LMFDB - Embedded newform 600.6.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,6,Mod(1,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("600.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	600.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:	$96.2302918878$
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} - 71x^{2} - 30x + 360 $ x^4 - 71x^2 - 30x + 360
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.61388$ of defining polynomial
Character	$\chi$	$=$	600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} -242.714 q^{7} +81.0000 q^{9} +142.542 q^{11} -1080.72 q^{13} -1329.42 q^{17} +1551.52 q^{19} -2184.42 q^{21} -1087.52 q^{23} +729.000 q^{27} +4374.67 q^{29} +2623.69 q^{31} +1282.88 q^{33} +2081.24 q^{37} -9726.46 q^{39} -15998.1 q^{41} -4083.17 q^{43} +4819.20 q^{47} +42102.9 q^{49} -11964.7 q^{51} +20686.4 q^{53} +13963.7 q^{57} -12562.3 q^{59} +41811.0 q^{61} -19659.8 q^{63} +60479.5 q^{67} -9787.71 q^{69} -27123.5 q^{71} +57295.5 q^{73} -34596.9 q^{77} -27152.5 q^{79} +6561.00 q^{81} +74516.5 q^{83} +39372.0 q^{87} +80563.6 q^{89} +262305. q^{91} +23613.2 q^{93} -88537.3 q^{97} +11545.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 36 q^{3} + 6 q^{7} + 324 q^{9} + 342 q^{11} - 222 q^{13} + 618 q^{17} + 1520 q^{19} + 54 q^{21} + 752 q^{23} + 2916 q^{27} + 4002 q^{29} + 3012 q^{31} + 3078 q^{33} + 14142 q^{37} - 1998 q^{39} - 18900 q^{41}+ \cdots + 27702 q^{99}+O(q^{100}) $ 4 * q + 36 * q^3 + 6 * q^7 + 324 * q^9 + 342 * q^11 - 222 * q^13 + 618 * q^17 + 1520 * q^19 + 54 * q^21 + 752 * q^23 + 2916 * q^27 + 4002 * q^29 + 3012 * q^31 + 3078 * q^33 + 14142 * q^37 - 1998 * q^39 - 18900 * q^41 + 11424 * q^43 + 7572 * q^47 + 17184 * q^49 + 5562 * q^51 + 47374 * q^53 + 13680 * q^57 + 43014 * q^59 - 19480 * q^61 + 486 * q^63 + 25620 * q^67 + 6768 * q^69 + 81036 * q^71 + 111324 * q^73 - 45748 * q^77 + 84988 * q^79 + 26244 * q^81 + 83200 * q^83 + 36018 * q^87 + 226488 * q^89 + 305228 * q^91 + 27108 * q^93 - 32016 * q^97 + 27702 * 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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−242.714	−1.87219	−0.936093	−	0.351752i	$-0.885586\pi$
$7$	−242.714	−1.87219	−0.936093	+	0.351752i	$0.885586\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	142.542	0.355190	0.177595	−	0.984104i	$-0.443168\pi$
$11$	142.542	0.355190	0.177595	+	0.984104i	$0.443168\pi$
$12$	0	0
$13$	−1080.72	−1.77359	−0.886797	−	0.462160i	$-0.847075\pi$
$13$	−1080.72	−1.77359	−0.886797	+	0.462160i	$0.847075\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1329.42	−1.11568	−0.557838	−	0.829950i	$-0.688369\pi$
$17$	−1329.42	−1.11568	−0.557838	+	0.829950i	$0.688369\pi$
$18$	0	0
$19$	1551.52	0.985995	0.492998	−	0.870031i	$-0.335901\pi$
$19$	1551.52	0.985995	0.492998	+	0.870031i	$0.335901\pi$
$20$	0	0
$21$	−2184.42	−1.08091
$22$	0	0
$23$	−1087.52	−0.428666	−0.214333	−	0.976761i	$-0.568758\pi$
$23$	−1087.52	−0.428666	−0.214333	+	0.976761i	$0.568758\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	4374.67	0.965940	0.482970	−	0.875637i	$-0.339558\pi$
$29$	4374.67	0.965940	0.482970	+	0.875637i	$0.339558\pi$
$30$	0	0
$31$	2623.69	0.490352	0.245176	−	0.969479i	$-0.421154\pi$
$31$	2623.69	0.490352	0.245176	+	0.969479i	$0.421154\pi$
$32$	0	0
$33$	1282.88	0.205069
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2081.24	0.249930	0.124965	−	0.992161i	$-0.460118\pi$
$37$	2081.24	0.249930	0.124965	+	0.992161i	$0.460118\pi$
$38$	0	0
$39$	−9726.46	−1.02398
$40$	0	0
$41$	−15998.1	−1.48631	−0.743155	−	0.669120i	$-0.766671\pi$
$41$	−15998.1	−1.48631	−0.743155	+	0.669120i	$0.766671\pi$
$42$	0	0
$43$	−4083.17	−0.336764	−0.168382	−	0.985722i	$-0.553854\pi$
$43$	−4083.17	−0.336764	−0.168382	+	0.985722i	$0.553854\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4819.20	0.318222	0.159111	−	0.987261i	$-0.449137\pi$
$47$	4819.20	0.318222	0.159111	+	0.987261i	$0.449137\pi$
$48$	0	0
$49$	42102.9	2.50508
$50$	0	0
$51$	−11964.7	−0.644136
$52$	0	0
$53$	20686.4	1.01157	0.505785	−	0.862659i	$-0.331203\pi$
$53$	20686.4	1.01157	0.505785	+	0.862659i	$0.331203\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	13963.7	0.569265
$58$	0	0
$59$	−12562.3	−0.469827	−0.234913	−	0.972016i	$-0.575481\pi$
$59$	−12562.3	−0.469827	−0.234913	+	0.972016i	$0.575481\pi$
$60$	0	0
$61$	41811.0	1.43869	0.719343	−	0.694655i	$-0.244443\pi$
$61$	41811.0	1.43869	0.719343	+	0.694655i	$0.244443\pi$
$62$	0	0
$63$	−19659.8	−0.624062
$64$	0	0
$65$	0	0
$66$	0	0
$67$	60479.5	1.64597	0.822984	−	0.568065i	$-0.192308\pi$
$67$	60479.5	1.64597	0.822984	+	0.568065i	$0.192308\pi$
$68$	0	0
$69$	−9787.71	−0.247490
$70$	0	0
$71$	−27123.5	−0.638557	−0.319278	−	0.947661i	$-0.603441\pi$
$71$	−27123.5	−0.638557	−0.319278	+	0.947661i	$0.603441\pi$
$72$	0	0
$73$	57295.5	1.25838	0.629192	−	0.777250i	$-0.283386\pi$
$73$	57295.5	1.25838	0.629192	+	0.777250i	$0.283386\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−34596.9	−0.664983
$78$	0	0
$79$	−27152.5	−0.489488	−0.244744	−	0.969588i	$-0.578704\pi$
$79$	−27152.5	−0.489488	−0.244744	+	0.969588i	$0.578704\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	74516.5	1.18729	0.593645	−	0.804727i	$-0.297688\pi$
$83$	74516.5	1.18729	0.593645	+	0.804727i	$0.297688\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	39372.0	0.557686
$88$	0	0
$89$	80563.6	1.07811	0.539056	−	0.842270i	$-0.318781\pi$
$89$	80563.6	1.07811	0.539056	+	0.842270i	$0.318781\pi$
$90$	0	0
$91$	262305.	3.32050
$92$	0	0
$93$	23613.2	0.283105
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−88537.3	−0.955426	−0.477713	−	0.878516i	$-0.658534\pi$
$97$	−88537.3	−0.955426	−0.477713	+	0.878516i	$0.658534\pi$
$98$	0	0
$99$	11545.9	0.118397
$100$	0	0
$101$	−97292.1	−0.949018	−0.474509	−	0.880251i	$-0.657374\pi$
$101$	−97292.1	−0.949018	−0.474509	+	0.880251i	$0.657374\pi$
$102$	0	0
$103$	−93587.8	−0.869213	−0.434607	−	0.900620i	$-0.643113\pi$
$103$	−93587.8	−0.869213	−0.434607	+	0.900620i	$0.643113\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−218604.	−1.84586	−0.922929	−	0.384971i	$-0.874212\pi$
$107$	−218604.	−1.84586	−0.922929	+	0.384971i	$0.874212\pi$
$108$	0	0
$109$	−107862.	−0.869562	−0.434781	−	0.900536i	$-0.643174\pi$
$109$	−107862.	−0.869562	−0.434781	+	0.900536i	$0.643174\pi$
$110$	0	0
$111$	18731.2	0.144297
$112$	0	0
$113$	125811.	0.926875	0.463438	−	0.886130i	$-0.346616\pi$
$113$	125811.	0.926875	0.463438	+	0.886130i	$0.346616\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−87538.2	−0.591198
$118$	0	0
$119$	322667.	2.08875
$120$	0	0
$121$	−140733.	−0.873840
$122$	0	0
$123$	−143983.	−0.858121
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−114751.	−0.631318	−0.315659	−	0.948873i	$-0.602226\pi$
$127$	−114751.	−0.631318	−0.315659	+	0.948873i	$0.602226\pi$
$128$	0	0
$129$	−36748.5	−0.194431
$130$	0	0
$131$	125378.	0.638329	0.319164	−	0.947699i	$-0.396598\pi$
$131$	125378.	0.638329	0.319164	+	0.947699i	$0.396598\pi$
$132$	0	0
$133$	−376576.	−1.84597
$134$	0	0
$135$	0	0
$136$	0	0
$137$	144988.	0.659982	0.329991	−	0.943984i	$-0.392954\pi$
$137$	144988.	0.659982	0.329991	+	0.943984i	$0.392954\pi$
$138$	0	0
$139$	414038.	1.81762	0.908810	−	0.417210i	$-0.136992\pi$
$139$	414038.	1.81762	0.908810	+	0.417210i	$0.136992\pi$
$140$	0	0
$141$	43372.8	0.183725
$142$	0	0
$143$	−154048.	−0.629963
$144$	0	0
$145$	0	0
$146$	0	0
$147$	378926.	1.44631
$148$	0	0
$149$	127390.	0.470077	0.235038	−	0.971986i	$-0.424478\pi$
$149$	127390.	0.470077	0.235038	+	0.971986i	$0.424478\pi$
$150$	0	0
$151$	−156233.	−0.557609	−0.278805	−	0.960348i	$-0.589938\pi$
$151$	−156233.	−0.557609	−0.278805	+	0.960348i	$0.589938\pi$
$152$	0	0
$153$	−107683.	−0.371892
$154$	0	0
$155$	0	0
$156$	0	0
$157$	472338.	1.52934	0.764669	−	0.644424i	$-0.222903\pi$
$157$	472338.	1.52934	0.764669	+	0.644424i	$0.222903\pi$
$158$	0	0
$159$	186178.	0.584031
$160$	0	0
$161$	263957.	0.802543
$162$	0	0
$163$	175546.	0.517514	0.258757	−	0.965942i	$-0.416687\pi$
$163$	175546.	0.517514	0.258757	+	0.965942i	$0.416687\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−102842.	−0.285350	−0.142675	−	0.989770i	$-0.545570\pi$
$167$	−102842.	−0.285350	−0.142675	+	0.989770i	$0.545570\pi$
$168$	0	0
$169$	796658.	2.14563
$170$	0	0
$171$	125674.	0.328665
$172$	0	0
$173$	309138.	0.785302	0.392651	−	0.919687i	$-0.371558\pi$
$173$	309138.	0.785302	0.392651	+	0.919687i	$0.371558\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−113060.	−0.271254
$178$	0	0
$179$	774268.	1.80617	0.903086	−	0.429461i	$-0.141296\pi$
$179$	774268.	1.80617	0.903086	+	0.429461i	$0.141296\pi$
$180$	0	0
$181$	−142119.	−0.322446	−0.161223	−	0.986918i	$-0.551544\pi$
$181$	−142119.	−0.322446	−0.161223	+	0.986918i	$0.551544\pi$
$182$	0	0
$183$	376299.	0.830626
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−189498.	−0.396278
$188$	0	0
$189$	−176938.	−0.360303
$190$	0	0
$191$	−557037.	−1.10484	−0.552422	−	0.833565i	$-0.686296\pi$
$191$	−557037.	−1.10484	−0.552422	+	0.833565i	$0.686296\pi$
$192$	0	0
$193$	127912.	0.247183	0.123591	−	0.992333i	$-0.460559\pi$
$193$	127912.	0.247183	0.123591	+	0.992333i	$0.460559\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	916132.	1.68187	0.840935	−	0.541136i	$-0.182006\pi$
$197$	916132.	1.68187	0.840935	+	0.541136i	$0.182006\pi$
$198$	0	0
$199$	−443136.	−0.793240	−0.396620	−	0.917983i	$-0.629817\pi$
$199$	−443136.	−0.793240	−0.396620	+	0.917983i	$0.629817\pi$
$200$	0	0
$201$	544316.	0.950300
$202$	0	0
$203$	−1.06179e6	−1.80842
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−88089.4	−0.142889
$208$	0	0
$209$	221158.	0.350216
$210$	0	0
$211$	496576.	0.767856	0.383928	−	0.923363i	$-0.374571\pi$
$211$	496576.	0.767856	0.383928	+	0.923363i	$0.374571\pi$
$212$	0	0
$213$	−244111.	−0.368671
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−636805.	−0.918031
$218$	0	0
$219$	515659.	0.726529
$220$	0	0
$221$	1.43672e6	1.97876
$222$	0	0
$223$	94509.5	0.127266	0.0636332	−	0.997973i	$-0.479731\pi$
$223$	94509.5	0.127266	0.0636332	+	0.997973i	$0.479731\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.16373e6	−1.49895	−0.749477	−	0.662030i	$-0.769695\pi$
$227$	−1.16373e6	−1.49895	−0.749477	+	0.662030i	$0.769695\pi$
$228$	0	0
$229$	820259.	1.03362	0.516811	−	0.856099i	$-0.327119\pi$
$229$	820259.	1.03362	0.516811	+	0.856099i	$0.327119\pi$
$230$	0	0
$231$	−311372.	−0.383928
$232$	0	0
$233$	1.22012e6	1.47236	0.736178	−	0.676788i	$-0.236629\pi$
$233$	1.22012e6	1.47236	0.736178	+	0.676788i	$0.236629\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−244372.	−0.282606
$238$	0	0
$239$	−1.02798e6	−1.16410	−0.582048	−	0.813154i	$-0.697748\pi$
$239$	−1.02798e6	−1.16410	−0.582048	+	0.813154i	$0.697748\pi$
$240$	0	0
$241$	−755090.	−0.837445	−0.418723	−	0.908114i	$-0.637522\pi$
$241$	−755090.	−0.837445	−0.418723	+	0.908114i	$0.637522\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.67676e6	−1.74875
$248$	0	0
$249$	670648.	0.685482
$250$	0	0
$251$	1.87333e6	1.87686	0.938428	−	0.345476i	$-0.112283\pi$
$251$	1.87333e6	1.87686	0.938428	+	0.345476i	$0.112283\pi$
$252$	0	0
$253$	−155018.	−0.152258
$254$	0	0
$255$	0	0
$256$	0	0
$257$	55931.3	0.0528229	0.0264114	−	0.999651i	$-0.491592\pi$
$257$	55931.3	0.0528229	0.0264114	+	0.999651i	$0.491592\pi$
$258$	0	0
$259$	−505146.	−0.467915
$260$	0	0
$261$	354348.	0.321980
$262$	0	0
$263$	−613008.	−0.546483	−0.273242	−	0.961945i	$-0.588096\pi$
$263$	−613008.	−0.546483	−0.273242	+	0.961945i	$0.588096\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	725072.	0.622448
$268$	0	0
$269$	452657.	0.381407	0.190704	−	0.981648i	$-0.438923\pi$
$269$	452657.	0.381407	0.190704	+	0.981648i	$0.438923\pi$
$270$	0	0
$271$	−550844.	−0.455623	−0.227811	−	0.973705i	$-0.573157\pi$
$271$	−550844.	−0.455623	−0.227811	+	0.973705i	$0.573157\pi$
$272$	0	0
$273$	2.36075e6	1.91709
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−992154.	−0.776926	−0.388463	−	0.921464i	$-0.626994\pi$
$277$	−992154.	−0.776926	−0.388463	+	0.921464i	$0.626994\pi$
$278$	0	0
$279$	212519.	0.163451
$280$	0	0
$281$	−731905.	−0.552954	−0.276477	−	0.961021i	$-0.589167\pi$
$281$	−731905.	−0.552954	−0.276477	+	0.961021i	$0.589167\pi$
$282$	0	0
$283$	459405.	0.340981	0.170490	−	0.985359i	$-0.445465\pi$
$283$	459405.	0.340981	0.170490	+	0.985359i	$0.445465\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.88296e6	2.78265
$288$	0	0
$289$	347487.	0.244734
$290$	0	0
$291$	−796836.	−0.551615
$292$	0	0
$293$	364973.	0.248366	0.124183	−	0.992259i	$-0.460369\pi$
$293$	364973.	0.248366	0.124183	+	0.992259i	$0.460369\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	103913.	0.0683564
$298$	0	0
$299$	1.17531e6	0.760279
$300$	0	0
$301$	991041.	0.630486
$302$	0	0
$303$	−875629.	−0.547916
$304$	0	0
$305$	0	0
$306$	0	0
$307$	307960.	0.186487	0.0932434	−	0.995643i	$-0.470277\pi$
$307$	307960.	0.186487	0.0932434	+	0.995643i	$0.470277\pi$
$308$	0	0
$309$	−842291.	−0.501841
$310$	0	0
$311$	−1.30555e6	−0.765405	−0.382703	−	0.923872i	$-0.625007\pi$
$311$	−1.30555e6	−0.765405	−0.382703	+	0.923872i	$0.625007\pi$
$312$	0	0
$313$	2.66546e6	1.53784	0.768921	−	0.639344i	$-0.220794\pi$
$313$	2.66546e6	1.53784	0.768921	+	0.639344i	$0.220794\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.23744e6	1.25056	0.625278	−	0.780402i	$-0.284986\pi$
$317$	2.23744e6	1.25056	0.625278	+	0.780402i	$0.284986\pi$
$318$	0	0
$319$	623574.	0.343093
$320$	0	0
$321$	−1.96743e6	−1.06571
$322$	0	0
$323$	−2.06262e6	−1.10005
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−970755.	−0.502042
$328$	0	0
$329$	−1.16968e6	−0.595771
$330$	0	0
$331$	2.09233e6	1.04969	0.524844	−	0.851199i	$-0.324124\pi$
$331$	2.09233e6	1.04969	0.524844	+	0.851199i	$0.324124\pi$
$332$	0	0
$333$	168581.	0.0833100
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.01282e6	0.485802	0.242901	−	0.970051i	$-0.421901\pi$
$337$	1.01282e6	0.485802	0.242901	+	0.970051i	$0.421901\pi$
$338$	0	0
$339$	1.13230e6	0.535132
$340$	0	0
$341$	373986.	0.174168
$342$	0	0
$343$	−6.13967e6	−2.81780
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.64429e6	0.733084	0.366542	−	0.930402i	$-0.380542\pi$
$347$	1.64429e6	0.733084	0.366542	+	0.930402i	$0.380542\pi$
$348$	0	0
$349$	898528.	0.394882	0.197441	−	0.980315i	$-0.436737\pi$
$349$	898528.	0.394882	0.197441	+	0.980315i	$0.436737\pi$
$350$	0	0
$351$	−787843.	−0.341328
$352$	0	0
$353$	−853045.	−0.364364	−0.182182	−	0.983265i	$-0.558316\pi$
$353$	−853045.	−0.364364	−0.182182	+	0.983265i	$0.558316\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.90400e6	1.20594
$358$	0	0
$359$	−2.05799e6	−0.842767	−0.421384	−	0.906882i	$-0.638455\pi$
$359$	−2.05799e6	−0.842767	−0.421384	+	0.906882i	$0.638455\pi$
$360$	0	0
$361$	−68869.5	−0.0278137
$362$	0	0
$363$	−1.26659e6	−0.504512
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−4.38094e6	−1.69786	−0.848930	−	0.528505i	$-0.822753\pi$
$367$	−4.38094e6	−1.69786	−0.848930	+	0.528505i	$0.822753\pi$
$368$	0	0
$369$	−1.29585e6	−0.495437
$370$	0	0
$371$	−5.02088e6	−1.89385
$372$	0	0
$373$	456550.	0.169909	0.0849545	−	0.996385i	$-0.472926\pi$
$373$	456550.	0.169909	0.0849545	+	0.996385i	$0.472926\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.72778e6	−1.71318
$378$	0	0
$379$	935559.	0.334559	0.167280	−	0.985909i	$-0.446502\pi$
$379$	935559.	0.334559	0.167280	+	0.985909i	$0.446502\pi$
$380$	0	0
$381$	−1.03276e6	−0.364492
$382$	0	0
$383$	2.78295e6	0.969413	0.484707	−	0.874677i	$-0.338926\pi$
$383$	2.78295e6	0.969413	0.484707	+	0.874677i	$0.338926\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−330737.	−0.112255
$388$	0	0
$389$	−2.18050e6	−0.730604	−0.365302	−	0.930889i	$-0.619034\pi$
$389$	−2.18050e6	−0.730604	−0.365302	+	0.930889i	$0.619034\pi$
$390$	0	0
$391$	1.44577e6	0.478252
$392$	0	0
$393$	1.12841e6	0.368539
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.05032e6	−0.334462	−0.167231	−	0.985918i	$-0.553483\pi$
$397$	−1.05032e6	−0.334462	−0.167231	+	0.985918i	$0.553483\pi$
$398$	0	0
$399$	−3.38919e6	−1.06577
$400$	0	0
$401$	−2.63701e6	−0.818937	−0.409469	−	0.912324i	$-0.634286\pi$
$401$	−2.63701e6	−0.818937	−0.409469	+	0.912324i	$0.634286\pi$
$402$	0	0
$403$	−2.83547e6	−0.869685
$404$	0	0
$405$	0	0
$406$	0	0
$407$	296664.	0.0887727
$408$	0	0
$409$	−2.72798e6	−0.806368	−0.403184	−	0.915119i	$-0.632096\pi$
$409$	−2.72798e6	−0.806368	−0.403184	+	0.915119i	$0.632096\pi$
$410$	0	0
$411$	1.30490e6	0.381041
$412$	0	0
$413$	3.04903e6	0.879603
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3.72634e6	1.04940
$418$	0	0
$419$	−5.94093e6	−1.65318	−0.826589	−	0.562806i	$-0.809722\pi$
$419$	−5.94093e6	−1.65318	−0.826589	+	0.562806i	$0.809722\pi$
$420$	0	0
$421$	474301.	0.130422	0.0652108	−	0.997872i	$-0.479228\pi$
$421$	474301.	0.130422	0.0652108	+	0.997872i	$0.479228\pi$
$422$	0	0
$423$	390355.	0.106074
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.01481e7	−2.69349
$428$	0	0
$429$	−1.38643e6	−0.363710
$430$	0	0
$431$	3.17898e6	0.824319	0.412159	−	0.911112i	$-0.364775\pi$
$431$	3.17898e6	0.824319	0.412159	+	0.911112i	$0.364775\pi$
$432$	0	0
$433$	2.09103e6	0.535970	0.267985	−	0.963423i	$-0.413642\pi$
$433$	2.09103e6	0.535970	0.267985	+	0.963423i	$0.413642\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.68732e6	−0.422662
$438$	0	0
$439$	−758627.	−0.187874	−0.0939371	−	0.995578i	$-0.529945\pi$
$439$	−758627.	−0.187874	−0.0939371	+	0.995578i	$0.529945\pi$
$440$	0	0
$441$	3.41034e6	0.835028
$442$	0	0
$443$	−940168.	−0.227613	−0.113806	−	0.993503i	$-0.536304\pi$
$443$	−940168.	−0.227613	−0.113806	+	0.993503i	$0.536304\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.14651e6	0.271399
$448$	0	0
$449$	3.83158e6	0.896937	0.448469	−	0.893799i	$-0.351970\pi$
$449$	3.83158e6	0.896937	0.448469	+	0.893799i	$0.351970\pi$
$450$	0	0
$451$	−2.28040e6	−0.527923
$452$	0	0
$453$	−1.40610e6	−0.321936
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2.01368e6	−0.451024	−0.225512	−	0.974240i	$-0.572406\pi$
$457$	−2.01368e6	−0.451024	−0.225512	+	0.974240i	$0.572406\pi$
$458$	0	0
$459$	−969144.	−0.214712
$460$	0	0
$461$	−152998.	−0.0335300	−0.0167650	−	0.999859i	$-0.505337\pi$
$461$	−152998.	−0.0335300	−0.0167650	+	0.999859i	$0.505337\pi$
$462$	0	0
$463$	−5.40324e6	−1.17139	−0.585696	−	0.810531i	$-0.699179\pi$
$463$	−5.40324e6	−1.17139	−0.585696	+	0.810531i	$0.699179\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.41706e6	−0.300674	−0.150337	−	0.988635i	$-0.548036\pi$
$467$	−1.41706e6	−0.300674	−0.150337	+	0.988635i	$0.548036\pi$
$468$	0	0
$469$	−1.46792e7	−3.08156
$470$	0	0
$471$	4.25104e6	0.882963
$472$	0	0
$473$	−582023.	−0.119615
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.67560e6	0.337190
$478$	0	0
$479$	−2.46163e6	−0.490213	−0.245106	−	0.969496i	$-0.578823\pi$
$479$	−2.46163e6	−0.490213	−0.245106	+	0.969496i	$0.578823\pi$
$480$	0	0
$481$	−2.24924e6	−0.443274
$482$	0	0
$483$	2.37561e6	0.463348
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.88009e6	0.550280	0.275140	−	0.961404i	$-0.411276\pi$
$487$	2.88009e6	0.550280	0.275140	+	0.961404i	$0.411276\pi$
$488$	0	0
$489$	1.57991e6	0.298787
$490$	0	0
$491$	−2.55154e6	−0.477637	−0.238819	−	0.971064i	$-0.576760\pi$
$491$	−2.55154e6	−0.477637	−0.238819	+	0.971064i	$0.576760\pi$
$492$	0	0
$493$	−5.81575e6	−1.07768
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.58324e6	1.19550
$498$	0	0
$499$	−8.58040e6	−1.54261	−0.771305	−	0.636466i	$-0.780396\pi$
$499$	−8.58040e6	−1.54261	−0.771305	+	0.636466i	$0.780396\pi$
$500$	0	0
$501$	−925574.	−0.164747
$502$	0	0
$503$	7.22134e6	1.27262	0.636309	−	0.771434i	$-0.280460\pi$
$503$	7.22134e6	1.27262	0.636309	+	0.771434i	$0.280460\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.16993e6	1.23878
$508$	0	0
$509$	−73769.6	−0.0126207	−0.00631035	−	0.999980i	$-0.502009\pi$
$509$	−73769.6	−0.0126207	−0.00631035	+	0.999980i	$0.502009\pi$
$510$	0	0
$511$	−1.39064e7	−2.35593
$512$	0	0
$513$	1.13106e6	0.189755
$514$	0	0
$515$	0	0
$516$	0	0
$517$	686938.	0.113029
$518$	0	0
$519$	2.78224e6	0.453395
$520$	0	0
$521$	6.33407e6	1.02232	0.511162	−	0.859484i	$-0.329215\pi$
$521$	6.33407e6	1.02232	0.511162	+	0.859484i	$0.329215\pi$
$522$	0	0
$523$	411025.	0.0657074	0.0328537	−	0.999460i	$-0.489540\pi$
$523$	411025.	0.0657074	0.0328537	+	0.999460i	$0.489540\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.48797e6	−0.547074
$528$	0	0
$529$	−5.25364e6	−0.816246
$530$	0	0
$531$	−1.01754e6	−0.156609
$532$	0	0
$533$	1.72895e7	2.63611
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6.96841e6	1.04279
$538$	0	0
$539$	6.00144e6	0.889782
$540$	0	0
$541$	239548.	0.0351884	0.0175942	−	0.999845i	$-0.494399\pi$
$541$	239548.	0.0351884	0.0175942	+	0.999845i	$0.494399\pi$
$542$	0	0
$543$	−1.27907e6	−0.186164
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.55307e6	0.221933	0.110967	−	0.993824i	$-0.464605\pi$
$547$	1.55307e6	0.221933	0.110967	+	0.993824i	$0.464605\pi$
$548$	0	0
$549$	3.38669e6	0.479562
$550$	0	0
$551$	6.78741e6	0.952412
$552$	0	0
$553$	6.59028e6	0.916413
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.19125e7	1.62691	0.813455	−	0.581627i	$-0.197584\pi$
$557$	1.19125e7	1.62691	0.813455	+	0.581627i	$0.197584\pi$
$558$	0	0
$559$	4.41275e6	0.597283
$560$	0	0
$561$	−1.70548e6	−0.228791
$562$	0	0
$563$	−1.67271e6	−0.222408	−0.111204	−	0.993798i	$-0.535471\pi$
$563$	−1.67271e6	−0.222408	−0.111204	+	0.993798i	$0.535471\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.59244e6	−0.208021
$568$	0	0
$569$	1.26524e7	1.63829	0.819146	−	0.573585i	$-0.194448\pi$
$569$	1.26524e7	1.63829	0.819146	+	0.573585i	$0.194448\pi$
$570$	0	0
$571$	−5.80638e6	−0.745273	−0.372637	−	0.927977i	$-0.621546\pi$
$571$	−5.80638e6	−0.745273	−0.372637	+	0.927977i	$0.621546\pi$
$572$	0	0
$573$	−5.01333e6	−0.637881
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.15669e7	−1.44636	−0.723182	−	0.690658i	$-0.757321\pi$
$577$	−1.15669e7	−1.44636	−0.723182	+	0.690658i	$0.757321\pi$
$578$	0	0
$579$	1.15121e6	0.142711
$580$	0	0
$581$	−1.80862e7	−2.22283
$582$	0	0
$583$	2.94869e6	0.359300
$584$	0	0
$585$	0	0
$586$	0	0
$587$	829454.	0.0993566	0.0496783	−	0.998765i	$-0.484180\pi$
$587$	829454.	0.0993566	0.0496783	+	0.998765i	$0.484180\pi$
$588$	0	0
$589$	4.07072e6	0.483485
$590$	0	0
$591$	8.24519e6	0.971028
$592$	0	0
$593$	−1.42831e7	−1.66796	−0.833978	−	0.551798i	$-0.813942\pi$
$593$	−1.42831e7	−1.66796	−0.833978	+	0.551798i	$0.813942\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.98823e6	−0.457977
$598$	0	0
$599$	−1.51569e7	−1.72601	−0.863007	−	0.505192i	$-0.831421\pi$
$599$	−1.51569e7	−1.72601	−0.863007	+	0.505192i	$0.831421\pi$
$600$	0	0
$601$	−4.20502e6	−0.474877	−0.237439	−	0.971403i	$-0.576308\pi$
$601$	−4.20502e6	−0.474877	−0.237439	+	0.971403i	$0.576308\pi$
$602$	0	0
$603$	4.89884e6	0.548656
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−242069.	−0.0266666	−0.0133333	−	0.999911i	$-0.504244\pi$
$607$	−242069.	−0.0266666	−0.0133333	+	0.999911i	$0.504244\pi$
$608$	0	0
$609$	−9.55613e6	−1.04409
$610$	0	0
$611$	−5.20819e6	−0.564396
$612$	0	0
$613$	−1.13279e7	−1.21759	−0.608793	−	0.793329i	$-0.708346\pi$
$613$	−1.13279e7	−1.21759	−0.608793	+	0.793329i	$0.708346\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.54069e7	1.62931	0.814654	−	0.579947i	$-0.196927\pi$
$617$	1.54069e7	1.62931	0.814654	+	0.579947i	$0.196927\pi$
$618$	0	0
$619$	1.49269e7	1.56583	0.782914	−	0.622129i	$-0.213732\pi$
$619$	1.49269e7	1.56583	0.782914	+	0.622129i	$0.213732\pi$
$620$	0	0
$621$	−792804.	−0.0824968
$622$	0	0
$623$	−1.95539e7	−2.01843
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.99042e6	0.202197
$628$	0	0
$629$	−2.76683e6	−0.278841
$630$	0	0
$631$	1.64446e7	1.64418	0.822091	−	0.569357i	$-0.192808\pi$
$631$	1.64446e7	1.64418	0.822091	+	0.569357i	$0.192808\pi$
$632$	0	0
$633$	4.46919e6	0.443322
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.55014e7	−4.44300
$638$	0	0
$639$	−2.19700e6	−0.212852
$640$	0	0
$641$	3.47616e6	0.334160	0.167080	−	0.985943i	$-0.446566\pi$
$641$	3.47616e6	0.334160	0.167080	+	0.985943i	$0.446566\pi$
$642$	0	0
$643$	−1.62205e7	−1.54716	−0.773581	−	0.633697i	$-0.781537\pi$
$643$	−1.62205e7	−1.54716	−0.773581	+	0.633697i	$0.781537\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7.87206e6	−0.739312	−0.369656	−	0.929169i	$-0.620524\pi$
$647$	−7.87206e6	−0.739312	−0.369656	+	0.929169i	$0.620524\pi$
$648$	0	0
$649$	−1.79065e6	−0.166878
$650$	0	0
$651$	−5.73125e6	−0.530025
$652$	0	0
$653$	1.37332e7	1.26034	0.630171	−	0.776457i	$-0.282985\pi$
$653$	1.37332e7	1.26034	0.630171	+	0.776457i	$0.282985\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.64094e6	0.419461
$658$	0	0
$659$	4.18405e6	0.375305	0.187652	−	0.982236i	$-0.439912\pi$
$659$	4.18405e6	0.375305	0.187652	+	0.982236i	$0.439912\pi$
$660$	0	0
$661$	1.95785e6	0.174291	0.0871455	−	0.996196i	$-0.472226\pi$
$661$	1.95785e6	0.174291	0.0871455	+	0.996196i	$0.472226\pi$
$662$	0	0
$663$	1.29305e7	1.14244
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.75755e6	−0.414066
$668$	0	0
$669$	850586.	0.0734772
$670$	0	0
$671$	5.95983e6	0.511008
$672$	0	0
$673$	8.85319e6	0.753463	0.376732	−	0.926322i	$-0.377048\pi$
$673$	8.85319e6	0.753463	0.376732	+	0.926322i	$0.377048\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.29523e6	−0.611740	−0.305870	−	0.952073i	$-0.598947\pi$
$677$	−7.29523e6	−0.611740	−0.305870	+	0.952073i	$0.598947\pi$
$678$	0	0
$679$	2.14892e7	1.78874
$680$	0	0
$681$	−1.04736e7	−0.865422
$682$	0	0
$683$	8.10456e6	0.664779	0.332390	−	0.943142i	$-0.392145\pi$
$683$	8.10456e6	0.664779	0.332390	+	0.943142i	$0.392145\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7.38233e6	0.596762
$688$	0	0
$689$	−2.23562e7	−1.79411
$690$	0	0
$691$	2.30304e7	1.83487	0.917437	−	0.397882i	$-0.130255\pi$
$691$	2.30304e7	1.83487	0.917437	+	0.397882i	$0.130255\pi$
$692$	0	0
$693$	−2.80235e6	−0.221661
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.12681e7	1.65824
$698$	0	0
$699$	1.09811e7	0.850065
$700$	0	0
$701$	1.60749e7	1.23553	0.617766	−	0.786362i	$-0.288038\pi$
$701$	1.60749e7	1.23553	0.617766	+	0.786362i	$0.288038\pi$
$702$	0	0
$703$	3.22910e6	0.246430
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.36141e7	1.77674
$708$	0	0
$709$	2.10037e7	1.56921	0.784605	−	0.619996i	$-0.212866\pi$
$709$	2.10037e7	1.56921	0.784605	+	0.619996i	$0.212866\pi$
$710$	0	0
$711$	−2.19935e6	−0.163163
$712$	0	0
$713$	−2.85332e6	−0.210197
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−9.25180e6	−0.672091
$718$	0	0
$719$	−1.42531e7	−1.02822	−0.514112	−	0.857723i	$-0.671878\pi$
$719$	−1.42531e7	−1.02822	−0.514112	+	0.857723i	$0.671878\pi$
$720$	0	0
$721$	2.27150e7	1.62733
$722$	0	0
$723$	−6.79581e6	−0.483499
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−4.59136e6	−0.322185	−0.161092	−	0.986939i	$-0.551502\pi$
$727$	−4.59136e6	−0.322185	−0.161092	+	0.986939i	$0.551502\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	5.42822e6	0.375720
$732$	0	0
$733$	4.59016e6	0.315550	0.157775	−	0.987475i	$-0.449568\pi$
$733$	4.59016e6	0.315550	0.157775	+	0.987475i	$0.449568\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.62087e6	0.584632
$738$	0	0
$739$	307856.	0.0207365	0.0103683	−	0.999946i	$-0.496700\pi$
$739$	307856.	0.0207365	0.0103683	+	0.999946i	$0.496700\pi$
$740$	0	0
$741$	−1.50908e7	−1.00964
$742$	0	0
$743$	6.44444e6	0.428266	0.214133	−	0.976805i	$-0.431307\pi$
$743$	6.44444e6	0.428266	0.214133	+	0.976805i	$0.431307\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.03583e6	0.395763
$748$	0	0
$749$	5.30581e7	3.45579
$750$	0	0
$751$	−5.56856e6	−0.360282	−0.180141	−	0.983641i	$-0.557655\pi$
$751$	−5.56856e6	−0.360282	−0.180141	+	0.983641i	$0.557655\pi$
$752$	0	0
$753$	1.68600e7	1.08360
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−7.47540e6	−0.474127	−0.237064	−	0.971494i	$-0.576185\pi$
$757$	−7.47540e6	−0.474127	−0.237064	+	0.971494i	$0.576185\pi$
$758$	0	0
$759$	−1.39516e6	−0.0879062
$760$	0	0
$761$	1.31205e7	0.821278	0.410639	−	0.911798i	$-0.365306\pi$
$761$	1.31205e7	0.821278	0.410639	+	0.911798i	$0.365306\pi$
$762$	0	0
$763$	2.61795e7	1.62798
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.35763e7	0.833281
$768$	0	0
$769$	−1.56508e7	−0.954378	−0.477189	−	0.878801i	$-0.658344\pi$
$769$	−1.56508e7	−0.954378	−0.477189	+	0.878801i	$0.658344\pi$
$770$	0	0
$771$	503382.	0.0304973
$772$	0	0
$773$	−2.07890e7	−1.25137	−0.625683	−	0.780078i	$-0.715179\pi$
$773$	−2.07890e7	−1.25137	−0.625683	+	0.780078i	$0.715179\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−4.54631e6	−0.270151
$778$	0	0
$779$	−2.48215e7	−1.46549
$780$	0	0
$781$	−3.86624e6	−0.226809
$782$	0	0
$783$	3.18913e6	0.185895
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−3.31541e7	−1.90809	−0.954047	−	0.299656i	$-0.903128\pi$
$787$	−3.31541e7	−1.90809	−0.954047	+	0.299656i	$0.903128\pi$
$788$	0	0
$789$	−5.51707e6	−0.315512
$790$	0	0
$791$	−3.05360e7	−1.73528
$792$	0	0
$793$	−4.51859e7	−2.55165
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.00986e6	0.112078	0.0560389	−	0.998429i	$-0.482153\pi$
$797$	2.00986e6	0.112078	0.0560389	+	0.998429i	$0.482153\pi$
$798$	0	0
$799$	−6.40671e6	−0.355033
$800$	0	0
$801$	6.52565e6	0.359370
$802$	0	0
$803$	8.16702e6	0.446966
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.07392e6	0.220205
$808$	0	0
$809$	3.55183e7	1.90801	0.954006	−	0.299787i	$-0.0969156\pi$
$809$	3.55183e7	1.90801	0.954006	+	0.299787i	$0.0969156\pi$
$810$	0	0
$811$	8.99587e6	0.480276	0.240138	−	0.970739i	$-0.422807\pi$
$811$	8.99587e6	0.480276	0.240138	+	0.970739i	$0.422807\pi$
$812$	0	0
$813$	−4.95760e6	−0.263054
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−6.33514e6	−0.332048
$818$	0	0
$819$	2.12467e7	1.10683
$820$	0	0
$821$	3.39427e7	1.75747	0.878736	−	0.477308i	$-0.158387\pi$
$821$	3.39427e7	1.75747	0.878736	+	0.477308i	$0.158387\pi$
$822$	0	0
$823$	−1.15878e7	−0.596348	−0.298174	−	0.954511i	$-0.596378\pi$
$823$	−1.15878e7	−0.596348	−0.298174	+	0.954511i	$0.596378\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	743207.	0.0377873	0.0188937	−	0.999821i	$-0.493986\pi$
$827$	743207.	0.0377873	0.0188937	+	0.999821i	$0.493986\pi$
$828$	0	0
$829$	2.94662e7	1.48915	0.744574	−	0.667540i	$-0.232653\pi$
$829$	2.94662e7	1.48915	0.744574	+	0.667540i	$0.232653\pi$
$830$	0	0
$831$	−8.92938e6	−0.448558
$832$	0	0
$833$	−5.59723e7	−2.79486
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.91267e6	0.0943683
$838$	0	0
$839$	3.18433e7	1.56175	0.780877	−	0.624685i	$-0.214773\pi$
$839$	3.18433e7	1.56175	0.780877	+	0.624685i	$0.214773\pi$
$840$	0	0
$841$	−1.37342e6	−0.0669596
$842$	0	0
$843$	−6.58714e6	−0.319248
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.41578e7	1.63599
$848$	0	0
$849$	4.13465e6	0.196865
$850$	0	0
$851$	−2.26340e6	−0.107136
$852$	0	0
$853$	9.10862e6	0.428627	0.214314	−	0.976765i	$-0.431249\pi$
$853$	9.10862e6	0.428627	0.214314	+	0.976765i	$0.431249\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.41175e6	−0.205191	−0.102596	−	0.994723i	$-0.532715\pi$
$857$	−4.41175e6	−0.205191	−0.102596	+	0.994723i	$0.532715\pi$
$858$	0	0
$859$	−9.64496e6	−0.445982	−0.222991	−	0.974821i	$-0.571582\pi$
$859$	−9.64496e6	−0.445982	−0.222991	+	0.974821i	$0.571582\pi$
$860$	0	0
$861$	3.49467e7	1.60656
$862$	0	0
$863$	1.95068e7	0.891579	0.445790	−	0.895138i	$-0.352923\pi$
$863$	1.95068e7	0.891579	0.445790	+	0.895138i	$0.352923\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.12739e6	0.141297
$868$	0	0
$869$	−3.87037e6	−0.173861
$870$	0	0
$871$	−6.53613e7	−2.91928
$872$	0	0
$873$	−7.17152e6	−0.318475
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−5.18933e6	−0.227831	−0.113915	−	0.993490i	$-0.536339\pi$
$877$	−5.18933e6	−0.227831	−0.113915	+	0.993490i	$0.536339\pi$
$878$	0	0
$879$	3.28476e6	0.143394
$880$	0	0
$881$	−1.43979e7	−0.624972	−0.312486	−	0.949922i	$-0.601162\pi$
$881$	−1.43979e7	−0.624972	−0.312486	+	0.949922i	$0.601162\pi$
$882$	0	0
$883$	2.09749e7	0.905311	0.452656	−	0.891685i	$-0.350477\pi$
$883$	2.09749e7	0.905311	0.452656	+	0.891685i	$0.350477\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.54403e7	−1.51248	−0.756238	−	0.654296i	$-0.772965\pi$
$887$	−3.54403e7	−1.51248	−0.756238	+	0.654296i	$0.772965\pi$
$888$	0	0
$889$	2.78517e7	1.18195
$890$	0	0
$891$	935218.	0.0394656
$892$	0	0
$893$	7.47710e6	0.313765
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.05778e7	0.438947
$898$	0	0
$899$	1.14778e7	0.473651
$900$	0	0
$901$	−2.75009e7	−1.12859
$902$	0	0
$903$	8.91937e6	0.364011
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−2.52614e7	−1.01962	−0.509812	−	0.860286i	$-0.670285\pi$
$907$	−2.52614e7	−1.01962	−0.509812	+	0.860286i	$0.670285\pi$
$908$	0	0
$909$	−7.88066e6	−0.316339
$910$	0	0
$911$	1.45292e7	0.580022	0.290011	−	0.957023i	$-0.406341\pi$
$911$	1.45292e7	0.580022	0.290011	+	0.957023i	$0.406341\pi$
$912$	0	0
$913$	1.06217e7	0.421714
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.04310e7	−1.19507
$918$	0	0
$919$	2.82570e7	1.10366	0.551832	−	0.833955i	$-0.313929\pi$
$919$	2.82570e7	1.10366	0.551832	+	0.833955i	$0.313929\pi$
$920$	0	0
$921$	2.77164e6	0.107668
$922$	0	0
$923$	2.93128e7	1.13254
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−7.58061e6	−0.289738
$928$	0	0
$929$	4.09229e6	0.155570	0.0777852	−	0.996970i	$-0.475215\pi$
$929$	4.09229e6	0.155570	0.0777852	+	0.996970i	$0.475215\pi$
$930$	0	0
$931$	6.53238e7	2.47000
$932$	0	0
$933$	−1.17499e7	−0.441907
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.13115e7	1.90926	0.954632	−	0.297790i	$-0.0962494\pi$
$937$	5.13115e7	1.90926	0.954632	+	0.297790i	$0.0962494\pi$
$938$	0	0
$939$	2.39892e7	0.887873
$940$	0	0
$941$	−1.60558e7	−0.591096	−0.295548	−	0.955328i	$-0.595502\pi$
$941$	−1.60558e7	−0.591096	−0.295548	+	0.955328i	$0.595502\pi$
$942$	0	0
$943$	1.73983e7	0.637130
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.82296e7	1.38524	0.692620	−	0.721302i	$-0.256456\pi$
$947$	3.82296e7	1.38524	0.692620	+	0.721302i	$0.256456\pi$
$948$	0	0
$949$	−6.19203e7	−2.23186
$950$	0	0
$951$	2.01369e7	0.722009
$952$	0	0
$953$	4.76886e7	1.70092	0.850458	−	0.526044i	$-0.176325\pi$
$953$	4.76886e7	1.70092	0.850458	+	0.526044i	$0.176325\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5.61217e6	0.198085
$958$	0	0
$959$	−3.51907e7	−1.23561
$960$	0	0
$961$	−2.17454e7	−0.759555
$962$	0	0
$963$	−1.77069e7	−0.615286
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−2.70295e6	−0.0929548	−0.0464774	−	0.998919i	$-0.514800\pi$
$967$	−2.70295e6	−0.0929548	−0.0464774	+	0.998919i	$0.514800\pi$
$968$	0	0
$969$	−1.85636e7	−0.635115
$970$	0	0
$971$	−2.87085e7	−0.977153	−0.488577	−	0.872521i	$-0.662484\pi$
$971$	−2.87085e7	−0.977153	−0.488577	+	0.872521i	$0.662484\pi$
$972$	0	0
$973$	−1.00493e8	−3.40292
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.40555e6	0.147660	0.0738302	−	0.997271i	$-0.476478\pi$
$977$	4.40555e6	0.147660	0.0738302	+	0.997271i	$0.476478\pi$
$978$	0	0
$979$	1.14837e7	0.382935
$980$	0	0
$981$	−8.73679e6	−0.289854
$982$	0	0
$983$	2.90145e7	0.957705	0.478853	−	0.877895i	$-0.341053\pi$
$983$	2.90145e7	0.957705	0.478853	+	0.877895i	$0.341053\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.05272e7	−0.343968
$988$	0	0
$989$	4.44054e6	0.144359
$990$	0	0
$991$	1.86483e7	0.603190	0.301595	−	0.953436i	$-0.402481\pi$
$991$	1.86483e7	0.603190	0.301595	+	0.953436i	$0.402481\pi$
$992$	0	0
$993$	1.88310e7	0.606037
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−6.75822e6	−0.215325	−0.107663	−	0.994187i	$-0.534337\pi$
$997$	−6.75822e6	−0.215325	−0.107663	+	0.994187i	$0.534337\pi$
$998$	0	0
$999$	1.51723e6	0.0480990

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.6.a.w.1.1		4
5.2	odd	4		120.6.f.b.49.1	✓	8
5.3	odd	4		120.6.f.b.49.5	yes	8
5.4	even	2		600.6.a.v.1.4		4
15.2	even	4		360.6.f.c.289.8		8
15.8	even	4		360.6.f.c.289.7		8
20.3	even	4		240.6.f.f.49.1		8
20.7	even	4		240.6.f.f.49.5		8
60.23	odd	4		720.6.f.o.289.7		8
60.47	odd	4		720.6.f.o.289.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.6.f.b.49.1	✓	8	5.2	odd	4
120.6.f.b.49.5	yes	8	5.3	odd	4
240.6.f.f.49.1		8	20.3	even	4
240.6.f.f.49.5		8	20.7	even	4
360.6.f.c.289.7		8	15.8	even	4
360.6.f.c.289.8		8	15.2	even	4
600.6.a.v.1.4		4	5.4	even	2
600.6.a.w.1.1		4	1.1	even	1	trivial
720.6.f.o.289.7		8	60.23	odd	4
720.6.f.o.289.8		8	60.47	odd	4

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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	600.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} -242.714 q^{7} +81.0000 q^{9} +142.542 q^{11} -1080.72 q^{13} -1329.42 q^{17} +1551.52 q^{19} -2184.42 q^{21} -1087.52 q^{23} +729.000 q^{27} +4374.67 q^{29} +2623.69 q^{31} +1282.88 q^{33} +2081.24 q^{37} -9726.46 q^{39} -15998.1 q^{41} -4083.17 q^{43} +4819.20 q^{47} +42102.9 q^{49} -11964.7 q^{51} +20686.4 q^{53} +13963.7 q^{57} -12562.3 q^{59} +41811.0 q^{61} -19659.8 q^{63} +60479.5 q^{67} -9787.71 q^{69} -27123.5 q^{71} +57295.5 q^{73} -34596.9 q^{77} -27152.5 q^{79} +6561.00 q^{81} +74516.5 q^{83} +39372.0 q^{87} +80563.6 q^{89} +262305. q^{91} +23613.2 q^{93} -88537.3 q^{97} +11545.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 36 q^{3} + 6 q^{7} + 324 q^{9} + 342 q^{11} - 222 q^{13} + 618 q^{17} + 1520 q^{19} + 54 q^{21} + 752 q^{23} + 2916 q^{27} + 4002 q^{29} + 3012 q^{31} + 3078 q^{33} + 14142 q^{37} - 1998 q^{39} - 18900 q^{41}+ \cdots + 27702 q^{99}+O(q^{100}) \) 4 * q + 36 * q^3 + 6 * q^7 + 324 * q^9 + 342 * q^11 - 222 * q^13 + 618 * q^17 + 1520 * q^19 + 54 * q^21 + 752 * q^23 + 2916 * q^27 + 4002 * q^29 + 3012 * q^31 + 3078 * q^33 + 14142 * q^37 - 1998 * q^39 - 18900 * q^41 + 11424 * q^43 + 7572 * q^47 + 17184 * q^49 + 5562 * q^51 + 47374 * q^53 + 13680 * q^57 + 43014 * q^59 - 19480 * q^61 + 486 * q^63 + 25620 * q^67 + 6768 * q^69 + 81036 * q^71 + 111324 * q^73 - 45748 * q^77 + 84988 * q^79 + 26244 * q^81 + 83200 * q^83 + 36018 * q^87 + 226488 * q^89 + 305228 * q^91 + 27108 * q^93 - 32016 * q^97 + 27702 * q^99

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