{67} + 19332 q^{69} + 44772 q^{71} - 22470 q^{73} + 49086 q^{75} + 75286 q^{79} + 26244 q^{81} + 17418 q^{83} + 139252 q^{85} - 49788 q^{87} + 28944 q^{89} + 7470 q^{93} + 144120 q^{95} + 216678 q^{97} + 37422 q^{99}+O(q^{100}) $ 4 * q + 36 * q^3 + 324 * q^9 + 462 * q^11 + 602 * q^13 + 228 * q^17 + 358 * q^19 + 2148 * q^23 + 5454 * q^25 + 2916 * q^27 - 5532 * q^29 + 830 * q^31 + 4158 * q^33 + 3914 * q^37 + 5418 * q^39 + 8316 * q^41 - 14518 * q^43 + 41700 * q^47 + 2052 * q^51 - 22164 * q^53 - 3892 * q^55 + 3222 * q^57 + 32886 * q^59 + 83732 * q^61 + 93192 * q^65 + 80034 * q^67 + 19332 * q^69 + 44772 * q^71 - 22470 * q^73 + 49086 * q^75 + 75286 * q^79 + 26244 * q^81 + 17418 * q^83 + 139252 * q^85 - 49788 * q^87 + 28944 * q^89 + 7470 * q^93 + 144120 * q^95 + 216678 * q^97 + 37422 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	31.9616	0.571746	0.285873	−	0.958267i	$-0.407716\pi$
$5$	31.9616	0.571746	0.285873	+	0.958267i	$0.407716\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	260.884	0.650077	0.325039	−	0.945701i	$-0.394623\pi$
$11$	260.884	0.650077	0.325039	+	0.945701i	$0.394623\pi$
$12$	0	0
$13$	−769.735	−1.26323	−0.631616	−	0.775281i	$-0.717608\pi$
$13$	−769.735	−1.26323	−0.631616	+	0.775281i	$0.717608\pi$
$14$	0	0
$15$	287.654	0.330098
$16$	0	0
$17$	1553.32	1.30358	0.651791	−	0.758399i	$-0.274018\pi$
$17$	1553.32	1.30358	0.651791	+	0.758399i	$0.274018\pi$
$18$	0	0
$19$	−750.049	−0.476656	−0.238328	−	0.971185i	$-0.576599\pi$
$19$	−750.049	−0.476656	−0.238328	+	0.971185i	$0.576599\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	754.854	0.297539	0.148769	−	0.988872i	$-0.452469\pi$
$23$	754.854	0.297539	0.148769	+	0.988872i	$0.452469\pi$
$24$	0	0
$25$	−2103.46	−0.673106
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	6008.93	1.32679	0.663395	−	0.748269i	$-0.269115\pi$
$29$	6008.93	1.32679	0.663395	+	0.748269i	$0.269115\pi$
$30$	0	0
$31$	6420.04	1.19987	0.599934	−	0.800049i	$-0.295193\pi$
$31$	6420.04	1.19987	0.599934	+	0.800049i	$0.295193\pi$
$32$	0	0
$33$	2347.95	0.375322
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4775.73	−0.573502	−0.286751	−	0.958005i	$-0.592575\pi$
$37$	−4775.73	−0.573502	−0.286751	+	0.958005i	$0.592575\pi$
$38$	0	0
$39$	−6927.62	−0.729327
$40$	0	0
$41$	5423.27	0.503850	0.251925	−	0.967747i	$-0.418936\pi$
$41$	5423.27	0.503850	0.251925	+	0.967747i	$0.418936\pi$
$42$	0	0
$43$	−11896.4	−0.981171	−0.490585	−	0.871393i	$-0.663217\pi$
$43$	−11896.4	−0.981171	−0.490585	+	0.871393i	$0.663217\pi$
$44$	0	0
$45$	2588.89	0.190582
$46$	0	0
$47$	17428.0	1.15081	0.575404	−	0.817869i	$-0.304845\pi$
$47$	17428.0	1.15081	0.575404	+	0.817869i	$0.304845\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	13979.9	0.752623
$52$	0	0
$53$	37650.6	1.84112	0.920561	−	0.390599i	$-0.127732\pi$
$53$	37650.6	1.84112	0.920561	+	0.390599i	$0.127732\pi$
$54$	0	0
$55$	8338.26	0.371679
$56$	0	0
$57$	−6750.44	−0.275198
$58$	0	0
$59$	−22078.1	−0.825717	−0.412859	−	0.910795i	$-0.635470\pi$
$59$	−22078.1	−0.825717	−0.412859	+	0.910795i	$0.635470\pi$
$60$	0	0
$61$	8173.38	0.281240	0.140620	−	0.990064i	$-0.455090\pi$
$61$	8173.38	0.281240	0.140620	+	0.990064i	$0.455090\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−24602.0	−0.722248
$66$	0	0
$67$	13001.7	0.353846	0.176923	−	0.984225i	$-0.443386\pi$
$67$	13001.7	0.353846	0.176923	+	0.984225i	$0.443386\pi$
$68$	0	0
$69$	6793.69	0.171784
$70$	0	0
$71$	−12349.6	−0.290742	−0.145371	−	0.989377i	$-0.546438\pi$
$71$	−12349.6	−0.290742	−0.145371	+	0.989377i	$0.546438\pi$
$72$	0	0
$73$	−43600.3	−0.957596	−0.478798	−	0.877925i	$-0.658927\pi$
$73$	−43600.3	−0.957596	−0.478798	+	0.877925i	$0.658927\pi$
$74$	0	0
$75$	−18931.1	−0.388618
$76$	0	0
$77$	0	0
$78$	0	0
$79$	76749.5	1.38359	0.691796	−	0.722093i	$-0.256820\pi$
$79$	76749.5	1.38359	0.691796	+	0.722093i	$0.256820\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−21893.6	−0.348836	−0.174418	−	0.984672i	$-0.555804\pi$
$83$	−21893.6	−0.348836	−0.174418	+	0.984672i	$0.555804\pi$
$84$	0	0
$85$	49646.5	0.745318
$86$	0	0
$87$	54080.4	0.766023
$88$	0	0
$89$	136967.	1.83291	0.916454	−	0.400139i	$-0.131038\pi$
$89$	136967.	1.83291	0.916454	+	0.400139i	$0.131038\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	57780.4	0.692744
$94$	0	0
$95$	−23972.8	−0.272527
$96$	0	0
$97$	93050.1	1.00412	0.502062	−	0.864832i	$-0.332575\pi$
$97$	93050.1	1.00412	0.502062	+	0.864832i	$0.332575\pi$
$98$	0	0
$99$	21131.6	0.216692
$100$	0	0
$101$	−122302.	−1.19297	−0.596484	−	0.802625i	$-0.703436\pi$
$101$	−122302.	−1.19297	−0.596484	+	0.802625i	$0.703436\pi$
$102$	0	0
$103$	−74801.3	−0.694730	−0.347365	−	0.937730i	$-0.612924\pi$
$103$	−74801.3	−0.694730	−0.347365	+	0.937730i	$0.612924\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	70772.4	0.597592	0.298796	−	0.954317i	$-0.403415\pi$
$107$	70772.4	0.597592	0.298796	+	0.954317i	$0.403415\pi$
$108$	0	0
$109$	−142262.	−1.14689	−0.573446	−	0.819243i	$-0.694394\pi$
$109$	−142262.	−1.14689	−0.573446	+	0.819243i	$0.694394\pi$
$110$	0	0
$111$	−42981.5	−0.331112
$112$	0	0
$113$	198899.	1.46533	0.732667	−	0.680588i	$-0.238275\pi$
$113$	198899.	1.46533	0.732667	+	0.680588i	$0.238275\pi$
$114$	0	0
$115$	24126.3	0.170117
$116$	0	0
$117$	−62348.6	−0.421077
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−92990.7	−0.577399
$122$	0	0
$123$	48809.4	0.290898
$124$	0	0
$125$	−167110.	−0.956592
$126$	0	0
$127$	219619.	1.20826	0.604131	−	0.796885i	$-0.293521\pi$
$127$	219619.	1.20826	0.604131	+	0.796885i	$0.293521\pi$
$128$	0	0
$129$	−107068.	−0.566479
$130$	0	0
$131$	−54622.1	−0.278093	−0.139047	−	0.990286i	$-0.544404\pi$
$131$	−54622.1	−0.278093	−0.139047	+	0.990286i	$0.544404\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	23300.0	0.110033
$136$	0	0
$137$	72942.8	0.332033	0.166017	−	0.986123i	$-0.446909\pi$
$137$	72942.8	0.332033	0.166017	+	0.986123i	$0.446909\pi$
$138$	0	0
$139$	110668.	0.485831	0.242915	−	0.970047i	$-0.421896\pi$
$139$	110668.	0.485831	0.242915	+	0.970047i	$0.421896\pi$
$140$	0	0
$141$	156852.	0.664419
$142$	0	0
$143$	−200811.	−0.821199
$144$	0	0
$145$	192055.	0.758587
$146$	0	0
$147$	0	0
$148$	0	0
$149$	327025.	1.20675	0.603373	−	0.797459i	$-0.293823\pi$
$149$	327025.	1.20675	0.603373	+	0.797459i	$0.293823\pi$
$150$	0	0
$151$	174490.	0.622771	0.311386	−	0.950284i	$-0.399207\pi$
$151$	174490.	0.622771	0.311386	+	0.950284i	$0.399207\pi$
$152$	0	0
$153$	125819.	0.434527
$154$	0	0
$155$	205195.	0.686020
$156$	0	0
$157$	−440292.	−1.42558	−0.712790	−	0.701377i	$-0.752569\pi$
$157$	−440292.	−1.42558	−0.712790	+	0.701377i	$0.752569\pi$
$158$	0	0
$159$	338856.	1.06297
$160$	0	0
$161$	0	0
$162$	0	0
$163$	56491.9	0.166540	0.0832698	−	0.996527i	$-0.473464\pi$
$163$	56491.9	0.166540	0.0832698	+	0.996527i	$0.473464\pi$
$164$	0	0
$165$	75044.3	0.214589
$166$	0	0
$167$	688014.	1.90900	0.954501	−	0.298209i	$-0.0963891\pi$
$167$	688014.	1.90900	0.954501	+	0.298209i	$0.0963891\pi$
$168$	0	0
$169$	221200.	0.595755
$170$	0	0
$171$	−60754.0	−0.158885
$172$	0	0
$173$	−282558.	−0.717782	−0.358891	−	0.933379i	$-0.616845\pi$
$173$	−282558.	−0.717782	−0.358891	+	0.933379i	$0.616845\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−198703.	−0.476728
$178$	0	0
$179$	151782.	0.354069	0.177034	−	0.984205i	$-0.443350\pi$
$179$	151782.	0.354069	0.177034	+	0.984205i	$0.443350\pi$
$180$	0	0
$181$	322258.	0.731151	0.365575	−	0.930782i	$-0.380872\pi$
$181$	322258.	0.731151	0.365575	+	0.930782i	$0.380872\pi$
$182$	0	0
$183$	73560.4	0.162374
$184$	0	0
$185$	−152640.	−0.327898
$186$	0	0
$187$	405235.	0.847429
$188$	0	0
$189$	0	0
$190$	0	0
$191$	538521.	1.06812	0.534059	−	0.845447i	$-0.320666\pi$
$191$	538521.	1.06812	0.534059	+	0.845447i	$0.320666\pi$
$192$	0	0
$193$	303440.	0.586380	0.293190	−	0.956054i	$-0.405283\pi$
$193$	303440.	0.586380	0.293190	+	0.956054i	$0.405283\pi$
$194$	0	0
$195$	−221418.	−0.416990
$196$	0	0
$197$	656475.	1.20518	0.602591	−	0.798050i	$-0.294135\pi$
$197$	656475.	1.20518	0.602591	+	0.798050i	$0.294135\pi$
$198$	0	0
$199$	−842204.	−1.50759	−0.753797	−	0.657107i	$-0.771780\pi$
$199$	−842204.	−1.50759	−0.753797	+	0.657107i	$0.771780\pi$
$200$	0	0
$201$	117016.	0.204293
$202$	0	0
$203$	0	0
$204$	0	0
$205$	173336.	0.288074
$206$	0	0
$207$	61143.2	0.0991796
$208$	0	0
$209$	−195675.	−0.309864
$210$	0	0
$211$	1.01133e6	1.56382	0.781908	−	0.623394i	$-0.214247\pi$
$211$	1.01133e6	1.56382	0.781908	+	0.623394i	$0.214247\pi$
$212$	0	0
$213$	−111147.	−0.167860
$214$	0	0
$215$	−380228.	−0.560981
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−392403.	−0.552869
$220$	0	0
$221$	−1.19564e6	−1.64673
$222$	0	0
$223$	−957124.	−1.28886	−0.644431	−	0.764663i	$-0.722906\pi$
$223$	−957124.	−1.28886	−0.644431	+	0.764663i	$0.722906\pi$
$224$	0	0
$225$	−170380.	−0.224369
$226$	0	0
$227$	−471480.	−0.607293	−0.303647	−	0.952785i	$-0.598204\pi$
$227$	−471480.	−0.607293	−0.303647	+	0.952785i	$0.598204\pi$
$228$	0	0
$229$	268196.	0.337959	0.168980	−	0.985620i	$-0.445953\pi$
$229$	268196.	0.337959	0.168980	+	0.985620i	$0.445953\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.54722e6	1.86708	0.933538	−	0.358478i	$-0.116704\pi$
$233$	1.54722e6	1.86708	0.933538	+	0.358478i	$0.116704\pi$
$234$	0	0
$235$	557027.	0.657970
$236$	0	0
$237$	690745.	0.798817
$238$	0	0
$239$	−937542.	−1.06169	−0.530843	−	0.847470i	$-0.678125\pi$
$239$	−937542.	−1.06169	−0.530843	+	0.847470i	$0.678125\pi$
$240$	0	0
$241$	1.28486e6	1.42499	0.712497	−	0.701675i	$-0.247564\pi$
$241$	1.28486e6	1.42499	0.712497	+	0.701675i	$0.247564\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	577339.	0.602128
$248$	0	0
$249$	−197042.	−0.201401
$250$	0	0
$251$	−709769.	−0.711103	−0.355552	−	0.934657i	$-0.615707\pi$
$251$	−709769.	−0.711103	−0.355552	+	0.934657i	$0.615707\pi$
$252$	0	0
$253$	196929.	0.193423
$254$	0	0
$255$	446819.	0.430309
$256$	0	0
$257$	1.82016e6	1.71900	0.859502	−	0.511132i	$-0.170774\pi$
$257$	1.82016e6	1.71900	0.859502	+	0.511132i	$0.170774\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	486723.	0.442263
$262$	0	0
$263$	359187.	0.320208	0.160104	−	0.987100i	$-0.448817\pi$
$263$	359187.	0.320208	0.160104	+	0.987100i	$0.448817\pi$
$264$	0	0
$265$	1.20337e6	1.05265
$266$	0	0
$267$	1.23270e6	1.05823
$268$	0	0
$269$	−1.15767e6	−0.975446	−0.487723	−	0.872998i	$-0.662172\pi$
$269$	−1.15767e6	−0.975446	−0.487723	+	0.872998i	$0.662172\pi$
$270$	0	0
$271$	780175.	0.645311	0.322655	−	0.946516i	$-0.395424\pi$
$271$	780175.	0.645311	0.322655	+	0.946516i	$0.395424\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−548757.	−0.437571
$276$	0	0
$277$	−326331.	−0.255540	−0.127770	−	0.991804i	$-0.540782\pi$
$277$	−326331.	−0.255540	−0.127770	+	0.991804i	$0.540782\pi$
$278$	0	0
$279$	520023.	0.399956
$280$	0	0
$281$	364094.	0.275073	0.137536	−	0.990497i	$-0.456082\pi$
$281$	364094.	0.275073	0.137536	+	0.990497i	$0.456082\pi$
$282$	0	0
$283$	64019.2	0.0475165	0.0237582	−	0.999718i	$-0.492437\pi$
$283$	64019.2	0.0475165	0.0237582	+	0.999718i	$0.492437\pi$
$284$	0	0
$285$	−215755.	−0.157343
$286$	0	0
$287$	0	0
$288$	0	0
$289$	992939.	0.699324
$290$	0	0
$291$	837451.	0.579731
$292$	0	0
$293$	−398020.	−0.270855	−0.135427	−	0.990787i	$-0.543241\pi$
$293$	−398020.	−0.270855	−0.135427	+	0.990787i	$0.543241\pi$
$294$	0	0
$295$	−705651.	−0.472101
$296$	0	0
$297$	190184.	0.125107
$298$	0	0
$299$	−581038.	−0.375860
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.10071e6	−0.688760
$304$	0	0
$305$	261234.	0.160798
$306$	0	0
$307$	638841.	0.386854	0.193427	−	0.981115i	$-0.438040\pi$
$307$	638841.	0.386854	0.193427	+	0.981115i	$0.438040\pi$
$308$	0	0
$309$	−673212.	−0.401103
$310$	0	0
$311$	−2.48188e6	−1.45506	−0.727529	−	0.686077i	$-0.759331\pi$
$311$	−2.48188e6	−1.45506	−0.727529	+	0.686077i	$0.759331\pi$
$312$	0	0
$313$	−1.47653e6	−0.851889	−0.425944	−	0.904749i	$-0.640058\pi$
$313$	−1.47653e6	−0.851889	−0.425944	+	0.904749i	$0.640058\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.23126e6	−1.24710	−0.623551	−	0.781783i	$-0.714310\pi$
$317$	−2.23126e6	−1.24710	−0.623551	+	0.781783i	$0.714310\pi$
$318$	0	0
$319$	1.56763e6	0.862516
$320$	0	0
$321$	636952.	0.345020
$322$	0	0
$323$	−1.16506e6	−0.621360
$324$	0	0
$325$	1.61910e6	0.850289
$326$	0	0
$327$	−1.28036e6	−0.662159
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.41122e6	−1.20967	−0.604836	−	0.796350i	$-0.706761\pi$
$331$	−2.41122e6	−1.20967	−0.604836	+	0.796350i	$0.706761\pi$
$332$	0	0
$333$	−386834.	−0.191167
$334$	0	0
$335$	415556.	0.202310
$336$	0	0
$337$	−1.79586e6	−0.861388	−0.430694	−	0.902498i	$-0.641731\pi$
$337$	−1.79586e6	−0.861388	−0.430694	+	0.902498i	$0.641731\pi$
$338$	0	0
$339$	1.79009e6	0.846011
$340$	0	0
$341$	1.67488e6	0.780007
$342$	0	0
$343$	0	0
$344$	0	0
$345$	217137.	0.0982169
$346$	0	0
$347$	2.20731e6	0.984103	0.492051	−	0.870566i	$-0.336247\pi$
$347$	2.20731e6	0.984103	0.492051	+	0.870566i	$0.336247\pi$
$348$	0	0
$349$	−148188.	−0.0651252	−0.0325626	−	0.999470i	$-0.510367\pi$
$349$	−148188.	−0.0651252	−0.0325626	+	0.999470i	$0.510367\pi$
$350$	0	0
$351$	−561137.	−0.243109
$352$	0	0
$353$	−1.97040e6	−0.841623	−0.420811	−	0.907148i	$-0.638255\pi$
$353$	−1.97040e6	−0.841623	−0.420811	+	0.907148i	$0.638255\pi$
$354$	0	0
$355$	−394714.	−0.166231
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.43395e6	0.996723	0.498362	−	0.866969i	$-0.333935\pi$
$359$	2.43395e6	0.996723	0.498362	+	0.866969i	$0.333935\pi$
$360$	0	0
$361$	−1.91353e6	−0.772799
$362$	0	0
$363$	−836917.	−0.333362
$364$	0	0
$365$	−1.39354e6	−0.547502
$366$	0	0
$367$	4.30348e6	1.66784	0.833920	−	0.551886i	$-0.186092\pi$
$367$	4.30348e6	1.66784	0.833920	+	0.551886i	$0.186092\pi$
$368$	0	0
$369$	439285.	0.167950
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.20305e6	−0.447726	−0.223863	−	0.974621i	$-0.571867\pi$
$373$	−1.20305e6	−0.447726	−0.223863	+	0.974621i	$0.571867\pi$
$374$	0	0
$375$	−1.50399e6	−0.552289
$376$	0	0
$377$	−4.62529e6	−1.67604
$378$	0	0
$379$	5.11488e6	1.82910	0.914550	−	0.404473i	$-0.132545\pi$
$379$	5.11488e6	1.82910	0.914550	+	0.404473i	$0.132545\pi$
$380$	0	0
$381$	1.97657e6	0.697590
$382$	0	0
$383$	−5.01055e6	−1.74537	−0.872687	−	0.488280i	$-0.837625\pi$
$383$	−5.01055e6	−1.74537	−0.872687	+	0.488280i	$0.837625\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−963609.	−0.327057
$388$	0	0
$389$	−4.08326e6	−1.36815	−0.684073	−	0.729413i	$-0.739793\pi$
$389$	−4.08326e6	−1.36815	−0.684073	+	0.729413i	$0.739793\pi$
$390$	0	0
$391$	1.17253e6	0.387866
$392$	0	0
$393$	−491599.	−0.160557
$394$	0	0
$395$	2.45304e6	0.791063
$396$	0	0
$397$	1.07300e6	0.341682	0.170841	−	0.985299i	$-0.445352\pi$
$397$	1.07300e6	0.341682	0.170841	+	0.985299i	$0.445352\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.89948e6	−0.589894	−0.294947	−	0.955514i	$-0.595302\pi$
$401$	−1.89948e6	−0.589894	−0.294947	+	0.955514i	$0.595302\pi$
$402$	0	0
$403$	−4.94173e6	−1.51571
$404$	0	0
$405$	209700.	0.0635274
$406$	0	0
$407$	−1.24591e6	−0.372821
$408$	0	0
$409$	−1.93577e6	−0.572198	−0.286099	−	0.958200i	$-0.592359\pi$
$409$	−1.93577e6	−0.572198	−0.286099	+	0.958200i	$0.592359\pi$
$410$	0	0
$411$	656486.	0.191699
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−699754.	−0.199446
$416$	0	0
$417$	996012.	0.280495
$418$	0	0
$419$	−3.93448e6	−1.09484	−0.547422	−	0.836856i	$-0.684391\pi$
$419$	−3.93448e6	−1.09484	−0.547422	+	0.836856i	$0.684391\pi$
$420$	0	0
$421$	−5.16927e6	−1.42142	−0.710712	−	0.703483i	$-0.751627\pi$
$421$	−5.16927e6	−1.42142	−0.710712	+	0.703483i	$0.751627\pi$
$422$	0	0
$423$	1.41167e6	0.383603
$424$	0	0
$425$	−3.26734e6	−0.877448
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.80730e6	−0.474119
$430$	0	0
$431$	−2.12951e6	−0.552187	−0.276093	−	0.961131i	$-0.589040\pi$
$431$	−2.12951e6	−0.552187	−0.276093	+	0.961131i	$0.589040\pi$
$432$	0	0
$433$	−3.20879e6	−0.822473	−0.411237	−	0.911529i	$-0.634903\pi$
$433$	−3.20879e6	−0.822473	−0.411237	+	0.911529i	$0.634903\pi$
$434$	0	0
$435$	1.72849e6	0.437971
$436$	0	0
$437$	−566177.	−0.141824
$438$	0	0
$439$	−6.99077e6	−1.73126	−0.865632	−	0.500680i	$-0.833083\pi$
$439$	−6.99077e6	−1.73126	−0.865632	+	0.500680i	$0.833083\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.46008e6	1.80607	0.903034	−	0.429568i	$-0.141334\pi$
$443$	7.46008e6	1.80607	0.903034	+	0.429568i	$0.141334\pi$
$444$	0	0
$445$	4.37768e6	1.04796
$446$	0	0
$447$	2.94323e6	0.696715
$448$	0	0
$449$	−5.61620e6	−1.31470	−0.657350	−	0.753586i	$-0.728323\pi$
$449$	−5.61620e6	−1.31470	−0.657350	+	0.753586i	$0.728323\pi$
$450$	0	0
$451$	1.41484e6	0.327542
$452$	0	0
$453$	1.57041e6	0.359557
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.81271e6	−1.97387	−0.986936	−	0.161111i	$-0.948492\pi$
$457$	−8.81271e6	−1.97387	−0.986936	+	0.161111i	$0.948492\pi$
$458$	0	0
$459$	1.13237e6	0.250874
$460$	0	0
$461$	−3.24339e6	−0.710798	−0.355399	−	0.934715i	$-0.615655\pi$
$461$	−3.24339e6	−0.710798	−0.355399	+	0.934715i	$0.615655\pi$
$462$	0	0
$463$	−5.70039e6	−1.23581	−0.617906	−	0.786252i	$-0.712019\pi$
$463$	−5.70039e6	−1.23581	−0.617906	+	0.786252i	$0.712019\pi$
$464$	0	0
$465$	1.84675e6	0.396074
$466$	0	0
$467$	−2.18014e6	−0.462585	−0.231292	−	0.972884i	$-0.574295\pi$
$467$	−2.18014e6	−0.462585	−0.231292	+	0.972884i	$0.574295\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−3.96263e6	−0.823059
$472$	0	0
$473$	−3.10358e6	−0.637837
$474$	0	0
$475$	1.57770e6	0.320840
$476$	0	0
$477$	3.04970e6	0.613707
$478$	0	0
$479$	−3.93616e6	−0.783852	−0.391926	−	0.919997i	$-0.628191\pi$
$479$	−3.93616e6	−0.783852	−0.391926	+	0.919997i	$0.628191\pi$
$480$	0	0
$481$	3.67605e6	0.724466
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.97403e6	0.574104
$486$	0	0
$487$	7.00176e6	1.33778	0.668890	−	0.743361i	$-0.266770\pi$
$487$	7.00176e6	1.33778	0.668890	+	0.743361i	$0.266770\pi$
$488$	0	0
$489$	508428.	0.0961517
$490$	0	0
$491$	−9.37114e6	−1.75424	−0.877119	−	0.480273i	$-0.840537\pi$
$491$	−9.37114e6	−1.75424	−0.877119	+	0.480273i	$0.840537\pi$
$492$	0	0
$493$	9.33378e6	1.72958
$494$	0	0
$495$	675399.	0.123893
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−9.75932e6	−1.75456	−0.877280	−	0.479978i	$-0.840645\pi$
$499$	−9.75932e6	−1.75456	−0.877280	+	0.479978i	$0.840645\pi$
$500$	0	0
$501$	6.19213e6	1.10216
$502$	0	0
$503$	1.69667e6	0.299004	0.149502	−	0.988761i	$-0.452233\pi$
$503$	1.69667e6	0.299004	0.149502	+	0.988761i	$0.452233\pi$
$504$	0	0
$505$	−3.90895e6	−0.682075
$506$	0	0
$507$	1.99080e6	0.343959
$508$	0	0
$509$	3.73566e6	0.639106	0.319553	−	0.947568i	$-0.396467\pi$
$509$	3.73566e6	0.639106	0.319553	+	0.947568i	$0.396467\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−546786.	−0.0917326
$514$	0	0
$515$	−2.39077e6	−0.397209
$516$	0	0
$517$	4.54668e6	0.748114
$518$	0	0
$519$	−2.54302e6	−0.414412
$520$	0	0
$521$	−7.32718e6	−1.18261	−0.591306	−	0.806447i	$-0.701388\pi$
$521$	−7.32718e6	−1.18261	−0.591306	+	0.806447i	$0.701388\pi$
$522$	0	0
$523$	9591.43	0.00153331	0.000766653	−	1.00000i	$-0.499756\pi$
$523$	9591.43	0.00153331	0.000766653		1.00000i	$0.499756\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.97236e6	1.56413
$528$	0	0
$529$	−5.86654e6	−0.911471
$530$	0	0
$531$	−1.78832e6	−0.275239
$532$	0	0
$533$	−4.17448e6	−0.636479
$534$	0	0
$535$	2.26200e6	0.341671
$536$	0	0
$537$	1.36604e6	0.204422
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3.75692e6	−0.551872	−0.275936	−	0.961176i	$-0.588988\pi$
$541$	−3.75692e6	−0.551872	−0.275936	+	0.961176i	$0.588988\pi$
$542$	0	0
$543$	2.90032e6	0.422130
$544$	0	0
$545$	−4.54692e6	−0.655731
$546$	0	0
$547$	1.03391e7	1.47746	0.738729	−	0.674002i	$-0.235426\pi$
$547$	1.03391e7	1.47746	0.738729	+	0.674002i	$0.235426\pi$
$548$	0	0
$549$	662044.	0.0937467
$550$	0	0
$551$	−4.50699e6	−0.632423
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.37376e6	−0.189312
$556$	0	0
$557$	−1.00476e7	−1.37222	−0.686108	−	0.727500i	$-0.740682\pi$
$557$	−1.00476e7	−1.37222	−0.686108	+	0.727500i	$0.740682\pi$
$558$	0	0
$559$	9.15708e6	1.23945
$560$	0	0
$561$	3.64712e6	0.489263
$562$	0	0
$563$	8.49380e6	1.12936	0.564678	−	0.825311i	$-0.309000\pi$
$563$	8.49380e6	1.12936	0.564678	+	0.825311i	$0.309000\pi$
$564$	0	0
$565$	6.35713e6	0.837799
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.81531e6	0.494025	0.247012	−	0.969012i	$-0.420551\pi$
$569$	3.81531e6	0.494025	0.247012	+	0.969012i	$0.420551\pi$
$570$	0	0
$571$	1.22133e7	1.56762	0.783812	−	0.620998i	$-0.213272\pi$
$571$	1.22133e7	1.56762	0.783812	+	0.620998i	$0.213272\pi$
$572$	0	0
$573$	4.84669e6	0.616678
$574$	0	0
$575$	−1.58780e6	−0.200275
$576$	0	0
$577$	2.29471e6	0.286938	0.143469	−	0.989655i	$-0.454174\pi$
$577$	2.29471e6	0.286938	0.143469	+	0.989655i	$0.454174\pi$
$578$	0	0
$579$	2.73096e6	0.338547
$580$	0	0
$581$	0	0
$582$	0	0
$583$	9.82243e6	1.19687
$584$	0	0
$585$	−1.99276e6	−0.240749
$586$	0	0
$587$	−7.53391e6	−0.902454	−0.451227	−	0.892409i	$-0.649014\pi$
$587$	−7.53391e6	−0.902454	−0.451227	+	0.892409i	$0.649014\pi$
$588$	0	0
$589$	−4.81534e6	−0.571925
$590$	0	0
$591$	5.90827e6	0.695812
$592$	0	0
$593$	−7.23521e6	−0.844918	−0.422459	−	0.906382i	$-0.638833\pi$
$593$	−7.23521e6	−0.844918	−0.422459	+	0.906382i	$0.638833\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−7.57984e6	−0.870410
$598$	0	0
$599$	3.33856e6	0.380182	0.190091	−	0.981766i	$-0.439122\pi$
$599$	3.33856e6	0.380182	0.190091	+	0.981766i	$0.439122\pi$
$600$	0	0
$601$	−1.48297e7	−1.67474	−0.837370	−	0.546636i	$-0.815908\pi$
$601$	−1.48297e7	−1.67474	−0.837370	+	0.546636i	$0.815908\pi$
$602$	0	0
$603$	1.05314e6	0.117949
$604$	0	0
$605$	−2.97213e6	−0.330126
$606$	0	0
$607$	6.37473e6	0.702247	0.351124	−	0.936329i	$-0.385800\pi$
$607$	6.37473e6	0.702247	0.351124	+	0.936329i	$0.385800\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.34149e7	−1.45374
$612$	0	0
$613$	−1.54869e7	−1.66461	−0.832306	−	0.554317i	$-0.812980\pi$
$613$	−1.54869e7	−1.66461	−0.832306	+	0.554317i	$0.812980\pi$
$614$	0	0
$615$	1.56003e6	0.166320
$616$	0	0
$617$	−4.55324e6	−0.481513	−0.240756	−	0.970586i	$-0.577396\pi$
$617$	−4.55324e6	−0.481513	−0.240756	+	0.970586i	$0.577396\pi$
$618$	0	0
$619$	1.16422e7	1.22126	0.610628	−	0.791918i	$-0.290917\pi$
$619$	1.16422e7	1.22126	0.610628	+	0.791918i	$0.290917\pi$
$620$	0	0
$621$	550289.	0.0572613
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.23221e6	0.126178
$626$	0	0
$627$	−1.76108e6	−0.178900
$628$	0	0
$629$	−7.41822e6	−0.747607
$630$	0	0
$631$	1.07436e7	1.07417	0.537087	−	0.843527i	$-0.319525\pi$
$631$	1.07436e7	1.07417	0.537087	+	0.843527i	$0.319525\pi$
$632$	0	0
$633$	9.10195e6	0.902870
$634$	0	0
$635$	7.01938e6	0.690819
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.00032e6	−0.0969140
$640$	0	0
$641$	1.23138e7	1.18371	0.591857	−	0.806043i	$-0.298395\pi$
$641$	1.23138e7	1.18371	0.591857	+	0.806043i	$0.298395\pi$
$642$	0	0
$643$	1.67944e7	1.60191	0.800955	−	0.598724i	$-0.204325\pi$
$643$	1.67944e7	1.60191	0.800955	+	0.598724i	$0.204325\pi$
$644$	0	0
$645$	−3.42205e6	−0.323882
$646$	0	0
$647$	−1.99108e7	−1.86994	−0.934970	−	0.354726i	$-0.884574\pi$
$647$	−1.99108e7	−1.86994	−0.934970	+	0.354726i	$0.884574\pi$
$648$	0	0
$649$	−5.75981e6	−0.536780
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.10990e7	1.93633	0.968163	−	0.250319i	$-0.0805355\pi$
$653$	2.10990e7	1.93633	0.968163	+	0.250319i	$0.0805355\pi$
$654$	0	0
$655$	−1.74581e6	−0.158999
$656$	0	0
$657$	−3.53163e6	−0.319199
$658$	0	0
$659$	1.75070e7	1.57035	0.785177	−	0.619271i	$-0.212572\pi$
$659$	1.75070e7	1.57035	0.785177	+	0.619271i	$0.212572\pi$
$660$	0	0
$661$	−1.80169e7	−1.60389	−0.801947	−	0.597395i	$-0.796203\pi$
$661$	−1.80169e7	−1.60389	−0.801947	+	0.597395i	$0.796203\pi$
$662$	0	0
$663$	−1.07608e7	−0.950737
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.53587e6	0.394771
$668$	0	0
$669$	−8.61412e6	−0.744124
$670$	0	0
$671$	2.13230e6	0.182828
$672$	0	0
$673$	1.48593e7	1.26462	0.632312	−	0.774714i	$-0.282106\pi$
$673$	1.48593e7	1.26462	0.632312	+	0.774714i	$0.282106\pi$
$674$	0	0
$675$	−1.53342e6	−0.129539
$676$	0	0
$677$	−7.18320e6	−0.602347	−0.301173	−	0.953569i	$-0.597378\pi$
$677$	−7.18320e6	−0.602347	−0.301173	+	0.953569i	$0.597378\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−4.24332e6	−0.350621
$682$	0	0
$683$	1.65092e6	0.135417	0.0677087	−	0.997705i	$-0.478431\pi$
$683$	1.65092e6	0.135417	0.0677087	+	0.997705i	$0.478431\pi$
$684$	0	0
$685$	2.33137e6	0.189839
$686$	0	0
$687$	2.41377e6	0.195121
$688$	0	0
$689$	−2.89810e7	−2.32576
$690$	0	0
$691$	−777444.	−0.0619404	−0.0309702	−	0.999520i	$-0.509860\pi$
$691$	−777444.	−0.0619404	−0.0309702	+	0.999520i	$0.509860\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.53713e6	0.277772
$696$	0	0
$697$	8.42406e6	0.656809
$698$	0	0
$699$	1.39250e7	1.07796
$700$	0	0
$701$	1.65671e6	0.127336	0.0636681	−	0.997971i	$-0.479720\pi$
$701$	1.65671e6	0.127336	0.0636681	+	0.997971i	$0.479720\pi$
$702$	0	0
$703$	3.58203e6	0.273364
$704$	0	0
$705$	5.01324e6	0.379879
$706$	0	0
$707$	0	0
$708$	0	0
$709$	4.99901e6	0.373481	0.186740	−	0.982409i	$-0.440208\pi$
$709$	4.99901e6	0.373481	0.186740	+	0.982409i	$0.440208\pi$
$710$	0	0
$711$	6.21671e6	0.461197
$712$	0	0
$713$	4.84619e6	0.357007
$714$	0	0
$715$	−6.41825e6	−0.469517
$716$	0	0
$717$	−8.43788e6	−0.612965
$718$	0	0
$719$	1.81384e7	1.30851	0.654253	−	0.756276i	$-0.272983\pi$
$719$	1.81384e7	1.30851	0.654253	+	0.756276i	$0.272983\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.15637e7	0.822721
$724$	0	0
$725$	−1.26395e7	−0.893070
$726$	0	0
$727$	−1.17918e7	−0.827454	−0.413727	−	0.910401i	$-0.635773\pi$
$727$	−1.17918e7	−0.827454	−0.413727	+	0.910401i	$0.635773\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.84789e7	−1.27904
$732$	0	0
$733$	4.96469e6	0.341297	0.170648	−	0.985332i	$-0.445414\pi$
$733$	4.96469e6	0.341297	0.170648	+	0.985332i	$0.445414\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.39194e6	0.230027
$738$	0	0
$739$	1.86292e7	1.25483	0.627413	−	0.778686i	$-0.284114\pi$
$739$	1.86292e7	1.25483	0.627413	+	0.778686i	$0.284114\pi$
$740$	0	0
$741$	5.19605e6	0.347639
$742$	0	0
$743$	1.77727e7	1.18109	0.590544	−	0.807006i	$-0.298913\pi$
$743$	1.77727e7	1.18109	0.590544	+	0.807006i	$0.298913\pi$
$744$	0	0
$745$	1.04523e7	0.689953
$746$	0	0
$747$	−1.77338e6	−0.116279
$748$	0	0
$749$	0	0
$750$	0	0
$751$	4.66059e6	0.301537	0.150769	−	0.988569i	$-0.451825\pi$
$751$	4.66059e6	0.301537	0.150769	+	0.988569i	$0.451825\pi$
$752$	0	0
$753$	−6.38792e6	−0.410556
$754$	0	0
$755$	5.57699e6	0.356067
$756$	0	0
$757$	−2.36520e6	−0.150013	−0.0750064	−	0.997183i	$-0.523898\pi$
$757$	−2.36520e6	−0.150013	−0.0750064	+	0.997183i	$0.523898\pi$
$758$	0	0
$759$	1.77236e6	0.111673
$760$	0	0
$761$	1.76946e7	1.10759	0.553794	−	0.832654i	$-0.313180\pi$
$761$	1.76946e7	1.10759	0.553794	+	0.832654i	$0.313180\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	4.02137e6	0.248439
$766$	0	0
$767$	1.69943e7	1.04307
$768$	0	0
$769$	7.90324e6	0.481936	0.240968	−	0.970533i	$-0.422535\pi$
$769$	7.90324e6	0.481936	0.240968	+	0.970533i	$0.422535\pi$
$770$	0	0
$771$	1.63814e7	0.992468
$772$	0	0
$773$	−2.22460e7	−1.33907	−0.669534	−	0.742782i	$-0.733506\pi$
$773$	−2.22460e7	−1.33907	−0.669534	+	0.742782i	$0.733506\pi$
$774$	0	0
$775$	−1.35043e7	−0.807639
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.06771e6	−0.240163
$780$	0	0
$781$	−3.22181e6	−0.189005
$782$	0	0
$783$	4.38051e6	0.255341
$784$	0	0
$785$	−1.40724e7	−0.815070
$786$	0	0
$787$	−2.49897e7	−1.43822	−0.719108	−	0.694898i	$-0.755449\pi$
$787$	−2.49897e7	−1.43822	−0.719108	+	0.694898i	$0.755449\pi$
$788$	0	0
$789$	3.23269e6	0.184872
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−6.29134e6	−0.355271
$794$	0	0
$795$	1.08304e7	0.607750
$796$	0	0
$797$	−1.40897e7	−0.785696	−0.392848	−	0.919603i	$-0.628510\pi$
$797$	−1.40897e7	−0.785696	−0.392848	+	0.919603i	$0.628510\pi$
$798$	0	0
$799$	2.70712e7	1.50017
$800$	0	0
$801$	1.10943e7	0.610970
$802$	0	0
$803$	−1.13746e7	−0.622512
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.04190e7	−0.563174
$808$	0	0
$809$	−2.54305e7	−1.36611	−0.683053	−	0.730369i	$-0.739348\pi$
$809$	−2.54305e7	−1.36611	−0.683053	+	0.730369i	$0.739348\pi$
$810$	0	0
$811$	−1.97155e6	−0.105258	−0.0526291	−	0.998614i	$-0.516760\pi$
$811$	−1.97155e6	−0.105258	−0.0526291	+	0.998614i	$0.516760\pi$
$812$	0	0
$813$	7.02158e6	0.372570
$814$	0	0
$815$	1.80557e6	0.0952184
$816$	0	0
$817$	8.92288e6	0.467681
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.11451e6	−0.0577067	−0.0288534	−	0.999584i	$-0.509186\pi$
$821$	−1.11451e6	−0.0577067	−0.0288534	+	0.999584i	$0.509186\pi$
$822$	0	0
$823$	2.79838e6	0.144015	0.0720073	−	0.997404i	$-0.477060\pi$
$823$	2.79838e6	0.144015	0.0720073	+	0.997404i	$0.477060\pi$
$824$	0	0
$825$	−4.93882e6	−0.252632
$826$	0	0
$827$	1.56547e6	0.0795940	0.0397970	−	0.999208i	$-0.487329\pi$
$827$	1.56547e6	0.0795940	0.0397970	+	0.999208i	$0.487329\pi$
$828$	0	0
$829$	1.33721e7	0.675791	0.337895	−	0.941184i	$-0.390285\pi$
$829$	1.33721e7	0.675791	0.337895	+	0.941184i	$0.390285\pi$
$830$	0	0
$831$	−2.93698e6	−0.147536
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2.19900e7	1.09146
$836$	0	0
$837$	4.68021e6	0.230915
$838$	0	0
$839$	−4.37187e6	−0.214418	−0.107209	−	0.994236i	$-0.534191\pi$
$839$	−4.37187e6	−0.214418	−0.107209	+	0.994236i	$0.534191\pi$
$840$	0	0
$841$	1.55961e7	0.760372
$842$	0	0
$843$	3.27685e6	0.158813
$844$	0	0
$845$	7.06989e6	0.340621
$846$	0	0
$847$	0	0
$848$	0	0
$849$	576173.	0.0274336
$850$	0	0
$851$	−3.60498e6	−0.170639
$852$	0	0
$853$	1.45910e7	0.686612	0.343306	−	0.939224i	$-0.388453\pi$
$853$	1.45910e7	0.686612	0.343306	+	0.939224i	$0.388453\pi$
$854$	0	0
$855$	−1.94179e6	−0.0908422
$856$	0	0
$857$	1.31276e7	0.610565	0.305283	−	0.952262i	$-0.401249\pi$
$857$	1.31276e7	0.610565	0.305283	+	0.952262i	$0.401249\pi$
$858$	0	0
$859$	−1.42131e7	−0.657211	−0.328605	−	0.944467i	$-0.606579\pi$
$859$	−1.42131e7	−0.657211	−0.328605	+	0.944467i	$0.606579\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.66170e7	1.21656	0.608278	−	0.793724i	$-0.291861\pi$
$863$	2.66170e7	1.21656	0.608278	+	0.793724i	$0.291861\pi$
$864$	0	0
$865$	−9.03101e6	−0.410389
$866$	0	0
$867$	8.93646e6	0.403755
$868$	0	0
$869$	2.00227e7	0.899441
$870$	0	0
$871$	−1.00079e7	−0.446990
$872$	0	0
$873$	7.53706e6	0.334708
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.42383e7	−0.625114	−0.312557	−	0.949899i	$-0.601186\pi$
$877$	−1.42383e7	−0.625114	−0.312557	+	0.949899i	$0.601186\pi$
$878$	0	0
$879$	−3.58218e6	−0.156378
$880$	0	0
$881$	1.51921e7	0.659445	0.329722	−	0.944078i	$-0.393045\pi$
$881$	1.51921e7	0.659445	0.329722	+	0.944078i	$0.393045\pi$
$882$	0	0
$883$	1.03997e7	0.448868	0.224434	−	0.974489i	$-0.427947\pi$
$883$	1.03997e7	0.448868	0.224434	+	0.974489i	$0.427947\pi$
$884$	0	0
$885$	−6.35086e6	−0.272567
$886$	0	0
$887$	−2.02129e7	−0.862618	−0.431309	−	0.902204i	$-0.641948\pi$
$887$	−2.02129e7	−0.862618	−0.431309	+	0.902204i	$0.641948\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.71166e6	0.0722308
$892$	0	0
$893$	−1.30718e7	−0.548540
$894$	0	0
$895$	4.85119e6	0.202437
$896$	0	0
$897$	−5.22934e6	−0.217003
$898$	0	0
$899$	3.85776e7	1.59197
$900$	0	0
$901$	5.84834e7	2.40005
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.02999e7	0.418033
$906$	0	0
$907$	−4.35643e6	−0.175838	−0.0879189	−	0.996128i	$-0.528022\pi$
$907$	−4.35643e6	−0.175838	−0.0879189	+	0.996128i	$0.528022\pi$
$908$	0	0
$909$	−9.90643e6	−0.397656
$910$	0	0
$911$	1.99500e7	0.796427	0.398214	−	0.917293i	$-0.369630\pi$
$911$	1.99500e7	0.796427	0.398214	+	0.917293i	$0.369630\pi$
$912$	0	0
$913$	−5.71168e6	−0.226771
$914$	0	0
$915$	2.35111e6	0.0928368
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−9.55172e6	−0.373072	−0.186536	−	0.982448i	$-0.559726\pi$
$919$	−9.55172e6	−0.373072	−0.186536	+	0.982448i	$0.559726\pi$
$920$	0	0
$921$	5.74957e6	0.223350
$922$	0	0
$923$	9.50594e6	0.367275
$924$	0	0
$925$	1.00455e7	0.386028
$926$	0	0
$927$	−6.05891e6	−0.231577
$928$	0	0
$929$	1.26025e7	0.479090	0.239545	−	0.970885i	$-0.423002\pi$
$929$	1.26025e7	0.479090	0.239545	+	0.970885i	$0.423002\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−2.23370e7	−0.840078
$934$	0	0
$935$	1.29520e7	0.484514
$936$	0	0
$937$	−1.00216e7	−0.372897	−0.186448	−	0.982465i	$-0.559698\pi$
$937$	−1.00216e7	−0.372897	−0.186448	+	0.982465i	$0.559698\pi$
$938$	0	0
$939$	−1.32888e7	−0.491838
$940$	0	0
$941$	1.54113e6	0.0567369	0.0283684	−	0.999598i	$-0.490969\pi$
$941$	1.54113e6	0.0567369	0.0283684	+	0.999598i	$0.490969\pi$
$942$	0	0
$943$	4.09377e6	0.149915
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.07046e7	1.11257	0.556287	−	0.830990i	$-0.312226\pi$
$947$	3.07046e7	1.11257	0.556287	+	0.830990i	$0.312226\pi$
$948$	0	0
$949$	3.35607e7	1.20967
$950$	0	0
$951$	−2.00813e7	−0.720014
$952$	0	0
$953$	−4.39769e7	−1.56853	−0.784263	−	0.620428i	$-0.786959\pi$
$953$	−4.39769e7	−1.56853	−0.784263	+	0.620428i	$0.786959\pi$
$954$	0	0
$955$	1.72120e7	0.610692
$956$	0	0
$957$	1.41087e7	0.497974
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.25878e7	0.439684
$962$	0	0
$963$	5.73257e6	0.199197
$964$	0	0
$965$	9.69841e6	0.335261
$966$	0	0
$967$	−4.20732e7	−1.44690	−0.723452	−	0.690375i	$-0.757445\pi$
$967$	−4.20732e7	−1.44690	−0.723452	+	0.690375i	$0.757445\pi$
$968$	0	0
$969$	−1.04856e7	−0.358743
$970$	0	0
$971$	−2.08006e7	−0.707992	−0.353996	−	0.935247i	$-0.615177\pi$
$971$	−2.08006e7	−0.707992	−0.353996	+	0.935247i	$0.615177\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.45719e7	0.490915
$976$	0	0
$977$	3.74338e7	1.25466	0.627332	−	0.778752i	$-0.284147\pi$
$977$	3.74338e7	1.25466	0.627332	+	0.778752i	$0.284147\pi$
$978$	0	0
$979$	3.57324e7	1.19153
$980$	0	0
$981$	−1.15232e7	−0.382297
$982$	0	0
$983$	2.41962e7	0.798664	0.399332	−	0.916806i	$-0.369242\pi$
$983$	2.41962e7	0.798664	0.399332	+	0.916806i	$0.369242\pi$
$984$	0	0
$985$	2.09820e7	0.689058
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.98005e6	−0.291936
$990$	0	0
$991$	−2.52080e7	−0.815369	−0.407684	−	0.913123i	$-0.633664\pi$
$991$	−2.52080e7	−0.815369	−0.407684	+	0.913123i	$0.633664\pi$
$992$	0	0
$993$	−2.17010e7	−0.698404
$994$	0	0
$995$	−2.69182e7	−0.861962
$996$	0	0
$997$	−4.95511e7	−1.57876	−0.789379	−	0.613907i	$-0.789597\pi$
$997$	−4.95511e7	−1.57876	−0.789379	+	0.613907i	$0.789597\pi$
$998$	0	0
$999$	−3.48150e6	−0.110371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.6.a.p.1.3		4
7.2	even	3		588.6.i.o.361.2		8
7.3	odd	6		84.6.i.c.37.3	yes	8
7.4	even	3		588.6.i.o.373.2		8
7.5	odd	6		84.6.i.c.25.3	✓	8
7.6	odd	2		588.6.a.n.1.2		4
21.5	even	6		252.6.k.f.109.2		8
21.17	even	6		252.6.k.f.37.2		8
28.3	even	6		336.6.q.i.289.3		8
28.19	even	6		336.6.q.i.193.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.i.c.25.3	✓	8	7.5	odd	6
84.6.i.c.37.3	yes	8	7.3	odd	6
252.6.k.f.37.2		8	21.17	even	6
252.6.k.f.109.2		8	21.5	even	6
336.6.q.i.193.3		8	28.19	even	6
336.6.q.i.289.3		8	28.3	even	6
588.6.a.n.1.2		4	7.6	odd	2
588.6.a.p.1.3		4	1.1	even	1	trivial
588.6.i.o.361.2		8	7.2	even	3
588.6.i.o.373.2		8	7.4	even	3

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	588.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +31.9616 q^{5} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} +31.9616 q^{5} +81.0000 q^{9} +260.884 q^{11} -769.735 q^{13} +287.654 q^{15} +1553.32 q^{17} -750.049 q^{19} +754.854 q^{23} -2103.46 q^{25} +729.000 q^{27} +6008.93 q^{29} +6420.04 q^{31} +2347.95 q^{33} -4775.73 q^{37} -6927.62 q^{39} +5423.27 q^{41} -11896.4 q^{43} +2588.89 q^{45} +17428.0 q^{47} +13979.9 q^{51} +37650.6 q^{53} +8338.26 q^{55} -6750.44 q^{57} -22078.1 q^{59} +8173.38 q^{61} -24602.0 q^{65} +13001.7 q^{67} +6793.69 q^{69} -12349.6 q^{71} -43600.3 q^{73} -18931.1 q^{75} +76749.5 q^{79} +6561.00 q^{81} -21893.6 q^{83} +49646.5 q^{85} +54080.4 q^{87} +136967. q^{89} +57780.4 q^{93} -23972.8 q^{95} +93050.1 q^{97} +21131.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 36 q^{3} + 324 q^{9}+O(q^{10}) \) 4 * q + 36 * q^3 + 324 * q^9	\( 4 q + 36 q^{3} + 324 q^{9} + 462 q^{11} + 602 q^{13} + 228 q^{17} + 358 q^{19} + 2148 q^{23} + 5454 q^{25} + 2916 q^{27} - 5532 q^{29} + 830 q^{31} + 4158 q^{33} + 3914 q^{37} + 5418 q^{39} + 8316 q^{41} - 14518 q^{43} + 41700 q^{47} + 2052 q^{51} - 22164 q^{53} - 3892 q^{55} + 3222 q^{57} + 32886 q^{59} + 83732 q^{61} + 93192 q^{65} + 80034 q^{67} + 19332 q^{69} + 44772 q^{71} - 22470 q^{73} + 49086 q^{75} + 75286 q^{79} + 26244 q^{81} + 17418 q^{83} + 139252 q^{85} - 49788 q^{87} + 28944 q^{89} + 7470 q^{93} + 144120 q^{95} + 216678 q^{97} + 37422 q^{99}+O(q^{100}) \) 4 * q + 36 * q^3 + 324 * q^9 + 462 * q^11 + 602 * q^13 + 228 * q^17 + 358 * q^19 + 2148 * q^23 + 5454 * q^25 + 2916 * q^27 - 5532 * q^29 + 830 * q^31 + 4158 * q^33 + 3914 * q^37 + 5418 * q^39 + 8316 * q^41 - 14518 * q^43 + 41700 * q^47 + 2052 * q^51 - 22164 * q^53 - 3892 * q^55 + 3222 * q^57 + 32886 * q^59 + 83732 * q^61 + 93192 * q^65 + 80034 * q^67 + 19332 * q^69 + 44772 * q^71 - 22470 * q^73 + 49086 * q^75 + 75286 * q^79 + 26244 * q^81 + 17418 * q^83 + 139252 * q^85 - 49788 * q^87 + 28944 * q^89 + 7470 * q^93 + 144120 * q^95 + 216678 * q^97 + 37422 * q^99

Introduction

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