LMFDB - Embedded newform 504.6.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,6,Mod(1,504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("504.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	504.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:	$80.8334451857$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1129}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 282 $ x^2 - x - 282
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-16.3003$ of defining polynomial
Character	$\chi$	$=$	504.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+67.2012 q^{5} -49.0000 q^{7} +O(q^{10})$	$q+67.2012 q^{5} -49.0000 q^{7} -117.201 q^{11} -267.610 q^{13} -374.396 q^{17} +2681.62 q^{19} -4634.86 q^{23} +1391.00 q^{25} -2514.04 q^{29} -2858.47 q^{31} -3292.86 q^{35} -10078.9 q^{37} -14085.7 q^{41} +14615.3 q^{43} +11153.1 q^{47} +2401.00 q^{49} -34902.3 q^{53} -7876.06 q^{55} +28830.9 q^{59} +30231.0 q^{61} -17983.7 q^{65} +40476.2 q^{67} -45618.5 q^{71} +4003.90 q^{73} +5742.86 q^{77} -82327.7 q^{79} -7535.39 q^{83} -25159.9 q^{85} -46686.5 q^{89} +13112.9 q^{91} +180208. q^{95} +16776.0 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 98 q^{7}+O(q^{10}) $ 2 * q - 98 * q^7	$ 2 q - 98 q^{7} - 100 q^{11} + 540 q^{13} - 1152 q^{17} + 2944 q^{19} - 2684 q^{23} + 2782 q^{25} - 996 q^{29} + 2616 q^{31} - 3492 q^{37} - 17016 q^{41} + 13640 q^{43} - 11832 q^{47} + 4802 q^{49} - 37548 q^{53} - 9032 q^{55} + 6320 q^{59} + 39764 q^{61} - 72256 q^{65} - 27376 q^{67} - 35460 q^{71} + 64188 q^{73} + 4900 q^{77} - 110088 q^{79} + 5896 q^{83} + 27096 q^{85} - 167160 q^{89} - 26460 q^{91} + 162576 q^{95} - 11876 q^{97}+O(q^{100}) $ 2 * q - 98 * q^7 - 100 * q^11 + 540 * q^13 - 1152 * q^17 + 2944 * q^19 - 2684 * q^23 + 2782 * q^25 - 996 * q^29 + 2616 * q^31 - 3492 * q^37 - 17016 * q^41 + 13640 * q^43 - 11832 * q^47 + 4802 * q^49 - 37548 * q^53 - 9032 * q^55 + 6320 * q^59 + 39764 * q^61 - 72256 * q^65 - 27376 * q^67 - 35460 * q^71 + 64188 * q^73 + 4900 * q^77 - 110088 * q^79 + 5896 * q^83 + 27096 * q^85 - 167160 * q^89 - 26460 * q^91 + 162576 * q^95 - 11876 * 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
$5$	67.2012	1.20213	0.601066	−	0.799200i	$-0.294743\pi$
$5$	67.2012	1.20213	0.601066	+	0.799200i	$0.294743\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−117.201	−0.292045	−0.146023	−	0.989281i	$-0.546647\pi$
$11$	−117.201	−0.292045	−0.146023	+	0.989281i	$0.546647\pi$
$12$	0	0
$13$	−267.610	−0.439181	−0.219590	−	0.975592i	$-0.570472\pi$
$13$	−267.610	−0.439181	−0.219590	+	0.975592i	$0.570472\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−374.396	−0.314202	−0.157101	−	0.987583i	$-0.550215\pi$
$17$	−374.396	−0.314202	−0.157101	+	0.987583i	$0.550215\pi$
$18$	0	0
$19$	2681.62	1.70417	0.852086	−	0.523402i	$-0.175337\pi$
$19$	2681.62	1.70417	0.852086	+	0.523402i	$0.175337\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4634.86	−1.82691	−0.913454	−	0.406941i	$-0.866595\pi$
$23$	−4634.86	−1.82691	−0.913454	+	0.406941i	$0.866595\pi$
$24$	0	0
$25$	1391.00	0.445120
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2514.04	−0.555107	−0.277553	−	0.960710i	$-0.589524\pi$
$29$	−2514.04	−0.555107	−0.277553	+	0.960710i	$0.589524\pi$
$30$	0	0
$31$	−2858.47	−0.534232	−0.267116	−	0.963664i	$-0.586071\pi$
$31$	−2858.47	−0.534232	−0.267116	+	0.963664i	$0.586071\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3292.86	−0.454363
$36$	0	0
$37$	−10078.9	−1.21035	−0.605175	−	0.796093i	$-0.706897\pi$
$37$	−10078.9	−1.21035	−0.605175	+	0.796093i	$0.706897\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−14085.7	−1.30864	−0.654318	−	0.756220i	$-0.727044\pi$
$41$	−14085.7	−1.30864	−0.654318	+	0.756220i	$0.727044\pi$
$42$	0	0
$43$	14615.3	1.20542	0.602709	−	0.797961i	$-0.294088\pi$
$43$	14615.3	1.20542	0.602709	+	0.797961i	$0.294088\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11153.1	0.736463	0.368232	−	0.929734i	$-0.379963\pi$
$47$	11153.1	0.736463	0.368232	+	0.929734i	$0.379963\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−34902.3	−1.70673	−0.853364	−	0.521316i	$-0.825441\pi$
$53$	−34902.3	−1.70673	−0.853364	+	0.521316i	$0.825441\pi$
$54$	0	0
$55$	−7876.06	−0.351077
$56$	0	0
$57$	0	0
$58$	0	0
$59$	28830.9	1.07827	0.539135	−	0.842219i	$-0.318751\pi$
$59$	28830.9	1.07827	0.539135	+	0.842219i	$0.318751\pi$
$60$	0	0
$61$	30231.0	1.04023	0.520113	−	0.854097i	$-0.325890\pi$
$61$	30231.0	1.04023	0.520113	+	0.854097i	$0.325890\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−17983.7	−0.527953
$66$	0	0
$67$	40476.2	1.10157	0.550785	−	0.834647i	$-0.314328\pi$
$67$	40476.2	1.10157	0.550785	+	0.834647i	$0.314328\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−45618.5	−1.07398	−0.536989	−	0.843589i	$-0.680438\pi$
$71$	−45618.5	−1.07398	−0.536989	+	0.843589i	$0.680438\pi$
$72$	0	0
$73$	4003.90	0.0879379	0.0439690	−	0.999033i	$-0.486000\pi$
$73$	4003.90	0.0879379	0.0439690	+	0.999033i	$0.486000\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5742.86	0.110383
$78$	0	0
$79$	−82327.7	−1.48415	−0.742076	−	0.670316i	$-0.766158\pi$
$79$	−82327.7	−1.48415	−0.742076	+	0.670316i	$0.766158\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−7535.39	−0.120063	−0.0600316	−	0.998196i	$-0.519120\pi$
$83$	−7535.39	−0.120063	−0.0600316	+	0.998196i	$0.519120\pi$
$84$	0	0
$85$	−25159.9	−0.377712
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−46686.5	−0.624765	−0.312383	−	0.949956i	$-0.601127\pi$
$89$	−46686.5	−0.624765	−0.312383	+	0.949956i	$0.601127\pi$
$90$	0	0
$91$	13112.9	0.165995
$92$	0	0
$93$	0	0
$94$	0	0
$95$	180208.	2.04864
$96$	0	0
$97$	16776.0	0.181034	0.0905168	−	0.995895i	$-0.471148\pi$
$97$	16776.0	0.181034	0.0905168	+	0.995895i	$0.471148\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	47377.8	0.462138	0.231069	−	0.972937i	$-0.425778\pi$
$101$	47377.8	0.462138	0.231069	+	0.972937i	$0.425778\pi$
$102$	0	0
$103$	−94613.3	−0.878738	−0.439369	−	0.898307i	$-0.644798\pi$
$103$	−94613.3	−0.878738	−0.439369	+	0.898307i	$0.644798\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	39361.1	0.332359	0.166180	−	0.986095i	$-0.446857\pi$
$107$	39361.1	0.332359	0.166180	+	0.986095i	$0.446857\pi$
$108$	0	0
$109$	−71562.4	−0.576924	−0.288462	−	0.957491i	$-0.593144\pi$
$109$	−71562.4	−0.576924	−0.288462	+	0.957491i	$0.593144\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−147347.	−1.08554	−0.542771	−	0.839881i	$-0.682625\pi$
$113$	−147347.	−1.08554	−0.542771	+	0.839881i	$0.682625\pi$
$114$	0	0
$115$	−311468.	−2.19618
$116$	0	0
$117$	0	0
$118$	0	0
$119$	18345.4	0.118757
$120$	0	0
$121$	−147315.	−0.914710
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−116527.	−0.667039
$126$	0	0
$127$	17624.9	0.0969654	0.0484827	−	0.998824i	$-0.484561\pi$
$127$	17624.9	0.0969654	0.0484827	+	0.998824i	$0.484561\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−85899.7	−0.437334	−0.218667	−	0.975800i	$-0.570171\pi$
$131$	−85899.7	−0.437334	−0.218667	+	0.975800i	$0.570171\pi$
$132$	0	0
$133$	−131399.	−0.644117
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−118301.	−0.538501	−0.269251	−	0.963070i	$-0.586776\pi$
$137$	−118301.	−0.538501	−0.269251	+	0.963070i	$0.586776\pi$
$138$	0	0
$139$	−356167.	−1.56357	−0.781783	−	0.623551i	$-0.785690\pi$
$139$	−356167.	−1.56357	−0.781783	+	0.623551i	$0.785690\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	31364.2	0.128261
$144$	0	0
$145$	−168946.	−0.667311
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−144959.	−0.534907	−0.267454	−	0.963571i	$-0.586182\pi$
$149$	−144959.	−0.534907	−0.267454	+	0.963571i	$0.586182\pi$
$150$	0	0
$151$	−427345.	−1.52523	−0.762617	−	0.646850i	$-0.776086\pi$
$151$	−427345.	−1.52523	−0.762617	+	0.646850i	$0.776086\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−192093.	−0.642217
$156$	0	0
$157$	343889.	1.11345	0.556724	−	0.830698i	$-0.312058\pi$
$157$	343889.	1.11345	0.556724	+	0.830698i	$0.312058\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	227108.	0.690507
$162$	0	0
$163$	38752.9	0.114244	0.0571222	−	0.998367i	$-0.481808\pi$
$163$	38752.9	0.114244	0.0571222	+	0.998367i	$0.481808\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	278212.	0.771942	0.385971	−	0.922511i	$-0.373866\pi$
$167$	278212.	0.771942	0.385971	+	0.922511i	$0.373866\pi$
$168$	0	0
$169$	−299678.	−0.807120
$170$	0	0
$171$	0	0
$172$	0	0
$173$	193333.	0.491124	0.245562	−	0.969381i	$-0.421028\pi$
$173$	193333.	0.491124	0.245562	+	0.969381i	$0.421028\pi$
$174$	0	0
$175$	−68159.0	−0.168240
$176$	0	0
$177$	0	0
$178$	0	0
$179$	22270.9	0.0519523	0.0259761	−	0.999663i	$-0.491731\pi$
$179$	22270.9	0.0519523	0.0259761	+	0.999663i	$0.491731\pi$
$180$	0	0
$181$	−208008.	−0.471936	−0.235968	−	0.971761i	$-0.575826\pi$
$181$	−208008.	−0.471936	−0.235968	+	0.971761i	$0.575826\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−677317.	−1.45500
$186$	0	0
$187$	43879.7	0.0917613
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−753867.	−1.49524	−0.747621	−	0.664126i	$-0.768804\pi$
$191$	−753867.	−1.49524	−0.747621	+	0.664126i	$0.768804\pi$
$192$	0	0
$193$	59536.9	0.115052	0.0575258	−	0.998344i	$-0.481679\pi$
$193$	59536.9	0.115052	0.0575258	+	0.998344i	$0.481679\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−159066.	−0.292020	−0.146010	−	0.989283i	$-0.546643\pi$
$197$	−159066.	−0.292020	−0.146010	+	0.989283i	$0.546643\pi$
$198$	0	0
$199$	800348.	1.43267	0.716335	−	0.697757i	$-0.245818\pi$
$199$	800348.	1.43267	0.716335	+	0.697757i	$0.245818\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	123188.	0.209811
$204$	0	0
$205$	−946576.	−1.57315
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−314289.	−0.497696
$210$	0	0
$211$	812277.	1.25602	0.628012	−	0.778204i	$-0.283869\pi$
$211$	812277.	1.25602	0.628012	+	0.778204i	$0.283869\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	982168.	1.44907
$216$	0	0
$217$	140065.	0.201921
$218$	0	0
$219$	0	0
$220$	0	0
$221$	100192.	0.137992
$222$	0	0
$223$	−14706.2	−0.0198033	−0.00990166	−	0.999951i	$-0.503152\pi$
$223$	−14706.2	−0.0198033	−0.00990166	+	0.999951i	$0.503152\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	847267.	1.09133	0.545665	−	0.838004i	$-0.316277\pi$
$227$	847267.	1.09133	0.545665	+	0.838004i	$0.316277\pi$
$228$	0	0
$229$	460043.	0.579708	0.289854	−	0.957071i	$-0.406393\pi$
$229$	460043.	0.579708	0.289854	+	0.957071i	$0.406393\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.03965e6	−1.25458	−0.627291	−	0.778785i	$-0.715836\pi$
$233$	−1.03965e6	−1.25458	−0.627291	+	0.778785i	$0.715836\pi$
$234$	0	0
$235$	749502.	0.885326
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−582709.	−0.659868	−0.329934	−	0.944004i	$-0.607027\pi$
$239$	−582709.	−0.659868	−0.329934	+	0.944004i	$0.607027\pi$
$240$	0	0
$241$	−729432.	−0.808988	−0.404494	−	0.914541i	$-0.632552\pi$
$241$	−729432.	−0.808988	−0.404494	+	0.914541i	$0.632552\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	161350.	0.171733
$246$	0	0
$247$	−717627.	−0.748439
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−795585.	−0.797081	−0.398540	−	0.917151i	$-0.630483\pi$
$251$	−795585.	−0.797081	−0.398540	+	0.917151i	$0.630483\pi$
$252$	0	0
$253$	543211.	0.533540
$254$	0	0
$255$	0	0
$256$	0	0
$257$	965362.	0.911712	0.455856	−	0.890054i	$-0.349333\pi$
$257$	965362.	0.911712	0.455856	+	0.890054i	$0.349333\pi$
$258$	0	0
$259$	493868.	0.457469
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.24337e6	−1.10843	−0.554217	−	0.832372i	$-0.686982\pi$
$263$	−1.24337e6	−1.10843	−0.554217	+	0.832372i	$0.686982\pi$
$264$	0	0
$265$	−2.34548e6	−2.05171
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.83372e6	−1.54509	−0.772544	−	0.634961i	$-0.781016\pi$
$269$	−1.83372e6	−1.54509	−0.772544	+	0.634961i	$0.781016\pi$
$270$	0	0
$271$	−1.22033e6	−1.00938	−0.504688	−	0.863302i	$-0.668393\pi$
$271$	−1.22033e6	−1.00938	−0.504688	+	0.863302i	$0.668393\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−163027.	−0.129995
$276$	0	0
$277$	1.24381e6	0.973990	0.486995	−	0.873405i	$-0.338093\pi$
$277$	1.24381e6	0.973990	0.486995	+	0.873405i	$0.338093\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.40312e6	−1.81555	−0.907776	−	0.419455i	$-0.862221\pi$
$281$	−2.40312e6	−1.81555	−0.907776	+	0.419455i	$0.862221\pi$
$282$	0	0
$283$	2.28221e6	1.69390	0.846952	−	0.531670i	$-0.178435\pi$
$283$	2.28221e6	1.69390	0.846952	+	0.531670i	$0.178435\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	690199.	0.494618
$288$	0	0
$289$	−1.27968e6	−0.901277
$290$	0	0
$291$	0	0
$292$	0	0
$293$	768737.	0.523129	0.261564	−	0.965186i	$-0.415762\pi$
$293$	768737.	0.523129	0.261564	+	0.965186i	$0.415762\pi$
$294$	0	0
$295$	1.93747e6	1.29622
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.24033e6	0.802343
$300$	0	0
$301$	−716152.	−0.455605
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.03156e6	1.25049
$306$	0	0
$307$	2.33067e6	1.41135	0.705674	−	0.708536i	$-0.250644\pi$
$307$	2.33067e6	1.41135	0.705674	+	0.708536i	$0.250644\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.00213e6	1.76007	0.880033	−	0.474912i	$-0.157520\pi$
$311$	3.00213e6	1.76007	0.880033	+	0.474912i	$0.157520\pi$
$312$	0	0
$313$	−2.65017e6	−1.52902	−0.764511	−	0.644611i	$-0.777019\pi$
$313$	−2.65017e6	−1.52902	−0.764511	+	0.644611i	$0.777019\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.38294e6	0.772957	0.386479	−	0.922298i	$-0.373691\pi$
$317$	1.38294e6	0.772957	0.386479	+	0.922298i	$0.373691\pi$
$318$	0	0
$319$	294648.	0.162116
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.00399e6	−0.535455
$324$	0	0
$325$	−372245.	−0.195488
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−546502.	−0.278357
$330$	0	0
$331$	102433.	0.0513891	0.0256945	−	0.999670i	$-0.491820\pi$
$331$	102433.	0.0513891	0.0256945	+	0.999670i	$0.491820\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.72005e6	1.32423
$336$	0	0
$337$	968896.	0.464732	0.232366	−	0.972628i	$-0.425353\pi$
$337$	968896.	0.464732	0.232366	+	0.972628i	$0.425353\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	335017.	0.156020
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−681115.	−0.303666	−0.151833	−	0.988406i	$-0.548518\pi$
$347$	−681115.	−0.303666	−0.151833	+	0.988406i	$0.548518\pi$
$348$	0	0
$349$	−2.54000e6	−1.11627	−0.558135	−	0.829750i	$-0.688483\pi$
$349$	−2.54000e6	−1.11627	−0.558135	+	0.829750i	$0.688483\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.07238e6	0.885183	0.442592	−	0.896723i	$-0.354059\pi$
$353$	2.07238e6	0.885183	0.442592	+	0.896723i	$0.354059\pi$
$354$	0	0
$355$	−3.06562e6	−1.29106
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.78554e6	1.55022	0.775108	−	0.631829i	$-0.217695\pi$
$359$	3.78554e6	1.55022	0.775108	+	0.631829i	$0.217695\pi$
$360$	0	0
$361$	4.71499e6	1.90420
$362$	0	0
$363$	0	0
$364$	0	0
$365$	269067.	0.105713
$366$	0	0
$367$	−4.24909e6	−1.64676	−0.823381	−	0.567490i	$-0.807915\pi$
$367$	−4.24909e6	−1.64676	−0.823381	+	0.567490i	$0.807915\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.71021e6	0.645082
$372$	0	0
$373$	4.02274e6	1.49710	0.748549	−	0.663080i	$-0.230751\pi$
$373$	4.02274e6	1.49710	0.748549	+	0.663080i	$0.230751\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	672780.	0.243792
$378$	0	0
$379$	5.21539e6	1.86504	0.932521	−	0.361116i	$-0.117604\pi$
$379$	5.21539e6	1.86504	0.932521	+	0.361116i	$0.117604\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.81885e6	1.67860	0.839298	−	0.543672i	$-0.182966\pi$
$383$	4.81885e6	1.67860	0.839298	+	0.543672i	$0.182966\pi$
$384$	0	0
$385$	385927.	0.132695
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.56102e6	−1.19316	−0.596582	−	0.802552i	$-0.703475\pi$
$389$	−3.56102e6	−1.19316	−0.596582	+	0.802552i	$0.703475\pi$
$390$	0	0
$391$	1.73527e6	0.574019
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5.53252e6	−1.78414
$396$	0	0
$397$	4.67478e6	1.48862	0.744312	−	0.667832i	$-0.232777\pi$
$397$	4.67478e6	1.48862	0.744312	+	0.667832i	$0.232777\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.73495e6	0.538797	0.269399	−	0.963029i	$-0.413175\pi$
$401$	1.73495e6	0.538797	0.269399	+	0.963029i	$0.413175\pi$
$402$	0	0
$403$	764955.	0.234624
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.18126e6	0.353477
$408$	0	0
$409$	4.42517e6	1.30804	0.654021	−	0.756477i	$-0.273081\pi$
$409$	4.42517e6	1.30804	0.654021	+	0.756477i	$0.273081\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.41271e6	−0.407548
$414$	0	0
$415$	−506387.	−0.144332
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.57342e6	0.994373	0.497187	−	0.867644i	$-0.334366\pi$
$419$	3.57342e6	0.994373	0.497187	+	0.867644i	$0.334366\pi$
$420$	0	0
$421$	4.36682e6	1.20077	0.600386	−	0.799711i	$-0.295014\pi$
$421$	4.36682e6	1.20077	0.600386	+	0.799711i	$0.295014\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−520785.	−0.139858
$426$	0	0
$427$	−1.48132e6	−0.393168
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.62468e6	1.45849	0.729247	−	0.684250i	$-0.239870\pi$
$431$	5.62468e6	1.45849	0.729247	+	0.684250i	$0.239870\pi$
$432$	0	0
$433$	78517.4	0.0201255	0.0100627	−	0.999949i	$-0.496797\pi$
$433$	78517.4	0.0201255	0.0100627	+	0.999949i	$0.496797\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.24289e7	−3.11337
$438$	0	0
$439$	−605066.	−0.149845	−0.0749223	−	0.997189i	$-0.523871\pi$
$439$	−605066.	−0.149845	−0.0749223	+	0.997189i	$0.523871\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.28586e6	−0.795499	−0.397749	−	0.917494i	$-0.630209\pi$
$443$	−3.28586e6	−0.795499	−0.397749	+	0.917494i	$0.630209\pi$
$444$	0	0
$445$	−3.13739e6	−0.751050
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6.81460e6	−1.59523	−0.797617	−	0.603164i	$-0.793906\pi$
$449$	−6.81460e6	−1.59523	−0.797617	+	0.603164i	$0.793906\pi$
$450$	0	0
$451$	1.65086e6	0.382181
$452$	0	0
$453$	0	0
$454$	0	0
$455$	881200.	0.199547
$456$	0	0
$457$	−4.12929e6	−0.924880	−0.462440	−	0.886651i	$-0.653026\pi$
$457$	−4.12929e6	−0.924880	−0.462440	+	0.886651i	$0.653026\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7.86347e6	1.72330	0.861652	−	0.507500i	$-0.169430\pi$
$461$	7.86347e6	1.72330	0.861652	+	0.507500i	$0.169430\pi$
$462$	0	0
$463$	1.47332e6	0.319406	0.159703	−	0.987165i	$-0.448946\pi$
$463$	1.47332e6	0.319406	0.159703	+	0.987165i	$0.448946\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−102489.	−0.0217462	−0.0108731	−	0.999941i	$-0.503461\pi$
$467$	−102489.	−0.0217462	−0.0108731	+	0.999941i	$0.503461\pi$
$468$	0	0
$469$	−1.98333e6	−0.416355
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.71294e6	−0.352037
$474$	0	0
$475$	3.73014e6	0.758561
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.03754e6	−0.405758	−0.202879	−	0.979204i	$-0.565030\pi$
$479$	−2.03754e6	−0.405758	−0.202879	+	0.979204i	$0.565030\pi$
$480$	0	0
$481$	2.69722e6	0.531562
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.12737e6	0.217626
$486$	0	0
$487$	−5.22108e6	−0.997558	−0.498779	−	0.866729i	$-0.666218\pi$
$487$	−5.22108e6	−0.997558	−0.498779	+	0.866729i	$0.666218\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	511328.	0.0957185	0.0478592	−	0.998854i	$-0.484760\pi$
$491$	511328.	0.0957185	0.0478592	+	0.998854i	$0.484760\pi$
$492$	0	0
$493$	941246.	0.174416
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.23531e6	0.405925
$498$	0	0
$499$	4.42818e6	0.796111	0.398055	−	0.917361i	$-0.369685\pi$
$499$	4.42818e6	0.796111	0.398055	+	0.917361i	$0.369685\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.77759e6	−0.489496	−0.244748	−	0.969587i	$-0.578705\pi$
$503$	−2.77759e6	−0.489496	−0.244748	+	0.969587i	$0.578705\pi$
$504$	0	0
$505$	3.18385e6	0.555551
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.97867e6	−1.70718	−0.853588	−	0.520949i	$-0.825578\pi$
$509$	−9.97867e6	−1.70718	−0.853588	+	0.520949i	$0.825578\pi$
$510$	0	0
$511$	−196191.	−0.0332374
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.35813e6	−1.05636
$516$	0	0
$517$	−1.30716e6	−0.215081
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.35311e6	0.541195	0.270597	−	0.962693i	$-0.412779\pi$
$521$	3.35311e6	0.541195	0.270597	+	0.962693i	$0.412779\pi$
$522$	0	0
$523$	−4.25132e6	−0.679625	−0.339812	−	0.940493i	$-0.610364\pi$
$523$	−4.25132e6	−0.679625	−0.339812	+	0.940493i	$0.610364\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.07020e6	0.167857
$528$	0	0
$529$	1.50456e7	2.33760
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.76947e6	0.574727
$534$	0	0
$535$	2.64511e6	0.399540
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−281400.	−0.0417208
$540$	0	0
$541$	−1.30343e7	−1.91467	−0.957336	−	0.288977i	$-0.906685\pi$
$541$	−1.30343e7	−1.91467	−0.957336	+	0.288977i	$0.906685\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−4.80908e6	−0.693538
$546$	0	0
$547$	8.59959e6	1.22888	0.614440	−	0.788963i	$-0.289382\pi$
$547$	8.59959e6	1.22888	0.614440	+	0.788963i	$0.289382\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.74169e6	−0.945997
$552$	0	0
$553$	4.03406e6	0.560956
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.89818e6	0.532383	0.266192	−	0.963920i	$-0.414235\pi$
$557$	3.89818e6	0.532383	0.266192	+	0.963920i	$0.414235\pi$
$558$	0	0
$559$	−3.91120e6	−0.529396
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.70888e6	0.626104	0.313052	−	0.949736i	$-0.398649\pi$
$563$	4.70888e6	0.626104	0.313052	+	0.949736i	$0.398649\pi$
$564$	0	0
$565$	−9.90192e6	−1.30496
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.47706e7	1.91257	0.956285	−	0.292435i	$-0.0944656\pi$
$569$	1.47706e7	1.91257	0.956285	+	0.292435i	$0.0944656\pi$
$570$	0	0
$571$	−1.00682e7	−1.29229	−0.646145	−	0.763214i	$-0.723620\pi$
$571$	−1.00682e7	−1.29229	−0.646145	+	0.763214i	$0.723620\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.44709e6	−0.813194
$576$	0	0
$577$	−1.49919e7	−1.87464	−0.937318	−	0.348475i	$-0.886700\pi$
$577$	−1.49919e7	−1.87464	−0.937318	+	0.348475i	$0.886700\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	369234.	0.0453797
$582$	0	0
$583$	4.09059e6	0.498442
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.41615e6	0.409205	0.204603	−	0.978845i	$-0.434410\pi$
$587$	3.41615e6	0.409205	0.204603	+	0.978845i	$0.434410\pi$
$588$	0	0
$589$	−7.66534e6	−0.910423
$590$	0	0
$591$	0	0
$592$	0	0
$593$	84972.3	0.00992294	0.00496147	−	0.999988i	$-0.498421\pi$
$593$	84972.3	0.00992294	0.00496147	+	0.999988i	$0.498421\pi$
$594$	0	0
$595$	1.23283e6	0.142762
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.17468e7	1.33768	0.668838	−	0.743408i	$-0.266792\pi$
$599$	1.17468e7	1.33768	0.668838	+	0.743408i	$0.266792\pi$
$600$	0	0
$601$	4.79610e6	0.541629	0.270815	−	0.962632i	$-0.412707\pi$
$601$	4.79610e6	0.541629	0.270815	+	0.962632i	$0.412707\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.89974e6	−1.09960
$606$	0	0
$607$	−3.53150e6	−0.389034	−0.194517	−	0.980899i	$-0.562314\pi$
$607$	−3.53150e6	−0.389034	−0.194517	+	0.980899i	$0.562314\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.98468e6	−0.323440
$612$	0	0
$613$	3.78238e6	0.406550	0.203275	−	0.979122i	$-0.434842\pi$
$613$	3.78238e6	0.406550	0.203275	+	0.979122i	$0.434842\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.18399e7	1.25209	0.626046	−	0.779786i	$-0.284672\pi$
$617$	1.18399e7	1.25209	0.626046	+	0.779786i	$0.284672\pi$
$618$	0	0
$619$	−1.68015e7	−1.76247	−0.881237	−	0.472675i	$-0.843288\pi$
$619$	−1.68015e7	−1.76247	−0.881237	+	0.472675i	$0.843288\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.28764e6	0.236139
$624$	0	0
$625$	−1.21776e7	−1.24699
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.77352e6	0.380295
$630$	0	0
$631$	−1.23029e6	−0.123008	−0.0615042	−	0.998107i	$-0.519590\pi$
$631$	−1.23029e6	−0.123008	−0.0615042	+	0.998107i	$0.519590\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.18441e6	0.116565
$636$	0	0
$637$	−642530.	−0.0627401
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.36169e6	−0.227027	−0.113514	−	0.993536i	$-0.536211\pi$
$641$	−2.36169e6	−0.227027	−0.113514	+	0.993536i	$0.536211\pi$
$642$	0	0
$643$	1.25756e6	0.119950	0.0599750	−	0.998200i	$-0.480898\pi$
$643$	1.25756e6	0.119950	0.0599750	+	0.998200i	$0.480898\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.45141e6	−0.605890	−0.302945	−	0.953008i	$-0.597970\pi$
$647$	−6.45141e6	−0.605890	−0.302945	+	0.953008i	$0.597970\pi$
$648$	0	0
$649$	−3.37901e6	−0.314904
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.84525e7	1.69345	0.846726	−	0.532029i	$-0.178570\pi$
$653$	1.84525e7	1.69345	0.846726	+	0.532029i	$0.178570\pi$
$654$	0	0
$655$	−5.77256e6	−0.525733
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.98016e6	0.357015	0.178508	−	0.983939i	$-0.442873\pi$
$659$	3.98016e6	0.357015	0.178508	+	0.983939i	$0.442873\pi$
$660$	0	0
$661$	−1.93255e7	−1.72039	−0.860195	−	0.509965i	$-0.829658\pi$
$661$	−1.93255e7	−1.72039	−0.860195	+	0.509965i	$0.829658\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.83020e6	−0.774313
$666$	0	0
$667$	1.16522e7	1.01413
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.54311e6	−0.303793
$672$	0	0
$673$	5.72688e6	0.487394	0.243697	−	0.969851i	$-0.421640\pi$
$673$	5.72688e6	0.487394	0.243697	+	0.969851i	$0.421640\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−105639.	−0.00885831	−0.00442915	−	0.999990i	$-0.501410\pi$
$677$	−105639.	−0.00885831	−0.00442915	+	0.999990i	$0.501410\pi$
$678$	0	0
$679$	−822024.	−0.0684243
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.96129e7	−1.60875	−0.804377	−	0.594119i	$-0.797501\pi$
$683$	−1.96129e7	−1.60875	−0.804377	+	0.594119i	$0.797501\pi$
$684$	0	0
$685$	−7.94997e6	−0.647350
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9.34018e6	0.749562
$690$	0	0
$691$	−7.94444e6	−0.632949	−0.316474	−	0.948601i	$-0.602499\pi$
$691$	−7.94444e6	−0.632949	−0.316474	+	0.948601i	$0.602499\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.39348e7	−1.87961
$696$	0	0
$697$	5.27364e6	0.411176
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.62349e6	−0.124783	−0.0623914	−	0.998052i	$-0.519873\pi$
$701$	−1.62349e6	−0.124783	−0.0623914	+	0.998052i	$0.519873\pi$
$702$	0	0
$703$	−2.70279e7	−2.06264
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.32151e6	−0.174672
$708$	0	0
$709$	−1.62774e6	−0.121610	−0.0608051	−	0.998150i	$-0.519367\pi$
$709$	−1.62774e6	−0.121610	−0.0608051	+	0.998150i	$0.519367\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.32486e7	0.975993
$714$	0	0
$715$	2.10771e6	0.154186
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8.91335e6	−0.643012	−0.321506	−	0.946908i	$-0.604189\pi$
$719$	−8.91335e6	−0.643012	−0.321506	+	0.946908i	$0.604189\pi$
$720$	0	0
$721$	4.63605e6	0.332132
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3.49702e6	−0.247089
$726$	0	0
$727$	1.16823e7	0.819773	0.409886	−	0.912137i	$-0.365568\pi$
$727$	1.16823e7	0.819773	0.409886	+	0.912137i	$0.365568\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.47193e6	−0.378745
$732$	0	0
$733$	5.24074e6	0.360274	0.180137	−	0.983642i	$-0.442346\pi$
$733$	5.24074e6	0.360274	0.180137	+	0.983642i	$0.442346\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.74385e6	−0.321709
$738$	0	0
$739$	−2.78382e6	−0.187513	−0.0937564	−	0.995595i	$-0.529887\pi$
$739$	−2.78382e6	−0.187513	−0.0937564	+	0.995595i	$0.529887\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.59091e6	0.570909	0.285455	−	0.958392i	$-0.407855\pi$
$743$	8.59091e6	0.570909	0.285455	+	0.958392i	$0.407855\pi$
$744$	0	0
$745$	−9.74139e6	−0.643029
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.92869e6	−0.125620
$750$	0	0
$751$	−1.13278e7	−0.732905	−0.366452	−	0.930437i	$-0.619428\pi$
$751$	−1.13278e7	−0.732905	−0.366452	+	0.930437i	$0.619428\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−2.87181e7	−1.83353
$756$	0	0
$757$	−7.45021e6	−0.472529	−0.236265	−	0.971689i	$-0.575923\pi$
$757$	−7.45021e6	−0.472529	−0.236265	+	0.971689i	$0.575923\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.67264e7	1.04698	0.523492	−	0.852031i	$-0.324629\pi$
$761$	1.67264e7	1.04698	0.523492	+	0.852031i	$0.324629\pi$
$762$	0	0
$763$	3.50656e6	0.218057
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7.71541e6	−0.473555
$768$	0	0
$769$	2.52349e7	1.53881	0.769407	−	0.638759i	$-0.220552\pi$
$769$	2.52349e7	1.53881	0.769407	+	0.638759i	$0.220552\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.74123e7	−1.65005	−0.825025	−	0.565097i	$-0.808839\pi$
$773$	−2.74123e7	−1.65005	−0.825025	+	0.565097i	$0.808839\pi$
$774$	0	0
$775$	−3.97614e6	−0.237797
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.77725e7	−2.23014
$780$	0	0
$781$	5.34654e6	0.313650
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.31098e7	1.33851
$786$	0	0
$787$	−1.24214e7	−0.714880	−0.357440	−	0.933936i	$-0.616350\pi$
$787$	−1.24214e7	−0.714880	−0.357440	+	0.933936i	$0.616350\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.22002e6	0.410296
$792$	0	0
$793$	−8.09010e6	−0.456847
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.08301e7	−0.603929	−0.301965	−	0.953319i	$-0.597642\pi$
$797$	−1.08301e7	−0.603929	−0.301965	+	0.953319i	$0.597642\pi$
$798$	0	0
$799$	−4.17568e6	−0.231398
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−469262.	−0.0256819
$804$	0	0
$805$	1.52619e7	0.830080
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.18990e7	1.17639	0.588196	−	0.808718i	$-0.299838\pi$
$809$	2.18990e7	1.17639	0.588196	+	0.808718i	$0.299838\pi$
$810$	0	0
$811$	8.37206e6	0.446972	0.223486	−	0.974707i	$-0.428256\pi$
$811$	8.37206e6	0.446972	0.223486	+	0.974707i	$0.428256\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.60424e6	0.137337
$816$	0	0
$817$	3.91928e7	2.05424
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.88483e7	−0.975918	−0.487959	−	0.872866i	$-0.662258\pi$
$821$	−1.88483e7	−0.975918	−0.487959	+	0.872866i	$0.662258\pi$
$822$	0	0
$823$	2.87140e7	1.47773	0.738864	−	0.673855i	$-0.235363\pi$
$823$	2.87140e7	1.47773	0.738864	+	0.673855i	$0.235363\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.72989e7	0.879537	0.439769	−	0.898111i	$-0.355060\pi$
$827$	1.72989e7	0.879537	0.439769	+	0.898111i	$0.355060\pi$
$828$	0	0
$829$	3.48523e7	1.76135	0.880675	−	0.473721i	$-0.157090\pi$
$829$	3.48523e7	1.76135	0.880675	+	0.473721i	$0.157090\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−898926.	−0.0448860
$834$	0	0
$835$	1.86962e7	0.927976
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.81074e7	−0.888080	−0.444040	−	0.896007i	$-0.646455\pi$
$839$	−1.81074e7	−0.888080	−0.444040	+	0.896007i	$0.646455\pi$
$840$	0	0
$841$	−1.41908e7	−0.691857
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.01387e7	−0.970265
$846$	0	0
$847$	7.21843e6	0.345728
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.67145e7	2.21120
$852$	0	0
$853$	−9.76976e6	−0.459739	−0.229870	−	0.973221i	$-0.573830\pi$
$853$	−9.76976e6	−0.459739	−0.229870	+	0.973221i	$0.573830\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.45112e6	0.207022	0.103511	−	0.994628i	$-0.466992\pi$
$857$	4.45112e6	0.207022	0.103511	+	0.994628i	$0.466992\pi$
$858$	0	0
$859$	−3.74490e7	−1.73164	−0.865820	−	0.500356i	$-0.833202\pi$
$859$	−3.74490e7	−1.73164	−0.865820	+	0.500356i	$0.833202\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.92400e7	−0.879384	−0.439692	−	0.898149i	$-0.644912\pi$
$863$	−1.92400e7	−0.879384	−0.439692	+	0.898149i	$0.644912\pi$
$864$	0	0
$865$	1.29922e7	0.590395
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.64890e6	0.433439
$870$	0	0
$871$	−1.08318e7	−0.483788
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.70982e6	0.252117
$876$	0	0
$877$	−9.26947e6	−0.406964	−0.203482	−	0.979079i	$-0.565226\pi$
$877$	−9.26947e6	−0.406964	−0.203482	+	0.979079i	$0.565226\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3.31518e7	1.43902	0.719510	−	0.694482i	$-0.244366\pi$
$881$	3.31518e7	1.43902	0.719510	+	0.694482i	$0.244366\pi$
$882$	0	0
$883$	−2.29544e7	−0.990751	−0.495375	−	0.868679i	$-0.664969\pi$
$883$	−2.29544e7	−0.990751	−0.495375	+	0.868679i	$0.664969\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.99207e6	−0.127692	−0.0638459	−	0.997960i	$-0.520337\pi$
$887$	−2.99207e6	−0.127692	−0.0638459	+	0.997960i	$0.520337\pi$
$888$	0	0
$889$	−863619.	−0.0366495
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.99084e7	1.25506
$894$	0	0
$895$	1.49663e6	0.0624535
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.18631e6	0.296556
$900$	0	0
$901$	1.30673e7	0.536258
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.39784e7	−0.567329
$906$	0	0
$907$	2.98331e7	1.20415	0.602075	−	0.798439i	$-0.294341\pi$
$907$	2.98331e7	1.20415	0.602075	+	0.798439i	$0.294341\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.49263e6	0.139430	0.0697150	−	0.997567i	$-0.477791\pi$
$911$	3.49263e6	0.139430	0.0697150	+	0.997567i	$0.477791\pi$
$912$	0	0
$913$	883156.	0.0350639
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.20908e6	0.165297
$918$	0	0
$919$	−1.83584e7	−0.717045	−0.358523	−	0.933521i	$-0.616719\pi$
$919$	−1.83584e7	−0.717045	−0.358523	+	0.933521i	$0.616719\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.22079e7	0.471670
$924$	0	0
$925$	−1.40198e7	−0.538751
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−3.58149e7	−1.36152	−0.680760	−	0.732506i	$-0.738350\pi$
$929$	−3.58149e7	−1.36152	−0.680760	+	0.732506i	$0.738350\pi$
$930$	0	0
$931$	6.43857e6	0.243453
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.94877e6	0.110309
$936$	0	0
$937$	1.66372e7	0.619057	0.309529	−	0.950890i	$-0.399829\pi$
$937$	1.66372e7	0.619057	0.309529	+	0.950890i	$0.399829\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.78598e7	1.39381	0.696905	−	0.717163i	$-0.254560\pi$
$941$	3.78598e7	1.39381	0.696905	+	0.717163i	$0.254560\pi$
$942$	0	0
$943$	6.52852e7	2.39076
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.17041e7	0.786441	0.393221	−	0.919444i	$-0.371361\pi$
$947$	2.17041e7	0.786441	0.393221	+	0.919444i	$0.371361\pi$
$948$	0	0
$949$	−1.07148e6	−0.0386206
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.43716e7	0.869265	0.434632	−	0.900608i	$-0.356878\pi$
$953$	2.43716e7	0.869265	0.434632	+	0.900608i	$0.356878\pi$
$954$	0	0
$955$	−5.06608e7	−1.79748
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.79675e6	0.203534
$960$	0	0
$961$	−2.04583e7	−0.714596
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.00095e6	0.138307
$966$	0	0
$967$	4.39697e7	1.51213	0.756063	−	0.654499i	$-0.227121\pi$
$967$	4.39697e7	1.51213	0.756063	+	0.654499i	$0.227121\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.36883e7	1.14665	0.573325	−	0.819328i	$-0.305653\pi$
$971$	3.36883e7	1.14665	0.573325	+	0.819328i	$0.305653\pi$
$972$	0	0
$973$	1.74522e7	0.590972
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.00132e6	0.100595	0.0502974	−	0.998734i	$-0.483983\pi$
$977$	3.00132e6	0.100595	0.0502974	+	0.998734i	$0.483983\pi$
$978$	0	0
$979$	5.47172e6	0.182460
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.00031e7	0.660257	0.330128	−	0.943936i	$-0.392908\pi$
$983$	2.00031e7	0.660257	0.330128	+	0.943936i	$0.392908\pi$
$984$	0	0
$985$	−1.06894e7	−0.351046
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.77400e7	−2.20219
$990$	0	0
$991$	−8.92502e6	−0.288686	−0.144343	−	0.989528i	$-0.546107\pi$
$991$	−8.92502e6	−0.288686	−0.144343	+	0.989528i	$0.546107\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.37843e7	1.72226
$996$	0	0
$997$	6.12645e6	0.195196	0.0975979	−	0.995226i	$-0.468884\pi$
$997$	6.12645e6	0.195196	0.0975979	+	0.995226i	$0.468884\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.6.a.o.1.2		2
3.2	odd	2		168.6.a.j.1.1	✓	2
4.3	odd	2		1008.6.a.bn.1.2		2
12.11	even	2		336.6.a.t.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.6.a.j.1.1	✓	2	3.2	odd	2
336.6.a.t.1.1		2	12.11	even	2
504.6.a.o.1.2		2	1.1	even	1	trivial
1008.6.a.bn.1.2		2	4.3	odd	2

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:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	504.a (trivial)

\(f(q)\)	\(=\)	\(q+67.2012 q^{5} -49.0000 q^{7} +O(q^{10})\)	\(q+67.2012 q^{5} -49.0000 q^{7} -117.201 q^{11} -267.610 q^{13} -374.396 q^{17} +2681.62 q^{19} -4634.86 q^{23} +1391.00 q^{25} -2514.04 q^{29} -2858.47 q^{31} -3292.86 q^{35} -10078.9 q^{37} -14085.7 q^{41} +14615.3 q^{43} +11153.1 q^{47} +2401.00 q^{49} -34902.3 q^{53} -7876.06 q^{55} +28830.9 q^{59} +30231.0 q^{61} -17983.7 q^{65} +40476.2 q^{67} -45618.5 q^{71} +4003.90 q^{73} +5742.86 q^{77} -82327.7 q^{79} -7535.39 q^{83} -25159.9 q^{85} -46686.5 q^{89} +13112.9 q^{91} +180208. q^{95} +16776.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 98 q^{7}+O(q^{10}) \) 2 * q - 98 * q^7	\( 2 q - 98 q^{7} - 100 q^{11} + 540 q^{13} - 1152 q^{17} + 2944 q^{19} - 2684 q^{23} + 2782 q^{25} - 996 q^{29} + 2616 q^{31} - 3492 q^{37} - 17016 q^{41} + 13640 q^{43} - 11832 q^{47} + 4802 q^{49} - 37548 q^{53} - 9032 q^{55} + 6320 q^{59} + 39764 q^{61} - 72256 q^{65} - 27376 q^{67} - 35460 q^{71} + 64188 q^{73} + 4900 q^{77} - 110088 q^{79} + 5896 q^{83} + 27096 q^{85} - 167160 q^{89} - 26460 q^{91} + 162576 q^{95} - 11876 q^{97}+O(q^{100}) \) 2 * q - 98 * q^7 - 100 * q^11 + 540 * q^13 - 1152 * q^17 + 2944 * q^19 - 2684 * q^23 + 2782 * q^25 - 996 * q^29 + 2616 * q^31 - 3492 * q^37 - 17016 * q^41 + 13640 * q^43 - 11832 * q^47 + 4802 * q^49 - 37548 * q^53 - 9032 * q^55 + 6320 * q^59 + 39764 * q^61 - 72256 * q^65 - 27376 * q^67 - 35460 * q^71 + 64188 * q^73 + 4900 * q^77 - 110088 * q^79 + 5896 * q^83 + 27096 * q^85 - 167160 * q^89 - 26460 * q^91 + 162576 * q^95 - 11876 * q^97

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