LMFDB - Embedded newform 900.6.a.x.1.2

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:	$3$
Coefficient field:	3.3.535753.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 186x - 432 $ x^3 - x^2 - 186*x - 432
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 60)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-11.7229$ 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)

$f(q)$	$=$	$q-12.5817 q^{7} +O(q^{10})$	$q-12.5817 q^{7} -164.713 q^{11} +849.118 q^{13} +1608.11 q^{17} -446.747 q^{19} -802.223 q^{23} -4477.98 q^{29} +5200.90 q^{31} +8248.56 q^{37} +6250.10 q^{41} -11314.2 q^{43} -29842.0 q^{47} -16648.7 q^{49} -9195.04 q^{53} -9948.84 q^{59} +41550.9 q^{61} +49074.0 q^{67} +44977.6 q^{71} +65643.2 q^{73} +2072.36 q^{77} -67157.8 q^{79} -54883.8 q^{83} +53984.2 q^{89} -10683.3 q^{91} +100814. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 88 q^{7}+O(q^{10}) $ 3 * q + 88 * q^7	$ 3 q + 88 q^{7} - 148 q^{11} - 868 q^{17} + 3000 q^{19} + 1052 q^{23} - 7962 q^{29} + 132 q^{31} + 11944 q^{37} - 8170 q^{41} + 26188 q^{43} - 37892 q^{47} + 13827 q^{49} - 11076 q^{53} + 46228 q^{59} + 3126 q^{61} + 126332 q^{67} + 80400 q^{71} + 42904 q^{73} - 234608 q^{77} - 64476 q^{79} - 126268 q^{83} + 38030 q^{89} - 49200 q^{91} + 331016 q^{97}+O(q^{100}) $ 3 * q + 88 * q^7 - 148 * q^11 - 868 * q^17 + 3000 * q^19 + 1052 * q^23 - 7962 * q^29 + 132 * q^31 + 11944 * q^37 - 8170 * q^41 + 26188 * q^43 - 37892 * q^47 + 13827 * q^49 - 11076 * q^53 + 46228 * q^59 + 3126 * q^61 + 126332 * q^67 + 80400 * q^71 + 42904 * q^73 - 234608 * q^77 - 64476 * q^79 - 126268 * q^83 + 38030 * q^89 - 49200 * q^91 + 331016 * 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
$7$	−12.5817	−0.0970496	−0.0485248	−	0.998822i	$-0.515452\pi$
$7$	−12.5817	−0.0970496	−0.0485248	+	0.998822i	$0.515452\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−164.713	−0.410436	−0.205218	−	0.978716i	$-0.565790\pi$
$11$	−164.713	−0.410436	−0.205218	+	0.978716i	$0.565790\pi$
$12$	0	0
$13$	849.118	1.39351	0.696755	−	0.717310i	$-0.254627\pi$
$13$	849.118	1.39351	0.696755	+	0.717310i	$0.254627\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1608.11	1.34957	0.674783	−	0.738016i	$-0.264237\pi$
$17$	1608.11	1.34957	0.674783	+	0.738016i	$0.264237\pi$
$18$	0	0
$19$	−446.747	−0.283908	−0.141954	−	0.989873i	$-0.545338\pi$
$19$	−446.747	−0.283908	−0.141954	+	0.989873i	$0.545338\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−802.223	−0.316210	−0.158105	−	0.987422i	$-0.550538\pi$
$23$	−802.223	−0.316210	−0.158105	+	0.987422i	$0.550538\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4477.98	−0.988752	−0.494376	−	0.869248i	$-0.664603\pi$
$29$	−4477.98	−0.988752	−0.494376	+	0.869248i	$0.664603\pi$
$30$	0	0
$31$	5200.90	0.972019	0.486009	−	0.873954i	$-0.338452\pi$
$31$	5200.90	0.972019	0.486009	+	0.873954i	$0.338452\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8248.56	0.990544	0.495272	−	0.868738i	$-0.335068\pi$
$37$	8248.56	0.990544	0.495272	+	0.868738i	$0.335068\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6250.10	0.580667	0.290334	−	0.956925i	$-0.406234\pi$
$41$	6250.10	0.580667	0.290334	+	0.956925i	$0.406234\pi$
$42$	0	0
$43$	−11314.2	−0.933152	−0.466576	−	0.884481i	$-0.654513\pi$
$43$	−11314.2	−0.933152	−0.466576	+	0.884481i	$0.654513\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−29842.0	−1.97053	−0.985265	−	0.171035i	$-0.945289\pi$
$47$	−29842.0	−1.97053	−0.985265	+	0.171035i	$0.945289\pi$
$48$	0	0
$49$	−16648.7	−0.990581
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9195.04	−0.449639	−0.224820	−	0.974400i	$-0.572179\pi$
$53$	−9195.04	−0.449639	−0.224820	+	0.974400i	$0.572179\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9948.84	−0.372085	−0.186043	−	0.982542i	$-0.559566\pi$
$59$	−9948.84	−0.372085	−0.186043	+	0.982542i	$0.559566\pi$
$60$	0	0
$61$	41550.9	1.42974	0.714868	−	0.699259i	$-0.246487\pi$
$61$	41550.9	1.42974	0.714868	+	0.699259i	$0.246487\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	49074.0	1.33556	0.667781	−	0.744357i	$-0.267244\pi$
$67$	49074.0	1.33556	0.667781	+	0.744357i	$0.267244\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	44977.6	1.05889	0.529445	−	0.848344i	$-0.322400\pi$
$71$	44977.6	1.05889	0.529445	+	0.848344i	$0.322400\pi$
$72$	0	0
$73$	65643.2	1.44173	0.720863	−	0.693077i	$-0.243746\pi$
$73$	65643.2	1.44173	0.720863	+	0.693077i	$0.243746\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2072.36	0.0398327
$78$	0	0
$79$	−67157.8	−1.21068	−0.605339	−	0.795968i	$-0.706963\pi$
$79$	−67157.8	−1.21068	−0.605339	+	0.795968i	$0.706963\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−54883.8	−0.874478	−0.437239	−	0.899345i	$-0.644044\pi$
$83$	−54883.8	−0.874478	−0.437239	+	0.899345i	$0.644044\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	53984.2	0.722423	0.361212	−	0.932484i	$-0.382363\pi$
$89$	53984.2	0.722423	0.361212	+	0.932484i	$0.382363\pi$
$90$	0	0
$91$	−10683.3	−0.135240
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	100814.	1.08791	0.543956	−	0.839114i	$-0.316926\pi$
$97$	100814.	1.08791	0.543956	+	0.839114i	$0.316926\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−35164.5	−0.343005	−0.171503	−	0.985184i	$-0.554862\pi$
$101$	−35164.5	−0.343005	−0.171503	+	0.985184i	$0.554862\pi$
$102$	0	0
$103$	−26256.9	−0.243865	−0.121933	−	0.992538i	$-0.538909\pi$
$103$	−26256.9	−0.243865	−0.121933	+	0.992538i	$0.538909\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−192949.	−1.62923	−0.814615	−	0.580003i	$-0.803052\pi$
$107$	−192949.	−1.62923	−0.814615	+	0.580003i	$0.803052\pi$
$108$	0	0
$109$	210504.	1.69705	0.848523	−	0.529159i	$-0.177492\pi$
$109$	210504.	1.69705	0.848523	+	0.529159i	$0.177492\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	69072.9	0.508875	0.254438	−	0.967089i	$-0.418110\pi$
$113$	69072.9	0.508875	0.254438	+	0.967089i	$0.418110\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−20232.8	−0.130975
$120$	0	0
$121$	−133921.	−0.831542
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	137071.	0.754115	0.377057	−	0.926190i	$-0.376936\pi$
$127$	137071.	0.754115	0.377057	+	0.926190i	$0.376936\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	161384.	0.821644	0.410822	−	0.911716i	$-0.365242\pi$
$131$	161384.	0.821644	0.410822	+	0.911716i	$0.365242\pi$
$132$	0	0
$133$	5620.83	0.0275532
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−236746.	−1.07766	−0.538829	−	0.842415i	$-0.681133\pi$
$137$	−236746.	−1.07766	−0.538829	+	0.842415i	$0.681133\pi$
$138$	0	0
$139$	139568.	0.612703	0.306352	−	0.951918i	$-0.400892\pi$
$139$	139568.	0.612703	0.306352	+	0.951918i	$0.400892\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−139861.	−0.571946
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	169441.	0.625249	0.312624	−	0.949877i	$-0.398792\pi$
$149$	169441.	0.625249	0.312624	+	0.949877i	$0.398792\pi$
$150$	0	0
$151$	141483.	0.504966	0.252483	−	0.967601i	$-0.418753\pi$
$151$	141483.	0.504966	0.252483	+	0.967601i	$0.418753\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−356278.	−1.15356	−0.576780	−	0.816900i	$-0.695691\pi$
$157$	−356278.	−1.15356	−0.576780	+	0.816900i	$0.695691\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	10093.3	0.0306881
$162$	0	0
$163$	492593.	1.45218	0.726088	−	0.687602i	$-0.241337\pi$
$163$	492593.	1.45218	0.726088	+	0.687602i	$0.241337\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	558396.	1.54936	0.774678	−	0.632356i	$-0.217912\pi$
$167$	558396.	1.54936	0.774678	+	0.632356i	$0.217912\pi$
$168$	0	0
$169$	349709.	0.941868
$170$	0	0
$171$	0	0
$172$	0	0
$173$	104434.	0.265294	0.132647	−	0.991163i	$-0.457652\pi$
$173$	104434.	0.265294	0.132647	+	0.991163i	$0.457652\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	439589.	1.02545	0.512725	−	0.858553i	$-0.328636\pi$
$179$	439589.	1.02545	0.512725	+	0.858553i	$0.328636\pi$
$180$	0	0
$181$	66771.5	0.151494	0.0757469	−	0.997127i	$-0.475866\pi$
$181$	66771.5	0.151494	0.0757469	+	0.997127i	$0.475866\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−264877.	−0.553911
$188$	0	0
$189$	0	0
$190$	0	0
$191$	907090.	1.79915	0.899574	−	0.436768i	$-0.143877\pi$
$191$	907090.	1.79915	0.899574	+	0.436768i	$0.143877\pi$
$192$	0	0
$193$	−992517.	−1.91798	−0.958991	−	0.283436i	$-0.908526\pi$
$193$	−992517.	−1.91798	−0.958991	+	0.283436i	$0.908526\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	520497.	0.955548	0.477774	−	0.878483i	$-0.341444\pi$
$197$	520497.	0.955548	0.477774	+	0.878483i	$0.341444\pi$
$198$	0	0
$199$	−861410.	−1.54198	−0.770988	−	0.636850i	$-0.780237\pi$
$199$	−861410.	−1.54198	−0.770988	+	0.636850i	$0.780237\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	56340.6	0.0959580
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	73584.9	0.116526
$210$	0	0
$211$	1.08168e6	1.67261	0.836304	−	0.548266i	$-0.184712\pi$
$211$	1.08168e6	1.67261	0.836304	+	0.548266i	$0.184712\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−65436.2	−0.0943341
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.36548e6	1.88063
$222$	0	0
$223$	−294324.	−0.396336	−0.198168	−	0.980168i	$-0.563499\pi$
$223$	−294324.	−0.396336	−0.198168	+	0.980168i	$0.563499\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	223019.	0.287261	0.143631	−	0.989631i	$-0.454122\pi$
$227$	223019.	0.287261	0.143631	+	0.989631i	$0.454122\pi$
$228$	0	0
$229$	−75342.8	−0.0949408	−0.0474704	−	0.998873i	$-0.515116\pi$
$229$	−75342.8	−0.0949408	−0.0474704	+	0.998873i	$0.515116\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.04814e6	1.26482	0.632411	−	0.774633i	$-0.282065\pi$
$233$	1.04814e6	1.26482	0.632411	+	0.774633i	$0.282065\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−871642.	−0.987060	−0.493530	−	0.869729i	$-0.664294\pi$
$239$	−871642.	−0.987060	−0.493530	+	0.869729i	$0.664294\pi$
$240$	0	0
$241$	228926.	0.253894	0.126947	−	0.991910i	$-0.459482\pi$
$241$	228926.	0.253894	0.126947	+	0.991910i	$0.459482\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−379341.	−0.395628
$248$	0	0
$249$	0	0
$250$	0	0
$251$	594245.	0.595362	0.297681	−	0.954665i	$-0.403787\pi$
$251$	594245.	0.595362	0.297681	+	0.954665i	$0.403787\pi$
$252$	0	0
$253$	132136.	0.129784
$254$	0	0
$255$	0	0
$256$	0	0
$257$	499436.	0.471679	0.235840	−	0.971792i	$-0.424216\pi$
$257$	499436.	0.471679	0.235840	+	0.971792i	$0.424216\pi$
$258$	0	0
$259$	−103781.	−0.0961320
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−760007.	−0.677530	−0.338765	−	0.940871i	$-0.610009\pi$
$263$	−760007.	−0.677530	−0.338765	+	0.940871i	$0.610009\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.88976e6	1.59230	0.796150	−	0.605099i	$-0.206866\pi$
$269$	1.88976e6	1.59230	0.796150	+	0.605099i	$0.206866\pi$
$270$	0	0
$271$	2.23595e6	1.84943	0.924717	−	0.380655i	$-0.124302\pi$
$271$	2.23595e6	1.84943	0.924717	+	0.380655i	$0.124302\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.82310e6	−1.42761	−0.713806	−	0.700343i	$-0.753030\pi$
$277$	−1.82310e6	−1.42761	−0.713806	+	0.700343i	$0.753030\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.28839e6	0.973381	0.486690	−	0.873575i	$-0.338204\pi$
$281$	1.28839e6	0.973381	0.486690	+	0.873575i	$0.338204\pi$
$282$	0	0
$283$	497215.	0.369044	0.184522	−	0.982828i	$-0.440926\pi$
$283$	497215.	0.369044	0.184522	+	0.982828i	$0.440926\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−78636.8	−0.0563535
$288$	0	0
$289$	1.16617e6	0.821329
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−445745.	−0.303332	−0.151666	−	0.988432i	$-0.548464\pi$
$293$	−445745.	−0.303332	−0.151666	+	0.988432i	$0.548464\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−681182.	−0.440641
$300$	0	0
$301$	142352.	0.0905620
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	947164.	0.573560	0.286780	−	0.957996i	$-0.407415\pi$
$307$	947164.	0.573560	0.286780	+	0.957996i	$0.407415\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.79602e6	1.63923	0.819615	−	0.572914i	$-0.194187\pi$
$311$	2.79602e6	1.63923	0.819615	+	0.572914i	$0.194187\pi$
$312$	0	0
$313$	−559111.	−0.322580	−0.161290	−	0.986907i	$-0.551565\pi$
$313$	−559111.	−0.322580	−0.161290	+	0.986907i	$0.551565\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.91082e6	1.06800	0.534000	−	0.845485i	$-0.320688\pi$
$317$	1.91082e6	1.06800	0.534000	+	0.845485i	$0.320688\pi$
$318$	0	0
$319$	737581.	0.405819
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−718419.	−0.383152
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	375463.	0.191239
$330$	0	0
$331$	−564319.	−0.283110	−0.141555	−	0.989930i	$-0.545210\pi$
$331$	−564319.	−0.283110	−0.141555	+	0.989930i	$0.545210\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.79332e6	0.860166	0.430083	−	0.902789i	$-0.358484\pi$
$337$	1.79332e6	0.860166	0.430083	+	0.902789i	$0.358484\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−856655.	−0.398951
$342$	0	0
$343$	420929.	0.193185
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.22394e6	−1.88319	−0.941595	−	0.336748i	$-0.890673\pi$
$347$	−4.22394e6	−1.88319	−0.941595	+	0.336748i	$0.890673\pi$
$348$	0	0
$349$	1.31539e6	0.578084	0.289042	−	0.957316i	$-0.406663\pi$
$349$	1.31539e6	0.578084	0.289042	+	0.957316i	$0.406663\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.39771e6	1.45128	0.725638	−	0.688077i	$-0.241545\pi$
$353$	3.39771e6	1.45128	0.725638	+	0.688077i	$0.241545\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.61103e6	1.88826	0.944131	−	0.329570i	$-0.106904\pi$
$359$	4.61103e6	1.88826	0.944131	+	0.329570i	$0.106904\pi$
$360$	0	0
$361$	−2.27652e6	−0.919396
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.16806e6	1.61536	0.807679	−	0.589623i	$-0.200723\pi$
$367$	4.16806e6	1.61536	0.807679	+	0.589623i	$0.200723\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	115689.	0.0436373
$372$	0	0
$373$	873693.	0.325152	0.162576	−	0.986696i	$-0.448020\pi$
$373$	873693.	0.325152	0.162576	+	0.986696i	$0.448020\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.80234e6	−1.37784
$378$	0	0
$379$	2.00655e6	0.717551	0.358775	−	0.933424i	$-0.383194\pi$
$379$	2.00655e6	0.717551	0.358775	+	0.933424i	$0.383194\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.34193e6	−1.16413	−0.582064	−	0.813143i	$-0.697755\pi$
$383$	−3.34193e6	−1.16413	−0.582064	+	0.813143i	$0.697755\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.41873e6	0.475364	0.237682	−	0.971343i	$-0.423612\pi$
$389$	1.41873e6	0.475364	0.237682	+	0.971343i	$0.423612\pi$
$390$	0	0
$391$	−1.29006e6	−0.426746
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.31507e6	−0.418766	−0.209383	−	0.977834i	$-0.567146\pi$
$397$	−1.31507e6	−0.418766	−0.209383	+	0.977834i	$0.567146\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.01811e6	1.55840	0.779200	−	0.626775i	$-0.215626\pi$
$401$	5.01811e6	1.55840	0.779200	+	0.626775i	$0.215626\pi$
$402$	0	0
$403$	4.41618e6	1.35452
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.35864e6	−0.406555
$408$	0	0
$409$	−314814.	−0.0930563	−0.0465282	−	0.998917i	$-0.514816\pi$
$409$	−314814.	−0.0930563	−0.0465282	+	0.998917i	$0.514816\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	125173.	0.0361107
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.54817e6	−1.26561	−0.632807	−	0.774310i	$-0.718097\pi$
$419$	−4.54817e6	−1.26561	−0.632807	+	0.774310i	$0.718097\pi$
$420$	0	0
$421$	4.05291e6	1.11445	0.557227	−	0.830360i	$-0.311865\pi$
$421$	4.05291e6	1.11445	0.557227	+	0.830360i	$0.311865\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−522781.	−0.138755
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.02007e6	−1.30172	−0.650858	−	0.759200i	$-0.725591\pi$
$431$	−5.02007e6	−1.30172	−0.650858	+	0.759200i	$0.725591\pi$
$432$	0	0
$433$	−4.54995e6	−1.16624	−0.583118	−	0.812387i	$-0.698168\pi$
$433$	−4.54995e6	−1.16624	−0.583118	+	0.812387i	$0.698168\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	358390.	0.0897744
$438$	0	0
$439$	−3.75909e6	−0.930938	−0.465469	−	0.885064i	$-0.654114\pi$
$439$	−3.75909e6	−0.930938	−0.465469	+	0.885064i	$0.654114\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.67476e6	−1.61594	−0.807972	−	0.589220i	$-0.799435\pi$
$443$	−6.67476e6	−1.61594	−0.807972	+	0.589220i	$0.799435\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.57947e6	0.603831	0.301916	−	0.953335i	$-0.402374\pi$
$449$	2.57947e6	0.603831	0.301916	+	0.953335i	$0.402374\pi$
$450$	0	0
$451$	−1.02947e6	−0.238327
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.78248e6	0.399240	0.199620	−	0.979873i	$-0.436029\pi$
$457$	1.78248e6	0.399240	0.199620	+	0.979873i	$0.436029\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.57073e6	0.563383	0.281691	−	0.959505i	$-0.409105\pi$
$461$	2.57073e6	0.563383	0.281691	+	0.959505i	$0.409105\pi$
$462$	0	0
$463$	5.50764e6	1.19402	0.597012	−	0.802232i	$-0.296354\pi$
$463$	5.50764e6	1.19402	0.597012	+	0.802232i	$0.296354\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.10049e6	0.233504	0.116752	−	0.993161i	$-0.462752\pi$
$467$	1.10049e6	0.233504	0.116752	+	0.993161i	$0.462752\pi$
$468$	0	0
$469$	−617434.	−0.129616
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.86359e6	0.382999
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.85993e6	−0.569530	−0.284765	−	0.958597i	$-0.591916\pi$
$479$	−2.85993e6	−0.569530	−0.284765	+	0.958597i	$0.591916\pi$
$480$	0	0
$481$	7.00401e6	1.38033
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	988304.	0.188829	0.0944144	−	0.995533i	$-0.469902\pi$
$487$	988304.	0.188829	0.0944144	+	0.995533i	$0.469902\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.57116e6	0.481310	0.240655	−	0.970611i	$-0.422638\pi$
$491$	2.57116e6	0.481310	0.240655	+	0.970611i	$0.422638\pi$
$492$	0	0
$493$	−7.20110e6	−1.33439
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−565895.	−0.102765
$498$	0	0
$499$	3.15722e6	0.567614	0.283807	−	0.958881i	$-0.408402\pi$
$499$	3.15722e6	0.567614	0.283807	+	0.958881i	$0.408402\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.90234e6	1.21640	0.608200	−	0.793784i	$-0.291892\pi$
$503$	6.90234e6	1.21640	0.608200	+	0.793784i	$0.291892\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8.99803e6	−1.53941	−0.769703	−	0.638402i	$-0.779596\pi$
$509$	−8.99803e6	−1.53941	−0.769703	+	0.638402i	$0.779596\pi$
$510$	0	0
$511$	−825903.	−0.139919
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4.91535e6	0.808776
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−8.75002e6	−1.41226	−0.706130	−	0.708082i	$-0.749561\pi$
$521$	−8.75002e6	−1.41226	−0.706130	+	0.708082i	$0.749561\pi$
$522$	0	0
$523$	−2.22239e6	−0.355276	−0.177638	−	0.984096i	$-0.556846\pi$
$523$	−2.22239e6	−0.355276	−0.177638	+	0.984096i	$0.556846\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.36364e6	1.31180
$528$	0	0
$529$	−5.79278e6	−0.900011
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.30707e6	0.809165
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.74225e6	0.406570
$540$	0	0
$541$	2.93950e6	0.431798	0.215899	−	0.976416i	$-0.430732\pi$
$541$	2.93950e6	0.431798	0.215899	+	0.976416i	$0.430732\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−6.12529e6	−0.875303	−0.437651	−	0.899145i	$-0.644190\pi$
$547$	−6.12529e6	−0.875303	−0.437651	+	0.899145i	$0.644190\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.00052e6	0.280714
$552$	0	0
$553$	844959.	0.117496
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.19245e6	0.299428	0.149714	−	0.988729i	$-0.452165\pi$
$557$	2.19245e6	0.299428	0.149714	+	0.988729i	$0.452165\pi$
$558$	0	0
$559$	−9.60709e6	−1.30036
$560$	0	0
$561$	0	0
$562$	0	0
$563$	44998.2	0.00598307	0.00299154	−	0.999996i	$-0.499048\pi$
$563$	44998.2	0.00598307	0.00299154	+	0.999996i	$0.499048\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.09576e6	0.141884	0.0709420	−	0.997480i	$-0.477399\pi$
$569$	1.09576e6	0.141884	0.0709420	+	0.997480i	$0.477399\pi$
$570$	0	0
$571$	−1.11639e7	−1.43294	−0.716468	−	0.697620i	$-0.754242\pi$
$571$	−1.11639e7	−1.43294	−0.716468	+	0.697620i	$0.754242\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−8.62132e6	−1.07804	−0.539019	−	0.842294i	$-0.681205\pi$
$577$	−8.62132e6	−1.07804	−0.539019	+	0.842294i	$0.681205\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	690531.	0.0848678
$582$	0	0
$583$	1.51454e6	0.184548
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.34006e6	−0.160520	−0.0802599	−	0.996774i	$-0.525575\pi$
$587$	−1.34006e6	−0.160520	−0.0802599	+	0.996774i	$0.525575\pi$
$588$	0	0
$589$	−2.32349e6	−0.275964
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.08044e7	−1.26172	−0.630860	−	0.775897i	$-0.717298\pi$
$593$	−1.08044e7	−1.26172	−0.630860	+	0.775897i	$0.717298\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.92132e6	−0.788174	−0.394087	−	0.919073i	$-0.628939\pi$
$599$	−6.92132e6	−0.788174	−0.394087	+	0.919073i	$0.628939\pi$
$600$	0	0
$601$	−6.16706e6	−0.696453	−0.348226	−	0.937410i	$-0.613216\pi$
$601$	−6.16706e6	−0.696453	−0.348226	+	0.937410i	$0.613216\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.18721e7	−1.30785	−0.653924	−	0.756560i	$-0.726878\pi$
$607$	−1.18721e7	−1.30785	−0.653924	+	0.756560i	$0.726878\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.53394e7	−2.74595
$612$	0	0
$613$	−2.27563e6	−0.244597	−0.122298	−	0.992493i	$-0.539026\pi$
$613$	−2.27563e6	−0.244597	−0.122298	+	0.992493i	$0.539026\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.34715e7	1.42463	0.712315	−	0.701860i	$-0.247647\pi$
$617$	1.34715e7	1.42463	0.712315	+	0.701860i	$0.247647\pi$
$618$	0	0
$619$	1.36512e7	1.43200	0.716000	−	0.698100i	$-0.245971\pi$
$619$	1.36512e7	1.43200	0.716000	+	0.698100i	$0.245971\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−679213.	−0.0701109
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.32646e7	1.33681
$630$	0	0
$631$	−6.51822e6	−0.651711	−0.325856	−	0.945420i	$-0.605652\pi$
$631$	−6.51822e6	−0.651711	−0.325856	+	0.945420i	$0.605652\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.41367e7	−1.38038
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.25104e6	−0.793165	−0.396583	−	0.917999i	$-0.629804\pi$
$641$	−8.25104e6	−0.793165	−0.396583	+	0.917999i	$0.629804\pi$
$642$	0	0
$643$	−8.10283e6	−0.772875	−0.386437	−	0.922316i	$-0.626294\pi$
$643$	−8.10283e6	−0.772875	−0.386437	+	0.922316i	$0.626294\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.23767e6	−0.491901	−0.245950	−	0.969282i	$-0.579100\pi$
$647$	−5.23767e6	−0.491901	−0.245950	+	0.969282i	$0.579100\pi$
$648$	0	0
$649$	1.63870e6	0.152717
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.53455e6	0.324377	0.162189	−	0.986760i	$-0.448145\pi$
$653$	3.53455e6	0.324377	0.162189	+	0.986760i	$0.448145\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−2.78813e6	−0.250092	−0.125046	−	0.992151i	$-0.539908\pi$
$659$	−2.78813e6	−0.250092	−0.125046	+	0.992151i	$0.539908\pi$
$660$	0	0
$661$	−7.98219e6	−0.710589	−0.355294	−	0.934754i	$-0.615619\pi$
$661$	−7.98219e6	−0.710589	−0.355294	+	0.934754i	$0.615619\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.59234e6	0.312653
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.84396e6	−0.586815
$672$	0	0
$673$	1.01108e7	0.860496	0.430248	−	0.902711i	$-0.358426\pi$
$673$	1.01108e7	0.860496	0.430248	+	0.902711i	$0.358426\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.25054e6	0.524138	0.262069	−	0.965049i	$-0.415595\pi$
$677$	6.25054e6	0.524138	0.262069	+	0.965049i	$0.415595\pi$
$678$	0	0
$679$	−1.26842e6	−0.105581
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.86733e7	1.53169	0.765843	−	0.643027i	$-0.222322\pi$
$683$	1.86733e7	1.53169	0.765843	+	0.643027i	$0.222322\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−7.80768e6	−0.626576
$690$	0	0
$691$	−1.93615e7	−1.54257	−0.771285	−	0.636490i	$-0.780386\pi$
$691$	−1.93615e7	−1.54257	−0.771285	+	0.636490i	$0.780386\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.00509e7	0.783649
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.67337e6	−0.512921	−0.256460	−	0.966555i	$-0.582556\pi$
$701$	−6.67337e6	−0.512921	−0.256460	+	0.966555i	$0.582556\pi$
$702$	0	0
$703$	−3.68502e6	−0.281223
$704$	0	0
$705$	0	0
$706$	0	0
$707$	442429.	0.0332885
$708$	0	0
$709$	4.97370e6	0.371590	0.185795	−	0.982589i	$-0.440514\pi$
$709$	4.97370e6	0.371590	0.185795	+	0.982589i	$0.440514\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−4.17228e6	−0.307362
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.76815e7	−1.27555	−0.637773	−	0.770225i	$-0.720144\pi$
$719$	−1.76815e7	−1.27555	−0.637773	+	0.770225i	$0.720144\pi$
$720$	0	0
$721$	330356.	0.0236670
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.84948e7	1.29782	0.648908	−	0.760867i	$-0.275226\pi$
$727$	1.84948e7	1.29782	0.648908	+	0.760867i	$0.275226\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.81945e7	−1.25935
$732$	0	0
$733$	1.93313e6	0.132893	0.0664464	−	0.997790i	$-0.478834\pi$
$733$	1.93313e6	0.132893	0.0664464	+	0.997790i	$0.478834\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.08311e6	−0.548163
$738$	0	0
$739$	1.17689e7	0.792730	0.396365	−	0.918093i	$-0.370271\pi$
$739$	1.17689e7	0.792730	0.396365	+	0.918093i	$0.370271\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.34755e6	−0.0895516	−0.0447758	−	0.998997i	$-0.514257\pi$
$743$	−1.34755e6	−0.0895516	−0.0447758	+	0.998997i	$0.514257\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.42762e6	0.158116
$750$	0	0
$751$	1.32950e7	0.860179	0.430089	−	0.902786i	$-0.358482\pi$
$751$	1.32950e7	0.860179	0.430089	+	0.902786i	$0.358482\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.12779e7	0.715302	0.357651	−	0.933855i	$-0.383578\pi$
$757$	1.12779e7	0.715302	0.357651	+	0.933855i	$0.383578\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−833469.	−0.0521708	−0.0260854	−	0.999660i	$-0.508304\pi$
$761$	−833469.	−0.0521708	−0.0260854	+	0.999660i	$0.508304\pi$
$762$	0	0
$763$	−2.64849e6	−0.164698
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.44774e6	−0.518504
$768$	0	0
$769$	1.29320e7	0.788590	0.394295	−	0.918984i	$-0.370989\pi$
$769$	1.29320e7	0.788590	0.394295	+	0.918984i	$0.370989\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	160225.	0.00964457	0.00482228	−	0.999988i	$-0.498465\pi$
$773$	160225.	0.00964457	0.00482228	+	0.999988i	$0.498465\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.79221e6	−0.164856
$780$	0	0
$781$	−7.40839e6	−0.434606
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.19076e7	−0.685309	−0.342654	−	0.939462i	$-0.611326\pi$
$787$	−1.19076e7	−0.685309	−0.342654	+	0.939462i	$0.611326\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−869054.	−0.0493862
$792$	0	0
$793$	3.52816e7	1.99235
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.19475e7	−1.22388	−0.611941	−	0.790903i	$-0.709611\pi$
$797$	−2.19475e7	−1.22388	−0.611941	+	0.790903i	$0.709611\pi$
$798$	0	0
$799$	−4.79893e7	−2.65936
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.08123e7	−0.591736
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.12570e7	1.14191	0.570953	−	0.820983i	$-0.306574\pi$
$809$	2.12570e7	1.14191	0.570953	+	0.820983i	$0.306574\pi$
$810$	0	0
$811$	−4.03970e6	−0.215674	−0.107837	−	0.994169i	$-0.534392\pi$
$811$	−4.03970e6	−0.215674	−0.107837	+	0.994169i	$0.534392\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5.05458e6	0.264929
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7.62547e6	−0.394829	−0.197414	−	0.980320i	$-0.563254\pi$
$821$	−7.62547e6	−0.394829	−0.197414	+	0.980320i	$0.563254\pi$
$822$	0	0
$823$	−2.50958e7	−1.29152	−0.645760	−	0.763541i	$-0.723459\pi$
$823$	−2.50958e7	−1.29152	−0.645760	+	0.763541i	$0.723459\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.79557e6	0.498042	0.249021	−	0.968498i	$-0.419891\pi$
$827$	9.79557e6	0.498042	0.249021	+	0.968498i	$0.419891\pi$
$828$	0	0
$829$	−1.01425e7	−0.512578	−0.256289	−	0.966600i	$-0.582500\pi$
$829$	−1.01425e7	−0.512578	−0.256289	+	0.966600i	$0.582500\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.67730e7	−1.33686
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.72767e7	−1.82824	−0.914119	−	0.405447i	$-0.867116\pi$
$839$	−3.72767e7	−1.82824	−0.914119	+	0.405447i	$0.867116\pi$
$840$	0	0
$841$	−458827.	−0.0223696
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.68495e6	0.0807009
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−6.61718e6	−0.313220
$852$	0	0
$853$	−1.56786e7	−0.737794	−0.368897	−	0.929470i	$-0.620265\pi$
$853$	−1.56786e7	−0.737794	−0.368897	+	0.929470i	$0.620265\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.90275e7	−1.35007	−0.675037	−	0.737784i	$-0.735872\pi$
$857$	−2.90275e7	−1.35007	−0.675037	+	0.737784i	$0.735872\pi$
$858$	0	0
$859$	−1.72480e7	−0.797546	−0.398773	−	0.917050i	$-0.630564\pi$
$859$	−1.72480e7	−0.797546	−0.398773	+	0.917050i	$0.630564\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.50082e7	−1.14302	−0.571511	−	0.820594i	$-0.693643\pi$
$863$	−2.50082e7	−1.14302	−0.571511	+	0.820594i	$0.693643\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.10617e7	0.496906
$870$	0	0
$871$	4.16696e7	1.86112
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.21258e7	0.532366	0.266183	−	0.963923i	$-0.414238\pi$
$877$	1.21258e7	0.532366	0.266183	+	0.963923i	$0.414238\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.32403e7	1.00879	0.504397	−	0.863472i	$-0.331715\pi$
$881$	2.32403e7	1.00879	0.504397	+	0.863472i	$0.331715\pi$
$882$	0	0
$883$	2.03793e7	0.879604	0.439802	−	0.898095i	$-0.355049\pi$
$883$	2.03793e7	0.879604	0.439802	+	0.898095i	$0.355049\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.73639e6	−0.330164	−0.165082	−	0.986280i	$-0.552789\pi$
$887$	−7.73639e6	−0.330164	−0.165082	+	0.986280i	$0.552789\pi$
$888$	0	0
$889$	−1.72459e6	−0.0731866
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.33318e7	0.559449
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.32896e7	−0.961085
$900$	0	0
$901$	−1.47867e7	−0.606818
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	3.25270e7	1.31288	0.656442	−	0.754377i	$-0.272061\pi$
$907$	3.25270e7	1.31288	0.656442	+	0.754377i	$0.272061\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.69652e7	0.677270	0.338635	−	0.940918i	$-0.390035\pi$
$911$	1.69652e7	0.677270	0.338635	+	0.940918i	$0.390035\pi$
$912$	0	0
$913$	9.04006e6	0.358917
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.03049e6	−0.0797402
$918$	0	0
$919$	−2.34135e7	−0.914485	−0.457243	−	0.889342i	$-0.651163\pi$
$919$	−2.34135e7	−0.914485	−0.457243	+	0.889342i	$0.651163\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.81913e7	1.47557
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.30115e7	1.25495	0.627475	−	0.778637i	$-0.284089\pi$
$929$	3.30115e7	1.25495	0.627475	+	0.778637i	$0.284089\pi$
$930$	0	0
$931$	7.43775e6	0.281234
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.91088e7	1.45521	0.727604	−	0.685997i	$-0.240634\pi$
$937$	3.91088e7	1.45521	0.727604	+	0.685997i	$0.240634\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	6.31958e6	0.232656	0.116328	−	0.993211i	$-0.462888\pi$
$941$	6.31958e6	0.232656	0.116328	+	0.993211i	$0.462888\pi$
$942$	0	0
$943$	−5.01397e6	−0.183613
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.24853e7	1.17710	0.588548	−	0.808462i	$-0.299700\pi$
$947$	3.24853e7	1.17710	0.588548	+	0.808462i	$0.299700\pi$
$948$	0	0
$949$	5.57389e7	2.00906
$950$	0	0
$951$	0	0
$952$	0	0
$953$	4.86404e6	0.173486	0.0867431	−	0.996231i	$-0.472354\pi$
$953$	4.86404e6	0.173486	0.0867431	+	0.996231i	$0.472354\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.97867e6	0.104586
$960$	0	0
$961$	−1.57974e6	−0.0551794
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.42868e7	−0.491324	−0.245662	−	0.969356i	$-0.579005\pi$
$967$	−1.42868e7	−0.491324	−0.245662	+	0.969356i	$0.579005\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.36593e7	0.464924	0.232462	−	0.972606i	$-0.425322\pi$
$971$	1.36593e7	0.464924	0.232462	+	0.972606i	$0.425322\pi$
$972$	0	0
$973$	−1.75601e6	−0.0594626
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.24945e7	−0.753945	−0.376972	−	0.926225i	$-0.623035\pi$
$977$	−2.24945e7	−0.753945	−0.376972	+	0.926225i	$0.623035\pi$
$978$	0	0
$979$	−8.89189e6	−0.296509
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.82995e7	0.934103	0.467051	−	0.884230i	$-0.345316\pi$
$983$	2.82995e7	0.934103	0.467051	+	0.884230i	$0.345316\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.07650e6	0.295072
$990$	0	0
$991$	4.01846e6	0.129980	0.0649898	−	0.997886i	$-0.479299\pi$
$991$	4.01846e6	0.129980	0.0649898	+	0.997886i	$0.479299\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−3.39350e7	−1.08121	−0.540604	−	0.841277i	$-0.681804\pi$
$997$	−3.39350e7	−1.08121	−0.540604	+	0.841277i	$0.681804\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.6.a.x.1.2		3
3.2	odd	2		300.6.a.j.1.2		3
5.2	odd	4		180.6.d.d.109.2		6
5.3	odd	4		180.6.d.d.109.1		6
5.4	even	2		900.6.a.w.1.2		3
15.2	even	4		60.6.d.a.49.3	✓	6
15.8	even	4		60.6.d.a.49.6	yes	6
15.14	odd	2		300.6.a.i.1.2		3
20.3	even	4		720.6.f.m.289.1		6
20.7	even	4		720.6.f.m.289.2		6
60.23	odd	4		240.6.f.d.49.3		6
60.47	odd	4		240.6.f.d.49.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.d.a.49.3	✓	6	15.2	even	4
60.6.d.a.49.6	yes	6	15.8	even	4
180.6.d.d.109.1		6	5.3	odd	4
180.6.d.d.109.2		6	5.2	odd	4
240.6.f.d.49.3		6	60.23	odd	4
240.6.f.d.49.6		6	60.47	odd	4
300.6.a.i.1.2		3	15.14	odd	2
300.6.a.j.1.2		3	3.2	odd	2
720.6.f.m.289.1		6	20.3	even	4
720.6.f.m.289.2		6	20.7	even	4
900.6.a.w.1.2		3	5.4	even	2
900.6.a.x.1.2		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	900.a (trivial)

\(f(q)\)	\(=\)	\(q-12.5817 q^{7} +O(q^{10})\)	\(q-12.5817 q^{7} -164.713 q^{11} +849.118 q^{13} +1608.11 q^{17} -446.747 q^{19} -802.223 q^{23} -4477.98 q^{29} +5200.90 q^{31} +8248.56 q^{37} +6250.10 q^{41} -11314.2 q^{43} -29842.0 q^{47} -16648.7 q^{49} -9195.04 q^{53} -9948.84 q^{59} +41550.9 q^{61} +49074.0 q^{67} +44977.6 q^{71} +65643.2 q^{73} +2072.36 q^{77} -67157.8 q^{79} -54883.8 q^{83} +53984.2 q^{89} -10683.3 q^{91} +100814. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 88 q^{7}+O(q^{10}) \) 3 * q + 88 * q^7	\( 3 q + 88 q^{7} - 148 q^{11} - 868 q^{17} + 3000 q^{19} + 1052 q^{23} - 7962 q^{29} + 132 q^{31} + 11944 q^{37} - 8170 q^{41} + 26188 q^{43} - 37892 q^{47} + 13827 q^{49} - 11076 q^{53} + 46228 q^{59} + 3126 q^{61} + 126332 q^{67} + 80400 q^{71} + 42904 q^{73} - 234608 q^{77} - 64476 q^{79} - 126268 q^{83} + 38030 q^{89} - 49200 q^{91} + 331016 q^{97}+O(q^{100}) \) 3 * q + 88 * q^7 - 148 * q^11 - 868 * q^17 + 3000 * q^19 + 1052 * q^23 - 7962 * q^29 + 132 * q^31 + 11944 * q^37 - 8170 * q^41 + 26188 * q^43 - 37892 * q^47 + 13827 * q^49 - 11076 * q^53 + 46228 * q^59 + 3126 * q^61 + 126332 * q^67 + 80400 * q^71 + 42904 * q^73 - 234608 * q^77 - 64476 * q^79 - 126268 * q^83 + 38030 * q^89 - 49200 * q^91 + 331016 * q^97

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