LMFDB - Embedded newform 252.6.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,6,Mod(1,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, 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("252.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	252.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:	$40.4167225929$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{5569}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1392 $ x^2 - x - 1392
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$37.8129$ of defining polynomial
Character	$\chi$	$=$	252.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+77.6257 q^{5} -49.0000 q^{7} +O(q^{10})$	$q+77.6257 q^{5} -49.0000 q^{7} -477.380 q^{11} -63.7544 q^{13} -1037.63 q^{17} -667.018 q^{19} -3251.63 q^{23} +2900.75 q^{25} -2300.97 q^{29} +3717.05 q^{31} -3803.66 q^{35} +12245.9 q^{37} +1829.65 q^{41} -20794.2 q^{43} +4283.37 q^{47} +2401.00 q^{49} -25718.4 q^{53} -37057.0 q^{55} +2838.71 q^{59} +16803.2 q^{61} -4948.98 q^{65} -62535.1 q^{67} -72301.0 q^{71} -55676.9 q^{73} +23391.6 q^{77} -3989.19 q^{79} +46092.2 q^{83} -80546.5 q^{85} -135385. q^{89} +3123.97 q^{91} -51777.7 q^{95} +142878. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{5} - 98 q^{7}+O(q^{10}) $ 2 * q + 6 * q^5 - 98 * q^7	$ 2 q + 6 q^{5} - 98 q^{7} + 90 q^{11} + 768 q^{13} - 1926 q^{17} + 2248 q^{19} - 6354 q^{23} + 4906 q^{25} - 10572 q^{29} - 3312 q^{31} - 294 q^{35} + 2104 q^{37} - 1266 q^{41} - 5768 q^{43} - 15612 q^{47} + 4802 q^{49} - 16512 q^{53} - 77696 q^{55} + 13140 q^{59} - 5796 q^{61} - 64524 q^{65} - 56116 q^{67} - 11022 q^{71} - 85384 q^{73} - 4410 q^{77} - 19620 q^{79} + 44424 q^{83} - 16916 q^{85} - 211218 q^{89} - 37632 q^{91} - 260568 q^{95} + 44864 q^{97}+O(q^{100}) $ 2 * q + 6 * q^5 - 98 * q^7 + 90 * q^11 + 768 * q^13 - 1926 * q^17 + 2248 * q^19 - 6354 * q^23 + 4906 * q^25 - 10572 * q^29 - 3312 * q^31 - 294 * q^35 + 2104 * q^37 - 1266 * q^41 - 5768 * q^43 - 15612 * q^47 + 4802 * q^49 - 16512 * q^53 - 77696 * q^55 + 13140 * q^59 - 5796 * q^61 - 64524 * q^65 - 56116 * q^67 - 11022 * q^71 - 85384 * q^73 - 4410 * q^77 - 19620 * q^79 + 44424 * q^83 - 16916 * q^85 - 211218 * q^89 - 37632 * q^91 - 260568 * q^95 + 44864 * 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
$5$	77.6257	1.38861	0.694306	−	0.719680i	$-0.255712\pi$
$5$	77.6257	1.38861	0.694306	+	0.719680i	$0.255712\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−477.380	−1.18955	−0.594775	−	0.803892i	$-0.702759\pi$
$11$	−477.380	−1.18955	−0.594775	+	0.803892i	$0.702759\pi$
$12$	0	0
$13$	−63.7544	−0.104629	−0.0523145	−	0.998631i	$-0.516660\pi$
$13$	−63.7544	−0.104629	−0.0523145	+	0.998631i	$0.516660\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1037.63	−0.870800	−0.435400	−	0.900237i	$-0.643393\pi$
$17$	−1037.63	−0.870800	−0.435400	+	0.900237i	$0.643393\pi$
$18$	0	0
$19$	−667.018	−0.423890	−0.211945	−	0.977282i	$-0.567980\pi$
$19$	−667.018	−0.423890	−0.211945	+	0.977282i	$0.567980\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3251.63	−1.28168	−0.640842	−	0.767673i	$-0.721415\pi$
$23$	−3251.63	−1.28168	−0.640842	+	0.767673i	$0.721415\pi$
$24$	0	0
$25$	2900.75	0.928241
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2300.97	−0.508061	−0.254031	−	0.967196i	$-0.581756\pi$
$29$	−2300.97	−0.508061	−0.254031	+	0.967196i	$0.581756\pi$
$30$	0	0
$31$	3717.05	0.694696	0.347348	−	0.937736i	$-0.387082\pi$
$31$	3717.05	0.694696	0.347348	+	0.937736i	$0.387082\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3803.66	−0.524846
$36$	0	0
$37$	12245.9	1.47057	0.735284	−	0.677759i	$-0.237049\pi$
$37$	12245.9	1.47057	0.735284	+	0.677759i	$0.237049\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1829.65	0.169984	0.0849920	−	0.996382i	$-0.472914\pi$
$41$	1829.65	0.169984	0.0849920	+	0.996382i	$0.472914\pi$
$42$	0	0
$43$	−20794.2	−1.71503	−0.857513	−	0.514463i	$-0.827991\pi$
$43$	−20794.2	−1.71503	−0.857513	+	0.514463i	$0.827991\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4283.37	0.282840	0.141420	−	0.989950i	$-0.454833\pi$
$47$	4283.37	0.282840	0.141420	+	0.989950i	$0.454833\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−25718.4	−1.25764	−0.628818	−	0.777553i	$-0.716461\pi$
$53$	−25718.4	−1.25764	−0.628818	+	0.777553i	$0.716461\pi$
$54$	0	0
$55$	−37057.0	−1.65182
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2838.71	0.106167	0.0530837	−	0.998590i	$-0.483095\pi$
$59$	2838.71	0.106167	0.0530837	+	0.998590i	$0.483095\pi$
$60$	0	0
$61$	16803.2	0.578186	0.289093	−	0.957301i	$-0.406646\pi$
$61$	16803.2	0.578186	0.289093	+	0.957301i	$0.406646\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4948.98	−0.145289
$66$	0	0
$67$	−62535.1	−1.70191	−0.850955	−	0.525238i	$-0.823976\pi$
$67$	−62535.1	−1.70191	−0.850955	+	0.525238i	$0.823976\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−72301.0	−1.70215	−0.851077	−	0.525042i	$-0.824050\pi$
$71$	−72301.0	−1.70215	−0.851077	+	0.525042i	$0.824050\pi$
$72$	0	0
$73$	−55676.9	−1.22283	−0.611417	−	0.791308i	$-0.709400\pi$
$73$	−55676.9	−1.22283	−0.611417	+	0.791308i	$0.709400\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	23391.6	0.449608
$78$	0	0
$79$	−3989.19	−0.0719146	−0.0359573	−	0.999353i	$-0.511448\pi$
$79$	−3989.19	−0.0719146	−0.0359573	+	0.999353i	$0.511448\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	46092.2	0.734400	0.367200	−	0.930142i	$-0.380316\pi$
$83$	46092.2	0.734400	0.367200	+	0.930142i	$0.380316\pi$
$84$	0	0
$85$	−80546.5	−1.20920
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−135385.	−1.81173	−0.905867	−	0.423562i	$-0.860780\pi$
$89$	−135385.	−1.81173	−0.905867	+	0.423562i	$0.860780\pi$
$90$	0	0
$91$	3123.97	0.0395460
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−51777.7	−0.588619
$96$	0	0
$97$	142878.	1.54183	0.770914	−	0.636939i	$-0.219800\pi$
$97$	142878.	1.54183	0.770914	+	0.636939i	$0.219800\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−44467.1	−0.433746	−0.216873	−	0.976200i	$-0.569586\pi$
$101$	−44467.1	−0.433746	−0.216873	+	0.976200i	$0.569586\pi$
$102$	0	0
$103$	202619.	1.88186	0.940931	−	0.338598i	$-0.109953\pi$
$103$	202619.	1.88186	0.940931	+	0.338598i	$0.109953\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−99525.9	−0.840382	−0.420191	−	0.907436i	$-0.638037\pi$
$107$	−99525.9	−0.840382	−0.420191	+	0.907436i	$0.638037\pi$
$108$	0	0
$109$	220930.	1.78110	0.890551	−	0.454883i	$-0.150319\pi$
$109$	220930.	1.78110	0.890551	+	0.454883i	$0.150319\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−29623.1	−0.218240	−0.109120	−	0.994029i	$-0.534803\pi$
$113$	−29623.1	−0.218240	−0.109120	+	0.994029i	$0.534803\pi$
$114$	0	0
$115$	−252410.	−1.77976
$116$	0	0
$117$	0	0
$118$	0	0
$119$	50843.7	0.329131
$120$	0	0
$121$	66840.8	0.415029
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−17407.2	−0.0996448
$126$	0	0
$127$	−264132.	−1.45315	−0.726577	−	0.687086i	$-0.758890\pi$
$127$	−264132.	−1.45315	−0.726577	+	0.687086i	$0.758890\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	77850.1	0.396352	0.198176	−	0.980166i	$-0.436498\pi$
$131$	77850.1	0.396352	0.198176	+	0.980166i	$0.436498\pi$
$132$	0	0
$133$	32683.9	0.160215
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−372403.	−1.69516	−0.847581	−	0.530666i	$-0.821942\pi$
$137$	−372403.	−1.69516	−0.847581	+	0.530666i	$0.821942\pi$
$138$	0	0
$139$	−274550.	−1.20527	−0.602636	−	0.798016i	$-0.705883\pi$
$139$	−274550.	−1.20527	−0.602636	+	0.798016i	$0.705883\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	30435.1	0.124461
$144$	0	0
$145$	−178615.	−0.705500
$146$	0	0
$147$	0	0
$148$	0	0
$149$	312982.	1.15492	0.577462	−	0.816417i	$-0.304043\pi$
$149$	312982.	1.15492	0.577462	+	0.816417i	$0.304043\pi$
$150$	0	0
$151$	432095.	1.54219	0.771093	−	0.636723i	$-0.219710\pi$
$151$	432095.	1.54219	0.771093	+	0.636723i	$0.219710\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	288539.	0.964662
$156$	0	0
$157$	−84603.7	−0.273930	−0.136965	−	0.990576i	$-0.543735\pi$
$157$	−84603.7	−0.273930	−0.136965	+	0.990576i	$0.543735\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	159330.	0.484431
$162$	0	0
$163$	306303.	0.902989	0.451494	−	0.892274i	$-0.350891\pi$
$163$	306303.	0.902989	0.451494	+	0.892274i	$0.350891\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	606514.	1.68287	0.841433	−	0.540362i	$-0.181713\pi$
$167$	606514.	1.68287	0.841433	+	0.540362i	$0.181713\pi$
$168$	0	0
$169$	−367228.	−0.989053
$170$	0	0
$171$	0	0
$172$	0	0
$173$	288481.	0.732828	0.366414	−	0.930452i	$-0.380585\pi$
$173$	288481.	0.732828	0.366414	+	0.930452i	$0.380585\pi$
$174$	0	0
$175$	−142137.	−0.350842
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−148858.	−0.347248	−0.173624	−	0.984812i	$-0.555548\pi$
$179$	−148858.	−0.347248	−0.173624	+	0.984812i	$0.555548\pi$
$180$	0	0
$181$	93377.8	0.211859	0.105930	−	0.994374i	$-0.466218\pi$
$181$	93377.8	0.211859	0.105930	+	0.994374i	$0.466218\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	950594.	2.04205
$186$	0	0
$187$	495342.	1.03586
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−246915.	−0.489738	−0.244869	−	0.969556i	$-0.578745\pi$
$191$	−246915.	−0.489738	−0.244869	+	0.969556i	$0.578745\pi$
$192$	0	0
$193$	481437.	0.930349	0.465175	−	0.885219i	$-0.345992\pi$
$193$	481437.	0.930349	0.465175	+	0.885219i	$0.345992\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	548236.	1.00647	0.503237	−	0.864149i	$-0.332142\pi$
$197$	548236.	1.00647	0.503237	+	0.864149i	$0.332142\pi$
$198$	0	0
$199$	158179.	0.283150	0.141575	−	0.989928i	$-0.454783\pi$
$199$	158179.	0.283150	0.141575	+	0.989928i	$0.454783\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	112748.	0.192029
$204$	0	0
$205$	142028.	0.236042
$206$	0	0
$207$	0	0
$208$	0	0
$209$	318421.	0.504238
$210$	0	0
$211$	283510.	0.438392	0.219196	−	0.975681i	$-0.429657\pi$
$211$	283510.	0.438392	0.219196	+	0.975681i	$0.429657\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.61416e6	−2.38150
$216$	0	0
$217$	−182136.	−0.262570
$218$	0	0
$219$	0	0
$220$	0	0
$221$	66153.2	0.0911109
$222$	0	0
$223$	−651135.	−0.876817	−0.438409	−	0.898776i	$-0.644458\pi$
$223$	−651135.	−0.876817	−0.438409	+	0.898776i	$0.644458\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	378294.	0.487264	0.243632	−	0.969868i	$-0.421661\pi$
$227$	378294.	0.487264	0.243632	+	0.969868i	$0.421661\pi$
$228$	0	0
$229$	−22332.8	−0.0281420	−0.0140710	−	0.999901i	$-0.504479\pi$
$229$	−22332.8	−0.0281420	−0.0140710	+	0.999901i	$0.504479\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	908940.	1.09685	0.548423	−	0.836201i	$-0.315229\pi$
$233$	908940.	1.09685	0.548423	+	0.836201i	$0.315229\pi$
$234$	0	0
$235$	332500.	0.392755
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.05363e6	−1.19315	−0.596573	−	0.802559i	$-0.703471\pi$
$239$	−1.05363e6	−1.19315	−0.596573	+	0.802559i	$0.703471\pi$
$240$	0	0
$241$	1.05233e6	1.16710	0.583550	−	0.812077i	$-0.301663\pi$
$241$	1.05233e6	1.16710	0.583550	+	0.812077i	$0.301663\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	186379.	0.198373
$246$	0	0
$247$	42525.3	0.0443512
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−972876.	−0.974705	−0.487352	−	0.873205i	$-0.662037\pi$
$251$	−972876.	−0.974705	−0.487352	+	0.873205i	$0.662037\pi$
$252$	0	0
$253$	1.55226e6	1.52463
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.77948e6	−1.68058	−0.840290	−	0.542137i	$-0.817615\pi$
$257$	−1.77948e6	−1.68058	−0.840290	+	0.542137i	$0.817615\pi$
$258$	0	0
$259$	−600047.	−0.555822
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−21959.6	−0.0195765	−0.00978827	−	0.999952i	$-0.503116\pi$
$263$	−21959.6	−0.0195765	−0.00978827	+	0.999952i	$0.503116\pi$
$264$	0	0
$265$	−1.99641e6	−1.74637
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.69111e6	1.42492	0.712461	−	0.701712i	$-0.247581\pi$
$269$	1.69111e6	1.42492	0.712461	+	0.701712i	$0.247581\pi$
$270$	0	0
$271$	−467863.	−0.386986	−0.193493	−	0.981102i	$-0.561982\pi$
$271$	−467863.	−0.386986	−0.193493	+	0.981102i	$0.561982\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.38476e6	−1.10419
$276$	0	0
$277$	906169.	0.709594	0.354797	−	0.934943i	$-0.384550\pi$
$277$	906169.	0.709594	0.354797	+	0.934943i	$0.384550\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	781388.	0.590338	0.295169	−	0.955445i	$-0.404624\pi$
$281$	781388.	0.590338	0.295169	+	0.955445i	$0.404624\pi$
$282$	0	0
$283$	−1.49843e6	−1.11217	−0.556085	−	0.831125i	$-0.687697\pi$
$283$	−1.49843e6	−1.11217	−0.556085	+	0.831125i	$0.687697\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−89652.8	−0.0642479
$288$	0	0
$289$	−343190.	−0.241707
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.56070e6	1.06206	0.531031	−	0.847352i	$-0.321805\pi$
$293$	1.56070e6	1.06206	0.531031	+	0.847352i	$0.321805\pi$
$294$	0	0
$295$	220357.	0.147425
$296$	0	0
$297$	0	0
$298$	0	0
$299$	207305.	0.134101
$300$	0	0
$301$	1.01891e6	0.648219
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.30436e6	0.802875
$306$	0	0
$307$	−889308.	−0.538525	−0.269263	−	0.963067i	$-0.586780\pi$
$307$	−889308.	−0.538525	−0.269263	+	0.963067i	$0.586780\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.44826e6	−1.43535	−0.717674	−	0.696380i	$-0.754793\pi$
$311$	−2.44826e6	−1.43535	−0.717674	+	0.696380i	$0.754793\pi$
$312$	0	0
$313$	−2.73102e6	−1.57567	−0.787834	−	0.615887i	$-0.788798\pi$
$313$	−2.73102e6	−1.57567	−0.787834	+	0.615887i	$0.788798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	249603.	0.139509	0.0697545	−	0.997564i	$-0.477778\pi$
$317$	249603.	0.139509	0.0697545	+	0.997564i	$0.477778\pi$
$318$	0	0
$319$	1.09844e6	0.604364
$320$	0	0
$321$	0	0
$322$	0	0
$323$	692115.	0.369124
$324$	0	0
$325$	−184936.	−0.0971209
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−209885.	−0.106903
$330$	0	0
$331$	646617.	0.324397	0.162199	−	0.986758i	$-0.448142\pi$
$331$	646617.	0.324397	0.162199	+	0.986758i	$0.448142\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.85433e6	−2.36329
$336$	0	0
$337$	−1.02782e6	−0.492994	−0.246497	−	0.969144i	$-0.579280\pi$
$337$	−1.02782e6	−0.492994	−0.246497	+	0.969144i	$0.579280\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.77445e6	−0.826375
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.64916e6	0.735256	0.367628	−	0.929973i	$-0.380170\pi$
$347$	1.64916e6	0.735256	0.367628	+	0.929973i	$0.380170\pi$
$348$	0	0
$349$	2.21201e6	0.972128	0.486064	−	0.873923i	$-0.338432\pi$
$349$	2.21201e6	0.972128	0.486064	+	0.873923i	$0.338432\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.10158e6	−0.470520	−0.235260	−	0.971932i	$-0.575594\pi$
$353$	−1.10158e6	−0.470520	−0.235260	+	0.971932i	$0.575594\pi$
$354$	0	0
$355$	−5.61242e6	−2.36363
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.29940e6	0.532116	0.266058	−	0.963957i	$-0.414279\pi$
$359$	1.29940e6	0.532116	0.266058	+	0.963957i	$0.414279\pi$
$360$	0	0
$361$	−2.03119e6	−0.820317
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.32196e6	−1.69804
$366$	0	0
$367$	1.55669e6	0.603305	0.301652	−	0.953418i	$-0.402462\pi$
$367$	1.55669e6	0.603305	0.301652	+	0.953418i	$0.402462\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.26020e6	0.475341
$372$	0	0
$373$	−3.12660e6	−1.16359	−0.581796	−	0.813335i	$-0.697650\pi$
$373$	−3.12660e6	−1.16359	−0.581796	+	0.813335i	$0.697650\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	146697.	0.0531579
$378$	0	0
$379$	−2.96497e6	−1.06029	−0.530143	−	0.847908i	$-0.677862\pi$
$379$	−2.96497e6	−1.06029	−0.530143	+	0.847908i	$0.677862\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.74774e6	−0.957149	−0.478574	−	0.878047i	$-0.658846\pi$
$383$	−2.74774e6	−0.957149	−0.478574	+	0.878047i	$0.658846\pi$
$384$	0	0
$385$	1.81579e6	0.624330
$386$	0	0
$387$	0	0
$388$	0	0
$389$	611343.	0.204838	0.102419	−	0.994741i	$-0.467342\pi$
$389$	611343.	0.204838	0.102419	+	0.994741i	$0.467342\pi$
$390$	0	0
$391$	3.37397e6	1.11609
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−309664.	−0.0998615
$396$	0	0
$397$	2.16794e6	0.690352	0.345176	−	0.938538i	$-0.387819\pi$
$397$	2.16794e6	0.690352	0.345176	+	0.938538i	$0.387819\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.14644e6	0.977143	0.488572	−	0.872524i	$-0.337518\pi$
$401$	3.14644e6	0.977143	0.488572	+	0.872524i	$0.337518\pi$
$402$	0	0
$403$	−236978.	−0.0726852
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.84593e6	−1.74931
$408$	0	0
$409$	5.58165e6	1.64989	0.824943	−	0.565215i	$-0.191207\pi$
$409$	5.58165e6	1.64989	0.824943	+	0.565215i	$0.191207\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−139097.	−0.0401275
$414$	0	0
$415$	3.57794e6	1.01980
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.25054e6	−0.626257	−0.313128	−	0.949711i	$-0.601377\pi$
$419$	−2.25054e6	−0.626257	−0.313128	+	0.949711i	$0.601377\pi$
$420$	0	0
$421$	3.45914e6	0.951180	0.475590	−	0.879667i	$-0.342235\pi$
$421$	3.45914e6	0.951180	0.475590	+	0.879667i	$0.342235\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.00990e6	−0.808313
$426$	0	0
$427$	−823356.	−0.218534
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.55926e6	1.70083	0.850417	−	0.526109i	$-0.176350\pi$
$431$	6.55926e6	1.70083	0.850417	+	0.526109i	$0.176350\pi$
$432$	0	0
$433$	5.05669e6	1.29612	0.648062	−	0.761587i	$-0.275580\pi$
$433$	5.05669e6	1.29612	0.648062	+	0.761587i	$0.275580\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.16889e6	0.543293
$438$	0	0
$439$	4.23225e6	1.04812	0.524059	−	0.851682i	$-0.324417\pi$
$439$	4.23225e6	1.04812	0.524059	+	0.851682i	$0.324417\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.05525e6	1.46596	0.732981	−	0.680249i	$-0.238128\pi$
$443$	6.05525e6	1.46596	0.732981	+	0.680249i	$0.238128\pi$
$444$	0	0
$445$	−1.05093e7	−2.51579
$446$	0	0
$447$	0	0
$448$	0	0
$449$	299186.	0.0700368	0.0350184	−	0.999387i	$-0.488851\pi$
$449$	299186.	0.0700368	0.0350184	+	0.999387i	$0.488851\pi$
$450$	0	0
$451$	−873438.	−0.202204
$452$	0	0
$453$	0	0
$454$	0	0
$455$	242500.	0.0549140
$456$	0	0
$457$	3.75177e6	0.840322	0.420161	−	0.907450i	$-0.361974\pi$
$457$	3.75177e6	0.840322	0.420161	+	0.907450i	$0.361974\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.94525e6	−1.52207	−0.761036	−	0.648709i	$-0.775309\pi$
$461$	−6.94525e6	−1.52207	−0.761036	+	0.648709i	$0.775309\pi$
$462$	0	0
$463$	−9.13226e6	−1.97982	−0.989910	−	0.141697i	$-0.954744\pi$
$463$	−9.13226e6	−1.97982	−0.989910	+	0.141697i	$0.954744\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	424136.	0.0899940	0.0449970	−	0.998987i	$-0.485672\pi$
$467$	424136.	0.0899940	0.0449970	+	0.998987i	$0.485672\pi$
$468$	0	0
$469$	3.06422e6	0.643262
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.92673e6	2.04011
$474$	0	0
$475$	−1.93485e6	−0.393472
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.78605e6	1.55052	0.775262	−	0.631640i	$-0.217618\pi$
$479$	7.78605e6	1.55052	0.775262	+	0.631640i	$0.217618\pi$
$480$	0	0
$481$	−780727.	−0.153864
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.10910e7	2.14100
$486$	0	0
$487$	2.32259e6	0.443763	0.221881	−	0.975074i	$-0.428780\pi$
$487$	2.32259e6	0.443763	0.221881	+	0.975074i	$0.428780\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6.01036e6	−1.12512	−0.562558	−	0.826758i	$-0.690183\pi$
$491$	−6.01036e6	−1.12512	−0.562558	+	0.826758i	$0.690183\pi$
$492$	0	0
$493$	2.38755e6	0.442420
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.54275e6	0.643353
$498$	0	0
$499$	3.37385e6	0.606562	0.303281	−	0.952901i	$-0.401918\pi$
$499$	3.37385e6	0.606562	0.303281	+	0.952901i	$0.401918\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.22068e6	0.215120	0.107560	−	0.994199i	$-0.465696\pi$
$503$	1.22068e6	0.215120	0.107560	+	0.994199i	$0.465696\pi$
$504$	0	0
$505$	−3.45179e6	−0.602304
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.56897e6	−0.268423	−0.134212	−	0.990953i	$-0.542850\pi$
$509$	−1.56897e6	−0.268423	−0.134212	+	0.990953i	$0.542850\pi$
$510$	0	0
$511$	2.72817e6	0.462188
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.57285e7	2.61318
$516$	0	0
$517$	−2.04480e6	−0.336452
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.06779e7	−1.72342	−0.861708	−	0.507404i	$-0.830605\pi$
$521$	−1.06779e7	−1.72342	−0.861708	+	0.507404i	$0.830605\pi$
$522$	0	0
$523$	−1.21007e7	−1.93444	−0.967219	−	0.253943i	$-0.918272\pi$
$523$	−1.21007e7	−1.93444	−0.967219	+	0.253943i	$0.918272\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.85691e6	−0.604941
$528$	0	0
$529$	4.13673e6	0.642714
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−116648.	−0.0177852
$534$	0	0
$535$	−7.72577e6	−1.16696
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.14619e6	−0.169936
$540$	0	0
$541$	−3.19805e6	−0.469778	−0.234889	−	0.972022i	$-0.575473\pi$
$541$	−3.19805e6	−0.469778	−0.234889	+	0.972022i	$0.575473\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.71499e7	2.47326
$546$	0	0
$547$	3.97811e6	0.568471	0.284235	−	0.958755i	$-0.408260\pi$
$547$	3.97811e6	0.568471	0.284235	+	0.958755i	$0.408260\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.53479e6	0.215362
$552$	0	0
$553$	195470.	0.0271812
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.14113e7	−1.55847	−0.779236	−	0.626731i	$-0.784392\pi$
$557$	−1.14113e7	−1.55847	−0.779236	+	0.626731i	$0.784392\pi$
$558$	0	0
$559$	1.32572e6	0.179441
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.46973e6	0.860231	0.430115	−	0.902774i	$-0.358473\pi$
$563$	6.46973e6	0.860231	0.430115	+	0.902774i	$0.358473\pi$
$564$	0	0
$565$	−2.29952e6	−0.303051
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−717028.	−0.0928444	−0.0464222	−	0.998922i	$-0.514782\pi$
$569$	−717028.	−0.0928444	−0.0464222	+	0.998922i	$0.514782\pi$
$570$	0	0
$571$	6.00286e6	0.770491	0.385246	−	0.922814i	$-0.374117\pi$
$571$	6.00286e6	0.770491	0.385246	+	0.922814i	$0.374117\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9.43217e6	−1.18971
$576$	0	0
$577$	1.55712e7	1.94707	0.973537	−	0.228531i	$-0.0733922\pi$
$577$	1.55712e7	1.94707	0.973537	+	0.228531i	$0.0733922\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.25852e6	−0.277577
$582$	0	0
$583$	1.22775e7	1.49602
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.84621e6	−0.939863	−0.469931	−	0.882703i	$-0.655721\pi$
$587$	−7.84621e6	−0.939863	−0.469931	+	0.882703i	$0.655721\pi$
$588$	0	0
$589$	−2.47934e6	−0.294475
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.33248e7	−1.55605	−0.778026	−	0.628232i	$-0.783779\pi$
$593$	−1.33248e7	−1.55605	−0.778026	+	0.628232i	$0.783779\pi$
$594$	0	0
$595$	3.94678e6	0.457036
$596$	0	0
$597$	0	0
$598$	0	0
$599$	7.77916e6	0.885861	0.442930	−	0.896556i	$-0.353939\pi$
$599$	7.77916e6	0.885861	0.442930	+	0.896556i	$0.353939\pi$
$600$	0	0
$601$	8.62898e6	0.974480	0.487240	−	0.873268i	$-0.338004\pi$
$601$	8.62898e6	0.974480	0.487240	+	0.873268i	$0.338004\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.18857e6	0.576314
$606$	0	0
$607$	−7.09353e6	−0.781431	−0.390715	−	0.920512i	$-0.627772\pi$
$607$	−7.09353e6	−0.781431	−0.390715	+	0.920512i	$0.627772\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−273084.	−0.0295932
$612$	0	0
$613$	4.52640e6	0.486521	0.243261	−	0.969961i	$-0.421783\pi$
$613$	4.52640e6	0.486521	0.243261	+	0.969961i	$0.421783\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.38009e6	0.886208	0.443104	−	0.896470i	$-0.353877\pi$
$617$	8.38009e6	0.886208	0.443104	+	0.896470i	$0.353877\pi$
$618$	0	0
$619$	−168342.	−0.0176590	−0.00882949	−	0.999961i	$-0.502811\pi$
$619$	−168342.	−0.0176590	−0.00882949	+	0.999961i	$0.502811\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.63385e6	0.684771
$624$	0	0
$625$	−1.04161e7	−1.06661
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.27066e7	−1.28057
$630$	0	0
$631$	1.66208e7	1.66180	0.830899	−	0.556423i	$-0.187826\pi$
$631$	1.66208e7	1.66180	0.830899	+	0.556423i	$0.187826\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.05034e7	−2.01786
$636$	0	0
$637$	−153074.	−0.0149470
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.02473e6	−0.0985068	−0.0492534	−	0.998786i	$-0.515684\pi$
$641$	−1.02473e6	−0.0985068	−0.0492534	+	0.998786i	$0.515684\pi$
$642$	0	0
$643$	−1.22962e7	−1.17286	−0.586428	−	0.810001i	$-0.699466\pi$
$643$	−1.22962e7	−1.17286	−0.586428	+	0.810001i	$0.699466\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.16537e6	0.203363	0.101682	−	0.994817i	$-0.467578\pi$
$647$	2.16537e6	0.203363	0.101682	+	0.994817i	$0.467578\pi$
$648$	0	0
$649$	−1.35515e6	−0.126292
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.22501e7	1.12424	0.562119	−	0.827056i	$-0.309986\pi$
$653$	1.22501e7	1.12424	0.562119	+	0.827056i	$0.309986\pi$
$654$	0	0
$655$	6.04317e6	0.550379
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.18607e6	−0.106389	−0.0531944	−	0.998584i	$-0.516940\pi$
$659$	−1.18607e6	−0.106389	−0.0531944	+	0.998584i	$0.516940\pi$
$660$	0	0
$661$	−1.45382e7	−1.29421	−0.647107	−	0.762399i	$-0.724021\pi$
$661$	−1.45382e7	−1.29421	−0.647107	+	0.762399i	$0.724021\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.53711e6	0.222477
$666$	0	0
$667$	7.48190e6	0.651174
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.02151e6	−0.687781
$672$	0	0
$673$	−5.99405e6	−0.510132	−0.255066	−	0.966924i	$-0.582097\pi$
$673$	−5.99405e6	−0.510132	−0.255066	+	0.966924i	$0.582097\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.08389e7	−1.74744	−0.873721	−	0.486428i	$-0.838300\pi$
$677$	−2.08389e7	−1.74744	−0.873721	+	0.486428i	$0.838300\pi$
$678$	0	0
$679$	−7.00102e6	−0.582756
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−4.55565e6	−0.373679	−0.186840	−	0.982390i	$-0.559824\pi$
$683$	−4.55565e6	−0.373679	−0.186840	+	0.982390i	$0.559824\pi$
$684$	0	0
$685$	−2.89080e7	−2.35392
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.63966e6	0.131585
$690$	0	0
$691$	1.68542e6	0.134281	0.0671404	−	0.997744i	$-0.478612\pi$
$691$	1.68542e6	0.134281	0.0671404	+	0.997744i	$0.478612\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.13122e7	−1.67365
$696$	0	0
$697$	−1.89849e6	−0.148022
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.45349e6	−0.188577	−0.0942887	−	0.995545i	$-0.530058\pi$
$701$	−2.45349e6	−0.188577	−0.0942887	+	0.995545i	$0.530058\pi$
$702$	0	0
$703$	−8.16820e6	−0.623359
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.17889e6	0.163940
$708$	0	0
$709$	−1.32314e7	−0.988530	−0.494265	−	0.869311i	$-0.664563\pi$
$709$	−1.32314e7	−0.988530	−0.494265	+	0.869311i	$0.664563\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.20865e7	−0.890380
$714$	0	0
$715$	2.36255e6	0.172828
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.35337e7	−0.976324	−0.488162	−	0.872753i	$-0.662333\pi$
$719$	−1.35337e7	−0.976324	−0.488162	+	0.872753i	$0.662333\pi$
$720$	0	0
$721$	−9.92835e6	−0.711277
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.67455e6	−0.471604
$726$	0	0
$727$	5.29416e6	0.371502	0.185751	−	0.982597i	$-0.440528\pi$
$727$	5.29416e6	0.371502	0.185751	+	0.982597i	$0.440528\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.15766e7	1.49344
$732$	0	0
$733$	−2.25014e7	−1.54685	−0.773426	−	0.633886i	$-0.781459\pi$
$733$	−2.25014e7	−1.54685	−0.773426	+	0.633886i	$0.781459\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.98530e7	2.02451
$738$	0	0
$739$	1.60739e7	1.08271	0.541353	−	0.840795i	$-0.317912\pi$
$739$	1.60739e7	1.08271	0.541353	+	0.840795i	$0.317912\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.31604e6	0.153913	0.0769563	−	0.997034i	$-0.475480\pi$
$743$	2.31604e6	0.153913	0.0769563	+	0.997034i	$0.475480\pi$
$744$	0	0
$745$	2.42955e7	1.60374
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.87677e6	0.317634
$750$	0	0
$751$	−7.77263e6	−0.502885	−0.251442	−	0.967872i	$-0.580905\pi$
$751$	−7.77263e6	−0.502885	−0.251442	+	0.967872i	$0.580905\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.35417e7	2.14150
$756$	0	0
$757$	−1.59716e7	−1.01300	−0.506498	−	0.862241i	$-0.669060\pi$
$757$	−1.59716e7	−1.01300	−0.506498	+	0.862241i	$0.669060\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.84940e7	−1.15763	−0.578815	−	0.815459i	$-0.696485\pi$
$761$	−1.84940e7	−1.15763	−0.578815	+	0.815459i	$0.696485\pi$
$762$	0	0
$763$	−1.08256e7	−0.673193
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−180980.	−0.0111082
$768$	0	0
$769$	−2.33524e7	−1.42402	−0.712009	−	0.702170i	$-0.752214\pi$
$769$	−2.33524e7	−1.42402	−0.712009	+	0.702170i	$0.752214\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−7.25262e6	−0.436562	−0.218281	−	0.975886i	$-0.570045\pi$
$773$	−7.25262e6	−0.436562	−0.218281	+	0.975886i	$0.570045\pi$
$774$	0	0
$775$	1.07823e7	0.644845
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.22041e6	−0.0720546
$780$	0	0
$781$	3.45151e7	2.02480
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.56742e6	−0.380383
$786$	0	0
$787$	1.18935e6	0.0684497	0.0342248	−	0.999414i	$-0.489104\pi$
$787$	1.18935e6	0.0684497	0.0342248	+	0.999414i	$0.489104\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.45153e6	0.0824871
$792$	0	0
$793$	−1.07128e6	−0.0604949
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.14041e7	−1.19358	−0.596790	−	0.802397i	$-0.703558\pi$
$797$	−2.14041e7	−1.19358	−0.596790	+	0.802397i	$0.703558\pi$
$798$	0	0
$799$	−4.44453e6	−0.246297
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.65790e7	1.45462
$804$	0	0
$805$	1.23681e7	0.672686
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−9.60341e6	−0.515886	−0.257943	−	0.966160i	$-0.583045\pi$
$809$	−9.60341e6	−0.515886	−0.257943	+	0.966160i	$0.583045\pi$
$810$	0	0
$811$	2.62263e7	1.40018	0.700091	−	0.714054i	$-0.253143\pi$
$811$	2.62263e7	1.40018	0.700091	+	0.714054i	$0.253143\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.37770e7	1.25390
$816$	0	0
$817$	1.38701e7	0.726982
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.65685e7	−1.89343	−0.946715	−	0.322073i	$-0.895620\pi$
$821$	−3.65685e7	−1.89343	−0.946715	+	0.322073i	$0.895620\pi$
$822$	0	0
$823$	−7.73488e6	−0.398065	−0.199032	−	0.979993i	$-0.563780\pi$
$823$	−7.73488e6	−0.398065	−0.199032	+	0.979993i	$0.563780\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.47172e7	0.748277	0.374138	−	0.927373i	$-0.377939\pi$
$827$	1.47172e7	0.748277	0.374138	+	0.927373i	$0.377939\pi$
$828$	0	0
$829$	−7.41886e6	−0.374931	−0.187465	−	0.982271i	$-0.560027\pi$
$829$	−7.41886e6	−0.374931	−0.187465	+	0.982271i	$0.560027\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.49134e6	−0.124400
$834$	0	0
$835$	4.70811e7	2.33685
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−116538.	−0.00571559	−0.00285780	−	0.999996i	$-0.500910\pi$
$839$	−116538.	−0.00571559	−0.00285780	+	0.999996i	$0.500910\pi$
$840$	0	0
$841$	−1.52167e7	−0.741874
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.85064e7	−1.37341
$846$	0	0
$847$	−3.27520e6	−0.156866
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.98190e7	−1.88480
$852$	0	0
$853$	−1.91763e7	−0.902387	−0.451193	−	0.892426i	$-0.649002\pi$
$853$	−1.91763e7	−0.902387	−0.451193	+	0.892426i	$0.649002\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.62548e6	0.0756012	0.0378006	−	0.999285i	$-0.487965\pi$
$857$	1.62548e6	0.0756012	0.0378006	+	0.999285i	$0.487965\pi$
$858$	0	0
$859$	−1.65931e7	−0.767265	−0.383632	−	0.923486i	$-0.625327\pi$
$859$	−1.65931e7	−0.767265	−0.383632	+	0.923486i	$0.625327\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.30415e7	1.05314	0.526568	−	0.850133i	$-0.323479\pi$
$863$	2.30415e7	1.05314	0.526568	+	0.850133i	$0.323479\pi$
$864$	0	0
$865$	2.23935e7	1.01761
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.90436e6	0.0855460
$870$	0	0
$871$	3.98689e6	0.178069
$872$	0	0
$873$	0	0
$874$	0	0
$875$	852954.	0.0376622
$876$	0	0
$877$	1.84493e7	0.809993	0.404996	−	0.914318i	$-0.367273\pi$
$877$	1.84493e7	0.809993	0.404996	+	0.914318i	$0.367273\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3.70548e7	1.60844	0.804219	−	0.594333i	$-0.202584\pi$
$881$	3.70548e7	1.60844	0.804219	+	0.594333i	$0.202584\pi$
$882$	0	0
$883$	−5.28466e6	−0.228095	−0.114047	−	0.993475i	$-0.536382\pi$
$883$	−5.28466e6	−0.228095	−0.114047	+	0.993475i	$0.536382\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.98545e7	0.847326	0.423663	−	0.905820i	$-0.360744\pi$
$887$	1.98545e7	0.847326	0.423663	+	0.905820i	$0.360744\pi$
$888$	0	0
$889$	1.29425e7	0.549240
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.85708e6	−0.119893
$894$	0	0
$895$	−1.15552e7	−0.482193
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8.55283e6	−0.352948
$900$	0	0
$901$	2.66861e7	1.09515
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.24852e6	0.294190
$906$	0	0
$907$	−2.00483e7	−0.809207	−0.404603	−	0.914492i	$-0.632590\pi$
$907$	−2.00483e7	−0.809207	−0.404603	+	0.914492i	$0.632590\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	765753.	0.0305698	0.0152849	−	0.999883i	$-0.495134\pi$
$911$	765753.	0.0305698	0.0152849	+	0.999883i	$0.495134\pi$
$912$	0	0
$913$	−2.20035e7	−0.873605
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.81465e6	−0.149807
$918$	0	0
$919$	−1.87846e7	−0.733692	−0.366846	−	0.930282i	$-0.619562\pi$
$919$	−1.87846e7	−0.733692	−0.366846	+	0.930282i	$0.619562\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.60951e6	0.178094
$924$	0	0
$925$	3.55222e7	1.36504
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.56290e7	1.73461	0.867304	−	0.497778i	$-0.165851\pi$
$929$	4.56290e7	1.73461	0.867304	+	0.497778i	$0.165851\pi$
$930$	0	0
$931$	−1.60151e6	−0.0605557
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.84513e7	1.43841
$936$	0	0
$937$	6.67800e6	0.248484	0.124242	−	0.992252i	$-0.460350\pi$
$937$	6.67800e6	0.248484	0.124242	+	0.992252i	$0.460350\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.42716e7	−1.26171	−0.630857	−	0.775899i	$-0.717297\pi$
$941$	−3.42716e7	−1.26171	−0.630857	+	0.775899i	$0.717297\pi$
$942$	0	0
$943$	−5.94933e6	−0.217866
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.28766e7	−1.91597	−0.957984	−	0.286821i	$-0.907402\pi$
$947$	−5.28766e7	−1.91597	−0.957984	+	0.286821i	$0.907402\pi$
$948$	0	0
$949$	3.54965e6	0.127944
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−5.33439e6	−0.190262	−0.0951311	−	0.995465i	$-0.530327\pi$
$953$	−5.33439e6	−0.190262	−0.0951311	+	0.995465i	$0.530327\pi$
$954$	0	0
$955$	−1.91669e7	−0.680056
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.82477e7	0.640711
$960$	0	0
$961$	−1.48127e7	−0.517398
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.73719e7	1.29189
$966$	0	0
$967$	−1.93877e7	−0.666744	−0.333372	−	0.942795i	$-0.608187\pi$
$967$	−1.93877e7	−0.666744	−0.333372	+	0.942795i	$0.608187\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1.07463e7	−0.365772	−0.182886	−	0.983134i	$-0.558544\pi$
$971$	−1.07463e7	−0.365772	−0.182886	+	0.983134i	$0.558544\pi$
$972$	0	0
$973$	1.34530e7	0.455550
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.41051e7	0.807926	0.403963	−	0.914775i	$-0.367632\pi$
$977$	2.41051e7	0.807926	0.403963	+	0.914775i	$0.367632\pi$
$978$	0	0
$979$	6.46300e7	2.15515
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4.05038e7	1.33694	0.668470	−	0.743739i	$-0.266950\pi$
$983$	4.05038e7	1.33694	0.668470	+	0.743739i	$0.266950\pi$
$984$	0	0
$985$	4.25573e7	1.39760
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.76149e7	2.19812
$990$	0	0
$991$	−4.01588e7	−1.29896	−0.649482	−	0.760377i	$-0.725014\pi$
$991$	−4.01588e7	−1.29896	−0.649482	+	0.760377i	$0.725014\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.22788e7	0.393185
$996$	0	0
$997$	−2.72276e7	−0.867505	−0.433753	−	0.901032i	$-0.642811\pi$
$997$	−2.72276e7	−0.867505	−0.433753	+	0.901032i	$0.642811\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.6.a.h.1.2		2
3.2	odd	2		84.6.a.c.1.1	✓	2
4.3	odd	2		1008.6.a.bo.1.2		2
12.11	even	2		336.6.a.x.1.1		2
21.2	odd	6		588.6.i.l.361.2		4
21.5	even	6		588.6.i.i.361.1		4
21.11	odd	6		588.6.i.l.373.2		4
21.17	even	6		588.6.i.i.373.1		4
21.20	even	2		588.6.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.a.c.1.1	✓	2	3.2	odd	2
252.6.a.h.1.2		2	1.1	even	1	trivial
336.6.a.x.1.1		2	12.11	even	2
588.6.a.k.1.2		2	21.20	even	2
588.6.i.i.361.1		4	21.5	even	6
588.6.i.i.373.1		4	21.17	even	6
588.6.i.l.361.2		4	21.2	odd	6
588.6.i.l.373.2		4	21.11	odd	6
1008.6.a.bo.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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	252.a (trivial)

\(f(q)\)	\(=\)	\(q+77.6257 q^{5} -49.0000 q^{7} +O(q^{10})\)	\(q+77.6257 q^{5} -49.0000 q^{7} -477.380 q^{11} -63.7544 q^{13} -1037.63 q^{17} -667.018 q^{19} -3251.63 q^{23} +2900.75 q^{25} -2300.97 q^{29} +3717.05 q^{31} -3803.66 q^{35} +12245.9 q^{37} +1829.65 q^{41} -20794.2 q^{43} +4283.37 q^{47} +2401.00 q^{49} -25718.4 q^{53} -37057.0 q^{55} +2838.71 q^{59} +16803.2 q^{61} -4948.98 q^{65} -62535.1 q^{67} -72301.0 q^{71} -55676.9 q^{73} +23391.6 q^{77} -3989.19 q^{79} +46092.2 q^{83} -80546.5 q^{85} -135385. q^{89} +3123.97 q^{91} -51777.7 q^{95} +142878. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5} - 98 q^{7}+O(q^{10}) \) 2 * q + 6 * q^5 - 98 * q^7	\( 2 q + 6 q^{5} - 98 q^{7} + 90 q^{11} + 768 q^{13} - 1926 q^{17} + 2248 q^{19} - 6354 q^{23} + 4906 q^{25} - 10572 q^{29} - 3312 q^{31} - 294 q^{35} + 2104 q^{37} - 1266 q^{41} - 5768 q^{43} - 15612 q^{47} + 4802 q^{49} - 16512 q^{53} - 77696 q^{55} + 13140 q^{59} - 5796 q^{61} - 64524 q^{65} - 56116 q^{67} - 11022 q^{71} - 85384 q^{73} - 4410 q^{77} - 19620 q^{79} + 44424 q^{83} - 16916 q^{85} - 211218 q^{89} - 37632 q^{91} - 260568 q^{95} + 44864 q^{97}+O(q^{100}) \) 2 * q + 6 * q^5 - 98 * q^7 + 90 * q^11 + 768 * q^13 - 1926 * q^17 + 2248 * q^19 - 6354 * q^23 + 4906 * q^25 - 10572 * q^29 - 3312 * q^31 - 294 * q^35 + 2104 * q^37 - 1266 * q^41 - 5768 * q^43 - 15612 * q^47 + 4802 * q^49 - 16512 * q^53 - 77696 * q^55 + 13140 * q^59 - 5796 * q^61 - 64524 * q^65 - 56116 * q^67 - 11022 * q^71 - 85384 * q^73 - 4410 * q^77 - 19620 * q^79 + 44424 * q^83 - 16916 * q^85 - 211218 * q^89 - 37632 * q^91 - 260568 * q^95 + 44864 * q^97

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