LMFDB - Embedded newform 900.6.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,6,Mod(1,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, 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("900.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	900.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:	$144.345437832$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{241}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 60 $ x^2 - x - 60
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 180)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$8.26209$ of defining polynomial
Character	$\chi$	$=$	900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-53.1450 q^{7} +O(q^{10})$	$q-53.1450 q^{7} -585.725 q^{11} +1085.16 q^{13} -1592.90 q^{17} +2410.90 q^{19} +2671.45 q^{23} -8435.80 q^{29} -5024.15 q^{31} -1828.70 q^{37} -7971.30 q^{41} -10556.8 q^{43} +29386.0 q^{47} -13982.6 q^{49} +23064.2 q^{53} +2214.89 q^{59} +21931.4 q^{61} +410.167 q^{67} -53657.9 q^{71} +17564.8 q^{73} +31128.4 q^{77} +59507.8 q^{79} -22914.7 q^{83} -56700.9 q^{89} -57670.9 q^{91} +151477. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 80 q^{7}+O(q^{10}) $ 2 * q + 80 * q^7	$ 2 q + 80 q^{7} - 240 q^{11} + 680 q^{13} + 540 q^{17} + 1096 q^{19} + 3480 q^{23} - 9420 q^{29} + 2992 q^{31} + 7520 q^{37} - 27120 q^{41} - 4720 q^{43} + 38280 q^{47} - 13062 q^{49} + 53580 q^{53} - 46800 q^{59} + 21508 q^{61} + 10880 q^{67} - 45840 q^{71} - 9580 q^{73} + 77160 q^{77} + 91072 q^{79} - 14160 q^{83} - 35160 q^{89} - 111616 q^{91} + 98780 q^{97}+O(q^{100}) $ 2 * q + 80 * q^7 - 240 * q^11 + 680 * q^13 + 540 * q^17 + 1096 * q^19 + 3480 * q^23 - 9420 * q^29 + 2992 * q^31 + 7520 * q^37 - 27120 * q^41 - 4720 * q^43 + 38280 * q^47 - 13062 * q^49 + 53580 * q^53 - 46800 * q^59 + 21508 * q^61 + 10880 * q^67 - 45840 * q^71 - 9580 * q^73 + 77160 * q^77 + 91072 * q^79 - 14160 * q^83 - 35160 * q^89 - 111616 * q^91 + 98780 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−53.1450	−0.409938	−0.204969	−	0.978769i	$-0.565709\pi$
$7$	−53.1450	−0.409938	−0.204969	+	0.978769i	$0.565709\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−585.725	−1.45953	−0.729764	−	0.683699i	$-0.760370\pi$
$11$	−585.725	−1.45953	−0.729764	+	0.683699i	$0.760370\pi$
$12$	0	0
$13$	1085.16	1.78088	0.890442	−	0.455097i	$-0.150395\pi$
$13$	1085.16	1.78088	0.890442	+	0.455097i	$0.150395\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1592.90	−1.33680	−0.668400	−	0.743802i	$-0.733021\pi$
$17$	−1592.90	−1.33680	−0.668400	+	0.743802i	$0.733021\pi$
$18$	0	0
$19$	2410.90	1.53213	0.766065	−	0.642764i	$-0.222212\pi$
$19$	2410.90	1.53213	0.766065	+	0.642764i	$0.222212\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2671.45	1.05300	0.526499	−	0.850176i	$-0.323504\pi$
$23$	2671.45	1.05300	0.526499	+	0.850176i	$0.323504\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8435.80	−1.86265	−0.931325	−	0.364188i	$-0.881346\pi$
$29$	−8435.80	−1.86265	−0.931325	+	0.364188i	$0.881346\pi$
$30$	0	0
$31$	−5024.15	−0.938985	−0.469492	−	0.882936i	$-0.655563\pi$
$31$	−5024.15	−0.938985	−0.469492	+	0.882936i	$0.655563\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1828.70	−0.219603	−0.109802	−	0.993954i	$-0.535022\pi$
$37$	−1828.70	−0.219603	−0.109802	+	0.993954i	$0.535022\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7971.30	−0.740576	−0.370288	−	0.928917i	$-0.620741\pi$
$41$	−7971.30	−0.740576	−0.370288	+	0.928917i	$0.620741\pi$
$42$	0	0
$43$	−10556.8	−0.870682	−0.435341	−	0.900266i	$-0.643372\pi$
$43$	−10556.8	−0.870682	−0.435341	+	0.900266i	$0.643372\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	29386.0	1.94042	0.970209	−	0.242271i	$-0.0778922\pi$
$47$	29386.0	1.94042	0.970209	+	0.242271i	$0.0778922\pi$
$48$	0	0
$49$	−13982.6	−0.831951
$50$	0	0
$51$	0	0
$52$	0	0
$53$	23064.2	1.12784	0.563922	−	0.825828i	$-0.309292\pi$
$53$	23064.2	1.12784	0.563922	+	0.825828i	$0.309292\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2214.89	0.0828365	0.0414183	−	0.999142i	$-0.486812\pi$
$59$	2214.89	0.0828365	0.0414183	+	0.999142i	$0.486812\pi$
$60$	0	0
$61$	21931.4	0.754644	0.377322	−	0.926082i	$-0.376845\pi$
$61$	21931.4	0.754644	0.377322	+	0.926082i	$0.376845\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	410.167	0.0111628	0.00558141	−	0.999984i	$-0.498223\pi$
$67$	410.167	0.0111628	0.00558141	+	0.999984i	$0.498223\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−53657.9	−1.26324	−0.631622	−	0.775276i	$-0.717611\pi$
$71$	−53657.9	−1.26324	−0.631622	+	0.775276i	$0.717611\pi$
$72$	0	0
$73$	17564.8	0.385777	0.192889	−	0.981221i	$-0.438214\pi$
$73$	17564.8	0.385777	0.192889	+	0.981221i	$0.438214\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	31128.4	0.598315
$78$	0	0
$79$	59507.8	1.07277	0.536384	−	0.843974i	$-0.319790\pi$
$79$	59507.8	1.07277	0.536384	+	0.843974i	$0.319790\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−22914.7	−0.365105	−0.182553	−	0.983196i	$-0.558436\pi$
$83$	−22914.7	−0.365105	−0.182553	+	0.983196i	$0.558436\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−56700.9	−0.758779	−0.379389	−	0.925237i	$-0.623866\pi$
$89$	−56700.9	−0.758779	−0.379389	+	0.925237i	$0.623866\pi$
$90$	0	0
$91$	−57670.9	−0.730051
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	151477.	1.63462	0.817311	−	0.576197i	$-0.195464\pi$
$97$	151477.	1.63462	0.817311	+	0.576197i	$0.195464\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	113395.	1.10609	0.553045	−	0.833151i	$-0.313466\pi$
$101$	113395.	1.10609	0.553045	+	0.833151i	$0.313466\pi$
$102$	0	0
$103$	−16700.2	−0.155106	−0.0775531	−	0.996988i	$-0.524711\pi$
$103$	−16700.2	−0.155106	−0.0775531	+	0.996988i	$0.524711\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−174972.	−1.47744	−0.738719	−	0.674014i	$-0.764569\pi$
$107$	−174972.	−1.47744	−0.738719	+	0.674014i	$0.764569\pi$
$108$	0	0
$109$	93282.6	0.752029	0.376014	−	0.926614i	$-0.377294\pi$
$109$	93282.6	0.752029	0.376014	+	0.926614i	$0.377294\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	103539.	0.762792	0.381396	−	0.924412i	$-0.375443\pi$
$113$	103539.	0.762792	0.381396	+	0.924412i	$0.375443\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	84654.8	0.548005
$120$	0	0
$121$	182023.	1.13022
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10725.7	0.0590088	0.0295044	−	0.999565i	$-0.490607\pi$
$127$	10725.7	0.0590088	0.0295044	+	0.999565i	$0.490607\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	115954.	0.590349	0.295174	−	0.955443i	$-0.404622\pi$
$131$	115954.	0.590349	0.295174	+	0.955443i	$0.404622\pi$
$132$	0	0
$133$	−128127.	−0.628077
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−120066.	−0.546535	−0.273267	−	0.961938i	$-0.588104\pi$
$137$	−120066.	−0.546535	−0.273267	+	0.961938i	$0.588104\pi$
$138$	0	0
$139$	−136471.	−0.599106	−0.299553	−	0.954080i	$-0.596838\pi$
$139$	−136471.	−0.599106	−0.299553	+	0.954080i	$0.596838\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−635606.	−2.59925
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	307357.	1.13417	0.567085	−	0.823659i	$-0.308071\pi$
$149$	307357.	1.13417	0.567085	+	0.823659i	$0.308071\pi$
$150$	0	0
$151$	41402.9	0.147771	0.0738853	−	0.997267i	$-0.476460\pi$
$151$	41402.9	0.147771	0.0738853	+	0.997267i	$0.476460\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	363308.	1.17632	0.588161	−	0.808744i	$-0.299852\pi$
$157$	363308.	1.17632	0.588161	+	0.808744i	$0.299852\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−141974.	−0.431663
$162$	0	0
$163$	−361620.	−1.06606	−0.533032	−	0.846095i	$-0.678947\pi$
$163$	−361620.	−1.06606	−0.533032	+	0.846095i	$0.678947\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	127049.	0.352518	0.176259	−	0.984344i	$-0.443600\pi$
$167$	127049.	0.352518	0.176259	+	0.984344i	$0.443600\pi$
$168$	0	0
$169$	806280.	2.17155
$170$	0	0
$171$	0	0
$172$	0	0
$173$	388638.	0.987257	0.493629	−	0.869673i	$-0.335670\pi$
$173$	388638.	0.987257	0.493629	+	0.869673i	$0.335670\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	365985.	0.853751	0.426875	−	0.904310i	$-0.359614\pi$
$179$	365985.	0.853751	0.426875	+	0.904310i	$0.359614\pi$
$180$	0	0
$181$	−747391.	−1.69571	−0.847855	−	0.530228i	$-0.822106\pi$
$181$	−747391.	−1.69571	−0.847855	+	0.530228i	$0.822106\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	933002.	1.95110
$188$	0	0
$189$	0	0
$190$	0	0
$191$	726043.	1.44006	0.720028	−	0.693946i	$-0.244129\pi$
$191$	726043.	1.44006	0.720028	+	0.693946i	$0.244129\pi$
$192$	0	0
$193$	215930.	0.417273	0.208636	−	0.977993i	$-0.433097\pi$
$193$	215930.	0.417273	0.208636	+	0.977993i	$0.433097\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	707885.	1.29956	0.649781	−	0.760122i	$-0.274861\pi$
$197$	707885.	1.29956	0.649781	+	0.760122i	$0.274861\pi$
$198$	0	0
$199$	1.04820e6	1.87635	0.938174	−	0.346164i	$-0.112516\pi$
$199$	1.04820e6	1.87635	0.938174	+	0.346164i	$0.112516\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	448321.	0.763570
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.41213e6	−2.23618
$210$	0	0
$211$	975085.	1.50777	0.753887	−	0.657004i	$-0.228177\pi$
$211$	975085.	1.50777	0.753887	+	0.657004i	$0.228177\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	267009.	0.384925
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.72855e6	−2.38069
$222$	0	0
$223$	−129078.	−0.173816	−0.0869079	−	0.996216i	$-0.527699\pi$
$223$	−129078.	−0.173816	−0.0869079	+	0.996216i	$0.527699\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.10942e6	−1.42900	−0.714502	−	0.699634i	$-0.753347\pi$
$227$	−1.10942e6	−1.42900	−0.714502	+	0.699634i	$0.753347\pi$
$228$	0	0
$229$	555967.	0.700585	0.350292	−	0.936640i	$-0.386082\pi$
$229$	555967.	0.700585	0.350292	+	0.936640i	$0.386082\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−107780.	−0.130062	−0.0650310	−	0.997883i	$-0.520715\pi$
$233$	−107780.	−0.130062	−0.0650310	+	0.997883i	$0.520715\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−691144.	−0.782661	−0.391331	−	0.920250i	$-0.627985\pi$
$239$	−691144.	−0.782661	−0.391331	+	0.920250i	$0.627985\pi$
$240$	0	0
$241$	601871.	0.667515	0.333758	−	0.942659i	$-0.391683\pi$
$241$	601871.	0.667515	0.333758	+	0.942659i	$0.391683\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.61621e6	2.72854
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−114860.	−0.115076	−0.0575380	−	0.998343i	$-0.518325\pi$
$251$	−114860.	−0.115076	−0.0575380	+	0.998343i	$0.518325\pi$
$252$	0	0
$253$	−1.56474e6	−1.53688
$254$	0	0
$255$	0	0
$256$	0	0
$257$	924496.	0.873117	0.436558	−	0.899676i	$-0.356197\pi$
$257$	924496.	0.873117	0.436558	+	0.899676i	$0.356197\pi$
$258$	0	0
$259$	97186.5	0.0900236
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.73745e6	1.54890	0.774448	−	0.632638i	$-0.218028\pi$
$263$	1.73745e6	1.54890	0.774448	+	0.632638i	$0.218028\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	108682.	0.0915751	0.0457875	−	0.998951i	$-0.485420\pi$
$269$	108682.	0.0915751	0.0457875	+	0.998951i	$0.485420\pi$
$270$	0	0
$271$	−193643.	−0.160169	−0.0800847	−	0.996788i	$-0.525519\pi$
$271$	−193643.	−0.160169	−0.0800847	+	0.996788i	$0.525519\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.70076e6	1.33182	0.665909	−	0.746033i	$-0.268044\pi$
$277$	1.70076e6	1.33182	0.665909	+	0.746033i	$0.268044\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−131461.	−0.0993184	−0.0496592	−	0.998766i	$-0.515814\pi$
$281$	−131461.	−0.0993184	−0.0496592	+	0.998766i	$0.515814\pi$
$282$	0	0
$283$	−2.11575e6	−1.57036	−0.785178	−	0.619271i	$-0.787428\pi$
$283$	−2.11575e6	−1.57036	−0.785178	+	0.619271i	$0.787428\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	423635.	0.303590
$288$	0	0
$289$	1.11748e6	0.787035
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−874729.	−0.595257	−0.297629	−	0.954682i	$-0.596196\pi$
$293$	−874729.	−0.595257	−0.297629	+	0.954682i	$0.596196\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.89895e6	1.87527
$300$	0	0
$301$	561040.	0.356925
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.26331e6	0.765002	0.382501	−	0.923955i	$-0.375063\pi$
$307$	1.26331e6	0.765002	0.382501	+	0.923955i	$0.375063\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.33506e6	0.782709	0.391355	−	0.920240i	$-0.372007\pi$
$311$	1.33506e6	0.782709	0.391355	+	0.920240i	$0.372007\pi$
$312$	0	0
$313$	1.61091e6	0.929414	0.464707	−	0.885465i	$-0.346160\pi$
$313$	1.61091e6	0.929414	0.464707	+	0.885465i	$0.346160\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.94530e6	1.08727	0.543636	−	0.839321i	$-0.317047\pi$
$317$	1.94530e6	1.08727	0.543636	+	0.839321i	$0.317047\pi$
$318$	0	0
$319$	4.94106e6	2.71859
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.84033e6	−2.04815
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.56172e6	−0.795450
$330$	0	0
$331$	−1.89097e6	−0.948670	−0.474335	−	0.880344i	$-0.657311\pi$
$331$	−1.89097e6	−0.948670	−0.474335	+	0.880344i	$0.657311\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.25968e6	1.56351	0.781754	−	0.623587i	$-0.214325\pi$
$337$	3.25968e6	1.56351	0.781754	+	0.623587i	$0.214325\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.94277e6	1.37047
$342$	0	0
$343$	1.63631e6	0.750986
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.70076e6	−0.758264	−0.379132	−	0.925343i	$-0.623777\pi$
$347$	−1.70076e6	−0.758264	−0.379132	+	0.925343i	$0.623777\pi$
$348$	0	0
$349$	3.20230e6	1.40734	0.703669	−	0.710528i	$-0.251544\pi$
$349$	3.20230e6	1.40734	0.703669	+	0.710528i	$0.251544\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.41449e6	0.604175	0.302088	−	0.953280i	$-0.402316\pi$
$353$	1.41449e6	0.604175	0.302088	+	0.953280i	$0.402316\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.37669e6	0.563768	0.281884	−	0.959448i	$-0.409041\pi$
$359$	1.37669e6	0.563768	0.281884	+	0.959448i	$0.409041\pi$
$360$	0	0
$361$	3.33634e6	1.34742
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.04063e6	−0.403303	−0.201652	−	0.979457i	$-0.564631\pi$
$367$	−1.04063e6	−0.403303	−0.201652	+	0.979457i	$0.564631\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.22575e6	−0.462345
$372$	0	0
$373$	−2.17929e6	−0.811040	−0.405520	−	0.914086i	$-0.632909\pi$
$373$	−2.17929e6	−0.811040	−0.405520	+	0.914086i	$0.632909\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.15420e6	−3.31716
$378$	0	0
$379$	−1.98879e6	−0.711197	−0.355599	−	0.934639i	$-0.615723\pi$
$379$	−1.98879e6	−0.711197	−0.355599	+	0.934639i	$0.615723\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	312912.	0.109000	0.0544999	−	0.998514i	$-0.482644\pi$
$383$	312912.	0.109000	0.0544999	+	0.998514i	$0.482644\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.07497e6	1.03031	0.515154	−	0.857098i	$-0.327735\pi$
$389$	3.07497e6	1.03031	0.515154	+	0.857098i	$0.327735\pi$
$390$	0	0
$391$	−4.25536e6	−1.40765
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.03308e6	1.28428	0.642141	−	0.766586i	$-0.278046\pi$
$397$	4.03308e6	1.28428	0.642141	+	0.766586i	$0.278046\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.60232e6	−0.497608	−0.248804	−	0.968554i	$-0.580038\pi$
$401$	−1.60232e6	−0.497608	−0.248804	+	0.968554i	$0.580038\pi$
$402$	0	0
$403$	−5.45201e6	−1.67222
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.07112e6	0.320517
$408$	0	0
$409$	−2.89467e6	−0.855640	−0.427820	−	0.903864i	$-0.640718\pi$
$409$	−2.89467e6	−0.855640	−0.427820	+	0.903864i	$0.640718\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−117710.	−0.0339578
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−6.70270e6	−1.86515	−0.932577	−	0.360970i	$-0.882446\pi$
$419$	−6.70270e6	−1.86515	−0.932577	+	0.360970i	$0.882446\pi$
$420$	0	0
$421$	1.76525e6	0.485401	0.242701	−	0.970101i	$-0.421967\pi$
$421$	1.76525e6	0.485401	0.242701	+	0.970101i	$0.421967\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.16555e6	−0.309357
$428$	0	0
$429$	0	0
$430$	0	0
$431$	374860.	0.0972022	0.0486011	−	0.998818i	$-0.484524\pi$
$431$	374860.	0.0972022	0.0486011	+	0.998818i	$0.484524\pi$
$432$	0	0
$433$	−2.42835e6	−0.622430	−0.311215	−	0.950339i	$-0.600736\pi$
$433$	−2.42835e6	−0.622430	−0.311215	+	0.950339i	$0.600736\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.44060e6	1.61333
$438$	0	0
$439$	2.03256e6	0.503363	0.251682	−	0.967810i	$-0.419016\pi$
$439$	2.03256e6	0.503363	0.251682	+	0.967810i	$0.419016\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.85613e6	−0.691462	−0.345731	−	0.938334i	$-0.612369\pi$
$443$	−2.85613e6	−0.691462	−0.345731	+	0.938334i	$0.612369\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.95580e6	1.16011	0.580053	−	0.814579i	$-0.303032\pi$
$449$	4.95580e6	1.16011	0.580053	+	0.814579i	$0.303032\pi$
$450$	0	0
$451$	4.66899e6	1.08089
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.09688e6	−0.245679	−0.122840	−	0.992427i	$-0.539200\pi$
$457$	−1.09688e6	−0.245679	−0.122840	+	0.992427i	$0.539200\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.76958e6	0.606962	0.303481	−	0.952837i	$-0.401851\pi$
$461$	2.76958e6	0.606962	0.303481	+	0.952837i	$0.401851\pi$
$462$	0	0
$463$	4.28413e6	0.928773	0.464387	−	0.885633i	$-0.346275\pi$
$463$	4.28413e6	0.928773	0.464387	+	0.885633i	$0.346275\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.94130e6	1.04845	0.524226	−	0.851579i	$-0.324355\pi$
$467$	4.94130e6	1.04845	0.524226	+	0.851579i	$0.324355\pi$
$468$	0	0
$469$	−21798.4	−0.00457606
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.18336e6	1.27078
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−7.29605e6	−1.45294	−0.726472	−	0.687196i	$-0.758841\pi$
$479$	−7.29605e6	−1.45294	−0.726472	+	0.687196i	$0.758841\pi$
$480$	0	0
$481$	−1.98444e6	−0.391088
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6.28904e6	1.20161	0.600803	−	0.799397i	$-0.294848\pi$
$487$	6.28904e6	1.20161	0.600803	+	0.799397i	$0.294848\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5.01098e6	−0.938035	−0.469018	−	0.883189i	$-0.655392\pi$
$491$	−5.01098e6	−0.938035	−0.469018	+	0.883189i	$0.655392\pi$
$492$	0	0
$493$	1.34374e7	2.48999
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.85165e6	0.517851
$498$	0	0
$499$	796409.	0.143181	0.0715904	−	0.997434i	$-0.477193\pi$
$499$	796409.	0.143181	0.0715904	+	0.997434i	$0.477193\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5.08033e6	0.895306	0.447653	−	0.894207i	$-0.352260\pi$
$503$	5.08033e6	0.895306	0.447653	+	0.894207i	$0.352260\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.77725e6	0.304056	0.152028	−	0.988376i	$-0.451420\pi$
$509$	1.77725e6	0.304056	0.152028	+	0.988376i	$0.451420\pi$
$510$	0	0
$511$	−933483.	−0.158144
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.72121e7	−2.83209
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.17893e6	0.351681	0.175840	−	0.984419i	$-0.443736\pi$
$521$	2.17893e6	0.351681	0.175840	+	0.984419i	$0.443736\pi$
$522$	0	0
$523$	−1.87227e6	−0.299305	−0.149652	−	0.988739i	$-0.547815\pi$
$523$	−1.87227e6	−0.299305	−0.149652	+	0.988739i	$0.547815\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.00298e6	1.25524
$528$	0	0
$529$	700305.	0.108805
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−8.65014e6	−1.31888
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8.18996e6	1.21426
$540$	0	0
$541$	−1.45286e6	−0.213417	−0.106709	−	0.994290i	$-0.534031\pi$
$541$	−1.45286e6	−0.213417	−0.106709	+	0.994290i	$0.534031\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.82156e6	0.689001	0.344500	−	0.938786i	$-0.388048\pi$
$547$	4.82156e6	0.689001	0.344500	+	0.938786i	$0.388048\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.03379e7	−2.85382
$552$	0	0
$553$	−3.16254e6	−0.439768
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.26764e7	−1.73125	−0.865624	−	0.500695i	$-0.833078\pi$
$557$	−1.26764e7	−1.73125	−0.865624	+	0.500695i	$0.833078\pi$
$558$	0	0
$559$	−1.14558e7	−1.55058
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.29998e7	1.72848	0.864240	−	0.503080i	$-0.167800\pi$
$563$	1.29998e7	1.72848	0.864240	+	0.503080i	$0.167800\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.42964e7	−1.85117	−0.925587	−	0.378535i	$-0.876428\pi$
$569$	−1.42964e7	−1.85117	−0.925587	+	0.378535i	$0.876428\pi$
$570$	0	0
$571$	−5.40555e6	−0.693824	−0.346912	−	0.937898i	$-0.612770\pi$
$571$	−5.40555e6	−0.693824	−0.346912	+	0.937898i	$0.612770\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.04395e7	−1.30539	−0.652694	−	0.757621i	$-0.726361\pi$
$577$	−1.04395e7	−1.30539	−0.652694	+	0.757621i	$0.726361\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.21780e6	0.149670
$582$	0	0
$583$	−1.35093e7	−1.64612
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.61958e6	−0.792931	−0.396465	−	0.918050i	$-0.629763\pi$
$587$	−6.61958e6	−0.792931	−0.396465	+	0.918050i	$0.629763\pi$
$588$	0	0
$589$	−1.21127e7	−1.43865
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8.50905e6	−0.993675	−0.496838	−	0.867843i	$-0.665506\pi$
$593$	−8.50905e6	−0.993675	−0.496838	+	0.867843i	$0.665506\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4.67122e6	−0.531940	−0.265970	−	0.963981i	$-0.585692\pi$
$599$	−4.67122e6	−0.531940	−0.265970	+	0.963981i	$0.585692\pi$
$600$	0	0
$601$	1.77737e6	0.200721	0.100360	−	0.994951i	$-0.468000\pi$
$601$	1.77737e6	0.200721	0.100360	+	0.994951i	$0.468000\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.29679e7	−1.42855	−0.714277	−	0.699863i	$-0.753244\pi$
$607$	−1.29679e7	−1.42855	−0.714277	+	0.699863i	$0.753244\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.18885e7	3.45566
$612$	0	0
$613$	1.21367e7	1.30451	0.652257	−	0.757998i	$-0.273822\pi$
$613$	1.21367e7	1.30451	0.652257	+	0.757998i	$0.273822\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.42427e6	−0.362122	−0.181061	−	0.983472i	$-0.557953\pi$
$617$	−3.42427e6	−0.362122	−0.181061	+	0.983472i	$0.557953\pi$
$618$	0	0
$619$	−1.52375e7	−1.59841	−0.799203	−	0.601061i	$-0.794745\pi$
$619$	−1.52375e7	−1.59841	−0.799203	+	0.601061i	$0.794745\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.01337e6	0.311052
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.91294e6	0.293566
$630$	0	0
$631$	215301.	0.0215265	0.0107632	−	0.999942i	$-0.496574\pi$
$631$	215301.	0.0215265	0.0107632	+	0.999942i	$0.496574\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.51734e7	−1.48161
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.80128e6	0.557671	0.278836	−	0.960339i	$-0.410051\pi$
$641$	5.80128e6	0.557671	0.278836	+	0.960339i	$0.410051\pi$
$642$	0	0
$643$	9.02804e6	0.861125	0.430562	−	0.902561i	$-0.358315\pi$
$643$	9.02804e6	0.861125	0.430562	+	0.902561i	$0.358315\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.95365e6	−0.183478	−0.0917392	−	0.995783i	$-0.529243\pi$
$647$	−1.95365e6	−0.183478	−0.0917392	+	0.995783i	$0.529243\pi$
$648$	0	0
$649$	−1.29732e6	−0.120902
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8.67419e6	−0.796060	−0.398030	−	0.917372i	$-0.630306\pi$
$653$	−8.67419e6	−0.796060	−0.398030	+	0.917372i	$0.630306\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.18122e7	1.05954	0.529768	−	0.848143i	$-0.322279\pi$
$659$	1.18122e7	1.05954	0.529768	+	0.848143i	$0.322279\pi$
$660$	0	0
$661$	4.24221e6	0.377649	0.188824	−	0.982011i	$-0.439532\pi$
$661$	4.24221e6	0.377649	0.188824	+	0.982011i	$0.439532\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.25358e7	−1.96137
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1.28458e7	−1.10142
$672$	0	0
$673$	2.05653e7	1.75024	0.875119	−	0.483907i	$-0.160783\pi$
$673$	2.05653e7	1.75024	0.875119	+	0.483907i	$0.160783\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.36927e6	0.701804	0.350902	−	0.936412i	$-0.385875\pi$
$677$	8.36927e6	0.701804	0.350902	+	0.936412i	$0.385875\pi$
$678$	0	0
$679$	−8.05025e6	−0.670093
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.11826e7	−0.917256	−0.458628	−	0.888628i	$-0.651659\pi$
$683$	−1.11826e7	−0.917256	−0.458628	+	0.888628i	$0.651659\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.50284e7	2.00856
$690$	0	0
$691$	−1.27525e7	−1.01601	−0.508007	−	0.861353i	$-0.669618\pi$
$691$	−1.27525e7	−1.01601	−0.508007	+	0.861353i	$0.669618\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.26975e7	0.990001
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8.25558e6	−0.634530	−0.317265	−	0.948337i	$-0.602765\pi$
$701$	−8.25558e6	−0.634530	−0.317265	+	0.948337i	$0.602765\pi$
$702$	0	0
$703$	−4.40882e6	−0.336461
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.02639e6	−0.453428
$708$	0	0
$709$	4.88990e6	0.365329	0.182665	−	0.983175i	$-0.441528\pi$
$709$	4.88990e6	0.365329	0.182665	+	0.983175i	$0.441528\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.34218e7	−0.988749
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.13547e6	−0.154053	−0.0770267	−	0.997029i	$-0.524543\pi$
$719$	−2.13547e6	−0.154053	−0.0770267	+	0.997029i	$0.524543\pi$
$720$	0	0
$721$	887535.	0.0635839
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.58055e7	−1.10911	−0.554554	−	0.832148i	$-0.687111\pi$
$727$	−1.58055e7	−1.10911	−0.554554	+	0.832148i	$0.687111\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.68159e7	1.16393
$732$	0	0
$733$	2.28769e6	0.157267	0.0786335	−	0.996904i	$-0.474944\pi$
$733$	2.28769e6	0.157267	0.0786335	+	0.996904i	$0.474944\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−240245.	−0.0162924
$738$	0	0
$739$	4.49661e6	0.302883	0.151441	−	0.988466i	$-0.451609\pi$
$739$	4.49661e6	0.302883	0.151441	+	0.988466i	$0.451609\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.69385e6	0.644205	0.322103	−	0.946705i	$-0.395610\pi$
$743$	9.69385e6	0.644205	0.322103	+	0.946705i	$0.395610\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.29890e6	0.605657
$750$	0	0
$751$	−1.50258e7	−0.972163	−0.486082	−	0.873913i	$-0.661574\pi$
$751$	−1.50258e7	−0.972163	−0.486082	+	0.873913i	$0.661574\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.42039e7	−0.900884	−0.450442	−	0.892806i	$-0.648734\pi$
$757$	−1.42039e7	−0.900884	−0.450442	+	0.892806i	$0.648734\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.32337e6	−0.0828358	−0.0414179	−	0.999142i	$-0.513188\pi$
$761$	−1.32337e6	−0.0828358	−0.0414179	+	0.999142i	$0.513188\pi$
$762$	0	0
$763$	−4.95751e6	−0.308285
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.40351e6	0.147522
$768$	0	0
$769$	−2.33698e7	−1.42508	−0.712540	−	0.701632i	$-0.752455\pi$
$769$	−2.33698e7	−1.42508	−0.712540	+	0.701632i	$0.752455\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.74390e7	1.04972	0.524860	−	0.851188i	$-0.324118\pi$
$773$	1.74390e7	1.04972	0.524860	+	0.851188i	$0.324118\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.92180e7	−1.13466
$780$	0	0
$781$	3.14288e7	1.84374
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	7.58390e6	0.436472	0.218236	−	0.975896i	$-0.429970\pi$
$787$	7.58390e6	0.436472	0.218236	+	0.975896i	$0.429970\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.50256e6	−0.312697
$792$	0	0
$793$	2.37991e7	1.34393
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−368007.	−0.0205216	−0.0102608	−	0.999947i	$-0.503266\pi$
$797$	−368007.	−0.0205216	−0.0102608	+	0.999947i	$0.503266\pi$
$798$	0	0
$799$	−4.68089e7	−2.59395
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.02882e7	−0.563052
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−2.68278e7	−1.44116	−0.720582	−	0.693370i	$-0.756125\pi$
$809$	−2.68278e7	−1.44116	−0.720582	+	0.693370i	$0.756125\pi$
$810$	0	0
$811$	−3.07164e7	−1.63991	−0.819953	−	0.572431i	$-0.806000\pi$
$811$	−3.07164e7	−1.63991	−0.819953	+	0.572431i	$0.806000\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.54513e7	−1.33400
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−9.59084e6	−0.496591	−0.248295	−	0.968684i	$-0.579870\pi$
$821$	−9.59084e6	−0.496591	−0.248295	+	0.968684i	$0.579870\pi$
$822$	0	0
$823$	−3.42513e7	−1.76270	−0.881349	−	0.472466i	$-0.843364\pi$
$823$	−3.42513e7	−1.76270	−0.881349	+	0.472466i	$0.843364\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.36507e6	0.120249	0.0601243	−	0.998191i	$-0.480850\pi$
$827$	2.36507e6	0.120249	0.0601243	+	0.998191i	$0.480850\pi$
$828$	0	0
$829$	−8.63277e6	−0.436279	−0.218139	−	0.975918i	$-0.569999\pi$
$829$	−8.63277e6	−0.436279	−0.218139	+	0.975918i	$0.569999\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.22729e7	1.11215
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.58597e7	−0.777842	−0.388921	−	0.921271i	$-0.627152\pi$
$839$	−1.58597e7	−0.777842	−0.388921	+	0.921271i	$0.627152\pi$
$840$	0	0
$841$	5.06516e7	2.46947
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−9.67362e6	−0.463320
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.88529e6	−0.231242
$852$	0	0
$853$	2.27282e7	1.06953	0.534764	−	0.845001i	$-0.320400\pi$
$853$	2.27282e7	1.06953	0.534764	+	0.845001i	$0.320400\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.71180e7	0.796163	0.398081	−	0.917350i	$-0.369676\pi$
$857$	1.71180e7	0.796163	0.398081	+	0.917350i	$0.369676\pi$
$858$	0	0
$859$	3.00031e7	1.38734	0.693669	−	0.720293i	$-0.255993\pi$
$859$	3.00031e7	1.38734	0.693669	+	0.720293i	$0.255993\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.61288e6	−0.119424	−0.0597122	−	0.998216i	$-0.519018\pi$
$863$	−2.61288e6	−0.119424	−0.0597122	+	0.998216i	$0.519018\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.48552e7	−1.56573
$870$	0	0
$871$	445097.	0.0198797
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	9.15655e6	0.402006	0.201003	−	0.979591i	$-0.435580\pi$
$877$	9.15655e6	0.402006	0.201003	+	0.979591i	$0.435580\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.61189e7	−1.13375	−0.566873	−	0.823805i	$-0.691847\pi$
$881$	−2.61189e7	−1.13375	−0.566873	+	0.823805i	$0.691847\pi$
$882$	0	0
$883$	−9.12268e6	−0.393750	−0.196875	−	0.980429i	$-0.563079\pi$
$883$	−9.12268e6	−0.393750	−0.196875	+	0.980429i	$0.563079\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.28277e6	0.353482	0.176741	−	0.984257i	$-0.443445\pi$
$887$	8.28277e6	0.353482	0.176741	+	0.984257i	$0.443445\pi$
$888$	0	0
$889$	−570018.	−0.0241899
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.08466e7	2.97297
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.23828e7	1.74900
$900$	0	0
$901$	−3.67390e7	−1.50770
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.09147e7	0.844176	0.422088	−	0.906555i	$-0.361297\pi$
$907$	2.09147e7	0.844176	0.422088	+	0.906555i	$0.361297\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4.29585e6	−0.171496	−0.0857479	−	0.996317i	$-0.527328\pi$
$911$	−4.29585e6	−0.171496	−0.0857479	+	0.996317i	$0.527328\pi$
$912$	0	0
$913$	1.34217e7	0.532881
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.16239e6	−0.242006
$918$	0	0
$919$	1.79222e7	0.700007	0.350004	−	0.936748i	$-0.386180\pi$
$919$	1.79222e7	0.700007	0.350004	+	0.936748i	$0.386180\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.82274e7	−2.24969
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.18638e7	0.451009	0.225505	−	0.974242i	$-0.427597\pi$
$929$	1.18638e7	0.451009	0.225505	+	0.974242i	$0.427597\pi$
$930$	0	0
$931$	−3.37107e7	−1.27466
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−3.19444e7	−1.18863	−0.594313	−	0.804234i	$-0.702576\pi$
$937$	−3.19444e7	−1.18863	−0.594313	+	0.804234i	$0.702576\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.48231e7	0.913866	0.456933	−	0.889501i	$-0.348948\pi$
$941$	2.48231e7	0.913866	0.456933	+	0.889501i	$0.348948\pi$
$942$	0	0
$943$	−2.12949e7	−0.779825
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.38376e7	−1.22610	−0.613049	−	0.790045i	$-0.710057\pi$
$947$	−3.38376e7	−1.22610	−0.613049	+	0.790045i	$0.710057\pi$
$948$	0	0
$949$	1.90606e7	0.687024
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.02206e7	−0.721211	−0.360605	−	0.932718i	$-0.617430\pi$
$953$	−2.02206e7	−0.721211	−0.360605	+	0.932718i	$0.617430\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.38090e6	0.224045
$960$	0	0
$961$	−3.38703e6	−0.118307
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	5.30981e7	1.82605	0.913025	−	0.407904i	$-0.133740\pi$
$967$	5.30981e7	1.82605	0.913025	+	0.407904i	$0.133740\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.52073e7	−1.19835	−0.599176	−	0.800617i	$-0.704505\pi$
$971$	−3.52073e7	−1.19835	−0.599176	+	0.800617i	$0.704505\pi$
$972$	0	0
$973$	7.25277e6	0.245596
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.08861e7	1.37037	0.685187	−	0.728368i	$-0.259721\pi$
$977$	4.08861e7	1.37037	0.685187	+	0.728368i	$0.259721\pi$
$978$	0	0
$979$	3.32112e7	1.10746
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.29571e7	0.757762	0.378881	−	0.925445i	$-0.376309\pi$
$983$	2.29571e7	0.757762	0.378881	+	0.925445i	$0.376309\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.82019e7	−0.916827
$990$	0	0
$991$	1.09601e7	0.354511	0.177256	−	0.984165i	$-0.443278\pi$
$991$	1.09601e7	0.354511	0.177256	+	0.984165i	$0.443278\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.29419e6	−0.0412346	−0.0206173	−	0.999787i	$-0.506563\pi$
$997$	−1.29419e6	−0.0412346	−0.0206173	+	0.999787i	$0.506563\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.6.a.t.1.1		2
3.2	odd	2		900.6.a.u.1.1		2
5.2	odd	4		900.6.d.k.649.2		4
5.3	odd	4		900.6.d.k.649.3		4
5.4	even	2		180.6.a.g.1.2	yes	2
15.2	even	4		900.6.d.n.649.2		4
15.8	even	4		900.6.d.n.649.3		4
15.14	odd	2		180.6.a.f.1.2	✓	2
20.19	odd	2		720.6.a.bg.1.1		2
60.59	even	2		720.6.a.bc.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.6.a.f.1.2	✓	2	15.14	odd	2
180.6.a.g.1.2	yes	2	5.4	even	2
720.6.a.bc.1.1		2	60.59	even	2
720.6.a.bg.1.1		2	20.19	odd	2
900.6.a.t.1.1		2	1.1	even	1	trivial
900.6.a.u.1.1		2	3.2	odd	2
900.6.d.k.649.2		4	5.2	odd	4
900.6.d.k.649.3		4	5.3	odd	4
900.6.d.n.649.2		4	15.2	even	4
900.6.d.n.649.3		4	15.8	even	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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	900.a (trivial)

\(f(q)\)	\(=\)	\(q-53.1450 q^{7} +O(q^{10})\)	\(q-53.1450 q^{7} -585.725 q^{11} +1085.16 q^{13} -1592.90 q^{17} +2410.90 q^{19} +2671.45 q^{23} -8435.80 q^{29} -5024.15 q^{31} -1828.70 q^{37} -7971.30 q^{41} -10556.8 q^{43} +29386.0 q^{47} -13982.6 q^{49} +23064.2 q^{53} +2214.89 q^{59} +21931.4 q^{61} +410.167 q^{67} -53657.9 q^{71} +17564.8 q^{73} +31128.4 q^{77} +59507.8 q^{79} -22914.7 q^{83} -56700.9 q^{89} -57670.9 q^{91} +151477. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 80 q^{7}+O(q^{10}) \) 2 * q + 80 * q^7	\( 2 q + 80 q^{7} - 240 q^{11} + 680 q^{13} + 540 q^{17} + 1096 q^{19} + 3480 q^{23} - 9420 q^{29} + 2992 q^{31} + 7520 q^{37} - 27120 q^{41} - 4720 q^{43} + 38280 q^{47} - 13062 q^{49} + 53580 q^{53} - 46800 q^{59} + 21508 q^{61} + 10880 q^{67} - 45840 q^{71} - 9580 q^{73} + 77160 q^{77} + 91072 q^{79} - 14160 q^{83} - 35160 q^{89} - 111616 q^{91} + 98780 q^{97}+O(q^{100}) \) 2 * q + 80 * q^7 - 240 * q^11 + 680 * q^13 + 540 * q^17 + 1096 * q^19 + 3480 * q^23 - 9420 * q^29 + 2992 * q^31 + 7520 * q^37 - 27120 * q^41 - 4720 * q^43 + 38280 * q^47 - 13062 * q^49 + 53580 * q^53 - 46800 * q^59 + 21508 * q^61 + 10880 * q^67 - 45840 * q^71 - 9580 * q^73 + 77160 * q^77 + 91072 * q^79 - 14160 * q^83 - 35160 * q^89 - 111616 * q^91 + 98780 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more