LMFDB - Embedded newform 588.6.a.n.1.4

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:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 699x^{2} - 686x + 59664 $ x^4 - 2x^3 - 699x^2 - 686*x + 59664
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2}\cdot 7 $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$9.18135$ 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)

$f(q)$	$=$	$q-9.00000 q^{3} +78.6718 q^{5} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} +78.6718 q^{5} +81.0000 q^{9} +691.519 q^{11} +818.732 q^{13} -708.046 q^{15} +1109.28 q^{17} -573.046 q^{19} +2517.60 q^{23} +3064.25 q^{25} -729.000 q^{27} -3258.19 q^{29} +10119.1 q^{31} -6223.67 q^{33} +4869.62 q^{37} -7368.59 q^{39} -13094.3 q^{41} -9303.64 q^{43} +6372.41 q^{45} +12905.8 q^{47} -9983.48 q^{51} -19540.2 q^{53} +54403.0 q^{55} +5157.42 q^{57} -25120.5 q^{59} -31362.2 q^{61} +64411.1 q^{65} +55943.8 q^{67} -22658.4 q^{69} +20501.0 q^{71} -67649.9 q^{73} -27578.2 q^{75} +14079.9 q^{79} +6561.00 q^{81} -77129.1 q^{83} +87268.6 q^{85} +29323.7 q^{87} +320.793 q^{89} -91071.9 q^{93} -45082.6 q^{95} -112009. q^{97} +56013.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{3} + 324 q^{9}+O(q^{10}) $ 4 * q - 36 * q^3 + 324 * q^9	$ 4 q - 36 q^{3} + 324 q^{9} + 462 q^{11} - 602 q^{13} - 228 q^{17} - 358 q^{19} + 2148 q^{23} + 5454 q^{25} - 2916 q^{27} - 5532 q^{29} - 830 q^{31} - 4158 q^{33} + 3914 q^{37} + 5418 q^{39} - 8316 q^{41} - 14518 q^{43} - 41700 q^{47} + 2052 q^{51} - 22164 q^{53} + 3892 q^{55} + 3222 q^{57} - 32886 q^{59} - 83732 q^{61} + 93192 q^{65} + 80034 q^{67} - 19332 q^{69} + 44772 q^{71} + 22470 q^{73} - 49086 q^{75} + 75286 q^{79} + 26244 q^{81} - 17418 q^{83} + 139252 q^{85} + 49788 q^{87} - 28944 q^{89} + 7470 q^{93} + 144120 q^{95} - 216678 q^{97} + 37422 q^{99}+O(q^{100}) $ 4 * q - 36 * q^3 + 324 * q^9 + 462 * q^11 - 602 * q^13 - 228 * q^17 - 358 * q^19 + 2148 * q^23 + 5454 * q^25 - 2916 * q^27 - 5532 * q^29 - 830 * q^31 - 4158 * q^33 + 3914 * q^37 + 5418 * q^39 - 8316 * q^41 - 14518 * q^43 - 41700 * q^47 + 2052 * q^51 - 22164 * q^53 + 3892 * q^55 + 3222 * q^57 - 32886 * q^59 - 83732 * q^61 + 93192 * q^65 + 80034 * q^67 - 19332 * q^69 + 44772 * q^71 + 22470 * q^73 - 49086 * q^75 + 75286 * q^79 + 26244 * q^81 - 17418 * q^83 + 139252 * q^85 + 49788 * q^87 - 28944 * q^89 + 7470 * q^93 + 144120 * q^95 - 216678 * q^97 + 37422 * q^99

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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	78.6718	1.40732	0.703662	−	0.710535i	$-0.251547\pi$
$5$	78.6718	1.40732	0.703662	+	0.710535i	$0.251547\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	691.519	1.72315	0.861574	−	0.507633i	$-0.169479\pi$
$11$	691.519	1.72315	0.861574	+	0.507633i	$0.169479\pi$
$12$	0	0
$13$	818.732	1.34364	0.671821	−	0.740714i	$-0.265512\pi$
$13$	818.732	1.34364	0.671821	+	0.740714i	$0.265512\pi$
$14$	0	0
$15$	−708.046	−0.812518
$16$	0	0
$17$	1109.28	0.930930	0.465465	−	0.885066i	$-0.345887\pi$
$17$	1109.28	0.930930	0.465465	+	0.885066i	$0.345887\pi$
$18$	0	0
$19$	−573.046	−0.364171	−0.182086	−	0.983283i	$-0.558285\pi$
$19$	−573.046	−0.364171	−0.182086	+	0.983283i	$0.558285\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2517.60	0.992353	0.496177	−	0.868222i	$-0.334737\pi$
$23$	2517.60	0.992353	0.496177	+	0.868222i	$0.334737\pi$
$24$	0	0
$25$	3064.25	0.980559
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−3258.19	−0.719419	−0.359710	−	0.933064i	$-0.617124\pi$
$29$	−3258.19	−0.719419	−0.359710	+	0.933064i	$0.617124\pi$
$30$	0	0
$31$	10119.1	1.89120	0.945601	−	0.325330i	$-0.105475\pi$
$31$	10119.1	1.89120	0.945601	+	0.325330i	$0.105475\pi$
$32$	0	0
$33$	−6223.67	−0.994860
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4869.62	0.584778	0.292389	−	0.956300i	$-0.405550\pi$
$37$	4869.62	0.584778	0.292389	+	0.956300i	$0.405550\pi$
$38$	0	0
$39$	−7368.59	−0.775752
$40$	0	0
$41$	−13094.3	−1.21653	−0.608265	−	0.793734i	$-0.708134\pi$
$41$	−13094.3	−1.21653	−0.608265	+	0.793734i	$0.708134\pi$
$42$	0	0
$43$	−9303.64	−0.767329	−0.383664	−	0.923473i	$-0.625338\pi$
$43$	−9303.64	−0.767329	−0.383664	+	0.923473i	$0.625338\pi$
$44$	0	0
$45$	6372.41	0.469108
$46$	0	0
$47$	12905.8	0.852198	0.426099	−	0.904676i	$-0.359887\pi$
$47$	12905.8	0.852198	0.426099	+	0.904676i	$0.359887\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−9983.48	−0.537473
$52$	0	0
$53$	−19540.2	−0.955521	−0.477760	−	0.878490i	$-0.658551\pi$
$53$	−19540.2	−0.955521	−0.477760	+	0.878490i	$0.658551\pi$
$54$	0	0
$55$	54403.0	2.42503
$56$	0	0
$57$	5157.42	0.210254
$58$	0	0
$59$	−25120.5	−0.939501	−0.469751	−	0.882799i	$-0.655656\pi$
$59$	−25120.5	−0.939501	−0.469751	+	0.882799i	$0.655656\pi$
$60$	0	0
$61$	−31362.2	−1.07915	−0.539575	−	0.841938i	$-0.681415\pi$
$61$	−31362.2	−1.07915	−0.539575	+	0.841938i	$0.681415\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	64411.1	1.89094
$66$	0	0
$67$	55943.8	1.52253	0.761264	−	0.648443i	$-0.224579\pi$
$67$	55943.8	1.52253	0.761264	+	0.648443i	$0.224579\pi$
$68$	0	0
$69$	−22658.4	−0.572935
$70$	0	0
$71$	20501.0	0.482647	0.241323	−	0.970445i	$-0.422419\pi$
$71$	20501.0	0.482647	0.241323	+	0.970445i	$0.422419\pi$
$72$	0	0
$73$	−67649.9	−1.48580	−0.742900	−	0.669402i	$-0.766550\pi$
$73$	−67649.9	−1.48580	−0.742900	+	0.669402i	$0.766550\pi$
$74$	0	0
$75$	−27578.2	−0.566126
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14079.9	0.253823	0.126912	−	0.991914i	$-0.459494\pi$
$79$	14079.9	0.253823	0.126912	+	0.991914i	$0.459494\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−77129.1	−1.22892	−0.614459	−	0.788949i	$-0.710626\pi$
$83$	−77129.1	−1.22892	−0.614459	+	0.788949i	$0.710626\pi$
$84$	0	0
$85$	87268.6	1.31012
$86$	0	0
$87$	29323.7	0.415357
$88$	0	0
$89$	320.793	0.00429289	0.00214644	−	0.999998i	$-0.499317\pi$
$89$	320.793	0.00429289	0.00214644	+	0.999998i	$0.499317\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−91071.9	−1.09189
$94$	0	0
$95$	−45082.6	−0.512507
$96$	0	0
$97$	−112009.	−1.20871	−0.604355	−	0.796715i	$-0.706569\pi$
$97$	−112009.	−1.20871	−0.604355	+	0.796715i	$0.706569\pi$
$98$	0	0
$99$	56013.0	0.574382
$100$	0	0
$101$	67254.0	0.656016	0.328008	−	0.944675i	$-0.393623\pi$
$101$	67254.0	0.656016	0.328008	+	0.944675i	$0.393623\pi$
$102$	0	0
$103$	−116635.	−1.08327	−0.541635	−	0.840614i	$-0.682195\pi$
$103$	−116635.	−1.08327	−0.541635	+	0.840614i	$0.682195\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−160645.	−1.35646	−0.678232	−	0.734848i	$-0.737254\pi$
$107$	−160645.	−1.35646	−0.678232	+	0.734848i	$0.737254\pi$
$108$	0	0
$109$	119337.	0.962072	0.481036	−	0.876701i	$-0.340261\pi$
$109$	119337.	0.962072	0.481036	+	0.876701i	$0.340261\pi$
$110$	0	0
$111$	−43826.6	−0.337622
$112$	0	0
$113$	−110893.	−0.816972	−0.408486	−	0.912765i	$-0.633943\pi$
$113$	−110893.	−0.816972	−0.408486	+	0.912765i	$0.633943\pi$
$114$	0	0
$115$	198064.	1.39656
$116$	0	0
$117$	66317.3	0.447881
$118$	0	0
$119$	0	0
$120$	0	0
$121$	317147.	1.96924
$122$	0	0
$123$	117849.	0.702364
$124$	0	0
$125$	−4779.64	−0.0273603
$126$	0	0
$127$	315184.	1.73402	0.867012	−	0.498287i	$-0.166038\pi$
$127$	315184.	1.73402	0.867012	+	0.498287i	$0.166038\pi$
$128$	0	0
$129$	83732.7	0.443018
$130$	0	0
$131$	85664.5	0.436137	0.218068	−	0.975933i	$-0.430024\pi$
$131$	85664.5	0.436137	0.218068	+	0.975933i	$0.430024\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−57351.7	−0.270839
$136$	0	0
$137$	−32632.2	−0.148541	−0.0742703	−	0.997238i	$-0.523663\pi$
$137$	−32632.2	−0.148541	−0.0742703	+	0.997238i	$0.523663\pi$
$138$	0	0
$139$	206590.	0.906925	0.453462	−	0.891275i	$-0.350189\pi$
$139$	206590.	0.906925	0.453462	+	0.891275i	$0.350189\pi$
$140$	0	0
$141$	−116152.	−0.492017
$142$	0	0
$143$	566169.	2.31529
$144$	0	0
$145$	−256328.	−1.01246
$146$	0	0
$147$	0	0
$148$	0	0
$149$	483711.	1.78493	0.892463	−	0.451121i	$-0.148976\pi$
$149$	483711.	1.78493	0.892463	+	0.451121i	$0.148976\pi$
$150$	0	0
$151$	−167221.	−0.596826	−0.298413	−	0.954437i	$-0.596457\pi$
$151$	−167221.	−0.596826	−0.298413	+	0.954437i	$0.596457\pi$
$152$	0	0
$153$	89851.3	0.310310
$154$	0	0
$155$	796087.	2.66153
$156$	0	0
$157$	−14108.5	−0.0456806	−0.0228403	−	0.999739i	$-0.507271\pi$
$157$	−14108.5	−0.0456806	−0.0228403	+	0.999739i	$0.507271\pi$
$158$	0	0
$159$	175862.	0.551670
$160$	0	0
$161$	0	0
$162$	0	0
$163$	654376.	1.92912	0.964558	−	0.263869i	$-0.0849988\pi$
$163$	654376.	1.92912	0.964558	+	0.263869i	$0.0849988\pi$
$164$	0	0
$165$	−489627.	−1.40009
$166$	0	0
$167$	182945.	0.507610	0.253805	−	0.967255i	$-0.418318\pi$
$167$	182945.	0.507610	0.253805	+	0.967255i	$0.418318\pi$
$168$	0	0
$169$	299030.	0.805374
$170$	0	0
$171$	−46416.7	−0.121390
$172$	0	0
$173$	−292008.	−0.741788	−0.370894	−	0.928675i	$-0.620949\pi$
$173$	−292008.	−0.741788	−0.370894	+	0.928675i	$0.620949\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	226084.	0.542421
$178$	0	0
$179$	194768.	0.454343	0.227172	−	0.973855i	$-0.427052\pi$
$179$	194768.	0.454343	0.227172	+	0.973855i	$0.427052\pi$
$180$	0	0
$181$	−256634.	−0.582261	−0.291130	−	0.956683i	$-0.594031\pi$
$181$	−256634.	−0.582261	−0.291130	+	0.956683i	$0.594031\pi$
$182$	0	0
$183$	282260.	0.623047
$184$	0	0
$185$	383102.	0.822971
$186$	0	0
$187$	767085.	1.60413
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−293949.	−0.583027	−0.291513	−	0.956567i	$-0.594159\pi$
$191$	−293949.	−0.583027	−0.291513	+	0.956567i	$0.594159\pi$
$192$	0	0
$193$	−69130.1	−0.133590	−0.0667950	−	0.997767i	$-0.521277\pi$
$193$	−69130.1	−0.133590	−0.0667950	+	0.997767i	$0.521277\pi$
$194$	0	0
$195$	−579700.	−1.09173
$196$	0	0
$197$	331748.	0.609037	0.304518	−	0.952506i	$-0.401505\pi$
$197$	331748.	0.609037	0.304518	+	0.952506i	$0.401505\pi$
$198$	0	0
$199$	574428.	1.02826	0.514130	−	0.857712i	$-0.328115\pi$
$199$	574428.	1.02826	0.514130	+	0.857712i	$0.328115\pi$
$200$	0	0
$201$	−503494.	−0.879031
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.03015e6	−1.71205
$206$	0	0
$207$	203925.	0.330784
$208$	0	0
$209$	−396272.	−0.627521
$210$	0	0
$211$	−383999.	−0.593777	−0.296889	−	0.954912i	$-0.595949\pi$
$211$	−383999.	−0.593777	−0.296889	+	0.954912i	$0.595949\pi$
$212$	0	0
$213$	−184509.	−0.278656
$214$	0	0
$215$	−731933.	−1.07988
$216$	0	0
$217$	0	0
$218$	0	0
$219$	608850.	0.857827
$220$	0	0
$221$	908200.	1.25084
$222$	0	0
$223$	−347165.	−0.467492	−0.233746	−	0.972298i	$-0.575098\pi$
$223$	−347165.	−0.467492	−0.233746	+	0.972298i	$0.575098\pi$
$224$	0	0
$225$	248204.	0.326853
$226$	0	0
$227$	361281.	0.465351	0.232676	−	0.972554i	$-0.425252\pi$
$227$	361281.	0.465351	0.232676	+	0.972554i	$0.425252\pi$
$228$	0	0
$229$	−752983.	−0.948848	−0.474424	−	0.880297i	$-0.657344\pi$
$229$	−752983.	−0.948848	−0.474424	+	0.880297i	$0.657344\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	714397.	0.862085	0.431042	−	0.902332i	$-0.358146\pi$
$233$	714397.	0.862085	0.431042	+	0.902332i	$0.358146\pi$
$234$	0	0
$235$	1.01532e6	1.19932
$236$	0	0
$237$	−126719.	−0.146545
$238$	0	0
$239$	343587.	0.389083	0.194541	−	0.980894i	$-0.437678\pi$
$239$	343587.	0.389083	0.194541	+	0.980894i	$0.437678\pi$
$240$	0	0
$241$	−76305.0	−0.0846272	−0.0423136	−	0.999104i	$-0.513473\pi$
$241$	−76305.0	−0.0846272	−0.0423136	+	0.999104i	$0.513473\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−469171.	−0.489316
$248$	0	0
$249$	694162.	0.709516
$250$	0	0
$251$	196249.	0.196618	0.0983090	−	0.995156i	$-0.468657\pi$
$251$	196249.	0.196618	0.0983090	+	0.995156i	$0.468657\pi$
$252$	0	0
$253$	1.74096e6	1.70997
$254$	0	0
$255$	−785418.	−0.756398
$256$	0	0
$257$	−1.70911e6	−1.61412	−0.807062	−	0.590467i	$-0.798944\pi$
$257$	−1.70911e6	−1.61412	−0.807062	+	0.590467i	$0.798944\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−263914.	−0.239806
$262$	0	0
$263$	−791525.	−0.705627	−0.352814	−	0.935694i	$-0.614775\pi$
$263$	−791525.	−0.705627	−0.352814	+	0.935694i	$0.614775\pi$
$264$	0	0
$265$	−1.53726e6	−1.34473
$266$	0	0
$267$	−2887.13	−0.00247850
$268$	0	0
$269$	1.15900e6	0.976572	0.488286	−	0.872684i	$-0.337622\pi$
$269$	1.15900e6	0.976572	0.488286	+	0.872684i	$0.337622\pi$
$270$	0	0
$271$	223134.	0.184562	0.0922811	−	0.995733i	$-0.470584\pi$
$271$	223134.	0.184562	0.0922811	+	0.995733i	$0.470584\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.11898e6	1.68965
$276$	0	0
$277$	−469046.	−0.367296	−0.183648	−	0.982992i	$-0.558791\pi$
$277$	−469046.	−0.367296	−0.183648	+	0.982992i	$0.558791\pi$
$278$	0	0
$279$	819647.	0.630400
$280$	0	0
$281$	−1.51447e6	−1.14418	−0.572090	−	0.820191i	$-0.693867\pi$
$281$	−1.51447e6	−1.14418	−0.572090	+	0.820191i	$0.693867\pi$
$282$	0	0
$283$	−1.07061e6	−0.794627	−0.397314	−	0.917683i	$-0.630057\pi$
$283$	−1.07061e6	−0.794627	−0.397314	+	0.917683i	$0.630057\pi$
$284$	0	0
$285$	405743.	0.295896
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−189365.	−0.133369
$290$	0	0
$291$	1.00808e6	0.697849
$292$	0	0
$293$	1.45450e6	0.989794	0.494897	−	0.868952i	$-0.335206\pi$
$293$	1.45450e6	0.989794	0.494897	+	0.868952i	$0.335206\pi$
$294$	0	0
$295$	−1.97627e6	−1.32218
$296$	0	0
$297$	−504117.	−0.331620
$298$	0	0
$299$	2.06124e6	1.33337
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−605286.	−0.378751
$304$	0	0
$305$	−2.46732e6	−1.51871
$306$	0	0
$307$	−2.23869e6	−1.35565	−0.677824	−	0.735224i	$-0.737077\pi$
$307$	−2.23869e6	−1.35565	−0.677824	+	0.735224i	$0.737077\pi$
$308$	0	0
$309$	1.04972e6	0.625426
$310$	0	0
$311$	−1.50139e6	−0.880223	−0.440111	−	0.897943i	$-0.645061\pi$
$311$	−1.50139e6	−0.880223	−0.440111	+	0.897943i	$0.645061\pi$
$312$	0	0
$313$	−1.78013e6	−1.02705	−0.513525	−	0.858074i	$-0.671661\pi$
$313$	−1.78013e6	−1.02705	−0.513525	+	0.858074i	$0.671661\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.47100e6	0.822176	0.411088	−	0.911596i	$-0.365149\pi$
$317$	1.47100e6	0.822176	0.411088	+	0.911596i	$0.365149\pi$
$318$	0	0
$319$	−2.25310e6	−1.23966
$320$	0	0
$321$	1.44581e6	0.783155
$322$	0	0
$323$	−635666.	−0.339018
$324$	0	0
$325$	2.50880e6	1.31752
$326$	0	0
$327$	−1.07403e6	−0.555453
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.82975e6	−0.917953	−0.458977	−	0.888448i	$-0.651784\pi$
$331$	−1.82975e6	−0.917953	−0.458977	+	0.888448i	$0.651784\pi$
$332$	0	0
$333$	394439.	0.194926
$334$	0	0
$335$	4.40120e6	2.14269
$336$	0	0
$337$	3.18627e6	1.52830	0.764148	−	0.645041i	$-0.223160\pi$
$337$	3.18627e6	1.52830	0.764148	+	0.645041i	$0.223160\pi$
$338$	0	0
$339$	998035.	0.471679
$340$	0	0
$341$	6.99755e6	3.25882
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.78257e6	−0.806305
$346$	0	0
$347$	2.47790e6	1.10474	0.552370	−	0.833599i	$-0.313724\pi$
$347$	2.47790e6	1.10474	0.552370	+	0.833599i	$0.313724\pi$
$348$	0	0
$349$	1.81720e6	0.798618	0.399309	−	0.916816i	$-0.369250\pi$
$349$	1.81720e6	0.798618	0.399309	+	0.916816i	$0.369250\pi$
$350$	0	0
$351$	−596856.	−0.258584
$352$	0	0
$353$	3.15355e6	1.34699	0.673493	−	0.739193i	$-0.264793\pi$
$353$	3.15355e6	1.34699	0.673493	+	0.739193i	$0.264793\pi$
$354$	0	0
$355$	1.61285e6	0.679240
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.70209e6	1.10653	0.553266	−	0.833004i	$-0.313381\pi$
$359$	2.70209e6	1.10653	0.553266	+	0.833004i	$0.313381\pi$
$360$	0	0
$361$	−2.14772e6	−0.867379
$362$	0	0
$363$	−2.85433e6	−1.13694
$364$	0	0
$365$	−5.32214e6	−2.09100
$366$	0	0
$367$	−3.61147e6	−1.39965	−0.699824	−	0.714315i	$-0.746738\pi$
$367$	−3.61147e6	−1.39965	−0.699824	+	0.714315i	$0.746738\pi$
$368$	0	0
$369$	−1.06064e6	−0.405510
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−433518.	−0.161338	−0.0806688	−	0.996741i	$-0.525706\pi$
$373$	−433518.	−0.161338	−0.0806688	+	0.996741i	$0.525706\pi$
$374$	0	0
$375$	43016.8	0.0157965
$376$	0	0
$377$	−2.66759e6	−0.966642
$378$	0	0
$379$	−2.12163e6	−0.758704	−0.379352	−	0.925252i	$-0.623853\pi$
$379$	−2.12163e6	−0.758704	−0.379352	+	0.925252i	$0.623853\pi$
$380$	0	0
$381$	−2.83666e6	−1.00114
$382$	0	0
$383$	−4.24679e6	−1.47932	−0.739662	−	0.672978i	$-0.765015\pi$
$383$	−4.24679e6	−1.47932	−0.739662	+	0.672978i	$0.765015\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−753595.	−0.255776
$388$	0	0
$389$	−3.52177e6	−1.18001	−0.590006	−	0.807399i	$-0.700875\pi$
$389$	−3.52177e6	−1.18001	−0.590006	+	0.807399i	$0.700875\pi$
$390$	0	0
$391$	2.79271e6	0.923811
$392$	0	0
$393$	−770981.	−0.251804
$394$	0	0
$395$	1.10769e6	0.357212
$396$	0	0
$397$	−734050.	−0.233749	−0.116874	−	0.993147i	$-0.537288\pi$
$397$	−734050.	−0.233749	−0.116874	+	0.993147i	$0.537288\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−222270.	−0.0690272	−0.0345136	−	0.999404i	$-0.510988\pi$
$401$	−222270.	−0.0690272	−0.0345136	+	0.999404i	$0.510988\pi$
$402$	0	0
$403$	8.28483e6	2.54110
$404$	0	0
$405$	516165.	0.156369
$406$	0	0
$407$	3.36743e6	1.00766
$408$	0	0
$409$	−3.50758e6	−1.03681	−0.518405	−	0.855135i	$-0.673474\pi$
$409$	−3.50758e6	−1.03681	−0.518405	+	0.855135i	$0.673474\pi$
$410$	0	0
$411$	293690.	0.0857600
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−6.06788e6	−1.72948
$416$	0	0
$417$	−1.85931e6	−0.523613
$418$	0	0
$419$	−1.97040e6	−0.548302	−0.274151	−	0.961687i	$-0.588397\pi$
$419$	−1.97040e6	−0.548302	−0.274151	+	0.961687i	$0.588397\pi$
$420$	0	0
$421$	−7.19318e6	−1.97795	−0.988976	−	0.148079i	$-0.952691\pi$
$421$	−7.19318e6	−1.97795	−0.988976	+	0.148079i	$0.952691\pi$
$422$	0	0
$423$	1.04537e6	0.284066
$424$	0	0
$425$	3.39909e6	0.912832
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−5.09552e6	−1.33673
$430$	0	0
$431$	524182.	0.135922	0.0679608	−	0.997688i	$-0.478351\pi$
$431$	524182.	0.135922	0.0679608	+	0.997688i	$0.478351\pi$
$432$	0	0
$433$	1.49326e6	0.382750	0.191375	−	0.981517i	$-0.438705\pi$
$433$	1.49326e6	0.382750	0.191375	+	0.981517i	$0.438705\pi$
$434$	0	0
$435$	2.30695e6	0.584541
$436$	0	0
$437$	−1.44270e6	−0.361386
$438$	0	0
$439$	2.65670e6	0.657932	0.328966	−	0.944342i	$-0.393300\pi$
$439$	2.65670e6	0.657932	0.328966	+	0.944342i	$0.393300\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.68798e6	−1.13495	−0.567474	−	0.823391i	$-0.692079\pi$
$443$	−4.68798e6	−1.13495	−0.567474	+	0.823391i	$0.692079\pi$
$444$	0	0
$445$	25237.3	0.00604148
$446$	0	0
$447$	−4.35340e6	−1.03053
$448$	0	0
$449$	−3.45899e6	−0.809716	−0.404858	−	0.914379i	$-0.632679\pi$
$449$	−3.45899e6	−0.809716	−0.404858	+	0.914379i	$0.632679\pi$
$450$	0	0
$451$	−9.05496e6	−2.09626
$452$	0	0
$453$	1.50499e6	0.344578
$454$	0	0
$455$	0	0
$456$	0	0
$457$	718279.	0.160880	0.0804401	−	0.996759i	$-0.474367\pi$
$457$	718279.	0.160880	0.0804401	+	0.996759i	$0.474367\pi$
$458$	0	0
$459$	−808662.	−0.179158
$460$	0	0
$461$	5.07998e6	1.11329	0.556647	−	0.830749i	$-0.312088\pi$
$461$	5.07998e6	1.11329	0.556647	+	0.830749i	$0.312088\pi$
$462$	0	0
$463$	3.40607e6	0.738416	0.369208	−	0.929347i	$-0.379629\pi$
$463$	3.40607e6	0.738416	0.369208	+	0.929347i	$0.379629\pi$
$464$	0	0
$465$	−7.16479e6	−1.53664
$466$	0	0
$467$	1.84268e6	0.390984	0.195492	−	0.980705i	$-0.437370\pi$
$467$	1.84268e6	0.390984	0.195492	+	0.980705i	$0.437370\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	126977.	0.0263737
$472$	0	0
$473$	−6.43364e6	−1.32222
$474$	0	0
$475$	−1.75595e6	−0.357091
$476$	0	0
$477$	−1.58276e6	−0.318507
$478$	0	0
$479$	9.11427e6	1.81503	0.907514	−	0.420022i	$-0.137978\pi$
$479$	9.11427e6	1.81503	0.907514	+	0.420022i	$0.137978\pi$
$480$	0	0
$481$	3.98692e6	0.785732
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8.81192e6	−1.70105
$486$	0	0
$487$	4.38440e6	0.837699	0.418850	−	0.908056i	$-0.362433\pi$
$487$	4.38440e6	0.837699	0.418850	+	0.908056i	$0.362433\pi$
$488$	0	0
$489$	−5.88939e6	−1.11378
$490$	0	0
$491$	−9.50592e6	−1.77947	−0.889735	−	0.456478i	$-0.849111\pi$
$491$	−9.50592e6	−1.77947	−0.889735	+	0.456478i	$0.849111\pi$
$492$	0	0
$493$	−3.61423e6	−0.669729
$494$	0	0
$495$	4.40664e6	0.808342
$496$	0	0
$497$	0	0
$498$	0	0
$499$	3.37566e6	0.606887	0.303444	−	0.952849i	$-0.401864\pi$
$499$	3.37566e6	0.606887	0.303444	+	0.952849i	$0.401864\pi$
$500$	0	0
$501$	−1.64651e6	−0.293069
$502$	0	0
$503$	7.82645e6	1.37926	0.689628	−	0.724164i	$-0.257774\pi$
$503$	7.82645e6	1.37926	0.689628	+	0.724164i	$0.257774\pi$
$504$	0	0
$505$	5.29099e6	0.923227
$506$	0	0
$507$	−2.69127e6	−0.464983
$508$	0	0
$509$	−7.42437e6	−1.27018	−0.635090	−	0.772438i	$-0.719037\pi$
$509$	−7.42437e6	−1.27018	−0.635090	+	0.772438i	$0.719037\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	417751.	0.0700848
$514$	0	0
$515$	−9.17590e6	−1.52451
$516$	0	0
$517$	8.92461e6	1.46846
$518$	0	0
$519$	2.62807e6	0.428272
$520$	0	0
$521$	−1.06142e6	−0.171314	−0.0856570	−	0.996325i	$-0.527299\pi$
$521$	−1.06142e6	−0.171314	−0.0856570	+	0.996325i	$0.527299\pi$
$522$	0	0
$523$	−1.91352e6	−0.305899	−0.152950	−	0.988234i	$-0.548877\pi$
$523$	−1.91352e6	−0.305899	−0.152950	+	0.988234i	$0.548877\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.12249e7	1.76058
$528$	0	0
$529$	−98058.3	−0.0152351
$530$	0	0
$531$	−2.03476e6	−0.313167
$532$	0	0
$533$	−1.07207e7	−1.63458
$534$	0	0
$535$	−1.26382e7	−1.90898
$536$	0	0
$537$	−1.75291e6	−0.262315
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1.16628e7	1.71321	0.856603	−	0.515976i	$-0.172571\pi$
$541$	1.16628e7	1.71321	0.856603	+	0.515976i	$0.172571\pi$
$542$	0	0
$543$	2.30971e6	0.336168
$544$	0	0
$545$	9.38842e6	1.35395
$546$	0	0
$547$	−6.36659e6	−0.909785	−0.454893	−	0.890546i	$-0.650322\pi$
$547$	−6.36659e6	−0.909785	−0.454893	+	0.890546i	$0.650322\pi$
$548$	0	0
$549$	−2.54034e6	−0.359717
$550$	0	0
$551$	1.86710e6	0.261992
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3.44791e6	−0.475143
$556$	0	0
$557$	9.72442e6	1.32808	0.664042	−	0.747695i	$-0.268840\pi$
$557$	9.72442e6	1.32808	0.664042	+	0.747695i	$0.268840\pi$
$558$	0	0
$559$	−7.61719e6	−1.03102
$560$	0	0
$561$	−6.90376e6	−0.926145
$562$	0	0
$563$	6.30266e6	0.838017	0.419008	−	0.907982i	$-0.362378\pi$
$563$	6.30266e6	0.838017	0.419008	+	0.907982i	$0.362378\pi$
$564$	0	0
$565$	−8.72413e6	−1.14974
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.16487e6	1.05723	0.528614	−	0.848862i	$-0.322712\pi$
$569$	8.16487e6	1.05723	0.528614	+	0.848862i	$0.322712\pi$
$570$	0	0
$571$	−1.28771e7	−1.65283	−0.826416	−	0.563060i	$-0.809624\pi$
$571$	−1.28771e7	−1.65283	−0.826416	+	0.563060i	$0.809624\pi$
$572$	0	0
$573$	2.64554e6	0.336611
$574$	0	0
$575$	7.71453e6	0.973061
$576$	0	0
$577$	1.14177e7	1.42771	0.713856	−	0.700293i	$-0.246947\pi$
$577$	1.14177e7	1.42771	0.713856	+	0.700293i	$0.246947\pi$
$578$	0	0
$579$	622171.	0.0771282
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1.35124e7	−1.64650
$584$	0	0
$585$	5.21730e6	0.630313
$586$	0	0
$587$	2.12781e6	0.254882	0.127441	−	0.991846i	$-0.459324\pi$
$587$	2.12781e6	0.254882	0.127441	+	0.991846i	$0.459324\pi$
$588$	0	0
$589$	−5.79871e6	−0.688721
$590$	0	0
$591$	−2.98574e6	−0.351627
$592$	0	0
$593$	−8.57378e6	−1.00123	−0.500617	−	0.865669i	$-0.666894\pi$
$593$	−8.57378e6	−1.00123	−0.500617	+	0.865669i	$0.666894\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5.16986e6	−0.593667
$598$	0	0
$599$	1.12874e7	1.28537	0.642686	−	0.766130i	$-0.277820\pi$
$599$	1.12874e7	1.28537	0.642686	+	0.766130i	$0.277820\pi$
$600$	0	0
$601$	−1.58260e7	−1.78725	−0.893625	−	0.448814i	$-0.851847\pi$
$601$	−1.58260e7	−1.78725	−0.893625	+	0.448814i	$0.851847\pi$
$602$	0	0
$603$	4.53145e6	0.507509
$604$	0	0
$605$	2.49506e7	2.77135
$606$	0	0
$607$	9.74366e6	1.07337	0.536686	−	0.843782i	$-0.319676\pi$
$607$	9.74366e6	1.07337	0.536686	+	0.843782i	$0.319676\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.05664e7	1.14505
$612$	0	0
$613$	6.15048e6	0.661086	0.330543	−	0.943791i	$-0.392768\pi$
$613$	6.15048e6	0.661086	0.330543	+	0.943791i	$0.392768\pi$
$614$	0	0
$615$	9.27137e6	0.988453
$616$	0	0
$617$	−3.36084e6	−0.355414	−0.177707	−	0.984083i	$-0.556868\pi$
$617$	−3.36084e6	−0.355414	−0.177707	+	0.984083i	$0.556868\pi$
$618$	0	0
$619$	6.88659e6	0.722399	0.361200	−	0.932488i	$-0.382367\pi$
$619$	6.88659e6	0.722399	0.361200	+	0.932488i	$0.382367\pi$
$620$	0	0
$621$	−1.83533e6	−0.190978
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−9.95179e6	−1.01906
$626$	0	0
$627$	3.56645e6	0.362299
$628$	0	0
$629$	5.40175e6	0.544387
$630$	0	0
$631$	3.59545e6	0.359484	0.179742	−	0.983714i	$-0.442474\pi$
$631$	3.59545e6	0.359484	0.179742	+	0.983714i	$0.442474\pi$
$632$	0	0
$633$	3.45599e6	0.342818
$634$	0	0
$635$	2.47961e7	2.44033
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.66058e6	0.160882
$640$	0	0
$641$	−1.98287e7	−1.90611	−0.953056	−	0.302795i	$-0.902080\pi$
$641$	−1.98287e7	−1.90611	−0.953056	+	0.302795i	$0.902080\pi$
$642$	0	0
$643$	−8.10851e6	−0.773417	−0.386708	−	0.922202i	$-0.626388\pi$
$643$	−8.10851e6	−0.773417	−0.386708	+	0.922202i	$0.626388\pi$
$644$	0	0
$645$	6.58740e6	0.623469
$646$	0	0
$647$	2.02746e6	0.190411	0.0952053	−	0.995458i	$-0.469649\pi$
$647$	2.02746e6	0.190411	0.0952053	+	0.995458i	$0.469649\pi$
$648$	0	0
$649$	−1.73713e7	−1.61890
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.51427e6	0.138969	0.0694847	−	0.997583i	$-0.477864\pi$
$653$	1.51427e6	0.138969	0.0694847	+	0.997583i	$0.477864\pi$
$654$	0	0
$655$	6.73938e6	0.613785
$656$	0	0
$657$	−5.47965e6	−0.495267
$658$	0	0
$659$	−3.22358e6	−0.289151	−0.144576	−	0.989494i	$-0.546182\pi$
$659$	−3.22358e6	−0.289151	−0.144576	+	0.989494i	$0.546182\pi$
$660$	0	0
$661$	−1.43908e7	−1.28109	−0.640546	−	0.767920i	$-0.721292\pi$
$661$	−1.43908e7	−1.28109	−0.640546	+	0.767920i	$0.721292\pi$
$662$	0	0
$663$	−8.17380e6	−0.722171
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.20281e6	−0.713918
$668$	0	0
$669$	3.12449e6	0.269907
$670$	0	0
$671$	−2.16875e7	−1.85953
$672$	0	0
$673$	7.33478e6	0.624237	0.312118	−	0.950043i	$-0.398961\pi$
$673$	7.33478e6	0.624237	0.312118	+	0.950043i	$0.398961\pi$
$674$	0	0
$675$	−2.23384e6	−0.188709
$676$	0	0
$677$	−3.05285e6	−0.255996	−0.127998	−	0.991774i	$-0.540855\pi$
$677$	−3.05285e6	−0.255996	−0.127998	+	0.991774i	$0.540855\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−3.25153e6	−0.268671
$682$	0	0
$683$	1.59759e7	1.31043	0.655213	−	0.755444i	$-0.272579\pi$
$683$	1.59759e7	1.31043	0.655213	+	0.755444i	$0.272579\pi$
$684$	0	0
$685$	−2.56724e6	−0.209045
$686$	0	0
$687$	6.77685e6	0.547817
$688$	0	0
$689$	−1.59982e7	−1.28388
$690$	0	0
$691$	−2.46533e6	−0.196417	−0.0982087	−	0.995166i	$-0.531311\pi$
$691$	−2.46533e6	−0.196417	−0.0982087	+	0.995166i	$0.531311\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.62528e7	1.27634
$696$	0	0
$697$	−1.45252e7	−1.13250
$698$	0	0
$699$	−6.42958e6	−0.497725
$700$	0	0
$701$	1.74893e7	1.34424	0.672122	−	0.740440i	$-0.265383\pi$
$701$	1.74893e7	1.34424	0.672122	+	0.740440i	$0.265383\pi$
$702$	0	0
$703$	−2.79052e6	−0.212959
$704$	0	0
$705$	−9.13790e6	−0.692427
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−2.18431e7	−1.63192	−0.815961	−	0.578107i	$-0.803792\pi$
$709$	−2.18431e7	−1.63192	−0.815961	+	0.578107i	$0.803792\pi$
$710$	0	0
$711$	1.14047e6	0.0846078
$712$	0	0
$713$	2.54758e7	1.87674
$714$	0	0
$715$	4.45415e7	3.25837
$716$	0	0
$717$	−3.09228e6	−0.224637
$718$	0	0
$719$	1.91174e7	1.37913	0.689566	−	0.724223i	$-0.257801\pi$
$719$	1.91174e7	1.37913	0.689566	+	0.724223i	$0.257801\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	686745.	0.0488596
$724$	0	0
$725$	−9.98391e6	−0.705433
$726$	0	0
$727$	−4.08900e6	−0.286933	−0.143467	−	0.989655i	$-0.545825\pi$
$727$	−4.08900e6	−0.286933	−0.143467	+	0.989655i	$0.545825\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.03203e7	−0.714330
$732$	0	0
$733$	−3.63268e6	−0.249728	−0.124864	−	0.992174i	$-0.539849\pi$
$733$	−3.63268e6	−0.249728	−0.124864	+	0.992174i	$0.539849\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.86862e7	2.62354
$738$	0	0
$739$	2.17957e6	0.146811	0.0734056	−	0.997302i	$-0.476613\pi$
$739$	2.17957e6	0.146811	0.0734056	+	0.997302i	$0.476613\pi$
$740$	0	0
$741$	4.22254e6	0.282507
$742$	0	0
$743$	1.21595e7	0.808060	0.404030	−	0.914746i	$-0.367609\pi$
$743$	1.21595e7	0.808060	0.404030	+	0.914746i	$0.367609\pi$
$744$	0	0
$745$	3.80544e7	2.51197
$746$	0	0
$747$	−6.24745e6	−0.409639
$748$	0	0
$749$	0	0
$750$	0	0
$751$	2.32495e7	1.50423	0.752115	−	0.659032i	$-0.229034\pi$
$751$	2.32495e7	1.50423	0.752115	+	0.659032i	$0.229034\pi$
$752$	0	0
$753$	−1.76624e6	−0.113517
$754$	0	0
$755$	−1.31556e7	−0.839928
$756$	0	0
$757$	−1.49939e7	−0.950985	−0.475492	−	0.879720i	$-0.657730\pi$
$757$	−1.49939e7	−0.950985	−0.475492	+	0.879720i	$0.657730\pi$
$758$	0	0
$759$	−1.56687e7	−0.987252
$760$	0	0
$761$	−1.27319e7	−0.796952	−0.398476	−	0.917179i	$-0.630461\pi$
$761$	−1.27319e7	−0.796952	−0.398476	+	0.917179i	$0.630461\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	7.06876e6	0.436707
$766$	0	0
$767$	−2.05669e7	−1.26235
$768$	0	0
$769$	2.34653e6	0.143090	0.0715451	−	0.997437i	$-0.477207\pi$
$769$	2.34653e6	0.143090	0.0715451	+	0.997437i	$0.477207\pi$
$770$	0	0
$771$	1.53820e7	0.931915
$772$	0	0
$773$	2.71863e7	1.63644	0.818222	−	0.574903i	$-0.194960\pi$
$773$	2.71863e7	1.63644	0.818222	+	0.574903i	$0.194960\pi$
$774$	0	0
$775$	3.10074e7	1.85443
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.50364e6	0.443025
$780$	0	0
$781$	1.41768e7	0.831671
$782$	0	0
$783$	2.37522e6	0.138452
$784$	0	0
$785$	−1.10994e6	−0.0642874
$786$	0	0
$787$	4.30273e6	0.247632	0.123816	−	0.992305i	$-0.460487\pi$
$787$	4.30273e6	0.247632	0.123816	+	0.992305i	$0.460487\pi$
$788$	0	0
$789$	7.12372e6	0.407394
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.56772e7	−1.44999
$794$	0	0
$795$	1.38354e7	0.776378
$796$	0	0
$797$	1.92325e7	1.07248	0.536241	−	0.844065i	$-0.319844\pi$
$797$	1.92325e7	1.07248	0.536241	+	0.844065i	$0.319844\pi$
$798$	0	0
$799$	1.43161e7	0.793337
$800$	0	0
$801$	25984.2	0.00143096
$802$	0	0
$803$	−4.67812e7	−2.56025
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.04310e7	−0.563824
$808$	0	0
$809$	−4.76367e6	−0.255900	−0.127950	−	0.991781i	$-0.540840\pi$
$809$	−4.76367e6	−0.255900	−0.127950	+	0.991781i	$0.540840\pi$
$810$	0	0
$811$	1.05529e7	0.563404	0.281702	−	0.959502i	$-0.409101\pi$
$811$	1.05529e7	0.563404	0.281702	+	0.959502i	$0.409101\pi$
$812$	0	0
$813$	−2.00821e6	−0.106557
$814$	0	0
$815$	5.14809e7	2.71489
$816$	0	0
$817$	5.33141e6	0.279439
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.51066e7	−0.782186	−0.391093	−	0.920351i	$-0.627903\pi$
$821$	−1.51066e7	−0.782186	−0.391093	+	0.920351i	$0.627903\pi$
$822$	0	0
$823$	2.48152e7	1.27708	0.638541	−	0.769588i	$-0.279538\pi$
$823$	2.48152e7	1.27708	0.638541	+	0.769588i	$0.279538\pi$
$824$	0	0
$825$	−1.90709e7	−0.975518
$826$	0	0
$827$	−5.89505e6	−0.299726	−0.149863	−	0.988707i	$-0.547883\pi$
$827$	−5.89505e6	−0.299726	−0.149863	+	0.988707i	$0.547883\pi$
$828$	0	0
$829$	−9.16246e6	−0.463048	−0.231524	−	0.972829i	$-0.574371\pi$
$829$	−9.16246e6	−0.463048	−0.231524	+	0.972829i	$0.574371\pi$
$830$	0	0
$831$	4.22141e6	0.212058
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.43926e7	0.714372
$836$	0	0
$837$	−7.37682e6	−0.363962
$838$	0	0
$839$	−1.65127e7	−0.809868	−0.404934	−	0.914346i	$-0.632705\pi$
$839$	−1.65127e7	−0.809868	−0.404934	+	0.914346i	$0.632705\pi$
$840$	0	0
$841$	−9.89532e6	−0.482436
$842$	0	0
$843$	1.36302e7	0.660593
$844$	0	0
$845$	2.35252e7	1.13342
$846$	0	0
$847$	0	0
$848$	0	0
$849$	9.63545e6	0.458778
$850$	0	0
$851$	1.22597e7	0.580306
$852$	0	0
$853$	−2.99081e7	−1.40740	−0.703699	−	0.710498i	$-0.748470\pi$
$853$	−2.99081e7	−1.40740	−0.703699	+	0.710498i	$0.748470\pi$
$854$	0	0
$855$	−3.65169e6	−0.170836
$856$	0	0
$857$	1.32675e7	0.617072	0.308536	−	0.951213i	$-0.400161\pi$
$857$	1.32675e7	0.617072	0.308536	+	0.951213i	$0.400161\pi$
$858$	0	0
$859$	−2.20488e7	−1.01954	−0.509769	−	0.860312i	$-0.670269\pi$
$859$	−2.20488e7	−1.01954	−0.509769	+	0.860312i	$0.670269\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.57732e7	0.720930	0.360465	−	0.932773i	$-0.382618\pi$
$863$	1.57732e7	0.720930	0.360465	+	0.932773i	$0.382618\pi$
$864$	0	0
$865$	−2.29728e7	−1.04394
$866$	0	0
$867$	1.70429e6	0.0770007
$868$	0	0
$869$	9.73652e6	0.437375
$870$	0	0
$871$	4.58030e7	2.04573
$872$	0	0
$873$	−9.07270e6	−0.402903
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.15132e7	0.944507	0.472253	−	0.881463i	$-0.343441\pi$
$877$	2.15132e7	0.944507	0.472253	+	0.881463i	$0.343441\pi$
$878$	0	0
$879$	−1.30905e7	−0.571458
$880$	0	0
$881$	−8.57354e6	−0.372152	−0.186076	−	0.982535i	$-0.559577\pi$
$881$	−8.57354e6	−0.372152	−0.186076	+	0.982535i	$0.559577\pi$
$882$	0	0
$883$	−5.13016e6	−0.221426	−0.110713	−	0.993852i	$-0.535314\pi$
$883$	−5.13016e6	−0.221426	−0.110713	+	0.993852i	$0.535314\pi$
$884$	0	0
$885$	1.77864e7	0.763362
$886$	0	0
$887$	−3.61335e6	−0.154206	−0.0771030	−	0.997023i	$-0.524567\pi$
$887$	−3.61335e6	−0.154206	−0.0771030	+	0.997023i	$0.524567\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4.53706e6	0.191461
$892$	0	0
$893$	−7.39562e6	−0.310346
$894$	0	0
$895$	1.53227e7	0.639408
$896$	0	0
$897$	−1.85511e7	−0.769820
$898$	0	0
$899$	−3.29700e7	−1.36057
$900$	0	0
$901$	−2.16755e7	−0.889523
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.01898e7	−0.819429
$906$	0	0
$907$	2.69438e7	1.08753	0.543765	−	0.839238i	$-0.316998\pi$
$907$	2.69438e7	1.08753	0.543765	+	0.839238i	$0.316998\pi$
$908$	0	0
$909$	5.44757e6	0.218672
$910$	0	0
$911$	−3.89588e7	−1.55528	−0.777641	−	0.628708i	$-0.783584\pi$
$911$	−3.89588e7	−1.55528	−0.777641	+	0.628708i	$0.783584\pi$
$912$	0	0
$913$	−5.33362e7	−2.11761
$914$	0	0
$915$	2.22059e7	0.876829
$916$	0	0
$917$	0	0
$918$	0	0
$919$	3.46434e7	1.35311	0.676553	−	0.736394i	$-0.263473\pi$
$919$	3.46434e7	1.35311	0.676553	+	0.736394i	$0.263473\pi$
$920$	0	0
$921$	2.01482e7	0.782684
$922$	0	0
$923$	1.67848e7	0.648504
$924$	0	0
$925$	1.49217e7	0.573409
$926$	0	0
$927$	−9.44746e6	−0.361090
$928$	0	0
$929$	−3.52341e7	−1.33944	−0.669721	−	0.742613i	$-0.733586\pi$
$929$	−3.52341e7	−1.33944	−0.669721	+	0.742613i	$0.733586\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.35125e7	0.508197
$934$	0	0
$935$	6.03479e7	2.25753
$936$	0	0
$937$	7.21950e6	0.268632	0.134316	−	0.990939i	$-0.457116\pi$
$937$	7.21950e6	0.268632	0.134316	+	0.990939i	$0.457116\pi$
$938$	0	0
$939$	1.60212e7	0.592968
$940$	0	0
$941$	3.73332e7	1.37443	0.687213	−	0.726456i	$-0.258834\pi$
$941$	3.73332e7	1.37443	0.687213	+	0.726456i	$0.258834\pi$
$942$	0	0
$943$	−3.29662e7	−1.20723
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.17014e7	−1.51104	−0.755520	−	0.655126i	$-0.772616\pi$
$947$	−4.17014e7	−1.51104	−0.755520	+	0.655126i	$0.772616\pi$
$948$	0	0
$949$	−5.53872e7	−1.99638
$950$	0	0
$951$	−1.32390e7	−0.474683
$952$	0	0
$953$	5.07681e7	1.81075	0.905374	−	0.424614i	$-0.139590\pi$
$953$	5.07681e7	1.81075	0.905374	+	0.424614i	$0.139590\pi$
$954$	0	0
$955$	−2.31255e7	−0.820507
$956$	0	0
$957$	2.02779e7	0.715721
$958$	0	0
$959$	0	0
$960$	0	0
$961$	7.37671e7	2.57664
$962$	0	0
$963$	−1.30123e7	−0.452155
$964$	0	0
$965$	−5.43859e6	−0.188004
$966$	0	0
$967$	−2.00129e7	−0.688246	−0.344123	−	0.938925i	$-0.611824\pi$
$967$	−2.00129e7	−0.688246	−0.344123	+	0.938925i	$0.611824\pi$
$968$	0	0
$969$	5.72099e6	0.195732
$970$	0	0
$971$	1.48724e7	0.506213	0.253107	−	0.967438i	$-0.418548\pi$
$971$	1.48724e7	0.506213	0.253107	+	0.967438i	$0.418548\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.25792e7	−0.760670
$976$	0	0
$977$	−4.86941e6	−0.163207	−0.0816037	−	0.996665i	$-0.526004\pi$
$977$	−4.86941e6	−0.163207	−0.0816037	+	0.996665i	$0.526004\pi$
$978$	0	0
$979$	221834.	0.00739727
$980$	0	0
$981$	9.66627e6	0.320691
$982$	0	0
$983$	−4.33110e7	−1.42960	−0.714800	−	0.699329i	$-0.753482\pi$
$983$	−4.33110e7	−1.42960	−0.714800	+	0.699329i	$0.753482\pi$
$984$	0	0
$985$	2.60992e7	0.857111
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.34228e7	−0.761461
$990$	0	0
$991$	1.03036e7	0.333277	0.166638	−	0.986018i	$-0.446709\pi$
$991$	1.03036e7	0.333277	0.166638	+	0.986018i	$0.446709\pi$
$992$	0	0
$993$	1.64677e7	0.529981
$994$	0	0
$995$	4.51913e7	1.44710
$996$	0	0
$997$	−3.88437e7	−1.23761	−0.618803	−	0.785546i	$-0.712382\pi$
$997$	−3.88437e7	−1.23761	−0.618803	+	0.785546i	$0.712382\pi$
$998$	0	0
$999$	−3.54995e6	−0.112541

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.6.a.n.1.4		4
7.2	even	3		84.6.i.c.25.1	✓	8
7.3	odd	6		588.6.i.o.373.4		8
7.4	even	3		84.6.i.c.37.1	yes	8
7.5	odd	6		588.6.i.o.361.4		8
7.6	odd	2		588.6.a.p.1.1		4
21.2	odd	6		252.6.k.f.109.4		8
21.11	odd	6		252.6.k.f.37.4		8
28.11	odd	6		336.6.q.i.289.1		8
28.23	odd	6		336.6.q.i.193.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.i.c.25.1	✓	8	7.2	even	3
84.6.i.c.37.1	yes	8	7.4	even	3
252.6.k.f.37.4		8	21.11	odd	6
252.6.k.f.109.4		8	21.2	odd	6
336.6.q.i.193.1		8	28.23	odd	6
336.6.q.i.289.1		8	28.11	odd	6
588.6.a.n.1.4		4	1.1	even	1	trivial
588.6.a.p.1.1		4	7.6	odd	2
588.6.i.o.361.4		8	7.5	odd	6
588.6.i.o.373.4		8	7.3	odd	6

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)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +78.6718 q^{5} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} +78.6718 q^{5} +81.0000 q^{9} +691.519 q^{11} +818.732 q^{13} -708.046 q^{15} +1109.28 q^{17} -573.046 q^{19} +2517.60 q^{23} +3064.25 q^{25} -729.000 q^{27} -3258.19 q^{29} +10119.1 q^{31} -6223.67 q^{33} +4869.62 q^{37} -7368.59 q^{39} -13094.3 q^{41} -9303.64 q^{43} +6372.41 q^{45} +12905.8 q^{47} -9983.48 q^{51} -19540.2 q^{53} +54403.0 q^{55} +5157.42 q^{57} -25120.5 q^{59} -31362.2 q^{61} +64411.1 q^{65} +55943.8 q^{67} -22658.4 q^{69} +20501.0 q^{71} -67649.9 q^{73} -27578.2 q^{75} +14079.9 q^{79} +6561.00 q^{81} -77129.1 q^{83} +87268.6 q^{85} +29323.7 q^{87} +320.793 q^{89} -91071.9 q^{93} -45082.6 q^{95} -112009. q^{97} +56013.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{3} + 324 q^{9}+O(q^{10}) \) 4 * q - 36 * q^3 + 324 * q^9	\( 4 q - 36 q^{3} + 324 q^{9} + 462 q^{11} - 602 q^{13} - 228 q^{17} - 358 q^{19} + 2148 q^{23} + 5454 q^{25} - 2916 q^{27} - 5532 q^{29} - 830 q^{31} - 4158 q^{33} + 3914 q^{37} + 5418 q^{39} - 8316 q^{41} - 14518 q^{43} - 41700 q^{47} + 2052 q^{51} - 22164 q^{53} + 3892 q^{55} + 3222 q^{57} - 32886 q^{59} - 83732 q^{61} + 93192 q^{65} + 80034 q^{67} - 19332 q^{69} + 44772 q^{71} + 22470 q^{73} - 49086 q^{75} + 75286 q^{79} + 26244 q^{81} - 17418 q^{83} + 139252 q^{85} + 49788 q^{87} - 28944 q^{89} + 7470 q^{93} + 144120 q^{95} - 216678 q^{97} + 37422 q^{99}+O(q^{100}) \) 4 * q - 36 * q^3 + 324 * q^9 + 462 * q^11 - 602 * q^13 - 228 * q^17 - 358 * q^19 + 2148 * q^23 + 5454 * q^25 - 2916 * q^27 - 5532 * q^29 - 830 * q^31 - 4158 * q^33 + 3914 * q^37 + 5418 * q^39 - 8316 * q^41 - 14518 * q^43 - 41700 * q^47 + 2052 * q^51 - 22164 * q^53 + 3892 * q^55 + 3222 * q^57 - 32886 * q^59 - 83732 * q^61 + 93192 * q^65 + 80034 * q^67 - 19332 * q^69 + 44772 * q^71 + 22470 * q^73 - 49086 * q^75 + 75286 * q^79 + 26244 * q^81 - 17418 * q^83 + 139252 * q^85 + 49788 * q^87 - 28944 * q^89 + 7470 * q^93 + 144120 * q^95 - 216678 * q^97 + 37422 * q^99

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