LMFDB - Embedded newform 900.6.a.n.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ 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)

$f(q)$	$=$	$q-169.745 q^{7} +O(q^{10})$	$q-169.745 q^{7} +290.235 q^{11} -639.980 q^{13} +129.765 q^{17} +971.196 q^{19} +849.765 q^{23} +8049.53 q^{29} +5300.25 q^{31} -4515.41 q^{37} -9063.76 q^{41} -1527.39 q^{43} -19055.6 q^{47} +12006.4 q^{49} +36584.9 q^{53} -16973.8 q^{59} +24534.3 q^{61} +49557.2 q^{67} -45154.1 q^{71} -29221.3 q^{73} -49266.0 q^{77} +13479.1 q^{79} -81951.4 q^{83} -67754.4 q^{89} +108634. q^{91} +101821. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 22 q^{7}+O(q^{10}) $ 2 * q - 22 * q^7	$ 2 q - 22 q^{7} - 372 q^{11} - 10 q^{13} + 1212 q^{17} - 1550 q^{19} + 2652 q^{23} + 1812 q^{29} + 10918 q^{31} - 3316 q^{37} - 19080 q^{41} - 12262 q^{43} - 15252 q^{47} + 17028 q^{49} + 20784 q^{53} - 18708 q^{59} + 35734 q^{61} + 98162 q^{67} - 114120 q^{71} - 109876 q^{73} - 147108 q^{77} + 95536 q^{79} + 3732 q^{83} - 93600 q^{89} + 201710 q^{91} + 91886 q^{97}+O(q^{100}) $ 2 * q - 22 * q^7 - 372 * q^11 - 10 * q^13 + 1212 * q^17 - 1550 * q^19 + 2652 * q^23 + 1812 * q^29 + 10918 * q^31 - 3316 * q^37 - 19080 * q^41 - 12262 * q^43 - 15252 * q^47 + 17028 * q^49 + 20784 * q^53 - 18708 * q^59 + 35734 * q^61 + 98162 * q^67 - 114120 * q^71 - 109876 * q^73 - 147108 * q^77 + 95536 * q^79 + 3732 * q^83 - 93600 * q^89 + 201710 * q^91 + 91886 * 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
$7$	−169.745	−1.30934	−0.654669	−	0.755915i	$-0.727192\pi$
$7$	−169.745	−1.30934	−0.654669	+	0.755915i	$0.727192\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	290.235	0.723217	0.361608	−	0.932330i	$-0.382228\pi$
$11$	290.235	0.723217	0.361608	+	0.932330i	$0.382228\pi$
$12$	0	0
$13$	−639.980	−1.05029	−0.525144	−	0.851014i	$-0.675988\pi$
$13$	−639.980	−1.05029	−0.525144	+	0.851014i	$0.675988\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	129.765	0.108902	0.0544508	−	0.998516i	$-0.482659\pi$
$17$	129.765	0.108902	0.0544508	+	0.998516i	$0.482659\pi$
$18$	0	0
$19$	971.196	0.617196	0.308598	−	0.951193i	$-0.400140\pi$
$19$	971.196	0.617196	0.308598	+	0.951193i	$0.400140\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	849.765	0.334949	0.167475	−	0.985876i	$-0.446439\pi$
$23$	849.765	0.334949	0.167475	+	0.985876i	$0.446439\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8049.53	1.77736	0.888680	−	0.458528i	$-0.151623\pi$
$29$	8049.53	1.77736	0.888680	+	0.458528i	$0.151623\pi$
$30$	0	0
$31$	5300.25	0.990587	0.495293	−	0.868726i	$-0.335061\pi$
$31$	5300.25	0.990587	0.495293	+	0.868726i	$0.335061\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4515.41	−0.542242	−0.271121	−	0.962545i	$-0.587394\pi$
$37$	−4515.41	−0.542242	−0.271121	+	0.962545i	$0.587394\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9063.76	−0.842072	−0.421036	−	0.907044i	$-0.638333\pi$
$41$	−9063.76	−0.842072	−0.421036	+	0.907044i	$0.638333\pi$
$42$	0	0
$43$	−1527.39	−0.125974	−0.0629868	−	0.998014i	$-0.520063\pi$
$43$	−1527.39	−0.125974	−0.0629868	+	0.998014i	$0.520063\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−19055.6	−1.25829	−0.629143	−	0.777290i	$-0.716594\pi$
$47$	−19055.6	−1.25829	−0.629143	+	0.777290i	$0.716594\pi$
$48$	0	0
$49$	12006.4	0.714369
$50$	0	0
$51$	0	0
$52$	0	0
$53$	36584.9	1.78901	0.894505	−	0.447058i	$-0.147528\pi$
$53$	36584.9	1.78901	0.894505	+	0.447058i	$0.147528\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−16973.8	−0.634816	−0.317408	−	0.948289i	$-0.602813\pi$
$59$	−16973.8	−0.634816	−0.317408	+	0.948289i	$0.602813\pi$
$60$	0	0
$61$	24534.3	0.844207	0.422104	−	0.906548i	$-0.361292\pi$
$61$	24534.3	0.844207	0.422104	+	0.906548i	$0.361292\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	49557.2	1.34871	0.674357	−	0.738405i	$-0.264421\pi$
$67$	49557.2	1.34871	0.674357	+	0.738405i	$0.264421\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−45154.1	−1.06304	−0.531522	−	0.847044i	$-0.678380\pi$
$71$	−45154.1	−1.06304	−0.531522	+	0.847044i	$0.678380\pi$
$72$	0	0
$73$	−29221.3	−0.641789	−0.320895	−	0.947115i	$-0.603984\pi$
$73$	−29221.3	−0.641789	−0.320895	+	0.947115i	$0.603984\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−49266.0	−0.946936
$78$	0	0
$79$	13479.1	0.242992	0.121496	−	0.992592i	$-0.461231\pi$
$79$	13479.1	0.242992	0.121496	+	0.992592i	$0.461231\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−81951.4	−1.30575	−0.652877	−	0.757464i	$-0.726438\pi$
$83$	−81951.4	−1.30575	−0.652877	+	0.757464i	$0.726438\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−67754.4	−0.906697	−0.453348	−	0.891333i	$-0.649771\pi$
$89$	−67754.4	−0.906697	−0.453348	+	0.891333i	$0.649771\pi$
$90$	0	0
$91$	108634.	1.37518
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	101821.	1.09878	0.549388	−	0.835567i	$-0.314861\pi$
$97$	101821.	1.09878	0.549388	+	0.835567i	$0.314861\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−188000.	−1.83381	−0.916903	−	0.399111i	$-0.869319\pi$
$101$	−188000.	−1.83381	−0.916903	+	0.399111i	$0.869319\pi$
$102$	0	0
$103$	−182762.	−1.69743	−0.848715	−	0.528850i	$-0.822623\pi$
$103$	−182762.	−1.69743	−0.848715	+	0.528850i	$0.822623\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	43480.9	0.367146	0.183573	−	0.983006i	$-0.441234\pi$
$107$	43480.9	0.367146	0.183573	+	0.983006i	$0.441234\pi$
$108$	0	0
$109$	77373.9	0.623775	0.311888	−	0.950119i	$-0.399039\pi$
$109$	77373.9	0.623775	0.311888	+	0.950119i	$0.399039\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	85548.0	0.630251	0.315126	−	0.949050i	$-0.397953\pi$
$113$	85548.0	0.630251	0.315126	+	0.949050i	$0.397953\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−22026.9	−0.142589
$120$	0	0
$121$	−76814.5	−0.476958
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−135885.	−0.747586	−0.373793	−	0.927512i	$-0.621943\pi$
$127$	−135885.	−0.747586	−0.373793	+	0.927512i	$0.621943\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−194884.	−0.992196	−0.496098	−	0.868267i	$-0.665234\pi$
$131$	−194884.	−0.992196	−0.496098	+	0.868267i	$0.665234\pi$
$132$	0	0
$133$	−164856.	−0.808118
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−305080.	−1.38871	−0.694356	−	0.719631i	$-0.744311\pi$
$137$	−305080.	−1.38871	−0.694356	+	0.719631i	$0.744311\pi$
$138$	0	0
$139$	−5844.46	−0.0256571	−0.0128286	−	0.999918i	$-0.504084\pi$
$139$	−5844.46	−0.0256571	−0.0128286	+	0.999918i	$0.504084\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−185745.	−0.759585
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	333916.	1.23217	0.616086	−	0.787679i	$-0.288718\pi$
$149$	333916.	1.23217	0.616086	+	0.787679i	$0.288718\pi$
$150$	0	0
$151$	311794.	1.11282	0.556411	−	0.830907i	$-0.312178\pi$
$151$	311794.	1.11282	0.556411	+	0.830907i	$0.312178\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−97454.7	−0.315539	−0.157770	−	0.987476i	$-0.550430\pi$
$157$	−97454.7	−0.315539	−0.157770	+	0.987476i	$0.550430\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−144243.	−0.438562
$162$	0	0
$163$	438778.	1.29353	0.646764	−	0.762690i	$-0.276122\pi$
$163$	438778.	1.29353	0.646764	+	0.762690i	$0.276122\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16203.3	0.0449585	0.0224792	−	0.999747i	$-0.492844\pi$
$167$	16203.3	0.0449585	0.0224792	+	0.999747i	$0.492844\pi$
$168$	0	0
$169$	38281.8	0.103104
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−178810.	−0.454230	−0.227115	−	0.973868i	$-0.572929\pi$
$173$	−178810.	−0.454230	−0.227115	+	0.973868i	$0.572929\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	743560.	1.73454	0.867269	−	0.497841i	$-0.165874\pi$
$179$	743560.	1.73454	0.867269	+	0.497841i	$0.165874\pi$
$180$	0	0
$181$	−757167.	−1.71789	−0.858945	−	0.512067i	$-0.828880\pi$
$181$	−757167.	−1.71789	−0.858945	+	0.512067i	$0.828880\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	37662.3	0.0787595
$188$	0	0
$189$	0	0
$190$	0	0
$191$	838607.	1.66332	0.831658	−	0.555288i	$-0.187392\pi$
$191$	838607.	1.66332	0.831658	+	0.555288i	$0.187392\pi$
$192$	0	0
$193$	−336567.	−0.650396	−0.325198	−	0.945646i	$-0.605431\pi$
$193$	−336567.	−0.650396	−0.325198	+	0.945646i	$0.605431\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−153157.	−0.281171	−0.140586	−	0.990069i	$-0.544899\pi$
$197$	−153157.	−0.281171	−0.140586	+	0.990069i	$0.544899\pi$
$198$	0	0
$199$	882692.	1.58007	0.790035	−	0.613062i	$-0.210062\pi$
$199$	882692.	1.58007	0.790035	+	0.613062i	$0.210062\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.36637e6	−2.32717
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	281875.	0.446366
$210$	0	0
$211$	−1.05268e6	−1.62776	−0.813881	−	0.581031i	$-0.802649\pi$
$211$	−1.05268e6	−1.62776	−0.813881	+	0.581031i	$0.802649\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−899692.	−1.29701
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−83046.9	−0.114378
$222$	0	0
$223$	−580506.	−0.781708	−0.390854	−	0.920453i	$-0.627820\pi$
$223$	−580506.	−0.781708	−0.390854	+	0.920453i	$0.627820\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	185542.	0.238989	0.119494	−	0.992835i	$-0.461873\pi$
$227$	185542.	0.238989	0.119494	+	0.992835i	$0.461873\pi$
$228$	0	0
$229$	−1.26781e6	−1.59759	−0.798795	−	0.601604i	$-0.794529\pi$
$229$	−1.26781e6	−1.59759	−0.798795	+	0.601604i	$0.794529\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.57668e6	−1.90263	−0.951313	−	0.308227i	$-0.900264\pi$
$233$	−1.57668e6	−1.90263	−0.951313	+	0.308227i	$0.900264\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−21421.6	−0.0242582	−0.0121291	−	0.999926i	$-0.503861\pi$
$239$	−21421.6	−0.0242582	−0.0121291	+	0.999926i	$0.503861\pi$
$240$	0	0
$241$	−444538.	−0.493022	−0.246511	−	0.969140i	$-0.579284\pi$
$241$	−444538.	−0.493022	−0.246511	+	0.969140i	$0.579284\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−621546.	−0.648233
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.61211e6	−1.61514	−0.807569	−	0.589773i	$-0.799217\pi$
$251$	−1.61211e6	−1.61514	−0.807569	+	0.589773i	$0.799217\pi$
$252$	0	0
$253$	246632.	0.242241
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8294.39	−0.00783343	−0.00391671	−	0.999992i	$-0.501247\pi$
$257$	−8294.39	−0.00783343	−0.00391671	+	0.999992i	$0.501247\pi$
$258$	0	0
$259$	766469.	0.709978
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.01603e6	−1.79724	−0.898622	−	0.438723i	$-0.855431\pi$
$263$	−2.01603e6	−1.79724	−0.898622	+	0.438723i	$0.855431\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.61429e6	1.36020	0.680098	−	0.733121i	$-0.261937\pi$
$269$	1.61429e6	1.36020	0.680098	+	0.733121i	$0.261937\pi$
$270$	0	0
$271$	−1.77120e6	−1.46502	−0.732510	−	0.680756i	$-0.761651\pi$
$271$	−1.77120e6	−1.46502	−0.732510	+	0.680756i	$0.761651\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	683027.	0.534858	0.267429	−	0.963578i	$-0.413826\pi$
$277$	683027.	0.534858	0.267429	+	0.963578i	$0.413826\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−549974.	−0.415505	−0.207753	−	0.978181i	$-0.566615\pi$
$281$	−549974.	−0.415505	−0.207753	+	0.978181i	$0.566615\pi$
$282$	0	0
$283$	−10829.9	−0.00803820	−0.00401910	−	0.999992i	$-0.501279\pi$
$283$	−10829.9	−0.00803820	−0.00401910	+	0.999992i	$0.501279\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.53853e6	1.10256
$288$	0	0
$289$	−1.40302e6	−0.988140
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−631657.	−0.429845	−0.214923	−	0.976631i	$-0.568950\pi$
$293$	−631657.	−0.429845	−0.214923	+	0.976631i	$0.568950\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−543833.	−0.351793
$300$	0	0
$301$	259267.	0.164942
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2.43383e6	−1.47382	−0.736910	−	0.675991i	$-0.763716\pi$
$307$	−2.43383e6	−1.47382	−0.736910	+	0.675991i	$0.763716\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.04321e6	−0.611604	−0.305802	−	0.952095i	$-0.598925\pi$
$311$	−1.04321e6	−0.611604	−0.305802	+	0.952095i	$0.598925\pi$
$312$	0	0
$313$	−3.40290e6	−1.96331	−0.981655	−	0.190666i	$-0.938935\pi$
$313$	−3.40290e6	−1.96331	−0.981655	+	0.190666i	$0.938935\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.91165e6	−1.62739	−0.813695	−	0.581293i	$-0.802547\pi$
$317$	−2.91165e6	−1.62739	−0.813695	+	0.581293i	$0.802547\pi$
$318$	0	0
$319$	2.33626e6	1.28542
$320$	0	0
$321$	0	0
$322$	0	0
$323$	126027.	0.0672136
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.23460e6	1.64752
$330$	0	0
$331$	3.54116e6	1.77654	0.888272	−	0.459319i	$-0.151906\pi$
$331$	3.54116e6	1.77654	0.888272	+	0.459319i	$0.151906\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.88459e6	0.903946	0.451973	−	0.892032i	$-0.350720\pi$
$337$	1.88459e6	0.903946	0.451973	+	0.892032i	$0.350720\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.53832e6	0.716409
$342$	0	0
$343$	814880.	0.373988
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.65065e6	−0.735923	−0.367961	−	0.929841i	$-0.619944\pi$
$347$	−1.65065e6	−0.735923	−0.367961	+	0.929841i	$0.619944\pi$
$348$	0	0
$349$	2.90451e6	1.27647	0.638233	−	0.769843i	$-0.279665\pi$
$349$	2.90451e6	1.27647	0.638233	+	0.769843i	$0.279665\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.92279e6	−0.821287	−0.410644	−	0.911796i	$-0.634696\pi$
$353$	−1.92279e6	−0.821287	−0.410644	+	0.911796i	$0.634696\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−708712.	−0.290224	−0.145112	−	0.989415i	$-0.546354\pi$
$359$	−708712.	−0.290224	−0.145112	+	0.989415i	$0.546354\pi$
$360$	0	0
$361$	−1.53288e6	−0.619070
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.31973e6	0.899027	0.449514	−	0.893273i	$-0.351597\pi$
$367$	2.31973e6	0.899027	0.449514	+	0.893273i	$0.351597\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.21011e6	−2.34242
$372$	0	0
$373$	−482246.	−0.179472	−0.0897360	−	0.995966i	$-0.528602\pi$
$373$	−482246.	−0.179472	−0.0897360	+	0.995966i	$0.528602\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.15154e6	−1.86674
$378$	0	0
$379$	872200.	0.311902	0.155951	−	0.987765i	$-0.450156\pi$
$379$	872200.	0.311902	0.155951	+	0.987765i	$0.450156\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−300328.	−0.104616	−0.0523082	−	0.998631i	$-0.516658\pi$
$383$	−300328.	−0.104616	−0.0523082	+	0.998631i	$0.516658\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.50760e6	1.51033	0.755164	−	0.655536i	$-0.227557\pi$
$389$	4.50760e6	1.51033	0.755164	+	0.655536i	$0.227557\pi$
$390$	0	0
$391$	110270.	0.0364765
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.02564e6	0.326603	0.163302	−	0.986576i	$-0.447786\pi$
$397$	1.02564e6	0.326603	0.163302	+	0.986576i	$0.447786\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.50832e6	−0.778973	−0.389486	−	0.921032i	$-0.627347\pi$
$401$	−2.50832e6	−0.778973	−0.389486	+	0.921032i	$0.627347\pi$
$402$	0	0
$403$	−3.39206e6	−1.04040
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.31053e6	−0.392158
$408$	0	0
$409$	513908.	0.151907	0.0759534	−	0.997111i	$-0.475800\pi$
$409$	513908.	0.151907	0.0759534	+	0.997111i	$0.475800\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.88121e6	0.831190
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.81805e6	−1.34071	−0.670357	−	0.742039i	$-0.733859\pi$
$419$	−4.81805e6	−1.34071	−0.670357	+	0.742039i	$0.733859\pi$
$420$	0	0
$421$	4.57343e6	1.25758	0.628792	−	0.777573i	$-0.283550\pi$
$421$	4.57343e6	1.25758	0.628792	+	0.777573i	$0.283550\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.16458e6	−1.10535
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.62028e6	−1.19805	−0.599025	−	0.800730i	$-0.704445\pi$
$431$	−4.62028e6	−1.19805	−0.599025	+	0.800730i	$0.704445\pi$
$432$	0	0
$433$	−4.91309e6	−1.25932	−0.629658	−	0.776872i	$-0.716805\pi$
$433$	−4.91309e6	−1.25932	−0.629658	+	0.776872i	$0.716805\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	825288.	0.206729
$438$	0	0
$439$	3.37313e6	0.835357	0.417679	−	0.908595i	$-0.362844\pi$
$439$	3.37313e6	0.835357	0.417679	+	0.908595i	$0.362844\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.83759e6	0.444877	0.222438	−	0.974947i	$-0.428598\pi$
$443$	1.83759e6	0.444877	0.222438	+	0.974947i	$0.428598\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.39574e6	−1.02900	−0.514501	−	0.857490i	$-0.672023\pi$
$449$	−4.39574e6	−1.02900	−0.514501	+	0.857490i	$0.672023\pi$
$450$	0	0
$451$	−2.63062e6	−0.609000
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.41704e6	0.989329	0.494665	−	0.869084i	$-0.335291\pi$
$457$	4.41704e6	0.989329	0.494665	+	0.869084i	$0.335291\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.73823e6	0.819245	0.409623	−	0.912255i	$-0.365660\pi$
$461$	3.73823e6	0.819245	0.409623	+	0.912255i	$0.365660\pi$
$462$	0	0
$463$	3.61581e6	0.783886	0.391943	−	0.919990i	$-0.371803\pi$
$463$	3.61581e6	0.783886	0.391943	+	0.919990i	$0.371803\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.45043e6	0.944299	0.472150	−	0.881518i	$-0.343478\pi$
$467$	4.45043e6	0.944299	0.472150	+	0.881518i	$0.343478\pi$
$468$	0	0
$469$	−8.41210e6	−1.76592
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−443303.	−0.0911062
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.39917e6	−0.278632	−0.139316	−	0.990248i	$-0.544490\pi$
$479$	−1.39917e6	−0.278632	−0.139316	+	0.990248i	$0.544490\pi$
$480$	0	0
$481$	2.88977e6	0.569510
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.40058e6	0.649727	0.324863	−	0.945761i	$-0.394682\pi$
$487$	3.40058e6	0.649727	0.324863	+	0.945761i	$0.394682\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	534631.	0.100081	0.0500403	−	0.998747i	$-0.484065\pi$
$491$	534631.	0.100081	0.0500403	+	0.998747i	$0.484065\pi$
$492$	0	0
$493$	1.04455e6	0.193557
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.66469e6	1.39189
$498$	0	0
$499$	−636912.	−0.114506	−0.0572529	−	0.998360i	$-0.518234\pi$
$499$	−636912.	−0.114506	−0.0572529	+	0.998360i	$0.518234\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−8.63601e6	−1.52192	−0.760962	−	0.648796i	$-0.775273\pi$
$503$	−8.63601e6	−1.52192	−0.760962	+	0.648796i	$0.775273\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.40751e6	0.240801	0.120400	−	0.992725i	$-0.461582\pi$
$509$	1.40751e6	0.240801	0.120400	+	0.992725i	$0.461582\pi$
$510$	0	0
$511$	4.96017e6	0.840319
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5.53062e6	−0.910013
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.65164e6	0.589377	0.294689	−	0.955593i	$-0.404784\pi$
$521$	3.65164e6	0.589377	0.294689	+	0.955593i	$0.404784\pi$
$522$	0	0
$523$	7.20035e6	1.15106	0.575532	−	0.817779i	$-0.304795\pi$
$523$	7.20035e6	1.15106	0.575532	+	0.817779i	$0.304795\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	687786.	0.107877
$528$	0	0
$529$	−5.71424e6	−0.887809
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.80063e6	0.884417
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.48468e6	0.516643
$540$	0	0
$541$	−8.62099e6	−1.26638	−0.633190	−	0.773996i	$-0.718255\pi$
$541$	−8.62099e6	−1.26638	−0.633190	+	0.773996i	$0.718255\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−503512.	−0.0719518	−0.0359759	−	0.999353i	$-0.511454\pi$
$547$	−503512.	−0.0719518	−0.0359759	+	0.999353i	$0.511454\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.81767e6	1.09698
$552$	0	0
$553$	−2.28800e6	−0.318159
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.74889e6	−0.785138	−0.392569	−	0.919722i	$-0.628414\pi$
$557$	−5.74889e6	−0.785138	−0.392569	+	0.919722i	$0.628414\pi$
$558$	0	0
$559$	977501.	0.132309
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7.40870e6	0.985079	0.492540	−	0.870290i	$-0.336069\pi$
$563$	7.40870e6	0.985079	0.492540	+	0.870290i	$0.336069\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.09761e7	−1.42124	−0.710618	−	0.703578i	$-0.751585\pi$
$569$	−1.09761e7	−1.42124	−0.710618	+	0.703578i	$0.751585\pi$
$570$	0	0
$571$	−5.43630e6	−0.697771	−0.348885	−	0.937165i	$-0.613440\pi$
$571$	−5.43630e6	−0.697771	−0.348885	+	0.937165i	$0.613440\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−2.86923e6	−0.358778	−0.179389	−	0.983778i	$-0.557412\pi$
$577$	−2.86923e6	−0.358778	−0.179389	+	0.983778i	$0.557412\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.39108e7	1.70967
$582$	0	0
$583$	1.06182e7	1.29384
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.37986e7	1.65288	0.826439	−	0.563027i	$-0.190363\pi$
$587$	1.37986e7	1.65288	0.826439	+	0.563027i	$0.190363\pi$
$588$	0	0
$589$	5.14759e6	0.611386
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5.73216e6	0.669394	0.334697	−	0.942326i	$-0.391366\pi$
$593$	5.73216e6	0.669394	0.334697	+	0.942326i	$0.391366\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.32335e6	0.720078	0.360039	−	0.932937i	$-0.382763\pi$
$599$	6.32335e6	0.720078	0.360039	+	0.932937i	$0.382763\pi$
$600$	0	0
$601$	−1.32003e7	−1.49073	−0.745365	−	0.666657i	$-0.767725\pi$
$601$	−1.32003e7	−1.49073	−0.745365	+	0.666657i	$0.767725\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−736256.	−0.0811067	−0.0405534	−	0.999177i	$-0.512912\pi$
$607$	−736256.	−0.0811067	−0.0405534	+	0.999177i	$0.512912\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.21952e7	1.32156
$612$	0	0
$613$	9.65998e6	1.03831	0.519153	−	0.854682i	$-0.326248\pi$
$613$	9.65998e6	1.03831	0.519153	+	0.854682i	$0.326248\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.73440e6	−0.712174	−0.356087	−	0.934453i	$-0.615889\pi$
$617$	−6.73440e6	−0.712174	−0.356087	+	0.934453i	$0.615889\pi$
$618$	0	0
$619$	1.27653e6	0.133908	0.0669539	−	0.997756i	$-0.478672\pi$
$619$	1.27653e6	0.133908	0.0669539	+	0.997756i	$0.478672\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.15010e7	1.18717
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−585941.	−0.0590510
$630$	0	0
$631$	2.70192e6	0.270147	0.135073	−	0.990836i	$-0.456873\pi$
$631$	2.70192e6	0.270147	0.135073	+	0.990836i	$0.456873\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−7.68385e6	−0.750292
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.37157e6	0.131848	0.0659241	−	0.997825i	$-0.479000\pi$
$641$	1.37157e6	0.131848	0.0659241	+	0.997825i	$0.479000\pi$
$642$	0	0
$643$	2.83409e6	0.270325	0.135162	−	0.990823i	$-0.456844\pi$
$643$	2.83409e6	0.270325	0.135162	+	0.990823i	$0.456844\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.73438e6	0.444633	0.222317	−	0.974975i	$-0.428638\pi$
$647$	4.73438e6	0.444633	0.222317	+	0.974975i	$0.428638\pi$
$648$	0	0
$649$	−4.92638e6	−0.459110
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.17931e6	−0.383549	−0.191775	−	0.981439i	$-0.561424\pi$
$653$	−4.17931e6	−0.383549	−0.191775	+	0.981439i	$0.561424\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.28157e7	1.14955	0.574776	−	0.818311i	$-0.305089\pi$
$659$	1.28157e7	1.14955	0.574776	+	0.818311i	$0.305089\pi$
$660$	0	0
$661$	1.01872e7	0.906884	0.453442	−	0.891286i	$-0.350196\pi$
$661$	1.01872e7	0.906884	0.453442	+	0.891286i	$0.350196\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.84021e6	0.595326
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.12072e6	0.610545
$672$	0	0
$673$	1.60486e7	1.36584	0.682919	−	0.730494i	$-0.260710\pi$
$673$	1.60486e7	1.36584	0.682919	+	0.730494i	$0.260710\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.02747e7	−0.861584	−0.430792	−	0.902451i	$-0.641766\pi$
$677$	−1.02747e7	−0.861584	−0.430792	+	0.902451i	$0.641766\pi$
$678$	0	0
$679$	−1.72837e7	−1.43867
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.00989e7	0.828365	0.414182	−	0.910194i	$-0.364068\pi$
$683$	1.00989e7	0.828365	0.414182	+	0.910194i	$0.364068\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.34136e7	−1.87897
$690$	0	0
$691$	−193942.	−0.0154517	−0.00772586	−	0.999970i	$-0.502459\pi$
$691$	−193942.	−0.0154517	−0.00772586	+	0.999970i	$0.502459\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.17616e6	−0.0917030
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.61596e7	1.24204	0.621020	−	0.783795i	$-0.286719\pi$
$701$	1.61596e7	1.24204	0.621020	+	0.783795i	$0.286719\pi$
$702$	0	0
$703$	−4.38535e6	−0.334669
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.19120e7	2.40107
$708$	0	0
$709$	−1.11955e7	−0.836426	−0.418213	−	0.908349i	$-0.637343\pi$
$709$	−1.11955e7	−0.836426	−0.418213	+	0.908349i	$0.637343\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4.50397e6	0.331796
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.35356e7	−0.976463	−0.488232	−	0.872714i	$-0.662358\pi$
$719$	−1.35356e7	−0.976463	−0.488232	+	0.872714i	$0.662358\pi$
$720$	0	0
$721$	3.10229e7	2.22251
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.18185e7	0.829326	0.414663	−	0.909975i	$-0.363899\pi$
$727$	1.18185e7	0.829326	0.414663	+	0.909975i	$0.363899\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−198202.	−0.0137187
$732$	0	0
$733$	5.35463e6	0.368103	0.184052	−	0.982917i	$-0.441079\pi$
$733$	5.35463e6	0.368103	0.184052	+	0.982917i	$0.441079\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.43833e7	0.975413
$738$	0	0
$739$	1.20704e7	0.813037	0.406518	−	0.913643i	$-0.366743\pi$
$739$	1.20704e7	0.813037	0.406518	+	0.913643i	$0.366743\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.18812e7	1.45412	0.727059	−	0.686575i	$-0.240887\pi$
$743$	2.18812e7	1.45412	0.727059	+	0.686575i	$0.240887\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−7.38067e6	−0.480719
$750$	0	0
$751$	−1.04766e7	−0.677830	−0.338915	−	0.940817i	$-0.610060\pi$
$751$	−1.04766e7	−0.677830	−0.338915	+	0.940817i	$0.610060\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−8.68279e6	−0.550706	−0.275353	−	0.961343i	$-0.588795\pi$
$757$	−8.68279e6	−0.550706	−0.275353	+	0.961343i	$0.588795\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.02185e7	−1.26558	−0.632788	−	0.774325i	$-0.718090\pi$
$761$	−2.02185e7	−1.26558	−0.632788	+	0.774325i	$0.718090\pi$
$762$	0	0
$763$	−1.31338e7	−0.816733
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.08629e7	0.666740
$768$	0	0
$769$	−3.99032e6	−0.243328	−0.121664	−	0.992571i	$-0.538823\pi$
$769$	−3.99032e6	−0.243328	−0.121664	+	0.992571i	$0.538823\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.81960e7	1.09529	0.547644	−	0.836711i	$-0.315525\pi$
$773$	1.81960e7	1.09529	0.547644	+	0.836711i	$0.315525\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8.80269e6	−0.519723
$780$	0	0
$781$	−1.31053e7	−0.768812
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3.02382e7	1.74028	0.870141	−	0.492803i	$-0.164028\pi$
$787$	3.02382e7	1.74028	0.870141	+	0.492803i	$0.164028\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.45214e7	−0.825213
$792$	0	0
$793$	−1.57015e7	−0.886660
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.02752e7	−1.13063	−0.565315	−	0.824875i	$-0.691245\pi$
$797$	−2.02752e7	−1.13063	−0.565315	+	0.824875i	$0.691245\pi$
$798$	0	0
$799$	−2.47275e6	−0.137029
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−8.48105e6	−0.464153
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−2.70426e7	−1.45271	−0.726353	−	0.687321i	$-0.758786\pi$
$809$	−2.70426e7	−1.45271	−0.726353	+	0.687321i	$0.758786\pi$
$810$	0	0
$811$	2.52093e7	1.34589	0.672944	−	0.739693i	$-0.265029\pi$
$811$	2.52093e7	1.34589	0.672944	+	0.739693i	$0.265029\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1.48340e6	−0.0777504
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7.52947e6	−0.389858	−0.194929	−	0.980817i	$-0.562448\pi$
$821$	−7.52947e6	−0.389858	−0.194929	+	0.980817i	$0.562448\pi$
$822$	0	0
$823$	−9.12105e6	−0.469402	−0.234701	−	0.972068i	$-0.575411\pi$
$823$	−9.12105e6	−0.469402	−0.234701	+	0.972068i	$0.575411\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.21234e7	−1.63327	−0.816634	−	0.577156i	$-0.804162\pi$
$827$	−3.21234e7	−1.63327	−0.816634	+	0.577156i	$0.804162\pi$
$828$	0	0
$829$	2.06186e7	1.04201	0.521007	−	0.853552i	$-0.325557\pi$
$829$	2.06186e7	1.04201	0.521007	+	0.853552i	$0.325557\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.55801e6	0.0777959
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−8.39078e6	−0.411526	−0.205763	−	0.978602i	$-0.565968\pi$
$839$	−8.39078e6	−0.411526	−0.205763	+	0.978602i	$0.565968\pi$
$840$	0	0
$841$	4.42838e7	2.15901
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.30389e7	0.624499
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.83704e6	−0.181624
$852$	0	0
$853$	2.50001e7	1.17644	0.588220	−	0.808701i	$-0.299829\pi$
$853$	2.50001e7	1.17644	0.588220	+	0.808701i	$0.299829\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.65503e7	−1.23486	−0.617429	−	0.786627i	$-0.711826\pi$
$857$	−2.65503e7	−1.23486	−0.617429	+	0.786627i	$0.711826\pi$
$858$	0	0
$859$	−2.88348e7	−1.33332	−0.666659	−	0.745363i	$-0.732276\pi$
$859$	−2.88348e7	−1.33332	−0.666659	+	0.745363i	$0.732276\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.92889e7	1.33868	0.669340	−	0.742956i	$-0.266577\pi$
$863$	2.92889e7	1.33868	0.669340	+	0.742956i	$0.266577\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.91210e6	0.175736
$870$	0	0
$871$	−3.17157e7	−1.41654
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.64075e7	−0.720352	−0.360176	−	0.932884i	$-0.617283\pi$
$877$	−1.64075e7	−0.720352	−0.360176	+	0.932884i	$0.617283\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.60570e7	−0.696986	−0.348493	−	0.937311i	$-0.613306\pi$
$881$	−1.60570e7	−0.696986	−0.348493	+	0.937311i	$0.613306\pi$
$882$	0	0
$883$	3.68062e6	0.158862	0.0794309	−	0.996840i	$-0.474690\pi$
$883$	3.68062e6	0.158862	0.0794309	+	0.996840i	$0.474690\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.66570e6	−0.113763	−0.0568817	−	0.998381i	$-0.518116\pi$
$887$	−2.66570e6	−0.113763	−0.0568817	+	0.998381i	$0.518116\pi$
$888$	0	0
$889$	2.30658e7	0.978844
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.85068e7	−0.776608
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.26646e7	1.76063
$900$	0	0
$901$	4.74744e6	0.194826
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	7.78292e6	0.314141	0.157070	−	0.987587i	$-0.449795\pi$
$907$	7.78292e6	0.314141	0.157070	+	0.987587i	$0.449795\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−933741.	−0.0372761	−0.0186381	−	0.999826i	$-0.505933\pi$
$911$	−933741.	−0.0372761	−0.0186381	+	0.999826i	$0.505933\pi$
$912$	0	0
$913$	−2.37852e7	−0.944343
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.30806e7	1.29912
$918$	0	0
$919$	3.80765e7	1.48720	0.743599	−	0.668626i	$-0.233117\pi$
$919$	3.80765e7	1.48720	0.743599	+	0.668626i	$0.233117\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.88977e7	1.11650
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.54032e7	−0.965715	−0.482858	−	0.875699i	$-0.660401\pi$
$929$	−2.54032e7	−0.965715	−0.482858	+	0.875699i	$0.660401\pi$
$930$	0	0
$931$	1.16606e7	0.440905
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.04763e7	1.50609	0.753047	−	0.657966i	$-0.228583\pi$
$937$	4.04763e7	1.50609	0.753047	+	0.657966i	$0.228583\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.37903e7	1.61214	0.806072	−	0.591817i	$-0.201589\pi$
$941$	4.37903e7	1.61214	0.806072	+	0.591817i	$0.201589\pi$
$942$	0	0
$943$	−7.70207e6	−0.282051
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.60998e7	−0.583373	−0.291687	−	0.956514i	$-0.594216\pi$
$947$	−1.60998e7	−0.583373	−0.291687	+	0.956514i	$0.594216\pi$
$948$	0	0
$949$	1.87011e7	0.674063
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.23728e7	−1.15465	−0.577323	−	0.816516i	$-0.695902\pi$
$953$	−3.23728e7	−1.15465	−0.577323	+	0.816516i	$0.695902\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.17858e7	1.81830
$960$	0	0
$961$	−536449.	−0.0187379
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−2.08002e7	−0.715322	−0.357661	−	0.933851i	$-0.616426\pi$
$967$	−2.08002e7	−0.715322	−0.357661	+	0.933851i	$0.616426\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−2.46372e7	−0.838576	−0.419288	−	0.907853i	$-0.637720\pi$
$971$	−2.46372e7	−0.838576	−0.419288	+	0.907853i	$0.637720\pi$
$972$	0	0
$973$	992069.	0.0335938
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.92231e7	−0.979469	−0.489734	−	0.871872i	$-0.662906\pi$
$977$	−2.92231e7	−0.979469	−0.489734	+	0.871872i	$0.662906\pi$
$978$	0	0
$979$	−1.96647e7	−0.655738
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4.62575e7	1.52686	0.763429	−	0.645892i	$-0.223514\pi$
$983$	4.62575e7	1.52686	0.763429	+	0.645892i	$0.223514\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.29792e6	−0.0421948
$990$	0	0
$991$	7.99315e6	0.258544	0.129272	−	0.991609i	$-0.458736\pi$
$991$	7.99315e6	0.258544	0.129272	+	0.991609i	$0.458736\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−4.93207e7	−1.57142	−0.785708	−	0.618597i	$-0.787701\pi$
$997$	−4.93207e7	−1.57142	−0.785708	+	0.618597i	$0.787701\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.6.a.n.1.1		2
3.2	odd	2		300.6.a.g.1.1	✓	2
5.2	odd	4		900.6.d.j.649.1		4
5.3	odd	4		900.6.d.j.649.4		4
5.4	even	2		900.6.a.r.1.2		2
15.2	even	4		300.6.d.f.49.3		4
15.8	even	4		300.6.d.f.49.2		4
15.14	odd	2		300.6.a.h.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.6.a.g.1.1	✓	2	3.2	odd	2
300.6.a.h.1.2	yes	2	15.14	odd	2
300.6.d.f.49.2		4	15.8	even	4
300.6.d.f.49.3		4	15.2	even	4
900.6.a.n.1.1		2	1.1	even	1	trivial
900.6.a.r.1.2		2	5.4	even	2
900.6.d.j.649.1		4	5.2	odd	4
900.6.d.j.649.4		4	5.3	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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	900.a (trivial)

\(f(q)\)	\(=\)	\(q-169.745 q^{7} +O(q^{10})\)	\(q-169.745 q^{7} +290.235 q^{11} -639.980 q^{13} +129.765 q^{17} +971.196 q^{19} +849.765 q^{23} +8049.53 q^{29} +5300.25 q^{31} -4515.41 q^{37} -9063.76 q^{41} -1527.39 q^{43} -19055.6 q^{47} +12006.4 q^{49} +36584.9 q^{53} -16973.8 q^{59} +24534.3 q^{61} +49557.2 q^{67} -45154.1 q^{71} -29221.3 q^{73} -49266.0 q^{77} +13479.1 q^{79} -81951.4 q^{83} -67754.4 q^{89} +108634. q^{91} +101821. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 22 q^{7}+O(q^{10}) \) 2 * q - 22 * q^7	\( 2 q - 22 q^{7} - 372 q^{11} - 10 q^{13} + 1212 q^{17} - 1550 q^{19} + 2652 q^{23} + 1812 q^{29} + 10918 q^{31} - 3316 q^{37} - 19080 q^{41} - 12262 q^{43} - 15252 q^{47} + 17028 q^{49} + 20784 q^{53} - 18708 q^{59} + 35734 q^{61} + 98162 q^{67} - 114120 q^{71} - 109876 q^{73} - 147108 q^{77} + 95536 q^{79} + 3732 q^{83} - 93600 q^{89} + 201710 q^{91} + 91886 q^{97}+O(q^{100}) \) 2 * q - 22 * q^7 - 372 * q^11 - 10 * q^13 + 1212 * q^17 - 1550 * q^19 + 2652 * q^23 + 1812 * q^29 + 10918 * q^31 - 3316 * q^37 - 19080 * q^41 - 12262 * q^43 - 15252 * q^47 + 17028 * q^49 + 20784 * q^53 - 18708 * q^59 + 35734 * q^61 + 98162 * q^67 - 114120 * q^71 - 109876 * q^73 - 147108 * q^77 + 95536 * q^79 + 3732 * q^83 - 93600 * q^89 + 201710 * q^91 + 91886 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more