LMFDB - Embedded newform 588.6.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,6,Mod(1,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, 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("588.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	588.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:	$94.3056860500$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.1904704.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 87x^{2} + 88x + 1838 $ x^4 - 2x^3 - 87x^2 + 88*x + 1838
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.36093$ of defining polynomial
Character	$\chi$	$=$	588.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -35.3849 q^{5} +81.0000 q^{9} +43.5572 q^{11} +648.252 q^{13} +318.465 q^{15} +598.639 q^{17} -640.987 q^{19} -271.648 q^{23} -1872.91 q^{25} -729.000 q^{27} -5289.56 q^{29} -443.101 q^{31} -392.015 q^{33} -8887.62 q^{37} -5834.27 q^{39} +7487.34 q^{41} +3604.32 q^{43} -2866.18 q^{45} -3015.33 q^{47} -5387.75 q^{51} -7113.47 q^{53} -1541.27 q^{55} +5768.88 q^{57} +32802.9 q^{59} -19163.3 q^{61} -22938.4 q^{65} +14991.9 q^{67} +2444.83 q^{69} +25088.7 q^{71} +11488.2 q^{73} +16856.1 q^{75} -54037.5 q^{79} +6561.00 q^{81} -12148.2 q^{83} -21182.8 q^{85} +47606.0 q^{87} +89045.0 q^{89} +3987.91 q^{93} +22681.3 q^{95} +35813.0 q^{97} +3528.13 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{3} + 324 q^{9} + 1872 q^{17} + 1728 q^{19} - 3648 q^{23} - 3996 q^{25} - 2916 q^{27} - 1248 q^{29} + 3888 q^{31} - 12032 q^{37} - 9072 q^{41} - 2128 q^{43} + 19872 q^{47} - 16848 q^{51} - 22248 q^{53}+ \cdots + 320544 q^{97}+O(q^{100}) $ 4 * q - 36 * q^3 + 324 * q^9 + 1872 * q^17 + 1728 * q^19 - 3648 * q^23 - 3996 * q^25 - 2916 * q^27 - 1248 * q^29 + 3888 * q^31 - 12032 * q^37 - 9072 * q^41 - 2128 * q^43 + 19872 * q^47 - 16848 * q^51 - 22248 * q^53 + 21168 * q^55 - 15552 * q^57 - 5040 * q^59 + 8640 * q^61 - 83880 * q^65 - 86160 * q^67 + 32832 * q^69 - 6720 * q^71 + 18144 * q^73 + 35964 * q^75 - 19648 * q^79 + 26244 * q^81 + 262512 * q^83 + 20168 * q^85 + 11232 * q^87 + 49680 * q^89 - 34992 * q^93 + 112032 * q^95 + 320544 * q^97

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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−35.3849	−0.632985	−0.316493	−	0.948595i	$-0.602505\pi$
$5$	−35.3849	−0.632985	−0.316493	+	0.948595i	$0.602505\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	43.5572	0.108537	0.0542685	−	0.998526i	$-0.482717\pi$
$11$	43.5572	0.108537	0.0542685	+	0.998526i	$0.482717\pi$
$12$	0	0
$13$	648.252	1.06386	0.531931	−	0.846788i	$-0.321466\pi$
$13$	648.252	1.06386	0.531931	+	0.846788i	$0.321466\pi$
$14$	0	0
$15$	318.465	0.365454
$16$	0	0
$17$	598.639	0.502392	0.251196	−	0.967936i	$-0.419176\pi$
$17$	598.639	0.502392	0.251196	+	0.967936i	$0.419176\pi$
$18$	0	0
$19$	−640.987	−0.407348	−0.203674	−	0.979039i	$-0.565288\pi$
$19$	−640.987	−0.407348	−0.203674	+	0.979039i	$0.565288\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−271.648	−0.107075	−0.0535373	−	0.998566i	$-0.517050\pi$
$23$	−271.648	−0.107075	−0.0535373	+	0.998566i	$0.517050\pi$
$24$	0	0
$25$	−1872.91	−0.599330
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−5289.56	−1.16795	−0.583975	−	0.811772i	$-0.698503\pi$
$29$	−5289.56	−1.16795	−0.583975	+	0.811772i	$0.698503\pi$
$30$	0	0
$31$	−443.101	−0.0828131	−0.0414065	−	0.999142i	$-0.513184\pi$
$31$	−443.101	−0.0828131	−0.0414065	+	0.999142i	$0.513184\pi$
$32$	0	0
$33$	−392.015	−0.0626639
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8887.62	−1.06729	−0.533644	−	0.845709i	$-0.679178\pi$
$37$	−8887.62	−1.06729	−0.533644	+	0.845709i	$0.679178\pi$
$38$	0	0
$39$	−5834.27	−0.614221
$40$	0	0
$41$	7487.34	0.695614	0.347807	−	0.937566i	$-0.386927\pi$
$41$	7487.34	0.695614	0.347807	+	0.937566i	$0.386927\pi$
$42$	0	0
$43$	3604.32	0.297270	0.148635	−	0.988892i	$-0.452512\pi$
$43$	3604.32	0.297270	0.148635	+	0.988892i	$0.452512\pi$
$44$	0	0
$45$	−2866.18	−0.210995
$46$	0	0
$47$	−3015.33	−0.199109	−0.0995543	−	0.995032i	$-0.531742\pi$
$47$	−3015.33	−0.199109	−0.0995543	+	0.995032i	$0.531742\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−5387.75	−0.290056
$52$	0	0
$53$	−7113.47	−0.347850	−0.173925	−	0.984759i	$-0.555645\pi$
$53$	−7113.47	−0.347850	−0.173925	+	0.984759i	$0.555645\pi$
$54$	0	0
$55$	−1541.27	−0.0687024
$56$	0	0
$57$	5768.88	0.235182
$58$	0	0
$59$	32802.9	1.22682	0.613412	−	0.789763i	$-0.289797\pi$
$59$	32802.9	1.22682	0.613412	+	0.789763i	$0.289797\pi$
$60$	0	0
$61$	−19163.3	−0.659394	−0.329697	−	0.944087i	$-0.606947\pi$
$61$	−19163.3	−0.659394	−0.329697	+	0.944087i	$0.606947\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−22938.4	−0.673409
$66$	0	0
$67$	14991.9	0.408010	0.204005	−	0.978970i	$-0.434604\pi$
$67$	14991.9	0.408010	0.204005	+	0.978970i	$0.434604\pi$
$68$	0	0
$69$	2444.83	0.0618195
$70$	0	0
$71$	25088.7	0.590653	0.295327	−	0.955396i	$-0.404572\pi$
$71$	25088.7	0.590653	0.295327	+	0.955396i	$0.404572\pi$
$72$	0	0
$73$	11488.2	0.252317	0.126158	−	0.992010i	$-0.459735\pi$
$73$	11488.2	0.252317	0.126158	+	0.992010i	$0.459735\pi$
$74$	0	0
$75$	16856.1	0.346023
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−54037.5	−0.974153	−0.487076	−	0.873359i	$-0.661937\pi$
$79$	−54037.5	−0.974153	−0.487076	+	0.873359i	$0.661937\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−12148.2	−0.193561	−0.0967804	−	0.995306i	$-0.530854\pi$
$83$	−12148.2	−0.193561	−0.0967804	+	0.995306i	$0.530854\pi$
$84$	0	0
$85$	−21182.8	−0.318006
$86$	0	0
$87$	47606.0	0.674316
$88$	0	0
$89$	89045.0	1.19161	0.595806	−	0.803129i	$-0.296833\pi$
$89$	89045.0	1.19161	0.595806	+	0.803129i	$0.296833\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3987.91	0.0478122
$94$	0	0
$95$	22681.3	0.257845
$96$	0	0
$97$	35813.0	0.386466	0.193233	−	0.981153i	$-0.438103\pi$
$97$	35813.0	0.386466	0.193233	+	0.981153i	$0.438103\pi$
$98$	0	0
$99$	3528.13	0.0361790
$100$	0	0
$101$	−30876.4	−0.301178	−0.150589	−	0.988596i	$-0.548117\pi$
$101$	−30876.4	−0.301178	−0.150589	+	0.988596i	$0.548117\pi$
$102$	0	0
$103$	20780.1	0.192999	0.0964994	−	0.995333i	$-0.469235\pi$
$103$	20780.1	0.192999	0.0964994	+	0.995333i	$0.469235\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−109596.	−0.925413	−0.462707	−	0.886511i	$-0.653122\pi$
$107$	−109596.	−0.925413	−0.462707	+	0.886511i	$0.653122\pi$
$108$	0	0
$109$	190846.	1.53857	0.769286	−	0.638905i	$-0.220612\pi$
$109$	190846.	1.53857	0.769286	+	0.638905i	$0.220612\pi$
$110$	0	0
$111$	79988.6	0.616199
$112$	0	0
$113$	193044.	1.42220	0.711099	−	0.703092i	$-0.248198\pi$
$113$	193044.	1.42220	0.711099	+	0.703092i	$0.248198\pi$
$114$	0	0
$115$	9612.24	0.0677766
$116$	0	0
$117$	52508.4	0.354621
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−159154.	−0.988220
$122$	0	0
$123$	−67386.1	−0.401613
$124$	0	0
$125$	176851.	1.01235
$126$	0	0
$127$	124379.	0.684286	0.342143	−	0.939648i	$-0.388847\pi$
$127$	124379.	0.684286	0.342143	+	0.939648i	$0.388847\pi$
$128$	0	0
$129$	−32438.8	−0.171629
$130$	0	0
$131$	19381.2	0.0986737	0.0493368	−	0.998782i	$-0.484289\pi$
$131$	19381.2	0.0986737	0.0493368	+	0.998782i	$0.484289\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	25795.6	0.121818
$136$	0	0
$137$	61952.4	0.282005	0.141002	−	0.990009i	$-0.454967\pi$
$137$	61952.4	0.282005	0.141002	+	0.990009i	$0.454967\pi$
$138$	0	0
$139$	381525.	1.67489	0.837444	−	0.546523i	$-0.184049\pi$
$139$	381525.	1.67489	0.837444	+	0.546523i	$0.184049\pi$
$140$	0	0
$141$	27138.0	0.114955
$142$	0	0
$143$	28236.0	0.115469
$144$	0	0
$145$	187171.	0.739295
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−84025.7	−0.310061	−0.155030	−	0.987910i	$-0.549548\pi$
$149$	−84025.7	−0.310061	−0.155030	+	0.987910i	$0.549548\pi$
$150$	0	0
$151$	53510.3	0.190983	0.0954916	−	0.995430i	$-0.469558\pi$
$151$	53510.3	0.190983	0.0954916	+	0.995430i	$0.469558\pi$
$152$	0	0
$153$	48489.7	0.167464
$154$	0	0
$155$	15679.1	0.0524195
$156$	0	0
$157$	−406847.	−1.31729	−0.658646	−	0.752453i	$-0.728871\pi$
$157$	−406847.	−1.31729	−0.658646	+	0.752453i	$0.728871\pi$
$158$	0	0
$159$	64021.2	0.200831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−457868.	−1.34980	−0.674902	−	0.737907i	$-0.735814\pi$
$163$	−457868.	−1.34980	−0.674902	+	0.737907i	$0.735814\pi$
$164$	0	0
$165$	13871.4	0.0396653
$166$	0	0
$167$	492535.	1.36661	0.683307	−	0.730132i	$-0.260541\pi$
$167$	492535.	1.36661	0.683307	+	0.730132i	$0.260541\pi$
$168$	0	0
$169$	48937.6	0.131803
$170$	0	0
$171$	−51919.9	−0.135783
$172$	0	0
$173$	756539.	1.92184	0.960918	−	0.276833i	$-0.0892849\pi$
$173$	756539.	1.92184	0.960918	+	0.276833i	$0.0892849\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−295226.	−0.708307
$178$	0	0
$179$	605856.	1.41331	0.706654	−	0.707559i	$-0.250204\pi$
$179$	605856.	1.41331	0.706654	+	0.707559i	$0.250204\pi$
$180$	0	0
$181$	491062.	1.11414	0.557070	−	0.830465i	$-0.311925\pi$
$181$	491062.	1.11414	0.557070	+	0.830465i	$0.311925\pi$
$182$	0	0
$183$	172469.	0.380701
$184$	0	0
$185$	314488.	0.675577
$186$	0	0
$187$	26075.0	0.0545281
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−430396.	−0.853660	−0.426830	−	0.904332i	$-0.640370\pi$
$191$	−430396.	−0.853660	−0.426830	+	0.904332i	$0.640370\pi$
$192$	0	0
$193$	32051.2	0.0619371	0.0309685	−	0.999520i	$-0.490141\pi$
$193$	32051.2	0.0619371	0.0309685	+	0.999520i	$0.490141\pi$
$194$	0	0
$195$	206445.	0.388793
$196$	0	0
$197$	283226.	0.519957	0.259978	−	0.965614i	$-0.416285\pi$
$197$	283226.	0.519957	0.259978	+	0.965614i	$0.416285\pi$
$198$	0	0
$199$	528549.	0.946134	0.473067	−	0.881026i	$-0.343147\pi$
$199$	528549.	0.946134	0.473067	+	0.881026i	$0.343147\pi$
$200$	0	0
$201$	−134927.	−0.235565
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−264939.	−0.440313
$206$	0	0
$207$	−22003.5	−0.0356915
$208$	0	0
$209$	−27919.6	−0.0442123
$210$	0	0
$211$	701155.	1.08420	0.542098	−	0.840315i	$-0.317630\pi$
$211$	701155.	1.08420	0.542098	+	0.840315i	$0.317630\pi$
$212$	0	0
$213$	−225799.	−0.341014
$214$	0	0
$215$	−127539.	−0.188168
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−103394.	−0.145675
$220$	0	0
$221$	388069.	0.534476
$222$	0	0
$223$	222810.	0.300036	0.150018	−	0.988683i	$-0.452067\pi$
$223$	222810.	0.300036	0.150018	+	0.988683i	$0.452067\pi$
$224$	0	0
$225$	−151705.	−0.199777
$226$	0	0
$227$	568884.	0.732755	0.366377	−	0.930466i	$-0.380598\pi$
$227$	568884.	0.732755	0.366377	+	0.930466i	$0.380598\pi$
$228$	0	0
$229$	569349.	0.717447	0.358724	−	0.933444i	$-0.383212\pi$
$229$	569349.	0.717447	0.358724	+	0.933444i	$0.383212\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.14235e6	−1.37851	−0.689254	−	0.724520i	$-0.742061\pi$
$233$	−1.14235e6	−1.37851	−0.689254	+	0.724520i	$0.742061\pi$
$234$	0	0
$235$	106697.	0.126033
$236$	0	0
$237$	486337.	0.562427
$238$	0	0
$239$	955201.	1.08168	0.540842	−	0.841124i	$-0.318106\pi$
$239$	955201.	1.08168	0.540842	+	0.841124i	$0.318106\pi$
$240$	0	0
$241$	371670.	0.412206	0.206103	−	0.978530i	$-0.433922\pi$
$241$	371670.	0.412206	0.206103	+	0.978530i	$0.433922\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−415521.	−0.433362
$248$	0	0
$249$	109334.	0.111752
$250$	0	0
$251$	−791754.	−0.793242	−0.396621	−	0.917982i	$-0.629817\pi$
$251$	−791754.	−0.793242	−0.396621	+	0.917982i	$0.629817\pi$
$252$	0	0
$253$	−11832.2	−0.0116216
$254$	0	0
$255$	190645.	0.183601
$256$	0	0
$257$	293629.	0.277310	0.138655	−	0.990341i	$-0.455722\pi$
$257$	293629.	0.277310	0.138655	+	0.990341i	$0.455722\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−428454.	−0.389317
$262$	0	0
$263$	999585.	0.891108	0.445554	−	0.895255i	$-0.353007\pi$
$263$	999585.	0.891108	0.445554	+	0.895255i	$0.353007\pi$
$264$	0	0
$265$	251710.	0.220184
$266$	0	0
$267$	−801405.	−0.687977
$268$	0	0
$269$	−327391.	−0.275858	−0.137929	−	0.990442i	$-0.544045\pi$
$269$	−327391.	−0.275858	−0.137929	+	0.990442i	$0.544045\pi$
$270$	0	0
$271$	754837.	0.624353	0.312176	−	0.950024i	$-0.398942\pi$
$271$	754837.	0.624353	0.312176	+	0.950024i	$0.398942\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−81578.5	−0.0650495
$276$	0	0
$277$	−1.83345e6	−1.43572	−0.717861	−	0.696187i	$-0.754879\pi$
$277$	−1.83345e6	−1.43572	−0.717861	+	0.696187i	$0.754879\pi$
$278$	0	0
$279$	−35891.2	−0.0276044
$280$	0	0
$281$	826428.	0.624366	0.312183	−	0.950022i	$-0.398940\pi$
$281$	826428.	0.624366	0.312183	+	0.950022i	$0.398940\pi$
$282$	0	0
$283$	696237.	0.516763	0.258381	−	0.966043i	$-0.416811\pi$
$283$	696237.	0.516763	0.258381	+	0.966043i	$0.416811\pi$
$284$	0	0
$285$	−204132.	−0.148867
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.06149e6	−0.747603
$290$	0	0
$291$	−322317.	−0.223126
$292$	0	0
$293$	−1.74281e6	−1.18599	−0.592994	−	0.805207i	$-0.702054\pi$
$293$	−1.74281e6	−1.18599	−0.592994	+	0.805207i	$0.702054\pi$
$294$	0	0
$295$	−1.16073e6	−0.776561
$296$	0	0
$297$	−31753.2	−0.0208880
$298$	0	0
$299$	−176096.	−0.113913
$300$	0	0
$301$	0	0
$302$	0	0
$303$	277888.	0.173885
$304$	0	0
$305$	678091.	0.417387
$306$	0	0
$307$	3.25919e6	1.97362	0.986810	−	0.161881i	$-0.0517560\pi$
$307$	3.25919e6	1.97362	0.986810	+	0.161881i	$0.0517560\pi$
$308$	0	0
$309$	−187021.	−0.111428
$310$	0	0
$311$	1.04271e6	0.611311	0.305656	−	0.952142i	$-0.401124\pi$
$311$	1.04271e6	0.611311	0.305656	+	0.952142i	$0.401124\pi$
$312$	0	0
$313$	−1.33094e6	−0.767889	−0.383945	−	0.923356i	$-0.625435\pi$
$313$	−1.33094e6	−0.767889	−0.383945	+	0.923356i	$0.625435\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	311977.	0.174371	0.0871855	−	0.996192i	$-0.472213\pi$
$317$	311977.	0.174371	0.0871855	+	0.996192i	$0.472213\pi$
$318$	0	0
$319$	−230398.	−0.126766
$320$	0	0
$321$	986365.	0.534288
$322$	0	0
$323$	−383719.	−0.204648
$324$	0	0
$325$	−1.21411e6	−0.637604
$326$	0	0
$327$	−1.71762e6	−0.888295
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.57687e6	1.29277	0.646387	−	0.763009i	$-0.276279\pi$
$331$	2.57687e6	1.29277	0.646387	+	0.763009i	$0.276279\pi$
$332$	0	0
$333$	−719898.	−0.355762
$334$	0	0
$335$	−530489.	−0.258264
$336$	0	0
$337$	−47287.5	−0.0226815	−0.0113408	−	0.999936i	$-0.503610\pi$
$337$	−47287.5	−0.0226815	−0.0113408	+	0.999936i	$0.503610\pi$
$338$	0	0
$339$	−1.73739e6	−0.821106
$340$	0	0
$341$	−19300.3	−0.00898829
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−86510.1	−0.0391308
$346$	0	0
$347$	329954.	0.147106	0.0735529	−	0.997291i	$-0.476566\pi$
$347$	329954.	0.147106	0.0735529	+	0.997291i	$0.476566\pi$
$348$	0	0
$349$	−3.37497e6	−1.48322	−0.741612	−	0.670830i	$-0.765938\pi$
$349$	−3.37497e6	−1.48322	−0.741612	+	0.670830i	$0.765938\pi$
$350$	0	0
$351$	−472576.	−0.204740
$352$	0	0
$353$	496314.	0.211992	0.105996	−	0.994367i	$-0.466197\pi$
$353$	496314.	0.211992	0.105996	+	0.994367i	$0.466197\pi$
$354$	0	0
$355$	−887763.	−0.373875
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.93333e6	1.61074	0.805368	−	0.592775i	$-0.201968\pi$
$359$	3.93333e6	1.61074	0.805368	+	0.592775i	$0.201968\pi$
$360$	0	0
$361$	−2.06523e6	−0.834068
$362$	0	0
$363$	1.43238e6	0.570549
$364$	0	0
$365$	−406510.	−0.159713
$366$	0	0
$367$	−2.81544e6	−1.09114	−0.545571	−	0.838065i	$-0.683687\pi$
$367$	−2.81544e6	−1.09114	−0.545571	+	0.838065i	$0.683687\pi$
$368$	0	0
$369$	606475.	0.231871
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.48843e6	0.553930	0.276965	−	0.960880i	$-0.410671\pi$
$373$	1.48843e6	0.553930	0.276965	+	0.960880i	$0.410671\pi$
$374$	0	0
$375$	−1.59166e6	−0.584482
$376$	0	0
$377$	−3.42897e6	−1.24254
$378$	0	0
$379$	−1.54843e6	−0.553724	−0.276862	−	0.960910i	$-0.589294\pi$
$379$	−1.54843e6	−0.553724	−0.276862	+	0.960910i	$0.589294\pi$
$380$	0	0
$381$	−1.11941e6	−0.395073
$382$	0	0
$383$	3.18741e6	1.11030	0.555150	−	0.831750i	$-0.312661\pi$
$383$	3.18741e6	1.11030	0.555150	+	0.831750i	$0.312661\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	291950.	0.0990901
$388$	0	0
$389$	3.41619e6	1.14464	0.572319	−	0.820031i	$-0.306044\pi$
$389$	3.41619e6	1.14464	0.572319	+	0.820031i	$0.306044\pi$
$390$	0	0
$391$	−162619.	−0.0537934
$392$	0	0
$393$	−174430.	−0.0569693
$394$	0	0
$395$	1.91211e6	0.616624
$396$	0	0
$397$	−1.81398e6	−0.577638	−0.288819	−	0.957384i	$-0.593263\pi$
$397$	−1.81398e6	−0.577638	−0.288819	+	0.957384i	$0.593263\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.28650e6	0.710084	0.355042	−	0.934850i	$-0.384467\pi$
$401$	2.28650e6	0.710084	0.355042	+	0.934850i	$0.384467\pi$
$402$	0	0
$403$	−287241.	−0.0881017
$404$	0	0
$405$	−232161.	−0.0703317
$406$	0	0
$407$	−387120.	−0.115840
$408$	0	0
$409$	−3.00982e6	−0.889676	−0.444838	−	0.895611i	$-0.646739\pi$
$409$	−3.00982e6	−0.889676	−0.444838	+	0.895611i	$0.646739\pi$
$410$	0	0
$411$	−557571.	−0.162816
$412$	0	0
$413$	0	0
$414$	0	0
$415$	429864.	0.122521
$416$	0	0
$417$	−3.43372e6	−0.966997
$418$	0	0
$419$	−6.78379e6	−1.88772	−0.943860	−	0.330347i	$-0.892834\pi$
$419$	−6.78379e6	−1.88772	−0.943860	+	0.330347i	$0.892834\pi$
$420$	0	0
$421$	−2.60696e6	−0.716852	−0.358426	−	0.933558i	$-0.616686\pi$
$421$	−2.60696e6	−0.716852	−0.358426	+	0.933558i	$0.616686\pi$
$422$	0	0
$423$	−244242.	−0.0663696
$424$	0	0
$425$	−1.12119e6	−0.301098
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−254124.	−0.0666658
$430$	0	0
$431$	3.72540e6	0.966006	0.483003	−	0.875619i	$-0.339546\pi$
$431$	3.72540e6	0.966006	0.483003	+	0.875619i	$0.339546\pi$
$432$	0	0
$433$	−846651.	−0.217013	−0.108506	−	0.994096i	$-0.534607\pi$
$433$	−846651.	−0.217013	−0.108506	+	0.994096i	$0.534607\pi$
$434$	0	0
$435$	−1.68454e6	−0.426832
$436$	0	0
$437$	174123.	0.0436166
$438$	0	0
$439$	−3.77924e6	−0.935930	−0.467965	−	0.883747i	$-0.655013\pi$
$439$	−3.77924e6	−0.935930	−0.467965	+	0.883747i	$0.655013\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.99728e6	1.93612	0.968061	−	0.250714i	$-0.0806655\pi$
$443$	7.99728e6	1.93612	0.968061	+	0.250714i	$0.0806655\pi$
$444$	0	0
$445$	−3.15085e6	−0.754272
$446$	0	0
$447$	756232.	0.179014
$448$	0	0
$449$	6.53115e6	1.52888	0.764441	−	0.644694i	$-0.223015\pi$
$449$	6.53115e6	1.52888	0.764441	+	0.644694i	$0.223015\pi$
$450$	0	0
$451$	326128.	0.0754999
$452$	0	0
$453$	−481593.	−0.110264
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.96167e6	1.55928	0.779638	−	0.626231i	$-0.215403\pi$
$457$	6.96167e6	1.55928	0.779638	+	0.626231i	$0.215403\pi$
$458$	0	0
$459$	−436408.	−0.0966853
$460$	0	0
$461$	−8.54527e6	−1.87272	−0.936362	−	0.351037i	$-0.885829\pi$
$461$	−8.54527e6	−1.87272	−0.936362	+	0.351037i	$0.885829\pi$
$462$	0	0
$463$	−5.93376e6	−1.28640	−0.643202	−	0.765696i	$-0.722394\pi$
$463$	−5.93376e6	−1.28640	−0.643202	+	0.765696i	$0.722394\pi$
$464$	0	0
$465$	−141112.	−0.0302644
$466$	0	0
$467$	250738.	0.0532019	0.0266010	−	0.999646i	$-0.491532\pi$
$467$	250738.	0.0532019	0.0266010	+	0.999646i	$0.491532\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	3.66162e6	0.760539
$472$	0	0
$473$	156994.	0.0322649
$474$	0	0
$475$	1.20051e6	0.244135
$476$	0	0
$477$	−576191.	−0.115950
$478$	0	0
$479$	6.55253e6	1.30488	0.652440	−	0.757841i	$-0.273746\pi$
$479$	6.55253e6	1.30488	0.652440	+	0.757841i	$0.273746\pi$
$480$	0	0
$481$	−5.76142e6	−1.13545
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.26724e6	−0.244627
$486$	0	0
$487$	2.78809e6	0.532702	0.266351	−	0.963876i	$-0.414182\pi$
$487$	2.78809e6	0.532702	0.266351	+	0.963876i	$0.414182\pi$
$488$	0	0
$489$	4.12081e6	0.779310
$490$	0	0
$491$	−2.79773e6	−0.523724	−0.261862	−	0.965105i	$-0.584337\pi$
$491$	−2.79773e6	−0.523724	−0.261862	+	0.965105i	$0.584337\pi$
$492$	0	0
$493$	−3.16653e6	−0.586768
$494$	0	0
$495$	−124843.	−0.0229008
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−3.44767e6	−0.619833	−0.309917	−	0.950764i	$-0.600301\pi$
$499$	−3.44767e6	−0.619833	−0.309917	+	0.950764i	$0.600301\pi$
$500$	0	0
$501$	−4.43281e6	−0.789015
$502$	0	0
$503$	−9.58331e6	−1.68887	−0.844434	−	0.535660i	$-0.820063\pi$
$503$	−9.58331e6	−1.68887	−0.844434	+	0.535660i	$0.820063\pi$
$504$	0	0
$505$	1.09256e6	0.190641
$506$	0	0
$507$	−440438.	−0.0760966
$508$	0	0
$509$	775432.	0.132663	0.0663314	−	0.997798i	$-0.478871\pi$
$509$	775432.	0.132663	0.0663314	+	0.997798i	$0.478871\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	467279.	0.0783941
$514$	0	0
$515$	−735303.	−0.122165
$516$	0	0
$517$	−131339.	−0.0216107
$518$	0	0
$519$	−6.80886e6	−1.10957
$520$	0	0
$521$	−3.38464e6	−0.546284	−0.273142	−	0.961974i	$-0.588063\pi$
$521$	−3.38464e6	−0.546284	−0.273142	+	0.961974i	$0.588063\pi$
$522$	0	0
$523$	190727.	0.0304900	0.0152450	−	0.999884i	$-0.495147\pi$
$523$	190727.	0.0304900	0.0152450	+	0.999884i	$0.495147\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−265258.	−0.0416046
$528$	0	0
$529$	−6.36255e6	−0.988535
$530$	0	0
$531$	2.65703e6	0.408941
$532$	0	0
$533$	4.85368e6	0.740037
$534$	0	0
$535$	3.87805e6	0.585773
$536$	0	0
$537$	−5.45270e6	−0.815974
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3.31351e6	−0.486738	−0.243369	−	0.969934i	$-0.578253\pi$
$541$	−3.31351e6	−0.486738	−0.243369	+	0.969934i	$0.578253\pi$
$542$	0	0
$543$	−4.41956e6	−0.643249
$544$	0	0
$545$	−6.75309e6	−0.973893
$546$	0	0
$547$	−9.56473e6	−1.36680	−0.683399	−	0.730045i	$-0.739499\pi$
$547$	−9.56473e6	−1.36680	−0.683399	+	0.730045i	$0.739499\pi$
$548$	0	0
$549$	−1.55222e6	−0.219798
$550$	0	0
$551$	3.39054e6	0.475762
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2.83039e6	−0.390045
$556$	0	0
$557$	−1.10322e6	−0.150670	−0.0753348	−	0.997158i	$-0.524003\pi$
$557$	−1.10322e6	−0.150670	−0.0753348	+	0.997158i	$0.524003\pi$
$558$	0	0
$559$	2.33650e6	0.316255
$560$	0	0
$561$	−234675.	−0.0314818
$562$	0	0
$563$	6.68581e6	0.888962	0.444481	−	0.895788i	$-0.353388\pi$
$563$	6.68581e6	0.888962	0.444481	+	0.895788i	$0.353388\pi$
$564$	0	0
$565$	−6.83085e6	−0.900230
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.59056e6	0.464923	0.232461	−	0.972606i	$-0.425322\pi$
$569$	3.59056e6	0.464923	0.232461	+	0.972606i	$0.425322\pi$
$570$	0	0
$571$	−6.87106e6	−0.881929	−0.440964	−	0.897525i	$-0.645363\pi$
$571$	−6.87106e6	−0.881929	−0.440964	+	0.897525i	$0.645363\pi$
$572$	0	0
$573$	3.87357e6	0.492861
$574$	0	0
$575$	508770.	0.0641730
$576$	0	0
$577$	−6.36182e6	−0.795503	−0.397752	−	0.917493i	$-0.630209\pi$
$577$	−6.36182e6	−0.795503	−0.397752	+	0.917493i	$0.630209\pi$
$578$	0	0
$579$	−288461.	−0.0357594
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−309843.	−0.0377546
$584$	0	0
$585$	−1.85801e6	−0.224470
$586$	0	0
$587$	−604781.	−0.0724441	−0.0362220	−	0.999344i	$-0.511532\pi$
$587$	−604781.	−0.0724441	−0.0362220	+	0.999344i	$0.511532\pi$
$588$	0	0
$589$	284022.	0.0337337
$590$	0	0
$591$	−2.54903e6	−0.300197
$592$	0	0
$593$	2.84581e6	0.332330	0.166165	−	0.986098i	$-0.446862\pi$
$593$	2.84581e6	0.332330	0.166165	+	0.986098i	$0.446862\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.75694e6	−0.546251
$598$	0	0
$599$	1.42538e7	1.62317	0.811587	−	0.584232i	$-0.198604\pi$
$599$	1.42538e7	1.62317	0.811587	+	0.584232i	$0.198604\pi$
$600$	0	0
$601$	3.70613e6	0.418538	0.209269	−	0.977858i	$-0.432892\pi$
$601$	3.70613e6	0.418538	0.209269	+	0.977858i	$0.432892\pi$
$602$	0	0
$603$	1.21435e6	0.136003
$604$	0	0
$605$	5.63165e6	0.625528
$606$	0	0
$607$	1.17915e6	0.129897	0.0649485	−	0.997889i	$-0.479312\pi$
$607$	1.17915e6	0.129897	0.0649485	+	0.997889i	$0.479312\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.95469e6	−0.211824
$612$	0	0
$613$	−8.43306e6	−0.906429	−0.453214	−	0.891402i	$-0.649723\pi$
$613$	−8.43306e6	−0.906429	−0.453214	+	0.891402i	$0.649723\pi$
$614$	0	0
$615$	2.38445e6	0.254215
$616$	0	0
$617$	1.68786e7	1.78494	0.892471	−	0.451105i	$-0.148970\pi$
$617$	1.68786e7	1.78494	0.892471	+	0.451105i	$0.148970\pi$
$618$	0	0
$619$	716117.	0.0751204	0.0375602	−	0.999294i	$-0.488041\pi$
$619$	716117.	0.0751204	0.0375602	+	0.999294i	$0.488041\pi$
$620$	0	0
$621$	198031.	0.0206065
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−405020.	−0.0414741
$626$	0	0
$627$	251276.	0.0255260
$628$	0	0
$629$	−5.32048e6	−0.536196
$630$	0	0
$631$	−564276.	−0.0564181	−0.0282090	−	0.999602i	$-0.508980\pi$
$631$	−564276.	−0.0564181	−0.0282090	+	0.999602i	$0.508980\pi$
$632$	0	0
$633$	−6.31040e6	−0.625961
$634$	0	0
$635$	−4.40114e6	−0.433143
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.03219e6	0.196884
$640$	0	0
$641$	1.77239e7	1.70378	0.851889	−	0.523722i	$-0.175457\pi$
$641$	1.77239e7	1.70378	0.851889	+	0.523722i	$0.175457\pi$
$642$	0	0
$643$	1.52159e7	1.45134	0.725671	−	0.688042i	$-0.241530\pi$
$643$	1.52159e7	1.45134	0.725671	+	0.688042i	$0.241530\pi$
$644$	0	0
$645$	1.14785e6	0.108639
$646$	0	0
$647$	1.56560e7	1.47035	0.735174	−	0.677879i	$-0.237101\pi$
$647$	1.56560e7	1.47035	0.735174	+	0.677879i	$0.237101\pi$
$648$	0	0
$649$	1.42880e6	0.133156
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−564378.	−0.0517949	−0.0258974	−	0.999665i	$-0.508244\pi$
$653$	−564378.	−0.0517949	−0.0258974	+	0.999665i	$0.508244\pi$
$654$	0	0
$655$	−685801.	−0.0624590
$656$	0	0
$657$	930546.	0.0841055
$658$	0	0
$659$	−1.79285e7	−1.60816	−0.804082	−	0.594518i	$-0.797343\pi$
$659$	−1.79285e7	−1.60816	−0.804082	+	0.594518i	$0.797343\pi$
$660$	0	0
$661$	−6.47211e6	−0.576159	−0.288079	−	0.957607i	$-0.593017\pi$
$661$	−6.47211e6	−0.576159	−0.288079	+	0.957607i	$0.593017\pi$
$662$	0	0
$663$	−3.49262e6	−0.308580
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.43690e6	0.125058
$668$	0	0
$669$	−2.00529e6	−0.173226
$670$	0	0
$671$	−834698.	−0.0715687
$672$	0	0
$673$	1.60916e7	1.36950	0.684751	−	0.728778i	$-0.259911\pi$
$673$	1.60916e7	1.36950	0.684751	+	0.728778i	$0.259911\pi$
$674$	0	0
$675$	1.36535e6	0.115341
$676$	0	0
$677$	1.54481e7	1.29540	0.647700	−	0.761896i	$-0.275731\pi$
$677$	1.54481e7	1.29540	0.647700	+	0.761896i	$0.275731\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−5.11995e6	−0.423056
$682$	0	0
$683$	1.27267e7	1.04391	0.521957	−	0.852972i	$-0.325202\pi$
$683$	1.27267e7	1.04391	0.521957	+	0.852972i	$0.325202\pi$
$684$	0	0
$685$	−2.19218e6	−0.178505
$686$	0	0
$687$	−5.12414e6	−0.414218
$688$	0	0
$689$	−4.61132e6	−0.370064
$690$	0	0
$691$	2.02504e7	1.61338	0.806692	−	0.590972i	$-0.201256\pi$
$691$	2.02504e7	1.61338	0.806692	+	0.590972i	$0.201256\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.35002e7	−1.06018
$696$	0	0
$697$	4.48221e6	0.349470
$698$	0	0
$699$	1.02811e7	0.795881
$700$	0	0
$701$	1.86857e7	1.43619	0.718097	−	0.695943i	$-0.245013\pi$
$701$	1.86857e7	1.43619	0.718097	+	0.695943i	$0.245013\pi$
$702$	0	0
$703$	5.69685e6	0.434757
$704$	0	0
$705$	−960276.	−0.0727651
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.86282e7	−1.39173	−0.695867	−	0.718171i	$-0.744980\pi$
$709$	−1.86282e7	−1.39173	−0.695867	+	0.718171i	$0.744980\pi$
$710$	0	0
$711$	−4.37703e6	−0.324718
$712$	0	0
$713$	120367.	0.00886717
$714$	0	0
$715$	−999131.	−0.0730899
$716$	0	0
$717$	−8.59681e6	−0.624510
$718$	0	0
$719$	1.89245e7	1.36522	0.682611	−	0.730782i	$-0.260844\pi$
$719$	1.89245e7	1.36522	0.682611	+	0.730782i	$0.260844\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−3.34503e6	−0.237988
$724$	0	0
$725$	9.90684e6	0.699987
$726$	0	0
$727$	1.55979e7	1.09454	0.547270	−	0.836956i	$-0.315667\pi$
$727$	1.55979e7	1.09454	0.547270	+	0.836956i	$0.315667\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	2.15768e6	0.149346
$732$	0	0
$733$	8.55919e6	0.588400	0.294200	−	0.955744i	$-0.404947\pi$
$733$	8.55919e6	0.588400	0.294200	+	0.955744i	$0.404947\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	653006.	0.0442842
$738$	0	0
$739$	2.83372e6	0.190873	0.0954367	−	0.995435i	$-0.469575\pi$
$739$	2.83372e6	0.190873	0.0954367	+	0.995435i	$0.469575\pi$
$740$	0	0
$741$	3.73969e6	0.250201
$742$	0	0
$743$	1.04924e7	0.697274	0.348637	−	0.937258i	$-0.386645\pi$
$743$	1.04924e7	0.697274	0.348637	+	0.937258i	$0.386645\pi$
$744$	0	0
$745$	2.97325e6	0.196264
$746$	0	0
$747$	−984005.	−0.0645202
$748$	0	0
$749$	0	0
$750$	0	0
$751$	8.66877e6	0.560864	0.280432	−	0.959874i	$-0.409522\pi$
$751$	8.66877e6	0.560864	0.280432	+	0.959874i	$0.409522\pi$
$752$	0	0
$753$	7.12578e6	0.457979
$754$	0	0
$755$	−1.89346e6	−0.120890
$756$	0	0
$757$	1.08369e7	0.687332	0.343666	−	0.939092i	$-0.388331\pi$
$757$	1.08369e7	0.687332	0.343666	+	0.939092i	$0.388331\pi$
$758$	0	0
$759$	106490.	0.00670971
$760$	0	0
$761$	−398694.	−0.0249562	−0.0124781	−	0.999922i	$-0.503972\pi$
$761$	−398694.	−0.0249562	−0.0124781	+	0.999922i	$0.503972\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.71581e6	−0.106002
$766$	0	0
$767$	2.12645e7	1.30517
$768$	0	0
$769$	1.38915e7	0.847096	0.423548	−	0.905874i	$-0.360785\pi$
$769$	1.38915e7	0.847096	0.423548	+	0.905874i	$0.360785\pi$
$770$	0	0
$771$	−2.64266e6	−0.160105
$772$	0	0
$773$	−2.26688e7	−1.36452	−0.682259	−	0.731111i	$-0.739002\pi$
$773$	−2.26688e7	−1.36452	−0.682259	+	0.731111i	$0.739002\pi$
$774$	0	0
$775$	829887.	0.0496323
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.79929e6	−0.283356
$780$	0	0
$781$	1.09279e6	0.0641078
$782$	0	0
$783$	3.85609e6	0.224772
$784$	0	0
$785$	1.43963e7	0.833827
$786$	0	0
$787$	6.91349e6	0.397887	0.198944	−	0.980011i	$-0.436249\pi$
$787$	6.91349e6	0.397887	0.198944	+	0.980011i	$0.436249\pi$
$788$	0	0
$789$	−8.99626e6	−0.514481
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1.24226e7	−0.701504
$794$	0	0
$795$	−2.26539e6	−0.127123
$796$	0	0
$797$	1.94035e6	0.108202	0.0541009	−	0.998535i	$-0.482771\pi$
$797$	1.94035e6	0.108202	0.0541009	+	0.998535i	$0.482771\pi$
$798$	0	0
$799$	−1.80509e6	−0.100031
$800$	0	0
$801$	7.21265e6	0.397204
$802$	0	0
$803$	500395.	0.0273857
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.94652e6	0.159267
$808$	0	0
$809$	−2.29908e7	−1.23505	−0.617524	−	0.786552i	$-0.711864\pi$
$809$	−2.29908e7	−1.23505	−0.617524	+	0.786552i	$0.711864\pi$
$810$	0	0
$811$	2.57027e7	1.37223	0.686116	−	0.727492i	$-0.259314\pi$
$811$	2.57027e7	1.37223	0.686116	+	0.727492i	$0.259314\pi$
$812$	0	0
$813$	−6.79353e6	−0.360470
$814$	0	0
$815$	1.62016e7	0.854406
$816$	0	0
$817$	−2.31032e6	−0.121092
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.47646e7	−0.764474	−0.382237	−	0.924064i	$-0.624846\pi$
$821$	−1.47646e7	−0.764474	−0.382237	+	0.924064i	$0.624846\pi$
$822$	0	0
$823$	9.50592e6	0.489209	0.244605	−	0.969623i	$-0.421342\pi$
$823$	9.50592e6	0.489209	0.244605	+	0.969623i	$0.421342\pi$
$824$	0	0
$825$	734207.	0.0375564
$826$	0	0
$827$	2.00621e7	1.02003	0.510014	−	0.860166i	$-0.329640\pi$
$827$	2.00621e7	1.02003	0.510014	+	0.860166i	$0.329640\pi$
$828$	0	0
$829$	−1.92254e7	−0.971601	−0.485801	−	0.874070i	$-0.661472\pi$
$829$	−1.92254e7	−0.971601	−0.485801	+	0.874070i	$0.661472\pi$
$830$	0	0
$831$	1.65011e7	0.828914
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.74283e7	−0.865046
$836$	0	0
$837$	323021.	0.0159374
$838$	0	0
$839$	1.80768e7	0.886577	0.443289	−	0.896379i	$-0.353812\pi$
$839$	1.80768e7	0.886577	0.443289	+	0.896379i	$0.353812\pi$
$840$	0	0
$841$	7.46827e6	0.364108
$842$	0	0
$843$	−7.43785e6	−0.360478
$844$	0	0
$845$	−1.73165e6	−0.0834295
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−6.26613e6	−0.298353
$850$	0	0
$851$	2.41430e6	0.114279
$852$	0	0
$853$	1.58070e7	0.743837	0.371919	−	0.928265i	$-0.378700\pi$
$853$	1.58070e7	0.743837	0.371919	+	0.928265i	$0.378700\pi$
$854$	0	0
$855$	1.83718e6	0.0859483
$856$	0	0
$857$	1.61495e7	0.751115	0.375557	−	0.926799i	$-0.377451\pi$
$857$	1.61495e7	0.751115	0.375557	+	0.926799i	$0.377451\pi$
$858$	0	0
$859$	−1.45107e6	−0.0670972	−0.0335486	−	0.999437i	$-0.510681\pi$
$859$	−1.45107e6	−0.0670972	−0.0335486	+	0.999437i	$0.510681\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.67359e7	−0.764932	−0.382466	−	0.923969i	$-0.624925\pi$
$863$	−1.67359e7	−0.764932	−0.382466	+	0.923969i	$0.624925\pi$
$864$	0	0
$865$	−2.67701e7	−1.21649
$866$	0	0
$867$	9.55340e6	0.431629
$868$	0	0
$869$	−2.35372e6	−0.105732
$870$	0	0
$871$	9.71855e6	0.434066
$872$	0	0
$873$	2.90085e6	0.128822
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.77411e7	−0.778900	−0.389450	−	0.921048i	$-0.627335\pi$
$877$	−1.77411e7	−0.778900	−0.389450	+	0.921048i	$0.627335\pi$
$878$	0	0
$879$	1.56853e7	0.684731
$880$	0	0
$881$	−3.78264e6	−0.164193	−0.0820967	−	0.996624i	$-0.526162\pi$
$881$	−3.78264e6	−0.164193	−0.0820967	+	0.996624i	$0.526162\pi$
$882$	0	0
$883$	−3.93853e7	−1.69993	−0.849967	−	0.526836i	$-0.823378\pi$
$883$	−3.93853e7	−1.69993	−0.849967	+	0.526836i	$0.823378\pi$
$884$	0	0
$885$	1.04466e7	0.448348
$886$	0	0
$887$	7.17846e6	0.306353	0.153177	−	0.988199i	$-0.451050\pi$
$887$	7.17846e6	0.306353	0.153177	+	0.988199i	$0.451050\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	285779.	0.0120597
$892$	0	0
$893$	1.93279e6	0.0811064
$894$	0	0
$895$	−2.14382e7	−0.894603
$896$	0	0
$897$	1.58486e6	0.0657675
$898$	0	0
$899$	2.34381e6	0.0967216
$900$	0	0
$901$	−4.25840e6	−0.174757
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.73762e7	−0.705234
$906$	0	0
$907$	2.19478e7	0.885877	0.442938	−	0.896552i	$-0.353936\pi$
$907$	2.19478e7	0.885877	0.442938	+	0.896552i	$0.353936\pi$
$908$	0	0
$909$	−2.50099e6	−0.100393
$910$	0	0
$911$	−2.60249e7	−1.03895	−0.519474	−	0.854486i	$-0.673872\pi$
$911$	−2.60249e7	−1.03895	−0.519474	+	0.854486i	$0.673872\pi$
$912$	0	0
$913$	−529142.	−0.0210085
$914$	0	0
$915$	−6.10282e6	−0.240978
$916$	0	0
$917$	0	0
$918$	0	0
$919$	2.18157e7	0.852078	0.426039	−	0.904705i	$-0.359909\pi$
$919$	2.18157e7	0.852078	0.426039	+	0.904705i	$0.359909\pi$
$920$	0	0
$921$	−2.93327e7	−1.13947
$922$	0	0
$923$	1.62638e7	0.628374
$924$	0	0
$925$	1.66457e7	0.639657
$926$	0	0
$927$	1.68319e6	0.0643329
$928$	0	0
$929$	1.24759e7	0.474276	0.237138	−	0.971476i	$-0.423791\pi$
$929$	1.24759e7	0.474276	0.237138	+	0.971476i	$0.423791\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−9.38438e6	−0.352941
$934$	0	0
$935$	−922663.	−0.0345155
$936$	0	0
$937$	−3.37205e7	−1.25471	−0.627357	−	0.778732i	$-0.715863\pi$
$937$	−3.37205e7	−1.25471	−0.627357	+	0.778732i	$0.715863\pi$
$938$	0	0
$939$	1.19785e7	0.443341
$940$	0	0
$941$	−2.49886e7	−0.919956	−0.459978	−	0.887930i	$-0.652143\pi$
$941$	−2.49886e7	−0.919956	−0.459978	+	0.887930i	$0.652143\pi$
$942$	0	0
$943$	−2.03392e6	−0.0744825
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.18165e7	1.15286	0.576430	−	0.817146i	$-0.304445\pi$
$947$	3.18165e7	1.15286	0.576430	+	0.817146i	$0.304445\pi$
$948$	0	0
$949$	7.44727e6	0.268430
$950$	0	0
$951$	−2.80779e6	−0.100673
$952$	0	0
$953$	−3.79217e6	−0.135256	−0.0676279	−	0.997711i	$-0.521543\pi$
$953$	−3.79217e6	−0.135256	−0.0676279	+	0.997711i	$0.521543\pi$
$954$	0	0
$955$	1.52296e7	0.540354
$956$	0	0
$957$	2.07358e6	0.0731883
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.84328e7	−0.993142
$962$	0	0
$963$	−8.87728e6	−0.308471
$964$	0	0
$965$	−1.13413e6	−0.0392052
$966$	0	0
$967$	2.32765e7	0.800482	0.400241	−	0.916410i	$-0.368926\pi$
$967$	2.32765e7	0.800482	0.400241	+	0.916410i	$0.368926\pi$
$968$	0	0
$969$	3.45347e6	0.118154
$970$	0	0
$971$	−2.66165e7	−0.905946	−0.452973	−	0.891524i	$-0.649637\pi$
$971$	−2.66165e7	−0.905946	−0.452973	+	0.891524i	$0.649637\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.09270e7	0.368121
$976$	0	0
$977$	−1.55013e7	−0.519555	−0.259777	−	0.965669i	$-0.583649\pi$
$977$	−1.55013e7	−0.519555	−0.259777	+	0.965669i	$0.583649\pi$
$978$	0	0
$979$	3.87855e6	0.129334
$980$	0	0
$981$	1.54586e7	0.512857
$982$	0	0
$983$	4.61036e7	1.52178	0.760889	−	0.648882i	$-0.224763\pi$
$983$	4.61036e7	1.52178	0.760889	+	0.648882i	$0.224763\pi$
$984$	0	0
$985$	−1.00219e7	−0.329125
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−979104.	−0.0318301
$990$	0	0
$991$	1.82281e7	0.589599	0.294799	−	0.955559i	$-0.404747\pi$
$991$	1.82281e7	0.589599	0.294799	+	0.955559i	$0.404747\pi$
$992$	0	0
$993$	−2.31918e7	−0.746384
$994$	0	0
$995$	−1.87027e7	−0.598889
$996$	0	0
$997$	8.74841e6	0.278735	0.139367	−	0.990241i	$-0.455493\pi$
$997$	8.74841e6	0.278735	0.139367	+	0.990241i	$0.455493\pi$
$998$	0	0
$999$	6.47908e6	0.205400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.6.a.m.1.1	✓	4
7.2	even	3		588.6.i.q.361.4		8
7.3	odd	6		588.6.i.p.373.1		8
7.4	even	3		588.6.i.q.373.4		8
7.5	odd	6		588.6.i.p.361.1		8
7.6	odd	2		588.6.a.o.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.6.a.m.1.1	✓	4	1.1	even	1	trivial
588.6.a.o.1.4	yes	4	7.6	odd	2
588.6.i.p.361.1		8	7.5	odd	6
588.6.i.p.373.1		8	7.3	odd	6
588.6.i.q.361.4		8	7.2	even	3
588.6.i.q.373.4		8	7.4	even	3

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:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	588.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -35.3849 q^{5} +81.0000 q^{9} +43.5572 q^{11} +648.252 q^{13} +318.465 q^{15} +598.639 q^{17} -640.987 q^{19} -271.648 q^{23} -1872.91 q^{25} -729.000 q^{27} -5289.56 q^{29} -443.101 q^{31} -392.015 q^{33} -8887.62 q^{37} -5834.27 q^{39} +7487.34 q^{41} +3604.32 q^{43} -2866.18 q^{45} -3015.33 q^{47} -5387.75 q^{51} -7113.47 q^{53} -1541.27 q^{55} +5768.88 q^{57} +32802.9 q^{59} -19163.3 q^{61} -22938.4 q^{65} +14991.9 q^{67} +2444.83 q^{69} +25088.7 q^{71} +11488.2 q^{73} +16856.1 q^{75} -54037.5 q^{79} +6561.00 q^{81} -12148.2 q^{83} -21182.8 q^{85} +47606.0 q^{87} +89045.0 q^{89} +3987.91 q^{93} +22681.3 q^{95} +35813.0 q^{97} +3528.13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{3} + 324 q^{9} + 1872 q^{17} + 1728 q^{19} - 3648 q^{23} - 3996 q^{25} - 2916 q^{27} - 1248 q^{29} + 3888 q^{31} - 12032 q^{37} - 9072 q^{41} - 2128 q^{43} + 19872 q^{47} - 16848 q^{51} - 22248 q^{53}+ \cdots + 320544 q^{97}+O(q^{100}) \) 4 * q - 36 * q^3 + 324 * q^9 + 1872 * q^17 + 1728 * q^19 - 3648 * q^23 - 3996 * q^25 - 2916 * q^27 - 1248 * q^29 + 3888 * q^31 - 12032 * q^37 - 9072 * q^41 - 2128 * q^43 + 19872 * q^47 - 16848 * q^51 - 22248 * q^53 + 21168 * q^55 - 15552 * q^57 - 5040 * q^59 + 8640 * q^61 - 83880 * q^65 - 86160 * q^67 + 32832 * q^69 - 6720 * q^71 + 18144 * q^73 + 35964 * q^75 - 19648 * q^79 + 26244 * q^81 + 262512 * q^83 + 20168 * q^85 + 11232 * q^87 + 49680 * q^89 - 34992 * q^93 + 112032 * q^95 + 320544 * q^97

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